dn ? ï y r-W.'i • a § v; 1 > i . Vi Î . A; «:• «b A. 'iÉ %* » i . - T « . * ■ (Km * "'y *ÿ w '• « -'V V’-'V •k'V ’ 1 • ■ ff * * ■■ . % K * - * F &•*<■ , ../' 1 4 ‘ % ,V. v* ¥ '•* ■ ■’■ ,W • , v.t % mjiÀ ■ « . * 1 . f-*. », '* ■# T. * • . î> Ü9 ‘/T **''.>*■ t- i*' iv. * ; % ,H ^.-.. • # #■<--. *• $s « * K » *»& « h É$ * ^ * «. .* $ *' *s ‘î** ÿ: ’jf1' ., • \***i!£ %,'. V* $ Ü* è , g. «y /» ' 1! # ^V# , ... •*Y> " ■».* f* .%^:/^\Wl5r'^vTu^\u, "%*% v*y #* *: V- h ■* ♦ J- t H V / V ÜT ir ♦- .V # <9 <1 \K- J. K-*' * 1 . *. «* *.4 * M Ÿ> A 4 /. ■y b> M". * m / \ *> 4 'C •A ' « y* V ■-•S*. i / \^V » t 4 * J •K 'S I * ¥ / V V «* fli V t î ' •Ä < K «♦ * f / / / V ^ *A ) ? ' A' J A / ry -2.r. / 1 >7 ' *A M,-/?/ I ‘ ^7 "W?'J f } 1C ^ (. > V M w >J/ -> »- / 7 z JOHN WILEY & SONS PUBLISH r The American House Carpenter. A Treatise on the Art of Building, comprising Styles of Architecture, Strength of Materials, and the Theory and Practice of the Construction of Floors, Framed Girders, Hoof Trusses, Rolled Iron Beams, Tubular Iron Girders, Cast-Iron Girders, Stairs, Doors, Windows, Mouldings and Cornices, together with r. Compend of Mathematics. A manual for the practical use or architects, carpenters, and stair-builders. By It. G. Hatfield, architect. Re-written and enlarged. Numerous fine wood engravings. 8vo, cloth, $5. The Theory of Transverse Strains, And its Application to the Construction of Buildings, including a full discussion of the Theory and Construction* of Floor Beams, Girders, Headers, Carriage Beams, Bridging, Rolled Iron Beams, Tubular Iron Girders, Cast-Iron Girders, Framed Girders and Roof Tresses, with Tables’ calculated expressly for the work, etc. By R. G. Hatfield, architect. Fully illustrated. Second edition with additions. Svo, cloth, $5. Cottage Residences. New edition. A series of Designs for Rural Cottages and Cottage Villas, and their Garden Grounds. By A. J. Downing. Containing a revised list of Trees, Shrubs, and Plants, and the most recent and best-selected Fruit, with some account of the new style of Cardens. By Henry Winthrop Sargent and Charles Downing. With many new designs in rural architecture. By George Haruey, architect. Svo, cloth, Carpenters’ and Joiners’ Handbook. Containing a Complete Treatise on Framing Hip ami Valley Roofs, together with much valuable instruction for all mechanics and amateurs, useful Rules, 'fables never before published, etc. By 11. W. Holly. New edition with additions. lSrno, cloth, 75 cents. The Architect’s and Builder’s Handbook-.Containing Original Tables and Valuable Information for Architects,, Builders, Engineers, and Contractors, fully illustrated with plates. By F. E. Kidder. Put up in pocket-book form. Third edition, revised and enlarged. Morocco Haps. $3.50. V Mailed, prepaid, on the receipt of the price. Catalogue of our publications gratis.m*** m WOODEN ROOF-TRUSS KS. I . . PLATE L JTHE ARCHITECT’S AND BUILDER’S POCKET-BOOK OP MENSURATION, GEOMETRY, GEOMETRICAL PROBLEMS, TRIGO' INO METRICAL FORMULAS AND TABLES. STRENGTH AND STABILITY OF FOUNDATIONS, WALLS. BUTTRESSES, PIERS, ARCHES, POSTS, TIES, BEAMS, GIRDERS, TRUSSES, FLOORS, ROOFS, ETC. IN addition to which is A GREAT AMOUNT OF CONDENSED INFORMATION: STATISTICS AND TABLES RELATING TO CARPENTRY, MASONRY, DRAINAGE, PAINTING AND GLAZING, PLUMBING, PLASTERING, ROOFING, HEATING AND VENTILATION, WEIGHTS OF MATERIALS, CAPACITY AND DIMENSIONS OF NOTED CHURCHES, THEATRES, DOMES, TOWERS, SPIRES, ETC., WITH A GREAT VARIETY OF MISCELLANEOUS IS FORMATION. BY FRANK EUGENE KIDDER, C.E., CONSULTING ARCHITECT. DENVER, COLO. ILLUSTRATED WITH 423 ENGRAVINGS, MOSTLY FROM ORIGINAL DESIGNS, EIGHTH EDITION, REVISED AND ENLARGED. NEW YORK : JOHN WILEY AND SONS, 53 East Tenth Street, Second door west of Broadway. 1890.Copyright, F. E. KIDDER, 1884. Press of J. J. Little & Co, Astor Place, New York.] Cfjts Booft I IS RESPECTFULLY DEDICATED TO THOSE WHOSE KINDNESS HAS ENABLED ME TO PRODUCE IT. TO MY PARENTS, WHO GAVE ME THE EDUCATION UPON WHICH IT IS BASED; TO MY WIFE, FOR HER LOVING SYMPATHY, ENCOURAGEMENT, AND ASSIST- { ANCE; ' '] TO ORLANDO W. NORCROSS OP WORCESTER, MASS., WHOSE SUPERIOR PRACTICAL KNOWLEDGE OF ALL THAT PERTAINS TO BUILDING HAS GIVEN ME A MORE INTELLIGENT AND PRACTICAL VIEW'OF THE SCIENCE OF CONSTRUCTION THAN I SHOULD OTHERWISE HAVE OBTAINED.PREFACE. In preparing the following pages, it has ever been the aim of the author to give to the architects and builders of this country a reference book which should be for them what Trautwine’s “Pocket-Book” is to engineers, — a compendium of practical facts, rules, and tables, presented in a form as convenient for application as possible, and as reliable as our present knowledge will permit. Only so much theory has been given as will render the application of the formulas more apparent, and aid the student in understanding, in some measure, the principles upon which the formulas are based. It is believed that nothing has been given in this book but what has been borne out in practice. As this book was not written for engineers, the more intricate problems of building construction, which may fairly be said to Tome within the province of the civil engineer, have been omitted. Desiring to give as much information as possible likely to be of service to architects and builders, the author has borrowed and ouoted from many sources, in most cases with the permission of the authors. Much practical information has been derived from the various handbooks published by the large manufacturers of rolled-iron beams, bars, etc,; and the author has always found the publishers willing to aid him whenever requested. Although but very little has been taken from Trautwine’s “ Pocket-Book for Engineers,” yet this valuable book has served the author as a model, which he has tried to imitate as well as the difference in the subjects would permit; and if his work shall prove of as much value to architects and builders as Mr. Trautwine’s has to engineers, he will feel amply rewarded for his labor. vvi PREFACE. As it is impossible for the author to verify all of the dimensions and miscellaneous information contained in Part III., he cannot speak for their accuracy, except that they were in all cases taken from what were considered reliable sources of information. The tables in Part II. have been carefully computed, and it is believed are free from any large errors. There are so many points of information often required by architects and builders, that it is difficult for one person to compile them all; and although the present volume is by no means a small one, yet the author desires to make his work as useful as possible to those for whom it has been prepared, and he will therefore be pleased to receive any information of a serviceable nature pertaining to architecture or building, that it may be inserted in future editions should such become necessary, and for the correction of any errors that may be found. The author, while compiling this volume, has consulted a great number of works relating to architecture and building; and as he has frequently been asked by students and draughtsmen to refer them to books from which they might acquire a better knowledge of construction and building, the following list of books is given as valuable works on the various subjects indicated by the titles: — “Notes on Building Construction,” compiled for the use of the students in the science and art schools, South Kensington, England. 3 vols. Eivingtons, publishers, Loudon. “ Building Superintendence,” by T. M. Clark, architect and professor of architecture, Massachusetts Institute of Technology. J. It. Osgood & Co., publishers, Boston. “ The American House Carpenter” and “ The Theory of Transverse Strains,” both by Mr. R. G. Hatfield, architect, formerly of New York. “Graphical Analysis of Roof-TruSses,” by Professor Charles E. Green of the University of Michigan. “The Eire Protection of Mills,” by C. J. II. Woodbury, inspector for the Factory Mutual Fire Insurance Companies. J olm Wiley & Sons, publishers. New York.PREFACE. vii “House Drainage and Water Service,” by Janies C. Bayles, editor of “The Iron Age” and “The Metal Worker.” David Williams, publisher, New York. “ The Builders’ Guide and Estimators’ Price-Book,” and “Plaster and Plastering, Mortars, and Cements,” by Fred. T. Hodgson, editor of “ The Builder and Wood Worker.” Industrial Publication Company, New York. “Foundations and Concrete Works” and “Art of Building,”, by E. Dobson. Weale’s Series, London. It would be well if all of the above books might be found in every architect’s office; but if the expense prevents that, the ambitious student and draughtsman should at least make himself acquainted with their contents. These works will also be found of great value to the enterprising builder. i I II 4 •I *PREFACE TO THE FOURTH EDITION. It is now a little more than two years since “ The Architect’s and Builder’s Pocket-Book” was first introduced to the public. During that time the author has received so many encouraging words and suggestions from a large number of architects and builders, that he desires to acknowledge their kindness, and to express the hope that the book will always merit their commendation. When preparing the book for publication, especial care and thought were given to the second part of the book; trusting that, if once well done, it would need but little revision for a number of years. The first part, also, it is believed, is quite complete in its way. For Part III., however, the author found time merely to compile such matter as he believed to be of practical value to architects or builders, thinking that, should the bock prove a success, this part could be easily revised and enlarged; and, since the second edition was published, the author has devoted such time as he could command to revising such portions as upon investigation seemed to require it, and preparing additional matter. It is the intention of the author, seconded by the publishers, to make each edition of the book more complete and perfect than the one preceding, in the hope that it may in time become to the architects of the present day what Gwilt’s “Encyclopaedia” was to those of former days. The great diversity of information, however, required by an architect, or those having to do with the erection and construction of buildings, renders it exceed ingly difficult for one person, having but a limited amount ixX PREFACE. of time to devote to the work, to make such a book as complete as could be desired. In the Preface to the first edition it was requested that those who might have information or suggestions which would increase the value of the hook would kindly send them to the author, or advise him of any errors that should be discovered. Several persons generously replied to this invitation; and several small errors have been corrected, and additional information given, as the result. It is believed, however, that there are yet many who have thought, at times, of various ways in which the book could be improved, or have in their private note-books practical data or suggestions which others in the profession would be glad to possess; and it is hoped all such will feel it for the interest of the profession to forward such items to the author. Any records or reports of tests of the strength of building materials of any kind will be especially appreciated. To the list of books given in the former Preface the author would add the following, which have been of much assistance in the preparation of the pages on steam-heating, and in his professional practice: — “The Principles of Heating and Ventilation, and their Practical Application,” by John S. Billings, M.D., LL.D., Sanitary Engineer, New York. “Steam-IIeating for Buildings; or, Hints to Steam-Fitters, by William J. Baldwin, M.E. John Wiley cfc Sons, New York. “Steam.” Babcock & Wilcox Company, New York and Glas- gow.CONTENTS PART I. PAGE Arithmetical Signs and Characters................... 3 Involution......................................... 3 Evolution, Square and Cure Root, Rules, and Tables . 4 Weights and Measures................................25 The Metric System...................................30 Scripture and Ancient Measures and Weights .... 33 Mensuration.........................................35 Geometrical Problems.............................. 68 Table of Chords ....................................85 Hip and Jack Rafters................................94 Trigonometry, Formulas and Tables...................95 PART II. Introduction............................... . . . 123 CHAPTER I. Definitions of Terms used in Mechanics.............125 CHAPTER II. Foundations....................................... 130 CHAPTER III. Masonry Walls......................................143 CHAPTER IV. Composition and Resolution of Forces. — Centre of Gravity........................................152 xiCONTENTS. xii CHAPTER Y. PAGE Retaining Walls................................161 CHAPTER VI. Strength of Masonry............................165 CHAPTER VII. Stability of Piers and Buttresses..............178 CHAPTER VIII. The Stability of Arches........................185 CHAPTER IX. Resistance to Tension..........................197 CHAPTER X. #■ Resistance to Shearing.........................213 CHAPTER XI. Strength of Posts, Struts, and CoCumns.........217 CHAPTER XII. Bending-Moments.............................. 250 CHAPTER XIII. Moments of Inertia and Resistance, and Radius of Gyration ....................................257 CHAPTER XIV. General Principles of the Strength of Beams, and Strength of Iron Beams . ..................280 CHAPTER XV. Strength of Cast-Iron, Wooden, and Stone Beams. — Solid Built Beams..........................307 CHAPTER XVI. Stiffness and Deflection of Beams..............318CONTENTS. Mil CHAPTER XVII. Strength and Stiffness of Continuous Girders .... 327 CHAPTER XVIII. F> itch Plate Girders........................336 CHAPTER XIX. Trussed Beams................................339 CHAPTER XX. Riveted Plate-Iron Girders...................345 CHAPTER XXI. Strength of Cast-Iron Arch-Girders......... 350 CHAPTER XXII. Strength and Stiffness of Wooden Floors . . 484 1G048 4.C904158 2.8020293 .045454545 23 529 12107 47958315 2.8438670 .042418261 24 576 ■ 13824 4.8989795 2.8844991 .041666667 25 625 15025 5.0000000 2.5240177 .046000000 2G 676 17576 5.0990195 2.5024960 .038461538 27 729 19083 5.1961524 3.CCCCOOO .031031037 23 734 21952 5.2415026 3.C2C5889 .035714286 29 841 .24389 5. oojI (/i8 3.0723168 .034482759 30 900 27000 5.4772256 3.1072325 .033333333 31 961 29791 5.5077614 3.1412806 .032258065 33 1021 32768 5.C508542 3.1748021 .033250000 33 1039 35937 5.1445026 3.8075343 .030303030 34 1156 39304 5.8305519 3.23C6118 .029411165 35 1225 42875' 5.9160798 8.271C663 .028571429 8G 1296 40056 6.CGOOOOO 3.2019272 .021711118 37 1309 50053 6.C327C25 3.3322218 .027027027 38 1444 55nm 6.1044140 3.2G19754 .026315189 39 1521 59319 6.2449980 3.3912114 .025641026 40 1600 64000 C.3245553 3.4199519 .025000000 41 1081 68921 6.4031242 3.44G2172 .024390244 43 1704 74088 0 . ‘lov/l lOl 3.47CC2C6 .023809524 43 1319 79507 6.5574385 3.5022981 .023255814 44 1926 85184 6.000:490 3.52C2483 .022727273 45 2025 91125 6.7C820;:9 3.5oC89o3 022222222 40 2116 97336 6.782GC00 3.5820479 .021739130 47 2209 103823 6.0550546 3.6CGC261 .021276600 48 2304 110592 • 6.0232032 3.G342411 .020833333 49 2401 117649 7.0000000 3.6593057 .020408163 50 2500 125000 7.0710678 3.6840314 .020000000 51 2001 —1 7.1414284 3.7C84258 * .019607813 52 2704 140008 7.2111C26 3.7325111 .019230769 53 2809 148877 7.2801099 8.73G2858 .018867925 54 2916 15 <*4 04 7.34! >4092 3.7797G31 .018518519 55 3025 160375 7.4101985 3.8029525 .018181818 5G 8136 175616 7.4833148 3.8258G24 .017851143 57 3219 185193 7.54QS344 3.8485011 .017543800 58* 3304 195112 7.0157731 3.8708766 .017241319 59 3481 205379 7.6811457 3.8929965 .016949153 CO 3000 216000 7.7459667 3.9148676 .016666607 61 3721 220981 7.8102:97 3.9364972 .016393443 63 3844 238328 7.8740079 3.9578915 .016129032CUBE ROOTS, AM) RECIPROCALS tv No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 63 3969 250047 7.9372539 3.9790571 .015873016 64 4095 262141 8.0000000 4.0000900 .015625000 65 4225 274625 8.032.2577 4.0207256 .015381615 65 4356 257496 8.1240384 4.0412401 .015151515 67 4439 300763 8.1853528 4.0615480 .014925373 03 4624 314432 8.2402113 4.0810551 .014705882 69 4761 325509 8.3066239 4.1015661 .014492754 70 4900 343000 8.3666003 4.1212S53 .01-428571-4 71 m 5041 337Oil 8.4261498 4.1408178 .01408450/ 72 5184 373243 8.4852814 4.1601676 .013888889 73 5329 339017 8.5-140037 4.1793390 .013698630 74 5176 405224 8.0023253 4.1983364 .013513514 75 5525 421875 8.66025-10 4.2171633 AlmmmB™ > .Ul000 131044 47437928 19.0262976 7.1269300 .0027C2431 £03 131769 47832147 19.0525589 7.1334925 .002754821 304 132496 48228544 .19.0787840 7.1400370 .002747253 305 133225 48627125 19.1049732 7.14G5G95 .002739726 3GG 1,33956 49027896 19.1311265 7.1530901 .002732240 307 134089 49430SG3 39.1572441 7.1595988 .002724796 3G8 135424 4983G032 39.18332G1 7.16C0057 .002717391 309 1361G1 50243409 19.2093727 7.1725809 .002710027 370 136900 50653000 39.2353841 7.17905-44 .002702703 371 137641 51004811 19.2013G03 7.10551C2 .002095418 372 138384 51478848 19.2373015 7.191CC53 .002G88172 __^14 SQUARES, CUBES, SQUARE ROOTS, No. 373 374 375 376 377 378 379 380 331 383 383 384 385 386 387 388 389 390 391 393 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 42-1 425 426 427 423 429 430 431 432 4. >3 431 Squares. . Cubes. 139129 51895117 139876 62313624 140025 62«o l • j 15 141376 6315«376 142129 5:35820:33 142884 64010152 143641 54439939 144400 5-1872000 145101 55300:341 145924 55742908 140689 50181887 147456 5002:3164 148225 67000025 148996 57512456 149709 67 90uG03 150544 5:'411072 151321 68803809 152100 59319000 Mi 5^7«6171 158004 002:30288 154449 CC098457 155236 01102984 150025 01029875 150816 02099136 157009 02570773 ' 158404 03044192 159201 63521199 1G0000 64000000 100801 64481201 101004 64964808 102409 €5450827 103216 65939264 104025 66430125 101836 60923116 105049 C7419143 tmIM 07917312 107281 08417929 1G8100 68921000 108921 094*20531 109744 09934528 170509 70444997 171396 70957944 172225 714731375 173056 71991296 173889 72511713 174724 7:36340:32 175501 73500059 176400 74C«88000 177241 74G184G1 178084 75151448 178929 75080907 1«9776 70225024 180625 70705025 181476 77308776 182329 77854483 183184 78402752 184611 78953589 184900 79507000 185761 80062991 180024 80021508 1874S9 81182737 1.:.... 6 8174G504 Square Roots. Cube Roots 19.3132079 7.1984050 19.3390790 7.2048322 19.3049107 7.2112479 19.3907194 7.2170522 19.4164878 7.2240450 19.4422221 7.2:304208 19.4079223 7.2307972 19.4935887 7.2431505 19.5192213 7.2435045 19.5448203 7.2558415 19.5703858 7.2G21075 19.5959179 7.2084824 19.6214109 7.2741864 19.6408827 7.2810794 19.0728156 7.2873617 19.0977156 7.29363:40 19.7230829 7.2998936 19.7484177 7.3061436 19.7137199 7.3123828 19.7989899 7.3180114 19.8242276 7.3248295 19.8434332 7.3310309 19.8740009 7.3372339 19.8997487 7.3434205 19.9248588 7.8495906 19.9439373 7.8557024 19.9749844 7.3019118 20.0000000 7.3080630 20.0249844 7.3141979 20.0439377 7.3803227 20.0748599 7.3864373 20.0991512 i. o9iCo418 20.1240118 7.3986303 20.1494417 7.4041206 20.1742410 7.4101950 20.1990099 7.4108595 20.2231484 7.4229142 20.24845C7 7.4289589 20.2781349 7.4:3499:48 20.2917831 7.4410189 20.3224014 7.4470342 20.3409899 7.4530399 20.3715488 7.4590359 20.3960781 7.4050223 20.4205719 7 4709991 20.445618:4 7.47G9GG4 20.4694895 7.4829242 20.4930015 7.4888724 20.5182845 7.4948113 20.5420:486 7.5007406 20.56G9038 7.5C66007 20 5912003 7.5125715 20.0155281 7.51817:40 20.6397074 7.5243052 20.0039783 7.5302482 20.0881009 7.5361221 20.7123152 7.5419807 20.7304414 7.5478423 20.7005395 7.5530888 20.7840097 7.5595203 20.8086520 7.5058548 20.8:420007 7 5711743 Reciprocals. .0026809G5 .002015197 .C02C06GG7 .C02G59574 .CC2G52520 .C02645503 .002008522 .0C2G34579 .0C2G24G72 .002G17801 .C02G1C9G6 .C02G041G7 .OC2597403 .CC2590G74 .002583919 .002577320 .002570694 .002564103 .002557545 .002551020 .002544529 .002538071 .002531610 .002525253 .002518892 .CC2512063 .002500206 .002500000 .002493706 .002487502 .002481390 .002475248 .002109136 .002463054 .002457002 .002450980 .002444988 .002439024 .002433090 .002427184 .002421308 .002415459 .002409639 .002403846 .002398082 .002392344 .002380635 .002380952 .002375297 .002369668 .002364006 .002358491 .002352941 .002347418 .002:341920 .002336449 .002331002 .002P25581 .002320186 .002314815 .00230940)9 .002364147CUBE ROOTS, AND RECIPROCALS. n i". 1 Squares. Cubes. Square Roots. Cube Roots. [ Reciprocals. 2)5 130225 82312375 '20.8566536 7.5769849 .00229SG51 4)3 155096 82ool&>5 20.8606130 7.5827865 .002293578 437 83453453 20.9045450 7.5885793 .002288380 4)3 1515-44 84027672 20.9284495 7.5943633 .002283105 43d 192721 84604519 20.9523268 7.6001385 .002277904 410 193600 85184000 20.9761770 7.6059049 .002272727 441 194481 85766121 21.0000000 7.6116626 .00226 <‘574 443 195364 86350888 21.0237960 7.6174116 .002202443 443 190249 86938307 21.0475652 7.6231519 .002257336 444 197136 87528384 21.0713075 7.6288837 .002252252 445 193025 88121125 21.0950231 7.6346007 .002247191 446 198916 88716536 21.1187121 7.6403213 .002242152 447 199S09 89314623 21.1423745 7.6460272 .002237136 443 200704 89915392 21.1660105 7.6517247 .002232143 449 201601 90518849 21.1896201 7.6574133 .002227171 450 202500 91125000 21.£132034 7.6630043 .002222222 451 203401 91733851 21.2)67606 7.6687665 .002217295 452 204304 928454)8 21.2802916 7.6744303 .002212389 453 205209 92959677 21.2837907 7.6300857 .002207506 454 206116 9357G6G4 21.3072753 7.G857323 .002202643 455 207025 94196375 21.3307290 7.6913717 .002197802 456 207936 94318816 21.35415G5 7.6970023 .002192982 457 203849 95443993 21.3775583 7.7020246 .002188184 458 209764 96071912 21.4009316 7.7082-388 .00218:4106 459 210681 90702579 21.4242853 7.7138443 .002178649 460 211600 97836000 21.4476106 7.7194426 .002173913 431 212521 97972181 21.4709106 7.72503^.0 .002169197 432 218444 98011128 21.4941853 7.730G141 .002164502 ■ 433 2143G9 99252347 21.5174343 7.7361877 .002159827 464 215296 99397344 21.540G592 7.7417532 .002155172 4G5 210225 100544025 21.5638587 7.7473109 .002150533 406 217156 101194096 21.5870331 7.7528006 .002145923 467 218039 10184755) 21.6101838 7.7584023 .002141328 463 219034 102503232 21.038)077 7.7639361 .002136752 469 219951 103161709 21.G5G4078 7.7694620 .002132196 470 220900 103823000 21.6794834 7.7749801 .002127660 471 221841 104487111 21.7025344 7.7804904 .0)2123142 472 222784 105154043 21.7255610 7.7859928 .002118644 473 223729 105823817 21.7485632 7.7914875 .002114165 474 224676 106490424 21.7715411 7.7909745 .002109705 475 225625 107171875 21.7944947 7.8024538 .002105263 476 226576 107850176 21.8174242 7.8079254 .002100840 477 227529 1085318)3 21.8403297 7.8133892 .002096436 478 223434 109215852 21 86:32111 7.8188456 .002092050 479 229441 109902239 21.8860686 7.8242942 .002087683 480 230400 110592000 21.9089023 7.8297353 .002088333 481 231361 111234041 21.9317122 7.8351688 .002079002 482 232324 111980168 21.9544984 7.8405949 .002074689 483 233289 112078587 21.9772610 7.84601:84 .002070393 484 234256 118379904 22.0000090 7.8514244 .002006116 485 235225 314084125 22.0227155 7.8568281 .002061856 486 236196 114791256 22.0454077 7.8622242 .002057613 487 237169 115501803 22.0680765 7.8G7C130 .002053383 488 238144 116214272 22.0907220 7.8729944 .002049180 489 239121 116930169 22.1133444 7.8783684 .002044990 490 240100 117649000 22.1359436 7.8837-352 .002040816 491 241031 118370771 22.1585198 7.8890946 .002030600 492 242004 119095488 22.1810730 7.8944463 .002032520 493 243019 119823157 22.2036083 7.8997917 .002028398 494 244036 120553784 22.2261108 7.9051294 .002021291 495 245025 121287375 22.2485955 7.9104599 .002020202 496 i 243016 122023936 22.2710575 7.9157832 .002016129 SQUARES, CUBES, SQUARECUBE ROOTS, AND RECIPROCALS 17 No. [ Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 559 312481 i 174676879 23 6431808 8.2376014 001788909 5G0 313000 175010000 23 6643191 8.2425706 .001785714 551 314721 176558481 23.0854380 8.2474740 .001782521 502 315844 177504323 23.7005392 8.2523715 .001779359 503 310909 1184o*iO“i i 23.7276210 8.2572633 .001776199 504 318096 179406144 23.7486842 8.2621492 .001772050 505 319225 180302125 23.7097286 8.2670294 .001709912 506 320350 181321496 23.7907545 8.2719039 .001700784 507 321489 182284203 23.8117018 8.2767726 .001703008 50.3 322024 183250432 23.8327500 8.2810355 .001700503 509 323701 184220000 2_>. bou * wU) 8.2864928 .001757409 570 324900 185193000 2:3.8746728 8.2913444 .001754386 571 320041 180109411 23.8950063 8.2961903 .001751313 572 327184 187149248 23.9165215 8.2010304 .001748252 573 323:329 ISO)£2517 23.9374184 8.£058051 .001745201 574 329476 180119224 23.9582971 8.3100941 .001742160 575 330025 190109G75 23.9791576 8.3155175 .001739130 576 331776 191102976 24.0000000 8 2203353 .001736111 577 332929 192100033 24.0208243 8.2251475 .001733102 578 334084 193100552 24.0416306 8.2299542 .001730104 579 335241 194104539 24.0024188 8.3347553 .001727116 580 336400 195112000 24.68:31891 8.3395509 .001724138 581 337561 196122941 24.1039416 8.3443410 .001721170 582 338724 197137368 24 1246702 8.3491256 .001718213 583 339889 198155287 24.1453929 8.27X9047 .001715266 584 841056 199176704 24.1000919 8.2786784 .001712329 585 342225 200201G25 24.1807732 8XC244G6 .001719402 586 343396 201230056 24.2074309 8.2082095 001700485 587 344569 202262003 24.2280829 8.3729008 .COI703578 588 345744 203297472 24.2487113 8.377 71S3 .CO1700080 589 340921 204336409 24.2093222 8.2824053 .001097793 590 348100 205379000 24.2899156 8.3872005 .001694915 591 349281 206425071 24.310491G 8.2919423 .001092047 592 350464 207474688 24.3310501 8.2900729 .001089189 593 351649 208527857 24.3515913 8.4013981 .101080341 594 352836 209584584 24.3721152 8.4061180 .001083502 595 354025 210044875 24.3920218 8.4108326 CO:080672 596 355216 211708736 24.4131112 8.4155419 .C01077852 597 350409 212770173 24.4335834 8.4202400 .001075042 598 357004 213847192 24.4540:185 8.4249448 .CO1072241 599 358801 214921*99 24.4744705 8.4296383 .001009449 coo 360000 21GOOCCOO 24.4948974 8.43432C7 .001G6CCG7 COI 361201 217081801 24.5153013 8.4390098 .001003894 C92 3G2404 218167208 24.5350883 8.4436877 .001001130 C >3 363609 219256227 24.5500533 8.4483C05 .001058375 C04 364816 220:348864 24.5764115 8.47X0281 .001055639 G05 300025 221445125 24.5967478 8.4570906 .001052893 006 £07236 222545016 24.617CC73 8.4023479 .0ol050105 0)7 368449 223G48513 24.6373.00 8.467CC01 .001047410 COB 309664 22-1755712 24.C57C5G0 8.471C471 .C01044737 C09 3* 0881 225806523 24.C779254 8.4702892 .COI042030 610 372100 22GC8irro 24 0981781 8.4809201 .001639344 Cll — 225005131 24.7184142 8.4855579 .001030001 012 £71544 229220028 24.7386338 8.4901848 .001033987 013 oioi09 230346397 24.7588308 8.4948005 .001031321 014 370996 231475544 24.7790234 8.4994233 .001028604 015 378225 232608375 | 24.7991935 8.5040350 .001020010 C1G 379456 233744S9G | 24.8193473 8.5086417 .001023377 017 380689 234885113 i 24 8394847 8.5132435 .0010207-: 6 618 381924 230029032 24.8596058 8.5178403 .001018123 619 383161 2*37176659 24.8797106 8.5224321 .001615309 620 384400 1 238328000 ! 24.8997992 8.5270189 .001012903SQUARES, CUBES, SQUARE ROOTS O i- tO — O — JO l- »O to to — to ig ig ic o •f tO t-* ig PJ ?? T 5b N CO ^ Cl V) iQ OO^OOCOCCOO^ OXWi'^XOONH NKUO'tC-QiCC'r CO tO -+ 55 »ft cxNt-xQ^oci'i5 o^cc — osxaCcs — co c ^ C3 r. ^ J i - jo o o o i- 3 c< jo 31?5 o *r> cc-r.txxJ 1.7 x ci t- -f xo? X COi-CH-CJXr?xW35 I'JOXCM-W J5ic:»x 'TOi-XOI-XON't' — XiC ^Ot-OOC-i* OXCXX-rT»'»*). *o — c> i-*r:o i- -t* ci os i—r- ci ct <- -r c)5^*coo(-o?:o x to eg — 3 so-a* ci 051- LC^ox'JXr'r.^’f :>oxicx-C‘.i-r^ o x to oaox coxxM-Ni'OXd c ic ic i: o c? r: ^ r: Cf cj c^cf « »h ^ i o cr. r. r. ~ ~ x x xxf-i-H- i«*3o §38 83333SSS33 83SSSSSS8S 8838 'I 4.7 O 1“ £££<£ 566-565 888 888 OQ-'XN^ONN OXOt-Clt-CH-O X Q i’ O « t- JO X CO X O’-iNWX’tCiiOO — to O iO 05 -t* X CO X CO CO -f **r “T lO *•''0 tO tO N(»XXf,** CO 1.0 1.0 lO lO lO . ' O iO O lO B iO iO lO lO lO tO W OCO^NXQOirOO XXXO^Xi5Nf I' x oi g t c? x i- oi oj x qd *o *g o oo> o co i- v — o x p ci o5 -r io o -* co oi o x Cl © -f f- t- 3* 3? 11- X Tf '-O tO r? w SB 'mi - -• - - I - ci co 05 — x — oti! 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W -/wwwwww.-.-.- Cl M Cl Cl CM Cl Cl Cl Cl & Cl Cl Cl C$ 7l Cj C^ Cl Cl 6l Cl Cl Cl Cl Cl Cl Cl Cl Cl O-TN ^JOOXOJO'OXCJO Oi-O-iONTOO J»JO to to to to to to ?-)->- g-xr- C X C^** 7 0> C. O JO 7 O 1.7 C> C5 O Cl 05 toglg i _ »^ . r, x x x x X Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl to NX 7 ^ 0) to X 05 c? ci ci 5i ?i ?i ci ci ?i Ci 7«/ f—i ~ i—i T/ t - t- TO 1-7 V7 Ct-CCiCXNNl-X eg — -r — c> to 1.7 x 17 to ci t- to ci x -j x to -r i-^-'-rx — i7C5Ci*3p ocicc-^toi-xo — eg C5JOWCOJOKCOWWSO C7X Ci-cc ci »r X CC »—•^,05tOi7tor5-*i^ — xcic-oMt-cix^ eg to ci v cox — ectox — 5? to lOOXOC-JO-O XXXCC005 0-- 0 ig to — : — ^ ^ c> 05 to ^ 0 — O — *<• 05 tO pill 05 — — O — — 05 tO i7 tO 05 -t — CO — ~*C*5t0iCt005 .r— O — — *2 X O Cl — to 05 C> OX C> to 07 — X Cl l-CM-Cl X — Ct tO 7 7. t> 7 O. t> 1* T *) V O» iO — — * - <»M r**' 1 _ OMJOCOWCOJOJCffl JOCOJO'C"tfC77 77 -X -7" -T "»T '»T *t» "T" X '*T ^ 'X '*T jig X Cl »7 05 ci 1.7 05 c> to 05 ec to X 05 — C> — jC to X O: — — 1.7 iC 47 »"“ i7 47 47 47 tO ^CT'C’tTCC’ri' — c? x — »7 to r- tr 05 ct — c? to — »7 to(-x 05 o —ci Cl Cl C> C> Cl Cl Cl Cl Cl — CO CC CC 7^ ^ C^ C7 77 — — — coocoooooo o o to to to to to to to to to to to O — Cl CO — 17 CO l - X 17 i7 i7 i7 i7 17 i7 17 i7 M CC — 17 tO i- X C5 CO - 0 jO to to ]0 W to to 1-1; MJC^l^tONXCt CO — r io io iz £ io ^ xx G£3 | 4G51.yl I 3175H5GSCUBE ROOTS, AND RECIPROCALS. 19 1 fio. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 083 1 488489 318311937 26.1342087 8.8065722 .001464129 oSl 407858 320913.X 4 20.1539937 8.8108681 .001401983 685 409225 321413125 20.1725047 8.8151598 .001459854 680 470590 322o2885(| 20.1918017 8.8194474 .001457720 687 471909 321242703 28.2100848 8.8237307 .001455004 088 473344 32500007.» 20.2297541 8.8230099 .001453483 689 474721 32<083, v9 20.2488095 8.8322850 .001451379 G90 478100 328509000 26.2078511 8.8365559 .001449275 GDI 477481 329939071 26.2808489 8.8408227 .001447173 092 478804 331373088 26.3058929 8.8-450854 .001445087 G93 480249 3-32812557 20.3248932 8.8493440 .001443001 694 481038 334255384 26.3438797 8.8535985 .001440922 695 483025 335702375 26.3828527 8.8578489 ‘ .001438849 698 484110 337151530 26.3818119 8.8020952 .00143G782 697 485809 33X08873 26.4007578 —1 .001434720 693 487201 340008392 26.4196896 8.8,05757 .001432005 699 488001 541532099 26.4380081 8.8748099 .001430015 700 490000 343000000 20.4575131 8.8790400 .001428571 701 491401 3444 ,'2101 26.4764046 8.8832001 .001420534 702 492804 345948408 20.4952826 8.8874882 .001424501 703 494209 34 ,'428927 20.5141472 8.8917003 .001422475 704 495010 345913904 20.5029989 8.8959204 .091420455 705 497025 350402025 20.5518301 8.9001304 .001118440 703 4JJ436 351895810 20.5700005 8.9043306 .001110431 707 499849 35-13932 43 20.5894710 8.9085387 .051114427 703 501204 3548942'13 >. 0082094 HBii .001 112429 709 502081 .350400829 2 0 02< 0® 8.9109311 .00111043r 710 .501100 557911'530 20.8458252 8.9211214 .001103151 711 535521 559825151 20.0045833 8.9X50,3 .001-100170 712 500944 300944128 20.0833281 8.9294902 .001104494 713 503X9 302407097 20.7020598 8.9330087 .001402525 714 509790 303994344 26.7207784 8.9378433 .091400500 715 511225 3G5525875 26.7394839 8.9420140 .001398001 718 512856 3G7061696 26.7581763 8.9401809 .001390648 717 514089 308001813 26.7768557 8.95034:33 .001394700 718 515524 310140232 20.7955220 8.95-45029 .001392758 719 510981 3 »1094959 20.8141754 8.9580581 * .001390821 720 518400 373248030 26.8328157 8.9028095 .001388889 721 519811 3,1805301 26.8514432 8.9009570 3 01386963 m : 521284 3, <>30 » 048 26.8700577 8.9711007 .001:185042 723 522729 377933007 26.8886593 8.9752406 .001383126 724 524178 379503124 26.9072481 8.9793766 .001381215 725 525025 581078125 26.9258240 8.9835089 .001379310 728 527070 382G57170 26.9443872 8.9870373 .001377410 727 528529 581240583 26.9629375 8.9917620 .001375516 728 529984 385828352 20.9314751 8.9958829 .001373026 729 531441 387420489 27.0000000 9.0000000 .001371742 730 532900 589017000 27.0185122 9. COil 13 4 .001369863 731 534301 390017891 27.03Ì0117 9.0032229 .001307989 782 535824 392223108 27.0554985 9.0123283 .001300120 733 537289 393832837 27.0739727 9.0104309 .001364256 734 538756 39-5-140904 27.0024344 9.0205293 .001302:398 735 540225 397005375 27.1108834 9.C24G2S9 .001300544 738 541090 398088250 27.1293199 9.0287149 .00135809G 737 543189 400315553 27.1477439 9.0328021 .C91350852 738 544644 401947272 27.1601554 9.0-308857 .CO1355014 739 540121 40:3585419 27.1&45544 9.0409055 .051:453180 740 547800 405224900 27 2029410 9.0450419 .001351351 741 549081 400809021 27.2213152 9.0491142 .001349528 742 550584 Hi 27.2396769 9.0531831 .001347709 743 552049 410172407 27.2580203 9.0572482 .001345895 744 553536 411830734 27.2703634 9.0013098 .00134408620 SQUARES, CUBES, SQUARE ROOTS No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 745 555025 413493625 27.2946881 9.0653677 .001342282 740 556516 415100930 27.3130006 9.0694220 .001:310-18:3 747 556009 410832723 27. ail3007 9.0734726 .001:338688 74a 559504 418508992 ::7.3495887 9.0775197 .00ia*36898 749 561001 420189749 2 <’.367 8644 9.0815631 .001335113 750 562500 421875000 27.3861279 9.0856030 .0013333a3 751 564001 423504751 27.4043792 9.0896392 .0i 1331558 752 565501 425259008 27.4226184 9.0536719 .001329737 753 567009 426957777 27.4408455 9.0977010 .001328021 754 568516 428661064 27.4590604 9.1017265 .001526260 755 570025 430308875 27.4772033 9.1057485 .001324503 756 571536 432081216 27.4954542 9.1097669 .001322751 757 573019 433798093 27.5130330 9.1137818 .001321004 758 574564 435519512 27.5317998 9.1177931 .001819201 759 576081 437245479 27.5499546 9.1218010 .001317523 7G0 577600 438970000 27.5680975 9.1258053 .001815789 701 579121 4 407'11081 27.5802284 9.1298061 .101314060 H 580544 412450728 27.6043475 9.ia38034 .001312:336 763 582169 411194947 27.02-4546 9.1377971 .001310616 764 583696 445943744 27.6405199 9.1417874 .001308901 765 585225 417*697 1 25 27. (>586334 9.1457742 .001807190 766 586 756 419455096 27.0707050 9.1497576 .001305483 767 53S2S9 451217663 27.6947618 9.1537375 .001803781 768 580821 452984832 27.7128129 9.1577139 .001502083 769 591361 454750009 27.7308492 9.1616869 .001300390 770 592900 456533000 27.7488739 9.1656565 .001298701 771 501111 458311011 27.7668808 9.1696225 .001297017 772 595984 460099048 27. <'848880 9.1735852 .C01295337 US 597529 461889917 27.8028775 9.1775445 .001293601 774 599076 463684824 2<.8208555 9.1815003 .001291990 775 600625 4654843<5 27.8:388218 9.1854527 .001290:323 776 602176 46<2S8.)7 6 27.85677 66 9 1894018 .001288660 4 é 4 603729 409097433 27.8747197 9.19:33474 .001287001 778 605284 470910952 27.8926514 9.1972897 .001285:347 779 606811 472729139 27.9105715 9.2012286 .001283697 7*80 608-100 47455200(4 27.9281801 9.2051641 .001282051 7*81 609961 47 037 9. >41 27.9103772 9.2090962 .001280410 7’S2 611.>24 478211768 27.9642629 9 2130250 .001278772 IBB 613089 480018687 27.9821372 9.2169505 .001277139 ■ 614656 481890304 28.0000000 9.2208720 .001275510 785 616225 4837 36625 28.0178515 9.2247914 .001273885 7*8(5 617796 485587 6; >6 28.0:356915 9.2287068 .001272265 787 619369 4874434(54 28.0535263 9 2326189 .001270648 788 620911 48930:1872 28.0713877 9 2805277 . 0012( >90:36 789 622521 491169069 28 0891438 9 2404333 .001267427 790 624100 493039000 28.1000386 9.244:3855 .0012(55823 791 625681 491913671 28.1247222 9 2482344 .( 01264223 792 627264 4967930S8 28.1421946 9.2521300 .001*262626 793 628819 498677257 28.100*57 9.2500224 .0012610:34 79 5 630136 500566184 28.1780056 9.2599114 .001259446 795 (>32025 502459875 28.1957444 9 2637973 .(10 257862 766 633610 504358330 28 21:34720 9.26®98 .001256281 797 035209 506261573 28.9311884 9.2715592 .001254705 798 030804 508109592 28.2488938 9.2754352 .001253ia3 799 638401 510082:399 28.2005881 9.2793081 .01)1251564 890 610000 512000000 28.28-12712 9.2831777 .00125(XXX) 801 041001 513922401 28.30104:34 9.2870410 .(X)l 248439 802 613201 515819608 28.3196045 9.2909072 .001246883 803 644809 517781627 28. $37 2546 9.2947671 .(X)1245a30 804 646410 519718464 28.3548038 9 2986239 .001243781 805 806 648025 619636 521660125 523606616 28 8725219 28.3901391 9.8024775 9.30C>3278 .001242236 1 .001240695 jCUBE ROOTS, AND RECIPROCALS 21 No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. 807 651 £49 525557943 23.4077454 9.3101750 .001239157 803 652804 527514112 28.4253403 9.3140190 .001237024 809 654481 529475129 28.4429253 9.3178599 .001286094 810 656100 531441000 28.4604989 9.3216975 .001234568 811 657721 533411731 28.47'80017 9.3255320 .0012:33046 812 659344 535387328 28.4956137 9.3293634 .001231527 813 600969 5371307797 28.5131549 9.3331916 .00123001.2 814 662596 539:353144 28.5306852 9.3370167 .001.228501 815 604225 541343375 28.5482048 9.3408386 .001226994 816 665856 543338196 28.5657137 9.3446575 .00i225490 817 667489 54533S513 28.5832119 9.3484731 .001223990 813 669124 547343432 28.6006993 9.3522857 .001222494 819 670761 549353259 28.0181760 9.3560952 .001221001 820 672400 551368000 28.6356421 9.3599016 .001219512 831 674041 553:387001 28.0530076 9.3637049 .001218027 822 675684 555412248 28.6705424 9.3675051 .001216545 833 677329 5574417(57 28.0879706 9.3713022 .001215067 824 678976 5594702.24 28.7054002 9.3750963 .001213592 825 680625 501515025 28.7228132 9.3788873 .001212121 826 682276 563559976 28.7402157 9.3826752 .001210654 827 683929 505609283 28.7576077 9.3864600 .001209190 823 685584 567663552 28.7749891 9.3902419 .001207729 829 687241 569722789 28.7923001 9.3940206 .001206273 830 688900 571787000 23.8097205 9.3977964 .001204819 831 690561 573856191 28.8270706 9.4015691 .001208309 832 692224 575930303 28.8444102 9.4053387 .001201923 833 693889 578009537 28.8617394 9.4091054 .001200480 834 695556 580093704 28.8790582 9.4128690 .001199041 8:35 697225 582182375 23.8963666 9.4160297 .001197605 836 693890 584277056 28.9130646 9.4203873 .001196172 837 700569 580370.353 28.9309523 9.4241420 .001194743 833 702244 588480472 28.9482297 9.4278036 .001193317 839 703921 590589719 28.9654907 9.4316423 .001191895 840 705600 592704000 28.9827535 9.4353880 .001190476 841 707281 594823321 29.0000000 9.4301307 .001189001 842 708904 596947688 29.0172363 9.4423704 .001187648 843 710049 599077107 20.0344623 9.446G072 .001186240 844 712336 601211584 29.0516781 9.4503410 .001184834 845 714025 603351125 29.0688837 9.4540719 .00118:3432 846 715716 6054957:36 29.0860791 9.4577999 .001182033 847 717409 607645423 29.1032644 9.4015249 .001180638 843 719104 609800192 29.1204396 9.4652470 .001179245 849 720801 611960049 29.1376046 9.4689661 .001177856 850 722.500 614125000 29.1547595 9.4726824 .001176471 851 724201 610295051 29.1719043 9.4763957 .001175088 852 725904 618470208 29.1890390 9.4801061 .00J173709 853 727009 620050477 29.2061637 9.4838136 .001172333 854 729316 622335864 29.2232784 9.4875182 .001170960 855 731025 625020375 29.2403830 9.4912200 .001169591 856 732736 627222016 29.2574777 9.4919188 .001108224 857 734449 629422793 29.2745623 9.4086147 .001160861 853 730104 631028H2 29.2916370 9.5023078 .001165501 859 737831 633839779 29.30S7018 9.5059980 .001104144 860 739609 636056000 29.3257566 9.5096854 .001102791 801 741321 638277381 29.3428015 9.5133699 .001161440 862 743044 640503928 29.3598305 9.5170515 .001160093 803 744769 642735047 29.3708616 9.5207303 .001158749 804 746496 644972544 29.3938769 9.5244063 .001157407 805 748225 647214025 29.4108823 9.5280794 .001156069 806 749956 649461896 29.4278779 9.5317497 .001154734 807 751089 651714363 29.4448637 9.5354172 .001153403 008 753424 053972032 29.4018397 9.5390818 ,00115207422 SQUARES, CUBES, SQUARE ROOTS, No. Squares. Cubes. Square Hoots. Cube Roots. Reciprocals 860 755161 656334900 20.4788050 0.5427437 .001150748 870 750900 058503000 20.4057624 . 0.5464027 .001149425 871 758641 CG07V6oll 29.5127001 9.5500589 .001148106 872 760384 663054548 20.5200401 9.5507123 .001140780 873 762129 665338617 20.54657:34 9.55736:30 .001145475 874 763876 667627624 20.50:14010 9.5610108 .001144165 875 765625 660021875 29.5803589 9.5646550 .001142857 876 767376 672221376 20.5572572 9.5682982 .001141553 877 760129 6745261133 29.0141858 9.5710377 .001140251 878 770884 676836152 20.6.310048 9.5755745 .001138952 870 772641 670151439 20.0470342 9.5792085 .001137656 880 774400 681472000 29.6647039 9.5828397 .001136364 881 776161 683797841 20.6816442 9.5864682 .001135074 882 777924 680128068 20.6084848 9.5900939 .001133787 883 770689 688465387 20.7153159 9.5037160 .0-. 1132503 884 781456 61X1807'104 20.7321375 9.5073373 .001131222 885 780225 693154125 29.7489493 9.6009548 .001120044 886 784006 695506456 20.7657521 9.6045696 .001128008 887 •786769 69;864103 20.7825452 0.6081817 .001127396 gss 788544 700227072 29.700:3280 9.6117011 .001126126 880 700321 702595360 29.8101030 9.6153077 .001124850 890 792100 704060000 20.8328678 9.6190017 .001123596 801 703881 707347971 20.8400231 0.0220030 .0011223:34 8-11 795664 700732288 20.806:3600 9.6262016 .001121076 803 797440 712121057 20.8831056 9.6207075 .(X)ll 10821 804 799236 714516084 20.8008328 9.0333907 .041118568 805 801025 716017375 20.0165506 9.6360812 .001117318 806 802816 7103231:36 20.0332501 9.6405600 .001116071 807 804600 721734273 20.9400583 9.6441542 .001114827 808 806404 734150702 29.0066481 9.6477567 .001113586 800 808201 726572699 29.9833287 9.6513106 .001112347 000 810000 720000000 30.0000000 9.6548038 .001111111 001 811801 7731432701 30.0166620 9.0554(84 .001109878 003 813604 733870808 30.0:3.3.3148 9.0C20403 .001108647 903 815400 736.314327 30.0400584 9.0050096 .001107420 904 817216 7:38763264 30.0005928 9.6001762 .0011( 6105 905 819035 741217625 30.0832170 1.6727403 .001104072 906 820836 743677416 30.9)98330 9.0763017 .(X)110.3753 907 823749 746142643 30.1104407 9.0798604 .(K)l 1(82530 908 834464 748613312 30.1:330383 9. (>834166 .001101322 900 820281 751080429 30.1406260 9.0869701 .001100110 910 828100 75.3571000 30.1662063 9.6005211 .001008001 911 829021 756058)31 30.1827765 9.0040004 .001007005 913 83174 4 758550528 3(3.190:3377 9. (>076151 .001090191 913 83.3569 761048107 30.2158800 9.7011583 .001095290 911 835306 703551044 30.2324329 9.7046989 .(X) 1(8)4(802 915 837225 706060875 30.2480660 9.7082300 .001 (8.82806 916 83905(5 768575206 30.2654010 9.711772:3 .001091703 917 840880 771095213 30.2820079 0.7153051 .001(8.X)513 918 842724 773020032 80.2085148 9.7188.3M . (XU 080:325 910 844561 770151559 30.3150128 0.7223(>31 .001088130 020 846100 778688000 30.3315018 9.7258883 .001086057 931 848241 781220961 30.3470818 9.7204100 .001085776 023 850084 783777448 30.8644520 9.7329309 .001084599 933 851929 786:330467 80.3800151 0.7364484 .001083423 924 853776 7SSSS<.X>24 30.307368:3 9.73000:34 .00108x8251 025 855025 70145.3135 30.4138127 0.7434758 .001081(881 026 857476 704022776 80.4302481 0.7460857 .001070014 927 859329 700597083 30.4466747 0.75040:30 .001078740 928 861184 700178752 80.4630924 0.7530070 .(X)1077586 929 86304] 801765080 80.4705013 9.7575002 .(X) 1076426 930 864000 804357000 30.4059014 9.7010001 .001075260CUBE ROOTS, AND RECIPROCALS. 23 No. Squares. Cubes. Square Roots. 1 Cube Roots. 1 Reciprocals. 931 860701 806954491 30.5122926 9.7644974 .001074114 932 868024 809557568 30.5^86750 9.7079922 .001072961 933 870489 812100237 30.5450487 9.7714845 .001071811 sm 872350 814780504 30.5614136 9.7749743 .001070664 933 874225 817400375 30.5777697 9.7784616 .001009519 930 870096 820025856 30 5941171 9.7819466 .001008376 937 877909 822650953 30.6104557 9.7854288 .0010(57236 938 879844 825293672 30.6267857 9.7889087 .001000098 939 881721 827930019 30.6431069 9.7928801 .001004903 aio 883600 830584000 30.6594194 9.7958611 .001063830 941 885481 833257021 30.6757233 9.79933S6 .001002099 942 887:364 835896888 30.6920185 9.8028036 .001001571 943 889249 838501807 30.7083051 9.8062711 .001000445 944 891130 841232584 30.72458.30 9.8097302 .001059322 945 893025 843908625 30.7408523 9.81319H9 .(01058201 940 894916 840590530 30.7571130 9.8166591 .001(57082 947 890809 849278123 30.7733651 9.8201169 .001055966 948 898704 851971392 30.7896(86 9.8285723 .001054352 949 900001 854070349 30.8058436 m—hh .001058741 950 902500 857375000 30.8220'00 9.8304757 .001052632 951 904401 860085351 30.8382879 9.8339238 .001051525 952 900:404 862801408 30.8544972 9 8873095 .001050420 953 908209 8òo»)(v*) 1 i i 30.8706981 9.8408127 .001049318 954 910110 868250004 30.8h68904 9.8442586 .001048218 955 912025 87C983875 30.9030743 9.84769-20 .001047120 950 913930 873722816 30.9192197 9.8511280 .001046025 957 915849 876467493 30.9354166 9.8545617 .601044932 958 917704 879217912 30.9515751 9.8579929 .001043841 959 919081 881974079 £0.9677251 9.8614218 .001042753 960 921000 884736000 30.9838668 9.8648483 .001041667 961 923521 887503681 31.0600000 9.8082724 .001040583 9(32 925444 890277128 31.0161248 9.8716941 .001089501 903 927369 893050347 31.0322413 9.8751135 .001088422 904 929296 8958411414 31.0483494 9.8785305 .001037344 965 931225 898632125 31.0644491 9.8819451 .C0103G269 966 933156 901428690 31 0805405 9.8853574 .001035197 967 935089 904231063 31.0966236 1 9.8887673 .COi034126 968 937024 907039232 81.1126984 9.8921749 .001038058 909 938961 909853209 81.1287648 9.8955801 .001031992 970 940900 912673000 81.1448230 0.8989830 .C01C3C928 971 942841 915498611 81.160729 9.9023835 .001029866 972 944784 918330048 SI.17 CO145 9.9057817 .001028807 973 946729 921167317 31.1929479 9.9091776 .001027749 974 948676 924010424 31.2089731 9.9125712 .COI026694 975 950025 920859375 31.2249900 9.9159024 .001025641 976 952570 929714176 81.2409987 9.9193513 .001024590 977 954529 932574*33 81.2569992 9.9227 37 9 nm:i 978 95(4484 9-35441352 81.2729915 9.9201222 .001022495 979 958441 938313739 SI.2889757 9 9295042 .001021450 980 960400 941192000 31.8049517 9.9328839 .C01C2C4G8 981 962361 944076141 man ' 9.9302013 .001013308 982 964324 946966168 31.3368792 0.9396303 .001018880 983 966289 949862087 31.3528308 9.948CC92 .001017294 984 908250 9527C-3904 31.8687743 9.9403797 .001016260 985 970225 955671(525 31.3847097 9.9497479 .001015228 986 172100 958585256 31.4000869 9.9531138 .001014199 987 974169 961504803 31 4165561 9 9504775 .001018171 983 976144 964430272 31.4021073 9 9598389 .001012146 983 978121 907301009 31.4488704 9.9031981 .001011122 oro 980100 970299000 31.4642654 9.9CC5549 Hn 391 9*2081 973242271 31.4801525 9 9099095 .colorerà 992 984064 970191488 31.4960315 9.9732619 .OCJCtf?24 SQUARES, CUBES, SQUARE ROOTS, ETC. No. Squares. Cubes. Square Roots. Cube Roots. Reciprocals. m 98G049 979140057 31.5119025 9.9700120 .001007049 994 9.88030 932107784 31.5277055 9.9799599 .001006030 995 990025 985074875 31.5430209 9.9833055 .001095025 990 992010 938047930 31.5594077 9.9800488 .001034910 997 991009 991020973 31.5753003 9.9899900 .001003009 HB 990004 934011992 31.5911380 9.9933289 .001002004 9)9 99.8941 9)7002499 31.0009013 9.996G05G .001001091 100J 1039090 1900009099 31.0227700 10.0000000 .001000090 1091 1002901 1003003091 31.03S5840 10.0033322 .0009990010 1943 10)4044 1000012003 31.6543830 10.0000025 .0009980040 ■Rb 1090009 1009027027 31.0701752 10.0099890 .0009970000 1041 1008010 1012 '48004 31.0859590 10.0133155 .0009900159 1095 1010025 1015075125 31.7017349 10.0100389 .0009950249 1000 1012030 101810821(5 31.7175039 10.0199001 .0009940358 1007 1014049 1021147 34 5 31.7382935 10.0232791 .0009930487 1003 101G004 1024192512 31.7499157 10.0205958 .0009920035 1009 1018081 1027243729 31.7047003 10.0299104 .0009910803 1010 • 1020100 1039301009 31.78049« 10.0332223 .0009900990 1011 HB 1033304331 31.7932232 10.0305330 . 0009891197 1012 1021114 1030135723 31.8119174 10.0398410 .0009881423 1013 1020169 1039504197 31.8270049 10.0431409 .0009871003 1014 1028190 1012594744 31.8433006 10.0404500 .0009861933 1015 1030225 1015078375 31.8590040 10.0497521 .0009852217 1016 1032250 1918772440 31.8747549 10.0530514 .0009842520 1017 1034289 1051871913 31.8904374 10.0503485 .0009832812 1018 1030324 1051977832 31.9001123 10.0590435 .0009823183 1019 1038-561 1053039359 31.9217794 10.0029304 .0009813543 1020 1040449 1001208099 31.9374388 10.000*2271 0009803922 1021 1042141 100433-2201 31.9530900 10.0095156 .0009794319 1022 10111.84 1007102048 31.9087317 10.0728020 .0009784736 1023 1010529 1070599107 31.9843712 10.0700801 .0009775171 1021 104 3570 1073741S24 32.(090909 10.07935)84 .0009705025 1025 1050325 1070894025 32 0150212 10.0820181 .0009750098 1020 1052.JÎ0 108044557(5 32 051*2318 10.085925-2 .0009740589 1027 1051729 ■ÜB 32.0108497 10.0892919 .0009737098 1028 1050781 10-4037.5952 32.0024391 10.0924755 .00097270*26 1029 1058841 101 >547339 32.0789298 10.0957409 .0009718173 1030 1000990 1042727099 3.2.0930131 10.0990103 .0009708738 1031 1062901 1095412741 32.1091887 10.102-2835 .00090993*21 1012 1005021 1044101703 3-2.1*247503 10.1055187 .00 90899*22 1033 1007089 11025029 57 32.1403173 10.10881171 .000908054*2 1031 1009150 1105507891 32.1558704 10.1120720 .0009671180 1035 1071225 1103717375 32.1714159 10.1153314 .0009001836 1030 107321)3 111î 93 4050 32.1869539 10.1185882 .0009052510 1037 1075309 1115157053 3*2.2021814 10.1*2184*28 .0009643202 1038 1077414 1118530872 3*2.2180074 10.1*250953 .0009033911 1039 1079521 1121022519 32.2335*229 10.1*283457 .0009024039 1010 1081009 1124801040 32.2490310 10.1315941 .0009615385 1011 1083031 1128U1021 32.2045310 10.1318403 .0009000148 1012 10357.)! 1131300038 3*2.2800243 10.1380845 .00095909:29 1013 1087.349 11310-20507 3*2.2955105 10.1413*206 .0009587738 1044 1084940 113789*5184 32.3109S8S 10.1445007 .0009578;) 44 1015 1092425 1141100125 3*2.3*204598 10.1478047 .0009509378 1010 1041110 114 4445 53(5 32.3419233 10 1510400 ,0009500229 1017 1090209 1147730823 3*2 3573794 10.154*2744 .0009551098 1018 1098304 1151022592 3*2 37*28481 10.1575062 .0009541985 1019 1100101 1154320049 32.3882095 10.1007359 .0009532888 1050 1102509 1157025000 32.4037035 10 1039030 .0009523810 1051 1104001 1100935051 82.4191301 10.1071893 .0009514718 105 > 1100704 1101252003 82.4345495 10.17041*29 .0009505703 1053 1108809 11077)77)877 3*2.4499015 I 10.1736344 .0009490070 1454 1110910 1170905404 32.4053002 1 10.1708539 0009487066WEIGHTS AND MEASURES, 25 WEIGHTS AND MEASURES. Measures of Length. 12 inches = 1 foot. 3 feet = 1 yard = 36 inches. 5^ yards = 1 rod = 198 inches = 104 ft. 40 rods = 1 furlong = 7920 inches p 660 ft. = 220 yds. 8 furlongs = 1 mile = 63360 inches — 5280 ft. = 1700 yds. 1 yard = 0.0005682 of a mile. 1 = = 320 rod*. gunter's chain. 7.92 inches = 1 link. 100 links = 1 chain = 4 rods = 06 feet. 80 chains = 1 mile. EOT ES ANI) CABLES. 6 feet = 1 fathom. 120 fathoms = 1 cable’s length. Table showing Inches expressed in Decimals of a Foot. In. i 0 i ! 2 | 3 4 J 5 ] 6 | 7 | 8 | 9 | 10 | '11 In. 0 Foot ■ > I61l' .25001.3333! .4167! ,500o! ,5833 .6667] .7300 .8333! .9167 0 1-32 .0020 .0859 1693 .2526 .3359 .4193 .5026! .5859!.669375261.8359|.9193 1-32 1-16 .0052 .0885 1719 2552 .33S5i.4219| .5052] .5885j.6719[.75521.83851.9219' 1-16 8-32 .007S .0911' 1745 .2578 .3411 .4245' .5078; .5911 '.67451.7578 .84111.9245 3-32 1-8 .0104 .0938 .17711.2604- .3438 .42711.5104) .5938!.6771 i.7604 .84381.9271 1-8 5-32 .0130 .0964, 1797I .2630 .346.4297' .5130! .5964 .6797; .7630; .84641.9297 5-32II 3-16 .0156 .0990' .1823] .2656’ .31901.4323 .5156; .5990j .6823;.7656;.84901.9323 3-16 .0182 .lUlG, .1849 .2682 .3516, .4349j .5182! .6016..6849 .7682(.8516 .9349 7-32 , 1-4 .0208 .1042 .lS75l.2708l.3542 .4375 .5208 .004216875 .7708 :85*2 .■9375 1-4 9-32 .02:14 .1068 .lOOlj.2734!.3568;.4401 !.5234j.00081.69011.7734'.85681.0401 9-32 5-16 .0260 .1094 .1927 .2760 .3-594;.4427 .5260 .6094 .6927,.7760'.8594j.9427 5-16 n-32 .0286 .1120 ,1953|.2786].362o|.4453] ,5286!.6120|.6953|.7786! .8620|.0453 11-32 EM .0313 .1146 .1979 .2813 .3646 .4479 .5313 .614G1.6979!.7813 .8646!.9479 3-8 : 1 *> io .0339 .1172 .2005 .28391.3672' .45051.5339;.6172' .70051.7839 .86721.9595 13-3,*2 7-16 .0365 .1198 .2031 .2865‘.3698'.4531!.5365 .6198 .70311.78651.86981.9531 7-16 15-32 4)391 .1224 .20571,289lj .3724' .45571.5391 j .6224 .70571.78911.87241.9557 15-32 1 1-2 .0417 .1259 .2083] .2917 .37501.45831.54171.6250] .70831.79171.8750] .0583 1-2 17-82 .0443 .1276 .21 OH j .2943 .3776; .46< )91.5443 i.6276 .7109!.7943 .8776! .96' )9 17-32 9-16 .0469 .1302 .2135j.29693802j,4035|.540o|.6302j.7135 .7969],8802j.9635 9-16 1 19-32 .0495 .1328 .21611.2995i .3828 .4601 j ,5495| .6328] .71011.7995: .8828! .9601 19-12 i 5-8 .0521 .1354 .2188 .8021'.3854 .4688 .5521 .6.354'.7188Ls02l!.8854].9688 5-8 21-32 .0547 .1380 .2214 .3047].3880:.4714 .5547i.6380j.7214 .8047'.8880 .9714 21-32 11-16 .0573 .1406 .2240:.3073!.3906 .4740 .5573 . 6406'.724o'.8073|.8906|.0740 11-16 | 23-32 .0599 .1432 .2206 .3099 .3932 i .47661.5599 .64321.7206 j .80991.8932 .9760 23-32 3-4 .0625 .1458 .2292 .3125 .3958 .4792 .5625 .6458 .72921.8125 .8958 .9792 3-1 25-32 .0651 .1484 .2318 .31511.3984 '4818 .5651 .6484,.7318,.8151l.8984|.9818 25-° ’ 13-16 .0077 .1510 .2344'.3177|.4010 .4844 .5677 ,651o!.73441.8177!.9010].9844 13-1 j 27-32 .0703 .1536 .2370] .3203 .4036 .4870: .5703' .6536 .73701.8203] .9036|.9870 27-32 7-8 .0729 .1563 .2396 .3220 .4063 .4896 .5720 .6563 .7306 .8229,.00631.9806 7-8 29-32 1.0755 .1589 .2422 .3255!.4039 .4022 .5755 .6-530 .7422 .8255l.9089|.9922 mamm 15-16 1.0781 .1015 .24481.32811 .*11* .4948 .5781 .6615 .7448: .82811.9115 .9948 15-10 j 81-S2 .0817 .1641 .2474. .3307! .4141: .4974 .5807 j .6041 .74741.83071.9141,1.9974 31-32 1 0 1 2 | 8 J 4 6 j 6 7 j 8 j 9 1 10 11 26 MEASURES OF SURFACE AND VOLUME. GEOGRAPHICAL, ANI) NAUTICAL. 1 degree of a great circle of the earth = 69.77 statute miles. 1 mile = 2046.58 yards. shoemakers’ measure. No. 1 is 4.125 inches in length, and every succeeding number is -*.333 of an inch. There are 28 numbers or divisions, in two series of numbers, viz., worn 1 to 13, and 1 to 15. MISCELLANEOUS. 1 palm = 3 inches. 1 span = 9 inches. 1 hand = 4 inches. 1 meter = 3.2809 feet. Measures of Surface. 144 square inches = 1 square foot. 9 square feet = 1 square yard — 1296 square inches. 100 square feet — 1 square (architects’ measure). LAND. 30 i square yards — 1 square rod. 40 square rods — 1 square rood = 1210 square yards. 4 square roods | = 1 acre = 4840 square yards. 10 square chains 1 = 160 square rods. 640 acres = 1 square mile = 3097600 square yards = 208.71 feet square = 1 acre. [102400 sq. rods = 2560 sq. roods. A section of land is a square mile, and a quarter-section is 160 acres. Measures of Volume. 1 gallon liquid measure = 231 cubic inches, and contains 8.339 avoirdupois pounds of distilled water at 39.8° F. 1 gallon dry measure = 268.8 cubic inches. 1 bushel ( Winchester) contains 2150.42 cubic inches, or 77.627 pounds distilled water at 39.8° F. A heaped bushel contains 2747.715 cubic inches. DRY. 2 pints = 1 quart = 67.2 cubic inches. 4 quarts = 1 gallon = 8 pints = 268.8 cubic inches. 2 gallons = 1 peek = 16 pints = 8 quarts = 537.6 cubic inches. 4 pecks = 1 bushel:= 64 pints — 32 quarts = 8 gals. — 2150.42 1 chaldron = 36 heaped bushels = 57.244 cubic feet. [cu. in. 1 cord of wood = 128 cubic feet.MEASURES OE VOLLME AND WEIGHT. 27 LIQUID. 4 gills — 1 pint. 2 pints “ I quart = 8 gills. 4 quarts - 1 gallon — 32 gills = 8 pints. In the United Status and Groat Britain 1 barrel of wine or bran 'y - oil gallons, anil contains 4.211 cubic feet. A hogshead is 03 gallons, but this term is often applied to casks of various capacities. Cubic Measure. 1728 cubic inches = 1 foot. 27 cubic feet = 1 yard. In mettsurinu »rood, a pile of wood cut 4 feet long, piled 4 feet high, and 8 feet on the ground, making 128 cubic feet, is called a cord. 10 cubic feet make one cord foot. rA perch of atone is 10 d feet long, 1 foot high, and Id feet thick, and contains 243 cubic feet. A perch of stone is, however, often computed differently in dif-' ferent localities; thus, in Philadelphia, 22 cubic feet are called a perch, and in some of the New-England States a perch is computed V at 10d cubic feet. A ton, in computing the tonnage of ships and other vessels, is 100 cubic feet of their internal space. Fluid Measure. 60 minims — 1 fluid draclun. 8 fluid drachms ” 1 ounce. 10 ounces = 1 pint. 8 pints = 1 gallon. Miscellaneous. Butt of Sherry — 108 gals. Puncheon of Brandy, 110 to 120 gals. Pipe of Port = 115 gals. Puncheon of Rum, 100 to 1!0 gals. Butt of Malaga = 105 gals. Hogshead of Brandy, 55 to 00 gals. Puncheon of Scotch Wins- Hogshead of claret, 40 gals, key, 110 to 130 gals. Measures of WeiR'lit. The standard avoirdupois pound is the weight of 27.7015 cubic inches of distilled water weighed in air at 30.83°, the barometer at 30 inches.28 MEASURES OF WEIGHT. Avoirdupois, or Ordinary Commercial Weight. 10 drachms = 1 ounce, (oz.). 10 ounces = 1 pound, (lb.). 100 pounds — 1 hundred weight (cwt.). 20 hundred weight ■ 1 ton. In collecting duties upon foreign goods at the United States custom-houses, and also in freighting coal, and selling it by wholesale, — 28 pounds m 1 quarter. 4 quarters, or 112 lbs. = 1 hundred weight. 20 hundred weight = 1 long ton = 2240 pounds. A stone = 14 pounds. A quintal = 100 pounds. The following measures are sanctioned by custom or law : 32 pounds of oats = 1 bushel. 45 pounds of Timothy-seed — 1 bushel. 48 pounds of barley — 1 bushel. 50 pounds of rye = 1 bushel. 50 pounds of Indian corn = 1 bushel. 50 pounds of Indian meal = 1 bushel. 00 pounds of wheat ~ 1 bushel. 00 pounds of clover-seed = 1 bushel. 00 pounds of potatoes = 1 bushel. 50 pounds of butter = 1 firkin. 100 pounds of meal or flour = 1 sack. 100 pounds of grain or flour = 1 cental. 100 pounds of dry fish = 1 quintal. 100 pounds of nails = 1 cask. 100 pounds of flour = 1 barrel. 200 pounds of beef or pork =s 1 barrel. Troy Weight! USET> IN WEIGHING GOLD Oli SILVER. 24 grains W 1 pennyweight (pwt.). 20 pennyweights = 1 ounce (oz.). 12 ounces = 1 pound (lb.). A carat of the jewellers, for precious stones, is, in the United States, 3.2 grains: in London, 3.IT grains, in Paris, 3.18 grains are divided into 4 jewellers' grains. In troy, apothecaries’, and avoirdupois weights, the grain is the same.MEASURES OF VALUE AND TIME. 29 Apothecaries’ Weight. USED IN COMPOUNDING MEDICINES, AND IN PUTTING UP MEDICAL PRESCRIPTIONS. 20 grains (gr.) = 1 scruple (3). 8 drachms = 1 ounce (oz.). 3 scruples = 1 drachm ( 3 ). 12 ounces = 1 pound (lb.). Measures of Value. UNITED STATES STANDARD. 10 mills = 1 cent. I 10 dimes = 1 dollar. 10 cents = 1 dime. I 10 dollars p 1 eagle. The standard of gold and silver is 900 parts of pure metal and 100 of alloy in 1000 parts of coin. The fineness expresses the quantity of pure metal in 1000 parts. The remedy of the mint is the allowance for deviation from the exact standard fineness and weight of coins. Weight of Coin. Double eagle = 516 troy grains. Eagle = 258 troy grains. Dollar (gold) = 25.8 troy grains. Dollar (silver) = 412.5 troy grains. Half-dollar = 192 troy grains. 5-cent piece (nickel) : 77.16 troy grains. 3-cent piece (nickel) : 30 troy grains. Cent (bronze) = 48 troy grains. Measure of Time. 00 seconds = I minute. j 3(35 days = 1 common year. 00 minutes = 1 hour. 3(36 days — 1 leap year. 24 hours = 1 day. A solar day is measured by the rotation of the earth upon its a is with respect to the sun. in (istronomicfd computation and in nautical time the day commences at noon, and in the former it is counted throughout the 24 hours. In civil computation the day commences at midnight, and is divided into two portions of 12 hours each. A solar year is the time in which the earth makes one revolution around the sun; and its average time, called the mean solar year, is 365 days, 5 hours, 48 minutes, 49.7 seconds, or nearly 305i days. A mean lunar month, or lunation of the moon, is 29 days, 12 hours, 44 minutes, 2 seconds, and 5.24 thirds.30 THE CALENDAR. —ANGULAR MEASURE The Calendar, Old and New Style. The Julian Calendar was established by Julius C;esar, 44 B.C., and by it one (lay was inserted in every fourth year. This was the same, thing as assuming that the length of the solar year was 305 days, (5 hours, instead of the value given above, thus introduein -an accumulative error of 11 minutes, 12 seconds, every year. Tli's calendar was adopted by the church in 325 A.D., at the Council of Nice. In the year 1582 the annual error of 11 minutes, 12 seconds, had amounted to a period of 10 days, which, by order of Pope Gregory XIII., was suppressed in the calendar, and the 5th of October reckoned as the 15th. To prevent the repetition of this error, it was decided to leave out three of the inserted days every 400 years, and to make this omission in the years which are not exactly divisible by 400. Tims, of the years 1700, 1800, 1900, 2000, all of which are leap years .according to the Julian Calendar, only the last is a leap year according to the Reformed or Gregorian Calendar. This Reformed Calendar was not adopted by England until 1752, when 11 days were omitted from the calendar. The two calendars are now often called the Old Style. and the New Style. The latter style is now adopted in every Christian country except Russia. Circular and Angular Measures. USED FOB MEASURIXG AXGI.ES AXD ARCS, AXD FOR DETEB-MIXIXG LATITUDE AXD LOXGJTUDE. (50 seconds (") = 1 minute ('). (50 minutes — 1 degree (°). 3(50 degrees = 1 circumference (C.). Seconds are usually subdivided into tenths and hundredths. A minute of the circumference of the earth is a geographical mile. Degrees of the earth’s circumference on a meridian average (59.16 common miles. THE METRIC SYSTEM. The metric system is a system of weights and measures based upon a unit called a meter. The meter is one ten-millionth part of the distance from the equator to either pole, measured oil the earth’s surface at the level of the sea.THE METRIC SYSTEM. 31 The names of derived metric denominations are formed by prefixing to the name of the primary unit of a measure — Mull (mille), a thousandth, Centi (sent'e), a hundredth, I)eci (des'e), a tenth, I)eka (dek'a), ten, This system, first adopted by i by other countries, and is much It was legalized in 1800 by Cong and is already employed by the by the Mint and the General Po llecto (liekto), one hundred, Kilo (kil'o), a thousand, Myria (mir'ea), ten thousand. I 'ranee, has been extensively adopted used in the sciences and the arts, ress to be used in the United States, Coast Survey, and, to some extent, st-OHice. Linear Measures. The meter is the primary unit of lengths. Table. 10 millimeters (mm.) 10 centimeters 10 decimeters 10 meters 10 dekameters 10 hectometers 10 kilometers = 1 centimeter (cm.) — 1 decimeter = 1 METER — 1 dekameter = 1 hectometer = 1 K1IX»METER (km.) = 1 myriameter = 0.3937 in. = 3.937 in. = 39.37 in. — 393.37 in. = 328 ft. 1 in. = 0.02137 mi. = 6.2137 mi. The meter is used in ordinary measurements; the centimeter or millimeter, in reckoning very small distances; and the kilometer, for roads or great distances. A centimeter is about l of an inch; a meter is about 3 feet 3 inches and jj; a kilometer is about 200 rods, or f of a mile. Surface Measures. • The square meter is the primary unit of ordinary surfaces. The are (air), a square, e^ph of whose sides is ten meters, is the unit of land measures. Table. 100 square millimeters (sq. mm.) = 1 square l — q 155 sr. jncp centimeter (sq. cm.) 1 100 square centimeters = 1 square decimeter = 15.5 sq. inches. 100 square decimeters ■ 1 square 1 _ 155Q inJ or l m sq. y(Js. meter (sq. 111.) >32 THE METRIC SYSTEM. Also 100 centiares, or sq. meters, = 1 ark (ar.) = 110.6 sq. yds. 100 ares — jl hectare (ha.) = 2.471 acres. A square meter, or one centiare, is about 10i| square feet, oi 1$ square yards, and a hectare is about 2.) acres. Cubic Measures. The cubic meter, or stere (stair), is the primary unit of a volume. Table. 3000 cubic millimeters (cu. mm.) = 1 cubic centimeter (cu. cm.) = 10.061 cubic inch. 1000 eunic centimeters — 1 cubic decimeter = 61.022 cubic inches. 1000 cubic decimeters = 1 cubic meter (cu. m.) - 35.314 cu. ft. The stere is the name given to the cubic meter in measuring wood and timber. A tenth of a stere is a decislere, and ten stores are a dekastere. A cubic meter, or stere, is about lj cubic yards, or about 2j cord feet. Liquid and Dry Measures. The liter (leeter) is the primary unit of measures of capacity, and is a cube, each of whose edges is a tenth of a meter in length. The hectoliter is the unit in measuring large quantities of grain, fruits, roots, and liquids. Table. 10 milliliters (ml.) — 1 centiliter (cl.) 10 centiliters = 1 deciliter 10 deciliters 10 liters 10 dekaliters 10 hectoliters — 1 UTER (1.) = 1 dekaliter = 1 HECTOLITER (111.) — 1 kiloliter — 0.338 fluid ounce. = 0.845 liquid gill. = 1.0507 liquid quarts. = 2.0417 gallons. — 2 bushels 3.35 pecks. = 28 bushels H pecks. A centiliter is about \ of a fluid ouqce; a liter is about 1 nt liquid quarts, or y'> of a dry quart) a hectoliter is about 2§ bushels; and a kiloliter is one cubic meter, or stere. Weights. The rjram is the primary unit of weights, and is the weight in a vacuum of a cubic centimeter of distilled water at the temperature of 3,0.2 degrees Fahrenheit.ANCIENT MEASURES AND WEIGHTS. 33 10 milligrams (mg.) 10 centigrams 10 decigrams 10 grams 10 dekagrams If) hectograms .10 kilograms 10 myriagrains 10 quintals Table. = 1 centigram - 1 decigram = 1 GItAM (g.) = 1 dekagram = 1 hectogram = 1 KILOGIiAM (k.) = 1 myriagram = 1 quintal = 1 toxxeau (t.) 0.1543 troy grain. 1.543 troy grains. 15.432 troy grains. 0.3527 avoir, ounce. 3.5274 avoir, ounces. 2.2046 avoir, pounds. 22.046 avoir, pounds. 220.46 avoir, pounds. 2204.6 avoir, pounds. The gram is used in weighing gold, jewels, letters, and small quantities of things. The kilogram, or, for brevity, kilo, is used by grocers; and the tonneau (tonno), or metric ton, is used in finding the weight of very heavy articles. A gram is about 15i grains troy; the kilo about 2-J pounds avoirdupois; and the metric ton, about 2205 pounds. A kilo is the weight of a liter of water at its greatest density; and the metric ton, of a cubic meter of water. Metric numbers are written with the decimal-point (.) at the right of the figures denoting the unit; thus, 15 meters and 3 centimeters are written, 15.03 m. When metric numbers are expressed by figures, the part of the expression at the left of the decimal-point is read as the number of the unit, and the part at the right, if any, as a number of the lowest denomination indicated, or as a decimal part of the unit; thus, 46.525 m. is read 46 meters and 525 millimeters, or 46 and 525 thousandths meters. In writing and reading metric numbers, according as the scale is 10, 100, or 1000, each denomination should be allowed one, two, or three orders of figures. SCRIPTURE AND ANCIENT MEASURES AND WEIGHTS. Digit Palm Span Scripture Long- Measures. Inches. = 0.912 Cubit = 3.648 Fathom 110.944 Feel. Inches. — 1 9.888 — 7 3.552 Egyptian Long Measures. Nahud cubit = 1 foot 5.71 inches. Iloyal cubit = 1 foot 8.66 Inches.34 ANCIENT MEASURES AND WEIGHTS. 1 Grecian Long Measures. Digit Pous (foot) Cubit Feet. Inches. = 0.7554 = 1 0.0875 = 1 1.59841 Stadium Mile Feet. = 604 = 4835 Jnchett 4.5 Jewish Lon g Measures. Cubit = 1.824 ft. Sabbath-day’s journey = 3648 ft. Mile Day’s journey = 7296 feet. = 33.164 mi'es. Roman Lon g Measures. Digit Uncia (inch) Ves (foot) Inches. = 0.72575 = 0.967 = 11.604 Cubit Passus Mile (miliarium Feet. = 1 = 4 = 4842 Inches. 5.406 10.02 Roman Weight. Ancient libbra = 0.7094 pound. Ancient Weights. Attic obolus Attic drachma Egyptian mina Ptolemaic mina Lesser mina Greater mina Talent Pound Troy Grains. _i 8-2 ~ I 9.1 C 51.9 = -i 54.6 l 69 = 8.326 = 8.985 I 3.892 Troy Grains. Alexandrian mina = 9.992 Denarius ( Homan ) = ( 51.9 Ì 62.5 Denarius Nero = 54 r 415.1 j 437.2 Ounce = l 431.2 Drachm = 146.5 = TVi of (Irarltma. = 60 mime = 56 pounds avoirdupois. = 12 Homan ounces. In this last table, where two or more values are given for the same weight, they are from different authorities on the subject. Arabian foot Babylonian foot Egyptian finger Miscellaneous. Feet. I Feet. = 1.095 Hebrew foot ~ 1.212 = 1.140 Hebrew cubit =1.817 = 0.06145 I Hebrew sacred cubit = 2.002MENSURATION. - DEFIN FIIONS. 35 Fig.l A Curved Line. MENSURATION. Definitions. A point is that which has only position. A plane is a surface in which, any two points being taken; ths straight line joining them will be wholly in the surface. A curved line is a line of which no portion is straight (Fig. 1). Parallel lines are such as are wholly in the same plane, and have the same direction (Fig. 2). A broken line is a line composed of a series of dashes; thus,---------------. Fig.2 An angle is the opening between two Parallel Lines, lines meeting at a point, and is termed a right angle when the two lines are perpendicular to each other, an acute angle when it is less or sharper than a right angle, and obtuse when it is greater than a right angle. Thus, in Fig. 3, A A A A are acute angles, OOOO are obtuse angles, It li R II are right angles. 6» lyS Fig.3 0 HRh R R R R Polygons. A polygon is a portion of a plane bounded by straight lines. A triangle is a polygon of three sides. A scalene triangle has none of its sides equal; an isosceles triangle has two of its sides equal; an equilateral triangle has all three of its sides equal. A right-angle triangle is one which has a right angle. The side opposite the right Fig. 4. angle is called the hypothewuse; the side on Right-angle Triangle, which the triangle is supposed to stand is called its base, and the other side, its altitude. Fig. 6. Irtosoelort TriangU Fig. 5. Scalene Triangle.GEOMETRICAL TERMS. 36 | A quadrilateral is a polygon of four sides. Quadrilaterals are divided into classes, as follows, — the trape• I zinm (Fig. 8), which has no two of its sides parallel; the trapezoid 1 (Fig. 9), which has two of its sides parallel; and the parallelogram (Fig. 10), which is bounded by two pairs of parallel sides. Fig. 8. Fig. 9. Fig. 10. A parallelogram whose sides are not equal,,and its angles not right angles, is called a rhomboid (Fig. 11); when the sides are all equal, but the angles are not right angles, it is called a rhombus (Fig. 12); and, when the angles are right angles, it is called a rectangle (Fig. 13). A rectangle whose sides are all equal is called a square (Fig. 14). Polygons whose sides are all equal are called regular. Fig. 11. Fig. 12. Fig. 13. Fig. 14. Besides the square and equilateral triangles, there are The pentagon (Fig. 15), which has five sides; The hexagon (Fig. 10), which has six sides; The heptagon (Fig. 17), which has seven sides; The octagon (Fig. IS), which has eight sides. “A J Fig. 16. Fig. 17. Fig. 13. The enneagon has nine sides. The decagon has ten sides. The dodecagon has twelve sides. For all polygons, Ihe side upon which it is supposed to stand is called its base; the perpendicular distance from the highest side orGEOMETRICAL TERMS. 37 angle to the base (prolonged, if necessary) is called the altitude; and a line joining any two angles not adjacent is called a diagonal. A perimeter is the boundary line of a plane figure. A circle is a portion of a plane bounded by a curve, all the points of which are equally distant from a point within called the centre (Fig. lit). The circumference is the curve which bounds the circle. A radius is any straight line drawn from the centre to the cir-' cumference. Any straight line drawn through the centre to the circumference on each side is called a diameter. An arc of a circle is any part of its circumference. A chord is any straight line joining two points of the circumference, as bd. A segment is a portion of the circle included between the arc and its chord, as A in Fig. 10. A sector is the space included between an arc and two radii drawn to its extremities, as B, Fig. 19. In the figure, ab is a radius, cd a diameter, and dh is a chord subtending the arc bed. A tangent is a right line which / ■ in passing a curve touches without cutting it, as fg, Fig. 19. Fig. 19. Volumes. A prism is a volume whose ends are equal and parallel polygons, and whose sides are parallelograms. A prism is triangular, rectangular, etc., according as its ends are triangles, rectangles, etc. A cube, is a rectangular prism all of whose sides are squares. A cylinder is a volume of uniform diameter, bounded by a curved surface and two equal and parallel circles. A pyramid is a volume whose base is a polygon, and whose sides are triangles meeting in a point called the vertex. A pyramid is triangular, quadrangular, etc., according as its base is a triangle, quadrilateral, etc. A cone is a volume whose base is a circle, from which the remaining surface tapers uniformly to a point or vertex (Fig. 20). Fi9- 20- ('onic scriioiis are the figures made by a plane cutting a cone.38 MENSURATION. An ellipse is the section of a cone when cut by a plane passing obliquely through both sides, as at ab, Fig. 21. A parabola is a section of a cone cut by a plane parallel to its side, as at cd. A hyperbola is a section of a cone cut by a plane at a greater angle through the base than is made by the side of the cone, as at eh. Mensuration and volumes. In the ellipse, the transverse axis, or tony diameter, is the longest line that can be drawn through it. The conjugate axis, or short diameter, is a line drawn through the centre, at right angles to the long diameter. A frustum of a pyramid or cone is that which remains after cutting off the upper part of it by a plane parallel to the base. A sphere is a volume bounded by a curved surface, all points of which are equally distant from a point within, called the centre. treats of the measurement of lines, surfaces, ItUXES. To compute the area of a square, a rectangle, a rhombus, or a rhomboid. Eri.K.—Multiply the length by the breadth or height; thus, in either of Figs. 22, 2d, 24, the area = ab X be. Fig.22 To compute the area of a triangle. c Rule. — Multiply the base by the alti- / tude, and divide by 2; thus, in Fig. 25, / ah X cd / area of abc —----~p • a -------^---------- 6 To find the length of the hypothenuse of a Fig.25 right-angle triangle when both sides are known.MENSURATION. — POLYGONS. 39 Rule. — Square Uie length of each of the sides making the right angle, add their squares together, and take the square root of their sum. Thus (Fig. 20), the length of hc = 3, and of t>c = 4; then ab = 3 X 3 =î 9 + (4 X 4) = 9 + 10 = 25. V^25 = 5, or ab — 5. To find the length of the base or altitude of a right-angle triangle, when the length of the hypothenuse and one side is known. Rule. — From the square of the length of the hypothenuse subtract the square of the length of the other side, and take the square root of the remainder. To find the area of a trapezium. Rule. — Multiply the diagonal by the sum of the two perpendiculars falling upon it from the opposite angles, and divide the product by 2. Or, ab X (ce + di) -------2-------= area (Fig. 2T). To find the area of a trapezoid (Fig. 28). Rule. — Multiply the sum of the two parallel sides by the perpendicular distance between them, and divide the product by 2. Fig. 28. To compute the area of an irregular polygon. Rule.—Divide the polygon into triangles by means of diagonal lines, and then add together the areas of all the triangles, as A, 1), and C (Fig. 29). To find the urea of a regular polygon. Rule. — Multiply the length of a side by the perpendicular distance to the centre (as c/o, Fig. 30), and that product by the number of sides, and divide the result by 2. To compute the area of a regular polygon when the length of a side only is given. Rule. —Multiply the square of the side by the multiplier opposite to the name of the polygon in column A of the following table: — Fig.20 a40 MENSURATION.-POLYGONS AND CIRCLES. Name of Polygon. No. of sides. A. Area. B. Radius of circumscribing circle. C. Length of the side. D. Radius of iuscribed circle. Triangle . . . 3 0.433013 0.5773 1.732 0.2887 Tetragon . . -1 1 0.7071 1.4142 0.5 M . . . 5 1.720477 0.8506 1.1756 0.6882 Hexagon . . . 6 *2.598076 1 1 0.S66 1 Heptagon . . . 7 3.633912 1.1524 0.8677 1.0383 Octagon . . . 8 4.828427 1.3066 0.7653 1.2071 | Nonagon . . . 9 G.1S1S24 1.4619 0.684 1.3737 Decagon . . 10 7.694209 1.618 0.618 1.5383 Undecagon . . 11 9.36564 1.7747 0.5634 1.7028 Dodecagon . . n 11.196152 1.9319 0.5176 1.866 To compute the radius of a circumscribing circle when the length of a side only is given. Rule. — Multiply the length of a side of the polygon by the number In column B. Example.—What is the radius of a circle that will contain a hexagon, the length of one side being 5 inches ? A ns. 1x1 = 3 inches. To compute the length of a side of a polygon that is contained in a given circle, when the radius of the circle is given. Rule. — Multiply the radius of the circle by the number opposite the name of the polygon in column C. Example. —What is the length of the side of a pentagon contained in a circle 8 feet in diameter ? Ans. 8 ft. diameter t 2 = 4 ft. radius, 4 X 1.1756 = 4.7024 ft. To compute the radius of a circle that can be inscribed in a given polygon, when the length of a side is given. Rule. — Multiply the length of a side of the polygon by the number opposite the name of the polygon in column D. Example. — What is the radius of the circle that can he inscribed in an octagon, the length of one side being 6 inches. ylas. 6 X 1.2071 = 7.2426 inches. Circles. To compute the circumference of a circle. Rule. — Multiply the diameter by 3.1416; or, for most purposes, by 3{ is efficiently accurate. Example. — What is the circumference of a circle 7 inches in diameter ? vliis. 7 X 3.1416 — 21.0912 inches, or 7 X 31 = 22 inches, the error in Ibis last being 0.0088 of an inch.MENSURATION. — CIRCLES. 41 To find the diameter of a circle when the circumference is given. Rule. —Divide the circumference by 3.1410, or for a very near approximate result multiply by 7 and divide by 22. To find the radius of an arc, when the chord and rise or versed sine are given. Rule.—Square one-half the chord, also square tlie rise; divide their sum by twice tlie rise; the result will be the radius. Example.—The length of the chord ac, Fig. 30£, is 48 inches, and the rise, bo, is 6 inches. What is the radius of the arc ? Ans. Rad I °f '2+±rf = g= 51 ins. Fi9- 30*-2bo 12 To find the rise or versed sine of a circular arc, when the chord and radius are given. Rule. —Square the radius; also square one-half the chord; subtract the latter from the former, and take the square root of the remainder. Subtract the result from the radius, and the remainder will be the rise. Example.—A given chord has a radius of 51 inches, and a chord of 48 inches. What is the rise ? Ans. Rise = rad — 'Ad2 — Ichord2 = 51 — ^21 >01 — 570 = 51 — 45 = G inches = rise. To compute the area of a circle. Rule. —Multiply the square of the diameter by 0.7854, or multiply the square of the radius by 3.1410. Example. —What is the area of a circle 10 inches in diameter? Ans. 10 x 10 X 0.7854 = 78.54 square inches, or 5 X 5 X 3.1410 = 78.54 square inches. The following tables win be found very convenient for finding the circumference and area of circiei. —42 MENSURATION. — CIRCLES AREAS AND CIRCUMFERENCES OF CIRCLES. . For Diameters from iV to 100, advancing by Tenths. Diarn. Area. Circuin. Diam. Area. Circum. Diam. Area. Circum. 0.0 5.0 19.0350 15.7080 10.0 78.5398 31.4159 .1 0.007854 0.31410 *1 2( ».428*2 10.0221 .1 80.1185 31.7301 0.031410 0.02832 21.23.72 16.3303 2 81.7128 32.0442 .3 0.070086 0.94248 .3 22.0018 10.0504 o •o 83.3229 32.3584 .4 0.1*2506 1.2500 .4 22.9022 16.9046 .4 84.9487 32.0720 .5 0.19635 1.5708 .5 23.7583 17.2788 .5 86.5901 32.9807 .6 0.‘28*274 1.8850 .0 24.0301 17.5929 .6 88.2473 33.3009 .7 0.38485 2.1991 .7 25.5170 17.9071 . i S9.9*202 33.0150 .8 (».50206 2.5133 .8 20.4203 18.2212 .8 91.6088 33.9292 .9 0.03017 2.8274 .9 27.3397 18.5354 .9 93.3132 34.2434 1.0 0.7854 3.1410 6.0 28.2743 18.8490 11.0 95.0332 34.5575 .1 0.9503 3.4558 .1 29.2247 19.1037 .1 90.7089 34.8717 1.1310 3.7099 .2 30.1907 19.4779 o 98.5203 35.1858 .3 1.3*273 4.0841 .3 31.1725 19.7920 .3 100.2875 35.5000 .4 1.5394 4.3982 .4 32.1099 20.1062 .4 102.0703 35.8142 .5 1.7071 4.7124 .5 33.1831 20.4204 .5 103.8689 36.1283 .6 2.0100 5.0205 .0 34.2119 20.7345 .0 105.0832 30.4425 .7 2.2098 5.3407 . i 35.2505 21.0487 . i 107.513,2 30.7500 .8 2.5447 5.0549 .8 30.3108 21.3028 .8 109.3588 37.0708 .9 2.8353 5.9690 .9 37.3928 21.0770 .9 111.2202 37.3850 2.0 3.1410 6.2832 7.0 38.4845 21.9911 12.0 113.0273 37.6991 .1 3.4030 6.5973 .1 39.5914 22.3053 .1 114.9901 38.0133 o 3.8013 6.9115 •» 40.7150 22.6195 •> 110.8987 38.3274 1.3 4.1548 7.2257 • O 41.8539 2*2.93: »0 .3 118.8229 38.0410 .4 4.5239 7.5398 .4 43.0084 23.2478 .4 120.7028 o8.9551 .5 4.9087 7.8540 .5 44.1780 23.5019 •O 122.7185 39.2099 .0 5.3093 8.1081 .0 45.3010 23.8701 .0 124.(>898 39.5841 .7 5.7256 8.4823 .7 40.'>603 24.1903 . i 120.0709 39.8982 .8 6.157;» 8.790.» .8 47.7830 24.5044 .8 128.0796 40.2124 .9 0.0052 9.1106 .9 49.0107 24.8186 .9 130.0981 40.5205 3.0 7.00.80 9.4248 8.0 50.2655 25.1327 13.0 132.7323 40.8407 .1 7 .'>47 7 9.7389 .1 51.5300 25.4409 .1 134.7822 41.1549 .2* 8.0425 10.0531 .2 52.8102 25.7011 •i 130.8478 41.4090 .3 8.5530 19.3073 .3 54.1001 26.075*2 o • O 138.9291 41.7832 .4 9.0792 10.0814 .4 55.4177 ‘20.3894 .4 141.0201 42.0913 ,5 9.6211 10.9950 .5 50.7450 2o. 1035 H 143.1388 42.4115 1 10.1788 11.3097 .0 58.088!» *27.0177 .0 145.2072 42.7257 . i 10.7521 11.0239 .7 59.4468 *27.3319 • 1 147.4114 43.0398 .8 11.3411 11.93*1 .8 04.8*212 *27.0400 .8 149.5712 43.35-10 11.9459 12.2522 .9 02.2114 27.900*2 .9 151.7408 43.0081 4.0 12.5601 12.5004 9.0 03.0173 28.2743 14.0 153.9380 43.9823 .1 13.2( »25 12.8805 .1 05.0388 *28.5885 .1 156.1450 44.2905 .2 13.8544 13.1947 •> 00.4701 28.90*27 158.3077 44.0100 .3 14.5220 13.5088 .3 07.9291 29.2108 .5 100.0001 44.9248 .4 15.2053 13.8230 .4 09.3978 29.5310 .4 102.8602 45.2389 .5 15.9043 14.1372 .5 70.88*22 29.8451 5. 105.1300 45.5531 .6 10.0190 14.4513 .6 72.38*23 30.1593 6. 167.4155 45.8673 .7 17.3494 14.7055 .7 73.8981 3*0.4734 i. 109.7107 40.1814 .8 18.0950 15.0796 .8 75.4296 2)0.7876 8. 172.0336 40.4956 .9 18.8574 15.3938 .9 70.9709 31.1018 9. 174.3062 40.8097 MENSURATION. — CIRCLES, 43 AREAS AND CIRCUMFERENCES OF CIRCLES. (Advaticiny by Tenths.) Diam. Area. Circum. Diam. Area. Circum. Diam Area. Circum. 15.0 176.7146 47.1239 20.0 314.1593 62.8319 25.0 490.8739 78.5398 .1 179.0786 47.4380 .1 317.3087 63.1460 .1 494.8087 78.85-10 2 181.4584 47.7522 2 320.4739 63.4602 498.7592 79.1681 .3 183.8539 48.0i)i34 .3 323.6547 03.7743 .3 502.7255 79.4823 .4 186.2650 48.3805 .4 326.8513 64.0885 ; .4 506.7075 79.7965 .5 188.6919 48.6947 .5 330.0636 64.4026 .5 510.7052 80.1106 .6 191.1345 49.00S8 .6 333.2916 64.7168 .6 514.7185 80.4248 .7 193.5928 49.3230 .7 336.5353 65.0310 .7 518.7476 80.7389 .8 196.0668 49.6372 .8 339.7947 65.3451 .8 622.7924 81.0531 .9 198.5565 49.9513 .9 343.0698 65.6593 .9 526.8529 81.3672 16.0 201.0619 50.2655 21.0 346.3606 65.9734 26.0 530.9292 81.6814 .1 203.5831 50.5796 .1 349.6671 66.2876 .i 535.0211 81.9956 .2 206.1199 50.8938 .2 352.9894 66.6018 .2 539.1287 82.3097 .3 208.6724 51.2080 .3 356.3273 66.9159 .3 543.2521 82.6239 .4 211.2407 51.5221 .4 359.6809 67.2301 .4 547.3911 82.9380 .5 213.8246 51.8363 .5 363.0503 67.5442 .5 551.5459 83.2522 .6 216.4243 52.1504 .6 366.4354 67.8584 .6 555.7163 83.5664 .7 219.0397 52.4646 .7 369.8361 68.1726 .7 559.9025 83.8805 .8 221.6708 52.7788 .8 373.2526 68.4867 .8 564.1044 84.1947 .9 224.3176 53.0929 .9 376.6848 68.8009 .9 568.3220 84.5088 17.0 226.9801 53.4071 22.0 380.1327 69.1150 27.0 572.5553 84.8230 .1 229.6583 53.7212 .1 383.5963 69.4292 .1 576.8043 85.1372 .2 232.3522 54.0354 .2 387.0756 69.7434 581.0690 85.4513 .3 235.0618 54.3496 .3 890.5707 70.0575 .3 585.3494 85.7655 .4 237.7871 54.6637 .4 894.0814 70.3717 .4 589.6455 86.0796 .5 240.5282 54.9779 .5 397.6078 70.6858 .5 693.9574 86.3938 .6 243.2849 55.2920 .6 401.1500 71.0000 .6 598.2849 86.7080 . 7 246.0574 55.6062 .7 404.7078 71.3142 .7 602.6282 87.0221 .8 248.8456 55.9203 .8 408.2814 71.6283 .8 606.9871 87.3363 .9 251.6494 56.2345 .9 411.8707 71.9425 .9 611.3618 87.6504 18.0 254.4690 56.5486 23.0 415.4756 72.2566 28.0 615.7522 87.9646 .1 257.3043 56.8628 .1 419.0963 72.5708 .1 620.1582 88.2788 .2 260.1553 57.1770 .2 422.7327 72.8849 .2 624.5800 88.5929 .3 263.0220 57.4911 .3 426.3848 73.1991 .3 629.0175 88.9071 .4 265.9044 57.8053 .4 430.0526 73.5133 .4 633.4707 89.2212 .5 268.8025 58.1195 .5 433.7361 73.8274 .5 637.9397 89.5354 .6 271.7164 58.4336 .6 437.4354 74.1416 .6 642.4243 89.8495 .7 274.6459 58.7478 .7 441.1503 74.4557 .7 646.9246 90.1637 .8 277.5911 59.0619 .8 444.8809 74.7699 .8 651.4407 90.4779 .9 280.5521 59.3761 .9 448.6273 75.0841 .9 655.9724 90.7920 19.0 283.5287 50.60(13 24.0 452.3893 75.3982 29.0 660.5199 91.1062 .1 286.5211 60.0044 .1 456.1671 75.7124 .1 665.0830 91.4203 .2 289.5292 60.3186 .2 459.9606 76,0265 .2 669.6619 91.7345 .3 292.5530 60.6327 .3 463.7698 76.3407 .3 674.2565 92.0487 .4 295.5925 60.9469 .4 467.5947 76.6549 .4 678.8668 92.3628 .5 298.6477 61.2611 .5 471.4352 76.9690 .5 683.4928 92.6770 .6 301.7186 61.5752 .6 475.2916 77.2832 .6 688.1345 92.9911 .7 304.8052 61.8894 .7 479.1636 77.5973 .7 692.7919 93.3053 .8 307.9075 62.2035 .8 483.0513 77.9115 .8 697.4650 93.6195 .0 311.0255 62.5177 .9 486.9547 78,2257 .9 702.1538 93.933644 MENSURATION. — CIRCLES AREAS AND CIRCUMFERENCES OF CIRCLES (Advanriny by Tenths.) il ia in. Area. Circum. Diam Area. Circum. Diam. Area. Circum. 30.0 706.8583 94.2478 35.0 962.1128 109.9557 40.0 1256.6371 125.6637 .1 711.5786 94.5619 .1 967.6184 110.2699 .1 1262.9281 125.9779 2 716.3145 94.8761 973.1397 110.5841 .2 1269.2348 126.2920 .3 721.0662 95.1903 .3 978.676)8 110.8982 .3 1275.5573 126.6062 .4 725.8336 95.5044 .4 984.2296 111.2124 .4 1281.8955 126.9203 .5 730.6167 95.8186 .5 989.79S0 111.5265 .5 1288.2493 127.2345 .G 735.4154 96.1327 .6 995.3822 111.8407 .6 1294.61S9 127.5487 .7 740.2299 96.4469 . i 1000.9821 112.1549 .7 1301.0042 127.8628 .8 745.0601 96.7611 .8 1006.5977 112.4690 .8 1307.4052 128.1770 .9 749.9060 97.0752 .9 1012,2290 112.7832 .9 1313.8219 128.4911 31.0 754.7676 97.3894 36.0 1017.8760 113.0973 41.0 1320.2543 128.8053 .1 759.6450 97.70:15 .1 1023.5387 113.4115 .1 1326.7024 129.1195 .2 —80 98.0177 .) 1029.2172 113.7257 o 1333.1663 129.4336 .3 769.4467 98.3319 .3 MB—13 114.0398 .3 1339.6458 129.7478 .4 774.3712 98.6460 .4 1040.6212 114.3540 .4 1346.1410 130.0619 .5 779.3113 98.9602 .5 1046.3467 114.66S1 .5 1352.6520 130.3761 .6 784.2672 99.2743 .6 1052.0880 114.9823 .6 1359.17S6 130.6903 j . 7 789.2388 99.5885 . 7 1057.«8449 115.2965 1365.7210 131.0044 1 .8 794.2260 99.9026 .8 1063.6176 115.6106 .8 1372.2791 131.3186 1 .9 799.2290 100.2168 .9 1069.4060 115.9248 .9 1378.8529 131.6327 I 32.0 804.2477 100.5310 37.0 1075.2101 116.2389 42.0 1385.4424 131.9469 .1 809.2821 100.8451 .1 1081.0299 116.5531 .1 1392.0476 132.2611 .2 814.3322 101.1593 ‘> 1086.86)54 116.8672 «> 1398.6685 132.5752 .3 819.3980 101.4734 .3 1092.7166 117.1814 .3 1405.3051 132.8894 .4 824.4796 101.7876 .4 1098.5835 117.4956 1411.9574 133.2035 •5 829.5768 102.1018 .5 1104.4662 117.8097 .5 141S.6254 133.5177 .6 834.6898 102.4159 .6] 1110.36)45 118.1239 .6 1425.3092 133.8318 .7 839.8185 102.7301 .7 1116 27S6> 118.4380 . i 1432.0086 134.1460 .8 844.9628 103.0442 .8 1122.2083 118.7522 .8 1438.7238 134.4602 .9 850.1229 103.3584 .9 1128.1538 119.0664 .9 1445.4546 134.7743 33.0 855.2986 103.6726 38.0 1134.1149 119.3805 43.0 1452.2012 135.0885 .1 860.4902 103.9867 .1 | 1140.0918 119.6947 .1 1458.9635 135.4026 •) 8G->.6973 104.3009 .2 l 1146.0844 120.0088 .2 1465.7415 135.716S .3 870.9202 104.6150 .3 I 1152.0927 120.3230 .3 1472.5352 136.0310 .4 8 < 0.loSS 104.9292 .4 j 1158.1167 120.6372 .4 I 1479.3446 136.3451 | ,5 j 881.4131 105.2434 .5 1164.1564 120.9513 .5 1486.1697 136.6593 886.6831 mam.* .6 1170.21 IS 121.2655 .6 1493.0105 136.9734 | MM 891.9688 105.8717 • i 1176.2830 121.5796 .7 1 1499.«8670 137.2876 ,S i 897.2 «03 106.1858 .8 l 1182.3698 121.«8938 .8 1506.7393 137.6018 .9 i 902.5874 105.5000 .9 | 1188.4724 122.2080 .9 I 1513.6272 137.9159 34.0 907.9203 106.8142 39.0 1194.5906 122.5221 44.0 j 1520.5308 138.2301 ■ 913.26SS 107.1283 .1 1200.7246) 122. «8363 .1 ! 1527.4502 138.5442 Ij 918.6331 107.4425 •> 1206.8742 123.1504 »> j 1534.3853 138.8584 .3 924.0131 107.7566 .3 1213.0396 123.4640 .3 ! 1 .>41.3360 139.1726 •4i 929.4088 108.0705 •1 , 1219.2207 123.7788 .4 1548.3025 139.4867 .5 934.8202 108.3849 .5 1225.4175 12«) 29 .5 I 1555.2847 139.8009 .6 940.2473 108.6991 fj§ 1231.6300 124.4071 .6 1502.2«826 140.1153 .7. 945.6901 109.0133 • « 1237.8582 124.7212 • > 1 1569.2962 140.4292 .8 951.1486 109.3274 .8 i 1244.1021 125.0354 .8 | 1576.3255 140.7434 .9 956.6228 109.6416 .9 1250.3617 125.3495 .9 i 1583.3706 141.0575MENSURATION. — CIRCLES, 45 AREAS AND CIRCUMFERENCES OF CIRCLES. (Advanciny by Tenths.) Diam. Area. Cireura. Diam. Area. Circtim. Diam. Area. Circum 45.0 1590.4313 141.3717 50.0 1963.4954 157.0796 55.0 2375.8294 172.7876 .1 1597.5077 141.6858 .1 1971.3572 157.3938 .1 23.84.4767 173.1017 .2 1601.5999 142.0000 .2 1979.2348 157.7080 o 2393.1396 173.4159 m :tm.7077 142.3142 .3 1987.1280 158.0221 .3 2401.81S3 173.7301 .4 161S.8313 142.62S3 .4 1995.0370 158.3363 .4 2410.5126 174.0442 .5 1625.9705 142.9425 .5 2002.9617 158.6504 .5 2419.2227 174.3584 .6 1633.1255 143.2566 .6 2010.9020 158.9646 .6 2427.9485 174.6726 .7 1640.2962 143.5708 .7 2018.8581 159.2787 .7 2436.6899 174.9867 .8 1647.4826 143.8849 .8 2026.8299 159.5929 .8 2445.4471 175.3009 .9 1654.6817 144.1991 .9 2034.8174 159.9071 .9 2454.2200 175.6150 46.0 1661.9025 144.5133 51.0 2042.8206 160.2212 56.0 2463.0086 175.9292 .1 1669.1360 144.8274 .1 2050.8395 160.5:154 .1 2471.8130 176.2133 .2 1676.3853 145.1416 .2 2058.8742 160.8495 .2 2480.6330 176.5575 .3 1683.6502 145.4557 .3 2066.9245 161.1637 .3 2489.4687 176.8717 .4 1690.9308 145.7699 .4 2074.9905 161.4779 .4 2498.3201 177.1858 .5 1698.2272 146.0841 .5 2083.0723 161.7920 .5 2507.1873 177.5000 .6 1705.5392 146.3982 .6 2091.1097 162.1062 .6 2516.0701 177.8141 1712.8670 146.7124 .7 2099.2829 162.4203 .7 2524.9687 178.1283 .8 1720.2105 147.0265 .8 2107.4118 162.7345 .8 2533.8830 178.4425 .9 1727.5697 147.3407 .9 2115.5563 163.0487 .9 2542.8129 178.7566 47.0 1734.9445 147.6550 52.0 2123.7160 163.3628 57.0 2551.7586 179.0708 .1 1742.3351 147.9600 .1 2131.8926 163.6770 .1 2560.7200 170.3849 .2 1749.7414 148.2832 .2 2140.0843 163.9911 .2 2509.6971 ' 179.6991 .3 1757.1635 148.5973 .3 2148.2917 164.3053 .3 2578.6899 180.0133 .4 1764.6012 148.9115 .4 2156.5149 164.6195 .4 2587.6985 180.3274 .5 1772.0646 149.2257 .5 2164.7537 164.9336 .5 2596.7227 180.6416 .6 1779.5237 149.5398 .6 2173.0082 165.2479 .6 2605.7626 180.9557 .7 1787.0086 149.8540 .7 2181.2785 165.5619 .7 2614.8183 181.2699 1 .8 1794.5091 150.1681 .8 21S9.5644 165.8761 .8 2623.8896 181.5841 .9 1802.0254 150.4823 .9 2197.8601 166.1903 .9 2032.9767 181.8982 48.0 1809.5574 150.7961 53.0 2206.1834 166.5044 58.0 2642.0794 182.2124 .1 1817.1050 151.1106 .1 2214.5165 166.8186 .1 2651.1979 182.5265 .2 1824.6684 151.4248 .2 2222.8653 167.1327 .2 2660.3321 182.8407 .3 1832.2475 151.73-89 .8 2231.2298 167.4469 .3 2669.4820 183.1549 .4 1839.8423 152.0531 .4 2239.6100 167.7610 .4 2678.6476 183.4690 .5 1847.4528 152.3672 .5 2248.0059 168.0752 .5 2087.8289 183.7832 ■ .6 1855.0790 152.6814 .6 2256.4175 168.3894 .6 2(597.0259 184.0973 .7 1862.7210 152.9956 .7 2261.8448 168.7035 .7 2706.2386 184.4115 .8 1870.3786 153.3097 .8 2273.2879 169.0177 .8 2715.4670 184.7256 .9 1878.0519 153.6239 .9 2281.7466 169.3318 . .9 2724.7112 185.0398 49.0 1885.7409 153.9380 54.0 2290.2210 169.0460 69.0 2733.9710 185.3540 .1 1893.4457 154.2522 .1 2298.7112 169.9602 .1 2743.2466 185.6681 .2 1901.1662 154.5664 .2 2307.2171 170.2743 .2 2752.5378 185.9823 .3 1908.9024 154.8805 .3 2315.7386 MB» .3 2761.8448 186.2964 .4 1916.6543 155.1947 .4 2324.2759 170.9026 .4 2771.1675 166.6106 , .5 1924.4218 155.5088 .5 2332.8289 171.2168 .5 2780.5058 186.9248 .6 1932.2051 155.8230 .6 2341.3976 171.5310 .6 2789.8599 187.2389 .7 1940.0042 156.1372 .7 2349.9820 171.8451 .7 2799.2297 187.5531 .8 1947.8189 156.4513 .8 2358.5821 172.1593 .8 2808.6152 187.8672 .9 1955.0493 156.7655 .9 2367.1979 172.4735 .9 2818.0165 188.181446 MENSURATION. — CIRCLES AREAS AND CIRCUMFERENCES OF CIRCLES. (Advancing by Tenths.) Dim Area. Circum. Diam. Area. Circum. Diam. Area. Cireum. 60.0 2827.4334 188.4956 65.0 3318.3072 204.2035 70.0 3848.4510 219.9115 .1 2836.8)60 188.8097 .1 3328.5253 204.5176 .1 3859.4544 220.2256 .2 2846.8144 189.1239 .2 3338.7590 204.8318 .2 3870.4736 220.5398 .3 2855.7784 189.4380 .3 3349.0085 205.1460 .3 3881.5084 220.8540 .4 2865.2582 189.7522 .4 3359.2736 205.4602 .4 3892.5590 221.16SI .5 2874.7536 190.0664 .5 3369.5545 205.7743 .5 3903.6252 221.4823 .6 2881.2648 190.3805 .6 3379.8510 206.0885 .6 3914.7072 221.7964 .7 2893.7917 190.6947 .7 3390.1653 206.4926 .7 3925.8049 222.1106 .8 2903.8343 191.0088 .8 3400.4913 206.71C>H .8 3936.9182 222.4248 .9 2912.8926 191.3230 .9 3410.8350 207.0310 .9 3948.0473 222.7389 61.0 2922.1666 191.6372 66.0 3421.1944 207.3451 71.0 3959.1921 223.0531 .1 2932 0563 191.9513 .1 3431.5695 207.6593 .1 3970.3526 223.3672 .2 29419)617 192.2655 f>) 3441.9603 207.9734 3981.5289 223.6)814 R 2951.2828 192.5796 .3 3452.3669 208.2870 .3 31)92.7208 223.9956 .4 2960.9197 192.8938 .4 3462.7891 208.6017 .4 4003.9284 224.3097 .5 2970.5722 193.2079 .5 3473.2270 208.9159 .5 4015.1518 224.6230 .6 2980.2405 193.5221 .6 3483.6807 209.2301 .0 4026./»908 224.9380 .7 2989.9244 193.8363 .7 3494.1500 209.5442 .7 4937.6456 225.2522 .8 2999.6241 194.1504 .8 3504.6351 209.8584 .8 4048.9160 225.5664 .9 3009.3395 194.4616 .9 3515.1359 210.1725 .9 4060.2022 225.S8U5 62.0 3019.0705 194.7787 67.0 3525.6524 210.4867 72.0 4971.5041 226.1947 .1 3028.8173 195.0929 .1 3536.1845 210.8909 .1 40S2.8217 226.50SS .2 3038.5798 195.4071 9 3546.7324 211.1150 4994.1550 226.8230 ■ 3048.3580 195.7212 .3 3557.2960 211.4292 .3 4105.5040 227.1371 .4 3058.1520 196.0354 .4 3567.8751 211.7433 .4 4116.8687 227.4513 R 3)67.9616 196.3495 .5 3578.4701 212.0575 .5 4128.2491 227.7655 .6 3077.7869 196.6637 .6 3589.0-nI l 212.3717 .6 4139.6452 228.0796 .7 3087.6279 196.9779 3599.7075 212.6858 .7 4151.0571 228.3938 .8 3097.4817 197.2920 .8 3610.5407 213.0000 .8 4162.4846 228.7079 .91 3107.3->71 197.0052 .9 3621.0075 213.3141 .9 4173.9279 229.0221 63.0 3117.2453 197.9203 68.0 3631.6811 213.6283 73.0 4185.3868 229.3363 ] .1! 3127.1492 198/2315 .1 3642.3701 213.9425 .1 4196.8615 229.6504 H 3137.0688 19S.51ST .2 5653.0751 214.2566 o 4208.3519 2^9.9646 ■ 3147.0040 198.8028 .3 3663.7960 214.5708 .3 4219.8579 230.2787 .4 3156.9550 193.1770 .4 3674.5324 214.8849 .4 4231.3797 230.5929 1 3166.9217 Mi .5 3685.284? 215.1991 .5 4242.9172 280.9071 .6 3176.9043 I9.).8ir>3 .6 3696.w523 215.5133 .6 4254.4704 231.2212 3186.9023 200.1195 • ( 8706.835'. 215.8274 . i 4266.0394 231.5354 .8 3196.9161 200.4336 .8 3717.635.1 216.1416 .8 4277 6240 231.8495 .9 3206.9456 200.7478 3728.4500 216.4556 .9 4289.2243 232.1637 64.0 3216.9909 201.0620 69.0 3739.2807 216.7699 74.0 4300.8403 232.4779 .1 3227.0518 201.3761 .1 8759.1270 217.0841 .1 4312.4721 232.7920 .2 3237.1285 201.6902 9 8760.9891 217.3982 o 4324.1195 233.1082 .3 3247.2222 202.0043 .3 3771.866') 217.7124 .3 4335.7827 233.4203 .4 3257.3289 202,3186 .4 3782.7693 218.0205 .4 4347.4616 233.7345 .5 3267,4527 202.6327 .5 3793.6695 218.3407 .5 4359.1562 234.0487 .6 3277.5922 202.9469 .6 3891.5914 218.0518 .6 4370.8664 234.3628 .7 3287.7474 203.2610 .7 3815,5350 218.9690 .7 4382.5924 234.6770 .8 3297.9183 203.5752 .8 3826.4913 219.283,2 .8 4394.3341 234.9911 .9 3308.1049 203.8894 .0 3837.4633 219.5973 .9 4406.0916 235.3053MENSURATION. — CIRCLES. 47 AREAS AND CIRCUMFERENCES OF CIRCLES. (Advancing by Tenths.) I Diam. Area. Circmn. Diam. Area. Circum. Diam Area. Circmn. 74.0 4417.8617 235.6194 80.0 5026.5482 251.3274 85.0 5674.5017 267.0354 .1 4429.653') 235.9336 .1 5039.1225 251.6416 .1 5687.8614 267.3495 .2 4441.4580 236.2478 .2 5051.7124 251.9557 .2 5701.2367 267.6637 .3 4453.2783 236.5619 .3 5064.3180 252.2699 .3 5714.6277 267.9779 .4 4405.1142 236.8761 .4 5076.9394 252.5840 .4 5728.0345 268.2920 .5 4476.9659 237.1902 .5 5089.5764 252.8982 .5 5741.4569 268.6062 .6 4488.8:532 237.5044 .6 5102.2292 253.2124 .6 5754.8951 268.9203 .7 4500.7163 237.8186 .7 5114.8977 253.5265 .7 5768.3490 269.2345 .8 4512.6151 238.1327 .8 5127.5819 253.8407 .8 5781.8185 269.5486 .9 4524.5296 238.4409 .9 5140.2818 254.1548 .9 5795.3038 269.8628 76.0 4536.4598 238.7010 81.0 5152.9973 254.4690 86.0 5808.S048 270.1770 .1 4548.4057 239.0752 .1 5165.72S7 254.7832 .1 5822.3215 270.4911 239.3894 5178.4757 255.0973 9 270.8053 .3 4572.3446 239.7035 .3 5191.2384 255.4115 .3 5849.4020 271.1194 .4 4584.3377 240.0177 .4 5204.0168 255.7256 .4 5862.9659 271.4336 .5 4596.3464 240.3318 .5 5216.81 if 256.0398 .5 5876.5454 271.7478 .6 4608.3708 240.6160 .6 5229.020S 256.3540 .6 5890.1407 272.0619 .7 4620.4110 240.9602 .7 5242.4463 256.66S1 .7 5903.7516 272.3761 .8 4632.4609 241.2743 .8 5255.2876 256.9823 .8 5917.3783 272.6902 .9 4644.5384 241.5885 .9 5268.1446 257.2966 .9 5931.0206 273.0044 77.0 4656.6257 241.9026 82.0 5281.0173 257.6106 87.0 5044.6787 273.3186 .1 4668.7287 242.2168 .1 5293.9056 257.9247 .1 5958.3525 273.6327 .2 4680.8474 242.5310 5306.8097 258.23S0 .2 5972.0420 273.9469 .3 4692.9818 242.8451 .3 5319.7295 258.5531 .3 5985.7472 274.2610 .4 4705.1319 243.1592 .4 5332.6650 258.8672 .4 5999.4681 274.5752 .5 4717.2977 243.4734 .5 5345.6162 259.1814 .5 6013.2047 274.8894 ».6 4729.4792 243.7876 .6 5358.5832 259.4956 .6 6026.9570 275.2035 .7 4741.6765 244.1017 .7 5371.5658 259.8097 .7 6040.7250 275.5177 .8 4753.8894 244.4159 .8 5384.5611 260.1239 .s1 6054.5088 275.8318 .9 4766.1181 244.7301 .9 5397.5782 260.4380 .9 6068.3082 276.1460 78.0 4778.3624 245.04421 83.0 5410.6079 260.7522 88.0 6082.1234 270.4602 .1 4790.6225 245.3584 .1 5423.6534 261.0663 .1 6095.9542 276.7743 .2 4802.8983 245.6725 .2 5436.7146 261.3805 .2 6109.8008 277.08S5 .3 4815.1897 245.9867 .3 5449.7915 261.6947 .3 6123.6631 277.4026 .4 4827.4969 246.3009 .4 5462.8840 262.0088 .4 6137.5411 277.7168 .5 4839.8198 246.6150 .5 5475.9923 262.3230 .5 6151.4348 278.0309 .6 4852.1584 246.9292 .6 5489.1163 262.6371 .6 6165.3442 278.3451 .7 4864.5128 247.213:’» .7 5502.2561 262.9513 .7 6179.2093 278.6593 .8 4876.8828 247.5575 .8 5515.4115 263.2655 .8 6193.2101 278.9740 .9 4889.2685 247.8717 .9 5528.5826 263.5796 .9 6207.1666 279.2870 79.0 4901.6699 248.1858 84.0 5541.7694 263.8938 89.0 6221.1389 279.6017 .1 4914.0871 248.5000 .1 5554.9720 264.2079 .1 6235.1268 279.9159 .2 4926.5199 248.8141 .2 5568.1902 264.5221 .2 6249.1304 280.2301 .3 4938.9685 249.1283 .3 5581.4242 264.8363 .3 6263.1498 280.5442 .4 4951.4328 249.4425 .4 5594.6739 265.1514 .4 6277.1849 280.8584 .5 4963.9127 249.7560 .5 5607.9392 265.4646 .5 6291.2356 281.1725 .6 4976.4084 250.0708 .6 5621.2203 265.7787 .6 6305.3021 281.4867 .7 4988.9198 250.3850 .7 1 5634.5171 266.0929 .7 6319.3843 281.8009 .8 5001.4469 250.6991 .S J 5647.8296 266.4071 .8 6333.4822 282.1150 .9 5013.9 S97 251.0133 .9 ! 1 5661.1578 266.7212 .9 6347.5958 282.429248 MENSURATION. — CIRCLES, AREAS AND CIRCUMFERENCES OF CIRCLES. (Advancing by Tenths.) Dium Area. Circum. l*iam. Area. Circum. Diam. Area. Circum. 90.0 6361.7251 282.7433 93.5 6860.1471 293.7389 97.0 7389.8113 304.7345 .1 6370.8701 283.0575 .6 0880.8419 294.0531 :! 7405.0559 305.0480 .2 0390.0:509 283.3717 .7 6895.5524 294.3672 7420.3162 305.3628 .3 6404.2073 283.0858 .8 0910.2780 294.6814 .3 7435.5922 305.6770 .4 6418.3995 284.0000 .9 6925.0205 294.9956 •4 7450.8839 305.9911 .5 0432.6073 284.3141 94.0 6939.7782 295.3097 .5 7466.1913 306.3053 .6 HiliHfl 284.0283 .1 0954.5515 295.6239 .6 7481.5144 306.6194 .7 6461.0701 284.9425 .2 6969.3106 295.9380 .7 7496.8532 306.9336 .8 6475.3251 285.2560 .3 0981.1453 296.2522 .8 7512.2078 307.2478 .9 6489.5958 285.5708 .4 6998.9638 296.5663 .9 7527.5780 307.5619 91.0 6503.8822 285.8849 .5 7013.8019 200.8805 98.0 7542.9640 307.8T01 .1 a— 280.1991 .6 7028.0538 297.1947 .1 7558.0050 308.1902 .2 6532.5021 286.5133 .7 7043.5214 297.5088 .2 7573.7830 308.5044 .3 6540.8356 280.8274 .8 7058.4047 207.8280 .3 7589.2161 308.8186 .4 6561.1848 287.1416 m 7073.3033 298.1371 •4 7004.0048 309.1327 .5 6575.5498 287.4557 95.0 7088.2184 298.4513 .5 7020.1293 309.4469 .6 6589.9304 287.7099 .1 7103.1488 208.7055 .6 7035.6095 309.7610 .7 6604.3208 288.0840 .2 7118.1950 200.0700 .7 7651.1054 310.0732 .8 6618.7388 288.3982 .3 7133.0508 299.3938 .8 7000.0170 310.3894 .9 6633.1660 288.7124 .4 7148.0343 299.7079 .9 7682.1444 310.7035 92.(4 6647.0101 289.0265 .5 7163.0270 300.0221 99.0 7697.0*vQ3 311.0177 .1 6602.0692 289.3407 .0 7178.0366 . 300.3:103 .1 7713.2461 311.3318 .2 6676.5441 289.6548 .7 7193.0612 800.0504 .2 7728.8206 311.6400 .3 6691.0347 289.9090 .8 7208.1016 300.9646 .3 7744.4107 311.9602 .4 6705.5410 290.2832 .9 7223.1577 301.2787 .4 7760.0166 312.2743 .5 6720.0030 290.5973 90.0 7238.2295 301.5929 .5 7775.6382 312.5885 .6 6734.6008 290.9115 .1 7253.3170 301.9071 .6 7791.2754 312.9026 .7 6749.1542 291.2256 .2 7268.4202 302.2212 .7 7806.9284 313.2168 .8 6763.7233 291.5398 .3 7283.5391 302.5354 .8 7822.5971 313.5309 .9 6778.3082 291.8540 .4 7298.6737 302.8405 .9 7838.2815 313.8451 93.0 6792.9087 292.1681 .5 731,3.8240 303.1637 100.0 7853.9S16 314.1593 .1 oBn 292.4823 .6 7328.9901 303.4779 ■ BBHhk9 292.7904 .7 7344.1718 303.7920 .3 68:46.8040 293.1106 .8 7359.3693 304.1062 .4 6851.4080 293.4248 .9 7374.5824 304.4203 MENSURATION, r- CIRCLES. 49 AREAS OF CIRCLES. {ADVANCING BY EIGHTHS.) AREAS. bia m 0.0 04 o-i 0 i (4 0 4 O.f H 0 0.0 0.0122 0.0190 0 1101 0.1963 0.306$ 0.4417 0.6013 1 0.7854 0.9940 1.2*47 1.484 1.767 2.073 2.405 2.761 2 3.1416 3.546 3.976 4.430 4.908 5.411 5.939 6.491 3 7.068 7.669 8.295 8.946 9.621 10.32 11.04 11.79 4 12.56 13.36 14.18 15.03 15.90 16.80 17.72 38.66 5 19.63 20.62 21.04 22.69 23.75 24.85 25.96 27.10 6 28.27 29.46 ■ 30.67 31.91 33.18 34.47 35.78 37.12 ■ 7 38.48 39.87 41.28 42.71 44.47 45.66 47.17 48.70 8 50.26 51.84 53.15 55.08 56.74 • 58.42 60.13 61.86 9 63.61 65.39 67.20 69.02 70.88 72.75 74.66 76.58 10 78.54 80.51 82.51 81.54 86.59 88.66 90.76 92.88 11 95.03 97.20 99.40 101.6 103.8 106.1 108.4 110.7 12 113.0 115.4 117.8 120.2 122.7 125.1 127.6 130.1 13 132.7 135.2 137.8 140.5 143.1 145.8 148.4 151.2 14 153.9 156.6 159.4 102.2 165.1 167 9 170.8 373.7 15 176.7 179.6 182.6 185.6 188.6 191.7 194.8 197.9 16 201.0 201.2 207.3 210.5 213.8 217.0 220.3 223.6 17 226.9 230.3 233.7 237.1 24 0 5 243.9 247.4 250.9 18 254.4 258.0 201.5 20').1 268.8 272.4 276.1 279.8 19 283.5 287.2 291.0 291.3 298.6 302.4 306.3 310.2 20 314.1 318.1 322.0 3-26.0 330.0 334.1 338.1 342.2 21 316.3 350.4 351.6 358.8 363.0 307.2 371A 375.8 • ')'> 380.1 384.4 3S8.8 393.2 397.6 402.0 405.4 410.9 23 415.4 420.0 421.5 429.1 433.7 438.3 443.0 447.6 24 452.3 457.1 401.3 46 i.6 471.4 476.2 481.1 485.9 25 490.8 495.7 509.7 505.7 510.7 515.7 520.7 525.8 26 530.9 536.0 511.1 5 46.3 551.5 556.7 562.0 567.2 27 572.5 577.8 583.2 588.5 593.9 599.3 604.8 610.2 28 C 15.7 621.2 626.7 632.3 637.9 643.5 649.1 654.8 29 6(50.5 606.2 671.9 677.7 * 683.4 6S9.2 695.1 700.9 30 703.8 712.7 . 718.6 724.6 730.6 736.6 742.6 748.6 31 754.8 760.9 767.0 773.1 779.3 7S5.5 791.7 798.0 32 804.3 810.6 816.9 823.2 829.6 836.0 842.4 848.8 33 855.3 861.8 808.3 874.9 881.4 838.0" 894.6 901.3 31 907.9 914.7 921.3 928.1 934.8 941.6 918.4 955.3 33 962.1 969.0 975.9 9S2.8 989.8 996.8 10. >3.8 1010.8 36 1017.9 1025.0 1032.1 1039.2 1046.3 10'>3.5 1060.7 1068.0 3 i 1075.2 1082.5 1089.8 1097.1 1101.5 1111.8 1119.2 1126.7 33 1134.1 1141.6 1149.1 1156.6 1161.2 1171.7 1179.3 1186.9 39 1191.6 1202.3 1210.0 1217.7 1225.4 1233.2 1241.0 1248.8 40 1236.6 1264.5 . 1272.4 1280.3 1288.2 1296.2 1304.2 1312.2 41 1320.3 1328.3 1336.4 1344.5 1352.7 1360.8 1369.0 1377.2 42 1385.4 1393.7 1402.0 1410.3 1418.6 1427.0 1435.4 1443.8 43 1452.2 1460.7 1469.1 1477.6 1486.2 1494.7 1503.3 1511.9 44 1520.5 1529.2 1537.9 1546.6 1555.3 1564.0 1572.8 1581.6 45 1590.4 1599.3 1608.2 1017.0 1626.0 1634.9 1643.9 1652.950 MENSURATION. — CIRCUMFERENCES CIRCUMFERENCES OF CIRCLES. (advancing by EIGHTHS. I CIRCUMFERENCES. Piani. 0.0 04 0.| m 0-1 O.f 0 l G. 8 0 0.0 0.3927 0.7854 1.178 1.570 1.963 2.356 2.748 l 3.141 3.534 3.927 4.319 4.712 5.105 5.497 5.890 2 6.2X3 6.675 7.068 7.461 7.854 8.246 8.639 9.032 3 9.421 9.817 10.21 10.60 10.99 11.38 11.78 12 17 4 12.56 12.95 13.35 13.74 14.13 14.52 14.92 15.31 5 15.70 16.10 16.49 16.88 17.27 17.67 18.06 18.45 6 18.84 19.24 19.63 20.02 20.42 20.81 21.20 21.59 7 21.99 22.38 22.77 23.16 23.56 23.95 24.34 24.74 8 25.13 25.52 25.91 26.31 26.70 27.09 27.48 27.88 9 28.27 28.66 29.05 29.45 29.84 30.23 30.63 31.02 10 31.41 31.80 32.20 32.59 32.98 33.37 AO OO. i è 34.16 11 34.55 34.95 35.34 35.73 36.12 36.52 36.91 37.30 12 37.69 38.09 38.48 38.87 39.27 39.66 40.05 40.44 13 40.84 41.23 41.62 42.01 42.41 42.80 43.19 43.58 14 43.98 44.37 44.76 45.16 45.55 45.91 46.33 46.73 15 47.12 47.51 47.90 48.30 48.69 49.OS 49.48 49.87 16 50.26 50.65 51.05 51.44 51.83 52.22 52.62 53.01 ■ 17 53.40 53.79 54.19 54.58 54.97 55.37 55.76 56.15 18 56.54 56.94 57.33 57.72 58.11 58.51 58.90 59.29 19 59.69 60.08 60.47 60.86 61.26 61.65 62.04 62.43 20 62.83 63.22 63.61 64.01 64.40 64.79 65.18 65.58 21 65.97 66.36 66.75 67.15 67.54 67.93 68.32 68.72 22 69.11 69.50 69.90 70.29 70.68 71.07 71.47 71.86 23 72.25 72.64 73.0 4 73.43 73.82 74.22 74.61 75.00 24 75.39 75.79 76.18 76.57 76.96 77.36 77.75 78.14 25 78.54 78.93 79.32 79.71 80.10 80.50 80.89 81.28 26 81.68 82.07 82.46 82.85 83.25 8.3.64 84.03 84.43 -1 84.82 85.21 85.60 86.00 86.39 86.78 87.17 87.57 28 87.96 88.35 88.75 89.14 89.53 89.92 90.32 90.71 29 91.10 91.49 91.89 92.28 92.67 93.06 93.46 93.85 30 94.24 94.64 95.03 95.42 95.81 96.21 96.60 96.99 31 97.39 97.78 98.17 98.57 98.96 99.35 99.75 100.14 32 100.53 100.92 101.32 101.71 102.10 102.49 102.89 103.29 33 103.67 104.07 10 4.46 10 4.85 105.24 105.61 106.03 106.42 34 105.81 107.21 107. H 107.99 108.39 108.78 109.17 109.56 35 109.96 110.35 110.74 111.13 111.53 111.92 112.31 112.71 36 113.10 113.49 113.88 114.28 114.67 115.06 115.45 115.85 37 116.24 116.63 117.0-2 117.42 117.81 118.20 118.6,0 118.99 38 119.38 119.77 120.17 120.56 120.95 121.34 121.74 122.13 39 122.52 122.92 123.31 123.70 124.09 124.49 124.88 125.27 40 125.66 126.06 126.45 126.84 127.24 127.63 128.02 128.41 41 128.81 129.20 127.59 129.98 130.38 130.77 13,1.16 131.55 42 131.95 132.34 132.73 133.13 133.52 133.91 134.30 134.70 415 135.09 135.48 135.87 136.27 136.66 137.05 137.45 137.84 -14 138.23 138.62 139.02 139.41 139.80 140.19 140.59 140.98 45 141.37 141.76 142.16 142.55 142.94 143.34 143.73 144.12MENSURATION. — CIRCLES. 51 AREAS AND CIRCUMFERENCES OF CIRCLES. From I to 50 Feet. (advancing by one inch.) Diam. Area. Circum. Diam. Area. Circum. Diam. Area. Circum. Ft. jFeet. Ft. In. Ft. Feet. Ft. In. Ft. Feet. Ft. In. 1 0 0.7854 3 5 0 19.635 15 9 0 63.6174 28 3.1 1 0.9217 3 A o 4S 1 20.2947 15 1® 1 64.8006 28 6* 2 1.069 3 8 2 20.9656 16 0:* -5 2 65.9951 * 28 9à 3 1.2271 3 11 3 21.6475 16 m 3 67.2007 29 6 1 4 1.3962 4 21 4 22.34 16 9 4 68.4166 29 or? 5 1.5761 4 5j| 5 23.0437 17 “5 5 69.644 29 7 6 1.7671 4 Slf 6 23.7583 17 31 6 70.8823 29 m 7 1.9689 4 EÎ 7 24.4835 17 6* ' 7 72.1309 30 M 8 2.1816 5 •73 8 25.2199 17 9* 8 73.391 30 9 2.4052 5 51 9 25.9672 18 3 2| 9 74.662 30 7A 10 2.6398 5 9 10 26.7251 18 3| 10 75.9433 30 11 11 2.8852 6 21 11 27.4943 18 "a 11 77/2362 31 2 0 3.1416 6 38 6 0 28.2744 18 10J 10 0 78.54 31 5 1 3.4087 6 65 1 29.0649 19 1| 48 1 79.854 31 1} 2 3.6869 6 9| 2 29.8668 19 2 81.1795 31 ■ 3 3.976 7 K * 3 30.6796 19 71 3 82.516 32 2g 4 4.276 7 07 "a 4 31.5029 19 10* 4 83.8627 32 5 4.5869 1 f 5 32.3376 20 if 5 85.2211 32 8* 6 4.9087 7 m 6 33.1831 20 41 6 86.5903 32 m 7 5.2413 8 n 7 34.0391 20 *8 7 87.9697 33 n 8 5.585 8 4 I 8 34.9065 20 m 8 89.3608 33 6* 9 5.9395 8 9 35.7847 21 if 9 90.7627 33 n 10 6.3049 8 m 10 36.6735 21 ok 10 92.1749 34 s 6 11 6.6813 9 n 11 37.5736 21 H 11 93.5986 34 ° 5 3 0 7.0686 9 5 7 0 38.4846 21 HI 3 11 0 95.0334 34 I 1 7.4666 9 8| 1 39.406 22 1 96.4783 34 Bl 3 2 7.8757 9 m 2 40.3388 22 61 2 97.9347 35 3 8.2957 10 2? 3 41.2825 22 n 3 99.4021 35 4 8.7265 10 5j? 4 42.2367 23 3 a 4 100.8797 35 71 5 9.1683 10 88 5 43.2022 23 21 5 102.3689 35 10| 6 9.6211 10 ni 3 6 44.1787 23- 65 6 103.8691 36 1Ï < 10.0846 11 7 45.1656 23 93 7 105.3794 36 4| 8 10.5591 11 «1 8 46.1638 24 n 8 106.9013 36 41 9 11.0446 11 n 9 47.173 24 41 9 108.4342 36 101 10 11.5409 12 1 10 48.1962 24 4 A 10 109.9772 87 BSE 11 12.0481 12 3* 11 49.2236 24 io§ 11 111.5319 37 51 4 0 12.5664 12 68 8 0 50.2656 25 n 12 0 113.0976 37 81 1 13.0952 12 08 1 51.3178 25 4* 1 114.6732 37 Ilf 2 13.6353 13 1 *> 52.3816 25 71 2 116.2607 38 B 3 BBS 13 41 3 53.4562 25 11 3 117.859 38 4 14.7479 13 71 4 54.5412 26 B “8 4 119.4674 38 *3 5 15.3206 13 101 5 55.6377 26 5 121.0876 39 0 6 15.9043 14 if 6 56.7451 26 H 0 122.7187 39 31 7 16.4986 14 48 7 57.8628 26 lié 7 124.3598 39 6jj 8 17.1041 14 n 8 58.992 27 If 8 126.0127 39 9| 9 17.7205 14 11 9 60.1321 27 9 127.6765 40 6 10 18.3476 15 *3 10 61.2826 27 9 10 129.3504 40 33 11 18.9858 15 54 11 62.4445 28 1 » 11 131.036 40 6352 MENSURATION. — CIRCLES Areas and Circumferences of Circles (Feet and Inches). Diam. Area. Circnm. Diaiu. Area. Circum. Plain. Area. Circnm. Ft. Feet. Ft. In. n. Feet. Ft. In. Ft. Feet. Ft. In. 13 0 132.7320 40 10 18 0 254.4090 50 6 è 23 0 415.4700 72 3 l 134.4391 41 1} i 250 8303 50 n 1 418.4915 72 ■ 2 130.1574 41 2 259.2033 57 1 2 421.5192 72 m 3 137.8807 41 -1 1 .7 o 20].5872 57 4 3 424.5577 ^ 3 n 4 139.020 41 i<| 4 203.9807 57 7rf 4 427.0055 73 3g f> 141.3771 42 ill 5 200.3804 57 n 5 430.0058 73 0} G 143.1391 42 si 0 208.8031 58 I 6 433.7371 73 f 144.9111 42 8 7 271.2293 58 4* 7 430.8175 74 1 8 140.0949 42 nj 8 273.6078 58 7| 8 439.9100 74 4J 9 14S.4890 43 ‘M 9 270.1171 58 Hi 9 443.0140 74 1 10 150.2943 43 5* 10 278.5701 58 2 10 446.1278 74 10gl 11 152.1109 43 11 281.0472 59 11 449.2530 75 1| 14 0 153.9384 43 113 19 0 283.5294 59 sj 24 0 452.3904 75 4j 1 1*55.7758 44 •)? 1 280.021 59 pH 1 455.5362 75 71 2 157.025 44 0 o 288.5249 60 21 2 458.6948 75 11 3 159.4852 44 n 3 291.0397 00 •'8 3 401.8642 70 2J 4 101.3553 45 i 4 4 293.5041 00 sg 4 405.0428 76 .51 5 103.2373 45 H 5 290.1107 60 5 408.2341 70 G 105.1303 45 oa 0 298.0483 00 3g 6 471.4303 76 llg 7 107.0331 45 m 7 301.2054 61 6} 7 474.0476 77 2g 8 108.9479 40 i 8 8 303.7747 61 R 8 477.8710 77 5$ 9 170.8735 40 4 9 300.355 61 9 481.1005 77 9 10 172.8091 46 7! 10 308.9448 61 10 484.3506 78 J 11 174.7505 40 ni 11 311.5409 02 ci 11 487.6073 18 8 g 15 0 170.715 47 14 20 0 314.10 02 ■ 25 0 490.875 78 65 1 178.0832 47 4§ 1 310.7824 02 1 1 494.1516 78 ■ 2 180.0034 47 7-4 2 319.4173 63 4.1 2 497.4411 79 g 3 182.0545 47 1087 3 3.22.003 03 7g 3 500.7415 79 B 4 184.0555 48 Ü 4 324.7182 03 id 4 504.051 79 7g 5 180.0084 48 5J 5 327.3858 63 1§ 5 507.3732 7!) Ill 6 188.0923 48 1 0 330.0043 64 4g 6 510.7003 80 lg 7 190.720 48 HI 7 332.7522 64 n 7 514.0484 SO 4 i 8 192.7710 49 ») '> 8 335.4525 64 n 8 517.4034 80 7§ 9 194.8282 49 sj 9 338.1037 65 • )1 as 9 520.7092 80 log 10 190.8940 49 10 340.8844 65 4 10 524.1441 si 1 11 198.973 50 0 11 343.6174 65 11 527.5318 81 5 16 0 201.0024 50 3g 21 0 340.3014 65 11» 26 0 530.9304 SI 8J 1 203.1015 50 Ü 1 349.1147 60 1 534.3379 si ng 2 205.2720 50 | 2 351.8804 60 2 537,7583 82 2§ 3 207.8940 51 3 354.0571 60 9 3 541.1890 82 5g 4 209.5204 51 33 4 357.4432 60 i 5 4 544.0299 82 8J 5 211.0703 51 oi 5 300.2417 67 B 5 548.083 83 111 6 213.8251 51 10 0 303.0y 11 67 61 6 551.5471 83 3 7 215.9890 52 ij 7 305.809S 67 If 1 555.0201 83 0g 8 218.1002 52 4j 8 308.7011 os Ï 8 55S.5059 83 9$ 9 220.3537 52 9 371.5432 68 9 502.0027 S4 g 10 222.551 52 ioX 10 374.3947 68 I 10 565.5084 84 3g 11 224.7003 53 ii 11 377.2587 68 11 569.027 S4 17 0 220.9800 53 43 22 0 380.1330 69 1! 27 0 572.5500 84 91 1 229.2105 53 8 1 383.0177 69 1 576.0949 85 1 2 231.4525 53 m *> 385.9144 09 J 2 579.6403 85 4g 3 233.7055 54 o 3SS.S22 69 8 583.2085 85 81 4 235.9082 54 &i 4 391.7389 70 n 4 580.7796 85 llg 5 238.243 54 si 5 394.0083 70 5 6 590.3037 80 U 6 240.5287 54 UK 0 397.0087 70 fl 6 593.9587 86 4g 7 242.8241 55 2| i 400.5583 70 4 r? 11 7 597.5625 86 7| 8 245.1310 55 0 8 403.5204 71 8 001.1793 80 11 9 247.45 55 91 9 400.4935 71 9 604.807 87 2ft 10 249.7781 50 I 4 10 409.4759 71 sg 10 ' 008.4436 87 5g 11 252.1184 50 11 412.4707 71 HI 11 612.0931 87 8JMENSURATION. — CIRCLES. 53 Areas and Circumferences of Circles (Feet and Inches). Diam. Area. Cirrum. Diam. Area. Circum. Diam. Area. i Circum. 1 Ft. Feet. Ft. In. Ft. Feet. Ft. In. Ft. Feet. Ft. In. 1 28 0 615.7536 87 11 h 3:5 0 855.301 103 8 38 0 1134.118 119 4* i 1 619.4228 88 •)6 1 859.624 103 HI 1 1139.095 119 7§ j 2 623.105 88 5Ï 2 863.961 104 “4 2 1144.087 119 104J 3 626.7982 8S 9 3 868.309 104 3 1149.089 120 2 4 630.5002 89 t 4 872.665 104 4 1154.110 120 I 5 634.2152 89 3* 5 877.035 104 ni 5 1159.124 120 I 6 637.9411 89 m 6 881.415 105 21 6 mm 120 m i 641.6758 89 9* 7 885.804 105 6 7 1169.202 121 24 8 645.4235 90 6 8 890.206 105 n 8 1174.259 121 °8 9 649.1821 90 9 894.619 106 1 4 9 1179.327 121 1$ 10 652.9495 90 1 10 899.041 106 3| 10 1184.403 121 Ilf 11 656.73 90 1U 11 903.476 106 6| 11 1189.493 122 »I. *29 0 660.5214 91 H 34 0 907.922 106 95 39 0 1194.593 122 1 1 664.3214 91 4g 1 912.377 107 I 1 1199.719 122 94 *2 668.1346 91 7* 9 916.844 107 4 2 1204.824 123 f 3 671.9587 91 3 921.323 107 7 8 3 1209.958 123 Of. 4 675.7915 92 ll 4 925.810 107 10* 4 1215.099 123 6* 5 679.6375 92 41 5 930.311 108 n 5 1220.254 123 m 6 683.4943 92 K 6 934.822 108 n . 6 1225.420 124 ii 7 687.3598 92 ill Y 939.342 108 7.1 7 12:50.594 124 4* 8 691.2385 93 25 8 943.875 108 10s 8 1235.782 124 7g 9 695.1028 93 Ü 9 948.419 109 2 9 1240.981 124 101 10 699.0263 93 10 952.972 109 10 1246.188 125 n 11 702.9377 93 n* 11 957.538 109 8* 11 1251.408 125 30 0 706.86 94 Of 35 0 962.115 109 n§ 40 0 1256.64 125 1 710.791 94 6 1 966.770 110 ok 1 1261.879 125 11 2 714.735 94 94 2 971.299 110 2 1267.133 126 01 -.j 3 718.69 95 3 975.908 110 «1 3 1272.397 126 4 722.654 95 3* 4 980.526 111 0 4 1277.669 126 Hi 5 726.631 95 6* 5 985.158 111 3g 5 1282.955 126 HI 6 730.618 95 m 6 989.803 111 6* 6 1288.252 127 m i 734.615 96 I 7 994.451 111 n 7 1293.557 127 51 8 738.624 96 4 8 999.115 112 à 8 1298.876 127 9 9 742.645 96 7* 9 1003.79 112 1 9 1304.206 128 1 4 10 746.674 96 103 10 1008.473 112 01 10 1309.543 128 3g 11 750.716 97 là 11 1013.170 112 10 11 1314.895 128 64 31 0 754.769 97 4f 36 0 1017.878 113 1* 41 0 1320.257 128 9§ 1 758.831 97 1 1022.594 113 4* 1 1325.628 129 3 4 2 762.906 97 101 2 1027.324 113 7g 2 1331.012 129 07 °8 3 766.992 98 2 3 1032.064 113 101 3 1336.407 129 7 * 4 771.086 98 5* 4 1036.S13 114 n 4 1341.810 129 10* 5 775.191 98 83 5 1041.576 114 41 5 1347.227 130 n 6 779.313 98 Hi 6 1046.349 114 8 6 1352.655 130 44:. 7 783.440 99 2| 7 1051.130 114 11* 7 1358.091 130 1 8 8 787.581 99 8 1055.926 115 2* 8 1363.541 130 10.Î 9 791.732 99 81 9 1060.731 115 9 1369.001 131 11 10 795.892 100 0 10 1065.546 115 9* 10 1374.47 131 5 11 800.065 100 3g 11 1070.374 115 Ht 11 1379.952 131 32 0 804.25 100 63 37 0 1075.2126 116 21 42 0 1385.446 131 118 1 808.442 100 9è 1 1080.059 116 6 1 1390.247 132 24 2 812.648 101 6 g 2 1084.920 116 91 2 1396.462 132 5t 3 816.865 101 3jJ 3 1089.791 117 1 4 3 1401.988 132 8jJ 4 821.090 101 61 4 1094.671 117 m 4 1407.522 132 ni 5 825.329 101 10 5 1099.564 117 61 5 1413.07 133 3 6 829.579 102 n 6 1104.469 117 9§ 6 1418.629 133 6* i 833.837 102 4g 7 1109.381 118 i 4 7 1424.195 133 9* 8 1 838.108 102 7* 8 1114.307 118 4 8 1429.776 134 1 5 9 842.2,91 102 m 9 1119.244 118 7g 9 1435.367 134 3| 10 846.681 103 n 10 1124.189 118 10* 10 1440.967 134 6* 11 850.985 103 41 11 1129.148 119 n 11 1446.580 134 9154 MENSURATION. — CIRCULAR ARCS Areas and Circumferences of Circles (Feet and Inches). Siam. Area. Circum. Siam. Area. Circum. Siam. Area. Circum. Ft. Feet. ■ In. Ft. Feet. Ft. In. Ft. Feet. Ft. In. 43 0 1452.*205 135 1 40 0 1001.900 144 1} 49 0 1885.745 153 11 ^ 1 1457.830 135 4£ 1 1007.931 144 n 1 1892.172 154 2J O 14(13.483 135 fi 9 1073 97 145 n s o 1 S98.504 154 5.4 1 t> 1409.14 135 loi 3 1680.02 145 3.4 3 1905.037 154 m 4 1474.804 130 IK 4 10X0.077 145 4 1911.497 154 iij 5 14X0.4S3 130 4} 5 1092.148 145 |l 5 1917.901 155 •)7 0 148G.173 130 3 0 1098.231 140 u 6 1924.420 155 6* 7 1491.870 130 11 7 1704.321 140 H i 1930.919 155 91 8 1497.582 137 Ol 8 1710.425 146 8 1937.310 156 i <> 150.3.305 137 M 9 1710.511 140 m 9 1943.914 150 34 10 1509.035 137 h 10 1722.003 147 u 10 1950.439 156 of 11 1514.779 137 11 1728.801 147 n 11 1956.909 156 41 0 1520.534 138 93 47 0 1734.947 147 7| 50 0 1963.5 157 7 1 1520.297 138 u 1 1741.101 147 n 2 15323*74 138 9 2 1747.274 148 O! “8 •) O 1537.802 139 h Q *> 1 i o3.4o;> 148 |l 4 1543.058 139 1 4 1759.043 148 H f> 1549.478 139 M 5 Î 705.845 148 1U 0 1555.2SS 139 n 0 1772.059 149 0 6 -8 4 1501.110 140 a 3 7 1778.28 149 H s 1500.959 140 8 1784.515 149 H 9 1572.812 140 7i 9 1790.701 150 5 10 1578.073 141 mi 10 17973*15 150 11 11 1584.549 141 U 11 1803.283 150 pi 45 0 1590.435 141 4J 48 0 1809.502 150 1 1590.329 141 7$ 1 1815.84S 151 6 g o 1002.237 141 10j o 1822.149 151 l| 3 1604.155 142 n a 1828.400 151 6| 4 1014.082 142 5 4 1834.779 151 10J 5 1020.023 142 n r> 1841.173 152 u G 1025.974 142 hi 0 1847.457 152 7 1031.933 143 “8 . 7 IS53.809 152 i i 8 1037.907 143 f>A 8 1800.175 152 10? 9 1043.891 143 §3 9 1X00.552 153 i| 10 1< >49.883 143 HI 10 1872.937 153 4| 11 1655.889 144 3 11 1879.335 153 ii Circular Arcs. To find the length of a circular arc when its chord and height, or versed sine is given; by the following table. Rule. —Divide the height by the chord; find in the column of heights the number equal to this quotient. Take out the corresponding number from the column of lengths. Multiply this number by the given chord. Example. — The chord of an arc is 80 and its versed sine is SO, what is tlie length of the arc ? Ans. BO t 80 — 0.375. The length of an arc for a height of 0.375 we find from table to be L3-KXW, 80 X 1.340(53 = 107.2504 =r-lengt h of arc.MENSURATION. — CIRCULAR ARCS.. i)0 TABLE OF CIRCULAR ARCS. 1 Hghts. Lengths. lights. Lengths. lights. Lengths. lights. Lengths. lights. Length*. .001 1.00001 .062 1.01021 .123 1.03987 .184 1.08797 .245 1.15308 .002 1.00001 .063 1.01054 .124 1.04051 .185 1.08890 .246 1.15428 .00:5 1.00002 .064 1.0loss ill 1.04116 .186 1.08984 .247 1.15549 .004 1.00004 .065 1.01123 .126 1.04181 .187 1.09079 .248 1.15670 .005 1.00007 .066 1.01158 .127 1.04247 .188 1.09174 .249 1.15791 .000 1.00010 .067 1.01193 .128 1.04313 .189 1.09269 .250 1.15912 .007 1.00013 .068 1.01228 .129 1.04380 .190 1.09365 .251 1.16034 .008 1.00017 .069 1.01264 .130 1.94447 .191 1.09461 .252 1.16156 .000 1.00022 .070 1.01301 .131 1.04515 .192 1.09557 .253 1.16279 .010 1.00027 .071 1.01338 .132 1.04584 .193 1.09654 .254 1.16402 .011 1.00032 .072 1.01376 .133 1.04652 .194 1.09752 .255 1.16526 .012 1.00038 .073 1.01414 .134 1.04722 .195 1.09850 .256 1.16650 .013 1.00045 .074 1.01453 .135 1.04792 .196 1.09949 .257 1.16774 .014 1.00053 .075 1.01493 .136 1.04862 .197 1.10048 .258 1.16899 .015 1.00061 .076 1.01533 .137 1.04932 .198 1.10147 .259 1.17024 .016 1.00069 .077 1.01573 .138 1.05003 .199 1.10247 .260 1.17150 .017 1.00078 .078 1.01614 .139 1.05075 .200 1.10347 .261 1.17276 .018 1.00087 .079 1.01656 .140 1.05147 .201 1.10447 .262 1.17403 .019 1.00097 .080 1.01698 .141 1.05220 .202 1.10548 .263 1.1753<» .020 1.00107 .081 1.01741 .142 1.05293 .203 1.10650 .264 1.1700339 7 0.0020302 1 0.1221730 67 1.1693706 8 0.0000388 8 0.0023271 8 0.1396263 68 1.1868239 9 0.0000436 9 0.00261.80 9 0.1570796 69 1.2042772 10 0.0000485 10 0.0029039 10 0.1745329 70 1.2217305 11 0.0000533 11 0.0031998 11 0.1919862 71 1.2391838 12 0.0000582 12 0.0034907 12 0.2094395 72 1.2566371 13 0.0000630 13 0.0037815 13 0.2268928 73 1.2740904 14 0.0O0O679 14 0.0040724 14 0.2443461 74 1.2915436 15 0.0000727 15 0.0043633 15 0.2617994 75 1.3089969 16 0.0000776 16 0.0046542 16 0.2792527 76 1.3264502 17 0.0000824 17 0.0049451 17 0.2967060 77 1.3439035 18 0.0000873 18 0.0052360 18 0.3141593 78 1.3613568 19 0.0000921 19 0.0055269 19 0.3310126 79 1.3788101 20 0.0000970 20 0.0058178 20 0.3490659 80 1.3962634 21 0.0001018 21 0.0061087 21 0.3665191 81 1.4137167 \ 22 0.0001067 22 0.0063995 22 0.3839724 82 1.4311700 0;( 0.0001115 ■ 0.0066901 23 0.4014257 83 1.4486233 ! 24 0.0001164 24 0.0069 SI 3 24 0.4188790 84 1.4660766 j 25 0.0001212 25 0.0072722 25 0.4303323 85 1.4835299 1 0.0001261 26 0.0075631 26 0.4537856 86 1.5009832 27 0.0001309 27 0.0078540 27 0.4712389 87 1.5184364 j 28 0.0001357 28 0.0081449 28 0.4886922 88 1.5358897 | 29 0.0001406 29 0.0084358 29 0.5061455 89 1.553:4430 I 30 0.0091454 30 0.0087266 30 0.5235988 90 1.5707963 i 31 •0.0001503 31 0.0090175 31 0.5410521 91 1.5882496 ! 32 0.0001551 32 0.00930S4 32 0.5585054 92 1.6057029 ! •>•> 0.0001600 33 0.0095993 33 0.5759587 93 1.6231562 : 34 0.0001618 34 0.0098902 34 0.5934119 94 1.6406095 3 5 0.0001697 35 0.0101811 35 0.6)108652 95 1.6580628 I 36 0.O001745 36 0.0104720 36 0.6283185 96 1.6755161 37 0.0001794 o7 0.0107629 37 0.6457718 97 1.6929694 38 0.O0O1S42 38 0.0110538 38 0.6632251 98 1.7104227 39 0.0001891 39 0.0113446 39 0.6806784 99 40 0.0001939 40 0.0116355 40 0.6981317 100 1.7453293 1 41 0.0001983 41 0.0119264 41 0.7155850 101 1.7627825 42 0.0002036 SI 0.0122173 42 0.7330383 102 1.7802358 43 O.Ot >02085 43 0.0125082 43 0.7501916 loi 1.7976891 44 0.00021: »3 44 0.0127991 44 0.7679449 104 1.8151424 45 0.0002182 45 0.0130900 45 0.7853982 105 1.8325957 46 0.0 >02230 46 0.0133809 46 0.8028515 106 1.8500490 47 0.000227'.> 47 0.0136717 47 0.8203047 107 1.8675023 48 0.0002327 48 0.0139626 48 0.8377580 108 1.8849556 49 0.0)02376 49 0.0142-535 49 0.8552113 109 1.9024089 50 0.0002424 50 0.0145444 50 0.8726646 110 1.9198622 • 51 0.00: >2473 51 0.0148353 51 0.8901179 111 52 0.0002521 52 0.0151202 52 0.9075712 112 1.9547688 53 0.00)2570 53 0.0154171 53 0.9250245 113 1.9722221 54 0.00026 IS 51 0.0157080 54 0.9424778 114 1.9896753 B 0.0402666 on 0.0159989 0) 0.9599311 115 2.0071286 56 0.0 >02715 56 0.0162897 56 0.9773844 116 2.0245819 57 0.000276)3 0) 0.0165806 57 0.9948377 117 2.0420352 58 0.0002812 58 0.016871.» ■ 1.0122910 118 2.0594885 59 0.0 HUH 59 0.0171024 59 1.0297443 119 2.0769418 60 0.0002909 69 0.0174)33 60 1.0471976 120 2.094395158 MENSURATION. —LENGTHS OF CHORDS. To compute the chord of an arc when the chord of half the arc and the versed sine are given. (The versed sine is the perpendicular bo, Fig. 31.) a/ I Rui.h.—From the square of the chord of / 3| half the arc subtract the square of the versed sine, and take twice the square root of the remainder. Exampi.k. — The chord of half the arc is GO, and the versed sine 30, what is the Length of the chord of the arc ? Ans. GO'2 — 3G2 = 2304, and 02304 = 48, and 48 X 2 = 90, the chord. To compute the chord of an arc iclien the diameter and versed sine are given. Multiply the versed sine hy 2, and subtract the product from the diameter; then subtract the square of the remainder from the square of the diameter, and take the square root of that remainder. Example. — The diameter of a circle is 100, and the versed sine of an are 8G, what is the chord of the arc ? Ans. 30 X 2 = 72. 100 — 72 = 28. 1002 — 282 - 921G. 0)216 = 00, the chord of the arc. To compute the chord of half an arc when the chord of the arc and the versed sine are given. ' Rule. — Take the square root of the sum of the squares of the versed sine and of half the chord of the arc. Example. —The chord of an arc is 90, and the versed sine 36, what is the chord of half the arc ? Ans. 0j62 + 482 = 00. To compute the chord of half an arc when the diameter and versed sine are given. Rule. —Multiply the diameter hy the versed sine, and take the square root of their product. To compute a diameter. Rule 1.—Divide the square of the chord of half the arc hy the versed sine. Rule 2.—Add the square of half the chord of the arc to the square of the versed sine, and divide this sum by the versed sine.MENSURATION.—ARCS AND VERSED SINES. 59 Exampi.e. —Wliat is the radius of an arc whose chord is 90, and whose versed sine is 30 ? Mi if!. 4S2 + 3G2 = 3000. 3000 -r 36'= 100, the diameter, and radius = 50. To compute the versed sine. Rule.—Divide the square of the chord of half the arc by the diameter. To compute the versed sine when the chord of the arc and the diameter are yiven. Rule.—From the square of the diameter subtract the square of the chord, and extract the square root of the remainder; subtract this root from the diameter, and halve the remainder. To compute the lenyth of an arc of a circle when the number of deyrees and the radius are yiven. Rule I. — Multiply the number of degrees in the arc by 3.1410 multiplied by the radius, and divide by 180. The result will be the length of the arc in the same unit as the radius. Rule 2. —Multiply the radius of the circle by 0.01745, and the product by the degrees in the arc. Example. — The number of degrees in an arc is 00, and the radius is 10 inches, what is the length of the arc in inches ? Am. 10 X 3.1410 X 00 = 1884.90 T-180 = 10.47 inches; or, 10 X 0.01745 X 00 = 10.47 inches. To compute the lenytli of the arc of a circle when the lenytli is yiven in deyrees, minutes, and seconds. Rule 1.—Multiply the number of degrees by 0.01745329, and the product by the radius. Rule 2. —Multiply the number of minutes by 0.00029, and that product by the radius. RuleS.—Multiply the number of seconds by 0.00000448 times the radius. Add together these three results for the length of the arc. See also table, p. 57. Example.—What is the length of an arc of G0° 10' 5", the radius being 4 feet ? Ans. ■ 1. 60° x 0.01745329 x 4 = 4.188789 feet. 2. 10' X 0.00029 X 4 = 0.0116 feet. 3. 5" X 0.0000048 X 4 = 0.000096 feet. 4.200485 feet.60 MENSURATION.--CIRCULAR SEGMENTS, ETC. To compute the area of a sector of c circle when the degrees of the p.g 30 arc and the radius are given (Fig. -12). ^— ------ Rui.e.—Multiply the number of degrees In li the arc by the area of the whole circle, and di- \ / vide by 300. \ / Example. —What is the area of a sector of \ / a circle whose radius is 5, and the length of the o arc is 00°? Ans. Area of circle = 10 X 10 X 0.7854 = 78.54. 78.5 X 00 Then area of sector = —ng—— = 13.09. 300 If the length of the arc is given in degrees and minutes, reduce it to minutes, and multiply by the area of the whole circle, and divide by 21000. To compute the urea of a sector of a circle when the length of the arc and radius are given. Rule. —Multiply the length of the arc by half the length of the radius, and the product is the area. 'To compute the area of a segment of a circle xvhen the chord and versed sine of the arc and the radius or diameter of the circle are given. Note. — The versed sine is the distance cd (Fig. 32). Rule 1 (ivhen the segment is less than a semicircle).—Ascer tain the area of tint sector having the same arc as the segment, then ascertain the area of a triangle formed by the chord of the segment and the radii of the sector, and take the difference of these areas. Rule 2 (ivhen the segment is greater than a semicircle).—Ascertain by the preceding rule the area of the lesser portion of the circle, subtract it from the area of the whole circle, and the remainder will give the area. To compute the convex surface of a sphere. Rule. — Multiply the diameter by the circumference, and the product will give the surface. Example. —What is the convex surface of a sphere of 10 inches diameter ? Ans. Circumference of sphere — 10 X 3.1410 = 31.410 inches; 10 X 31.dll — 314.10 sq. in., the surface of sphere.MENSURATION. —SPHERES AND SPHEROIDS. Cl To compute the surface of a segment of a sphere. Rule.—Multiply the height (he, Fig. 33) by the circumference of the sphere, and add the product to the area of the base. To find the area of the base, we have the diameter of the sphere and the length of the versed sine of the arc ahd, and we can find the length of the chord ad by the rule on p. 56. Having, then, the length of the chord ad for the diameter of the base, we can easily find the area. 1) Example. —The height, he, of a segment ahd, is 36 inches, and the diameter of the sphere is 106 inches. What is the convex surface, and what the whole surface? Ans. 100 X 3.1416 =-314.16 inches, the circumference of sphere. 36 X 314.16 = 11300.76, the convex surface. The length of ad — 100 — 36 X 3 = 28. \l 100'2 — 282 = 96, the chord ad. 962 X 0.7854 - 7238.2404, the area of base. 11309.76 + 7238.2464 = 18548.0064, the total area. To compute the surface of a spherical zone. Rule. — Multiply the height (<:d, Fig. 34) a by the circumference of the sphere for the convex surface, and add to it the area of the two ends for the whole area. fig. 34 Spheroids, or Ellipsoids. Definition.—Spheroids, or ellipsoids, are figures generated by the revolution of a semi-ellipse about one of its diameters. When tlie revolution is about the. short diameter, they are prolate ; and, when it is about the long diameter, they are oblate. To compute the surface of a spheroid when the spheroid is prolate. Rule.—Square the diameters, and multiply the square root of half their sum by 3.1416, and this product by the short diameter. Example. — A prolate spheroid has diameters of 10 and lq Inches, what is its surface ? Ans. 102 = 100, and 142 = 196._ /296 Their sum = 296, andip^- = 12.1655. 12.1655 X 3.1416 X 10 = 382.191 square inches.62 MENSURATION.--CONES AND PYRAMIDS. To compute the surface of a spheroid when the spheroid is oblate. Rule. — Square the diameters, and multiply the square root of half their sum by 3.1416, and this product by the long diameter. To compute the surface of a cylinder. Rule. — Multiply the length by the circumference for the con-vex surface, and add to the product the area o> the two ends for the whole surface. To compute the sectional area of a circular ring (Fig. 35). Rule. — Find the area of both circles, and subtract the area of the smaller from the area of the larger: the remainder will be the area of Fi9 • 35 the ring. To com}>ute the surface of a cone. Rule. —Multiply the perimeter or circumference of the base by one-half the slant height, or side of the cone, for the convex area. Add to this the area of the base, for the whole area. Example. —The diameter of the base of a cone is 3 inches, and the slant height 15 inches, what is the area of the cone ? Ans. 3 X 3.1-116 = 9.4248 = circumference of base. 9.4248 X — 70.686 square inches, the convex surface. 3 X 3 X 0.7854 = 7.068 square inches, the area of base. Area of cone — 77 754 square inches. pig 26 To compute the area of the surface of the frus- tum of a cone. Rule. — Multiply the sum of the perimeters of the two ends by the slant height of the frustum, and divide by 2, for the convex surface. Add the area of the lop and bottom surfaces. To compute the surface of a pyramid. Rule. — Multiply the perimeter of the base by one-half the slant height, and add to the product, the area of the base. To compute the surface of the frustum of a pyramid. Rule. —Multiply the sum of the perimeters of the two ends by the slant height of the frustum, halve the product, and add to the result the area of the two ends.MENSURATION. — PRISMS. 63 MENSURATION OF SOLIDS. To compute the volume of a prism. Rule. —Multiply the area of the hase by the height. This rule applies to any prism of any shape on the base, as long as the top and bottom surfaces are parallel. To compute the volume of a prismoid. Definition.—A prismoid is a a solid having parallel ends or bases dissimilar in shape with quadrilateral sides. Rule.—To the sum of the areas of the. two ends add four times the area of the middle section parallel to them, and multiply this sum by one-sixth of the perpendicular height. Example.—What is the volume of a quadrangular prismoid, as in Fig. 37, in which ah = 6", cd — 4", ac = he = 10", ce = 8", cf = 8", and ih — G" ? 0 4 4 Ans. Area of top = —— X 10 = 50. 8 + 6 Area of bottom = w— X 10 = 70, G + G Area of middle section = —x— X 10 = 60. 150 + 70 + (4 X 00)1 x V = 000 cubic inches. Note. — The length of the end of the middle section, as mil in Fig. 37 = eil + ef To find the volume of a prism truncated obliquely. Rule. —Multiply the area of the base by the average height of the edges. Example. — What is the volume of a truncated prism, as in Fig. 38, where ef = G inches,//; = 10 inches, ea = 10, ci — 12, dll = 8, and fh = 82 Ans. Area of base = G X 10 Average height of edges : Fig.38 GO square inches. 10+12 + 8 + 8 = 9b inches. 69 X 9| = 970 cubic inches.64 MENSURATION. — POLYHEDRONS. To compute the volume of a wedge when the ends are parallel anil equal. IIui.k. —Multiply the area of one end by the length of the wedge. To compute the volume of a wedge when the ends are not parallel. Rule. — Add together the lengths of the three edges, ab, cd, and ef; multiply their sum by the perpendicular height of the wedge, and then by the breadth of the back, and divide the product by 6. Polyhedrons. Definition. — A regular body is a solid contained within a certain number of similar and equal plane faces, all of which are equal regular polygons. The whole number of regular bodies which can possibly be found is five. They are: — 1. The tetrahedron, or pyramid. 2. The hexahedron, or cube, which has six square faces. 3. The octahedron, which has eight triangular faces. 4. The dodecahedron, which has twelve pentagonal faces. 5. The icosahedron, which has twenty triangular faces. To compute the volume of a regular polyhedron. Rule 1 (ivlien the radius of the circumscribing sphere is given).— Multiply the cube of the radius of the sphere by the multiplier opposite to the body in column 2 of the following table. Rule 2 (when the radius of the inscribed sphere is-given).— Multiply the cube of the radius of the inscribed sphere by the multiplier opposite to the body in column 3 of the following table. Rule 3 (ivhen the surface is given). — Cube the surface given, extract the square root, and multiply the root by the multiplier opposite to the body in column 4 of the following table. Figure. i. No. of Sides. Q Volume by radius of circum scribing sphere. 3. Volume by radius of inscribed circle. 4. Volume by surface. Tetrahedron . . 4 0.5132 13.85641 0.0517 IIexahedron . . 6 1.5306 8.0000 0.06804 Octahedron . . 8 1.33833 6.9282 0.07311 Dodecahedron . 12 2.78517 5.55029 0.0S169 Icosahedron . . 20 2.53615 5.05406 0.0856MENSUltATlON.® CONES, PYRAMIDS, ETC, Go To compute the volume of a cylinder. Rule. — Multiply the area of the base by the height. To compute the volume of a cone. To compute the volume of the frustum of a cone. Rule.—Add together the squares of the diameters of the two ends and the product of the two diameters; multiply this sum by 0.7S54, and this product by the height, and then divide this last product by 3. Example. —What is the volume of a frustum of a cone 9 inches high, 5 inches diameter at the base, and 3 inches at the top ? Rule.—Multiply the area of the base by the perpendicular height, and take one-tliird of the product. Fig. 40. Am. 5- + 32 = 34. 3 X 5 = 15. 15 + 34 = 49 the sum of the squares and product of the diameters. 49 X 0.7S54 = 38.4846. 88.484« X 9 ----o-----= 115.4538 cubic inches. O To comjntlc the volume of a pyramid. Rule.—Multiply the area of the base by the perpendicular height, and take one-tliird of the product. To compute the volume of the frustum of a pyramid. Rule. — Find the height that the pyramid would be if the top were put on, and then compute the volume of the completed pyramid and the volume of the part added; subtract the latter from the former, and the remainder will lie the volume of the frustum. To compute the volume of a sphere. Rule. — Multiply the cube of the diameter by 0.5236. To compute the volume of a segment of a sphere. Rule 1. —To three times the square of the radius of its base add the square of its height; multiply this sum by the height, and the product by 0.5236. Rule 2. —From three times the diameter of the sphere subtract twice the height of the segment; multiply this remainder by the square of the height, and the product by 0.5236. Example. —The segment of a sphere has a radius, ac (Fig. 41), of 7 inches for its base, and a height, cb, of 4 inches: what is its volume ? * ylits. (by Rule 1). 