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EMBRACING THE PRINCIPLES OF CONSTRUC- 
 TION AS APPLIED TO PRACTICAL DESIGN 
 
 WITH NUMEROUS ILLUSTRATIONS OF TOPOGRAPHICAL, 
 
 MECHANICAL, ENGINEERING, ARCHITECTURAL, 
 
 PERSPECTIVE, AND FREE-HAND DRAWING 
 
 EDITED BY 
 
 W. E. WORTHEN, C. E. 
 
 OF 1 
 
 UN: 
 
 NEW YORK 
 D. APPLETON AND COMPANY 
 
 1896 
 
COPYRIGHT, 1885, 1896, 
 BY D. APPLETON AND COMPANY. 
 
PREFACE. 
 
 " AT the suggestion of the publishers, this work was undertaken to form 
 one of their series of dictionaries and cyclopaedias. In this view, it has 
 been the intention to make it a complete course of instruction and book of 
 reference to the mechanic, architect, and engineer. It has not, therefore, 
 been confined to the explanation and illustration of the methods of projec- 
 tion, and the delineation of objects 'which might serve as copies to the 
 draughtsman, matters of essential importance for the correct and intelligible 
 representation of every form ; but it contains the means of determining the 
 amount and direction of strains to which different parts of a machine or 
 structure may be subjected, and the rules for disposing and proportioning 
 of the material employed, to the safe and permanent resistance of those 
 strains, with practical applications of the same. Thus, while it supplies 
 numerous illustrations in every department for the mere copyist, it also 
 affords suggestions and aids to the mechanic in the execution of new 
 designs. And, although the arranging and properly proportioning alone 
 of material in a suitable direction, and adequately to the resistance of the 
 strains to which it might be exposed, would produce a structure sufficient 
 in point of strength for the purposes for which it is intended, yet, as in 
 many cases the disposition of the material may be applied not only practi- 
 cally, but also artistically. . . . 1857." 
 
 " There are no changes in the principles of projection as applied to 
 drawing, and no marked improvement in drawing-instruments ; but in the 
 present practice finished drawings in shade and colour are exceptional. It 
 is sufficient, for almost every purpose, for the draughtsman to make accu- 
 rate projections with pencil on paper, and trace them afterward on cloth. 
 The pencil-drawings can be readily altered or amended, and, where there 
 are many repetitions of the same parts, but a single one may be drawn. 
 On the tracing they are made complete, and these are preserved as originals 
 in the office, while sun-prints of them are used for details of construction 
 in the shop, or distributed as circulars among customers. 
 
 " Of late years the science of ' graphics ' has become of great impor- 
 tance, and is here fully illustrated in its varied applications, showing not 
 only this method of recording the facts of the statistician, and affording 
 comparisons of circumstances and times, the growth of population, the 
 
 iii 
 
IV 
 
 PREFACE. 
 
 quantities and cost of agricultural and mechanical production, and of their 
 transport, movements of trade, fluctuations of value, the atmospheric con- 
 ditions, death-rates, etc., but also in its application to the plotting of 
 formulae for their ready solution, by the draughtsman and designer. For 
 many of the rules in this work the results of the formulae of various 
 authors have been plotted graphically, and the rule given one deemed of 
 the greatest weight, not always by average, but most consistent with our 
 own experience. 
 
 " In astronomical calculations every decimal may have its importance. 
 It is not so in those of the mechanical or architectural designer ; solutions by 
 graphics are sufficient for their purpose, and, simpler than mathematical 
 calculations, they are thus less liable to error ; it is a very good practice to 
 use one as a check on the other. It is to be remarked that inaccuracy in 
 facts, and carelessness in observation, are not eliminated from an equation 
 by closeness of calculation, and when factors are not established within the 
 limits of units it is useless to extend the results to many places of decimals. 
 It is of the utmost importance to know at first well the object and pur- 
 poses of the design, the stresses to which its parts are to be subjected, and 
 the strength and endurance of the materials of which it is to be composed. 
 In establishing rules for ourselves, be sure of the facts, and that there are 
 enough of them for a general application. Rules are necessary, but their 
 application and usefulness depend largely on the experience of the user, 
 and life must be a record of applications and effects. It is comparatively 
 easy to make a work strong enough ; but to unite economy with propor- 
 tion is difficult. . . . 1886." 
 
 The first edition of this work was suggested by " The Engineer and 
 Machinist's Drawing Book " of Messrs. Blackie & Son, 1855, from which, 
 with the consent of the publishers, much of the text and illustrations were 
 taken. Since then, in the many editions, it has been the aim to keep up 
 with mechanical progress, and matter has been drawn from all sources. 
 Credit, as far as possible, has been given to mechanics for their designs and 
 to experimenters for their results. 
 
 Geometrical problems and examples of orthographic projection of the 
 first work are largely retained, but examples of mechanical and archi- 
 tectural construction are brought up to the present age of steel, with the 
 latest illustrations of the applications of steam, and some of electricity. 
 Isometry is retained, perspective has been more fully illustrated, and free- 
 hand drawing now includes the recent processes by which, through pho- 
 tography, the mechanical labour of sketching is diminished, adding to the 
 correctness of detail and improving the effectiveness on paper. This may 
 be called an age of illustration, and the processes have enabled a work like 
 the present drawing book to give better and more illustrations, less text, 
 more comprehensiveness, and greater certainty of detail. 
 
 Mr. Robert E. Hawley, brought up in my office, has had charge of the 
 new drawings and has acted as co-editor. AV. 
 
CONTENTS. 
 
 PAGES. 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS 1-82 
 
 Drawing of lines straight, curved, perpendicular, and parallel : angles, ai'cs, 
 and circles, 13. Triangles, polygons and circles, inscribed and described ; 
 polygonal angles ; use of protractor, 21. Use of the triangle and square ; areas 
 of figures ; scales, 25. Similar triangles, squares of proportionate sizes, 30. El- 
 lipse, parabola, hyperbola, spiral, 35. Drawing-board, table ; straight-edges ; 
 T-squares: parallel rulers ; curves, variable and adjustable; splines and weights ; 
 thumbtacks; drawing pens; dotting instrument; compasses; dividers; plot- 
 ting scales; protractors; sector; pantographs, 51. Drawing paper; tracing 
 paper; tracing cloth ; heliographic paper ; damp stretching and mounting, 55. 
 Use of instruments; representation of surfaces; enlarging and reduction of 
 drawings ; designs in lines, 62. Lettering ; profile and cross-section paper, 77. 
 Ornamental designs in straight and curved lines, 82. 
 
 PLOTTING ....* 83-94 
 
 Scales ; plotting for surveys, plans and maps ; plotting by protractor, by lati- 
 tudes and departures, by triangles, by offsets ; United States division of public 
 lands, 94. 
 
 TOPOGRAPHICAL DRAWING 95-120 
 
 Conventional signs ; representation of hills ; chart from United States Sur- 
 vey, 102. Railway, 103. Hydrometrical chart, geological and section, 109. 
 Transferring field notes, 110. Map projections, 116. Colored topography; 
 pen and brush work, 119. Meridians and borders, 120. 
 
 ORTHOGRAPHIC PROJECTION 121-146 
 
 Point ; straight line ; solid ; simple bodies : pyramid ; prism, 127. Conic 
 sections, 130. Intersection of solids, 139. The helix, 141. Development of 
 surfaces, 144. Shade lines, 146. 
 
 SHADES AND SHADOWS 147-166 
 
 Shadow of a point, of a straight line, of a solid, of a pyramid, of a cylinder, 
 of a hollow hemisphere; niche, 154. Lines of shade on sphere ; ring; grooved 
 pulley ; screw, 159. Manipulation shades, surfaces in light, in shade by flat 
 tints, by softened tints. Examples in plates. Conventional tints, 166. 
 
 MATERIALS 167-185 
 
 Earth and rocks; woods, 170. Masonry, technical terms for; stones, gra- 
 nitic, argillaceous, sand, lime, 174. Artificial building material ; brick, sizes of ; 
 fire ; enamelled tile ; terra cotta, 175. Mortars ; concrete ; plastering ; weight 
 of masonry, 177. Metals, conventional signs of, properties of; alloys, strength 
 of, graphic representation by Prof. Thurston ; sulphur: glass; rubber; paints; 
 coal ; flame, 185. 
 
 v 
 
VI. 
 
 CONTENTS. 
 
 MECHANICS 
 
 Force : centres of gravity ; mechanical powers ; parallelogram of forces ; 
 toggle joint ; hydraulic press ; statics ; dynamics ; velocity of falling bodies, 
 196. Friction, 201. Mechanical work ; unit of force ; the force of animals, 
 water, steam, and their application ; the steam-pressure indicator and cards, 
 211. Motion, example of the path of; parts of machines ; of the crank ; the 
 Stanhope lever ; Whitworth's quick return ; parallel motion ; car coupler, 218 ; 
 valve diagram ; Corliss cut off : link motion ; valve gear, 228. 
 
 HAGES 
 
 186-228 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTION 
 
 Stress ; strain ; dead load ; factor of safety ; safe load of columns, cast and 
 wrought iron ; shearing, torsional and transverse stress ; graphic diagrams, 
 239. Table of safe load on yellow-pine beams, on cast iron, on wrought iron, 
 on steel ; box girders ; composite, 250. Bolts and nuts ; screws ; washers, 257. 
 Shafts and axles, cast and wrought iron, 262. Pillow blocks ; standards ; 
 hangers ; steps ; suspension and thrust bearings, 273. Couplings ; clutches, 
 282. Pulleys, wooden and iron plates ; cone, 287. Belts, plain, twist, and 
 cross. Strength of rope driving, 299; chain driving; leather links, 301. 
 Gearing, spur, rack, and pinion ; bevel. Form of teeth ; cycloid ; hypocycloid ; 
 involute. 310. Diagram of stress on teeth ; diameter of pitch circle. Adcock's 
 table of arcs for gear teeth ; mortise wheels, 316. Projections of a spur, bevel, 
 and worm wheels ; screws, 330. Frictional and wedge gearing, 333. Blocks for 
 running rigging ; chains ; chain couplings ; wire rope ; sockets ; hooks ; 337. 
 Levers, cranks, 342. Eccentrics; wiper; stamp mill, 347. Connections, cot- 
 ters, pins, rods, 348. Eccentrics and straps ; crossheads, 354. Working beam ; 
 guide bars, 358. .Steam cylinders; pistons of pumps. Water pumps, 362. Wood 
 and cup packing, 364. Steam jacket ; air chamber ; Thames Ditton pump ; 
 Reidler ; Worthington, 366. The injector, 368. Clearances ; piston rods ; stuff- 
 ing boxes, 370. Steam ports : Valves, cylindrical, balanced, automatic, 
 disk, rubber, ball, poppet, flap, 376. Valves controlled by hand, cocks^ 
 plug. Valves : compression, air, globe, gate, damper rotary, safety, 382. Hy- 
 drants, 383. Riveted joints, 389. Boilers : tubular, marine ; water tube ; flue : 
 locomotive, vertical, 398. Connections of steam and water pipes ; wrought- 
 iron pipes ; couplings ; unions ; coils ; joints for submerged pipes, 406. Gov- 
 ernor ; fly-wheels ; air chambers ; accumulators ; hydraulic press ; jack screw ; 
 housings. 
 
 229-414 
 
 ENGINEERING DRAWING 
 
 Foundations ; concrete base ; crib work ; New York dock ; Thames embank- 
 ment ; breakwater ; screw piles ; masonry curbs ; steel caissons ; Poughkeepsie 
 bridge pier ; pneumatic piles ; caissons and air lock ; freezing process, 435. 
 Retaining walls, 436. Dams : earth, crib, masonry, 444. Gates : head, waste, 
 451. Canals, navigation, power. Locks of canals ; flumes and conduits, 459. 
 Reservoirs ; sheet-iron pipe ; water tanks, 462. Water mains for city service ; 
 specials ; inspection, 466. Sewers : brick, vitrified pipe, circular, egg-shaped, 
 concrete, man-holes, 471. Gas supply, 472. Roads and highways ; street pave- 
 ments : granite, asphalt, wooden, block, 479. Railroads ; road bed : rails ; elec- 
 tric conduit, 483. Roofs and bridges ; principle of bracing ; frames, wooden, 
 iron ; trestles, 498 ; truss bridges : wooden, iron, combination, 510. Turn ta- 
 bles ; ferry-landing bridge ; high wrought-iron trestles ; masonry piers ; arch 
 bridge, 522. Boiler setting, horizontal, tubular, 526 ; chimneys, 530. Loca- 
 tion of machines ; foundation, 535. Tunnels : principles of timbering ; Hoo- 
 sac ; bar timbering, 539. Railway rolling stock ; box car ; standard passenger ; 
 locomotive frame, 545. Wave-line principle of ship construction, 547. 
 
 415-547 
 
CONTENTS. vii 
 
 PAGES 
 
 ARCHITECTURAL CONSTRUCTION 548-693 
 
 Plans and elevations, 553 ; details of construction ; timber frames and 
 floors, 561. Examples of fire-proofing, old, recent ; skeleton frames, fire-re- 
 tarding construction of mills, 571 ; windows ; stairs ; doors. Fireplaces ; 
 flues ; roofs ; gutters ; cornices, 587. Plastering ; mouldings, 590. Sizes of 
 rooms ; water appliances and accessories ; Ferguson's rules of proportion ; de- 
 signing of house ; illustration and details ; country and city, 609. Apartment 
 houses ; store and warehouses ; machine-shop ; school-houses ; churches ; 
 theatres; lecture rooms ; music and legislative halls; waves of sound; effect 
 of air currents ; space for seats : ancient and modern churches ; organs ; 628. 
 Theatres, dimensions of some, 630. Legislative halls, acoustic principles; 
 hospitals ; stables ; cowhouses ; greenhouses, 634. Ventilation and warming ; 
 stoves ; hot-air furnaces ; steam and hot water circulation, 647. Radiators ; 
 laying out of pipes, 649. Plumbing ; soil pipe ; fixtures for kitchens ; baths ; 
 water-closets ; traps and bends, 656. Lighting : gas, electric, wiring. Orders 
 of architecture : Greek and Roman, Romanesque, Byzantine, Gothic, the Re- 
 naissance. Arches ; domes and vaults ; buttresses ; towers ; bell cots ; spires, 
 674. Windows, lancet, traceried ; doorways, 679. Mouldings ; arch and 
 architrave ; capitals ; bases ; string courses and cornices, 682. Ornaments, 693. 
 
 ISOMETRICAL DRAWING ." . . . . . 694-705 
 
 PERSPECTIVE 706-725 
 
 Points and planes of perspective ; parallel and angular perspective of cubes 
 and other solids, of buildings, of an arched bridge, of an interior, of a staircase, 
 of reflection of objects in water, of shadows ; perspective as illustration of ad- 
 vertisements. 
 
 FREE-HAND DRAWING 726-764 
 
 Materials : paper, pencils, lithographic chalk, pens, ink. Proportions of Hu- 
 man Frame, Geometrical drawings of, " Dictionnaire Raisone par Viollet le 
 Due " and Dr. Rimmer's " Elements of Design." Half tones of photographs of 
 plaster models, " ecorche " of wash drawing of flowers, etc., P. de Lohgpre. Pen 
 and ink reproduction of photographs on plain salted paper, models "ecorche," 
 Sandow, manikins, Venus de Milo, and Dumas. Pumping Station after Emer- 
 son in toothpick and splatter work. Drawings on stipple paper or clayboard, Sal- 
 vini and Venetian fete on the Seine. Pen and ink drawing, hands, feet, heads, 
 Electioneer, Cow, Donkey from Landseer, hoofs, paws, muzzles, Espanola y 
 Americana, Erik Werenskiold and design by Fortuny, Alexandrian pilot, Head 
 of Sheik, Water Bearer, Donkey's Head, Deer, Ducks, Landscapes, Oak Trees, 
 Morning, Cattle going Home, Lady of the Woods, Elm, Cedar, Sketch in 
 chalk, Suez Canal and sea sketch. 
 
 APPENDIX ...... 765-861 
 
 Patent office, Requirements for drawings and Registration of prints and labels. 
 Mensuration, areas of surfaces, contents of solids ; measures, lineal, of surface ; 
 of capacity ; liquid : dry. Weights, apothecaries', Troy, avoirdupois, compari- 
 son of; Dynamic Table: cubic measure; shipping measure; register; ship- 
 ping ; carpenter's rule. Table of inches and parts in decimals of a foot ; elec- 
 trical units ; units of heat. Table of fifth powers of numbers ; weight of cast- 
 iron balls, of cast-iron pipe, weights of rolled iron, 773 ; weight of wrought. 
 Tables of dimensions and weight of wrought iron welded tubes ; nominal and 
 actual diameters of boiler tubes. Heavy pipe for driven wells ; spiral riveted 
 tubes, heavy and light ; weight of copper and brass rods ; rivets ; wrought spikes ; 
 cut nails and spikes ; wire nails, weight of. Galvanized telegraph wire ; 
 weights ; resistance in ohms ; sizes used. Standard Beams and Channels of Asso- 
 
vriii CONTENTS. 
 
 PAGES 
 
 ciation of American Steel Manufacturers ; grades of steel ; weights of lead pipe. 
 Weight of a cubic foot of water at different temperatures. Flow of water, 
 781-791, over weirs, through pipes and conduits ; graphic diagrams of Kutter 
 formulae. Table of equalizing the diameter of pipes ; flow of air ; comparison 
 of flow of water by the Kutter diagrams, with that of air; with that of gas, 
 and the products of combustion in chimneys. The Babcock and Wilcox boiler ; 
 the Green economizer ; the Heine boiler. Table of saturated steam, 798; ex- 
 pansive working of steam. Table of factors of evaporation. Electric Light 
 and Power Station, Twenty-eighth Street, New York city, 805. Diagram of 
 electric wiring ; lamp socket switch and Lundell motor ; Table of the density 
 of gases. Specific gravity of liquids, of earths, of woods, of metals, solders ; 
 alloys. Table of the circumferences and areas of circles, 819. Tables of 
 squares, cubes, and roots, 826 ; of reciprocals ; Latitudes and Departures, 835. 
 Natural sines and cosines, 845. Logarithms, 861. 
 
 SCRAPS 862-912 
 
 Compound steel cylinders ; manholes and covers ; compressed-air locomo- 
 tive ; cranks ; rudder frame ; boiler flues ; screw propeller ; spherical bearing ; 
 conventional signs of riveting ; mechanical stokers. Boilers : Stirling and 
 Abendroth and Root. Engines Corliss stationary: Deane steam pump; 
 Reidler valve ; Locomotives ; car springs ; elevated railroad ; cable grip ; der- 
 rick. Dams: canvas, earth, masonry, movable; Builder's hardware; hinges; 
 construction of safes. Mantels and fireplaces ; doors ; marquetry ; pediment ; 
 brackets ; railing ; summer house ; windows ; doorways ; porches ; house fronts ; 
 dormers and towers ; skeleton construction ; Broad Street Station, Philadel- 
 phia. Church spires ; churches ; perspective ; buildings of Centennial Exhi- 
 bition ; of World's Pair ; Coney Island. 
 
DESCRIPTION OF PLATES. 
 
 I. Shading of prism and cylinder by flat tints. Page 160. 
 II. Shading of cylinder and segment of hexagonal pyramid. Page 161. 
 Ill, IV. Finished shading and shadows of different solids. Page 163. 
 V. Shades and shadows on screws. Page 164. 
 VI. Example of topographical drawing, done entirely with the pen. 
 
 Page 101. 
 
 VII. The same, with the brush, in black. Page 117. 
 VIII. The same, with the brush, in colour. Page 118. 
 IX. Contoured map of Staten Island, shaded by superimposed washes, 
 the washes increasing in intensity or strength as required to pro- 
 duce the effect. Page 117. 
 X. Geological map of part of New Jersey, coloured to show the different 
 
 formations. Page 106. 
 XI, XII. Topographical maps of parts' of Massachusetts. 
 
 XIII. Plan and ceiling in colour. Page 548. 
 
 XIV. Perspective view of Gothic church, finished in colour. (Frontispiece.) 
 XV. Front elevation of a building, in colour. 
 
 XVI. Finished perspective drawing, with shades and shadows, of a large 
 
 bevel- wheel and two pinions, with shifting clutches. Page 160. 
 XVII. Plan, elevation, and section of bevel-wheel, pinion, and clutches, 
 shown in perspective Plate XVI. Page 160. 
 
 EEEATA. 
 
 Page 160, 19th line, omit Plate XI, and for Plates XII and XV, 
 substitute Plates XVI and XVII. 
 
APPLETONS' 
 
 CYCLOPAEDIA OF DRAWING. 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 MOST persons, at some time, have made use of the simple drawing instru- 
 ments, as pencils, straight edges or rulers, dividers and compasses with change- 
 able .points, and many suppose that there can be no skill in their use; but to 
 one critical in these matters there are great differences, even in common draw- 
 ings, in the straightuess and uniformity of the lines and in the care of the 
 surface of the paper. 
 
 Pencils are marked according to their hardness: H (hard), HH, H H H, 
 
 to 8 H ; or H, V (very) H, V V H, M (me- 
 dium), H M^ M B (black), S (soft), M S, V S, 
 V V S ; or by numerals, 1, 2, 3, to 8. 
 
 Select for the geometrical problems and 
 for usual drawings a No. 3. or H H H pencil. 
 It should be sharpened to a cone-point (Fig. 
 1). Where a pencil is used for drawing lines 
 only, some draughtsmen sharpen the pencil 
 to a wide edge, like a chisel. 
 In drawing a straight line, hold the ruler firmly with the left hand ; with 
 the right hand hold the pencil lightly but without slackness, and a little 
 inclined in the direction of the line to be drawn, keeping the pencil against 
 the edge of the ruler, and in the same relative position to the edge during the 
 whole operation of drawing the line. 
 
 2 (1) 
 
 FIG. 1. 
 
2 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 To draw a clean line and preserve the point of the pencil, the part of the 
 cone a little above the point of the pencil should bear against the edge of the 
 ruler, and the pencil should be carried steadily while drawing. Any oscilla- 
 tion will throw the point farther from or nearer the ruler, and the line will not 
 be straight (Fig. 2). 
 
 FIG. 2. 
 
 In the use of the compasses do not make a hole through the paper with 
 the needle or sharp point, but only into the paper sufficient to maintain the 
 position. 
 
 Keep the paper clean, and use rubber as little as possible. 
 
 A geometrical point, which is position only, is indicated in drawing by the 
 prick-mark of a needle or sharp point, or the dot of a pencil ; sometimes' it is 
 inclosed 0, sometimes designated by the intersection of two short lines X >. 
 A line, which is extension in length only, is indicated by a visible mark of 
 pencil or pen traced upon the paper. 
 
 Geometrically lines have but one dimension, length, and the direction of a 
 line is the direction from point to point of the points of which the line is com- 
 posed : in drawing, lines are visible marks of pencil or pen upon paper. 
 
 A straight line is such as can be drawn along the edge of the ruler, and is 
 one in which the direction is the same throughout. In drawing a straight line 
 through two given points, place the edge of the ruler very near to and at equal 
 distances from the points, as the point of the pencil or pen should not be in 
 contact with the edge of the ruler (Fig. 3). 
 
 FIG. 3. 
 
 Lines in geometry and drawing are generally of limited extent. A given 
 or known line is one established on paper or fixed by dimensions. Lines of 
 the same length are equal. 
 
 Curved Lines. For the pencil-points of compasses, whittle down the 
 stumps of pencils to suit. Insert the pencil-point in the compasses. With 
 the needle or sharp point resting on the paper describe a line with the pencil 
 around this point ; the line thus described is usually called a circle more 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 3 
 
 strictly it is the circumference of a circle the circle being the space inclosed. 
 A portion of a circumference is an arc. The point around which the circum- 
 ference is described is the centre of the circle (Fig. 4). 
 
 The line embraced between the two points of the compasses is called the 
 radius of the circle, and by mechanics a sweep; a line passing through the 
 centre and terminating at each end in the circum- 
 ference is a diameter, and is equal in length to 
 two radii ; any line not passing through the cen- 
 tre and limited by the circumference is less than 
 a diameter and is a chord. The space embraced 
 between a chord and its lesser arc is a segment. 
 The space embraced between two radii and its arc 
 is a sector; if the arc is the quarter of the cir- 
 cumference, the sector is distinguished as a quad- 
 rant. 
 
 It will be observed that arcs are lines which 
 are continually changing the directions, and are 
 called curved lines, but there are other curved lines than those described by 
 compasses, of which the construction will be explained hereafter. 
 
 Lines which can neither be drawn by rulers or compasses, representing the 
 directions of brooks and rivers, the margins of lakes and seas, points in which 
 are established by surveys, defined on paper, and connected by hand-drawing, 
 are irregular or crooked lines. 
 
 Where it is necessary to distinguish lines by names, we place at their 
 extremities letters or figures, as A B, 1 2 ; the line A B, or 1 2. 
 But in lines other than straight, or of considerable extent, it is often necessary 
 to introduce intermediate letters and figures, as a a a. 
 
 FIG. 4. 
 
 In the following problems, unless otherwise implied or designated, where 
 lines are mentioned, straight lines are intended. 
 
 If a straight line moves sideways in a single direction, it will sweep over a 
 surface which is called a plane. All drawings are projections on planes of 
 paper or board. 
 
 Two lines drawn on paper, and having the same direction, can never come 
 any nearer each other, and must always be at the same distance apart, however 
 far extended. Such lines are called parallel lines. 
 
 To draw a line parallel to a given line, and at a given distance from it 
 (Fig. 5). 
 
 Draw the line A B for the given .line, and take in the compasses the 
 distance A C the distance at which the other line is to be drawn. On A, 
 as a centre, describe an arc, and on B, as a centre, a similar arc ; draw the 
 line C D just touching these two arcs, which will be the parallel line re- 
 quired. 
 
4 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 To draw a line parallel to a given line through a given point outside this 
 line (Fig. 6). 
 
 Draw the given line A B, and mark the given point C. With C as a centre, 
 find an arc that shall only just touch A B ; and with B as a centre, and the 
 
 n 
 
 FIG. 5. 
 
 same radius, describe an arc D. Draw through the point C a line just touching 
 this last arc, and the line C D will be the parallel line required. 
 
 Two lines in the same plane, not parallel to each other, will come together 
 if extended sufficiently far. The inclination or intersection of two lines is 
 called an angle (Fig. 7). 
 
 If but two lines come together, the angle may be designated by a single 
 letter at the vertex, as the angle E. 
 
 But, if three or more lines have a common vertex, the angles are designated 
 by the lines of which they are composed, as the angle D B C of the lines 'D B 
 and B C ; the angle A B C of A B and B C ; the angle A B D of A B and B D. 
 The letter at the vertex must always be the central letter. 
 
 Describe a circle (Fig. 8). Draw the diameter A B. From A and B as 
 centres, with any opening of the compasses greater than the radius, describe 
 two arcs cutting each other as at D. Through the intersection of these arcs 
 and the centre C, draw the line D E. 
 D E makes, with the diameter A B, four 
 angles, viz., A C D, D C B, B C E, and 
 EGA. Angles are equal whose lines 
 
 
 Fid. 7. 
 
 have equal inclination to each other, and whose lines, if placed one on the 
 other, would coincide. By construction, the points C and D have, respectively, 
 equal distances from A and B ; the line D C can not, therefore, be inclined 
 more to one side than to the other, and the angle A C D must be equal to the 
 angle BCD. Such angles are called right angles. The four angles, formed 
 by the intersection of D E with A B, are equal, and are right angles. 
 
 The angles A C D and D C B, on the same side of A B, are called adjacent 
 angles ; as also DOB and B C E, on the same side of D E. 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 5 
 
 If the base line be parallel with the surface of still water, it is called an 
 horizontal line, and the line perpendicular to it is called a vertical line. 
 
 Draw the line C F. It will be observed that the angle F C B is less than 
 a right angle, and it is called an acute angle ; 
 the angle F C A is greater than a right angle, 
 and it is called an obtuse angle. It will be 
 observed that, no matter how many lines be 
 drawn to the centre, the sum of all the angles 
 on the one side of A B can only be equal to 
 two right angles, and, on both sides of A B, 
 can only be equal to four right angles. It will 
 be observed that the angles at the centre in- 
 clude greater or less arcs between their sides, 
 according to the greater or less inclination of 
 their sides to each other ; that the right angles 
 intercept equal arcs, and that, no matter how 
 large the circle, the proportion of the circle 
 intercepted by the sides of an angle is always 
 the same, and that the arcs can therefore be 
 taken as the measures of angles. For this 
 purpose the whole circumference is supposed 
 to be divided into three hundred and sixty de- 
 grees (360), each degree subdivided into sixty minutes (60'), and each minute 
 into sixty seconds (60"). Each right angle has for its measure one quarter of 
 the whole circumference (-^f^), or 90, and is called a quadrant. 
 
 To construct an angle equal to 
 a given angle (Fig. 9). 
 
 Draw any angle, as C A B, for 
 the given angle, and the line a b 
 
 A 
 
 FIG. 10. 
 
 \\i 
 
 k 
 FIG. 11. 
 
 as the base of the required angle. From A, with any suitable radius, describe 
 the arc B C, and from a, with the same radius, describe the arc b c. With 
 the compasses take the length of the chord B C, and, from b as centre, describe 
 an arc cutting the arc b c at c, and draw the line a c; cab is the required 
 angle. 
 
 To construct an angle of sixty degrees (Fig. 10). 
 
 Lay off any base line, and from A, with any radius, describe an arc, and 
 from B, with the same radius, describe another arc cutting the first, as at C. 
 Draw the line C A, and the angle CAB will be an angle of sixty degrees. 
 
f> 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 For if, on the circumference of any circle, chords equal to the radius are 
 stepped off successively, six will exactly complete the circle, making 360. 
 
 To draw a perpendicular to a line from a point without the line 
 (Fig. 11). 
 
 Draw a line, and mark the given point A. From A as a centre, with a 
 
 \F 
 
 B 
 
 A 
 FIG. 12. 
 
 FIG. 13. 
 
 suitable radius, describe an arc cutting the line at G and F. From G and 
 F, as centres, describe arcs cutting each other. The line drawn through 
 the point A, and the point of intersection E, will be perpendicular to the 
 line G F. 
 
 The radial line A E divides the chord G F and the arc G E F into two 
 equal parts ; and, conversely, the line perpendicular to the middle point of a 
 chord of a circle is radial passes through the centre of that circle. 
 
 To draw a perpendicular to a line from a point ivithin that line (Fig. 12). 
 1st Method. Draw a line, and take the point A in the line. From A, as 
 a centre, describe arcs cutting the line on each side at B and C. From B and 
 C, as centres, describe intersecting arcs at D. Draw a line through D and A, 
 and it will be perpendicular to the line B C at A. 
 
 2d Method (Fig. 13). Draw the line, and mark the point A as before. 
 From any centre F, without the line, and not directly over A, with a radius 
 equal to F A, describe more than a half-circle cutting the line, as at D. From 
 
 D, through the centre F, draw a 
 line cutting the arc at E. Draw 
 A E, and it will be the perpendicu- 
 lar to the line A D. 
 
 The line D E is the diameter 
 of a circle, and the angle DAE, 
 with its vertex at A in the circum- 
 ference, embraces with its sides half 
 a circle. It has been shown that 
 angles at the centre of a circle have 
 for their measure the arc embraced 
 by their sides. Angles with their 
 vertices in the circumference have 
 for their measure half the arc em- 
 braced by their sides; and, consequently, angles embracing half a circumfer- 
 ence are right angles. 
 
 To draw a perpendicular to the middle point of a line (Fig. 14). 
 
 -B 
 
 FIG. 14. 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. f 
 
 From the extremities A and B of the line, as centres, describe similar inter- 
 secting arcs above and below the line. Through these intersections draw the 
 line C D. It will be perpendicular to the line A B, and bisect or divide it into 
 two equal parts. 
 
 If the line A B be considered the chord of a circle, its centre would lie in 
 the line C D. 
 
 This construction is sometimes used merely to divide a line into two equal 
 parts, or bisect it; it can be more readily done with dividers (Fig. 15). 
 
 Place one point of the dividers on one end of the line, and open the 
 dividers to a space as near as may be half the line. Turn the dividers on the 
 
 FIG. 15. 
 
 central point ; if the other point then falls exactly on the opposite extremity 
 of the line, it is properly divided ; but, if the point falls either within or with- 
 out the extremity of the line, divide the deficit or excess by the eye, in 
 halves, and contract or extend the dividers by this measure. Then apply the 
 dividers as before, and divide deficit or excess till a revolution exactly covers the 
 length of the line. By accustoming one's self to this process, the eye is made 
 accurate, and one estimate is sufficient for a correct division of any deficit or 
 excess. By a similar process it is evident that a line can be divided into any 
 number of equal parts, by assuming an opening of the dividers as nearly as 
 possible to that required by the division, and, after spacing the line, dividing 
 the deficit or excess by the required number of parts, contracting or expanding 
 the dividers by one of these parts, and spacing the line again, and so on till it 
 is accurately divided. 
 
 To bisect a given angle (Fig. 16). 
 
 Construct an angle, and from its vertex A, as a centre, describe an arc 
 cutting the two sides of the angle at B and C. From B and C, as centres, de- 
 
 FIG. 16. 
 
 FIG. 17. 
 
 scribe intersecting arcs. Draw a line through A and the point of intersection 
 D, and this line will bisect the angle. 
 
 To bisect an angle when the vertex is not on the paper (Fig. 17). 
 
8 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 Let A B and E C be two lines inclined to each other ; at equal distances 
 and parallel to the above lines draw a b and a c, intersecting lines; bisect -the 
 angle b a c. A line a d drawn through the vertex and the point of bisection is 
 the required line. 
 
 Through two given points to describe an arc of a circle with a given radius 
 (Fig. 18). 
 
 From B and C, the two given points, with an opening of the dividers equal 
 to the given radius, describe two arcs intersecting at A. From A, as a centre, 
 with the same radius, describe an arc, and it will be the one required. 
 
 FIG. 18. 
 
 FIG. 19. 
 
 To find the centre of a given circle, or of an arc of a circle. 
 
 Of a circle (Fig. 19). Draw the chord A B. Bisect it by the perpen- 
 dicular C D, whose extremities lie in the circumference, and bisect C D. Gr, 
 the point of bisection, will be the centre of the circle. 
 
 To find the centre of an arc (Fig. 20). Select the points A, B, and C in 
 the arc, well apart. From A and B as centres, and then from B and C as 
 centres, describe arcs of equal radii cutting each other ; draw the two lines D E 
 and F G through their intersections. The point 0, where these lines meet, is 
 the centre required. 
 
 To describe a circle passing through three given points (Fig. 20). 
 
 Proceed, as in the last problem, to find the point 0. From 0, as a centre, 
 with a radius A, describe a circle, and it will be the one required. 
 
 X T. 
 
 FIG. 20. 
 
 FIG. 21. 
 
 To describe an arc of a circle passing through three given points, ivhere the 
 centre is not available (Fig. 21). 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 9 
 
 From the extreme given points A and B describe arcs A E and B D ; 
 through the third given point C draw lines from A and B, intersecting the arcs 
 at and ; from and cut the arcs in either direction by equal divisions, 
 1, 2, 3, and 1', 2' ; draw lines A 1, A 2 ; A 1', A 2' ; B 1, B 2 ; B 1', 
 B 2'. The successive intersections of A 1 by B 1, A 2 by B 2, A 1' by B 1' are 
 points in the required arc by the connection of which the problem will be 
 complete. 
 
 To describe this arc mechanically (Fig. 22). 
 
 Lay off on a piece of cardboard the three points A, C, B, and connect them 
 by lines extended beyond the points A and B ; and then cut out the cardboard 
 
 FIG. 22. 
 
 along these lines. Insert pins at the points A and B on the drawing, and 
 placing the cardboard templet against these pins, and the angle against the 
 point C, slide the templet each way, dotting in the drawing the angle C in its 
 different positions. These dots will be points in the curve, which are to be con- 
 nected. By extending the bisecting line in different positions of the templet 
 to the drawing, radial lines are given which will be useful in laying off voussoir 
 joints on segmental arches of large radius. Kadial lines are also necessary in 
 perspective drawing, for 
 which an instrument 
 called the centrolinead 
 (Fig. 23) is used. The 
 principle is similar to 
 that of the cardboard 
 templet. 
 
 To draw a tangent to 
 a circle from a given 
 point in the circumfer- 
 ence. 
 
 1st Method (Fig. 24). 
 Through the given 
 point A draw the radial 
 line A C. The perpen- 
 dicular F G to that line 
 will be the tangent re- 
 quired. FIG. 23. 
 
 2d Method (Fig. 25). 
 
 From the given point A set off equal arcs, A B and A I>. Join B and D. 
 Through A draw A E parallel to B D, and it will be the tangent required. 
 This method is useful when the centre is inaccessible. 
 
10 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 To draw tangents to a circle from a point without it (Fig. 26). 
 From the given point A draw A C to the centre of the circle. Bisect A C 
 to find the point D. From D, as a centre, describe an arc with a radius D C, 
 
 A 
 
 E 
 
 FIG. 24. 
 
 FIG. 25. 
 
 cutting the circle at E and F. Draw A E and A F, and they will be the tan- 
 gents required. 
 
 To' construct within the sides of an angle a circle tangent to these sides at a 
 given distance from the vertex (Fig. 27). 
 
 FIG. 26. 
 
 FIG. 27. 
 
 Let a and b be the given points equally distant from the vertex A. Draw 
 a perpendicular to A C at a, and to A B at b. The intersection of these per- 
 pendiculars will be the centre of the required circle. 
 
 In the same figure, to find the centre, the radius being given, and the 
 points a and b not known. Draw lines parallel to A C and A B, at a distance 
 equal to the given radius, and their intersection will be the centre required. 
 
 To describe a circle from a given point to touch a given circle (Figs. 28 
 and 29). 
 
 FIG. 28. 
 
 FIG. 29. 
 
 D E being the given circle, and B the given point, draw a line from B to 
 the centre C, and produce it, if the point B is within the circle, until it cuts 
 the circle at A. From B, as a centre, with a radius equal to B A, describe the 
 circle F G, touching the given circle, and it will be the circle required. 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 11 
 
 In all cases of circles tangent to each other, their centres and their points 
 
 of contact must lie in the same straight line. 
 
 To draw tangents to two given circles (Fig. 30). 
 
 Draw a straight line through the centres of the two given circles. From 
 the centres A and B draw parallel radii, A D and B E, in the same direction. 
 
 FIG. 30. 
 
 Draw a line from D to E, and produce it to meet the centre line at C ; and 
 from C draw tangents to one of the circles by Fig. 26. Those tangents will 
 touch loth circles as required. 
 
 To construct a circle through a given point tangent to a second circle at a 
 given point (Fig. 31). 
 
 Let A be the given point of a circle A D C, B the point through which the 
 required circle is to be drawn. Connect A and B, extend A O, bisect A B by 
 a perpendicular. The intersection of this perpendicular with A extended 
 will be the centre of the required circle. The same method of construction 
 would apply if the point B were inside the circle ADC. 
 
 Between two inclined lines to draiv a series of circles touching these lines 
 and touching each other (Fig. 32). 
 
 Fm. 31. 
 
 FIG. 32. 
 
 Bisect the inclination of the given lines A B and C D by the line N ; this 
 is the centre line of the circles to be inscribed. From a point, P, in this line, 
 draw P B perpendicular to the line A B ; and from P describe the circle B D, 
 touching the given lines, and cutting the centre line at E. From E draw E F 
 perpendicular to the centre line, cutting A B at F ; from F describe an arc, 
 with a radius F E, cutting A B at G ; draw G H parallel to B P, giving H the 
 centre of the second touching circle, described with the radius H E or H G. 
 By a similar process the third circle, I N", is described. And so on. 
 
 Inversely, the largest circle may be described first, and the smaller ones in 
 succession. 
 
12 
 
 CONSTRUCTION OP GEOMETRICAL PROBLEMS. 
 
 Note. This problem is of frequent use in scroll-work. 
 
 Between two inclined lines to draw a circular arc to fill up the angle 
 (Fig. 33). 
 
 Let A B and D E be the inclined lines. Bisect the inclination by the line 
 F C, and draw the perpendicular A F D to define the limit within which the 
 circle is to be drawn. Bisect the angles A and D by lines cutting at C, 
 
 Fio. 34. 
 
 and from C, with radius C F, draw the arc H F G, which will be the arc 
 required. 
 
 To Jill up the angle of a straight line and a circle, with a circular arc of a 
 given radius (Fig. 34). 
 
 On the centre C, of the given circle A D, with a radius C E equal to that 
 of the given circle plus that of the required arc, describe the arc E F. Draw 
 G F parallel to the given line H I, at the distance G H, equal to the radius 
 of the required arc, cutting the arc E F at F. Then F is the required centre. 
 Draw the perpendicular F I, and the line F C, cutting the circle at A ; and, with 
 the radius F A or F I, describe the arc A I, which will be the arc required. 
 
 To fill up the angle of a straight line and a circle, ivitli a circular arc to 
 join the circle at a given point (Fig. 35). 
 
 In the given circle B A draw the 
 radius to A, and extend it. At A 
 
 FIG. 36. 
 
 draw a tangent, meeting the given line at D. Bisect the angle A D E, so 
 formed, with a line cutting the radius, as extended at F ; and, on the cen- 
 tre F, with radius F A, describe the arc A E, which will be the arc required. 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 13 
 
 To describe a circular arc joining two circles, and to touch one of them at a 
 given point (Fig. 36). 
 
 Let A B and F G be the given circles to be joined by an arc touching one 
 of them at F. 
 
 (L r 
 
 FIG. 37. 
 
 FIG. 38. 
 
 Draw the radius E F, and produce it both ways ; set off F H equal to the 
 radius, A C, of the other circle ; join to H, and bisect it with the perpen- 
 dicular L I, cutting E F at I. On the centre I, with radius I F, describe the 
 arc F A, which will be the arc required. 
 
 To find the arc which shall be tangent to a given point on a straight line, 
 and pass through a given point outside the line (Fig. 37). 
 
 Erect at A, the given point on the giv- 
 en line, a perpendicular to the line. From 
 C, the given point outside the line, draw 
 C A, and bisect it with a perpendicular. 
 The intersection of the two perpendiculars 
 at a will be the centre of the required arc. 
 To connect two parallel lines by a re- 
 versed curve composed of two arcs of equal 
 radii, and tangent to the lines at given 
 points (Fig. 38). 
 
 Join the two given points A and B, and divide the line A B into two equal 
 parts at C ; bisect C A and C B by perpendiculars ; at A and B erect perpen- 
 diculars to the given 
 lines, and the intersec- 
 tions a and b will be 
 the centres of the arcs 
 composing the required 
 curve. 
 
 To join two given 
 points in two given 
 parallel lines by a re- 
 versed curve of two 
 equal arcs, whose cen- 
 tres lie in the parallels 
 (Fig. 39). 
 
 Join the two given 
 
 points A and B, and divide the line A B in equal parts at C. Bisect A C and 
 B C by perpendiculars ; the intersections a and b of the parallel lines, by these 
 perpendiculars, will be the centres of the required arcs. 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 On a given line, 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 on the perpen- 
 dicular, bisecting the given line, and to be tangent to the other two arcs 
 (Fig. 40). 
 
 Let A B be the given line, and C the given point. Draw C D perpen- 
 dicular to A B ; lay off A a, B b, and C c, each equal to the given radius of the 
 side arcs ; draw a c, and bisect it by a perpendicular ; the intersection of this 
 line with the perpendicular C D will be the required centre of the central arc 
 e C e'. Through a and b draw the lines D e and D e' ; from a and b, with the 
 given radius equal to a A or b B, describe the arcs A e and B e'. From D, as 
 a centre, with a radius equal to C D, and, consequently, by construction, equal 
 to D e and D e', describe the arc e C e'. The entire curve A e C e' B is the 
 compound curve required. 
 
 PROBLEMS ON POLYGONS AND CHICLES. 
 
 Three lines inclosing a space form a triangle (Fig. 41). If two of the sides 
 are of equal length, it is an isosceles triangle (Fig. 42) ; if all three are of equal 
 
 Fio. 41. 
 
 FIG. 42. 
 
 length, it is an equilateral triangle (Fig. 43). If one of the angles is a right 
 angle, it is a right-angled triangle (Fig. 44), and if no two of the sides are of 
 equal length, and not one of the angles a right angle, it is a scalene triangle. 
 
 To construct an isosceles triangle (Fig. 42). 
 
 Draw any line as a base, and, from 
 each extremity as a centre, with equal 
 radius, describe intersecting arcs. Draw 
 
 FIG. 43. 
 
 FIG. 44. 
 
 a line from each extremity of the base to this point of intersection, and the 
 figure is an isosceles triangle. 
 
 To construct an equilateral triangle (Fig. 43). 
 
 Draw a base line, and from each extremity as a centre, with a radius equal 
 
CONSTRUCTION OP GEOMETRICAL PROBLEMS. 
 
 15 
 
 to the base line, describe intersecting arcs. Draw lines from the extremi- 
 ties of the base to this point of intersection, and the figure is an equilateral 
 triangle. 
 
 To construct a right-angled triangle (Fig. 44). 
 
 Construct a right angle by any one of the methods before described. 
 Draw a line from the extremity of the one side to the extremity of the other 
 side, and the figure is a right-angled triangle. 
 
 It will be evident, in looking at any right-angled triangle, that the side 
 opposite the right angle is longer than either of the other or adjacent sides ; 
 this side is called the hypothenuse. 
 
 To construct a triangle equal to a given triangle ABC (Fig. 45). 
 
 1st Method (Fig. 46). Draw a base line, and lay off its length equal to 
 A B; from one of its extremities, as a centre, with a radius equal to A C, 
 describe an arc ; and, from its other extremity, with a radius equal to B C, 
 describe an arc intersecting the first. Draw lines from the extremities to the 
 point of intersection, and the triangle equal to A B C is complete. 
 
 FIG. 45. 
 
 FIG. 46. 
 
 2d Method (Fig. 47). Draw a base line, as before, equal to A B. At one 
 extremity construct an angle equal to C A B, and at the other an angle equal 
 to C B A. The sides of these angles will intersect, and form the required 
 triangle. 
 
 3d Method (Fig. 48). Construct an angle of the triangle equal to any angle 
 of A B C, say the angle A C B. On one of its sides measure a line equal to 
 C A, and on the other side one equal to C B ; connect the two extremities by a 
 line, and the triangle equal to A B C is complete. 
 
 FIG. 47. 
 
 FIG. 48. 
 
 From the above constructions it will be seen that, if the three sides of a 
 triangle, or two sides and the included angle, or one side and the two adjacent 
 angles are known, the triangle can be constructed. 
 
 Construct a triangle, ABC (Fig. 49). Extend the base to E, and draw 
 B D parallel to A C. As A C has the same inclination to C B that B D has, 
 the angle C B D is equal to the angle A C B. As A C has the same inclina- 
 tion to A E that B D has, the angle D B E is equal to C A B. That is, the 
 
16 
 
 CONSTRUCTION OP GEOMETRICAL PROBLEMS. 
 
 two angles formed outside the triangle are equal to the two inside at A and ; 
 and the three angles at B are equal to the three angles of the triangle, and 
 their sum is equal to two right angles. Therefore, the sum of the three angles 
 of a triangle is equal to two right angles. 
 
 On one side of a triangle (Fig. 50) construct a triangle equal to the first. 
 The exterior lines of the two triangles form a four-sided or quadrilateral 
 
 figure, of which the opposite sides are 
 equal and parallel, and the opposite an- 
 
 FIG. 49. 
 
 FIG. 50. 
 
 gles equal. This figure is called a parallelogram, and the line C B, extend- 
 ing between opposite angles, is a diagonal. 
 
 On the hypothenuse of a right-angled triangle (Fig. 51) construct another 
 equal to it, and the exterior lines form a parallelogram, which, as all the angles 
 are right angles, is called a rectangle. If the four sides are all equal, it is called 
 a square. 
 
 A parallelogram in which all the sides are equal, but the angles not right 
 angles, is called a rhombus (Fig. 52) ; if only the opposite sides are equal, it is 
 
 FIG. 51. 
 
 FIG. 52. 
 
 called a rhomboid (Fig. 50) ; if only two sides are parallel, the figure is a trape- 
 zoid (Fig. 53). 
 
 Take any figure (Fig. 54) bounded by straight lines and from any interior 
 point draw lines to all the angles. There will be as many triangles as sides, and 
 the sum of the angles of the figure will be equal to as many times two right 
 
 FIG. 53. 
 
 FIG. 54. 
 
 angles as sides less the four right angles at the centre, the sum of the angles 
 of any triangle being equal to two right angles. If a line be drawn from the 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 interior point to one side, another triangle is added to the collection and two 
 right angles to the sum of the angles. 
 
 Polygons are figures of many angles, which if equal and of equal sides are 
 
 FIG. 55. 
 
 FIG. 56. 
 
 FIG. 57. 
 
 FIG. 58. 
 
 called regular polygons, and are designated by the number of their sides, as 
 pentagons, hexagons, octagons, nonagons, decagons, etc. 
 
 To describe a circle about a triangle (Fig. 59). 
 
 Bisect two of the sides A B, A C, of the triangle at E, F ; from these points 
 draw perpendiculars cutting at K. From the centre K, with K A as radius, 
 describe the circle A B C, as required. 
 
 To inscribe a circle in a triangle (Fig. 60). 
 
 Bisect two of the angles A, C, of the triangle 
 A B C, by lines cutting at D ; from D draw a 
 perpendicular D E to any side, as A C ; and 
 with D E as radius, from the centre D, describe 
 the circle required. 
 
 When the triangle is equilateral, the centre 
 of the circle is more easily found by bisecting 
 two of the sides, and drawing perpendiculars. 
 Or, draw a perpendicular from one of the sides 
 to the opposite angle, and from the side set off 
 one third of the length of the perpendicular. 
 
 To inscribe a square in a circle ; and to describe a circle about a square 
 (Fig. 61). 
 
 To inscribe the square. Draw two diameters, A B, C D, at right angles, 
 and join the points A, C, B, D, to form the square as required. 
 
 FIG. 60. 
 
 FIG. 61. 
 
 To describe the circle. Draw the diagonals A B, C D, of the given square, 
 cutting at E ; on E as a centre, with E A as radius, describe the circle as required. 
 
 3 
 
18 
 
 CONSTRUCTION OP GEOMETRICAL PROBLEMS. 
 
 In the same way, a circle may be described about a rectangle. 
 
 To inscribe, a circle in a square ; and to describe a square about a circle 
 (Fig. 62). 
 
 To inscribe the circle. Draw the diagonals A B, C D, of the given square, 
 cutting at E ; draw the perpendicular E F to one of the sides, and with the 
 radius E F, on the centre E, describe the circle. 
 
 To describe the square. Draw two diameters A B, C D, at right angles, 
 
 ( \ '/ 
 
 x^ 
 
 >*' 
 
 
 X v - 
 
 
 XL J; 
 
 1 
 
 F 
 
 FIG. 62. 
 
 1 
 
 and produce them ; bisect the angle D E B at the centre by the diameter F G,. 
 and through F and G draw perpendiculars A C, B D, and join the points A, D, 
 B, C, where they cut the diagonals, to complete the square. 
 
 To inscribe a pentagon in a circle (Fig. 63). 
 
 Draw two diameters, A C, B D, at right angles ; bisect A at E, and from 
 E, with radius E B, cut A C at F ; from B, with radius B F, cut the circum- 
 ference at G and H, and with the same radius step round the circle. to I and K ; 
 join the points so found to form the pentagon. 
 
 To construct a regular hexagon upon a given straight line (Fig. 64). 
 
 From A and B, with a radius equal to the given line, describe arcs cutting 
 at g ; from g, with the radius g A, describe a circle ; with the same radius set 
 
 FIG. 64. 
 
 off from A the arcs A G, G F, and from B the arcs B D, D E. Join the 
 points so found to form the hexagon. 
 
 To inscribe a regular hexagon in a circle (Fig. 65). 
 
 Draw a diameter, A B ; from A and B as centres, with the radius of the 
 circle A C, cut the circumference at D, E, F, G ; draw straight lines A D, 
 D E, etc., to form the hexagon. 
 
 To describe a regular hexagon about a circle (Fig. 66). 
 
CONSTRUCTION OP GEOMETRICAL PROBLEMS. 
 
 19 
 
 Draw a diameter, A B, of the given circle. With a radius A D from A as a 
 centre, cut the circumference at C ; join A C, and bisect it with the radius 
 D E ; through E draw F G parallel to A C, and with the radius D F describe 
 
 FIG. 
 
 the circle F H. Within this circle describe a regular hexagon by the preceding 
 problem ; the figure will touch the given circle as required. 
 
 To construct a regular octagon upon a given straight line (Fig. 67). 
 
 Produce the given line A B both ways, and draw perpendiculars A E, B F ; 
 
 m I 
 
 FIG. 68. 
 
 E .- 
 
 - -F 
 
 bisect the external angles at A and B by the lines A H, B C, each equal to 
 A B ; draw C D and H G parallel to A E and equal to A B ; and from the 
 centers G, D, with a radius equal to A B, cut the 
 perpendiculars at E, F, and draw E F to complete 
 the octagon. 
 
 To make a regular octagon from a square 
 (Fig. 68). 
 
 Draw the diagonals of the square intersecting 
 at e ; from the corners A, B, C, D, with A e as 
 radius, describe arcs cutting the sides at g h, etc. ; 
 join the points so found to complete the octagon. 
 
 To inscribe a regular octagon in a circle (Fig. 
 69). 
 
 Draw two diameters, AC, B D, at right an- 
 gles ; bisect the arcs A B, B C, etc., at e,/, etc. ; and join A e, e B, etc., for the 
 inscribed figure. 
 
 To describe a regular octagon about a circle (Fig. 70). 
 
20 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 Describe a square about the given circle A B ; draw perpendiculars h k, etc., 
 to the diagonals, touching the circle. 
 
 Or, to find the points h, k, etc., cut the sides from the corners of the square, 
 as in Fig. 68. 
 
 To inscribe a circle within a regular polygon. 
 
 When the polygon has an even number of sides, as in Fig. 71, bisect two 
 
 FIG. 71. 
 
 Fio. 72. 
 
 opposite sides at A and B, draw A B, and bisect it at C by D E drawn between 
 opposite angles ; with the radius C A describe the circle as required. 
 
 When the number of sides is odd, as in Fig. 72, bisect two of the sides at A 
 and B, and draw lines A E, B D, to the opposite angles, intersecting at C ; 
 from C, with C A as radius, describe the circle as required. 
 
 To describe a circle about a regular polygon. 
 
 When the number of 
 sides is even, draw two 
 diagonals from opposite 
 angles, like D E (Fig. 
 71),,, to intersect at C; 
 and from C, with C D as 
 radius, describe the cir- 
 cle required. 
 
 When the number of 
 sides is odd, find the cen- 
 tre C (Fig. 72) as in last 
 problem, and, with C D 
 as radius, describe the 
 circle. 
 
 For the construction 
 of the regular polygons 
 Fig. 73 will be found 
 useful. 
 
 Divide the interior 
 circle into the number 
 
 of degrees corresponding Fja 73 
 
 to the proportion of the 
 
 sides of the polygon to the entire circle, e. g., -2-f -2- 72. With a radius of unity 
 describe an exterior circle and extend radii through the divisions of the in- 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 21 
 
 terior circle. The chords of th^ arcs intersected correspond to the sides of the 
 different polygons. 
 
 The figure gives the polygons such as are usually found in practice, but a 
 similar figure can be constructed increasing the number of sides as far as may 
 be required. 
 
 For the laying out of angles the protractor is used. In its simplest form it 
 consists of a semicircle of metal or horn of which the edge is divided into 180 
 degrees. 
 
 To lay off a given angle say 47 (Fig. 74) place the edge of the protractor, 
 a 5, along the given line and make the centre of the protractor coincide with 
 the vertex c of the angle to be laid off ; mark off on 
 the edge the division 47, remove the protractor, and 
 through this mark and the vertex c draw a line; 
 the angle a c d will be equal to 47, and b c d to 
 133. These two are supplements of each other, or 
 what each requires to make up the sum of 180. 
 
 Fig. 75 represents the terms used in defining 
 angles, and of which tables are given in the Appen- 
 dix by which angles may be constructed without the 
 use of the protractor. 
 
 Considering BCD the angle, the perpendicular 
 D H dropped from the radius at D and intersecting 
 
 the diameter at II is the sine, the line B A perpendicular to B C and inter- 
 secting the extended radius at A is the tangent, the extension of the radius C D 
 to the intersection of the tangent at A the secant ; the versed sine is the line 
 B II extending from the sine to the tangent. The cosine, cotangent, cosecant, 
 and coversed sine are respectively the sine, tangent, secant, and versed sine 
 of the angle D C F, the complement of B C D, having the number of degrees 
 necessary to complete the quadrant of 90 degrees. 
 
 FIG. 75. 
 
22 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 USE OF TKIANGLE AND SQUARE. 
 
 Right-angled triangles constructed of wood, hard rubber, celluloid, or metal 
 are very useful in connection with a straight-edge, or ruler, in drawing lines 
 parallel or perpendicular to other lines. 
 
 To draw lines parallel to each other, place any edge of the triangle in close 
 contact with the edge of the ruler. Hold the ruler (Fig. 76) firmly with the 
 
 FIG 
 
 thumb and little finger of the left hand, and the triangle with the other three 
 fingers ; with a pencil or pen in the right hand, draw a line along one of the 
 free edges of the triangle ; withdraw the pressure of the three fingers upon the 
 triangle, and slide it along the edge of the ruler, keeping the edges in close 
 contact ; a line drawn along the same edge of the triangle, as before, will be 
 parallel to the first line. If the edge of the hypothenuse of the triangle be 
 placed in contact with the ruler, lines drawn along one edge of the triangle will 
 be at right angles to those drawn along the other. 
 
 Through a given point to draw a line parallel to a given line (Fig. 77). 
 
 Place one of the shorter edges of the triangle along the given line A B, and 
 bring the ruler against the hypothenuse ; slide the triangle up along the edge 
 of the ruler until the upper edge of the ruler is sufficiently near to the given 
 point C to allow a line to be drawn through it. Draw the line, and it will be 
 parallel to A B. 
 
 If the triangle be slid farther up along the edge of the ruler, and a line be 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 23 
 
 drawn through C along the other edge of the triangle (Fig. 78), this line will 
 be perpendicular to A B. If the triangle be slid still farther up along the 
 
 FIG. 77. 
 
 edge of the ruler, and a third line be drawn touching A B, the figure con- 
 structed will be a rectangle ; and if E D be laid off on A B, equal to C E, the 
 figure inclosed is a square (Fig. 79). 
 
 It will be seen that the triangle and ruler afford a much readier way of 
 
 C C 
 f-\ f-\ 
 
 
 
 A 
 
 ,0, 
 
 B A 
 
 , ^-X- te 
 
 E D 
 
 B 
 
 _X 
 
 
 FIG. 78. 
 
 FIG. 79. 
 
 drawing parallel lines, and lines at right angles, than the compasses and ruler, 
 and may be used in solving the following problems : 
 
 The area of a figure is the 
 quantity of space inclosed by its 
 lines. 
 
 Construct a right angle (Fig. 
 SO). Divide the base and the 
 perpendicular by dividers into 
 any number of equal spaces ; for 
 instance, ten on the one and five 
 on the other. Construct a rec- 
 tangle with this base and perpen- 
 dicular, and through the points of division lay off lines parallel to the base and 
 perpendicular. The rectangle will be divided into fifty equal squares, and its 
 
 FIG. 80. 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 measure in squares will be the divisions ten in the base, multiplied by the five 
 in the perpendicular. If the division were inches, then the area of this rec- 
 tangle would be fifty square inches; if 
 feet, then fifty square feet. If there 
 
 D 
 
 Fio. 81. 
 
 FIG. 82. 
 
 were but five divisions in the base and five in the perpendicular, the surface 
 would be twenty-five squares. Therefore, a rectangle has for its measure the 
 base multiplied by its adjacent side or height. 
 
 Draw a diagonal, and the rectangle is divided into two equal triangles. 
 Each triangle must therefore have for its measure the base multiplied by half 
 the perpendicular, or, as is usually said, by half the altitude. 
 
 Take any triangle (Fig. 81), and from its apex draw a line perpendicular to 
 the base. The triangle is divided into two right-angled triangles, which must 
 have for their measure A D X i C D, and D B X i C I), and the sum of the 
 two must be A B X i C D. 
 
 If the perpendicular from the apex falls outside the triangle (Fig. 82), then 
 the triangle B D C and ADC will have for their measure B D X \ C D and 
 
 A D X \ C D, consequently their 
 difference, ABC, must have for 
 its measure A B X \ C D. Any 
 polygon can be divided into trian- 
 gles (see Fig. 54), and its area is 
 made up of the sum of the areas of 
 the triangles. By graphic con- 
 struction the sum of the areas of 
 the different triangles composing a 
 polygon may be resolved readily 
 into a single triangle and its area 
 taken. For instance, take a six- 
 sided polygon (Fig. 83), draw a line 
 from A to C, and a line parallel to A C, at B intersecting the extended base at 
 B', a diagonal drawn from A to B' will give one side of the triangle ; draw a 
 diagonal from A to E, extend the side E D of the polygon indefinitely, draw a 
 line at F parallel to A E, and intersect the extended side at e, draw the line 
 A D and a parallel to this at e, intersecting the extended base at E'. A di- 
 agonal drawn from A to E' will, with the side previously obtained and the base, 
 give a triangle equal in area to the polygon. 
 
 FIG. 83. 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 25 
 
 i 
 
 g 
 
 SCALES. 
 
 The distances given in Fig. 80 may represent feet, yards, miles, or any other 
 unit of measure. Thus, if they represent miles, the figure represents an area 
 of fifty square miles. With a scale of equal 
 parts, each part may represent any unit of 
 measure, and a drawing on paper to that scale 
 represents the object from which it is drawn 
 in a reduced form, from which measures in 
 detail by the scale may be more readily and 
 as accurately taken as from the natural ob- 
 ject in the shop or on the estate, and if de- 
 signs are made to a scale they can be exe- 
 cuted conformably and accurately in all their 
 parts in either enlarged or reduced size. 
 
 Practically a two-foot rule, with its divis- 
 ions into inches, halves, quarters, eighths, and 
 sixteenths, may be made use of as a scale of 
 equal parts, any division being taken as the 
 unit to represent a foot, a yard, or a mile ; but 
 among drawing instruments scales especially 
 adapted to the purpose are found in great 
 variety of forms, divisions, and material. Fig. 
 84 represents a convenient form of scale, as it 
 contains, in addition to the simply divided 
 scales, a protractor along its edges. 
 
 The simply divided scales consist of a series 
 of equal divisions of an inch, which are num- 
 bered 1, 2, 3, etc., beginning from the second 
 division on the left hand ; the upper part of 
 the left division in each is subdivided into 
 twelve equal parts, and the lower part into ten 
 equal parts. The scales are marked at the left 
 1 inch, , , , and when used in drawing the 
 scale is written as I inch, f , , or inch to a 
 foot, rod, or mile, or whatever may be the 
 unit of actual measure. When the unit is the 
 inch the first scale will be full size, the second 
 f, the third , and the fourth full size. If 
 the scale adopted is such a part of an inch to 
 the foot, then the upper subdivisions will 
 represent inches. 
 
 Above the simply divided scales there is a 
 scale marked C, which is a scale of chords; taking a radius equal to C-60, the 
 chords of the different angles are represented by the division ; thus an angle of 
 20 the chord will be C-20. 
 
 - 
 
 at 
 
 OUT 
 
 \ 
 
 s 
 
 Fia. 84. 
 
CONSTRUCTION OP GEOMETRICAL PROBLEMS. 
 
 SIMILAR TRIANGLES. 
 
 Triangles which are equiangular are similar, and have their homologous 
 sides that is, their sides adjacent to the equal angles proportional ; conversely, 
 two triangles which have their homologous sides proportional are equiangular. 
 
 Two triangles which have their sides parallel (Fig. 85) or perpendicular to 
 
 FIG 85. 
 
 FIG. 86. 
 
 each other (Fig. 86) are similar. A line c drawn parallel to one side c' of a 
 triangle (Fig. 87) forms a triangle a c b whose sides are proportional to the 
 original triangle. 
 
 In Fig. 88 a polygon is divided into triangles by lines from an interior 
 point to its angles and these lines intersected by lines parallel to the sides of 
 the polygon. The figure thus constructed is a polygon similar to the original 
 polygon composed of triangles similar to the triangles into which it was 
 divided. 
 
 In the figure the parallel lines are drawn across the sides of the triangles at 
 one half their length, and the areas of the small triangles are therefore equal to 
 the square of one half or to one quarter that of the original triangles ; conse- 
 quently the area of the interior 
 polygon is one quarter that of the 
 exterior one. As this construction 
 obtains at any intersecting length, 
 it affords a means of reducing the 
 scale of the original polygon. 
 
 To a scale of inch (Fig. 89) 
 
 lay off a line and divide from by equal units to 6 ; at 6, with a radius equal 
 to 6 on scale ( inch), describe an arc, and from with a scale of f inch, with 
 a radius equal to 6, intersect the previous arc. Complete the triangle through 
 this intersection, and draw lines parallel to 6, 6' through the divisions of the 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 27 
 
 first line ; the triangle will be divided into six similar triangles of which the 
 homologous sides are proportional and represented on their different scales by 
 the same number of units. 
 
 FIG 89. 
 
 Fig. 90 illustrates the application of scales to the measurement of lines 
 
 which are inaccessible. Thus the lines a b and a c, with their inclosed angle, 
 
 can be measured, and, if plotted to any scale, the line c b can be measured on 
 
 the same scale. 
 
 The height of an object may be obtained by the application of similar tri- 
 
 angles, or by the length of the shadow cast, which is merely another applica- 
 
 tion of the same method. The observer measures 
 
 off, say, 60 feet from a flag pole (Fig. 91), and a 
 
 rod is held at, say, 12 feet from the observer; a 
 
 sight is then taken to the top and bottom of the 
 
 flag pole at the 60-feet distance, and the points at 
 
 which the sights intersect the rod are found to be 
 
 10 feet apart. Then by construction the height of 
 
 the flag pole is found by scale to be 50 feet. By 
 
 means of shadows, if the length of the shadow is 
 
 found to be 40 feet and the shadow cast by a 10- 
 
 foot rod is 8 feet, then by plotting the height is found, as before, 50 feet. 
 
 The value of the above solution of geometrical problems depends on the 
 
 accuracy of the drawing. 
 
 To construct a square equal to one half of a given square (Fig. 92). 
 
 Let a b c d be the given 
 square, and draw diagonals 
 in it. The square, e b f d, 
 constructed on one half of 
 one of these diagonals, will 
 be equal to one half the' 
 given square. 
 
 FIG. 91. 
 
 To construct a -square equal to double a given square (Fig. 93). 
 Construct a square on one of the diagonals in the given square, or inclose 
 the square with parallels to the diagonals of the square. 
 
28 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 To construct a square equal to three times a given square (Fig. 94). 
 
 Extend the base of the given square to the length of its diagonal. Draw a 
 line from the point at which this line ends to the extreme angle of the square, 
 and upon this line erect a square, which will be the square required. 
 
 For a square four times the size of a given square, make the base double 
 that of the given square. 
 
 FIG. 94. 
 
 To construct a square equal to five times a given square (Fig. 95). 
 
 Extend the base of the given square, making the extension to d e equal to 
 c d. From e draw a line to #, and on this line construct a square, which will 
 be the square required. 
 
 FIG. 95. 
 
 Assuming the side of the given square in Figs. 92, 93, 94, and 95 to be the 
 radius (or diameter) (Fig. 96) of a given circle, then the side of the square to 
 be constructed half, twice, three, four, or five times the size of the given square 
 will be the radii (or diameters) of the circles half, twice, three, four, or five 
 times the size of the given circle. 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 29 
 
 It will be seen by Fig. 93 that the square constructed on the diagonal of a 
 square is equal to double that of the original square. 
 
 FIG 
 
 On any right-angled triangle A C B (Fig. 97) let fall a perpendicular from 
 the vertex of the right angle to the hypothenuse A B ; the triangle will be 
 divided into two similar triangles, similar to each other and to the original 
 triangle, and 
 
 AD:AC::AC:AB; that is, A C 2 = A D X A B ; 
 
 BD:BC::BC:AB; that is, B C 2 = B D X A B. 
 
 AD-(-BD = AB, and the sum of the two equations is A B 2 = A C 8 + 
 BC 2 . 
 
 Therefore the square constructed 
 on the hypothenuse of a right-angled 
 triangle is equal to the sum of the 
 squares of the other two sides (Fig. 
 98). 
 
 To determine how much is added 
 to a given square by extending its base 
 and constructing a square thereon 
 (Fig. 99). 
 
 Let a represent the side C D of the given square. The area of the square 
 is a X a or a 2 . 
 
 Extend the side C D by a length, D G, represented by b. Then the new 
 
 FIG. 97. 
 
30 
 
 CONSTRUCTION OP GEOMETRICAL PROBLEMS. 
 
 square (a -\- b) X (a + b) will be made up of the old square, or a 2 , and two rec- 
 tangles, D G E H and P E K L, or 2 (a X b) or 2 a b, and one square, E H K J, 
 
 b X b, or b 2 . The area 
 
 To determine how much 
 is taken from the area of a 
 given square, by reducing 
 its base and constructing a 
 square (Fig. 99). 
 
 Let a represent C G, 
 the side of the given square. 
 
 FIG. 99. 
 
 Reduce C G, the length G D, represented by b. The new square (a b) s is 
 the given square, or a 2 , diminished by two rectangles, D G J K and P L J H, 
 or 2 a b excepting one square, E H J K, b X b or -f- b 2 . The area (a b) 2 
 = a*-2ab + b 2 . 
 
 The last two constructions, in default of a table of squares, may often be 
 found of use. 
 
 CONSTRUCTION OF THE ELLIPSE, PARABOLA, HYPERBOLA, AND SPIRAL. 
 
 An ellipse is an oval-shaped curve (Fig. 100), in which, if from any point, P, 
 lines be drawn to two fixed points, F and F', called foci, their sum will always 
 
 be the same. The line A B pass- 
 >_ ing through the foci is the major 
 
 axis, and the perpendicular C D at 
 the centre of it is the minor axis. 
 
 To construct an ellipse, the axes 
 being known (Fig. 100). 
 
 1st Method. Let the two axes 
 be the lines A B and C D. From 
 C as a centre, with a radius equal 
 to E B (half the major axis), de- 
 scribe an arc cutting this axis at 
 two points, F and F', which are 
 the foci. Insert a pin in each of 
 the foci, and loop a thread upon them, so that, when stretched by a pencil 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 31 
 
 FIG. 101. 
 
 inside the loop, the point of the pencil will coincide with C. Move the pencil 
 round, keeping the loop evenly stretched, and it will describe an ellipse. This 
 construction follows the definition above given of an ellipse, that the sum of 
 the distances of every point of the curve from the foci is equal. It is seldom 
 used by the draughtsman, as it is difficult to keep a thread evenly stretched ; 
 but for gardeners, laying out beds or plots, it is very convenient and sufficiently 
 accurate. 
 
 There are many forms of ellipsographs for drawing ellipses, and various 
 sizes of ellipses in hard wood and rubber on sale. Pattern-makers usually lay 
 out ellipses by means of a trammel 
 (Fig. 101), which consists of a 
 rectangular cross, with guiding 
 grooves in which movable rods at- 
 tached to sliders on a bar are 
 fitted, so as to move easily and uni- 
 formly. In describing an ellipse 
 place the trammel with its grooves 
 on the lines of the axes with the 
 bar on the line of the major axis ; 
 set the pencil or marker on the 
 extremity of this axis, and slip the 
 outer rod to the crossing of the 
 grooves and clamp it to the bar. Now slide the rod down the minor axis, and, 
 with the pencil at the extremity of this axis, clamp the intermediate rod to the 
 bar at the crossing of the guides. Revolve the bar, the intermediate rod fol- 
 lowing the major-axis groove, and the extreme rod that of the minor axis, 
 and the pencil will describe the ellipse. Light trammels are made for the 
 
 use of draughtsmen, but, as 
 the necessity of drawing el- 
 lipses is not frequent, it can 
 be readily done by the use 
 of a strip of cardboard (Fig. 
 102). Lay off the major 
 and minor axes on the pa- 
 per ; these represent the 
 grooves of the trammel. 
 Now take a strip of card- 
 board with a straight edge, 
 lay it along the line of the 
 
 maior axis, and mark the 
 FIG. 102. J . 
 
 positions a at the extremity 
 
 of this axis, and c at the crossing of the axes ; place the mark a on the ex- 
 tremity of the minor axis, and mark on the edge of the card at b the cross- 
 ing of the axes. Revolve the card as described for the trammel, mark the posi- 
 tions of a by points, and connect them for the curve. 
 
 To construct an approximate semi-ellipse by means of Jive arcs of circles. 
 
 Let A B (Fig. 103) be the major axis, and D the semi-minor axis. Draw 
 the semicircles A C B and d D a". Divide these semicircles into equal parts 
 
32 
 
 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 by the radial lines e, Of, e', Of. From the points of intersection of 
 these radial lines with the semicircumference draw g b, h a, h' a', g' b', parallel 
 to the major axis. From e,f, e',f, intersections of the radial lines with the 
 
 semicircumference A C B, draw e b, fa, e' a', and f V parallel to the minor 
 axis. The intersections of these lines with b g, a h, etc., will be points on the 
 ellipse. Now through the three points a, D and a' construct an arc of a circle. 
 Connect a and b with a chord, bisect it with a perpendicular ; where this per- 
 pendicular intersects a S 
 
 at c is the centre of the 
 
 , 
 arc a b. 
 
 Connect b and c; d, 
 the intersection of b c with 
 A B, will be the centre of 
 the arc b A. Arcs through 
 a' b' and B can be con- 
 structed in the same way, 
 or the centres can be 
 transferred. 
 
 The ellipse can in the same way be made up of any number of arcs of 
 circles. 
 
 To draw a tangent to an ellipse through a given point in the curve (Fig. 104). 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 33 
 
 From the given point T d raw i straight lines to the foci, F, F' ; produce F T 
 beyond the curve to c, and bisect the exterior angle c T F' by the line T d. 
 This line T d is the tangent 
 required. 
 
 To draw a tangent to an 
 ellipse from a given point 
 without the curve (Fig. 105). 
 
 From the given point T 
 as a centre, with a radius 
 equal to its distance from 
 the nearest focus F, describe 
 an arc ; from the other focus 
 F', with the major axis as 
 radius, cut the arc at K, L, and draw K F', L F', touching the curve at M, N ; 
 then the lines T M, T N, are tangents to the curve. 
 
 The Parabola. 
 
 The parabola may be defined as an ellipse whose major axis is infinite ; its 
 characteristic is that every point in the curve is equally distant from the direc- 
 trix E N and the focus F (Fig. 106). 
 
 To construct a parabola when the focus and directrix are given. 
 
 1st Method (Fig. 106). Through the 
 focus F draw the axis A B perpendicular 
 to the directrix E N, and bisect A F 
 
 FIG. 106. 
 
 FIG. 107. 
 
 at e, the vertex of the curve. Through a series of points, C, D, E, on the di- 
 rectrix, draw parallels to A B ; connect these points, C, D, E, with the focus 
 F, and bisect by perpendiculars the lines F C, F D, F E. The intersections 
 of these perpendiculars with the parallels will give points, C', D', E', in the 
 curve, through which trace the parabola. 
 
 2d Method (Fig. 107). Place a straight-edge to the directrix E N, and 
 apply to it a square LEG; fasten at G- one end of a cord, equal in length to 
 4 
 
34 CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 E G ; fix the other end to the focus F ; slide the square steadily along the 
 straight-edge, holding the cord taut against the edge of the square by a pencil, 
 D, and it will describe the curve. 
 
 To construct a parabola when the vertex, the axis, and a point of the curve 
 are given (Fig. 108). 
 
 Let A be the vertex, A B the axis, and D the point in the curve. Con- 
 struct the rectangle A B D C ; divide D C into, say, four equal parts at 123, 
 and A C into the same number at 1' 2' 3' ; draw diagonals, A 3, A 2, A 1 ; and 
 
 parallels to the axis through 1' 2' 3'. The intersection of the diagonals A 3, 
 A 2, A 1 with the parallels 3', 2', 1' at G, F, E will be points in the required 
 curve. 
 
 Extend the axis to B', making A B'=A B ; draw perpendiculars to the axis 
 from G, F, E, D ; lay off toward B', a'=A. a, A '=A b, A c' = A c; and draw 
 B' D, c' E, V F and a' G. These lines will be tangents to the curve at D, E, 
 F, G, and lines perpendicular to the tangents at these points will be perpen- 
 dicular to the curve. 
 
 The Hyperbola. 
 
 An hyperbola is a curve from any point, P, in which, if two straight lines 
 be drawn to two fixed points, F, F', the foci, their difference will always be 
 the same. 
 
 To describe an hyperbola (Fig. 109). 
 
 From one of the foci, F, with an assumed radius, describe an arc, and from 
 the other, focus F', with another radius exceeding the former by the given 
 
 difference, describe two small arcs, cut- 
 ting the first as at Pandjt?/ Let this 
 operation be repeated with two new 
 radii, taking care that the second shall 
 exceed the first by the same difference 
 as before, and two new points will be 
 determined ; and this determination of 
 points in the curve may thus be con- 
 tinued till its track is obvious. By 
 making use of the same radii, but 
 transposing, that is, describing with the 
 greater about F, and the less about F', 
 FIG. 109. we have another series of points equal- 
 
CONSTRUCTION OF GEOMETRICAL PROBLEMS. 
 
 35 
 
 ly belonging to the hyperbola, and answering the definition ; so that the hyper- 
 bola consists of two separate branches. 
 
 The curve may be described mechanically (Fig. 110) by fixing a ruler to 
 one focus, F', so that it may be turned round on this point, and connecting the 
 other extremity of the ruler, R, to the 
 other focus, F, by a cord shorter than 
 the whole length of the ruler by the 
 given difference ; then a pencil, P, 
 
 FIG. 110. 
 
 FIG. 111. 
 
 keeping this cord always stretched, and at the same time pressing against the 
 edge of the ruler, will, as the ruler revolves around F', describe an hyperbola. 
 
 To draw a tangent to any point of an hyperbola (Fig. 111). 
 
 Let P be the point. On F' P lay off P p, equal to F P ; draw the line F p; 
 from P let fall a perpendicular on this line, P p, for the tangent. 
 
 To describe a spiral (Fig. 112 and Fig. 113, the primary on a larger scale). 
 
 Divide the circumference of the primary into any number of equal parts, 
 say not less than eight. To these points of division o, e,f, i, etc., draw tangents, 
 
 \ 
 
 FIG. 112. 
 
 FIG. 113. 
 
 and from these points draw a succession of circular arcs ; thus, from o lay off 
 o g, equal to the arc a o reduced to a straight line, and connect a and g by a 
 curve ; from e, with the radius e g, describe the arc g h; from / the next arc, 
 and so on. Continue the use of the centres successively and repeatedly to the 
 extent of the revolutions required. 
 
DRAWING INSTRUMENTS. 
 
 THE simple drawing instruments illustrated and applied in the construc- 
 tion of the preceding problems, together with scales of equal parts, a protractor, 
 and a drawing-pen, are all the instruments essential for topographical or me- 
 chanical drawing. It is often convenient, for facility in working, to have com- 
 passes of various sizes and modifications, and these, together with an assortment 
 of rulers, triangles, squares, scales, and protractors adapted to varied work, are 
 included in boxes of drawing instruments furnished by dealers. The smaller 
 rulers and triangles are generally of hard rubber, and the larger of wood. As 
 it is often inconvenient to carry long rulers, or straight-edges, and difficult to 
 
 procure them ready-made, the draughts- 
 man may have to depend on a carpenter 
 or joiner for them. 
 
 The drawing-board in its simplest 
 form consists merely of narrow strips of 
 thoroughly seasoned white-pine wood, 
 free from knots, closely joined and glued, 
 and held together either with a ledge at 
 each end or with battens screwed to the 
 back. For small boards, the former 
 kind is in some ways the best, as it ad- 
 mits of being planed on all four edges. 
 
 Fig. 114 is more elaborate and one 
 of the best drawing boards, possessing 
 all the qualities of a first-class board. It 
 is made of pine wood, glued up to the 
 required width, with the heart side of 
 each piece of wood at the surface. A 
 pair of hard-wood battens are screwed to 
 
 the back, the screws passing through the ledges in oblong slots that are bushed 
 with brass, which fit closely under the heads, and yet allows the screws to 
 move freely when drawn by the contraction of the board. To give the battens 
 power to resist the tendency of the surface to warp, a series of grooves are 
 
 36 
 
 FIG. 114. 
 
DRAWING INSTRUMENTS. 
 
 37 
 
 sunk, half the thickness of the board, over the entire back. These grooves 
 take the transverse strength out of the wood and allow it to be controlled by 
 the battens, leaving at the same time the longitudinal strength of the wood 
 nearly unimpaired. 
 
 To make the two working edges perfectly smooth, allowing an easy move- 
 ment of the square, a slip of hard wood is let into the end of the board. The 
 slip is afterward sawed apart at about every inch to admit of contraction. The 
 drawing-board should be truly rectangular and have perfectly straight sides, 
 for the use of the T square. Two sizes are sufficient for ordinary use 41 x 30 
 inches for double elephant paper, and 
 31X24 inches for imperial and smaller 
 sizes. Boards smaller than these are 
 too light, and unsteady in handling. 
 
 The drawing-table should be about 
 6 feet long and 4 feet wide, of 1 
 inch stuff, constructed similarly to 
 
 FIG. 115. 
 
 the drawing-board, and it is usually 
 supported by a pedestal the height 
 and inclination of which is adjusta- 
 ble, or on trestles, or a strong frame at 
 such height that the draughtsman 
 may not have to stoop to his work. 
 
 Fig. 115 shows an excellent form 
 of trestle ; the upper part of the horses is attached to hard- wood supports, 
 which slide through the body of the trestle and are provided with numerous 
 holes ; by means of strong pins passing through the body of the horses and 
 the holes the board may be set at various angles, the steel points in the top 
 preventing the drawing-board from sliding or slipping off. 
 
 Straight-edges are made of close-grained, thoroughly seasoned wood, such 
 as mahogany, maple, pear, etc. ; also of celluloid, hard rubber, steel, or German 
 silver. Those made of maple or pear wood answer every purpose and have the 
 advantage of soiling the paper less than rubber or metal. No varnish of any 
 description should be applied to any of the instruments used in drawing, as 
 varnish will retain dust and soil the paper. Use the wood in its natural state, 
 keeping it carefully wiped. Straight-edges should be about inch thick in the 
 square or slightly rounded edges and 1 to 2 inches wide, according to their 
 length. As the accuracy of a drawing depends greatly on the straightness of 
 the lines, the edge of the ruler should be perfectly straight. To test this, 
 place a sheet of paper on a perfectly smooth board; insert two very fine 
 needles in an upright position through the paper into the board, distant from 
 each other nearly the length of the ruler to be tested ; bring the edge of the 
 ruler against these needles, and draw a line from one needle to the other; 
 reverse the ruler, bringing the same edge on the opposite side and against the 
 needles, and again draw a line. If the two lines coincide, the edge is straight ; 
 but if they disagree, the ruler is inaccurate, and must be rcjointed. When 
 one ruler has been tested, others can be examined by placing their edges against 
 the correct one, and holding them between the eye and the light. 
 
 Triangles may be made of the same kinds of wood as the ruler, somewhat 
 
38 
 
 DRAWING INSTRUMENTS. 
 
 thinner, and of various sizes. They should be right-angled, with acute angles 
 of 45, or of 60 and 30. The most convenient size for general use measures 
 
 from 3 to 6 inches on the side. A larger size, from 8 to 10 inches long on the 
 side, is convenient for making drawings to a large scale. In the smaller trian- 
 gles circular openings are made in the body for the insertion of the end of the 
 finger, to give facility in sliding the triangle on the paper. Triangles are 
 sometimes made as large as 15 to 18 inches on the side ; but in this case they 
 
 are framed in three pieces, about 1 inch 
 wide, leading the centre of the triangle open. 
 The value of the triangle in drawing perpen- 
 dicular lines depends on the accuracy of the 
 right angle. To test this (Fig. 116), draw a 
 line with an accurate ruler on paper. Place 
 the right angle of the triangle near the 
 centre of this line, and make one of the ad- 
 jacent sides to coincide with the line ; now 
 draw a line along the other adjacent side, 
 
 which, if the angle is strictly a right angle, will be perpendicular to the first 
 line. Turn the triangle on this perpendicular side, bringing it into the position 
 ABC'; if now the sides of the triangle agree with the line B C' and A B, 
 the angle is a right angle, and the sides are straight. If they do not agree, 
 they must be made to do so with a plane, if right angles are to be drawn 
 by the triangle. The straightness of the hypothenuse or longest side can be 
 tested like a common ruler. 
 
 FIG. 117. 
 
 The T square is a thin straight-edge or ruler (Fig. 117), fitted at one end 
 with a stock, applied transversely at right angles. The stock being so formed 
 
DRAWING INSTRUMENTS. 
 
 39 
 
 as to fit and slide against one edge of the drawing-board, the blade reaches 
 over the surface, and presents an' edge of its own at right angles to that of the 
 board, by which parallel straight lines may be drawn upon the paper. The 
 stock should be long enough to give sufficient bearing on the edge of the board, 
 and heavy enough to act as a balance to the blade, and to relieve the operation 
 of handling the square. The blade should be sunk flush into the upper half of 
 the stock on the inside, and very exactly fitted. It should be inserted full 
 breadth, as shown in the figure ; notching and dovetailing is a mistake, as it 
 weakens the blade, and adds nothing to the security. The upper half of the 
 stock should be about ^ inch broader than the lower half, to rest firmly on the 
 board and secure the blade lying flatly on the paper. 
 
 One half of the stock c (Fig. 118) is in some cases made loose, to turn upon 
 a brass swivel to any angle with the blade a, and to be clenched by a screwed 
 
 FIG. 118. 
 
 nut and washer. The loose stock is useful for drawing parallel lines obliquely 
 to the edges of the board, such as the threads of screws, oblique columns, or 
 connecting-rods of steam-engines. 
 
 T squares are also made with a single movable head, shown in Fig. 119 ; 
 
 FIGS. 119, 120. 
 
 the blade, turning on , is clamped in position by the thumb-nut b. Fig. 120 
 illustrates a T square with a protractor at the head, convenient for laying off 
 lines of designated angles. 
 
 In many drawing-cases will be found the parallel ruler (Fig. 121), consist- 
 
 FIG. 121. 
 
DRAWING INSTRUMENTS. 
 
 ing of two rulers connected by two bars moving on pivots, so adjusted that the 
 rulers, as they open, form the sides of a parallelogram. The edge of one of the 
 rulers being retained in a position coinciding with, or parallel to, a given line, 
 when the other ruler is moved, lines drawn along its edge are also parallel to 
 the given line. This instrument is only useful in drawing small parallels, and 
 in accuracy and convenience does not compare with either the triangle and 
 ruler or T square. 
 
 Another form of parallel ruler (Fig. 122) consists of a strip of wood with 
 bevelled edges, having two holes to receive two broad wheels, a, a, which are 
 
 FIG. 122. 
 
 connected by an axle passing under the metal cover, b, #, and revolving in the 
 supports, c, c ; the wheels come slightly below the surface of the wood, as 
 shown in the end elevation. In drawing parallel lines the fingers are placed 
 with a firm pressure about the centre of the metal cover, and the ruler is 
 moved in the proper direction. This ruler is more easily applied than the 
 former, but is more liable to error. 
 
 VAKIABLE CURVES. 
 
 For drawing arcs of a large radius, beyond the range of ordinary compasses, 
 and lines varying in curvature, thin slips of wood, termed curves, are usually 
 employed. These forms are of very general application, but others of almost 
 every form, and made of hard rubber, pear wood, or celluloid, can be pur- 
 chased. Whatever be the nature of the curve, some portion of the instrument 
 will be found to coincide with its commencement, and it can be continued 
 throughout its extent by applying, successively, such parts as are suitable, care 
 being taken that the parts are tangent to each other, and that the continuity is 
 not injured by unskilful junction. 
 
 Fig. 123 shows an adjustable curve ruler, the main features of which are a 
 hard-rubber face, , which holds the form of the required curve by a bar of 
 
 FIG. 123. 
 
 soft lead, Z>, kept in contact with the rubber face by the fasteners, c, and a flat 
 spring inside these fasteners. This curve, while useful in the coarser kinds of 
 draughting, does not do as neat or accurate work" as the separate curves above 
 given. 
 
 Thin splines are also to be had, which, held in position by leaden weights, 
 serve admirably for a guide to the pen in describing curves (Fig. 124). For the 
 
DRAWING INSTRUMENTS. 41 
 
 same purpose a thin, hard-rubber ruler, with soft-rubber backing, answers well, 
 and, as it can be readily rolled up, is extremely portable. 
 
 The weights above shown are very convenient in holding the drawing-paper 
 on the board, but thumb-tacks (Fig. 125), steel points with large, flat heads, 
 are in general use. They can be readily forced into the wood, and as readily 
 raised, but thumb-tack lifters can be purchased. 
 
 in 
 
 Elliptic, parabolic, and hyperbolic (see above) curves are furnished 
 sets, but the draughtsman can make a model out of thick cardboard or cellu- 
 loid, with which he can draw a very uniform curve. 
 
 For the drawing of ellipses, very neat trammels or compasses with elliptic 
 guides or patterns may be purchased. 
 
42 
 
 DRAWING INSTRUMENTS. 
 
 Fio. 124. 
 
 FIG. 125. 
 
 The drawing or right-line pen (Fig. 126) consists of two blades with steel 
 points, fixed to a handle ; and they are so bent that a sufficient cavity is left 
 between them for the ink. The blades are set with the 
 points more or less open by means of a mill-headed screw, 
 so as to draw lines of any required fineness or thickness. 
 For red inks, the blades of the pen should be nickle-plated 
 or German silver. One of the blades is framed with a joint, 
 so that by taking out the screw the blades may be complete- 
 ly opened, and the points effectively cleaned after use. The 
 ink is put between the blades by a common pen. In using 
 the pen, it should be slightly inclined in the direction of 
 the line to be drawn, and care should be taken that both 
 points touch the paper. These observations equally apply 
 to the pen-points of the compasses. The drawing - pen 
 should be kept close to the ruler or straight- 
 edge, and in the same direction during the 
 whole operation of drawing the line. Care 
 must be taken to hold the straight-edge 
 firmly with the left hand, that it does not 
 change its position. 
 
 For drawing close parallel lines in me- 
 chanical and architectural drawings, or to 
 represent railroads, canals, or roads, a rail- 
 road pen (Fig. 127) is frequently used, a 
 double pen with an adjusting screw to set the pens to any required small 
 distance. This instrument is also made with pencil points (Fig. 128). 
 
 Border - pens (Fig. 
 129), for drawing broad 
 lines, are double pens 
 with an intermediate 
 blade, and are applicable 
 to the drawing of map- 
 borders. The same work 
 may be done by drawing 
 heavy outer lines with 
 the common drawing- 
 pen, and filling in with a 
 brush or writing-pen. 
 
 The curve-pen (Fig. 
 130) is especially de- 
 signed for the ready 
 drawing of curved lines. 
 The axis of this pen is 
 carried through the han- 
 dle and fastened by a 
 nut on top, allowing the 
 
 pen to revolve, and thus more easily follow the curve. This instrument, made 
 with two pens (Fig. 131), is called a railroad curve-pen. 
 
 FIG. 128. 
 
 FIG. 129. 
 
43 
 
 The dotting-pen (Fig. 132) has on the back blade a pivot, on which may be 
 placed a dotting- wheel, resembling the rowel of a spur ; the screw is for opeu- 
 
 FIG. 131. 
 
 ing the blades to remove the wheel for cleaning after use or replace it with one 
 of another character of dot. A variety of dotting-wheels accompanies the in- 
 strument, each producing a different-shaped dot. These are used as distin- 
 guishing marks for different classes of boundaries on maps ; for instance, one 
 kind of dot distinguishes county boundaries, another kind town boundaries, a 
 third kind distinguishes that which is both a county and a town boundary, etc. 
 
 1 
 
 FIG. 132. 
 
 In using this instrument, the ink must be inserted between the blades above 
 the dotting-wheel, so that, as the wheel revolves, the points pass through the 
 ink, each carrying with it a drop, and marking the paper as it passes. It 
 sometimes happens that the wheel will revolve many times before it begins to 
 deposit its ink on the drawing, thereby leaving the first part of the line blank, 
 and, when it is gone over again, the first-made dots are liable to get blotted. 
 This evil may be avoided by placing a piece of blank paper over the drawing 
 to the very point the dotted line is to commence at, and drawing the wheel 
 over the blank paper first, so that by the time it 
 reaches the proper point the ink begins to flow. 
 
 The dotting -instrument (Fig. 133) works on the 
 principle of the drawing-pen. The outer wheel is 
 rolled on the edge of a T square or straight-edge, 
 and turns a ratchet wheel which causes the pen to 
 move up and down. The flat point close to the 
 pen must slide on the paper. To change the pat- 
 tern of the dotted lines, the spring which holds the 
 wheels on the axle is thrown back, and the proper ratchet wheel inserted. 
 
 The best pricking -point is a fine needle held as in Fig. 134, and is used to 
 transfer drawings by pricking through at the points of a drawing into the paper 
 
 FIG. 133. 
 
 FIG. 134. 
 
 placed beneath. The handle of the ordinary drawing-pen often contains a 
 pricking-point, which may be used by unscrewing the pen where it is joined to 
 the handle. 
 
 When drawings are transferred by tracing a prepared black sheet being 
 placed between the drawing and the paper to receive the tracing the eye end 
 of the needle forms a good tracing-point. 
 
44: 
 
 DRAWING INSTRUMENTS. 
 
 The stylus (Fig. 135) is a piece of polished agate placed in a handle, and is 
 used as a tracing-point. 
 
 FIG. 135. 
 
 Compasses are fitted with ink-points and with lengthening bars for drawing 
 larger circles. Compasses should have joints in the legs, so that the points, 
 pencil, and pen may be set perpendicular to the planes in which the 
 circles are described (Fig. 136). Compasses of this general form may 
 be had in sizes of 3 to 7 inches. 
 
 For the measurement and laying off 
 of small spaces, and the describing of 
 small circles, there are small bow com- 
 passes (Fig. 137). These are sometimes 
 made with an adjusting screw between 
 the legs. 
 
 For the measurement or laying off 
 
 FIG. 136. 
 
 FIG. 137. 
 
 Fio. 138. 
 
 of distances the plain dividers are convenient, but for ready and close ad- 
 justment the hair dividers (Fig. 138) are most suitable. The only difference 
 is that in the hair dividers one of the points is attached to the body by a 
 
 FIG. 139. 
 
 spring, and by means of the screw b it can be moved a very little toward or 
 from the fixed point more accurately than by closing or opening the dividers. 
 In dividing a line into equal parts especially, it enables one to divide the excess 
 or deficit readily. 
 
DRAWING INSTRUMENTS. 
 
 For convenience of carrying in the pocket, there are portable or turn-in 
 compasses (Fig. 139). There is a small attachment for a common pencil 
 which enables it to be used like compasses. 
 
 FIG. 140. 
 
 Three-legged dividers (Fig. 140) are convenient, while transferring measures 
 from a drawing to a copy on an equal scale, for locating a third point when two 
 are established. 
 
 For setting off very long lines, or describing circles of large radius, beam 
 compasses are used (Fig. 141). These consist of a mere strip of wood, A, and 
 two brass or German silver boxes, B arid C, which can easily be attached to the 
 beam ; connected with the brass boxes are the two points of the instrument, 
 G and H. The object of this instrument is the nice adjustment of the points 
 G and H at any definite distance apart; at F is a slow-motion 
 screw, by which the point G may be moved any very minute dis- 
 tance after the distance from H to G has been adjusted as nicely 
 as possible by the hand alone. The wheel attachment, I, is to 
 carry the weight of the beam. The metal parts of this instru- 
 ment occupy but little space. 
 
 There are beam compasses in which the beam is graduated, 
 and in which the boxes corresponding to B and C are fitted with 
 vernier or reading plates, to afford the means of minutely subdi- 
 viding the divisions on the beam. 
 
 Beam compasses are also made of small round German-silver 
 bars, one screwing into the other, on which are slides adapted for 
 carrying pen or pencil and points. 
 
 16' 
 
 H 
 
 FIG. 142. 
 
 Proportional dividers (Fig. 142), for copying and reducing drawings, are 
 found in most cases of instruments. 
 
 Closing the dividers and loosening the screw C, the slide may be moved up 
 
46 DRAWING INSTRUMENTS. 
 
 in the groove until the mark on the index corresponds with the required num- 
 ber ; then clamping the screw, the space inclosed between the long points, A B, 
 will be as many times that between the short points, E D, as is shown by the 
 number opposite the index. If the lines are to be reduced, the distances are 
 measured with the long points, and set off by the short ones ; if the lines are 
 to be enlarged, then vice versa. 
 
 Proportional dividers are also used for dividing the circumference of a 
 circle into a number of parts. A special scale along the graduated edge, 
 marked circles, is used, it being only necessary to move the slider to the 
 proper number on this scale to obtain a chord of the proper length. It 
 often happens that the length of the points becomes reduced by use or acci- 
 dent. In this case it is only necessary to loosen the screw holding the short- 
 ened point, take it out, grind to a point, and set to its former length. 
 
 Scales. The application of simple scales to the construction of diagrams 
 has been explained ; but among drawing instruments scales especially adapted 
 to plotting are to be found in great varieties of form, divisions, and material. 
 It is usual, especially in topographical drawings, for the draughtsman to con- 
 struct a scale upon the finished sheet on account of its ready application to the 
 determination of measures, and when the drawing is to be reduced or enlarged 
 by photographing it is indispensable. Moreover, paper expands and contracts 
 under hygrometric changes ; the scale should be subject to those same changes. 
 To remedy this inconvenience Mr. Charles Holzapfel has introduced paper 
 scales, which are portable and cheap; but as all kinds of paper are noi 
 equally susceptible to changes of condition on the atmosphere, the detached 
 paper scale affords only a partial correction. 
 
 The scale should be written or drawn in all drawings ; also the date of com- 
 pletion and name or initials of the draughtsman, as these data may be of value 
 in the identification of the drawing. 
 
 In all working architectural and mechanical drawings, use as large a scale 
 as possible ; and even then do not depend upon the mechanics employed in the 
 construction measuring correctly, but write in the dimensions as far as prac- 
 ticable. For architectural plans, the scale of ^ of an inch to the foot is in very 
 general use and is convenient for the mechanic, as the common two-foot rule 
 carried by all mechanics is subdivided into ths, iths, and sometimes sixteenths 
 of an inch, and the distances on a drawing to this scale can therefore be easily 
 measured by them. This fact should not be lost sight of in working drawings. 
 When the dimensions are not written, make use of such scales that the dis- 
 tances may be measured by the subdivisions of the common two-foot rule ; 
 thus, in a scale of ^ or ^ full size, 6 inches or 3 inches represent one foot ; 
 in a scale of an inch to the foot or twelfth full size, each an inch repre- 
 sents 6 inches, of an inch, 3 inches ; but when or fa an inch to the foot, 
 or any similar scale, is adopted, it is evident that these divisions can not be 
 taken by the two-foot rule. 
 
 Plotting scales (Fig. 143) are scales of equal parts, with the divisions usu- 
 ally on a bevelled edge, by which any length may be marked off on the paper 
 without using dividers. There are also small offset scales, for use of which see 
 " Topographical Drawing." 
 
 Sometimes these scales are made with edges bevelled on both sides, and 
 
DRAWING INSTRUMENTS. 
 
 47 
 
 graduated to four different scales. Sometimes the section of the scale is tri- 
 angular (Fig. 144), with six scales on the different edges. To avoid confusion 
 
 NrLL 
 
 1 II II 1 i 1 II II 1 1 Ml II II 1 1 1 
 
 ill]' 1 ,! Ml II 1 1 1 1 1 I! 1 1 1 II 
 
 \ 
 
 ao 
 
 
 \ f Oil- 6 1 It 19 
 
 IS Itr IE |Z It 
 
 FIG. 143. 
 
 from having many scales on one ruler, the triangular scale has a small slip of 
 metal, A, readily put on, which covers partially the scales not in use. 
 
 FIG. 145. 
 
 FIG. 144. 
 
 To divide a given line into any number of equal parts (Fig. 145). 
 Let A B be the line, and the number of parts be ten. Draw a perpendicu- 
 lar at one extremity, A, of the line ; with a plotting scale place the zero at the 
 other extremity, B, of the line ; make the mark 10 
 on the scale coincide with the perpendicular ; draw 
 a line along the edge of the scale, and mark the 
 line at each division of the scale 1 to 9 ; draw per- 
 pendiculars through these marks to the line A B, 
 and they will divide A B into ten equal parts. 
 
 The above figure illustrates the construction of 
 diagonal scales. The simply divided scales give 
 only two denominations, primaries and tenths, or 
 
 twelfths ; but more minute subdivision is attained by the diagonal scale, which 
 consists of a number of primary divisions, one of which is divided into tenths, 
 
 and subdivided into 
 hundredths by diago- 
 nal lines (Fig. 146). 
 This scale is con- 
 structed in the follow- 
 ing manner : Eleven 
 parallel horizontal lines 
 are ruled, inclosing ten 
 equal spaces ; from one 
 end set off the primary 
 unit divisions, 0, 1, 2, 3, 
 and draw vertical lines 
 through these points; 
 subdivide the extreme unit to the left on the upper and lower lines into ten 
 equal parts, 1, 2, 3, etc. ; connect on the upper line with 1 on the lower line 
 
 FIG. 146. 
 
48 
 
 DRAWING INSTRUMENTS. 
 
 A 
 
 by a diagonal, and draw lines parallel to it through the other subdivisions. To 
 take a measurement of, say, 168, we place one foot of the dividers on the pri- 
 mary 1, and carry it down to parallel 8, and then extend the other foot to the 
 intersection of the diagonal which falls from the subdivision 6 with this par- 
 allel. The primaries may, of course, be considered as yards, feet, or inches ; 
 and the subdivisions as tenths and hundredths of these respective denomina- 
 tions. If the number of parallel spaces be eight and the subdivision be 
 twelve, we can measure feet, inches, and eighths. In the diagonal scale the 
 vertical subdivisions are often omitted. 
 
 The diagonals may be applied to a scale where only one subdivision is re- 
 quired. Thus, if seven lines be ruled (Fig. 147), inclosing six equal spaces, 
 
 and the length be divided 
 into primaries, as A B, B C, 
 etc., the first primary, A B, 
 may be subdivided into 
 twelfths by two diagonals 
 running from 6, the mid- 
 dle of A B, to 12 and 0. 
 We have here a very con- 
 venient scale of feet and inches. From C to 6 is 1 foot 6 inches ; and from 
 C on the several parallels to the various intersections of the diagonals we obtain 
 1 foot and any number of inches from 1 to 12. 
 
 For the designing of machinery, it is very convenient to have some scale of 
 reference by which to proportion the parts ; for this purpose a vertical and 
 horizontal scale may be drawn on the walls of the room. 
 
 Vernier scales are preferred by some to the diagonal scale already described. 
 To construct a vernier scale (Fig. 148) by which a number to three places may 
 be taken, divide all the primary divisions into tenths, and number these sub- 
 
 7/V 
 
 
 
 
 2 v 
 
 
 
 
 <>L^- 
 
 
 
 
 f / \r 
 
 
 
 
 / \, 
 
 
 
 
 / \ 
 
 
 
 
 0/2 
 FIG. 147. 
 
 10 
 
 2 4} 6 8 
 
 f 
 
 
 f 
 
 
 f 
 
 
 II I I II 
 
 I I I I 
 
 I 
 
 I I I I I I 
 
 I 
 
 I I i I I II I 
 
 
 wn I ' I I I I i i I 
 
 
 
 
 
 wv |/0 8 6 \ t 2 
 
 FIG. 148. 
 
 divisions 1, 2, 3, from left to right. Take off now with the compasses eleven 
 of these subdivisions, set the extent off backward from the end of the first 
 primary division, and it will reach beyond the beginning of this division, or 
 zero point, a distance equal to one of the subdivisions. Now divide the extent 
 thus set off into ten equal parts, marking the divisions on the opposite side of 
 the divided line to the lines marking the primary divisions and the subdivisions, 
 and number them 1, 2, 3, etc., backward from right to left. Then, since the 
 extent of eleven subdivisions has been divided into ten equal parts, so that 
 these ten parts exceed by one subdivision the extent of ten subdivisions, each 
 one of these equal parts, or, as it may be called, one division of the vernier 
 scale, exceeds one of the subdivisions by a tenth part of a subdivision, or a 
 hundredth part of a primary division ; thus, if the subdivision be considered 
 10, then from to the first division of the vernier will be 11 ; to the second, 
 22 ; to the third, 33 ; to the fourth, 44 ; to the fifth, 55, and so on, 66, 77, 
 88, 99. 
 
DRAWING INSTRUMENTS. 
 
 49 
 
 To take off the number 253 fr,om this scale, place one point of the dividers 
 at the third division of the vernier ; if the other point be brought to the pri- 
 mary division 2, the distance embraced by the dividers will be 233, and the 
 dividers must be extended 
 to the second subdivision of 
 tenths to the right of 2. 
 If the number were 213, 
 then the dividers would have 
 to be closed to the second 
 subdivision of tenths to the 
 left of 2. The number, thus 
 taken, may be 253, 25'3, 
 2'53, according as the pri- 
 mary divisions are taken as 
 hundreds, tens, or units. 
 
 The construction of this 
 scale is similar to that of the 
 verniers of theodolites and 
 surveying instruments. 
 
 The sector in its old 
 form carried several scales 
 on its faces. As given in 
 Fig. 149, there are only 
 double scales starting from 
 the centre joint, which, 
 without drawing, may be 
 applied to the solution of problems on similar triangles. 
 
 Let the lines A B, A C, represent the legs of the sector, and A D, A E, two 
 equal sections from the centre ; then, if the points B C and D E be connected, 
 the lines B C and D E will be parallel ; therefore, the triangles A B C, A D E, 
 
 FIG. 149. 
 
 FIG. 150. 
 
 will be similar, and, consequently, the sides A B, B C, A D, D E, proportional 
 that is, as A B : B C : : A D : D E ; so that if A D be the half, third, 
 5 
 
50 
 
 DRAWING INSTRUMENTS. 
 
 or fourth part of A B, then D E will be a half, third, or fourth part of 
 B C ; and the same holds of all the rest. Hence, if D E be the chord, 
 sine, or tangent of any arc, or of any number of degrees to the radius A D, 
 then B C will be the same to the radius A B. Thus, at every opening of 
 the sector, the transverse distances D E and C B from one ruler to another are 
 proportional to the lateral distances, measured on the lines A B, A C. It is to 
 
 FIG. 151. 
 
 be observed that all measures are to be taken from the inner lines, since these 
 
 only run accurately to the centre. 
 
 On the scale in common boxes of drawing instruments, the edges of the 
 
 sides are divided as a protractor (Fig. 84) for 
 the laying out of angles. The ordinary pro- 
 tractor consists of a semicircle of thin metal or 
 horn (Fig. 150), whose circumference is divided 
 into 180 degrees (180). In the larger protrac- 
 tors each of these divisions is subdivided. 
 
 Application of the protractor. To lay off a 
 given angle from a given point on a straight 
 line, let the straight line a b of the protractor 
 coincide with the given line, and the point c 
 with the given point ; now mark on the paper 
 against the division on the periphery coinciding 
 with the angle required ; remove the protractor, 
 and draw a line through the given point and the 
 mark. 
 
 Fig. 151 is a protractor with a straight-edge 
 revolving on a horn centre. Where the straight- 
 edge is intersected by the edge of the protractor 
 a vernier is attached, and will be found useful in 
 close work for dividing the degrees into tenths. 
 This protractor is often extended to full circles. 
 For plotting field-notes expeditiously, drawing-paper can be obtained with 
 
 FIG. 152. 
 
DRAWING INSTRUMENTS. 
 
 51 
 
 large, full circular protractors printed thereon, on which the courses can be 
 readily marked, and thus transferred to the part of the paper required by a 
 parallel ruler, or by triangle and ruler. These sheets are of especial use in 
 plotting at night the day's work, as, on account of the large size of the protrac- 
 tor, angles can be laid off with greater accuracy than by the usual protractor of 
 a drawing-instrument case, with less confusion of courses, and more expe- 
 ditiously. 
 
 The pantograph is used for the copying of drawings, either on the same 
 scale, on a reduced scale, or on an enlarged scale. 
 
 Fig. 152 shows its simplest form and its application. The lower left-hand 
 point, around which the frame is turned, is fixed, and the proportion of the 
 drawing is determined by the position of the screw eyes in the holes of the 
 arms. 
 
 Fig. 153 is another form, of more finished construction. It consists of a 
 set of jointed rulers, A, B, and another, C, D, about one half the length of the 
 
 FIG. 153. 
 
 former. The free ends of the smaller set are jointed to the larger at about the 
 centre. Casters are placed at a a, etc., to support the instrument and to allow 
 an easy movement. 
 
 DBAWING-PAPERS. 
 
 Papers adapted to drawing may be obtained of various qualities, thicknesses, 
 and dimensions, either in sheets, pads, or rolls. Machine-made papers are 
 generally used, and are to be had from stock in rolls up to 62 inches in width, 
 but to order much wider. They are generally made from cotton for the more 
 finished drawings ; but stronger papers for working drawings and details are of 
 manilla or of coarse heavy stock. 
 
 Eoll and sheet papers can be had mounted or backed with cotton cloth, 
 which prevents. them from being torn, and permits of their being hung to the 
 walls as maps. 
 
 Hand-made drawing-papers are usually made in certain standard sizes, about 
 as follows : 
 
 Cap 13 inches by 17 inches. 
 
 Demy 15 20 " 
 
 Medium 17 " 22 " 
 
 Royal 19 " 24 " 
 
 Super Royal 19 " 27 " 
 
 Tracing-paper is a prepared tissue paper, transparent and qualified to re- 
 ceive ink lines and tinting without spreading. When placed over a drawing 
 
 Imperial 22 inches by 30 inches. 
 
 Atlas 26 " 34 " 
 
 Double Elephant 27 " 40 " 
 
 Antiquarian 31 " 53 " 
 
52 DRAWING INSTRUMENTS. 
 
 already executed, the drawing is . distinctly visible through the paper, and it 
 may be copied directly or traced in pencil or ink. 
 
 Tracing-paper often becomes tender with age, is apt to break in the folds, 
 is not easily rolled. It is not suitable, therefore, for permanent drawings ; but 
 the tracing can be readily transferred to drawing-paper by means of transfer 
 paper. Place the fair sheet on the drawing-board, above it the transfer sheet 
 with the prepared face down, then the tracing, and steady the whole by weights 
 or by thumb-tacks fixed into the drawing-board. A fine, smooth point is then 
 passed over each boundary and line on the tracing with a pressure of the hand 
 sufficient to cause a clear line to be left by the transfer paper on the fair sheet. 
 Finish these lines in ink. The copyist should be careful in his manipulation 
 that no unnecessary lines or smutches be left on the fair sheet. Transfer paper 
 can be readily obtained in sheets, either in black, blue, vermilion, or graphite, 
 or it can be made by smearing with a piece of flannel one surface of thin paper 
 with a coating of lard and graphite and, after a day's drying, wiping off the 
 superfluous portion with a soft rag. 
 
 Parchment papers are much stronger than tracing-papers, and are usually 
 transparent enough to serve the same purpose ; the thicker kinds are well 
 adapted for drawing and engrossing. De La Rue's process for the manufac- 
 ture of parchment paper is to plunge unsized paper for a few seconds into 
 sulphuric acid, diluted with half to a quarter its bulk of water, the solution 
 being of the same temperature as the air, and afterward wash with 'weak 
 ammonia. 
 
 A drawing may be made to accompany a letter by saturating the letter 
 paper with benzine till it becomes transparent, then using it as tracing-paper, 
 copying the design in pencil, and finishing in copying-ink after the benzine 
 has evaporated, so that it can be transferred with the descriptive writing to the 
 letter book. 
 
 Transparent tracing-cloth can be had in wide and long rolls. It is much 
 stronger than tracing-paper, and serves a permanent purpose. Should the 
 tracing-cloth refuse to take ink lines well, almost any fine white powder will 
 remedy this, such as chalk, fuller's earth, pipe clay, or plaster of Paris sprinkled 
 on and rubbed in well. 
 
 It is usual to draw on the dull side of the cloth, except where colour is to 
 be put in, when the ink lines are drawn on the glossy side and the colour on 
 the dull back. Designs. and finished drawings made in pencil on paper are 
 traced on cloth in ink, and in this form are preserved as originals and can be 
 copied by the heliographic process, either wholes or details as needed. When a 
 white sheet of paper is placed behind the tracings, the drawing may be readily 
 photographed on a reduced or enlarged scale, and much more cheaply than by 
 any other process ; and such negatives may be used in process engraving for 
 book illustrations. 
 
 Heliographic paper can readily be had in sheets or rolls, and the mixture 
 for the preparation of the paper can also be purchased, or can be made by 
 dissolving If ounce of common citrate of iron in 8 ounces of water, and 1 
 ounce of red prussiate of potash in 8 ounces of water, and then mixing them 
 just previous to using. Papers and mixtures must be kept from the light or 
 they will lose their sensitiveness. The above is a mixture for the most com- 
 
DRAWING INSTRUMENTS 
 
 53 
 
 mon form of sun prints, called, the ferro-prussiate or blue process, in which 
 white lines are developed on a blue ground. By the cyanotype process blue 
 lines are developed on a white ground ; by the nigrosine process, black lines on 
 a white ground ; by the chromide dry process, dark lines on a tinted ground. 
 Papers for all of the above processes are on sale, with directions for use. If 
 none can be had, and it is desired to prepare some, use the ferro-prussiate 
 process as the simplest, of which a recipe has been given above, the paper 
 should be chemically neutral, of even material, and capable of being washed. 
 Inks are to be had especially adapted for the tracings in bottles and cakes. It 
 is necessary for a good print that the lines should be of a deep black. If not 
 sufficiently opaque, burnt Sienna, burnt umber, or gamboge added to the ink 
 improves the prints. 
 
 For the manipulation there is needed plate glass and a blanket a little larger 
 than the drawing, also a shallow tray, that the drawing can be placed in flat for 
 washing. 
 
 Lay down the blanket on the drawing-board, above that the ferro-prussiate 
 paper, next the drawing, and then the glass. Expose to the sunlight until the 
 background is a metallic gray. The length of exposure may be from five min- 
 utes up, depending on the intensity of the sunlight, the age of the prepared 
 paper, and the transparency of the tracing. Now lay the ferro-prussiate paper 
 in the tray, cover with water, and leave it for five to ten minutes ; wash thor- 
 oughly and dry. 
 
 The usual form of printing-frame, as purchased of dealers, shown in Fig. 
 154, consists of a frame into which fits a sheet of glass, preferably of plate glass, 
 
 - -O. rnacin 
 
 N g/ass 
 
 FIG. 154. 
 
 with a hinged backboard, to the inner side of which a piece of felt is glued in 
 the smaller sizes, while in the larger the felt is separate ; on the back of this 
 board are two brass springs fitting under the metal catch, making a close con- 
 tact with the glass. As shown in the small sectional drawing, the frame is 
 turned upside down. The glass is placed in first, then, successively, the tracing 
 with its face to the glass, the prepared paper with the prepared side to the 
 tracing, and the felt ; then the backboard is placed and held in position by the 
 springs, the frame is turned up, and the directions given above as to exposure 
 and washing are properly carried out. 
 
 When it is necessary to make additions and alterations on blue prints, a 
 
54 DRAWING INSTRUMENTS. 
 
 special ink can be procured; lines made with this preparation on the blue 
 ground turn white. (See FREE-HAND DRAWING.) 
 
 The helios process is useful for copying not only drawings, but contracts, 
 estimates, tables, etc., when they are written on transparent paper or cloth; and 
 so is the nigrosine process. 
 
 Bristol board is a cardboard with a very fine surface. It can be obtained of 
 various thicknesses and of the same dimensions as sheet drawing-papers. It is 
 adapted to water-colours, pen-and-ink sketches, and fine line-drawings ; the 
 Patent Office requires sheets 10 by 15 inches, and these can be obtained with 
 border and wording of witness, inventor, and attorney properly printed in. 
 
 Mouth glue, for the sticking of the edges 6f drawing-paper to the board, is 
 made of glue and sugar or molasses ; it melts at the temperature of the mouth, 
 and is convenient for the draughtsman. 
 
 Drawing-paper may be fixed down on the drawing-board by thumb-tacks at 
 the corners, by weights, or by gluing or pasting the edges. The first is 
 sufficient when no shading or colouring is to be applied, and if the sheet is 
 not to be a very long time on the board ; and it has the advantage of preserv- 
 ing the paper in its natural state. For shaded or tinted drawings, the paper 
 must be damped. 
 
 Damp-stretching is done as follows : The edges of the paper should first be 
 cut straight, and, as near as possible, at right angles to each other. The sheet 
 should be enough larger than the intended drawing and its margin to admit of 
 being afterward cut from the board, leaving the pasted or glued border. 
 
 The paper must first be placed on the drawing-board and thoroughly and 
 equally damped with a sponge and clean water on the side on which the draw- 
 ing is to be made. This done, lay a straight flat ruler on the paper, with its 
 edge parallel to, and about half an inch from, one of its edges. The ruler must 
 now be held firm, while the projecting half inch of paper is being turned up 
 along its edge ; then a piece of mouth glue, having its edge partially dissolved 
 by holding it in warm water for a few seconds, must be passed once or twice 
 along the turned-up edge of the paper, after which, by sliding the ruler over 
 the glued border, it will be again laid flat, and, the ruler being pressed down 
 upon it, that edge of the paper will adhere to the board. If sufficient glue has 
 been applied, the ruler may be removed directly, and the edge finally rubbed 
 down by an ivory book-knife or by the bow of a common key, rubbing it on a 
 slip of paper placed on the drawing-paper, so that the surface of the latter may 
 not be soiled ; this will firmly cement the paper to the board. Another edge of 
 the paper is then treated in like manner, and the remaining edges in succession. 
 Sometimes strong paste or mucilage is used instead of glue. 
 
 The wetting of the paper is done for the purpose of expanding it ; and the 
 edges, being fixed to the board in its enlarged state, act as stretchers upon the 
 paper, while it contracts in drying, which it should be allowed to do gradually. 
 All creases or undulations by this means disappear from the surface, and it 
 forms a smooth plane to receive the drawing. 
 
 After the drawing is finished, cut off the paper inside the pasted edge, and 
 remove the edge by warm water and the knife. 
 
DRAWING INSTRUMENTS. 
 
 55 
 
 MOUNTING PAPEE AN.D DBAWINGS, VAENISHING, ETC. 
 
 When paper of the requisite quality or dimension can not be purchased 
 already backed, it may be mounted on muslin. The cloth should be well 
 stretched upon a smooth flat surface, being damped for that purpose, and its 
 edges glued down, as was recommended in stretching drawing-paper. Then 
 with a brush spread strong paste, beating it in till the grain of the cloth be all 
 filled up ; for this, when dry, will prevent it from shrinking when subsequently 
 removed ; then, having cut the edges of the paper straight, paste one side of 
 every sheet, and lay them upon the muslin sheet by sheet, overlapping each 
 other slightly. If the drawing-paper is strong, it is best to let every sheet lie 
 five or six minutes after the paste is put on it, for, as the paste soaks in, the 
 paper will stretch, and may be better spread smooth upon the cloth ; whereas, 
 if it be laid on before the paste* has moistened the paper, it will stretch after- 
 ward and rise in blisters when laid upon the cloth. The paper should not be 
 cut off from its extended position till thoroughly dry, which should not be 
 hastened. Leave it in a dry room to do so gradually, if time permit ; if not, it 
 may be exposed to the sun ; in the winter season the help of a fire may be neces- 
 sary ; but it should not be placed too near a scorching heat. 
 
 In joining two sheets of paper together by overlapping, it is necessary, in 
 order to make a neat joint, to feather-edge each sheet ; this is done by care- 
 fully cutting with a knife half way through the paper near the edges on the 
 sides which/ are to overlap each other and then stripping off a feather-edged 
 slip from each, which, if done dexterously, will produce a very neat and effi- 
 cient joint. 
 
 For mounting and varnishing drawings or prints, stretch a piece of linen 
 on a frame, to which give a coat of isinglass or common size ; paste the back of 
 drawing, which leave to soak; and then lay it on the linen. When dry, give it 
 at least four coats of well-made isinglass size, allowing it to dry between each 
 coat. Take Canada balsam diluted with the best oil of turpentine, and with a 
 clean brush give it a full flowing coat. 
 
 When drawings are not mounted on muslin, the edges may be protected 
 from tearing by binding with gummed tape, or strips of paper which may be 
 cut or purchased. 
 
 Drawings, as far as possible, should be preserved flat in drawers, and this is 
 especially desirable for tracings which are to be often sun-printed. 
 
 The classification of drawings is varied. The common method is to devote 
 a separate drawer to the drawings of each machine, or of each group or class of 
 machine ; another is to have drawers of various sizes and arrange the drawings 
 according to sizes. 
 
 MANAGEMENT OF THE INSTEUMENTS. 
 
 In constructing preparatory pencil-drawings, it is advisable, as a rule of 
 general application, to make no more lines upon the paper than are necessary 
 to the completion of the drawing in ink ; and also to make these lines just 
 dark enough to be sufficiently distinct. 
 
 It is often beneficial to ink in one part of a drawing before touching other 
 parts at all ; it prevents confusion, makes the first part easy of reference, and 
 
56 DRAWING INSTRUMENTS. 
 
 allows of its being better done, as the surface of the paper inevitably contracts 
 dust and becomes soiled in the course of time, and therefore the sooner it is 
 done with the better. 
 
 Circles and circular arcs should, in general, be inked in before straight lines, 
 as the latter may be more readily drawn to join the others than can the former. 
 When a number of circles are to be described from one centre, the smaller ones 
 should be inked first, while the centre is in better condition. When a centre 
 is required to bear some fatigue, it should be protected with a thickness of stout 
 card glued or pasted over it, to receive the compass-leg. 
 
 India-rubber is the ordinary medium for cleaning a drawing and for cor- 
 recting errors of the pencil. For slight work it is quite suitable ; that sub- 
 stance, however, operates to destroy the surface of the paper ; and, by repeated 
 application, it so ruffles the surface as to spoil it for fine drawing, especially if 
 ink shading or colouring is to be applied. It is much better to leave trivial 
 errors alone, if corrections by the pencil may be made alongside without confu- 
 sion, and not clear away superfluous lines till the inking is finished. 
 
 W T hen ink lines have to be erased to any considerable extent, the best way 
 is to use an ink-erasing rubber. Single lines may be erased by cutting a long 
 narrow slit in a piece of thin cardboard or celluloid and erasing through it. 
 
 For cleaning a drawing, a piece of bread two days old is preferable to India- 
 rubber, as it cleans the surface well and does not injure it. A sponge rubber 
 may also be used for this purpose. For ordinary small erasures of ink lines, a 
 sharp rounded pen-blade, applied lightly and rapidly, does well, and the sur- 
 face may be smoothed down by the thumb-nail. In drawings intended to be 
 highly finished, particular pains should be taken to avoid the necessity for cor- 
 rections, as everything of this kind detracts from the appearance. 
 
 The best work can only be accomplished by keeping the instruments in 
 good order; their working parts should be carefully preserved from injury. 
 The scales must be kept scrupulously clean ; the inking tools should have 
 especial care, and the blades kept well set, for which a small oil-stone is con- 
 venient. 
 
 To dress up the tips of the blades of the pen or of the bows, as they usually 
 become worn unequally, they may be screwed up into contact in the first place, 
 and passed along the stone, turning upon the point in a directly perpendicular 
 plane, till they acquire an identical profile. Being next unscrewed and exam- 
 ined to ascertain the parts of unequal thickness round the nib, the blades are 
 laid separately upon their backs on the stone, and rubbed down at the points, 
 till they be brought up to an edge of uniform fineness. It is well to screw 
 them together again, and to pass them over the stone once or twice more, to 
 bring up any fault ; to retouch them also on the outer and inner side of each 
 blade, to remove barbs or fraying; and, finally, to draw them across the palm 
 of the hand. 
 
 India ink, which is commonly used for line-drawing, should be rubbed 
 down in water to the degree that avoids the sloppy aspect of light lining with- 
 out making the ink too thick to run freely from the pen. This medium degree 
 may be judged of after a little practice by the appearance of the ink on the 
 palette. The best quality of ink has a soft feel when wetted and smoothed^ 
 being then free from grit or sediment, and has a musky smell. 
 
DRAWING INSTRUMENTS. 
 
 57 
 
 Slabs of many forms and different materials are used in grinding down the 
 ink. The one shown in Fig. 155 is a square slab of slate, with a countersunk 
 circular recess and a well in the centre to hold the ink ; the cover is a piece of 
 heavy glass. 
 
 A quantity of ink may be prepared at one time, but it must be kept well 
 covered to exclude dust and prevent evaporation. The pen should be filled by 
 
 FIG. 155. 
 
 a narrow strip of paper, dipped in the ink and inserted between the blades. 
 
 India ink and ink of various colours can be purchased in bottles, and this 
 answers very satisfactorily for most work. Waterproof ink, which admits of 
 being washed over, can be bought in sticks or in bottles. 
 
 It is of primary importance to keep the blades of the inking tools free from 
 obstruction ; this may be readily accomplished without unscrewing the pen by 
 passing a slip of paper between the blades, or by drawing the point firmly over 
 a piece of paper or on the fleshy part of the hand. 
 
 EXERCISES WITH THE DRAWING-PEN. 
 
 Before proceeding to the construction of finished drawings, skill should be 
 acquired in the use of the drawing-pen, supplemented often by the writing-pen. 
 Beginning with lines, outlines of figures, alphabets, and the like, the draughts- 
 man should strive to acquire the habit of readily drawing clean, uniform lines, 
 without abruptness or breaks where straight lines connect with curved ones. 
 Draw straight lines of different grades : 
 
 as, fine 
 
 medium 
 
 coarse 
 
 at first, lines of indefinite length, taking care that they are drawn perfectly 
 straight and of uniform width or grade. Then draw lines of definite length 
 between assumed points, taking care to terminate the lines exactly at these 
 points. Lines as above are / till lines. The grades depend on the effect which 
 the draughtsman wishes to give. 
 
 Draw dotted lines, broken lines, and broken and dotted lines, of different 
 grades : 
 
58 
 
 DRAWING INSTRUMENTS. 
 
 Draw fine lines at uniform distances from each other : 
 
 FlO. 156. 
 
 To give uniform appearance, the lines must be of uniform grade and equally 
 spaced. Practice in lines of this sort is important, as they are much used in 
 drawing to represent sections, shades, and conditions, as soundings on charts, 
 density or characteristics of population, areas of rain, temperature, and the like. 
 Draw lines as in Fig. 156. These lines are diagonal with the border-lines, and 
 
 are used to represent sections 
 of materials. In the figure, 
 lines differently inclined .rep- 
 resent different pieces of the 
 same material. 
 
 Instruments called section- 
 liners are to be had for draw- 
 ing these lines, but for the usual needs of a drawing office the triangle and 
 straight-edge, with the drawing-pen, will be sufficient. 
 
 Sections of different materials may be represented in different kinds of 
 lines (see page 177). 
 
 To represent cylindrical surfaces (Fig. 157). 
 
 Draw a semi-circumference, and mark on it a 
 number of points, at equal distances apart, and 
 through these points draw lines perpendicular to 
 the diameter across the surface to be represented. 
 It is not absolutely necessary that the central space 
 should be equal to the others; it will be more 
 effective to leave out two of the lines, and make 
 it to this extent wider. 
 
 To construct a mass of equal squares (Fig. 
 158). 
 
 Lay off a right angle, and on its sides mark 
 as many points at equal distances apart as may 
 be necessary ; through these points draw lines 
 parallel to the sides. 
 Or, construct a rectangle ; mark on its sides as many points, at equal dis- 
 tances apart, as may be necessary ; through these points draw the lines. 
 To construct the squares diagonally to the base (Fig. 159). 
 
 FIG. 157. 
 
DRAWING INSTRUMENTS. 
 
 59 
 
 Mark on the sides of the right angle as many points, at distances apart 
 equal to the diagonal of the required squares, as may be necessary. Connect 
 these points by lines as shown, and 
 through the same points draw lines 
 at right angles to the others. 
 
 To cover a surface with equilat- 
 eral triangles (Fig. 160^. 
 
 Construct an angle of 60, and 
 mark on its sides points at dis- 
 tances apart equal to the side of 
 the triangle. Connect these points 5 
 and through these points draw lines 
 
 FIG. 158. 
 
 parallel to the sides of the angle. 
 
 Two such triangles joined at the 
 base form a lozenge. Six triangles may be arranged as a hexagon. The whole 
 surface may be arranged in lozenges or hexagons. 
 
 To cover a surface with octagons and squares (Fig. 
 161). 
 
 Lay off the surface in squares having sides equal to the 
 width of the octagons. Corner the outer squares to form 
 octagons (Fig. 68). Extend the sides of these octagons 
 across the other squares, and similar corners will be cut 
 off, and the octagons and squares required will be com- 
 plete. 
 
 With the aid of paper thus 
 covered with squares, triangles, 
 or lozenges, various geometrical 
 designs may be readily con- 
 structed, pleasing in their effect, 
 and affording good practice to 
 young draughtsmen. 
 
 Any design can be copied by 
 covering it and the clean sheet 
 with squares. Mark the positions of points in the design or the sides of cor- 
 
 C 
 
 F 
 
 FIG. 159. 
 
 FIG. 160. 
 
 FIG. 161. 
 
60 
 
 DRAWING INSTRUMENTS. 
 
 responding squares, and draw the connecting lines. To enlarge or reduce 
 the design, make the squares or triangles proportionately larger or smaller. 
 
 FIG 162. 
 
 FIG. 163. 
 
 FIG. 164. 
 
 In transferring designs and drawings from books or plates, on which squares 
 can not be drawn, it is very convenient to have a square of glass, with squares 
 upon it, which may be laid on the drawing, and thus serve the same purpose as 
 
 if squares had been drawn. The 
 glass may be readily prepared by 
 painting one of its surfaces with a 
 thin coat of gum, and drawing 
 squares upon it with the drawing- 
 pen ; if every fifth or tenth line be 
 made fuller or in a different colour, 
 it will be still more convenient for 
 
 reference. 
 
 Fig. 162 gives the front and side 
 views of an acanthus leaf, the surface being covered with squares, and on a 
 ground of like squares in Fig. 163 the side view is transferred, but in a re- 
 versed position. This is done by making the position of the outline and then 
 of the interior lines with reference to the squares, as designated by letters and 
 numerals. Fig. 164 is a transfer of both figures on a reduced scale. 
 
 Designs for woven goods, oil cloths, ceiling and wall ornamentation, and the 
 like are usually based on geometrical figures, and in certain proportions sym- 
 metry and subordination of one part to another fall within the term 
 of artistic. 
 
 The following are designs in which the ruling figure is a trefoil : 
 "In the equilateral triangle (Fig. 165), each side is divided by 
 a dot, and from the centre of the triangle lines are drawn to each 
 angle, and from the dot in the middle of each side to the opposite sides of the 
 figure. The geometrical plan of the design is thus laid out, and the figure is 
 easily filled in by drawing simple curves from the centre of the form to the 
 
DRAWING INSTRUMENTS. 
 
 61 
 
 dot on each side of it, and, lastly, filling in the form of the trefoil a little below 
 the point of each corner of the triangle. 
 
 "The square (Fig. 166), which is the next form, is developed in much the 
 same manner. The sides are bisected, and from a point in the centre lines are 
 
 FIG. 165. 
 
 FIG. 166. 
 
 FIG. 167. 
 
 carried to each angle, and to all the dots on the sides. As in the preceding 
 figure, slight curves are made on either of 'the side-lines, and the trefoil is 
 added to each angle, with the base of the middle leaf touching the transverse 
 working-lines between the sides. 
 It will be seen that the penta- 
 gon (Fig. 167) and the hexagon 
 (Fig. 168) also are formed in 
 the same general manner, but 
 the proportion of the top of 
 the trefoil varies from its sides. 
 " In drawing the circular 
 rosette (Fig. 169), the circum- 
 ference should be constructed 
 on a vertical and a horizontal diameter, with two other diameters bisecting 
 it at equal angles, which divide it into eight sections, the half diameters, upon 
 
 FIG. 168. 
 
 FIG. 169. 
 
 FIG. 170. 
 
 all of which curved lines and the top of the trefoil are made. A series of 
 arcs may be added at the pleasure of the designer. In the two pieces of 
 moulding (Figs. 170 and 171) the trefoil is inserted vertically to the sides in 
 one and horizontally in the other. In the latter, a half of the trefoil is added 
 upon the sides to enrich the elementary figure ; and the double line and the 
 transverse lines which form the squares are repeated for the sake of symmetry, 
 and as affording an impression of agreeable repose. 
 
 " It is from such a basis as this that all these various patterns are derived, 
 and they produce a result which an inexperienced eye, unaccustomed to analyze 
 designs, could scarcely resolve into its elements." 
 
62 
 
 DRAWING INSTRUMENTS. 
 
 Figs. 172-175 are other illustrations of the same principle, of varieties of 
 rosettes constructed on a similar plan. 
 
 FIG. 171. 
 
 All of these designs can be constructed mechanically, but more grace is 
 given to the design by the filling in with free hand, and it is an excellent prac- 
 
 FIG. 172. 
 
 Fio. 173. 
 
 FIG. 174. 
 
 FIG. 175. 
 
 tice in the execution of the more elaborate Saracenic and Moorish diaper. 
 In all of these where there are repetitions of the same figures it is usual to 
 draw but one, and then transfer this, with the finish in crayon or pencil. 
 
 LETTERING. 
 
 In Fig. 176 are examples of block letters constructed on squares, a rudimen- 
 tary form of mechanical letters, which can be made with the aid of cross-sec- 
 tion paper. 
 
 Although lettering admits of an endless variety of forms, the draughtsman 
 should comprehend that there are rules on which letters should be constructed 
 before he undertakes the free-hand method. 
 
 Fig. 177 gives the designation of various parts of a letter to which reference 
 is made in the description. In the Roman letters the square is taken as the 
 scale of construction. Fig. 178 gives the scale of proportionate width. W 
 takes the whole square, its height and width being equal ; I is one quarter as 
 wide ; A, five sixths, etc. To obtain the width of any letter according to this 
 scale, the height may be marked off on the vertical 12. Where the horizontal 
 line from this point intersects the diagonals of the desired letter the width is 
 measured. 
 
 The thickness of the body stroke of the letters is about one fifth the height; 
 the thickness of the body curve is slightly in excess of this, and the excess is 
 added outside the letter ; otherwise in comparison with the straight body 
 strokes the curved stroke would appear too thin. 
 
 All letters are of the same height, except those curved or pointed at the top or 
 
DRAWING INSTRUMENTS. 
 
 63 
 
 bottom, such as C, G, J, 0, S, U, A, V, W. When the curved or pointed parts 
 are at the top they must extend a little above the line and when at the bottom 
 
 
 below the line, otherwise they will look smaller than those of square outline. 
 The lower feet of letters and the feet of T extend about one third the height, 
 but the upper feet are a trifle smaller. 
 
 The intermediate horizontal hair stroke in B, E, F, H, and E is a little 
 
 N 
 
 D _l> 
 
 z u. x 
 
 UJ > 
 
 r 
 
 a 
 
 Fia. 177. 
 
 above the centre, P slightly below the centre, and A about one third of the, 
 height above the lower line. 
 
 The hair strokes and outlines are first put in ; the outline is then filled with 
 a writing-pen, toothpick, or brush. 
 
DRAWING INSTRUMENTS. 
 
 Fio. 179. 
 
 However well proportioned the letters may be, an even effect is not produced 
 unless a proper space is made between them. In letters of square form the 
 spacing is equal, but where such combinations as LT, AT, AA, and numerous 
 others occur the spacing must be less. No general rule 
 can be given for this, and it must be left to the practised 
 eye of the draughtsman. 
 
 The only rule necessary for the construction of small 
 Koman letters is that for ascertaining their height as 
 compared with the capitals : Let a vertical line a b (Fig. 
 179), equal to the height of the -capital, be drawn, and 
 a line at right angles at the top of this line, equal to one 
 half its length ; connect d to ec, and lay off the length 
 of the line d e, equal to b d ; then a e will be the height 
 required. When the learner has acquired some dexterity 
 in lettering, the upper and lower line alone will be nec- 
 essary for his guidance ; he may then attempt the ex- 
 ecution of the curves, without the compass, by free 
 hand. 
 
 In Italic letters the proper angle for their slant 
 is 23 from the vertical ; the proportions are the same 
 as in the Roman letters 
 
 In stump letters the capitals are the same as Italics. The small letters differ 
 somewhat, and are made with one bold stroke of the pen, the hair line gliding 
 imperceptibly into the body stroke. 
 
 Block letters, of which an example has been given, constructed on squares, 
 are one of the most valuable types, being very distinct and readily drawn by 
 the drawing-pen. The letters are of the same proportional width as the Roman, 
 except that the M is a square. The height and width of the letters are varied 
 to suit their application. There are lettering 
 triangles (Fig. 180) made to give the angles 
 with the verticals of inclined letters. 
 
 Old English and German text and other let- 
 ters of a similar character may be quickly and 
 neatly written by the use of a wooden toothpick 
 for the body strokes, the hair lines and termina- 
 tions being afterward put in with a fine pen. 
 
 An old style of writing that has lately gained 
 considerable popularity in this country is round 
 writing. This resembles ordinary writing in 
 that one letter is joined to the next, and each 
 word written as a whole. There are special pens 
 made for this writing, which are very useful; 
 but in place of these the ordinary stub pen can 
 be used. This writing consists of a very few 
 elements, merely shaded semicircles, straight 
 shaded lines, and diagonal hair strokes ; the pen is held in such a manner that 
 you are able to draw a fine hair line at an angle of 45 with both nibs of the 
 pen, and the pen is held in this position for all letters. 
 
 NXY 
 
 AMV 
 
 W 
 
 FIG. 180. 
 
DRAWING INSTRUMENTS. 
 
 HOMAET 
 
 ABCDEFGHIJK 
 LMNOPQRSTW 
 
 abcdeWXYZ fghij 
 klmnopqrstuvwxyz 
 
 ITALIC 
 
 LMNOPQRSTUV 
 
 alcdeWXYZ fghij 
 klm n op qrs t uvwxyz 
 
 acefg h ik ms u wxy z 
 i n m iv v vi yn w ix XLC DM 
 
 O12 34 5 6 780 
 
 V r. 
 
66 DRAWING INSTRUMENTS. 
 
 ABCDEGHJKLMNOPRSTUWY 
 ABCDEGHJKLMNOPRSTUVWY 
 
 ABCDEFGHJKLMNOPRSTUVWr 
 ABCDEFGHIJKLMNOPQBSTUVWY 
 
 
DRAWING INSTRUMENTS. 67 
 
 ENGLISH GOTHIC. 
 
 ABC DE FGH IJ KLMN OP 
 
 QRST UV WX YZ 
 
 1234567890 
 
 ITALIC. 
 
 ABC DE FGH IJ KLMN OP QRST 
 
 UV WX YZ 
 
 a be de fgh ij klmn op qrst uv f ivx yz 
 
 TUSCAN. 
 
 ABC DE FGH IJ KLMN OP QRST 
 
 UV WX YZ 
 
 1234567890 
 
 ABC DE; PGH IJ 
 KLMUOPQEST 
 
 UVWX YZ 
 
 ate de fgh. ij klmn op 
 
 qrst uvwxyz 
 
 
 
DRAWING INSTRUMENTS. 
 
 ENGLISH CHURCH TEXT. 
 
 ic / 
 
 iC fl ' 
 abr fa fglj ij klntn np qrst tin rax 
 
 g 
 
 MEDIAEVAL. 
 
 O 
 
 adr bp fg| ij hlran 09 qrsf ufi fDf g? 
 
 131 K3LWB 
 
 WX f 25 
 
 lic d.e fgh ij ttlmn np qrst utr wx 
 
 COAST CHART No. 20 
 
 NEW YORK BAY AND HARB OR 
 
 NEW YORK 
 
DRAWING INSTRUMENTS. 
 
 69 
 
 LJ 
 
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70 DRAWING INSTRUMENTS. 
 
 Cxmfrart 
 
 PLAN FRONT ELEVATION 
 REAR SIDE TRANSVERSE 
 LONGITUDINAL SECTION 
 
 The character and size of the letters should be in accordance with the draw- 
 ing on which they are to appear. Thus in engineering or mechanical drawing 
 nothing is so appropriate as the block and Koman letter. 
 
 In topographical or map drawing several styles of various sizes are used ; 
 Koman capitals of different sizes are used in designating States, countries, and 
 cities, while State boundaries and towns, large villages, summits, and boundaries 
 of countries are written with an initial capital and small Roman letters. Large 
 bodies of water are denoted in Italic capitals, smaller bodies, such as streams, 
 creeks, small lakes, and ponds, with an initial Italic capital and stump letters, 
 the general direction of the letters following the course of the stream. Capital 
 block letters differing in size represent ranges of mountains, railroads, stations, 
 streets, and prominent objects on a map. Oblique capital block letters repre- 
 sent railroads and canals. 
 
 Old English and German text are the styles most employed for the en- 
 grossing of certificates and similar uses. 
 
 Letters on topographical drawings are written horizontally, so as to be read 
 from the lower right-hand corner of the map, except such as follow the course 
 of a stream or a railroad ; and these can usually be arranged to be read from 
 the same direction. 
 
 A number of specimen titles are given, illustrating the use of the different 
 styles of letters, but it will be found that the plainer the letters the better the 
 effect produced, and for this reason it is generally well to omit fancy letters. 
 No better example of neat and dignified titles can be found than those on the 
 maps issued by the United States Coast Survey, the chief beauty of which con- 
 sists in the admirable adjustment of the sizes of letters, the few styles, and the 
 almost perfect execution. 
 
 The illustrations of letters and titles given are taken from those in general 
 use, but the draughtsman should make a large collection of titles of maps and 
 books, drawings of machinery, advertisements, and business cards, copying 
 carefully such as may suit him, by which he will gain ease of manipulation 
 and taste in selection, to give character and finish to his own drawings. 
 
DRAWING INSTRUMENTS. 
 
 71 
 
 PROFILE AND CROSS-SECTION PAPER. 
 
 Paper printed in squares is used by designers of figures for calicoes, silks, 
 and woollens. For the engineer, there is a class of papers called profile and 
 cross-section papers, sold in sheets or rolls, and of various scales. In the first, 
 which is almost entirely applicable to lines of surveys of railroads and high- 
 ways, the vertical scale is to the horizontal as 20 to 1. This is the usual dis- 
 tortion to make grades, with the cuts and fills apparent. The latter originally 
 
 
 UNITED STATES UNITS. 
 FIG. 181. 
 
 intended, as the name implies, for cross-sections of railway or canal cuts, 
 bat now extensively employed by the architectural and mechanical designer 
 for the rough sketches of works either executed or to be executed ; by the 
 
72 
 
 DRAWING INSTRUMENTS. 
 
 sanitarian, for the plotting of death-rates ; for thermometric and hygrometric 
 readings ; by the broker and merchant, for the graphic representation of 
 the prices of gold, stocks, or articles of merchandise, during a term of years ; 
 by the railway superintendent, for the movement of trains ; and for a multitude 
 of other uses. Cross-section papers most generally applicable are in. divisions 
 of tenths ; but as mechanics are more conversant with the two-foot rule, of 
 which the divisions are in eighths, paper with like divisions are more convenient, 
 and designs on it more intelligible to them. 
 
 Fig. 181 shows a graphical method of determining the equivalent values of 
 the metric system of measurements in United States units, or vice versa. The 
 vertical scale represents the metric units, and the horizontal the common or 
 United States units. 
 