3 X 72 — 147, and 147 + 42 = 163, three times6G MENSURATION. — SPHEROIDS, PARABOLOIDS, ETC. tlie square of the radius of the base plus the square of the height. b 103 X 4 X 0.5230 = 341.3872 cubic inches vol- ume. Second Solution. — I’y the rule for finding the diameter of a circle when a chord and its versed sine are given, we find that the diameter of the sphere in this case is 10.25 inches; then, by Rule2, (3 X 10.25) — (2 X 4) = 40.75, and 40.75 X 4a X 0.5236 = 341.3872 cubic inches, the volume of the segment. To compiile the volume of a spherical zone. Definition. — The part of a sphere included between two parallel planes (Fig. 42). Rule. — To the sum of the squares of the radii of the two ends add one-third of the square of the height of the zone; multiply this sum by the height, and that product by 1.5708. To compute the volume of a spheroid. Fig.43 Rule. — Multiply the square of the revolving axis by the fixed axis, and this product by 0.5230. To compute the volume of a paraboloid of revolution (Fig. 43). Rule. —Multiply the area of the base by half the altitude. • To compute the volume of a hyperboloid of revolution (Fig. 44). Rule. — To the square of the radius of the base add the square of the middle diameter; multiply this sum by the height, and the product by 0.5230. To compute the volume of any fiyure of revolution. Rule. —Multiply the area of the generating surface by the circumference described by its centre of gravity. To compute the volume of an excavation, ivhere the ground is irregular, and the bottom of the excavation is level (Fig. 45). Rule.—Divide the surface of the ground to be excavated into equal squares of about 10 feet on a side, and ascertain by meansMENSURATION. — EXCAVATIONS. 67 of a level tha height of each corner, a, a, a, b, b, b, etc., above the level to which the ground is to be excavated. Then add together the heights of all the corners that only come into one square. Next take twice the sum of the heights of all the corners that come in two squares, as b, b, b ; next three times the sum of the heights of all the corners that come in three squares, as c, c, c ; and then four times the sum of the heights of all the corners that belong to four squares, as <7, d, d, etc. Add together all these quantities, and multiply their sum by one-fourth the area of one of the squares. The result will be the volume of the excavation. Example. —Let the plan of the excavation for a cellar be as in the figure, and the heights of each corner above the proposed bottom of the cellar be as given by the numbers in the figure, then the volume of the cellar would be as follows, the area of each square being 10 X 10 = 100 square feet: — Volume = \ of 100 (a’s + 2 b’s + 3 c’s + 4 d’s). The a's in this case = 4 + 0 + 3 + 2 + 1 + 7 + 4 = 27 2 X the sum of the b's = 2 X (3 + 0 + 1 + 4 + 3 + 4) = 42 3 X the sum of the c’s = 3 X (1 + 3 + 4) =24 4 X the sum of the iZ’s = 4 X (2 + 3 + 0 + 2) =52 145 Volume = 25 X 145 = 3625 cubic feet, the quantity of earth to be excavated.68 GEOMETRICAL PROBLEMS. GEOMETRICAL PROBLEMS. Tig.46 Problem 1. — To fixed, (Fig. 46). From a and b, with any radius greater than half of ah, deseribe ares intersecting in c and d. The line cd, connecting these intersections, will bisect ah, and be perpen-1 dicular to it. Problem 2. — To draw a perpendicular to a given straight line from a point without it. 1st Method (Fig. 47). — From the point a describe an arc with „ snllicient radius that it will cut the line be in two places, as e and /. From e and / describe two arcs, with the same ratlins, intersecting in g; then a line drawn from a to g will be perpendicular to the line be. 2d Method (Fig. 48). — From any two points, d and c, at some distance apart in the given line, and wilh radii da and ca respectively, describe arcs cutting at a and e. Draw ae, and it will be the perpendicular required. This method is useful Where the given point is opposite the end of the line, or nearly so. Problem 3. — To draw a perpendicular to a straight line from a given point, a, in that line. / ) m / / / / c (l Fig.48 A yt a Fig.40 1st Method (Fig. 49). — With any radius, from the given point a in the line, describe arcs cutting the line in the points b and c. Then with b and c as centres, and with any radius greater than ab or or, describe arcs cutting each other at <1. The line du will be the perpendicular desired.GEOMETRICAL PROBLEMS. 69 2n Method (Fig. 50, when the given point is at the end of the line). — From any point, b, outside of the line, and with a radius ba, describe a semicircle passing through a, and cutting the given line at <1. Through b and d draw a straight line intersecting the semicircle at e. The line ea will then be perpendicular to the line ac at the point a. \ Fig.SO ■ / V y d 3n Method (Fig. 51) or the 3, 4, and 5 Method. — From the point a on the given line measure off 4 inches, or 4 feet, or 4 of any other unit, and with the same unit of measure describe an arc, with a as a centre and 3 units as a radius. Then from b describe an arc, with a radius of 5 units, cutting the first arc in e. Then ca will be the perpendicular. This method is particularly useful in laying out a right angle on the ground, or framing a house where the foot is used as the unit, and the lines laid off by straight edges. In laying out a right angle on the ground, the proportions of the triangle may be 30, 40, and 50, or any other multiple of 3, 4, and 5; and it can best be laid out with the tape. Thus, first measure off, say 40 feet from a on the given line, then let one person hold the end of the tape at b, another hold the tape at the 80-foot mark at a, and a third person take hold of the tape at the 50-foot mark, with his thumb and finger, and pull the tape taut. The 50-foot mark will then be at the point c in the line of the perpendicular. Problem 4. — To draw a straight line parallel to a given line at a given distance apart (Fig. 52). I I I I I I a Fig.52 b From any two points near the ends of the given line describe two arcs about opposite the line. Draw the line cd tangent to these ares, and it will be parallel to ab.70 GEOMETRICAL PROBLEMS. Problem 5.— To construct an angle equal to a given angle. With the point A, at the apex of the given angle, as a centre, and any radius, describe the arc BC. Then with the point a, at the vertex of the new angle, as a centre, and with the same radius as before, describe an arc like BC. Then with BC as a radius, and b as a centre, describe an arc cutting the other at c. Then will cab be equal to the given angle CAB. Problem C. — From a point on a given line to draw a line making an angle of 60° with the given line (Fig. 54). Take any distance, as ah, as a radius, and, with a as a centre, describe the arc be. Then with b as a centre, and the same radius, describe an arc cutting the first one at c. Draw from a a line through c, and it will make with ab an angle of 00°. Problem 7. — From a given point, A, on a given line, AE, to draw a line making an angle of 45° with the given line (Fig. 55). Measure off from A, on AE, any distance, Ab, and at b draw a line perpendicular to AE. Measure off on this perpendicular be equal to Ab, and draw a line from A through c, and it will make an angle with AE of 45°. Problem 8. — From any point. A, on a given line, to draic a line ivhich shall make any desired angle with the given line (Fig. 56). To perform this problem we must have a table of chords at hand (such as is found on pp. 85-03), which we use as follows. Find in the table the length of chord to a radius 1, for the given angle. Then take any ra-D ^ Fig"56 b ^ dins, as large as convenient, describe an arc of a circle be with A as a centre. Multiply the chord of the angle, found in the table, by the length of the radius Ab, and with the product as a new radius, and b as a centre, describe a short arc cutting be in d. Draw a line from A through b, and it will make the desired angle with I)E.GEOMETRICAL PROBLEMS 71 Example.—Draw a line from A on BE, making an angle of 44° 40' with BE. Solution. — We find that the largest convenient radius for our arc is 8 inches: so with A as a centre, and 8 inches as a radius, we describe the arc be. Then, looking in the table of chords, we find the chord for an angle or arc of 44° 40' to a radius 1 is 0.76. Multiplying this by 8 inches, we have, for the length of our new radius, 6.08 inches, and with this as a radius, and b as a centre, we describe an arc cutting be in d. Ad will then be the line desired. Problem 0. — To bisect a given angle, as BA C (Fig. 57). With A as a centre, and any radius, \ ^ describe an arc, as cb. With c and b as | f centres, and any radius greater than ^ one-half of cb, describe two arcs inter-secting in d. Draw from A a line Fig.57 through d, and it will bisect the angle BA C. Problem 10. — To bisect the angle contained between two lines, as AB and CB, when the vertex, of the angle is not on the drawing (Fig. 58) Draw/e parallel to AB, and cd parallel to CD, so that the two lines will intersect each other, as at i. Bisect the angle eid, as in the preceding problem, and draw a line through i and o which will bisect the angle between the two given lines. Problem 11. — Through tvjo given points, B and C, to describe an arc of a circle with a given radius (Fig. 59). With B and C as centres, and a radius equal to the given radius, describe two arcs intersecting at A. With A as a centre, and the same radius, describe the arc be, which Fig.59 will be found to pass through the given points, B and C.72 GEOMETRICAL PROBLEMS. Froiu,em 12. — To'find the centre of a given circle (Fig. 60). Draw any chord in the circle, as ah, and bisect this chord by the perpendicular cd. This line will pass through the centre of the circle, and ef will be a diameter of the circle. Bisect ef, and the centre o will be the centre of the circle. Fig.60 iflkotiup 13. — To draw a circular arc through three given points, as A, II, and C (Fig. 61). Draw a line from A to 71 and from 71 to C. Bisect AB and BC by the lines a a and cc, and prolong these lines until they intersect at o, which will be the centre for the arc sought. With o as a centre, and Ao as a radius, describe the arc ABC. Pnoni.EM 14. — To describe a circular arc jiassing through three given points, tchen the centre is not available, by means of a triangle (Fig. 02). Let A, B, and C be the given points. Insert two stiff pins or nails at A and C. riace two strips of wood, as shown in the figure; one against A, the other against C, and inclined so that their intersection shall come at the third point, 71. Fasten the strips together at their intersection, and nail a third strip, T, to their other ends, so as to make a firm triangle. Place the pencil-point at 71. and, keeping the edges of the triangle against A and 71. move the triangle to the left and right, and the pencil will describe the arc sought.GEOMETRICAL PROBLEMS. 73 When the points A and C are at the same distance from B, if a strip of wood be nailed to the triangle, so that its edge de shall be at right angles to a line joining A and C as the triangle is moved one way or the other, the edge de will always point to the centre of the circle. This principle is used in the perspective linear d. Problem 15. — To find a circular arc which shall he tangent to a given point, A, on a straight line, and pass through a given point, C, outside the ■ line (Fig. 63). Draw from A a line perpendicular to the given line. Connect A and C by a straight line, and bisect it by the perpendicular ac. The point where these two perpendiculars intersect will be the centre of the circle. Problem 16. — To connect two parallel lines by a reversed curve composed of two circular arcs of equal radius, and tangent to the lines at given points, as A and B (Fig. 64). Join A and B, and di- A vide the line into two equal parts at C. Bisect CA and CB by perpen- h q diculars. At A and B erect perpendiculars to the given lines, and the _________ intersections a and b will be the centres of the Fig. 64 arcs composing the required curve. Problem 17. — On a given line, as AB, to construct a compound curve of three arcs of circles, the radii of the two side ones being equal and of a given length, and their centres in the given line; the central arc to pass through a given point, C, on the perpendicular bisecting the given line, and tangent to the other two arcs (Fig. 65). Draw the perpendicular CD. Lay off Aa, Bb, and Cc, each equal to the given radius of the side arcs; join74 GEOMETRICAL PROBLEMS. ac; bisect ac by a perpendicular. The intersection of this line with the perpendicular Cl) will be the required centre of the central arc. Through a and b draw the lines De and De'; from a and b, with the given radins, equal to A a, Jib, describe the arcs vie'and Be; from D as a centre, and CD as a radius, describe the arc eCe’ which completes the curve required. Problem 18. — To construct a triangle upon a given straight line or base, Ike length of the two sides being given (Fig. 06). First (an equilateral triangle, Fig. 00a).—With the extremities A and B of the given line as centres, and AB as a radius, describe arcs cutting each other at C. Join AC and BC. Second (when the sides are unequal, Fig. 06b). — Let AD be the given base, and the other two sides be equal to C and B. With 7) as a centre, and a radius equal to C, describe an indefinite arc. With A as a centre, and B as a radius, describe an arc cutting the. first at E. Join E with A and D, and it will give the required triangle. Problem 19. — To describe a circle about a triangle (Fig. 07). Bisect two of the sides, as AC and CB, of the triangle, and at their centres erect perpendicular lines, as ae and be, intersecting at e. With e as a centre, and eC as a radius, describe a circle, and it will be found to pass through A and B. C Problem 20. — To inscribe a circle in a triangle (Fig. 08). Bisect two of the angles, A and B, of the triangle by lines cutting each other at o. With o as a centre, and oe as a radius, describe a circle, which will be found to just, touch the other two sides.GEOMETRICAL PROBLEMS. iO Problem 21. — To inscribe a square in a circle, and to describe a circle about a square (Fig. GO). To inscribe the square. Draw two diameters, AB and CD, at right angles to each other. Join the points A, D, B, C, and we have the inscribed square. To describe the circle. Draw the diagonals as before, intersecting at E, and, with E as a centre and AE as a radius, describe the circle. Problem 22. ■— To inscribe a circle in a square, and to describe a square about a circle (Fig. 70). To inscribe the circle. Draw the diagonals AB and CD, intersecting at E. Draw the perpendicular EG to one of the sides. Then with E as a centre, and EG as a radius, describe a circle, which will be found to touch all four sides of the square. To describe the square. Draw two diameters, AB and CD, at ‘ right angles to each other, and prolonged beyond the circumference. Draw the diameter GF, bisecting the angle CEA or BED. Draw lines through G and F perpendicular to GF, and terminating in the diagonals. Draw AD and CB to complete the square. Problem 23. — To inscribe a pentagon in a circle (Fig. 71). Draw two diameters, AB and CD, at right angles to each other. Bisect /10 at E. With E as a centre, and EC as a radius, cut OB at F. With C as a centre, and CF as a radius, cut the circle at G and II. With these points as centres, and the same radius, cut the circle at I and J. Join I, J, II, G, and C, and we then have inscribed in the circle a regular pentagon7G GEOMETRICAL PROBLEMS. Problem 24.—To inscribe a rccjular hexagon in a circle (Fig. 72). Solution.—Lay olf on the circumference the radius of the circle six times, and connect the points. Problem 25. — To construct a regular hexagon upon a given straight line, AB (Fig. 73). From A and I>, with a radius equal to AB, describe arcs cutting at O. With 0 as a centre, and a radius equal to AB, describe a circle, and from A and B lay off the length AB on the circumference of the circle, and join the points thus obtained. The result will be a regular hexagon. Problem 2G. — To construct a regular octagon upon a given straight line, AB (Fig. 74). Produce the line AB both ways, and draw the perpendiculars Aa and Bh, of indefinite length. Bisect the external angles at A and B, and make the length of the lines equal to AB. From II and C draw lines parallel to Aa, and equal in length to AB; and from the centre! G and 1) describe arcs, with a radius AB, cutting the perpendiculars Aa and Bh in I'’and E. Join GF, FE, and El). Problem 27. — To make a regular octagon from a square (Fi".73)l Draw the diagonals AB and BC, and from the corners A, B, C, and I), with a radius equal to AO, describe arcs cutting theGEOMETRICAL PROBLEMS. 77 sides of the square in a, b, c, d, e,/, h, and i. Join these points to complete the octagon. Problem 28. — To inscribe a regular octagon in a circle (Fig. 70). Draw two diameters, ATI and CD, at right angles to each other. Bisect the angles AOD and AOC by the diameters EF and GIf. Join A, E, D, IT, B, etc., for the inscribed figure. A D E Fig. 76 ir'ig.77 Problem 29. — To inscribe a circle within a regular polygon. First (when the polygon has an even number of sides, as in Fig. 77).—Bisect two opposite sides at A and B, and drawyl7J, and bisect it at C by a diagonal, DE, drawn between two opposite angles. With the radius CA describe the circle as required. Second (when the number of sides is odd, as in Fig. 78). —Bisect two of the sides at A and B, and draw £ lines, A E and BD, to the opposite angles, intersecting at C. With C as a centre, and CA as a radius, describe the circle as required. Problem 30. — To describe a circle without a regular polygon. When the number of the sides is even, draw two diagonals from opposite angles, as ED and Gil (Fig. 77), intersecting at C; and from C', with CD as a radius, Fig.7a describe the circle required. When the number of sides is odd, find the centre, C, as in last problem; and with C as a centre, and CD (Fig. 78) as a radius, describe the circle required.GEOMETRICAL PROBLEMS. PROBLEMS ON THE ELLIPSE, THE PARABOLA, THE HYPERBOLA, AND THE CYCLOID. The Ellipse. Phobeem 31. —-To describe an ellipse, the length and breadth, or the two axes, being given. H 1st Method (Fig. TO, the two axes, A Ji and CD, being given),— On AH and CD as diameters, and from the same centre, 0, describe the circles AGBII and CLDK. Take any convenient number of points on the circumference of the outer circle, as b, b', b", etc., and from them draw lines to the centre, 0,-eutting the inner circle at the points a, a', u", etc., respectively. From the points b, b', etc., draw lines parallel to the shorter axis; and from the points a, a', etc., draw Fig.79 D Fig.80 lines parallel to the longer axis, and intersecting the first set of lines at c, c', c", etc. These last points will be points in the ellipse, and, by obtaining a sutficient number of them, the ellipse can easily be drawn. 2d Method (Fig..80). — Take the straight edge of a stiff piece of paper, cardboard, or wood, and from some point, as a, mark off ah equal to half the shorter diain-GEOMETRICAL PROBLEMS. 79 eter, and nc equal to half the longer diameter. Place the straight edge so that the point b shall be on the longer diameter, and the point c on the shorter: then will the point a be over a point in the ellipse. Make on the paper a dot at a, and move the slip around, always keeping the points b and c over the major and minor axes. In this way any number of points in the ellipse may be obtained, which may be connected by a curve drawn freehand. 3n Method (Fig. 81, given the two axes AB and CD.)—From the point D as a centre, and a radius MO, equal to one-lialf of AB, describe an arc cutting AB at F and F'. These two points are called the foci of the ellipse. [One property of the ellipse is, that the sum of the distances of any two points on the circumference from the foci is the same. Thus F'D + DF= F'E + EF or F'G + GF.] Fix a Fi9-81 couple of pins into the axis AB at F and F', and loop a thread or cord upon them equal in length, when fastened to the pins, to AB, so as, when stretched as per dotted line FDF', just to reach the extremity D of the short axis. Place a pencil-point inside the chord, as at E, and move the pencil along, always keepin" the cord stretched tight. In this way the pencil will trace the outline of the ellipse. Pkodlem 32. — To draw a tangent to an ellipse at a given point on the curve (Fig. Let it be required to draw a tangent at the point E on the ellipse shown in Fig. 82. First find the foci F and F', as in the third method for describing an ellipse, then from / \F 9 \ 1 A i i J80 GEOMETRICAL PROBLEMS. E draw lines EF and EF'. Prolong EE1 to a, so that Ea shall equal EF. Bisect the angle aEF as at b, and through b draw a line touching the curve at E. This line will be the tangent required. If it were desired to draw a line normal to the curve at E, as, for instance, the joint of an elliptical arch, bisect the angle FEF', and draw the bisecting line through E, and it will be the normal to the curve, and the proper line for the joint of an elliptical arch at that point. Problem 33. — To draw a tangent to an ellipse from a given ■ point without the curve (Fig. 83). From the point T as a centre, and a radius equal to the distance to the nearer focus F, describe a circle. From F' as a centre, and a radius equal to the length of the longer axis, describe arcs cutting the circle just described at a and b. Draw lines from F' to a and b, cutting the circumference of the ellipse at E and G. Draw lines from T through E and G, and they will be the tangents required. Problem 34. — To describe an ellipse approximately, by means of circular arcs. First (with arcs of two radii, Fig. 84). —Take half the difference of the two axes All and CD, and set it off from the centre 0 to a and c on OA and OC; draw ac, and setoff half ac to d; draw. (It parallel to ac; set off Oe equal to Od; join ei, and draw em and dm parallels to di and ie. On m as a centre, with a radius mC, describe an arc through C, terminating in 1 and ’2; and with i as a centre, and id as a radius, describe an arc through 1), terminating in points 3 and 4. On d and e as centres describe arcs through A and II. Connecting the points 1 and 4, 2 and 3. The four arcs thus dc-GEOMETRICAL PROBLEMS. 81 scribed form approximately an ellipse. This method does not apply satisfactorily when the conjugate axis is less.than two-tliirds of the transverse axis. Fig.84 D Second (with arcs of three radii, Fig. So). — On the transverse axis AB draw the rectangle AGEB, equal in height to DC, half the conjugate axis. Draw GD perpendicular to AC\ Set off OK equal to OC, and oh AK as a diameter describe the semicircle82 GEOMETRICAL PROBLEM!!. ANK. Draw a radius parallel to OC, intersecting the semicircle at N, and the line G'E at P. Extend OC to L and to D. Set off OM equal to PN, and on J) as a centre, with a radius 1)M, describe an arc. From A and I! as centres, with a radius OL, intersect this arc at a and b. The points II, a, 1), b, II', are the centres of the arcs required. Produce the lines all, Da, Db, bll', and the spaces enclosed determine the lengths of each arc. This process works well for nearly all ellipses. It is employed in striking out vaults, stone arches, and bridges. Note. — In this example the point IF happens to coincide with the point K, hut this need not necessarily he the case. The Parabola. Pkoui.em 35.— To construct a parabola when the vertex A, the axis AD, and a point, M, of the curve, are given (Fig. SC). Construct the rectangle ADMC. Divide MC into any number of equal parts, four for instance. Divide AC in like manner. Connect A\, A2, and A3. Through 1', 2', 3', draw parallels to the axis. The intersections I, II, and III, of these lines, are points in the required curve. Pnom,em 36. — To draw a tangent to a given point, II, of the parabola (Fig. 8G). From the given point II let fall a perpendicular on the axis at b. Extend the axis to the left of A. Make Aa equal to Ab. Draw «11, and it is the tangent required. The lines perpendicular to the tangent are called normals. To find the normal to any point I, having the tangent to any other point, II. Draw the normal lie. From 1 let fall a perpendicular Id, on the axis AB. Lay off de equal to be. Connect Ie, and we have the normal required. The tangent may be drawn at I by. laying off a perpendicular to the normal le at I.GEOMETRICAL PROBLEMS. 83 The Hyi>erhola. The hyperbola possesses the characteristic that if, from any point, P, two straight lines he drawn to two fixed points, F and F', the foci, their difference shall always be the same. Pnom.km 37. — To describe an hyperbola through a given vertex, with the given difference ah, and one of the foci, F (Fig. S7). Draw the axis of the hyperbola ATS, with the given distance ah and the focus F marked on it. From b lay off bF, equal to aF for the other focus. Take any point, as 1 on AH, and with al as a radius, and F as a centre, describe two short arcs above and below the axis. With IA as a radius, and F' as a centre, describe arcs'cutting those just described at P and P'. Take several points, as 2, 3, and 4, and obtain the corresponding points P2, P3, and P4 in the same way. Join these points with a curved line, and it will be an hyperbola. To draw a tangent to any point of an hyperbola, draw lines from the given point to each of the foci, and bisect the angle thus formed. The bisecting line will be the tangent required.The Cycloid CO 7 The cycloid is the curve described by a point in the circumference of a circle rolling in a straight line. Pkoblk.u 3S. — To describe a cycloid (Fig. 88). Draw the straight line All az the base. Describe the generating circle tangent to this line at the centre, and through the centre of the circle, C, draw the line EE parallel tc the base. Let fall a perpendicular from C upon the base. Divide the semi-circumference into any number of equal parts, for instance, six. Lay off on All and CE distances C'l', 1'2', etc., equal to the divisions of the circumference. Draw the chords Dl, 1)2, etc. From the points 1', 2', 3', on the line CE, with radii equal to the generating circle, describe arcs. From the points 1', 2', 3', 4', 5', on the line BA, and with radii equal respectively to the chords Dl, D2, D3, D4, Do, describe arcs cutting the preceding, and the intersections will be points of the curve required. GEOMETRICAL PROBLEMS 8i TABLE OF CHORDS; Radius = 1.0000. M. 0° 1° 2° 3° 4° 5° 6° 7° 8° 9° 10° M. 0' .0000 .0175 .0349 .0524 .0698 .0S72 .1047 .1221 .1395 .1569 .1743 0' 1 .0003 .0177 .0352 .0526 .0701 .0875 .1050 .1224 .1398 .1572 .1746 1 2 .0006 .0180 .0355 .0529 .0704 .0878 .1053 .1227 .1401 .1575 .1749 2 1 3 .0009 .0183 .0358 .0532 .0707 .0881 .1055 .1230 .1404 .1578 .1752 3 4 .0012 .0186 .0361 .0535 .0710 .OS84 .1058 .1233 .1407 .1581 .1755 4 5 .0015 .0189 .0364 .0538 .0713 .0887 .1061 .1235 .1410 .1584 .1758 5 1 6 .0017 .0192 .0366 .0541 .0715 .0S90 .1064 .1238 .1413 .1587 .1761 6 ! 7 .0020 .0195 .0369 .0544 .0718 .0393 .1067 .1241 .1415 .1589 .1763 7 8 .0023 .0198 .0372 .0547 .0721 .0896 .1070 .1244 .1418 .1592 .1766 8 9 .0026 .0201 .0375 .0550 .0724 .0399 .1073 .1247 .1421 .1595 .1769 9 10 .0029 .0204 .0378 .0553 .0727 .0901 .1076 .1250 .1424 .1598 .1772 10 11 .0032 .0207 .0381 .0556 .0730 .0904 .1079 .1253 .1427 .1601 .1775 11 12 .0035 .0209 .03S4 .0558 .0733 .0907 .1082 .1256 .1430 .1604 .1778 12 13 .0038 .0212 .0387 .0561 .0736 .0910 .1084 .1259 .1433 .1607 .1781 13 14 .0041 .0215 .0390 .0564 .0739 .0913 .1087 .1262 .1436 .1610 .1784 14 15 .0044 .0218 .0393 .0567 .0742 .0916 .1090 .1265 .1439 .1613 .1787 15 16 .0047 .0221 .0396 .0570 .0745 .0919 .1093 .1267 .1442 .1616 .1789 16 17 .0049 .0224 .0398 .0573 .0747 .0922 .1096 .1270 .1444 .1618 .1792 17 18 .0052 .0227 .0401 .0576 .0750 .0925 .1099 .1273 .1447 .1621 .1795 18 19 .0055 .0230 .0404 .0579 .0753 .0928 .1102 .1276 .1450 .1624 .1798 19 20 .0058 .0233 .0407 .0582 .0756 .0931 .1105 .1279 .145a .1627 .1801 20 21 .0061 .0236 .0410 .0585 .0759 .0933 .1108 .1282 .1456 .1630 .1804 21 22 .0064 .0239 .0413 .058S .0762 .0936 .1111 .1285 .1459 .1633 .1807 22 23 .0067 .0241 .0416 .0590 .0765 .0939 .1114 .1288 .1462 .1636 .1810 23 24 .0070 .0244 .0419 .0593 .0768 .0942 .1116 .1291 .1465 .1639 .1813 24 25 .0073 .0247 .0422 .0596 .0771 .0945 .1119 .1294 .1468 .1642 .1816 25 26 .0076 .0250 .0425 .0599 .0774 .0948 .1122 .1296 .1471 .1645 .1818 26 27 .0079 .0253 mm .0602 .0776 .0951 .1125 .1299 .1473 .1647 .1821 27 28 .0081 .0256 .0430 .0605 .0779 .0354 .1128 .1302 .1476 .1650 .1824 28 29 .0084 .0259 .0433 .0608 .07S2 .0957 .1131 .1305 .1479 .1653 .1827 29 30 . .0087 .0262 .0436 .0611 .0785 .0960 .1134 .1308 .1482 .1656 .1830 30 31 .0090 .0265 .0439 .0614 .0788 .0962 .1137 .1311 .1485 .1659 .1833 31 32 .0093 .0268 .0442 .0617 .0791 .096 j .1140 .1314 .1488 .1662 .1836 32 33 .0096 .0271 .0445 .0619 .0794 .0368 .1143 .1317 .1491 .1665 .1839 33 34 .0099 .0273 .044S .0622 .0797 .0971 .1145 .1320 .1494 .1668 .1842 34 35 .0102 .0276 .0451 .0625 .0800 .0974 .1148 .1323 .1497 .1671 .1845 35 36 .0105 .0279 .0454 .0628 .0803 .0977 .1351 .1325 .1500 .1674 .1847 30 * 37 .0108 .0282 .0457 .0631 .0806 .0980 .1154 .1328 .1502 .1676 .1850 37 38 .0111 .0285 .0160 .0634 .0808 .0983 .1157 .1331 .1505 .1679 .1853 38 1 m .0113 .0288 .0462 .0637 .0S11 .0986 .1160 .1334 .1508 .1682 .1856 39 I 40 .0116 .0291 .0465 .0640 .0814 .0989 .1163 .1337 .1511 .1685 .1859 40 41 .0119 .0294 .0468 .0643 .0817 .0992 .1166 .1340 .1514 .1688 .1862 41 42 .0122 .0297 .0471 .0646 .0820 .0994 .1169 .1343 .1517 .1691 .1865 42 43 .0125 .0300 .0474 .0649 .0823 .0997 .1172 .1346 .1520 .1694 .1868 43 i44 .0128 .0303 .0477 .0651 .0826 .1000 .1175 .1349 .1523 .1097 .1871 44 45 .0131 .0305 .0480 .0654 .0829 .1003 .1177 .1352 .1526 .1700 .1873 45 46 .0134 .0308 .0483 .0657 .0832 .1006 .1180 .1355 .1529 .1703 .1876 46 j 47 .0137 .0311 .0486 .0660 .0835 .1009 .1183 .13«) i .1531 .1705 .1879 47 1 m .0140 .0314 .04 S9 .0663 .0S38 .1012 .1186 .1860 .1534 .1708 .1882 48 49 .0143 .0317 .0492 .0666 .0840 .1015 .1189 .1363 .1537 .1711 .1885 49 50 .0145 .0320 .0194 .0669 .0843 .1018 .1192 .1366 .1540 .1714 .1888 50 51 .0148 .0323 .0197 .0672 .0846 .1021 .1195 .1369 .1543 .1717 .1891 51 52 .0151 .0326 .0500 .0675 .0849 .1023 . U 98 .1372 .1546 .1720 .1894 52 •Jo .0154 .0329 .0503 .0678 .0852 .1026 .1201 .1375 .1549 .1723 .1897 53 54 .0157 .0332 .0506 .0681 .0855 .1029 .1204 .1378 .1552 .1726 .1900 54 55 .0160 .0335 .0509 .0683 .0858 .1082 .1206 .1381 .1555 .1729 .1902 55 56 .0163 .0337 .0512 .0686 .0861 .1085 .1209 .1384 .1558 .1732 .1905 56 57 .0166 .0340 .0515 .0689 .0864 .1038 .1212 .1386 .1560 .1734 .1908 57 58 .0169 .0343 .0518 .0692 .0867 .1041 .1215 .1389 .1563 .1737 .1911 58 59 .0172 .0346 .0521 .0695 .0869 .1044 Mm .1392 .1566 .1740 .1914 59 60 1 .0175 .0349 .0521 .0698 .0872 .1047 .1221 .1395 .1569 .1743 .1917 6086 GEOMETRICAL TROBLEMS Table of Chords; Radius = 1.0000 (continued). M. 11° 12° 13° 14° 15° 16° 17° 18° 19° 20° 21° XI. O' .1917 .2091 .2264 .2437 .2611 .2783 .2956 .3129 .3301 .3473 .3615 O' 1 .1920 .2093 .2267 .2440 .2613 .2780 .2959 .3132 .3304 25476 .3648 1 *2 .1923 .2096 .2270 .2443 .2616 .27S9 .2962 .31:44 .3307 25479 .3650 2 g -ipp .2099 .2273 .2446 .2619 .2792 .2965 .3310 .&4S2 .3653 3 4 .1928 .2102 .2276 .2449 .2622 .2795 .2968 .3140 .3312 .3484 .3656 4 5 .1931 .2105 .2279 .2452 .2625 .2798 .2971 .3143 .3315 .3487 .3659 5 G .1934 .2108 .2281 .2455 .2628 .2801 .2973 .3146 .3318 .3490 .3662 6 7 .1937 .2111 .2284 .2458 .2631 .2804 .2976 .3149 .3321 .3493 .3665 7 8 .1940 .2114 .2287 .2460 .2634 .2807 .2979 .3152 .3324 .3496 .306$ 8 9 .1943 .2117 .2290 .2463 .2* >36 .2809 .2982 .3155 .3327 .3499 .3670 9 10 .1946 .2119 .2293 .2466 .2639 •2S12 .2985 .3157 .3330 .3502 .3673 10 11 .1949 .2122 .2296 .2469 .2642 .2815 .2988 .3160 .3333 .3504 .3676 11 12 .1952 .2125 .2299 .2472 .2645 .2S18 .2991 .3163 .3335 .3507 .3679 12 13 .1955 .2128 .2302 .2475 .2648 .2821 .2994 .3166 .3338 .3510 .3682 13 14 .1957 .2131 .2305 .2478 .2651 .2824 .2996 .2169 .3341 .3513 256S5 14 15 .1900 .2134 .2307 .2481 .2654 .2827 .2999 .3172 .3344 .3516 .3688 15 16 .1963 .2137 .2310 .24S4 .2657 .2S30 .3002 .3175 .3347 .3519 .3690 16 17 .1966 .2140 .2313 .2486 .2660 .2832 .3005 .3178 .3330 .3522 .3693 17 18 .1969 .2143 .2316 .2489 .2662 .2835 .3008 .3180 .3353 .3525 .36i>6 IS 19 .1972 .2146 .2:119 .2192 .2665 .2838 .3011 .3183 .3355 .3527 .3699 19 20 .1975 .2148 .23’^ .2495 .2668 .2841 .3014 251S6 .3358 .3530 .3702 20 21 .1978 .2151 .2325 .2498 .2671 .2844 .3017 .31S9 .3361 .3533 .3705 21 •>•> .1981 .2154 .2328 .2501 .2674 .2847 .3019 .3192 .3364 .3536 .3708 22 23 .1983 .2157 .2331 .2504 .2677 .2850 .3022 .3195 25367 .3539 .3710 23 24 .1980 .2160 .2333 .2507 .2680 .2853 .3025 .3198 .3370 .3542 .3713 24 25 .1989 .2163 .2336 .2510 .2683 .2855 .3028 .3200 .3373 .3545 .3716 25 26 .1992 .2166 .2339 .2512 .2685 .2S5S .3031 .3203 .3376 25547 .3719 26 27 .1995 .2169 .2342 .2515 .2688 .2861 .3034 .3206 •337S .3550 .3722 27 28 .1998 .2172 .2345 .2518 .2691 •2S64 .3037 .3209 .3381 .3553 .3725 28 29 .2001 .2174 .2348 .2521 .2694 .2867 .3040 .3212 .3384 25556 .372S 29 30 .2004 .2177 .2351 .2524 .2697 •2S70 .3042 .3215 .3387 .3559 25730 30 31 .2007 .2180 .2354 .2527 .2700 .2873 .3045 .3218 .3390 .3562 .3733 31 32 .2010 .2183 .2357 .2530 .2703 .2876 .3048 .3221 .3393 .3565 25736 32 33 .2012 .2186 .2359 .2533 .2706 .2878 .3051 .3223 .3396 .3567 25739 33 34 .2015 .2189 .2362 .2536 .2709 .2881 .3054 .3226 .3398 .3570 .3742 34 35 .2018 .2192 .2365 .253S .2711 •28S4 .3057 .3229 .3401 .3573 .3745 35 36 .2021 .2195 .2368 .2541 .2714 ■2SS7 .3060 .3232 .3404 .3576 .3748 36 37 .2021 .2198 .2371 .2544 .2717 .2890 .3063 .3235 .3407 .3579 25750 37 38 .2027 .2200 .2374 .2547 .2720 .2893 .3065 .3238 .3410 .3582 .3753 08 39 .2030 .2203 .2377 .2550 .2723 .2896 .3068 .3241 25413 .3585 23756 39 40 .2033 .2206 .2380 .2553 .2726 .2899 .3071 .3244 .3416 .3587 .3759 40 41 .2036 .2209 •23S3 .2556 .2729 .2902 .3074 .3246 .3419 .3590 .3762 41 42 .2038 .2212 .2385 .2559 .2732 .2904 .3077 .3249 .3421 .3593 .3765 42 43 .2041 .2215 .2388 .2561 .2734 .2907 .3080 .3252 .3424 .3596 .3768 43 4* .2044 .2218 .2391 .2564 .2737 .2910 .3083 .3255 .3427 .3599 .3770 44 45 .2047 .2221 .2394 .2567 .2740 .2913 .3086 .3258 .3430 .3602 .3773 45 Ug .2050 .2224 .2397 .2570 .2743 .2916 .3088 .3261 .3433 .3605 .3776 46 147 .2053 .2226 .2400 .2573 .2746 .2919 .3091 .3264 .3436 .3608 .3779 47 4$ .20.46 .2229 .2403 .2576 .2749 .2922 .3094 .3267 .3439 .3610 .3782 48 49 .2059 .2232 .2406 .2579 .2752 .2925 .3097 .3269 .3441 25613 ■37S5 49 50 .2062 .2235 .2409 .2582 .2755 .2927 .3100 .3272 .3 111 .3616 .37 SS 50 51 .2065 .2238 .2411 .2585 .2758 .2930 2)103 .3275 .3447 .3619 .3790 51 52 .2067 .2241 .2414 .2587 .2760 .2933 1*106 .3278 .3450 .3622 .3793 52 5> .2070 .2244 .2417 .2590 .2763 .2936 2! 109 .3281 .3453 25625 .3796 53 54 .2073 .2247 .2420 .2593 .2766 .2939 .3111 21284 .3456 .3628 .3799 54 55 ‘1076 .2250 .2423 .2596 .2760 .2942 .3114 .3287 .3459 25630 .3S02 55 56 .2079 .2253 .2426 .2599 .2772 .2945 .3117 .3289 .3462 .3653 •3S05 56 57 .2082 .2255 .2429 .2602 .2775 .2948 .3120 .3292 .3464 25636 .3808 57 5S .2085 .2258 .2432 .2605 .2778 .2950 .3123 .3295 .3467 .3639 .3810 58 59 .2088 .2261 .2434 .2608 .2781 .2953 .3126 .3298 .3470 .3642 .3813 59 60 .2091 .2264 .2437 .2611 .2783 .2956 .3129 .3301 .3473 .3645 25S16 60GEOMETRICAL PROBLEMS. 87 Table of Chords; Radius == 1.0000 (continued). M. 22° 23° 24° 25° 26° 27° 28° 29° 30° 31° 32° | M.j 0' .3816 mm .4158 .4329 .4499 .4669 .1838 .5008 .5176 .5345 .5513 0'| 1 .3819 .3990 .4161 .4332 .4502 .4672 .1811 .5010 .5179 .5348 .5516 i! 2 .3822 .3993 .4164 .4334 .4505 .4675 .4814 .5013 .5182 .5350 .5518 2 t 3 .3825 .3996 .4167 .4337 .4508 .4677 .4847 .5016 .5185 .5353 .5521 4 .3828 .3999 .4170 .4340 .4510 .4680 .4850 .5019 .5188 .5356 .5524 4 5 .3830 .4002 .4172 .4343 .4513 .4683 .4853 .5022 .5190 .5359 .5527 1 j 6 .3833 .4004 .4175 .4346 .4516 .4686 .1855 .5024 .5193 .5362 .5530 l; ; i .3836 .4007 .4178 .4349 .4519 .4689 .4858 .5027 .5196 .5364 ■ 8 .3834 .4010 .4181 .4352 .4522 .4692 .1861 .5030 .5199 .5367 .5535 8| 9 .3842 .4013 .4184 .4354 .4525 .4694 .4864 .5033 .5202 .5370 .5538 9 10 .3845 .4016 .4187 .435 i .4527 .4697 .4867 .5036 .5201 .5373 .5541 10 11 .3848 .4019 .4190 .4360 .4530 .4700 .4869 .5039 .5207 .5376 .5543 11 12 .3850 .4022 .4192 .4363 .4533 .4703 .1872 .5011 .5210 .5318 .5546 12 13 .3853 .4024 .4195 .4366 .4536 .4706 .4875 .5014 .5213 .5381 .5549 13 14 .3856 .4027 .4198 .4369 .4539 .4708 .1878 .5047 .5216 .5381 .5552 14 15 .3859 .4040 .4201 .4371 .4542 .4711 .4881 .5050 .5219 .5387 .5555 15 16 .3862 .4033 .4204 .4374 .4544 .4714 .1884 .5053 .5221 .5390 .5557 16 17 .3865 .4036 .4207 .4377 .4547 .4717 .4886 .5055 .5224 .5392 .5560 17 IS .3868 .4039 .4209 .4380 .4550 .4720 .4889 .5058 .5227 .5395 .5563 18 19 .3870 .4042 .4212 .4383 .4553 .4723 .1892 .5061 .5230 .5398 .5566 19 20 .3873 .4044 .4215 .4386 .4556 .4725 .4895 .5064 .5233 .5401 .5569 20 21 .3876 .4017 .4218 .4388 .4559 .4728 .4898 .5067 .5235 .5401 .5571 21 22 .3879 .4050 .4221 .4391 .4561 .4731 .1901 .5070 .5238 .5406 .5574 22 23 .3882 .4053 .4224 .4394 .4564 .4734 .4903 .5072 .5241 .5409 .5577 23 24 .3885 .4056 .4226 .4397 .4567 .4737 .4906 .5075 .5244 .5412 .5580 24 25 .3888 .4059 .4229 .4400 .4570 .4710 .4909 .5078 .5247 .5415 .5583 25 26 .3890 .4061 .4232 .4403 .4573 .4742 .4912 .5081 .5249 .5418 .5585 26 27 .3893 .4064 .4285 .4405 .4576 .4745 .4915 .5084 .5252 .5420 .5588 27 28 .3896 .4067 .4288 .4408 .4578 .4748 .4917 .5086 .5255 .5423 .5591 28 29 .3899 .4070 .4241 .4411 .4581 .4751 .4920 .5089 .5258 .5426 .5594 29 30 .3902 .4073 .4244 .4414 .4584 .4754 .4923 .5092 .5261 .5429 .5597 30 31 .3905 .4076 .4246 .4417 .4587 .4757 .4926 .5095 .5263 .5432 .5599 31 32 .3908 .4079 .4249 .4420 .4590 .4759 .4929 .5098 .5266 .5434 .5602 32 33 .3910 .4081 .4252 .4422 .4593 .4762 .4932 .5100 .5269 .5437 .5605 33 34 .3913 .4084 .4255 .4425 .4595 .4765 .4934 .5103 .5272 .5440 .5608 34 35 .3916 .4087 .4258 .4428 .4598 .4768 .4937 .5106 .5275 .5443 .5611 35 36 .3919 .4090 Hi .4431 .4601 .4771 .4940 .5109 .5277 .5446 .5613 36 37 .3922 .4093 .4263 .4434 .4604 .4773 .4943 .5112 .5280 .5448 .5616 37 3S .3925 .4096 .4266 .4437 .4607 .4776 .4916 .5115 .5283 .5451 .5619 38 39 .3927 .4098 .4269 .4439 .4609 .4779 .4948 .5117 .5286 .5454 .5622 39 40 .3930 .4101 .4272 .4442 .4612 .4782 .4951 .5120 .5289 .5457 .5625 40 141 .3933 .4101 .4275 .4445 .4615 .4785 .4954 .5123 .5291 .5460 .5627 41 42 .3936 .4107 .4278 .4448 .4618 .4788 .4957 .5126 .5294 .5462 .5630 42 43 .3939 .4110 .4280 .4151 .4621 .4790 .4960 .5129 .5297 .5465 .5(533 43 144 .3942 .4113 .4283 .4454 .4624 .4793 .4963 .5131 .5300 .5468 .5636 44 45 .3945 .4116 .4286 .4456 .4626 .4796 .4965 .5134 .5303 .5471 .5638 -to 16 .3947 .4118 .4289 .4459 .4629 .4799 .4968 .5137 .5306 .5474 .5641 4C I 147 .3950 .4121 .4292 .4462 .4632 .4802 .4971 .5140 .5308 .5476 .5644 47 ! 4S .3953 .4124 .4295 .4465 .4635 .4805 .4974 .5143 .9311 .5479 .5647 48; | 49 .3956 .4127 .4298 .4468 .1638 .4807 .4977 .5145 .5314 .5482 .5650 49 1 50 .3959 .4130 .4300 .4471 .4611 .4810 .4979 .5148 .5317 .5485 .5652 50 : 51 .3962 .4133 .4303 .4474 .4643 .4813 .4982 .5151 .5320 .5488 .5655 51 j 52 .3965 .4135 .43»)6 .4476 ,4616 .1816 .4985 .5154 .5322 .5490 .5658 52 53 .3967 .4138 .4309 .4479 .4619 .4819 .4988 .5157 .5325 .5493 .5661 53 54 .3970 .4141 .4312 .4482 .4652 .4822 .4991 .5160 .5328 .5496 .5664 54 55 .3973 .4144 .4315 .4485 .4655 .4824 .4991 .5162 .5331 .5499 .56(56 55 56 .3976 .4147 .4317 .4188 .4658 .4827 .4996 .5165 .5334 .5502 .5669 56 57 .3979 .4150 .4320 .1491 .4660 .4830 .4999 mm .5336 .5504 .5(572 57 58 .3982 .4153 .4323 .4193 .4633 .4833 .5002 .5171 .53:’9 .*>'•07 .5675 58 59 .3985 .4155 .4326 .4496 .1666 .4836 .5005 .5174 .5342 M .5678 •rl 60 .3987 .4158 . 1329 .149 ) .4 >69 .1838 .5098 .5176 .'*345 .5513 .5 580 GEOMETRICAL PROBLEMS 88 Table of Chords ; Radius = 1.0000 {con tim ted) M. 33° 34° 3.5° ry O Olf 0^-0 0 i 38° 39° 40° 41° 42° 43° M. 0' .5080 .5847 .0014 .0180 .0340 .6511 .6670 .0840 .7004 .7107 0' 1 .5083 .5850 .0017 .01 S3 .0340 .6514 .0079 .6843 -7007 .7170 .IO-» 1 o .508«) .5853 .0020 .OlSti .0352 .6517 .0082 .6846 .7010 .7173 .733') ‘2 3 .51)80 .5850 .0022 .0180 .0354 .6520 .0084 .6849 .7012 . i 1 < <* ,733s •> 4 .5001 .5850 .0025 .0101 .08)57 .6522 .0087 .6851 .7015 .7178 . 1 341 4 5 .50)04 .5801 .01 »28 .0194 .0)300 6525 .0090 .6854 .7018 .7181 .7344 0 .5007 .5804 .0031 .0107 .0303 .652-8 .0693 .6857 .7020 .7184 .734«) (i r» 4 .570t> .5807 .0034 .6200 .0305 .0531 .0005 .6800 .7023 .7 ISO .7340 ■ s .:)703 .5870 .0030 .0202 .080) > .0538 .069S .0802 .7020 .7180 .7352 8 : o ,5705 .5872 .033'.) .020 > .0371 .0530 .0701 .6805 .7029 .7192 .73)4 0 10 .570-8 .5875 .0042 .0208 .0814 .0530 .0704 .0808 .7031 .7195 •7o'-> 1 lo i 11 .5711. .5878 .0) 15 .0211 .0870 .0542 .0706 .6870 .7034 .7197 .7309 11 12 .5714 .5881 .0047 .0214 .6370 .0544 .0709 .6873 .7037 .7200 .7302 12 1 13 .5717 .5884 .0 »5.» .0210 .6882 .0547 .0712 .6876 .7040 .7203 .7305 1:, I 14 .5710 .5880 .0)053 .0219 .63 85 .G550 .0715 .0879 .7042 .7205 .7308 n 1 11 .5722 .5889 .0050 .0222 .6387 .6553 .6717 .6881 .7045 .7208 .73)1 15 Iti .5725 .5802 .0058 .0225 .68>0O .0555 .0720 .6884 .7048 .7211 .73i3 n 1 17 ,57-S .58*05 .0001 .0227 .6303 .0558 .0723 .6887 .7050 .7214 •7310 17 IS .5730 .5807 .0004 .0230 .6806 .0501 .6725 .6890 .7053 .7210 .7370 I 10 .5733 .500») .0 »0)7 .0233 .6398 .0504 .6728 .6892 .7050 .7210 .7381 u 20 .5003 .0070 .0230 .6401 .0500 .6731 .6895 .7059 7222 .7384 20 •21 .5730 .5000 .0072 .0238 .6404 .6509 .6734 .089S .7001 .7224 .7387 21 .5742 .5000 .(>975 .0241 .6407 .6572 .6736 .0901 .7004 .7227 .7300 •?*> 2') 1 23 .5744 .5011 .0 »7 8 .0244 .6410 .0575 .6739 .0903 .7007 .7230 .7302 24 .5747 .5014 .0081 .0247 .6412 .6577 .0742 .0900 .7069 .7232 .7305 04 1 23 .5750 .5017 .0083 .0249 .6415 .6580 .6715 .6909 .7072 .7235 .7308 m 20 ■ .5020 .6 IS j .0252 .6418 ,6583 .0747 .6911 .7075 .7238 .7400 2) 2 i .5750 .5022 .0080 .0255 .6421 .6586 .6750 .0014 .7078 .7241 .7403 97 2S .5758 .5025 .0002 .0258 .0,428 .6588 .6753 .0017 .7080 .7243 .7406 28 20 .57**1 .5028 .0005 .0260 .6426 .6501 .67 56 .0020 .7083 .7240 .7408 *20 30 .•*764 .5031 .0007 .6263 .6420 .6504 .0758 .6922 .7080 .7240 • 1411 30 31 .5707 .5931 .610» .0200 .6432 .6507 .6701 .0025 .7089 .7251 .7414 31 i 32 .5700 .5936 .6103 .0209 .6434 .6500 .0704 .0028 .7001 .7254 .7417 32 33 ■ .5939 .6100 .0272 .0437 .0002 .6707 .0031 .7004 .7257 .7410 90 OO 34 .5775 .5042 .0108 .0274 .0440 .60)05 .0709 .00:*3 .7007 .7260 .7422 34 3’> .5778 .5045 .6111 .0)277 .0443 .0008 .0772 .0030 .7000 .7202 .7425 35 30 .5781 .5047 .6114 .0280 .0445 .0010 .0« <5 .<>939 .7102 .7265 .7427 36 f) | .5781) .5950 .6117 .028: > .0*448 .0013 .0777 .0941 .7105 .7208 .7430 37 38 .578ti .5953 .6110 .0285 .0451 .0016 .0780 .0044 .7108 .7270 .7433 38 30 .5780 .5056 .6122 .0288 .6454 .6019 .0783 .0047 .7110 .7273 • < 43-) 30 40 .5702 .5059 ,6125 .6201 .0450 .6021 .0780 .0050 .7113 .7276 .7438 40 41 .5705 .5901 .6128 .C294 .6450 .6024 .<‘788 .0052 .7116 .7270 .7441 41 42 .5707 .5964 .6130 .6200 .6402 .<>027 .0791 .0055 .7118 .72S1 .7443» 42 43 .5800 .5007 .613»:i .0200 .0405 .0030 .<»794 .<»058 .7121 .7284 .7440 43 44 .5803 .5970 .6130 .0302 .6407 .0032 .0707 .0001 .7124 .7287 .7440 44 45 .5800 .5972 .6139 .0305 .0470 ,0085 .0709 .00<*3 .7127 .7280 .7452 45 40 .5808 .5975 .0142 .0307 .0473 .0088 .080*2 .0900 .7129 .7202 .7454 46 47 .5811 .5078 .0144 .0310 .0470 .0040 .081‘5 .0000 .7132 .7205 .7457 47 4S .5814 .5081 .6147 .6313 .<>478 .0043 .0808 .0071 .7135 .7208 .7400 48 40 .5817 .5084 .6150 .6316 .0481 .0040 .6810 .0074 .7137 .7300 .7402 40 50 .5820 .5086 .6153 ■6318 .0484 .0049 .0813 .0077 .7140 .7303 .7405 50 51 .5822 .5089 .615.5 .6321 .0487 .0051 .0810 .0081 .7143 .7306 .7408 51 52 .5825 .5002 .6158 .032 4 .0 480 .005 4 .<’819 .0*081 .7140 .7308 .7471 52 53 .5828 .5 »05 .0101 .68.27 .<>102 .00'7 .<>821 .«‘>085 .7148 .7311 .7473 53 54 .5881 .50. >7 .6104 .638)0 .0405 .0000 .<*>824 .0.188 .7151 .7314 • ) 4 « 0 54 5 5 .5804 .0000 .0106 .08,8.2 .0408 .0002 .0827 .0001 .7154 .7310 .7470 55 50 .58: i0 .0003 .0109 .0885 .0500 .0005 .0829 .0003 .7150 : .7310 .7481 56 57 .5880 .0006 .0172 . '888 ; .0508 .0008 .0832 9)996 .7159 1 .7322 .7484 57 1 58 .58 12 .01 (00 1 .011.) .08 !1 Ì .0500 .0*071 .0835 .0000 .7102 .7325 .7487 58 i 50 .58 15 .0011 . .0178 .08 18 1 .0509 .6078 .0838 .7<»01 .7105 .73*27 .7480 I 50 i 60 .5817 .00:4 1 .01 so .t,»4n * .0511 .0070* .08 10 .7004 .7107 1 ,7«*'>0 .7402 60 111 *geometrical problems 89 Table of Chords; Radius = 1,0000 (continued). M. 44° 45° 46° 47° 48° 49° 50° 51° 52° 53° 54° M. 0' .7492 .7654 .7815 .7975 .8135 .8294 .8452 .8610 .8767 .8924 .9080 0' 1 .7405 .7650 .7817 .7978 .8137 .8297 .8455 .8613 .S770 .8927 .9082 1 •> .7498 .7659 .782.) .7980 .8140 .8299 .845S .8615 .8773 .8929 .9085 2 ;; .7500 .7662 .7824 .7983 .8143 .8302 .8 460 .8618 .8775 .8932 .9088 3 4 .7503 .7644 .7825 .7986 .8145 .8304 .8463 .8621 .8778 .8934 .9090 4 .'> .7506 .7667 .7828 .7988 .8148 .8307 .846«» .8623 .8780 .8937 .9093 5 :> .7508 .7670 .7831 1 .7991 .8151 .8310 .8468 .81326 .8783 .8940 .9095 6 .7511 .7672 .78 »3 : .799 4 .8153 .8 >12 .8471 .8629 .8786 .8942 .9098 7 8 .7514 .7675 .783») .7996 .8156 .8315 .8473 .8631 .8788 .89-15 .9101 8 .7516 .7678 .7839 .7999 .SI 59 .8318 .8476 .8634 .8791 .8947 .9103 9 1U .7519 .7681 .7841 .8002 .8161 .8320 .8479 .S030 .8794 .8950 .9106 10 11 .7522 .7683 .7814 .8004 .8164 .8323 .8481 .8039 .8796 .8953 .9108 11 12 .7524 .7686 .7847 .8007 .8167 .8326 .8484 .8642 .8799 .8955 * .9111 12 13 .7527 .7689 .7849 .8010 .8169 .8328 .8487 .8644 .8801 .8958 .9113 13 14 .7530 .7691 .7852 .SDÌ 2 .SI 72 .8331 .84 S9 .8647 .8804 .8960 .9116 14 15 .7533 .7694 .7855 .8015 .8175 .8334 .8492 .8650 .8807 .896)3 .9119 15 16 .7535 .7697 .7857 i .8018 .8177 .8336 .8495 .8652 .8809 .8966 .9121 16 IT .7538 .7699 .786:) 1 .8020 .8180 .8339 .8497 .8655 .8812 .8968 .9124 17 18 .7541 .7702 .7864 ! .8023 .8183 .8341 .8500 .8657 .8814 .8971 .9126 18 in .7543 .7705 .7S65 ! .8026 .8185 .8344 .8502 .8660 .8817 .8973 .9129 19 20 .7546 .7707 .7868 j .8028 .8188 .8347 .8505 .8603 .8820 .8970 .9132 20 21 .7549 .7710 .7871 i .8031 .8190 .8349 .850S .8665 .8822 .8979 .9134 21 ‘22 .7551 .7713 .7873 .8 >34 .8193 .8352 .8510 .8668 .8825 .8981 .9137 22 24 .7554 .7715 .7876 .8)3» .8196 .8355 .8513 .8671 .8828 .8984 .9139 23 I 24 HËÜI .7718 .7879 i .8049 .8198 .8357 .8516 .8673 .8830 .8986 .9142 24 25 .7560 .7721 .7882 ! .8042 .8201 .83» >0 .8518 .8676 .8833 .8989 .9145 25 26 .7562 . 1124 .7884 : .8044 .8204 .8363 .8521 .8678 .8835 .8992 .9147 26 27 .750.» .7726 .7887 1 .8047 .8206 .836;» .8523 .8681 .8838 .8994 .9150 27 28 .756 S .7729 .7890 .SO')0 .8209 .8368 .852«) .8684 .8841 .8997 .9152 28 20 .7570 .7731 .7892 ! .805*2 .8212 .8371 .8529 .8081» .S843 .8999 .9155 29 30 .7573 .7734 .7895 I .8055 .8214 .83 i o .8531 .8689 .884») .9002 .9157 30 31 .7571» . i141 .7898 1 .S05S .8217 .8376 .85.34 .8692 .8848 .9005 .9160 31 3° .7578 .7740 .7900 .8060 .8220 .8378 .8537 .8694 .8851 .9007 .9163 32 34 .7581 .7742 .7903 .8063 .8222 .8381 .8539 .8697 .88:54 .9010 .91» >5 33 34 .7584 .7745 .7906 .806'» .8225 .8384 .8542 RÜ .8856 .9012 .91 «>8 34 Ü-) .7586 .7748 .7908 .8068 .8228 .838»» .8545 .8702 .8859 .9015 .9170 35 3*' .7589 .7750 .7911 ! .8071 .8230 .8389 .8547 .8705 .8861 .9018 .9173 36 .7592 • i i •>•’» .7914 ; .8074 .8233 .8392 .8550 .8707 .8864 .9020 .9176 37 38 .7595 .7756 .7916 .807») .8236 .8394 .8552 .8710 .8867 .9023 .9178 38 39 .7597 .7758 .7919 .8079 .8238 .8397 .8555 .8712 .8869 .9025 .9181 39 40 .7600 .7761 .7922 .8082 .8241 .8400 .8*558 .8715 .8872 .9028 mm 40 j 41 .7603 .7764 .7924 1 .8081 .8244 S4»V2 .8.500 .8718 .8874 .9031 .9186 41 j 42 .7605 . 1166 .7927 .8087 .8240 .8 405 .85» ,8720 .8877 .9033 .9188 42 43 .7608 .7769 .7930 .8090 .8249 .8408 .85«»«) .8723 .8880 .9036 .9191 43 : 44 .7611 .7772 .8092 .8251 .8410 .8568 .S72») .8882 .9038 .9194 44 : 45 .7613 .7774 .7935 .8095 .8254 .8413 .8571 .8728 .8885 ill .919») 45 1 ; 40 .7616 .1 IH .7938 .8098 .8257 .8415 .8573 .8731 .8887 .9044 .9199 46 1 47 .7619 .7780 .7940 .8100 .8259 .8418 .8 hO .8734 .8890 .9046 .92» >1 47 4S .7621 .7782 .7943 .8103 .8262 .8421 .8579 .8736 .8893 .9049 .9204 48 40 .7624 .7785 .7946 .8105 .8205 .8 423 .8581 .8739 .8895 .9051 .9207 49 ; 50 .7627 .7788 .7948 .8108 .S2»»7 .8426 .8584 .8741 .8898 .9054 .9209 5») 51 .7629 .7791 .7951 .8111 .827») .8429 .8587 .8744 .8900 .9056 .9212 ;51 52 .7632 .7793 .7954 .8113 .8273 .8-131 .8589 .8747 .8903 .9059 .9214 .-)•» 1 54 .703-» .7796 7956 .8116 .8275 .8434 .8592 .8749 .8906 .9062 .9217 53 51 .7648 .7799 .7959 .8119 .8278 .8437 .8594 .8752 .8908 .9064 .9219 54 : .) ) .7610 .781)1 .7902 .8121 .8281 ! .8439 .8597 .8754 .8911 .9067 .9222 0 > 56 .7014 .780 4 .7964 .8124 I.8283 I .8442 ,8(»O0 .*757 .8914 .9069 .9225 56 57 .7016 .7807 .7947 .8127 .8286 1 .814 4 .8302 .8760 .8916 .9072 .9227 «77 , 58 .7648 .7809 .7970 .8129 .8280 .8447 .8«j0.5 .8762 .8919 .9075 .923») 58 50 .7651 .7812 .7972 .8132 1 .8291 .8450 .8608 .8765 .8921 .0077 .923.2 69 E 60 .7654 .7815 .7975 .8135 ! .8294 .8452 .8610 .8767 .8924 .9080 .9235 s90 GEOMETRICAL PROBLEMS Table of Chords; Radius = 1.0000 (continued). M. 55 50° r-’n.O 57 58® 59° 60° Gl° 62° 63° 64° M. 0' .9235 .9389 .9543 .9696 .9848 1.0000 1.0151 1.0301 1.0450 1.0598 0' 1 .9238 .9392 .9546 .9(599 .9851 1.0003 1.0153 1.0303 1.0452 1.0601 1 »> .9240 .9395 .9548 .9701 .9854 1.0005 1.0156 1.0306 1.0455 1.0603 0 3 .92 43 .9397 .9551 .9704 .9856 1.0008 1.0158 1.0308 1.0457 1.0606 3 1 4 .9215 .9400 .9553 .9706 .9859 1.0010 1.0161 1.0311 1.0460 1.0608 4 .9248 .9402 .9556 .9709 .98(51 1.0013 1.0163 1.0313 1.0462 1.0611 5 r> .9250 .9405 .9559 .9711 .9864 1.0015 1.0166 1.0316 1.0465 1.0613 6 . .9253 .9407 .9561 .9714 .986(5 1.0018 1.0168 1.0318 1.0467 1.0016 7 8 .9256 .9410 .9564 .9717 .9869 1.0020 1.0171 1.0321 1.0470 1.0618 8 - 9 .9258 .9413 .956(5 .9719 .9871 1.0023 1.0173 1.0323 1.0472 1.0621 9 1 10 .9261 .9415 .9569 .9722 .9874 1.0025 1.0176 1.0326 1.0475 1.0623 10 j 11 .9263 .9418 .9571 .9724 .9876 1.0028 1.0178 1.0328 1.0477 1.0626 11 i 12 .9266 .9420 .9574 .9727 .9879 1.0030 1.0181 1.0331 1.0480 1.0628 12 13 .9268 .9423 .9576 .9729 .9881 1.0033 1.0183 1.0333 1.0482 1.0630 13 14 .9271 .9425 .9579 .9732 .9S84 1.0035 1.0186 1.0336 1.0485 1.0633 14 15 .9274 .9428 .9581 .9734 .9886 1 SUB 1.0188 1.0338 1.0487 1.0635 15 16 .9276 .9430 .9584 .9737 .9889 1.0040 1.0191 1.0341 1.0490 1.0638 16 17 .9279 .9433 .9587 .9739 .9891 1.0043 1.0193 1.0343 1.0492 1.0640 17 18 .9281 .9436 .9589 .9742 .9894 1.0045 1.0196 1.0346 1.0495 1.0643 18 19 .9284 .9438 .9592 .9744 .9897 1.0048 1.0198 1.0348 1.0497 1.0645 19 20 .9287 .9441 .9594 .9747 .9899 1.0050 1.0201 1.0351 1.0500 1.0648 20 21 .9289 .9443 .9597 .9750 .9902 1.0053 1.0203 1.0353 1.0502 1.0650 21 22 .9292 .9446 .9599 .9752 .9904 1.0055 1.0206 1.0356 1.0504 1.0653 22 23 .9294 .9448 .9602 .9755 .9907 1.0058 1.0208 1.0358 1.0507 1.0655 23 24 .9297 .9451 .9604 .9757 .9909 J. 00(50 1.0211 1.0361 1.0509 1.0658 24 25 .9299 .9454 .9607 .97(59 .9912 1.00(53 1.0213 1.0363 1.0512 1.0660 25 26 .9302 .9456 .9610 .97(52 .9914 1.00(55 1.0216 1.0366 1.0514 1.0662 26 27 .9305 .9459 .9612 .9765 .9917 1.00(58 1.0218 1.0368 1.0517 1.0665 27 28 .9307 .9461 .9615 .97(57 .9919 1.0070 1.0221 1.0370 1.0519 1.0667 28 29 .9310 .9464 .9617 .9770 .9922 1.007:5 1.0223 1.0373 1.0522 1.0670 29 30 .9312 .9460 .9620 .9772 .9924 1.0075 1.0226 1.0375 1.0524 1.0672 30 31 .9315 .9469 .9622 .9775 .9927 1.0078 1.0228 1.0378 1.0527 1.0675 31 32 .9317 .9472 .9625 .9778 .9929 1.0080 1.0231 1.0380 1.0529 1.0677 32 33 .9320 .9474 .9627 .9780 .9932 1.0083 1.0233 1.0383 1.0532 1.0680 33 34 .9323 .9477 .9(530 .9783 .9934 1.008(5 l .023(5 1.0385 1.0534 1.0682 34 35 .9325 .9479 .9633 .9785 .9937 1.0088 1.0238 1.0388 1.0537 1.0685 35 36 .9328 .9482 .9635 .9"88 .9939 1.0091 1.0241 1.0390 1.0539 1.0687 36 37 .9330 .9484 .9638 .9790 .9942 1.0093 1.0243 1.0393 1.0542 1.0690 37 38 .9333 .9487 .9640 .9793 .9945 1.0096 1.0246 1.0395 1.0544 1.0692 38 39 .9335 .9489 .9643 .9795 .9947 1.0098 1.0248 1.0398 1.0547 1.0694 39 40 .93: »8 .9492 .9645 .9798 .9950 1.0101 1.0251 1.0400 1.0549 1.0697 40 41 .9341 .9495 .9648 .9800 .9952 1.0103 1.0253 1.0403 1.0551 1.0699 41 42 .9343 .9497 .9650 .9803 .9955 1.010(5 1.0256 1.0405 1.0554 1.0702 42 43 .9346 .9500 .9653 .9805 .9957 1.0108 1.0258 1.0408 1.0556 1.0704 43 44 .9348 .9502 .9655 .9808 .99(50 1.0111 1.02(51 1.0410 1.0559 1.0707 44 45 .9351 .9505 .9658 .9810 .99(52 1.0113 1.0263 1.0413 1.0561 1.0709 45 46 .9353 .9507 .96(51 .9813 .9965 1.0116 1.02(5(5 1.0415 1.0564 1.0712 46 47 .9356 .9510 .9663 .9816 .99(57 1.0118 1.0268 1.0418 1.0566 1.0714 47 48 .9359 .9512 .96(5(5 .9818 .9970 1.0121 1.0271 1.0420 1.0569 1.0717 48 |) .9361 .9515 .9668 .9821 .9972 1.0123 1.0273 1.0423 1.0571 1.0719 49 5.) .9304 .9518 .9671 .9823 .9975 1.0126 1.0276 1.0425 1.0574 1.0721 50 51 mm .9520 .967:5 .9826 .9977 1.0128 1.0278 1.0428 1.0576 1.0724 51 52 .9369 .9523 .967(5 .9828 .9980 1.01:51 1.0281 1.0430 1.0579 1.0726 52 53 .9371 .9525 9(578 .9831 .9982 1.01:53 1.0283 1.0433 i 1.0581 1.0729 53 54 .937 4 3*528 .9681 .98 53 .9985 1.0136 1.028(5 1.0435 1 1.0584 1.0731 54 55 .9377 .9530 .9(583 .983(5 .99-i 7 1.0138 1.0288 1.0438 1.0586 1.0734 55 56 .9379 .9533 .9(58(5 MBB .9990 1.0141 1.0291 1.0440 1.0589 1.07:16 56 57 .9382 .9536 .9(589 .9811 .9992 1.0143 1.0293 1.0443 1.0591 1.07.“>9 57 58 .9384 .9538 .9(591 .9843 .9945 1.014(5 1.0296 1.0445 1.0593 1.0741 58 59 .9387 .9541 .9694 .9846 .9998 1.0148 1.0298 1.0447 1.0596 1.0744 59 60 .9389 .9543 .9696 .9848 1.0900 1.0151 1.0301 1.0450 1.0598 1.0746 600 EOM KT K1 " a Ii P UOHLE MS. 91 Table of Chords; Radius = 1.0000 (continved). M. (>5° 66° (>7° (58° (»9° 70° 71° 72° 73° M. j 0' 1.0746 1.0893 1.1039 1.1184 1.1328 1.1472 1.1614 1.1756 1.1 S96 0' 1 1.0748 1.0895 1.1041 1.1186 1.1331 1.1474 1.1616 1.1758 1.1899 1 2 1.0751 1.0898 1.1044 1.1189 1.1333 1.1476 1.1619 1.1760 1.1901 2 3 1.0753 1.0900 1.1016 1.1191 1.1335 1.1479 1.1621 1.1763 1.1903 3 1 4 1.0756 1.0903 1.1048 1.1194 1.1338 1.1481 1.1624 1.1705 1. loffi 4 I 5 1.0758 1.0905 1.1051 1.1196 1.1340 1.1483 1.1626 1.1767 1.1908 5 1 6 1.0761 1 0907 1.1053 1.1198 1.1342 1.1486 1.1628 1.1770 1.1910 6 I | 1.0763 1.0910 1.1056 1.1201 1.1345 1.1488 1.1631 1.1772 1.191:5 8 1.0766 1.0912 1.1058 1.1203 1.1347 1.1491 1.1633 1.1775 1.1915 8 9 1.0768 1.0915 1.1061 1.1206 1.1350 1.1493 1.1635 1.1777 1.1917 9 10 1.0771 1.0917 1.1063 1.1208 1.1352 1.1495 1.1638 1.1779 1.1920 10 11 1.0773 1.0920 1.1065 1.1210 1.1354 1.1498 1.1640 1.1782 1.1922 11 12 1.0775 1.0922 1.1068 1.1213 1.1357 1.1500 1.1642 1.1784 1.1924 12 13 1.0778 1.0924 1.1070 1.1215 1.1359 1.1502 1.1645 1.1786 1.1927 13 14 1.0780 1.0927 1.1073 1.1218 1.1362 1.1505 1.1647 1.1789 1.1929 14 15 1.0783 1.0929 mm 1.1220 1.1364 1.1507 1.1650 1.1791 1.1931 15 16 1.0785 1.0932 1.1078 1.1222 1.1366 1.1510 1.1652 1.1793 1.1934 16 17 1.0788 1.0934 1.1080 1.1225 1.1369 1.1512 1.1654 1.1796 1.1936 17 18 1.0790 1.0937 1.1082 1.1227 1.1371 1.1514 1.1657 1.1798 1.1938 18 19 1.0793 1.0939 1.1085 1.1230 1.1374 1.1517 1.1659 1.1800 1.1941 19 20 1.0795 1.0942 1.1087 1.1232 1.1376 1.1519 1.1661 1.1803 1.1943 20 21 1.0797 1.0944 1.1090 1.1234 1.1378 1.1522 1.1664 1.1805 1.1946 21 oo 1.0800 1.0946 1.1092 1.1237 1.1381 1.1524 1.1666 1.1807 1.1948 •2‘2 23 1.0802 1.0949 1.1094 1.1239 1.1383 1.1526 1.1668 1.1810 1.1950 23 24 1.0805 1.0951 1.1097 1.1242 1.1386 1.1529 1.1671 1.1812 1.1952 24 25 1.0807 1.0954 1.1039 1.1244 1.1388 1.1531 1.1673 1.1814 1.1955 25 26 1.0810 1.0956 1.1102 1.1246 1.1390 1.1533 1.1676 1.1817 1.1957 26 RE « i 1.0S12 1.0959 1.1104 1.1249 1.1393 1.1536 1.1678 1.1819 1.1959 27 28 1.0S15 1.0961 1.1107 1.1251 1.1395 1.1538 1.1680 1.1821 1.1962 28 29 1.0817 1.0963 1.1109 1.1254 1.1398 1.1541 1.1683 1.1824 1.1964 29 30 1.0820 1.0966 1.1111 1.1256 1.1400 1.1543 1.1685 1.1826 1.1966 30 31 1.0822 1.0968 1.1114 1.1258 1.1402 1.1545 1.1687 1.1829 1.1969 31 32 1.0824 1.0971 1.1116 1.1261 1.1405 1.1548 1.1690 1.1831 1.1971 32 : 33 1.0827 1.0973 1.1119 1.1263 1.1407 1.1550 1.1692 1.1833 1.1973 33 34 1.0829 1.0976 1.1121 1.1266 1.1409 1.1552 1.1694 1.1836 1.1976 34 35 1.0832 1.0978 1.1123 1.1263 1.1412 1.1555 1.1697 1.1838 1.1978 35 36 1.0834 1.0980 1.1126 1.1271 1.1414 1.1557 1.1699 1.1840 1.19S0 36 37 1.0837 1.0983 1.1128 1.1273 1.1417 1.1560 1.1702 1.1843 1.1983 37 38 1.0839 1.0985 1.1131 1.1275 1.1419 1.1562 1.1704 1.1845 1.1985 38 59 1.08 U 1.0988 1.1133 1.1278 1.1421 1.1564 1.1706 1.1847 1.1987 39 40 1.0S44 1.0990 1.1136 1.1280 1.1424 1.1567 1.1709 1.1850 1.1990 40 41 1.0846 1.0993 1.1138 1.1283 1.1426 1.1569 1.1711 1.1852 1.1992 41 42 1.0 S19 1.0995 1.1140 1.1285 1.1429 1.1571 1.1713 1.1854 1.1994 42 43 1.08*1 1.0937 1.1143 1.1287 1.1431 1.1574 1.1716 1.1857 1.1997 43 44 1.0854 1.10UO 1.1145 1.1290 1.1433 1.1576 1.1718 1.1859 1.1999 44 45 1.0856 1.1002 1.1148 1.1292 1.1436 1.1579 1.1720 1.1861 1.2001 45 46 1.0859 1.1005 1.1150 1.1295 1.1438 1.1581 1.1723 1.1804 1.2004 46 47 1.0861 1.1007 1.1152 1.1297 1.1441 1.1583 1.1725 1.1866 1.2006 47 48 1M 863 1.1010 1.1155 1.1299 1.1443 1.1586 1.1727 1.1863 1.2008 48 49 1.0866 1.1012 1.1157 1.1302 1.1445 1.1588 1.1730 1.1871 1.2011 49 50 1.0868 1.1014 1.1163 1.1304 1.1448 1.1590 1.1732 1.1873 1.2013 50 51 1.0871 1.1017 1.1162 1.1307 1.1450 1.1593 1.1735 1.1875 1.2015 51 52 1.0873 1.1019 1.1165 1.1309 1.1452 1.1595 1.1737 1.1878 1.2018 52 . 53 1.0876 1.1022 1.1167 1.1311 1.1455 1.1598 1.1739 1.1880 1.2020 53 54 1.0878 1.1024 1.1169 1.1314 1.1457 1.1600 1.1742 1.1882 1.2022 54 ÜO 1.0881 1.1027 1.1172 1.1316 1.1460 1.1602 1.1744 1.1885 1.2025 55 56 1.0883 1.1029 1.1174 1.1319 1.1462 1.1605 1.1746 1.1887 1.2027 56 57 1.0885 1.1031 1.1177 1.1321 1.1464 1.1607 1.1749 1.1889 1.2029 57 58 1.0888 1.1034 1.1179 1.1323 1.1467 1.1609 1.1751 1.1892 1.2032 58 59 1.0890 1.1036 1.1181 1.1326 1.1469 1.1612 1.1753 1.1894 1.2034 59 60 1.0893 1.1039 1.1184 1.1328 1.1472 1.1614 1.1756 1.1896 1.2036 6092 GEOMETRICAL PROBLEMS. Table of Chords; Radius = 1.0000 (continued). M. 74° 75° 76° <%o 77 78° 79° 0 o 00 81° 8'i° M. 0' 1.2036 1.2175 1.2313 1.2450 1.2586 1.2722 1.2856 1.2989 1.3121 0' 1 1.2039 1.2178 1.2316 1.2453 1.2589 1.2724 1.2858 1.2991 1.3123 1 2 1.2041 1.2180 1.2318 1.2455 1.2591 1.2726 1.2S60 1.2993 1.3126 •2 3 1.2043 1.2182 1.2320 1.2457 1.2593 1.2723 1.2862 1.2996 1.312S 3 4 1.2046 1.2184 1.2322 1.2459 1.2595 1.2731 1.2865 1.2998 1.3130 4 5 1.2048 1.2187 1.2325 1.2462 1.2598 1.2733 1.2867 1.3000 1.3132 5 6 1.2050 1.2189 1.2327 1.2464 1.2600 1.2735 1.2869 1.3002 1.3134 6 7 1.2053 1.2191 1.2329 1.2466 1.2602 1.2737 1.2871 1.3004 1.3137 i 8 1.2055 1.2194 1.2332 1.2468 1.2604 1.2740 1.2874 1.3007 1.3139 8 9 1.2057 1.2196 1.2334 1.2471 1.2607 1.2742 1.2876 1.3009 1.3141 9 10 1.2060 1.2198 1.2336 1.2473 1.2609 1.2744 1.2878 1.3011 1.3143 10 11 1.2062 1.2201 1.2338 1.2475 1.2611 1.2746 1.2880 1.3013 1.3145 11 12 1.2064 1.2203 1.2341 1.2478 1.2614 1.2748 1.2882 1.3015 1.3147 12 13 1.2066 1.2205 1.2343 1.2480 1.2616 1.2751 1.2S85 1.3018 1.3150 13 14 1.2069 1.2208 1.2345 1.2482 1.2618 1.2753 1.2887 1.3920 1.3152 14 15 1.2071 1.2210 1.2348 1.2484 1.2620 1.2755 1.2889 1.3022 1.3154 15 16 1.2073 1.2212 1.2350 1.2487 1.2623 1.2757 1.2891 1.3024 1.3156 16 17 1.2076 1.2214 1.2352 1.2489 1.2625 1.2760 1.2894 1.3027 1.3158 17 18 1.2078 1.2217 1.2354 1.2491 1.2627 1.2762 1.2896 1.3029 1.3161 18 19 1.2080 1.2219 1.2357 1.2493 1.2629 1.2764 1.2898 1.3031 1.3163 19 20 1.2083 1.2221 1.235S) 1.2496 1.2632 1.2766 1.2900 1.3033 1.3165 20 1 21 1.2085 1.2224 1.2361 1.249S 1.2634 1.2769 1.2903 1.3035 1.3167 21 22 1.2087 1.2220 1.2364 1.2509 1.2636 1.2771 1.2905 1.3038 1.3169 22 1 | 23 1.2090 1.2228 1.2366 1.2503 1.2633 1.2773 1.2907 1.3040 1.3172 23 : m 1.2092 1.2231 1.2308 1.2505 1.2641 1.2775 1.2909 1.3042 1.3174 24 i 25 1.2094 1.2233 1.2370 1.2507 1.2643 1.2778 1.2911 1.3044 1.3176 25 26 1.2097 1.2235 1.2373 1.2509 1.2645 1.2780 1.2914 1.3046 1.3178 26 27 1.2099 1.2237 1.2375 1.2512 1.2643 1.2782 1.2916 1.3049 1.3180 27 i 28 1.2101 1.2240 1.2377 1.2514 1.2650 1.2784 1.2918 1.3051 1.3183 28 i 29 1.2104 1.2242 1.2380 1.2516 1.2652 1.2787 1.2920 1.3053 1.3185 29 30 1.2106 1.2244 1.2382 1.25IS 1.2654 1.2789 1 M 1.3055 1.3187 30 31 1.2103 1.2247 1.2384 1.2521 1.2656 1.2791 1.2925 1.3057 1.3189 31 32 1.2111 1.2249 1.2386 1.2623 1.2659 1.2793 1.2927 1.3060 1.3191 32 33 1.2113 1.2251 1.2389 1.2525 1.2661 1.2795 1.2929 1.3062 1.3193 33 34 1.2115 1.2254 1.2391 1.2528 1.2663 1.2798 1.2931 1.3064 1.3196 34 35 1.2117 1.2256 1.2393 1.2530 1.2665 1.2800 1.2934 1.3066 1.3198 35 36 1.2120 1.2258 1.2396 1.2532 1.2668 1.2802 1.2936 1.306S 1.3200 36 37 1.2122 1.2260 1.2398 1.2534 1.2670 1.2804 1.2938 1.3071 1.3202 37 38 1.2124 1.2203 1.2400 1.2537 1.2672 1.2807 1.2940 1.3073 1.3204 38 j 39 1.2127 1.2265 1.2402 1.2539 1.2674 1.2809 1.2942 1.3075 1.3207 39 40 1.2129 1.2267 1.2405 1.2541 1.2677 1.2811 1.2945 1.3077 1.3209 40 4i 1.2131 1.2270 1.2407 1.2543 1.2679 1.2813 1.2947 1.3079 1.3211 41 42 1.2134 1.2272 1.2409 1.2546 1.2681 1.2816 1.2949 1.3082 1.3213 42 43 1.2136 1.2274 1.2412 1.2548 1.2633 1.2818 1.2951 1.3084 1.3215 43 44 1.2138 1.2277 1.2414 1.2550 1.2686 1.2820 1.2954 1.3086 1.3218 44 45 1.2141 1.2279 1.2410 1.2552 1.2088 1.2822 1.2956 1.3088 1.3220 45 1 46 1.2143 1.2281 1.2418 1.2555 1.2690 1.2825 1.2958 1.3090 1.3222 46 ; 47 1.2145 1.2283 1.2421 1.2557 1.2092 1.2827 1.296) 1.3093 1.3224 47 1 4S 1.2148 1.2286 1.2423 1.2559 1.2695 1.2829 1.2962 1.3095 1.3226 48 i 49 1.2150 1.22SS 1.2425 1.2562 1.2697 1.2831 1.2965 1.3097 1.3228 49 ! 50 1.2152 1.2290 1.2428 1.2564 1.2699 1.2833 1.2967 1.3099 1.3231 50 ! 51 1.2154 1.2293 1.2430 1.2566 1.2701 1.2836 1.2909 1.3101 1.3233 51 1 52 1.2157 1.2295 1.2432 1.2568 1.2704 1.2838 1.2971 1.3104 1.3235 52 1 53 1.2159 1.2297 1.2434 1.2571 1.2706 1.2840 1.2973 1.3106 1.3237 53 54 1.2161 1.2299 1.2437 1.2573 1.2708 1.2842 1.2976 1.2.108 1.3239 54 55 1.2164 1.2302 1.2439 1.2575 1.2710 1.2845 1.2978 1.3110 1.3242 55 56 1.2166 1.2304 1.2 441 1.2577 1.2713 1.2847 1.2980 mam 1.3244 56 57 1.2168 1.2306 1.2443 1.2580 1.2715 1.2843 1.2982 1.3115 1.3246 57 58 1.2171 1.2309 1.2446 1.2582 1.2717 1.2851 1.2985 1.3117 1.3248 58 59 1.2173 1.2311 1.2448 1.2584 1.2719 1.2854 1.2987 1.3119 1.3250 59 60 1.2175 1.2313 1.2450 1.2586 1.2722 1.2856 1.2989 1.3121 1.3252 60GEOMETRICAL PROBLEMS. 93 Table of Chords; Radius = 1.0000 (concluded). M. 83° 84° 85° 8(>° 87° 88° 89° M. 0' 1.3252 1.3383 1.3512 1.3640 1.3767 1.3393 1.4018 I 0'\ 1 1.3255 1.3385 1.3514 1.3642 1.3769 1.3895 1.4020 1 1 9 1.3257 1.3387 1.3516 1.3644 1.3771 1.3S97 1.4022 1 3 1.3259 1.3389 1.3518 1.3646 1.3773 1.3899 1.4024 1) j 4 1.3261 1.3391 1.3520 1.3048 1.3776 1.3902 1.4026 4 1 1.3263 1.3393 1.3523 1.3651 1.3778 1.3904 1.4029 5 6 1.3265 1.3390 1.3525 1.3653 1.3780 1.3906 1.4031 6 7 1.3268 1.3398 1.3527 1.3655 1.3782 1.3908 1.4033 1 8 1.3270 1.3400 1.3529 1.3657 1.3784 1.3910 1.4035 8 9 1.3272 1.3402 1.3531 1.3659 1.3786 1.3912 1.4037 9 I 19 1.3274 1.3404 1.3533 1.3661 1.3788 1.3914 1.4039 10 11 1.3276 1.3406 1.3535 1.3663 1.3790 1.3916 1.4041 11 12 1.3279 1.3409 1.3538 1.3665 1.3792 1.3918 1.4043 12 13 1.3281 1.3411 1.3540 1.3668 1.3794 1.3920 1.4045 13 14 1.3283 1.3413 1.3542 1.3670 1.3797 1.3922 1.4047 14 1 15 1.3285 1.3415 1.3544 1.3672 1.3799 1.3925 1.4049 15 16 1.3287 1.3417 1.3546 1.3674 1.3801 1.3927 . 1.4051 16 17 1.3289 1.3419 1.3548 1.3676 1.3803 1.3929 1.4053 17 IS 1.3292 1.3421 1.3550 1.3673 1.3805 1.3931 1.4055 18 19 1.3294 1.3424 1.3552 1.3,680 1.3807 1.3933 1.4058 19 20 1.3296 1.3426 1.3555 1.3682 1.3809 1.3935 1.4060 20 21 1.329S 1.3428 1.3557 1.3685 1.3811 1.3937 1.4062 21 22 1.3300 1.3430 1.3559 1.3687 1.3813 1.3939 1.4064 o-> 23 1.3302 1.3432 1.3561 1.3689 1.3816 1.3941 1.4066 23 24 1.330-5 1.3434 1.3563 1.3,691 1.3818 1.3943 1.4068 24 25 1.3307 1.3437 1.3505 1.3693 1.3820 1.3945 1.4070 25 26 1.3309 1.3439 1.3567 1.3695 1.3822 1.3947 1.4072 26 27 1.3311 1.3441 1.3570 1.3697 1.3824 1.3950 1.4074 27 28 1.3313 1.3443 1.3572 1.3699 1.3826 1.3952 1.4076 28 29 1.3315 1.3445 1.3574 1.3702 1.3828 1.3954 1.4078 29 1 30 1.3318 1.3447 1.3576 1.3704 1.3830 1.3956 1.40S0 30 i 31 1.3320 1.3149 1.3578 1.3706 1.3832 1.3958 1.4082 31 32 1.3322 1.3452 1.3580 1.3708 1.3834 1.3960 1.4084 32 oo 09 1.3324 1.3454 1.3582 1.3710 1.3837 1.3962 1.4086 33 34 1.3326 1.3456 1.3585 1.3712 . 1.3839 1.3964 1.4089 34 35 1.3328 1.3458 1.3587 1.3714 1.38 41 1.3906 1.4091 35 ! 36 1.3331 1.3460 1.3584 1.3716 1.3843 1.3968 1.4093 36 1 9* * 1.3333 1.3462 1.3591 1.3718 1.3845 1.3970 1.4095 37 38 1.3335 1.3465 1.3593 1.3721 1.3847 1.3972 1.4097 38 1 39 1.3337 1.3467 1.3595 1.3723 1.3849 1.3975 1.4099 39 j 40 1.3:139 1.3469 1.3597 1.3725 1.3851 1.3977 1.4101 40 41 1.3341 1.3471 1.3599 1.3,727 1.3853 1.3979 1.4103 41 j 42 1.3344 1.3173 1.3602 1.3729 1.3855 1.3981 1.4105 42 ( 43 1.3346 1.3475 1.300 4 1.3731 1.3858 1.3983 1.4107 43 1 44 1.3348 1.3477 1.3090 1.3133 1.386) 1.3,985 1.4109 44 ! 4:> 1.3350 1.3180 1.3608 1.3,735 1.3862 1.3987 1.4111 45 ; 46 1.3352 1.3182 1.36,10 1.3738 1.3864 1.3989 1.1113 46 ! 47 1.3354 1.3484 1.3612 1.3740 1.3866 1.3991 1.4115 47 j 48 1.3357 1.3186 1.3614 1.3742 1.3868 1.3993 1.4117 48 m 1.3359 1.3488 1.3617 1.3744 1.3870 1.3995 1.4119 49 I 50 1.3361 1.3490 1.3619 1.3746 1.3872 1.3997 1.4122 50 : 51 1 ..‘1363 1.3492 1.3621 1.3748 1.3874 1.3999 1.4124 51 : 52 1.3365 1.3495 1.3023 1.3750 1.3876 1.4002 1.4126 52 ! 53 1.336 i 1.3197 1.3025 1.3752 1.3879 1.4004 1.4128 53 . 54 1.3370 1.3499 1.3627 1.3754 1.3881 1.4006 1.4130 54 i 55 1.3372 1.3501 1.3029 1.3757 1.3883 1.4008 1.4132 55 56 1.3374 1.3503 1.3631 1.3759 1.3885 1.4010 1.4134 56 1 1.3376 1.3505 1.3634 1.3761 1.38S7 1.4012 1.4136 57 58 1.3378 1.3508 1.3636 1.3763 1.3889 1.4014 1.4138 58 59 1.33S0 1.3510 1.3638 1.3765 1.3891 1.4016 1.4140 59 60 Bfl CO cO CO l-H 1.3512 1.3640 1.3767 1.3S93 1.4018 1.4142 6094 HIP AND JACK RAFTERS. Lengths and Bevels of Hid and Jack Rafters. The lines ab and be in Fig. 89 represent the walls at the angle of a building; be is the seat of the liip-rafter, and nf of a jack-rafter. Draw eh at right angles to be, and make it equal to the rise of the roof; join b and h, and hb will be the length of the liip-rafter. Through e draw di at right angles to be. Upon b, with the radius bli, describe the arc hi, cutting di in i. Join b and i, and extend gj h Fig. 89. to meet bi injf l then yj will be the length of the jack-rafter. The length of each jack-rafter is found in the same manner,—by extending its seat to cut the line bi. From/ draw/t at right angles to fg, also fl at right angles to be. Make/fc equal to ft by the arc Uc, or make yk equal to yj by the arc jk; then the angle at j will be the top bevel of the jack-rafters, and the one at k the down bevel. Backing of the hip-rafter. At any convenient place in be (Fig. 89), as o, draw mn at right angles to be. From o describe a circle, tangent to bh, cutting be in a. Join m and a and n and s; then these lines will form at s the proper angle for bevelling the top of the liip-rafter.TRIGONOMETRY. 95 TRIGONOMETRY. It is not the purpose of the author to teach the use of trigonometry, or what it is; but, for the benefit of those readers who have already acquired a knowledge of this science, the following convenient formulas, and tables of natural sines and tangents, have been inserted. To those who know how to apply these trigonometric functions, they will often be found of great convenience and utility. These tables are taken from Searle’s “ Field Engineering,” John Wiley & Sons, publishers, by permission.96 TRIGONOMETRIC FORMULAS, Trigonometric Functions. ' Let A (Fig. 107) = angle BAG = arc BF, and let tlio radius AF *= AB t= All = L We then have » BO = AG = DF = IIG - AD sin A cos .1 tan A cot A sec A cosec A — AG versin A — CF = BE covei*s A = LX = JIB exsec A = BD coexsec A = BG chord A = BF chord 2 A = BI = In the right-angled triangle ABC (Fig. 107) Let AB = cf AG = bt and 2>C = a. We then have: 1. sin A 2. cos A 3. tan A 4. cot A 5. sec A 6. cosec A ?s vers -4 8. exsec ..4 = — = cos Is =: Sin B = cot 72 = tan B = cosec 2> = sec B c —b — =s covers B c - ss cooxsec B n j f —— Cl . J-, y. covers ^1 = — —= versin i> 10. coexsec A = — — = exsec I? a 11. a r;f sin ^4 = 6 tan A 12. b r=z c cosM = a cot A 13 c - a— = —' sin A coj -.4 14. a = c cosi> — 6 cot I? 15. b — c sin B ~ a tan B 1G. c 17. a = V (c + b) vc — b) 18» & I jfjc + a) (c — a) c « 4 a _ b cos 2? sin B 19. 20. C= 90* = .4 + 2? a b 21. area = —aTRIGONOMETRIC FORMULAS. 97 Solution op Oblique Triangles. c/ A/ >> No Fig. 108. GIVEN. SOUGHT. FORMULÆ. 22 A, B, a C,b,c C= 180°-^(A + R), 6 = -.— x ■ sin B, 1 ’ sm A c = .** . sin (A + R) sm A 23 A, a, b B, C, c Bin B = Eln ^ . 6, C= 180°-(A 4- R) 28 area X — y2ab sin C. 29 > a, 6, c M Lets=^(o + 6+c);sin^A=|/(S—^||| CO o ... /s(s — a) , M cos 34 A =|/-\c--;tan^Ac=|/-V(;_a)i 31 . 2 Vs (« _ a) (è—b} (k — c) 1 j sin A = t I ; be 9 2 (« — 5) (s — c) vers A = ■4**—, ——.—-be 32 area IC — Fs (s _ a) (s — b) (j — c) S3 A, B, C, a area ^ __ a2 sin B. sin 0 2 sm A98 TRIGONOMETRIC FORMULAS GENERAL FORMULAS. 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 1 = VI — cos2 A — tan A cos A cosec A 2 sin Y A cos A = vers A cot Y A sin A : sin A sin A = Y 14 vers 2 A = Y]4 (1 — cos 2 A) cos A : 1 sec A = y 1 — sin2 A = cot A sin A cos A cos A tan A tan A 1 — vers A , = 2 cos2 Y A — 1 = 1—2 sin2 Y A cos2 14 A — sin2 ]4 A = Y Y Y cos 2 A 1 cot A sin ^4 cos -4 ■/ cos2 A Y sec2 A — i 4/ 1 — cos2 A cos A Fin 2 A 1 + cos 2 tan A = cot A cot A = cot A = 1 — cos 2 A . sin 2 -4 1 vers 2^4 sin 2 A = exsec A cot}4 A tan A cos A sin  = V cosec2 A — 1 sin 2 A 1 — cos 2 A tan Y A exsec A • sin 2 ^4 vers 2 A 1 + cos 2 ^4 sin 2-4. vers A = 1 — cos yl = sin .4 tan Y A = 2 sin2 -4 vers A r= exsec cos A exsec A = sec A — 1 = tan A tan A = vers A cos A /1 — cos A y—1— sin Y A : sin 2 A = 2 sin A cos A / vers A - y—» cos Y A i 1 -f~ cos A 9 cos 2 A = 2 cos2 A — 1 = cczn A — sin2 A *= 1 — 2 sin^TRIGONOMETRIC FORMULAS. 99 General Formulas. „ . tan A . . . 1 — cos A _ , / f 53. tan 14 A = , , = cosec A — cot A — — —• — 4/ . 1 + secA sio^ r 1 54 tan 2 A = 1 + 2 tan A 1 — tan3 A — CCS + COS _ .... sn A 1 + cos A 1 55. cot. UA= - —j- = —A - — =-----------------I—; , vers A sm A cosec A — cot A I 56. cot 2 A = cot3 A — 1 2 cot A 57. versJéA= ■ 14 vers A 1 — cos A 1 + VI — J4 vers A 2 + V2 (1 + cos A) 58. vers 2 A = 2 sin3 A „ , . . 1 — cos A 59. exsec ]4 A = -—— 60. exsec 2 A — (1 + cos A) + V)> (l cos A) tan3 A 1 — tan3 A 61. sin (A ± B) = sin A. cos li ± sin B. cos A 62. cos (A ± B) — cos A. cos B T sin A. sin B 63. sin A -f- sin B — 2 sin \4 (A -f- B) cos J4(A — B) 64. sin A — sin B = 2 cos \4 (A + C) sin 14 (A — B) 65. cos A cos B = 2 cos }4 (A + B) cos \4 (A — B) 66. cos B — cos A = 2 sin % (A -f- B) sin 14 (A — B) 67. sin3 A — sin3 B = cos3 B — cos3 A = sin (A + B) sin (A — B) 68. cos3 A — sin3 B = cos (A + B) cos {A — B) sin (A + 73) 69. tan A + tan B = cos A . cos B , . , _ sm (A — B) i0. tan A — tan B — - -,--.— - cos A . cos B100 NATURAL SINES AND COSINES 0 » 1 2° 3° | 4 0 / Sine I Cosin Sine Cosin Sine Cosin Sine Cosin 1 Sine |Cosin 0 .00000 One. .01745 .99985 .03190 .99939 .05234 .99863 .06976 .99756 1 .00029 One. .01774 .99984 .03519 .99938 .05263 .99861 .07005 .997*54 2 .00058 One. .01803 .99984 .03548 .99937 .05292 .99860 .070« .99752 3 .00037 One. .‘01832 .99983 .03577 .99936 .05321 .99858 .07063 .99750 4 .00116 One. .01802 .99983 .03606 .99935 .05350 .99857 .07092 .99748 5 .00145 One. .01891 .99982 .03635 .99934 .05379 .99855 .07121 .99746 6 .00175 One. .01920 .99982 .03G64 .99933 .05408 .99854 .07150 .99744 7 .00204 One. .01949 .99981 .03093 .99932 .05437 .99852 .07179 .99742 8 .00233 One. .01978 .99980 .03723 .99931 .05406 .99851 .07208 .99740 9 .00262 One. .02007 .99980 .03752 .99930 .05495 .99849 .07237 .99738 10 .00291 One. .02036 .99979 .03781 .99929 .05524 .99847 .07266 .99736 11 .00320 .99999 .02065 .99979 .03810 .99927 .05553 .99846' .07295 .99731 B .00349 .99999 .02094 .99978 .03839 .99926 .05582 .99844 ! .07324 .99731 13 .00378 .999991 1.02123 .99977 .03SG3 .99925 i.05G11 .99842 .07353 .99729 14 .00407 .99999 I.02152 .99977 .03897 .99924 .05640 .99841 .07382 .997*27 15 .00430 .99999 .021^1 .99976 .03926 .99923 .05669 .99839 .07411 .99725 16 .00465 .99999 1.02211 .99976 .03955 .99922 .05698 .99838 .07440 .99723 17 .00495 .99999 1.02240 .99975 .0398-1 .99921 .05727 .99836 .07469 .99721 18 .00524 .99999 |.02269 .99974 .04013 .99919 .05756 .99834 .07498 .99719 19 .00553 .99998 .02298 .99974 .04042 .99918! !.05785 .99833 .07527 .99716 20 .00582 .99998 .02327 .99973 .04071 .99917 .05814 .99831 .07556 .99714 21 .00611 .99998 1.02356 .99972 .04100 .99916 .05844 .99829 .07585 .99712 oo .00610 #99998. .02385 .99972 .01129 .99915 |.05873 .99827 .07614 .99710 23 .00669 .99998] .02414 .99971 .04159 .99913 .05902 .99826! .(■7643 .99708 24 .00698 .999981 .02443 .99970 .04188 .99912 .05931 .998241 .07672 .99705 25 .00727 .99997 .02472 .99969 .04217 .99911 1.05960 .99822! .07701 .99703 26 .00756 .99997 .02501 .99969 1.04246 .99910 .05989 .99821 ! .07730 .99701 27 .00785 .99997 .02530 .99968 1.04275 .99909 .06018 .99819| .07759 .99699 28 .00814 .99997 .02560 .99967 1.04304 .99907 |.06047 .99817 .07788 .99696 29 .00844 .999961 .02589 .99966 .0-13:« .99906 |.06076 .99815! .07817 .99094! 30 .00873 .99996 .02618 .99966 .04362 .99905 .06105 .99813 .07846 .990921 31 .00902 .99996 .02647 .99965 .04391 .99904 .06134 .99812' .07875 .99689 32 .00931 .99996 .02676 .99964 .0442Q .99902 .06163 .99810: .07904 .99(587 33 .00960 .999"5 .02705 .99963 .04449 .99901 .06192 .99808' .07933 .99685 34 .00989 .99995 .02734 .99963 .04478 .99900 .06221 .99806| .07962 .99683 35 .01018 .99995 .02763 .99962 .04507 .99898 .06250 .99804! .07991 .99680 36 .01017 .99995 1102792 .99961 .04536 .99897 .06279 .998031 .08020 .99678 37 .01076 .99994 .02821 .99960 .04565 .99896 .06308 .99801 ! .08049 .99076 38 .01105 .99994 .02850 .99959 .0159-1 .99894 .06337 .99799' .08078 .99073 39 .01134 .99994 .02879 .99959 .04623 .99893 .06366 .99797! .08107 .99671 40 .01104 .99993 .02908 .99958 .04653 .99892 .06395 .99795 .08136 .99668 41 .01193 .99993 .02938 .99957 .04682 .99890 |.06424 .99793' .08165 .99606 42 .01222 .99993 .02967 .99956 .04711 .99889 .06453 .99792! .08194 .99664 43 .01251 .99992 .02996 .99955 .04740 .99888 .06482 .99790 .08223 .99661 44 .01280 .99992 .03025 .99954 1.04709 .99886 I.06511 .99788' .08252 .99659 45 .01309 .99991 .03054 .99953 1.04798 .90885 1.06540 .99786 .08281 .99657 40 .01338 .99991 .03083 .99952 .04857 1.99883 !.06569 .99784 .08310 .99654 47 .01367 .99991 .03112 .99952 .04856 | .99882 1.06598 .99782 .08339 .99652 48 .01396 .99990 .03141 .99951 .0-1885 .99881 .06627 .99780 .08368 .99(549 49 .01425 .99990 .03170 .99950 .0491-1 .99879 1.06056 .99778 .08397 .99647 50 .01451 .99989 .03199 .99949 .04943 .99878 j.06085 .99776. .08426 .99644 51 .01483 .99989 .03228 .00948 .04972 .99876 .06714 .99774| .0R455 .99612 flfi .01513 .99989 .03257 .99947 .05001 .99875 .06743 .99772 .08484 .99(539 53 .01512 .99988 BUM .99916 .05030 .99873 .00773 .99770 .08513 .99637 51 .01571 .99988 .03316 .99915 .05059 .99872 .06802 .99768 .08542 .99635 55 .01600 .99987 .03315 .99944 .05088 .99870 .06831 .99766 .08571 .99(532 56 .01029 '.99987 .03374 .99943 .05117 .99869 .06800 .99704 .08000 .99(530 57 .01658 .99980 .03403 .90942 .05146 .99807 .06889 .99762; .08629 .99(527 58 .01687 .99986 .03432 .99941 .05175 .99866 .06918 .99760) .08658 .99(525 59 .01716 .99985 .03461 .99940 .052Û5 .99864 .06947 .99758 .08087 .99(522 CO .01745 .99985 .03490 .99939 .05234 .99863 .06976 .99756' .08716 .99619 / Cosin Sine Cosin Sine Cosin Sine I Cosin Sine Cosin Sine - 89° 00 00 o 87® • 1 86° 85® GO 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 31 30 29 28 27 26 25 24 23 22 21 20 15 14 13 n ii 10 9 8 7 6 5 4 3 0 1 0NATURAL SINES AND COSINES 101 1 5 0 6 O 7 » 8 > 9 > Sine ! Cosin 1 Sine I Cosin ; Sine I Cosin Sine I Cosin Sine 1 Cosin / 0 1 .08716 .99619t .10453] .99452 .12187 .99255 .13917 .99027 .15643 ,98769 60 1 .08745| .99617j [.10482 .99449 .12216 .99251 .13946 .99023 .15672 .98764! 59 2 ,08774 .99614! 1.10511 .99446 .12245 .99248, .13975 .99019 .15701 .98760J 58 3 .088031 .99612! .10540 .99443 .12274 .99244: .14001 .99015 .15730 .987551 57 4 1 ,0883l| .99609! .10569 .99440 .I2302I .99210 .140:331 .99011 .15758! .98751i 56 5 ! .08860 .*99607 .10597 .99437 .123311 .99237! .14061 .99006 .15787] .98740! 55 6 .08889 .99604! .10626 .99434 ! .12360' .99233! .14090 .99002 .1581G .98741 ! r>4 r* 4 1 .08918 99G02, .10655 .99431 y .12389j .99230 .14119 .98998 .158451 .98737] 53 8 1 .08947 .995991 .10084 .99423 .12418; .99226 .14148! .98994 .15873Ì .98732 52 9 .08976 .995961 .10713 .99124 1 .12447 .99222! .14177i .98990 .15902 .98728 51 1° .09005 .99594 .10742 .99421: .12476 .992191 .14205 .98986 .15931 .98723 50 11 . 00034' .99591 ! .10771 .99418 .12504 .99215* .142341 .98982 .15959 .98718] 49 12 .09063 .995881 .108001 .99415 .12533 .99211 .14263 .98978 .15988 .98714 48 13 .09092 .99586; .10829 .99112 .12562] .99208| .14292' .98973 .16017 .98709, 47 14 .09121 .99583 ! .10858; .99409 .125911 .99204' .14320 .98969 !.16046 .98704 40 15 ! .09150 .99580| .10387] .99406 .12620 .99200 .14349 .98965 .16074 .98700 45 16 : .09179 .99578] .10916, .99402 .12649! .99197! .14378 .98961 !.16103 .98695 44 17 .09208 .99575 1.10945 .99399 ; .12678 .99193 .14407 .98957 .16132 .98690 43 18 , .09237 .99572 !.109731 .99396' .12706 .99189| .14436 .98953 .16160 .98686 42 19 .09266 .99570j :.11002: .99393 ! .127:351 .99186 .14464 .98948 .16189 .98681 41 20 .09295 .99567 j .11031 .99390 .12764 .99182: .14493 .98944 !.16218 .98670 40 21 .09324' .99564 !.11060'.99336 .12793 .99178* .14523 .98940] 1.16246 .98671 39 22 .09333 .995621 .11039 .99383 .128*2 .99175, .14551 .989361 .16275 .98667 38 23 .09382 .99559] j .11118 .99330 .12851 .99171' .14580 .989311 .16304 .98662 37 24 .09411 .99556| j.11147 .99377 j .12880 .99167] .14608 .98927 .16333 .98657 36 25 .09440 .905531 !.11176 .993741 .12908 .99163 .14637 .98923 .16361 .98652 35 26 .09469 .995511 1.11205 .99370 .12937 .99160] .14666 .98919 .16390 .98648 34 27 .09498 .995481 .11234 .99367 .12966 .99156: .14695 .98914] .16419 .98643 33 28 .09527 .995451 .11263 .99364 .12995 .99152! .14723 .98910 .16447 .986:18 32 29 .09556 .99542| .11291 .99360 .13024 .99148; .14752 .98906] .16476 .98633 31 30 .09585 .99540 .11320 .99357 .13053 .99144. .14781 .98902 .16505 .98629 30 31 .09614 .99537 1.11349 .99354 .13081 .99141 .14810 .98897 ].ìesas .98624 29 32 .03642 .99534 1.11378 .99351 : .13110 .99137] .14888 .98893 .16562 .98619 28 33 .09671 .99531 1.31407 .99347, .13139 .99133! .14867 .98889 1.16591 .98614 27 34 .00700 .99528 .11436 .99344 .13168 .991231 .14896 .98884 1.16620 .98009 26 35 .09729 .99526 .11465 .99341! .13197 .99125! .14925 .98880 1.16648 .98604 25 36 .09758 .99523 !.11494 .99337 .13226 .99122! .14954 .98876 .16677 .98600 24 37 .09787 .99520 .11523 .99334 .13254 .99118' .14982 .98871 .16706 .98595 23 38 .09816 .99517 !.11552 .99331 .13283 .99114! .15011 .98867 ;.16734 .98590 22 39 .09815 .99514 1.11580 .99327! .13312 .99110 ,.15010 .98863 .16763 .98585 21 40 .098 <4 .99511 j.11609 .99324 .13341 .99106: .15069 .98858 j.16792 .98580 20 41 .09903 .99508 .11638 .99320 .13370 .99102 .15097 .98854 1.16820 .98575 19 42 .099:12 .99506 .11667 .99317! .13399 .99093 .15126 .98849 !.16849 .98570 18 43 .099G1 .99503 !.11696 .99314 .13427 .99094; .15155 .98845 l.16878 .98565 17 44 .09990 .99500 I.11725 .99310 .1345G .99091i .15184 .98841 !.16906 .98561 16 45 .10019 .99497 !.11754 *99307 .13485 .99087! .15212 .98836 !.16935 .98556 15 46 .10018 .99494 .11783 .99:103 .13514 .99083! .15241 .98832 !.16964 .98551 14 47 .10077 .99491 1.11812 .99300! .ia>43 .99079' .15270 .98827 .16992 .98546 H 48 .10106 .99488 1.11840 .992971 .13572 .99075 .15.299 .98823 .17021 1.98541 112 49 .10135 .99485 i .11869 .99293 .13000 .99071 .15327 .98818 !.17050 .98536 ■ 50 .10164 .99482 !.11898 .99290, .13029 .99067 .15356 .98814 .17078 .98531 10 51 .10192 [.99479 !.11927 .99286* .13658 .99063 .15385 .98809 .17107 .98526 ! 9 52 .10221 .99476 .11956 .99283 .13687 .99059] .15414 .98805 .17136 i.98521 i 8 53 .10250 .99473 .li935 .99279 .13716 .99055 .15442 .98800 ;.17164 1.98516 7 54 .10279 .99470 ,.12014 .99276 .1374-4 .99051 .15471 .98796 .17193 ].98511 6 55 .10308 .99467 .12043 1.99272 .13773 .99047 .15500 .98791 1.17222 .98506 5 56 .10387 .99401 .12071 1.99269! .13802 .99043 .15529 .98787 i.17250 ! .98501 4- 57 10306 j.99401 .12100 1.99205 .18831 .99039 .15557 .98782 i.17279 1-.98490 3 58 .10395 .09458 .12129 '.99262 .1.3860 .99035! .15586 .98778 !.17308 .98491 2 59 .10424 .99455 .12158 1.99258 ! .1-3889 .99031 .15615 .98773 i .17a°>6 .98180 1 GO .10453 .99452 .12187 .99255 !.13917 .99027] .15643 .98769 :.17365 ‘.98481 0 / Cosin I Sine Cosinj Sine ! Cosin j Sine Cosin Sine I Cosin Sine / 1 00 ►f* 0 1 83* II 82° 01° 00 0 0 102 NATURAL SINES AND COSINES 10° 11° 12° 13 ° 14® 1 / Sine Cosin Sine Cosin Sine Cosin ■ Sine Cosin | Sine Cosin 1 "o .17305 798481 .10081 .98103, .20791 .978151 .22495 .974371 .24192 .970301 00 1 .17393 .98470 .19109 .98157 .20820 .97809! .22523 .97430, .24220 .970231 59 2 .17422 .98471 .19138 .981521 .20848 .97803! 2255° .97424 .24249 .97015! 58 3 .17451 .98406 .19167 .98146' .20877 .977971 .22580 .97417 .24277 .97008! 57 4 .17479 .98401 .19195 .98140 .20905 .977911 .22008 .97411| .24:405 .970011 50 5 .17508 .98155 .19224 .98135 .209:43 .97784! .22037 .97404| ! .24:433 .96991 55 0 .17537 .98150 .19252 .98129 .20902 .97778 .22005 4)7398 ! .24362 .96987 54 7 .17505 .98145 .19281 .98124' .20990 .97772 .22093 .97391| !.24390 .96980 53 8 .17594 .98440 .19:309 .98118 .21019 .97760 .22722 .97384! 1 .24418 .96973 52 9 .17023 .93435 .19338 .98112 .21047 .97700 .22750 .97378 , .24440 .90906 51 10 .17051 .98430 .19366 .98107! .21070 .977*4 .22778 .973711 i .24474 .90959 50 11 .17080 .98425 .19395 .98101 .21104 .97748 .22807 .973651 !.24503 .96952 49 12 .17708 .98120 .19423 .98090 .21132 .97742 .22835 .97358 .24531 .90945 48 li .17737 .98414 .19152 .93090 .21161 .97735 .22803 .97351| 1.24559 .90937 47 14 .17766 .98409 .19181 .98084 .21189 .97729 .22892 .97345 1.24587 .96930 40 15 .17794 .98404 .19509 .98070 .21218 .97723 .22920 .97338 ;.24015 .96923 45 IS .17823 .98399 .19538 .98073 .21216 .97717 .22948 .973311 .24044 .96910 44 17 .17852 .98394 .19500 .98067 .21275 .97711 .22977 .97325; 1 .24072 .90909 43 18 .17880 .98389 .19595 .98001 .21303 .97705 .23005 .97318 ! .24700 .96902 42 19 .17909 .98383 .19023 .98056 .21331 .97098 .23033 .97311 1 .24728 .96894 41 20 .17937 .98378 .19052 .93050 .21300 .97092, .23002 .97304 •24750 .90887 40 21 .17960 .98373 .19680 .98044 .21388 .97686 .23090 .97298 .24784 .96880 39 o> .17995 .98308 .19709 .98039 .21417 .97080 .23118 .97291 ! ! .24813 .96873 38 21 .18023 .98362 .19737 .980331 .21415 .97073 .23140 .97284 1 .24841 .968G0 37 21 .18052 .98357 .19700 .930271 .21174 .97067 .214175 .97278 1 .24809 .90858 30 25 .18081 .98352 .19794 .930211 .21502 .97GG1 .23203 .97271 1 .24897 .90851 35 26 .18109 .98347 .19823 .930161 .21530 .97G55 .23231 .97204! ! .24925 .96844 31 27 .18138 .98341 .19851 .930101 .21559 .97048 .2321)0 .97257 ! .24954 .96837 33 23 .18100 .98336 .19880 .930041 .21537 .97642 .23283 .97251 1 .24982 .96829 32 29 .18195 .98331 .19908 .97903! .21610 .97030 .23310 .97244 .25010 .90822 31 30 .18224 .98325 .19937 .97992 .21044 .97G30 .23345 .97237 .25038 .90815 30 31 .18252 .98320 .19965 .97987 .21072 .97023' .23373 .97230 .25066 .96807 29 32 .18281 .98315 .19994 .97981 .21701 .970171 .23401 .97223 .25191 .90800 28 33 .18309 .98310 .20022 .97975| .21729 .97011 .23429 .97217 .25122 .96793 27 31 .18338 .98304 .20051 .979091 .21753 .97004 .23458 .97210 .25151 .96786 20 35 .18307 .98299 .20079 .97903 .21786 .97593 .23486 .97203 .25179 .9G778 25 30 .18395 .98294 .20108 .97958! .21814 .97592! .23514 .97196 ! .25207 .90771 24 37 .18424 .98288 1.20130 .97952| .21843 .975S5 .23542 .97189 .25235 .90764 23 38 .18452 .98283 |.20105 .97940 .21871 .97579; .23571 .97182 ! .25203 .90750 o.» 39 .18481 .98277 .20193 .97940! .21899 .97573! .23599 .97170I !.25291 .90749 21 40 .18509 .98272 .20222 .97934 .21923 .97506 ! .23027 .97109! .25320 .90742 20 41 .18538 .98267 .20250 .97928! .21950 .97560 .23050 .97102! .25348 .96734 19 42 .18507 .93201 1.20279 .97922! .21985 .97553 ! .23084 .97155, .25370 .90727 38 43 .18595 .93250 |.20307 .97910| .22013 .97547 .23712 .97148: .25404 .90719 17 41 .18G24 .98250 I.20336 .979101 .22011 .97541 .23740 .97141! .25432 .90712 10 45 .18052 .93215 !.20301 .97905! !.22070 .97534. .23709 .97134, .25400 .90705 15 1G .18081 .93210 .20393 .97899! .22093 .97528! .23797 .97127 1 .25488 .96697 14 47 .18)710 .93234 |.20121 .97893| .22120 .975211 .23825 .97120, .25510 .96690 13 43 . mm .93229 |.20450 .97887| .22155 .97515! .23853 .97113 .25545 .96682 12 49 .18707 .93223 1.20178 .978811 .22183 .97508! .2:4882 .97100 .25573 .9GG75 11 50 .18795 .98218 I.20507 .97875| !.22212 .97502 ! .23910 .97100 .25001 .90067 10 51 .18824 .98212 1.20535 .978691 .22240 .97490 !.23938 .97093 .25629 .96660 9 52 .18852 .93207 .20503 .978G3 .22208 .97189; .239GG .97080i .25657 .90053 8 53 .18881 .93201 .20592 .97857 |.22297 1.974831 .23995 .97079| .25085 .90045 7 5-1 .18910 .98193 1.20620 .97851 .22325 i.97470 .24023 .97072 .25713 .9GG38 0 55 .18938 .93190 [.20649 .97845 .22353 97470 .24051 .97005 .25741 .90030 5 5G .18967 .93185 |.20677 .97839 .22:482 .97403 .24079 .97058 .25709 .90023 4 57 .18995 .98179 .20706 .97833 .22110 '.97457 .24108 .97051 .25798 .96615 3 53 .19024 .98174 .20734 .97827 .224:48 i.97450 ! .24130 :97044 .25820 .90008 0 59 .19052 .93108 |.20763 .97821 .22107 1.97444; .24164 .97037 .25854 .90000 1 GO .19081 .98103 .20791 .97815 |.22495 .97437 .24192 .97030 .25882 .90593 0 Cosin Sine Cosin 1 Sine 1 o 1 O m j ! Sine Cosin Sine Cosin Sine / 79° CO o 77° o O 75° NATURAL SINES AND COSINE«. .103 1 15° 16° I ) 7° 18° 19° Sine | Cosin Sine | Cosin Sine Cosin j Sine Cosin | Sine Cosin 0 .258821.96593 .275641.96126 ' .29237 .95630 !.30902 .95106 !.32557 .94552 60 1 .25910'.96585 .27592 .96118 1.29265 .95622 :.30929 .95097 .32584 .94542 59 2 .25938;.96578 !.27620 .96110 ).29293 .95613 .30957 .95088 ;.32612 .04533 58 3 .25966 .96570 !.27648 .96102 !.29321 .95605 .30985 .95079 i.82639 .94523 57 4 .25994 .96562 j .27676 .96094 !.29348 1.95596 .31012 .95070 1.32667 .94514 56 5 .26022 .1H)555 1.27704 .96086 :.29376 .95588 .31040 .95061 HI .94504 55 C 1.26050 .96547 !.27731 .96078 !.29404 .95579 .31068 .95052 1.32722 .94495 54 7 1.26079,. 96540 ).27759 *96070 .29432 .95571 .31095 .95043 .32749 .94485 53 8 .26107 .96532 .27787 .96062 ,.29460 .95562 .31123 .95033 .32777 .94476 52 9 I.26135 .96524 !.27815 .96054 .29487 .95554 .31151 .95024 ;.32804 .94466 51 10 .26163,.96517 !.27843 .96046 |.29515 .95545 .31178 .95015 .82832 .94457 50 11 .261911.96509 .278711.96037 .29543 .95536 '.31206 .95006 .32859 .94447 49 12 .26219 .96502 .27899' .96029 I.29571 .95528 j.31233 .94997 .32887 .94438 48 13 1.26247 .96494 | .279271.96021 |.29599 .95519 |.31261 .94988 .32914 .94428 47 14 |.26275i.96486 .27955 .96013 !.29626 .95511 .31289 .94979 .32942 .94418 46 15 .26303;. 96479 |.27983 .96005 .29654 .95502 .31316 .94970 .32969 .94409 45 36 .26331 .96471 1.280111.95997 .29682 .95493 .31344 .94961 .32997 .94399 44 17 .26359i.96463 1.280391.95989 .29710 .95485 .31372 .04952 ! .33024 .94390 43 18 .263871.96450 .280671.95981 .29737 .95476 1.31399 .94943 .33051 .94380 42 19 .26415 .96448 |.28095 .95972 !.29765 .95467 !.31427 .94933 .83079 .94370 41 20 .26443|.96440 .281231.95964 j.29793 .95459 .31454 .94924 .33106 .94361 40 21 .26471 .96433 .281501.95956 .29821 .95450 .31482 .94915 .83134 .94351 39 9‘> .26500 '.96425! .28178!.95948 .29849 .95441 .31510 .94006 .asioi .94342 38 23 .265281.96417 .28206|.95940 .29876 .95433 .31537 .94897 .33189 .94332 37 24 .265561.964101 .28234 .95931 !.29904 .95424 .31565 .94888 .33216 .94322 36 25 .265841.964021 .28262! .95923 |.29932 .95415 .31593 .94878 .33244 .94313 35 26 .266121.963941 .28290 ! .95915 .29960 .95407 .31620 .94869 .33271 '.94303 34 27 .26640|.96386| .283181.95907 .29987 .95398 .31648 .94800 .33298 .94293 33 28 .26668! .96379) .28346 .95898 .30015 .95389 1.31675 .94851 .33326 .94281 32 29 .26696).96371) .283741.95890 .30043 .95380 !.31703 .94842 |.33353 .94274 31 30 .26724 .96363 .28402:.95882 1.30071 .95372, .31730 .94832 .33381 .94204 SO 31 .26752'.9G355I .284291.95874' 1.30098 .95363 .31758 .94823 .33408 .94254 29 32 ,26780|.96347! .284571.95805 !.30126 .95354 .31786 .94814 .33436 .94245 28 33 .26808!.9G:340| .284851.95857 !.30154 .95345 .31813 .94805 .33463 .94235 27 34 .26836|.90332i .28513 .95849 .30182 .95337! .31841 .94795 .33490 .94225 26 35 .268641.96324! .285411.95841 1.30209 .953281 .31868 .94786 .33518 .94215 25 36 .208921.90316! .285091.95832 1.30237 .95319! .3189G .94777 •33545 .94206 24 37 .26920 .96308 .28597|.95824 |.30265 .95310; .31923 .94768 .83573 .94196 23 38 .26943!.963011 .28625|.95810 .30292 .95301 .31951 .94758 .33600 .94186 22 39 .26976).96293) .28652 .95807 |.30320 .952C3I I.31979 .94749 .33G27 .9417G 21 40 .270041.90285 .28680,.95799 .30348 .95284! .32006 .94740 .33655 .94167 20 41 .27032 '.902771 .28708 .95791 1.30376 .95275 .32034 .94730 .33682 .94157 19 42 .27060 .96269; .287301.95782 1.30403 .95206 |.32061 .94721 .a3710 .94147 13 43 .270881.96261) .28764!.95774| 1.304311 .95257 ; .32089 .94712 .a3737 .94137 17 44 .271161.902531 .287921.957661 1.30459 .95248! .32116 .94702 .33764 .94127 16 45 .27144 .96246' .28820 .95757 .30486! .95240! .3214-1 .94693 ,a3792 .94118 15 46 .271721.90238! .288471.957491 .30514i .95231| .32171 .94684 .33819 .94108 14 47 .27200|.96230| .288751.957401 .30542] .95222, .32199 .94674 .33846 .94098 13 43 .27228 .90222! .28903|.95732| .30570 .952131 .32227 .94665 .33874 .94088 12 49 .27256). 96214 .28931|.957241 .30597 .95204' .32254 .94656 .33901 .94078 11 50 .27284 .962061 .289591.95715 .30625 .95195! .32282 .94646 .33929 .94068 10 51 .273121.961981 .28987|.95707| .30653 .95186! .32309 .94637 .33056 .94058 9 52 .27340!.96190 .29015 .95698 .30080 .95177; .32337 .94627 .33983, .94049 8 53 .27368j.96182) .290421.95690 .80708 .951681 .32364 .94018 .3401l| .94039 7 54 .27396' .96174 .290701.95681 .30736 .95159 .32392 .94609 .34038| .94029 6 55 .27424 .961661 .29098!. 95673 .30763 .95150 .32419 .94599 .S40G51 .94019 5 56 .274521.96158 .29126!.95664 .30791 .95142 .32447 .94590 ,84uy3 .94009 4 57 .27480|.96150 .29154'.95656 .30819 .95133 | .32474 .93580 .34120 .93999 3 58 .27508.96142 .29182'.95647 .30846 .95121 ! .32502 .94571 .34147 93989 2 59 .27536 .96134 .292091.95639 .30874 95115 .32529 .94561 .34175 93979 1 60 .27564 . 96126 .292371.95630 .30902 95106 .32557 .94552 | .34202 939G9 0 / Cosint Sine Cosin| Sine C^sin Sinei Cosin Sine —J Cosin Sine 74° CO o *2 Q 71 ° II 70 o 104 NATURAL SINES AND COSINES / 20° 21° 22° to CO 0 24" Sine Cosin Sine Cosin Sine Cosin Sine I Cosin ! Sine Ì Cosin / 0 .34202 .93909 .35837 .93358 .374011 .92718 .39073- .920501 .40674 .91355' R 1 .34229 .93959 .35804 .93348 .37488 .92707 .39100] .92039 .40700 .91343 59 2 .34257 .939491 .35891 .93337 .37515 .92097 .39127j .92028 .40727 .91331 58 3 .34284 .93939 .359181 .93327 .37542 .92080 .39153 .92016S . 40753 ; .91319 57 - 4 .34311 .93929 .359451 .9:1316 .37509 .92075] .39180 .92005] .40780 .91307 56- 5 .34339 .93919 .35973 .93306 .87595 .92004 .89207 .91994 .40806! .91295 55 C .34300 .93909 .30000' .93295! .37022 .92053 .39234| .91982 .40833; .91283 54 7 .84393 .93899 .30027 .93285 .37049 .92042 .39260| .91971 .40800: .91272: 53 8 .34421 .93889 .3005-1 .93274 .37076 .92031 .89287 .91959 .40886; .91260 52 9 .34448 .93879 .30081 .93204 .87703 .92020 .39314 .919481 .40913! .91248 51 10 .34475 .93869 .30108, .93253 .37730 .92009 .89341 .919301 .40939 .91236, 50 11 .34503 .93859 .36135 .93243 .37757 .92598 .89367 .91925 .40966 .91224' 49 12 .34530 .93849 .30102 .93232 .37784 .92587 .89394 .91914| .40992 .91212 48 13 .34557 .93839 .30190 .93222 .37811 .92570 .39421 .91902 .41019 .91200 47 14 .34584 .93829 .3G217 .93211 .37838 .92505 .39448 .918911 .41045 .91188 46 15 .34012 .93819 .36244 .93201 .37805 .92554 .39474 .918791 .41072 .91176 45 1G .34039 .93809 .30271 .93190 .37892 .92543 .39501 .91868] .41098] .91104 44 17 .34000 .93799 .30298 .93180 .37919 .92532 .39528 .91856; .41125 .91152 43 18 .84091 .93789 .3G325 .931G9 .37940 .02521 .89555 .918451 .41151 .91140 42 W .34721 .93779 .36352 .93159 .37973 .92510 .39581 .91833, .41178 .91128 41 20 .34748 .93709 .30379 .93148 .87999 .92499 .39608 .91822 .41204 .91110 40 21 .34775 .93759] .80406 .93137 .38026 .92488 K39635 .91810' .41231 .91104 39 22 .34803 .93748j .30434 .93127 .88053 .92477 1.39001 .91799 1.41257 .91092 38 23 .34830 .93738 |.30401 .93116 .38080 .92400 !.39688 .91787, I.41284 .91080 37 24 .34857 .93728i .30488 .93100 .38107 .92-155 1.39715 .91775 .41310 .91008 36 25 .34884 .93718ì .80515 .93095 .38134 .92444 |.39741 .91764 .41337 .91056 35 2G .34912 .93708! !.36542 .93084 !.38101 .92432 .39708 .917521 ].41363 .91044 34 27 .34939 .93G98 ].30509 1.93074 i .38188 ].92421 .39795 .91741 .41390 .91032 33 28 .34900 .93688 !.30590 .93003 !.38215 1.92410 .39822 .91729] !.41416 .91020 32 29 .34993 .936771 !.36623 .93052' ! .:.8241 .92399 .39848 .91T18 !.41443 .91008 31 30 ..35021 .930071 .30050 ,.93042 1.38208 .92388 .39875 .91700 .41409 .90990 30 31 .35048 .9305?1 .36077 .93031 .38295 .92377 .39902 .91694 .41496 .90984 29 32 .35075 .93047 |.30701 .93020 .38322 .92300 .39928 .91083 .41522 .90972 28 33 .35102 .930371 1.36731 .93010 1.38349 1.92355 .89955 .91671 .41549 .90900 27 34 .35130 .93020 .30758 .92999 I.38370 1.92343 .39982 .91600 .41575 .90948 26 35. .35157 .93610' .36785 ! .92988 .38403 1.92332 .40008 .91648 .41002 .90936 25 30 .35184 .93000 .36812 ! .92978 .38-130 1.92321 .40035 ] .91630 1 .41028 .90924 24 37 .35211 93590 .30839 .92967 .38450 ‘.92310 .40002 1.91625 1.41655 .90911 23 88 .35239 .93585 .30807 .92956 1.38483 .92299 1.40088 ; .91613 !.41081 .90899 0,) 39 .35206 .93575 ;36894 .92945 1 .88510 .92287 .40115 .91G01 .41707 .90887 21 40 .35293 .93505 .30921 .92935 .38537 .92276 .40141 .91590 ,.41734 .90875 20 41 .35320 .935551 .36948 .92924 1.38504 .92265 .40168 .91578 .41760 .90863 19 42 .35347 .93544 .3097'5 .92913 .88591 .92254 .40195 .91566 .41787 .90851 18 43 .35375 .93534 .37002 .92902 .38017 .92243 .40221 ! .91555 I.41813 .90839 17 41 .35402 .93524 .37029 .92892 .38044 .92231 .40248 .91543 .41840 .90820 1 16 45 .35429 .93514 .37056 .92881 .38071 .922-20 .40275 .91531 |.41800 .90814 15 40 .35450 .93503 .37083 .92870 .38098 .92209 .40301 .91519 !.41892 .90802 14 47 .35484 .93493 .37110 .92859 .88723 .92198 .40328 .91508 .41919 .90790 13 48 .35511 .934S3 .37137 .92S49| .38752 .92180 .40355 .91490 ].41945 .90778 12 49 .35538 .93472 .37101 .92838 136778 .92175 .40381 1.91484 j .41972 .90700 11 50 .35505 .93402, .37191 .92827 .88803 .92104 .40408 ‘ .91472 .41998 .90753 10 51 .35592 .93452’ .87218 .92816 .38832 .92152 .40434 ! .91461 .42024 1.90741 9 52 .35019 .93441 .37245 .92805 .38859 .92141 .40401 i.91449 1.42051 .90729 8 53 .35047 .93431 .37272 .92794 .38880 .92130] .40488 .91437 .42077 .90717 7 51 .35074 .93420 .37299 .92784 .38912 .92119 .40514 .91425 .42104 .90704 6 55 .35701 .93410 .87320 .92773 .38939 .92107 .40541 .914141 .42130 ;90092 5 50 .35728 .93400 .87353 .92762 .38900 .92090 .40507 .91402 .42150 .90080 4 57 .35755 .93:389 .37380 .92751 .38993 .92085 .40594 .91390 .42183 .90008 3 58 .35782 .93379 .37407 .92740 !.39020 .92073 !.40021 .91378 .42209 .90655 2 59 .35810 .93308 .374:14 .92729 I.39040 .92002' .40047 .91300 |.42235 .90643 1 00 .35837 .93358' .37401 .92718 .39073 .920501 .40074 .91355 j.42262 .90631 0 / Cosili t Sine Cosin Sine | Cosin Sine Cosin Sine Cosin | Sine / 69° 68° 67° 660 65° NATURAL SINES AND COSINES 105 25° 26° 27° 0 CO 01 29° Sine Cosin Sine Cosin Sine Cosin Sine Cosin 1 Sine Cosin 0 .42262 .90631 j .43837 .89879 .45399 .89101 .46947 .88295! .48481 .87462 60 1 .42288 .90618 1.43803 .89807 .45425 .89087 .46973 .882811 .48506 .87448 59 o .42315 .90606 .43889 .89854 .45451 .89074 .46999 .882671 .48532 .87434 53 3 .42341, .90594 .43916 .89841 .45-177 .89061 .47024 .882541 .48557 .87420 57 * 4 .42367 .90582 .43942 .89828 .45503 .89048 .47050 .88240! .485S3 .87406 50 5 .42394 .90569 .43968 .89816 .45529 .89035 .47070 .88226! .48608 .87391 55 6 .42420 .90557 .43994 .89803 .45554 .89021 .47101 .88213'; .48634 .87377 54 7 .42446 .90545 1.44020 .89790 .45580 .89008 .47127 .88199 .48059 .87363 53 8 .42473 .90532 .44046 .89777 .45606 .88995 .47153 .88185! .48684 .87349 52 9 .42499 .90520 1.44072 .89764 .45632 .88981 .47178 .881721 .48710 .87335 51 10 .42525 .90507 .44098 .89752 .45658 .88968 .47204 .88158 ..48735 .87321 50 11 .42552 .90495 ! .44124 .89739 .45684 .88955 .47229 .88144 .48761 .87306 49 12 42578 .90183 !.44151 .89726 .45710 .88942 .47255 .881301 .48786 .87292 48 13 .42604 .90470 .44177 .897131 .45736 .88928 .47281 .881171 .48811 .87278 47 14 .42631 .90458 .44203 .897001 .45762 .88915 .47306 .881031 .48837 .87204 46 15 .42657 .90116 .44229 .896871 .45787 .88902 .47332 .880S9 .48862 .87250 45 1G .42083 .90433 .44255 .890741 .45813 .88888 .47358 .88075 .488S3 .87235 44 17 .42709 .90421) ! .44281 .896621 .45839 .88875 .47383 .88062 .48913 .87221 43 18 .42736 .904081 j .44307 .89649 .4.5865 .88862 .47409 .880481 .48938 .87207 42 19 .42762 .90396 I .44333 .89636 .45891 .8S848 .47434 .88034| 48964 .87193 41 20 .42788 .90383 1.44359 .89G23 .45917 .88835 .47460 .88020 .48989 .87178 40 21 .42815 .90371 .44385 .89610 .45942 .88822 .47486 .88006 .49014 .87164 39 oo .42841 .90358 .44411 .895971 .45968 .88S08 .47511 .87993 .49040 .87150 33 23 .42367 .90346 i .44437 .895841 .45994 .83795 .47537 .87979 .49065 .87130 37 24 .42894 .90334 1 .44464 .89571 ! .46020 .8S782 .475G2 .87905) .49090 .87121 06 25 .42920 .90321 !.44490 .895581 .46016 .88768 .47588 .87951 .49116 .87107 35 26 .42946 .90309 .44516 .89545 .4G072 .88755 .47614 .87937 .49141 .87093 34 27 .42972 .90296 .44542 .89532 .46097 .83741 .47639 .87923 .49166 .87079 33 23 .42999 .90284 .44568 .89519 .40123 .88728 .47665 .879091 .49192 .87064 32 23 .43025 .93271 .44594 .89506 .40149 .83715 .47090 .87896! .49217 .87050 31 SO .43051 .90259 .44620 .89493 .46175 .88701 .4771G .878821 .49212 .87036 30 31 .43077 .90246 .44646 .89480 .46201 .88688 .47741 .87868 .49268 .87021 29 32 .43104 .90233 .4-1072 .89467 .46226 .83674 .47767 .87854 .49293 .87007 23 33 .43130 .90221 .44698 .89454 .4G252 .8SG61 .47793 .87840 .49318 .86993 27 34 .43156 .902031 .44724 .89441 .4G278 .88047 .47818 .87826 .49344 .8G978 23 35 .43182 .90196 .44750 .89423 .40304 .8S034 .47844 .87812 .49369 .86964 25 33 .43209 .90183 .44776 .89415 .40330 .88620 .47809 .87798 .49394 .86949 21 Q~ O i .43235 .901711 .44802 .89402 .46355 .88607 .47895 .87784 .49419 .86935 23 33 .43261 .90158! .44828 .89389 .46381 .88593 .47920 .87770 .49445 .86921 22 39 .43287 .901461 .4-4854 .89376 .4G-107 .88580 .47910 .87756 .49470 .86906 21 40 .43313 .90133! .44880 .89363 .46433 .88566 .47971 .87743 .49495 .86892 20 41 .43340 .9OI20I .44906 .89350 .46458 .88553 .47997 .87729 .49521 .86878 19 43 .43366 .90108! .44932 .89337 .4G4S4 .83539 .48022 .87715 .49546 .8G863 18 43 .43392 .900951 .44958 .89324 .46510 .88526 .48048 .87701 .49571 .86849 17 44 .43418 .900821 .44934 .89311 .46536 .88512 .48073 .87687 .49596 .8G834 16 45 .4:3445 .90070 .45010 .89298 .4G5G1 .88499 .48099 .87673 .49022 .80820 15 4G .43471 .900571 .45036 .89285 .46587 .88485 .48124 .87659 .49647 .86805 14 47 .43497 .90045 .45002 .89272 .46613 .88472 .48150 .87645 .49672 .86791 13 43 .4:3523 .90032| .45038 .89259 .46639 .88458 .48175 .87G31 .49697 .86777 12 49 .43549 .90019! .45114 .89215 j .4GG64 .88445 .48201 .87617 .49723 .86762 11 50 .43575 .90007, .45140 .89232 .4C690 .88431 .48226 .87603 .49748 .86748 10 51 .43602 .89994 .45166 .89219 !.46716 .88417 .48252 .87589 .49773 .86733 9 K > .43628 .89981 .45192 .89206 .46742 .88404 .48277 .87575 1.49798 .86719 8 53 .43654 .899G8 .45218 .89193 .46767 .88390 .48303 .87561 .49824 .86704 7 54 .43680 .89956 .45243 .89180 .46793 .88377 .48328 .87546 .49849 .86690 6 55 .43706 .89943 .45209 .89167 1.46819 .88363 .48354 .87532 .49874 .86675 5 56 .43733 .899301 .45295 .89153 !.46844 .88349! .48379 .87518 .49899 .86661 4 57 .43759 .89918 .45321 .89140 ! .40870 .883361 .48405 .87504 .49924 .86646 3 53 .43785 .89905 .45347 .89127 Ì .46896 .883221 .48430 .87490 .49950 .86632 2 59 .43811 .89892 .45373 .89114 .46921 .88308! .48456 .87476 .49975 .86617 1 GO .43837 .898791 .45399 .89101 .40947 .88295 .48481 .8V4G2 .50000 .86603 0 / Cosin Sine 1 Cosin Sine Cosin Sine I Cosin Sine Cosin Sine t 64° 63° 62° 61° 60° 106 NATURAL SINES AND COSINES CO o o 31° 32° 33° 1 34® Sine Cosin Sine Cosin 1 Sine i Cosin Sine ! Cosin ! Sine 1 Cosin f 0 .50000 .86603[ .51504 .85717 .52992 .84805 .54464 .83867 .55919! .82904 60 1 .50025 .86588 .51529 .85702 .53017, .84789 .54488 .83851 .55943| .82887 59 2 .50050 .86573 .51554 .85687 .53041! .84774 .54513 .83835 ; .55968 .82871 58 3 .50076 .86559 .51579 .85672 .53066 .84759 .54537i .83819! .55992 .82855 57 4 .50101 .86544 ,.51604 .85657 .53091i .84743 .54561 .83804 .56016 .82839 56 5 .50126 .86530 .51628 .85642 .531151 .84728 .54586! .83788 .56040 .82822 55 6 .50151 .86515 .51653 .85627 .53140 .84712 .54610 .83772 .56064 .82806 54 7 .50176 .86501 .51678 .85612 .531G4 .84G97 .54635! .83756 .56088 .82790 53 8 .50201 .86486 .51703 .85597 .53189 .84681 .54059' .83740 .56112 .82773 52 9 .50227 .86471 .51728 .85582 .53214 .84006 .54683 .83724 .56136 .82757 51 10 .50252 .86457 .51753 .85567 .53238 .84650 .54708' .837u8| .56160 .82741 50 11 .50277 .86442 .51778 .85551 .53263 .84635f .54732 .83692 .56184 .82734 49 12 .50302 .86427 .51803 .85536 .53288 .84619 .5475G .83076 ! .50208 .82708 48 13 .50327 .86413 1.51823 .85521 .53312 .84004 .54781 .83660! 1.56232 .82692 47 14 .50352 .86398 .51852 .85506 .53337 .84588 .54805 .83645! .56256 .82675 46 15 .50377 .86384 .51877 .85491 .533G1 .84573 .54829 .83629; .56280 .82659 45 16 .50103 .86369 .51902 .85476 .53386 .84557 .54854 .83613 !.56305 .82643 44 17 .50428 .80354 .51927 .85461 .53411 .81542 .54878 .83597 !.56329 .82626 43 18 .50453 .86340 1.51952 .85446 .534:55 .84536 .54902 .83581 1.56353 .82610 42 19 .50478, .80325 |.51977 .85431 .53400 .84511 .54927 .83565 .56377 .82593 41 20 .50503 .60310 .52002 .85416 .53484 .84495 .54951 .835491 .56401 .82577 40 21 .50528 86295 .52026 .85401 .53509 .84480 .54975 .83533 .56425 .82561 39 oo .50553 .86281 I.52051 . 853iS5 .53534 .84164 .54999 .83517 .56449 .82544 38 23 .50578 .86266 !.52076 .85370 .5:J558 .84448 .55024 .83501 .56473 .82528 37 24 .50603 .86251 1.52101 .85355 .53583 .84433 .55048 .83485 .56497 .82511 36 25 .50628 .86237 1.52126 .85340 .53607 .81417 .55072 .83469 1 .50521 .82495 35 26 .50654 .86222 .52151 .85325 .53032 .84402 .55097 .83453 j .56545 .82478 34 27 .50679 .86207 .52175 .85310 . 53656 .84386 .55121 .83437 I.56569 .82462 33 28 .50704 .86192 .52200 .a>294 .53681 .84370 .55145 .83421 .56593 .82446 32 29 .50729 .86178 .52225 .85279' .53705 .84355 .55169 .83405 .56617 .82429 31 30 .50754 .86163 .52250 .852641 .53730 .84339 .55194 .83389 .56641 .82413 30 31 .50779 .86148 .52275 .85249 .53754 .84324 ' .55218 .83373 !.56665 .82396 29 32 .50804 .86133 .52299 .85234 .53779 .84308 .55242 .83356 1.56689 .82380 28 S3 .50829 .86119 .52324 .85218 .53804 .84292 .55206 .83340 1.56713 .82363 27 34 .50854 .86104 .52349 .85203 .53828 .84277, .55291 .83324 .82347 26 35 .50879 .86089 .52374 .85188| .53853 .84261 •55315 .83308, !.56760 .82330 25 36 .50904 .86074 .52399 .85173 .53877 .84245 .55339 .83292 ■.56784 .82314 24 37 .50929 .86059 .52423 .851571 .53902 .84230 .55363 .83276 1.56808 .82297 23 38 .50954 .86045 .52448 .85142' .53926 .84214 .55388 .83200 I.56832 .82281 00 39 .50979 .86030 .52473 .85127 .53951 .81198 .55412 .83244 !.56856 .82264 21 40 .51004 .80015 .52498 .85112, .53975 .ama •55436 .83228 !.56880 .82248 20 41 .51029 .86000 .52522 .85096' .54000 .84167ì .55460 .83212 .56904 .82231 19 42 .51054 .85985 .52547 .85081 .54024 .84151 .55484 . 83195 1.56928 .82214 18 43 .51079 .85970 .52572 .85006 .54049 .84135! .55509 .83179 .56952 .82198 17 44 .51104 .85956 .52597 .85051 .54073 .84120' . 6òi)"v3 .83163 .56976 .82181 16 45 .51129 .85941 .52621 .85035 .54097 .84104 .55557 .83147 .57000 .82165 15 46 .51154 .85926 .52616 .85020 .54122 .84088 .55581 .83131 .57024 .82148 14 47 .51179 .85911 .52671 .85005 .54146 .84072 .556(15 .83115 .57047 .82132 13 48 .51204 .85896 .52608 .81989 .54171 .84057 .55030 .83098 .57071 .82115 12 49 .51229 .85881 .52720 .84974'' .54195 .81041 .55654 .83082 .57095 .82098 11 50 .51254 .85866 .52745 .84959 j .54220 .84035 .55678 .83066 .57119 1.82082 10 51 .51279 .85851 .52770 .84943* .54244 .84009 '.55702 .83050 1.57143 .82065 9 m .51304 .85830 .52794 .84928 .54209 .83994 ; .55726 .81034 .57167 .82048 8 53 .51329 .85821 .52819 .84913 .54293 .83978 .55750 .83017! ! .57191 .82032 7 54 .51354 .85806 .52844 .84897 .5-1317 .83962, .55775 .83001 !.57215 .82015 6 55 .51379 .85792 .52869 .84882 .54342 .839461 .55799 .82985 I.57238 .81999 5 56 .51404 .85777 .52893 .84866 .54306 .83930! .55823 .82969 I.57202 .81982 4 57 .51429 .85762 .52918 .84851 .54391 .83915! .55847 .82953 1.57286 .81965 8 58 .51454 .85747 .52943 .84836 .54415 .83899, .55871 .82936 ! .67310 .81949 0 59 .51479 .85732 .52967 .84820 .54440 .83883! .55895 .82920 .57334 .81932 1 GO .51504 .85717 .52992 .84805 .54404 .83867! .55919 .82904 .57358 .81915 0 f Cosin Sine Cosin | Sine | Cosin Sine i Cosin Sine Cosin Sine 59° o oo io 57° 1 56° 55° NATURAL SINES AND COSINES, 107 35° 0 CO 00 3" r° 0 co 00 co co 0 Sine Cosin Sine Cosin Sine Cosin Sine Cosin Sine Cosin 0 .57358 .81915! .58779 .80902 .60182 .79864 .61566 .78801 .62932 .77715 €0 1 .57381 .818991 .58802 .80885| .60205 .79846 .61589 .78783 .62955 .77696 59 2 .57405 .81882' .58826 .80867 .C0228 .79829 .61612 .78765 .62977 .77678 58 3 .57429 .81805: .58849 .808501 .60251 .79811 .61635 .78747 .63000 .77660 57 4 .57453 .818481 .58873 .808331 .60274 .79793 .61658 .787291 .63022 .7764! 56 5 .57477 .818321 .58896 .808161 .60298 .797761 .61681 .7871l| .63045 .77623 55 6 .57501 .818151 .58920 .80799! .60321 .79758 .61704 .786941 .63068 .77605 54 r* 4 •5< • >24 .81798 .58943 .807821 .60344 .79741| .61726 .78676! .63090 .77586 53 8 .57548 .81782! .58967 .80765 .60367 .79723' .61749 .786581 .63113 .77568 52 9 .57572 .81765 .58990 .807481 .60390 .79706! .61772 .78640' .63135 .77550 51 10 .57596 .81748. .59014 .80730j .60414 .796881 .61795 .786221 .63158 .77531 50 11 .57619 .81731 .59037 .80713| .60437 .79671 .61818 .786041 .63180 .77513 40 12 .57643 .81714 .59061 .80696 .60460 .79653 .61841 .78586' .63203 .77494 48 13 .57667 .81698' .59084 .80679 .G0483 .79635 .61864 .78568, .63225 .77476 47 14 .57691 .81681| .59108 .80662 .C0506 .79618 .61887 .78550 ! .63248 .77458 46 15 .57715 .81064 .59131 .80644 .60529 .79600 .61909 .78532! .63271 .77439 45 16 .57738 .81647 .59154 .80627! .60553 .79583 .61932 .78514, .63293 .77421 44 17 .57762 .81631 .59178 .80610 .60576 .79565 .61955 .78496' .63316 .77402 43 18 .57786 .81614 .59201 .80593 .60599 .79547 .61978 .78478 .63338 .77384 42 19 .57810 .81597' .59225 .80576 .60622 .79530' .62001 .78460 .63361 .77366 41 20 .57833 .81580' .59248 .80558 .60645 .795121 .62024 .78442 .63383 .77347 40 21 .57857 .81503' .59272 .80541 .60668 .79494 !.62046 .78424 .03406 .77329 39 uy .57881 .81546 .59295 .80524 .60691 .79477 |.620G9 .78405| .63428 .77310 33 23 .57904 .81530; .59318 .80507i .60714 .79459 i .62092 .78387! .63451 .77292 37 24 .57928 .81513! .59342 .80489 .C0738 .79441 1.62115 .78369| .63473 .77273 36‘ 25 .57952 .81496 .59365 .80472 .60761 .79424 .62138 .78351j .63496 .77255 35 26 .57976 .81479, .59:589 .80455 .60784 .79406 1.62160 .78333! .63518 .77236 34 27 .57999 .81462 .59412 .80438' .60807 .79388 .62183 .78315| .63540 .77218 33 28 .58023 .81445! .59436 .80420 .C0830 .79371 .62206 .782971 .63563 .77199 32 29 .58047 .81428i .59459 .80403 60853 .79353! .62229 .78279 .63585 .77181 31 30 .58070 .81412 .59483 .80386 .60876 .79335 .62251 .782611 .63608 .77162 30 31 .58094 .813951 .59506 .80368 .60899 .793181 .62274 .78243 .63630 .77144 29 32 .58118 .81378 .59529 .80351 .60922 .79300! .62297 .78225! .63653 .77125 28 33 .58141 .81361; .59552 .80334 .60945 .79282 1.62320 .78206! .63675 .77107 27 34 .58165 .81344 .59576 .80316 .60968 .79264! !.62342 .78188! .63698 .77088 26 35 .58189 .81327! .59599 .80299 .60991 .79247ì 1.62365 .78170! .63720 .77070 25 36 .58212 .81310: .59622 .80282 .61015 .79229 !.62388 .78152 .63742 .77051 24 37 .58236 .81293! .59046 .802641 .61038 .792111 j .62411 .78134 .63765 .77033 23 38 .58260 .81276! .59069 .80247. .61061 .79193 1.62433 .78116 .63787 .77014 22 39 58283 .812591 .59693 .80230 .61084 .79176 1.62456 .78098 .63810 .7G99G 21 40 .58307 .81242! .59716 .80212 .61107 .79158, .62479 .78079; .63832 .76977 20 41 .58330 .81225! .59739 .801951 .61130 .79140' .62502 .78061 .63854 .76959 19 42 .58354 .81208 .59763 .80178 .61153 .79122' .62524 .78043' .63877 .7G940 18 43 .58378 .81191 .59786 .801G0 .6117C .79105! 1.62547 .78025 .63899 .76921 17 44 .58401 .81174 .59809 .80143! .61199 .79087 |.62570 .78007 .63922 .76903 16 45 .58425 .81157 .59832 .80125! .61222 .79069, 1.62592 .77988 .63944 .70884 15 46 .58449 .81140 .59856 .80108! .61245 .79051 .62615 .77970 .63966 .76866 14 47 .58472 .81123; .59879 .80091 .61208 .79033' .62638 .77952 .63989 .76847 13 48 .58496 .81106 .59902 .80073 .61291 .79016 .62660 .77934 .64011 .76828 12 49 .58519 .81089 .59926 .80056 j .61314 .78998' .62683 .77916 .64033 .76810 11 50 .58543 .81072, .59949 .80038 .61337 .78980 .62706 .77897, .64056 .76791 10 51 .58567 .81055 .59972 .80021: .61360 .78962 .62728 .77879' .64078 .76772 9 52 .58590 .81038! .59995 .800031 .61383 .78944 .62751 .77861 .64100 .76754 8 53 .58614 .81021 .60019 .79986' .61406 .78926 .62774 .77843; .64123 .76735 54 .58637 .81004 .60042 .79908 .61429 .78908 .62796 .77824 .64145 .76717 6 55 .58661 .80987 .60065 .79951 .61451 .78891 .62819 .77806I .64167 .76698 5 56 .58684 .80970 .60089 .79934 .61474 .78873 .62842 .(i ICO 1 .64190 .76679 4 57 .58708 .80953 .60112 .79916 .61407 .78855' .62864 .77769| .64212 .76661 V 58 .58731 .80936 .60135 .79899 .61520 .78837 .62887 .77751' .64234 .76642 2 59 .58755 .80919 .60158 .79881 .61543 .78819, .62909 .77733 .64256 .76623 1 CO .58779 .80902 .60182 .79864 .61566 .78801: .62932 . t i t lo .64279 .76604 0 t Cosin Sine j Cosin Sine Cosin Sine j Cosin Sine j Cosin ; Sine 9 54* 63° 1 52° 1 5P 50° 108 NATURAL SINES AND COSINES o o 1 41° | 42° | 43° | 44° / Sine Cosin 1 Sine Cosin | Sine Cosin 1 Sine Cosin Sine Cosin' / "5 .64279 .76604 .65606 .75471 .66913 .74314 .68200 .73135 i .69466 .71934 60 l .64301 .76586! .65628 .75452 .66935 .74295 .68221 .73116 .69487 .71914 59 2 .(>4323 .76567 .65650 .75433 .66956 .74276 .68242 .73096 .69503 .71894 58 3 .04346 .76548 .65672 .75414 .66978 .74256 .68264 .73076 .69529 .71873 57 4 .64368 .7G530 .65694 .75395 .66999 .74237 .68285 .73056 .69549 .71853 56 5 .64390 .76511 .65716 .75375 .67021 .74217 .68306 .73030 .695<0 .71833 55 6 .64412 .76492 .65738 .75356 .67043 .74198 .68327 .73016 .69591 .71813 54 7 .64435 .76473 .65759 .75337 .67064 .74178 .68349 .72996 .69012 .71792 53 8 .64457 .764551 .65781 .75318 .670SG .74159 .68370 .72976 .69633 .71772 52 9 .61479 .76433 ! .65803 .75299 .67107 .74139 .6S391 .72957 .69654 .71752 51 10 .64501 .76417 .65825 .75230 .67129 .74120 .68412 .72937 .69675 .71732 50 11 .64524 .76398 .65847 .75261 .67151 .74100 .68434 .72917 .69696 .71711 49 ■ .64546 .76330 .05369 .75241 .67172 .74080 .68455 .72887 .69717 .71091 48 13 .64568 .76361 .65891 .75222 .67194 .74061 .68476 .72877 .69737 .71671 47 14 .64590 .76312 .65913 .75203 .67215 .74041 .68497 .72857 .69758 .71G50 46 15 .64012 .76323 .65935 .75184 i .67237 .74022 .68518 .72S37 .69779 .71630 45 16 .61635 .76394 .65956 .75165 .67258 .74002 .6S539 .72817 .69800 .71610 44 17 .64657 .762361 .05978 .75146 .67280 .73983 .68561 .72797 .69821 .71590 43 18 .64679 .76267, .66990 .75123 .67301 .739(53 .68582 .72777 .69S42 .715G9 42 19 .61701 .70243 .66023 .751071 .67323 .73944 .68603 .72757 HHH .71549 41 20 .64723 .76229, .66044 .75088 .67344 .73924 .68624 .72737 .69883 .71529 40 21 .61746 .76210 .66066 .75039 ! .673G6 .73904 .68645 .72717 .69904 .71508 39 22 .04768 .76192, .63933 .75050 .61387 .73885 .68600 .72097 .69925 .71488 33 23 .64790 .76173, .63109 .75030| .67409 .73S03 .68683 .72G77 .69940 .71468 37 24 .64812 .76154' .63131 .75011 .67430 .7384G .68709 .72657 .699GG .71447 or» ÒU 25 .61834 .76135 .63153 .74932 i .67452 .73823 .68730 .72637 .69987 .71427 35 26 .64856 .76116 .66175 .749731 .67473 .73803 .68751 .72017 .70003 .71407 34 27 .64878 .76097, .66197 .74953! .67495 .73787 .68772 .72597 .70029 .713SG 33 28 .64901 .76078' .66218 .74934 .67516 .73767 .68793 .72577 .70049 .713C0 32 29 .64923' .76059! .65240 .74915 .67538 .73747 .68814 .72557 .70070 . 713-15 31 30 .64945 .70041 .06262 .74896i .67559 .73728 .68835 .72537, .70091 .71325 30 31 .64967 .76022 * .63284 .748761 .67530 .73703 .68857 .72517 .70112 .71305 29 32 .64989 .76003 .63 603 .74857 .C7G92 .73® .6S878 .72497 70132 .71284 23 33 .65011 .75984' .66327 .74833 .67623 .73669 .68899 .72477 .70153 .71264 34 .65033 .75965' .63349 .74818 .67615 .73649 .68920 .72457 .70174 .71248 23 35 .65055 .75946 .63371 .74799 .67033 .73629 .68941 .72437 .70195 .71223 35 36 .65077 .75927, .63393 .74780 .67333 .73010 .63962 .7241? .70215 .71203 21 37 .65100 .75903 .63414 .74760 .67709 .73599 .68983 .72397 .70230 .71182 23 38 .65122 .75889 .66436 .74741 .67730 73570 .69004 .72377 .70257 .71® 2> 39 .65144 .75870 .66 453 .74722 657752 .73551 .69025 .72357 .70277 .71141 21 40 .65166 .75851j .66480 .74703 .6«i <3 .73531 .69046 .723371 .70298 .71121 20 41 .65188 .75832 .66501 .74633 .67795 .73511 .69067 .72317 .70319 .71100 19 42 .65210 .75813 .66526 .74634 .67816 .73491 .69088 .72297 .70389 .71030 IS 43 .65232 .75794 . 665 45 .74344 .67837 .73472 .69109 .72277 .70300 .71059 17 44 .65254 .75775, .66566 .74625 .67859 .73452 .69130 .722571 .70381 .71039 10 45 .65276 .75756 .66588 .74695 .67830 .73432 .69151 1122361 .70401 .71019 15 46 .65298 .75738 .66610 .745SG .67901 .73413 .69172 .722161 .70422 .70908 14 47 .65320 .75719 .66662 .74567 .67923 .73393 .69193 .72196 .70443 .70978 13 48 .65342 .75700 .66656 .74543 .67944 .73373 .69214 .72176 .70403 .70957 12 49 .65364 .75680 .66675 .74528 .67965 .73353 .69235 .72156 .70484 .70937 11 50 .65386 .75661 .66697 .74509 .67987 .73333 .69256 .72136i .70505 .70916 10 51 .65408 .75642! .66718 .74489 .68008 .73314 .69277 .721161 .70525 .70896 9 52 .05430 .756231 .66740 .74470 .63029 .73294 .69298 .72095 .70.V16 .70875 8 53 .65452 .75604 .66762 .74451 i .68051 .73274 .69319 .72075 .70567 .70855 7 54 .65474 .75585 , 6678*6 .74431 .68072 .73254 .69340 .72055 .70587 .70834 6 55 .65496 .75566 .66805 .74412 .68093 .73234 .69361 .72035 .70608 .70813 5 56 .65518 .75547 ,66827 .74392 .681li .73215 .69382 .720151 .70628 .70793 4 57 .65540 .75528 .60848 .74373 1.68136 .73195 .69403 .71995! .70049 .70772 3 58 .65562 !.75509 .60870 .74353 .68157 .73175 .69424 .71974 .70670 .70752 o 59 .65584 | .75490 .66891 .74334 .68179 .73155 .69445 1.719541 .70690 .70731 1 CO .65606 .75471 .66913 .74314 |.68200 1.73135 .69466 .71934 .70711 .70711 0 r Cosin 1 Sine Cosin Sine ! Cosin | Sine | Cosin 1 Sine I Cosin Sine f 49° 48° 47° 1 46° 45° NATURAL TANGENTS AND COTANGENTS 109 1— i 0° 1° 2° J s Tang | Cotang Tang Cotang 1 Tang Cotang | Tang Cotang 0 .00000 Infinite. .01746 57.2900 .03492 28.6863 .05241 19.0811 GO ■4 1 .00029 3437.75 .01775 56.3506 .03521 23.3994 .05270 18.9755 59 2 .00058 1718.87 .01804 55.4415 .03550 28.1604 .05299 18.8711 58 3 .00087 1145.92 .01833 54.5613 .03579 27.9372 .05328 18.7078 57 4 .00116 859.436 .01863 53.7086 .03609 27.7117 .05357 18.6656 56 5 .00145 687.549 .01891 52.8821 .03G38 27.4899 .05387 18.5645 55 6 .00175 572.957 .01920 52.0807 .03CG7 27.2715 .05116 18.4645 54 r i .00204 491.106 .01949 51.3032 .03606 27.0506 .05145 18.3C55 53 8 .00233 429.718 .01978 50.54S5 .03725 26.8450 .05474 18.2677 52 9 .00262 381.974 .02007 49.8157 .03754 26.6367 .05503 18.1708 51 10 .00291 343.774 .02036 49.1039 .03783 26.4316 .05533 18.0750 50 11 .00320 312.521 .02066 48.4121 .03812 26.2296 .05562 17.9802 49 12 .00349 236.478 .02035 47.7395 .03342 26.0207 .05591 17.8803 48 13 .00378 264.441 .02124 47.0S53 .03871 25.8348 .05620 17.7934 47 14 .00407 245.552 .02153 46.4189 .03000 25.6418 .05649 17.7015 46 15 .00436 229.182 .02182 45.8294 .03929 25.4517 .05Ci8 17.6106 45 16 .00465 214.858 .02211 45.2261 .03958 25.26-14 .05708 17.5205 44 17 .00495 202.219 .02240 44.G386 .039S7 25.0798 .05737 17.4314 43 18 .00524 190.984 .02269 44.OGGI .04016 24.8978 .05706 17.3432 42 19 00553 180.932 .02298 43.5031 .0-1046 24.7185 .05795 17.2558 41 20 00582 171.885 .02328 42.9641 .04075 24.5418 .05824 17.1693 40 21 00C11 163.700 .02357 42.4335 .04104 24.3675 .05854 17.0837 39 22 00640 156.259 .02386 41.9158 .04133 24.1957 .05833 16.9990 33 23 00GG9 149.405 .02415 41.4106 .04162 24.0263 .05912 16.9150 37 21 00698 143.237 .02444 40.9174 .04191 23.8593 .05941 16.8319 36 25 .00727 137.507 .02473 40.4358 .04220 23.69-15 .05070 16.7496 35 26 .00756 332.219 .02502 39.9655 .04250 23.5321 .05999 16.CC81 34 27 .00785 127.321 .02531 39.5059 .01279 23.3718 .00029 1G.5874 33 28 .00815 122.774 .02560 39.0568 .04208 23.2137 .00058 16.5075 32 29 .00844 118.540 .02589 38.6177 .04337 23.0577 .0G037 16.4283 31 30 .00873 114.589 .02619 38.1885 .04366 22.9038 .00116 16.3499 30 31 .00902 110.892 .02648 37.7686 .04395 22.7519 .06145 16 9709 29 32 .00931 107.426 .02077 37.3579 .04424 22.6030 .06175 16.1052 23 33 .009G0 104.171 .02706 36.95G0 .04454 22.45-11 .06204 16.1190 27 31 .00989 101.107 .02735 36.5627 .04483 22.3031 .06233 16.0435 2G 35 .01013 93.2179 .02764 36.1776 .04512 22.1C40 .06262 15.9G87 25 36 .01017 95.4895 .02793 35.8006 .04541 22.0217 .06291 15.8945 21 37 .01076 92.9085 .02822 35.4313 .04570 21.8813 .06321 15.8211 23 38 .01105 90.4633 .02851 35.0095 .01599 21.7426 .06350 15.7483 °2 39 .01135 88.1436 .02881 34.7151 .04628 21.6056 .06379 15.6762 21 40 .01164 85.9398 .02910 34.3678 .04658 21.4704 .06408 15.6048 20 41 .01193 83.8435 .02939 34.0273 .04687 21.3309 .06437 15.5340 19 42 .01222 81.8470 .029G8 33.6935 .04716 21.2049 .06467 15.4638 18 43 .01251 79.9434 .02997 33.3GG2 .04745 21.0747 .06496 15.3943 17 44 .012S0 78.1263 .03026 83.0452 .04774 20.9400 .06525 15.3254 16 45 .01309 76.3900 .03055 32.7303 .01803 20.8188 .06554 15.2571 15 46 .C1338 74.7292 .03084 82.4213 .04833 20.6933 .06584 15.1893 14 47 .01367 73.1390 .03114 32.1181 .01862 20.5091 .06613 15.1222 13 48 .01396 71.6151 .03143 31.8205 .01891 20.4465 .06643 15.0557 42 49 .01425 70.15.33 .03172 31.5284 .04920 20.3253 .06071 14.9898 11 50 .01455 68.7501 .03201 81.2416 .04949 20.2056 .06700 14.9244 10 51 .01484 67.4019 .03230 30.9599 .04978 20.0872 .06730 14.8596 9 52 .01513 66.1055 .03259 30.6833 .05907 19.9702 .06759 14.7954 8 53 .01542 64.8580 .03288 80.4116 ! .05037 19.8546 .06788 14.7317 54 .01571 63.6567 .03317 30.1446 1 .05006 19.7403 .06817 6 55 .01G00 62.4992 .03346 29.8823 1 .05095 19.6273 .06847 Ï4.G059 5 56 .01629 61.3829 .03376 29.6245 .05124 19.5156 .0G876 14.5438 4 5 i .01658 60.3058 .03405 29.3711 | .05153 19.4051 .00905 14.4823 3 58 01687 59.2059 .03434 29.1220 ! .05182 19.2959 .06934 14.4212 Q 59 ! .01716 58.2612 .03463 28.8771 .05212 19.1879 .06903 14.3607 1 CO j .01746 57.2900 .03492 28.6363 ,05241 19.0811 .06993 14.3007 0 / Cotang Tang j Co tang Tang Cotang Tang Cotang Tang 1 89° i o 00 00 f 1 £-00 i 86° 110 NATURAL TANGENTS AND COTANGENTS. 4° S € 7 o 9 Tang Cotang 1 Tang Cotang Tang Cotang ; Tang j Cotang "0 .06993 14.3007 .08749 11.4301 .10510 9.51436 .12278 1 8.14435 1 .07022 14.2411 .08778 11.3919 .10540 9.48781 .12308 8.12181 2 .07051 14.1821 .08807 11.3540 .10509 9.40141 .123S8 8.1053G . 3 .07080 14.1235 .08837 11.3103 .10599 9.43515 .12307 8.08000 4 .07110 14.0055 .08800 11.2789 .10028 9.40904 .12397 8.06C74 5 .07139 14.0079 .08895 11.2417 .10057 9.38307 .12426 8.04750 6 .07108 13.9507 .08925 11.2048 .10087 9.35724 .12456 8.02848 7 .07197 13.8940 .08954 11.1081 .10716 9.33155 .12485 8.00948 8 .07227 13.8378 .08983 11.1310 .10740 9.30599 .12515 7.99058 9 .07256 13.7821 .09013 11.0954 .10775 9.28058 .12544 7.97170 10 .07285 13.7207 .09042 11.0594 .10805 9.25530 .12574 7.95302 11 .07314 13.0719 .09071 11.0237 .10834 9.23016 .12603 7.93138 12 .07344 13.0174 .09101 10.9882 .10803 9.20516 .12033 7.91582 13 .07373 13.5034 .09130 10.9529 .10893 9.18028 .12C02 7.89734 14 .07402 13.5098 .09159 10.9178 .10922 9.15554 .12092 7.87895 15 .07431 13.4506 .09189 10.8829 .10952 9.13093 .12722 7.80004 10 .07401 13.4039 .09218 10.8183 .10981 9.10G46 .12751 7.84242 17 .07490 13.3515 .09247 10.8139 .11011 9.0S211 .12781 7.82428 m .07519 13.2996 .09277 10.7797 .11040 9.05789 .12810 7.80022 19 .07518 13.2180 .09306 10.7457 .11070 9.03379 .12840 7.78825 20 .07578 13.1909 .09335 10.7119 .11099 9.00983 .12809 7.77035 21 .07007 13.1461 .09365 10.C783 .11128 8.98598 .12899 7.75254 22 .07030 13.0958 .09394 10.0450 | .11158 8.90227 .12929 7.73480 /wO .07005 13.0458 .09423 10.0118 .11187 8.93807 .12958 7.71715 21 .07095 12.9902 .09453 10.5789 .11217 8.91520 .12988 7.09957 25 .07724 12.9409 .09482 10.5102 .11240 8.89185 .13017 7.08208 26 .07i53 12.8981 .09511 10.5136 .11270 8.86802 .13047 7.00406 27 .07782 12.3496 .09541 10.4813 .11305 8.84551 .13070 7.04732 28 .07812 12.8014 .09570 10.4491 .11335 8.82252 .13100 7.63005 29 .07841 12.7536 .09000 10.4172 .11364 8.79904 .13136 7.01287 SO .07870 12.7002 .09029 10.3854 .11394 8.77089 .13165 7.59575 31 .07899 12.0591 .09058 10.3538 .11423 8.75425 .13195 7.57872 32 .07929 12.0124 .09088 10.3224 .11452 8.73172 .13224 7.56176 33 .07958 12.5000 .09717 10.2913 .11482 8.70931 .13254 7.54487 H .07987 12.5199 .09746 10.2002 .11511 8.08701 .13284 7.52800 35 .08017 12.4742 .09776 10.2294 .11541 8.60482 .13313 7.51132 36 .08046 12.4288 .09805 10.1988 .11570 8.04275 .13343 7.49405 37 .08075 12.3838 .09834 10.1083 .11600 8.62078 .13372 7.47800 38 .08104 12.3390 .09804 10.1381 .11629 8.59893 .13402 7.46154 39 .08134 12.29(6 .09893 10.1080 .11059 8.57718 .13432 7.44509 40 .08103 12.2505 .09923 10.0780 .11088 8.55555 .13401 7.42871 41 .08192 12.2007 .09952 10.0483 .11718 8.53402 .13491 7.41210 42 .08221 12.1032 .09981 10.0187 .11747 8.51259 .13521 7.3901G 43 .08251 12.1201 .10011 9.98931 .11777 8.49128 .13550 7.37999 44 .08280 12.0772 .10010 9.90007 .11806 8.47007 .13580 7.303S9 45 .08309 12.0346 .100(59 9.93101 j .11836 8.44890 .13009 7.34780 40 .08339 11.9923 .10099 9.90211 • .11805 8.42795 .13639 7.33190 47 .08308 11.9504 .10128 9.87338 j .11895 8.40705 .13609 7.31000 48 .08397 11.9087 .10158 9.8(482 .11924 8.38025 .13098 7.30018 49 .08427 11.8073 .10187 9.81011 .11954 8.30555 .13728 7.28442 50 .08456 11.8202 .10216 9.78817 .11983 8.SW96 .13758 7.20873 51 .08485 11.7853 .102(6 8.70009 .12013 8.32446 .13787 7.25310 52 .08514 11.7(48 .10275 9.73217 .12042 8.30406 .13817 7.23754 53 .08544 11.7045 .10305 9.70111 .12072 8.28376 .13846 7.22204 54 .08573 11.6645 .10334 9.07080 .12i01 8.26355 .13870 7.20001 55 .08002 11.6248 .10303 9.6(935 .12131 8.24345 .13906 7.19125 50 .08032 11.5853 .10393 9.(52205 .12100 8.22314 .13935 7.17594 57 .08001 11.5(01 .10(22 9.50490 .12190 8.20352 I .13905 7.10071 58 .08(590 11.5072 .1(V(52 9.50791 .12219 8.18370 .13995 7.14553 59 .08720 11.4085 .10481 9.54100 .12249 8.16398 .14024 7.13042 GO .08749 11.4301 .10510 9.51130 .12278 8.14135 ! .14054 7.11537 / Cotang Tang ! Co tang Tang ■ Cotang Tang Cotang Tang 1 L 85c 1 o CO 83° o 03 GO / GO 59 58 57 50 55 54 53 52 51 50 49 48 47 46 45 44 a 41 ko 39 I IS 8 32 31 30 29 28 27 26 25 124 123 ! O») 21 20 19 18 ! 17 116 15 14 113 112 11 10 9 8 7 6 5 4 3 2 1 _0 /NATURAL TANGENTS AND COTANGENTS 111 8° 9° 0 O H 11° / Tang I Cotang 1 Tang Cotang Tang Cotang Tang Cotang “o .14054 7.11537 .15838 6.31375 .17633 5.67128 .19438 5.14455 60 1 .14084 7.10038 .15868 6.30189 .17663 5.60165 .19468 5.13658 59 2 .14113 7.08546 .15898 6.29007 .17093 5.65205 .19498 5.12862 58 3 .14143 7.07059 .15928 6.27829 .17723 5.64248 .19529 5.12069 57 4 .14173 7.03579 .15958 6.26655 .17753 5.63295 .19559 5.11279 56 5 .14202 7.04105 .15988 6.25186 .17783 5.62344 .19589 5.10490 55 6 .14232 7.02637 .16017 6.243211 .17813 5.61397 .19619 5.09704 54 fj .14262 6.91174 .10047 6.23100 .17843 5.60452 .19649 5.08921 53 8 .14291 6.99718 .1G077 6.22003 .17873 5.59511 .19080 5.08139 52 9 .14321 6.98268 .16107 6.20851 .17903 5.58573 .19710 5.07360 51 10 .14351 6.96823 .10137 6.19703 .17933 5.57638 .19740 5.06584 50 n .14381 6.95385 .16167 6.18559 1 .17963 5.56706 .19770 5.05809 49 m .14410 6.93952 .10196 6.17419 ! .17993 5.55777 .19801 5.05037 48 13 .14440 6.92325 .16226 6.16283 .18023 5.54851 .19831 5.04257 47 ii .14470 6.91104 .16256 0.15151 I .18053 5.53927 .19801 5.0&199 46 15 .14199 6.8D6S8 .16236 6.14023 ! .180S3 5.53007 .19S91 5.02734 45 16 .14529 6.88278 .16316 6.12899 j .18113 5.52090 .19921 5.01971 44 17 .14559 6.8GS74 .16346 6.11779 1 .18143 5.51176 .19952 5.01210 43 18 .14588 6.85475 .10376 6.10664 .18173 5.50264 .19982 5.00451 42 19 .14618 6.84082 .16405 6.09552 .18203 5.49356 ,20012 4.99695 41 20 .14648 6.82694 .16435 6.08444 .18233 5.48451 .20042 4.98940 40 21 .14678 6.81312 .16405 6.07840 ! .18263 5.47548 .20073 4.98188 39 on, .11707 6.79936 .16495 6.06240 I .18293 5.46648 .20103 4.97433 38 23 .14737 6.78564 .16525 6.05143 i .18323 5.4jìol .20133 4.9GG90 37 21 .14767 6.77199 .16555 6.04051 ! .18353 5.44857 ,20104 4.95945 36 25 .14796 6.75838 .16585 6.029G2 ! .183S4 5.43966 .20194 4.95201 35 26 .14826 6.74-483 .16615 6.01878 1 .18414 5.43077 .20224 4.944G0 34 27 .14356 6.73133 .16645 6.00797 1 .18444 5.42192 .20254 4.93721 33 23 .14386 6.71789 .16674 5.99720 : .18474 5.41309 .20285 4.92984 32 2D .11915 6.70450 .16704 5.93346 , .18504 5.40429 .20315 4.92249 31 30 .14945 6;69116 .16734 5.97576 .18534 5.39552 .20345 4.91516 30 31 .14975 6.67787 .16764 5.90510 ! .18564 5.38677 .20376 4.90785 29 32 .15005 6.63463 .16794 5.95448 .18594 5.37805 .20406 4.90056 28 33 .15034 6.G5144 .16824 5.94390 .18G24 5.36936 .20436 4.89330 27 34 .15064 6.63831 .16854 5.93365 .18654 5.36070 .20466 4.88605 26 35 .15094 6.02523 .16884 5.92283 .18684 5.35206 .20497 4.878S2 25 36 .15124 6.61219 .16914 5.91236 .18714 5.34345 .20527 4.87162 24 37 .15153 6.59921 .16944 5.90191 ..18745 5.33487 .20557 4.86444 23 33 .15183 6.53627 .16974 5.89151 .18775 5.32031 .20583 4.85727 22 39 .15213 6.57339 .17094 5.88114 .18805 5.31778 .20318 4.85013 21 40 .15243 6.56055 .17033 5.87080 .18835 5.30928 .20648 4.84300 20 41 .15272 6.54777 .17063 5.86051 .18865 5.30080 .20679 4.83590 19 42 .15302 6.53503 .17093 5.85024 .10895 5.29235 .20709 4.82882 18 43 .15:332 6.52234 .17123 5.84001 .18925 5.28393 .20739 4.82175 17 44 .15362 6.50970 .17153 5.82982 .18955 5.27553 .20770 4.81471 16 45 .15391 6.49710 .17183 5.81966 .189S6 5.2G715 .20800 4.80769 15 46 .15421 6.48456 .17213 5.80953 .19016 5.258S0 .20830 4.80068 14 47 .15451 6.47206 .17243 5.79944 .19016 5.25048 ■.208G1 4.79370 13 48 .15481 6.45901 .17273 5.78938 .19076 5.24218 1 .20891 4.78673 12 49 .15511 6.44720 .17303 5.77936 .19106 5.23391 ! .20921 4.77978 11 50 .15540 6.43484 .17333 5.76937 .19136 5.22566 .20952 4.77286 10 51 .15570 6.42253 .17363 5.75941 .19166 5.21744 .20982 4.76595 9 52 .15600 6.41026 .17393 5.74949 .19197 5.20925 .21013 4.75906 8 53 .15630 6.39804 .17423 5.739G0 .19227 5.20107 .21043 4.75219 7 54 .15660 6.38587 .17453 5.72974 .19257 5.19293 .21073 4.74534 6 55 .15689 6.37374 .17483 5.71992 .19287 5.18480 .21104 4.73851 5 56 .15719 6.3G1G5 .17513 5.71013 .19317 5.17G71 .21134 4.73170 4 57 .15749 6.31961 .17543 5.70037 .19347 5.16863 .21161 4.72190 3 53 .15779 6.337G1 .17573 5.69064 .19378 5.16058 .21195 4.71813 2 59 .15809 6.325G6 .17603 5.68094 .19408 5.15256 .21225 4.71137 1 GO ! .15838 6.31375 1 .17633 5.67128 .19438 5.14455 .21256 4.70463 0 / Cotang Tang Cotang Tang Cotang Tang Cotang Tang / o *H 00 co o o 79° .78° 1 112 NATURAL TANGENTS AND COTANGENTS. 12° 13° | 14° 15° / 1 Tang | Co tan g Tang Cotang Tang Cotang Tang | Cotang "Ô .21256 4.70463 .23087 4.33148 .24933 4.01078 | .26795 | 3.73205 60 1 .21286 4.69791 .23117 4.32573 .24964 4.00582 | .26826 3.72771 59 2 .21316 4.69121 .23148 4.32001 | .24995 4.00086 .26857 1 3.72338 58 3 .21347 4.G8452 .23179 4.31430 .25026 3.99592 [ .26888 | 3.71907 57 4 .21377 4.C7786 .23209 4.30860 .25056 3.99099 1 .26920 3.71476 56 5 .21408 4.67121 .23240 4.30291 .25087 3.98G07 .26951 3.71046 55 6 .21438 4.GG458 .23271 4.29724 .25118 3.98117 .26982 3.70616 54 7 .21469 4.65797 .23301 4.29159 .25149 3.97627 .27013 3.70188 53 8 .21499 4.G5138 .23332 4.28595 .25180 3.97139 .27044 3.69761 52 9 .21529 4.64480 .23363 4.28032 .25211 3.96G51 .27076 3.69335 51 10 .21560 4.63825 .23393 4.27471 .25242 3.96165 .27107 3.68909 50 11 .21590 4.63171 .23424 4.26911 .25273 3.95680 .27138 3.684R5 49 12 .21621 4.62518 .23455 4.26352 .25304 3.95196 .27169 3.6S061 48 13 .21651 4.61868 .23485 4.25795 .25335 3.9-1713 .27201 3.67G3S 47 14 .21082 4.G1219 .23516 4.25239 .25306 3.94232 1 .27232 3.67217 46 15 .21712 4.G0572 .23547 4.24685 .25397 3.93751 1 .27263 3.66796 45 16 .21743 4.59927 .23578 4.24132 1 .25428 3.93271 ! .27294 3.G6376 44 17 .21773 4.59233 .23608 4.23580 | .25459 3.92793 .27326 3.65957 43 18 .21804 4.58641 .23639 4.23030 .25490 3.92316 .27357 3.65538 42 19 .21&34 4.58001 .23G70 4.22481 .25521 3.91839 .273« 3.G5121 41 20 .21864 4.57363 .23700 4.21933 .25552 3.91364 .27419 3.64705 40 21 .21895 4.56726 .23731 4.21387 .25583 3.90890 .27451 3.64289 39 22 .21925 4.56091 .23762 4.20842 .25G14 3.90417 .27482 3.63874 38 23 .21956 4.55458 .23793 4.20298 .25645 3.89945 .27513 3.634G1 37 24 .21986 4.51826 .23823 4.19756 ! .25676 3.89474 .27545 3.63048 36 25 .22017 4.54196 .23851 4.19215 1 .25707 3.89004 .27576 3.C2G36 35 26 .22047 4.53568 .23885 4.18675 .25738 3.88536 .27G07 3.62224 34 27 .22078 4.52941 .23916 4.18137 .25769 3.880G8 .276;« 3.61814 33 28 .22108 4.52316 .23946 4.17600 .25800 3.87601 .27670 3.61405 32 20 .22139 4.51693 .23977 4.170G4 .25831 3.87136 .27701 3.60990 31 30 .22169 4.51071 ,21008 4.16530 .25862 3,86671 .27732 3 60588 30 31 .22200 4.50451 .24039 4.15997 .25893 3.86208 .27764 3.60181 29 32 .22231 4.49832 .24069 4.15465 .25924 3.85745 .27795 3.59775 28 33 .22261 4.49215 .21100 4.14931 .25955 3.85284 .27826 3.59370 27' ■ 00009 4.48600 .21131 4.14405 .25986 3.84824 .27858 3.58906 26 35 .22322 4.47986 .21162 4.13877 .26017 3.84364 .27889 3.585G2 25 36 .22353 4.47374 .24193 4.13350 .26048 3.83906 .27921 3.581GO 21 37 .22383 4.46764 .24223 4.12825 .26079 3.83449 .27952 3.57758 23 38 .22-114 4.4G155 .24254 4.12301 .26110 3.82992 .27983 3.57357 oo 39 .22444 4.45548 .24285 4.11778 .26141 3.82537 .28015 3.56957 21 40 .22475 4.44942 .21316 4.11256 .26172 3.82083 .28046 3.56557 20 41 .22505 4.44338 .24347 4.10736 .26203 3.81630 .28077 3.56159 19 42 .22536 4.43735 .24377 4.10216 .26235 3.81177 .28109 3.55701 18 43 .22567 4.43134 .21408 4.09699 .26206 3.80726 .28140 3.55364 17 44 .22597 4.42534 .24139 4.09182 .26297 3.80276 .28172 3.54968 16 45 .22028 4.41936 .24470 4.08606 .2G328 3.79827 .28203 3.54573 15 46 .22658 4.41340 .21501 4.08152 .20359 3.79378 .28234 3.54179 14 47 .22GS9 4.40745 .24532 4.07639 .28390 3.78931 .28266 3.53785 13 48 .22719 4.40152 .24562 4.07127 .26421 3.78485 .28297 3.53393 112 49 .22750 4.39560 .21593 4.0G616 .26452 3.78040 .28329 3.53001 11 50 .22781 4.38909 .24624 4.06107 .26483 3.77595 .28360 3.52609 10 51 .22811 4.33381 .24655 4.05599 .26515 3.77152 .28391 3.52219 9 52 .228-12 4.37793 .21686 4.05092 .26546 3.76709 .28423 3.51829 8 53 .22872 4.37207 .24717 4.0-1586 .26577 3.76268 .28454 3.51441 7 54 .22903 4.30G23 .21747 4.04081 .26608 3.75828 .28486 3.51053 6 55 .22934 4.30040 .24778 4.03578 .26639 S.75388 .28517 3.50666 5 56 .229G4 4.35459 .24809 4.03076 .2G670 3.74950 .28549 3.50279 4 57 .22995 4.34879 .24840 4.02574 .26701 3.74512 .28580 3.49894 3 53 .23026 4.31300 | .2-1871 4.02074 .26733 3.74075 .28612 3.49509 *> 59 .23056 4.33723 1 .24902 4.01576 .26764 3.73C40 | .28643 3.49125 1 CO .23087 4.33148 1 .24933 4.01078 .26795 3.73205 .28675 3.48741 0 / Cotang Tang Cotang Tang 1 Cotang Tang 1 Cotang Tang / \ 77° ! 76° 75° 74“ NATURAL TANGENTS AND COTANGENTS'. 1 16° 17° _| 18° 19° / Tang Cotang Tang Cotang | Tang Cotang Tang Cotang "Ô .28675 3.48741 .30573 3.27085 ; .32492 3.07768 .34433 2.90421 1 .28706 3.48359 .30605 3.26745 | .32524 3.07464 .34465 2.90147 2 .28738 3.47977 .30637 3.20406 | .32556 3.07160 .34498 2.89873 3 .28769 3.47596 .30G69 3.26067 .32588 3.06857 .34530 2.89600 4 .28800 3.47216 .30700 3.25729 1 .32621 3.06554 .34563 2.89327 5 .28832 3.46837 .30732 3.25392 .32653 3.06252 .34596 2.89055 6 .28864 3.46458 .30764 3.25055 .32685 3.05950 .34628 2.88783 7 .28895 3.46080 .30796 3.24719 .32717 3.05649 .34661 2.88511 8 .28927 3.45703 .30828 3.21383 ! .32749 3.05349 .34693 2.88240 9 .28958 3.45327 .30860 3.24049 ! .32782 3.05049 .34726 2.87970 10 .28990 3.44951 .30891 3.23714 | .32814 3.04749 .34758 2.87700 11 .29021 3.4-1576 .30923 3.23381 | .32846 3.04450 .34791 2.87430 12 .29053 3.44202 .30955 3.23048 .32S78 3.04152 .34824 2.87161 13 .29084 3.43829 .30987 3.22715 .32911 3.03854 .34856 2.86892 14 .29116 3.43456 .31019 3.22384 .32943 3.03556 .34889 2.86G24 15 .29147 3.43084 .31051 3.22053 .32975 3.032G0 .34922 2.86356 16 .29179 3.42713 .31083 3.21722 .33007 3.02963 .34954 2.8G089 17 .29210 3.42343 .31115 3.21392 .33040 3.02667 .34987 2.85822 18 .29242 3.41973 .31147 3.21063 .33072 3.02372 .35020 2.85555 19 .29274 3.41604 .31178 3.20734 .33104 3.02077 .35052 2.85289 20 .29305 3.41236 .31210 3.20406 .33136 3.01783 .35085 2.85023 21 .29337 3.40869 .31242 3.20079 .33169 3.01489 .35118 2.84758 22 .29368 3.40502 .31274 3.19752 .33201 3.01196 .35150 2.84494 23 .29400 3.40136 .31306 3.19426 .33233 3.00903 .35183 2.8-1229 24 .29432 3.39771 .31338 3.19100 .33206 3.00C11 .35216 2.83965 25 .29463 3.39406 .31370 3.18775 .33298 3.00319 .35248 2.83702 26 .29495 3.39042 .31402 3.18451 .33330 3.00028 .35281 2.83439 27 .29526 3.38679 .31434 3.18127 .333G3 2.99738 .35314 2.