 Example. What is the equivalent value of seven kilometres in miles ? Read 
 upward on the metric scale to 7, then read on that horizontal line to the point 
 of intersection with the line designated " Miles and Kilometres," that is, at 
 the point on the United States scale of units representing 4*35, and you find 
 that seven kilometres are equal to 4-35 miles. 
 
 What is the value of five pounds in kilogrammes ? The process is the same 
 as the foregoing, except that, to change United States units into the metric 
 units, you first read horizontally, then upward. In this case five pounds is 
 found equal to 2*25 kilogrammes. The divisions may represent single units, 
 ten units, one hundred units, etc. Of late it has been common here and in 
 England, to write ft. Ibs., instead of Ibs. ft., but where French weight and 
 measures obtain, the rule is to say kilogrammetres, that is the weight before 
 the distance moved. 
 
 Fig. 182 shows the method of finding the average of a number of observa- 
 tions, to determine the velocity of a current of water. The figure represents 
 the path of a float in a wooden flume or channel, of rectangular section, from 
 
 WIDTH OF FLUME 
 1O' 20" 
 
 FIG. 182. 
 
 Francis's " Lowell Hydraulic Experiments." The width of the cut represents 
 the width of the flume, each abscissa being one foot ; the ordinates are the 
 speeds of float in divisions of O'l foot per second ; the small circles represent 
 the floats in their observed path and speed ; and the curved line shows the 
 average velocity in the different threads of the stream, from which the lines 
 of average velocities of the entire width of flume are deduced. 
 
 The velocities were taken by tin tubes loaded so as to float within about an 
 inch of the bottom of the flume, with the top plugged and projecting a few 
 inches above the surface of the water. The results were checked by flows 
 
DRAWING INSTRUMENTS. 73 
 
 measured over a weir ; but for all practical purposes the velocities as taken by 
 the floats may be considered averages on each thread of the stream. A full 
 set of tubes were prepared adapted to the depths of the water, and taken to 
 the flumes while experimenting. For general use a cylinder may be adapted 
 as a float with an open pipe sliding down it adjustable to the depth. 
 
 Fig. 183 is a diagram illustrating graphically the difference between the 
 charge on a ton of merchandise per mile on the New York Central and Hud- 
 son Eiver Railroad and the Erie Canal for every year between 1857 and 1880. 
 The higher values in every case represent the railroad rates and the lower the 
 canal rates. The black band shows the difference between them. 
 
 FIG. 183. 
 
 Fig. 184 is a graphic representation made up from the records of " The 
 Engineering and Mining Journal," exhibiting the amount of pig iron made in 
 the United States per year for thirty-one years. In constructing the diagram 
 
Y4 DRAWING INSTRUMENTS. 
 
 cross-section paper was used, but in tracing the vertical cross lines were 
 omitted. 
 
 FIG. 184. 
 
 Fig. 185 is made up from the time-table of the New York, New Haven and 
 Hartford Railroad, showing the movement of trains, two from New York and 
 two from New London, the horizontal lines being cut off on a scale of miles for 
 each station, and the vertical lines being a scale of hours. If the speed had 
 been uniform, the line showing the movement of trains would have been 
 straight, but the line represents the practical running time. 
 
DRAWING INSTRUMENTS. 
 
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 Fia. 185. 
 
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76 DRAWING INSTRUMENTS. 
 
 
 
 
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 Range of Temperature degrees daily 
 A Percentage of Humdty dally 
 
 A A 
 
 
 55.2 
 
 T. Total deaths from all causes 
 
 
 L. Deaths from Loca Diseases 
 
 
 nfant Mortal ty under 1 year of age 
 
 
 Z. Mortal ty from all Zymot c Diseases 
 
 
 
 
 
 
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 JUNE JULY 
 
 AUGUST SEPTEMBER 
 
 1870 
 
 Fio. 186. 
 
DRAWING INSTRUMENTS. ff 
 
 Fig. 186 is a graphic representation of the mortality and general classes of 
 diseases as registered by the New York City Board of Health for the months 
 of June, July, August, and September, 1870 ; with the daily records of tem- 
 perature and humidity ; to complete these diagrams there should be one of the 
 daily rainfalls. For meteorological purposes it is usual to take at observatories 
 the commencement, termination, and amount of rainfall. 
 
 Above is an ornamental design in straight lines on the bases of lines parallel 
 to the sides of an equilateral triangle. 
 
 In pages 78, 79 are given designs on the bases of lin'es ruled in rectangles 
 and lozenges. The figure, page 80, illustrates how colors may be represented 
 in a design. 
 
 In pages 81, 82 are illustrations in single lines of the tracery of Gothic 
 windows of which the lines of construction are left to assist the draftsman 
 in completion of the design, which will afford excellent practice in the intricate 
 work of making lines tangent to each other. 
 
DRAWING INSTRUMENTS. 
 
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80 
 
 DRAWING INSTRUMENTS. 
 
DRAWING INSTRUMENTS. 
 
 81 
 
82 
 
 DRAWING INSTRUMENTS. 
 
PLOTTING. 
 
 PLOTTING is the laying out on paper in plan, or horizontal projection, the 
 boundaries of portions of the earth's surface of greater or less extent, from the 
 notes of surveys or other records. When the extents are large, embracing de- 
 grees of latitude and longitude, the plots are designated as maps ; but if of 
 small extent, as lots, estates, and farms, they are usually designated as plans or 
 plots. After completing the outlines, it is usual to fill up the plot with the charac- 
 teristic features, geographical, geological, agricultural, industrial, and domestic, 
 which are expressed conventionally, as will be shown under the head of " Topo- 
 graphical Drawing." 
 
 Scales. The choice of the scale for the plot depends on the purpose for 
 which the drawing is intended. It should be large enough to express all de- 
 sirable details which it is intended to illustrate, and the place it is to oc- 
 cupy. 
 
 Plans of house-lots and small plots of farm surveys are usually so many 
 feet to the inch ; maps of surveys of States, so many miles to the inch ; and 
 maps of railway surveys, so many feet to the inch or so many inches to the 
 mile. Formerly the lines of farms were measured by the four-rod chain. Two 
 to three chains to the inch was then a very common scale. 
 
 On the United States Coast Survey all the scales are expressed fractionally 
 and decimally. The original surveys are generally on a scale of one to ten or 
 twenty thousand, but in some the scale is larger or smaller. The public sur- 
 veys embrace three general classes: 1. Small harbour charts. 2. Charts of 
 bays, sounds, etc. 3. General coast charts. 
 
 The scales of the first class vary from go ^ 6 to 60 ^ 00 , according to the nature 
 of the harbour and the objects to be represented. 
 
 The scale of the second class is usually fixed at -g-^^n,-. Preliminary charts, 
 are, however, issued of various scales, from -g-jj-J-j-o to 2 * . 
 
 Of the- third class the scale is fixed at 4 fa 6 for the general chart of the 
 coast from Gay Head to Cape Henlopen, although considerations of the prox- 
 imity and importance of points on the coast may change the scales of charts of 
 other portions of our extended coast. 
 
 On all plots of large surveys, it is very desirable that the scales adopted 
 should bear a definite numerical proportion to the linear measurement of the 
 ground to be mapped, and that this proportion should be expressed fractionally 
 on the plan, even if the scale be drawn or expressed some other way, as miles to 
 the inch. The decimal system has the most to recommend it, and is generally 
 adopted in government surveys. 
 
 For railroad surveys, the New York general railroad law directs the scale of 
 
 83 
 
84 PLOTTING. 
 
 map which is to be filed in the State Engineer's office to be 500 feet to ^ foot, 
 
 For canal maps, a scale of two chains to the inch, i g * 8 4 , is employed. In 
 England plans and sections for projected lines of inland communication, or 
 generally for public works requiring the sanction of the legislature, are re- 
 quired by a standing order to be drawn to scales not less than four inches to 
 the mile, 1 : 15,840, for the plan, and 100 feet to the inch, 1 : 1,200, for the 
 profiles. 
 
 In the United States engineer service the following scales are prescribed : 
 
 General plans of buildings ...... 10 feet to the inch, 1 : 120 
 
 Maps of ground with horizontal curves 1 foot apart . 50 " " 1 : 600 
 
 Topographical maps, 1 mile square .... 1 mile to 2 feet, 1 : 2,640 
 
 Topographical maps, 3 miles square .... 1 " 1 foot, 1 : 5,280 
 
 Topographical maps, between 4 and 8 miles . 1 " 6 in., 1 : 10,560 
 
 Topographical maps, 9 miles square . . . .1 " 4 " 1 : 15,840 
 
 Maps not exceeding 24 miles square ..... 1 " 2 " 1 : 31.680 
 
 Maps comprising 50 miles square . . . .1 " 1 inch, 1 : 63,360 
 
 Maps comprising 100 miles square . . . .1 " " 1 : 126,720 
 
 Surveys of roads and canals ...... 50 feet to 1 " 1 : 600 
 
 Many government maps are made on a scale of -^oVo- or 
 
 It is always desirable that the scale should be drawn on maps and plans, as 
 they are often reduced by photography. 
 
 In cities and towns, plots of lots and squares are generally rectangular, and 
 they can be readily plotted on any convenient scale. 
 
 Fig. 187 is a plan of the usual New York city lot, 25 X 100, on a scale of 20 
 feet to the inch, or -g-^. 
 
 Fig. 188 is a city square containing thirty-two of these lots, on a scale of 
 100 feet to the inch, or 12 1 00 . The most accurate way is to plot the large rec- 
 tangle 400x200 feet, and then subdivide it. 
 
 Fig. 189 is a plan of the city squares, with the inclosing streets, on a scale 
 200 feet to the inch, or -y^Vff- 
 
 But many lots and most estates are not rectangular, and these, the angles 
 being recorded, must be plotted by the aid of a protractor. 
 
 If the survey has been made by triangles, the principal triangles are first 
 laid down in pencil by the intersection of their sides, the length being taken 
 from the scale and described with compasses. In general, when the surveys 
 have been conducted without instruments, the positions of the points on paper 
 are determined by the intersection and construction of the same lines as has 
 been done in the field. 
 
 Surveys are mostly conducted by measuring the inclination of lines to a 
 meridian or to each other by the compass or by the transit. In the survey of 
 farms, where great accuracy is not required, the compass is mostly used. 
 The compass gives the direction of a line with reference to the magnetic 
 meridian. The true meridian can be obtained from the observations of the 
 polar star or from the magnetic meridian corrected from the records of the 
 variations by the geodetic surveys of the United States at the place and 
 time. 
 
 The plane table is very convenient for filling in the details of a survey when 
 
PLOTTING. 
 
 85 
 
 the principal points have been determined by triangulation, and its records are 
 readily transferred to the drawing. 
 
 At the left of Fig. 190 are the notes of a compass survey, from which the 
 figure is plotted by drawing a meridian through each station and laying off the 
 
 FIG. 187. 
 
 angle of deflection. In small drawings it is more convenient, as shown in Fig. 
 191, to plot all the bearings on a single meridian and then transfer them to the 
 places where they are wanted by any instrument for drawing parallel lines, or 
 to lay off on a single meridian as many bearings as convenient and then trans- 
 fer the meridian for another plot. 
 
 If, as in Fig. 192, the plot fails to close that is, if the termination a of the 
 last line does not join the commencement of the first line at 1, either the sur- 
 vey or the plotting is incorrect. If the latter be correct, the error of the sur- 
 vey must be balanced, or distributed through the lines and angles of the plot 
 (Fig. 192). Connect 1 with a, and draw lines parallel to 1 a through 2, 3, 4, 
 
86 
 
 PLOTTING. 
 
 5, of the plot. Draw an indefinite line, I a (Fig. 193), and on this, with any 
 convenient scale, lay off consecutively the lines of the survey, 1-2, 2-3, 3-4, 
 
 J 
 
 FIG. 189. 
 
 5 ? 5_#. Erect perpendiculars at the extremities of the lines, 2, 3, 4, 5, and 
 a. On the perpendicular a b, lay off 1 a from the plot and connect b 1. The 
 
 3.23 
 
 la 
 
 -(5)- 
 3.55 
 
 cd 
 
 -(4)- 
 222 
 
 -(8)- 
 1.29 
 
 -(2)- 
 2.70 
 
 -OH 
 
 FIG. 190. 
 
 intersections of the perpendiculars by this line will determine how much each 
 of the points of the plot is to be moved on the parallels to 1 a to distribute 
 the error. The dotted lines on the figure show the corrected outline. If the 
 amount by which the plot fails to close is large, the plot should be resurveyed. 
 
PLOTTING. 
 
 By the aid of the Traverse Table a plot of a survey may be balanced. 
 The Traverse Table (see appendix) is a table of differences of latitudes and 
 
 FIG. 192. 
 
 departures, the difference of latitude between two stations being the difference 
 north and south between them ; the difference of departure, the difference east 
 and west. 
 
 Thus, N S (Fig. 194) being the meridian, N 
 
 and A B the course, A C is the difference of 
 latitude, and A D the departure. 
 
 The differences vary according to the length 
 of A B, and the angle it makes with the merid- 
 ian. 
 
 Taking the field notes of the following sur- 
 
 FIG. 193. 
 
 vey, we make a table as follows of the stations, bearings, and distances, leaving 
 columns for latitudes and departures : 
 
 STATION. 
 
 Bearing. 
 
 Distance. 
 
 LATITUDES. 
 
 DEPARTURES. 
 
 N. 
 
 S. 
 
 E. 
 
 w. 
 
 1 
 
 N. 52 B. 
 S. 29f E. 
 S. 31| W. 
 N. 61 W. 
 
 1,063 
 410 
 769 
 713 
 
 655 
 346 
 
 356 
 654 
 
 &38 
 203 
 
 405 
 624 
 
 2 
 
 3 
 
 4 
 
 
 Find by the Traverse Table the number of degrees of the angle or bearing 
 on the left-hand side of the page if less than 45, and on the right-hand side if 
 more. The numbers on the same line running across the page are the latitudes 
 and departures for that angle and for the distances which may represent any 
 
88 
 
 PLOTTING. 
 
 unit, as feet, chains, links, metres, etc. The traverse table gives the latitudes 
 or departures for a single unit ; 10, 100, 1,000, or any other decimal quantity 
 may be obtained by moving the decimal point one, two, three, or more places 
 to the right. 
 
 Thus let us take station 1 in previous survey, in which the latitude and de- 
 partures of a course having a distance of 1,063 feet and a bearing of 52, and 
 then take out the latitude and departure for 1, 6, 3, and place them as below : 
 
 Distance. 
 1,000 
 60 
 
 1,063 
 
 Latitudes. 
 616-0 
 36-94 
 1-847 
 
 654-787 
 
 Departures. 
 
 788-0 
 47-28 
 2-364 
 
 837-644 
 
 If the survey has been accurately performed, the northings and southings 
 of the latitude and the eastings and westings of the departures will balance and 
 the survey will close. In the preceding survey they do not balance ; it there- 
 fore becomes necessary to balance it. This operation consists in correcting the 
 latitudes and departures of the courses so that the sums of the northings and 
 southings of the latitudes and the eastings and westings of the departures shall 
 be equal. This is done by distributing the difference of their sums among the 
 courses in proportion to their length. 
 
 The difference between the northings and southings of latitude, which is 9, 
 is divided by the total length, 2,955, which gives the amount per foot to be 
 added to the lesser column and subtracted from the greater column in propor- 
 tion to the length of the courses to cause the northings and southings to 
 balance. The departures are balanced in a similar manner. 
 
 The following table gives the original latitudes and departures and the cor- 
 rected ones : 
 
 i^. 09 to M Stations. 
 
 Bearings. 
 
 Dis- 
 tances. 
 
 LATITUDES. 
 
 DEPARTURES. 
 
 CORRECTED 
 LATITUDES. 
 
 CORRECTED 
 DEPARTURES. 
 
 N. 
 
 S. 
 
 E. 
 
 W. 
 
 N. 
 
 S. 
 
 E. 
 
 vv. 
 
 N. 52 B. 
 S. 29J E. 
 S. 31f W. 
 N. 61 W. 
 
 1,063 
 410 
 769 
 713 
 
 655 
 "346 
 
 "356 
 654 
 
 838 
 203 
 
 
 658 
 
 "355 
 651 
 
 834 
 
 201 
 
 408 
 627 
 
 405 
 624 
 
 "348 
 
 
 
 2,955 
 
 1,001 
 
 1,010 
 
 1,041 
 
 1,029 
 
 1,006 
 
 1,006 
 
 1,035 
 
 1,035 
 
 After the latitudes and departures have been corrected, it is necessary to 
 make a drawing of the plot. How this is done will be readily seen by the accom- 
 panying illustration of this survey (Fig. 195). This figure also illustrates a 
 very convenient and accurate method of determining the area of the survey 
 mathematically. By means of the latitudes and departures the area of the 
 full parallelogram is taken, and the triangles on the four sides of the plot are 
 deducted from it, leaving 489,245 square feet as the area of the figure. 
 
 The use of the compass is now confined to the surveying of land areas of 
 large extent and little value, or as a means of checking the bearings as taken by 
 the theodolite or transit. Forcing should not be attempted with the latter in- 
 struments. If the survey does not balance almost exactly, it should be resur- 
 
PLOTTING. 
 
 89 
 
 veyed ; otherwise it could not be used in a court of law except as approximately 
 correct. The steel tape divided decimally is almost exclusively used in accu- 
 
 rate work. 
 
 The system of plotting by traverse is the same for the survey by transit as 
 
 Pep. 834 
 
 *> Dep. 20l' 
 
 Dep. 627' 
 
 Fio. 195. 
 
 by compass, except as the angles are taken to minutes and seconds. The latitudes 
 and departures are to be taken by logarithmic sines and cosines ; if these are 
 not obtainable, the table for natural sines and cosines in the appendix will give 
 minutes, and seconds can be obtained by interpolation of differences. 
 
 Fig. 196 is the plot of a survey by transit. Any side of the plot may be 
 assumed as a meridian. The bearings are always taken to the right of it.. 
 After it has been plotted it can be checked by the traverse. Meridians de- 
 scribed through each of the angles will show the meridional angle, 
 tudes and departures may be obtained as follows : 
 
 The lati- 
 
 Angle. 
 
 Nat. sine. 
 
 
 Distance. 
 
 Departure. 
 
 41 40' 
 
 0-66480 
 
 X 
 
 350 
 
 = 233 
 
 Angle. 
 
 Nat. cosine. 
 
 
 Distance. 
 
 Latitude. 
 
 41 40' 
 
 0-74703 
 
 X 
 
 350 
 
 = 261 
 
 In the third line of the third column the angle, instead of 120, is set 
 down at 30, because, when an angle is in excess of 90, 180, or 270, the ex- 
 cess is the angle of which sine and cosine are to be found. 
 
 There can be no division of distance into latitude and departure when the 
 course is either at right angles to or parallel with the meridian. It is then 
 
90 
 
 PLOTTING. 
 
 FIG. 196. 
 
 either all departure or all latitude. In the tables the latitudes are identical for 
 angles and for their supplements, and so are the departures. Reference to Fig. 
 75 will illustrate this, calling the sine latitude and the cosine departure. 
 
 Course. 
 
 Azimuth. 
 
 Angle. 
 
 Dis- 
 tance. 
 
 Latitudes. 
 
 Depar- 
 tures. 
 
 Triangles deducted. 
 
 Rectangles deducted. 
 
 0-1 
 
 41 40' 
 
 41 40' 
 
 350 
 
 261 
 
 23R 
 
 261 x 233 _ 30106 
 
 
 1 2 
 
 85 30 
 
 85 30 
 
 280 
 
 22 
 
 279 
 
 2 
 22 x 279 _ g 069 
 
 233x22 = 5,126 
 
 2-3 
 
 120 00 
 
 30 00 
 
 320 
 
 160 
 
 277 
 
 2 
 277 x 160 _ 00160 
 
 
 3-4 
 
 154 29 
 
 64 29 
 
 300 
 
 271 
 
 129 
 
 2 
 129 x 271 _ 17479 
 
 129 x 160 = 20,640 
 
 4-5 
 
 228 00 
 
 48 00 
 
 700 
 
 469 
 
 520 
 
 2 
 
 
 5-6 
 
 288 30 
 
 18 30 
 
 420 
 
 133 
 
 3Q8 
 
 2 
 
 133 x 398 O g 16? 
 
 
 6-0 
 
 360 00 
 
 00 00 
 
 484 
 
 484 
 
 000 
 
 2 
 
 
 
 
 
 
 
 
 221,521 
 
 25,766 
 
 Circumscribed rectangle, 918 x 900 = 826,200 
 Deduct 221,521 + 25,766 = 247,287 
 
 578,913 square feet. 
 
PLOTTING. 
 
 91 
 
 When it is difficult to measure along the boundaries of an estate, the sur- 
 veys should be made along more convenient and accessible lines, which may be 
 by a closed plot, as in Fig. 196, or by base lines from which the intersections of 
 boundaries are established by offsets carefully determined by measures and 
 angles from points of the survey. 
 
 Fro. 197. 
 
 The first work of the draughtsman is to complete the plot along the lines 
 of the survey; work up the estimate of contents by traverse, and check by 
 measures on -the plot. 
 
 Fig. 197 is an illustration. 
 
 Find the cosine 49 40' distance 215 = 140-1 sine 163-1 
 " 55 25' " 152= 86-3 " 125-1 
 
 226-4 
 Line 525226-4 = 298-6 
 
 = 301-0 the line of plot. 
 
 38 
 
 = 12708 = Bine of 7' 18' 
 
 90 -7 18' = 82 42' 90 - 49 40' = 40 20'. 
 
 By similar construction 34 30' and 78 30' are calculated. 
 
 360- (82 42' + 40 20' + 34 30' + 78 30') = 123 58'. 
 
 The line of the plot, 301, and the adjacent angle, 123 58', are thus ob- 
 tained, and all lines and angles of the plot are established in the same way. 
 The plot is then transferred to a clean sheet, with lines and angles as calcu- 
 lated, but without any lines of survey. 
 
 When the lines of a plot are irregular, as in Fig. 198, divide the plot into 
 equal spaces, and draw parallel lines across the figure through the points of 
 division, add together the two extreme lines, divide the sum by two, and to 
 this dividend add the lengths of the other lines and multiply their sum by the 
 
92 
 
 PLOTTING. 
 
 equal vertical distance between the parallel 
 lines, which will give very closely the entire 
 area. 
 
 Having completed the plot, that is, the 
 main lines of the survey, the filling of other 
 points may in general be done on paper in 
 the same way as they have been established in the field. Intersections of the 
 main lines by roads, streams, fences, and the like are measured off ; other points 
 
 FIG. 198. 
 
 FIG. 199. 
 
 not intersecting, are usually fixed by triangles or by offsets, or lines run on 
 purpose by angles from the main lines. 
 
 In case of unimportant lines, as the crooked 
 brook, for instance (Fig. 199), offsets are taken to 
 the most prominent angles, as a a a, and the inter- 
 mediate bends are sketched by eye into the field- 
 book, and similarly on the plan. 
 
 The most rapid way of plotting the offsets is by 
 the use of a plotting and offset scale (Fig. 200), the 
 one being fixed parallel to the line A B from which 
 the offsets are to be laid off, at such a distance from 
 it, that the zero-line on the movable or offset scale 
 coincides with it, while the zero of its own scale is 
 on a line perpendicular to the position of the station 
 A from which the distances were measured. In the 
 field-book all the offsets are referred to the point of 
 beginning on any one straight line. Move the offset 
 scale to the first distance by the scale at which an 
 offset has been taken, and mark off the length of the 
 offset on its corresponding side of the line ; establish 
 thus repeated points, and join the points by lines as 
 they are on the ground. It may not always be pos- 
 sible to obtain the same scales as those of the plan ; 
 but they may be made of thick drawing-paper or 
 pasteboard. For extensive plotting, as in govern- 
 ment surveys, the offset scale may be made to slide 
 in a groove upon the plotting scale. 
 
 In protracting the triangles of an extended trigo- 
 nometrical survey in which the sides have been cal- 
 culated or measured, it is better to lay down the 
 triangles from the length of their sides ; for ordina- 
 ry surveys, the triangulation is most expeditiously 
 plotted by the means of a protractor. 
 
PLOTTING. 
 
 93 
 
 The outlines of the survey having been balanced and plotted in, with the 
 subsidiary points, as established by offsets and by triangles, the filling in of the 
 interior detail, with the natural features of the ground, from the skeleton or 
 suggestions in the field-book or other records, is done according to conventional 
 signs, to be shown under " Topographical Drawing." 
 
 The public lands of the United States are surveyed, mapped, and divided 
 into nearly square tracts, according to the following system : 
 
 Ranges. Standard lines must first be determined from which to measure. 
 Accordingly, in each land-district some meridian line is run due north and 
 south ; this is called the principal meridian. From some point of the principal 
 meridian is also run a line due east and west, called the base line. 
 
 Other lines are then run in the same direction as the principal meridian, at 
 distances of six miles, measured on the base line, on each side of it. The strip 
 between the principal meridian and the first line, thus run east of it, is known 
 as Range 1 East, the second strip as Range 2 East, etc. And so on the west. 
 This division is shown in Fig. 201. 
 
 FIG. 201. 
 
 
 
 j 
 
 ? 
 
 
 
 
 
 Tp.2 
 
 
 
 
 
 
 
 North 
 
 
 
 tn 
 
 . 
 
 . 
 
 S 
 
 
 
 
 
 S 3 
 
 a 
 
 w 
 
 i? 
 
 
 
 
 
 
 
 
 a 
 
 
 
 
 
 ^ 
 
 * 
 
 1 
 
 1 
 
 
 Tp.l 
 
 
 
 * 
 
 K* 
 
 * 
 
 * 
 
 
 North 
 
 
 W 
 
 
 
 
 
 
 
 
 
 
 BASE 
 
 LINK 
 
 
 
 
 
 
 _j 
 
 
 
 . 
 
 
 Tp.l 
 
 
 
 I 
 
 ! 
 
 i 
 
 * 
 
 
 South 
 
 
 
 
 MERIDIAN 
 
 PRINCIPAL 
 
 
 
 Tp.2 
 South 
 
 
 
 FIG. 202. 
 
 Townships. In like manner, lines are run north and south of the base 
 line at intervals of six miles. These lines cut at right angles those which 
 separate the ranges, and with them form squares six miles on each side, called 
 toiunslrips. Each township contains thirty-six square miles. 
 
 The township nearest the base line on the north is known as Township 1 
 North, of the particular range it is in ; the next farther north is Township 2 
 North, of that range, and so on. In like manner, going south from the base 
 line, we have in succession Township 1 South, Township 2 South, etc. 
 (Fig. 202). 
 
 Sections. Each township is divided into thirty-six squares, called sections, 
 each one mile long and one mile wide, and therefore having an area of one 
 square mile. The sections of a township are numbered 1, 2, 3, etc., up to 36, 
 
PLOTTING. 
 
 beginning at the northeast, and running alternately from right to left and from 
 left to right, as shown in Fig. 203. 
 
 A section may be subdivided into half-sections, quarter-sections, eighths, and 
 sixteenths, designated as in the example that follows : 
 
 Let F G (Fig. 204) be Section 3 of Township 2 North, in Eange 1 West ; 
 then 
 
 6 
 
 5 
 
 4 
 
 3 
 
 2 
 
 1 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 18 
 
 17 
 
 16 
 
 15 
 
 14 
 
 13 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 30 
 
 29 
 
 28 
 
 27 
 
 26 
 
 25 
 
 31 
 
 32 
 
 33 
 
 34 
 
 35 
 
 36 
 
 1 mile. 
 
 D 
 
 E 
 
 FIG. 203. 
 
 FIG. 204. 
 
 A is N. (north) of Section 3, Township 2 North, Range 1 West. 
 
 B is S. W. (southwest) of Section 3, Township 2 North, Eange 1 West. 
 
 C is W. (west) of S. E. (southeast) of Section 3, Township 2 North, 
 Range 1 West. 
 
 D is N. E. i of S. E. i of Section 3, Township 2 North, Range 1 West. 
 
 E is S. E. i of S. E. i of Section 3, Township 2 North, Range 1 West. 
 
 Correction Lines. If the meridian lines were parallel to each other, the 
 townships and sections would be exact squares. But as these lines gradually 
 converge toward the north, meeting at the pole, the townships deviate some- 
 what from squares, being narrower on the north than on the south ; and the 
 northern sections of a township are a little smaller than the southern ones. In 
 order that the townships of a range may not thus keep getting smaller and 
 smaller as we go toward the north, a new base line, called a correction line, is 
 taken at intervals, differing in length in different land-districts, and new north- 
 and-south lines are run at distances of six miles measured on the correction 
 lines. 
 
 The system of survey described above is not used in Texas, the public lands 
 there being State property. 
 
TOPOGRAPHICAL DRAWING. 
 
 TOPOGRAPHICAL DRAWING is the delineation of the surface of a locality, 
 with the natural and artificial objects, as houses, roads, rivers, hills, etc., upon 
 it in their relative dimensions and positions, giving, as it were, a miniature 
 copy of the farm, field, district, etc., as it would be seen by the eye moving 
 over it. Many of the objects thus to be represented can be defined by regular 
 and mathematical lines, but many other objects, from their irregularity of out- 
 line, it would be very difficult thus to distinguish; nor are the particular 
 irregularities necessary for the expression. Certain conventional signs have 
 therefore been adopted in general use among draughtsmen, some of which 
 resemble, in some degree, the objects for which they stand, while others are 
 purely conventional. These signs may be expressed by lines, by tints, or by 
 botb. 
 
 Figs. 205 and 206 represent meadow or grass land, the short lines being 
 
 FIG. 205. 
 
 FlO. 206. 
 
 FIG. 208. 
 
 supposed to represent tufts of grass ; the bases of the tufts should always be 
 parallel to the base of the drawing, whatever may be the shape of the in- 
 closure. 
 
 Figs. 207, 208, 209, and 210 give various methods of representing trees. 
 Figs. 207 and 208 represent in plan a forest and an orchard respectively. The 
 
 FIG 209. 
 
 FIG. 210. 
 
 FIG. 211. 
 
 method of Figs. 209 and 210, showing the same in elevation, while it is not 
 consonant with the projection of the plan, to many is more expressive and in- 
 telligible. 
 
 95 
 
96 
 
 TOPOGRAPHICAL DRAWING. 
 
 Fig. 211 represents cultivated land. The lines are supposed to represent 
 plough-furrows, and adjacent fields should be distinguished from each other 
 by different inclinations of lines. 
 
 Figs. 212 and 213 represent marsh or bog land. Fig. 212 is the more ordi- 
 nary mode of representing fresh- water bog ; Fig. 213, salt marsh. 
 
 FIG. 213. 
 
 FIG. 214. 
 
 Fig. 214 represents a river, with mud and sand banks. Sand is designated 
 by tine dots, made with the point of the pen ; mud by a series of short parallel 
 lines. Gravel is represented by coarser dots, and stones by irregular angular 
 forms. 
 
 Water is almost invariably represented, except in connection with bogs, by 
 drawing a line parallel to the shore, following its windings and indentations 
 closely, then another parallel a little lighter and a little more distant, a third 
 still more so, and so on. Brooks, and even rivers when the scale is small, are 
 represented by one or two lines. Fig. 215 gives a plan and sectional view of 
 water, in which the white curves represent the character and direction of the 
 
 FIG. 215. 
 
 Fio. 216. 
 
 flow of streams, retarded at bottom and sides, and more rapid near the surface 
 and at centre. The direction of the current may also be shown by arrows. 
 Fig. 216 represents a bold shore bounded by cliffs. 
 
TOPOGRAPHICAL DRAWING. 
 
 97 
 
 Fig. 217 represents a turnpike. . If the toll-bar and marks for a gate be 
 omitted, it is a common highway. Fig. 218 represents a road as sunk or cut 
 through a hill ; Fig. 219, one raised upon an embankment. Fig. 220 is a rail- 
 
 Via. 217. 
 
 FIG. 218. 
 
 FIG. 219. 
 
 FIG. 220. 
 
 road, often represented without the cross-ties by two heavy parallel lines, some- 
 times by but one. 
 
 Fig. 221 represents a bridge with a single pier ; Fig. 222, a swing or draw 
 bridge ; Fig. 223, a suspension bridge ; and Fig. 224, a ford. Fig. 225 is a lock 
 
 FIG. 221. 
 
 FIG. 222. 
 
 FIG. 223. 
 
 FIG. 224. 
 
 of a canal. Canals may be represented like roads, except that in the latter the 
 side from the light is the shaded line ; in the former, the side to the light. Or 
 
 by 
 
 Conventional signs for the more important ob- 
 jects that are likely to need representation on a 
 map are : 
 
 Saw-mill, 
 
 FIG. 225. 
 
 Signal of Survey, A 
 
 Telegraph, $MP 
 
 Court-house, III 
 
 Post-office, jjj% 
 
 Tavern, ^J^ 
 
 Blacksmith's shop, ^L 
 
 Guide-board, 
 
 Quarry, ^ 
 
 Grist-mill, 
 
 Wind-mill, 
 
 Steam-mill, 
 
 Furnace, 
 
 Woolen-factory, 
 
 Cotton-factory, 
 
 Dwellings, 
 
 Churches, 
 
 Grave-yards, 
 
 HE CURRENT 
 
 Anchorage for ships, 
 Anchorage for coasters, J^, 
 
 Rocks always covered, ^ 
 8 
 
 Buoys, f \ 
 Wrecks, 
 Harbors, 
 
 Light-house, 
 
 Signal-house, 
 
 Channel-marks, 
 
98 
 
 TOPOGRAPHICAL DRAWING. 
 
 The localities of mines may be represented by the signs of the planets, which 
 were anciently associated with the various metals, and a black circle is used 
 for coal, thus, $ Mercury, ? Copper, ^ Lead, D Silver, Gold, $ Iron, 21 
 Tin, Coal. 
 
 The Representation of Hills. The two methods in general use for rep- 
 resenting with pen or pencil the slopes of ground are known as the vertical 
 and the horizontal. In the former (Fig. 226), the strokes of the pen follow the 
 course that water would take in running down these slopes. In the second 
 (Fig. 227), they represent horizontal lines traced round them, such as would be 
 shown on the ground by water rising progressively by stages, 1, 2, 3, 4, 5, 6, up 
 the hill. The last is the more correct representation of the general character 
 and features of the ground, and, when vertical levels or contours have been 
 traced by level at equal vertical distances over the surface of the ground, they 
 should be so represented ; or when, by any lines of levels, these contours can 
 be traced on the plans with accuracy, the horizontal system should >be adopted : 
 but where, as in most plans, the hills are but sketched in by the eye, the verti- 
 cal system should be adopted ; it affords but proximate data to judge of the 
 slope, whereas, by the contour system, the slope may be measured exactly. It 
 is a good maxim in topographical drawing not to represent as accurate any- 
 thing which has not been rigorously established by surveys. On this account, 
 for general plans, when the surface of the ground has not been levelled, nor is 
 required to be determined with mathematical precision, use the vertical system 
 of representing slopes. 
 
 On drawing hills on the vertical system, it is very common to draw contour- 
 lines in pencil as guides for the vertical strokes. If the horizontal lines be 
 traced at fixed vertical intervals, and vertical strokes be drawn between them 
 in the line of quickest descent, they supply a sufficiently accurate representa- 
 tion of the face of the country for ordinary purposes. It is usual to make the 
 vertical strokes heavier the steeper the inclination, and systems have been pro- 
 posed and used by which the inclination is defined by the comparative thick- 
 ness of the line and of the intervening spaces. 
 
 FIG. 226. 
 
 Fio. 227. 
 
TOPOGRAPHICAL DRAWING. 
 
 99 
 
 In describing ground with the pen, the light is generally supposed to 
 descend in vertical rays, and the illumination received by each slope is di- 
 minished in proportion to its divergence from the plane of the horizon. Thus, 
 in Fig. 228, it will be seen that a horizontal surface receives an' equal portion 
 of light with the inclined surface 
 resting upon it, and, as the inclined 
 surface is of greater extent, it will be 
 darker than the horizontal in propor- 
 tion to the inclination and conse- 
 quent increase of the surface, and on 
 this principle varied forms of ground 
 are represented by proportioning the FIG. 228. 
 
 thickness of stroke to the steepness of the slope. 
 
 In the German system proposed by Major Lehmann for representing the 
 slopes of ground by a scale of shade, the slope, at an angle of 45, is indicated 
 by black, the horizontal plane by white. 
 
 A modification of Lehmann's method, proposed by the United States Coast 
 
 FIG. 229. 
 
 Survey, has the advantage of discriminating between slopes of greater inclina- 
 tion than 45. The table gives the proportions of black and white for different 
 inclinations, and the construction may easily be un- 
 derstood from Fig. 229. 
 
 Contour - Lines. Conceive a hill to be com- 
 pletely covered with water. Then suppose the water 
 to be drawn down, say five feet at a time. Each 
 line of contact of the hill and the water will be a 
 contour-line, or a line every point of which is at the 
 same height or level above a fixed horizontal plane, 
 called the datum-plane. For a small hill, stake out 
 the ground in squares of say fifty feet to the side, 
 and take levels at each point of these squares, and as 
 
 many intermediates as the change of slope makes necessary. To draw the map, 
 lay off these squares to a scale, and mark the elevation of each point and the 
 intermediates in pencil. Then by the eye draw in the contours at such vertical 
 distances apart as the requirements of the map call for. For a large survey, 
 say of a mountain, such a method is impracticable. In this case, the surveyor 
 fixes a number of points at the same level, the points being absolutely estab- 
 lished by the transit or compass, so that they can be plotted accurately. Con- 
 nect all points on each level, and fill in the distances between by the eye, on the 
 supposition that the slope is uniform between these lines. The lines absolutely 
 
 Slope. 
 
 Propor 
 Black. 
 
 tion of 
 White. 
 
 2 
 
 1 or 2f 
 
 1 
 
 10 
 
 5 
 
 or 6 
 
 2 
 
 9 
 
 10 
 
 or 11 
 
 3 
 
 8 
 
 15 
 
 or 18 
 
 4 
 
 7 
 
 25 
 
 or 26 
 
 5 
 
 6 
 
 35 
 
 
 6 
 
 5 
 
 45 
 
 
 7 
 
 4 
 
 60 
 
 
 8 
 
 3 
 
 75 
 
 
 9 
 
 2 
 
100 
 
 TOPOGRAPHICAL DRAWING. 
 
 established and those merely sketched in must not be confounded, and should 
 be distinguished apart either by colour, by size of lines, or by dotting. The 
 contour-lines denoting every five, ten, etc., feet above the datum or plane of 
 reference may be numbered with such height. This is an effective way of rep- 
 
 resenting hills, but is only to be recommended when lines have been traced 
 and it becomes a record of facts. Fig. 230 represents, on double the scale, the 
 
 FIG. 230. 
 
 half of the hill (Fig. 227), with one half completed by drawing the interme- 
 diate contour-lines. 
 
 The objection to the drawing of hills by any system is that the depths of 
 shade representing different slopes conflict with the lights and shades of the 
 
TOPOGRAPHICAL DRAWING. 
 
 101 
 
 drawing, and are therefore confusing. The plan adopted by Von Eggloff stein 
 in his maps was to form a model by cutting out of sheet-wax under the needle 
 of a sewing machine, on the lines of contours, and then properly superimposing 
 them on one another. A mould was then taken from them in plaster. A 
 
 FIG. 231. 
 
 model from the mould, also in plaster, was then taken. This was watered while 
 fresh by a vertical rain from a water-pot, which broke down the vertical edge 
 of the contours, and gave natural lines of watershed. This model was then 
 photographed under an inclined light, and gave an admirable projection. 
 
 Fig. 231 is a contoured map of Greenwood Cemetery and vicinity, Brook- 
 lyn, N. Y. 
 
 Fig. 232 is a map of the harbour and city of New Haven, reduced from the 
 charts of the United States Coast Survey. 
 
 Plate VI is a map of a farming country. These two maps illustrate^the 
 practical applications of topographical conventionalities. 
 
 r-, 
 
 /A : 
 
102 
 
 TOPOGRAPHICAL DRAWING. 
 
 FIG. 232. 
 
TOPOGRAPHICAL DRAWING. 
 
 103 
 
 Railway Surveys are usually plotted by tangents. The curves are then put 
 in, and the topographical features' for the width necessary. The curves are 
 designated by degrees, as a curve of 1, 2, 3, etc., 
 according as the angle subtended at the centre by a 
 100-feet chord is 1, 2, 3, etc. 
 
 Knowing the tangent points, it is easy to plot in 
 the curve, as the centre of the curve must be the 
 intersection of the perpendiculars to the tangents 
 at these points ; or, with one point of tangency, 
 erect a perpendicular at this point, and lay off the 
 radius on it to get the centre of the curve. 
 
 When the curves are larger than can be de- 
 scribed by the dividers or beam compasses, they can 
 be plotted as shown in geometrical problems, or 
 
 points of a curve may be obtained by calculation of their ordinates, and the 
 curves drawn from point to point by variable curves. Knowing the central 
 ordinate of the curve between two points, the central ordinate of one half that 
 curve will be approximately one quarter of the first ; but the greater the num- 
 ber of degrees in the arc, the less near to the truth is the rule. 
 
 ^, 3l.43PeAEattperSlilt ^, Level 
 
 Degree. 
 
 Radii, ft. 
 
 Central 
 ordinate. 
 
 1 
 
 5729-65 
 
 0-218 
 
 2 
 
 2864-93 
 
 0-436 
 
 3 
 
 1910-08 
 
 0-655 
 
 4 
 
 1432-69 
 
 0-873 
 
 5 
 
 1146-28 
 
 1-091 
 
 6 
 
 955-37 
 
 1-309 
 
 7 
 
 819-02 
 
 1-528 
 
 8 
 
 716-78 
 
 1-746 
 
 9 
 
 637-27 
 
 1-965 
 
 10 
 
 573-69 
 
 2-183 
 
 FIG. 234. 
 
 Fig. 234 represents a plot of a railway line ; in this plot the curve is repre- 
 sented as a straight line, the radius of curvature being written in. This method 
 is sometimes adopted when it is desirable to confine the plot within a limited 
 space upon the sheet, and it is convenient plotted thus directly beneath the 
 profile or longitudinal section (Fig. 233). 
 
 In plotting the section, a horizontal or base line is drawn, on which are laid 
 off the stations or distances at which levels have been taken ; at these points 
 perpendiculars, or ordinates, are erected ; upon them are marked the heights of 
 the ground above the base ; and the marks are joined by straight lines. 
 
 Where borings or soundings have been made, and it is necessary to indicate 
 the character and define the limit of the material, the rock may be shown by 
 diagonal hatchuring, streams as in Figs. 222 and 223, and other substances by 
 a combination of lines and'dots, resembling as nearly as possible the material 
 
104 
 
 TOPOGRAPHICAL DRAWING. 
 
 which it is to represent, and the name inserted. If there is a bog or mud in 
 which soundings have been made, the position and depth of soundings should 
 be given ; but when work is to be done by contract, characteristics, unless well 
 established, should not be definitely marked. 
 
 Since it would be in general impossible to express the variations of the sur- 
 face of the ground in the same scale as that adopted for the plan, it is cus- 
 tomary to make the vertical scale larger than the horizontal, usually in the 
 proportion of 10 or 20 to 1. Thus, if the horizontal scale of the plan be 400 
 
 feet to the inch, the vertical scale would be 40 or 20 feet to the inch. For the 
 purpose of facilitating the plotting of profiles, profile-paper can be obtained 
 from stationers, on which are printed horizontal and vertical lines. 
 
 In the plotting of sections across the line which are extended but little 
 beyond the line of the cut or embankment, equal vertical and horizontal scales 
 are adopted ; these plots are mostly to determine the position of the slope, or 
 to assist in calculating the excavation. When cross sections are extended to 
 show the grade of cross-road, or changes of level at considerable distance from 
 the line of rail, the same scales, vertical and horizontal, are adopted as in the 
 longitudinal section or profile. 
 
 In Fig. 233 the upper or heavy line represents the line of the rail, the 
 grades being written above ; this is the more usual way, but sometimes, as in 
 Fig. 235, the profile and plan are combined ; that is, the heights and depths 
 above and below the grade line of the road are transferred to the plan, and re- 
 
 Fio. 235. 
 
 ferred to the line in plan, which becomes thus a representation both in plan 
 and elevation. 
 
 Cross sections, for grades of cross-roads, etc., are usually plotted beneath or 
 above the profile or across the line when plan and profile are combined. 
 
 Besides the complete plans, as above, giving the details of the location, land 
 plans are required, showing the position and direction of all lines of fences and 
 boundaries of estates, with but very few of the topographical features. The 
 centre line of the road is represented in bold line, and at each side, often in 
 red, are represented the boundaries required for the purposes of way. In gen- 
 
TOPOGRAPHICAL DRAWING. 
 
 105 
 
 eral, a width of 100 feet is the amount of land set off, lines parallel to the cen- 
 tral line being at a distance of 50 feet on each side ; but when, owing to the 
 depth of the cut or embankment, the slopes run out beyond this limit, the ex- 
 tent is determined by plotting a cross section and transferring the distances 
 thus found to the plan, and inclosing all such points somewhat within the 
 limits as set off for railway purposes. These plans are generally filed in the 
 register's office for the county through which the line passes. 
 
 Hydrometrical or Marine Surveys. In plotting hydrometrical or marine 
 surveys, the depths of soundings are seldom expressed by sections, but by 
 figures written on the plan, expressing the sounding or depth below a datum 
 
 FIG. 236. 
 
 line, generally that of high water, the low-water line being usually represented 
 by a single continued line. The soundings are expressed in fathoms or in feet. 
 Fig. 236 is a map of Cape Cod Bay plotted by this method. The depths are 
 expressed in feet, and the dotted lines are contour-lines or lines of equal depths. 
 
106 
 
 TOPOGRAPHICAL DRAWING. 
 
 An effective way of making a marine chart is to express the different depths 
 by lines varying in direction, distance apart, width, etc. Fig. 237 is a chart of 
 the Isle of Wight and the surrounding water, with the depths expressed as 
 shown at the bottom of the cut. Sections are often used for rivers, especially 
 rivers like those of the West, that have a very changeable bottom. By plot- 
 ting sections, taken at different times, over one another, distinguishing 
 
 Depth under 5 Fathoms. 5 to 10 Fathoms. 
 
 10 to 20 Fathoms. 
 
 Over 20 Fathoms. 
 
 5 Miles. 
 
 FIG. 237. 
 
 them apart by a difference in colour and variety of line, the changes that take 
 place in the bottom of the river, and the erosion of the banks, are boldly shown. 
 
 In a geological profile the different rocks or formations are sometimes dis- 
 tinguished by colours, explained by marginal notes and squares, but more often 
 by marks, dots, or cross-hatchings. 
 
 The geological and statistical features of a country may be expressed simi- 
 larly and graphically in lines, as in Fig. 238, a preliminary survey of Kentucky, 
 illustrating the principal geological features; and in Fig. 239, in a broader 
 form, giving a larger extent of country, including the portion of the United 
 States east of the Eocky Mountains and the southeasterly portion of Canada. 
 
 These maps give the larger geological divisions, and are suited for books, 
 but are not as effective nor comprehensive of the smaller subdivisions, of which 
 Plate X is an example of a portion of a map of the State of New Jersey. 
 
TOPOGRAPHICAL DRAWING. 
 
 107 
 
 '4\ 
 
 
 
108 
 
 TOPOGRAPHICAL DRAWING. 
 
 FIG. 239. 
 
 STRATIFIED ROCKS 
 
 
 
 1 . 1.1, L-l 
 
 Sections of horizontal and inclined strata. 
 Fio. 240. 
 
TOPOGRAPHICAL DRAWING. 
 
 109 
 
 Psycho 
 
 Quater- 
 nary. 
 
 Tapir, Peccary. Bi 
 Equus. Megatheri 
 
 , Llama. 
 
 Equus Beds. 
 
 /fjjaiu, Tapiriu, Elefhat. 
 
 Pliohippus Beds. 
 
 P/iohipput, Mastodon, B<a, ete. 
 
 Miohippus Beds. 
 
 , alherium, TAinohyui. 
 
 Oreodon Beds. 
 
 Edentates, Hyanodon, Hyracodon. 
 
 Brontotherium Beds. 
 
 Afesokippus, Menodwt, Kbitlierium. 
 
 Diplacodon Beds. 
 
 hpMppus, Amynodon. 
 
 Dinoceras Beds. 
 
 Tinixeras, Uintatherium, Limnohyut, 
 OroUppju, Helaletft, Colonocerat. 
 
 Coryphodon Beds. 
 
 Eohipput, Monkeys, Carnivores, Ungulates, Tillodonts, Rodents, 
 Serpents. 
 
 Lignite Series. 
 
 HtfdrcuauruSj Drypkaaurvs. 
 
 Pteranodon Beds. 
 
 Birds with Teeth, Ifesptn 
 
 Birds with Teeth, llesperornit 
 Pterodactyls, Plesiosaure. 
 
 is, Ichthyornia. 
 
 Dakota Group. 
 
 Atlantosaurus Beds. 
 
 Dinosaurs, Apatosaurus, Allot 
 
 a, Nanosa 
 
 Connecticut River Beds. 
 
 First Mammals (Marsupials), (Dromatherw 
 Dinosaur Foot-prints, Amphitaurun. 
 Crocodiles (Belodon). 
 
 Permian. 
 
 Coal-Measures. 
 
 First Reptiles (). 
 
 Sub-carboniferous. 
 
 First known Amphibians (Labyrinthodonts). 
 
 Corniferous. 
 
 Schoharie Grit. 
 First known Fishes. 
 
 Upper Silurian. 
 Lower Silurian. 
 
 Primordial. 
 
 Huronian. 
 
 Laurentian. 
 
 No Vertebrates known. 
 
 Section of the earth's crust to illustrate vertebrate life in America. 
 Fio. 241. 
 
 Fig. 241 is an ideal section from Le Conte, in which all the most important 
 American strata occurring in different places are brought together and ar- 
 ranged in the order of time. 
 
 In the diagram the different rock-systems are placed one on top of the 
 other, and the vertical black spaces represent by their breadth the relative 
 dominance of different classes at different times. 
 
 The subdivisions of these again into periods and epochs are founded on 
 more local unconformities, and especially on less important changes in the 
 species. 
 
HO TOPOGRAPHICAL DRAWING. 
 
 Transferring. It is usual, in plotting from a field-book, to make first a 
 rough draft, and then a finished copy on another sheet. 
 
 In the copy, only the established points and lines for the outlines are to be 
 transferred. Many lines of construction, balances of surveys, and trial lines of 
 the rough draft are to be omitted, and it is well to sketch in roughly the natural 
 features on the rough draft for aid in the completion of the finished copy. 
 
 The most common way of transferring, for a fair copy, is by superposition 
 of the plan above the sheet intended for the copy, and pricking through every 
 intersection of lines on the plan and all points necessary to preserve. The 
 clean paper should be laid and fastened smoothly on the drawing-board ; the 
 rough draft should be laid on smoothly, and retained in its position by weights, 
 glue, or tacks. The needle must be held perpendicular to the surface of the 
 plan, and pressed through both sheets ; begin at one side and work with sys- 
 tem, so as to prick through each point but once, nor omit any ; make the 
 important points a trifle the larger. For the irregular curves, as of rivers, make 
 frequent points, but very small ones. On removing the plan, select the impor- 
 tant points, those defining leading lines ; draw in these, and the other points 
 will be easily recognised from their relative position to these lines. When any 
 point has not been pricked through, its place may be determined by taking any 
 two established points adjacent to the one required, and with radii equal to 
 their distance on the plan from the point required, describing arcs on the copy 
 on the same side of both points ; the arcs will intersect at the point de&ired. 
 In this way, as in a trigonometrical survey, having established the two extremes 
 of a base, a whole plan may be copied. In extensive drawings it is very com- 
 mon to prick off but a few of the salient points, and fill in by intersections, as 
 above, or by copying detached portions on tracing-paper and transferring them 
 to the copy ; the position of each sketch being determined by the points pricked 
 off, the transfer is made by pricking through, as above, or by transfer-paper 
 placed between the tracing and the copy. 
 
 Plans may be copied, on a reduced or enlarged scale, by means of the pan- 
 tagraph or by the method of squares. 
 
 Map Projections. For a farm or other small survey, the surface of the earth 
 is conceived to be flat, and the map as a horizontal projection of the plane sur- 
 face on a reduced scale. But for large maps, as of countries, States, rivers, or 
 the like, where the meridians and parallels of latitude are represented, such a 
 system would be erroneous. Tfre surface of the earth being a sphere, it is in- 
 capable of development on a plane, so that it becomes necessary to make the 
 best approximation possible in form, relation, and proportional area of the por- 
 tions to be represented on a map or chart. There are many different kinds of 
 projection, all more or less imperfect. 
 
 In general, map projections may be divided into two classes : perspective 
 and developments. The most useful of the perspective projections are the 
 orthographic, the stereographic, and the globular or equidistant. 
 
 Orthographic Projection (Figs. 242 and 243). In this projection the eye is 
 supposed to be at an infinite distance, and the plane of projection perpendicu- 
 lar to the line of sight. Let the circle A L A' M (Fig. 242) represent a hemi- 
 sphere ; draw the diameters A A' and L M at right angles ; divide the arc A L 
 and A M into nine parts, making the parallels thus 10 apart ; and through 
 
TOPOGRAPHICAL DRAWING. 
 
 Ill 
 
 these divisions draw lines across the sphere parallel to A A' ; mark off consecu- 
 tively on a strip of paper the points where the parallels intersect the central 
 meridian L M, and apply and mark off these points along the equatorial diam- 
 eter A A'. Draw meridians with centres on the extension of the line of the 
 equator intersecting the points marked off on this line, and meeting at the poles. 
 Stenographic Projection (Figs. 244 and 245). As in the last example, draw 
 a circle with a central and an equatorial diameter (Fig. 244) ; on the central 
 meridian describe a circle E D F, with a diameter equal to the radius of the 
 
 FiQ. 243. 
 
 larger circle ; through the centre of this smaller circle draw a diameter E F, and 
 divide the arcs E and F into nine equal parts ; and from the pole D, 
 through each of these points, draw lines intersecting the equator. These lines 
 divide the equator into eighteen parts, each containing 10 of longitude, and 
 through these points and the poles meridians are described. Mark off the equa- 
 torial divisions on a slip of paper and transfer them to the central meridian ; 
 
 FIG. 244. 
 
 FIG. 245. 
 
 divide each quarter of the circumference of the sphere into nine equal parts, 
 though these points and the divisions on the central meridian describe arcs. 
 Globular or Equidistant Projection (Fig. 246). A good practical projec- 
 
112 
 
 TOPOGRAPHICAL DRAWING. 
 
 tion can be made by drawing a circle with a diameter for the equator and 
 another at right angles for the central meridian, dividing each quadrant and 
 each radius into nine equal parts. Meridians, with centres on the line of the 
 equatorial diameter, can now be drawn passing through the equatorial divisions, 
 and the poles and parallels of latitude on the line of the central meridian 
 through the divisions of this line and those of the quadrant. 
 
 The foregoing perspective projections in their application to astronomy and 
 geography are usually confined to the representation of the hemisphere, and 
 but rarely to smaller surfaces ; they are therefore of but little practical use to 
 
 the engineer or surveyor, and for land and 
 sea charts on a large scale and of limited 
 extent they are not well adapted. To ren- 
 der developments possible a cylindric or 
 conic surface is substituted in place of the 
 ordinary plane of projection, which surface 
 is afterward developed in a plane. The 
 eye is supposed either at the centre of the 
 sphere or else its position is arbitrary. This 
 conception gives rise to two kinds of de- 
 veloped projections, one kind employing 
 a cylinder, tangent generally at the equa- 
 tor, the other employing a cone, tangent 
 no. 246. generally at the middle parallel. 
 
 The more useful of these projections 
 are Mercator's, the conic, Bonne's, and the polyconic. 
 
 Mercator's Projection (Fig. 247). This is especially valuable to the navi- 
 
 /6O /go /60 140 120 /OO <3O 6(? 
 
 IOO /2O 
 
TOPOGRAPHICAL DRAWING. 
 
 113 
 
 gator, as by it he can lay off his course on a straight line. In this projection 
 the surface of the sphere is developed on a tangent cylinder. Conceive the 
 .outlines of the continents to be drawn on this, and afterward the surface to be 
 unrolled and laid flat ; the result is a chart on Mercator's projection. 
 
 Let P Q P' R (Fig. 248) be the projection of a sphere, P P' and R Q its 
 diameters, and U Y X Z a portion of the tangent , cylinder in the axis of which 
 
 . X 
 U 
 
 60 120 
 
 FIG. 248. 
 
 180 
 
 240 
 
 oOO 
 
 00 
 
 00 
 
 70 
 300 
 
 is contained the axis of the sphere. Divide the quadrant P Q into 10 parts, a 
 line drawn from C through these points, intersecting the tangent X Z, will be 
 the point for the parallel of that degree. Make the equator of the projection 
 3'14 times the diameter of the cylinder, and divide it into twenty-four equal 
 parts ; these points will be 15 apart, or hour distances. Verticals are drawn 
 through these points, giving us parallels of latitude. 
 
 Conic Projection. Instead of on the surface of a 
 sphere, the map is projected on the surface of a cone. 
 The projection may be developed either on a tangent 
 cone or an intersecting cone. The developed arc of the 
 middle latitude is employed for the graduation of longi- 
 tudes. Fig. 249 gives an illustration of the development 
 on a tangent cone in which P B is the sphere and 
 A M is the distance from the apex of the cone to the 
 middle parallel and point of contact ; A M will be the 
 radius for the central parallel and A the point from which all parallels are 
 described. Fig. 250 shows the cone developed on a plane surface. The 
 length of a degree on a meridian of the earth is 69*41 miles at the poles and 
 68 - 70 at the equator. 
 
 Bonne's Projection. In this projection a central meridian and a central 
 parallel are employed, the latter being the development of the circle of contact 
 of the tangent cone. The other parallels are concentric arcs, as in the simple 
 9 
 
 Fio. 249. 
 
114 
 
 TOPOGRAPHICAL DRAWING. 
 
 conic projection drawn through the graduation of the central meridian. Each 
 parallel, however, is divided in accordance with the varying lengths of a degree 
 of longitude in the different latitudes (see table) and an arc passed through 
 
 FIG. 250. 
 
 these points. The map is slightly distorted at the corners on account of the 
 parallels, as projected, being concentric arcs. The great advantage of Bonne's 
 projection is that true proportions of areas are preserved. This method is al- 
 
 most universally employed for the detailed topographical maps based on the 
 trigonometrical surveys of the different countries of Europe. 
 
TOPOGRAPHICAL DRAWING. 
 
 115 
 
 Polyconic Projection. This employs a tangent cone for every parallel. 
 Each parallel of latitude therefore is independently developed. This has the 
 effect of increasing the length of the degrees of latitude in proportion as we 
 recede from the central meridian. To draw a map according to the tables, 
 lay off (Fig. 251) on the straight line N" S, representing the middle meridian^ 
 the lengths representing the 10 of latitude between 20, 30, 40, etc. 
 Through these points draw circular arcs with the radii designated by R in the 
 table. On these arcs lay off the lengths of 10 of longitude for each corre- 
 sponding 10 of latitude on each side of the central meridian. Through the 
 points thus formed draw the meridians, which will be found slightly concave 
 toward the middle one. If the scale is so large that it is impossible to draw 
 the circular arcs with beam compasses, erect perpendiculars at the points 20, 
 30, 40, and 50, and on them lay off the values D M from the tables. At 
 each of the points so found erect perpendiculars, and set off on them the corre- 
 sponding values of D P. Through the points thus found draw the parallels 
 and meridians. By this projection there is little distortion at any portion of 
 the map ; a scale of degrees and minutes of the parallels and meridians, by 
 means of which positions, determined by their latitudes and longitudes, may 
 be inserted in the maps ; the use of a linear scale in any portion or direction ; 
 and the intersection of parallels and meridians at nearly right angles. 
 
 Co-ordinates of Curvature in Miles for Maps of Large Extent. 
 
 
 Latitude 20. 
 
 Latitude 24. 
 
 Latitude 28. 
 
 Latitude 32. 
 
 LONGI- 
 
 
 
 
 
 TUDE. 
 
 
 
 
 
 
 
 
 
 
 D. M. 
 
 D. P. 
 
 D. M. 
 
 D. P. 
 
 D. M. 
 
 D. P. 
 
 D. M. 
 
 D. P. 
 
 2 
 
 130-0 
 
 0-8 
 
 126-4 
 
 0-9 
 
 122-2 
 
 1-0 
 
 117-4 
 
 1-1 
 
 4 
 
 260-0 
 
 3-1 
 
 252-8 
 
 3-6 
 
 244-4 
 
 4-0 
 
 234-8 
 
 4-3 
 
 6 
 
 390-0 
 
 6-9 
 
 379-2 
 
 8-1 
 
 366-5 
 
 9-0 
 
 352-0 
 
 9-8 
 
 8 
 
 520-0 
 
 12-4 
 
 505-5 
 
 14-4 
 
 488-6 
 
 16-0 
 
 469-3 
 
 17-3 
 
 10 
 
 649-8 
 
 19-4 
 
 631-7 
 
 22-4 
 
 610-4 
 
 25-0 
 
 586-3 
 
 27-1 
 
 12 
 
 779-7 
 
 27-8 
 
 757-9 
 
 32-2 
 
 732-4 
 
 36-0 
 
 703-5 
 
 39-1 
 
 14 
 
 909-2 
 
 38-0 
 
 883-6 
 
 43-9 
 
 853-7 
 
 49-0 
 
 819-6 
 
 53-1 
 
 16 
 
 1039-2 
 
 49-6 
 
 1009-9 
 
 57-4 
 
 975-7 
 
 64-1 
 
 936-8 
 
 69-5 
 
 18 
 
 1168-1 
 
 62-8 
 
 1134-8 
 
 72-6 
 
 1096-0 
 
 80-9 
 
 1051-9 
 
 87-8 
 
 20 
 
 1298-0 
 
 77-6 
 
 1261-2 
 
 89-7 
 
 1218-8 
 
 100-1 
 
 1169-2 
 
 108-6 
 
 R. 
 
 10892 
 
 8905 
 
 7458 
 
 6348 
 
 
 Latitude 36. 
 
 Latitude 40. 
 
 Latitude 44. 
 
 Latitude 48. 
 
 LONGI- 
 
 
 
 
 
 TUDE. 
 
 
 
 
 
 
 
 
 
 
 D. M. 
 
 D. P. 
 
 D. M. 
 
 D. P. 
 
 D. M. 
 
 D. P. 
 
 D. M. 
 
 D. P. 
 
 2 
 
 112-0 
 
 1-2 
 
 106-1 
 
 1-2 
 
 99-7 
 
 1-2 
 
 92-7 
 
 1-2 
 
 4 
 
 224-0 
 
 4-6 
 
 212-2 
 
 4-8 
 
 198-9 
 
 4-8 
 
 185-4 
 
 4-8 
 
 6 
 
 335-9 
 
 10-3 
 
 318-1 
 
 10-7 
 
 298-7 
 
 10-9 
 
 277-9 
 
 10-8 
 
 8 
 
 447-7 
 
 18-4 
 
 423-9 
 
 18-9 
 
 398-0 
 
 19-3 
 
 370-3 
 
 19-2 
 
 10 
 
 559-2 
 
 28-7 
 
 529-4 
 
 29-7 
 
 497-1 
 
 30-2 
 
 462-3 
 
 30-0 
 
 12 
 
 670-5 
 
 41-3 
 
 634-7 
 
 42-8 
 
 595-9 
 
 43-4 
 
 554-1 
 
 43-2 
 
 14 
 
 781-6 
 
 56-2 
 
 739-7 
 
 58-2 
 
 694-3 
 
 59-1 
 
 645-6 
 
 58-8 
 
 16 
 
 892-3 
 
 73-4 
 
 844-3 
 
 76-0 
 
 792-3 
 
 77-1 
 
 736-5 
 
 76-7 
 
 18 
 
 1002-6 
 
 92-8 
 
 948-5 
 
 96-1 
 
 889-9 
 
 97-5 
 
 827-0 
 
 97-0 
 
 20 
 
 1112-5 
 
 114-5 
 
 1052-3 
 
 118-5 
 
 986-9 
 
 120-2 
 
 916-9 
 
 119-6 
 
 R. 
 
 5461 
 
 4729 
 
 4110 
 
 3575 
 
116 
 
 TOPOGRAPHICAL DRAWING. 
 
 Length of a Degree of Longitude at Different Latitudes, and at Sea-Level. 
 
 Deg. 
 of 
 
 Miles. 
 
 Deg. 
 of 
 
 Miles. 
 
 Deg. 
 of 
 
 Miles. 
 
 Deg. 
 of 
 
 Miles. 
 
 Deg. 
 of 
 
 Miles. 
 
 Deg. 
 of 
 
 Miles. 
 
 Lat. 
 
 
 Lat. 
 
 
 Lat. 
 
 
 Lat. 
 
 
 Lat. 
 
 
 Lat. 
 
 
 
 
 69-16 
 
 14 
 
 67-12 
 
 28 
 
 61-11 
 
 42 
 
 51-47 
 
 56 
 
 38-76 
 
 70 
 
 23-72 
 
 2 
 
 69-12 
 
 16 
 
 66-50 
 
 80 
 
 59-94 
 
 44 
 
 49-83 
 
 58 
 
 36-74 
 
 72 
 
 21-43 
 
 4 
 
 68-99 
 
 18 
 
 65-80 
 
 32 
 
 58-70 
 
 46 
 
 48-12 
 
 60 
 
 34-67 
 
 74 
 
 19-12 
 
 6 
 
 68-78 
 
 20 
 
 65-02 
 
 34 
 
 57-39 
 
 48 
 
 46-36 
 
 62 
 
 32-55 
 
 76 
 
 16-78 
 
 8 
 
 68-49 
 
 22 
 
 64-15 
 
 36 
 
 56-01 
 
 50 
 
 44-54 
 
 64 
 
 30-40 
 
 78 
 
 14-42 
 
 10 
 
 68-12 
 
 24 
 
 63-21 
 
 38 
 
 54-56 
 
 52 
 
 42-67 
 
 66 
 
 28-21 
 
 80 
 
 12-05 
 
 12 
 
 67-66 
 
 26 
 
 62-20 
 
 40 
 
 53-05 
 
 54 
 
 40-74 
 
 68 
 
 25-98 
 
 82 
 
 9-66 
 
 COLOTJKED TOPOGRAPHY. 
 
 Topographical features may be represented effectively and expeditiously by 
 means of the brush and water-colours, either by India ink alone, or by various 
 tints, or by the union of both. (For preparing colours for tints and their ap- 
 plication, see page .) 
 
 The most important colours for conventional tints are, besides India ink, 
 indigo, carmine or crimson lake, and gamboge, used separately or compounded, 
 and burnt sienna, yellow ochre, and vermilion, generally used alone. 
 
 The following conventional colours are used by the French military engineers 
 in their coloured topography : Woods, yellow, using gamboge and a very little 
 indigo ; grass land, green, made of gamboge and indigo ; cultivated land, brown, 
 made of lake, gamboge, and a little India ink or burnt sienna will answer. 
 Adjoining fields should be slightly varied in tint. Sometimes furrows are in- 
 dicated by strips of various colours. Gardens are represented by small rec- 
 tangular patches of brighter green and brown; uncultivated land, marbled 
 green and light brown ; brush, brambles, etc., marbled green and yellow ; heath, 
 furze, etc., marbled green and pink ; vineyards, purple, composed of lake and 
 indigo ; sands, light brown, made of gamboge and lake, or yellow ochre will do ; 
 lakes and rivers, light blue, with a darker tint on their upper and left-hand sides ; 
 seas, dark blue, with a little yellow added ; marshes, the blue of water, with 
 spots of grass green, the touches all lying horizontally ; roads, brown, between 
 the tints for. sand and cultivated ground, with more India ink ; hills, greenish 
 brown, made of gamboge, indigo, lake, and India ink. Woods may be finished 
 up by drawing the trees and colouring them green, with touches of gamboge 
 
TOPOGRAPHICAL DRAWING. 
 
 in 
 
 toward the light (the upper and left-hand side), and of indigo on the opposite 
 side. 
 
 In addition to the conventional colours, an imitation of the conventional 
 signs is introduced in colour with the brush, and shadows are almost invaria- 
 bly introduced. The light is usually supposed to come from the upper left- 
 hand corner, and to fall sufficiently oblique to allow of a decided light and 
 shade to the slopes of hills, trees, etc. After the shadow has been painted, the 
 outline of the object is strengthened by a heavy black line on the side opposite 
 the light. The flat tints are first laid on as above, and then the conventional 
 signs are drawn in with a pencil and coloured in with appropriate and more 
 intense tints ; the shadows are generally represented in India ink. 
 
 For the shading of hills, wash the surface first with the proper flat tint, 
 trace in with a pencil the outlines, then lay on in India ink tints proportioned 
 in intensity to the height of the hills and steepness of the slopes. To soften the 
 tints, a water brush is used ; the tints are laid on with the colour brush, and 
 softened by passing the water brush rapidly along the edges. The water brush 
 must not have too much water, as it would in that case lighten the tint more 
 than is intended, and leave a ragged, harsh edge. Tints may be applied in 
 very light shades, one over another, with the boundary of the upper tint not 
 reaching the extreme limit of the tint below it. 
 When depth of shade is required, it is best produced 
 by application of several light tints in succession ; 
 no tint is to be laid over the other until the first is 
 dry. 
 
 In shading by contours it is usual to increase the 
 intensity of the shade beyond that .of the mere super- 
 position of one shade on another; where there are 
 
 high altitudes and numerous contours, the lower ones should be put in with 
 a different tint, as burnt sienna or sepia, with increasing shades, and above 
 these graduated shades of India ink, beginning at the same intensity as that of 
 the colour last put on (Plate IX). 
 
 When woods have to be represented, the shading used for the trees, instead 
 of interfering with the shadows due to the slopes, may be made to harmonize 
 with them, and contribute to the general effect by presenting greater or less 
 depth, according to the position of the woods on the sides or summits of the 
 hills. 
 
 An expeditious and effective way of representing hills with a brush, imitat- 
 ing hills drawn with a pen on the vertical system, is effected by pressing out 
 the brush flat to a comb-like edge, drawing this over a nearly dry surface of 
 India ink, and then brushing lightly or more heavily between the contours, 
 according to the steepness of the slope, each of the comb- like teeth making its 
 mark (Plate VII). 
 
 Kivers and masses of water may be shaded in with a colour and water brush ; 
 or, by superposition of light tints, a shadow may be thrown from the bank 
 toward the light, and the outline of this bank strengthened with a heavy black 
 line ; the tints are to be in indigo. 
 
 Topographical drawings may be made in water-colour with but one tint, as 
 India ink, or ink mixed with a little sepia. The conventional signs are made 
 
TOPOGRAPHICAL DRAWING. 
 
 in imitation of pen drawings, the hills in softened tint, or drawn with the comb- 
 edged brush. 
 
 Artistic and effective drawings are made of hills as they would appear in 
 nature under an oblique light, the sides of the hills next the light receiving it 
 more brilliantly according as they are inclined nearer to right angles with its 
 rays, and the shades on the sides removed from the light increasing in intensity 
 as the slopes increase in steepness. 
 
 Having damp-stretched the paper upon the drawing-board, first draw in the 
 lines in pencil, and afterward repeat them with a very light ink-line ; a soft 
 sponge, well saturated, should then be passed quickly over the surface of the 
 drawing in order to remove any ink that would be liable to mix with the tint 
 and mar its uniformity. 
 
 The moistened surface will prevent the tint from drying too rapidly at the 
 edges. In tinting, never allow the edge to dry until the whole surface is cov- 
 ered ; leave a little superfluous colour along the edge while filling the brush. In 
 applying a flax tint to large surfaces, let the drawing-board be inclined at an 
 angle of five or six degrees, to allow the colour to flow downward over the sur- 
 face. With a moderately full brush, commence at the upper outline, and carry 
 the colour along uniformly from left to right and from right to left in horizon- 
 tal bands, taking care not to overrun the outlines, in approaching which the 
 point of the brush should be used, and at the lower outline let there be only 
 sufficient colour in the brush to complete the tinting. 
 
 The colour should not be allowed to accumulate in inequalities of the paper, 
 but should be evenly distributed over the whole surface. 
 
 Too much care can not be given to the first application of colour ; attempt- 
 ing to remedy a defect by washing or applying fresh tints generally makes bad 
 worse. 
 
 Erasers should never be used on a tinted drawing, as the paper, when 
 scratched, receives the tint more readily, and retains a larger portion of colour 
 than other parts, thereby causing a darker tint. 
 
 Marbling is done by using two separate tints, and blending them at their 
 edges. A separate brush is required for each tint ; before the edge of the first 
 is dry, pass the second tint along the edge, blending one tint into the other, and 
 continue with each tint alternately. 
 
 In reference to general effect in tinted topographical drawings, intensity and 
 everything else should be subordinate to clearness. No tint should be prom- 
 inent or obtrusive. Tints that are of small extent must be a little more intense 
 than large surfaces, or they will appear lighter in shade. Keep a general tone 
 throughout the whole drawing. Beginners will find it best to keep rather low 
 in tone, strengthening their tints as they acquire boldness of touch. 
 
 Plate VIII gives an example of coloured topography. 
 
 The plan is usually so drawn that the top may represent the north ; the 
 upper left-hand corner is then the northwest. 
 
 In inking in, commence first with the light lines, since a mistake in these 
 lines may be covered by the shade-lines. Describe all curves before drawing 
 the straight lines, for it is easier to join neatly a straight line to a curve than 
 the opposite. Ink in with system, commencing, say, at the top ; ink in all light 
 lines running easterly and westerly, then all light lines running northerly and 
 
TOPOGRAPHICAL DRAWING. 
 
 southerly, then commence in the same way and draw in the shade lines. Ele- 
 vated objects have their southern and eastern outline shaded, while depressions 
 have the northern and western ; thus, in conventional signs, roads and canals 
 are shaded on opposite sides. Having inked in all lines that are drawn with a 
 ruler or described with compasses, commence again at one corner to fill in the 
 detail, keeping all parts of the plan except what you are actually at work 
 upon covered with paper to preserve it from being soiled. The curved lines of 
 brooks, fences, etc., are sometimes drawn with a drawing-pen, sometimes with 
 a steel pen or goose-quill. 
 
 Boundary lines of private properties, of townships, of counties, of States, 
 etc., are indicated by various combinations of short lines and dots, thus : 
 
 All plans should have meridian lines drawn on them ; also scales, and the 
 dates on which the plans were finished. Page 120 gives several designs for 
 meridians and borders. In these diagrams it will be observed that both true 
 and magnetic -meridians are drawn ; this is desirable when the variation is 
 known, but in many surveys merely the magnetic meridian is taken ; in these 
 cases this line is simply represented with half of the barb of the arrow at 
 the north point, and on the opposite side of the line from the true meridian. 
 
 Scales on drawings which are to be reproduced by photography should 
 always be drawn in ; on others, the proportion of the plan to the ground should 
 be expressed decimally, as ? oW TUOW or ^J stating the number of feet, 
 chains, etc., to the inch. 
 
120 
 
 TOPOGRAPHICAL DRAWING. 
 
ORTHOGRAPHIC PROJECTION. 
 
 ARCHITECTURAL and mechanical drawings are usually the delineation of 
 bodies by orthographic projection, which is the representation on a sheet of 
 paper, having only two dimensions, length and breadth, of solids, having three, 
 length, breadth, and thickness, on such scales that dimensions can be taken 
 from the parts, and structures and machines constructed therefrom. 
 
 Place any surface, for instance, a sheet of paper or a drawing-board, at 
 right angles to the sun's rays. This may be readily done by inserting a pin 
 into the surface, and making it vertical to the surface in every direction by a 
 right-angled triangle ; then place the surface in the direct rays of the sun, and 
 in such a position that there will be no shadow on the surface from the pin ; 
 the sun's rays are then perpendicular to the surface. Take a wafer or a circu- 
 lar bit of paper and hold it over the paper by means of a long pin or wire, and 
 we obtain shadows, as above, varying with the inclination of the wafer to the 
 plane of the paper. When parallel with the plane, the shadow is a complete 
 circle ; when at right angles, a line ; and it varies between them as the wafer is 
 inclined. These shadows are the orthographic projections of the wafer; no 
 line can be longer than it is naturally, but, if inclined or vertical, it is reduced 
 in length till it becomes a point only. The orthographic projection of the pin 
 which has determined the position of the surface is merely the shadow of the 
 head. If the pin be inclined at all, the body of the pin is projected as a 
 shadow by a line ; if the pin be laid on the surface, its projection is the whole 
 length of the pin.. The sun's rays act as perpendiculars, which will be here- 
 after spoken of as projecting the points of an object upon a surface which 
 will represent the object itself in drawing; and, should any confusion occur to 
 the draughtsman of how an object is to be projected or drawn, if he can make 
 the outline of the object on any convenient scale in wire and get its shadows 
 by the sun's vertical rays on a plane, he can readily see how the object should 
 be drawn. 
 
 Since the surfaces of all bodies may be considered as composed of points, 
 the first step is to represent the position in space of a point, by referring it to 
 planes whose position is established. The projection of a point upon a plane 
 is the foot of the perpendicular let fall from the point to the plane. 
 
 If on two planes not parallel to each other, whose positions are known, we 
 
 121 
 
ORTHOGRAPHIC PROJECTION. 
 
 have the projection of a point, the position of this point is determined by erect- 
 ing perpendiculars from each plane at the projected points : their intersection 
 will be the point. 
 
 If from every point of an indefinite straight line, A B (Fig. 252), placed in 
 any manner in space, perpendiculars be let fall on a plane, L M N 0, whose 
 position is given, then all the points in which these perpendiculars meet the 
 plane will form another indefinite 
 straight line, a I : this line is called the 
 projection of the line A B on this 
 plane. It is only necessary to project 
 two points of the line, and the straight 
 line drawn through the two projected 
 points will be the projection of the 
 
 FIG. 252. 
 
 Fio. 253. 
 
 given line. The projection of a straight line, itself perpendicular to the plane, 
 is the point in which it meets the plane. 
 
 If the projections a b and a' V of a straight line on the two planes L M N 
 and L M P Q (Fig. 253) are known, this line A B i& determined ; for if, through 
 one of its projections, a #, a plane be drawn perpendicularly to L M N 0, and 
 if through a' b' another plane be drawn perpendicular to L M P Q, the inter- 
 section of the two planes will be the line A B. 
 
 To delineate a solid, it must be referred to three planes, at right angles to 
 each other. 
 
 Thus, let a b c (Fig. 254) be a parallelepiped in an upright position, of 
 which the plane a b is horizontal, and the planes a c and c b vertical. Let d e, 
 df, and d g be the planes of projection. The sides of the body being parallel 
 to these planes, each to each, let the figure of the parallelepiped be projected 
 on them. Draw parallel lines from the angles of the body perpendicular to the 
 planes, as indicated by the dotted lines ; then upon the plane d e, a' b' is the 
 projection of the surface a b, the plan of the object; upon the plane df, a' c', 
 the projection of the surface a c, the front elevation; and upon the plane d g, 
 the projection b' c' of the surface b c, the side elevation. Thus, three distinct 
 views of the regular solid a b c are delineated on plane surfaces, any two of 
 which are a sufficient description of the object. From the two figures a' c', V c\ 
 for example, the third figure a' b' may be compounded, by drawing the vertical 
 lines c' h b' i and a' k, c' I to meet the plane d e, and by producing them hori- 
 zontally till they meet and form the figure a' b'. Similarly, the figure b' c' may 
 
ORTHOGRAPHIC PROJECTION. 
 
 123 
 
 be deduced from the other two by the aid of the lines h, i, from a! V and the 
 lines m, n, from a' c' . 
 
 It is in this way that a third view of any piece of machinery is to be found 
 from two given views ; and in many cases two elevations, or one elevation and 
 
 FIG. 254. 
 
 FIG. 255. 
 
 a plan, may afford a sufficiently complete idea of the construction of a machine. 
 When parts are inclosed by others, views of the interior are required, in which 
 case the machine is supposed to be cut across by planes, vertical, horizontal, or 
 inclined, to reveal its structure. Such views are termed sections, and distin- 
 guished, with reference to the planes of section, as vertical or horizontal or 
 inclined sections. 
 
 In practice, the drawings are done upon one common surface, the plane of 
 paper. Suppose the plane d g (Fig. 254) revolved back into the position d a', 
 and d e also moved to d e', both of these positions being in the plane of d f. 
 This done, the three views are depicted on one plane surface (Fig. 255) ; d I 
 and d m are the ground and vertical lines ; the positions of the same points in 
 a' c' and a 1 V are in the same perpendicular from the ground-line ; and the 
 position of a point in the plane may be found by applying the edge of the 
 square to the same point as represented in the elevation. The same is true as 
 between the two elevations, and establishes a method of drawing several views of 
 one machine upon the same surface of paper in strict agreement with each other. 
 
 PKOJECTIONS OF SIMPLE BODIES. 
 
 Right projections of a regular hexagonal pyramid (Fig. 256). Two distinct 
 geometrical views are necessary to convey a complete idea of the form of the 
 object: an elevation to represent the sides of the body, and to express its 
 height ; and a plan to express the form horizontally. 
 
124 
 
 ORTHOGRAPHIC PROJECTION. 
 
 Draw a horizontal line L T through the centre of the sheet to represent the 
 ground line. Then draw a perpendicular to the ground line, S S', to represent 
 the axis of the pyramid. 
 
 To construct the plan, from any point, S', on the line S S', as a centre, con- 
 struct the hexagonal base ; the lines A' S', B' S', etc., represent the projections 
 of its edges in the plan. 
 
 Since the base of the pyramid rests upon the horizontal plane, it must 
 
 be projected vertically upon the ground line. From each of the angles at 
 A', B', C', and D' erect perpendiculars to that line. The points of intersec- 
 tion, A, B, C, and D, are the true positions of all the angles of the base ; and 
 it only remains to lay off the height of the pyramid, from the point G to S, and 
 to draw S A, S B, S C, and S D, which are the only edges of the pyramid visi- 
 ble in the elevation ; S A and S D, being in the vertical plane, are seen in their 
 true length ; the points F' and E' being situated in the lines B B' and C C', the 
 lines S B and S C are each the projections of two edges of the pyramid. 
 
 To construct the projections of the same pyramid, having its base set in an 
 inclined position, but with its edges S A and S D still in the vertical plane 
 (Fig. 257). 
 
 With the exception of the inclination, the vertical projection of this solid is 
 
ORTHOGRAPHIC PROJECTION. 
 
 125 
 
 precisely the same as in the preceding example, and it is only necessary to copy 
 that elevation. To do this, fix the position of the point D upon the ground 
 line, through which draw D A, making with L T the desired inclination of the 
 base of the pyramid. Make D A equal to the A D of the preceding figure, and 
 on this erect the vertical projection S A D of that figure. 
 
 Since the edges S A and S D are still in the vertical plane, and the point 
 D remains unaltered, the projection A' of the point A will still be in the line 
 A' S'. The remaining points, B', C', etc., in the projection of the base, are 
 found by the intersections of perpendiculars let fall from the corresponding 
 points in the elevation, with lines drawn parallel to A' S', at a distance equal to 
 the width of the base. Joining all the contiguous points, we obtain A' B' C' D' 
 E' F', the horizontal projection of the base, two of its sides being concealed by 
 the body of the pyramid. The vertex S having been similarly projected to S', 
 and joined by straight lines to the several an- 
 gles of the base, the projection of the solid 
 is completed. 
 
 To find the horizontal projection of a trans- 
 verse section of the same pyramid, made by a 
 plane perpendicular to the vertical, but in- 
 clined at an angle to the horizontal plane of 
 projection, letting all the sides of the base be 
 inclined to the ground line (Fig. 258). 
 
 Since none of the sides of the base are to 
 be parallel with the ground line, draw a line 
 A' D' making the required angle with that 
 line, and from the points A' and D' set out the 
 angular points of the hexagon. To obtain the 
 projections of the edges of the pyramid, join 
 the angular points which are diametrically 
 opposite and project the figure thus obtained 
 upon the vertical plane, as shown in the eleva- 
 tion. 
 
 If the cutting plane be represented by the 
 line a d in the elevation, it will expose, as the 
 section of the pyramid, a polygon whose an- 
 gular points, being the intersections of the va- 
 rious edges with the cutting plane, will be 
 projected in perpendiculars drawn from the 
 points where it meets these edges respective- 
 ly ; from the points a, f, b, etc., let fall the 
 perpendiculars a a', //', b b', etc. ; join their 
 contiguous points of intersection with the 
 lines A' D', F' C', B' E', etc. ; and the resulting 
 six-sided figure represents the section required. The edges F S and E S, being 
 concealed in the elevation, but necessary for the construction of the plan, are 
 expressed by dotted lines, as also is the portion of the pyramid situated above 
 the cutting plane, supposed to be removed, but necessary in order to draw the 
 lines representing the edges. The ordinary method of expressing sections in 
 
 Fio. 258. 
 
126 
 
 ORTHOGRAPHIC PROJECTION. 
 
 purely line drawings is by filling up the spaces comprised within their outlines 
 with a number of parallel straight lines drawn at equal distances, called section 
 lines. 
 
 To represent in plan and elevation a regular six-sided prism in an upright 
 position (Fig. 259). 
 
 Lay down the ground line Gr K and draw the axis of the prism S S'. De- 
 scribe the hexagonal plan A' B' C' D' E' F', as in the previous example. From 
 each of the angular points, A', B', etc., erect perpendiculars, and on one of these 
 perpendiculars set off A G, the height of the prism, and draw a parallel to the 
 ground line, A D, which completes the vertical projection. The face, B C H I, 
 being parallel to the vertical plane, is seen in its true size, while Gr H and I K 
 
 are each equal to one half of H I, which enables us to draw the elevation with- 
 out constructing the plan a fact to be remembered in the drawing of nuts, 
 bolt-heads, etc., in machine drawing. 
 
 To form the projections of the same prism, supposing it to have been moved 
 round the point G in a plane parallel to the vertical plane (Fig. 260). 
 
ORTHOGRAPHIC PROJECTION. 
 
 127 
 
 Copy the elevation (Fig. 259) on the inclined base G K ; let fall perpen- 
 diculars from all the angles in the elevation ; and join the contiguous points of 
 intersection with the horizontal lines appropriate to these points respectively. 
 The plan remaining the same width as before, the polygon A' B' C' D' E' F' is 
 the projection of the upper surface, 
 and G' H' I' K' L' M' that of the base 
 of the prism. All the edges are rep- 
 resented in the horizontal projection 
 by equal straight lines, as D' K', A' G', 
 etc., and the sides A' B ', G' H', etc., 
 remain still parallel to each other, 
 which affords the means of verifying 
 the accuracy of the drawings. The 
 upper surface and the base, seen ob- 
 liquely in this projection, do not ap- 
 pear as true hexagons in the plan. 
 
 Required the projections of the 
 same prism set into a position in- 
 clined to loth planes of projection 
 (Fig. 261). 
 
 Assuming the inclination of the 
 prism upon the horizontal plane to be 
 as in the preceding figures, copy the 
 plan of Fig. 260 on an axis A' K' in- 
 clined to the vertical plane of projec- * 
 tion. Since the prism preserves its 
 former inclination to the horizontal 
 plane, every point in it, as A, in as- 
 suming its new position, simply moves 
 in a horizontal plane, and will there- 
 fore be at the same distance above the 
 ground line that it was in the eleva- 
 tion (Fig. 260), and it will also be in 
 the perpendicular A' A ; the point of 
 intersection A is, therefore, its projec- 
 tion in the elevation. Determine the 
 remaining angular points in this view and join the contiguous points and the 
 corresponding angles of the upper and lower surface and the figure is complete. 
 
 CONIC SECTIONS. 
 
 The plan of the cone (Fig. 262) is simply a circle, described from the centre 
 S', with a diameter equal to that of the base. Its elevation is an isosceles 
 triangle, obtained by drawing tangents A' A, B' B, perpendicular to and 
 intersecting the ground line; then set off upon the centre line the height 
 C S, and join S A, S B. These lines are called the exterior elements of the 
 cone. 
 
 Given the projections of a cone, and the direction of a plane X X, cutting it 
 perpendicularly to the vertical, and obliquely to the horizontal plane ; required 
 
 FIG. 261. 
 
128 
 
 ORTHOGRAPHIC PROJECTION. 
 
 to find, first, the horizontal projection of this section; and, secondly, the out- 
 line of the ellipse thus formed (Figs. 262, 263). 
 
 Through the vertex of the cone draw a line S E to any point within the 
 base A B ; let fall a perpendicular from E, cutting the circumference of the base 
 
 in E', and join E' S' ; then 
 another perpendicular let 
 fall from e will intersect 
 E' S' in a point e', which 
 will be the horizontal pro- 
 jection of a point in the 
 curve required ; and so on 
 for any number of points. 
 
 The exterior genera- 
 trices A S and B S being 
 both projected upon the 
 line A' B', the extreme lim- 
 its of the curve sought will 
 be at the points a' and V on that line, which are the projections of the points 
 of intersection a and b of the cutting plane with the outlines of the cone. 
 And since the line a' V divides the curve symmetrically into two equal parts, 
 the points/', g', h', etc., will be obtained by setting off above that line, and on 
 their respective perpendiculars, the distances d' d 3 , e' e z , etc. A sufficient num- 
 ber of points having thus been determined, the curve drawn through them 
 (which will be found to be an ellipse) will be the outline of the section re- 
 quired. 
 
 This curve may be obtained by another method, depending on the principle 
 that all sections of a cone by planes parallel to the base are circles. Thus, let 
 the line F G represent such a cutting plane ; the section which it makes with 
 the cone will be denoted on the horizontal projection by a circle drawn from 
 
ORTHOGRAPHIC PROJECTION. 
 
 129 
 
 the centre S', with a radius equal to half the line P G ; and by projecting the 
 point of intersection H of the horizontal and oblique planes by a perpendicular 
 H H', and noting where this line cuts the circle above referred to, two points 
 H' and I' are determined in the curve. Additional points are obtained similarly. 
 The preceding methods exhibit the section as fore-shortened. To solve the 
 second question proposed, let the cutting plane X X be conceived to turn 
 upon the point b, so as to coincide with the vertical line b k, and (to avoid con- 
 
 fusion of lines) let b Tc be trans- 
 ferred to a' b'j which will repre- 
 sent, as before, the extreme limits 
 of the curve required ; take any 
 point, as d, in this new position 
 of the cutting plane, it will be - B\ 
 represented by d", and, if the cut- 
 ting plane were turned upon a' V 
 (Fig. 263) as an axis till it is 
 parallel to the vertical plane, the 
 point which had been projected 
 at d" would then have described 
 round a' V an arc of a circle, 
 whose radius is the distance d' d* 
 
 (Fig. 262). This distance, therefore, set off at d' and/' on each side of a' V ^ 
 gives two points in the curve sought. The curve drawn through any number 
 of points thus obtained will be an ellipse of the true form and dimensions of 
 the section. 
 
 To find the horizontal projection and actual outline of the section of a cone, 
 made ~by a plane Y Y parallel to one side or element, and perpendicular to the 
 vertical plane (Figs. 264, 265). 
 
 Determine by the second method laid down in the preceding problem any 
 10 
 
 FIG. 264. 
 
130 
 
 ORTHOGRAPHIC PROJECTION. 
 
 number of points, as F', G', J', K', etc., in the curve representing the horizon- 
 tal projection of the section specified. The horizontal plane passing through 
 M gives only one point M', which is the vertex of the curve sought. 
 
 To determine the actual outline of this curve, suppose the plane Y Y to turn 
 as upon a pivot at M, until it has assumed the position M B, and transfer M B 
 to the parallel M 9 B" (Fig. 265). The point F will thus have first described 
 the arc F E till it reaches the point E, which is then projected to E 4 ; suppose 
 the given plane, now represented by M" B', to turn upon that line as an axis, 
 until it assumes a position parallel to the vertical plane, the point E 9 , which is 
 distant from the axis M' B' by the distance F' S' (Fig. 264), will now be pro- 
 jected to F a and G", two points in the curve required, which is a, parabola. 
 
 To draw the vertical projection of the sections of two opposite cones made by 
 a plane parallel to their axis (Fig. 266). 
 
 D 
 
 Let C E D and C B A be the two cones, and X X the position of the 
 cutting plane. Project in plan either of the cones, as b E' D' ; from its centre, 
 with a radius equal to L H, describe a circle, and draw the tangent 1) a ; b a 
 will be the horizontal projection of the cutting plane. Draw the line H' M' 
 parallel to the cutting plane ; H', M' corresponding in position to the inter- 
 sections H, M, of the plane with the cones. From H' and M' lay off distances 
 equal to L K, K I, and the length of the cone, and through these points draw 
 perpendiculars, as/V, d" c', b' a', etc., which must be made equal to the chords 
 / e, d c, b a, made by the cutting plane a 5, with circles whose radii are G K, 
 I F, and the radius of the base of the cone. Through the points ', c', e', H',/', 
 d', b', draw the curve for the projection required. A similar construction will 
 give the sectional projection of the opposite cone at M'. The curve thus found 
 is the hyperbola. 
 
 PENETRATIONS OR INTERSECTIONS OF SOLIDS. 
 
 Represent the projections of two cylinders of unequal diameters meeting 
 each other at right angles (Fig. 267). The one, the rectangle ABED for its 
 vertical, and the circle A' H' B' T' for its horizontal projection ; the other, being 
 
ORTHOGRAPHIC PROJECTION. 
 
 131 
 
 horizontal, is indicated in the former by the circle L P I N, and in the latter by 
 the rectangle L' I' K' M'. From the position of these two solids the curves 
 formed by their junction will be projected horizontally in the curves 0' H' P', 
 B' T' S', and vertically in L P I N. 
 
 But, if the position of these bodies be changed into that represented by Fig. 
 268, the lines of their intersection will assume in the vertical projection a 
 totally different aspect, and may be determined as follows : 
 
 Through any point taken upon the plan of Fig. 268 draw a horizontal line 
 a' #', indicating a plane cutting both cylinders parallel to their axes; this 
 
 B 
 
 FIG. 267. 
 
 FIG. 268. 
 
 plane would cut the vertical cylinder in lines drawn perpendicularly through 
 the points c' d'. To find the vertical projection of its intersection with the 
 other cylinder, conceive its base I' L', after being transferred to I 2 L", to be 
 revolved about I* L a as an axis parallel to the horizontal plane ; this is expressed 
 in part by simply drawing a semicircle of the diameter I* L a . Produce the 
 line a' b' to a' ; then set off the distance a" e' on each side of the axis I K, and 
 draw straight lines through these points parallel to it. These lines a b, g li, 
 
132 
 
 ORTHOGRAPHIC PROJECTION. 
 
 denote the intersection of the plane a' V with the horizontal cylinder, and 
 therefore the points c, d, m, o, where they cut the perpendiculars c c' ', d d', are 
 points in the curve required. By passing other horizontal planes similar to a' b' 
 through both cylinders, and operating as before, any number of points may be 
 obtained. The vertices i and k of the curves are projected directly from i' and 
 &', the intersections of the outlines of both cylinders. When the cylinders are 
 of unequal diameters, as in the present case, the curves of penetration are hy- 
 perbolas. 
 
 When the diameters of the cylinders are equal (Fig. 269), and when they 
 
 r 
 
 FIG. 269. 
 
 FIG. 270. 
 
 cut each other at right angles, the curves of penetration are projected vertically 
 in straight lines perpendicular to each other. 
 
 To delineate the intersections of two cylinders of equal diameters at right 
 angles, when one of the cylinders is inclined to the vertical plane (Fig. 270). 
 
 Suppose the two preceding figures to be drawn, the projection c of any 
 point, as <?', must be situated in the perpendicular c' c. Since the distance of 
 
ORTHOGRAPHIC PROJECTION. 
 
 133 
 
 this point (projected at c in Fig. 269) from the horizontal plane remains un- 
 altered, it must also be in the horizontal line c c. Upon these principles all 
 the points indicated by literal references in Fig. 270 are determined ; the 
 curves of penetration resulting therefrom intersect each other at two points 
 projected upon the axial line L K, of which that marked q alone is seen. The 
 ends of the horizontal cylinder are ellipses. 
 
 To find the curves resulting from the intersection of two cylinders of un- 
 equal diameters, meeting at any angle (Fig. 271). 
 
 Suppose the axes of both cylinders to be parallel to the vertical plane, and 
 let A B E D and N Q P be their projections upon that plane. In construct- 
 ing, in the first place, their 
 horizontal projection, ob- 
 serve that the upper end 
 A B of the larger cylinder is 
 represented by an ellipse 
 A' K' B' M', which may easi- 
 ly be drawn by the help of 
 the major axis K' M' equal 
 to the diameter of the cylin- 
 der, and of the minor A' B', 
 the projection of the diame- 
 ter. The visible portion of 
 the base of the cylinder be- 
 ing similarly represented by 
 the semi-ellipse L' D' H', its 
 entire outline will be com- 
 pleted by drawing tangents 
 L' M' and H' K'. The up- 
 per extremity P N of the 
 smaller cylinder will also be 
 projected in the ellipse p' N'. 
 Conceive a plane, as 
 a' g', to pass through both 
 cylinders parallel to their 
 axes ; it will cut the surface 
 of the larger cylinder in 
 two straight lines, passing 
 through the points /' and g' 
 on the upper end of the cyl- 
 inder ; these lines will be 
 represented in the elevation 
 by projecting the points/' 
 and g' to /, g, and drawing 
 a f and c g parallel to the axis. 
 
 V-s ^"^ 
 
 ff ? -- : ^, 
 
 K 
 
 FIG. 271. 
 
 The plane a' g' will in like manner cut the 
 
 smaller cylinder in two straight lines, which will be represented in the verti- 
 cal projection by d h and e t, and the intersections of these lines with a/ 
 and c g will give four points I, &, m, and n, in the curves of penetration 
 these points, one only, I, is visible, /', in the plan. 
 
 Of 
 
134 
 
 ORTHOGRAPHIC PROJECTION. 
 
 To find the curves of penetration in the elevation without the aid of the plan 
 (Fig. 271). 
 
 Let the bases D E and Q of both cylinders be revolved parallel to the 
 vertical plane after being transferred to any convenient distance, as D a E a and 
 Q a O a , from the principal figure ; they will then be vertically projected in the 
 
 circles D a H a E a and Q a G' O a . Draw 
 a c a parallel to D E, and at any suitable 
 distance from the centre I ; this line 
 will represent the intersection of the 
 base of the cylinder with a plane parallel 
 to the axes of both, as before. The in- 
 tersection of this plane with the base of 
 the smaller cylinder will be found by 
 setting off from R a distance R p, equal 
 to I o, and drawing through the point p 
 a straight line parallel to Q 0. The 
 intersection of the supposed plane with 
 the convex surfaces of the cylinders will 
 be represented by the lines af, eg, and 
 d A, e i; and, consequently, the inter- 
 sections of these lines indicate points in 
 the curves sought. These points may 
 be multiplied by conceiving other planes 
 to pass through the cylinders. 
 
 To find the curves of penetration of 
 a cone and sphere (Fig. 272). 
 
 Let D S be the axis of the cone, 
 A' L' B' the circle of its base, and the 
 triangle A B S its projection on the ver- 
 tical plane ; and let C, C', be the projec- 
 tions of the centre, and the equal circles 
 E' K' F' and E G F those of the cir- 
 cumferences of the sphere. 
 
 This problem can be solved only by 
 the aid of imaginary intersecting planes. 
 Let a b represent the projection of a 
 horizontal plane ; it will cut the sphere 
 in a circle whose diameter is a b, and 
 
 which is partially drawn from the centre C' in the plan, as a' f b'. Its in- 
 tersection with the cone is also a circle described from the centre S' with the 
 diameter c d as c' f d' ; the points e' and /"', where these two circles cut each 
 other, are the horizontal projections of two points in the lower curve, which 
 is entirely hidden by the sphere. The points referred to are projected ver- 
 tically upon the line a b at e and/. The upper curve, which is seen in both 
 projections, is obtained by a similar process ; but it is to be observed that the 
 horizontal cutting planes must be taken in such positions as to pass through 
 both solids in circles which shall intersect each other. In this respect it 
 will be necessary, first, to determine the vertices m and n of the curves of 
 
 FIG. 272. 
 
ORTHOGRAPHIC PROJECTION. 
 
 135 
 
 penetration. For this purpose, conceive a vertical plane passing through the 
 axis of the cone and the centre of the sphere ; its horizontal projection will 
 be the straight line C' L', joining the centres of the two bodies. Suppose 
 this plane to be turned upon the line C C' as on an axis, until it becomes 
 parallel to the vertical plane ; the points S' and L' will now have assumed 
 the positions S 3 and L 2 , and consequently the axis of the cone will be projected 
 vertically in the line D' S 8 , and its side in S* L 8 , cutting the sphere at the 
 points p and r. Conceive the solids to have resumed their original relative 
 positions, the vertices or adjacent limiting points of the curves of penetration 
 must be in the horizontal lines p o and r q, drawn through the points deter- 
 mined as above ; their exact positions on these lines may be ascertained by pro- 
 jecting vertically the points m' and n', where the arcs described by the points p 
 and r, in restoring the cone to its first position, intersect the line S' L'. 
 
 It is of importance, further, to ascertain the points at which the curves of 
 penetration meet the 
 outlines A S and S B 
 of the cone. The plane 
 which passes through 
 these lines, being pro- 
 jected horizontally in 
 A' B', will cut the sphere 
 in a circle whose diame- 
 ter is i' j' ; this circle, 
 described in the eleva- 
 tion from the centre C, 
 will cut the sides A S 
 and S B in four points, 
 at which the curves of 
 penetration are tangent 
 to the outlines of the 
 cone. 
 
 To find the lines of 
 penetration of a cylin- 
 der and a cylindrical 
 ring, or annular torus 
 (Fig. 273). 
 
 Let the circles A' E' 
 B', F' G' K', represent 
 the horizontal, and the 
 figure A C B D the ver- 
 tical projection of the 
 torus, and let the circle 
 II' /' L',.and the rectan- 
 gle H I M L be the 
 analogous projections of 
 the cylinder, which 
 
 passes perpendicularly through it. Conceive, as before, a plane, a 5, to 
 horizontally through both solids ; it will cut the cylinder in a circle which 
 
 pass 
 will 
 
136 ORTHOGRAPHIC PROJECTION. 
 
 be projected in the base H'/' L' itself, and the ring in two other circles, of 
 which one only, part of which is represented by the arc/' b* b' ', will intersect 
 the cylinder at the points/' and i s , which, being projected vertically, will give 
 two points /and b 9 in the upper curve of penetration. 
 
 Another horizontal plane, taken at the same distance below the centre line 
 A B as that marked a b is above it, will cut the ring in circles coinciding with 
 those already obtained ; consequently the points/' and b 9 indicate points in the 
 lower as well as in the upper curves of penetration, and are projected vertically 
 at d and e. Thus, by laying down two planes at equal distances on each side 
 of A B, four points in the curves required are determined. 
 
 To determine the vertices m and n, following the method explained in the 
 preceding problem, draw a plane n , passing through the axis of the cylinder 
 and the centre of the ring, and conceive this plane to be revolved about the 
 point until it has assumed the position B', parallel to the vertical plane ; 
 the point n', representing the extreme outline of the cylinder in plan, will now 
 be at r', and, being projected vertically, that outline will cut the ring in two 
 points p and r, which would be the limits of the curves of penetration in the 
 supposed relative position of the two solids ; and by drawing the two horizontal 
 lines r n and p m, and projecting the point n' vertically, the intersections of 
 these lines, m and n, are the vertices of the curves in the actual position of the 
 penetrating bodies. 
 
 The points at which the curves are tangents to the outlines H I and L M of 
 the cylinder may readily be found by describing arcs of circles from the centre 
 through the points H' and L', which represent these lines in the plan, and 
 then proceeding, as above, to project the points thus obtained upon the eleva- 
 tion. Lastly, to determine the points, as j, z, etc., where the curves are tan- 
 gents to the horizontal outlines of the ring, draw a circle P' s' j' with a radius 
 equal to that of the centre line of the ring, namely, P D ; the points of inter- 
 section z' and/ are the horizontal projections of the points sought. 
 
 Required to represent the section which would be made in this ring by a 
 plane, N' T', parallel to the vertical plane. 
 
 Such a section will be represented in its actual form and dimensions in the 
 elevation. To determine its outlines, let two horizontal planes, g q and i k, 
 equidistant from the centre line A B, be supposed to cut the ring ; their lines 
 of intersection with it will have their horizontal projections in the two circles 
 g' o' and h 1 q', which cut the given plane N' T' in o' and q'. These points being 
 projected vertically to 0, <?, &, etc., give four points in the curve required. The 
 line N' T' cutting the circle A' E' B' at N', the projection N of this point is 
 the extreme limit of the curve. 
 
 The circle P' s'f, the centre line of the rim of the torus, is cut by the planes 
 N' T' at the point *', which, being projected vertically upon the lines D P and 
 C I, determines s and I, the points of contact of the curve with the horizontal 
 outlines of the ring. Finally, the points t and u are obtained by drawing from 
 the centre a circle, T' v', tangent to the given plane, and projecting the point 
 of intersection v' to the points v and x, which are then to be replaced upon C D 
 by drawing the horizontals v t and x u. 
 
 Required to delineate the lines of penetration of a sphere and a regular hex- 
 agonal prism whose axis passes through the centre of the sphere (Fig. 274). 
 
ORTHOGRAPHIC PROJECTION. 
 
 137 
 
 The centres of the circles forming, the two projections of the sphere being 
 upon the axis C C' of the upright prism, which is projected horizontally in the 
 regular hexagon D' E' F' G' H' I', and all the lateral faces of the prism being 
 equidistant from the centre of the sphere, their lines of intersection with it will 
 be circles of equal diameters. The perpendicular face, represented by the line 
 E' F' in the plan, will meet the surface of the sphere in two circular arcs, E F 
 and L M, described from the centre 0, with a radius equal to c' b' or a 1 c'. 
 And the intersections of the two oblique faces D' E' and F' G' will obviously 
 be each projected in two arcs of an ellipse, whose major axis dg is equal to 
 e'/', and minor axis the vertical projection of e' f. As it is necessary to 
 draw small portions only of these curves, the following method may be em- 
 ployed : Draw D G through the points E, F; divide the portions E F and 
 F G respectively into the same number of equal parts, and, drawing perpen- 
 diculars through the points of division, set off from F G the distances from 
 
 FIQ. 274. 
 
 FIG. 275. 
 
 the corresponding points in E F to the circular arc E C F, as points in the 
 elliptical arc required. The remaining elliptical arcs can be traced by the 
 same method. 
 
138 
 
 ORTHOGRAPHIC PROJECTION. 
 
 Required to draw the lines of penetration of a cylinder and a sphere, the 
 centre of the sphere being without the axis of the cylinder (Fig. 275). 
 
 Let the circle D' E' L' be the projection of the base of the given cylinder, 
 and let A B be the diameter of the given sphere. If a plane, as c' d', be drawn 
 parallel to the vertical plane, it will cut the cylinder in two straight lines, 
 
 G G', H H'. This plane will also cut 
 the sphere in a circle described from 
 the centre C with a radius of half the 
 line c' d' ; its intersection with the lines 
 G G' and H H' will give so many points 
 in the curves sought, viz., G, H, I, K. 
 
 The planes a' V and e' f, which are 
 tangents to the cylinder, furnish respect- 
 ively only two points in the curves ; of 
 these points, E and F alone are visible, 
 the other two, L and M, being concealed 
 by the solid ; therefore the planes drawn 
 for the construction of the curves must 
 be all taken between a' b 1 and e'f. The 
 plane which passes through the axis of 
 the cylinder cuts the sphere in a circle 
 whose projection upon the vertical plane 
 will meet at the points D, N, and g, h, 
 the outlines of the cylinder, to which 
 the curves of penetration are tangents. 
 
 To find the lines of penetration of 
 a frustum of a cone and a prism (Fig. 
 276). 
 
 The frustum is represented in the 
 plan by two circles described from the 
 centre C' ; and the horizontal lines M N 
 and M' N' are the projections of the 
 axis of a prism of which the base is 
 square, and the faces respectively par- 
 allel and perpendicular to the planes of 
 projection. 
 
 In laying down the plan of this solid, it is supposed to be inverted, in order 
 that the smaller end of the cone and the lines of intersection of the lower sur- 
 face, F G, of the prism may be exhibited. According to this arrangement, the 
 letters A' and B' ought, strictly speaking, to be marked at the points I' and H', 
 and conversely ; but, as the part above M' N' is exactly symmetrical with that 
 below it, the distribution of the letters of reference in the figures can lead to no 
 confusion. 
 
 The intersection of the plane F G with the cone is projected horizontally 
 in a circle described from the centre C', with the diameter F' G'. The arcs 
 I' F' A' and H' G' B' are the only parts of this circle which require to be 
 drawn. In the vertical projection the extreme points K, L, A, B need only 
 be found, for the lines of intersection are here projected straight. 
 
 Fio. 276. 
 
ORTHOGRAPHIC PROJECTION. 
 
 139 
 
 To describe the curves formed by the intersection of a cylinder with the 
 frustum of a cone, the axes of the two solids cutting each other at right angles 
 (Fig. 277). 
 
 The projections of the solids are laid down in the figure precisely as in the 
 preceding example. The intersections of the outlines in elevation furnish four 
 points in the curves of penetration. 
 These points are all projected horizontal- 
 ly upon the line A' B'. Now pass a 
 plane, as a b, horizontally through both 
 solids ; its intersection with the cone will 
 be a circle of the diameter c d, while the 
 cylinder will be cut in two parallel 
 straight lines, represented in the eleva- 
 tion by a b, and whose horizontal projec- 
 tion may be determined in the following 
 manner: Conceive a vertical plane, f g, 
 cutting the cylinder at right angles to 
 its axis, and let the circle g ef thereby 
 formed be described from the intersec- 
 tion of the axes of the two solids; the 
 line j h will now represent, in this posi- 
 tion of the section, the distance of one of 
 the lines sought from the axis of the cyl- 
 inder. Set off this distance on both sides 
 of the point A', and through the points 
 k and a' thus obtained, draw straight 
 lines parallel to A' B' ; the intersections 
 of these lines with the circle drawn from 
 the centre C' of the diameter c' d' will 
 give four points m',p', n, and o, which, 
 being projected vertically upon a b, deter- 
 mine two points, m and p, in the curves 
 required. 
 
 In order to obtain the vertices or ad- 
 jacent limiting points of the curves, draw 
 from the vertex of the cone a straight 
 line, t e, touching the circle g ef, and let a horizontal plane be supposed to 
 pass through the point of contact e. Proceed according to the method given 
 above to determine the intersections of this plane with each of the solids in 
 question the four points i', r', q, and s, projected vertically upon the line e r, 
 determine the vertices i and r required. 
 
 THE HELIX. 
 
 A helix is the curve described upon the surface of a cylinder by a point re- 
 volving round it, and at the same time moving parallel to its axis by a certain 
 invariable distance during each revolution. This distance is called the pitch of 
 the helix or screw. 
 
 Required to construct the helical curve described by the point A 1 upon a cyl- 
 
 r-US * JB 
 
 FIG 277. 
 
140 
 
 ORTHOGRAPHIC PROJECTION. 
 
 inder projected horizontally in the circle A' C' F', the pitch being represented 
 by the line A 1 A 3 (Fig. 278). 
 
 Divide the pitch A 1 A 3 into any number of equal parts, say eight ; and 
 through each point of division, 1, 2, 3, etc., draw straight lines parallel to the 
 ground line. Then divide the circumference A' C' F' into the same number of 
 equal parts ; the points of division, A', B', C', E', F', etc., will be the horizontal 
 
 projections of the differ- 
 ent positions of the given 
 point during its motion 
 round the cylinder. Thus, 
 when the point is at B' in 
 the plan, its vertical pro- 
 jection will be the point of 
 intersection B of the per- 
 pendicular drawn through 
 B' and the horizontal 
 drawn through the first 
 point of division. 'Also, 
 when the point arrives at 
 C' in the plan, its vertical 
 projection is the point C, 
 where the perpendicular 
 drawn from C' cuts the 
 horizontal passing through 
 the second point of divis- 
 ion, and so on for all the 
 remaining points. The 
 curve A 1 B C E F A 3 , 
 drawn through all the 
 points thus obtained, is 
 the helix required. 
 
 To draw the vertical 
 elevation of the solid con- 
 tained between two helical 
 surfaces and two concen- 
 tric cylinders (Fig. 278). 
 
 A helical surface is 
 generated by the revolu- 
 Fro 278. tion of a straight line 
 
 round the axis of a cylin- 
 der, its outer end moving in a helix, and the line itself forming with the axis 
 a constant and invariable angle. 
 
 Let A' C' 1^' and K' M' 0' represent the concentric bases of the cylinders, 
 whose common axis S T is vertical ; the curve of the exterior helix A 1 C E F A 3 
 is the first to be drawn according to the method above shown. Then, having set 
 off from A 1 to A" the thickness of the required solid, draw through A" another 
 helix equal and similar to the former. Now construct, as above, similar helices, 
 K C and K a C 9 s , of the same pitch as the last, but on the interior cylinder. 
 
ORTHOGRAPHIC PROJECTION. 
 
 141 
 
 The lines A' K', B' L', C' M', etc., represent the horizontal projections of the 
 various positions of the generating straight line, which, in the present example, 
 has been supposed to be 
 horizontal ; and these lines 
 are projected vertically at 
 A 1 K, B L, etc. 
 
 In the position A 1 K 
 the generating line is pro- 
 jected in its actual length, 
 and at the position C' M' 
 its vertical projection is the 
 point C. The same re- 
 mark applies to the genera- 
 trix of the second helix. 
 
 To determine the verti- 
 cal projection of the solid 
 formed by a sphere moving 
 in a helical curve (Fig. 
 279). 
 
 Let A' C' E' be the base 
 of a cylinder, upon which 
 the centre point C' of a 
 sphere whose radius is a' C' 
 describes a helix, which is 
 projected on the vertical 
 plane in the curve A C E J, 
 determined as before. From 
 the various points A, B, 
 
 C, D , in this curve, 
 
 as centres, describe circles 
 
 with the radius a' C' ; these 
 
 denote the various positions 
 
 of the sphere during its 
 
 helical motion ; and, if lines 
 
 be drawn touching them, 
 
 the curves thereby formed will constitute the figure required. One of these 
 
 curves disappears at 0, but reappears again at I. The exterior and interior 
 
 circles of the plan represent the horizontal projection of the solid in question. 
 
 The conical helix differs from the cylindrical one in that it is described on 
 the surface of a cone instead of on that of a cylinder ; but the construction 
 differs but slightly from the one described. By following out the same prin- 
 ciples, helices may be represented as lying upon spheres or upon any other 
 surfaces of revolution. 
 
 FIG. 279. 
 
 DEVELOPMENT OF SURFACES. 
 
 The development of the surface of a solid is the drawing or unrolling on a 
 plane the form of its covering, the form that cut out of paper would exactly fit 
 and cover the surface of the solid. Frequently, in practice, the form of the 
 
142 
 
 ORTHOGRAPHIC PROJECTION. 
 
 surface of a solid is found by applying paper or thin sheet-brass directly to the 
 solid and cutting it to fit. 
 
 To develop the surface of a cylinder formed ly the intersection of another 
 equal cylinder, as the knee of a stove-pipe (Fig. 280). 
 
 Let A B C D be the elevation of the pipe or cylinder. Above A. B describe 
 the semicircle A' 4' B' of the same diameter as the pipe ; divide this semicircle 
 
 
 ^ 
 
 V 
 
 ^ 
 
 ^ 
 
 ^\ 
 
 2 
 
 *X^ 
 
 
 r> 
 
 into any number of equal parts, 
 eight for instance ; through these 
 points, 1', 2', 3', etc., draw lines 
 parallel to the side A C of the pipe, 
 and cutting the line C D of the in- 
 tersection of the two cylinders. 
 Lay off A* B" equal to the semicir- 
 cle A' 4' B', and divided into the 
 same number of equal parts ; 
 through these points of division 
 erect perpendiculars to A" B", and 
 on these perpendiculars lay off the 
 distances A" C", 1" 1", 2" 2", 3" 3", 
 and so on, corresponding to A C, 
 
 1 1, 2 2, 3 3, etc., in preceding figure. Through the points C", 1", 2" , D", 
 
 draw connecting lines, which gives but one half of the surface of the pipe, the 
 
 other being exactly similar to it. 
 
 To develop the surface of a cylinder intersected ly another cylinder, as in 
 
 the formation of a T-pipe (Fig. 281). 
 
 The construction is similar to the preceding. 
 
 To develop the surface of a right cone (Fig. 282). 
 
 From C' as a centre, with a radius, C' A', equal to the inclined side A C of 
 
 FIG. 280. 
 
ORTHOGRAPHIC PROJECTION. 
 
 143 
 
 the cone, describe an arc of a circle, and on this arc lay off the distance 
 A' B' A", equal to the circumference of the base of the cone ; connect A' C' and 
 C' A", and A' B' A' C' is the developed surface required. 
 
 To develop the surface of the frustum of a cone, D A B E (Fig. 282). 
 D' E' D" is the development of the cut-off cone D E as shown by the 
 
 preceding construction, and A' B' A' 
 D" E' D' the developed surface of the 
 frustum. 
 
 To develop the surface of a frus- 
 tum of a cone, when the cutting plane 
 a b is inclined to the base (Fig. 282). 
 
 On A B, the base, describe the 
 semicircle A 3' B ; divide the semicir- 
 cle into any number of equal parts, 
 six for instance; from each point of 
 division, 1', 2', 3', 4', 5', let fall perpen- 
 diculars to the base at 1, 2, 3, 4, 5 ; 
 connect each of these last points with 
 
 B' 
 
 FIG. 282. 
 
 the apex C. Divide now the arc A' B' A", equal to the circumference of the 
 base A 3 B, into twelve equal parts ; each of these parts by the construction is 
 equal to the arc A 1', 1' 2' ; connect these points of division with the point C' ; 
 on C' A' take C' a' equal to C a, a being the point at which the plane cuts the 
 inclined side of the cone ; in the same way on C' B', lay off C' b' equal to C b. 
 
 All the lines connecting the apex C with the base, included within the two 
 inclined sides, are represented as less than their actual length, and must be 
 projected on the inclined sides to determine their absolute dimensions ; project, 
 therefore, the points 1*, 2*, 3*, 4', 5*, at which the cutting plane intersects the 
 lines 01, 02, C 3, C 4, C 5, by drawing parallels to the base through these 
 points to the inclined side C B. Now lay off C' 1", C' 2"", etc., equal to C 1' ', 
 C 2' ', etc. ; connect the points a', 1', 2', .... J', . a', and a' A' B' A" a" b' 
 is the developed surface. 
 
144 
 
 ORTHOGRAPHIC PROJECTION. 
 
 To develop the surface of a sphere or ball (Figs. 283, 284). 
 
 The surface can not be accurately represented on a plane, but only approxi- 
 mately by gores. Let CAB (Fig. 284) be the eighth of a hemisphere ; on C D 
 describe the quarter circle D A c ; divide this arc into any number of 
 equal parts, six for instance; from the points of division 
 1, 2, 3, ... let fall perpendiculars on C D, and from the 
 intersections with this line describe arcs 1' 1", 2' 2", 
 3' 3', ... cutting the line C B at 1", 2", 3*, .... 
 on the straight line C' D' (Fig. 283), lay off 
 C' D' equal to the arc D A c, with as many 
 
 A' 
 
 \ 
 
 i 
 i 
 
 I 
 
 II 
 
 FIG. 283. 
 
 Fio. 284. 
 
 equal divisions; then from either side of this line lay off 1'" 1"", 2'" 2"" .... 
 D' B' equal to the arcs 1' 1", 2' 2" .... D B (Fig. 284). Connect the points 
 C', 1"", 2 /|r ", .... and C' A' B' is approximately the developed surface. 
 
 In the preceding demonstrations the forms are described to cover the sur- 
 
 face only ; in construction, allowance is to 
 be made for lap by the addition of mar- 
 gins on each side. And on account of the 
 difficulty, in the formation of hemispheri- 
 cal ends of boilers, of bringing all the 
 gores together at the apex, it is usual to 
 make them, as shown (Fig. 285), by cut- 
 ting short the gores, and surmounting the 
 centre with a cap-piece. 
 
 For small boilers and air chambers the 
 spherical ends are dished. 
 
 SHADE LINES. 
 
 The effectiveness of outline drawings is 
 increased by the use of shade lines ; the 
 method is wholly conventional. The rule 
 used in preparing drawings for the United 
 
 States Patent Office is the simplest and the one generally used in this country. 
 In this the light falls from the upper left-hand corner of the sheet of drawing- 
 paper, and at an angle of 45 ; hence, the right-hand and lower edge of a 
 projection (Fig. 286) and the left-hand and upper edge of a recess appear in 
 heavy lines (Fig. 287). In the case of a circular object the shade is ter- 
 
 FlQ 
 
ORTHOGRAPHIC PROJECTION. 
 
 145 
 
 ] I 
 
 LJ 
 
 FIG. 286. 
 
 FIG. 288. 
 
 FIG. 289. 
 
 11 
 
146 
 
 ORTHOGRAPHIC PROJECTION. 
 
 initiated by a diameter inclined at an angle of 45, the shade being a graduated 
 one, as shown in Fig. 288. 
 
 Fig. 289 is an example of this method. 
 
 In a second method the elevation is shaded, as in the foregoing, but in the 
 plan the light falls from the lower left-hand corner of the drawing, making the 
 upper and right-hand edges of projections and the lower and left-hand side of 
 
 Elevation. 
 
 FIG. 290. 
 
 recesses shaded. Examples of this method are shown in Fig. 290. In the 
 orthographic projection of solids the boundary lines between surfaces in the 
 light and shadow are made heavy. 
 
 Still another method is described in the following chapter on " Shades and 
 Shadows," where the light falls over the left shoulder, and this is the one gen- 
 erally used in the casting of shadows and often in topographical drawings. 
 
SHADES AND SHADOWS. 
 
 RAYS of light are diffused through space in straight lines, and the direct 
 rays of the sun may be regarded as parallel. The light on an object may come 
 directly from the source of light, or by reflection from other objects or surfaces 
 exposed to light. The surfaces of an object under direct light are the most 
 strongly illuminated, while other surfaces, from their form or position receiv- 
 ing less light, are in shade or direct shadow. Shadows are cast by an object 
 upon another object by the interception of light, or upon any surface by pro- 
 jections or undercutting. The limit of the line of direct shadow is called the 
 line of shade. 
 
 In the delineation of shadows, the most convenient mode of regarding the 
 rays of light is, in all cases, as falling in the direction of the diagonal of a cube, 
 of which the sides are parallel to the planes of projection. The projections of 
 the ray form each an angle of 45 with the ground line. This is not true of 
 the ray itself in space, for that forms an angle of 54 44' with the ground line, 
 and an angle of 35 16' with each of the planes of projection. 
 
 To find the shadow of a point, as A, A' (Fig. 291), on either plane of pro- 
 jection, the vertical, for instance, draw a line through the horizontal projection 
 
 FIG. 291. 
 
 FIG. 292. 
 
 of the point A' at an angle of 45 with the ground line, L T, and at the point 
 of intersection of those lines, a', erect a perpendicular to intersect the vertical 
 projection of the ray through A, which will be at the point a, the shadow in 
 question. 
 
 The converse of this method determines the shadow of the point on the 
 
 147 
 
148 
 
 SHADES AND SHADOWS. 
 
 horizontal plane. The shadow thrown by B of the given line falls at J, and the 
 straight line a b, which joins these two points, is the shadow required. 
 
 The line a b is equal and parallel to the given line A B ; this results from 
 the circumstance that the latter is parallel to the vertical plane. 
 
 tn- ^///y\ 7 /A 
 
 FIG. 293. 
 
 The shadows of a rectangular slip of paper or wood, A B C D, cast upon the 
 same vertical plane and parallel to it, is the rectangle A B C D (Pig. 291). 
 
 When the object is not parallel to the given plane, the shadow cast is no 
 longer a figure equal and similarly placed, but the method of determining it is 
 similar (Fig. 292). 
 
 By a combination of the foregoing principles, the shadow of a slip of mould- 
 ing placed on the vertical plane (Fig. 293) is determined. 
 
 FIG. 295. 
 
 Fig. 294 shows the shadow cast by a rectangular slip of cardboard at right 
 angles to the vertical plane ; Fig. 295, the shadow cast by a slip parallel to the 
 ground plane and at right angles to the vertical plane, which is itself set at a 
 
SHADES AND SHADOWS. 
 
 149 
 
 horizontal angle ; Fig. 296, the shadow of the slip cast on a concave surface ; 
 Fig. 297, the shadow cast on a vertical plane by a circle parallel to it ; Figs. 
 298, 299, 301, the shadows on a vertical plane by circles in various positions 
 
 A' <f B' 
 
 FIG. 297. 
 
 ^ / // 
 
 FIG. 298. 
 
 relative to this plane, in all which the shadow assumes the form of an ellipse ; 
 Fig. 300, the shadow cast by a circle horizontal to the ground plane on a verti- 
 cal convex surface ; Fig. 302, the shadow cast by a circle parallel to the vertical 
 
 ! 
 \ 
 
 ^'-^a 
 
 / ' ~]\t*/ . * 
 
 J) 
 
 m 
 
 FIG. 299, 
 
 FIG. 300. 
 
 plane of the projection, the shadow being thrown upon two plane surfaces at 
 an angle to each other. 
 
150 
 
 SHADES AND SHADOWS. 
 
 In every drawing where the shadows are to be inserted, it is of the utmost 
 importance that the projections which represent the object whose shadow is 
 required, and the surface upon which this shadow is cast, should be exactly de- 
 
 FIG. 301. 
 
 FIG. 302. 
 
 fined by lines drawn lightly in India ink, and, to prevent mistakes, erase all 
 pencil marks before proceeding to the operations determining the lines of 
 shadow. 
 
 MEN 
 
 FIG. 303. 
 
 FIG. 304. 
 
 In Fig. 303 the three sides of the hexagonal pyramid, A' B' F', A' B' C', and 
 A' C' D', alone receive the light ; consequently the edges A' F' and A' D' are 
 the lines of shade. To determine the shadow cast by these two lines, draw 
 from the projections of the vertex of the pyramid the lines A b and A' a' paral- 
 
SHADES AND SHADOWS. 
 
 151 
 
 FIG. 305. 
 
 lei to the ray of light ; then erect at the point J a perpendicular to the ground 
 line, which gives at a' the shadow of the vertex on the horizontal plane on the 
 other side of the ground line ; and finally join this last point, a', with the points 
 
 D' and F' ; the lines D' a' and F' a' are the 
 
 outlines of the required shadow on the 
 horizontal plane. But, as the pyramid is 
 sufficiently near the vertical plane to throw 
 a portion of its shadow upon it, this por- 
 tion may be found by erecting at the point 
 c, where the line A' a' cuts the ground 
 line, a perpendicular c a, intersecting the 
 line AZ> in a; the lines ad and ae join- 
 ing this point with those where the hori- 
 zontal part of the shadow meets the ground 
 line, will be its outline upon the vertical 
 plane. 
 
 Fig. 304 represents the shade on a cyl- 
 inder placed vertically, and its shadow cast 
 on two planes of projection. 
 
 Draw the tangents D' d' and C' c 1 parallel to the rays of light ; these are the 
 outlines of the shadow cast upon the horizontal plane. Through the point of 
 contact C' draw the vertical line C C' ; this line denotes the line of shade upon 
 the surface of the cylinder. 
 
 If the shadow of this cylinder were entirely cast upon the horizontal plane, 
 it would terminate in a semicircle drawn from the centre 0', with a radius equal 
 to that of the base ; but a part of the shadow of the upper part is thrown upon 
 the vertical plane, and its outline is defined by an ellipse drawn in the manner 
 indicated in Fig. 298. 
 
 Fig. 305 represents the line of shade and the shadow of a horizontal cylinder 
 inclined to the vertical plane. The construction in this case is the same as that 
 explained by Fig. 304. Of the horizontal lines of shade, C D alone is visible 
 in the elevation, while A B alone is seen in the plan, where it may be found 
 by drawing a perpendicular from A meeting the base F' G' in A'. The line 
 A' E', drawn parallel to the axis of the cylinder, is the line of shade required. 
 Project the shadow of the line A B on the vertical plane, as in previous exam- 
 ples, to define the outline of the shadow of the cylinder. 
 
 The example here given presents the particular case in which the bases of 
 the cylinder are parallel to the direction of the rays of light. In this case, to 
 determine the line A' E' lay off the angle A' L A\ equal to 35 16', so that the 
 side A 2 L shall be tangent to the circle F' A" G', representing the base of the 
 cylinder laid down on the horizontal plane ; through the point of tangency, 
 A", draw a line, A' E', parallel to the axis of the cylinder, for the line of 
 shade. 
 
 Fig. 306 represents a cylinder upon which a shadow is thrown by a rec- 
 tangular cap, of which the sides are parallel to the planes of projection. The 
 shadow in this case is derived from the edges A' D' and A' E', the first of which, 
 being perpendicular to the plane of projection, gives a straight line at an angle 
 of 45 for the outline of its shadow, whereas the side A' E' being parallel to 
 
152 
 
 SHADES AND SHADOWS. 
 
 that plane, its shadow is determined by a portion of a circle, a b c, described 
 from the centre, 0. 
 
 If the cap be hexagonal (Fig. 307), or circular (Fig. 308), the mode of con- 
 struction is similar. Commence by finding the points which indicate the main 
 
 FIG. 306. 
 
 FIG. 307. 
 
 FIG. 308. 
 
 direction of the outline. To ascertain the point a at which the shadow com- 
 mences, draw from a' the line a' A' at an angle of 45, and projected vertically 
 to a A. Then the highest point b (Fig. 308) is determined by the intersection 
 of the radius 0' B', drawn parallel to the ray, with the circumference of the 
 base of the cylinder on which the shadow is cast ; the point c, where the outline 
 of the cast shadow intersects the line of shade, is determined by a like process. 
 In Figs. 309,310, and 311 a hexagonal prism is substituted for the cylinder. 
 
 c" Si 
 
 FIG. 309. 
 
 FIG. 310. 
 
 FIG. 311. 
 
 Fig. 312 shows the section of a steam-cylinder by a plane passing through 
 its axis, with its piston and rod in full. To define its shadows, conceive the 
 piston P to be removed ; the shadow cast into the interior of the cylinder will 
 then consist of that projected bjf the vertical edge B C, and by a portion of the 
 horizontal edge B A. To find the first, draw through B' a line, B' #', at an 
 angle of 45 with B' A' ; the point b', where this line meets the interior surface 
 of the cylinder, being projected vertically, gives the line bf as the outline of 
 
SHADES AND SHADOWS. 
 
 153 
 
 SEF 
 
 k A 
 
 the shadow sought. Then, parallel to the direction of the light, draw a tangent 
 at F' to the inner circle of the base ; its point of contact, being projected to F 
 in the elevation, marks the commencement of the outline 
 of the shadow cast by the upper edge of the cylinder. 
 The point #, where it terminates, will be the intersection 
 of the straight line fb, already determined, with a ray, 
 B b, from the upper extremity of the edge B C ; and any 
 intermediate point in the curve, as e, may be found in the 
 same way. The outline of the shadow required will then 
 be the curve F e b and the straight line b f. Insert the 
 piston P and its rod T into the cylinder, as shown. The 
 lower surface of the piston will then cast a shadow upon 
 the interior surface of the cylinder, of which the outline 
 D d h o may be formed as above. The piston-rod T, 
 being cylindrical and vertical, casts a shadow into the in- 
 terior of the cylinder, consisting of the rectangle ij I Tc 
 drawn parallel to the axis. 
 
 In Fig. 313 is shown the interior of a cylinder closed 
 at the top with the exception of a central apertur.e 
 through which the light is admitted. Follow previous construction, but to 
 determine what parts of the upper and lower edges of the central aperture 
 cast their shadows into the interior of the cylinder, establish the position of 
 the point of intersection, c, of the two curves b c /and ace, shadows of these 
 edges, which is the cast shadow of the lowest point, C, in the curve D C, pre- 
 viously laid down in the circular opening of the cover. 
 
 For the shadow cast in the interior of a cylinder, in section, inclined to the 
 horizontal plane (Fig. 314), on a convenient part of the paper draw the diag- 
 
 FIG. 312. 
 
 FIG. 313. 
 
 FIG. 314. 
 
 FIG. 315. 
 
 onal m o, parallel to the line of light A' E, and construct a square, mnop (Fig. 
 315) ; from one of the extremities, o, draw the line o r, parallel to A' B', and 
 through the opposite extremity, m, draw a perpendicular, r s, to this line, and 
 set off on the perpendicular the distance r s equal to the side of the square, and 
 join s o. Now, draw through the point A', in the original figure, a line, A' a', 
 
154: 
 
 SHADES AND SHADOWS. 
 
 FIG. 316. 
 
 parallel to s o, intersecting the cir- 
 cle A' a' B' in the point a', which, 
 being projected by a line parallel to 
 the axis of the cylinder, and meet- 
 ing the line drawn from A at an 
 angle of 45?, gives the first point 
 a in the curve C d a. The other 
 points are obtained in like manner, 
 by drawing at pleasure other lines, 
 such as D' d', parallel to A' a'. 
 
 For the shadow cast into the in- 
 terior of a hollow hemisphere (Fig. 
 316), let A B C D represent the 
 horizontal projection of a concave 
 hemisphere. Draw through the 
 centre a line, A C, perpendicular to 
 the ray of light ; the points B and 
 
 D will at once give the extremities of the curves sought. On any point of B D 
 produced, as 0', construct the semicircle A' a' C' with a radius, A' 0', equal to 
 
 A 0. At A' draw the line 
 B * A' #', making an angle of 35 
 
 f /'^g 16' with A' C'. The ' angle 
 
 made by the ray of light in 
 space with the planes of pro- 
 jection, a', the point of inter- 
 section of the line with the 
 semicircle, projected to , gives 
 a point of the outline of the 
 shadow. Similar sections, as EF 
 parallel to A C, will give other points. As this 
 outline is an ellipse whose axes are B D and twice 
 a, it may be constructed, when the point a is 
 determined, by the ordinary methods for ellipses. 
 In a niche (Fig. 317) the shadow of the cir- 
 cular outline upon the spherical portion is part 
 of an ellipse, e c D, found as in the previous 
 example. The point e, where this ellipse cuts 
 the horizontal diameter A F, limits the cast 
 shadow upon the spherical surface ; therefore 
 all the points beneath it must be determined 
 upon the cylindrical part. Through A' in the 
 plan draw the line A' a' parallel to the ray of 
 light; project a' till it intersects the line of 
 light A a in the elevation at a. The line of 
 shadow below a is the shadow of the edge of 
 the cylinder, and is a straight line. The line 
 of shadow between a and e is the outline of the 
 317- circular part falling on a cylindrical surface. 
 
 0' 
 
 >E 
 
SHADES AND SHADOWS. 
 
 157 
 
 exterior of the ring. Draw rays of Alight through E* F 4 , draw a diagonal ray 
 tangent at <?, transfer e 1 C 1 to C* e* ; C 3 e* will be half the minor axis, and E a F a 
 the major axis of the ellipse which defines 
 the limit of shadow for the interior of the 
 ring. 
 
 The shadow on the surface of a grooved 
 pulley (Fig. 320) is cast by the circumfer- 
 ence of the edge A' B'. Draw central lines 
 through C of the plan and describe a circle 
 C b with the radius of the least diameter of 
 the pulley ; from b draw a line b a at an 
 angle of 45 ; project a to a', and from a' 
 draw a' b' as a limiting point in the line of 
 shade, for another point draw a horizontal 
 line from b' intersecting the centre line at 
 b* ; project c and d on the plan to the ele- 
 vation, and intersect the projection of d at 
 d' by a ray of light from c' ; d 1 is the high- 
 est point in the curve. Take any horizon- 
 tal line E F in the elevation and describe 
 from C on the plan an arc with a radius 
 
 equal to one half this line ; draw from C' a ray intersecting E F at e, which 
 project to e', and from e' as a centre describe an arc with a radius equal to C B ; 
 the point of intersection of this arc with the circumference of the plan E F is 
 projected to/'. The limit of shade is then drawn through the points #', d', #*, 
 /', and g. 
 
 FIG. 320. 
 
 FIG 
 
 Fig. 321 represents the projections of a screw with a single square thread, 
 and placed in a horizontal position, A' a' being the direction of the ray of 
 light. The shadow is simply that cast by the outer edge, A B, of the thread 
 upon the surface of the inner cylinder ; its outline is delineated in the same 
 
158 
 
 SHADES AND SHADOWS. 
 
 manner as in treating of a cylinder surmounting another of smaller diameter 
 (Fig. 308). 
 
 The shadow cast by the helix ABC upon the concave surface of the square- 
 threaded nut is a curve, a C (Fig. 322), determined in the same way as that 
 in the interior of a hollow cylinder. The same rule applies to the edges A A 2 
 
 / 
 
 V\^\\ 
 
 ;< ^V \ 
 
 K -^ 
 
 A- 
 
 ! ' J \ I 
 
 L f\ / 
 
 #"- \ / 
 )^'- 
 
 k x 
 
 
 / 
 
 FIG. 32a. 
 
 and A" E, as well as to those of the helix F G H and the edge H I. The shadow 
 of the two edges J K and K L, thrown upon an inclined helical surface, of 
 which A L is the generatrix, follows the rules given in Fig. 323. 
 
 FIG. 323. 
 
SHADES AND SHADOWS. 
 
 159 
 
 - In the case of a triangular-threaded screw (Fig. 323) the outer edge, A' C D, 
 of the thread projects its shadow upon a helical surface inclining to the left, of 
 which the generatrix is known. 
 
 Describe from the centre a number of circles representing the bases of so 
 many cylinders, on the surfaces of which suppose helical lines to be traced of 
 the same pitch as those which form the exterior edges of the screw. Draw any 
 line parallel to the ray of light, as B' E', cutting the circles described in the 
 plan in the points B', F', G', E', which are projected to their corresponding 
 helical lines in the elevation at B 3 , F, G, and E. Transferring the point B' to 
 its appropriate position B on the edge A' C D,and drawing through the latter a 
 line, B i, at an angle of 45, its intersection with the curve B 3 G E will give one 
 point in the curve of the shadow required. By constructing other curves, as 
 H J K, the remaining points in the curve, as h, may be found. 
 
 C- 
 
 x \ 
 
 
 J) 
 
 L 
 
 FIG. 324. 
 
 The same processes are requisite to determine the outlines of the shadows 
 cast into the interior surfaces of the corresponding nut (Fig. 324). These 
 shadows are derived not only from the helical edge, A B D, but also from that 
 of the generatrix, A C. 
 
 The principles here laid down and illustrated are a sufficient guide for the 
 delineation of the shades and shadows of nearly all ordinary forms and com- 
 binations of machinery and architecture. Students should not copy the figures 
 as here represented, but should adopt some convenient scale somewhat larger, 
 and construct drawings according to the description. 
 
 MANIPULATION" OF SHADING AND SHADOWS, AND METHODS OF TINTING. 
 
 The intensity of a shade or shadow is regulated by the forms of bodies, the 
 position that they occupy in relation to the light, and the distance from 
 the eye. 
 
160 SHADES AND SHADOWS. 
 
 Surfaces in the Light. Flat surfaces wholly exposed to the light, and at 
 all points equidistant from the eye, receive a uniform tint. 
 
 Of two parallel surfaces, the one nearer the eye receives a lighter tint. Every 
 surface exposed to the light, but not parallel to the plane of projection, having 
 no two points equally distant from the eye, receives unequal tints, gradually 
 increasing in depth as the parts recede from the eye. Of two surfaces un- 
 equally exposed to the light, the one more nearly perpendicular to the rays re- 
 ceives the fainter tint. 
 
 Surfaces in Shade. When a surface entirely in the shade is parallel to the 
 plane of projection, it receives a uniform dark tint. Of two objects in the 
 shade parallel to each other, the one nearer the eye receives the darker tint. 
 On a surface in the shade, inclined to the plane of projection, the parts nearest 
 the eye receive the deepest tint. 
 
 If two surfaces exposed to the light, but unequally inclined to its rays, have 
 a shadow cast upon them, the part of it which falls upon the more lighted sur- 
 face should be darker than where it falls upon the other surface. 
 
 The methods of shading generally adopted are either by the superposition 
 of any number of flat tints, or by tints softened off at their edges. 
 
 (See Plates I, II, III, IV, V, XI, XII, and XV.) 
 
 In the shading of a prism by flat tints (Plate I, Fig. 4), the face, a b c d, 
 being parallel to the plane of projection, receives a uniform tint usually of India 
 ink or sepia. When the surface to be tinted is large, put on a very light tint 
 first, and then go over the surface with a tint sufficiently dark to give the 
 desired tone. The face, b g h c, being inclined to the plane of projection, re- 
 ceives a graduated tint from the line b c to the line g h. This is obtained by 
 laying on a succession of flat tints. Divide the plan V g' into equal parts, as at 
 the points 1', 2', and from these points project lines upon, and parallel to the 
 sides of, the face b g h c. These lines should be drawn very lightly in pencil, 
 as they merely serve to circumscribe the tints. A grayish tint is then spread 
 over the portion of the face between the lines b c and 1, 1 (Fig. 2). When this 
 is dry, the same tint is laid on again, and extended over the space comprised 
 within the lines b c and 2, 2 (Fig. 3). A third tint, covering the whole surface 
 be Tig (Fig. 4), imparts the desired graduated shade to that side of the prism. 
 The number of tints in the graduated shade depends upon the size of the sur- 
 face, and the depth of tint must vary according to the number used. The 
 greater the number, the softer the appearance and the less harsh the lines which 
 border the different tints. This method is preferable to that sometimes em- 
 ployed of first covering the whole surface b g h c with a faint tint, then putting 
 on a second tint b 2 2 c, followed by a narrow wash Z> 1 1 c, because the outline 
 of each wash remains untouched and presents a prominence and harshness. 
 
 The face a dfe, also inclined to the plane of projection, should, as it is en- 
 tirely in the light, be covered by a series of much fainter tints than the surface 
 b g h C, which is in the shade, darkening, however, toward the line ef. The 
 gradation of tint is effected as on the face b g h c. 
 
 In shading a cylinder by means of flat tints (Figs. 5-12), the line of separa- 
 tion between the light and shade, a b (Fig. 6), is determined by the radius a' 
 (Fig. 5), drawn perpendicular to the rays of light R 0. The part of the eleva- 
 tion of the cylinder which is in the shade is comprised between the lines a b 
 
SHADES AND SHADOWS. 
 
 and c d. This portion, then, should be shaded as a surface in the shade in- 
 clined to the plane of projection. All the remaining part that is visible of the 
 cylinder presents itself to the light ; but, in consequence of its curvature, the 
 rays of light form angles varying at every part of its surface, which should re- 
 ceive a graduated tint. Determine the part of the surface that is most directly 
 affected by the light, situated about the line ei (Fig. 12). The visual rays are 
 perpendicular to the vertical plane and parallel to V ; the part which ap- 
 pears clearest to the eye may be limited by the line T 0, which bisects the 
 angle V R. Project the points e' and w', and draw the lines e i and m n 
 (Fig. 12) ; the surface comprised between these lines will represent the lightest 
 part of the cylinder. 
 
 This part should have no tint upon it whatever if the cylinder is polished 
 as a turned iron shaft or a marble column ; but if the surface of the cylinder 
 be rough, as a cast-iron pipe, then a very light tint, considerably lighter than 
 on any other part, may be given' it. 
 
 Divide the half- plan of the cylinder/' m' a' c' into any number of equal 
 parts by faint pencil-lines, and begin the shading by laying a tint over a c d b 
 (Fig. 6). When this is dry, put on a second tint covering the line a b of sepa- 
 ration of light and shade, and extending over one division (Fig. 7). Proceed 
 in this way, as shown in Figs. 6-12, until the whole of that part of the cylinder 
 which is in the shade is covered. Treat in a similar manner the part feig 
 (Fig. 12), and complete the operation by covering the whole surface of the cyl- 
 inder, excepting the surface e m n *', with a very light tint. 
 
 In shading the frustum of a hexagonal pyramid (Plate II), the face abed 
 should receive a uniform flat tint, as in Plate I, or the tint may be slightly deep- 
 ened toward the top of the pyramid, as that surface is not quite parallel to the 
 vertical plane. 
 
 The face b g h c, being inclined and in the shade, should receive a dark tint ; 
 darkest where it meets the line b c, and gradually becoming lighter as it ap- 
 proaches the line g h. To produce this effect, apply a narrow strip of tint at 
 be (Fig. 6), and then, qualifying the tint in the brush with a little water, join 
 another strip to this, and finally, by means of another brush moistened with 
 water, soften off this second strip toward the line 1, 1, which may be taken as 
 
 the limit of the first tint. 
 
 \ 
 
 When the first tint is dry, cover it with a second, which must be similarly 
 treated, and should extend up to the line 2,2 (Fig. 7). Proceed in this manner 
 with other tints, until the whole face b g li c is shaded (Fig. '8). In the same 
 way the face e a dfis to be covered, though with a considerably lighter tint, for 
 the rays of light fall upon it almost perpendicularly. 
 
 The tint on these two faces should be slightly graduated from ea tofd, and 
 from cli to b g ; but this graduation may be disregarded until some proficiency 
 in shading has been acquired. 
 
 In shading a cylinder by means of softened tints (Plate II), the boundary 
 of each tint being indicated, 'as in Plate I, the first strip of tint must cover the 
 line of extreme shade a b, and then be softened off on each side. Other succes- 
 sively wider strips of tint follow, and receive the same treatment as the one first 
 put on. 
 
 If, after shading a figure by the foregoing method, any very apparent iue- 
 12 
 
162 SHADES AND SHADOWS. 
 
 qualities present themselves in the shades, such defects may be remedied, in 
 some measure, by washing off excesses of tint with a brush or a damp sponge, 
 and by supplying a little colour to those parts which are too light. 
 
 Dexterity in shading figures by softened tints is facilitated by practising 
 upon large surfaces. 
 
 Whatman's best rough-grained drawing-paper is better adapted for receiving 
 colour than any other. Of this paper, the double-elephant size is preferable, as 
 it possesses a peculiar consistence and grain. The face of the paper to be used 
 is the one on which the water-mark is read. correctly. 
 
 The paper for a coloured drawing ought always to be damp-stretched 
 (page 54). 
 
 The size of the brushes used depends upon the scale to which the drawing 
 is made. Long, thin brushes, however, should be avoided. Those possessing 
 corpulent bodies and fine points are to be preferred, as they retain a greater 
 quantity of colour, and are more manageable. 
 
 Sable brushes are more durable and better than camel's hair, but more ex- 
 pensive. Economy may be practised by using camel's hair for the larger sizes 
 and red or black sable for the smaller. 
 
 During the process of laying on a flat tint, if the surface be large, the draw- 
 ing may be slightly inclined, and the brush well charged with colour, so that 
 the lower edge of the tint may be kept moist until the whole surface is covered. 
 In tinting a small surface, the brush should never have much colour in it, other- 
 wise the surface will unavoidably present coarse, ragged edges, and an uneven 
 appearance throughout. 
 
 All objects with curved outlines have a certain amount of reflected light 
 imparted to them. Bodies, whatever may be their form, are affected by re- 
 flected light; but, with a few exceptions, this light is only appreciable on 
 curved surfaces. 
 
 In proportion to its degree of polish or brightness is the amount of reflected 
 light which a body spreads over adjacent objects, and also its own susceptibility 
 of illumination under the reflection from other bodies. A polished steam- 
 cylinder or a white porcelain vase both receives and imparts more reflected light 
 than a rough casting or a stone pitcher. 
 
 Shade, even the most inconsiderable, ought never to extend to the outline 
 of any smooth circular body. On a polished sphere the shade should be deli- 
 cately softened off just before it meets the circumference, and, when the shad- 
 ing is completed, the body-colour intended for the sphere may be carried on to 
 its outline. This will give a clearness to the part of the sphere influenced by 
 reflected light which it could not have possessed if the shade-tint had been ex- 
 tended to its circumference. Very little shade should be suffered to reach the 
 outlines even of rough circular bodies, lest the colouring present a harsh, 
 unreal appearance. 
 
 Shadows become lighter as they recede from the bodies which cast them, 
 and appear to increase in depth as their distance from the spectator diminishes. 
 In Nature this increase is only appreciable at considerable distances. But it is 
 important for the effective representation of machinery that the variation in 
 the distance of each part from the spectator should strike the eye, and an exag- 
 geration in expressing the varying depths of the shadows is one means of effect- 
 
SHADES AND SHADOWS. 163 
 
 ing that object. The shadows on the nearest and most prominent parts of a 
 machine should be made as dark as colour can render them, the colourist being 
 thus enabled to exhibit a marked difference in the shadows on the other parts 
 of the machine as they recede from the eye. The same direction is applicable 
 to shades. The shade on a cylinder, situated near the spectator, ought to be 
 darker than on one more remote. As a general rule, the colour on a machine 
 should become lighter as the parts on which it is placed recede from the eye. 
 
 Plates III and IV present some examples of finished shading and shadows 
 of the solids given under the head of " Orthographic Projection." 
 
 The direction of the shades and shadows in the elevation is from the upper 
 left-hand corner, and in the plan from the lower left-hand corner. 
 
 The shadow on a concave surface is darkest toward its outline, and becomes 
 lighter as it nears the edge of the object. Reflection from the part of the sur- 
 face on which the light falls causes this gradual diminution in the depth of 
 the shadow, the greatest amount of reflection being opposite the greatest 
 amount of light. No brilliant or extreme light should be left on concave sur- 
 faces, as such lights tend to render it doubtful whether the objects presented 
 are concave or convex. After the body-colour has been put on, a faint wash 
 should be passed very lightly over the whole concavity. This will not only 
 modify and subdue the light, but soften asperities in the tinting, which are 
 particularly unsightly on a concave surface. 
 
 The lightest part of a sphere is confined to a mere point, around which the 
 shade gradually increases as it recedes. This point is not indicated on the 
 figure because the shade-tint on a sphere ought not to be spread over a greater 
 portion of its surface than is shown there. The very delicate and hardly per- 
 ceptible progression of the shade in the immediate vicinity of the light-point 
 should be effected by means of the body-colour, either by lightening it toward 
 the light point for polished or light-coloured curved surfaces, or deepening the 
 body-colour for unpolished surfaces from the light point until it meets the 
 shade tint over which it is spread uniformly. 
 
 To shade a sphere effectively, put on two or three softened-off tints in the 
 form of crescents converging toward the light-point, the first one being carried 
 over the point of deepest shade. 
 
 A ring (Plate IV, Fig. 7) is a difficult object to shade. To change with ac- 
 curate and effective gradation the shade from the inside to the outside of the 
 ring, to leave with regularity a line of light upon its surface, and to project its 
 shadow with precision, requires attention and care. 
 
 It should be noted that the depth of a shadow on any object is in propor- 
 tion to the degree of light which it encounters on the surface of that object. 
 In the plan (Fig. 6) the shadow of the apex of the cone falls upon the lightest 
 point of the sphere and is the darkest part of the shadow. The deepest portion 
 of the shadow of the cone on the cylinder in the plan (Fig. 4) is where it coin- 
 cides with the line of extreme light. Flat surfaces are similarly affected, the 
 shadows thrown on them being less darkly expressed according as their inclina- 
 tion to the plane of projection increases. The body-colour on a flat surface 
 should, on the contrary, increase in depth as the surface becomes more inclined 
 to this plane. 
 
 Reflected lisrht is incident to shadows as well as to shades. This is observ- 
 
164 SHADES AND SHADOWS. 
 
 able where the shadow of the cone falls upon the cylinder, though to a less ex- 
 tent, on other parts of these figures. The reflected light on the cone from the 
 sphere or cylinder adds greatly to the effect of the shadows and to the appear- 
 ance of the objects themselves. 
 
 The peculiarities and effects of light, shade, and shadow may be seen in the 
 examples of screws (Plate V). 
 
 In topographical and architectural drawing artistic effect may often be in- 
 troduced, but in mechanical drawing distinctness of outline and accuracy of 
 expression are essential ; though, to maintain harmony in the colouring and to 
 equalize the appearance of the drawing, large shades should be coloured less 
 dark than small ones, at equal distances from the eye, and no very dark shad- 
 ing is permissible. 
 
 In preparing colours for tints, great care should be used in grinding. The 
 end of the cake should be slightly wetted and rubbed on a porcelain palette, 
 then transferred by a wet brush to another saucer, and water added to bring 
 to the required tint. Mixed colours should be intimately blended by the brush. 
 Grind more than enough of the tint required, and let it stand in the saucer 
 till the grosser particles have settled and the liquid is of clear, uniform tint. 
 It is common to make little boxes or bags of waste drawing-paper to hold the 
 colours instead of saucers ; the gross matters, settling on the bottom, are not 
 then so readily disturbed. 
 
 Instead of hard cakes of colour, moist colours are used, either in pans or 
 tubes, which saves the trouble of grinding. For flat tints or washes, aniline 
 colours, dissolved in water and kept bottled, are the readiest means of colour- 
 ing, but are not applicable to finished work. 
 
 If the surface of the paper is greasy and resists colours, dissolve a piece of 
 ox-gall, the size of a pea, in a tumbler of water, and use this solution with the 
 colours instead of plain water. 
 
 When the brush is too full, as it comes toward the limit of the tint, take up 
 the surplus moisture on a wet sponge or piece of cloth or blotting-paper. 
 
 An expeditious way of shading a cylinder, or of delineating the shores of a 
 stream or lake, is by drawing with a brush full of the darkest tint along the 
 sides, and then, with a wet brush, modifying this tint toward the light from 
 the sides, so as to give a shaded appearance. For this purpose, two brushes 
 will be necessary one with colour, the other with water ; also, a tumbler of 
 water, and a piece of blotting-paper, to take up the excess of moisture from 
 paper or brush. Often a single line of dark colour blended this way will ex- 
 press all that is necessary, but the effect may be improved by a sort of stippling 
 with the colour-brush and by extending the line of shade. 
 
 The same effect is obtained better by drawing two faint pencil-lines on the 
 elevation of the cylinder, for instance, to indicate the extremes of light and 
 shade on its surface and passing the brush, moderately full of the darkest tint, 
 down the line of deepest shade, spreading the colour more or less on either side, 
 according to the diameter of the cylinder ; then, before this layer of tint is dry, 
 if possible, toward the line of extreme light, beginning at the top, and encroach- 
 ing slightly over the edge of the first tint, lay on another not quite so dark, but 
 about double its width. Put on the second tint before the first is thoroughly 
 dry, that its edges may be softened by the application of the second. While 
 
SHADES AND SHADOWS. 
 
 165 
 
 this second tint is still damp, with a much lighter colour in the brush, pro- 
 ceed in the same manner with a third tint, and so on to nearly the line of ex- 
 treme light. Eepeat this process on the other side of the first tint, approach- 
 ing the outline of the cylinder with a very faint wash, so as to represent the 
 reflected light which progressively modifies the shade as it nears that line. 
 Then let a darkish narrow strip of tint meet and pass along the outline of 
 the cylinder on the other side of the extreme line of light, after which gradu- 
 ally fainter tints should follow, treated in the manner already described, and 
 becoming almost imperceptible just before arriving at the line of light. 
 
 But it is not possible, by the above-described means alone, to impart a suf- 
 ficient degree of well-regulated rotundity to the appearance of such an object. 
 It may be necessary to equalize the superfluities and deficiencies of colour to 
 some extent by a species of gross stippling. This is done by spreading a little 
 colour over the parts where it is deficient, and then passing the brush; supplied 
 with a very light wash, very lightly over nearly the whole width of the shade. 
 This process may be repeated to suit the degree of finish which it is desired to 
 give the drawing. The shading of all curved surfaces is treated in the same 
 manner. 
 
 The shades having been put in, the shadows follow. Draw the outline of 
 the shadow in pencil, and along its inner line, the line which forms a portion 
 of the figure of the object whose shadow is to be represented, lay on a strip of 
 the darkest tint, wide or narrow, according to the width of the shadow, and 
 then, before it is dry, soften off its outer edge. 
 
 The finish is made by a light wash or two of the body-colour, and in pass- 
 ing over the shades and shadows care must be taken to manoeuvre the brush at 
 such parts quickly and lightly. 
 
 The shades and shadows of a machine being modified in intensity as their 
 distance from the eye increases, its body-colour should be treated in a similar 
 manner, becoming less bright as the parts of the machine which it covers recede 
 from the spectator. 
 
 When the large circular members of a machine have been shaded, the shad- 
 ows, and even the body-colour on those parts farthest removed from the eye, are 
 to follow, and the proportion of India ink in the tint used should increase as 
 the part to be coloured becomes more remote. A little washing, moreover, of 
 the most distant parts is allowable, as it gives a pleasing appearance of atmos- 
 pheric remoteness, or depth, to the colour thus treated. 
 
 The amount of light and reflection on the members of a machine should 
 diminish in intensity as the distance of such objects from the spectator in- 
 creases. As it is necessary, for effect, to render on the parts of a machine near- 
 est the eye the contrast of light and shade as intense as possible, so, for the 
 same object, the light and shade on the remotest parts should be subdued and 
 blended according to the extent or size of the machine. 
 
 To add to the definiteness of a coloured mechanical drawing, it is well to 
 make the lines of light and shade distinct. 
 
 After having marked in pencil the position of the extreme light, take #ie 
 drawing-pen, filled with a just perceptible tint, and draw a line of colour on 
 one side of the line of light, almost touching it ; then with the brush, filled 
 with similar light tint, join this line of colour while still wet, and fill 'up the 
 
166 SHADES AND SHADOWS. 
 
 space unoccupied by the shade-tint, within which the very light colour in the 
 brush will disappear. Treat the part of the object on the other side of the line 
 of light in the same way. The extreme depth of shade may, with great effect, 
 be indicated by filling the pen with dark shade-tint, and drawing it exactly 
 over the line representing the deepest part of the shade. On either side, join- 
 ing this strip of dark colour, another, of lighter tint, is to be drawn. Others 
 successively lighter follow, until, on one side, the line of the body is joined, and 
 on the other the lightest part of the body is nearly reached. The line of light 
 is then to be shown, and the faint tint used for this to be spread with the brush 
 lightly over the whole of the part of the body that is situated on either side of 
 this line, thus blending into smooth rotundity the graduated strips of tint drawn 
 by the pen. 
 
 In all tinted drawings the important parts should be more conspicuously 
 expressed than the mere adjuncts. Thus, if the drawing be to explain the 
 construction of the machine, the tint of the edifice may be more subdued than 
 those of the machine ; and if the machine be unimportant, it may be repre- 
 sented in mere outline, while the edifice is brought out conspicuously. 
 
 With regard to washings, the soft sponge is an excellent means of correct- 
 ing great errors in drawing or colouring, but care must be taken not to rub 
 the surface. In removing or softening colour, for large surfaces, use the sponge ; 
 for small spots, the brush. While colouring, keep a clean, moist brush by you 
 to remove or modify a colour. 
 
 The immediate effect of washing is to soften a drawing, an effect often very 
 desirable in architectural and mechanical drawings, and the process is simple 
 and easily acquired ; keep the sponge or brush and the water clean ; after the 
 washing is complete, take up the excess of moisture with the sponge or brush 
 or with a piece of clean blotting-paper. Where vigour is required, let the 
 borders of the different tints be distinct. 
 
 There are no conventional tints that draughtsmen have agreed upon to be 
 uniformly used to represent different materials. India ink is not a black, but 
 a brown, making with a blue a greenish cast, and with gamboge a smear. A 
 coloured drawing is better without the use of any India ink at all ; any depth of 
 colour may be as well obtained with blue as with black. There is the objection 
 to gamboge that it is gummy, and does not wash well, and a better effect is ob- 
 tained with yellow ochre. For the reds, the madder colours are the best, as 
 they stand washing. For the shade-tint of almost any substance a neutral tint 
 is required, such as Payne's gray, or madder brown subdued with indigo. 
 
MATERIALS. 
 
 VARIED materials enter into the composition of structures and machines, or form 
 their supports, which are to be represented by the draughtsman. That of the earths and 
 rocks, in their natural position, are 
 shown under the head of " Topo- 
 graphical Drawing," or by a closer 
 imitation of Nature, with or without 
 colour. 
 
 Fig. 325 represents a plan and 
 section of an earth-bank of a canal, 
 with a paved rock- slope. 
 
 A base of rock may be represented 
 by stratifications (Fig. 326). 
 
 Rocks, gravels, sands, muds, etc., 
 either in their natural or structural 
 positions, are shown in "Engineering 
 Drawing." 
 
 Earth, when first dug, occupies 
 more space than when in its natural 
 condition, but, after a time, it shrinks 
 and becomes more compact. The 
 earth dug out of a hole, when settled, 
 will not fill the hole. Sand, gravel, 
 loam, and clay, will occupy from 8 to 12 per cent less space than when in the natural 
 cut. 
 
 Loose, dry sand weighs from 90 to 100 pounds per cubic foot; compacted, 110; 
 gravel, about the same ; clay, in the bank, 120 pounds. Sands and gravels are excellent 
 material for embankments and fills. The slopes in cuts and fills are usually H horizon- 
 tal to 1 perpendicular. Sands and gravels are readily drained, and, when dry, are but 
 little affected by frost. The clays are hard to drain, heave with the 
 frost when wet, and, under the influence of a thaw or excess of 
 water, become fluid, but well rammed they are used as puddle walls 
 in the centre of reservoir embankments for stanchness. In these 
 positions . it is recommended that the clay should be compacted 
 dry. Very fine sand, with gravel, and perhaps some admixture of 
 clay, a glacier till, is known as hard-pan by engineers, very difficult 
 to be moved with the pick, and often requiring blasting. The same 
 material wet, but without gravel, forms a quicksand a jelly-like 
 material from which, if a spadeful be taken out, the hole closes up 
 at once, and excavation shows but little visible sign of a depression, the space being 
 made good from the entire mass. There is another material, called quicksand, which is 
 rather a running sand even when not wet, it rests with a very flat slope; the particles 
 are very fine, and flow like the sands in an hour-glass. 
 
 107 
 
 FIG. 325. 
 
 FIG. 326. 
 
168 
 
 MATERIALS. 
 
 Sands and gravels are large components of mortars, letons, and concrete ; and burnt 
 with clay, of brick, tile, and pottery. 
 
 BUILDING MATERIALS. , - 
 
 The natural building materials of civilized communities are wood and stone, which 
 are to be worked or fashioned to the purposes to which they are to be applied. 
 
 FIG. 327. 
 
 FIG. 328. 
 
 Fig. 327 shows various sections of timber in which the rings or yearly growth are 
 very strongly shown, and the effect of shrinkage by black margins and distortion, accord- 
 ing to the form of cut. 
 
 The strongest timber lies about one third the radius from the pith in the butt log ; 
 in the top log the heart position seems strongest; but in important bridge and floor tim- 
 bers the heart should be excluded (Fig. 328) ; exterior rings or sap are soft and liable to 
 decay. 
 
 For beams or girders, the timber should be cut so that the stress will be parallel with, 
 
 and not across, grain. For posts in 
 
 compression, and lengthwise of the 
 fibre, the section may be from any 
 part. 
 
 Figs. 329, 330, and 331 are draw- 
 ings of wood, longitudinal and sec- 
 tional, in which the grain of the 
 wood is imitated, but wood is more 
 often represented in plain outline, 
 and the cross-section of a timber 
 thus (Fig. 332), or by mere hatch- 
 ing. When distinguished by col- 
 our, burnt sienna is used commonly 
 
 FIG. 329. 
 
 Fm. 330. 
 
 Fia. 331. 
 
 FIG. 332. 
 for wood, but sometimes the colour and grain of the wood is imitated. 
 
 The draughtsman, for his designs, will probably have to confine himself to the timber 
 within his reach. But he should know what is best for his purpose, reference being had 
 to economy in cost and maintenance. For most purposes, wood should be seasoned, so 
 that joints may not open under this operation after the material is in the structure. But, 
 for work under water, wood should be but slightly seasoned, as a swelling of the wood 
 may be disastrous. Seasoning of timber may be done by exposure for a time to outer 
 
MATERIALS. 169 
 
 air-currents; if in a kiln, it can be done speedily with heated air, or by steam. For 
 beams, girders, and the like, there should be few knots, especially on the outer edges 
 for posts, small ones are not objectionable ; while for sidings and under-floors, firm, large 
 knots do not impair the work ; but no smooth work can be made with knotty lumber. In 
 most specifications, lumber is "to be square-edged, without sap, and large or loose knots." 
 
 In selecting lumber for a permanent structure, the life and endurance of the material 
 are to be considered. Most of the woods, sheltered from the wet and exposed to air- 
 currents, will last for a very long time ; but many will check and warp and become dis- 
 torted. All lumber in earth beneath the level of water will last indefinitely. In salt 
 water, above the earth, all are subject to the attacks of the worm the Teredo and Lim- 
 noria and, where the water is pure, the destruction is very rapid. Sewer-water and 
 fresh water are both destructive to the worm. Green or wet timber, in positions from 
 which the air is excluded, soon fail through dry rot, and even seasoned timber under un- 
 favourable conditions. 
 
 The life of timber exposed to wet or dry rot can be prolonged by filling the pores 
 with creosote, pyrolignite of iron, solutions of chlorides of mercury or zinc and various 
 other antiseptics. There are works in which the timber is first steamed in close cylinders 
 to remove the sap and rarefy the air in the pores, and then injecting the preserving 
 fluids. Open tanks of plank can be readily constructed, in which the soaking of lumber 
 in cold solution, especially that of corrosive sublimate, is effective. 
 
 CHARACTERISTICS AND USE. 
 
 White Pine. A wood of the most general application in the market; is light, stiff, 
 easily worked, nails are easily driven into it, and takes paint well, warps and checks but 
 little in seasoning, endures well in exposed situations ; clear stuff, of best quality, useful 
 for patterns and models, for interior finish of houses, doors, window sashes, furniture. It 
 forms the base or inner core of the best veneered work, holds glue well, and the composite 
 structure is better than single solid wood. The cheaper kinds of pine are used for frames 
 of buildings, posts, girders, and beams. Even with large knots is well adapted for board- 
 ings, and is extensively used for goods-boxes. 
 
 Southern Pine. A heavy, strong, resinous, lasting wood, clear and mostly without 
 knots, hard to be worked by hand-tools, and when seasoned difficult to nail. The sur- 
 faces, from their resinous character, do not hold paint well. It is used very largely for 
 girders, beams, and posts of mills and warehouses, and for floors of the same, when ex- 
 posed to heavy work or travel. For the first, it can be obtained of almost any dimension 
 to suit ; for floors, it is sold in long strips, from two to six inches wide, of varied lengths, 
 tongued and grooved, and when laid is blind-nailed, toeing the nail through the tongue, 
 so that the nail-head does not show. 
 
 Experiments of the Forestry Division of the United States Department of Agriculture 
 prove that the extracting of the turpentine from the long-leaf yellow-pine trees does not 
 in any material sense injure them for use as lumber. The bled timber is heavier in the 
 bottom cuts by about two pounds per cubic foot. 
 
 Canadian Red, Norway, and Silver Pines are resinous woods, like the Southern pine, 
 and are used for similar purposes, but are not as valuable woods less straight in the 
 grain, and with more knots. 
 
 Spruce. A light, straight-grained wood, with but few knots, which are small and 
 often decayed. It does not last well exposed to the weather, and checks and warps 
 badly in seasoning. It is the most common wood here for floor-beams and common 
 floors, but it must be well braced and nailed, and is not fitted for joiner- work. 
 
 Hemlock is similar to the spruce, and, when selected, is less liable to check and twist 
 in seasoning. It is often of a very poor quality, Irash and shaky. Exposed, it is but 
 little better, if any, than the spruce. For stables, it is well adapted for grain-boxes, as 
 the fibre prevents the gnawing of rats. 
 
MATERIALS. 
 
 Ash. Some of the ashes are of exceeding toughness. A straight, close-grained wood. 
 It is used for carriage and machine frames, and for interiors, doors, wainscot, floors, 
 when no paint is used. 
 
 Chestnut. Somewhat like the ash in appearance, but coarser-grained, and very en- 
 during in exposed positions. It is most largely used for cross-ties of railways. As a 
 roof-frame exposed in the inside, and in general interior finish without paint, the effect 
 is very good. The closer-grained woods are very often thus used. 
 
 Black Walnut is, in the trunk, a straight-grained, gummy wood, clogging the plane 
 a little in its working; the knots are useful for veneer. Were the wood 'cheap enough, 
 it would undoubtedly make a good frame. It is used here for desks and counters, for 
 furniture and interior finish, as an ornamental wood. 
 
 Butternut. Similar to the black walnut, less commonly used, but fully equal as an 
 ornamental wood. 
 
 Hickory. A strong, tough wood; is used for cogs of mortise-wheels, handspikes, axe- 
 helves, and wheelwrights' work. 
 
 Beech. A close-grained wood, but of little application in this market. Sometimes 
 used for cogs of wheels, for small tool-handles, and in marquetry. 
 
 Oak, Live. A very strong, tough, enduring wood, used industrially almost entirely 
 for ship-building. Ornamentally, in marquetry and panels. 
 
 Oak, White. A very valuable, strong, tough wood, with great endurance. It is 
 heavy, and hard to work, and was formerly used largely for the frames of houses, but 
 has been superseded by the white pine. It is used in ship-yards and in water-works 
 for the frames of flumes, penstocks, and dams, and for the planking of the latter, for 
 dock-buffers and piles, and for railway and warehouse platforms. The red and black 
 oaks may in general be considered a cheaper and poorer quality of the white oak. All 
 have a handsome grain, that adapts them to ornamental work. 
 
 Bass, Poplar, White-wood, are light woods, mostly used in the manufacture of fur- 
 niture, for drawer-bottoms, cabinet-backs, panels; they are very clear stock, easily 
 worked, and can be readily obtained in thin, wide boards. 
 
 Cedar. A straight-grained, light wood, of great endurance, valuable for posts, sills, 
 shingles ; used for pails and domestic utensils. The red variety, from its odour, is ad- 
 mirable for drawers and chests, preserving their contents from moths. 
 
 Locust is in the market only in small sticks ; is of extreme endurance. It is used 
 almost invariably here for the sills of the lowest floors of buildings, where there can be 
 no ventilation, and for treenails of ship-planks. 
 
 Elm. Although a tree of wide diffusion, is but little used as lumber. It is kept for 
 an ornamental tree, beyond its usefulness for any other purpose but fuel. Well selected, 
 it is said to be an enduring timber, useful for piles and places exposed to wet. 
 
 Maples are tough, close-grained woods, rather to be considered among the ornamental 
 woods, for furniture and interior finish. The same may be said of the cherry, plum, and 
 apple tree, of which the denser woods are admirably adapted for the handles of small 
 tools, for bushings of spools and bobbins. 
 
 The list of imported woods is extremely large, mostly for ornamental purposes; but 
 the mahogany is one of the very best of woods for patterns and small models, as it 
 changes but little in seasoning; and the lignum-vitae, a very hard and heavy wood, is 
 used for pulley-sheaves, packing-rings of pumps, water-wheel steps, and shaft-bushings. 
 
 Timbers of the same kind vary much in their weight, strength, and endurance, ac- 
 cording to the localities in which they are grown, the season at which they are cut, and 
 how seasoned. Tables are given in the Appendix in detail, their varied resistance under 
 stresses and their specific gravities. 
 
 Of late years paper in sheets and pulp has been used instead of woods, and serves 
 well many purposes on account of its little shrinkage, incombustibility, strength, and 
 endurance. 
 
MATERIALS. 
 
 171 
 
 Earth. 
 
 Concrete. 
 
 Brick. 
 
 ^///YA/////^//,. 
 y///A//////////// 
 
 ////A//. 
 
 ' 
 
 Rubble. 
 
 Timber. 
 
 STONES. 
 
 In selecting the form of construction, and the stones of which it is to be composed, 
 the draughtsman must be governed by the fitness for the purpose and the cost. He must 
 select from what he can readily get, and arrange the form to suit the material. He must 
 know what is to be the exposure, and what the effect will be on the stones. Almost any 
 stone will stand in a protected wall, but many of the sandstones and slates disintegrate 
 and exfoliate under the influence of the weather, heat, cold, frost, and moisture. Even 
 the granites are liable to serious decomposition when the feldspars are alkaline ; and the 
 limestones (dolomites), of which the English Houses of Parliament are composed, have 
 failed in the sulphurous air of London smoke, while at Southwell Minster they have 
 stood for over 800 years. Chemical tests of stone to determine endurance are deceptive. 
 The safe way is to see how the material has stood in like situations to the one in which 
 it is to be employed, or go to the quarry, and see how the stones have weathered. 
 
 The strength of stones to resist crushing, as determined by experimental cubes, is 
 even in the weaker stones much in excess of what would be required in structures, but 
 most stones are weak under cross-strains, and failures in construction are more likely to 
 occur by faulty workmanship or design, by which the stones are subjected to unequal 
 strains, and for which they are not adapted. The weight should not be brought on the 
 outer edges or arrises, as the faces will chip readily; nor should most stones be used for 
 wide-span lintels, unless relieved by the masonry above the opening. 
 
 TECHNICAL TERMS OF MASONRY. 
 
 Agreeably to the nomenclature recommended in "Transactions of the American 
 Society of Civil Engineers," November, 1877: 
 
 Rubble masonry includes all stones which are used as they come from the quarry, pre- 
 pared at the work by roughly knocking off their corners. It is called uncoursed rul>Ue 
 (Fig. 333) when it is laid without any attempt at regular courses ; coursed rubble, when 
 levelled off at specified heights to a horizontal surface (Fig. 334). 
 
 Square-stoned Masonry. Square stones cover all stones that are roughly squared and 
 roughly dressed on bed and joints. 
 
 Quarry-faced stones are those which are left untouched as they come from the quarry. 
 
 Pitch-faced stones are those on which the arris is clearly defined beyond which the 
 rock is cutting away by pitching-tool. 
 
172 
 
 MATERIALS. 
 
 Drafted stones are those in which the face is surrounded by a chisel-draft. 
 If laid in regular courses of about the same rise throughout, it is range-work (Fig. 
 335). If laid in courses that are not continuous, it is broken range (Fig. 336). 
 
 FIG. 334. 
 
 Cut stones or ashlar covers all squared stones with smoothly- dressed bed and joints. 
 Generally, all the edges of cut stone are drafted, if the face is not entirely fine cut, but 
 
 WTfH 
 
 ICT*;TI 
 
 r^ ' ltw - -- 
 
 X. 
 
 Fio. 335. 
 
 FIG. 336. 
 
 they may be quarry-faced or pitch-faced ; as a rule, the courses are continuous (Figs. 337, 
 338), but, if broken by the introduction of smaller stones of the same kind, it is called 
 broken ashlar (Fig. 336 j. If the courses are less than one foot in height, it is small ash- 
 lar (Fig. 337). 
 
 3H akgir'4?^^l^ 
 
 t'"'"-"; : tl v i rcrpM^ifeiffrTi'''^^!^')-^ 
 ,:^-''^H^ :; " ;; "i4hfe:i::^ 
 
 Fio. 337. 
 
 ,!!! J't 
 
 FIG. 338. 
 
 Squared-stoned Masonry. The joints in one course should not come directly over 
 those of another; there should be a lap or bond, and, in connecting the front or face with 
 the backing, headers must be introduced for bond. Headers are stones extending into 
 the wall, stretchers running with the face. 
 
 The backing is of rubble, sometimes laid dry, but as from its many and large joints it 
 settles more than the face, it should be set in mortar to provide uniformity of support. 
 
 In addition there is a class of ornamental stone work, specified as "close jointed, 
 hammer-dressed rubble," in which there are no courses and no pinners. The joint of 
 each stone is carefully fitted to those beneath it. 
 
 For rubble-work, all varieties of sound stone are used, and of almost any size. In dry 
 work, for foundations and for heavy revetment-walls, the stones are laid with derricks, 
 but they must have fair beds and builds. If boulders, they must be split, and cobbles 
 in the filling are worse than useless, as they are unstable, and in settlement act as wedge 
 to increase the movement. 
 
Drafted Quarry Face. 
 
 MATERIALS. 
 
 Bush Hammered. 
 
 Axed. 
 
 173 
 
 GRANITIC STONES. 
 
 Granite and syenite are by builders classed as granites. The granite in general rifts 
 in any direction, and works well under the hammer and points. From these circum- 
 stances it is more desirable than the syenites, which are much harder to be worked. Both 
 are admirable stones for heavy dock-walls, bridge-abutments, river-walls, either as 
 rubble-squared stones or cut work, and are very enduring. They are also used for the 
 faces of important buildings, either as fine-cut, quarry, or pitched-face. Ornamental 
 work of the simpler kind is readily produced ; more elaborate is expensive, but it is about 
 the only stone in this climate in which foliage and sharp undercut work will stand the 
 weather without exfoliating. These stones, especially the syenites, admit of a high 
 polish, and are used considerably for columns and panels in buildings, and in monumen- 
 tal work. Gneiss is of the granitic order, but a cheaper, poorer stone. It splits with 
 difficulty, except parallel with line of bed. It has a foliated structure, and is not adapted 
 for ashlar, but is very good for squared-stone masonry and rubble-work, and often used 
 for sidewalk-covers of vaults. 
 
 ARGILLACEOUS STONES. 
 
 The slates or stones thus designated by builders were formerly in very common use 
 as roofing material, and were almost entirely from Wales, but latterly they are taken from 
 Vermont and Pennsylvania, and other parts of the United States. They are also used, 
 in thicknesses of one inch and above, for floors, platforms, facing of walls, mantels, and 
 for wash-tubs by plumbers. Soap-stone maybe classed under the clay stones; also, used 
 for tubs, for stoves, and for the lining of grates and furnaces. 
 
 The Ulster, or North River blue stone of this market, is a coarser slate, a very strong 
 and enduring stone; it can be quarried of varying thickness up to twelve inches, and of 
 any dimension that can be transported. It can be readily cut, hammer-dressed, axed, 
 planed, and rubbed. Is generally used for sidewalks under these various forms. It is 
 used as bond-stones in brick piers, for caps, sills, and string-courses. 
 
 THE SANDSTONES. 
 
 Sandstones, called also freestones, from the ease with which they are worked; and 
 from their colours, are very popular for the fronts of edifices. In general, they are not 
 very enduring stones, and when laid must be set parallel to their natural beds, as other- 
 wise they flake off under the influence of the weather. The sandstones are not all of the 
 same quality; those in which the cementing material is nearly pure silex, are strong, 
 enduring stones, but not those in which the cementing material is alumina, or lime. By 
 examining a fresh fracture, the character of the stone can generally be detected. A clay, 
 shining surface with sharp grains indicate a good stone; while rounded grains, a dull, 
 mealy surface, indicate a soft, perishable stone. None of the sandstones in this locality 
 are used for heavy pier or abutment work and the like, but there are sandstones in other 
 localities adapted to it. 
 
 LIMESTONE. 
 
 The coarser calcareous stones are of great variety; some are well adapted for 
 building stones, being hard and compact, while others are soft and friable. They are 
 more easily worked than granite, but are not considered as enduring. They are well 
 
174 
 
 MATERIALS. 
 
 adapted to the same class of heavy work, and the locks of the Erie and Northern Canals 
 and the dam across the Mohawk, at Cohoes, are built from limestone on the line of the 
 canals. 
 
 The finer kinds of limestones are classed under the head of marbles. They are easily 
 worked, sawed, turned, rubbed, and polished. Marble is not popular as a building 
 material, although more enduring than most sandstones, but is susceptible to the action 
 of sulphurous gases in the smoky air of cities ; and it is said that the Capitol at Wash- 
 ington, D. C., built of marble, is suffering from disintegration. But, for interior finish, 
 as tiles, wainscots, architraves, mantels, linings of walls, it is admirably adapted, and 
 from its richness, cleanliness, and variety of colour, it is very ornamental and effective. 
 
 ARTIFICIAL BUILDING MATERIAL. 
 
 The most common and useful are bricks. They are generally made of clay, with an 
 admixture of sand, well incorporated together, and mixed with water to the consistence 
 of a smooth, strong, viscous mud, pressed into moulds, dried, and burned, the best quality 
 being those in the interior of the kiln. The exteriors are light, friable bricks adapted 
 to walls supporting but little weight and not exposed to wet. The brick forming the 
 arches are very hard-burned, dark in colour, often swelled and cracked ; but, by proper 
 selection, they can be used for foot-walks. A good brick is well burned throughout; 
 when struck, it gives a ringing sound, and is of uniform shape. 
 
 Bricks vary somewhat in size and weight in different localities from 8 to 8J inches 
 long x 3 to 4 inches broad x 2 to 2 inches thick; in general, the thickness of a wall 
 with the joints is called some multiple of 4", as 8", 12", 16" . . . walls, and the num- 
 ber of bricks in such wall is estimated by multiplying the number of brick in a square 
 foot of face by 2, 3, 4, ... In some cases bricks are laid by the thousand and there are 
 custom allowances for corners, openings, and indents. The best face or front brick are 
 pressed and are of great variety of colours, but uniform in tint. Of late there has been a 
 class of face brick of which the edges are taken off by a set, leaving the face like rock- 
 faced ashlar without draft. 
 
 For variety, Pompeian face brick 1" x 4" x 12", Roman 2" x 4" x 10", faces plain 
 glazed and enamelled. Moulded brick for base, cornices, caps, sills, corners, and arches 
 are in common use. 
 
 Bricks are laid in mortar, of lime, lime and cement, or cement only all with an ad- 
 mixture of sand ; in common walls, in lime ; in walls of heavy buildings, above-ground, 
 in lime and cement; beneath, and in wet, exposed positions, in cement only. The com- 
 mon bond of the different courses of brick is by header-courses every fifth or seventh 
 course. When bricks are laid in arches they are set on edge, and turned in 4-inch rings, 
 without any bond between the different rings; or with a bond of brick lengthways, 
 
MATERIALS. 175 
 
 when two courses come on the same line, or with radial joints, or alternate 4" and 8" 
 courses of brick work in masses laid header and stretcher; however laid, the strength 
 depends on full joints in strong cement mortar. 
 
 Bricks set on edge, as in arches or in a level course, are here termed rowlocks. 
 
 Arch brick, between iron beams, to reduce the weight, are often made hollow, and 
 laid in flat arches ; that is, the joints are radial, but the upper and lower surfaces are 
 level. Hollow brick are also used for walls and partitions. 
 
 Fire-brick can be made of any size and pattern, but are usually 9 x 4^ x 2f . They 
 are used for the lining of furnaces, flues, and chimneys, exposed to the action of flame or 
 great heat. Fire-clay, with an admixture of sawdust, which is burned out in the firing, 
 leaves a light, porous, spongy mass, which can be sawed in sheets or strips, and is well 
 adapted for covering the exposed parts of iron beams and girders, and, as it admits of 
 nailing, is convenient for partitions. 
 
 Enamelled Brick. The English size is that of fire-brick the American is that of com- 
 mon brick. The brick, on the faces to be exposed, are covered with glaze of varied colors 
 and designs, and fired. They make a handsome ornamental face for walls, do not absorb 
 moisture, and can be washed. 
 
 Tile are a species of brick, with or without enamel. The latter were originally used 
 for roof-covering, but now are used in flooring walks and the like. The enameled or 
 encaustic tile are generally in squares, 4" x 4", 6" x 6", 8" x 8", but there are smaller 
 ones for tessellation, and rectangular strips for borders. They can be obtained of any 
 color or design, forming beautifully ornamented floors and wall-panels. 
 
 Terra-cotta, a kind of brick, is now largely used for exterior decoration. It is molded 
 in every variety of capitals, cornices, caps, friezes, and panels. It is a good, strong brick, 
 with all the good qualities of such a material. 
 
 Mortars. Brick are never laid dry, except in the under part of drains, to admit of the 
 removal of ground- water. Stone- work, except in rough, heavy, rubble-work, is also gen- 
 erally laid in mortar. Where cut-work is backed with rubble, the joints in the latter 
 should be as close as possible, and full of mortar, that the settling of the wall in itself 
 may not be more in the backing than in the face. Some lay the rubble dry, and fill in 
 with cement grout, or cement mortar made liquid to flow into the interstices, but the sand 
 is apt to separate and get to the bottom of the course. 
 
 By mortar, is usually understood a mixture of quicklime and sand, but mortar may 
 have-an addition of cement to the lime, or it may be cement only with sand. 
 
 Lime, or properly quicklime, is made by the calcination of limestone, shells, and sub- 
 stances composed largely of carbonate of lime, carbonic-acid gas, water of crystallization, 
 and organic coloring-matter. Quicklime, brought in contact with water, rapidly absorbs it, 
 with a great elevation of temperature, and bursting of the lime into pieces, reducing it to 
 a fine powder, of from two to three and a half times the volume of the original lime. This 
 is slaked lime. It may be slaked slowly by exposure to the air, from which it will take 
 the moisture. This is air-slaked lime. Barrels of lime exposed to rain often take fire 
 from the heat caused by slaking. The paste of slaked lime may be kept uninjured for a 
 considerable time, if protected from the air, and this may readily be done by a covering of 
 sand, and it is customary, in some places, to hold it over one season, as an improvement 
 to the uniformity of quality in the paste. But, in general, the lime is used soon after 
 slaking, and is thoroughly mixed with sand, in various proportions, generally about two 
 of sand to one of lime. The theory of the mixture is, that the lime should fill the void 
 spaces in the sand, and the space occupied by the mortar is a little in excess of that occu- 
 pied by the sand alone. 
 
 The sand should be sharp, clean, silicious grains, from one twelfth to one sixtieth of an 
 inch in diameter. Close brick -joints do not admit of as coarse sand as those of cnt stone 
 work, and, in rubble-work, sand coarser than the above can be used, and there will be 
 considerable saving of lime in using a mixture of coarse and fine sand. 
 
176 MATERIALS. 
 
 The hydraulic limes contain a small proportion of silica, alumina, and magnesia ; slake 
 with but little heat, and small increase of volume ; are more or less valuable, according 
 to the property which they have for hardening under water; but, in this particular, are 
 not equal to the hydraulic cements. 
 
 There are two great distinctions of cements natural, as quarried from the rock and 
 burnt, and artificial, like Portland, definite proportions from known materials, generally 
 preferred from their uniformity of composition ; but in this country the natural cements, 
 from old quarries and responsible makers, are satisfactory. 
 
 Cement is used in all masonry in exposed and wet situations. With a small admix- 
 ture of lime, it works better under the trowel, and for brick-work it does not sensibly 
 impair its value. Cement adds to the strength of lime-mortar. 
 
 The value of a cement depends very largely on its fineness; the residue thrown out 
 from a 2,500-per-inch mess should not exceed 10 per cent, the diameter of the wire is 
 about one half the width of the mesh. This residue is reckoned as sand. The voids in 
 the sand should be filled by the cement. To determine the amount of voids in the sand, 
 .fill a tight box of known capacity with the sand and then pour in all the water that it 
 will hold. The whole may be considered as unity, and the quantity of water the per- 
 centage of voids. In the same way the voids of gravel or broken stone may be deter- 
 mined for concrete. In mixing cement mortar, it is important that it should be thorough, 
 that the sand should be clean and damp, and that each particle should be covered with 
 cement, and if to be used for concrete or leton, that the stone or gravel should be clean 
 and damp, and their surfaces should be completely covered with the mortar. Suppose 
 the proportions be as below : 
 
 1 part cement, without voids 1.0 
 
 3 parts sand, 30 per cent voids 2.1 
 
 6 parts broken stone, 50 per cent voids 3.0 
 
 Parts in mixture 6.1 
 
 The parts of sand and broken stone to cement are larger than in common use, or, say : 
 
 Portland 1, 3, 5 
 
 Natural. 1, 2, 5 
 
 But the increase is rather as an offset to coarseness of cement and defects in mixing. As 
 a rule, from the time the water is added to the concrete, it should be kept damp for a 
 month after it is laid. 
 
 The Danish patent sand cement is sand ground Math Portland cement to a uniform 
 fineness, and then used like pure cement with great economy of construction and without 
 impairing the strength of the mortar. 
 
 Concrete is used for the base course or foundations of walls, and is formed in situ, 
 that is, depositing and ramming it in the trench where it is to be left ; or by forming in 
 moulds, in immense blocks, for docks or break- water, or in the forms of brick. 
 
 The bituminous cements are formed of natural bitumens, or artificial from coal-tar 
 mixed with various proportions of gravel and inert material. The mixture is usually 
 heated, put down in layers, and rolled or rammed. It is used for roads and sidewalks, 
 and for water-proof covering of vaults. For the covering of roofs, coarse paper, sat- 
 urated with bitumen, is put on in layers, one over the other, breaking joints, cemented 
 with the bitumen, the last coat being of bitumen, in which gravel is imbedded. For 
 an anti-damp course in a wall, or for the joints in the bricks of a wet cellar-floor, or on 
 top of a roof, hot bitumen is used as a cementing material with dry bricks. 
 
 Plastering. Coarse- stuff is nothing more than common brick-mortar, with an admix- 
 ture of bullock's hair. When time can not be given for the setting it is ganged, that is, 
 mixed with some plaster of Paris. Fine-stuff is made of pure lump-lime with an admix- 
 ture of fine sand, and perhaps plaster of Paris. Hard-finish is composed of fine-stuff and 
 
MATERIALS. 
 
 177 
 
 plaster of Paris. One-coat work is of coarse-stuff, which may be rendered, that is, put on 
 masonry, or laid on laths. Two-coat work is a coat of coarse-stuff, or scratch-coat; that 
 is, after the coat is partially dry it is scratched for a back for the fine coat. In three 
 coats, the first coat is a scratch coat, the second the brown-coat, and the third hard -finish. 
 Keene's cement, for the last finish, gives a hard surface, which admits of washing. 
 
 A single brick weighs between 4 and 5 pounds; but a cubic foot, well laid in cement, 
 with full joints, will weigh about 112 pounds. They have resisted, in an experimental 
 test, as high as 13,000 pounds to the square inch, but 12 tons should be the limit to the 
 load per square foot ; and the brick should be uniform, well burned, and closely laid in 
 cement. In lime mortar, the load should not exceed 3 tons per square foot. 
 
 The granites weigh from 160 to 180 pounds per cubic foot; the limestones from 150 
 to 175; the sandstones from 130 to 170; the slates from 160 to 180; mortar, set, about 
 100 pounds ; masonry, laid full in mortar, according to the quality of the stone and the 
 percentage of mortar, from 150 to 170 pounds. But, for practical purposes, common 
 mortar-rubble is not equal in strength to a brick wall, as it is seldom laid with equal care, 
 with joints not as well filled or the load as evenly distributed ; but cut stones will sustain 
 more, and ashlar, up to 50 tons per square foot for sound, strong stones. 
 
 METALS. 
 
 Metals are often to be shown distinctively by the draughtsman. If he can use colour, 
 he will in a measure imitate that of the material. For cast-iron, India-ink, with indigo, 
 and a slight admixture of lake ; for wrought-iron, the same colours, with stronger pre- 
 dominance of the blue; steel, in Prussian- blue ; brass, in a mixture of gamboge and 
 burnt sienna; copper, gamboge and crimson lake. In drawings where no colour is ad- 
 missible as for photographing, or to be reproduced in printing, some conventional 
 hatchings are used to represent sections of metals, but none have been so established as 
 to have a universal application. The following are submitted to represent the most com- 
 mon industrial metals: 
 
 Brass or Bronze. 
 
 Lead.' 
 
 Copper. 
 
 Steel. 
 
 Wrought Iron. 
 
 Cast Iron. 
 
 Under the term iron may be included cast-iron, wrought-iron, and steel, differing from 
 each other in the percentage of carbon contained, and in the uses to which they are applied. 
 Cast-iron contains more carbon than the others, say from two to five per cent. It can be 
 13 
 
178 
 
 MATERIALS. 
 
 cast in varied forms in moulds, but can not be welded or tempered. The usual moulds are 
 in sand or loam, in which the pattern is imbedded, and when drawn out the space is filled 
 with molten metal. The drawing of patterns for molding involves a knowledge of the art 
 of founding. The shrinkage of the metal, usually about one per cent, for which provision 
 must be made in increased size of pattern, is provided for by the pattern-maker, the 
 draughtsman giving finished sizes, but the draughtsman must know whether the pattern 
 can be drawn from the sand, and by what system of cores voids can be left ; or it may 
 often happen that castings, designed as a whole, will have to be made in a number of 
 pieces, involving flanges and bolts. In cooling, the shrinkage takes place the soonest in 
 the thinnest parts, and, if great care be not taken by the molder in exposing the thicker 
 parts to the air first, the parts will shrink unequally, and there will be a strain induced 
 which will materially weaken the casting, and it may even break in the mold. The 
 draughtsman, in his design, should make the parts of as uniform thickness as possible. 
 
 Castings cool from the outside inward, in annular crystals perpendicular to the face, 
 as in Figs. 339 and 340. Now, if the casting consist of a right angle (Fig. 341), there will 
 evidently be a weak place along the line A B, but, if the angle be eased by a curve, the 
 
 FIG. 339. 
 
 FIG. 340. 
 
 FIG. 341. 
 
 Fiq. 342. 
 
 crystallization takes place as in Fig. 342, and the line of weakness is avoided. This is 
 effected by a very small easement of the angle, and a cove is almost invariably introduced. 
 In castings, in almost all metals, the same effects result from cooling, and therefore the 
 changes of direction should not be abrupt. 
 
 "When castings are ordered for important structures, iron of certain tensile strength is 
 called for, and specimens of the metal, in small rectangular bars, are required, cast at the 
 same time and under as nearly the same conditions as the casting which may be subjected 
 to test. 
 
 If the casting be made in dry sand, it cools slowly, and the surface is comparatively 
 soft ; if in greensand sand somewhat moist the surface becomes harder ; but if cast 
 on an iron plate, or chill, some irons become as hard as the hardest steel, useful in 
 surfaces exposed to heavy wear, as the treads of railway-wheels. Cast-iron, in general, 
 is brittle under the blows of a hammer, but some mixtures, under a process of annealing, 
 become malleable iron, used largely for steam-fittings, parts of agricultural machines, forms 
 requiring the toughness of wrought-iron, but difficult to forge. 
 
 Wrought-iron is produced from cast-iron by removing the carbon and impurities by 
 puddling, squeezing, heating, and rolling. As a material, it is sold in all sizes of wire, 
 rods, shafts, bars, plates, shapes girders and beams, chains and anchors. Its applica- 
 tion industrially is well known. When hot, it can be welded, forged, drawn, and swaged 
 into almost any required shape. Under the *team-hammer, the largest shafts, anchors, and 
 cranks can be built, or by hand or by machinery it can be wrought into tacks, nuts, bolts, 
 nails, or drawn into the finest wire. 
 
 For shafts of mills it is generally turned in a lathe and polished, but of late it can be 
 bought, up to six inches diameter, cold-rolled, which adds very considerably to the strength, 
 and is ready for use. 
 
 Bessemer and Siemens-Martin metals are made by burning out the carbon from a 
 melted iron, and then reintroducing a known quantity, say from 0'03 to 0'6 per cent of car- 
 bon. There are other patents covering somewhat different irons, but the above are the best 
 known. All are commonly classed as steel, but by many are called homogeneous metal ; 
 
MATERIALS. 
 
 first-class iron, of very uniform texture and great strength, but not equal to that of the best 
 steel. 
 
 Steel is produced from pure wrought-iron by what is called cementation heating the 
 bars in contact with charcoal, by which a certain amount of carbon is taken up. The bars, 
 when taken out, are covered with blisters, apparently from the expansion of minute bub- 
 bles within ; hence called Mistered steel. From this shear-steel can be produced by piling, 
 heating, and hammering, or cast-steel from melting in a crucible. 
 
 Steel, when broken, does not show the fibrous character of wrought-iron. The frac- 
 ture of shear-steel is fine, with a crystalline appearance. The fracture of cast-steel is very 
 fine, requiring very close inspection to show the crystals or granulations; its appearance 
 is that of a fine, light, slaty-gray tint, almost without luster. Steel is stronger than any of 
 the other iron products, and especially applicable for the piston-rods of steam-engines, and 
 positions requiring great strength and stiffness, with the minimum of space. But it is the 
 way in which steel can be hardened and tempered which adapts it to its peculiar appli- 
 cations. 
 
 When the malleable metals are hammered or rolled, they generally increase in hard- 
 ness, elasticity, and denseness, and some kinds of steel springs are made by the process 
 of hammer hardening ; but the usual process of hardening and tempering is by heating 
 the steel to a degree required by the use to which it is to be applied, and cooling it 
 more or less suddenly by immersing in water or oil. The greater the difference between 
 the heated steel and the cooling medium, the greater the hardness, but too much heat 
 may burn the steel, and too sudden cooling make it too brittle. Steel, in tempering, is 
 heated from 430 Fahr. to 630. The temperature is shown by the color from a pale 
 yellow to deeper yellow, light purple to a dark purple, dark blue to a light blue, with a 
 greenish tinge. 
 
 Steel is used for the edges of all cutting-tools, faces of hammers and anvils, and is gen- 
 erally welded to bodies of wrought-iron, but often composing the entire tool ; for saws, 
 springs, railway tires, pins, and can be bought in the form of wire, rods, bars, sheets, and 
 plates, in varied forgings and castings. 
 
 All irons are very liable to rust, and must be protected where exposed to moisture. 
 Polished surfaces are kept wiped and oiled, others painted, others galvanized or plated 
 with some less oxidizable metal, generally tin, zinc, or nickel. Of late, a process has 
 been introduced of coating them with black oxide, but is yet of no general application. 
 
 Antimony, bismuth, copper, lead, tin, and zinc, are used more or less industrially, and 
 alloys of them are extremely useful. They may be hardened somewhat by the process 
 of rolling and hammering, but can not be welded. Joinings are made by soldering or 
 brazing or burning that is, melting together. 
 
 Antimony expands by cooling. With tin, in equal proportions, it makes speculum- 
 metal, and is used, with lead, to make type. Type metal makes a very good bearing for 
 shafts and axles. 
 
 Bismuth is chiefly used as a constituent of fusible metal : 3 bismuth, 5 lead, and 3 tin, 
 is an alloy which melts at 212. Other mixtures are made, increasing the melting-point 
 to adapt the metal for fusible plugs in boilers, or lowering the melting-point, so that, in 
 case of fire in a building, a heat of say 140 melts the joint made by the metal, and lets 
 water through sprinklers, to automatically put out the fire. 
 
 Copper is very malleable and ductile. In sheets, it is used for the cover of roofs, gut- 
 ters, leaders, lining of bath-tubs, kettles, stills, and kitchen utensils. It is worked more 
 easily than iron, and is stronger than lead or zinc, but it is much more costly than either 
 of these metals, and its oxide is so poisonous that, without great care and cleaning, it can 
 not be used to transmit or contain anything that may be used as food, without a cover of 
 tin. It oxidizes slowly, and is used extensively for ships' fastenings and for bottom-sheath- 
 ing. It is the most important element in all the brass and bronze alloys. 
 
180 MATERIALS. 
 
 Brass, in common use, covers most of the copper alloys, no matter what the other 
 components are, whether zinc, tin, or lead, or all three. 
 
 Copper and zinc will mix in almost any proportions, The ordinary range of good 
 yellow brass is from 4^ to 9 ounces of zinc to the pound of copper. With more zinc it 
 becomes more crystalline in its structure, but, as zinc is very much cheaper than copper, 
 the founder is apt to increase the percentage of zinc, with the addition of a small per- 
 centage of lead. Muntz metal, in its best proportion, contains lOf ounces of 'zinc to the 
 pound of copper. 
 
 Copper and tin mix in almost any proportion. The composition of ancient bronzes is 
 from 1 to 3 ounces of tin to the pound of copper. Ten parts of tin to 90 of copper is the 
 usual mixture for field-pieces, and this is used in steam-engine work, often under the 
 name of composition. Bell-metal is from 4 to 5 ounces of tin to the pound of copper ; Bab- 
 bit-metal, for journal-boxes, 90 of tin to 10 of copper. 
 
 Copper and lead mix in any proportion up to nearly one half lead, when they separate 
 in cooling. 
 
 An addition of from one quarter to one half ounce of tin to the pound of yellow brass 
 renders it sensibly harder. A quarter to one half ounce of lead makes it more malleable. 
 
 German-silver is 50 copper, 25 zinc, and 25 nickel. 
 
 Holzapfel gives the following alloys : 
 
 1^ ounce tin, ounce zinc, to 16 ounces copper, for works requiring great tenacity. 
 
 1^ to If ounces tin, 2 ounces brass, to 16 ounces copper, for cut wheels. 
 
 2 ounces tin, 1^ ounce brass, to 16 ounces copper, for turning-work. 
 
 2J ounces tin, 1-j- ounce brass, to 16 ounces copper, for coarse-threaded nuts and bearings. 
 
 2J ounces tin, 2 ounces zinc, to 16 ounces copper, Sir F. Chantry's mixture, from which 
 a razor was made, nearly as hard as tempered steel. 
 
 Professor R. H. Thurston, at the Stevens Technological Institute, tested various 
 alloys of copper, tin, and zinc, and, by a graphic method (Fig. 343), exhibits the re- 
 sults, enabling others to judge the probable strength of other mixtures. The apices of 
 the triangle marked copper, tin, and zinc, represent the points of pure metal, 100 per 
 cent. The lines opposite the apex of any metal represent the of such metal thus the 
 base opposite copper represents an alloy of tin and zinc only, without any copper, and 
 every line drawn above this base, and parallel to it, will contain a percentage of copper 
 increasing by regular scale, from th'e base to the apex, and so with lines opposite tin and 
 zinc ; the first contains only copper and zinc, the latter tin and copper, and the percent- 
 ages of tin and zinc increase with the distance from their opposite lines to their vertices. 
 The intersections of these percentage parallels define the percentages of each metal, their 
 sum always making 100 per cent. If the strength of such alloy, as obtained by test, rep- 
 resent an ordinate or elevation, on any convenient scale, at its opposite intersection of 
 percentage, a contour map, as in the figure, may be drawn and a model made from it. 
 The summit, 65,000 on the figure, represents the position of the strongest alloy found: if 
 through the scales marked copper on each side, we find the parallel to the base, which 
 passes through this summit, it will be found to be about 55 that is, 55 per cent copper. 
 In like manner, the parallel to the o zinc base, intersecting this summit, will be about 43 
 per cent zinc ; and, in the same way, tin is 2 per cent. 
 
 To find the probable strength of any mixture, it is only necessary to find the contour 
 intersected by the triple parallels representing the percentages. It is said probable 
 strength, because the care and manipulation of the founder are such important factors in 
 the result. 
 
 Aluminum is about two and a half times the specific gravity of water. With the re- 
 duction in the cost of manufacture, its uses have become more extended, and its alloys 
 now occupy a high position in practical industry. With a few per cent of silver, 
 titanium or copper aluminum wire can be made of a tensile strength of 80,000 pounds to 
 the square inch, and an electrical conductivity, weight for weight with copper wire, of 
 
MATERIALS. 
 
 181 
 
 170 to 100. A few per cent of aluminum added to other metals gives valuable proper- 
 ties; with 8 to 12 per cent of copper, it makes one of the finest and strongest metals 
 
 ff t 'o' vt 'M>r 
 
 i 06 
 
 FIG. 343. 
 
 known; a 10-per-cent aluminum bronze can be made to fill a specification of 180,000 
 pounds tensile strength and 5 per cent elongation in 8 inches ; and in the proportion of 
 one third to three fourths of a pound to a ton of steel prevents blow holes and unsound 
 tops of ingots. 
 
 There are various other alloys, as phosphate bronze, aluminum bronze, Sterro-metal, 
 of which the strength are given hereafter in the appendix. 
 
 Lead is a very soft metal, that can be readily rolled into sheets and drawn into pipes, 
 and is so flexible that it can be readily fitted in almost any position. It is, therefore, 
 especially adapted to the use of plumbers, for the lining of cisterns and tanks, and for pipes 
 for the conveyance of water and waste. For pipes for conveying pure water for drinking 
 purposes, or for cisterns containing it, it is objectionable, as it oxidizes, and the oxide is a 
 dangerous and a cumulative poison, but, in common waters which are more or less hard, 
 the insides of the pipes become covered with a deposit which protects them. It is well, 
 before drinking from a lead pipe in which the water has stood for a time, to draw off all 
 the water, and, in lead-lined cisterns exposed more or less to the air, to protect them by a 
 coating of asphalt varnish. Lead expands readily, and has so little tenacity that, in many 
 positions, if heated, it has not strength in cooling to bring it back to its original position. 
 It remains in wrinkles on roofs, and, for pipes conveying hot water, unless continuously 
 supported, it will hang down in loops, continuously increasing under variations of tern- 
 
182 MATERIALS. 
 
 perature, to rupture. But it makes a very good plating for sheet-iron for roofs, and its 
 oxides are the most valuable of all pigments. 
 
 Tin, in a pure state, is used for domestic utensils, as block-tin, and has also been used 
 for pipes in the conveyance of water by parties who feared the poisonous qualities of lead 
 pipe. But its chief use is for the covering of sheet-iron, which is sold under the name 
 of tin or tin-plate, and is of universal application for architectural, industrial, and do- 
 mestic purposes. It is so little affected by air and moisture that roofs, in many places, 
 covered with it, need no painting, and oxidization takes place in the iron beneath only 
 from deficiency in the plating, or from the abrasion or breaks in it. 
 
 Zinc, in the pure form of spelter, is crystalline and brittle, but at a temperature be- 
 tween 210 and 300 it is so ductile and malleable that it can be readily rolled into 
 sheets, and of late has been used as a cheap substitute for sheet-copper ; but, under con- 
 siderable variations of temperature, as for lining of bath-tubs, it takes permanent 
 wrinkles, and, for coverings of roofs, suitable provision must be made for its expansion. 
 But as a plating of iron, under the name of galvanizing, it affords an admirable protec- 
 tion, cheaply, and extends the use of iron in sheets, bolts, and castings, where it would 
 not otherwise be applicable. Zinc, as a pigment, does not discolour, like lead, under 
 the action of sulphuretted hydrogen, but it is objected toby painters for its want of body 
 or cover. 
 
 Sulphur, when used in sufficiently large masses as to show on a drawing, may be rep- 
 resented by a reddish-yellow tint, or some distinctive hatching. It melts at 248 Fahr., 
 and, from its fluidity, answers admirably for the filling of joints between stones, beneath 
 the base plates of iron columns, between wood and stone, and around anchor-bolts in stone, 
 forming, when cold, a strong, uniform bearing, and adapting itself to the roughness of 
 the material, and is detached with difficulty. It is used largely for the bases of engines, 
 and for the joints of the cap-stones of dams. On the dam across the Mohawk, at Cohoes, 
 N. Y., many tons were used in these joints, the depth of sulphur being about 6 inches, and 
 now, after about twenty years' use, it has been renewed, and there has been no injurious 
 effect from the sulphur on the limestone, of which the apron or capping is composed. 
 It is better for most of the above purposes than lead, being cheaper, more fluid when 
 molten, shrinks less in cooling, is less affected by temperature, does not creep under 
 pressure, and its crushing strength is adequate to any of the positions of use above, but 
 it is brittle under blows. It sometimes rusts the bolts or iron with which it is brought 
 in contact, but this is prevented by an addition of about 20 per cent of coal tar. This 
 mixture is used as a cement to fasten lights in illuminating tile and vault covers. 
 
 When heated to about 300, sulphur begins to grow viscid, and at 428 it has the 
 consistence of thick molasses. Above this, it begins to grow thin again. Heated to 
 518, and thrown into cold water, it becomes for a time plastic, and is used for taking 
 moulds or casts. 
 
 Sulphur, in powder, mixed in proportions of one sal-ammoniac, two sulphur, and 
 fifty ,of iron-filings, makes a mastic which is used for calking the joints of iron pipes, 
 especially gas- pipes. The joint is called a rust-joint. 
 
 Pure silver is too soft for general purposes ; it alloys readily with lead, zinc, bismuth, 
 gold, and copper. The last is the most important in an industrial point of view in its 
 use for plate, coin, and ornaments, which invariably contain a certain amount of copper. 
 
 Silver is used for plating by the electro-plate process, or by fire plating, in which the 
 sheet is soldered or sweated to some other metal, as iron, German silver, composition, 
 copper, etc. 
 
 Gold is used industrially for chemical laboratories ; sometimes as plate or for orna- 
 ment, but mostly for gilding the baser metals by electro-plating or by fire gilt. As gold 
 leaf, it is very largely employed for ornamental purposes ; leaf can be beaten beyond any 
 requirements in the arts ; a single grain of gold was spread to the extent of 75 square 
 inches, and the same weight of silver to 98 square inches. Platinum is used largely for 
 
MATERIALS. 
 
 183 
 
 various apparatus in chemical laboratories, as it withstands high temperatures and is proof 
 against a large number of chemicals. It is to be had in wire sheets, crucible and dishes, 
 and in stiles of large capacity for the concentration 
 of sulphuric acid. 
 
 Glass, in drawing, is represented by a bluish tint 
 or by different shades or hatchings, expressive of the 
 effect of light upon it, whether the light is reflected 
 or transmitted. 
 
 Fig. 344 represents a portion of a mirror when the 
 light is reflected. The exterior of windows is often 
 represented in the same way, but with deeper shades, 
 and often with a piece of curtain behind in white 
 with dim outline. A window viewed from inside 
 is represented in shades less than in the figure, or as 
 transparent, which is conveyed by the dimness 6f 
 outline of figures or skies seen beyond. 
 
 Fig. 345 represents a glass flask. # 10 . 344. 
 
 Fig. 346 represents a glass box with glass sides. 
 
 Fig. 347 represents a glass jar containing fluids of different densities. 
 
 Figs. 348 and 349 represent spars, which may be taken for any transparent sub- 
 stances, as glass, ice, and the like. 
 
 Common window-glass is blown in the form of cylinders (hence called cylinder- 
 glass), flatted out, and cut in lights of varying dimensions, from 6x8 up to 30 x 30 
 
 FIG. 345. 
 
 FIG. 346. 
 
 FIG. 347. 
 
 inches, and put up in boxes containing about fifty 
 square feet. It is classed as single-thick (bout ^ 
 inch) and double-thick (i inch). When the squares 
 are large, or used for sky-lights, they should be the 
 latter. Plate-glass polished plate is used for win- 
 dows of stores and first-class buildings. It can be got of almost any dimensions, and of 
 a thickness from -^ to of an inch. Rough plate is largely used for floor-lights and 
 sky-lights. It is cut to required sizes, and of a thickness from f to one inch. 
 
 Single thick cylinder-glass cuts off from about 8 to 15 per cent of the light. 
 
 Double-cylinder, from 12 to 20 per cent of the light. 
 
 Polished plate, three sixteenths inch thick, from 5 to 7 per cent of the light. 
 
 Rough plate, one half inch thick, from 20 to 30 per cent of the light. 
 
 Rough plate, one inch thick, from 30 to 40 per cent of the light. 
 
 Ribbed glass, one eighth inch thick, known as "factory glass," gives by diffusion a 
 more effective illumination than clear, plain glass or the crystal-ribbed glass when either 
 is screened with shades. On any exposure but the southern, the crystal-ribbed glass 
 may be used without window-shades. 
 
184 MATERIALS. 
 
 This is when the glass is clean ; but there is always a film of moisture on its surface, 
 which attracts dust, and impairs very much the transmitted light. Rough plate more 
 readily retains the dirt, and, when it is used as floor-lights, becomes scratched. It is 
 therefore usual, in the better class of buildings, to use a cast white glass, set in iron 
 frames. In outer, or platform lights, these lights are in the form of lenses, set in cast- 
 iron frames with an asphalt putty, or resting on iron frames and imbedded in Portland 
 cement. 
 
 FIG. 348. FIG. 349. 
 
 Rubber, mixed and ground with sulphur, subjected to heat, becomes vulcanized, and 
 is not affected by moderate variations in temperature. Soft rubber, most extensively 
 used for industrial purposes, is subjected to a heat of from 265 to 300, aud for a time 
 can withstand a temperature a little below this without losing its elasticity ; after a time 
 it will harden. Soft rubber is classed as pure rubber, and fibrous rubber, or rubber with 
 cloth. Pure rubber contains about fifty per cent of rubber and fifty per cent of com- 
 pound, white lead and sulphur. It is used for the buffers and springs of railway- 
 carriages, and for the faces of valves and seats of water-pumps, but it is not well suited 
 for the pumping of hot water, especially above 212, as it is liable to lose its elasticity ; 
 and, although some valves may stand a considerable time, it is almost impossible to 
 secure uniformity in the rubber. Fibrous rubber rubber ground with cotton or other 
 fibre, or spread on cloth, on more or less thicknesses is used for the packing of faced 
 joints of pipes and gaskets for water or steam. It makes a stanch joint, and, even when 
 hardened under heat, it still preserves it. Rubber cloth is also used for belting and 
 hose-pipes. When used for the conveyance of steam, the inner coat is the first affected, 
 and it may be some time before the whole pipe suffers. In buying rubber, explain the 
 purpose to which it is to be applied, and depend on the guarantee of the vender. Rubber 
 is often to be designated by the draughtsman, which it may be by a bluish-black tint, or 
 by lines across it parallel to its length. 
 
 Paints are used for a twofold purpose for covering and preserving the material to 
 which they are applied, and for ornamentation. The best and the most general is white- 
 lead ground with linseed-oil, either used by itself or mixed with various other pigments, 
 as ochre, chrome, lamp-black, etc. It is often adulterated with barytes. For the cover- 
 ing of iron, or for the packing of close joints in it, nothing is better than pure red-lead, 
 but many of the oxides of iron, red or yellow, form good covers of iron, and, as cheap 
 and good paints, are used on tin roofs. All the leads and pigments are ground in oil: 
 if the oil is raw, it dries slowly; driers, as litharge, are added to hurry the process, but, 
 with boiled oil, no drier is necessary. Almost any inert substance, as cement, chalk, or 
 sand, if fine enough, can be ground with oil for a paint, and make a good cover, and for 
 these fish-oil will answer. The general specification for painting is "paint with good 
 coats of white-lead, of sucli colour as may be directed." The priming-coat of new wood- 
 work requires more oil than paint. For the next coats, one-half pound of paint to the 
 square yard would be considered a good coat. If the paint is on old work, or that 
 which has been already painted, there will be a little less lead required. Wood should 
 be fairly dry before the application of paint, so that it may properly adhere and not in- 
 
MATERIALS. 
 
 185 
 
 close moisture that may rot the wood. The knots should bc.Mlled, that is, covered with 
 shellac varnish or similar preparation, to prevent the exuding of the resin. The heads 
 of nails should be sunk, and the holes and cracks filled with putty, and the surface of 
 the wood smoothed. 
 
 Coals and other minerals are represented like rocks or stones, in varied shades of 
 tones and colours. Fig. 350 represents the fire-box of a locomotive, with coal in the 
 state of ignition in its usual type. In colour, flame is represented in streaks of red- 
 yellow, with dark tints for smoke. Water occupies the lower half of the boiler; but, as 
 
 FIG. 350. 
 
 FIG. &>L. 
 
 steam under pressure is invisible like gas, the space occupied by it is shown as empty. 
 If the direction of its movement is desired, it is indicated by arrows. Steam issuing 
 into atmosphere, or boiling in an open kettle, has the appearance of a very light smoke 
 or cloud (Fig. 351). 
 
 There are many substances used in such masses in construction, or to be shown in the 
 processes of manufacture, that must be graphically represented by the draughtsman by a 
 general imitation of their natural appearance, or conventionally with explanatory mar- 
 ginal blocks and legends. 
 
MECHANICS. 
 
 THE draughtsman in designing a structure should be conversant not only with the 
 nature of the material, but also with the forces to which it is to be subjected their 
 magnitude, direction, and points of application, and their effects; that is, he should 
 know the first principles of mechanics, the science of rest, motion, and force to wit, 
 Statics, Dynamics, and Kinematics. Statics treats of balanced forces, or rest ; dynamics, 
 of unbalanced forces, where motion ensues; and kinematics, of the comparison of mo- 
 tions with each other. Considering statical forces simply in the abstract, the bodies to 
 which they are applied are assumed to be perfectly rigid, without breaking, binding, 
 twisting, or in any wise changing upon the application of such forces. 
 
 Force is a cause tending to change the condition of a body as to rest or motion. Force 
 is measured by weight. In England and the United States the unit of force is the pound ; 
 on the Continent, the gramme. All bodies fall, or tend to fall, to the earth. This force 
 is called the attraction of gravitation. Its direction is that of a string from which a 
 weight is suspended (Fig. 352). It is called a vertical line, and its di- 
 rection is toward the centre of the earth. Practically, all such lines 
 are considered parallels. Let a mass, P (Fig. 353), be suspended by a 
 cord. Each particle is acted on by gravity, and the resultant of all 
 these parallel forces is the force resisted by the cord, or the entire 
 weight of the body. If a mass (Fig. 354) be suspended from two dif- 
 ferent points, P and Q, the directions of the string will meet at a point 
 C, which is called the centre of gravity, where all the weight may be 
 considered to be concentrated. When a body of uniform density has 
 
 rHr- 
 
 Fio. 352. 
 
 FIG. 353. 
 
 Fm. 354. 
 
 FIG. 355. 
 
 a centre of symmetry (a point which bisects all straight lines drawn through it), this 
 point coincides with the centre of gravity. The middle of a straight line, the centre of 
 a circle, the intersection of the diagonals of a parallelogram, the intersection of lines 
 drawn from any two angles of a triangle to the centres of the opposite sides, are the cen- 
 tres of symmetry ; in solids, the centre of a sphere, the middle point of the axis of a 
 cylinder, and the intersection of the diagonals of a parallelepiped. 
 
 The centre of gravity of the triangular pyramid (Fig. 355) is in the straight line A E, 
 connecting the apex A with the centre of symmetry of the base triangle BCD, and dis- 
 tant J of the length of the line A E from E. 
 
 186 
 
MECHANICS. 
 
 187 
 
 The centre of gravity of solids which may be divided into symmetrical figures and 
 pyramids, as for all practical purposes most may be, can be found by determining the 
 centre of gravity of each of the solids of which it is composed, and then compounding 
 them. The centre of gravity of bodies inclosed by more or less regular contours, as a 
 ship, for instance, is determined by dividing it into parallel and equidistant sections, 
 finding the centre of gravity of each, and compounding them. 
 
 The centre of gravity of a body may be determined practically, as shown above, by 
 its suspension from different points. It can be done generally more readily by balancing 
 the body in horizontal positions on different lines of support ; the centre of gravity will 
 lie in the intersection of planes perpendicular to these lines. A body, unless the vertical 
 line from the centre of gravity falls within the base of support (Fig. 356), will fall over 
 (Fig. 357). A person carrying a weight insensibly throws a portion of the body forward, 
 backward, or laterally, to balance the load. Thus, in Fig. 358, the body is thrown 
 
 FIG. 356. 
 
 FIG. 357. 
 
 FIG. 358. 
 
 back, so that the vertical from the centre of gravity g, compounded of the centre of 
 gravity G of the woman and H of the load, falls within the base of the feet. 
 
 When a figure rests in such a position that its centre of gravity is in its lowest posi- 
 tion, it is said to be in stable equilibrium. A ball may rest in any position, as the centre 
 of gravity is neither depressed nor raised by movement ; but in the toy (Fig. 359) any 
 movement tends to raise the centre of gravity, and, on the cessation of the force, the 
 body returns to its original position. The ellipsoidal form (Fig. 360), placed on its 
 pointed end, is balanced, but the slightest movement lowers the centre of gravity, and, 
 without the application of an outside force, it can not be raised, and therefore falls. 
 
 FIG. 359. 
 
 FIG. 360. 
 
 FIG. 361. 
 
 This is called unstable equilibrium, while in the position shown in Fig. 361 it is in stable 
 equilibrium. In the toy (Fig. 362) the body of the figure is light, and the weight of the 
 balls brings the centre below the point of support. This will admit of great oscillation, 
 and return to its original position. 
 
 When two parallel forces, F F', are applied at the extremities of a straight line (Fig. 
 363), they have a resultant, R, equal to their sum, and acting at a point, C, which 
 divides the line inversely in proportion to the forces. If the forces are equal, the point 
 C will be at the centre of the line; if the force F is double that of F ; , C A will be equal 
 to one half C B. This is called the principle of the lever. 
 
188 
 
 MECHANICS. 
 
 Levers, in practice, are called of the first (Fig. 864), the second (Fig. 365), and the 
 third class (Fig. 366), according to the positions of the three forces, the weight, W, the 
 power applied, P, and the fulcrum, or support or turning-point, F, of the lever. The 
 two extreme forces must always act in the same direction ; the 
 middle one must act in the opposite direction, and be equal to 
 the sum of the other two ; and the magnitude of the extreme 
 forces is inversely proportional to their distances from the mid- 
 dle one. Let the middle force, c, be measured by a spring-bal- 
 ance (Fig. 367), it will mark the sum of the weights a and b. 
 
 FIG. 362. 
 
 F 
 
 FIG 
 
 FIG. 364. 
 
 Call the distance from a to c, x, and from & to c, y, then the weight a will be to the 
 weight & as y is to x, or a x = b y. Suppose the weight a to be 6 pounds and at J 3 
 pounds, at c it will be 9 pounds, and x will be to y as 6 to 3, or, if the lever is 48 inches, 
 5 c will be 16 inches and ac 32 inches. 
 
 W F 
 
 FIG. 367. 
 
 FIG. 366. 
 
 FIG. 368. 
 
 To find graphically the fulcrum, or point, at which a lever should be supported to 
 sustain in equilibrium weights, or equivalent forces, acting at the extremities of the 
 lever. Let A B (Fig. 368) be the lever. At A and B let fall and erect perpendiculars to 
 the lever. Lay off from A, on any convenient scale, A B', corresponding to the weight 
 applied at B ; and at B, on the same scale, B A', the weight applied at A ; draw the line 
 A' B' ; its intersection, F, with the lever will be the position of the fulcrum. This is on 
 the hypothesis that there is no weight to the lever, or that, after determining the posi- 
 
MECHANICS. 
 
 189 
 
 F 
 
 c' 
 
 FIG. 369. 
 
 tion of the fulcrum, the lever itself is balanced on the point by the addition of weight 
 on the short arm F A, or the reduction of weight on the long one F B. If the lever is 
 of uniform weight, on perpendiculars to C, the centre 
 of the lever (Fig. 369), and to F, the fulcrum, as before 
 determined, lay off F C', the weight of the lever, and 
 C F', the sum of the weights applied at A and B ; draw 
 C' F'. Its intersection, F", will be the actual fulcrum, 
 taking into consideration the weight of the lever in ad- 
 dition to the weights suspended at the extremities. 
 
 The Wheel and Axle. If a weight, P, be suspended 
 from the periphery of a wheel (Fig. 370). while another 
 weight, "W, is suspended on the opposite side of a bar- 
 rel or axle attached to the wheel, the principle of ac- 
 tion is the same as that of the lever. P multiplied by 
 its length of lever, the radius ca of the wheel, is equal 
 to W multiplied by its length of lever, the radius of the " 
 axle c&; the axle c is the fulcrum. If a movement 
 downward be communicated to P, as shown by the 
 dotted line, a rotary motion is given to the wheel and 
 axle ; the, cord of P is unwound while that of W is 
 wound up, while P is still suspended from a and W from 5; the leverage, or distance 
 from the fulcrum, of each is the same as atx first. The wheel and axle is a lever of con- 
 tinuous action. Since the wheel has a larger circumference than the axle, by their revo- 
 lution more cord will be unwound from the former than is wound up on the latter; P 
 
 will descend faster than W is raised in the proportion of 
 the circumference of the wheel to that of the axle, or of 
 their radii that is, as c a to c b. When P has reached the 
 position P', W will have reached W. If c a be four times 
 c J, then P will have moved four times the distance that 
 W has. The movement is directly as the length of the 
 levers, or the radii of the points of suspension. Therefore, 
 
 \ to move a large weight by the means of a smaller one, the 
 
 smaller must move through the most space, and that the 
 spaces described are as the opposite ends of the lever, or 
 inversely as the weights. 
 
 It is the fundamental principle of the action of all me- 
 chanical powers, that whatever is gained in power is lost in 
 space travelled ; that, if a weight is to be raised a certain 
 number of feet, the force exerted to do this must always 
 be equal to the product of the weight by the height to 
 which it is to be raised; thus, if 200 pounds are to be 
 raised 50 feet, the force exerted to do this must be equal 
 to a weight which, if multiplied by its fall, will be equal 
 to the product 200x50, or 10,000; this force may be a 
 weight of 10,000 pounds falling 1 foot, or of 1 pound fall- 
 ing 10,000 feet. 
 
 It is now common to refer all forces exerted to a unit of 
 pounds-feet (see p. 72, Fig. 181), that is, 1 pound falling 1 
 foot ; and the effect to the same unit of pounds-feet, 1 pound 
 raised 1 foot. Thus, in the example above, the force exerted or power is 10,000 pounds- 
 feet falling; the effect 10,000 pounds-feet raised. In practice, the pounds-feet of force 
 exerted must always be more than the pounds-feet of effect produced; there must be some 
 excess of the former to produce movement and to overcome resistance and friction of parts. 
 
 FIG. 370. 
 
190 
 
 MECHANICS. 
 
 The measure of any force, as represented by falling weight, is termed the absolute power 
 of that force ; the resulting force, or useful effect for the purposes for which it is applied, 
 is called the effective power. 
 
 The Pulley. The single fixed pulley (Fig. 371) consists of a single grooved wheel 
 movable on a pin or axis, the strap through which the pin passes being attached to some 
 fixed object. A rope passes over the wheel in the groove ; on one side the force is 
 exerted, and on the other the weight is attached and raised. It may be considered a 
 wheel and axle of equal diameters, or as a lever in which the two sides are equal, the pin 
 being the fulcrum. P, the force exerted, must therefore be equal to the weight W, raised ; 
 and, if movement takes place, W will rise as much as P descends. 
 
 The fixed pulley is used for its convenience in the application of the force ; it may be 
 easier to pull down than up, for instance ; but the pounds-feet of force must be equal 
 to the pounds-feet of effect. The tension on the rope is equal to either the force or 
 weight. 
 
 Fig. 372 is a combination of a fixed pulley, A, and a movable pulley, B. The sim- 
 plest way to arrive at the principle of this combination is to consider its action. Let P 
 be pulled down, say, two feet ; the length of rope drawn to this side of the pulley must 
 be furnished from the opposite side. On that side there is a loop, in which 
 the movable pulley, with the weight W attached, is suspended. Each side 
 of this loop, 2 and 3, must go to make up the two feet for the side or end 
 1. Cords 2 and 3 will therefore furnish each one foot. 
 As these cords are shortened one foot, the w/eight W is 
 
 FIG. 371. 
 
 FIG. 374. 
 
 raised one foot, and, as the movement of W is but one foot for the two feet of P, W 
 must be twice P. 
 
 In the combination of pulleys (Fig. 373), let P be pulled, say, three feet; then this 
 length of rope, drawn from the opposite side of the pulley, is distributed over the three 
 cords, 2, 3, 4, and the weight W is raised one foot ; consequently, W is three times P. 
 The cord 1 supports P, the cords 2, 3, 4, the weight W, or three times P; consequently, 
 the tension on every cord is alike. The same rope passing freely around pulleys must 
 have the same tension throughout ; so that, to determine the relation of W to P, count 
 the number of cords which sustain the weight. Thus, in Fig. 374, the weight is sus- 
 tained by four cords ; consequently, it is four times the tension of the cord, or four times 
 the force P. In order not to confuse the cords, the pulleys are represented as in the 
 figures; but, in construction, the pulleys, or sheaves, are usually of the same diameter, 
 and when connected, as A and B, and C and D, they run on the same pin. 
 
 The Inclined Plane. To support a weight by means of a single fixed pulley, the force 
 must be equal to the weight. Suppose the weight, instead of hanging freely, to rest 
 upon an inclined plane & d (Fig. 375) ; if motion ensue, to raise the weight W the height 
 a &, the rope transferred from the weight side of the pulley will be equal to b d, and P 
 will have, consequently, fallen this amount ; thus, if & d be six feet, and a ft one foot, 
 
MECHANICS. 
 
 191 
 
 while "W is raised one foot, P has descended six feet ; and, as pounds-feet of power must 
 equal pounds-feet of effect, P will be one sixth of W ; thus, P is to W as a ft is to & d, or 
 as the height of the incline is to its length. If the end of the plane d be raised, till it 
 becomes horizontal, the whole weight would rest on the plane, and no force would be 
 necessary at P to keep it in position ; if the plane be revolved on &, till it becomes per- 
 
 FIG. 375. 
 
 FIG. 376. 
 
 pendicular, then the weight is not supported by the plane at all, but it is wholly depend- 
 ent on the force P, and is equal to it. Between the limits, therefore, of a level and a 
 perpendicular plane, to support a given weight W, the force P varies from nothing to 
 an equality with the weight. 
 
 The construction (Fig. 376) illustrates the principle of the wedge, which is but a 
 movable inclined plane; if the wedge be drawn -forward by the weight P, and the weight 
 W be kept from sliding laterally, the fall 'of P a distance equal to a d will raise the 
 weight W a height c 6. P will therefore be to "W as c & is to a d. For example, if the 
 length of the wedge a d be ten feet, and the back c I two feet, then P will be to W as 
 two to ten, or one fifth of it. 
 
 Let the inclined plane a b d (Fig. 376) be bent round, and attached to the drum A 
 (Fig. 377), to which motion of revolution on its axis is given, by the unwinding of the 
 turns of a cord from around its periphery, through the action of a weight P suspended 
 from a cord passing over a pulley. If the weight W be retained in its vertical position, 
 
 by the revolution of the drum it will be forced up 
 the incline, and when the cord has unwound one half 
 turn from the drum, and consequently the weight P 
 descended a distance, c e, equal to one half the cir- 
 cumference of the drum, the weight W has been 
 
 raised to the height a b 
 by the half revolution 
 of the plane; P must 
 therefore be to W as 
 a & is to one half the 
 circumference. Extend 
 the inclined plane so as 
 to encircle the drum 
 (Fig. 378). The figure 
 illustrates the mechan- 
 ism of the screw, which 
 may be considered as formed by wrapping a fillet-band or thread around a cylinder at a 
 uniform inclination to the axis. In practice, the screw or nut, as the case may be, is 
 moved by means of a force applied at the extremity of a lever; a complete revolution 
 raises the weight the distance from the top of one thread to the top of the one above, or 
 the pitch. If the force be always exerted at right angles to the lever (Fig. 379), the 
 lever may be considered the radius of a wheel, at the circumference of which the force 
 is applied. Thus, if the lever be three feet long, the diameter of the circle would be six 
 feet, and the circumference 6 x 3'1416, or 18^ feet; if the pitch be one'inch, or one 
 
 FIG. 377. 
 
 FIG. 378. 
 
192 
 
 MECHANICS. 
 
 twelfth of a foot, then the force would be to the weight as one twelfth is to 18 - 85; and 
 if the force be one pound, the weight would be 226 '20 pounds. 
 
 The resultant of two forces of exertion, as has been shown, is their sum, and counter- 
 balances the force of re- 
 sistance, which must be 
 applied at a point inter- 
 mediate between, and 
 distant from each of 
 them inversely as the 
 forces exerted. 
 
 The resultant of any 
 number of parallel forces 
 acting in one direction 
 is equal to their sum act- 
 ing in the same direc- 
 tion at some intermedi- 
 ate point; that is, the 
 effect of all the forces 
 is just the same as if 
 there were but one force, equal to their sum, acting at this point, and is balanced by an 
 equal force acting in the opposite direction. This central point may be determined by 
 finding the resultant, i. e., the sum, and the point 
 of application for any two of the forces, as shown 
 graphically in Figs. 368, 369, and then of the other 
 two, the resultants thus determined being again 
 added together like simple forces. 
 
 Inclined Forces are those whose directions are in- 
 clined to each other. When two men of equal 
 strength pull directly opposite to each other, the resultant is nothing. Let a third take 
 hold of the centre of the rope (Fig. 380), and pull at right angles to the rope ; he will 
 make an angle in the rope, and the other two now pull in directions inclined to each 
 other. The less the force exerted at the centre, the less the flexure in the rope ; but 
 
 FIG. 379. 
 
 FIG. 380. 
 
MECHANICS. 
 
 193 
 
 when it becomes equal to the sum of tiie forces at the ends, the two, to balance it, 
 must pull directly against it, bringing the ends of the rope together, and acting as paral- 
 lel forces. Between the smallest force and the largest that can be exerted at the centre 
 and maintain a balance or equilibrium, the ends of the rope assume all varieties of angles, 
 which angles bear definite relations to the forces. 
 
 Represent these forces by weights (Fig. 381). Let P and P' be the extreme forces 
 acting over the pulleys M and N, and tending to draw the rope straight, which the 
 weight P" prevents. Lay off 
 the weight of P (90 pounds) 
 along A B, and the weight of 
 P' (60 pounds) along A C. 
 Draw B D parallel to A C, 
 and C D parallel to A B. 
 Connect D with A. If this 
 is measured with the same 
 scale that A B and A C were 
 laid off with, it will be found 
 that it equals 120 pounds, 
 which will be found to be Fl 
 
 Fio. 384. 
 
 the same as the weight P". A D, therefore, gives the amount and direction of the re- 
 sultant of the two forces P and P', which resultant is balanced by P". In the same 
 way the resultant of any number of inclined forces (Fig. 382) may be found by com- 
 pounding the resultant of any two forces with a third, and so on. 
 
 As two forces may be compounded into a single resultant, so conversely one force 
 may be resolved into two components; thus, let the weight P (Fig. 383) be supported by 
 two inclined rafters, C A and C B. Each resists a part of the force exerted by the weight 
 P. To find the force exerted against the abutments A and B, in the direction of C A 
 and C B, draw c A' (Fig. 384) parallel to C A, c B' to C B, and c d, a parallel to the line 
 C P, the direction in which the weight P acts ; lay off c d from a scale of equal parts, a 
 length which will represent the number of pounds, or whatever unit of weight there may 
 be in the weight P ; draw d a parallel to c B', and d & parallel to c A' ; c a, measured on the 
 scale of equal parts adopted, will represent the pounds or units of weight exerted against 
 A in the direction of C A, and c b the pounds or units of weight exerted against B in the 
 direction of C B. 
 
 This method of finding the resultant of two forces, or the components of one force, is 
 called the parallelogram of forces. If two sides of a parallelogram represent two forces 
 in magnitude and direction, the resultant of these two forces will be represented in mag- 
 nitude and direction by the diagonal of the parallelogram and conversely. 
 
 The sum of a c and c & is greater than c d; that is, the weight P exerts a greater force 
 
 in the direction of the lines C A and C B, against A and B, than its own weight; but the 
 
 down pressure upon A and B is only equal to the weight of P and of the rafters which 
 
 support it, which last, in the present consideration, is neglected. Resolve c &, the force 
 
 14 
 
194 
 
 MECHANICS. 
 
 acting on B in the direction of c B', into gb or c e the downward pressure, and eg or eb 
 the horizontal thrust on the abutment B, and c a into cf and fa. To decompose a force, 
 
 form a triangle, with the direction of the other 
 forces, upon the line representing the magni- 
 tude and direction of the given force ; c e repre- 
 sents the weight on B, c/the weight on A; cd, 
 or ce + de, the whole weight P; therefore, the 
 weight upon the two abutments A and B is 
 equal to the whole weight of P. 
 
 The steelyard (Fig. 385) is a lever, from the 
 short arm of which a dependent hook or scale 
 supports the article to be weighed; while, on 
 Fio. 385. the long arm, a fixed weight, P, is slid in cr 
 
 out from the fulcrum till it balances the article ; 
 
 the distance as marked on a scale on the long arm determines the weight. In plat- 
 form-scales, when very heavy weights are balanced by small weights on a graduated arm, 
 
 Fio. 386. 
 
 combinations of levers are used, the principle of which can be understood from Fig. 386. 
 Thus, suppose P F to be 7", a F 2", a F' 9", I F' 2", b F" 11", F" W 3". 
 
 P is to force a as a F to P F, or as 2 to 7 
 
 Force a is to 5 as & F' to a F, or as 2 to 9 
 
 I is to W as F" W to & F", or as 3 to 11 
 
 P is to W as 
 
 12 to 693 
 
 The differential axle, or Chinese capstan, consists of an axle with two different diam- 
 eters (Fig. 387), the weight W being suspended in the loop of a cord wound around 
 
 FIG. 387. 
 
 these axles in opposite directions by a single turn of the axle. The weight is only raised 
 or lowered by the difference between these two circumferences; one takes up while the 
 other lets out, and the P, to balance W, must be as these differences of circumference of 
 axles is to the circumference of the wheel from which P is suspended. 
 
 The differential screw (Fig. 388) consists of an exterior screw, A, and an interior 
 screw, B. By the revolution of the arm, the screw A is moved through the plate D in 
 
MECHANICS. 
 
 195 
 
 proportion to its pitch, but the interior -screw B moves inward its pitch, and the move- 
 ment of W is only the pitch of A less that of B, and the power applied is to the weight 
 moved as the difference of these pitches is to the circumference described by the power. 
 As the lever (Fig. 889) moves under the action of power or weight, the lever be- 
 comes inclined to the direction of the forces, but the forces are still parallel. The rela- 
 tions of the forces to each other are not changed, but the absolute action of each is only 
 that due to the length a & and & c, to which the directions of the forces are perpendicular. 
 In the bent lever (Fig. 390) the action of the forces is estimated on lengths of arms, 
 
 r 
 
 C 
 
 m 
 
 
 
 FIG. 391. 
 
 FIG. 389. FIG. 390. 
 
 determined by the perpendiculars a b and & c let fall from the fulcrum on the directions 
 
 of the forces. 
 
 The toggle-joint (Fig. 391) is much used for presses. The force is exerted in the 
 
 direction of the arrow at C, and the effective force is to 
 
 separate the plates A and B. The action is as shown in 
 
 Fig. 391a. Equal movements, as C-l, 1-2, 2-3, corre- 
 spond to unequal movements at A and B, as A a', a' a?, 
 
 a? a 3 . The nearer the force C is to the line A B, the 
 
 less the movement <z a a 3 ; and, consequently, the force C 
 
 exerts greater effects in intensity, but the latter is less in 
 
 movement. 
 
 Fig. 392 exhibits the principle of the hydraulic press. 
 
 The small plunger or piston may be considered the application of the force, and the 
 
 large one the weight to be raised to balance each other; the pressure per square inch of 
 
 surface must be the same, 
 and the force must be to 
 weight as the surface of 
 its piston is to that of 
 the weight - piston. If 
 motion takes place, the 
 force will move through 
 space corresponding to 
 the area of weight-pis- 
 ton, while the weight will 
 move that of the area of 
 
 the force-piston. And this is the great principle of all mechanism in the transmission 
 
 of force : there can be no total gain. What is gained in force is lost in movement, and 
 
 in many complicated machines the theoretical comparison of force applied and resultant 
 
 force may be ascertained by the measures of their movements. 
 
 FIG. 391o. 
 
196 
 
 MECHANICS. 
 
 1 Ib. 
 
 FIG. 392. 
 
 The resultant effects of forces, as heretofore treated, 
 have been without motion, or static. But when motion 
 is produced, the forces are called dynamic. A weight sus- 
 pended or supported exerts a force, which is balanced by 
 the resistance of the suspending or supporting medium; 
 but a falling weight acquires an increasing velocity with 
 every unit of time or space passed. All bodies would fall 
 with the same velocities were it not for the different re- 
 sistances from the air due to their different bulk in pro- 
 portion to their weight. Dense articles, as stones and met- 
 als, acquire a velocity in this latitude of about 32 '2 feet in 
 each second, called the intensity of gravity, or g. The 
 value of g at the equator is 32-088; at the poles, 32 -253. 
 A body 
 
 Starting with a velocity 
 
 Falls during the 1st second 
 
 Acquiring a velocity of 
 
 falls during the 2d second 
 
 Acquiring a velocity of twice 32, or 
 
 Falls during the 3d second 
 
 Acquiring a velocity of 3 X 32 = 
 
 Falls during the 4th second 
 Arxjuiring a velocity of 4 X 32 = 
 
 Falls during the 5th second 
 Acquiring a velocity of 5 X 32 = 
 
 .0 Ft. Tot. Fall. 
 
 16\= , 16 16 
 
 \32 feet per second. 
 
 \64 feet per second. 
 
 48 64 
 
 16\= 80 144 
 
 96 v feet per second. 
 
 16\= 112 256 
 
 128 feet per second. 
 
 16\ = 
 
 .. 144 400 
 
 160 feet per second. 
 
 Calling s the space passed over, V the terminal velocity in feet, t the time in seconds 
 of falling, s = $gt*, v = gt or = ^/64.4s. In determining the velocity of issuing water 
 under a head A, corresponding to 8 in the equation, it is generally near enough to reckon 
 v as eight times the square root of the head (^h). 
 
 The motion of falling bodies is a uniformly accelerated one, but there are also uni- 
 formly retarded motions in which the velocity is decreased by equal losses in equal times. 
 There are also uniform motions when bodies are impelled by a constant force and op- 
 posed by constant resistances. 
 
 In Fig. 393, o s represents the trace of a body impelled horizontally by a uniform, 
 but falling through the action of gravity with an accelerated, force. This curve, a 
 parabola, represents approximately the curve of the thread of stream issuing from an 
 orifice, or flowing. 
 
 It will be seen that to produce twice the velocity the body must fall through four 
 times the space ; that there is four times the force stored in the body. But to maintain 
 this velocity uniformly, only twice the force is necessary. The momentum of a body is 
 its mass multiplied by its velocity, but its inertia is as the square of the velocity. It is 
 an established principle of mechanics that the results must be proportional to the causes : 
 if a body has to be raised four feet to obtain a double velocity in falling, the destructive 
 result of that fall must also be four times. 
 
 Under statics, it has been shown that forces may be resolved and compounded. The 
 same may be done dynamically that which has been treated as weight must now be 
 considered as momentum. 
 
MECHANICS. 
 
 197 
 
 In treating of dynamic forces the re"sultants have been considered as equal to the exer- 
 tion, without any losses by resistances. This never happens in practice ; the resistances 
 are a very large element. Resistances from the medium 
 in which the bodies are moved are from the surfaces on 
 which the bodies are supported ; resistances due to the 
 displacement of the fluid in which the bodies move, and 
 frictional resistances, or what is termed skin-resistances, 
 of bodies moving through air or water; and the surface- 
 resistance of bodies sliding or rolling on each other. 
 Suppose a weight to rest on a horizontal surface it will 
 take a certain force to move the insistent weight depend- 
 ing on the amount of this weight and the kind of sur- 
 faces in contact, and the force that will just cause motion 
 overcomes the friction, or frictional force, and is equal to 
 it. The frictional force is only a percentage of the in- 
 sistent force of the body, and this percentage is called 
 tbe coefficient of friction. If the horizontal surface of sup- 
 port be raised at one end, so as to make the surface in- 
 clined, it will after a time become so steep that the insist- 
 ent body will slide down the surface. Thus, in Fig. 394, 
 
 if the body Q is ready to slip on the surface A B, the angle BAG, which represents the 
 angle of the surface with the horizontal, is called the angle of repose, or limiting angle 
 of frictional resistance; or thus (Fig. 395), if the force acting in the direction P"Mis 
 
 FIG. 
 
 just sufficient to produce motion of the mass M along the plane F Q, the angle P M P" 
 is the limiting angle of resistance. 
 
 General Morin has made an elaborate course of experiments on friction, some of the 
 results of which are given in the table on the following page. 
 
 In Morin's experiments the surfaces of the woods were first planed and those of the 
 metals filed and polished with the utmost care. When the friction without lubrication 
 was to be determined, any unctuosity was especially provided against. For unctuous 
 surfaces, the unguent was carefully wiped off, so that no layer of it should prevent their 
 intimate contact. 
 
 When lubricated, the resistance from the viscidity of the lubricant may be overlooked, 
 compared with the friction. As respects the nature of the substance used in lubrication, 
 it was observed, by comparison of the coefficient of the friction of motion, that with 
 hog's lard and olive oil surfaces of wood on metal, wood on wood, metal on wood, and 
 metal on metal, had all very nearly the same friction between 0'07 and 0'08. With tal- 
 low the coefficient was the same except in the case of metals upon metals. 
 
 Practically the sliding of wooden surfaces on each other or on metal surfaces is con- 
 fined within small limits. As far as possible, sliding surfaces in most machines or me- 
 chanical appliances are of metals, and when surfaces are liable to be affected by rust and 
 adhere, they are made of brass or bronze. But in the moving of houses or the launching 
 of ships the ways are of wood, and in this country of Southern pine, as being without 
 knots and extremely stiff. 
 
198 
 
 MECHANICS. 
 
 Sliding Friction of M. Morin's Experiments on Plane Surfaces. 
 
 Sliding 
 surface. 
 
 8. 
 
 Surface 
 at rest. 
 
 s'. 
 
 STATE OF THE SURFACES. 
 
 FRICTION OF 
 MOTION. 
 
 FRICTION OF 
 QUIESCENCE. 
 
 II 
 il 
 
 tl 
 
 2 ,2 a 
 
 .W 
 
 il 
 
 ioi 
 
 B-fi 
 
 B<H 
 
 OtM 
 
 Oo 
 
 S M.5S 
 
 "3 M 01 
 
 M 4 S 
 
 K 
 
 Btl 
 
 <ij <w 
 
 o^ 
 
 G iCW 
 
 "1 03 
 
 h-1 d o> 
 
 Oak 
 
 Oak 
 
 Fibres s s' parallel to 
 motion. 
 Fibres s perpenclic- j 
 ular to motion.. . \ 
 Fibres s s' perpen- 
 dicular to motion. 
 Fibres s s' parallel ( 
 to motion. 
 Fibres s s' parallel j 
 to motion. { 
 Fibres s perpendicu- 
 lar to motion. 
 Fibres s s' parallel j 
 to motion . ( 
 
 Without lubrication 
 
 Without lubrication 
 Unctuous 
 
 0-478 
 
 0-324 
 0-143 
 0-336 
 
 0-246 
 0-136 
 0-432 
 0-119 
 0-450 
 
 0-360 
 0-330 
 0-490 
 0-107 
 0-372 
 
 0-195 
 0-125 
 0-138 
 0-177 
 0-194 
 
 2533' 
 
 17 58 
 8 9 
 18 35 
 
 13 50 
 7 45 
 23 22 
 6 48 
 24 16 
 
 19 48 
 18 16 
 26 7 
 6 7 
 20 25 
 
 11 3 
 7 8 
 7 52 
 10 3 
 10 59 
 
 0-625 
 
 0-540 
 0-314 
 
 0-376 
 
 0-694 
 0-420 
 0-570 
 
 0-530 
 
 o-ioo 
 
 0-137 
 
 0-194 
 0-118 
 
 0-100 
 0-162 
 
 0-100 
 
 o-ioo 
 
 0-164 
 
 32 1' 
 
 28 23 
 17 26 
 
 20 37 
 
 34 46 
 22 47 
 29 41 
 
 27 56 
 5 43 
 
 7 49 
 
 10 59 
 6 44 
 
 5 43 
 9 13 
 
 5 43 
 9 19 
 
 Oak . . . 
 
 Oak 
 
 Oak 
 
 Oak 
 
 Without lubrication 
 
 Without lubrication 
 Unctuous 
 
 Oak . 
 
 Elm 
 
 Elm 
 
 Beech 
 Cast-iron.. 
 Oak 
 
 Oak... J 
 
 Oak 
 Oak .... 
 
 Without lubrication 
 Unctuous 
 
 Without lubrication 
 
 Without lubrication 
 Unctuous 
 
 Fibres s s' parallel j 
 to motion . | 
 
 Without lubrication 
 Unctuous 
 
 Cast-iron . 
 
 Elm 
 
 Wrought- 
 iron. 
 
 Cast-iron . 
 
 Wrought- 
 iron. 
 
 Cast-iron . 
 
 Bronze . . . 
 
 Wrought- 
 iron. 
 Bronze . . . 
 
 Cast-iron . 
 Bronze . . . 
 Cast-iron . 
 
 Fibres of the wood 
 perpendicular to 
 motion. 
 Fibres of the wood ( 
 parallel to motion. ( 
 Fibres s s' parallel j 
 to motion | 
 
 Without lubrication 
 
 Without lubrication 
 Surface unctuous. . . 
 Without lubrication 
 Surfaces unctuous.. 
 Without lubrication 
 Surfaces unctuous.. 
 Lubricated with 
 olive oil 
 
 Cast-iron.. 
 
 Wrought- 
 iron. 
 
 Wrought- 
 iron. 
 
 Cast-iron.. 
 
 Cast-iron . . 
 
 Wrought- 
 iron. 
 Bronze.. . . 
 
 Cast-iron.. 
 Bronze.. . . 
 Bronze.. . . 
 Steel 
 
 Fibres s s' parallel J 
 to motion ] 
 
 Fibres s s' parallel \ 
 to motion 1 
 
 0-066 
 0-143 
 0-152 
 0-144 
 0-314 
 0-197 
 0-100 
 0-070 
 0-064 
 
 0-055 
 0-172 
 0-160 
 0-161 
 0-166 
 0-147 
 0-132 
 0-217 
 0-107 
 0-201 
 0-134 
 0-202 
 
 
 Surfaces unctuous.. 
 Without lubrication 
 Surfaces unctuous.. 
 ' water 
 soap 
 
 8 9 
 8 39 
 8 12 
 
 5 43 
 
 Fibres s s' parallel \ 
 
 Lubri- tallow 
 cated { lard 
 
 to motion 1 
 
 Fibres s s' parallel 
 to motion \ 
 
 with olive oil. : 
 lard and 
 (_ plumbago 
 Without lubrication 
 Surfaces unctuous. . 
 Without lubrication 
 Surfaces unctuous.. 
 Without lubrication 
 Surfaces unctuous.. 
 Without lubrication 
 Surfaces unctuous.. 
 Without lubrication 
 Surfaces unctuous.. 
 Without lubrication 
 
 9 46 
 9 6 
 9 9 
 9 26 
 
 8 22 
 7 22 
 12 15 
 6 7 
 11 22 
 7 38 
 11 26 
 
 Fibres s s' parallel j 
 to motion | 
 
 Fibres s s' parallel { 
 to motion | 
 
 Fibres s s' parallel 
 to motion 
 
 Fibres s s' parallel 
 to motion 
 
 Fibres s s' parallel to 
 motion. 
 
 In the launching of a vessel, the width of the ways must be proportioned to its 
 weight, not to exceed three tons per square foot. A slope of f " to the foot permits the 
 control of the vessel if necessary. On a slope of " to the foot, a vessel will move 
 steadily and gently down without attaining a high speed, and is the one adopted where 
 other circumstances do not affect the choice. A slope of |" to the foot is often used 
 for large vessels; the speed attained, however, is very high, and in narrow water both 
 dangerous and inconvenient. A slope of 1 in 12 may be used where the distance to be 
 traversed is short, and the ship launched broadside in. 
 
MECHANICS. 
 
 199 
 
 Before striking the dog-shore, which prevents the ship from sliding down the ways, 
 they are thoroughly slushed with some greasy mixture, like tallow and soap. 
 
 It was formerly held that friction was directly as the weight, without regard to the 
 amount of surface or velocity of movement, and Morin's experiments, referred mostly to 
 the friction of quiescence and slow movements, come within this rule ; the results give 
 coefficients that may be considered maxima, but in practice it has been found that the 
 coefficient of friction with unguents is reduced by increase of velocity and of tempera- 
 ture, that extent of surface may be prejudicial, and that careful selection of unguents, 
 according to the work to be done, will reduce friction. 
 
 Axle and Rolling Friction. Axle friction has been generally supposed to follow the 
 laws of sliding friction, with the exception that its coefficient is a smaller fraction of the 
 total pressure applied. There are but few experimental data relating to this branch of 
 the subject. The results of some experiments by Morin are given in the following table : 
 
 Ratios of Friction to Pressure for Axles in Motion in their Bearings. 
 
 I. ACCORDING TO MORIN'S EXPERIMENT. 
 
 DESIGNATION OF SURFACES 
 IN CONTACT. 
 
 State of surfaces and nature of lubrication. 
 
 Dry or slightly 
 greasy. 
 
 Greasy, and wet 
 with water. 
 
 Lubricated, and 
 wet with water. 
 
 Oil, tallow, or 
 hogs 1 lard. 
 
 Purified very 
 soft grease. 
 
 Hogs 1 lard with 
 plumbago. 
 
 Greasy, very 
 soft to the touch. 
 
 Supplied 
 in the 
 ordinary 
 manner. 
 
 The 
 grease 
 continu- 
 ally 
 renewed. 
 
 I'.ronze on bronze 
 
 
 
 
 0-079 
 
 
 
 
 
 Bronze on cast-iron 
 
 
 
 
 0-049 
 0-054 
 0-054 
 0-054 
 0-054 
 
 
 
 
 1 ron on bronze 
 
 0-251 
 
 0-189 
 
 
 075 
 0-075 
 0-075 
 0-075 
 0-125 
 
 o-ioo 
 
 0-116 
 
 0-090 
 
 0-111 
 
 
 Iron on cast-iron ... 
 
 
 
 Cast-iron on cast-iron 
 
 
 0-137' 
 0-161 
 
 0-079 
 
 
 
 
 Cast-iron on bronze 
 
 0-194 
 0-188 
 0-185 
 
 0-065 
 
 
 0-1-37 
 0-166 
 0-140 
 .0-153 
 
 Iron on lignum-vitae.. . 
 
 
 
 Cast-iron on lignum-vitae 
 Lignum-vitjB on cast-iron 
 
 
 
 0-92 
 
 
 0-109 
 
 
 
 
 Lignum-vitas on lignum-vitae.. 
 
 
 
 
 0-170 
 
 
 
 
 
 
 
 
 It is well known that the obstruction which a cylinder meets in rolling along a smooth 
 plane is quite distinct in its character, and far less in amount, to that which is produced 
 by the friction of the same cylinder drawn lengthwise along a plane. 
 
 In the composition of machines attrition should be avoided as much as possible, and 
 rolling motions substituted. 
 
 On this principle depend the advantages of the application of friction-wheels and fric- 
 tion-rollers. The extremity of an axle c, Fig. 395a, instead of 
 resting in a cylindrical socket, is made to rest on the circum- 
 ferences of two wheels, A and B, to the axles of which, a and &, 
 the friction is transferred, and consequently diminished in the 
 ratio of the radius of the wheel A to the radius of the axle a. 
 
 At speeds and pressures usual in machinery, the resistance 
 from friction decreases as the journals and bearings heat up to 
 limits which the engineer and mechanic would consider safe. 
 
 With very heavy pressures and slow speeds, the lubricant 
 may be forced out and journal and lubricant be brought into 
 
 close bearing, while with light pressure and low speeds the journals float on the film of 
 fluid, and the frictional resistance is independent of the materials of which journal and 
 bearing are composed. 
 
 Mr. C. J. H. Woodbury, in his experiments on the driving of cotton spindles, found 
 the coefficient of friction to be from 7 to SO^er cent, the load being from 1 to 5 pounds 
 
 FIG 395a. 
 
200 MECHANICS. 
 
 per square inch, while Prof. Thurston, with heavy loads of 1,000 pounds per square inch, 
 as on the crank-pins of the North River steamboat engines, found the coefficient of fric- 
 tion was one half of one per cent, the unguent being sperm oil. Practically, it may be 
 said that the coefficient of friction for light-running spindles should not exceed 10 per 
 cent, and for the usual work in shops, of say 100 to 200 pounds, should not exceed from 
 2 to 3 per cent. 
 
 For lubricants under heavy pressure and slow speeds, use graphite, soapstone, tallow, 
 lard, and greases. For the journals of calender rolls in paper mills, it is common to lay 
 on the top of the journals a large piece of salt pork, skin up. 
 
 Cast iron holds oil better than any other metal or alloy, and is the best metal to use 
 for light bearing, perhaps for heavy. 
 
 It has been proved by Mr. Waite's experiments that a highly polished bearing is 
 more liable to friction than any surface finely lined by filing. The lines left by the file 
 serve as reservoirs for the oil, while the high polish leaves no room for the particles be- 
 tween the metal surfaces. 
 
 Mr. Waite's experiments on very heavy bearings at Manchester, N. H., go far "to 
 prove that a considerable quantity of thin, fine oil keeps the bearing much cooler, and 
 requires less power than a smaller quantity of thick, viscous oil. No vegetable oil is fit 
 to use as a lubricant, and castor oil is the worst of all, because the most viscous, and all 
 vegetable oils in connection with fine vegetable fibre, like cotton, are liable to spontaneous 
 combustion. 
 
 " The rule of best lubrication is to use an oil that has the greatest adhesiveness to metal 
 surfaces, and the least adherence as to its own particles. Fine mineral oils stand first in 
 this respect, sperm second, neat's-foot third, lard fourth." 
 
 Experiments on rolling friction usually include that of the axles. The resistance 
 of a wheel rolling on a smooth and resistant plane is very different, and much less 
 than that of the same wheel fixed and sliding on the same plane; a moving railroad 
 train is brought to a stop by the brake reducing or stopping the rotation of the 
 wheels. 
 
 Between the usual diameters of wheels the resistance has not been found to increase 
 inversely as the diameter, but rather as their square roots of the diameter, and less on 
 hard and well-graded roads. 
 
 Of the resistance dependent on the material and character of the roads there have 
 been many experiments. The following formula and table is from Prof. Thurston on 
 "Rolling Friction." 
 
 W 
 
 R=/_ in which R = resistance, W the total weight, and r the radius of the wheel: 
 r 
 
 Kind of road. Value of friction. 
 
 Well paved 0'02 
 
 Hard, smooth ground 0'02 
 
 Well macadamized and rolled , O'OIS 
 
 Smooth wooden pavement O'Ol 
 
 Ordinary railroads '003 
 
 Best possible railroad O'OOl 
 
 "On railway trains a minimum resistance is reached usually at a speed between 10 
 and 15 miles per hour, but the increase is not great at higher speeds where the common 
 system of lubrication is practised ; in ordinary work, the resistance varies as low as from 
 4 pounds per ton of train up to 25, and sometimes above 30 pounds. " 
 
 At a speed of 25 miles per hour, Chanute makes the increase of resistance due to 
 curvature about 0'4 pound per degree per ton. 
 
 The amount of resistance is measured by the weight resting on the axles, multiplied 
 by c, which is dependent on the condition of the surface of the rails : 
 
MECHANICS. 201 
 
 Perfectly dry . ._. c = 
 
 Average condition \ to ^ 
 
 Damp in rain, fog, or tunnel ^ 
 
 Greasy or iced ^ 
 
 With head winds the resistance is increased with the square foot of front exposed and 
 as the square of the velocity in miles per hour. 
 
 Side winds, by forcing the flanges of the wheels against the rails, adds very seriously 
 to the resistance of draught. 
 
 On street railways the resistance to the movement of cars is greater than upon rail- 
 roads, and may be considered from 3 to 5 times as much. 
 
 From his experiments in 1840, Morin states the resistance applied at the circumfer- 
 ence of the wheel on pavements and macadamized roads to be inversely proportional to 
 the diameter of the wheel independent of the breadth of the tire and increasing with the 
 velocity. 
 
 By the Studebacker experiments in South Bend, Ind., it was found that there was no 
 reduction in resistance on hard roads in increasing the width of the tire, but rather the 
 contrary, nor in soft mud or slush ; but on sandy roads or across fields the resistance is 
 very much less with a broad tire. 
 
 In many States there are laws regulating the width of tire in proportion to the load, 
 and even on the harder roads it is supposed that the broad tire deteriorates the track 
 less than the narrow one, and in fact acts as a roller and improves the roadway, and for 
 the same purpose the front wheels are of less gauge than the hind ones, that they may 
 not track in the same rut. 
 
 It is of importance that the line of draught should be horizontal, as with an oblique 
 pull the effort of the animal may either be exerted in carrying the load or by increasing 
 the resistance of it on the roadway. 
 
 MECHANICAL WORK OR EFFECT. 
 
 Mechanical work is the effect of the simple action of a force upon a resistance which 
 is directly opposed to it, and which it continuously destroys, giving motion in that di- 
 rection to the point of application of the resistance ; it is therefore the product of two 
 indispensable qualities or terms : 
 
 First. The effort, or pressure exerted. 
 
 Second. The space passed through in a given time, or the velocity. 
 
 The unit of force in England and here is represented by the pound, and the unit of 
 space by the foot. 
 
 The amount of mechanical work increases directly as the increase of either of these 
 terms, and in the proportion compounded of the two when both increase. If, for 
 example, the pressure exerted be equal to 4 pounds, and the velocity one foot per second, 
 the amount of work will be expressed by 4 x 1 = 4. If the velocity be double, the work 
 becomes 4x2 = 8, or double also ; and if, with the velocity double, or 2 feet per second, 
 the pressure be doubled as well that is, raised to 8 pounds the work will be 8 x 2 = 16 
 pounds feet. It is more usual to write foot-pounds ; but as explained before the Conti- 
 nental idiom of Tcilogrammetre is followed, in which the unit of force, kilogramme, pre- 
 cedes that of space, the metre, it should be pounds-feet. 
 
 In comparison of motors with each other, it is usual to speak of them as so many 
 horse-power equivalent to 550 pounds-feet per second, or 33,000 pounds-feet per minute. 
 The Continental horse-power is equal to 75- kilogrammetres or 542 '48 pounds-feet per 
 second. 
 
 It is very common to use other units of force and space, as tons-miles; and train- 
 miles, in railway practice. 
 
 The time must also be expressed or understood. It is impossible to express or state 
 
202 MECHANICS. 
 
 intelligibly an amount of mechanical effect without indicating all the three terms force, 
 space, and time. 
 
 The motors generally employed in manufactures and industrial arts are of two kinds 
 living, as men and animals ; and inanimate, as water and steam. 
 
 What may be termed the amount of a day's work, producible by men and animals, is 
 the product of the force exerted, multiplied into the distance or space passed over, and 
 the time during which the action is sustained. There will, however, in all cases be a 
 certain proportion of effort, in relation to the velocity and duration, which will yield the 
 largest possible product or day's work for any one individual, and this product may be 
 termed the maximum effect. In other words, a man will produce a greater mechanical 
 effect by exerting a certain effort at a certain velocity than he will by exerting a greater 
 effort at a less velocity, or a less effort at a greater velocity, and the proportion of 
 effort and velocity which will yield the maximum effect is different in different indi- 
 viduals. 
 
 In the manner and means in which the strength of men and animals is applied there 
 are three circumstances which demand attention: 
 
 1. The power, when the strength of the animal is exerted against a resistance that is 
 at rest. 
 
 2. The power, when the stationary resistance is overcome, and the animal is in mo- 
 tion. 
 
 3. The power, when the animal has attained the highest amount of its speed. 
 
 In the first case the animal exerts not only its muscular force or strength, but at the 
 same time a very considerable portion of its weight or gravity. The power, therefore, 
 from these causes must be the greatest possible. In the second case some portion of the 
 power of the animal is withdrawn to maintain its own progressive motion ; consequently 
 the amount of useful labour varies with the variations of speed. In the third case the 
 power of the animal is wholly expended in maintaining its locomotion ; it therefore can 
 carry no weight. 
 
 Weisbach calls the mean effort of an animal one fifth its weight, which may serve as 
 a general rule, but, in practice, will be considerably modified, when applied to the indi- 
 vidual, depending upon the exertions, and the conditions and circumstances under which 
 it is made. A man-power is usually estimated at one sixth of a horse-power (H. P.); 
 yet, if the muscular force of a man be required for an effort of short duration, it will ex- 
 ceed one horse-power. Thus, a horse-power is equal to 33,000 pounds-feet per minute, 
 or 550 pounds-feet per second; and, if a man weighing 150 pounds move upstairs at the 
 rate of four feet per second, he exerts a force of 600 pounds-feet, which he can readily 
 double for a few seconds. 
 
 The force of a man is utilized mechanically through levers, as in pumping or rowing, 
 or at a vertical capstan, or at a crank, carrying or dragging loads, shovelling, etc. In 
 continuous work at the lever he will exert from 25 to 30 pounds ; at the crank, from 15 
 to 20 pounds. 
 
 The muscular force of horses is utilized in the draught of carriages, in hoisting, and 
 in horse-powers, either moving in a circle round a central shaft or on a revolving plat- 
 form, or on an endless belt. The draught of a horse varies with the speed of movement 
 and its duration. Trautwine gives the draught of a horse at two and a half miles per 
 hour for 10 hours per day, 100 pounds; 8 hours, 125 pounds; 6 hours, 166f pounds; 5 
 hours, 200 pounds. The omnibus -horses here average nearly six miles per hour, and 
 make 18 to 24 miles per day; the average will not exceed 16 miles. At the Manhattan 
 Gas Works a span of horses hoist from the lighter 200 tons gross in 10 hours to the 
 height of say 25 feet, with charges of 6 to the ton, in a bucket weighing 150 pounds, the 
 rope passing over a single block and through a snatch-block. On a horse-power, the 
 force exerted by a single horse is from 125 to 175 pounds, at an average speed of about 
 three milos per hour, and for eight hours per day. Beyond a speed of four miles per 
 
MECHANICS. 
 
 203 
 
 hour the pounds-foot of work of a horse will decrease in an increasing ratio up to the 
 limits of his speed, when the whole work done will be used up in locomotion. In pro- 
 portioning levers, cranks, traces, chains, through which animal force is transmitted to 
 machines, or for mechanical purposes, it is not safe to estimate the stress as the average 
 force ; there are impulses and stresses in action which will exceed the weight of the 
 animal. 
 
 Water-Power. Water, used for the purposes of power, moves machinery either by its 
 weight, by pressure, by impact, or by reaction, and is applied through various forms of 
 wheels. 
 
 Fig. 396 is what is termed a tub wheel with a vertical shaft, the wheel running in a 
 bottomless, wooden case or tub to which the water is conveyed by a wooden trunk or spout 
 
 and acting on the floats or buckets by im- 
 pact. The wheel was commonly used for 
 grist mills, and the power was applied di- 
 rectly to the stones through the vertical 
 shaft. The same wheel on a horizontal 
 
 FIG. 390. 
 
 FIG. 397. 
 
 shaft, without the tub, was called a flutter wheel, and the power was transferred by a 
 crank on the shaft through a wooden connecting rod or pitman to the saw frame of a 
 saw mill. A similar wheel, but of 
 larger diameter, was suspended over 
 the surface of a stream, the floats be- 
 ing dipped into the water sufficiently 
 to give motion and power. 
 
 Fig. 397 is the section of the up- 
 per part of a breast wheel, in which 
 the water is admitted through the 
 gates g g g, acting after its delivery 
 into the buckets through gravity ; the 
 power is transferred through the cir- 
 cumferential gear to a jack shaft, usu- 
 ally placed below the gates ; a plank 
 case called the breast is set outside 
 the wheel with but little clearance 
 from the gates to within a few feet 
 of the bottom of the wheel. Wheels 
 of this class, where water is brought 
 over the top of the wheel and dis- 
 charged into the buckets without a 
 breast, is called an overshot wheel, 
 and when the water is discharged be- 
 neath the wheel, an undershot. The 
 overshot and breast, when properly FIG. 398. 
 
204 
 
 MECHANICS. 
 
 proportioned, give a good percentage of effect, but occupy considerable space, and the; 
 speed of the gear is slow. 
 
 Fig. 398 is one of the earliest improvements of the reactor, the Scotch wheel ; the 
 principle is that of the sky-rocket. It is very simple in construction; the central curve 
 of the hollow arm or adjutage is constructed like a cam, in which the horizontal move- 
 ment is a percentage of the radial velocity due to the head, and the cross sections of the 
 arm a contracted vein. 
 
 The above wheels belong rather to history, but there are positions and circumstances 
 which make them the most available. 
 
 The wheels now in use are mostly of cast iron or bronze, occupying small space, and 
 the shafts giving a velocity nearly proportioned to the speed of the machinery which it 
 is to drive. 
 
 . Fig. 399 is a Fourneyron, called in this country a Boyden wheel, from the improve- 
 ments which he made upon it. The water flowing downward through a pipe is diverted 
 by guides at the bottom, which give it an outward direction with a tangential whirl as it 
 strikes the buckets of the wheel, which are formed to give a reactive impulse. 
 
 In Fig. 400, the Jonval, there are fixed guides in the tube that give motion to the 
 water against guides in the wheel beneath and a resulting reaction. 
 
 FIG. 399. 
 
 FIG. 400. 
 
 FIG. 401. 
 
 Fig. 401 is the type of wheel commonly used ; the fixed guides are at the outside of 
 the wheel and the motion of the water is inward and downward against the buckets. 
 The shaft of this wheel is generally vertical, and the power transmitted generally through 
 bevel gears, but it is often preferable to set the shaft horizontally and transmit the power 
 through belting. These wheels, whether horizontal or vertical, can be set above the tail 
 race and the entire fall utilized through draught tubes below the wheels, and admit of 
 the use of belts. The greatest length of draught tube from the centre of horizontal 
 shaft to tail water at Manchester, N. H., is 26 feet. 
 
 Wheels can be furnished by many makers from stock from 6 to 84 inches in diameter, 
 the smaller ones for very extreme heads of water, and the larger to the limit of 100 feet. 
 The power for the different wheels depends on the head, the speed of the shaft on the 
 size of the wheels. For the best percentage of effect, the wheel must be of good de- 
 sign, material, and workmanship, and the makers will give their opinion as to which of 
 their wheels they consider best adapted for any given position. 
 
 Fig. 402 is the elevation of a Pelton wheel, in which the action of the water on the 
 bucket is by impact, and Fig. 403 a section of the bucket showing the action of the 
 water upon it; they work under heads as high as 1,000 feet and to 2,000 H. P., giving n 
 large percentage of efficiency. 
 
 However used, the mechanical effect inherent in water is the product of its weight 
 into the height from which it falls; but there are many losses incurred in its application, 
 
MECHANICS. 
 
 205 
 
 so that only a portion of the mechanical effect becomes available ; and the comparative 
 efficiency of any water-wheel or motor is represented by this percentage of the absolute 
 effect of the water applicable to power. 
 
 The watershed of the Merrimac River at Lowell is 4,085 square miles; and the per- 
 manent water equal to one cubic foot per second per square mile of shed during working 
 
 FIG. 402. 
 
 hours ; gross fall, 34 feet ; loss in canal, say 1 foot ; and in flumes and races, 1 foot ; 
 
 4. t 11 on t 4 4,085 x 62-4 pounds x 32 feet 
 
 net fall, 32 feet; = 14,830 H. P. 
 
 550 pounds 
 
 Taking the effective power on the wheel at 80 per cent, which is obtained from the 
 wheels at full gate, 14,830 x '80 = 11,864 effective H. P. A very close rule to deter- 
 mine the E. H. P. is, multiply the number of cubic feet per second by the fall in feet 
 
 and divide by 11, thus: 4 > 85 x 32 = 11,884. 
 
 Wind is applied for the purposes of power ; but, as there is no constancy in its action, 
 its use is mostly confined to the purpose of raising water by means of pumps into cisterns 
 or reservoirs. 
 
 FIG. 403. 
 
 FIG. 404. 
 
 Steam is the elastic fluid into which water is converted by a continuous application 
 of heat. It is used to produce mechanical action almost invariably by means of a piston 
 movable in a cylinder. Thus, in Fig. 404, the steam entering through the lower 
 
206 
 
 MECHANICS. 
 
 channel-way, or port, presses against the under side of the piston in the direction 
 of the arrow, the piston is forced upward, the steam above the piston escaping 
 through the exhaust-channel o. When the piston reaches the top of the cylinder, 
 the valve is changed by mechanism, the steam enters above the piston, and the 
 steam below it escapes through the exhaust ; in this way a reciprocating motion is es- 
 tablished. 
 
 It is not practicable at a single instant to let the whole pressure of the steam in the 
 boiler into the cylinder, nor maintain it uniform during the whole stroke; there are 
 losses of pressure due to friction through the pipes, valves, and channel-ways, and back 
 pressure from the same causes in the exhaust, condensation from exposed surfaces, and 
 consequent reduction of pressure and volume of steam, and there are gains of effect by 
 cutting off the introduction of steam to the cylinder at some point of a partial stroke, 
 which is completed by the expansion of the inclosed steam, but with a constantly di- 
 minishing pressure. 
 
 Under expansion, the volume of a confined gas is inversely proportional to the pres- 
 sure to which it is exposed. This is called the law of Mariotte, or the law of expansion, 
 under equal temperatures for all gases, and is shown graphically in Fig. 405. 
 
 FIG. 405. 
 
 To determine the action of the steam within a steam-engine cylinder, an indicator is 
 used. It consists of a small cylinder connected by a steam pipe with the main cylinder, 
 with its piston controlled by a graduated spring, and recording by a pencil the varying 
 pressures in the main cylinder on a paper card wrapped around a reciprocating cylin- 
 drical tube ; to this, movement is given by a cord attached to some mechanism connected 
 with the steam-engine piston, but with a reduced motion. 
 
 Indicator cards show the efficiency of a steam engine by a comparison with theoreti- 
 cal ones, the mean effective power exerted (M. E. P.), the volume of steam expended, 
 losses of pressure by imperfect passages or channels, leaks in valves and around pistons 
 and by radiation. 
 
 Fig. 406 is an indicator card from a non-condensing engine. The steam exhausts 
 directly into atmosphere, and the back pressure, 2 to 3 pounds, is shown by the line just 
 above that of the atmosphere line. 
 
MECHANICS. 
 
 207 
 
 In Fig. 407 the exhaust is into the condenser, where a vacuum is maintained by the 
 air pump of from 2 to 3 pounds. The M. E. P. of the condensing engine is greater than 
 
 ABOVE AT. BELOW AT. 
 
 Atmospheric Line 
 FIG. 406. 
 
 FIG. 407. 
 
 that of the non-condensing, but this is not obtained without expense of power in running 
 the air pump, and more condensation in the cylinder due to the lower temperature at 
 which the steam leaves it. See Table of Temperatures of Steam, Appendix. 
 
 The ends of both cards are rounded by the early opening of the steam valve in the 
 out stroke, and by the earlier closing of the exhaust valve in the return stroke. This 
 last shuts up the balance of steam in the cylinder, as shown by the curve of compression, 
 and retains a volume of some importance in the next stroke, especially in quick-running 
 engines regulated by movable eccentrics, and cushions the piston at the end of the 
 stroke. 
 
 Both cards are divided into ten equal spaces, and subdivided for the average pres- 
 sures, which are given in figures on the scale of pounds of the indicator. The sum of 
 the pressures are then divided by 10, which gives the M. E. P., which, multiplied by the 
 area in square inches of the piston by the travel in feet per minute and divided by 33,000, 
 gives the H. P. 
 
 Assume the diameter of the cylinder in Fig. 406 
 to be 15" (area, 176-7); stroke, 3 feet; revolutions, 
 90 per minute. 90 x 3 x 2 = 540 feet travel per 
 minute. 
 
 Then, 
 
 64-7 x 176-7 x 540 
 
 = 187 H. P. 
 
 33,000 
 
 The indicator card may be divided into twenty 
 equal spaces, and the first, third, fifth, . . . nine- 
 teenth taken as subdivision ordinates, or it may be 
 divided by a flexible grid ; but where ordinates are 
 not required the M. E. P. may be taken by plani- 
 meter. 
 
 The M. E. P. of a steam cylinder can be esti- 
 mated approximately from the initial pressure and 
 the cut-off. The reduction of pressure by the cut- 
 off is shown in the diagram (Fig. 408) constructed 
 from the formula of Rankine for dry saturated 
 steam. Multiply the initial pressure by the decimal 
 on the vertical line of the diagram at its intersection 
 by the curve of cut-off, and the result is the M. E. P. 
 of the entire card to 0; from which to find the 
 effective pressure as given by the indicator card 
 there must be subtracted the loss of pressure be- 
 tween the lower line of card and the line. Assume the initial pressure in Fig. 406 
 to be 124 pounds above atmosphere, and the vacuum 14 -7, or, together, 138 '7 cut-off 
 
208 
 
 MECHANICS. 
 
 l, 
 
 i.oo 
 
 1.00 
 
 Q.90 
 
 o 
 
 to 0.80 
 
 0.70 
 
 0.60 
 
 < 0.50 
 
 O 
 Ul 
 
 Q 
 
 ? 0.40 
 id 
 
 (. 
 
 0.30 
 
 ZERO 0, 
 
 LINE 
 
 
 
 1.11 
 
 0.90 
 
 EXPANSIONS 
 1.25 1.43 
 
 0.80 0.70 
 
 1.67 
 
 0.60 
 
 CUT, 
 
 0.10 
 10 
 
 0.20 0.30 
 
 5 3.33 
 
 EXPANSIONS 
 Fio. 408. 
 
 2. 
 
 0.50 
 1.00 
 
 0.90 
 
 0.70 
 
 0.60 
 
 0.50 
 
 0.40 
 
 0.20 
 
 0.10 
 
 0.40 
 2.5 
 
 0.50 
 2. 
 
 at J, or -25 of the stroke, 4 expansions by diagram -582 x 138-7 = 80-7 pounds, less 
 
 80-7 
 
 17 - 7 
 3 pounds above atmosphere and 14 - 7 below, 
 
 63 x 176-7 x 540 
 33,000 
 
 63-0 
 = 182 '2 H. P. A result a little less than that of the card. 
 
MECHANICS. 
 
 209 
 
 In the appendix will be found a table of both dry and moderately moist steam from 
 the same authority. 
 
 At the commencement of each stroke there is a space between the piston and the 
 cylinder head which, together with the space between it and the valve, are called clear- 
 ances, and, as it causes an expenditure of steam, is reckoned in percentages of the 
 stroke. 
 
 To enable one to judge of the economy of the steam engine, it is necessary to know 
 how much weight of steam is used. In the example the pressure at cut-off is taken at 
 139 pounds, and the weight of a cubic foot of steam at this tension, by tables of satu- 
 rated steam in appendix, is -3092 pound ; the area of cylinder = 1 -227 square feet. 
 
 Stroke 3 x '25 cut-off = -75 
 
 Clearance, 3 per cent of stroke = '09 
 
 84 foot. 
 
 1-227 x -84 x -3092 = -319. '319 x 180 strokes x 60 minutes = 3,445 pounds per 
 hour. If the evaporation be 8 pounds of water per pound of coal, then the consump- 
 tion of coal would be 431 pounds, and -- = 2'30 pounds of coal per H. P. per hour. 
 
 187 
 
 Of late years, the principle of expansion has been very much extended by the con- 
 struction of compound engines. Fig. 409 shows the general arrangement, without valves, 
 of a simple compound engine of two cylinders a 
 high-pressure (h. p. c.) and a low-pressure (1. p. c.) 
 one. The h. p. c. (ABC D) draws its steam from 
 the boiler and exhausts into the 1. p. c. (A', B', 
 C', D') ; the top of the h. p. c. into the bottom 
 of the 1. p. c., and vice versa, so that the pressure 
 on the pistons of the two cylinders is in the same 
 direction. 
 
 To determine whether a steam engine is work- 
 ing properly, it is necessary to compare the abso- 
 lute card with the theoretical one. 
 
 Figs. 410 and 411 represent indicator cards, 
 
 taken from a condensing and non-condensing engine. On these are shown the con- 
 struction of the isothermal curve. It will be observed that there is a line, A B, to the 
 top of the card. The space between this and the card represents the clearance, which, 
 estimated in percentages of the capacity of the cylinder, is plotted on the indicator card. 
 On the indicator card, as taken by the instrument, the absolute can not be taken, but 
 only that of the atmosphere, the will be at a distance below this, corresponding to the 
 
 FIG. 409. 
 
 FIG. 410. 
 
 FIG. 411. 
 
 barometric pressure, usually 14-7 pounds. Draw the line parallel to the atmospheric 
 line, the clearance line perpendicular to it, a line parallel to the line, at the height of 
 the initial pressure, and a line parallel to the clearance line at the point of cut-off on the 
 initial pressure line. Any point on the expansion line, 'as l a , 2 a , 3 a , may be determined 
 
 15 
 
210 MECHANICS. 
 
 by drawing lines B 1, B 2, B 3, and then horizontal lines la li, 2 a 2i, 3 3 3i from their in- 
 tersections li, 2!, 3i. With the cut-off line, parallel to the line, and perpendiculars 
 from 1, 2, 3, the intersections of these two lines, 1 2 , 2 2 , 3 2 , will be the points in the 
 curve. Inversely, the indicator card may be tested by the construction of parallelo- 
 grams on the curve, and if their diagonals intersect at a point on the line it may be 
 considered that this represents the amount of clearance. The curves in the outline of 
 the cards, at the times of admission, cut-off, and exhaust, show the action of the valves 
 and time occupied in change of condition. The stroke commences at A, cuts off at C, 
 commences to exhaust at E ; about D the exhaust valve closes. 
 
 To enable a draughtsman to comprehend the action of the steam in multiple cylinders 
 and calculate approximately the diameter and capacity of the different cylinders, a curve 
 of expansion is constructed, Fig. 405. The ratio of expansion is on the Mariotte curve 
 and is as the reciprocal of the pressures. Taking as the unit the abscissa of the pressure 
 of 100 pounds, at 120 it is '833, at 40 pounds 2 '50, and on this principle the curve is 
 drawn, taking cross-section paper, for the ease with which it may be laid down and as 
 more intelligible in its explanation. 
 
 In Fig. 405 is illustrated a triple compound with an initial pressure (I. P.) of 120 
 pounds and a final expansion of twenty-one times. To divide the expansions between 
 the three cylinders, V^T = 2'76 expansions in the first cylinder, 2'76 x -833 = 2'30 
 abscissa at end of stroke of first cylinder. 
 
 (1) f^Wa = 43 '48 final pressure, ordinate corresponding to an abscissa of 2 '30. 
 2"7o. 
 
 120 
 
 (2) 2-76' = 7-617. 7-617 x -833 = 6-345 = 15-76 pounds. 
 
 7"u4 
 
 120 
 
 (3) 2-76 3 = 21. 21 x -833 = 17'5 r = 5'71 pounds. 
 
 til 
 
 With a cut-off at -833, and an expansion of 2-76, the stroke will be 2 -3, which is the 
 uniform stroke in the three cylinders. Calling the diameter of the H. P. cylinder 15" 
 A 176-7 n", the area of the I. P. cylinder will be 176-7 x 2-76 = 487-69 square inches 
 = 25" diameter, and that of the low pressure 487-69 x 2'76 = 1,346 square inches 41|" 
 diameter. Taking the M. E. P. of the different cards by averages or by planimeter, in 
 which the area of the card in square inches is divided by the length of card in inches 
 and multiplied by the vertical scale, we find the M. E. P. for first cylinder 42 '2, second 
 cylinder 15 '4, third cylinder 5'6. The pounds-feet per stroke will be: 
 
 First cylinder, 176 -7 x 2 -3 x 42 -2 = 17,150 
 Second " 487-69 x 2-3 x 15 '4 = 17,274 
 Third " 1,346- x 2'3 x 5-6 = 17,336 
 
 The M. E. P. of the whole card was 16'6, stroke 17-5, area 176-7 = - - = 17,110 pounds- 
 
 o 
 
 feet, a result very nearly that of the single cylinder, but as the areas measured by the 
 planimeter includes only that between the lines of final pressure, while the absolute card 
 in the low-pressure cylinder should be carried to the line of exhaust, or say 3 pounds 
 above 0, the pounds-feet of effect should be more than in either of the others, which it 
 would be impossible to equalize with the same stroke and the same rates of expansion. 
 All these results are obtained from the plotted isothermal curve, but in expanding steam 
 does not maintain the same temperature, and occupies less space than shown by the 
 above curve, and making allowance for this a curve is developed which is called the 
 adiabatic curve. 
 
 To plot this on the diagram take the abscissa at 120 as equal to 3 -65 (or the volume 
 of steam at this pressure), construct a scale taking the volumes at the different pressures 
 from the table in the Appendix ; but if the stroke and the areas of cylinders is considered 
 the same as in the previous calculations, although the area of the cards is less, yet as the 
 
MECHANICS. 211 
 
 length of card is less, the M. E. P. remains nearly the same, and consequently the pounds- 
 feet of results, estimating pounds-feet of work done in the several cylinders by the mean 
 pressures as taken from Rankine's tables in the Appendix. In these cases the average 
 pressure is taken from the 0, from which has been subtracted the initial pressure in the 
 next cylinder. 
 Thus: 
 
 Average pressure in first cylinder 88 - 04 
 
 Initial pressure in second cylinder 43 - 5 
 
 M. E. P. in first cylinder 44 -54 
 
 Average pressure in second cylinder 31-8 
 
 Initial pressure in third cylinder 15-8 
 
 M. E. P. in second cylinder 16 '00 
 
 Average pressure in third cylinder 11-56 
 
 Pressure at end of expansion 5'7 
 
 M. E. P. in third cylinder . If86 
 
 For dry steam the results are very nearly the same as by the Mariotte curve. In none 
 of the calculations is the difference as great as will be found in practice, where it is 
 found desirable to jacket the steam cylinders and heat the steam in the intermediate 
 chambers to prevent condensation from the walls of the cylinders and passages, and to 
 restore the heat which has been converted into work. 
 
 The above card refers to an engine in which there is an intermediate chamber between 
 the cylinders. 
 
 MOTION. 
 
 In the designing of machines it is often important to show the changes in the moving 
 parts, that they may not conflict with each other in the practical working, and to obviate 
 the effect of dead-points through which motion may have to be transferred. 
 
 To describe the path of the crosshead of the piston of the steam engine, its connect- 
 ing-rod, and crank, Fig. 412, draw the path of the crosshead which, controlled by 
 guides, is a straight line; divide into equal parts to 6; on the same line lay off the 
 length of the connecting-rod from to 0', and the length of the crank to the centre C ; 
 from C as a centre with a radius equal to the length of the crank describe a circle which 
 will be the path of the crank-pin. From the points 1, 2, 3, 4, 5, and a radius equal to 
 that of the connecting-rod, describe arcs cutting the crank path at 1', 2', 3', 4', 5', 0' 
 and 6' on the line of the crosshead path, the commencement and end of the strokes. 
 The path of the crosshead is divided into equal spaces, but that of the crank-path is not. 
 The point 3' corresponding to the mid stroke 3 of the crossbead is not that of the half 
 circumference of the crank-path. These irregularities are due to the angularity of the 
 connecting-rod, which is here made less than its usual proportion to that of the crank. 
 
212 
 
 MECHANICS. 
 
 Moreover, when the crank-pin and the axle-centre are in the line with the connecting- 
 rod, as at 0' or 6 6', the driving force passes through the fixed axis and no motion is 
 
 54 '3210 
 
 FIG. 412. 
 
 possible. Some other mechanism is necessary to pass over the dead-points; usually this 
 is by means of the fly-wheel, in which the force stored in the mass by rotation continues 
 the motion, and rightly proportioned nearly with uniformity, and the irregularities are 
 transferred to the stroke, Fig. 413. 
 
 654 3 A 2 10 
 
 FIG. 413. 
 
 If the crank is the driver there will be no dead-point ; the circular movement of the 
 crank-pin will be converted into the rectilinear and reciprocating one of the crosshead. 
 
 In locomotives the cranks from the cylinders are put at right angles, that there may 
 be no dead-point. 
 
 STANHOPE LEVERS. 
 
 In the Stanhope hand press (Pig. 414) the extreme power is obtained by a combina- 
 tion of levers in which the greatest pressure is exerted when the platen has reached the 
 type. A is a fixed point on which the bell-crank lever B revolves, and to the extremity 
 of which the power is applied, and which describes the circle designated by equal arcs, 
 0-10 ; at the angle & of B there is a connection C with the lever D revolving on a fixed 
 centre, d; under this action the point of the lever D, at which the pressure is exerted, 
 passes through the small arc 0-10, corresponding to the arcs of the extremity of the 
 
MECHANICS. 
 
 213 
 
 lever B, but always with decreasing lengths, as shown on the figure, and consequently 
 increasing pressure. 
 