&3176 28 .29558 3.38317 .31466 3.17804» .33395 2.99447 .35346 2.82914 29 .29590 3.37955 .31498 3.17481 .33427 2.99158 .35379 2.82653 30 .29021 3.37594 .31530 3.17159 .33460 2.98868 .35412 2.82391 31 .29653 3.37234 .31562 3.16838 .33492 2.98580 .35445 2.82130 32 .29685 3.3G875 .31594 3.16517 .33524 2.98292 .35477 2.81870 33 .29716 3.3G516 .31626 3.16197 .33557 2.98004 ! .35510 2.81610 34 .29748 3.36158 .31658 3.15877 .33589 2.97717 .35543 2.81350 35 .29780 3.35800 .31690 3.15558 .33621 2.97430 .35576 2.81091 36 .29811 3.35443 .31722 3.15240 .33654 2.97144 .35G08 2.80833 37 .29843 3.35087 .31754 3.14922 .33686 2.96858 .35641 2.80574 38 .29875 3.34732 .31786 3.14605 .33718 2.96573 i .35674 2.80316 39 .29906 3.31377 .31818 3.14288 .33751 2.96288 .35707 2.80059 40 .29938 3.34023 .31850 3.13972 .33783 2.960041 .35740 2.79802 41 .29970 3.83670 .31882 3.13656 .33816 2.95721 .35772 2.79545 42 .30001 3.83317 .31914 3.13341 1 .33848 2.95437 1 .35805 2.79289 43 .30033 3.32965 .31946 3.13027 .33881 2.95155 | .35838 2.790:33 44 .30065 3.32614 .31978 3.12713 .33913 2.94872 1 .35871 2.78778 45 .30097 3.32264 .32010 3.12400 .33945 2.94591 | .35904 2.78523 46 .30128 3.31914 .32042 3.12087 .33978 2.94309 j .35937 2.78269 47 .30160 3.31565 .32074 3.11775 .34010 2.94028 1 .35969 2.78014 48 .30192 3.31216 .32106 3.11464 .34043 2.93748 • .36002 2.77761 49 .30224 3.30868 .32139 3.11153 .34075 2.934G8 .3G035 2.77507 50 .30255 3.30521 »32171 3.10842 .34108 2.93189 .36068 2.77254 51 .30287 3.30174 .32203 3.10532 .34140 2.92910 .36101 2.77002 52 .30319 3.29829 .32235 3.10223 .34173 2.92632 .36134 2.76750 53 .30351 3.29-183 .32267 3.09914 1 .34205 2.92354 .36167 2.76498 54 .30382 3.29139 .32299 3.09606 .34238 2.92076 .36199 2.76247 55 .30414 3.28795 .32331 3.09298 .34270 2.91799 ■HI 2.75996 56 .30446 3.28452 .32363 3.08991 .34303 2.91523 .3G2G5 2.75746 57 .30478 3.28109 .32396 3.08685 .34335 2.91216 .36298 2.75496 58 .30509 3.27767 .32428 3.08379 .34308 2.90971 .36331 2.75246 59 .30541 3.27426 .32460 3.08073 .34400 2.90696 .36364 2.74997 60 .30573 3.27085 .32492 3.07768 .34433 2.90421 .36397 2.74748 Cotang Tang Cotang Tang Cotang Tang Cotang Tang 73° 72° 71° 1 70° 113 i--1 / CO 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 J) /114 NATURAL TANGENTS AND COTANGENTS. o O e* 21° 1 22° 23° —— / / Tang Cot an g Tang Cotang Tang Cotang Tang Cotang “Ô .36397 2.74748 .38386 2.00509 .40403 2.47509 I .42147 2.35585 60 i .36430 2.74499 .38420 2.G0283 .40436 2.47302 .42482 2.35395 59 2 .36463 2.74251 .38453 2.C0057 .40470 2.47095 .42516 2.35205 58 s .36496 2.74004 .38487 2.59831 .40504 2.40888 | .42551 2.35015 57 4 2.73756 .38520 2.59606 .40538 2.46G82 .42585 2.34825 56 5 .36562 2.73509 .38553 2.59381 .40572 2.46476 .42619 2.34636 55 6 2.73263 .38587 2.59156 .40606 2.46270 .42654 2.34147 54 7 .36628 2.73017 .38620 2.58932 .40640 2.460G5 .42G88 2.34258 53 8 .30661 2*72771 .38654 2.58708 .40674 2.458G0 .42722 2.34069 52 9 .36694 2.72526 .38687 2.58484 .40707 2.45055 .42757 2 33881 51 10 .30727 2.72281 .38721 2.58261 .40741 2.45451 .42791 2.33693 50 11 .36760 2.72036 .38754 2.58038 .40775 2.45246 .42826 2.33505 49 12 .36793 2.71792 .38787 2.57815 .40809 2.45043 .428G0 2.33317 48 13 .30826 2.71548 .38821 2.57593 .40843 2.4-1839 .42894 2.33130 47 14 .36859 2.71305 .38854 2.57371 .40877 2.44636 .42929 2.32943 46 15 .30892 2.71002 .38888 2.57150 .40911 2.44433 .42963 2.32756 45 16 .30925 2.70319 .38921 2.56928 .40945 2.44230 .42998 2.32570 44 17 .30958 2.70577 .38955 2.56707 .40979 2.44027 .43032 2.32383 « 18 .36991 2.70335 .88988 2.56487 .41013 2.43825 .43067 2.32197 42 19 .37024 2.70094 .39022 2.56206 .41047 2.43G23 .43101 2.32012 « 20 .37057 2.69853 .39055 2.56046 .41081 2.43122 .43136 2.31826 40 21 .37090 2.69612 .39089 2.55827 .41115 2.43220 .43170 2.31641 39 op .37123 2.C9371 .39122 2.55008< .41149 2.43019 .43205 2.31156 38 23 .37157 2.60131 .39156 2.55389 .41183 2.42319 .43239 2.31271 37 24 .37190 2.63392 .39190 2.55170 .41217 2.42318 .43274 2.31086 36 25 .37223 2.GSG53 .39223 2.54952 .41251 2.42418 .43308 2.30902 35 26 .37256 2.63114 .39257 2.54734 .41285 2.42218 .43343 2.30718 34 27 .37289 2.63175 .39290 2.54516 .41319 2.42019 .43378 2.30534 33 28 .37322 2 67037 .39324 2.54299 .41353 2.41819 .43412 2.30351 32 29 .37355 2.67700 .39357 2.54032 .41387 2.41G20 .43447 2.301G7 31 30 .37388 2.67402 .39391 2.53805 .41421 2.41421 .43481 2.29984 30 31 .37422 2.67225 .39425 2.53648 .41455 2.41223 .43516 2.29801 29 32 .37455 2 63989 .39458 2.53432 .41490 2.41025 .43550 2.29619 28 33 .37488 2.60752 .39492 2.53217 .41524 2.40827 .43585 2.29437 27 34 .37521 2.06516 .39526 2.53001 .41558 2.40G29 .43620 2.29254 26 35 .37554 2.GG281 .39559 2.52786 .41592 2.40432 .43654 2.29073 23 36 .37588 2.GG946 .39593 2.52571 .41626 2.40235 .43689 2.28891 24 37 .37621 2.G5811 .39626 2.52357 .416G0 2.40038 .43724 2.28710 23 38 .37654 2.65576 .39660 2.52142 .41694 2.89841 .43753 2.28528 oo 39 .37687 2.65342 .39694 2.51929 .41728 2.39645 .43793 2.28348 21 40 .37720 2.65109 .39727 2.51715 .41763 2.39449 .43828 2.28167 20 41 .37754 2.64875 .39761 2.51502 .41797 2.39253 .43862 2.27987 '19 42 .37787 2.64042 .39795 2.51289 .41831 2.39058 .43897 2.27806 18 43 .37820 2.61410 .39829 2.51076 .418G5 2.38863 .43932 2.27626 17 44 .37853 2.04177 .398G2 2.50864 41899 2.38663 .43966 2.27447 16 45 .37887 2.63915 .39896 2.50652 .41933 2.38473 .44001 2.27267 15 46 .37920 2.63714 .39930 2.50440 .419G8 2.38279 .44036 2.27088 14 47 .37953 2.63483 .39963 2.50229 .42002 2.38084 .44071 2.2G909 13 48 .37986 2.G3252 .39997 2.50018 .42036 2.37891 .44105 2.26730 12 49 .38020 2.03021 .40031 2.49807 .42070 2.37697 .44140 2.26552 111 50 .38053 2.02791 .40065 2.49597 .42105 2.37504 .44175 2.26374 10 51 .33086 2.62561 .40098 2.49386 .42139 2.37311 .44210 2.26196 9 52 .38120 2.62332 .40132 2.49177 .42173 2.37118 .44244 2.26018 8 53 .38153 2.62103 .40166 2.48967 .42207 2.36925 .44279 2.25840 7 54 .33186 2.61874 .40200 2.48758 .42242 2.36733 .44314 2.25603 6 55 .33220 2.61646 .40234 2.48549 .42276 2.36541 .44349 2.25486 5 50 .33253 2.61418 .40267 2 48340 .42310 2.36349 .44384 2.25S09 4 57 .38286 2.61190 .40301 2.48132 .42345 2.36158 .44418 2.25132 3 58 .38320 2.60963 .40335 2.47924 .42379 2.35967 .44453 2.24956 2 59 .38353 2.60736 .40369 2.47716 .42413 2.35776 .44188 2.24780 1 GO .38386 2.60509 .40403 2.47509 .42447 2.35585 .44523 2.24604 0 Cotang Tang, Cotang | Tang 1 Cotang Tang .Cotang ; Tang / i 69° • 08° 1 C7° 66° NATURAL TANGENTS AND COTANGENTS 115 to o 25° 26° 27° t Tang Cotang Tang Cotang Tang Cotang Tang Cotang § .44523 2.2-4604 .46631 2.14451 .48773 2.05030 .50953 1.96261 60 1 .44558 2.24428 .46666 2.14288 .48809 2.04879 .50989 1.96120 59 2 .44593 2.24252 .46702 2.14125 .48845 2.04728 .51026 1.95979 58 3 .44627 2.21077 .46737 2.13963 .48881 2.04577 .51063 1.95838 57 4 .44662 2.23902 .46772 2.13S01 .48917 2.04426 .51099 1.95698 56 5 .44697 2.23727 .46808 2.13639 .48953 2.04276 .51136 1.95557 55 6 .44732 2.23553 .46843 2.13477 .48989 2.04125 .51173 1.95417 54 7 .44767 2.23378 .46879 2.13316 .49026 2.03975 .51209 1.95277 53 8 .44802 2.23204 .46914 2.13154 | .49062 2.03325 .51246 1.95137 52 9 .44837 2.23030 .46950 2.12993 .49098 2.03675 .51283 1.94997 51 10 .44872 2.22857 .46985 2.12832 .49134 2.03526 .51319 1.94858 50 11 ?44907 2.22683 .47021 2.12671 .49170 2.03376 .51356 1.94718 49 12 .44942 2.22510 .47056 2.12511 .49206 2.03227 .51393 1.94579 48 13 .44977 2.22337 .47092 2.12350 .49242 2.03078 .51430 1.94440 47 14 .45012 2.22164 .47128 2.12190 .49278 2.02929 .51467 1.94301 46 15 .45047 2.21992 .47163 2.12030 .49315 2.02780 .51503 1.94162 45 1G .45082 2.21819 .47199 2.11871 .49351 2.02031 .51540 1.94023 44 17 .45117 2.21647 .47234 2.11711 .49387 2.02483 .51577 1.93885 43 18 .45152 2.21475 .47270 2.11552 .49423 2.02:335 .51614 1.93746 42 19 .45187 2.21304 .47305 2.11392 .49459 2.02187 .51651 1.93608 41 20 .45222 2.21132 .47341 2.11233 .49495 2.02039 .51688 1.93470 40 21 .45257 2.20961 .47377 2.11075 .49532 2.01891 .51724 1.93332 39 22 .45202 2.20790 .47412 2.10916 .49568 2.01743 .51761 1.93195 38 23 .45327 2.20619 .47448 2.10758 .49604 2.01596 .51798 1.93057 37 24 .45362 2.20449 .47483 2.10600 .49640 2.01449 .51835 1.92920 36 25 .45397 2.20278 .47519 2.10442 .49677 2.01302 .5187*2 1.92782 35 20 .45432 2.20108 .47555 2.10284 .49713 2.01155 .51909 1.92645 34 27 .45467 2.19938 .47590 2.10126 .49749 2.01008 .51946 1.92508 33 28 .45502 2.19769 .47626 2.09969 .49786 2.00862 .51983 1.92371 32 29 .45538 2.19599 .47662 2.09811 .49822 2.00715 .52020 1.92235 31 30 .45573 2.19430 .47698 2.09654 .49858 2.00569 .52057 1.92098 30 31 .45608 2.19261 .47733 2.09498 .49894 2.00423 .52094 1.91962 29 32 .45643 2.19092 .47769 2.09341 .49931 2.00277 .52131 1.91826 28 33 .45678 2.18923 .47805 2.09184 .49967 2.00131 .52168 1.91690 27 34 .45713 2.18755 .47840 2.09028 .50004 1.99986 .52205 1.91554 26 35 .45748 2.18587 .47876 2.08872 .50040 1.99811 .52242 1.91418 25 36 .45784 2.18419 .47912 2.C8716 .50076 1.99695 .52279 1.91282 24 37 .45819 2.18251 .47948 2.08560 .50113 1.99550 .52316 1.91147 23 38 .45854 2.18084 .47984 2.08405 .50149 1.99406 .52353 1.91012 22 39 .45889 2.17916 .48019 2.03250 .50185 1.99261 .52390 1.90876 21 40 .45924 2.17749 .48055 2.08094 .50222 1.99116 .52427 1.90741 20 41 .45960 2.17582 .48091 2.07939 .50258 1.98972 .52464 1.90607 19 42 .45095 2.17416 .48127 2.07785 .50295 1.98828 .52501 1.90472 18 43 .46030 2.17249 .48163 2.07630 .50331 1.98684 .52538 1.90337 17 44 .46065 2.17083 .48198 2.07476 .50368 1.98540 .52575 1.90203 16 45 .46101 2.16917 .48234 2.07321 .50404 1.98396 .52613 1.90069 15 46 .46136 2.16751 .48270 2.07167 .50441 1.98253 .52650 1.89935 14 47 .46171 2.16585 .48306 2.07014 .50477 1.98110 .52687 1.8C801 13 48 .46206 2.16420 .48342 2.06S60 .50514 1.97966 .52724 1.89667 12 49 .46242 2.16255 .48378 2.0G706 .50550 1.97823 .52761 1.89533 11 50 .46277 2.16090 .48414 2.06553 .50587 1.97681 .52798 1.89400 10 51 .46312 2.15925 .48450 2.06400 .50623 1.97538 .52836 1.89266 9 52 .46348 2.15760 .48486 2.06247 .50660 1.97395 .52873 1.89133 8 53 .46383 2.15596 [48521 2.06094 .50696 1.97253 .52910 1.89000 7 54 .46418 2.15432 .48557 2.05942 .50733 1.97111 .52947 1.88867 6 55 .46454 2.15268 .48593 2.05790 .50769 1.96969 .52985 1.88734 5 56 .46489 2.15104 .48629 2.05G37 .50806 1.96827 .53022 1.88602 4 57 .46525 2.14940 .48665 2.05485 .50843 1.96685 .53059 1.88469 3 58 .4G560 2.14777 .48701 2.05333 .50879 1.965-44 .53096 1.88337 2 59 .46595 2.14614 .48737 2.05182 .50916 1.96402 .53134 1.88205 1 60 .46631 2.14451 .48773 2.05030 .50953 1.96261 .53171 1.88073 _0 / Cotang Tang ! Cotang Tang Cotang Tang Cotang Tang r 65° 64° 63° 62® 116 NATURAL TANGENTS AND COTANGENTS. 0 00 ©a 29° I 80° 0 rH CO 1 Tan# Cotang 1 Tang Cotang i Tang j Cotang I Tang Cotang *0 .53171 1.88073" .55431 1.80405 j • 5 4 t OO 1.73205 Ì .60086 1.66428 60 1 .53208 1.87941 .55469 1.80281 .57774 1.73089 | .60126 1.60318 59 g .53246 1.87809 i .55507 1.80158 1 .57813 1.72973 j .60165 1.66209 58 3 .53283 1.87677 .55545 1.80034 .57851 1.72857 .60205 1.66099 57 4 .53320 1.87546 .55583 1.79911 1 .57890 1.72741 .60245 1.65990 56 5 .53358 1.87415 .55621 1.79788 Ì .57929 1.72625 1 .60284 1.65881 55 6 .53395 1.87283 .55659 1.79665 .57968 1.72509 ! .60324 1.65772 54 rt .53432 1.87152 .55697 1.79542 | .58007 1.72393 1 .00364 1.65663 53 8 .53470 1.87021 .55736 1.79419 1 .58046 1.72278 .60403 1.65554 52 9 .53507 1.86891 .55774 1.79296 1 1 .58085 1.72163 .60443 1.65445 51 Î0 .53o45 1.80760 .55812 1.79174 .58124 1.72047 .60483 1.65337 50 11 .53582 1.86630 .55850 1.79051 .58162 1.71932 .60522 1.65228 49 12 .53620 1.86499 .55888 1.78929 .58201 1.71817 .60562 1.65120 48 13 .53657 1.86369 .55926 1.78807 .58240 1.71702 .60602 1.65011 47 14 .53694 1.86239 .55904 1.78085 Î .58279 1.71588 .60642 1.64903 46 15 .53732 1.86109 .56003 1.78563 ; .5S318 1.71473 .60681 1.64795 45 16 .53769 1.85979 .56041 1.78441 I .58357 1.71358 .60721 1.64687 44 17 .53807 1.85850 .56079 1.78319 1 .58396 1.71244 1 .60761 1.64579 43 18 .53844 1.85720 .56117 1.78198 1 .58435 1.71129 .60801 1.64471 42 19 .53882 1.85591 .56156 1.78077 ! .58474 1.71015 .60841 1.64363 41 20 .53920 1.85402 .56194 1.77955 | .58513 1.70901 .60881 1.64256 40 21 .53957 1.85333 .56232 1.77834 .58552 1.70787 .60921 1.64148 39 oo .53995 1.85204 .56270 1.77713 .58591 1.70673 1 .60960 1.64041 38 23 .54032 1.85075 .56309 1.77592 , .58631 1.70560 .61000 1.63934 37 24 .54070 1.81946 .56317 1.77471 .58670 1.70446 1 .61040 1.63826 36 25 .51107 1.84818 .56385 1.77351 .58709 1.70332 .61080 1.63719 35 26 .54145 1.84689 .56424 1.77230 1 .58748 1.70219 .61120 1.63612 34 27 .54183 1.84561 .56462 1.77110 .58787 1.70106 1 .61160 1.63505 33 28 .51220 1.84433 .56501 1.7G990 .58826 1..C9992 j .61200 1.63398 32 29 .51258 1.84305 .56539 1.76869 I .58865 1.69879 | .61240 1.63292 31 30 .54296 1.84177 .56577 1.7G749 .58905 1.69766 .61280 1.63185 30 31 .51333 1.81019 .56616 1.76629 .58944 1.69653 .61320 1.63079 29 32 .51371 1.83922 .56654 1.76510 .58983 1.69541 | .61360 1.62972 28 33 .51409 1.83794 .56693 1.76390 I .59022 1.69428 ! .61400 1.62866 27 34 .54446 1.83067 .56731 1.76271 j .59061 1.69316 1 .61440 1.62760 26 35 .54184 1.83540 .50769 1.76151 i .59101 1.69203 .61480 1.62654 25 36 .54522 1.83413 .56808 1.76032 ! .59140 1.69091 .61520 1.62548 24 37 .51560 1.83286 .56846 1.75913 I .59179 1.68979 .61561 1.62442 23 38 .5-1597 1.83159 .56885 1.75794 1 .59218 1.68866 .61601 1.62336 22 39 .51635 1.83033 .56923 1.75675 1 .59258 1.68754 .61641 1.62230 21 40 .54673 1.62906 .56962 1.75556 .59297 1.68643 .61681 1.62125 20 41 .54711 1.82780 .57000 1.75437 .59336 1.68531 .61721 1.62019 19 42 .54748 1.82654 .57039 1.75319 | .59376 1.6&419 .61761 1.61914 18 43 .51786 1.82528 .57078 1.75200 .59415 1.68308 .61801 1.61808 17 44 .51824 1.82402 .57116 1.75082 ! .59454 1.68196 .61842 1.61703 16 45 .54862 1.82276 .57155 1.74964 ; .59494 1.68085 .61882 1.61598 15 46 .54900 1.82150 .57193 1.74S46 I .59533 1.67974 .61922 1.61493 14 47 .54938 1.82025 .57232 1.74728 I .59573 1.67863 .61962 1.61388 13 48 .51975 1.81899 .57271 1.74610 1 .59612 1.67752 .62003 1.61283 12 49 .55013 1.81774 | .57309 1.74492 ; .59651 1.67641 .62043 1.61179 11 50 .55051 1.81649 .57348 1.74375 | .59691 1.67530 .62083 1.61074 10 51 .55089 1.81524 .57386 1.74257 .59730 1.67419 .62124 1.60970 9 52 .55127 1.81399 .57425 1.74140 1 .59770 1.67309 .62164 1.60S65 8 53 .55165 1.81274 1 .57464 1.74022 .59809 1.67198 .62204 1.60761 rt i 54 .55203 1.81150 .57503 1.73905 .59849 1.67088 .62215 1.60657 6 55 .55241 1.81025 1 .57541 1.73788 1 .59888 1.66978 .62285 1.60553 5 56 .55279 1.80901 I .57580 1.73671 1 .59928 1.66867 .62325 1.60449 4 57 .55317 1.80777 .57619 1.73555 1 .59967 1.66757 .62366 1.60345 3 58 .55355 1.80653 .57057 1.73438 .00007 1.66647 .62406 1.60241 2 59 .55393 1.80529 .57696 1.73321 1 .60046 1.66538 .624-16 1.60137 1 60 55431 1.SO105 .57735 1.73205 ; .60086 1.06428 .62487 1.60033 0 Cotang Tang Cotang Tang Cotang Tang l Cotaug Tang 61« 60° 59° 1 58° NATURAL TANGENTS AND COTANGENTS 117 CO r ! CO CO o o ** CO 35° Tang | Cotang | ! Tang Cotang Tang Cotang | Tang Cotang 0 .62487 1.00033 I ! .64941 1.53986 .67451 1.48256 | .70021 1.42815 60 1 .62527 1.599:40 | i .64982 1.53888 .67493 1.48163 1 .7006-1 1.42726 59 2 .62568 1.59826 1 .65024 1.53791 .67536 1.48070 .70107 1.42638 58 3 .62608 1.59723 I .65005 1.53693 .67578 1.47977 j .70151 1.42550 57 4 .62649 1.59620 .65106 1.53595 .67620 1.47885 .70194 1.42462 56 5 .62689 1.59517 .65148 1.53497 .67663 1.47792 .70238 1.42374 55 C .62730 1.59414 .65189 1.53400 .67705 1.47699 .70281 1.42286 54 V .62770 1.59311 .65231 1.53302 .67748 1.47607 .70325 1.42198 53 8 .62811 1.59208 .65272 1.53205 .67790 1.47514 .70368 1.42110 52 9 .62852 1.59105 .65314 1.53107 .67832 1.47422 .70412 1.42022 51 10 .62892 1.59002 .65355 1.53010 .67875 1.47330 .70455 1.41934 50 11 .629.33 1.5S900 .65397 1.52913 .67917 1.47238 .70499 1.41847 49 12 .62973 1.58797 Ì .65438 1.52816 .67960 1.47146 .70542 1.41759 48 13 .63014 1.58695 ! .65180 1.52719 .68002 1.47053 .70586 1.41672 47 14 .63055 1.58593 | .65521 1.52622 .68045 1.40062 .70629 1.41584 46 15 .63095 1.58190 .65563 1.52525 .68088 1.46870 .70673 1.41497 45 16 .63136 1.58388 | .65604 1.52129 .68130 1.46778 .70717 1.41409 44 17 .63177 1.58286 1 .65616 1.52832 .68173 1.46086 .70760 1.41322 43 18 .63217 1.58184 .65688 1.52235 .68215 1.46595 .70804 1.41235 42 19 .63258 1.58083 .65729 1.52139 .68258 1.46503 .70848 1.41148 41 20 .63299 1.57981 .65771 1.52043 ! .68301 1.46411 .70891 1.41061 40 21 .63340 1.57879 .65813 1.51046 | .68343 1.46320 .70935 1.40974 39 22 .633S0 1.57778 .65S51 1.51850 | .68386 1.46229 .70979 1.40887 38 23 .63421 1.57676 .65896 1.51754 .68429 1.46137 .71023 1.40800 37 24 .63462 1.57575 .63933 1.51C58 .68471 1.46046 .710G6 1.40714 36 25 .63503 1.57474 .659S0 1.5J5G2 .68514 1.45955 .71110 1.40627 35 26 .63544 1.57372 .66021 1.5X406 .68557 1.45864 .71154 1.40540 34 27 .63584 1.57271 .66003 1.51370 .68600 1.45773 .71198 1.40454 33 28 .63625 1.57170 .66105 1.51275 .68642 1.45GS2 .71242 1.40367 32 29 .63G66 1.57069 .66147 1.51179 .68085 1.45592 .71285 1.40281 31 30 .63707 1.56969 .66189 1.51084 .68728 1.45501 .71329 1.40195 30 31 .63748 1.56868 .66230 1.50988 .68771 1.45-110 .71373 1.40109 29 32 .63789 1.56767 .66272 1.50893 .68814 1.45320 .71417 1.40022 28 33 .63830 1.56067 .CG314 1.50797 .68857 1.45229 .71461 1.39936 27 31 .63871 1.56566 .60356 1.50702 .G8900 1.45139 .71505 1.39850 2G 35 .63912 1.564G0 .60398 1.50007 .G8942 1.45049 .71849 1.39764 25 36 .63953 1.5G3G3 .66440 1.50512 .68985 1.44953 .71593 1.39679 m (£ .63991 1.5G2G5 .66482 1.50417 .69028 1.448G8 .71637 1.39593 23 .64035 1.50105 .60524 1.50322 .69071 1.44778 .71681 1.39507 22 39 .64076 1.5G0G5 .6G5G6 1.50228 .69114 1.44688 .71725 1.39421 21 40 .64117 1.559G6 .66608 1.50133 .69157 1.44598 .71769 1.39336 20 41 .64158 1.55866 .66650 1.50038 .69200 1.44508 .71813 1.39250 19 42 .64199 1.55766 .66692 1.49944 .69243 1.44418 .71857 1.39165 18 43 .64240 1.55666 .06734 1.49849 .G92S6 1.44329 .71901 1.39079 17' 44 .64281 1.55567 .66776 1.49755 .69329 1.44239 .71946 1.33994 16 45 .64322 1.55467 j .66818 1.49G61 .69372 1.44149 .71990 1.38909 15 46 .61363 1.55303 .66860 1.49566 .69416 1.44000 .72034 1.38824 14 47 .61404 1.55269 | .66902 1.49472 .69459 1.43970 .72078 1.38738 13 48 .61446 1.55170 .66944 1.49378 .69502 1.43881 .72122 1.38G53 12 49 .64487 1.55071 .66986 1.49284 .69545 1.43702 .72167 1.385C8 11 50 .64528 1.54972 .67028 1.49190 .69588 1.43703 .72211 1.38484 10 51 .64569 1.54873 .67071 1.49097 .69631 1.43614 .72255 1.38399 9 52 .64610 1.54774 .67113 1.49003 .69G75 1.4:3525 .72299 1.38314 8 53 .64652 1.54675 .67155 1.48909 .69718 1.43436 .72344 1.38229 7 .64C93 1.54576 1 .67197 1.48316 .69761 1.43347 .72388 1.38145 6 •JO .61734 1.54473 ! .67239 1.43722 .69804 1.43258 .72432 1.38060 5 56 .t)4i i D 1.54379 .67282 1.48G29 .698-17 1.43169 .72477 1.37976 4 o/ .64817 1.54281 1 .67324 1.48536 .69S91 1.43080 .72521 1.37891 3 58 .61858 1.54183 .67306 1.484-12 .69931 1.42902 .72565 1 37807 2 59 .64399 1.54085 .67409 1.48349 .69977 1.42903 .72010 1.37722 1 60 .64941 1.53986 .67451 1.48256 .70021 1.42S15 .7265-1 1.37G38 0 ; Cotang I Tang Cotang Tang Cotang Tang Cotang Tang L £ 7° 53° 55° 54» 118 NATURAL TANGENTS AND COTANGENTS. 36° 37° 38° 39° Tang Co tan g Tang Cotang 1 Tang Cotang Tang Cotang / 0 .72654 1.37638 175355 1.32704 1 .78129 1.27994 .80978 1.23490 GO 1 .72699 1.37554 1 .75401 1.32624 ! .78175 1.27917 .81027 1.23416 59 2 .72743 1.37470 1 .75447 1.32514 ! .78222 1.27841 .81075 1.23343 58 3 .72788 1.37386 .75492 1.32464 j .78269 1.27764 .81123 1.23270 57 4 .72832 1.37302 .75538 1.32384 | 1781316 1.27688 .81171 1.23196 56 5 .72877 1.37218 .75584 1.32304 .78363 1.27611 .81220 1.23123 55 6 .72921 1.371134 .75629 1.32224 .78110 1.27535 .81268 1.23050 54 7 .72966 1.37050 ! .75675 1.32144 .78457 1.27458 .81316 1.22977 53 8 .73010 1.36967 1 .75721 1.32064 j .78504 1.27382 .81364 1.22904 52 9 .73055 1.36883 1 .75767 1.31984 .78551 1.27306 .81413 1.22*31 51 10 .73100 1.36800 .75812 1.31904 j .78598 1.27230 .81461 1.22758 50 11 .73144 1.36716 .75858 1.31825 : .78645 1.27153 .81510 1.22685 49 12 .73189 1.36633 | .75904 1.31745 ; .78692 1.27077 .81558 1.22612 4S 13 .73234 1.36519 .75950 1.31666 i .78739 1.27001 .81606 1.22539 47 14 .73278 1.36466 | .75996 1.31586 j .78786 1.26925 ! .81655 1.22407 46 15 .73323 1.36383 .76042 1.31507 .78834 1.26849 | .81703 1.22394 45 16 .73368 1.36300 .76088 1.31427 I .78881 1.26774 .81752 1.22321 44 17 .73413 1.36217 .76134 1.31348 .78928 1.26698 ! .81800 1.22249 43 18 .78457 1.36134 .76180 1.31269 j .78975 1.26622 .81849 1.22176 42 19 .73502 1.36051 ! .76226 1.31190 ! .79022 1.26546 | .81898 1.22104 41 20 .73547 1.35968 .76272 1.31110 , .79070 1.20471 .81946 1.22031 40 21 .73592 1.35885 .76318 1.31031 .79117 1.26395 .81995 1.21059 39 QO .73637 1.35802 ! .76364 1.30952 .79104 1.26319 . 82044 1.21886 38 23 .73681 1.35719 | .76410 1.30873 1 .79212 1.26244 .82092 1.21814 37 24 .73726 1.35637 .76456 1.30795 i .79259 1.26169 .82141 1.21742 36 25 .73771 1.35554 .76502 1.30716 : .79306 1.26093 .82190 1.21670 35 26 .73816 1.35472 .76518 1.30637 i .79354 1.26018 ! .82238 1.21598 34 27 .73861 1.35389 | .7(i594 1.30558 j .79401 1.25943 | .82287 1.21520 33 28 .73906 1.35307 .76640 1.30480 .79449 1.25867 I .82336 1.21454 90 29 .73951 1.35224 .76686 1.30401 .79496 1.25792 . .82385 1.21382 31 30 .73996 1.35142 .76733 1.30323 .79541 1.25717 .82434 1.21310 30 31 .74041 1.35060 .76779 1.30244 .7*9591 1.25642 1 .82-1&3 1.21238 29 32 .74083 1.34978 .76825 1.30166 .79039 1.25507 .82531 1.21166 2S 33 .74131 1.34896 .76871 1.30087 .79686 1.25492 .82o80 1.21094 27 34 .74176 1.34814 .76918 1.30009 .79734 1.25417 .82029 1.21023 26 35 .74221 1.34732 .76904 1.29931 .79781 1.25343 ! .82678 1.20951 25 36 .74267 1.34650 .77010 1.29853 .79829 1.25268 1 .82727 1.20879 24 37 .74312 1.31568 | .77057 1.29775 .79877 1.25193 .82776 1.20808 23 38 .74357 1.34187 .77103 1.29696 .79924 1.25118 .82825 1.20736 22 39 .74402 1.31405 .77149 1.29618 .79972 1.25044 .82874 1.20665 21 40 .74447 1.34323 .77196 1.29541 .80020 1.24969 j .82923 1.20593 20 41 .74492 1.34242 .77242 1.29463 .80067 1.24895 .82972 1 205*22 19 42 .74538 1.34160 .77289 1.29385 .80115 1.248.20 .83022 1.20451 18 43 .74583 1.34079 | .77335 1.29307 .80163 1.24746 .83071 1.20379 17 44 .74628 1.33998 j .77382 1.29229 .80211 1.24672 .83120 1.20308 16 45 .74674 1.33916 ! .77428 1.29152 .80258 1.24597 .83169 1.20237 15 46 .74719 1.33835 j .7 «475 1.29074 .80306 1.24523 .83218 1.20166 14 47 .74764 1.33754 | .77521 1.2S997 .80354 1.24449 I .83268 1.20095 13 48 .74810 1.33673 1 .77568 1.28919 1 .80402 1.24375 ! .83317 1.20024 12 49 .74855 1.33592 I .77615 1.28842 .80450 1.24301 ! .83366 1.19953 11 50 .74900 1.33511 .77661 1.28764 .80498 1.24227 .83415 1.19SS2 10 51 .74946 1.33430 ! .77708 1.28687 .80546 1.24153 .83465 1.19811 9 'V> .74991 1.33349 ! .77754 1.28610 .80594 1.24079 .83514 1.197*40 8 53 .75037 1.33268 .77801 1.28533 .80642 1.24005 .83564 1.19669 r* ê 54 .75082 1.33187 .77848 1.28456 .80090 1.23931 ! .83613 1.19599 6 55 .75128 1.33107 .77895 1.28379 .80738 1.23858 I .83662 1.19528 5 56 .75173 1.33026 .77941 1.28302 .80786 1.23784 .83712 1.19457 4 57 .75219 1.32946 .77988 1.28225 .a)834 1.23710 .83761 1.19387 3 58 .75264 1.32805 .78035 1.28148 .80882 1.23637 1 .83811 1.19316 o 59 .75310 1.32785 .78082 1.28071 ,80930 1.23563 | .83860 1.10246 1 (X) .75355 1.32704 .78129 1.27994 .80978 1.23490 .83910 1.19175 0 j Cotang to Cotang Tang Cotang Tang Cotang Tang / 53° II 52° il 51° II 50°NATURAL TANGENTS AND COTANGENTS 119 40° 41° 42* 4* CO O , i / Tang 1 Cotang 1 Tang Cotang Tang | Cotang Tang I Cotang "Ó .83910 I 1.19175 .86929 1.15037 .90040 1.11061 .93252 1.07237 60 1 .83960 : 1.19105 .86980 1.14969 .90093 1 10996 .93306 1.07174 59 A .84009 1.19035 .87031 1.14902 .90146 1.10931 .93300 1.07112 58 3 .81059 1.18964 .87082 inn .90199 1.10867 .9:3415 1.07049 57 4 .84108 1.18894 1 .»87133 1.14767 .90251 1.10802 .93469 1.06987 56 5 .84158 1 1.18824 .87184 1.14699 .90304 1.10737 .93524 1.00925 55 6 .84208 1 1.18754 ! .87236 1.14632 .90357 1.10072 .93578 1.06862 54 .84258 I 1.18684 .87287 1.14565 .90410 1.10607 .93633 1.00800 53 8 .84307 1.18614 .87338 1.14498 .90403 1.10543 .93688 1.06738 52 9! .84357 | 1.18544 .87389 1.144Î10 .90516 1.10478 .93742 1.06676 51 10 j .84407 1.18474 .87441 1.14363 .90569 1.10414 .93797 1.06613 50 11 .84457 1.18404 .87492 1.14296 .90621 1.10349 .93852 1.06551 49 12 .84507 1.18334 .87543 1.14229 .90674 1.10285 .93906 1.06489 48 13 .84556 1.18264 .87595 1.14162 .90727 1.19220 .93961 1.06427 47 14 .84606 1.18194 I .87646 1.14095 .90781 1.10156 .94016 1.06365 46 15 .84656 1.18125 .87698 1.14028 .90834 1.10091 .94071 1.06303 45 16 .84706 1.18055 ! .87749 1.13961 .90887 1.10027 .01125 1.0C241 44 17 .84756 1.17986 j .87801 1.13894 .90940 1.09963 ! .94180 1.06179 43 18 .84806 1.17916 1 .87852 1.13828 .90993 1.09899 j .94235 1.06117 42 19 .84856 1.17846 .87904 1.13761 .91046 1.09834 .94290 1.06056 41 20 .84906 1.17777 .87955 1.13694 .91099 1.09770 .94345 1.05994 40 21 .84956 1.17708 .88007 1.13627 .91153 1.09706 .94400 1.05932 39 22 .85006 1.176:38 1 .88059 1.13561 .91206 1.09042 .94455 1.05870 38 23 .85057 1.17569 .88110 1.13494 .91259 1.09578 .94510 1.05809 37 24 .85107 1.17500 .88162 1.13428 .91313 1.09514 .94565 1.05747 36 25 .85157 1.17430 .88214 1.13361 .91366 1.09450 .94620 1.05685 35 26 .85207 1.17361 ! .88265 1.13295 .91419 1.09:386 .94676 1.05624 34 27 .85257 1.17292 1 .88317 1.13223 .91473 1.09322 i .94731 1.05562 33 28 .85308 1.17223 .88369 1.13162 .91526 1.09258 ; .94786 1.05501 32 29 .85358 1.17154 .88421 1.13096 .91580 1.09195 .94841 1.05439 31 30 .85408 1.17085 .88473 1.13029 .91633 1.09131 .9-4896 1.05378 30 31 .85458 1.17016 .88524 1.12963 .91687 1.09067 .94952 1.05317 29 32 .85509 1.16947 .8857G 1.12897 .91740 1.09003 .95007 1.05255 28 33 .85559 1.16878 .88628 1.12831 .91794 1.08040 .95062 1.05194 27 31 .85609 1.16S09 .88680 1.12765 .91847 1.08S76 .95118 1.05133 26 35 .85660 1.16741 .88732 1.12G99 .91901 1.0SS13 .95173 1.05072 25 36 .85710 1.16672 .88784 1.12633 .91955 1.08749 .95229 1.05010 24 37 .85761 1.16603 .88836 1.12567 .92008 1.08686 .95284 1.04949 23 38 .85811 1.16535 .88888 1.12501 .92062 1.0SG22 .95340 1.04888 22 39 .85862 1.16466 .88940 1.12435 .92116 1.08559 .95395 1.04827 21 40 .85912 1.16398 .88992 1.12369 .92170 1.08496 .95451 1.04766 20 41 .85963 1.1G329 .89045 1.12303 .92224 1.08432 .95506 1.04705 19 42 .86014 1.16261 .89097 1.12238 .92277 1.08369 .95562 1.04644 T8 43 .86064 1.16192 .89149 1.12172 .92331 1.08306 .95G18 1.04583 17 41 .86115 1.16124 .89201 1.12106 .92385 1.08243 .95673 1.04522 10 45 .86166 1.16056 .89253 1.12041 .92439 1.08179 .95729 1.04461 15 4G .66216 1.15987 .89306 1.11975 .92493 1.08116 .95785 1.04401 14 47 .86267 1.15919 .89358 1.11909 .92547 1.08053 .95841 1.04340 13 48 .86318 1.15851 .89410 1.11844 .92601 1.07990 .95897 1.04279 12 49 .66368 1.15783 .89463 1 11778 .92655 1.07927 .95952 1.04218 11 50 .86419 1.15715 .89515 1.11713 .92709 1.07804 .9G008 1.04158 10 51 ! .86470 1.15647 .89567 1.11648 .92763 1.07801 .9C064 1.04097 9 52 .86521 1.15579 .89620 1.11582 .92817 1.07738 .96120 1.04036 8 53 .86572 1.15511 .89672 1.11517 .92872 1.07676 .96176 1.03976 7 54 .86623 1.15443 .89725 1.11452 .92926 1.07613 .96232 1.03915 6 55 .86674 1.15375 1 .89777 1.11387 .92980 1.07550 .96288 1.03855 5 56 .86725 1.15308 .89830 1.11321 .930:34 1.07487 .96344 1.03794 4 57 .86776 1.15240 .89883 1.11256 .93088 1.07425 .96400 1.03734 3 58 .86827 1.15172 .899:15 1.11191 .93143 1.07362 .96457 1.03G74 o 59 .86878 1.15104 .89988 1.11126 .93197 1.07299 .96513 1.03C13 1 60 .86929 1.15037 f ,.90040 1.11061 I .93252 1.07237 .96569 1.03553 0 / Cotang | Tang , Cotang | Tang Cotang Tang ! Cotang Tang $ 49° |l 48° 11 47° Jt 46°120 NATURAL TANGENTS AND COTANGENTS. 0 44° Tang Cotang Tang Cotang 0 ■JR 1.03553 60 20 .97700 1.02355 1 .90025 1.03493 59 21 .97750 1.02295 2 .90081 1.03433 58 o.) .97813 1.02230 3 .907:18 1.03372 57 23 .97870 1.02170 4 .90791 1.03312 50 24 .97927 1.02117 5 .90850 1.03252 55 25 .97984 1.02057 6 .90907 1.03192 54 1 20 .98041 1.01998 r* 4 .909(53 1.03132 53 27 .98098 1.01939 8 .97020 1.03072 52 28 .98155 1.01879 9 .97070 1.03012 ■ 29 .98213 1.01820 10 .97133 1.02952 50 30 .98270 1.01701 11 .97189 1.02892 49 PI .98327 1.01702 \2 .97240 1.02832 48 32 .98384 1.01042 13 .97302 1.02772 47 33 .98441 1.01583 14 .97359 1.02713 40 34 .98499 1.01524 15 .97410 1.02053 45 35 .98550 1.01465 /6 .97472 1.02593 44 30 .98013 1.01400 1? .97529 1.02533 43 37 .98071 1.01347 18 .97580 1.02474 42 38 .98728 1.01288 19 .97013 1.02414 41 39 .98780 1.01229 m .97700 1.02355 40 40 .98843 1.01170 f Cotang Tang / / Cotang Tang 45° 46! I 1 44o Tang Cotang 40 40 .98843 1.01170 20 39 141 .98901 1.01112 19 38 ' 42 .98958 1.01053 18 37 1 43 .99010 1.00994 17 30! i44 .99073 1.00935 16 m 45 .99131 1.00870 15 34 ! 40 .99189 1.00818 U 33 ; 47 .99247 1.00759 13 32 48 .99304 1.00701 12 31 i 49 .99302 1.00642 11 30 ;5o .99420 1.00583 10 29 51 .99478 1.00525 9 28 ! 52 .99530 1.00407 8 27 I 53 .99594 1.00408 r* 1 1 j 54 .99052 1.00:450 0 25 1 i 55 .99710 1.00291 5 21 i 50 .99708 1.00233 4 23 1 i 57 .99820 1.00175 3 1 58 .99884 1.00110 0 21 j ,59 .99942 1.00058 1 j 00 1.00000 1.00000 0 t t Ootang Tang / __ 45» iPART IL Strength of Materials, and Stability of Structures.~,e——----------------------- 1 t JINTRODUCTION. 123 INTRODUCTION. IM the chapters constituting this part of the book, the author has endeavored to present to architects and builders handy and reliable rules and tables for determining the strength or stability of any piece of work they may have in hand. Every pains has been taken to present the rules in the simplest form consistent with their accuracy; and it is believed that all constants and theories advanced are fully up to the knowledge of the present day, some of the constants on transverse strength having but recently been determined. The rules for wrouglit-iron columns have lately been slightly changed by some engineers; but as the question of the strength of wrouglit-iron columns has not yet been satisfactorily settled, and as the formulas herein given undoubtedly err on the safe side if at all, we have thought best not to change them, especially as they are still used by many bridge engineers. The question of the wind-pressure on roofs has not been taken up in as thorough manner as would be needed for pitch roofs of very great span; but for ordinary wooden roofs, and iron roofs not exceeding one hundred feet span, the method given in Chap. XVI11. is sufficiently accurate. Any one wishing to study the most accurate method of obtaining the effect of the wind-pressure on roofs will find it in Professor Green’s excellent work on “ Graphical Analysis of Roof Trusses.” In conclusion, the author recommends these chapters as presenting accurate and modern rules, especially adapted to the requirements of American practice.124 EXPLANATION OF SIGNS AND TERMS. EXPLANATION OF SIGNS AND TERMS USED IN THE FOLLOWING FORMULAS. Besides the usual arithmetical signs and characters in general use, the following characters and abbreviations will frequently be used:— The sign y/ means square root of number behind. y means cube root of number behind. • ( ) means that all the numbers between are to be taken as one quantity. . means decimal parts; 2.5 = 2-fa, or .46 = t$>. The letter A denotes the co-efficient of strength for beams one inch square, and one foot between the supports. C denotes resistance, in pounds, of a block of any material to crushing, per square inch of section. E denotes the modulus of elasticity of any material, in pounds per square inch. e denotes constant for stiffness of beams. F denotes resistance of any material to shearing, per square inch. R denotes the modulus of rupture of any material. S denotes a factor of safety. T denotes resistance of any material to being pulled apart, in pounds, per square inch of cross-section. Breadth is used to denote the least side of a rectangular piece, and is always measured in inches. Depth denotes the vertical height of a beam or girder, and is always to be taken in inches, unless expressly stated otherwise. Length denotes the distance between supports in feet, unless otherwise specified. Abbreviations. — In order to shorten the formulas, it has often been found necessary to use certain abbreviations; such as bet. for between, bot. for bottom, dist. for distance, diam. for diameter, hor. for horizontal, sq. for square, etc., which, however, can in no case lead to uncertainty as to their meaning. Where the word “ton” is used in this volume, it always means 2000 pounds.DEFINITIONS tfF TERMS. 125 CHAPTER I. DEFINITIONS OF TERMS USED IN MECHANICS. The following terms frequently occur in treating of mechanical construction, and it is essential that their meaning be well understood. Mechanics is the science which treats of the action of forces. Applied Mechanics treats of the laws of mechanics which relate to works of human art; such as beams, trusses, arches, etc. IJest is the relation between two points, when the straight line joining them does not change in length or direction. A body is at rest relatively to a point, when any point in the body is at rest relatively to the first-mentioned point. Motion is the relation between two points, when the straight line joining them changes in length or direction, or in both. A body moves relatively to a point, when any point in the body moves relatively to the point first mentioned. Force is that which changes, or tends to change, the state of a body in reference to rest or motion. It is a cause regarding the essential nature of which we are ignorant. We cannot deal with forces properly, but only with the laws of their action. Equilibrium is that condition of a body in which the forces acting upon it balance or neutralize each other. Statics is that part of Applied Mechanics which treats of the conditions of equilibrium, and is divided into: — «. Statics of rigid bodies. b. Hydrostatics. In building we have to deal only with the former. Structures are artificial constructions in which all the parts are intended to be in equilibrium and at rest, as in the case of a bridge or roof-truss. They consist of two or more solid bodies, called pieces, which are connected at portions of their surfaces called joints. There are three conditions of equilibrium in a structure; viz.: — I. The forces exerted on the whole structure must balance each other. These forces are: — a. The weight of the structure. b. The load it carries.DEFINITIONS OF TERMS I 126 c. The supporting pressures, or resistance of the foundations, called external forces. II. The forces exerted on each piece must balance each other. These forces are: — a. The weight of the piece. b. The load it carries. c. The resistance of its joints. III. The forces exerted on each of the parts into which any piece may be supposed to be divided must balance each other. Stability consists in the fulfilment of conditions I. and II., that is, the ability of the structure to resist displacement of its parts. Strength consists in the fulfilment of condition III., that is, the ability of a piece to resist breaking. Stiffness consists in the ability of a piece to resist bending. The theory of structures is divided into two parts; viz.: — I. That which treats of strength and stiffness, dealing only with single pieces, and generally known as strength of materials. II. That which treats of stability, dealing with structures. Stress. — The load or system of forces acting on any piece of material is often denoted by the term “stress,” and the word will be so used in the following pages. The intensity of the stress per square inch on any normal sur-I face of a solid is the total stress divided by the area of the section in square inches. Thus, if we had a bar ten feet long and two inches square, with a load of 8000 pounds pulling in the direction of its length, the stress on any normal section of the rod would be 8000 pounds; and the intensity of the stress per square inch would be 8000 -T- 4, or 2000 pounds. Strain. — When a solid body is subjected to any kind of stress, an alteration is produced in the volume and figure of the body, and this alteration is called the “ strain.” In the case of the bar given above, the strain would be the amount that the bar would stretch under its load. The Ultimate Strength, or Breaking Load, of a body is the load required to produce fracture in some specified way. The Safe Load is the load that a piece can support without j impairing its strength. - Factors of Safety. — When not otherwise specified, a factor of safety means the ratio in which the breaking load exceeds the safe load. In designing a piece of material to sustain a certain load, it is required that it shall be perfectly safe under all circumstances; and hence it is necessary to make an allowance for any defects in the material, workmanship, etc. It is obvious, that, forUSED IN MECHANICS. 127 materials of different composition, different factors of safety will be required. Thus, iron being more homogeneous than wood, and •css liable to defects, it does not require so great a factor of safety. And, again, different kinds of strains require different factors If safety. Thus, a long wooden column or strut requires a greater factor of safety than a wooden beam. As the factors thus vary for different kinds of strains and materials, we will give the proper factors of safety for the different strains when considering the resistance of the material to those strains. Distinction between Dead and Live Load. — The term “dead load,” as used in mechanics, means a load that is applied by imperceptible degrees, and that remains steady; such as the weight of the structure itself. A “live load” is one that is applied suddenly, or accompanied with vibrations; such as swift trains travelling over a railway-bridge, or a force exerted in a moving machine. It has been found by experience, that the effect of a live load on a Iteam or other piece of material is twice as severe as that of a dead load of the same weight: hence a piece of material designed to carry a live load should have a factor of safety twice as large as one designed to carry a dead load. The load produced by a crowd of people walking on a floor is usually considered to produce an effect which is a mean between that of a dead and live load, and a factor of safety is adopted accordingly. The Modulus oi' Rupture is a constant quantity found in the formulas for strength of iron beams, and is eighteen times the value of the constant “ A.” Modulus of Elasticity. — If we take a bar of any elastic material, one inch square, and of any length, secured at one end, and to the other apply a force pulling in the direction of its length, we shall find by careful measurement that the bar has been stretched or elongated by the action of the force. Now, if we divide the total elongation in inches by the original length of the bar in inches, we shall have the elongation of the bar per unit of length; and, if we divide the pulling-force per square inch by this latter quantity, we shall have what is known as the modulus of elasticity. Hence we may define the modulus of tiajtlicift/ ax the pulling or compressing force per unit of section divided by the elongation or compression per unit of length. As an example of the method of determining the modujus of elasticity of any material, we will take the following: — Nup]>osc we have a bar of wrought-iron, two inches square and128 DEFINITIONS OF TERMS. ten feet long, securely fastened at one end, and to the other end we apply a pulling-force of 40,000 pounds. This force causes the bar to stretch, and by careful measurement we find the elongation to be 0.0414 of an inch. Now, as the bar is ten feet, or 120 inches, long, if we divide 0.0414 by 120, we shall have the elongation of the bar per unit of length. Performing this operation, w'e have as the result 0.00034 of an inch. As the bar is two inches sciuare, the area of cross-section is four square inches, and hence the pulling-force per square inch is 10,000 pounds. Then, dividing 10,000 by 0.00034, we have as the modulus of elasticity of the bar 29,400,000 pounds. This is the method generally employed to determine the modulus of elasticity of iron ties; but it can also be obtained from the deflection of beams, and it is in that way that the values of the modulus for most woods have been found. Another definition of the modulus of elasticity, and which is a natural consequence of the one just given, is the number of pounds that would be required to stretch or shorten a bar one inch square by an amount equal to its length, provided that the law of perfect elasticity would hold good for so great a range. The modulus of elasticity is generally denoted by E, and is used in the determination of the stiffness of beams. Moment.— If we take any solid body, and pivot it at any point, and apply a force to the body, acting in any direction except in a line with the pivot, we shall produce rotation of the body, provided the force is sufficiently strong. This rotation is produced by what is called the moment of the force; and the moment of - a force .about any given point or pivot is the product of the force into the perpendicular distance from the pivot to the line of action of the force, or, in common phrase, the -product of the force into the arm with which it acta. Tlie Centre of Gravity of a body is the point through which the resultant of the weight of the body always acts, no matter in what position the body be. If a body be suspended at its centre of gravity, and revolved in any direction, it will always be in equilibrium. (For centre of gravity of surfaces, lines, and solids, see Chap. IV. ICLASSIFICATION OF STRAINS. 129 CLASSIFICATION OF STRAINS WHICH MAY BE PRODUCED IN A SOLID BODY. The different strains to which building-materials may be exposed are: — I. Tension, as in the case of a weight suspended from one end of a rod, rope, tie-bar, etc., the other end being fixed, tending to stretch or lengthen the fibres. II. Shearing“ Strain, as in the case of treenails, pins in bridges, etc., where equal forces are applied on opposite sides in such a manner as to tend to force one part over the adjacent one. III. Compression, as in the case of a weight resting on top of a column or post, tending to compress the fibres. IY. Transverse or Cross Strain, as in the case of a load on a beam, tending to bend it. Y. Torsion, a twisting strain, which seldom occurs in building-construction, though quite frequently in machinery.130 FOUNDATIONS. CHAPTER II. FOUNDATIONS. Tiik following chapter on Foundations is intended to furnish the reader with only a general knowledge of the subject, and to enable him to be sure that he is within the limits of safety if he follows what is here given. For foundations of large works, or buildings upon spil of questionable firmness, the compressibility of the soil should be determined by experiments. The term “foundation” is used to designate all that portion of any structure which serves only as a basis on which to erect the superstructure. This term is sometimes applied to that portion of the solid material of the earth upon which the structure rests, and also to the artificial arrangements which may be made to support the base. In the following pages these will be designated by the term “ foundation-bed.” Object of Foundations. — The object to be obtained in the construction of any foundation is to form such a solid base for the superstructure that no movement shall take place after its erection. But all structures built of coarse masonry, whether of stone, or brick, Mill settle to a certain extent; and, Mitli a feur exceptions, all soils M ill become compressed under the u'eight of almost any building. Our main object, therefore, is not to prevent settlement entirely, but to insure that it shall be uniform; so that, after the structure is finished, it M’ill have no cracks or flaM'S, liouever irregularly it may be disposed over the area of its site. Foundations Classed. — Foundations maybe divided into two classes: — Class I. — Foundation}* constructed in situations where the natural soil is sufficiently firm to bear the weiyht of the intended structure. Cl,ass II. —Foundations in situations where an artificial bearing-stratum must be formed, in consequence of the softness or looseness of the soil.FOUNDATIONS. B Each of these two great classes may be subdivided into two divisions: — a. Foundations in situations where water offers no impediment to the execution of the work. b. Foundations under water. It is seldom that architects design buildings whose foundations are under water; and, as this division of the subject enters rather deeply into the science of engineering, we shall not discuss it here. Borins'. — Before we can decide what kind of foundation it will be necessary to build, we must know the nature of the subsoil. If not already known, this is determined, ordinarily, by digging a trench, or making a pit, close to the site of the proposed works, to a depth sufficient to allow the different strata to be seen.. For important structures, the nature of the subsoil is often determined by boring with the tools usually employed for this purpose. When this method is employed, the different kinds and thickness of the strata are determined by examining the specimens brought up by the auger used in boring. Foundations of tlie First Class. — The foundations included under this class may be divided into two cases, according to the different kinds of soil on which the foundation is to be built: — Case I. — Foundations on soil composed of materials whose stability is not affected by saturation with water, and ichich are firm enough to support the weight of the structure. Under this case belong, — Foundations on Iiock. — To prepare a rock foundation for being built upon, all that is generally required is to cut away the loose and decayed portions of the rock, and to dress the rock to a plane surface as nearly perpendicular to the direction of the pressure as is practicable; or, if the rock forms an inclined plane, to cut a series of plane surfaces, like those of steps, for the wall to rest on. If there are any fissures in the rock, they should be filled with concrete or rubble masonry. Concrete is better for this purpose, as, when once set, it is nearly incompressible under any thing short of a crushing-force; so that it forms a base almost as solid as the rock itself, while the compression of the mortar joints of the masonry is certain to cause some irregular settlement. If it is unavoidably necessary that some parts of the foundation shall start from a lower level than others, care should be taken to keep the mortar joints as close as possible, or to execute the lower portions of the work in cement, or some hard-setting mortar: otherwise the foundations will settle unequally, and thus cause much injury to the superstructure. The load placed on the rock should at no time exceed one-eighth of that necessary to crush it. 1’ip-132 FOUNDATIONS. fessor Rankine gives the following examples of the actual intensity of the pressure per square foot on some existing rock foundations : — Average of ordinary cases, the rock being at least as strong as the strongest red bricks............................2(KXX) Pressures at the base of St. Rollox chimney (450 feet below the summit)---------- On a layer of strong concrete or beton, 6 feet deep .... 6670 On sandstone below the beton, so soft that it crumbles in the hand................................................... 4000 The last example shows the pressure which is safely borne in practice by one of the weakest substances to which the name of rock can be applied. M. Jules Gaudard, C.E., states, that, on a rocky ground, the Roquefavour aqueduct exerts a pressure of 26,S00 pounds to the square foot. A bed of solid rock is unyielding, and appears at first sight to offer all the advantages of a secure foundation. It is generally found in practice, however, that, in large buildings, part of the foundations will not rest on the rock, but on the adjacent soil; and as the soil, of whatever material it may be composed, is sure to be compressed somewhat, irregular settlement will almost invariably take place, and give much trouble. The only remedy in such a case is to make the bed for the foundation resting on the soil as firm as possible, and lay the wall, to the level of the rock, in cement or hard-setting mortar. Foundation on Compact Stony Earths, such as Gravel or Sand. — Strong gravel may be considered as one of the best soils to build upon; as it is almost incompressible, is not affected by exposure to the atmosphere, and is easily levelled. Sand is also almost incompressible, and forms an excellent foundation as long as it can be kept from escaping; but as it has no cohesion, and acts like a fluid when exposed to running water, it should be treated with great caution. The foundation bed in soils of this kind is prepared by digging a trench from four to six feet deep, so that the foundation may be started below the reach of the disintegrating effects of frost. The bottom of the trench is levelled; and, if parts of it are required to be at different levels, it is broken into steps. Care should be taken to keep the surface-water from running into the trench; and, if necessary, drains should be made at the bottom to carry away the water. The weight resting on the bottom of the trench should be proportional to the resistance of the material forming the bed.FOUNDATIONS. 133 Mr. Gaudard says that a load of 16,500 to 18,300 pounds per square foot lias been put upon close sand in the foundations of Gorai Bridge, and on gravel in the Lock Ken Viaduct at Bordeaux. In the bridge at Nantes, there is a load of 15,200 pounds to the square foot on sand; but some settlement has already taken place. Rankine gives the greatest intensity of pressure on foundations in firm earth at from 2500 to 3500 pounds per square foot. In order to distribute the pressure arising from the weight of the structure over a greater surface, it is usual to give additional breadth to the foundation courses: this increase of breadth is called the spread. In compact, strong earth, the spread is made one and a half times the thickness of the wall, and, in ordinary earth or sand, twice that thickness. Case II.—Foundations on soils firm enough to support the leeight of the structure, but whose stability is affected by water. The principal soil under this class, with which we have to do, is a clay soil. In this soil the bed is prepared by digging a trench, as in rocky soils; and the foundation must be sure to start below the frost-line, for the effect of frost in clay soils is very great. The soil is also much affected by the action of water; and hence the ground should he well drained before the work is begun, and the trenches so arranged that the water shall not remain in them. And, in general, the less a soil of this kind is exposed to the air and weather, and the sooner it is protected from exposure, the better for the work. In building on a clay bank, great caution should be used to secure thorough drainage, that the clay may not have a tendency to slide during wet weather. The safe load for stiff clay and marl is given by Mr. Gaudard at from 5500 to 11,000 pounds per square foot. Under the cylindrical piers of the Szegedin Bridge in Hungary, the soil, consisting of clay intermixed with fine sand, bears a load of 13,300 pounds to the square foot; but it was deemed expedient to increase its supporting power by driving some piles in the interior of the cylinder, and also to protect the cylinder by sheeting outside. Mr. McAlpine, M- Inst. C.E., in building a high wall at Albany, N.Y.j succeeded in safely loading a wet clay soil with two tons to the square foot, but with a settlement depending on the depth of the excavation. In order to prevent a great influx of water, and consequent softening of the soil, he surrounded the excavation with a puddle trench ten feet high and four feet wide, and he also spread a layer of coarse gravel on the bottom. Foundations in Soft Earths. — There are three materials in general use for forming an artificial bearing-stratum in soft soils.134 FOUNDATIONS. Whichever material is employed, the bed is first prepared by excavating a trench sufficiently deep to place the foundation-courses below the action of frost and rain. Great caution should be used in cases of this kind to prevent unequal settling. The bottom of the trench is made level, and covered with a bed of stones, sand, or concrete. Stones. —When stone is used, the bottom of the trench should be paved with rubble or cobble stones, well settled in place by ramming. On this paving, a bed of concrete is then laid. Sand. — In all situations where the ground, although soft, is of sufficient consistency to confine the sand, this material may be used with many advantages as regards both the cost and the stability of the work. The quality which sand possesses, of distributing the pressure put upon it, in both a horizontal and vertical direction, makes it especially valuable for a foundation bed in this kind of soil; as the lateral pressure exerted against the sides of the foundation pit greatly relieves the bottom. There are two methods of using sand; viz., in layers and as piles. In forming a stratum of sand, it is spread in layers of about nine inches in thickness, and each layer well rammed before the next one is spread. The total depth of sand used should be sufficient to admit of the pressure on the upper surface of the sand being distributed over the entire bottom of the trench. Sand-piling is a very economical and efficient method of forming a foundation under some circumstances. It would not, however, be effective in very loose, wet soils; as the sand would work into the surrounding ground. Sand-piling is executed by making holes in the soil, or in the bottom of the trench, about six or seven inches in diameter, and about six feet deep, and filling them with damp sand, well rammed so as to force it into every cavity. In situations where the stability of piles arises from the pressure of the ground around them, sand-piles are found of more service than timber ones, for the reason that the timber-pile transmits pressure only in a vertical direction, while the sand-pile transmits it over the whole surface of the hole it fills, thus acting on a large area of bearing-surface. The ground above the piles should be covered with planking, concrete, or masonry, to prevent its being forced up by the lateral pressure exerted by the piles; and, on the stratum thus formed, the foundation walls may be built in the usual manner. Foundations on Piles. — Where the soil upon which we wish to build is not firm enough to support the foundation, one of the most common methods of forming a solid foundation bed isFOUNDATIONS. 135 by driving wooden piles into the soil, and placing the foundation walls upon these. The piles are generally round, and have a length of about twenty times their mean diameter of cross-section. The diameter of the head varies from nine to eighteen inches. The piles should be straight grained, and free from knots and ring strokes. Fir, beach, oak, and Florida yellow-pine are the best woods for piles; though spruce and hemlock are very commonly used. Where piles are exposed to tide-water, they are generally driven with their bark on. In other cases, it is not essential. Piles which are driven through hard ground, generally require to have an iron hoop fixed tightly on their heads to prevent them from splitting, and also to be shod with iron shoes, either of cast or wrought iron. Long piles may be divided into two classes, —those which transmit the load to a firm soil, thus acting as pillars; and those where the pile and its load are wholly supported by the friction of the earth on the sides of the pile. In order to ascertain the safe load which it will do to put upon a pile of the first class, it is only necessary to calculate the safe crushing-strength of the wood; but, for piles of the second and more common class, it is not so easy to determine the maximum load which they will safely support. Many writers have endeavored to give rules for calculating the effect of a given blow in sinking a pile; but investigations of this kind are of little practical value, because we can never be in possession of sufficient data to obtain even an approximate result. The effect of each blow' on the pile will depend on the momentum of the blow, the velocity of the ram, the relative weights of the ram and the pile, the elasticity of the pile-head, and the resistance offered by the ground through which the pile is passing; and, as the last-named conditions cannot well be ascertained, any calculations in w'hich they are only assumed must of necessity be mere guesses. Load on Piles. — Professor Rankine gives the limits of the safe load on piles, based upon practical examples, as follows: — For piles driven till they reach the firm ground, 1000 pounds per square inch of area of head. For piles standing in soft ground by friction, 200 pounds per square inch of area of head. But as, in the latter case, so much depends upon the character of the soil in which the piles are driven, such a general rule as the above is hardly to be recommended. Several rules for the bearing-load on piles have been given,136 FOUNDATIONS. founded upon practical experience; and they are probably the best that we can rely upon, with our present knowledge of the subject. Perhaps the rule most commonly given is that of Major Sanders, United-States Engineer. He experimented largely at Fort Delaware, and in 1851 gave the following rule as reliable for ordinary pile-driving. Sanders’s Rule for determining the load for a common wooden pile, driven until it sinks through only small and nearly equal distances under successive blows: — weight of hammer in lbs. X fall in inches Safe load in lbs. = ------^“sinking at last blow-------- Mr. John C. Trautwine, C.E., in his pocket-book for engineers, gives a rule which appears to agree very well with actual results. Ilis rule is expressed as follows: — cube root of x weight of x 1 ™ Extreme load in _ fad hi feet hammer in lbs. tons of 2240 lbs. — Last sinking in inches + 1 For the safe load he recommends that one-half the extreme load should be taken for piles thoroughly driven in firm soils, and one-fourtli when driven in river-mud or marsh. According to Mr. Trautwine, the French engineers consider a pile safe for a load of 25 tons when it refuses to sink under a hammer of 1344 pounds falling 4 feet. The test of a pile having been sufficiently driven, according to the best authorities, is that it shall not sink more than one-fifth of an inch under thirty blows of a ram weighing 800 pounds, falling 5 feet at each blow. A more common rule is to consider the pile fully driven when it does not sink more than one-fourth of an inch at the last blow of a ram weighing 2500 pounds, falling 30 feet. In ordinary pile-driving for buildings, however, the piles often sink more than this at the last blow; but, as the piles are seldom loaded to their full capacity, it is not necessary to be so particular as in the foundations of engineering structures. A common practice with architects is to specify the length of the piles to be used, and the piles are driven until their heads are just above ground, and then left to be levelled off afterwards. Example of Pile Foundation. — As an example of the method of determining the necessary number of piles to support a given building, we will determine the number of piles required to support, the side-walls of a warehouse (of which a vertical sec-FOUNDATIONS. 137 tion is shown in Fig. 1). The walls are of brick, and the weight may be taken at 110 pounds per cubic foot of masonry. The piles are to be driven in two rows, two. feet on centres; and it is found that a pile 20 feet long and 10 inches at the top will sink 1 ROOF 30 LBS. PER O V/j 11 TT- 1 iv J • j u V/7- - w\ > 1 VQ 90 LBS. per n § M iv 1 1 1 1 1 Vt77\ Wi B 90 LBS. m. U ii 1 m m m © m ii i II '2 0/ Z/A I mm 10O LBS. E 175 LBS. 175 LBS. 200 LBS. m i ~TT V 1 IC3; m H Vi m U ■ vm. m m HK n Fig. 1. ua one inch under a 1200-pound hammer falling 20 feet after the pile has been entirely driven into the soil. What distance should the piles be on centres lengthwise of the wall ?138 FOUNDATIONS. By calculation we find that the wall contains 157 J cubic feet of masonry per running foot, and hence weighs 17,300 pounds. The load from the floors which comes upon the wall is: — From the first floor ........................ 1500 lbs. From the second floor.........................13S0 lbs. From the third floor......................... 1380 lbs. From the fourth floor......................... 790 lbs. From the fifth floor.......................... 720 lbs. From the sixth floor.......................... 720 lbs. From the roof................................. 240 lbs. Total................................... 6730 lbs. Hence the total weight of the wall and its load per running foot is 24,036 pounds. The load which one of the piles will support is, by Sanders’s rule, 1200 X 240 —a — = 36000 pounds. By Trautwine’s rule, using a factor of safety of 2.5, the safe load would be (^20 X 1200 X 0.023 —2 5 x~(T+~l)-------= 1'”* f°ns (°f 2240 pounds), or 33600 pounds. Then one pair of piles would support 72,000, or 67,200 pounds, according to which rule we take. Dividing these numbers by the weight of one foot of the wall and its load, we find, that, by Sanders’s rule, one pair of piles will support 3 feet of the wall, and, by Trautwine’s rule, 2.8 feet of wall: hence the piles should be placed 2 feet 9 inches or 3 feet on centres. In very heavy buildings, heavy timbers are sometimes bolted to the tops of the piles, and the foundation walls built on these. In Boston, Mass., a large part of the city is built upon made land, and hence the buildings have to be supported by pile foundations. The Building Laws of the city require that all buildings “exceeding thirty-five feet in height (with pile foundation) shall have not less than two rows of piles under all external and party walls, and the piles shall be spaced not over three feet on centres in the direction of the length of the wall.” As an example of the load which ordinary piles in the made land of Boston will support, it may be stated that the piles under Trinity Church in Boston support two tons each, approximately. For engineering works, various kinds of iron piles are used; but they are too rarely used for foundations of buildings to come within the scope of this chapter. For a description of theseFOUNDATIONS. 139 piles the reader should consult some standard work on engineering. A very good description of iron piles is given in “Wheeler’s Civil Engineering,” and also in “ Trautwine’s Handbook.” Concrete Foundation Beds. — Concrete is largely used for foundation beds in soft soil, and is a very valuable material for this purpose; as it affords a iirm solid bed, and can be spread out so as to distribute the pressure over a large area. Concrete is an artificial compound, generally made by mixing lime or cement with sand, water, and some hard material, as broken stone, slag, bits of brick, earthenware, burnt clay, shingle, etc. If there is any choice of the materials forming the base of the concrete, the preference should be given to fragments of a somewhat porous nature, such as pieces of brick or limestone, rather than to those with smooth surfaces. (See page 637.) The broken material used in the concrete is sometimes, for convenience, called the aggregate, and the mortar in which it is incased, the matrix. The aggregate is generally broken so as to pass through a \\ or 2 inch mesh. In damp ground or under water, hydraulic lime should of course be used in mixing the concrete. Laying: Concrete. — A very common practice in laying concrete is to tip the concrete, after mixing, from a height of six or eight feet into the trench where it is to be deposited. This process is objected to by the best authorities, on the ground that the heavy and light portions separate while falling, and that the concrete is therefore not uniform throughout its mass. The best method is to wheel the concrete in barrows, immediately after mixing, to the place where it is to be laid, gently tipping it into position, and carefully ramming into layers about twelve inches thick. After each layer has been allowed to set, it should be swept clean, wetted, and made rough, by means of a pick, for the next layer. Some contractors make the concrete courses the exact width specified, keeping up the sides with boards, if the trench is too wide. This is a bad practice; for when the sides of the foundation pits are carefully trimmed, and the concrete rammed up solidly against them, the concrete is less liable to be crushed and broken before it has entirely consolidated. It is therefore desirable that the specifications for concrete work should require'that the whole extent of the excavation be filled, and that, if the trenches are excavated too wide, the extra amount of concrete be furnished at the contractor’s expense. Concrete made with hydraulic lime is sometimes designated as be ton, or beton.140 FOUNDATIONS. The pressure allowed on a concrete bed should pot exceed one-tenth part of its resistance to crushing. Trautwine gives as the average crushing-strength of concrete, forty tons per square foot. Foundations in Compressible Soil. — The great difficulty met with in forming a firm bed in compressible soils arises from the nature of the soil, and its yielding in all directions under pressure. There are several methods which have been successfully employed in soils of this kind. I. When the compressible material is of a moderate depth, the excavation is made to extend to the firm soil beneath, and the foundation put in, as in firm soils. The principal objection to this method is the expense, which ■ would often be very great. II. A second method is to drive piles through the soft soil into the firm soil beneath. The piles are then cut off at a given level and a timber platform laid upon tbe top of the piles, which serves as a support for the foundation, and also ties the tops of the piles together. III. A modification of the latter method is to use shorter piles, which are only driven in the compressible soil. The platform is made to extend over so large an area that the intensity of the pressure per square foot is within the safe limits for this particular soil. IV. Another modification of the second method consists in using piles of only five or six inches in diameter, and only five or six feet long, and placing them as near together as they can be driven. A platform of timber is then placed on the piles, as in the second method. The object of the short piles is to compress the soil, and make it firmer. “This practice is one not to be recommended; its effect being usually to pound up the soil, and to bring it into a state which can best be described by comparing it to batter-pudding.” 1 V. Still another method is to surround the site of the work with sheet-piling (fiat piles driven close together, so as to form a sheet), to prevent the escape, of the, soil, which is then consolidated by driving piles into it at short distances from each other.- The piles are then sawn off level, and tbe ground excavated between them for two or three feet, and filled up with concrete: the whole is then planked over to receive the superstructure. The great point to be attended to in building foundations in soils of this kind is to distribute the weight of the structure equally 1 Dobson on FouuiUcious.FOUNDATIONS. 141 over the foundation, which will then settle in a vertical direction, and cause little injury; whereas any irregular settlement would rend the work from top to bottom. Planking; for Foundation Beds. — In erecting building? on soft ground, where a large bearing-surface is required, planking may be resorted to with great advantage, provided the timber can be kept from decay. If the ground is wet and the timber good, there is little to fear in this respect; but in a dry situation, or one exposed to alternations of wet and dry, no dependence can be placed on unprepared timber. There are several methods employed for the preservation of timber, such as kyanizing or creo-soting; and the timber used for foundations should be treated by - one of these methods. The advantage of timber is, that it will resist a great cross-strain with very trifling flexure; and therefore a wide footing may be obtained without any excessive spreading of the bottom courses of the masonry. The best method of employing planking under walls is to cut the stuff into short lengths, which should be placed acrosN the foundation, and tied longitudinally by planking laid to the width of the bottom coui'se of masonry in the direction of the length of the wall, and firmly spiked to the bottom planking. Another good method of using planking is to lay down sleepers on the ground, and fill to their top with cement, and then place the planking on the level surface thus formed. For the cross-timbers, four-inch by six-inch timber, laid flatwise, will answer in ordinary cases. Foundations for Chimneys. — As examples of the foundations required for very high chimneys, we quote the following from a treatise on foundations, in the latter part of a work on “ Foundations and Foundation Walls,” by George T. Powell. Fig. 2 represents the base of a chimney creeled in IX.“)!) for the Chicago Refining Company, 151 feet high, and 12 feet square at the142 FOUNDATIONS. foot. The base, merely two courses of heavy dimension stone, as shown, is bedded upon the surface-gravel near the mouth of the river, there recently deposited by the lake. The mortar employed in the joint between the stone is roofing-gravel in cement. The area of the base is 250 square feet, the weight of chimney, inclusive of base, 025 tons, giving a pressure of 84 pounds to the square inch. This foundation proved to be perfect. Fig. 3 represents the base of a chimney erected in 1872 for the McCormick Reaper "Works, Chicago, which is 100 feet high, 14 feet square at the foot, with a round flue of 0 feet 8 inches diameter. The base covers 025 square feet; the weight of the chimney and base is approximately IKK) tons; the pressure upon the ground (dry hard clay) is therefore 24j pounds to the square inch. This foundation also proved to be perfect in every respect. TABLE Showing the jun'missible loads upon various kinds of foundation beds, per square foot. Rock foundations . . . 4000 to 40,000 lbs., average, 20,000 lbs. Coarse gravel and sand........................... 2500 to 3o00 lbs. Clay ................................................. 4000 lbs. Concrete........................................... • 8000 lbs. Piles in artificial soil, for each pile............... 4000 lbs. Piles in firm soil, for each pile..............30,000 to 140,000 lbs. /MASONRY WALLS. 143 CHAPTER III. MASONRY WALLS. Footing1 Courses. — In commencing the foundation walls of a building, it is customary to spread the bottom courses of the masonry considerably beyond the face of the wall, whatever be the character of the foundation bed, unless, perhaps, it he a solid rock bed, in which case the spreading of the walls would he useless. These spread courses are technically known as “footing courses.” They answer two important purposes: — 1st, By distributing the weight of the structure over a larger area of bearing-surface, the liability to vertical settlement from the compression of the ground is greatly diminished. 2d, By increasing the area of the base of the wall, they add to its stability, and form a protection against the danger of the work being thrown out of “plumb” by any forces that may act against it. Footings, to have any useful effect, must be securely bonded into the body of the work, and have sufficient strength to resist the violent cross-strains to which they are exposed. Footings of Stone Foundations. — As, the lower any stone is placed in a building, the greater the weight it has to support and the risk arising from any defects in the laying and dressing of the stone, the footing courses should be of strong stone laid on bed, with the upper and lower faces dressed true. By laying on bed is meant laying the stone the same way that it lay before quarrying. In laying the footing courses, no back joints should be allowed beyond the face of the upper work, except where the footings are in double courses; and every stone should bond into the body of the work several inches at least. Unless this is attended to, the footings will not receive the weight of the superstructure, and will be useless, as is shown in Fig. 1. In proportion to the weight of the superstructure, the projection of each footing course beyond the one above it must be reduced, or the cross-strain thrown on the projecting portion of the masonry will rend it from top to bottom, as shown in Fig. 2. In building laj'jjft, massed sof work, such as the abutment« of144 MASONRY WALLS. bridges and the like, the proportionate increase of bearing-surface obtained by the footings is very slight, and there is generally great risk of the latter being broken off by the settlement of the body of the work, as in Fig. 2. It is therefore usual in these cases to give very little projection to the footing courses, and to bring up the work with a battering-face, or with a succession of very slight offsets, as in Fig. 3. 4 i mLé Fig. 3. Footings of undressed rubble built in common mortar should never be used for buildings of any importance, as the compression of the mortar is sure to cause movements in the superstructure, if rubble must be used, it should be laid with cement mortar, Ju that the whole will form a solid mass; in which case, the size and shape of the stone, are of little consequence. In general, footing stones should be at least two by three feet on the bottom, and eight inches thick. The Building Laws of the city of New York require that the footing under all foundation walls, and under all piers, columns, posts, or pillars resting on the earth, shall be of stone or concrete. Under a foundation wall the footing must be at least twelve Inches wider mail the bottom width of the wall, and under piers, columns.MASONRY WALLS. 145 posts, or pillars, at least twelve inches wider on all sides than the bottom width of the piers, columns, posts, or pillars, and not less than eighteen inches in thickness; and, if built of stone, the stones shall not be less than two by three feet, and at least eight inches thick. All base-stones shall be well bedded, and laid edge to edge; and, i if the walls are built of isolated piers, then there must be inverted arches, at least twelve inches thick, turned under and between the piers, or two footing courses of large stone, at least ten inches thick in eacli course. The Boston Building Laws require that the bottom course for all foundation walls resting upon the ground shall be at least twelve inches wider than the thicknjBSs given for the foundation walls. Footings of Brick Foundations.—In building with brick, the special point to be attended to in the footing courses is to keep the hack joints as far as possible from the face of the work; and, in ordinary cases, the best plan is to lay the footings in single courses; the outside of the work being laid all headers, an« no course projecting more than one-fourtli brick beyond the on) above it, except in the case of an eight-inch wall.14G MASONRY WALLS. Figs. 4, 5, 6, and 7 show footings for walls varying from one brick to three bricks in thickness. Fig. 7. The hricks used for footings should he the hardest and strongest that can be obtained. The bottom course should in all cases be a double one. Too much care cannot be bestowed upon the footing courses of any building, as upon them depends much of the stability of the work. If the bottom courses be not solidly bedded, if any rents or vacuities are left in the beds of the masonry, or if the materials themselves be unsound, or badly put together, the effects of such carelessness are sure to show themselves sooner or later, and almost always at a period when remedial efforts are useless. Inverted Arelies. — In structures where the weight of the superstructure is sustained by a number of piers, it is often advantageous to connect the base of the piers by means of inverted arches; as they serve to distribute the weight of the structure evenly over the foundation bed. The form of the arch is commonly that of a semi-ellipse, or approaching to it. The arches, if of brick, should he at least twelve inches thick. In using inverted arches, care should be taken that the outer arches have sufficient abutments, otherwise the thrust of the arch may push the wall against which it abuts out of a perpendicular. Foundation Walls. — Foundation walls should start below the reach of frost, and should be carefully bonded together, and made as solid and compact as possible. The bottom courses are often laid dry, and the remainder in cement mortar. If made of stone, they should not.be less than twenty inches thick, and, if of brick, never less than twelve inches in thickness. In ordinary foundations it is only necessary to provide a wall that shall not he crushed by the weight of theMASONRY WALLS. 147 superstructure. The working-strength of the foundation wall can easily be determined by multiplying the area of its upper surface in square feet by six tons for brick-work, two and a half tons for common rubble, and, for good coursed rubble, by one-fifteenth of the crushing-strength of the stone it is built with. For wooden buildings, an eighteen or twenty inch rubble wall, or twelve-inch brick wall, is generally used. In soils of sand, gravel, or loam, the wall is generally built with both sides vertical: in clay soils either the inside or outside of the j Mall is general'y battered. In such a case it Mrould, of course, be better to batter the wall on the inside, if the room is of no value. For brick and stone buildings, the foundation walls are generally from eight to tMrelve inches thicker than the wall next above them. In New-York City the laws require that all foundations shall be built of stone or brick, laid in cement mortar. In Boston, rubble stone is permitted for foundations of building^ ] of moderate height. The thickness required for foundation wall^in the cities of Boston and New York, is shown by the tables on pp. 149, 150, and 151. Brick and Stone Walls.— Very little is known regarding the stability of Malls of buildings, beyond Mdiat has been gained by practical experience. The only strain which comes upon any horizontal section of such a wall, which can be estimated, is the direct M’eiglit of the wall above, and the pressure due to the floors and roof. But it is generally found necessary to make the M'all thicker than the considerations of the crushing-strength alone would require. With the same amount of material, a hollow wall is more stable than a solid one, and it also possesses many other advantages over solid M alls. The strength of a brick wall depends very much upon the bond. In this country it is a general rule among masons to use as few headers, or bond brick, as they can possibly get along with. The common custom is, to make every ninth or tenth course \ of headers, and build the remainder of the wall of stretchers. Brick bearing-walls of buildings should never be less than twelve inches thick below the top floor, and stone walls not less than sixteen inches. The thickness of the walls required by the laws of the cities of New Y'ork and Boston are shown by the tables on pp. 149-151. The New-York Law further requires, .that “all foundation walls shall be built of stone or brick, and shall be laid in cement mortar,148 MASONRY WALLS. . . . and shall be increased four inches in thickness for every additional five feet in depth below the depth of ten feet from the curb level.” Hollow Walls. — “ In all the walls that are built hollow, the same amount of stone or brick shall be used in their construction as if they were solid; and no hollow walls shall be built unless the two walls are connected by proper ties, either of brick, stone, or galvanized iron, placed not over twenty-four inches apart.” In addition to the requirements indicated in the table, the Boston Building Laws provide as follows: — “ If the owner shall elect, the amount of material herein specified for external walls may be used either in piers or buttresses, provided the external walls between said piers and buttresses shall not be less than twelve inches thick in buildings less than fifty feet in height ; if in excess of fifty feet, and not over a hundred feet in height, the external walls between said piers and buttresses shall be not less than sixteen inches thick. “ In all brick buildings over twenty-five feet in width, not having either brick partition walls, or girders supported by columns running from front to rear, and the entire height of the building, the external and party walls shall be increased four inches in thickness for every additional twenty-five feet in the width of said building. “Vaulted walls of the same thickness, independent of withes, may be used instead of solid walls ; and the walls on either side of air-space shall be not less than eight inches thick, and tied together perpendicularly with continuous withes of hard-burned brick of good quality, or other approved material, which shall be not more than three feet apart, and the air-space shall be smoothly plastered. “All brick walls and buttresses shall be of merchantable, wellshaped bricks, well laid and bedded, with well-filled joints, in lime or cement mortar, and well flushed up at every course with mortar; and all brick used during the warm months shall be well wet at the time they are .laid, and shall be dry at the time they are laid in the cold months.” “ Every ninth course at least of a brick wall shall be a heading or bonding course, except where walls are faced with face-brick, in which every ninth course shall be bonded with Flemish headers, or by cutting the course of the face-brick, and putting in diagonal headers behind the same. “All walls of a brick building meeting at an angle shall be anchored to each other every ten feet in their height by tie-anchors, made of at least one and a quarter inch by three-eighths of an inchMASONRY WALLS. 149 wrought-iron, which shall be securely built into the side or partition walls not less than thirty-six inches, and into the front and rear walls at least one-half the thickness of the wall. “ Walls may be made with a facing of stone, or other approved material, securely tied to a backing of not less than eight inches of hard brick-work laid in mortar, by means of metal clamps; but the thickness of facing and backing, taken together, shall not be less than the thickness required for a brick wall of the same height.’’ THICKNESS OF WALLS REQUIRED IN BOSTON. (Laws op 1SS5.) DWELLING HOUSES. ' Thickness of Foundation Walls. Height of Walls Above.l Block Stone. Rubble Stone. Brick. Not exceeding S5 feet IS in. 22)£ in. 10 in. Exceeding 85 and not exceeding 60 feet .... , showing that b must be halfway between a and c, as was proved before. Tlie Principle of the Lever.— This principle, is based upon the two preceding propositions, and is of great importance and convenience. VI I. If three jtarallel forces acting in one place bahnice each other, then each force must be proportional to the distance between the other two. Thus, if we have a rod All (Figs. 9a, 9b, and 9c), with three forces, P,, P2, P3, acting on it, that the rod shall be balanced, we must have the 10 PJ 15 12 Fig.9a P3156 COMPOSITION OF FORCES. following relation between the forces and their points of applica-tion; viz.,— P, I\ 5%. CB ■ AB 'AC' or Pl : P2 : P3 :: PC : AB : AC. This is the case of the common lever, and gives the means of determining how much a given lever will raise. p Fig.9 b u A't C -iB ft Po F*i Fig.9 o The proportion is also true for any arrangement of the forces (as shown in Figs, a, b, and c), provided, of course, the forces are lettered in the order shown in the figures. Example. —Let the distance AC be 0 inches, and the distance CB be 12 inches. If a weight of 500 pounds is applied at the point B, how much will it raise at the other end, and wliat support wili be required at C (Fig. 9b)? Ans. Applying the rule just given, we have the proportion: — JP, : P, :: AC : CB, or 500 : (P,) :: 6 : 12. Hence P, = 1000 pounds; or 500 pounds applied at B will lift 1000 suspended at A. The supporting force at C must, by proposition V.J be equal to the sum of the forces Pi and /';J, or 1500 pounds in this case. Centre of Grnvity. — The lines of action of the force of gravity converge towards the centre of the earth; but the distance of the centre of the earth from the bodies which we have occasion to consider, compared with the size of those bodies, is so great, that we may consider the lines of action of the forces as parallel. The number of the forces of gravity acting upon a body may be considered as equal to the number of part icles composing the body. The centre of ;jraviti/ of a body may be defined as the Joint through which the resultant of the parallel forces of gravity, acting upon the body, passes in every position of the body.CENTRES OF GRAVITY. 157 If a body be supported at its centre of gravity, and be turned about that point, it will remain in equilibrium in all positions. The resultant of the parallel forces of gravity acting upon a body is obviously equal to the weight of the body, and if an equal force be applied, acting in a line passing through the centre of gravity of the body, the body will be in equilibrium. Examples of Centres of Gravity. — Centre of Gravity of Lines. Straight Lines.—By a line is here meant a material line whose transverse section is very small, such as a very fine wire. The centre of gravity of a uniform straight line is at its middle point. This proposition is too evident to require demonstration. The centre of gravity of the perimeter of a triangle is at the centre of the circle inscribed in the lines joining the centres of the sides of the given triangle. Thus, let ABC (Fig. 10) be the given triangle. To find the centre of gravity of its perimeter, find the middle points, D, E, and F, and connect them by straight lines. The centre of the circle inscribed in the triangle formed by these lines will be the centre of gravity sought. Symmetrical Lines.—The centre of gravity of lines which are symmetrical with reference to a point will be at that point. Thus the centre of gravity of the circumference of a circle or an ellipse is at the geometrical centre of those figures. The centre of gravity of the perimeter of an equilateral triangle, or of a regular polygon, is at the centre of the inscribed circle. The centre of gravity of the perimeter of a square, rectangle, or parallelogram, is at the intersection of the diagonals of those figures. Centre of Gravity of Surfaces. Definition. — A surface here means a very thin plate or shell. Synnnetrical Surfaces. — If a surface can be divided into two symmetrical halves by a line, the centre of gravity will be on that line: if it can be divided by two lines, the centre of gravity will be at their intersection. The centre of gravity of the surface of a circle or an ellipse is at the geometrical centre of the figure; of an equilateral triangle or a regular polygon, it is at the centre of the inscribed circle; of a parallelogram, at the intersection of the diagonals; of the surface of a sphere, or an ellipsoid of revolution, at the geometrical centre of the body; of the convex surface of a right cylinder at the middle point of the axis of the cylinder. Irregular Figures.—Any figure may be divided into rectanglesIB CENTRES OF GRAVITY. and triangles, and, the centre of gravity of each being found, the centre of gravity of the whole may be determined by treating the centres of gravity of the separate parts as particles whose weights are proportional to the areas of the parts they represent. Trianyle.—To find the centre of gravity of a triangle, draw a line from each of two angles to the middle of the side opposite: the intersection of the two lines will give the centre of gravity. Quadrilateral.— To find the centre of gravity of Any quadrilat-eral, draw diagonals, and, from the end of each farthest from their intersection, lay off, toward the intersection, its shorter segment: the two points thus fornir with the point of intersection will form a triangle whose centre of gravity is that of the quadrilateral. Thus, let Fig. 11 be a quadrilateral whose centre of gravity is sought. Draw the diagonals AD and 13C, and from A lay off A E — ED, and from li lay off BlI — EC. From E draw a line to the middle of Eli, and from Fa line to the middle of Eli. The point of intersection of these two lines is the centre of gravity of the quadrilateral. This is a method commonly used for finding the centre of gravity of the voussoirs of an arch. Table of Centres of Grarity.—Let a denote a line drawn f 'om the vertex of a figure to the middle point of the base and D the distance from the vertex to the centre of gravity. Then B Segment. In an isosceles triangle . In a segment of a circle . In a sector of a circle, the ver- D D D B x 3« chord* 12 X area 2 x chord :> x arc Sector. D = 0A2r>II I) — * li D — 0.420a tex being at the centre ) In a semicircle, vertex being at ) the centre ) In a quadrant of a circle In a semi-ellipse, vertex being ) at the centre S In a par'bola, vertex at intersection of ( axis with curve) ) In a cone or pyramid........................._ 2) = :Jf( In a frustum of a cone or pyramid, let h ~ height of complete cone or pyramid, h* — height of frustum, and the vertex he at apex D « of complete eone or pyramid; then 7) d(/d 4 (/** 7/CENTRES OF GRAVITY. 159 The common centre of gravity of two figures or bodies external to each other is found by the following rule: — Multiply the smaller area or weight by the distance between centres of gravity, and divide the product by the sum of the areas or weights: the quotient will be the distance of the common centre of gravity from the centre of gravity of the larger area. Example.—As an example under the above s' >v rule and tables, let us find the common centre of f \ gravity of the semicircle and triangle shown in /_______k_/_____\ Fig. 12. I / We must first find the centres of gravity of the \ ^ / two parts. \]/ The centre of gravity of the semicircle is 0.425 Ii Fig. 12 from Ay or 2.975. The centre of gravity of the triangle is ^ of 8", or 2.006" from A ; and hence the distance between the centre of gravity is 2.975" + 2.000", or 5.041". 3| x 49 The area of the semicircle is approximately —^—, or 77 square inches. The area of the triangle is 7X8, or 50 square inches. The sum of the areas is 133 square inches. Then, by the above rule, the distance of the common centre of gravity from the centre 50 X 5.041 of gravity of the semicircle is---pr;----= 2.37". or 2.975 — 2.37 = 0.005 inches from A. Centre of Gravity of Heavy Particles* — Centre of Gravity of Two Particle#. — Let P be the P*W weight of a particle at A (Fig. 13), and W that at C. The centre of gravity will be at some point, By on the line joining A and C. The point B must be so situated, that if the two particles were held together by a stiff wire, and were supported at B by a force equal to the sum of P and IF, the two particles would be in equilibrium. The problem then comes under the principle of the lever, and hence we must have the proportion, PH- IF : P :: AC : BC, or P x AC k'=7TF' If TF= P, then BC — AB, or the centre of gravity will be half- Fig.13 -© W160 CENTRES OF GRAVITY. way between the two particles. This problem is of great importance, for it presents itself in many practical examples. Centre of Gravity of Several Heavy Particles. — Let IFt, W2, IF3, IF4 and 1F6 (Fig. 14) be the weights of the particles. Join IF] and 1F2 by a straight line, and find their centre of gravity A, as in the preceding problem. Join A with 1F3, and find the centre of gravity B, which will be the centre of gravity of the three weights IF,, IF?, and IF3. Proceed in the same way with each weight, and the last centre of gravity found will be the centre of gravity of all the particles. In both of these cases the lines joining the particles are supposed to be horizontal lines, or else the horizontal projection of the real straight line which would join the points.RETAINING WALLS. 161 CHAPTER V. RETAINING WALLS. A Retaining AVall is a wall for sustaining a pressure of earth, sand, or other filling or backing deposited behind it after it is built, in distinction to a brest or face wall, which is a similar structure for preventing the fall of earth which is in its undisturbed natural position, but in which a vertical or inclined face has been excavated. Fig. 1 gives an illustration of the two kinds of wall. Retaining "Walls.—A great deal has been written upon the theory of retaining walls, and many theories have been given for computing the thrust which a bank of earth exerts against a retaining wall, and for determining the form of wall which affords the greatest resistance with the least amount of material. There are so many conditions, however, upon which the thrust exerted by the backing depends,—such as the cohesion of the earth, the dryness of the material, the mode of backing up the wall, etc.,—that in practice it is impossible to determine the exact thrust which will be exerted against a wall of a given height. It is therefore necessary, in designing retaining walls, to be guided by experience rather than by theory. As the theory of retaining walls is so vague and unsatisfactory, we shall not offer any in this article, but rather give such rules and cautions as have been established by practice and experience. In designing a retaining wall there are two things to be considered,— the backing and the wall. The tendency of the backiny to slip is very nmcli less when it is1G2 RETAINING WALLS. in a dry state than when it is filled with water, and hence every precaution should be taken to secure good drainage. Besides surface drainage, there should be openings left in the wall for the water which may accumulate behind it to escape and run off. The manner in which the material is filled against the wall also affects the stability of the backings. If the ground be made irregular, as in Fig. 1, and the earth well rammed in layers inclined from the wall, the pressure will be very trifling, provided that attention be paid to drainage. If, on the other hand, the earth be tipped, in the usual manner, in layers sloping towards the wall, the full pressure of the earth will be exerted against it, and it must be made of corresponding strength. O Fig.2 The Wall. — Retaining walls are generally built with a battering (sloping) face, as this is the strongest wall for a given amount of material; and, if the courses are inclined towards the back, the tendency to slide on each other will be overcome, and it will not be necessary to depend upon the adhesion of the mortar. The importance of making the resistance independent of the adhesion of the mortar is obviously very great ; as it would otherwise be necessary to delay backing up a wall until tbe mortar was thoroughly set, which might require several months.RETAINING WALLS. 163 The Back of the Wall should he left Bough.— In brickwork it would be well to let every third or fourth course project an inch or two. This increases the friction of the earth against the back, and thus causes the resultant of thinforces acting behind the wall to become more nearly vertical, and to fall farther within the base, giving increased stability. It also conduces to strength not to make each course of uniform height throughout the thickness of the wall, but to have some of the stones, especially near the back, sufficiently high to reach up through two or three courses. J!y this means the whole masonry becomes more effectually interlocked or bonded together as one mass, and less liable to bulge. Where deep freezing occurs, the back of the wall should be sloped forwards for three or four feet below its top, as at OC (Fig. 2), which should be quite smooth, so as to lessen the hold of the frost, and prevent displacement. Figs. 3, 4, 5, and 6 show the relative sectional areas of walls of different shapes that would be required to resist the pressure of a bank of earth twelve feet high (“Art of Buildingjj E. Dobson, p. 20). The first three examples are calculated to resist the maximum thrust of wet earth, while the last shows the modified form usually adopted in pract ice. Buies for the Thickness of the Wall. — As has been stated, the only practical rules for retaining walls which we have are empirical rules based upon experience and practice. Mr. John C. Trautwine, C.E., who is considered authority on engineering subjects, gives the following table in his “Pocket-Book for Engineers,” for the thickness at the base of vertical retaining walls with a sand-backing deposited in the usual manner. C The first column contains the vertical height CD (Fig. 1) of the earth as compared with the vertical height of the wall; which latlcr164 RETAINING WALLS. is assumed to be 1, so that the table begins with backing of the same height as the wall. These vertical walls may be battered to any extent not exceeding an inch and a half to a foot, or 1 in 8, without affecting their stability, and without increasing the base. Proportion of Retaining1 Walls. Total height of the earth compared with the height of the wall above ground. Wall of cut stone in mortar. Good mortar, rubble, or brick. Wall of good, dry rubble. 1 0.35 0.40 0.50 1.1 0.42 0.47 0.57 1.2 0.46 0.51 0.61 1.3 0.49 0.54 0.64 1.4 0.51 0.56 0.66 1.5 0.52 0.57 0.67 1.6 0.54 0.59 0.69 1.7 0.55 0.60 0.70 1.8 0.56 0.61 0.71 2 0.58 0.63 0.73 - 2.5 0,60 0.65 0.75 3 0.62 0.67 0.77 4 0.63 0.68 0.78 6 0.64 0.69 0.79 Brest Walls (from Dobson's “Art of Building”).—Where the ground to be supported is firm, and the strata are horizontal, the office of a brest wall is more to protect than to sustain-the earth. It should be borne in mind that a trifling force skilfully applied to unbroken ground will keep in its place a mass of material, which, if once allowed to move, would crush a heavy wall; and therefore great care should be taken not to expose the newly opened ground to the influence, of air and wet for a moment longer than is requisite for sound work, and to avoid leaving the smallest space for motion between the back of the wall and the ground. The strength of a brest wall must be proportionately increased when the strata to be supported inclines towards the wall: where they incline from it, the wall need be little more than a thin facing to protect the ground from disintegration. The preservation of the natural drainage is one of the most important points to be attended to in the erection of brest walls, as upon this their stability in a great measure depends. No rule can be given for the best manner of doing this: it must be a matter for attentive consideration in each particular case.STRENGTH OF MASONRY. 16*5 CHAPTER VI. STRENGTH OF MASONRY. By the term “strength of masonry’’ we mean its resistance to a . crushing-force, as that is the only force to which masonry should be subjected. The strength of the different stones and materials used in masonry, as determined by experiment, is given in the following table: — Average Ultimate Crushing-Load in Pounds per Square Inch for Stones, Mortars, and Cements. Stones, Etc. Crushing weight in pounds per square inch. Brick, common (Eastern) 10,000 a Brick, best pressed 12,000 a Brick (Trautwine) 77C to 4,660 Brickwork, ordinary 0 300 to 500 Brickwork, good in cement 450 to 1,000 *Q0 Brickwork, tirst-class, in cement Concrete (1 part lime,‘3 parts gravel, three weeks old) 620 •Lime mortar, common 770 Portland dement, best English — Pure, three months old 3,760 Pure, nine months old 5,960 1 part sand, 1 part cement — Three months old * . 2,480 Nine months old 4,520 Granites, 7750 to 22,750 12,000 Blue granite, Fox Island, Me 14,875 Blue granite, Staten Island, N.Y Gray granite, Stony Creek, Cornu 22,250 15,750 North River (N.Y.) flagging 13,425 Limestones, 11,000 to 25,000 12,000 Limestones from Glen Falls, N.Y 11,475 Lake limestone, Lake Champlain, N.Y. . . . 1 25,000 White limestone, Marblehead, O White limestone from Joliet, 111 11,225 * 12,775 Marbles — From East Chester, N.Y. 12,950 Common Italian * 11,250 Vermont (Sutherland Falls Company).... 10,750 Vermont, Dorset, Vt 7,612 Drab, North Bay Quarry, Wis 20,025 Sandstones — Brown, Little Falls, N.Y 6,000 9,850 Brown, Middletown, Conn 6,950 Red, Haverstraw, N.Y 4,350 Red-brown, Seneca freestone, Ohio 9,687 Freestone, Dorchester, N.B. . . 9,150 Longmeadow sandstone, from Springfield, Mass. 8000 to 14,000 a a From tests made for the author at the United-States Arsenal, Watertown, M ass.166 STRENGTH OF MASONRY. The stones in this table are supposed to be on bed, and the height to be not more than four times the least side. Of the strength of rubble masonry, Professor ltankine states that “the resistance of r/ood coursed rubble masonry to crushing is about four-tenths of that of single blocks of the stone it is built with. The resistance of common rubble to crushing is not much greater than that of-the mortar which it contains.” Stones generally commence to crack or split under about one-half of their crusliing-load. Criiishing'-Heiglit of Brick ami Stone. — If we assume the weight of brickwork to be 112 pounds per cubic foot, and that it would crush under 450 pounds per square inch, then a vertical uniform column 580 feet high would crush at its base under its own weight. Average sandstones at 145 pounds per cubic foot would require a column 5950 feet'high to crush it; and average granite at 165 pounds per cubic foot would require a column 10,470 feet high. The Merchants’ sliot-tower at Baltimore is 246 feet high, and its base .sustains a pressure of six tons and a half (of 2240 pounds) per square foot. The base of the granite pier of Saltash Bridge (by Brunei) of solid masonry to the height of 96 feet, and supporting the ends of two iron spans of 455 feet each, sustains nin-? tons and a half per square foot. The highest pier of Rocquefavour stone aqueduct, Marseilles, is 305 feet, and sustains a pressure at the base of thirteen tons and a half per square foot. Working-Strength of Masonry. — The working-strength of masonry is generally taken at from one-sixtli to one-tenth of the crusliing-load for piers, columns, etc., and in the case of arches a factor of safety of twenty is often recommended for computing the resistance of the voussoirs to crushing. Mr. Trautwine states that it cannot be considered safe to expose even first-class pressed brickwork in cement to more than thirteen or sixteen tons’ pressure per square foot, or good hand-moulded brick to more than two-thirds as much. Sheet lead is sometimes placed at the joints of stone columns with a view to equalize the pressure, and thus increase the strength of the column. Experiments, however, seem to show that the effect is directly the reverse, and that the column is materially weakened thereby.1 Piers—Masonry that is so heavily loaded that it is neeessary to proportion it in regard to its strength to resist crushing, is, as a general rule, iu the form of piers, either of brick or stone. -Vs 1 T rant wine’s Tucket-book, p. 176.STRENGTH OF MASONRY. 167 these piers are often in places where it is desirable that they should occupy as little space as possible, they are often loaded to the full limit of safety. The material generally used for building piers is brick: block or cut stone is sometimes used, for the sake of appearance; but rubble-work should never be used for piers which are to sustain posts, ' pillars, or columns. Brick piers more than six feet in height : could never be less than twelve inches square, and should have properly proportioned footing courses of stone not less than a foot thick. The brick in piers should be hard and well burned, and should be laid in cement, or cement mortar at least, and be well wet before being laid, as the strength of a pier depends 1very much upon the mortar or cement with which it is laid: those piers which have to sustain very heavy loads should be built up with the best of the Rosendale cements. The size of the pier should be determined by calculating the greatest load which it may ever have to sustain, and dividing the load by the safe resistance of one square inch or foot of that kind of masonry to crushing. Example. —In a large storehouse the floors are supported by a girder running lengthwise through the centre of the building. The girders are supported every twelve feet by columns, and the lowest row of columns is supported on brick piers in the basement. The load which may possibly come upon one pier is found to be 65,000 pounds. What should be the size of the pier ? Ans. The masonry being of good quality, and laid in cement mortar, wre will assume that its crushing-strength is 0C0 pounds per square inch; and, taking one-sixth of this as the working-load, we find that the pier must contain 65000 k- 100, or 650 square inches. This would require a pier about 24 X 27 inches. It is the custom with many architects to specify bond stones in brick piers (the full size of the section of the pier) every three or four feet in the height of the pier. These bond stones are generally about four inches thick. The object in using them is to | distribute the pressure on the pier equally through the whole mas-. Many first-class builders, however, consider that thljpiers are stronger without the bond stone; and it is the opinion of the writer that a pier will be just as strong if they are not used. Section •'! of the Building Laws of the city of New York requires that every isolated pier less than ten superficial feet at the base, and all piers supporting a wall built of rubble-stone or brick, or under any iron beam or arch-girder, or arch on which a wall rests, or lintel supporting a wall, shall, at intervals of not less than thirty inches jn height, have built into it a bond stone not less thanSTRENGTH OF MASONRY. •68 four inches thick, of a diameter each way equal to the diametei of the pier, except that in piers on the street front, above the curb, the bond stone may be four inches less than the pier in diameter. Piers which support columns, posts, or pillars, should have the top covered by a plate of stone or iron, to distribute the pressure over the whole cross-section of the pier. In Boston, it is required that “ all piers shall be built of good, hard, well-burned brick, and laid in clear cement, and all bricks used in piers shall be of the hardest quality, and be well wet when laid. “ Isolated brick piers under all lintels, girders, iron or other columns, shall have a cap-iron at least two inches thick, or a granite cap-stone at least twelve inches thick, the full size of the pier. “ Piers or columns supporting walls of masonry shall have for a footing course a broad leveller, or levellers, of block stone not less than sixteen inches thick, and with a bearing surface equal in area to the square of the width of the footing course plus one foot required for a wall of the same thickness and extent as that borne by the pier or column.” For the Strength of Masonry IFalls, see Chap. III. The following tables give the results of some tests oil brick, brick piers, ami stone, made upder the direction of the author, in behalf of the Massachusetts Charitable Mechanics Association. The specimens were tested in the government testing-machine at Watertown, Mass., and great care was exercised to make the tests as perfect as possible. As the parallel plates between which the brick and stone were crushed are fixed in one position, it is necessary that the specimen tested should have perfectly parallel faces. The bricks which were tested were rubbed on a revolving bed until the top and bottom faces were perfectly true and parallel. The preparation of the bricks in this way required a great deal of time and expense; and it was so difficult to prepare some of the harder brick, that' they had to be broken, and only one-lialf of lie brick prepared at a time.STRENGTH OF MASONRY. 169 TABLE Showing the Ultimate and Cracking Strength of the Brick, the Size and Area of Face. Name of Brick. Size. Area of face in sq. ins. Commenced to crack under lbs. per sq.iuch. Net strength lbs. per sq. inch, j Philadelphia Face Brick . . . Whole brick 33.7 4,303 6,062 (< << «< Whole brick 82.2 3,400 5,S3l (« << «( Whole brick 84.03 ' 2,879 5,862 Average 3,527 5,918 Cambridge Brick (Eastern) . Half brick . 10.89 3,670 9,825 “ “ “ Whole brick 25.77 7,760 12,941 “ «< “ Half brick . 12.67 3,393 11,681 t( <( Half brick . 13.43 3,797 14,296 Average 4,655 12,186 Boston Terra-Cotta Co.’s Brick, Half brick . 11.46 11,518 13,839 <( (i (( bi. H 1 ~ H i X 77 - o p. ^ 1 lbs. lbs. lbs. Lime mortar 150,000 833 1,502 j Lime mortar, 3 parts; Portland cement, 1 part. 290,000 1,875 3,020 Lime mortar, 3 parts; Newark and Ro6endale cements, 1 part 2,552 Lime mortar, 3 parts: Roman cement, 1 part . . 105,000 1,041 2,030 Portland cement, 1 part, sand, 2 parts .... 1,302 2,500 Newark and Rosendale cements, 1 part; sand, 2 parts 708 2,135 Roman cement, 1 part; sand, 2 parte . . . . . 185,000 1,770 1,927 For first-rate brick-work in cement, Mr. Trautwine gives numbers which correspond to 770 to 1088 pounds to the square inch. These numbers are also very much less than those obtained from the tests of the piers laid in cement. The Portland cement used in building these piers was the kind known as Brooks, Slioobridge, & Co.’s cement; and this with the other brands were furnished for the tests by Messrs. Waldo Brothers, Boston, Mass. Actual Tests of the Crushing-Strength of Marble anti Santis tong (made under the direction of the author for the Massachusetts Charitable Mechanics’ Association).—These tests were made with the government testing-machine at the United-States Arsenal, Watertown, Mass., and every precaution was taken to secure accurate and reliable results. Sutiieiu.and Fai.ls Yeumont Makbi.e (white).—This marble is quarried by the "Vermont Marble Company, Centre Rutland, Yt. Block No. 1.—Xo. 1 quality, “Y” layer. Weight 180 pounds per cubic foot. Size 0" X 6.081 X 5.90". Sectional area 35.9 square inches. Ultimate strength 404,000 pounds (11,250 pounds per squar? inch).174 STRENGTH OF MASONRY. This block commenced to crack around the edges under 350.00ft pounds’ pressure (0750 pounds per square inch), and failed suddenly under 404.000 pounds. Block No. 2. —Same kind and quality as No. 1. Size 0" X 0" x 0.02". Sectional area 3(i. 12 square inches. Ullimalc strciujth eiO.OOO pounds (10,243 pounds per sijmirc inch). This block did not crack at all until it gave way entirely under a pressure of 370,000 pounds. Bkowxstone Ftio.M the Bav of Ffxdy Qf a keying Company.— Samples of the three following varieties of brown sandstone were furnished for testing by the Boston agents of the Bay of Fundy Quarrying Company. Maiiy's Point (N.B.) Stone. —This stone is a fine-grain dark-brown sandstone. lllock No. 1 measured 4.04" X 4.04" X 8". Sectional area 10.3 square inches. This stone commenced to crack at the corners and along tin' edges under 108,000 pounds’ pressure, and continued cracking until it suddenly broke into a number of pieces under 127,600 pounds pressure, or 7828 pounds per square inch. lllock No. 2 measured 4" X 3.75" X 7". Sectional area 15 square inches. The stone commenced to crack near one corner under a pressure of 31,000 pounds, and' under 77,000 pounds it was badly cracked. It flew from the machine in fragments when the pressure reached 113.800 pounds, or 7580 pounds per square inch. Wood's Point (N.B.) Sandstone. — This stone is of about the same color as the Mary's Point stone, but it has a much coarser grain, and is not very hard. lllock No. 1 measured 4.03" X 4.03" X 8". Sectional area 10.2 square inches. Commenced to crack at 50,000 pounds, on the corners, and continued cracking, along the edges and at the corners, until it was crushed under 80,000 lbs.' pressure, or 4032 lbs. per square inch. Block No. 2 measured 4" X 3.08" X 7.25". Sectional area 15.02 square inches. This stone commenced to crack under a pressure of 44.000 pounds, and failed under a pressure of 02.500 pounds, or 3070 pounds per square inch. Longmeadow Stone.—The Bay of Fundy Quarrying Company also quarry a variety of the Longmeadow (Mass.) sandstone, which is a reddish-brown in color. Block No. 1 measured 3.80" X 3.87" X 7.42". Sectional area 14.71 square inches.STRENGTH OF MASONRY. 175 This stone showed no cracks whatever until the pressure had reached 152,000 pounds, when it commenced to crack at the cornel's. When the pressure reached 200,000 pounds, the stone suddenly flew from the machine in fragments, giving an ultimate strength of 13,590 pounds per square inch. This stone did not fit into the machine very perfectly. Block No. 2 measured 3.39" X 3.97" X 7.5". Sectional area 15.6 square inches. The stone commenced to crack along the edges under a pressure of 47,000 pounds. Under 78,000 pounds, a large piece of the stone split off from the bottom of the block, and under 142,300 pounds’ pressure, the stone failed, cracking very badly. Ultimate strength per square inch 9121 jiounds. Brown Sandstone from East Longmeadow, Mass. —Quarried by Xorcross Brothers & Taylor of East Longmeadow. This firm works several quarries, the stone differing in the degree of hardness, and a little in color, which is a reddish brown. The different varieties take the name of the quarry from which they come. Soft Safest,fry Brownstone. — This stone is one of the softest varieties quarried by this firm, although it is about as hard as the ordinary brownstones. The specimens tested were selected by the foreman of the stone-yard without knowing the purpose for which they were to be used, and were rather below the average of this stone in quality. Block No. 1 measured 4" X 4" X 7.5S". Area of cross-section 16 square inches. Ultimate strength 141,000 pounds, or 8812 qwunds per square inch. .Stone did not commence to crack until the pressure had reached 130.000 pounds. Block No. 2 measured 4" X 4" x 7.85". Area of cross-section 16 square inches. Ultimate strength 129,000 pounds, or 8062 pounds per square inch. There were no cracks in the specimen when it was under 100,090 pounds’ pressure. Hard Saflsbfry Brownstone. — This is one of the hardest and finest of the Longmeadow sandstones. * Block No. 1 measured 4.16" X4.10" X 8". Sectional area 17.3 square inches. Ultimate strength 233,900 pounds, or 13,520 pounds per square inch. Stone did not commence to crack until the wqessure had reached 220.000 pounds, almost the crushing-strength. Block No. 2 measured 4.15" X 4.15" X 8". Sectional area 17.2 square inches. Ultimate strength 252,000pounds, or 14,050ptuouZ* per square inch.176 STRENGTH OF MASONRY. This specimen did not commence to crack until the pressure had reached 240,000 pounds, or 13,053 pounds to the square inch. The following table is arranged to show the sectional area and strength of each specimen, and the average for each variety of stone. The cracking-strength, so to speak, of the stone, is of considerable importance, for, after a stone has commenced to crack, its permanent strength is probably reached; for, if the load which caused it to crack were allowed to remain on the stone, it would probably in time crush the stone. In testing the blocks, however, some inequality in the faces of the block might cause one corner to crack when the stone itself had not commenced to weaken. Name op Stone. Sectional area. • trja s o p ~ * £ 3 S* 2 ® 5 0^3 T3 ||| sT ^ § p Z u y OC “ V | = S2 ifi CC -1 'p H ® Marble. sq.ins. lbs. lbs. lbs. Block No. 1 35.90 11,250 „ 9,750 404,000 Block No. *2 36.12 10,243 10,243 370,000 Average 10,746 9,996 - Sandstones. — Mary's Point. Block No. 1 16.30 7,S2S 6,626 127,600 Block No. 2 15.00 7,586 2,066 113,800 Average 7,707 2,346 - Wood's Point. Block No. 1 16.20 4,932 3,044 80,000 Block No. 2 15.92 3,976 2,763 62,500 Average 4,454 2,903 - Longrneadow Stone (Bay of Fun-day Quarrying Company’s). Block No. 1 14.71 13,596 10,333 200,000 Block No. 2 15.60 9,121 3,012 142,300 Average 11,358 6,672 - Soft Saitlsbury Stone. Block No. 1 16.00 8,812 8,250 141,000 Block No. 2 16.00 8,062 6,500 129,000 Average 8,437 7,375 - Hard Saulsbury. Block No. 1 17.30 13,520 12,716 233,900 Block No. 2 17.20 14,650 13,953 252,000 Average 14,085 13,334 — Gen. Q. A. Gillmore, a few years ago, tested the strength of many varieties of sandstone by crushing two-inch cubes. The re-STRENGTH OF MASONRY. 177 suits obtained by him ranged from 4350 pounds to 0850 pounds per square incli. Comparing the strength of the stones tested by the author with these values, we find that the specimens of Hard Saulsbury sandstone had a strength far above the average for sandstones. and the other specimens have about the same, values as those obtained by Gen. Gillmore. We should expect, however, smaller values from blocks 4" X 4" X 8" than from two-inch cubes; for, as a rule, small specimens of almost any material show a greater strength than large specimens. It is interesting to note the mode of fracture of the blocks of brownstone, which was the same for each specimen. The blocks fractured by the sides bursting off; and, when taken from the machine, the specimens had the form of two pyramids, with their apexes meeting at the centre, and having for their bases the compressed ends of the block. The pyramids were more clearly shown in some specimens than in others, but it was evident that the mode of fracture was the same for all specimens. Kin be Sandstone.—In 1883 the writer superintended the testing of two six-inch cubes of the Kibbe variety of Longmeadow sandstone, quarried by Xoreross Brothers. One block withstood a pressure of 12,590 pounds to the square inch before cracking, and the other did not commence to crack until the pressure had reached 12,185 pounds to the square inch. The ultimate strength of the first block was 12,019 pounds, and of the second 12,874 pounds, per square inch. Strength and Weight of Colorado Building Slone* **. The following are the most reliable data obtainable of the strength and weight of the stones most extensively used for building in Colorado. * Red Granite from Platte Canon. Crushing strength per square inch, 14,000 pounds. Weight per cubic foot, 164 pounds. Red Sandstone from Pike’s Peak Quarry, Manitou. Crushing strength, 0.000 pounds per square inch. Red Sandstone from Greenlee & Son’s quarries, Manitou (adjacent to the Pike’s Peak quarries'). Crushing weight, 11,000 pounds per square inch on bed. Weight, 140 pounds per cubic foot. * Gray Sandstone from Trinidad. Crushing weight, 10,000 pounds per square inch. Weight, 145 pounds per square inch. * From tests made for the Board of Capitol Managers (of Colorado) by State Engineer E. S. Nettleton. in 1885, on two-inch cubes. ** Tested at U. S. Arsenal, Watertown, Mass.177a COLORADO BUILDING STONES. f Lava Stone, Gurry’s Quarry, Douglas County. Crushing strength, 10,675 pounds per square inch. Weight, 119 pounds per cubic foot. * Fort Collins, gray sandstone (laminated), much used for foundations. Crushing strength, bed 11,700 pounds, edge 10,700 pounds per square inch. Weight, 140 pounds per cubic foot. (One ton of this stone measures just a perch in the wall.) * St. Vranis, light red sandstone (laminated), excellent stone for foundations. Very hard. Crushing strength, bed 11,505 pounds, edge 17,181 pounds per square inch. Weight, 150 pounds per cubic foot. * From tests made for the Board of Capitol Managers (of Colorado) by State Engineer E. S. Nettleton, in 1885, on two-inch cubes. t From tests made by Denver Society of Civil Engineers, in 1884, also on two-inch cubes.178 STABILITY OF PIERS AND BUTTRESSES. CHAPTER VII. STABILITY OF PIERS AND BUTTRESSES A pi eh or buttress may be considered stable when the forces acting upon it do not cause it to rotate or “tip over,” or any course of stones or brick to slide on its bed. When a pier has to sustain only a vertical load, it is evident that the pier must be stable, although it may not have sufficient strength. It is only when the pier receives a thrust such as that from a rafter or an arch, that its stability must be considered. Ill order to resist rotation, we must have the condition that the moment of the thrust of the pier about any point in the outside of the pier shall not exceed the moment of the weight of the pier about the same point. To illustrate, let us take the pier shown in Fig. 1. Let us suppose that this pier receives the foot of a rafter, which exerts a thrust T in the direction AB. The tendency of this thrust will be to cause the pier to rotate about the outer edge 61; and the moment of the thrust about this point will be T X flj&t, a^l being the arm. Now. that the pier shall be just in equilibrium, the moment of the weight of the pier about the same edge must just equal T x The weight of the pier will, of course, act through the centre of gravity of the pier (which in this case is at the centre), and in a vertical direction; and its arm will be b[C, or one-half the thickness of the pier. Hence, to have equilibrium, we must have the equation, T X alfm = 11' X />,<*. But under this condition the least additional thrust, or the crushing off of the outer edge, would cause the pier to rotate: hence'» to have the pier in safe equilibrium, we must use some factor of safety. This is generally done by making the moment of the weight equal to that of the thrust when referred to a point in the bottom of the pier, a certain distance in from the outer edge. This distance for piers or buttresses should not be less than one« fourth of tin* thickness of the pier.STABILITY OF PIERS AND BUTTRESSES. 179 Representing this point in the figure by b, we have the necessary equation for the safe stability of the pier, T X ah = IF X \t, l. denoting the width of the pier. We cannot from this equation determine the dimensions of a pier to resist a given thrust; because we have the distance ah. 1. and IF, all unknown quantities. Hence, we must first guess at the size of the pier, then find the length of the line ah, and see if the moment of the pier is equal to that of the thrust. If it is not, we must guess again. Graphic Method of determining' the Stability of a Pier or Buttress. — When it is desired to determine if a given pier or buttress is capable of resisting a given tlmist, the problem can easily be solved graphically in the following manner. Let ABCI) (Fig. 2) represent a pier which sustains a given thrust T at B. To determine whether the pier will safely sustain this thrust, we proceed as follows. Draw the indefinite line BX in the direction of the thrust. Through the centre of gravity of the pier (which in this case is at the centre of the pier) draw a vertical line until it intersects the line of the thrust at c. As a force may be considered to act anywhere in its line of direction, we may consider the thrust and the weight to act at the point e; and the resultant of these two forces can be obtained by laying off the thrust T from e on e.X, and the weight of the pier IF, from e on the line eY, both to the same scale (pounds to the inch), completing the parallelogram, and drawing the diagonal. If this diagonal prolonged cuts the base of the pier at less than one-fourth of the width of the base from the outer edge, the pier will be unstable, and its dimensions must be changed. The skibiUty of a piet; may be increased by adding to its weight180 STABILITY OF PIERS AND BUTTRESSES. (by placing some heavy material on top), or by increasing its width at the base, by means of “set-offs,” as in Fig. 3. Figs. 3 (A and B) show the method of determining the stability of a buttress with offsets. The first step is to find the vertical line passing through the centre of gravity of the whole pier. This is best done by dividing the buttress up into quadrilaterals, as ABCD, I)EFG, and C.lllK (Fig. 3A), finding the centre of gravity of each quadrilateral by tlie method of diagonals, and then measuring the perpendicular distances X,, A'2, Xit from the different centres of gravity to the line KI. Multiply the area of each quadrilateral by the distance of its centre of gravity from the line KI, and add together the areas and the products. Divide the sum of the latter by the sum of the former, and the result will be the distance of the centre of gravity of the whole buttress from KI. This distance we denote by AT0. Example I. — Let the buttress shown in Fig. 3A have the dimensions given between the cross-marks. Then the area of the quadrilaterals and the distances from their centres of gravity to KI would be as follows: 1st area — 35 sq. ft. A', = O'.05 1st area X X\ = 33.25 2d area — 23 sq. ft. A'* — 2'. 95 2d area X X2 = 07.S5 3d area = 11 sq. ft. A'3 — A.95 3d area X A':) = 54.45 'total area, 09 sq. ft. Total moments, 155.55 The sum of the moments is 155.55; and, dividing this by the total area, we have 2.25 as the distance X0. Measuring this to the scale of the drawing from KI, we have a point through which the vertical line passing through the centre of gravity must pass. Fig.3 A Fig.3 bSTABILITY OF PIERS AND BUTTRESSES. 181 After this line is found, the method of determining the stability of the pier is the same as that given for the pier in Fig. 2. Fig. 3B also illustrates the method. If the buttress is more than one foot thick (at right angles to the plane of the paper), the cubic contents of the buttress must be obtained to lind the weight. It is easier, however, to divide the real thrust by the thickness of the buttress, which gives the thrust per foot of buttress. * Line of Resistance. — Definition. The line of resistance or of pressures, of a pier or buttress, is a line drawn through the centre of pressure of each joint. The centre of pressure of any joint is the point where the resultant of the forces acting on that portion of the pier above the joint cuts it. The line of pressures, or of resistance, when drawn in a pier, shows how near the greatest stress on any joint comes to the edges of that joint. It can be drawn by the following method. Let A BCD (Fig. 4) be a pier whose line of resistance we wish to draw. First divide the pier in height, into portions two or three feet high, by drawing horizontal lines. It is more convenient to make the portions all of the same size. Prolong the line of the thrust, and draw a vertical line through the centre of gravity of the pier, intersecting the line of thrust at the point a. From a lay off to a scale.the thrust T and the weights of the different portions of the pier, commencing with the weight of the upper portion. Thus, w ] represents the weight of the portion above the first joint; iv2 represents the weight of the second portion; and so on. The sum of the to’s will equal the whole weight of the pier. Having proceeded thus far, complete a parallelogram, with Tand v) i for its two sides. Draw the diagonal, and prolong it. Where it cuts the first joint will be a point in the line of resistance. Draw another parallelogram, with T and W\ + iv2 for its two sides. Draw the diagonal intersecting the second joint at 2. Proceed in182 STABILITY OF PIERS AND BUTTRESSES. this way, when the last diagonal will intersect the base in 4. Join the points 1, 2, 3, and 4, and the resulting line will be the line of resistance. We have taken the simplest case as an example; but the same principle is true for any case. Should the line of resistance of a pier at any point approach the outside edge of the joint nearer than one-quarter the width, of the joint, the pier should be considered unsafe. As an example embracing all the principles given above, we will take the following case. Example II. — Let Fig. 5 represent the section of a side Avail of a church, with a buttress against it. Opposite the buttress, on the inside of the wall, is a liainmer-beam truss, which we will suppose exerts an outward thrust on the Avails of the church amounting to about 9600 pounds. We Avill further consider that the resultant of the thrust acts at P, and at an angle of 60° Avith a horizontal. The dimensions of the Avail and buttress are given in Fig. 5A, and the buttress is tAvo feet thick. Question. — Is the buttress sufficient to enable the Avail to Avithstand the thrust of the truss ? The first point to decide is if the line of resistance cuts the joint CD at a safe distance in from C. To ascertain this, A\’e must find the centre of gravity of the wall and buttress above the joint CD. We can find this easiest by tlie method of moments around KM (Fig. 5A), as already explained. The distance AJ is, of course, half the thickness of the Avail, or one foot. We next find the centre of gravity of the portion CEFG (Fig. 5A), by the method of diagonals, and, scaling the distance A'2, Ave find it to be 2.95 feet. The area of CEFG = A._, = 10 square feet; and of GlliL ~ A, — 2G square feet. 'Then Ave have, A', = 1 X2 = 2.95 Or the centre of gravity is at a distance 1.5 foot from the line ED (Fig. 5). Then on Fig. 5 measure the distance Xa = 1.5 foot, and through the point a draAv a vertical line intersecting the line of the thrust prolonged at O. Noav, if the thrust is 9600 pounds for a buttress tAvo feet thick, it would he half that, or 4800 pounds, for a buttress one foot thick. We Avill call tlie Aveight of the A i = 26 A, X A', = 26 A 2 = 10 A 2 X A's = 29.5 86 86 ) 55.5 A'n = 1.5STABILITY OF PIERS AND BUTTRESSES. 183 masonry of which the buttress and wall is built 150 pounds per cubic foot. Then the thrust is equivalent to 4800 *r- 150, or 32 cubic feet of masonry. Laying this off to a scale from O, in the direction of the thrust and the area of the masonry, 36 square feet from O on the vertical line, completing the rectangle, and drawing the diagonal, we find it cuts the joint CD at by within the limits of safety. We must next find where the line of resistance cuts the base AI). K First find the centre of gravity of the whole figure, which is found by ascertaining the distances AV, AY, iu Fig. 5A, and making the following computation: X ' = V A t= 40 AM x o II H“ AY = 2'. 98 A./ = 24 AM * AY = 71.52 Ay = 4'. 95 A,' = 12 Al x AY ~ 59.40 70 70 ) 170.92 AV = 2.25 Then from the line ED (Fig. 5) lay off the distance A,/ =? 2'.25, and draw through m a vertical line intersecting the line of the thrust at O'. On this vertical from O' measure down the whole area 70, and from its extremity lay off the thrust T = 32 at the184 STABILITY OF PIERS AND BUTTRESSES. proper angle. Draw the line O'e intersecting the base at c. This is the point where the line of resistance cuts the base; and, as it is at a safe distance in from A, the buttress has sufficient stability. If there were more offsets, we should proceed in the same way, finding where-the line of resistance cuts the joint at the top of each offset. The reason for doing this is because the line of resistance might cut the base at a safe distance from the outer edge, while higher up it might come outside of the buttress, so that the buttress would be unstable. The method given in these examples is applicable to piers of any shape or material. Should the line of resistance make an angle less than 30° with any joint, it might cause the stones above the joint to slide on their bed. This can be prevented either by dowelling, or by inclining the joint. It is very seldom in architectural construction that such a case would occur, however.THE STABILITY OK ARCHES. 185 CHAPTER Yin. THE STABILITY OF ARCHES. The arch is an arrangement for spanning large openings by means of small blocks of stone, or other material, arranged in a particular way. As a rule, the arch answers the same purpose as the beam, but it is widely different in its action and in the effect that it has upon the appearance, of an edifice. A beam exerts merely a vertical force upon its supports, but the arch exerts both a vertical load and an outward thrust. It is this thrust which requires that the arch should be used with caution where the abutments are not abundantly large. Before taking np the principles of the arch, we will define the many terms relating to it. The distance ec {Fig. 1) is called the .span of the arch; ai, its rise; b, its crown; its lower boundary line, ear, its soffit or intrados; the outer boundary line, its back or extrados. The terms “soffit” and “back” are also applied to the entire lower and upper curved surfaces of the whole arch. The ends of the arch, or the sides which are seen, are called its faces. The blocks of which the arch itself is composed are called coussoirs: the centre one, K, is called the keystone; and the lowest ones, SS, the springers. In segmental arches, or those whose intrados is not a complete semicircle, the springers generally rest upon two stones, as HR, which have their upper surface cut to receive them: these stones are called skewbaeks. The line connecting the lower edges of the springers is called the springing-line; the sides of the arch are called the haunches; and the load in the triangular space, between the haunches and a horizontal line drawn from the crown, is called the spandrel. The blocks of masonry, or other material, which support two successive arches, are called ]>iers: the extreme blocks, which, in the case of stone bridges, generally support on one side embankments of earth, are called abutments. A pier strong enough to withstand the thrust of either arch, should the other fall down, is sometimes called an abutment pier. Besides their own weight, arches usually support a permanent load or surcharge of masonry or of earth. In using arches in architectural constructions, the form of the186 THE STABILITY OF ARCHES. arch is generally governed by the style of the edifice, or by a limited amount of space. The semicircular and segmental forms of arches are the best as regards stability, and are the simplest to construct. Elliptical and three-centred arches are not as strong as circular arches, and should only be used where they can be given all the strength desirable. The xtrenr/th of an arch depends very much upon the care with which it is built and the quality of the work. In stone arches, special care should be taken to cut and lay the beds of the stones accurately, and to make the bed-joints thin and close, in order that the arch may be strained as little as possible in settling. To insure this, arches are sometimes built dry, yrout or liquid mortar being afterwards run into the joints; but the advantage of this method is doubtful. Urick Arches may be built either of wedge-shaped bricks, moulded or rubbed so as to fit to the radius of the soffit, or of bricks of common shape. The former method is undoubtedly the best, as it enables the bricks to be thoroughly bonded, as in a wall; but, as it involves considerable expense to make the bricks of the proper shape, this method is very seldom employed. Where bricks of the ordinary shape are used, they are accommodated to the curved figure of the arch by making the bed-joints thinner towards the, intrados than'towards the extrados; or, if the curvature is sharp, by driving thin pieces of slate into the outer edges of those joints; and different methods are followed for bonding them. The most common way is to build the arch in concentric rings, each half a brick thick; that is, to lay the bricks all stretchers, and to depend upon the tenacity of the mortar or cement for the connection of the several rings. This method is deficient in strength, unless the bricks an? laid in cement at least as tenacious as themselves. Another way is to introduce courses of headers at intervals, so as to connect pairs of half-brick rings together. This may be done either by thickening the joints of the outer of a pair of half-brick rings with pieces of slate, so that there shall be the same number of courses of stretchers in each ring between two courses of headers, or by placing the courses of headers at such distances apart, that between each pair of them there shall be one course of stretchers more in the outer than in the inner ring. The former method is best suited to arches of long radius; the latter, to those of short radius. Hoop iron laid round the arch, between half-brick rings, as well as longitudinally and radially, is very useful for strengthening brick arches. 'The bands of hoop iron which traverse the arch radially may also be bent, and prolonged in the bed-joints of the backing and spandrels,.THE STABILITY OF ARCHES. 