 To obtain the positions of the lever from the path of the lever B, from A as a centre, 
 and with a radius equal to A 0, describe an arc the path of the extremity of B, and 
 
 FIG. 414. 
 
 divide into equal arcs 0-10 ; on A as a centre describe another arc, A &, and from any 
 point, say 5, with a radius equal to 60, describe an arc 5", intersecting the arc &; from 
 d as a centre, with a radius equal to d c, describe an arc, and from 5", with a radius equal 
 to &c, intersect the previous arc, and through this point draw a radial line from d; with 
 a radius equal to d 0, describe an arc for the path of the extremity of the lever D ; the 
 intersection of the radial with this arc will give the point 5 on the smaller arc, corre- 
 sponding to 5 of the larger. In the same manner other positions can be determined. 
 
 
 
 WHITWORTH'S QUICK RETURN MOTION. 
 
 Fig. 415 is known as Whitworth's Quick Return Motion, and consists of a driven 
 crank, a ; moving at a uniform speed, to the end of the crank is a block, d, sliding in a 
 slotted link;, this link is fastened to a fixed centre, e, and has a vibrating motion. A 
 
 FIG. 415. 
 
 tangent drawn from each side of the crank-path to the point e will give the extreme 
 movements of the link c, and a line at right angles to this line and from the centre of the 
 crank-shaft on each side will divide the crank-path in two parts. The tool-holder, /, or 
 planer platen, travels over an equal space while the crank moves from d' to d as from d to 
 d' ; it follows, therefore, as the crank is moving uniformly, the motion of the tool-holder 
 while cutting is slow and the return quick. 
 
214 
 
 MECHANICS. 
 
 FIG. 416. 
 
 FIG. 417. 
 
MECHANICS. 
 
 WATT'S PARALLEL MOTIQJ*. 
 
 Watt's parallel motion is shown in Fig. 416, the 
 object of this motion being to obtain a rectilinear mo- 
 tion of the piston-rod and air-pump rod under the 
 varying angularity of the beam. 
 
 In the figure the line CD indicates the centre of 
 the beam ; from the point H, situated midway between 
 C and D, and from D, are suspended two links, H I 
 and D E, of equal length. Connect E and I by a rod 
 equal in length to D H ; the point I is then attached 
 to the fixed centre at O, and is equal in length to E I, 
 with an angle E I O equal to the angular motion of the 
 beam. By the movement of the beam the point H is 
 thrown outward from C and I inward an equal amount ; 
 the centre point, K, connected with the air pump, takes 
 a nearly vertical movement, and the same may be said 
 of the point E ; the heavy lines illustrate the position 
 of the various levers when the beam is in the middle 
 of its travel. 
 
 It is the practice in this country to use crossheads 
 and guides for the piston rather than parallel motions. 
 
 Fig. 417 represents another form of parallel motion 
 to preserve the perpendicularity of the piston - rod 
 against the varying angle of the connecting-rod. 
 
 This motion consists of two pairs of equal radial 
 bars, O I and O' I', moving in planes parallel to that of 
 the circle of revolution of the crank, and on opposite 
 sides of the piston-rod and connected by a link 1 1', 
 while the centres O and O' are fixed at points at equal 
 distances from the centre of motion and on opposite 
 sides; the communication with the piston-rod is at C, 
 which point moves nearly perpendicularly. 
 
 The dotted lines show the positions of rods and 
 levers when the crank is at the extreme point of its 
 revolution. 
 
 JANNEY CAR COUPLER. 
 
 Fig. 418 is a drawing of the Janney car-coupling 
 device, the lower portion being in section to explain 
 the internal mechanism, and the upper portion a top 
 view, showing the exterior appearance. 
 
 The section and top view of the coupler, in dotted 
 lines, represent the couplers approaching each other 
 for coupling, and the section and portion in solid lines 
 show the coupling completed. 
 
 The knuckle B represents a pinion with two teeth 
 B' B a , and the drawhead A and tooth or nose, B 3 , of 
 the other coupler representing the rack. When the 
 couplers are brought together the teeth engage each 
 other, the hub revolves, and the long tooth B' is carried 
 around to a locking position, the catch C being forced 
 back by the circular end of the tooth B' in passing, 
 after which the catch is returned to its locking position 
 by the catch spring F. 
 
 \ 
 
216 
 
 MECHANICS. 
 
 The uncoupling is effected by throwing over a platform lever, which in turn forces 
 around the coupler lever D and catch C, allowing the tooth B' to revolve outward. 
 
 VALVE MOTION. 
 
 To establish the reciprocating motions of the piston, valves must be moved so as to 
 alternately let the steam into one end of the cylinder and permit it to escape at the other. 
 The mechanism by which the valves are moved are usually by means of an eccentric on 
 the crank-shaft, and a strap and rod connecting it with the valve rod. 
 
 The motion of the rod is the same as if the eccentric were a small crank. 
 
 FIG. 419 
 
 Fig. 419 shows six positions of one of the old slide valves and piston, together with 
 the position of the crank and eccentric for each movement of the valve. 
 
MECHANICS. 
 
 217 
 
 In Fig. 419 the valve is merely sufficient to cover the ports, and at the slightest 
 movement in either direction passages wfll be opened for steam to the cylinder, and the 
 escape of the exhaust from it ; under these conditions there can be no cut-off, and con- 
 sequently no economy from expansion of steam. 
 
 FIG. 420. 
 
 For common engines the cut-off is effected by the lap and lead of the valve (Fig. 
 420). When the valve is placed centrally over the ports, the portion of the valve over- 
 lapping the steam ports is known as lap : when on the steam side, it is designated as 
 outside lap, and as inside lap when on the exhaust side. In quick-running engines, both 
 steam and exhaust ports are opened before the completion of the stroke; on the steam 
 side there is a cut-off or closing of the valve before the completion of the stroke to ad- 
 mit of expansion, and before the completion of exhaust for compression, thus saving heat 
 and relieving the pressure and, consequently, the friction of the slide valve. 
 
 The channels, usually three in number, alternately exposed and covered during the 
 movement of the valves, are called ports. The ones admitting steam to the ends of the 
 cylinder are the steam ports ; the central one giving exit of the steam from the cylinder is 
 the exhaust port ; the spaces between the ports are called bridges. The working surfaces 
 on valve and cylinder are valve and valve seat faces. 
 
 The width of opening of the steam ports at the beginning of the stroke for the ad- 
 
218 
 
 MECHANICS. 
 
 mission or release of steam is termed lead ; on the steam side outside lead ; on the ex- 
 haust side inside lead. When the valve is placed at half stroke over the ports, the 
 amount which it overlaps each port, either internally or externally, is the lap ; on the 
 steam side outside lap , on the exhaust side inside lap. Laps and leads by themselves 
 refer to outside laps and leads. The amount by which the valve has travelled beyond 
 its middle position when the piston is at the end of the stroke is the linear advance. 
 
 The proper understanding of the positions of valve under the different positions 
 during the stroke are of great importance to the draughtsman. It is very common in a 
 shop to have a model of an eccentric in which the throw can be readily changed and 
 applied with a small rod for connection to a simple section of valve and ports. But the 
 valve motion may be illustrated by diagram. 
 
 VALVE DIAGRAMS. 
 
 The separate drawings of the movements of a valve may be shown in one general 
 diagram containing the continuous movement of the valve ; for illustration, the valve of 
 the last engraving is taken. 
 
 Draw a perpendicular line (Fig. 421), E D, called the datum line ; from the centre, C, 
 of this line describe a circle equal in radius to that of the eccentric ; for the first position 
 
 FIG. 421. 
 
 of the eccentric draw a line perpendicular to D E of one inch and five sixteenths (one 
 inch lap and five sixteenths lead) till it intersects the path of the eccentric at 1. Draw 
 a line from C to 1 ; this is the first position of the eccentric ; from this point divide half 
 the circle into six equal parts, representing six positions of the eccentric ; from each of 
 these points draw a horizontal line till it intersects the datum line ; these lines give the 
 linear movement of the eccentric!, and therefore the travel of the valve. From the centre 
 C draw a circle equal in radius to that of the crank; commencing at 1', this being the 
 first position of the crank, divide half the circle into six equal parts, the positions num- 
 bered 1' 2' 3' on the crank-path corresponding to 1, 2, 3 of the eccentric path ; on the crank 
 
MECHANICS. 
 
 219 
 
 circle construct a square, F, G, H, I, and through each of the points of the crank-path 
 draw lines parallel to the side of the square. Draw a horizontal dotted line A B through 
 C, and from A as the centre of the exhaust port lay off the ports and bridges. Above 
 and below this draw parallel lines abed ef across the square, indicating the limits of 
 the exhaust opening and ports on both sides of the drawing, also horizontal lines g and<?' 
 to define the position of the valve when placed centrally over the ports, and draw a sec- 
 tion of the valve according to the dimensions given. Lay off the distance from D E to 
 the first position of the eccentric below g' on the line F H, and draw an outline of the 
 valve as moved into this position. Lay off on each successive perpendicular line 2', 3', 
 etc., from <?', the corresponding positions of the eccentric as measured from DE, and 
 draw a curve through these points. Lay off the position of the edges of the valve hijk 
 on the lines 2', 3', etc., above the determined points 2 2 , 3 2 , etc., and draw curves through 
 these points. 
 
 In this position of the valve the steam port has been opened by a distance equal to 
 pic; through^? draw a horizontal line intersecting the elliptical curve at a', draw shade 
 lines from a to the curve below ; this shade represents the continuance of the steam in 
 the cylinder; the intersection a' is the point of cut-off and commencement of expansion. 
 On the other side the cylinder is wide open to the exhaust, as shown by M, and is closed 
 where the elliptical curve intersects the line e. The waste steam becomes compressed and 
 is shown at N, and is utilized on the return stroke, as shown at L. The period of admis- 
 sion against the piston is shown where the line of the curve intersects a at o, and is 
 shown by the small space O at the commencement of the back stroke and at P for the 
 forward stroke. 
 
 This diagram entirely disregards the angularity of the connecting-rod. 
 
 With a single valve it is difficult to secure economy in the cut-off; it is usual in 
 the larger sizes of stationary engines to have steam and exhaust valves moved by separate 
 mechanisms, and independent of each other, and to regulate the engine by a direct con- 
 nection of the steam valve with the governor. 
 
 Among this class of engines, the widest known is the CORLISS, of which Fig. 422 is 
 a side elevation of one of the simplest and earliest forms. 
 
 FIG. 422. 
 
 The exhaust valves at the bottom of the cylinder are connected positively with the 
 wrist-plate w, vibrated by a hooked connection with an eccentric on the engine shaft. 
 Connected with the wrist plate are two vibrating levers, R R', to the upper ends of which 
 
220 
 
 MECHANICS. 
 
 lever pawls, pp, are attached, which rest on the stems, s, of the steam valves. On the 
 stems are notches against which the pawls strike, push back the stems, compress the in- 
 closed spring, and open the valves, and this continues till the outer end of the pawl 
 lever, coming in contact with the head of the lever a, controlled by the governor, releases 
 the spring which closes the valve. The cases, 5, are cylindrical, with air cushions at the 
 ends. 
 
 Fig. 813 shows sections of the valves of which the ports are very long and narrow. 
 In their construction, the valves may be considered cylindrical plugs, of which portions 
 near the ports are cut away for the prompt admittance and exhaust of steam ; the valves 
 are fitted on the lathe and the seat by boring. The motion given to the valves is rock- 
 ing, but the valves are not firmly connected to the rocking shaft ; in the figure the valves 
 are shown shade lined, and the shaft or stem plain ; the valves are not affected by the 
 packing of the valve stem, but always rest upon the face of the ports. 
 
 FIG. 423. 
 
 Since the lapse of the Corliss patent, it may be considered the most popular form of 
 design, and its manufacture has been taken up by many shops, which, while in the 
 matter of valves and cut-off coming within the claims of the old patent, have been much 
 improved in the details of mechanism. The dash-pot is now set vertically ; the plunger 
 acts without springs by gravity. 
 
 Fig. 423 is the side elevation of a Corliss engine of the Fishkill Landing Machine 
 Company, of which the details of the dash-pot are given in Fig. 424. It consists of a 
 cylinder, A, of two different diameters, to which is fitted the double-diameter plunger, 
 with grooved air packing, the one in the cylinder, the other on the lower plunger. At 
 the commencement of the lift the plunger is at the bottom ; as it is raised, a vacuum is 
 formed beneath the plunger, partial, as air is supplied from the annular space x x through 
 
MECHANICS. 
 
 221 
 
 the channelways e e'. but the vacuum increases with the increase of the space y at the 
 cut-off. 
 
 The plunger's descent is rapidly retarded by a partial vacuum in e e', and at the end 
 of its stroke by the compression of the air in y, and seats without pounding. By the 
 
 FIG. 424. 
 
 valve v in the passage e e' the flow of air between the upper and lower end of the cylinder 
 is controlled to produce this effect. 
 
 Figs. 425, 426, and 427 are details of the releasing mechanism of the same engine. 
 
 To the valve stem A is attached a single-armed lever, to which is suspended the 
 dash-pot connection Y, and at the extremity the steel catch-plate c. The double crank 
 
 FIG. 425. 
 
 FIG. 426. 
 
 C C' rocks on the valve stem ; to its lower end there is a connection X with the wrist- 
 plate, from which it receives motion ; at its upper extremity C there is a small rocking 
 
222 
 
 MECHANICS. 
 
 lever and hook E which is pressed inward by the spring /, and the hook E, engaged 
 with the catch-plate c, gives the motion of the wrist-plate to C C', opens the valve, and 
 
 FIG. 427. 
 
 raises the piston of the dash-pot. The governor-rod Z moves the triple crank H H' H 3 , 
 rocking on the valve stem A ; at the extremity of H' there is a roller R which is thrown 
 by the governor into such a position that it comes in contact with the roller R' on the 
 hook E in its motion upward, and is thrown outward, disengaging the hook E from the 
 catch c; this arm, which is connected with the dash-pot, instantly falls, the valve closes, 
 
 FIG. 428. 
 
MECHANICS. 
 
 223 
 
 and steam is cut off sharply; the return stroke again 
 engages the catch and hook, and the release is again 
 effected by the position of the governor. 
 
 On the arm H a there is an adjustable cam W, 
 which serves as an automatic safety stop-motion. 
 When the engine is at its lowest normal speed, and 
 the hook E is at the point of engagement with the 
 valve lever B, the roller R' comes nearly in contact 
 with the camW. Should the balls fall below the 
 point corresponding to the lowest normal speed, the 
 bell- crank H will move in the direction of the arrow; 
 the cam W will come in the way of the roller R', 
 which will ride on the top of it, thus preventing the 
 hook E from engaging with the end of the valve 
 lever B, and the valve will remain closed. No steam 
 being admitted, the engine will stop. 
 
 Link motion (Fig. 428) is a mechanism by which 
 the travel of the valve is changed and the motion of 
 the piston reversed at the option of the engineer. 
 It consists of two eccentrics at nearly opposite points 
 on the crank-shaft, each of which is connected by 
 rods to the opposite ends of a shifting link A; with- 
 in the link is a sliding block B ; the block B is at- 
 tached to a rocker R. The link is suspended from a 
 bell-crank C, controlled by the engineer through the 
 rod a. As the link A is raised or lowered it slides 
 on the block B, of which the movement becomes 
 more or less, or is changed in direction according to 
 its position in the link, and this motion is transferred 
 to the valve through the rod r. 
 
 This is a common form of link motion, but the 
 same effect is obtained by a fixed link and a movable 
 block without a rocker shaft connected by a link to 
 the valve rod. 
 
 DIAGRAM OP LINK MOVEMENT. 
 
 The horizontal and vertical movement of the dif- 
 ferent positions of the sliding block of the stationary 
 link in connection with the eccentrics is shown in 
 Fig. 429. 
 
 The link is attached to a radius rod attached to a 
 centre O, which compels the centre of the link to vi- 
 brate in an arc c c'. The crank-path is divided into a 
 number of equal parts, say 12, commencing at its first 
 position, and the forward and backward eccentric 
 paths are similarly divided, also commencing at the 
 first position. The length of the eccentric rod is 
 then taken in the compasses and an arc described 
 from the point 1 of the fore eccentric, and another 
 from 1' of the back eccentric. The arcs for the fore 
 eccentric are struck above the centre horizontal line, 
 and for the back eccentric are struck below this line. 
 An arc is now struck from the centre of the link, 
 
224 
 
 MECHANICS. 
 
MECHANICS. 
 
 225 
 
 16 
 
226 MECHANICS. 
 
 with a radius equal to the height at which the sliding block is in the link ; where this 
 arc intersects that struck from the fore eccentric will indicate the point at which the 
 sliding block will be when the gear is full forward, and the same arc described, cutting 
 the arc from the back eccentric, will give the point of the sliding block in the link 
 when in full gear backward; and an arc described through these two points, with a 
 radius equal that of the link,, will give the centre line of the link when the eccentrics 
 are in the position indicated. 
 
 The other positions of the link are obtained in a similar manner. 
 
 In describing the centre line of the link, numerous trials are necessary to get the 
 proper centre of the arc passing through the two points, and where many positions of 
 the link are necessary it will expedite the work by making a templet of the link in card- 
 board, which can be readily applied to the several series of points. 
 
 The upper line on the link movement connecting the series of points indicates full 
 forward gear. The arc cc', mid-gear. The lower line, full gear backward. 
 
 This diagram also shows the position of the valve in forward and backward stroke. 
 
 JOY'S VALVE GEAR. 
 
 Figs. 430 and 431 are drawings of Joy's valve gear in different positions. This gear 
 differs from the link motion in that the valve motion, instead of being given by means of 
 eccentrics, is imparted directly from the connecting-rod. 
 
 At a point A of the connecting-rod a link B is attached, the movement of this link 
 being controlled by the radius rod C attached to a stationary point I of the engine ; from 
 a point D of the link, movement is given to the lever E; from the upper end E' of this 
 lever, motion is given to the valve spindle G. The centre F has also a vertical movement 
 due to the vibration of the connecting-rod at A ; the centre or fulcrum F of the lever 
 E is therefore carried in blocks sliding in slots of the link K, which has a radius equal 
 to the length of the valve spindle G. In Fig. 431 the link is shown at mid-gear; this 
 link can be partially rotated by a point on the outside of the link corresponding to the 
 point F of the lever E, shown in Fig. 430. When the link is thus inclined, as shown at X 
 and Y, the vertical movement of E causes the blocks of the link and the centre F to trav- 
 erse a path inclined to a vertical line. The centre F has therefore a horizontal move- 
 ment, the amount of which depends on the obliquity of the link. It is by this means 
 that the cut-off of steam and the forward and backward movement of the engine is con- 
 trolled. 
 
 The dotted ellipse A N indicates the path of the point A of the connecting-rod, and 
 the dotted curve beneath shows the path of the pin D, which partakes of the motion of 
 the point A of the connecting-rod and of the extremity of the radius rod E. 
 
 WALSCHAERT VALVE GEAR. 
 
 Figs. 432, 433, 434, 435 are drawings of the Walschaert valve gear. In this gear 
 the valve derives its motion partly from the piston-rod crosshead and partly from a single 
 eccentric moving a vibrating link. 
 
 To the crosshead A is attached a fixed arm B; to this arm is attached the link C, 
 which communicates its motion to the combination arm D. To the upper end of this 
 arm is connected the valve stem d; just below this connection the combination arm is 
 pivoted to a centre e ; the valve derives the other element of its motion from the radius 
 rod E attached to the centre e and to the slider in the vibrating slotted link F, which is 
 hung externally at its centre/*; the link receives its motion from the eccentric rod G 
 attached to its lower end and connecting with the eccentric g. 
 
 The sliding of the rod F in the slotted link by means of the levers, shown in dotted 
 lines, is the means adopted for giving an earlier or later cut-off and forward or back- 
 ward gear. 
 
 Skeleton diagrams (Figs. 433 to 435) illustrate the movement of this gear : 
 
MECHANICS. 
 
 227 
 
228 
 
 MECHANICS. 
 
 
MACHINE DESIGN AND MECHANICAL 
 CONSTRUCTIONS. 
 
 IN the designing of new machines and mechanical constructions, the draughtsman 
 must draw from his knowledge of well-known forms and parts, and combine them ; but, 
 to proportion them properly, and adapt them to the purposes required, he must under- 
 stand the stresses to which they are to be subjected, and the action and endurance of the 
 material to be used, to withstand these stresses. 
 
 In the present technical application of the term, stress is confined to a force exerted 
 between two bodies or parts of a body, such as a pull, push, or twist. Strain is the 
 alteration produced by a stress. Stress is the cause, strain the effect ; the first is meas- 
 ured by the load, the latter by the deformation of the body produced by the first. A 
 stress, not greater than the elastic limit of the material acted upon, produces a strain 
 which disappears as soon as the load is removed : up to this limit the strain is propor- 
 tional to the stress ; beyond, the strain increases faster than the stress, up to the point 
 of rupture. The elastic limit is a percentage of the breaking strain, varying with the 
 kind of material and application of stress. Stress is usually designated as load, meaning 
 thereby the sum of all the external forces acting on the member or structure, together 
 with its weight. 
 
 Dead load, or weight, is a steady, unchangeable load. Live loads are variable, alter- 
 nately imposed and removed, or varying in intensity or direction. It is usual, in design- 
 ing constructions, to proportion the parts to resist a much greater load than will be 
 brought on them in the structure ; the load is multiplied by a factor termed factor of 
 safety, as a security against imperfections in material and workmanship, contingencies of 
 settlement, and other incidental stresses. But it must be observed that these imperfec- 
 tions are such as can not be seen and met ; there can be no factor of safety to provide 
 for poor and unknown material and defective workmanship. 
 
 The factor of safety adopted for dead loads varies but little with the same kind of 
 material ; but for live loads the factor varies not only with the material, but with the 
 character of the stresses, whether they are applied and relieved gradually or suddenly ; 
 whether they only vary in intensity, or also in direction, alternately compressive or ten- 
 sile. In this latter case the load should never be considered less than the sum of the 
 stresses, with a large factor of safety. Vibrations, shocks, and changes in the direction 
 of stresses concentrate the strains at the weakest point of the construction, and rupture 
 takes place at these points, which would be adequate to the strain if the form through- 
 out were uniform with that at these points. Thus, boiler-plates show wear just at the 
 "edge of the lap of the sheets, and ear-axles (Fig. 436), with sharp angles at the journals, 
 are known to break after a time, while under the same stresses an axle of uniform size 
 with the journal would not break; nor if a slight cove J inch radius (Fig. 437) be made 
 in the angle to distribute stress. 
 
 Besides provisions for strength, the draughtsman should understand the necessities 
 of the construction, and the character of the material to be used. He should know 
 what parts of the design are to be forged, cast, framed, and how it is to be done. He 
 should know what wear is to be met, and what waste, as rust or rot, to be provided for. 
 
 229 
 
230 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 This knowledge can only be arrived at by reference to examples of practice and by ob- 
 servation of results under similar conditions of use and time. 
 
 FIG. 436. 
 
 The stresses to which constructions and parts of constructions are subjected are the 
 tensile or stretching stress, tending to lengthen a body in the direction of the stress ; the 
 compressive or crushing stress, tending to shorten a body in the direction of the stress; 
 the shearing or cutting stress, tending to elongate, compress, and deflect ; the torsional or 
 twisting stress, the effect being an angular deflection of the parts of the body ; and the 
 transverse or lateral stress, tending to bend the body or break it across. 
 
 In Appendix is given a table of the strength of various metals to resist compression 
 and tensional stresses, and examples will hereafter be given of varied constructions, with 
 their usual or required factors of safety ; but, for a practical rule for the common neces- 
 sities of the above stresses, under dead loads, 10,000 pounds per square inch for wrought- 
 iron and 12,000 for steel may be considered perfectly safe. 
 
 Posts in structures are subjected to compressive stresses ; but, as the action is modi- 
 fied somewhat by a tendency to bend, depending on the proportion of the length to the 
 diameter, and the material of which they are composed, the usual tables of crushing 
 strength are not generally applicable, and the formulae to be depended on are those de- 
 duced from practical tests. The best tests of wooden posts are those made by Professor 
 Lanza, for the Boston Manufacturers' Mutual Fire-Insurance Company, and the following 
 are the results : 
 
 "That the strength of a column of hard pine or oak, with flat ends, the load being 
 uniformly distributed over the ends, is practically independent of the length, such 
 columns giving way by direct crushing, the deflection, if any, being very small. Tests 
 were on columns 6" to 10" diameter x 12 feet. The average crushing strength of very 
 highly seasoned, hard pine was 7,386 pounds per square inch. Some very slow-growth 
 and highly seasoned, 9, 339 pounds; very wet and green, 3,015 pounds; seasoned about 
 three months, 3,400 pounds; not very well seasoned and not very green, 4,400 to 4,700 
 pounds. The average of two specimens of thoroughly seasoned white-oak, 7, 150 pounds ; 
 for green and knotty, average, 3,200 pounds. Spruce, nearly 5,000 pounds. White- 
 wood, 3,000 pounds. 
 
 "That it is a mistake to turn columns, taper, or even turn them at all, square col- 
 umns being much stronger, cheaper, and better, and that accuracy of fitting is of great 
 consequence, that the stress may be directly vertical." The professor recommends that 
 longitudinal holes be bored through the centre of columns to allow of the circulation of 
 air (in the experiments the holes were 1'7" diameter), and that iron caps be used instead 
 of wooden bolsters, as the wooden bolster will fail at a pressure far below that which 
 the column is capable of resisting, and the unevenness of pressure brought about by the 
 bolster is sometimes so great as to crack the column. He also recommends horizontal 
 holes in the iron caps to connect the longitudinal ones in the column with the outer air. 
 
 From the whole of the experiments, we estimate the safe load, for fair-grained, well- 
 seasoned oak or yellow-pine columns, to be from 1,000 to 1,500 pounds per square inch; 
 for the more imperfect and green specimens, from 300 to 500 pounds ; for good speci- 
 mens of whitewood, about 300 pounds ; and of spruce, about 500 pounds. 
 
 Cast-Iron. For the columns of buildings where the load is dead, cast-iron is very 
 generally used. They are, in interiors, mostly of circular section, but for outer columns 
 forms are used suited to the necessities of their position or style of architecture. They 
 admit of considerable ornamentation and finish direct from the mould; but, as they are 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 231 
 
 liable to defects not readily detected in the process of casting, the factor of safety is 
 usually taken as high as 5. To protect them against the effects of fire and water in con- 
 flagrations, they should be covered with a shell of light refractory material, as porous 
 brick or tile. 
 
 The experiments of Hodgkinson are the usual basis of all formula? on the strength of 
 circular cast-iron columns, and the ends of all columns are now required to be faced by 
 architects and by the rules of building departments, since Mr. Hodgkinson states this 
 rule, that ' ' in all long columns, of the same dimensions, the resistance to fracture by 
 flexion is three times greater when they are flat and firmly bedded than when they are 
 rounded and capable of moving." 
 
 Table of the safe load of solid cylindrical columns, with flat ends calculated with a 
 factor of safety of 5. 
 
 TABLE OF SAFE LOADS FOE SOLID CAST-IKON COLUMNS, WITH FLAT ENDS. 
 
 Diam. 
 
 8' 
 1,000 
 
 11.5. 
 
 9' 
 1,000 
 Ibs. 
 
 10' 
 1,000 
 Ibs. 
 
 ll' 
 
 1,000 
 Ibs. 
 
 12' 
 
 1,000 
 Ibi. 
 
 13' 
 
 1,000 
 Ibs. 
 
 14' 
 1,000 
 Ibs. 
 
 15' 
 1,000 
 Ibs. 
 
 16' 
 1,000 
 Ibs. 
 
 17' 
 
 1,000 
 Ibs. 
 
 18' 
 1,000 
 Ibs. 
 
 19' 
 1,000 
 Ibs. 
 
 20' 
 1,000 
 Ibs. 
 
 21' 
 1,000 
 Ibs. 
 
 22' 
 1,000 
 lb. 
 
 23' 
 
 1,000 
 Ibs. 
 
 24' 
 
 1,000 
 
 11*. 
 
 8" 
 
 29- 
 
 23- 
 
 20- 
 
 17- 
 
 14- 
 
 18- 
 
 11- 
 
 10- 
 
 9- 
 
 8- 
 
 7- 
 
 7- 
 
 6- 
 
 6- 
 
 6- 
 
 5- 
 
 4- 
 
 3}" 
 
 40- 
 
 31- 
 
 26- 
 
 22- 
 
 19- 
 
 17- 
 
 lS- 
 
 13- 
 
 12- 
 
 11- 
 
 10- 
 
 9- 
 
 8- 
 
 7- 
 
 7- 
 
 6- 
 
 6- 
 
 3*" 
 
 50- 
 
 41- 
 
 84- 
 
 29- 
 
 25- 
 
 22- 
 
 19- 
 
 17- 
 
 15- 
 
 14- 
 
 12- 
 
 ii- 
 
 10- 
 
 10- 
 
 9- 
 
 8- 
 
 8- 
 
 3J" 
 
 63" 
 
 54- 
 
 48- 
 
 37- 
 
 82' 
 
 23- 
 
 24- 
 
 2^- 
 
 19 
 
 18- 
 
 16- 
 
 is- 
 
 18- 
 
 12- 
 
 11- 
 
 11' 
 
 10- 
 
 4" 
 
 77- 
 
 66- 
 
 54- 
 
 46 
 
 40' 
 
 85- 
 
 81- 
 
 27- 
 
 24- 
 
 22- 
 
 20- 
 
 18- 
 
 17- 
 
 15- 
 
 14- 
 
 18- 
 
 12- 
 
 41" 
 
 92- 
 
 80- 
 
 70- 
 
 57' 
 
 49" 
 
 43- 
 
 88- 
 
 34- 
 
 80- 
 
 27- 
 
 25 
 
 23- 
 
 21 
 
 19- 
 
 18- 
 
 16- 
 
 15- 
 
 44" 
 
 no- 
 
 96- 
 
 84- 
 
 74- 
 
 61' 
 
 53- 
 
 47- 
 
 41- 
 
 87- 
 
 83- 
 
 30- 
 
 28" 
 
 25- 
 
 23- 
 
 22- 
 
 20- 
 
 19- 
 
 H" 
 
 130- 
 
 113- 
 
 99- 
 
 88- 
 
 73- 
 
 64- 
 
 5B- 
 
 50- 
 
 45' 
 
 41- 
 
 87- 
 
 84 
 
 81- 
 
 28- 
 
 26- 
 
 24' 
 
 23- 
 
 5" 
 
 152- 
 
 133- 
 
 117- 
 
 103- 
 
 92' 
 
 77' 
 
 68- 
 
 60- 
 
 54- 
 
 49- 
 
 44- 
 
 4(1- 
 
 87- 
 
 84- 
 
 81- 
 
 29- 
 
 27' 
 
 51" 
 
 176- 
 
 154- 
 
 136- 
 
 121- 
 
 108- 
 
 97" 
 
 81- 
 
 72- 
 
 64- 
 
 S8- 
 
 58- 
 
 48- 
 
 44- 
 
 40- 
 
 87- 
 
 85- 
 
 32- 
 
 54" 
 
 201- 
 
 177- 
 
 157- 
 
 14:f 
 
 125- 
 
 113 
 
 95- 
 
 85- 
 
 76- 
 
 68- 
 
 62' 
 
 57' 
 
 52- 
 
 48- 
 
 44- 
 
 41- 
 
 88- 
 
 5J" 
 
 230- 
 
 203- 
 
 ISO- 
 
 161- 
 
 144' 
 
 180- 
 
 118- 
 
 99- 
 
 89" 
 
 80- 
 
 73- 
 
 66- 
 
 61- 
 
 56- 
 
 52- 
 
 48- 
 
 45- 
 
 6 ' 
 
 260- 
 
 230' 
 
 205- 
 
 18}' 
 
 165- 
 
 149- 
 
 J35- 
 
 115- 
 
 103- 
 
 98- 
 
 84' 
 
 77- 
 
 7r 
 
 65- 
 
 60- 
 
 56- 
 
 52- 
 
 6J" 
 
 292- 
 
 260- 
 
 232- 
 
 203- 
 
 187' 
 
 169- 
 
 154- 
 
 140- 
 
 119- 
 
 108- 
 
 98- 
 
 89- 
 
 82' 
 
 75" 
 
 69- 
 
 64- 
 
 60- 
 
 6i" 
 
 327- 
 
 292- 
 
 2H1- 
 
 234- 
 
 212- 
 
 192- 
 
 174- 
 
 159- 
 
 146- 
 
 124- 
 
 112- 
 
 102- 
 
 94- 
 
 86- 
 
 80- 
 
 74- 
 
 C9- 
 
 6J" 
 
 864- 
 
 326- 
 
 292- 
 
 263 : 
 
 238- 
 
 216- 
 
 197- 
 
 180- 
 
 .163- 1 141- 
 
 128- 
 
 117' 
 
 107- 
 
 99- 
 
 91- 
 
 85- 
 
 79- 
 
 7" 
 
 404- 
 
 362- 
 
 325- 
 
 293- 
 
 266" 
 
 242- 
 
 221- 
 
 202- 
 
 186- IT!' 
 
 146- 
 
 133- 
 
 122- 
 
 112- 
 
 104- 
 
 96- 
 
 90- 
 
 71" 
 
 445- 
 
 40D- 
 
 361- 
 
 326- 
 
 296" 
 
 269- 
 
 24G- 
 
 2-26- 
 
 208- 
 
 192- 
 
 177" 
 
 151- 
 
 138- 
 
 127- 
 
 118- 
 
 109- 
 
 101- 
 
 q 
 
 489' 
 
 441- 
 
 3 8- 
 
 361- 
 
 328" 
 
 299' 
 
 274- 
 
 251- 
 
 231- 
 
 214- 
 
 198- 
 
 no- 
 
 156- 
 
 148- 
 
 133- 123- 
 
 114- 
 
 7t" 
 
 536- 
 
 434- 
 
 43^- 
 
 898- 
 
 86^' 
 
 331- 
 
 303- 
 
 5!78- 
 
 257- 
 
 237- 
 
 220- 
 
 204- 
 
 175' 
 
 Ibl- 
 
 149- 
 
 138- 
 
 128- 
 
 8" 
 
 581' 
 
 5^9- 
 
 430- 
 
 436- 
 
 393- 
 
 864' 
 
 a34- 
 
 808- 
 
 284- 
 
 263- 
 
 244- 
 
 227- 
 
 196- 
 
 180- 
 
 167- 
 
 155- 
 
 144- 
 
 8" 
 
 '639- 
 
 626- 
 
 571- 
 
 521- 
 
 477' 
 
 487- 
 
 402- 
 
 871- 
 
 343- 
 
 318- 
 
 296- 
 
 275- 
 
 257- 
 
 241- 
 
 207- 
 
 192- 
 
 178- 
 
 9" 
 
 802- 
 
 733- 
 
 670- 
 
 614- 
 
 564- 
 
 519- 
 
 479 
 
 442- 
 
 410- 
 
 881- 
 
 3S4- 
 
 331- 
 
 809- 
 
 290- 
 
 272- 
 
 235- 
 
 218- 
 
 94" 
 
 926- 
 
 849- 
 
 780- 
 
 717- 
 
 66D- 
 
 603- 
 
 563- 
 
 522- 
 
 484- 
 
 461- 
 
 420- 
 
 393- 
 
 867- 
 
 345- 
 
 324- 
 
 805- 
 
 265- 
 
 Id" 
 
 1058- 
 
 975- 
 
 898- 
 
 829- 
 
 765- 
 
 708- 
 
 656- 
 
 6'I9- 
 
 566- 
 
 528- 
 
 493- 
 
 461- 
 
 432- 
 
 406- 
 
 382- 
 
 360- 
 
 340- 
 
 104" 
 
 1195- 
 
 1108- 
 
 1026- 
 
 957- 
 
 892- 
 
 8-W 
 
 779- 
 
 74H- 
 
 693- 
 
 653- 
 
 610- 
 
 580- 
 
 546- 
 
 511- 
 
 485- 459- 
 
 433- 
 
 ii" 
 
 1359- 
 
 1264- 
 
 1159- 
 
 1033- 
 
 1017- 
 
 950- 
 
 889- 
 
 846- 
 
 793- 
 
 751- 
 
 708' 
 
 660- 
 
 627" 
 
 589- 
 
 561- 542- 
 
 513- 
 
 114- 
 
 1517- 
 
 1413- 
 
 1319- 
 
 122fi- 
 
 1147- 
 
 1080- 
 
 1018- 
 
 956- 
 
 904- 
 
 852- 
 
 810- 
 
 758- 
 
 727- 
 
 691- 
 
 665- 
 
 613- 
 
 587- 
 
 12" 
 
 1674- 
 
 15S8- 
 
 1470- 
 
 1830- 
 
 1289- 
 
 1221 
 
 1142- 
 
 1074- 
 
 1018- 
 
 973- 
 
 916- 
 
 871- 
 
 746- 
 
 701- 
 
 667' 
 
 645- 
 
 611- 
 
 Solid columns are very seldom used in constructions; they are almost invariably made 
 hollow, the shell being \" to 2" thick. To determine the safe load of a hollow column, 
 it will be sufficiently accurate to take from the table the safe load of a column equal to 
 that of the exterior diameter, and subtract from this the safe load of a column of a 
 diameter equal to the core. 
 
 Example. -To find the safe load of a column 12 feet long, 8" exterior diameter, 
 shell |". 
 
 Safe load of 8" column 398,000 Ibs. 
 
 " " "6V " 212,000" 
 
 " " required column 186,000 " 
 
 For square box-columns, it will be safe to estimate that a square column will support 
 as much as a round one, the side of the one being equal to the diameter of the other, 
 and the thickness of shell the same. 
 
 For a star-column (Fig. 438), the load should be about \ less than on a cylindrical 
 column of same diameter and same area of section. 
 
 There is a convenience in the use of cast-iron that the brackets for the support of 
 
232 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 beams and girders can be readily cast on, but it must be done with care in the design 
 and in the moulding. It will be seen (Fig. 341) that weak places occur in the cooling 
 when the junctures are at right angles, which must be avoided by easements, or coves, 
 in the patterns, and by intelligence in the moulding, pouring, and cooling. For many 
 years cast-iron was the only metal used for posts, and with but few accidents from im- 
 perfect construction or from conflagrations. 
 
 Wrought-Iron Columns. With the decrease in the cost of the manufacture of shapes 
 in wrought-iron, columns of this material have largely superseded those of cast-iron in 
 
 FIG. 439. 
 
 FIG. 440. 
 
 constructions liable to varying loads and shocks. Fig. 439 shows the section of a 
 Phoenix column, Fig. 440 of the Keystone. 
 
 TABLE OF PHCENIX COLUMNS. 
 
 MAKK OF COLUMN. 
 
 
 Thick D ess in 
 inches. 
 
 Area in square 
 inches. 
 
 Weight in pounds 
 per foot. 
 
 Internal diam- 
 eter. 
 
 A 
 
 i 
 
 2'8 
 
 9-3 
 
 
 4 segments 
 
 A 
 
 5-8 
 
 19'4 
 
 f 
 
 B 
 
 y> 
 
 <K 
 
 5'0 
 
 16'7 
 
 
 4 segments 
 
 1" 
 
 ]4'8 
 
 51. 
 
 4rt 
 
 B 2 
 
 A 
 
 5'8 
 
 19-4 
 
 
 4 segments. . 
 
 & 
 
 17- 
 
 58-6 
 
 Btl 
 
 C.. 
 
 A 
 
 8'8 
 
 30-3 
 
 
 4 segments. . 
 
 1A 
 
 40' 
 
 138. 
 
 f& 
 
 D..! 
 
 i 
 
 14-0 
 
 48'2 
 
 
 5 segments. . 
 
 4 
 
 26- 
 
 89'7 
 
 * 
 
 B..? 
 
 j 
 
 I8- 
 
 55'2 
 
 
 6 segments 
 
 i, a 
 
 60- 
 
 207' 
 
 11 
 
 F 
 
 .a 
 
 24'5 
 
 84-5 
 
 
 7 segments 
 
 i 
 
 36'4 
 
 125-6 
 
 13 
 
 G 
 
 A 
 
 24- 
 
 82'8 
 
 
 8 segments 
 
 1&. 
 
 80- 
 
 276- 
 
 14f 
 
 
 
 
 
 
 BUILT COLUMNS. 
 
 Open columns should be used in positions exposed to dampness and rust on account 
 of their accessibility to painting and inspection. Built columns present advantages in 
 the facility with which connections can be made to floor beams or bracing rods. 
 
 In determining upon the proper sections for these columns, the material should be so 
 disposed that the tendency to yield will not be greater in one direction than in the other 
 to secure the full strength of the column. 
 
 The rivets at the ends should be pitched not more than three inches apart, and the 
 maximum distance between rivets in the line of stress should not exceed sixteen times 
 the thickness of the plate. 
 
 Figs. 459 and 460 are plan and elevation of a latticed column. 
 
 Spacing of the Lattice or Lacing Bars. The object of these bars is to join the two 
 channels composing the post or chord, and thus cause them to act together; they should 
 be attached at intervals so close that there shall be no danger of failure of the channels 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 233 
 
 between the points of attachment, which never should be more than 0'6 of the interval, 
 and lacing bars are not allowed to make an angle of more than 60 with each other, or 
 less than 60 with the flanges. 
 
 FIG. 441. 
 
 FIG. 442. 
 
 FIG. 443. 
 
 FIG. 444. 
 
 FIG. 445. 
 
 Fio. 446. 
 
 > C 
 
 --D -- 
 
 FIG. 447. 
 
 i D 
 
 !& 
 
 * 
 
 ! 
 
 
 
 FIG. 451. 
 
 FIG. 452. 
 
 FIG. 453. 
 
 FIG. 454. 
 
 FIG. 455. 
 
 FIG. 456. 
 
 FIG. 457. 
 
 FIG. 458. 
 
 On the Strength of Wrought-Iron Columns. In the 
 former editions of this work the strength of wrought y,,,,,. 
 columns was shown by curves of the average breaking 
 loads (Fig. 461), as determined by experiments on the 
 Phosnix, Keystone, Piper, and open columns with flat 
 ends. Vertical distances represent the pounds of load rpz 
 per square inch, the horizontal the proportion of the * 
 
 length of the columns to the diameter ; D is taken as 
 
 the least outside diameter as marked on the varied sections above. The lower curves 
 represent the safe loads, under factors of safety of 3, 4, and 5. In looking at these 
 curves, it will be observed that, within the common limits of prac- 
 tice, of 15 to 35 - these lines may be considered straight ; that 
 
 diameter 
 
 with iron of a breaking strength of 52,000 pounds per square inch, 
 and within the above limits, and a factor of safety of 3, the safe load 
 may be taken at 11,000 per square inch; with a factor of safety of 4, 
 at 8,000 pounds; with a factor of safety of 5, at 6,500 pounds; and 
 that for common and usual purposes 10,000 pounds per square inch 
 is a safe load. 
 
 At present there are few wrought-iron columns 
 manufactured; they are almost invariably steel, but 
 the diagram still represents in their working propor- 
 tions the action of columns under stress. The steel 
 
 When, as ob- 
 
 FIG. 459. 
 
 FIG. 460. 
 
 column is about 20 per cent stronger. 
 
234 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 served above, the common and usual safe load on iron is 10,000 pounds per square inch, 
 that on steel is 12,000 pounds. 
 
 For struts, angle irons, or a combination of them forming an angle or cross (Figs. 441 
 
 L 
 
 D 
 
 and 442), with or without separators, are generally used, and in these cases for D in 
 
 take D equal to 0-8 of the shortest leg or legs. 
 
 It has generally been considered that columns with pin or cylindrical ends had about 
 
 ZO 30 
 
 FIG. 461. 
 
 three fourths of the resisting strength of flat ends, but if the pin ends are closely fitted, 
 so that the strains are uniformly in the direction of the length of the column, the differ- 
 ence is but little between the two kinds of ends. 
 
 Shearing Stresses. Parts of machines and of constructions subjected to these stresses 
 have often the resistances modified by friction, combined with other stresses. The sizes 
 of parts necessary to resist such stresses practically, as in the cases of bolts, rivets, and 
 the like, will be hereafter illustrated by examples and determined by particular rules. 
 In general, the strength to resist shearing stress is, in wrought-iron and steel, from 70 to 
 80 per cent of its tensile strength; in cast-iron, about 40 per cent of its crushing strength. 
 The softer woods, as spruce, white pine, hemlock, resting on walls or girders, will safely 
 sustain a load of 200 to 300 pounds per square inch of bearing surface, and the harder 
 woods, as oak and Southern pine, 300 to 500 pounds. By experiment, oak treenails, 1" 
 to If" diameter, were found to have an ultimate shearing strength of about two tons per 
 square inch of section; but, according to Rankine, the planks thus connected together 
 should have a thickness of at least three times the diameter of the treenails. In 3" planks, 
 If" treenails bore only 1-43 tons per square inch of section; in 6" plank, 1'73 tons. 
 
 Torsional Stress.' Every shaft through which power is transmitted, whether through 
 gears, cranks, or pulleys, is subjected to a torsional stress, of which the power acting 
 tangentially to the shaft in one direction is resisted by the load in an opposite direction. 
 When this stress exceeds a certain limit depending on the material, the fibres are twisted 
 asunder, but much below this limit the elasticity of the shaft may be too great to trans- 
 mit power uniformly. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 235 
 
 The length of the axle subjected to torsion does not affect the actual amount of pres- 
 sure required to produce rupture, but only the angle of torsion which precedes rupture, 
 and therefore the space through which the pressure must be made to act. 
 
 A practical limit of torsional deflection is 1 in a length equal to twenty diameters 
 
 FIG. 462. 
 
 Fio. 463. 
 
 of the shaft or T ^ part of a full turn. D. K. Clark gives the following rule : "To 
 find the diameter of a shaft capable of transmitting a given torsional stress within gaod 
 working limits. Divide the torsional stress in foot-pounds by 18 '5 for cast-iron; 27'7 
 for wrought-iron; and 57'2 for steel. The cube root of the quotient is the diameter of 
 the shaft in inches. 
 
 Example. On the teeth of a 4^-foot gear, the force exerted is 2,800 pounds. What 
 should be the diameter of a wrought-iron shaft to transmit this force safely ? 
 
 The torsional stress will be 2,800 pounds multiplied by the radius of leverage, 2Jfeet, 
 
 or 6,300 foot-pounds = ( 2 = 228, /228 = 6-1. 
 
 Transverse Stress. The strength of a beam is influenced by the manner in which its 
 ends are supported. Where the ends are simply supported that is, resting upon the 
 abutments and the beam loaded with a weight W (Fig. 462), the beam is subjected to a 
 bending movement, or transverse stress, composed of a tensile stress on the lower part of 
 the beam and compressive on the upper part. In addition, the weight of the beam and 
 its supported load act on the abutments as shearing stresses. 
 
 If the ends of a beam are fixed in the wall, however, the transverse stresses are con- 
 siderably relieved throughout the beam, which is thereby capable of sustaining a heavier 
 load. Figs. 464 to 471 are examples of beams with both ends simply supported and 
 both ends fixed, or one end supported and one end fixed, and a comparison of their 
 strength for a centre load and a load uniformly distributed. 
 
 The strength of a square or rectangular beam to resist transverse stress is as the 
 breadth and the square of the depth; and inversely as the length, or the distance from or 
 between the points of support. Thus a beam twice the breadth of another, other pro- 
 portions being alike, has twice the strength ; or twice the depth, four times the strength ; 
 but twice the length, only half the strength. 
 
 It is evident, therefore, that, with the same area of section, the deeper a beam the 
 stronger it will be, if the breadth is sufficient to prevent lateral buckling. 
 
 To cut the best beam from a log, Fig. 463, the section of which is a circle : draw a 
 diameter, divide it into three equal parts, erect perpendiculars at the points of division 
 1, 2, and they will intersect the circumference at the corners of the beam, of which the 
 extremities of the diameter are the other two. 
 
 For the transverse strength of rectangular beams the general formula is W = j , in 
 
 which W is the breaking weight; 8, a number determined by experiment on different 
 materials ; ft, the breadth, and d, the depth in inches ; and ?, the length in feet. 
 
 Figs. 464 to 471 represent the usual methods of loading beams, and the loads as 
 
236 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 drawn represent the comparative strength of beams under these different conditions. 
 Thus, in Fig. 464, the beam supports but one unit of load, while Fig. 465 supports twice as 
 
 FIG. 464. 
 
 FIG. 465. 
 
 much. The formulae given represent the safe dead loads with a factor of safety of 6, 
 deduced from experiments of Mr. C. J. H. "Woodbury on Southern pine. For spruce the 
 
 FIG. 466. 
 
 FIG. 467. 
 
 coefficient would be about \ less, and for live loads the factor of safety should be 12. 
 Beams fixed at one end and loaded at the other (Fig. 464). 
 
 fid* 
 Safe load = 30 -=-. 
 
 Beams fixed at one end and load distributed uniformly, not as represented in the 
 figure, as the two units of weight would be spread over the whole length of the beam 
 (Fig. 465). 
 
 7> fl* 
 Safe load = 60 -f-. 
 
 L 
 
 Beams supported at the extremities and loaded at the middle (Fig. 466). 
 
 ft/7 2 
 
 Safe load = 120 -=-. 
 l 
 
 Beams supported at the extremities and the load uniformly distributed (Fig. 467). 
 
 & d? 
 Safe load = 240 - 
 
 FIG. 468. 
 
 FIG. 469. 
 
 Beams, one end firmly fixed, the other supported, and loaded at the middle (Fig. 468). 
 
 bd? 
 Safe load = 160 -p 
 
 Beams with one end fixed, the other supported, and load uniformly distributed 
 (Fig. 469). 
 
 Id* 
 Safe load = 240-^-. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 Beams with both ends fixed, and loaded at centre (Fig. 470). 
 
 237 
 
 Safe load = 240 
 
 I ' 
 
 FIG. 470. 
 
 FIG. 471. 
 
 Beams with both ends fixed, and load uniformly distributed (Fig. 471). 
 
 bd' 2 
 Safe load = 360 -y-. 
 
 If the loads on a beam be neither at the centre nor uniformly distributed, 
 but at intermediate points, it is important to determine the load which placed 
 at the centre will produce an equivalent stress. 
 
 Thus on a beam of 30 feet span, if a weight of 800 pounds be placed at 10 
 feet from an end. Lay off on any convenient scale (Fig. 472) a horizontal line 
 
 FIG. 472. 
 
 of 30 feet, and at a point 10 feet from one end, draw a vertical or ordinate from 
 this point, and at the ends A and B. 
 
 From any convenient point on the ordinate beneath A draw a line parallel 
 to A B and set off to 0, a distance equal to of the span for the pole distance 
 of the force polygon about to be constructed, which is established at i of the 
 span, in order to determine readily the equivalent central load. From the 
 point a in the ordinate lay off on any scale a a' = 800 pounds, draw lines a 
 and a', and the force polygon is complete. 
 
 From a extend the line a till it intersects the weight ordinate from D at 
 E. Draw E F parallel to a' to intersect the ordinate B ; connect F with a ; 
 F a is called the closing line of the equilibrium polygon. 
 
 The weight 800 pounds is supported by the abutments A and B. Draw a 
 line c in the force polygon parallel to F a ; the distance above this line in 
 
238 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 pounds will be the amount of the load on A, 533 pounds, c a', below the line, 
 267 pounds will be that on B. 
 
 If the weight be placed centrally (Fig. 473) and its material sufficiently 
 elastic to be considered uniformly distributed for its whole length ; to determine 
 its effect under these conditions, draw the equilibrium polygon a E F as if the 
 
 FIG. 475. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 239 
 
 weight of 800 pounds were suspended from a single central point ; divide the 
 weight into any number of equal parts (say 8, of 100 pounds each), and draw 
 ordinates through the centre of these parts. Draw through 1' a line parallel 
 to 1 and consecutively through 2' 3' 4' 5' 6' 7' 8' lines parallel to 2, 3, etc. 
 The central weight as measured on the longest ordinate is 690 pounds. 
 
 If the weight 1,200 pounds (Fig. 474) be distributed on a beam of 40 feet 
 40 -4- say 8 = 5. Lay off at each extremity one half part or 2^ feet, and for the 
 intermediate lengths, 7 parts of 5 feet each, and draw ordinates through these 
 divisions ; follow the diagram construction as in the preceding, each weight 
 being 150 pounds, the longest ordinate is 600 pounds, or one half the total 
 weight of 1,200 pounds, and each abutment load is 600 pounds. 
 
 In Fig. 475 the weights are many and unequally distributed; construct dia- 
 gram as before, always making the pole distance equal to a quarter of the span ; 
 measure the longest ordinate, which is 595 pounds, the equivalent central load. 
 Draw c parallel to the closing line a F ; the total of the weights 1,100 pounds 
 will be supported by abutment A, 500 pounds, and by abutment B, 600 pounds. 
 In all the figures the weight of the beam itself is not taken into account, but, 
 if considered, it is almost invariably of one section and therefore distributed, 
 and its central load will be one half the total weight. 
 
 The following table for the strength of yellow-pine beams is calculated for 
 a safe central load, as determined by the graphic constructions or by taking one 
 half the uniformly distributed load if this is given. 
 
 TABLE OF THE SAFE CENTEAL LOAD OF YELLOW-PINE BEAMS, CALCULATED 
 FROM THE FORMULA 120*^!. 
 
 6 
 
 Span in 
 
 DEPTH IN INCHES OF YELLOW-PINE BEAMS, ONE INCH WIDE. 
 
 feet. 
 
 3 ins. 
 
 4 ins. 
 
 5 ins. 
 
 6 ins. 
 
 Tins. 
 
 8 ins. 
 
 9 ins. 
 
 10 ins. 
 
 11 ins. 
 
 12 ins. 
 
 13 ins. 
 
 14 ins. 
 
 15 ins. 
 
 16 ins. 
 
 4 
 
 270- 
 
 480- 
 
 750" 
 
 1080- 
 
 1470- 
 
 1920' 
 
 2430" 
 
 3000' 
 
 3630- 
 
 4320' 
 
 5070- 
 
 5880' 
 
 6750- 
 
 7680- 
 
 5 
 
 216- 
 
 384- 
 
 600' 
 
 864- 
 
 1176- 
 
 1536- 
 
 1944' 
 
 2400- 
 
 2904- 
 
 3456' 
 
 4056" 
 
 4704- 
 
 5400' 
 
 6144- 
 
 6 
 
 180- 
 
 320- 
 
 500- 
 
 720- 
 
 980- 
 
 1280- 
 
 1620' 
 
 2000- 
 
 2420- 
 
 2880' 
 
 3380' 
 
 3920- 
 
 4500- 
 
 5120- 
 
 7 
 
 154- 
 
 274- 
 
 430- 
 
 616" 
 
 840- 
 
 1097" 
 
 1389" 
 
 1714- 
 
 2074- 
 
 2469' 
 
 2897' 
 
 3360' 
 
 3857' 
 
 4388- 
 
 8 
 
 135- 
 
 240- 
 
 375- 
 
 540- 
 
 735- 
 
 960' 
 
 1215' 
 
 1500- 
 
 1815- 
 
 2160- 
 
 2535' 
 
 2940- 
 
 3375- 
 
 3840- 
 
 9 
 
 120- 
 
 213- 
 
 333- 
 
 480- 
 
 53- 
 
 853' 
 
 1080- 
 
 1333' 
 
 1613. 
 
 1920- 
 
 2253- 
 
 2613' 
 
 3000- 
 
 3413- 
 
 10 
 
 108' 
 
 192' 
 
 300- 
 
 432- 
 
 588' 
 
 768- 
 
 972- 
 
 1200- 
 
 1452. 
 
 1728- 
 
 2028- 
 
 2352' 
 
 2700- 
 
 3072- 
 
 11 
 
 
 175- 
 
 273- 
 
 392- 
 
 535- 
 
 700- 
 
 882- 
 
 1092- 
 
 1320- 
 
 1571- 
 
 1844. 
 
 2140" 
 
 2457" 
 
 2793- 
 
 12 
 
 
 160- 
 
 250- 
 
 360- 
 
 490- 
 
 640- 
 
 810- 
 
 1000- 
 
 vio- 
 
 1440' 
 
 1690' 
 
 I960- 
 
 2250- 
 
 2560- 
 
 13 
 
 
 
 230- 
 
 332- 
 
 452' 
 
 592- 
 
 747- 
 
 923- 
 
 lin- 
 
 1328' 
 
 1560- 
 
 1808- 
 
 2070- 
 
 2363- 
 
 14 
 
 
 
 215- 
 
 308- 
 
 420- 
 
 548' 
 
 693- 
 
 860" 
 
 1037' 
 
 1234- 
 
 1448' 
 
 1680- 
 
 1928- 
 
 2192- 
 
 15 
 
 
 
 
 288- 
 
 392' 
 
 512- 
 
 648- 
 
 800' 
 
 968- 
 
 1155- 
 
 1352- 
 
 1568- 
 
 1800- 
 
 2048- 
 
 16 
 
 
 
 
 270- 
 
 368- 
 
 480- 
 
 607- 
 
 748- 
 
 907' 
 
 1080' 
 
 1267' 
 
 1470- 
 
 1688- 
 
 1920- 
 
 17 
 
 
 
 
 254- 
 
 346- 
 
 452- 
 
 566- 
 
 704- 
 
 854- 
 
 1016' 
 
 1193- 
 
 1384- 
 
 1588- 
 
 1808- ' 
 
 18 
 
 
 
 
 
 327- 
 
 427* 
 
 540" 
 
 668- 
 
 806- 
 
 960- 
 
 1126- 
 
 1307- 
 
 1500- 
 
 1707- 
 
 19 
 
 
 
 
 
 
 404- 
 
 512' 
 
 632- 
 
 764- 
 
 909- 
 
 1067' 
 
 1238- 
 
 1422- 
 
 1616- 
 
 20 
 
 
 
 
 
 
 384- 
 
 486- 
 
 600- 
 
 726- 
 
 864- 
 
 1014- 
 
 1176- 
 
 1350- 
 
 1536' 
 
 21 
 
 
 
 
 
 
 
 463- 
 
 572- 
 
 691- 
 
 823- 
 
 966- 
 
 1120' 
 
 1287- 
 
 1463- 
 
 22 
 
 
 
 
 
 
 
 442- 
 
 546' 
 
 660- 
 
 785- 
 
 922- 
 
 1070- 
 
 1228- | 
 
 1395' 
 
 23 
 
 
 
 
 
 
 
 
 522- 
 
 631' 
 
 752- 
 
 882- 
 
 1023- 
 
 1178- 
 
 1329- 
 
 24 
 
 
 
 
 
 
 
 
 500- 
 
 605- 
 
 720' 
 
 845- 
 
 980- 
 
 1125- 
 
 1280- 
 
 25 
 
 
 
 
 
 
 
 
 
 581- 
 
 691' 
 
 811' 
 
 940- 
 
 1080- 
 
 ^so- 
 
 30 
 
 
 
 
 
 
 
 
 
 558' 
 
 665- 
 
 780- 
 
 904- ' 
 
 1035- 
 
 il 82- 
 
240 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 The table gives the load which the beam can sustain, allowing a certain 
 factor of safety, but the strength given is in excess of stiffness, and in perma- 
 nent construction it is necessary to proportion the beams to bear its load with 
 a certain limited deflection. Cross-lines on the tables represent those limits, 
 but, as usually the weight of construction is much less than that of the change- 
 able loads, and as the change of deflection is that to be guarded against, the 
 weight of construction may be neglected, and it will only be necessary to con- 
 sider the amount of movable loads above these lines. 
 
 The table is deduced from Mr. Woodbury's experiments on yellow pine, 
 of good quality and practical sizes. For spruce he takes loads of about one 
 fifth less. 
 
 The table is intended to be used as a unit of width by which the strength 
 of timber of usual depths and spans can be estimated, by multiplying by such 
 widths as are found in practice ; widths of less than two inches are not used. 
 Mr. Woodbury established the limit of deflection in wooden beams at three 
 
 432 W I 3 
 quarters of an inch for 25-feet span, and his formula is E = , in which 
 
 E, the modulus of elasticity per square inch, is for Southern pine 2,000,000, and 
 for spruce 1,200,000 : W central load in pounds, I the span in feet, b the breadth, 
 h the depth, and d the deflection of beam, all in inches. Using this formula, 
 marks are drawn in each column of depth, above which the loads will be sup- 
 ported stiffly, and below less than recommended ; the formula is only applicable 
 to seasoned wood. Mr. Woodbury's results are confirmed by late experiments 
 on the long-leaved pine by Prof. Johnson for the Forestry Division of the 
 United States Department of Agriculture. 
 
 He gives the shearing strength measured by the cross-section of the beam at 
 about 600 pounds to the square inch, and the crushing strength across the grain 
 at about 1,150 pounds. In the resting of the beam on abutments of masonry it 
 will be difficult to exert a clean shearing stress, and the strength would be in 
 excess of any likely to occur, but the stress would be a crushing one, and be met 
 by the area in square inches resting upon the wall. 
 
 In the above tables there is a factor of safety of six that is, the rupture 
 should not take place except at a load six times that given. This is to provide 
 for some unknown weakness of the timber, or some sudden excess of load, de- 
 preciation by age, etc., but this factor does not make up for want of inspection 
 or knowledge of the material. It is well to load some of the timbers, as a test 
 to the elastic limit, which can be done by placing two timbers at convenient dis- 
 tances apart and loading with pig iron or barrels of sand. 
 
 The early cast-iron beams used in framing have been almost entirely super- 
 seded by wrought ones, as cheaper and more reliable. In the application of the 
 former, the sections were adapted to the stresses, as shown in Figs. 476-478. 
 Practical examples of cast-iron beams are given, as there may be conditions 
 under which it may be necessary to make use of them. 
 
 A beam subjected to a transverse stress, as shown in Fig. 462, one side is 
 compressed, while the other side is extended ; and therefore, where extension 
 terminates and compression begins, there is a lamina or surface, g h, which is 
 neither extended nor compressed, called the neutral surface. As the strains 
 are proportional to the distance from this surface, the material of which the 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 241 
 
 beam is composed should be concentrated as much as possible at the outer sur- 
 faces, as can readily be done in beams of cast and wrought iron. Mr. Hodgkin- 
 
 FIG. 476. 
 
 FIG. 477. 
 
 FIG. 478. 
 
 son found that the strength of cast-iron to resist compression is about six times 
 that to resist extension ; the top web is therefore made only one sixth" the area 
 of the lower one. The depth of the beam is generally about one sixteenth of 
 its length, the deeper of course the stronger ; the thickness of the stem or the 
 upright part should be from an inch to 1 inch, according to the size of the 
 beam. The rule for finding the ultimate strength of beams of the above sec- 
 tion is : Multiply the sectional area of the bottom flange in square inches by 
 the depth of the beam in inches, and divide the product by the distance be- 
 tween the supports in feet, and 2'42 times the quotient will be the breaking 
 weight in tons (2,000 pounds). The section thus determined is that of the 
 greatest strain, and can be reduced toward the points of support, either by 
 reducing the width of the flanges to a parabolic form (Fig. 479), or by reducing 
 the thickness of the bottom flange ; the reduction of the girder in depth is not 
 in general as economical or convenient. 
 
 FIG. 479. 
 
 For railway structures subject to an impulsive force, Mr. Joseph Cubitt, 
 C. E., recommends that the section of the upper flange should be one third 
 that of the lower. 
 
 1-7 
 
 FIG. 480. 
 
242 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Fig. 480 is side elevation, plan, and section of cast-iron girder, adopted by 
 him for railway purposes, a pair of girders for each track, the rails being sup- 
 ported on wooden cross-beams. 
 
 DIMENSIONS FOR DIFFEEENT SPANS. 
 
 Opening. 
 
 Bearing on 
 abutment 
 
 Height of girder 
 at center. 
 
 Top flange. 
 
 Bottom flange 
 at center. 
 
 At end. 
 
 Thickness of 
 middle web. 
 
 12ft. 
 
 l'-6" 
 
 l'-3" 
 
 3" X 11" 
 
 l'-4" X If 
 
 l'-8" X 11" 
 
 11" 
 
 30ft. 
 
 2''6" 
 
 3'- 
 
 5" x 2" 
 
 l'-6" X 2" 
 
 l'-10" x 2" 
 
 2" 
 
 45ft. 
 
 2'-9" 
 
 3'-9" 
 
 7" x 21" 
 
 2'' x 2" 
 
 2'' X 21" 
 
 2" 
 
 Rolled I-beams (Fig. 481) may be taken as the type in wrought-iron and 
 steel sections. The depths of the beams, A, and the widths, B, of bottom and 
 top flanges do not vary much with the different makers for the 
 same class of beams, but the thickness of the stems varies more. 
 The tables of the Strength of Wrought-iron and Steel Beams 
 (pages 243, 244) have been made up by comparison of the 
 tables of different makers, the safeload is taken in units of 100 
 pounds, and 00 are to be added to the tabulated figures to give 
 the safe distributed load in pounds. 
 
 It is assumed in these tables that proper provision is made 
 for preventing the beam from deflecting sideways. They should 
 be held in position at distances not exceeding twenty times the 
 width of the flange ; this is usually effected by the brick arches 
 between the beams, or the wooden joists resting on them. The beams will 
 support the loads as given in the tables, but the deflection may be too much 
 for the purposes to be served. A line is drawn in each column in the tables, 
 at which the deflection is -j^ beyond that due to the weight of the beams, or 
 one inch for every thirty feet of span, beyond which the deflection is apt to 
 crack the plastering of ceilings. 
 
 To find the sectional area of a beam, plate or rod from its weight, multiply 
 weight per foot by 3 and divide by 10 ; and, conversely, to determine the weight 
 multiply the sectional area by 10 and divide the product by 3. 
 
 Thus, if a steel bar 12 feet long weigh 480 pounds, or 40 pounds per foot, 
 
 FIG. 481. 
 
 its sectional area will be 
 
 inches section will weigh 
 
 3 X 40 
 
 10 
 9 X 10 
 
 , or 12 square inches ; and a bar of 9 square 
 = 30 pounds per foot. 
 
 For naval constructions, deck-beams (Fig. 482) are from 3" to 12" deep, with 
 varied widths of flanges and thicknesses of stem ; lighter than the 
 grades of heavy and light I-beams, but heavier can be rolled to order ; 
 bulb angles with terminal bulbs similar to those of deck-beams, on 
 long legs of from 5" to 10", and short legs of 2" to 3" can be had 
 of varied thicknesses. Properly proportioned, they are equal in 
 strength to the I-beams. 
 
 Coupled I-Beams. When the load is beyond the strength of a 
 single I-beam, two or more may be united, as shown in Fig. 483. 
 
 Fio. 4852. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 243 
 
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 10 * 
 
 ^j* to os 
 
 CM to r-b- CO 
 
 OO5 OSOO 00 
 
 OS1OCM Osb-^CMOCO 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 PH 
 
 
 04 
 
 r 
 
 g 
 
 S3 
 
 CO tO rHrH 
 
 too 
 
 OS OS rH OS 
 CO CM CM rH 
 
 to o o o o 
 
 O O OS OS OO 
 
 CM OSO CMOb- 
 
 
 
 
 
 
 
 
 
 
 
 
 n 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 
 -* 
 
 b 
 
 fi 
 
 
 
 g 0? 
 
 5^ 
 
 SS 838 
 
 CM OO * O b- 
 00 t- b-t- to 
 
 to to o 10 o 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 02 
 
 
 
 
 9 
 
 CM 
 
 So 
 
 b-O 00-* 
 b-O COCM 
 
 coos 
 
 * ' O 
 
 1O OS OS CO 
 OS 00 CO t- 
 
 gg SS g 
 
 ooo 
 
 rH 
 
 
 
 
 
 
 
 
 
 
 
 5^1 
 
 
 
 
 
 
 
 
 
 
 
 rH 
 rH 
 t-H 
 
 
 OS 
 
 CO 
 
 b 
 
 OO 
 
 Sb2 
 
 SrH tOO 
 OS rH 
 
 5 
 
 JJO gtg 
 
 Sgiggg 
 
 OOtOT* 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 cc 
 fi 
 
 
 CO 
 
 * 
 b 
 
 00 
 rH 
 
 CM to 
 1OCM 
 
 coo -*to 
 
 OS 03 
 
 to to 
 
 COTf rH b- 
 
 C C-1 O 00 '^ 
 
 
 r?5 
 
 
 
 
 
 
 
 
 
 
 
 <) 
 
 
 co 
 
 b 
 
 CO 
 
 So 
 
 b23 
 
 3 
 
 s 3 
 
 CO CO 3| CO r-< 
 
 CO CO CO CO CO 
 
 
 
 
 
 CO 
 
 
 
 
 
 
 
 
 O 
 
 r5 
 
 
 CO 
 
 CO 
 
 b 
 
 CO 
 rH 
 
 b-CM 
 
 OS 00 
 
 OSO COO) 
 
 to to o * 
 
 33 
 
 COOS OS CM 
 
 s 
 
 
 H 
 
 
 
 
 
 
 
 
 
 
 
 w 
 
 H 
 
 GQ 
 
 
 CM 
 
 b 
 
 O 
 
 3 
 
 to os co os 
 
 to os 
 coco 
 
 o oo to o 
 
 OS CM CMC<I 
 
 |M 
 
 
 H 
 
 
 
 CO 
 
 
 
 
 
 
 
 
 t> 
 ^ 
 
 -- 
 
 
 CO 
 
 CO 
 
 b 
 
 
 g 
 
 CMO oto 
 1O-* * 
 
 CM OS 
 
 COCM 
 
 b-lO CO 
 (MCM O 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 r^ 
 
 " 
 
 s 
 
 CM 
 
 b 
 
 O 
 
 gg 
 
 CM b- CO OS 
 
 * CO CO CM 
 
 to * 
 
 IN CM 
 
 CMCS1 CSI 
 
 
 
 ?! 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 s 
 
 b 
 
 00 
 
 S3 
 
 1O rH 00 1C 
 
 coos CM CM 
 
 CM 
 
 OS 00 b- 
 
 
 
 w 
 
 
 
 OS 
 
 
 
 
 
 
 
 
 s 
 H 
 
 
 
 
 CM 
 
 CO 
 
 b 
 
 OS 
 
 coco 
 
 b--* rH OS 
 CM CM CM rH 
 
 b-tO 
 rH rH 
 
 
 
 
 PH 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 O 
 
 w 
 
 
 CO 
 CM 
 
 b 
 
 t- 
 
 OSb- 
 O3 CM 
 
 -f V. O 
 CM CM rHrH 
 
 0* 
 
 
 
 
 hJ 
 
 CQ 
 
 
 sanoii 
 
 [ 
 
 *ri 
 
 . 
 
 t* 00 OS O 
 
 Sr- 
 
 23 SS 
 
 H-00 OiO T-I 
 
 C^CO* OtOb-COOSO ^JC^ICOTPOtOb-OOOS 
 CMCMCM CM CM IN CM CM CO OS CO CO CO CO CO CO CO CO 
 
 <3 
 
 EH 
 
 } 
 
 Width of Flnge. 
 
 Thickness of Stem. 
 
 Weight per foot. 
 
 
 
 
 'sxaojc 
 
 iL^TUkt 
 
 ras X3a^4 
 
 NI 
 
 iaa aonvxsia 
 
244 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 8 
 
 t- 
 
 i> 
 
 
 
 o oo 
 
 T-( O Tt 1 
 
 03 t-CO 
 
 ss 
 
 O-* 
 
 m&t 
 
 So 
 
 
 Tf GO 
 
 OOO CO 00 
 
 ooriooc ^> 
 
 t- I- {^ i- O CO 
 
 COCOCOOiCOOiOO'^ 
 
 
 
 
 
 
 a 
 
 CO 
 
 g 
 
 S 
 
 
 Hi 
 
 11 
 
 11 
 
 oo 
 
 coo 
 
 OOOOJ^l- 
 
 w*co-*o 
 
 CO O t- CO i 
 1- 1- O CD O O 
 
 
 
 
 
 
 
 
 8 
 
 t- 
 
 S 
 
 S 
 
 
 CO *O GO 
 
 2 
 
 ii 
 
 1^; 
 CO 
 
 sE 
 
 gggl 
 
 :r; i -'":? -^ 
 
 mmfim 
 
 
 
 
 a 
 
 
 
 s 
 
 s 
 
 
 ill 
 
 OCO 
 
 St- 
 
 oo 
 
 ScO 
 
 Sl-l 
 o 
 
 lisi 
 
 
 
 
 
 
 
 o 
 
 s 
 
 
 
 ill 
 
 7? 
 o 
 
 15 
 
 coo 
 
 OTTO 
 
 1111 
 
 
 
 s 
 
 o 
 
 o 
 
 3 
 
 
 
 o * o 
 
 Ii 
 
 ! 
 
 O CO 
 
 coco 
 
 COCO 
 
 Igia 
 
 ~T ?? 7/ i i O O 
 CXOJCXCMINM 
 
 oSoSSSo'Sgo'o 
 
 
 c* 
 
 s 
 
 SB 
 
 
 
 
 
 35 
 
 cc m 
 
 CO CO 
 
 CO CO 
 
 S5 
 
 ii 
 
 00 t- t 00 
 CO M T-I O 
 MMMM 
 
 SoSSSo 000^TO?=S 
 
 
 o 
 
 * 
 
 
 
 
 
 111 
 
 11 
 
 ils 
 
 6<^j 
 
 11 
 
 o?SSco 
 
 o o r ^rcoco 
 
 t^-r^OCDCOOt^TfT-fO 
 
 BteteiTxiHiMeooS 
 
 
 
 o 
 
 8 
 
 
 
 8 
 
 
 
 co co M 
 
 OCO 
 CNSN! 
 
 C5 O 
 
 653 
 
 is 
 
 co?2 
 
 J* CO O CO 
 
 00 W t CO O O 
 
 ^-OOTf^HOOOCO'-ICOO 
 T-HOOOOOO5OGOGO 
 
 
 
 e 
 
 
 
 S 
 
 
 
 ill 
 
 CO QO 
 
 gcS 
 
 !8^ 
 
 ) 
 
 S8S2 
 
 O O O O O CO 
 
 owoccoeO'-ioooo 
 oooooBt>{^i--t-ooo 
 
 
 
 
 
 
 
 
 S 
 
 CO 
 
 
 
 
 
 ii 
 
 12 
 
 S3 
 
 3^ 
 
 00 TH 
 
 coco 
 
 S2o 
 
 oo ooooo 
 
 aefcs c8fe8 
 
 
 
 
 
 
 s 
 
 
 
 <N 
 
 
 
 O 00 CO 
 
 r>CO 
 
 co 
 
 s^ 
 
 8 
 
 O *-* t- CO 
 
 oococo 
 
 cot-t-Scoco 
 
 SSSSoSSSSS 
 
 
 
 CO 
 
 ? 
 
 fr 
 
 1 
 
 
 
 
 
 
 
 ggg^s So? S3 
 
 ^co 
 
 Si 
 
 2o 
 
 s 
 
 >-* I- CO O 
 
 05 CO CO OD 
 
 1> t- l^ O O O 
 
 
 
 
 
 
 S 
 
 s 
 
 oo 
 
 
 
 
 
 
 
 OO t^- O CO 1-* Tf OOO 
 
 coo 
 
 g 
 
 lO 
 
 o cc 
 
 55*5 
 
 oc t- 
 
 12668 
 
 o-ofeSSS 
 
 t- 
 
 s 
 
 s 
 
 8 
 
 
 
 
 
 
 
 
 CO C* CO O5 00 1-1 00 CO 
 OtO'-'OO COO COM 
 
 t-oo 
 
 ^* o 
 
 oo 1 
 
 coco 
 
 oo 
 
 001- 
 
 (NOOCO 
 1- O CD CO 
 
 T~ OO CO "^f M O 
 COiOiOtOiOiO 
 
 
 
 
 8 
 
 5 
 
 CD 
 
 
 
 
 
 
 
 
 ifl CO OD t- T-^ OO t'' GO 
 
 CO^CO'* CO i-l OO5 
 
 8S 
 
 COTX 
 
 t-t- 
 
 o o 
 
 oo 
 
 (NO 
 
 CO O 
 
 sss? 
 
 ^ii9sa 
 
 o 
 
 g 
 
 CO 
 
 o 
 
 OJ 
 
 O 
 
 
 
 
 
 
 
 
 
 Ils^si^s 
 
 coco 
 
 00 * 
 OCO 
 
 80 
 o 
 
 ^T-I 
 
 o o 
 
 ooo-*w 
 
 ^ O CO CO O Tf 
 
 s 
 
 CO 
 
 s 
 
 CO 
 
 
 
 
 
 
 
 
 
 CO CO i ' O O5QO i>t- 
 
 33 
 
 o yt 
 oo 
 
 OCO 
 Tf T 
 
 -*<N 
 
 S8SS 
 
 COOJ i-iOOOO 
 COCOCOCOSXOJ 
 
 
 
 CO 
 CO 
 
 s 
 
 CO 
 
 
 
 
 
 
 
 
 
 gscisscssg 
 
 0$ 
 
 583 
 
 
 
 gi 
 
 8 
 
 CO 
 
 p 
 
 O 
 
 
 
 
 
 
 
 
 
 COOO OCO CO'* 
 CO 1- O O O * * 
 
 ^00 
 TtCO 
 
 ss 
 
 coS 
 
 ss 
 
 T* 
 
 o 
 t- 
 
 OJ 
 
 a 
 
 o 
 
 
 
 
 
 
 
 
 
 (MOOOT- o^-i t^-rC 
 ooooo * f coco 
 
 g?? 
 
 feS 
 
 ss 
 
 s 
 
 
 
 OJ 
 
 8 
 
 oo 
 
 
 
 
 
 
 
 
 
 OO^CO COCO UJOJ 
 
 S$*1 
 
 -^o 
 
 (NS< 
 
 s^ 
 
 fcS 
 
 SHHQNI 
 
 saq 
 
 OOt-CO 00 -HCV 
 
 COT). 
 
 oo 
 
 t-00 
 
 00 
 
 SSS5S 
 
 ss^sss 
 
 coc^coScoc^cSco 
 
 
 
 
 
 
 
 Depth of Beam. 
 
 Widlh of Flanges. 
 
 1 
 
 H 
 
 Weight per foot. 
 
 
 8J 
 
 -HOc 
 
 ans 
 
 x: 
 
 ! N3 
 
 ia^ NI 
 aAvxaa 
 
 aoNVisia 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 245 
 
 
 A cast-iron block, or separator, is inserted between the beams, and two bolts, 
 through them and the block, add lateral strength. The bolt-holes, placed at 
 some distance from the centre of the span, do not reduce the transverse strength. 
 To strengthen an I-beam or box-girder composed of I-beams, rivet 
 plates on the top and bottom flanges (Figs. 484 and 485), thus 
 adding to the section by the area of the plate, less the rivet- 
 holes. Box-girders, except of the larger sizes, are preferably 
 composed of channel-beams (Fig. 486). 
 
 Fio. 483. 
 
 FIG. 484. 
 
 FIG. 485. 
 
 FIG. 486. 
 
 The rivet spacing at the ends of a box-girder not over 3" at the middle 6". 
 Channel-beams can be furnished of depths the same as I-beams, from three 
 to fifteen inches, of varied grades of light and heavy. 
 
 COMPARATIVE STRENGTH AND STIFB^NESS OF STEEL BOX GIRDERS. 
 
 FIG. 492. 
 
 Depth of Girder. 
 
 Inches 
 
 10" 
 
 12" 
 
 15" 
 
 20" 
 
 Weight of [s per foot. 
 
 Lbs. 
 
 15 
 
 24 
 
 20 
 
 30 
 
 32 
 
 51 
 
 64 
 
 80 
 
 Size of Plates. 
 
 Inches 
 
 12 x* 
 
 14 x* 
 
 16 x i 
 
 18 xf 
 
 
 10 
 
 694- 
 
 758- 
 
 1004- 
 
 1096- 
 
 1836- 
 
 2052- 
 
 4030- 
 
 4500- 
 
 
 11 
 
 631- 
 
 689- 
 
 912- 
 
 997- 
 
 1669- 
 
 1866- 
 
 3664- 
 
 4091- 
 
 
 12 
 
 578- 
 
 631- 
 
 837- 
 
 913- 
 
 1530- 
 
 1710- 
 
 3359- 
 
 3750- 
 
 
 13 
 
 534- 
 
 583- 
 
 772- 
 
 843- 
 
 1412- 
 
 1579- 
 
 3100- 
 
 3462- 
 
 
 14 
 
 496- 
 
 541- 
 
 717- 
 
 783- 
 
 1311- 
 
 1466- 
 
 2879- 
 
 3214- 
 
 
 15 
 
 463- 
 
 505- 
 
 669- 
 
 731- 
 
 1224- 
 
 1368- 
 
 2687- 
 
 3000- 
 
 
 16 
 
 434- 
 
 473- 
 
 627- 
 
 685- 
 
 1147- 
 
 1283- 
 
 2519- 
 
 2813- 
 
 
 17 
 
 408- 
 
 446- 
 
 591- 
 
 645- 
 
 1080- 
 
 1207- 
 
 2371- 
 
 2647- 
 
 CO 
 
 18 
 
 386- 
 
 421- 
 
 558- 
 
 609- 
 
 1020- 
 
 1140- 
 
 2239- 
 
 2500- 
 
 E 
 
 19 
 
 365- 
 
 899- 
 
 528- 
 
 577- 
 
 966- 
 
 1080- 
 
 2121- 
 
 2368- 
 
 rM 
 
 o 
 
 20 
 
 347- 
 
 379- 
 
 502- 
 
 548- 
 
 918- 
 
 1026- 
 
 2015- 
 
 2250- 
 
 ft 
 
 P-I 
 
 21 
 
 330- 
 
 361- 
 
 478- 
 
 522- 
 
 874- 
 
 977- 
 
 1919- 
 
 2143- 
 
 i 
 
 22 
 
 815- 
 
 344- 
 
 456- 
 
 498- 
 
 835- 
 
 933- 
 
 1832- 
 
 2045- 
 
 BO 
 
 23 
 
 302- 
 
 329- 
 
 436- 
 
 477- 
 
 798- 
 
 892- 
 
 1752- 
 
 1957- 
 
 
 
 24 
 
 289- 
 
 316- 
 
 418- 
 
 457- 
 
 765- 
 
 855- 
 
 1679- 
 
 1875- 
 
 W |y 
 
 ^ fq 
 
 25 
 
 278- 
 
 303- 
 
 402- 
 
 438- 
 
 734- 
 
 821- 
 
 1612- 
 
 1800- 
 
 
 26 
 
 267- 
 
 291- 
 
 386- 
 
 422- 
 
 706- 
 
 789- 
 
 1550- 
 
 1731- 
 
 | 5 
 
 27 
 
 257- 
 
 281- 
 
 372- 
 
 406- 
 
 680- 
 
 760- 
 
 1493- 
 
 1667- 
 
 w 
 
 28 
 
 248- 
 
 271- 
 
 359- 
 
 391- 
 
 656- 
 
 733- 
 
 1439- 
 
 1607- 
 
 g 
 
 29 
 
 239- 
 
 261- 
 
 346- 
 
 378- 
 
 633- 
 
 708- 
 
 1390- 
 
 1552- 
 
 
 ^ 
 
 30 
 
 231- 
 
 253- 
 
 335- 
 
 365- 
 
 612- 
 
 684- 
 
 1343- 
 
 1500- 
 
 H 
 
 2 
 
 31 
 
 224- 
 
 244- 
 
 324- 
 
 354- 
 
 592- 
 
 662- 
 
 1300- 
 
 1452- 
 
 R 
 
 32 
 
 217- 
 
 237- 
 
 314- 
 
 343- 
 
 574- 
 
 641- 
 
 1259- 
 
 1406- 
 
 
 33 
 
 210- 
 
 230- 
 
 304- 
 
 332- 
 
 556- 
 
 622- 
 
 1221- 
 
 1364- 
 
 ( 
 
 34 
 
 204- 
 
 223- 
 
 295- 
 
 322- 
 
 540- 
 
 604- 
 
 1185- 
 
 1324- 
 
 
 35 
 
 198- 
 
 216- 
 
 287- 
 
 313- 
 
 525- 
 
 586- 
 
 1152- 
 
 1286- 
 
 
 36 
 
 193- 
 
 210- 
 
 279- 
 
 304- 
 
 510- 
 
 570- 
 
 1120- 
 
 1250- 
 
 
 37 
 
 188- 
 
 205- 
 
 271- 
 
 296- 
 
 496- 
 
 555- 
 
 1089- 
 
 1216- 
 
 
 38 
 
 183- 
 
 199- 
 
 264- 
 
 288- 
 
 483- 
 
 540- 
 
 1061- 
 
 1184- 
 
 
 39 
 
 178- 
 
 194- 
 
 257- 
 
 281- 
 
 471- 
 
 526- 
 
 1033- 
 
 1154- 
 
 
 40 
 
 173- 
 
 189- 
 
 251- 
 
 274- 
 
 459- 
 
 513- 
 
 1008- 
 
 1125- 
 
246 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 TABLE GIVING INCREASE OR DECREASE OF STRENGTH, IN PER CENT, 
 ABOVE OR BELOW THAT GIVEN BY PRECEDING TABLE, FOR DIFFER- 
 ENT THICKNESSES OF PLATES IN STEEL BOX GIRDERS. 
 
 Thickness ;of Plate. 
 
 r 
 
 V 
 
 i" 
 
 i" 
 
 i" 
 
 l" 
 
 
 12" 
 
 
 10" 
 
 15* 
 
 -19 
 
 
 
 + 20 
 
 + 41 
 
 
 
 i 
 
 12" 
 
 42 
 
 10" 
 
 '24* 
 
 -18 
 
 
 
 + 18 
 
 + 38 
 
 
 
 i 
 
 14" 
 
 I 
 
 12" 
 
 20* 
 
 
 
 
 + 19 
 
 + 40 
 
 
 
 f-i 
 
 14" 
 
 1 
 
 A 
 
 12" 
 
 30* 
 
 
 
 
 + 18 
 
 + 36 
 
 
 
 *o 
 
 16" 
 
 
 
 15" 
 
 "32* 
 
 
 -15 
 
 
 
 + 15 
 
 + 30 
 
 
 a 
 
 16" 
 
 o 
 
 15" 
 
 51* 
 
 
 -13 
 
 
 
 + 13 
 
 + 27 
 
 
 g 
 
 18" 
 
 fi 
 
 20" 
 
 "64* 
 
 
 
 -10 
 
 
 
 + 10 
 
 + 20 
 
 
 18" 
 
 CO 
 
 20" 
 
 '80* 
 
 
 
 -9 
 
 
 
 + 9 
 
 + 20 
 
 It may be desirable, on account of position, to finish a box-girder as in Fig. 
 487 ; in this case the dimensions must be such as to admit of a helper inside to 
 hold the rivets. Fig. 488 shows a closed box-beam made of channel-bars and 
 plates. The lower channel is first riveted, and the u-pper one afterward. This 
 
 FIG. 487. 
 
 FIG. 488. 
 
 FIG. 489. 
 
 FIG. 490. 
 
 FIG. 492. 
 
 form gives a dean surface below, but the lower channel-bar can be reversed and 
 riveted the same as the upper. 
 
 Where the purpose can be served by I-beams, either single, or coupled, as in 
 Fig. 483, or in numbers, they afford the best and cheapest construction. But, 
 where the spans are large and loads heavy, it is often economical to obtain 
 greater depth by means of plate-girders, as in Figs. 489, 490, 491, 492, or per- 
 haps from requirements of position, as in Fig. 493, subject as above to the 
 
 necessities of large inside dimen- 
 sions. These girders are made up 
 of plates of uniform thickness, 
 and angle-irons riveted together. 
 
 Angle-irons are made of varied 
 dimensions, and are classed as 
 
 FIG. 493. 
 
 FIG. 494. 
 
 FIG. 495. 
 
 equal-legs (Fig. 494), unequal-legs 
 (Fig. 495), and square-root angles 
 when the thickness of the iron is uniform throughout, and consequently the 
 interior angle a complete right angle without rounding. 
 
 Angle irons are manufactured of equal legs, of f ", ", ", 1", li", H", If, 
 If", 2", 2V, 2V, 2|", 3", 3" , 4", 5", 6", of varied thicknesses, according to the 
 size and necessities of construction, from V on the smaller sizes to f" on the 
 larger ones. 
 
 Angles of unequal legs 7" X 3|", 6V X 4", 6" X 4", 6" X 3V, 5" X 4", 
 5" X 3V, 5" X 3", 4V X 3", 4" X 3|", 4" X 3", 3" X 3",3V X 2|", 3i"X 2", 
 3" X 2", 3" X 2", 2V X 2", 2" X H", 3 " X If", If" X 1", with the same 
 variety of thickness as the equal legs, up to 1 inch for the largest size. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 247 
 
 DIMENSIONS AND WEIGHTS OP Z BARS. 
 
 Thickness 
 of metal 
 in inches. 
 
 SIZE IN INCHES. 
 
 Weight 
 per foot. 
 
 Thickness 
 of metal 
 in inches. 
 
 SIZE IN INCHES. 
 
 Weight 
 per foot. 
 
 Flange. 
 
 Web. 
 
 Flange. 
 
 Flange. 
 
 Web. 
 
 Flange. 
 
 f 
 
 34. 
 
 6 
 
 84 
 
 15-6 
 
 if 
 
 3| 
 
 54 
 
 31 
 
 28-3 
 
 A 
 
 3ft 
 
 6ft 
 
 3l% 
 
 18-3 
 
 i 
 
 8ft 
 
 4 
 
 8ft 
 
 8-2 
 
 | 
 
 8f 
 
 64 
 
 3* 
 
 21-0 
 
 A 
 
 8* 
 
 4ft 
 
 84 
 
 10-3 
 
 A 
 
 84 
 
 6 
 
 4 
 
 22-7 
 
 i 
 
 8ft 
 
 44 
 
 8ft 
 
 12-4 
 
 i 
 
 8ft 
 
 6ft 
 
 8ft 
 
 25-4 
 
 A 
 
 8ft 
 
 4 
 
 3ft 
 
 13-8 
 
 A 
 
 31 
 
 84 
 
 31 
 
 28-0 
 
 i 
 
 34 
 
 4ft 
 
 34 
 
 15-8 
 
 i 
 
 si 
 
 6 
 
 84 
 
 29-3 
 
 A 
 
 8ft 
 
 44 
 
 8ft 
 
 17-9 
 
 ft 
 
 8ft 
 
 6ft- 
 
 8ft 
 
 32-0 
 
 i 
 
 8ft 
 
 4 
 
 8ft 
 
 18-9 
 
 i 
 
 8| 
 
 H 
 
 9% 
 
 34-6 
 
 H 
 
 8* 
 
 4ft 
 
 84 
 
 20-9 
 
 A 
 
 Si 
 
 5 
 
 3i 
 
 11-6 
 
 i 
 
 8ft 
 
 44 
 
 8ft 
 
 22-9 
 
 i 
 
 8ft 
 
 5ft 
 
 8ft 
 
 13-9 
 
 I 
 
 s}| 
 
 3 
 
 m 
 
 6-7 
 
 A 
 
 H 
 
 H 
 
 8f 
 
 16-4 
 
 A 
 
 2f 
 
 8ft 
 
 2f 
 
 8-4 
 
 i 
 
 8 
 
 5 
 
 3i 
 
 17-8 
 
 i 
 
 W 
 
 3 
 
 m 
 
 9-7 
 
 A 
 
 8ft 
 
 8ft 
 
 8ft 
 
 20-2 
 
 ft 
 
 2f 
 
 8ft 
 
 2f 
 
 11-4 
 
 t 
 
 3f 
 
 &i 
 
 3| 
 
 22-6 
 
 i 
 
 2*4 
 
 3 
 
 m 
 
 12-5 
 
 H 
 
 8 
 
 5 
 
 3i 
 
 23-7 
 
 A 
 
 2f 
 
 8ft 
 
 21 
 
 14-2 
 
 t 
 
 3& 
 
 5ft 
 
 3ft 
 
 26-0 
 
 
 
 
 
 
 T-irons (Fig. 496) may be used for top and bottom flanges in the manufac- 
 ture of plate-girders by riveting a web on one side of the T, or on both sides, 
 with a separator between of the thickness of the stem E ; but, as 
 the areas of section of T-irons to be had are small, the flanges 
 will be too slight in proportion to the webs at depths above that 
 of rolled beams. Angle-irons are then to be preferred for flanges. 
 The T-irons are well adapted in many positions as struts or 
 FIG. 496. braces, and can be bought of varied dimensions and weights, 
 from widths, B, of from 2 to 5 inches, and equal or less depths, A, and thick- 
 nesses from -fa" to f ". 
 
 Rivets for plate-girders are usually from f " to " diameter, and pitched or 
 spaced not more than 6" nor less than 3" between centres. The number of 
 rivets through flange and stem are the same, but alternating. Usually angle 
 irons and plates can be had of the full length of girder, but, where joints are 
 necessary, they should be butt, with a splicing-piece to make the strength as 
 nearly as possible uniform. Stiffeners are often necessary for the webs, which 
 may be of band, angle, or T-iron, and one should always be placed at each end, 
 where the shearing stress is the greatest. 
 
 A common formula for determining the strength of a wrought-iron beam 
 g ' 
 
 or girder is W , in which W is the load in pounds, equally 
 
 .L 
 
 distributed on the beam, D the effective depth between the centres of gravity 
 of the flanges, and L the clear span, both in the same unit, feet or inches ; a 
 the area of the top or bottom flange in square inches ; a' the area of the stem. 
 To construct a diagram from the formula, in which the relation of the fac- 
 
248 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 tors may be shown. Let S be 10,000 for riveted girders of wrought iron 
 
 
 -g) 80,000. On a sheet of cross-sec- 
 
 (12,000 for steel), then W = y X 
 
 tion paper, from a corner, 0, lay off on the line of ordinates, 5, 10, 15, 20, 
 
 25, representing the factor a -f- . From the same 0, on the line of abscissas, 
 
 i iV> iV> fa &> TrV> * 
 
 > representing -. Suppose ^ = ^, then W = 
 
 ( a -(_ ^.) 2,000. If a + ^ be = 10, then W = 20,000. From the intersection 
 
 of ordiuate on line of ^, and abscissa line of 10, draw a line to the point 0. 
 This line will represent the safe distributed load W, and its intersections of the 
 
 m 
 
 P. / 
 
 17 /* 
 
 '10 
 
 FIG. 497. 
 
 ordinates and abscissas will represent the relative proportions of the two factors 
 
 -Y- and a -}- under this load. On the abscissa line 15, and ordinate 3 V, W = 
 Jj o 
 
 30,000, on line 20, 40,000, and so on, and lines drawn from these intersections 
 to will represent W. 
 
 In the diagram lines below 5 and above 30 on line of ordinates are erased, 
 as within these limits may be found most of the proportions required in 
 practice. 
 
 Application of the Diagram. "What will be the area of section + of a 
 
 girder, 40-foot span, depth 32", distributed load 90,000 pounds ? 
 
 D in the formula represents the distance between the centres of gravity of 
 the flanges, which will be somewhat less than the depth of beam. Approxi- 
 
 D 480" 
 mately assume it at 30", y- = -^r = -fa, and the intersection of the line 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 249 
 
 of load, 90,000, with the ordinate 
 
 , will be 18, on the line of a -) . 
 
 A fair 
 
 proportion of a to a' is 5 to 6, therefore 
 
 O 
 
 -f- 18 or a' = 18. 
 
 O 
 
 18 
 
 = 0-6" = 
 
 thickness of web, and a = f of 18 = 15, or weight per foot of one flange = 
 
 15 X 10 
 
 s -- = 50 pounds, which is slightly in excess of the weight of two angle- 
 o 
 
 irons 6 X 4 X f , compensated by thickness of web outside centres of gravity. 
 
 This calculation is sufficiently near for all practical purposes, but D can be 
 found more accurately by plotting the angle-irons, on thick card- board, cut- 
 ting out, and then balancing for the centre of gravity. 
 
 Composite Seams. Often, in constructions where the beams or girders are 
 of wood, and on account of extent of spans and loads, the stress is beyond the 
 strength and stiffness of beams of this material, of readily available dimen- 
 sions, it is usual to supplement by some application of iron. A simple form, in 
 which the iron is not exposed to view, is by bolting a plate or flitch of wrought- 
 iron between two beams, of the full length and depth of the beams, and of such 
 thickness as may be necessary. In bolting them together, let the bolt-holes 
 be so bored that the weight of the beam may primarily be on the wood ; the 
 stress will then be better adjusted between the two materials when in service. 
 It is usual to make the holes zigzag, in two lines about one quarter the depth 
 of beam from each edge, the holes closer together nearer the ends. The safe- 
 distributed load for the iron may be estimated from the formula : W. = 
 
 15000 bh*. .. . . , 
 
 j -- , o breadth, h depth, I length all in inches. 
 
 I 
 
 Fig. 498 represents a bracing truss of wrought-iron between two beams, 
 which should be let into the wood. As it is held firmly laterally, the factor of 
 
 FIG. 498. 
 
 safety may be considered about one third of the crushing resistance of the ma- 
 terial. The load on each inclined bar will be one half the load on the centre, 
 multiplied by the length of the bar and divided by the rise. Instead of wrought- 
 iron, cast-iron or wood is used. 
 
 Fio. 499. 
 
 In Fig. 499 the beams are strengthened by a tension-rod, of which the 
 strength may be determined by that of the material ; allowing the usual factor 
 of safety, the load is obtained as in the example above. The deeper the block 
 beneath the centre of the beam, the less the stress on the rods for the same 
 load. In construction, the beam should not be cambered by the screwing up 
 
250 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 of the rod ; but, if the beams are crowning, the convex side should be placed 
 upward, the nut turned by hand just to a bearing, and the tension put on by 
 the settlement of the beams under the load. 
 
 FIG. 500. 
 
 Fig. 500 represents the trussing of a beam by two struts and a tension-rod. 
 The stress on the tension-rod is the load on c, multiplied by the length a d, 
 divided by c d. 
 
 The theory of trusses will be treated and illustrated under " Bridges " and 
 " Hoofs," and the proportions of rivets and forms of plate-iron joints under 
 " Boiler Construction." 
 
 Bolts and nuts are of such universal application that their manufacture 
 forms the centre of large industries. Much thought has been given to their 
 
 FIG. 501. 
 
 proportions and the forms of thread, but without producing complete uni- 
 formity in the practice of different countries and makers. The old form of 
 thread was the triangular pitch (Fig. 502), still used by some, especially when 
 the threads are cut in a lathe. In this country the standard U. S. thread is 
 
 FIG. 503. 
 
 that recommended by the Franklin Institute in 1864 (Fig. 503). The angle is 
 60, with straight sides and flat surface at top and bottom, equal to one eighth 
 the pitch, or distance from centre to centre of threads. 
 
 In England, the standard thread for bolts and nuts is the Whitworth (Fig. 
 504) ; the angle is 55, with top and bottom rounded. 
 
 2.oo 
 
 The square and rounded threads (Figs. 505 and 506) are only made to order 
 and used in presses and the like as parts of machines. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 251 
 
 Figs. 507, 508, and 509 represent the proportions of the various parts of 
 English nuts to the diameters of bolts, as 1, or unity. Fig. 508 is a flange-nut, 
 in which a washer-like flange is forged with the nut. 
 
 FIG. 506. 
 
 Fig. 510 is a cap-nut, in which the thread does not go through the nut, to 
 prevent leaking along the thread, and a soft copper washer is introduced to pre- 
 vent leakage below the nut. 
 
 FIG. 507. 
 
 FIG. 508. 
 
 FIG. 509. 
 
 Figs. 511 and 512 are circular nuts, in one of which holes are drilled to 
 insert a rod for turning, and in the other grooves for a spanner. 
 
 Lock-nuts (Fig. 513) are intended to prevent the gradual unscrewing of 
 
 FIG. 510. 
 
 FIG. 511. 
 
 FIG. 512. 
 
 FIG. 513. 
 
 nuts subjected to vibration, which is to a great extent prevented by the use of 
 double nuts, the lock-nut being one half the thickness of the common nut. 
 The usual practice is as shown, the lock-nut being outside ; the better way is 
 inside. 
 
 The following figures are from trade circulars ; the limits of sizes given are 
 such as can usually be found in stock. 
 
 Figs. 514, 515, and 516 are machine-bolts, from %" to f" diameter, and 1" 
 to 4" long, but not flanged, as in Fig. 514, unless expressly ordered ; the dot- 
 ted line shows the radius of curvature of a finished head. The diagonal lines 
 beneath the head (Figs. 515 and 516) represent square bolts tapering into round 
 bolts, as shown by the curved lines. 
 
252 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Figs. 517 and 518 are tap-bolts and set screws, from " to f" diameter, and 
 from 1" to 3" long. 
 
 FIG. 514. 
 
 Fig. 519 is a carriage-bolt, from ^" to f" diameter, and from 1" to 16" long. 
 Fig. 520 is a plough-bolt, from f" to " diameter, and from 1" to 4" long. 
 
 FIG. 515. 
 
 FIG. 516. 
 
 Fig. 521 is a stove-bolt, from " diameter and from f " to 3" long. 
 
 Figs. 522 and 523 are machine-screws without nuts ; the holes in the metals 
 
 FIG. 518. 
 
 are tapped to receive them ; Fig. 522 is button-headed ; Fig. 523 a counter- 
 sunk head both slotted to admit of driving by a screw-driver. They are made 
 of various-sized wire and lengths, and sold by the gross like the common wood- 
 
 
 FIG. 519. 
 
 screw (Fig. 524). The wood-screw is for connecting pieces of wood together, 
 or metal to wood. They are of very great variety, usually with a gimlet-point, 
 
 FIG. 521. 
 
 FIG. 522. 
 
 FIG. 523. 
 
 FIG. 524. 
 
 FIG. 525. 
 
 FIG. 526. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 253 
 
 so that they can be screwed into the wood, without any holes being previously 
 made. When made of rods, with a square or hexagonal head (Figs. 525 and 
 526) to admit of the use of a wrench, they are called lag-screws. It will be 
 seen that wood-screws differ in their thread from bolts and machine-screws. 
 The thread is a very sharp V, flatter on the upper surface, and the flat space 
 
 FIG. 528. 
 
 FIG. 529. 
 
 between the threads wide as the thread, making it of easier introduction into 
 the wood, and retaining as much strength in the iron as in the wood. 
 
 Fig. 527 is a stud-bolt, which is screwed firmly into one of the pieces of 
 connected metal ; the other is bored so as to slip over the bolt, and the nut 
 then brought down upon it. It is in common use for holding on the bonnets 
 of steam-chests and water-chambers, the bolt remaining permanent. 
 
 Fig. 528 is a 7*oo&-bolt ; it relieves the necessity of a bolt through the bot- 
 tom-piece, and may be turned like a button, to loose or hold the bottom-plate. 
 
 Fig. 529 is another kind of button-bolt ; the lower end can revolve on a stud 
 or pin if the nut be raised enough to clear the cap or upper plate. By this 
 arrangement there is no necessity of taking off the nut entirely ; the bolt lies 
 in a slot in the cap, and the nut bears on three sides. 
 
 FIG. 530. 
 
 FIG. 531. 
 
 FIG. 532. 
 
 Figs. 530, 531, and 532 show expedients to prevent the bolt from turning 
 when the nut is being screwed on or off. 
 
 Fig. 533 is an anchor-bolt, flattened and jagged, introduced into a hole in 
 masonry, and firmly leaded or sulphured in; but later experiments with deep 
 holes have established that it was not necessary to flatten and jag the bolts, and 
 with holes If and If diameter and 3' 6" deep, and bolts 1* diameter when leaded 
 or sulphured in, did not develop as uniform adhesion as when the spaces were 
 filled in with Portland cement and allowed a set of two weeks, when the bolts 
 broke under the test. The resistance was 400 to 500 pounds per square inch of 
 surface of bolt exposed. 
 
 It is very common to split the bolt at the bottom (Fig. 534) and insert a 
 
254 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 wedge in the cleft and drive it against the bottom of the bolts, forcing open the 
 cleft and holding the bolt in the hole. 
 
 Fig. 535 is a bolt keyed into the stone, corresponding to a form of lewis for 
 raising stones in which the key (Fig. 536) is wedged in, and when the stone is 
 
 FIG. 533. 
 
 FIG. 534. 
 
 FIG. 536. 
 
 set the wedge is struck and the lewis withdrawn. Fig. 537 is a double or chain 
 lewis. 
 
 For small holes in brick or stone masonry drill a hole and insert a short 
 piece of lead pipe and force in a wood-screw of a little larger diameter than the 
 bore of the pipe, or drive a wooden pin into the hole, and screw into the' wood. 
 
 Take a clout-nail or picture-nail, turn up 
 the point, cast a lead petticoat on it, and 
 drive it into a loosely fitting hole ; by driv- 
 ing, the point and the wedge form of the 
 nail expands the lead and gives a firm hold. 
 Similar forms of points to anchors of copper 
 wire are convenient in holding stones to- 
 gether or face stones of a building to the 
 backing. 
 
 JZxpansion-kolts (Fig. 538) from %" to 1" 
 diameter are on sale and very convenient in 
 connecting architectural iron-work to ma- 
 sonry, and have this convenience : that they 
 can be removed when necessary; all that is required is a hole of uniform diam- 
 eter and of suitable size and depth to insert the bolt with nut and jaw ; then 
 by turning the head, the iron or composition nut is drawn outward, 
 opening the wedge-shape jaws and causing them to bind strongly 
 against the side. There is a small split-steel band around the jaws to 
 keep them together before insertion in the hole and to free the jaws 
 when they are to be removed. The bolts may have a head at the 
 top, or a screw with a nut, or the lower nut may be formed as a head. 
 FIG. 538. -pig. 539 is a bolt with a fang-nut or corner turned down and 
 driven into the wood to prevent turning ; the screwing to be done at the 
 head. 
 
 It is often convenient to use bolts with two nuts, as in Fig. 540, or collar- 
 bolts, which are readily made to order, and of any dimensions. 
 
 FIG. 537. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 255 
 
 Fig. 541 is a hanger-bolt, ; the Jag-screw part is screwed into the wooden 
 beam, the hanger then put over the bolt, and the nut put on. 
 
 FIG. 541 
 
 FIG. 539. 
 
 FIG. 540. 
 
 FIG. 542. 
 
 Figs. 542 and 543 represent forms of turn-buckles, and the swivel and pipe, 
 sometimes designated as swivels. Turn-buckles are very useful in straining 
 tierods, where neither end of the bolt can be got at. By turning the buckle, 
 the rod can readily be made longer or shorter. In the pipe-swivel, right and 
 left threads are cut on the bolts, so that each turn of the pipe shortens or 
 lengthens the tie by double the pitch of the screw. The turn-buckle is also 
 made in the same way, with two screws instead of a head at one end. 
 
 FIG. 543. 
 
 Screws, unless otherwise ordered, are made right-handed ; that is, turning 
 the nut to screw up, the hand moves from left to right, the apparent motion of 
 the sun. 
 
 On the Strength of Bolts. The strength of a bolt depends on its smallest 
 section that is, between the bottom of the threads. It is very common, there- 
 
 FIG. 544. 
 
 fore, to upset the screw-end, so that the screw may be cut entirely from this 
 extra boss, or re-enforce. Bolt-ends (Fig. 544) are sold either with or with- 
 out re-enforce, to be welded to bolts. It will be observed that the ends of the 
 pipe-swivel bolts (Fig. 543) are thus upset. 
 
 In the following table the sizes and dimensions of bolts and nuts are from 
 the United States standard, and the strength, or safe-load of the bolts, is com- 
 puted from the report of the committee on the test of wrought-iron and chain- 
 cables to the United States Government in 1879. Nuts and heads as furnished 
 are either hexagonal or square. Columns 4, 5, and 6 apply equally to either. 
 
 There is often an uncertainty in the determination of the load. The ef- 
 fective load due to the forces acting on the machine may be estimated with 
 tolerable accuracy. But that due to the forces used in tightening the nut is 
 uncertain. If the bolt is screwed up so as to develop a reaction between the 
 counected pieces, the additional load may be greater than the effective one. 
 
256 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Washers (Fig. 545) in common use to provide seatings for nuts which 
 would otherwise rest on rough metallic surfaces, and also to adapt bolts to 
 
 Diameter of 
 screw in 
 inches. 
 
 Diameter at 
 root of 
 thread. 
 
 Thread per 
 inch of 
 length. 
 
 Short diam- 
 eter of nut 
 and head. 
 
 Thickness of 
 nut. 
 
 Thickness of 
 head. 
 
 Safe-load of 
 upset bolts. 
 
 Safe-load of 
 plain bolts. 
 
 1 
 
 
 
 TV 
 
 1 
 
 & 
 
 
 
 TV 
 
 
 
 1 3 
 
 -nrn- 
 o & 
 
 TV 
 
 if 
 
 
 
 1 
 
 185 
 
 20 
 
 1 
 
 1 
 
 1 9 
 
 
 
 A 
 
 240 
 
 18 
 
 tf 
 
 5 
 
 
 
 
 f 
 
 294 
 
 16 
 
 frr 
 
 r 
 
 II 
 
 
 1,700 
 
 TV 
 
 344 
 
 14 
 
 25 
 ~5S 
 
 TV 
 
 If 
 
 
 1,900 
 
 1 
 
 400 
 
 13 
 
 
 
 1 
 
 TV 
 
 
 2,200 
 
 TV 
 
 494 
 
 12 
 
 fi 
 
 TV 
 
 u 
 
 
 2,500 
 
 f 
 
 507 
 
 11 
 
 1A 
 
 f 
 
 H 
 
 
 2,800 
 
 -u 
 
 
 
 
 
 H 
 
 1? 
 
 
 3,200 
 
 1 
 
 620 
 
 10 
 
 H 
 
 1 
 
 f 
 
 6,000 
 
 3,600 
 
 H 
 
 
 
 Hi 
 
 if 
 
 if 
 
 7,000 
 
 4,300 
 
 f 
 
 731 
 
 9 
 
 ITV 
 
 1 
 
 ft 
 
 8,000 
 
 5,100 
 
 i 
 
 837 
 
 . 8 
 
 If 
 
 i 
 
 if 
 
 10,000 
 
 7,000 
 
 H 
 
 940 
 
 7 
 
 if! 
 
 il 
 
 li 
 
 12,000 
 
 9,000 
 
 it 
 
 1-065 
 
 7 
 
 2 
 
 11 
 
 1 
 
 15,000 
 
 11,000 
 
 if 
 
 1-160 
 
 6 
 
 2 TV 
 
 if 
 
 HV 
 
 18,000 
 
 13,500 
 
 H 
 
 1-284 
 
 6 
 
 2f 
 
 11 
 
 ITV 
 
 21,000 
 
 16,000 
 
 if 
 
 1-389 
 
 51 
 
 2A 
 
 if 
 
 ^A 
 
 24,000 
 
 19,000 
 
 it 
 
 1-490 
 
 5 
 
 21 
 
 if 
 
 if 
 
 28,000 
 
 22,300 
 
 1* 
 
 1-615 
 
 5 
 
 2yf 
 
 H 
 
 115 
 
 32,000 
 
 25,500 
 
 2 
 
 1-712 
 
 41 
 
 8 1 
 
 2 
 
 ITV 
 
 36,000 
 
 29,300 
 
 2fr 
 
 
 
 Mr 
 
 21 
 
 14 
 
 40,000 
 
 33,000 
 
 21 
 
 1-962 
 
 41 
 
 31 
 
 21 
 
 If" 
 
 45,000 
 
 37,000 
 
 2f 
 
 
 
 Hi 
 
 2f 
 
 1|| 
 
 50,000 
 
 41,500 
 
 21 
 
 2-175 
 
 4 
 
 3i 
 
 21 
 
 
 55,000 
 
 46,000 
 
 2f 
 
 
 
 4rV 
 
 2f 
 
 2^V 
 
 
 
 2f 
 
 2-425 
 
 4 
 
 41 
 
 2f 
 
 21 
 
 
 
 2* 
 
 
 
 47-5- 
 
 2| 
 
 
 
 
 3 
 
 2-629 
 
 31 
 
 4f 
 
 3 
 
 2 TV 
 
 
 
 31 
 
 2-879 
 
 31 
 
 5 
 
 31 
 
 21 
 
 
 
 31 
 
 3-100 
 
 31 
 
 5f 
 
 31 
 
 2H 
 
 
 
 3 
 
 3-317 
 
 3 
 
 5| 
 
 3| 
 
 
 
 
 4 
 
 3-567 
 
 3 
 
 61 
 
 4 
 
 3 T V 
 
 
 
 41 
 
 3-798 
 
 2& 
 
 61 
 
 41 
 
 31 
 
 
 
 41 
 
 4-028 
 
 24 
 
 6& 
 
 41 
 
 3 T V 
 
 
 
 41 
 
 4-255 
 
 2f 
 
 71 
 
 4f 
 
 3f 
 
 
 
 5 
 
 4-480 
 
 21 
 
 7f 
 
 5 
 
 311 
 
 
 
 51 
 
 4-730 
 
 21 
 
 8 
 
 51 
 
 4 
 
 
 
 51 
 
 5-053 
 
 2| 
 
 8f 
 
 51 
 
 4 T \ 
 
 
 
 5| 
 
 5-203 
 
 2f 
 
 8| 
 
 5f 
 
 H 
 
 
 
 6 
 
 5-423 
 
 21 
 
 91 
 
 6 
 
 
 
 
 shorter spaces than their lengths are sold for bolts up to 2" diameter. Cir- 
 cular in form, their diameter is slightly in excess of that of the largest diam- 
 eter of the nut, and the hole that of the bolt, and thickness from -fa" to *, 
 according to the diameter of the bolt. 
 
 Lock-nut Washers. Nuts subject to jars are apt to unscrew of themselves; 
 to prevent which many expedients are adopted, as cupping the washer or turn- 
 ing up one corner of a square washer against a face of the nut. The simplest is 
 Shaw's Lock-nut, in which one side of a circular washer is cut through, one 
 edge pressed up and the other down, to form a spring by which a pressure is 
 brought on the thread when the nut is screwed home. In the Billing's lock- 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 25T 
 
 nut by cupping a circular washer a similar effect is obtained through the elas- 
 ticity of the cup. 
 
 The square washer is used under both head and nut on surfaces of wood, 
 
 FIG. 545. 
 
 FIG. 546. 
 
 and of dimensions suited to the stress. That they may neither sink into the 
 wood nor bend or break, cast-iron is frequently used, and often, as shown in 
 Fig. 546, for roof-frames. 
 
 Shafts and Axles. Short shafts, revolving in bearings or boxes, or fastened 
 with pulleys, drum, or wheels revolving on them, are called axles ; but long 
 or heavy revolving bars are usually termed shafts. They may be independent ; 
 that is, a single shaft, revolving in its bearings, or coupled, forming what is 
 termed a line of shafting. The small shafts, as in clock-work and spinning- 
 machinery, are termed pins and spindles. 
 
 Shafts and axles are made of wood and metal, and of varied sections and 
 form. 
 
 Wooden shafts are polygonal, circular, or square section (Fig. 547). 
 
 Wrought metal, iron, or steel shafts, are almost invariably circular in sec- 
 tion, but sometimes square. 
 
 Cast-iron is used in great variety of section and form for shafts (Fig. 548) ; 
 without uniformity longitudinally, but adapted to their position and load. 
 
 FIG. 548. 
 
 Formerly, either wood or cast-iron was invariably used for water-wheel 
 shafts ; but a change of motors, from the breast, over-shot and under-shot 
 wheels to reactors or turbines, has involved an entire change of construction, 
 and now only wrought-iron is used. Wooden shafts are often used in ma- 
 chines subject to wet and shocks, or from the greater convenience in obtaining 
 the material, and from this last necessity the journals and boxes are sometimes 
 of wood; but for wooden shafts it is the usual practice to insert cast-iron jour- 
 nals with boxes or bearings of the same material. 
 18 
 
258 
 
 is. 
 
 FIG. 549. 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Fig. 549 is a side view of a wooden shaft with 
 one end in section, and Fig. 550 an end view of 
 the shaft. On the journal B is cast four wings, 
 c c, and a small spindle, I. The ends of the shaft 
 are bored for the spindle and grooved to receive 
 the wings ; the casting is then drawn into place, 
 hooped with hot ferules, a a, and after this hard 
 wood wedges are driven on each side of the wings 
 and iron spikes are sometimes driven into the 
 end of the wood ; most millwrights omit the spin- 
 dle a. 
 
 Figs. 551, 552, and 553 represent different 
 views of a cast-iron shaft of a water-wheel. Fig. 
 551 is an elevation of the shaft, with one half in 
 section to show the form of the core ; Fig. 552, an 
 end elevation ; Fig. 553, a section on the line c c 
 across the centre. The body is cylindrical and 
 hollow, and cast with four feathers, c c, disposed 
 at right angles to each other, and near the extrem- 
 ities of these feathers four projections, for the at- 
 tachment of the bosses of the water-wheel or pul- 
 ley. These projections are made with facets, so 
 as to form the corners of a circumscribing square 
 (Fig. 552), and are planed to receive the keys 
 by which they are fixed to the naves which are 
 grooved to receive them. The shaft is cast in 
 one entire piece, the journals turned, and the 
 feathers of an external parabolic outline to stiffen 
 the shaft. 
 
 Shafts like the examples given are for pur- 
 poses where their loads are nearly constant and 
 for moderate speeds, and cast-iron gives satisfac- 
 tory results. 
 
 The usual length of such journals is from one 
 to one half times the diameters, and the safe load 
 500 pounds to the square inch, taking the area 
 as d*, the square of the diameter, the diameter 
 and length of journal being considered equal. 
 
 To determine the size of a shaft, considered as 
 a beam merely, but with a shifting load as by 
 the revolution of the shaft each longitudinal line 
 of surface has to undergo successively tension and 
 compression. The safe load of wrought-iron is 
 estimated at 6,000 pounds per square inch, and 
 the formula on which the graphic diagram (Fig. 
 554) is constructed is d '06 \J~wl, d being di- 
 ameter, I = length between bearings, both in 
 Flo 550 inches, w the load in pounds ; the load is not only 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 259 
 
 FIG. 552. 
 
 the weight of shaft and pulleys or gears, but also the 
 stress in transmitting the power. 
 
 Use of Diagram. Suppose w = 50,000 pounds, 
 and I 6 feet = 72", then w"l = 3,600,000, the or- 
 dinate of 3*6 cuts the curve 
 on the abscissa 9*2, which 
 is the required diameter of 
 the shaft in inches. 
 
 Fly - wheel and crank 
 shafts are of forged iron or 
 steel, often forged in steps 
 (Fig. 555) with the largest 
 boss beneath the wheel hub, 
 and sufficiently raised above 
 the next to admit of the 
 planing of the key seats. 
 
 The transverse stress upon the shaft, due to the 
 transmission of power, is equal to the H. P. divided 
 by the velocity of surface, 
 whether of belt or of gear, 
 by which it is transmitted, 
 and the same acts by torsion 
 through the leverage of the 
 radius of the pulley or gear. 
 This stress is seldom calcu- 
 lated, as it is sufficiently met 
 by the tables and diagrams 
 for the determination of 
 the diameters of shafts. 
 
 Keys are pieces of met- 
 al, usually steel, employed to secure the hubs of pul- 
 leys, gears, and couplings to shafts. They may be 
 sunk keys (Fig. 556), flat keys (Fig. 557), and 
 hollow keys (Fig. 558). The shaded circle repre- 
 sents the shaft. The breadth of the key (Fig. 559) 
 is uniform, but the thickness is tapered about 
 one eighth of an inch per foot. The shoulder h 
 is for the purpose of drawing out the key. Sunk 
 keys are not necessarily taper. Some prefer them 
 of uniform section, and to force the hub on over 
 the key. 
 
 It is good practise in fitting keys that they shall 
 always bind tight sideways, but not necessarily 
 touch either at the bottom of the key-seat or the 
 top of the slot cut in the hub. Such keys depend 
 upon a forcing fit of the wheel upon the shaft so 
 tight as to require screw-pressure to put the wheel 
 in place upon the shaft. 
 
 FIG. 553. 
 
 FIG. 551. 
 
260 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 PROPORTIONS OF SUNK KEYS. 
 
 DIAMETER OF SHAFT, 
 IN INCHES. 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 ll 
 
 12 
 
 Breadth of key 
 
 | 
 
 | 
 
 
 
 u 
 
 1* 
 
 1| 
 
 U 
 
 2-t 
 
 *ft 
 
 OB 
 
 ' 2Jr 
 
 31 
 
 Thickness of key 
 Depth sunk in shaft. . 
 Depth sunk in wheel . 
 
 25 
 10 
 
 15 
 
 34 
 125 
 215 
 
 43 
 15 
 
 28 
 
 52 
 175 
 345 
 
 61 
 20 
 
 41 
 
 71 
 225 
 485 
 
 80 
 25 
 55 
 
 89 
 275 
 615 
 
 98 
 30 
 68 
 
 1-07 
 325 
 745 
 
 1-16 
 35 
 
 81 
 
 1-25 
 375 
 
 875 
 
 14" 
 
 12 
 
 11 
 
 s 9 
 
 456789 
 Product of Load and Span in Millions. 
 
 FIG. 554. 
 
 10 
 
 11 
 
 12,000,000 
 
 Car-Axles. Fig. 560 is the form and dimensions of axle adopted as stand- 
 ard by the American Master Car-Builders' Association for wrought-iron and 
 
 steel. 
 
 FIG. 555. 
 
 Shafting. Thus far, independent shafts or axles have been treated of, and 
 the dimensions have been established mostly by the load acting transversely; 
 
 Flu. 550. 
 
 FIG. 557. 
 
 FIG. 558. 
 
 FIG. 559. 
 
 but, in transferring power to machines, lines of shafting are necessary, almost 
 invariably of wrought-iron or steel bars, which are subject not only to trans- 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 261 
 
 verse but also torsional stress. When there are no pulleys or gears on the 
 shafts between the bearings, and the couplings are close to the bearings, there 
 is still an amount of deflection due to the weight of the shaft. James B. 
 Francis, C. E., puts the maximum distances between bearings for shafts of 
 wrought-iron or steel, under these conditions, as follows : 
 
 Diameter 
 of shaft. 
 
 Distance between 
 bearings. 
 
 Diameter Distance between 
 of shaft. bearings. 
 
 Diameter 
 of shaft. 
 
 Distance between 
 bearings. 
 
 1" 
 
 12ft. 
 
 5" 
 
 21 ft. 
 
 9" 
 
 26 ft. 
 
 2 
 
 15 
 
 6 
 
 22 
 
 10 
 
 27 
 
 3 
 
 18 
 
 7 
 
 24 
 
 11 
 
 28 
 
 4 
 
 20 
 
 8 
 
 25 
 
 12 
 
 28 
 
 The diagram (Fig. 561) is one established by J. T. Hen- 
 thorn, M. E., to determine the size of wrought-iron shaft- 
 ing, to transmit a fixed amount of horse-power. 
 
 Use of Table. To find the size of a shaft making 150 
 revolutions, and transmitting 350 horse-power. 
 
 The intersection of the ordinate of 350 with the abscissa 
 of 150 is between the diagonals 5 and 5^, and the diameter 
 of the shaft may be taken safely at 5". 
 
 Mr. Francis has constructed a table from his own experi- 
 ments, of which the following is a synopsis : 
 
 " The following table gives the power which can be 
 safely carried by shafts making 100 revolutions per minute. 
 The power which can be carried by the same shafts at 
 any other velocity may be found by the following simple 
 rule : 
 
 " Multiply the power given in the table by the number 
 of revolutions made by the shaft per minute; divide the 
 product by 100 ; the quotient will be the power which can 
 be safely carried." 
 
 The diagram and table given are applicable to shafts 
 which are called second movers, subject to no sudden 
 shock. For first movers, Mr. Francis takes but one half 
 the horse-power given in the table for any diameter of 
 shafts. Of late, cold-rolled shafts can be procured in the 
 market, which are much stiffer than turned shafts, but not 
 equal to that given for steel in the table. 
 
 It is usual to make the shafts of second and third movers 
 throughout manufactories and shops of uniform diameter, 
 without reduction at the journals, the end-slip being pre- 
 vented by collars keyed or fastened by set-screws. The 
 usual length between bearings is from 7 to 10 feet; but 
 that they may run smooth, and not spring intermediately, 
 it is desirable that they should never be less than 2 inches 
 diameter, and that the pulleys or gears through which the 
 power is transmitted to the next mover or to the machine 
 should be as near as possible to the bearing. 
 
 FIG. 
 
262 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Horse-power which can be safely transmitted by shafts making 100 revolutions per minute, in which the 
 transverse strain, if any, need not be considered ; if of 
 
 Diameter 
 in inches. 
 
 Wrought- 
 iron. 
 
 Steel. 
 
 Diameter 
 in inches. 
 
 Wrought- 
 iron. 
 
 Steel. 
 
 Diameter 
 in inches. 
 
 Wr ought- 
 iron. 
 
 Steel. 
 
 1- 
 
 2-0 
 
 32 
 
 4-5 
 
 182- 
 
 291- 
 
 7-5 
 
 843- 
 
 1350- 
 
 1-5 
 
 6-7 
 
 10-7 
 
 5- 
 
 250- 
 
 400- 
 
 8- 
 
 1024- 
 
 1638- 
 
 2- 
 
 16-0 
 
 25-6 
 
 5-5 
 
 332- 
 
 532- 
 
 8-5 
 
 1228- 
 
 1965- 
 
 2-5 
 
 31-2 
 
 50- 
 
 6- 
 
 432- 
 
 691- 
 
 9- 
 
 1458- 
 
 2332- 
 
 3- 
 
 54-0 
 
 86-4 
 
 6-5 
 
 549- 
 
 878- 
 
 9-5 
 
 1714- 
 
 2743- 
 
 3-5 
 
 85-7 
 
 137- 
 
 7- 
 
 686- 
 
 1097- 
 
 10- 
 
 2000- 
 
 3200- 
 
 4- 
 
 128- 
 
 204- 
 
 
 
 
 
 
 
 Fig. 562 represents a line of shafting. A is an upright shaft ; a a, bevel- 
 gears ; b 1), bearings for the shafts ; c, coupling or connection of the several 
 pieces of shafting. These shafts are of wrought-iron or steel, of uniform sec- 
 tion. As the power is distributed from this line of shafting, the torsional 
 strain diminishes with the distance from the bevel-gears or first movers, and 
 the diameter of each piece of shafting may be reduced consecutively, if neces- 
 sary ; but uniformity will generally be found to be of more importance than a 
 small saving of material. The drawing given is of a scale large enough to 
 order shafting by, but the dimensions should be written in. 
 
 In laying out lines of shafting, the position of the bearings is usually fixed, 
 and the lengths of shafts must be determined thereby, with as few couplings as 
 possible. When there is no necking or reduction of the shafts, which is usually 
 the case, the orders given for shafting will be so many lengths and of such 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 263 
 
 diameters, and so many couplings and hangers. When there is to be a neck- 
 ing, the sketch for the order may be very simple, showing length and diameter 
 of shaft, and position, length, and diameter of bearing. 
 
 The couplings and pulleys are to be placed as near the bearings as possible. 
 It frequently happens, therefore, that the coupling and pulley are needed at 
 
 FIG. 562. 
 
 the same point ; to remedy this, as the position of the pulley depends on the 
 machine which it is required to drive, it frequently can not be moved without 
 considerable inconvenience or loss of room ; the shaft will have, therefore, to 
 be lengthened or shortened, to change position of coupling ; or, if the coup- 
 lings are plate couplings, they may be made with faces for belts. 
 
 FIG. 564. 
 
 When a horizontal shaft is supported from beneath, its bearing is usually 
 called a pillow- or plumber-block, or standard ; if suspended, the supports are 
 called hangers. 
 
 Figs. 563 and 564 are the elevation and plan of a pillow-block. It consists 
 of a base plate, A, the body of the block B, and the box C. The plate is bolted 
 
 
264 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 FIG. 566. 
 
 securely to its base, the surface on which the block B rests being horizontal. 
 A and B are connected by bolts passing through oblong holes to adjust the 
 position in either direction laterally. The box or bush C is of composition, in 
 two parts or halves, extending through the block, and forming a collar by 
 which it is retained in its place. The cap of the block is retained by the screws 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 265 
 
 FIG. 568. 
 
 ooo ; in the figure there are two screws on one side and one on the other ; 
 often four are used, two on each side, but most frequently but one on each 
 side. 
 
 The standard is for the support of horizontal shafts at a considerable dis- 
 tance above the foundation-plate. Fig. 565 is a front elevation ; Fig. 566, a 
 plan ; and Fig. 567, an end elevation of a standard. Like the pillow-block, the 
 plate A is fastened to the foundation itself, and the upper surface is placed per- 
 fectly level in both directions. On these bearing surfaces, a a a, the body of the 
 standard rests, and can be adjusted in position horizontally, and then clamped 
 by screws to the foundation-plate, or keyed at the ends. 
 
 Elevations and plan are usually drawn in such positions to each other that 
 lines of construction can be continued from one to the other, which not only 
 simplifies the drawings, but makes them more readily intelligible. Letters and 
 dotted lines in these figures illustrate this sufficiently. 
 
 The sides of the elevations are represented as broken ; this is often done 
 in drawing, when the sides are uniform, and economy of space on the paper is 
 required. 
 
 Hangers. Figs. 568, 569, and 570 are the plan side and front elevation of 
 a side hanger especially adapted to a position in which the strain is in one di- 
 rection and against the upright 
 part. 
 
 Figs. 571, 572, and 573 are side 
 elevation, plan, and section on 
 line A B of a centre hanger of an 
 old pattern, but simple, adapted to 
 any strain, and if adjusted to a 
 position where the shaft is not 
 likely to be moved, the form is 
 strong and economical. 
 
 Fig. 574 is of a later pattern, 
 in which the shaft can be readily 
 adjusted or removed. 
 
 Hangers are bolted to the floor- 
 timbers, or to strips placed to 
 sustain them, the centres of the 
 boxes being placed accurately in 
 line, both horizontally and later- 
 ally. 
 
 Figs. 575-578 represent differ- 
 ent views of what may be called a 
 yoke-hanger. A is the plate which 
 is fastened to the beam, E is the 
 yoke, and B the stem of the yoke, cut with a thread so as to admit of a vertical 
 adjustment ; the box D of the shaft C is supported by two pointed set-screws 
 passing through the jaws of the yoke ; this affords a very flexible bearing, and 
 a chance for lateral adjustment. 
 
 The last hangers are of the design of William Sellers & Co., who have made 
 improvements in the designs for bearings, pillow-blocks, hangers, shafting, and 
 
 FIG. 569. 
 
 FIG. 570. 
 
266 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 couplings, which are in use in their own shops and have been extensively copied 
 by others. Some of the distinctive forms are further illustrated. 
 
 FIG. 571. 
 
 o 
 
 J 
 
 \ 
 
 o 
 
 FIG. 572. 
 
 Figs. 579 and 580 are the front and side elevations of a pillow-block, one 
 half of each being in section. The length of the box is about four diameters 
 
 FIG. 574. 
 
 of the shaft. The centre bearings are spherical and fit in corresponding recesses 
 in the block and cap, permitting the bearing to adjust itself to the journal of 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 26? 
 
 the shaft. Lubrication is ordinarily through the centre of the cap ; but the 
 upper box has two cups containing a mixture of oil and tallow which is usually 
 
 FIG. 577. 
 
 FIG. 578. 
 
 solid but melts when the bearing heats. The maximum pressure allowed is 
 50 pounds per square inch, the diameter multiplied by the length of bearing 
 or4D. 
 
 FIG. 580. 
 
 Another feature of the Sellers bearings is the spherical-shaped drip-pan to 
 catch the waste oil. By the distribution of the oil the metal of the shaft will 
 
 
268 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 not touch that of the box. Any metal can be used for the box ; cast-iron is the 
 cheapest and best if the surface is kept oiled, but the poorest if allowed to run 
 
 dry. The oil cup at the centre of the cap for a 
 shaft %%" in diameter making 120 revolutions per 
 
 FIG. 582. 
 
 minute has a capacity of 2'2 fluid ounces, which is sufficient for six months' 
 run. The above revolutions per minute are Messrs. Sellers & Co.'s -practice 
 for machine shops, for wood-working machinery 250, for that of cotton and 
 woollen mills 300 to 400 per minute. 
 
 FIG. 583. 
 
 Fig. 581 is a ball-and-socket hanger the construction of which is similar to 
 the pillow-block. The centre spherical bearings are adjusted vertically by the 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 269 
 
 screws d and e, the interior of which are made in hexagonal form and into 
 which a key is fitted for the operation of the screw. 
 
 Fig. 582 is a view of a side hanger adapted to a counter-shaft ; the square 
 slot a is for the shipping bar. 
 
 Fig. 583 represents the elevation of a bracket, or the support of a shaft 
 bolted to an upright ; the box is movable, and is adjusted laterally by the set- 
 screws. Fig. 584 is a front elevation of the back-plate cast on the post ; it will 
 be seen that the holes are oblong, to admit of the vertical adjustment of the 
 bracket. The Sellers pillow-block (Figs. 579 and 580) may be used for the same 
 purpose. 
 
 For Upright Shafts. Footstep, or Step, for an Upright Shaft. Fig. 585 
 represents a half elevation and section of the step. It consists of a foundation 
 or bed-plate, A, a box, B, and a cup or socket, C. The plate A is firmly fast- 
 
 
 FIG. 585. 
 
 ened to the base on which it rests ; in the case of heavy shafts, often to a base 
 of granite. The box B is placed on A, the bearing surface being accurately 
 levelled, and fitted either by planing or chipping and filing ; the bearing sur- 
 faces b are commonly called chipping-pieces, which are the bearing surfaces of 
 the bottom of B. A and B are held together by two screws ; the holes for these 
 are cut oblong in the one plate at right angles to those of the other ; this ad- 
 mits of the movement of the box in two direc- 
 tions to adjust nicely the lateral position of the 
 shaft, after which, by means of the screws, the 
 two plates are clamped firmly to each other. C, 
 the cup or bushing, which should be made of 
 brass, slips into a socket in B. Frequently circu- 
 lar plates of steel (Figs. 586 and 587) are dropped 
 into the bottom of this cup for the step of the 
 shaft. The cup C, in case of its sticking to the 
 shaft, will revolve with the shaft in the box B ; if 
 plates are used, these also admit of movement in 
 the cup. 
 
 Fig. 588 represents the elevation of a bearing for an upright shaft, in which 
 the shaft is held laterally by a box and bracket above the step. The step B is 
 made larger than the shaft, so as to reduce the amount of wear incident to a 
 heavy shaft. The end of the shaft and the cup containing oil are shown in 
 
 FIG. 586. 
 
 FIG. 587. 
 
270 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 dotted line. The bed-plate A rests on pillars, between which is placed a pil- 
 low-block or bearing for horizontal shaft. 
 
 Figs. 589 and 590 represent the elevation and vertical section of the suspen- 
 sion bearing used by Mr. Boyden for the support of the shaft of his turbine- 
 wheels. It having been found difficult to supply oil to the step of such wheels, 
 it was thought preferable by him to suspend the entire weight of wheel and 
 shaft, where it could be easily attended to. The shaft (see section) is cut into 
 necks, which rest on corresponding projections cast in the box b ; the spaces in 
 the box are made somewhat larger than the necks of the shaft, to admit of Bab- 
 bitting, as it is termed, the box ; that is, the shaft being placed in its position 
 
 Fio. 589. 
 
 in the box, Babbitt, or some other soft metal melted, is poured in round the 
 shaft, and in this way accurate bearing surfaces are obtained ; projections or 
 holes are made in the box to hold the metal in its position. The box is sus- 
 pended by lugs b, on gimbals c, similar to those used for mariners' compasses, 
 which give a flexible bearing, so that the necks may not be strained by a slight 
 sway of the shaft. The screws ee support the gimbals, consequently the shaft 
 and wheel ; by these screws the wheel can be raised or lowered, so as to adjust 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 271 
 
 its position accurately ; beneath the box will be seen a movable collar, to adjust 
 the lateral position of shafts. 
 
 No weight rests on the foot of the shaft, but a cast-iron plate is firmly bolted 
 to the floor of the wheel pit, with side flanges, and set screws by which iron or 
 wooden cushions can be adjusted to preserve the shaft in its central position. 
 
 The great care in design and mechanical construction of his wheels and their 
 details enabled Mr. Boyden to obtain large percentages of effect, and led to the 
 general introduction of turbines. In form and construction they have been 
 much simplified, and with economy. Some makers still retain a form of upper 
 hangers and bottom guides, but wooden steps (Fig. 591) are now almost uni- 
 versally adopted. They are made either conical or a portion of a sphere, of 
 various woods, usually lignum-vitas, but oak and poplar are preferred by some. 
 
 FIG. 591. 
 
 FIG. 594. 
 
 The load is from 50 to 75 pounds per square inch. The fibres of the wood are 
 placed vertically, and afford an excellent bearing surface. Water is sometimes 
 introduced into the centre of the wood, or into a box around it, from the upper 
 level of water. When cast-iron or steel is used for the step, it is usual to incase 
 the box and supply oil by leading a pipe, sufficiently high above the surface of 
 the water, to force the oil down. 
 
 For long, upright shafts, it is very usual to suspend the upper portion by a 
 suspension-box, and to run the lower on a step, connecting the two portions 
 by a loose sleeve or expansion coupling, to prevent the unequal meshing of the 
 bevel-wheels, incident to an alteration of the length of shaft by variations of 
 temperature. The suspension is frequently made by a single collar at the top 
 of the shaft. 
 
272 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Fig. 592 is a one half outside end view, and one half transverse. Fig. 593 
 is a section on the centre line of axle, and Fig. 594 sectional plan of box on 
 centre line of axle with a plan of journal and journal bearing of the standard 
 journal box adopted by the Master Car-Builders' Association, 1874. 
 
 Thrust Bearings for Screw Propeller Shafts. The thrust along the shaft 
 of a steamship propelled by a screw is taken up by collar bearings, and through 
 
 them transmitted to the 
 ship. Small shafts up to 
 8" in diameter have gen- 
 erally one thrust collar, 
 but sometimes compara- 
 tively small shafts have 
 several thrust collars. In 
 the latter case the bearing 
 may have a brass bush in 
 halves containing grooves 
 to receive the collars on 
 the shaft. The general 
 
 595. 
 
 FIG. 596. 
 
 practice is to fit between 
 the collars cast-iron or 
 
 cast-steel horseshoe - shaped pieces, clamped between two nuts, which are 
 threaded on a screwed steel bar, supported at its ends by solid bearings cast 
 on the block, as shown in Figs. 595 and 596. In this design each horseshoe 
 piece may be adjusted separately by the nuts on each side, or they may be all 
 moved together by means of the nuts at the ends of the bars. 
 
 A useful diagram (Fig. 597), with accompanying explanation, by George K. 
 Bate, Assoc. M. Inst. C. E., is presented in the Practical Engineer for Nov. 2, 
 1894. " In its construction the effective horse power, or the power actually 
 employed in propelling the ship, has been assumed to be equal to two thirds 
 the indicated H. P. of the engines, so that 
 
 " I. H. P. = the indicated H. P. of the engines. 
 
 " E. H. P. = effective H. P. = I. H. P. x |. 
 
 " K = speed of the vessel in knots. 
 
 " T total thrust, or load on thrust block in pounds. 
 
 " P = pressure on thrust collars, pounds per square inch. 
 
 " S = surface of thrust collar, square inches per I. H. P. ; then 
 
 gTr = K X 101 -3 = speed in feet per minute and work done per 
 
 minute in foot-pounds. 
 
 " = T x K X 101-3,so that T X K X 101-3 = E. H. P. X 33,000 ; therefore 
 E. H. P. X 33,000 
 
 K X 101-3 
 
 " Again, if E. H. P. = f I. H. P., we may write 
 _ 21 H. P. X 33,000 _ I. H. P. x 22,000 
 
 I. H. P. x 217 
 
 3 K X 101-3 K X 101-3 K 
 
 therefore S, the surface of thrust collars in square inches per I. H. P. = 
 
 217 
 K x P- 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 273 
 
 " The diagram gives the value of .S with P, varying from 30 to 80 pounds 
 per square inch. In ordinary practice this pressure is 50 to 60 pounds per 
 square inch in naval, and 40 to 50 pounds per square inch in mercantile 
 steamers, although, in cases where white metal is fitted, it is found that these 
 loads may be safely increased by 25 per cent. 
 
 " As an example, suppose the case of a thrust block for a vessel having en- 
 gines of 3,000 I. H. P., driving her at a speed of 18 knots, to determine the 
 necessary surface of thrust collars that the pressure on them may not exceed 
 60 pounds per square inch. At the point marked 18 knots on the scale for 
 speed of vessel follow the ordinate up till it cuts the line marked 60 on the 
 scale of pressures on the thrust collars to the left of the diagram, at which 
 point of intersection follow to the right, and read off the corresponding surface 
 of collars per I. H. P. i. e., 0-201 square inches ; then 0-201 x 3,000 = 603 
 square inches, the total thrust surface required for E. H. P. = 3,000 x f = 
 
 2,000, and - ' = T, the load on the block equal to 36,180 pounds ; 
 
 -lo X -LU.L" o 
 
 then, if pressure per square inch between surfaces is not to exceed 60 pounds, 
 we have total surface of block = ^ = 603, as per diagram." 
 
 60 
 
 30 
 
 [40 
 
 50 
 
 
 080 
 
 .553 
 
 1 O 
 
 10 11 12 14 16 18 20 25 
 
 SPEED OF VESSEL, KNOTS. 
 
 FIG. 597. 
 
 30 
 
 Couplings are the connections of shafts, and are varied in their construction 
 and proportions, often distinctive of the mechanic making them. 
 
 The Face Coupling (Fig. 598), the one in general use for the connecting of 
 
 wrought-iron shafts, consists of two plates or disks with long, strong hubs, 
 
 through the centre of which holes are accurately bored to fit the shaft ; one 
 
 half is drawn on to the shaft and tightly keyed ; the plates are faced square 
 
 19 
 
274 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 with the shaft, and the two faces are brought together by bolts. The number 
 and size of the bolts depend upon the size of the shaft ; never less than 4 for 
 shafts less than 3 inches diameter, and more as the diameter increases ; the 
 size of the bolts varies from f to 1 inch in diameter. The figure shows a 
 usual proportion of parts for shafts of from 2 to 5 inches diameter ; for larger 
 than these, the proportion of the diameter of the disk to that of the shaft is 
 too large. 
 
 Fig. 599 is a rigid sleeve coupling for a cast-iron shaft ; it consists of a 
 solid hub or ring of cast-iron hooped with wrought-iron ; the shafts are made 
 
 with bosses, the coupling is slipped 
 on to one of the shafts, the ends 
 of the two are then brought to- 
 
 FIG. 598. 
 
 FIG. 599. 
 
 gether ; and the coupling slipped back over the joint, and firmly keyed. This 
 
 is an extremely rigid connection. Some makers use keys without taper, and 
 
 force the couplings on the shafts. 
 
 Fig. 600 is a screw coupling for the connecting of the lighter kinds of 
 
 shafts. It will be observed that this coupling admits of rotation but in one 
 
 direction the one tending to bring 
 the ends of the shafts toward each 
 
 other ; the reverse motion tends to 
 
 .-, ,1 -, 
 
 unscrew, throw them apart, and un- 
 couple them. 
 
 Figs. 601 and 602 is a double-cone 
 vice coupling ; B is the outer shell or 
 sleeve, C C the two cones, and D the bolts. The sleeve is cylindrical outside, 
 but bored with a double taper inside, smallest at the centre. The cones are 
 bored to fit the shaft, and turned outside to fit the interior cones of the sleeve. 
 
 FIG. 600. 
 
 FIG. 601. 
 
 FIG. 602. 
 
 There are three bolt grooves in the cones and sleeves, and one is cut through to 
 give elasticity to the cones. The sleeves and cones are adjusted over the joint 
 of the shafts, leaving it an easy fit, some f inch between the ends of the cones. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 2T5 
 
 If the bolts be introduced and screwed up, the cones are brought nearer to each 
 other and the shafts are securely clamped together. 
 
 Fig. 603 is a clamp coupling for a square shaft. 
 
 In many cases it occurs that rigid couplings, such as have been given, are ob- 
 jectionable; they necessarily imply that, to run with the least strain possible, 
 
 FIG. 
 
 FIG 
 
 the bearings should be in accurate line ; any displacement involves the spring- 
 ing of the shaft, heating of the journals, and loss by friction. Wherever, from 
 any cause, the 'alignment can not be very nearly accurate, some coupling that 
 admits of lateral movement 
 should be adopted. The 
 simplest of these is the box 
 or sleeve coupling (Fig. 
 604), sliding over the end 
 of two square shafts, keyed 
 to neither, sometimes held 
 
 i , FIG. 605. 
 
 in place by a pin passing 
 
 through the coupling into one of the shafts. For round shafts, the loose 
 sleeve Coupling is a. pipe or hub, generally 4 to 6 times the diameter of the 
 shaft in length, sliding on keys fixed on either shaft. 
 
 FIG. 606. 
 
 Fig. 605 is a horned coupling. The two parts of the coupling are counter- 
 parts, each firmly keyed to its respective shaft, but not fastened to each other ; 
 
276 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 FIG. 607. 
 
 the horns of the one slip into the spaces of the other, and, if accurately fitted, 
 it affords an excellent coupling, and is not perfectly rigid. 
 
 It often happens that some portion of a shaft or machine is required to be 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 277 
 
 stopped while the rest of the machinery continues in motion. It is evident 
 that, if one half of a horned coupling be permitted to slide lengthwise on the 
 key the key being fixed in the shaft, forming in this case what is more 
 usually called a feather when the horns of one half are out of the spaces of 
 the other, communication of motion will cease between the shafts. 
 
 Fig. 606 represents a coupling of this sort for a large shaft, from the Cor- 
 liss Steam-Engine Company. The horns are 8 in number on each part, and 
 are thrown readily in or out of action by the handle h turning nut in the loose 
 part of the clutch on the screw cut on the shafts. 
 
 Fig. 607 is another form of disengaging a large pulley from a main shaft, 
 from the Corliss Steam-Engine Company. The pulley is fastened to a cast- 
 iron pipe or sleeve p through which the main shaft s passes. The two are 
 attached by means of the coupling c, one half 
 of which is attached to the shaft and the other 
 to the sleeve. When bolted together, the pul- 
 ley and main shaft move together ; but if the 
 bolts be removed, then the pulley becomes sta- 
 tionary even if the shaft is running. Shaft 
 and sleeve have independent bearings. A 
 section of the coupling c on a larger scale 
 shows the strong taper of the bolts without 
 head. 
 
 It is difficult to maintain shafts in exact 
 line, and slight disarrangements are met by 
 the elasticity of the shafts but, as a further 
 precaution, flexible couplings are used, of which 
 Fig. 608, called Oldham's Coupling, makes a 
 strong connection and admits of considerable 
 variation in the lines of the coupled shafts. 
 It consists of two heads fastened to their several shafts, across the face of these 
 heads two grooves are cut, and between these faces an intermediate plate is in- 
 serted with two tongues at right angles to each other, slide-fitted to the grooves 
 in the heads, thus coupling the two shafts. 
 
 In Fig. 609 the coupling admits of more motion. The grooves are in the in- 
 termediate plate, and one of the tongues is fitted in its head with a T or dove- 
 tail groove held in position by a set screw or pin, by the removal of which the 
 tongue can be withdrawn and the shaft uncoupled. 
 
 FIG. 608. 
 
 FIG. 609. 
 
 FIG. 610. 
 
 Hooke's Joint or Universal Coupling is used to connect two shafts whose 
 axes intersect, and it has the advantage that the angle between the shafts may 
 
278 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 FIG. 611. 
 
 be varied while they are in motion. Fig. 610 shows the entire coupling partly 
 in section ; the shafts to be coupled are forked at their ends ; these forked ends 
 carry between them a cross the arms of which are at right angles to each other. 
 
 The arms of the cross are jointed to 
 the forks so that they may turn freely 
 about their axes. The angular veloci- 
 ties of the shafts will be unequal ex- 
 cept at every quarter revolution, but 
 by using a double Hooke's joint, as 
 
 shown in Fig. 611, the two shafts will have the same angular velocities, if 
 they make equal angles with the intermediate shaft and are in the same plane 
 with it. 
 
 It is often necessary to engage shafts, when one is in motion, or disengage 
 when both are in motion. One of the oldest forms of clutch for this purpose 
 is the slide or clutch couplings, when the motion is required but in one direc- 
 tion (Fig. 612). A represents the half of the coupling that is keyed to the 
 shaft, B the sliding half, c the handle or lever which communicates the sliding 
 movement ; the upper end of the lever terminates in a fork, inclosing the hub 
 of the coupling, and fastened by two bolts or pins to a collar round the neck 
 of the hub ; to support B the end of its shaft extends a slight distance into the 
 coupling A. Shafts can not be engaged with this form of coupling while the 
 
 driving shaft is in rapid 
 ,._ .,_ ,_ . motion without shock. 
 
 r- I 
 
 To 
 
 obviate this, other forms of 
 coupling are requisite ; one 
 of these is represented (Fig. 
 613). On the shaft B is 
 fixed a drum or pulley, 
 
 FIG. 612. 
 
 which is embraced by a friction 
 this band consists of two straps 
 ends projecting on either side ; 
 is the common form of bayonet 
 affords a guide to the prongs or 
 ping these prongs forward, they 
 friction band ; the shaft A being 
 
 FIG. 613. 
 
 band as tightly as may be found necessary ; 
 of iron, clamped together by bolts, leaving 
 the portion of the coupling on the shaft A 
 clutch ; the part cc is fixed to the shaft, and 
 bayonets b b, as they slide in and out. Slip- 
 are thrown into gear with the ears of the 
 in motion, the band slips round on its pulley 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 2T9 
 
 till the friction becomes equal to the resistance, and the pulley gradually attains 
 the motion of the clutch. 
 
 But of all slide couplings, to engage and disengage with the least shock and 
 at any speed, the friction-cone coupling (Fig. 614) is by far the best. It con- 
 sists of an exierior and interior cone, a, b ; a is 
 fastened to the shaft- A, while Z slides in the usual 
 way on the feather of the shaft B ; pressing b 
 forward, its exterior surface is brought in contact 
 
 FIG. 614. 
 
 FIG. 615. 
 
 with the interior conical surface of a; this should be done gradually; the 
 surfaces of the two cones slip on each other till the friction overcomes the 
 resistance, and motion is transmitted comparatively gradually and without 
 danger to the machinery. The longer the taper of the cones, the more difficult 
 the disengagement ; but the more blunt the cones, the more difficult to keep 
 the surfaces in contact. An angle of 8 with the line of shaft is a very good 
 one for surfaces of cones of cast-iron on cast-iron. When thrown into gear, 
 the handle of the lever or shipper (Fig. 615) is slipped into a notch, that it 
 may not be thrown out by accident. 
 
 The objection to this coupling is 
 that it will work out of gear unless the 
 shipper - handle is held firmly in its 
 position, and producing considerable 
 friction against the collar. To ob- 
 viate this the shipper is made to act 
 on a toggle-joint fastened to the shaft, 
 and, once thrown, the pressure is self- 
 continued and preserved without any 
 
 action of the shipper, and without fric- FIG. eie. 
 
 tion. 
 
 Fig. 616 represents a double-friction clutch, of the Weston-Capen patent. 
 The clutch G is slid over the toggle, and the friction cone is forced into the 
 pulley and engaged therewith. In the figure, D' is thus engaged with A', while 
 D and A are not in contact. 
 
280 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 When from any cause, as in rolling mills, the gearing is subject to sudden 
 shocks, which might be injurious unless some means were adopted to modify 
 the blow, friction couplings may be introduced of which Fig. 617 is an illus- 
 
 Fio. 617. 
 
 tration, in which the frictional resistance is sufficient to transmit the required 
 power, but under sudden shocks yields and slips. It consists of two hubs, both 
 keyed to their shafts ; one of the hubs has a wood-lined groove into which the 
 plate of the other is inserted and friction is produced between the two, by the 
 bolts, which bring together the parts of the groove ; loosening the bolts removes, 
 the frictional contact and disengages the clutch. 
 
 Fig. 618 is a longitudinal section and Fig. 619 a transverse section of the 
 Weston clutch. The five iron disks A engage with solid keys on the long boss 
 of the spur wheel E, within which the driving shaft C turns freely when no 
 coupling pressure is applied to the disks. The drum D containing the six in- 
 termediate wood disks B slides on feathers on the shaft C, and the groove Gr on 
 the outer end of the drum receives the forked end of a lever by which the coup- 
 ling pressure is applied, compressing the disks against the fixed collar F on the 
 shaft, and thereby coupling the spur wheel E to the shaft C. 
 
 FIG. 618. 
 
 FIG. 619. 
 
 In Fig. 620 is shown the cylinder-friction clutch of Koechlin. In this case 
 the clutch movement takes place readily. The part A is a hollow cylinder in 
 which three internal clamp pieces are fitted, each being provided with a bronze 
 shoe. These are thrown in and out of action by means of a sliding collar B' 
 which operates right- and left hand-screws by means of the lever b. The clamps 
 slide in radial grooves. The nuts for the right- and left-hand screws can be 
 closely adjusted and clamped by set screws, so that a radial movement of less 
 than -" is sufficient to throw the clutch in or out of action. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 281 
 
 Figs. 621 and 622 are sections and elevation of a friction clutch in which 
 the piece A is in the form of a ring of unequal thickness and divided at its thin- 
 
 FIG. 620. 
 
 nest part. From the thickest part of the ring A a strong arm proceeds to a 
 central boss which is keyed to the shaft. This ring, arm, and boss are all cast 
 in one piece. An outer ring or shell B is bored or turned to fit easily over the 
 
 FIG. 621. 
 
 Fm. 622. 
 
 ring A. On the back of the piece B is a boss which serves to carry a wheel or 
 pulley. To use the arrangement as a shaft coupling, the boss on the back of 
 B is keyed to one shaft, the boss of A being keyed to the other. 
 
 To take the place of fast and loose pulleys, as shown in the figures, the piece 
 B rides loose on the shaft, except when it is bound to the ring A. The pieces 
 A and B are bound together by the expansion of the former caused by the rota- 
 tion of right- and left-handed screws working in two nuts which fit into sockets 
 in the ring A, one on each side of the line of division. By pushing the sliding 
 boss C along the shaft, movement of rotation is communicated to the screw 
 through the link D and lever E. The lever E is secured to the middle portion 
 
282 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 of the screw by a grooved key, which is held by a set screw. At the back of 
 this key there is a clearance space sufficient to allow of the key being with- 
 drawn clear of the grooves, so that the lever may be turned on the screw into 
 another position and take up the wear of the screw threads. 
 
 Bovet's magnetic coupling (Fig. 623) consists of a block or head keyed to 
 the power shaft, on the face of which is a groove containing the iron-wire core, 
 connected with the brushes which bear on the rings. 
 
 FIG. 623. 
 
 FIG. 624. 
 
 The shaft to be clutched is provided with a block F, capable of sliding along 
 and approaching C D until it is in contact. It follows, therefore, that when the 
 wire B is traversed by a current, F is attracted against C D and participates in 
 the motion of A. Adhesion is obtained without any external reaction. 
 
 Fig. 624 is a spring hub used on a large rope-driving wheel to take up the 
 shock of starting. The wheel W runs loose on the driving shaft S and is pro- 
 vided with lugs a, a, a, which project into the spring hub H keyed to the driv- 
 ing shaft S. Three heavy springs interposed between the lugs and the hub al- 
 low a circumferential movement of four feet on a six-foot-diameter wheel. 
 
 Pulleys are used for the transmission of motion from one shaft to another 
 by the means of belts ; by them every change of velocity may be effected. The 
 speeds of two shafts will be to each other in the inverse ratio of the diameter 
 of their pulleys. Thus, if the driving shaft make 100 revolutions per minute, 
 and the driving pulley be 18 inches in diameter, while the driven pulley is 12 
 inches, then, 
 
 12 : 18 :: 100 : 150; 
 
 that is, the driven shaft will make 150 revolutions per minute without allow- 
 ance for slip. Where there is a succession of shafts and pulleys, to find the 
 velocity of the last driven shaft : Multiply together all the diameters of the 
 driving pulleys by the speed of the first shaft, and divide the product by the 
 product of the diameters of all the driven pulleys. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 283 
 
 Pulleys are made of cast-iron and of every diameter, from 2 inches up to 
 20 feet. The number of arms v,ary according to the diameter ; for less than 
 8 inches diameter the plate pulley is preferable (Fig. 625) ; that is, the rim is 
 attached to the hub by a plate ; for pulleys of larger diameters, those with arms 
 are used, never less than four in number. The arms are made usually straight 
 (Fig. 626), sometimes curved (Fig. 627). 
 
 FIG. 625. 
 
 FIG. 626. 
 
 FIG. 627. 
 
 Fig. 628 represents a portion of the elevation of a pulley sufficient to show 
 the proportion of the several parts, and Fig. 629 a section of the same. The 
 parts may be compared proportionately with the diameter of shaft ; thus the 
 
 FIG. 629. 
 
 thickness of the hub is about the diameter of the shaft ; this proportion is 
 also used for the hubs of couplings ; the width of the arms from f to full diam- 
 eter ; the thickness half the width ; the thickness of the rim from -J- to the 
 diameter ; the length of hub the same as the width of face. 
 
 Fig. 630 is a large pulley of the Southwark Foundry pattern. The hub is 
 cast with four divisions, to admit of contraction in cooling, and the rim is in 
 halves, to admit of the pulley being put on the shaft without removing it from 
 its bearings, a very common practice with large pulleys. Wrought-iron rim- 
 pulleys consist of a spider that is, the hub and arms of cast-iron, and a 
 wrought-iron plate-rim is bolted to flanges on the extremities of the arms. 
 
 Fig. 631 represents a faced coupling pulley, an expedient sometimes adopted 
 when a joint occurs where a pulley is also required ; the two are then combined ; 
 
284 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 the pulley is cast in halves two plate pulleys, with plates at the side instead of 
 central, faced and bolted together. 
 
 Wooden pulleys called drums are used for pulleys of very wide face. Fig. 
 632 represents one form of construction in elevation and longitudinal section. 
 
 It consists of two cast-iron pulleys A A, with nar- 
 row rims ; they are keyed on to the shaft at the 
 required distance from each other, and plank or 
 lagging is bolted on the rims to form the face of 
 the drum ; the heads of the bolts are sunk beneath 
 the surface of the lagging, and the face is turned. 
 
 .Jt? 
 
 FIG. 631. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 285 
 
 Fig. 633 represents a wooden plate pulley, consisting of sectors of inch 
 boards firmly glued and nailed together, the joints of the boards being always 
 broken. The face is formed in a similar way by nailing and gluing arcs of 
 board one to another to the required width of face ; these last should be of 
 
 FIG. 632. 
 
 clear stuff. The whole is retained on the shaft by an iron hub, cast with a 
 plate on one side, and another separate plate sliding on to the hub ; the hub is 
 placed in the centre of the pulley, the two plates are brought in contact with 
 the sides of the pulley, and bolted through ; and the pulley turned. A similar 
 arrangement of hub is used for the hanging of grindstones. 
 
 Fig. 634 is an elevation of Chase's pulley similar in its rim to that of Fig. 
 633, but an iron spider supplies the place of the wooden plate. They are built 
 
 FIG. 633. 
 
 FIG. 634. 
 
 up with solid rims or split, as in the figure. They are made of the usual pulley 
 dimensions up to 15 feet in diameter, and any desirable width of face, and able 
 to transmit any amount of H. P. at any speed safe for a belt, 
 
 Fig. 635 is a perspective of a pulley with a wrought-iron rim. It is shown 
 
286 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 with two spiders, but it can be made with a single one or with any number, ac- 
 cording to the width of belt required. They are also made in halves, adapted 
 to any position, and safe under all practical requirements. 
 
 A ctmnter-shaft is one distinct from the main shaft, but connected with it 
 by a belt for the purpose of driving a machine. On the counter-shaft there is 
 
 an arrangement of fixed and loose 
 pulleys by which motion can be 
 communicated to it or cut off. 
 
 In Fig. 636, B is the belt from 
 the main shaft which is being 
 shifted on or off the fast pulley F 
 on the counter by the fingers f f 
 on the shipper bar. The belt is 
 shifted to a full position on either 
 the pulleys F or L. When the belt 
 is on the fixed pulley F, the motion 
 of the main shaft is communicated 
 to the counter ; when on the loose 
 pulley L, the counter-shaft remains 
 still and the pulley revolves upon 
 it. It sometimes happens that the 
 
 FIG. ess. friction of the loose pulley upon 
 
 the counter induces its revolution ; 
 
 to prevent which an arrangement is made, as shown in section (Fig. 637), by 
 which the loose pulley moves on a fixed sleeve. 
 
 The shipper handle, not shown, is held positively by notches, or by some 
 arrangement attached to the shipper bar. The faces of the pulleys should be 
 flat or but slightly rounded. The fixed sleeve should be kept oiled. 
 
 FIG. 636. 
 
 FIG. 637. 
 
 At the other end of the shaft cone pulleys are shown which correspond to 
 similar cones on the machine but reversed so that the speed of the machine 
 can be changed by shifting the belt from one set of cones to the other when 
 the machine is stopped. 
 
 Cone pulleys may be made continuous (Fig. 638), thus becoming conoids 
 upon which the belt can be shifted to any line by an adjusting guide. 
 
 It is often necessary to reverse the motion of a machine. This is readily 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 28T 
 
 done by a system of fast and loose pulleys, as shown in the plan and elevation 
 (Figs. 639 and 640), in which A is a" drum or wide-faced pulley on the driving- 
 shaft, B a fast pulley on the driven shaft, and C 
 and D loose pulleys on the same. The move- 
 ment is indicated by the direction of the arrows. 
 The driving-shaft revolves always in the same 
 direction, but on the driven shaft the loose pul- 
 ley of the straight belt is drawn from the bottom, 
 and partakes of the same motion as the driving- 
 pulley ; while by the cross-belt the draft is at the 
 top of its pulley, and the motion reversed. If 
 the straight or open belt be shipped on to the 
 fast pulley B, the motion given to the shaft is 
 like that of the driving-shaft ; if the cross-belt be 
 shipped on to the fast pulley, the motion of the 
 shaft is reversed. In the elevation, the lower side 
 of the open belt is straight, while there is a sag 
 in the upper; the first is the tight or leading 
 belt, through which the power is transmitted, 
 while the upper side is the loose or slack belt. 
 
 When the belt is shifted, while in motion, to a new position on a drum or 
 pulley, or from fast to loose pulley, or vice versa, the lateral pressure must be 
 applied on the advancing side of the belt, on the side on which the belt is ap- 
 
 FIQ. 638. 
 
 FIG. 640. 
 
 preaching the pulley, and not on the side on which it is running off. It is only 
 necessary that a belt, to maintain its position, should have its advancing side 
 in the plane of rotation of that section of the pulley on which it is required to 
 remain, without regard to the retiring side. On this account, the shipper yoke 
 or pins must be on opposite sides of the shipper bar. 
 
 When the main shaft is connected directly with a machine, and it has to be 
 thrown out, the belt is often slipped from the pulley (Fig. 641), and hangs 
 loosely from the shaft, by which the belt is worn and often becomes entangled 
 
288 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 with the shaft or with couplings. It is better to have a hook suspended from 
 the ceiling to catch the belt when thrown off ; or, still better, iron suspended 
 bows (Fig. 641), from which it is easier to slip the belt on the pulley again. 
 
 FiG. 641. 
 
 Motion may be transmitted by belts to shafts at right angles to each other. 
 
 Figs. 642 and 643 is a plan and elevation in which A is the driving-shaft 
 and pulley and B the driven one, at ; right angles to each other. The arrows 
 show the advancing sides of the belts and their position with regard to the face 
 of the pulleys. 
 
 FIG. 643. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 289 
 
 In the case of inclined axis (Fig.- 644) the leading line falls in the middle 
 plane of each pulley, but the following side of the helt does not, hence such 
 systems can only be run in one direction. The leaving points in the figures 
 
 
 FIG. 644. 
 
 FIG. 645. 
 
 are at a and b. The arrangement gives an open belt when the angles between 
 the planes of the pulleys = 0, and at cross belt = 180. In the intermediate 
 positions a partial crossing at the belt is pro- 
 duced, the angle = 90 ; the belt is quarter 
 twist (Fig. 645) ; if = 45, it is quarter crossed. 
 The maximum leading-off angle is 25, which 
 occurs when the distance between the axis is 
 equal to twice the diameter of the largest pul- 
 ley. 
 
 Guide pulleys are very useful in belt trans- 
 mission for shafts at varied angles, and the 
 proper direction is obtained when each guide 
 pulley is placed at the point of departure of 
 its plane with that of the next following pul- 
 ley. 
 
 Fig. 646 is an arrangement adopted in 
 portable grist-mills for driving the vertical 
 shafts, #, #, of mill-stones, from pulleys on a 
 horizontal shaft. Here it is thought necessary 
 to use guide-pulleys. 
 
 Figs. 647 and 648 are the elevation and 
 20 
 
290 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 plan of another arrangement of pulleys and guide-pulleys ; a b is the intersec- 
 tion of the middle plane of the principal pulleys. Select any two points a and 
 b on this line, and draw tangents a c, b d, to the principal pulleys. Then c a c 
 
 F 
 
 FIG. 017. 
 
 FIG. 648. 
 
 and e b d are suitable directions for the belt. The guide-pulleys must be placed 
 with their middle planes coinciding with the planes cacandebd. The belt 
 will run in either direction. 
 
 In Figs. 649 and 650 are parallel axes with two guide-pulleys. In the first 
 the guide-pulleys are placed in planes tangent to both operating pulleys, and 
 hence driving may occur in either direction. Usually, however, it is required 
 to provide for motion in .but one direction, in which case the second form is 
 used as being simpler. The pulley B may be used as one of the guide-pulleys, 
 in which case it may be placed loose upon the same shaft as A, and <D or D be 
 made drivers or driven. 
 
 It is necessary to stretch the belt over the pulleys to prevent its slip while 
 conveying power. But if the belt is very heavy, and runs nearly horizontally, 
 
 FIG. 649. 
 
 FIG. 650. 
 
 its weight will supply a portion of the adhesion which diminishes with the in- 
 clination of the belt till it becomes vertical, when the friction of the stretch is 
 the only factor of the adhesion of the belt to the lower pulley ; and, as the belt 
 lengthens by use, the value of this friction becomes nothing. This position of 
 pulleys should not obtain if it can be avoided ; but if not, the friction-stress 
 should be by means of an idler or binder (Fig. 651) on the loose belt; distinc- 
 tively the idler rests in a loose frame against the belt, acting by gravity, while 
 the binder is forced against the belt by mechanical appliances. By the relief 
 of the binder the belt becomes slack, the friction of the belt on the pulleys be- 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 291 
 
 comes nothing, and motion stops on "the driven pulley ; but that of the belt may 
 continue by its friction on the driver, from which it can be raised by a rope 
 attachment or by a more positive contrivance. This is not found necessary 
 when the arrangement is used for the engaging or disengaging of machines, 
 but the driver pulley is provided with flanges so that the 
 belt can not slip off. Idlers or binders are of necessity 
 when the two pulleys are near to each other, as in steam 
 hoist engines, either to increase the bearing surface on the 
 pulleys or make up for the slight weight of a short belt. 
 
 Belts run the best when their length and position are 
 such as to give the frictional stress without much stretching 
 on the pulleys, and without binders, and for this purpose 
 the tight side of the belt that is, the one approaching 
 the driver pulley should be at its under side. 
 
 In determining the necessary length of a belt for any 
 position, the simplest way is to measure it, if the construc- 
 tion is complete ; if not, to make a drawing of the pulleys 
 in position to a scale, and measure on the drawing. 
 
 The width of the belt should always be a little less 
 than the face of the pulley ; both are to be determined by 
 the power to be transmitted and the velocity of movement. 
 
 Allowance should be made in calculation of speed of a 
 driven pulley for the slip of the belt, which is always some- 
 thing, but should not fall behind more than one per cent 
 that of the driver ; the friction, with too tight a belt is too 
 much, and the slip, with a too slack one. Too small dimen- 
 sion of pulley or too little of arc of contact increases slip. It does no particu- 
 lar harm to have a belt unnecessarily wide, but it does to have it too narrow. 
 If the diameter of the pulleys be increased, the speed of the belt is also in- 
 creased, and for transmitting the same power, its width decreased. 
 
 By experiments of H. B. Gale at the Washington University, St. Louis, the 
 practical limits of speed of belt may be taken at from 3,000 to 7,000 feet per 
 minute. The flesh side of leather possesses much greater tension than the 
 grain or hair side, and on this account, and in the practice of most mill- 
 wrights, the grain side is put in contact with the pulley, and in double belts 
 the flesh side is placed centrally. 
 
 FIG. 651. 
 
 John T. Henthorn's formula for double belts is 
 
 D x TT x R 
 
 = H - R or 
 
 450 450 
 
 per inch in width, in which D is the diameter of pulley in feet, R the revolu- 
 tions per minute. This is expressed graphically in Fig. 652. 
 
 Use of Diagram. To find the horse power that can be transmitted by a 
 24" belt on a 20-foot pulley making 100 revolutions per minute : The abscissa 
 line 100 intersects the diagonal 20 on the ordinate line 14 ; 14 x 24 = 336 = 
 horse power transmissible. 
 
 To find the belt necessary to transmit 100 horse power through a 10-foot 
 pulley and 120 revolutions per minute of shaft: The abscissa 120 cuts the 
 
 diagonal 10 on the ordinate line 8J; -.(- = 12" width of belt. If the pulley 
 
 i 
 
292 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 were 12-foot instead of 10, it will be seen by the diagram that the intersection 
 of diagonal would be at 10, and the width of belt - = 10". 
 
 This rule referred to single belts capable of transmitting one half the power 
 of the double belt, would require a velocity of 900 feet per minute for a belt of 
 
 Diameter of Pulleys, shown by Diagonals. 
 
 567 8 9 10 u 12 Is 
 Horse-Power per Inch of Width. 
 FIG. 652. 
 
 1" in width to transmit 1 horse power, and if the belt were triple, and running 
 with the same velocity, it would transmit 3 horse power, which is a rule in 
 common use. 
 
 The above rules are applicable to India rubber and canvas belts, which are 
 largely used. They are made of plies of closely woven duck, stitched and cross- 
 stitched together, with or without rubber between the plies and on the outer 
 surfaces; the rubber belts are the only ones that can be run in wet places. 
 The plain duck belts depend largely on the close stitching of the plies, and can 
 be used as cross belts, or in any place where leather belts can be used. 
 
 There is a great difference among mechanics as to the amount of power 
 that may be transmitted by a belt with economy, but the rules as given by 
 Henthorne above are within limits of practice. The Amoskeag Manufacturing 
 Co., Manchester, N. H., run two double belts each 40" wide, and one 24" wide, 
 on a 30' fly-wheel pulley, of 110" face, making 61 revolutions per minute with 
 1,950 indicated H. P. on the steam engine and transmitted through the belts, 
 say 1,800 H. P., gives 17 - 3 H. P. for each inch in width of belt, and Henthorne 
 12-8. The belts were considered heavily loaded but not overtaxed. 
 
 Samuel Webber, in the " American Machinist," February, 1894, reports the 
 case of a belt 30" wide, f " thick, running for six years at a velocity of 3,900 feet 
 per minute on a pulley of 5 feet diameter and transmitting 556 H. P., which 
 gives a velocity of 210 feet per minute per inch width per H. P. By Mr. Fred. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 2 ( J3 
 
 W. Taylor's rule it would be used to" -transmit only 123 H. P., who as Mem. A. S. 
 C. E., in Vol. XV. of its "Transactions," has given conclusions from his prac- 
 tical use of belts in the running of a machine shop day and night for nine 
 years, a very long life if estimated in day's work of ten hours each, with ample 
 opportunity for repairs, and for narrow belts transmitting power to machinery. 
 He finds by testing the tension of belts by a spring balance between two clamps 
 attached to the ends of the belt whi.e stretching, the most economical average 
 total load for double belting to be 200 to 225 pounds per square inch of sec- 
 tion, that a total load of 111 pounds per inch width corresponds to a pulling 
 power of 65 pounds, of 54 pounds to 26 pounds, and that the maximum speed 
 for economy should be from 4,000 to 4,500 feet per minute. 
 
 " Belts are more durable and work more satisfactorily when made narrow 
 and thick than wide and thin. Minimum diameter of pulley for a double belt 
 12", for a triple 20", for a quadruple 30". The ends of a belt should be fastened 
 together by splicing and cementing instead of lacing, wiring, or using hooks or 
 clamps of any kind." 
 
 Leather belts may be purchased from stock from 1" to 48" in width, round 
 belts from '' to " in diameter. 
 
 Rubber 3-ply is equal to single leather. The following are stock sizes : 2-ply 
 from 1" to 28" width ; 6-ply up to 60" ; 7- and 8-ply to order. 
 
 Full rolls contain 400 to 450 feet, and endless belts are made to order. Solid 
 cotton belts, 2-ply I" to 6" wide ; 4-ply 1" to 22". 
 
 The use of endless ropes instead of belts is of very old application, by single 
 lines of rope with outdoor exposure, and large pulleys at considerable distances 
 apart. In Fig. 653 an arrangement is shown for the transfer of a reciprocat- 
 
 FIG. 653. 
 
 ing power ; one end of the rope is attached to and wound on one barrel while 
 the other end is wound in an opposite direction on another barrel, so that as 
 the ropes are unwound from one barrel they are taken up by the other, the 
 length of the reciprocating movement being the length of transfer from one 
 barrel to the other. 
 
 This arrangement is sometimes applied to hoists, and with chains instead of 
 ropes to the old type of planers. 
 
 Of late the use of ropes for the t7'ansmission of power has increased very 
 rapidly both in this country and in England, on account of their economy in 
 first cost and maintenance, in transmitting large amounts of power to consider- 
 able distances with simplicity in changes of direction and distribution of power, 
 smooth running, and absence of slip. 
 
 There are two forms of arrangement, in one of which a single spliced endless 
 rope by its tension gives the necessary adhesion (Fig. 654), and as the rope grows 
 slack by use, taking it up by a fresh splice ; in the other the slack is taken up 
 by a tension carriage. In both forms increase of power is met by multiple 
 grooves or pulleys and in the number of loops of rope. 
 
294 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Fig. 655 shows the tension carriage as applied to the driving cable on the 
 Brooklyn Bridge. A is the pulley connected with the steam engine, on which 
 
 FIG. 654. 
 
 there are four grooves and lines of rope for adhesion ; one line passes over the 
 large standing 10-foot sheave B, thence round the tension pulley C on a weighted 
 car moving in inclined rails, thence over the sheave D to the line of bridge, over 
 
 Fio. 655. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 295 
 
 the bridge beyond track, and returning the other to the pulley A. The cable 
 in diameter and general arrangement is similar to that for cable roads, and is 
 only given as an illustration of an extreme size of tension car. For transmis- 
 sion of power to machinery the diameter of rope does not exceed 2 inches, and 
 the tension car is usually a light grooved pulley, sliding on vertical or horizon- 
 
 FIG. 656. 
 
 tal tracks, with weights attached and moving vertically to give the requisite 
 adhesion. 
 
 With most makers the pulleys are single multiple grooves, but the Link Belt 
 Engineering Co. make light pulleys, with a single groove to 
 each, which can be bolted together to make a multiple grooved 
 sheave of the requisite number of ring sections. 
 
 " The diameter of the pulleys has an important effect on the 
 wear of the rope. The larger the sheaves, the less the fibres of 
 the rope slide on each other, and consequently there is less in- 
 ternal wear of the rope. The pulleys should not be less than 
 forty times the diameter of the rope for economical wear, and 
 as much larger as it is possible to make them. This rule applies 
 also to the idle and tension pulleys as well as to the main driv- 
 ing-pulley." 
 
 Fig. G56 is a view of a horizontal tension carriage ; Fig. 657, 
 a half turn with vertical tension ; Fig. G58, a portion of the line 
 of shaft of the factory of the same company. 
 
 The usual material for rope gearing of mills is either hemp, 
 manilla, or cotton. The ropes are untarred hawser laid that 
 is, formed with three strands twisted together right-handed ; a 
 strand is made by twisting yarns together left-handed. 
 
 The " Stevedore " rope of the Link Belt Engineering Co. is a 
 4-strand rope, manufactured from long-fibre manilla laid in tallow mixed with 
 plumbago (to reduce the friction in the bending of the strands passing around 
 
 FIG. G57. 
 
296 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 the sheaves), which renders it nearly waterproof, therefore suitable for out-of- 
 door work. 
 
 Fig. 659 represents a section of the grooves of a pullc-y as designed by E. 
 D. Leavitt, M. E. 
 
 FIG. 658. 
 
 For the strength of a rope, the assumption of Mr. C. W. Hunt is that " a 
 rope one inch in diameter should have a working strain of 200 pounds at all 
 speeds. This is about one twentieth of the strength of the splice. This large 
 
 \ 
 
 FIG. 659. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 297 
 
 ROPE DRIVING. 
 
 Horse power of manilla 
 rope at various speeds. 
 
 2i I 
 
 30 40 50 60 70 80 90 100 110 
 
 VELOCITY OF DRIVING ROPE IN FEET PER SECOND. 
 
 FIG. 660. 
 
 130 140 
 
 ROPE' DRIVING. 
 
 The curves show the sag of the 
 ropes when transmitting the normal 
 amount of power. It is the same at 
 all speeds for the driving part, but 
 variable for the slack part. The sag 
 for the slack part is computed for 
 speeds of 40, 60 and 80ft. per sec. 
 
 margin is to enable the rope to perform a great amount of useful work before 
 
 it is so weakened by wear that it is necessary to be renewed. There are many 
 
 strains which can not be computed 
 
 owing to the irregularities of the 
 
 power and the work. The diagram 
 
 (Fig. 660) takes into consideration 
 
 the effects of the centrifugal force 
 
 so that the strain on the rope is con- 
 
 stant on the driving side in trans- 
 
 mitting the tabular horse power, no 
 
 matter what the speed may be. It 
 
 shows also the power a rope trans- 
 
 mits at various speeds, illustrating 
 
 the rapidity with which the horse 
 
 power decreases when the speed gets 
 
 beyond about eighty feet per sec- 
 
 ond." 
 
 It is desirable in all cases of rope 
 transmission to so arrange the drive 
 that the slack side of the rope shall 
 be on the upper part of the pulley, 
 thus increasing the arc of contact, as 
 the two sides will then approach 
 each other when in motion. 
 
 In order that the desired tensions 
 shall be attained in the two parts 
 of a rope, the deflection or sag must 
 be of predetermined values. ' 
 
 Fig. 601 is another diagram of 
 
 E8 
 no. eei. 
 
298 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Mr. Hunt's showing the sag of the rope. The rope is supposed to have the 
 strain constant at all speeds on the driving side and in direct proportion to the 
 area of cross section, hence the catenary of the driving side is not affected by 
 the speed or by the diameter of the rope. 
 
 The deflection between the pulleys on the slack side varies with each change 
 of load or change of speed. 
 
 Having determined the sag of the rope from the diagram, lay off the pulleys 
 as in Fig. 662, draw a horizontal line A B, and from the centre of this line and 
 normal to it the line E C equal to the sag of the rope, and construct a parabola 
 
 by dividing the line C D into, say, five equal parts and the line A D into the same 
 number of equal parts; the intersections of the lines Cl, C2, C3, etc., by the per- 
 pendiculars at 1', 2', 3', etc., give the points of the curve. 
 
 The determination of the sag of ropes whose points of contact with their 
 pulleys are not level may be determined by calculation ; but as it can only be 
 for one condition of speed and load, it will be sufficient to determine it practi- 
 cally by taking a cord equal to the whole distance between the points of con- 
 tact, and that of the sag as given by the Hunt diagram, when the sag is central. 
 Fix one end of the cord and raise the other to the level of the position it is to 
 occupy in running, and the amount of sag and its position will be defined by 
 the cord. If it is necessary to represent it by drawing, construct two parabolas. 
 
 Wire-rope power transmission is applicable for distances of from 50 to 400 
 feet, and withstands weather exposure. It is of much less diameter than hemp 
 rope to transmit the same power and has more endurance. For endurance the 
 diameters of wheel and rope and sag must be proportioned to each other. The 
 table below is from the circular of the Trenton Iron Works Co., gives the pro- 
 portionate diameters of wheels and ropes, and the H. P. transmission at 100 
 revolutions : 
 
 Diameter 
 
 Diameter 
 
 
 Diameter 
 
 Diameter 
 
 
 Diameter 
 
 Diameter 
 
 
 of wheel, 
 in feet. 
 
 of rope, 
 in inches. 
 
 H. P. 
 
 of wheel, 
 in feet. 
 
 of rope, 
 in inches. 
 
 H. P. 
 
 of wheel, 
 in feet. 
 
 of rope, 
 in inches. 
 
 H. 1 
 
 3 
 
 1 
 
 7 
 
 7 
 
 ft 
 
 36 
 
 9 
 
 4 
 
 82 
 
 4 
 
 * 
 
 9 
 
 6 
 
 f 
 
 38 
 
 10 
 
 f 
 
 91 
 
 5 
 
 1 
 
 11 
 
 7 
 
 
 
 44 
 
 8 
 
 I 
 
 ft 
 
 5 
 
 ft 
 
 16 
 
 8 
 
 
 
 51 
 
 9 
 
 I 
 
 112 
 
 5 
 
 i 
 
 20 
 
 7 
 
 ft 
 
 54 
 
 10 
 
 I 
 
 124 
 
 6 
 
 \ 
 
 24 
 
 8 
 
 1 1 
 'i fi 
 
 01 
 
 8 
 
 l 
 
 13C 
 
 5 
 
 A 
 
 26 
 
 9 
 
 1! 
 
 69 
 
 9 
 
 1 
 
 14( 
 
 G 
 
 ft 
 
 31 
 
 8 
 
 4 
 
 73 
 
 10 
 
 1 
 
 102 
 
 In applying this table for a given amount of horse power, preference should 
 be given to the larger wheels as most serviceable. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 299 
 
 If more H. P. is to be transmitted Jt may be obtained by increasing the revo- 
 lution in a direct proportion np to the limit of 80 feet per second, but not by 
 the increase of tension of rope as shown by sag, which may be taken for a span 
 of 100 feet at ? feet, and for other spans directly as their squares that is, for 
 200 feet it would be ? X 4 = 2'8 feet. The sag as given is that measured from 
 a horizontal line through the point at which the rope leaves the wheel. If the 
 two wheels are not on the same level the sag must be measured from the level 
 of the point of contact with the lower wheel, and the span to be used in deter- 
 
 FIG. 663. 
 
 FIG. 664. 
 
 mining the sag below this level is the distance along the horizontal line from 
 the wheel to the point at which it again intersects the rope. This point may 
 be ascertained by hanging a wire in place on the wheels before splicing the rope. 
 
 If the difference of level is very great, run the rope over intermediate carry- 
 ing sheaves so placed as to give a level stretch of rope of which the sag can be 
 taken from the rule of sag. 
 
 Intermediate supporting pulleys should be avoided as far as practicable, as 
 each one increases the wear on the rope. When they can not be dispensed with 
 they should not be less than one half to two thirds the diameter of the main 
 wheels. 
 
 The sag of the rope when doing full work will be one half that when at rest. 
 The driving-sheaves should always be lined with some elastic material, as rubber, 
 leather, or wood. Figs. 663 and 664 give the section of grooves as made at dif- 
 ferent works. 
 
 Power-transmission by Chains. Fig. 665 is a sectional perspective of a pul- 
 ley in which there is a groove for the vertical links of the chains and a face for 
 
300 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 the horizontal ones, which serve merely as guides for hoisting in cranes or saw- 
 mills and the like. By the introduction of ribs on the face (Fig. 666) adapted 
 to the length of the link or pitch 
 of the chain the motion is deter- 
 minate and is used to transmit 
 power. 
 
 Fig. 667 is a sprocket wheel for 
 
 FIG. 665. 
 
 FIG. 663. 
 
 FIG. 687. 
 
 punched links with teeth between each link, especially adapted to position 
 where the stress is great and the movement slow, for which they can readily be 
 proportioned. ' .- 
 
 The same class of wheel and chains are made light and used as in bicycles, 
 and driven at considerable speeds with little friction. 
 
 FIG. 668. 
 
 FIG. 669. 
 
 FIG. 670. 
 
 The link belting is made with malleable iron links and detachable, so that 
 the belt can be readily lengthened or shortened. The sprocket wheels are 
 machine finished to pitch. 
 
 Figs. 668, 669, and 670 are drawings of links of different forms. They are 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 301 
 
 made of all dimensions to suit the purposes and stresses required. At the 
 
 usual speeds of leather belts, link belting is noisy and wears rapidly, but at 
 
 moderate speeds for conveyors of 
 
 grain, clay, and the like they are 
 
 admirably adapted, and in such 
 
 positions and conditions of speed 
 
 can be used to transmit power. 
 
 Leather link belting consists 
 of links made of leather con- 
 nected by iron or steel pins. A 
 belt of this design can be made of 
 any width, works freely on a pul- 
 ley of small diameter, and can be 
 driven at very high speed, which, 
 combined with its great strength, 
 make this form of belt very suita- / 
 
 FIG. 671. 
 
 ble for driving dynamos. When 
 the link belt runs between guides, 
 or on flanged pulleys, the rivet 
 heads are faced on the outside 
 links with leather after the belt is 
 riveted up. 
 
 When a link belt of considerable width works on a curved pulley, either 
 there is contact at the centre only, or the pins are bent where the band is in 
 contact with the pulley. The belt in this case is in two or more longitudinal 
 strips, hinged together, as shown in Fig. 671. 
 
 GEARING. 
 
 The term gearing, in a general sense, is applied to all arrangements for the 
 transmission of power; but, in a particular sense, to toothed gearing, which 
 may in general be divided into three classes spur, bevel, and screw. In the 
 former the axis of the driving and driven wheels are parallel to each other; in 
 the bevel they may be situated at any angle ; if of equal size and at right angles, 
 they are called mitre gears. In screw gearing a toothed wheel is driven by a 
 screw wifh their axes usually at right angles to each other. Spur wheels are 
 termed external or internal, according to the disposition of their teeth with re- 
 gard to the rim of the wheel. 
 
 Rack gear and pinion are employed to convert a rotary into a rectilinear 
 motion, or vice rersa. In this arrangement the pinion is a spur wheel, acting 
 on teeth placed along a straight bar (Fig. 680). 
 
 Bevel gearing consists of toothed wheels formed to work together in differ- 
 ent planes, their teeth being disposed at an angle to the plane of their faces. 
 
 Trundle pins or Avheels (Fig. 672) are constructed with cylindrical pieces 
 called staves or pins, instead of teeth. A pinion with double plates is called a 
 lantern ; the wheel, a face or crown wheel ; this construction is very useful 
 when iron gears can not be easily obtained or repaired. 
 
 Fundamental principle. In order that two circles A and B (Fig. 673) may 
 be made to revolve bv the contact of the surfaces of the curves m m and n n of 
 
302 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 their teeth precisely as they would by the friction of their circumferences, it is 
 necessary and sufficient that a line drawn from the point of contact t of the 
 teeth to the point of contact c of the circumferences (pitch-circles) should, in 
 
 every position of the point , be perpendicular or normal to the surfaces of con- 
 tact at that point to both the curves m m and n n, a particular property of that 
 curve known as the cycloid. 
 
 When one wheel conducts the other, it is called the driver or leader, and the 
 other the driven or follower. The angular velocities of the pitch circles is the 
 same, but the number of revolutions of the wheels is inversely as their diam- 
 eters taken at the pitch circles. If the driver is 24" diameter, making 120 
 revolutions per minute, and the driver is required to make 200 revolutions per 
 minute, then the diameter of the driven gear would be 
 
 120 x 24 4" 
 200 I4 10 ' 
 
 Fig. 674 gives the designation of the various parts of a spur wheel by whicli 
 names they will hereafter be called. 
 
 Pitch Cirple_L._ 
 
 FIG. 074. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 There is considerable variation in the proportion of teeth, as 
 Thickness of teeth, from '45 to '48 pitch. 
 Space between teeth, from -55 to '52 pitch. 
 
 303 
 
 Height of teeth outside pitch circle, from -2 to -3 pitch. 
 
 Depth of teeth inside pitch circle, from *3 to *4 pitch. 
 
 The above is for cast teeth, used without finishing ; the teeth are made 
 narrower than the space, arid the height less than the depth on account of the 
 irregularities of a rough casting. The teeth and space, when machine cut, may 
 be made the same or very nearly so. 
 
 The cycloid (Fig. 675) is a curve described by any point on the circumfer- 
 ence of a circle on a straight line as a base which is the pitch line of the rack 
 in its application to the formation of teeth of a rack and pinion. 
 
 Divide the circumference of the generating or rolling circle A into a mini- 
 
304: 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 her of equal parts, say 12 ; draw chords from each of these points to ; divide 
 the base line into an equal number of parts of the same length as the arcs of 
 the generating circle numbered 0, 1', 2', etc., and from each of these points 
 erect a perpendicular intersecting the centre of the circle A B at 1", 2", etc. ; 
 from each of these points describe arcs with a radii equal to the generating 
 circle. From the points 1', 2', 3', 4', etc., on the base line, and with radii equal 
 successively to the chords 1, 2, 3, 4, etc., describe arcs cutting the pre- 
 ceding, the intersections will be points of the required curve. 
 
 The epicycloid (Fig. 676) is a cycloid formed on the circumference of a 
 circle as base, which, in its application to the teeth of wheels, is the pitch circle 
 of external gearing. 
 
 The path of the centre of the generating circle is concentric with the base 
 circle. Divide the generating circle from and the arc of the base circle into 
 the same number of equal parts. Eadiai lines from C, passing through 1', 2', 
 etc., and intersecting A B, give the points from which the arcs of the generat- 
 ing circle are described ; from the points 1', 2', 3', etc., on the base circle, and 
 with radii equal successively to the chords 1, 2, 3, etc., describe arcs cut- 
 ting the preceding; the intersections are points in the required curve. 
 
 The hypocycloid (Fig. 677) is a cycloid with a base of the interior circum- 
 ference of a circle corresponding to the pitch circle of internal gearing. 
 
 5' 6' 
 
 Proceed as in the previous example and divide the generating circle and 
 base line into equal parts, describe the path of the centre of the generating 
 circle A B concentric with the base circle, and from each division of the base 
 line draw radial lines from C, intersecting the line A B at 1", 2", etc. ; from 
 these centres describe arcs of a radius equal to the generating circle ; from the 
 points 1', 2', 3', etc., on the base circle, and with radii equal successively to the 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 305 
 
 chords 1, 2, 3, etc., describe arcs- cutting the preceding; the intersections 
 are points of the required curve. 
 
 INVOLUTE. 
 
 The involute (Fig. 678) is the curve described by the end of a string being 
 unwound from the circumference of a circle. 
 
 FIG. 678. 
 
 Divide the circumference of the given circle into any number of equal parts, 
 as 0, 1, 2, etc. At each of these points draw tangents to the given circle ; on 
 the first of these lay off the distance 1-1', equal to the arc 0-1 ; on the second 
 lay off 2-2', equal to twice the arc 0-1 or the arc 0-2 ; establish in a similar 
 way the points 3', 4', 5', as far as may be necessary, which are points in the 
 required curve. 
 
 The involute curve may be described mechanically in several ways. Thus, 
 let A (Fig. 679) be the centre of a wheel for which the form of involute teeth 
 is to be found. Let m n a be a 
 thread lapped round its circum- 
 ference, having a loop-hole at its 
 extremity, a; in this fix a pin, 
 with which describe the curve or 
 involute a b . . . . h, by unwinding 
 the thread gradually from the 
 circumference, and this curve will 
 be the proper form for the teeth 
 of a wheel of the given diameter. 
 
 In all the problems in which 
 curves have been determined by 6 _ 9 
 
 the position of points, the more 
 numerous the points the more accurately can the curve be drawn. 
 
 Spur Wheel and Rack (Fig. 680). The pitch-circle of the spur wheel is 
 drawn and the proper curve for the flank and face of the teeth obtained by 
 21 
 
306 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 rolling a generating circle on the inside and outside of the pitch-circle ; thus 
 the point A' on the generating circle gives for the flank of the teeth the hypo- 
 cyeloid A' B', and for the face the epicycloid A' C'. 
 
 FIG. 680. 
 
 The curve for the teeth of the rack is obtained by rolling the same generat- 
 ing circle on the upper and lower side of the rack pitch-line, giving two cy- 
 cloids, A C and A B, for the face and flank of the teeth. The diagram showing 
 the construction of above curves is, to avoid confusion, separate from the draw- 
 ing of the spur wheel and rack. 
 
 The diameter of the generating circle is to a certain extent arbitrary. In this 
 and the following examples it is taken at a little less than the radius of the 
 smallest spur wheel ; if taken at exactly the radius, the flanks of the teeth of 
 this wheel will be radial lines, which is not usually as satisfactory as where the 
 flanks are of hypocycloidal form, as the teeth, being narrower at the root, have 
 a tendency to break at this point. 
 
 When a number of wheels are intended to gear together, the same size of 
 generating circle and the same pitch and dimensions of the teeth must be main- 
 tained. The generating circle should never be larger than the radius of the 
 smallest wheel of the set. 
 
 Where a drawing of a whole wheel is to be made, the circumference can be 
 divided by radial lines into the same number of parts as there are teeth. The 
 curve of one tooth having been found, a templet can be made and applied 
 successively; or find an arc of a circle corresponding as closely as possible 
 to the cycloidal curve, and apply this at the proper divisions ; the latter way 
 is the more common. 
 
 Spur Wheels. Fig. 681 shows two wheels gearing together, one of ten and 
 the other of thirty teeth. The diameter of the generating circle is less than 
 the radius of the pitch-circle of the smaller wheel ; when the generating circle 
 rolls on the exterior and interior of both pitch-circles, it will in the former 
 case generate the faces of the teeth as shown at A C and A' C', and in the latter 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 307 
 
 case their flanks A B and A' B'. In Bother respects the construction is similar 
 to the former example. 
 
 The simplest illustration of the action of epicycloidal teeth is when they are 
 employed to drive a trundle, as represented in Fig. 672. Let it be assumed that 
 
 FIG. 681. 
 
 the staves of the trundle have no sensible thickness ; that the distance of their 
 centres apart, that is their pitch, and also their distance from the centre of the 
 trundle, that is their pitch-circle, are known. The pitch-circles of the trundle 
 and wheel being then drawn from their respective centres B and A, set off the 
 pitches upon these circumferences, corresponding to the number of teeth in the 
 wheel and number of staves in the trundle ; let five pins, a b c, etc., be fixed into 
 the pitch-circle of the trundle to represent the staves, and let a series of epicy- 
 cloidal arcs be traced with a describing circle, equal in diameter to the radius 
 of the pitch-circle of the trundle, and meeting in the points klmn, etc., alter- 
 nately from right and left. If motion be given to the wheel in the direction 
 of the arrow, then the curved face m r will press against the pin b, and move 
 it in the same direction ; but as the motion continues, the pin will slide up- 
 ward till it reaches w, when the tooth and pin will quit contact. Before this 
 happens, the next pin a will have come into contact with the face a I of the next 
 tooth, which, repeating the same action, will bring the succeeding pair into con- 
 tact ; and so on continually. 
 
 To allow of the required thickness of staves, it is sufficient to diminish the 
 
308 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 size of the teeth of the wheel by a quantity equal to the radius of the staves 
 (sometimes increased by a certain fraction of the pitch for clearance) by draw- 
 ing within the primary epicycloids, at the required distance, another series of 
 curves parallel to these. In practice, a portion must be cut from the points of 
 the teeth, and also a space must be cut out within the pitch-circle of the driver, 
 to allow the staves to pass ; but no particular form is requisite ; the condition to 
 be attended to is simply to allow of sufficient space for the staves to pass with- 
 out contact. 
 
 Internal Gearing (Fig. 682). Draw the spur wheel as in the previous exam- 
 ples ; the face of the teeth of the internally geared wheel will be the hypocycloid 
 
 FIG. 682. 
 
 A 0, formed by the generating circle rolling on the interior of the pitch-circle, 
 and the flanks the epicycloid A B, formed by the generating circle rolling on the 
 exterior of the pitch-circle. 
 
 INVOLUTE TEETH. 
 
 Fig. 683 is a pair of spur wheels showing the mode of drawing involute teeth. 
 The addendum, pitch, and root-circles of both wheels are first drawn, then the 
 base-circles of the two wheels, which must bear the same proportion to each other 
 as the pitch-circles. To arrive at this proportion a semicircle is constructed on 
 the line A C B with a diameter equal to the radius A of the pinion ; a per- 
 pendicular erected where the larger addendum-circle intersects the line A C B 
 intersects the semicircle at E ; a line is then drawn from A to E and a normal to 
 it at E of indefinite length ; then from B a parallel to A E intersecting the nor- 
 mal at E' ; circles concentric to the pitch-circles drawn through E and E' give 
 the base-circles. 
 
 All teeth on the line E E' will be in contact at the same time, called, there- 
 fore, the line of contact. The involute curves of the teeth of both wheels are 
 drawn from their respective base-circles, which correspond to the given circle of 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 309 
 
 Fig. 678. Both involutes in Fig. 683 start from the point of contact E, hence 
 one curve is described forward, the other backward. 
 
 Thus on the normal E' E there are eight parts, and from the commencement 
 of the next normal seven parts are marked off, the next six, and so on until 
 finally the involute reaches the base-circle. The part of the tooth within the 
 base-circle may be a radius to it and tangent to the involute, and small fillets 
 should be drawn connecting their roots to the root-circle. 
 
 Fio. 683. 
 
 A pinion gearing into a rack is shown in Fig. 684. The faces of the teeth of 
 the rack can be taken at an angle of 23 with the perpendicular ; this angle 
 gives the longest line of contact, which may be considered the pitch-line of the 
 rack, and, the pitch-circle of the pinion being indicated, the base-circle for the 
 construction of the involute curves is obtained by drawing a line through the 
 point C' where the two pitch-lines come together at an angle of 23 with 
 the horizontal and where a normal from this line intersects the centre C of the 
 pinion a line is drawn, and through its junction E of the line of contact a 
 base-circle is described ; at this point E the involute is drawn as in the pre- 
 vious example. As in cycloidal teeth, the exact curve may be laid down for one 
 tooth, and an arc corresponding as closely as possible to the involute used to 
 describe the remainder. 
 
 All involute wheels whose teeth have the same pitch and obliquity to the 
 line of contact work well together, but no wheel should have less than twelve 
 teeth to work well. 
 
 Involute wheels can not be made with very long teeth, because the obliquity 
 of the line of contact will be too great. 
 
 The diameters of spur wheels are in proportion to the number of their revo- 
 
310 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 lutions per minute, but the relative sizes of a pair of bevel wheels is determined 
 by a division of the angle included between the two axes inversely as the ratio 
 
 FIG. 684. 
 
 of their angular velocities. Let B and C (Fig. 685) be the position of the two 
 given axes, and let them be prolonged till they meet in a point A. Further, let 
 it be required that C makes seven revolutions while B makes four. From any 
 points D and E in the lines A B, A C, and perpendicular to them, draw D d 
 and E e of lengths (from a scale of equal parts) inversely as the number of rev- 
 olutions which the axes are severally required to make in the same unit of time. 
 Thus, the angular velocity of axis B being 4, and that of the axis C being 7, 
 the line D d must be drawn = 7, and the line E e 4. Then through d and e 
 parallel with the axes A B and A C draw d c and e c till they meet in c. A 
 straight line drawn from A through c will then make the required division 
 
 of the angle BAG, and define the 
 line of contact of the two cones, 
 by means of which the two rolling 
 frusta may be projected at any 
 convenient distance from A. 
 
 If the relative perimeters, di- 
 ameters, or radii, of the pair have 
 been determined, then the lines D 
 d and E e are to each other direct- 
 ly as these quantities. B F and 
 C F are radii of the pitch-circle. 
 
 The case in which the axes are 
 
 FIG. ess. neither parallel nor intersecting 
 
 admits of solution by means of a 
 
 pair of bevels upon an intermediate axis, so situated as to meet the others in 
 any convenient points. 
 
 When the contiguity of the shafts is such as to permit of their being con- 
 nected by a single pair, skewed bevels (Fig. 702) are sometimes employed. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 311 
 
 When the axes are at right angles to each other, and do not intersect, the 
 wheel and screw may be employed to connect them. The velocity of motion is 
 in this arrangement immediately deduced from that of the screw, its number 
 of threads, and the number of teeth in its gearing-wheel. Thus, if it be re- 
 quired to transmit the motion of one shaft to another, contiguous and at right 
 angles to it the angular motions being as 20 to 1 then, if the screw be a 
 single-threaded one, the wheel must have 20 teeth ; but if double-threaded, the 
 number of teeth will be increased to 40, for 2 teeth will be passed at every revo- 
 lution. If the screw have few threads compared with the number of teeth of 
 the wheel, it must always assume the position of driver on account of the ob- 
 liquity of the thread to the axis ; and in this respect its action is analogous to 
 that of a travelling rack, moving endwise one tooth, while the screw makes one 
 revolution on its axis. 
 
 If the pitch-circle be divided into as many equal parts as there are teeth to 
 be given to the wheel, the length of one of these parts is termed the pitch of 
 the teeth. 
 
 The pitch depends on the power to be transmitted or the stress on each tooth. 
 The diagram (Fig. 686) is by John T. Henthorn, M. E., in which pitch and 
 
 1,000 2,000 
 
 3,000 4,000 5,000 
 Stress in Pounds. 
 
 Fio. 686. 
 
 6,000 
 
 7,000 8,000 9,000 
 
 face, represented by multiples of the pitch, are proportioned to the stress in 
 pounds. 
 
 If the pitch be known, the number of teeth in a wheel can be determined 
 approximately by dividing the circumference of the wheel by the pitch, but 
 there must be no remainder in the quotient there can be no fraction of a pitch 
 either the pitch or diameter of wheel must be changed to produce this re- 
 sult ; generally the latter, as gears are usually made of determinate inches and 
 fractions, as given in the table, by which also calculation for diameters and 
 number of teeth is much simplified. 
 
312 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Example 1. Given a wheel of 88 
 teeth, 2^-inch pitch, to find the di- 
 ameter of the pitch-circle. Here the 
 tabular number in the second col- 
 umn answering to the given pitch is 
 7958, which multiplied by 88 gives 
 70-03 for the diameter required. 
 
 2. Given a wheel 33 inches diam- 
 eter, If-inch pitch, to find the num- 
 ber of teeth. The corresponding 
 factor is 1-7952, which, multiplied 
 by 33, gives 59-242 for the number 
 of teeth that is, 59 teeth nearly. 
 Now 59 would here be the nearest 
 whole number, but as a wheel of 60 
 teeth may be preferred for conven- 
 ience of calculation of speeds, we 
 may adopt that number, and find 
 the diameter corresponding. The 
 factor in the second column answer- 
 ing to If pitch is -557, and this mul- 
 tiplied by 60 gives 33-4 inches as the 
 diameter which the wheel ought to 
 have. 
 
 Another mode of sizing wheels in 
 relation to their pitches, diameters, 
 and number of teeth, is adopted, in 
 some machine shops, by dividing the 
 diameter of the pitch-circle into as 
 many equal parts as there are teeth 
 to be given to the wheel. To illus- 
 trate this by an arithmetical exam- 
 ple, let it be assumed that a wheel of 
 20 inches diameter is required to have 
 
 40 teeth ; then the diametral pitch, 
 20 1 
 
 77: = = 4- inch ; 
 40 m 2 
 
 that is, the diameter being divided into equal parts corresponding in number 
 to the number of teeth in the circumference of the wheel, the length of each 
 of these parts is % an inch, consequently m = 2 ; and according to the phrase- 
 ology of the workshop, the wheel is said to be one of two pitch. 
 
 A decided advantage is obtained by the use of the diametral-pitch sys- 
 tern, since circular pitch = di *-* f^ 6 , diam . = circ. pitch X No. of teeth 
 
 
 D = x N. 
 
 N = -p- x D. 
 
 PITCH IN 
 
 RULE. To find the 
 
 RULE. To flnd the 
 
 INCHES 
 AND 
 
 diameter in inches. 
 
 number of teeth, 
 
 PARTS OF 
 
 multiply the number 
 
 multiply the given 
 
 AN INOII. 
 
 of teeth by the tabu- 
 
 diameter in inches 
 
 
 lar number answer- 
 
 by the tabular num- 
 
 
 ing to the given 
 
 ber answering to the 
 
 
 pitch. 
 
 given pitch. 
 
 Values of 
 
 p. 
 
 P 
 
 Values of 
 
 Values of-p- 
 
 6 
 
 1-9095 
 
 5236 
 
 5 
 
 1-5915 
 
 6283 
 
 4| 
 
 1-4270 
 
 6981 
 
 4 
 
 1-2732 
 
 7854 
 
 3* 
 
 1-1141 
 
 8976 
 
 3 
 
 9547 
 
 1-0472 
 
 2f 
 
 8754 
 
 1-1333 
 
 2^ 
 
 7958 
 
 1-2566 
 
 2* 
 
 7135 
 
 1-3963 
 
 2 
 
 6366 
 
 1-5708 
 
 i| 
 
 5937 
 
 1-6755 
 
 If 
 
 5570 
 
 1-7952 
 
 If 
 
 5141 
 
 1-9264 
 
 H 
 
 4774 
 
 2-0944 
 
 If 
 
 4377 
 
 2-2848 
 
 IJ 
 
 3979 
 
 2-5132 
 
 H 
 
 3568 
 
 2-7926 
 
 i 
 
 biss 
 
 3-1416 
 
 1 
 
 2785 
 
 3-5904 
 
 i 
 
 2387 
 
 4-1888 
 
 i 
 
 1989 
 
 5-0266 
 
 i 
 
 1592 
 
 6-2832 
 
 i 
 
 1194 
 
 8-3776 
 
 * 
 
 0796 
 
 12-5664 
 
 No. of teeth 
 
 which always brings out the diameter as a number with an inconvenient frac- 
 tion if the pitch is in even inches or simple fractions of an inch. By the di- 
 ametral-pitch system the diameter may be in even inches or convenient frac- 
 tions, and the number of teeth is usually an even multiple of the number of 
 inches in the diameter. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 313 
 
 RELATION OP DIAMETRAL TO CIRCULAR PITCH. 
 
 Diametral 
 pitch. 
 
 Circular 
 pitch. 
 
 Diametral 
 pitch. 
 
 Circular 
 pitch. 
 
 Circular 
 pitch. 
 
 Diametral 
 pitch. 
 
 Circular 
 pitch. 
 
 Diametral 
 pitch. 
 
 
 In 
 
 
 In. 
 
 
 In. 
 
 
 In 
 
 1 
 
 3-142 
 
 11 
 
 0-286 
 
 3 
 
 1-047 
 
 fl 
 
 3 "SSI 
 
 H 
 
 2-094 
 
 12 
 
 0-262 
 
 24 
 
 1-257 
 
 1 
 
 3-590 
 
 2 
 
 1-571 
 
 14 
 
 0-224 
 
 2 
 
 1-571 
 
 IS 
 
 3-867 
 
 2i 
 
 1-396 
 
 16 
 
 0-196 
 
 11 
 
 1-676 
 
 1 
 
 4-189 
 
 24 
 
 1-257 
 
 18 
 
 0-175 
 
 If 
 
 1-795 
 
 
 
 4-570 
 
 dt 
 
 1-142 
 
 20 
 
 0-157 
 
 If 
 
 1-933 
 
 f 
 
 5-027 
 
 3 
 
 1-047 
 
 22 
 
 0-143 
 
 H 
 
 2-094 
 
 A 
 
 5-585 
 
 8| 
 
 0-898 
 
 24 
 
 0-131 
 
 We 
 
 2-185 
 
 4 
 
 6-283 
 
 4 
 
 0-785 
 
 26 
 
 0-121 
 
 If 
 
 2-285 
 
 ft 
 
 7-181 
 
 5 
 
 0-628 
 
 28 
 
 0-112 
 
 1A 
 
 2-394 
 
 t 
 
 8-378 
 
 6 
 
 0-524 
 
 30 
 
 0-105 
 
 1* 
 
 2-513 
 
 A 
 
 10-053 
 
 7 
 
 0-449 
 
 32 
 
 0-098 
 
 1A 
 
 2-646 
 
 * 
 
 12-566 
 
 8 
 
 0-393 
 
 36 
 
 0-087 
 
 H 
 
 2-793 
 
 A 
 
 16-755 
 
 9 
 
 0-349 
 
 40 
 
 0-079 
 
 1A 
 
 2-957 
 
 i 
 
 25-133 
 
 10 
 
 0-314 
 
 48 
 
 0-065 
 
 1 
 
 3-142 
 
 A 
 
 50-266 
 
 To find the outside diameter of spur-gear blanks, add two parts of the pitch to 
 the pitch diameter. Thus for an 8-pitch gear of 40 teeth the outside diameter 
 of blank is ^-, equal to 5 inches ; for a 12-pitch gear of 36 teeth the outside 
 diameter of blank is ^f , equal to 3 inches ; for a 16-pitch gear of 46 teeth the 
 outside diameter of blank is ff, equal to 3 inches. 
 
 To obtain the distance between the centres of two gears, add the number of 
 teeth together and divide half the sum by the diametral pitch. Thus if two 
 gears have 40 and 30 teeth respectively and are 5 pitch, add 40 and 30, making 
 70, divide by 2, and then divide this quotient 35 by the diametral pitch 5, and 
 the result, 7 inches, is the distance between centres. 
 
 It is a common practice of shops to take as the diameter of the rolling circle 
 the radius of the smallest pinion which will ever be used for gears of this pitch, 
 and constructing the epicycloids for different diameters of this pitch, and allow- 
 ing arcs of circles corresponding very closely to these epicycloids. On this 
 principle Robert Adcock, C. E., constructed a table of radii for these arcs, for 
 rolling circles of pinions of 8, 10, and 12 teeth. We give the last only as best 
 suited to the usual conditions of practice : 
 
 TABLE OP RADII OP ARCS OP CIRCLES FOR GEAR TEETH. 
 
 3 
 
 CJ 
 
 SMALLEST PINION, 
 
 3 
 
 0) 
 
 SMALLEST PINION, 
 
 3 
 
 <u 
 
 SMALLEST PINION, 
 
 I 
 
 *s2 
 
 TWELVE TEETH. 
 
 3 
 
 =1 
 
 TWELVE TEETH. 
 
 
 jjj 
 
 *! 
 
 TWELVE TEETH. 
 
 o 
 
 aa 
 
 Radii of the Eadii of the 
 
 "3 
 
 V 
 
 .2^ 
 
 Radii of the 
 
 Radii of the 
 
 
 s| 
 
 Radii of the 
 
 Radii of the 
 
 X 
 
 '3 
 
 faces of i flanks of 
 
 ,-; 
 
 ^5 -ij 
 
 faces of 
 
 flanks of 
 
 
 ^s 
 
 faces of 
 
 flanks of 
 
 st 
 
 So, 
 
 teeth. teeth. 
 
 
 
 a -a 
 
 teeth. 
 
 teeth. 
 
 1 
 
 M ft 
 
 teeth. 
 
 teeth. 
 
 12 
 
 1-93 
 
 1-88 0-75 
 
 
 27 
 
 4-31 
 
 4-23 
 
 0-84 
 
 4-63 
 
 1-41 
 
 42 
 
 6-69 
 
 6-60 ! 0'89 
 
 6-91 
 
 1-20 
 
 13 
 
 2'09 
 
 2-040-76 7-45 
 
 7-14 
 
 28 
 
 46 
 
 39 
 
 85 
 
 37 
 
 38 
 
 43 
 
 85 
 
 76 
 
 89 
 
 7-06 
 
 20 
 
 14 
 
 2-25 
 
 19 '77 4-86 
 
 4-27 
 
 29 
 
 62 
 
 55 
 
 85 
 
 92 
 
 36 
 
 44 
 
 7-01 
 
 92 
 
 89 
 
 22 
 
 19 
 
 15 
 
 2-40 
 
 35 
 
 78 3-92 
 
 3-04 
 
 30 
 
 78 
 
 70 
 
 86 
 
 5-07 
 
 34 
 
 45 
 
 17 
 
 7-07 
 
 89 
 
 38 
 
 18 
 
 16 
 
 2-56 
 
 50 
 
 78 -62 
 
 3-53 
 
 31 
 
 94 
 
 86 
 
 86 
 
 21 
 
 32 
 
 46 
 
 33 
 
 23 
 
 90 
 
 53 
 
 18 
 
 17 
 
 2-72 
 
 66 
 
 79 -58 
 
 2-22 
 
 32 
 
 5-10 
 
 5-02 
 
 86 
 
 37 
 
 30 
 
 47 
 
 49 
 
 39 
 
 90 
 
 09 
 
 17 
 
 18 
 
 2-88 
 
 82 -80 -59 
 
 2-02 
 
 33 
 
 26 
 
 18 
 
 86 
 
 52 
 
 29 
 
 48 
 
 64 
 
 55 
 
 90 
 
 84 
 
 16 
 
 19 
 
 3-04 
 
 97 -81 1 -63 
 
 1-87 
 
 34 
 
 42 
 
 34 
 
 87 
 
 67 
 
 28 
 
 49 
 
 80 
 
 71 
 
 90 
 
 8-00 
 
 16 
 
 20 
 
 3-20 
 
 3-13 
 
 81 1 -73 
 
 1-76 
 
 35 
 
 58 
 
 49 
 
 87 
 
 82 
 
 26 
 
 50 
 
 96 
 
 86 
 
 90 
 
 96 
 
 16 
 
 21 
 
 3-35 
 
 29 
 
 82 -83 
 
 1-68 
 
 36 
 
 74 
 
 65 
 
 87 
 
 97 
 
 25 
 
 51 
 
 8-12 
 
 8-02 
 
 91 
 
 31 
 
 16 
 
 22 
 
 3-51 
 
 44 
 
 82 -95 
 
 1-61 
 
 37 
 
 90 
 
 81 
 
 88 
 
 6-13 
 
 24 
 
 52 
 
 28 
 
 18 
 
 91 
 
 47 
 
 If 
 
 23 
 
 3-67 
 
 60 
 
 83' 4-07 
 
 1-56 
 
 38 
 
 6-05 
 
 97 
 
 88 
 
 29 
 
 23 
 
 53 
 
 44 
 
 34 
 
 91 
 
 63 
 
 11 
 
 24 
 
 3-83 
 
 76 
 
 83 -21 
 
 1-51 
 
 39 
 
 21 
 
 6-13 
 
 88 
 
 44 
 
 23 
 
 54 
 
 60 
 
 50 
 
 91 
 
 79 
 
 14 
 
 25 
 
 3-99 
 
 91 
 
 84 -34 
 
 1.47 
 
 40 
 
 37 
 
 28 
 
 88 
 
 60 
 
 22 
 
 55 
 
 76 
 
 66 
 
 91 
 
 95 
 
 14 
 
 26 
 
 4-15 
 
 4-07J -84J -48 
 
 1-44 
 
 41 
 
 53 
 
 44 
 
 89 
 
 75 
 
 21 
 
 56 
 
 92 
 
 81 
 
 91 
 
 9-10 
 
 14 
 
314 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 jj 
 
 ,2 SMALLEST PINION, 
 
 ^ _ | SMALLEST PINION, || 
 
 ,2' SMALLEST PINION, 
 
 I 
 
 fc. g /TWELVE TEETH. 
 
 o ! 
 
 ! J 
 
 V. g ! TWELVE TEETH. 
 
 - g TWELVE TEETH. 
 
 53 
 
 6 
 
 j.g Radii of the 
 g *; faces of 
 
 Radii of the 
 flanks of 
 
 4-1 
 
 O 
 
 o 
 
 II 
 
 Radii of the 
 faces of 
 
 Radii of the *S 
 flanks of 
 
 J .g Radii of the 
 "3 *j faces of 
 
 Radii of the 
 flanks of 
 
 
 (2 - teeth. 
 
 teeth. 
 
 
 
 "" '-' 
 
 teeth. 
 
 teeth. szj 
 
 K A teeth. 
 
 teeth. 
 
 67 
 
 9-08 8-97 
 
 0-91 
 
 9-26 1-13 
 
 120 19-10 
 
 18-99 
 
 1910 
 
 183 29-13 29-00 
 
 97 
 
 29-27 
 
 
 68 
 
 23 9-13 
 
 91 
 
 42 
 
 13 
 
 121 
 
 26 
 
 1915 
 
 42 
 
 1-06 
 
 184 
 
 28: -IB 
 
 i 
 
 43 
 
 1-03 
 
 69 
 
 37| '29 
 
 92 
 
 57 
 
 13 
 
 122 
 
 42 
 
 30 
 
 95 
 
 57 
 
 
 185 -44 -32 
 
 
 59 
 
 
 60 
 
 55 -45 
 
 92 
 
 73 
 
 13 
 
 123 
 
 58 
 
 46 
 
 
 73 
 
 06 
 
 186 -60 
 
 48 
 
 97 
 
 74 
 
 1-03 
 
 61 
 
 71 -61 
 
 92 
 
 89 
 
 12 
 
 124 
 
 74 
 
 62 
 
 
 89 
 
 
 187 
 
 76 
 
 64 
 
 
 90 
 
 
 62 
 
 87 
 
 77 
 
 92 
 
 10-05 
 
 12 
 
 125 
 
 90 
 
 78 
 
 95 
 
 20-05 
 
 06 
 
 188 
 
 92 
 
 80 
 
 
 30-06 
 
 
 63 
 
 10-03 -92 
 
 92 
 
 20 
 
 12 
 
 126 
 
 20-05 
 
 94 
 
 
 21 
 
 
 189 
 
 30-08 
 
 96 
 
 98 
 
 22 
 
 1-03 
 
 64 
 
 19 10-08 
 
 92 
 
 36 
 
 12 
 
 127 
 
 21 
 
 2010 
 
 
 37 
 
 05 
 
 190 
 
 24 
 
 3012 
 
 
 38 
 
 
 65 
 
 35 -24 
 
 92 
 
 62 
 
 12 
 
 128 
 
 37 
 
 26 
 
 96 
 
 53 
 
 
 191 
 
 40 
 
 28 
 
 98 
 
 54 
 
 1-02 
 
 66 
 
 51 -40 
 
 92 
 
 68 
 
 11 
 
 129 
 
 53 
 
 42 
 
 
 69 
 
 05 
 
 192 
 
 55 
 
 48 
 
 
 70 
 
 
 67 
 
 67 '56 
 
 92 
 
 84 
 
 11 
 
 130 
 
 68 
 
 58 
 
 
 84 
 
 
 193 
 
 71 
 
 59 
 
 
 86 
 
 
 6S 
 
 83 
 
 72 
 
 92 
 
 99 
 
 11 
 
 131 
 
 84 
 
 74 
 
 96 
 
 21-00 
 
 05 
 
 194 
 
 87 
 
 75 
 
 98 
 
 31-02 
 
 1-02 
 
 69 
 
 98 
 
 88 
 
 93 
 
 11-15 
 
 11 
 
 132 
 
 21-00 
 
 89 
 
 
 16 
 
 
 196 
 
 31-03 
 
 91 
 
 
 18 
 
 
 70 
 
 11-14 11-04 
 
 93 
 
 31 
 
 11 
 
 133 
 
 17 
 
 21-05 
 
 
 32 -05 
 
 196 
 
 19 
 
 31-07 
 
 
 83 
 
 1-02 
 
 71 
 
 30 11-20 
 
 93 
 
 47 
 
 1-10 
 
 !l34 
 
 33 
 
 21 
 
 96 
 
 48 
 
 197 
 
 86 
 
 23 
 
 98 
 
 49 
 
 
 72 
 
 46 -35 
 
 93 
 
 63 
 
 10 135 
 
 49 
 
 37 
 
 
 64 -05 ' 198 
 
 61 
 
 39 
 
 
 65 
 
 1-02 
 
 73 
 
 62 -51 
 
 93 
 
 79 
 
 10,1136 
 
 65 
 
 53 
 
 96 
 
 80 
 
 199 
 
 67 
 
 55 
 
 
 81 
 
 
 74 
 
 78 
 
 67 
 
 93 
 
 95 
 
 10 137 
 
 81 
 
 69 
 
 
 96 -05 
 
 200 
 
 83 
 
 71 
 
 98 
 
 97 
 
 
 75 
 
 94 
 
 83 
 
 93 
 
 12-10 
 
 10 138 
 
 96 
 
 85 
 
 22-11 
 
 201 
 
 99 
 
 87 
 
 3213 
 
 
 76 
 
 12-1011-99 
 
 93 
 
 26 -09 139 
 
 2212 
 
 22-01 -96 
 
 27 -05 202 
 
 32-15 
 
 32-02 
 
 
 29 
 
 
 77 
 
 26 12-15 
 
 
 42 
 
 09 
 
 140 
 
 28 
 
 17 
 
 
 43 -05 
 
 203 
 
 30 
 
 18 
 
 
 45 
 
 
 78 
 
 42 -30 
 
 93 
 
 58 
 
 09 141 
 
 44 
 
 33 
 
 
 59 
 
 204 
 
 46 
 
 34 
 
 
 61 
 
 
 79 
 
 58 
 
 47 
 
 
 74 
 
 09 142 
 
 60 
 
 48 -96 
 
 ' -75 -05 1 205 
 
 62 
 
 50 
 
 
 77 
 
 
 80 
 
 73 
 
 63 
 
 93 
 
 90 
 
 09 143 
 
 76 
 
 64 
 
 911 
 
 206 
 
 78 -66 
 
 
 92 
 
 
 81 
 
 89 
 
 79 
 
 
 13-06 
 
 09 144 
 
 92 
 
 80 
 
 
 23-07 
 
 05 
 
 207 
 
 94 
 
 82 
 
 
 33-08 
 
 
 82 
 
 13-05 
 
 94 
 
 93 
 
 22 
 
 09 145 
 
 23-08 
 
 96 -96 
 
 23 
 
 
 208 
 
 33-10 
 
 98 
 
 
 24 
 
 
 83 
 
 21 
 
 13-10 
 
 
 38 
 
 09 146 
 
 24 
 
 2312 
 
 38 1-04 209 
 
 26 
 
 33-14 
 
 
 40 
 
 
 84 
 
 37 
 
 26 
 
 94 
 
 53 
 
 08 
 
 147 
 
 40 
 
 28 
 
 64 |210 
 
 42 
 
 30 
 
 
 56 
 
 
 85 
 
 53 
 
 42 
 
 94 
 
 69 
 
 08 148 
 
 56 
 
 44 
 
 70 -04;'211 
 
 58 
 
 46 
 
 
 72 
 
 
 86 
 
 69 
 
 58 
 
 
 85 
 
 08 149 
 
 "7*2 
 
 60 -96 
 
 86 212 
 
 74 
 
 61 
 
 
 88 
 
 
 87 
 
 85 
 
 74 
 
 94 
 
 14-01 
 
 08I|150 
 
 87 
 
 76 
 
 24-02; -04 213 
 
 90 
 
 77 
 
 
 34-04 
 
 
 88 
 
 14-01 -90 
 
 94 
 
 17 
 
 08 
 
 151 
 
 24-03 
 
 92 
 
 18 1214 
 
 34-06 
 
 93 
 
 
 20 
 
 
 89 
 
 1714-06 
 
 
 33 
 
 08 
 
 152 
 
 19 
 
 24-07 '96 
 
 34 
 
 215 
 
 21 
 
 34-09 
 
 
 36 
 
 
 90 
 
 33 -22 
 
 94 
 
 49 
 
 08 
 
 153 
 
 35 
 
 23 
 
 50 
 
 04 216 
 
 37 
 
 25 
 
 
 51 
 
 
 91 
 
 49 -38 
 
 94 
 
 65 
 
 08 
 
 154 
 
 51 
 
 39 
 
 65 
 
 
 217 
 
 53 
 
 41 
 
 
 67 
 
 
 92 
 
 ' -64 -53 
 
 
 81 
 
 08 
 
 155 
 
 67 
 
 55 '96 
 
 81 
 
 04 218 
 
 69 
 
 57 
 
 
 83 
 
 
 93 
 
 80 -69 
 
 94 
 
 97 
 
 08 
 
 156 
 
 83 
 
 71 
 
 98 
 
 219 
 
 85 
 
 73 
 
 
 99 
 
 
 94 
 
 96 -85 
 
 94 
 
 15-12 
 
 07 
 
 |157 
 
 99 
 
 87 
 
 2513 
 
 04 220 
 
 35-01 
 
 89 
 
 
 3515 
 
 
 95 
 
 15-14 15-01 
 
 
 30 
 
 07 
 
 158 
 
 25-15 
 
 25-03 -97 
 
 29 
 
 221 
 
 17'35'05 
 
 
 31 
 
 
 96 
 
 28| -17 
 
 94 
 
 44 
 
 07 
 
 159 
 
 31 
 
 19 
 
 45 
 
 04 222 
 
 33 -20 
 
 
 47 
 
 
 47 
 
 44 -33 
 
 
 60 
 
 07 
 
 160 
 
 47 
 
 35 
 
 61 
 
 223 
 
 40 
 
 36 
 
 
 63 
 
 
 98 
 
 60 -49 
 
 94 
 
 76 
 
 07 
 
 161 
 
 62 
 
 51 '97 
 
 77 
 
 i 224 
 
 65 
 
 52 
 
 
 79 
 
 
 99 
 
 76 
 
 65 
 
 
 92 
 
 -07 
 
 162 
 
 78 
 
 66 
 
 93 
 
 04 225 
 
 80 
 
 68 
 
 
 95 
 
 
 100 
 
 92 
 
 81 
 
 95 
 
 16-08 
 
 07 
 
 163 
 
 94 
 
 82 
 
 26-09 
 
 226 
 
 96 
 
 84 
 
 3610 
 
 
 101 
 
 16-08 
 
 97 
 
 
 24 
 
 07 
 
 164 
 
 26-10 
 
 98 -97 
 
 25 
 
 04 227 
 
 3612 
 
 36'00 
 
 26 
 
 
 102 
 
 24 
 
 16-13 
 
 
 40 
 
 
 165 
 
 262614 
 
 42 
 
 : 228 
 
 28 
 
 16 
 
 42 
 
 
 103 
 
 39 
 
 28 
 
 95 
 
 56 
 
 07 
 
 166 
 
 42 -30 
 
 56 
 
 04: 229 
 
 44 
 
 32 
 
 
 58 
 
 
 104 
 
 55 
 
 44 
 
 
 72 
 
 
 167 
 
 58 
 
 46 
 
 72 
 
 
 230 
 
 59 
 
 48 
 
 
 74 
 
 
 105 
 
 71 
 
 60 
 
 
 87 
 
 07 
 
 168 
 
 74 
 
 62 -97 
 
 88 
 
 03 
 
 231 
 
 75 
 
 64 
 
 
 90 
 
 
 106 
 
 87 
 
 76 
 
 95 
 
 17-03 
 
 
 169 
 
 90 
 
 78 
 
 27-04 
 
 
 232 
 
 9! 
 
 79 
 
 
 37-06 
 
 
 107 
 
 17-03 
 
 92 
 
 
 19 
 
 06 
 
 170 
 
 27-06 
 
 94: 
 
 22 
 
 03 
 
 233 
 
 37-08 
 
 95 
 
 
 22 
 
 
 108 
 
 19 
 
 17-08 
 
 
 35 
 
 Ijl71 
 
 2227-10 -97 
 
 36 
 
 
 234 
 
 24 
 
 3711 
 
 
 58 
 
 
 109 
 
 35 
 
 24 
 
 95 
 
 51 
 
 06 
 
 172 
 
 38 -25 
 
 
 62 
 
 03 
 
 235 
 
 40 
 
 27 
 
 
 54 
 
 
 110 
 
 51 
 
 40 
 
 
 67 
 
 
 173 
 
 53 
 
 41 
 
 
 68 
 
 
 236 
 
 56 
 
 43 
 
 
 69 
 
 
 111 
 
 67 
 
 66 
 
 
 83 
 
 06 
 
 174 
 
 69 
 
 57 
 
 97 
 
 84 
 
 
 237 
 
 72 
 
 59 
 
 
 85 
 
 
 112 
 
 83 
 
 71 
 
 95 
 
 99 
 
 
 175 
 
 85 
 
 73 
 
 
 1-00 
 
 03 
 
 1238 
 
 87 
 
 75 
 
 
 38-01 
 
 
 113 
 
 99 
 
 ' -87 
 
 
 18-15 
 
 06 
 
 176 
 
 28-01 
 
 89 
 
 
 28-16 
 
 
 239 
 
 38-03 
 
 91 
 
 
 17 
 
 
 114 
 
 18-1518-03 
 
 95 
 
 80 
 
 
 177 
 
 17 
 
 28-05 
 
 97 
 
 31 
 
 03 
 
 240 
 
 19 
 
 38-07 
 
 
 33 
 
 
 L15 
 
 30 -19 
 
 
 46 
 
 06 
 
 178 
 
 33 
 
 21 
 
 
 47 
 
 
 241 
 
 85 
 
 23 
 
 
 69 
 
 
 116 
 
 46 -35 
 
 
 26 
 
 
 179 
 
 48 
 
 37 
 
 
 63 
 
 
 242 
 
 51 
 
 88 
 
 
 fc5 
 
 
 117 
 
 62 -51 
 
 95 
 
 62 
 
 06 
 
 180 
 
 64 
 
 53 
 
 97 
 
 79 
 
 03 
 
 243 
 
 67 
 
 54 
 
 
 39-01 
 
 
 118 
 
 78 -67 
 
 
 78 
 
 
 181 
 
 80 
 
 69 
 
 
 95 
 
 
 244 
 
 83 
 
 70 
 
 
 17 
 
 
 119 -94 -83 
 
 95 
 
 94 
 
 1-06 
 
 182 
 
 97 '84 
 
 
 2911 
 
 1-03 
 
 245 
 
 99 
 
 86 
 
 
 33 
 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 315 
 
 No. of teeth. ' 
 
 Radius of 
 pitch-circle. 
 
 SMALLEST PINION, 
 TWELVE TEETH. 
 
 No. of teeth. 
 
 Radius of 
 pitch-circle. 
 
 -SMALLEST PINION, 
 TWELVE TEETH. 
 
 3 
 
 
 *o 
 
 I 
 
 Radius of 
 pitch-circle. 
 
 SMALLEST PINION, 
 TWELVE TEETH. 
 
 Radii of the 
 faces of 
 teeth. 
 
 Radii of the 
 flanks of 
 teeth. 
 
 Radii of the 
 faces of 
 teeth. 
 
 Radii of the 
 flanks of 
 teeth. 
 
 Radii of the 
 faces of 
 teeth. 
 
 Radii of the 
 flanks of 
 teeth. 
 
 246 
 
 39-15 
 
 3902 
 
 
 39-28 
 
 
 265 
 
 42-17 42-04 
 
 
 42-31 
 
 
 284 
 
 45-19 
 
 45-06 
 
 
 45-33 
 
 
 247 
 
 31 
 
 18 
 
 
 44 
 
 
 266 
 
 33 -20 
 
 
 46 
 
 
 285 
 
 35 
 
 .22 
 
 
 49 
 
 
 248 
 
 47 
 
 34 
 
 
 60 
 
 
 267 
 
 49 '36 
 
 
 62 
 
 
 286 
 
 51 
 
 38 
 
 
 64 
 
 
 249 
 
 64 
 
 -50 
 
 
 76 
 
 
 268 
 
 64 '52 
 
 
 78 
 
 
 287 
 
 67 
 
 54 
 
 
 80 
 
 
 250 
 
 78 
 
 86 
 
 
 92 
 
 
 269 
 
 80 
 
 68 
 
 
 94 
 
 
 288 
 
 83 
 
 70 
 
 
 96 
 
 
 251 
 
 94 
 
 82 
 
 
 40-08 
 
 
 270 
 
 97 
 
 84 
 
 
 43-10 
 
 
 289 
 
 99 
 
 86 
 
 
 46-12 
 
 
 252 
 
 40-10 
 
 97 
 
 
 24 
 
 
 271 
 
 43-13 
 
 1-00 
 
 
 26 
 
 
 290 
 
 46-15 
 
 46-02 
 
 
 28 
 
 
 253 
 
 26 
 
 40-13 
 
 
 40 
 
 
 272 
 
 28 
 
 43-15 
 
 
 42 
 
 
 291 
 
 30 
 
 17 
 
 
 44 
 
 
 254 
 
 42 
 
 30 
 
 
 56 
 
 
 273 
 
 44 
 
 31 
 
 
 58 
 
 
 292 
 
 46 
 
 33 
 
 
 60 
 
 
 255 
 
 59 
 
 45 
 
 
 72 
 
 
 274 
 
 60 
 
 47 
 
 
 74 
 
 
 293 
 
 62 
 
 49 
 
 
 76 
 
 
 256 
 
 74 
 
 61 
 
 
 87 
 
 
 275 
 
 76 
 
 63 
 
 
 90 
 
 
 294 
 
 78 
 
 65 
 
 
 82 
 
 
 257 
 
 90 
 
 77 
 
 
 41-03 
 
 
 276 
 
 92 
 
 79 
 
 
 44-05 
 
 
 295 
 
 94 
 
 81 
 
 
 98 
 
 
 258 
 
 41-06 
 
 93 
 
 
 20 
 
 
 277 
 
 44-08 
 
 96 
 
 i 
 
 21 
 
 
 296 
 
 47-10 
 
 97 
 
 
 47-13 
 
 
 259 
 
 22 
 
 41-09 
 
 
 36 
 
 
 278 
 
 24 
 
 44-11 
 
 
 37 
 
 
 297 
 
 25 
 
 47-13 
 
 
 29 
 
 
 260 
 
 38 
 
 25 
 
 
 51 
 
 
 279 
 
 40 
 
 27 
 
 
 53 
 
 
 298 
 
 42 
 
 29 
 
 
 45 
 
 
 261 
 
 53 
 
 41 
 
 
 67 
 
 
 280 
 
 56 
 
 43 
 
 
 69 
 
 
 299 
 
 58 
 
 45 
 
 
 61 
 
 
 262 
 
 69 
 
 56 
 
 
 83 
 
 
 281 
 
 71 
 
 59 
 
 
 85 
 
 
 300 
 
 74 
 
 61 
 
 
 77 
 
 
 263 
 
 85 
 
 72 
 
 
 99 
 
 
 282 
 
 87 
 
 74 
 
 
 45-01 
 
 
 Rack 
 
 
 129 
 
 1-00 
 
 0-129 
 
 1-00 
 
 264 
 
 42-01 
 
 89 
 
 
 42-15 
 
 
 283 
 
 45-03 
 
 90 
 
 
 17 
 
 
 
 
 
 
 
 
 Rule. Seek in the first column of the table for the number of teeth it is 
 proposed that the wheel shall contain. In a line with such number of teeth 
 take from columns 2, 3, 4, 5, and 6 the numbers that are in them ; and in 
 every case multiply such numbers by the pitch. The products will be the 
 number of inches and parts of inches to which the compasses must be opened 
 to describe the circles and parts of circles that are required. 
 
 Example. Suppose that a wheel (Fig. 687) is to be made to contain thirty 
 teeth, and that the pitch of the teeth is to be 2 inches, proceed as follows : 
 
 FIG. 687. 
 
 Seek in column 1 for 30, the number of proposed teeth, and take from col- 
 umn 2 the numbers 4-78, which multiplied by 2 inches, the product will be 
 11-957". Open the compasses, therefore, to this radius and describe a circle, 
 which will be the " pitch-circle." On an arc of this circle lay off 2'5 X '48 = 
 1-2" for the thickness of a tooth, and 2 -5 x -52 = 1-3" for the space. Having 
 
316 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 determined the number of teeth and pitch, next, in column 3, and in the same 
 line with 30 teeth, will be found the numbers 4'70, which multiply by 2 
 inches the product will be 11'75. With the compasses opened to this dis- 
 tance, and from the same centre as the last, describe another circle, which will 
 be the paths of centres for the curves of the faces of the teeth. From column 
 4 similarly take the numbers 0'86 and multiply by 2 inches. The product is 
 2 - 15, to which distance the compasses must be opened to describe the faces of 
 the teeth. 
 
 FIG. 688. 
 
 Again, in column 5, multiply 5 - 07 X 2 - 5 = 12-675", and from the centre, 
 with this radius, describe another circle for the paths of centres of flanks of the 
 teeth, from column 6, 1'34 X 2'5 = 3'35, the radius of the flanks of the teeth. 
 
 For the height of a tooth a common proportion is -fa of pitch outside of 
 pitch-circle, and ^ of pitch within, which leaves -fo pitch for clearance at the 
 bottom, where usually small arcs are described to connect the teeth with the 
 wheel. 
 
 Having described a few teeth of any gear to its full size, the rest may be laid 
 off from a templet, or cutters made by which the teeth may be accurately 
 formed. In the illustration the teeth and spaces are proportioned to a common 
 form (see page 303). 
 
 It is not uncommon to make one of the set of gears with wooden teeth, 
 mortices being cast in the periphery of the wheel for the insertion of these 
 teeth hence called mortise or core ; the elasticity of the wood diminishes the 
 effect of shocks, and they run with less noise. 
 
 The usual proportions and construction of mortise wheels are shown in Fig. 
 688, a section across and with the rim of the wheel. The figures represent the 
 proportions to pitch as unity ; b is from 2 to 3 p. The teeth are held in posi- 
 tion by wooden dovetailed keys. 
 
 Fig. 689 is a section across the rim of mortised bevel-gear ; the figures are 
 
 FIQ. 689. 
 
n 
 
 \ 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 317 
 
 as before in ratios to p. In this illustration the teeth are held in by pins, com- 
 mon also to spur-mortise-gears. 
 
 It is unusual in drawings to complete gears with teeth according to the ex- 
 amples given ; it is sufficient for the purposes of pattern-making that the pitch- 
 circle, pitch and form of one tooth be given. For 
 lines of shafting, spur-gears may be represented, like 
 plain pulleys, tangent to each other, of the diameters 
 of the pitch-circle, with the pitch and number of 
 teeth written in : bevel-gears, as in Fig. 690. In fin- 
 ished drawings, detail is necessary. The following 
 simple forms of describing spur-and bevel-gears will 
 in general answer the purpose, but if more accuracy 
 is required use Adcock's tables. 
 
 Projections of a Spur Wheel. To draw side ele- FtG 690 
 
 vation (Fig. 691), an edge view (Fig. 692), and a ver- 
 tical section (Fig. 693) of a spur wheel with 54 teeth and a pitch of two inches : 
 
 Determine the radius of the pitch-circle from the table, page 312 
 (-6366 X 54 = 34-376 = D. E = 17-19) ; draw the central line A C B and the 
 perpendicular D E; on C as a centre, with a radius 17-19, describe the pitch- 
 circle, and divide it into 54 equal parts. To effect this division, without defa- 
 cing by repeated trials that part of the paper on which the teeth are to be repre- 
 sented, describe from the same centre c i with any convenient radius, a circle 
 abed] with the same radius divide its circumference into six equal parts, and 
 subdivide each sixth into nine equal parts, and draw radii to the centre c ; these 
 radii will cut the pitch-circle at the required number of points. Divide the 
 pitch (2 inches) into 10 equal parts ; mark off beyond the pitch-circle a dis- 
 tance equal to 3 of these parts, and within it a distance equal to 4 parts, and 
 from the centre C describe circles passing through these points ; these circles 
 are projections of the cylinders bounding the points of the teeth and the roots 
 of the spaces respectively. 
 
 In forming the outlines of the teeth, the radii, which, by their intersections 
 with the pitch-circle, divide into the required number of parts, may be taken 
 as the centre lines of each tooth. The thickness of the tooth, measured on the 
 pitch-circle, is -46 pitch x 2* = -92, and the width of the space is equal to 
 '54: p x 2" = 1-08. These distances being set off, take in the compasses the 
 length of the pitch, and from the centre g describe a circular arc h i ; and from 
 the centre j, with the same radius, describe another arc h k touching the 
 former ; these arcs, being terminated at the circles bounding the points of the 
 teeth and the bottoms of the spaces respectively, form the curve of one side of 
 a tooth. The other side is formed in a similar manner, by drawing from the 
 centre I the arc m n, and from the centre p the arc m 0, and so on for all the 
 rest of the teeth. 
 
 The teeth having been completed, proceed to the delineation of the rim, 
 arms, and eye of the wheel. The thickness of the rim is usually made equal to 
 that of the teeth, say one half of the pitch, which distance is accordingly set off 
 on a radius within the circle of the bottoms of the spaces, and a circle is de- 
 cribed from the centre C through the point q thus obtained. Within the rim, 
 a strengthening feather q r, in depth about three fourths of the thickness of the 
 
318 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 JB 
 

 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 319 
 
 rim, is generally formed, as shown in the plate. Describe the eye, or central 
 aperture for the reception of the shaft, to the specified diameter, as also the 
 circle representing the thickness of metal round the eye, usually equal to the 
 pitch of the teeth. 
 
 To draw the arms, from the centre C, with the radius C u equal to the 
 pitch, describe a circle ; draw all the radii, as C L, which are to form the centre 
 lines of the arms, and set off the distance L #, equal to one third pitch, on each 
 side of these radii at the inner circumference of the rim ; and through all the 
 points thus obtained draw tangents to the circle passing through u. The con- 
 tiguous arms are rounded off into each other by arcs of circles (Fig. 691). 
 Draw the central web of the arm by lines parallel to their radii, making the 
 thickness about f inch for wheel of about this size. 
 
 Having completed the elevation for the edge view and vertical section, 
 draw the perpendiculars F G and H I (Figs. 692 and 693) as central lines in 
 the representations ; set off on each side of these lines half the breadth of the 
 teeth, and draw parallels ; project the teeth of Fig. 691 upon Fig. 692, by 
 drawing through all the visible angular points straight lines parallel to A B, 
 and terminated at either extremity by the verticals representing the outlines of 
 the breadth of the wheel ; project in like manner the circles of the hub ; lay 
 off half length on each side of F G-, and draw parallels to it. The section (Fig. 
 693) is made on the line D E of the elevation ; project, as in Fig. 692, those 
 portions which will be visible in this section, and shade those parts which are 
 in section. The arms are made tapering in width, and somewhat less than the 
 face of the wheel (Fig. 694) ; a cross-section of one of them is made by a plane 
 passing through XX' and Y Y'. The points y, z, in Fig. 691, and correspond- 
 ing lines in Fig. 693, represent the edges of key-seat. 
 
 Oblique Projection of a Spur Wheel. In drawing an object in an oblique 
 position with respect to the vertical plane of projection, lay down in the first 
 place the elevation and plan as if it were parallel to that plane (Figs. 695 and 
 696). Then transfer the plan to Fig. 698, giving it the same inclination with 
 the ground line which the wheel ought to have in relation to the vertical plane ; 
 and assuming that the horizontal line A B represents the axis of the wheel, 
 both in the parallel and oblique positions, the centre of its front face in the 
 latter position will be determined by the intersection of a perpendicular raised 
 from the point C' (Fig. 697) with that axis. Take any point, as a in Fig. 695, 
 and the projection of that point on Fig. 696 must be in the line a a, parallel to 
 A B ; this point being projected at a' (Fig. 698), must be in the perpendicular 
 a' a ; therefore the intersection of these two lines is the point required. Thus 
 all the remaining points J, c, d, etc., may be obtained by the intersections of the 
 perpendiculars raised from the points #', c', d', etc. (Fig. 698), respectively, with 
 the horizontals drawn through the corresponding points in Fig. 695. Since the 
 points e and/, in the further face of the wheel, have their projections in a and 
 b (Fig. 695), their oblique projections will be situated in the lines a a and b J, 
 but they are also at e and/; consequently the lines e a and fb are the oblique 
 projections of the edges a' e' and b'f. 
 
 All the circles which, in the rectangular elevation (Fig. 695), have been em- 
 ployed in the construction of this wheel are projected in the oblique view into 
 ellipses ; thus, since the plane F' G', in which these circles are situated, is verti- 
 
320 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 321 
 
 oal, the major axes of all the ellipses in question are perpendicular to the line 
 A B, and equal to the diameters of the circles of which they are respectively the 
 projections ; and the minor axes, representing the horizontal diameters, will all 
 coincide with the line A B. 
 
 To obtain the ellipse into which the pitch-circle is projected, set off upon 
 the vertical D E (Fig. 697) above and below C the radii of the pitch-circle for 
 the major axis, and project i' f (Fig. 698) to the horizontal axis at i and/ (Fig. 
 697) for the minor axis. 
 
 The intersection of the horizontal lines g g, h h, etc., with the projection of 
 the pitch-circle gives the thickness of the teeth at the pitch-line ; in the same 
 manner the circles bounding the extremities and roots of the teeth may be de- 
 termined. If strict accuracy is required, a greater number of points is neces- 
 sary for the construction of the curvature of the teeth, and two additional cir- 
 cles, 77i n and op, may be drawn on Fig. 695 and projected to Fig. 697, and the 
 points of their intersection with the curves of the teeth projected to correspond- 
 ing points as indicated by the same letters. 
 
 Projections of a Bevel Wheel. Fig. 699 is an edge view, Fig. 700 a face view, 
 and Fig. 701 a vertical transverse section. For the determination of the divi- 
 sion of the angle of inclination of the axes of a pair of bevel wheels, see Fig. 
 685 ; for their size and proportion, the rules given for spur wheels ; thus, con- 
 sider the base of the cone A B (Figs. 700 and 701) as the diameter of the pitch- 
 circle of a spur wheel, and proportion the pitch, form, and breadth of teeth, ac- 
 cording to the stress to which they are to be subjected. 
 
 Having determined and laid down, according to the required conditions, the 
 axis S of the primitive cone, the diameter A B of its base, the angle A S 
 which the side of the cone makes with the axis, and the straight lines A 0, D o', 
 perpendicular to A S, and representing the sides of two cones, between which 
 the breadth of the wheel (or length of the teeth) is comprised, the first opera- 
 tion is to divide the primitive circle, described with the radius A C, into a num- 
 ber of equal parts corresponding to the number of teeth or pitch of the wheel. 
 Then upon the section (Fig. 701) draw with the radius o A or o B, moving par- 
 allel to itself, outside the figure, a small portion of a circle, upon which con- 
 struct the outlines of a tooth M, and of the rim of the wheel, with the same 
 proportions and after the same manner in reference to spur wheels ; set off from 
 A and B the points a, d, and /, denoting respectively the distances from the 
 pitch-line to the points and roots of the teeth, and to the inside of the rim, and 
 join these points to the vertex S of the primitive cone, terminating the lines of 
 junction at the lines D o', E o' ; the figure a 1) c d will represent the lateral form 
 of a tooth, and the figure c dfe a section of the rim of the wheel, by the aid of 
 which the face view (Fig. 700) is constructed. 
 
 The points a, J,.c, d, and e, having been projected upon the vertical diame- 
 ter A' B', describe from the centre C' a series of circles passing through the 
 points thus obtained, and draw any radius, as C' L, passing through the centre 
 of a tooth. On either side of the point L set off the distances L &, L I, making 
 the thickness of the tooth M at the point, and indicate, in like manner, upon 
 the circles passing through the points B' and d', its thickness at the pitch-line 
 and root ; then draw radii through the points i, /, k, g, m, etc., terminating 
 them respectively at the circles forming the projections of the corresponding 
 22 
 
322 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 323 
 
 parts at the inner extremity of the teeth ; these radial lines will represent the 
 rectilinear edges of all the teeth. The curvilinear outlines may be delineated 
 by arcs of circles, tangents to the radii g C' and i C', and passing through the 
 points obtained by the intersections of the radii and the various concentric cir- 
 cles. The radii of these circular arcs may in general, as in the case of spur 
 wheels, be taken equal to the pitch, and their centres upon the interior and ex- 
 terior pitch-circles ; thus the points g and i, n and o, for example, are the centres 
 for the arcs passing through the corresponding points in the next adjacent teeth, 
 and vice versa. 
 
 The drawing of the teeth in the edge view (Fig. 699), and of such portions 
 of them as are visible in the section (Fig. 701), are explained by inspection of 
 the lines of projection. In the construction observe that every point in the 
 principal figure from which they are derived is situated upon the projection of 
 the circle drawn from the centre C', and passing through that point. Thus the 
 points g and i, for example, situated upon the exterior pitch-circle, will be de- 
 termined in Fig. 699 by the intersection of their lines of projection with the 
 base A B of the primitive cone ; and the points k and I will be upon the straight 
 line passing through 
 a a (Fig. 701), and so 
 on. Further, as the 
 lateral edges of all the 
 teeth in Fig. 699 are 
 radii of circles drawn 
 from the centre 0', so 
 in Fig. 700 they are 
 represented by lines 
 drawn through the va- 
 rious points found as 
 above for the outer ex- 
 tremities of the teeth, 
 and converging toward 
 the common apex S ; while the cen- 
 tre lines of the exterior and interior 
 extremities themselves all tend to 
 the points o and o' respectively. 
 
 Skew- Bevels. When the axes of 
 wheels are inclined to each other, 
 and yet do not meet in direction, 
 and it is proposed to connect them 
 by a single pair of bevels, the teeth 
 must be inclined to the base of the 
 frusta to allow them to come into 
 contact. Set off a e (Fig. 702) equal 
 to the shortest distance between the 
 
 axes (called the eccentricity), and Fm 702 
 
 divide it in e, so that a c is to e c as 
 
 the mean radius of the frustum to the mean radius of that with which it is to 
 work ; draw cm d perpendicular to a e. The line cm d gives the direction of 
 
324 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 the teeth ; and, if from the centre a, with radius a c, a circle be described, the 
 direction of any tooth of the wheel will be a tangent to it, as at c. Draw the 
 line d e perpendicular to cm d, and with a radius d e equal to c e describe a 
 circle ; the direction of the teeth of the second wheel will be tangents to this 
 last, as at d. 
 
 System composed of a Pinion driving a Rack (Fig. 703). The pitch-line 
 M N of the rack and the pitch-circle A B D of the pinion being laid down 
 touching one another, divide the latter into twice the number of equal parts 
 that it is to have teeth, and set off the common distance of these parts upon 
 the line M N, as many times as may be required ; this marks the thickness of 
 
 FIG. 703. 
 
 the teeth and width of the spaces in the rack. Perpendiculars drawn through 
 all these points to the solid part of the rack will represent the flanks of the teeth 
 upon which those of the pinion are to be developed in succession. The curva- 
 ture of these latter should be an involute A c of the circle A B D. The teeth 
 might be cut off at the point of contact d upon the line M N, for at this posi- 
 tion the tooth A begins its action upon that of the rack E ; but it is better to 
 allow a little more length ; in other words, to describe the circle bounding the 
 points of the teeth with a radius somewhat greater than C d. 
 
 For the form of the spaces in the rack, set off from M N, as at the point e, 
 a distance slightly greater than the difference A a of the radius of the pitch- 
 circle, and that of the circle limiting the points of the teeth, and through this 
 point draw a straight line F G- parallel to M N. From this line the flanks of 
 all the teeth of the rack spring, and their points are terminated by a portion 
 of a cycloid A 5, which, however, may in most instances be replaced by an arc of 
 a circle. As the depth of the spaces in the pinion depends upon the height 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 325 
 
 of this curved portion of the teeth ; their outline is formed by a circle drawn 
 from the centre C, with a radius a little less than the distance from this point 
 to the straight line bounding the upper surface of the teeth of the rack. 
 
 System composed of a Rack driving a Pinion. In this case the construc- 
 tion is identical with that of the preceding example, except that the form 
 proper to be given to the teeth of the rack is a cycloid generated by a point A 
 in the circumference of the circle A E C rolling on the line M N. The curva- 
 ture of the teeth is an involute as before. 
 
 System composed of an Internal Spur Wheel driving a Pinion (Fig. 704). 
 The form of the teeth of the driving-wheel is in this instance determined by 
 the epicycloid described by a point in the circle A E C, rolling on the concave 
 
 FIG. 704. 
 
 circumference of the primitive circle MAN. The points of the teeth are to 
 be cut off by a circle drawn from the centre of the internal wheel, and passing 
 through the point E, indicated by the contact of the curve with the flank of 
 the driven tooth. 
 
 The wheel being supposed to be the driver, the curved portion of the teeth 
 of the pinion may be very small. This curvature is a part of an epicycloid 
 generated by a point in the circle MAN rolling upon BAD. 
 
 System composed of an Internal Wheel driven T)y a Pinion. Fig. 705 
 involves a different mode of treatment from that employed in the preceding 
 cases. The epicycloidal curve A a, generated by a point in the circle having 
 the diameter A 0, the radius of the circle MAN, and which rolls upon the 
 circle BAD, can not be developed upon the flank A 5, the line described by 
 the same point in the same circle in rolling upon the concave circumference 
 
326 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 MAN; and for the reason that that curve is situated without the circle 
 BAD, while the flank, on the contrary, is within it. In order that the pinion 
 
 FIG. 705. 
 
 may drive the wheel uniformly according to the required conditions, form the 
 teeth so that they shall act always upon one single point in those of the wheel. 
 By taking for the curvature of the teeth of the pinion the epicycloid A rf, de- 
 scribed by the point A in the circle MAN rolling over the circle B A D, as in 
 the preceding examples, the tooth E of the pinion begins its action upon the 
 tooth F of the wheel at the point of contact of their respective primitive cir- 
 cles, and it is unnecessary to be continued beyond the point c, because the suc- 
 ceeding tooth H will then have been brought into action upon G ; consequently 
 the teeth of the wheel might be bounded by a circle passing through the point 
 c. One of the practical advantages which this species of gearing has over 
 wheels working externally is that the surfaces of contact of the wheel and 
 pinion admit of being more easily increased ; and, by making the teeth some- 
 what longer than simple necessity demands, the strain may be distributed over 
 two or more teeth at the same time. The flanks of the teeth of the wheel are 
 formed by radii drawn to the centre 0, and their points are rounded off to en- 
 able them to enter freely into the spaces of the pinion. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 327 
 
 DRAWING OF SCREWS. 
 
 Projections of a Triangular-threaded Screw and Nut (Fig. 706). Draw the 
 ground line A B, and the centre lines C c' of the figures, from as a centre, 
 with a radius equal to that of the exterior cylinders, describe the semicircle 
 
 C 
 
 FIG. ror. 
 
 #36; and with the radius of the interior cylinder the semicircle bee. Draw 
 the perpendiculars a a" and 6 6', b b" and e e" , to represent the vertical projec- 
 tions of the exterior and interior cylinders ; divide the semicircle a 3 6 into any 
 
328 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 number of equal parts, say 6, and through each part draw radii to divide the 
 interior. On the line a' a" set off the length of the pitch as many times as re- 
 quired ; and through the points of division draw lines parallel to the ground 
 line A B. Divide the pitch into twice the number of equal parts that the 
 semicircles have been divided into, and following instructions laid down (page 
 140), construct the helix a! 3' 6 both in the screw and nut. 
 
 Having obtained the point #', by the intersection of the horizontal line pass- 
 ing through the middle division of a' a with the perpendicular b b", describe the 
 helix b' c' e', which will represent the bottom of the groove. The apparent out- 
 lines of the screw and its nut will then be completed by drawing the lines b' a', 
 a' ', etc., to the curves of the helices ; these are not, strictly speaking, straight 
 lines, but their deviation from the straight line is, in most instances, so small 
 as to be imperceptible. 
 
 When a long series of threads have to be delineated, they should be drawn 
 mechanically, by means of a templet constructed in the following manner : Take 
 a small slip of thin wood or pasteboard, and draw upon it the helix a' 3' 6 to the 
 same scale as the drawing, and cut the slip carefully and accurately to this line. 
 Applying this templet upon Fig. 706, so that the points a' and 6 on the plate 
 shall coincide with a' and 6 on the drawing, the curve a' 3' 6 can be drawn 
 mechanically, and so on for the remaining curves of the outer helix. The same 
 templet may be employed to draw the corresponding curves in the screw-nut 
 by simply inverting it; but for the interior helix a separate one must be cut, 
 its outlines being laid off in the same manner. 
 
 Projections of a Square-threaded Screw and Nut (Fig. 707). The depth 
 of the thread is equal to its thickness, and this latter to the depth of the 
 groove. The construction is similar to the preceding ; the same letters and 
 figures mark relative parts. The parts of the curve concealed from view are 
 shown in dotted lines. 
 
 M 
 
 
 
 FIG. 708. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 329 
 
 System composed of a Wheel and- Tangent, or Endless Screw. In laying out 
 the work, the pitch of the teeth is to be determined by the required stress, as 
 for spur wheels, and the number of the teeth in the wheel by the number of 
 turns of the screw for each revolution of the wheel. With these determined, 
 take C (Fig. 708) to be the centre of the wheel, E F the axis of the screw, C A the 
 radius of the pitch-circle of the wheel, and G A that of the pitch-cylinder of 
 the screw ; the line M N drawn through A, parallel to E F, will be the gen- 
 eratrix of that cylinder, which will serve the purpose of determining the form 
 of the teeth. The section is made through the axis, and is the case of a rack 
 driving a pinion ; consequently the curve of the teeth, or rather thread, of the 
 screw should be simply a cycloid generated by a point in the circle AEG, de- 
 scribed upon A C as a diameter, and rolling upon the straight line M N". The 
 outlines of the teeth are helical surfaces described about the cylinder forming 
 the screw, with the pitch A b equal to the distance, measured upon the primi- 
 tive scale, between the corresponding points of two contiguous teeth. These 
 curves are expressed by dotted lines. The teeth of the wheel are set at angle 
 to the plane of its face, and with surfaces corresponding to the inclination and 
 
 FIG. 709. 
 
 FIG. 710. 
 
 helical form of the thread of the screw. Usually the points of the teeth and 
 bottoms of the spaces are formed of a concave outline, adapted to the convexity 
 
330 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 of the screw, in order to pre- 
 sent as much bearing surface 
 as possible to its action. In 
 this kind of gearing it is al- 
 most invariably the screw that 
 imparts the motion. 
 
 A screw is designated as 
 right or left hand, according as 
 the forward motion is imparted 
 by turning the screw to the 
 right or left. 
 
 In the proportions adopted 
 by the Yale & Towne Manu- 
 facturing Co. for worm gear- 
 ing, the wheel under the weight 
 will revolve the screw slowly.. 
 This angle (slightly less than 
 the angle of quiescence, see 
 Morin's table, page 199) of the 
 teeth is found to be the best 
 
 FIG. 711. 
 
 FIG. 712. 
 
 adapted for economy of power. In the wheel the teeth in section are those of 
 a spur wheel, cut with a chasing cutter, and in the screw turned in a lathe. 
 
 Figs. 709 and 710 are two views, worm and wheel, with such lines of con- 
 struction dotted as will explain the manner of drawing. 
 
 Fig. 711 is the Albro worm and worm wheel as used on the cruisers of the 
 United States Government for ship steering and for heavy windlass work. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 331 
 
 Frictional Gearing. When motion is not continuous for a long time, but 
 frequently stopped and started or" reversed, frictional gearing is very often 
 used. The starting is with as little shock as with belting, as by the usual ap- 
 pliances this pressure may be applied gradually and is fully as positive. The 
 simplest form of frictional gearing is that in which the surfaces in contact cor- 
 respond to that of the pitch-circles (Fig. 712). 
 
 Fig. 713 is a plan of a bevel frictional gear. One half is shown in section. 
 The surface of the large or driven gear is of cast-iron, that of the pinion of 
 paper, in washers compressed by a hydraulic press and firmly held together by 
 bolts. The bevel in section is in contact with the large wheel-surface, the other 
 is disengaged. A slight motion to the right will throw the one in contact out, 
 and not throw in the other, and motion ceases in the large driven wheel ; a still 
 further motion throws in the left pinion, and the motion of the driven wheel is 
 reversed. 
 
 The mode in which this is done is shown in the elevation (Fig. 714). The 
 shipper consists of a bell-crank, controlled by a 
 screw. The screw works in a stand, on the top of 
 which is a hand wheel ; the hand wheel can be 
 moved in either direction, and any desirable pres- 
 sure can be brought upon the frictional surfaces 
 by means of the screw. It is not unusual, instead 
 of two pinions to have one pinion, with a little 
 clearance on each side, revolving between two 
 wheels, a slight lateral motion, in either direction, 
 bringing it iji 
 contact with one 
 or the other of 
 the wheels. Some 
 provision, by a 
 
 loose coupling or otherwise, must be made to admit of this lateral movement 
 in the pinion shaft. Straight pulleys, or what would correspond to spur-gears 
 without teeth, are constructed, as in the example given, and are thrown in or 
 out of gear by a lateral motion of the pinion. 
 
 In proportioning the face of the pulleys it has been found safe to consider 
 it the same as belts, given in the table (page 292). The pressure can be ap- 
 plied according to the requirements of driving, and there is no falling off in 
 the friction. The frictional surfaces are not always paper ; wood, leather, and 
 prepared rubber are frequently used. 
 
332 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Wedge Gearing, or Robertson Grooved- Surf ace Frictional Gearing. Fig. 
 715 is the cross-section of the rims of two wheels of this gearing. The angle 
 
 recommended by Robertson is 50 (usually not 
 over 30 in our practice), and the pitch to vary 
 somewhat with the velocity and power to be 
 transmitted. For the adhesion, it is safe to 
 make the horizontal face equal to that of a belt 
 under the same circumstances of transfer of 
 power. 
 
 Fig. 716 shows the application of wedge 
 gearing to a hoist drum d. The power is ap- 
 Fl - 715 - plied through the shaft s ; one end of the drum 
 
 shaft rests in a swivel box, the other in an eccentric box. Motion is given 
 through the eccentric handle by which the drum gear can be engaged with that 
 
 FIG. 716. 
 
 of the pinion of the driving shaft ; the drum is revolved and the load raised. 
 In lowering, the drum is thrown out by the eccentric, and the brake b applied, 
 which is controlled by a system of levers brought within reach of the man at 
 the eccentric lever. 
 
 When applied to the driving of a rotary fire pump, the friction pinion on 
 the pump shaft is thrown into gear with the friction wheel of the main shaft 
 by a sliding frame of iron with a screw and hand wheel. With this apparatus 
 the pump can be started without shock or jar and without reducing the speed 
 of the main shaft. 
 
 For large, straight pulleys, wooden rims from 6" to 8" deep, and built up in 
 segments from 1* to 2' thick, so placed that the direction of the fibres shall 
 follow the circumference of the wheel as nearly as possible. The segments are 
 firmly clamped together and secured by glue joints, nails, and bolts, and the 
 rim is bolted to the arms of the pulley with additional outside and deeper rings 
 secured to the rim. The whole is then turned and finished. This pulley may 
 be used with a belt as a friction gear. 
 
 Figs. 717 and 718 are sections of wooden bevel friction gears, showing the 
 way in which they are formed by wooden disks planed and fitted, glued, nailed, 
 and bolted, and then turned to the proper angle and face. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 333 
 
 For the transmission of small powers, the combination of one conical wheel 
 and one narrow disk wheel with rounded edges, both of iron, is sometimes used 
 
 FIG. 717. 
 
 FIG. 718. 
 
 FIG. 719. 
 
 (Fig. 719). The pressure is applied to the disk wheel, and is so arranged that 
 it can be shifted along its axis for a variable speed motion. The surfaces in 
 
 contact are limited, the diameters, therefore, should 
 be as large as possible, with high velocity. 
 
 Another variable speed 
 gear is made by using, instead 
 of a cone, a crown plate on 
 which the disk rests which 
 can be moved inward or out- 
 ward from the centre of the 
 driving plate. 
 
 Rope for running rigging 
 for derricks and general hoist- 
 ing work is made of hemp or manilla, but rope of iron or steel wire, mostly 
 with hemp centres, is preferred for many positions. 
 
 The shells of the common and lighter blocks are made of solid wood, mor- 
 tised for the reception of the sheaves, and rope-strapped (Fig. 720). Larger 
 and better blocks are made of separate pieces of wood, riveted or bolted to- 
 gether top and bottom, and iron-strapped (Fig. 721) ; the sheaves (Fig. 722) are 
 
 usually of lignurn-vitae, metal-bushed ; 
 the pin is fastened to the shell, and 
 the sheaves revolve on the pin. To 
 diminish friction under heavy weights, 
 friction rolls are introduced (Fig. 723). 
 
 FIG. 720. 
 
 FIG. 721. 
 
 FIG. 722. 
 
 To strengthen the pin, blocks are inside-strapped (Fig. 724), or with 
 wrought-iron straps or malleable-iron shells (Fig. 725) and cast-iron sheaves, 
 entirely without woodwork. Most blocks (Figs. 724 and 725) have beckets 
 attached. 
 
 In the Appendix will be found a table of the strength and weight of hemp, 
 steel-, and iron- wire rope and chain ; but as a simple rule of their working 
 strength, multiply the square of the girth or circumference of a hemp rope by 
 
334: 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 100, of an iron-wire rope by 600, of steel-wire rope by 1,000, and the square of 
 the diameter of the rods of which a chain is made by 12,000. 
 
 FIG. 723. 
 
 FIG. 724. 
 
 FIG. 725. 
 
 The following table of sizes are from the Boston and Lockport Block Com- 
 pany : 
 
 DIMENSIONS. 
 
 DIMENSIONS. 
 
 Size sheave. 
 
 For dia. rope. 
 
 Size shell. 
 
 Size sheave. 
 
 For dia. rope. 
 
 Size shell. 
 
 Six fxf 
 
 i 
 
 4 inches. 
 
 8 xlfxf 
 
 li 
 
 12 inches. 
 
 3 x fxf 
 
 $ 
 
 5 " 
 
 9 xl|xf 
 
 li 
 
 13 " 
 
 3xl x 
 
 i 
 
 6 " 
 
 9ixlxf 
 
 If 
 
 14 " 
 
 4jxl xj 
 
 i 
 
 7 " 
 
 10 xlfxl 
 
 H 
 
 15 " 
 
 4fx.lixf 
 
 i 
 
 8 " 
 
 11 x2|xl 
 
 2 
 
 16 " 
 
 5 x 1 x | 
 
 i 
 
 9 " 
 
 12 x2fxl 
 
 2f 
 
 18 " 
 
 6ixl*xi 
 
 H 
 
 10 " 
 
 14 x2fxli 
 
 21 
 
 20 " 
 
 7J x 1 x f 
 
 H 
 
 11 " 
 
 15 x3xli 
 
 3 
 
 22 " 
 
 
 
 
 16 x3xl 
 
 3| 
 
 24 " 
 
 Gin-blocks (Fig. 726) are made with wrought- and malleable-iron frames 
 and wrought swivel-hook. 
 
 Winding-drums or barrels must have their 'diameters pro- 
 portioned to the diameters of the rope or chain to be used (see 
 table of sheaves above), and their length to the length of rope 
 or chain to be taken in, and when the coils or turns of the rope 
 are numerous, provision must often be made for keeping the 
 rope or chain so that one coil may not ride on another. This 
 is done by spiral grooves in the barrel, or shifting the barrel or 
 the rope-guide automatically. 
 
 FIG. 726. Fig. 1806 shows the way in which a chain cable is taken in 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 335 
 
 with but few coils on the barrel. The coils are sufficient for the friction of 
 taking up the cable ; the tight cable is wound on the larger part of the barrel, 
 and as the coils are unwound on the slack side the tight coil slips down to a 
 smaller diameter ; the weight of the chain on the slack side, as it drops into 
 the locker, is sufficient to preserve the friction ; but with a rope and a few 
 turns on the barrels the force of a man is sufficient, and he can readily slack 
 and hold the load in position or lower without changing the direction of mo- 
 tion or the speed of the barrel. 
 
 FIG. 728. 
 
 FIG. 727. 
 
 Chain-wheels with pockets are especially applicable to the purpose of hoist- 
 ing, requiring a width only slightly greater than that of the chain, and a diam- 
 eter sufficient to give the proper engagement with it. 
 
 Yule & Towne Manufacturing Company have made a pitch chain, of com- 
 mon form but of uniform links, especially adapted to hoists, and Figs. 727, 
 728, and 729 illustrate the construction of their chain-wheel. A is a pocketed 
 chain-wheel, made of soft cast-iron, mounted on a frame B. C is the chain- 
 guide enveloping the lower half of the chain-wheel. The inner curved surface 
 of the chain-guide is grooved, and is of such a shape as to leave a space be- 
 tween it and the periphery of the chain-wheel merely sufficient to admit the 
 
 FIG. 730. 
 
 FIG. 731. 
 
 FIG. 732. 
 
 FIG. 7:33. 
 
 FIG. 734. 
 
 FIG. 735. 
 
 chain ; it must then enter properly and continue engaged with the chain-wheel. 
 E is a chain-guide roller, that delivers the slack chain into the box or locker. 
 
336 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 FIG. 736. 
 
 D is the chain-stripper, bolted also to the plate B, with a tongue or rib pro- 
 jecting into the centre groove of the wheel which disengages the chain. 
 
 Forms of chain-cables are rep- 
 
 , resented by the open circular link 
 
 (Fig. 730), the open oval. (Fig. 731), 
 oval with pointed stud (Fig. 732), 
 oval with broad-headed stud (Fig. 
 733), an obtuse - angled stud-link 
 (Fig. 734), and the parallel-sided 
 stud-link (Fig. 735). The usual 
 proportions of chain -links are 6 
 diameters of the iron in length by 
 3 in width. The end links, which 
 terminate each 15 fathoms of chain, 
 are 6 - 5 in length to 4'1 in breadth, 
 and the iron about 1*2 the diameter 
 of the rest of the chain. 
 
 Chain Couplings. Fig. 736 is 
 an ordinary coupling link of an 
 anchor chain. The link is of 
 wrought-iron, the bolt and pin of 
 steel, both galvanized. The next 
 link i g ma de somewhat longer than 
 
 FIG 737. other links of the chain, so that 
 
 the coupling link may be more 
 
 readily introduced. Fig. 737 is a coupling link in which one part is a swivel, 
 so that the chain may have a rotation about its axis of length without twist- 
 
 -4,5 e ~*0~*!**fc-f* -a* 
 
 FIG. 738. 
 
 FIQ. 739. 
 
 FIG. 740. 
 
 FIG. 741. 
 
 ing. Figs. 738, 739, and 741 are sockets for wire-rope connections, shown in 
 section, Fig 739, with a wrought-iron conical key ; often the wires of the rope 
 are spread with wrought-iron wedges or nails, and in addition lead is poured in. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 337 
 
 In Fig. 739 the conical cone is. formed over an iron core by unravelling the 
 rope, passing the wires over it, and serving the ends to the rope below ; an iron 
 
 socket fits over the core. Fig. 740 is 
 a thimble in which the end of the 
 rope is either spliced in or loose and 
 served to the rope, with the end 
 wires turned down. Fig. 741 is an 
 open socket with a swivel hook, to 
 prevent the twisting and untwisting 
 of the rope and thereby weakening it. 
 Fig. 742 is a loop-clamp. 
 
 Hooks. Figs 743 and 744 (from 
 Kedtenbacher) represent two wrought-iron hooks. The proportions of Fig. 
 743 are : assuming the neck of the hook as 1, the diameter of journals of the 
 
 FIG. 
 
 FIG. 743. 
 
 FIG. 744. 
 
 FIG. 745. 
 
 traverse are I'l ; width of traverse at centre, 2 ; distance from the centre of the 
 hook to the centre of the traverse, 7'5 ; interior circle of the hook, 3-4; great- 
 est thickness of the hook, 2'8. Assuming (Fig. 744) the diameter of the wire 
 of the chain as 1 : interior circle of hook is 3'2, and greatest thickness' of 
 hook 3-5. 
 
 Fig. 745 represents a hook of the Yale & Towne Manuf'g Company as fitted 
 for a cross-head ; the diameter at A is that of iron from which the hook is 
 forged, and the section shown hatched is equal to that of the round iron. Hooks, 
 of the proportions but with a much greater load than given in the table below, 
 yield by the gradual opening of the jaw, giving ample notice before rupture. 
 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Ton. 
 
 Capacity of hook 
 
 -1- 
 
 \ 
 
 1 
 
 1 
 
 H 
 
 9, 
 
 8 
 
 4 
 
 5 
 
 6 
 
 8 
 
 10 
 
 
 In. 
 
 In. 
 
 In. 
 
 In. 
 
 ID. 
 
 In. 
 
 In. 
 
 In. 
 
 In. 
 
 In. 
 
 In. 
 
 In. 
 
 Dimensions of A 
 
 4 
 
 if 
 
 4 
 
 1A 
 
 1} 
 
 1ft 
 
 If 
 
 2 
 
 <H 
 
 n 
 
 2 
 
 3* 
 
 Dimensions of D 
 
 li 
 
 1* 
 
 H 
 
 13- 
 
 9 
 
 H 
 
 ?, 
 
 3* 
 
 H 
 
 4} 
 
 51- 
 
 6* 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 23 
 
338 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 All parts of the hook are drawn in parts of A. 
 
 Levers. Figs. 746 and 747 are side and front elevations of an ordinary 
 straight lever on a shaft ; both are shown broken, either because the length is 
 indefinite, or because it is inconvenient to put on the p#per. The handle 
 should be from 5 to 6 inches long, and 1* diameter. The bar beneath the 
 
 T 
 
 X. 
 
 i 
 
 - ( 
 
 i. FIG 
 
 cr: 
 
 
 
 . r-i 
 
 FIG. 741 
 
 to- 1 
 
 FIG. 748. 
 
 FIG. 749. 
 
 FIG. 750. 
 
 FIG. 751. 
 
 handle to be square, and of uniform width on one side of the lever and a taper 
 on the other, as shown, of about " in 4 feet on each side. The sides of the 
 square at the handle to be \f length in inches, or say f" for 30" lever, " for 
 4 feet, and 1* for 5 feet. The neck of the shaft to be, as proportioned in the 
 drawing, about ^ of the greatest width of the lever, and the diameter of hub 
 1-j^-. The stress exerted by a man may be from 75 to 100 pounds, and the size 
 of the shaft will depend on the torsional stress between the hub of the lever 
 and the point of resistance. 
 
 Fig. 748 is a hand-lever forming one arm of a bell-crank a bolt passing 
 through a slot in the frame and the arm of the lever, and the two are clamped 
 together by a thumb-nut, n, by which the lever can be held in any position. 
 
 The same purpose is often effected by notches in 
 / ^ the frame, into which the arm of the lever is caught, 
 
 or by spring latches, as in Fig. 749. 
 
 Figs. 750 and 751 are side view and plan of a 
 foot-lever. The foot-plate is 8" X 5" X f", and as 
 the lever is subject to double the stress of the hand- 
 lever above, the dimension should be somewhat in- 
 creased. The side of square next the foot-plate 
 should be, say for a lever of 30", 1"; of 4 feet, \\'\ 
 of 5 feet, 14/; the form and taper as in the hand- 
 lever. 
 
 Figs. 752 and 753 are views of a hand-crank. The diameter of the handles, 
 for convenience in grasping, should not be less than 1 " ; if for the force of two 
 men, 1|", and from the diameter of the handle the rest may be proportioned as 
 
 FIG. 752. 
 
 FIG. 753. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 339 
 
 fe 1.4- 
 
 in the figure. The length of handle for a single man should be from 10" to 
 12"; for two men, from 20" to 24"; the crank from 15" to 18", and the height 
 of shaft above the foot support for the men from 2' 10" to 3' 2". 
 
 In the construction of steam engines the makers adopt simple rules of pro- 
 portion between the diameter of cylinder and its connections. Thus, if that of 
 the diameter of 
 cylinder be 1, that 
 of the crank-shaft 
 at the journal is 
 50, of crank-pin 
 25, of crank-pin 
 eye *6, that of the 
 cylinder may be 
 altered moderate- 
 ly without change in that of con- 
 nections which may be kept in 
 stock. In large quick-running en- 
 gines (Fig. 791) the crank-pin is of 
 larger proportions than above. 
 
 Figs. 754 and 755 are two views 
 of a wrought-iron crank, and Figs. 
 756 and 757 of a cast-iron crank,* 
 both proportioned in their parts to 
 the diameter of the large eye as 
 unity, but, as shown by the diagram 
 and rule following, these figures can only apply to a single throw of crank, as 
 the diameters of the two eyes vary as their distances apart. 
 
 Taking the diameter of the large eye of the crank as the unit, Eedtenbacher 
 gives in the table the relative sizes of cen- 
 tral and end eyes of cranks, depending on 
 the proportion between the length of crank 
 and the diameter of central eye. The first 
 column exhibits the number of times the 
 diameter of eye is contained in the 'length 
 of crank ; the second and third columns 
 give the suitable diameters of crank-pins. 
 
 The diameters of crank-pins as above 
 given are on the basis of a length of from 1 
 to 1-J of the diameter ; if the length be in- 
 creased beyond this, the diameter should be 
 increased in the ratio of 1 to the square root 
 of the diameter. 
 
 Disk-cranks are circular disks of cast- 
 iron, with crank-pins of iron or steel, and 
 as much strength of metal around the pin as in the crank. They are better 
 than the crank, in that there is no unbalanced crank, and part of the weight 
 
 FIG. 754. 
 
 FIG. 755. 
 
 DIAMETER OF EYE, BEING UNITS. 
 
 I 
 
 d 
 
 For wrought- 
 iron shafts. 
 
 Cast-iron 
 shafts. 
 
 2 
 
 0-85 
 
 0-62 
 
 8 
 
 0-69 
 
 0-51 
 
 4 
 
 0-60 
 
 0-44 
 
 5 
 
 0-54 
 
 0-39 
 
 6 
 
 0-49 
 
 0-36 
 
 7 
 
 0-45 
 
 0-33 
 
 8 
 
 0-42 
 
 0-31 
 
 9 
 
 0-40 
 
 0-29 
 
 10 
 
 0-38 
 
 0-28 
 
 11 
 
 0-36 
 
 0-26 
 
 12 
 
 0-34 
 
 0-25 
 
 13 
 
 0-33 
 
 0-24 
 
 * " Elements of Machine Design," Unwin. 
 
340 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 I.ZF 
 IS 
 1..5 
 
 
 of the connection can be balanced by a proper disposition of metal within 
 the area of the disk. 
 
 Fig. 758 is a plan of a double crank-axle, although by the projection the 
 lower axle A appears as a straight shaft. The dimensions given are from an 
 axle in use. In 
 construction the 
 cranks are rectan- 
 gular in section, 
 of which the 
 width is T V the 
 depth, and the 
 depth 1-5 the di- 
 ameter of crank- 
 journal. Double cranks are usually 
 forged solid, and the slot for the 
 crank cut out ; that shown in the 
 figure was cast in steel for a double 
 compound engine, 7 X 15 X 15, and 
 has long worked satisfactorily. 
 
 Fig. 759 is a drawing of a crank- 
 axle adapted to a machine. 
 
 Fig. 760 is an elevation and sec- 
 tion of the driving wheel of a loco- 
 motive, with a counterbalance op- 
 posite the crank-pin, and Fig. 761 is a section of the trailing wheel. The crank- 
 pin of the driver has two journals one for the main connecting rod, the 
 other for the coupling rod, for which there is a journal on the trailing wheel. 
 
 Fig. 762 is a front and side elevation of a return crank, returning back and 
 
 FIG. 756. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 341 
 
342 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 having rotation about the main crank-shaft, used on small steam engines in- 
 stead of an eccentric to give motion to the valve-rod. With its centre on the 
 
 FIG. 759. 
 
 central line of the main shaft there will be no motion, and used for the posi- 
 tion of an oil cup, the oil flowing down the moving arm for the lubrication of 
 the crank- journal. 
 
 Eccentrics. An eccentric is a modified crank ; the crank-pin is enlarged so 
 
 as to include the crank-shaft ; motion 
 is conveyed through the crank to the 
 eccentric, and not through the eccen- 
 tric to the shaft. 
 
 Fig. 763 represents a front view, 
 Fig. 764 the side view, and Fig. 765 a 
 section, of a form of eccentric usually 
 adopted in steam engines for giving 
 motion to the valves regulating the ac- 
 tion of the steam upon the piston. 
 A ring or hoop, eccentric strap, is ac- 
 curately fitted within projecting ledges 
 on the outer circumference of the eccentric, so that the latter may revolve 
 freely within it ; this ring is connected by an inflexible rod with a system of 
 levers, by which the valve is moved. By the revolution of the shaft to which 
 
 FIG. 762. 
 
 FIG. 7G3. 
 
 FIG. 765. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 343 
 
 the eccentric is fixed an alternating rectilinear motion is impressed upon the 
 rod, its amount being determined- by the eccentricity, or distance between the 
 centre of the shaft and that of the exterior circle. The throw of the eccentric 
 is twice the eccentricity C E ; or the diameter of the circle described by the 
 point E. The alternating motion is identical with that of the crank. 
 
 The term eccentric is generally confined to the circular eccentric, all others, 
 with exception of that last described, being called cams or wipers, but eccentric 
 is often applied to curves composed of points situated at unequal distances 
 from a central point or axis. 
 
 Fig. 766. To draw the eccentrical symmetrical curve called the heart, which, 
 when revolving with a uniform motion on its axis, communicates to a movable 
 point A, a uniform rectilinear motion of ascent and descent. 
 
 Let C be the axis or centre of rotation upon which the eccentric is fixed, and 
 which is supposed to revolve uniformly ; and let A A' be the distance which the 
 point A is required to traverse dur- 
 ing a half revolution of the eccen- 
 tric. From the centre C, with radii 
 respectively equal to C A and C A', 
 describe two circles ; divide the 
 outer one into, say, 16 equal parts, 
 and through these points of divis- 
 ion draw the radii C 1, C 2, C 3, etc. ; 
 divide the line A A' into the same 
 number of equal parts as in the 
 semicircle, and through all the 
 points 1', 2', 3', etc., draw circles 
 concentric with the former ; the 
 points of their intersection B, D, E, 
 etc., with the respective radii C 1, 
 C 2, C 3, etc., are points in the 
 curve required, its vertex being at 
 the point 8. 
 
 When the axis, in its angular motion, shall have passed through one division, 
 the radius C 1 coincides with C A', the point A, urged upward by the curvature 
 of the revolving body on which it rests, takes the position indicated by 1' ; and 
 further, when the succeeding radius C 2 shall have assumed the same position, 
 the point A will have been raised to 2', and so on till it arrives at A', after a 
 half revolution of the eccentric. The remaining half, A G F 8, of the eccen- 
 tric, symmetrical with the other, enables the point A to descend in the same 
 manner as it was elevated. 
 
 If the eccentric is turned vertically and the point of a weighted lever rests 
 upon the curve, the lever will take a uniform motion from the curve ; but if a 
 groove be cut in the face of a wheel with its outer edge like that of the curve 
 and its inner one parallel to the outer one and a friction roll be inserted in the 
 groove and attached to a lever or rod, the motion of roller and rod will be similar 
 to that of the weighted lever resting on the eccentric. This construction is of 
 frequent use, applicable to a great variety of motions, and is designated as agrooved 
 cam. The grooves may be cut either in the face of a plate or of a cylinder. 
 
344 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 In the " Transactions of the American Society of Mechanical Engineers " is 
 an illustration of a grooved cam cut on a machine of W. A. Gabriel, M. E., the 
 motions being first laid out on paper and transferred to wooden or metallic forms, 
 which act as guides to the cutting out of the grooves. The difficulty for the 
 
 draughtsman is to comprehend 
 the motions required and lay 
 them out on the paper, which in 
 lack of a machine may be trans- 
 ferred to the metal surface and 
 cut by hand, with the aid of 
 drills, chisels, and files. 
 
 Fig. 767. To draw a double 
 and symmetrical eccentric curve, 
 such as to cause the point A to 
 move in a straight line, and with 
 an unequal motion ; the velocity 
 of ascent being accelerated in a 
 given ratio from the starting- 
 point to the vertex of the curve, 
 and the velocity, of descent being 
 retarded in the same ratio. 
 
 As in Fig. 76G, take C.as the 
 
 centre of motion and A A' the distance to which the point A is to be moved by 
 a half revolution; with C as centre and a radius C A' describe a circle and di- 
 vide the semicircle into eight equal arcs, and draw the radii C 1, C 2, C 3, etc. 
 On A A' as a diameter describe a semicircle and divide it also into eight equal 
 arcs and project the points of division 1', 2', 3', and 4' ; on the diameter A A', 
 with C as a centre through the several points projected, describe circles ; the 
 
 FIG. 767. 
 
 FIG. 769. FIG. 770. 
 
 intersection severally of these circles with the radii C 1, 02, C 3, will give 
 points of the eccentric curve required. 
 
 Fig. 768. To construct a double and symmetrical eccentric, which shall pro- 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 345 
 
 FIG. 771. 
 
 :0* 
 
 
346 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 duce a uniform rectilinear motion, with periods of rest at the points nearest to, 
 and farthest from, the axis of rotation. 
 
 The lines in the figure above referred to indicate the construction of the 
 
 FIQ. 772. 
 
 curve in question, which is simply a modification of the eccentric represented 
 at Fig. 766. In the present example the eccentric is adapted to allow the mov- 
 able point A to remain in a state of rest 
 during the first quarter of a revolution 
 B D ; then, during the second quarter, to 
 cause it to traverse, with a uniform mo- 
 tion, a given straight line A A', by means 
 of the curve D G ; again, during the next 
 quarter E F G, to render it stationary at 
 the elevation of the point A' ; and finally, 
 to allow it to subside along the curve B E, 
 with the same uniform motion as it was 
 elevated, to its original position, after hav- 
 ing performed the entire revolution. 
 
 Fig. 769 represents an edge view of 
 this eccentric, and Fig. 770 a vertical sec- 
 tion of it. 
 
 If but one side of this were constructed, 
 and the motion only equal to that of the 
 arc and reciprocating, it would raise and 
 lower every point resting on it, and would 
 be called a wiper. The wiped surface is 
 generally flat, an arm extending out from 
 the rod to be raised, and a curve D. G may 
 be formed adapted to the height of lift, 
 and action during the lift. 
 
 Fig. 771 is a partial vertical section of 
 the valve motion of one of the high-service 
 pumping engines at St. Louis, Mo., show- 
 ing the wiper W, or lifting toe, with its 
 connection with the valves. The wiper 
 shaft has a reciprocating motion by which 
 the valve stem is raised and dropped. The valves are double-beat balanced ; 
 the cut shows the arrangement of the steam jacket J, a space for live steam in- 
 closing the cylinder, to aid in preserving the temperature and pressure within it. 
 
 FIG. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 347 
 
 Fig. 772 exhibits on a larger scale a portion of a section in plan and eleva- 
 tion of the steam piston and mode~of packing of the above engine. 
 
 Fig. 773 is the elevation of a stamp mill in which a double wiper on a rotary 
 shaft raises the stamp by its contact with a hub or collar on its shaft, and lets 
 it fall suddenly by its weight. 
 
 Machines like the above are often used in bleacheries to give a watered 
 finish to goods. The battery of stamps is equal to the width of the cloth which 
 turns in a roll beneath it, and the blows follow in quick succession. 
 
 Connections. Figs. 774 and 775 are sections of cottered joints of wrought- 
 iron bars, the first made with a socket and the end of one of the bars ; the 
 
 latter by a sleeve connecting the two bars. The bars in the socket and sleeve 
 are upset, to give more section than the bars themselves, so that the slots cut 
 for the cotters c c will not reduce the strength below that of the bars. The 
 cotters must have sufficient shearing strength and bearing surface, and at the 
 same time diminish as little as possible the section of the parts connected. 
 The proportions given in the figures are drawn to a scale of the diameter of 
 the enlarged part as the unit, and the 
 proportions given in figures are such as 
 obtain in practice for wrought-iron. 
 If the cotter be of steel, its breadth 
 may be three fourths of that given, 
 preserving the other dimensions the 
 same ; the thickness is '25 of the unit. 
 
 The knuckle-joint (Fig. 776) is 
 given in dimensions of the bar as a 
 unit, and adapted to usual work. If 
 there is much motion at the joint, the 
 wearing-surface should be larger, by 
 increasing the width of the eyes and Fio 
 
 the length of the pin. The pin in the 
 drawing is through the collar ; usually the pin is extended, and the pin passes 
 through the bolt outside the collar. 
 