187 By the aid of hoop-iron bond, Sir Marc-Isambard Brunei built a half-arch of bricks laid in strong cement, which stood, projecting from its abutment like a bracked, to the distance of sixty feet, until it was destroyed by its foundation being undermined. The New-York City Building Laws make the following requirements regarding brick arches: — “ All arches shall be at least four inches thick. Arches over four foot span shall be increased in thickness toward the haunches by additions of four inches in thickness of brick. The first additional thickness shall commence at two and a half feet from the centre of the span; the second addition, at six and one-half feet from the centre of the span; and the thickness shall be increased thence four inches for every additional four feet of span towards the haunches. “The said brick arches shall be laid to a line on the centres with a close joint, and the bricks shall be well wet, and the joints filled with cement mortar in proportions of not more than two of Bind to one of cement by measure. The arches shall be well grouted and pinned, or chinked with slate, and keyed.” Rule for Radius of Brick Arche*. —A good rule for the radius of segmental brick arches over windows, doors, and other small openings, is to make the radius equal to the width of the opening. This gives a good rise to the arch, and makes a pleasing proportion to the eye. It is often desirable to span openings in a wall by means of an arch, when there is not sufficient abutments to withstand the thrust or kick of the arch. In such a case, the arch can be formed on two east-iron skewbacks, which are held in place by iron rods, as is shown in Fig. 2. When this is done, it is necessary to proportion the size of the rods to the thrust of the arch. The horizontal thrust of the arch is very nearly represented by the following formula: — load on arch x span Horizontal thrust - £ X^-ise^ arch In tat If two tension*rods are used, as is generally the case, the diameter of each rod can be determined by the following rule: — / total load on arch x span Diameter in inches — \ / t~~•, ~ ■ -• . Y y 10 X rise or arch in teet X188 THE STABILITY OF ARCHES. If only one rod is used, 8 should be substituted in the place of Hi, in the denominator of the above rule; and, if three rods are used, 24 should be used instead of 1(5. Centres for Arches. — A centre is a temporary structure, generally of timber, by which the voussoirs of an arch are supported while the arch is being built. It consists of parallel frames or ribs, placed at convenient distances apart, curved on the outside to a line parallel to that of the soffit of the arch, and supporting a series of transverse planks, upon which the arch stones rest. The most common kind of centre is one which can be lowered, or struck all in one piece, by driving out wedges from below it, so as to remove the support from every point of the arch at once. The centre of an arch should not be struck until the solid part of the backing has been built, and the mortar has had time to set and harden; and, when an arch forms one of a series of arches with piers between them, no centre should be struck so as to leave a pier with an arch abutting against one side of it only, unless the pier has sufficient stability to act as an abutment. When possible, the centre of a large brick arch should not be struck for two or three months after the arch is built. Mechanical Principles of the Arch. — In designing an arch, the first question to be settled is the form of the arch; and in regard to this there is generally but little choice. Where the abutments are abundantly large, the segmental arch is the strongest form; hut, where it is desired to make the abutments of the arch as light as possible, a pointed or semicircular arch should he used. Depth of Keyxtone. — Having decided upon the form of the arch, the depth of the arch-ring must next he decided. This is generally determined by computing the required depth of keystone, and making the whole ring of the same or a little larger depth. In considering the strength of an arch, the depth of the keystone is considered to be only the distance from the extrados to the intra-dos of the arch; and if the keystone projects above the arch-ring, as in Fig. 1, the projection is considered as a part of the load on the arch. There are several rules for determining the depth of the keystone, but all are empirical; and they differ so greatly that it is difficult to recommend any particular one. Professor Eankine's Rule is often quoted, and is probably true enough for most arches. It applies tq both circular and elliptical arches, and is as follows: — Kankine’s Rule.— For the depth of the keyxtone, take a mean proportional between the inside radius at the crown, and 0.12 of a foot for a single arch, and 0.17 of a foot for an arch forming one of a series. Or, if represented by a formula,THE STABILITY OF ARCHES. 189 Depth of keystone for a single arch, in feet = V (0.12 X radius at crown). Depth of keystone for an arch of a series, in feet = V (0.17 X radius at crown). This rule seems to agree very well with actual cases in arches of a certain kind. By it, however, the depth of keystone is the same for spans of any length, provided the radius is the same; and in this particular, it seems to us, the rule is not satisfactory. Trautwine’s Rule. — Mr. Trautwine, from calculations made on a large number of arches, has deduced an original rule for the depth of keystone, which is more agreeable to theory than Ran-kine’s. His rule is, for cut stone, Depth of key, in feet = Radiushalf span^ + Q 2 foot_ For second-class work, this depth may be increased about one-eighth part, or, for brick or fair rubble, about one-fourth. The following table gives a few examples of the depth of keystone of some existing bridges, together with the depth which would be required by Trautwine’s or Rankine’s Rule. From this table it will be seen that both rules agree very well with practice. TABLE I. Showing Depth of Keystone of Some Existing Arches. O o Calculated depth of key. BRIDGE (circular arc). 1 1 ca 1 Radius. 1 1 'S co ■ ■ I“ 1 JZ -- £4 Engineer. Cabin John, Washington n. ft. ft. ft. ft. ft. Aqueduct . . . . . Grosvenor Bridge, Chester, 2*20.0 57.25 134.25 4.60 4.11 4.00 Meigs. Eng 200.0 42.00 140.00 4.00 4.07 4.10 Hartley, j Dora Riparia, Turia, Italy . 148.0 18.00 160.10 4.92 4.03 4.38 Mosca. Tongueland, England. . . Dean Bridge, Scotland, in a 11S.0 38.00 64.80 3.50 3.00 2.79 Telford, j Falls Bridge, Philadelphia 90.0 30.00 48.90 3.00 2.62 2.88 Telford. and Reading Railroad . . Chestnut-street Bridge, Phil- 78.0 25.00 43.00 3.00 2.46 2.27 Steele. adelphia, brick in cement. Philadelphia and Reading 60.0 18.00 34.00 2.50 2.20 2.00 Kneass. Railroad Philadelphia and Reading 44.0 8.00 34.30 2.50 2.08 2.02 Steele. Railroad 31.2 5.00 26.80 1.66 ■ 1.79 Steele.190 THE STABILITY OF ARCHES. Table II., taken from Trautwine’s “Civil Engineers’ Handbook,” gives the depth of keystone for arches of first-class cnt-stone, according to Trautwine’s Rule. For second-class cut-stone,'add about one-eigbtli part, and, for good rubble or brick, about one-fourtli part. TABLE II. Tabic of Keystones for Arches of First-Class (Jnt-SItmr. Span Rise in Parts or the Span. feet. H i 1 1 1 l 1 f 3 f 5 6 8 1 0 key. ft. key. ft. key. ft. key. ft. key. ft. key. ft. key. ft. »7 0.55 0.56 0.58 0.00 0.61 0.61 o.os 4 0.70 0.72 0.74 0.70 0.79 0.83 0.88 6 0.81 0.83 0.80 0.S9 0.92 0.97 1.03 8 0.01 0.93 0.96 1.00 1.03 1.09 1.16 10 m 1.01 1.04 1.07 1.11 1.18 1.26 15 1.17 1.19 .1.22 1.26 1.30 1.40 1.50 ‘JO 1.32 1.35 1.38 1.43 1.48 1.59 1.70 m 1.45 1.48 1.53 1.51 1.61 1.76 1.88 30 1.57 1.60 1.65 1.71 1.78 1.91 2.04 IR) 1.08 1.70 1.76 1.83 1.90 2.04 2.19 40 1.78 1.81 1.88 1.95 2.03 2.18 2.33 50 1.07 2.00 2.08 2.16 2.25 2.41 2.58 GO 2.14 2.18 2.26 2.35 2.44 2.02 2.80 so 2.44 , 2.49 2.58 2.08 2.78 2.98 3.18 100 2.70 2.75 2.S6 2.97 3.09 3.32 3.55 120 2.94 2.99 3.10 3.22 3.35 3.01 3.88 140 3.16 3.21 3.33 3.46 3.60 3.87 4.15 MM 3.36 3.44 3.58 3.72 3.87 4.17 180 3.56 3.03 3.75 3.90 4.06 4.38 mm 3.74 3.81 3.95 4.12 4.29 ‘2*20 . 3.91 4.00 4.13 4.30 4.48 ‘240 . 4.07 4.15 4.30 4.48 - ‘200 4.23 4.31 4.47 4.66 2so 4.38 4.46 4.63 300 4.53 4.02 4.80 Having decided what the thickness of the arcli-ring will be, it remains to determine whether such an arch would be stable if built. The following example will illustrate the method of determining diis point: — Example I. — Unloaded semicircidar arch of 20 foot span. First, to find the depth of keystone, we will take Rankine’s Rule, liul by it we have, Depth of key = ^0.12 x 10 = = 1.1 foot. Trautwine’s Rule would give nearly the same, or, + 0.2 foot = 1.3 foot. y/lO + 10 4THE STABILITY OF ARCHES. 191 But, if we should compute the stability of a semicircular arch cf 20 foot span, and 1.3 foot depth of keystone, we should find that the arch was very unstable: hence, In this case, we must throw the rule aside, and go by our own judgment. In the opinion of th ; author, such an arch should have at least 2* feet depth of arch-ring, and we will try the stability of the arch with that thieknee\ In all calculations on the arch, it is customary to consider th i arch to be one foot thick at right angles to its face; for it is eviden\ that, if an arch one foot thick is stable, any number of arches of the same dimensions built alongside of it would be stable. Graphic Solution of the Stability of the Arch.— The most convenient method of determining the stability of the arch is by the graphic method, as it is called. 1st Step. — Draw one-half the arch to as large a scale as convenient, and divide it up into voussoirs of equal size. In this example, shown in Fig. 3, we have divided the arch-ring into ten equal voussoirs. (It is not necessary that these slioujd be the actual voussoirs of which the arch is built.) The next step is to find the area of each voussoir. Where the arch-ring is divided into voussoirs of equal size, this is easiest done by computing the area of the arch-ring, and dividing by the number of voussoirs. Rule for area of one-haf of arch-ring is as follows: — Area in square feet = 0.7854 X (outside radius squared — inside radius squared). In this example the whole area equals 0.7854 X (12.52 —102) = 44.2 square feet. As there are ten equal voussoirs, the area of each voussoir is 4.4 square feet. Having drawn out one-half of the arch-ring, we divide each joint into three equal parts; and from the point A (Fig. 3) we lay off to a scale the area of each voussoir, one below the other, commencing Fig.3192 THE STABILITY OP ARCHES. with the top voussoir. The whole length of the line AE will equal the whole area drawn to same scale. The next step is to find the vertical line passing through the centre of gravity of the whole arch-ring. To do this, it is first necessary to draw vertical lines through the centre of gravity of each voussoir. The centre of gravity of one voussoir may be found by the method of diagonals, as in the second voussoir from the top (Fig. 3}. Having the centre of gravity of one voussoir, the centres of gravity of the others can easily be obtained fh>m it. Next, from A and E (Fig. 3) draw lines at 45° with AE, intersecting at O. Draw 01, 02, 03, etc. Then, where AO intersects the first vertical line at a, draw a line parallel to 01, intersecting the second vertical at 6. Draw be parallel to 02, c Step. —Draw a horizontal line through A (the upper part of the middle third), and a vertical line through I); the two lines intersecting at C (Fig. 3). Now, that the arch shall be stable, it is considered necessary that it shall be possible to draw a line of resistance of the arch within the middle third. We will, then, first assume that the line of resistance shall act at A, and come out at B. Then draw the line CB, and a horizontal line opposite the point 10, between Q and P. This horizontal line represents the horizontal thrust at the crown. Draw AP equal to QP, and the lines PI, P2, P3, etc. Then, from the point where AC prolonged intersects the first vertical, draw a line to the second vertical, parallel to PI; from this point a line to the third vertical, parallel to P2 ; and so on. The last line should pass through B. If these lines, which we will call the line of resistance, all lie within the middle third, the arch may bo considered to be stable. Should the line of resistance pass outside of the arch-ring, the arch should be considered unstable. In Fig. 3 this line does not all lie in the middle third, and we must see if a line of resistance can yet be drawn within that limit. 2d Tuiai.. —The line of resistance in Fig. 3 passes farthest from the middle third at the seventh joint from the top; and we will next pass a line of resistance through A and where the lower line of the middle third cuts the seventh joint, or at 1) (Fig. 4). To do this, we must prolong the line on in practice. Hence the load which this description of iron can suspend with safety may approach much more nearly the limit of its breaking-strain than can be attempted with the harder or coarser sorts, where a greater margin must necessarily be left. “ As a necessary corollary to what we have just endeavored to establish, the writer now submits, in addition, that the working-strain should be in proportion to the breaking-strain per square inch of fractured area, and not to the breaking-strain per square inch of original area, as heretofore. Some kinds of iron experimented on by the writer will sustain with safety more than double the load that others can suspend, especially in circumstances where the load is unsteady, and the structure exposed to concussions, as in a ship or railway bridge.” Eye-Bars and Screw-Ends. Iron ties are generally of fiat or round bars attached by eyes and pins, or by screw-ends. In either case, it is essential that the proportion of the eyes or screw-ends shall be such that the tie will not break at the end sooner than in the middle. In important work, eyes are forged on the ends of flat or round bars, by hydraulic pressure, in suitably shaped dies; and, while the risk of a welded eye is thus avoided, a solid and well-formed eye is made from the iron of the bar itself. A similar process is adopted for enlarging the screw-ends of long rods; so that, when the screw is cut, the diameter of the screw at the root of the thread is left a little larger than the body of the rod. Frequent trial with such rods has proven that they will pull apart in tension anywhere else but in the screw; the threads remaining perfect, and the nut turning freely after having been subjected to such a severe test. By this means the net section required in tension is made available with the least excess of material, and noRESISTANCE TO TENSION. 203 more (lead weight is put upon the structure than is actually needed to carry the loads imposed. Fig. 1 shows the proportions for the eye : and pins of tie-bars, and the screw-ends of rods, usually employed. When the tic-bar is round, the diameter of the pin should be 1^ times the diameter of the bar, and the other dimensions of the eye should be in proportion. The thickness of flat bars should be at least one-fourth of the ■width in order to secure good bearing surface on the pin; and the metal at the eyes should be as thick as the bars on which they are upset. Table IV., following, gives the proportion for upset screw-ends for different sixes of rods. Cast-Iron lias only about one-third the tensile strength of wrought-iron; and as it is liable to air-holes, internal strains from unequal contraction in cooling, and other concealed defects, reducing its effective area for tension, it should never be used where it is subject to any great tensile stress. Tables. The following tables give the strength of iron rods, bars, steel and iron wire ropes, manila ropes, and dimensions of upset screw-ends. The diameter in Table II. is the least diameter of the rod; and, if the screw is cut into the rod without enlarging the end, the effective diameter between the threads of the screw should be used in calculating the strength of the rod. Table V. was compiled from data furnished by the John A. Uool»ling’s Sons Company of New York. The ropes with nineteen wires to the strand are the most pliable, aud are generally used for hoisting and running rope. The ropes with seven wires to the strand are stiffer, and are better adapted for standing-rope, guys, and rigging. Table VI. is taken from Trautwine’s “ Pocket-Book for Engineers.”204 RESISTANCE TO TENSION. Table VII. gives the weight and proof, or safe strength, of chains manufactured by the New-Jersey Steel and Iron Company. TABLE II. Strength of Iron Rods. Safe Tensile Strengths of Round Wrougiit-Iron Rods J to 4 Inches in Diameter, and the Weights per Foot, the Safe Strength being taken at 10,000 Pounds per Square Inch. Diameter in inches. Weights per foot. Safe strengths in lbs. Diameter in inches. Weights per foot. Safe strength in lbs. i 8 • • 0.041 123 n. . 11.95 35,460 1 4 * * 0.165 491 9» -4 * • 13.39 39,760 i • • 0.372 1,104 28 . . 14.92 44,300 i 5 . . 0.661 1,963 91 -3 • • 16.53 49,080 6 8 • • 1.04 3,068 2| . . 18.23 54,110 i ■ ■ . •. . 1.49 4,418 9* . . 20.01 59,390 i ■ ■ 2.03 6,013 2J . . 21.87 64,910 1 . . 2.65 7,854 3 . . 23.81 70,680 1J • • 3.35 9,940 3j . . 25.83 76,690 n • • 4.13 12,272 3$. . 27.94 82,950 H • 5.00 14,840 35 . . 30.13 89,460 Hi • •' 5.95 17,670 J| • • 32.41 96,210 m - - 6.9S 20,730 3| • * 34.76 103,200 U- • 8.10 24?050 3.1 . . 37.20 110,150 ju • • 9.30 27,610 3? • ■ 39.72 117,930 ! 8 i ’ ’ 1 10.5S 31,420 4 . . 42.33 125,660RESISTANCE TO TENSION, 205 TABLE III. Safe Strength of Flat Boiled Iron Bars. CO • 02 00 a> o Width in inches. 1" U" H" ir 2" OX'- 4 OX" *2 03" 4 3" H” lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 1 Td 630 780 940 1,090 1,250 1,410 1,560 1,720 1,880 2,030 l 8 1,250 1,560 1,880 2,190 2,500 2,810 3,130 3,440 3,750 4,060 3. 16 1,880 2,340 2,810 3,280 3,750 4,220 4,690 5,160 5,630 6*,090 1 A 2,500 3,130 3,750 4,380 5,000 5,630 6,250 6,880 7,500 8,130 JC 16 3,130 3,910 4,690 5,470 6,250 7,030 7,810 8,590 9,380 10,200 J1 8 3,750 4,690 5,630 6,560 7,500 8,440 9,380 10,300 11,300 12,200 A 4,380 5,470 6,560 7,660 8,750 9,840 10,900 12,000 13,100 14,200 5,000 6,250 7,500 8,750 10,000 11,300 12,500 13,800 15,000 16,300 9 T& 5,630 7,030 8,440 9,840 11,300 12,700 14,100 15,500 16,900 18,300 5. 8 6,250 7,810 9,380 10,900 12,500 14,100 15,600 17,200 18,800 20*300 1L Tt5 6,880 8,590 10,300 12,000 13,800 15,500 17,200 18,900 20,600 22,300 5. 4 7,500 9,380 11,300 13,100 15,000 16,900 18,800 20,600 22,600 24,400 1H T6 8,130 10,200 12,200 14,200 16,300 18,300 20,300 22,300 24,400 26,400 7 8 8,750 10,900 13,100 15,300 17,500 19,7QP 21,900 24,100 26,300 28,400 1 5 T6 9,380 11,700 14,100 16,400 18,800 21,100 23,400 25,800 28,100 30,500 1 10,000 12,500 15,000 17,500 20,000 22,500 25,000 27,500 30,000 32,500 i| 10,600 13,300 15,900 18,600 21,300 23,900 26,600 29,200 31,900 34,600 n 11,300 14,100 16,900 19,700 22,500 25,300 28,100 30,900 33,800 36,600 11,900 14,800 17,800 20,800 23,800 26,700 29,700 32,700 35,600 38,600 U 12,500 15,600 18,800 21,900 25,000 28,100 31,300 34,400 37,500 40,600 l| 13,800 17,200 20,600 24,100 27,500 30,900 34,400 37,800 41,300 44,700 O 15,000 18,800 22,500 26,300 30,000 33,800 37,500 41,300 45,000 48,800 If 16,300 20,300 24,400 28,400 32,500 36,600 40,600 44,700 48,800 52,800 1 3 17,500 21,900 26,300 30,600 35,000 39,400 43,800 48,100 52,500 56,900 U 18,800 26,400 28,100 32,800 37,500 42,200 46,900 51,600 56,300 60,900 2 20,000 25,000 30,000 35,000 40,000 45,000 50,000 55,000 60,000 65,000 Notk.—Computed at 10,000 lbs. per square inch.206 RESISTANCE TO TENSION TABLE III. (concluded). Safe Strength of Flat Boiled Iron Bara. ao . 00 aa 0) c A Width in inches. M O o a 2 ■ [H .S 3£" 3f" 4" 41" ■*4 4!" 43" ■*4 5" 5i" G" Gi" lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. tV 2,190 2,340 2,500 2,660 2,810 2,970 3,130 3,440 3,750 4,060 i 4,380 4,690 5,000 5,310 5,630 5,940 6,250 6,880 7,500 8,130 6,560 7,030 7,503 7,970 8,440 8,910 9,380 10,300 11,300 12,200 £ 8,750 9,380 10,000 10,600 11,300 11,900 12,500 13,800 15,000 16,300 TV 10,900 11,700 12,500 13,300 14,100 14,800 15,600 17,200 18,S00 20,300 § 13,100 14,100 15,000 15,900 16,900 17,800 18,800 20,600 22,500 24,400 * 15,300 16,400 17,500 18,600 19,700 20,800 21,900 24,100 26,300 1 2S,400| £ 17,500 18,800 20,000 21,300 22,500 23,800 25,000 ‘27,500 30,000 32,500 o Iff 19,700 21,100 22,500 23,900 25,300 26,700 28,100 30,900 33,800 36,600 . i 8 21,900 23,400 25,000 26,600 28,100 29,700 31,300 34,400 37,500 40,600 1 1 Iff 24,100 25,800 27,500 29,200 30,900 32,700 34,400 37,800 41,300 44,700 1 26,300 28,100 30,000 31,900 33,800 35,600 37,500 41,300 45,000 48,800 i§ 28,400 30,500 32,500 34,500 36,600 38,600 40,600 44,700 48,800 52,800 2 8 30,600 32,800 35,000 37,200 39,400 41,600 43,800 48,100 52,500 56,900 ■ Iff 32,800 35,200 37,500 39,800 42,200 44,500 46,900 51,600 56,300 60,900 1 35,000 37,500 40,000 42,500 45,000 47,500 50,000 55,000 60,000 65,000 l.'o 37,200 39,800 42,500 45,200 47,800 50,500 53,100 58,400 63,800 69,100 u- 39,400 42,200 45,000 47,800 50,600 53,400 56,300 61,900 67,500 73,100 lj 41,600 44,500 47,500 50,500 53,400 56,400 59,400 65,300 71,300 77,200 n 43,800 46,900 50,000 53,100 56,300 59,400 62,500 68,800 75,000 81,300 n 48,100 51,600 55,000 58,400 61,900 65,300 68,800 75,600 82,500 89,400 ii 52,500 56,300 60,000 63,800 67,500 71,300 75,000 82,500 90,000 97,5001 it 56,900 60,900 65,000 69,100 73,100 77,200 ■1,300 89,400 97,500 105,000 j n 61,300 65,600 70,000 74,400 78,800 83,100 87,500 90,300 105,000 113,800 n 65,600 70,300 75,000 79,700 84,400 89,100 93,800 103,100 112,500 121,900 2 70,000 75,000 80,000 85,000 90,000 95,000 100,00ojll0,000|120,000 130,000RESISTANCE TO TENSION 207 TABLE IV. Upset Screie-Ends for Hound and Square Bars. Standard Proportions of the Keystone Bridge Company. o X Round Bars. Square Bars. ~ a O C p * X •3 w o f l. ✓ - ^ ^ o of screw s of thread. O Or £4 X O ^ <♦- O ro u, « X •m — O P H fffi 2 ? 0 — x <2 0'S 0 M 0. X sc ^ 12 0 0 5 ai o r* « 5 s <° • —j a 0 ert O .3 s- a P si M O x ^ © CJ c3 « O g 0 C CO £0 c3 O a 0 i- ■ A *2° O) eS O P H ws P 5 H M * inches. inches. inches. no. per cent. inches. inches. no. per cent. i 4 0.620 10 54 3 i 8 0.620 10 21 ‘1 d 0.620 10 21 0.731 9 33 t 1 8 0.731 9 37 1 0.837 8 41 It Tb 1 0.837 8 48 1 0.837 8 17 4 4 1 0.837 8 25 1« 0.940 7 23 i! 0.940 7 34 14 1.065 7 35 1 ,8 U 1.065 7 48 1 1.160 6 38 H 1.065 7 29 if 1.160 6 20 1 11 1.160 6 35 14 1.284 6 29 Ife if 1.160 6 19 4 1.389 34 34 j* H 1.284 6 'SO 4 1.389 54 20 n 1.284 6 17 4 1.490 5 24 j4l if 1.389 23 4 1.615 5 31 Ire i! 1.490 5 29 11 1.615 5 19 1| If 18 1.499 5 18 2 1.712 44 22 l.7,; 1.615 5 26 24 1.837 28 u 2 1.712 n 30 m 1.837 44 18 1 2 1.712 4 20 24 1.962 4è 24 1| 91 ™ 1.837 44 28 ■ 2.087 44 30 U* : 91 ■“8 1.837 4 18 H * 8 2.087 44 20 ’I 91 9l “ 1 1.962 44 26 91 — •> 2.170 4 21 1.962 44 IT 2.300 4 26 11 1 T ft A 1 b 93 ■^8 2.087 44 24 9& -8 2.300 4 18 21 2.175 '4 26 ■^4 2.425 4 23 2 2* 2.175 4 18 2j 2.550 4 28 ** «1 2.300 4 24 2g 2.550 4 20 9l ^8 2f 2.300 4 17 3 2.629 34 20 2,3o 92 2.425 4 23 Ql ^8 2.754 3i 24 1ocp-W» 0Cfc7*M>- xiu^iX xp to to —I-OOH crfj> to to —‘-*4» ®tw to t o —t~0C|CJ» 0+- to to o\~~ to to “H0*8 to to j®~ inches. Diameter of round or side of square bar. 4- 4— Jk|VX|U« 4- 4» 4* 44 xh CO CO xh*+o X CO X CO N*-afco X CO X CO «H*-OOb— CO CO XH to to XMXM 3* Diameter of upset screw-end. r& X 4- H io —1 JJ« C* V» W 4* CO © ~T I X tc ro ro »—* CO X to to to to 3— to to to to © to to C» © to to to to X CO to to to X o Excess of effective area of screw-end over bar. 4- 4—■ 4» 4-afeoxt— 4* 4— Xb- CO CO XHXH CO CO •Hwxfp* ccfw*ib- co co X{WX}W CO CO o S' <0 X Diameter of upset screw-end. 4- 44 & © 0» to CO X CO CO to © co to CO co o C* to ^4 4- 4-4» 4— to to CO CO CO to ►— to •4 Cl CO CO to 1 to o c* o 3.004 3.004 to to X 41 4| Cl © 4^ s’ o S' 0) X Diam. of screw at root of thread. CO I d ► i to ro ^•lo:4oo to CO xm CO CO CO CO CO CO CO CO CO CO CO CO 3 Threads per inch. B > s X t—‘ to CO ►— to 4- © rc io 4-* >—* to x co to »—1 o © to to 4- 1-1 K) o o to 1—1 to X rt) o w r* w Go C5 O) S t§J Co <1 CS o OS 03 208 RESISTANCE TO TENSION.RESISTANCE TO TENSION. 209 TABLE V. Strength of Iron and Steel Wire Hopes, Manufactured by tiie John A. Roebling’s Sons Co., New York. Trade number. Diameter in inches. Weight per foot in lbs. of rope with hemp centre. Iron. Cast-Steel. Breaking strain in tons. Proper working load in tons. Breaking strain in tons. Proper workmg load in tons. With Nineteen Wires to the Strand. 1 2? 8.00 74.00 15.00 130.0 26.0 2 2 6.30 65.00 13.00 100.0 21.0 3 it 5.25 54.00 11.00 78.0 17.0 4 If 4.10 44. (X) 9.00 64.0 13.0 5 If 3.65 39.00 8.00 55.0 11.0 |i if 3.00 33.00 6.50 - - 6 A 2.50 27.00 5.50 39.0 8.0 7 A 2.00 20.00 4.00 30.0 6.0 8 1 1.58 16.00 3.00 24.0 5.0 9 i & 1.20 11.50 2.50 20.0 4.0 10 i 0.88 8.64 1.75 13.0 3.0 10i I 8 0.70 5.13 1.25 9.0 2.0 10* H 0.44 4.27 0.75 6.5 1.5 10J i 0.35 3.48 0.50 5.5 1.0 With Seven Wires to the Strand. 11 if 3.37 36.00 9.00 67.0 16.00 12 A 2.77 30.00 7.50 55.0 12.50 13 if 2.28 25.00 6.25 45.0 10.00 14 A 1.82 20.00 5.00 30.0 8.00 15 1 1.50 16.00 4.00 30.0 6.50 10 t 1.12 12.30 3.00 22.0 5.00 17 a 4 0.88 8.80 2.25 17.0 3.50 18 ft 0.70 7.60 2.00 13.5 3.00 19 f 0.57 5.80 1.50 10.0 2.25 20 A 0.41 4.10 1.00 8.0 1.75 21 i 0.31 2.83 0.75 6.0 1.25 22 A 0.23 2.13 0.50 — - 23 f 0.19 1.65 - 4 1.00 24 Ä 0.16 1.38 - 3 0.75 25 0.125 1.03 — 210 RESISTANCE TO TENSION. Ropes, Hawsers, and Cables. (IIASWELL.) Hopes of hemp fibres are laid with three or four strands of twisted fibres, and run up to a circumference of twelve inches. Hawsers are laid with three strands of rope, or with four rope strands. Cables are laid with three strands of rope only. Tarred ropes, hawsers, etc., have twenty-five per cent less strength than white ropes: this is in consequence of the injury the fibres receive from the high temperature of the tar, — 290°. Tarred hemp and manila ropes are of about equal strength. Manila ropes have from twenty-five to thirty per cent less strength than white ropes. Hawsers and cables, from having a less proportionate number of fibres, and from the increased irregularity of the resistance of the fibres, have less strength than ropes; the difference varying from thirty-five to forty-five per cent, being greatest with the least circumference. Ropes of four strands, up to eight inches, are fully sixteen per cent stronger than those having but three strands. Hawsers and cables of three strands, up to twelve inches, are fully ten per cent stronger than those having four strands. The absorption of tar in weight by the several ropes is as follows : — Bolt-rope .... 18 per cent Cables..................21 per cent Shrouding . . 15 to 18 per cent Spun-yarn . . 25 to 80 per cent White ropes are more durable than tarred. The greater the degree of twisting given to the fibres of a rope, etc., the less its strength, as the exterior alone resists the greater portion of the strain. To compute the Strain that can be borne with Safety by New Ropes, Hawsers, and Cables, deduced from tlie Experiments of the Russian Government upon the Relative Strength of Different Circumferences of Ropes, Hawsers, etc. The United-States navy test is 4%00 pounds for a while rope, of three strands of best liiga hemp, of one and three-fourths inches in circumference (i.e., 17,000 pounds per square inch); but in the following table 14,000 pounds is taken as the unit of strain that can be borne with safety. Rule. —Square the circumference of the rope, hawser, etc., and multiply it by the following units for ordinary ropes, etc.RESISTANCE TO TENSION 211 TABLE Showing the Units for computing the Safe Strain that may be borne by Ropes, Hawsers, and Cables. Uoi »ES. Hawsers. Cables. White. Tarred. White Tar’d. __ i White. Tar’d. Description. 1 •z £ » rz 02 -t <£ 02 X 02 'ti if! CO to j- 02 CO tn ”2 02 CO !» 02 Circumference in ins. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. White rope, 2.5 to 6 ins. 1,140 HB - - 600 - ' - - White rope, 6 to 8 ins. . 1,0.10 1,260 - - 570 - 510 - White rope, 8 to 1*2 ins. 1,045 880 - - 530 - 530 - White rope, 12 to IS ins. - - - - 550 - 550 - White rope, 18 to 26 ins. - - - - - - 560 - Tarred rope, 2.5 to 5 ins. - - 855 1,005 - 460 - - Tarred rope, 1 to 8 ins. . - - 825 940 - 480 - - Tarred rope, 8 to 12 ins. - - 780 820 - 505 - 505 Tarred rope, 12 to IS ins. - - - - - - - fl| Tarred rope, 18 to 26 ins. - - - - - - - 550 Manila rope, 2.5 to 6 ins. 810 950 - - 440 - - - Manila rope, 1 to 12 ins. 760 835 - 465 - 510 - Manila rope, 12 to IS ins. - - - - - - 535 - Manila rope, 18 to 26 ins. 560 “ 1 When it is required to ascertain the iveight or strain that can be borne by ropes, etc., in general use, tlie above units should be reduced one-third, in order to meet the reduction of their strength by chafing, and exposure to the weather. TABLE VI. Strength and Weight of Manila Rope. . Weight per ft. j Breaking load. Diameter. o 5 Weight per ft. i Breaking load. ins. ins. lbs. lbs. tons. ins. ins. lbs. lbs. tons. I 0.239 a 4 .019 560 0.2S0 1.92 6 1.19 25,536 12.77 0.318 1 .0:i3 784 0.39 2.07 0* 1.39 29,120 14.50 0.477 1| .074 1,568 0.78 2.23 1.02 32,704 16.35 0.636 2 .132 2,733 1.36 2.39 7* 1.86 36,288 18.14 i ! 0.795 2| .206 4,278 2.14 2.55 8 2.11 39,872 19.93 j 0.955 3 .297 6,115 3.06 2.86 9 2.67 47,040 23.52 ' 1.11 O I .404 8,534 4.27 3.18 10 3.30 54,208 27.10 I 1.27 4 .528 11,558 5.78 3.50 11 3.99 61,376 30.69 1.43 4h .668 14,784 7.39 3.82 12 4.75 68,544 34.27 ! 1.5) 5 .S25 18,308 9.18 4.14 13 5.58 75,712 37.86 1.7 5 5i .998 21,952 10.97 4.45 14 "6.47 82,880 41.44212 RESISTANCE TO TENSION TABLE VII. Weight and Proof Strength of Chain. Manutactured by the New-Jersey Steel and Iron Company. Stud Chain Short Link Chain. X.B. Crane Chain. Average Average weight Size. weight per Proof. Size. Proof. Proof. fathom. per fathom. inches. lbs. tons. inches. lbs. tons. tons. 3 4 QO oo 10 h 2f j 4 - n lo 1 8 38 43 12 14 4 5 7 4 1 — 1 ft Tt> 50 16 I 9i 2 3 1 58 18 T2^ 12 2£ 4 ItV 65 20 i 15 H 4* H 72 23 u Tff 19 M 80 26 25 5* 4 8 88 28 jo ■ 30 7 Si H 98 31 2 4 35 8 10 1| 110 34 13 1 6 40 H ni lìV 114 37 1 8 47 11 13 H 127 41 1 ft 54 m 14i H 138 44 1 61 14 16 if 150 48 ItV 69 16 19 H 157 52 h 76 18 21 IS 170 56 85 20 23 1±3 1 1 6 184 60 n 95 22 25 11 -Ljg 200 64 103 24 27 214 68 if 113 26 29 2 230 72 Its 123 28 31 oX 250 80 133 30 oo OO 21 290 88 RESISTANCE TO SHEARING. 213 CHAPTER X. RESISTANCE TO SHEARING. By shearing is meant the pushing of one part of a piece by the other. Thus in Fig. 1, let abed be a beam resting upon the supports SS, which are very near together. If a sufficiently heavy load were placed upon the beam, it would cause the beam to break, not by bending, but by pushing the whole central part of the beam through between the ends, as represented in the figure. This mode of fracture is called “shearing.” The resistance of a body to shearing is, like its resistance to tension, directly proportional to the area to be sheared. Hence, if we denote the resistance of one square inch of the material to shearing by F, we shall have as the safe resistance to shearing, Safe shearing 1 area to be sheared X F strength J ~ S' ’ • ( ) «S' denoting factor of safety, as before. A piece of timber may be sheared either longitudinally or transversely; and, as the resistance is not the same in both cases, the value of F will be different in the two cases. Hence, in substituting values for F, we must distinguish whether the force tends to shear the piece longitudinally (lengthwise), or transversely (across). Table I. gives the values of F, as determined by experiment, for the most common materials employed in architectural construction.214 RESISTANCE TO SHEARING. TABLE I.m . Showing the Resistance of Materials to Shearing, both Longitudinally and Transversely, or the Values of F. Materials. Values of F. • Longitudinally. 1 ransverselv. j lbs. 1 lbs. Cast-iron - 27,700 a Wrought iron - 5.0.000 E Steel .... - 63,7401 White ash - 1,400a Reech - 5,200 c Kirch - 5,600 e Hemlock 5401 2,700 c Locust 1,180 cl * 7,000 c White oak 780 d 4,400 c White pine 490 d 2,480 e Yellow pine 510 d 5,700 c Spruce 470 d 3,255 c Rlack walnut - 2,000 a Oregon pine 840 e - Oregon spruce 310e - Oregon ash 1 . . 784 e - Oregon maple . 732 e There are but few cases in architectural construction in which the resistance to shearing has to be provided for. The one most frequently met with is at the end of a tie-beam, as in Fig. 2. The rafter 7? exerts a thrust which tends to push or shear off the piece A 7>C7), and the area of the section at CD should offer enough resistance to keep the rafter in place. This area is equal to CD a Uankine. . l> Kirkaldy. c Trautwine. d Ilatiield. e United-States Covern-W.ile:town Arsenal. iiH.n atRESISTANCE TO SHEARING. 215 times the breadth of the tie-beam; and, as the breadth is fixed, we have to determine 'the length, CD. If we let 11 denote the horizontal thrust of the rafter, then, by a simple deduction from formula 1, we have the rule: — Length of CD in inches Sx II breadth of beam X F’ (2) F, in this case, being the resistance to shearing longitudinally. Example I. — The horizontal thrust of a rafter is 20,000 pounds, the tie-beam is of Oregon pine, and is ten inches wide: how far should the beam extend beyond the point D? Ann. In this case II = 20,000 pounds, and from Table I. we find that F — 840; S we will take at 5. Then 5 X 20000 CD — ~~iq~x~s4(7» or nearly 12 inches. Practically a large part of the thrust is generally taken up by an iron bolt or strap passed, through or over the foot of the rafter and tie-beam, as at A (Fig. 2). When this is done, the rod or strap should be as obliquely inclined to the beam as is possible; and, whenever it can be done, a strap should be used in preference to a rod, as the rod cuts into the wood, and thus weakens it. . Another common case in which the resistance to shearing should be considered is in the case of iron pins and wooden tree-nails. If we have three bars fastened together by a pin, and each pulling in the direction indicated by the arrows in Fig. 3, they will tend to shear off the pin at the sections a a. If the pull exerted by the tie II be denoted by IT, then each section of the pin will have to resist one-lialf II, as there are216 RESISTANCE TO SHEARING. two sections to resist the whole. Then, from Rule 1, we deduce the following: — I S X II Diameter of wooden pin in inches = iC 570$ x F (-°') Diameter of wrouglit-iron pin in inches = In formula 3, F is the resistance to shearing transversely. Example II.—Suppose the bar B is pulling with a force of 141,372 pounds: what should be the diameter of an iron pin to resist it ? , ' 141372 Ans. Diameter = square root of = square root of 9, or 3 inches. These are about the only two cases in which rupture by shearing is liable to take place in architectural construction. The strength of riveted joints involves the consideration of shearing; but the architect seldom has occasion to calculate the strength of such joints: and, as the question of rivets is a rather complicated one, it will not be discussed in this chapter. A description of the more common forms of riveted joints will, however, be found in Chap. XXIX. Occasionally a large beam of short span, such as a ten by ten inch beam, two feet long, needs to be computed with reference to shearing at the points of support: but such beams do not occur in building construction; and, where they do occur, they can be computed by formula 1. 1 H 15708 <4>STRENGTH OF l’OSTS, STRUTS, AND COLUMNS. 2l. CHAPTER XI. STRENGTH OP POSTS, STRUTS, AND COLUMNS. As the strength of a post, strut, or column, depends primarily upon the resistance of the given material to crushing, we must first determine the ultimate crushing-strength of all materials used for this purpose. The following table gives the strength for all materials used in building, excepting brick, stone, and masonry, which will be found in Chap. VI. TABLE I. Average Ultimate Crushing-Loads, in Pounds per Square Inch, for BuiIding-Materials. Crushing Crushing Material. weight, Hi lbs. Material. weight, in lbs. per sq. iuch. per sq. inch. c. c. For Stone, Brick, Woods (continued). and Masonry, see Beech 9,300 a Chap. VI. Birch ] 1,600 a Cedar 6,500 a Metals. Hemlock 5,400 b Cast-iron .... 80,000 Locust 11,720 b Wroiurht-iron . . . 36,000 Black walnut . . . 5,690 .Steel (cast) .... 225,000 a White oak .... 3,150 to 7,000 Yellow pine . . . 4,400 to 6,000 Woods. Ash 8,600 a White pine .... Spruce j 2,800 to 4,500 The values given for wrought and cast iron are those generally used, although a great deal of iron is stronger than this. The values for white oak, yellow pine, and spruce, are derived from experiments on full-size posts, made with the government testing-machine at Watertown, Mass.; the smaller value representing the strength of such timber as is usually found in the market, and the larger value, the strength of thoroughly seasoned straightgrained timber. For these woods a smaller factor of safety may be a Traut wine. b Hatfield.218 STRENGTH OF WOODEN POSTS AND COLUMNS. used than for the others, the strength of which was derived from experiments on small pieces. The values for wood are for dry timber. .Wet timber is only about one-half as strong to resist compression as dry timber, and this fact should he taken into account when using green timber. The strength of a column, post, or strut depends, in a large measure, upon the proportion of the length to the diameter or least thickness. Up to a certain length, they break simply by compression, and above that they break by first bending sideways,-and then breaking. Wooden Columns. For wooden columns, where the length is not more than fifteen times the least thickness, the strength of the column or strut may be computed by the rule, Safe load = area of cross-section X C factor of safety (1) where C denotes the strength of the given material as given in Table I. The factor of safety to be used depends upon the place where the coluntn or strut is used, the load which comes upon it, the quality of tlie material, and, in a large measure, upon the value taken for C. Thus for white oak, yellow pine, and spruce, the value C is the actual crushing-strength of fulhsize posts of ordinary quality : lienee we need not allow a factor of safety for these greater than four. For the other woods, we should use a factor of safety of at least six. If the load upon the column or post is such as comes .upon the floor of a macliinc-sliop", or where heavy machinery is used, or if the strut is for a railway-bridge, a larger factor of safety should be used in all cases. . If the quality of the timber is exceptionally good, we may use the larger values for the constant C, in the case of the last four woods given in the table. For ordinary hard pine or oak posts, multiply the area of cross-section in inches by 1000; for spruce, by 800, and for white pine, by 700 pounds. Example I. —What is the safe load for a liard-pine post 10 by 10 inches, 12 feet long? Ans. Area of cross-section = 10 X 10 = 100 square inches; ICO X 1000 H 100,000 pounds.'95 Ki^ovwìiAla [rom rc^ulV^ o[ iVtftav'tbiVr\ $YSev\c\l /v^orc Ycliinb\e l^awx ’<30tV 5_.'l‘lo8o^ ÒOZ^ry Iptyo L —:—: \ 3 7«qoOC7m 1 ' J .■ • {'{'"# to' itf$*(»p^S2oo,fy/o{+^ J&boO')7$ooJé>-7 — 4>iktfìjgXn*H ;3o8)o kf^r/ | j . v g" alte3àjj£ffUl& 2/^0 //vujo Uì7fcv Ln<&ar_ I y V /O* ; Jjooo ^ 200 j f'fci** óLSSo S5Sot> M77.Q | 4gobpj 2b^OO Ti+y-O 708^0 é>qifLo E»fr,,frg °fp ktf^d/oV*« C.3S' o 55-773 ^/7/ Ifid 57^5 b fof2.i.//p20 aJA - - ♦ **" < |\ IO < to' pooo ! *Joooo 7Jot+o y^-99o 7ò8oo ‘“J'iTf'Z.o JoS^o bStf-oo /^VlX" logooo 10B000 *f2/uo 9f82o & % !T4>o S'Jòoo £2.0$ o ! j /*>'*£ » l y^2o 7W7 n*AV J_____ |_XX^y lV /2j<*oo /2*jL*o ,/io Io9*joo toLX^o'Jt> liSù j7?4*7& | /7 ' V ' n-L>fóo i2^o2u> llcxjtito'iiH-qrT- i ll'tlU’ nx*oo tyxtoo' n2ìc>0 itfa£flJ*+L$6o ttJ-SOo/Wl’Lo lìtlif-ò./3/32/ ! 1 /lf*V /f !/7£*^oo HLHdo pUiioo r7tlA«otl53^»f» ./5>f7o /^?2io //fufeo j/V-022^i I Uk' |23o^oa 2 io loo 2.Ò0 +UV 2J0 ifoòr2o(* f" *2 0‘ \$ùoooa ^t>o0o\èùoo*o’ÒClOOvtJ‘3(>oooo 3&6ooo v?2//ZO P/&**0 \3c#/<>0 » WT ; J f'Htytf -CvJ V rv1* *: f; V / j ''df*i / ’>> ^ rtVK'JiV 'Vs fO p;'P \ ■ •^* > Ì ^ ij *r* ‘V f ...4 • >t fvt -if ♦ • ft i^vfC Ol ' M * <>. Vi O ^Oi ,*ì(:j-: V isT^ •» f* .• j. ;> / V . 'I >. f ^ • i I i * i ">/ •'i A'^i /H «i y i‘ , ÿ > ‘ . V# * ‘ , •” ' ‘ • •• ■ *• i Si ^ /.. *i > il r* J f . n *, t * o c> o « \ ^ / 4 • #* * * o ; i *i V * ’I t A V ' I ' f ^ • % è '.j .. y 'V> jo.-vr ' r ‘a *>c . n \ *> , > V ' V, . • i i V. «> >fi> u i u-.' *V‘. u\v v' y »s N ' N Oy.f ’\ c •' 'Ì r\ V rw'v* -X ■\ \fc£ V > > J MJ v •* Ô > V • * f 4iV' o - Y-» o o » . STRENGTH OF WOODEN FOSTS AND COLUMNS. 219 Example II.—What is the safe load for a spruce strut 8 feet long, 0" X8;/? Am. Area of cross-section = 48; 48 X 800 = 38,400 pounds. Columns and struts over fifteen diameters long, or where the length is more than fifteen times the least thickness of the post or strut, are liable to break by both bending and crushing. In speaking of the length of a column as affecting its strength, we mean the greatest length of the column which is unsupported sideways. Thus we may have a column forty feet long ; but if it is securely braced Ivory ten feet, so that it cannot bend, it will be just as strong as though it were but ten feet long. Till column must, however, be prevented from bending in any direction; for, if it were only braced in two ways, it would be no stronger than if not braced at all. For columns over this length, the following formula, deduced by Mr. Lewis Gordon from Mr. Hodgson’s experiments, has been for a long time in use, and it probably answers better than any other formula: — For rectangular ) breaking- * _ C X area of cross-section pillars and struts f load, in lbs. ~ sq. of length (2) For cylindrical ) breaking- _ C X are^of cross-section columns > load, in lbs. ~~ / sq. of length \ vm ‘ 1.7(1 + w 71—tr x 0.004) m \ sq. or breadth / In these two formulas the length is taken in inches. The value of C is the crushing-strength of the particular kind of timber used for the strut. Mr. C. Shaler Smith, a prominent engineer, determined from his own experiments that tin* value of C for hard pine should be about 5000, and that is the value generally used by engineers. For white pine or spruce we should use 3000 for C. With these values, a factor of safety not larger than six may be used for almost any case in building-construction, and in many cases a factor of safety as low as four might be used. Example 1IL—What would be the safe load fora hard-pine strut 15 feet long, 0" X 8" ? ' A ns. The ratio of length to least thickness is in this case 30: so we should employ formula 2. Then 5000 X 48 Breaking-weight = ----------------- = 52176 pounds. 1 -j- x 0.004 (jl .220 STRENGTH OF HARD PINE AND OAK STRUTS. Taking one-sixtli of this, we have for the safe load, 8096 pounds. Example IV. — What would be the safe load for a turned white-pine column 12 feet long, and 0 inches in diameter at its smallest part ? Hfis. Using formula 3, we have, Breaking-load 3000 X 28.3 —^442---------\ F 1-5133 pounds. 1.7U + X O.OO4) As in this case the pine would probably be of very good quality, and dry, and the load not being very severe, we may use a factor of safety of four, which would give a safe load of 3783 pounds. Tables for the Strength of Hard-Pine and Oak Struts. The application of Table II. is obvious. To use Table III., find first the ratio of the length to the diameter of the column, both being taken in inches. Look In the table, and find the safe load per square inch for that ratio, and multiply by the area of the smallest cross-section of the post. Example. — Take the same case as in Example III. Here the ratio was 30, and looking in Table III., opposite 30, we find 1087. Multiplying this by 48, the cross-section of the post, we have 52,176 pounds for the breaking-load, as before. Fig 7ron crips for timber pillars are often used in important constructions, and are an excellent invention, as they serve to distribute the thrust evenly through the pillar, and also form a bracket, which is often desirable, for supporting the ends of girders where a second post rests on top of the tirst. Eig. 1 shows the section of one of the best forms of caps.STRENGTH OF WOODEN POSTS AND COLUMNS. 221 TABLE II. Hard-Pine and Oak Posts. Breakiny-Loads, in Tons (of 224-0 pounds), of Square Posts or moderately Seasoned Hard Pine or Oak, firmly fixed and equally loaded. Side of square post or strut in inches. Height iu feet. 6 8 39.7 9 35.0 10 30.9 11 27.4 12 24.3 13 21.7 14 19.4 15 17.5 16 15.8 17 14.3 18 13.0 19 11.9 20 10.9 22 9.2 24 7.9 20 o.s; 2S 5.9 30 5.2 7 8 02.4 90. S 50.0 83.5 50.3 75.3 45.1 68.9 40.6 62.5 30.0 57.2 33.1 51.9 30.0 47.0 27.3 43.4 24.9 40.0 22.7 30.6 20.9 33.8 19.2 31.1 10.3 26.8 14.1 23.2 12.2 20.1 10.7 17.7 9.4 15.3 9 10 124.4 103.0 115.1 152.5 105.8 142.0 96.4 132.0 87.0 122.0 81.5 113.5 70.0 105.0 70.0 97.5 04.0 90.0 59.5; 84.0 55.0 78.0 51.0 73.0 47.0 68.0 40.0 59.0 30.0 52.0 30.0 49.0 2S.0 41.0 25.0 34.0 li 12 207.0 255.0 195.0 242.5 183.0 230.0 171.5 217.0 100.0 204.0 150.0 192.0 140.0 180.0 131.0 169.5 122.0 159.0 114.0 149.5 100.0 140.0 99.5 132.0 93.0 124.0 82.0 109.0 72.0 97.0 64.0 87.0 57.0 78.0 51.0 70.0 14 16 367.0 500.0 353.0 4S3.0 339.0 406.0 323.0 449.0 307.0 432.0 292.0 414.5 277.0 397.0 263.5 $80.0 250.0 303.0 237.0 347.0 224.0 331.0 212.5 310.0 201.0 301.(; 182.0 271.0 163.0 24&0 148.0 220.0 133.0 200.0 121.0 -189.0222 STRENGTH OF WOODEN POSTS AND^OLUMNS. TABLE III. IIard-Pine 1’ii.lars and Posts. lireaking-Loads, in Pounds per Square Inch of Cross-Sections, for IIard-Pine Pillars and Posts whose Heights are measured by the Diameter or Least Side. Length in inches divided by least thickness in inches. Round. Rectan- gular. Length ill inches divided by least thickness in inches. Round. Rectan- gular. 15 1548 2631 34 523 8S9 16 1453 2470 35 498 847 17 1364 2319 36 476 S09 IS 1281 2178 37 454 772 19 1203 2046 38 434 73S j 20 1131 1923 • 39 415 706 21 1064 1809 40 398 676 22 1002 1703 41 ■ 381 647 23 944 1005 42 365 621 24 890 1513 43 351 596 25 841 1429 44 336 572 26 794 1350 45 • 323 550 27 750 1276 46 311 528 - 28 711 1209 47 299 50S 29 674 1146 48 287 489 30 640 1087 49 27S 472 ' 31 607 1032 50 268 455 32 577 9S1 51 258 439 QQ OO 548 933 52 249 423 LSTRENGTH OF CAST-IRON COLUMNS. 223 Cast-Iron Columns. For cast-iron columns, where the length is not more than six or eight times the diameter or breadth‘of column, the safe load may he obtained by simply multiplying the metal area of cross-section by 6| tons, which will give tons for the answer. Above this proportion, that, is, where the length is more than eight times the breadth or diameter, the following formulas should be used. These formulas are known as Gordon's and llankine’s. Fokmulas— For solid cylindrical cast-iron columns, Safe load in lbs. = Metal area X 13330 sq. of length in inches 1 q----------- - - f— --------T—e sq. of diam. in inches X 20b For hollow cylindrical columns of cast-iron, Safe load in lbs. = Metal area X 103:10 sq. of length in inches 1 J------•--------5------ 400 X sq. of diam. in inches (4) (5) For hollow or solid rectangular pillars of cast-iron, ei mu Safe load in lbs. Metal arca X 13030 sq. of length in inches (6) 1 + F >00 X sq. of least side in inches For cast-iron jtosls, the cross-section being a cross of kjuuI arris. + Safe load in lbs. = 1 + Metal area x 13330 sq. of length in inches 133 x sq. of total breadth in inches Example I.- What is the safe load for a hollow cylindrical cast-iron column, 10 feet long, fi inches external diameter, and 1" thickness of shell ? ,ln.s. We must first find the metal area of the cross-section of the column, which we obtain by subtracting the area of a circle of four inches in diameter from the area of one six inches in diameter. The remainder will be the area of the metal. The area of a six-incli circle is 28.27 square inches, and of a four-inch, 12.56 square inches; and the metal area of the column is 15.71 square inches.224 STRENGTH OF CAST-IRON COLUMNS. Then, substituting known values in formula, 5, we have 15.71 x 13330 Safe load =-------14400” = pounds. 1 + 400 x15(5 There is no use in carrying the residt farther than the nearest hundred pounds, because the accuracy of our formulas will not warrant it. Example II. — IVliat is the safe load for a cast-iron column 12 feet long, the cross-section being a cross with equal arms, one inch thick, the total breadth of two arms being 8" ? Ans. The area of cross-section would be 8 + 7 = 15 square inches. Then, by formula 7, Safe load in lbs. = 15 X 13330 20730 1 + 133 X 04 = 58300 pounds. Projecting1 Caps. Hollow columns calculated by the foregoing formulas should not be cast with heavy projecting mouldings round the top or bottom, as in Fig. 2, at « and b. It is obvious that these are weak, and Would break off under a load much less tliau would be required toSTRENGTH OF CAST-IRON COLUMNS. 225 crush the column. When such projecting ornaments are deemed necessary, they should be cast separately, and be attached to a prolongation of the column by iron pins or screws. Ordinarily it is better to adopt a more simple base and cap, which can be cast in one piece with the pillar, without weakening it, as in Fig. 3. In all the rules and formulas given for cast-iron * columns, it is supposed that the ends have bearings planed true, and at right angles to the axis of the column. Sometimes it is desirable to have cast-iron struts in trusses, secured at one or both ends by a pin passing through an eye in the end of the strut. Where the strut is hinged at one end only, the strength should be taken at about two-thirds of what the above rules would give, and, when binged at both ends, only one-third the safe load given by the formulas. Tables of Cast-Iron Columns. By an inspection of the foregoing formulas for cast-iron columns, it will be seen, that, all other conditions being the same, the strength per square inch of cross-section of any column varies only with the ratio of the length to the diameter or least thickness. Thus a column 15 feet long and 10 inches diameter would carry the same load per square inch as a similar column 9 feet long and 0 inches diameter, both having the ratio of length to diameter as 18 to 1. Owing to this fact, tables can be prepared giving the safe load per square inch for columns having their ratio of length to diameter less than 40. Oil this principle Table IV. has been computed, giving the loads per square inch of cross-section for hollow cylindrical and rectangular cast-iron columns. To use this table, it is only necessary to divide the length of the column in inches by the least thickness or diameter, and opposite tin* number in column I. coming nearest to the quotient find the safe strength per square inch for the column. Multiply this load by the metal area in the cross-section of the column, and the result will be the safe load for the column. Exa.mi’i.h III.—What is the safe load fora 10-inch cylindrical cast-iron column 15 feet long, the shell being 1 inch thick? Mas. The length of the column divided by the diameter, both in inches, is 18, and opposite 18 in Table IV. we find the safe load per square inch for a cylindrical column to be 700(3 pounds. The metal area of the column we find to be 28.27 inches; and, multiplying these two numbers together, we have for the safe load .of the column 208,200 pounds, or about 104 tons.226 STRENGTH OF CAST-IRON COLUMNS. Besides tliis table, we have computed Table V. following, which gives at a glance the safe load fora cast-iron column coming within the limits of the table, and of a thickness there shown. Thus, to find the safe load for the column given in the last example, we have only to look in the table for columns having a diameter of 10 inches and a thickness of shell of 1 inch, and opposite the length of the column we find the safe load to be 104 tons, the same as found above. The safe load in both tables is one-sixth of the breaking-load. TABLE IY. Strength of Hollow Cylindrical or liectangxdar Cast-Iron Pillars., (Calculated by Formulas 5 and 6.) Length divided by external Freaking-weight in pounds per square inch. Safe load in pounds per square inch. breadth or diameter. Cylindrical. Hec tango lar. Cylindrical. Rectangular. 5 75,204 76,190 12,549 12,698 6 73,395 74,627 12,232 12,438 7 71,269 72,859 11,878 12,143 8 68,905 70,922 11,494 11,820 ’ 9 66,528 68,846 11.088 11,474 10 64,000 60,000 10.06) 11,111 11 61,420 64,412 10,230 10,735 12 58,823 62,111 9,80 4 HB1 13 50,239 59,790 9,373 9,905 14 53,859 57,471 8,970 9,578 15 51,200 55,172 8,5:53 9,195 10 48,780 62,910 8,130 8,317 17 46,444 50,697 .. 7,741 8,449 18 44,198 48,543 7,360 8,0U0 19 42,050 40,457 7,008. 7,743 20 40,000 44,444 6,000 7,407 21 38,050 42,508 6,341 7*085 22 36,200 40,650 6,033 0,775 23 34,455 38,872 5,742 6,479 24 32,787 ' 37,174 5,404 3BB 25 31,219 35,555 6,203 5,920 20 29,741 34,014 4,957 5,009 27 28,343 32,547 4,724 5,423 28 . 27,027 31,152 4,504 •5,192 29 25,785 29,S2S 4,297 4,971 30 . 24,615 25,571 ’ 4,102 4.701 31 23,512 27,310 3,918 4,818 32 22,472 26,246 3,145 4,374 33 21,491 25,172 3,581 4,195 34 20,505 24,154 3,427 4,026 35 19,692 23,188 3,282. 3,864STRENGTH OF CAST-IRON COLUMNS, 227 TABLE Y. Showing Safe Load in Tons for Cylindrical Cast-Iron Columns. Tuickness op Shell | Inch. ! Length Diameter of column (outside). • I | column. 6 ins. ' 7 ins. 8 ins. 9 ins. 10 ins. 11 ins. 12 ins. 13 ins. j Feet. Tone. Tone. Tons. Tons. Tons. Tons. Tons. Tons. 1 6 60.0 7S.1 94.0 110.8 128.6 144.9 161.7 180.0 - 7 55.7 72.2 88.9 106.9 124.2 140.1 156.4 176.0 8 50.7 66.3 83.8 101.1 117.7 135.2 151.1 170.3 9 45.8 61.9 78.7 95.2 113.4 130.4 145.8 164.5 10 40.8 56.0 73.5 89.4 106.8 123.2 140.5 158.7 11 37.1 - 51.5 68.4 83.6 100.1 118.3 135.2 153.0 « 12 33.4 #7.1 63.3 79.1 95.9 113.5 129.9 147.2 13 30.9 44.2 58/1 73.9 89.4 106.3 124.6 141.4 14 ‘27,2 39.8 54.7. 70.0 85Ù 101.4 119.2 135.6 m ‘24.7* 36.8 49.6J» 64*1 78.5 96.6 114.0 129.9 1.6 22.3 33.9 46.2. 60.3 71.9 91.8 108.7 124.1 18 - 29.0 1 41.0 52.5 67.6 84.5' 103 f'4 118.3 ‘20 24.4 36.0 A 44.7 63.3 77.2 98.1 112:5 • Metal area of cross-section. eq. ins. sq. ins. sq. ins. . sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. 12.37 14.73 17.10 19.44 21,80 24.15 26.51 28.86 Thickness op Shell 1 Inch. Length Diameter of column (outside). column. 6 ins. 7 ins. 8 ins. 9 ins. 10 ins. 11 ins. 12 ins. 13 ins. Feet. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 6 77 100 121 143 167 188 211 ’ 234 7 71 92 118 138 161 182 204 230 8 64 85 ' 108 m 153 176 197 222 9 58 79 101 123 147r- 170 190 215 10 52 72 95 116 138,* 161 183 207 11 47 66 8$ 108 S 130v- 154 175 200 1*2 42 60 81 102. 124 147 169 192 ! 13 39 57' 75 95 116 138 162 184 i 14 35 52 i 69. 90 110 132 155 177 i ~~lf> 31 47 - 64r 83 104 126 148 170 16 - 28 43 59 78 96 119 142 162 18 - 25 39 1 53 68 88 " 105 12# 151 28* 22 35 ' 46 58 79 94 114 136 ! Metal area of cross-section. sq. ins. sq. HI ins. sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. 15.71 ,18.82 ‘22.00 25.14 ‘28.27 31.41 34.56 37.70228 STRENGTH OF CAST-IRON COLUMNS, TABLE V. (continued). Safe Load in Tons for Cylindrical Cast-Iron Columns. Thickness or Shell 1J Inches. Length Diameter of column (outside). of ■ | column. 6 ins. 7 ins. 8 ins. 9 ins. 10 ins. 11 ins. 12 ins. 13 ins. Feet. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 6 91 119 145 173 203 230 257 286 7 84 111 137 167 196 222 249 2S1' 8 76 102 130 158 186 214 241 272 9 69 94 122 149 179 206 232 263 10 62 86 114 140 168 198 224 253 11 56 79 106 131 158 188 215 245 12 50 73 98 123 151 180 207 235 13 45 68 90 116 141 170 199 • 226 14 41 62 83 109. 134 161 190 217 15 37 56 77 100 127 153 181 208 16 33 52 71 94 I UT, 144 173 198 18- _ 45 61 82' 106* 129 156 184 20 - — 56 69 96 114 139 166 Metal area of cross-section. sq.ins. % sq.ins. sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. sq. ins. 18.65 22.58 26.52 30.44 34.36 38.29 42.22 46.14 Thickness of Shell 1J Inches. Length Diameter of column (outside). of column. 8 ins. 9 ins. 10 ins. 11 ins. 12 ins. 13 ins. 14 ins. 15 ins. Feet. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 6 168 201 236 269 302 336 365 401 7 159 194 228 ■ 259 292 330 359 391 8 150 184 219 251 282 320 348 381 9 141 173 211 242 272 309 338 369 10 132 163 196 230 262 298 330 359 41 123 152 1S4 219 252 287 319 353 12 . 113 143 176 210 242 276 310 343 13 104 134 164 197 232 265 302 331 14 96 126 156 188 223 255 288 318 15 89 117 148 179 213 244 278 810 16 83 110 « 936 170 203 233 264 299 18 73 95 124' 149 183 217 245 279 20 64 81 112 134 163’ 195 224 258 Metal area of cross-section. sq. ins. sq.ins. sq. ins. sq.ins. sq.ins. sq.ins. sq.ins. sq. ins. 30.63 35.34 40.06 44.77 49.48 54.19 58.90 63.62 x \N ♦ 4 ♦ A # ISTRENGTH OF BQ T he forniula in most gener al us is that aft r i feat ed to Prof. Lewis his name. W ith t he use of t he pr cas e, it giv •es results agree ing-Jjj ed in pract ice w it-h columns Of S' ari.e •ers) as are commonly us co effic ients for the given ornately with averages obtair breaking load In lbs per sa.in. of area or cross section of col* ■ i 4b iHvh i Cll ■f" is a co efficient depending upon the nature of the material and to some extent upon the chape or cross section or the col. It is orten taken approximaetly enough,as being the ult crushing st rength or short blocks or the given material. For good American wrought iron, such as is used ror cols -'40000 is generally used; ror cast iron 30000. Mr. Cleeman round f or mildHteel.15 carbon 53D00 and for hard steel .56! carbon 33000.Mr. C. whaler Smith gives 5000 for pine. «a* for wrot iron is usually taken as follows: when both ends of the col are flat or fixed 36000 to 40000 when both ends of the col are hinged 13000 to 20000 When to s' end flat Or fixed & other hi J)f;ed 24000 to 30000. For cast iron about One eight? and for pine about One twelft? f . , €■ t1 t' Cl ' (so Q a L*r 11 I ■ H M SM of these xigures is generally useda r\ ' S : - r*. « /: * •> ^ - r Y* î f • i r+ A • * • l. * L ■ -- * • c e-r 1 , 41 **■ .X 4 f r r r-. *• A X ' A - • f Ik • 1 4 0 1 r 'r ^ fi r 4 • r «K ft • r V ._/ ' L • . 1 *• ' • t. • 1 0 f a rv « ^ «* f - . ft f» , * ? r X r* ) 3 *> •- • <_. ^ • 1 J v 5 V t * . » 'J A » ft ; . 5 Ui/j j : 4- * r •■ *» ^ • * r. 3 • r* I * . , . J '' rr ■V r i k ft— 2 72 . » i r r. ’ r * ^ * W / . .... J c rf •A / l > 1 • ■■■ *t -, Vr V * >1 ' r r* I O'i - ^ *• — • 1 4 ft J- • ^ i • » % •ì 1 r * • in r . $ •• • 4 4« a A L j . j > ( i « 1 * .. ä 4- r ■> A f V- • V f *- A • i * J * 1 A. «• I r • f * Ï k 1 - * ; :4 - r ri f * 1 n A : , • r 4* ti *»â «*♦ 4^ l 1 r < A 1 ft ;n A ^ 1 ■ % • ft r X x • A ft V. »* V 4 *.•*. - • ?* r v-p / - ’ « A 4 r > r * *- • - * * » i « ft* v-f v‘ . i . - - * ft * 1 4 A ~ :( or f *r.' rs r \ * J *■ ^ j 1 rf." 1 r-A ** \ — s. « - | • u- ■ i » '■> 0 a r r • o r r 2 . /') ( r • ’\ n rr r 1 f 'f*' rr ' 1" » j i. ' 4 * V f * » * « 4 . ft A • « » r. » t-~ 1 t i f V ; 4 ; r . , i n rr ft f 4 - • > • ■ « •_ •*- • . 4. 1 « V • V ^ i 4 •* V 4 % t r SS. r * * J m~y r-r r rr 4 ' • » - g ' * >3- a >. a • € » r . » r • M. ft 1 <4 - r f r *. / V * 1 1 r ^ r 1 rr ’ i r* V « ft\ r è* 9 « f 'T .O i Oj fr s r O'j - * X r .0 . a L w- a c - r rrO A f f,A A i r ' *• « i » V* / V • f r ;Hm r 4 ^ » » / ' -, rî O 4 ^ •rO * iftl* is th* length or theffc 0I1I n. If the column has between its ends, supports which prevent t from yieldinq^ldewis e, the length is to be measured betweenlsuch supports. *rR is the least radius of gyration of the cross section of the pillar. and "r* must be In the same unit; agfboth in fe or b ot h in i nch e c. * kr * /*> \ i r* 4 » c f ! & X * V f * r 3 _ j [ r ^ V J f f à * t c +* r r ^ 4 ( 4 A 1 4 r 0 u. - p* # V» J / .*.j 4. I ^ » V/ r f - i* • * • i X ft Î 1 . J « i r I 4 f; r *. * -/ A. . r •» r ~ r r\ » « . • A J P M 4 j i i iSTRENGTH OF CAST-IRON COLUMNS. 229 TABLE Y. (concluded). Safe Load in Tons for Cylindrical Cast-Iron Columns. Thickness op Shell 2 Inches. Length Diameter of column (outside). of column. 8 ins. 9 ins. 10 ins. 11 ins. 12 ins. 13 ins. 14 ins. 15 ins. Feet. Tons. Tons. Tons. Tons. Tons Tons. Tons. Tons. 6 207 251 296 339 383 428 467 514 7 200 242 286 328 371 421 459 502 8 185 229 271 316 358 408 445 490 • 9 173 215 258 305 345 393 432 474 10 162 202 245 291 333 380 422 462 11 151 189 231 277 320 366 409 450 12 141 178 221 265 307 352 396 438 13 131 167 206 249 295 339 383 424 14 122 158 196 237 283 325 369 412 15 Tl2 145 183 226 270 311 354 398 16 102 136 170 215 °57 297 339 384 18 90 119 155 189 232 276 314 359 20 79 101 142 170 207 249 286 332 Meta area of cross-section. sq. ins. sq. ins. sq. ins. sq ins. sq. ins. sq. ins. sq. ins. sq.ins. 37.70 43.98 50.266 56.55 62.34 69.11 75.40 81.68 Note. — If the breaking-load is desired, multiply the safe load by 6. WrougTit-Iron Posts and Columns. In trusses, roofs, etc., built of wrouglit-iron, the pieces in compression are usually made of wrouglit-iron, as well as the pieces which are in tension. 'Wrouglit-iron is also used, to some extent, for columns in buildings; the column being made up of three or more sections of wrouglit-iron bolted together. In using wrouglit-iron to resist compression, any shape may be used which offers a resistance to bending. Four angle-irons bolted together in the form of a cross make a good strut, or two channel-irons bolted together, back to back. In the latter case, the greatest strength proportionate to the weight of the bar is obtained when the channels are separated, so that the distance from the outside edge of one to the outside edge of the other shall be equal to the width of the channel. I-beams are also suitable to resist .compression, but have the objection that they cannot readily be riveted to other parts of the truss or frame.230 STRENGTH OF WROUGHT-IRON POSTS. The formulas for wrouglit-iron posts are similar to those for .cast-iron columns, though the constants are different. There is good reason to believe that we have no formulas for wrouglit-iron posts that we may consider as giving the absolute breaking-weight; but, for want of better formulas, the following are in general use, and appear to err on the safe side. Strength of Wrouglit-iron Posts. Solid rectangular posts. Metal area X 30000 Safe load in lbs. = / sq. of length in inches\ ^ V 3000 X sq. of least side/ (8) Hollow rectangular j)Osts. Safe load in lbs. = Metal area X 36000 sq. of length in inclies\ ^6000 X sq. of least side/ (9) Solid cylindrical pillars. Safe load in lbs. = Metal .area x 36000 (10) / sq. of length i.i inches \ \ 3000 X sq. of diarn. in inches/ Hollow cylindrical posts Safe load in lbs. — —r Metal area x 36000 / sq. of length in inches \ 41 1 + -JT—1------=■------1 o (11; Cross of equal arms. Safe load in lbs. = 4000 X sq. of diam. in inches/ Metal area X 36000 ' . (1-) + 11 + sq. of 1500 X sq, f length in inches \ :i. of breadth in inches/ Angle-iron of equal or unequal arms. Metal area X 36000 Safe load in lbs. L ..(131 / sq. of length in inches \ 1500 X sq. of least arm in inches/STRENGTH OF WHOUG HT-IRON POSTS. 231 For other sections than the above, their strength as a column may be found by first finding the radius of gyration (see Chap. XIII.) of the cross-section, and then substituting in the following formula: — In which r denotes the radius of gyration.. This is considered by many to be the most accurate formula for computing the safe load of irregularly shaped sections (such as angles, tees, channels, etc.) which we have; and Messrs. A. and P. Roberts & Co., of the Pencoyd Iron-Works, have published in their recent handbook, entitled “ Wrought-lron and Steel in Construction,” the following tables, based upon the above formula, and experiments made at the Pencoyd Iron-Works by Mr. Janies Christie, giving the ultimate and safe loads per square inch of cross-section for wrought-iron struts, for ratios of length to radius of gyration, varying from 20 to 4S0. These tables, by their permission, are given on the following pages. The radius of gyration of all of their sections of beams, channels, angles, and tees, are given in Chap. XIII., and also of all of the Union Iron Mills’ shapes. To use these tables, it is only necessary to divide the length of the strut by the least radius of gyration, if the strut is free to bend either way, and from the tables find safe or ultimate loads per square inch corresponding to this ratio. The area of the cross-section multiplied by the load taken from the table will give the • greatest safe or ultimate load on the strut, according to the table used. The numbers in the first column of these tables correspond to the length of the column in inches divided by the radius of gyration. Example.—What is the greatest safe load for an 8-incli 00-pound 1 beam (Union Iron Mills’ make), 10 feet long, fixed at both ends, with fiat bearings, and free to bend in either direction Mas. The least radius of gyration for this section is 0.83 inches: 120 * hence length of post divided by least radius of gyration is «"go, or 1-14. From Table VI I. we find that the safe strength fora ratio halfway between 140 and 150 would be about 5080 pounds, and multiplying this by the area of the cross-section of the beam, 0.0 square inches, we have for the safe load 33,528 pounds. Note.—In lining Table« VI. and VII. the length of column, and radius of gyration, inuat both be in tile same unit. Metal area x 36000 (14) • Safe load in lb°232 STRENGTH OF WROUGHT-1RON STRUTS. TABLE VI. WROUGHT IRON STRUTS. ULTIMATE PRESSURE IX LBS. PER SQUARE IXCII. LENGTn LEAST RADIUS OF GYRATION. Flat Ends. Fixed Ends. Hinged Ends. Round Ends. 20 46.000 46,000 46,000 44.000 30 43,000 4 3.000 43.000 40.250 40 40.000 40.000 40,000 36,500 50 38,000 38.000 38,000 33,500 60 36,000 36,000 3J.U00 30,5)0 70 34,000 34,000 33.750 27.750 80 32,000 32,000 31.500 25. (400 90 30,900 31.000 29.750 22.750 100 29,800 30.000 28,000 20.500 110 23,050 29,000 26,150 18.500 120 26,300 28.000 24.300 16.500 130 24.900 26,750 22,650 14 650 140 23,500 25,500 21.000 12.800 150 21,750 24,250 18.750 11.150 160 20,000 23.000 16,500 9,5 0. 170 flgUOft 21,500 14.650 8.500 180 16,800 20,000 12.800 7,500 x 190 15,650 18,750 11,800 6.750 200 14.500 17,500 10.800 6.000 210 13,600 16 250 9.800 5,500 220 12,700 15.000 8.800 5.000 230 11,950 14.000 8.150 4,650 240 11,200 13,000 7,500 4,300 250 10,500 12,000 7.000 4,050 260 9.800 11,000 6,500 3.800 270 9,150 10.500 6.100 3.500 280 8,500 10.000 5,700 3.200 290 7,850 9,500 5.350 3,000 300 7,200 9 000 5.000 2,800 310 6,600 8,500 4,750 2,650 320 6,000 8,000 4,500 2,500 &30 5,550 7,500 4,250 2.300 340 5,100 7,600 4.000 2.100 350 4,700 6,750 3,750 2.000 360 4,300 6.500 3.500 1.900 370 3.900 6.150 3.250 1.800 380 3,500 5.800 3.000 1.700 390 3,250 5.5*0 2.750 1,000 400 3.000 5.200 2.500 1.500 410 2,750 5.0'»0 2.400 1.400 420 2.500 4.800 2,300 1,300 430 2,350 4.550 2.2(H) 440 2,200 4,300 2.100 450 2,100 4.050 2.(XX) 460 2,000 3,8(0 1,900 470 1,950 1.850 4S0 1,900 1,800 STRENGTH OF WROUGHT-IKON STRUTS 233 TABLE VII. GREATEST SAFE LOADS ON STRUTS. Greatest safe load in lbs. per square inch of cross section for vertical stmts. Hoth ends are supposed to be seemed as indicated at the head of each column. If both ends are not secured alike, take a mean pr porlio al between the values given for the classes to which each end belongs. If the strut is hinged by any uncertain method so that the centres of p.ns and axis of strut may not coincide, or the pins may be relatively small and looely fitted, it is best in such cases to consider the strut as “ round ended.” Length Flat Ends. Fixed Ends. Hinged Ends. Round Ends. LEAST RADIUS OP GYRATION. 20 14,380 14,380 13,940 13,330 30 13,030 13/ 30 12,460 11,670 40 11,700 11.760 11.110 10,140 50 10,860 10,860 10,130 8,930 60 10,000 10,000 9,230 7,820 70 9,190 9,190 8,330 6,850 80 8,420 8,420 7,500 5^950 90 7,920 7,950 6,840 • 5*230 300 7,450 7,500 6,220 4^560 110 6,840 7.070 5.620 3,980 120 6,260 6,670 5.060 3*440 130 5,790 6,220 4,580 2,960 140 5,340 5,800 4,120 2*510 150 4,830 5,390 3.570 2,120 160 4,350 5,000 3,060 1,760 170 3,920 4.570 2,640 l’530 180 • 3,500 4,170 2,250 1*310 190 3,190 3.830 2,029 1*150 • 200 2,900 3,500 1,800 Looo 210 2,670 3,190 1,590 890 220 2,440 2,880 1,400 790 230 2,250 2,640 1,260 720 240 2,070 2,410 1,140 650 250 1,910 2,180 1,040 600 260 1,750 1,960 940 550 270 1,610 1,840 870 500 280 1,460 1,720 790 440 290 1,330 1,610 730 410 300 1,200 1,500 670 370 310 1,080 1,390 620 350 320 970 1,290 580 320 830 880 1,190 540 290 340 800 1.090 490 260 &50 720 1,040 450 240 360 650 980 420 230 370 580 920 380 210 380 510 850 340 200 390 470 800 310 80 400 430 740 280 70234 STRENGTH OF WROUGHT-IRON STRUTS. Strength of Trenton I-Beams, Channels, T-Bars, Angle-Irons, and Deck-Beams, used as Posts. The four largest companies which manufacture rolled-iron beams and bars in the United States, — viz., the Phoenix Iron Company of Philadelphia, the New-Jersey Steel and Iron Company of Trenton, N. J., Carnegie Brothers & Co. of Pittsburgh, Penn., and A. and P. Roberts & Co. of the Pencoyd Iron-Works, Philadelphia, Penn., — each publish a book giving complete information concerning all the sections of beams and bars which they manufacture, and also the strength of the beams, channels, and most of the angle and T-bars. When it is desired to use wrought-iron In construction, it is generally cheaper and more convenient to use some of the standard shapes to be found in the market; and hence tables giving the strength.of all these standard sections are of very great assistance to both the builder and the architect. As it would require too much space to give the strength of the different sections of iron rolled by each of the four different companies mentioned, the author has given the strength of the Trenton beams and bars as a sample; and beams and bars of other make, having the same outside dimensions and of the same weight, should have essentially the same strength. The Trenton forms are those manufactured by the New-Jersey Steel and Iron Company. The following tables, giving the strength of the different sections used as posts, are made up from tables furnished by.the manufacturers. To use these tables, it is only necessary to substitute in the fbr-mula at the head of the table the proper values of a and r for the particular section to be used, together with the square of the span, and perform the operation indicated. Examim.k.— What is the safe load for'a 9-inch 50-pound channel, used as a post, 10 feet long, and not stayed in either direction? A ns. We have for the safe load in tons the expression, 4 x a x )• L‘z + r We find from the table, that, for a 9-incli 50-pound channel, a = 5.08, and r — 124 (we must take the least value of r when the post is free to bend in either direction). Substituting these values in the above expression, we have, 4 X 5. OS X 124 100+124 Safe load in tons = = 11.2 tons.235 ^ STRENGTH OF WROUGHT-IRON STRUTS. If the post were stayed so that the channel must bend edgewise if it bends at all, we could use the larger value of r, which would give us 19.6 tons for the safe load. The table on p. 237 shows the ratio of length of posts unsupported edgewise to length unsupported sidewise, giving'the same strength. Thus a 15-inch 200-pound beam used as a post should be braced every five feet, so that it cannot bend sideways, if the whole strength of the iron is to be utilized. Table VIII., p. 238, gives the values of a and r for channels, used as posts, riveted together in the various ways there described. Strength of Trenton Rolled Beams and Channels used as Struts. Columns with faced or fixed ends. 4 X n x r Safe load in tons = _j_ r - ’ Columns with hinged ends. 4 X a X j Safe load in tons =------ L°- + 4 in which L2 = square of the length in feet. r. r. Beams. a. Channels. a. Side Edge- Bide- Edge-i wise. wise. wise. wise. 15 inch, 200 lbs. 20.02 343 8,830 15 inch, 190 lbs. 18.85 428 7,762 15 “ 100 t< 15.04 256 8,702 15 44 120 44 12.00 301 7,833 121 1 170 t< 1G.77 382 5,832 m 1 140 44 14.10 317 5,170 12\ “ 125 (( 12.33 235 5,777 iil I 85 “ 8.62 172 *5,275 m 130 If 13.3G 300 4,372 104 “ 60 44 6.00 160 3,685 luA “ 105 <( 10.44 230 4,445 9 44 70 44 7.02 100 2,925 !> “ 125 t< 12.33 227 3,015 9 44 50 44 5.08 124 2,892 | 0 “ 85 4 { 8.32 170 3,1ST 8 “ 45 44 4.48 142 . 2,480 9 “ 70 U 6.53 136 3,277 8 44 33 “ 3.30 100 2,493 1 8 41. 80 (( 8.03 240 2,002 7 44 36 44 3.60 136 1,883 S 44 65 tc G.37 180 2,632 7 “ 25J 1 2.54 82 1,700 7 44 55 It 5.50 177 2,011 6 44 45 “ 4.32 123 1,257 6 44 120 l( 11.84 393 1.3.75 6 44 33 “ 3.20 101 1,343 6 44 00 ft 8.70 310 1,431 6 44 22., “ 2.25 77 1,403 0 44 50 4.01 141 1,475 5 44 19 44 1.02 57 930 G 44 40 u 4.01 102 1,452 4 44 164 “ 1.05 49 597 5 44 40 II 3.00 109 086 3 4.4 15 “ 1.45 51 341 <( 9% 4 44 37 (( 3.06 121 634 Deck-Beams. 4 44 30 «( 2.91 96 644 8 inch, 65 lbs. 6.29 147 2,177 4 44 18 It 1.77 36 639 7 44 55 44 5.35 ios 1,640236 STRENGTH OF WROUGHT-IRON STRUTS. Table IX., p. 239, also shows the distance required between the backs of channels used as posts to give equal stiffness parallel with and at right angles to the web. Thus if we wish to use two 9-inch 70-pound channels, placed back to back, for posts, they should be placed 4.9 inches apart to have the same stiffness in either direction. In this way we get the most strength out of the iron used. Strength of Trenton Angle and T-Bars used as Struts. With faced or fixed ends. 4 X a X r Safe load in tons = —T.—> L- -r r in which L denotes the length of the strut in feet. With hinged ends. 4 X « X Safe load in tons = i2+4 Angle-Bars with Even Legs. Designation Weight per Designation Weight per of bar. foot in lbs. a. r. of bar. foot in lbs. a. r. 6" x 6" 19 to H 5.75 865 mm x B 31 to 41 0.94 92.0 x 4 Y' 12' to 20| 3.75 4 SO 111 x 13" 2 to 31 0.02 72.0 4" x 4" 9.1 to IS 2.86 381 U" x 1|" ia to 23 0.53 52.0 "1 B y *j1 ** Si to 141 2.4S 2$$ 11" x l|" 1 to 11 0.30 37.0 3" x 3" 4.8 to 121 1.44 216 1" X 1" J to 1 i 0.23 23.0 on'/ V 03'/ -•1 -J 5.4 to 91 1.62 177 if* i" 0.6 to 1 0.20 17.5 2A" x ÜB :s.j to 71 1 19 148 To to 0.8 0.17 12.5 2J" x il" 05 to 6 1.06 no Axgi.e-Bars with Uneven Legs. Designation of bar. Weight per foot in pounds. a. r. 6 inches X 4 inches 14.00 to 23 4.18 t 923 f 335 6 4 inch way 5 “ X Ol «( 10.20 to 191 3.05 i 638 1 261 5 3A (i 1« 41 » X 3 “ 9.00 to 14.j ,2.67 513 \ 186 41 3 << 4 “ X 3 •• 7.00 to 141 2.09 i 403 \ 197 4 3 It (( 44 3J “ X 1J inch * 4.01 1.19 i 316 i 36 31 l| 44 44 44 44 3 “ X 2.j inches 48 to 9.| 1.31 1 2-23 i 142 3 01 -5 44 44 44 44 3 “ X O K 4.00 to *-1 ‘5 1.19 ) 82 3 2 44 44STRENGTH OF WROUGHT-IRON STRUTS. 237 Tee-Bars. Designation of bar. Weight per foot in pounds. a. r. 4 inches X 4 inches 124 3.75 t 371 ) 175 Vertically Sidewise 3* n X 34 “ 9.6 and 10.8 2.87 284 } 133 Vertically Sidewise 3 a X 3 “ 7.0 and 9\ 2.11 { 209 ( 97 Vertically Sidewise 24 a X 91 «« 5.0 and 5| 1.46 i f 68 Vertically Sidewise 2 a X 2 “ 3J and 3J 0.94 92 I 43 Vertically Sidewise 5 a X 24 “ 11.7 3.50 \ SI f 363 Vertically Sidewise 3 a X 2 " 4.8 and 5.8 1.45 81 1 117 Vertically Sidewise 2 a X 14 “ 3.00 0.91 ) 1 Vertically Sidewise 2* a X 1| “ 2.40 0.74 1 61 Vertically Sidewise 2 a X 1 inch 2.15 0.65 | 1 Vertically Sidewise 14 a X 1 1 1.86 0.56 ) H Vertically Sidewise If it is desired to use Phoenix or Pencoyd beams or bars, or those of the Union Iron Mills, for reason either of cost or convenience, the strength of those beams or bars should be the same as Trenton beams or bars of the same weight and outside dimensions* The strength of iron struts of the same outside dimensions is proportional to their weight. The tables for Trenton beams and bars used as posts are based upon the metal resisting with safety 8000 pounds to the square inch for short pieces. Italic of Length of Posts unsupported Edgewise to Length unsupported Sidewise, giving Same Strength. Beams. Size. Ratio Size. Ratio 15 in. 200 lbs. 5.07 8 in. 65 lbs. 3.82 15 “ 150 (t 6.S3 7 << 55 “ 3.37 12*, “ 170 <1 3.91 6 “ 120 1.87 12} “ 125 << 4.96 6 m 90 “ 2.15 101 “ 135 << 3.82 6 “ 50 “ 3.23 104 “ 105 y 4.40 6 “ 40 “ 3.77 10£ 1 90 (f 4.48 5 I 40 “ 3.01 9 m 125 3.04 5 30 1 3.38 9 I? 85 3.90 4 “ 37 1 2.29 9 i 70 4.37 4 “ 30 “ 2.59 8 “ 80 3.29 4 “ 18 “ 4.21 Channels. Size. Ratio Size. Ratio 15 ill. 190 lbs. 4.26 6 in. 45 il>8. 3.20 15 " 120 “ 5.10 6 1 33 “ 3.65 12} m 140 “ 4.04 6 1 224 4.27 if “ 70 “ 5.56 5 1 19 4.04 104 ®* 60 “ 4.80 4 1 164 3.49 9 « 70 « 3.92 3 “ 15 2.59 9 “ 50 “ 4.83 8 « 45 «« 4.18 Strut Bars. 8 <• 33 ■ 4.78 7 “ 36 i 3.72 5 in. 22 bs. 3.00 7 “ 254“ 4.55 5 “ 16 3.22 © ’x Designation of Bar. ’S) O od [Note. - - a I area of cross- ». section in square inches, r = 250 times the square of the radius of gyration ■ X — >> bi. F3 z ? of the section.! CO 1. 125 inch heavy channel j 14.10 317.00 H “ light “ |; 7 00 180.00 9 1 heavy 1 1? 7.02 190.00 9 “ light « ) « i r 5.08 124.00 6 “ heavy « ) i ( r 4.32 123.00 6 » light 1 ii 3.20 101.00 5 “ 19 lbs. « ( I 1 r ' J .92 57.00 4 “ 164 IkB - l;: 1.64 49.20 3 “ 15 lbs. (( \ 1 / I 1.45 51.00 1 1 “ heavy strut J 2.15 47.70 5 “ light \ ft ( V 1.55 43.60 OQO s- Z Hsl 1 t mm % -0 2 ~ X 1 x «'S X! i 1 IMI 1 ï © h j m jî ü ^ c cû I J O O O ^ 2 ® i ë ^ A**l “ 30 • JZ « XS æ’c.c|| $ B £ 1 1 2 M >TS gœ 0 5) ^ », S- s •« X ^ © 0 *■" 1 ü cî • 1 bfi 1 -P 5 ? Ö ï .= £ 2 Ä * 0= _2 Ja-^ j; cô © .0 ’So II. III. IV. 28.20 28.20 26.49 630.00 5,170.00 669.00 14.00 14.00 13 17 322.00 5,470.00 342.00 14.04 14.04 12.96 371.00 2,925.00 401.00 10.16 - 10.16 9.33 230.00 2,892.00 250.00 8.64 8.64 7.84 248.00 1,257.00 272.00 6.40 6.40 5.84 197.00 1,343.00 215.00 3.84 3.84 3.62 111.00 930.00 118.00 3.29 3.29 3.07 102.00 597.00 110.00 2.89 2.89 2.67 116.00 341.00 125.00 4.30 4.30 3.67 132.00 433.00 153.00 3.11 3.11 2.61 105.00 457.00 124.00 Two bars riveted together, but separated from each other as below, allowance made for rivet-holes in stem of channels and in llanges of strut-bars; 5-inch rivets in 6-inch and smaller bars, {-inch in larger bats. Strength sidewise; GC bO CS s p ac ÎZ Im V. 26.49 5.357.00 13.17 5.675.00 12.96 3.061.00 9.33 3.036.00 7.84 1.326.00 5.84 1.420.00 3.62 987.00 3.07 640.00 2.67 351.00 3.67 356.00 2.61 376.00 i inch. VI. 26.49 746.00 13.17 395.00 12.96 461.00 9.33 297.00 7.S4 324.00 5.84 260.00 3.62 152.00 3.07 144.00 2.67 163.00 3.67 196.00 2.61 164.00 5 inch. VII. 26.49 831.00 13.17 456.00 12.96 529.00 9.33 352.00 7.S4 384.00 5.84 314.00 3.62 194.00 3.07 186.00 2.67 209.00 3.67 248.00 2.61 211.00 1 inch. VIII. 26.49 1,024.00 13.17 603.00 12.96 687.00 9.33 485.00 7.84 525.00 5.84 443.00 3.62 301.00 3.07 294.00 2.67 324.00 3.67 376.00 2.61 329.00 P «—< C » » o Hi $ P 0 *1 2 cr g P 0 ft S x 2, r* » O p c- 2 rt- I w s 03 > w t-* w < K> CO CO STRENGTH OF WROUGHT-IRON STRUTS.STRENGTH OF WROUGHT-IRON STRUTS 239 Example.—Wliatis the safe load, as a strut, of a pair of five-inch nineteen-pound channels, six feet long, riveted together one inch apart, and having fixed ends ? 4 x 3.62 X 304 Aiis. Safe load = ;—r,,. M,,- F 12.95 tons. oh 4- 304 TABLE IX. Showing Distance required from Back to Back of Channels forming Posts to give Equal Stiffness Parallel with and at Bight Angles to the Web. Flanges turned out- Flanges turned in* ward | |. Ills- ward \ J. Dia- tance apart ill the tance out o out cleiu in inches. in inches. Size op Channel. >4 ° = ? c -c x g £ £ M 'e n > t-i 5/) o u. 5f)g.O O •g * X IB O O • £ 0E 8m iH * **£ — X cr 5 Jn 7 s-s rj6 B w Of w 15 inch heavy, 190 lbs. per yard. 8.32 2.40 10 64 13.36 7.44 24.68 15 <( light, 120 “ “ 9.OS 3.20 20.38 12.88 7.06 24.18 121 << heavy, HO “ “ 6.5S 1.70 15.80 11.06 6.18 20.28 121 tc light, 70 “ “ it 7.70 2.86 17.14 10.72 5.88 mm 101 (1 00 “ “ tt 6.28 2.30 14.02 8.80 4.S2 16.54 0 ti heavy, 70 “ “ tt 4.90 1.24 11 .SO 8.30 4.64 15.26 9 it light, f>0 “ “ ü 5.40 1.84 12210 7.92 4.36 14.82 •8 ii light, 45 “ “ tt 4.00 1.20 11.00 7.64 4.30 14.04 s it ex. light, 33 “ »1 ü 5.02 1.70 11.40 7.34 4.02 13.72 7 «« light, 36 “ “ ti 3. S6 0.90 9.40 0.70 3.74 12.30 7 it ex. light, 25J “ “ ii 4.OS 1.32 9.34 6.12 3.36 11.38 6 it heavy, 45 “ I ti 2. SO 0.30 7.42 5.72 3.22 10.34 6 it light, 33 “ m ti 3.20 0.88 7.94 5 72 3.20 10.46 6 t< ex. light, oat ü ü ii 3.52 1.00 8.32 5.68 3.16 10.48 5 it ex. light, 19 “ 44 ii 2. S3 0.70 6.80 4.69 2 62 8.66 4 ex. light, 184 1 " ti 2.03 0.34 5.16 3.87 2.18 7.00 3 ex. light, 15 « “ it 1.16 0.00 0.08 3.20 1.78 5.62 * Fosls formed of channels meeting ilange tu llunge will have a width out to out exceeding that here required.240 PHŒN1X \V RO U G Hï-I HON COLUMNS. Phoenix Wrouglit-Iron Columns. inch of metal than cast-iron, and fen» less liable to contain any flaw in the metal. They can also he painted on the inside, before the column is put up, 0 to protect them from any dampness in the atmosphere, which might cause rust. These columns are made up of the rolled segments “ C,” which are riveted together, by rivets about six along their sides, as shown at “ A.’’ Between every two segments an iron bar is frequently inserted, through which the rivets pass. These bars, or “ flats ” as they are called, in-• ■<* crease the area of the cross-section, : ~ and contribute much to the strength of the pillar. The columns are fitted with cast-iron caps and bases. Table columns rolled by the Pliœnix Iron Company, as published in their book of sections. These columns vary from the small four-segment column (3? inches inside diameter, and capable of supporting a load of 7if tons with safety when twenty feet long) t G (14? inches inside diameter, with strength sufficient to bear safely 400 tons when of the same length). The streni/th of these columns may be computed by means of the formula for hollow cylindrical wrought-iron The rolled segment column of the Phoenix Iron Company is coming into quite general use for vertical struts in iron bridges, and for columns in buildings. For columns whose length is more than fifteen times the diameter, they possess greater strength per square inches apart, by means of flanges X., pp. 242, 243, gives the sizes of the to the massive eight-segment tube columns, or the table on p. 244, which is used the same as the similar table for cast-iron columns.KEYSTONE WltOUGHT-lRON COLUMNS. 241 Example. —What is the safe load for the four-segment column B®, inches internal diameter, ^ inch thickness of shell, and 20 feet long ? Ah*. From Table X., p. 242, we find the outside diameter of this column, with a thickness of half an inch, to be (5.9 inches, which would make the ratio of the length to the diameter of the column about 35 to 1. Then, from 'Fable XI., p. 244. we find the safe load per square inch for a cylindrical column of that length to be 7077 pounds. The metal area of cross-section of the column is 13.S square inches: hence the safe load for the column is 13.8 X 7077 = 103,002 pounds, or 51.S tons. If we compare this with a cast-iron column of the same dimensions, we find, from Table A'., for cast-iron columns, p. 227, that a seven-inch column three-fourths of an inch thick (which is as thin as columns should be cast), and twenty feet long, will only support 24.4 tons, or less than half what the wrought-iron column supports with less metal. Messrs. Carnegie Brothers & Co. also manufacture wrouglit-iron columns of two patterns, which answer the same purpose as the Phoenix wrouglit-iron columns. The first pattern is that known as the Keystone Octagon column, of which a section is shown in Fig. 5. They are rolled in segments as shown, and fastened together by rivets. Table XII. gives the diameters, areas, and weights of these columns as rolled. In computing their strength, they should be considered as square columns, the diameter given in the table being the outside diameter of the columns.242 PHCENIX WROUGHT-IRON COLUMNS. TABLE X. Sizes of Phoenix Columns. One Seument. One Column. Mark. I,east out-i-ide diameter In inches. i Thickness in inches. Weight in pound« per yard. Area in sq. inches. Weight in pounds per foot. A _3„ 1 t> M 3.8 12.6 4.0(H) 4 segmen t. l 4 12 144 4.8 5.8 10.0 19.3 4.125 | 4.250 3§' | inter, diam. 3 H 17 0.8 22.0 4.375 1 4 10 0.4 21.3 5.310 B1 fi T6 194 7.8 20.0 5.440 3 i 23 9.2 30.0 5.500 4 segment. 175 m 10.8 35.3 5.090 inter, diam. i tk 30 334 12.0 13.4 40.0 44.0 5.S10 5.940 h. 8 37 14.8 49.3 0.000 X 4 184 7.4 24.0 0.440 B2 TrJ 3 8 224 204 9.0 10.8 30.0 35.3 0.500 8.870 4 segment. 304 12.2 40.0 0.810 o\§" inter, diam. i A 344 384 13.8 15.4 40.0 51.3 0.940 7.000 fi 8 424 17.0 50.0 7.190 4 4 25 10.0 33.3 7.090 5 Tt> 30 12.0 40.0 7.810 35 14.0 40.0 7.940 T2^ 40 10.0 53.3 8.080 45 18.0 00.0 8.190 C 1 1 t> 48 19.2 04.0 8.310 4 segment. l> 11 TB 53 58 21.2 23.2 70.0 77.3 8.440 8.500 inter, diam. 3 03 25.2 84.0 8.090 +S 08 27.2 90.0 8.810 7 73 29.2 97.3 8.930 i 83 33.2 110.0 9.190 ii 93 37.2 124.0 9.440 l* 103 41.2 137.3 9.090PHCENIX WROUGHT-IRON COLUMNS, 243 TABLE X. — Concluded. Sizes of Phoenix Columns. One Seoment. One Column. Mark. Least Outside | Thickness in inches. Weight in pounds per yard. Area in sq. indies. Weight in pounds per foot. diameter i in inches. 1 4 28 14.0 46.6 0.625 I> 5 Tt> 32 16.0 53.3 0.750 2 36 18.0 66.0 0.870 5 segment. Vb 40 20.0 66.6 lo.ooo 9J" inter, diam. 1 'i 44 22.0 tr•> •) 10.125 . Tb 48 24.0 80.0 10.250 i 28 16.8 56.0 11.500 B| 32 10.2 64.0 11.625 i 36 21.6 72.0 11.750 40 24.0 80.0 11.875 E 1 « 44 26.4 88.0 12.000 0 segment. TB 48 53 28.8 31.8 06.0 10(5.0 12.125 12.250 11" inter, diam. i l lii 58 34.8 116.0 12.875 3 4 63 37.8 126.0 12.500 n 68 40.8 136.0 12.625 7 8 73 43.8 146.0 12.750 1 83 40.8 166.0 13.000 H 30 24.0 80.0 15.000 3 8 35 28.0 98.3 15.125 ■fs 40 82.0 106.6 15.250 i 45 36.0 120.0 15.375 1 (T 50 40.0 138.3 15.500 G f» 8 55 44.0 146.6 15.625 8 segment. ft 3 4 60 65 48.0 52.0 160.0 178.3 15.750 15.875 14J" inter, diam. 1 3 11) 70 56.0 186.6 16.000 7 8 75 60.0 200.0 16.125 1 85 68.0 226.6 16.375 H 05 76.0 253.3 16.625 H 105 84.0 280.0 16.875 12 115 02.0 306.6 17.125 Note. — The weight of rivet-heads adds from two to live per cent to the weight of finished columns.244 STRENGTH OF WROUGHT-IRON COLUMNS. TABLE XI. Strength of Hollow Cylindrical or Rectangular Wrought-Iron Pillars. (Calculated by Formulas 9 and II.) length divided by external Breaking weight in pound* per nquare inch. Safe load in pouiid* per square inch. breadth or diameter. Cylindrical. Rectangular. Cylindrical. Rectangular. ■ 8 35,495 35,620 8,874 8,905 9 35,369 35,520 8,842 8,SS0 10 35,217 35,410 8,804 8,852 11 35,057 35,288 8,764 8,822 12 34,883 35,156 8,721 8,789 13 34,697 35,013 8,674 8,753 14 34,497 34,861 8,624 8,715 15 34,286 34,698 8,571 8,674 16 34,062 34,527 8,515 8,632 17 33,827 34,346 8,457 8,586 IS 35,582 34,155 8,395 8,539 19 S3,.327 33,957 8,332 8,4S9 20 33,061 33,750 8,265 8,437 21 32,787 33,535 8,197 8,384 22 32,504 33,313 8,126 8,328 23 32,213 33,083 8,053 8,271 24 31,915 32,846 7,979 8,211 25 31,610 32,604 7,902 8,151 26 31,298 32,354 7,824 8,088 27 30.9S1 32,100 7,770 8,025 28 30,659 31,840 7,665 7,960 29 30,331 31,574 7,583 7,893 30 30,000 31,304 7,500 7,826 31 29,665 31,030 7,416 32 29,326 30,758 7,331 7,689 33 28,1*85 30,469 7,246 7,677 34 28,042 30,184 7,160 7,546 35 28,297 29,896 7,077 7,476 36 27,950 29,605 6,9S7 7,401 37 27,603 29,312 6,901 7,328 38 27,254 29,017 6,813 7,254 39 26,906 28,719 6,726 7,179 40 26,557 28,421 6,639 7,105 41 26,209 28,121 6,552 7,030 42 25,KtV> ‘27,8*21 6,465 6,955 43 25,515 27,511 6,378 6,S77 44 25,171 27,218 6,293 6,804 45 24,827 26,916 6,206 6,729 46 24,486 26,614 6,121 6,653 24,146 26,312 6,036 6,578 48 23,809 26,011 5,952 6,533 49 23,475 25,711 5,869 6,428 50 23,143 25,412 5,786 6,353 51 22,814 25,113 5,703 6,278 52 22,488 24.S16 5,02*2 6,204 53 22,164 24,520 5,541 6,130 54 21,845 24,226 5,461 6,056 55 21,5*28 23,934 5,3S2 5,9S3 56 21.215 23,642 5,304 5,910 57 20,906 23,354 5,226 5,838 58 20,600 23,067 5,150 5,767 59 20,293 22.7S2 5,074 5,695 60 20,(XX) 22,500 5,000 5,625Thickness, KEYSTONE OCTAGON COLUMNS. THICKNESSES AND CORRESPONDING AREAS AND WEIGHTS PER FOOT. 10-Inch Column. 8 Inch Column. 6-Inch Column. 4-Inch Column. 00 CD o> 4 Segments. Weight of one 4 Segments. Weight of one 4 Segments. Weight of one 4 Segments. Weight of one X W Area. Weight. segment. Area. Weight. segment. Area. Weight. segment. Area. Weight. segment. t-t sq. in. lbs. lbs. sq. in. lbs. lbs. sq. in. lbs. lbs. sq. in. lbs. lty. Inch. _ — — - 5. GO 18.7 4.7 3.91 13.0 3.3 ft - - - 9.78 32.6 8.2 7.13 23.8 5.9 4.98 16.6 4.2 '4 14.2*2 47.4 11.9 11.80 39.3 9.8 8.C6 28.9 7.2 6.05 20.2 5.0 5 . ‘1 ft 16.58 55.3 13.8 13.81 46.0 11.5 10.20 34.0 8.5 7.12 23.7 5.9 8 18.04 63.1 15.8 15.83 52.8 13.2 11.73 39.1 9.8 8.20 27.3 6.8 H 21.30 71.0 17.8 17.85 59.5 14.9 13.26 44.2 11.1 9.27 30.9 7.7 i 23.60 78.0 19.7 19.86 66.2 16.6 14.79 49.3 12.3 - - - ;» Til 26.01 86.7 21.7 21.88 72.9 18.2 16.32 54.4 13.6 - - — 8 28.37 04.6 23.6 23.89 79.6 19.9 — _ _ - - - 30.73 102.4 25.6 25.91 86.4 21.6 - - - - - — 33.00 110.3 27.6 _ _ _ _ _ _ — - - - 35.45 118.2 29.5 — " KEYSTONE WROUGHT-IRON COLUMNS. 245ThickndiS PIPER’S PATENT RIVETLESS COLUMNS. THICKNESSES AND CORRESPONDING AREAS AND WEIGHTS PER 'FOOT. 10-Inch Column 8 Inch Column 6-Inch Column 4-Inch Column. Thickness. 4 segments, incl. batten«. Weight of one HCg- ment. Weight of one batten. 4 segments, incl. batten«. Weight of one sog-rncnt. Weight of one batten. 4 segments, incl. battens. Weight of one segment. Weight of one batten. 4 segments, incl. battens. Weight of one segment. Weight of one batten. Ami. i Weight A rea. Weight Area. Weight Area. Weight 1 sq.in. lbs. lbs. 11)6. sq.in. lb«. lbs. lbs. sq.in. lbs. lbs. lbs. sq.in. lbs. lbs. lbs. Inch. _ _ - - - - — - — 5.21 17.4 2.5 1 _ — 10.98 30.0 0.1 7.30 24.3 4.2 ' 0.00 20.0 3.1 1 T ► 1.87 10.00 53.3 0.3 12.50 41.7 7.3 8.43 28.1 5.2 0.80 22.7 3.8 11.87 Ttf 17.00 50.7 10;0 14.08 40.8 8.0 0.55 31.8 0.1 7.00 25.3 4.5 ‘i 6 Ì9M 00.0 12.5 15.55 51.8 9.0 ’3.1 10.08 35.6 7.0 to CO 8.39 28.0 5.1 . 7, 21.70 72.8 14.1 17.08 50.0 11.1 11.81 39.4 8.0 . - - - - [■4.0 23.00 78.7 15.7 18.00 02.0 12.4 - - - - - - - - - 25.50 85.0 17.3 20.18 07.1 18.7 — — — — — — — — - 27.40 01.3 18.8 — - - - - - - - - - - - - 20.50 07.7 20.4 > — 246 RIVETLESS WR0UGI1T-IR0N COLUMNS.RIVETLESS WROUGHT-IRON COLUMNS. 247 Piper’s Patent Rivetless Column. Fiper’s patent kivetless column, also manufactured by Messrs. Carnegie Brothers & Co., is a wrouglit-iron column rolled in segments, but fastened together by means of grooved battens fitting over the flanges of the segments. The object of this is to get a column which shall be more pleasing in appearance than the riveted columns. Both of these forms of columns may, of course, be fitted with cast-iron caps and base^J Fig. 6 and the table following show the form of the column, and the diameters, areas, thickness, and weights, rolled. These columns should be considered as hollow cylindrical wrouglit-iron columns, the diameter given in the table being for the outside of the column. Columns exposed to the weather should be kept thoroughly covered with a good thick coat of paint, that mixed with red lead being preferable. As it is impossible to repaint the inner surface of closed columns, or, at best, this is attended with much difficulty and expense, such columns should preferably be used only in the interior of buildings, where the changes in temperature are not considerable, and the air is comparatively dry. In places exposed to the extremes of temperature, and unprotected from the rain, the paint on the inner surface of the columns will sooner or later cease to be a protection to the iron from the moisture of the atmosphere; corrosion will set in, and, once begun, will continue as long as there is unoxidized metal left in the column. Figs. 7, 8, and 9 represent types of columns with open sections, which admit of repainting at any time, and are therefore suitable for outdoor work. Wrouglit-iron columns fail, either by deflecting bodily out of the straight line, or by the buckling of the metal between rivets or other points of support. Both actions may take place at the same time; but, if the latter occurs by itself, it is an indication that the rivet-spacing or £ Fig. 8 the thickness of metal is insufficient, provided, however, that the248 FIRE-PROOFING WROUGHT-IRON COLUMNS. length of column is greater than twelve diameters, as columns of shorter length fail generally by the buckling of the metal. The rule has been deduced from actual experiments, that the distance between centres of rivets in columns should not exceed, in the line of stress, sixteen times the thickness of metal of the parts joined, and that the distance between rivets or other points of support at right angles to the line of stress should not exceed thirty times the thickness of metal. Method of Fire-Proofing Wrouglit-Iron Columns. The Phoenix columns are made thoroughly secure from expansion, bending, or buckling, which- may be caused by fire in the contents or combustible materials of buildings, by being incased in a non-conducting and incombustible covering: they may also, by the same means, be given any desired form, and prepared for an exterior finish of Portland, Keene’s, or other cements. The Keene cement finish may be in any desired color, and will take a polish equal to marble. Fireproof blocks are made of lime of Teil concrete, or of porous terracotta. They are secured by countersunk iron plates hooked to the rivet-heads of the columns. 2 (Fig. 10) is a perspective view of such a column, showing it in the various stages of completion. 1 is a section on the line ab. 3 is a view of one of the plates enlarged, and 4 shows one of the blocks. This work is done by Fig. 10.COMPARATIVE RESISTANCE OF IRON AND STEEL. 249 the Fire-Proof Building Company of New York in all the seaboard States, and by P. B. Wight, 73 Dearborn Street, Chicago, in all the . Western States. Comparative Resistance of Iron, Mild Steel, ami Hard Steel. The comparative resistance of struts or columns of iron, mil 1 steel, and hard steel, is clearly shown by the following diagram, taken, by permission, from the work on “ Wrought-lron and Steel in Construction,” referred to on p. 231. The lines in the diagram indicate the breaking-loads for the columns or struts.250 BENDING-MOMENTS. CHAPTER XI f. BENDING-MOMENTS. Tiie bending-moment of a beam or truss represents tlie destructive energy of the load on the beam or truss at any point for which the bending-moment is computed. The moment of a force around any given axis is the product of the force into the perpendicular distance between the line of action of the force and the axis, or the product of the force into its arm. In a beam the forces or loads are all vertical and the arms horizontal. The bending-moment at any cross-section of a beam is the algebraic sum of the moments of the forces tending to turn the beam around the horizontal axis passing through the centre of gravity of the section. Example. — Suppose we have a beam with one end securely fixed into a wall, and the other end projecting from it, as in Fig. 1. Let us now suppose we have a weight, which, if placed at the end of the beam, will cause it to break at the point of support. Then, if we were to place the weight on the beam at a point near the wall, the beam would support the weight easily; but, as we move the weight towards the outer end of the beam, the beam bends more and more; and, when the w'eiglit is at the end, the beam breaks, as shown by the dotted lines, Fig. 1. Now, it is evident that the destructive energy of the weight is greater, the farther the weight is removed from the wall-end of the beam, though the weight itself remains the same all the time. The reason for this is, that the moment of the weight tends to turn the beam about the point A, and thus produces a pull on the upper fibres of the beam, and compresses the lower fibres. As the weight is moved out oil the beam, its moment becomes greater, and hence also the pull and compression on •be fibres; and, when theBENDING-MOMENTS. 251 moment of the weight produces a greater tension or compression on the fibres than they are capable of resisting, they fail, and the beam breaks. Before the fibres break, however, they commence to stretch, and this allows the beam to bend: hence the name “bending-moment ” has been given to the moment which causes a beam to bend, and perhaps ultimately to break. There may, of course, be several loads on a beam, and each one having a different moment, tending to bend the beam; and it may also occur that some of the weights may tend to turn the beam in different directions: the algebraic sum of their moments (calling those tending to turn the beam to the right +, and the others —) would be the bending-moment of the beam. Knowing the bending-moment of a lx?am, we have only to find the section of the beam that is capable of resisting it, as is shown in the general theory of beams, Chap. XIV. To determine the bending-moments of beams mathematically, requires considerable training in mechanics and mathematics; but, as most beams may be placed under some one of the following cases, we shall give the bending-moment for these cases, and then show how the bending-moment for any other methods of loading may be easily obtained by a scale diagram. Examples of Bemling-Moments. Case I. Beam fixed at one end, and loaded with concentrated load 111 ; Bending-moment = IT X L. (L IT may, or may not, be the whole lei ngth of the beam, according to where the U - weight is located.) m Case II. -L----->1 Fig. 2 w Beam, fixed at one end, loaded with a distributed load IT. Bending moment = IT X a*252 BENDING-MOMENTS. Case III. Beam fixed at one end, loaded with both a concentrated and a distributed load. Case IV. Beam supported at both ends, loaded with concentrated load at centre. W Bend i ng-moment Beam supported at both ends, loaded with a distributed load \V. Case VI. Bending-moment,. 1C x L s' Beam supported at both ends, loaded with concentrated load not al. centre. Bending-moment, = W x ?iLX-g. LBENDING-MOMENTS. 253 Case VII. Benin supported at both ends, loaded with two equal concentrated loads, equally distant from the centre. From these examples it will be seen that all the quantities which enter into the bending-moment are the weight, the span, and the distance of point of application of concentrated load from each end. The bendinq-moment for any case other than the above may easily be obtained by the graphic method, which will now be explained. Graphic Method of Determining' Bending-Moments. The bending-moment of a beam supported at both ends, and loaded with one concentrated load, may be shown graphically, as follows : — Let IF be the weight applied, as shown. Then, by rule under Case VI., the bending- |<-n - moment directly under M* = m X n x Ur* Draw the beam, with the given span, accurately to scale, and then measure down Jhe line AB equal to tlie bending - moment. Connect B with each end of the beam. If, then, we wished to find the bending-moment at any other point of the beam, as at o, draw the vertical line y to BC; and its length, measured to the same scale as AB, will give the bending-moment at o. Beam loitli two concentrated loads. To draw the bending-moment for a beam with two concentrated loads, first draw the dotted lines A Bl) and A CD, giving the outline254 BENDING-MOMENTS. of the bending-moment for each load separately; EB being equal m X n r X s to W X —£—> and FC equal to P X — Now, the bending-moment at the point E equals EB, due to the load W, and Eb, due to the load P: hence the bending-moment at E should be drawn equal to EB + Eb = EB, ; and at F the bending, moment should equal FC + Fc = FC\. The outline for the bending-moment due to both loads, then, would be the line ABiCiD, and the greatest bending-moment would in this particular case be FCX. Beam with three concentrated loads. Proceed as in the last case, and draw the bending-moment for each load separately. Then make AD ~ A1 + M2 + .43, BE — j>l j>2 + B'-i, and CF — Cl + C2 + C3. The line HDEFI will then be the outline for the bending-moment due to all the weights. The bending-moment for a beam loaded with any number of concentrated weights mav be drawn in the same way. Fig-IIBENDING-MOMENTS. 25 5 Beam with uniformly distributed load. B Draw the beam with the given span, accurately to a scale, as before, and at the middle of the beam draw the vertical line A II L equal to II X —» It representing the whole distributed load. Then connect the points C, 11. D by a parabola, and it will give the outline of the bending-moments. If, now, we wanted the bending-moment at the point a, we have only to draw the vertical line ah, and measure it to the same scale as - 111. and it will be the moment desired. Methods for drawing the parabola may be found in “ Geometrical Problems,” Part I. Beam loaded with both distributed and concentrated loads. To determine the bending-moment in this case, we have only to combine the methods for concentrated loads and for the distributed load, as shown in the accompanying figure. The bending-moment at any point on the beam will then be limited by the line ABC on top, and CDEFA on the "P bottom ; and the ^ greatest bending-moment will be' the longest vertical line that can be drawn between J Fig. 13 these two bounding lines. For example, the bending-moment at X would be BE. The position of the greatest bending-moment will depend upon the position of the concentrated loads, and it may and may not occur at the centre.256 BENDING-MOMENTS. Example. —What is the greatest bending-moment in a beam of 20 feet span, loaded with a distributed load of 800 pounds and a concentrated load of 500 pounds 0 feet from one end, and a concentrated load of 000 pounds 7 feet from the other end ? L Mns. 1st, The moment due to the distributed load is 11 X —> o 800 X 20 500 X 6 X 14 the concentrated load of 500 pounds is---^------, or 2100 pounds. Hence we draw E'2 = 2100 pounds, to the same scale as J31, and then draw the lines AE and CE. 3d. The bending-moment for the concentrated load of 600 pounds 000 X 7 X 13 is ----—~----, or 2730 pounds; and we draw D'-i = 2730 pounds, and connect D with A and C. 4th, Make EII = 2 — 4, and DG =3 — 5, and connect G and II with C and A and with eacli other. The greatest bending-moment will be represented by the longest vertical line which can be drawn between the parabola ABC and the broken line AIIGC. In this example we find the longest vertical line which can be drawn is xy ; and by scaling it we find the greatest bending-moment to be 5550 pounds, applied 10 feet 11 inches from the point A. In this case, the position of the line JFy was determined by drawing the line TT\ parallel to IIG, and tangent to ABC. The line Xy is drawn through the point of tangency.MOMENTS OF INERTIA AND RESISTANCE. 257 CHAPTER XIII. MOMENTS OF INERTIA AND RESISTANCE, AND RADIUS OF GYRATION Moment of Inertia. Tiie strength of sections to resist strains, either as girders or as posts, depends not only on the area, but also on the form of the cross-section. The property of the section which represents the effect of the form upon the strength of a beam or postTs its moment of inertia, usually denoted by I. The moment of inertia for any cross-section is the sum of the products obtained by multiplying the area of each particle in the cross-section by the square of its distance from the neutral axis. Note. — The neutral axU of a beam is the line on which there is neither tension nor compression; and, for wooden or wrought-iron beams or xwsts, it may, for all practical purposes, be considered as passing through the centre of gravity of the cross-section. For most forms of cross-section the moment of inertia is best found by the aid of the calculus; though it may be obtained by dividing the figure into squares or triangles, and multiplying their areas by the squares of the distance of their centres of gravity from the neutral axis. Moment of Resistance. The resistance of a beam to bending and cross-breaking at any given cross-section is the moment of the two equal and opposite forces, consisting of the thrust along the longitudinally compressed layers, and the tension along the longitudinally stretched layers. This moment, called “the moment of resistance,” is, for any given cross-section of a beam, equal to moment of inertia extreme distance from axis In the general formula for strength of columns, given on p. 231, the effect of the form of the column is expressed by the square of the radius of gyration, which is the moment of inertia of the section divided by its area; or ~r = r2. The moments of inertia of the principal elementary sections, and a few common258 MOMENTS OF INERTIA AND RESISTANCE. forms, are given below, which will enable the moment about any given neutral axis for any other section to be readily calculated by merely adding together the moments about the given axis of the elementary sections of which it is composed. In the case of hollow or re-entering sections, the moment of the hollow portion is to be subtracted from that of the enclosing area. Moments of Inertia and Resistance, and Radii of Gyration. I = Moment of inertia. R ~ Moment of resistance. G — Radius of gyration. A — Area of the section. Position of neutral axis represented by broken line. I—i) H t ! I = R = 12' b(P ö I é—J 3 i ■ i fÖ i 1 I 1 1 A •v iq i i 1 1 ■ •** y V 20 1 d _rL’ h,n - bä? 1 =- 12 " ' 21 i G2 ~ bd — b,d/ I-Beam (another formula). Let a denote area of one flange, a' denote area of web, d' — effective depth between centres of gravity of flanges; / a'\4r4. È = 0.78o4r3. 7 = 0.7Sr,4(^-r/). 7? = Ö.T854260 MOMENTS OF INERTIA. For the sections of rolled iron beams and bars to be found in the market, the moments of inertia are given in the “Book of Sections” published by the manufacturers. The following tables give the moments of inertia for the beams, channels, angle and T bars, manufactured by A. & P. Roberts & Co., the New-Jersey Steel and Iron Company, the Phoenix Iron Company, and Messrs. Carnegie Brothers & Co. The weights for the beams and channels are in all cases in 2)oiinds per yurd for the angle and T bars, in some qasea in pounds per foot. MOMENTS OF INERTIA, AND RADII OF GYRATION, OF PENCOYD BEAMS. I. II. III. 1 IV. V. Weight Area of Moments of inertia. Radii of gyration. Size in per yard. section. iilcllC6* lbs. sq.ins. Axis A B. | Axis C D. Axis A B. Axis C D. 15 200.0 19.90 682.0S j 28.50 5.86 1.20 15 145.0 14.55 521.19 16.91 5.98 1.08 12 108.0 16.89 371-98 28.19 4.69 1.17 12 120.0 11.95 272.86 12.22 4 78 1.01 10i 119.0 11.89 206.55 i 14.24 4.17 1.09 10i 97.0 9.70 168.28 9.29 4.16 0.98 10 112.0 11.17 178.58 10.64 8.94 0.98 10 90.0 9.04 148.81 8.09 4.05 0.95 9 90.0 9.07 118.81 8.44 8.62 0.96 9 70.0 6.98 94.44 5.59 O.G8 0.89 8 81.0 8.14 88.98 7.28 8.21 0.94 8 65.0 6.58 69.17 5.02 3.25 0.88 7 65.0 0.58 49.78 4.15 2.75 0.79 7 52.0 5.14 48.08 ! 8.48 2.89 0.82 6 50.0 5.04 26.92 ! 2.15 2.81 0.05 6 40.0 4.08 24.10 I 1.80 2.43 0.66 5 84.0 8.88 18.40 | 1.21 1.99 0.60 5 80.0 2.94 12.50 1.09 2.06 0.60 4 28.0 2.90 7.69 1.17 1.63 0.68 4 18.5 1.90 5.14 : 0.49 1.65 0.51 [| 28.0 2.25 .“>.29 ! 0.77 1.21 0.59 *> 4 17.0 1.71 2.66 | 0.48 1.25 0.53 jRADII OF GYRATION. 261 We have given in each case all the information furnished by the respective companies; and although it would be very desirable to have the same information for all the sections that is given for the Pencoyd sections, the author has thought it best not to complete the tables by his own work. Knowing the strengtli of a Pencoyd section, it will be very easy to judge of the strength of another similar section of the same extreme dimensions and area, or weight. For the Pencoyd and Union Iron Mills sections, the radii of gyration are also given. MOMENTS OF INERTIA, AND RADII OF GYRATION, OF PENCOYD DECK-BEAMS. I. II. HI. IV. V. VI. Weight per Area of cross- Moments of inertia. Radii of gyration. Distance d from base to neutral axis. inches. yard. lbs. section, sq. ins. Axis A B. Axis C D. Axis A B. L_, M I ?5T 1 12 104 10.38 221.98 9.33 4.02 0.95 5.24 11 91 9.00 104.09 7.04 4.25 0.92 4.08 10 80 8.02 118.22 6.19 3.S4 0.87 4.27 0 72 7.17 84.77 4.92 3.44 0.83 4.00 8 61 0.11 57.00 3.03 3.07 0.77 3.50 7 52 5.21 34.40 2.59 2.57 0.71 3.20 o 42 4.18 21.95 1.04 2.20 0.63 2.65 5 34 3.37 12 04 0.98 1.89 0.54 2.22262 MOMENTS OF INERTIA. MOMENTS OF INERTIA, AND RADII OF GYRATION, OF PENCOYD CHANNELS. i. II. III. IV. V. VI. Size in inches. Weight per yard. Ibs. Area of cross- Moments of inertia. Radii of gyration. Distance d from section, sq. ins. Axis A 15. Axis C 1). Axis A 15. Axis 0 I). base to neutral axis. 15 148.00 14.86 451.51 19.05 5.51 1.13 0.95 12 88.50 8.83 182.71 7.42 4.55 0.92 0.71 12 60.00 5.94 123.71 3.22 4.56 0.74 0.62 10 60.00 5.99 92.08 4.29 3.92 084 0.75 10 49.00 4.89 73.91 2.33 3.89 0.69 0.64 9. 54.00 5.40 04.34 2.47 3.45 0.6S 0.67 9 37.00 3.72 43.65 1.31 3.43 0.59 0.55 8 43.00 4.25 40.00 2.17 3.06 0.71 0.60 8 30.00 2.96 28.23 1.06 3.09 0.60 0.50 fr 7 41.00 4.10 29.51 1.71 2.68 0.65 0.65 7 26.00 2.04 IS. 46 0.90 2.64 0.5S 0.48 0 33.00 3.29 18.37 1.46 2.36 0.67 0.66 6 23.00 2.27 11.07 0.59 2.27 0.51 0.46 27.30 2.73 10.29 0.80 • 1.93 0.56 0.61 5 19.00 1.88 6.67 0.37 1.88 0.45 0.42 0 4 21.50 2.15 5.16 0.54 1.55 0.50 0.53 4 17.50 1.75 4.14 0.41 1.54 0.48 0.45 3 15.00 1.52 2.03 0.32 1.16 0.46 0.51 2i 11.30 1.13 0.80 0.21 0.85 0.43 0.46 2 8.75 0.S8 0.48 0.08 0.74 0.31 0.37RADII OF GYRATION. MOMENTS OF INERTIA, AND RADII OF GYRATION, OF PENCOYD ANGLE-BARS. II. III. IV. V. VI. Size, in inches. Weight per yard. Moments of inertia. Radii of gyration. Distance d from base to neutral axis. Axis A B. Axis C D. Axis A B. Axis C D. 6 X 6 X 50.6 17.68 7.16 1.87 1.19 1.66 6 X 6 X 1 110.0 35.46 15.00 1.80 1.17 1.86 5 X 5 X A 41.8 10.02 4.16 1.55 1.00 1.41 5 X 5 X 1 90.0 19.64 8.67 1.48 0.98 1.61 4 X 4 X 4 8 28.6 4.36 1.86 1.24 0.81 1.14 4 X 4 X 5 54.4 7.67 3.45 1.19 0.80 1.27 o 1 X Q 1 OJ X i 24.8 2.87 1.20 1.07 0.70 1.01 Q1 o? X ql X 4 « 39.8 4.33 1.85 1.04 0.69 1.10 o o X 3 X 1 4 14.4 1.24 0.51 0.93 0.60 0.84 o o X 3 X 33.6 2.62 1.15 0.88 0.59 0.98 2Î X 2Î X A. 4 13.1 0.95 0.39 0.85 0.55 0.78 2Î X 2? X 1 V 25.0 1.67 0.72 0.82 0.54 0.87 2i X 2* X 1 4 11.9 0.70 0.29 0.77 0.50 0.72 h X a X J 2 22.5 1.23 0.54 0.74 0.49 0.81 2* X n X 1 4 10.6 0.50 0.21 0.69 0.45 ; 0.65 Çl X 2i X A 17.8 0.79 0.34 0.67 0.44 0.72 2 X 2 X ■ 7.1 0.27 0.11 0.62 0.40 0.57 2 X 2 X 1 8 13.6 0.50 0.21 0.61 0.39 0.64 n X It X n 1 <) 6.2 0.18 0.08 0.53 0.36 0.51 H X If X 2 8 11.7 0.31 0.14 0.51 0.35 ; 0.57 n X u X A 5.3 0.11 0.05 0.46 0.31 0.44 X It X 4 8 9.8 0.19 0.09 0.44 0.31 0.51 u X n X 1 8 3.0 0.05 0.02 0.41 0.26 0.30 li X if X 1 4 5.6 0.08 0.04 0.38 0.26 0.40 1 X i X 1 2.3 0.02 0.01 0.29 0.20 0.30 1 X i X 1 i 4.4 0.04 0.02 0.29 0.20 0.35264 MOMENTS OF INERTIA. MOMENTS OF INERTIA, AND RADII OF GYRATION, OF PENCO YD ANGLE-BARS. It. lit. IV. V. VI. VI. Size, in inches. m CL . 43 b Moments of inertia. Uadii of gyration. Distance front base to neutral axes. {t Axis A B. Axis 0 1). Axis E F. Axis A B. Axis C D. Axis E F. d. 1. 6 X 4 X 1 41.8 15.46 5.60 3.55 1.92 1.16 0.92 1.96 0.96 6 X 4 X 1 90.0 30.75 10.75 7.46 1.85 1.09 0.91 2.17 1.17 5 X 4 X i 32.3 8.14 4.60 2.47 1.59 1.20 0.87 1.53 1.03 5 X 4 X 1 80.0 18.17 10.17 6.10 1.51 1.13 0.86 1.75 1.25 5 X 34 X 3 x 30.5 7.78 3.23 1.95 1.60 1.03 0.80 1.61 0.86 5 X :n X .3 4 58.1 13.92 5.55 3.72 1.55 0.98 0.79 1.75 1.00 5 X 3 X 3 K 28.0 7.37 2.04 1.42 1.61 0.85 0.70 1.70 0.70 5 X 3 X i 54.4 13.15 3.51 2.58 1.55 0.80 0.69 1.84 0.84 44 X 3 X 2 X 20.7 5.50 1.98 1.27 1.44 0.86 0.69 1.49 0.74 44 X O ») X X 43.0 8.44 2.98 2.04 1.40 0.83 0.68 1.58 0.83 •4 X O 1. X 2 X 20.7 4,17 2.99 1.44 •1.25 1.06 0.74 1.20 0.95 4 X 1 i HR X X 43.0 6.37 4.52 2.34 1.22 1.03 0.73 1.29 1.04 4 X o *) X 3 X 24. S 3.90 1.92 1.10 1.26 0.88 0.67 1.28 0.78 4 X • ) X 1 39.8 0.03 2.87 1.09 1.23 0.85 0.65 1.37 0.87 •> 1 X X ) 1 A 1 21.2 2.53 1.72 0.80 1.09 0.90 0.64 1.07 0.82 1 i •> ; X ♦1 X n 30.7 4.11 2.81 1.49 1.06 0.87 0.64 1.17 0 93 18 X •Jt X H 10.2 1.42 0.90 0.47 0.94 0.74 0.54 0.03 0.68 3 X 2 J X j 2 25.0 2.08 1.30 0.72 0.91 0.72 0.54 1.00 0.75 •) m X 2 X I 11.9 1.09 0.39 0.25 0.96 0.58 0.46 0.99 0.49 3 X 2 X i 22.5 1.92 0.07 0.47 0.92 0.55 0.46 1.08 0.58 O l ») i X 24 X ■ 17.8 2.19 0.94 0.50 1.11 0.73 0.56 1.14 0.64 •) Î •) > X 21 X 4 27.5 3.24 1.30 0.87 1.08 0.70 0.56 1.20 0.70 6 X 34 X I7*» 39.(5 14.70 3.81 2.68 1.93 0.98 0.82 2.06 0.81 Lo X 11 X l 85.0 29.24 7.21 5.75 1.86 0.92 0.81 2.26 1.01 X 4 X lTt> 44.0 19.29 5.72 3.87 2.09 1.14 0.94 2.18 0.93 (it X 4 X l 95.0 38.00 11.00 8.35 2.02 1.08 0.93 2.38 1.13 5Ì X 3| X 3 gd| 10.12 3.27 2.14 1.77 1.05 0.81 1.82 0.82 X 34 X s 52.3 15.73 4.90 3.35 1.73 0.97 0.80 1.91 0.91 7 X *> l •>/ X x 01.7 30.25 5.28 4.45 2.21 0.92 0.85 2.57 0.82 1 X X l 95.045.37 7.53 6.70 2.19 0.88 0.84 «• 0.96RADII OF GYRATION. 265 MOMENTS OF INERTIA, AND RADII OF GYRATION, OF PENCOYD T-BARS. c EVEN LEGS. II. III. IV. V. VI. Size, in inches. Weight per yard. Moments of inertia. Radii of gyration. Distance d from base to neutral axis. Axis A B. Axis C D. Axis A B. Axis C D. 4 X 4 X 4 36.50 5.26 2.55 1.20 0.84 1.14 O l »>2 X Q.t Q>2 X * h II 31.00 3.47 1.70 1.06 0.74 1.60 O X 3 X 1 1 SI 26.00 2.10 1.01 0.90 0.62 0.90 ! 2* X 2 2 X -ï-16 19.50 . 1.12 0.58 0.78 0.55 075 2-| X 2i X 3 8 17.52 0.97 0.49 0.75 0.53 0.75 2] X 2} X 1 4 11.75 0.52 0.30 0.65 0.50 0.61 2* X 2* X <♦ Pi 12.00 0.54 0.27 0.67 0.47 0.65 2 X 2 X y à? 10.50 0.38 0.19 0.60 0.43 0.60 1« X n X * 7.10 0.21 0.10 0.54 0.37 0.50 H X U x ■h 6.00 0.13 0.06 0.46 0.32 0.45 n X li X a 4.50 0.07 0.04 0.37 0.27 0.37 1 ‘ X 1 X 3.00 0.03 0.02 0.30 0.26 0.30 I266 MOMENTS OF INERTIA. MOMENTS OF INERTIA, AND RADII OF GYRATION, OF PENCOYD T-BARS. C D UNEVEN LEGS. II. III. IV. V. VI. Size,in inches. Weight per yard. Moments of inertia. Radii of gyration. Distance 11 from base to A\i< A I>. Axis C I). Axis A B. Axis C D. rientrai axis. 4* X 3j * 44.50 5.27 3.66 1.09 0.91 1.16 j 4 X 31 41.80 4.65 3.23 1.05 0.88 1.09 ! 1 5 X 21 30.70 1.61 4.01 0.72 1.14 0 67 5 X 2* 33.00 1.63 4.58 0.70 1.17 0.64 4 X 3 25.90 1.94 2.18 0.86 0.92 0.77 4 X 3 25.25 2.09 1.69 0.91 0.82 0.84 4 X 2 20.40 0.68 1.68 0.58 0.91 0.54 •> U X 3^ 28.25 3.12 1.06 1.05 . 0.61 1.10 •J X 21 23.80 1.38 0.94 0.76 0.63 0.82 3 X u 11.20 0.19 0.56 0.41 0.71’ 0.37 oJL X li 9.10 0.10 0.33 0.33 0.60 0.32 2 X n 8.75 0.16 0.18 0.43 0.45 0.43 2 X 1 7.00 0.05 0.17 0.26 0.49 0.27 2 X vV 5.88 0.01 0.17 0.13 0.54 0.17 22 X 12 18.75 0.56 0.62 0.55 0.58 0.66 22 X 2 21.00 0.83 0.63 0.63 0.55 0.75 5 X 31 48.44 5.37 5.31 1.05 1.04 1.05 5 X 4 44.10 6.24 5.25 1.19 1.09 1.08 X y Iff 6.50 0.01 0.24 0.12 0.61 0.18RADII OF GYRATION. 267 MOMENTS OF INERTIA OF TRENTON BEAMS. 1a I? Id I. ii. III. Size, in inches. Weight per yard. lbs. Area of cross-section. Moments of inertia. sq. ins. Axis A B. Axis 0 D. 15 200 20.02 707.1 27.46 15 120 15.04 523.5 15.29 m 170 16.77 391.2 25.41 121 125 12.33 288.0 11.54 10i 135 13.36 233.7 15.80 loi 105 10.44 185.6 9.43 10* 90 8.90 164.0 8.09 9 125 12.33 150.8 11.23 9 85 8.50 111.9 7.35 9 70 7.00 93.9 4.92 8 80 8.03 83.9 7.55 8 65 6.37 67.4 4.55 7 55 5.50 44.3 3.90 6 120 11.84 64.9 18.59 6 90 8.70 49.8 10.78_ 0 50 4.91 29.0 2.74 6 40 4.01 23.5 1.61 5 40 3.90 15.4 1.68 5 30 2.99 12.1 1.04 4 37 3.06 9.2 1.74 4 30 2.91 7.5 1.11 4 18 1.77 4.5 0.31268 MOMENTS OF INERTIA. MOMENTS OF INERTIA OF TRENTON CHANNELS. ;B I. H. III. VI. Size, in inches. Weight per yard. lbs. Area of cross-section. sq. ins. Moments of inertia. Distance d from base | to neutral axis, in inches. Axis A B. Axis C I). 15 190 18.85 586.0 32.25 1.260 15 120 12.00 376.0 14.47 0.950 12} 140 14.10 291.6 17.87 1.120 12} 70 7.00 153.2 5.04 0.755 10} 60 6 00 88.4 3.84 0.628 9 70 7-02 82.1 5.35 0.850 9 50 5.08 58.8 2.53 0.630 8 45 4.48 44.5 2.54 0.760 8 33 3.30 32.9 1.44’ 0.580 7 36 3.60 27.1 1.96 0.715 7 251- 2.54 17.3 0.S3 0.511 6 45 4.32 21.7 2.12 0.725 6 33 3.20 17.2 1.30 0.630 6 22 2 2.25 12.6 0.70 0.540 5 19 1.92 7.2 0 44 0.464 4 16 £ 1.65 3.9 0.32 0.460 Q »> 15 1.45 2.0 0.29 0510 Deck- Beams. s 65 6.29 54.7 3.70 1 # 55 5.35 35.1 3.60 — Note. — The weights of the channels for each size given above may b 54.0 5.4 2S.4 2.51 2.30 0.68 5 30.0 3.0 12.3 1.0S 2.03 0.60 5 39.0 3.9 14.2 1.34 1.91 0.59 4 24.0 2.4 6.19 0.71 1.61 0.55 4 30.0 3.0 6.99 0.87 1.53 0.54 3 21.0 2.1 3.09 0.55 1.21 0.55 *> o 27.0 2.7 3.54 ' 0.84 1.15 0.56RADII OF GYRATION, 273 MOMENTS OF INERTIA, AND RADII OF GYRATION, OF UNION MILLS CHANNEL-BARS. ;b 1. II. IV. VI. Si7.0, in Weight per yard, in lbs. Total area of section. Moments of inertia. Radii of gyration. Distance of centre of gravity from outside of web. inches. Axis A B. Axis A B. 15 120.0 12.(X) 359.00 5.47 0.82 15 180.0 18.00 471.00 5.12 0.88 12 00.0 6.00 119.00 4.46 0.69 12 77.5 6.75 140.00 4.56 0.74 12 90.0 9.00 168.00 4.31 0.72 12 90.0 9.00 176.00 4.42 0.72 12 150.0 15.00 248.00 4.07 0.S3 10 48.0 4.80 62.50 3.61 0.55 10 52.5 5.25 75.50 3.79 0.63 10 90.0 9.00 106.80 3.44 0.66 10 60.0 6.00 89.40. 3.86 0.70 10 105.0 10.50 126.90 3.48 0.65 9 43.5 4.35 47.40 3.30 0.58 9 54.0 5.40 64.80 3.46 0.68 9 90.0 9.00 89.10 3,15 0.73 8 37.5 3.75 34.50 3.03 0.53 8 46.5 4.65 39.20 2.90 0.53 8 48.0 4.80 45.30 3.07 0.66 8 84.0 8.40 64.50 2.77 0.73 7 31.5 3.15 22.40 2.67 0.52 7 40.5 4.05 26.10 2.54 0.52 7 42.0 4.20 30.60 2.70 0.66 7 60.0 6.00 37.90 2.51 0.68 6 22.5 2.25 12.10 2.32 0.48 6 28.5 2.85 13.90 2.21 0.47 6 30.0 3.00 16.60 2.35 0.60 6 48.0 4.80 22.00 2.14 0.62 5 19.5 1.95 '7.00 1.90 0.44 5 25.5 2.55 8.25 1.80 0.44 5 27.0 2.70 10.22 1.94 0.61 5 42.0 4.20 13.35 1.78 0.64 4 18.0 1.80 4.11 1.51 0.46 4 21.0 2.10 4.51 1.47 0.46 4 21.0 2.10 4.98 1.54 0.54 4 27.0 2.70 5.78 1.46 0.56 3 15.0 1.50 2.04 1.17 0.51 3 18.0 1 80 2.27 1.12 0.52 I274 MOMENTS OF INERTIA MOMENTS OF INERTIA, AND RADII OF GYRATION, OF UNION MILLS ANGLE-IRONS. For minimum and maximum thickness and weight. 1. VI. II. IV. Size,in inches- Weight per foot, in lbs. Area of cross-section, in sq.ins. Distance of centre of gravity from outside of flange, in ins. Moments of inertia. Radii of gyration. Axis A B. Axis A B. Min. _ . Max. Min. Max. Min. Max. I Min. 1 Max. Min. Max. 6x6 19.2 39.2 5.75 11.75 1.68 1.96 19.900 43.100 1.90 1.90 4x4 9.5 19.5 2.86 5.86 1.14 1.35 4.360 9.550 1.20 1.30 34 x 34 8.3 17.0 2.48 5.11 1.01 1.22 2.870 6.380 1.10 1.10 3J x 34 7.7 15.8 2.30 4.73 0.95 1.16 2.270 5.100 0.99 1.00 3X3 5.9 12.2 1.78 3.65 0.86 1.01 1.510 3.350 0.92 0.96 2} x 2J 5.4 8.8 1.62 2.65 0.80 0.91 1.150 1.990 0.84 0.87 2i X 24 4.9 8.0 1.46 2.39 0.74 0.85 0.850 1.490 0.76 0.79 2i x 2| 3.5 7.3 1.06 2.19 0.66 0.79 0.500 1.130 0.69 0.72 2X2 3.1 5.6 0.94 1.69 0.59 0.70 0.350 0.680 0.61 0.63 n *13 2.1 5.0 0.62 1.50 0.51 0.64 0.180 0.480 0.54 0.56 id n 1.8 3.6 0.53 1.09 0.44 0.55 0.110 0.250 0.46 0.48 U *ij 1.0 2.0 0.30 0.61 0.35 0.43 0.044 0.098 0.38 0.40 ij x n 0.9 1.8 0.27 0.55 0.32 0.39 0.032 0.071 0.34 0.36 1 x 1 0.8 1.2 0.23 0.36 0.30 0.33 0.022 0.035 0.30 0.31RADII OF GYRATION. 275 MOMENT’S OF INERTIA. AND RADII OF GYRATION, OF UNION MILLS ANGLE-IRONS. For minimum and maximum thickucss and weight. (Moments of inertia, and radii of gyration, for axes AB and CD only.) Upper figures, axis AB; lower figures, axis CD. 1. VI. II. IV. Size, in inches. Weight per foot, in lbs. Area of cross-section, in sq. ins. Distance of centre of gravity from outside of llangc, in ins. Moments of inertia. Radii of gyration. Min. Max. Mill Max. Min. Max. Min. Max. Min. Max. 6x4 13.0 26.4 4.18 7.93 } 1.96 / 0.96 2.17 1.17 15.500 5.610 30.700 11.500 1.90 1.20 2.00 1.20 5x4 10.8 22.0 3.23 6.61 1 y-> j 1.03 1.74 1.24 8.160 4.670 17.500 10.300 1.60 1.20 1.00 1.20 5 x 10.2 20.8 3.05 6.23 1 1.61 j 0.86 1.82 1.07 7.780 3.180 16.700 7.090 1.00 1.00 1.60 1.10 5x3 9.5 19.5 2.86 5.86 f 1.70 \ 0.70 1.91 U.91 7.370 2.040 15.870 4.660 1.60 0.S4 1.60 0.89 4 x 3j 8.0 18.3 2.67 5.48 ) 1.20 1 0.96 1.41 1.16 4.180 2.990 9.140 6.650 1.25 1.06 1.29 1.10 4X3 8.3 17.0 2.48 5.11 t 1.28 i 0.78 1.49 0.99 3.960 1.920 8.700 4.380 1.26 0.88 1.30 0.93 3j x ? 7.7 15.8 2.30 4.73 i 1.08 i 0.83 1.29 1.01 2.720 1.S5U 6.070 4.210 1.09 0.90 1.13 I 0.94 3J x 2 4.2 8.5 1.25 2.56 \ 1.10 f 0.18 1.21 0.1,1 1.360 0.400 2.930 0.910 1.04 0.57 1.07 0.60 3 x 4.4 9.0 1.31 2.69 S 0.91 t 0.66 1.05 0.SO 1.170 0.740 2.540 1.640 0.94 0.75 0.97 0.78 3X2 4.0 8.1 1.19 2.44 \ 0.99 j 0.49 1.13 0.63 1.090 0.390 2.360 0.S9O 0.96 0.57 <».99 0.60 21 x ‘2 “i £ 3.5 7.3 1.06 2.19 i 0.79 j 0.54 0.92 0.67 0.650 0.370 1.440 0.S50 0.78 0.59 0.S1 0.G2 ■1 K]| 2.6 4.0 0.78 1.20 } 0.69 i 0.37 0.76 0.44 0.310 0.120 0.500 0.200 0.63 0.39 0.65 0.41276 MOMENTS OF INERTIA PROPERTIES OF UNION MILLS T-IRONS. I. VI. II. IV. III. 1 V. X) 0 0 c Axis A B. Axis C D. o ï %-» w &/).-5 — cT mm GQ u 4> Æ * .Sf— 'S a B ** • X X 2 ~ <-» - X X ~ — cj ^ 1 u— . 0 c X w IB O c 0 0 .2 ' *-*-• Cj £ 0 0 — X rt 0 x 0 ? 9 SC w ~ >» « Ö0 5X3 13 3.90 0.73 2.50 1.10 0.80 5.70 2.30 1.21 5 x 2J 10J 3.0S 0.58 1.40 0.71 0.66 4.60 1.80 1.21 4^ x 34 15 4.50 1.13 5.20 2.18 1.07 3.90 1.70 0.93 4x5 14 4.20 1.57 10.50 3.05 1.57 2.70 1.40 O.SO 4 x 4J 13.4 4.05 1.37 7.80 2.48 1.39 2.70 1.40 0.82 4X4 12 3.60 1.18 5.40 1.91 1.22 2.60 1.30 0.84 4x3 2.78 0.80 2.10 0.96 0.87 2.30 1.10 0.90 4 x 2.} V4 2.25 0.62 1.10 0.60 0.70 2.00 1.00 0.93 4X2 64 1.95 0.46 0.54 0.35 0.53 1.80 0.91 0.96 3J X 4 Hi 3.38 1.24 5.15 1.87 1.23 1.80 1.00 0.72 3i x 3i 10 3.00 1.04 3.34 1.36 1.05 1.60 0.93 0.73 3j x 3 n 2.78 0.85 . 2.14 1.00 0.88 1.60 0.93 0.77 3x4 12.1 3.68 1.35 5.55 2.10 1.24 1.30 0.87 0.60 3 x 3J 11J 3.53 1.15 3.93 1.67 1.06 1.40 0.92 0.62 3 x | 7.6 2.28 0.90 1.89 0.90 0.91 0.94 0.63 0.64 3 x 2\ 6 1.80 0.69 0.96 0.53 0.73 0.77 0.51 0.66 2] x 3 64 1.95 0.96 1.66 0.81 0.93 0.50 0.40 0.51 x 2§ 6.6 1.98 0.86 1.39 0.74 0.84 0.55 0.44 0.53 2* x 24 5.4- 1.62 0.75 0.91 0.43 0.75 0.46 0.37 0.53 24 x U 3 0.90 0.30 0.09 0.10 0.32 0.33 0.26 0.61 Note. — The moments of inertia and resistance, and radii of gyration, in this table, are close approximations only. The table does not include all sizes manufactured.RADII OF GYRATION. 277 For compound sections made up of two or more beams or bars, the moments of inertia are found by combining those of the several shapes as given in the preceding tables. Thus: — I =■ Twice the moment of inertia for beam a (col. II.) + that for beam b (col. III.). _________________I________________ k sum of areas of beams a and b (col. I.) I = Twice area of beam a (col. I.) x d2 + twice moment of inertia for beam a (col. III.) + that for beam b (col. II.). I — --------------------------------. d + i width flange of beam a I. Q2 — --------------------------------- sum of areas of beams a and b (col. I.) I — Twice area of channels (col. I.) X d2 4- moment of inertia (col. III.), in which d = distance of centre of gravity of the channel from centre line of the combination. ________________7_______________ ^ area of the two channels (col. I.) I = Twice the moment in col. II. G2 = Same as for single channel. H Ua When a section is employed alone, either as girder or post, the neutral axis passes through its centre of gravity. When rigidly connected with other sections forming part of a compound section, the neutral axis passes through the centre of gravity of the com-278 MOMENTS OF INERTIA. pound section; and therefore the moment of inertia of the elementary section will not be that around its own centre of gravity, but around an axis at a distance from that point. The moment of inertia of a section about an axis other than that through its centre of gravity is equal to the moment about the axis through its centre of gravity plus the product of the area of the section by the square of the distance of its centre of gravity from the axis about which the moment of inertia is sought. The first step, then, in finding the moment of inertia, is to find the position of the centre of gravity of the section. For all symmetrical sections, this, of course, lies at the middle of the depth. For triangles, it is found on a line parallel with the base, and distant one-third the height of the triangle above the base. For other sections, it is found by supposing the area divided up into elementary sections, and multiplying the area of each such section by the distance of its centre of gravity from any convenient line. The sum of these products divided by the total area of the section will give the distance of the centre of gravity from the line from which the distances were measured. Example.' — Find the neutral axis of a X section having the following dimensions : width, 8 inches ; depth, 10 inches ; thickness of metal, 2 inches. The area of the vertical flange, considering it as running through to the bottom of the section, would be 10 X 2, or 20 square inches; and the distance of its centre of gravity above the bottom line, 5 inches. The product of these quantities, therefore, is 100. The area of the bottom flange, not included in the vertical flange as above taken, is (5 times 2, or 12 square inches; the distance of its centre of gravity above the bottom line, 1 inch; and the product of the two, therefore, 12. The sum of these products 112 divided by the total area is 7^-, or 3.5 inches, which is the distance of the centre of gravity above the bottom line of the section. Having found the neutral axis of this section, its moment of inertia is readily found by the formula before given. Thus, in the case just supposed, d would be 10 — 3.5 = 6.5; il, — 3.5; d„~ 1.5; anil the moment would be (see p. 259), = (2 X 0.53) + (3 x 3.5») - (6 X 1.5») = 1 3 The moment of resistance of this section as a girder would be 290.3 r: .or 44’,; and if a strain on the fibres of the iron of 12,000 (>.•) poun Is per square inch be allowed, then, since the moment of resistance of the girder multiplied by strain per square inch mustMOMENT OF RESISTANCE. 279 equal the bending-moment of the load, it will be able to support a load whose bending-moment is 44i times 12,000 pounds, or 536,000; i.e., if used as a girder secured rigidly at one end, and loaded at the other, it would support a load, in pounds, of 536000 length in inches Or if supported at both er Is, and the load uniformly distributed over the span, it would support a load eight times as great; the bending-moment in such case being one-eighth that in the former case (see pp. 251, 252). Note. — The formulas and figures on pp. 258, 250, and 277, are taken, by permission of The New-Jersey Steel and Iron Company, from a handbook which they publish, entitled “ Useful Information for Engineers and Architects,” and containing full information pertaining to the forms of iron which they manufacture.280 PRINCIPLES OF THE STRENGTH OF BEAMS. CHAPTER XIV. GENERAL PRINCIPLES OF THE STRENGTH OF BEAMS, AND STRENGTH OF IRON BEAMS. By the term “beam” is meant any piece of material which supports a load whose tendency is to break the piece across, or at right angles to, the fibres, and which also causes the piece to bend before breaking. When a load of any kind is applied to any beam, it will cause it to bend by a certain amount; and as it is impossible to bend a piece of any material without stretching the fibres on the outer side, and compressing the fibres on the inner side, the bending of the beam will produce tension in its lower fibres, and compression in its upper ones. This tension and compression are also greatest in those fibres which are the farthest from the neutral axis of the beam. The neutral axis is the line along which the fibres of the beam are neither lengthened nor shortened by the bending of the beam. For beams of wrouglit-iron and wood the neutral axis practically passes through the centre of gravity of the cross-section of the beam. To determine the strength of any beam to resist the effects of any load, or series of loads, we must determine two things: first, the destructive force tending to bend and break the beam, which is called the “bending-moment;” and, second, the combined resistance of all the fibres of the beam to being broken, which is called the “ moment of resistance.” The methods for finding the bending-moments for any load, or series of loads, have been given in Chap. XII.; and rules for finding the moment of resistance, which is equal to the moment of inertia divided by the distance of the most extended or compressed fibres from the neutral axis, and the quotient multiplied by the strength of the material, have been given in Chap. XIII., together with tables of the moment of inertia for rolled iron sections of the usual patterns. Now, that a beam shall just be able to resist the load, and not break, we must have a condition where the bending-moment in the beam is equal to the moment of resistance multiplied by the strength of the material. That the beam may be abundantly safe to resist the given load, the moment of resistance multiplied byPRINCIPLES OF THE STRENGTH OF BEAMS. 281 strength of material must be several times as great as the bending-moment; and the ratio in which this product exceeds the bending-moment, or in which the breaking-load exceeds the safe load, is known as the “ factor ” of safety. By “the strength of the material” is meant a certain constant quantity, which is determined by experiment, ami which is known as the 44 Modulus of Rupture.” Of course this value is different for each different material. The following fable contains the values of this constant divided by the factor of safety, for most of the materials used in building-construction. The moment of resistance multiplied by these values will give the safe resisting-power of the beam. Modulus of Rupture for Safe Strength. Material. Value of m ?n lbs. Material. Value of R, iu lbs- Gael Iron 5544 American white pine . . 1440 I | Wrought-iron 1*2,000 American yellow pine . . 2250 Xk Stcet / *. . • . . /(■<.< 24,000 American spruce. . . . 1620 / 2 American ash 2000 Michigan pine 1530 American red beech . . 1800 Bluestone flagging (Hud- American yellow birch 1620 son River) 375 American white cedar . . 1000 Granite, average .... 300 American elm 1400 Limestone 270 New- 1Cngland flr . . . . 1500 Marble 300 Hemlock 1200 Sandstone 150 American white oak . . : m*) Slate 900 The above values of R for wrought-iron and steel are one-fourth that for the breaking-loads; for cast-iron, one-sixth; for wood, one-third; and for stone, one-sixth. The constants for wood are based upon the recent, tests made at the Massachusetts Institute of Technology upon full-size timbers of the usual quality found in buildings. The figures given in the above table are believed to be amply safe for beams in floors of dwellings, public halls, roofs, etc.; but, for floors in mills and warehouse-floors, the author recommends that not more than two-thirds of the above values be used. The safe loads for the Trenton, Phcenix, and Union Iron Mills sections, uted as beams, are all computed with 12,000 pounds for the safe value- of or with 12,000 pounds fibre strain, as it is generally called. There are certain cases of beams which most frequently occur in building-construction, for which formulas can be given by which the safe loads for the beams may be determined directly; but it often happens that we may have either a regularly shaped beam282 PRINCIPLES OF THE STRENGTH OF'PRAMS. irregularly loaded, or a beam of irregular section, but with a common method of loading, or both; and in such cases it is necessary to determine the bending-moment, or moment of resistance, and find the beam whose moment of resistance multiplied by R is equal to this bending-moment, or what load will give a bending-moment equal to the moment of resistance of a beam multiplied by Ii. For example, suppose we have a rectangular beam of yellow pine loaded at irregular points with irregular loads: what dimensions shall the beam be to carry these loads ? We will suppose that we have found the bending-moment caused by these loads to be 480,000 inch pounds. Then, as bending-moment equals moment of resistance multiplied by R, BX 7)2 480,000 pounds = —^— x 2250 = B X D2 X 075; or „ ■ 480000 BX Z)2 = = 1280. 0(0 12S0 If we assume D = 12 inches, then B = -rpr = 9 inches, or the beam should be 9 inches by 12 inches. If, instead of a liard-pine beam, we should wish to use an iron beam to carry our loads in the above example, we must find a beam whose moment of resistance multiplied by 12,000 equals 480.000 inch pounds. We can only do this by trial, and for the first trial we will take the Trenton rag-inch 125-pound beam. The moment of inertia of this beam is given as 2S8; and its moment of resistance is one-sixth of this, or 48. Multiplying this by 12,000, we have 576,000 pounds as the resisting-force of this beam, or 96.000 pounds over the bending-moment. Hence we should probably use this beam, as the next lightest beam would probably not be strong enough. In this way we can find the strength of a beam of any cross-section to carry any load, however irregularly disposed it may be. Strength of Wrought-iron Beams, Channels, Angle and T Bars. It is very seldom that one needs to compute the strength of wrought-iron beams, channels, etc.; because, if he uses one of the regular sections to be found in the market, the computations have already been made by the manufacturers, and are given in their handbook. There might, however, be cases whore it would be necessary to make the calculations for any particular beam; and to meet such cases we give the following formulas.PRINCIPLES OF THE STRENGTH OF BEAMS. 283 Beams fixed at one end, and loaded dt the other (Fig. 1). Safe load ill pounds = ^ 1000 X moment of inertia length in feet X y Beams fixed at one end, loaded with uniformly distributed load (Fig. 2). Safe load in pounds = > 2000 X moment of inertia length in feet X y Fig. 2. Beams supported at both ends, loaded at middle (Fig. 3). W pi Safe load in pounds Fig. 3. __ 4000 x moment of inertia span in feet X y (3> Brants supported at both ends, load uniformly distributed span in feet X y284 PRINCIPLES OF THE STRENGTH OF BEAMS. Beams supported at both ends, loaded icith concentrated load not at centre (Fig. 5). 1000 X moment of inertia X span in feet Safe load in pounds - — - ^ ^ Beams supported at both ends, loaded with W pounds, at a distance m from each end (Fig. 6). Fig. 6. Safe load IF, in pounds at each point = 1000 X moment of inertia m in feet Xy ^ The letter y in the above formulas is used to denote the distance of the farthest fibre from the neutral axis; and, in beams of symmetrical section, y would be one-lialf the height of the beam in inches. These formulas apply to beams of any form of cross-section, from an I-beam to an angle or T bar. Weight of Beam to be subtracted from its Safe Load. As the weight of iron beams often amounts to a considerable proportion of the load which they can carry, the weight should always be subtracted from the maximum safe load: for beams with concentrated loads, and for beams with distributed loads, one-half the weight of the beam should be subtracted. Example 1. — What is the safe load fora Trenton 12-j-incli light I-beam, 125 pounds per yard, having a clear span of 20 feet, the load being concentrated at a point 5 feet from one end ? 1000 X I X span 1000 X 288 X 90 _ Ans. Safe load (For. 5) = —“ 5 X 15 X 6* ' ~ 12,500 pounds.STRENGTH OF IRON BEAMS. 285 Example 2.—A 12-incli heavy Union Iron Mills channel-bar, weighing 90 pounds per yard, and having a clear span of 24 feet, supports a concentrated load at two points, 6 feet from each end. What is the maximum load that can be supported at each point consistent with safety ? J I , , 1000 X 108 Has. Safe load at each point = —^ „— 4000 pounds. The moment of inertia for channels and angle-bars, and other sections, will be found in Chap. XIII. Deepest Beam always most Economical. Whenever we have a large load to carry with a given span, it will be found that it can be carried with the least amount of iron by using the deepest beams, provided the beams are not too strong for the load. Thus, suppose we wish to support a load of 9 tons with a span of 20 feet, by means of Trenton beams. We could do this either by one 124-inch beam at 125 pounds per yard, or by two 9-inch beams at 85 pounds per yard. But the 12|-inch beam, 21 feet long, would weigh only 875 pounds, while the tuo 9-inch beams would weigh 1190 pounds; so that, by using the deeper beam, we save 315 pounds of iron, worth from three to five cents per pound. , B The following table, under the heading gives the relative strength of Trenton beams in proportion to their weight, thus exhibiting the greater economy of the deeper patterns. Trenton Boiled I-Beams. Strength of each Beam in Proportion to its Weight. Beam. C IT 15 inch heavy .... 37.41 15 it light 36.76 12* H heavy .... 2S.41 m it light 30.64 101 ( t heavy .... 26.64 10* it light 27.20 104 it extra litrlit . . . 27.78 9 it extra heavy . . 21.44 •9 it heavy .... 23.41 9 i t light 23.86 8 it heavy .... 20.99 Beam. c if' 8 inch, light 20.75 7 “ 55 pounds . . . 18.37 6 “ 120 I ... 14.33 6 “ 00 “ ... 14.07 6 “ heavy 15.36 6 “ light 15.65 5 “ heavy 12.27 5 “ light 12.00 4 “ heavy 9.95 4 1 light 10.03 4 “ extra light . . . 