 Connecting-rods, in their application to steam engines, are the rods connect- 
 ing the piston through the cross-head to the crank. When two cranks are 
 connected it is called a coupling-rod. 
 
34:8 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Fig. 777 is the side elevation of the eccentric and strap of a Reynold's 
 20* X 48* Corliss engine. The eccentric rod is made with a shoulder, the stem 
 is fitted to the If* socket, and held firmly by a nut in the 4" groove. 
 
 Fig. 778 is a double eccentric, used 
 in giving motion to the link in the cut- 
 off. The eccentrics are made in halves 
 for convenience in putting them on the 
 shaft. 
 
 Governors are used on quick-run- 
 ning engines (page 407) to control the 
 cut-off through movable eccentrics, but 
 eccentrics adjustable by hand are con- 
 venient as applied to pumps in modi- 
 fying the stroke. With a uniform ac- 
 tion of the motor the stroke of the 
 pump is reduced, giving greater pres- 
 sure on the plunger, say from a domes- 
 tic to a fire pressure. 
 
 Figs. 779 and 780 are plan and sec- 
 tion of such an eccentric. The crank-pin is fastened on a slide-rest moving in 
 guides attached to the power shaft by a long hub. To hold the slide firmly to 
 the shaft-hub there are three key-bolts, n n n. To move the crank-pin slide 
 these bolts must be slacked, and by means of a hand lever (Fig. 781), using as 
 a fulcrum the pins on one of the guides, and bringing the short end of the 
 lever against a projection,^, attached to the crank-pin, it may be moved to any 
 desirable position or stroke and clamped to the hub firmly by the key-nuts. 
 
 FIG. 777. 
 
 1 
 
 ; i 
 
 t-f--+ 
 
 '-'v-A j 
 
 1 
 
 t i 
 
 L/ * 
 
 i 'N_ ' - 
 
 ^ 
 
 T 
 
 , . 
 
 
 l n- 1 1 
 
 i | 
 
 1 i T 
 
 i_J 1 
 
 f*~~\~t T rf" 
 
 --;-- 
 
 
 
 n-i"- :< rVv i i 
 
 
 
 --- 
 
 
 FIG. T78. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 349 
 
 FIG. 779. 
 
 FIG. 
 
 Fig. 782 is the elevation, plan, and section of an eccentric strap of a locomo- 
 tive engine, of cast-iron, with very long bolts, and a very rigid construction. 
 The strap forms a cup-section over the projecting ring on the eccentric, and 
 retains the oil at the bottom for more efficient lubrication and prevents drip. 
 
 Figs. 783, 784, and 785 is another cast-iron eccentric strap, in which a box 
 is inserted, fitted with metallic disks, to prevent frequent oiling. The bolts at 
 the large end are bored up to a sectional area of that of the screwed portion to 
 secure equal elasticity. 
 
 FIG. 782. 
 
350 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 FIG. 784. 
 
 U 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 351 
 
 In many marine engines the boxes of both crank and cross-head pins are 
 connected with strong and heavy bolts without any other rod. 
 
 Fig. 786 is a strap-end of a connecting-rod, from the Corliss Steam-Engine 
 Company. The peculiarity is the adjusting-screws connected with the boxes. 
 
 Fig. 787 is the strap-end of a locomotive connecting-rod in which the wear 
 of the boxes is taken up by a cotter at the end of the strap. 
 
 t2T 
 
 FIG. 788. 
 
 FIG. 78 
 
 In Fig. 788 the key is between the bolts ; the weakness from bolt-holes or 
 cotter-slot is compensated by the width of the strap. 
 
 Fig. 789 is the box-end of a locomotive-rod, in which the strap gives place 
 to a box forged on the end of the connecting-rod. 
 
352 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 353 
 
 Figs. 790, 791, and 792 are side, plan, and end views of a connecting-rod of 
 the Southwark Foundry, of Philadelphia, used on their fast-running Porter- 
 Allen engines. 
 
 The cross-head end is a strap-end, while that of the crank is a box-end, and 
 of larger diameter than the former on account of the extra wear and size of the 
 crank-pin. The length of the page does not admit of the representation of the 
 full length of the connecting-rod on the scale ; it is therefore shown broken, 
 with the dimensions figured in. The sections of the two ends are drawn in on 
 the rods ; the circular section A is the same as that of the piston-rod, and both 
 are represented in the conventional hatching of cast-iron, but it is of wrought- 
 iron. The gib g and key or cotter v at the strap-end are of steel, and the key 
 is fastened when in position by a set-screw through the head. At the box- 
 end, a wedge and screw forces the box into position. 
 
354: MACHINE DESIGN AND MECHANICAL CONSTEUCTIONS. 
 
 Figs. 793 and 794 are the sides and top elevations of the connecting- and 
 coupling-rods of the express locomotives of the New York Central and Hud- 
 son Eiver Kailroad, designed by Mr. Buchanan, and built at the Schenectady 
 Locomotive Works, of which the drawing of driving wheels will be found 
 at Fig. 760, of the frame Fig. 1301, and the boiler Fig. 926, with the gen- 
 eral weights and dimensions of the locomotive. Fig. 795 is a section of the 
 coupling-rod. 
 
 Fig. 796 is a connecting-rod of the Eider hot-air engine, in which the ends, 
 made of gun metal, are connected together by a tube in which is fitted a rod, 
 extending from the upper to the lower brass, and so arranged that one key, E, 
 capable of nice adjustment by nuts, at once takes up the lost motion on both 
 upper and lower brasses. 
 
 FIG. 796. 
 
 Fig. 797 is the stub end of a coupling-rod. The bushes are solid, of brass, 
 and kept from turning round by taper-pins, which are secured by set-screws 
 pressing on the larger end ; taper, -fa in 3 inches. 
 
 Cast-iron connecting-rods are now very seldom used. In some cases of 
 vertical-beam pumping engines it is necessary that the weight of the steam 
 pistons should be counterbalanced by some mass of material, and it may be 
 convenient to make use of a heavy pump-connection. 
 
 The wrought-iron crank connections of American river-boat engines are 
 peculiar in their construction. They are made as light as possible, with very 
 great stiffness. Fig. 798 represents the side elevation of such a connecting- 
 rod. The means adopted to give the required stiffness consist of a double- 
 truss brace, a a, of round iron, which is fastened by bolts to the rod near each 
 end ; struts b b, cut with a screw, and furnished with nuts, pass through the 
 centre of the brace, by which means the braces are tightened. The connecting- 
 rod at its smallest part near the extremities is of the same diameter as the pis- 
 ton-rod ; the boss in the centre is from one to two inches more. 
 
 Fig. 799 is the front view of the forked end of the rod, which is fitted with 
 the usual straps, gibs, and cotters. Fig. 800 is the side view of the brace-rod. 
 
 The working-beam (Figs. 801 and 802), of similar framed construction, was 
 at one time largely used for river boats, but may still be applicable in many 
 places for its lightness and stiffness. It is composed of a skeleton frame of 
 cast-iron round which a wrought-iron strap is fitted and fastened. The strap 
 is forged in one piece, and its extreme ends are formed into large eyes, which 
 are bored to receive the end-pins or journals. The skeleton frame is a single 
 casting, and contains the eyes for the main centre and air-pump journals ; the 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 355 
 
 centre hub is strengthened by wrought-iron hoops shrunk 
 upon it. At the points of contact of the strap and skele- 
 ton, key-beds are prepared. Small straps connect the frame 
 and main-strap at these points, keyed to the frame keys 
 riveted over. The frame is further braced by wrought-iron 
 straps, C C, which tie the middle of the long arms to the 
 extremities of the shorter ones. The following are the gen- 
 eral dimensions : From centre to centre of end-journals, 26 
 feet; this is somewhat less than the usual proportion to 
 length of stroke, being slightly less than double the 
 stroke ; length of centre hub, 26", a a ; diameter of 
 main centre eye c, 15|" ; of air-pump journal-eye d, 6f; 
 of end-journals e e, 8". 
 
 Double plates or flitches of wrought-iron are often 
 used in the construction of working-beams and side-levers. 
 Fig. 803 is the section between the two plates of a beam 
 
 FIG. 79 
 
 FIG. 797. 
 
 of this kind, attached to the compound pumping engines 
 at Milwaukee, Wis. The plates are each 30 feet long, 
 by 6' 4" deep at centre, by If" thick. The connections 
 between the two, shown in section in the figupe, are 
 
 cast-iron pipes with wide flanges at 
 
 each end riveted or bolted to the 
 
 plates. The main centre and other 
 
 small journal - pins are rods of 
 
 wrought-iron, passing through the 
 
 pipes, and extending outside the 
 
 plates to form the journals ; c is 
 
 the section of the pin for crank 
 
 connection, p for that of pump, h 
 
 for that of high- pressure cylinder, I 
 
 for that of low-pressure cylinder, FIG. 799. 
 
 FIG. 800. 
 
356 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 m for main centre-pin, and g for the parallel-motion links. This last is usu- 
 ally the position of the air-pump centre, but in this engine the air-pump is 
 
 FIG. 802. 
 
 below the high-pressure cylinder, and its piston-rod is extended to the air- 
 pump piston. The dimensions are H. P., cylinder 36" X 62" ; L. P., cylin- 
 
 FIG. 803. 
 
 der 58" X 8 feet. The heads of the two cylinders are kept at the same levels 
 by increasing the length of the H. P. piston. 
 
 Working-beams with vertical steam engines are now seldom used except in 
 
 Fia. 804. 
 
 connection with large pumps. For power the horizontal engine is now sub- 
 stituted and connected closely with the fly-wheel. Fig. 804 represents such 
 an engine and frame. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 357 
 
 The pin p is the connection in.the cross-head (Fig. 804) between the piston- 
 rod of the steam engine and the connecting-rod to the crank. 
 
 Figs. 805, 806, and 807 represent the plan, end view, and section of the 
 cross-head adopted by the Southwark Foundry for their high-speed engines. It 
 is of cast-iron, with large, 
 flat faces, the pin p for 
 the connecting-rod being 
 in the middle of the 
 length. This pin is of 
 wrought-iron, large and 
 flattened on top and bot- 
 tom, so that the boxes of 
 the rod can never bind on 
 the pin at the extreme of 
 the vibrations of the rod ; 
 usually these pins are 
 round. The pin is formed 
 with large squares at the 
 ends, by which it is fitted 
 into the jaws of the cross- 
 head, where it is secured FIG. so?. 
 by a steel pin passing 
 
 through the cross-head. The bearing surfaces of the head and those of the 
 guide-bars are finished by scraping to true planes ; there are no means of ad- 
 justment, as there is no wear if kept clean. 
 
 FIG. 808. 
 
 1 
 
 FIG. 809. 
 
 It is to be understood that the piston-rod moves in a straight line, and that 
 the stress on the connecting-rod pin is mostly oblique. Guides are to be pro- 
 vided, between which the cross-head slides, to take the oblique stress off the 
 piston-rod. 
 
358 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Pigs. 808 and 809 are elevation and plan of guide-bars suited to the cross- 
 head above, which are in common use for both vertical and horizontal engines. 
 Lugs or ears are cast on the steam-cylinders, and on the frames to which the 
 bars are bolted, and between which the cross-head slides. The grooves or 
 notches across the guide-bars, at the ends of the stroke, are to throw off any 
 grease or dirt that may be carried along by the head and prevent their accumu- 
 lation. The stress on the guide-bars is due to the pressure of the steam on the 
 piston acting obliquely on the crank through the connecting-rod, and is the 
 greatest when the crank is at right angles to the piston. It can be determined 
 by multiplying the pressure on the piston by the length of the crank, and divid- 
 ing the product by the length of the connecting-rod, which will be the stress 
 tending to separate the guides. If the connecting-rod be 3 times the stroke, or 
 6 times that of the crank, which is the usual proportion, then the stress is 
 the pressure on the piston. Sometimes the proportion of connecting-rod to 
 
 FIG. 810. 
 
 stroke is 2 to 1. When a portion of the force of the steam is opposed directly 
 to the resistance, as in direct-acting pumps, and only the irregularities in the 
 
 steam-pressure are transmitted through the connect- 
 ing-rods, the proportion of rod to stroke may be still 
 smaller. 
 
 Fig. 810 is the cross-head of the Harris-Corliss 
 engine in which the bearing surfaces of guides are 
 adjusted by bolts passing through wedges. 
 
 On locomotives it is not unusual to have the 
 guide on one side, as in Fig. 811, where the side-bars are of wrought-iron and 
 the slide-block is fastened between the two plates of the cross-head by bolts. 
 It is the most common practice in this country to use guides with vertical en- 
 gines, even when the connection is with working-beams. 
 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 359 
 
 Steam- Cylinders. Fig. 812 is a sectional plan of a common form of small 
 steam-cylinder. A is the cylinder, B the piston, b the piston-rod, D the slide- 
 valve, d the valve-rod, C the valve- 
 chest, c the chest-cover, s s the 
 steam-ports, e the exhaust-port, S the 
 stuffing-box of the piston-rod, s' that 
 of the valve-rod. H is the front 
 head and H' the back head of the 
 cylinder. The bolts attaching the 
 heads to the body of the cylinder are 
 not shown. 
 
 Length of Cylinder. It is the 
 present practice, in the construction 
 of stationary engines for driving ma- 
 chinery, to make the stroke not over 
 twice the diameter of the cylinder, and for diameters above 24" about 1 time the 
 diameter of the cylinder, and invariably to place the cylinders horizontally with 
 a direct connection with the crank, without the intervention of a working-beam. 
 
 Fig. 813 is a front view partly in section, and Fig. 814 is a transverse section 
 through the centre of a Fishkill Corliss steam-engine cylinder, giving the steam- 
 
 Fia. 812. 
 
 FIG. 813. 
 
 valves in section and outside view showing their connection with the wrist-plate 
 and its motion, also the piston-rod and stuffing-box with a scale of reference. 
 The thickness of shell Mr. Henthorne finds, by many examples in Corliss's large 
 practice, to conform to the formula t = -268 Vd, t and d being in inches. 
 
360 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Thus the thickness of the shell of a 16" cylinder will be V 16 X '268 = 4 X '268 = 
 1-072, a little more than 1". The thickness of flanges should exceed that of 
 the shell by | to % its thickness. The bolts should not be less than f " and sel- 
 dom more than 1". It is better to increase the number of bolts than their di- 
 
 Fio. 814. 
 
 ameter, the breadth of flange to be about 3 times the diameter of the bolts, and 
 the pitch of the bolts, or the distance between centres, about 6 times the diam- 
 eter of the bolts. 
 
 Cast-iron is the material chiefly used for pistons, but those of wrought-iron, 
 brass, bronze, and cast-steel are common. The wrought-iron pistons recently 
 introduced in American locomotive practice follow the designs of the older 
 cast-iron pistons. 
 
 Fig. 815 is the cast-iron piston of a locomotive. The spring or snap-rings 
 forming the packing are of cast-iron, 1^-" wide by %" thick, of uniform section. 
 The split is made with a half lap, and 
 the splits of the two rings are on oppo- 
 site sides of the piston. The outsides of 
 the rings are turned to a diameter slightly 
 in excess of that of the cylinder, and are 
 sprung into recesses of the piston fitted 
 to receive them. 
 
 Fig. 816 is a half plan and half sec- 
 tion and Fig. 817 a full cross-section of 
 piston of a steam-cylinder of Leavitt's 
 design ; the packing is Wheelock's, in which the rings are in sections joined to 
 adjust themselves to cylinders that have become worn. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 361 
 
 FIG. 816. 
 
 SECTION E F 
 
 FIG. 817. 
 
 Large cast-iron pistons are made hollow and strengthened by internal radial 
 ribs. In Fig. 772 is shown the packing of the steam piston of the cylinder of 
 one of the St. Louis pumping-engines. Fig. 818 is a design from an English 
 manual for a large cast-iron piston. The dimensions marked on the figure are 
 
 in terms of the unit ** , in which D is the diameter of the piston in inches 
 
362 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 and P the initial pressure per 
 square inch. The junk-ring is 
 secured by wrought-iron or steel 
 bolts and brass nuts. The diam- 
 eter of the junk-ring bolts may be 
 
 28TVP 
 
 ^-~ 1- 1 inch, and they may 
 
 be placed at a pitch of seven to 
 ten times their diameter. The 
 number of ribs or webs may be 
 
 about -=r + 2 and their thickness 
 
 4D.yp 
 
 100 ' 
 
 The size of the space 
 
 FIG. 818. 
 
 for the packing will depend on 
 the design of packing adopted. 
 Fig. 819 is a sectional plan and Fig. 820 is a sectional elevation of the inte- 
 rior of a piston-ring, showing another common form of ring packing, which 
 consists of a single interior ring r and two exterior rings r" r", and each cut in 
 
 FIG. 819. 
 
 FIG. 820. 
 
 two and so fastened that the joints are always broken. The packing is set out 
 by springs s s, one of which is shown. F is the follower, which can be taken 
 off for the admission of the rings and springs, and then replaced and bolted to 
 the piston, making a close joint with the end of the rings. The depth of the 
 piston at the exterior is from 3" to 9", varying with the diameter of the piston. 
 Figs. 821, 822, and 823 are sections of the exterior rings of pistons adapted 
 
 FIG. 821. 
 
 FIG. 822. 
 
 FIG. 823. 
 
 more particularly to water-pumps. Fig. 821 depends on the closeness of fit of 
 the exterior of the piston with the inner surface of the cylinder, and when accu- 
 rately turned and fitted the leak is very inconsiderable, and by the use of grooves 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 363 
 
 (Fig. 822) it is still less. In Fig. 823 the joint between the piston and the cyl- 
 inder is made tight by a gasket, usually of hemp, compressed by a joint ring or 
 follower, a, in the pocket between piston and cylinder. 
 
 Wood-packing put in short staves, as shown in Fig. 824, is often used for 
 pump-pistons and buckets. Make the diameter of the wood-packing a little 
 less than the diameter of the barrel of the pump, to allow for the swelling 
 which takes place when the wood becomes saturated with water. 
 
 FIG. 825. 
 
 FIG. 824. 
 
 FIG. 826. 
 
 Fig. 825 is a cup leather packing, and Fig. 826 is a U-packing for small 
 cylinders of hydraulic presses. The application of the first will be understood 
 from Fig. 827, in which the piston is packed with two cup leathers, in this case 
 to withstand pressure in both directions. Were the piston single-acting, but 
 one cup would be necessary and if from beneath the piston, this would be the 
 lower cup. The flexible flange is pressed against the inside of the cylinder, and 
 the joint is perfectly tight. 
 
 FIG. 827. 
 
 FIG. 829. 
 
364: 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Fig. 828 shows the application of the U-packing ; it is put into a recess in 
 the cylinder by bending the packing into a saddle-bag form, and then allowing 
 it to spring back into the recess. In English practice hemp packings serve 
 the same purpose, and are necessary when the temperature of the water ex- 
 ceeds 90. 
 
 Packings can be obtained from hydraulic-pump and press manufacturers, 
 and are kept in stock of all the usual sizes. Their depths are from 1" to 1" 
 for diameters varying from 4" to 14", and the space occupied by the thickness 
 in the U from \" to f ". A filling of flat braided hemp is placed inside the U 
 to keep it tight when not under pressure. The packings are made by steeping 
 the leather in warm water, forcing them into a mould, and leaving them to dry 
 and harden. The moulds (Fig. 829) are made of either metal or wood ; fre- 
 quently the rings are of metal, and the piston over which the cup is formed, of 
 wood. 
 
 The outer cylinder forming the exterior shell of the steam-jacket is fastened 
 to the steam-cylinder, but is allowed to move under changes of temperature. 
 In Figs. 830 and 831 are given longitudinal and transverse sections of a steam- 
 cylinder with jacket, as constructed by E. D. Leavitt, M. E., in which the upper 
 and lower end of jacket is cast with the cylinder, and the connection between 
 the two is by a copper U-ring, which admits the necessary expansion and con- 
 traction. All steam-cylinders, whether with or without jackets, should be 
 clothed that is, covered with some preparation to prevent the escape of heat 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 365 
 
 from contact with air. The usual clothing is hair felt, with a lagging that is, 
 an exterior shell of some wood, usually black walnut. 
 
 FIG. 832. 
 
 Fig. 832 is a section and partial elevation of the air-chamber of one of 
 Leavitt's pumps. It is of the Thames Ditton variety, in which the whole charge 
 of water is drawn into the lower chamber, and a portion corresponding to the 
 difference between the two plungers raised to the full head in the upper cham- 
 ber j in the down stroke a quantity of water equal to the displacement of the 
 upper plunger is raised to the full head. The packing of the lower plunger is 
 by grooves in the exterior ring ; this form of packing by proper fitting makes a 
 tight joint without friction. With this packing the lower chamber of the pump 
 can be drawn off and examined, leaving the water-load in the upper piston. The 
 dash-pot of a Fishkill Landing Corliss engine, with a similar groove packing, 
 
366 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 was tested to a water pressure of 140 pounds and leaked very little. Like pack- 
 ings are used in air-pump pistons with satisfaction. 
 
 The pump-valves are not shown in the drawing, but are Riedler's controlled 
 valves, which are opened by the movement of the water and closed positively by 
 a mechanical connection. 
 
 Fio. 834. 
 
 FIG. 833. 
 
 Fig. 833 is a section of a single Worth- 
 ington steam-pump, one of a pair always 
 placed side by side (hence called duplex), 
 combined to act reciprocally on the steam- 
 valves of each other. 
 
 In the form of pump shown, the length 
 of barrel is about equal to the diameter 
 of the piston, but the length of the piston 
 is equal to that of the stroke of the pump 
 plus that of the length of the barrel. 
 
 In Fig. 834 the pump barrel is long 
 and the piston short. 
 
 The duplex was H. R. Worthington's 
 invention and is now the general type of 
 most makers for boiler feed pumps and 
 small water supplies. For the boiler feed 
 especially of locomotives the injector is 
 largely used, and as supplementary to the 
 pump. 
 
 The injector is an apparatus in which 
 the momentum of a jet of steam issuing 
 from the boiler is transferred to a body 
 of water, producing a resulting velocity 
 sufficient to force the water into the same 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 367 
 
 boiler. Many shops manufacture, injectors. The section shown (Fig. 835) is 
 from William Sellers & Co., one of the earliest makers, and the explanation of 
 its working illustrates the principles of a good and successful machine. 
 
 To start the injector the valve G is raised to permit the admittance of water 
 into the chamber J ; the lever F is then opened sufficiently to allow the steam 
 through the opening d of, while the plug // still keeps the forcing nozzle a 
 closed, thus admitting steam into the annular steam nozzle i, which, passing 
 into the water-chamber J and the combining tube through the overflow cham- 
 ber I and into the overflow E, producing a strong vacuum in the water-chamber 
 J, into which the water from the source of supply is forced by the atmospheric 
 pressure, the characteristic of a lifting injector. The combined jet of water 
 and steam passes through the combining tube ; the valve F is then fully raised, 
 admitting steam into the forcing nozzle a ; this steam, uniting with the jet in 
 the combining tube, accelerates its velocity to such an extent that the valve g 
 is forced down, thus allowing the passage of water to the boiler. 
 
 n 
 1 
 
 i 
 
 FIG. 835. 
 
 The issuing steam and water maintains its velocity by reduction of the areas 
 of channel discharge till the last channel in connection with the boiler, which 
 increases till it meets the valve g. 
 
 The lower the temperature of the feed-water the greater the capacity of the 
 injector. As the steam strikes the water its momentum is checked and trans- 
 ferred to heat and at once absorbed by the feed, all the energy apparently 
 wasted reappearing in the latent and sensible heat transferred to the feed. It 
 is this absorption of heat that gives the great economy over the feed-pump, but 
 for raising water it is not to be used except where simplicity of construction 
 and convenience of application may offset the waste of power. 
 
 To use the injector as a heater to prevent the freezing of the water in the 
 water-tank, the valve G is opened and the eccentric lever H closed ; the steam 
 will then have free admittance into the chamber I and the water-tank. 
 
368 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Each size of injector is named from the diameter of the delivery tube in 
 millimetres. In the table given by Strickland L. Kneass the diameter runs 
 from 3'2 to 11'5 millimetres; the steam pressure from 30 to 150 pounds, and 
 the discharge 23'5 to 550 cubic feet per hour ; height of lift, 1 foot ; tempera- 
 ture of water, 60. 
 
 Ejectors are also used as pumps for the raising of water by suction and 
 pressure, of which Figs. 836 and 837 are of the simplest form. The steam is 
 
 FIG. 836. 
 
 FIG. 837. 
 
 FIG. 838. 
 
 ejected through a central nozzle a, while the water is raised and discharged 
 through the pipe inclosing it. The same principle of induced current is 
 applied through a current of water, steam, or air in discharging earth from a 
 caisson, the ashes from the boiler hold of a marine engine, water from founda- 
 tions or from driven wells. 
 
 Fig. 838 is a section of a Korting blower to improve the draft in the ash- 
 pit, fire-box, or chimney of a boiler, and Fig. 839 is that of a larger size, show- 
 ing the construction of the jet in which the force of the steam drawing the 
 air through compound nozzles reduces the intensity of its flow, but increases 
 its quantity to suit the purposes of draft. 
 
 M. Mondesir, in the French Exhibition of 1867, effected the ventilation by 
 means of reservoirs of compressed air. Around the exterior of the building 
 there was a large underground gallery into which the exterior air was intro- 
 duced by sixteen vertical shafts symmetrical in position, and from the main 
 gallery the air was distributed by radial galleries into the interior of the build- 
 ing, into which the air current was introduced by numerous jets of condensed 
 air from the reservoirs. The air was supplied to the jets at a pressure of 29| to 
 3l inches of water. The vitiated air escaped through a ventilator in the roof. 
 
 Clearances in cylinders' include, in general signification, not only the spaces 
 between the piston and cylinder-heads at the ends of the stroke, but also the 
 spaces between the cylinder and the valves ; and as those spaces are voided in 
 a steam-cylinder at each stroke for which adequate work from the steam is not 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 369 
 
 obtained, they are usually made as small as possible. If 
 the steam is fairly dry, from %' to'-l" will be sufficient 
 for end-clearances that is, minimum distance between 
 piston and cylinder-head. 
 
 Piston-rods are proportioned to the stress on them, 
 usually one square inch of section to each 5,000 pounds 
 of stress. In Fig. 815 the tapered end fits a taper hole 
 in the piston, and is riveted over. It is more usually 
 held by a nut, some use a shoulder on the inner end of 
 the piston-rod instead of a taper, and the nut brings the 
 piston strongly up against this shoulder. 
 
 Piston-rods are made either of steel or hammered 
 iron, some makers of engines preferring one and some 
 the other material. 
 
 Stuffing-boxes are the mechanisms to prevent the 
 leakage of steam, air. or water, in the movement of the 
 piston or other rod out of the cylinder or chest. They 
 consist of an annular chamber around the rod, general- 
 ly filled with gaskets of hemp, which is forced down by 
 a ring or gland into close contact with the rod and the 
 sides of the box. In Fig. 812 there are two stuffing- 
 boxes shown, one for the main piston-rod, the other for 
 the valve-rod. In the latter the cap of the gland is 
 fitted with a screw to connect it with the side of the 
 stuffing-box, by which the gasket may be more or less 
 compressed. This is the general form of stuffing-box 
 for small stems or rods, sometimes with a ring or follower on the top of the 
 gasket, which is forced down by the gland without turning the ring or gasket. 
 
 In the figure the stuffing-box is made of brass, 
 and screwed into the end of the steam- or valve- 
 chest. 
 
 The stuffing-box to the piston is cast with the 
 
 FIG. 839. 
 
 FIG. 841. 
 
370 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 head of the cylinder, and is bored out, .and a brass bushing fitted and driven 
 into the end of the box. The hole through the bushing in most boxes fits the 
 
 piston-rod accurately. The gland is of cast- 
 iron, turned to fit the stuffing-box, and bored 
 to fit the piston-rod ; after packing the box 
 the gland is forced in and retained by screws. 
 Fig. 840 is the plan and section of a com- 
 mon stuffing-box, in which the thickness of 
 packing is from " to l", and the depth from 
 1 to 2 times the diameter of the piston-rod. 
 The number of bolts vary with the diameter 
 of the piston seldom more than four, and 
 usually but two. 
 
 Fig. 841 is the section of a stuffing-box of 
 the proportions adopted by the Southwark 
 Foundry. Taking the diameter of piston-rod 
 A as the unit, B is 2, C 3, D 2, all scant up 
 Fl - 842 - to a 3" rod, or 22" cylinder. For a 28" X 42' 
 
 A = 4, with an allowance of -fa" for clearance, B 6|, 9, D 6". 
 
 Fig. 842 is a modification with double-cone grooves in the glands and stuf- 
 fing-box by which the loose fibrous packing is prevented from 
 getting into the joint spaces. 
 
 Besides hemp gaskets, there are a very great variety of 
 packings, patented or otherwise, which are very good, adapted 
 to common stuffing-boxes, and easily procured. 
 
 Valves Steam-cylinder Valves. The simplest and most common is the 
 slide D, of which the action is described under the chapter on Motion, pages 
 216-218. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 3Y1 
 
 Of the Size of Ports or Openings. Under " Steam-pipes " will be given the 
 formula for the flow of steam, but the general rule of proportioning the ports 
 of a cylinder is to consider the velocity of steam 100 feet per second, and of the 
 exhaust 80 feet per second. With the slide-valve the opening and closing are 
 made gradually, thereby throttling the flow of the steam. To avoid this, Mr. 
 Corliss in his engine has made the ports long and narrow ; the steam-valves open 
 quickly and close at once by a drop ; the exhaust- valves move rapidly from the 
 wrist connection. From the great size and form of the common slide-valve there 
 ensues a great pressure on the surface ; various expedients have been adopted 
 to relieve this pressure, which is especially desirable in quick-running engines. 
 
 Fig. 843 is a horizontal section of cylinder, through steam- and exhaust- 
 valves, of a Porter- Allen engine, and Fig. 844 a vertical cross-section through 
 
 FIG. 844. 
 
 cylinder and valves. The valves are four in number, one for admission and one 
 for exhaust, at each end of the cylinder, and on opposite sides. They stand 
 vertically to drain the cylinder. The valves work between opposite parallel 
 seats ; the exhaust-valves nearly and the admission-valves wholly in equilibrium. 
 The action of the back plate, and how the wear is taken up, will be understood 
 from the section (Fig. 844), which passes through the middle of one pressure- 
 plate. It is made hollow, and most of the steam supplied to two of the open- 
 ings passes through it. It is arched to resist the pressure of the steam without 
 deflection. It rests on two inclined supports, one above and the other below 
 the valve. These inclines are so steep that the plate will move down under 
 steam pressure ; and that it may be closed up to the valve with only a small 
 vertical movement, the pressure-plate is held in its correct position by projec- 
 tions in the chest on one side and tongues from the cover in the other, which 
 bear against it at the near end, as shown. Between these guides it is capable 
 of motion up and down and back and forth from i y to |". The pressure of 
 the steam on this plate tends to force it down the inclines to rest on the valve. 
 By the means of the screw the plate is forced up and away from the valve, and 
 
372 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 can be so nicely adjusted that the valve works freely and perfectly steam-tight. 
 When the pressure is greater in the cylinder than in the chest, the pressure- 
 plate is forced back, to the instant relief of the cylinder. 
 
 Cylindrical Valves. Fig. 845 represents the section of the steam-cylinder 
 of an Armington & Sims steam engine with a cylindrical valve. The steam- 
 
 JTI 
 
 FIG. 845. 
 
 chest S is central and incloses the valve ; the exhaust chambers E E are at the 
 ends of the valve, and are connected through the hollow stem or body of the 
 valve. The valve depends on its accuracy of fit for its tightness. The valve- 
 chamber is bored out and ground, the valve is turned, ground, and carefully 
 worked by hand, to so close a fit that there is no loss of steam in action, and 
 the valve is completely balanced. 
 
 There is a form of balanced valves, called the double-beat, much used both 
 for steam and water valves. Fig. 846 is a sectional elevation of a steam valve 
 
 Fia. 846. 
 
 FIG. 847. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 373 
 
 of this kind, and Fig. 847 a plan of the lower seat a, with the valve-guides g g 
 in section. There are two seats, flf and b, and two faces on the valve corre- 
 sponding to them. The balance depends upon the relative diameters of the 
 bearing-lines of the two faces. In the figure, if the exterior of the bearing at 
 
 y*j ^j i^j y,j 
 
 FIG. 848. 
 
 \' 
 
 Z> and the interior at ft are both tight, the valve is balanced under any pressure, 
 except as to its own weight ; s is the valve-stem, and the hole r is for a bolt to 
 fasten the valve-seat to the casting of the steam- chest. The scale is full size. 
 Fig. 848 is another form of balance, consisting of two equal poppet-valves 
 connected together the steam passage to the cylinder being central, and the 
 steam-chest at each end, connected. 
 
 Automatic valves, that are moved by the action of the fluid in which they are 
 placed. 
 
 The double-beat valve (Fig. 846) is sometimes used in large pumping-en- 
 gines. From its two beats, the lift is about one half that of a plain valve. 
 There must be difference enough in the faces to admit of the lift of the valve 
 
 by the pressure of water acting on this 
 difference. The seats of the valves are 
 often made of wood, set endways. 
 
 Large valves, from their great weight 
 and flow of water through them, are noisy 
 in both seating and lifting. This is met 
 in the balanced valves by slower move- 
 ments of the piston ; but present practice 
 is to obtain outlets by increasing the num- 
 ber of valves, the total area of the aper- 
 tures, and the speed. 
 
 Fig. 849 is a single-beat direct lift- valve 
 guided by three feathers on its under side, 
 
 which slide in the cylindrical part of the pipe. The feathers are of a screw 
 form, by which a rotary motion is given to the valve through the flow of the 
 water, which prevents its beating on the same parts of the seat ; usually the 
 
 FIG. 849. 
 
 FIG. 8so. 
 
374 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 FIG. 851. 
 
 FIG. 852. 
 
 feathers are straight, and often four instead of three. Fig. 850 is a poppet-valve 
 guided by a central stem ; both these valves have conical faces and seats, with 
 generally an inclination of 45 to the axis of the valve. In many valves the 
 faces and seats are flat, one or the other of which is of soft metal or rubber. 
 
 Figs. 851 and 852 are elevation and section of a rubber disk- valve in very 
 common use in direct-acting pumps and small pumping- engines ; sometimes 
 
 with a thimble in the rubber as a 
 guide ; usually, as in the figure, 
 with a metallic plate on top of 
 the rubber for the bearing of the 
 spring ; valve-seat generally of 
 composition, with spindle riveted 
 or screwed into it. Sometimes 
 the rubber is held in a metallic 
 plate or cup. The springs at 
 their backs cushion the blow on 
 the lift, and start the valve down- 
 ward promptly on the check of 
 the waterflow at the end of the 
 stroke. The great desideratum 
 of water - valves is that there 
 should be little lift, but ample 
 water-way. 
 
 Fig. 853 is a section of Field's pump-valves, an English design for high-water 
 service, as fire-pumps, of which the flaps are rubber disks. For lower pressures, 
 as for the pumping of half-stuff in paper-mills, valves are made in the shape 
 of a bishop's mitre slit lengthwise 
 at the top and partly down the 
 sides. The bottom flanges should 
 be held loosely without bolts 
 through them. 
 
 FIG. 853. 
 
 FIG. 854. 
 
 FIG. 855. 
 
 Fig. 854 is a ball- valve, guided in its movement by an open guide-cage, c, 
 which is held down by a set screw in the cover. Globe-valves with this form 
 are on sale. Ball- valves are usually small metallic balls on metallic or wooden 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 375 
 
 seats, or rubber balls on metallic seats ; and cylindrical valves have been made 
 of the same section as in the figure 4 the body of the valves of brass pipes with 
 rubber jackets. 
 
 In Fig. 855 the ball is of rubber and the seat is of composition, screwed 
 into the pipe with a cage of the same metal, screwed 
 to the seat. 
 
 Fig. 856 is a section of a poppet-valve; the body 
 is of cast-iron, but the valve and seat are of brass. 
 The valve is guided by three feathers. The lift of 
 the valve may be controlled by a set screw in the 
 cover which admits of adjustment to varied lifts. 
 
 Figs. 857 and 858 are the plan and section of a 
 disk- valve for the air pump of a condensing steam 
 engine. The valve consists of a disk of rubber ly- 
 ing on a flat grating or perforated plate of brass, 
 held in position between the grating and a spheri- 
 cal guard by a central bolt. The shape of the guard 
 gives a uniform flexure to the rubber in lifting, and 
 an easy flow to the current of air and water. The rubber is not closely clamped 
 
 FIG. 856. 
 
 between the guard and plate, as will be seen in the figure, 
 being screwed home, is riveted, and 
 the upper nut usually pinned to pre- 
 vent turning. The size of the aper- 
 tures in the grating are adapted to the 
 thickness of the rubber. With an ex- 
 ternal diameter of opening of 6", and 
 
 The lower nut, after 
 
 ,wga 
 
 >W G' 
 
 FIG 857. 
 
 FIG. 858. 
 
 rubber " thick, the exterior ring of openings may be f " by f ", the lands or 
 
 spaces between openings " wide, and exterior lap of the rubber % inch. With 
 
 larger diameters and larger openings thicker rubber must be used. This valve 
 
 is often made of a long strip or flap of rubber, on a suitable grating, with a 
 
 curved guard attached on one side. For 
 
 the common air-pump pressure, f" rubber 
 
 is sufficient for apertures 1" X 4". With 
 
 the use of backing and face plates on the 
 
 rubber or leather flaps, gratings may be 
 
 dispensed with (Fig. 859). Valves of this 
 
 description duplicate (Fig. 860) beneath 
 
 the central pin and half circular in plan 
 
 are often used in pumps, and are called 
 
 butterfly valves. It is the best practice to 
 
 insert thimbles in the rubber (Fig. 859), FIG. 859. 
 
376 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 and the rivets connecting the plates pass through these thimbles, so that the 
 rubber may be held but not tightly fastened a rule applicable to all such valves. 
 Check-valves (Figs. 861 and 862) are placed outside of large pumps to pre- 
 vent the return of water in cases of accident to the pumps, and for facility of 
 
 SPACE OCCUPIED BY THE 
 VALVES. 
 
 FIG. 860. 
 
 
 Measure- 
 
 Measure- 
 
 SIZE. 
 
 ment from 
 face to face 
 
 ment from 
 end to end 
 
 
 of flange. 
 
 of hub. 
 
 Inches. 
 
 
 
 4 
 
 11* 
 
 1*| 
 
 5 
 
 iif 
 
 16 
 
 6 
 
 -14| 
 
 16 
 
 8 
 
 17f 
 
 19 
 
 10 
 
 ail 
 
 24 
 
 12 
 
 24i 
 
 26| 
 
 16 
 
 29 
 
 31 
 
 18 
 
 33 
 
 35 
 
 20 
 
 35J 
 
 38 
 
 24 
 
 39f 
 
 39 
 
 their examination. Valves of this kind open from the pressure of water be- 
 neath, and, from a state of rest, with some suddenness and shock. To prevent 
 this in large valves, there is a valve and small by-pass pipe, from one -side to 
 the other of the valves, by opening which the pressure on the two sides of the 
 valve may be equalized, and the excess due to the starting of the pump dis-' 
 tributed. At many pumping works the by-pass is kept open except when 
 necessary to get at the pumps. In case of accident to the pumps the flow 
 through the by-pass would be comparatively small, and readily shut off. 
 
 FIG. 861. 
 
 FIG. 862. 
 
 Valves controlled by Hand. Fig. 863 represents a side view of a water bib- 
 cock, called a hose-bib, because the outlet end is fitted with a screw to adapt it 
 to a hose. Without this screw it is a plain bib. If both ends of the cock are 
 in the same line, it is called a stop-cock. The ends may not be fitted with 
 screws, as in the figure ; the screws are sometimes female screws, and often 
 with taper ends, to solder lead pipe to, or to drive into a cask. These cocks 
 come under the common designation of plug-cocks, from their interior con- 
 struction, which will be readily understood from the section given in Fig. 864. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 377 
 
 FIG. 863. 
 
 FIG. 864. 
 
 FIG. 865. 
 
 They are used in both steam and water pipes, but not in the former when the 
 use is frequent and daily, and then usually not over 2" in diameter of passage. 
 
 FIG. 866. 
 
 FIG. 867. 
 
 Fig. 865 is the side view of a compression water-bib, used when the pres- 
 sure of the water is great. The section is somewhat similar to that of Fig. 
 870, in which a rubber disk is forced against a metallic seat to shut off the 
 flow. 
 
 Fig. 866 is a side view of a common air-cock for boilers and steam work ; 
 they are plugs in their construction, as are the cocks used in gas-fitting ; size of 
 vent of air-cocks, to \ inch diam- 
 eter. 
 
 Fig. 867 is an air-cock in which 
 the valve is a plug of Jenkins's pat- 
 ent composition, mostly rubber and 
 graphite, on a flat seat of small sur- 
 face. 
 
 Figs. 868 and 869 are front 
 views of globe - valves, so called 
 from the shape of the body inclos- 
 ing the valve. Fig. 868 is an an- 
 gle globe- valve ; Fig. 869, a cross 
 globe- valve used for cutting off the 
 steam supply through the vertical FIG. scs. FIG. seo. 
 
378 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 pipe from the horizontal one ; the same purpose is attained by putting a 
 straightway valve on the vertical pipe. The valve seats of all globe-valves, 
 kept in stock and on sale, are now made of soft metal or of rubber, or of some 
 mixture of it vulcanized ; soft rubber for cold water, hard rubber for hot water 
 or steam. Fig. 870 is a front view of a globe- valve, with one side turned off 
 
 DIMENSIONS OF GLOBE-VALVES 
 IN COMMON USE, WITH SOFT 
 SEATS. 
 
 Diameter 
 of opening 
 in seat. 
 
 Length 
 over all. 
 
 Diameter 
 of globe. 
 
 Body 
 metal. 
 
 4 
 
 24 
 
 If 
 
 Brass. 
 
 i 
 
 2i 
 
 If 
 
 
 1 
 
 24 
 
 14 
 
 
 4 
 
 2f 
 
 2* 
 
 
 1 
 
 3f 
 
 24 
 
 
 H 
 
 4i 
 
 3 
 
 
 14 
 
 4f 
 
 if 
 
 
 a 
 
 5f 
 
 4| 
 
 
 24 
 
 8 
 
 
 Iron. 
 
 3 
 
 9f 
 
 7* 
 
 
 34 
 
 10 
 
 
 
 4 
 
 114 
 
 9* 
 
 
 5 
 
 13f 
 
 10 
 
 
 6 
 
 14| 
 
 124 
 
 
 7 
 
 16* 
 
 14 
 
 
 8 
 
 17i 
 
 14 
 
 
 c 
 
 FlG . 
 
 to show its interior construction. 
 
 The diaphragm d d is in the form 
 
 of a | l,which divides the interior 
 
 of the globe ; through the flat part 
 
 is the aperture for the passage of 
 
 the fluid covered by the valve, controlled by the handle and screw on its stem. 
 
 The arrows show the direction of flow. 
 
 Fig. 871 is a perspective of the valve, showing the grooves through which 
 it is slipped on to the head on the stem and held. The composition or rubber 
 is in the form of a ring slipped into a circular groove in the bottom of the 
 valve. In some valves the rubber ring is slid into a straight groove and re- 
 tained by a nut on the stem, and the head of the stem is held to the valve by a 
 nipple. 
 
 When the seat is of soft metal it is run into a groove in the case, faced, and 
 the valve formed with a circular chisel edge is forced down by the hand wheel 
 into the soft metal. On account of the loss of head by the change of direction 
 in flow through the valve aperture it is better to make it of a little larger 
 diameter than that of the pipe. 
 
 Figs. 872 and 873 are the plan and section of a steam valve of the South- 
 wark Foundry pattern ; the seats and faces are of metal ground to a fit. The 
 valve is guided by three wings, w w. The flow through globe-valves, as will be 
 seen by their sections, have three changes of direction ; to avoid this, straight- 
 way gates are almost invariably used on water mains. 
 
 If the double-beat valve (Fig. 846 or 848) be mounted with a screw (like the 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 379 
 
380 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 above), as the pressure on the valve may be nearly balanced throughout its 
 movement, it can be made of large area, and still be under the control of a 
 hand wheel. 
 
 GATE-VALVES. 
 
 When the pressure is always on one side of the valve, it may be made as a 
 plain gate, sliding in grooves, and raised up into a chamber ; but it is the usual 
 practice to make these gates double, each being forced positively against faces. 
 
 The earliest of these 
 forms of gates are the 
 Peet (Fig. 874) and the 
 Coffin valve (Fig. 875). 
 
 The first were usually 
 made of small sizes, and 
 for steam - pipes, it is 
 shown in the figure shut ; 
 
 FIG. 874. 
 
 the two side valves or disks are forced against their respective faces by the cone 
 suspended between the disks, forced upward by its stem coming in contact with 
 the bottom of the case ; as the gate is raised, the cone drops, the pressure is re- 
 leased, and the valve easily drawn up into the chamber above, giving an un- 
 obstructed passage through the body of the valve. 
 
 In the Coffin valve the disks are suspended by two pivots in their backs to 
 a wedge connected with the same. The wedge by its downward movement 
 forces the disks outward against the seats, while by the upward motion this 
 pressure is relieved. 
 
 In the Pratt and Cady valve (Fig. 876) the seats are of soft metal which 
 are cast in a mould and forced into position, making a tight fit with the body 
 of the valve and a seat for the valve with smooth faces. In case of a cutting 
 of the soft metal, it can be readily withdrawn and replaced by a new one. 
 
 The above forms of gates, especially those of large sizes, are used as water 
 gates. Steam-valves are mostly of the globe pattern. 
 
 Between the boilers and the steam-chest of an engine there should be a 
 valve that can be shut promptly. The simplest is the damper-valve (Fig. 877), 
 which is also used for the control of the draft in the smoke-pipe, but for a 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 381 
 
 steam-pipe it is made with a closer 
 fit; it never can be close, but 
 sufficiently so to control or slow 
 down the engine. A better valve 
 is the usual Corliss steam-cylin- 
 der valve, moved by a handle, or 
 the register - valve (Fig. 878), 
 which is held down by a spring, 
 and is opened and shut like a 
 register. These valves may be 
 connected to a governor. 
 
 Fig. 879 is a valve used in 
 Nasmyth's works for steam ham- 
 mers. Opposite the ports there 
 are false ports or slight recesses 
 in the shell. The steam enters 
 at the end of the valve into the 
 spaces a a ; the endwise pressure 
 is received by a thrust bearing. 
 This valve is so nearly balanced 
 that it is readily moved by hand. 
 To prevent excessive pres- 
 sures, either of steam or water, a 
 safety-valve is used (Fig. 880), 
 which consists of a poppet-valve 
 held down by a lever and weight. 
 To determine the weight coun- 
 terbalancing the pressure, put 
 the valve and lever in position, 
 attach the valve to the stem, and 
 with a spring balance attached to 
 the lever at its connection with 
 the stem, find how much weight it takes to lift the valve-stem and lever ; this 
 is a constant, which if divided by the area of valve at its lowest bearing diame- 
 
 Fia. 877. 
 
 FIG. 878. 
 
382 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 ter will give the constant pressure per square inch. The movable weight is the 
 
 P of a steelyard, and is to be estimated by multiplying fp by the weight and 
 
 dividing it by fs and the 
 
 area of the valve, to which 
 
 is to be added the constant 
 
 weight per square inch 
 
 found above. 
 
 FIG. 879. 
 
 FIG. 880. 
 
 The safest and simplest way is to put a blank flange on below the valve and 
 with a force pump inject water to certain pressure, as shown by a steam gauge ; 
 balance this pressure by a weight P on the lever and mark its distance from 
 the fulcrum, which will give the weight per square inch on the valve at the 
 position of the P ; in the same way determine other points on the lever. 
 
 Fig. 881 is what is termed a 
 pop safety-valve ; the steam issu- 
 ing as the valve rises, impinges on 
 a cup surface to force the valve 
 farther open. The valve is held 
 down by a spring, but can be 
 raised by the lever I. Valves of 
 this kind are often inclosed in a 
 locked box, that they may not be 
 tampered with. 
 
 . To determine the weight on 
 the spring, test it by raising the 
 pressure in the boiler, as shown by 
 the gauge, to the height which is 
 deemed safe ; adjust the valve to 
 the lifting point by the nuts on 
 the side bolts. If not in position 
 on a boiler, test by a pump. 
 
 Hydrants. For water-service 
 in connection with high-pressure 
 mains. 
 
 Fig. 882 is a section of a post- 
 hydrant. The valve v consists of 
 a series of leather disks bolted together and turned conical, which is brought in 
 contact with a corresponding seat by the valve-rod and its screw at the top of 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 383 
 
 the hydrant. The valve is opened by being forced down into the cavity of a 
 branch of the pipe-main ; n is the- nozzle for the coupling of the hose ; outside 
 
 the main pipe of the hydrant there is a 
 case, extending from near the line of valve 
 to the ground line, called the hydrant 
 or frost case, which prevents the hydrant 
 from being lifted by the frost. Were 
 the water left in the hydrant, it would 
 freeze in most exposures during winter; 
 the hydrant, when not in use, is therefore 
 kept empty. This is effected by a small 
 hole at a, which, when the valve is closed, 
 is opened, and the water in the hydrant, if 
 any, is discharged. This vent is closed by 
 a slide attached to the valve-rod, when this 
 last is moved down to open the main valve. 
 Instead of leather for the valve-face, many 
 valves are fitted with rubber ; there is also 
 a great variety of valves for hydrant pur- 
 poses slides, poppets, disks but in the 
 arrangement of hydrants the illustration is 
 the common one, although often without 
 the frost case. 
 
 FIG. 882. 
 
 FIG. 888. 
 
 Fm. 889. 
 
 Riveted Joints, as used in the Construction of Boilers. Figs. 883-889 are 
 forms of rivets with their proportions referred to the diameters next the heads. 
 The thickness of the plate connected by rivets will be given in tables here- 
 
384 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 FIG. 890. 
 
 after. Figs. 884 and 885 are the usual finish of rivets in hand-riveting ; Figs. 
 886 and 887, when made by machines, in which, as the rivet-hole is slightly 
 
 counter-sunk, the strength of the head is increased. 
 Fig. 888 is a counter-sunk rivet, the head being 
 flush with the outside of the plate. Fig. 889 is the 
 head of a rivet, in which a narrow strip at the edge 
 is burred down by a chisel, or calked, to make the 
 joint between rivet and plate tight. 
 
 Fig. 890 is a plan and section of a single-riveted 
 
 lap-joint. Joints of this kind fail from the tear of the plate on the line of 
 rivets if the rivets are too close, or the distance of the rivet to the outside of 
 the plate too small, or by the shear of the rivets if they are too few. 
 
 Rivet-holes. Punching has an injurious effect upon plates ; but this injury 
 (if the plates are not cracked by the process) is removed by afterward annealing 
 them, or by rymering or drilling to the extent of ^ inch on the diameter. 
 
 It is difficult to insure the correct spacing of the holes when they are made 
 by punching. In the best boiler work the rivet-holes are drilled after the 
 plates have been bent or flanged and put together in their proper places. This 
 insures that the corresponding holes in the different plates shall be exactly op- 
 posite to one another. After drilling, the plates are taken asunder and any 
 
 burr that has been formed at the edges of the 
 holes is removed. Riveting may be performed 
 either by hand hammering or by a machine. 
 Hydraulic riveting machines are the best. 
 
 For wrought-iron plates a tenacity of 47,000 
 pounds is estimated per square inch in the di- 
 rection of the fibre, and 40,000 pounds per 
 square inch across the fibre ; steel plates, a te- 
 nacity of 65,000 pounds per square inch. 
 
 Shearing resistance of wrought-iron rivets 
 is about equal to the tenacity of wrought-iron 
 plates ; the shearing resistance of steel rivets is 
 about eight tenths the tenacity of steel plates. 
 
 Fig. 891 shows the connection of two plates 
 by means of single-riveted lap-joint. AVhen 
 the plates are arranged as shown in the lower 
 section, the tension in the plates causes a bend- 
 ing action on them at the lap. To avoid this, 
 the plates have sometimes a set at the lap, as shown in the upper section. 
 
 DIMENSIONS OP SINGLE-RIVETED LAP-JOINTS FOR BOILER WORK. 
 
 FIG. 891. 
 
 
 IRON PLATES AND IRON RIVETS. 
 
 STEEL PLATES AND STEEL RIVETS. 
 
 THICKNESS OF PLATE. 
 
 Diam. of rivet. 
 
 Pitch. 
 
 Diam. of rivet. 
 
 Pitch. 
 
 A 
 
 1 
 
 li 
 
 
 1A 
 
 * ! 
 
 ! 
 
 il :* 
 
 l f 
 
 j2 
 
 A 
 
 H / 
 
 2i 2 
 
 1 1 J 
 
 2i g** 
 
 H * 
 
 1A 
 
 H 
 
 14 
 
 5 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 385 
 
 The efficiency of the joint in the percentage of the strength of the plate is 
 to be taken at the lowest figure, wKether in tear or shear. In the above table 
 it is 55 per cent. The distance between the edge of the plate and the centre 
 of the outer rivet, as shown in the figures by Z, is invariably one and a half time 
 the diameter of the rivet. 
 
 FIG. 892. 
 
 FIG. 893. 
 
 The arrangement of the rivets in Fig. 892 is known as zigzag riveting, while 
 that of Fig. 893 is chain riveting. 
 
 DIMENSIONS OP DOUBLE-RIVETED LAP-JOINTS. 
 
 t. 
 
 IRON PLATES AND IRON RIVETS. 
 
 STEEL PLATES AND STEEL RIVETS. 
 
 d. 
 
 P- 
 
 c. 
 
 cl. 
 
 d. 
 
 p- 
 
 c. 
 
 cl. 
 
 1 
 
 H 
 
 n 
 
 1ft 
 
 11 
 
 f 
 
 2 
 
 If 
 
 2 
 
 A 
 
 t 
 
 2f 
 
 If 
 
 2 
 
 it 
 
 2f 
 
 1ft 
 
 2i 
 
 i 
 
 H 
 
 21 
 
 1* 
 
 24 
 
 1 
 
 m 
 
 H 
 
 2i 
 
 ft 
 
 1 
 
 3 
 
 1ft 
 
 8* 
 
 H 
 
 21 
 
 1ft 
 
 2i 
 
 1 
 
 If 
 
 3i 
 
 If 
 
 2f 
 
 i 
 
 3 
 
 l| 
 
 21 
 
 to 
 
 i 
 
 3 
 
 If 
 
 21 
 
 ift 
 
 3i 
 
 lii 
 
 2| 
 
 i 
 
 ift 
 
 3ft 
 
 m 
 
 2| 
 
 H 
 
 8J 
 
 if 
 
 2f 
 
 n 
 
 H 
 
 3ft 
 
 11 
 
 2f 
 
 1ft 
 
 3| 
 
 m 
 
 21 
 
 i 
 
 1ft 
 
 3f 
 
 2 
 
 21 
 
 i 
 
 31 
 
 m 
 
 3 
 
 H 
 
 1J 
 
 31 
 
 2rV 
 
 3 
 
 1ft 
 
 8f 
 
 2 
 
 8t 
 
 i 
 
 1ft 4 
 
 2fV 
 
 3i 
 
 if 
 
 8| 
 
 2i 
 
 3J 
 
 The efficiency of joints in the above table is for iron plates 68 per cent, 
 and steel, 64 per cent. 
 
 In all riveted joints the distance between adjacent rivets, measured from cen- 
 tre to centre, whether in the same or different rows, should not be less than 2d. 
 
 On the results of experiments on riveted joints, Professor Kennedy has stated 
 that the net section of metal in the plate, measured zigzag, should be from 30 
 to 35 per cent, in excess of that measured straight across. This gives a diago- 
 
 nal pitch of . 
 
 o 
 26 
 
386 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Treble-riveted Lap-joints. In Fig. 894 the riveting is zigzag, and in Fig. 
 895 chain. 
 
 FIG. 894. FIG. 895. 
 
 DIMENSIONS OF TREBLE-RIVETED LAP-JOINTS. 
 
 t. 
 
 IRON PLATES AND IRON RIVETS. 
 
 STEEL PLATES AND STEEL RIVETS. 
 
 d. 
 
 p- 
 
 c. 
 
 cl. 
 
 d. 
 
 p- 
 
 c. 
 
 cl. 
 
 1 
 
 H 
 
 8J 
 
 If 
 
 24 
 
 i 
 
 34 
 
 If 
 
 8J 
 
 tt 
 
 i 
 
 8J 
 
 If 
 
 2i 
 
 H 
 
 3f 
 
 If 
 
 2| 
 
 1 
 
 H 
 
 3H 
 
 1* 
 
 2| 
 
 i 
 
 3* 
 
 HI 
 
 2* 
 
 If 
 
 i 
 
 8| 
 
 Itt 
 
 2* 
 
 iiV 
 
 8H 
 
 lit 
 
 2f 
 
 i 
 
 1A 
 
 44 
 
 2iV 
 
 H 
 
 H 
 
 31 
 
 2 
 
 24 
 
 it 
 
 1* 
 
 4A 
 
 2A 
 
 21 
 
 1& 
 
 4 
 
 2A 
 
 21 
 
 i 
 
 1A 
 
 4i 
 
 2J 
 
 2i 
 
 ii 
 
 *ft 
 
 2A 
 
 3 
 
 IA 
 
 H 
 
 4tt 
 
 2f 
 
 3 
 
 1A 
 
 4f 
 
 2 
 
 34 
 
 H 
 
 iA 
 
 *i 
 
 2i 
 
 3i 
 
 if 
 
 4* 
 
 2f 
 
 8J 
 
 The efficiency of joints for iron plates is 74 per cent, and that of steel 70 per 
 cent. The strength of a treble-riveted joint (Figs. 896 and 897) may be increased 
 by making the pitch of the inner row of rivets one half that of the outer. 
 
 d = 1-27 1 for iron plates and iron rivets. 
 
 d = l-59 for steel plates and steel rivets. 
 
 The pitch of a quadruple lap-joint will be the same as in the last example. 
 
 A butt joint with a single cover-strap (Fig. 898) is composed of two lap- 
 joints, and is proportioned by the rules previously given for lap-joints. With 
 this form of joint, the tension on the plates will tend to bend the cover-strap. 
 For that reason the cover-strap is made thicker than the plates. If ^, = thick- 
 ness of cover-strap and t thickness of plates, then #, = 1#. For single- 
 riveted butt-joints (Fig. 899), with double cover-straps, the usual rule for the 
 thickness of each butt-strap is t 1 = f t. 
 
 The diameter of the rivets for different thicknesses of plates may be as fol- 
 lows : 
 
 d = t -\- \ for iron plates and iron rivets. 
 
 d = t -j- ^ for steel plates and steel rivets. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 387 
 
 FIG. 901. 
 
388 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 Double-riveted Suit-joints with Double Cover-straps (Fig. 900). The pitch 
 of the rivets are the same in each row. If alternate rivets in the outer rows be 
 removed, as in Fig. 901, the arrangement is stronger. 
 
 The diameter of the rivets (Fig. 900) for different thicknesses of plates may 
 be as follows : 
 
 d = t -f- T ^- for iron plates and iron rivets. 
 
 d = t -\- % for steel plates and steel rivets. 
 
 The diameter of the rivets (Fig. 901) for different thicknesses of plates may 
 be as follows : 
 
 d = t + for iron plates and iron rivets. 
 
 d = t + y^- f or steel plates and steel rivets. 
 
 Fig. 902 is a triple-riveted butt-joint with double cover-plate, as butt-joints 
 with double covers, one on each side of the plates, increase the shearing resist- 
 
 FIG. 902. 
 
 FIG. 905. 
 
 ance of the rivets, so that rupture always takes place in the plates ; and as these 
 can not bend, and there is considerable frictional resistance between the plates, 
 the strength of the joint has been found to be more than that due to the net 
 section of the plates between the rivets. 
 
 Fig. 903 is a plan and section of a combined lap- and butt-joint. The pitch 
 of the exterior rows is double that of the central one ; for a f" plate, 4" for the 
 former and 2" for the latter. 
 
 Fig. 904 is a section of joint, showing a better arrangement than Fig. 903, 
 requiring less work, more easily calked, and of as much strength. 
 
 Fig. 905 is the plan and section of a butt-joint when the cover is of T-iron 
 a not uncommon form of strengthening flues to resist collapse. 
 
 Junction of more than Two Plates, shown in Plans and Sections (Figs. 906, 
 907, and 908). These become necessary when cross-joints intersect longitudi- 
 nal ones. At these joints one or more of the plates are thinned or drawn out 
 by forging. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 389 
 
 Fig. 909 is the plan .and section of an angular connection of plates by the 
 means of angle-iron ; this should be a little thicker than the plates, and its 
 width four times the diameter of the rivets. 
 
 C 
 
 .kj iv 
 
 ryl K ^ 
 
 IxxXl [^3 
 
 FIG. 906. 
 
 FIG. 907. 
 
 FIG. 908. 
 
 Figs. 910, 911, and 912 are sections of angular connections by flanging the 
 plates. The iron should be good and the curvature easy ; inside radius at least 
 four times the thickness of the plates. 
 
 FIG. 909. 
 
 FIG. 910. 
 
 FIG. 911. 
 
 FIG. 912. 
 
 Figs. 913 and 914 are sections of joints of cylinders of unequal diameters, 
 or surfaces not in line with each other. 
 
 Figs. 915, 916, and 917 are sections of fire-box legs. 
 
 ) ( 
 
 FIG. 913. 
 
 FIG. 914. 
 
 FIG. 915. 
 
 FIG. 916. FIG. 917. 
 
 In all connections provisions are to be made for the means of holding the 
 head of the rivet, and for riveting and for calking the joints. 
 
 Fig. 918 is the perspective view of a horizontal tubular boiler, very largely 
 used with anthracite as a fuel, but with bituminous coal the tubes should be of 
 the larger diameters. 
 
 The proportions of the boiler vary with the requirements of their position, 
 and with the views of the mechanical engineer or maker constructing them. 
 Many use a dome, but it is the better practice to increase the diameter of the 
 boiler an inch or two for more steam space, if necessary, and insert a dry pipe 
 in the space. Those in most extensive use are with shells of 4 to 5 feet inside 
 diameter and 3" to 3" tubes, 14 to 16 feet long. The line of the top of the 
 upper tubes is usually about y 1 ^ of the diameter of the boiler above its centre ; 
 tubes arranged in vertical rows, with distance between tubes $ of their diam- 
 
390 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 eter. By keeping the average distance the same, but making them farther 
 apart at the top row, say % diameter, and the lowest diameter, so that the line 
 of tubes is radial instead of vertical, the outside of the tube will meet the flame 
 better, and at the same time be more readily cleaned with a brush. 
 
 FIG. 918. 
 
 The following table is from Barr, showing the greatest number of tubes 
 which should be put in a given head, no tube to come nearer to the shell than 
 2" for boilers of small diameter, 2" for medium, and 3" for the larger series : 
 
 Diameters of 
 
 
 bodies inside, 
 in inches. 
 
 3 in. 
 
 3iin. 
 
 3i in. 
 
 3} in. 
 
 4 in. 
 
 4} in. 
 
 5 in. 
 
 36 
 
 26 
 
 23 
 
 20 
 
 19 
 
 16 
 
 12 
 
 10 
 
 40 
 
 34 
 
 34 
 
 25 
 
 23 
 
 20 
 
 14 
 
 14 
 
 44 
 
 48 
 
 36 
 
 32 
 
 25 
 
 25 
 
 20 
 
 16 
 
 48 
 
 50 
 
 38 
 
 36 
 
 30 
 
 26 
 
 21 
 
 18 
 
 52 
 
 57 
 
 50 
 
 48 
 
 38 
 
 32 
 
 26 
 
 21 
 
 56 
 
 72 
 
 57 
 
 55 
 
 48 
 
 41 
 
 32 
 
 23 
 
 60 
 
 80 
 
 68 
 
 62 
 
 55 
 
 46 
 
 36 
 
 30 
 
 A is the man-hole, to enable the mechanic to get into the boiler to examine 
 it. It consists of a cast-iron frame, bolted to the shell of the boiler, with an 
 elliptical opening usually 9" X 15" in the clear ; the valve laps about 1" on each 
 side. In closing the opening the valve is passed down into the boiler, and is 
 brought up against the valve-seat, where it is held by its stem passing up 
 through a movable yoke, and brought up tight by a nut and screw. The joint 
 is made with a gasket or with sheet-rubber. The man -hole is often placed in 
 one of the boiler heads, or at one side, above the tubes, for convenience of 
 access. B is the hand-hole, of the same general construction as the man-hole, 
 but smaller, to enable the fireman to clean the boiler. Formerly this hand- 
 hole was quite small, but of late the practice is to place a hand-hole at the rear 
 end of the boiler and a man-hole at the front in the position B. This is for the 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 391 
 
 readier cleaning and repairs of the boiler ; it reduces the number of tubes by 
 four, but without detriment to the ^evaporation of the boiler ; and by taking off 
 both covers one can look directly through the boiler. As the hand-hole is ex- 
 posed to the flame and products of combustion, it is well to make it small, say 
 3" X 5" ; III are lugs by which the boilers are supported on the brickwork, 
 but they are in the way in getting the boiler through a confined space, and 
 rest so solidly on the brickwork that it often becomes cracked by the expansion 
 of the boiler. It is preferable that the boiler should be hung as in Fig. 
 1252. In the head above the tubes there are rivet-heads, and also in the sides 
 back of the first seams at each end. These are for the attachment of diagonal 
 stays. The tubes themselves serve as stays in the lower part of the boiler, but 
 above, the flat surface needs something to prevent the head from moving out 
 under pressure. The stays are made of round or flat iron (see Fig. 1249), 
 bolted directly to the shell, the round part being flattened, and connected by 
 a yoke and pin to a crow-foot or piece of angle-iron attached to the head. 
 The stays are from f " to 1^" diameter or equivalent sections. 
 
 To determine the diameter of stays in square inches multiply the area sup- 
 ported by the stays and divide the product by 7,000 for wrought-iron stays not 
 welded ; and for steel stays, under same condition, by 9,000 ; but if welded or 
 otherwise worked after heating, take three fourths of above. 
 
 Forms of Boiler Stays. Fig. 919 is a direct stay in which a hole is drilled 
 through the head of a boiler ; the stay has an outside nut of a thickness equal 
 
 tt-J 
 
 FIG. 919. 
 
 FIG. 920. 
 
 FIG. 922. 
 
 to the diameter of the screwed part, and the inside 
 
 or locked nut three fourths of this thickness. The 
 
 plates are stiffened by inside and outside washers. 
 
 If the stay is diagonal it is usual to increase its area in the proportion of the 
 
 length of the diagonal to that of the horizontal. 
 
 The flat parallel surfaces surrounding the fire-boxes of locomotive and 
 marine boilers are secured by means of screwed stays, so called because they 
 are screwed into the plates (Figs. 920, 921, and 922). After being screwed 
 into the plates their ends are riveted. The fracture of the stays is detected by 
 the escape of steam through the small holes which are sometimes drilled through 
 the screwed parts. The screwed stays for locomotive boilers are usually placed 
 about 4 inches apart, centre to centre, and vary in diameter from f- inch to 1 
 inch. 
 
 In marine boilers the screwed stays are made of steel, and they vary in 
 diameter from 1^- inch to If inch. They are provided with washers and 
 nuts at each end, as shown at Fig. 919. The nuts have a thickness of from 
 five eighths to three fourths the diameter of the stay. 
 
392 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 FIG. 923. 
 
 In the steel-screwed stays the ends of these stays 
 are drilled as shown, and after they are screwed into 
 place a steel drift is driven into the holes by slight 
 blows to expand the ends tightly into the plates to 
 make steam-tight joints. 
 
 Fig. 923 is a gusset stay, used in angles, consist- 
 ing of a triangular plate with the edges flanged and 
 riveted to the shell. 
 
 BARK'S PROPORTIONS FOR STAY-BOLTS FOR FLAT SURFACES. 
 
 CENTRE TO CENTRE OF STAY-BOLTS IN SQUARE INCHES. 
 
 JL 1 l^ODlll V3 jyt'A 
 
 square inch. 
 
 \" plate. 
 
 T V' plate. 
 
 f" plate. 
 
 T y plate. 
 
 i" plate. 
 
 60 
 
 5f 
 
 61 
 
 7* 
 
 8* 
 
 9 
 
 80 
 
 4f 
 
 5 
 
 8* 
 
 7* 
 
 71 
 
 100 
 
 4i 
 
 4f 
 
 5J 
 
 6| 
 
 7 
 
 120 
 
 81 
 
 4i 
 
 5 
 
 H 
 
 6f 
 
 140 
 
 8| 
 
 4* 
 
 4f 
 
 5J 
 
 6 
 
 00 
 
 oooo 
 
 'OOOOOO 
 
 oooooo 
 
 OOOOOO 
 
 oooooo; 
 ooo. 
 
 poooo 
 
 oooooo , 
 ooooooo 
 ooooooo 
 ooooooo 
 
 OOOOOOOj 
 
 oooooo. 
 
 Stationary boilers as 
 designed and built un- 
 der the direction of 
 John E. Codman, M. E., 
 for the Philadelphia 
 Water- Works, of which 
 Fig. 924 is a transverse 
 section and one half 
 cross - section through 
 fire-box, and one half 
 front view without the 
 doors. Fig. 925 is a 
 longitudinal section. 
 These boilers have inside 
 fire-boxes, and the out- 
 side is protected by a 
 covering of brick or 
 some clothing of a non- 
 conducting material to 
 prevent radiation. The 
 corrugated furnaces are 
 made in this country 
 by the Continental Iron 
 Works, Brooklyn, of an 
 inside diameter of from 
 30" to 60" and up to 32 
 feet long. 
 
 Kules for calculat- 
 ing the pressure allowa- 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 393 
 
394 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 395 
 
 ble on corrugated furnaces adopted by the Board of United States Supervising 
 Inspectors : 
 
 Corrugations to be 6 inches pitch and 1 inch deep. 
 
 14000 _ , . 
 
 y^ X T = working pressure in pounds per square inch. 
 
 T = thickness in inches. 
 
 D = mean diameter in inches. 
 
 Example. Given a corrugated 
 furnace 40 inches mean diameter 
 to carry 175 pounds working pres- 
 sure, required the thickness of 
 metal. 
 
 P. XD. 
 
 = thickness. 
 
 175 X 40 
 
 FIG. 927. 
 
 14000 14000 
 
 = | inch thickness of metal. 
 
 Figs. 926 to 929 are drawings 
 of a locomotive boiler as designed 
 and constructed by Mr. Buchanan 
 for express passenger locomotives 
 for the N. Y. C. and H. R. R. R, 
 Fig. 926 is a longitudinal section 
 Fig. 927 a transverse section of 
 one half the fire-box and an eleva- 
 tion of one half of that end. Fig. 
 928 of the fire-box with cover off, 
 and showing one half of the tubes. 
 Fig. 929 are details of 'the riveting. 
 
 Figs. 930 and 931 are drawings 
 of a marine boiler of the United 
 States steamer Minneapolis, show- 
 ing longitudinal and cross-section 
 of fire-box end. When locomo- 
 tive or marine boilers are used as 
 stationary their outsides should be 
 protected as the Codman boiler 
 (page 392). 
 
 Water-tube boilers are now in 
 extensive use, economical in evap- 
 oration, and popular from the 
 comparative safety from explosion. 
 Some of the numerous and varied 
 forms will be found illustrated in 
 the Appendix. 
 
 Flue Boilers. Where bitumi- 
 nous coal is used, small tubes be- 
 come clogged with soot; it was 
 therefore customary to construct boilers with large tubes or flues of boiler-iron 
 riveted together, which sometimes failed from collapse. It may be considered 
 
 -A.. 
 
396 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 ample to make the tubes subject to outside stress 50 per cent thicker than for 
 bursting, especially for the large drawn tubes now made. Mr. Fairbairn, from 
 his experiments, considered it necessary to make the joints of tubes subject to 
 collapse as in Figs. 932 and 933. 
 
 1 Spaces in bottom Row 
 "- ftTPAtcB Bolts in Ring li'deep 
 
 3 Equal Spaces in bottom Ro 
 j'Patch bolts Ij'deep in RiJg 
 
 FIG 929. 
 
 Fig. 934 is a section of the Shapley boiler, as made by the Knowles Steam- 
 Pump Works a good form of upright boiler, with the head of the boiler 
 stayed by rods directly to the crown-sheet, beneath which short tubes or nip- 
 ples connect the fire-box with a cast iron smoke-box around the boiler and the 
 draft is downward through vertical tubes to a smoke-box in the base. The 
 crown-sheet and downward draft tubes are well covered by water. It is an ad- 
 mirable illustration for the draughtsman of how a boiler in action may be 
 represented. 
 
 The usual form of upright boiler consists of a fire-box, extending a little 
 above the door, and tubes extending from the crown-sheet to the top-head, 
 over which there is a bonnet to receive the smoke, which is led off by a smoke- 
 
MACHINE DESIGN AND 'MECHANICAL CONSTRUCTIONS. 
 
 397 
 
 
 >Hfl 
 
 ':^-''-~'~~~- booba!.| 
 - r--' oooooooooci.'h'x 
 
 oiOOOOOOOOOOOOQV^V 
 o!0000000000000!N3* 
 
 o'cooooooooooocj.:' 
 
 oiOOOOOOOOOOOOO*.* 
 
 !oooocoooooooo!r 
 
 o !OOOOOOOOOOOOOir 
 o|0 00000 OOOOOOOi.*' 
 |OOOOOOOOOOOOO|.' 
 
 o!ooooooooooooo!; 
 e' -?-?^2 o -9j5P_P_9 Q.QV*: 
 
 o [6 o"6o o o o o o o o 5 cjoTV 
 
 "^oiOOOOOOOOOOOOOOOOO/'vO 
 
 o'Oooooooooooooooojaoodi. 
 
 ? !OOOOOOOOOOC|OOOOQfOOOO''. 
 |OOOOOOOOOOOpOOQOOOOO . 
 
 iooooooooooolooqpooooo'': 
 
 o lOOooOOOOOOOpOGSOOOOOOiI* 
 
 o b oooo oo 0000096000000!-: 
 Oolooooooooooolo/jooooooo j;. 
 
 IQQ.QQ.QP ooo oop 
 
 4 iOQ D.Q.OJ OO O OO C 
 
 lO'OO'O'Oo'oooooo 
 p-o-ooe-q ooo 
 
 oooooooooo 
 
398 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 pipe. It is a convenient form for furnishing steam for a small power, but not 
 as economical in combustion, and apt to prime that is, take up water with 
 the steam, and leak at the top of the tube exposed in the smoke-box. 
 
 The common vertical boilers are 
 from 2 feet 6" to 4 feet 6" outside 
 diameter of shell, with water space in 
 legs of 2" to 3" ; extreme height of 
 boiler from 2 to 2|- times the outside 
 diameter of fire-box ; tubes from 2" to 
 2" diameter, and spaced from 1" to 
 1" apart Water-line from 10" to 15" 
 above crown-sheet. 
 
 There is supposed to be a propor- 
 tion between the tube sectional area 
 and the grate-surface, say from ^ in 
 the horizontal to in the verti- 
 cal ; but this rule is entirely em- 
 pirical. There is also a propor- 
 tion of grate to heating surface ; 
 but only the same class of boilers 
 can be compared with each other, 
 as fire-box surface and that ex- 
 posed directly to the flame is 
 much more effective than that of 
 the tubes, and the products of 
 
 FIG. 932. 
 
 FIG. 934. 
 
 FIG. 
 
 combustion escape at much different temperatures in different boilers. 
 
 Flange Connections for Steam and Water Pipe. Fig. 935 is a section of a 
 flanged connection of a cast-iron pipe of the most usual form, but some thick- 
 en or re-enforce the pipe a little for 1" to 2" in length next the flange ; but if 
 there is a good fillet in the angle of the flange it is unnecessary. 
 
 The flanges are almost invariably faced, and joints made with red and white 
 lead, or a sheet-rubber washer, or with corrugated copper gaskets (Fig. 936) 
 of very thin sheet copper, which are used of full diameter of flanges on rough 
 boiler joints and red-lead putty ; but for faced surfaces thin paint will insure a 
 perfect joint inside the bolts. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 399 
 
 FIG. 935. 
 
 FIG. 936. 
 
 DIMENSIONS OF PIPE FLANGES AND CAST-IRON PIPES. 
 
 J. E. Codman, M. E., Pro. Engrs. Club, Philadelphia. 
 
 Diam. 
 of 
 pipe. 
 
 DIAMETER OP 
 FLANGE. 
 
 Diameter 
 of bolt 
 circle. 
 
 Diam. 
 of 
 bolt. 
 
 No. of 
 bolts. 
 
 Thick- 
 ness of 
 flange. 
 
 THICKNESS OF PIPE. 
 
 Weight per 
 foot with- 
 out flange. 
 
 Weight 
 of flange 
 and bolts. 
 
 Diagr. 
 
 Form. 
 
 Inches. 
 
 Dec. 
 
 2 
 
 k 
 
 a* 
 
 44 
 
 i 
 
 4 
 
 1 
 
 1 
 
 0-373 
 
 6-96 
 
 4-41 
 
 3 
 
 71 
 
 
 H 
 
 f 
 
 4 
 
 1 
 
 H 
 
 0-396 
 
 11-16 
 
 5 93 
 
 4 
 
 9 
 
 8| 
 
 7 
 
 i 
 
 6 
 
 h 
 
 1% 
 
 0-420 
 
 15-84 
 
 7-66 
 
 5 
 
 9* 
 
 
 8 
 
 i 
 
 6 
 
 f 
 
 A 
 
 0-443 
 
 21-00 
 
 9-63 
 
 6 
 
 lOf 
 
 11 
 
 H 
 
 f 
 
 8 
 
 * 
 
 rt 
 
 0-466 
 
 26-64 
 
 11-82 
 
 8 
 
 18} 
 
 .... 
 
 ill 
 
 * 
 
 8 
 
 If 
 
 i 
 
 0-511 
 
 39-36 
 
 16-91 
 
 10 
 
 li 
 
 154 
 
 18* 
 
 t 
 
 10 
 
 i 
 
 A 
 
 0-557 
 
 54-00 
 
 23-00 
 
 12 
 
 17| 
 
 
 15f 
 
 1 
 
 12 
 
 if 
 
 H 
 
 0-603 
 
 70-56 
 
 30-13 
 
 14 
 
 20 
 
 
 18 
 
 1 
 
 14 
 
 i 
 
 H 
 
 0-649 
 
 89-04 
 
 38-34 
 
 16 
 
 22 
 
 22* 
 
 20 
 
 1 
 
 16 
 
 iA 
 
 H 
 
 0-695 
 
 109-44 
 
 47-70 
 
 18 
 
 24 
 
 24* 
 
 22* 
 
 i 
 
 16 
 
 11 
 
 f 
 
 0-741 
 
 131-76 
 
 58-23 
 
 20 
 
 27 
 
 26f 
 
 24i 
 
 1 
 
 18 
 
 1A 
 
 M 
 
 0-787 
 
 156-00 
 
 70-00 
 
 22 
 
 88f 
 
 29 
 
 26* 
 
 1 
 
 20 
 
 li 
 
 H 
 
 0-833 
 
 182-16 
 
 83-05 
 
 24 
 
 81* 
 
 
 28| 
 
 1 
 
 22 
 
 1A 
 
 i 
 
 0-879 
 
 210-24 
 
 97-42 
 
 26 
 
 33* 
 
 33J 
 
 31 
 
 1 
 
 24 
 
 if 
 
 II 
 
 0-925 
 
 240-24 
 
 113-18 
 
 28 
 
 35J 
 
 35f 
 
 33i 
 
 1 
 
 24 
 
 i* 
 
 14 
 
 0-971 
 
 272-16 
 
 130-35 
 
 30 
 
 38 
 
 .... 
 
 35i 
 
 1 
 
 26 
 
 1A 
 
 1 
 
 1-017 
 
 306-00 
 
 149-00 
 
 32 
 
 40 
 
 40* 
 
 37| 
 
 H 
 
 28 
 
 if 
 
 1A 
 
 1-063 
 
 341-76 
 
 169-17 
 
 34 
 
 42* 
 
 
 40 
 
 11 
 
 30 
 
 itt 
 
 11 
 
 1-109 
 
 379-44 
 
 190-90 
 
 36 
 
 45 
 
 44f 
 
 42 
 
 H 
 
 32 
 
 it 
 
 16 
 
 1-155 
 
 419-04 
 
 214-26 
 
 38 
 
 47 
 
 
 44 
 
 H 
 
 32 
 
 lie 
 
 1A 
 
 1-201 
 
 460-56 
 
 239-27 
 
 40 
 
 49 
 
 49 
 
 46 
 
 H 
 
 34 
 
 11 
 
 ii 
 
 1-247 
 
 504-00 
 
 266-00 
 
 42 
 
 51* 
 
 5H 
 
 48i 
 
 H 
 
 34 
 
 HI 
 
 1A 
 
 1-293 
 
 549-36 
 
 294-49 
 
 44 
 
 53$ 
 
 .... 
 
 50i 
 
 li 
 
 36 
 
 2 
 
 i 
 
 1-339 
 
 596-64 
 
 324-78 
 
 46 
 
 55$ 
 
 56 
 
 52f 
 
 11 
 
 38 
 
 A 
 
 if 
 
 1.385 
 
 645-84 
 
 356-94 
 
 48 
 
 58 
 
 58i 
 
 55 
 
 li 
 
 40 
 
 2i 
 
 1A 
 
 1-431 
 
 696-96 
 
 391-00 
 
 Fig. 937 is a section of the joint used by Sir William Armstrong for the 
 pipes of his accumulator. For a working pressure of 800 pounds per square 
 inch, pipes of 5" diameter are made 1" thick and tested to 3,000 pounds per 
 square inch. The flange is elliptical, and there are but two bolts ; one pipe 
 slightly enters the other, forming a dovetailed recess in which is placed a gutta- 
 percha ring " in diameter. 
 
 Figs. 938 and 939 are sections of two other forms of cast-iron flanged pipes, 
 both with projections fitting into grooves. The packing in Fig. 939 is a ring 
 of lead. In Siemens's air reservoirs, where the pressure sustained by steel rings 
 is 1,000 pounds per square inch, the joint is made by turning a V-groove in 
 
400 
 
 MACHINE DBSiaN AND MECHANICAL CONSTRUCTIONS. 
 
 the face of the rings, and placing in it a ring of annealed copper 1 5 F " diameter. 
 This form is adopted by many mechanics for forming flanged joints even for 
 steam purposes. 
 
 FIG. 937. 
 
 FIG. 938. 
 
 FIG. 939. 
 
 Figs. 940, 941, and 942 are steam-pipe joints, as used at the works of 
 Narragansett Electric Lighting Company, where it is essential to maintain 
 
 the 
 the 
 
 CALKING EDGE 
 
 FIG. 940. 
 
 FIG. 941. 
 
 FIG. 942. 
 
 full boiler pressure permanently. Fig. 940 is a joint between two wrought-iron 
 
 pipes; Fig. 941 that between a wrought-iron and cast-iron pipe. In "both these 
 the joints are calked. In Fig. 942, between two cast- 
 iron pipes, the joint is made by a gasket of vulcanized 
 asbestos placed in a recess of the female joint. 
 
 Fig. 943 is a connection between wrought-iron plates, 
 in which the joint is made by a copper ring brazed to- 
 gether. 
 
 Wrought-iron Pipe Connections. With the present 
 cost of wrought-iron pipes, they are almost invariably 
 used for the conveyance of steam, but are more liable 
 to rust for water purposes than cast iron. Wrought- 
 iron pipes are either butt-welded or lap-welded. It is a 
 
 mere question of manufacture. It is difficult to make a lap-welded tube less 
 
 than 1" diameter, and therefore below this size they are usually butt-welded ; 
 
 but this size and above, lap- welded. 
 
 Wrought-iron pipes of the smaller 
 
 diameters are connected by socket-sleeve 
 
 couplings (Fig. 944), of wrought-iron, 
 
 of large diameters, by cast-iron flanges 
 
 screwed to the ends of the pipes to be 
 
 coupled. The screw in the coupling is 
 
 tapped parallel usually, but the ends of 
 
 FIG. 943. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 401 
 
 the tubes are cut with a taper thread, uniform with all makers, of 1 in 32 to 
 the axis. The length of the screwed portion varies with the diameter. 
 
 FIG. 944. 
 
 FIG. 945. 
 
 Fig. 945 is the longitudinal section of tapering tube-end with the screw 
 thread as actually formed, and considered standard by the late Eobert Briggs, 
 C. E., in his " Treatise on Warming Buildings by Steam." It is shown in the 
 figure double full size for a nominal 2" tube. 
 
 DIMENSIONS OF WROUGHT TUBES AND COUPLINGS. 
 
 DIAMETER OF TUBE. 
 
 CIRCUMFERENCE. 
 
 SCREWED ENDS. 
 
 Weight 
 per foot 
 in length. 
 
 COUPLINGS. 
 
 Nomi- 
 nal in- 
 side. 
 
 Actual in- 
 side. 
 
 Actual out- 
 side. 
 
 Inside. 
 
 Outside. 
 
 No. of 
 threads 
 perend. 
 
 Length of 
 screw. 
 
 Outside 
 diameter. 
 
 Length. 
 
 In. 
 
 In. 
 
 In. 
 
 ID. 
 
 In. 
 
 lu. 
 
 In. 
 
 Us. 
 
 In. 
 
 ID. 
 
 i 
 
 0-27 
 
 0-41 
 
 0-85 
 
 1-27 
 
 27 
 
 0-19 
 
 0-24 
 
 0-55 
 
 1 
 
 i 
 
 0-36 
 
 0-54 
 
 1-14 
 
 1-70 
 
 18 
 
 0-29 
 
 0-42 
 
 0-70 
 
 1 
 
 1 
 
 0-49 
 
 0-67 
 
 1-55 
 
 2-12 
 
 18 
 
 0-30 
 
 0-56 
 
 0-83 
 
 1 
 
 4 
 
 0-62 
 
 0-84 
 
 1-96 
 
 2-65 
 
 14 
 
 0-39 
 
 0-84 
 
 1-01 
 
 1A 
 
 1 
 
 0-82 
 
 1-05 
 
 2-59 
 
 3-30 
 
 14 
 
 0-40 
 
 1-13 
 
 1-24 
 
 if 
 
 1 
 
 1-05 
 
 1-31 
 
 3-29 
 
 4-13 
 
 1H 
 
 0-51 
 
 1-67 
 
 1-53 
 
 if 
 
 li 
 
 1-38 
 
 1-66 
 
 4-33 
 
 5-21 
 
 114 
 
 0-54 
 
 2-26 
 
 1-89 
 
 i| 
 
 14 
 
 1-61 
 
 1-90 
 
 5-06 
 
 5-97 
 
 114 
 
 0-55 
 
 2-69 
 
 2-17 
 
 2 
 
 2 
 
 2-07 
 
 2-37 
 
 6 49 
 
 7-46 
 
 llj 
 
 0-58 
 
 3-67 
 
 2-68 
 
 2i 
 
 24 
 
 2-47 
 
 2-87 
 
 7-75 
 
 9-03 
 
 8 
 
 0-89 
 
 5-77 
 
 3-19 
 
 2f 
 
 3 
 
 3-07 
 
 3-50 
 
 9-64 
 
 11-00 
 
 8 
 
 0-95 
 
 7-55 
 
 3-87 
 
 3 
 
 34 
 
 3-55 
 
 4-00 
 
 11-15 
 
 12-57 
 
 8 
 
 1-00 
 
 9-06 
 
 4-40 
 
 3 
 
 4 
 
 4-03 
 
 4-50 
 
 12-65 
 
 14-14 
 
 8 
 
 1-05 
 
 10-73 
 
 4-99 
 
 O l 
 Of 
 
 44 
 
 4-51 
 
 5-00 
 
 14-15 
 
 15-71 
 
 8 
 
 1-10 
 
 12-49 
 
 5-49 
 
 8| 
 
 5 
 
 5-04 
 
 5-56 
 
 15-85 
 
 17-47 
 
 8 
 
 1-16 
 
 14-56 
 
 6-19 
 
 
 6 
 
 6-06 
 
 6-62 
 
 19-05 
 
 20-81 
 
 8 
 
 1-26 
 
 18-77 
 
 7-24 
 
 3f 
 
 7 
 
 7-02 
 
 7-62 
 
 22-06 
 
 23-95 
 
 8 
 
 1-36 
 
 23-41 
 
 8-36 
 
 4 
 
 8 
 
 7-98 
 
 8-62 
 
 25-08 
 
 27-10 
 
 8 
 
 1-46 
 
 28-35 
 
 9-49 
 
 4 
 
 9 
 
 9-00 
 
 9-69 
 
 28-28 
 
 30-43 
 
 8 
 
 1-57 
 
 34-08 
 
 10-54 
 
 4i 
 
 10 
 
 10-02 
 
 10-75 
 
 31-47 
 
 33-77 
 
 8 
 
 1-68 
 
 40-64 
 
 11-72 
 
 5 
 
 Figs. 946, 947, 948, also from Briggs's treatise, give the dimensions of the 
 parts of elbows, tees, crosses, and branches. Fig. 947 shows the parts of an 
 elbow designated by letters in Fig. 940, and Fig. 948 shows the applicability of 
 the same to tees and crosses. The scale is one quarter full size ; if much used, 
 it would be better for the draughtsman to construct one of full size. The 
 dimensions are obtained by measuring from the base or zero to the inclined 
 lines, on ordinates corresponding to the inside diameter of pipe required. 
 
 When pipes are thus put together in lengths, with couplings, it is frequently 
 impossible to take out a length of pipe for repairs or alterations without break- 
 27 
 
402 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 ing a coupling or fitting ; provision is made for disconnections by the insertion 
 of a union or unions in the line. 
 
 FIG. 946. 
 
 Fig. 949 is an exterior view, and Fig. 950 a section, of the common mal- 
 leable-iron union ; p and p' are the halves into which the tube is screwed, and 
 the joint is made by a male and female coupling. The male, b, turning on a 
 flange on the tube jo, is screwed to the other half of the coupling, and the joint 
 is made tight by a rubber washer, shown in black. These unions are used only 
 
 FIG. 949. 
 
 FIG. 951. 
 
 in the smaller sizes of pipes. The flange coupling (Fig. 951) is preferred by 
 most fitters, and they are made of diameters up to 14" ; the thickness is about 
 one half that of the length of a coupling of the same diameter. The bolts are 
 from -" to f ", and spaced somewhat larger than that given for cast-iron flanges. 
 The width of flange is such as to admit of the head and nut of the bolt without 
 projection beyond the edge of the flange. 
 
 Fig. 952 is a common cast-iron flange, and with about the same proportions 
 as in Fig. 951. When the lines are long, and provision can not be made by 
 bends for the expansion and contraction of pipes under changes of temperature, 
 a fitting like a stuffing-box is often used, the end of one of the tubes being 
 attached to the box, and the other sliding in and out like a piston-rod ; some- 
 times expansion is permitted by two flexible flanges, admitting of a sort of bel- 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 403 
 
 lows-like movement; sometimes by^a U-connection between pipes, as in Fig. 
 933, or a succession of corrugations. 
 
 Fig. 953 is a soldering union ; the ring, &, is like that of the male coupling 
 (Fig. 950), which is screwed directly to the wrought-iron pipe, while a is a 
 brass tube, with a shoulder on the bottom on which the coupling, a, turns, and 
 a lead pipe is soldered to the tube. If it is not necessary to break the joints, a 
 soldering nipple (Fig. 954) only is necessary, one end of which is screwed into 
 the wrought-iron pipe, and the other soldered to the lead pipe. 
 
 FIG. 952. 
 
 FIG. 953. 
 
 FIG. 954. 
 
 FIG. 955. 
 
 Fig. 955 is a close nipple ; Fig. 956 is a shoulder nipple. 
 
 If the uncut part of the tube is longer than in the figure, it is called a long 
 nipple ; they serve the purpose of short pipes. 
 
 Fig. 957 is a bushing. There is a thread cut inside. It is screwed into a 
 coupling, and the pipe that is screwed into the bushing must be smaller in 
 diameter than that connected with the coupling. The service of the bushing 
 
 FIG. 956. 
 
 FIG. 957. 
 
 FIG. 958. 
 
 FIG. 959. 
 
 is to connect pipes of different diameters, but the reduction of one side or arm 
 of a coupling tee, or cross is better. 
 
 Fig. 958 is a plug to close up the end of a pipe by screwing it into the 
 coupling ; caps are used for the same purpose ; half-couplings with one end 
 closed, or blank flanges that is, flanges covering the aperture in the pipe 
 bolted to a flange on the end of a pipe. 
 
 It will be seen by Fig. 947 that the cast-iron elbow makes a very short turn, 
 with considerable obstruction to the flow of the fluid through it. Fig. 959 is 
 an elbow in which the obstruction is very much reduced. It consists of a 
 piece of wrought-iron pipe curved to an easy radius ; and, as a general rule, it 
 may be said that for the connection of pipes not in a line with each other, it is 
 better to bend the pipe, if possible, than make angles by cast-iron elbows. 
 
 Figs. 960, 961, and 962 are oblong, spiral, and flat coils, showing the extent 
 to which pipe can be bent by machinery, and are used largely for heaters and 
 in refrigerating plants. 
 
404 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 
 FIG. 960. 
 
 Figs. 963 and 964 are a tee and a cross, as used in connections of hydraulic 
 presses, made of composition. The tubes are of wrought-iron, extra thick. 
 The usual dimensions for such are as follows : 
 
 Outside diameter. 
 Inside diameter . 
 
 de diameter f " .*-" J 
 
 n Da 
 
 The joints are made by leather washers, square ends on square seats. 
 
 ^-.----^ 
 
 FIG. 965. 
 
 FIG. 966. 
 
 FKI. 
 
 In Leland's lead-pipe coupling (Figs. 965 and 966) a double-cone ring is 
 inserted between the ends of the pipes to be coupled, and they are brought to- 
 
 FIG. 967. 
 
 FIG. 970. 
 
 gether by the common wrought-iron pipe-union, the inner surfaces of which 
 are adapted to the surfaces of the lead pipe, compressing it between the pipe 
 and the cone. 
 
 Figs. 967 and 968 are a section and plan for a similar joint of steel water 
 pipe. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 405 
 
 Figs. 969 and 970 are Petit's pipe joint used in the water system of the camp 
 at Chalons. A rubber ring is inserted in the short bell, one clamp being con- 
 nected ; the other is brought over and secured, compressing the rubber, making 
 a tight and a slightly flexible joint, easily taken apart and put together. 
 
 Fig. 971 is a flexible steam joint consisting of a ball and socket carefully 
 turned and then ground to a close fit, to be connected with wrought-iron pipe. 
 
 The earlier flexible joints for water mains were turned and fitted, but about 
 
 FIG. 971. 
 
 1870 John T. Ward, C. E., introduced a ball-and-socket pipe in which the 
 socket was turned but the ball was fitted by a lead packing run in. There was 
 some danger to the socket by the stress of the ball if the movement was in ex- 
 cess of that contemplated. Mr. Ward in his late designs made stops o o (Fig. 
 972) to prevent this. Other engineers have re-enforced the ball by a hoop, of 
 which Fig. 973 is an example, from the " Transactions " A. S. C. E., designed 
 by James C. Duane, C. E., and laid beneath the East River from New York 
 city to Ward's Island. 
 
 Fig. 974 is a section and elevation of a flexible joint for a submerged steel 
 water main used at Toronto, Ontario. This joint is 5 feet in diameter, con- 
 
 NOTE: 
 
 A BAND SEAT TO BE TURNED 
 TRULY CYLINDRICAL AND 
 BAND SECURELY SHRUNK ON 
 
 B CORNERS TO BE ROUNDED 
 OFF AS SHOWN 
 
 FIG. 973. 
 
 sisting of two parts, one part having a turned spherical section riveted to the 
 straight pipe ; the other part is a socket, on the inside of which are two U- 
 shaped sections, one riveted to the sheet-iron of the socket and the other to a 
 flange fastened to another flange on the end of the socket ; these U-shaped 
 
406 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 rims are filled with soft pig lead projecting -J- of an inch beyond the rim ; the 
 lead joint bears against the spherical section and makes a close yet flexible 
 
 Wood feck/ntf 
 
 *, / rt > *5 
 
 C&sf Iron f/an$e 
 
 o 
 
 
 
 i 1 ' 
 
 
 
 
 
 
 J \ 
 
 
 
 : \ \ 
 
 O 
 
 o| V \ 
 
 
 \ 
 
 
 
 
 | \ \ 
 |o' v \ 
 
 
 
 |0| \ \ 
 
 c 
 o 
 
 i3 \ \ o 
 
 o 
 
 io! \ x |0 
 
 
 
 
 
 
 1 ' \ x p 
 
 1 1 \ f 
 
 o 
 
 
 _ 
 
 IS 
 
 
 4 
 
 frl~ __ ^p 
 
 ^ 
 
 | 
 
 -1 
 f> 
 
 o 
 
 
 
 o 
 
 S 
 
 
 t 
 
 \a 
 
 o 
 
 
 
 
 , 
 
 f> 
 
 
 
 
 
 
 
 o 
 
 
 
 ^^ 
 
 t 
 
 1 
 
 
 
 (ft 
 
 
 
 
 o 
 
 Oft 
 
 
 
 o 
 
 
 c 
 
 
 O 
 
 
 
 
 
 O 
 
 
 Q 
 
 [ 
 
 I OO OOOO OOOO 
 
 
 
 ^ 
 
 o 
 o c 
 
 o 
 
 b 
 
 
 
 
 
 <g 
 
 oC 
 
 b 
 
 o 
 
 I i 
 
 
 
 
 
 D 
 
 
 ot 
 
 b 
 
 o 
 
 
 
 
 t 
 
 p 
 
 
 
 
 o 
 
 
 
 
 
 ol 
 
 b 
 
 o 
 
 o c 
 
 
 
 
 o 
 
 o o 
 
 xa^e 
 
 
 5. 
 
 1 
 
 1 
 
 FIG. 974. 
 
 joint. The space between the two flanges on the socket end is made tight 
 by a wood packing. 
 
 Fig. 975 is the section of a ball-joint used as a connection for the 12" steel 
 pipe for a temporary supply of water to Liverpool and sunk beneath the Mer- 
 sey, which was connected for a length of 
 800 feet on the shore and drawn across 
 the river by the united forces of horses 
 and a steam winch in twenty-eight min- 
 utes. The ball is of cast-iron, turned, and 
 has a socket of the same material, with a 
 joint of cast lead, for which two holes are 
 shown one for running the metal and 
 the other for the vent. 
 
 Governors. In the running of all ma- 
 chinery there are variations of speed, due to varying powers and resistances 
 caused by increase or decrease in the pressure producing the power, as of 
 steam or water, or in the resistances of the machinery, from more or less being 
 brought into action, or through inequalities of work done. To maintain the 
 speeds at as much uniformity as possible, governors are used, which, applied 
 to steam engines or water-wheels, open or close valves or gates, and increase or 
 reduce the supply of steam or water to the cylinders or wheels, according to 
 the varying necessities. The ordinary governor (Fig. 976) consists of two 
 heavy balls, suspended by links from a spindle, and caused to revolve by some 
 connection with the shaft of the motor. In the figure the governor is driven 
 by a belt-connection to the pulley, jo, bevel-geared to the governor. When at 
 rest, the balls hang close to the spindle, but when in motion the balls rise by 
 
 FIG. 975. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 407 
 
 the centrifugal force. When the motor is running at its established speed, for 
 which the pulley is to be adjusted, the links assume a position nearly at 45 
 with the spindle. If the speed falls off, the balls fall, and, acting on the lever, 
 as shown in side view, open the valve or gate controlling the passage of steam 
 to the cylinder or water to the wheel ; if the speed rises, the balls rise and close 
 the valve or gate. The lever does not always connect directly with the gate, 
 nor is there always a lever, but the rise or fall of the balls acts on some mech- 
 anism which performs the function of reducing or increasing the supply of 
 steam or water. / 
 
 The size of the balls depends somewhat on the work to be done, the resist- 
 ance to be overcome in the movement of the gate and connections, and may be 
 much reduced if this work is thrown on some other mechanism, which is usu- 
 ally the case in the regulation of water-wheels ; while for steam engines the 
 
 FIG. 976. 
 
 FIG. 977. 
 
 work to be done by the governor is reduced by balancing the steam-valve, or 
 to the merely releasing of a trip, cuts off the movement of the valve at any 
 point of stroke. 
 
 The common governor (Fig. 976) is sufficient for the regulation of drop 
 cut-off engines like the Corliss (Figs. 422, 423), but for slide-valve engines with 
 throttle regulation the Porter governor (Fig. 977) is better adapted; the balls 
 of this governor are comparatively light, but they are connected to a heavy cen- 
 tral weight by levers, the same as those connecting the balls with the spindle. 
 
 Of late it has become very common to run engines at very high speeds, be- 
 yond that possible to be obtained by drop cut-offs; and light slide-valves are 
 used in which the governors act by shifting the cams (actuating the valves) 
 placed within the pulley or fly-wheel. Fig. 978 is an elevation of one of this 
 class of governors the Westinghouse. The disk A is cast solid and keyed to 
 one of the cranks. The loose eccentric C is suspended by the arm c from the 
 pin (7, around which it has a motion of adjustment ; B B are the governor 
 weights, pivoted on the pins b b ; one of the weights is connected to the ec- 
 
408 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 centric by the link /", and both weights are connected to operate in unison by 
 the link e. Coil springs, D D, furnish the centripetal or returning force. The 
 eccentric encircles the shaft S, the opening being elongated to admit of the 
 
 FIG. 978. 
 
 FIG. 979. 
 
 proper motion. The stops ss limit the motion of the weights. Fig. 979 shows 
 the governor in the position of latest cut-off. The governor weights are shown 
 in the position of rest, whereby the eccentric is thrown over to its position of 
 greatest eccentricity, giving a maximum travel to the valve corresponding to a 
 cut-off of about f stroke. The parts of the governor remain in this position 
 till the engine is within a few revolutions of its full speed. The centrifugal 
 force of the weights then overbalances the tension of the springs, and the 
 weights move outward, reducing the travel of the eccentric and valve, conse- 
 quently shortening the cut-off and closing the exhaust earlier, thus increasing 
 the compression curve and preserving greater economy in running the engine. 
 
 Fly- Wheels. In most machinery there are great inequalities of movements, 
 from the great difference in power exerted or resistances overcome, and in the 
 application of the force, as through cranks. To obviate this, fly-wheels are 
 used, which absorb energy in one part of their revolution and give it out at 
 another, or by their mass in movement overcome resistances, as in the punch- 
 ing, shearing, and rolling of metal, which comes only periodically, and is much 
 in excess of that usually required. In addition, fly-wheels give governors time 
 to act, and consequently the motion is more uniform and constant. 
 
 All shafting, pulleys, and machines in movement act as regulators, and 
 where the resistances vary largely on machines they require independent fly- 
 wheels. In addition, friction, hygrometric, and other conditions vary so much 
 at different times, even with the same engines, that it is impossible to get data 
 for an estimate by any mathematical formula embracing the conditions. From 
 the experience of the best mechanical engineers, and from published examples 
 of constructions, are deduced the following rules, applicable to common prac- 
 tice for the fly-wheels of steam engines : The diameter of fly-wheel to be 4 
 times that of the stroke of the engine, and the entire weight of the wheel 40 
 times the square root of the diameter, its exterior velocity being about 5,000 
 feet per minute ; if less or more, increase or reduce the weight inversely as the 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 409 
 
 velocity. The rim is generally a little less than of the whole weight, but the 
 arms should be made strong in view of the fact that a great strain may be 
 produced in them by any suddenly interposed obstacle. For rolling-mill en- 
 gines, Prof. C. B. Richards takes the weight of the fly-wheel at 60 times the 
 square of the diameter of the cylinder, and the diameter of the wheel 5 times 
 that of the stroke, and rim velocity not to exceed 125 feet per second. 
 
 FIG. 980. 
 
 In most stationary engines the fly-wheel is a pulley or band-wheel or gear 
 driving the machinery, but often the fly-wheel is independent. Fig. 980 is the 
 elevation and section of a fly-wheel built by the Southwark Foundry. The 
 construction will be understood from the drawings, but the wrought-iron links 
 
410 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 411 
 
 connecting the segments, shown pn a larger scale (Fig. 981), do not project, 
 but are countersunk in the sides of the rim. 
 
 The cast-iron fly-wheel of a steam engine 36" X 72", built for the Amos- 
 keag Manufacturing Company in 1883, 30 feet diameter, 110" face, with one 
 set of 12 arms, and total weight 116,000 pounds, making 61 revolutions per 
 minute ; exploded in 1891, and was replaced by a wooden-rimmed pulley with 
 two sets of arms (Figs. 982 and 983). The counterbalance of the cranks and 
 connecting-rods was obtained by placing heavy cast-iron plugs in the outer ends 
 of the three arms directly opposite each crank. Though the total weight of 
 the wheel is not much less than that of the old one, the weight of the rim 
 (31,855 pounds) is only about one half, but has shown itself ample for a very 
 steady speed at much less cost and greater security. 
 
 Fly-wheels, and in fact all wheels that have a rapid motion, should be bal- 
 anced, and if driven by cranks their connections should be balanced dynamic- 
 ally to make the motion as uniform as possible. 
 
 Air- Chambers. The action of the air-chamber is very similar to that of a 
 fly-wheel ; it tends to make the outflowing or inflowing pressure of the fluid 
 uniform, and cushions or prevents the reaction that takes 
 place from the fluid in reciprocating pumps, especially 
 crank-pumps ; but pumps in which the pistons or plung- 
 ers start very slowly and stop equally so require but little 
 air-chamber. Cornish engines are usually provided with 
 a stand-pipe instead of an air-chamber that is, a vertical 
 pipe of considerably larger diameter than that of the 
 pump, and high enough to contain the water-column. 
 
 Fig. 984 is the section of a copper air-chamber for the 
 smaller size of steam or hand-pumps. It is screwed into 
 the top of the pump-chamber. Fig. 985 is the elevation of 
 an air-chamber for power pumps of larger size of cast-iron 
 or a cast-iron base with a copper chamber. A flange is 
 cast on the top of the pump-chest, and the chamber is 
 bolted to it. 
 
 Fig. 986 is the elevation of an air-chamber of one of 
 the older Brooklyn pumping-engines. 
 
 The lower end of the small air-chamber (Fig. 984) is 
 necked, or of smaller diameter than the main part of the 
 chamber. This prevents a too sensitive reaction of the air 
 and retards its escape ; for the same purpose a diaphragm 
 perforated with holes is put across the inside of the cham- 
 ber. When the inlet column is long, whether suction or 
 under pressure, put an air-chamber on it. 
 
 Air-chambers should be from ten to fifteen times the 
 capacity of the pump-cylinder, with glass gauges to show 
 the quantity of air in them for large pumps, and some pro- 
 vision to supply and maintain the air at such levels as will be found by experi- 
 ment suited to the easiest working of the pump. A large air-chamber can in 
 this way be reduced in capacity, while that of a too small chamber can not be 
 increased. Air-chambers are made of wrought-iron or steel and the heads 
 
 FIG. 985. 
 
412 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 
 are bumped up or dished. The thickness of the cylindrical part is to be deter- 
 mined by the rules on riveting (page 383). The dished head, struck up on a 
 
 spherical radius (Fig. 987) equal 
 to the diameter of the cylinder 
 and of the same thickness of 
 plate, is of equal strength. 
 
 In the earlier application of 
 water to the distribution of power 
 the air-chamber held the reserve 
 of force, but since the use of 
 power in this form has become of 
 general application, the accumu- 
 lator (Fig. 988) is the form 
 adopted, in which the pressure of 
 the water in the pipe A raises the 
 piston B from which is suspended 
 the dead weight C sufficient to 
 maintain the pressure required. 
 For the distribution of power 
 throughout the city of London 
 the pressure is about 700 pounds 
 to the inch. At the Forth "Bridge 
 Works there are two forms of 
 accumulators in use through 
 which the high-pressure water is 
 pumped ; in one form a 16" cylin- 
 der is loaded with dead weights, in 
 the other an 8" cylinder is loaded 
 with steam. 
 
 rfi ffi ffi fft rfi ftiffifflffli 
 
 FIG 
 
 FIG. 987. 
 
 To make a hydraulic riveting machine that could be introduced into some 
 of the more complex parts of the structure, it was necessary to increase the 
 pressure of the riveter and reduce its dimensions. This was effected by the 
 multiplier (Fig. 989), in which the usual high pressure is introduced into a large 
 
MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 413 
 
 FIG. 991. 
 
 u 
 
414 
 
 MACHINE DESIGN AND MECHANICAL CONSTRUCTIONS. 
 
 cylinder with its piston connected with a plunger of smaller diameter, and the 
 pressure from the smaller cylinder is connected with the riveter. 
 
 For hydraulic cylinders the common rules are for cast-iron to a pressure up 
 to 2,000 pounds per square inch and cast-steel to 6,000 pounds per square inch, 
 rarely exceeding 8,000 pounds. For the resistance of thick shells Rankine gives 
 the bursting strain 
 
 --> 
 
 in which /is the tenacity of the metal, E and r exterior and interior diameter 
 respectively. 
 
 Figs. 990 and 991 are the elevation and plan of a common form of hydraulic 
 press for the baling of goods. The dimensions of the bolts at the four corners 
 should be estimated from the hydraulic pressure^ with a factor of safety of 5. 
 
 As tools the hydraulic punch, riveter, and jack are in general use. 
 
 Yin. 992. 
 
 FIG. 993. 
 
 FIG. 994. 
 
 FIG. 
 
 Fig. 992 is the section of a common jack-screw in which the screw is turned 
 by a lever of which the hole beneath shows where its end is inserted. The whole 
 weight is taken by the extended base. 
 
 Figs. 993 and 994 are side and end views of a cast-iron housing ; the screw 
 exerts a pressure on the roll box journals to be resisted by the frame. 
 
 Fig. 995 is the side elevation of a cam punch and shear, the action being to 
 spring the jaws, which are reinforced. 
 
 The above drawings of machines, not shown elsewhere in the work, are given 
 as illustrations of forms adapted to the stresses for which they are designed, 
 
ENGINEERING DRAWING. 
 
 THERE is no part of engineering more important than that of securing a 
 good foundation for the structure. Where likely to be disturbed by frost, the 
 structure should start below it, unless, as in the extreme northern regions where 
 frost is permanent at certain depths, the support should be in it. In preparing 
 the foundation for any structure, there are two sources of failure which must 
 be carefully guarded against : viz., inequality of settlement, and lateral escape 
 of the supporting material ; and if these radical defects can be guarded against, 
 there is scarcely any situation in which a good foundation may not be obtained. 
 It is therefore important that, previous to the commencement of the work, 
 soundings should be taken to ascertain the nature of the soil and the lay of the 
 strata, to determine the kind of foundation ; and the more important and 
 weighty the superstructure, the more careful and deeper the examination. But 
 it must be understood that in general it is not an unyielding but a uniformly 
 yielding stratum that is required, and that a moderate settlement is not objec- 
 tionable, but an inequality of settlement. 
 
 In good sand or gravel, the load on foundations per square foot is usually 
 from three to five tons. Many soils are very compressible, not supporting one 
 ton per square foot ; if the structure is important, the bearing resistance of the 
 strata should be tested by experiment. The base of the wall is extended to 
 secure the requisite area of bearing-surface, either by a base-stone (Fig. 996), 
 by a bed of concrete (Fig. 997), or by extending the wall by steps (Fig. 998), 
 
 
 
 
 
 FIG. 996. 
 
 FIG. 997. 
 
 FIG. 
 
 with or without concrete base, or the weight may be distributed by inverted 
 arches between walls and piers. 
 
 Wooden platforms are often used for foundations, but must be laid beneath 
 the water line where they are kept wet, otherwise rot will take place. These 
 platforms may be of a single course of plank or plank and timber, as in Fig. 999, 
 or may be very much extended, forming rafts or grillages of many courses. A 
 similar foundation has been introduced in Chicago, where the great depth of 
 clay requires extended areas of foundation, but instead of timber it consists 
 
 415 
 
4:16 
 
 ENGINEERING DRAWING. 
 
 of a combination of masonry with common rails or I-beams, affording greater 
 length of offsets and less depth than by timber or masonry. The length of 
 
 offsets may be calculated from the insistent weight, 
 by the formula of beams fixed at one end and uni- 
 formly loaded or one quarter that given in the tables 
 (pages 243 and 244). 
 
 Fig. 1000 is a section of one of these foundations 
 beneath a pier, but equally applicable to the support 
 of walls. The space between the beams is filled 
 with cement, mortar, or concrete, which adds to the 
 stiffness of the structure and is a preservative of the 
 metal. The iron grillages for piers are distinct, and 
 each proportioned in bearing-surface toVits proposed load. In astronomical 
 observatories it is especially necessary that the foundations for the large and 
 
 FIG. 999. 
 
 STEEL I BEAMS 
 
 . STEEL RAILS' 
 
 'TTTTTTTTTTTTT^ f 
 
 JXXXXJ.XXX J.XX JTT T T T ,T T T.XXS 
 
 
 FIG. 1000. 
 
 delicate instruments should be detached from those of the building, and also 
 where the noise and jar of the machinery might interfere with the occupancy of 
 the building for other purposes. 
 
 Walls on party lines and confined to these lines are often partially supported 
 by cantilever beams reaching over interior posts. Such a construction is shown 
 in Figs. 1001 and 1002. 
 
 It is the safer construction to lay walls in air and open to inspection, and 
 
ENGINEERING DRAWING. 
 
 therefore important that their foundations should be freed from water, which 
 can be done by inclosing them with a bank of earth or by a curb of sheet-piling 
 (Figs. 1003 and 1004). Sheet- 
 piling is usually of plank two to 
 three inches thick, set or driven. -^g~ 
 For driving, the bottom of the 
 plank should be sharpened to a 
 chisel-edge, a little out of centre 
 toward the ranging timber side, 
 and cornered slightly at the outer 
 edge, that it may hug the timber -^ 
 and the plank while being driven- 
 Fig. 1005 is the section of a 
 timber sheet-piling, in which a 
 tongue and groove forms the 
 guide, the grooves being either 
 
 FIG. 1003. 
 
 FIG. 100-1. 
 
 FIG. 1005. 
 
 made in the timber, as shown at 
 
 a a, or planted on, b b. The pile should be of uniform thickness, but the 
 widths may be random ; six inches thick is a good practical thickness, driving 
 well under short and frequent blows of a ram ; the tongue should be of hard, 
 straight-grained wood, 2 inches by 2 inches, and well spiked to the pile. 
 
 Frequently, to secure the foundation from water, a wall is constructed of 
 two rows of sheet-piling, driven one within the other, and the space between the 
 
 two filled with clay or some compact earth. 
 This is called a coffer-dam ; the two rows 
 of piling are stayed to each other by bolts, 
 and if the wall is wide enough no other 
 stays or braces will be necessary. 
 
 Pile-drivers are now constructed to drive 
 sheet-piling in panels of 6 to 8 feet wide, which serves to preserve the line and 
 make tight joints. 
 
 In quicksands it is very common to secure a foundation by consolidation 
 with small stones or rubble worked down by iron bars or driven by rams of a 
 pile-driver, till sufficient resistance is secured for the structure. Some success- 
 ful experiments have been made in compacting such sands by forcing down 
 cement mortar through pipes. * In soft earths piles are generally used for this 
 purpose, and if a firm bottom can be secured at a reasonable depth they are the 
 most economical expedient. 
 
 Piles are used either as posts or columns driven through soft earth to a hard 
 bottom, or depending on their skin resistance to give the necessary support, 
 either in earth naturally compact or made so by the driving of the piles. In, 
 the first case care must be taken that the piles be driven sufficiently deep into 
 the lower strata to secure their ends from slipping laterally, and soundings 
 should be made carefully to ascertain the dip and character of these strata. In 
 many places, from the hardness and the inclined position of the lower strata, 
 this kind of foundation is inapplicable and unsafe. 
 
 For a foundation where no firm bottom can be found within an available depth, 
 piles are driven, to consolidate the mass, a few feet apart over the whole area of 
 28 
 
418 
 
 ENGINEERING DRAWING. 
 
 the foundation, which is surrounded by a row of sheet-piling to prevent the escape 
 of the soil ; the space between the pile-heads is then filled to the depth of several 
 feet with stones or concrete, and the whole is covered with a timber platform. 
 
 It is very difficult to establish a rule of general application for the load which 
 a pile will sustain. It is well in untried soil to drive a few piles, noting the set- 
 tlement under blows, and then load the piles in excess of what they will be re- 
 quired to bear, noting the results from time to time ; and if settlement con- 
 tinues, drive deeper, or more piles with less spaces. 
 
 Major Saunders, in the " Journal of the Franklin Institute," gave a rule for 
 the safe load of piles depending on the skin resistance, which has been of general 
 application, but of which the factor of safety is unnecessarily large, and is given 
 below modified to 4 instead of 8. 
 
 Multiply the weight of the ram in pounds by the distance in which it falls 
 in inches at the last blow, and divide the product by four times the depth 
 driven in inches at this blow. 
 
 Weight of ram 2,000 pounds, fall 15' or 180", set If. 
 
 2000 X 180 
 
 = 60,000 pounds. 
 
 Safe load = 
 
 X 4: 
 
 In driving piles the effect of the ram should be carefully noted, the rate of 
 fall under successive blows, the brooming or splitting of the head of the pile, 
 and the rebound of the ram after the blow. Especial note should be taken 
 toward the close of the driving to determine the last set when the .formula 
 above given is used. 
 
 The usual weight of the ram or hammer employed on our public works va- 
 ries from 1,400 to 2,400 pounds, and the height of leaders or fall from 20 to 35 
 feet ; but there is a great advantage in reducing the fall, increasing the weight 
 of the hammer, and the frequency of the blows. As generally driven, and of 
 average size, when the whole weight is to be supported by the pile, ten tons 
 
 may be considered a usual load, but when 
 additional support is received from com- 
 pacted earth, broken stone, or concrete be- 
 tween piles and caps, this bearing surface 
 should also be taken into consideration. 
 
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 FIG. 1006. 
 
 FIG. 1007. 
 
 Figs. 1006 and 1007 represent plan and elevation of a pile foundation ; the 
 piles are usually from 10" to 14" top diameter, and driven at about 3 feet be- 
 tween centres. The tops are cut off square and capped with timber the caps 
 
ENGINEERING DRAWING. 
 
 419 
 
 treenailed or ragbolted. to the piles, and plank spiked to the timber. In the 
 figure a sheet-piling, s s, is shown, inclosing the piles ; the spaces between piles 
 and timbers are often filled with concrete, small stone, or closely packed earth. 
 
 In compacting some soils it has been found that good results may be ob- 
 tained by drawing the pile and filling the hole with sand. It would seem that 
 the best result would be obtained by forming these piles of a uniform taper 
 downward, as the consolidation would be more uniform, the withdrawal easier, 
 and less disturbance of the sides of the hole. The consolidation might be still 
 further increased by ramming the sand in thin layers, owing to the ability of 
 the latter to transmit pressure laterally. The sand should be fine, sharp, clean, 
 and the grains of uniform size. 
 
 It is often difficult to obtain or drive piles of sufficient length, and they 
 must be spliced. After the first pile is driven, its head is cut level ; a wooden 
 dowel or iron pin, penetrating each pile about a foot, is inserted in the centre ; 
 the upper pile is then fitted to the lower one, or a cast-iron collar or ring of 
 plate iron 6" to 12" wide may be used to strengthen the joint and protect the 
 pins from splitting either head of the piles. 
 
 Spliced in this way, the pile is deficient in lateral stiffness. In most posi- 
 tions it is safer to re-enforce the splice by flatting the sides of the piles and 
 nailing on with, say, 8-inch spikes, four pieces 2 or 3 inches thick, 4 or 5 inches 
 wide, and 4 to 6 feet long. In the erection of the bridge over the Hudson 
 River at Poughkeepsie, N. Y., two piles were thus spliced together to form a 
 single one 130 feet long. 
 
 Piles may be made of any required length or cross-section by bolting and 
 fishing together, sidewise and lengthwise, a number of squared timbers. Hol- 
 low-built piles 40 inches in diameter and 80 feet long were used as guide- 
 piles in constructing the St. Louis bridge. To protect the head of the pile 
 against brooming or splitting it is usual to drive on a tight-fitting wrought- 
 iron ring or cap (Fig. 1008) by a blow of 
 the hammer. In driving into compact 
 gravel or shingle the point of the pile 
 should be protected with iron straps, as 
 shown in Fig. 1009. 
 
 Fig. 1010 is a section of a bulkhead 
 wall on the North Eiver front, in positions 
 where the mud is deep, as designed and 
 constructed by George S. Greene, Jr., Chief 
 Engineer for the Department of Docks, 
 New York city. 
 
 FIG. 1008. 
 
 FIG. 1009. 
 
 The site of the wall is first dredged to hard mud compacted with sand. The vertical- 
 piles are then driven, and small cobble-stones mixed with coarse gravel put around and 
 among the piles to the height of the under side of the binding frames, and rip-rap stone 
 placed outside the piles, in front and rear. 
 
 The binding frames, then slid down to their places, were made of two pieces of spruce 
 plank 5 x 10 inches, placed edgewise one over the other, and running from front to 
 rear of the piles between the rows. An oak beam 8x8 inches is let through these 
 planks in front of the front row and in rear of the rear row of piles, and an oak wedge 
 block fitted and placed by the divers between the oak beam and each pile nearest it. 
 
420 
 
 ENGINEERING DRAWING. 
 
 5$ 
 
 >y c* 
 
 rt ^ 
 
 
 
ENGINEERING DRAWING. 
 
 421 
 
 These frames hold the front rows of piles firmly, in case there should be any tendency in 
 them to tilt outward. More cobble-stone is then put in to the height of the bottom of 
 the base blocks of the wall, weighting the binding frames and preventing any tendency 
 to floating. 
 
 The bracing piles are then driven on a slope of 6 inches horizontal to 12 inches ver- 
 tical, between the rows of vertical piles, and spaced 3 feet from centre to centre longi- 
 tudinally and transversely. All the piles are staylathed and adjusted in position as soon 
 as they are driven. 
 
 The bracing piles are cut off at right angles to their axis, about 1 foot below mean 
 low water, and capped with 12-inch square timber, running longitudinally. The sides 
 of the caps are kept horizontal and vertical, and a sloping recess or notch made to re- 
 ceive the head of each bracing pile, and give it a good bearing. 
 
 The six rear rows of vertical piles are cut off at 2 inches above mean low water, and 
 notched front and rear to give an 8-inch -wide bearing across their tops for the trans- 
 verse caps. 
 
 The three front rows of vertical piles are cut off by a circular saw, suspended in the 
 ways of a pile-driver, at 15 -3 feet below mean low- water mark, to receive the concrete 
 base blocks of the wall. It being impossible to cut off piles at this distance below the 
 surface of the water to exactly the same height, and as the bottom of the concrete base- 
 blocks would rest only upon the highest piles of those under them, a mattress of burlap, 
 containing freshly mixed soft mortar, in a layer about 2 inches thick, placed on a net- 
 work of marline stuff, supported by a plank frame about its edges, is lowered upon the 
 tops of these piles immediately before setting the base-blocks upon them. The diver 
 then cuts the netting between the edge of the mattress and the plank frame, and the 
 frame floats to the surface of the water. 
 
 The base-block is then immediately placed in position upon the mattress of mortar 
 resting on the piles, and the excess of mortar is pressed out from between the head of the 
 pile and the bottom of the base-block, until each pile has a well and evenly distributed 
 portion of the load to carry. 
 
 The concrete base-blocks for this section are 7 feet wide at the bottom and 5 feet 
 wide at the top; on the front the vertical height is 13 feet, and on the rear 14 feet. The 
 top has a step on the rear of 1 foot height and 1 foot wide, extending the entire length 
 of the block, for the purpose of giving the mass concrete backing of the granite super- 
 structure a good hold upon the block. For handling, grooves for chains are moulded in 
 the end, and a longitudinal hole, 2 feet in the clear above the 
 bottom, connects them, with the corners rounded, to enable the 
 chain to render easily. The face is curved inward, to save ma- 
 terial while giving a broad base; their length is 12 feet. 
 
 After the blocks are set, the vertical chain-grooves in each 
 block, coming opposite to each other, are filled in with concrete 
 in bags, well rammed into place. This closes the joints between 
 the blocks, and also acts as a tongue set into the grooves in the 
 blocks. 
 
 Fig. 1011 shows a block in section with the central- 
 and side-groove spaces, which are deep enough to admit 
 of the easy slipping in and withdrawal of the chain. 
 
 T 
 
 FIG. 1011. 
 
 As soon as the base-blocks are set, and the groove filled in, 
 the cross-caps resting on the tops of the vertical piles, and on 
 the longitudinal caps of the bracing piles, reaching about half 
 way across the base-blocks, are placed and fastened. Oak treenails are used in all fast- 
 enings. The small cobbles are then filled in around and among the piles to the top of 
 the caps, and the rip-rap placed in the rear of them. 
 
422 
 
 ENGINEERING DRAWING. 
 
 Figs. 1012 and 1013 are the elevation and plan of a crib with dock or pier. 
 Below the level of the water, as here shown, the logs are round and locked 
 
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 2 
 
 V. 
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 FIG. 1013. 
 
 to the cross-timbers ; above the water the timber is squared, the exterior walls 
 presenting a tight, smooth surface into which the cross-timbers are dovetailed. 
 
 A quarantine station, built for the port of New York, on the west bank con- 
 sists of an exterior wall of cribwork, of which Fig. 1014 is a section ; the in- 
 closed space is filled with sand, comprising an extent of about 228' X 448'. The 
 base to low water consists of cribs about 80' in length, sunk, and loaded with 
 stone; above it he construction is continuous. The base timbers 14" X 14", 
 upper 12" X 12", interties 12" X 12", at intervals of 7 feet centres ; the 
 ranging timbers to be secured at every joint to these below, and at every cross- 
 ing by 1" square bolts 21" long. The exterior is close-fendered with oak plank, 
 iron bolted, with three iron bands at the corners. 
 
 Fig. 1015 is a transverse section of the river-wall Thames embankment, 
 Middlesex side ; a wall of concrete, etc., faced with granite, with a sewer and 
 subway within the same, lined with brickwork. The different material is rep- 
 resented by different shadings and letters : g, granite ; b b, brickwork ; c c, 
 concrete. 
 
ENGINEERING DRAWING. 
 
 423 
 
 Extracts from specifications : 
 
 "The embankment- wall is to be formed within iron caissons or coffer-dams. As 
 soon as the excavations shall have been made to the requisite depths, and the works 
 cleared of water, the trenches shall be filled up with concrete to a level of 12J feet be- 
 low datum, and a bed dressed to the proper slope and level for the footings of the brick 
 wall. This wall to be formed thereon, generally in courses at right angles to the face 
 of the wall. The subway shall be formed 7 feet 6 inches high by 9 feet wide in the 
 clear, generally ; the side-walls to be 18 inches, the arch 1 foot 1 inch thick. The sub- 
 way sewer and river- wall shall be tied into each other, at intervals of 6 feet, by cross or 
 
 FIG. 1014. 
 
 counterfort walls 18 inches thick, extending from the brickwork of the wall to a vertical 
 line 9 inches beyond the side of the sewer farthest from the said wall, and from footings 
 9 feet below datum, which are to be bedded on a concrete foundation 12 inches thick, 
 up to the under side of the subway. The upper arch of the subway, and all other simi- 
 lar arches, shall be coated on their outside circumference with a 1" layer of Claridge's 
 patent Seyssel asphalt. The whole of the stones above 11 V datum to be dowelled to- 
 gether in bed and joints with slate-dowels, not less than 5 for every foot run of wall ; 
 each 2J inches square at least, let fully 2 inches into each stone, very accurately fitted, 
 and run in with neat cement; the stones to be bedded and jointed in cement, and the 
 joints struck with neat cement." 
 
 Fig. 1016 is an isometrical view of the overflow and outlet of the Victoria 
 and Regent Street sewers in the Thames embankment. S is the main sewer, 
 and W the subway shown in Fig. 1015 ; s s s the street-sewers, discharging into 
 the overflow basin ; w w the weirs over which the water is discharged into 
 the weir-chamber c c ; p is the penstock-chamber, which is but a continuation 
 of the weir-chamber. Whenever, from storms, the discharge from the street- 
 sewers (s s s) is greater than can be carried off by the main sewer (S), the water 
 rises in the overflow-chamber (0), passes over the weirs (w ^v) down into the 
 weir-chamber (c), then into the penstock-chamber, and through the flap-gates 
 (</) into the river. 
 
424 
 
 ENGINEERING DRAWING. 
 
 Extracts from the specifications : 
 
 "The foundation to be of concrete, not 
 less than 2 feet in thickness ; upon this 
 brick-work shall be built for the flooring of 
 
 the chambers, and for the side- 
 end and weir- walls. The weir- 
 chamber shall be divided in the 
 direction of its length, by a brick 
 wall, into two rectangular over- 
 flow-channels, covered with cast- 
 iron plates, 6 feet 8J inches long, 
 3 feet wide by | inch general 
 thickness, with strong ribs and 
 flanges on the under side, prop- 
 erly bolted together and jointed 
 with iron cement, and bolted 
 
 FIG. 1015. 
 
 down to stones which are to be built into the under side of the brickwork of the base- 
 ment chamber. Arches on either side, running parallel thereto, and communicating with 
 this chamber and with the weirs which are to be formed, upon which weir-walls, divided 
 so as to correspond with these arches, are to be built in brickwork, capped with granite 
 
ENGINEERING DRAWING. 
 
 425 
 
426 
 
 ENGINEERING DRAWING. 
 
 blocks, 4 feet long, 2 feet deep, and 2 feet 3 inches in the bed. The floor of the pen- 
 stock-chamber to be formed with York landings, 6 inches thick, having a fall of 3 inches 
 to the river. The outlets for the penstock-chamber through the river-wall shall be 
 formed by an arch-recess in granite, and fixed with two tidal flaps, well hung, and firm- 
 ly secured to the masonry by strong bolts and screws. 
 
 "The subway is to be continued over the low-level sewer, and across the overflow- 
 chamber, by cast-iron plates, curved to the form of the arch, J inch general thickness, 
 with strong ribs and flanges on the upper side, properly bolted together, and strongly 
 bolted down to the brickwork ; jointed with iron cement, and covered with brickwork, 
 to form the floor of the subway. From a point of 10 feet 3 inches on either side of the 
 central longitudinal line of the chamber, where the sewer and subway are farthest from 
 the river-wall, these are again to be brought into their general position by two curves, 
 each not less than 80 feet in length. 
 
 "The whole of the cast-iron shall receive one coat priming of red lead and linseed 
 oil, and three coats best coal-tar, before fixing; and the accessible surfaces one further 
 coat best coal-tar, when fixed." 
 
 Fig. 1017 is the section of the dike or jetty forming a breakwater for the 
 harbour of Boulogne, France. It may be considered in two distinct parts, cor- 
 
 responding to the substructure and the superstructure. The substructure is 
 formed by a mass of natural and artificial rip-rap, with a central core of stones 
 weighing about two hundred and fifty pounds each, resting on the bottom, and 
 rising to a level of one metre above low tide. The shore side is protected by a 
 pitching of stone, each about twelve hundred pounds weight ; the sea-side 
 slope, by one of heavy rubble of about seven tons each, and covered by beton- 
 blocks weighing uniformly thirty-three tons each ; on the above is built the 
 masonry superstructure. On each ide of the wall, on a level with the lower 
 platform, the slopes are consolidated by masonry bermes formed of isolated 
 blocks, which protect the foot of the wall and afford a path for the workmen 
 and materials at low tide. 
 
 The water-jet is extensively employed on sandy shores for the sinking of 
 piles for foundations of lighthouses, wharves, etc., and in the Southern States 
 it renders the palmetto available, which resists the ravages of the teredo, but is 
 too soft to withstand the blows of the pile-driver. In a simple and effective 
 application it consists of a pile to which a small iron pipe is attached, extend- 
 ing below the bottom of the pile. A flexible hose is attached to the top of the 
 
ENGINEERING DRAWING. 
 
 427 
 
 FIG. 1018. 
 
 pipe and, water being forced through, the earth is washed away from the bot- 
 tom and side of the pile, which falls by its own or superadded weight or light 
 blows of a maul or hammer. 
 
 The jet has also been employed in a great variety of ways to facilitate the 
 passage of screw and disk piles, cylin- 
 ders, etc., through earthy material, and 
 as an ejector to remove earth from the 
 inside of caissons, and relieving stranded 
 vessels by removing the sand from their 
 bottoms and sides. 
 
 Cast-iron or wrought-iron is used for 
 piles, and, at the present prices of iron, 
 with economy. They can be driven by 
 the pile-driver, the interior earth removed 
 by an augur, by sand-pump, or water-jet. 
 The blows of the ram should be cush- 
 ioned by a wooden block. For the con- 
 struction of the iron pier at Coney 
 Island, N. Y., to the bottom of the 
 wrought-iron pipes 8f" in diameter, in 
 lengths of from 12 to 20 feet, cast-iron 
 disks were fastened by set screws and 
 sunk by an inside water-jet to a depth of 
 17 feet in the sand. 
 
 In Chili iron piles were sunk 14f" 
 diameter with bottom flange of 42"; the 
 pump discharged 12,000 gallons per hour 
 through two 2" pipes extending 8" below 
 the base, and would sink two piles 28 feet 
 on an average in 18 hours. The bottom 
 was of coarse compact sand and the pile 
 was worked down by an endless cable 
 passing around a pulley on the pile and 
 giving it a motion of rotation. 
 
 The screw pile usually consists of a 
 wrought shaft from 3" to 8" diameter to 
 which is keyed a cast-iron screw from 2 
 to 5 feet in diameter sunk by turning 
 the shaft by hand or power, mostly used 
 for marine purposes, for the foundations 
 of lighthouses, or anchors for buoys 
 
 where they resist the upward motion of the waves. As they are sunk without 
 jar or disturbance of the soil, they are adapted to positions where the neigh- 
 bouring structure might be injured by other methods of sinking foundations. 
 
 Figs. 1018, 1019, and 1020 are sections of masonry curbs sunk by water-jets 
 for a quay at Calais, France. 
 
 The base is of concrete made in a mould, the rest of the masonry laid in 
 cement mortar. 
 
 FIG. 1019. 
 
 FIG. 1020. 
 
428 
 
 ENGINEERING DRAWING. 
 
 The sand surface beneath the blocks was exposed to the action of strong 
 jets, and the mixture of sand and water was pumped up by centrifugal pumps ; 
 the suction-pipe nozzle was just below the level of the bottom of the block ; 
 care was taken that the quantity of water forced in should be the same as that 
 pumped out, and the level of the water in the curb should be just below that 
 in the surrounding sand. When the curb had reached the bottom after a de- 
 scent of about 15 feet, the sand was allowed to settle and the opening was filled 
 with concrete up to a level where it could be filled with masonry. 
 
 The blocks 1, 3, 5 are sunk alternately, 
 and then 2, 4, 6, the blocks being cemented 
 together as shown in Fig. 1021. 
 
 It has long been common in India to 
 sink brick wells in clusters by hand labour, 
 
 24 WALL 
 
 FIG. 1021. 
 
 FIG. 1023. 
 
 excavating and pumping for the foundations of bridges, but by substitution of 
 bucket-dredges to remove the inner cores of earth greater .facility in work and 
 depth of sinking has been secured. This system has been successfully applied 
 to the sinking of iron caissons. 
 
 In the sinking of small curbs for wells it is common to make circular plank 
 curbs supported by segmental ribs inside, and load them so that they will sink 
 as the earth is removed from the inside and then lining with masonry. 
 
 Fig. 1022 is a partial section of a shoe of the 50-foot diameter well sunk at 
 Long Island City, N. Y. It consists of timber segments bolted together on 
 the top of which a brick wall was laid ; on the outside is a board sheathing. 
 As the earth is removed from the inside the curb settles from the weight of the 
 brick masonry, which is built up and settles again and again till the required 
 depth is reached. As the boards are set at a batter of 2" to 1 foot, the struc- 
 ture settles readily as the earth is removed. Were it not for the batter the 
 earth would press against the sheathing and masonry, and vertical bolts would 
 be necessary in the latter to anchor the courses together and prevent rupture. 
 The sheathing of boards is planed on the outside and firmly attached to the 
 shoe with bound wooden segments above to preserve the form, which are re- 
 moved as the brickwork reaches them. 
 
 Fig. 1023 is a partial section of a steel caisson, exhibiting the cutting edge 
 re-enforced by steel plates and supported by beams to the bottom girder. 
 
 Foundations for the abutments of piers and bridges may be constructed on 
 
ENGINEERING DRAWING. 
 
 429 
 
 FIG. 1023. 
 
 any of the systems illustrated, but as it is generally necessary that they should 
 
 not extend in width so as to obstruct the current of the stream and increase its 
 
 velocity, it is usual to inclose the site of 
 
 the foundations within a coffer-dam, 
 
 freeing it from water and preparing the 
 
 foundation ; but this is expensive in deep 
 
 water, and other means are adopted. 
 
 Figs. 1024, 1025, and 1026 are plans 
 and sections of the foundations of one 
 of the Poughkeepsie bridge piers, which 
 may be designated as a crib, although its 
 walls are tighter than usual in such con- 
 structions. 
 
 It is built of 12" X 12" white hemlock 
 timber, except the bottom or cutting 
 course of the shoe, which is white oak. 
 The shoe is carried up in a triangular 
 section of solid lumber around the sides 
 and a central longitudinal division to 
 the height of 20 feet. Top surface at 
 sides 10 feet wide, at ends 9 feet, central 
 16 feet ; above the shoe there are hollow 
 spaces. Cross-timber walls 2 feet thick 
 divide the spaces into pockets, of which those above the shoe are weighting 
 pockets and those open at the bottom are dredging pockets D. The earth is 
 dredged from the pockets D and discharged into the pockets W, and acts as a 
 weight to sink the crib, care being taken to distribute the load to equalize 
 the sinking. As the earth is dredged from the pockets D the crib sinks, and 
 when it reaches the" bottom they are cleared out and the space filled with con- 
 crete lowered into place in boxes of one cubic yard capacity and unloaded by a 
 trip. 
 
 The top of the crib was sunk to a level of 7 feet below water mark and the 
 concrete brought to within 2 feet of this level ; this space was*lled with broken 
 stone and levelled by divers; a floating caisson, or tight box with timber bottom 
 and sides, was floated over the crib and sunk and the masonry begun. When 
 complete above water the sides of the caisson was removed. 
 
 The Chinese anchors used in mooring the cribs of the Poughkeepsie bridge 
 as shown in Fig. 1027 is composed of 3" X 8" hemlock planks 10 feet ]ong 
 piled to make the interior dimensions 6' X 6' X 6', which is filled with broken 
 stone,, each anchor holding 8 cubic yards ; the eye-bolts shown at the corners 
 serve the double purpose of holding the framework together and carrying the 
 slings to which the cable is fastened. 
 
 Open caissons, as shown in the description of the Poughkeepsie bridge, are 
 useful when a suitable foundation can be secured either in a uniformly yield- 
 ing material or by preparing one by means of piles or divers. 
 
 Figs. 1028 and 1029 are illustrations of the means adopted by G. A. Parker, 
 C. E., in lowering the caissons for the erection of some of the piers of the Sus- 
 quehanna bridge. 
 
430 
 
 ENGINEERING DRAWING. 
 
 He commenced by dredging away as much as possible of the material in the bed of 
 the river at the pier site. A f-inch-thick boiler-iron curb was then sunk and secured in 
 place. The curb was about 30 feet wide and 50 to 60 feet long, and of sufficient 
 height to reach above the bed of the river. The material was then pumped by sand- 
 pumps out of the curb, which gradually undermined, and settled down to the estab- 
 lished depth, or to the bed-rock. When stumps, logs, or boulders were met with they 
 were removed by divers working in a bell. After the rock had been thoroughly 
 cleaned off, it was brought to a uniform level by a solid bed of concrete extending over 
 a greater space than the size of the bottom of the pier by the use of the diving bell. 
 
 Three guide-piles on each side, and one at each end, were fixed firmly in position. A 
 strong platform of solid timber, the size of the bottom of the pier, was then placed in 
 position over the curb and at the surface of the water. On this was placed a caisson of 
 iron large enough to contain the pier, and with sides and ends high enough to reach to 
 
 the level of high water after the caisson 
 was landed on the bottom. The caisson 
 was then made water-tight. The bottom 
 was then floored over with masonry and 
 stone, and laid in mortar up the sides of 
 the caisson to the top, thus constituting a 
 stone caisson inside of an iron one. This 
 was secured to the guide piles, and the 
 masonry of the pier proper was laid up, the caisson sinking as the weight of masonry 
 inside increased, until it finally settled upon the bottom which had been prepared for 
 it. At some of the piers (Figs. 1028 and 1029) screw-rods were used to suspend the 
 pier and gearing attached, governed by one man, who could raise or lower without assist- 
 ance the whole pier. Wooden piles were driven for some piers and cut off by machinery 
 just above the ground, and the platform, with its masonry, lowered upon them. 
 
 Fm. 1026. 
 
ENGINEERING DRAWING. 
 
 431 
 
 Piers are sometimes made by sinking a wrought-iron curb, extending from 
 the bottom to above the level of the- water, driving within it the usual propor- 
 tion of piles, and then filling the 
 spaces entirely with concrete. 
 This process was adopted in form- 
 ing some of the piers of the bridge 
 of the Shore Line Eailroad across 
 the Connecticut River at Say- 
 brook, under Oushing's patent. 
 
 Pneumatic Piles. The vac- 
 uum process in which the hollow 
 pile of cast- or wrought-iron was 
 sunk, by capping it and exhaust- 
 ing the air within, and thus load- 
 ing it with the pressure of the ex- 
 terior atmosphere, gave place to 
 the plenum process, which was 
 adopted for the old piers of the 
 Third Avenue bridge across the 
 Harlem River, of which Fig. 1030 
 is a section. It conists of a pipe 
 
 in two sections ; the upper one is called the air-lock, which can be connected 
 with either the lower chamber or the atmosphere. The lower chamber being 
 under pressure and shut off from the air-lock, the workman passes from the 
 outer air into the lock, closes the door, opens the pipe connection between the 
 two compartments, and, when the pressure becomes equal, opens the lower door 
 and passes down. 
 
 1027 _ 
 
 y 
 
 i M i i i i rr 
 
 FIG. 1028. 
 
 FIG. 1029. 
 
 Mr. McAlpine enlarged the bottom of the cylinder to a conical form, and 
 also added largely to the bearing surface by poling-boards driven obliquely out- 
 ward. After the excavation was completed, a strong course of concrete was 
 laid, and, when set, the air-lock was taken off and the balance of the concrete 
 filling was done in open air. 
 
432 
 
 ENGINEERING DRAWING. 
 
ENGINEERING DRAWING. 
 
 433 
 
 The pneumatic caisson has superseded the pneumatic pile. The system of 
 air-lock is the same, but, instead of sinking several piles and then combining 
 them with a structure, the caisson em- 
 braces the whole pier and starts from 
 the bottom. The pneumatic caisson 
 is an inverted boat or diving bell of 
 timber, which forms the working or 
 air chamber, connected with the upper 
 air by pipes and air-locked shafts for 
 the ascent and descent of the work- 
 men and for the removal of waste 
 from, and the delivery of material 
 into, the working chamber. 
 
 Figs. 1031, 1032, and 1033 are the 
 plan and partial sections of a late form 
 of the framing of a caisson. The ceil- 
 ing of the working chamber is of 
 plank, and between the first and sec- 
 ond courses of beams there is a double- 
 plank floor, which are calked and then 
 coated with pitch, and the spaces be- 
 tween the timbers filled with concrete. 
 