10.00286 STRENGTH OF IRON BEAMS. Another important advantage in the use of deeper beams is their greater stiffness. By referring to the tables, it will be seen that a beam twenty feet long, under its safe load, if 6 inches deep, will deflect 9 inches deep, will deflect 12i inches deep, will deflect 15 inches deep, will deflect 0.95 inch. 0.G3 inch. 0.46 inch. 0.38 inch. A floor or structure formed of deep beams will therefore be much more rigid than one of the same strength formed of smaller sections. There are, of course, cases where the use of deep beams would be inconvenient, either from increasing the depth of the floor, or from the fact, that, with a light load and short span, they woidd have to be placed too far apart for convenience. In general, however, it will be best to employ the deep beams. Inclined Beams. —The strength of beams inclined to the horizon may be computed, with sufficient accuracy for most purposes, by using the formulas given for horizontal beams, taking the horizontal projection of the beam as its span. Steel Beams. — Steel beams are now used to a large extent, especially in the Western States. They are cheaper to manufacture than iron beams, as they are made directly from the ore and in one process; while with iron beams the ore is first converted into cast iron, then into wrought iron, and then rolled. Steel beams, however, are not apt to be of uniform quality. Some may be even very brittle; they are, however, fully twenty-five per cent stronger than iron; but as their deflection is only about 9, three per cent less than that of iron beams, there is but very little economy of material possible in their use. If steel beams are used, they can be spaced one-quarter distance farther .apart than iron beams; or they will safely carry one-quarter more load than given in the tables for iron beams; but in no case, where full load is allowed, must the span in feet of steel beams exceed twice the depth in inches. With full safe loads, the deflection of steel beams will always be about one-eighth greater than that of iron beams. Strength of Trenton, Pencoyd, Plioenix, and Union Iron 3Iills Rolled Beams, Channels, Angle and T Bars. The following tables give the strength and weight of the various sections to be found in the market, together with the general dimensions of the I-beams. The tables are in all cases made up from data published by the respective manufacturers. The deflection of the beams under their maximum safe distributed load is also given in the last four tables. These latter tables will be found very convenient, for they can bo used for the spans indicated, without any computations whatever.STRENGTH OF IRON BEAMS, 287 STRENGTH, WEIGHT, AND DIMENSIONS OF TRENTON ROLLED I-BEAMS. 1 I. II. HI. IV. V. I)esi gnation of beam. Weight per yard, in lbs. Safe distributed load for one foot of span, in lbs. Moment of inertia. Neutral axis perpendicular to web. Width of flange, in ins. Area of cross-section^ in ins. 20 inch, heavy . . 272 1,320,000 - 6.75 27.20 20 << light . . . 200 990,000 - 6.00 20.00 15 (t heavy . . 200 748,000 707.1 5.75 20.02 15 n light . . . 150 551,000 523.5 5.00 15.04 121 n heavy . . 170 511,000 391.2 5.50 16.77 121 (< light . . . 125 377,000 288.0 4.79 12.33 101 it heavy . . 135 360,000 233.7 5.00 13.36' m it light . . . 105 286,000 185.6 4.50 10.44 10£ il extra light . 90 250,000 164.0 4.50 8.90 9 (I extra heavy, 125 268,000 150.8 4.50 12.33 9 It heavy . . 85 199,000 111.9 4.50 8.50 9 It light . . . 70 167,000 93.9 4.00 7.00 8 II heavy. . . 80 168,000 83.9 4.50 8.03 8 II light . . . 65 135,000 67.4 4.00 6.37 7 It 55 11)8. . . 55 101,000 44.3 3.75 5.50 6 II 120 “ 120 172,000 64.9 5.25 11.84 0 i i 90 “ . . 90 132,000 49.8 5.00 8.70 0 II heavy . . 50 76,800 29.0 3.50 4.91 0 II light . . . 40 62,600 23.5 3.00 4.01 5 (( heavy . . 40 49,100 15.4 3.00 3.90 5 II light . . . 30 38,700 12.1 2.75 2.99 4 II heavy . . 37 36,800 9.2 3.00 3.66 4 4 ( light . . . 30 30,100 7.5 2.75 2.91 4 ii extra light. 18 18,000 4.5 2.00 1.77288 STRENGTH OF IRON BEAMS, STRENGTH, WEIGHT, AND DIMENSIONS OF TRENTON CHANNEL-BARS AND DECK-BEAMS. % n. III. IV. V. Designation of bar. Weight per yard, in lbs. Safe distributed load, in lbs., for one foot of span. Moment of inertia I. Width of flange, in ins. Area of cross-section, in ins. Channel-Bars. 15 inch, heavy . . 190 625,000 586.0 43 18.85 15 “ light . . . 120 401,000 376.0 4 12.00 12$ “ heavy . . 140 381,000 291.6 4 14.10 12$ “ light . . . 85 200,100 181.9 3 8.62 l°i “ heavy . . 60 134,750 88.4 23 6.00 9 “ heavy . . 70 146,000 82.1 3J 7.02 9 “ light . . . 50 104,000 58.8 24 5.08 8 “ light . . . 45 88,950 44.5 2$ 4.48 8 “ extra light 33 65,800 32.9 2.2 3.30 7 “ light . . . 36 62,000 27.1 24 3.60 7 “ extra light . 25$ 39,500 17.3 2 2.54 6 “ heavy . . 45 58,300 21.7 2$ 4.32 6 “ light . . . 33 45,700 17.2 23 3.20 6 “ extra light . 22$ 33,680 12.6 1|. 2.25 5 “ extra light . 19 22,800 7.2 If 1.92 4 “ extra light . 16$ 15,700 3.9 1$ 1.65 3 “ extra light . 15 10,500 2.0 1$ 1.45 , . • Deck-Beams. 8 inch 65 91,800 54.7 4$ I 6.29 7 «( 55 63,500 35.1 44 5.35STRENGTH OF IRON BEAMS 289 STRENGTH, WEIGHT, AND DIMENSIONS OF TRENTON ANGLE AND T BARS. I. II. I. II. Safe Safe Designation of Weight distributed Designation of Weight distributed per foot, load for one per foot, load for one bar. in lbs. foot of span. bar. In lbs. foot of span, in lbs. in lbs. Angles Even Legs. Angles Unequal Legs. 6 in. x 6 in. 19.00 36,900 6 in. x 4 in. 14.00 ( 30,680 44 « x 4| “ 124 18,000 ( 14,750 4 “ x 4 “ 94 12,184 5 “ x 34 “ 10.20 f 18,353 34 " x 34 “ 84 9,200 ( 9,651 3 « X3 “ 4.80 4,611 44 “ x 3 “ 9.00 ( 14,580 2| “ x 2J “ 5.40 4,710 ( 7,020 2| “ x 24 “ 3.90 3,156 4 “ X3 ** 7.00 ( 9,850 2* «1 x 21 “ 3.50 2,530 ( 5,871 2 “ x 2 “ 3.13 1,970 34 « x 14 “ 4.00 f 5,515 13 “ x 13 “ 2.00 1,150 ( 1,148 14 | x 14 “ 1.75 832 • 3 “ x 24 “ 4.37 f 4,490 1» “ x 1> “ 1.00 393 ( 3,233 1 « x 1 “ 0.75 246 3 “ x 2 “ 4.00 ( 4,334 7 (( v 7 « 8 A 0.00 186 ( 2,080 1“ X J » 0.56 133 T-Bars. 4 in. x 4 in. 12.50 15,800 3 in. x 2 in. 4.80 2,540 34 “ x 34 “ 9.60 10,550 2 “ x 14 “ 3.00 1,355 3 “ x3 “ 7.00 6,680 *)I «« vll (< 2.40 604 24 “ x 24 “ 5.00 3,850 2 “xi “ 2.15 457 2 " X2 “ 3.13 1,970 14 “ xi “ 1.86 ; 421 5 “ x 24 “ 11.70 6,34 [ Note. — Nearly all of the above bars can be rolled with greater thickness if desired, and the strength would increase in proportion. The above values are for the lightest weight of the bars.290 STRENGTH OF IRON BEAMS STRENGTH, WEIGHT, AND DIMENSIONS OF UNION IRON-MILLS ROLLED I-BEAMS. | h. hi. IV. V. Designation of beam. Weight per yard, in lbs. Safe distributed load for one foot of span, in lbs. Moment of inertia. Neutral axis perpendicular to web. Width of flange, in ins. Area of cross-section, in ins. 15 inch, light . . . 150 565,000 530.00 5.03 15.0 15 it heavy . . 195 655,000 614.00 5.33 19.5 15 it light . . . 200 722,000 677.00 5.55 20.1 15 «( heavy . . 240 800,000 750.00 5.81 24.0 12 <1 light . . . 126 367,000 275.00 4.64 12.6 12 “ heavy . . 180 454,000 340.00 5.09 18.0 10J <( light . . . 95 251,000 165.00 4.54 9.5 10J II heavy . . 135 306,000 201.00 4.92 13.5 10 (< light . . . 90 240,000 150.00 4.32 9.0 10 {< heavy . . 135 300,000 187.00 4.77 13.5 9 (( light . . . 70 174,000 97.50 4.01 7.0 9 << heavy . . 99 208,000 117.00 4.33 9.9 9 <4 extra light . 135 282,000 159.00 4.94 13.5 9 <« extra heavy, 150 300,000 169.00 5.10 15.0 8 tl light . . . 66 140,000 69.90 3.81 6.6 8 H heavy . . 105 181,000 90.40 4.29 10.5 7 it light . . . 54 105,000 45.80 3.61 5.4 7 H heavy . . 75 124,000 54.30 3.91 7.5 6 it light . . . 41. 65,300 24.50 3.24 4.1 6 (I heavy . . 54 75,800 28.40 3.46 5.4 5 If light . . . 30 39,500 12.30 2.73 3.0 5 (« heavy . . 39 45,500 14.20 2.91 3.9 4 U light . . . 24 24,800 6.19 2.48 2.4 4 it heavy . . 30 28,000 6.99 2.63 3.0 3 <« light . . . -, 10,500 3.09 2.32 2.1 3 •• heavy . . I 18,90) 3.54 2.52 2.7STRENGTH OF IRON BEAMS 291 STRENGTH, WEIGHT, AND DIMENSIONS OF UNION IRON-MILLS CIIANNEL-BA RS. Designation of bar. Weight per yard, in lbs. Safe distributed load for one foot of span, in lbs. Designation of bar. Weight per yard, in lbs. Safe distributed load for one foot of span, in lbs. 15 in., light . . . 120.0 382,000 7 in. light . 31.5 51,300 15 “ heavy . . 180.0 502,000 7 “ heavy 40.5 59,700 12 “ one weight, 60.0 159,000 7 “ light . 42.0 69,800 12 “ light . . . 67.5 187,000 7 1« heavy 60.0 86,400 12 “ heavy . . 90.0 223,000 6 << light . 22.5 32,300 12 “ light. . . 90.0 235,000 6 <( heavy 28.5 37,100 12 “ heavy . . 150.0 331,000 6 «• light . 30.0 44,200 10 “ one weight, 48.0 100,000 6 It heavy 48.0 58,600 10 “ light. . . 52.5 121,000 5 light . 19.5 22,400 10 “ heavy . . 90.0 171,000 5 “ heavy 25.5 26,400 10 “ light . . . 60.0 143,000 5 “ light . 27.0 32,700 10 “ heavy . . 105.0 203,000 5 “ heavy 42.0 42,700 9 “ one weight, 43.5 84,000 4 ti light . 18.0 16,500 9 “ light. . . 54.0 115,200 4 “ heavy 21.0 18,100 9 “ heavy . 90.0 158,400 4 “ light . 21.0 19,900 8 | light . . . 37.5 68,900 4 “ heavy 27.0 23,100 8 “ heavy . . 46.5 78,500 3 “ light. 15.0 10,600 8 “ light . . . 48.0 90,400 3 “ heavy 18.0 12,100 g “ heavy . . 84.0 129,100 STRENGTH, WEIGHT, AND DIMENSIONS OF UNION IRON-MILLS ANGLE-IRONS. ANGLES WITH EQUAL LEGS. Size. Weight per foot, in lbs. Safe distributed load ter one foot of span. Size. Weight per foot, in lbs. Safe distributed load for one foot of span. 6 inch x 6 inch 19.2 36,800 24 inch x 24 inch 3.5 2,600 4 “ x 4 << 9.5 12,000 2 “ x 2 I 3.1 2,000 H “ "x 34 << 8.3 9,600 13 “ X 13 “ 2.1 1,120 34 « x 3J ti 7.7 7,900 14 1 X 14 1.8 800 3 “ x 3 K 5.9 5,700 14 « X 14 « 1.0 400 Im “ x 2J «( 5.4 4,700 1J “ X 14 “ 0.9 320 ‘>1 ^5 ■ x 24 4.9 3,800 1 ‘«xl “ 0.8 240292 STRENGTH OF IRON BEAMS STRENGTH, WEIGHT, AND DIMENSIONS OF UNION IRON-MILLS ANGLE-IRONS. ANGLES WITH UNEQUAL LEGS. Weight Safe dis Weight Safe dis Size. tributed load Size. tributed load per fool, for one foot per foot, for one fool in lbs of span. in lbs. of span. 6 in. x 4 in. 13.9 \ 30,400 { 14,400 3* m x 3 in. 7.7 t 9,040 1 6,800 5 “ x 4 (i 10.8 \ 19,200 1 12,800 3.1 «« x 2 r 4.2 5,040 1 2,080 5 ** x 3j < < 10.2 18,400 j 9,000 3 >• x 2* “ 4.4 t 4,480 ) 3,200 5 “ x 3 <( 9.5 t 17,000 7,120 3 <* x 2 “ 4.0 4,320 j 2,080 4 “ x 3J •• 8.9 \ 11,920 ) 9,440 “ x 2 r 3.5 3,040 | 2,000 4 “ x 3 «1 8.3 1 11,680 ( 6,960 *2 “ x 13 “ 2.6 1,840 1 960 Note. — The above are for the least thickness that the angle-irons are lolled. STRENGTH, WEIGHT, AND DIMENSIONS OF PHCEXIX ROLLED I-BEAMS. I. ' ii. III. IV. V. Designation of beam. Weight per yard, in lbs. Safe distributed load for one foot of span, in lbs. Moment of inertia. Width of llauge, in ins. Area of cross-section, in ins. Neutral axis perpeu dicularto web. 15 inch, heavy . . 200 820,000 707.5 5.30 20.0 15 (< light . . . 150 604,000 531.0 4.75 15.0 12 heavy . . 170 5S4.000 398.5 5.50 17.0 12 << light . . . 125 416,000 2S8.0 4.75 12.5 10* << heavy . . 135 356,000 239.0 5.00 13.5 10* «( light . . . 105 310,000 1S9.0 4.50 10.5 9 (( extra heavy, 150 394,000 194.0 5.37 15.0 9 ft heavy . . 84 216,000 112.5 4.00 8.4 9 «( light . . . 70 184,000 96.5 3.50 7.0 8 (i heavy . . 81 188,000 87.5 4.50 8.1 8 If light . . . 65 63.5 4.00 6.5 7 «« heavy . . 69 144,000 57.0 4.00 6.9 7 tf light . . . 55 108,0tXl 43.5 3.50 5.5 6 (( heavy . . 50 90,000 31.0 3.50 5.0 6 << light . . . 40 70,000 24.5 2.75 4.0 5 II heavy . . 36 50,000 15.0 3.00 3.6 5 II light . . . 30 42.(XX) 12.5 2.75 3.0 4 << heavy . . 30 36,000 8.0 2.75 3.0 4 II light . . . 18 20,000 4.5 2 00 1.8STRENGTH OF IRON BEAMS. 293 STRENGTH, WEIGHT, AND DIMENSIONS OF PENCOYD I-BEAMS. 1. II. 111. IV. V. Safe dls- Moment of inertia. Width of flange, iu ins. Area of cross-section, in ins. Designation of beam, in ins. Weight per yard, ip lbs. tributed load for one foot of span,in net tons. Neutral axis perpen dicular to web. 15 X §4 heavy sect’n. 200.0 424.41 682.0S 53 19.90 15 X 1 u «4 233.0 5 n 23.30 15 X A light 145.0 324.30 521.19 54 14.55 15 X H ik 201.0 H 20.10 12 X H heavy (4 108.0 289.32 371.98 *>2 16.89 12 X 5 t« {4 194.0 *>32 19.40 12 X 2 I light 44 120.0 212.22 272.80 <51 lit 11.95 12 X n it <4 163.0 *>-h 10.30 10i X A heavy 44 119.0 183.59 200.55 45 11. S9 104 X 8 “ 44 105.0 5ft 10.50 ioi X 3 8 light 4 4 97.0 149.54 108.23 44 9.70 1G| X 3 4 “ 44 136.0 45 13.00 10 X 1 2 heavy 4 4 112.0 102.02 173.58 11.17 10 X 3 4 ik 4k 137.0 ^8 13.70 10 X si light 4 4 90.0 138.43 14S.31 4? 9.04 10 X 1 It 4 4 100.0 455 10.00 9 X It heavy 4 4 90.0 123.21 118.81 4$ 9.07 9 X 3 4 •• 44 122.0 m 12.20 9 X J 1 h 4 light . 44 70.0 97.94 94.44 45 0.98 9 X l s •• 4 4 88.0 4 „4 8. SO 8 X 11 3 2 heavy 4 4 81.0 97.92 83.93 41 8.14 8 X 3 4 it 44 109.0 4 & 10.90 8 X T6 light 44 05.0 80.70 09.17 4 0.53 8 X 176 it 44 75.0 45 7.50204 STRENGTH OF IRON BEAMS. STRENGTH, WEIGHT. AND DIMENSIONS OF PENCOYD I-BEAMS (concluded). 1 1 1 II. III. IV. V Safe dis Moment of inertia Width of flange* iu ins. Area of cross-section, in ms. Designation of beam, in ins. Weight per yard, m lbs. tnbuted load for one foot of span,in net tons. Neutral axis perpen-dicu larto web < X 1*6 heavy sectTi. 65.0 GG.3S 49.7S 3^ 6.58 7 X 3 4 «t ss.o 44 s.so 7 X ii light 4 4 51.0 57.44 43.0S m 5.14 7 X 3 4' SS.O 4i S.SO 6 X ii heavy 44 50.0 41.87 2G.92 O ft or* 5.04 G X h 8 «t 44 63.0 n 6.30 G X 1 4 light 4» 40.0 37.49 24.10 n •> 4.OS G X $ 8 44 Go.O 33 6.30 5 X ft Tb heavy 44 34.0 25.01 13.40 2H 3,38 5 X 1*5 44 44 40.0 m 4.00 5 X '32 light 4 4 30.0 23.33 12.50 2* 2.94 5 X Ytf 4 4 44 40.0 9:} \ ^3i 4.00 4 X i 4 heavy 44 2S.0 17.04 7.69 2 3 ~ 4 2.90 4 X 1 2 44 38,0 o O S.SO | 4 X l i t>4 light 44 IS. 5 12.00 5.14 2i 1.90 4 X 1 4 44 4 4 21.5 2.15 6 X i 4 heavy 44 23.0 10.24 3.29 2| 2.25 3 X Vb 44 4 4 28. G 2U 2.SG »■> X & light 44 n.o S.2S 2.66. 2i 1.71 •> f X A 4 44 21.7 2:i3 2.17 Note. — Minimum and maximum thicknesses given . any intermediate section can be rolled. Safe loads given are for iron .* steel beams eau also be rolled.STRENGTH OF IRON BEAMS] 29$ STRENGTH, WEIGHT, AND DIMENSIONS OF PENCOVD DECK-REAMS. I. II III. IV V Designation of bar. Weight per yard, in lbs. Safe dis tributed load for one foot of span,in net tons. Moment of inertia 1. Width , of iiangc, in ius. A rea of cross-section, in ius. 12 inches X it inch. 101.0 172.60 221.98 53 10.40 12 44 X HP 1 b 44 138.0 oa 13.80 11 (4 X 3 a 44 91.0 139.50 164.09 5i 9.06 11 4» X l» a 44 118.0 5.3 11.80 10 (4 X 3 a 44 80.0 110.30 118.22 51 8.02 10 <4 X l, a 44 105.0 L 10.50 9 If X 3 a 44 72.0 87.90 84.77 5 7.17 9 4 4 X i a 4 4 94.0 51 9.40 8 (4 X it 44 61.0 C7.30 57.66 n 6.11 8 44 X 44 84.0 4 it 8.40 r- t 44 X 44 52.0 45.80 34.40 41 5.21 7 44 X 5 a 44 72.0 4il 7.20 1 c 4» X .*> T6 44 42.0 34.20 21.95 33 4-18 6 44 X u lb 44 57 0 4 5.7U 5 44 X s T6 44 34.0 22.40 12.04 H i ;>.;>7 5 44 X 4 5.40 7 X l1 6* heavy light <( 18.80 12.45 6.67 n 1.8S 5 X i “ u 32.90 m 3.29 4 X | heavy a 21.50 12.03 5.16 m 2.15 4 X * a 31.50 m 3.15 4 X A light <6 17.50 9.65 4.14 1.75 4 X 3 “ 8 a 23.70 n? 2.37 3 X fa 15.20 6.32 2.03 m 1.52 3 X H 18.90 m 1.89 Oi ^4 X i 11.30 3.33 0.80 i? 1.13 2 X 7_ .J* 8.75 2.25 0.48 iä 0.88 9 X 0 10.00 Hz 1.00 H X 3V 3.50 ii 0.35 Note. — Minimum and maximum thicknesses given : any intermediate see tioiiw can be rolled. 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CO CO tO 10 tO ‘ ►—4 ►—* *—11 deflection, in — i—' 4- OO to © H-* O' © O' © O' —* © to -X> O' —4 CO O' CO i—* —1 to © GO © to © lO ‘ to O' © -4 —J © 4- to to 4* CO 4« inches. o IJMMhIMM . ►_ _i _i >_i fc_l O' 4- 4- CO CO to ' to »—•>—*©©©© OO CO © © O' O’ © O' © O' © O' © O' © O' © O' © O' © O' © O' © O' © o © © © © © ©©©©©©©©©©©©©©© Weight of beam, in IbR. , . i_i )_1 >—I ua u-t w_l to to to to to © © © © t—‘ i—• to to CO CO O' © X © © to 4- © © tt © 4- oc to to © © © 4- to to CO 4- © 4— co © to tons. i-* >—* l—l © © © © © © © © © © © © © 0 (^1 © © Corresponding CO to © rO © oc “1 © O' 4- 4- co cc to to to 1—1 1—1 deflection, in feffi 4- —1 CO co to -“1 to •^o 4- 4- 4- © -t cc to 00 © •4 00 —1 O' 4- 00 © © © “~t —1 inches. 1—4 © i—i I >—i 4- 4- CO to *—1 1 £ © © © 00 © © © Weight of beam, © on CO -1 © © 4- © CO Iv © © O' © Cc to © © CO —t © CO -4 © CO *4 o CO -4 <-> co "I © CO —i © CO —1 © CO 00 © •—* 4k *-1 © CO t—ji—i>—i>-4i—IMH•*—i l-il—itO 00 © ©©©©•—4tCCOG04-©--IOC© ^1© 4-© 4-© © to © bo © © co oc oo Safe load, net tons. >mL tit • 5* to ©t ys 0.903 0.977 1.053 1.131 1.211 1.294 © © ©©©©©©p©©ppp© bot-t ©cib'©4-iucocoiototOH-i»-i CO © © CO —t © © © to CO 4- © —1 4* toco GO©GOCO©CX©0'tOCO-44—4-^ Corresponding deflection, in inches. 1—l >—i 1—1 )—1 w ►—* to to >-* >-* © © O' ® © tO CO 4- © 00 “0 © co to Q © © 00 00 -1 —1 *'1 © © © © © 4» 4ik © O' *-* -1 CO © © © © to CO 4- © © »-* ©CO “^©COtO©©~4 0'C3tO©CO—1 Weight of beam, in lbs. © © © © © *4 -4 © K-4 CO © Q0 H-* 4^^-J 0000p©©0^^t0p^©^4 J-* © © bo © b« h-» oo ^-4 bo to bo Safe load, net tons. 4* © • 3 —* OS © c 35 0.784 0.856 0.928 1.000 1.080 1.160 1.240 1.330 ocooppppopcpp © b> O' '4- 4- CO C0 to to to to O' © CO GO O0 00 00 © O' —* CC 4. ►-iGlOO'COl-ii—‘ © CO ■-* i— 4-©© Corresponding deflection, in inches. H-IMMIM M M I—1 —1 CO CO to . to 1—* I—* . © © O' © © •-* -1 to CC CO © O' © © © © © O' ©©©CC00—1~-1©©©©4^4*-© 4^ © O' -J © tO ~-l CO 00 4- © O’ © O' © O' © O' © O' © O' © O' © Weight of beam, in lbs. O' © © © © © © © ►“».CO ©^t ©to © ^t 1—i r— _i |—i i—i —i ^ -1 -I X X O t! O ^ “ tO 4 © ©bo^4H-‘©'-4*-ltOH-‘QO©©© Safe load, net tons. © M- • 3* o CN ae • t—IMM411—l (-1 © © iu bo to h-1 © co bo O' © --'I GO © tO 4- © O’ © t-l -4 O' © to oppoopcoppppo C5 © O' 4. 4- CC bo tO tO t-1 cc i—» O’ cc to •—* ©. to --i o; © ©> X © lO © X C 4- 4- 10 X 5i -1 4* Corresponding deflection, in inches. V—1 —i ©©©CO©© coco l.-iX4-*-4 4- © © © © O' © O' © O' -•t -1 -1 © © O’ © © 41- 4* 4- CO CO —t co © © co © © to © o« to cc O' O O' © O' © O’ © O' O O’ © © © Weight of beam, in lbs. CO to to to to to to to t-itOtO'-'Mk-ii-iMi-iMu.Mk-i © © GC —1 © O’ 4- CO tOH-*©©GO-4©.Oi4-COtO>—'© Clear span,In feet. STCY38 NCrtTI XIN5DHJ JO HXONJHXS SAFE DISTRIBUTED LOADS AND DEFLECTIONS OF PIICENIX I-BEAMS.© to 1C tO VC 13 tO tO tC 10 tO <—» ^^x-i^C'4-c4tc--ca ^>—• l—l >—< k—* >-_« t—< —. CC —1 © O' 4- © 1C *—• © Clear span, in feel. f—i 0 r. -1 -*1 m x x a tc c c 0^ «x v wv Weight of beam, 3§ C C> © Cl © c © c- © ©w©.».©ww©w©_w W w © © 1 ^©-„W 1 CC CO CO W 4- 4- *. 5- f' C O'' H-J CCOMMXCCO Safe load, net © -‘i be © I— 05 on ©> g +- -1 © 05 —1 to -1 05 © ©- 50 tons. c0 II ►— t—IMW.*1^0000 —«cccoc-»® c; c x j. *i<5 c " iv « c. to ©©©©©©©©-i©aocn ocoopooco 05 O'» 4. 4. 05 C3 tO 10 S*1 li C' ® w *1 10 *J C5 '.o *9 © 00 tO C5 4- 05 —• IO Concsponding dcflectiou, in inches. r OD OOOC^IMM-I o: 05 c. C? Cl C U'^*.4iWM&3«KI Wl™ht Of beam, f- r-> oc on to ©-1 4- — © © I o -i 4- ic © 05 © c 00 * O tv |J. a X O ti 4* Cl X 9 1; ! 4- Ci X C IO 4- C. X O in lbs. p2p5 02 ©G2C-2C54.4-4-4-4-i— tO 05 4- b« © be © tO 4- ©. be CICIO'C. C*-1*ICCO -* 4- -1 ic H C -14- 1C Safe load, net tons. wm-mmmiJmCOCC “~1 05 4- 03 tC 1— © O © OC —1 ©. C*v O O GO ^1 ^ vC1 *-—* IO 4— v^ w OwOO^OOOw'lO^O ©COO©©©©© C; b> 4- 4- 02 05 tO IO ►-* —i4-4-C5CC—‘--IC5© On 0> 00 tO © QC 0' >-* © Corresponding deflection, In inches. — 1©©©©C»CnCnCn4k4k4k c *1 ci ;; c x a. w - e o; 4- CQUCCwCCwCS^ 4. c-3 t: co w c: u. »0 10, O '1 C' IO C X 0)' 00 © © 05 O © 05 O © 05 Weight of beam, j ! in lbs. 03 0 2 03 03 05 05 05 4- 4- 4k 4k On On On —* IO 05 b’ © —J © >—4 io Cn —1 © to bn On © © —I -1 00 © © to H io » i)' i* S{ifc load, net tons. © -t ©5 b' 4- 05 tO l-4 © CC © -1 G5 © *-» 3C O' -1 4- to 05 4- to O' 02« OC> © —» C © © © © © © © © to O 05 03 —1 © © © © © © © O’ 4» 4- 03 03 to to 4- -1 tO ©. © O' —' © oo c p-> co oo on Corresponding deflection, iu inches. GO 0C -f -I -1 -1 05 © © C* on o» on 4- 4- —* 30 0» 10. © -I 4* 1C © 05 4- —* 0C' C GD C-~ C*» c£ Iv O' 00 »—* 4— O' cc CO 0« Cv CO tv to :o o to w -1 to m CO — ** -1 O Weight of beam,! in lbs. to K tc to IsG tc w to cc cc CC c: 4^ 4^ 4— v* O«* CO CO m tO 4— CT» •-4 w »—i CC 4k 4k O’ cr> © © -1 © © 02 -I io be 4- Safe load, net tons. oc in be —1 05 O' 44 03 io © © -00 -1 © 3V ^ I s' g g ^ t * i ^ ? © © © c © © © b' 4- 4k 02 05 to to •4 1 IO 05 — © — ©> O' 4- O' tO 4- O' C01 responding deflection, in inches. © © © On On © On 4k 4k 4. 4* 4k 09 03 i • IO © TC © 4- to © -1 O' 03 -J © © © 3C -J O' © t© © OC -1 O' © tO G OC c: Co 03 to to to to 4k to © CC 05 03 —4 -1 O’* 02 to © 00 © Weight of beam, in lbs. / * to to to to to tc 02 o: 05 02 02 CO 4- 4k 4-4. 4- o- © —i b © O'—* to 4- © be © to cn hr. cn on © © —i -j O’» © on tO Safe load, net tons. »J © toioto—‘O©© © © to ^ b x *-i 1 4- b: "—* »-* G © b - T bb -14.CJ. to © O’ tO © © -< -* —4 tO 4— © ccoc©cccccc4-x4- o* a c © © © © 4- *4- 03 03 to © to © © on r-J +- 05 © to Corresponding deflection, in inches. © © © © C' © on © © 4k 4k 4- 4- 03 © 03 © © ■*- to *© -1 m to © x © or —• © © 4- C-M- -1 J. O' to © © 02 © -1 4 1 a O' 02 IO IO to to to © -1 on © to © © M © Weight of beam, in lbs. 02 to IO to to to to to to to to — — —* >-* — —* •—1 ■-* —» •-* © © v/5 -1 © Oi 4- 03 M r— © © x: -1 © © 4— © to <—* © Clear span, in feet. t1 ■ 1 l)G? mnrati Btoh xmmj -jq hj.onshlls SAFE DISTRIBUTED LOADS AND DEFLECTIONS OF PIICENIX I-BEAMS.SAFE DISTRIBUTED LOADS AND DEFLECTIONS OF PIKEN1X I-BEAMS. 1 7-in. 55 lbs. 6-in. 50 lbs. 6- ill. 40 lbs. 5-in. 36 lbs. 5- in. 30 ibs. 4-in. 30 lbs. £ S. ! W CJ O • '■+- «•* Cj 12 o © o »- u— Q rZ tf) a §1. m Zzz — 5 o *■* Cj d 6 ~ © o — o r7i be a §.2 . © g 0-5.2 g **■« o , — ® -a -o bi,~ 3 ® o 3 Q £ ^ bfi 2 o! . U O JS £ X ^ 0-73 .S (J c8 Cj A o . ♦J X, J3 JO .bp— Qj ^ Is d o © o 3 «+- o d «—1 m bf, 2 m. © ■ O d 9 JO «M O . ® — JO bf;— fc- ** a d ® d o 0/ 0) cT 10 5.4 0.248 183 4.5 0.290 167 3.5 0.286 133 2.50 0.337 120 2.10 0.336 100 1.80 0.448 100 10 11 4.8 0.293 201 4.1 0.352 183 3.2 0.348 146 2.30 0.413 132 1.90 0.405 no 1.63 0.545 no n 220 €\ m 200 9 Q 0.410 160 4.0 2 00 0 466 144 1 70 0 471 120 1 50 0 643 120 12 id 4.2 0.423 238 3.4 0.481 217 2.7 0.486 173 1.90 ■HI 156 1.60 0.563 130 1.38 0.752 130 13 14 3.9 0 491 256 3.2 0.566 233 2.5 0.562 186 1.80 0.667 168 1.50 0.660 140 1.28 0.872 140 14 1 70 0 774. 180 1 40 150 1.20 1 000 150 15 15 3.6 0.558 275 3.0 0.653 250 2.3 0.636 200 1.60 0.885 192 1.30 0.854 160 1.12 1.130 160 16 16 3.4 0.651 293 2.8 0.740 267 2.2 0.738 213 1.50 0.995 204 1.20 0.945 170 1.06 1.290 170 17 17 3.2 0.722 311 2.6 0.824 283 2.0 0.805 226 1.40 1.100 216 1.20 1.120 ISO 1.00 1.440 180 18 18 3.0 0.803 330 2.5 0.940 300 1.9 0.907 240 1.30 1.200 228 1.10 1.210 190 0.95 1.020 190 19 19 2.8 0.882 348 2.4 1.060 317 1.8 1.010 253 1.20 1.290 240 1.00 1.280 200 0.90 1.790 200 20 20 2.7 0.992 366 2.2 1.130 333 1.7 1.110 266 1.20 1.500 252 1.00 1.450 210 0.85 1.950 210 21 21 2.5 1.060 385 2.1 1.250 350 1.6 1.210 ■i 1.10 1.580 264 0.95 1.620 220 0.81 2.140 220 22 22 2.4 1.170 403 2.0 1.370 367 1.6 1.390 293 1.10 1.800 276 0.90 1.750 230 0.78 2.350 230 23 23 2.3 1.280 421 1.9 1.490 383 1.5 1.490 306 1.00 1.800 288 0.85 1.880 240 0.75 2.570 210 24 24 2.2 1.390 440 1.8 1.600 400 1.5 1.580 320 1.00 2.110 300 0.82 2.050 250 0.72 2.790 250 25 25 2.1 1.500 458 1.8 mm 417 1.4 1.790 333 0.95 2.250 312 0.80 2.250 260 0.69 3.010 260 26 26 2.1 1.690 476 1.7 1.920 433 1.3 1.870 346 0.92 2.440 324 0.77 2.430 270 0.06 3.260 270 27 27 2.0 1.800 495 1.6 2.020 450 1.3 2.090 360 0.90 2.660 336 0.75 2.610 280 0.64 3.510 280 28 28 1.9 1.900 513 1.6 2.200 467 1.2 2.150 373 0.86 2.S30 3*8 0.72 2.810 290 0.62 3.770 290 29 29 1.8 2.010 531 1.5 2.360 483 1.2 2.390 386 0.83 3.200 360 0.70 3.030 300 0.00 4.020 300 30 30 1.8 2.230 550 1.5 2.610 500 1.1 2.430 400 300 STRENGTH OF PHOENIX IRON BEAMS.STRENGTH OF PENCOYD IRON BEAMS. 301 SAFE DISTRIBUTED LOADS AND DEFLECTIONS OF PENCOYD BEAMS. Safe loads in net tons evenly distributed. Fora concentrated load in middle, allow one-half of that given in table below. O £ Ï w r -a*0B Length of span, In feet. § or 10 12 14 10 18 20 22 24 2G 28 30 15 200.0 | 42.44 35.37 30.31 26.53 23.58 21.22 19.29 17.68 16.32 15.16 14.15 0.11 0.15 0.21 0.27 0.34 0.42 0.51 0.61 0.72 0.83 0.96 15 145.0 \ 22.10 22.10 22.10 20.27 18.02 16.21 14.74 13.51 12.47 11.58 10.81 0.07 0.12 0.20 0.27 0 34 0.42 0.51 0.61 0.72 0.83 0.95 12 168.0 j 28.93 24.11 20.67 18.08 16.07 14.47 13.15 12.05 11.13 10.33 9.64 0.13 0.19 0.26 0.34 0.43 0.53 0.64 0.77 0.90 1.05 1.20 12 120.0 j 21.22 17.69 15.16 13.26 11.79 10.01 9.65 8.84 8.16 7.58 7.07 0.13 0.19 0.26 0.34 0.43 0.53 0.64 0.77 0.90 1.06 1.20 10| 119.0 j 18.35 15.30 13.11 11.47 10.20 9.18 8.35 7.65 7.06 6.56 6.12 0.15 0.22 0.30 0.39 0.49 0.61 0.74 0.88 1.03 1.19 1.37 10! 97.0 | 14.95 12.46 10.68 9.35 8.31 7.48 6.80 6.23 5.75 5.34 4.98 0.15 0.22 0.30 0.39 0.49 0.61 0.74 0.88 1.03 1.19 1.37 10 112.0 | 16.20 13.50 11.57 10.13 9.00 8.10 7.36 6.75 6.23 5.79 5.40 0.16 0.23 0.31 0.41 0.52 0.64 0.78 0.92 1.08 1.26 1.44 10 90.0 j 13.84 11.54 9.89 8.65 7.69 6.92 6.29 5.77 5.32 4.94 4.61 0.16 0.23 0.31 0.41 0.52 0.64 0.78 0.92 1.08 1.26 1.44 9 90.0 { 12.32 10.26 8.80 7.70 6.84 6.16 5.60 5.13 4.74 4.40 4.11 0.18 0.26 0.35 0.46 0.58 0.71 0.86 1.02 1.21 1.40 1.61 9 70.0 j 9.79 8.16 7.00 6.12 5.44 4.90 4.45 4.08 3.77 3.50 3.26 0.18 0.26 0.35 0.46 0.58 0.71 0.86 1.02 ■ 1.21 1.40 1.61 81.0 | 9.79 8.16 6.99 6.12 5.44 4.89 4.45 4.08 3.77 3.50 3.26 j o 0.20 0.29 0.39 0.51 0.65 0.80 0.97 1.16 1.36 1.67 8 65.0 j 8.00 0.20 6.72 0.29 5.76 0.39 5.04 0.51 4.48 0.65 4.03 0.80 3.67 0.97 3.36 1.16 3.10 1.36 2.88 1.57 2.69 j 1.80\ 65.0 J G.63 5.53 4.74 4.15 3.69 3.32 3.02 2.77 2.55 2.37 2.21 1 7 0.23 0.33 0.44 0.58 0.74 0.90 1.09 1.32 1.55 1.80 2.06\ 7 52.0 j 5.74 4.79 4.10 3.59 3.19 2.87 2.61 2.39 2.21 2.05 1.91 1 0.23 0.33 0.44 0.58 0.74 0.90 1.09 1.32 1.65 1.80 2.06 50.0 | 4.18 3.49 2.99 2.62 2.33 2.09 1.90 1.74 _ — - ! o 0.27 0.38 0.52 0.69 0.87 1.07 1.29 1.54 - - 40.0 j 3.74 3.12 1 2.68 2.34 j 2.08 1.87 1.70 1.56 - - o 0.27 0.38 , 0.52 0.69 0.87 1.07 1.29 1.54 - - - I 34.0 j 2.50 2.08 1.79 1.56 1.39 1.25 1.14 1.04 _ — - — a 0.32 0.46 0.63 0.82 1.04 1.29 1.56 1.86 - - - | 30.0 J 2.33 1.94 1.67 1.46 \ 1.30 1.17 1.06 0.97 - - - ; 0 0.32 0.46 0.63 0.82 1.04 1.29 1.56 1.85 - - - i 4 28.0} 1.79 0.40 1.49 0.58 1.28 0.79 1.12 1.03 1.00 1.31 0.90 1.61 - - - - - 18.5 | 1.19 1.00 0.86 0.75 0.67 0.60 - - - - - j 4 0.40 0.58 0.79 1.03 1.31 1.61 - - - - 3 23.0 J 1.02 0.53 0.S5 0.77 0.73 1.05 0.64 1.37 _ - - - - - 17.0 j 0.82 0.69 ! 0.59 1 0.52 - - - - - - - 1 0.53 0.77 1.05 7 0 7 1 .0 1 « Note. — The figures i;i Italics arc the deflections in inches corresponding to the loads above. For the deflections of greatest safe loads in middle, take four-fifths ( !) of the tabular figures in Italics.302 STRENGTH OF TRENTON IRON BEAMS. SAFE DISTRIBUTED LOADS AND DEFLECTIONS OF TRENTON BEAMS. Safe loads in net tons evenly distributed (in addition to weight of beam). For a concentrated load in middle, allow one-half of that given in table below. © 8 2 (C Length of spat , in feet. O (9 m Oil N J - £ s: a..*- 10 12 14 16 18 20 22 24 26 28 30 15 200 g 30.84 30.77 26.25 22.84 20.18 18.03 16.27 14.78 13.52 12.42 11.47 Ì 0.09 0.14 0.19 0.24 0.31 0.38 0.46 0.66 0.64 0.76 0.86 15 150 . 24.80 22.66 19.33 16.82 14.85 13.27 11.97 10.88 9.95 9.14 8.43 i 0.09 0.14 0.19 0.24 0.31 0.38 0.46 0.66 0.64 0.76 0.86 12* J 25.27 20.95 17.85 15.52 13.68 12.21 10.99 9.96 9.09 8.33 7.67 llU ) 0.12 0.16 0.23 0.30 0.38 0.46 0.66 0.67 0.79 0.91 1.06 12* 125 18.64 15.46 13.17 11.45 10.10 9.01 8.11 7.35 6.71 6.15 5.66 Ì 0.12 0.17 0.23 0.30 0.38 0.46 0.66 0.67 0.79 0.91 1.06 10* 135 i 16.14 14.73 12.54 10.89 9.59 8.55 7.69 6.96 6.34 5.80 5.32 Ì 0.14 0.19 0.27 0.36 0.44 0.64 0.66 0.78 0.92 1.07 1.22 10* 105 12.83 11.71 9.97 8.66 7.63 6.80 6.11 5.54 5.04 4.62 4.24 i 0.14 0.19 0.27 0.36 0.44 0.64 0.66 0.78 0.92 1.07 1.22 10* 90 10.27 10.24 8.72 7.57 6.67 5.95 5.35 4.85 4.42 4.04 3.72 Ì 0.14 0.19 0.27 0.36 0.44 0.64 0.66 0.78 0.92 1.07 1.22 Q 125 I 13.19 10.92 9.28 8.01 7.07 6.28 5.63 5.08 4.61 4.20 3.84 I 0.16 0.23 0.31 0.40 0.61 0.63 0.77 0.91 1.07 1.24 1.43 9 85 1 9.81 8.12 6.91 5.99 5.27 4.69 4.21 3.81 3.46 3.16 2.89 1 0.16 0.23 0.31 0.40 0.61 0.63 0.77 0.91 1.07 1.24 1.43 9 70 1 Ì 8.23 0.16 6.82 0.23 5.80 0.31 5.03 0.40 4.43 0.61 3.94 0.63 3.54 0.77 3.20 0.91 2.91 1.07 2.66 1.^4 2.43 1.43 8 80 f 8.27 6.84 5.81 5.04 4.43 3.93 3.52 3.18 2.88 2.63 2.40 I 0.18 0.26 0.36 0.46 0.68 0.71 0.86 1.03 1.20 1.40 1.61 8 65 6.6 [ 5.49 4.67 4.04 3.55 3.16 2.83 2.55 2.31 2.10 1.93 Ì 0.18 0.26 0.36 0.46 0.68 0.71 0.86 1.03 1.21 1.40 1.61 f 4.96 4.10 3.48- 3.01 2.64 2.34 2.09 1.88 1.70 1.55 1.41 t DO I 0.20 0.29 0.40 0.62 0.66 0.82 0.99 1.17 1.38 1.60 1.84 6 120 l 8.40 6.92 5.86 5.05 4.42 3.90 3.47 3.10 2.79 2.51 ) 0.24 0.34 0.47 0.61 0.77 0.96 1.16 1.37 1.61 1.87 2.14 6 90 1 i . 6.45 5.32 4.50 3.88 3.40 3.00 2.67 2.39 2.15 1.94 1.75 0.24 0.34 0.47 0.61 0.77 0.96 1.16 1.37 1.61 1.87 2.14 6 50 3.76 3.10 2.63 2.27 1.98 1.75 1.56 1.40 - - 1 0.24 0.34 0.47 0.61 0.77 0.96 1.16 1.37 - - 6 40 1 f mm 0.24 »> eira 0.34 2.14 0.47 1.85 0.61 1.02 0.77 1.13 0.96 1.2« 1.16 - _ — 5 40 I 1 2.39 0.28 1.97 0.41 1.66 0.66 1.43 0.73 1.21 0.92 1.04 1.14 - - - ~ 30 I 1 1.88 1.55 1.31 1.13 0.9 > 0.87 - - — - _ 0 0.28 0.41 0.66 0.73 0.92 1.14 - - - • - - A 37 I 1.78 1.46 1.23 1.05 0.91 _ — - - _ _ 1 1 0.36 0.61 0.70 0.91 1.16 - - « - - - - 30 1.45 1.19 1.00 0 86 0.75 - - - - - - j 0.36 0.61 0.70 0.90 1.16 - - - - - j. 18 I 0.87 0.72 0.60 0.52 - - - - - - i 0.36 0.61 0.70 0.91 - Note. — The figures in Italics are the deflections, in inches, corresponding to the loads above. For the deflections of greatest safe load* in middle, take four-fifths of the tabular figures in Italics.STRENGTH OF UNION MILLS IRON BEAMS. 303 SAFE DISTRIBUTED LOADS AND DEFLECTIONS OF UNION IRON-MILLS BEAMS. Safe loads In not tons evenly distributed (including weight of beam). For a concentrated load in middle, allow one-lialf of that given in table below. . R • Lenath of span, in feet. o a ; mm i V ÖL ^ N i — i/ T! y Ü _ y** /“> — BB io ! 12 14 10 18 20 22 24 26 28 30 1 1 150 | 28.24 ■ 20.17 17.65 15.69 14.12 12.84 11.77 10.86 10.09 9.41 1.) O.O!) 0.13 0.13 0.24 0.30 0.37 0.43 0.53 0.62 0.72 0.83 196 j 32.76 27.30 23.40 20.48 18.20 16.38 14/89 13.65 12.60 11.70 10.92 0.09. 0.13 0.18 0.24 0.30 0.37 0.43 0.53 0.62 0.72 0.83 15 200 j 36 >12 30.10 25.80 22.58 20.07 18.06 16.42 15.05 n.89 12.90 12.04 0.00 0.13 0.13 0.24 0.30 0.37 0.45 0.63 0.62 0.72 0.83 15 240 | 40.00 33.33 28.57 25.00 22.22. 20.00 18.18 16.67 15.38 14.29 13.33 0.00 0.13 0.18 0.24 0.30 0.37 0.43 0.53 0.62 0.72 0.83 12 136 | 18.36 115.30 13.11 11.48 10.20 9.18 8.35 7.65 7.06 6.56 6.12 0.12 0.17 0.23 0.30 0.37 0.46 0.56 0.66 0.78 0.90 1.04 12 180 j 22.68 18.90 116.20 14.18. 12.60 11.34 10.31 9.45 8.72 8.10 7.56 0.12 0.17 0.23 0.30 0.37 0.46 0.56 0.66 0.78 0.90 1.04 10* 94^ j 12.56 10.47 8.97 7.85 6.98 6.28 5.71 5.23 4.83 4.49 4.19 0.13 0.10 0.20 0.34 0.43 0.33 0.64 0.76 0.89 1.03 1.19 10* -! 15.32 0.13 12.77 0.10 10.94 0.20 9.58 0.34 8.51 0.43 7.66 0.33 6.96 0.64 6.38 0.76* 5.89 0.89 5.47 1.03 5.11 1.19 10 90 j 12.00 10.00 8.57 7.50 G.67 6.00 5.45 5.00 4.62 4.29 4.00 O.I4 O.20 0.27 0.33 0.43 0.33 0.67 0.80 0.94 1.09 1.23 10 -i 15.00 12.50 10.71 9.3S 8.33 7.50 6.82 6.25 mm 5.36 5.00 0.14 0.20 0.27 0.33 0.43 0.33 0.67 0.80 4). 94 1.09 1.25 9 ™àj 8.68 7.23 6.20 5.41 4.82 4.34 3.95 3.62 3.34 3.10 2.89 0.7.7 0.22 0.30 0.40 0.30 0.62 0.75 0.89 1.04 1.20 1.39 9 99 ! ( 10.40 .8.67 7.43 6.50 5.78 5.20 4.73 4.33 4.00 3.71 3.47 0.15 0.22 0.30 0.40\ 0.30 0.62 0.75 0.89 1.04 1.20 1.39 8 r>6 S t 7.00 5.83 5.00 4.38 3.89 3.50 3.18 2.92 2.69 2.50 2.33 0.17 0.23 0.31 0.44 0.36 0.69 0.84 1.00 7.77 1.36 1.56 105 9.04 7.53 6.45 5.65 5.02 4.62 4.11 3.77 3.48 3.23 3.01 m l ? 0.23 0.34 0.44 0.36 0.69 0.84 1.00 1.17 1.30 1.66 54 j 5.24 4.37 3.74 3.28 2.91 2.62 2.38 2.18 2.02 1.87 / 0.20 0.23 0.3!) 0.31 0.64 0.79 0.96 1.14 1.34 1.65 - 75 j 6.20 5 17 4.43 3.88 3.44 3.10 2.82 2.58 2.38 2.21 / 0.20 0.23 0.30 0.31 0.64 0.79 0.96 1.14 1.34 1.55 - 6 40* j 3.26 0.23 2.72 0.33 2.33 0.43 2.04 0.39 1.81 0.73 1.63 0.92 1.48 7.77 1.36 1.33 ~ — — I «1 3.79 3.16 2.71 2.37 2.11 1.90 1.72 1.58 - - - 0.23 0.33 0.43 0.39 0.73 0.92 1.11 1.33 - - - 30 j 1.98 1.65 1.41 1.24 1.10 0.99 — - - ■ - - 0 -JS 0.40 0.33 0.71 0.90 7.77 - - - - - 5 89 J 2.3S 0.2S 1.90 0.40 1.63 0.33 1.42 0.71 1.26 0.90 1.14 7.77 ~ - - - — o 1 \ tM 1.03 0.89 0.78 0.69 - - - - - - ~l ( 0.33 O.30 0.08 0.89 1.13 - - - - - - an 1 1.10 1.17 1.00 0.88 0.78 - - - - - - 0.33 0.30 O.08 0.89 1.13 - - - - - - 21 I 0.82 0.69 0.59 — — _ - - - - - O.40 0.07 0.91 ' - - - - - - - 1 t 0.94 0.79 0.67 - - - - - - - - ■ i O.40 0.67 0.91 - " ' « < Note. — The figures in Italics are the deflections, in inches, corresponding to the loads above. For the deflections of greatest safe loads in middle, take four* fifths () of the tabular figures in Italics.304 BEAMS SUPPORTING BRICK WALLS. Beams Supporting Brick Walls. In the case of iron beams supporting brick walls having no openings, and in which the bricks are laid with the usual bond, the prism of wall that the beam sustains will be of a triangular shape, the height being one-fourtli of the span. Owing to frequent irregularities in the bonding, it is best to consider the height as one-third of the span. Fig. 7. The greatest bending-stress at the centre of the beam, resulting from a brick wall of the above shape, is the same as that caused by a load one-sixth less, concentrated at the centre of the beam, or two-thirds more, evenly distributed. The weight of brickwork is very nearly ten pounds per square foot for one inch in thickness ; and from this data we find that the bending-stress on the beams would be the same as that caused by a uniformly distributed load equal to 2o X square of span in feet X thickness in inches ii Having ascertained this load, we have merely to determine from the, proper tables the size of beams required to carry a distributed load of this amount. Example. — It is proposed to support a solid brick wall 12 inches thick, overall opening 12 feet wide, on rolled iron beams: what should be the size and weight of beams ? Aittt, P>y tin' rule given above, the uniformly distributed loadFRAMING AND CONNECTING IRON BEAMS. 305 which would produce the same bending-stress on the beam as the wall, equals 25 X 144 X 12 —--------- 4S00 pounds. As the wall is twelve inches thick, it would be best to use two beams placed side by side to support it, as they would give a greater area to build the brick ou ; then the load on each beam would be 2400 pounds, or 1.2 tons. From the preceding tables for safe distributed loads on beams, we find that a 4-inch heavy beam would just about support this load; but as a 5-incli light beam would not weigh any more, and would be much stiffer, it would be better for us to use two 5-incli light beams to support our wall. If a wall has openings, such as windows, etc., the imposed weight on the beam may be greater than if the wall is solid. For such a case consider the outline of the brick which the beam sustains to pass from the points of support diagonally to the outside corners of the nearest openings, then vertically up the outer line of the jambs, and so on, if other openings occur above. If there should be no other openings, consider the line of imposed brickwork to extend diagonally up from each upper corner of the jambs, the intersection forming a triangle whose height is one-third of its base, as described above. H7(Oi benmare used to support a wall entirely (that is, the beams run under the. whole length of the wall), and the wall is more than sixteen or eighteen feet long, the whole weight of the wall should be taken as coming upon the beams; for, if the beams should bend, the wall would settle, and might push out the supports, and thus cause the whole structure to fall. Framing and Connecting' Iron Beams. When beams are used to support walls, or as girders to carry floor-beams, they are often placed side by side, and should in such Fig. 8. Fig. 9. Fig. 10. Fig. 41. cases be furnished with cast-iron separators fitting between the flanges, so as to firmly combine the two beams. These separators may be placed from four to six feet apart. Such an arrangement is shown by Figs. 8 and 10, Figs. 9 and 11 showing forms of sepa-FRAMING AND CONNECTING IRON BEAMS. ivs usually employed; that with two bolt-holes being used for 15-inch and T2\-inch beams, ami that with a single hole for iller sizes. When beams are required to be framed together, it is usually done as shown by the accompanying cuts, in which Fig. 12 shows two beams of the same size fitted together. Fig. 13 shows a beam fitted flush with the bottom flange of a beam of larger size. Fig. 14 shows a smaller beam fitted to the stem of a larger beam, above the lower flange. Wooden bean)ft may be secured to an iron girder in the same manner as an iron beam, by framing the end, and securing it by an angle-bracket; or an angle-iron may be riveted to the neb of the "iron girder to afford a flat bearing on which the wooden beam may x rest, as in Fig. 15.STRENGTH OF CAST-IRON BEAMS. 307 CHAPTER XY STRENGTH OF CAST-IRON. WOODEN, AND STONE BEAMS—SOLID BUILT BEAMS. Cast-Iron Beams. — Most of our knowledge of the strength of east-iron beams is derived from the experiments of Mr. Eaton llodgkinson. From these experiments he found that the form of cross-section of a beam which will resist the greatest transverse strain is that shown in Fig. 1, in which the bottom flange contains six times as much metal as the top flange. When east-iron beams are subjected to very light, strains, the areas of the two flanges ought to be neatly equal. As in practice it is usual to submit beams to strains le§s than the ultimate load, and yet beyond a slight strain, it is found, that when the flanges are as 1 to 4, we have a proportion which approximates very nearly the requirements of practice. The thickness of the three parts — web, top'flange, and bottom flange — may with advantage, be made in proportion as 5, 0, and 8. If made in this proportion, the width of the top flange will be equal to one-third of that of the bottom flange. As the result of his experiments, Mr. llodgkinson gives the following rule for the breaking-weight at the centre for a cast-iron beam of the above form : — Area of hot. flange x depth x 0 in square inches in ins. (jj Breaking-load in tons * deaTTpan in feet “ Cast-iron beams should always be tested by a load equal to that which they are designed to carry. Wooden Beams. — Wooden beams are almost invariably sqiian' or rectangular shaped timbers, and we shall therefore consider only that shape m the following rules and formulas.308 STRENGTH OF WOODEN BEAMS. For beams with a rectangular cross-section, we can simplify our formulas for strength by substituting for the moment of inertia b x d3 its value, viz., ——> '"'here b = breadth of beam, and <* its depth. Then, substituting this value in the general formulas for beams, we have for rectangular beams of any material the following formulas : — Beams fixed at one end, and loaded at the other (Fig. 2). Safe load in pounds = 4 X length in feet Breadth in inches = 4 X load X length in feet square of depth X A ‘ (2) (3) Beams fixed at one end, and loaded with uniformly distribided load (Fig. 3). breadth, x square of depth X A Safe load in pounds = -----2 X leng^hTfcel------’ ,4) 3 X length in feet X load square of deptli X . I Breadth in inches (5/STRENGTH OF WOODEN REAMS. 309 Beams supported at both ends, loaded at middle (Fig. 4). W Safe load in pounds or Breadth in inches ' = breadth X square of depth X Am span in feet span in feet X load square of depth X A' (0) C7) Beams supported at both ends, load uniformly distributed (Fig. 5). Fig. 5. 2 X breadth x square of depth X A Safe load in pounds = * span in feet---(8) or span in foot x load Breadth in inches = 2 x square of depth x A' (9) Beams supported at both ends, loaded with concentrated load NOT AT CENTRE (Fig. 0). breadth x sq. of depth x span X A Safe load in pounds =-------------v—^- j---------------’ (10) or 4 x load x ui x H Breadth in inches ■ “7~—---------------------^ ' i- (III # square of depth X span X .1310 STRENGTH OF WOODEN BEAMS. Beams supported at both ends, and loaded with W pounds at a distance m from each end (Fig. 7). . Safe load H' in pounds _ breadth X square of depth X A at each point 4 X m or 4 X load at one point X m Breadth in inches = sq. of depth X A (12) (13) Note. — In the laHt two cases the lengths denoted by m and n should b* takeu in feet, the same as the spans. Values of tiie Constant A. The letter A denotes the safe load for a unit beam one inch square and one foot span, loaded at the centre. This is also one-eighteenth of the modulus of rupture for safe loads. The following are the values of A, which are obtained by dividing the moduli of rupture in Chap. XIV. by 18. TABLE I. Values of A.—Co-efficient for Beams. Material. A lbs Material. A lbs. Cast-iron 308 • American white pine . . 80 Wrought-iron 666 American yellow pine 125 ~ Steel 1333 American spruce . • . Michigan pine .... 85 American ash 111 V > L Jl >i 11 >• ' American red beech . . 100 Bluestone flagging (Hud- American yellow birch 90 son River) 21 American white cedar . . 55 Granite, average . . . 17 American elm 77 Limestone 15 New England hr . . . . 83 Marble 17 Hemlock Sandstone 8 American white oak . . / ' 105 Slate 50 These values for the «-efficient A are one-tliird of the breaking-weight of timbers of the same size and quality as that used in first-class buildings. This is a sufficient allowance for timbers in roof t russes, and beams which do not have to carry a more severe load than that on a dwelling-house floor, and small halls, etc. Where there is likely to he very much vibration, as in the floor of a mill, or a gymnasium-floor, or floors of large public halls, tin* author recommends that only l'o:;r fifths of the above values of A he used.RELATIVE STRENGTH OF BEAMS. 311 Exampi.e 1. —What load will a hard-pine beam, S inches by 12 inches, securely fastened into a brick wall at one end, sustain with safety, 6 feet out from the wall ? Ans. Safe load in pounds (Formula 2) equals . 8 X 144 x 125 4 x (} = 6000 lbs. Example 2. — It is desired to suspend two loads of 10,000 pounds each, 4 feet from each end of an oak beam 20 feet long. What should be the size of the beam ? Ans. Assume depth of beam to be 14 inches; then (Formula 13) 4 X 10000 X 4 breadth = —190 x loir*- = ® inches, nearly : therefore the beam should be 8 X 14 inches. Relative Strength of Rectangular Beams. From an inspection of the foregoing formulas, it will be found that the relative strength of rectangular beams in different cases is as follows : — Beam supported at both ends, and loaded with a uniformly distributed load............................................1. Beam supported at both ends, and loaded at the centre ... i Beam fixed at one end, and loaded with a uniformly distributed ' load.........................................................i Beam fixed at one end, and loaded at the other . . ... . i Also the following can be shown to be true : — Beam firmly fixed at both ends, and loaded at the centre- . . 1 Beam fixed at both ends, and loaded with distributed load . . 1£ These facts are also true of a uniform beam of any form of cross-section. When a square beam is supported on its edge, instead of on its side, —that is, has its diagonal vertical, — it will bear about seven-tenths as great a breaking-load. The strongest beam which can be cut out of a round log is one in which the breadth is to the deptli as 5 to 7, very nearly, and can be found graphically, as shown in margin. Draw any diagonal, as ah. and divide it into three equal parts by the points c and d; from these points draw perpendicular lines, and connect the points e. and/ with a and b, as shown. Fig. 8. Cylin'düical Beams. — A cylindrical beam is only as812 STRENGTH OF WOODEN BEAMS. strong as a square beam whose side is equal to the diameter of the circle. Hence, to find the load for a cylindrical beam, first find the proper load for the corresponding square beam, and then divide it by 1.7. The hearing of the ends of a beam on a wall beyond a certain amount does not strengthen the beam any. In general, a beam should have a hearing of four inches, though, if the beam he very short, the bearing may be less. Weight of the Beam itself to be taken into Account. — The formulas we have given for the strength of beams do not take into account the weight of the beam itself, and hence the safe load of the formulas includes both the external load and the weight of the material in the beam. In small wooden beams, the weight of the beam is generally so small, compared with the external load, that it need not be taken into account. But in larger wooden beams, and in metal and stone beams, the weight of the beam should be subtracted from the safe load if the load is distributed; and if the load is applied at the centre, one-lialf the weight of the beam should be subtracted. The weight per cubic foot for different kinds of timber maybe found in the table giving the Weight of Substances, Part III. Tables for tlie strength of hard pine, spruce, and oak beams, are given below, for beams one inch wide. To find the strength of a given beam of any other breadth, it is only necessary to multiply the strength given in 'the table by the breadth of the given beam. Example. — What is the safe load at the centre for a yellow-pine beani, supported at both ends, 8 inches by 12 inches, 20 feet clear span ? Ans. From Table II., safe load for one inch thickness is 900 pounds. 900 X 8 = 7200 pounds, safe load for beam. For a distributed load, multiply these figures by 2. To find the size of a beam that will support a given load with a given span, find the safe load for a beam of an assumed depth one inch wide, and divide the given load by this strength. Example. —What size spruce beam will be required to carry a distributed load of 8040 pounds for a clear span of IS feet? >Dks. This load would correspond to a load of 4320 pounds at the e'entre of the beam. From the table we find that a beam 12 inches deep and 1 inch thick, 18 feet span, will support 720 pounds; and dividing the load, 4320 pounds, by 720, we have 6 for the breadth of the beam in inches: hence the beam should be 0 by 12 inches, to carry a distributed load of 8640 pounds with a span of 18 feet.TABLE 1L HARD-PINE BEAMS. Table of safe quiescent loads for horizontal rectangular beams of Southern hard pine, one inch broad, supported at both ends, and loaded at the centre. Depth of bourn. Span, 4 feet. Span, 0 feet. Span, 8 feet. Span, 10 feet. Span, 12 feet. Span, 14 feet. Depth of beam. Span, 16 feet. Span, 18 feet. Span, 20 feet. Span, 22 feet. Span, 24 feet. Span, 25 feet. ins. . lbs. lbs. lbs. lbs. lbs. lbs. ins. lbs. lbs. lbs. lbs. ibs. lbs 4. 500 333 250 • 200 166 142 6 281 250 225 204 187 180 5 7S1 521 . 390 312 260 223 7 383 340 306 278 255 245 6 1125 750 562 450 375 321 8 500 444 400 364 333 320 t 1531 1021 765 612 510 437 9 633 562 506 460 422 405 8 2000 1333 1000 800 666 571 10 781 694 625 568 520 500 • 9 2531 . 1687 1265 1012 843 723 .12 1125 1000 900 816 748 720 10 3125 2083 1562 1250 1042 893 14 1532 1361 1225 1112 1020 1080 12 4600 3000 2250 1800 1500 1285 15 1758 1562 1406 1278 1172 1125 14 6125 4083 3062 2450 2042 1750 16 2000 1778 1600 1456 1332 1280 15 7031 4687 3515 2812 2343 2007 18 2532 2248 2024 ' 1840 16S8 1620 16 8000 5333 4000 3200 2666 2285 20 3124 2776' 2500 2272 2080 2000 STRENGTH OF IIARD-P1NE BEAMS. 313OAK BEAMS. ^ Table of safe quiescent loads for horizontal rectangular beams of white oak, one inch broad, snppotted at both ends, and loaded at'the • , , centre. x Depth! of beam. Span, 4 feet. Span, 6 feet. Span, 8 feet. Span, 10 feet. Span, 12 feet. 1 Span, 1 14 feet, j Depth of beam. Span, 16 feet. Span, ljL feet. Span, 20. feet. Span, 22 feet. Span, 24 feet. Span, 25 feet i:iS. lbs. ♦ ibs. t. "7“— Ibs. Ibs. lbs. lbs. ins. lbs. lbs. Ibs. ibs. lbs. lbs. 4 420 '280 210 168* 140 120 6 236 210 189 172 . 157 151 5 036 437 328 263 -«£18 187 7 321 286 . 257 234* 214 206 6 945 630 472 378 315 270 8 420 373 336 305 280 269 7 41286 '857 | 643 515 428 367 9 531 472 '425 • 387 354 340 8 I Ms. Em ' 840 672 560. . 480 10 656 583 525 477 i 437 402 Si 1 |l 1417 1063 . , 850 . • 708 607 12 945 840 756 688 630 604 JO *• •^262o,; 1750 1312 10o04 ’ 875 748 14 1286 1143 1029 936 857 824 lV 37&0 : ' 2520 isoo 1512 . 1260 10S0 15 1476 1312 1181 1074 984 945 14 * I. /5146 3430 2572 # 2058 1715 1468 # ! 16 1680. 1493 1344 1220 1120 1076 ,15 •> ^ \ 1 1 • ;2362 1908 10S7 1 2124 1888 , 1700 1548 1416 - 1360 16 !■ \ 6720., 4480 3360 2688 224Qr ! : 1920 1 -M 2624 2332 2100 1908 1748 • 1008 STRENGTH OF OAK BEAMS.7 • & TABLE IV. SPSEJfiEr-SEAMS. r t Table of safe quiescent loads for horizontal rectangular beam« of spruce, one inch broad, supported at both ends, and loaded at the centre. Depth of Span, Span, Span, 8 feet. . • Span, Span, Span, Depth 1 of Span, . Span, Span, Span, Span, Span, beam. , 4 feet. . 6 feet. 10 feet. . 12 feet. 14 feet. beam. 16 fe$t. 18 feet. 20 feet. 22 feet. 24 feet. .25 feet. ins. lbs. lbs. lbs. lbs. • lbs. lbs. ! ins. lbs. ibs. Ibs. lbs. lbs. lbs. '4 360 240 180 144 120 103 6 202 * 18p, 162 147 135 130 5 : 562 375 . 281 225 187 161 7 255 ‘ 245 220 200 183 176 6. 81Q 540 405 324 270 230 8 360 320 288 262 240 ‘ 230 • 7 1 1102 73o 551 441 367 315 9 455 405 364 331 303 291 8 / 1440 mo 72Q 576. 4§0 411 10 562 500 450 409 ^ 375 360 , 1 1822 I \ ' on 72fr 007 520 12 810 720 648 „ M • . 588 540 520 10 2250. ’ lb00;‘ : lt24 m ... 1 750 . 644 • | i Pst. 1102 980 f‘-%32 ' 800 7^5 x 704 ■■ t^240 zip 162ft icj^ 920 ■ i$$§ V 12651 ■ "1125 ' i012 920 843 810 4108 . 2940 ;; >204 '^1764; : jM* 16 ' 1440 . 1280 1152 1048 960 920 15 ; ; $062 3375 V‘253l, ' 2025 Ip . 1820 1620 1456 1324 1212 1164 16 3840 2880 ;V^S9^'r' '; \f4 \ ■ 20 2248 2000 .1800 1636 1500 • r 1440 ‘ %■- V'-'1 K JV/ STRENGTH OF SPRUCE BEAMS. I 3 If)SOLID BUILT WOODEN BEAMS. ^olC ' The foregoing tables for strength of beams are computed from the data given in this chapter, taking the strength of the wood as given in the table for co-efficients. These tables are as accurate as can be computed with our present knowledge; and, as the co-efficients are based upon experiments on full-sized timbers, it would seem as though the tables should be almost absolutely correct. • Stone Beams. — The same formulas apply to stone as to wooden beams, oniy the values of the co-efficient A are only from one-sixth to one-tenth of breaking loads. Sandstone beams should never be subjected to any very heavy loads; but, where used as lintels, the stone should be relieved by iron beams or brick arches back of the stone. Solid Built Beams. [From Wheeler’s “Civil Engineering.”] . A solid beam is oftentimes required of greater breadth or thickness than that of any single piece of timber. To provide such ajaeam, it is necessary to use a combination of pieces, consisting of several layers, of timber laid in juxtaposition, and firmly fastened together by bolts, straps, or other means, so that, the whole shall act as a single piece. This is termed a solid built beam. When two pieces of timber are built into one beam having twice the depth of either, keyn of hard wood are used to resist the shearing-strain along the joint, as shown in Fig. 9. Tredgold' gives the rule, that the breadth of the key should be twice its depth, and the sum of the depths should be equal to once anil a third the total depth of the beam. It has been recommended to have the bolts and the keys on the right of the centre make an angle of forty-five degrees with the axis of the beam, and those on the .left to make the supplement of this angle. The keys are sometimes made of two wedge-shaped pieces, for the purpose of making them fit the notches more snugly, and, in cascof shrinkage in the timber, to allow of easy re-adjustment. When the depth of the beam is required to be less than the sum of the depths‘of the two pieces, they are often built into one by indenting them, the projections of the one fitting accurately into the notches blade in the other, and the. two firmly fastened together by bolts or straps. The built beam shown in Fig. 10 illustrates this method. In this particular example the beam tapers slightly from tile middle to the ends; so that the iron bands may be slipped on over the ends, and driven tight with mallets.SOLID BUILT WOODEN BEAMS. 317 When a beam is built of several pieces in length as well as in depth, they should break joints with each other. The layers below the neutral axis should be lengthened by the scarf or fish joints used *or resisting tension; and the upper ones should have the ends abut against each other, using plain butt joints. [zz-.zz: jl L rs [ j I L 1 r j I L = IN l 1 :::: 1 L i 1 i r j 1 J 1 l i r.... 1 1 — 1 L i 1 1 ' i Many builders prefer using a built beam of selected timber to a single solid one, on account of the great difficulty of getting the latter, when very large, free from defects: moreover, the strength of the former is to be relied upon, although it cannot be stronger than the corresponding solid one, if perfectly sound.318 STIFFNESS AND DEFLECTION OF BEAMS. CHAPTER XYI. STIFFNESS AND DEFLECTION OF BEAMS. In Chaps. XIY. and XY. we have considered the strength of beams to resist breaking only ; but in all first-class buildings it is desired that those beams which show, or which support a ceiling, should not only have sufficient strength to carry the load with safety, but should do so without bending enough to present a bad appearance to the eye, or to crack the ceiling : hence, in calculating the dimensions of such beams, we should not only calculate them with regard to their resistance to breaking, but also to bending. Unfortunately, we have at present no method of combining the two calculations in one operation. A beam apportioned by the rules for strength will not bend so as to strain the fibres beyond their elastic limit, but will, in many cases, bend more than a,due regard for appearance will justify. The amount which a beam bends, under a given load is called its deflection, and its resistance to bending is called its stiffness: hence the stiffness is inversely as the deflection. The rules for the stiffness of beams are derived from those for the deflection of beams; and the latter are derived partly from mathematical reasoning, and partly from experiments. We can find the deflection at the centre, of any beam not strained beyond the elastic limit, by the following formula: — load in lbs. X cube of span in inches X c Def. in inches — nioljuius 0f elasticity x moment of inertia' ^ ^ The values of c are as follows : — Beam supported at both ends, loaded at centre . “ “ “ uniformly loaded . “ fixed at one end, loaded at the other. . . “ “ “ uniformly loaded . . . 0.021 0.013 0.3:]:} 0.125 By making the proper substitutions in Formula 1, we derive theSTIFFNESS AND DEFLECTION OF BEAMS.. 819 following formula for a rectangular beam supported at both ends, and loaded at the centre: — load X cube of span X 1728 Def. m inches ■ , m.------... '/i.r**—.—-r—7,, (2) 4 X breadth X cube of depth X E ' the span being taken in feet. From this formula the value of the modulus of elasticity, K, for different materials, has been calculated. Thus beams of known dimensions are supported at each end, and a known weight applied at the, centre of the beam. Tliel deflect Bn of the beam is then carefully measured; and, substituting those known quantities in Formula 2, the value of E is easily obtained. , - 1728 formula 2 may be simplified somewhat by representing ^ by 1 p, which gives us the formula IF X /,* Def. in inches = j, x nifftr (3) For a distributed load the deflection will be five-eighths of this. Note.—The constant F corresponds to Hatfield’s F, In his Transverse Strains. If we wish to find the load which shall cause a given deflection, we can transpose Formula 2 so that the load shall form the left-hand member. Thus : — Load at centre _ 4 X breadth X cube of depth X def. in ins. X E in pounds — cube of span X 1728 Now, that this formula may be of use in determining the load to put upon a beam, the value of the deflection must in some way be fixed. This is generally done by making it a certain proportion of the span. 'Thus Tredgold and many other authorities say, that, if a floor-beam deflects more than one-fortieth of an inch for every foot of span, it is liable to crack the ceiling on the under side; and hence this is the limit which is generally given to the deflection of beams in first-class buildings. 'Then, if we substitute for “deflection” the value, length in feet y 40, in the above formula, we have, breadth X cube of depth X e Load at centre = —^uare of length---------------(5> letting e = Many engineers and architects think that one-tliirtieth of an inch . per foot of span is not too much to allow for the deflection of floorO320 STIFFNESS AND DEFLECTION OF BEAMS. beams, as a floor is seldom subjected to its full estimated load, and then only for a short time. If we adopt this ratio, we shall have as our constant for deflec- E Don, e, — 12960" In either of the above cases, it is evident that the values used for E, F, e, or e,, should be derived from tests on timbers of the same size and quality as those to be used. It has only been within the last three or four years that we have had any accurate tests on the strength and elasticity of large timbers, although there had been several made on small pieces of various woods. The values of the various constants for the first three woods in the following table have been derived from tests made by Professor Lanza and his students at the Massachusetts Institute of Technology, and the values for the other woods are about six-sevenths of the values derived from Mr. Hatfield’s experiments. The author believes that the values given in this table may be relied upon for timber such as is used in first-class construction. TABLE I. Values of Constants for Stiffness or Deflection of Beams. E ~ Modulus of elasticity, pounds per square inch. F = Constant for deflection of beam, supported at both ends, and loaded at the centre. e = Constant, allowing a deflection of one-fortietli of an inch per foot of span. et = Constant, allowing a deflection of one-thirtieth of an inch per foot of span. Material. E. E *=«•■ E e~ 17280" E ei “ 12960' Cast iron 15,700,000 36,300 907 1210 Wrought-iron .... 26,000,000 60,000 1500 2000 .Steel 31,000,000 71,760 1794 2358 Yellow pine 1,780,000 4,120 103 137 Spruce 1,294,000 3,000 75 100 White oak 1,240,000 2,870 72 95 White pine 1,073,000 2,4S0 62 82 Hemlock 1,045,000 2,420 60 80 Whilewood 1,278,000 2,960 74 98 Chestnut 944,(MX) 2,180 54 1,482,000 3,430 86 114 Maple 1,902,000 4,400 110 146STIFFNESS AND DEFLECTION OF BEAMS. 321 Rules for Stiffness of Beams. Knowing. the deflection caused by a weight at the centre of a beam, and the ratio of other deflections, caused hy different modes of loading and supporting, we can easily deduce the formulas for the different eases considered under the strength of rectangular beams. These cases are — Beams Suppohted at Both Ends. Loaded at the centre, breadth X cube of depth X e »Safe load =--------------77-----I------, (0) square of length ' ' or, load X square of length Breadth =------;—, ,, v---• (7) cube ol depth X e ' ' Loaded at a point other than the centre, m and n being the segmentx into which the beam ix divided, breadth X cube of depth X square of length X' e Safe load = —■--------------—2--------------------------- , (8) 10 X nr X ir or, load x m2 X n* 1 X Hi ~~ cube of depth X square of length X e ^ Load uniformlg distributed, 8 X breadth X cube of depth X e Safe load —-------BH--------I,----I------, (10) 5 X square of length ' ' or, 5 x load x square of length Breadth = -—,—7-, v-----------------• (11) 8 X cube ol depth X« ' > Inclined beam, loaded at the centre,' breadth X cube of depth X e Safe load — x j10r_ tnst,. between supports’ or, load x length x hor. dist. between supports Breadth I --------------7—rj—b—3-------:----------• (13) cube of depth X e ' ' Beams Fixed at One End. Loaded at extreme end, breadth X cube of depth X e Safe load ■---------------——zirasSBPWfc# (14) 10 X square of length * ' v 1 1 Tredgold’e Elements of Carpentry, p. 65.322 • STIFFNESS AND DEFLECTION OF BEAMS, or, 1(5 X load X square of length readtli — cube of depth x e Load uniformly distributed, breadth X cube of depth X e Safe load = ----ttt:-------H----it---’ (> X squaie ot length or, Ox load X square of length Breadth =-------,--v.—.—-.—I—• cube ot depth X e Tables. Tables II., III., and IV. have been prepared so as to show at a glance the greatest load that a beam one inch thick will support without exceeding the limit of deflection or the safe strength. They give the same results as would be obtained by using the above formulas. (15| (10) (IT) Ratio of the Stiffness of Beams. If the stiffness of a beam, supported at both ends, and loaded at the centre be called.........................................1 Then that of the same beam, with the same load uniformly distributed, will be.........................................$ Firmly fixed at both (aids, and loaded at the centre, according to Moseley...................................................5 Firmly fixed at both ends, and uniformly loaded.................8 Fixed at one end, and loaded at the other....................tV Fixed at one end, and uniformly loaded....................... The stiffest rectangular beam containing a given amount of material is that in which the ratio of depth to breadth is as 10 to 6: hence, in designing beams, the depth and breadth should be made to approach as near this ratio as is practicable. Example 1. — What, is the greatest distributed load that an 8 by 10 inch wliite-pine girder of 12 foot clear span will support, without deflecting at the centre more than V 14,002 9,374 7030 5624 4686 3780 2889 2281 18.50 1525 1282 15 10 10,000 10,006 * 8000 6400 5332 4570 3506 2709 2244 1851 1536 16 Loads above horizontal lines are calculated by formula for stiffness ; those betoWj by formula for .strength. STIFFNESS AND DEFLECTION OF BEAMS.. 323/V fi# f/Ò fJ-r 0 'V' / ?// /-fniA' tr. k „ r ^ cr>vv et.- WCk Sj ^-V^H TABLE Haï. SPRUCE BEAMS. * /Ut?u rv* /2 Table of maximum distributed loads which can be supported by horizontal rectangular beams of spruce timber, one inch broad, and supported at both ends, with safety and without deflecting more than one-thirtieth of an inch per foot of span. Depth of beam. Span, 4 feet. Span, 6 feet. Span, 8 feet. Span, 10 feet. CO O £ fD — Span, 14 feet. Span, 16 feet. Span, 18 feet. Span, 20 feet. Span, 22 feet. Span, 24 feet. Depth - of beam. ins. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. * lbs. bs. lbs. ins. 0 1,620, 960 540, 345 240 176 135 106 86 71 60 6 1 2,880 \ /.1920 1280 819 569 . 417 320 251 205 170 142 8 9 3,644 2430 1822 1166 810 584 455 '359 292 240 202 9 10 4,500 3000 2248 1600 1141 816 625 « 493 400 330 277 10 12 [ 6,480 4320 4240 2592 1920 Hilo 10Ç0 <851; 691 ‘ JV 57Q 480 12 i4*V 8,816 5880 4408 3528 2239 ' 1715 1352 1098 905 762 p 14 1) 10,124 . 6750 5062 4050 3374 2734 2109 ' 1663 1350 1114 935 15 16 11,520 7680 576Ô 4008 3840 3288 2560 2019 1638 1352 1134. 1 16 0 T Loads above horizontal lines are calculated by formula for HtjffyievH,* those below, by formula for ntrength. 324 STIFFNESS AND DEFLECTION OF BEAMS.TABLE n\ OAK BEAMS. Table of maximum distributed loads which can be supported by horizontal rectangular beams of white oak, one inch broad, and supported at both ends, with safety and without deflecting more than one-thirtieth of an inch per foot of span. Depth of Span, Span, Span, Span, 10 feet. Span, 12 feet. Span, 14 feet. ' Span, Span, 18 feet. Span, 20 feet. Span, Span, 24 feet. Depth beam. 4 feet. 6 feet. 8 feet. 16 feet. 22 feet. beam. ins. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. It)8. ins. 6 1,890 911 511 328 226 . 167 128 101 82 68 57 6 8 3,360 2160 1213 778 537 397 303 240 194 160 * 135 8 9 4,252 2834 1727 110.8 765 565 432 ■ 342 277 229 192 9 10 5,250 . 3500 2370 1520 1050 775 593 470 380 314 264 10 12, 7,560 5040 3780 2626 1814 1339 1Q24 812 656 542 456 12 14 10,290 6860 5144 4170 2881 2126 1627 1289 1042 862 724 14 15 11,812 7874 5906 4724 3543 2615 2001 1586 1282 1060 891 15 16 13,440 - 8960 6720 5376 ’ 4480 3174 2429 1925 1556 1286 1081 16 Loads above horizontal lines are calculated by formula for stiffness; those below, by formula for strength. STIFFNESS AND DEFLECTION OF BEAMS. 325326 . STIFFNESS AND- DEFLECTION OF BEAMS Example 2. —What should be the dimensions of a yellow-pine beam of 10 foot span, to support a concentrated ioad of 4250 pounds, without deflecting more than £ of an inch at the centre ? A ns. A deflection of £ of an inch in a span of 10 feet is in the proportion of jV, of an inch per foot of span; and as the load is concentrated, and applied at the centre, we should use Formula 7, employing for e the value given in the fourth column, opposite yellow pine. Formula 7 gives the dimensions of the breadth, and to obtain it we must assume a value for the depth. For this we will first try 8 inches. Substituting in Formula 7, we have, 4250 x 100 Breadth = 5p-> x 137 = ® inches, nearly. This would give us a beam 6 by 8 inches. Example 3. — What is the largest load that an inclined spruce beam 8 by 12 inches, 12 feet long between supports, will carry at the centre, consistent with stiffness, the horizontal distance between the supports being 10 feet ? Ans. Formula 12 is the one to be employed, and we will use the value of e given in the third column, opposite spruce. Making the proper substitutions, we have, 8 x 1728 X 75 Safe load = —f^ x To— = P°unds. Cylindrical Beams. For cylindrical beams the same formulas may be employed as for rectangular beams, only, instead of c, use 1.7 X e; that is, a cylindrical beam bends 1.7 times as much as the circumscribing rectangle. Deflection of Iron Beams. For rolled-iron beams the deflection is most accurately obtained by Formula 1. The following approximate formula gives the deflections quite accurately for the maximum safe loads, square of span in feet Deflection in inches = 7,, x the depth of beam' The deflections for the Phoenix, Peneoyd, Trenton, and Union Iron-Mills beams, are given In the tables for strength of beams, in Chap. XIY. In using iron beams, it should be remembered that the deepest beam is always the most economical; and tbe stiffness of a floor is always greater when a suitable number of deep beams are used.CONTINUOUS GIRDERS.. 327 CHAPTER XVII. STRENGTH AND STIFFNESS OF CONTINUOUS GIRDERS "Girders resting upon three or more supports care of quite frequent occurrence in building construction; and a great variety of opinions is held as to the relative strength and stiffness of continuous and non-continuous girders; very few persons, probably, having any correct knowledge of the subject. In almost every building of importance, it is necessary to employ girders resting upon piers or columns placed from eight to fifteen feet apart; and in many cases girders can conveniently be obtained which will span two and even three of the spaces between the piers or columns. When this is the case, the question arises, whether it will be better construction to use a long continuous girder, or to have each girder of only one span. Most architects are probably aware that a girder of two or more spans is stronger and stiffer than a girder of the same section, of only one span, but just how much stronger and stiffer is a question they are unable to answer. As it is seldom that a girder of more than three spans is employed in ordinary buildings, we shall consider only these two cases. In all structures, the first point which should be considered is the resistance required of the supports; and we will first consider the resistance* offered by the supports of a continuous girder. In this chapter we shall not go into the mathematical discussion of tin* subject, but refer any readers interested in the derivation of the formulas for continuous girders to an article on that subject, by the author, in the June (1881) number of Van Nostrand’s “ Engineering Magazine.” Supporting* Forces. Girder.s of Tivo Spans, loaded at the Centre of Each Span. — If a girder of two spans, l and is loaded at the centre of the span l328 CONTINUOUS GIRDERS. with IT pounds, and at tlio centre of /, with IK, pounds, the re-action of the support It, will be represented by the formula the re-action of the support I?a by It, = ^(1K+1K,), (2) and the re-action of the support It4 by the formula 13 IK, — i W 4 I---------32------• u If 1K = IK,, then each of the end supports would have to sustain Yfc of one of the loads, and the centre support V* of IK. Were the girder cut so as to make two girders of one span each, then the end supports would carry i or IK, and the centre support Is IK; hence we see, that, by having the girder continuous, we do not require so much resistance from the end supports, but more from the central support. Fig t Girder of Two Spans, uniformly Distributed Load over Each Span. — Load over each span equals w pounds per unit of length, lie-action of left support, ?»r k. = aU Re-action of central support, Re-action of right support, lf + l31 4 Ul + f)]- (4) 1 r3 1 w (5) ifl-w3 1 41, (l + /,)J’ (6) When both spans are equal to l, the re-action of each end support, is i wl\, and of the central support tol: hence the girder, by being continuous, reduces the re-action of the end supports, and increases (ha't of the central support by one-fourtli, or twenty-five per cent.CONTINUOUS GIRDERS. 3*29 Continuous Girder of Three Equal Sjmns, Concentrated Load oj W Pounds at Centre of Each Span. Re-action of either abutment, Ii i — — is',) ID; - (7) Re-action of either central support, R, = Hi = |3 lU; (8) or the re-action of the end supports is lessened three-tenths, and that of the central supports increased three-twentieths, of that which they would have been, had three separate girders of the same cross-section been used, instead of one continuous girder. Continuous Girder of Three Equal Spans uniformly loaded with w Pounds per Unit of Length. Re-action of either end support, if,* P^-lwl; (9) Re-action of either central support, Ii2 = = (10) hence the re-actions of the end supports are one-fifth less, and of the central supports one-tenth more, than if the girder were not continuous. Strength of Continuous Girders. — Having determined the reaction of the supports, we will now consider the strength of the girder. The strength of a beam depends upon the material and shape of the beam, and upon the external conditions imposed upon the beam. The latter give rise to the bending-moment of the beam, or the amount by which the external forces (such as the load and supporting forces) tend to bend and break the beam. it is this bending-moment which causes the difference in the bearing-strength of continuous and lion-continuous girders of the same cross-section. Continuous Girders of Two Spans.—When a rectangular beam is at the point of breaking, we have the following conditions :— I Rendim?- Mod. of rupture X breadth X scp of depth_ moment — [ 6 ~ E * and, that the beam may carry its load with perfect safety, we must divide the load by a proper factor of safety.330 CONTINUOUS GIRDERS. Hence, if we can determine the bending-moment of a beam under any conditions, we can easily determine the required dimensions of the beam from Formula 11. The greatest bending-moment for a continuous girder of two spans is .almost always over the middle support, and is of the opposite kind to that which tends to break an ordinary beam. Distributed Load. — The greatest bending-moment in a continuous girder of two spans, Z and Z1, loaded with a uniformly distributed load of to pounds per unit of length, is xoP + wl,* Bending-moment = -g ^ f. i \ ’ (12) V/lien l =■ l [, or both spans are equal, to!'2 Bending-moment = -g-, (12«) which is the same as Hie bending-moment of a beam supported at both ends, and uniformly loaded over its whole length: hence a continuous yirder of two spans uniformly loaded is no stronyer than if non-continuous. Concentrated Load. — The greatest bending-moment in a continuous girder of two equal spans, each of length Z, loaded with \V pounds at centre of one span, and with IF, pounds at the centre of the other span, is • Bending-moment = 33^ Z (IF + ll'i). (13) When IT = 11'I or the two loads are equal, this becomes Bending-moment = tk IT Z, (13«) or one-fourth less than what it would be were the beam cut at the middle support. Continuous Girder of Three. Spans, Distributed Load. — The greatest bending-moment in a continuous girder of three spans loaded with a uniformly distributed load of w pounds per unit of length, the length of each end span being Z, and of the middle span Z, is at either of the central supports, and is represented by the formula, icZ3 4- trZi8 Bending-moment = (14) When the three spans are equal, t his becomes wl- Bending-moment = -Jq, (14«) or one-fiftli less than what it would be were the beam not continuous.CONTINUOUS GIRDERS. 331 Concentrated Loads. — The greatest bending-moment in a continuous girder of three equal spans, each of a length l, and each loaded at the centre with IK pounds, is Bending-moment r= 11' l, (15) or two-fifths less than that of a non-continuous girder. Deflection of Continuous Girders. Continuous Girder of Two Ei/uid Spans. — The greatest deflection of a continuous girder of two equal spans, loaded with a uniformly distributed load of w pounds per unit of length, is Hj Deflection = 0.00541(3 (16) (E denotes modulus of elasticity; 7, moment of inertia;) The deflection of a similar beam supported at both ends, and uniformly loaded, is 1014 Deflection = 0.013020 Hence the deflection of the continuous girder is only about two-fifths that of a non-continuous girder. The greatest deflection in a continuous girder is also not at the centre of either span, but between the centre and the abutments. The greatest, deflection of a continuous girder of two equal spans, loaded at the centre of one span with a load of IK pounds, and at the centre of the other span with IK, pounds, is, for the span with load 11 , Deflection = (23 IK —9 IK,) Is 1536 El ’ (17) for the span with load IK,, Deflection (23 IK, —9IK) P 1536 El (17a) When both spans have the same load, 7 TKf3 Deflection = sttx -Htf • 768 El (17 b) The deflection of a beam supported at both ends, and loaded at the centre with IK pounds, is IK Is Deflection = jgTicv or the deflection of the continuous girder is only seven-sixteenths of the non-continuous one.332 CONTINUOUS GIRDERS. Continuous Girder of Three Equal Spans. — Uniformly distributed load of w pounds per unit of length, wli Deflection at centre of middle span = 0.00052 (18) Greatest deflection in end spans = 0.00(5884 -jjjf (19) or the greatest deflection in the girder is only about one-lialf that of a non-continuous girder. Concentrated load of )V pounds at centre of each span, Deflection at centre of middle span = 4$q~ei~ (20) 11 WP Deflection at centre of end spans = ofioTuT (21) or only eleven-twentieths of the non-continuous girder. Several Observations and Formulas for Designing: Continuous Girders. From the foregoing we can draw many observations and conclusions, which will be of great use in deciding whether it will be best in any given case to use a continuous or non-continuous girder. First as to the Suj>ports. — We see from the formulas given for the re-action of the supporting forces in the different cases, that in all cases the end supports do not have as much load brought upon them when the girder is continuous as when it is not; but of course the difference must be made up by the other supports. This might often be desirable in buildings where the girders run across the building, the ends resting on the side walls, and the girders being supported at intermediate points by columns or piers. In such a case, by using a continuous girder, part of the load could be taken from the walls, and transferred to the columns or piers. lint there is another question to be considered in such a case, and that is vibration. Should the building be a mill or factory in which the girders had to support machines, then any vibration given to the middle span of the beam would be carried to the side walls if the beam were continuous, while if separate girders were used, with their ends an inch or so apart, but little if any vibration would be carried to the side walls from the middle span. In all cases of important construction, the supporting forces should be carefully looked after. • Strength, — As the relative strength of continuous and non-continuous girders of the same size and span, And loaded in the same way, is as their bending-moments, we can easily calculate theCONTINUOUS GIRDERS. 333 strength of a continuous girder, knowing the formulas for its bending-moment. From the values given for the bending-moments of tin; various cases considered, we see that the portion of tliegirdef' most strained is that which comes over the middle supports; also that, except In the single case of a girder of two spans uniformly loaded, the strength of a girder is greater if it is continuous than if it is not. I »ut the gain in strength in some instances is not Very great, although it is generally enough to pay for making the girder continuous. Stiffness. —The stiffness of a girder is indirectly proportional to its dellection; that is, the less the deflection under a given load, the stitfer tlie girder. Now, from the values given for the deflection of continuous girders, we see that a girder is rendered very much stitfer by being made continuous ; and this may be considered as the principal advantage in the use of such girders. It is often the case in building-construction, that it is necessary to use beams of much greater strength than is required to carry thi! superimposed load, because the deflections would be too great if the beam were made smaller. But, if we can use continuous girders, we may make the beams of just the size required for strength; as the deflections will be lessened by the fact of the girders being continuous. It should therefore be remembered, that, where great stillness is required, continuous beams or girders should he used if possible. Formula for Strength and Stiffness. For convenience we will give the proper formulas for calculating I lie strength and stillness of continuous girders of rectangular cross-section. The formulas for strength are deduced from the formula, r , BX D2 X R Bending-moment = —------^------’ (22) where It is a constant known as the modulus of rupture, and is eighteen times what is generally known as the co-eflieient of strength. SriiuxfiTn. — Continuous girder of two equal sjtans, loaded uniformly over each .spun, 2 X B X D2 X A Breaking-weight = --------j-------’ (23) where B denotes tins breadth of the girder, I) the depth of the girder (both in inches), and L the length of one span, in feet. The334 CONTINUOUS GIRDERS. values of the constant A are three times the values given in- Table: I., p. 310. For yellow pine, 375 pounds; for spruce, 270 pounds; and for white pine, 240 pounds, — may be taken as reliable values for A. Continuous girder of two equal spans, loaded equally at the centre of each span, 4 lì x If x A Breaking-weight — 3 x ------f------* (24) Continuous girder of three equal spans, loaded uniformly over each sjian, 5 li x D 2 X A Breaking-weight = ^ x ------7------1 (25) Continuous girder of three equal spans, loaded equally at the centre of each span, 5 lì x 7)2 X A Breaking-weight = 0 X---------y-----• (26) Stiffness. — The following formulas give the loads which the .beams will support without deflecting more than one-thirtieth of an inch per foot of span. Continuous girder of TWO equal spans, loaded uniformly over each spun, Iix If x e Load on one span = -q-^, x l2 ' (27) Continuous girder of two equal spans, loaded equally at centre of each span, 16 lì X D3 x e Load on one span = y X----y-j---• (2S) Continuous girder of three equal spans, loaded mtiformly over each span, , /»’ X D * X e Load on one span = -7..,.,,. • U.00 X L* (29) Continuous girder of three equal spans, loaded equally at the centre of each span, 20 77 X If x c (30) Load on one span — y. X L'1 The value of the constant e is 12,960, divided by tile modulus of elasticity; and, for the three woods most commonly used as beams, the following values may be taken : — Yellow pine, 137; white pine, 82; spruce, 100.CONTINUOUS GIRDERS. 335 For iron beams we may find the bending-moment by the formulas given, and, from tallies giving the strength and sections of rolled beams, find the beam whose moment of inertia =■ bending-moment X depth of beam 2000 when the bending moment is in foot pounds. For example, we have a continuous 1-beam of three equal spans, loaded over each span, with 2000 pounds per foot, distributed. Each span being 10 feel, then, from formula 14«, we have 2000 X 100 Bending-moment =--------jÿ----- = 20000. 20000 Moment of inertia = X depth of beam; 20,000 .- 2000 = 10, and we must find a beam whose depth multiplied by ten will equal its moment of inertia. If we try a ten-inch beam, we should have 10 X 10 = 100; and we see from Tallies, pp. 200-272, that no ten-inch beam has a moment of inertia as small as 100: so we will take a nine-inch beam. 9 X 10 = 90, and the lightest nine-inch beam has a moment of inertia of 93: so we will use that beam. In the case of continuous I-beams of three equal spans, equally loaded with a distributed load, we may take four-fifths of the load on one span, and find the iron beam which would support that load if with only one span. Kxamui.k.— if we have an I-beam of three equal spans of 10 feet eaeli loaded with 20,000 pounds over each span, what size beam should we use ? An$, I of 20,o00 = 10,000. The equivalent load for a span of one foot would lie 10,000 X 10= 100,000. From Tables, Chap. XIV., we find that the beam whose co-efficient is nearest to this is the nine-inch light beam, — the same beam which we found to carry the same load in the preceding example. For beams of two equal spans loaded uniformly, the strength of the beam is the same as though the beam were not continuous. The formulas given for the re-actions of the supports and for the dellections of continuous girders with concentrated loads, were verified by the author by means of careful experiments on small steid bars. The other formulas have been verified by comparison with other authorities, where it was possible to do so; though one or two of the cases given, the author has never seen discussed in any work on the subject.336 FLITCH PLATE GIRDERS. CHAPTER XVIII. FLITCH PLATE GIRDERS. In framing large buildings, it often occurs that the floors must be supported upon girders, which themselves rest upon columns; and it is required that the columns shall he spaced farther apart than would be allowable if wooden girders were used. In such cases the Flitch Plate girder may be iron plate used, oftentimes with advan- The different pieces are bolted together every two feet by three-fourths-inch bolts, as shown in elevation. It has been found by practice that the thickness of the iron plate should he about one-twelfth of the whole thickness of the beam, or the thickness of the wood should be eleven times the thickness of the iron. As the elasticity of iron is so much greater than that of wood, we must proportion the load on the wood so that it shall bend the same amount as the iron plate: otherwise the whole strain might be thrown on the iron plate. The modulus of elasticity of wrought-iron is about thirteen times that of hard pine; ora beam of hard pine one inch wide would bend thirteen times as much as a plate of iron of the same size under the same load. Hence, if we want the hard-pine beam to bend the same as the iron plate, we must put only one-thirteenth .as much load on it. If the wooden beam is eleven times as thick as the iron one, we should put eleven-thirteenths of its safe load on it, or, what amounts to the same thing, use a constant only eleven-thirteenths of the strength of the wood. On this basis the following formulas have been made up for the strength of Flitch Plate girders, in which the thickness of the iron is one-twelftli of the breadtli of the beam, approximately : — tage. A section and elevation of a Flitch Plate girder is shown in Fig. 1. ■ Fig. t.FLITCH FLATE GIRDERS. 337 Let I) — Depth of beam. B = Total thickness of wood. L — Clear span in feet. I = Thickness of iron plate. f _ i 100 pounds for hard pine. I 73 pounds for spruce. 11' = Total load on girder. 77/0), for beauts supported at both ends, Safe load at centre, in pounds 7)2 — (y7?-f*7501), (1) Safe distributed load, in pounds 2D2 — ^ i/B + 750«). (2) For distributed load, _ H»u \2J'B + 1500«' (3) For load at centre 1 ]VL D~\fB + 7501' (4) As an example of the use of this kind of girder, we will take the case of a railway-station in which the second story is devoted to olliees, and where we must use girders to support the second floor, of twenty-five feet span, and not less than twelve feet on centres, if we can avoid it. This would give us a floor area to be supported by tin* girder of 12 X 25 = 300square feet; and, allowing 105 pounds per square foot as the weight of the superimposed load and of the floor itself, we have 31,500 pounds as the load to be supported by the girder. Now we lind. by computation, that if we were to use a solid girder of hard pine, it would require a seventeen-inch by fourteen-inch beam. If we were to use an iron beam, we find that a lifteen-ineh heavy iron beam would not have the requisite strength for this span, and that we should be obliged to use two twelve-inch beams. We will now see what size of Flitch Plate girder we would require, should we decide to use such a girder. We will assume I he total breadth of both beams to be twelve inches, so that we can use t wo six-inch timbers, which we will have hard pine. The thickness of the iron will be one inch and one-eighth. Then, substituting in Formula 3, we have, / 31500 X 25 ,— ^ ~ \ 2 X 100 X 12 + 1500 X 1J “ V192, or 14 inches. Hence we shall require a twelve-inch by fourteen-inch girder. Now,338 FLITCH-PLATE GIRDERS. for a comparison of the cost of the three girders we have considered in this example. The seventeen-inch by fourteen-inch hard-pine girder would contain 515 feet, board measure, which, at five-cents a foot, would amount to $25.75. Two twelve-inch iron beams 25 feet 8 inches long will weigh 2083 pounds; and, at four cents a pound, they would cost 883.32. The Flitch-Plate girder would contain 304 feet, board measure, which would cost $18.20. The iron plate, would weigh 1312-J pounds, which would cost $52.50; making the total cost of the girder $70.70, or $13 less than the iron beams, and $45 more than the solid hard-pine beams. Flitch-Plate beams also possess the advantage that the wood almost entirely protects the iron; so that, in case of a fire, the heat would not probably affect the iron until the wooden beams were badly burned.TRUSSED BEAMS« 339 CHAPTER XIX. TRUSSED BEAMS Whenever we wish to support a floor upon girders having a span of more than thirty feet, we must use either a trussed girder, a riveted iron-plate girder, or two or more iron beams. The cheapest and most convenient way is, probably, to use a large wooden girder, and truss it, either as in Figs. 1 and 2, or Figs. 8 and 4. I n all t hose forms, it is desirable to give the girders as much depth as the conditions of the ease will permit; as, the deeper the girder, the less strain there is in the pieces. In the belly-rod truss we either have two beams, and one rod which runs up between them at the ends, or three beams, and two rods running up between the beams in the same way. The beams should be in one continuous length for the whole span of the girder, if they can be obtained that length. The requisite dimensions of the tic-rod, struts, and beam, in any given case, must be determined by first finding the stresses which come upon these pieces, and then the area of eross-seetion required to resist these stresses. Fou si no i.k strut i»lll v-KOD trusses, such as is represented by Fig. f, the strain ujion the pieces may be obtained by the following formulas : — For DiSTRinuTED load IF over ivhole girder, Tension in T 8 „ length of T 10 ^ X length of C‘ U) Compression in C — tt IF. 3 Compression in B = Jq IF ^ X length of C length of B (3) (2)340 TRUSSED BEAMS. For concentrated load W over C, Tension in T W length of T — 2 X length of B (4) c- IF. ir length of B B = ¥ x length of C (5) For girder trussed as represented in Fig. 2 under a distributed load W over whole girder, 3 length of S Compression in S = ^ W X Tension in R m* 3 length of B Tension in B = JqW X jensth 0fê (6) (7) For concentrated load, W at centre, W length of S Compression in 8 = y X length of jj- Tension in R Tension in B = IF. W length of B ~ 2 X length of C (8) (9) For double strut belly-rod truss (Fig. 3), with distributed load W over whole girder, Compression in C = 0.367 IF. length of B Comp, in B or D = 0.367 W X ]eno.th 0f q- (IDTRUSSED BEAMS. 341 For CONCENTRATED LOAD Tension in T Compressicm in C m over each of the struts C, _ „ .length of T ~~ ^ X length of O’ = m Comp, in B or tension in J) — IT x length of B length of O* (12) 03) For girder trussed, as in Fig. 4> under a distributed load II' over whole girder, 2 3 _ . _ length of 8 Compression in S = 0.367 TF X <14> Tension in R = 0.367 IK. _ . . ^ length of» ,. ^. Tension in B or comp, in D = 0.367 W X ,——tt* (h>) ^ length of R Under concentrated loads W applied at 2 and 3. _ .... length of 5 Compression in S = Wm i—Ti—rr/ 1 length of R Tension in R = IK. __ length of B Tension in B or comp. In D = IK K ;--n—ft»* 1 . length of R (10) (17) Trusses such as shown in Figs. 3 and 4 should he divided so that the rods R, or the struts C1, shall divide the length of the girder into three equal or nearly equal parts. The lengths of the pieces 7\ ff, B, R, S, etc., should be measured on the centres of the pieces. Thus the length of R should be taken from the centre of the tie-beam B to the centre of the strut 1); and the length of C should be measured from the centre of the rod to the centre of the strut-beam B. After determining the strains in the pieces by these formulas, we may compute the area of the cross-sections by the following rules : — comp, in strut Area of cross-section of strut =-------------• (18) Diameter of single tie-rod /tension in rod ~ \ 9425 (19) 1 Allowing 12,000 pounds safe tension per square inch in the rod.342 TRUSSED BEAMS. Diameter of each of two tie-rods 1= For the beam B we must compute its necessary area of cross-section as a tie or strut (according to which truss we use), and also the area of cross-section required to support its load acting as a beam, and give a section to the beam equal to the sum of the two sections thus obtained. Area of cross-section of B to [ _ tension comp. resist tension or compression ( T 01 C ' * In trusses 1 and 2, In these formulas, C = 1000 pounds per square inch for hard pine and oak, 800 pounds per square inch for spruce, 700 pounds per square inch for white pine, 13.000 pounds per square inch for cast-iron. T = 2000 pounds per square inch for hard pine, 1800 pounds per square inch for spruce, 1500 pounds per square Inch for white pine, 10.000 pounds per square inch for wrought-iron. A — 125 pounds per square inch for hard pine, 105 pounds per square inch for oak, 90 pounds per square inch for spruce, 80 ponds per square inch for white pine. Examples. — To illustrate the method of computing the dimensions of the different parts of girders of this kind, we will take two examples. 1. — Computation for a girder such as is shown in Fig. 1, for a span of 30 feet, the truss to be 12 feet on centres, and carrying a floor for which we should allow 100 pounds per square foot. The girder will consist of three strut-beams and two rods. We can allow the belly-rod 7' to come two feet below the beams B, and we will assume that the depth of the beams B will be 12 inches; then till) length of C (which is measured from the centre of the beam) would be 30 inches. The length of B would, of course, be 15 feet, ami by computation, or by scaling, we lind the length of T to be 15 feci 2^ inches. Breadth of B (as a beam) = IF X L 2 X D- X A' (22) In trusses 3 and 4, 2 x IF X L Breadth of B (as a beam) = 5 x D- x A ‘TRUSSED BEAMS. 343 The total load on the girder equals the span multiplied by the distance of girders on centres, times 100 pounds = 30 X 12 X 100 = 30000 pounds. Then we find, from Formula 1, _ , . 1824 inches Tension in rod = of 30000 X 30 incites = pounds; and, from Formula 20, /05004 Diameter of each rod = XjlSS-'B = inches, nearly. The strut-beams we will make of spruce. The compression in the two strut-beams = ^ of 30000 X -'/u0 = 04S00 pounds, or 21000 pounds for each strut. To resist this compression would require 21000 , or 27 square inches of cross-section, which corresponds to a beam 2| inches by 12 inches. The load on 71 — J of 30000, or 18000 pounds: and, as there are three beams, this gives but 0000 pounds’ load on each beam. Then, from Formula 22, 0000 X 15 li 2 X 144 X 00 * inches, and, adding to this the 2 j inches already obtained for compression, we have for the strut-beams three 55-inch by 12-inch spruce beams. The load on C — | (F, or 22500 pounds. If we are to have a number of trusses all alike, it would be well to have a strut of cast-iron; but, if we are to build but one, we might make the strut of oak. If 22500 of cast-iron, the strut should have pwjy» 01' 1-3 square inches of cross-section at its smallest section, or about 1 inch by 2 inches. If 22500 of oak, it would require a section equal to or 224 square inches, = 44 inches by 5 inches, at its smallest section. Thus we* have found, that for our truss we shall require three strut-beams 0 inches by 12 inches (of spruce), about 31 feet long, two belly-rod? II inches diameter, and a cast-iron strut 1 inch by 2 inches at the smallest end, or else an oak strut 44 inches by 5 inches. 2. — It is desired to support a floor over a lecture-room forty feet wide, by means of a trussed girder; and, as the room above is to be used for electrical purposes, it is desired to have a truss with very little iron in it, and hence we use a truss such as is shown in Fig. 4. Where the girders rest on the wall, there will be brick pilasters having a projection of six inches, which will make the span of the truss 39 feet; and we will spape the rods Ii li so as to divide the tie-beam into three equal spans of In fect each. The tie-beam will344 TRUSSED BEAMS. consist of two liaril-pine beams, with the struts coming between them. We will have two rods, instead of one, at /?, coming down each side of the strut, and passing through an iron casting below the beams, forming supports for them. The height of truss from centre to centre of timbers we must limit to IS inches, and we will space the trusses S feet on centres. Then the total floor-area supported by one girder equals S feet by 39 feet, equal to 312 square feet. The heaviest load to which the floor will be subjected will be the weight of students, for which 75 pounds per square foot will be ample allowance; and the weight of the floor itself will be about 25 pounds; so that the total weight of the floor and load will be 100 pounds per square foot. This makes the total weight liable to come on one girder 31,200 pounds. Then we find, Formula 14, 157 ins. Compression in struts = 0.3G7 W X — 106S00 pounds. 156 ins. Tension in both tie-beams = 0.36711’ X w . = 106000 pounds. lb ms. ^ Tension in both rods It = 0.367 IF = 11450 pounds. The timber in the truss will be hard pine, and hence we must have 106800 —100(T ’ 01 square inches, area of cross-section in the strut, which is equivalent to a 9-incli by 12-incli timber; or, as that is not a merchantable size, we will use a 10-incli by 12-incli strut. The tie-beams will each have to carry one-half of 106000, or 53000 53000 pounds ; and the area of cross section to resist this equals 2qqo ~ 27 inches, or 2$ inches by 12 inches. The distributed load on one section of each tie-beam coming from the floor-joist equals 13 X 8 X 100 = 10400 pounds; and from Formula 23 we have 2 X 11’ X 1j 2 X 10400X13 «ll = ■; x ])- x A I 5 x 144 x120— = *nc^ies‘ Then the breadth of each tie-beam must be 3^ inches + 2£ inches = 5| inches, or say 6 inches: hence the tie-beams will be 6 inches by 12 inches. Each rod will have to carry 5725 pounds, and their diameter /5725 I will be = 4 inch, nearly. Thus we have found, for the dimensions of the various pieces of the girder: — Two tie-beams 6 inches by 12 inches; two rods at each joint, inch diameter; and strut-pieces 10 inches by 12 inches.1UVKTLD l’LATllltON U1KDUUS. 813 CHAPTER XX. RIVETED PLATE-IRON GIRDERS. Wiiexeveh the load upon a girder or the span is too great to admit of using an iron beam, and the use of a trussed wooden g’i\ler is impracticable, we must employ a riveted iron-plate girder, (iirders of this kind are quite commonly used at the present day ; as they can easily be made of any strength, and adapted to any span. They are not generally used in buildings for a greater span than sixty feet. These girders are usually made either like Fig. 1 or Fig. 2, in section, with vertical stiffeners riveted to the web-plates every few feet along their length. The vertical plates, called “ web-plates,” are made of a single plate of wrought-iron, rarely less than one-fourth, or morn than five-eighths, of an inch thick, an 1 generally three-eighths of an inch thick. Under a distributed load, the web of three-eighths of an inch thick is generally sufficiently strong to resist the shearing-stress in the girder without huckling, provided that two vertical pieces of angle-iron are riveted to the web, near each end of the girder. These vortical pieces of angle-iron or T-iron, whichever is used, are called “stiffeners;” and when the girder is loaded at the centre, and sometimes when Fig. 1 Fig. 2.346 RIVETED PLATE-IRON GIRDERS. under a distributed load, it is necessary to use the stiffeners for the whole length of the girder, placing them a distance apart equal to the height of the girder. The web is only assumed to resist tlie shearing-stress in the girder. The top and bottom plates of the giider, which have to be proportioned to the loads, span, and height, , are fastened to the Web by means of angle-irons. It has been found, that in nearly all cases the best proportions for the angle-irons is 3 inches by 3 inches by i inch, which gives the sectional area of two angles five and a half square inches. The two angles and the plate taken together form the flange; the upper ones being called the “upper flange,” and the lower ones the “ lower flange.” • Rivets. — The rivets with which the plates and angle-irons are joined together should be three-fourths of an inch in diameter, unless the girder is light, when five-eighths of an inch may he sufficient. The spacing ought not to exceed six inches, and should be closer for heavy flanges; and in all cases it should not be more than three inches for a distance of eighteen inches or two feet from the end. Rivets should also not be spaced closer than two and a half times their diameter. Rules for the Strength of Riveted Girders. In calculating the strength of a riveted girder, it is customary to consider that the flanges resist the transverse strain in the girder, and that the web resists the shearing-strain. To calculate the strength of a riveted girder very accurately, we should allow for , the rivet-holes in the flanges and angle-irons ; but we can compute the strength of the girder with sufficient accuracy by taking the strength of the iron at ten thousand pounds per square inch, instead of twelve thousand pounds, which is used for lolled beams, • and disregarding the rivet-holes. Proceeding on this consideration, we have the following rule for the strength of the girder : — 10 X area of one flange X height Safe load in tons = --------3 x spanjnl^t —* <1 > Area of one flange I _ 3 X load x span in feet in square inches J — 10 X height of web in inches' The height of the girder is measured in inches, and is the height of the web-plate, or the distance between the flanged-plates. The web we may make either one-half or tlirce-eighths of an inch thick; and, if the girder is loaded with a concentrated load at flic centre or any other point, we should use vertical stiffeners the whole length of the girder, spaced the height of the girder apart.RIVETED PLATE-IRON GIRDERS. 347 If the load is distributed, divide one-fourth of the whole load on tlie girder, in tons, by the vertical sectional area of the web-plate; and if the quotient thus obtained exceeds the figure given in the following table, under the number nearest that which would 1.4 X height of girder be obtained by the following expression, “ thickne^ofweb ’ then stiffening pieces will be required up to within one-eighth of the span from the middle of the girder. 30 35 40 45 50 55 60 65 ?0 75 80 85 90 . 95 100 3.08 2.84 2.01 2.39 2.1Sf 1.99 1.82 1.66 1.52 if 1.28 1.17 1.08 l.oo 0.92 Kxami'i.h.—A brick wall 20 feet in length, and weighing 40 tons, is to be supported by a riveted plate-girder with one web. The girder will be 24 inches high. What should be the area of each Range, and Lhe thickness of the web ? X 40 X 20 A ns. Area of one flange = 10 X 24— = square inches. Subtracting 5 square inches for the area of two 3-incli by 3-inch angle-irons, we have 5 square inches as the area of the plate. If we make the plate 8 inches wide, then it should be 5 -r 8, or § of an inch thick. The web we will make f of an inch thick, ami put two stiffeners at each end of the girder. To find if it will be necessary to use more stiffeners, we divide j of 40 tons, equal to 10 tons, by the area of the vertical section of the web, which equals $ of an inch X 21 inches = 0 square inches, and we obtain 1.11. The expression 1.1 x height, of girder , ,• i , in this case, equals 80.0. The number near- Lhickness of web ’ ’ 1 est tins in tlut table is 00, and the figure under it is 1.08, which is a little less than l.ll ; showing that we must use vertical stiffeners up to within 3 feet of the centre of the girder. These vertical stiffeners we will make of 2§-inch by 2|-inch angle-irons. From the formula for the area of flanges, the following table has been computed, which greatly facilitates the process of finding the necessary area of flanges for any given girder.RIVETED PLATE-IRON GIRDER*. 318 Table of Co-efficient of Flanges for Riveted Girders. Co-efficient for determining the area required in flanges, allowing 10,000 pounds per square inch of cross-section fibre strain : — Uclb. — Multiply the load, in tons of 2000 pounds uniformly distributed, by the co-efficient, and divide by 1000 pounds. The quotient will be the gross area, in square inches, required for each flange. i Distance Depth of gilder, out to out of web, in inches, between supports, in foot. 12 14 16 18 20 22 24 26 28 30 32 34 36 1 10 250 214 1SS 167 150 136 125 115 107 100 94 88 S3 11 275 236 206 1S3 165 150 138 127 118 no 103 97 92 12 300 257 225 •J00 ISO 164 150 138 129 120 113 106 mo 13 325 270 244 217 195 177 163 150 13'.» 130 122 115 108 14 350 300 203 233 210 191 175 162 150 140 131 124 117 15 375 321 281 250 225 205 1SS 173 161 150 141 132 125 is 400 343 mm 267 240 218 200 185 171 160 150 141 133 IT 425 36 4 310 283 o.V, 232 213 196 1S2 170 159 150 142 18 450 3S6 33S 300 270 245 225 208 193 ISO 169 159 150 10 475 407 356 317 2S5 259 238 219 204 190 178 168 158 20 500 429 375 333 300 273 250 O3I 214 200 1SS 176 167 31 525 450 394 350 315 286 263 242 225 210 197 185 175 '2‘2 550 471 413 367 33,0 300 275 254 236 220 206 194 183 23 575 493 431 383 345 314 288 265 246 230 216 203 192 24 000 514 450 400 360 327 300 277 257 240 mm 212 200 25 625 536 469 417 375 341 313 2S3 26S 250 234 221 208 2i» 050 557 488 433 300 355 325 soo 279 260 244 220 2fr 21 075 570 506 450 405 3«>8 338 312 2S9 270 253, 23 S 225 28 700 0OD 525 467 420 382 350 323 300 GB) 263» 247 mm 29 725 021 544 483 435 395 363 335 oil 2'J0 272 256 242 30 750 643 563 500 450 409 375 346 321 300 284 265 250 31 775 604 581 517 465 423 3 8.8 358 332 310 291 271 2-58 «.) 800 08f) 000 533 480 436 HUH 369 343 320. 3.00 2x2 20 7 33 825 707 010 550 495 450 413 381 354 33u 309 291 275 34 850 720 638 507 510 464 425 My 364 340 319 300 05 875 750 656 ■ •>81 525 477 438 404 375 350 328 309 222 36 000 771 675 (mo 540 491 450 415 :;s6 360 3,3, s 31 s lit 025 703 604 617 555 505 463 427 H 370 317 326 308 38 050 S14 713 63,3 570 518 475 438 407 38 J 356 33,5 317 39 075 836 731 650 585 532 4.S3 440 418 390 36) > 314 325 Example.—Let us take the same girder that we have just computed. Here the span was 20 feet, and the depth of girder 24 inches. From the table we find the co-efficient to be 2o0; and multiplying this by the load, 40 tons, and dividing by 1000, we have 10 square inches as the area of one flange, being the same result as that obtained before.RIVETED PLATE-IRON GIRDERS; 349 Girders intended to carry plastering should be limited in depth font to out of web) to one-twenty-fourth of the span-length, or half an inch per foot of span: otherwise the deflection is liable to cause the plastering to crack. In heavy girders, a saving of iron may often be made by reducing the thickness of the flanges towards the ends of the girder, where the strain is less. The bending-moment at a number of points in the length of the girder mav bo determined, and the area of the flange at the. different points made proportional to the bending-moments at those points. The thickness of the flanges is easily varied, as required by forming them of a suflirient number of plates to give the greatest thickness, and allowing them to extend on each side of the centre, only to such distances as may he necessary to give the required thickness at each point. The deflection of girders so formed will he greater than those of uniform cross-section throughout. TABLES OP SAFE LOADS FOR RIVETED PLATE-IRON GIRDERS. The tables given on pp. 319« and 3495 have been computed according to the formula on p. 34(5, to give an idea of the size of girder that will he required for a given load, of the heights and spans indicated. If it is remembered that the strength of a girder depends directly as the area of its flanges and its height, the width and thickness of the flange plate may he changed, provided ih<‘ area remains the sunn , without altering its strength. Thus a girder 36""high, with flanges formed of 4\r' x 4i" X angles, and £" X 24" plate, would he as strong as one with the same angles and l" x 12" plate, provided the web plates are properly stiffened, as described on p. 347. In computing the weight of the girders in the tables, no allowance has been made for stiffeners. In computing the strength of riveted girders, it will be convenient to know that — The an a of tw o 3" x 3" X angle-irons = 5.5 square inches. “ 3£" X x i” “ ~ 0.4 “ 4" X 4" x |" “ =7.4 “ “ 4i" X 4i" x ■“ = 8.4 “STRENGTH OF RIVETED WROUGHT-IRON GIRDERS. Span in Ft. Safe Load in Tons of 2,000 Lbs., Equally Distiubuted. Span in Ft. 20 41 50 62 75 46 56 70 84 61 73 92 110 20 22 38 45 57 68 42 51 64 76 56 67 84 100 22 24 35 41 52 62 39 47 58 70 51 61 77 92 24 25 33 40 50 60 37 45 56 67 49 59 74 88 25 26 32 38 48 58 36 43 54 64 47 57 71 85 26 28 . 30 36 44 53 33 40 50 60 44 53 66 78 28 30 28 33 41 50 31 37 47 56 41 49 61 73 30 32 26 31 39 46 29 35 44 52 38 46 57 69 32 34 24 29 37 44 27 33 41 49 36 43 54 65 34 36 23 28 34 42 26 31 39 47 34 41 51 61 36 38 22 26 33 40 24 29 37 44 32 38 48 58 38 40 20 25 31 37 23 28 35 42 30 37 46 55 40 42 23 30 35 — 26 33 40 - - 44 52 42 45 28 mm oo _ — 31 37 - - - 49 45 48 31 — 35 - - - 46 48 50 ~ 30 - - — 33 — — — 44 50 349a RIVETED WROUGHT-IRON GIRDERS.STRENGTH OF RIVETED WROUGHT-IRON GIREEF.S. ' — ; -T X t-- < I f— T. ■■ ■*HI .. _ ' ’ _ S 3 ^ . c=3*_j 5 SJJ^ 5) s Tjb - a» i Span in Ft. Safe Load in Tons of 2,000 Lbs. , Equ ALLY Distributed. Span in Ft. 20 58 70 87 105 72 86-. 108 130 85 102 128 154 20 24 48 58 73 87 60 72 90 108 71 85 106 128 24 26 45 54 67 81 55 66 83 100 65 79 98 118 26 28 42 50 62 75 51 61 77 93 61 73 91 110 28 30 39 47 58 70 48 57 72 86 57 68 85 102 30 32 36 44 55 66 45 54 67 . 81 53 64 80 96 32 34 34 41 51 62 42 51 63 76 50 60 75 90 34 36 32 39 48 58 40 48 60 72 47 57 71 85 36 38 30 37 46 55 38 45 57 68 44 54 67 81 38 40 29 35 44 52 36 43 54 65 42 51 64 77 40 42 - QQ *.)o 42 50 - 41 51 62 — 49 61 73 42 44 - 32 40 48 — 39 49 59 — 47 58 70 44 46 - - 38 46 - — 47 56 _ — 56 • 67 46 48 - - - 44 - — ~ 54 — — 64 48 50 — - - 42 - — 52 — — _ 61 50 RIVETED WROUGHT-IRON GIRDERS. 34%350 CAST-IRON ARCH-GIRDERS. CHAPTER XXI. STRENGTH OF CAST-IRON ARCH-GIRDERS, WITH WROUGHT-IRON TENSION-RODS. Cast-iron arch-girders are now quite extensively employed to support the front or rear walls of brick buildings. Fig. 1 shows the usual form of such a girder, the section of the casting and rod being shown in Fig. 2. o Fig. 2. The casting is made in one piece with box ends, the latter having grooves and seats to receive the wrought-iron tie-rod. The tie-rod is made from one-eiglith to three-eighths of an inch shorter than the casting, and has square ends forming shoulders so as to fit into the castings. The rod has usually one weld on its length, and great care should be taken that this weld be perfect. The rod is expanded by heat, and then placed in position in the casting, and allowed to contract in cooling; thus tying the two ends of the casting together to form abutments for receiving the horizontal thrust of the arch. If the rod is too long, it will not receive the full proportion of the strain until the cast-iron has so far deflected, that its lower edge is subjected to a severe tensile strength, which cast-iron can feebly resist. If the tie-rod is made too short, the casting is cambered up, and a severe initial strain put upon both the cast and wrought iron, which enfeebles both for carryingCAST-IRON ARCH-GIRDERS. 351 a load. The girders should have a rise of about two feet six inches on a length of twenty-live feet.1 Rules for Calculating Dimensions of Girder and Rod] A east-iron arch-ginler is considered'as a long column, subject to a certain amount of bending-strain ; and the resistance will be governed by tin* laws affecting the strength of beams, as well as by those relating to the strength of columns. Fig. 3. If we regard the arch as flexible, or as possessing no inherent stillness, and the rod as a chord without weight, we can deduce the following formula for the horizontal thrust or strain: — llor. thrust, _ B per foot of span X span in feet, squared or strain ~ 8 X rise of girder in feet ’ From tins rule we can calculate the required diameter of the tension-rod, which may be expressed thus : — VIoail on girder X span in feet ---™ ■ —r—-o- ■---------------(2) 8 X rise in feet X i8o4 ' ' Toe rule generally used, however, in proportioning the wrought- . iron tii! to the cast-iron arch is to allow one square inch of cross-section of tic-rod for every ten net tons of load imposed vpon the span of the arch. /The following table, taken from Air. Fryer’s book on “ Areliitec- 1 Architectural Iron-Vvrork for Unihline;- — William J. Fi.Vkr, Jim. t’p- 38.352 CAST-IRON ARCH-GIRDERS. tural Iron-Work,” shows the section of the cast-iron arch required to support solid brick walls, and having a span of from 13 to 26 feet. Height of wall. Thickness of wall. Dimensions of Section. Top flange. Centre web. Bulb. 40 feet. 50 “ 40 « 50 « 12 inches. 12 f* 16 “ 16 “ 12" x V' 12" x 12" x 1»" 16" x MM 12" x l" 12" x l" 12" x |" 12" x 1" 3" x 2" 3" x 2" Z\" x 2" 4" x 2" Substitute for Cast-iron Arcli-Gircler. In the cast-iron arch-girder with wrought-iron tension-rod, the casting only serves to resist compression. Its place can as well be filled by a brick arch footed on a pair of cast-iron skewbacks, which are themselves held in position by a pair of tie-rods, as in Fig. 8. In this case, Formula 1 will still give the horizontal pull to he resisted by the tie-rods ; but, as we must have two rods instead of one, the diameter of each will be obtained by the formula, Diameter of each _ / total loa(l on arel> x sP;ul _ (f-) rod in inches \ Hi X rise of arch in feet X 7854 N.B. — The rise is measured from the centre of the rod to the centre of the arch. It will also he remembered that the span is to be alicays taken in feet, unless otherwise specified. Example 1. — It is desired to support a 12-inch brick wall 40 feet high over an opening 20 feet wide, with a cast-iron arcli-girder. What should be the dimensions of the girder ? For the casting, we find from the table that the cross-section of the flange should be 12 inches by 1 inch ; of the web, 12 inches by f inch ; and of the bulb, 3 inches by 2 inches. We will make the rise of the girder 2 feet and (5 inches, and from Formula 2 we find1 Diam. of rod in inches I V weight of wall X span 8 X rise of arch in feet X 7854 (20 X 20 X 112) X 20 rr- b X 2-i X 7854 ~ 1 2f ins. VpMdUeiiw that the girder woijld only support about twenty feet of the wall in height, the wall above that supporting itself.WOODEN FLOORS. 358 CHAPTER XXII STRENGTH AND STIFFNESS OF WOODEN FLOORS Btr<‘tlgtli of Floors. — In calculating the strength of floor-beams, the first thing to he decided is the span of the beams, which is generally determined by the size of the opening to he covered ; and the second is the load which is to come upon the floor. Wooden (loor-beams should not have a span of more than twenty-live feet (if it can be so arranged); for, if they are of a greater length than this, it is difficult to stiffen them sufficiently to prevent vibration under a heavy or moving load When the distance between the bearing-walls of a building is greater than the above limit, partition walls should be built, or else the beams should he supported by iron or wooden girders resting upon iron or wooden columns. Tin* Building Laws of the cities of New York and Boston require that in all buildings more than thirty feet in width, except churches, theatres, seboolhouses, car-stables, and other public buildings, the space between any two of the hearing-walls shall not be over twenty-live feet, unless girders are substituted in place of the partition-wall. Floor'-beams, when supported at three or more points, should always be made continuous if possible, as the strength of i-ach portion of the beam is thereby greatly increased. Sii pei'ini posed Loads. — There is some difference of opinion among authorities as to what should be allowed for the superimposed io.ul upon the floor of a dwelling or upon the floors of public buildings. The New-York Building Law requires that in all buildings every floor shall have sufficient strength to bear safely upon every superficial foot of its surface seventy-live pounds, and, if used as a place of public assembly, one hundred and twenty pounds. Indwelling-houses, where the maximum load consists of nothing but ordinary furniture and the weight of some ten or twelve.people, it is not necessary to allow more than forty pounds per square foot for the superficial load; and, in most casdS, eighty pounds per square foot is ample allowance for the weight of an assemblage of people. Only ip rases where people are liable to be jammed together during35 4b WOODEN FLOORS. Material. Measurements. Weights. Dye Stuffs, etc,—ConVd. Floor space. Cubic feet. Gross. Per eq. ft. Per cubic ft. Barrel Rosin 3.0 9.0 430 143 4S “ Lard Oil . . * . . 4.3 12.3 422 98 34 Rope ... - - 42 Miscellaneous. Box Tin 2.7 0.5 139 99 278 “ Glass - _ _ 60 Crate Crockery ..... 9.9 39.6 1600 162 40 Cask Crockery 13.4 42.5 600 52 14 Bale Leather 7.3 12.2 190 26 16 *• Goatskins 11.2 16.7 300 27 18 u Raw Hides 6.0 30.0 400 67 13 " “ “ compressed, 6.0 30.0 700 117 23 “ Pole Leather .... 12.6 8.9 200 ‘2*2 16 Pile Pole Leather .... _ _ _ 17 Barrel Granulated Sugar. . 3.0 7.5 317 106 42 “ Brown Pillar . . . 3.0 7.5 310 113 45 Cheese ..... ... — 30 Weight of the Floor itself. — Having decided upon the span of the floor beams and upon the superimposed load, we must next consider the weight of the floor itself. Wooden floors in dwellings weigh, on the average, from seventeen to twenty two pounds per square foot of floor, including the weight of the plastering on the under side. For ordinary spans the weight may be taken at twenty pounds per square foot." and, for long spans, twenty two pounds per square foot. For floors in public buildings, I the weight per sqtiare foot seldom exceeds twenty-five pounds, and ; if may safely be assumed at that amount. In warehouse floors, which have to sustain very heavy loads, the weight per square foot may sometimes be as great as forty or fifty jt pounds; and in such cases the approximate weight of the floor per j square foot should be first calculated. Factor of Safety to be used.—in.considering the load i on a floor, it should be remembered that the effect of a load sud ricnly applied upon a beam is t wice as great as that of the same load gradually applied; and hence the factor of safety used for tlu* [ former should be twice as great as that for the latter. The load caused by a crowd of people is usually considered to produce an effect which is a mean between that of the same load when gradu-ally and when suddenly applied; and hence a factor of safety is employed which is a mean between that for a live and for a dead load. Till factors of safety for floor-timbers adopted by the best engineers vary from 3 to 5. For short spans in ordinary dwellings, I t puHi<* building*, and stores. :i is probably aiWy sufficient tor ^WOODEN FLOORS. 335 strength; but for long spans, and floors in factories and machine-shops, a factor of safety of 5 should often be used.1 Kules for the Strength of Floor-beams. — In considering the strength of a floor, we assume it to be equally loaded over its whole surface, as this would be the severest strain to which the timbers could be subjected. Hence, in calculating the dimensions of the floor beams, we use the formula for a distributed load. That formula for rectangular beams, 2 X breadth X depth squared X A Safe load - span in feet X .S <1 > S being tin; factor of safety. For floor beams the safe load is represented by the superimposed load and weight of floor supported by each beam. The area of floor supported by each beam equals the length of beam multiplied by the distance between centres. If we let / denote the weight of the superimposed load per square foot of floor surface, and f the weight of one square foot of the floor itself, then the total weight per square foot will be (/+/') pounds, and the total load on each beam will equal Length of beam X distance between centres X (/ +/')• Now, if we substitute this expression in place of the safe load in the above formula, and solve for the depth, we shall have, Square of _ ^ x het. centres X length squared X (/+/') depth ~ 2 X breadth X A ^ or, if we solve for the distance between centres, we shall have, Distance between _ 2 x breadth X depth squared X A (3) centres in feet S X length squared X (/-{-/') U.H, — The length and distance between centre« must be taken m feet, and On* length mean« omy the distance between support«, or the clear span. The values of the constant A for the four woods in general use art* as inflows : — Spruce ..... 270 Oak...................»315 Hurd pine............375 White pine .... 2-K) Formulas'2 and 8 apply to all floors supported by rectangular brains, whatever be the factor of safety employed, the weight of 1 I’ntil vety recently it lisis been out custom to use factors of safety twice as great as these, hut, a« we have had occasion to reduce the constants for strength to ahoul one-half of that formerly used, we have reduced the factors of safely accordingly. It will he found that the result is the same as that obtained by the rule« of other writers.356 WOODEN FLOORS. the superimposed load, or of the floor itself. To illustrate the application of these formulas, we will give two examples such as are constantly occurring in practice. ExAMrLE 1. — What should be the dimensions of the spruce floor-beams in a dwelling, the beams to have a span of 15 feet, and to be placed 16 inches, or 1^ feet, on centres ? Ana. In this case we would use a factor of safety of 5: / should be taken at 40 pounds, f at 20 pounds, and A is 270 pounds. Assume 2 inches for the breadth. Then, by Formula 2, 5 X H X ‘225 X 60 Square of depth = —2 X 2 X 270— = ® The depth = ^83 = a little over 9 inches. Hence, to have the requisite strength, the beams should be 2 X 10 inches. Example 2. — It is desired to use 2 by 10 inch yellow-pine beams in the floor of a church, the beams to have a span of 16 feet. What distance should they be spaced on centres ? Ans. Here S = 5, f — 100 pounds, f — 25 pounds, and A = 375 pounds. Then, by Formula 3, we have 2 x 2 x 100 x 375 Distance between centres = —5 x 256 x 125— = or H ins. Hence the floor will be sufficiently strong if the beams are placed 11 inches on centres. Bridging- of Floor-beams. — By “bridging” is meant a system of bracing floor-beams, either by means of small struts, as in Fig. 1, or by means of single pieces of boards at right angles to the joists, and fitting in between them. The effect of this bracing is decidedly beneficial in sustaining any concentrated weight upon a floor; but it does not materially strengthen a floor to resist a uniformly distributed load. The bridging also stiffens the joists, and prevents them from turning sidewise. It is customary to insert rows of cross-bridging at from every five to eight feet in the length of the beams ; and to be effective they should be in straight lines along the floor, so that each strut may abut directly uponWOODEN FLOORS. 357 those, adjacent to it. The method of bridging shown in Fig. 1, and known as “ cross-bridging,” is considered to be by far the best, as it allows the thrust to act parallel to the axis of the strut, and not across the grain, as must be the case where single pieces of hoard are used. The bridging should he of lj-incli by 3-inch stock. Carriage-beams, Headers, and Tail-beams. — Fig. 2 represents the plan of the timbers of a floor, having a stairway opening on each side. The short beams, as KL, are called the “ tail-beams: ” the beams EF and GII, which support the tail-beams, are called the “ headers: ” and the beams A H and CD, the “ carriage-beams,” or “ trimmers.’’ The tail-beams are calculated in the same way as ordinary floor-joist; but it is evident that the headers and trimmers will require separate computations. It would he very difficult to give formulas that would serve for every case of trimmers and headers ; and the best way in any case is to find the load which the trimmer lias to carry, and then, from the formulas already given, determine the required dimensions. In a floor such as is represented in Fig. 2, it is evident that the floor-area supported by EF or GII — o j 4.OS } 6.07 { 60.15 14 7 26.16 ( 2.43 i 5.33 ) l 61.40 16 8 31.63 ) ( 3.00 ) ( 85.40 12 6 17.42 ) 75 | 4.OS 7.40 86.48 14 7 21.82 J 2.96 ( 5.33 1 l 87.73 16 8 26.46 ) 3.00 111.55 12 6 15.25 100 \ 4.08 } 8.55 \ 112.03 14 7 19.12 ( 8.42 l 5.33 ) l 113.88 16 8 23.23 ) ( 3.00 ) ( 137.55 1 6 13.73 125 4.08 9.55 138.63 14 7 17.23 J 3.82 l 5.33 ) i 139.88 16 8 20.96 ) ( 3.00 ) ( 163.45 12 6 12.59 i 50 ; 4.08 } 10.45 \ 104.53 14 7 15.S2 [ 4.18 i 5.33 S l 165.78 10 8 19.25 ) ( 3.00 189.26 12 6 11.71 ) 175 4.08 j 11.20 | 100.34 14 7 14.70 { 4.51 \ 5.33 ) 1 191.59 10 8 17.91 ) ( 3.00 ) ( 215.05 m 6 10.98 ) EH ; 4.08 210.13 14 7 13.80 1 4.82 i 5.33 ) ( 217.38 16 8 16.81 ) 3 no ‘240.75 12 0 10.38 ) 1.0S { 12.75 j 241.S3 14 i 13.00 5.11 l r. RB9 •”'» 1 ( 213.03 10 S 15.00 I ( 3.00 ) ( 200.45 12 6 9.80 1 250 ; 4.08 13.45 207.53 14 7 12.40 ■-38 i S l 208.78 10 8 15.08 J 3.00 ) ( 201.55 12 6 9.43 ) <>- 1 4.OS 13.55 { 292.03 14 7 11.80 } 5.02 ( 5.33 ) 1 203.88 16 8 14.40 j 3.00 ) ( 317.72 12 6 9.03 ) :ioo \ 4.0S 14.72 318.80 14 7 11.36 } 5.89 l 5.33 J 1 320.05 10 8 13.85 ) Stiffness of Wooden Floors. Floors in first-class buildings should possess something more than mere strength to resist fracture: they should have sufficient stiffness to prevent the floor from bending, under any load, enough to cause, the ceiling to crack, or to present a bad appearance to the eye. To obtain this desired quality in floors, it is necessary to calculate the requisite dimensions of the beams by the formulas for stiffness ; and, if the dimensions obtained are larger than those362 WOODEN FLOORS. obtained by the formulas for strength, they should be adopted, instead of those obtained by the latter fornmlas. The only way in which we can be sure that a beam is both strong enough and stiff enough to bear a given load is to calcnlate the required dimensions by both the] formula for strength and the formnla for stiffness, and take the larger dimensions obtained. As a general rule, those beams in which the proportion of depth to lemjth is veiy small should be calculated by the formulas for slretu/fh, and vice versa. Formula 10, Chap. XVI., gives the load which a given beam will carry without deflecting more than one-fortieth or one-thirtieth of an inch per foot of span, according to the value of e which we use. Formula 11, Chap. XVI., gives the dimensions of the beam to carry a given load under the same conditions. In the ease of floor-beams, the load is given, and is represented, as we saw under the Strength of Floors, by the expression, Distance between centres in feet X length in feet X (/+ f'). Then, if we substitute this expression in place of the load in Formula 11, Chap. XVI., we shall have the formula, f> X dist. between centres X'cube of length X (/+/') Breadth = ^^ube^ofdopthxT^ or 8 X breadth X cube of depth X e Dist. between centres = yx cube of length X (/ +7V <8) The proper values for/ and/' have been given under the Strenyth of floors in the preceding part of this chapter, and the value of e for any given case may be found in Chap. XVI.1 In ordinary floors, when the values off used are those recommended above, a deflection of one-thirtieth of an inch per foot of span may safely be allowed, as the floors would probably be very rarely loaded to their utmost capacity, and then but for a short time; so that it wouid have no injurious effects. As an example showing the application of Formula 7, we Mill take Example 1 under the strength of wooden floors. In this example, the beams were to have a span of la feet, and be place'll l1, feet on centres; ./'was taken at 40 pounds, andJ' at 20 pounds. What should be the dimensions of the beams, that they may safely carry the load upon them without deflecting more than 3*0 of an inch per foot of span ? 1 The value« for e, for spruce, hard pine, and oak, arp, Def.^y/., Def.= 5yz, Spruce.................................E'*1 ?■* Hard pine...............................1ST M3 Oak......................................»o nWOODEM EEOOJUS. 563 Ans. We have simply to substitute our known quantities in Formula 7, assuming the depth at 10 inches, and taking' the value of e at 100 pounds, the beams being of spruce. Performing the operation, we have, _ 5 X 1l, x I.t1 x (40 + 20) Iireadth - jfx 1(P xloo = LCS inchcs‘ 'I’lds gives us about the same dimensions that we obtained when considering tile beam in regard to its strength only: hence a beam two by ten inches would fulfil both the conditions of strength and stiffness, in the case of headers, stringers, etc., where the joist has to carry not only a distributed load, but also one or more concentrated loads applied aL different points of the beam, the required dimensions can best be obtained by considering the beam to be made up of a number of pieces of the same depth, placed side by side, and computing the required Iireadth of beams of that depth to carry each of the loads singly, and then taking the sum of the breadths for the breadth required. The formula for stiffness of plank-floors has already been given on p. 351).FIRE-PROOF FLOORS. CHAPTER XXIII. FIRE-PROOF FLOORS. The term “ fire-proof floor” is here understood to mean a floor constructed of some fire-proof material, supported on or between rolled-iron I-beams. The various kinds of fire-proof material in general use are brick, hollow terra-cotta tile, the hollow blocks manufactured by the jFire-Proof Building Company of New York, from the liydrauli* lime of toil, porous terra-cotta, corrugated iron, plaster and ashes, magneso calcite, selenetic cement. Fig. i. \ J o r-^-1 f ] r J The first five of the above are used in the shape of arches lmilt between iron beams, the others being used as a protection to tho floor after the frame of the floor is up. In designing a floor of either of these materials, it is only necessary, generally, to calculate the size of the I-beam; as the arches between the beams will stand any load which will not injure the beams. Before, however, considering the method of calculating the size of the I-beams, we will describe briefly the usual modes of building fire-proof floors. Iroh beams arc generally laid in floors as shown in Fig. 1, theFIRE-PROOF FLOORS. 865 joists either resting on top of the girders, as in Fig. 2, or bolted to the sides of the girders. Fig. 3 shows the detail of connection when the under sides are made flush ; Fig. 4, the joint to bring the upper sides flush; and Fig. 5 shows the form usually adopted when the beams are of the same size, or the centre lines are brought together. Arrangements of this kind are also used to connect the trimmer-beams of hatchways, jambs, and stairways.1 Fig. 6. The wail ends of the joists and girders should be provided with shoes or hearing plates of iron or stone, as the brickwork is apt to crush under the ends of beams, unless the load is distributed by this means over a sufficient surface. Anchor-straps should be bolted to the end of each girder and to the wall end of every alternate joist, binding the walls firmly from falling outwards in the event of fine or other accident. Several simple modes of anchorage are shown in Figs. 8, 4, and 5. When one beam does not give sufficient strength for a girder, it is customary to bolt together two or more with cast separators between them, as shown in Fig. 6. 1 The details of the councctious and framing of iron beams are more clearly shown on pp. 305, 306.366 FIRE-PROOF FLOORS. Brick Arches. The common way of making a fire-proof floor of brick is to fill the space between the joists with brick arches, resting on the lower flanges against terra-cotta or brick skewbacks. When this method is pursued, care should be taken that the bricks of which the arches are composed are of good shape, and very hard. They should be laid in contact with each other, without mortar; and all the joints should be filled with the best cement grout, and be keyed with slate. The arches need not be over four inches thick for spans between six and eight feet, except for about a foot at each springing, where they should be eight inches thick for spans between six and eight feet, care being taken to form the skewbacks quite solid, and square to the line of pressure. The rise of the arch should be abort one-eighth of the span, or an inch and a half to the foot;1 and the most desirable span is between four and six feet. Above the arch the space is filled with cement concrete, in which wooden strips, three inches by four inches, are embedded for nailing the flooring to. The thrust of the arches is taken up by a series of tie-rods, usually from three-quarters to one inch in diameter, placed in lines from six to eight feet apart, and running from beam to beam from one end of the building to the other, being anchored into each end Avail with stout washers; an 1 Hatfield’s Transverse Strains, p. 345.FIRE-PROOF FLOORS. 367 angle-bar or channel serving as a wall-plate for distributing the strain produced by the thrust of the first arch (Fig. 7). The weiyht of a brick arch with cement filling is about seventy pounds per superficial foot of floor. Arches of Hollow Brick, Teil Hollow Blocks, and Hollow Tile. Owing to the great weight of brick arches and the necessary filling, together with the expense of furring to obtain a flat ceiling, flat arches of hollow brick, or of hollow blocks of fire-proof material, are now quite generally used. These arches, being much lighter than brick, admit of the use of lighter beams, and thus effect a saving in the thickness and cost of the floor. They also give a level coiling, dispensing with the necessity for furring or lathing, and possess several other advantages, described in Chap. XXV. If segmental arches are desired, however, they can be made of either of these materials. The voussoirs of the flat arches are cemented together with joints inclined to a common centre, as in a segmental arch. The skewbacks take the form of the iron beams against which they rest, and each block keys with the adjacent; ones no two joints being allowed to be parallel, as this would endanger the safety of the flat arch. The lower surface of the flat arch descends about one-lialf»or three-quarters of an inch below the flanges of the iron beams; and the bottom of the latter is cemented over to protect them from fire, and to form a flat ceiling, which is then ready for plastering. The danger of cracks in the ceiling at the joint of the flat arch and iron beam is avoided, as the beam is cemented over before the plastering is put on. Fig. 10. Fig. 10 shows a section of a flat arch of the usual form.1 1 The cuts used in this chapter are taken, with their permission, from the Pho* nix Iron Company's book of I'seful Information for Architects and Engineers.368 FIRE-PROuF FLOORS. Among the largest manufacturers of hollow blocks for floor-arclies are The Fire-Proof Building Company of New York; the liar Han Hollow and Porous Brick Co)npany, also of New York; the Pioneer Fire-Proof Construction Company of Chicago; and the Wight Fire-Proofing Company of Chicago and New York. These companies manufacture blocks for various depths and spans of arches, and of various weights per square foot. The following tables, compiled from data obtained from the printed circulars of the various companies, show the sizes of arches manufactured, their weight per square foot, and in some cases the loads which they will sustain. FLAT ARCHES, Manufactured by the Fire-Proof Building Company of New York. Teil Hollow Blocks. Hollow Fire-clay Bricks. Spans between Depth of Weight Spans between Depth of Weight iron beams. arch. per sq. ft. iron beams. arch. per sq. ft. Up to 2 ft. 0 in. 4 in. 16 lbs. Up to 3 ft. 0 in. 4 in. 20 lbs. Up to 2 “ 9 “ 5 1 20 “ Up to 5 “ 0 “ a “ • 30 I Up to 3 “ 0 “ 6 “ 22 I Up to 6 “ 0 “ 7 “ 38 “ Up to 4 “ 9 1 7 m 24 “ Up to 7 “ 0 “ 8 “ 45 “ Up to 5 “ 0 “ s 1 20 ■ Up to 5 “ 6 “ 9 “ 20 1 Up to 0 sB 3 “ 10 “ 35 1 Up to 7 “ 0 “ 11 I 39 “ Up to S “ 0 “ 12 “ 43 1 Up to 10 “ 0 “ 15 “ 52 “ FLAT ARCIIES OF HOLLOW FIRE-CLAY- ERICK, Manufactured by the Raritan IIollow and Porous Brick Company. Span between beams. Depth of arch. Weight per sq. ft. Safe load per sq. ft. 3 ft. 6 in 6 in. 34 lbs. 1000 lbs. 3 “ 6 r to 4 ft. 6 in 7 fl 37 1 1000 “ ; 4 “ 6 “ to 5 “ 6 “ . . . . 8 1 41 “ 1000 “ 6 “ 0 “ to 6 “ 6 “ . . . . 10 ■ 48 “ 1000 M 6 0 r to 7 “ 0 c< . | • • 12 “ 54 “ 1000 “ FLAT ARCIIES OF IIOLLOW FIRE-CLAY TILE, Manufactured by the Wight Fire-Proofing Company. Spans between beams. Depth of arch. Weight per sq. ft. 7 in* 24 lbs. 5 to 7 “ 9 “ 80 I 7 to 9 “ 12 “ 35 «370 FIRE-PROOF FLOORS. If the floor were constructed of the flat arches represented in the tables given above, tin* weight of the arch per square foot should bo substituted in place of tin* number 70 in the parenthesis of the above expression Having obtained the value of this expression, less the weight of the beam, we must proceed to find the size of the beam which will carry this load. The safe distributed load in pounds for a span of one foot of Trenton, Union Mills, Plnenix, and Pencoyd beams, is given in the tables in Chap. XIV.; and, to obtain the safe load for any span, it is only necessary to divide the safe load for one foot by the given span in feet. The safe distributed load for a span of one foot we will call the co-gflicient of the beam ; then, from the considerations given above, we have, C'o-cflicient__weight of _ dist. between x span x(f+”0f (11 in jMnnids beam centres squared T * ' ' ' and Dist. between _ co-efficient in pounds — weight of beam centres span squared X (/+70) As an illustration of the method of calculating the required si«e of the I-beams, we will take the following : — Kxamim.k 1.