 Fig. 1240 is a section of the pier 
 of the Bismarck bridge. The sand 
 was removed from the air chambers 
 by water electors. As it is removed 
 
 , , , , . . FIG. 1030. 
 
 from the chamber, the masonry sinks 
 
 the caisson, and when it reaches the bottom the space is filled with concrete 
 or with sand. If the top of the caisson, when first sunk, does not reach the 
 surface of the water, a curb is formed on the top to a height sufficient to per- 
 
 w 
 
 I 
 
 1 
 
 \ 
 
 \ 
 
 X 
 
 
 1 
 
 1 
 
 <4/'r Si//7/y/v 
 
 FIG. 1034. 
 
 FIG. 1035. 
 
 mit the construction of the masonry in the open air. To preserve the position 
 of the air-lock during the whole construction there is an offset in the pipe at 
 29 
 
434 
 
 ENGINEERING DRAWING. 
 
 FIG. 1036. 
 
 FIG. 1037. 
 
ENGINEERING DRAWING. 
 
 435 
 
 this point, and necessarily an inconvenience in the change of movement of ma- 
 terial and workmen, but in later constructions this is avoided. 
 
 Fig. 1034 is a recent plan and section of a main shaft, through which access 
 to the caisson is obtained by the workmen ; each length consists of a cylindrical 
 shell 4 feet in diameter and 8 feet long, flanged at the bottom, with a head hav- 
 ing an opening in it; these lengths are added as fast as the caisson sinks 8 feet. 
 The head of the top and second section are provided with doors (Fig. 1035), 
 thus making the air-lock. When a length has been added the top door is re- 
 moved and placed on the top of the new length ; the lower door may then be 
 taken off and placed where the other has been removed. 
 
 Figs. 1036 and 1037 are plan and section of the Barr-Moran air-lock on the 
 excavating shaft. The top doors are double slides, tight-fitting and working 
 on cast-iron guides, operated by pistons driven by steam or by compressed air 
 from the caisson. The lower door is a flap-valve, balanced by a counterweight 
 and operated through a rocker-shaft extending through the air-lock, and a 
 quadrant and chain attached to the piston-rod of a steam or of a pneumatic 
 cylinder in connection with the caisson. In the figures both doors are shut. 
 Under atmospheric pressure the upper doors can be opened for the clear move- 
 ment of the bucket by a rope attached to the bail, the rope passes through a 
 stuffing-box, around which the doors close when shut. With the upper doors 
 shut, and the air in the lock brought to caisson pressure, the flap-door can be 
 opened and the bucket lowered to the bottom. At the bottom of the shaft 
 there is a flap-door which can be raised when it is necessary to repair the air- 
 lock or to raise it by adding another length of pipe. 
 
 The air-lock at the top of the shaft adds to the security of the workmen 
 and to the easier supervision of the machinery, while the single straight pipe 
 gives greater facility to the movement of men and materials. 
 
 In the freezing process, invented by F. H. Poetsch, of Austria, the site of 
 the foundation is inclosed with large vertical pipes sunk by a water jet ; within 
 these pipes, which are closed at the lower end and sunk to the proper depth, a 
 small pipe open at the bottom is inserted, through which is forced a freezing 
 mixture (as chloride of calcium), returning through the outer pipe. In this 
 way a curb of ice is formed, inside which the earth is removed for the foundation. 
 In Fiusterwalde, Austria, 12 
 tubes of 8" diameter were 
 sunk through 115 feet of 
 quicksand, in a circle of 
 about. 14 feet diameter, for a 
 shaft 8 feet diameter. After 
 the brick lining was laid the 
 tubes were withdrawn, a hot 
 circulation freeing them from 
 the ice. 
 
 This process has been used 
 successfully in this country. 
 
 Retaining -\\a\\s are such 
 as sustain a lateral pressure 
 from an embankment or head of water (Figs. 1038 and 1039). The width of 
 
 FIG. 1038. 
 
 FIG. 1039. 
 
436 ENGINEERING DRAWING. 
 
 a retaining- wall depends upon the height of the embankment which it may 
 have to sustain, the kind of earth of which it is composed (the steeper the nat- 
 ural slope at which the earth would stand, the less the thrust against the wall), . 
 and the comparative weight of the earth and of the masonry. The formula 
 given by Morin for ordinary earths and masonry is b = 0-285 Ti -\-li' \ that is, to 
 find the breadth of a wall laid in mortar, multiply the whole height of the em- 
 
 285 
 bankment above the footing by ; for dry walls make the thickness one 
 
 fourth more. 
 
 Most retaining-walls have an inclination or batter to the face, sometimes 
 also the same in the back, but offsets (Fig. 1038) are more common. The 
 usual batter is from one to three inches horizontal for each foot vertical. To 
 determine the thickness of a wall having a batter, " determine the width by 
 the rule above, and make this width at one ninth of the height above the base." 
 
 The formulas for the thickness of retaining-walls are very complicated. En- 
 gineers make use of some general rules as above, and depend on their experi- 
 ence for any modification. The top of the wall should not be less than 2 feet, 
 and in climates subject to frost it will be impossible to secure permanently the 
 upper part. Where the soil is saturated with water, it is usual to put weep- 
 holes at or near the bottom to relieve the pressure against the back of walls laid 
 in mortar. 
 
 Buttresses and counterforts in the rear of a wall of which the construction 
 requires a uniformity of thickness are only considered equivalent to increasing 
 the strength by the mean amount added to the horizontal section of the wall ; 
 but when the buttresses are on the line of the bridge trusses, they add to the 
 strength by the better distribution of the weight of the truss and by the secu- 
 rity which the weight of the truss gives at this point to the wall. The but- 
 tresses should be well bonded to the wall. 
 
 Dams are constructed to pond water for the supply of cities and towns ; for 
 inland navigation, by deepening the water over shoals, and the feeding of ca- 
 nals ; for power in its application to mills and workshops ; and for irrigation. 
 To whatever purpose the water is to be applied, there are two questions to be 
 settled : Whether the level will be raised high enough by the construction, and 
 whether the flow of the stream is sufficient for the purpose required ; and, fur- 
 ther, it may often be important to know how large a pond will be thus formed, 
 how ample a reservoir to balance unequal flows or intermittent use. If the 
 pond be small, so that the water can not be retained, and the supply is only 
 the natural run of the stream, then the minimum flow of the stream ds the 
 measure of its capacity. 
 
 The rule that obtains on the Merrimac Eiver, at Lowell and Lawrence, 
 where the pondage is more than the average, is that 1 cubic foot per second 
 per day of 12 hours per square mile of water-shed can be depended on for per- 
 manent mill-power. On small streams it happens that comparatively large 
 pondage may be secured, and the supply be equal to one half the rainfall. 
 
 Blodgett, in his " Climatology of the United States," says that " in this 
 sense of permanence as a physical fact we may consider the quantity of rain 
 for a year as a surface-stratum, on the Alantic slope and in the Central States 
 of 3 feet, which may be diminished to half this quantity, or increased to twice 
 
ENGINEERING DRAWING. 
 
 437 
 
 as great a depth in the extreme years. The evaporation from a water surface 
 is now usually considered equal to that of a rainfall ; therefore in the estimate 
 of the water-shed available for pondage the area of the reservoir is not taken 
 into account; the quantity of rain falling upon it is offset by the evaporation. 
 
 The usual form of dams for small streams and but little fall is to build a rub- 
 ble wall across the stream and secure the up-stream side with an earth of loamy 
 gravel puddled with water to the consistence of mortar or rammed in, the top 
 where the water is to flow over or through being protected by tight planking, 
 around or beneath which the water can not leak. 
 
 Lake McMillan dam, built across Pecos River, Colorado, intercepting its 
 entire flow for the purpose of irrigation, is 1,686 feet long, 54 feet high, with 
 an estimated water capacity of 1,000,000,000 cubic feet. Fig. 1040 is a section 
 
 FIG. 1040. 
 
 showing its construction, a heavy rock fill with a puddle slope in reservoir 
 front. There is an ample spillway or waste channel by which all surplus water 
 will be discharged, with no flow over the dike. The South Fork, Pa., dam, 
 similar in construction to the above, gave way from a flow over the top of the 
 dam, due to an insufficient waste and an extreme freshet. 
 
 In the " Transactions " of the A. S. C. E., December, 1892, James D. Schuy- 
 ler, a member, gives a description of the asphalt lining of reservoirs for the city 
 of Denver. The excavation consisted of 2' to 3' sandy loam, 12' to 15' hard clay 
 impregnated with alkali, and 8' to 12' of shale. On the completion of the em- 
 bankment the slopes were sprinkled and rolled with a roller of 5 tons, drawn up 
 and lowered by an engine. Beginning at the bottom, the slopes were laid in 
 horizontal strips of asphalt 10' wide and about If" thick spread with hot rakes, 
 tamped with hot tampers, and smoothed with hot smoothing-irons. While the 
 sheet was still warm anchor spikes of strap iron 1" X ", 7" to 8" long, were 
 driven through it into the bank, in rows about 12" centres, alternately flush 
 with the surface and projecting 1" ; lumber strips of 2" X 4" were placed loosely 
 above them on the slope for the workmen. After the finishing coat was applied, 
 the projecting spikes were driven flush and painted over. The bottom thick- 
 ness was 1", spread, tamped, and rolled with a cold roller of 5 tons. The finish- 
 ing coat of refined Trinidad asphalt, fluxed with residuum oil, was poured on 
 from hot buckets, and ironed over with smoothing-irons heated to a cherry-red. 
 
 Dikes or dams over which there is no flow of water can be made entirely of 
 earth. It is sufficient that the material be made more compact than the natu- 
 ral earth in which the dam is built, that it be of sufficient section to resist the 
 pressure, and width to obstruct the flow of water through it, and reduce the 
 percolation to a safe and economical limit, the passage of water between the 
 particles of earth being like that through very small broken and crooked pipes. 
 
438 
 
 ENGINEERING 
 "5 
 
 DRAWING. 
 
 Dikes across salt marshes are 
 made of material taken from the 
 marsh at some distance from the 
 site of the dike, well packed in thin 
 layers on a base prepared on the 
 soil without excavation. Sand and 
 gravel, being heavier than the moist 
 material, break through it and set- 
 tle to the bottom, involving often 
 the construction of a large em- 
 bankment, while by the use of a 
 homogeneous material the founda- 
 tion is not displaced but compressed. 
 
 Fig. 1041 is a section of the 
 dike or embankment for the Ashti 
 Tank or Keservoir, constructed for 
 retaining water for irrigation pur- 
 poses in India. The following is 
 an abstract of the description of 
 the work given in the " Minutes 
 of the Proceedings of the Insti- 
 tute of Civil Engineers," vol. Ixxvi : 
 
 "The net supply available for irri- 
 gation may be calculated thus : 
 Available capacity 
 
 of tank 1,348,192,450 cub. ft. 
 
 Deduct loss by 
 
 evaporation, etc. 233,220,240 " 
 
 Net supply availa- 
 ble for irrigation 1,114,972,210 " 
 
 " Area of catchment basin nearly 92 
 square miles." 
 
 The total length of the dam is 
 12,709 feet ; the breadth at the top, 
 which is uniform throughout, 6 
 feet ; breadth at full supply-level, 
 42 feet; height of the top of the 
 dam above full supply-level, 12 feet; 
 greatest height of dam, 58 feet. 
 The seat of the dike throughout 
 was cleared of vegetable mould, 
 stones, and loose material, all trees 
 and shrubs with their roots being 
 completely grubbed or dug out. 
 The puddle-trench laid in the nat- 
 ural ground is rectangular in cross- 
 section, 10 feet in width, excavated 
 
ENGINEERING DRAWING. 
 
 439 
 
 through various materials to a compact water-tight bed, and then filled in with 
 puddle material, consisting of two parts of sand and three parts of black soil, 
 carefully mixed and worked by treading with the feet, and then kneaded into 
 balls and thrown or dashed into the trench in layers up to 12 inches in thick- 
 ness. The puddle was brought to a level of 1 foot above the ground. Across 
 the river the trench was cut down to the rock and filled with concrete. 
 
 The general distribution of the material of the dam is shown in the figure. 
 The central core is formed of the best black soil attainable ; on each side, ex- 
 tending to the surface of the mixed material, brown, reddish, or white earth is 
 used. The outer part of the dike is formed of a mixture of equal parts of 
 black soil and sand. The black soil may be described as a clayey earth, tena- 
 cious and adhesive when wet a product of the decomposition of volcanic rock. 
 The brown and reddish soils are of a clayey nature, but contain admixtures of 
 fine sand, nodules, and thin layers of fine grains of lime. The white soil con- 
 sists of finely powdered particles of 
 a grayish color, similar to wood 
 ashes, which when dry possesses lit- 
 tle adhesion, but when wet is ad- 
 hesive. 
 
 The various soils were laid in the 
 work in layers 8 inches in thickness, 
 every layer being thoroughly watered 
 and rolled with iron rollers. The 
 outer slope was protected by a mix- 
 ture of equal parts of soil and sand, 
 
 and with sods of grass, laid about 3 feet apart, which in time extended over the 
 whole slope. 
 
 The inner slope is protected from the action of the waves by being pitched 
 or faced with dry stone, set by hand, and laid on a layer of coarse sand. The 
 stones of the pitching were bedded on the slope, and were laid with their 
 broadest end downward (Fig. 1042), each stone being roughly squared with the 
 hammer, and touching for at least 3 or 4 inches. The interstices were then 
 packed with small stone-chippings, and finished off with sand. 
 
 P 
 
 FIG. 1042. 
 
 FIG. 1043. 
 
 Fig. 1043 is the section of a crib-dam in northeastern Colorado for the 
 pondage of water for the purposes of irrigation. The crib-work is of round 
 logs, 10" at least in diameter, joined at the end as in ordinary log huts, with 
 
440 ENGINEERING DRAWING. 
 
 dovetail or tongue. Each crib is 18 feet long on the face, and the fastenings 
 are 2" X 18" treenails. The cribs are set radially, forming a curve up-stream 
 of 200 to 238 feet radius. The crib gives the stability, but the water-tightness 
 depends on a shutter, p, or vertical panel of timber, and the filling of earth on 
 the up-stream side. 
 
 For dikes where water does not flow over the top a construction similar to 
 Fig. 1043 is very strong and in many places the most economical, but without 
 any wood, which is likely to decay or rot when exposed to the air. In con- 
 struction it consists of a mass of masonry laid dry, with a nearly vertical up- 
 stream face pinned and pointed with cement mortar and again faced with a 
 concrete or cement wall in close connection with the wall face and mortar 
 bonded with it. The face of the concrete or cement wall is plastered with a 
 light coat of cement and protected against the wash of the water or thrust of 
 the ice by an earth embankment. This embankment adds to the security of 
 
 FIG. 1044. 
 
 the dam by cutting off seams in the rock foundations and to the stanchness 
 of the cement wall. 
 
 Fig. 1044 is a section of the dam across the Connecticut River, at Holyoke, 
 Mass. This dam is 1,017 feet long between abutments, averages 30 feet high 
 by a base of 80 feet and is constructed of timber crib-work, loaded with stone 
 for about one third its height. The foot of each rafter is bolted to the ledge, 
 and all their intersections are treenailed together with 2" white-oak treenails. 
 The inclined plank face is loaded with gravel, and the joint at the ledge cov- 
 ered with concrete. The lower or base tier of ranging timbers are 15" X 15", 
 the other timbers 12" X 12". The rafters are placed vertically over each othe,r, 
 in bents of 6 feet between centres. The planking is of hemlock 6" thick, with 
 oak cross planking at crest of dam 4" thick at bottom and 8" at top. The crest 
 is plated with iron " thick, 5 feet wide. During the construction the dam 
 was planked first about 30 feet on the incline ; a space was then left of about 
 16 feet width by sufficient length, through which the water flowed ; and the 
 balance of the dam was then completed. A plank flap was then made for the 
 opening, and when everything was ready it was shut down and the pond filled. 
 The objection to the flap construction is that the space left for the waterway 
 through the dam, after its completion, serves as a duct for air from below 
 which softens and rots the timber-work. 
 
ENGINEERING DRAWING. 
 
 441 
 
 When the dam was first contemplated the longer time and extra cost re- 
 quirud for a masonry construction turned the scale in favour of crib-work. 
 Some twenty-five years after its completion it was found that the water over- 
 fall from the crest was cutting out the ledge beneath the dam, and a crib-apron 
 was constructed entirely across the river, sheathed with plank on a slope of 
 about 2| to 1 from the crest downward. From the decay and weakening of the 
 timbers of the old darn, continual repairs have been required, and it is now de- 
 cided to put in a masonry dam, the wooden dam having served its purpose for 
 some fifty years. 
 
 Fig. 1045 is a section of the dam across the Croton Eiver, constructed under 
 the direction of Mr. John B. Jervis, for the supply of the aqueduct for the city 
 of New York. This dam was built on an earth foundation, with curved roll in 
 
 FIG. 1045. 
 
 cut stone, extending by a timber-apron some 50 feet, supported by strong crib- 
 work. Originally there was a small supplementary dam farther down to set the 
 water back on to the crib-apron, but this was washed out, and the crib- work is 
 replaced by heavy rock pave. In the erection of this dam all loose material was 
 removed, and then the cribs C and D were built up and the tops were planked ; 
 on this planking were carried up the cribs F and G. Between these piers the 
 space E, as well as e below and on the cribs, was filled in with concrete ; on this 
 the body of the dam was erected in stone-masonry, laid in cement. The face- 
 work of granite is cut to admit of a joint, not exceeding T 3 ^ of an inch. The 
 radius of the granite face is 55 feet, and the dam 38 feet high from level of 
 apron to crest of dam. This dam has been in use fifty years and is in very good 
 condition and tight. 
 
 Fig. 1046 is a section of a part of the dam across the Merrimac River, at 
 Lowell, built under the direction of Mr. James B. Francis. It was laid dry, with 
 the exception of the upper face and coping, which was laid full in cement. 
 The horizontal joints at the crest were run in with sulphur. The coping- 
 stones were dowelled to the face and together, and clamped to an inclined stone 
 
442 
 
 ENGINEERING DRAWING. 
 
 on the lower slope ; the end-joint between these stones was broken by making 
 every alternate lower stone longer, and the upper shorter, than shown in the 
 drawings. 
 
 The Cohoes dam (Fig. 1047) was built by W. E. Worthen, C. E., directly 
 below an old dam of somewhat similar construction to that of Holyoke. The 
 
 SCALE : } inch = 1 foot. 
 
 old dam had become very leaky and worn, and the overfall had in many places 
 cut deep into the rock, and in some places within the line of the dam. It was 
 therefore proposed to make the new dam, as a roll to the old one, to discharge 
 the water as far from the foot of the dam as possible, and to keep the old dam 
 for the protection of the new. The exterior of the dam was of rock-faced ash- 
 lar ; the caps were in single lengths of 10 feet, and none less than 15" thick and 
 2 feet wide ; they were dowelled together with two galvanized wrought-iron 
 dowels each. The whole work was laid full in cement ; the 20" face was laid 
 first to divert the leak of the old dam from the new work. The whole was 
 brought up to the outline, to receive the cap-stones, which were bedded in ce- 
 ment ; the top-joints were then run or grouted in neat cement, to within about 
 6" of the top of the stone, which was afterward run in with sulphur. Entire 
 length of overfall, 1,443 feet ; average depth below crest of dam, 12 feet. 
 
 Where the body of water which may at any time discharge over the dam is 
 largo and the fall high, it is especially desirable to secure a location where the 
 overfall can be upon solid rock. If there be a ledge at the side of the river, and 
 none can be found in the channel, it is often better to make a solid dike across 
 the river and above the level of freshets, and cut the overfall out of the bank. 
 
ENGINEERING DRAWING. 
 
 443 
 
 FIG. 1047. 
 
 When the dam can have only an earth foundation, an artificial apron, or plat- 
 form of timber or rock, is to be made, on which the water may fall, or the high 
 fall may be broken up by a succes- 
 sion of steps. In some cases a roll 
 or incline, like that given in the 
 Croton dam, is extended to the bed 
 of the stream, and continued by an 
 apron. The water thus rolls or 
 slides down, and takes a direction, 
 as it leaves the apron, parallel with 
 that of the bed of the stream. But FIG. 1043. 
 
 care must be taken to protect the outer extremity of the apron by sheet-piling 
 and heavy paving, as the current, by its velocity, takes with it gravel and all 
 small rocks, and undermines the apron. 
 
 FIG. 1049. 
 
444 
 
 ENGINEERING DRAWING. 
 
 FIG. 1051. 
 
 To retain the flow of rivers in dry seasons when the ponding will have little 
 or no effect on works farther up the stream, flash-boards are used, which usually 
 
 consist of an iron bolt driven 
 into the crest of the dam, 
 against which common boards 
 are raised to be swept off when 
 the river rises unexpectedly. 
 To control the levels of the ca- 
 nal, hand flash-boards are used, 
 as in Fig. 1048, sliding in per- 
 manent grooves. 
 
 Figs. 1049, 1050, and 1051 
 are plan, elevation, and section 
 of the Beetaloo dam, South 
 Australia. The dam is built of 
 concrete, 580 feet long, 110 feet 
 
 high, 110 feet thick on a level 
 with the bed of the creek, and 
 14 feet thick at the top. The 
 cross-section is in accordance with Prof. Kankine's formula, the horizontal 
 curvature having a radius of 1,414 feet. The structure is founded on rock 
 and has a spillway 200 feet long by 5 feet deep, its channel below being di- 
 vided by walls into three sections, as shown by the drawings. 
 
 Head-gates are constructions necessary to control the flow from the river- 
 pond or reservoir into the canal or conduit by which the water is to be con- 
 veyed and distributed for the purposes to which it is to be applied. The top 
 of the works should therefore be entirely above the level of the highest freshets, 
 that no water may pass, except through the gates ; and it is better that the 
 opening of the gates should be entirely below the level of the top of the dam, 
 to prevent as much as possible the passage of drift and ice, which are often ex- 
 cluded by booms and racks placed outside the gates. 
 
 Figs. 1052, 1053, and 1054 are drawings, in plan and detail, of the head- 
 gates, and the machinery for hoisting them, at the Cohoes Company's dam. 
 There are ten gates, four 8' x 6' 6" and six 8' x 9' in the clear ; all can be 
 hoisted by machinery connected with a turbine- wheel at a, or separately by hand. 
 At b there is an overfall, at the same height as the dam, over which any drift 
 that is brought against the gate-house is carried. At c there is a similar over- 
 fall within the gates, and another at d, by which any sudden rise of the level 
 of the canal is prevented. At e there is a gate for drawing down the pond, and 
 another at /, for drawing off by the canal, both raised and lowered like the 
 head-gates. The head-gates are of solid timber bolted together, moving in 
 cast-iron guides set in grooves in the stone ; in front of these grooves there is 
 another set of grooves (g g}, which are intended for slip-planks or gates, to be 
 put in whenever it is necessary to shut off the water from the gates themselves 
 in case of repairs. 
 
 Hoisting Apparatus. To each gate there are strongly bolted two cast-iron 
 racks, geared into two pinions on a shaft extending across the gate-space, and 
 supported on cast-iron standards on the piers. At one extremity of this shaft 
 
ENGINEERING DRAWING. 
 
 445 
 
 a? 
 
 H 
 ^ 
 fei 
 "1 
 
 Cl 
 
446 
 
 ENGINEERING DRAWING. 
 
ENGINEERING DRAWING. 
 
 447 
 
 there is a worm-wheel, driven by a worm or screw on a shaft perpendicular to 
 the pinion-shaft. The worm-shaft can be driven either by a hand-wheel at one 
 end, or by ihe friction-bevel at the other. The friction-bevel can be driven in 
 either direction by being brought in contact with one or other of the friction- 
 bevels on a shaft extending the whole length of the gate-house, and in gear 
 directly with the small turbine at a. The small turbine draws its supply through 
 a pipe, built in the walls, and opening into the space between the gates and the 
 slip-plank groove. 
 
 In the guard gates at Lowell, instead of racks attached to the gates they are 
 supported by strong rods with screws cut at the upper ends and are raised by 
 
 FIG. 1055. 
 
 nuts in the hubs of a pair of horizontal gears driven by a pinion between the 
 two on an upright shaft, connected with the gearing of a turbine water-wheel. 
 In late constructions the gates are raised by hydraulic jacks in connection 
 with the city water mains. 
 
 Small gates in canals are usually made of wood, with wooden starts on which 
 a rack or racks are planted and hoisted by hand through a crank and pinion 
 shaft. Fig. 1055 is a perspective of the hoisting apparatus of such a gate with 
 a single stem or start. The pinion is driven through a horn wheel and lever. 
 The lever is pressed down and the gate raised ; when this movement is stopped 
 a dog catches a ratchet on the back of the wheel and the gate is held. The 
 lever is slipped outward, and is brought in contact with another horn, with 
 another depression and raise of gate. The fulcrum of the lever is the centre 
 pin of the wheel. In dropping the gate the lever holds it in position, the dog 
 is thrown out, the lever thrown out from the horn, and the gate drops from its 
 own weight ; if it sticks, it can be forced down by the reverse movement of the 
 lever. 
 
 Gates seldom used are raised by chains over a barrel by handspikes, and 
 held by ratchet and dog and dropped as above ; but for gates at the head of 
 
448 
 
 ENGINEERING DRAWING. 
 
ENGINEERING DRAWING. 
 
 449 
 
 flumes leading to wheels the provision for their movement must be positive in 
 both directions, as they are of frequent use. 
 
 Figs. 1056 and 1057 are the elevation and section of flume head-gates as 
 manufactured and used at Holyoke, Mass., for such positions. G Gr are plank 
 gates sliding laterally, moved by two pinions working into racks at the top and 
 bottom of the gates, turned by a capstan bar on a horn wheel head. F is the 
 flume, circular, of wooden staves or wrought-iron plates. P is a paddle gate by 
 which the flume must be filled slowly, and A the pipe for escape of air from 
 the flume during filling. 
 
 The Cheney head-gates (Figs. 1058 and 1059), as applied to several of the 
 water gates of the canals at Lowell, are gates supported by wheels, which run 
 
 FIG. 1058. 
 
 FIG. 1059. 
 
 on upright cam shafts on each side of the gate ; when the gate is moved either 
 up or down, the cam shafts are turned to raise the gate from the flat-closing 
 surface, and take the whole water pressure, which is, in a measure, relieved by 
 the opening of this joint, and the whole friction is transferred to the wheels 
 and the gate readily raised. 
 
 Fig. 1060 is the elevation of a circular tube of plate-iron, as used for a 
 waste gate in the canal of the Connecticut River Company at Windsor Locks. 
 It is of the form of a hollow plug, largely used for bath tubs, and now called a 
 standard waste. The tube is 8 feet diameter and 9 feet high. The joint be- 
 tween the plug and the pipe extending to the river, is made by angle irons A A. 
 The movement of the plug vertically is controlled by radius bars working in 
 30 
 
450 
 
 ENGINEERING DRAWING. 
 
 FIG. 1060. 
 
ENGINEERING DRAWING. 
 
 451 
 
 centres on the back wall ; they are made of channel bars, each set braced hori- 
 zontally. The plug is raised by a differential pulley-block suspended at C from 
 a wooden beam across the water. The hoist chain is carried by a yoke to the 
 two radii bars at B ; when eased, the plug drops readily to its seat. The level 
 of the canal is about 35 feet above that of the river, and the discharge is so 
 large that it produces a scour in the canal. When the level of the canal rises 
 above the crest of the tube it forms an overfall ; a depth of a little over 2 feet 
 will take the whole capacity of the tube. 
 
 Figs. 1061 and 1062 are the front elevation and section of the gates of Farm 
 Pond, Sudbury River Conduit, Boston Water- Works. The main web or plate 
 
 FIG. 1061. 
 
 Flo. 1062. 
 
 of the gate is 1" thick, the ribs 6" deep, the gate-stems 2" diameter. The 
 nuts by which the gates are raised are geared together, and actuated by a 
 double crank. The gates and guides are faced with brass, about -fo" thick. 
 
 Similar gates are very common, plates of cast-iron strengthened by ribs ; 
 the guides are also of cast-iron, bolted to the masonry, with faces of the gates 
 and guides usually of brass plates, as iron faces become rusty and stick. 
 
 Canals. The sections of canals depend upon the purposes to which they 
 are to be applied, whether for navigation or for power ; if for navigation, ref- 
 
452 
 
 ENGINEERING DRAWING. 
 
 erence must be had to the class of boas for which they are intended ; if for 
 power, to the quantity of water to be supplied, and sundry precautions of con- 
 struction. 
 
 Fig. 1063 is a section of the Erie Canal: width at water-line, 70 feet; at 
 bottom, 28 feet; depth of water, 7 feet; width of tow-path, 14 feet. The 
 
 FIG. 1063. 
 
 slopes are gravelled and paved, the edge of the tow-path is paved with cobble- 
 paving, and the path gravelled. The smaller canals of this State and of Penn- 
 sylvania are generally 40 feet wide at water-line, and 4 feet deep ; the Delaware 
 and Earitan, 75' x 7' ; the Chesapeake and Delaware, 66' x 10' ; the ship-canals 
 of Canada, 10 feet deep and from 70 to 190 feet wide. 
 
 The dimensions for canals for the supply of mills depend first, on the 
 quantity of water to be delivered. Their area of cross-section should be such 
 that the average velocity of flow should not exceed two to three feet per second, 
 and in northern climates less velocity than this would be still better ; it should 
 always be such that during the winter the canals may be frozen over ; and re- 
 
 FJG. 1064. 
 
 main so, to prevent the obstruction from drift and anchor-ice in the water- 
 wheels. The usual depths of the larger canals are from 10 to 15 feet ; with 
 such depths the cover of ice which reduces the section by the amount of its 
 thickness does not materially decrease the velocity of flow, nor diminish very 
 perceptibly the available head. 
 
 Fig. 1064 is a section of the Northern Canal, at Lowell, Mass., which may 
 be considered a model for large works. The width at water-line is 103 feet 
 and the depth 16', and is intended for an average flow of 2,700 cubic feet per 
 second. The fall in the whole length of 4,300 feet is between 2" and 3" ; when 
 covered by ice, about 4". The sides are walled in dry rubble, and coped by 
 split granite. The portion above, and about three feet below, the water-line, or 
 between the limits of extreme fluctuations of level, is laid plumb, that the ice 
 may have as free a movement as possible vertically. 
 
 Fig. 1065 is a section, on a scale of -J" = 1 foot, of the river-wall of this 
 same canal, where the canal passes out into and occupies a portion of the river 
 channel. The main wall is in dry masonry, faced on river-side with rough- 
 faced ashlar, pointed beds and end-joints. The inside lining is of two courses 
 of cement-wall, to the dry rubble wall pointed with cement, against which is 
 
ENGINEERING DRAWING. 
 
 453 
 
 laid the first cement lining, plastered on the inside, and the interior wall is 
 then laid ; the granite inside wall, above lining, is laid in cement. 
 
 FIG. 1065. 
 
 Locks of CanaU, Figs. 1066 and 1067 are portions of plan and vertical 
 section of locks, taken from the general plans for timber locks on the Chemung 
 Canal. They represent the half of upper gates. Fig. 1068 is a section of one 
 side of the lock of the same. Fig. 1069 is the plan of a portion of one of the 
 
 FIG. 1066. 
 
 enlarged locks of the Erie Canal, showing one of the upper gates and the side- 
 walls. Fig. 1070 is a cross-section of one of the same locks, showing the cul- 
 
454: 
 
 ENGINEERING DRAWING. 
 
 vert in the centre between the locks, used for the supply of the waste of the 
 lower level. The proper height of water in this level is controlled by gates in 
 the upper level. 
 
 FIG. 1067. 
 
 FIG. 1068. 
 
 SCALE : -f," - 1 foot. 
 
 FIG. 1069. 
 
ENGINEERING DRAWING. 
 
 455 
 
 Fig. 1071 is a drawing, in outline, of the hollow quoin of the lock-gate, on 
 a scale of ^ full size (Chemung Canal). 
 
 FIQ. loro. 
 
 SCALE : T V = 1 foot. 
 
 Fig. 1072 is a plan and elevation of pintal for heel-post of lock, with a sec- 
 tion of the bottom of the post. The pintal is imbedded in bottom timber or 
 stone, as the case may be. 
 
 FIG. 1071. 
 
 A full size. 
 
 FIQ. 1072. 
 
 Fig. 1073 is a plan and elevation of the strap for the upper part of heel-post. 
 Extracts from lock specifications (" New York State Canals," 1854) : 
 
 " Locks to be composed of hydraulic stone masonry, placed on a foundation of tim- 
 ber and plank. The chamber to be 18' wide at the surface of the water in the lower 
 level, and 110' long between the upper 
 and lower gate-quoins. The side-walls 
 to extend 21' above the upper gate- 
 quoins, and 14' below lower gate- 
 quoins. If the bottom is of earth, 
 and not sufficient to support the foun- 
 dation, then bearing-piles of hard 
 wood, not less than 10" diameter at 
 small end, shall be driven. The foun- 
 dation timbers to be 12" x 12", and of 
 such lengths as will extend from and 
 cover the outside piles, and to be tree- 
 nailed with a 2" white-oak or white- 
 elm treenail, 24" long, to each pile. FIG. 1073. 
 
456 ENGINEERING DRAWING. 
 
 "If without bearing-piles, the foundation to be composed of timber, 12" thick and 
 not less than 10" wide, counterhewed on upper side, placed at uniform distance, accord- 
 ing to their width, to occupy or cover at least of the area of the foundation, and 
 under the lower mitre-sill to be placed side by side : in all cases to be of sufficient 
 length to extend across the lock to the back line of the centre buttresses, and at the 
 head and foot to the rear or back line of wing-walls. The timber under the lower 
 mitre-sill to be of white oak, white elm, or red beech, the other foundation and apron 
 timber to be of hemlock. The foundation to be extended 3' above the face of the main 
 wall at the head of the lock, and at the foot from 25' to 30' below the exterior wing 
 that portion of the spaces between the timbers in all cases to be filled with clean coarse 
 gravel, well rammed in, or concrete. Where rock composes the bottom of the lock, and 
 is of such a character that timber is not required for the foundation, the same shall be 
 excavated smooth and level, and the first course of stone well fitted to the rock. 
 
 " Sheet-Piling. In all cases where rock does not occur, there shall be a course at the 
 head of the foundation, under each mitre-sill, and at the lower end of the wings, and at 
 the lower end of the apron, to be from 4' to 6' deep as may be required in each to ex- 
 tend across the whole foundation. 
 
 "Flooring. A course of 2^" pine or hemlock plank to be laid over the whole of the 
 foundation timbers, except a space, 3' wide, under the face-line of each wall to be 2" 
 white oak; the whole to be well jointed, and every plank to be treenailed with two 
 white-oak treenails at each end, and at every 3' in length, to enter the timber at least 5", 
 or with wrought-iron spikes, treenails to fill 1" bore. Platform for the upper mitre-sill 
 to be 5' 10" wide, and 6' high above foundation, and to extend across from side-wall to 
 side-wall, to be composed of masonry, coped with white-oak timbers, extending 6" into 
 each side-wall. Mitre-sills to be of best white-oak timber, 9" thick, to be well jointed, 
 and bolted to the foundation or platform timbers with bolts 20" long, 1" x 1", well 
 ragged and headed. 
 
 "Masonry. The main walls, for 21' 6" in length, from wing-buttresses at the head, 
 and 32' at lower end, to be 9' 8" thick, including recesses, and for the intermediate 
 space, 7' 8J" thick, with three buttresses projecting back 2', and 9' long at equal dis- 
 tances apart. The quoin-stones, in which the heel-post is to turn, shall not be less than 
 5' 6" in length in line of the chamber, to be alternately header and stretcher. The 
 recesses for the gates to be 20" wide at top of wall, 12' long, with sub-recesses, 9" wide, 
 6' high, 10' long, for the valve-gates. 
 
 " Culvert between Locks. The sluice-way shall be made in the head- wall with cut- 
 stone jambs, grooves to be cut in the jambs for the sluice-gates, the bottom of the aper- 
 ture to be of cut stone, with lower corner beveled off, over which the water will fall into 
 the well, the bottom of which shall be covered with a sheeting of cut stone, 6" thick. 
 
 " Second flooring of seasoned 2" first-quality white-pine plank, to be well jointed, and 
 laid on the foundation between the walls, from the breast-wall to lower end of main wall, 
 and also on the floor of the wall. 
 
 " Gates. The framing to be made of best quality white-oak timber; the cross-bar to 
 be framed into heel and toe posts with double tenons, each tenon to be 7" long, and se- 
 cured with wrought-iron Ts, well bolted. The heel and toe posts to be framed to the 
 balance-beam by double tenons, and secured by a wrought-iron strap and balance-rod, 
 from the top of the beam to the under side of the upper bar. The lower end of the 
 heel-posts to be banded with wrought-iron bars ; the collar and other hangings to be of 
 wrought-iron, secured together with a double nut and screw, and to the coping by bed- 
 ding the depth of the iron in, and by screw-bolts fastened with sulphur and sand-cement. 
 The pivots and sockets which support the heel -posts to be of best cast-iron; a chilled 
 cast-iron elliptical ball, 2i" horizontal, and 1" vertical diameter, to be placed on the 
 pivot and in the socket of each heel-post, gates planked with seasoned 2" white-pine 
 plank, jointed, grooved, and tongued tongues of white oak." 
 
ENGINEERING DRAWING. 
 
 457 
 
 Water, ponded by dams, and conveyed by canals for use as mill-power, is 
 carried within the workshops or manufactories, to be applied on water-wheels, 
 by some form of covered channels usually designated as flumes. The common 
 form of a flume for the conveyance of water to breast, overshot, or undershot 
 wheels, is of a rectangular section, framed with sills, side-posts, and cap, and, 
 if a large section is required, with intermediate posts. The sills are set, and 
 earth well rammed in the spaces between them ; the bottom plank is then laid, 
 posts and cap framed with tenon and mortice, set and pinned, and the plank is 
 then firmly spiked on the outside of posts and caps. The planks are usual- 
 ly partially seasoned and 
 brought to close joints; 
 the size of timbers will 
 depend on the depth be- 
 neath the soil, or the in- 
 sistent load. Within the 
 mill, and just above the 
 wheel, the flume is framed 
 without a cover, and the 
 posts and side-planks are 
 
 brought above the level of 
 
 FIG. 1074. 
 
 the water. This open 
 
 flume is termed the penstock, especially necessary, in the class of wheel above 
 
 referred to, to secure the full head of water. 
 
 For the conveyance of water to turbine-wheels, wrought-iron pipes are 
 almost invariably used. Cast-iron is also sometimes used, with flange, or hub 
 and spigot-joints. Plank-pipes (Fig. 1074) are also used, made with continu- 
 ous staves, and hooped with wrought-iron; these constructions are much 
 cheaper, and serve a very good purpose. The head-gates of flumes are placed 
 at the head of the flumes, in a recess back from the face of the canal, with 
 racks in front to prevent the passage of any drift that might obstruct or injure 
 the wheel. The total area of passages through the racks should liberally exceed 
 
 the area of cross-section of the flume, not 
 only on account of the extra lateral fric- 
 tion of the rack-bars, but also on account 
 of their liability to become obstructed. 
 Sometimes two sets of racks are placed 
 in front of the flumes, especially for 
 turbines and reacting wheels : a coarse 
 rack outside, with wide spaces, say 2" 
 and a finer one inside, say of f" to f" 
 spaces. 
 
 Conduits for the supply of water to 
 cities and towns are of masonry, or cast- 
 or wrought-iron pipes. Their capacity to 
 deliver the required quantity depends upon 
 the area and form of cross-section, and 
 
 the velocity of flow due to the loss of head or of pressure permissible ; this 
 velocity being due primarily to gravity, but largely modified by conditions of 
 
 FIG. 1075. 
 
458 
 
 ENGINEERING DRAWING. 
 
 structure, as the kind and amount of wetted surface, and length and directness 
 of line. 
 
 Fig. 1075 is a cross-section of the main conduit of the Nassau Water-Works 
 for the supply of the city of Brooklyn, Long Island. The width is 10' at the 
 springing of the arch; the side- walls 3 feet in height; versed sine. of invert, 
 8" ; height of conduit in centre, 8' 8" ; fall or inclination of bottom, 1 in 
 10,000. 
 
 The foundations to be of concrete, 15' wide, on earth, or, if the water was trouble- 
 some, on a platform of plank. The side-walls of stone, with an interior lining of 4'' 
 brick; arch brick, 12", and the invert 4" thick. The outside of arch, and each wall, 
 were plastered over on the outside with a thick coat of cement mortar. In both cuttings 
 and embankments the arch was covered with 4' of earth, with a width of 8' at top, 
 and slopes on each side of 1 to 1, covered with soil and seeded with grass. 
 
 Fig. 1076 is a section of the conduit of the Boston Water- Works. The 
 inside section is equal to a circle 8 feet diameter, and is uniform throughout 
 except in tunnels. The section given is the general one, resting on concrete, 
 brick lining at sides and invert at bottom, with an 8" arch at top for a 4' cover, 
 and 12" for exceptional depths or under railway-tracks. The lower corners 
 were of special brick. 
 
 The inclination of the conduit is 1 foot per mile, and the flow 80,000,000 
 gallons per 24 hours when full, or 5 feet above centre of invert. The maximum 
 flow takes place when the depth of 
 water is 7' 2", the delivery then being 
 109,000,000 gallons. 
 
 Fig. 1077 represents a section of 
 
 FIQ. 1076. 
 
 FIG. 1077. 
 
 the old Croton Aqueduct, in an open rock-cut. The width at spring of arch, 
 7' ; versed sine of invert, 6" ; height of conduit, 8' 6" ; fall or inclination of 
 bottom, about 1 in 5,000. 
 
 Fig. 1078 is a two half section of the new Croton Aqueduct, of which one 
 is in earth and rock, and the other entirely in rock. The foundation is in con- 
 crete, the lining and arches in brick. At the junction of the invert and side 
 linings there is a special angle block in brick ; exterior of brickwork is plastered. 
 
 Before the construction of High Bridge water was conveyed for the supply 
 of the city through siphon pipes beneath the Harlem Kiver, afterward through 
 two 3' cast-iron pipes in the masonry of the bridge. As the demand for water 
 increased in the city, the obstruction caused by lack of capacity in these pipes 
 made the introduction of a larger pipe necessary (Fig. 1079), which has been 
 
ENGINEERING DRAWING. 
 
 459 
 
 made of wrought-iron, fa" thick and 7' Qfa" in diameter, supported by cast- 
 iron columns which admit of a rocking movement, and slip-joints in the 
 pipe to compensate for any ex- 
 pansion or contraction. The 
 pipes are inclosed in a long cham- 
 ber, extending the whole length 
 of the bridge, covered by a brick 
 arch, laid in cement with a cover 
 of asphalt, and a brick pavement 
 over all. A A are the arch-stones 
 of the bridge. 
 
 In large works, where there 
 is considerable length of conduit, 
 receiving reservoirs, within or 
 near the limits of the city, are 
 necessary as a precaution to guard 
 against accidents which might 
 
 happen to conduit or dam, and cut off the supply, and also as a sort of balance 
 against unequal or intermittent draught among the consumers. The size of 
 these reservoirs must depend on the necessities of the case, and on the facilities 
 
 Fia. 1079. 
 
 for construction. At Eidgewood, Brooklyn, there are three reservoirs in con- 
 nection, of total capacity of about 325,000,000 gallons. The capacity of the 
 upper Croton reservoir in Central Park, New York, in two compartments, is 
 about 1,000,000,000 gallons. 
 
 Fig. 1080 is a section of the division-bank of the Croton reservoir, made of 
 earth, with a puddled ditch in the centre. 
 
460 
 
 ENGINEERING DRAWING. 
 
 Extracts from the specification : 
 
 "All the banks will have the inner and outer slopes of 1 base to 1 perpendicular. 
 All the inner or water-slopes will be covered with 8" of broken stone, on which will be 
 placed the stone pavement, 1J foot thick. The outer slopes will be covered with soil 1" 
 
 FIG. 1080. 
 
 foot thick. The banks, when finished, to be 15 feet on top, exclusive of the soil on the 
 outer slope. The top of the outer bank to be 4 feet above water-line ; the top of the 
 division-bank to be 3 feet below water-line. In the centre of all the banks a puddle- 
 bank will be built, extending from the rock to within 2 feet of the top of the outer bank. 
 It will be 6' 2" wide at top in division-bank, and 14' wide at top in exterior bank, and 
 16' wide at a plane 38' feet below top of exterior bank. In the middle of the division- 
 bank there will be built a concrete wall 4' high, 20" wide ; 8" thickness of concrete to be 
 laid on the top of the bank, on each side of, and connected with, this wall. On the 
 pavement 18" thick will be laid in concrete. The slope-wall on each side of the division- 
 bank, 10' in width, to be laid in cement. 
 
 "Puddle-ditches are to be excavated to the rock under the centre of all embank- 
 ments. The earth within the working-lines of interior slopes to be excavated to the 
 depth of 40' below top of exterior bank, rock 36'. The puddle-ditch to be formed of 
 clay, gravel, sand, or earth, or such admixture of these materials, or any of them, as the 
 engineer may direct, to be laid in layers of not more than 6", well mixed with water, 
 and worked with spades by cutting through vertically, in two courses at right angles 
 with each other; the courses to be 1" apart, and each spading to extend 2" into the 
 lower course or bed. The puddle to extend to all the masonry and pipes, as the engineer 
 may direct. 
 
 "The embankments to be formed in layers of not more than 6", well packed by cart- 
 ing and rolling, and, where rollers can not be effectually used, by ramming. The em 
 bankments to be worked to their full width as they rise in height, and not more than 2' 
 in advance of the puddle. The interior slopes of all the banks to be covered with 8" 
 thickness of stone, broken to pass through a 2" ring. On this to be laid the paving, 18" 
 in thickness, of a single course of stones set on edge at right angles with the slope, laid 
 dry, and well wedged with pinners." 
 
 Fig. 1081 is a drawing of a sheet-iron water pipe as used in the States of the 
 Pacific slope. The bottom joint is a slip-joint of the stove-pipe iron order, 
 which Mr. Hamilton Smith considers good for pressures not exceeding 380 
 feet. For pressures greater than this, the lead joint, as shown in Fig. 1082 
 in section, should be used ; it consists of an inner sleeve riveted to the inside 
 
ENGINEERING DRAWING. 
 
 461 
 
 of one pipe and an outer sleeve covering the joint with a f " space, into which 
 lead is run and calked as for cast-iron pipes (see page 463). 
 
 FIG. 1081. 
 
 The pipe as drawn is on an incline and is anchored by a wire cable. Un- 
 der heavy heads and on inclines the pipes should be wired together, and at 
 angles lead joints should be used. 
 
 The pipes are invariably coated with asphalt, inside and out, by immersing 
 them for some minutes in the hot liquid. The pipes are double riveted, and 
 the thickness of the iron very much less than used here ; the Cherokee pipe, 30 
 inches diameter up to 203 feet head, is No. 14, or -083" thick ; head 900 feet, 
 |" thick. 
 
 At first slip-joints were made for expansion, but were found unnecessary. 
 Time has shown no practical depreciation in the pipes. 
 
 For conduits of large size, and under extreme pressure, wrought-iron or 
 steel may be considered the rule in the United States ; they are more econom- 
 ical in cost and safer than masonry, less disturbance from settlement, practi- 
 cally without leak, and admitting of prompt and easy connections and repairs. 
 
 In construction pipes are made up at shops and riveted like boilers, and are 
 put together in such lengths as can be readily transported and laid. Welded 
 longitudinal joints on the separate lengths can be obtained without much in- 
 crease in cost, and flanges may be turned on them for bolting or riveting. 
 
 For water supplies to locomotives on our railroads, the usual tanks are 
 wooden cisterns strongly hooped and resting on timber frames. Fig. 1083 is a 
 tank of wrought-iron on the Orleans Railway, containing 22,000 gallons, and 
 applicable to small services of towns and villages. The inverted-dome bottom 
 has no supports except at the circumference, where there is a strong iron plate 
 with extra angle irons, as shown in detail. The pipes pass through the bottom, 
 but give it no support. This form has been applied at the Liverpool water- 
 works in the Norton Tower, where the tank is 82 feet in diameter and contains 
 651,000 gallons. 
 
4:62 
 
 ENGINEERING DRAWING. 
 
 Water-works for small towns often draw their supply from driven wells, 
 and for this service stand-pipes or tanks of iron are preferable to reservoirs in 
 earth ; but waters from these sources are usually deficient in oxygen, and con- 
 fervae develop rapidly in open reservoirs exposed to the light and air. They 
 must therefore be of small dimensions, so that there be no stagnation ; the 
 water must circulate and be changed often. These inconveniences, together 
 
 Fie. 1083. 
 
 with warmth in such metallic tanks in summer weather, has led to the making 
 of tight tanks as in the elevator service, under a pressure of about 100 pounds 
 to the square inch and earth covered. 
 
 Distribution. Figs. 1084 to 1088 are sections of the spigot and faucet ends 
 of cast-iron water-mains as used in Brooklyn and Philadelphia. 
 
 Below is given a table of dimensions, thickness, etc., of pipes, which may 
 be considered fair averages for water-mains ; 4" pipe is the smallest diameter 
 to be used in distribution, and if a hydrant is placed at the extremity of 500 
 feet of such pipe, the head will be very much reduced in case of fire service. 
 
ENGINEERING DRAWING. 
 
 463 
 
 In cities the minimum diameter is usually 6", except in the 4" connection of 
 dead ends for circulation. The thicknesses given in the table are as low as ad- 
 
 FIG. 1084. 
 
 FIG. 1085. 
 
 FIG. 1086. 
 
 FIG. 1087. 
 
 missible for tapping, and if the pipe is not of uniform thickness, is to be put 
 the thickest side up for this purpose. 
 
 The diameters given in the table are such as can 
 be found in stock, but other sizes will be made to 
 
 FIG. 1088. 
 
 Diameter. 
 
 Thick- 
 ness. 
 
 Weight per 
 foot laid. 
 
 Lead-joint 
 weight. 
 
 Safe, feet 
 head. 
 
 4 
 
 0-42 
 
 19-7 
 
 4 
 
 1,609 
 
 6 
 
 0-47 
 
 32-6 
 
 7 
 
 1,200 
 
 8 
 
 0-47 
 
 42-4 
 
 10 
 
 900 
 
 12 
 
 0-61 
 
 82-2 
 
 16 
 
 800 
 
 16 
 
 0-67 
 
 119-7 
 
 23 
 
 680 
 
 20 
 
 0-75 
 
 164-7 
 
 31 
 
 600 
 
 24 
 
 0-80 
 
 210-3 
 
 40 
 
 520 
 
 30 
 
 0-98 
 
 320-0 
 
 57 
 
 520 
 
 36 
 
 1-15 
 
 450-0 
 
 75 
 
 500 
 
 48 
 
 1-44 
 
 757-0 
 
 120 
 
 480 
 
 order ; if ordered of extra thickness the bore is preserved uniformly, and the 
 extra metal is put on the outside. 
 
 After laying, a hemp gasket is forced down to the lower end of the bell to 
 prevent the molten lead from escaping into the pipe ; the end of the pipe is 
 then stopped by a clay roll, or a rope covered with clay, or clay alone, or with 
 a metal clamp containing clay, and the molten lead is then poured in through 
 an aperture or gate at the top ; after cooling, the joint is made secure and tight 
 by compacting the lead with calking tools made for this purpose. 
 
 Wrought-iron pipes have also been used for distribution, generally with 
 cast-iron bells, riveted at one end and a small ring on the other (Fig. 1089) or 
 by flanges secured on both ends and bolted together when laid. 
 
 Specials. All parts of a main except the straight pipes are called special 
 castings. 
 
 Fig. 1090 is a 12" X 8" 4- way branch, shown full and in section, diagonally. 
 
464 
 
 ENGINEERING DRAWING. 
 
 The horns on the branch are for straps to hold in a plug, cap, or a connected 
 short or curved pipe. The 4- way branches are often called crosses, and the 
 
 FIG. 1091. 
 
 FTG. 1090. 
 
 FIG. 1092. 
 
 3-way T's, or single branches. The branches may be 
 of any appropriate size. In ordering, designate'diam- 
 eter of main pipe first, and then that of the branches. 
 It is very common in these pipes to make all of the 
 ends bell ends it saves sleeves when pipes are cut, as 
 they usually are at street intersections. 
 
 Fig. 1091 is a section of a sleeve for uniting cut 
 pipes or uncut spigot-ends ; a kind of double hub is 
 often used for the former. Sometimes sleeves are 
 made in halves, and bolted together. 
 
 Fig. 1092 is the section of a reducer for the con- 
 nection of pipes of unequal diameters. 
 
 Fig. 1093 is the section of a bend ; the horns on 
 the outer circle are for straps between the pipes, as 
 the pressure is unbalanced. 
 
 FIG. 1093. 
 
ENGINEERING DRAWING. 
 
 465 
 
 House-services are mostly lead pipes ; the taps on the mains for house-con- 
 nections in New York city are usually 1". 
 
 Fig. 1094 is a clamp sleeve. 
 Figs. 1095 and 1096 are globe specials 
 of the Builders' Iron Foundry, Providence, 
 
 FIG. 1094. 
 
 FIG. 1095. 
 
 FIG. 1096. 
 
 a T and a cross branch, which serve the purpose of Fig. 1090 and are much 
 lighter, and Figs. 1097 to 1102 are specials of the same makers. 
 
 FIG. 1098. 
 
 FIG. 1097. 
 
 FIG. 1099. 
 
 FIG. 1100. 
 
 FIG. 1101. 
 
 FIG. 1102. 
 
 According to the specifications of " Cast-Iron Distribution- Pipes and Pipe- 
 Mains, with their Branches," etc., Brooklyn, L. I. : 
 
 Every pipe-branch and casting to pass a careful hammer-inspection, to be subject 
 thereafter to a proof by water-pressure, and while under the required pressure to be 
 rapped with a hand-hammer from end to end, to discover whether any defects have been 
 overlooked. The pipes were to be carefully coated inside and outside with coal-pitch 
 and oil, according to Dr. R. A. Smith's process, as follows : 
 31 
 
466 ENGINEERING DRAWING. 
 
 "Every pipe must be thoroughly dressed and made clean from sand and free from 
 rust. If the pipe can not be dipped promptly, the surface must be oiled with linseed-oil 
 to preserve it until it is ready to be dipped ; no pipe to be dipped after rust has set in. 
 The coal-tar pitch is made from coal-tar, distilled until the naphtha is entirely removed 
 and the material deodorized. The mixture of 5 or 6 per cent of linseed-oil is recom- 
 mended by Dr. Smith. Pitch, which becomes hard and brittle when cold, will not an- 
 swer. The pitch must be heated to 300 Fahr., and maintained at this temperature 
 during the time of dipping. Every pipe to attain this temperature before being removed 
 from the vessel of hot pitch. It may then be slowly removed and laid upon skids to 
 drip." 
 
 Seivers. For the removal of waste water from houses and rainfall, sewers 
 are very convenient in towns and cities, even before the construction of water- 
 works ; but, after the introduction of a liberal and regular supply of water, 
 sewers are indispensable. The ruling principle in the establishment of sewer- 
 age-works is, that each day's sewage of each street and of each dwelling should 
 be removed from the limits of city and town promptly before decomposition 
 begins, and that it should not be allowed to stagnate in the sewers, producing 
 noxious gases prejudicial to health. To attain this end, the refuse fluids must 
 be sufficient in quantity to float and carry off the heavier matters of sewage. 
 
 There has been considerable discussion of late whether sewage and rainfall 
 should be carried off by a single system of pipes. This must depend largely on 
 the location, economy of construction, and the financial ability to carry out 
 the design. If the rainfall can be provided for by street gutters, the sewers may 
 be for the conveyance of house- waste only and very small. If the sewage is to 
 be discharged into a river of so large a flow that practically it will not pollute 
 it, or into large bodies of water not used for domestic purposes, it is cheaper 
 and better to discharge rainfall and sewage by one channel. The dimensions 
 of the sewer will depend on the quantity to be discharged and the grade for 
 which graphic sheets will be found in the Appendix. The rainfall is here 
 estimated at 1" in depth per hour for the whole area drained, but this is in 
 many places exceeded for excessive rainfalls, and it is well at such times to 
 have a partial relief by the street gutters. For the sewage flow it is usual to 
 calculate it from the water that is or may be furnished on any length, and 
 that the maximum that may be delivered at any one time is 50 per cent greater 
 than the average flow, and further that at that time the sewer should be one 
 half full. 
 
 The value of sewers depends on the correctness of their lines, uniformity of 
 descent, and smoothness of interior surface. 
 
 For Washington, Brooklyn, and New York, and many other large cities 
 discharging into large rivers, there is but one system of sewers for rainfall and 
 sewage, and for the great areas drained very large sewers are necessarily con- 
 structed. Fig. 1103 is a section of a Washington sewer, the largest in the 
 United States. The bottom course of the sewer, which is exposed to a strong 
 current, is of stone ; the ring courses are of brick three for the 13-foot sewers 
 and two for the 7-foot. 
 
 Fig. 1104 is a section of the largest Brooklyn sewer. In general there is no 
 base or side supports of concrete or wall, except where the bottom is of quick- 
 sand. The sewer at the upper end is of circular section 10' diameter, brick- 
 
ENGINEERING DRAWING. 
 
 467 
 
 work 12" ; next size 12', then 14', und 15' all brickwork 16". At the dis- 
 charge into Gowamis Creek the section is nearly rectangular with top of iron 
 beams and brick arches, brick side-walls, and inverts. 
 
 The smaller sewers in all cities and towns are of vitri- 
 fied ware or cement of circular section, or of cement of 
 
 FIG. 1103. 
 
 egg-shaped form, but beyond the diameter of 24 inches or its equivalent section 
 brick is generally used. The vitrified pipe is laid either with (Fig. 1105) or 
 
 without concrete base and side sup- 
 ports, depending somewhat on the 
 kind of earth through which it is 
 
 FIG. 1104. 
 
 FIG. 1105. 
 
468 
 
 ENGINEERING DRAWING. 
 
 laid and the desirability of uniformity or security of construction, but the 
 joints are laid full in cement with a re-enforce outside and wiped clean on the 
 inside. Up to 15" the pipes are usually bell and spigot, larger they are laid 
 with collars. The egg-shaped cement pipe is usually constructed with a sole 
 plate, which rests on a base of plank double thickness at the joints. The joints 
 of the pipe are struck with cement mortar inside and out, made very strong 
 
 FIG. 1106. 
 
 FIG. 1107. 
 
 outside and. very smooth inside, but usually without concrete backing, but 
 packed solidly in earth. Fig. 1106 is a section of an English form of sewer 
 cheap and strong. The bricks are laid with close joints inside and an exterior 
 coating of cement plaster. 
 
 This form of construction is not uncommon here with a central pipe of 
 vitrified stoneware. The concrete backing is very desirable to withstand the 
 shock of heavy moving loads, especially for pipes of large diameter. 
 
 Fig. 1107 is the usual form of 
 egg-shaped pipe, of which the pro- 
 portions are given in the table. 
 
 Thickness of brickwork, 8"; 
 boards shown at bottom only used 
 in cases of soft earth for convenience 
 
 EGG-SHAPED SEWERS. 
 
 Equal to a 
 circular 
 sewer of 
 
 D inches. 
 
 D and R' 
 inches. 
 
 R" inches. 
 
 Dimter. 
 
 
 
 
 24 in. 
 
 19-8 
 
 29-8 
 
 5-0 
 
 30 
 
 24-8 
 
 37-2 
 
 6-2 
 
 36 
 
 29-8 
 
 44.7 
 
 7-4 
 
 42 
 
 34-8 
 
 52-1 
 
 8-7 
 
 48 
 
 39-7 
 
 59-5 
 
 9-9 
 
 54 
 
 44-6 
 
 67-0 
 
 11-2 
 
 60 
 
 49-6 
 
 74-4 
 
 12-4 
 
 of construction. For area of egg- 
 shaped sewer of above section, multi- 
 ply R s by 4-6. 
 
 In general, brick sewers are only 
 well packed in earth, but when the 
 soil is loose and it is desirable to secure an extended base and side supports, 
 the form Fig. 1108 is adopted, which may be in rubble masonry or concrete, 
 and if the earth cover is light and the structure exposed to heavy moving loads 
 the masonry should be carried above the arch. At the bottom, where the arch 
 is of small radius and it is difficult to get good joints in the brickwork, it is 
 very common to place inverts of vitrified ware which are sold for this purpose. 
 Fig. 1109 is a section of a sewer constructed in situ at Mount Vernon, N. Y,, 
 through a long rock-cut. Concrete was rammed in around an inverted centre, 
 and after it was withdrawn the face was plastered with a very thin coat of 
 
ENGINEERING DRAWING. 
 
 469 
 
 FIG. 1108. 
 
 Portland cement ; the invert is of vitrified ware. A plumb J, is inserted across 
 the sewer (Fig. 1110) with points for centres at different heights, the radius 
 is uniform, and the arch turned as high as convenient for the rock sides and 
 
 FIG. 1111. 
 
 FIG. 1109. 
 
 FIG. 1110. 
 
470 
 
 ENGINEERING DRAWING. 
 
 street grades, either on levels and offsets or by inclines, the object being to se- 
 cure space in the sewer, and save the rock excavated for other city purposes. 
 Fig. 1111 represents the first joint from the sewer on house branches curved 
 
 and with a cover by means of which any 
 obstruction in sewer or branch could 
 be better reached and removed. 
 
 Man-holes are built along the line 
 of sewers, usually from 200 to 400 feet 
 apart, and at every junction and change 
 of direction, to give access to the sewers 
 for purposes of inspection and removal 
 of deposit. 
 
 
 FIG. 1112. 
 
 SCALE : J * = 1 foot. 
 FIG. 1113. 
 
 In the large Washington sewer the man-hole is constructed on the side of 
 the sewer ; in the Brooklyn one the aperture of the man-hole comes within the 
 exterior wall. In most sewers it is directly over them. 
 
 Figs. 1112 and 1113 are section and 
 plan of the man-hole at present used by 
 the Croton Sewer Department. It consists 
 of a funnel-shaped brick wall, oval at the 
 bottom 4', circular at top 2' diameter. 
 
 FIG. 1114. 
 
 Side-walls, 8" thick, through which the pipe-sewers pass at the bottom of the 
 well. Across the open space the sewer is formed in brick, whose bottom section 
 corresponds to that of pipe, side-walls carried up perpendicular to top of sewer ; 
 
ENGINEERING DRAWING. 
 
 471 
 
 the flat spaces at the sides of sewer are flagged. The top of the sewer is a 
 heavy cast-iron frame, fitted with a" strong cover, which may or may not be per- 
 forated, for ventilation. In the figure the main sewer is 12" pipe, with a 12" 
 branch entering at an acute angle, as all branches and connections with a 
 sewer should. The short lines on the left vertical wall represent sections of U 
 staples, built in to serve for a ladder. 
 
 When the T is at right angles to the through pipe it is very convenient to 
 form the base, as in Fig. 1114, in concrete to a wooden mould, with the inside 
 plastered. This form enables the sewer to be seen in all directions by a light 
 
 from man-hole to man-hole. Curved pipes 
 are now seldom laid ; angles are made at the 
 
 FIG. 1116. 
 
 FIG. 1117. 
 
 man-holes, with straight pipe between. When necessary, catch-basins (of 
 which Fig. 1115 is a vertical section, Fig. 1116 a half plan and a half sectional 
 plan) are placed at the corner of the street or at depressions of the street gut- 
 ters, to catch any heavy matter that might clog the sewer or fill the dock slips, 
 and is removed by hand. The cast-iron hood makes a seal, preventing the es- 
 cape of sewer gas through the sewer branch. Fig. 1117 is a perspective draw- 
 ing of the basin head at the corner of the street. 
 
 Where, by the difference of grades between the main line of sewer and the 
 branches, the latter are at a considerable height above the base of the man-hole, 
 it is best to lead the branch down by a vertical cast-iron pipe having a cross at 
 the top for access to the branch, and an elbow at the bottom to protect the ma- 
 sonry from the effects of the discharge. 
 
 Gas-Supply. Next in importance to the necessities of a city or town for 
 water-supply and sewerage is the luxury of gas-supply. The gas-works should 
 always be placed remote from the thickly populated part of the city, for under 
 the best regulations some gas (offensive and deleterious) will escape in the manu- 
 facture. They should be placed at the lowest level, for gas, being light, readi- 
 ly rises, and the portions of the city below the works are supplied at less pres- 
 sure than those above. Gas-mains, like those for water, are of cast-iron, and put 
 together in the same way ; but, as they have to resist no pressure beyond that of 
 the earth in which they are buried, they are never made of as great thickness as 
 those of water-pipes, but drips must be provided, and the pipe laid with such in- 
 clination to them that the condensed tar may be received in them and pumped out. 
 
 Owing to the reduction in price complete systems of wrought-iron gas-pipe 
 are now laid, with the usual screw-coupling and tested to a pressure much in 
 excess of that of the gas. 
 
4:72 
 
 ENGINEERING DRAWING. 
 
 WEIGHT OP GAS-PIPES PER RUNNING FOOT. 
 
 3* laibs. 
 
 4" 16 ' 
 
 6" 27 " 
 
 8".. ..,.. .40 " 
 
 10" 501bs. 
 
 12" 62 " 
 
 16" 103 " 
 
 20".. . 150 " 
 
 The holes for access to the valves on gas-pipes usually consist of two pipes, 
 the lower one with a saddle base resting on the pipe, the upper with a cap at 
 the top which is adjustable to the height of the pavement by screws cast on 
 both pipes, the upper one being the extension. 
 
 Roads and highways are terms applied distinctively to routes of land travel 
 in the country, though in general they may include city streets and avenues. 
 Lines of canal, river, lake, and ocean transit are often designated as highways. 
 
 The first requirements in the opening up of a country are good roads, which 
 must naturally be of the cheapest kind, but with the increase of population 
 they must be made more and more permanent, and the reflex action extends to 
 farming and mechanical facilities and increase in population. 
 
 For constant and continued use there can be no good road without drain- 
 age ; there may be seasons when hauling is practicable, but no road will stay 
 good unless the surplus water is got rid of, and if the road is well drained it 
 will pay the farmer to extend the drainage to the fields adjacent. 
 
 Fig. 1118 is a section of a dirt road in which there is a central drain, and 
 above it a 4" layer of coarse straw, hay, or stubble, the bed for which has been 
 excavated on slopes toward the drain of 1" to 2" to the foot. The earth should 
 then be filled above in layers and rolled, with slopes to the sides sufficient to 
 
 
 Fio. 1119. 
 
 carry off all the water from the centre. The width of a common road should 
 be 16 feet, slopes of about 8 inches to each side, grade not to exceed 9' to the 
 hundred, or about 5, and this not continuous, but broken by changes of grade. 
 Fig. 1119 is a section of a gravel road similar to the dirt road, but, as illus- 
 trative of a road with a steep grade, the cut across the turf borders are diagonal 
 to turn the water more readily into the ditches, while in the level road the cuts 
 are at right angles. If not convenient to construct the drains in the centre of 
 the road, they may be placed at the sides with the sub-slopes discharging into 
 them. Throughout our country there is at the present time a strong move- 
 
ENGINEERING DRAWING. 
 
 473 
 
 ment in favour of good roads, and machinery has been designed and constructed 
 by which the cost of construction has been very much reduced, with great im- 
 provement in the character of the roads. 
 
 Across marsh lands it is important that the surface of sods and roots should 
 not be broken, but that the weight of the roadway should be further distrib- 
 uted by a layer of brush, fagots, or poles, on which the layers of earth should 
 be sods from the marsh taken distant from the roadway, compacted in layers 
 and finished like the dirt road, but without gutters, or at such distance and 
 depth as not to weaken the road float. , 
 
 For country roads, wagons subject to heavy loads should have wide tires, 
 say 4" tires for the front wheels and 6" for the rear ones, and the paths of the 
 wheels should lap but not track, the axles of the rear wheels being longer than 
 those of the fore. 
 
 Near the sea, where the soil is sand or gravel and stone is not to be had 
 readily, oyster shells make an admirable road ; they are arranged in layers like 
 Macadam, and grinding together beneath rolling and travel, they make a com- 
 pact and solid cover. 
 
 Macadam was the first to reduce the construction of broken-stone roads to 
 a science, and has given the name, in his own and this country, to all this class 
 of roads. He says that " the whole science of artificial road-making^consists in 
 making a dry, solid path on the natural soil, and then keeping it dry by a dura- 
 ble water-proof covering." The road-bed, having been thoroughly drained, 
 must be properly shaped, and sloped each way from the centre, to discharge any 
 water that may penetrate it. Upon this bed a coating of 3" of clean broken 
 stone, free from earth, is to be spread on a dry day. This is then to be rolled, 
 or worked by travel till it becomes almost consolidated ; a second 3"-layer is 
 then added, wet down so as to unite more readily with the first; this is then 
 rolled, or worked, and a third and fourth layer, if necessary, added. Macadam's 
 standard for stone was 6 ounces for the maximum weight, corresponding to a 
 cube of 1", or such as would pass in any direction through a 2" ring. The 
 Telford road is of broken stone, supported on a bottom course or layer of stone 
 set by hand in the form of a close, firm pavement. 
 
 Early in the construction of the New York Central Park, after trials of the 
 Macadam and Telford roads, gravel was adopted and still maintains its position 
 for these roads. The gravel road, of which a cross-section of one half is shown 
 (Fig. 1120), consists of a layer of rubble-stones, about 7" thick, on a well-rolled 
 or packed bed, with a covering of 5' of clean gravel. C is the catch-basin for 
 
 FIG. 1120. 
 
474 
 
 ENGINEERING DRAWING. 
 
 the reception of water and deposit of silt from the gutters ; S is the main 
 sewer or drain, and s a sewer-pipe leading to a catch-basin on opposite side of 
 the road. In wider roads each side has its own main drain, and there is no 
 cross-pipe s. The road-bed was drained by drain-tiles of from 1" to 4" bore, 
 at a depth of 3' to 3' below the surface. The maximum grade of the Park 
 roads is 1 in 20. 
 
 In New York the streets above Fourteenth Street running north and south 
 are called avenues, and those at right angles, streets, and boulevard is applied 
 to very wide avenues in which there are rows of trees. The terms street and 
 avenue, as laid out, are the established bounds within which no buildings 
 may be erected. The street, therefore, technically includes the street or trav- 
 elled way for carriages, and the side- 
 walks and front areas. New York 
 streets above Fourteenth Street are 60 
 and 100 feet wide, avenues 100 feet, 
 of which the carriage-way occupies 
 one half (Fig. 1121), and the sidewalks 
 and area one quarter on each side. 
 The space occupied by areas is from 
 
 5 to 8 feet, which may be inclosed by iron fence ; the space beneath it and the 
 sidewalk to the curb and used for vaults can be leased from the city. The 
 stoop-line extends into the sidewalk beyond the area-line some 1' to 18",. fixing 
 the limit for the first step and newel to a high stoop or platform. The boule- 
 vard in the old line of upper Broadway and the Bloomingdale Eoad is 150 
 feet wide, of which 100 feet are to be carriage-way, and 25 feet on each side for 
 sidewalk and area, the latter not to exceed 7 feet ; one row of trees to be set 
 within the sidewalk, about 2 feet from the curb. 
 
 The best grade is from 1 in 50 to 1 in 100 ; this gives ample descent for the 
 flow of water in the gutters. Many of our street-gutters have a pitch not ex- 
 ceeding 1 foot in the width of a block, or 200 feet. 
 
 The grade of a road is described as 1 in so many ; so many feet to the mile, 
 or such an angle with the horizon : 
 
 FIG. 1121. 
 
 Inclination. 
 
 Feet per mile. 
 
 Angle. 
 
 Inclination. 
 
 Feet per mile. 
 
 Angle. 
 
 linlO 
 
 528 
 
 5 43' 
 
 1 in 30 
 
 176 
 
 1 55' 
 
 1 " 11 462 
 
 5 
 
 1 " 40 
 
 132 
 
 1 26' 
 
 1 " 14 
 
 369 
 
 4 
 
 1 " 50 
 
 106 
 
 1 9' 
 
 1 " 20 
 
 264 
 
 2 52' 
 
 1 " 57 
 
 92 
 
 1 
 
 1 " 29 
 
 184 
 
 2 
 
 1 " 100 
 
 53 
 
 35" 
 
 The foot-walks in this city and vicinity are generally formed of flags, or 
 what is here termed blue-stone, laid on a bed of sand or cement-mortar. The 
 flags are from 2" to 4" thick. Stone thicker, select in quality, upper surface 
 bush hammered or planed, close jointed, and covering the whole width of the 
 sidewalk in one stone, add much to the exterior finish of large houses. Bricks 
 are often used in towns, or places where good flagging can $ot be readily ob- 
 tained, usually laid flatways on a sand-bed. Granite is often employed for side- 
 walks in lengths equal to the width of the sidewalk, and making a cover for the 
 
ENGINEERING DRAWING. 
 
 475 
 
 vault beneath ; the objection to it is that by wear and under certain atmospheric 
 conditions it becomes slippery. 
 
 Cement face, with a base of concrete and laid in squares to admit of expan- 
 sion, makes a good and permanent walk. For the country a composition of 
 coal tar, or asphalt and gravel, is economical and satisfactory. 
 
 In Paris there is no area (Fig. 1122) ; the sidewalk comes up to the house 
 or street-line, and there is a space for trees between sidewalk and street-curb. 
 
 This space is available for pedestrians, a part being a gravel, asphalt, or flagged 
 walk. The following are the dimensions according to the law of June 5, 1856 : 
 
 Entire width of 
 boulevard and 
 
 Width of 
 
 Width of 
 
 Width for 
 
 Rows of 
 
 DISTANCE OF 
 
 ROW FROM 
 
 avenues. 
 
 
 
 
 
 Street-line. 
 
 Street-curb. 
 
 Metrei. 
 
 26 to 28 
 
 Metrei. 
 12 
 
 Metres. 
 
 Metrei. 
 
 1 
 
 Metres. 
 
 5-5 to 6-5 
 
 Metres. 
 
 1-5 
 
 30 " 34 
 
 14 
 
 
 
 1 
 
 6-5 " 8-5 
 
 1-5 
 
 36 " 38 
 40 
 
 12 to 13 
 14 
 
 3-5 
 
 3-5 
 
 8-0 to 8-5 
 9-5 
 
 2 
 2 
 
 5-0 " 5-5 
 6-5 
 
 1-5 
 1-5 
 
 1 metre = 3 -281 feet. 
 
 GRANITE BLOCK PAVEMENT WITH FOUNDATION AGREEABLY TO NEW YORK 
 CITY SPECIFICATIONS (Fig. 1123). 
 
 Stone blocks of granite, measuring on the upper surface not less than 8" nor 
 more than 12" in length, not less than 3" nor more than 4" in width, not 
 
 FIG. 1123. 
 
 less than 7" nor more than 8" in depth, blocks to be dressed to form when laid, 
 close-end joints, and side joints not exceeding 1" in width. 
 
 Bridge or Crossing Stones. Each stone to be not less than 4' nor more than 
 8' long and 2' wide; thickness, from 6" to 8"; dressed to an even face on 'top, 
 bottom bedded, sides square and full, ends cut to a bevel of 6" in 2' not paral- 
 lel with the line of vehicle travel. 
 
476 ENGINEERING DRAWING. 
 
 Curbstones shall not be less than 3' in length, 5" thick, 20" deep, matched 
 width, the top cut to a bevel of 1", front cut to a fair line to a depth of 14", 
 ends from top to bottom squared. 
 
 The subsoil to be excavated and removed to a depth of 16" below the top 
 line of the proposed pavement; roadbed shall be truly shaped and trimmed to 
 the required grade, and rolled to ultimate resistance with a roller weighing not 
 less than ten tons. When the roller can not reach any portion of the roadbed 
 it shall be tamped or rolled with a small roller. Upon the foundation a 6" bed 
 of concrete, except in the space between the rails known as the horse- ways, 
 where no concrete shall be laid. Concrete of Portland cement one part of 
 cement, three parts of sharp sand, seven parts of broken stone by measure ; or 
 one part of cement, three parts of sand, four of broken stone, and three of gravel. 
 Of Rosendale cement one part of cement, two of sand, and four of broken 
 stone ; or one of cement, two of sand, two of broken stone, and two of gravel. 
 
 On this concrete foundation, and on the foundations of the horse-ways of 
 street railways (where no concrete shall be laid) shall be laid a bed of clean, sharp 
 sand, perfectly free from moisture, not less than 1" thick, to the depth neces- 
 sary to bring the pavement to the proper grade when properly rammed. Upon 
 this bed of sand cross-walks to be laid, stone blocks in courses at right angles 
 with the line of street, except in intersections of streets, when the courses shall 
 be laid diagonally. Each course of blocks shall be of uniform depth and width, 
 so laid that all end joints shall be broken by a lap of at least 3" ; joints be- 
 tween courses not more than 1" in width. As the blocks are laid they shall be 
 covered immediately with clean, hard, hot, dry gravel, of proper size, which 
 shall be brushed into the joints until all the joints become filled ; gravel shall 
 be free from sand. Blocks shall be thoroughly rammed to an unyielding bear- 
 ing, with uniform surface, true to the roadway grade. Before pouring the 
 paving cement, the joints and gravel filling must be made dry and free from 
 dirt. Paving cement shall be poured into the joints while the gravel is still 
 hot (paving cement to be heated to 300 Fahr.) until the joints are filled flush 
 with the top of the blocks. 
 
 Paving cement to be composed of twenty parts refined Trinidad asphalt 
 and three parts of residuum oil, mixed with a hundred parts of coal tar. 
 
 Gutter stones are not now used, the paving extending to the curbstone. 
 
 Granite Pavement ivithout Concrete Bed Similar to the above, excepting 
 that the subsoil shall be excavated and removed to the depth of 10" below 
 the top line of the proposed pavement when rolled or rammed. Upon this 
 foundation shall be laid a bed of clean sharp sand, or clean fine gravel, to the 
 depth necessary to bring the pavement and cross-walks to the proper grade. 
 No ramming shall be done within 25 feet of the face of the work being laid. 
 Whenever the pavement shall have been constructed it shall be covered with a 
 good and sufficient second coat of clean sharp sand, and immediately thoroughly 
 rammed until the work is made solid and secure; no paving cement in the 
 joints of the blocks. 
 
 Asphalt Pavement with Concrete Foundation (Fig. 1124). The subsoil or 
 other matter shall be excavated and removed to the depth of 9 inches below the 
 top line of the proposed Trinidad asphalt pavement, which shall have a crown 
 not to exceed the rate of 5 inches on a roadway of 30 feet, 9 inches below the 
 
ENGINEERING DRAWING. 
 
 477 
 
 top of the proposed asphalt pavement, and 13 inches below the top of the 
 stone block pavement adjoining the- rails, man-hole heads, and stop-cock boxes, 
 
 the entire road-bed to be thoroughly rolled with a heavy roller. Upon the 
 foundation thus prepared shall be laid a bed of hydraulic cement concrete, 6 
 inches in thickness, which, if necessary, must be protected from the action of 
 the sun and wind until set. Upon this foundation must be laid a fine bitumi- 
 nous concrete or binder of clean, broken stone not exceeding 1 inch in their 
 largest dimensions, thoroughly screened, and coal-tar residuum, commonly 
 known as No. 4 paving composition. The stone to be heated by passing 
 through revolving heaters, and thoroughly mixed by machinery with the pav- 
 ing composition in the proportion of 1 gallon of paving composition to 1 cubic 
 foot of stone. The binder to be spread with hot iron rakes to true grade of 
 the pavement and to such thickness that, after being thoroughly compacted 
 by tamping and hand rolling, the surface shall have a uniform grade and cross- 
 section, and the thickness of the binder at any point shall be not less than three 
 fourths of an inch, the upper surface to be parallel with the surface of the 
 pavement to be laid. Upon this foundation must be laid the wearing surface, 
 the basis of which, and of paving cement, must be pure asphaltum, unmixed 
 with any of the products of coal tar. The wearing surface to be composed of 
 refined asphaltum, heavy petroleum oil, fine sand containing not more than 
 one per cent of hydrosilicate of alumina and fine powder of carbonate of lime ; 
 asphalt from the Pitch Lake, on the Island of Trinidad ; petroleum oil to be 
 freed from all impurities and brought to a specific gravity of from 18 to 22 
 Beaume, and a fire test of 250 Fahr. The cement from these two hydro- 
 carbons shall have a fire test of 250 Fahr., to be composed of 100 parts of pure 
 asphalt and from 15 to 20 parts of petroleum oil. 
 The pavement mixture to be composed of : 
 
 Asphaltic cement 12 to 15 parts. 
 
 Sand 83 to 70 " 
 
 Pulverized carbonate of lime 5 to 15 " 
 
 Sand and asphaltic cement to be heated separately to about 300 Fahr., the 
 carbonate of lime while cold to be mixed with the hot sand and then with the 
 asphaltic cement. The pavement mixture, at a temperature of about 250, shall 
 then be carefully spread by means of hot iron rakes in such manner as to give 
 a uniform and regular grade. The surface shall then be compressed by hand 
 rollers, after which a small amount of hydraulic cement shall be swept over it, 
 and it shall then be thoroughly compressed by a steam roller weighing not less 
 than 250 pounds to the inch run, the rolling to be not less than five hours for 
 
4:78 
 
 ENGINEERING DRAWING. 
 
 every 1,000 yards of surface. After its ultimate compression the pavement 
 must have a thickness of not less than 2 inches. 
 
 Of the powdered carbonate of lime, 5 to 15 per cent shall be of an impalpa- 
 ble powder, the whole of it to pass a No. 26 screen ; of the sand, none to pass 
 a No. 80 screen ; and the whole of it, a No. 10 screen. 
 
 Pavements, Salt Lake City, Utah. The streets are 92 feet from curb to 
 curb. The pavement is practically in two portions an asphalted central por- 
 tion and two stone-block surfaces arranged as shown in Fig. 1125. The 
 foundation of the block pavement is 10 feet wide on each side, is composed of 
 
 coarse sand laid on a foundation compacted by a 10- 
 ton steam roller ; the sand stratum is 6" thick. The 
 blocks are of very hard sandstone, and vary from 
 
 7" X 3f X 7" to 10" X 4" X 7" ; they form a pavement 7" thick, with 
 joints not over f" wide, filled with fine-screened gravel and grouted .pitch. 
 Before the joints are poured the blocks are rammed with hammers weighing 
 from 75 to 80 pounds until brought to a firm bed at the proper level. The 
 asphalted portion of the street is laid on a concrete foundation 6" thick. 
 
 Methods of laying Brick Pavements (Fig. 1126). After excavation the 
 ground is compacted with rollers weighing about 2 tons. A foundation course 
 
 
 of sand or broken stone is thoroughly compressed with the roller to the exact 
 form and crown of the finished pavement. On this foundation a layer of 
 brick is laid flatwise, with the longest dimension longitudinal with the road- 
 way ; on this layer sufficient sand is spread to fill all joints and then rolled or 
 rammed ; then a cushion of 1" or 2" of sand, and on this a layer of brick is set 
 on edge, with the longest dimension across the roadway, which layer is either 
 rolled or rammed. After rolling is completed, any broken bricks are replaced 
 with whole ones ; this is also done in the first layer. By some a foundation is 
 preferred composed of 8" to 15" of broken stone, made compact with rollers 
 weighing from 12 to 15 tons, and upon this is spread a 3" cushion of sand. A 
 crown of 4" is allowed in a roadway 50 feet wide, and 3" in a roadway 40 feet 
 wide. 
 
 Modulus of rupture in compression : First-quality brick, 1,700 pounds per 
 square inch ; absorption, 1-6 per cent; abrasion under test to be equal to granite. 
 
ENGINEERING DRAWING. 
 
 479 
 
 Modulus of rupture in compression : Second-quality, 1,500 pounds per square 
 inch ; absorption, 5 per cent ; abrasion not to exceed twice that of granite. 
 Of wooden pavements, cedar block is the most general (Fig. 1127). The 
 
 CEDAR BLOCKS 
 
 2 12LANK 
 1"X 12"STRINGER 
 
 FIG. 1127. 
 
 curbstone is set to the true grade, and the ground between is graded to the 
 true cross-section of the street. After this the surface is rolled by a heavy 
 steam roller ; next, 3" of coarse sand is spread over the entire surface. String- 
 ers 1" x 12" are then laid across the street from curb to curb, usually 8" apart 
 and bedded in the sand to the true section of the roadway ; after the stringers 
 are in place and well tamped, sand is levelled between flush to the top of the 
 stringers. On the stringers and sand the foundation planks, 1" to 2" thick, 
 well seasoned and dry, are laid close together lengthwise of the street, the ends 
 abutting on the stringers. Sometimes the foundation boards are dipped in hot 
 coal tar before laying. The paving blocks are sawed from peeled cedar fence 
 posts, and run from 4' to 8* diameter. The standard length is 6", but some 
 cities use blocks 7* or 8* long. 
 
 The cedar blocks are packed on end close together upon the plank founda- 
 tion, and with the spaces between adjoining blocks three-sided rather than 
 more sides. Against the curb, or any other straight vertical surface, each 
 alternate block should be split in halves and the straight side placed against 
 the curb. After the blocks are in place the spaces between them are filled with 
 gravel rammed down by iron rods. Some require that a paving cement, com- 
 posed of coal tar and asphalt, be poured over the whole surface of the pave- 
 ment, and run into the spaces between the blocks. The entire surface of the 
 pavement is then covered with about I" of fine roofing gravel. 
 
 RAILROADS. 
 
 The necessity of a well-drained road-bed is as important beneath rails as on 
 a highway. The cuts should be excavated to a depth of at least 2 feet below 
 grade, with ditches at the sides still deeper, for the discharge of water. The 
 embankments should not be brought within 2 feet of grade; this depth to be 
 left in cut and on embankment for the reception of ballast. The best ballast 
 is Macadam stone, in which the cross-ties are to be bedded, and finer broken 
 stone packed between them. Good coarse gravel makes very good ballast ; but 
 sand, although affording filtration for the water, is easily disturbed by the pas- 
 sage of the trains, raising a dust, an annoyance to travellers, and an injury to 
 the rolling-stock by getting into boxes and bearings. The average length of 
 sleepers on the 4' 8" gauge railways is about 8 feet; bearing surface, 7" ; dis- 
 tance between centres, 2' to 2' 6", except at joints, where they are as close to 
 
480 
 
 ENGINEERING DRAWING. 
 
 each other as the necessity of tamping beneath them will admit. Average 
 width of New York railways of same gauge as above, for single lines, in cuts 
 18', banks 13' ; for double lines, cuts 31', banks 26'. 
 
 Figs. 1128 and 1129 are two standard sections of the permanent way of the 
 
 CROSS SECTION GRAVEL BALLAST. 
 FIQ. 1129. 
 
 Pennsylvania Kailroad, in which the width of cuts and top of embankments are 
 the same, 31' 4", and other dimensions equally ample. 
 
 Rail sections are of infinite variety and weights, adapted to the class of 
 railroads on which they are to be used, and the loads and speed of trains to 
 which they are to be subjected. For roads of the common gauge, the weight of 
 rails is from 56 to 100 pounds per yard. The joints are made with a fish-plate. 
 
 Figs. 1130, 1131, and 1132 are the elevation, section, and plan of the 
 standard rail-joint of the West Shore Railroad. 
 
 FIG. 1130. 
 
 -. F3~T~ ""-YT 
 
 "--... L .? J ..- - % 
 
 Sfin 
 
 
 
 | *4J" / ^-- 
 
 
 ^^ 
 
 67 POUND STB 
 
 
 j 
 
 
 
 FIG. 1181. 
 
 With the increase of speed on railways, the tracks on the larger roads are 
 being laid with rails of 100 pounds to the yard, and wrought-iron ties are being 
 tested, but experiments on them with very heavy trains and high speeds have 
 not yet been conclusive as to their adoption. 
 
ENGINEERING DRAWING. 
 DIMENSIONS OF STANDARD RAILS.* 
 
 481 
 
 POUNDS PER YARD. 
 
 A. 
 
 B. 
 
 c. 
 
 D. 
 
 E. 
 
 F. 
 
 G. 
 
 40 
 
 3* 
 
 3| 
 
 14 
 
 i 
 
 
 1JL 
 
 25 
 
 45 
 
 3U 
 
 8U 
 
 2 
 
 21 
 
 Hi 
 
 1JL 
 
 f? 
 
 27 
 
 50 
 
 84 
 
 3* 
 
 21 
 
 8 
 
 Sh% 
 
 
 ?t 
 7 
 
 55 
 
 iX 
 
 4 
 
 84 
 
 *4 
 
 2U 
 
 111 
 
 re 
 
 JS 
 
 60 
 
 4i 
 
 4 
 
 24 
 
 M 
 
 m 
 
 1JL 
 
 37 
 
 il 
 
 65 
 
 4* 
 
 4 A 
 
 013 
 
 %$ 
 
 21 
 
 1A 
 
 6i 
 1 
 
 70 
 
 4f 
 
 44 
 
 9rt 
 
 If 
 
 glf 
 
 It* 
 
 
 75 
 
 411 
 
 4|f 
 
 2M 
 
 si 
 
 2ff 
 
 1|| 
 
 ii 
 
 80 
 
 5 
 
 5 
 
 2i 
 
 
 2f 
 
 1| 
 
 if 
 
 85 
 
 &A 
 
 5& 
 
 2A 
 
 *{ 
 
 2f 
 
 ifl 
 
 s 
 
 90 
 
 54 
 
 5| 
 
 2f 
 
 || 
 
 g4A 
 
 lit 
 
 A 
 
 95 
 
 5-A- 
 
 5ft 
 
 2}4 
 
 ! 
 
 2SJ 
 
 li 
 
 ? 
 
 100 
 
 5i 
 
 54 
 
 2f 
 
 i 
 
 S-/4 
 
 Itl 
 
 s 
 
 
 
 
 
 
 
 
 
 * Dimensions not given in table are constant, and are to be taken from the standard sec- 
 tion of a 70-pound rail. 
 
 Fig. 1132 is a drawing of a standard 70-pound rail. 
 
 Fia. 1132. 
 
 Figs. 1133, 1134, and 1135 are the front elevation, plan, and side elevation 
 of a New York city cable conduit adapted to Love's electric traction, one half 
 of each, showing the construction of the hand holes h on each side at every 
 third yoke about 15 feet centres, the other half showing the intermediate con- 
 struction. Details of hand hole are shown in Figs. 1136, 1137, and 1138. 
 
482 
 
 ENGINEERING DRAWING. 
 
 There are two copper-wire conductors supported and well insulated beneath 
 each hand hole, which affords access to the insulator and wire. The support 
 for the conductor is formed by two iron rods made fast inside the yoke, carry- 
 ing the mica block into which the hook-shaped conductor is inserted. 
 
 
 
 ! 
 
 - -,, , 
 
 1 
 
 ! h 
 
 ! 
 
 ( 1 
 
 
 ^ " - 
 
 FIG. 1135. 
 
 FIG. 1134. 
 
 FIG. 1133. 
 
 Under the main body of the car is a thin, broad plough (Figs. 1139 and 
 1140), side and front elevation, that passes through the slot between the rails 
 with wires (as shown in figures) leading to the trolley, which is brought in 
 contact with the conductor by making connections with both wires as it is 
 
 FIG. 1137. 
 
ENGINEERING DRAWING. 
 
 483 
 
 pressed against them. The 
 current forms a complete cir- 
 cuit, passing up through one 
 wire and one blade of the 
 plough to the motor; thence it 
 returns through the other blade 
 and wire to the power house. 
 None of the current returns 
 through the rails or escapes 
 through the ground. 
 
 The pulleys shown (Fig. 
 1133) are those belonging to 
 the cable traction, see Appen- 
 dix. Manholes into the con- 
 duit are placed at intervals for 
 inserting the lower bar of the 
 plow, the tongue of which is 
 raised through the slot, and 
 fastened by the pin p to the 
 head attached to the car. 
 
 FIG. 1139. 
 
 ROOFS AND BRIDGES. 
 
 At pages 490 and 491 will be found illustrations of the trussing of wooden 
 beams. These are simple forms, which may be used in roofs or bridges, and 
 rules are given for the proportion of parts. Rolled I-beams or plate-girders 
 will senve also for floor-beams and moderate spans, but with modern necessities 
 much more complicated structures are required. 
 
 On the General Principles of Bracing. Let Fig. 1141 be the elevation of 
 a common roof-truss, and let a weight, W, be placed at the foot of one of the 
 suspension-rods. Now, if the construction consisted merely of the rafter C' B, 
 and the collar-beam 0' C, resting against some fixed point, then the point B 
 would support the whole downward pressure of the weight ; but in consequence 
 of the connection of the parts of the frame, the pressure must be resolved into 
 components in the direction C' A and C' B ; C' b will represent the pressure in 
 
 the direction C' B, C' w the portion 
 of the weight supported at B, C' a 
 
 FIG. 1141. 
 
 FIG. 1142. 
 
 the pressure in the direction C' A, and w W the portion of the weight sup- 
 ported on A. The same resolution obtains to determine the direction and 
 amount of force exerted on a bridge-truss of any number of panels, by a weight 
 placed at any point p of its length (Fig. 1142). In either case, the effect of the 
 oblique form C' A upon the angle C is evidently to force it upward ; that is, a 
 weight placed at one side of the frame has, as in case of the arch, a tendency 
 
484: 
 
 ENGINEERING DRAWING. 
 
 to raise the other side. The effect of this upward force is a tension on a por- 
 tion of the braces, according to the position of the weight ; but as braces, from 
 the manner in which they are usually connected with the frame, are not capa- 
 ble of opposing any force of extension, it follows that the only resistance is 
 that which is due to the weight of a part of the structure. 
 
 Figs. 1143 and 1144 illustrate the effects of overloading at single points such 
 forms of construction. Such an unequal loading on trusses requires that a 
 
 Fio. 1143. 
 
 FIG. 1144. 
 
 FIG. 1145. 
 
 portion of the load W be transferred to each point of support inversely propor- 
 tionate to the distances of the weight from each support. The above trusses 
 are not prepared to transfer this weight to but one support. To remedy the 
 difficulty, it will be necessary to add braces running in the opposite direction 
 
 as shown by dotted lines (Fig. 1145), at every point 
 
 subject to the above distortion. These are called 
 
 counter-braces. 
 
 To prevent the braces from becoming loose 
 
 when the counter-braces are in action, it is always 
 
 customary to give the braces and counter-braces an initial compression, by put- 
 ting a moderate tension on the suspension-rods. In this case, therefore, the 
 passage of a load produces no additional strain upon any of the timbers, but 
 tends to relieve the counters. The counter-braces do not assist in sustaining 
 the weight of the structure ; on the contrary, the greater the weight of the 
 structure itself, the more will the counter-braces be relieved. 
 
 If, instead of the counter-braces, the braces themselves are made to act 
 both as ties and struts, as in some iron bridges and trusses, then the upward 
 force will be counteracted by the tension of the brace. 
 
 If a system be composed of a series of suspension-trusses (Fig. 1146) in which 
 
 FIG. 1146. 
 
 the load is uniformly distributed, represent the load at each of the points, 4, 3, 
 2, 1, 2', etc., by 1, then the load at 4 will be supported ^ upon a and upon 
 3 ; hence the strut 3 will have to support a load of 1 + -5 To ; of this 
 will be supported by 2 and -J- by a ; f of 1-5 = 1, I -f- 1 = 2, load on strut 2 ; 
 f of this load, or 1-5, will be supported at 1, and since from the opposite side 
 
ENGINEERING DRAWING. 435 
 
 there is an equal force exerted at.l, therefore the strut 1 supports 1 -f- 1-5 -f- 
 1-5 = 4. 
 
 c a 
 
 The tension on the rod c-2 = 2 - 
 
 c 1 
 
 " " 2-3 = 2 " 
 
 " " 3-4 = 3 " 
 
 If this construction be reversed as in a roof-truss, the parts which now act 
 as ties become braces, and the braces ties. The force exerted on the several 
 parts may be estimated in a similar way as for the suspension-truss. Neither 
 of these constructions would serve for a bridge-truss, subject to the passage of 
 heavy loads, but are only fit to support uniform and equally distributed loads. 
 
 To frame a construction so that it may be completely braced that is, under 
 the action of any arrangement of forces the angles must not admit of alterd- 
 tion, and consequently the shape can not. The form 
 
 FIG. 1148. FIG. 1149. 
 
 should be resolvable into either of the following elements (Figs. 1147, 1148, 
 and 1149) : 
 
 In these figures, lines - represent parts required to resist compres- 
 
 sion ; lines - parts to resist tension only ; lines parts to re- 
 
 sist both tension and compression. 
 
 In a triangle (Fig. 1147), an angle can not increase or diminish without 
 the opposite angles also increasing or diminishing. In the form Fig. 1148 a 
 diagonal must diminish; in Fig. 1149 a diagonal must extend, in order that 
 any change of form may take place. Consequently, all these forms are com- 
 pletely braced, as each does not permit of an effect taking place, which would 
 necessarily result from a change of figure. Hence, also, any system composed 
 of these forms, properly connected, breaking joint as it were into each other, 
 must be braced to resist the action of forces in any direction ; but as in general 
 all bridge-trusses are formed merely to resist a downward pressure, the action 
 on the top chord being always compression, it is not necessary that these chords 
 should act in both capacities. 
 
 Most city roofs are flat ; the timbers are placed like those of the floor be- 
 neath, but at greater distances between centres, as they have less load to sustain. 
 The roof covering is inclined to discharge the rain water ; and the timbers be- 
 neath are arranged to admit of this inclination, which may be quite small, 
 of an inch to the foot ; but it should be positive and uniform, to prevent the re- 
 tention of the water in puddles. It is an advantage to hold the snowfall, as it 
 relieves the street of a great encumbrance. 
 
 Figs. 1150, 1151, and 1152 represent elevations or portions of elevations of 
 the usual form of framed roofs. The same letters refer to the same parts in 
 all the figures. T T are the tie-beams, R R the main rafters, r r the jack-raf- 
 ters, P P the plates, p p the purlins, K K the Icing-posts, k k king-bolts, q q 
 queen-bolts both are called suspension-bolts C the collar or straining beam, 
 B 13 braces or struts, b b ridge-boards, e corbels. 
 
486 
 
 ENGINEERING DRAWING. 
 
 The pitch of the roof is in the inclination of the rafters, and is usually des- 
 ignated in reference to the span, as , , f , etc., pitch ; that is, the height of 
 
 FIG. 1150. 
 
 the ridge above the plate is , , f , etc., of the span of the roof at the level of 
 the plate. The steeper the pitch of the roof, the less the thrust against the 
 
 FIG. 1151. 
 
 side- walls, the less likely the snow or water to lodge, and consequently the 
 tighter the roof. For roofs covered with shingles or slate, in this portion of 
 the country, it is not advisable to use less than pitch ; above that, the pitch 
 should be adapted to the style of architecture adopted. The pitch in most 
 common use is the span. 
 
 Fig. 1150 represents a simple form of framed roof ; it consists of rafters, 
 resting upon a plate framed into the tie- or ceiling-beam ; this beam is sup- 
 ported by a suspension-rod, k, from the ridge, if supported from below, the rod 
 is omitted. If neither ceiling nor attic floor are necessary, an iron tie-rod con- 
 necting the plates is sufficient. The rafters are to be spaced from 1 to 2 feet 
 centres, and the tie-beams at intervals of from 6 to 8 feet ; the roof cover to be 
 of boards nailed to the rafters. This form of construction is sufficient for any 
 roof of less than 25 feet span, and of the usual pitch, and may be used for a 
 40-foot span by increasing the depth of the rafters; deep rafters, should always 
 be bridged. By the introduction of a purlin extending beneath the centre 
 
ENGINEERING DRAWING. 
 
 487 
 
 of the rafter, supported by a brace to the foot of the suspension-rod, as shown 
 in dotted line, the depth of the rafters may be reduced. If the tie-beam, 
 
 FIG. 1152. 
 
 which is also a ceiling and floor beam, be below the plate some 2 to 4 feet the 
 thrust of the roof is resisted (Fig. 1153) by bolts, b b, passing through the plate 
 and the beam, and by a collar-plank, C, spiked on the sides of the rafters, high 
 enough above the beam for head-room. For roofs f pitch and under 20 feet 
 span the bolts are unnecessary, the collar alone being sufficient. 
 
 Fig. 1151 represents a roof, a larger span than Fig. 1150 ; the frame may 
 be made very strong and safe for roofs of 60 feet span. King-bolts or suspen- 
 sion-rods are now oftener used than posts, with a small 
 triangular block of hard wood or iron, at the foot of 
 the bolts, for the support of the braces. The objec- 
 tion to this form of roof is that the framing occupies 
 all the space in the attic ; on this account the form, 
 Fig. 1152, is preferred for roofs of the same span, and 
 is also applicable to roofs of at least 75 feet span, by 
 
 the addition of a brace to the rafter from the foot of the queen-bolt. The col- 
 lar-beam (Fig. 1155) is also trussed by the framing similar to Fig. 1151. 
 
 In many church and barn roofs 
 the tie-beam is cut off (Fig. 1154), 
 
 T 
 
 FIG. 1153. 
 
 FIG. 1154. 
 
 FIG. 1155. 
 
488 
 
 ENGINEERING DRAWING. 
 
 the queen-post being supported on a post, or itself extending to the base, with 
 a short tie-rod framed into it from the plate. 
 
 Figs. 1156 and 1157 are representations of the feet of rafters on an enlarged 
 scale. In Fig. 1156 the end of the rafter does not project beyond the face of 
 the plate ; the cove is formed by a small tri- 
 angular, or any desirable form of plank, se- 
 
 FIG. 1156. 
 
 FIG. 1157. 
 
 FIG. 1158. 
 
 cured to the plate. The form given to the foot of the rafter is called a crow- 
 foot. In Fig. 1157 the rafter itself projects beyond the plate to form the cov- 
 ing. Fig. 1158 represents a front and side elevation and plan of the foot of 
 a main rafter, showing the form of tenon, in this case double ; a bolt, passing 
 through the rafter and beam, retains the foot of the 
 former in its place. Fig. 1159 represents the foot of a 
 main rafter, with a wooden shoe too short at a, outside 
 of the rafter ; it should be framed as in Fig. 1158. 
 
 FIG. 1159. 
 
 r , _,, ir^ri 
 
 Jo n o I 
 
 / ' ^' ' - 1 ' t 
 
 FIG. 1160. 
 
 FIQ 1161. 
 
 FIG. 1162. 
 
 Roofs may be very neatly and strongly framed by the introduction of cast- 
 iron shoes and abutting plates for the ends of the braces and rafters. Fig. 1160 
 
 FIG. 1163. 
 
 FIG. 1164. 
 
 represents the elevation and plan of a cast-iron king-head for a roof similar to 
 Fig. 1151 ; Fig. 1161, that of the brace-shoe ; Fig. 1162, that of the rafter-shoe 
 
ENGINEERING DRAWING. 
 
 489 
 
 FIG. 1165. 
 
 for the same roof ; Fig. 11G3, the front and side elevation of the. queen-head 
 of roof similar to Fig. 1152 ; and Fig. 1164, elevation and plan of queen brace- 
 shoe. Fig. 1165 represents the section of a rafter-shoe 
 for a tie-rod ; the side flanges are shown in dotted line. 
 
 On the size and the proportions of the different 
 members of a roof : Tie-beams, usually serving a double 
 purpose, are affected by two strains : one in the direc- 
 tion of their length, from the thrust of the rafters ; the 
 other a cross-strain, from the weight of the floor and 
 
 ceiling. In estimating the size necessary for the beam the thrust need not be 
 considered, because it is always abundantly strong to resist this strain, and the 
 dimensions are to be determined as for a floor-beam merely, each point of sus- 
 pension being a support. When tie-rods are used, the strain is in the direction 
 of their length only, and their dimensions can be calculated, knowing the pitch, 
 span, and weight of the roof per square foot, and the distance apart of the ties, 
 or the amount of surface retained by each tie. 
 
 The weight of the wood-work of the roof may be estimated at 40 pounds 
 per cubic foot ; slate at 7 to 9 pounds, shingles at 1^- to 2 pounds per square 
 foot. The force of the wind may be assumed at 15 pounds per square foot. 
 The excess of strength in the timbers of the roof, as allowed in all calculations, 
 will be sufficient for any accidental and transient force beyond this. Knowing 
 the weights, pressures, and their directions on parts of a roof, their stresses may 
 be determined by the parallelogram of forces and dimensions proportioned to 
 the strength of the materials of which the roof is composed. It will generally 
 be sufficient for the draughtsman to have practical examples of construction to 
 draw from. Dimensions are therefore given of the parts of wooden roofs already 
 illustrated. Beams are usually proportioned to the weight that they are to sus- 
 tain in floors and load, but where tie-rods are used, the stress upon them may 
 be determined by the following rule : 
 
 Rule. Multiply one half the weight of the roof and load by one half 
 the span, and divide the product by the rise or height of ridge above 
 eaves. 
 
 Gwilt, in his "Architecture," recommends the following dimensions for por- 
 tions of a roof : 
 
 Span. 
 
 Form of Roof,, 
 
 Rafters. 
 
 Braces. 
 
 Posts. 
 
 Collar-beams. 
 
 Feet. 
 
 
 Inches. 
 
 Inches. 
 
 Inchei. 
 
 Inchtl. 
 
 25 
 
 Fig. 1151, 
 
 5x4 
 
 5x3 
 
 5x5 
 
 
 30 
 
 i< 
 
 6x4 
 
 6x3 
 
 6x6 
 
 
 35 
 45 
 
 Fig. 1152, 
 
 5x4 
 6x5 
 
 4x2 
 5x3 
 
 4x4 
 6x6 
 
 7x4 
 7x6 
 
 50 
 
 2 sets of queen-posts, 
 
 8x6 
 
 5x3 
 
 j 8x8 
 ] 8x4 
 
 
 9x6 
 
 60 
 
 it u 
 
 8x8 
 
 6x3 
 
 j 10x8 
 ] 10x4 
 
 
 11x6 
 
 These dimensions, for rafters, are somewhat less than the usual practice in 
 this country ; no calculations seem to have been made for using the attic. An 
 average of framed roofs here would give the following dimensions nearly : 30 
 feet span, 8x5 inches ; 40 feet, 9 x 6 ; 50 feet, 10 x 7 ; 60 feet, 11x8; col- 
 
490 
 
 ENGINEERING DRAWING. 
 
 lar-beams the same size as main rafters. Roof -frames from 8 to 12 feet from 
 centre to centre. 
 
 Dimensions for jack-rafters, 15 to 18 inches apart : 
 
 For a bearing of 12 feet 6x3 inches. 
 
 " 10 " . . 9x3 " 
 
 For a bearing of 8 feet. . . 4x3 inches. 
 20 " . . 10x3 " 
 
 Purlins : 
 
 Length of bearing. 
 
 Distances apart in Feet. 
 
 Feet. 
 8 
 
 10 
 12 
 
 6 
 7x5 
 9x5 
 10x6 
 
 8 
 8x5 
 10x5 
 11x6 
 
 10 
 9x5 
 10x6 
 12x7 
 
 12 
 9x6 
 11x6 
 13x8 
 
 The pressure on the plates is transverse from the thrust of the rafters, but 
 in all forms except Fig. 1150, owing to the notching of the rafters on the pur- 
 lins, this pressure is inconsiderable. The usual size of plates for Figs. 1150 
 and 1151 is 6 X 6 inches. The dimensions of rafters depend on the distances 
 between their supports and between centres. The depth in all such cases to be 
 greater than the width ; 2 to 6 inches may be taken as the width, 8 to 12 for 
 the depth. 
 
 In the framing of roofs it is now customary, for roofs of mills, to omit pur- 
 lins, jack-rafters, and plates, and make the roof-boards of plank stiff enough to 
 supply their places, from 2" to 3" thick (according to the space between the 
 frames), tongued and grooved, and strongly spiked to the main rafters. There 
 is no need of plates ; the plank forms a deep beam, and, if the ends of the 
 frame are secured, there is no need of intermediate ties. 
 
 Joinings. As timber can not always be obtained of sufficient lengths for 
 the different portions of a frame, or to tie the walls of a building, it is often 
 necessary to unite two or more pieces together by the ends, called scarfing or 
 lapping. Fig. 1166 is a most common means of lapping or halving employed 
 when there is not much longitudinal stress, and when a post is to be placed 
 beneath the lower joint. 
 
 For beams with a butt joint under the last condition joint bolts (Fig. 1167) 
 are often used in which the ends of the timber- are squared and held together 
 by bolts inserted in holes central of the beams with the nuts in side-pockets, in 
 which one is screwed up by turning the bolt and the other by a cold chisel. 
 
 Fig. 1168 is a long scarf, in which the parts are bolted through and strapped, 
 suitable for tie-beams. Joints (Figs. 1169, 1170, and 1171) are also. often made 
 
 V 
 
 FIG. 1166. 
 
 FIG. 1167. 
 
 FIG. 1168. 
 
 by abutting the pieces together and bolting splicing-pieces on each side ; still 
 further security is given by cutting grooves in both timbers and pieces and 
 driving in keys, k k. 
 
 Iron Roofs. Roofs of less than 30 feet span are often made of corrugated 
 iron alone, curved into a suitable arc, and tied by bolts passing through -the 
 iron about 2 to 4 feet above the eaves. 
 
ENGINEERING DRAWING. 
 
 491 
 
 Fig. 1172 represents the half eleyation of an iron roof of a forge at Paris ; 
 Figs. 1173, 1174, and 1175, details on" a larger scale. This is a common type of 
 
 FIG. 1169. 
 
 f ! 
 
 8 i 1 - 
 
 a T 
 
 
 ,i i 1 
 
 1; 
 !' 
 
 m ^ 
 
 ;! 
 
 5 It U 
 
 9 j! E 
 
 >\ 
 
 ' 
 
 HI 
 
 e o 
 
 o o o 
 
 FIG. 1170 
 
 FIG. 1171. 
 
 iron roof, consisting of main rafters, R, of the I-section (Fig. 1175), trussed by 
 a suspension-rod, and tied by another rod. The purlins are also of I-iron, se- 
 cured to the rafters by pieces of angle-iron on each side ; and the roof is cov- 
 ered with either sheet-iron resting on jack-rafters, or corrugated iron extending 
 
 from purlin to purlin. The rafter-shoe, A, and the strut, S, are of cast- 
 iron ; all the other portions of the roof are of wrought-iron. In Ameri- 
 can practice it is usual to make the strut of wrought-iron, with a single 
 pin connection at its foot, instead of as in the figure. 
 The surface coverec 1 . by this particular roof is 53 metres (164 feet) long and 
 
 30 metres (98 feet) w'de. There are eleven frames, including the two at the 
 
 ends, which form the gables. 
 
492 
 
 ENGINEERING DRAWING. 
 
 The following are the details of the dimensions and weights of the different 
 parts : 
 
 Pounds. 
 
 2 rafters, 0'72 feet deep ; length together, 99-1 feet 1,751 
 
 5 rods, 0-13 feet diameter ; length together, 131-4 feet 882 
 
 16 bolts, 0-13 feet diameter 79 
 
 8 bridle-straps, 0-24 x 0-05 123 
 
 2 pieces, 0-46 thick, connecting the rafters at the ridge, i g g 
 
 4 pieces, 0'46 thick, at the foot of the strut ) " 
 
 4 pieces, 0*36 thick, uniting the rafters at the junction in the strut together 
 
 with their bolts and nuts 176 
 
 2 cast-iron struts 308 
 
 2 rafter-shoes 287 
 
 Total of one frame 3,694 
 
 16 purlins, 1 ridge-iron, each 0'46 deep, 17'2 long 2,985 
 
 Bolts for the same 64 
 
 16 jack-rafters, I-iron, (M6 deep 2,489 
 
 "Weight of iron covering, including laps, per square foot 2-88 
 
 Eoofs are sometimes made with deep corrugated main rafters with flat iron 
 between, or purlins and corrugated iron for the covering. The great objec- 
 tion to iron roofs lies in the con- 
 densation of the interior air by the 
 outer cold, or, as it is termed, 
 sweating ; on this account, they 
 are seldom used for other buildings 
 
 than boiler-houses or depots, except a ceil- 
 ing be made below to prevent the contact 
 of the air inside with the iron. 
 
 Fig. 1176 is an elevation of one of the 
 
 FIG. 1174. 
 
 s 
 
 FIG. 1175. 
 
 three panels of one of 
 
 the cast-iron girders fo/ connecting the columns, and carry- 
 ing the transverse main gutters, which supported the roof 
 of the Crystal Palace of the English Exposition of 1851. 
 Figs. 1177 to 1181 are sections of various parts on an en- 
 larged scale. The depth of the girder was 3 feet, and its 
 length was 23 feet 3f inches. Tin sectional area of the 
 bottom rail and flange in the centre (Fig. 1178) was 6f 
 square inches ; the width of both bottom and top rail (Fig. 
 
ENGINEERING DRAWING. 
 
 493 
 
 1177) was reduced to 3 inches at their extremities. The weight of these girders 
 was about 1,000 pounds, and they were proved by a pressure of 9 tons, dis- 
 tributed on the centre panel. 
 
 FIG. lire. 
 
 A second series of girders were made of similar form, but of increased di- 
 mensions in the section of their parts. Their weight averaged about 1,350 
 
 FIG. 1179. 
 
 FIG. 1180. 
 
 FIG. 1181. 
 
 pounds, and were proved to 15 tons. A third series weighed about 2,000 
 pounds, and were proved to 22% tons. 
 
 Figs. 1182 and 1183 are sections and details of the trusses for sustaining the 
 roof and floor of the English High and Latin School Gymnasium, Boston, 
 Massachusetts. The object of sustaining the gymnasium -floor by rods was to 
 secure a drill-hall for the military exercises of the school, and trusses were de- 
 signed to have sufficient strength to resist the vibration of the floor. As the 
 trusses were to be in sight, a central cloumn of cast-iron was introduced to sus- 
 tain the centre of the top chord, with lattice between the main diagonals to 
 enable them to act as counters, and a 3^-inch gas-pipe for horizontal bracing- 
 struts. The floor-sustaining rods all have upset ends, and their tops pass 
 through ornamental foliated castings, but their connection with the trusses is 
 wholly of wrought-iron. 
 
 The top chords consist of two 9-inch channel-irons weighing 50 pounds per 
 
494 
 
 ENGINEERING DRAWING. 
 
 yard, and one plate 12 X f inches. The end-posts have the same section. 
 The bottom chord consists of four bars 2$ X I inch at the shallow end of the 
 
 Fm. 1183. 
 
 truss, and four bars 2 X f of an inch at the deep end of the truss. The diago- 
 nals are two bars 3x1 inch at deep end of truss, and two bars 3 X inch at 
 shallow end of truss. The pins are all 2^ inches diameter. The trusses were de- 
 signed and constructed by D. H. Andrews, C. E., of the Boston Bridge Works. 
 
 In order to secure free space in the room beneath the roof, a roof or bridge 
 truss may be constructed above, and the roof framed as a floor suspended from 
 it, with such pitch as is requisite to shed rainfall. 
 
 Figs. 1184 to 1189 are the elevation and details for an iron roof-truss, for 
 wood, slate, or corrugated iron covers, built by the Missouri Valley Bridge and 
 Iron Works, A. S. Tulloch, engineer. 
 
 Fig. 1190 is a half cross-section of a two-story freight-shed for the New 
 York, Lake Erie and Western Eailroad, a simple and cheap construction of 
 wood, readily framed and put together. The shed rests upon a pile-dock. 
 The platform for the reception of freight is 4 feet above the dock-planking 
 and about 26 feet wide, with occasional inclined runs for the transfer of freight 
 to or from vessels. 
 
ENGINEERING DRAWING. 
 
 495 
 
 
 TY 
 
496 
 
 ENGINEERING DRAWING. 
 
 CROSS-SECTION OF ONE HALF OF A FREIGHT-SHED, NEW YORK, LAKE ERIE 
 AND WESTERN RAILROAD. 
 
ENGINEERING DRAWING. 
 
 497 
 
 Fig. 1191 is a section of the Baltimore and Ohio coaling bins for loco- 
 motives. 
 
 Piers. Fig. 1192 is an elevation of a pile-pier for a bridge. Tenons are 
 cut on the top of the piles, and a cap (a) mortised on. The two outer piles 
 
 FIG. 1191. 
 
 driven in an inclined position, and the heads bolted to the piles adjacent. The 
 piles are made into a strong frame laterally by the planks b and c, and plank- 
 braces dd on each side of the piles, bolted through. The string-pieces of the 
 
 n n rr 
 
 n n n. 
 
 5^1 
 
 ii 
 
 
 J c 
 
 FIQ. 1192. 
 
 bridge rest on the cap. Longitudinal braces are often used, their lower ends 
 resting on the plank b which should be, then, notched on to the piles and 
 their upper ends coming together, or with a straining-piece between, beneath 
 33 
 
498 
 
 ENGINEERING DRAWING. 
 
 the string-pieces, acting not only as supports to the load, but also as braces to 
 prevent a movement forward of the frames. As the tendency of a moving 
 train is to push forward the structure on which it is supported, in railway- 
 bridges especially, great care is taken to brace the structure in every way ver- 
 tically and horizontally, laterally and longitudinally. If the plank c be a tim- 
 ber-sill, and the piles beneath be replaced by a masonry-pier, the structure will 
 represent a common form of trestle. 
 
 Fig. 1193 is an illustration of a trestle-bent supporting a span of the Brook- 
 lyn Elevated Kailroad over its coal-yard. The timber is of yellow pine of the 
 
 FIG. 1193. 
 
 dimensions shown ; the ties are extended beyond the guard-rail on one side to 
 support a plank walk. The bent is about 15 feet high. 
 
 Bridge- Trusses. Whatever maybe the form of truss or arrangement of 
 the framing, provided its weight is supported only at the ends, the tension of 
 the lower chord, or the compression of the upper chord at centre, may be de- 
 termined by this common rule : 
 
 Rule. The sum of the total weight of the truss, and the maximum dis- 
 tributed load which it will be called on to bear, multiplied by the length of the 
 span, and divided by 8 times the depth of the truss in the middle, the quo- 
 
ENGINEERING DRAWING. 
 
 499 
 
 tient will V the tension of lower chord and compression of upper at the mid- 
 dle. In ne t rly all the forms of diagonal bracing, if the uniform load be con- 
 sidered as acting from the centre toward each abutment, each tie or brace 
 sustains the whole weight between it and the centre, and the strain is this 
 weight multiplied by the length of tie or brace, divided by its height. Any 
 diagonals, equally distant from the centre, sustain all the intermediate load : if 
 rods, as in Fig. 1195, by tension ; if braces, as in Fig. 1194, by compression. 
 It follows, therefore, that in all these trusses the upper and lower chords 
 
 Fio. 1194. 
 
 FIG. 1195. 
 
 should be stronger at the centre than at the ends, while diagonals should be 
 largest at the abutments. Unless the weight of the bridge is great compared 
 with the moving loads, counter-braces become necessary. 
 
 The general rule adopted in the construction of the Howe truss is, to make 
 the height of the truss one eighth of the length up to 60 feet span ; above this 
 span the trusses are 21 feet high, to admit of a system of lateral bracing, with 
 head clearance for a person standing on the top of a freight-car. From 175 
 feet to 250 feet span, height of truss gradually increased up to 25 feet. Moving 
 load for single-track railroad-bridge calculated at 2 tons per running foot. 
 
 Wooden Truss- Bridges. Fig. 1194 is the elevation of a few panels of a 
 Howe truss, and Fig. 1195 of a Pratt truss. The Howe truss is by far the most 
 popular of all wooden trusses, being readily framed and put together, uniting 
 great strength with simplicity of construction. 
 
 Fig. 1196 is the side elevation of three of the five panels of a Howe truss 
 highway-bridge of the New York, Lake Erie and Western Railroad. Fig. 1197 
 is a cross-section. There is a section of 3" plank laid close, and another beneath, 
 laid with spaces ; these planks are laid diagonally across the floor-beams, and 
 at right angles to each other, to act as lateral bracing. Fig. 1198 is the details 
 of the abutment end of bridge ; the foot of the brace rests on a cast-iron shoe. 
 The length over all that is, including the portions on the abutment is 81' 2", 
 or 75 feet between abutments, usually designated as the span. 
 
 Figs. 1199, 1200, and 1201 are the side elevation, floor cross-section, and! 
 plan of floor and bottom chord of three of the twelve panels of a single-track 
 railway Howe truss. Their length is each 10' 10^-". The centre braces are- 
 two, 7" X 10" ; the centre rods three, 1" diameter. The counters, each one- 
 6" X 8" ; lateral brace, top and bottom, 6" X 6" ; rods !" ; top chord, four 
 pieces, 7" X 12" ; bottom chord, four pieces, 7" X 15" ; floor-beams, 7" X 16". 
 The shoes, splices, and blocks between chord-timbers are of cast-iron. In the 
 earlier practice the angle-blocks were of oak, and the splices made as in Fig.. 
 1202. Both were satisfactory. 
 
500 
 
 ENGINEERING DRAWING. 
 
ENGINEERING DRAWING. 
 
 501 
 
 77 
 
 \ 
 
 FIG. 1201. 
 
502 
 
 ENGINEERING DRAWING. 
 
ENGINEERING DRAWING. 
 
 503 
 
 ) 
 
 i 
 
 
 _i 
 
 
 
 
 Cj ' ' ' "~ 
 
 Combination Truss. Fig. 1203 is the elevation and details of a combina- 
 tion or composite truss, which owes its name to the use of the two materials, 
 wood and wrought-iron, the 
 tension members being of 
 iron and the compression of 
 wood. This class is entirely 
 American in practice, and 
 embodies an essentially Amer- Fio. 1202. 
 
 ican feature, pin connections. 
 
 The bridge illustrated is a double intersection combination through bridge, de- 
 signed by L. L. Buck, 0. E., for the Northern Pacific Eailroad crossing the 
 Yakima River in the State of Washington. The span is 300 feet between cen- 
 tres of end pins, and is divided into 15 panels of 20 feet each. The height from 
 centre to centre of chords is 40 feet, and width 20 feet from centre to centre. 
 
 The cheapness of timber and the distance from which iron had to be 
 secured made it advisable to construct this form of bridge. The timbers 
 are heavy and framed, so that no two pieces are in direct contact, and provision 
 has been made by which iron can at any time be substituted for the wood. 
 
 Iron Bridges. When the span is of moderate extent, the load can be safely 
 carried by beams put together at the works and transferred to the road in 
 complete form. Plate or lattice girders are used, put together with rivets. 
 
 Fig. 1204 is a plan side and end elevation of a plate-girder bridge on the 
 New York, New Haven and Hartford Railroad. The bridge is for a single line 
 of rails, crossing a highway on a skew. The following is the bill of material : 
 
 LIST OF MATERIAL FOR PLATE-GIRDER BRIDGE, WOODMONT, CONN. 
 
 NUMBER. 
 
 Section. 
 
 Length. 
 
 Weight, pounds. 
 
 32 
 
 6"x6" xft'Ls. 
 
 47' 1" 
 
 40,530 
 
 8 
 
 5" x 3|" x |" Ls. 
 
 15' 10" 
 
 2,140 
 
 4 
 
 5" x 3" x jY Ls. 
 
 19' 9" 
 
 884 
 
 8 
 
 4"x3" xf" Ls. 
 
 15' 10" 
 
 1,063 
 
 68 
 
 3"x3" xf" Ls. 
 
 15' 10" 
 
 OQ' 1A" 
 
 7,538 
 
 O w-ij' 
 
 14 
 4 
 
 U 4t U 1( 
 
 yy 10 
 23' 0" 
 
 -6,0/CO 
 
 644 
 
 4 
 
 11 it 
 
 27' 0" 
 
 756 
 
 4 
 
 3" x -f-J " bars. 
 
 24' 6" 
 
 786 
 
 38 
 
 3" xf}' " 
 
 24' 6" 
 
 6,405 
 
 
 Total Ls and bars 
 
 
 63,572 
 
 16 
 
 8 
 
 48" x T 7 / plate. 
 48" xf" 
 
 15' 8i" 
 
 15' sr 
 
 17,590 
 7,419 
 
 16 
 
 20" x 1" 
 
 2' 0* 
 
 1,880 
 
 1 
 
 15" xf" 
 
 25' 6" 
 
 478 
 
 1 
 
 
 10' 3" 
 
 192 
 
 6 
 
 14" xf" 
 
 18' 3" 
 
 1,916 
 
 1 
 
 
 23' 0" 
 
 402 
 
 16 
 
 i 
 
 21' 7" 
 
 6,043 
 
 16 
 
 4 
 
 4' 1" 
 
 1,143 
 
 16 
 
 4 
 
 1' 5" 
 
 396 
 
 16 
 
 14" XrY 
 
 26' 10" 
 
 8,766 
 
 32 
 
 *- ** b 
 
 23' 5f" 
 
 15,342 
 
 2 
 
 4 
 
 29' 0" 
 
 1,184 
 
 4 
 
 1*" x - 7 " 
 
 24' 6" 
 
 1,786 
 
 2 
 
 12' xf" 
 
 22' 6" 
 
 675 
 
 Total weight 
 
 of bridge, exclusive of rivet he 
 
 
 128,784 
 
 
504: 
 
 ENGINEERING DRAWING. 
 
 o Oooooooooooo o 
 
 5(0,8 
 
 ,5(0,8 xj% x,8 "'mi Z 
 ",.% ',8 '.8 T o 
 
 E E K 
 
ENGINEERING DRAWING. 
 
 505 
 
 Figs. 1205, 1206, and 1207 are plans, elevations, and details of a plate-girder 
 river bridge built for the New York, New Haven and Hartford Railroad by the 
 Berlin Iron Bridge Company within the limits of the builders' yards. It is a 
 single-track bridge 102' 9" from end to end of girders fixed at one end and sup- 
 ported on rollers at the other. 
 
 Figs. 1208, 1209, and 1210 are respectively the outside elevation, top lateral 
 bracing, and cross-frame of a single-track lattice-deck span designed by the 
 Pho3nix Bridge Company. The bridge is 70 feet long over all, and divided 
 into four panels, two of which are shown in the figure ; there are five cross- 
 frames attached to the vertical posts. Each truss is built in two sections, 
 spliced as shown, the blackened circles indicating holes through which the 
 connecting rivets will be driven in the field. The girders are 7' 11" deep, and 
 placed 6' 6" apart. The ties of the railroad rest immediately on the top chord. 
 
 BILL OF MATERIAL. 
 
 Two lattice girders and bracing (Figs. 1208, 1209, 1210). 
 
 DESCRIPTION. 
 
 Sections. 
 
 Sizes. 
 
 Length. 
 
 Weight 
 per foot. 
 
 No. of 
 pieces. 
 
 Remarks. 
 
 Weight, 
 pounds. 
 
 Top chords 
 
 Angles. 
 
 a 
 
 Plates. 
 
 t 
 t 
 
 i 
 
 it 
 u 
 
 u 
 H 
 
 at 4 
 
 6x4 
 6x4 
 6x4 
 6x4 
 6x4 
 5x31 
 5x3 
 4x3 
 5 x 3^ 
 5x31 
 
 5 x 31 
 3x3 
 3x3 
 3x3 
 3x3 
 3x3 
 12x f 
 12x f 
 12x f 
 12x f 
 12x f 
 12 x ft 
 llx f 
 8x i 
 6x H 
 12x f 
 10 x f 
 10 x | 
 10 x ft 
 9x f 
 9x | 
 9x ft 
 7x ft 
 10 x f 
 9x f 
 9x ft 
 6x H 
 6x i 
 3x ft 
 
 32' 4" 
 37' 7" 
 32' 4" 
 37' 7" 
 11' 7" 
 11' 3" 
 11' 3" 
 7' 10" 
 9' 4" 
 9' 4" 
 
 0' 9" 
 5' 5" 
 9' 7" 
 9' 3" 
 8' 9" 
 6' 2" 
 32' 5" 
 37' 8" 
 15' 9" 
 21' 0" 
 2' 0" 
 2' 2" 
 2' 2" 
 2' 2" 
 11' 6" 
 1' 8' 
 1' 4" 
 1' 2" 
 
 r o" 
 
 1' 4" 
 0' 6" 
 1' 0" 
 0' 8" 
 2' 1" 
 2' 1" 
 1' 9" 
 0' 9" 
 1' 0" 
 0' 9" 
 
 16-3 
 16-3 
 15-0 
 15-0 
 16-7 
 9-0 
 9-0 
 6-7 
 14-0 
 11-3 
 
 10-0 
 6-0 
 6-0 
 6-0 
 6-0 
 6-0 
 30-0 
 30-0 
 30-0 
 30-0 
 30-0 
 17-5 
 13-8 
 13-3 
 13-8 
 30-0 
 12-5 
 12-5 
 10-4 
 11-3 
 11-3 
 9-4 
 7-3 
 12-5 
 11-3 
 9-4 
 13-8 
 10-0 
 1-9 
 
 4 
 4 
 4 
 4 
 8 
 8 
 8 
 20 
 2 
 6 
 
 2 
 4 
 2 
 6 
 10 
 10 
 2 
 2 
 2 
 2 
 4 
 4 
 8 
 8 
 8 
 4 
 2 
 2 
 20 
 4 
 4 
 6 
 5 
 2 
 2 
 4 
 8 
 8 
 8 
 
 
 2,106 
 2,452 
 1,938 
 2,256 
 1,547 
 792 
 792 
 1,049 
 261 
 633 
 
 14 
 
 129 
 115 
 333 
 525 
 372 
 1,945 
 2,262 
 945 
 1,260 
 240 
 152 
 242 
 234 
 1,270 
 204 
 25 
 23 
 208 
 38 
 16 
 40 
 24 
 52 
 39 
 66 
 83 
 80 
 13 
 593 
 
 
 
 Bottom chords 
 
 
 u u 
 
 
 Diagonals 
 
 Cut. 
 
 
 u 
 
 Verticals 
 
 Top laterals 
 
 Cut. 
 
 
 
 Top lateral. 
 
 
 Connecting angles for 
 end bottom laterals. 
 Struts 
 
 Bottom laterals 
 
 
 
 Cross bracing 
 
 
 Top and bottom struts. 
 Web for top chord 
 
 Web for bottom chord. 
 
 U U It .1 
 
 Splice plates 
 
 
 
 
 
 
 
 
 u u 
 
 
 U !( 
 
 
 Diagonals 
 
 
 Wall plates 
 
 
 Gussets 
 
 Cut. 
 
 H 
 
 Sway frame. 
 Cut. 
 Top lateral cut. 
 Bottom lateral cut. 
 Sway frame. 
 Top lateral. 
 Top lateral cut. 
 Bottom lateral. 
 
 
 <t 
 
 i 
 
 
 
 
 
 
 
 Fillers.. . 
 
 
 
 
 
 
 1,482 pairs rivet heads 
 Total 
 
 
 
 25,368 
 
 
506 
 
 ENGINEERING DRAWING. 
 
 %<x *-. 
 
 .*.tinu 
 
 *. si Su'iailps I. 
 
 I, ^^^BiBB^adboEES 
 
 T 
 
 S 
 
 "\ G 
 2X" 
 
 n 
 
 ^3 
 
 o 
 o 
 
 Z jri 
 
ENGINEERING DRAWING. 
 
 507 
 
 3 
 
 / 
 
 
 'QA 'cLJH "1 
 
 
508 
 
 ENGINEERING DRAWING. 
 
ENGINEERING DRAWING. 
 
 509 
 
 
510 
 
 ENGINEERING DRAWING. 
 
 Fig. 1211 is a partial plan and elevation of a highway bridge built by the 
 King Bridge Company. It is pin connected, with floor-beams riveted to the 
 posts above the pin. Fig. 1212 is a section, and Fig. 1213 the end post, with 
 part of the portal bracing. The lateral bracing of both top and bottom chords 
 is designed to take the stresses arising from wind pressure. Each system forms 
 with the respective chords of both trusses an independent truss, for which the 
 stresses are calculated and the members proportioned. The wind stresses of 
 the upper system are carried through the portal bracing and end posts to the 
 abutments, and those of the lower lateral bracing immediately to the end shoes 
 and abutments ; the floor-beams are compression members. 
 
 Fig. 1214 contains drawings in detail of a railroad bridge through span on 
 the Lima and Oroya Kailroad, and is a good example of the usual American 
 practice. It is a single-intersection Pratt truss through span, built of iron and 
 steel. The length from centre to centre of end pins is 180 feet, with eight 
 panels, each 22 feet 6 inches ; depth of truss 26 feet and distance between 
 trusses 16 feet. The bridge is of the pin-connected type, the tension members 
 being steel eye-bars, and the compression members built sections of plates and 
 angles. The floor- beams are riveted to the posts above the pins. 
 
 Fig. 1215 is a side elevation of an angle connection of end brace and top chord. 
 
 Conventional signs of riveting will be found in the Appendix. 
 
 Fig. 1216 is a standard pin-nut, on sale in sizes from 1||" to 7-^'' diameter 
 
 Fio. 1215. 
 
 FIG. 1216. 
 
 of pin; Fig. 1215 is an example of its use. Fig. 1217 is a side and top eleva- 
 tion of a standard clevis, in sizes, changing by eighths from " to 3" diam- 
 eter of bar. 
 
 Figs. 1218, 1219, and 1220 are elevation, plan, and detail of a turn-table as 
 
 made by the Passaic Boiling Mill Company. 
 The table is entirely centre-bearing, and 
 rests on a hard steel base. The girders are 
 strongly built of plates and angles and cover- 
 plates, and connected to each other by 
 frames and lateral angle bracing. The sizes 
 vary from 40 to 60 feet in length. 
 
 Figs. 1221, 1222, and 1223 are section 
 and details of a turn-table as made by the 
 Greenleaf Manufacturing Company. The 
 conspicuous feature of this centre is the 
 nest of rollers, shown on a larger scale in Fig. 1222. The rolls and bearings 
 are made of good quality tool steel and are 2^ inches in length and of the 
 
 FIG. 1217. 
 
ENGINEERING DRAWING. 
 
 511 
 
 FIG. 1219. 
 
 same diameter on the large end and 1^- inch in diameter on the small end, 
 and run free in the. annular groove. Fig. 1223 is the end carriage of the same 
 table. 
 
 FIG. 1220. 
 
 Fio. 1223. 
 
 
512 
 
 ENGINEERING DRAWING. 
 
 1 1 1 ! 11 
 
 I Jill i 
 o 
 
 i 
 
ENGINEERING DRAWING. 
 
 513 
 
 Figs. 1224 to 1228 are illustrations of a landing-bridge common at New 
 York city ferries. Fig. 1224 is a longitudinal section, showing a section of the 
 
 float,/, with its lever shown at one 
 side of the float and stone coun- 
 terpoise to balance the weight of the 
 bridge. Fig. 1225 is the front ele- 
 vation, and Fig. 1226 the plan, one 
 half in planking and one half in 
 framing. There are two chain- 
 barrels, on each side of the bridge, 
 
 FIG. 1229. 
 
 FIG. 1230. 
 
 worked by hand-wheels ; on the outer ones are the chains by which the boat is 
 drawn up to the bridge ; on the inner ones the chains by which the bridge is 
 adjusted to the load on the boat, and by which a part of the weight of the 
 bridge is held, the upper ends of the chains being attached to the frame over- 
 34 
 
514 
 
 ENGINEERING DRAWING. 
 
 head. The details (Figs. 1227 and 1228) in section and plan explain the con- 
 struction of the land-hinge ; a cushion of rubber is introduced into the joint 
 to modify the shock caused by the boat striking the bridge, and a flap of 
 wrought-iron to cover the joint, for protection to travel, and security from dirt. 
 
 FIG. 1231. 
 
 For motives of economy in cost or space, it has become very common among 
 engineers to construct bridges with frequent wrought-iron piers or trestles of 
 great height rather than extended truss spans ; one of the most remarkable of 
 these is the Kinzua Viaduct, of which Fig. 1229 represents a single pier in per- 
 spective, with details, Fig. 1230. 
 
 Fig. 1231 is a section of the roadway of the Eivermont Bridge, Lynchburg, 
 Va., supported by similar piers. 
 
 Fig. 1232 is the elevation of a bridge over the Rio Galisteo, Apache Canon, 
 on the line of the N. M. & S. P. R. R. The ends of the bridge rest on abut- 
 ments on opposite sides of the canon. The centre line of the track is on a ten- 
 degree curve, and is 14 inches off the centre line of the plate girders at the 
 centre and at both ends of the span. 
 
 Fig. 1233 is an elevation of one bent of the Third Avenue Elevated Road, 
 and Fig. 1234 of post and foundation on a line of cross-section. 
 
 The foundations for the columns supporting the iron structure are simply 
 blocks of concrete, capped with a block of granite, to which the bases of the 
 wrought-iron columns are bolted. In soft material nine piles were driven, 
 capped with 12-inch timbers, and concrete filled in around them. On sandy 
 bottom concrete was placed about 6 feet below the surface. On a portion of 
 the line a right of way 50 feet wide was secured ; the tracks were supported on 
 plate girders resting upon piers of brick masonry, with foundations of concrete 
 below the surface of the ground. The brick piers were capped with granite. 
 
ENGINEERING DRAWING. 
 
 515 
 
 Fig. 1235 is an elevation and. half plan of the foundations of a post on the 
 Brooklyn Elevated Kailroad, with a base of cast-iron, and Fig. 1236 a similar 
 
 FIG. 1232. 
 
 one, with a wrought-iron base. In neither is there any cap except that con- 
 nected with the post base, and the anchorage is to cast-iron plates wholly with- 
 in the masonry, which is of concrete. 
 
 Fig. 1237 is a plan of one of the stone piers of the railway bridge across 
 the Susquehanna at Havre de Grace. To lessen as much as possible the ob- 
 
 c 
 
 1 II II I/ 
 
 J 
 
 \ 
 
 7~ ~^i 
 
 "\ f 
 
 S i 
 
 15- >! 13.5 * 
 
 -< 23: - 
 
 
 
 FIG. 1233. 
 
 
 struction to the flow of the stream, it is usual to make both extremities of the 
 piers pointed or rounded. Sometimes the points are right angles ; sometimes 
 angles of 60; often a semicircle, the width of the pier being the diameter; 
 occasionally pointed arches, of which the radii are the width of the pier, the 
 centres being alternately in one side, and their arcs tangent to the opposite 
 side. None of the stoneg break joint at the angle ; this is important in oppos- 
 ing resistance to drift-wood and ice. It is not unusual, in very exposed places, 
 to make distinct ice-breakers above each pier usually of strong crib-work, with 
 a plank-slope upstream of 45, and with a width somewhat greater than that 
 of the pier. 
 
 Fig. 1238 is the plan and Fig. 1239 the side elevation of a pier of a bridge 
 across the Missouri, on the Northern Pacific Railroad at Bismarck, designed and 
 
516 
 
 ENGINEERING DRAWING. 
 
 constructed by George S. Morison, C. E. In this design both ends of the pier 
 are rounded, but the upper extremity is extended beyond the main body of the 
 pier, and is inclined and plated with iron between low- and high-water mark. 
 
 FIG. 1235. 
 
 FIG. 1236. 
 
 FIG. 1234. 
 
 FIG. 1237. 
 
ENGINEERING DRAWING. 
 
 517 
 
 FIG. 1239. 
 
518 
 
 ENGINEERING DRAWING. 
 
 This is intended not only to turn aside drift, but as an ice-breaker ; the ice, 
 moving up the incline, is broken by its own weight. 
 
 Fig. 1240 is a section of the foundation of the Bismarck bridge, showing 
 the construction of the inverted caisson, similar to that used for the Brooklyn 
 
 FIG. 1240. 
 
 Bridge piers. The caissons are 74' long, 26' wide, and 17' high outside; the 
 working-chamber 7 feet high. The caissons are built of pine, sheathed with 
 two thicknesses of 3" oak-plank. Above is crib-work filled in with Portland 
 cement concrete; a a are the air-locks. The sand was removed from the 
 caissons by water-ejectors. 
 
 Arch bridges are of stone, brick, concrete, or metal ; the parts of the arch 
 exert a direct thrust upon the abutments, resisted by the inherent weight of 
 the latter, or its absolute fixed mass, as in the case of natural-rock abutments. 
 
 Arch bridges, in masonry, are arcs of circle, semicircular (Fig. 1241), seg- 
 mental (Fig. 1242), elliptical, or described from three or five centres (page 32). 
 
 FIG. 1241. 
 
 FIG. 1242. 
 
 The stones forming the arch are called voussoirs, or arch-stones ; those form- 
 ing the exterior face are called ring-stones, the inner line of arch the intrados, 
 exterior line the extrados. The stones at the top, which are those set last and 
 complete the arch, are key-stones. - The courses from which the arches spring 
 are called skew-backs, and the first course the spring ing-course. The masonry 
 on the shoulders of the arch is called the spandrel-courses, or spandrel-backing. 
 The weight at the crown of a semicircular arch tends to raise the haunches. 
 
ENGINEERING DRAWING. 519 
 
 This is counteracted by the spandrel-backing, and by the earth-load, which 
 should be carefully distributed on each side of the arch. 
 
 To determine the depth of the key-stone, Eankine gives the following em- 
 pirical rule, which applies very well to most of the above examples : 
 
 Depth at key, for an arch of a series, in feet, = <y/'17 X radius at crown. 
 For a single arch, = -v/'12 X radius at crown. 
 
 To find the radius at crown of a segmental arch, add together the square of 
 half the span and the square of the rise, and divide their sum by twice the rise 
 
 ' 
 
 2r 
 
 Thus, the Blackwall Eailway bridge has a span of 87 feet, and a rise of 16 
 43-5" + 16 a _ 1892-25 + 256 _ 
 
 2 X 16 ~~32~~ 
 
 To find the radius of an elliptical arch, on the hypothesis that it is an arch 
 of five centres (Fig. 103, page 32), the half-span is a mean proportional be- 
 tween the rise and the radius. Thus, for example, the Great Western Railway 
 bridge is 128' span, and 24-25' rise 
 
 64 a = 24-25 X R 
 4096 
 
 2425 
 To find the depth of key-stone, by rule above, as in one of a series 
 
 d = V'l? * 169 = V28^3 = 5-33. 
 
 The depth of the voussoir is increased in most bridges from the key-stone 
 to the springing-course, but not always ; it is safer to increase the depth. 
 
 If an arch be loaded too heavily at the crown, the lines of pressure pass 
 above the extrados of the crown and below the line of intrados at the haunches, 
 depressing the crown and raising the haunches, separating the arch into four 
 pieces, and vice versa if the arches are overloaded at the haunches. To pre- 
 vent such effects, especially from moving loads, in construction the arches are 
 loaded with masonry and earth, and this constant load in such excess that there 
 will be no dangerous loss of equilibrium by accidental changes of load. 
 
 The horizontal thrust may be determined, according to Rankine, by the 
 following approximate rule, which seldom errs more than 5 per cent : 
 
 The horizontal thrust is nearly equal to the weight supported between the 
 crown and that part of the soffit whose inclination is 45. This thrust is to be 
 resisted by the masonry of the abutment and the earth-load behind it. 
 
 Thus, if Fig. 1243 be a section of an abutment of an arch, the horizontal 
 thrust exerted at T is resisted by the mass of masonry of the abutment ; the 
 tendency is to slide back the abutment on its base A C, or turn it over on the 
 point A. The sliding motion is resisted by friction, being a percentage, say 
 from | to | of the weight of the abutment and of half the arch which is sup- 
 ported by this base ; but, in turning over the abutment on the point A, the ac- 
 tion may be considered that of a lever, the force T acting with a lever T C to 
 raise the weight of the abutment on a lever A B (G being the centre of gravity, 
 and G B the perpendicular let fall on the base), and the weight of half the 
 arch on the lever A C. That is, to be in equilibrium, the horizontal thrust 
 
520 
 
 ENGINEERING DRAWING. 
 
 T X T C must be less than the sum of the weights of the abutment multiplied 
 
 by A B, and the weight of the arch multiplied by A C. 
 
 Skew bridges are those in which the abutments are 
 parallel, but not at right angles to the centre line, 
 and the arches oblique. To construct these in cut 
 stone requires intelligence and education both in the 
 designer and stone-cutter; but, when the work is 
 laid full in cement, so that the joints are as strong as 
 the material itself, this refinement of stone-cutting is 
 not necessary. The arch may safely be constructed 
 as a regular cylinder of a diameter equal to the rec- 
 tangular distance between the abutments, with its 
 extremity cut off parallel to the upper line of road. 
 