—What sized Trenton I-beam would be required in the floor of a first-class store, the beams to have a span of 12 feet, to be 4 feet apart, and to be filled in between with brick arches ? Anx. Ily Formula 1 we find that the co-efficient minus weight of the beam = 4 x 144 x (got) + 70) = 184,320 pounds. Now, if we look in the table giving the strength of Trenton beams, we shall find that the beam whose co-efficient comes next above 184,320 pounds is the “0-inch heavy,” whose co-efficient is 100,000 pounds. The weight of the beam would be 340 pounds: hence the 0-inch heavy beam is amply strong enough for the purpose. Iron beams for floors with arches should have a bearing on the wall of at least six inches. To save the trouble of going through the above work every tinu it is desired to find the size of beam needed In a so-called first-class store, the following tabic, showing the size of rolled I-beams for floors in that class of buildings, for distances between centres of four, five, and six feet, has been calculated. The application of this table is too simple to require explanation.FIRE-PROOF FLOORS. 371 Tables siiowin" the Required Size of Rolled I-Rcams for Brick Arched Floors in First-class Stores and 'Warehouses. Calculated to sustain a load on the Jloor of 250 pounds per super-Jidal foot, the strenyth of the beams beiny those published by the manufacturers. TRENTON ROLLED I-BEAMS. Distances between Centres of Beams, in Feet. 4 5 0 -71 Designation Weight, Designation Weight, Designation Weight, of beam. in lbs. of beam. in lbs. of beam. in lbs. 6 5 inch heavy. 80.0 6 inch li^rht . 80.0 6 inch heavy, 100.0 7 C “ light . 93.3 6 “ heavy, 116.7 6 “ 90 lbs. 210.0 8 6 “ 9u 11«. 240.0 6 “ 9. wire-lathing. Figs. 0 and 7 show forms of porous terra-cotta blocks manufacture 1 by the Raritan Hollow and Porous Brick Company for protecting iron girders. Other companies also manufacture blocks for the same purpose.FIRE-PROOF CONSTRUCTION FOR BUILDINGS. 389 The Raritan Hollow and Porous Brick Company of New York manufacture, for the purpose of fire-proof partitions, hollow burnt-clay bricks of various dimensions and thicknesses, to answer the different requirements. They are light, vermin-proof, do not transmit cold, heat, or sound, and can be set up by any bricklayer. Fig. 7. rOROUS TERRA-COTTA GIRDER-FURRING. When these bricks are used for partitions, small strips of wood are inserted at intervals to receive nails for the securing of baseboards, wainscoting, etc.; or else one or more rows of porous terra-cotta blocks are used. The standard sizes of these bricks are those shown in Figs. 8, 9, and 10. The weight per square390 FIRE-PROOF CONSTRUCTION FOR BUILDINGS. partitions formed of these brick, plastered both sides with seven-eighths inch plastering, is, for Fig. 8, forty pounds, Fig. 9, thirty pounds, and for Fig. 10, twenty-five pounds. The same company also manufacture hollow brick of porous terra-cotta, of the same sizes as the burnt-clay bricks shown in the figures mentioned. Fig. 11 shows the porous terra-cotta ceiling-blocks manufactured by this company. rOHOUS TEIÏKA-COTTA CEILINGS ON T-IKON8. APPENDIX. W ire Tfflfldnff- — An improved form of wire lathing is now made by the .Stanley Corrugated Fire-Proof Lathing Company of New York, which has a narrow corrugation every six inches, which gives satlieient space for a continuous key, and yet brings the plaster so near to the beams that they are actually sealed, and all possibility of currents of air between is prevented. This lathing can readily be applied to ceilings, beams, columns, etc., without the necessity of furring. The manufacturers claim that their lathing has been largely used with satisfaction, both in this country and by the Metropolitan Board of Works of London. 892 WOODEN ROOF-TRUSSES, CHAPTER XXVI. WOODEN ROOF-TRUSSES, WITH DETAILS.1 Whenever it is required to roof a liall, room, or building, where the clear span is more than twenty-five feet, the roof should be supported by a truss of some form. The various forms of trusses used for this purpose have certain features and principles in common, differing from those in bridge and floor trusses, which have led to grouping them in one class, called “roof-trusses.” Xearly all roof-trusses in churches, and halls of like character, and the larger proportion of trusses used in all kinds of buildings, are constructed principally of wood, with only iron tie-rods and bolts ; and, as wooden trusses are of more interest to the architect and builder than iron trusses, they have been more completely described, and a greater variety of forms are given than for iron 1 The parts of the various trusses shown are all drawn slightly out of scale, in order to show how they are joined together. The trusses thus look heavy in proportion to the span; but it should be borne in mind that the figures are lie* to show the size of the timber, but the relation of the various parts.WOODEN ROOF-TRUSSES. 393 roof-trusses, which are discussed in another chapter. In the Northern States and Canada, where there are often heavy snowstorms, experience has taught that the best form of roof for a building, except, perhaps, in large cities, is the A, or pitch roof. The inclinations of the roof may vary from twenty-six degrees, or six inches to the foot, to sixty degrees, or twenty-one inches to the foot, but should not be less than six inches to the foot for roofs covered with slate or shingles. For roofs covered with composition roofing, tin, or copper, the inclination may be as little as five-eighths of an inch to the foot. at their lower ends by the wall-plate, and holding themselves up at the top by their own stiffness and strength. A piece of board, called the “ ridgo-plate,” is generally placed between the upper ends of the rafters, and the rafters are nailed to it. In some localities this ridge-piece is not used, but the upper ends of each pair of rafters are held together by a piece of board nailed to the side of the rafters before they are raised. The walls of the building are prevented from being pushed outward by the floor or ceiling beams, which are nailed to the plate. The rafters are placed about two feet, or twenty inches, on centres, and the lxjarding is najled directly on the rafters. The horizontal joists support the attic-floor and the ceiling of the room below. Such a roof can only be used, however, when the distance between the wall-plates is not more than twenty-four feet ; for with a greater span the rafters, unless made extremely heavy, will sag very considerably.394 WOODEN ROOF-TRUSSES. KiiiS Post Truss. — Whenever we wish to roof a building in which the wall-plates are more than tvventy-fbur feet apart, we must adopt some method for supporting the rafters at the centre. The method generally employed (shown in Fig. 2) is to use trusses like that shown in the figure, spaced about twelve feet apart in the length of the building, and on those place large beams, called “pur- lins,” which support the roof, or jack-rafter’s. As the distance from one purlin to the next is not generally more than six or eight feet, the jack-rafters may be made as small as two inches by six inches. When the span of the truss is more than thirty-four feet, two purlins might be placed on each side of the truss, or at A and A. It is always best, however, to place the purlins only over the end of a brace, or at a joint, when it can be so arranged. The ceiling of the room covered by the roof is framed with light joists supported by the tie-beam of the trass. These ceiling-joists should not be mortised into the tie-beam, but should rest on a two-inch by four-inch strip, bolted to the tie-beam as shown in Fig. 3. When the span of the truss exceeds thirty-five feet, it is difficult to get spruce timbers long enough for the tie-beam without splicing; and in that case one of the best methods of building up the tie-WOODEN ROOF-TRUSSES. 395 beam is to make it of two-inch plank bolted together, the pieces breaking joint, that no two joints shall be opposite each o'her. This form of truss is very rarely used where the timbers may be seen from the room below, and they are therefore generally left rough. If they were to be planed, and made a part of the finish of the room below, it would be necessary to use solid tie-beams spliced together, or else build the truss of hard pine, of which wood, timbers may be obtained fifty or sixty feet long. The form of truss shown in Fig. 2 is the modern form of the old king post truss, shown in Fig. 4, which was made wholly of wood, excepting the iron straps used to connect the pieces at the joints. Queen Post Truss. — When the span to be roofed is between thirty-five and forty-five feet, a truss such as is shown in Fig. 5 is preferable, for several reasons, to the king post truss. It consists of a horizontal straining-beam, separating the upper ends of the principal rafters, and a rod at each end of* the straining-beam, leaving a large space in the centre of the beam clear. This is a great advantage in many cases where it is desired to utilize the attic for rooms. This form of truss should not be used for a span of over forty feet. For spans from forty feet to fifty feet, another form of the same truss, shown in Fig. (>, should be used. This is a very strong form of truss, and leaves considerable clear space in the centre. In this truss the principal rafter should be made of two pieces, — one running to the top, the other only to* the straining-beam. This gives the greatest economy in construe-396 WOODEN ROOF-TRUSSES. tion, and allows of forming a proper joint at B. It should be borne in mind that the strength of a truss depends largely upon the way in which the pieces are joined together, and that a truss may fail, simply through bad and improper joints. Fig. 7 shows a truss used in one of the old buildings in London over a room sixty feet wide. It is built of oak, and has wooden truss shown in Fig. 6. The actual dimensions of the various pieces of the truss are given in the figure. Fig. 8 shows a queen post truss supporting a portion of the roof of the Massachusetts Charitable Mechanics’ Association building in Boston, Mr. William C. Preston, architect. The timbers, which are of hard pine, have the dimensions shown in the figure. > r—i ,V:W 9 '9 'tzjJ Detail of joint at A (Fig. 6). Fig. 9 shows a queen post roof-truss, adapted to the suspension of a floor from the joints of the truss. As it was necessary to have the centre rod tq support the floor beneath, it then became necessary to put in the braces B, />’. The braces C C would only come into play when one of the extreme rods was loaded, and none of the others. These braces are called “counterbraces.” The manner in which the foot of the principal rafter is secured to the tie-beam is rather uncommon,WOODEN ROOF-TRUSSES. 397 and an enlarged detail of it is shown in Fig. 10. This truss is from the Museum of Fine Arts, St. Louis, Mo., Messrs. Peabody & Stearns, architects, Boston, Mass. For spans of from forty to eighty feet, a truss such as is shown in Fig. 11 is one of the best forms to adopt, where a pitch roof is desired. The struts should be largest towards the centre, and the tie-rods also. The main rafter, on the contrary, and the tie-beam, have the greatest strain at the joint A. Figs. 12 and 13 show details of joints A and C.398 WOODEN ROOF-TRUSSES. The trusses which have thus far been given are the simplest forms of modern trusses for spanning openings up to sixty or seventy-five feet in width, or even greater, where it is de- DETAIL OF JOINT "A” FIG.9 sired to have a pitch roof. At the present day, however, flat roofs are very extensively used; and, when it is desired to carry a flat roof, a different form of truss will be found more economical. SUITABLE FOR SPANS UP TO 80 FT.WOODEN ROOF-TRUSSES. 399 The form of truss generally employed for flat roofs is that shown in Figs. 14 and 15. This truss may be adapted to any span from not to be less than one-eighth of the span, and, if possible, should be made one-seventh, as the higher the truss, the less will be the strain on the chords.1 It should be noticed, that in this truss the braces are inclined in the opposite direction to that in which they are placed in the 1 The two horizontal pieces are called the “chords; ” the top one, the upper chord, which is always in compression; and the bottom one, the lower chord, which is always in tension.400 WOODEN ROOF-TRUSSES. trusses previously shown. The distances between the vertical rods should be so arranged that the braces shall not make an angle of more than forty-five degrees with a horizontal line. Fig. 16 shows the best method of forming the joints, A, A, A, B, II, B, etc. (Fig. !•"))> although not very frequently used in roof-trusses. For spans over forty feet, the tie-beam should be made up of plank bolted together, as shown in Fig. ?>, unless it is possible to have the tie-beam In one piece. This is a good form of truss for theatres, and large halls where there is a horizontal ceiling. Counter-Braces, — If it is desired to load the truss at any point other than the centre with a concentrated load,—as, for instance, suspending a gallery by means of rods from the roof-WOODEN ROOF-TRUSSES. 401 trasses,—the trass should have additional braces, called “counter-braces,” slanting in the opposite direction to the braces shown. These counter-braces need only be used when the trass is unsyni-metrically loaded. Wooden Trusses with Iron Ties. — In all trusses where the tie-beam of the truss is not horizontal, but higher in the centre ban at the ends, it is better to substitute an iron tie for the wooden tie-beam. Fig. 17 shows a form of truss very well suited for the roofs of carriage-houses, stables, or any place where it is desired to have considerable height in the centre of the room, and a ceiling is not desired. The horizontal iron rod is fastened to the two struts at their ends, and the other two rods are fastened only at their ends, and merely ran over the end of a strut in a groove. The iron rods are tightened by means of the turn-buckles shown on the drawing. Fig. 18 shows a detail of the upper joint A. A better way of making the joint would be to have an iron box cast to receive the end of the rafters, and fasten the ends of the tie. Arched Trusses with Iron Tie-Kods.— For buildings where it is desired to have the trusses and roof-timbers show, with no ceiling but that formed by the roof, a very pretty and graceful form of truss is obtained by the use of arched ribs, either for the principal chords of the truss, or for braces. In such trusses an iron tie-rod adds to the grace and apparent, lightness of the truss, and maybe very conveniently used. Fig. Id shows a form of truss used to support the roof of the Metropolitan Concert Ilall, New-York City, George B. Post, architect. The span of the truss iu402 WOODEN IlOOF-TRUSSES. that building is about fifty-four feet, and the proportions are about as shown in Fig. 19. The arcli between the rafter and the raised rib is ornamented with sawed work. The truss has a very light aifil airy appearance, besides embodying all the strength that can be desired in it. The tie-rod is kept from sagging by a vertical rod from the centre of the arch. Fig. ---------------------s a-------------------->i 20 shows a design for an open timber truss, with an arched rib, instead of braces, and an iron tie-rod from A to B. This tie-rod may either be let into the wooden beams C and D, for a distance of only about two feet, or, what would be better, run clear through the wooden beams to the outside of the truss. For trusses of very con siderable span, wooden arched trusses may be used with economy and effect. E Figs. 21 and 22 are good examples of this form of truss. The arched ribs support all the load that comes upon the truss, and the tie-rods prevent the ends of the arch from spreading, as would be the case if there were no tie-rods. The bracing between the arched ribs is simply to unite them, and distribute the stresses arising from the load proportionately over the two ribs. The frame-work shown above the arch in Fig. 21 is simply toWOODEN ROOF-TRUSSES. 403 support tlic purlins and rafters, and only carries the load directly to the arch. It does not assist the truss in any „way in carrying the load. The method of supporting the roof of the Fifth Avenue Riding-School, New-York City, is slightly peculiar and very ingenious; and, as it is an excellent example of the advantage of the arched foj^n of truss, we shall give a brief description of the construction of the roof and its supports. A plan of the riding-room is represented by Fig. 23. The room is one hundred and six feet six inches long, and seventy-three feet wide. <5 This space is kept en- k tirely clear of posts or columns; and the entire roof is supported by two large trusses, one of which is shown in Fig. 22. The roof between the trusses and on either side is supported by smaller trusses resting on these large trusses ; but each of the large trusses eventually carries a roof-area equal to about 2U30 square feet, and a great amount of extra frame-work. It was de-si red to provide for the thrust of these large arches without having rods showing in the room, and the method adopted is very ingenious. Opposite the upper ends of the iron posts which receive the arched ribs are oak404 WOODEN ROOF-TRUSSES. struts, which are held in place by iron tie-bars and heavy iron beams, which together form a horizontal truss at each end. These two trusses are prevented from being pushed out by two tliree-inch by one-incli tie-bars in each side wall shown in the plan (Fig. 23). r > Z (/> X o :§ 2 O T) 5 z o -o R* O w Tl o o 2 in H 73 C O The bottoms of the two iron posts are tied together by iron rods running wider the floor the whole length of the room. Altogether this gives for the tie-rods of each truss two bars three inches by one inch, and an inch and a half iron rod, which would be equivalent to two tie-bars three inches and tliree-fourths by one inch. Enlarged sections of the ribs, uprights, and braces, are shown in Fig, 22, It should be noticed that the uprights act bothWOODEN llOOF-TRUSSES. 405 as struts and tics, by having Iron rods through their centre holding the two ribs together. Fig. 24 shows a detail, or enlarged view, of the Iron skewback and post at each end of the truss shown in Fig. 22. Fig. 25 shows the method adopted for supporting the roof anti gallery of the City Armory at Cleveland, O. Open-Timber Trusses. — One of the principal characteristics of tin* Gothic style of architecture is that of making the structural portions of the building ornamental, and exposing the whole construction of an edifice to view ; and, as the pointed a rebel and steep, roofs were developed, the roof-truss became an important feature in the ornamentation of the interior of the Gothic churrhes. These trusses were built almost entirely of wood, and generally of very heavy timbers, to give the appearance of great strength. One of the simplest forms of these trusses is shown ill Fig. 20. As may be seen in the figure, the truss is really not much more than a40G WOODEN ROOF-TRUSSES. beam braced by brackets at the ends, though it does not, in its appearance to the eye, offer any suggestion of a beam. Fig. 27 shows another form of a small roof-truss ornamented according to the Gothic style. This is in principle a king post truss with brackets at the supports. Fig. 28 ' shows an early form of what is known as the “hammer - beam truss,” from the beam II, called the “ hammer-beam.” This truss differs in principle from all the trusses we have thus far described, in that it has no tie-beam, or no sidjstitute for one. The rafters are connected near the top of the truss by a short tie - beam ; but this would offer but little resistance to the rafters spreading at their lower ends: hence the truss must depend upon some outside force to keep it intact. This outside force is generally the resistance of the masonry FT/' „1 walls which support ^ the truss. These walls are generally very heavy, and are often re-enforced on the outside by buttresses built against the wall directly opposite the roof-trusses. In most cases such a wall possesses sufficient stability to withstand the thrust of the truss, and hence the tie-beam may be dispensed with ; but in a wooden building the walls offer no resistance whatever to being thrust out, unless tied at the top, and hence no truss which exerts an outward thrust on the wallsWOODEN ROOF-TRUSSES 407408 WOODEN ROOF-TRUSSES. should be used in such a building. It is therefore impracticable to use a hammer-beam truss in a wooden building. In roofs where this form of truss is used, the ceiling is generally formed of matched sheathing nailed to the under side of the jack-rafters between the purlins, thus allowing them to be seen. The purlins are generally decorated; and false ribs are often placed vertically between them, so as to divide the ceiling into panels. Fig. '2d, Plat« T., shows a hammer-beam truss, the same in principle as that shown in Fig. 28, only a little more developed, and more highly ornamented. This figure shows the ends of the hammer-beams carved to represent different kings.WOODEN liOOE-TIiUSSES. 40»410 WOODEN ROOF-TRUSSES. Figs. 26-29 represent trusses taken from old English churches; hut the lianimer-beam truss is also frequently used in this country to support the roof of Gothic churches. Fig. 30 represents half of one of the trusses in the First Church, Boston, Mass., Messrs. Ware & Van Brunt, architects. The truss is finished in black walnut, aiid has the effect of being very strong and heavy. Fig. 31 shows the framing of the same truss without any casing or falsework. It should be noticed that inside t)WOODEN ROOF-TRUSSES. 411 turned column, at the upper part of the truss (Fig. 30), there is an iron rod (Fig. 31) which holds up the joint A.1 In this form of truss the outward thrust of the arch enters the wall just above till corbel, K; and, as the direction of the thrust is inclined only about thirty degrees from a vertical, the tendency which it has to overthrow the wall is not very great, and may be easily resisted by a wall twenty inches or two feet thick, re-*u forced by a buttress on the outside. In trusses of this kind, the pieces should be securely fastened together whenever they cross or touch each other, and the whole truss made as rigid as possible. No dependence for extra strength should be made on the casings and panel-work. Fig. 32 shows a truss derived from a hammer-beam truss, in which the ceiling is made to take the form of a vault. Trusses of 1 The main rafters of this truss are two five-inch by thirteen-inch hard-pine timbers.412 WOODEN ROOF-TRUSSES. this kind, where there is no bracket under the hammer-beam, are not as stable as that shown in Fig. 30. Fig. 33 shows a form of truss used in Emmanuel Church at Shelburne Falls, Mass., Messrs. Van Brunt & Howe, architects, Boston. This truss was probably derived from the hammer-beam truss, and possesses an advantage over that truss in that it has in effect a trussed rafter, so that there is no danger of the rafter being broken; and, if the truss is securely bolted together at all its joints, it exerts but very little thrust on the walls. The rafters and cross-tie are formed of two pieces of timber bolted together, and the small upright pieces run in between them. The trusses in the church ait Shelburne Falls have the hammer-beams carved to represent angels.WOODEN ROOF-TRUSSES. 413 Fig. 34 shows a form of liammer-beam truss sometimes used in wooden churches. The braces 7HI are carried down nearly to tne floor, so that no outward thrust is exerted on the walls. It is generally better, however, in wooden buildings, to use a truss with a tie-rod; and, if an iron rod is used, it will not mar the effect of the height of the room seriously. If the roof-trusses are placed only about eight feet apart, the roof may be covered with two and a half inch spruce plank laid directly from one truss to the other without the intervention of jack-rafters or purlins. The planking can then be covered with slate or shingles on the outside, and sheathed within. Fig. 34 shows the roof covered in this way. Purlins are put in, however, flush with the rafters of the truss to divide the ceiling into panels. Fig. 35 shows a section through the roof of St. James’s Church, Great Yarmouth, Eng. The span is thirty-three feet, and the trusses are spaced about eight feet apart from centres.414 WOODEN ROOF-TRUSSES, The size of the scantlings are as follows : — Principals: Rafters.....................12 inches X 9 inches. Collars . . . . ... 9 U X 9 “ Ridge ... 12 m X 5 “ Purlins . . . . ... 8 u X 5 “ Cradling . . . . It X “ The roof is constructed of Memel timber.IRON ROOFS AND ROOF-TRUSSES. 415 CHAPTER XXVII. IRON ROOFS AND ROOF-TRUSSES, WITH DETAILS OF CONSTRUCTION. Owing to the increasing cost of lumber, and the necessity of erecting buildings as nearly fire-proof, and with as little inflammable material in the roof, as possible, it is becoming quite a common practice to roof large and expensive buildings with iron roofs, which, of course, involves the use of iron roof-trusses: hence it is important that the architect and progressive builder should have a general idea of the construction and principles involved in iron roof-trusses, and be familiar with the best forms of trusses for different spans, conditions of loading, etc. I-Beam. Deck-Beam. Channel-Bar. T-Bars. Resides being non-combustible, iron roof-tmsses are superior to wooden trusses in that they may be built much stronger and lighter, and are much more durable. Various forms of trusses have been constructed to suit different416 IRON ROOFS AND ROOF-TRUSSES. conditions of span, load, height, etc., and of these the following examples have been found to be the best and most economical. Before proceeding to describe these various forms of trusses, we would call the reader’s attention to the sections of beams, angle-irons, T and channel bars, shown in Fig. 1. It will frequently be necessary to refer to these sections; as they are the principal shapes of rolled iron entering into the construction of iron roofs, and it is of great importance that an architect or builder be familiar with their forms and names. For convenience In describing the different forms of iron roofs, we shall divide them into the following classes : — 1st, Truss-roofs with straight rafters, which are simply braced frames or girders. 2d, Bowstring-roofs tvith curved rafters of small rigidity, and with a tie-rod and bracing. 3d, Arched roofs, in which the rigidity of the curved rafter is sufficient to resist the distorting influence of the load without additional bracing. Trussed Hoofs. — For small spans, the most economical and simplest form of truss is that represented in Fig. 2. (Owing to the Fig. 2. small scale to which it is necessary to draw these figures, we have represented the pieces by a single line, which has been drawn heavy for strut-pieces, and light for ties and rods.) This truss was built by the Phoenix Iron Company for the' roof of a furnace-building. It consists of two straight rafters of channel or T bars, two struts supporting the rafters at the centre, a main tie-rod, and two inclined ties assisting the tie-rod to support the end of the struts. The lines on the top of the truss represent the section of a monitor on the roof, which is not a part of the truss, but only supported by it. One of the great merits of this truss is that it has but few pieces in compression, viz., the rafters and two struts : whieh is a condi-IRON ROOFS AND ROOF-TRUSSES. 417 tion very desirable in iron trusses, owing to the fact that wrought-iron resists a tensile strain much better than a compressive one, and hence it is more economical to use wrought-iron in the form of ties than in the form of struts. It should be borne in mind that for ties, rods or flat bars of iron are the most suitable; while for struts, it is necessary to use some form of section that offers considerable resistance to bending, such as a T-iron, or four angle-irons riveted together in the form of a cross; for wrought-iron struts always fail by bending or buckling, and not by direct crushing. In Figs. 2-10 the pieces which are struts, or resist a compressive strain, are drawn with heavy lines, and those pieces which act as ties are drawn with a light line. Fig. 3 represents a truss similar to that in Fig. 2, but having two struts instead of one, which is more economical where the span is over fifty-six feet, for the reason that it allows the rafters to be made of lighter iron. For spans of from seventy to a hundred feet, the form of truss shown in Fig. 4 has been found to be about the most economical and satisfactory in every respect. Fig. 4. The rafters in this truss, for moderate spans, may be T-irons; and for larger spans, channel-bars and the ties and struts may be bolted to the vertical rib. For very large spans, channel-bars may be used, placed back to back, with the ends of the bracing bars between them. I-beams are also used for the rafters, but they have the objection of not being in a shape to connect readily 'with the other forms of iron. The flanges of an I-beam do not offer so good an opportunity for riveting as do those of angle and T irons and418 IRON ROOFS AND ROOF-TRUSSES. channel-bars. The ties are rods of round iron or flat bars; and the struts, commonly T-irons or angle-irons bolted together. Fig. 5. Another form of truss, shown in Fig. 5, derived from the wooden queen post truss, is very commonly used for spans of from sixty to a hundred and forty feet. A modification of this truss is shown in Fig. 6, in which both struts and ties are inclined, instead of only the Fig. 6. struts, as in Fig. 5. The truss in Fig. 6 has the advantage that the struts are shorter, more nearly perpendicular to the rafters, and less strained. Bowstring-Roofs.—In designing iron roofs, it is sometimes desired to vary the ordinary straight pitch roof by using a curved rafter. Two examples of such roofs are shown in Figs. 7 and S, Fig. 7. which were constructed by the Phoenix Iron Company of Philadelphia. These may be considered as the simplest forms of bowstring-roofs. The principal use of the bowstring-roof proper is for roofingIRON ROOFS AND ROOF-TRDSSES. 419 very large areas in one span, snch as is often desired in railway-stations, skating-rinks, riding-scliools, drill-halls, etc. < MARKET-HOUSE, TWELFTH AND MARKET STREETS, PHILADELPHIA. Fig. 8. Fig. 9 represents the diagram of a bowstring-truss of a hundred and fifty-three feet span. The trusses in this particular case are spaced twenty-one feet six inches apart. The arched rafter consists of a wrought-iron deck-beam nine inches deep, with a plate, ten inches by an inch and a fourth, riveted to its upper flange. Towards the springing, this rib was strengthened by plates, seven inches by seven-eighths of an inch, riveted to the deck-beam on each side. The struts are wrought-iron I-beams seven inches deep. The tie-rods have six and a half square inches area, and the diagonal tension-braces are an inch and a fourth diameter. These trusses are fixed at one end, and rest on rollers at the other, permitting free expansion and contraction of the iron under the varying heat of the sun. Fig. 10 shows a similar truss having a span of two hundred and twelve feet. It consists of bowstring principals spaced twenty-420 IRON ROOFS AND ROOF-TRUSSES. four feet apart. The rise is one-fiftli the span, the tie-rod rising seventeen feet in the middle above the springing, and the curved rafter rising forty feet and a half. The rafter is a fifteen-inch wrought-iron I-beam. The tie is a round rod in short lengths, four inches diameter, thickened at the joints. The tension-bars of the bracing are of plate-iron, five inches to three inches in width, and five-eighths of an inch thick. The struts are formed of bars having the form of a cross. The following table, taken from Unwin’s “ Wrought-iron Bridges and Roofs,” gives the principal- proportions of some notable bowstring-trusses, mostly in England: — PROPORTIONS OF BOWSTRING-ROOFS. Location. Clear span. 1 Rise of rafter. Depth of truss. o li Rafter. 1 o c5 1 cS -3 j O § CO i £ £ B £ t- ci X X p 5 ft. ft. ft. ' ft. in. in. sq. in. j sq. 91 Lime Street .... m 30 12 m 1 10 14 a 0 h Birin Inghaui .... 211 40j 23 24 15 ■ 35 12* 1 Cannon Street .... mo 60 30 331 21 14 28 (’haring Cross .... 164 45 20 35 IS 12.4. 25 - Blackfriar’s Bridge . . 87.', od 9 ■ 6 IH - - London Bridge . . . 88 27 18 18 7 104 5 Amsterdam 120 30 13 For spans much exceeding a hundred and twenty or a hundred and thirty feet the bowstring-truss is much the most economical, and advantageous to use. Arched Root's. — These roofs consist of trusses in the form of an arch, having braced ribs, which possess sufficient rigidity in themselves to resist the load upon them. The thrust of these large ribs, however, has to be provided for, as in the case of masonry arches, either by heavy abutments or by tie-rods. As these trusses embrace the most difficult problems of engineering, and are rarely used, we have thought best not to give any examples of such trusses. If any reader should have occasion to visit the Boston and Providence Railroad Depot at Boston, he can there see an admirable example of this form of truss. At springing twenty-five square inches.IRON ROOFS AND ROOF-TRUSSES. 421 Details of Iron Trusses. After deciding upon the form of truss which it will be best to use, the shape of the iron to form the different members is a matter to he considered. There are many practical reasons which make it desirable to use certain shapes of iron in constructing iron trusses, even though those shapes may not be the most desirable in regal'd to strength; so that a knowledge of the details of iron trusses is requisite for any one who wishes to become a master of building construction. By far the best way to study the details of construction is to observe work already built and that which is in process of construction; but this requires considerable time, and often the thing one wants cannot be found at hand. The following details of the various ways of joining the different members of iron tiuiscs will be found useful. There are two general methods of constructing iron trusses. One is to make all the parts of the truss of combinations of angle-irons, channel-bars, and flat plates, and rivet them together at the joints, so that the truss will consist of a frame-work of iron bare all riveted together. The other method is to use channel-bars, T-irons, l-beams, etc., for the rafters and struts, and rods for the ties, which are connected at the joints by eyes and pins. Fig. 11. In the first method the ties are either made of flat bars or angle-irons. Fig. 11 shows two ways in which the tie-rod is secured to the foot of the rafter in the second method of construction. A casting, forming a sort of “shoe,” is made, in which the rafter fits, and the tie is secured to the “shoe” by means of an eye-end and pin; or a plate may be bolted to each side, and the whole rest on an iron plate. Of course the tie must in either case consist of two bars, one on each side of the shoe.422 IRON ROOFS AND ROOF-TRUSSES. Fig. 12 illustrates two ways of fastening the upper ends of the struts to the rafters. In the first method the casting is made to fit inside the strut, and is bolted to the bottom of the rafter. Fig. 13 shows the joints at the foot of the struts, as made in the second method. The peaks in either method are secured by means of fish-plates riveted to both rafters (Fig. 14). Fig. 14. Fig. 15 shows the proportions for eyes and screw ends for tension- bars as used in this method of construction.IRON ROOFS AND ROOF-TRUSSES. 423 Figs. 16 and 17 show the manner of forming the joints in the first method of construction. Fig. 10 represents the joint at the Fig. 16. bottom of the main rafter; and Fig. 17, the joint where a rafter, straining-beam, tie, and strut come together. All the pieces are securely riveted to a piece of plate-iron, which thus holds them together. The' other joints are formed in a similar way. Which is the better method of construction depends very much on circum- stances.. In roofs of wide span, provision for expansion of the iron, due to changes of temperature, may be made by resting the skewback of one end of the truss on a cast wall-plate, with rollers interposed to permit of the sliding of the shoe without straining the wall, as in Fig. 18; but tills precaution is not necessary in roofs of sixty feet span or less. Careful experiments have proved that an iron rod one hundred feet long will vary about a tenth of a foot for a change of temperature of a hundred and fifty degrees F.; and, as this is the greatest range to which iron beams and rods in a building would probably be subjected in this climate, compensation to that amount would be sufficient for all purposes. For sixty feet span,424 IRON ROOFS AND ROOF-TRUSSES. the vibration of each wall would then he only fifteen-thousandths of a foot either way from the perpendicular, — a variation so small, and so gradually attained, that there is no danger in imposing it upon the side-walls by firmly fastening to them each shoe of the rafter. Expansion is also provided against by fastening down one »hoe with wall-bolts, and allowing the other to slide to and fro on ihe wall-plate without rollers. After the trusses are up, there are various ways of constructing the roof itself. If the roof is to be of slate, it is best to space the trusses about seven feet apart, and use light angle-irons for purlins, •which are spaced from seven to fourteen inches apart, according to the size of the slate. On the iron purlins the slate may be laid directly, and held down by copper or lead nails clinched around the angle-bar; or a netting of wire may be fastened to the purlins, and a layer of mortar spread on this, in which the slates are bedded. When greater intervals are used in spacing rafters, the purlins may be light beams fastened on top or against the sides of the principalsIRON ROOFS AND ROOF-TRUSSES. 425 'with brackets, allowance always being made for longitudinal ex-« pansion of the iron by changes of temperature. On these- purlins are fastened wooden jack-rafters, carrying the sheathing-hoards of laths, on which the metallic or slate covering is laid in the usual manner; or sheets of corrugated iron may be fastened from purlin to purlin, and the whole roof be entirely composed of iron. When the rafters are spaced at such intervals as to cause too much deflexion in the purlins, they may be supported by a light beam placed midway between the rafters, and trussed transversely with posts and rods. These rods pass through the rafters, and have bevelled washers, screws, and nuts at, each end for adjustment. By alternating the trusses on each sid< of the rafter, and slightly increasing the length of the purlins above them, leaving all others with a little play in the notches, sufficient provision will be made for any alteration of length in tin roof, due to changes of temperature. When wooden purlins are employed, they may be put between the rafters, and held in place by tie-rods on top^and fastened to tiie rafters by brackets; or liook-liead spikes may be driven up into tlie purlin, the head of the spike hooking under the llange of the beam, spacing-pieces of wood being laid on the top of the beam from purlin to purlin. The slieatliing-boards and covering are then nailed down on top of all in the usual manner. Fig. 20.426 THEORY OF ROOF-TRUSSES. £ * r ' ^ I CHAPTER XXVIII. THEORY OF ROOF-TRUSSES. In this chapter it is proposed to give practical methods for computing the weight of the roof with its load, and the proportion of the truss and its various parts. The first step in all calculations for roofs is to find the exact load which will come upon each truss, and the load at the different joints. The load carried by one truss will be equal to the weight of a section of the roof of a width equal to the distance between the trusses, together with the weight of the greatest load of snow that is ever likely to come upon the roof. In warm climates, of course, the weight of snow need not be provided for.1 It is a very common practice to assume the maximum weight of the roof and its load at from forty to sixty pounds per square foot of surface ; but, while this may be sufficiently accurate for wooden roofs, it would hardly answer for iron roofs, where the cost of the iron makes it desirable to use as little material in the truss as will enable it to carry the roof with safety, and no more. The weight of the roof itself can be easily computed, and a sufficiently accurate allowance can be made for the weight of the truss ; and, if the roof is to be in a climate where snow falls, a proper allowance must be made for thatand, lastly, the effect of the wind on the roof must also be taken into account. Mr. Trautwine says, that within ordinary limits, for spans not exceeding about seventy-jive feet, and with trusses seven feet apart, the total load per square foot, including the truss itself, purlins, etc., complete, may be safely taken as follows : — Roof covered with corrugated iron, unboarded ... 8 pounds. If plastered below the rafters....................18 “ Roof covered with corrugated iron oi’iboards .... 11 “ If plastered below the rafters....................18 “ Roof covered with slate, unboarded, as on laths . . .13 “ Roof covered with slate on boards H inches thick . . 16 “ Roof covered with slate, if plastered helow the rafters . 26 “ Roof covered with shingles on laths....................10 “ If plastered below the x’afters, or below tie-beam . 20 “ Roof covered with shingles on J-incli board .... 13 “ /428 THEORY OE ROOE-TRUSSES. wind: hence the resultant of the wind pressure must act in a direction normal (at right angles) to the face of the roof. In this [country the wind seldom blows with a pressure of more than forty pounds per square foot on a surface at right angles to the direction of the wind ; and it is considered safe to use that number as the greatest wind pressure.1 But the pressure .on the roof is generally much less than this, owing to the inclination of the roof. The following table gives the normal wind pressure per square foot on. surfaces inclined at different angles to the horizon, for a horizontal wind pressure of forty pounds per square foot. NORMAL WIND PRESSURE. Angle or ltoor. Normal pressure. Angle or Roof. Normal pressure. Degrees. Rise in one foot. Degrees. Rise in one foot. ! 5 1 inch. 5.2 lbs. 35 8J inches. 30.1 lbs. 10 2) inches. . 0.0 “ 40 10 33.4 “ 15 :H “ 14.0 “ 45 12 30.1 “ 20 42 “ 18.8 “ 50 iiA “ 1 (j 38.1 “ 25 54 “ 22.5 55 171 “ i 1 A 30.0 “ :>o n 1 << o-nx 20.5 “ 00 20 J “ 40.0 Until of late years it has been the general custom to add the wind pressure in with the weight of snow and roof ; and, although this is evidently not the proper way to do, yet for wooden trusses it gives results which are perhaps sufficiently accurate for all practical purposes; and, if caution is taken to put in extra bracing l wherever any four-sided figure occurs, this method will answer i very well for wooden trusses." For iron trusses, however, the ! strains in the truss due to the vertical load on the truss, and those due to the wind pressure, should be computed separately, and then , combined, to give the maximum strains in the various pieces of the truss. It should be borne in mind that a horizontal wind pressure of forty pounds per square foot is quite an uncommon occurrence, and, when it does occur, generally is of short duration ; so that a truss which would not withstand this pressure, if applied for a long 1 At tile observatory, Didst on, Liverpool, the following wind pressures per square foot have been registered. ISPS, Fob. 1, 70 pounds; Feb. £2, Go pounds; Dec. '27, 80 pounds. 1870, Sept. 10, Go pounds; Oet. IS, 05 pounds. 1871, March 9, 90 pounds. 1875, Slept. 27, 70 pounds. 1877, Jan. 30, 03 pounds; Nov. 23, 03.5 pounds. — Aukuk'AN Ahchitkct, vol. xv. p. 237.THEORY OF ROOF-TRUSSES. 1 429 time, may possess sufficient elasticity to withstand the strain for a' short time without injury. In very exposed positions, such as on high hills or mountains, where the force of the wind is unobstructed, the roofs of all lugu buildings should be especially designed to withstand its powerful effects. tira phical Analysis of 1'oof-Trusses. — The simplest, and in most cases the readiest, way of computing the strains in trusses, is by the graphic method, which consists in representing the loads and strains by lines drawn to a given scale of pounds to the fraction of an inch. We think the graphic analysis of roof-trusses may be best shown by examples, and hence shall give a sufficient variety to show the method of procedure for most of the trusses already described ini these articles. Example 1. — As the simplest case, we will take the truss shown in Fig. 4, Chap. XXVI. If we should draw a line through the centre of each piece of this truss, we should have a diagram such as is shown in Fig. 1. We will suppose that this truss has a span of :14 feet, and the rafters have an inclination of 45° with a horizontal line. Theft the length of the rafter would be 24 feet; and, if the trusses were 12 feet apart, one truss would support a roof-area of 12 x 24 X 2 = 570 square feet. Now, if we look at Fig. 1, we can see that the purlin or plate at A or E would carry one-lialf of the roof from A to IS. The purlin at 11 would carry the roof from a point midway between A an 1 II to a point midway between 11 and C, which would be one-fourth the area of roof supported by each truss.THEORY OF ROOF-TRUSSES. 427 For spans of from seventy-five to one hundred and fifty feet, it | will suffice to add four pounds to each of these totals. The weight of an ordinary latli-and-plaster ceiling is about ten pounds per square foot; and that of an ordinary floor of an inch and one-fourth boards, together with the usual three by twelve joist, fifteen inches apart from centre to centre, is from ten to twelve pounds per square foot. White-pine timber, if dry, may be considered to weigh about twenty-five pounds; Northern yellow pine, thirty-five pounds; and Southern yellow pine, forty-five pounds per cubic foot. Ordinary spruce may be considered to weigh twenty-six pounds per cubic foot. Oak may be reckoned at from forty I to fifty pounds; cast-iron, at four hundred and fifty pounds; and ! wrought-iron, at four hundred and eighty pounds per cubic foot. For flat roofs, the weight per square foot of the various roofing I materials on seven-eighths inch boards, not including the rafters or joist, may be taken as follows : — Roof covered with tar and gravel over 4 thicknesses of felt,. 9!> lbs. “ “ “ M. F. tin . .•..................3£ “ “ “ “ cotton duck (12 ounce)..........2f “ j “ “ “ 16-ounce copper.................3J “*l!| From this data the weight of the roof itself may be easily computed. and we have then only the weight of the snow and effect of winds to allow for. t Snow. — Any allowance for the weight of snow must depend j upon the latitude. It may accumulate in considerable quantities, becoming saturated with water, and turning to ice. The weight of a cubic foot is very various. Freshly fallen snow may weigh from five to twelve pounds. Snow and hail, sleet or ice, may weigh I from thirty to fifty pounds per cubic foot; but the quantity on a I roof will usually be small. Snow saturated with water will usually ’ slide olf from roofs of ordinary pitch. An allowance of from twelve to fifteen pounds per square foot of roof will suffice for most latitudes. j 1 Wooden trusses frequently support an attic-floor, and in such I cases the weight of the floor and its greatest probable load should ■ be considered as applied at the joints of the truss. Wi ml Pressure. —The load on the roof, thus far considered, is a steady dead load, which of course acts in a vertical direction, < Rut roofs are frequently subjected to great pressure from the force I of the wind ; and, as this can act only on one side of the roof at a time, it is an unsymmetrical load, and moreover it does not act i vertically. The pressure of the wind on an inclined surface is always normal to the «jiirfactvno matter what the direction® the 1THEORY OF ROOF-TRUSSES. T 430 The purlins C and D would also support the same amount of roof. If we consider the roof to he slated on hoards an inch and a fourth thick, we shall have for the weight of one square foot 16 pounds ; allowing for snow, 15 pounds ; normal pressure of wind, 36; total weight or load on one square foot, 67 pounds ; total weight supported by one truss, 67 X 576 = 38,592 pounds ; total load coming at each of the points li, C, and I), one-fourth of 38,592 = 9648 pounds. The load coming at A and E is supported directly by the walls of the building, and need not be considered as coming on the truss at all. If, now, we draw a vertical line on our paper, and, commencing at the upper end, lay off 9648 pounds at some convenient scale, say |5000 pounds to the inch (in the following figures different scales have been used to keep the diagrams within the limits of the page, but were first drawn to a large scale to get thes tresses more accurately), and then one-half of 9648 pounds* or 4824 pounds, to the same scale, we shall have the line ac (Fig. la) representing just half the load on the truss, or the load coming on each of the supports. Now, that the forces acting in the rafter and tie-beam, and the supporting forces, all coming together at the point A, shall balance each other, they must be in such a proportion, that if we draw a line from a parallel to the rafter, and a line through c parallel to the tie-beam, the line ad must represent the thrust in the lower part of the rafter, and the line dc, the pull in the tie-beam. If we next consider the forces acting on the joint li, commencing with the rafter, and going around to the right, we find that the first force which we know, is the force in the rafter, represented in Fig. la by the line da. Next we have the weight, 9648 pounds, acting down, represented by the line ah, and there remain two unknown forces,—that in the upper part of the rafter and the force in the strut. To obtain these forces, draw a line through b (Fig. la), parallel to the rafter, and a line through d, parallel to the strut. These two lines will intersect in e; and the line be will represent the force in the rafter, and the line ed the force in the strut. Furthermore, if we follow the direction in which the forces act, we shall see that the force da acts up : hence the rafter is in compression. The remaining forces must act around in order : hence ab acts down. be acts towards the joint, and ed acts up towards the joint, so that both pieces are in compression. Next take the forces acting at the point ('. The first force we know is eh. which acts up ; next we have (lie weight, 9648 pounds,THEORY OF ROOF-TRUSSES. 431 which would extend beyond c to /; then there remain the forces in the rafter to the right, and the vertical tie, which are determined by drawing a line through / parallel to the rafter, and a line tlirough e parallel to the tie. These two lines intersect in i; and the line if will represent the force in the rafter, and ei will represent, the pull in the tie. We have now only to measure the lines hi our diagram of forces, and we have the forces acting in every part of till truss; as, of course, the corresponding pieces on the different sides of the truss would be similarly strained. Measuring the different force-lines by the same scale we used in laying off the weight, we find the strains as shown by the figures on the lines, Fig. In. Having found the strain-pressure in the different parts of the truss, it is very easy to determine wliati should be their dimensions. Thus tlie compression in the foot of the rafter is 20,750 pounds. Now, if we wish to make it of hard pine, we know that hard pine will safely bear 1000 pounds to the square inch; and lienee we shall need Yi/imP = 21 square inches area in the rafter. This would require only a 3 by 7 timber ; but, as the rafter will need to be cut into more or less, we will give it more area, and call it a 6 by 6. Till short struts have a pressure of 6,900 pounds, and hence need not be larger than a 2 by 4, except, that, being so thin, it is liable to bend; and so we will make it 4 inches by 6 inches. The tie-beam resists a pull of 14,700 pounds; and, as hard pine will safely withstand a tensile strain of 2000 pounds, we should only need about eight square inches of area: but, while this would resist the pull, we must add enough more to allow for cutting into the tie at the joints, and for sagging under its own weight; so that we w ill make the beam out of a 6 inch by 6 inch timber. The centre t ie, which has to resist a pull of 9648 pounds, we will make of wrought-iron instead of wood, as shown in Fig. 4, Chafl XXVI.; and, as wrought-iron may be safely trusted with a pull of 10,000 pounds to the. square inch of cross-section, we -shall need a rod having a sectional area of not quite one square inch, or a rod of an inch and an eighth, or an inch and a fourth, in diameter. If tin1 rafter and strut had been of spruce, we should have divided the strain by 800 pounds, or 700 if of white pine ; and for the tie we should have divided the pull by 1800 if spruce was to be used, anil by 1500 if we intended to use white pine. It will be noticed, that, while we determine the size of our timbers mathematically, it often happens that we must make them considerably larger to prevent their bending under their own weight, and to allow for cutting, boring, splicing, etc.; so that it w ill not do to depend entirely upon mathematical deductions, but432 THEORY OF ROOF-TRTJSSES. these should be supplemented by a practical knowledge of the subject. The methods of determining the strains in this truss applies to all trusses properly put together, and which do not exert an outward thrust on their supports. Example 2. — For further illustration we will take the truss shown in Fig. 5, Chap. XXVI., and of which a diagram is given in Fig. 2. We will assume that it has a span of 45 feet, and other dimensions as given in the figure ; also that the trusses are placed 12 feet apart from centres. I3y glancing at Fig. 5, Chap. XXV L, it will be seen that the purlin at 2 (Fig. 2) carries the weight of that portion of the roof extending from halfway between purlins 1 and 2 to the ridge of the roof, and in this case equal to 13A X 12 = 102 square feet. The purlin at 1 supports the roof for 4$ feet each side of it, or 9 X 12 = 108 square feet. This would bring a pressure of 10,854 pounds at the joint 2, and 7230 pounds at joint 1. Besides this, we have a ceiling suspended from the tie-beams of the truss, which would weigh about twenty pounds to theTHEORY OF ROOF-TRUSSES. 433 square foot more. This weight would he supported one-third at each of the joints 3 and 4, and one-sixth at each end of the truss. The weight of the ceiling, coming at joints 3 and 4, may be assumed to be hung from joints 2 and 5 by means of the vertical rods: so we can add the weight coming from the ceiling to the weight of the roof, and consider it as applied at the points 2 and 5. The whole area of the ceiling is 12 X 45 = 540 square feet, and its weight about 9000 pounds ; making 3000 pounds applied at 3 and 4, and the total load at 2 and 5, 13,854 pounds. The load at 1 we have already determined to be 7236 pounds. This gives us sufficient data with which to draw our diagram of strains. As in Example 1, first lay off the loads on a vertical line, to some convenient scale ; thus, ad (Fig. 2a), load at first purlin, L and de, the loads at 2 and 3 combined. Then ae represents half the weight supported by the truss, and also the load coming upon each support. To draw the strains, first draw ab (Fig. 2a) parallel to AB (Fig. 2), and a horizontal line through e, intersecting ab in b; next go to the joint 1, and we have the force ba, acting upwards ; then the load ad; then from d, the stress in DC, which must act in a direction parallel to it, and the stress in BC, also acting parallel to it. These last two stresses are found by drawing a line through d parallel to DC, and a line through b parallel to BC. Note. — In Fig. 2 the lines are denoted by the letters either side of them; thus the bottom of the rafter on the left is called AB, and the brace BC; the left upright tie is denoted CF, and the right one FG. In the diagram of strains, the line representing the strain in any piece is denoted by the same letters as the piece, with the difference that small letters are used for the strain diagram, and the letters come at the ends of the lines. This method of notation (known as “Bow’s Notation”) is very convenient, and aids greatly in following out the strains. Next take the strains in the pieces at the joint 3. We know already the strains kb and be, and drawing the line cf parallel to C F, and kc parallel to KC, we have the strains in the remaining pieces. It will be noticed that the line ef lies over the line eb; but it should be kept in mind that they represent two separate strains, and should be measured separately. Considering next the strains at joint 2, we find we already have/c, cd, and de (13,254 pounds), leaving only kf to close the figure ; thus showing that the strain in the beam EF is the same as that in the tie FK, though the former is a compressive strain, and the latter a pulling one. We now have the strains in all the pieces of the truss, represented by the corresponding lines in Fig. 2a, and, measuring these434 THEORY OF ROOF-TRUSSES. by our scale of pounds, we find them to be as shown by the figures on the lines in the strain diagrams. Then, if the truss were to be built of spruce, we should need 37800 ■yjjo“ = 47 square inches of section, at least in the main rafter, 20800 _ . ,31300 ■gOQ' = 33 square inches, in the straining-beam, and = 18 square inches, in the end of the tie-beam. Knowing these least dimensions, we can modify them to allow for cutting, joints, sagging, etc., according to our judgment. Thus we would make the rafters and straining-beam 0 inches by 10 inches, the tie-beam 6 inches by 10 inches, and the braces 0 inches by 6 inches. The rods have a strain of 3700 pounds plus the direct pull of 3000 pounds; making 6700 pounds’ pull on the rod, which would require a rod one inch in diameter. 12 7,200 1 r cx 2< D. yB\ 240 12 F 5 1 ,240 's.K 7f00 4 - / LX /A \h g\ 0 0 —^-----= 11,733 pounds. This would require a rod an inch and one-fourth in diameter with the screw-ends upset. The rod should have a turn-buckle at T. 6f* 4800 The beam B would have a compressive strain ---------------= O 10,000 pounds, which would require a beam about 1}- inch by 8 inches; but, as the beam has also to carry the weight of the jack-Fig. 15. 4 52 theory of KOOF-TKU ssks. rafters between two points of support, we slia.ll lie OTill "0(1 to use n. six-incb by eiglit-ineli timber for tlie straining-beam of our x^tii'liii- ess ioTHEORY OF ROOF-TRUSSES. 453 The struts we will make, as shown in Fig. 14, of cast-iron, the thickness of the iron being one-half of an inch. Iron Truss. — For an iron truss required to meet the conditions of this case, a form such as is shown in Fig. 15 will be found the most economical. The tie-rod is raised slightly in the centre, and the bracing is arranged so as to bring the struts nearer at right angles with the rafters. The length of the rafters, position of the purlins, and distance between trusses from centres, are the same as in the case of the wooden truss: hence the loads on the truss will be the same. As we have slightly changed the position of the tie and bracing, however, we shall be obliged to make a new strain sheet. The method of drawing it is so similar to that for the wooden truss, that we shall not describe the necessary steps. It should be noticed, that, where we had one vertical tie at the centre of the wooden truss (Fig. 11), we have two in the iron truss, slightly inclined: hence the strain on each one is only fi (Fig. 15a), a little more than one-half of the strain on the rod FI (Fig. 12). The stresses in the different pieces of the truss are indicated by the figures on the corresponding lines (Fig. 15a). In Fig. 15 the heavy lines denote the pieces which are in compression, and the light lines those which are in tension. The dotted lines refer only to the measurements. Taking first the pieces which are in compression, we find the greatest compression on any part of the rafter is 85,300 pounds on the lower length, and 58,200 pounds on the upper length. We shall probably find that the best way in which to build up the rafter will be to use two ][ back to back, which are capable of resisting the compression on the middle length, and bolting a plate on each side to give the additional strength required in the lower length. For this truss we will use the Trenton shapes of rolled iron. To compute the required dimensions of the channels for our rafter, we have the formula (p. 235), 4 X a X r Safe load — _|_ • The values of a and r we shall find in the last column of the table on p. 238, as we shall keep the channels an inch apart. The length of each section of the rafter is 11.5 feet.. We wMl first assume two heavy nine-inch channels, and see if they will answer. For these we have, 4 X 12.90 X 687 Safe load, in tons = 25 + 687 = ^ tons, or 86,000 pounds. This is a little stronger than we require: so we will try two nine-454 THEORY OF ROOF-TRUSSES. inch light channels. These we find, only have a safe load of 28 tons, or 50,000 pounds: so we must use the nine-inch heavy channels, which are strong enough for any part of the rafter. To be absolutely sure that our rafter has sufficient strength, however, we will rivet a six-incli by three-eighths of an inch plate to the upper flanges of the channels, for the lower length of the rafter. The strut EF has 17,000 pounds’ compression upon it, and is ten feet long. • Assuming two four-inch sixteen and a half pound channels, placed an inch apart, back to back, we find their 4 X 3.07 X 294 Safe load, in tons = —ioo + 294— = ® tons, or 1S00 pounds. These we will therefore use. The strut AD has but 13,100 pounds’ compression upon it, and is only seven feet long; so that, for this, two tliree-incli channels placed back to back will be amply sufficient. For the ties we will use angle-irons. The greatest strain in the main tie is 74,500 pounds, which requires only seven square inches of cross-section; and hence, if we use for our main tie two light four-inch by six-inch angles, we shall have ample strength under all circumstances. For the tie FI we will use two bars three inches by one-half inch, giving a total cross-section of three square inches. For the t ie DE we will use two bars two inches by three-eighths of an inch. The joints of the truss will be formed by riveting a thick piece of iron plate between the channels, and riveting the struts and ties to that, after the method shown in Figs. 10 and 17, Chap. XXVII. With this example we leave the subject of roof-tritsses. As we have stated before, the method of finding the strains due to wind pressure alone we have not given, because, in the trusses which come especially before the architect or builder, the methods here given we believe sufficiently accurate. Any one wishing to learn t he method of drawing the diagram of strains due to wind pressure alone will find it fully explained in Green's “ Graphical Analysis of Roof-Trusses.” 1 1 Published by Jolm Wiley & Sons, New York.JOINTS. 455 CHAPTER XXIX. JOINTS. The stability of any piece of frame-work depends in a very great measure upon the manner in which the joints are made. It is therefore very important, that in drawing trusses, or framework of any kind, the designer should have a good knowledge of the fundamental principles upon which every joint should be constructed, and of the most approved methods of forming the principal joints found in frame-work.1 Joints are the surfaces at which the pieces of a frame touch each other. They are of various kinds, according to the relative positions of the pieces and to the forces which the pieces exert oil each other. Joints should he made so as to give the largest bearing-surfaces consistent with the best form for resisting the particular strains which they have to support, and particular attention should be paid to the effects of contraction and expansion in the material of which they are made. In planning them the purpose they are to serve must he kept in mind, for the joint most suitable in one case would oftentimes be the least suitable in another. JOINTS IN TIMBER-WORK. In frames made of timber, the pieces may be joined together in three ways, —by connecting them; 1. End to end; 1 As the author could think of no better way in which to present the subject, he has taken, by permission of Professor Wheeler and of the publjshers, the following chapter on joints from the text-book, on Civil Engineering, prepared by Professor Wheeler for the use of the cadets of the United-States Military Academy, and published by John Wiley & Soii£ of New York. The author heartily recommends Professor Wheeler’s work to the architect or builder who wishes to obtain a thorough knowledge of construction and the materials employed therein.,456 JOINTS. » 2. The end of one piece resting upon or notched into the face of another; and 3. The faces resting on, or notched into each other. I. »Joints of Beams united End to End, the axes of the beams being in the same straight line. First, Suppose the pieces are required to resist strains in the direction of their length. This case occurs, when,.in large or long frames, a single piece of the required length cannot be easily procured. The usual method of lengthening is in this case by fishing or scarfing, or by a combination of the two. Eisli-Joints.— When the beams abut end to end, and are connected by pieces of wood or iron placed on each side, and firmlyJOINTS. 457 boiled to the timbers, the joint is called a fish-joint, and the beam is said to be fished. This joint is shown in Fig. 1, and makes a strong and simple connection. When the beams are used to resist a strain of compression, tire tish-pieces should be placed on all four sides, so as to prevent any lateral movement whatever of the beams. If the strain be one of tension, it is evident that the strength of the joint depends principally upon the strength of the bolts, assisted by the friction of the fish-pieces against the sides of the timber. The dependence upon the bolts may be much lessened by notching the fish-pieces upon the beams, as shown on the upper side of the piece in Fig. 3 ; or by making use of keys or blocks of hard wood inserted in shallow notches made in both the beam and fish-piece, as shown on the lower side of the piece in tlie same figure.458 JOINTS. •Care should be taken not to place the bolts too near the ends of the pieces. The sum of the areas of cross-sections of the bolts should not be less than one-fiftli that of the beam. Scarf-Joints. — In these joints the pieces overlap each other, and are bolted together. The form of lap depends upon the kind of strain to which the beam is to be subjected. Fig. 4 is an example of a simple scarf-joint that is sometimes used when the beam is to be subjected only to a slight strain of extension. A key or folding wedge is frequently added, notched equally in both beams at the middle: it serves to bring the surfaces of the joint tightly together. This joint is often made by cutting the beams in such a manner as to form «projections which fit into corresponding indentations. A good example, in which two of these notches are made is shown in Fig. 5.JOINTS. 459 Tli« total lap shown in this figure is ten times the thickness of the timber, and the depth of the notches at A and /1 are each equal to one-fourth that of the beam. The bolts are placed at right angles to the principal lines of the joint. This is a good joint where a strain of tension of great intensity is to be resisted, as, by the notches at A and B, one-lialf of the crosc-section of the beam resists the tensile strain. Combination of Fisli and Scarf «Joints. — The joint shown in Fig. 6 is a combination of the fish and scarf joints, and is much used to resist a tensile strain. Second, Suppose the pieces are required to resist a transverse strain. In this case the scarf-joint is the one generally used, and it is then formed sometimes by simply halving the beams near their ends, as shown in Fig. 6. The more usual and the better form of joint for this case is sliown.in Fig. 7. In the upper portion of this joint the abutting surfaces are perpendicular to the length of the beam, and extend to a depth of at least one-tliird, and not exceeding one-lialf, that of the beam. In the bottom portion they extend one-tliird of the depth, and are perpendicular to the oblique portion joining the upper and lower ones. 'I'lie lower side of the beam is fished by a piece of wood of iron plate secured by belts or iron hoops, so as to better resist the tensile strain to which this portion of the beam is subjected. Fig. 8. Itopresonts a scarf-joint arranged to resist a cross-strain and one of extension. Tile bottom of the joint is fished by an iron plate; and a folding wedge inserted ut c serves to bring all the surfaces of the joint to their bearings. Third, Suppose the piece required to resist cross-strains combined with a tensile strain. The joint frequently used in this case is shown in Fig. 8. In the previous cases the axes were regarded as being in the same straight line. If it be required to unite the ends, and have the axes make an angle with each other, this maybe done by halving the beams at the ends, or by cutting a mortise in the centre of one, shaping the end of the other to fit, and fastening the ends together460 JOINTS. by pins, bolts, straps, or other devices. The joints used in the latter case are termed “ mortise” and “ tenon joints.” Their form will depend upon the angle between the axes of the beams. II. Joints of Beams, the axes of the beams making an angle with each other. Mortise and Tenon Joints. — When the axes are perpendicular to each other, the mortise is cut in the face of one of the beams, and the end of the other beam is shaped into a tenon to fit the mortise, as shown in Fig. 9. Fig. 9. Represents a mortise and tenon joint when the axes of the beams are perpendicu. lar to each other, a, tenon on the beam A ; b, mortise in the beam B ; c, pin to hold the parts together. When the axes are oblique to each other, one of the most common joints consists of a triangular notch cut in the face of one of the beams, with a shallow mortise cut in the bottom of the notch, the end of the other beam being cut to fit the notch and mortice, as shown in Fig. 10. Fig. 10. Represents a mortise and tenon joint when the axes of the beams are oblique to each other« In a joint like this the distance ah should not be less than one-lialf the depth of the beam A ; the sides ah and he should be per-JOINTS. 461 pcndicular to each other when practicable; and the thickness of the tenon d should be about one-fifth of that of the beam A. The joint should be left a little open at c to allow for settling of the frame. The distance from b to the end 1) of the beam should be sufficiently great to resist safely the longitudinal shearing-strain caused by the thrust of the beam A against the mortise. Suppose the axes of the beams to be horizontal, and the beam A to be subjected to a cross-strain; the circumstances being such that the end of the beam A is to be connected with the face of the other beam B. In this case a mortise and tenon joint is used, but modified in form from those just shown. To weaken the main or supporting beam as little as possible, the mortise should be cut near the middle of its depth; that is, the centre of the mortise should be at or near the neutral axis. In order that the tenon should have the greatest strength, it should be at or near the under side of the joint. Since both of these conditions cannot be combined in the same jont, a modification of both is used as shown in Fig. 11. A, the cross-beam; B, cross-section of main beam; t, the tenon. The tenon has a depth of one-sixth that of the cross-beam A, and a length of twice this, or of one-third the depth of the beam. The lower side of the cross-beam is made into a shoulder, which is let into the main beam one-half the length of the tenon. Double tenons have been considerably used in carpentry. As a rule, they should never be used, as both are seldom in bearing at the same time. III. Joints used to connect Beams, the Faces resting on or notched into Each Other. — The simplest and strongest joint in this case is made by cutting a notch in one or both beams, and fastening the fitted beams together. If the beams do not cross, but have the end of one to rest upon the other, a dove-tail joint is sometimes used. In this joint, a notch, trapezoidal in form, is cut in the supporting beam, and the end of the other beam is fitted into this notch. A Fig. 11JOINTS. 462 On .account of tlie shrinkage of timber, the dove-tail joint should never be used, except in cases where the shrinkage in the different parts counteract each other. It is a joint much used in joiner’s work. The joints used in timber-work are generally composed of plane surfaces. Curved ones have been recommended for struts, but the experiments of Ilodgkinson would hardly justify their use. The simplest forms are, as a rule, the best, as they afford the easiest means of fitting the parts together. Fastening's.— The pieces of a frame are held together at the joints by fastenings, which may be classed as follows: — 1. Pins, including nails, spikes, screws, bolts, and wedges; 2. Straps and tie-bars, including stirrups, suspending-rods, etc.; and 3. Sockets. These are so well known that a description of them is unnecessary. General Rules to be observed in tlie Construction of Joints. In planning and executing joints and fastenings the following general principles should be kept in view: — I. To arrange the joints and fastenings so as to weaken as little as possible the pieces which are to be connected. II. In a joint subjected to compression to place the abutting surfaces as nearly as possible perpendicular to the direction of the strain. III. To give to such joints as great a surface as practicable. IV. To proportion the fastenings so that they will be equal in strength to the pieces they connect. V. To place the fastenings so that there shall be no danger of the joint giving way by the fastenings shearing, or crushing the timber. JOINTS FOR IRON-WORK. The pieces of an iron frame are ordinarily joined by means of rivets, pins, or nuts and screws. Riveted Joints. — A rivet is a short headed bolt or pin, of iron or other malleable material, made so that it can be inserted into holes in the pieces to be fastened together, and that the point of the bolt can be spread out or beaten down closely upon the piece by pressure or hammering. This operation is termed “riveting,” and is performed by hand or by machinery. By hand, it is doneJOINTS. 463 with a hammer by a succession of blows : by machinery, as ordinarily used, the heated bolt is both pressed into the hole, and riveted, by a single stroke. If a machine uses a succession of blows, the operation is then known as “snap-riveting.” By many it is claimed that machine riveting possesses great superiority over that by hand, for the reason that the rivets more completely fill the holes, and in this way become an integral part of the structure. It is doubtful if it possesses the advantage of superior strength to any marked degree. It does certainly possess, however, the advantage of being more (piicldy executed without damage to the heads of the rivets. The holes are generally made by punching, are about one-twentieth of an inch larger than the diameter of the rivet, and are slightly conical. The diameter of the rivet is generally greater than the thickness of the plate through which the hole is to be punched, because of the difficulty of punching holes of a smaller size. Punching injures the piece when the latter is of a hard variety of iron; and for this reason engineers often require that the holes be drilled. Drilling seems to be the better method, especially when several thicknesses of plates are to be connected, as it insures the precise matching of the rivet-lioles. The appearance of the iron around a hole made by punching gives a very fair test of the quality of the iron. When two or more plates are to be riveted, they are placed together in the proper position, with the rivet-holes exactly over one another, and screwed together by temporary screw-bolts inserted through some of the holes. The rivets, heated red-hot, arc then inserted into the holes up to the head, and by pressure or hammering the small end is beaten down fast to the plate. In a good joint, especially when newly riveted, the friction of the pieces is very great,-being sufficient to sustain the working-load without calling into play the shearing-resistance of the rivets. In calculating the strength of the frame, the strength due to friction is not considered, as it cannot be relied on after a short time in those cases where the frame is subjected to shocks, vibrations, or great changes of temperature. Number and Arrangement of Rivets. The*general rule determining the number is, that the suvi of the areas of the cross-sections of the rivets shall be equal to the effective sectional area of the plate after the holes have been punched. This rule is based on the theory that the resistance to shearing-strain in the rivet is equal to the tenacity of the plate. \o determine the proper distance between the rivets in the direction of any row> so that the strength of the rivéts in any single row464’ JOINTS. shall be equal to the strength of the section of the plate along this row after the holes have been punched, let • d, be the diameter of the rivet; c, the distance from centre to centre of the rivets; a, the area of cross-section of the rivet; A', the effective area, between two consecutive rivets of the cross-section of the plate along the row of rivets; and t,. the thickness of the iron plate. T, the tenacity of iron. S, the shearing strength of rivets. It has been assumed that and the rule requires that T=S, Ta TA = S X a, or = ~p = Î. We have whence \tuT2 A'~ t(c- d) . ird2 Bfi c — + a for the distance from centre to centre of the consecutive rivets in any one row. English engineers, in practice, use rivets whose diameters are f, f, f, 1, li, and 1£ inches, for iron plates f, ft, f, j, |, and % inches thick respectively, and take the distance from centre to centre at two diameters for a strain of compression, and two and a half diameters for extension. The distance of the centre of the extreme rivet from the edge of the plate is taken between one and a half and two diameters. Instead of assuming the resistance to shearing in the rivet equal to the tenacity of the iron plate, a better rule would be to make the product arising from multiplying the sum of the areas of the cross-sections of the rivets by the amount of shearing-strain allowed on each unit, equal to the maximum strain transmitted through the joint. If the strain was one of compression in the plates, and the ends exactly fitted, the only riveting required would be that necessary to keep the plates in position. As the workmanship rarely, if ever, admits of so exact fitting, the rivets should be proportioned by the rules just given. The size of the head of a rivet depends upon the diameter of the rivet. It is usually circular in form, and should have a diameterJOINTS. 4G5 not less than twicey and a thickness at the centre not less than one-lialf, the diameter of the rivet. Various methods are used in the arrangement of the rivets. The arrangement often used for lengthening a plate is shown in Fig. 12. This method is known as “ chain-riveting.” Fig. 13 shows another method used for the same purpose, in which the number of rivets is the same as in the previous example; but there is a better disposition of them. Figs. 14 and 15 show the arrangement of the rivets often used to fasten ties to a plate. Figs. 16, 17, and 18 show in plan the forms of several kinds of riveted joints. ■ Fig. 16 shows the single shear-joint or single lap-joint.4(50 JOINTS. Fig. 17 shows the ordinary fish-joint. In this joint the fish or cover plates are placed on each side, and have a thickness of half that of the plates to be connected: sometimes only one cover-plate is used, and then the connection is known as the “butt-joint.” When several plates are to be fastened together, the method shown in Fig. 18 is the one ordinarily used. The proportions for eyes and pins, and for screio ends and nuts, for tension-rods, will be found in Chap. IX.A.RT III. Rules, Memoranda, and Tables USEFUL IN Designing, Estimating, and Building. V' CLASSICAL MOULDINGS. Moulding'S are so called because they are of the same shape throughout their length as though the whole had been cast in the same mould or form. The regular mouldings, as found in remains of classic architecture, are eight in number, and are known by the following names : — L—i L ) Annulet, bund, cincture, fillet, Astragal, or bead, listel, or square. Ovolo, quarter-round, or echinus. Scotia, trochilus, or mouth* Inverted cymatium, or cyma-reversa. The last two are both called “ ogee.” Some of these terms are derived thus : Fillet, from the French word fit, “thread;” astragal, from astragalos^ “a bone of the heel,” or “the curvature of the heel;” bead, because this moulding, when properly carved, resembles a string of beads; torus, or tor® the Greek for rope, which it resembles when on the base of a column ; scotia, from skotia, “darkness,” because of the strong shadow which its depth produces, and which is increased by the projection of the torus above it; ovolo, from ovum, “an egg,” 469470 HEAT AND VENTILATION. which this member resembles, when carved, as in the Ionic capital; cavetto, from cavus, “hollow;” cymatium, from Icmna-ton, “ a wave.” Characteristics of Moulding’s. — Neither of these mouldings is peculiar to any one of the orders of architecture; and although each has its appropriate use, yet it is by no means confined to any certain position in an assemblage of mouldings. The use of the fillet is to bind the parts, as also that of the astragal and torus, which resemble ropes. The ovolo and cyma-reversa are strong at their upper extremities, and are therefore used to support projecting parts above them. The cyina-recta and cavetto, being weak at their upper extremi ties, are not used as supporters, but are placed uppermost to covei and shelter the upper parts. The scotia is introduced in the base of a column to separate the upper and lower torus, and to produce a pleasing variety and relief. The form of the bead and that of the torus is the same: the reasons for giving distinct names to. them are, that the torus, in every order, is always considerably larger than the bead, and is placed among the base mouldings, whereas the bead is never placed there, but on the capital or entablature.' The torus, also, is seldom carved, whereas the bead is ; and while the torus, among the Greeks, is frequently elliptical in its form, the bead retains its •circular shape. While the scotia is the reverse of the torus, the cavetto is the reverse of the ovolo, and the cyma-recta and cyma-reversa are combinations of the ovolo and cavetto. HEAT AND VENTILATION. The causes of the loss of heat in ventilated rooms are, (1) units of heat required to warm the passing air, (2) units of heat absorbed by walls, (3) units of heat absorbed by ceiling, (4) units of heat absorbed by floor, and (5) units of heat absorbed by windows. The sources of heat in rooms are, (1) units of heat generated by the occupants, (2) units of heat generated by lights, and (3) units of heat generated by fires or heating apparatus. An adult man requires for respiration and transpiration hourly 215 cubic feet of atmospheric air, or 215 X 0.077 = 10.5 pounds, and generates about 290 units of heat, 100 units of which go in the formation of vapor; the other 190 units being dissipated by radiation to the surrounding objects, and contact with the colder air. The amount of air required, and the heat generated, by gaslights may be, approximatedHEAT AND VENTILATION. 471 - sufficiently near for practical purposes thus ; The specific gravity : of gas is about half that of atmospheric air, or 0.038 pounds per cubic foot, and requires for complete combustion 0.038 X 17 = 0.05 0.05 i pounds of air, or = 8.44 cubic feet. Each cubic foot of gas burned emits-about GOO units of heat. An oil-lamp with a good wick consumes about 154 grains per hour, equal to 35 lamps per pound. Each pound of oil requires 150 cubic feet of air for complete combustion, and generates about 10,000 units of heat, or 400 per lamp. Candles six to the pound may be reckoned the same as a lamp consuming oil, each candle burning about 170 grains per hour. TABULATED IN ROUND NUMBERS. An adult man vitiates per hour (cubic feet) .... Each cubic foot of gas burned “ “ . . . . Each pound of oil burned “ “ .... Each pound of candles “ “ . , . . Units of heat generated by a man per hour . . . . Units of heat generated by one cubic foot of gas . v Units of heat generated by one pound of oil or candles . 215 . 8.5 . 150. . 160 . 190 . 600 . 16,000 An average gas-burner consumes about four feet of gas per hour. Windows, as ordinarily constructed, will admit about eight cubic feet of air per minute. Ventilation of Theatres. Exampi.es of Theatre Ventilation. — The plenum principle (of forcing pure air into the building, and driving out the fmpure air) has been introduced into the Metropolitan Opera House, New York, where every fixed chair in the house has air admitted to it. A very full and interesting,account of the ventilation of this house is given in the “Sanitary Engineer” of Dec. 6, 1883. The object being to have an excess of air entering the building beyond that escaping by the regularly provided foul-air outlets, thus insuring an internal atmospheric pressure slightly in excess of the air without the building. This maintains an outward current through crevices of doors, windows, and .other openings, to accomplish which in a practical manner a blowing engine is used, and the supply of air is almost unlimited. A controlling valve is fitted in the centre of the auditorium ceiling, by the adjustment of which the pressure within the house is regulated, and the condition of plenum maintained under the varying pressures induced by the changing speed of the fan.472 CHIMNEYS. The plenum principle is also employed in the Madison-Square Theatre, New York. The inlet for fresh air is by a descending flue, which is six feet square, lined with wood, and in this is placed a conical cheese-cloth hag forty feet deep. This filters the air, which afterwards passes over ice in summer, four tons being used each night, — two tons before, and two tons after, the air passes the fan at the bottom of the inlet shaft. The fan forces the air into a brick duct, from which sheet-iron pipes lead the air into four brick casings, which surround the steam radiators used for warming the air in the winter. The auditorium has four sections of ninety seats each, and from the steam chambers direct to each of these seats a four-inch tin circular pipe conveys the air. In addition to these, special ducts from the fan are used in the summer to pour an extra supply of cooled air to various parts of the auditorium. Tests used to prove the efficiency of this system have given satisfactory results. A temperature without draughts, and not exceeding sixty-five degrees F., is the most desirable in a theatre; and it should be the aim of every theatre-builder to attain this result, which will at times necessitate cooling the incoming air during the summer months. CHIMNEYS. [From the “Building and Engineering Times.”] The object of a chimney is to produce the draught necessary for the proper combustion of the fuel, as well as to furnish a means of discharging the noxious products of combustion into the atmos- " pherc at such a height from the ground that they may not be considered a nuisance to people in the vicinity of the chimney. Regarding the second of the above purposes for which chimneys are built, it need only be said, that it is of secondary importance only, and that where due attention is given to the proper methods of setting boilers, and proportionating grate areas, furnaces, and rate of combustion, the smoke nuisance is comparatively unknown, and is of no practical importance whatever. The main points to be considered in designing chimneys are the right proportions to insure, first, a good and sufficient draught, and, second, stability. Without entering into any demonstration of the velocity of the flow of the heated gases through the furnace and flues leading into and up the chimney, we will briefly state a few of the principles governing the dimensions of chimneys. The motive power or forceCHIMNEYS. 473 which produces the draught is the action of gravity upon the difference in the specific gravities of the heated column of the gases of combustion inside the chimney, and the atmosphere at its normal temperature outside of the chimney, by which the former is forced tip the flue; and the laws governing its velocity are the same as those governing the velocity of a falling body; anil it can be proved that its velocity, and, consequently, the amount or volume of air drawn into the furnace, and which constitutes the draught, is in proportion to the square root of the height of the chimney. It is a common error that the force of the draught is in direct proportion to the height; so that, with two chimneys of the same area of flue, one being twice the height of the other, the higher one would produce a draught twice as strong as the other. The intensity of draught under these circumstances would be in the proportion of the square root of 1 to the square root of 2, or as 1 to 1.42. To double the draught-power of any given chimney by adding to the height, it would be necessary to build it to four times the original height. Practically there is a limit to the height of a chimney of any given area of flue, beyond which it is found that the additional height increases the resistance duo to the velocity and friction more rapidly than it increases the flow of cold air into the furnace. For chimneys not over forty-two inches in diameter, the maximum admissible height is about three hundred feet. From an investigation of the same laws we find that the velocity of the flow of cold air into the furnace is in proportion to the square, root of the ratio between the density of the outside air and the difference in the densities of the outside air and the heated gases' in the chimney; from which we may deduce the fact that very *• little increase of draught is obtained by increasing the temperature of the gases in the chimney above 550 or 600 degrees F. By raising the temperature of the flue from 600 to 1200 degrees we would increase the draught less than twenty per cent, while the waste of heat would be very considerable. Conversely, we may reduce the temperature of the flue about one-half, when the tern-I>erature is as high as six hundred degrees, by means of an economizer or otherwise, and the reduction of drauglit-foroe would be only about twenty per cent, as before. It is found that the principal causes which act to impair the draught of a chimney, and which vary greatly with different types of boilers and settings, are the resistance to the passage of the air offered by the layer of fuel, bends, elbows, and changes in the dimensions of the flues, roughness of the masonry of brick flues, holes in the passages which allow the entrance of cold air, and, generally, any variation from a straight, air-tight passage of uniform474 CHIMNEYS. size from combustion-chamber to chimney-flue; anil the resistance to draught is in direct proportion to the magnitude and number of such variations. In designing a chimney, it is, therefore, always necessary to consider the type of boiler, method of setting, arrangement of boilers and flues, location of chimney, and every thing which will be likely to in any way interfere with its efficient performance. Much, of course, depends upon the judgment and experience of the designer, and it would be impossible to give any general rule which would cover all cases. When only one boiler discharges into a chimney, for instance, the chimney requires a much larger area per pound of fuel burned thkn when several similar boilers discharge into a chimney of the same height; and, taking all these varying circumstances into consideration, a great deal of judgment is, in many cases, required to determine the proper dimensions. It is a common idea that a “chimney cannot be too large in other words, the larger the area of the flues, the better the draught will be. But this is not always the case. In many cases where a chimney has been built large enough to serve for future additions-to the boiler-power, the draught has been much improved as additional boilers have been set at work. The cause of this is to be found in the increased steadiness of draught where several boilers are at work and are fired successively, as well also as in the better maintenance of the temperature of the flue; as the velocity of the gases necessarily increases with the increased amount required to be discharged, and they do not have time to cool off to so great an extent as when they move more slowly. General Rules for Brick Chimneys. [From Moleswo'rth’s “ Pocket-Book.”] The diameter at the base should be not less than one-tenth of the height. Batter of chimneys, 0.3 inch to the foot. Thickness of brick-work, a brick from top to twenty-five feet from ditto. A brick and a half from twenty-five to fifty feet from the top, increasing by half a brick for each twenty-five feet from the top. . ... If the inside diameter at the top exceeds four feet- six inches, the top length should be a brick and a half thick. iCHIMNEYS. 475 Velocity of Artificial Draught. IT = Height of chimney in feet. T = Temperature of air supplying the chimney. t — Temperature of air at top of chimney. V — Velocity in feet per second. V = 0.365^1/(2’—f). Area of Chimneys. F = Quantity of coal consumed per hour, in pounds. h = Height of chimney, in feet. 1IP — Horse-power of engine (indicated). A — Area of chimney at top, in square inches. _ 15JF _ 150IIP ~ \fh ~ \Jh‘ Proportions for Boiler Chimneys. [From “The Builder.”] For marine boilers the general rule is to allow fourteen square inches area of chimney for each nominal horse-power: for stationary boilers the area of the chimneys should be one-fifth greater than the combined area of all the flues or tubes. In boilers provided with any other means of draught, such as a steam-jet or a fan-blower, the dimensions of the chimney are not so important as in cases where the draught is produced solely by the chimney. Rule for finding the Required Area for any Chimney. — Multiply the nominal horse-power of the boiler by 112, and divide the product by the square root of the height of the chimney in feet. The quotient will be the required area, in square inches, at top of chimney. TABLE SHOWING DIAMETER AND HEIGHT OF CHIMNEY FOR ANY BOILER. IIoiHC- , Height of Interior Horse- Height of Interior power of chimney, diameter at power of chimney, diameter at boiler. in feet. • top. ■ .boiler. in feet. top. 10 60 14 inches. 70 120 30 inches. 11 75 14 “ 90 120 34 “ 16 90 16 1 120 135 • 38 i H 99 17 “ 160 150 43 “ 30 105 21 ¥ 200 165 47 “ m 120 26 “ 250 180 52 1 GO 120 27 “ 1 S80 195 57 “ | /476 CHIMNEYS. DRAUGHT OF CHIMNEYS FOR DIFFERENT HEIGHTS. [From “The Builder.”] Height of chimney, in feet. Pressure due to 116° F., in lbs. per foot. ' Velocity of draught, in miles per hour. 30 1.2 16 40 1.6 18 50 2.0 20 60 2.4 22 70 2.8 24 80 3.2 26 Wrong’llt-Iron Chimneys. The Pennsylvania Steel Company had the following wrought-iron chimney-stacks in use in 1883; and, according to Mr. Bent the superintendent, they have proved both durable and economical. Blast furnace No. 2, Whitewell stove-stack, 6 feet 6 inches diameter; 165 feet high; lined with nine-inch fire-brick for 30 feet, four-inch red brick for 135 feet. Erected 1877. O. II. furnace stack, diameter 7 feet; height 135 feet; lined same as above. Erected 1880. Rail Mill boiler-stack No. 2, 6 feet diameter; 112 feet high. Erected 1881. Whitewell stove-stack, 6 feet 6 inches diameter; height 170 feet; lined same as above. Erected 1881. Forge boiler-stack, 7 feet diameter; inside shell lined with nine-inch fire-brick for 30 feet, then four-inch red brick to top; 110 feet high from base-plate. Erected 1S69. Rail Mill boiler-stack, 7 feet diameter; 110 feet high; lined as above. Erected 1874. Rail Mill gas-furnace stack, same as Rail Mill boiler-stack. Erected 1875. Rail Mill gas-furnace stack, same as Rail Mill. Erected 1876.478 STAIRS. STAIRS. Wooden stairs are generally built with two-ineh plank stringers notched out on the upper side to form the steps, and covered with pieces of boards, whose length is equal to the width of the stairs. The horizontal boards upon which the feet are placed are called the treads; and the vertical boards, the risers. In first-class work, the treads should be an inch and a quarter thick, and the risers seven-eighths of an inch thick, and both should be of some hard wood. The strihgers Should not be placed over sixteen inches apart from centres, and twelve inches is better. Tbe treads generally project an inch and a half beyond the face of the risers, forming a nosing. A good rule for the proportion of risers and treads is that the sum of the rise and tread shall equal seventeen inches and a half. Thus, if the rise is six inches, the tread should be eleven inches and a half (plus the width of the nbsing); or, if the rise is eight inches, the tread should be but nine incites and a half. The rise is always measured from top to top of treads; and the tread, from face to face of risers. The following table shows at a glance how many risers or treads there will be in any given distance. Example. — In a certain building the height from the top of the first floor to the top of the second is IS feet. IIow many risers will be required, and what will they be ? Ams. Find in the table the heights coming nearest to IS feet, and then notice the height and number of risers necessary to attain •this height. Thus, in the column headed Tj inches, at the bottom we find IS feet 14 inches, showing that :>(,) risers 7i inches each will give 18 feet 14 inches. If we used a rise of 74 inches, 25* risers would also give us IS feet l j inches. Hence we shall need either 2!) of 00 risers, according as we wish our rise 71 or 74 inches. If we use a rise of 7| inches, we shall only require 28 risers. Tbe number of treads in a given distance can be found in the same way.03 to »0 to to to to to to to to , o o go - j © C' if- c; io h o CC -1C- O' 10 ^ © © 00 -7 © O’ tfa 05 tO M O O X 05 - I -I Ci 05 O! O' *- 4- W Iv IV H »-* o 000.00 o.o cc.oocooooa -©.© 'O C ic 00 “t -t O 05 O' in A* if* W W 10 15 ^ H O -7 »-* -I ® ® © O —‘/-*— *— fcr-t h-a O O' C0 C CO -1 S © Cl O’ c 4- 03 05 tO 10 r-L »-* o Feet. Cw © 10 -1^0 CO cc h—i S B o v* p Jo to -1 © A—»a-*4-- taa* i4<3 M OC ►a—443» Inches. —4 -1 £ P O' b'< 4- 4- 05 15.10 I-- r-* O O © CO CO -4 -4 0101C 4- 4- 05 to to 1-4 O Feet. * ~© £ *»OK.*4 ■ »• A-4« O O' 0 03 GO'W •—* Ci i—* *- © to -7 o O’ O 05 OOMQH 4* O IO M Inches. -( “1 Cl Cl P» 4- 4- 05 CC 1 3 l-a o b C« CO —7 -t Cl O’ o* 4- 4- 05 10 to i— o Feet. 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[From Haswcll’s “Engineers’ and Mechanics’ Pocket-Book.”] The flow of gas -is determined by the same rules as govern, that of the flow of water. The pressure applied is indicated and estimated in inches of water. DIAMETER AND LENGTH OF GAS-PIPES TO TRANSMIT GIVEN VOLUMES OF GAS TO BRANCH PIPES. [Dr. Ure.] Volume per hour, in cu. ft. Diameter, in ins. Length, in feet. Volume per hour, in cu. ft. Diameter, in ins. Length, in feet. 50 0.40 100 2000 5.32 2000 250 1.00 200 2000 6.33 4000 500 1.97 600 2000 7.00 6000 700 2.65 looo 6000 7.75 1000 1000 3.16 1000 6000 9.21 2000 1500 3.87 1000 8000 8.95 1000 The volumes of gases of like specific gravities discharged in equal times by a horizontal pipe under the same pressure, and for different lengths, are inversely as the square roots of the lengths. The loss of volume of discharge by friction, in a pipe six inches in diameter and one mile in length, is estimated at ninety-five per cent. Gas Memoranda. Tn distilling fifty-six pounds of coal, the volume of gas produced in cubic feet, when the distillation was effected in three hours, was 41.3; in seven hours, 47.5; in twenty hours, 33.5; and, in twenty-live hours, 31.7. A retort produces about six hundred cubic feet of gas in five hours, with a charge of about one and a half hundred-weight of coal, or 2800 cubic feet in twenty-four hours. A cubic foot of good gas, from a jot one-thirty-tliird of an inch in diameter and a flame of four inches, will burn sixty-five minutes. Internal lights require four cubic feet, and external lights about five cubic feet, per liour. When large or Argand burners are used, from six to ten cubic feet will be required.'480 SEATING-SPACE IN THEATRES. SEATING-SPACE IN THEATRES. [From Condon “ Building Times.”] The question of seating is one upon-which a manager and the public are apt to differ. The requirements of the Metropolitan Board of Works with respect to seating are, that “ the area to be assigned to each person shall not be less than one foot eight inches by one foot six inches, in the gallery, nor less than two feet four inclie^yume foot eight inches, in the other parts of the house, rooniCgp («her place of public resort.” These conditions it is perhapPi^dly necessary to say are not complied with in any theatre under the jurisdiction of the Board. Until theatres are licensed to hold a certain number, or other legal restrictions enforced, an architect, in calculating the seating-capacity for the cheaper parts of his theatre, must be guided by past experience. In the upper circle, pit, and gallery, where the seats are not divided off, the audience will pack itself in an astonishing manner, when a calculation is made of the space in inches occupied by each person. From average calculations made in London theatres, the width of seat required in the unnumbered parts of a theatre is as follows: upper circle, eighteen inches; pit, sixteen inches; amphitheatre, sixteen inches; gallery, fourteen inches. It is not intended to advocate a minimum space for the seats: on the contrary, there cannot be a doubt but that, if the minimum of eighteen inches were strictly enforced, it would be a most desirable innovation. The several divisions of the auditorium are provided with more or less luxuriant seats according to the price paid for admission. Tlie stalls are usually fitted with arm-chairs, or fauteuils. The width of seat, and the space allowed between each row, vary considerably, according to the degree of comfort and convenience. In any case, the space allotted to each seat in the stalls is greater than that given in any other part of a theatre. The width of the seats adopted varies from twenty inches to twenty-four inches; and the distance from back to back, from three feet to five feet. The stall-seats should be the very embodiment of an easy arm-chair. A very frequent fault results from the seat being too high, and the back not sufficiently inclined. It should not be forgotten that the occupants of the stalls have to look up towards the stage. They should be able to recline easily in the chair at an angle suited to the line of vision. To sit in some stalls is to insure a stiff neck. The discomfort of stall-seats may arise from two causes, which the architect should endeavor to avoid. Firstly, the lloor of the stallsSPACES OCCUPIED BY SCHOOL-SEATS. 481 should not be sunk too low. It should never be more than four feet below the highest point of the stage-floor. Secondly, the seat should not be too high, and the back sufficiently inclined for the occupant to accommodate himself to the angle of vision. .As instances of comfortable stall-chairs, the following dimensions are those of seats in two representative theatres. Width, twenty-one inches; depth, sixteen inches; height of seat from floor, sixteen inches; height from floor to top of back rail, two feet ten inches; distance from back to back, three feet ten inches. In the other case the seats are continuous, and “ tip up.” Width from centre to centre of arms, twenty-three inches; depth, twenty-four inches; height from floor, sixteen inches; inclination of back, 115 degrees; and the distance from back to back, three feet. Di •ess-Circle. — The seats in this part are similar to those in the stalls; but the inclination of the backs should be slightly less, unless the circle is low, and not much in height above the stage-level. It is also advantageous to make the seat one or two inches higher than the stall-chairs. In the theatre previously alluded to, the dress-circle seats are twenty inches wide, eighteen inches deep, eighteen inches high, and inclination of back 115 degrees. The width of the steps upon which the seats are fixed ranges from three feet to three feet six inches. Upper Circle.— The steps in this part maybe reduced to two feet six inches. This reduction in width is imperative at each level: otherwise the height of the steppings would be inconvenient. The seats should be divided by arm-rests, and have back rails. They should be eighteen inches wide, fifteen inches deep, eighteen inches high, and about 100 degrees inclination of the backs. SPACES OCCUPIED BY SCHOOL-SEATS. SIZES OF CHAIRS AND DESKS FOR SCHOOLS AND ACADEMIES. Age of scholar. Height of chair. Height of desk (next scholar). Space occupied by desk and chair (back to back of desk;. 10 to 18 years. 163 inches. 29* inches. 2 feet 0 inches. 14 to 10 i 154 «< 28 “ 2 “ 9 9 m to 14 1 15£ (< 27* 1 2 “ 8 “ 10 to 12 “ i*l (< 2G£ i 2 “ 7 1 8 to 10 “ 131 n 251 2 “ 5 1 7 to 8 “ 12} u 24 “ 2 “ 4 “ 6 to 7 “ 11} << 2?t “ 2 1 3 1 5 to 6 “ 10* << 21 “ 2 “ 2 “ 4 to 6 “ << 19 “ 2 “ 0 “ Desks for two scholars are three feet ten inches long, and for a single scholai two feet long.482 SYMBOLS FOR THE APOSTLES AND SAINTS. SYMBOLS FOR THE APOSTLES AND SAINTS. From the co nstant occurrence of symbols in the edifices of the middle ages an 1 many of the cathedrals of the present day, the following list of symbols, as commonly attached to the apostles and saints, may be found useful: —: Holy Apostles. • - St. Peter. —Bears a key, or two keys with different wards. St. Andrew.—Leans on a cross so called from him; called by heralds the saltire. St. John the Evangelist.—With a chalice, in which is a winged serpent. When this symbol is used, the eagle, another symbol of him, is never given. St. Bartholomew..— With a flaying knife. St. James the Less.—A fuller’s staff, bearing a small square banner. St. James the Greater. —A pilgrim’s staff, hat, and escalop-shell. St. Thomas. — An arrow, or with a long staff. St. Simon. — A long saw. ,SL. Jude. —A club. ■ St. Matthias.—A hatchet. St. Philip. —Leans on a spear, or lias a long cross in the shape of a T. St. Matthew. —A knife, or dagger. St. Mark. — A winged lion. ' St. Luke. — A bull. St. John. —An eagle. St. Paul. —An elevated sword, or two swords in saltire. St. John the Baptist. — An Agnus Dei. St. Stephen. —AVith stones in his lap. Saints. St. Agnes. — A lamb at her feet. St. Cecilia.—AVith an organ. St. Clement..— AVith an anchor. St. David. —Preaching on a hill. St. penis. —With his head in his hands. St. George. —AVith the dragon. Si. Nicholas.—AVitli three naked children in a tub, in the end whereof rests his pastoral staff. Vincent. —On the rack.WEIGHT OF BELLS. — DIMENSIONS OF DOMES. 483 WEIGHT OF BELLS. Bells. Pounds. Bells. Pounds. City Hall, New York . . Fire Alarm, Thirty-third - 22,300 Rouen, France .... 40,000 St. Paul’s, London . . . 11,470 Street, New York . . 21,612 St. Ivan’s, Moscow . . . 127,830 Linden, Germany . . . 10,854 St. Peter’s, Rome . . . 18,600 Lewiston, Me 10,233 Vienna 40,200 Montreal, Canada . . . 28,560 Westminster, M Big Ben,” Moscow, Russia .... 432,000 England 30,350 Oxford, “Great Tom,” Worcester, England . . 6,600 England Olmutzi Bohemia . . . 18,000 40,000 York, England .... 6,381 . DIMENSIONS OF THE PRINCIPAL DOMES. LIST OF THE PRINCIPAL DOMES IN THE WORLD. Their diameter, and height from the ground. [Gwilt’e Encydopaidia.J Place. Diameter, feet. Height, feet. Pantheon, at Rome 142 143 Duomo, or Sta. Maria del Fiore, at Florence, 139 310 St. Peter’s, at Rome 139 330 Sta. Sophia, at Constantinople 115 201 Baths of Caracal la (ancient) 112 116 St. Paul’s, London 112 215 Mosque of A eh met 92 120 Chapel of the Medici 91 199 Baptistery, at Florence 86 110 Church of the Invalids, 'at Paris 80 173 Minerva Medica, at Rome 78 97 Madonna della'Salute, Venice 70 133 St. Gemevieve, at Paris (Pantheon) .... 67 190 Duomo, at Sienna 57 148 I>uomo, at Milan 57 254 St. Vitali’s, at Ravenna 55 94 Val de Grace, at Paris ......... 55 133 San Marco, Venice 44 United-States Capitol, Washington .... 124| 484 HEIGHTS OF COLUMNS, TOWERS, AND DOMES. HEIGHTS OP COLUMNS, TOWERS, DOMES, SPIRES, ETC. COLUMNS. Name. Place. Feet. Alexander St. Petersburg . . . 175 Bunker Hill Charlestown, Mass. . 22H Chimney (St. Rollox). . . . Glasgow 455^ Chimney (Musprat’s). . . . Liverpool . .- . . 406 City London 202 July Paris 157 Napoleon Paris 132 Nelson’s Dublin 134 Nelson’s London 171 Place Vendôme Paris 136 Pompey’s Pillar Egypt 114 Trajan Rome 145 Washington Washington . . . 555 York London 138 TOWERS AND DOMES. Name. Place. Feet. Tower Babel 680 Tower Baalbec 500 Capitol Washington . . . 287* 1 Cathedral Antwerp 476 Cathedral Cologne 501 Cathedral Cremona 392 Cathedral Escurial 200 Cathedral Florence..... 384 Cathedral Milan 438 Cathedral St. Petersburg. . . 363 Leaning Tower Pisa 188 Porcelain China 200 St. Paul’s London 366 Strasbourg Venice . . . . '. 328 St. Mark’s City Ilall, Philadel- Utrecht pliia . . . . . 537-iCAPACITY OF CHURCHES, THEATRES, ETC. 485 HEIGHT OF SPIRES. Name. Place. Feet. Cathedral, new New York .... 325 Grace Church New York .... 216 Cathedral Salisbury .... 450 St. John’s New York .... 210 St. Paul’s New YqBs . . . . 200 St. Mary’s Liibeck 404 St. Peter’s Rome 391 St. Stephen’s Vienna 465 Trinity Church New York .... 286 Balustrade of Notre Dame Paris 216 Hotel des Invalides .... Paris 344 Pyramid of Cheops .... Egypt 520 Pyramid of Sakara .... Egypt 356 St. Peter’s lVUJiltJ ...... 518 CAPACITY OF SEVERAL CHURCHES, THEATRES, AND OPERA-HOUSES, Estimating a person to occupy an area of 19.7 inches square. CHURCHES. St. Peter’s Milan Cathedral St. Paul’s, Rome .... St. Paul’s, London .... St. Petronio, Bologna . . . Florence Cathedral . . . Antwerp Cathedral . . . St. Sophia’s, Constantinople, St. John Lateran .... 37.000 32.01 >0 25.000 24,400 24,300 24.000 23.000 22,900 Notre Dame, Paris .... Pisa Cathedral St. Stephen, Vienna . . . St. Dominic’s, Bologna . . St. Peter’s, Bologna . . . Cathedral of Sienna . . . St. Mark’s, Venice .... Spurgeon’s Tabernacle . . 21,000 13.000 12.400 12.000 11.400 11,000 7.000 7.000 THEATRES AND OPERA-HOUSES. Carlo Felice, Genoa . . . 2500 Opera-House, Berlin . . . 1636 Opera-House, Munich . . 2370 New-York Academy . . . 1326 Alexander, St. Petersburg . 23'>2 Philadelphia Academy . . 3124 San Carlos, Naples . . . 2240 Boston Theatre, Boston . . Imperial, St. Petersburg . . 2100 Madison-Square Theatre, La Scala, Milan 2113 New York '. Academy of Paris .... 2092 Metropolitan Opera-House, Drury Lane, London . . . 1948 New York 3500 Covent Garden, London . . 1084 Globe Theatre, Boston . . 486 DIMENSIONS OF THEATRES, ETC. DIMENSIONS OF THEATRES AND OPERA-HOUSES. The following are the dimensions of some of the prominent theatres in this country and in Europe: — Distance in Feet. Height, in Feet. Name and Location. Between boxes and footlights. Between footlights and curtain. 5 1 §> £ 50 0 — > Ü r* 0 1 +-> 'O H 0> & ? ■2 . CÎ *■> "X o Breadth Qf curtain. a> « o — he «8 & £ X © u-t 'S o ® Is ° o j_ +3 r* From floor of pit to cornice. From floor of pit to centre of ceiling. Alexander, St. Petersburg 65 11 81 58 56 75 53 58 , Berlin 62 16 76 51 41 92 43 47 La Scala, Milan . . . 77 _ 18 78 71 49 86 60 64. San Carlo, Naples . . 77 18 74 74 52 66 SI 83 Grand Theatre, Bordeaux 46 10 69 47 37 80 50 57 Salle Lepelletier, Paris . 67 9 82 66 43 78 66 Oovent Garden, London, 66 a _ 55 51 32 • 86 54 - Drury Lane, London 64 a _ 80 56 32 48 60 - Boston Theatre, Boston, 53 18 68 _ 46 87 55.5 58 Academy of M usic, New York 74 13 71 62 48 83 74 _ Pike’s Opera-House,. New York .... 54 el O.J 63* 48 44 76 52 67 Opera-House, Philadelphia 61 17 72 66 48 90 64* 74 Globe Theatre, Boston . Museum, Boston . . . 65 h _ 38 60 c 30 62 - 61 b — 46 68 c 31 68 " . • a These dimensions include the distance between the footlights and curtain. b Total depth of auditorium . c Total width of auditorium.PRINCIPAL DIMENSIONS OF THE ENGLISH CATHEDRALS. [Gwilt.] Cathedral. Total internal length, in ft. Naves and Aisles. Choirs. Tran- septs. Spires and Towers. Length, in ft. Breadth, in ft. Height, in ft. Length, ^ in ft. Breadth, in ft. Height, in ft. Breadth, in ft. Height, in ft. Winchester .... 545 247 86 78 138 73 186 Ely 517 327 73 70 101 73 70 178 Tower . . . 210 Canterbury .... 514 214 70 80 150 74 80 154 Tower . . . 235 Old St. Paul’s . . . 500 335 91 102 165 42 88 248 Spire . . . 534 York 49S 264 109 99 131 - 99 222 Tower . . . 234 Lincoln 498 _ 83 83 - - - 227 Tower . . . 260 Westminster . . . 489 130 96 101 152 - 151 189 Peterborough . . . 480 231 78 78 138 - 78 203 Louvre . . . 150 Salisbury . . ... 452 246 76 84 140 - 84 210 Spire . . . 387 Durham 420 _ _ - 117 33 71 176 Tower . . . 214 Gloucester .... 420 174 84 67 140 - 86 144 Tower . . . 225 Lichfield 411 213 67 - 110 - 67 • - Spire, 258 W., 183 Norwich 411 230 71 - 165 - - 191 Spire . . . 317 Worcester .... 410 212 78 _ 126 - 74 130 Tower . . . 196 Chichester .... 401 205 91 61 100 - - 131 Spire . . . 267 Exeter 390 173 74 69 131 - 69 140 Tower . . . 130 Wells 371 191 67 67 306 - 67 135 Tower » . . 160 Hereford (ancient) . 370 144 68 68 105 - 64 140 Chester 348 _ 73 73 - - - - - Tower . . . 127 Rochester .... 306 150 65 - 156 - - 122 Spire . . . 156 Carlisle ..... 213 - 71 71 137 71 - - Bath 210 136 72 78 - - - 126 Tower . . . 162 Bristol 175 100 75 73 100 - - 328 Tower . . . 127 Oxford 154 74 54 41 80 " o~ 1 3,î 102 Spire . . . 1S4 DIMENSIONS OF ENGLISH CATHEDRALS. 487488 DIMENSIONS OF VARIOUS OBELISKS DIMENSIONS OP THE VARIOUS OBELISKS EXIST-ING AT THE PRESENT TIME. [Gwilt’s Encyclopaedia.] Situation. Height, in English feet. Thickness, in English feet. At top. Relow. Two large obelisks mentioned by Diodorus Siculus 158.2 7.9 ll.S Two obelisks of Nuncoreus, son of Sesostris, according to Herodotus, Diodorus Siculus, and Pliny 121.8 6.6 10.5 Obelisk of Rhameses, removed to Rome by Con-stantius 118.4 6.2 10.2 Two obelisks, attributed by Pliny to Smerres and Eraphius 106.0 5.9 ' 9.8 Obelisks of Nectanabis, erected near the Tomb of Arsinoe by Ptolemy Philadelphia 105.5 5.3 9.2 I Obelisk of Constantius, restored and erected in front of S. Giovanni Laterano, at Rome . . . 105.5 6.2 9.6 1 Part of one of the obelisks of the son of Sesostris, in the centre of the piazza in front of St. Peter's, 82.4 5.8 9.t | Two at Luxor 79.1 5.3 Obelisk of-Augustus, from the Circu® Maximus, now in the Piazza del Popola at Rome 0 78.2 4.5 7.4 Two in the ruins at Thebes, still remaining . . . 72.8 5.0 7.5 Obelisk of Augustus, raised by Pius VI. in the Piazza di Monte Citorio 71.9 4.9 7.9 Two obelisks: one at Alexandria, vulgarly called Cleopatra's Needle, and the other at Heliopolis . 67.1 5.1 8.1 Obelisk by Pliny, attributed to Sothis 63.3 4.5 5.1 Two obelisks in the ruins at Thebes 63.3 4.5 5.1 Great obelisk at Constantinople 59.7 4.5 7 ** Obelisk in the Piazza Xavona, removed from the Circus of Caracal la 54.9 2.9 4.5 Obelisk at Arles 50.1 4.5 7.4 Obelisk from the Mausoleum of Augustus, now in front of the Church of Sta. Maria Maggiore, at Rome . . . . • 4S.3 2.9 i O 1 4.0 1 Obelisk in the Gardens of Sallust, according to Mercati 48.3 2.9 4.3 Obelisk at Bijije, in Egypt 42.9 2.6 4.2 Small obelisk at Constantinople, according to Gyl-lius 34.2 3.9 5.9 30.0 0 0 3.9 Obelisk of the Villa Mattel . 26.4 n *> 2.7 Obelisk in the Piazza della Rotunda 20.1 2.1 2.4 Obelisk in the Piazza di Minerva ....... 17.6 2.0 2.6 Obelisk of the Villa Medici 16.1 1.9 2.4MISCELLANEOUS MEMORANDA. 489 MISCELLANEOUS MEMORANDA. Weight of Men and Women. — The average weight of twenty thousand men and women weighed at Boston, 1S64, was,—men, 141j pounds; women, 124y pounds. Smallest Convenient Size of slab for a 14-inch wash-bowl, 21 by ' 24 inches. Height of slab from floor, 2 feet 6 inches. Very small (12-inch) corner wash-bowl; slab, 1 foot 11 inches each side. Urinals should be 2 feet 2 inches between partitions; partitions 6 feet high. Space occupied by Water-Closets, 2 feet 6 inches wide, 2 feet deep. Dimensions of Double Bed. — 6 feet 6 inches by 4 feet 6 inches. Dimensions of Single Beds (in dormitories). — 2 feet 8 inches by C feet 6 inches. Dimensions of a Bureau. — 3 feet 2 inches wide, 1 foot 6 inches deep, and upwards. Dimensions of a Washstand (common chamber-sets).—2 feet’ 4 inches wide, 1 foot 6 inches deep. Dimensions of a Barrel. — Diameter of head, 17 inches; bung, 19 inches; length, 28 inches; volume, 7680 cubic inches. Dimensions of Billiard-Tables (Collender). —4 feet by 8 feet, 4 feet 2 inches by 9 feet, and 5 feet by 10 feet. Size of room required, if feet by 17 feet, 14 feet by 18 feet, and 15 feet by .20 feet respectively. Ilorsc-Stalls.—Width, 3 feet 10 inches to 4 feet, or else 5 feet or over in width, 9 feet long. Width should never be between 4 and 5 feet, as in such cases the horse is liable to cast 'himself. Dimensions of Drawings for Patents (United States).—8.5 by 12 inches. Pitch of Tin, Copper, or Tar-and-Gravel Roof. — Five-eighths of an inch to the foot, and upwards. A fall of one-tenth of an inch in a mile will produce a current in rivers. Melted snow produces from one-fourtli to one-eighth of its bulk in water. At the depth of forty-five feet, the temperature of the earth is uniform throughout the year. A spermaceti candle 0.85 of an inch in diameter consumes an inch in length in an hour. Velocity of sound in water, 4708 feet per second. Avenues of City of New York run 28° 50’ 30" east of north. A rerage Height of Hand Rail to Stairs in Dwellings.—2 feet 7 inches from top of stop on line with riser. ' - r . fo, . ^ ? y ly r Si ft? .. 4 L frtr* 1 ft c, i. i-ttv .... ft490 LEAD MEMORANDA.—WEIGHT OF IRON. LEAD MEMORANDA. For roofs and gutters use 7-pouiul lead. For hips and ridges use 6-pound lead. For flashings use 4-pound lead. Gutters should have a fall of at least one inch in 10 feet. No sheet of lead should be laid in greater length than ten or twelve feet without a drip to allow of expansion. A pig of lead is about three feet long, and weighs from a hundred-weight and a fourth to a hundred-weight and a half. Spanish pigs are about a hundred-weight. Joints to lead pipes require a pound of solder for every inch in diameter. WEIGHT OF WROUGHT-IRON. Geneial Rules for determining- flic Weight of any Piece of Wrough t-Iron. One cubic foot of wrouglit-iron weighs................ 480 lbs. One square foot one inch thick “.................-*r2Q or 40 lbs. One square inch one foot long “..............or 3^ lbs. One square inch one yard long “ . . . . 3^ X 3 or 10 lbs. Thus it appears that the weight of any piece of wrouglit-iron in pounds per yard is equal to ten times its area in square inches. Example. — The area of a bar 4 inches X 1 inch = 4 square inches, and its weight is 40 lbs. per yard. For round iron, the weight per foot may be found by taking the diameter iii quarter-inches, squaring it, and dividing by 6. Example. —What is the weight of 2-incli round iron ? 2 inches 8 quarter-inches. 8- = 64. V — 10§ lbs. per foot of 2-incli round. Example. —What is the weight of »inch round iron ? | inch — 3 quarter-inches. 8'2 = 9. g = lbs. per foot of Blnch round. The above rules are very convenient, and enable mental calculations of weight to he quickly obtained with accuracy.WEIGHT OF FLAT AND BAR IRON. 491 WEIGHT PER FOOT OF FLAT, SQUARE, AND ROUND WROU GHT-IRON. Thickness or Diameter. Weight of Weight per Foot. In inches. In decimals of a foot. a square foot, in lbs. Square bar, in lbs. Round bar, in lbs. - A 0.0026 1.263 0.0033 0.0026 A 0.0052 2.526 0.0132 0.0104 A 0.0078 3.789 0.0296 0.0233 tj 0.0104 5.052 0.0526 0.0414 A 0.0130 6.315 0.0823 0.0646 de 0.0156 7.578 0.1184 0.0930 7_ t'l 0.0182 8.841 0.1612 0.1266 i. 4 0.0208 10.100 0.2105 0.1653 9 3*? 0.0234 11.370 0.2665 Bfl o 1 8 A 1 4 a 8 ■fd X 2 <) T6 Ft 8 1 1 lti 1 0.208 0.417 0.625 0.833 1.04 1.25 1.46 1.67 1.88 2.08 2.29 11 0.234 0.469 0.703 0.938 1.17 1.41 1.64 1.87 2.11 2.34 2.58 ■ y* 0.260 0.521 0.781 1.040 1.30 1.56 1.82 2.08 2.34 2.60 2.86 0.2S6 0.573 0.859 1.150 1.43 1.72 2.01 2.29 2.58 2.86 3.15 H 0.313 0.625 0.938 1.250 1.56 1.88 2.19 2.50 2.81 3.13 3.44 0.339 0.677 1.020 1.360 1.69 2.03 2.37 2.71 3.05 3.39 3.73 H 0.365 0.729 1.090 1.460 1.82 2.19 2.55 2.92 3.28 3.65 4.01 n 0.391 0.781 1.170 1.560 1.95 2.34 2.73 3.12 3.51 3.91 4.30 2 0.417 0.833 1.250 1.670 2.08 2.50 2.92 3.33 3.75 4.17 4.58 ■ 0.443 0.886 1.330 1.770 2.21 2.65 3.10 3.54 3.98 4.43 4.87 ■ 0.469 0.938 1.410 1.880 2.34 2.81 3.28 3.75 4.22 4.69 5.16 2S 0.495 0.990 1.480 1.9S0 2.47 2.97 3.46 3.96 4.46 4.95 5.44 H 0.521 1.040 1.560 2.080 2.60 3.13 3.65 4.17 4.69 5.21 5.73 06 0.547 1.090 1.640 2.190 2.73 3.28 3.83 4.38 4.92 5.47 6.02 25 0.573 1.150 1.720 2.290 2.86 3.44 4.01 4.58 5.16 5.73 6.30 2| 0.599 1.200 1.800 2.400 3.00 3.60 4.20 4.79 5.39 5.99 6.59 3 0.625 1.250 1.8S0 2.500 3.13 3.75 4.38 5.00 5.63 6.25 6.88 35 0.677 1.350 2.030 2.710 3.39 4.06 4.74 5.42 6.09 6.77 7.45 3£ 0.729 1.460 2.190 2.920 3.65 4.38 5.10 5.83 6.56 7.29 8.02 35 0.781 1.560 2.340 3.130 3.91 4.69 5.47 6.25 7.03 7.81 8.59 4 0.833 1.670 2.500 3.330 4.17 5.00 5.83 6.67 7.50 8.33 9.17 ■ 0.885 1.770 2.660 3.540 4.43 5.31 6.20 7.08 7.97 8.85 9.74 ■ 0.938 1.880 2.810 3.750 4.69 5.63 6.56 7.50 8.44 9.38 10.31 4| 0.9D0 1.980 2.970 3.960 4.95 5.94 6.93 7.92 8.91 9.90 10.89 5 1.042 2.080 3.130 4.170 5.21 6.25 7.29 8.33 9.38 10.42 11.46 ■ 1.090 2.190 3.280 4.380 5.47 6.56 7.66 8.75 9.84 10.94 12.03 4 1.150 2.290 3.440 4.580 5.73 6.88 8.02 9.17 10.31 11.46 12.60 K °4 1.200 2.400 3.590 4.790 5.99 7.19 8.39 9.58 10.78 11.98 13.18 6 1.250 2.500 3.750 5.000 6.25 7.50 8.75 10.00 11.25 12.50 13.75 ■ 1.300 2.600 3.910 5.210 6.51 7.81 9.11 10.42 11.72 13.02 14.32 6 è 1.350 2.710 4.060 5.420 6.77 8.13 9.4S 10.83 12.19 13.54 14.90 65 1.410 2.810 4.220 5.630 7.03 8.44 9.84 11.25 12.66 14.06 15.47 9? 1 1.460 2.920 4.380 5.830 7.29 8.75 10.21 11.67 13.13 14.58 16.04 7$ 1.510 3.020 4.530 6.040 7.55 9.06 10.57 12.08 13.59 15.10 16.61 74 1.560 3.130 4.690 6.250 7.81 9.38 10.94 12.50 14.06 15.63 17.19 75 1.610 3.230 4.840 6.460 8.07 9.69 11.30 12.92 14.53 16.15 17.76 8 1.670 3.330 5.000 6.670 8.33 10.00 11.67 13.33 15.00 16.67 18.33 84 1.720 3.440 5.160 6.880 8.59 10.31 12.03 13.75 15.47 17.19 1S.91 84 1.770 3.540 5.310 7.080 8.85 10.63 12.40 14.17 15.94 17.71 19.48 84 1.820 3.650 5.410 7.290 9.11 10.94 12.76 14.58 16.41 18.23 20.05 9 1.880 3.750 5.630 7.500 9.38 11.25 13.13 15.00 16.88 18.75 20.63 91 1.930 3.850 5.780 7.710 9.64 11.56 13.49 15.42 17.34 19.27 21.20 91 1.9S0 3.960 5,940 7.920 9.90 11.88 13.85 15.83 17.SI 19.79 21.77 n 2.030 4.060 6.090 8.130 10.16 12.19 14.22 16.25 18.28 20.31 22.34 10 2.080 4.170 6.250 8.330 10.42 12.50 14.58 16.67 18.75 20. S3 22.92 II 2.140 4.270 6.410 8.540 10.68 12.81 14.95 17.08 19.22 21.35 23.49 2.190 4.3S0 6.560 8.750 10.94 13.13 15.31 17.50 19.69 21.88 24.06 105 2.240 4.480 6.720 8.960 11.20 13.44 15.68 17.92 20.16 22.40 24.64 11 2.290 4.580 6.880 9.170 11.46 13.75 16.04 18.33 20.63 22.92 25.21 ul 2.340 4.690 7.030 9.380 11.72 14.06 16.41 18.75 21.09 23.44 25.78 114 2.400 4.790 7.190 9.580 11.98 14.38 16.77 19.17 21.56 23.96 26.35 •114 2.450 4.900 7.340 9.790 12.24 14.69 17.14 19.58 22.03 24.48 26.93 12 2.500 5.000 7.500 10.000 12.50 15.00 17.50 20.00 22.50 25.00 27.50WEIGHT OF FLAT IRON 495 WEIGHT, PER FOOT, OF FLAT IRON. mt ® o Thickness, in Fractions of Inches. a ~ t-. a 4 -L3 T b 7 8 J s lb 1 1* iJ wv It6« n 1 2.50 2.71 2.92 3.13 3.33 3.54 3.75 3.96 4.17 4.37 4.58 lj 2.81 3.05 3.28 3.52 3.75 3.98 4.22 4.45 4.69 4.92 5.10 H 3.13 3.39 3.65 3.91 4.17 4.43 4.69 4.95 5.21 5.47 5.73 n 3.44 3.72 4.01 4.30 4.58 4.87 5.16 5.44 5.73 6.02 6.30 n 3.75 4.06 4.38 4.69 5.00 5.31 5.63 5.94 6.25 6.56 6.88 if 4.00 4.40 4.74 5.08 5.42 5.75 6.09 6.43 6.77 7.11 7.45 i| 4.38 4.74 5.10 5.47 5.83 6.20 6.56 6.93 7.29 7.06 8.02 n 4.69 5.08 5.47 5.86 6.25 6.64 '7.03 7.42 7.81 8.20 8.59 o 5.00 5.42 5.83 6.25 6.67 7.08 7.50 7.92 8.33 8.75 9.17 if 5.31 5.75 6.20 6.64 7.08 7.52 7.97 8.41 mm 9.30 9.74 <)1 5.63 6.09 6.56 7.03 7.50 7.97 8.44 8.91 9.38 9.84 10.31 4 5.94 6.43 6.93 7.42 7.92 8.41 8.91 9.40 9.90 10.39 10.89 21 6.25 6.77 7.29 7.81 8.33 8.85 9.38 9.90 10.42 10.94 11.46 06 6.56 7.11 7.66 8.20 8.75 9.30 9.84 10.39 10.94 11.48 12.03 H 0.88 7.45 8.02 8.59 9.17 9.74 10.31 10.89 11.40 12.03 12.60 97 -a 7.19 7.79 8.39 8.98 9.58 10.18 10.78 11.38 11.98 12.58 13.18 3 7.50 8.13 8.75 9.38 10.00 10.63 11.25 11.88 12.50 13.13 13.75 8.13 8.80 9.48 10.16 10.83 11.51 12.19 12.86 13.54 14.22 14.90 3| 8.75 9.48 10.21 10.94 11.67 12.40 13.13 13.85 14.58 15.31 10.04 on 9.38 10.16 10.94 11.72 12.50 13.28 14.06 14.84 15.63 10.41 17.19 4 10.00 10.83 11.67 12.50 13.33 14.17 15.00 15.83 16.67 17.50 18.33 4$ 10.63 11.51 12.40 13.28 14.17 15.05 15.94 16.82 17.71 18.59 19.48 4.} 11.25 12.19 13.13 14.06 15.00 15.94 16.88 17.81 18.75 19.09 20.63 4g 11.88 12.86 13.85 14.84 15.83 16.82 17.81 18.80 19.79 20.78 21.77 5 12.50 13.54 14.58 15.63 16.67 17.71 18.75 19.79 20.83 21.88 22.92 5} 13.13 14.22 15.31 16.41 17.50 18.59 19.69 20.78 21.88 22.97 24.06 5i 13.75 14.90 16.04 17.19 18.33 19.48 20.03 21.77 22.92 24.06 25.21 4 14.38 15.57 16.77 17.97 19.17 20.36 21.56 22.76 23.96 25.16 26.35 6 15.00 16.25 17.50 18.75 20.00 21.25 22.50 23.75 25.00 20.25 27.50 0| 15.03 10.93 18.23 19.53 20.83 22.14 23.44 24.74 20.04 27.34 28.05 0-3 10.25 17.00 18.96 20.31 21.07 23.02 24.38 25.73 27.08 28.44 29.79 oa 16.88 18.28 19.69 21.09 22.50 23.91 25.31 20.72 28.13 29.53 30.94 7 17.50 18.96 20.42 21.88 23.33 24.79 26.25 27.71 29.17 30.62 32.08 | 18.13 19.04 21.15 22.66 24.17 25.68 27.19 28.70 30.21 31.72 33.23 18.75 20.31 21.88 23.44 25.00 26.56 28.13 29.69 31.25 32.81 34.38 n 19.38 20.99 22.00 24.22 25.83 27.45 29.06 30.08 32.29 33.91 35.52 8 20.00 21.07 23.33 25.00 26.C7 28.33 30.00 31.67 33.33 35.00 36.67 « 20.03 22.34 24.0 5 25.78 27.50 29.22 30.94 32.06 34.3S 36.09 37.81 si 21.25 23.02 24.79 26.56 28.33 30.10 31.88 33.65 35.42 37.19 38.96 *1 21.88 23.70 25.52 27.34 29.17 30.99 32.81 34.64 36.46 38.28 40.10 1 22.50 24.38 26.25 28.13 30.00 31.88 33.75 35.63 37.50 39.38 41.25 ®J 9§ 23.13 25.05 20.98 28.91 30.83 32.76 34.69* * 36.61 38.54 40.47 42.40 23.75 25.73 27.71 29.69 31.07 33.65 35.63 37.60 39.58 41.50 43.54 »3 24.38 20.41 28.44 30.47 32.50 34.53 36.56 38.59 40.03 42.66 44.09 10 25.00 27.08 29.17 31.25 33.33 35.42 37.50 39.58 41.07 43.75 45.83 10} 25.02 27.70 29.90 32.03 34.17 30.30 38.44 40.57 42.71 44.84 40.98 20.25 28.44 30.03 32.81 35.00 37.19 39.38 41.56 43.75 45.94 48.13 \4 20.88 29.11 31.35 33.59 35.83 38.07 40.31 42.55 44.79 47.03 49.27 11 27.50 29.79 32.08 34.38 30.07 38.96 41.25 43.54 45.83 48.13 50.42 11} 28.13 30.47 32.81 35.16 37.50 39.84 42.19 44.53 40.88 49.22 51.50 ll! 2 8.75 31.15 33.;>4 35.94 38.33 40.73 43.13 45.52 47.92 50.31 52.71 ■ 29.38 31.82 34.27 36.72 39.17 41.01 44.06 46.51 48.90 51.41 53. S 5 “ j 8 ».00 3i.50 35.00 37.50 40.00 42.50 45.00 47.50 50.00 52.50 55.00496 WEIGHT OF CAST-IRON PLATES, WEIGHT OF CAST-IRON PLATES. WEIGHT, IN POUNDS, OF CAST-IRON PLATES ONE INCH THICK. (Calculated at 450 lbs. per cubic foot.) . • X it "3 Width, in Inches. 6 8 10 12 14 16 18 20 24 30 4 6.25 8.3 10.4 ' 12.5 4 16.6 1S.7 20.S 25 31 6 9*37 12.5 15.6 18.7 21.8 25.0 28.1 31.2 3S 47 8 12.50 16.6 20.8 25.0 29.1 oo o 00.0 37.4 41.6 50 62 10 15.60 20.8 26.0 31.2 36.4 41.6 46.8 52.0 63 78 12 18.70 25.0 3lB 37.5 43.7 49.9 56.2 62.4 75 04 14 21.80 29.2 36.4 43.7 51.0 58.2 65.5 72 .S 88 109 16 24.90 Of> o O 0*0 41.6 50.0 5S.2 66.6 74.9 83.2 100 125 18 28.10 3t.5 46.8 56.2 65.5 74.9 84.2 93.6 113 140 i 20 31.20 41.6 52.0 62.5 72.8 83.2 93.6 104.0 125 f ; 156 ! 22 34.30 45.8 57.2 6S.6 80.1 91.5 103.0 114.4 138 ; ' 1721 24 37.50 50.0 62.4 75.0 87.4 99.8 112.3 124.8 150 : i 1S7 j ‘ 26 40.00 54.0 67.6 SI .2 94.6 10S.2 121.7 13521 163 203 j 28 43.60 5S.2 72. S 87.5 101.9 116.5 131.0 145.6 175 ! 218 [ 30 46.S0 62.4 78.0 93.7 109.2 124.8 140.4 156.0 1SS j 234 | ; 32 49 .SO 66.6 83.2 100.0 116.5 133.1 150.3 i 166.4 200 1 250 j 36 56.10 75.0 93.6 112.5 131.0 150.0 16S.4 1S7.2 225 281 jWEIGHT OF ROLLED LEAD, iOPPER, AND BRASS, SHEET AND BA I V, Thickness or diameter, in inches. Lead. Copper. Brass. Thickness or diameter, in inches. Sheets, per square foot. Square bars 1 foot long. Hound bars 1 foot long. Sheets, per square foot. Square bars 1 foot long. Round bars 1 foot long. Sheets, per square foot. Square bars 1 foot long. Round bars 1 foot long. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 1-32 1.86 0.005 0.004 1.44 0.004 0.003 1.36 0.004 0.003 1-32 1-10 3.72 0.019 0.015 2.89 0.015 0.012 2.71 0.014 0.011 1-16 3-32 5.58 0.044 0.034 4.33 0.034 0.027 4.06 0.032 0.025 3-32 1-8 • 7.44 0.078 0.001 5.77 0.060 0.047 5.42 0.056 0.044 1-8 5-32 9.30 0.121 0.095 7.20 . 0.094 0.074 6.75 0.088 0.069 5-32 3-10 11.20 0.174 0.137 8.66 0.135 0.106 8.13 0.127 0.100 3-16 7-32 13.00 0.237 0.187 10.10 0.184 0.144 9.50 0.173 0.136 7-32 1-4 14.90 0.310 0.244 11.50 0.240 0.189 10.80 0.226 0.177 1-4 5-10 18.60 0.485 0.381 14.40 0.376 0.295 13.50 0.353 0.277 5-16 3-8 *22.30 0.698 0.548 17.30 0.541 0.425 16.30 0.508 0.399 3-8 7-10 *26.00 0.950 0.740 20.20 0.736 0.578 19.00 0.691 0.543 7-16 1-2 29.80 1.240 0.974 23.10 0.962 0.755 21.70 * 0.903 0.709 1-2 9-10 33.50 1.570 1.230 26.00 1.220 0.955 24.30 1.140 0.900 9-16 5-8 37.20 1.940 1.520 28.90 1.500 1.180 27.10 1.410 1.110 5-8 11-10 40.90 2.340 1.840 31.70 1.820 1.430 29.80 1.700 1.340 11-16 3-4 44.00 2.790 2.190 34.60 2.160 1.700 32.50 2.030 1.6(H) 3-4 13-10 48.30 3.270 2.570 37.50 2.550 1.990 35.20 2.380 1.870 13-16 7-8 52.10 3.800 2.9S0 40.40 2.940 2.310 37.90 2.760 2.170 7-8 15-10 56.00 4.370 3.420 43.30 3.380 2.650 40.60 3. ISO 2.490 15-16 1 59.50 4.960 3.900 46.20 3.850 3.020 43.30 3.610 2.810 1 1 1-8 60.90 6.270 4.920 52.00 4.870 3.820 48.70 4.570 3.6(H) 1 1-8 1 1-4 74.40 7.750 6.090 57.70 6.010 4.720 54.20 5.640 4.43,0 1 14 1 3-3 81.80 9.370 7.370 63.50 7.280 5.720 59.00 6.820 5.370 1 3-8 1 1-2 89.30 11.200 8.770 69.30 8.650 6.800 65.00 8.120 6.380 1 1-2 1 5-8 90.70 13.100 10.30 75.10 10,200 . 7.980 70.40 9.530 7.490 1 5-8 1 3-4 104.00 15.200 11.90 80.80 11.800 9.250 75.90 11.100 8.680 1 3-4 1 7-8 112.00 17.500 13.70 86.00 13.500 10.600 81.30 12.700 9.070 1 7-8 •> 119.00 19.800 15.60 92.30 15.400 12.100 86.70 14.400 11.300 2 WEIGHT OF LEAD, COPPER, AND BRASS. 497498 WEIGHT OF BOLTS, NUTS, AND BOLT-HEADS WEIGHT OF ONE HUNDRED BOLTS WITH SQUAB HEADS AND NUTS. [Iloopes & Townsend’s list.] Length Diameter of Bolts. under head to point. \ in. AiH- 3 . 7f 1». _7 . 16 111 • \ in. 5 • -g in. 4- in. i in. 1 in. j lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. n 4.00 7.00 10.50 15.20 22.50 39.50 63.00 - - n 4.35 7.50 11.25 16.30 23.82 41.62 66.00 - - 2 4.75 8.00 12.00 17.40 25.15 43.75 69.00 109.00 163 5.15 8.50 12.75 18.50 26.47 45.88 72.00 113.25 169 ■ 5.50 9.00 13.50 19.60 27.80 48.00 75.00 117.50 174 ■ 5.75 9.50 14.25 20.70 29.12 50.12 78.00 121.75 180 3 6.25 10.00 15.00 21.80 30.45 52.25 81.00 126.00 185 3$ 7.00 11.00 16.50 24.00 33.10 56.50 87.00 134.25 196 4 7.75 12.00 18.00 26.20 35.75 60.75 93.10 142.50 207 4§ 8.50 13.00 19.50 28.40 38.40 65.00 99.05 151.00 218 5 9.25 14.00 21.00 30.60 41.05 69.25 105.20 159.55 229 r;i 10.00 15.00 22.50 32.80 43.70 73.50 111.25 168.00 240 6 10.75 16.00 24.00 35.00 46.35 77.75 117.30 176.60 251 - - 25.50 37.20 49.00 82.00 123.35 185.00 262 7 - - 27.00 39.40 51.65 86.25 129.40 193.65 273 7i - - 28.50 41.60 54.30 90.50 135.00 202.00 284 8 - - 30.00 43.80 59.60 94.75 141.50 210.70 295 9 - - - 46.00 64.90 103.25 153.60 227.75 317 10 - - - 48.20 70.20 111.75 165.70 224.80 339 11 - - - 50.40 75.50 120.25 177.80 261.85 360 12 - - - 52.60 80.80 128.75 189.90 278.90 382 13 - - - - 86.10 137.25 202.00 295.95 404 14 - - - - 91.40 145.75 214.10 313.00 426 15 - - - - 96.70 154.25 226.20 330.05 448 16 - - - - 102.00 162.75 238.30 347.10 470 17 - - - - 107.30 171.00 250.40 364.15 492 18 - — - _ 112.60 179.50 262.60 381.20 514 19 — - — - 117.90 188.00 274.70 398.25 536 20 - - - - 123.20 206.50 286.80 415.30 558 L'crinch ) addit'l \ 1.37 2.13 3.07 4.18 5.45 8.52 12.27 16.70 21.82 WEIGHTS OF NUTS AND BOLT-IIEADS, IN POUNDS. For calculating the weight of longer bolts. ■ Diameter of bolt, in inches. 1 t 3 IT 4 & 3 8 T 7 8 Weight of hexagon nut and head - 0.017 0.057 0.128 0.267 0.43 0.73 Weight of square nut and head - 0.021 0.069 0.164 0.320 0.55 0.88 Diameter of bolt, in inches. 1 H B if 0 Ol 3 Weight of hexagon nut and head Weight of square nut and 1.10 2.14 3.78 5.6 8.75 17 28.8 head 2.56 4.42 7.0 10.50 21 36.4499 WEIGHT OF IRON RIVETS. IRON RIVETS.—WEIGHT PER HUNDRED. Length Diameters. under lead. 1 4 3 8 i 1 3 4 1 8 1 i 1.895 4.848 0.966 16.79 26.49 39.3 55.2 H 2.067 5.235 10.340 17.86 27.99 41.4 57.9 \i 2.238 5. GIG 11.040 18.96 29.61 43.5 60.7 if 2.410 6.003 11.730 20.03 31.13 45.6 63.4 if 2.582 6.402 12.430 21.04 32.74 47.8 66.2 if 2.754 6.789 13.120 22.11 34.25 49.9 68.9 if 2.926 7.179 13.810 23.21 35.86 52.0 71.7 ï| 3.098 7.566 14.500 24.28 37.37 54.1 74.4 2 3.269 7.956 15.190 25.48 38.99 56.3 77.2 E 3.441 8.343 15.8S0 26.56 40.40 58.4 79.9 n 3.613 8.733 16.570 27.65 42.11 60.5 82.7 93. 3.785 9.120 17.260 28.73 43.67 62.6 85.4 ft 3.957 9.511 17.950 29.82 45.24 64.8 88.2 It 4.129 9.898 18.640 30.90 46.80 66.9 90.9 03 ■ 4.301 10.290 19.330 31.99 48.36 69.0 93.7 91 4.473 10.670 20.020 33.08 49.92 71.1 96.4 3 4.644 11.060 20.710 34.18 51.49 73.3 99.2 li °8 4.816 11.440 21.400 35.27 53.05 75.4 101.9 oi °4 4.988 11.840 22.090 36.35 54.61 77.5 104.7 03 ,J8 5.160 12.230 22.780 37.44 56.17 79.6 107.4 Si 5.332 12.620 23.480 38.52 57.74 81.8 110.2 5.504 13.010 24.170 39.60 59.30 83.9 112.9 3f 5.676 13.390 24.860 40.69 60. S6 86.0 116.7 Ql °8 5.848 13.780 25.550 41.78 62.42 88.1 119.4 4 6.019 14.170 26.240 42.87 63.99 90.3 121.2 41 6.191 14.560 26.930 43.94 65.55 92.4 123.9 6.363 14.950 27.620 45.01 67.11 94.5 126.6 100 heads. 0.519 1.74 4.14 8.10 13.99 22.27 33.15 Length of rivet required to make one head = 1} diameters of round bar.500 NAILS AND SPIKES. NAILS AND SPIKES. SIZE, LENGTH, AND NUMBER TO THE POUND. Cumberland Nail and Iron Company. Ordinary. Clinch. Finishing. Size. Length, No. to Length, No. to Size. Length, No. to in inches. pound. in inches. pound. in inches. pound. 2 a X 8 716 2 152 4 d if 384 3(7 fine ItV 588 m 133 5 d 13 ■fl 3 d 448 fl 92 6(7 2 204 4d it 336 i 72 8(7 1 102 5 d it 216 3 60 10(7 3 80 6(7 2 166 m 43 12(7 0 5 65 Id 2i 118 20(7 46 8 d m 94 10 d n 72 Fence. 12 d | 50 Core. 20 d 0 32 | 96 66 30(7 20 m 40 d D 17 m 2i 56 6(7 2 143 50(7 5 14 50 8(7 i 68 60 d 5i 10 3 40 10(7 8 60 12(7 II 42 Light. Spikes. 20(7 30(7 40(7 8 4 4| 25 18 14 4 a 5 d (id H lì 2 373 272 196 W 02 4 él 19 15 13 WH W II L H 2Ì 69 72 10 o Brads. n 0-2 6 9 7 Slate. 3(7 B 288 C)d 2 163 8(7 1 Od ■ 96 74 Boat. 4(7 iA 244 2| 5(7 if 187 12 d O 1 •I« 50 i| 206 6(7 2 146 TACKS. Number 4^ Number H Number Size. il) to Size. to Size. to o k—1 pound. o pound. || M pound. 1 oz. 1 8 16000 4 oz. m 1 (T 4000 14 oz. Hi 1 6 1143 ■ 1 A 10066 6 “ y "l t> 2666 16 I 1 1000 i 4 8000 8 1 A 8 2000 18 44 m 888 ft 1 1 (T 6400 10 1 tt 1600 20 “ i 800 3 “ » 5333 12 “ 3. 4 1333 22 fl 727WEIGHT OF PLAIN CAST-IRON PIPES. 501 WEIGHT OF PLAIN CAST-IRON PIPES. WEIGHT OF A LINEAR FOOT WITHOUT JOINTS. Thickness of Metal, in Inches. s, a i T 3 ■5 l 5 8 3 4 7 8 1 n lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. lbs. 2 5.5 8.7 12.3 16.1 20.3 24.7 29.5 34 5 39.9 21 0.8 10.6 14.7 19.2 24.0 29.0 34.4 40.0 46.0 o ') 7.0 12.4 17.2 22.2 27.6 32.3 39.3 45.6 52.2 ;u 0.2 14.3 19.6 25.3 31.3 37.6 44.2 51.0 58.3 4 10.4 10.1 22.1 28.4 35.0 41.9 49.1 56.6 64.4 11.7 18.0 24.5 31.5 38.7 46 2 54 0 62.1 70.6 5 12.1) 19.8 27.0 34.5 42.3 50.5 59.9 67.7 76.7 •51 14.1 21.0 29.5 37.6 46.0 54.8 63.8 73.2 82.9 6 15.3 23.5 31.9 40.7 49.7 59.1 68.7 78.7 89.0 7 17.8 27.2 36.9 46.8 57.1 67.7 78.5 89.8 101.0 8 20.3 30.8 41.7 52.9 64.4 76.2 88.4 101.0 114.0 9 22.7 34.5 46.0 59.1 71.8 84.8 98.2 112.0 126.0 10 25.2 38.2 51.5 05.2 79.2 93.4 108.0 123.0 138.0 11 27.0 41.9 50.5 71.3 80.5 102.0 118.0 134.0 150.0 m 30.1 45.0 01.4 77.5 93.9 111.0 128.0 145.0 163.0 14 32.5 49.2 00.3 83.6 101.0 119 0 138.0 156.0 175.0 ! 14 35.0 52.9 71.2 89.7 109.0 128.0 147.0 167.0 187.0 15 37.4 50.0 70.1 95.9 116.0 136.0 157.0 178.0 199.0 | 10 39.1 00.3 81.0 102 0 123.0 145.0 167.0 189.0 212.0 IS 44.8 07.7 90.9 114.0 138.0 162.0 187.0 211.0 236.0 20 49.7 75.2 101.0 127.0 153.0 179.0 206.0 233.0 261.0 1 22 54.0 82.6 111.0 139.0 168.0 197.0 226.0 255.0 285.0 24 59.0 89.9 120.0 151.0 182.0 214.0 245.0 278.0 310.0 20 04.5 97.3 131.0 164.0 198.0 231.0 266.0 300.0 335.0 28 09.4 105.0 140.0 176.0 212.0 249.0 286.0 323.0 360.0 30 74.2 112.0 150.0 188.0 1 227.0 266.0 305.0 345.0 384.0 Note. — For each joint, atkl a foot to length of pipe.502 WEIGHT OF CAST-IllON PIPES IN GENERAL. WEIGHTS, PER FOOT, OF CAST-IRON PIPES IN GENERAL USE, INCLUDING SOCKET AND SPIGOT ENDS. [Dennis Long & Co.] Diameter. Thickness. Weight per toot. Diameter. Thickness. Wei per trht oot. o inches. \ + inch. S lhs. 14 inches. 1 inch. 148 lbs. 2 | 3 “ 8 0i “ 10 66 1 66 2 85 2 66 1 66 Y 14 “ 10 66 ft 66 108 B o 66 i + “ 11 “ 10 66 3 66 4 120 O O 66 3 44 8 144 | 10 66 X 66 152 Q o 66 1 P 2 18 “ 10 66 1 1 175 o o “ 5 6 6 8 24 B IS “ £ 66 114 4 66 $ + “ io^ 3 18 44 3 4 147 4 66 i “ 24 3 18 “ 1_ 66 101 4 “ £, 66 8 41 | 20 44 ft 6 6 8 142 (5 “ 3 44 8 25 “ 20 44 % 100 (3 “ £ *>•> 6 6 *>o 20 44 8 U 107 0 66 ü 44 8 42'- ■ 20 44 1 “ 215 0 66 3 44 4 52 “ 24 44 b 6 6 150 S 66 3 << 8 40 “ 24 44 3 66 4 100 s 66 1 44 Y 44} “ 24 66 L 66 224 8 66 5. 44 8 50 “ 24 66 1 ■“ 257 8 66 3 44 4 08 P 40 66 3 66 4 247 10 66 Vti+ “ 50 “ 40 66 7 6 6 277 10 B I 66 '1 54 “ 40 1 1 410 10 "8 OS “ 40 1| | 300 10 “ 3. 6 *. 4 80 “ 40 44 .7 6 6 442 12 « I << Y 07 “ 40 44 1 “ 381 44 12 i /» u 8 82 “ 40 “ H fl 420 12 “ 3 66 4 00 “ 30 44 H 1 470 12 7. 4t 8 117 “ 48 l “ 512 14 | X 44 2 74 “ 48 “ if " 584 14 1 5 66 8 04 “ 48 44 Ü 1 685 14 66 3 44 T 114 “ 48 U “ 775 WEIGHT OF CAST-IRON WATER-PIPES. 503 WEIGHTS OF CAST-IRON WATER-PIPES. In pounds, per foot run, including bells and spigots. Diameter. Philadel- phia.1 Chicago.2 Cincinnati.2 Stand- ard.2 Light.2 Weight. Thickness 2 ins. _ - - - 7 6 Q U o 15.000 - 17 l ~i inch. 15 13 4 “ 21.111 24.167 23 l 7 u 22 20 6 “ 30.106 36.666 50 3 4 a 33 30 8 “ 40.683 50.000 65 3 4 u 42 40 10 “ 52.075 65.000 80 3. 4 a 60 55 12 “ 69.162 oo ooo oo.o*>) 100 3 4 u 75 70 10 “ 102.522 125.000 130 3 4 u - - 20 “ 147.681 - 200 1 8 a - - 24 “ - 250.000 224 . 7_ 8 a - - 30 “ - - 300 1 u - - 36 “ - 450.000 430 H ! u - - Water-pipe is usually tested to three hundred pounds’ pressure per square inch before delivery, and a hammer test should be made while the pipe is under pressure. The Philadelphia lengths for each section are, for three and four inch pipe, 9 feet; all larger sizes, 12 feet 3t inches in length. The Cincinnati lengths are uniform for all diameters, —12 feet. Chicago, same as Cincinnati. Standard lengths are, for two-incli pipe, 8 feet, and all other sizes, 12 feet. The thickness of the lead joint ranges from one-fourtli inch on small sizes to one-lialf inch on the large sizes. WEIGHTS OF LEAD AND GASKET FOR PIPE JOINTS. [Dennis, Long, & Co.] Diameter of pipe. Lead. Gasket. Diameter of pipe. Lead. Gasket. 2 inches. 2.5 lbs. 0.125 lbs. 12 inches. 15 lbs. 0.250 lbs. 3 “ 3.5 “ 0.170 “ 14 i 18 1 0.375 “ 4 | 4.5 “ 0.170 1 16 “ 22- m 0.500 “ 6 “ 6.5 « 0.200 “ 18 1 26 “ 0.500 “ 8 “ 9.0 i 0.300 | 20 “ 33 " 0.625 “ 10 “ 13.0 “ 0.250 I 1 From Trautwine. 2 Dennis, Long, & Co., Louisville, Ky.WROUGHT-IRON WELDED TUBES, FOR STEAM, GAS, OR WATER. inch and below, butt welded; proved to three hundred pounds j 1J inch and above, lap welded: proved to five hundred pounds per per square inch, hydraulic pressure. j square inch, hydraulic pressure. TABLE OF STANDARD DIMENSIONS (Monnis, Taskeh, & Co., Limited). Inside diameter, in inches. Actual outside diameter, in inches. Thickness in inches. Actual inside diameter, in inches. Internal circum- ference, in inches. External circum- ference, in inches. 1 j , Length of pipe per square foot of inside surface, in feet. Length of pipe per square foot of outside surface, in feet. Internal area, in inches. External area, in inches. Length of pipe containing one cubic foot, in feet. Weight per foot of length, iii pounds. Number of threads -per inch of screw. 1 0.405 0.068 0.270 0.848 1.272 14.150 9.440 0.0572 0.129 2500.00 0.243 27 1 0.540 0.088 0.364 1.144 1.696 10.500 7.075 0.1041 0.229 1385.00 0.422 18 g 0.075 0.091 0.494 1.552 2.121 7.670 5.657 0.1916 0.358 751.50 0.561 18 1 0.840 0.109 0.623 1.957 2.652 6.130 4.502 0.3048 0.554 472.40 0.845 14 1.050 0.113 0.824 2.589 3.299 4.635 3.637 0.5333 0.866 270.00 1.126 14 1 1.315 0.134 1.048 3.292 4.134 3.679 2.903 0.8627 1.357 166.90 1.670 1U 1J 1.060 0.140 1.380 4.335 5.215 2.768 2.301 1.4960 2.164 96.25 2.258 ll| 1£ 1.900 0.145 T 5.061 5.969 2.371 2.010 2.0380 2.835 70.65 2.694 115 2 2.375 0.154 2.067 6.494 7.461 1.848 1.611 3.3550 4.430 42.36 3.667 11k 2£ 2.875 0.204 2.468 7.754 9.032 1.547 1.328 4.7830 6.491 30.11 5.773 8 0 3.500 0.217 3.067 9.636 10.996 1.245 1.091 7.3880 9.621 19.49 7.547 8 Ol 4.000 0.226 3.548 11.146 12.566 1.077 0.955 9.8870 12.566 14.56 9.055 8 4 4.500 0.237 4.026 12.648 14.137 0.949 0.849 12.7300 15.904 11.31 10.728 8 4k 5.000 0.247 4.508 14.153 15.708 0.848 0.765 15.9390 19.635 9.03 12.492 8 5 5.563 0.259 5.045 15.849 17.475 0.757 0.629 19.9900 24.299 7.20 14.564 8 6 6.625 0.280 6.065 19.054 20.813 0.630 0.577 28.8890 34.471 4.98 18.767 8 7.625 0.301 7.023 22.063 23.954 0.544 0.505 38.7370 45.663 3.72 23.410 8 8 8.625 0 322 7.982 25.076 27.096 0.478 0.444 50.0390 58.426 2.88 28.348 8 9 9.688 0.344 9.001 28.277 30.433 0.425 0.394 63.6330 73.715 2.26 34.077 8 10 10.750 0.366 10.019 31.475 33.7 < 2 0.381 0.355 78.8380 90.762 1.80 40.641 8 Taper of threads, 1 to 32 ou each side. 504 WROUGIlT-IItON WELDED TUBES.WROUGHT-IItON AND LAP-WELDED TUBES, 505 WBOUGHT-IRON WELDED TUBES, EXTRA STRONG. Table of standard dimensions. Nominal diameter, in inches. Actual outside diameter, in inches. Thickness, in Inches. 1 Actual inside Diameter, in Inches. 1 Extra strong. Double extra strong. Extra strong. Double extra | strong. i 0.405 0.100 _ 0.205 - i 0.540 0.123 - 0.294 i 0.675 0.127 - 0.421 - i 0.840 0.149 0.298 0.542 0.244 $ 1.050 0.157 0.314 0.736 0.422 1 1.315 0.1S2 0.364 0.951 0.587 n 1.660 0.194 0.388 1.272 0.884 if 1.900 0.203 0.406 1.494 1.088 2 2.375 0.221 0.442 1.933 1.491 91 2.875 0.280 0.560 2.315 1.755 n O 3.500 0.304 0.60S 2.892 2.284 3 k 4.000 0.321 0.642 3.358 2.716 4 4.500 0.341 0.682 3.818 3.136 LAP-WELDED AMERICAN CHARCOAL IRON BOILER- TUBES. Standard dimensions (Table of Morris, Tasker, & Co., Limited). o +-> o "Xj * ,— o C V w Standard thickness, in inches. Internal diameter, in inches. Internal circumfer-ence, iu inches. 0 y "y 1 § 'z 2 I- O Ci **-> «-> A ° ü s- O P |y|jg o •—î o g o %_ .£ H § ® o — 'S) O CL ’uL Internal area, in inches. ■'* cS jD 5 Ì5 « — O ■2'S w = o 0 Jx 1 1 w *>- 1 0.072 0.856 2.689 3.142 4.460 3.819 0.575 0.785 0.70S H 0.072 1.106 3.474 3.927 3.455 3.056 0.960 1.227 0.900 I 0.083 1.334 4.191 4.712 2.863 2.517 1.396 1.767 1.250 Pi 0.095 1.560 4.901 5.498 2.418 2.183 1.911 2.405 1.665 *2 0.098 1.804 5.667 6.283 2.113 1.909 2.555 3.142 1.9S1 B 0.098 2.0-54 6.484 7.063 1.850 1.698 3.314 3.976 2.238 BK 0.109 2.283 7.172 7.854 1.673 1.528 4.094 4.909 2,755 O'i 0.109 2.533 7.957 8.639 1.50:8 1.390 5.033 5.940 3.045 3 0.109 2.783 8.743 9.425 1.373 1.273 6.083 7.069 8.333 3', 0.119 3.012 9.462 10.210 1.268 1.175 7.125 8.296 3.958 0.119 3.262 10.218 10.995 1.171 1.091 8,357 9.621 4.272 K 0.119 3.512 11.033 11.781 1.0S8 1.018 9.687 11.045 4.590 4 0.130 3.741 11.753 12.566 1.023 0.955 10.992 12.5-66 5.320 0.130 4.241 13.323 14.137 0.901 0.849 14.126 15.904 6.010 5 0.140 4.720 14.818 15.708 0.809 0.764 17.437 19.635 7.226 6 0.151 5.699 17.901 18.849 0.670 0.637 25.503 28.274 9.346 7 0.172 6.657 20.914 21.991 0.574 0.545 34.805 38.484 12.435 8 0.182 7.636 23.9S9 25.132 0.500 0.478 45.795 50.265 15.109 9 0.193 8.615 27.055 2S.274 0.444 0.424 58.291 63.617 18.002 10 0.214 9.573 30.074 31.416 0.399 0.382 71.975 78.540 22.190 1 In estimating the effective steam-heating or boiler surface of tubes, the surface in contact with air, or gases of combustion (whether internal or external to the tubes), is to be taken. For heating liquids by steam, superheating steam, or transferring heat from one liquid or one gas to another, the mean surface of the tubes is to be taken.506 GALVANIZED AND BLACK IRON. AMERICAN AND BIRMINGHAM WIRE GAUGES be Thickness, in Incues. Q> bb 9 Thickness, in Inches. No. of gauge. Thickness, in Inches. <+-4 o 6 54 American gauge. Birming- ham gauge. be o 6 M American gauge. Birming- ham gauge. American gauge. Birming- ham gauge. 0000 0.4000 0.454 11 0.0907 0.120 25 0.0179 0.020 000 0.4096 0.425 12 0.080S 0.109 26 0.0160 0.018 00 0.3648 0.380 13 0.0719 0.095 27 0.0142 0.016 0 0.3248 0.340 14 0.0641 © © 28 0.0126 0.014 1 0.2893 0.300 15 0.0570 • 0.072 29 0.0112 0.013 2 0.2576 0.284 16 0.0508 0.065 30 0.0100 0.012 Q o 0.2294 0.259 17 0.0452 0.058. 31 0.0089 0.010 4 0.2043 0.238 18 0.0403 0.049 32 0.0079 0.009 5 0.1819 0.220 19 0.0359 0.042 33 0.0070 0.008 6 0.1620 0.203 20 0.0319 0.035 34 O b co 0.007 7 0.1443 0.1S0 21 0.0284 0.032 35 0.0056 0.005 8 0.1285 0.165 22 0.0253 0.028 36 0.0050 0.004 1 0.1144 0.148 23 0.0225 0.025 10 0.1019 0.134 24 0.0201 0.022 GALVANIZED AND BLACK IRON. Weight, in pounds, per square foot of galvanized sheet-iron, both flat and corrugated. The numbers .and thicknesses are those of the iron before it is galvanized. When a flat sheet (the ordinary size of which is from two feet to two feet and a half in width, by six to eight feet in length) is converted into a corrugated one, with corrugations five inches wide from centre to centre, and about an inch deep (theCORRUGATED IRON. 507 common size), its width 13 thereby reduced about cne-tenth part, or from thirty to twenty-seven inches; and consequently the weight per square foot of area covered is increased about one-ninth part. When the corrugated sheets are laid upon a roof, the overlapping of about two inches and a half along their sides, and of four inches along their ends, diminishes the covered area about one-seventh part more, making their weight per square foot of roof about one-sixth part greater than before. Or the weight of corrugated iron per square foot, in place on a roof, is about one-tliird greater than that of the flat sheets of above sizes of which it is made. Weight of Ikon pek Squake Foot. p Black. Galvanized. 5 Sc Flat. Corrugated. Flat. Corrugated. 6 ' Lbs. On roof. Lbs. On roof. Lbs. On roof. Lbs. On roof. 30 0.485 0.566 0.539 0.647 0.818 0.954 0.896 1.08 29 0.526 0.614 0.583 0.701 0.859 1.000 0.954 1.14 28 0.565 0.659 0.628 0.753 0.898 1.040 0.997 1.20 27 0.646 0.754 0.718 0.861 0.979 1.140 1.090 1.30 26 0.722 0.842 0.802 0.963 1.060 1.240 1.180 1.41 25 0.808 0.942 0.897 1.070 1.140 1.330 1.270 1.52 24 0.889 1.040 0.907 1.180 1.220 1.420 1.360 1.62 23 1.010 1.180 1.120 1.350 1.3*0 1.560 1.490 1.79 I V>*> 1.130 1.310 1.260 1.510 1.460 1.700 1.620 1.95 21 1.290 1.500 1.430 1.720 1.630 1.900 1.810 2.17 20 1.410 1.640 1.560 1.880 1.750 2.040 1.940 2.33 19 1.690 1.970 1.880 2.250 2.030 2.370 2.260 2.71 18 1.980 2.310 2.200 2.640 2.320 2.700 2.580 3.09 17 2.340 2.730 2.600 3.120 2.680 3.120 2.980 3.57 16 2.630 3.070 2.920 3.510 2.960 3.450 3.290 3.95 15 2.910 3.390 3.230 3.880 3.250 3.790 3.610 4.33 14 3.360 3.920 3.730 4.480 3.690 4.300 4.100 4.92 13 3.840 4.480 4.270 5.120 4.180 4.870 4.640 5.57 Note. — The galvanizing of sheet-iron adds about one-third of a pound to its weight per square foot. KEYSTONE BRIDGE COMPANY’S CORRUGATED IRON. The Keystone Bridge Company’s corrugations are 2.425 inches long, measured on the straight line. They require a length of iron of 2.725 inches to make one corrugation, and the depth of corrugation is || inch. One corrugation is allowed for lap in the width of the sheet, and six inches in the length, for the usual pitch of roof508 CORRUGATED IRON. of two to one. Sheets can be corrugated of any length not exceeding ten feet. The most advantageous width is thirty inches and a half, which, allowing a half-inch for irregularities, will make eleven corrugations, equal to thirty inches, or, making allowance for laps, will cover twenty-four inches and a fourth of the surface of the roof. By actual trial it was found that corrugated iron No. 20, spanning six feet, will begin to give a permanent deflection for a load of thirty pounds per square foot, and that it will collapse with a load of sixty pounds per square foot. The distance between centres of purlins should therefore not exceed six feet, and preferably be less than this. The following table is calculated for sheets thirty inches and a half wide before corrugating: — Xnmber by Birmingham gauge. 00 o o 00 a o [o Weight per square foot, Oat, in pounds. II m o s -4-3 o £ Jn fen o *o o a Weight per square of one hundred square feet, when laid, allowing six inches’ lap in length, and two inches and a half, or one corrugation, in width of sheet, for sheet-lengths of P 4. ~ ® * in C Irj k*- K*- 5 feet. r> feet. 7 feet. 8 feet. feet. 10 feet. 16 0.065 2.61 3.28 365 358 353 350 348 346 2.95 18 0.019 1.97 2.18 275 270 267 261 262 261 2.31 20 0.035 1.10 1.76 196 192 190 1SS 189 185 1.71 22 0.028 1.12 1.11 156 151 152 150 149 118 1.16 21 0.022 0.88- 1.11 123 121 119 118 117 117 1.22 26 0.018 0.72 0.91 101 09 97 97 96 95 1.06 Results of Test of a corrugated sheet No. 20, two feet wide, six feet long between supports, loaded uniformly with fire-clay. Load per square foot, in pounds. Deflection at centre, under load, in inches. Permanent deflection load removed, in inches. Load per square foot, in pounds. Deflection at centre, under load, in inches. Permanent deflection load removed, in inches. 5 1 5 35 9' 1 5 10 4 - 40 “8 i 15 l - 45 f§§ l| 20 y i 50 i A 25 n _ 55 6* •.Not noted 30 ii Ù 60 Broke down (<MEMORANDA FOR EXCAVATORS, ETC. 509 MEMORANDA FOR EXCAVATORS AND WELL- DIGGERS. Excavating is generally done by the cubic yard, or square; a cubic yard being twenty-seven cubic feet; and a square is generally reckoned as eight yards, or a cube six feet by six feet by six feet. % Weill 3 feet clear diameter and £ brick thick will require the net excavation, per foot in depth of . 11 cubic feet. a o feet 6 inches diameter, { brick thick Hi it ti tt 4 it a 1 V ti 1"? it ii a 4 “ 6 a n 1 2 it it 213 it it tt 5 tt it 1 2 it it 2G it ti it 5 “ 6 u a 1 2 it it 30 ti ti a 0 u it J 2 it it 30 it it u G “ 6 a a 1 it a 50 i- ti ti a 7 it a 1 i t n 5G it ii it 7 “ G a it 1 it a C3i it it u 8 t i it 1 it “ 71 it it it 8 “ 6 a it 1 it “ 78^ it ii it 9 it it 1 ti tt 86i it ii a 10 tt it 1 it “ 104 it it it 10 “ G a a 1 it it 113 it ii a 11 a u 1 it a 1223 ti ii tt 11 “ G u it 1 it a 132J it ii tt 12 it it 1 a a 143i a ii From 11 1 to 15 cubic feet of chalk Y to 19 a a clay 18 to 24 a it eartli t = 1 ton weight. 18 to 20 a a grave 19 to 25 a a sand Or an average for general calculations may be taken as follows: — 14 cu. feet of chalk weigh 1 ton 119 cu. feet of gravel weigh 1 ton 15 “ “ clay “ 1 “ 122 “ “ sand “ 1 “ 21 “ “ earth “ 1 “ i A cubic yard of earth in original position will occupy from a cubic yard and a fourth to a cubic yard and a half, when dug. A single load of sand or loam should contain 22 cubic feet; a double load, 44 cubic feet. When buying by the load, the size of load should always be specified.510 MEMORANDA FOR BRICKLAYERS. MEMORANDA FOR BRICKLAYERS. QUANTITY OF BRICK-WORK IN BARREL-DRAINS AND WELLS . Including wastage in clipping around the curves. ** Biametei in clear. Thickness of work. brick- Superficial feet of brick-work in one linear yard. Number of bricks required for one linear yard. ! 1 ft. 0 ins. 0 ft. 44 ins. 10 ft, G ins. 115 1 ii 6 1. 0 ll i. 21 u 2 m 148 2 n 0 ll 0 44 a 25 “ 10 r 181 2 ii 0 a 0 ll 9 a 33 “ 0 “ 4G2 2 a 6 i l 0 ll 9 a 37 1 8 ‘1 528 2 a G ll 1 ll 1 n 43 “ 2 “ 90G 3 a 0 II 0 9 a 42 B 6 “ 594 3 ii 0 11 1 ll 1 a 47 “ 10 “ 1004 3 n 0 ll 0 11 9 ii 47 a i a 659 3 a G li 1 ll 1 ii 52 “ 7 “ 1104 4 n 0 ll 0 ll 9 ii 51 “ 10 * 725 4 a 0 ll 1 ll 1 a 57 “ 3 1203 5 a 0 ll 0 ll 9 ii 61 “ 3 “ 857 5 a 0 ll 1 ll 1 a 6(1 “ 9 “ 1402 0 a 0 ll 1 ll 1 a 70 “ 1 “ 1597 7 a 0 ll 1 ll 1 n 85 “ 0 “ 1795 Note.—In the Eastern States, the thickness would be four inches, eight inches, and twelve inches, instead of those given in the tabic, as the brick are smaller. A load of mortar measures a cubic yard, or twenty-seven cubic feet; requires a cubic yard of sand and nine bushels of lime, and will fill thirty hods. A bricklayer’s hod, measuring 1 foot 4 inches by 9 inches by 9 inches, equals 1296 cubic inches in capacity, and contains twenty bricks. A single load of sand and other materials equals a cubic yard, or twenty-seven cubic feet; and a double load equals twice that quantity. A measure of lime is a single load, or cubic yard. One thousand bricks closely stacked occupy about fifty-six cubic feet. One thousand old bricks, cleaned and loosely stacked, occupy about seventy-two cubic feet. One superficial foot of gauged arches requires ten bricks. One superficial foot of facings requires seven bricks.DRAIN-PIPE. 511 One yard of paving requires thirty-six stock bricks laid flat, or fifty-two on edge, and thirty-six paving bricks laid flat, or eighty-two on edge. The bricks of different makers vary in dimensions, and those of the same maker vary also, owing to the different degrees of heat to which they are subjected in burning. The memoranda given above, for brick-work are therefore only approximate. The following table gives the usual dimensions of the bricks in various parts of the country: — Description. Inch?s. Description. Inches. Baltimore front Philadelphia front, Wilmington front. Trenton front . . Croton Colabaugh . . . 1 8* x 4J x 2| 1. 81 x 4 x 2J 8j x S| x 28 Maine Milwaukee . . . North River . . Trenton .... Ordinary . . . T h x | x 28 85 X x 8 x 3i x 2> 8 x 4' x S i x x 2$ t 8g x 4* x 2% Fire-brick Valentine's (Woodbridge, N.J.) Downing’s (Allentown, Penn.) . 8J x 4| x 2J ins. 9 B 41 x 24■ The weight of the, smaller sized bricks is about four pounds on the average, and of the larger about six pounds. Dry bricks will absorb about one-fifteenth of their weight in water. DRAINPIPE. There are three kinds of drain-pipe offered in the market; viz., “Salt Glazed Vitrified Clay-Pipe,” “Slip Glazed Clay-Pipe,” and “Cement Pipe.” The name of the latter sufficiently indicates what it is without any description. The “Slip Glazed Clay-Pipe” is made of what is known as “fire” (such as fire-brick) clay, which retains its porosity when subjected to the most intense heat. It is glazed with another kind of clay, known as “slip,” which, when subjected to heat, melts, creating a very thin glazing, which, being a foreign substance to the body of the pipe, is liable to wear or scale off. “ Salt Glazed Clay-Pipe ” is made of a clay, which, when subjected to an intense heat, becomes vitreous or glass-like; and is glazed by the vapors of salt, the salt being thrown in the fire, thereby creating a vapor which unites chemically with the clay, and forms a glazing, which will not scale or wear off, and is impervious to tlie action of acids, gases, steam, or any other known substance. It unites with the clay in such a manner as to form part of the I ndy of the pipe, and is therefore indestructible.512 DRAIN-PIPE. Salt-glazed pipe can only be made from clay that will vitrify, that is, when subjected to an intense heat will come to a bard, compact body, not porous. And it should be borne in mind that “slip glazing” is only resorted to when the clays are of such a nature that they will not vitrify. The material of drain-pipes should be a bard, vitreous substance; not porous, since this would lead to the absorption of the impure contents of the drain, would have less actual strength to resist pressure, would be more affected by the frost, or by the formation of erystals in connection with certain chemical combinations, or would be more susceptible to the chemical action of the constituents of the sewerage. “ Much experience with cement sewer-pipes seems to demonstrate that they are not sufficiently uniform in quality, nor sufficiently strong and durable, to be used with confidence in any important work, whether public or private. Seioer-pipes should he salt ylazed, as this requires them to be subjected to a much more intense heat than is needed for ‘ slip glazing, and thus secures a harder material.” The standard salt glazed sewer and drain pipe manufactured by the Akron Sewer Pipe Company of Akron, O., lias been found to answer all requirements, and is one of the best drain-pipes to be found in the market. The follow hi S <5 2 ® l§ S O Size of 0) ’—'O ^ O — o «♦H *** Pipe. ~ "3 ~ -3 C! ? P «S o V-. O P 'S ^ "3 Æ'ë ■g 9 ’S tî "3 fl s§j fm JJ o C O S O 3 3 3 G 2 •i-> r3 ~ 3 3 3 G D |§j 3 •ri 3 HlC a Width, in Inches. 15 10 17 18 19 20 21 22 23 ft. in. ft. in. ft: in. ft. in. 1 1 3 1 .4 1 5 1 0 2 2 0 2 8 2 10 3 0 3 3 9 4 0 4 3 4 6 4 5 0 5 4 5 8 0 0 5 0 3 0 8 7 1 7 6 6 7 0 8 0 8 0 9 0 7 8 9 9 4 9 11 10 6 8 10 0 10 8 11 4 12 0 9 11 3 12 0 12 9 13 6 10 12 0 13 4 14 2 15 0 11 13 9 14 8 15 7 10 0 12 15 0 10 0 17 0 18 0 13 10 3 17 4 18 5 19 6 14 17 0 18 8 19 10 21 0 13 18 9 20 0 21 3 22 6 10 20 0 21 4 22 8 24 0 17 21 3 22 8 24 1 25 0 18 22 6 24 0 25 0 27 0 19 23 9 25 4 20 11 28 6 20 25 0 20 8 28 4 30 0 21 20 Q O 28 0 29 9 31 0 22 27 0 29 4 31 2 33 0 23 28 9 30 8 32 7 34 0 24 30 0 32 0 34 0 36 0 25 31 3 33 4 35 5 37 0 20 32 0 34 8 30 10 39 0 27 33 9 30 0 88 3 40 0 28 35 0 37 4 39 8 42 0 29 30 »> tj 38 8 41 1 43 0 30 37 0 40 0 42 0 45 0 31 38 9 41 4 43 11 40 6 ft. in. ft. in. ft. in. ft. in. ft. ill. 1 7 1 8 1 9 1 10 1 11 Q O 2 3 4 3 0 3 8 3 10 4 9 5 0 5 q o 5 6 5 9 0 4 6 8 7 0 7 4 7 8 7 11 8 4 8 9 9 2 9 7 9 6 10 0 10 0 11 0 11 0 11 1 11 8 12 3 12 10 13 5 12 8 13 4 14 0 14 8 15 4 14 3 15 0 15 9 16 6 17 3 15 10 10 8 17 0 18 4 19 2 17 5 18 4 19 3 20 2 21 1 19 0 20 0 21 0 22 0 23 0 20 7 21 8 22 9 23 10 24 11 22 2 23 4 24 6 25 8 20 10 23 9 25 0 26 3 27 6 28 9 25 4 20 8 28 0 29 4 30 8 26 11 28 4 29 9 31 2 32 7 28 6 30 0 31 6 33 0 34 6 30 1 31 8 33 3 34 10 36 5 31 8 33 4 35 0 36 8 38 4 33 3 35 0 36 9 38 6 40 Q O 34 10 36 8 38 6 40 4 42 2 30 5 38 4 40 3 42 2 44 i 3S 0 40 0 42 0 44 0 46 0 39 7 41 8 43 9 45 10 47 11 41 2 43 4 45 6 47 8 49 10 42 9 45 0 47 Q O 49 6 51 9 44 4 40 8 49 0 51 4 53 8 45 11 48 4 50 9 53 2 55 7 47 0 50 0 52 6 55 Ö 57 0 49 1 51 8 54 3 56 10 59 5SCANTLINGS REDUCED TO BOARD MEASURE. 515 Scantlings reduced to Board Measure. Explanation of Table. —At the left-hand of the page will !>* found the length of each scantling, in feet. At the head of each of the remaining columns will be found the sizes, being the width and thickness, in inches ; and opposite the given length of each will be found the contents of each scantling. SU. Z> 1x2 inches. 2x2 inches. 2x3 inches. 2x4 inches. 2x5 inches. 2x6 inches. 2x7 inches. 2x8 inches. n. in. a. in. n. in. ft. in. ft. in. feet. ft. in. ft. in. 2 0 4 0 8 l 0 i 4 i 8 2 2 4 2 8 Q O 6 l 0 i 6 2 0 2 G 3 3 G 4 0 4 8 l 4 2 0 2 8 3 4 4 4 8 5 4 5 10 l 8 2 G o o 4 4 2 5 5 10 6 8 6 1 0 2 0 3 0 4 0 5 0 6 7 0 8 0 7 1 2 2 4 3 0 4 8 5 10 7 8 2 9 4 8 1 4 2 8 4 0 5 4 G 8 8 9 4 10 8 i 9 1 G 3 0 4 G G 0 7 6 9 10 6 12 0 10 1 8 3 4 5 0 6 8 8 4 10 11 8 13 4 11 1 10 3 8 5 G 7 4 9 2 11 12 10 14 8 12 9 0 4 0 6 0 8 0 10 0 12 14 0 10 0 13 9 9 4 4 G G ■ 8 8 10 10 13 15 2 17 4 14 9 4 4 8 7 0 9 4 11 8 14 16 4 18 8 15 2 6 5 0 'J G 10 0 12 G 15 17 6 20 0 16 f 8 5 4 8 0 10 8 13 4 10 18 8 21 4 IT 9 10 5 8 8 G 11 4 14 2 17 19 10 22 8 1 18 3 0 G 0 9 0 12 0 15 0 18 21 0 24 0 19 3 2 0 4 9 0 12 8 15 10 19 22 2 25 4 20 3 4 0 8 10 0 13 4 1G 8 20 23 4 20 8 2! 3 6 7 0 10 G 14 0 17 . G 21 24 6 2S 0 1 Hi 3 8 7 4 11 0 14 8 18 4 22 25 8 29 4 1 *•*.’» 3 10 8 11 G 15 4 19 2 23 26 10 GO 8 24 4 o 8 0 12 0 10 0 20 0 24 28 0 0 21 1 9 8. 4 12 0 10 8 20 10 25 29 2 0*1 4 2(! 4 4 8 8 13 0 17 4 21 8 20 30 4 34 8 27 1 6 9 0 13 G 18 0 22 0 27 31 6 36 0 1 28 1 8 9 4 14 0 18 8 23 4 28 82 8 87 4 ! 29 \ 10 9 8 14 G 19 4 24 2 29 33 10 |g 8 30 o 10 0 15 0 20 0 25 0 30 35 0 40 0 1 31 2 10 4 15 G 20 8 25 10 31 3G 2 41 4 32 5 4 10 8 10 0 21 4 26 8 32 37 4 42 8516 SCANTLINGS REDUCED TO BOARD MEASURE. Scantlings Reduced, etc. (Continued). pE w fl) 3 2x9 inches. 2 x lo inches. 2 x ll inches. 21 x 5 inches. 25 X 6 inches. 2J x 7 inches. 2i x 8 inches. 24 x {) inches. ft. in. ft. in. ft. in. ft. in. ft. in. ft. in. ft. in. ft. in. •> 3 0 B o 4 Q o 8 2 1 2 0 2 11 8 4 0 0 9 1 •> 4 0 5 0 5 0 »> o 2 3 9 4 5 5 0 5 8 0 0 0 8 7 4 4 2 5 0 5 10 0 8 i 0 5 7 0 8 4 9 2 5 I fj 0 3 7 4 8 4 9 5 0 9 0 10 0 11 0 0 o o 7 0 8 9 10 0 11 0 7 10 0 11 8 12 10 7 4 8 9 10 0 0 11 8 13 9 8 12 0 13 4 14 8 8 4 10 0 11 8 13 4 15 0 9 13 0 15 0 10 0 9 5 11 3 13 2 15 0 10 11 10 15 0 10 8 18 4 10 5 12 0 14 7 10 8 18 9 11 10 0 18 4 20 2 11 0 13 9 10 1 18 4 20 8 12 18 0 20 0 22 0 12 6 15 0 17 6 20 0 22 0 13 19 6 21 8 23 10 13 7 10 3 19 0 21 8 24 5 14 21 0 23 4 25 8 14 7 17 0 20 5 28 4 26 3 15 22 0 25 0 27 0 15 8 18 9 21 11 25 0 28 2 i 10 24 0 20 8 29 4 10 8 20 0 23 4 20 8 80 0 17 25 0 28 4 31 2 17 9 21 3 24 10 28 4 31 11 18 27 0 30 0 oo 0 18 9 22 0 20 0 0 30 0 00 00 9 19 28 6 31 8 34 10 19 10 28 9 27 9 31 8 85 8 20 30 0 33 4 80 8 20 10 25 0 29 2 38 4 O'- O l 0 21 31 0 35 0 88 0 21 11 20 3 30 8 35 0 39 5 22 33 0 30 8 40 4 22 11 27 6 32 1 30 8 41 0 0 23 34 0 38 4 42 2 24 0 28 9 00 'JO 1— < 38 4 48 2 24 36 0 40 0 44 0 25 0 30 0 35 0 40 0 45 0 25 37 0 41 8 45 10 20 1 Q Ol Ö 36 6 41 8 40 11 26 39 0 43 4 47 8 27 1 32 0 37 11 43 4 48 9 27 40 0 45 0 49 0 28 2 88 9 39 5 45 0 50 8 28 42 0 40 8 51 4 29 2 35 0 40 10 40 8 52 0 20 43 0 48 4 53 2 80 o p 80 3 42 4 48 4 54 5 30 45 0 50 0 55 0 81 o *> 87 0 43 9 50 0 '50 0 0 31 40 0 51 8 50 10 4 88 9 45 2 51 8 58 0 32 48 0 53 t -jt 58 8 •>•> 4 41 0 40 ( 53 4 00 1 21 x 10 2Ä x 11 2' x 12 3x3 3x4 0 • 0 < 5 3 x <*, 3 1 * inches. inches. inches. inches. inches. inches. inches. inches. ft. in. ft. in. ft. in. ft. in. feet. ft. in. ft. in. ft. in. 2 4 o 4 7 5 0 i 0 •> AJ 2 0 8 0 •) •) 0 »> o 0 o o 0 11 7 0 o 0 0 0 0 9 4 0 5 0 0 4 8 4 9 2 10 0 *> i 0 4 5 0 0 0 i 0 5 10 5 11 0 12 0 *> 9 5 0 0 0 7 0 8 9 0 12 0 13 9 15 0 4 0 0 7 0 9 0 10 6 7 14 I 10 1 17 0 5 »> u 7 8 9 10 0 12 8 8 10 8 18 4 20 0 0 0 8 10 0 12 0 14 0 9 18 9 20 8 »>•> 0 0 9 9 11 •> 0 18 0 i 15 9SCANTLINGS REDUCED TO BOARD MEASURE. 517 Scantlings Reduced, etc. (Continued). 0C uplgj n* 10 2* x ll 2 .^ x 12 3x3 3x4 3x5 3x6 3x7 44 49 0 55 0 00 0 GO 29 4 30 8 44 23 40 51 9 57 0 03 3 09 30 8 38 4 40 24 48 54 0 00 0 00 0 72 32 0 40 0 48 25 50 50 3 02 0 08 9 75 33 4 41 8 50 20 52 58 0 05 0 71 0 78 34 8 43 4 52 27 54 00 9 07 0 74 3 81 30 0 45 0 54 28 50 03 0 70 0 77 0 84 37 4 46 8 50 29 58 05 3 72 0 79 9 87 38 8 48 4 58 .30 00 07 0 75 0 82 0 90 40 0 50 0 00 31 02 09 9 77 0 85 3 93 41 4 51 8 02 32 04 72 0 80 0 8S 0 . 90 42 8 53 4 04 ä 4x7 4x8 4x9 4 x 10 4x11 4 x 1*2 5 x 5 5x6 inches. inches. inches. inches. inches. inches. inches. inches. ft. in. ft. in. feet. ft. in. ft. in. feet. ft. in. ft. in. 2 4 8 5 4 0 0 8 7 4 8 4 2 5 0 o »> 7 0 8 0 9 10 0 11 0 12 0 3 7 0 4 9 4 10 s 12 13 4 14 8 10 8 4 10 0 5 11 8 13 4 15 10 8 18 4 20 10 5 12 0 0 14 0 10 0 18 20 0 22 0 24 12 0 15 0 7 16 4 18 8 21 23 4 25 8 28 14 7 17 0 8 18 8 21 4 24 20 8 29 4 32 10 8 20 0 I 21 0 24 0 27 30 0 33 0 30 18 9 22 0 10 23 4 20 8 30 33 4 30 8 40 20 10 25 0 11 25 8 29 4 00 30 8 40 4 44 22 11 27 0 12 28 0 32 0 30 40 0 44 0 48 25 0 30 0 13 30 4 34 8 39 43 4 47 8 52 27 1 32 0 14 32 8 37 4 42 40 8 51 4 50 29 2 35 0 15 35 0 40 0 45 50 0 55 0 00 31 3 37 0 1(5 37 4 42 8 48 53 4 58 8 04 33 4 40 0 17 39 8 45 4 51 50 8 02 4 08 35 5 42 0 18 42 0 48 0 54 00 0 00 0 72 37 0 45 0 19 44 4 50 8 57 03 4 09 8 70 39 7 47 0 20 40 8 53 4 00 00 8 73 4 80 41 8 50 0 21 49 0 50 0 <’>:] 70 0 77 0 84 43 9 52 0 22 51 4 58 8 00 73 4 80 8 88 45 10 55 0 23 53 8 01 4 09 70 8 84 4 92 47 11 57 0 24 50 0 04 0 72 80 0 88 0 90 50 0 60 0 25 58 4 00 8 75 83 4 91 8 100 52 1 02 0 20 00 8 09 4 78 80 8 95 4 104 54 2 05 0SCANTLINGS REDUCED TO BOARD MEASURE. 519 Scantlings Reduced, etc. (Continued). II 4X7 4x8 4x9 4 x 10 4x11 4 x 12 5x5 5x6 3 c inches. inches. inches. inches. inches. inches. inches. inches. ft. in. ft. in. feet. ft. in. ft. in. feet. ft. in. ft. in. 27 63 0 72 0 81 90 0 99 0 108 56 3 67 6 28 65 4 74 8 84 93 4 102 8 112 58 4 70 0 29 67 8 77 4 87 96 8 106 4 116 60 5 72 6 80 70 0 80 0 90 100 0 110 0 120 62 6 75 0 31 72 4 82 8 93 103 4 113 8 124 64 7 77 6 32 74 8 85 4 96 106 8 116 4 128 66 8 80 0 *0 11: 5x7 5x8 5x9 5 x 10 6x6 6x7 6X8 7x7 o - inches. inches. inches. inches. inches. inches. inches. inches. ft. in. ft. in. ft. in. ft. in. feet. ft. in. feet. ft. Br 2 5 10 6 8 7 6 8 4 6 7 0 8 8 2 3 8 9 10 0 11 3 12 6 9 10 6 12 12 3 4 11 8 13 4 15 0 16 8' 12 14 0 16 16 4 5 14 7 16 8 18 9 20 10 15 17 6 20 20 5 6 17 6 20 0 22 () 25 0 -18 21 0 24 24 6 7 20 5 23 4 26 3 29 2 21 24 6 28 28 7 8 23 4 26 8 30 0 33 4 24 28 0 32 32 8 9 26 3 30 0 33 9 37 6 27 31 6 36 36 9 10 20 2 33 4 37 6 41 8 30 35 0 40 40 10 11 32 1 36 8 41 3 45 10 33 38 6 44 44 11 12 35 0 40 0 45 0 50 0 36 42 0 48 49 0 18 37 11 43 4 48 9 54 2 39 45 6 52 53 1 14 40 10 46 8 52 6 58 4 42 49 0 56 57 2 15 43 9 50 0 56 3 62 6 45 52 6 60 61 3 16 46 8 53 4 60 0 66 8 48 56 0 64 65 4 17 49 7 56 8 63 9 70 10 51 59 6 68 69 5 18 52 6 60 0 67 6 75 0 54 63 0 72 73 6 19 55 5 63 4 71 3 79 2 57 66 6 76 77 7 20 58 4 66 8 75 0 83 4 60 70 0 80 81 8 21 61 3 70 0 78 9 87 6 63 73 6 84 85 9 22 64 2 73 4 82 6 91 8 66 77 0 88 89 10 28 67 1 76 8 86 3 95 10 69 80 6 92 93 11 24 70 0 80 0 90 0 100 0 72 84 0 96 98 0 25 72 11 83 4 93 9 104 2 75 87 6 100 102 1 26 75 10 86 8 97 6 108 4 78 91 0 104 106 2 • 27 78 9 90 0 101 3 112 6 81 94 6 108 110 3 28 81 8 93 4 105 0 11« 8 84 98 0 112 114 4 29 84 7 96 8 108 9 120 10 87 101 6 116 118 5 30 87 6 100 0 112 6 125 0 90 105 0 120 122 6 81 90 5 103 4 116 3 129 2 93 108 6 124 126 7 32 93 4 106 8 120 0 133 4 96 112 0 128 130 8520 SCANTLINGS REDUCED TO BOARD MEASURE Scantlings Reduced, etc. (Continued). Ul I I ‘qAr] 7X8 inches. 7x9 inches. 8x8 inches. 8x9 inches. 8 x 10 inches. 9 x 9 inches. 9 x 10 inches. 9 x 1 i | inches. ft. in. ft. in. ft. in. feet. ft. in. feet. ft. in. ft. in. : 2 9 4 10 0 10 8 12 10 4 10 0 10 0 16 6 •> is 14 0 15 9 10 0 18 20 0 20 0 22 0 24 9 i 4 18 8 21 0 21 4 24 20 8 27 0 30 0 30 0 | 5 20 4 20 0 20 8 00 33 4 30 9 37 6 41 0 j 0 28 0 01 0 02 0 00 40 0 40 0 45 0 49 0 1 02 8 00 9 37 4 42 4(i 8 47 3 52 0 57 9 8 07 4 42 0 42 8 48 50 4 54 0 00 0 00 0 9 42 0 47 8 48 0 54 00 0 00 9 07 0 74 0 10 40 8 52 0 50 4 00 00 8 07 0 75 0 82 0 11 51 4 57 9 58 8 00 73 4 74 3 82 0 90 9 12 r>() o 00 0 (>4 0 72 80 0 81 0 90 0 99 0 m (>0 8 08 0 09 4 78 80 8 87 9 97 0 107 0 14 05 4 70 0 74 8 84 90 4 94 0 105 0 115 0 15 70 0 78 9 80 0 90 100 0 101 3 112 0 123 9 10 74 8 84 0 85 4 90 100 8 108 0 120 0 102 0 IT 79 4 89 0 90 8 102 110 4 114 9 127 0 140 0 18 84 0 94 0 90 0 108 120 0 121 0 105 0 148 0 19 88 8 99 9 101 4 114 120 8 128 0 142 0 156 9 20 90 4 105 0 100 8 120 130 4 105 0 150 0 105 0 21 98 0 110 0 112 0 120 140 0 141 9 157 0 170 0 22 102 8 115 0 117 4 102 140 8 148 0 165 0 181 0 20 107 4 120 9 122 8 108 150 4 155 0 172 0 189 9 24 112 0 120 0 128 0 144 100 0 102 0 180 0 198 0 25 110 8 10 i 0 100 4 150 10)0 8 108 9 187 0 200 0 20 121 4 100 0 108 8 150 173 4 175 0 195 0 214 0 27 128 0 141 9 144 0 102 180 0 182 0 202 0 222 9 28 100 8 147 0 149 4 108 180 8 189 0 210 0 201 0 29 105 4 152 0 154 8 174 190 4 195 9 217 0 209 0 00 140 0 157 0 100 0 180 200 0 202 0 225 0 247 0 01 144 8 102 9 105 4 180 200 8 209 0 202 0 255 9 02 149 4 108 0 170 8 192 210 4 210 0 240 0 204 0PLANK MEASURE. 521 Plank Measure. Board measure is the basis of plank measure; that is, a plank two inches thick, and thirteen feet long, and ten inches wide, contains evidently twice as many square feet as if only one inch thick: therefore, in estimating the contents of any plank, we first find the contents of the surface taken one inch thick, and then, if the plank be one inch and a quarter thick, we add one-quarter of the contents to itself, which gives the contents (in board measure) of the plank. Contents of Planks in Board Measure. Thickness, Inches. 5V IDTII, IN Inches. V o 6 7 8 9 10 11 12 13 14 15 10 17 18 19 20 ft. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. 10 6 7 8 9 10 ii 12 14 15 16 17 18 1!» 20 21 11 7 8 b 10 n 13 14 15 16 17 18 20 21 22 23 12 8 9 10 11 12 14 15 16 17 19 20 21 22 24 25 13 8 0 11 12 14 15 16 18 19 20 22 23 24 26 27 14 9 10 12 13 15 16 17 19 20 22 23 25 26 28 20 15 9 11 12 14 16 17 19 20 22 23 25 27 28 30 31 Hi 10 12 13 15 17 18 20 22 23 25 27 28 30 mm ms i 17 11 12 14 10 18 19 21 23 25 27 28 30 32 34 35 IS 11 13 15 17 19 21 22 24 26 28 30 32 34 3(i 37 1!) 12 14 10 18 20 22 24 26 28 30 ‘•>0 34 36 38 40 20 13 15 17 19 21 23 25 27 29 31 ')») 35 37 40 42; 21 13 15 17 20 22 24 26 28 31 gggg 35 37 39 42 44 i 2 2 14 10 18 21 23 25 27 30 32 34 37 39 41 44 46 ; 23 14 17 19 22 24 26 29 31 34 36 3S 41 43 46 48 j 24 15 17 20 22 25 27 30 32 35 O'7 • ) 1 40 42 45 47 50 1 25 10 18 21 23 26 29 31 34 30 39 42 44 47 49 52 J 2(i 10 19 22 24 27 30 32 35 38 41 43 40 49 51 54 | 27 17 20 22 25 28 31 34 37 39 42 45 48 51 53 56 i 28 17 20 23 26 29 32 35 38 41 44 47 50 52 55 58 j 29 18 21 24 27 30 33 36 39 42 45 48 51 54 ~~57 60 ! 30 19 22 25 28 31 34 37 41 44 47 50 53 56 59 62 31 19 23 26 29 32 36 39 42 45 48 52 55 58 61 05 ( 32 20 23 27 30 oo OO 37 40 43 47 50 53 57 60 03 07 33 21 24 27 31 34 38 41 45 48 52 55 58 62 65 69 34 21 25 28 32 35 39 42 46 50 53 57 60 64 67 71 «>5 22 26 29 33 36 40 44 47 51 55 58 ■ 66 69 73 j 1522 PLANK MEASURE PLANK MEASURE (Continued). Contents of Planks in Boakd Measuke. Thickness, li Inches. bf> O Width , IN Inches. if P*4 0 7 8 0 10 11 12 13 14 15 10 17 18 10 *20 ft. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. 10 7 0 10 11 10 14 15 10 17 10 20 21 22 24 25 11 8 10 11 12 14 15 10 18 10 21 22 20 25 20 27 12 0 10 12 10 15 10 18 10 21 22 24 25 27 28 00 13 10 11 10 15 10 18 10 21 20 24 20 28 20 11 •><> oo 14 11 12 14 10 17 10 21 20 24 20 28 00 01 OQ »JO 05 15 1L 10 15 17 10 21 22 24 20 28 00 02 04 00 08 10 12 14 10 18 20 99 Ami 24 20 28 00 02 04 00» 08 40 17 10 15 17 10 21 20 25 28 00 02 04 00 08 41 40 18 14 10 18 20 22 25 27 20 01 04 00 08 40 40 45 10 14 17 10 21 24 20 28 01 •>.» *)»■> 00 08 40 42 40 48 20 15 17 20 22 25 27 00 02 05 08 40 42 45 48 50 21 10 18] 21 24 20 20 01 04 07 40 42 44 47 50 50 22 10 10 oo 25 27 00 »>*> »>♦> 05 08 42 44 40 40 50 »55 20 17 20 20 20 20 02 04 07 40 44 40 48 51 55 58 24 18 21 24 27 00 Q ■> »)•> 00 00 42 45 48 51 54 57 00 25 10 22 25 28 01 05 :n 40 44 47 50 5:! 50 00 00 20 20 20 20 20 02 00 00 42 45 40 52 55 58 02 05 27 20 24 27 00 04 08 40 40 47 51 54 57 00 04 08 28 21 24 28 01 05 00 42 45 40 50 50 50 00 07 70 20 22 25 20 »■>•> 00 40 40 47 50 55 58 01 05 00 1^0 iO 00 22 20 00 04 07 42 45 48 52 57 00 00 07 72 75 01 20 27 01 07 00 40 40 50 54 50 02 05 00 74 78 0,2 24 28 02 00 40 44 48 52 50 00 04 08 72 70) 80 Hi 25 20 »>•> 07 41 45 40 50 r>7 02 00 70 74 78 80 04 20 00 04 08 42 47 51 55 50 04 08 72 70 81 85 05 20 01 05 00 44 48 52 50 01 00 70 74 78 88 88 Contents ok Planks in Roakd Measuke. Thickness, 2 Inches. o W I DTI 0 7 8 0 10 11 1*2 ft. feet. feet. feet. feet. feet. feet 10 10 1 1 10 15 17 18 20 11 11 10] 15 17 18 20 22 12 12 14 10 18 20 22 24 10 10 15 17 20 22 24 20 , in Inches. 1.0 H 15 10 17 IS 10 feet. feet. feet. feet. feet. feet. feet. 22 20 25 27 28 00 02 24 20 27 29 01 OO oo 05 20 28 00 02 04 80 08 28 00 •JO OO 05 •07 09 41 20 feet. 00 13 ♦> i 40 m IPLANK MEASURE. 523 PLANK MEASURE (Continued). Contents of Planks in Boaki> Measure. Thickness, 2 Inches. . | 2? Width, in Inches. Z) 0 7 8 9 10 . . 11 12 13 14 15 10 17 18 • 19 20 ft. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. 14 14 16 19 21 28 26 28 30 33 35 87 40 42 44 47 15 15 18 20 23 25 28 30 33 35 38 40 43 45 48 50 16 16 19 21 24 27 29 32 35 37 40 43 45 48 51 58 n 17 20 23 26 28 31 34 37 40 43 45 48 51 54 57 18 18 21 24 27 30 QQ OO 36 89 42 45 48 51 54 57 60 19 19 22 25 29 32 35 38 41 44 48 51 54 57 60 63 20 20 23 27 30 oo •>■) 37 40 48 47 50 53 57 60 63 67 21 21 25 28 32 35 39 42 46 49 53 56 60 63 67 70 22 22 26 29 33 37 40 44 48 51 55 59 62 66 70 73 2;] 28 27 81 35 38 42 46 50 54 58 61 05 69 <3 77 24 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 25 25 29 33 38 42 46 50 54 58 63 67 71 75 79 88 2(5 26 30 35 39 43 48 52 56 61 65 69 74 78 82 87 27 27 32 36 41 45 50 54 59 63 68 72 77 81 86 90 28 28 33 37 42 47 51 56 61 65 70 75 79 84 89 98 29 29 34 39 44 48 53 58 68 68 73 1i 82 87 92 97 80 30 35 40 45 50 B 60 65 70 75 80 85 90 95 100 81 31 36 41 47 52 57 62 67 72 78 83 88 93 98 103 37 43 48 53 59 64 69 75 80 85 91 96 101 107 ')•) oo 89 44 50 55 61 66 72 77 83 88 94 99 105 110 84 34 40 45 51 57 62 68 74 79 85 91 96 102 108 113 85 35 41 47 53 58 64 70 76 82 88 93 99 105 111 117 Contents of Planks in Board Measure. Thickness, 2$ Inches. Width, in Inches. o 0 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ft. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. ■ feet. feet. 10 11 13 15 17 19 21 28 24 26 28 80 32 34 36 88 11 12 14 17 19 21 23 25 27 29 31 oo oo 35 37 39 41 12 18 16 18 20 23> 25 27 29 32 34 36 38 41 48 45 18 15 17 20 22 24 27 29 32 34 37 39 41 44 46 49 14 16 IS 21 28 26 29 32 34 37 39 42 45 47 50 53 15 17 20 28 25 28 31 34 37 39 42 45 48 51 58 56 16 18 21 24 27 30 oo Oo 86 39 42 45 48 51 54 57 60 17 19 22 26 28 32 35 38 41 45 48 51 54 57 61 64524 PLANK MEASURE PLANK MEASURE (Continued). Contents of Planks in Boakd Mkasuhe. Thickness, 2| Inches. Width, in Inches. 1 I o ! »! 7 8 9 1« 11 12 13 14 l 15 V] 16 17 18 19 20 1 ft. feet. Feet.i feet. feet. "eet.ifeet. 'ect. feet. ■KM.. 'eet. feet.1 feet. Feet, j is •j) 24 27 30 34 37 41 44 47 51 54 57 bJ | 04 1 08 19 21 25 20 1 32 30 39 43 46 501 53 57 61 04 08 1 71 20 22 26 so 33 88 41 45 49 H 56 60 64 08 711 75 21 ' is 28 32 35 39 43 47 51 55 59 63 67 71 75 79 22 ' 25 29 33 36 41 45 50 54 58 62 66 70 74 78 83 2:! 20 SO 85 38 43 47 52 56 60 05 69 73 78 82 86 24 1 27 32 36 40 45 50 54 59 63 08 72 77 81 80 90 25 28 •>o BM 38 41 47 52 50 61 66 70 75 80 84 89: 94 2(5 29 34 39 43 49 54 59 63 68 73 78 83 88 93 98 27 SO 85 41 45 51 50 01 66 71 70 81 86 91 90 101 28 SI 87 42 46 58 58 63 68 74 79 84 89 95 100 105 29 3S 38 44 48 54 00 65 71 70 82 87 92 98 103 109 SO S4 41 45 49 56 02 68 73 79 84 90 90 101 107 113 SI 85 41 47 51 58 04 70 76 81 87 93 99 105 110 116 S2 so 42 48 58 00 00 72 78 84 90 96 102 108 114 120 M S7 43 50 54 02 68 74 80 87 93 99 105 111 118 124 S4 S8 45 51 56 04 70 77 83 89 96 102 108 115 121 128 85 39 40 58 58 00 72 79 85 92 98 105 112 118 125 131 Contents OF Planks IN Boaiid Measuiie. Thickness L 2* Inches bf) Width IN Inches. O D 6 7 8 9 10 11 12 13 14 15 10 ! 17 18 10 20 ft. feet. feet. feet feet. feet. feet. feet feet feet. feel 'feet fee, feet. feet. 10 12 15 17 19 21 23 25 27 29 31 o*> 35 1 1 40 42 11 14 10 18 21 23 25 27 30 32 34 87 39 41 44 47 12 15 18 20 23 25 28 30 83 35 88 40 43 45 48 ■ IS 10 19 22 24 27 30 58 35 88 41 43 40 49 51 54 14 17 20 23 20 29 32 85 38 41 44 47 50 58 55 58 15 19 22 25 28 31 34 38 41 44 47 50 53 56 59 63 u] 20 23 27 30 <>.l »» 87 40 43 47 50 53 57 60 63 07 17 21 25 28 32 35 39 48 40 50 53 57 00 64 67 71 IS 22 26 30 34 38 ! 41 45 49 53 56 ! 60 64 68 71 19 24 28 32 80 40 i 44 48 51 55 59 i 63 67 71 75 79 20 25 29 88 88 42 40 50 54 58 63 67 71 75 79 83 21 20 31 85 39 44 48 58 57 61 66 i 70 74 79 83 88PLANK MEASURE. 525 TLANK MEASURE (Continued). Contexts of Planks in Board Measure. Thickness, 2$ Inches. ?! Width, in Inches. £ 6 7 8 9 10 11 12 13 u 15 16 17 18 1 29 34 38 43 48 53 58 02 07 72 77 81 80 91 90 24 30 35 40 45 50 55 00 05 70 75 80 85 90 95 100 2.5 31 30 42 47 52 57 03 os 73 78 83 89 94 99 104 2 6 7 8 9 10 11 12 13 14 15 1G 17 18 19 20 ft. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. 10 17 20 23 26 29 32 35 OS 41 44 47 50 52 55 58 li 10 22 24 29 32 05 38 41 45 47 51 54 57 61 64 12 20 25 28 32 35 39 42 46 40 • 1 56 60 w 67 70 13 23 27 30 34 38 42 46 40 53 57 61 64 68 72 76 14 25 20 00 07 41 45 40 53 57 61 65 69 74 78 82 15 26 31 05 39 44 48 53 57 61 66 70 74 70 83 88 16 28 33 37 42 47 51 56 61 05 70 75 70 84 80 03 17 30 35 40 45 50 55 60 64 69 74 79 84 80 04 00 18 32 37 42 47 53 58 63 08 74 79 84 80 05 100 105 19 33 30 44 50 55 61 67 72 78 83 89 04 1(K) 105 111 20 05 41 47 53 58 64 70 76 82 88 93 00 105 111 117 21 37 43 49 55 61 67 74 80 86 92 98 104 110 116 123 22 38 45 51 58 64 71 77 83 90 96 103 100 116 122 128 23 40 47 54 60 67 74 81 87 94 101 107 114 121 127 134 24 42 49 56 63 70 77 84 01 98 105 112 110 126 133 140 25 44 51 58 66 73 80 88 05 102 109 117 124 131 130 146 26 45 53 61 68 76 83 01 90 106 114 121 129 137 144 152 27 47 55 63 71 79 87 05 102 110 118 126 134 142 150 158 28 40 57 (>5 74 82 90 08 106 114 123 131 139 147 155 163 20 51 59 68 76 85 93 102 no 118 127 135 144 152 161 160 30 52 61 70 79 88 06 105 114 123 131 140 149 158 166 175NAILING MEMORANDA. 527 PLANK MEASURE (Concluded). Contents of Planks in Board Measure. Thickness, 3^ Inches. tfj Width IN [NCHES. Feet |. c 7 8 9 10 u 12 13 14 15 10 17 IS 10 *20 ft. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. feet. 31 54 63 72 81 90 99 109 118 127 136 145 154 163 172 181 32 56 05 75 84 93 103 112 121 131 140 149 159 168 177 187 *>•> 58 67 77 87 96 106 116 125 135 144 154 164 173 183 193 34 59 69 79 89 99 109 119 129 139 149 159 169 179 188 198 35 61 71 82 92 102 112 123 133 143 153 163 174 184 194 204 NAILING MEMORANDA. [From u Builder’s Guide, and Estimator’s Price-Book.”] Quantity of Nails for Different Kinds of Work. For 1000 shingles allow ....... 3i to 5 pounds 4cZ. nails, or Q d to 3/ U 3d. “ 1000 latlis about 6 m 3d. fine 1000 feet clapboards U 18 u Gd. box 1000 “ covering boards. . . U 20 (( 8d. common 1000 “ “ “ . . . U 25 a 10 d. “ 1000 | upper floors, sq. edged, a 38 ite 10(Z. floor 1000 “ “ “ a 41 a 12 (Z. “ im „ „ {-‘«»aj u 35 u 10d. “ 1000 “ “ “ “ a 42 u 12(Z. “ H ( studs or ) 10 partitions { stuiUling } u 1 a 10tZ. common 1000 “ furring, 1 by 3 . . . u 45 u lOcZ. 1000 “ “ 1 by 2 . . . u 65 u lOtZ. “ 1000 “ pine finish .... a 30 u 8d. finish528 MEMORANDA FOR PLASTERERS. MEMORANDA FOR PLASTERERS. Measuring- Plasterers’ Work. The following paragraphs, taken from one of our leading jour nals, describe the usual method of measuring plasterers’ work : — Plastering is always measured by the square yard for all plain work, by the superficial foot for all cornices of plain members, and by the linear foot for enriched or carved mouldings in cornices. “ By ‘plain work’ is meant straight surfaces (like ordinary walls an| ceilings), without regard to the style or quality of finish put upon the job. Any panelled work, whether on walls or ceilings, run with a mould, would be rated by the foot superficial. “ Different methods of valuing plastering find favor in different portions of the country. The following general rules are believed to be equitable and just to all parties: — “ First, Measure on all walls and ceilings the surface actually plastered, without deducting any grounds or any openings of less extent than seven superficial yards. “ Second, Returns of chimney-breasts, pilasters, and all strips of plastering less than twelve inches in widtn, measure as twelve inches wide; and where the plastering is finished down to the base, surbase, or wainscoting, add six inches to height of walls. “ Third, In closets, add one-half to the measurement. Raking ceilings, and soffits of stairs, add one-half to the measurement; circular or elliptical work, charge two prices; domes or groined ceilings, three prices. “ Fourth, For each twelve feet of interior work done farther from the ground than the first twelve feet, add five per cent; for outside work, add one per cent for each foot that the work is. done above the first twelve feet. “ Stucco-work is generally governed by the following rules; viz., mouldings less than one foot girt are rated as one foot, over one foot, to be taken superficial. When work requires two moulds to run same cornice, add one-fifth. For each internal angle or mitre, add one foot to length of cornice, and, for each external angle, add two feet. All small sections of cornice less than twelve inches long measure as twelve inches. For raking cornices, add one-half; circular or elliptical work, double price; domes and groins, three prices. For enrichments of all kinds a special price must be charged. The higher the work is above ground, the higher the charge must be ; add to the rate of five per cent for every twelve feet above the first twelve feet.”MEMORANDA FOR ROOFERS. Ò29 Useful Memoranda. The following facts may often prove of use to the plasterer: — One hundred yards of plastering will require fourteen hundred laths, four bushels and a half of lime, four-fifths of a load of sand, nine pounds of hair, and five pounds of nails, for two-coat work. Three men and one helper will put on four hundred and fifty yards, in a day’s work, of two-coat work, and will put on a hard finish for three hundred yards. A load of mortar measures one cubic yard, or twenty-seven cubic feet, requires one cubic yard of sand and nine bushels of lime, and will fill thirty hods. A single load of sand and other materials equals one cubic yard, or twenty-seven cubic feet; and a double load of sand equals twice that quantity. . A measure of lime is a single load, or cubic yard. A bushel of hair weighs, when dry, about fifteen pounds. MEMORANDA FOR ROOFERS. Slate Roofs. The pitch of a slated roof should be about one in height to four in length. The usual lap is about three inches, but it is sometimes four inches. Each slate should be fastened by two threepenny slate-nails, either of galvanized iron, copper, or zinc. On roofs of gas-houses the nails should be of copper or yellow-metal. A square of slate is one hundred superficial feet, allowances being made for the trouble of cutting the slates at the hips, eaves, round chimneys, etc. The sides and bottom edges of the slates should be trimmed, and the nail-lioles punched as near the head as possible. They should be sorted in sizes, when they are not all of one size, and the smallest placed near the ridge. The thickness of slates varies from three-sixteenths to five-sixteenths of an inch, and their weight from 2.6 to 4.56 pounds per square foot. Elastic Cement. — In first-class work, the top course of slate on ridge, and the slate for two to four feet from all gutters, and one foot each way from all valleys and hips, should be bedded in elastic oement. Roofingf-Paper.— Roof-boards should be covered with one or two thicknesses of tarred felt roofing-paper, before the slate are laid. No dry or rosin-sized felt should be used on roofs.530 MEMORANDA FOR ROOFERS. Flashing's.— By “ flashings” are meant pieces of tin, zinc, or copper, laid over slate, and up against walls, chimneys, copings, etc. Counter-flashings are of lead or zinc, and are laid between the courses in brick, and turned down over the flashings. In flashing against stone-work, grooves or reglets often have to be cut to receive thl counter-flashings. Close and Open Valleys. — A close valley is where the slate are mitred and flashed in each course, and laid in cement. In such valleys no metal can be seen. Close valleys should only be used for pitches above forty-five degrees. An open valley is where the valley is formed of sheets of copper or zinc fifteen or sixteen inches wide, and the slate laid over these. Rule for computing the Number of Slates in a Square. Subtract three inches, or the amount of head-cover, from the length of the slate, multiply the remainder by the width, and divide by two. This will give the number of square inches covered per slate ; divide 14,400 (the number of square inches in a square/ by the number so found, and the result will be the number of slates required. The following table gives the number of slates per square for the usual sizes, allowing three inches for head-cover : — Number of Slates per Square. Size, in inches. Pieces per square. Size,in inches. Pieces per square. Size, in inches. Pieces per square. 6 x 12 533 8 x 16 277 12 x 20 141 7 x 12 457 9 x 16 246 14 x 20 121 8 x 12 400 10 x 16 221 11 x 22 137 9 x 12 355 9 x 18 213 12 x 22 126 7 x 14 374 10 x 18 192 14 x 22 108 8X14 327 12 x 18 160 12 x 24 114 » x 1 1 291 10 x 20 109 14 x 24 98 10" x 14 261 11 x 20 154 16 x 24 86 The weight of slate per cubic foot is about 174 pounds, or, per square foot of various thicknesses, as follows : — Thickness, in inches . . . i 1 3 1 3 1 Weight, in pounds .... . . 1 1.É1 2.71 3.62 5.43 7.25 Tin' weight of slating laid per square foot of surface covered will, of course, depend on the size used. The weight of 10 by 18MEMORANDA FOR ROOFERS. 531 siate, three-sixteenths of an inch thick, for example, per square foot of roof, would be 5.80 pounds. An experienced roofer will lay, on an average, two squares of slate in ten hours. Ordinary roofing-paper weighs about fifteen pounds per square, and averages about fifty pounds in a roll. At the present time |1884| the additional cost of laying slate in elastic cement varies from thirteen to fifteen per cent. Shingles. The average width of a shingle is four inches : hence, when shingles are laid four inches to the weather, each shingle averages sixteen square inches, and 900 are required for a square of roofing. If 4i inches to the weather, 800 will cover a square. 5 “ “ “ 720 “ “ 5£ “ “ “ 055 “ “ 6 “ “ “ 000 “ “ This is for common gable-roofs. In hip-roofs, where the shingles are cut more or less to fit the roof, add five per cent to above figures. A carpenter will carry up and lay on the roof from fifteen hundred to two thousand shingles per day, or two squares to two squares and a half of plain gable-roofing. One thousand shingles laid four inches to the weather will require five pounds of shingle-nails to fasten them on. Six pounds of fourpenuy nails will lay one thousand split pine shingles. Roofing-Tiles. Tiles are thin slabs of baked clay. They are extensively used in Europe for roofs, gutters, and liouse-siding, and, to some extent, in this country. Plain roofing-tiles are usually made £ of an inch in thickness, 10-i inches long, and 6^ inches wide. They weigh from 2 to 2i pounds each, and expose about one-half to the weather. 740 tiles cover 100 superficial feet. They are hung upon the lath by two oak pins inserted into holes made by the moulder. Plain tiles are now made with grooves and fillets on the edges, so that they arc laid without overlapping very far, the grooves leading the water. This is economical of tiles, and saves half of the weight, but is subject to leak in drifting rains, and to injury by hard frosts. Pan-tiles, first used in Flanders, have a wavy surface, lapping under, and being overlapped by, the adjacent tiles of the same rank.532 MEMORANDA FOR ROOFERS. They are made 14^ by 10?, expose ten inches to the weather, and weigh from 5 to 5i pounds each. 170 cover 100 square feet of surface. Crown, ridge, hip, and valley tiles are semi-cylindrical, or segments of cylinders, used for the purposes indicated. A gutter-tile has been introduced in England, forming the lower course, being nailed to the lower sheathing-board or lath. Siding-tiles are used as a substitute for weather-boarding. Holes arc made in them when moulding, and they are secured to the lath by flat-headed nails. The gage, or exposed face, is sometimes indented to represent courses of brick. Fine mortar is introduced between them when they rest upon each other. Siding-tiles are sometimes called “ weather-tiles ” and “ mathematical tiles.” These names are derived from their exposure or markings. They are variously formed, having curved or crenated edges, and various ornaments, either raised or encaustic. The glazed tiles are inferior to slate, as they imbibe about one-seventh of their weight of water, and tend to rot the lath on which they are laid. Good roofing-slate only imbibes one two-liundredth part of its weight, and is nearly waterproof. Tin Hoofs. [Revised for Fourth Edition.] A tin roof of good material, properly put on, and kept properly painted, will last from thirty to forty years. It should not be painted for the first time until it has been well washed by rain, to get the grease olf the tin; and all rosin, if used, should be carefully scraped off. One or more layers of felt-paper should be placed under the tin, to serve as a cushion, and also to deaden the noise produced by the rain striking the tin. Fora steep roof, the tin should be put on with a standing groove, and with the cross seams double-locked and soldered. A very common and cheaper method for steep roof is, to double-lock both the vertical and cross seams, and fill the joint with white lead instead of soldering; but the other method is much the best. For flat roofs, the tin should be locked and soldered at all joints, and secured by tin cleats, and not by driving the nails through the tin itself. In soldering the joints, rosin as a flux is generally preferred; although some roofers recommend the use of diluted chloride of zinc. Roofing-plates are made of steel or iron, and covered with a mixture of lead and tin, and are designated as “tern,” “leaded,” or “ roofing tin,” in distinction from plates coated only with tin, and therefore called “bright tin.” Roofing-plates are coated byMEMORANDA FOR ROOFERS. 532a two Methods. The original manner of coating the plates was by dipping the black plate inio the mixture of tin and lead, and allowing the sheets to absorb all the coating that was possible; ami several brands of roofing-tin are still made by this process. Till other process, by which the majority of roofing-plates are now made, is known as the “ Patent-roller Process,’’ by which the plates are put into a bath of tin and lead, and are passed through rolls, the pressure of which leaves on the iron or steel a thickness of coating which, to a great extent, determines the value of the plate. These rolls can be so adjusted as to leave a good amount of coating on the plate, an ordinary coating, or a very scant one; the heavier the coating, the more valuable the plate. There have been only two sizes of roofing-plates made for a number of years; namely, 14 X 20 and 20 x 28: and of these two sizes, the larger is more generally used, from the fact that, being double the size of the smaller plate, it requires less seams on the roof, and consequently cheapens the cost of putting on. Resides thise two sizes, there is another size, 10 X 20, which is used for gutters and leader-pipe. A better roof will be obtained by using the 14 X 20 than the 20 X 28, because the seams are closer together, thus making the roof stronger; and, if put on with a standing seam, there is more allowance for expansion and contraction. For steep roofs with standing groove, the tin should be laid with the smallest dimension for the width; as it makes the roof stronger, and allows a greater amount of expansion and contraction. Unfortunately, it is much cheaper to lay them the other way., as less cleats, solder, nails, and labor are required. Hence the architect should always specify in what way the sheets are to be laid, For a Hat roof with fiat seams, it does not make any difference which way the plates are laid, as the entire roof is practically a solid sheet. There are two thicknesses of roofing-plates: namely, 10, or No. 20 gauge, which represents a weight of eight ounces to the square foot; and IX, or No. 27 gauge, which represents a weight of ten ounces to the square foot. The IO plates are far more extensively used than the IX, because they are less expensive; and as the question of cheapness has more or less bearing on the subject of tin roofs, the IC has therefore been given the larger preference. Rut at the same time it should not be forgotten, that, for the extra cost of a box of IX over TO, there is an excess of weight given over that on an IC plate of two ounces per square foot; and this is fully worth the difference in price. The value of one brand of roofing-plates as against another5326 MEMORANDA FOR ROOFERS. brand is dependent on five things: 1st, the quality of the material of which the plate is made; 2u, the coating, or the thickness of the tin and lead that is upon the plate, and which can only he determined hy trying with a knife; 3d, the net weight of the hundred and twelve sheets in the box; 4th, the squareness of the sheets; 5th, assortment of the sheets. It is impossible to determine the value of any brand by the sample sheets; in fact, all plates arc rolled hot, in packs of eight sheets each. The two outside sheets cooling more rapidly than the six inside sheets, the latter are always slightly thinner; consequently there are twenty-eight sheets in every box heavier than the other eighty-four. As thousands of sheets are rolled at a time, then tinned, and indiscriminately packed, there is lawfully a slight difference in weight of the different boxes of the same brand, while there is a greater difference between the weights of the boxes of different brands. The standard weight of an ordinary IC plate, 14 X 20 size, is 108 pounds to the hundred and twelve sheets; while there are many boxes imported of IC 14 X 20 that run all the way from 90 to 120 pounds in weight. The standard weight of a box of IX 14 X 20 is 13(3 pounds; or IX 20 X 28, 272 pounds. There are IX 14 X 20 plates imported that do not weigh over 120 pounds per box, while others weigh as much as 150 pounds for the same size. It may he, that the lighter sheets have as heavy a coating of lead and tin as the heavier sheets; hut the probability is, that they have not. Ail roofing-plates were formerly made of coke or charcoal iron, while now they are made of either Martin Siemen’s or the Bessemer steel. Of these two makes, the former can always he depended upon as a perfectly reliable article; while this cannot at all times he said of the latter. In all tin roofs, whether the main body of the roof is made of ordinary rolled or dipped plates, the gutters, Hashings, etc., should he made of the heaviest coated or dipped plates, and should always he of IX thickness. In the manufacture of all rooling-plate;., there“ arise imperfect sheets, having corners and edges broken, spots uncovered with tin, etc. These are packed hy themselves in separated boxes, and denominated as “wasters;” while the perfect sheets are denominated “prime” plates. The boxes containing “ wasters,” or imperfect sheets, are marked “ICW,” or “IXW.” according to the thickness; so that, where the letter |W” appears on a box, it shows that the box contains imperfect sheets, and should not he accepted when “prime ” tin is specified. To make sure of obtaining the best tin in the market, and getting the article specified for, the architect should specify that “every sheet be stamped with the name of the brand and thickness” (IC or IX, asPLUMBING. 533 the ease may be), as all the best plates are now coming out with every sheet stamped. In this way only can the architect or owner be sure that he is getting what the specifications call for; and it also protects the honest contractor against those who estimate with the intention of using a cheaper article. An exceilent paint for tin roofs is composed of 10 pounds Venetian red, 1 pound red lead, 1 gallon pure linseed oil. The tin should be painted again at the end of the first year, and once in four years afterwards. PLUMBING. The following is a portion of “xVu Ordinance for the Regulation of Plumbing,” passed by the Boston City Government, March, 1883. It is recommended as worthy of observance by architects and plumbers elsewhere. “ Section 3. —Every building shall be separately and independently connected with the public sewer when such sewer is provided, and, if such sewer is not provided, with a brick and cement cesspool of a capacity to be approved by the inspector. “ Sect. 4. ■— Drains and soil-pipes through which water and sewage is used and carried shall be of iron, when within a building, and for a distance of not less than five feet outside of the foundation walls thereof. They shall be sound, free from holes and other defects, of a uniform thickness of not less than g of an inch for a diameter of four inches or less, or of an inch for a diameter of five or six inches, with a proportional increase of thickness for a greater diameter. They shall be securely ironed to walls, iaid in ■trenches of uniform grade, or suspended to floor-timbers by strong iron hangers, as the said inspector may direct. They shall be supplied with a suitable trap, placed, with an accessible elean-out, either outside or inside the foundation wall of the building. They shall have a proper fall towards the drain or sewer ; and soil-pipes shall be carried out through the roof, open, and undiminished in si/.e, to such height as may be directed by the said inspector: but no soil-pipe shall be carried to a height less than two feet above the roof. Changes in direction shall be made with curved pipes, and connections with horizontal pipes shall be made with Y branches. “Sect. 5.— Rain-water leaders, when connected with soil or drain pipes, shall be suitably trapped. “Sect. 6. —Sewer, soil-pipe, or waste-pipe ventilators shall not be constructed of brick, sheet-metal, or earthenware, and chimney-flues shall not be used as such venlilators.031 rLUMBING. “Skct. 7, — Iron pipes, before being put in place, shall be first tested by the water or kerosene test, and then coated inside and out with coal-tar pitch applied hot, or with paint, or with some equivalent substance. Joints shall be run with molten lead, and thoroughly calked and made light. Connections of lead pipes with iron pipes shall be made with brass ferrules properly soldered and calked to the, iron. ‘•Skct. s. — Every sink, basin, bath-tub, water-closet, slop-hopper, and each set of trays, and every fixture having a waste-pipe, shall be furnished with a trap, which shall be' placed as near as practicable to the fixture that it serves. Traps shall be protected from siphonage or air-pressure by special air-pipes of a size not less than the waste-pipe ; but air-pipes for water-closet traps shall be of not less than two-inch bore for thirty feet or less, and of not less than three-inch bore for more than thirty feet. Air-pipes shall be run as direct as practicable, and shall be of not less than four-inch bore where they pass through tlie roof. Two or more air-pipes may be connected together or with a soil-pipe; but, in every case of connection with a soil-pipe, such connection shall be above the upper fixture of the building. r Sect.jE — Drip or overflow pipes from safes under water-closets and other fixtures, or from tanks or cisterns, shall be run to some place in open sight ; and in no case shall any such pipe be connected directly with the drain, waste-pipe, or soil-pipe. “.Skct. 10.— Waste-pipes from refrigerators, or other receptacles tu which provisions are stored, shall not be connected with a drain, soil-pipe, or other waste-pipe, unless such waste-pipes are provided with traps suitably ventilated ; and in every case there shall be an open tray between the trap and refrigerator. ••Skct. 11. —Every water-closet, or line of water-closets, on the same floor, shall be supplied with water from a tank or cistern; and the flushing-pipe shall not be less than one inch in diameter.” Hydraulics ol Plumbing. The following pages on the hydraulics of plumbing are. with the consent of the author, taken directly from the fifth edition (1SS-1) Of an excellent work on “ House-Drainage and Water-Service,”1 by James C. IJayles, Esq., editor of ” The Iron Age” and ” The Metal-Worker.” II'ater is practically an incompressible liquid, weighing, at the 1 Published by David Williams, t& lteude Slieet, New York. The author recommends this work to architects and plumbers as a thorough aud able treatise on the plumbing of city and couuliy housesPLUMBING. average temperature of sixty degrees F., about 62.3 pounds to the cubic foot, and 8.3 pounds to the gallon. These figures are subject to slight variations incident to changes in temperature. A column of water twelve inches high exerts a downward pressure of about 0.43 of a pound to the square inch. A column two feet high exerts a pressure of about 0.86 of a pound, or just twice that exerted by a column one foot high. This pressure per square inch, due to head,1 is irrespective of volume, or any thing else except vertical height of column. With these figures in mind, the calculation of the pressure per square inch due to any head is a simple matter. The following rules will he found valuable for reference : — To FIND PRESSURE IN FOUNDS PER SQUARE INCH EXERTED ijy a Column of Water.—Multiply the height of the column, in feet, by 0.43. To find tiie Head.—Multiply the pressure, in pounds per square inch, by 2.31. Pressure of Water. — The weight of water or of other liquids is as the quantity, but the pressure exerted is as the vertical height. Fluids press equally in all directions: lienee any vessel or conduit containing a fluid sustains a pressure on the bottom equal to as many times the weight of the column of greatest height of that fluid as the area of the vessel is to the sectional area of the column. Lateral Pressure.®- The lateral pressure of a fluid on the sides of the vessel or conduit in which it is contained is equal to the product of the length multiplied by half the square of the depth and by the weight of the fluid in cubic unit of dimensions. The following formula is simple and satisfactory : multiply the submerged area in inches by the pressure due to one-half the depth. 1 iy “submerged area’’ is meant the surface upon which the water presses ; for example, to find the lateral pressure upon the sides of a tank twelve feet long by twelve feet deep : 144 X 144 ■' 20736 indies of side. The pressure at the bottom will be 12 X 0.43 = 5.10 pounds, while the pressure at the top is 0, giving us, say, 2.6 pounds as the average : therefore 20736 x 2.0 = 53014 pounds. Discharge of Water.—The quantity of water discharged during a given time from a given orifice, under different heads, is nearly as the square roots of the corresponding heights of the water in the reservoir or containing vessel above the surface of the orifice. JSmall orifices, on account of friction, discharge proportionately 1 A head of water equals the height that the water rises above the oriiice.536 PLUMBING. less than those which are larger and of the same shape under the same pressure. Circular apertures are the most efficacious, having less surface in proportion to area than any other form. If a cylindrical horizontal tube through which water is discharged be of greater length than its diameter, the discharge is much increased. It can be lengthened with advantage to four times the diameter of the orifice. To find the Number of United-States Gali.ons contained in a Foot of Pipe of any Diameter.—Square the diameter of the pipe in inches, and multiply the square by 0.0408. Velocity of Flow of Water. — Water which has a chance to flow downward does so with a velocity in exact proportion to its head. The following table gives the velocity of flow of water due to heads of from one to forty feet: — Velocity in Feet per Second due to Heads of from 1 to 40 Feet.1 Head. Velocity. Head. Velocity. Head. Velocity. Head. Velocity. 0.5 5.67 10.5 25.98 20.5 36.31 30.5 44.29 1.0 8.02 11.0 26.60 21.0 30.75 31.0 44.65 1.3 9.82 11.5 27.19 21.5 37.IS 31.5 45.01 2.0 11.34 12.0 27.78 22.0 37.61 32.0 45.37 2.5 12.68 12.5 28.35 22.5 38.04 32.5 45.72 3.0 13.89 13.0 28.91 23.0 38.46 33.0 46.07 3.5 15.00 13.5 29.46 23.5 38.88 33.5 46.42 4.0 16.04 14.0 30.00 24.0 39.29 34.0 46.76 4.5 17.01 14.5 30.54 24.5 39.69 34.5 47.10 5.0 17.93 15.0 31.06 25.0 40.10 35.0 47.44 575 IS.81 15.5 31.57 25.5 40.50 35.5 47.78 (>.() 19.64 16.0 32.08 20.0 40.89 36.0 48.12 r>.5 20.44 16.5 32.58 26.5 41.28 ■ 36.5 48.45 7.0 21.22 17.0 33.00 27.0 41.67 37.0 48.78 i .•> 21.96 17.5 33. do 27.5 42.05 37.5 49.11 8 0 2 2. OS 1S.0 34.02 28.0 42.44 3S.0 49.44 8.5 2: ».38 18.5 34.49 28.5 42.81 38.5 49.76 0.0 21.0 i 19.0 34.90 29.0 43.19 39.0 50.08 0.5 24.72 19.5 35.41 29.5 43.56 39.5 50.40 10.0 25.30 20.0 35.80 30.0 43.92 40.0 50.72 In plumbing-work we cannot secure this velocity in the flow of water through pipes, because of the friction which constantly tends to diminish it. The longer the pipe, the greater the friction and consequent retardation of the flow. In the following table we have the head of water consumed by friction in pipes one yard long and from one to four inches in diameter. This table shows the head of water required to produce a given flow per minute, lly means of the rules given on p. 538 it is made applicable to any length of pipe; and a variety of problems relating to lengths and diameters of pipe, discharge in gallons, and head in feet, are solved by it. 1 I>ox'h Hydraulics.PLUMBING. 537 Head of Water consumed by Friction in Pqies one Yard Lony.1 I Diameter of the Pipe, in Inches. 1 U 2 3 3 h 4 w Head of Water, in Feet. 1 0.0041 0.00054 0.00012 0.000042 0.000016 0.0000078 0.000004 o 0.01(54 0.00216 0.00051 0.000168 0.000067 0.0000313 0.000016 3 0.0370 0.00487 0.00115 0.000379 0.000152 0.0000705 0.000036 4 0.0658 0.00867 0.00205 0.000674 0.000271 0.0001250 0.000064 5 0.1028 0.01354 0.00821 0.001053 0.000423 0.000195 0.000100 6 0.1481 0.01950 0.00463 0.001517 0.000609 0.000282 0.000144 i 0.2016 0.02655 0.00630 0.002064 0.000830 0.000383 0.000196 8 0.2633 0.0346S 0.00S23 0.002696 0.001084 0.000501 0.000257 9 0.33:33 0.04389 0.01041 0.003413 0.001372 0.000634 0.000325 H 0.4110 0.05410 0.01286 0.004210 0.001690 0.000783 0.000401 -0 1.64 0.21670 0.05140 0.016850 0.006770 0.00313 0.001600 30 3.70 0.4S770 0.115 0.037920 0.0152 0.00707 0.003610 40 6.58 0.86700 0.205 0.067420 0.0271 0.01253 0.006430 50 10.28 1.35 0.321 0.1053 0.0423 0.01958 0.010040 00 14.81 1.95 0.463 0.1517 0.0C)O9 0.02820 0.014460 70 20.16 2.65 0.630 0.2064 0.0830 0.03839 0.019690 80 26.33 3.46 0.823 0.2696 0.10S4 0.05014 0.025720 00 33.33 4.38 1.041 0.3413 0.1372 0.06346 0.032550 1(H) 41.1 5.4 1.28 0.421 0.169 0.078 0.0401 110 40.7 6.5 1.55 0.509 0.205 0.094 0.0486 120 50.2 7.8 1.85 0.606 0.243 0.112 0.0578 130 69.5 9.1 2.17 0.712 0.286 0.132 0.0679 140 80.6 10.6 2.52 0.825 0.332 0.153 0.0788 150 02.5 12.1 2.89 0.948 0.381 0.176 0.0904 1(30 105.3 13.8 3.29 1.078 0.433 0.200 0.1028 170 118.9 15.6 3.71 1.217 0.485 0.226 0.1161 180 133.3 17.5 4.16 1.365 0.549 0.253 0.1312 lyo 148.5 19.5 4.64 1.521 0.611 0.282 0.1450 200 164.6 21.6 5.14 1.685 0.677 0.313 0.1607 210 181.4 23.8 5.67 1.858 0.747 0.345 0.1772 220 100.1 26.2 6.22 2.039 0.819 0.379 0.1945 230 217.6 28.6 6.80 2.229 0.896 0.414 0.2126 2*0 237.0 31.2 7.40 2.427 0.975 0.451 0.2314 250 257.1 33.8 . 8.03 2.633 1.058 0.489 0.2511 200 278.1 36.6 8.69 2.848 1.145 0.529 0.2716 270 200.9 39.5 9.37 3.071 1.234 0.571 0.2929 280 322.6 42.4 10.08 3.303 1.328 0.614 0.3150 200 346.0 45.5 10.81 3.54-1 1.424 0.658 0.3379 30.) 370.3 48.7 11.58 3.792 1.524 0.705 0.3617 310 305.4 52.0 12.35 4.049 1.627 0.752 0.3162 320 421.3 DO.D 13.16 4.215 1.734 0.802 0.4115 330 448.1 59.0 14.00 4.589 1.844 0.853 0.4376 340 475.6 62.6 14.87 4.871 1.958 0.905 0.4645 850 504.0 66.3 15.75 5.162 2.075 0.959 0.4923 300 .58:’,.3 70.2 16.66 5.461 2.196 1.015 0.5248 870 568.3 74.1 17.60 5.769 2.336 1.072 0.5502 380 504.2 78.2 18.57 6.0S5 2.446 1.131 0.5803 8,00 625.8 82.4 19.56 6.108 2.576 1.191 0.6112 400 658.4 86.7 20.57 6.742 2.710 1.253 0.6430 410 601.7 91.0 21.61 7.0 vi 2.847 1.317 0.6755 420 725.8 95.5 22.68 7,433 2.988 1.382 0.7089 480 760.8 100.1 23.80 7.790 3.130 - 1.448 0.7430 440 796.6 104.9 24.80 8.150 3.270 1.516 0.7780 450 833.2 109.7 26.00 8.530 3,430 1.586 0.8130 too 870.7 114.6 27.20 8.910 3.580 1.657 0.8500 470 000.0 119.7 28.40 9.300 3.740 1.730 0.8870 -J SO 9 48.0 124.8 29.60 9.700 3.900 1.805 0.9250 400 088.0 130.1 30.80 10.110 4.060 1.881 0.9640 500 1028.7 135.4 32.10 10.530 4.230 1.958 1.0040PLUMBING. 538 The practical application of this table will be found in the following rules : — To find the Head of Water, when Diameter and Length of Pipe, and Number of Gallons discharged per Minute, are known. — In the above table the head due to a length of one yard is found opposite the number of gallons. Multiply that number by tbe given length in yards, and we have the required head in feet. Thus, to find the head necessary to deliver 130 gallons per minute by a pipe 4 inches in diameter, 500 yards long: opposite 130 gallons in the table, and under 4 inches in diameter, is 0.679, which, multiplied by 500, gives 339 5 feet, the head sought. To find the Diameter of the Pipe, when Head, Length of Pipe, and the Number of Gallons discharged per Min- ute, are known. — Divide the head of water in feet by the length of the pipe in yards, and the number nearest to this in the table opposite the number of gallons will be found under the required diameter. To find the Number of Gallons discharged, when the Head, Length of Pipe and its Diameter, are known. — Divide the head of water in feet by the given length in yards, and the nearest number thereto in the table under the diameter will be found opposite the required number of gallons. To find the Length, when the Head, Number of Gallons per Minute, and Diameter of Pipe, are known. — Divide the given head by the head for one yard, found in the table under the given diameter and opposite the given number of gallons, and the result is the required length. The actual discharge of pipes is easily calculated with approximate accuracy by Prony’s formula. In using this formula, find the discharge in gallons per minute by multiplying the head in inches by the diameter of the pipe in inches, and divide the product by the length of the pipe in inches In the following table, find the number nearest to the quotient thus obtained in the first column, and the discharge in gallons per minute will be found opposite it, under the diameter of the pipe used. The discharge of small pipes may be calculated with sufficient accuracy for practical purposes from the following convenient table, showing the quantity of water that will ilow through a pipe 500 feet long in 24 hours, with a pressure due to a head of ten feet : — jj-inch boro . . . 570 gallons. I ij-inch bore . . 3,200 gallons, i-inch “ . . . 1,150 “ 1-inch “ . . 0,624 f-incli “ . . . 2,010 “ | 1 j-inch “ . . 10,000 “PLUMBING. 539 Discharge of Pipes by Proni/s Formula. Velocity in feet per second. Diameter of the Pit E, IN Inches. 1 î 1 2 oi 3 J 4 5 0 L Gallons discharged per Minute. . I O.O001V2402 0.025 0.0511 0.1150 0.2045 0.3196 0.4602 0.626 0.818 1.278 1.841 o. nono5137 0.05 0.1022 0.2301 0.4091 0.6392 0.9204 1.252 1.636 2.556 3.682 0.000 Mil OS 0.075 0.1534 0.34501 0.6136 0.9588 1.381 1.87S 2.454 3.834 5.523 0.0001341 0.100 0.2045 0.4602 0.8182 1.278 1.841 2.504 3.273 5.113 7.363 0.0001.Soft 0.125 0.2556 0.5750 1.023 1.598 2.301 3.130 4.090 0.390 9.205 1 0.0002301 0.15 0.3067 0.6900 1.227 1.917 2.761 3.756 4.90S 7.668 11.05 0.0003016 0.175 0.3578 0.8053 1.432 2.237 3.221 4.3S2 5.728 8.947 12.83 0.0003702 0.2 0.4090 0.9201 1 636 2.557 3.682 5.008 6.546 10.23 14-73 0.0001452 0.225 0.4601 1.035 1.841 2.876 4.142 5.634 7.363 11.50 16.57 0.0005266 0.25 0.5112 1.150 2.045 3.196 4.602 6.260 ÜM 12.78 18.41 0 0006140 0.275 0.5624 1.265 1 2.250 3.515 5.002 6.886 9 000 14.00 20.25 0.00070S0 0.3 0.6135 1.381 . 2.454 3.S35 5.522 7.512 MÜ 15.34 22.00 0 000S0S7 0 325 0.6616 1.496 j 2.659 4.154 5.9S2 8.138 10.64 16.62 23.93 0.0000154 0.35 0.7157 1.611 Î 2.864 4.474 6.443» 8.764 11.46 17.89 25.77 0.001028ft 0.375 0.7609 1.726 3.068 4.794 6.903 9.390 12.27 19.17 27.61 0.00114S0 0.4 0.8180 1.841 3.273 5.113 7.363 10.02 13.09 20.45 29.45 0.001274 0.425 0.S691 1.955 ' 3.477 5.433 7.823 10.64 13.91 21.73 31.29 0.001406 0 45 0.9202 2.071 3.682 5.757 S.2S4 11.27 14.73 23.01 33.13 0.001545 0.475 0 9713 2.186 3.886 6.077 8 744 11.89 15.55 24.29 34.97 0.001690 0.5 1.023 2.301 ; 4.091 6.392 9.204 12.52 16.37 25.57 36. S2 0.002 0.55 1.125 2.531 ! 4.500 7.031 10.12 13.77 IS 00 28.12 40.50 0.00233 0.6 1.227 2.761 ■ 4.909 7.670 11.04 15.02 19.64 30.68 44.18 0.0O2093 jO.65 ÜB 2.9911 5.31 S 8.309 11.90 16.28 21.27 33.23 47.86 0.003079 0.7 1.431 3.221 5.727 8.948 12.88 17.53 •>2.91 35.79 51.54 0 003490 10.75 11.533 3.450 6.136 9.5SS 13.81 18.7S 24.54 38.34 55.23 0.003926 j-o.s 1.636 3.682 6.544 10.23 14.73 20.03 26.18 40.90 58.90 0.0043N8 |0.s5 1.73S 3.912 6.954 10.S6 15.65 21.29 27.82 43.46 62.59 0.004876 0.9 1.841 4.142 7.363 11.51 1ft. 57 22.53 29.46 40.02 66.27 0.005928 1.0 *2.045 4.602 8.182 12.78 18.41 25.04 32.73 51.13 73.63 0.00648 j 1.05 2.147 4.S32 8.591 13.42 19.33 26.29 34.34 53.69 77.31 o.00703 ]1.1 2.249 5.062 9.000 14.06 20.25 27.54 36.00 56.24 SO. 99 0.007691 11.15 2.351 5.292 9.409 14.70 21.15 2S.80 37.64 58.80 84.67 0.008338 1.2 2.454 5.522 9.818 15.34 22.09 30.05 39.28 61.36 88.36 0.009 1.25 2.556 5.753 10.23 15.96 23.01 31.30 40.91 63.91 92.04 Having determined the pressure due to head with which he has to deal, and the size of the pipe needed to discharge «a given quantity in a given time, the plumber must calculate the strength which his pipe must possess to resist this pressure under all conditions. This he need not do with absolute accuracy, for the reason that he must use the pipe he rinds in the market; but the strength of the sizes in the market is known, and on the basis of this knowledge lie can determine the weight of pipe he requires. In all such calcinations, however, there should oe a iioerai margin for safety. The pipe may corrode, external influences may weaken- it, and extraordinary pressures may be brought to bear upon it, —as by the suuuen closing of a cock, which, owing to the incompressible nature or water, causes it to strike a powerful biow\ due to the suddeniv arrested momentum of the entire coiumn of water in the pipe.o 10 PLUMBING. This often bursts pipes which are. amply strong to resist a great deal more than the normal pressure to which they are subjected. Ollier causes also operate to increase the pressure, and tax the resisting powers, of the pipe; and it must be strong enough to bear these without straining. The following table gives the relation of size and thickness to strength in standard lead pipes. These lig-ures are compiled from the results of careful tests. Weiyht and Strenyth of Lead Pipes. 1 £ tl O & cc j, 9 x b - 1 X X ir j-fafe working pressure. o n n r «2 m 0Q X “ .o ?' ? . JM i ills. lb oz. ins. lbs. 11)H. ins. lb OZ. ins. lbs. lbs. i AAA 1 12 0.18 1968 492 1 A 4 0 0.21 857 214 A A 1 5 0.15 1027 405 1 B 3 4 0.17 745 186 1 s A 1 2 0.13 1381 347 1 C 2 8 0.14 562 140 1 .'i B 1 0 0.12) 1342 335 1 P 2 4 0.125 518 129 X 0 0 14 0.11 1187 296 1 E 2 0 0.10 475 118 3 a - 0 10 0.087 1085 271 1 - i 8 0.09 325 88- 7 T t> - 0 ‘4 0.03 775 193 H AAA 6 12 0.275 962 240 l i AAA 3 0 0.25 1787 446 1 A A 5 12 0.25 823 20") i 5 - 2 8 0.225 1055 413 I A 4 11 0.21 {S35 171 j 1 y A A 2 0 0.18 1393 04S3 4 B O 11 0.17 546 136 1 L Y A i 10 0.16 1285 321 y (; 3 0 0.135 420 M 1 Y B i 3 0.125 930 245 p 2 8 0.125 350 87 1 Y C i 0 0.10 782 195 4 - 2 0 0.095 322 80 i 1 Y P 0 9 0.065 468 117 4 AAA 8 0 0.29 742 185 i 1 Y 0 10 0.07 550 139 A A 7 0 0.25 700 175 l Y - 0 12 0.0.) 025 156 1 A 6 4 0.22 628 157 ! 1 A A A 3 8 0.28 1548 387 B 5 0 0.18 506 126 j £ A A ‘> 12 0.21 13S0 345 4 C 4 4 0.15 430 107 s A 2 8 0.18 1152 288 N P 3 8 0.14 315 78 | h B 2 0 0.16 987 246 n ■ - 3 0 0.12 245 61 j *> c i 7 0.117 795 19S if B 5 0 - - lit) j ‘j’ I) i 4 0.10 708 177 4 C 4 0 - - 93 A AAA 4 14 0.20 1462 30") n3 P 3 10 0.125 318 79 | A A A O 8 0.225 1225 306 2 AAA 10 11 0.30 611 152 4 A 3 0 0.19 1072 208 2 A A 8 14 0.25 511 127 4 B 2 3 0.15 865 216 2 A 7 0 0.21 405 101 1 4 *c 1 12 0.125 782 195 2 B 6 0 0.19 360 90 | p 1 3 0.00 505 126 2 C 5 0 0.16 260 65 1 AAA 6 0 o.ao 1230 307 •4 I> 4 0 0.09 200 50 1 A A 4 8 0.23 910 227 ]MEMORANDA FOR PAINTERS. 541 Wrouglit-iron pipes suitable for water service range in diameter from half an inch to sixteen inches. The tables on pp. 504, 505, show the weight of the various sizes manufactured. Messrs. Tasker & Co., of the Pascal Iron-Works, Philadelphia, subject the pipes which they manufacture to the following tests:— One-half to one and one-fourtli inch, butt-welded, 300 pounds hydraulic pressure per square inch. One and one-lialf to ten inch, lap-welded, 500 pounds hydraulic psessure per square inch. Practically they are strong enough to bear any pressure with which the plumber has to deal. The same is true of drawn brass and copper pipes. The pressures to be dealt with in American plumbing practice vary through a wide range. In cities supplied by what are known as gravity-works — i.e., where dependence is placed on natural head at the distributing reservoir, as in New York — the pressure of water is often very light. Where pumping machinery is used, and a high head is maintained in tall stand-pipes, or the pumps deliver directly into the mains, we sometimes get pressures of one hundred pounds to the square inch, and upward. * MEMORANDA FOR PAINTERS. [From “Builders’ Guide and Price Book.”] Painting. Painters’ work is generally estimated by the yard, and the cost depends upon the number of coats applied, besides the quality of the work, and the material to be painted. One coat, or priminy, will take, for 100 yards of painting, 20 pounds of lead and 4 gallons of oil. Two-coat work, 40 pounds of lead and 4 gallons of oil. Tliree-coat, the same quantity as two coats ; so that a fair estimate for 100 yards of tliree-coat work would be 100 pounds of lead and 10 gallons of oil. 1 gallon priming color 1 “ white zinc 1 “ white paint 1 “ lead color 1 “ black paint 1 “ stone color will cover 50 superficial yards. u 50 66 66 66 44 u 66 a 50 66 66 u 50 u 66 66 44 it 66542 WINDOW-GLASS. 1 gallon yellow paint wi 1 “ blue color 1 “ green paint 1 “ bright emerald green 1 “ bronze green will cover 44 superficial yards. 75 One pound of paint will cover about 4 superficial yards the first coat, and about 6 each additional coat. One pound of putty, for stopping, every 20 yards. One gallon of tar and 1 pound of pitch will cover 12 yards superficial the first coat, and 17 yards each additional coat. A square yard of new brick wall requires, for the first coat of paint In oil, of a pound ; and for the second, 3 pounds ; and for the third, 4 pounds. A day’s work on the outside of a building is 100 yards of first coat, and SO yards of either second or third coat. An ordinary door, including casings, will, on both sides, make 8 to 10 yards of painting, or about 5 yards to a door without the casings. An ordinary window makes about 2j or 3 yards. Fifty yards of common graining is a day’s work for a grainer and one man to rub in. In painting blinds of ordinary size, 12 is a fair day’s work for one coat, and 0 pounds of lead and 1 gallon of oil will paint them. Polished French plate window-glass, which is considered to be the highest grade of window-glass in the market, may be obtained in lights varying -in size from a piece one inch square to a light eight feet wide and fourteen feet long. Owing to the extra cost of rolling large lights, the price per square foot of large lights is sometimes twice that of smaller lights; so that the cost of plate-glass must be estimated by a price-list, giving the cost of every different size of light. Such a price-list is given below. This list remains the same from year to year, and is known as the “ standard” list for polished plate-glass. The fluctuations in the price of gl.iBj are arranged by means of a discount, which is the same for all sizes. At the present time the discount on large lots of plate-glass is about fifty per cent. WINDOW-GLASS.WINDOW-GLASS 543 Piuce-List of Polished Plate-Glass. Sizes, in inches; prices, i:i dollars and cents. 12 14 16 IS 20 I — 24 26 28 *24 2.00 2.35 2.70 3.05 4.05 4.45 4.90 _ 20 2.20 2.55 2.90 3.95 4.40 4.85 5.30 5.70 - 28 2.35 2.75 3.80 4.25 4.75 5.20 5.70 8.15 8.75 30 2.55 2.95 4.05 4.55 5.10 5.60 6.10 8.70 9.40 m 2.70 3.80 4.35 4.90 5.40 5.95 8.60 9.30 10.00 34 2.85 4.05 4.60 5.20 5.75 8.35 9.10 9.90 10.65 30 3.05 4.25 4.90 5.50 6.10 8.85 9.65 10.45 11.25 38 3.85 4.50 5.15 5.80 8.50 9.35 10.20 11.05 11.90 40 4.05 4.75 5.40 6.10 8.95 9.85 10.75 mm 12.50 42 4.25 5.00 5.70 8.45 9.40 10.35 11.25 12.20 13.15 44 4.45 5.20 5.95 8.85 9.85 10.80 11.80 12.80 13.To 46 4.70 5.45 8.20 9.20 10.30 11.30 12.35 13.35 14.40 48 4.90 5.70 8.60 9.65 10.75 11.80 12.90 13.95 15.05 50 5.10 5.95 8.95 10.05 11.15 12.30 13.40 14.55 15.65 52 5.30 8.15 9.30 10.45 11.65 12.80 13.95 mm 21.55 54 5.50 8.45 9.65 10.85 12.03 13.30 14.50 15.70 22.35 56 5.70 8.75 10.00 11.25 12.50 13.75 15.05 21.55 23.20 58 5.90 9.10 10.35 11.65 12.95 14.25 15.55 22.30 24.00 60 6.10 9.40 10.75 12.05 13.4 14.75 16.10 23.10 24.85 62 8.30 9.70 11.10 12.45 13.85 15.25 22.00 23.85 25.70 64 S.GJ 10.00 11.45 12.90 14.30 15.75 22.70 24.60 26.50 60 8.85 10.35 11.80 13.30 14.75 21.50 23.45 25.40 27.35 68 9.10 10.65 12.15 13.70 15.20 22.15 24.15 26.15 28.15 70 9.40 10.95 12.55 14.10 15.65 22.80 24.85 26.90 29.00 72 9.65 11.25 12.90 14.50 10.10 23.45 25.55 27.70 29.80 74 9.95 11.00 13.25 14.90 21.90 24.10 26.25 28.45 30.65 70 10.20 11.90 13.60 15.30 22.50 24.75 27.00 29.25 31.50 78 10.45 12.20 14.05 15.70 23.10 25.40 27.70 30.00 32.30 80 10.75 12.50 14.30 76.10 23.05 26.03 28.40 30.75 33.15 82 11.00 12.85 14.65 21.83 24.25 20.70 28.60 31.55 33.95 St 11.25 13.15 15.03 22.33 24.85 27.35 29.80 32.30 34.80 SO 11.55 13.45 15.40 22.90 25.45 28.00 30.55 33.10 35.60 88 11.80 13.75 15.75 23.45 26.05 28.65 31.25 33.85 36.45 90 12.05 14.10 10.10 23.95 26.65 29.30 31.95 34.60 37.30 92 12.35 14.40 21.80 24.50 27.20 29.95 32.65 35.40 38.10 94 12.00 14.70 25.05 27.80 30.00 33.35 36.15 38.95 90 12.90 15.05 22.70 25.55 28.10 31.25 34.10 36.90 39.75 98 13.15 15.35 23.20 20.10 29.00 31.90 34.80 37.70 40.60 100 13.40 15,05 23.65 20.05 29.60 32.55 35.50 38.45 41.40 102 13.70 15.95 24.15 27.15 30.20 33.20 36.20 39.25 42.25 lo4 13.95 21.55 24.00 27.70 30.75 33.85 36.90 40.00 43.10 105 14.20 21.95 25.10 28.20 31.35 34.50 37.65 40.75 43.90 108 14.50 22.35 25.53 2S.75 31.95 35.15 38.35 41.55 44.75 110 14.75 22.80 26.05 29.30 32.55 35.80 39.05 42.30 45.55 112 15.05 23.20 20.50 29.80 33.15 36.45 39.75 43.10 46.40 114 15.30 23.00 27.00 30.35 33. To 37.10 40.45 43.85 47.20 110 15.55 24.00 27.45 30.90 34.30 37.75 41.20 44.60 48.05 118 15.85 24.45 27.95 31.10 34.90 38.40 41.90 45.40 48.85 120 16.10 24.85 i 28.40 31.95 35.50 39.05 42.60 46.15 49.70 122 21.65 25.25 | 28.90 32.50 30.10 39.70 43.30 46.90 50.55 124 22.00 25.70 29.35 33.00 30.70 40.35 44.00 47.70 51.35 120 22.35 20.10 29.80 33.55 37.30 41.00 44.75 48.45 52.20 128 22.70 26.50 30.30 34.10 37.85 41.65 45.45 49.25 53.00 12 1 14 16 18 20 22 24 26 28544 WINDOW-GLASS Price-List of Polished Plate-Glass (Continued). Sizes, in inches; prices, in dollars and cents. 30 32 34 3G 38 40 42 44 46 48 34 11.40 12.15 12.90 _ _ _ ! 30 12.05 12.90 13.70 14.50 - - - _ - 1 38 12.75 13.00 14.45 15.30 21.35 - - - - - 40 13.40 14.30 15.20 16.10 22.50 23.05 _ _ 42 14.10 15.05 15.95 22.35 23.00 24.85 20.10 - _ 44 14.75 15.75 22.15 23.45 24.75 26.05 27.35 28.65 _ 40 15.45 21.80 23.15 24.50 25.85 27.20 28.00 29.95 31.30 _ 48 10.10 22.70 24.15 25.55 27.00 28.40 29.80 31.25 32.65 34.10 I 50 22.20 23.05 25.15 26.65 28.10 29.60 31.05 32.55 34.00 35.50 1 52 23.10 24.00 20.15 27.70 29.25 30.75 32.30 33.85 35.40 30.90 | 54 23.95 25.55 27.15 28.75 80.35 31.95 33.55 35.15 36.75 38.85 50 24.85 20.50 28.15 29.80 31.50 33.17) 34. SO 36.45 38.10 39.75 1 58 25.75 27.45 29.15 30.90 32.00 34.30 36.05 37.75 39.45 41.20 00 20.05 28.40 30.20 31.95 33.75 35.50 37.30 39.05 40.85 42.00 i 02 27.50 29.35 31.20 38.00 34.85 30.70 38.50 40.35 42.20 44.00 | 04 28.40 30.30 32.20 34.10 30.00 a» o* .No 39.75 41.05 43.55 45.45 | 00 29.30 31.25 33.20 35.15 37.10 39.05 41.00 42.95 44.90 40.85 i 08 30.20 32.20 34.20 30.20 38.20 40.25 42.25 44.25 40.25 48.30 70 31.05 33.15 35.20 37.30 39.35 41.40 43.50 45.55 47.65 49.70 72 31.95 34.10 30.20 38.35 40.45 42.00 44.75 46.87) 49.00 51.10 74 32.85 35.05 37.20 39.40 41.00 43.80 45.95 48.15 50.35 52.55 70 33.75 30.00 38.20 40.45 42.70 44.95 47.20 49.45 51.70 58.95 78 34.00 36.90 39.25 41.55 43.85 46.15 48.45 50.75 53.05 55.40 80 35.50 37.85 40.25 42.00 44.95 47.35 49.70 52.05 54.45 56.80 82 30.40 38.80 41.25 43.05 40.10 48.50 50.95 53.35 55.80 5S.20 84 37.30 39.75 42.25 44.75 47.20 49.70 52.20 54.65 57.15 59.05 80 38.15 40.70 43.25 45.80 48.35 50.90 53.45 55.95 58.50 61.05 88 39.05 41.65 44.25 46.85 49.45 52.05 54.05 57.30 59.90 62.50 90 39.95 42.00 45.25 47.95 50.00 53.25 55.90 58.60 61.25 63.90 92 40.85 43.55 40.25 49.00 51.70 54.45 57.15 59.90 62.60 65.30 94 41.70 44.50 47.30 50.05 52.85 55.00 58.40 61.20 63.95 66.75 90 42.00 45.45 48.30 51.10 53.95 50.80 59.05 62.50 65.30 68.15 98 43.50 40.40 49.30 52.20 55.10 58.00 60.90 63.80 66.70 69.60 100 44.40 47.35 50.30 53.25 50.20 59.15 02.15 65.10 68.05 71.00 102 45.25 48.30 51.80 51.30 7)7.87 00.35 03.37) 60.40 69.40 72.40 104 40.15 49.25 52.30 55 40 58.45 01.55 04.00 67.70 70.75 78.85 100 47.05 50.20 53.30 50. p|| 59.00 02.70 05.87) 69.00 72.15 75.25 108 47.95 51.10 54.30 57.7)0 00.70 03.90 07.10 70.30 73.50 76.70 110 48.80 52.05 55.30 58.00 01.85 05.10 08.35 71.60 74.85 78.10 112 49.70 53.00 56.35 59.05 02.95 00.25 09.00 72.90 70.20 79.50 114 50.00 53.95 57.35 00.70 04.10 07.45 70.80 74.20 77.55 80.95 110 51.50 54.90 58.35 01.77) 05.20 08.05 72.05 75.50 78.95 82.85 ns 52.35 53. 85 59.35 02.85 00.85 09.80 73.30 76.80 80.30 88. SO m 53.25 50.80 00.35 03.90 07.47) 71.00 74.55 7S.10 81.05 85.20 122 54.15 57.75 61.35 0 4.95 08.60 72.20 75.80 79.40 83.00 91.00 124 mam 58.70 02.35 00.05 09.70 73.87) 71.05 80.70 84.35 92.. 00 j 120 55.90 59.05 08.35 07.10 70.80 74.55 78.30 82.00 90.00 94.00 12S 50.80 60.00 01.40 08.15 71.95 75.75 79.50 83.30 92.00 96.00 130 57.70 01.55 65.40 69.25 78.0') 70.90 SO. 75 84.00 93.00 97.00 132 58.60 02.50 00.10 70.30 74.20 78.10 82.00 90.00 94.00 99.00 134 59.45 03.45 07.40 71.05 75.30 79.;>0 83.25 92.00 90.00 100.00 130 00.3 > 01.40 08.40 72.40 70.45 80.45 84.50 93.00 97.00 102.00 138 01.25 05.80 O.UO 78.50 ) i • •)«) 81.05 90.00 94.00 99.00 108.00 140 02.15 00.25 70.10 74.55 7S.70 82.85 91.00 90.00 100.00 105.00 i14- 03.00 07.20 71.40 75.00 79.80 84.00 93.00 97.00 102.00 106.00 1 144 03.90 08.15 72.10 70.70 80.95 85.20 94.00 99.00 103.00 108.00 30 34 m 3S 40 42 44 46 4SWINDOW-GLASS. 545 Price-List of Polished Plate-Glass (Continued). Sizes, in inches; prices, in dollars and cents. 50 52 54 50 58 00 02 Oi 66 68 70 72 50 37 _ _ _ _ _ _ _ _ 52 3S 40 - - _ _ _ — — _ — _ 51 40 42 43 - - _ _ _ _ _ _ 56 41 43 45 46 _ _ _ _ _ _ _ 58 43 45 46 4S 50 - - - - - - - 60 44 46 48 50 51 53 _ _ _ _ .. _ 62 46 48 50 51 53 55 57 _ _ _ _ _ 61 47 49 51 53 55 57 59 61 - _ - _ 66 49 51 53 00 57 59 61 62 64 _ _ _ 68 50 52 54 56 58 60 62 64 66 68 - - 70 52 54 56 58 60 62 64 66 68 70 72 _ 72 53 O) 58 60 62 64 66 68 70 72 75 77 74 55 57 59 61 63 66 68 70 72 74 77 79 76 56 58 61 63 65 67 70 72 74 76 79 81 78 58 60 62 65 67 69 72 74 76 78 81 S3 80 59 62 64 66 69 71 13 76 78 80 83 85 82 61 63 60 68 70 73 75 78 80 82 85 92 81 62 65 67 70 72 75 77 80 82 84 91 94 86 64 60 6) 71 74 76 79 81 84 91 94 96 88 65 68 70 73 76 78 81 83 90 93 96 99 90 67 69 72 75 77 80 83 85 92 95 98 101 92 68 71 73 76 79 82 84 92 94 97 100 103 94 70 72 75 78 81 83 91 94 97 99 102 105 96 71 74 77 80 82 85 93 96 99 102 105 108 98 72 75 78 81 84 91 95 98 101 104 107 110 1 100 74 77 80 83 90 93 96 100 103 106 109 112 10 2 75 78 81 84 92 95 98 102 105 108 111 114 lot 77 80 83 91 94 97 100 104 107 110 113 116 ■ 78 82 85 92 96 1 99 102 106 109 112 115 119 108 80 83 91 94 97 101 104 108 111 114 118 121 110 81 85 92 96 99 103 100 110 113 116 120 123 112 83 91 94 98 101 105 308 112 115 118 122 125 114 84 92 96 99 103 106 110 113 117 121 124 128 116 90 94 97 101 105 108 112 115 119 123 126 130 I 118 92 95 99 103 106 110 114 117 121 125 128 132 1 120 93 97 101 105 108 112 116 119 123 127 131 134 122 95 99 102 106 110 114 118 121 125 129 133 137 121 96 100 104 108 112 116 120 123 127 131 135 139 124 98 102 106 110 114 118 122 125 129 133 137 141 128 100 104 108 112 115 119 123 127 131 135 139 143 13) 101 105 109 113 117 121 125 129 133 138 142 146 | 132 103 107 111 115 119 123 127 131 136 140 144 148 i bH 104 108 113 117 121 126 129 133 138 142 146 150 136 106 110 114 118 123 127 131 135 140 144 148 152 138 107 112 116 120 125 129 133 137 142 146 150 155 140 109 113 118 122 126 131 135 139 144 148 152 157 142 110 115 119 124 128 333 137 141 146 150 155 159 144 112 116 121 125 130 134 139 143 14S 152 157 161 146 114 118 123 127 132 136 141 145 150 154 159 164 148 115 120 124 129 134 13S 143 147 152 157 161 166 150 117 121 126 131 135 140 145 149 154 159 163 168 152 128 133 138 143 148 153 158 163 169 174 179 184 1 154 129 135 140 145 150 155 160 16.) 171 176 181 186 156 131 136 142 147 152 157 363 108 173 178 184 189 1 158 133 138 143 149 154 159 165 170 175 181 186 191 160 134 140 145 151 156 161 167 172 177 183 188 194 162 136 142 147 152 158 163 169 174 180 185 191 196 164 138 143 149 154 160 165 171 176 182 187 193 198 1 50 52 54 56 58 00 02 64 I 68 70 72546 WINDOW-GLASS Price-List of Polished Plate-Glass (Concluded). Sizes, in inches; prices, in dollars. 74 76 78 80 82 84 86 88 «0 92 94 96 74 81 _ _ _ _ . _ 70 S3 90 — - - - - - - - - _ 78 90 92 95 - - - - - *- - - - SO 92 95 97 100 _ _ _ _ _ _ _ _ 82 91 97 99 102 105 - - - - - - _ SI 97 99 102 105 107 no - - - _ — _ SO 99 102 104 107 110 112 115 - - _ _ _ ss 101 104 107 110 112 115 118 120 - - - 90 104 100 109 112 115 118 120 123 120 _ - 1 92 log 100 112 114 117 120 123 126 120 132 - - i 94 103 lit 114 117 120 123 120 129 132 135 137 — 90 111 113 116 110 122 125 128 131 134 137 140 143 98 113 110 119 122 12o 128 131 134 137 140 143 146 100 115 118 121 124 128 131 134 137 140 143 140 140 102 117 121 124 mm * i 130 133 130 140 143 140 149 152 104 120 123 120 129 133 130 139 142 140 149 152 155 106 122 125 120 132 135 139 142 145 148 152 155 158 10S 121 123 131 124 138 141 144 14S 151 155 158 101 110 127 130 133 137 140 144 147 151 154 157 101 104 im 120 132 130 130 143 140 150 153 157 100 104 107 114 131 135 ■ 138 142 145 140 153 150 100 103 107 170 l III 134 137 141 144 148 152 155 150 102 100 170 173 ) 11S 130 140 143 147 151 154 158 102 165 109 173 170 120 138 142 140 149 153 157 101 164 108 172 175 170 122 140 141 148 152 150 150 103 107 171 175 178 107 12i 143 147 150 154 158 102 100 170 174 177 100 2u0 120 145 140 153 157 10)1 105 109 172 170 105 100 203 12S 147 151 155 159 103 107 171 175 179 • 108 ■ -07 100 150 154 158 102 105 170 174 178 107 201 205 210 102 152 150 10)0 104 108 172 177 195 200 201 200 213 134 154 158 103 107 171 175 104 Ss 203 207 212 216 130 157 101 105 10)9 173. 178 107 201 200 210 215 219 138 159 103 107 172 170 105 100 204 209 213 218 223 140 101 b)0) 170 174 179 108 202 207 212 210 221 220 142 103 108 172 177 100 2; 0 205 210 215 220 224 220 1 144 100 170 175 179 108 203 208 213 218 !).)•) 227 140 108 173 177 190 201 200 211 210 221 220 231 230 i 148 170 175 194 199 204 200 214 KB 224 229 fa* 239 ! 150 173 177 107 202 207 212 217 ooo gag OO'J’ 237 242 i 152 ISO 194 100 BE 200 215 220 225 230 235 240 304 | 154 102 107 202 207 212 217 223 228 233 23.8 302 208 ! 150 104 100 204 210 215 220 225 231 230 241 300 312 1 158 100 202 207 212 218 223 228 234 239 203 309 310 ! 100 100 204 210 215 220 220 231 237 242 307 313 320 102 201 207 212 218 223 229 234 240 304 310 317 324 104 204 200 215 220 220 o*i*> 237 301 307 314 321 328 100 200 212 218 223 220 234 240 304 311 318 325 3.»2 10S 200 215 220 220 23’> 237 301 308 315 322 320 336 170 211 217 223 229 235 240 305 312 319 320 333 340 | 74 76 78 80 82 84 86 88 90 92 94 96 | The above table was kindly furnished the author by Messrs. Hills, Turner, & (/<>., Host Oil, Mass., importers and dealers in French and American window-glass.WINDOW-GLASS. —GLASS FOll SKYLIGHTS 547 Ordinary Window-Glass. M'indtitij-i/lass is sold by the. box, which contains, as nearly as may be, fifty square feet, whatever may be the size of the panes. The thickness of ordinary, or “single thick,” window-glass, is about one-sixteentli of an inch, and, of “double thick,Mnearly one-eighth of an inch. The tensile strength of common glass varies from 2000 pounds to 3000 pounds per square inch, and its crushing strength from 6000 pounds to 10,000 pounds. The following table gives the number of panes of window-glass in one box, or fifty feet: — Size, in inches. cr c? P - c * ® Size, in . inches. Panes in box. Size, in inches. Pane? in box. Size, in inches. Panes in box. 6x8 150 12 x 19 32 16 x 20 23 24 x 44 7 7X9 115 12 x 20 30 16 x 22 20 24 x 50 6 8 x 10 90 12 x 21 29 16 x 24 19 24 X 56 5 8 x ll 82 12 x 22 27 16 X 30 15 26 X-86 8 8 x 12 75 12 x 23 26 16 x 36 12 26 x 40 7 9 x 10 80 12 x 24 25 16 x 40 11 26 x 48 6 9 x ll 72 13 X 14 40 18 x 20 20 26 X 54 5 9 x 12 67 13 x 15 37 18 x 22 18 28 x 34 8 9 x 13 62 IS x 16 35 18 x 24 17 2S X 40 6 9xU 57 13 x 17 33 18 x 26 15 28 X 46 6 9 x i:> 53 13 x IS 31 18 x 34 12 28 x 50 5 9 X 16 50 13 x 19 29 18 x 36 11 30 x 40 6 10 x 10 7.) 13 x 20 28 18 x 40 10 30 x 44 4 10 x 12 60 13 x 21 26 18 x 44 9 30 x 4S 5 ü x 13 00 13 x 22 25 20 x 22 16 30 x 54 5 10 x 14 52 13 x 24 23 20 x 24 15 32 x 42 5 10 x 15 48 14 x 15 34 20 x 25 14 32 x 44 5 10 X 16 45 14 x 16 32 20 x 26 14 32 x 46 5 10 x 17 42 14 x 18 29 20 X 28 13 32 x 48 5 10 X IS 40 14 x 19 27 20 x 30 12 32 x 50 4 11 x 11 59 14 x 20 26 20 x 34 11 32 x 54 4 11 x 12 00 14 x 22 23 20 x 36 10 32 x 56 4 11 x 13 50 14 x 24 22 20 x 40 9 32 x 60 4 11 x 14 47 14 x 28 18 20 X 44 8 34 X 40 5 11 x 15 44 14 x 32 16 20 x 50 7 34 x 44 5 11 x Ü 41 14 x 36 14 22 x 24 14 34 x 46 5 11 x 17 39 14 x 40 13 22 x 26 13 34 x 50 4 11 x IS 36 15 x 16 30 22 x 28 12 34 x 52 4 Ü x 12 50 15 x 18 27 22 x 36 9 34 x 56 4 12 x 13 46 15 x 20 24 22 x 40 8 36 x 44 5 ll x 14 43 15 x 22 22 22 x 50 7 36 x 50 4 m x 15 40 15 x 24 20 24 x 28 11 36 x 56 4 12 x 16 3S 15 x 30 16 24 x 80 10 36 x 60 3 EÜ x 17 35 15 x 32 15 24 x 32 9 36 x 64 3 12 x is 33 16 x 18 25 24 x 36 8 . 40 x 60 0 Glass for Skylights. Where skylights are glazed with clear or double thick glass, it c. may be used in lengths of from sixteen to thirty inches by a width » of from nine to fifteen inches. A lap of at least an inch and a half548 ASPHALTUM. 1 is necessary for all joints. This is the cheapest mode of glazing. The best, however, for skylight purposes, is fluted or rough plate-glass. The following thicknesses are recommended as proportionate to sizes: — 12 inches by 48 inches is the extent for glass T\ inch thickness. 15 66 60 “ 66 u 1 66 ? 66 20 u 100 “ 66 66 3 66 IS 66 94 u 156 “ u 66 1 66 1 66 Weight of Bough Glass per Square Foot. Thickness • - - i tW i t ^ f f 1 inch. Weight.............2 2| 5 7 8| 10 12J pounds. ASPHALTUM. (For Kock Asphalt, see page 591.) Asphaltum is used extensively for composition roofing, for the same purpose as tar. Asphaltum, or solid bitumen, is a natural pitch, found in different countries. The most accessible and economical for use in the United States is obtained from the “ Great Pitch Lake,” a remarkable and inexhaustible deposit in the island of Trinidad. It is impervious to water, and is one of the most unchangeable and durable substances known, — qualities which, together with its tenacity, adhesiveness, and resistance to the effects of the most extreme changes of heat and cold, make it a cementing material of the greatest value for roofs, pavements, and various other purposes. The principal advantages claimed for asphaltum as a roofing material over pitch and coal-tar, arise from the fact that the bituminous matter of the asphalt is not volatile at any temperature of the sun's heat, and is therefore permanent; while in all materials manufactured from coal-tar there are volatile oils, which slowly evaporate ou exposure to the sun and air, destroying the flexibility and life of the material. The fact is now well known, that any pitch or cement manufactured from coal-tar thus gradually deteriorates, until, in the course of years, it becomes brittle, and crumbles away; and that felt saturated with coal-tar in like manner hardens, until it becomes brittle and finally worthless. Asphalted sheathing-felt, for roofing purposes, and for laying under shingles, slates, clapboards, etc., is also made in a similar manner to the tarred papers more commonly used for the above purposes. Dots these materials may be found in the market, in a condition ready for use.WEIGHT OF CUBIC FOOT OF SUBSTANCES. 549 CAPACITY OP FREIGHT CARS. [From the “American Architect.”] A car-load is nominally 20,00(1 pounds. It is also 70 barrels of salt, 70 of lime, 00 of flour, 00 of whiskey, 200 sacks of flour, 0 cords of soft wood, 18 to 20 head of cattle, 50 to 60 head of hogs, 80 to 100 head of sheep, 9000 feet of solid hoards, 17,000 feet of siding, 13,000 feet of flooring, 40,000 shingles, one-half less of hard lumber, one-fourth less of green lumber, one-tenth of joists, scantling, and all other large timbers, 340 bushels of wheat, 400 of corn, iOSO of oats, 400 of barley, 300 of flax-seed, 360 of apples, 430 of Irish potatoes, 360 of sweet potatoes, 1000 bushels of bran. WEIGHT OF A CUBIC FOOT OF SUBSTANCES. Names op Substances, Avcrage weight, in lbs. ' Anthracite, solid, of Pennsylvania................ “ broken, loose............................... “ “ moderately shaken................. “ heaped bushel, loose......................... * Ash, American white, dry......................... Asplialtum........................................ ] Brass (copper and zinc), cast................... . “ rolled........................................ 1 Brick, best pressed............................... “ common hard................................... “ soft, inferior................................ I Brickwork, pressed brick.......................... “ ordinary ................................... ) Cement, hydraulic, ground, loose, American, Kosen- dale................................... “ hydraulic, ground, loose, American, Louisville ......................................... “ hydraulic, ground, loose, English, Portland, > Cherry, dry...................................... ► Chestnut, dry.................................... • Coal, bituminous, solid...................... “ “ broken, loose.......................... “ heaped bushel, loose.................. ■, Coke, loose, of good coal........................ “ “ heaped bushel.......................... 93 54 58 80 38 87 504 524 150 125 100 140 112 56 50 90 42 41 84 49 74 27 3S550 WEIGHT OF CUBIC FOOT OF SUBSTANCES. IVeUjht of Cubic Foot of Substances {Continued). Names or Substances. Average weight, ia lbs. Copper, cast..................................... rolled.................................. Earth, common loam, dry, loose................... “ “ ■ “ “ moderately rammed . . “ as a soft flowing mud....................... Ebony, dry....................................... Elm, dry. . ................................... Flint............................................ Glass, common window •........................... Gneiss, common................................... Gold, cast, pure or 24-carat . :................. “ pure, hammered . . ..................... Granite.......................................... Gravel, about the same as sand................... Hemlock, dry..................................... Hickory, dry..................................... Hornblende, black................................ Ice.............................................. Iron, cast....................................... “ wrought, purest ............................ “ “ average ............................. Ivory ........................................... Lead............................................. Lignum vitic, dry................................ Lime, quick, ground, loose, or in small lumps . . |‘ ' “ “ “ thoroughly shaken . . . “ “ “ “ per struck bushel . . Limestones and marbles........................... “ “ “ loose, in irregular fragments, Mahogany, Spanish, dry........................... “ Honduras, dry............................. Maple, dry....................................... Marbles. (See Limestones.) Masonry, of granite or limestone, well-dressed . . “ “ mortar rubble........................ “ “ dry nibble........................... “ r sandstone, well-dressed................. Mercury, at 32° Fahrenheit....................... f;42 548 7d 1)5 108 70 35 102 157 108 1204 1217 170 90 to 100 58.7 450 485 480 114 711 83 53 75 60 108 90 53 «>) 49 105 154 138 144 849WEIGHT OF CUBIC FOOT OF SUBSTANCES. 550a Weight of Cubic Foot of Substances (Concluded). Names or Substances. Average weight, in lbs. Mica.......................................... Mortar, hardened.............................. 1S3 103 Mud, dry, close............... “ wet, fluid, maximum . . Oak, live, dry................ “ white, dry................. “ other kinds................ Petroleum .................... Pine, white, dry.............. “ yellow, Northern. . . . “ “ Southern. . . . Platinum...................... Quartz, common, pure .... Rosin......................... Salt, coarse, Syracuse, N.Y. . . “ Liverpool, fine, for table use Sand, of pure quartz, dry, loose. “ well shaken................ “ perfectly wet.............. Sandstones, fit for building . . SO to 110 120 59 52 32 to 45 55 25 34 45 1342 105 09 45 49 90 to 106 99 to 117 120 to 140 151 Shales, red or black 102 Silver 055 Slate . . ............................... Snow, freshly fallen....................... “ moistened and compacted by rain. . Spruce, dry................................ Steel ..................................... Sulphur ................................... Sycamore, dry.............................. Tar........................................ Tin, cast.................................. Turf or peat, dry, impressed. . . . . . Walnut, black, dry......................... Water, pure rain or distilled, at 60 degrees F. “ sea.................................... Wax, bees’ ................................ Zinc or spelter............................ 175 5 to 12 15 to 50 25 491) 125 *>7 02 459 20 to 30 38 02 J 04 60.5 437 Greet) timbers usually weigh from oue-tifth to otic-half more than dry.550b DIMENSIONS OF CHURCH BELLS. DIMENSIONS AND WEIGHT OF CHURCH BELLS Manufactured by William Blake & Co., Boston. Wekjht. Tone. Diameter. Size of frame, outside. Horizontal dimensions. Diameter of vertical wheel. lbs. 200 21 in. 42 X 32 in. 34 in. 250 901 '—'1 in. 40 X 30 in. 38 in. 300 E 24 in. 40 X 30 in. no OO in. 350 ns 20 in. 40 X 30 in. 3S in. 400 i) 27^ in. 53 X 40 in. 44 in. 500 h 20 in. 53 X 40 in. 44 in. 000 c OI ol in. 00 X 48 in. 40 in. 700 B oo «Jo in. 00 X 48 in. 40 in. 800 A« 341 in. 00 X 48 in. 40 in. 000 30 in. 70 X 54 in. 58 in. 1000 A 0*7 o i in. 70 X 54 in. 58 in. 1100 us 38i in. 70 X 57 in. 04 in. 1200 30 in. 70 X 57 in. 04 in. 1 1300 40 in. 70 X 57 in. 04 in. | 1400 G 41 in. 70 X 57 in. 04 in. 1500 42 in. 70 X 57 in. 04 in. 1000 4:51 in. 80 X 03 in. 72 in. 1700 Si 44 i in. 80 X 03 in. 72 in. 1850 F 40 in. 80 X 03 in. 72 in. 2000 47 in. 01 X 07 in. 75 in. 2200 K 48 in. 01 X 07 in. 75 in. 2500 ns 51 in. 100 X 70 in. 84 in. 3000 53 in. 112 X <3 ill. 00 in. 0200 L 55 in. 112 X Bo 4*> in. 00 in. j 4000 ( $ 58 in. 124 X 78 in. 108 in. 5000 C 03 in. 124 X 78 in. 108 in. | 1 Size of Roim 2 for Bells. for bolls of loss than 500 pounds . . . . y inch diameter. 44 “ 500 to 800 pounds . . . i 44 4 * 4 44 800 to 1800 pounds . ^ u u 44 abovo 1800 pounds . . The actual weights usually exceed above from two to three per cent.WEIGHT AND COST OF BUILDINGS. 551 WEIGHT OF BUILDINGS. [From the “ American Architect.”] It has been calculated that the pressure per square foot of the superstructure upon the foundation walls of a few of the best-known buildings is as follows : — Dome of United-States Capitol at Washington, 13,477 pounds Girard College, Philadelphia................. 13,440 “ St. Peter’s, Rome............................ 33,330 “ St. raid’s, London............................30,450 “ St. Genevieve, Paris......................... 60,000 “ Le Toussaint, Angers ........................ 90,000 “ while the pressure upon the earth per square foot in the case of St. Paul’s, London, is 42,950 pounds. COST OF PUBLIC BUILDINGS. An experienced architect and surveyor, on the 19th of February, 1879, prepared, and presented to Gen. Meigs, Quartermaster-General, the estimate which follows of the cost of various public an! private buildings in this country, the comparison being by cubic feet, external dimensions : — Sub-Treasury and Post-Office, Boston, Mass....$2,080,507 United-States Branch Mint, San Francisco, Cal. . . . 1,500,000 Custom and Court House and Post-Office, Cairo, 111. . 271,081 . Custom and Court House and Post-Office, Columbia, S.C.. ................................................. 381,900 LTiited-States building, Des Moines, Io............... 221,437 United-States building, Knoxville, Tenn.................. 398,847 United-States building, Madison, Wis..................... 329,389 United-States building, Ogdensburg, N.Y.................. 216,576 United-States building, Omaha, Neb....................... 334,000 United-States building, Portland, Me..................... 392,215 German Bank, Fourteenth Street, Newport, R.I. . . . 475,000 Staats-Zeitung, New-York City............................ 475,100 Western Union Telegraph, New-York City................. 1,400,000 Masonic Temple, New-York City.......................... 1,900,000 Centennial building, Shepherd’s, corner Twelfth and Pennsylvania Avenues, Washington, D.C.................. 246,073 Add to this the United-States National Museum, fireproof building, at Washington, D.C....................... 250,000552 WEAR AND TEAR OF BUILDING MATERIALS THE WEAR AND TEAR OF BUILDING MATERIALS. At the tenth annual meeting of the Fire Underwriters’ Associsv tion of the North-west, held at Chicago in September, 1879, Mr. A. W. Spalding read a paper on the wear and tear of building materials, and tabulated the result of liis investigations in the following form : — Brick Frame dwelling. dwelling (shingle roof. ) Frame store. Brick store, (shingle roof.) Material in aT C o £ oT O o j; aT «M . O Q £ oT **-*•—. 0 0 p Building. ♦a « i ’•g s SR U) ■ r 'o z: § « cs O to bf) £ - u : 9 o ci Bp 0 3 0 ï cî c3 ct ^ ~ . C5 si 7— T, — Cj 7— 7“ 3 —1 T ■—1 O'O C- > ** > "M >>* 0 'w fu << < Brick 75 n 66 li Plastering .... 20 5 30 3J 16 6 30 i Tainting, outside 5 20 7 14 5 20 6 16 Tainting, inside . . I 14 7 14 5 20 6 16 Shingles 10 6 16 6 16 6 16 6 Cornice 40 ■ 40 01 -2 30 ii 40 •>1 -2 Weather-boarding . 30 35 - 30 01 - Sheathing .... 50 o 50 2 40 50 2 Flooring 20 5 20 5 13 8 13 8 Doors, complete . . 30 n 30 3 5 25 4 30 35 Windows, complete 30 35 30 3J 25 4 30 35 Stairs and newel 30 3J 30 35 20 5 20 5 Base 40 PI 40 ®| -2 30 35 30 3a Inside blinds . . . 30 3a 30 35 30 |] **3 30 3J Building hardware . 20 5 20 5 13 8 13 8 Piazzas and porches 20 5 20 5 20 5 20 5 Outside blinds . . 16 6 16 6 16 6 16 6 Sills and first - floor joints 25 4 40 -i 25 4 30 °»1 ♦»a Dimension lumber . 50 2 75 H 40 •21 66 1‘ These figures represent the averages deduced from the replies made by eighty-three competent builders unconnected with fire-insurance companies, in twenty-seven cities and towns of the. eleven Western States.CAPACITY OF ROUND TANKS AND CISTERNS, IN GALLONS. Diameter in Inches. Height of Tank. Diameter in Inches. 1 in. 2 in. 3 in. 4 in. 5 in. 6 in. 7 in. 8 in. 9 in. 10 in.. 11 in. 1 ft. 2 ft. 1 0.003 0.007 0.010 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.08 1 2 0.014 0.027 0.040 0.05 0.07 0.08 0.10 0.11 0.12 0.14 0.15 0.16 0.32 2 3 0.031 0.061 0.092 0.12 0.15 0.18 0.21 0.24 0.27 0.31 0.34 0.37 0.73 3 4 0.054 0.108 0.163 0.22 0.27 0.53 0.38 0.44 0.49 0.54 0.60 0.65 1.30 4 ' 5 0.085 0.170 0.255 0.34 0.43 0.51 0.59 0.68 0.77 0.85 0.93 1.02 2.04 5 6 0.122 0.244 0.367 0.49 0.61 0.73 0.87 0.98 1.10 1.22 1.34 1.47 2.94 6 7 0.166 0.333 0.500 0.67 0.83 1.00 1.17 1.33 1.50 1.67 . 1.83 2.00 4.00 7 8 0.218 0.435 0.653 0.87 1.09 1.31 1.52 1.74 1.96 2.18 2.39 2.61 5.22 8 9 0.275 0.550 0.826 mm 1.38 1.64 1.93 2.20 2.48 2.75 3.03 3.28 6.57 9 10 0.340 0.680 1.020 1.36 1.70 2.04 2.38 2.72 3.06 3.40 3.74 4.08 8.16 10 11 0.411 0.823 1.234 1.65 2.06 2.47 2.88 3.30 3.70 4.11 4.52 4.94 9.87 11 12 0.490 0.979 1.469 1.96 2.45 2.94 3.43 3.92 4.41 4.90 5.39 5.87 11.73 12 13 0.575 1.149 1.723 2.31 2.87 3.45 4.02 4.62 5.17 5.75 6.32 6.90 13.79 13 14 0.666 1.332 1.999 2.67 3.33 4.00 4.66 5.33 6.00 6.66 7.33 8.00 15.99 14 15 0.765 1.530 2.259 3.06 3.82 4.59 5.36 6.12 6.88 7.65 8.41 9.18 18.36 15 16 0.870 1.740 2.621 3.48 4.35 5.22 6.09 6.96 7.83 8.70 9.57 10.44 20.89 16 17 0.983 1.965 2.947 3.93 4.91 5.90 6.88 7.86 8.84 9.83 10.81 11.79 23.58 17 18 1.102 . 2.203 3.304 4.41 5.51 6.60 9.71 8.81 9.91 11.02 12.12 13.22 26.44 18 19 1.223 2.455 3.682 4.91 6.14 7.36 8.59 9.82 11.05 12.27 13.50 14.73 29.54 19 20 1.360 2.720 4.080 5.44 6.80 8.16 9.52 10.88 12.34 13.60 14.96 16.32 32.64 20 21 1.499 2.999 4.498 6.00 7.50 9.00 10.50 11.99 13.50 14.99 16.49 17.99 35.98 21 22 1.646 3.291 4.936 6.58 8.23 9.87 11.52 13.16 14.81 16.46 18.10 19.75 39.49 22 23 1.799 3.598 5.397 7.19 8.99 10.79 12.59 14.40 16.19 17.99 19.78 21.58 43.17 23 24 1.958 3.916 5.874 7.83 9.79 11.75 13.71 15.67 17.63 19.58 21.54 23.50 47.00 24 25 2.125 4.250 6.375 8.50 10.63 12.75 14.87 17.00 19.13 21 25 23.37 25.50 51.00 25 26 2.298 4.596 6.894 9.19 11.49 13.79 16.09 18.40 20.68 22.98 25.28 27.58 55.16 26 27 2.479 4.958 7.437 9.91 12.39 14.87 17.35 19.83 22.31 24.79 27.26 29.74 59.49 27 28 2.665 5.330 7.995 10.66 13.33 15.99 18.66 21.32 23.99 26.66 29 32 31.99 63.97 28 29 2.859 5.718 8.677 11.44 14.30 17.16 20.01 22.87 25.73 28.59 31.45 34.31 68.62 29 30 3.060 6.120 9.180 12.24 15.30 18.36 21.42 24.48 27.54 30.60 33.66 36.72 73.44 30 CAPACITY OF CISTERNS AND TANKS. 5f)3Capacity of Round Tanks and Cisterns, in Gallons (Concluded). Diameter in Height of Tank. Diam- eter in Inches. 3 in. 6 in. 9 in. 1 ft. 2 ft. 3 ft. 4 ft. 4 ft. 6in. 5 ft. 5 ft. 6 in. 6 ft. 6 ft. 6 in. 7 ft. Inches. 31 9.7 19.4 29.1 38.8 77 116 154 173 193 212 232 251 270 31 32 10.4 20.8 31.2 41.6 83 125 166 187 208 229 250 271 292 32 33 HI 22.2 33.3 44.4 89 133 178 200 222 244 266 288 310 33 34 11.8 23.6 35.4 47.2 94 141 188 212 235 258 282 306 330 34 35 12.5 25.0 37.5 50.0 100 150 200 225 250 275 300 325 350 35 36 13.2 26.4 39.6 52.8 105 158 210 236 263 289 316 342 369 36 37 13.9 27.8 41.7 55.6 111 167 222 250 278 306 334 362 390 37 38 14.7 29.4 44.1 58.8 117 176 234 263 293 322 352 382 412 38 39 15.5 31.0 46.5 62.0 124 186 248 279 310 341 372 403 434 39 40 16.3 32.6 48.9 65.2 - 130 195 260 293 325 357 390 422- 455 40 41 17.1 34.2 51.3 68.4 137 205 274 308 342 376 410 444 378 41 42 18.0 36.0 54.0 72.0 144 216 288 324 360 396 432 468 504 42 43 18.9 37.8 56.7 75.6 151 227 302 340 378 416 454 492 530 43 44 19.7 39.4 59.1 78.8 158 236 316 355 394 433 472 511 551 44 45 20.6 41.2 61.8 82.4 165 247 330 371 412 453 494 535 576 45 46 21.6 43.2 64.8 Si.l 173 *259 346 389 432 475 518 561 604 46 47 22.5 45.0 67.5 90.0 180 270 360 405 450 495 540 585 630 47 48 23.5 47.0 70.5 94.0 188 282 376 423 470 517 564 611 658 48 49 24.5 49.0 73.5 98.0 196 294 392 441 490 539 588 637 686 49 50 25.5 51.0 76.5 102.0 204 306 408 459 510 561 612 663 714 50 51 26.5 53.0 79.5 106.0 212 318 424 477 530 583 636 689 742 51 52 27.6 55.2 82.8 110.4 221 331 442 497 552 607 662 717 .772 52 53 28.6 57.2 85.8 114.4 229 343 458 515 572 629 686 743 800 53 54 29.7 59.4 89.1 118.8 238 356 476 535 594 653 712 771 830 54 55 30.8 61.6 92.4 123.2 246 369 492 554 615 677 738 800. 862 55 36 32.0 64.0 96.0 128.0 256 384 512 576 640 704 768 832 896 56 57 Q‘J 1 OO. J. 66.2 99.3 132.4 265 397 530 596 662 728 794 860 926 57 58 34.3 68.6 102.9 137.2 274 412 548 617 686 755 824 893 962 58 59 35.5 71.0 106.5 142.0 284 426 568 639 710 781 852 923 994 59 60 36.7 73.4 110.1 146.8 • 294 440 588^ 661 734 807 880 953 1 1028 60 553a capacity of cisterns and tanks.CAPACITY OF CISTERNS AND TANKS 553b CAPACITY OF CISTERNS AND TANKS. XuMBF.ii of Barrels (31£ Gals.) in Cisterns and Tanks. Diameter, in Feet. £.5 5 6 7 8 9 10 11 12 13 5 23.3 .33.6 45.7 59.7 75.5 93.2 11*2.8 134.3 157.6 0 28.0 40.3 54.8 71.7 90.6 111.9 135.4 161.1 189.1 4 32.7 47.0 64.0 83.6 105.7 130.6 15N.0 188.0 220.6 8 37.3 53.7 73.1 95.5 1*20.9 149.2 180.5 214.S *25*2.1 9 42.0 60.4 82.2 107.4 136.0 167.9 IS» 241.7 283.7 10 46.7 67.1 91.4 119.4 151.1 ISO. 5 2*25.7 2t)8.6 315.2 11 51.3 73.9 100.5 131.3 166.2 205.1 248.2 295.4 340.7 12 56.0 80.6 109.7 143.2 181.3 2*23.8 270.8 322.3 378.2 13 60.7 87.3 118.8 155.2 196.4 242.4 293.4 349.1 409.7 14 65.3 94.0 127.9 167.1 211.5 261.1 315.9 376.0 441.3 15 70.0 100.7 137.1 179.0 226.0 289.8 338.5 402.8 472.8 16 74.7 107.4 146.2 191.0 241.7 298.4 361.1 429.7 504.3 17 79.3 114.1 155.4 202.9 256.8 317.0 3S3.6 456.6 535.8 18 84.0 120.9 164.5 214.8 27*2.0 335.7 406.2 483.4 567.3 19 88.7 127.6 173.6 226.8 287.0 354.3 4*28.8 510.3 598.0 20 93.3 134.3 182.8 23S.7 302.1 373.0 451.3 53i .1 030.4 •5 t Diameter, in Feet. 14 15 16 17 18 19 20 21 22 5 182.8 209.8 238.7 269.5 302.1 336.6 373.0 411.2 451.3 6 219.3 251.8 286.5 323.4 36*2.6 404.0 447.6 493.5 541.6 i 255.9 293.7 334.2 3 i 7.3 4*23.0 471.3 5*22.*2 575.7 631.9 8 292.4 335.7 382.0 431.2 483.4 538.6 590. S 058.0 722.1 9 329.0 377.7 429.7 485.1 543.8 605.9 671.4 740.2 812.4 10 365.5 419.6 477.4 539.0 604.3 673.3 746.0 8*22.5 902.7 11 402.1 461.6 525.2 592.9 667.7 740.6 8*20.6 904.7 99*2.9 12 438.6 503.5 572.9 646.8 725.1 807.9 895.2 987.0 1083.2 13 475.2 545.5 620.7 700.7 785.5 875.2 969.8 1009.2 1173.5 14 511.8 587.5 668.2 754.6 846.0 942.6 1044.4 1151.5 1*263.7 15 548.3 629.4 716.2 808.5 906.4 1009.9 1119.0 12.83.7 1354.0 16 584.9 671.4 773.9 862.4 966.8 1077.2 1193.6 1315.9 1444.3 17 621.4 713.4 811.6 916.3 1027.2 1044.6 12 f'»s. 2 1398.2 1534.5 18 65S.0 755.3 859.4 970.2 1087.7 1*211.9 1342.8 1480.4 1624.8 19 694.5 797.3 907.1 1024.1 1148.1 1*279.2 1417.4 1502.7 1715.1 20 731.1 839.3 954.9 1078.0 1*208.5 1346.5 1492.0 1044.9 1805.3 11 Diameter, in Feet. — .z 23 24 25 26 2: 28 29 30 c 5 493.3 537.1 582.8 630.4 679.8 731.1 784.2 839.3 6 592.0 644.5 699.4 756.5 815.8 877.3 941.1 1007.1 4 690.6 752.0 815.9 882.5 951.7 10*23.5 1097.9 1175.0 8 • 789.3 859.4 932.5 1008.6 1087.7 1169.7 1254.8 1:54*2.8 9 887.9 966.8 1049.1 1134.7 12*23.6 1316.0 1411.6 1510.7 10 986.6 1074.2 1165.6 1260.8 1359.6 146*2.2 1568. *2 1678.5 11 1085.2 1181.7 1282.2 1386.8 1495.6 1008.7 17*23.0 1846.4 12 1183.9 1289.1 1398.7 1512.9 1631.5 1754.6 1882. *2 *2014.2 1 13 1282.6 1396.5 1515.3 1639.0 1767.5 1900.8 2039.0 *2182.0 14 1381.2 1503.9 1631.9 1765.1 1903.4 *2047.1 ■ 2195.9 234:5.9 15 1479.9 1611.4 1748.4 1891.1 2039.4 *2193.3 2352.7 *2517.8 16 1578.5 1718.8 1865.0 2017.2 2175.4 2:539.5 2509.6 2685.6 17 1677.2 1826.2 1981.6 2143.3 2311.3 2485.7 2666.4 2853.5 IS 1775.9 1933.6 2098.1 2269.4 2447.3 2631.9 2823.3 3021.3 19 1874.5 2041.1 2214.7 2395.4 2583.2 2778.1 2980.1 3189.2 20 1973.2 2148.5 2321.2 2521.5 2719.2 2924.4 3137.0 3357.0 For tank* that are tapering, measure the diameter four-tenths from large end.554 COMPARISON OF THERMOMETERS. WEIGHT AND COMPOSITION OF AIR. 1 cubic foot of air at 32 degrees F., under a pressure of 14.7 pounds per square inch, weighs 0.0S0728 of a pound. Therefore 1000 cubic feet = S0.72S pounds. 1 cubic foot = 1.292 ounces. 1 cubic foot of air contains . 1 cubic foot of air contains . 53.85 cubic feet of air contain Carbonic acid (7 — o. O = 8. ( 23 per cent oxygen. ( 77 per cent nitrogen. ( 0.29716 ounce oxygen. ( 0.99484 ounce nitrogen. 1.29200 total weight. ( 0.0185725 pound oxygen. I 0.0621555 pound nitrogen. 0.080728 pound, j 1.000 pound oxygen. I 3.347 pounds nitrogen. 4.347 pounds. = CO.i = 22. Oz = 16. 6 + 16 = 22. For combustion to carbonic acid, 1 pound of coal requires 2| pounds of oxygen, or 143.6 cubic feet of air, supposing all of the oxygen to combine with the coal. 280 to 300 cubic feel of air per pound of coal is tiie usual allowance for imperfect combustion. 11.59 pounds of air for perfect combustion. 24.00 pounds of air for imperfect combustion. COMPARISON OF THERMOMETERS. To convert the degrees of different thermometers from one into the other, use the following formula: — F stands for degrees of Fahrenheit, or 212° ) C “ “ Celsius,1 or 100° r boiling-point. H “ “ Reamur, or 80°) 97? 9(7 W F - hr + 32, and F = + 32 for degrees above freezing-point. OR C = 5(F—32) 9 5(F-:-32) 9 9(7 and F = -v— 32 for degrees below freezing-point, o 4(F—32) W . , and 7? — -——------for degrees above freezing-point. 4(F-r 32) , and li = ----q----for degrees below freezing-point. 1 Often e&l.'eil tviitigrade.COLORS OF IRON CAUSED BY HEAT. 555 Zero of Celsius or Reamur = + 32° Fahrenheit. Zero of Fahrenheit = — 17.77° C, or — 14.22° 11. 1. How much is 8° Celsius above Zero in Fahrenheit ? 9X8 72 F = Mfl HI 14.4 + 32 = 4(5.4° above, o o 2. How much is 8° Celsius below Zero in Fahrenheit ? 9 X 8 72 F - —F— = -r = 14.1 — 32 ■ 17.0° above, o o IN CASES WIIEKE THE PRODUCT IS SMALLER THAN 32, IT INDICATES THAT THE DEGREE IS ABOVE ZERO OF FAHRENHEIT; SEE EXAMPLE 2. 3. How much is 19° Celsius below Zero in Fahrenheit ? 9 X 19 F = —g-------32 = 34.2 — 32 = 2.2 below Fahrenheit. DIFFERENT COLORS OF IRON CAUSED BY HEAT. [Pouillet.] 0. 210° 221 256 261 370 000 525 700 800 900 1000 1100 1200 1300 1400 1500 1600 Fah. 410° 430 493 502 Î 690fj 932 Color. 977 1292 1472 1657 ’ 1832 2012 I 2192 2372 2552 2732 M 2912( Pale yellow. - • Dull yellow. Crimson. iViolet, purple, and dull blue; between 261° and 1 370° C. it passes to bright blue, to sea-green, and then disappears. Commences to be covered with a light coating of oxide, loses a good deal of its hardness, becomes a good deal more impressible to the hammer, and can be twisted with ease. Becomes nascent red. Sombre red. Nascent cherry. Cherry. Bright cherry. Dull orange. Bright orange. White. Brilliant white, welding heat. Dazzling white.556 MELTING-POINT AND EXPANSION OF METALS. MELTING-POINT OF METALS. Name. Fah. Fall. Authority. Platina . . . 4593° Antimony . . 955 S42 i. Lowthian inn. Bismuth . . . 487 507 t » Tin (average) . 475 — Lead “ 022 020 u Zinc .... 772. 782 u Cast-iron . . . 2780 j 1922 to 2012, white 1 ) 2012 to 2192, gray J Pouillet. Wrought-iron . 2552 2733, welding heat. H Copper (average), 2174 LINEAR EXPANSION OF METALS. Between 0° and 100" C. For V C. For 1° F. Zinc 0.00294 Lead 0.00284 - - Tin 0.00222 — - Copper, yellow .... 0.00188 - - u red 0.00171 — - Forged iron1 0.00122 0.0000122 0.00000677 Steel4 0.00114 0.0(100114 0.00000633 Cast-iron1 0.00111 0.0000111 0.00000616 For a change of 100° F. a bar of iron 1475 feet long will extend one foot. Similarly, a bar 100 feet long will extend 0.0078 of a foot, or 0.8130 of an inch. According to the experiments of Dnlong & Petit, we have the mean expansion of iron, copper, and platinum between 0° and 100° C., and 0° and 300° C.j as below. From 0° to 100° C. 0° to 300° C. 1 ron 0.00180 0.00146 Copper 0.00171 0.00188 Platinum . 0.00884 0.00918 1 Lu^lucc X Lavoisier. 2 lîamsdcn.THE PROPERTIES OF WATER. 557 The law for the expansion of iron, steel, and cast-iron at very high temperatures, according to Rinman, is as follows: — From 25° to 525° C., red heat, — 500° C. For 1° C. 1° Fah. Iron 0.00714 0.0000143 = 0.0000080 Steel ..... 0.01071 0.0000214 = 0.0000119 Cast-iron . . . 0.01250 0.0000250 flü.0000139 From 25° to 1300°, nascent white, = 1275c 7.27 II 3 “ 34.81 tt 44.38 O 1 tt 40.52 tt 52.07 O 1 a 47.2(> tt GO. 82 tt 54.00 tt 60. oo tt 4 “ Gl. 93 tt WEIGHT OF LUMBER PER THOUSAND (M) FEET. BOAIU> MKASrilE. Dry. Tartly seasoned. Green. Pine and hemlock .... Norway and yellow pine . . Oak and walnut Ash and maple 2500 lbs. 3000 “ 4000 “ 3500 “ 2700 lbs. 4000 “ 5000 “ 4000 “ 3000 lbs. 5000 lbs.WEIGHTS OF CORD WOOD,-*- EXPLOSIVES. 579 WEIGHTS OF CORDWOOD. Lbs. Car- bon. Lbs. Car- bon. 1 cord hickory . . 4468 100 1 cord Canada pine 1870 1 42 1 “ hard maple . 2864 58 1 “ yellow oak . . 1 2920 61 1 I beech . . . 3234 64 1 “ white oak . . 1870 81 1 “ ash ... . 3449 79 1 « Lombardy pop- 1 “ birch . . . 2368 49 lar .... 1775 41 1 “ pitch-pine 1903 43 1 “ red oak . . . 3255 70 EXPLOSIVE FORCE OF VARIOUS SUBSTANCES 15? ~ ““ 3TING, ETC. Book. — Hodgson.) Volume of gas. Heat. explosive force. 509 0.173 litre. 88 G08 0.225 1 137 041 0.210 “ 139 704 0.248 “ 190 972 0.318 “ 309 590 0.801 “ 472 087 0.780 “ 530 Gun-cotton mixed witn cmoraLe 578 0.585 “ 080 of potash Picric acid mixed with chlorate 1420 0.484 “ 080 of potash Pierate mixed with chlorate of 1424 0.40S “ 582 potash 1422 0.337 I 478 Xitro-glycerine 1320 0.710 “ 939 The above table is by the celebrated M. Bertlielot, who further t describes nitro-glycerine “as really tbe ideal of portable force. It a burns completely without residue; in fact, gives an excess of oxy-jfgen; it develops twice as much heat as powd'er, three and a half titimes' more gas, and has seven times the explosive force, weight ilfor w eight, and, taken volume for volume, it possesses twelve times f more energy.” From the extreme danger of the wrork, none but a i*competent chemist should attempt to manufacture it.578 DIMENSIONS OF CARRIAGES, ETC. Dimensions of Carriages. — Covered Buggy (Goddan — Length over all, 14 feet; width, 5 feet; height, 7 feet 4 indie Will turn in space from 14 to 20 feet square, according to skill. Coupé. — Length over all, 18 feet; width, 0 feet; height, 6 fee 0 inches! Buygy (Piano Box). — Length over all, 14 feet; width, 4 feet 11 inches. Landau. — Length over all, 19 feet 0 inches: width, 6 feet 3 inches; height, 0 feet 3 inches; length of pole, 8 feet 0 inches. Stanhope GUj (2 11®5|. — Length over all, 10 feet 0 indies; width, 5 feet 8 inches; height, 7 feet 0 inches] Victoria. — Length, without pole, 9 feet 0 inches; length of pole, 8 feet; width over all, 5 feet 4 inches. Lir/ht Brouyham.— Length, without Dole nr «h*** n * feet; width over all, 5 feet 4 i; WEIGHT, PER FOOT, OF LEAD SA 1 inch. o I *->? 1 LL 6 li n Si if a HÌ 2 inches. is| ■ -4 L L IM 9t LL Is 1 LL 28.03 3 LL 34.81 I 1 m LL 40.52 01 warn Li 47.2(5 h LL -54.00 4 LL 01.03 Piano, 3.’ 3”x6. ’ 6” Bathtub, 6.’o”-6.’6”x2. 0 Foot bath, 2.’ 6”x2. ’xi6” [high.] Urinals, 26” between partitions. Wash stand, [at least] 2.’o”x22” Water closet, 30”xq* 6” [at least]. Flour barrel, i7”-i9”x28” Laundry tubs, 2’x2’x2.’ Double bed, 4.’6”x6.’6” Singlebed, 2.’8”x6.’6” » Seats [upholstered] 8”-ir” high. Seats [not upholstered] i4”-i6” high. I Billiard Table, 4. ’ 4”xS.’ 39 Room required, [at least] I3’xi7’ (( n (C 44. 38 52.07 00.82 69. So WEIGHT OF LUMBER PER THOUSAND (M) FEET liOAitn MEAsrnrc. Dry. Tartly I _ I seasoned. I Green. Pine and hemlock' . Norway and yellow pine . Oak and walnut .... Ash and maple .... Ej 2 ".?• 1300011«. I...... SI p0001^ 00OO “ 1 4()00 ((WEIGHTS OF COltD WOOD.—EXPLOSIVES. 579 WEIGHTS OF CORDWOOD. Lbs. Car- bon. Lbs. Car- bon. 1 cord hickory . • 4463 100 1 cord Canada pine 1870 42 1 M hard maple . 2S64 58 1 B yellow oak . . 1 2920 61 1 1 beech . . . 3234 64 • 1 1 white oak . . 1870 81 1 | ash .... 3449 79 1 « Lombardy pop- 1 u birch . . . 2368 49 lar .... 1775 41 1 “ pitch-pine 1903 43 1 « red oak . . . 3255 70 EXPLOSIVE FORCE OF VARIOUS SUBSTANCES USED FOR BLASTING, ETC. (Builders’ Guide and Price Book. — Hodgson.) Substances. Ileat. Volume of gas. Estimated explosive force. Blasting-powder 509 0.173 litre. 88 Artillery powder. . . . . . COS 0.225 “ 137 Sporting-powder Powder, nitrate of soda for its 041 0.210 “ 139 base Powaler, chlorate of potash for 704 0.248 “ 190 its base 972 0.318 “ 309 Gun-cotton 0.801 “ 472 Picric acid 087 0.780 “ 5.‘)() Picrate potash Gun-cotton mixed with chlorate 578 0.585 “ oso of potash Picric acid mixed with chlorate 1420 0.484 “ oso of potash Picrate mixed with chlorate of 1424 0.408 “ 582 potash 1422 0.337 “ 478 Xitro-glycerine 1320 0.710 “ 939 The above table is by the celebrated M. Bertlielot, who further describes nitro-glycerine “as really the ideal of portable force. It burns completely without residue; in fact, gives an excess of oxygen ; it develops twice as much heat as powd'er, three and a half times' more gas, and has seven times the explosive force, weight for weight, and, taken volume for volume, it possesses twelve times more energy.” From the extreme danger of the wrork, none but a competent chemist should attempt to manufacture it.580 FORCE OF THE WIND.-THE FIVE ORDERS. FORCE OF THE WIND. (Builders’ Guide and Price Book.) Miles Feet Feet Force, in lbs., Description. PER per minute. per second. per Hour. square foot. 1 88 1.47 0.005 Hardly perceptible. 2 Q o 176 264 2.93 4.4 0.020 1 0.044 } Just perceptible. 4 5 352 440 5.87 7.38 0.079 l 0.123 S Gentle breeze. 10 15 880 1320 14.G7 22 0.492 ) 1.107 ) Pleasant breeze. 20 25 1760 2260 29.3 26.0 1.970 1 3.067 ) Brisk gale. 80 85 2640 3080 44 51.3 4.429 1 6.027 \ High wind. 40 45 3520 3960 5S.G (50 7.870 i 9.900 ( Very high wind. 50 4400 T*» Ü <*>.o 12.304 Storm. 60 70 5280 OlliO 88 102.7 17.733 > 24.153 1 ■j Great storm. SO 100 7040 8800 117.3 140.6 31.490 ) 49.200 i Hurricane. THE CLASSICAL ORDERS. The term “ order,” in its architectural moaning, refers to the system of colunmiation practised by the Greeks and Romans, and is employed to denote the columns and entablature together. These two divisions combined constitute an order, and so far all orders are alike; but, as there were certain distinct styles of columns and entablatures employed by the Greeks and Romans, tin1! orders have been divided into live classes, which are commonly known as the Five Orders, The plainest and simplest of the orders is the Tuscan' Order, which was used by the early Romans, and supposed to have been borrowed by them from the Etruscans; the next three orders, viz., the Dome, Ionic, and Cokintiiian, were originated and perfected by the Greeks; and the last, or Composite Order, was the work of the Roman artists, who endeavored to improve upon the Greek Corinthian.THE FIVE ORDERS. 581 The ancient Greeks and Romans, using these orders continually, brought them to peifeotion; an 1 the best examples of the different orders have in modern times served as guides in designing classical buildings. As has been stated, an order consists of two divisions, the column and entablature; and each of these is subdivided into three distinct parts or members, — viz., the column, into base, shaft, and capital; the entablature, into architrave, frieze, and cornice. That those who wish to employ any of the orders in their designs may readily draw them in the right proportions, the different orders have been analyzed, and a certain size given to each part in terms of the diameter of the column. For this purpose the lower diameter of the column is taken as the proportional measure for all the other parts and members of an order, for which purpose it is subdivided into sixty parts, called minutes. Being proportional measures, diameters and minutes are not fixed ones like feet and inches, but are variable as to the actual dimensions which they express, — larger or smaller, according to the actual size of the diameter of the column. For example, if the diameter be just five feet, a minute, being one-sixtieth, will be exactly one inch. In the following engravings which are taken from Hatfield’s “ House Carpenter,” the numbers in column II denote the height of the parts opposite them in minutes; and the numbers in column P denote the projection of the corresponding part from the axis of the column, also in minutes. Some writers give the proportions of the parts in diameters, modules, and >ninutes; the module being half a diameter, or thirty minutes. Its use, however, rather complicates the measurements, instead of simplifying them. The following definition of the five orders is taken from “ The House Carpenter” (John Wiley & »Sons, publishers), and corresponds with what is generally given in other architectural works. Tii:; Tuscan Ohdei; (Fig. 1) is said to have been introduced of the Romans by the Etruscan architects, and to have been the only s'y'.e used in Italy before the introduction of the Grecian orders. d hi is the plainest order used by the Romans, it having but few mouldings, and no carving or enrichments. The shaft was more slender than the Doric, and had a base consisting of a plinth and superincumbent torus, connected with the body of the shaft by a fillet. Although the capital had the same individual mouldings as. the Doric, they did not project nearly as far. The use of this order in Italy was very limited, owing to its rudeness; find all that is582 THE FIVE ORDERS. known concerning it is from Vitruvius, no remains of buildings in this style being found among ancient ruins. Fig. 1.— Modified Tuscan Order. Tiik Dome Okdku (Fig. 2) is the oldest and simplest of the Greek orders. Its principal features, as well as its mouldings and ornaments, are simple; its character is severe, and it bears throughout the impress of repose, solidity, and strength. The Doric columns, which are short, powerful, and closely ranged together, in order to support the weight of the massive entablature, consist ofTHE FIVE ORDERS. 583 the shaft and the capital, and rest immediately, without base, on the upper step, which serves as the ground floor of the temple. The shaft is channelled perpendicularly into twenty flutes, which have a sharp edge or arris; and is greatly diminished towards the top, so that the diameter above is much less than at the base.584 THE FIVE ORDERS. This tapering does not take place in a straight line, but by a grad-u il decrease in a gentle parabolic curve, -which is known as the entasis. The architrave is a rectangular block separated by a projecting fillet from the frieze. The frieze of the Doric Order is not taken up with sculpture in uninterrupted succession; but it occurs in groups, at regular intervals, separated by features called triglyphs, which are quadrangular projecting slabs, higher than they are broad, with perpendicular channels, and are to be considered as supports of the cornice. They are distributed in such a way that one occurs over the middle of each column, and of each intervening space; in the case of the corner columns, however, the triglyphs are introduced at the corners, and not over the centre of the column. The spaces formed between the triglyphs arc called metopes. They are either squares, or oblongs of greater breadth than height, and were originally open. After they were closed, alto-reliefs were generally introduced, which in the larger temples represented the deeds of gods and heroes, and in the smaller ones the skulls of animals. The Doric was much more largely used in Italy and Sicily than either of the other orders, and in the classical buildings of modern times it is very commonly found. It is very suitable for the lower story of a façade which has two or more orders, one above the other. Tub Ionic Ohdei: (Fig. 3) did not come into use until the Doric had been perfected and in use for a long time. According to historians, it was invented by Hermogenes of Alabanda; and he being a native of Caria, then in the possession of the Ionians, the order was called the Ionic. The distinguishing features of this order are the volutes or spirals of the capital, and the dentils among the bed-mouldings of the cornice; although, in some instances, dentils are wanting. The Ionic Order also has more mouldings than the Doric; its forms are richer and more elegant; and, as a style, it is lighter and more graceful than the Doric. The Doric Order has been compared to the male and the Ionic to the female figure. The Ionic column has a less diminished shaft, and a smaller parabolic curve, than the Doric. It is like the Doric, channelled; the Hidings, which are twenty-four in number, are separated by annulets, ami are therefore narrower but at the same time deeper than the Doric, and are terminated at the top and bottom by a final curvature. 'This order differs from the Doric, also, in having a base, which is generally of the Attic form, as shown in Fig. 3.THE FIVE ORDERS. 585 To describe tiie Ioxic VoeuteI — Draw a perpendicular from a to s (Fig. 4), and make as equal to 20 pin., or to y of the whole height ac; draw so at right angles to sa, and equal to 1{ min.; upon o, with 2i min. for radius, describe the eye of the volute; about o, the centre of the eye» draw the square rtl2, with sides equal to half the diameter of the eye, viz.', 2\ min., and divide it586 THE FIVE ORDERS. into 144 equal parts, as shown at Fig. 5. The several centres in rotation are at the angles formal hy the heavy lines, as figured, 1, 2, 3, 4, 5, 0, etc. The position of these angles is determined by commencing at the point 1, and making each heavy line one part less in length than the preceding one. Xo. 1 is the centre for the arc 'ih (Fig. 4); 2 is the centre for the arc be; and so on to the last. c Fig. 4. — Ioxic Volute. The inside spiral line is to be described from the centres x, x, x, etc. (Fig. 5), being the centre of the first small square towards the middle of the eye from the centre for the outside arc. The breadth of the fillet at aj is to be made equal to 2,s(> min. This is for a spiral of three revolutions: but one of any number of revolutions, as 4 or G, may be drawn, by dividing of (Fig. 5) into a corresponding number of equal parts. Then divide the part nearest the centre o into two parts, as at It; join o and 1, also o and 2; draw /til parallel to ol, and h-i parallel to o2; then the lines ol, o2, Ii3, It4THE FIVE ORDERS. 587 will determine the length of the heavy lines, and the place of the • centres. *■ The Corinthian Order (Fig. 6) is in general like the Ionic, though its proportions are lighter and more slender, and the individual parts are more rich and elegant. The distinguishing feature of the order is its beautiful capital, which has the shape of an expanded calyx, its form being borrowed from organic nature. The acanthus, or brank-ursine, is imitated in the leaves, as well as in the buds and stalks. The abacus is square in shape, with its sides curved into a retreating semicircle, and its truncated comers covered by the volutes shown in the engraving. The Attic base is often used with this order, the same as with the Ionic, although a different base is shown in the cut. Grecian Orders modified by tiie. Romans. — The orders of Greece were introduced into Rome in all their perfection. But the luxurious Romans, not satisfied with the simple elegance of their refined proportions, sought to improve upon them by lavish displays of ornament. They transformed in many instances the true elegance of the Grecian art into a gaudy splendor, better suited to their less refined taste. The Romans remodelled each of the orders. The Doric was modified by increasing the height of the column to eight diameters; by changing the echinus of the capital for an ovolo, or quarter-round, and adding an astragal and neck below it; by placing the centre, instead of one edge, of Gie first triglyph over the centre of the column; and introducing horizontal instead of inclined intitules in the cornice, and in some instances dispensing with them altogether. The Ionic was modified by diminishing the size of the vo’utes, an 1. hi (some specimens, introducing a new capital in whlcu the volu.es588 THE FIVE ORDERS. were diagonally arranged. This new capital has been termed modern Ionic. The favorite order 0 feet thick, and perfectly solid. The immense stones of which these, as well as Egyptian wallsHOCK ASPHALT. 591 generally, were built, had botli their inside and outside surfaces faced, and the joints throughout the body of the wall as perfectly close as upon the outer surface. The dimensions and extent of the buildings may be judged from the Temple of Jupiter at Thebes, which was 1400 feet long and TOO feet wide, exclusive of the porticos of which there was a great number. A great dissimilarity exists in the proportion, form, and general features of Egyptian columns. For practical use the column shown in Fig. 8 may be taken as a standard of the Egyptian style. ROCK ASPHALT. Rock asphalt is a lime ore impregnated naturally, by a geological phenomenon still but imperfectly explained, with bitumen in the proportion of 0 to 10 for 100. it is found in strata likflcoal. It exists In Europe in many places, and is a material relatively rare, but of great value. It is mined principally at Seyssel and Pyri-mont, in the valley of the Rhone, France, and in the Yal de Travers, canton of Xeucli&tel, Switzerland. If a piece of asphaltic ore be exposed to a temperature of from 80° to 100° centigrade, it will become powder. The bitumen, which serves to keep together the molecules of lime, softened by the beat, begins to melt; and, their cohesion thus destroyed, the grains of lime, each coated with a pellicle of bitumen, separate, and form a chocolate-colored powder. If, while it is yet hot, tlxis powder is put into a mould, it will re-assume, as soon as it is cold, its former consistency; and the block of ore will have been reconstructed with ts same grains, and, in general, its same properties. It is upon ihis singular property that the principle of laying roadways of compressed asphalt is founded. If, instead of treating asphalt as explained above, it should after being broken be heated in kettles (in which a little bitumen has been first put to serve as a foundation) for five or six hours, there will be obtained a sort of black semi-liquid paste, which is mastic asphalt. This is the material which, being mixed with a little gravel, is used for laying walks, floors, roofs, etc. In this operation the bitumen which is first put in the kettle plays the same part as grease in a frying-pan. It stops the asphalt from burning before it is melted, while at the same time it restores the bitumen that the asphalt has lost by evaporation. The paste referred to is then put in the moulds, varying in shape according to the use for which the asphalt is prepared. Generally each of the cakes thus formed bparp the manufacturer’s mark,592 ROCK ASPHALT. which is of great use in detecting frauds, for nothing is easier to imitate than real mastic asphalt. With a little bitumen and some macadam powder, any one can make a block that the most practised eye cannot tell from the genuine. It is time alone which denounces the imposture, and often at a disastrous price to those so deluded. Compressed asphalt has been long used in Europe for carriageways, sidewalks, and courtyards, subject to considerable traffic. Mr. E. P. North, C.E., member of A.S.C.E., in his report on the pavements of London and Paris (“ Transactions American Society of Civil Engineers,” clxxx. p. 12G), says “ From a sanitary point of view, asphalt is without a peer. Its surface is smooth, regular, and non-absorbent, with no cavities or cracks of any kind to retain the infected mud and dust of the streets, and the soil beneath it is kept dry. It is more thoroughly cleaned, either by sweeping or washing, than any other pavement. Its freedom from noise, and its other excellences, are fast placing it in all the business and banking streets of the city of London, where it seems to be superseding all other pavements. In comparison with granite, its great economy is to brain-workers and the owners of horses.” In an article in Johnson’s Cyelopanlia, (Jen. Q. A. Gillmore says of the “natural rock asphalt,” — “It must be conceded that nothing has yet been discovered which can replace with entire satisfaction the bituminous limestones of Seyssel and Val de Travers. In the natural asphaltic rock, the calcareous matter is so intimately and impalpably combined with the bitumen, resists so thoroughly the action of air and water and even muriatic acid, is so entirely free from moisture, —properties due, perhaps, to the vast pressure and intense heat under which the ingredients have been incorporated by nature, — that we are forced to attribute the excellence of this material to the existence of certain natural conditions which the most skilful artificial methods fail to reproduce.” Mastic asphalt is used for floors of cellars, stores, breweries, •malt-houses, hotel kitchens, stables, laundries, conservatories, public buildings, carriage-factories, sugar-refineries, mills, rinks, etc.; and for any place where a hard, smooth, clean, 3rJ, fire and water proof, odorless, and durable covering of a light color is required, either in basement or upper stories. It can be laid either over cement concrete, brick, or wood, in one sheet without scams; also over cement concrete for roofs, for fire-proof buildings. For dwelling-house cellars, especially on moist or filled land, this material is especially adapted, being water-tight, non-absorbent, free from mould or dust, impervious to sewer-gases, and for sanitary purposes invaluable.COST OF ROOFING-SLATES. 593 Mastic asphalt is also valuable for damp courses over foundations, and for covering vaults and arches underground. The use of asphalt for roofs is extending, many of the principal buildings in London and a large number in this country being covered with it. It possesses especial advantages for this purpose from the fact that it is both fireproof and fire-resisting. Architects and builders desiring to employ asphalt for any of the above purposes should be careful to secure tliBgenuine Fod-cZH Travers or Seyssel rock asphalt, as there are imitations which are of but little value. For floors of cellars, courtyards, etc., laid on the ground, a base of cement concrete 3 inches thick should first be laid; and over this is put a layer of asphalt from | to 1-J inch thick, according to the use to which it is to be put. For ordinary cellar floors, the asphalt need not be more than f inch thick: for yards on which heavy teams are to drive, it should be H inches thick. In specifying asphalt pavement, both the thickness of the concrete and of the asphalt should be given: it should also be remembered, “ asphalt pavement ” does not include the concrete foundation unless so specified. In laying asphalt over plank or boards, a layer of stout, dry (not tarred) slieatliing-paper should first be put down, and the asphalt laid on this. Asphalt floors for stables should be at least 1 inch thick. The cost of rock asphalt in the large cities varies from 20 to 25 cents per square foot in jobs of 2000 feet and over. This does not include the concrete foundation. Imitation asphalts are laid for as little as one-half this amount, and German and other cheap asphalts for about two-tliirds the above price. Comparative Cost of Different Sizes of Koofing- Slate. The following table slynvs the prices for No. 1 Monson (Maine) roofing-slates delivered on wharf in Boston, May 20, 1885. It will be seen that the medium sizes, such as 1G X 10, 10 X 8, 18 X 10, cost the most; and, as the sizes increase or diminish from these, the price decreases. The price of Brownville (Maine) slates are in all cases 81 per square more than the Monson slates. The price of Bangor (Pennsylvania) slates in Boston, at the same date, is very nearly the same as for Monson slates, except for 10 X 8’s, which are §1 a square less. Red slates cost from $12 to $12.50 per square.MEASUREMENT OF STONE WORK. 594 PRICES OF MONSON (MAINE) SLATES. Size. Price per square. IM ice per 1 SlZF i ^1"r-* • square. Size. Price per square. Size. Price per square. 24 x 14 $5 75 ! 20 x 10 ! $0 50 10 x 9 $7 00 12 x 9 $5 75 24 x 12 6 00 IS x 12 0 25 10 x 8 7 50 12 x 8 a oo 22 x 14 5 75 18 x li 6 50 14 x 12 6 00 : 12 x 7 5 50 2:> x 12 6 00 18 x 10 6 75 14 x 10 6 50 1 12 x 0 5 00 22 x 11 G 00 18 x 9 6 50 14 x 9 6 50 1 11 x 8 5 50 20 x 14 6 00 J 10 x 12 6 25 14 x 8 6 75 j 11 x 7 5 00 | x 12 6 25 | 10x11 J 6 T>0 14 x 7 6 50 10 x 8 5 00 20 X 11 6 25 10 x 10 7 00 ■ X 10 5 75 Measurement of Stone Work. Stone walls are generally measured by the perch, which is 16 feet 6 inches long, IS inches thick, and 12 inches high, and contains 24j cubic feet. It is generally reckoned, however, as 25 cubic feet. In some localities, 22 cubic feet, or 16 feet 6 inches long X 16 inches wide X 12 inches high, is called a perch, when measured in the wall. Occasionally stone work is measured by the cubic yard of 27 cubic feet. Net measurement is that where all openings through the walls are deducted, and 24J cubic feet allowed to one perch. Gross measurement is that where no openings under one perch are deducted, and 25 cubic feet allowed to one perch. When openings are deducted, it is generally agreed to allow a compensation for plumbing and squaring the jambs, and for sills and lintels. Stone walls less than 16 inches thick are reckoned as if 16 inches thick by masons, and over 1(5 inches thick each additional inch is counted. Rubble walls are sometimes measured by the cord of 128 cubic feet. Footing courses are always measured extra. Fare work of a superior kind of rubble masonry is measured separately and described. (Jnoin stones of selected stones are allowed as block stone, and other dressings in a similar manner. Walthi'l of block stone is charged at per gubic foot, according to description, similar to ashlar prepared and set, including all beds and joints; but the face is charged extra per foot superficial, according to the way in which it may be dressed. Granite, freestone, limestone, etc., used for trimming, is generally sobl in rough blocks by the cubic foot. Ashlar, platforms, etc., are generally •measured by the square foot; belt courses, strings, etc., by the lineal foot; the price depending upon the number of mouldings, etc. Marble, bluestone, and slate are sold by the square foot, the price varying according to the thickness.MEASUREMENT OF BRICKWORK. 595 Measurement of Brickwork. Brickwork is generally measured by the one thousand bricks, laid in the wall, and sometimes by the cubic foot. In estimating by the one thousand, the contractor figures on what the bricks will cost delivered at the site of the building, and adds to this the cost of laying in the wall, including the cost of the mortar. The general custom in measuring the exterior brick walls of buildings is to compute the total number of brick in the wall, anil ‘-lien the number of face or outside brick that will be required. The difference will be the number of common brick. The outside brick generally cost more than those used for the interior, have to be culled, and the labor in laying costs more. In measuring brickwork, it is customary to deduct all openings for doors, windows, archways, etc.; but not for small flues, ends of joists, boxes of window frames, sills, or lintels, etc., on account of the wastage of material in clipping around or filling in such parts of the work, and the increased amount of time required'. There are different methods of computing the number of brick in any given quantity of work. Some contractors will compute the total number of cubic feet of brickwork in the building, and. multiply by the number of brick contained in a cubic foot, allowing for wastage, etc. This is probably as accurate a method as can be followed. The larger number of masons, however, compute the superficial area of the walls, and multiply by the number of brick in the wall to one square foot of surface; the number, of course, depending upon the thickness of the wall. In the Eastern States, the follow ing scale will be a fair average: — 4-inch Avail, or ■4-brick . ... 74 bricks per superficial fool 66 66 66 8-inch a u 1 -brick . ... 15 66 12-inch a 66 14-brick . 224 66 .6 6 66 66 16-incli u 66 2 -brick . ! ! 1 30 66 66 66 66 20-inch u 66 24-brick . ... 374 66 66 66 66 24-inch a 66 3 -brick . ... 45 66 66 66 66 In the Middle and Western States, the bricks are larger, and the following scale will be more correct for that section of the country: — 44-inch Avail, or 4-brick . 7 bricks per superficial foot 9 -inch “ “ 1 -brick . . 14 2 66 66 66 13 -inch “ “ 14-brick . . 21 1 66 6 6 6 6 18 -inch “ “ 2 -brick . . 28 B 66 66 66 22 -inch I I 24-brick . . 35 1 66 66 66 And seven bricks additional for each half-brick added to thickness. The following table shows the number of bricks in any given wall, from 4 inches to 24 inches in thickness, and for from 1 to KXKhsuperficial feet.596 TABLE OF NUMBER OF BRICKS IN A WALL. TABLE TO FIND NUMBER OF BRICKS IN A WALL Applicable to Eastern States; for Western States, reduce by one-fifteenth. Super- ficial Number or Bricks to Thickness or feet of wall. 4 in. 8 in. 12 in. 16 in. 20 in. 24 in. 1 s 15 23 30 38 45 2 15 30 45 60 75 90 3 23 45 08 90 113 135 4 30 (50 90 120 150 180 5 38 75 111 150 188 225 6 45 90 135 180 225 270 7 Ç8 105 158 210 2()3 315 8 (50 120 180 240 3(H) 360 0 (58 135 203 270 338 405 10 75 150 225 300 373 450 20 150 300 450 000 750 900 30 225 450 075 900 1125 1350 40 300 (500 900 1200 1500 1S00 50 :p 750 1125 1500 1875 2250 (50 450 9(H) 1350 1800 2250 27(H) 70 525 1050 1575 2100 2625 3150 80 (5(H) 1200 1800 2400 3000 3(500 00 (575 1350 2025 2700 3375 4050 100 750 1500 2250 3000 3750 45(H) 200 1500 3000 4500 6000 7500 9000 300 2250 4500 (5750 9000 11250 13500 400 30(H) (5000 9000 12000 15000 18000 500 3750 75(H) 11250 15(HH) 18750 225(H) (500 4500 9000 155500 1S000 22500 27000 7(H) 5250 10500 15750 21000 2(5250 31500 800 0000 12000 18000 24000 ,'JOOOO 3(5000 000 0750 13500 20250 270(H) 33750 40500 1000 7500 15000 22500 30000 375(H) 45000 2000 150(H) 300(H) 45000 60000 75000 0)000 3000 225(H) 45000 (57500 90000 112500 135000 4000 30000 (50000 90000 1200(H) 1.500(H) 180000 5000 37500 75000 112500 15(HK!0 187500 225000 (>000 45000 90000 135000 pOQOO 225000 270000 7(X)0 52500 105000 157500 210000 2(52500 315000 8000 (>0000 120000 181HHH) 2400(H) 30(H)00 360000 0000 (575(H) 135000 2025(H) 270000 337->00 405000 10000 75(HX> 15(H)00 225000 300000 375000 450000 Application of Taker. — How many bricks will there be in 084(5 superficial feet of wall 1(5 inches thick? Answer. — In 0000 square feet there are 270000 bricks. “ 800 “ 1 “ * 240(H) “ I 40 “ K “ “ 12(H) “ “ (? “ “ “ “ ISO “ In 084(5 square feet there are 295380 bricks.BRICKS REQUIRED IN SETTING BOILERS 597 TABLE OF NUMBER OF BRICKS REQUIRED IN THE SETTING OF HORIZONTAL TUBULAR BOILERS. Fj-inished by Mr. Arthur Walworth, Engineer of the Walworth Manufacturing Company, Boston. The number of bricks are for double 8-inch side and rear walls, with air space between. If one of the 8-inch side walls be omitted, deduct the number of bricks in the last column. Diameter of Boiler, 24 Inches. Length of Boiler. Length of Grate. Bricks in outside wall. 2 ft. 2 ft. 6 in. 3 ft. 3 ft. 6 in. 4 ft. Feet. Bricks. Bricks. Bricks. Bricks. Bricks. G mm 2407 2387 2367 2347 535 7 2728 2708 2688 2668 2048 010 8 8029 3009 29S9 2969 2949 685 9 3330 3310 3290 3270 3250 760 10 8631 3611 3591 3571 3551 835 11 3932 3912 3392 3872 3852 910 Firebrick. VJ7 143 7 59 176 191 — Diameter of Boiler, 30 Inches. Length of | Length of Grate. Bricks in one outside wall. Boiler. 2 ft. 6 in. 3 ft. 3 ft. C in. 4 ft. 4 ft. 6 in.I 5 ft. Feet. Bricks. Bricks. Bricks. Bricks. Bricks. ! Bricks. 6 3307 3344 3321 3298 8275 8252 099 7 3755 3732 3709 8686 86! >3 8640 797 8 4143 4120 4097 4074 4051 4028 895 9 4531 4508 4485 4462 4489 4628 993 10 4919 4896 4873 4850 4827 4804 1091 11 5307 5284 5201 5238 5215 5192 1189 12 5695 5672 5649 5626 5603 5580 1287 13 6083 6060 6037 6014 5991 5968 1385 14 6471 6448 6425 6402 6879 6356 1483 Firebrick. 178 197 216 236 264 273 ■ Diameter of Boiler, 36 Ixeni :s. Length of Grate. Bricks in one outside wall. | Length of Boiler. 2 ft. 6 in. 3 ft. 3 ft. 6 in. 4 ft. 4 ft. 6 in. 5 ft. Feet. Bricks. Bricks. Bricks. Bricks. Bricks. Bricks. 8 4296 4270 4244 421S 4192 4166 905 9 4691 4665 4639 4613 4587 4561 1005 10 5086 5060 5034 5008 4982 4956 1105 11 5481 5455 5429 5403 5337 5151 1205 12 5876 5850 5824 5798 5772 5746 1305 13 6271 1 6245 6219 6193 6167 6141 1405 14 6666 6640 6614 6588 6562 6536 1505 15 7061 7035 |009 6983 6957 6931 1605 10 7456 7430 7404 7378 7352 7326 1705 | Firebrick. 220 241 262 283 304 326 —598 BRICKS REQUIRED IN SETTING BOILERS Table of Bricks required in Setting Boilers (Concluded). Diameter of Boiler, 42 Inches. Length of Length of Grate. Bricks in one out- I side wall, j Boiler. 1 3 ft. 3 ft. 6 in. 4 ft. 4 ft. 6 in. 5 ft. 5 ft. 6 in. Feet. Bricks. ! Bricks. Bricks. Bricks. Bricks. Bricks. 10 5777 5749 5721 5693 5665 5637 1227 11 6210 61SS 0169 6132 6104 6076 loo i 12 6055 6027 0599 6571 6543 6515 1447 13 7094 7006 7038 7010 6984 6954 1557 14 7533 7505 7477 7449 7421 7393 1667 15 7972 7944 7916 7888 7800 7832 1777 10 8411 8383 8355 8327 8299 8271 1887 17 8S50 SS22 8994 8766 8738 8710 1997 Firebrick. 277 300 3-3 346 309 392 " DlAM ET El of Boiler, 4S Inches. Length of Length of Grate. Bricks in one outside wall. Boiler. È ft. 6 in. 4 ft. 4 ft. 6 in . 5 ft. 5 ft. 6 in. 6 ft. Feet. Bricks. Bricks. Bricks. Bricks. Bricks. Bricks. 10 6721 6390 00)59 6628 6597 i 6566 1366 11 7202 7171 7140 7109 7078 7047 14S7 12 7683 7052 7021 7590 7559 752S 1008 13 SI04 8133 8102 8071 8040 8009 1729 14 8645 80)14 8583 8552 8521 < 8490 1850 15 9120 9095 900 4 9033 9002 8971 1971 10 9007 9570 9545 9514 9483 9452 2092 17 10088 10057 10020 9995 990 4 9932 2213 IS 10509 I 10538 10597 10476 10445 1 10414 2.'>34 Firebrick. ») T O O •> -» Oil) OO i 301 3 S3 409 433 Diameter of Boiler, 54 Inches. Length of Boiler. Length of Grate. Bricks in one outside wall. 4 ft. 4 ft. 6 in. 5 ft. 5 ft. 6 in. 6 ft. Feet. 'T. Bricks. Bricks. Bricks. Bricks. 10 7233 7200 7107 7134 7101 1455 11 7746 7713 7080 7647 7614 1585 12 8259 8226 8193 8160 8127 1715 t 13 8772 8739 8700 8673 8640 1845 IT 9285 9252 9219 9186 9153 1975 15 9798 9705 9732 9399 9066 2105 16 10311 10278 10245 10212 10179 2235 17 10824 10791 10758 10725 10092 2365 18 11337 11304 11271 11238 11205 2495 Firebrick. 374 401 4M 43 ö 4S2 REFRIGERATORS. 599 REFRIGERATORS.1 A Satisfactory Refrigerator caxxot be Adapted to a Badly Proportioned Space. as Refrigerators for kitchen use, with ice located in centre Fig. 1. Depth should not be over three feet, nor under two feet; height, four feet to seven feet; length of front, largely determining capacity, say from five feet to seven feet. Fig. 1 is a desirable arrangement for use in ordinary dwellings. Fig. 2 shows greater capacity, with ice al ng the back instead of in centre. It is 1 tetter adapted for use in large families, entertaining considerably, and for small clubs, boarding-houses, restaurants, private hospitals, etc. Such refrigerators should be built with two insulated partitions, permitting the use of one-third, two-thirds, or the whole refrigerator at one time, as demand upon it may require, with a proportionate supply of ice. A good arrangement is a window in the wall, at back of refrigerator, through which ice may. be passed into refrigerator. Such window with ice compartment, as Fig. 1, should have casing head five inches below refrigerator top. As in Fig. 2, sill a little above refrigerator top. Refrigerators over two feet three inches deep should he built in sections bolted together, rendering them easy to transport and handle in contracted space. Fig. .‘i is a refrigerator for use in butler's pantries, where economy of space is important. Ice-tank is arranged as a drawer. Height and depth arc uniform with other side furniture: height, about two feet eight inches ; depth, about two feet ; length of front determining capacity, hut never less than two feet ten inches. Such an arrangement gives a stationary top that may be utilize ! Fig. 2. 1 From Useful to Architects. B. O. McKay, l'ubU*her, Util Unaulway.600 REFRIGERATORS. Fig. 4. for disli-draining board, table use, or built upon for shelving, closets, etc. In every three feet or three feet six inches of front, one ice-drawer is .allowed. The finish, wood, trim, and hardware should correspond with other fittings. l)raiiia°'C.—A short, accessible, well-trapped drain is imperative, and should be as nearly under the centre of the ice . | compartment as possible. It is well to have f ■ ---- "~q, refrigerators on casters, so they are easily moved for cleaning about them. Fig. 4 shows a good drainage arrangement, permitting removal of refrigerator at will. A consultation with the refrigerator builder whose work may be called for, is always wise before deciding in relation to space to be occupied by large refrigerators, refrigerating rooms, freezers, etc. All large work in hospitals, hotels, etc., should be protected by iron about the base, brought up, say, six inches. Where a very low temperature is required, as for game or fish, carried in large quantities, or in medical colleges, where the object is to preserve bodies, it is absolutely necessary that ice should go into the tanks from top. number’s pan for reception of refrigerator drip should be countersunk in floor. Usual complement of refrigerators for use in ordinary families : one in kitchen ; one in butler's pantry. Large families same, with greater capacity. Small clubs, small restaurants, etc. : one general storage ; one wine ; one in or near kitchen, for cook’s use ; one fish. Large hotels, clubs, restaurants, etc. : one storage for large' meat; one in or near kitchen, for cook’s use ; one fish ; one milk and butter ; one in storeroom ; one ice-cream (in hotels) ; one wine. Private hospitals : one large storage ; one cook’s use in or near kitchen ; one milk and butter ; one iron-lined box for broken ice. Large hospitals same, but increased capacity, and a small refrigerator in each ward. Isolated hospitals should have large storage ice-houses in addition. Medical colleges, for preserving bodies, with accommodations for eight bodies. Dimensions : about eight feet six inches front, seven feet six inches deep, and nine feet high. Ice going into tanks from toilSTEAM-HEATING. 601 STEAM-HEATING. HEAT, FUEL, WATER, STEAM, AND AIR. Heat is measured in two ways: 1st, by the thermometer, as in ordinary practice; and 2d, by the work which it performs. The unit of heat (sometimes called the British thermal unit) is that quantity of heat which will raise the temperature of one pound of water at or near the freezing-point, 1° Fahrenheit. A French “ calorie” is the heat required to raise one kilogramme of water 1° Centigrade, and is equal to 3.96832 British thermal units. The equivalent in force of the unit of heat is the raising of 772 pounds avoirdupois one foot high, and is called the mechanical equivalent of heat. Various kinds of fuel contain a certain number of thermal units per pound; and the method of heating which will convey the largest number of units to the air to be wanned is the most economical, so far as fuel and heating are concerned. But no method has yet been devised which will utilize more than about 85 per cent of the heat units contained in the fuel. Fuel.1 — The value of any fuel is measured by the number of heat units which its combustion will generate. The fuels generally used in heating are composed of carbon and hydrogen, and ash, with sometimes small quantities of other substances not materially affecting its value. “Combustible” is that portion which will burn, the ash or residue varying from 2 to 36 per cent in different fuels. The following table gives, for the more common combustibles, the air required for complete combustion, the temperature with different proportions of air, the theoretical value, and the highest attainable value under a steam-boiler, assuming that the gases pass off at 320°, the temperature of steam at 75 pounds pressure, and the incoming draft to be at 60°. 1 From Steam, published by the Babcock & Wilcox Company, New York and Olusgow.tz zz ~ ^ ~ ^ cc TABLE OF COMBUSTILLES. Hp 1 9 P 1 <* 2 CD ® IS** £ » £-—< M» r- «<î Boiler. ** << CD c+ O ^ CD r* » r— p 5> CD >—< • cd P n O >-+a Kind of Combustible. P o O >a oc O-, o r_, g s ■ >» 03 — aCf ^ B ■ 3 1 0-> £ tu £ o tj S 5 £rn c3 >* Jig 9 ®o’ i« 2- ,— £ £ 72 à p O C^H p P p co 1C W CO M w f. CC >—1 tO 00 —I 4— O c-K f-3 1 5 rc ® P o ~ic«.£ ^ 1 ! «M ■** o H "O H s oz 5 G. n CP c *-»* co t^2 in 1 >» p X3 O ü ^ •4—> P3 * r* o g* pi x: c-1 o B 0) ;>2 'D O Q C-l ^ B ■ a £ r* CJ O* H i” K x < rn ^ § O Pa G «*H •—« O £§'< £ o<1 ^ XJ — j « > r9 H-( « «Î CP ^ CS c 5 1,2 s “ f « O O Hydrogen 36.00 5750 3860 2860 1940 62032 64.20 ^.-,i iH o 2 Petroleum 15.43 5050 3515 2710 1850 21000 21.74 18.55 19.90 S’ 2 £ • ^ H 8 cd ( Charcoal, ) o —1jjJEv IB p. CD Carbon < Coke, 12.13 4580 3215 2440 1650 14500 15.00 13.30 14.14 S? ° O O • • p4 ( Anthracite Coal, ) o i ® Coal — Cumberland . . . 12.00 4900 3360 2550 1730 15370 15.90 14.28 15.06 3 “ Coking Bituminous, 11.73 5140 3520 2680 1810 1.5837 16.00 14.45 15.19 o CD s “ Cannel 11.80 4850 3330 2540 1720 15080 15.60 14.01 14.76 Ou pi if Lignite 9.30 4600 3210 2490 1670 1 11745 12.15 10.78 11.46 •••••• r-* jj Peat — Kiln-dried .... 7.68 4470 3140 2420 1660 9660 10.00 8.92 9.42 m tf ( Air-dried, ) 5.76 4000 2S20 2240 1550 7000 7.25 6.41 6.78 M CD ( 25 per cent water, j p P Wood — Kiln-dried . . . (( j Air-dried, ) 6.00 40S0 2910 2260 1530 7245 7.50 6.64 7.02 M 1—• tC IO IO CO 1 cd o P- ►-* 4.80 3700 2670 2100 1490 5600 5.80 4.08 4.39 CC '5 -J C3 Ci 05 ose Wic. m -i cH’ «-<3 ( 20 per cent water, ( cn A—i Ei o c;« 5 in — = — — 602 STEAM-HEATING.STEAM-HEATING. 603 The following table of American coals lias been compiled from various sources: — AMERICAN COALS. Coal. State. Kind of Coal. o Theoretical Value. Coal. State. Kind of Coal. o Theoretical Value. Ü X £ X m u I « 1 n *4 O P-i $ * sp eo u *2 ^ • fl* a o £ o J’enn., Anthracite, 3.49 14,199 14.70 111., Bureau Co., 5.20 13,025 13.48 “ “ 6.13 13,535 14.01 ! “ Mercer Co., 5.00 13,123 13.58 “ “ 2.90 14,221 14.72 1 “ Montauk, 5.50 12,659 13.10 “ Can n el, 15.02 13,143 13.60 Ind., Block, 2.50 13,588 14.38 “ Connellsville, 6.50 13,368 13.S4 “ Caking, 5.66 14,146 14.61 “ Semi bi’nous, 10.77 13,155 13.62 1 “ Cannel, 6.00 13,097 13.56 “ Stone’s Gas, 5.00 14,021 14.51 Md., Cumberland, 18.98 12,226 12.65 “ Youghiogheny, 5.00 14,265 14.76 1 Ark., lAKHite, 5.00 9,215 9.54 “ Brown, 9.30 12,324 12.75 i Col., “ 9.25 13,562 14.04 Kentucky, Caking, 14,391 14.89 ! “ u 4.50 13,866 14.35 “ Cannei, 2.00 15,19S 16.76 Texas, “ 4.50 12,962 13.41 << “ 14.SO 13,360 13.84 Wash. Ter., “ 3.40 11,551 11.96 ** Lignite, 7.00 9,326 9.65 l’enn., Betroleum, 20,746 21.47 “Slack,” or the screenings from coal, when properly mixed,— anthracite and bituminous, — and burned by means of a blower on a grate adapted to it, is nearly equal In value of combustible to coal, but its percentage of refuse is greater. One pound of pure carbon, when completely burned, yields 14,500 heat units. Water and Steam.— The several conditions of water are usually stated as the solid, the liquid, and the gaseous. Two conditions are covered by the last term; and water should be understood as capable of existing in four different conditions, —the solid, the liquid, the vaporous, and the gaseous. At and below 32° F., water exists in the solid state, as ice; at 39° F., it reaches its maximum density. At the sea-level, water boils, or vaporizes, at 212° F.; the vapor given off being known as steam. Superheated Steam. — Steam which has a higher temperature than that normal to its pressure is termed “superheated,” or “gaseous.” Dr. Siemens found, that, when steam at 212° was heated separate from water, it increased rapidly in volume, up to 230°, after which it expanded uniformly, as a permanent gas. The use in any steam-boiler of superheating surface exposed to the heated gases of combustion, is highly objectionable, and is of doubtful efficiency. Steam cannot be superheated when in contact with water. Sensible and Latent Heat of Steam. — The temperature of steam, as shown by the thermometer, is called its sensibleSTEAM-HEATING G04 lieat, and this varies with every different pressure; but it is found that steam contains more heat than is shown by the thermometer, and this extra heat is called the latent heat of steam. The following table gives the number of British thermal units in a pound of water at different temperatures below the boiling-point. They are reckoned above o2° F.; for, strictly speaking. water does not exist below 32°, and ice follows another law. The table also gives the weight per cubic foot at each temperature, calculated by Rankine’s formula. HEAT UNITS IN WATE11 BETWEEN 32° AND 212° F., AND WEIGHT OF WATER PER CUBIC FOOT. Tem- pera- ture. Heat Unit«. Weight, lbs. per1 cub. ft. rr 1 (‘111 1 pera-1 ture. Heat Units. Weight, lbs. per cub. ft. ! Tem-: pera-| ture. Heat Units. Weight, lbs. per cub. ft. 32°F. 0. 02.42 123CF. 91.16 61.68 16S°F. 1 136.44 60.81 ■ «3» 62.42 124 92.17 61.07 j 109 137.45 60.79 40 8. 02.42 125 93.17 61.05 i 170 138.45 60.77 45 13. 02.42 126 94.17 61.03 171 139.46 60.75 50 18. 02.41 1 127 95.18 61.01 172 140.47 60.73 ■ 20. 62.40 j 128 90.18 61.60 1 173 141.48 6o.70 54 22.01 62.40 1 129 97.19 61.58 174 142.49 60.08 56 24.01 62.39 1 130 98.19 61.56 175 143.50 60.06 58 20.01 62.38 131 99.20 61.54 176 144.51 60.04 60 28.01 62.37 132 109.20 61.52 177 145.52 60 02 02 30.01 62.36 133 101.21 61.51 i 178 140.52 60.59 04 32.01 02.35 134 102.21 61.49 179 147.53 60.57 (50 34.02 62.34 135 103.22 61.47 180 148.54 60.55 08 30.02 62.33 136 104.22 61.45 181 149.55 60.53 70 38.02 62.31 137 105.23 61.43 182 150.56 60.50 72 40.02 62.30 138 106.23 61.41 1S3 151.57 60.48 74 42.03 62.28 139 107.24 61.39 184 152.58 60.46 70 44.03 62.27 140 108.25 61.37 185 153.59 60.44 78 40.03 62.25 141 109.25 61.36 186 154.60 60.41 80 48.04 62.23 142 110.26 61.34 187 155.61 60.39 82 50.04 62.21 143 111.26 61.32 188 156.62 60.37 84 52.04 02.19 144 112.27 61.30 | 189 157.63 60.34 80 54.05 62.17 145 113.2S 61.28 190 158.64 60.32 88 50.05 62.15 146 114.28 61.26 191 159.65 60.29 DO 58.00 62.13 147 115.29 61.24 j US 100.67 60.27 02 60.00 62.11 148 110.29 61.22 193 101.08 00.25 01 02.00 62.09 149 117.30 61.20 194 102.69 00.22 00 04.07 02.07 150 118.31 61.18 195 163».70 0O.20 OS 00.07 02.05 151 119.31 01.16 196 164.71 60.17 100 08.08 62.02 1 152 120.32 61.14 j 197 105.72 00.15 102 70.09 02.00 153 121.33 61.12 1 198 100.73 00.12 101 72.09 61.97 1 154 122.33 61.10 199 167.74 60. in 100 74.10 61.95 i 155 123.:’,4 01.08 200 108.75 60. u7 108 70.10 01.92 1 156 124.35 61.06 | 201 169.77 00. ()., 110 7S.11 61.89 i 157 125.35 61.04 1 202 170.78 60.02 112 80.12 61.80 153 126.36 01.02 1 203 171.79 00.00 114 82.13 61 .S3 I 159 127.37 61.04 204 172.80 59.97 115 83.13 61.82 100 128.37 60.98 205 173.81 59.9.) 110 84.13 01.80 j 161 129.38 00.90 206 174.83 59.92 117 85.14 61.78 102 130.39 60.94 207 175.81 59.89 118 80.14 61.77 103 131.40 00.92 208 170.8’.) 59.87 110 87.15 01.75 164 132.41 00.90 209 177.86 59.84 120 88.15 61.74 105 133.41 00.87 210 HR 59.82 121 89.15 61.72 100 131.42 00.85 211 179.89 59.79 122 90.10 61.70 ! 167 135.43 00.83 212 180.90 59.76STEAM-HEATING. 605 When a solid becomes a liquid, or a liquid becomes a vapor, heat is absorbed, more than was necessary to raise it to the temperature of conversion; and this latent heat does work in the destruction of the force of cohesion and other changes which take place, and must be absorbed from some other substance. In the case of steam in a boiler, it comes from the fuel during combustion. When steam or vapor is condensed, this same quantity of heat that was received, from whatever source, is again given off to any substance within its influence, — air, water, iron pipes, etc., — colder than itself; and it is this property, together with its great power of absorbing and retaining heat, which makes water and its vapor such a valuable medium for conveying heat from the furnace to the rooms to be warmed. The apacific heat (or heat-absorbing capacity) of water is not constant, but rises in an increasing ratio with the temperature; so that it requires more heat, the higher the temperature, to raise a given quantity of water from one temperature to another. Thus, the specific heat at 32° being 1, at 212° it is 1.013, and at 320° (the temperature of 75 pounds steam-pressure) it is 1.0294. The amount of heat rendered latent by each pound of water in becoming steam varies at different pressures, decreasing as the pressure increases. This latent heat, added to the sensible heat (or thermometric temperature), constitutes the “total heat.” The “total heat” being greater as the pressure increases, it will take more heat, and consequently more fuel, to make a pound of steam, the higher the pressure. The table given on the following page shows the properties of steam at different pressures, from 1 pound to 400 pounds “total pressure;” i.e., above vacuum. The gauge-pressure is about 15 pounds less than the total pressure; so that, in using this table, 15 must be added to the pressure as given by the steam-gauge. The column of “Temperatures” gives the tliermometric. temperature of steam and boiling-point at each pressure. The “Factor of Equivalent Evaporation” show's the proportionate cost, in heat or fuel, of producing steam «at the given pressure, as compared with atmospheric pressure. To ascertain the equivalent evaporation at any pressure, multiply the given evaporation by the factor of its pressure, and divide the quotient by the factor of the desired pressure. Each degree of difference, in temperature of feed-water makes a difference of 0.00104 in the amount of evaporation. Hence, to «ascertain the equivalent evaporation from any other temperature of feed than 212°, add to the factor given, as many times 0.00104 as tin* temperature of feed-water is degrees below 212°.606 STEAM-HEATING. For other pressures than those given in the table, it will be practically correct to take the proportion of the difference between the nearest pressures given in the table. TABLE OF PROPERTIES OF SATURATED STEAM.1 p a. V CL G D — d a 3 '-Æ U 4-J 5 ^ ^2 <£ fi-g | •4-> it a v £ d i s I'M -2Sh 1) c é I5 'w O X ^ jpl 1 £ G "X £ —> — TJ ! > 1 = t 1 r i £ > ; — c h !r? — C 1 102 1113.05 1042.964 0.0080 330.36 20620 0.965 •) 1*20.260 11*20.4') 1020.010 0.0058 172.08 10720 0.972 •> • m 141.022 1125.131 1015.254 0.0085 117.52 7320 0.977 4 158.070 1128.625 1007.229 0.0112 89.62 5600 0.981 5 102.830 1131.449 1000.727 0.0137 72.66 4535 0.984 6 170.123 1188.820 995.249 0.0103 61.21 3814 0.986 7 176.910 1135.890 990.471 0.0189 52.94 3300 , 0.988 s 1S2.910 1137.726 980.245 0.0214 46.69 2910 0.990 9 188.816 118.9.375 982.484 0.0284 41.79 2607 ! 0.992 10 193.240 1140.877 978.958 0.020)4 31.84 2360 0/994 15 218.025 1146.912 9 >4.978 0.0387 27.85 1612 1/ 00 m 227.917 1151.454 974.415 0.0711 19.72 1220.3 1.005 27 240.000 1155.189 945.8J5 0.0084 15.99 9S-£.S 1.008 30 250.245 1158.208 938.925 0.0775 18.46 826.8 1.012 35 259.176 1100.987 982.152 0.0 '75 11.67 718.4 1.015 40 207.120 110.3.410 920.472 0.0944 10.27 62'.2 1.017 45 274.206 1105.60 ) 921.884 0.1111 9.18 501.8 1.017 50 2S0.S54 1107.000 910.081 0.1227 8.31 508.5 1.021 55 280.89 7 1109.442 912.290 0.1: >43 7.01 404.7 1.023 00 292.520 1171.158 908.247 0.1457 7.01 428.5 1.025 65 297.777 1172.762 904.402 0.1509 6.49 397.7 1.027 70 302.718 1174.269 900.844 0.1081 6.07 371.2 1.028 ( ») 307.388 1175.092 897.526 0.1792 5.68 348.3 1.030 so 311.81-2 1177.012 894.880 0.1901 5.85 328.3 1.081 85 810.021 1178.320 891.280 0.2010 5.05 310..5 1 1.033 90 320.039 1179.551 888.375 0.2118 4.79 294.7 1.084 95 828.884 1180.724 885.588 0.2224 4.55 280.6 1.035 100 327.571 1181.849 883.914 0.2830 4.33 267.9 ! 1.036 105 381.113 1182.929 880.842 0.2484 4.14 207.5 1.087 110 304.0*23 1188.970 877.805 0.2537 3.97 246.0 1.088 115 387.814 1184.974 875.472 0.2040 3.80 236.3 1.089 1*20 840.995 1185.944 878.155 0.2742 3.65 227.6 1.040 125 344.074 1186.888 870.911 0.2812 3.51 219.7 1.041 180 347.059 1187.794 808.735 0.2942 3.38 212.3 1.042 140 352.757 1189.585 SO 4.5 00 0.8188 3.16 199.0 I 1.044 150 358.101 1191.180 800.621 0.8840 2.96 187.5 1.046 100 308.277 1192.741 850.874 0.8520 2.79 177.3 1.047 170 368.158 1194.228 858.294 0.3709 2.63 168.4 1.049 ISO 372.82*2 1195.650 849.809 0.8889 2.49 100.4 1.051 190 377.291 1197.018 840.584 0.4072 o 158.4 1.052 200 381.573 1198.319 843.482 0.4249 2.20) 117.1 1.053 250 401.072 1208.785 831.221 0.5404 1.88 114 1.059 300 418.225 1208.787 819.010 0.0486 1.54 90 1.004 350 481.956 1212.580 810.090 0.7498 1.83 S3 1.068 400 444.919 1217.094 800.198 0.8502 l.is 13 1.073 1 Bu mm, 14th et Habeoek 8: Wilcox | onipany. New York ami Ok isgow.STEAM-HEATING. 607 Air.—Air is a mechanical mixture of oxygen and nitrogen, the proportion for pure air being 77 per cent of nitrogen and 23 per cent of oxygen, by weight. It also contains about of its volume of carbonic-acid gas and some watery vapor, and is capable of absorbing any other gas or vapor to a certain extent, distributing them through the whole atmosphere by what is called the law 0/ diffusion of gases, — a property which gases have of mixing and diluting, which prevents gases of different specific gravities from stratifying for any considerable time. This property is of the utmost importance to air; for, if any noxious or poisonous gas Mere to remain separated in the atmosphere, any one breathing it Mould be instantly killed. Air at 00° F., and with the barometer at 30 inches, is taken as the standard for the comparison of the u’eight of gases, itself being considered as unity. At the temperature of 32°, 13,{ cubic feet of air weigh a few grains over one pound avoirdupois. The expansion of air is nearly uniform at all temperatures, expanding about of its hulk at 32°, and for each increase of one degree in temperature. The following table, giving the volume and M’eight of dry air, tension and M’eight of vapor, etc., M’ill be found useful for reference. In this table 1000 cubic feet of dry air is taken for a unit, and the co-efficient of expansion is taken at the air being under constant pressure of 30 inches of mercury. Column 5 is taken from Guyot’s tables, Regnault’s data.STEAM-HEATING. 608 VOLUME AND WEIGHT OF AIR, AND WEIGHT OF VAPOR IN SATURATED AIR. Tem- pera- ture. 1. Volume. 2. Number of Cubic Feet to 1 Found. 3. Weight of 1000 Cubic Feet Dry Air. 4. Tension of Vapor. 5. Weight of Vapor Saturated in 1000 Cubic Feet. 6. Weight of Air Displaced by Vapor. w 4 • 0 0.9340 11.460 87.260 0.04379 0.07930 0.1264 5 0.9449 11.591 86.289 0.05747 0.10289 0.1646 10 0.9551 11.726 85.251 0.07116 0.12588 0.2014 15 0.9653 11.8(59 84.317 0.08535 0.14932 0.23S9 20 0.9755 11.992 88.403 0.10748 0.18180 0.2909 25 0.9857 12.125 82.440 0.13367 0.22871 0.3(561 30 0.9959 12.258 81.566 0.16581 0.27491 0.4398 32 1.0000 mm 81.235 0.17989 0.29633 0.4741 36 1.0082 12.417 80.515 0.21066 0.35201 0.5632 40 1.0163 12.523 79.872 0.24604 0.40770 0.(3523 44 1.0244 12.629 79.176 0.28647 0.47070 0.7531 48 1.0326 12.735 78.493 0.33284 0.54204 0.8672 52 1.0408 12.841 77.825 0.38574 0.62282 0.9965 56 1.0489 12.947 77.220 0.44352 0.71063 1.1370 60 1.0571 13.053 76.628 0.51683 0.82173 1.3147 64 1.0652 13.159 75.988 0.59229 0.93390 1.4943 68 1.0734 13.265 75.357 0.67994 1.0631 1.7008 72 1.0816 13,.371 74.794 0.78018 1.21050 1.9368 76 1.0897 13.477 74.184 0.89103 1.31715 2.1076 80 1.0979 13.583 73.638 1.01669 1,5540 2.4864 .84 1.1060 13.689 73.046 1.15705 1.7536 2.8058 88 1.1142 13.795 72.4(54 1.31554 1.9772 3.1635 92 1.1223 13.901 71.942 1.49067 2.2257 3.5611 96 1.1305 14.007 71.377 1.69214 2.5060 4.0096 100 1.1387 14.113 70.872 1.91937 2.8220 4.5152 104 1.1468 14.219 70.323 2.14669 3.133 5.0138 10S 1.1550 14.325 09.784 2.43323 3.523 5.63(58 112 1.1631 14.431 69.3,00 2.72984 3.926 6.2826 116 1.1713 14.537 (58.77(5 3.05954 4.367 6.9882 120 1.1794 14.643 68.30(5 3.41728 4.843 7.7488 124 1.1876 14.749 67.797 3.81775 5.371 8.5940 128 1.1957 14.855 67.295 4.2(5073 6.088 9.7430 132 1.20:’>9 14.961 66.845 4.72888 6.559 10.4950 136 1.2121 15.0(57 66.357 5.25807 7.240 11.584 1-10 1.2202 15.173 65.919 5.81736 7.957 12.731 144 1.2284 15.279 65.442 6.48029 8.800 14.048 148 1.23(55 15.385 64.977 7.14323 9.630 15.408 152 1.2447 15.491 64.5(58 7.9104 10.595 16.952 156 1.2528 15.597 (54.102 8.6923 11.566 18.506 160 1.2(510 15.703» 63.694 9.5948 12.681 20.290 164 1.2(591 15.809 03.251 10.5579 13.828 22.125 168 1.2773 15.915 62.814 11.4673 14.950 23.920 172 1.2855 16.021 (52.422 12.7165 16.47 2(5.36 176 1.293,6 16.127 61.996 13.8(557 17.43 27.89 ISO 1.3,018 1(5.233 (51.614 15.2343 19.47 31.96 184 1.3,099 1(5.339 61.200 16.(5030 21.08 33.73 18S 1.3181 16.445 (50.790 18.1447 22.89 36.63 192 1.3262 1(5.551 60.423 19.7441 24.75 39.60 196 1.3344 16.657 (50.024 21.4297 2(5.69 42.71 200 1.3426 16.763 59.(566 23.2962 28.85 46.16 Watery Vapor in tlie Atmosphere. —Air is capable of lidding, or absorbing, a certain quantity of vapor of water, tlie proportion depending on the temperature of the air.STEAM HEATING. 609 The warmer it is, the larger quantity it will hold; and as it becomes cool again, it deposits it, or forms clouds or fogs, which condense on any thing colder than the air, leaving the air, upon raising its temperature, capable of taking up more moisture, to be again deposited in dew or rain. It is this property of air which gives it its drying qualities. An absolutely dry atmosphere is an almost impossibility. Air at 32° contains, when saturated with moisture, riti of its weight of water; at 59° it contains gV; at 86° it contains ; its capacity for moisture being doubled by each increase of 27° F. Air is said to be “ saturated ” when it has absorbed all the water it will hold at that temperature. The tension of vapors is the elastic force or pressure which they exert on the sides of vessels in which they are contained. Air, to be healthful, should contain about 75 per cent of the moisture required for saturation. It requires more heat to raise the temperature of a given quantity of moist air one degree than for dry air; but, unless the air is saturated, this difference is not of much practical importance. Columns 6 and 7 on opposite page give the weight of vapor in 1000 cubic feet of saturated air, and the weight of displaced air, for different temperatures from 0 to 200°. The numbers in column 6 are obtained by multiplying the corresponding numbers in column 4 by column 5, and the product by Column 7 is obtained from column 6, by multiplying value of column 6 by f. Specific Heat of Air. — The specific heat of any substance is the quantity of heat required to raise its temperature one degree, compared with the quantity of heat required to raise the temperature of one pound of water at the same temperature one degree. The specific heat of air, as determined by Eegnault, is 0.2374. Hence one thermal unit will raise the temperature of one pound of water or 4* pounds of dry air (equals 51.7 cubic feet at 32° F.) 1° F. As all air contains more or less moisture, which must also be warmed, 50 cubic feet is generally considered as the equivalent of one pound of water in heating. As one pound of steam at 0 (gauge) pressure condensed to water gives off 965 thermal units, it is therefore equivalent to warming about 48,000 cubic feet of air one degree.<»10 STEAM-HEATING. Heating' Apparatus. A steam-lieating plant may be divided Into three distinct parts: 1st. the boiler, or steam generator; 2d, the radiators; and 3d, the supply and return pipes connecting the two. In determining the size of a plant required for a given building, the customary practice is, to first determine the amount of radiating surface required to heat the different rooms and halls; then the size of boiler required to furnish sufficient steam for the radiating surface determined upon; and third, the arrangement and size of the piping. lladiators. — Radiators are generally made of iron, and may be of any shape that will allow of a good circulation of steam through them, and also permit the air to circulate freely about the outside. It is also desirable that the thickness of the metal shall be only sufficient to give sufficient strength. Probably over half the radiators now in use are made, either in whole or in part, of wrought-iron piping; and it is still preferred by many steam-fitters for radiators. There are also many varieties of cast-iron radiators, which will be described farther on. Classes of Radiators. — Radiators are divided into three classes: those affording, 1st, direct radiation; 2d, indirect radiation; 3d, direct-indirect radiation. Direct Radiating Surfaces embrace all heaters placed within a room or hall to warm the air nlrcady in the room. Indirect Radiating Surfaces embrace heating surfaces placed outside the rooms to be heated, and should only be used in connection with some system of ventilation. There are two distinct modes of indirect radiation, — one where all the heating surface is placed in a chamber having one side open to the atmosphere; and a fan located on the other side of the room draws the air through the radiating surfaces, and impels it through tubes or ducts to the various rooms in the building. Such a system is only practical where a thorough system of ventilation is provided, and power to propel the fan night and day. The other and more common method is to provide a separate radiator for each room, located at the bottom of vertical flues, leading to the room. The radiators are generally located in the basement, and provided with tin pipes to conduct the hot air to the rooms. Where the rooms are very large, it will generally be found best to divide the heating surface into two stacks, with separate pipes and registers. Direct-Indirect radiation is a mean between the other two methods. The radiators are placed in the rooms to be heated, asSTEAM HEATING. (m in the first method, and a supply of fresh air brought to them through openings in the outside wall of the room, or through a space under the lower sash of a window. Efficiency of Radiators* — The condensation of one pound of steam at 0, or pressure of one atmosphere to water at 21^°, gives out 005 thermal units. Hence, to determine the amount of heat given out by any radiator in a given time, it is only necessary to determine the amount of water in pounds which the radiator condenses in the same time, and multiply it by 965. The radiator which, under the same conditions of steam-pressure, and volume and temperature of surrounding air, will condense the most water in a given time, is the most efficient. Heating’ by Direct Radiation. — Direct radiation being much more economical than indirect radiation, it will always be much more commonly used for steam or hot-water heating; and in buildings not requiring a great amount of ventilation it offers a nearly perfect mode of heating. Measurement of Radiators. — liadiators are rated, or measured, not according to their size, but according to the amount of heating surface coming in contact with the air. The size of radiator for a given amount of heating surface will depend entirely upon the form or shape of the radiator. The cheapest direct radiator is one formed of wrought-iron pipes (1-inch pipes being generally preferred) placed against a wall, one above the other, and connected with return bends to afford a circulation. Till length of pipe required to make up a given amount of heating surface can easily be determined by the use of the table on p. 504. For rooms in which it is desirable that the heating apparatus shall present a neat appearance, and occupy as little space as possible, some form of upright radiator is generally employed. The most Tig. 1. —Direct Pipe Radiator.612 STEAM-HEATING. usual form is the pipe radiator, of which a small one is shown in Fig. 1. These radiators are formed of a number of short, upright, 1-inch tubes, from 2 feet S inches to 2 feet 10 inches long, screwed into a hollow cast-iron base or box, and are either connected together in pairs by return bends at their upper ends, or else each tube stands singly, with its upper end closed, and having a hoop-iron partition extending up inside it, from the nottom to nearly the top. The radiators are also made circulat in form, either in one piece, or in halves for encircling iron columns. The following table shows the dimensions of 1-inch pipe radiators for different heating surfaces: — TABLE OF VERTICAL PIPE RADIATORS. No. of Rows and Till »Of* Su rface, Le ngth. No. of Rows and Tubes Surface 1 Length "Width of 15ase. in hach Uow. in ^i- Ftp Width of Base. in Each Row. in Sq. Ft.1 i Ft. In. I Ft- III. X 4 4 0 101 X* 8 16 ! 1 6} .2 6 6 1 1)1 r~ 10 20 1 101 8 8 1 6} 12 24 2 01 to 10 1 105 14 2S j 2 S * 12 • > •> 1 16 qo 1 •? 101 -a 16 16 2 105 9 B 18 36 3 Oi -4 20 20 3 6f C - 20 40 3 «I c ° 24 24 4 24 4S 4 28 28 4 lol - — 28 56 4 ioi 32 32 5 64 "3 32 64 1 5 61 r'’ 38 38 6 38 76 6 61 * 8 24 1 64 -r 4 16 ; 0 10« x § 12 36 •> 01 . X X $ 32 1 61 c X 16 20 48 60 3 10J 61 * X 12 16 4S 64 ; 0 1 2 of “4 lot 24 7? 4 of -4 f 1 20 SO j 3 61 i- w JO 28 84 4 10J - ~ — 24 96 ; 4 32 96 5 1 .2 28 112 ; 4 I"! 38 114 6 s 32 128 ! 5 I 61 1 For radiators 35 inches high.STEAM-HEATING. 613 CIRCULAR RADIATORS. Surface, in Sq. Ft. | J ! Size, Diameter. | ! Valves Required. Surface, 1 in Sq. Ft. Size, Diameter. Valves Required. Supply. Return.; Supply. Return ! Ft. In. ! In. In. I In ha Ives to surrou ml IS l n i i 3 * co lum ns. .‘50 l (U ! 3 5 3 * Ft. In. In. In. f)4 l ill 1 i 56 2 •>> hi li 72 2 IBM j 1 l SO • 2 4* H n 102 2 m ! n 1 102 2 »h ■ ■ 130 i 3 2 1 n If 130 3 2 n ■ 100 ! 3 2 i 1 4 160 3 2 ■ ■ Cast-Iron Radiators.— Of late years various forms of east-iron radiators have been introduced for heating by direct radiation. Fig. 2 shows one pattern of these radiators, known as the Dundy Radiator, manufactured by the A. A. Gritting Company of Jersey City.614 STEAM-HEATING. These radiators are very extensively used throughout the United States. The Walker & Pratt Manufacturing Company of Boston also manufacture a very similar radiator. The dimensions of this radiator for various heating surfaces are shown by the table below. The dimensions are for radiators 30 inches high. For radiators 30 inches high, the heating surface will be one-sixth less. DIMENSIONS OF THE BUNDY LOOP CAST-IRON RADIATOR, 36 INCHES HIGH. Single Row of Loops. No. of Heating Surface. Length. Width. ! No. of Heating Surface. Length. Width. *_jOops. Sq. Ft. Ft. In. In. j Luu|)S. Sq. Ft. Ft. In. Iu. 3 9 1 1 II H ; 13 39 3 9 | 64 4 12 1 4 14 42 4 0 5 15 1 7 «4 6 i 15 45 4 3 64 6 18 1 ios 16 48 4 64 64 7 21 o i 64 18 54 5 1 4 8 24 2 M 64 i 20 60 5 7 64 9 27 2 8 64 22 66 6 M 64 10 30 2 11 64 24 72 6 8 6^ 11 33 3 3 64 26 78 7 2 64 12 36 3 5 64 Two Rows of Loops. 6 18 1 1 104 24 72 3 6 104 8 24 1 44 104 26 78 3 9 104 10 30 1 8 104 28 84 4 1 104 12 36 1 11 m 30 90 4 34 10.V 14 42 2 2 if! 32 96 4 64 104 16 48 2 5 104 40 120 5 7 104 18 54 2 8 104 44 132 6 1 104 20 60 2 11 104 48 144 6 8 104 22 66 3 S 104 52 156 7 01 ^5 10 V Three Rows of Loops. 9 27 1 1 14 27 81 2 8 14 12 36 1 44 14 30 90 3 0 14 15 45 1 7 14 OO OO 99 3 3 14 18 54 1 11 14 36 108 3 6 14 21 63 2 2 14 39 117 3 10 14 24 72 2 5 14 45 135 4 4 14 Four Rows of Loops. 16 48 I 1 5 18J 72 216 5 J 18J 20 60 ! 1 7 188 SO 240 5 74 11 24 72 | 1 11 ml 88 264 6 14 m 40 120 1 3 0 m 96 288 6 8 18gSTEAM-HEATING. 615 CIRCULAR RADIATORS, 36 INCHES HIGH. No. of Loops. Heating Su rface. Sq. Ft. Outside Diameter of Base. Ft. In. No. of Loops. Heating Surface. Sq. *'t. Outside Diameter of Base. Ft. In. 10 15 20 •>•> 30 45 60 66 1 1 1 1 4 11 26 31 50 72 78 93 150 216 o 2 2 3 a * n 65 n In Halves to Encircle Columns. No. of Heating Outside Inside | No. of Heating Outside Inside Surface. l)iam. Diam. Loops. Surface. Diam. Diam. Sq. Ft. Ft. In. Ft. In. Sq. Ft. Ft. In. Ft. In. 26 78 2 2 0 9 34 10*2 2 6 '5 1 H * 1 50 150 3 0 1 Ik Fig. 3 shows a form of extended-surface cast-iron radiator, manufactured by Ingalls & Kendricken of Boston, which is largely in use In Massachusetts. The table following shows the dimensions of the radiator.GIG DIMENSIONS OF CAST-IRON RADIATORS DIMENSIONS OF CLOGSTON S PATENT CAST-IRON RING RADIATOR. Manufactured by Ingalls Be Kenduicken, Boston. A rra nged for heating with No. Sur- of rrubes in face in Length rows. ' each row. sq. feet. 1 x 4 20 ft. in. 1 8 in. 0 1 x 5 25 2 0 1 1 X (5 ;>() 2 4 0 1 x 7 35 2 8 0 1 X 8 40 3 0 0 **• o 1 X 9 45 3 5 0 1 X 10 50 3 9 0 To 1X11 55 3 11 0 •—* ji 1 X 12 00 4 4 0 l X j:3 05 4 S 0 1 X 14 70 5 1 0 1 X 15 75 5 5 0 t X 10 80 5 0 0 igh or low pressure steam. ! No. \ of rows. Tubes in each row. Surface in sq. feet. f.e: gth. ft. in. 9 X 4 40 1 8 9 X 5 50 • 9 0 9 X 0 00 2 4 2 X 2 70 9 8 2 X 8 80 3 0 o 9 X 9 90 3 5 9 2 X 11 110 o f > 11 p 9 X 12 120 4 4 9 X 13 130 4 8 2 X 14 140 5 1 2 X 15 150 5 5 1 2 X 10 100 5 9 The heating-surface given above is for radiators 36 inches high. 30 inches high, take four-fifths of the above. For radiators Width.STEAM-HEATING. 017 Rules for determining: Direct Radiating Surface required for heating various classes of rooms and buildings. The common practice of determining the direct radiating surface required in heating, is to allow one square foot of radiating surface to a certain number of cubic feet to be warmed. The following proportions may be considered as an average of those recommended by different engineers and experts: — For dwellings, cold or exposed rooms, 1 foot heating surface to •% cubic feet; for dwellings, ordinary rooms, 1 foot heating surface to (50 or 70 cubic feet; for dwellings, warm, sunny rooms, 1 foot heating surface to 75 cubic feet; for stores, wholesale, 1 foot heating surface to 125 cubic feet; for stores, retail, 1 foot heating surface to 100 cubic feet; for offices, 1 foot heating surface to 75 cubic feet; for churches and audience-rooms, 1 foot heating surface to 125 to 150 cubic feet; for factories and workshops, 1 foot heating surface to 200 cubic feet. City houses require less heat than country houses, and brick houses less than wood. Upper rooms require less heat than those on the ground floor. Mr. William J. Baldwin, in his excellent work on “ Steam-Heating for Buildings,” 1 recommends the following rule, which he has used for several years, and which is not wholly empirical: — “Divide the difference in temperature between that at which the room is to be kept, and the coldest outside atmosphere, by the difference between the temperature of the steam-pipes and that at which you wish to keep the room; and the product will be the square feet, or fraction thereof, of plate or pipe surface to each square foot of glass, or its equivalent in wall-surface.” Tlie equivalent glass surface is found by multiplying the superficial area of the walls in square feet by the number opposite the substance in the following table, and dividing by 1,000 (the value of glass). The result is the equivalent of so many square feet of glass in cooling power, and should be added to the window surface. TABLE OF POWER OF TRANSMITTING HEAT OF VARIOUS BUILDING SUBSTANCES, COMPARED WITH EACH OTHER. Window Glass...........................1,000 Oak and Walnut........................... 66 White Pine............................... 80 Pitch Pine...............................100 Lath and Plaster..................75 to 100 1 Published by John Wiley & Sons of New York.618 STEAM-HEATING. Common Brick (rough)................... 200 to 250 Common Brick (whitewashed)...................200 Granite or Slate.............................250 Sheet-Iron........................... 1,030 to 1,110 It must be distinctly understood that the extent' of heating surface found in this way offsets only the windows and other cooling surfaces it is figured against, and does not provide for cold air admitted around loose windows, or between the boarding of poorly constructed wooden houses. These latter conditions, when they exist, must be provided for by additional heating surface. Examjilr. —What amount of heating surface should be supplied to the sitting-room of a wooden dwelling with two outside walls, one 11 feet by l) feet high, and the other 15 feet by 9 feet; the total window area being 54 square feet, the external temperature frequently being at 0 F., and the steam never exceeding 5 pounds pressure ? yliis.1 — Temperature of room, 70° — 0° = 70° ; temperature of steam-pipes at 5 pounds, 22S — 70° = 15S; 70 -r 158 = .443, ora little less than onc-half a square foot of heating surface to each square foot of glass or its equivalent. Area of outside walls = 14 X 9 = 120 + 15 X 9 = 126 + 135 = 201. Subtracting the glass area, 54, we have 207 square feet of lath and plaster. 207 x 100 “ 20,700 54 X 1,000 = 54,000 1,000)74,700 Equivalent glass area = 74. Multiplying this by .443, we have 33 as the number of square feet of radiating surface required to warm the room, or 1 foot of surface to 58 cubic feet of air-space. In practical work, it is well to determine the heating surface by both of the methods given, and then use the larger quantity. There can never be any bad results from having an excess of heating surface, while a deficiency will always result in cold r >oms in extreme cold weather. Direct-Indirect Radiation. The only difference between this method of heating and the direct method is, that external air is introduced into the room m such a way that it shall come in contact with the radiator, and, 1 It should be noticed that this proportion does not depend upon the sii f the room, but only upon the climate, pressure of the steam, and desired temperature of the room.STEAM-HEATING. 619 becoming heated, circulate through the room; and, unless other means are provided, pass out through the cracks around the doors and windows. There are several methods of arranging the radiators and cold-air inlets, although nearly all require that the radiator shall be located against an outside wall. A very simple and efficient method, where the radiator has an open base, is to set the radiator 6 inches or more from the floor, on blocks, with a galvanized iron or wooden pipe to convey the- air from a register in the outer wall under the radiator. The pipe should be provided with a damper, so that it can be closed when desired. Another method is, to place a sort of metal hood back of the radiator, with the air-duct connected with the centre of it, thus forcing the air up against the radiator. This, however, is objectionable in many cases, on account of appearance, and the collection of dust and dirt. One of the best methods with which the author is acquainted is that shown in Fig. 4. It consists of a compound coil radiator, enclosed in a box of either iron, marble, or wood, lined with tin, and provided with registers at the top for the escape of the heated air. The cold air enters through a hollow iron sill placed above the wooden sill of a window, down back of the radiator, through a galvanized iron pipe, to the space under the radiator. The cold-air inlet is provided with a damper, so that it can be closed; and looqiiudiysJX-iei/aiiorjofCod. registers are also placed at ^IG- 4- the base of the radiator casing, so that, in very cold weather, the cold-air inlet may be partially or wholly closed, and the air allowed to circulate through the bottom register, up through the radiator, and out of the top registers. Indirect Radiation. Heating by indirect radiation is, as has been previously stated, accomplished by two methods; the more general method being to have separate radiators for each room, located in the cellar or620 STEAM-HEATING. basement, incased with metal or wood lined with tin, and provided with a fresh-air inlet, and tin pipe to convey the hot air to the room to be heated. The other method is, to provide one cold-air inlet for the whole building, and place a large coil of steam-pipes behind it, so that all the air entering the building must pass through this coil. Such a method can only be used in connection with fan-ventilation. Fig. 5 shows the usual method of casing indirect radiators. The casing is generally of wood lined with tin, or of sheet-metal. The former is best when the cellar is to be kept cool, as there is a greater loss by radiation and conduction through metal cases; otherwise metal is best, as it will not crack, and, when put together with small bolts, can be removed to make repairs, without damage. The boxes should be fitted with a door on one of the sides, and the cold-air pipe should always be provided with a damper. The vertical air-ducts are usually tin Hues built into the wall when the building is going up. Sometimes they are only plastered;STEAM-HEATING. 621 but round, smooth metal linings with close joints give much the best results. The cross-section of an air-duct should be comparatively large, as a large volume of warmed air, with a slow velocity, gives the best result. There should be a separate vertical air-duct for every outlet or register. In branched vertical air-ducts one is generally a failure. The heated air from one heater may be taken to two or more vertical air-ducts, when they start directly over it; but one should not be taken from the top and the other from the side, or the latter will be a total failure, unless the room to which the flue runs is exhausted; i.e., the cold or vitiated air of the room is drawn out by a heated flue or otherwise. Inlet .or cold-air ducts are best when there is one for every coil or heater. Sometimes only one large-branched cokl-air duct is used, but this system will give trouble unless all the rooms are ventilated by forced ventilation. The Radiators. — For indirect radiation, a form of radiator is employed different from those used for direct heating. In this method the desideratum is, to have as many feet of heating surface in as little space as possible; appearance being of no importance. The earliest form used, and which is still used to a great extent, is the pipe-coil radiator, in which a coil of pipes connected at the ends with return bends is used for the radiator. This gives a much larger surfaoe for the same space than the vertical pipe radiator, and can be easily made by any steam-fitter. Fig. 6 represents an improved form of indirect pipe radiator, made by the Walworth Manufacturing Company. The regular sizes of this radiator are as follows: — No. of Tubes. Square Feet. Length of Base. Ft. In. Width of Base. Inches. 4X4 16 0 91 4X6 24 1 l£ 4X8 32 1 bk 4 x 10 40 1 9A 4 x 12 48 2 n 4 x 16 64 2 s 4 x 20 80 3 7f 4 x 24 96 4 u 4 x 28 112 4 91 (5-22 STEAM-HEATING. Fig. 7 represents six sections of Gold’s pin indirect radiator, manufactured by the II. B. Smith Company, Westfield, Mass. This is a cast-iron radiator, which is very extensively used throughout the country. As [here is now no patent on this radiator, and it is comparatively cheap, it is manufactured by many different companies. Fig. 7. — Gold’s Fin-Radiator. The radiator as made by the II. H Smith Company is made in sections of nominally 10 square feet of heating surface to a section, but actually 8.37 square. Each section is 6^ inches high, 41 inches long, and 3 inches wide, and contains 912 pins, each pin having a base of | inch, a top of -J inch, and a length of inch; the pins being in staggered rows, as shown in Fig. S. To find the floor-space for any number of sections, allow 3 inches for the width of each section, plus i inch for each outside section, and till thickness of the box twice. Fig. 8 represents six sections of C'logston’s patent cast-iron indirect radiator, manufactured by Ingalls & Kendricken of Boston. Similar forms of radiators are used by other firms. This particular pattern is made in sections 42 inches long, 8 inches high at the centre, and 4-g inches wide; each section containing 16 square feetSTEAM-HEATING. 623 few years a new of heating surface. The same firm also manufacture a smaller size of the same radiator, which is 30 inches long, 8 inches high, and 4i inches wide, and contains 10 square feet of heating surface. Each stack of indirect radiators should have a straightway valve in the supply and return pipes, and be supplied with an automatic air-valve, as shown in the cuts. Compound Coil Heaters.— Within radiator, called the compound coil heater,1 has been placed on the market. This is an extended surface radiator, made by winding coils of square wire of strong elastic tension around the 1-inch pipes of common pipe radiators. These radiators have been carefully tested in comparison with other radiators, and by actual work in heating, and have proved in all cases equal to any other radiator made. They are admirably adapted for indirect radiation, and the manufacturers claim that they are the cheapest radiator per foot of surface on the market. Fig. i) shows a section of pipe with the wire coiled around it. They are sold in boxes lined with tin, which are the only casing needed; hence they come all ready to put up. These heaters, when cased, are all 26 inches high, and 49 inches long over all. The width varies according to the amount of heating surface, as follows : — Heating Surface. Sq. Ft. Width over allj including Casing. In. i j Heating i Surface. ! Sq. Ft. i i ... Width over all, including Casing. ; In. 1 Heating Surface. Sq. Ft. Width over all, including Casing. In. 24 11 64 231 104 32 13* 72 112 38* 40 16 80 23S 120 41 48 B i 88 31 128 43* 56 21 96 33| i> 1 Manufactured by Edward E. Gold Sc Co. of NV .v York.STEAM-HEATING. G24 Nearly all indirect radiators can be used either for steam or hot water; and for this reason it is often advantageous to heat dwellings, etc., entirely by indirect, radiation, in which case the apparatus may be used for heating by hot water in moderate weather: and, by drawing off the water in cold weather, the pipes and radiators may be filled with steam. This method is now largely employed in first-class city houses. Rules for compiMing Indirect Heating1 Surfaces. It is quite a common custom among steam-fitters to double the direct radiating surface for indirect radiation, but this is an exceedingly loose method. In wanning by indirect radiation, a fresh supply of air is constantly passing over the radiator, and no air is heated twice. The heated air usually enters the room at from 110° to 130° in hot weather, and, coming in contact with walls, windows, furniture, etc., is quickly cooled to the desired temperature. It is therefore evident, that, if we can determine the amount of air to be warmed, and, by experiments, the quantity of air that one square foot of indirect radiator will heat under certain conditions, we can easily determine the radiating surface required. I3y careful study of the records of various experiments made on indirect heating, and by certain fundamental principles in steamheating, the author has computed the table following, showing the quantity of air which one foot of indirect radiating surface will warm in an hour, at various steam-pressures, and from 0 and 10° F. Divide the quantity of air to be heated per hour by the correspondin'! number in the table, and the result will be the amount of indirect radiatin'/ surface required in well-built brick buildings, and in which the window surface is not more than -f„ the cubic contents of the rooih. Where the window surface exceeds this proportion, increase the radiating surface from 10 to 20 per cent. For wooden buildings also, add 10 per cent. The numbers in the columns under “ Forced Draughtshould not be used unless the air in the room to be heated is changed at least six times an hour; and the quantity of air should never be taken at less than four times the cubic contents of the room. If the external temperature is liable to be at 0° for any length of time, the fourth and fifth columns should be used. The second and third columns are intended for comparatively warm climates.STEAM-HEATING. 625 QUANTITY OF AIR WARMED PER HOUR BY ONE SQUARE FOOT OF INDIRECT HEATING SURFACE, WITH NATURAL OR FORCED DRAUGHTS. Cubic Feet op Air Warmed per Hour. Steam Pressure above Atmosphere. 10° to 100° F. j 0° to 120° F. Natural Draught. Forced Draught. Natural Draught. Forced Draught. Lbs. Pipe and Pin. Pin. Pipe and Pin. Pin. 0 150 251 125 208 3 160 267 133 223 5 165 276 138 229 10 177 296 148 246 20 198 330 165 275 30 212 353 177 294 60 245 408 204 340 Example II, —As an example of indirect heating, we will take the same room as in Example I.: viz., room 15' X 14' X 9', with 54 square feet of window area; steam-pressure, 5 pounds; location, Massachusetts; wooden house. An$. —Cubic contents = 15 X 14 X 9 = 1,890. Multiplying this by 4, we have 7,560 cubic feet of air to be heated per hour. Dividing by 138, taken from column 4, we have 54 as the number of square feet of heating surface required to heat this amount of air. As the building is of wood, and the glass area exceeds of the cubic space, wl had better increase the heating surface 10 per cent, making it 60 square feet. Example III,—What should be the indirect heating surface in a schoolroom 24 X 32 X 12 feet, where the air is changed six times an hour; brick building, situated in Northern States; steam pressure, 5 pounds. Am. —24 x 32 x 12 = 9,216. Multiplying by 6, we have 55,296. Dividing this by 229, we have 242 square feet as the required heating surface. If the room had only natural ventilation, we would multiply the contents by 4, and divide by 138; and we have 260 square feet. Radiators are always more effective, the greater the quantity of air passing over them. Indirect Radiation, with Plenum Ventilation.— The plenum system of ventilation is produced by forcing warm, fresh air into all the rooms, and by causing a pressure slightly in excess of that of the external atmosphere, forcing the impure air from the room.626 STEAM-HEATING. This system requires that the whole air-supply of the building shall enter at one point, where it must pass through a large steam radiator, generally a stack of one-incli pipes, and from thence into one large duct, with branches to the various rooms, or into a plenum chamber in the cellar, from which it passes upward, through ducts provided for the purpose, into the rooms above. If the heated air passes directly into a main air-shaft, with branches to the various rooms, it must be heated to required degree before entering the duct, by the single large radiator referred to ; but if the air passes into a plenum chamber, it is generally only heated to about (50° by the main radiator, and smaller indirect radiators are located at the foot of the ducts leading to the rooms, to give the air entering the rooms the desired temperature. The latter is much the better way for large buildings, especially theatres, concert halls, churches, etc. In either case a fan will be required, which must be located just behind the large steam radiator, to draw the air through it, and produce the xfienum. Steam-Boilers. The capacity of steam-boilers for generating steam is generally designated by the number of horse-povoer of the boiler. Strictly speaking, there is no such thing as “horse-power” to a steam-boiler, as it is a measure applicable only to dynamic effect. But, as boilers are necessary to drive steam-engines, the same measure applied to steam-engines has come to be universally applied to the boiler, and cannot well be discarded. At tlie present time a horse-power is generally measured by the evaporation of 30 pounds of water per hour, at TO pounds pressure, from feed-water at 100°. For heating purposes it is more convenient to designate boilers by the square feet of heating surface which they contain. One square foot of heating surface in one form of boiler may, however, be much more efficient than in another style; and the value of a foot of heating surface must be determined by experiment. The following table gives an approximate list of square feet of heating surface per horse-power in different styles of boilers; the rate of combustion of coal per hour, per square foot of fire surface, required for that rating; the relative economy, and the rapidity of steaming: —STEAM-HEATING. 627 Type op Boiler. O ^ a> . o oO m Coal for each Sq. Ft. Relative Economy. Relative Rapidity of steaming. Authority. Water-tube .... 10 to 12 0.3 1.00 1.00 Isherwood. Tubular 14 to IS 0.25 0.91 0.50 << Flue 8 to 12 0.4 0.79 0.25 Prof. Trowbridge. Plain Cylinder. . . 6 to 10 0.5 0.69 0.20 « Locomotive .... 12 to 16 0.275 0.85 0.55 Vertical Tubular . . 15 to 20 0.25 0.80 0.60 In tubular boilers, 15 square feet of heating surface is generally taken as a horse-power. A horse-power in a steam-engine, or other prime mover, is 550 pounds raised 1 foot per second, or 33,000 pounds 1 foot per minute. For determining the capacity of a boiler for supplying a given amount of radiating surface, allow one square foot of boiler surface to from 7 to 10 square feet of radiating surface; the proportion depending upon the nature of the radiating surface and the efficiency and size of the boiler. Small boilers for house-use should be much larger proportionately than large plants. In average buildings in the Northern States, where the building is entirely heated by direct radiation, one square foot of surface in a horizontal tubular boiler, well set, and with the supply and return pipes properly run, will supply 8 square feet of radiating surface. If all indirect radiation is used, this number should be reduced to 6. Classes of Boilers. — There are a great many kinds of boilers manufactured for heating purposes, and especially for heating dwelling-houses. For dwellings, it is desirable that the boiler shall be safe, provided with automatic dampers, safety-valves, etc., and shall be as simple as possible, and designed to utilize the largest possible percentage of the heat generated by combustion. For heating large buildings, either a tubular or sectional boiler is generally employed. The former is so common as hardly to need description. It consists of a wrought-iron cylinder with closed ends, with the lower half filled with wrought-iron tubes, which pass through the ends, and are welded to it. When set and ready for use, the boiler is filled to a point a little above the highest row of tubes ; the boiler is set so that the products of combustion shall pass under the boiler, and back again through the tubes to the front of the boiler, from whence they pa#$ to the chimney.628 £ T E AM-H EAT IN G. Hence the heating surface in a horizontal tubular boiler consists of one-half the area of the shell and ends, and the total external area of the tubes. The heating surfaces for the various standard sizes manufactured by Kendall & Roberts, of Cambridge, Mass., are given in the table on pp. (133, 634. These surfaces would also apply to boilers of the same dimensions and number of tubes of any other manufacture. Upright tubular boilers are filled with tubes in the same way. Sectional Boilers are generally made of cast-iron, each section being a boiler by itself. The steam is collected in a common wrought-iron drum, and returned to another drum. The advantage of these boilers is, that no serious explosion can result from them; as, should an explosion occur, it would probably be confined to not more than two sections, which in most boilers can be easily replaced. These boilers are especially adapted to schools, churches, etc. Supply and Return Pipes. — The main supply-pipe should be not less than 4 feet above the water-line of the boiler in mediumsized buildings; and In buildings covering a larger area, the height should be as much more than this as- it is practical to make it. Where the condensed water is returned to the boiler, or where low pressure of steam is used, the diameter of the main in inches should be equal to one-tenth, of the square root of the radiating surface supplied. If the mains are not suitably covered with nonconducting material, their surface should be added to the radiating surface. Example. — What should be the size of main to supply 400 feet of radiating surface, itself included? dii.v.—y400 = 20. Divide by 10, and we have 2 inches as the diameter of our main. lleturn-pipes should be at least i{ inch in diameter, and never less than one-half the diameter of the main, — longer returns requiring larger pipe. A thorough drainage of steam-pipes will effectually prevent all cracking and pounding noises therein. Boss of Heat from Steam-Pipes.1 The following table shows the loss of heat from steam-pipes, naked, and clothed with wool or hair felt of different thicknesses. Steam pressure, 75 pounds. External air, 60°. 1 From Steam. Babcock & Wilcox Company, Yew York and Glasgow.STEAM-HEATING 629 0D O JS o Outside Diameter of Pipe , without Felt. hH 2 Inches I 4 Inches 6 Inches 1 Inches 12 Inches t£ Diameter. Diameter. Diameter. Diameter. i Diameter. ^ 31 cL cL 3 O ! Units per H ® B j >3 H Units per II CD X Sì) 'Z, W X MW £ 3 ! o n a g U~t O _I m e ° .— 3 = i^=! ■ g U-t o S o * 1 .2 H . MB ” . o * * OC-t-> .3 !*" ., CD o w * O j I O Gh CÇ O w O 1—( cc O X- O ^ Z E-1 1-1 * 1 \ Tl Js, H & K 1 i K Phh 0 219.0 l.Ooj 132 390.S 1.00 75 624.1 1.000 46 729.8 1.000 ! 40 1077.4 1.000 26 i 4 100.710.46 2S8 lS0.9l0.46 160 _ - - - - 1 - ! - - - I 5 65.7i0.30l 441 117.2'0.30 247 187.2 0.300 154 219.6 0.301 ! 132 301.7 0.280 92 1 43.8 0.20l 662' 73.910.18 392 111.0 0.178 261 128.3 0.176j 225‘ 185.3 0.172 157 2 28.4 0.13; 1020; 44.7j0.ll 64S 66.2 0.106 438 75.2 0.103! 385; 98.0 0.091 294 4 19.8,0.09 1464! 28.1 0.07 1031 41.2 0.066 703 46.0 0.063 i 630j 60.3 0.056 486 6 " Ì ‘ "1 23.4 0.06 1238 33.7 0.054 860 34.3 0.047 j 845 ^B^B 45.2 0.042 642 There is a wide difference in the value of different substances for protecting from radiation, their value varying nearly in the inverse ratio of their conducting power for heat, up to their ability to transmit as much heat as the surface of the pipe will radiate, after which they become detrimental rather than useful as covering. This point is reached nearly at baked clay or brick. Table of the conducting power of various substances, from Peclet: — Substance. Conducting Power. 1 Substance. Conducting Power. Blotting Paper .... 0.274 1 Wood, across fibre . . 0.83 Eiderdown 0.314 I Cork 1.15 Cotton or Wool, any ( 0.323 j Coke, Pulverized . . 1.29 density, t India Rubber .... 1.37 Ilemp, Canvas .... 0.418 Wood, with fibre . . 1.40 Mahogany Dust . . . 0.523 | Plaster of Paris . . . 3.86 Wood Ashes 0.531 1 Baked Clay .... 4.83 Straw | Glass. ...... 6.6 Charcoal Powder . . . 0.636 1 Stone 13.68 A smooth or polished surface is of itself a good protection; polished tin or Russia iron having a ratio, for radiation, of 53 to 100 for cast-iron. Mere color makes but little difference. Ilair or wool felt has the disadvantage of becoming soon charred from the heat of steam at high pressure, and sometimes of taking fire therefrom. This has led to a variety of “cements” for cover-630 DRYING BY STEAM. Ing pipes, composed generally of clay mixed with different sub-stances, as asbestos, paper fibre, charcoal, etc. A series of carefnl experiments, made at the Massachusetts Institute of Technology in 1871, showed the condensation of steam in a pipe covered by one of them, as compared with a naked pipe and one clothed with hair felt, was 100 for the naked pipe, 67 for the “ cement ” covering, and 27 for the hair felt. Table of relative value of non-conductors, from experiments by Charles E. Emery, Pli.D.: — Non-Conductor. Value. Non-Conductor. Value. Wool-Felt 1.000 Loam, dry and open . . 0.550 Mineral Wool No. 2 . . . 0.832 Slacked Lime .... 0.480 “ with Tar, 0.715 Gas-House Carbon . . 0.470 Sawdust 0.680 Asbestos 0.363 Mineral Wool No. X . . . 0.676 Coal Ashes 0.345 Charcoal 0.632 Coke in Lumps .... 0.277 l’ine Wood, across fibre . 0.553 Air Space, undivided 0.136 “ Mineral wool,” a fibrous material made from blast-furnace slag, is a good protection, and is incombustible. Drying- by Steam.1 There are three modes of drying by steam: 1st, by bringing wet substances in direct contact with steam-heated surfaces, as by passing cloth or paper over steam-heated cylinders, or clamping veneers between steam-heated plates; 2d, by radiated heat from steam-pipes, as in some lumber-kilns and laundry drying-rooms; 3d, by causing steam-heated air to pass over wet surfaces, as in glue-works, etc. The second is rarely used except in combination with the third. The first is most economical, the second less so, and the third least. Under favorable circumstances it may be estimated that one-horse power of steam will evaporate 24 pounds water by the first method, 20 by the second, and 15 by the third. The philosophy of drying or evaporating moisture by heated air rests upon the fact that the capacity of air for moisture is rapidly increased by rise in temperature. If air at 52° is heated to 72°, its capacity for moisture is doubled, and is four times what it was at 32°. The following table gives the weight of a saturated mixture of air and aqueous vapor at different temperatures up to 160°, — 1 From Steam. Babcock & Wilcox Company.DRYING BY' STEAM. 631 the practical limit of heating air by steam, — together with the weight of vapor, in pounds and percentage, and total heat, with the portion thereof contained in the vapor: — SATURATED MIXTURES OF AIR AND AQUEOUS VAPOR. i.d B s £ Es 9 O' I U«W © x a> o I ■ :ui c ° B H ! Per Cent of Water in Mixture. Heat Units in 100 Cub. Ft. of Mix-i ture. Per Cent of Ileat j in Vapor. 1 ■ 1 ^ j b* Ìd 8« o'Si |6j= ■eA O t>.= 1 Weight of Water in 100 Cub. Ft. of Mixture, in Lbs. | Per Cent of Water | in Mixture. ■ § «»-i 2° III 1 K o 11 Ofl £.2 35 8.004 0.034 0.42 42.8 86.69 ‘ 100 6.924 0.283 4.08 422.0 74.58 40 7.920 0.041 0.52 59.8 76.59 ■ 105 6.830 0.325 4.76 474.7 76.22 45 7.834 0.049 0.62 77.7 68.98 : no 6.741 0.373 5.23 533.9 77.88 50 7.752 0.059 0.76 97.6 66.29 115 6.650 0.426 6.41 599.1 79.52 55 7.688 0.070 0.91 118.3 64.58 , 120 6.551 0.488 7.46 672.4 81.14 60 7.589 0.082 1.08 140.1 64.31 . 125 6.454 0.554 8.55 750.5 82.62 65 7.507 0.097 1.29 164.9 64.76 • 130 6.347 0.630 9.90 839.4 84.13 70 7.425 0.114 1.49 189.7 66.21 135 6.238 0.714 11.44 936.7 85.57 75 7.342 0.134 1.79 221.6 66.74 | 140 6.131 0.806 13.14 1042.7 86.89 80 7.262 0.156 2.15 253.6 68.02 145 6.015 0.909 15.11 1160.6 88.18 85 7.178 0.182 2.54 289.7 69.66 I 150 5.891 1.022 17.33 1288.4 89.39 90 7.108 0.212 2.98 330.2 71.19 155 5.764 1.145 19.88 1427.4 90.53 C5 7.009 0.245 3.50 373.4 72.87 160 5.679 1.333 23.47 1638.7 91.93 By inspection of above table, it will be seen why it is more economical to dry at the higher temperatures. The atmosphere is seldom saturated with moisture, and in practice it will be found generally necessary to heat the air about 30° above the temperature of saturation. The best effect is produced where there is artificial ventilation, by fan or by chimney, and the course of the heated air is from above downwards. Temperature of Fire. By reference to the table of fuels (p. 602), it will be seen that the temperature of the fire is nearly the same for all kinds of combustibles under similar conditions. If the temperature is known, the conditions of combustion may be inferred. The following table, from M. Pouillet, will enable the temperature to be judged by the appearance of the fire : —632 TEMPERATURE OF FIRE. Appearance. I Temperature F. Appearance. Temperature F. Red, just visible .... 977° Orange, deep .... 2010° “ dull 1290° | “ clear .... 2190° “ Cherry, dull.... 1470° i White heat 2370° “ “ full .... 1600° “ bright .... 2550° “ “ clear . . . 1830° “ dazzling.... 2730“ To determine temperature by fusion of metals, etc., — Substance. Temperature F. Metal. Temperature F. Metal. Temperature F. Tallow, 92° Bismuth, 518“ Silver, pure, 1S30“• Spermaceti, 120“ Lead, 630“ Gold Coin, 2156“ Wax, white, 154° Zinc, 793° : Iron, Cast, rued., 2010“ Sulphur, Ml Antimony, 810“ 1 Steel, 2550“ Tin, 455° Brass, 1650® Wrought-Iron, 2910°634 STEAM HEATING. — BOILERS UPRIGHT TUBULAR BOILERS. Manufactured by Kendall & Roberts, Cambridgeport, Mass. Diameter of shell. Height of shell. Number of tubes. Diameter of tubes. Length of tubes. Heating surface. Horse- power. ins« ft. in. in. ft. in. ft. IS 4 0 40 n 0 33 — 18 4 6 40 0 39 — — 18 5 0 40 H 0 45 - - 24 5 0 25 2 3 0 52 31 24 5 6 25 2 4 0 58 24 6 0 25 2 4 0 04 31 30 5 0 45 2 3 0 80 5 30 5 6 45 2 3 0 90 0 30 6 0 45 2 4 0 102 6§ 30 6 6 45 2 4 (5 114 71 30 7 0 45 2 5 0 125 36 6 0 65 2 4 0 I4."> n 36 6 6 65 2 4 6 102 10f 36 7 0 65 2 5 0 180 12 36 7 6 65 2 5 0 195 13 36 8 0 65 *> 0 0 210 14 42 0 6 100 2 4 0 240 10 42 7 0 100 5 0 208 IS 42 i 6 100 2 5 0 293 m 42 8 0 100 2 6 0 318 21 48 i 0 120 2 5 0 320 21 48 7 6 120 2 5 6 350 23 48 8 0 120 2 0 0 380 25 54 8 6 186 0 6 000 40 54 0 0 ISO 7 0 075 45 54 1) 6 180 2 7 0 720 48 60 10 0 250 2 7 0 07."> 05 60 11 0 250 *> 8 6 IKK) *■ •» 4*> 00 12 0 250 2 9 0 1224 81REGISTERS AND VENTILATORS C> SÔ DIMENSIONS OF REGISTERS AND VENTILATORS, Made by the Tuttle & Bailey Mani facti i:i\<; Company. Size given List as on Opening to admit Body of Register. Extreme Dimensions of Register Face. Depth Reg Closed. of the ister. <)pen. ( >]H'ni'.g to admit 1 roll Bonier. 4| y 6] X 6f B X * 8 13 a>| - 4 X N 4 X s 5] x 9i î lì - 4 X 10 4 X 10 ri '*4 X Di i! - 4 X n 4 X 13 r.ô x 15 1* -4 - 4 X 15 4 X 15 ;>8 x 16] i! oi - \ - 4 X 1$ 4 X 17* r.6 °Ô X 193 m •> 1 -1 - G X 8 R X S] 1 1 X II 13 ■ H x 12* G X 9 61 X 9 1 à X fil If 03 -8 ■ x m G X 10 6} X 10 77 ‘ a X 12 1] 23 H X 14] 6 X 14 6] X 13$ 8 X la7 o> log X D] G X IG 61 x IG 8 X m n Ol 11! X |l G X 1S 6 X 1S 8 X 20 « 0». 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X M6 ““8 12 X 18 m X 18] 14 X 20 4] H X 23 12 X ti m X m 14] X 21 n “4 4 i 17] X Ri 12 X 24 12 X 24 m X |t 4] K X ■ 14 X 14 DI X 14.1 16g X m ©fi H X 20] 14 X 18 14* X 18] R X 20] 9# SI 20] X 24 14 X m X 22 16] X 24] il 4 20] X 27] 10 X 25 m X o-,i “’-a 17] X 273 B ■'a 5j} 22 X 32 IG X IG t@ X 10 m X 1*2 0 4] 00 X 22 IG X 20 IG- X 20] Df X 22] 0 4* 21] X o*. 1 â IG X 24 163 X 24] 1*3 X 27 3 41 H X 30 \ s 20 X 20 20 \ X 20] 221 X 22] K 5] 20] X 26] 20 X 24 20 X 24 22] X 2G 3 g r. 5 "8 20 X 29] 20 X 2G m X 20] 0*2,7 X 2SJ OG *>fì or 1 p* 4 X 3,3] 21 X 29 W X 29 HE —*-,£ x 31 i aro ••*4 ,J4 28 X 3G 24 X 24 24 X 24 HH ) - X 23,] - - BH X 27 27 X 27 X 29] P 34 X 34 27 X 38 27 X 38 î X 40] €1 ■ 34 X 45 m X 30 pi X ÜJ • * °-T * IL 4] 7 3i j y S7] STEAM HEATING.—BOILERS. 633 HORIZONTAL TUBULAR BOILERS. Manufactured by Kendall & Roberts, Cambridge, Mass. o 3 O **H * O Ü 5 ^ jn g Z IBB 'S O O ro aa O t£> s? o V- ~ list — f w T. Z ZJ “ X tx i 7Z ZJ Z/ X 0— w MR B - Z 0 a ^ — Vh £ * X ~ ■ 84 17 188 Q »> 16 7 2432 162 ’ 62 558 14592 84 1(5 188 •> *> 15 7 1 U 2270 151 58 522 13620 78 17 164 0 1 16 HB 2132 142 54 486 12792 78 16 164 1 15 Vs 2011 134 50 450 12066 72 17 140 o o 16 ü 1880 125 48 432 11280 72 1G 140 Q 1 15 3 x 1765 117 44 396 10590 GO 17 117 o .1 10 3 1538 103 40 360 9228 06 16 117 • » •> 15 3 1442 96 36 324 8052 GO 18 65 i l • ) 1 17 3 1074 72 28 232 6444 60 17 92 •) o 16 3 1229 82 32 288 7374 GO 16 92 15 n 1152 77 30 270 6912 GO 15 92 3 14 3 1075 72 28 252 6450 GO 14 92 •> o 13 3 998 67 26 234 5988 54 18 50 Q ! gg / 17 n 900 60 26 2:34 5400 54 17 72 3 16 r> T t> 977 65 26 234 5862 54 16 72 3 15 5 917 61 26 234 5502 54 15 72 •> > 14 Hi 857 57 24 216 5142 54 14 72 •> o 13 ■ 797 53 24 216 4782 54 13 72 *1 12 1 ‘1 it 735 49 24 216 4410 48 17 49 •> • ) 16 5 IS 084 46 20 180 4104 48 16 4!) •> *> 15 D 642 43 18 152 3852 48 15 49 3 14 r 1 It 600 40 18 152 3600 48 14 49 o • > 13 "I1*» 555 37 18 152 3330 48 13 49 •> *> 12 5 1 i) 513 34 18 152 3078 48 12 65 2v 11 5 Ti» 542 36 18 152 3252 42 16 38 •> *> 15 J 5( )8 34 16 144 3048 42 15 38 *) *> 14 IS 4 476 32 16 144 2856 42 14 3S *> ’> 13 441 30 16 144 2646 42 13 38 3 12 1 4 408 27 14 126 2448 42 12 45 2.; 11 J 4 31)0 26 14 126 2340 42 11 45 2} 10 1 '4 355 24 14 126 2130 42 10 45 ol 9 1 4 320 22 12 108 1920 42 9 45 24 S 1 4 285 19 m 108 1710 I m 13 28 3 12 1 4 306 20 12 108 1836 m 12 34 OL 11 _L 4 298 20 12 108 1788 36 11 34 10 i. 4 271 18 10 90 1626 36 10 34 2.1 9 1 4 244 16 10 IX) 1464 36 9 34 2.V 8 J 4 211 14 8 72 12(>6 36 8 34 24 7 i. 4 190 12 8 72 1140 30 9 30 2 8 _L 4 152 10 6 54 912 30 8 30 2 7 i 4 153 8 1 6 54 798 Thickness of heads one-eighth inch greater than thickness of shell. The last three columns added by the author.636 CAPACITY OF PIPES AND REGISTERS ESTIMATED CAPACITY OF PIPES A IMP REGISTERS. ROUND PIPES. 1 Diameter Area in Diameter Area in Diameter Area in of pipe. sq.inches. of pipe. sq.inches. of pipe. sq. inches. , 7 inches 38 12 inches. 113 22 inches. 380 8 “ 50 14 “ 154 24 “ 452 9 1 63 16 1 201 26 | 531 10 1 78 18 “ 254 28 i( 616 11 I 95 20 “ 314 30 “ 707 KECTANOULAR TIDES. Size Area in Size Area in Size Area in of pipe. sq. inches. of pipe. sq. inches. of pipe. sq. inches. 4x8 32 8 x 20 160 12 x 18 216 4 x 10 40 8 x 24 192 12 x 20 240 4 x m 48 10 x 12 120 12 x 24 288 4 x 10 64 10 x 15 150 14 x U 196 G x 10 60 10 x 16 160 14 x 16 224 G x 12 72 10 x 18 ISO 14 x 20 280 G x 16 96 10 x 20 200 16 x 16 256 8 x 10 80 12 x 12 144 16 x 18 2SS 8 x 12 96 12 x 15 180 16 x 20 320 8 x 16 128 12 x 16 192 16 x 24 384 REGISTERS. Size of Capacity in | Size of Capacity in Size of Capacity in opening. sq. inches. opening. sq. inches. opening. sq. inches. 6 x 10 40 10 x 14 93 20 x 20 267 8 x 10 53 10 x 16 107 20 x 24 320 8 x 12 64 12 x 15 120 20 x 26 347 8 x 15 80 12 x 19 152 21 x 29 406 9 x 12 i 2 14 x 22 205 27 x 27 4S6 9 x 14 84 15 x 25 250 27 x 38 6S4 10 x 12 80 16 x 24 256 30 x 30 600 KOI ND REGISTERS. Size of Capacity in Size of Capacity in Size of Capacity in opening. sq. inclies. opening. sq. incites. opening. sq. inches. 7 inches. 26 12 inches. 75 20 inches. 209 8 “ 33 | 14 “ 103 24 “ 301 9 I 42 i 16 “ 134 30 “ 471 10 “ 52 18 I 163 36 “ 679PROPORTIONS POR CONCRETE FOOTINGS. 637 Proportions for Concrete Footings Under Foundations. A good proportion for mixing concrete for footings is 1 barrel of cement, 2 barrels of coarse sand, 4 barrels of broken stone. Brick such as is found in many of the Western States is too soft to be used, though in the East it would answer very well. The proper proportions of sand, cement and gravel should be thrown dry into a mortar box and the water turned on, and the whole quickly and thoroughly worked together, and carried at once to the trenches. Concrete should be well rammed in layers not over 6 inches thick, and the trenches should not be wider than the desired width of the footings. The author considers concrete mixed in this proportion, with Portland cement, preferable to any but heavy dimension stone footing, and on soft soil it should rank first. The proportion of stone to cement is often made as great as 6 to 1.INDEX Air, weight and composition of .... PAGE . . 504 American and Birmingham wire gauges . • 506 Anchor irons for iron beams. . . . . . 4 • • 365 Ancient weights • • 34 Apostles, symbols for the • • 482 Arch-girders, cast-iron, strength of . . . • • 350 Arched roofs, iron • • 420 Arched roof-trusses • • 401 Arches, brick, for floors • • 366 Arches, inverted . .' • • 146 Arches, stability of 185, 191 Architects’ charges, scale of • • 570 Architects, list of noted • • 564 Area of circles, rule ; . . 41 44 44 44 tables 40, 4' ', 49 44 “ irregular polygons, rules for . . • • 39 44 “ regular polygons, rules .... • • 39 “ squares, rectangles, etc., rules for • • 36 “ “ trapeziums, rules • . 39 “ “ trapezoids, rules • • 39 “ “ triangles, rules • • 38 Asphaltum • • . • 548 Baptistery of Pisa • - 566 Bead, moulding • • 469 Beams, iron, cast . • 307 “ iron, wrought . • 282 “ strength of, general principles . . • • 280 “ wooden, strength of • • 307 Bearings of beams • • 312 Bedsteads, dimensions of . . 489 Bells, weight of . • 483 Belly-rod trusses • • 339 Bending-moments . . 250 44 “ examples of • • 251 “ “ graphical method . . . • • 253 44 44 of continuous girders . • . 329 Billiard tables and rooms, dimensions of . • • 489 Birmingham and American wire gauges • • • • • • • 506 639G40 INDEX. Blue-print copies of tracings, to make .... Bluestone, strength of......................... Board measure, table of ... . .... Boiler tubes................................... Bowstring roof-trusses......................... Box-girders.................................... Brest walls.................................... Brick arches for doors......................... Brick piers, strength of . . . ................ Brick walls.................................... Bricks, dimensions of.......................... 44 strength of.............................. Brick-work, strength of........................ Brick-work in drains and wells................. Bricklayers’ memoranda......................... Bridging of floor-beams . ..................... Brooklyn Bridge, the.............. .... Built beams, solid............................. Bureau, dimensions of a........................ Buttresses, stability of....................... Cables......................................... Calendar, the old and new...................... Canterbury Cathedral .......................... Capacity of elm relies, theatres, opera-houses, etc. 44 of cisterns and tanks................... 11 of drain-pipes....................... 44 of freight-cars...................... Carriage-beams................................. Cast-iron arch-girders, strength of............ Cast-iron beams, strength of................... Cast-iron columns, strength of................. Cast-iron pipes................................ Castings, shrinkage in......................... “ weight of............................ Cathedral, Amiens.............................. 44 Canterbury........................... 44 Lincoln........................•. . . 44 Salisbury............................ “ St. Paul’s.............................. 44 York................................. 44 of Chatres........................... 44 of Pisa ... ........................... 44 of Ivhenns........................... 44 of Toledo............................ Cathedrals, English, dimensions of............. Cements, strength of........................... Centre of gravity, definitions, etc. . . . . . 44 44 44 examples....................... Centres for arches............................. Chains, strength and weight of.................. Chimneys, boiler, proportions for . . . . . . 41 brick, rules for....................... TAGE . . . 500 . . . 105 . . . 513 . . . 505 . . . 418 . . . 345 . . . 104 . . . 306 ... 109 148, 149, 150 . . . 511 ... 109 . . . 105 . . . 510 ... 510 . . . 356 ... 574 ... 316 . . . 489 . . . 180 ... 210 ... 30 . . . 566 ... 485 . . . 553 ... 512 . . . 549 . . . 357 ... 350 ... 307 . 226, 227 . . . 502 . . . 572 . . . 572 .. . . 566 ... 566 . . . 506 . . . 566 . . . 508 . . . 505 . . .. 506 . . . 506 . 565, 566 . . . 566 . . . 487 . 165, 171 ... 156 . . . 157 . . . 188 ... 212 ... 475 ... 474INDEX. 641 Chimneys, foundation of.......................... “ general principles of..................... “ wrought-iron.............................. Chords, table of................................. Church of Notre Dame, Paris...................... Churches, capacity of.........................% Circles, area of................................. ** circumference of........................... Circular and angular measure..................... Circular arcs, length of......................... Circular sectors, area of.................... . Circumference of circles, rule................... “ “ “ tables.................... Cisterns and tanks, capacity of.................. Co-eflicient for beams, table of................. Co-eflicient of friction......................... Coin, weight of.................................. Colors of iron caused by heat.................... Columns, cast-iron, caps and bases............... “ cast-iron, strength of.................. “ Keystone, wrought-iron.................... “ monumental, height of..................... ** patent rivetless ........................ “ Thcenix, wrought-iron................... “ strength of............................. “ wooden, strength of....................... “ wrought-iron, strength of..............*. Comparative resistance to crushing of iron aud steel Comparison of thermometers....................... Composition of forces............................ Concrete...................................... Concrete, strength of............................ Cones, surface of . #.................... Construction of mills............................ Consumption of water in cities................... Continuous girders, strength and stiffness of . . . Corrugated sheet-iron............................ Cost of public buildings......................... Cost per square foot of factories . ............. Counter-braces................................... Counter-flashings................................ Croton aqueduct, N.Y...................« . . . . Crushing height of brick and stone............... Crushing strength of materials................... ** strength of stones, bricks, cements, etc. . Cube root, rule for determining.................. “ “ table of.............................. Cycloid, to describe a........................... C$ ma-recta ..................................... Cyma-reversa..................................... Dead load, definition of......................... Deflection of beams.............................. PAGE . . 141 . . 472 . . 476 . . Sii . . 566 . . 4S5 40, 47, 41) 40, 48, 49 . . 30 . 54-57 . . 60 . . 40 . 42-48 . . 553 . . 310 . . 560 . . 29 . . 555 . . 224 . . 223 241, 245 . . 484 . . 246 . . 240 . . 217 . . 218 . . 229 . . 249 . . 554 . . 152 . . 139 . . 165 . . 62 . . 375 . . 559 . . 327 . . 507 . . 551 . . 382 . - 400 . . 530 . . 573 . . 166 . . 217 . . 165 . . 4 84 469 469 127 318INDEX. Deflection of continuous girders . . . 44 of iron beams.................. Details of iron roofs and roof-trusses Dimensions of a barrel............ . 44 of beds ....................... 44 of billiard tables and rooms |* of bricks................... ‘1 of bureaus.................. 44 of drawings for patents . . 44 of English cathedrals . . . “ of horse-stalls............. 44 of iron beams for floors . . 44 of obelisks................. 44 of theatres and opera-houses 4< of the principal domes . . 44 of urinals and water-closets 44 of wash-bowls............... of wash-stands............. Discharge of water..................... Domes, dimensions of................... Drain-pipes............................ Draught Iff chimneys................... Drums and pulleys, speed of............ Egyptian long-measure.................. Elastic cement......................... Ellipse, to describe an................ Ellipsoids............................. Equilibrium, definition of............. Evolution.............................. Excavations, measuring................. Excavators’ ami well-diggers' memoranda Expansion of metals.................... Experiments on brick and tile floor-arches Extrados, definition of................ Eye-bars............................... Factor of safety....................... 44 44 44 for floors............. 44 44 44 for iron beams.... 44 “ 44 for tension............ 44 44 44 for wooden beams . . Fillet, moulding....................... Fire-proof ceilings.................... “ construction................ 44 floors, description of . . . 44 floors, strength of ... . 41 materials................... “ roofs ...................... Fire-proofing iron beams............... 44 4 4 columns............ Fish-joints............................ Flashings.............................. Flat arches for floors................. PAGE . 331 . 326 . 421 . 489 . 4 SO . 4S9 . 511 . 489 . 489 . 487 . 4S9 . 371 . 488 . 486 . 483 . 489 . 489 . 4S9 . 535 . 483 . 511 . 476 . 573 . 33 . 529 . 78 . 61 . 125 4 . 67 . 509 . 556 . 373 . 185 . 202 . 126 ; 354 .281 . 197 . 310 . 469 . 391 . 383 364, 3S5 . 369 383, 3S4 . 386 . 3S8 248, 3S7 . 456 . 530 . 368INDEX. 643 PAGE FI itch-plate girders...................................................336 Floors, solid or mill............................................. 359, 378 “ wooden, loads on.....................................................354 “ “ stiffness of................................................ 361 “ “ strength of........................................... 353, 355 Flow of gas in pipes......................................................477 “ of water...............................................................536 Footing courses...........................................................143 Force, definition of......................................................125 Forces....................................................................152 Foundation walls................................................. 146, 148, 149 Foundations...............................................................130 Framing and connecting iron beams.................................. 305, 363 French plate window-glass, price-list.....................................345 Friction, co-efficient of.................................................560 . Gas memoranda.......................................................... 477 Gas-pipes.................................................................504 “ flow of gas in....................................................477 Geometrical problems.......................................................68 Girders, continuous.......................................................327 “ flitch-plate..................................‘....................336 “ for floors . ....................................................359 ** riveted plate iron...............................................345 Glass for skylights .. .................................................. 547 Granite, strength of......................................................165 Graphical analysis of roof-trusses........................................429 Gravity, centre of....................................................... 128 Grecian long measures................................................ • 34 Grindstones, weight of................................................... 573 Gunter’s chain.........................................................< 25 Hammer-beam roof-tru^ses, analysis of.................................... 443 “ “ description.........................................406 Hardness of v,roods...................................................... 563 Haunches, definition of...................................................185 Hawsers...................................................................210 Headers...................................................................357 Heading courses...........................................................151 Heat and ventilation......................................................470 Heights of columns (monumental)..........................................484 “ of spires........................................................485 “ of towers and domes..............................................481 Hollow brick arches.......................................................367 “ brick partitions........................................... 385, 390 “ tile arches........................................................367 Horse-power.............................................................. 572 Horse-stalls, dimensions of.......................................... • 489 Hydraulics of plumbing....................................................534 Hyperbola, to describe an..................................................83 I Inches expressed in decimals of a foot, table...............................25 Inclined beams, strength of........................'....................286 Intrados, definition of...................................................185 Inverted arches..................................^......................146644 INDEX. Involution......................... Iron roofs and roof-trusses .... Jewish long measures............... Joints, definition of.............. “ for iron-work................. “ in timber work................ Keystone, definition of............ ** depth of, rule and table King post roof-truss............... Kirkaldy’s conclusions on wrought-iron Lead memoranda..................... Lead pipes, weight and strength of . Leaniug Tower of Pisa.............. Lever, the principle of............ Limestone, strength of............. Lincoln Cathedral.................. Line of resistance in masonry . • . List of noted architects........... Live load, definition of........... Load on roofs ..................... Loads, superimposed, on floors . • . Lumber grades in Boston............ Marble, strength of................ Masonry, strength of............... Masonry walls...................... Measure, circular and angular . . . “ cubic ....................• . “ fluid..................... “ shoemakers’............... Measures, Egyptian long............ ** geographical and nautical . “ Grecian long................. “ Jewish long............... “ * Roman long.............. “ . Scripture long.......... Measures of land . . .............. “ of length................ “ of surface ..... I . “ of time.................. “ of value . . . . . . . “ of volume ....... of weight................ “ of weight. Roman . . . . Mechanics.......................... Melting-point of metals............ Mensuration........................ Metric system, the................ “ “ cubic measures . . . “ “ linear measures . . . «* “ liquid and dry measures “ “ surface measures . . u weights. . . .♦ PAGE . 3 . 415 . 34 . 455 . 462 . 455 . 185 188, 189 . 394 . 200 . 490 . 540 . 566 • 155 . 165 . 566 . 1S1 . 564 . 127 . 426 . 353 . 564 165, 173 . 165 . 143 . 30 . 27 27 , . 26 33 , . 26 , . 34 . . 34 . . 34 . . 33 . . 26 . . 25 . . 26 . . 29 . . 29 . . 26 . . 27 . . 34 . . 125 . . 556 . 35-41 . . 30 . . 32 . . 31 . . 32 . . 31 a 32INDEX, 645 PAGE Mill construction......................................................... 375 Mill floors............................................................ 359,37$ Mineral wool...................................................... 562 Miscellaneous information................................................. 489 Modulus of elasticity................................................. 127, 320 “ of rupture...................................................... 127, 281 Moment of inertia..................................................... 257, 278 M oment of resistance ............................................... 257, 279 , Moments...................................................... 128,154 Momeuts, bending............................................................250 Mortars, strength of. ....................................... 165, 170 Mortise and tenon joints................................................... 460 Motion......................................................................125 Mouldings, classical...................................................... 469 Nails, quantity for different kinds of work.................................527 Nails and spikes, size, length, etc.........................................500 Neutral axis.............................................................. 280 Obelisks, dimensions of................................................... 488 Open-timber trusses........................................................ 405 Opera-houses, capacity of........................................... . 485 “ dimensions of................................................ 486 Painting memoranda..........................................................541 Parabola, to describe a......................................................82 Parallelogram of forces................................................... 153 Parthenon, the ........................................................... 564 Piers, brick, strength of..........................................166,169 Piers, stability of....................................................... 178 Piles . . . ............................................................134,135 Pitch of flat roofs.........................................................489 Prank measure............................................................. 521 Plasterers’ rules and memoranda.................................... 528, 529 Plate glass, French, price-list of..........................................543 Plate-iron girders..........................................................345 Plumbing....................................................................533 Polygon of forces...........................................................154 Polygons, definitions of.....................................................35 Porous terra-cotta..........................................................384 Pressure of water...........................'...........................535 Principle of the lever ................................................... 155 Principles of the arch..................................................... 188 Prisms, volume of............................................................63 Properties of water.........................................................557 Proportion of rectangular beams for maximum strength........................311 Purlins, trussed............................................................451 Pyramids, surface of.........................................................62 “ volume of.............................................................65 Queen post roof-truss.......................................................395 Radius of gyration................................................... 257-277 Rafters, hip and jack, length of 94 Relative hardness of woods .................................................563 Relative strength of rectangular beams......................................311 Resistance, line of.........................................................181646 INDEX, PAGE Resolution of forces.......................................................152 Rest.......................................................................125 Retaining-walls.......................................................... 161 Riveted joints.............................................................462 Riveted plate iron girders.................................................345 Rivets, iron, weight of....................................................409 Rods, wrought-iron, strength of............................................204 Roof-trusses, iron.........................................................415 “ theory of..........................................................426 “ wooden.............................................................392 Rooting paper................*...........................................529 “ slate.................................................................529 “ tiles.................................................................531 “ tin...................................................................532 Roofs, iron............................................................... 415 “ mill, description of..................................................380 Roman long measu res ..................................................... 34 ** weight, measures of...................................................34 Ropes, hemp, Manila, tarred, etc...........................................210 “ iron and steel wire, strength of......................................209 “ and cables, measures.................................................. 25 Saints, symbols for the ...................................................482 Sand..................................................................... 134 Sandstone, strength of..............................................165,173 Scale of architects’ charges...............................................570 Scantlings reduced to board measure...................................... 515 Scarf-joints . ............................................................458 School-scats, space occupied by............................................481 Scotia................................................................... 469 Screw-ends.............................................................. . 202 Screw-ends, upset, table of................................................207 Scripture long measures................................................... 33 Seating-space in schools...................................................481 P in theatres...................................................... 480 Shearing, resistance to....................................................213 Shearing-strength of materials........................................... 214 Sheet-iron, black and galvanized...........................................506 Shingles............................................................... 531 Shrinkage in castings.................................................... 572 Signs, arithmetical....................................................... 3 Skew’back, definition of................................................. 185 Slate.................................................................... 530 Slate roofs................................................................o29 Soffit, definition of......................................................1S5 Solid built beams........................................................ 316 Span, definition of .......................................................1$5 Speed of drums and pulleys.................................................573 Spheres, surface of............................................... • • • • 00 “ volume of..........................................................05 Spheroids . . . . ....................................................61,66 Spires, height of........................................................ 485 Springers, definition of..................................................,185G48 INDEX PAGE St. Paul’s Church, London..................................................506 St. Sophia, Constantinople.................................................565 Suspension bridge, Niagara...............................................573 Symbols for the apostles and saints......................................4S2 ble of board measure..................................................513 of bowstring roofs, proportion......................................420 of chords............................................................85 of circles, areas, and circumferences.............................42-54 of circular arcs, length.......................................55, 56 of co-eflieient of friction.........................................560 of crushing-strength of cast-iron posts per square inch.............226 44 of hard-pine and oak posts................ 221, 222 “ 44 of materials...................................165 “ of round cast-iron columns.....................227 “ “ of wrought-iron struts per square inch, 232, 233, 244 dimensions of Phce nix beams for floors . . . 372 tt of Trenton 44 44 44 . . inches expressed in decimals of a foot . . keystone wrought-iron columns . 245 keystones for arches, depth of . 190 moments of inertia of Pencoyd angles . . . 263 <( 44 <4 44 44 beams . . . 260 tt 44 <4 44 channels. . . 262 «4 44 44 44 44 T-bars . . «4 44 44 of Phtenix beams . . . . . • • • • . . 271 44 44 44 of Trenton angles . . . . 269 44 44 44 44 44 beams. . • 44 44 44 44 44 channels . . 44 44 44 44 44 T-bars . . 44 44 44 of Union Mills angles • 44 44 44 44 44 44 beams 970 4 4 44 44 44 44 44 channels. 44 «4 44 44 44 44 T-bars noted architects . Pluenix wrought-iron columns . 242 plank measure priee-lis t of plate-glass radii of gyration of Pencoyd angles . . • 44 44 44 44 44 beams..............................260 44 “ 44 channels..............................262 44 44 44 44 44 T-bars.............................265 44 44 44 of Union Mills angles...........................274 44 44 44 44 44 44 beams...........................272 44 44 44 44 44 44 channels.................«. . . 273 44 44 44 44 44 44 T-bars...........................276 of riveted girders, co-efllcient of flanges . ....................348 of shearing-strength of materials.................................214 of sines ami cosines, natural...................................100-108 of squares ami cubes, square root and cube root................... 7 of stiffness of hard-pine beams...................................323 44 44 of oak beams..............................................325 of spruce beamsINDEX, 649 PAGE Table of tangents and co-tangents, natural.............................109-120 44 of tensile strength of chains.......................................212 “ “ 44 44 of flat iron bars...............................205 14 “ 44 44 of iron rods....................................204 44 ** 44 •* of materials.....................................198 44 “ 44 • “ of ropes, hawsers, and cables....................211 “ of thickness of walls for buildings, Boston and New York . . 149, 150 “ of transverse strength of floor plank...............................*'»01 “ 14 of hard-pine beams............................313 “ 44 “ •* of materials.................................310 ®* “ 44 44 of oak beams.................................314 44 44 44 44 of Pencoyd beams and bars............ 293, KOI 44 44 44 44 of Phoenix beams........................ 298, 372 44 44 44 44 of spruce beams..............................315 44 44 44 44 of Trenton beams and bars .... 287, 302,371 44 44 44 44 of Union Mills beams and bars .... 290,303 44 of treads and risers.................................................479 44 of Trenton angle and T-bars as struts................................236 44 44 44 beams and channels as struts............................235 44 of upset screw-ends................................................ 207 Tacks, size, length, weight, etc...........................................500 Tail-beams.............................................................. . 357 Toil hollow' block arches..................................................367 Tensile strength of materials..............................................193 Tension, resistance to.................................................... 197 Theatre ventilation........................................................471 Theatres, capacity of......................................................485 “ dimensions of........................................................486 seating-space in.................................................480 Thermometers, comparison of.............................................. 554 Tie-rods for arches, formula for...........................................351 for floor-arches..................................................373 Tiles, rooflng.............................................................531 Time, measures of...........................................................29 Tin roofs..................................................................532 Tinned doors...............................................................384 Torus moulding.............................................................469 Towers, heights of....................................................... 484 Triangle of forces.........................................................153 Trigonometry, formulas and tables...........................................95 Trimmers.................................................................. 358 Trussed beams..............................................................339 purlins ..........................................................451 Tuileries..................................................................568 Ultimate strength, deflnition of......................................... 126 Upset screw-ends, table of................................................ 207 - Urinals, space for.........................................................489 Valleys, close and open....................................................530 Vaulted party-walls........................................................151 ^locity of flow of water.............................................. 536 v,* nation......................................................... 470 n*a, definitions of................................................. 37 / __L__________ _________________| -------------- «--------------«1--------------INDEX. 647 PAGE Square root, rule for determining . ........................................ 4 44 44 table of..................................................... 7 Stability, definition of................................................... 126 Stability of arches..........................................................185 ** of piers, buttresses, etc.........................................178 Stairs......................................................................478 Statics, definition of......................................................125 Steam-pipes.................................................................504 Stiffness of beams, general formula..........................................318 “ of beams, ratio of................................................322 44 of continuous girders.............................................334 44 of cylindrical beams..............................................326 44 of hard-pine beams................................................323 44 of oak beams......................................................325 44 of rectangular beams, formulas....................................321 “ of spruce beams...................................................324 Stirrup-irons...............................................................358 Strain, definition of.......................................................126 Strength of beams, general principles.......................................280 44 44 44 iron, formulas for...................................283 44 44 44 supporting brick wall................................304 44 of cast-iron beams................................................ 307 44 of cast-iron columns....................................... 226,227 44 of chains........................................................ 212 44 of continuous girders........................................ 329, 333 44 of cylindrical beams...........................................311 44 of flat rolled iron bars.......................................205 ** of hard-pine beams, table......................................313 “ of hemp and Manila ropes.......................................... 210 44 of inclined beams..............................................286 44 of iron and steel wire ropes...................................209 44 of masonry.....................................................165 44 of mortars.................................................165,170 44 6f oak beams, table............................... . ... 314 44 of Pencoyd beams, channels, etc. . . . . . . . . . . 293,301 44 of Phoenix iron beams...........................................298 44 of pins and tree-nails..........................................215 44 of posts, struts, and columns...................................217 44 of rectangular beams, diagonal, vertical........................311 44 of solid timber and plank floors................• 361 44 of spruce beams, table............................................315 44 of Trenton beams, channels, etc............................ 287, 302 44 44 44 44 proportional to weight........................285 44 of Union Mills beams, channels, etc........................ 290, 303 44 of wooden beams....................................................307 44 of wooden floors ..................................................353 44 of wrought-iron (tensile)......................................... . 199 44 of wrought-iron rods...............................................204 Stress, definition of......................................................126 Structures, definition of ...........................................125 Struts, hard-pine and oak, strength of.....................................221 44 wrought-iron, strength of..................................... 232-234 i650 INDEX. VouRsolrs, definition rf............... Walls, masonry......................... Wash-bowls, size of.................... Water-closets, space for............... Water, consumption of, in cities . . . “ properties of.................... Weal and tear of building materials . . Weight, apothecaries*.................. “ avoirdupois...................... “ troy............................. Weight and consumption of air . . . “ and strength of lead pipes . . “ of bells....................... “ of bolts, nuts, and bolt-heads “ of brass, lead, and copper . . “ of buildings................... “ of cast-iron pipes............. “ of cast-iron plates............ “ of cast-irou water-pipes . . . “ of castings.................... “ Of COiQ8 ...................... “ of copper, brass, and lead . . r of earth....................... “ of flat and bar iron........... “ of floors................ “ of grindstones................. “ of iron rivets................. “ of lead and gasket for pipe joints “ of lead, copper, and brass . . “ of men and women . . . . . “ of roofs....................... “ of snow........................ “ of substances per cubic foot . . “ of wrouglit-iron, rules for . . Weights, ancient....................... Well-diggers* memoranda................ Wind pressure.......................... Window-glass . ....................... . Wire gauges, American and Birmingham Wire lathing........................... Wooden beams, strength of .... « Wooden columns ........................ Wrouglit-iron chimneys................. Wrought-iron, fractured surface of . . Vf rought-iron posts and columns . . . water-pipes ..... “ welded tubes................... York Cathedral ........................ PAGE . . 185 143-150 . . 489 . . 4S9 . . 559 . . 557 . . 552 . . 29 . . 28 . . 28 . . 551 . . 540 . . 483 . . 498 . . 497 . . 551 §01, 502 , . 496 , . 503 , . 572 . . 29 , . 497 , . 509 491, 494 . 354 . 573 . 499 . 503 . 497 . 4S9 426, 427 . 427 . 549 . 490 . 34 . 509 . 427 . 547 . 506 385,391 . 307 . 218 . 476 . 200 . 229 r 504 . 504 • 565INDEX C51 INDEX TO ADDITIONS. PAGE Air -609 Asphalt rock 591 Bells, dimensions and weight of . 550b Blackboards, height of . . . , 577 Boiler setting, number of bricks required . 597 Boilers, steam . 626-628 Bundy Radiators . . . . . . 613 Capacity of pipes and registers . 636 Carriages, dimensions of . 578 Combustibles 602 Compound coil heaters . . . . 623 Cost of roofing-slates . . . . . 593 Dimensions of bells . . . 44 of carriages . . . 578 44 of Connecticut State Capitol..... 576 44 of fire-engines. . . 577 44 of hose-carriages . . 577 44 of ladder-wagons . 577 44 of Metropolitan Opera House . . 575 44 of Philadelphia City Hall............576 44 of radiators . . 611-623 44 of registers and ventilators . . . . 635 44 of schoolrooms . . 577 44 of Steinway pianos . 577 44 of Washington Monument ..................575 Direct steam radiation» rules for . 617 Drying by steam..................630 Expansion of air.................607 Explosives.......................579 .Eire, temperature of . . . 631, 632 Fire-engines, dimensions and weight of......................577 Force of the wind................580 Fuel...................... 601, 602 Heal.............................601 Horizontal tubular boilers . .628,633 Hose-carriages..................577 Indirect steam radiating surfaces, rules for .♦............ 624, 625 Indirect steam radiation. . . . 619 Ladder-wagons, dimensions of . 577 Latent heat of steam.............603 Measurement of brick-work * . 595 Measurement of stone-work . . 594 Metropolitan Opera House, dimensions of 575 Orders, the five ....*.. 580 PAGE Philadelphia City Hall, dimensions of..........................576 Pianos, dimensions of .... 577 Plenum ventilation.................625 Radiation, direct-indirect steam . 618 “ direct steam . . . 611-618 “ indirect steam . . . 619 Radiators, steam, classification . 610 “ 44 description . 611-623 44 44 dimensions of, 611-623 44 44 efficiency of . 611 Refrigerators............... 599, 600 Registers, capacity of .... 636 44 dimensions of : . . 635 Riveted plate-iron girders, tables of..................... 349a, 3496. Rock asphalt.......................591 Roofing tin (revised)..............532 Sash-weights, lead, dimensions of...........................578 Schoolrooms, dimensions of . . 577 Slate, roofing,' cost of.......593 Specific heat of air...........605 44 4 4 of water . . . 605 Steam........................ 603-605 44 boilers............. 626,633 44 drying.....................630 44 heating............... 601-629 44 heating surfaces, rules for, 617 44 pipes, loss of heat from . 628 44 pipes, mains, and returns, size of..................628 Stone-w'ork, measurement of . . 594 Superheated 6teara.............603 Temperature of fire . . . 631,632 Thermal unit, British .... 601 Tin roof (revised).............532 Tubular boilers, horizontal, 628, 633 “ 44 upright . . . 634 Unit of heat...................601 Washington Monument, dimensions of.................. 575 Water..................... 603,604 Weight of bells................. 550b 44 of compressed lead sash- weights ..............578 44 of cord-wood .... 579 44 of fire-engines .... 577 44 of hose-carriages . . . 577 44 of ladder-wagons . . . 577 44 of lumber per M. . . . 578 Wiud, force of............... • 580 TEXT-BOOKS AND INDUSTRIAL WORKS. ARCHITECTURE. ARCHITECT AND BUILDER’S HAND-BOOK. Carpentry and Building; Cottages, etc. IL JOHN WILEY & SONS, 58 East Tenth Street, ARCHITECTURE THE ARCHITECT’S AND BUILDER’S POCKET - BOOK of Mensuration, Geometry, Geometrical Problems, Trigonometrical Formulas and Tables, Strength and Stability of Foundations, Walls, Buttresses, Piers, Arches, Posts, Ties, Beams, Girders, Trusses, Floors, Roofs, etc. In addition to which is a great amount of condensed information—Statistics and Tables relating to Carpentry, Masonry, Draiuage, Painting ana Glazing, Plumbing, Plastering, Roofing, Heating and Ventilation, Weights of Materials, Capacity and Dimensions of Noted Churches, Theatres, Domes, Towers, Spires, etc. With a great variety of miscellaneous information. By Frank Eugene Kidder, C.E., Consulting Architect, Boston. Illustrated with 423 Engravings, mostly from original designs. 7th edition, with additions, 1890. Over 600 pages of text. Morocco flaps............$3 “ The book admirably fulfils its purpose of becoming an indispensable companion in the work of every architect, young or old.”—American Ai'chitect. COTTAGE RESIDENCES. New edition. A Series of Designs for Rural Cottages and Cottage Villas, and their Garden Grounds. By A. J. Downing. Containing a revised list of Trees, Shrubs, and Plants, and the most recent and best selected Fruit, with some account of the newer style of Gardens. By Henry Winthrop Sargent and Charles Downing. With many new designs in Rural Architecture, including 28 full-page plates, printed separately. Bv George Harney, Architect. 8vo, cloth .........$2 HINTS TO PERSONS ABOUT BUILDING IN THE COUNTRY. By A. J. Downing. AND HINTS TO YOUNG ARCHITECTS. Calculated to facilitate their practical operations. By George Wightwick, Architect. Wood-engravings. 8vo, cloth....................................$2 CARPENTERS’ AND JOINERS’ HAND-BOO K. 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Gravity Circulating Apparatus—Radiators and Heating Surfaces—Classes of Radiation—Heating Surfaces of Boilers—Boilers for Heating—Forms of Boilers—On Boiler Setting—Proportion of Heating Surfaces of Boilers to Surfaces of Buildings—Relations of Grates and Chimneys to Boilers—Safety Valves—Draft Regulators—Automatic Water Feeders—Air Valves on Radiators— Wrought Iron Pipe—Main Pipes—Steam—Heat of Steam—Air— High Pressure Steam—Exhaust Steam and Its Value—Boiling and Cooking by Steam and Apparatus—Dry by Steam—Steam Traps—Boiler Connections and Attachments, etc.—Specification for a Steam Heating Apparatus, including Cooking, Washing and Drying. With many Plates. * ■ Mr, Baldwin has supplied a want long felt for a practical work on Heating and Heating Apparatus.”—Sanitary Engineer. Eleventh Edition (1890). l2mo, Cloth. $2.50. 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