 For such an arch hard-burned brick is the best material, the outer voussoirs 
 
 being cut stone. 
 
 In the rules above given no consideration is paid to the strength of the 
 
 cement in which the stones are bedded. When the cement is thoroughly set, 
 
 the structure is in a measure monolithic, and the thrust is inconsiderable. 
 
 Fig. 1244 is the elevation of one of the stone arches of the Minneapolis 
 
 U niou Kail way Viaduct, with the timber centres on which the arch was turned. 
 
 The arch is nearly semicircular, 97'82 feet span, 50 feet rise ; width, 28 feet ; 
 
 B 
 FIG. 1243. 
 
 Piers 10' thick at springing line, 5 ribs to each arch. 
 
 FIG. 1244. 
 
 depth of arch at spring, 40" ; at key, 36". The piers are 10 feet thick at 
 springing line ; their up-stream ends are at angles to the main body of the 
 piers, and parallel to the thread of the stream. The whole length is 2,100 feet, 
 of which there are 3 arches of 40 feet span, 16 of 80 feet, and 4 of 100 feet. 
 Height above water, 65 feet ; total height, 82 feet. 
 
 The centres were very light frames, five to each arch ; the chords, timber 
 arches, and ties were each 12" X 12", the central braces 10" X 10", and the 
 shorter side-braces 8" X 8" ; the bolts, single, 1" diameter. 
 
ENGINEERING DRAWING. 
 
 521 
 
 The bridge was constructed after the designs and under the direction of 
 Charles C. Smith, 0. E., Chief Engineer of the St. Paul, Minneapolis and 
 Manitoba Kailway, and is an example of a very economical and stable con- 
 struction. The piers are of Minnesota granite, but above springing line the 
 masonry is of magnesian limestone. It was commenced in February, 1882, and 
 completed in November, 1883. 
 
 FIG. 1345. 
 
 Fig. 1245 is a half-elevation and half -section of the Cabin John Bridge used 
 for supporting the Potomac conduit on the line of supply to the city of Wash- 
 ington. 
 
 LOCATION. 
 
 Material. 
 
 Form of arch. 
 
 Span. 
 
 Rise. 
 
 Depth 
 at 
 crown. 
 
 Depth 
 at 
 spring. 
 
 Manchester and Birmingham Railroad. 
 
 (1 It 41 <( 
 
 London and Brighton Railroad 
 
 Brick. 
 
 Semicircular. 
 
 u 
 44 
 
 18 
 63 
 30 
 
 9 
 31-6 
 15 
 
 1-6 
 3 
 1-6 
 
 Uniform. 
 2-3 
 
 " " Blackwall " 
 
 H 
 
 Segmental. 
 
 87 
 
 16 
 
 4'H 
 
 Uniform. 
 
 Great Western Railroad 
 
 u 
 
 Elliptical. 
 
 128 
 
 24-3 
 
 5 
 
 7*U 
 
 Chestnut Street (Philadelphia) Railroad 
 High Bridge, Harlem River, New York 
 St. Paul, Minneapolis & Manitoba Rail- 
 road (largest arch), at Minneapolis. . 
 Cabin John Washington Aqueduct.. . . 
 Licking Aqueduct and Ohio Canal. . . . 
 Monocacy " 
 
 Stone. 
 
 u 
 
 Segmental. 
 Semicircular. 
 
 Segmental. 
 
 44 
 
 Elliptical. 
 
 60 
 
 80 
 
 97-8 
 220 
 90 
 54 
 
 18 
 40 
 
 50 
 57-3 
 15 
 9 
 
 2-6 
 
 2-8 
 
 3 
 4-2 
 2-10 
 2-6 
 
 3-4 
 
 Hutcheson " 
 
 it 
 
 Segmental. 
 
 79 
 
 13-6 
 
 3-6 
 
 4-6 
 
 Chemin du Per du Nord, sur 1'Oise 
 D'Enghien Railroad du Nord 
 
 tt 
 
 Semicircular. 
 
 82-5 
 24-4 
 
 13-5 
 12-2 
 
 4-6 
 1-4 
 
 
 Du Crochet Railroad 
 
 44 
 
 
 13-2 
 
 6-6 
 
 l-7i 
 
 
 Experimental arch, designed and built 
 by M. Vaudrav, Paris. . . 
 
 tf 
 
 Seermental. 
 
 124 
 
 6-11 
 
 2-8 
 
 3-7 
 
 The arch last in the list was a very bold specimen of engineering, built as 
 an experiment, preliminary to the construction of a bridge over the Seine. It 
 was made of cut stone, laid in Portland cement, with joints of ", and left to 
 set four mouths ; the arch was 12' wide ; the centres rested on posts in wrought- 
 iron boxes filled with sand, and, as the centring was eased by the running out 
 of the sand, the crown came down T V ; the joints of one of the skew-backs 
 opening -^^ during the first day, it came down T /. It was then loaded 
 with a distributed weight of 360 tons ; under this load the crown settled ^ 
 more. Since then nothing has stirred, although it was afterward tested by 
 allowing five tons to fall vertically 1' 6" on the roadway over the key-stone. 
 This bridge will not come within any of the rules laid down for other construe- 
 
522 
 
 ENGINEERING DRAWING. 
 
 tions. The rise is about ^ the span, although the usual practice for segmental 
 and elliptical arches is more than {-, or within the limits of and . 
 
 Melan concrete arch, Stockbridge, Mass. (Figs. 1246 and 1247), is a seg- 
 mental arch of 100 feet span and -fa rise. 
 
 The iron in the arch consisted of bent ribs of 7-inch, 15-lb. I-beams, spaced 
 28 inches apart and raised about 1 to 2 inches from the soffit. The concrete 
 
 in the abutments (1 cement, 3 sand, and 6 
 stone) was put in first with the adjoining 
 wing walls to such a height as to inclose 
 the ends of the iron ribs, which abutted 
 against an iron cross-angle previously 
 bolted to them. 
 
 The concrete in the arch is 9 inches 
 thick at the crown and 30 inches at the 
 haunches. The concreting of the arch 
 was done in dne day. Then the concrete 
 filling, the face and wing walls, and finally 
 the coping were built. 
 
 The bridge is a concrete monolith, the 
 coping forming one piece with the arch. 
 The iron railing is embedded in the con- 
 crete. 
 
 After striking the centres the arch settled about | inch at the abutments 
 and f inch at the centre. 
 
 The bridge was built for $1,475. 
 
 In suspension- bridges the platform of the bridge is suspended from cables, 
 or chains, the ends of which are securely anchored within the natural or arti- 
 ficial abutments. 
 
 The curve of a suspended chain is that known as the catenary, and, if the 
 whole weight of the structure were in the chain itself, this would be the curve 
 of the chains of a suspension-bridge ; but, as a large part of the weight and the 
 whole of the loading lies in the platform, the curve approaches that of the 
 parabola, and in all calculations it is so regarded. 
 
 FIG. 1847. 
 
ENGINEERING DRAWING. 
 
 523 
 
 Let Fig. 1248 represent a suspension-bridge, in which A, B, 0, are points in 
 a parabolic curve. 
 
 Rule. Add together four times the square of the deflection (E B) s and the 
 square of half-span (A E) a , and take the square root of this sum ; multiply this 
 
 FIG. 1248. 
 
 result by the total weight of one chain and all that is suspended from it, in- 
 cluding the distributed load, and divide this product by four times the deflec- 
 tion (E B) of the cable at the centre, and the result will be the tension on one 
 chain, at each point of support, A and C. The angles made by the chain at 
 the point of support viz., angle POL and the angle of the back-stays, or con- 
 tinuation of the chain (angle L C N) should be equal to each other, in order 
 that there be no tendency to overturn the tower L and A F. 
 
 BRIDGES. 
 
 Main 
 spans. 
 
 Deflection 
 of chain or 
 cable. 
 
 No. of 
 chains and 
 cables. 
 
 Total effective 
 section of 
 cable in 
 square inches. 
 
 Mean weight 
 of cable per 
 foot of span 
 (pounds). 
 
 Fixed load 
 per ft. of 
 span (Ibs.). 
 
 Breadthof 
 platform 
 in feet. 
 
 Menai 
 
 570 
 
 43 
 
 16 
 
 260 
 
 880 
 
 
 28 
 
 Chelsea 
 
 348 
 
 29 
 
 4 
 
 230 
 
 767 
 
 
 47 
 
 Pesth 
 
 666 
 
 47-6 
 
 4 
 
 507 
 
 1,690 
 
 9,892 
 
 46 
 
 Bamberg.. . . 
 
 211 
 
 14-1 
 
 4 
 
 40-2 
 
 137 
 
 1,581 
 
 30-5 
 
 Freyburg . . . 
 
 870 
 
 63 
 
 4 
 
 49 
 
 167 
 
 760 
 
 21-25 
 
 Niagara Falls 
 
 821 
 
 54 and 64 
 
 4 
 
 241-6 
 
 820 
 
 2,032 
 
 24 
 
 Cincinnati. . . 
 
 1,057 
 
 89 
 
 2 
 
 172-6 
 
 516 
 
 2,580 
 
 36 
 
 Brooklyn.. . . 
 
 1,595-5 
 
 128-5 
 
 4 
 
 188 
 
 501-3 
 
 5,865 
 
 85 
 
 BOILER-SETTING. 
 
 Fig. 1249 is a longitudinal section, Fig. 1250 a plan with section of wall, 
 and Fig. 1251 an elevation half-front and half-sectional of a boiler and setting 
 as recommended by the Hartford branch of the Hartford Steam-Boiler and In- 
 spection Company, showing the interior bracing, steam and water connections, 
 and brickwork. There are ten braces in each head, secured to pieces of T-iron, 
 placed radially, as shown in dotted lines (Fig. 1251). The feed-pipe is through 
 the front-head, just above the line of tubes, extending to the back of the boiler, 
 with a perforated branch across it, that the water may be warmed in its passage 
 and distributed. The front is a projecting cut-away front, the boiler-head being 
 nearly on a line with the front below, different from that given in Fig. 918, 
 where the lower part of the shell projects beyond the head of the boiler, and 
 the cast-iron front covers the end. The doors giving apcess to the tubes are 
 usually semicircular, and hung on the top diameter, but it will be found more 
 convenient to form them in two quadrants, and hung so as to move horizontally. 
 The boilers are to be protected against radiation by a covering of ashes, or a 
 brick arch, resting on the side- walls. 
 
524: 
 
 ENGINEERING DRAWING. 
 
 In the figure the man-hole frame is 
 riveted to the inside of the boiler ; fre- 
 quently it is on the outside. In many 
 positions the man - hole should be 
 placed in the back-head, as easier of 
 access. With the blow-off in the flue 
 (Fig. 1249) it is well to make it a circu- 
 lating pipe by connecting an inch-pipe 
 inside the valve with the upper water- 
 space of the boiler. 
 
 Figs. 1252, 1253, and 1254 are draw- 
 ings of boiler as made by the Fishkill 
 Landing Machine Company. The 
 boilers are suspended between vertical 
 side - walls. The side- walls here are 
 composed of bearing walls, air spaces, 
 and an inside 4" wall, which is exposed 
 
 FIG. 1251. 
 
ENGINEERING DRAWING. 
 
 525 
 
 FIG. 1253. 
 
 FIG. 1254. 
 
 OOOOOOOOO 
 
 OOOOOOODO 
 OOOOOOOOO 
 OOOOOOOOO 
 
 OOOOOOOOO 
 OOOOOOOOO 
 OOOOOOOOO 
 OOOOOOOOO 
 
 ooo oo 
 
526 ENGINEERING DRAWING. 
 
 to the heat of the gases from combustion. The fire-box is lined with fire-brick 
 to the height of the centre of the boiler, and also the exposed surface of the 
 bridge wall. A covering of thin corrugated iron extends from wall to wall over 
 the boiler, and supports a thick cover of ashes. 
 
 The boiler is built without dome, but of full diameter to give steam capa- 
 city ; the dry pipe of light iron (gutter or |J shape) extends the whole length 
 of the upper part of the boiler, to which it is riveted, but with a washer be- 
 tween it so as to leave open joints of about " for the admission of steam. The 
 pipes are laid out in fan lines to offer as much surface as possible to the flame 
 and to admit of ready cleaning. There are two man-holes, one above the tubes 
 where easiest of access, and one beneath in the front-head with a hand-hole in 
 the back-head. The breeching is of cast-iron, attached to the boiler, with doors 
 opening laterally ; the smoke-pipe leads out from the top with a damper at 
 the bottom ; the size of the smoke-pipe is proportioned to the number and size 
 of tubes, according to the tables given in the Appendix. 
 
 The furnace doors are hung to the sides of the frames to expose the full 
 width of opening for cleaning, and smaller doors are hung on the main doors 
 for firing. 
 
 CHIMNEYS. 
 
 Figs. 1255 and 1256 are sections of a small circular chimney, about 100 feet 
 high, in which the buttresses are within the outer shell, supporting but not at- 
 tached to a central flue ; these flues may be made of brick glazed ware or con- 
 crete. This chimney is without outside buttresses, panels, or ornaments, and 
 offers the least possible resistance to wind. 
 
 For chimneys of small diameter it is difficult to obtain circular brick, 
 but the chimney may be made octagonal of any face, with a strong bond, 
 by corner brick, from brickyards and terra-cotta works. Chimneys of the 
 above section, 100 feet high, with bottom corners rounded, will give ample 
 draft with an area of chimney-flue of two square inches for every pound of 
 anthracite per hour burned upon the grate. 
 
 If the chimney is of less height it is well to increase the section in propor- 
 tion to the reduction, but the top should be always above buildings, trees, etc., 
 and remote from obstructions that would check the draft. Chimneys should 
 have flues at least 16" diameter, and there is no objection to flues of larger area 
 than given by rule above ; but it is indispensable for a sure draft that all flues 
 should have corners rounded without abrupt changes in area or direction. 
 
 Fig. 1257 is a sectional elevation of a chimney 160 feet high, from John T. 
 Henthorne, M. E., with a cross-section (Fig. 1258) midway of the height. 
 
 Figs. 1259, 1260, and 1261 are sectional elevation, front elevation, and cross- 
 sections of a chimney of large dimensions, built by Mr. Henthorne for the 
 Narragansett Electric Light Works, Providence, E. I. 
 
 Fig. 1262 is a vertical section of a chimney at the Eidgewood Pumping- 
 engine House, Brooklyn, N. Y., and Fig. 1263 an elevation at the point where 
 the square base is changed into an octagonal. 
 
 Fig. 1264 is the cross-section of a buttressed chimney at 100 feet above base, 
 built for the Calumet & Hecla Mining Company, and designed by E. D. 
 Leavitt, Jr., M. E. The whole height of the chimney is 150 feet. The but- 
 tress walls are 16" and 12" thick, that of the body 12" and 8", and of the cen- 
 
ENGINEERING DRAWING. 
 
 iH'H 
 
 j. ->'--J,j 
 
 FIG. 1256. 
 
 ...J... 
 
 FIG. 1358. 
 
 FIG. 1255. 
 
 FIG. 1257. 
 
528 
 
 ENGINEERING DRAWING. 
 
 (V. 
 
 -h 
 
 \--b 
 
 FIG. 1239. 
 
 FIG. 1260. 
 
 FIG. 1261. 
 
ENGINEERING DRAWING. 
 
 529 
 
530 
 
 ENGINEERING DRAWING. 
 
 l_ 
 
 
 
 
 ' 
 
 
 
 
 i 
 
 
 1 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 ' 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 i 
 
 i 
 
 
 1 
 
 , 
 
 ID 
 
 FIG. 1266. 
 
 FIG. 12G7. 
 
 tral flue 8" and 4", offsetting into 
 each other by 1" offsets ; the taper 
 is 4 inches to 10 feet on each side. 
 Fig. 1265 is a half elevation and 
 half section of the cap and the 
 cover of the interior flue by which 
 its expansion is permitted. 
 
 Figs. 1266 and 1267 are eleva- 
 tion and section of a wrought-iron 
 chimney stack, such as are now in 
 common use ; they are brick-lined, 
 with a spread base well anchored 
 and without other stays. 
 
 In most chimneys an interior 
 and independent flue is used, 
 which may expand without dis- 
 turbing and cracking the exterior 
 shell. Chimneys are built of va- 
 rious sections, uniform throughout, 
 with flues tapering to the top, or 
 increasing in section, bell-mouthed, 
 some, like a lamp chimney, con- 
 tracted at the entrance ; all draw 
 well if without abrupt changes in 
 section and direction, and adapted 
 to the position. Chimneys are 
 less likely to be overturned by the 
 wind, the nearer the section to the 
 circular and the smoother in out- 
 line, and the fewer the projections. 
 
 ON THE LOCATION OF MACHINES. 
 
 The construction of buildings 
 for mills and manufactories (if 
 any aesthetic effect is intended) is 
 usually left to the architect, but 
 the necessities of the construction, 
 the weights to be supported, and 
 the space to be occupied, must be 
 supplied by the mechanical en- 
 gineer or millwright. 
 
 In the arrangement of a manu- 
 factory or workshop it is of the 
 utmost importance to know how 
 to place the machinery, both as to 
 economy of space and also of work- 
 ing. Where a new building is to 
 be constructed for a specific pur- 
 
ENGINEERING DRAWING. 
 
 531 
 
 pose of manufacture, it will be found best to arrange the necessary machines 
 as they should be, and then build the edifice to suit them. For defining the 
 position of a machine, the space it occupies in plan and elevation, the position 
 of the driven pulley or gear, of the operative, and the spaces for the working 
 and access to parts, are required. To illustrate this subject, take a two-story 
 weaving-room, of which Fig. 1268 is an elevation and Fig. 1269 a plan. 
 
 Lay down the outlines of an interior angle of the building, and dot in, or 
 draw in red or blue, the position and width of beams. This last is of impor- 
 tance, as no driving-pulley can come beneath the beam, and this is also the posi- 
 
532 
 
 ENGINEEKING DBA WING. 
 
 FIG. 1869. 
 
ENGINEERING DRAWING. 533 
 
 tion for the hanger. Lay off the width of the alleys and of the machines. The 
 first alley, or nearest the side- wall; is a back alley ; that is, where the operative 
 does not stand, and so on alternate alleys. Draw the lines of shafting central 
 to the alleys, as in this position the belts are least in the way. One operative * 
 usually tends four looms ; they are therefore generally arranged in sets of four, 
 two on each side of the main alley, where the operative stands ; the twos are 
 placed as close to each other as possible, say one inch between the lays, a small 
 cross-alley being left between them and the next set. Lay off next the alley 
 necessary at.the end of the room, and space off the length of two rows of looms 
 with alleys at the end of alternate looms, and mark the position of the pulleys. 
 Looms are generally rights and lefts, so that the pulleys of both looms come in 
 the space where there is no alley. Should the pulley come beneath a beam, 
 the loom must be either moved to avoid it, or the pulley may be shifted to the 
 opposite end of the loom. Parallel with the pulleys on the looms draw the 
 driving-pulleys on the shafts/ that is, k parallel 'with &, b with Z, / with /, and 
 so on. Draw the third and fourth rows of looms ; since the second and third 
 rows are driven from the same shaft ; if they are placed on the same line, it will 
 be impossible to drive both from the same end, and, as this is important, move 
 the third row the width of the pulley #, and, for the sake of uniformity, the 
 fourth row also. Lay off the length of looms and position of pulleys as before, 
 and parallel with the pulleys the driving-pulleys on the shaft, that is, c against 
 c, g against #, and so on. Having in this way plotted in all the looms, every 
 alternate set being on a line with the third and fourth row, if there are to be 
 looms on the story above, proceed to lay down the position of the looms on this 
 floor; and since for economy of shafting it is usual to drive from the lines in 
 the lower rooms, to avoid errors, interference of belts and pulleys, plot the 
 upper room on the same paper or board as the lower room, in two different 
 colored inks, or drawing the machines in one room in deep and in the other 
 in light line, as shown in Fig. 1269. If the width of the rooms is the same, 
 the lateral lines of looms and alleys are the same, and it is only necessary, 
 therefore, to fix the end lines. As the first loom in the outer row of looms, 
 in the lower room, occupies for its belt the position k on the shaft, the loom 
 in the upper room must be moved either one way or the other to avoid 
 this ; thus the position i of the pulley on the loom must be made parallel 
 to the pulley i on the shaft, so in the other looms a to a, e to e, d to d, 
 and h to h. 
 
 Besides the plan, it is often necessary, and always convenient, to draw a 
 sectional elevation (Fig. 1268) of the rooms, with the relative positions of the 
 driving-pulleys and those on the machines, to determine the length of the belts, 
 and also to see that their position is in every way the most convenient possi- 
 ble. In the figure, one of the lower belts should have been a cross-belt, and 
 one of the upper ones straight : had the belts to the second row of looms in the 
 upper story been drawn (as they should have been) straight, the belt would 
 have interfered a little with the alley, and it would have been better to have 
 moved the driving-shaft a trifle toward the wall. 
 
 From this illustration of the location of machines, knowing all the require- 
 ments, in a similar way any machinery may be arranged with economy of space, v 
 materials, power, and attendance. These last two items are of the more iui- * \ 
 
534 
 
 ENGINEERING DRAWING. 
 
 > * 
 
ENGINEERING DRAWING. 
 
 535 
 
 portance as they involve a daily expense, where the others are almost entirely 
 in the first outlay. 
 
 Machine Foundations. Figs. 1270, 1271, and 1272 are side, end elevations, 
 and plan of the foundation of the stationary steam-engine. F is the cast-iron 
 frame or bed-plate of the engine; B the granite bed of the engine, or coping of 
 foundation ; P the stone or brick pier, laid full in cement. The sides and sur- 
 faces of granite exposed are usually fine-hammered, the upper bed or build to 
 receive the engine-frame, hammer-dressed and set level. Strong wrought iron 
 bolts pass through frame, bed, and pier, with nuts at each end, and the whole 
 is strongly bolted together. Pockets are left in the pier near the bottom for 
 access to nuts, and these pockets are covered by granite caps or iron plates. 
 
 Few stationary steam engines are now built with bed-plates extending the 
 whole length of the engines, but the illustration is applicable to the partial 
 plates supporting the cylinder and pillow-block, and to engines and machines 
 for which heavy foundations are necessary. It is not an uncommon practice, 
 instead of granite caps, to use timber, to cushion the shocks and blows incident 
 to most machinery. 
 
 Tunnels. Figs. 1273 to 1282 are illustrations, with description, taken from 
 " Tunneling," a standard work on this subject by H. S. Drinker, C. E. 
 
 Fio. 1273. 
 
 Figs. 1273 to 1278 illustrate the principles of timbering applied to drivings 
 gallery through running material. Figs. 1273 and 1274 are parts of the con- 
 struction on a large scale, with the technical names of the parts. Each frame is 
 called a timber-set. Suppose a leading set (Figs. 1275 and 1276) is in place, 
 close to the face, and that the leading ends of the poling-boards resting above 
 this leading set are held up from the collar by wedges sufficiently high to allow 
 the insertion of the new poling-boards. In Fig. 1276 the sets e e, standing mid- 
 way between the front and the hind ends of the poling-boards, serve as middle 
 sets between the main sets dd. By referring to the plan (Fig. 1278) of a gal- 
 lery thus timbered it will be seen that the side-poling has to be wedged out at 
 its leading end, just as the roof-poling is wedged up, and the space to be filled 
 across the top by the roof-poling is wider over a front main-set than over a back 
 
536 
 
 ENGINEERING DRAWING. 
 
 one. The two outer top poling-boards (Fig. 1273) are therefore made wider at 
 their leading ends than at their back ends. The miners begin by inserting the 
 roof-poling at either corner of the face, removing the extreme end-wedges be- 
 tween the collars and the poling, and driving into this space the new poling- 
 boards (i. e., the ones shown in Fig. 1273). Though the wedges between the 
 collar and the poling-boards serve well enough to keep back the material, it 
 would be dangerous thus to take any of them out were there no other guard 
 for the poling, as the board just above the wedge removed would be pressed 
 down ; a run might also be started, and all the other wedges forced out, when 
 the poling-boards would snap down on the leading collar, and perhaps break 
 
 FIG. 1275. 
 
 off; in any event, it would be a matter of great trouble to get them wedged up 
 again. In order to guard against this trouble, a cross-board or plank a (Fig. 
 1274) is placed just under the poling-boards, and over the wedges. Then, when 
 one wedge is removed, this cross-connection holds in place the poling-board 
 
 that is immediately above the wedge 
 removed, until the new board can be 
 
 FIG. 1277. 
 
 put in ; it also stays the tendency to any general movement. The new poling- 
 board being inserted, it is driven ahead six or twelve inches, and then tern- 
 
ENGINEERING DRAWING. 
 
 537 
 
 FIG. 1279. 
 (Section of Fig. 1281, through A B, looking west.) 
 
 HOOSAC TUNNEL. 
 Timbering and arching through soft ground at the West End. Scale, 10' = 1". 
 
 FIG. 1280. 
 
538 
 
 ENGINEERING DRAWING. 
 
 West. 
 
 East. 
 
 FIG. 1281. 
 
 HOOSAC TUNNEL. 
 West End. Scale, 10' = 1". 
 
 FIG. 1282. 
 
ENGINEERING DRAWING. 539 
 
 porarily stayed by wedges, b (Fig. 1277). The corner roof-polings being thus 
 in place, the middle ones (Fig. 1273) are similarly inserted. Then the top 
 retaining-board in the face is cut out, and the material allowed to flow into the 
 heading through the space. As room is thus given ahead, the poling-boards 
 are gradually driven forward, say 24 or 30 inches, or about half the length of 
 a board. Whenever they are thus tapped, the wedges b (Fig. 1274) must be 
 loosened, and then tightened again after the driving. The side-poling is simi- 
 larly advanced ; as space is gained ahead, it must be protected by new face- 
 boarding, stayed by stretchers. Thus the work can be gradually carried down 
 to the floor of the heading by successively taking out the face-boards. Often 
 the floor of the gallery also has to be planked, and, in very extreme cases, to 
 be poled similarly to the roof and sides. 
 
 While inserting the new poling-board for half its length the boards have 
 been held in place by the double support offered by a and b. The face retain- 
 ing-boards are kept back by a vertical plank laid across them, and stayed by 
 stretchers. On this newly excavated chamber the outside pressure will be 
 great, acting, as it does, on the front half length of the poling-board c a, and, if 
 the remaining work is not rapidly executed, the front ends of the boards may 
 be snapped beyond a ; then, if it were attempted to drive the remaining portion 
 of the board on, as soon as its back end left b it would snap between a and b. 
 A middle set is therefore required at once. The middle set being in position, 
 the work of excavating the face can be proceeded with as before. The face- 
 boards are removed, one by one, from top to bottom, and the polings are driven 
 in to their full length ; then in the new length ahead the next main set is 
 erected. Such are the general principles of heading-driving through running 
 ground, or sheet-piling in tunnelling. 
 
 Figs. 1279 to 1282 show the English system of bar-timbering, as used at 
 the Hoosac Tunnel for the soft ground at the west end. The material was of 
 the worst character, and was exceedingly difficult to drive through. Figs. 1279 
 and 1280 are cross-sections, the one looking west from A B, the other east. 
 Fig. 1281 is a longitudinal section. Fig. 1282 is a cross-section of the tunnel 
 as completed with an invert, and the bars not drawn but bricked in. 
 
 Railway Stock. Figs. 1283 and 1284 are the elevation and plan of a stand- 
 ard box-car of the New York Central and Hudson Eiver Railroad. 
 
 Figs. 1285 to 1288 are the plan and elevations of the truck for the 
 same car. 
 
 Figs. 1289, 1290, and 1291 are end-elevations and cross-sections, Figs. 1292 
 and 1293 longitudinal sections, and Fig. 1294 plan of a standard passenger-car 
 of the Pennsylvania Eailroad. 
 
 Fig. 1295 is a plan and section of a pair of wrought-iron plates which sup- 
 port a car body in the centre of the truck. There are two the body centre- 
 plate and the truck centre- plate. The centre-pin or king-bolt, not shown, carries 
 none of the strain except in emergencies. 
 
 Figs. 1296 to 1299 are elevations, in full and parts, and Fig. 1300 a plan of 
 the trucks of the above car. 
 
 In the figures, both of standard box and passenger cars, the elevations and 
 plans are usually broken, to show the construction. When the two sides or 
 two ends of a car or truck are similar, it has not been considered necessary to 
 
540 
 
 ENGINEERING DRAWING. 
 
ENGINEERING DRAWING. 
 
 541 
 
 8 
 
 canters or gidp Door- 1 
 
 !<- Centers of Swing Hangers 4'-S"- -*> ! 
 
 K- Centers or SWinj | HAiiger Bearing 4'-3 ^J 
 
 < Spring PlanK 2'- lOVz" ,N n 
 
 K Transom 3-3" --X- Swing Doistor. 2 - io!i* 1 >i 
 
 < ' Ccnterof "pn"4ig 2-4'' >f 
 * Between Centers of Journal Bearing 6-3 > 
 
 FIG. 1288. ^. Axle e'-' 
 
 End Devation. 
 Track Gauge if'-D'/^' 
 
542 
 
 ENGINEERING DRAWING. 
 
 f**U \ I ^^jizz^C. 11} \^J, ?' 
 
 FIG. 1295. 
 
ENGINEERING DRAWING. 
 
 543 
 
 3-l'Y "* * -3-l'/' 
 
 FIG. 1297. ' 
 
 FIG. 1300. 
 
ENGINEERING DRAWING. 
 
 A.. ._iffl.-Xl- 
 
ENGINEERING DRAWING. 
 
 545 
 
 show both, but the figure is completed with a section of the other part, through 
 a different plane. 
 
 The following letters of reference and technical names of similar parts 
 apply equally to all the figures : 
 
 k. Draw- bar. 
 
 I, Journal-box. * 
 
 - m, Pedestal. 
 
 n, Pedestal tie-bar. 
 
 Sill. 
 
 End-sill. 
 
 Intermediate floor-timbers. 
 
 Centre floor-timbers. 
 
 Sill knee-iron or strap. o, 
 
 d, Body bolster. p, 
 
 e, Body bolster truss-rod. p' 
 /, Truck side-bearing. q, 
 g, Centre plate, body or truck.- r, 
 h, Check-chain on the truck, hooking into s, 
 h', Check-chain eye on the car. t, 
 i, Body truss-rod. to, 
 i, Body truss-rod queen-post. v, 
 j, Cross-frame tie-timber. 
 
 Pedestal stay-rod. 
 
 Pedestal arch-bar. 
 
 Pedestal inverted arch-bar. 
 
 Transom. 
 
 Truck bolster. 
 
 Spring-plank. 
 
 Swing-hanger. 
 
 Safety-beam. 
 
 Equalizing-bar. 
 
 Fig. 1301 is a plan and elevation of the frame of a locomotive as designed 
 by Mr. Robert Buchanan of the New York Central and Hudson River Railroad. 
 The distinguishing feature is the American " bar " frame, while the plate frame 
 may be called the English practice. The latter admits of a wider fire-box be- 
 tween the frames, but in the American example the fire-box is entirely above 
 the frames. 
 
 The Wave-line Principle of Ship- Construction, from Russell's " Naval Archi- 
 
 FIG. 1302. 
 
 tecture." The general doctrines arrived at by J. Scott Russell, F. R. S., from 
 numerous and long-continued experiments and practical tests, is " that the 
 form of least resistance for the water-line of the bow is horizontally the curve 
 of versed sines, and that the form of least resistance for the stern of the vessel 
 is the cycloid ; and you can either adopt the said cycloid vertically or horizon- 
 tally, or you can adopt it partly vertically and partly horizontally, according to 
 the use of the vessel or the depth of water." 
 
 " That the length of entrance, or fore body, should be f , and that of the 
 run, or after body, f ." 
 36 
 
546 
 
 ENGINEERING DRAWING. 
 
 " When it is required to construct the water-lines of the 
 bow of a ship of which the breadth and the length of the bow 
 are given, so as to give the vessel the form of least resistance 
 to passage through the water, or to obtain the highest velocity 
 with a given power : Take the greatest breadth, M M (Fig. 
 1302), on the main section of construction at midship-breadth, 
 and halve this breadth, MO; at right angles to M M at 
 draw the centre line of the length of the bow, X ; on each 
 half-breadth describe a half-circle, dividing its circumference 
 into, say, eight equal parts. Divide the length X into the 
 same number of equal parts. The divisions of the circle^ 
 reckoned successively from the extreme breadth, indicate the 
 breadths of the water-line at the successive corresponding 
 points of the line of length. Through .the divisions of the 
 
 FIG. 1303. 
 
 circles draw lines parallel to X, and through the divisions 
 of X lines parallel to M M. These, intersecting one an- 
 other, show the successive points in the required water-line. 
 The line traced through all these points is the wave water- 
 line of least resistance for a given length of bow and breadth 
 of body." 
 
 To construct the water-lines of the after body or run of a 
 ship (Fig. 1304), the mid-section (Fig. 1303) being given : 
 The bow is constructed as in Fig. 1302, but with 12 divisions' 
 on the centre line ; for the run lay off 8 divisions, each equal 
 to those of the bow ; divide the half circle into 8 equal parts,. 
 and draw chords to these divisions from to 1, 2, 3, 4. From 
 the point 1 on the centre line lay off an inclined line equal 
 and parallel to the chord 1 ; the point 1' will be in the water- 
 line. In the same way from the point 2 draw an inclined line 
 parallel and equal to the chord 2, for 2', and determine in 
 the same way the points 3', 4', 5', 6', 7'. The other circles 
 drawn in the figure are described on semi-diameters of the 
 mid-section at different levels, and the points of their wave- 
 lines are determined on the same inclined lines 1 1', 2 2', but 
 the lengths are those of the chords of the different circles. 
 In Fig. 1303, the elevations of the mid body, the curved lines 
 of sections are projected from the plan. 
 
 Fig. 1305 is a body plan of a vessel adapted to speed ; 
 Fig. 1306 of one adapted to freight. 
 
 " To determine the after body it is expedient to construct. 
 
ENGINEERING DRAWING. 
 
 547 
 
 a vertical wave-line on the run as well as a horizontal one, and in designing 
 shallow vessels to give more weight to the vertical wave-line." 
 
 " The wave system destroys all idea of any proportion of breadth to length 
 being required for speed. An absolute length is required for entrance and run, 
 but, these being formed in accordance with the wave principle for any given 
 speed, the breadth may have any proportion to that which the uses of the ship 
 and the intentions of the constructor require." 
 
 " The wave system allows us to give the vessel as much length as we please. 
 It is by this means that we can give to a vessel of the wave form the capacity 
 we may require, but which the ends may not admit. Thus, the Great Eastern, 
 
 FIG. 1305. 
 
 FIG. 1306. 
 
 which is a pure example of the wave form, has an entrance or fore body of 
 330', a run or after body of 220', and a middle body of 120', which was made 
 of this length merely to obtain the capacity required. The lengths of the fore 
 and after body are indicated by the required speed, and if the beam is fixed, 
 it is only by means of a due length of middle body that the required capacity, 
 stability, and such other qualities are to be given as will make a ship, as a whole, 
 suit its use." 
 
 Length of entrance of a vessel for a 10-mile speed should be 42 feet, of run 
 30 feet ; for a 20-mile speed, 168 and 120 feet ; that is, the lengths increase as 
 the squares of the speed. 
 
 Under Isometrical Drawing is given an illustration of a vessel constructed 
 on wave-lines. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 IT is the duty of an architect to design a building to be suitable and con- 
 venient for the purposes for which it is intended ; to select and dispose of the 
 materials of which it is composed to withstand securely and permanently the 
 stresses and wear to which they may be subjected ; to arrange the parts to pro- 
 duce the artistic effects consistent with the use of the building and its location, 
 and to apply such appropriate ornament as may express the purpose and har- 
 monize with the construction. 
 
 In domestic architecture, by far the most extensive branch of the profession, 
 most persons can give some idea of the kind of building which they wish to 
 have constructed, and perhaps express by line the general arrangement of 
 rooms ; but it is left to the architect to settle the style of building appropriate 
 to the position, to adapt the dimensions and positions of rooms and passages to 
 the requirements, to determine the thickness of walls and partitions, and arrange 
 for drainage, heating, and ventilating. The graphical representation is left to 
 the draughtsman, and his assistance is the more valuable if he is not only con- 
 versant with practical details, but understands the best proportions of parts, the 
 necessities of construction, and the requirements of building laws. 
 
 The draughtsman usually commences his education with the copying of 
 drawings. For this purpose, in Figs. 1307 to 1310, inclusive, are given plans 
 and elevations of a simple house, showing the drawings necessary to get a clear 
 comprehension of plans and elevation ; but for an estimate and for constructive 
 purposes it is necessary in addition to draw elevations of the remaining sides 
 and one or more longitudinal and transverse sections showing the framing and 
 general construction; details drawn to a large scale are also required, from which 
 and the specifications the building may be erected. 
 
 The size of the page has compelled the titles to be put within the body of 
 the drawings ; after copying, place them outside, and give ample margin. In 
 all scale drawings the scale should form a part of the title. On Fig. 1311 the 
 section and end-elevation are given together. This is also for economy of space, 
 but should be copied by the draughtsman in two distinct drawings, each of the 
 full width of the building. 
 
 Instead of hatching the walls and partitions, as in the examples given, these 
 are often indicated in solid black, or in colour, brick walls by carmine, wooden 
 walls and partitions by burnt sienna, other materials by colours as nearly repre- 
 senting them as possible, which may be purchased in pans under the name of 
 technical water colours. 
 
 Plate XIII is an illustration in colour of a plan and ceiling from a design 
 of Arthur Gilman, architect. 
 
 548 
 
ARCHITECTURAL DRAWING. 
 
 549 
 
550 
 
 ARCHITECTURAL DRAWING. 
 
ARCHITECTURAL DRAWING. 
 
 551 
 
552 
 
 AECHITEOTUEAL DRAWING. 
 
 _J 
 
 o 
 
 IP 
 
 
ARCHITECTURAL DRAWING. 
 
 553 
 
 SECTION. 
 
 FIG 1311. 
 
554 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Details of Construction. Foundations in variety and requirements are 
 treated fully under " Engineering Drawing," but for common structures the 
 draughtsman, if there are examples in the vicinity of the proposed structure, will 
 conform to the teachings of practice, or to the building laws, if there are any in 
 force. In general, for small buildings, cellar-walls, if of stone laid in mortar, 
 should not be less than 18" thick ; if of brick, 16", arid the base 6" to 12" wider. 
 
 OD D D D D O D 
 
 T 
 
 m 
 
 
 T 
 
 I 
 
 FIG. 1312. 
 
 FIG. 1313 
 
 FIG. 1314. 
 
 Figs. 1312 to 1314 represent the side and end elevations and plan of a tim- 
 ber frame building, of common construction, supported on brick or stone walls, 
 
ARCHITECTURAL CONSTRUCTION. 
 
 555 
 
 in which s s are the sides, V V the floor-beams, 1 1 trimmer-beams, and h the 
 header. The beams, J 5, framed into the header are called tail-beams; pp are 
 posts, which are distinguished by their position, as corners, intermediate and 
 window posts ; p' p' studs, g g girts, c c plates, d d braces, and r r rafters. 
 
 Usual dimensions of timber for frame of common dwelling-houses : sills 
 G" X 8", posts 4" X 8", studs 2" X 4" or 3" X 4", girts 6" X the depth of floor- 
 joists, plates 4" X 6" ; the floor-joists (J, Fig. 1315) are notched into the girts. 
 The posts and studs are tenoned into the sills and girts. Fig. 1316 represents a 
 
 FIG. 1315. 
 
 I c 
 
 FIG. 1316. 
 
 FIG. 1318. 
 
 tenon, b c, in side and end elevation, and mortise, a ; the portions of the end of 
 the stud resting on the beam are called the shoulders of the tenon. 
 
 The frame is covered with boards usually 1" thick, laid either horizontally 
 or diagonally, and nailed strongly to the posts or studs. Fig. 1317 is the ele- 
 vation of the end frame of a house, showing by breaks the diagonal cover of 
 boards and the inner lathing. The lower story is sheathed or ceiled with nar- 
 row boards, the upper shingled. 
 
 In the balloon or spike frame the stiffness depends on the nailing, and mor- 
 
556 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 CEILING LINE 
 
 
 3*X *" , . 
 
 ( 3 X 4 
 
 3*X 4* 
 
 
 Vx 4" 
 
 
 2*X 4* 
 
 
 
 FLOOR LINE 
 
 
 
 FIG. 1319. 
 
 
 FIG. 1320. 
 
 FIG. 1321. 
 
 N 
 
 tise and tenon are omitted, and the girts supplied by 
 a board a 3" or 4" x 1* let into the studs (Fig. 1318) 
 and firmly nailed to them. The studs are nailed to 
 the plates and sills. 
 
 The studs at all door-openings should be set at 
 least 2* wider and 3* higher than the size of the fin- 
 ished opening. It is not unusual to have double studs 
 (2" x 4") to inclose these openings (Fig. 1319). This 
 leaves the doorway more or less independent of the 
 partition. Partition studs are of smaller dimensions 
 than those of the frame, but are set like them and 
 usually 12" and 16" centres, adapted to the length of 
 the lath (48"). The sizes of the studs are generally 
 2 X 4, 3 X 5, or 3 X 6 inches, according to the height 
 of the partition ; for very high partitions, greater 
 depth may be required for the studs, but three inches 
 will be sufficient width. 
 
 Partitions are usually cut in between sills placed 
 on the floor-beams (Fig. 1320), and similar caps above, 
 beneath the beams. Where partitions of the second 
 story are directly above those on the first story it is 
 better to foot the studs on the caps of the latter, and 
 not on the beams (Fig. 1321). Where there are double 
 floors, the sills are placed on the bottom floor, or on 
 the floor without a sill. It may be important that the 
 partitions should be self-sustaining. This is effected 
 by simple bridging, well nailed to the studs, as shown 
 in Fig. 1322, or by herring-bone bridge, as shown in 
 plan of floor (Fig. 1330), or by a system of trussing, 
 as in Fig. 1323. This method of trussing must vary 
 with the position of opening. The foot of the braces 
 should rest on a positive support. The bridging 
 should be accurately cut and firmly nailed. Bridging 
 distributes the weight of the partition, but trussing 
 concentrates it at the ends of the braces. 
 
 Walls in Masonry. For walls above the cellar, it 
 will be found difficult to lay stone walls in mortar, 
 
 FIG. 1322. 
 
 FIG. 1323. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 55T 
 
 with fair bond and face, less than 16" thick. Brick walls may be as thin as 8" 
 for exteriors, and for partitions 4"., Brick walls are usually bonded by head- 
 ing-courses every fifth to seventh course. Where the outside 
 course is pressed or face brick, these are laid on stretchers, and 
 the bond with the backing may be thin strap-iron, laid in the 
 joints, or, by cutting off the interior corners of the face-course, 
 say every fifth course, and laying common brick diagonally of 
 the wall resting in this clipped corner (Fig. 1324). The face of 
 buildings is often built of thin ashlar, which is secured with 
 iron anchors to the brick backing. 
 
 In most cities there are building acts in force defining the 
 kinds of material and thickness of walls and foundations, to 
 which all constructions within their limits must conform. 
 
 Openings in masonry-walls are covered by lintels or arches, 
 or both. It is usual to place a stone or cast-iron lintel in the 
 exterior face over openings for doors and windows, with a wooden lintel inside 
 (Fig. 1325), and a relieving arch above. For larger openings, brick arches are 
 turned on cast-iron skew-backs, 
 of which the thrust is resisted 
 
 FIG. 1324. 
 
 FIG. 1325. 
 
 FIG. 1326. 
 
 by a tie-bolt (Fig. 1326), or cast-iron lintels, box girders, j^- or rolled I-beams. 
 
 But it is to be observed that, when the cement is set, there is little or no thrust 
 from the arch. The whole dead work, or masonry without 
 an opening, forms a monolithic beam, and, if there is depth 
 enough of this, the arch is of no account. It is the custom 
 in the north of Italy to construct flat lintels of brick, of con- 
 siderable span, depending entirely on the mortar for strength. 
 
 FIG. 1327. 
 
 FIG. 1328. 
 
 FIG. 1329. 
 
 To distribute the weight of piers over the foundation of walls with open- 
 ings, it is very common to construct inverted arches beneath spans or open- 
 ings. In old houses it was not unusual to make the exterior arches of an 
 
558 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 opening flat or rectangular in outline, with the joints radial. This is now rele- 
 gated to ornamental construction. 
 
 Concrete Walls. It is common in many places where brick and stone are 
 expensive and gravel is abundant to make walls of concrete, in proportions of 
 
 FIG. 1330. 
 
 one of cement to five to seven of gravel. The space requisite for the wall is 
 inclosed with plank, and is filled in with concrete, well rammed. Figs. 1327 
 and 1328 are plans of concrete walls with inclosing plank, and Fig. 1329 an 
 elevation. The planks are held by bolts passing through wall and plank, all of 
 which are removed after the wall is set, and the bolt-holes are then filled with 
 cement. The thickness of walls should be a little in excess of those of brick. 
 
 Flooring. The timbers which support the flooring-boards and ceiling of a 
 room are called the naked flooring. 
 
 The simplest form of flooring, and the one usually adopted in the construc- 
 tion of city houses and stores, is represented in plan and section (Fig. 1330). 
 It consists of a single series of beams or deep joists, reaching from wall to wall. 
 As a lateral brace between each set of beams a system of bridging is adopted, 
 of which the best is the herring-bone bridging, formed of short pieces of joists 
 about 2x3, crossing each other, and nailed securely to the tops and bottoms 
 of the several beams, represented by a and b ; and wherever a flue occurs, or a 
 stairway or well-hole prevents one or more joists from resting on the wall, a 
 header, H, is framed across the space into the outer beams or trimmer-beams 
 T T, and the beams cut off or tail-beams are framed into the header. 
 
 Whenever the distances between the walls exceed the length that can safely 
 be given to floor-joists in one piece, an intermediate beam or girder, running lon- 
 gitudinally, is introduced, on which the joist may be set (Fig. 1331), notched on 
 (Fig. 1332), or boxed in (Fig. 1333), or both boxed and notched. They may 
 also be framed in with tenon and mortise ; the best form is the tusJc-tenon 
 
ARCHITECTURAL CONSTRUCTION. 
 
 559 
 
 \ 
 
 FIG. 1331. 
 
 FIG. 1332. 
 
 FIG. 1333. 
 
 FIG. 1334. 
 
 (Fig. 1334). Flooring is still further varied, by framing with girders longi- 
 tudinally ; beams crosswise, and framed into or resting on the girders ; and 
 joists framed into the beams, running the same direction as the girders. When 
 the joists are not flush or level with the bottom of the 
 beams or girders, either in the finish the beams will 
 show, or ceiling-joists or furrings will have to be intro- 
 duced. 
 
 The width of beams as sold in the market is from 2" 
 to 6" ; those for common houses and spans are 2", but 
 for more important buildings and structures 3" to 4". 
 The depth of beams and distances between centres may be determined from 
 the weight to be supported (that is, load per square foot by the number of 
 feet, including that of the floor), and from the table, page 239 ; but this table 
 gives the central load, which is one half the distributed load. 
 
 The following table gives the load per square foot for different characters of 
 buildings ; for floors of 
 
 Dwellings 40 pounds per square foot. 
 
 Churches and public halls 80 " " " " 
 
 Warehouses and general merchandise 250 " " " " 
 
 Factories 200 to 400 " " " " 
 
 Snow, 30 inches deep 16 " " " " 
 
 Maximum wind pressure 50 " " " " 
 
 Roofs for wind and snow 30 " " " " 
 
 For slate 45 " " " 
 
 Plastering 8 " 
 
 Timber on sale is seldom found sawed to fractional dimensions of an inch 
 or in lengths to fractions of feet. 
 
 Trimmer-beams and headers should be of greater width than the other 
 beams, depending on the distance of the headers from the wall, and the num- 
 ber of tail-beams framed into the trimmers ; by the New York Building Act all 
 headers must be hung in stirrup-irons (Fig. 1335), and not framed. 
 
 Beams must be anchored to the walls and to each other when there are 
 two lengths meeting on girders. The straps to be not less than 1" X f", 
 spiked to sides, on top, or bottom of beam. The anchors and beams to make a 
 
 FIG. 1336. 
 
 FIG. 1335. 
 
 FIG. 1337. 
 
 tie across the building about every 6 feet. Figs. 1336 and 1337 are common 
 forms of ties between beams or between beams and walls. Fig. 1338 is an 
 
560 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 anchor between beams and embracing a column. Fig. 1339 is a spear-anchor 
 between beams and wall ; the angle is driven into the beam. In warehouses it 
 
 is usual to carry anchors 
 through the wall with large 
 washers, often ornamental, 
 and nut on the outside. 
 
 Wooden beams built into 
 walls must have an air space 
 
 FIG. 1338. 
 
 FIG. 1339. 
 
 round them with ends cut diagonally and anchored to the walls (Fig. 1340). 
 The air spaces at the sides are for ventilation, and with the diagonal cuts at the 
 ends permit the beams weakened by fire to fall without disturbing the walls. 
 Cast-iron boxes and plates for ends of beams can be purchased, with flanges 
 for anchors to walls and beams. 
 
 Floors. In New York it is usual to lay single floors of tongued and grooved 
 boards directly on the beams, but in the Eastern States double floors are more 
 common. The first floor consists of an inferior kind of boards, as hemlock, 
 unmatched, laid during the progress of the work as a sort of staging for the 
 carpenter and mason, and, in finishing, a second course is laid on them of 'better 
 material, generally tongued and grooved, but sometimes only jointed. Ceilings 
 should always be furred, and the laths be nailed to the strips. Furring-strips 
 usually are of inch board, 2" wide, and 12" from centre to centre, nailed cross- 
 wise from joist to joist. 
 
 In dwellings it is desirable to isolate the floors of each story, so that noise, 
 vermin, smells, and fire may be cut off. The first is usually done by deafening, 
 which consists (Fig. 1341) of a sub-floor, resting on cleats, nailed to the beams 
 
 and about 4" below the floor. This space 
 is filled with lime-mortar, weakened by a 
 mixture of sand or gravel, but strong 
 
 FIG. 1340. 
 
 FIG. 1341. 
 
 enough to set. If the timber is green, the contact of the mortar is apt to pro- 
 duce dry rot. 
 
 Deafening may be secured to great extent by double floors, with two or three 
 thicknesses of carpet paper or sheathing quilt between. When small vermin, 
 like cockroaches, can pass, stenches and fire can find their way. To prevent 
 which, begin at the cellar and cut off all access to space between plastering and 
 floor; if the ceiling is plastered, fill in between studs tightly with strips of 
 
ARCHITECTURAL CONSTRUCTION. 
 
 561 
 
 board, and above them a course of brick in cement. 
 Fibrous hemlock boards prevent the gnawing of rats 
 and mice. 
 
 The fireproof of old builders consisted simply of 
 plain cylindrical or groined arches (Fig. 1342) in 
 stone or brick masonry or concrete. 
 
 Figs. 1343 to 1347 are illustrations of Roman 
 constructions in masonry, from " Dictionnaire Rai- 
 sonne de 1'Architecture," par M. Viollet Le Due. 
 
 Fig. 1343 is a perspective view of a cylindrical arch in process of construc- 
 tion. The centres A and lagging B are quite light, as the full load of the arch 
 
 FIG. 1342. 
 
 1344. 
 
 FIG. 1343 
 
 is never borne by them. On the lagging, B, a cover of flat tile, C, is laid in ce- 
 ment, and above ribs, D D, and girts, E E, in brick masonry, shown on a larger 
 scale in Fig. 1344, with the plank P used for the support of the girt bricks E, 
 which is removed after the mortar is set. The panels are now filled with concrete. 
 37 
 
562 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Fig. 1345 represents rib and portions of 
 girts of a groin shown in plan, Fig. 1346, 
 efg h being that of the rib ; K, a timber of 
 the centre. 
 
 A similar construction also obtained for 
 domes, the girts being of the same width as 
 the ribs, and sunk panels formed by furr- 
 ing up on the wooden lagging of the cen- 
 tres. 
 
 Fig. 1347 is a perspective of a donre, in 
 which the brick skeleton, ribs, and girts 
 are curved, with panels, B B, of concrete. 
 
 In Italy ceilings are made in single 
 courses of brick, groined, and laid without 
 centres, the arcs being described on the 
 side-walls, and the bricks laid in plaster to 
 a line. The spandrels may be levelled up 
 with concrete, when rooms above are to be 
 occupied, but often there is only the 
 
 brick arch forming the ceiling of the principal rooms, with a light wooden 
 
 roof above. 
 
 FIG. 1346. 
 
 FIG. 1345. 
 
 FIG. 1347. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 563 
 
 The sizes of Italian brick are 2" X 4" X 10" and If X 4" X 12" ; when 
 there is no load above, these are laid flatwise in the arch. 
 
 In one of the warehouses of the Appleton Manufacturing Company at 
 Brooklyn the floor was constructed of groined arches in concrete, supported on 
 brick side- walls and piers. Piers were 2' square and 13' centres ; the arches 
 finished for a floor above were 21" deep at the spring and 9" at the key. Ceil- 
 ing was plastered and the room beneath occupied as a warehouse. In the room 
 above a wooden floor was laid and used as a printery, and has been occupied 
 for many years. There has been found one objection to the concrete that 
 water in quantity spilt on the upper floor sipes through, to prevent which there 
 should have been on the concrete floor a thin coat of Portland cement or of 
 asphalt. 
 
 The first fireproof buildings in use here were like the example above of the 
 Appleton warehouse, with supports and arches entirely in masonry. The latter 
 were usually cylindrical or segmental, plain or groined. The depth taken for 
 the construction and the floor space occupied by the supports were objection- 
 able, but as fireproof constructions they were the best. 
 
 The erection of high buildings, the large and valuable stocks often carried 
 in stores and warehouses, have involved the necessity of forms of fireproof con- 
 structions which will prevent the spreading of conflagrations and secure the 
 contents of a building, but it must be understood that no construction has yet 
 been invented that will prevent the destruction of a building by fire whose 
 contents are large masses of combustibles ; the buildings should be called fire- 
 resisting rather than fireproof. The first buildings designed for this purpose 
 consisted of iron I-beams, brick arches, and concrete spandrels, with wooden 
 floors above. This, aside from its weight, was satisfactory, but a skew-back tile 
 and plaster were necessary to protect the bottom flange of the iron beam. The 
 tie-rod, when necessary, was concealed in the arch. 
 
 In another form of construction, instead of using a movable centre, a 
 crimped sheet-iron arch was used, which was left in and made a part of the 
 structure, above which there was a concrete filling to the level of the top of the 
 beam. 
 
 To reduce the weight of brick and increase their fire resistance, brick are 
 now made hollow, with flat surfaces below for the reception of plastering, and 
 above for the wooden floors (Fig. 1348). Such floorings weigh only about one 
 half that of solid brick arches, and therefore admit of beams of less weight per 
 
 FIG. 1348. 
 
 square foot of floor, either by the reduction of the weight of the beam or an 
 increase of span. A thin layer of concrete is put on the top of the brick, and 
 wooden strips embedded to nail the floor to. 
 
 By mixing sawdust with the clay, which is burned out in the firing, the 
 product becomes a very light, porous, and firm substance, which can be cut 
 
564 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 with the saw and will hold a nail. Porous brick, hollow, with thicker ribs than 
 the tiles, are sawed to voussoir joints and laid between I-beams as flat arches. 
 
 Fio. 1349. 
 
 FIG. 1350. 
 
 For ceilings, porous tiles about 2" thick rest on iron straps suspended from 
 wooden beams (Fig. 1349), or on the flanges of j_-irons in a bed of mortar (Fig. 
 1350), the thickness of the plaster finish beneath 
 affording some protection to the iron straps and 
 flanges from fire. 
 
 Iron posts not protected are not as safe as wooden 
 ones. Cast-iron posts may be cast with a surface of 
 nails or projections for plaster of common mortar, or of 
 cement, with a finish of Keene's cement, which admits 
 of washing. Figs. 1351 and 1352 are elevation and 
 section of a Phoenix column with a porous terra-cotta 
 covering; Figs. 1353 and 1354, a similar covering ap- 
 plied to a square and round post. 
 
 Fig. 1355 shows in perspective the applications of 
 hollow brick to the lining of exterior walls bonding 
 with the common-sized brick and equal in crushing 
 strength ; they supply the place of a course, and the 
 moisture will not strike through. 
 
 FIG. 1351. 
 
 FIG. 1352. 
 
 FIG. 1353. 
 
 FIG. 1354. 
 
 FIG. 1355. 
 
 Fig. 1356 is another form applied di- 
 rectly to the face of a wall secured by 
 flat-headed nails driven into the joints of 
 the brickwork. 
 
 Fig. 1357 shows the mode of setting 
 hollow brick for a partition ; where it is 
 necessary that nails should be driven, 
 porous brick should be inserted. 
 
 Improvements in material, especially 
 the use of rolled steel instead of wrought- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 565 
 
 iroii, led to a skeleton construction which consists of rolled steel, wrought- or 
 cast-iron columns supporting a frame of rolled steel, beams and girders, set 
 and framed before any of the masonry except the foundation walls are laid. 
 
 The next in order of construction are 
 the side and end walls, of which the 
 masonry may be wholly or partially self- 
 sustaining or merely a screen or cur- 
 tain extending from the top of any 
 wall girder to the under side of the next 
 girders above. The outside posts in 
 
 FIG. 1357. 
 FlO. 1356. 
 
 the last case as well as the inside, therefore, sustain all the dead and live load. 
 The dead loads include material used in actual construction, or fixtures and 
 machinery which form a permanent part of the necessities of occupation. Live 
 loads are the weight of occupants, furniture, goods, stores, and movables. 
 
 Tie Kocfs 
 
 Co/. 
 
 
 -O 
 
 CJ 
 
 FIG. 1358. 
 
 Fig. 1358 is a floor plan of girders and beams of the standard form; columns 
 are preferably of rolled steel, but often of cast-iron. In all the walls, centrally 
 between walls and columns, and between columns, girders are framed on which 
 the beams rest and to which they are usually fastened. 
 
566 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Under the head of " Engineering Drawing " foundations are shown with the 
 bases of these columns and the iron grillage on which they are supported. All 
 the parts are proportioned to the loads which they are to sustain. 
 
 The iron surface of all columns, girders, and beams must be protected from 
 
 FIG. 1359. 
 
 fire, commonly by brick common, hollow, or porous, and of thickness not less 
 than 4". The floors are usually flat arches of hollow brick (Fig. 1348) in which 
 the skew-back extends below the flanges of beams and girders to sustain the 
 plaster protection. 
 
 Besides the flat arches of hollow brick and of porous brick, concrete has been 
 used for beams and flooring, but, as the material has comparatively little tensile 
 strength, the bottom tension of a concrete beam has been met by steel rods or 
 wire anchored and embedded in the material. Twisted bars of wrought-iron 
 from " to 2" square have been introduced by E. E. Ransome, of San Francisco, to 
 strengthen flat floors of considerable span, and, as they can not slip, the anchor- 
 age is well and uniformly distributed. Experiments by Kirkaldy for Mr. "Hyatt 
 demonstrated that the expansion and contraction of concrete and of iron is 
 equal under changes of temperature. 
 
 Metropolitan Fire Proofing Company of this city make use of the I-beam 
 frame and tension rods or wires of galvanized iron (Fig. 1359) which are em- 
 bedded in a concrete composed very largely of plaster-of-Paris cement and a 
 little sand, with crushed coke, cork, or sawdust. 
 
 Figs. 1360, 1361, and 1362 are plan 
 and elevations of one of the standard 
 connections of I-beams and Z-bars 
 from the hand-book of the Carnegie 
 Steel Company. 
 
 Figs. 1363 and 1364 is a cross-pin- 
 tle connection of girders and struts of 
 the Phoenix Iron Company with their 
 usual steel column. 
 
 Figs. 1365 and 1366 is a cast-iron 
 pintle connection of the same company. 
 Figs. 1367, 1368, and 1369 are 
 drawings of a cast-iron joint detail. 
 
 FIG. 1362. 
 
 - 
 
 
 Itr- 1 
 
 FIG. i860. 
 
 In the present form of framing 
 steel skeletons, rivets are preferred to 
 bolts, and much depends on their 
 strength in shear. 
 
 Fig. 1370 is a plate of the standard 
 connection of angles for I-beams from 
 the Carnegie Company. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 567 
 
 FIG. 1363. 
 
 FIG. 1365. 
 
 FIG. 1368. 
 
 Fio. 1367. 
 
 FIG. 1369. 
 
 FIG. 1371. 
 
568 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Fig. 1371 is a section of the wall of the New Street front of the Manhattan 
 Life Insurance Company of New York, showing the position of wall girders. 
 The front is 176' X 254' high, with five posts, including the corner ones. 
 
 The great objections made to the skeleton-frame construction are that the 
 
 STANDARD CONNECTION ANGLES. 
 FOR I BEAMS. 
 
 for 3 "lJ61bs. 
 for 4"! [? Ibs. 
 
 for 5" I j 10 Ibs. 
 Ibs. 
 
 CHANNELS, 
 for is''c 33 N>s> 
 
 for8"c jn Ibs. 
 for g'C J 14 Ibs. 
 
 Connections for 3,4,5, and 6 I-beams apply also to Channels. 
 All holes for % Bolts or Rivets. 
 FIG. 1370. 
 
 iron is liable to rust in position where it can not be seen, but it is met by care in 
 cleansing the iron, in boiling in oil and painting, or by the Smith process used 
 in protecting cast-iron pipes, and, second, the want of protection against wind 
 strains and a proper system of bracing without interfering with the occupancy 
 of the building. The section of walls (Figs. 1372 and 1373) shows the dimen- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 569 
 
 FIG. 1372. 
 
 FIG. 1373. 
 
 sions required by the Building Depart- 
 ment of New York city for the skeleton 
 construction and for plain walls with- 
 out it. 
 
 If the wall construction follows 
 promptly that of the steel frame, and 
 it is temporarily held by tie rods, a 
 mortar set will be secured which will 
 resist the usual wind stresses. For 
 buildings entirely of brick masonry, it 
 is usual to introduce wooden braces 
 as the work progresses which are re- 
 moved when fully closed in. 
 
 FIG. 1374. 
 
 FIG. 1375. 
 
 In New England there has been introduced what are called fire-retarding 
 constructions for mills, of which the beams, posts, and floors are of wood. The 
 
 FIG. 1376. 
 
570 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 principle of the construc- 
 tion is to consolidate the 
 material in such a way 
 that a fire can be held long 
 enough in any room in 
 which it may originate till 
 the water and appliances 
 under the management of 
 an established fire depart- 
 
 FIG. 1377. 
 
 ment connected with the mills or public can get it 
 under control. 
 
 Figs. 1374 and 1375 are elevation and section of 
 post, beams, and floors which show the details of con- 
 
 Fio. 1378. 
 
 Fio. 1379. 
 
 struction, and Fig. 1376 of a mill showing a floor and 
 a roof. The ceiling must be high posted and the 
 windows wide, and set well up to the ceiling to admit 
 of light into a wide mill. When there is plenty of 
 ground space, it is in most industries safer from fire 
 and more economical in management to have a one- 
 story mill (Fig. 1377), of which details are shown in 
 Figs. 1378 and 1379. A one-story mill may be of any 
 dimension required, as the central spaces are lighted 
 by monitor decks. The principle of all this construc- 
 tion is that there should be no spaces where dust may 
 collect ; in the floors the material is close, but if the 
 
ARCHITECTURAL CONSTRUCTION. 
 
 571 
 
 timber be not seasoned or kiln-dried, as there is no circulation of air, it may dry 
 rot. The posts have a bore through the centre and holes connecting with it at 
 
 FIG. 1380. 
 
 top and bottom to provide air circulation, and it is better not to paint any of 
 the exposed timber until all the moisture is dried out. 
 
 In mills sheathing is preferable to plastering, as this last may break and pieces 
 fall into the machinery; but Fig. 1380 shows a form of construction that has 
 served well for warehouses, and is safer and more ornamental than the usual 
 timber constructions. The lathing consists of wire cloth fastened with staples, 
 stapled through furring strips ' ' to ", and then the usual three-coat plaster. 
 
 Doors. In stud-partitions, the openings for doors are framed as in Fig. 
 1319, the door-frame being independent of the studs. 
 
 Fig. 1381 represents the elevation and Fig. 1382 the horizontal section of a 
 common inside-door. A A are the stiles, B, 0, H, D, the bottom, lock, parting, 
 and top rail, E the panels, and F the muntin ; the combination of mouldings 
 and offsets around the door, G, is called the architrave j in the section, a a are 
 the partition-studs, b b the door-jambs. 
 
 Fig. 1383 represents the forms of the parts of a door, and the way in which 
 they are put together. When the tenons are to be slipped into the mortises, 
 they are covered with hot glue, and, after being closed up, keys are driven in. 
 
 With regard to the proportions of internal doors, they should depend in 
 some degree on the size of the apartments ; in a small room a large door always 
 gives it a diminutive appearance, but doors leading from the same room or 
 passage, which are brought into the. same view, should be of uniform height. 
 The smaller doors which are found on sale are 2 feet 4 inches X 6 feet ; for 
 water-closets, or very small pantries, they are sometimes made as narrow as 15 
 inches, but any less height than 6 feet will not afford requisite head-room ; 
 2 feet 9 inches X 7 feet, 3 feet X 7 feet 6 inches, or 3 feet 6 inches X 8 feet, 
 are well-proportioned, six-panelled doors. But the apparent proportions of a 
 door may be varied by the omission of the parting-rail, making the door four- 
 panelled, or narrowed still more by the omission of the lock-rail, making a two- 
 panelled door. Sometimes the muntin is omitted, making but one panel ; but 
 this, of course, will not add to the appearance of width, but the reverse. Wide 
 panels are objectionable, as they are apt to shrink from the mouldings and 
 crack. The mouldings are generally planted on, and nailed to the stiles and 
 rails, but sometimes formed on the latter. 
 
572 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 When the width of the door exceeds four feet, it is generally made in two 
 parts, each part being hung to its side of the frame, or one part hung to the 
 
 other, so as to fold back like a 
 shutter; or the parts may be 
 made to slide back into pockets 
 
 
 
 
 
 
 G 
 
 
 
 
 D 
 
 
 
 
 
 E 
 
 
 
 
 
 H 
 
 
 
 
 F 
 
 E 
 
 O 
 
 
 
 
 
 
 C 
 
 A 
 
 
 
 
 A 
 
 
 
 S 
 
 
 FIG. 1381. 
 
 
 123 4-tt 
 
 FIG. 1382. 
 
 FIG. 1383. 
 
 or grooves in the partition. The doors may be supported on wheels, and run 
 on tracks at the floor-level; or the tracks may be above the doors, and the 
 doors suspended ; or they may be supported by levers, and be moved parallel 
 without rollers ; all these appliances can be purchased. 
 
 Figs. 1384, 1385, and 1386 are the elevation, vertical and horizontal sections 
 of a pair of slid ing-doors. There are no knobs, but countersunk pulls to the 
 doors, that they may be slid entirely within the pockets, with a special handle 
 in the locks at the edges of the doors for withdrawing them. 
 
 Figs. 1387 and 1388 are vertical and horizontal sections of the same doors 
 hung on butts or hinges. 
 
 Figs. 1389 and 1390 are the elevation and horizontal section of an antas- 
 finished outside-door, with the side-lights C C, and a top, fan, or transom light 
 B. The bar A is called a transom, and this term is applied generally to hori- 
 zontal bars extending across openings, or even across rooms. 
 
 Fig. 1391 is the elevation of an outside folding-door. The plan (Fig. 1392) 
 shows a vestibule, V, and an interior door. The outer doors open, as shown by 
 the arcs, and fold back into the pockets or recesses, p p, in the wall. This is a 
 very common form of doors for first-class houses in this city. The fan-lights 
 are made semicircular, and also the head of the upper panels of the door ; these 
 
ARCHITECTURAL CONSTRUCTION. 
 
 573 
 
 panels in the interior or vestibule door are of glass. Of late the outer doors are 
 extensively used as storm doors, glazed with plate glass exposing the vestibule, 
 and hung with spring butts and ornamental in finish. 
 
 FIG. 1386. 
 
 Windows are usually understood to be glazed apertures. The sashes may 
 be stationary, but for most positions they are made to open either by sliding 
 
574: 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 vertically, or laterally, or like doors. The first is the common form of window, 
 and the sashes are generally balanced by weights ; the second, except in a cheap 
 
 FIG. 1387. 
 
 FEET 
 
 form in mechanics' shops, are seldom used ; the third, often used 
 for access to balconies or between rooms, are called casements, or 
 French windows. 
 
 Figs. 1393 and 1394 are the outside elevation and horizontal 
 section of one half of a common 
 
 FIG. 1390. 
 
 FIG. 1392. 
 
 box-frame, and Fig. 1395 a vertical section of the same in a wooden 
 frame house. S is the sill of the sash-frame, W the frame-sill, 
 with a wash to discharge the water, B the bottom rail of the sash, 
 M the meeting rails, T the top rail, H the head of the sash-frame, 
 and A the architrave similar to that around doors. Instead of 
 two sills, S and W, one is often used, and inclined to form the wash. D is the 
 common outside blind. In the sectional plan (Fig. 1400), C C' are the win- 
 dow-stiles, F the pulley -stile, w w the sash-weights, p the parting strip, and D D 
 double-fold shutters. 
 
 Figs. 1396 and 1397 are the interior elevation and vertical section of a box- 
 frame window in a masonry wall ; Fig. 1398 is an exterior view of the same 
 
ARCHITECTURAL CONSTRUCTION. 
 
 D 
 
 FIQ. 1393. 
 
 M 
 
 <B 
 
 FEET 
 
 FIG. 1394. 
 
 Fio. 1395. 
 
576 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 window, and Pig. 1399 a horizontal section. In 
 masonry walls the sills are usually of stone, as 
 shown in Fig. 1397, with a lap of the window 
 frame on it. In setting the sill, one course of brick 
 is left out beneath the central portion to admit of 
 settlement in the walls without stress to the sill. 
 
 o 
 
 B : 
 
 E 3 
 
 FIG. 1396. 
 
 FIG. 1397. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 In brickwork, the 
 height of sill and 
 cap corresponds 
 to a determinate 
 number of courses, 
 so that it may not 
 be necessary to 
 split brick in set- 
 ting them. 
 
 Unless the win- 
 dows begin from, 
 or nearly from, 
 the floor, the point 
 a (Fig. 1395) may 
 be fixed at a height 
 of about 30 inches 
 above the floor, 
 and the top of the 
 window sufficient- 
 ly below the ceil- 
 ing to allow space 
 for the architrave 
 or other finish 
 above the window, 
 and for the cornice 
 of the room, if 
 there be any ; .a 
 little space be- 
 tween these adds 
 to the effect. For 
 common windows, 
 the width of the 
 sash is 4 inches 
 more than that of 
 the glass, and the 
 height G inches 
 more ; thus the 
 sash of a window 
 
 3 lights wide and 
 
 4 lights high, of 
 12" X 16" glass, is 
 3 feet 4 inches 
 wide and 5 feet 10 
 inches high. In 
 plate - glass win- 
 dows more width 
 is taken for the 
 stiles and rails. 
 
 38 
 
 FIG. 1398. 
 
 Fie. 1399. 
 
578 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 The usual sizes of cylinder glass are 7" X 9" up to 24" X 36", but single thick 
 glass may. be had up to 40" X 60" ; double thick, 48" X 62". Plate glass, 
 polished or rough, may be had of a size as large as 14 X 8 feet. 
 
 In Fig. 1393 the 
 blind D is hinged to 
 the hanging stile, and 
 is closed within the 
 opening in the mason- 
 ry. The slats are 
 
 . 1400. 
 
 1101. 
 
 FIG. 1402. 
 
 movable on pin tenons, 
 
 and those of each half, 
 
 upper and lower, are 
 
 connected by a central 
 
 bar, so that they are 
 
 moved together, and 
 
 adjusted at any angle to the light. In Fig. 1399 the blinds are inside, 4-fold, 
 
 and folding back into pockets. It is more usual to make the pockets for the 
 
 blinds inclined to the window, as in- Fig. 1400, giving to the interior more 
 
 light, or ampler space for curtains. 
 
 Fig. 1401 is the outside elevation of a French window or casement. 
 
 Fig. 1402 represents the sectional elevation of the same window, in broken 
 lines, and on a larger scale ; the same letters designate similar parts as in Fig. 
 1395. A transom-bar is often framed between the meeting-rails, and in this 
 case the upper sash may be movable; in Fig. 1402 it is fixed. An upright, 
 called a mullion, is often introduced in the centre, against which the sash shuts. 
 
 For use as doors, the lower sashes should not be less than 5 feet 6 inches 
 high. In these forms of sash the rails and stiles are wide, and for equal aper- 
 tures. French windows when closed admit light. The chief objection to this 
 window lies in the difficulty of keeping out the rain at the bottom in a driving 
 storm. To obviate this, the small moulding d, with a drip or undercut, is nailed 
 to the bottom rail ; but the more effectual means is the patent weather-strip, 
 the same as used on outside doors. 
 
 Dormer or attic windows are framed and set as in an upright stud-partition. 
 
 Stairs consist of the tread or step on which to set the feet, and risers, up- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 579 
 
 right pieces supporting the treads each tread and riser forms a stair. If the 
 treads are parallel they are called fliers ; if less at one end than the other, they 
 are called winders, f and w (Fig. 1409). The top step, or any intermediate 
 wide step, for the purpose of resting, is called a landing ; the height from the 
 top of the nearest step to the ceiling above the headway ; the rounded edge of 
 the step a nosing (a, Fig. 1403) ; and if a small hollow or cavetto (b) be glued 
 in the angle of the nosing and riser, it is called a moulded nosing. The pieces 
 which support the ends of the stairs are the strings (Fig. 1404) ; that against 
 the wall the wall-string, the other the 
 outer string. Besides these strings, 
 pieces of timber are framed and placed 
 beneath the fliers, when the stairs are 
 
 a ^ 
 
 FIG. 1404. 
 
 wide (Fig. 1405), called carriages. Sometimes the strings, instead of being 
 notched out to receive the steps, have the upper and lower edges parallel, with 
 
 grooves cut in the inner faces to 
 receive the ends of their steps and 
 
 FIG. 1405. 
 
 FIG. 1406. 
 
 risers (Fig. 1406). These are called housed strings. Steps and risers are se- 
 cured in the grooves by wedges covered with glue, and driven in. For the 
 rough, strong strings of warehouses the carriages are made of plank, with 
 grooves to receive plank-treads, and without risers. 
 
 Figs. 1407 and 1408 are elevation and plan of a straight run of stairs, both 
 partly in section. N is the newel-post, n a baluster, h the hand-rail, iv the 
 well. In the section of the floors, cleats are shown nailed to the beams ; on 
 these short boards are nailed to form a box for the reception of mortar for the 
 deafening. The opening represented in the plan (which must occur between 
 the outer strings, if they are not perpendicular over each other) is called the 
 well (W, Fig. 1409). 
 
 The breadth of tread in general use is from 9 to 12 inches; in the best 
 staircases, it should never be less than 11 inches, nor more than 15. The 
 height of the riser should be the more, the less the width of the tread ; for 
 a 15-inch tread the riser should be 5 inches high ; for 12 inches, 6.J- ; for 9 
 inches, 8. In laying out the plan of stairs, having determined the starting- 
 
580 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 point, either at bottom or top, as the case may be, find exactly the height of 
 the story ; divide this by the height you suppose the riser should be. Thus 
 
 Fro. 1408. 
 
 (Fig. 1410), if the height of the story and thickness of floor be 9 feet, and we 
 suppose the riser should be 7 inches high, then 108 inches, divided by 7 = 15f 
 
 It is clear that there must be a full number of steps, either 16 or 15 ; to 
 be ntjar the supposed height of the riser, adopt 15, then 
 ^Y- 7^g- inches, height of riser. 
 
 For this particular case, assume the breadth of the step as 10 inches, and 
 the length at 3 feet, a very usual length, seldom exceeding 4 feet in the stair- 
 cases of private houses. For the plan lay off the outside of the stairs, two 
 parallel lines 3 feet apart, and space off from the point of beginning 14 treads 
 of 10 inches each, and draw the cross-parallel lines. To construct the eleva- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 581 
 
 tion, project the lines of the steps in plan, and divide the height, either on a 
 perpendicular or by an inclined line, into the number of risers (15), and draw 
 
 FIG. 1409. 
 
 FIG. 1-110. 
 
 horizontal lines through these points; or the same points maybe determined by 
 intersection of the projections of the plan with a single inclined line drawn 
 along the^ nosing of top and bottom steps. The number of treads is always one 
 less than the number of risers, the reason of which appears in the elevation. 
 
 For the framing plan the drawing of the elevation of stairs is in general 
 necessary, to determine the opening to be framed in the upper floor, to secure 
 proper headway. Thus (Fig. 1410), the distance, a, between the nearest stair 
 and the ceiling should not be less than 6 feet 6 inches ; a more ample space 
 improves the look of the stairway ; but if confined in our limits, this deter- 
 mines the position of one header ; the other will be of course at the top of the 
 stairs. When one flight is placed over another, the space required for timber 
 and plastering, under the steps, is about 6 inches for ordinary stairs. 
 
 When the stairs are circular, or consist in part of winders and fliers, as in 
 Fig. 1409, the width of the tread of the winders should be measured on the 
 central line. The construction of the elevation is similar to that of the straight 
 run (Fig. 1410), dividing the space between the stories by a number of parallel 
 lines equal to the number of risers, and intersecting the parallels by projections 
 from the plan. The objection to all circular stairs of this form, or with a small 
 well-hole or a central shaft, is that there is too much difference between the 
 width of the tread, but a small portion being of a suitable size. The hand- 
 somest and easiest stairs are straight runs, divided into landings, intermediate 
 of the stories, and either continuing then in the same line, or making a full re- 
 turn at right angles. It is at times fashionable to make the newel a prominent 
 feature in the hall, often occupying valuable space. It is sufficient that it 
 
582 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 be large and stiff enough for a support to the hand-rail and may be equally 
 ornamental. 
 
 The top of the hand-rail should, in general, be about 2' 8" to 3' above the 
 nosing, and should follow the general line of the steps. The angles of the hand- 
 rail should always be eased off. A hand-rail, affording assistance in ascending 
 or descending, should not be wider than the grasp of the hand (Fig. 1411) ; but 
 where, for architectural effect, a more massive form may be necessary, it is very 
 convenient to have a sort of double form, with the hand-rail at the top (Fig. 
 1412), or as in Fig. 1413, with the groove outside. 
 
 To a draughtsman conversant with the principles of projection already given, 
 it will not be difficult to draw in the hand-rail of stairs, or to lay off the mould 
 
 FIG. 1411. 
 
 Fio. 1412. 
 
 FIG. 1413. 
 
 for its construction. It will follow the line of stair- nosing, and where there are 
 changes of pitch they are made to connect by curves tangent to these pitches, 
 except where the landings are square, and newels set at the head of the land- 
 
 Fro. 1414. 
 
 ings, the rail is framed into the newel. At the bottom the rail is curved to the 
 horizontal, when it comes into or upon top of the newel. 
 
 Balusters are of great variety usually turned forms attached to the treads 
 
ARCHITECTURAL CONSTRUCTION. 
 
 583 
 
 by dovetails, covered with the returned nosing, or with pin-ends and holes in 
 treads and under side of caps. Sometimes (especially in iron-work) the baluster 
 
 ELEVATION. 
 
 PLAN. 
 
 FIG. 1415. 
 
 FIG. 1416. 
 
 is set iii a bracket from the face of the string, as in Fig. 1415 ; or the balusters 
 may be cast with the bracket. 
 
 Fig. 1414 is the side elevation of a stairs with wrought-iron string and rail. 
 The string is made of wrought-iron knees, welded together continuously, with 
 
 a flat bottom-bar riveted across the lower angle of the knees, usually supported 
 by an intermediate round bar-post. Where posts can not be put in, it is better 
 
ARCHITECTURAL CONSTRUCTION. 
 
 that the bottom bar should be a carriage or beam of I or channel-iron, with 
 knees or cast-iron angle-blocks riveted on the top of the beam. It is not un- 
 usual to make housed strings of plate-iron, with angle-irons riveted on to 
 receive the treads and risers. If the plate-iron be wide enough to serve in- 
 stead of balusters, it makes a very strong and stiff carriage. 
 
 Figs. 1415 and 1416 are the plan and elevation of a cast-iron stairs, with a 
 central post or newel (this term is applied also to the first post of any stairs). 
 The newel-ring, tread, and riser of each step are cast in one piece, and they are 
 put together by placing one newel-ring upon that below and bolting the outer 
 extremity of the riser to the tread below. 
 
 Fig. 1417 is a form of cast-iron stairs with a well instead of a newel; the 
 step and riser are bolted together by the flanges. It will be seen that one tread 
 is wider than the others ; this is a landing. 
 
 Fireplaces. Fireplaces for wood are made with flaring jambs of the form 
 shown in plan (Fig. 1418) ; the depth from 1 foot to 15 inches, the width of 
 opening in front from 2 feet 6 inches 
 to 4 feet, according to the size of 
 the room to be warmed ; .height 2 
 
 feet 3 inches to 2 feet 9 inches, the 
 width of back about 8 inches less 
 
 than in front ; but at present fire- FIG ]419 
 
 places for wood are seldom used, 
 
 stoves and grates having superseded the fireplace. The space requisite for the 
 largest grate need not exceed 2 feet in width by 8 inches in depth. The requi- 
 site depth is given by the projection of the grate, and the mantel-piece. Eanges 
 require from 4 feet 4 inches to 6 feet 4 inches wide X 12 inches to 20 inches 
 deep ; jambs 8 inches to 12 inches. 
 
 Fig. 1419 represents the elevation of a mantel-piece of very usual propor- 
 tions. The length of the mantel is 5 feet 5 inches, the width at base 4 feet 6 
 
 inches, the height of opening 2 feet 7 inches, 
 and width 2 feet 9 inches. A portion of this 
 opening is covered by the iron sides or architrave 
 of the grate, and the actual open space would 
 not probably exceed 18 inches in width by 2 feet 
 in height. In brick or stone houses the flues are 
 formed in the thickness of the wall, but when 
 distinct they have an outside shell of a half-brick or 4 inches, and sometimes 
 8" (Fig. 1420) ; the witlis or division- walls .always 4". 
 
 The size of house flues is usually 8" X 8", but some are 4" X 8^, 4" X 12", 
 and 8" X 12". The flues of different fireplaces should be distinct. Those from 
 the lower stories pass up through the jambs of the upper fireplaces, and, keep- 
 ing side by side with but 4-inch brickwork between them, are topped out above 
 
 Fio. 1420. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 585 
 
 the roof, sometimes in a double and often in a single line 16 inches wide by a 
 
 breadth required by the number of flues, as in Fig. 1420 or in Fig. 1421. The 
 
 ._ . latter is an illustration of how far 
 
 flues may be diverted from a vertical 
 
 .mi ji li ll Ij jj{ line, but it is to be observed that the 
 
 construction must be stable, as any 
 settling or cracks not only injures the 
 draught of the chimney, but impairs 
 
 FIG. 1421. 
 
 FIG. 1422. 
 
 the security of the building against fire. Changes of direction of flues should 
 never be abrupt. The back of the fireplace may be perpendicular through its 
 whole height, but it is usual to incline the upper half inwardly toward the room, 
 making the throat to the flue long and narrow. It is very common to form 
 the upper 3" to 4" of the inclined back by an iron plate, which can be turned 
 back or forward to increase or diminish the draught. Fig. 1422 represents the 
 arrangement of frame and brick arch for the support of the hearth. The 
 chimney is generally capped with stone, sometimes with tile or cement pots. 
 As an architectural feature, the chimney is often made to add considerably to 
 the effect of a design. 
 
 Roofs. Framed roofs have been illustrated (page 486), City roofs are 
 usually flat, and timbered similarly to floors, but not so strongly, with a slight 
 pitch to discharge rainfall. Eoofs of country dwellings are usually framed like 
 stud-partitions, with inclined studs somewhat deeper than if they were ver- 
 tical, depending on the inclination from the vertical; if flat, depth like that 
 of a floor. The theory of the 
 construction of the gambrel or 
 Mansard roof (Fig. 1423), a 
 
 FIG. 1423. 
 
 FIG. 1424. 
 
 roof with two kinds of pitch, is that of the polygon of rods, and self-support- 
 ing; but, in general, they have central support from partitions, and their 
 outlines are much varied by curves in the lower rafters cut from plank. 
 
586 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Fig. 1424 is the plan of a roof as usually drawn, shaded strongly at the 
 ridges. The transept roof is hipped at A and B. 
 
 Gutters are generally formed in the cornice (Fig. 1425) ; sometimes on the 
 roof (Fig. 1426), and sometimes by raising a parapet (Fig. 1427) and forming 
 a valley. The intersection of two roofs as at v forms a valley. 
 
 FIG. 14:25. 
 
 FIG. 1426. 
 
 FIG. 1427. 
 
 Fig. 1425 represents a form of gutter very common to city buildings, the 
 roof boarding extending over the gutter; but it is preferable to make the roof 
 pitch from both rear and front to the centre of the building, and to carry the 
 leader down in the interior, where it may serve as a soil- 
 pipe for the water-closets, basins, and baths, affording 
 ventilation in fair weather and a scour in rains. 
 
 FIG. 1428. 
 
 FIG. 1429. 
 
 Fig. 1428 is a gutter of a cottage roof. 
 
 Fig. 1429 is the section of a Mansard roof, showing the side elevation of a 
 dormer-window, with the gutter below its sill. 
 
 The sheet-metal forming the gutter must extend well up or back beneath 
 the shingles or felt, or be soldered to the tin of the roof, to prevent water find- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 587 
 
 ing its way into the interior; and at the 
 sides flashings of tin must extend on the 
 walls above the roof and into the joints of 
 the brick. 
 
 Sheet-metal cornices, at their first intro- 
 duction, were put up on wood lookouts, cut 
 to the form of the cornice, but it is now the 
 practice to use metal supports and fasten- 
 ings to the entire exclusion of wood in 
 many cases cheaper and always safer against 
 fire and rotting, but the iron used must be 
 protected against rust by galvanizing or 
 heavy coats of paint or by both. Fig. 1430 
 shows a section of a galvanized cornice with 
 bar-iron frames anchored to the wall and FlG 1430 
 
 roof and riveted to the cornice ; the joints 
 
 of the cornice should be both riveted and soldered. The pitch of the gutter is 
 secured by variations in the bend of the roof braces. 
 
 Plastering. To prevent damp striking through the plastering of outer 
 walls, and cracks in ceilings, it is usual to fur. walls and beams; that is, to nail 
 
 FIG. 1431. 
 
 Fio. 1432. 
 
 vertical strips of wood to the walls, and across from beam to 
 beam. Furring-strips are from 1" to 2" wide, and about " 
 thick, nailed at distances of 12" or 16" centres (usually the 
 former), adapted to the length of the laths, which are 4 feet 
 long, and about 1|" x " = spaces between laths " to f". The 
 first coat of mortar is the scratch-coat, which is forced through 
 the interstices between the laths, to make a lock to retain it. 
 
 FIG. 1434. 
 
 FIG. 1435. 
 
 FIG. 1436. 
 
 This coat is about " thick. The next or brown coat is about 
 " thick, and if the last coat is a sand-finish, it will be less than 
 " thick ; while, if the last coat is a hard finish, its thickness 
 will be almost imperceptible. Figs. 1431 and 1433 are sections 
 of furring and plastering. 
 
588 ARCHITECTURAL CONSTRUCTION. 
 
 The brown coat is usually carried down to the floor. Over this is nailed 
 the base-board, A (Fig. 1433), for the finish around the bottom of the walls of 
 the room. To guard against the crack formed between the floor by the shrink- 
 ing of the base, a teuon is formed along the outer edge of the latter with a 
 groove for its reception in the floor. This is termed dadoing. Above the base 
 is a moulding forming a part of the base ; above this, there may be a moulded 
 rail, B, called the chair-rail, or surbase, and between a panel, termed a dado. 
 The walls of stores are generally ceiled up as high as the surbase. For the 
 finish of the angle of the wall and ceiling, it is usual in the better rooms to 
 form a cornice in plaster. The cornices are mouldings of varied forms, with 
 or without enrichments that is, plaster ornaments. Figs. 1434, 1435, and 
 1436 are sections of cornices. If the rooms are low, the cornice should extend 
 but little on the wall, but well out on the ceiling. 
 
 In all architectural finish mouldings are a necessity, the simpler forms of 
 which are taken from Greek or Koman examples. 
 
 Greek and Roman Mouldings. The regular Greek mouldings are eight in 
 number : the Fillet or Band, Torus, Astragal or Bead, Ovolo, Cavetto, Cyma 
 Ilec/ta or Ogee, Cyma Reversa or Talon, and Scotia. 
 
 The fillet (a, Fig. 1437) is a small rectangular member, on a flat surface, 
 whose projection is usually made equal to its height. 
 
 The torus and astragal are semicircles in form, projecting from vertical 
 diameters, as in Fig. 1438. The astragal is distinguished from the torus in 
 the same order by being made smaller. The torus is generally employed in the 
 bases of columns; the astragal, in both the base and capital. 
 
 FIG. 1437. FIG. 1433. FIG. 1439. 
 
 The ovolo is a member strong at the extremity, and intended to support. 
 The Roman ovolo consists of a quadrant or a less portion of a circle (Fig. 1439). 
 The Greek ovolo is elliptic. 
 
 To describe the Greek ovolo (Fig. 1440): Draw df from the lower end of 
 the proposed curve, at the required inclination ; draw the vertical g efto define 
 the projection, the point e being the extreme point of the curve. Draw e li 
 parallel to df, and draw the vertical d li k, such that d li 
 is equal to hk. Divide eh and e f into the same number 
 of equal parts ; from d draw straight lines to the points of 
 division in ef, and from k draw lines through the divi- 
 sions in e h to meet those others successively. The inter- 
 sections so found are points in the curve, which may be 
 FIG. 1440. traced accordingly. 
 
 The cavetto is described like the Roman ovolo by 
 
 circular arcs, as shown in Figs. 1441 and 1442. Sometimes it is composed of 
 two circular arcs united (Fig. 1443) ; set off be, two thirds of the projection, 
 draw the vertical b d equal to be, and on d describe the arc b i. Join e d and 
 produce it top; draw i n perpendicular to e d, set off no equal to ni, and 
 
ARCHITECTURAL CONSTRUCTION. 
 
 589 
 
 draw the horizontal line op meeting ep ; on p describe the arc io to complete 
 the curve. 
 
 The ogee, or cyma recta (Fig. 1444), is compounded of a concave and a con- 
 vex surface. Join a and 5, the extremities of the curve, and bisect a b at c; on 
 a and c, as centres, with the radius a c, describe arcs cutting 
 at d ; and on b and c, describe arcs cutting at e. On d and e, 
 t \ as centres, describe the arcs a c, cb, composing the moulding. 
 
 FIG. 1441. 
 
 FIG. 1442. 
 
 FIG. 1444. 
 
 The cyma reversa, or talon (Fig. 1445), is a compound curve, distinguished 
 from the ogee by having the convex part uppermost. 
 
 If the curve be required to be made quicker, a shorter radius than a c must 
 be employed. The projection of the moulding n b (Fig. 1444) is usually equal 
 to the height a n. 
 
 To describe the Greek talon : Join the extreme points a, b (Fig. 144G) ; 
 bisect a b at c, and on a c and c b, describe the semicircles b d c and c a. Draw 
 perpendiculars d o, etc., from a number of points in a c and 
 c b, meeting the circumferences ; and from the same points set 
 off horizontal lines equal to the respective perpendiculars : 
 o n equal to o f/, for example. The 
 curve line b n a, traced through the 
 ends of the lines, will be the contour 
 of the moulding. 
 
 To describe a scotia : Divide the 
 perpendicular a b (Fig. 1447) into three 
 equal parts, and with the first, a e, for 
 radius, on e as a centre, describe the arc 
 afh, in the perpendicular c o set off c I equal a e, join e I, and bisect it by the 
 perpendicular o d, meeting c o at o, on the centre 0, with o c for radius, complete 
 the figure by the arc c h. 
 
 These mouldings, and combinations of them, are stuck in wood, and are to 
 be purchased in every variety. Fig. 1448 represents some of the common forms 
 always to be had, and of suitable sizes. 
 
 Proportions and Distribution of Rooms and Passages. Rooms of dwell- 
 ing-houses are to be proportioned and arranged according to the necessities of 
 position and use, the space that can be occupied, the financial means available, 
 and of ten J;o suit the peculiar wishes of owners or occupants. In cities, the 
 limits of the lot restrict the arrangements to a small ground-space, and require 
 an increase in the number of stories. Use has established certain forms often 
 peculiar to different cities, beyond which there is little change ; but in the 
 country, where there is plenty of ground-space, and where many stories are 
 usually injurious to the* aesthetic effect, and where there are few canons in archi' 
 
 FIG. 1447. 
 
590 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 1448. 
 
ARCHITECTURAL CONSTRUCTION. 591 
 
 tecture to be observed, there is little limit to the variety of forms and arrange- 
 ments of country-houses. 
 
 In designing a country-house, where one is not restricted in space, it is often 
 convenient to mark out the rooms of the desired size on slips of paper, accord- 
 ing to some scale, then cut them out and arrange them in as convenient an 
 order as possible, and modify the arrangement by the necessities of construction 
 and economy. Thus, the greater the outside surface in proportion to the in- 
 cluded area, and the greater the number of chimneys and space used for pas- 
 sages, the greater the cost. The kitchen should be of convenient access to the 
 dining-room, both should have large and commodious pantries, and all rooms 
 should have an access from an entry, without being compelled to pass through 
 other rooms ; this is particularly applicable to the communication of the kitchen 
 with the front door. Outside doors for common and indiscriminate access 
 should not open into important rooms. All rooms to be occupied as living or 
 bedrooms should have flues opening directly or indirectly to the outer air for 
 ventilation. 
 
 As to the size of the different rooms, they must of course depend on the pur- 
 poses to which they are to be applied, the class of house, and the number of 
 occupants. The kitchen for the poorer class of houses is also used as an eat- 
 ing-room, and should therefore be of considerable size to answer ooth purposes; 
 for the richer houses, size is necessary for the convenience of the work. In the 
 older New York private houses the average was about 16 X 20 feet ; for me- 
 dium houses in the country generally less, say 12 X 16. A back kitchen, scul- 
 lery, or laundry, should be attached to the kitchen, and may serve as a passage- 
 way out. 
 
 The Dining or Eating Rooms. The common width of dining-tables varies 
 from 4 to 5 feet 6 inches ; the space occupied by the chair and person sitting 
 at the table is about 18 inches ; the table-space, for comfort, should be not less 
 than 2 feet for each person at the sides of the table, and considerable more at 
 the head and foot ; hence the space that will be necessary for the family and 
 its visitors at the table may be calculated. Allow a further space of 2 feet at 
 each side for passages, and some 3 to 5 at the head for the extra tables or chairs, 
 for the minimum of space required ; but, if possible, do not confine the dining- 
 room to meagre limits, unless for very small families ; let not the parties be lost 
 in the extent of space, nor let them appear crowded. 
 
 TJie parlours should be made according to the rules given below, not square, 
 but the length about once and a half the width ; if much longer than this, break 
 up the walls by transoms or projections. As to the particular dimensions, no 
 rules can be given; they must depend on every person's taste and means; 
 20 x 16 is a very fair size for a regular living-room parlour, not a drawing- 
 room. The same size is ample for a sleeping-room. The usual width of single 
 beds is 2' 8" to 3' 6" ; of three-quarter, 4' to 4' 6" ; of whole, 5 to 6 feet ; the 
 length, 6' 6" ; and as the other furniture may be made to consist of but very 
 few pieces, if adequate means of ventilation are provided parties may live 
 snugly in small quarters. The bed*should not stand too near the fire, nor 
 between two windows ; its most convenient position is head against an interior 
 wall, with a space on each side of at least 2 feet. To the important bedrooms 
 of first-class houses, dressing rooms should be attached, and, if there is water 
 
592 
 
 ARCHITECTURAL CONSTRUCTION. 
 
ARCHITECTURAL CONSTRUCTION. 593 
 
 and sewer service, fitted with set bowls and baths and water-closets. If pos- 
 sible, there should be windows opening to the outer air, but always with flue- 
 ventilation. 
 
 Pantries. Closets for crockery should not be less than 14 inches in depth 
 in the clear ; for the hanging up of clothes, not less than 18 inches, and should 
 be attached to every bedroom. For medium houses, the closets of large sleep- 
 ing-rooms should be at least 3 feet wide, with hanging-room, and drawers and 
 shelves. There should also be blanket-closets, for the storing of blankets and 
 linen ; these should be accessible from the entries, and may be in the attic. 
 Store-closets should also be arranged for groceries and sweetmeats. 
 
 Passages. Front, entries are usually 6 feet wide in the clear ; common pas- 
 sage-ways, 3 feet ; these are what are required, but ample passages give an im- 
 portant effect to the appearance of the house. The width of principal stairs 
 should be not less than 3 feet, and all first-class houses, especially those not pro- 
 vided with water-closets and slop-sinks on the chamber-floor, should have two 
 pairs of stairs, a front and a back pair ; the back stairs need not necessarily be 
 over 2 feet 6 inches in width. 
 
 TJie Height of Stories. It is usual to make the height of all the rooms on 
 each floor equal ; it can be avoided by furring down, or by the breaking up of 
 the stories, by the introduction of a mezzonine or intermediate story over the 
 smaller rooms. Both remedies are objectionable. 
 
 The average height of the stories for common city dwellings is : Cellar, 6 
 feet 6 inches ; common basement, 8 to 9 feet ; English basement, 9 to 10 feet ; 
 principal story, 12 to 35 feet ; first chamber floor, 10 to 12 feet ; other chamber- 
 floors, 8 to 10 feet all in the clear. For country-houses, the smaller of the 
 dimensions are more commonly used. Attic stories are sometimes but a trifle 
 over 6 feet in height, but are objectionable. 
 
 Privies, Water- Closets, and Out-Houses. The size of privies must depend 
 greatly on the uses of the building to which they are to be attached, its position, 
 and the character of its occupants. Allowing nothing for evaporation and ab- 
 sorption, the entire space necessary for the excrementitious deposits of each 
 individual, on an average, will be about seven cubic feet for six months, of 
 which seven eighths is fluid. In the country, vaults are usually constructed of 
 dry rubble-stone, and the fluid matters are expected to be filtered through the 
 earth, the same as in cesspool-waste ; but great care must be taken that they 
 neither vitiate the water-supply nor the air of the house. A brick and cement 
 vault, air and water tight, with a ventilating-pipe into a hot chimney-flue, is 
 the best preventive, and may even be built within the house. In all other cases 
 there should be free air-space between the house and privy. In the city, where 
 there is adequate water-supply and sewerage, the water-closet should be adopted. 
 The water-closet, or privy, with a single seat, should occupy a space not less 
 than 4' x 2' 6". The rise of seat should be about 17" high ; and the hole egg- 
 shaped, 11" x 8". The earth-closet, when properly taken care of, is an ex- 
 tremely useful appendage to a country-house, and the space requisite for it is 
 the same as that of a water-closet. 
 
 The forms of modern water appliances, and the means to get rid of house- 
 waste, will be illustrated hereafter, under the heads of Ventilation and 
 Plumbing. 
 
 39 
 
594: ARCHITECTURAL CONSTRUCTION. 
 
 For Wood or Coal Sheds or Bins. In estimating the size of these accesso- 
 ries, it may only be necessary to state that a cord of wood contains 128 cubic 
 feet, and a ton of anthracite egg coal occupies a space of about 40 cubic, feet, of 
 bituminous coal about 45 cubic feet, of coke from 45 to 50 cubic feet. 
 
 On the Size and Proportion of Rooms in general. " Proportion and or- 
 nament," according to Ferguson, " are the two most important resources at 
 the command of the architect, the former enabling him to construct ornamen- 
 tally, the latter to ornament his construction." A proportion to be good must 
 be modified by every varying exigence of a design ; it is of course impossible to 
 lay down any general rules which shall hold good in all cases ; but a few of its 
 principles are obvious enough. To take first the simplest form of the propo- 
 sition, let us suppose a room built, which shall be an exact cube of say 20 feet 
 each way such a proportion must be bad and inartistic ; and, besides, the 
 height is too great for the other dimensions. As a general rule, a square in 
 plan is least pleasing. It is always better that one side should be longer than 
 the other, so as to give a little variety to the design. Once and a half the 
 width has been often recommended, and with every increase of length an in- 
 crease of height is not only allowable, but indispensable. Some such rule as 
 the following meets most cases : " The height of the room ought to be equal to 
 half its width plus the square root of its length " ; but if the height exceed the 
 width the effect is to make the room look narrow. Again, by increasing the 
 length we diminish, apparently, the other two dimensions. This, however, 5 is 
 merely speaking of plain rooms with plain walls ; it is evident that it 'will be 
 impossible, in any house, to construct all the rooms and passages to conform to 
 any one rule of proportion, nor is it necessary, for in many rooms it would not 
 add to their convenience, which is often the most desirable end ; and, if re- 
 quired, the unpleasing dimensions may be counteracted by the art of the archi- 
 tect, for it is easy to increase the apparent height by strongly marked vertical 
 lines, or bring it down by horizontal ones. Thus, if the walls of two rooms of 
 the same dimensions be covered with the same strongly marked striped paper, 
 in one case the stripes being vertical and in the other horizontal, the apparent 
 dimensions will be altered very considerably. So also a deep, bold cornice 
 diminishes the apparent height of a room. If the room is too long for its other 
 dimensions, this can be remedied by breaks in the walls, by the introduction 
 of pilasters, etc. So also, as to the external dimensions of a wall, if the length 
 is too great it is to be remedied by projections, or by breaking up the lengths 
 into divisions. 
 
 Understanding the general necessities of a dwelling, the proportions of 
 rooms, forms of construction, and space to be occupied, the draughtsman is 
 prepared to undertake designing, and for this purpose cross-section paper will 
 be found of very great use. Taking the side of a small square as a unit one 
 foot, for instance he can readily pencil in rooms and passages, and alter and 
 modify at pleasure. 
 
 Figs. 1445 to 1456 are illustrations of this form of designing. Partitions 
 are to be as much as possible one over the other, and the posts or walls ar- 
 ranged in the cellar, for the support of these lines of partitions. For the 
 sketch, it is sufficient to make door and window openings 3 feet, unless for 
 some particular purpose double-fold doors or bow or mullioned windows are 
 
ARCHITECT ORAL CONSTRUCTION. 
 
 595 
 
 required. In arranging the stairs, the whole run may be taken as !- time the 
 height of the story between floor and floor ; as square landings have one riser, 
 
 FIG. 1457. 
 
 K 
 
 FIG. 1458. 
 
 FIG. 1459. 
 
 p 
 
 
 P 
 
 = = 
 
 ! 
 
 P 
 
 
 
 
 -i # 
 
 i 1 
 
 the rise will be equal to the square less one tread, say, for 
 the design 1 foot. The length of opening in the frame, 
 say 12 feet for a straight run. 
 
 Figs. 1447 to 1472 are plans of familiar forms of 
 houses, drawn to the scale of 32 feet to the inch, as 
 illustrations to the student and as examples to be copied 
 on a larger scale. The same letters of reference are used 
 on all the plans. Thus, K K designate kitchens, cooking- 
 rooms, or laundries ; D D eating-rooms ; S S sleeping- 
 rooms; PP drawing-rooms, parlours, or libraries; pp pantries, china or store 
 closets, or clothes-presses ; c c water-closets and bath-rooms. These last are 
 not shown in the plans of country houses, but are recognised as a necessity in 
 the best of this class. The space occupied by them is given on page 593. 
 
 Figs. 1457, 1458, and 1460 are first-story plans of houses of square out- 
 line. Fig. 1459 is the second story of Fig. 1458. 
 
 FIG. 1460. 
 
 J) 
 
 FIG. 1462. 
 
 . , 
 
 FIG. 1461. 
 
 FIG. 1463. 
 
 FIG. 1464. 
 
 This form of house has the greatest interior accommodations for the outside 
 cover, and, although not picturesque in its elevation, is a very convenient 
 and economical structure. The kitchen (Fig. 1460) is in the basement, and 
 the connection with the dining-room is by a dumb-waiter in the pantry (p). 
 In Fig. 1461 the plan is the same as in Fig. 1460, but the kitchen (K} is 
 
596 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 in an L attached to the house; there is a small opening between the pantry 
 (p) and kitchen, through which dishes are passed to and from the dining- 
 room. 
 
 Fig. 1462 is the plan of a very small but convenient floor, of prettier out- 
 line than the square ; V is a portico or veranda. No chimney is shown in the 
 
 . .ML, 
 
 I) 
 
 i: 
 
 i. 
 
 K 
 
 .> 
 
 s 
 
 I 
 
 ~n t 
 
 = 
 
 _/''_L 
 
 T 
 
 s 
 
 J" 
 
 
 i_ - 
 
 
 s 
 
 s 
 
 FIG. 1465. 
 
 FIG. 1466. 
 
 FIG. 1467. 
 
 FIG. 1468. 
 
 sleeping-room S', there should be one either against the stairs or the back 
 wall. 
 
 Figs. 1463 and 1464 are first-story plans of houses still more extensive. 
 All of the above are adapted to the country, dependent on lights on all sides, 
 and ample spaces. 
 
 In the cities, houses are mostly confined to one form in their general out- 
 
 iniiii l 
 
 | r- 
 
 
 FIT] 
 
 1 
 
 
 
 p 
 
 
 
 
 
 
 | 
 
 
 3 
 
 __ 
 
 
 
 
 \. 
 
 T- 
 
 
 
 T- 
 
 \ 
 
 mm m 
 
 K 
 
 
 1) 
 
 FIG. 1469. 
 
 FIG. 1470. 
 
 FIG. 1471. 
 
 s 
 
 FIG. 1472. 
 
 line a rectangle. Figs. 1465 and 1469 may be taken as the usual type of 
 New York city houses. Figs. 1465, 1466, and 1467 are the basement, first and 
 second floor plans of a three-rooms-deep, high-stoop house as the first floor is 
 reached by an outside flight of steps about 6 feet high. There is usually a 
 cellar beneath the basement, but in some cases there are front vaults, entered 
 beneath the steps to the front door ; the entrance to the basement itself is also 
 
ARCHITECTURAL CONSTRUCTION. 
 
 597 
 
 LIBRARY DINING. RDDM 
 
 FIR5T 5TDRY 
 
 FIG. 1473. 
 
 FIG. 1474 
 
598 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 1475. 
 
 FIG. 1476. 
 
ARCHITECTURAL CONSTRUCTION. 599 
 
 beneath the steps. The front room of the basement may be used as an eating- 
 room, for the servants' sleeping-room, billiards, or library. The usual dining- 
 room is on the first floor, a dumb-waiter being placed in the butler's pantry, p, 
 for convenience in transporting dishes to and from the kitchen. The objec- 
 tion to three-rooms-deep houses is that the central room is too dark, being 
 lighted by sash folding-doors between that and the front or rear rooms, or both. 
 Fig. 1468 is a modification to avoid this objection, the dining-room, or tea- 
 room, as it is generally called, being built as an L, so that there is at least one 
 window in the central room opening directly outdoors. This was an old 
 fashion here, and has lately been revived. 
 
 Figs. 1469 to 1470 are plans of the several floors of an English-basement 
 house, so called, distinguished from the former in that the principal floor is up 
 one flight of stairs. The first story or basement is but one or two steps above 
 the street, and contains the dining-room, with its butler's pantry and dumb- 
 waiter, a small sitting-room, with, in some cases, a small bedroom in the space 
 in the rear of it. The kitchen is situated beneath the dining-room, in the sub- 
 basement. The grade of the yard is in general some few steps above the floor 
 of the kitchen. Vaults for coal and provisions are excavated either beneath 
 the pavement in front or beneath the yard. The advantages of this form of 
 house are the small reception-room on the first floor, which in small families 
 and in the winter months is the most frequently occupied as a sitting-room of 
 any in the house ; the spaciousness of its dining-room and parlours in propor- 
 tion to the width of the house, which is often but 16 feet 8 inches in width, or 
 three houses to two lots, and not infrequently of even a less width. The ob- 
 jections to the house are the stairs, which it is necessary to traverse in passing 
 from the dining-rooms or kitchen to the sleeping-rooms, but this objection 
 would, of course, lie against any house of narrow dimensions, where floor-space 
 is supplied by height. 
 
 In New York, outside access to the kitchen is from the front, as there is no 
 back street or alley. In Philadelphia, where the lots are deeper, and there is 
 a street in the rear, the kitchen is usually in a rear L, on the level of the first 
 floor, with the dining-room above it on a mezzanine or half-story between the 
 first and second floors. 
 
 Figs. 1477 to 1483 are plans and elevations of a country-house in the Flem- 
 ish or Queen Anne style. 
 
 Fig. 1486 is the front elevation of a high-stoop house, in New York city, of 
 brown stone, a comparatively old but still popular design. 
 
 To accommodate the poor and people of small means in all cities, it was, 
 and to some extent still is, the custom to divide houses which were intended 
 for single families into small apartments for many, or to let rooms singly for this 
 purpose. This was found to be objectionable to both occupants and owners, 
 and houses have been constructed especially for parties of limited means. 
 Virtually, they are now nearly all apartment-houses, each family having distinct 
 rooms or suites to itself. But the term tenement-houses is applied to the cheaper 
 kind of apartments, occupied by the poorer class, and situated in the least ex- 
 pensive localities. The common form of tenement-house consists of two build- 
 ings, one in the front and one in the rear of the lot, with an outer or air space 
 between. A hall leads through the first story to the central area ; on each side 
 
600 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 PLAN OF FIKST FLOOK. - 
 B 
 
ARCHITECTURAL CONSTRUCTION. 
 
 601 
 
 PLAN OF SECOND FLOOR. 
 
602 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FKAMING-PLAN OF FIRST FLOOR. 
 
 FIG. 1479. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 603 
 
604: 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 ELEVATION OF CHIMNEY OF DINING-KOOM. 
 
 SECTION. 
 
 FIG. 1481. 
 
 FIG. 1482. 
 
 J FEET 
 
ARCHITECTURAL CONSTRUCTION. 
 
 605 
 
606 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 ENGLISH KURAL STYLE. 
 
 FIG. 1484. 
 
ARCHITECTURAL CONSTRUCTION. 
 ITALIAN VILLA, BY UPJOHN. 
 
 607 
 
 FIG. 1485. 
 
608 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Flo. 1486. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 609 
 
 of this hall there may be small stores and apartments. Stairs from the hall 
 lead to the apartments above. The 25 feet is divided in two, making two liv- 
 ing-rooms on each front ; these are the only rooms opening directly into the 
 outer air. Bedrooms are attached to each of these rooms, but take their light 
 and air from the staircases, or small light-wells. In the rear houses there are 
 two tenements to each story ; they take their light and air from the central and 
 back areas. Water-closets are in the central area. These tenements are 
 mostly occupied by work-people, largely of foreign birth, dependent directly 
 011 small wages. There is a large class, of limited means, to whom these 
 accommodations are insufficient ; parties who can not well afford an entire 
 house, but still wish for the privacy of one. Within the limits of a lot 
 25' X 100' it has been found difficult to secure all the necessaries of light and 
 ventilation, with the number of suites of apartments adapted to the means of 
 the occupants, and satisfactory as an investment to the owners. Fig. 1487 is 
 a plan of one of the best of these designs. It provides for four families 
 
 FIG. 1487. 
 
 on each story, although it will be observed by the plan of the stairs that 
 the front and rear tenements are not on the same level ; they are separated 
 by the half flight of stairs. By means of the cross-shaped court between the 
 adjacent houses, every room, including the bath-room, has a window to the 
 open air. This is the most commendable feature of the plan. It is remark- 
 able, also, however, for providing more conveniences than have been customary 
 in dwellings of this class, as, for instance, a small bath-tub as well as a water- 
 closet for each family, and two wash-tubs as well as a sink ; also, a dumb- 
 waiter (common to two families on each level) for bringing up fuel, provisions, 
 etc. The large rooms have recesses for beds, which provide for an extra bed- 
 room, while detracting but little from their value as parlors, as the recess may 
 be curtained off in the daytime, or the bed turned up. The dimensions of the 
 rooms, as marked on the plans, are the average length and breadth. These 
 suites are much too restricted for a very large class, but apartment-houses some- 
 what on this model are constructed in desirable localities, where the accom- 
 modations and conveniences are equal to those of any private house, and not 
 bounded by the limits of a single lot nor single story, many unsurpassed in 
 luxury of finish and appointments. 
 
 The larger apartment-houses are often designated as flats. The suites should 
 be supplied with water, gas, and steam heat ; should be entirely distinct in their 
 Tentilation and protected against fire ; some are now lighted by electric light. 
 40 
 
610 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Fig. 1488 is an illustration of a " flat " situated on the corner of a street, 
 and one suite takes its light exteriorly from the streets while the other depends 
 
 in a measure on the court, with a ventilating passage from the rear beneath 
 the fire-escape grating. Kitchens, in the figure, are attached to the suites ; 
 the laundries are in the upper story. Many flats are without kitchens or laun- 
 dries, and meals are furnished either from without or from restaurants in the 
 building. It then corresponds very nearly to a hotel without transient cus- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 611 
 
 SEDKWALK. 
 
 Fia. 1489. 
 
 FIG. 1490. 
 
612 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 torn, with ample and separate suites. It would seem that boarding-houses 
 might be built on such plans less extensive in their arrangements and adapted 
 to small families of moderate means ; but boarding-houses are almost invariably 
 private houses, but little modified for the more public use. 
 
 Stores and Warehouses. Fig. 1489 is the front elevation of a common type 
 of New York city store, occupying a single lot of 25 feet in width. It will be 
 observed that there are two stories beneath the level of the sidewalk, the base- 
 ment and sub-cellar, and this construction still obtains largely ; but deep base- 
 ments only are considered preferable by some, with extra stories at the top 
 rather than in the cellar. Fig. 1490 is a section of the front wall, showing 
 heights of stories, which of late years have been increased over former practice, 
 say to 16' for the first story, 13' for the second, and 12' and 11' for others, the 
 light for the interior being taken almost universally from the front and rear, 
 and sky lights done away with. 
 
 Fig. 1491 is a plan of the first-story floor, with basement in front dotted 
 in ; five feet of this space, or that usually allotted for areas, is covered with 
 
 B |A 
 
 FIG. 1491. 
 
 illuminating tile (Fig. 1492), that is, small glass lenses, set in iron frames, the 
 whole water-tight ; or lenses on a skeleton frame of cast-iron set in Portland 
 cement. In the extreme rear there is a small area, A, open to the air, of about 
 
 FIG. 1492. 
 
 5 feet, for light and air to the basement and cellar. The offices of the first 
 story are situated at B, over which there is usually a curved lean-to of illumi- 
 nating tile. The main wall above this story is on the line a b plain brick 
 with iron shutters. When shutters are used to close the first-story front they 
 are mostly rolling shutters of sheet-steel. The hoist-way to the upper stories 
 is at c, a position somewhat objectionable as interfering with the use of the 
 
613 
 
 ARCHITECTURAL CONSTRUCTION 
 
 
 H^ifW 
 
 I... 11 
 
614 
 
 stairs, when a common hoist-wheel is used ; but if it is a power-hoist, then it is 
 put close to the wall, guarded by a rail, with a passage round to the stairs. In 
 50-feet-front stores the hoist is put on the opposite corner from the stairs, as 
 
 FIG. 1493. 
 
 at D, but this cuts off considerable light from the first-story front. In some 
 the arrangement is as in Fig. 1493, in which the hoists c c are in the rear of 
 the stairs. The arrangement for offices in the rear of the first story is in a T, 
 with spaces at the sides for the ventilation and light of the lower stories. It 
 will be observed that there is no central door, as in the elevation (Fig. 1489), 
 which last most usually obtains for wholesale stores. Formerly illuminating 
 tile on iron quadrant frames over rear extensions of stores were common, but 
 were objectionable from the inside condensation and drip.- It is very common 
 to leave open areas at the sides, inclosed by brick walls (Fig. 1493), with the 
 windows protected by iron shutters. For deep stores the area should be at one 
 side and central, say from 30 to 40 feet long and 6 feet wide, which may be 
 covered in the first story with glass. If this recess is on the side occupied by 
 the staircases, it does not detract from the inside finish of the stores. 
 
 Hoists now in large stores are power-hoists that is, worked by either 
 steam, water, or electricity. The platform of a freight-hoist is usually 5 feet 
 square ; for passenger-hoists, in wholesale stores, somewhat less 4' X 5'. For 
 the raising of goods from the basement or sub-cellar to the sidewalk there is 
 a hatch in the front light platform, opposite some window, and the space is 
 like that of freight-hoists, 5' X 5' ; these may be power or hand hoists. For 
 the delivery of goods into these lower stories there is often a slide or incline, 
 iron-plated, ending at the bottom with an easy curve to the horizontal, down 
 which boxes and bales are slid. 
 
 Fig. 1494 is a perspective view of a city machine and blacksmith shop. It 
 was built for a purpose, and to express the purpose constructionally and eco- 
 nomically. As regards convenience and strength, it was found to be, on occu- 
 pation, all that could be wished. Posts, lintels, window-frames, sashes, and 
 ornamental letters were of iron, and painted a very deep green; the structure 
 was of brick, with sills and bands of rubbed Ulster bluestone, roof of Welsh 
 
ARCHITECTURAL CONSTRUCTION. 
 
 615 
 
ARCHITECTURAL CONSTRUCTION. 
 
 slate. The chimneys shown in front, although not dummies, were never used. 
 Power and heat were supplied by steam-boilers in the front vault, with a long 
 flue, slightly rising, leading to a chimney at the centre of the side blank wall. 
 On each side of this chimney, and separated from it by a thin with, there were 
 flues. Forges occupied all the exterior walls of the basement, front and side 
 areas, and the draught was upward and then down into the nearly horizontal 
 flues connected with the central flues, and the draught was invariably good. 
 Care was taken that all angles, horizontal and vertical, were rounded. 
 
 School- Houses. Figs. 1495 and 1496 are an elevation and plan of a country 
 district school-house, with seats for forty-eight scholars. There are two en- 
 trances, one for each sex, with ample accommodations of entry or lobby-room 
 for the hanging up of hats, bonnets, and cloaks. A side door leads from each 
 entry into distinct yards, and an inside door opens into the school-room. The 
 desk, T, of the teacher, is central between the doors, on a platform, P, raised 
 some 6" or 8" above the floor. In the rear of the teacher's desk is a closet or 
 small room, for the use of the teacher. The seats are arranged two to each 
 desk, with two alleys of 18" and a central one of 2'. The passages around the 
 room are 3'. 
 
 Figs. 1497 and 1498 are the elevation in perspective and plan of an English 
 country school-house, introduced as suggestive whether a one-story plan might 
 not be better suited, and of more beautiful effect in our own country towns, 
 where there is plenty of ground space, than the imitation of city edifices of 
 many stories. 
 
 On the Requirements of a School-House. Every scholar should have room 
 enough to sit at ease, his seat should be of easy access, so that he may go to and 
 fro, or be approached by the teacher without disturbing any one else. The 
 seat and desk should be properly proportioned to each other and to the size of 
 the scholar for whom it is intended, who should not sit in a cross-light, the 
 light should come from a single direction as near as possible over the left shoul- 
 der. The seats, as furnished by the different makers of school furniture, vary 
 from 9" to 14" in height ; and the benches from 17" to 28" ; measuring on the 
 side next the scholar. The average width of the desk is about 18", and it is 
 formed with a slope of from 1" to 2-J-", with a small 
 
 i p I p I p P horizontal piece of from 2" to 3" at top. There is a 
 ' shelf beneath for books, but it should not come within 
 
 I I I I n about 3" of the front. The width of the seat varies 
 p p from 10" to 14", with a sloping back, like that of a 
 chair ; it should, in fact, be a comfortable chair. In 
 
 a 
 
 QQ] 
 
 the figure, two scholars occupy one bench. Fig. 1499 
 represents another arrangement, in which each scholar 
 has a distinct bench ; this is more desirable, but not 
 quite so economical in room. In primary schools desks 
 ' [1 are not necessary; and in many of the intermediate 
 "FIG. i499~ schools the seat of one bench is formed against the 
 
 back of the next bench ; but seats distinct are preferable. 
 
 The teacher's seat is invariably on a raised platform, and had better be against 
 a dead wall than where there are windows. Blackboards and maps should be 
 placed along the walls. Care should be taken in the warming and ventilation ; 
 
 I 1 
 
ARCHITECTURAL CONSTRUCTION. 
 
 017 
 
 warm air should be introduced in proportion to the number of scholars, and 
 ventiducts should be formed to carry off the impure air. 
 
 FIG. 1498. 
 
 In cities and large towns it is almost indispensable to build school-houses 
 many stories in height, dividing the rooms in each story according to the neces- 
 
618 
 
 ARCHITECTURAL CONSTRUCTION. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 619 
 
 sities of their occupancy. The management of schools differs in different locali- 
 ties. This will be seen in the illustrations given below, showing the arrange- 
 ments of school-houses in the city of New York and of Cleveland, Ohio. 
 
 Fig. 1500 is an elevation in perspective of one of the largest of the New 
 York city schools, showing the yards around it. Fig. 1501 is the plan of the 
 
 FIG. 1501. 
 
 grammar-department floors of this house ; and Fig. 1502 the plan of the same 
 floors of another house of a different outline. 
 
 Figs. 1503 to 1506 are plans of school-houses, built at Cleveland, Ohio, a 
 type inaugurated under the supervision of the then superintendent, Mr. A. J. 
 Kickoff. Figs. 1503, 1504, and 1505 are plans of the High-School house. Fig. 
 1503 is the plan of the third story ; Figs. 1504 and 1505 of those portions of 
 the second and first stories which differ from that of the third. There is a rear 
 vestibule in the first story to correspond with the one in front, shown in the 
 figure. In the whole building there are 14 session-rooms, each 37' X 30' X 
 16' ; each having its connecting cloak-room ; one general assembly-room, 94' X 
 56' X 38' high, with a seating capacity for at least 1,000 persons ; one lecture- 
 room, with seats for 100, with an apparatus- room ; one room for drawing, 30' x 
 55', with a room for models, drawing-boards, etc. ; two rooms for the principal 
 and reception-room ; five rooms for library and recitation-rooms. 
 
620 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Fig. 1506, a plan of one half of one story of the Walton Avenue School, on 
 a larger scale, explains more fully the arrangement of seats and the ventilation. 
 Four ventilating educts, of 8 square feet of section each, may be heated to any 
 required temperature for the purposes of circulation by four upright 2" steam- 
 pipes ; six ducts of 1 square foot section lead from different points in the floor 
 
 OW)a]:C:D:G:D:D:DSH]:D:[tO: 
 
 Fio. 1502 
 
 of each session-room (as shown in dotted lines in the figure) into the ventilating 
 educts. There are besides other registers opening directly into the educts. 
 The building is heated by steam coils or radiators placed under the windows of 
 the rooms, with provision for the admission of fresh air under the stone sills 
 behind the radiators. The main light of every room is admitted at the left 
 hand of the pupil, so that in writing the shadow of the hand does not fall on 
 the space to be written on. There are none of the cross-lights that so seriously 
 impair the vision. The wall facing the pupil and behind the teacher is un- 
 broken by windows, affording large and convenient spaces for blackboards. 
 
 Churches, Theatres, Lecture- Roomx, Munic and Lcyixlative Halls. To the 
 proper construction of rooms or edifices adapted for these purposes some knowl- 
 edge of the general principles of acoustics, and their practical application, is 
 necessary. In the case of lecture-rooms and churches, the positions of the 
 speaker and the audience are fixed ; in theatres, one portion of the inclosed 
 space is devoted to numerous speakers and the other to the audience; in legis- 
 lative halls, the speakers are scattered over the greater part of the space, and 
 also form the audience. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 621 
 
622 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 -fHJ )- 
 
 [ipa aaaa l|p 
 jo a p.- a a D iji ja 
 la a a a a a q ja 
 
 to D D D a D d p 
 
 itpDDDDOdjD 
 
 |p a D D a D nip 
 a a a a D a D J D 
 
 ''' i i i i i FEET 
 
ARCHITECTURAL CONSTRUCTION. 623 
 
 The transmission of sound is by vibrations, illustrated by the waves formed 
 by a stone thrown into still water; but direction may be given to sound, so that 
 the transmission is not equally strong in every direction ; thus, Saunders found 
 that a person reading at the centre of a circle of 100 
 feet in diameter, in an open meadow, was heard most dis- ,.- ' ~ -- x 
 
 tinctly in front, not as well at the sides, but scarcely at 
 all behind. Fig. 1507 shows the extreme distance every 
 way at which the voice could be distinctly heard : 92 feet 
 in front, 75 feet on each side, and 31 feet in the rear. 
 The waves of sound are subject to the same laws as those 
 of light, the angles of reflection are equal to those of in- 
 cidence ; therefore, in every inclosed space there are re- 
 fleeted sounds, more or less distinct, according to the po- 
 sition of the hearer, and to the form and condition of the surfaces against 
 which the waves of sound impinge. Thus, of all the sounds entering a para- 
 bolic sphere, the reflected sounds are collected at the focus. Solid bodies 
 reflect sound, but draperies absorb it. As, in all rooms, the audience can never 
 be concentrated at focal points, nor is it possible in any construction to make 
 calculation for all positions, it is in general best to depend on nothing but the 
 direct force of the voice, and not to construct larger than can be heard directly 
 without aid from reflected sounds. 
 
 There is great difference in the strength of voice of different speakers ; the 
 limits as given in the figure are for ordinary reading in an open space. In in- 
 closed spaces, owing to the reflected sounds or some other cause, there are cer- 
 tain pitches or keys peculiar to every room, and to speak with ease the speaker 
 must adapt his tone to those keys. The larger the room, the slower and more 
 distinct should be the articulation. 
 
 It has been observed that the direction of the sound influences the extent to 
 which it may be heard. The direction of the currents of air through which 
 the sound passes affects the transmission of the sound, and this may be made 
 useful when the rooms are heated by hot air, by introducing the air near the 
 speaker and placing the ventilators or educts at the outside of the. rooms, and 
 by placing their apertures rather nearer the bottom of the room than at the 
 top. It would seem much better and easier to make a current of air a vehicle 
 of sound rather than depend on reflection. 
 
 In the " Baltimore Academy of Music," designed by Mr. J. Crawford Neil- 
 son, architect, the ventilation was arranged to obstruct the formation of air- 
 currents of unequal density. The whole supply of fresh air is admitted at the 
 back of the stage, is there warmed, crosses the stage horizontally, and passes 
 through the proscenium and then, somewhat diagonally toward the roof, across 
 the auditorium in one grand volume and with gentle motion so as to almost 
 entirely prevent the formation of minor air- currents. It is exhausted partly 
 by an outlet in the roof and partly by numerous registers in the ceilings of 
 the galleries. From this central outlet and from the large flues of the reg- 
 isters the air passes into the ventilating-tower over the great chandelier, which 
 supplies, in its heat, a part of the motive power of the circulation. It is 
 further expelled from the tower by means of large valves, offering no obstacle 
 to the egress of air, but completely cutting off" its entrance. 
 
624: 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 1508. 
 
 FIG. 1509. 
 
 The direction of the air-currents within the house was determined by thistle 
 balls, and the quantity, as found by anemometers, was about 15,000 cubic feet 
 per minute. This amount, sufficient to ventilate the house, is that required to 
 impress the proper movement on its atmosphere. It is amply sufficient for 
 ventilation, as is shown by the fact that the thermometers of the upper circle do 
 not vary perceptibly from those of the orchestra circle. The seating capacity 
 of the house is about sixteen hundred persons. The acoustics are satisfactory. 
 On the Space occupied by Seats in general. A convenient arm-chair occu- 
 pies about 20" X 20", the seat itself being about 18" in depth, and the slope of 
 
 the back 2" ; 18" more affords am- 
 ple space for passage in front of 
 the sitter. In churches the seats 
 are arranged by pews or stalls, the 
 width of each pew in general being 
 about 2' 10". In the arrangement 
 of theatre seats the bottom turns 
 up (Figs. 1508 and 1509), and 29" 
 only is allowed for both seat and 
 passage-way, and 18" for the width 
 of seat, which may be taken as the 
 average allowance in width to each 
 sitter in comfortable public rooms. 
 In lecture-rooms, benches and set- 
 tees are often used, the space there occupied by seat and passage being about 
 2' 6". 
 
 In the earlier churches, ceremonies and rites formed a very large part of the 
 worship, the sight was appealed to rather than the hearing, and for this pur- 
 pose churches were constructed of immense size, and with all the appliances 
 of ornament and construction, with pillars, vaults, groins, and traceried win- 
 dows. In the churches of this country, the great controlling principle in the 
 construction of a church is its adaptation to the comfortable hearing and seeing 
 the preacher. In this view alone, the church is but a lecture-room ; the ceiling 
 should be low, and pillars and transoms should be little used ; but since even 
 the character of the building may tend to devotional feelings in the audience, 
 and since certain styles and forms of architecture have long been used for 
 church edifices, it has been the custom to follow these time-honoured examples, 
 adapting them to modern requirements of church worship, with adequate means 
 of heating and ventilation. 
 
 Fig. 1511 is a plan of an ancient basilicon or Romanesque church. Fig. 
 1510 is a sectional elevation of the same. Fig. 1512 is a plan of a Gothic 
 church, in which C is the chancel, usually at the eastern extremity, T T the 
 transept, and N the nave. In general elevation the Gothic and Eomanesque 
 agree : a high central nave and low side aisles. In the later Eomanesque the 
 transept is also added. 
 
 The basilicas aggregated within themselves all the offices of the Romish 
 church. The circular end or apse, with the raised platform, or dais, in front 
 appropriated to the altar ; in the rear the confessional and the sacristy ; beneath 
 was the crypt, where were placed the bodies of the saints and martyrs, and pul- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 625 
 
 pits were placed in the nave, from which the services were said or sung by the 
 inferior order of clergy. 
 
 The plan (Fig. 1512) is that of the original Latin cross, the eastern limb 
 or chancel being the shortest, and the nave the longest. Sometimes the eastern 
 
 FIG. 1510. 
 
 Fui. 1511. 
 
 limb was made equal to that of the transept, sometimes even longer, but never 
 to exceed that of the nave. In the Greek cross all the limbs are equal. In 
 most of the French Gothic churches the eastern end is made semicircular, often 
 inclosed by three or more apsidal chapels, that is, semi-cylinders, surmounted 
 by semi-domes. 
 
 The Byzantine ohurch consisted internally of a large square or rectan- 
 gular chamber, surmounted in the centre by a dome, which rested upon 
 massive piers ; an apse was formed at the eastern end. Circular churches 
 were built in the earlier ages for baptisteries, and for the tombs of saints and 
 emperors. 
 
 The Greek, Roman, and English churches conform in their cathedrals and 
 larger edifices nearly to the Komanesque or Gothic models. But as the general 
 requirements for church services now are those of a lecture-room, modern 
 churches are constructed adapted to these purposes, and, in cities, to the size 
 and form of the lots, with some ecclesiastical accessories of towers and steeples, 
 windows and doors and interior finish. 
 
 Fig. 1513 is the plan of the English church at The Hague. 
 
 FIG. 1513. 
 
 Fig 1514 is the plan of a Wesleyan chapel in London ; the requirements of 
 the service have been well adapted to the necessities of the lot. 
 41 
 
626 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Fig. 1515 is the cross-section of a common form of small country church, 
 with nave w, aisles a a, and clear-story c. The effect, both inside and out, is 
 
 FIG. 1514. 
 
 good, but there are objections to large or masonry-columns, which cut off the 
 view of the desk and the altar from many sitters, and to the windows of the 
 
 FIG. 1515. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 627 
 
 clear-story, which in winter act as coolers to the air descending in draughts 
 upon the heads of the congregation beneath them. Neither columns nor clear- 
 story are constructively necessa- 
 ry ; the span can readily be met 
 by a single roof, and sufficient 
 light can be obtained from the 
 sides. 
 
 Figs. 1516, 1517, and 1518 
 are examples of open- timbered 
 Gothic roofs of churches. 
 
 The technical names (Fig. 
 1516) are: 1, Principals ; 2, Pur- 
 lines; 3, Collars; 4, Braces; 5, 
 Wall-pieces ; 6, Wall-plates ; 7, 
 Struts; 8, Rafters. 4 and 5 are 
 shown in section. 
 
 The length of pews is vari- 
 ous, being of two sizes, adapted 
 to either small or large families, 
 say from 7' 6" to 12' 6", 18" be- 
 ing allowed for each sitter. In 
 arrangement it is always consid- 
 ered desirable that there should 
 be a central aisle, and if but 
 four rows of pews (often of two 
 sizes in the same church), an 
 aisle against each wall ; if six 
 rows, one row on each side will 
 be wall-pews. Formerly it was 
 the universal practice to con- 
 struct pews with doors, but of 
 late it is more customary to omit 
 the doors, making the pews open 
 stalls. 
 
 Few churches are now with- 
 out an organ ; its dimensions 
 should of course depend on the 
 size of the church. In form it 
 may be adapted somewhat to the 
 place which may be appropriated 
 to it either in a gallery over 
 the main entrance or above the 
 pulpit or at the side of the chan- 
 cel, as in Fig. 1513. Sometimes 
 there are two, one at each ex- 
 tremity of the church, one organist playing on both by electric connection. In 
 general, it is oblong in form, the longer side being with the keys. The di- 
 mensions suited to a medium-sized church are about 9' X 15', and 12' in height. 
 
 FIG. 1518. 
 
f,28 ARCHITECTURAL CONSTRUCTION. 
 
 The vestry-room, if used for the purposes of its meetings, should be adapted 
 in size to the purpose ; but if only for a withdrawing or robing room for the 
 clergyman, it may be of very small dimensions, and should be accessible from 
 without. The Sunday-school room, in general, requires in plan about half 
 the area of the church. From motives of economy it is usually placed in the 
 basement of the church ; but, in the country especially, it is better that it 
 should be a separate building, and form one of the group of church, parson- 
 age, and Sunday-school house. 
 
 In elevation, city churches are generally Romanesque and Gothic, occasion- 
 ally Byzantine. The Greek have no tower, but often a spire above the portico ; 
 the Romanesque and Gothic generally one tower, over the central door of en- 
 trance, or at one corner ; sometimes two, one at each side of the principal door, 
 almost invariably surmounted by spires, high and tapering, usually of wood, but 
 in some instances of stone. 
 
 Theatres. In theatres and opera-houses it is not only necessary that the 
 audience should have a good position for hearing and seeing the performance 
 upon the stage, but also to see each other. The most approved form, now, for 
 the body of a dramatic theatre is a circular plan, the opening for the stage 
 occupying from one fourth to one fifth of the circumference, the sides of the 
 proscenium being short tangents; but for a lyric theatre, where music only -is 
 performed, and where, consequently, hearing is easier, the curve is elongated 
 into an ellipse, with its major axis toward the stage. 
 
 In the general position of the stage, proscenium, orchestra, orchestra seats, 
 parquette, and boxes, but one plan is followed. The line of the front of the 
 stage, at the footlights, is generally slightly curved, with a sweep, say, equal to 
 the depth of the stage, and the orchestra and parquette seats are arranged in 
 circles concentric with it : of the space occupied by seats we have already 
 spoken. The entrance to the parquette may be through the boxes, near the 
 proscenium, and centrally, but better at the sides, dividing the boxes into three 
 equal benches ; the seats in the boxes are usually concentric with the walls, and 
 more roomy than those of the parquette. The orchestra seats are of a height 
 to bring the shoulders of the sitter level with the floor of the stage, and the 
 floor of the parquette rises to the outside, 1 in 15 to 18. The floor of the first 
 row of boxes is some 2 to 3 feet above the floor of the parquette at the front 
 centre, and rises, by steps at each row, some 4 inches ; in the next tier of boxes 
 the steps are considerably more in height, and so on in the boxes above. In 
 general, three rows of boxes are all that is necessary ; in front, above the sec- 
 ond, the view of the stage is almost a bird's-eye view. The floor of the stage 
 descends to the footlights at the rate of about 1 in 50. In large theatres it is 
 of the utmost importance that all the lobbies or entries should be spacious, and 
 the means of exit numerous and ample the staircases broad, in short flights 
 and square landings, and not circular, as, in case of fright, the pressure of 
 persons behind may precipitate those in front the whole length of the flight. 
 Ladies' drawing-rooms should be placed convenient to the lobbies, of a size 
 adapted to that of the theatre, also rooms for the reception of gentlemen's 
 canes and umbrellas, both with usual water arrangements. The box-office 
 should be near the entrance and arranged to interfere as little as possible with 
 the approach to the doors of the house. At the entrance there should be a 
 
ARCHITECTURAL CONSTRUCTION. 
 
 629 
 
 FIG. 1519. 
 
 very spacious lobby, or hall, so that the audience may wait sheltered from the 
 weather ; if possible, there should be a long portico over the sidewalk, to cover 
 the approach to the carriages. Only single 
 entrances are necessary to distinct parts of 
 the house, but the greater the number of, 
 and the more ample places for exit at the 
 conclusion of the piece, or for the contin- 
 gency of fire, the better. 
 
 Fig. 1519 is a plan suggested by Fer- 
 guson of keeping the centre of the bal- 
 conies perpendicular over one another, and 
 then, by throwing back the sides of each 
 balcony till the last is a semicircle, the 
 whole audience would sit more directly 
 facing the stage, would look at it at a bet- 
 ter angle, and the volume of sound be con- 
 siderably increased by its freer expansion 
 immediately on leaving the stage. 
 
 Figs. 1520 and 1521 are a plan and section of Wagner's theatre. 
 In cities, the auditoria of dramatic theatres conforming to the shape of the 
 lots are rectangular in their outline, and seldom exceed a seating capacity of 
 
 1,500. Lyric theatres are 
 
 PLAN - much larger, both in seat- 
 
 ing and scenic capacity. 
 Lecture-rooms are usually 
 arranged with the au- 
 dience-floor flat, room rec- 
 tangular, with reading-desk 
 or platform raised, and 
 with or without galleries. 
 The same form usually ob- 
 tains for music-halls, only 
 they are much greater in 
 extent, the first being capa- 
 ble of containing from 500 
 to 1,000 persons ; whereas 
 some music-halls will con- 
 tain 2,500, and Ferguson 
 thinks that a music-hall 
 might be arranged so that 
 even 10,000 might hear as 
 well as in those of present 
 construction. The lecture 
 and music halls are seldom 
 devoted to a single purpose, 
 but are used for political 
 meetings, for fairs, and dances, and the constructon must be such as to serve 
 these other purposes. 
 
 FIG. 1520. 
 
 SECTION. 
 
 FIG. 1521. 
 
630 ARCHITECTURAL CONSTRUCTION. 
 
 COMPARATIVE TABLE OP THE DIMENSIONS OP A PEW THEATRES. 
 
 
 DISTANCE, IN FEET. 
 
 HEIGHT, IN FEET. 
 
 
 a 
 
 1 
 
 cS 
 
 |4 
 
 to 
 
 ll 
 
 '$ !? 
 
 ~ 03 
 
 9 
 
 a 
 9 
 
 rj 
 
 || 
 
 ft 
 
 *M V 
 
 
 ll 
 
 NAME AND LOCATION. 
 
 O bo 
 
 Ou 
 
 *s 
 
 
 
 O 
 
 "SS 
 
 o'o 
 
 "o S 
 
 
 ^ 
 
 *c " 
 
 S-g 
 
 _Q -FH 
 
 O 
 
 
 ^8 
 
 
 
 
 
 $" 
 
 ,3 
 
 s 
 
 O 
 
 T3 1. 
 S 
 
 So 
 
 1C g 
 
 
 J 
 
 & c3 
 
 4J 
 
 "S 
 
 
 
 s ^ 
 
 0" 
 
 S c3 
 
 
 g 
 
 E 
 
 W fl 
 
 
 
 -!_> 
 
 E 
 
 g 
 
 
 03 
 
 
 
 oj 
 
 o 
 
 n 
 
 
 
 ' 
 
 
 
 Alexandre, St. Petersburg 
 
 65 
 
 11 
 
 84 
 
 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 
 
 81 
 
 83 
 
 Grand Theatre, Bordeaux 
 
 46 
 
 10 
 
 69 
 
 47 
 
 37 
 
 80 
 
 50 
 
 57 
 
 Salle Lepelletier, Paris 
 
 67 
 
 9 
 
 82 
 
 66 
 
 43 
 
 78 
 
 52 
 
 66 
 
 Covent Garden, London 
 
 66* 
 
 
 55 
 
 51 
 
 32 
 
 86 
 
 54 
 
 
 Drury Lane, London 
 
 64* 
 
 
 80 
 
 56 
 
 32 
 
 48 
 
 60 
 
 
 Boston, Boston 
 
 53 
 
 18 
 
 68 
 
 
 46 
 
 87 
 
 554 
 
 58 
 
 Academy of Music, New York. . . . 
 
 74 
 
 13 
 
 71 
 
 62 
 
 48 
 
 83 
 
 74 
 
 
 Grand Opera-House, New York. . . 
 
 54 
 
 84 
 
 634 
 
 48 
 
 44 
 
 76 
 
 52 
 
 67 
 
 Opera-House, Philadelphia 
 
 61 
 
 17 
 
 72 
 
 66 
 
 48 
 
 90 
 
 644 
 
 74 
 
 
 * These dimensions include the distance between the footlights and curtain. 
 
 Legislative Halls. Although much has been written about their construc- 
 tion in relation to acoustic principles, there is great disagreement in practical 
 examples, and in the deductions of scientific men. The Chamber of French 
 Deputies was constructed, after a report of most celebrated architects, in a 
 semicircular form, surmounted by a flat dome, but as the member invariably 
 addresses the house from the tribune, at the centre, in its requirements it is but 
 a lecture-room. Mr. Mills, architect, of Philadelphia, recommends for legisla- 
 tive or forensic debate a room circular in its plan, with a very slightly concave 
 ceiling. Dr. Reid, on the contrary, in reference to the Houses of Parliament, 
 gave preference to the square form, with a low, arched ceiling. The Hall of 
 Representatives at Washington is 139 feet long by 93 feet wide, and about 36 
 feet high, with a spacious retiring gallery on three sides, and a reporters' gal- 
 lery behind the Speaker's chair. The members' desks are arranged in a semi- 
 circular form. The ceiling is flat, with deep-sunk panels, openings for ventila- 
 tion, and glazed apertures for the admission of light. The ventilation is 
 intended, in a measure, to assist the phonetic capacity of the hall, the air 
 being forced in at the ceiling and drawn out at the bottom. 
 
 In reviewing the general principles of acoustics, it will be found that those 
 rooms are the best for hearing in which the sound arrives directly to the ear, 
 without reflection ; that the sides of the room should neither be reflectors nor 
 sounding-boards, and that surfaces absorbing sound are less injurious than 
 those that reflect. Slight projections, such as ornaments of the cornices and 
 shallow pilasters, tend to destroy sound, but deep alcoves and recessed rooms 
 produce echoes. Let the ceiling be as low as possible, and slightly arched or 
 domed ; all large external openings should be closed ; as M. Meynedier ex- 
 presses it, in his description of an opera-house, " Let the hall devour the sound ; 
 as it is born there, let it die there." 
 
 Hospitals. In large cities, hospitals, by necessity, are confined to narrow 
 spaces, but they should be placed, if possible, on river fronts or on open parks, 
 
ARCHITECTURAL CONSTRUCTION. 631 
 
 to secure as much open-air ventilation as possible. They are usually many 
 stories in height, with large warils one above the other. Sir J. T. Simpson 
 alleges a very high rate of mortality in hospitals after surgical operations as 
 compared with the mortality after the same operations when performed at the 
 homes of the patients, and asserts that the mortality after operations performed 
 in hospitals containing more than 300 beds is in excess of that in hospitals con- 
 taining less; that great hospitals are great evils in exact proportion to their 
 magnitude, and suggests the construction of smaller hospitals. 
 
 Figs. 1522 and 1523 are an elevation and plan of an English country hos- 
 pital. 
 
 Stables. Under this general name are included the barn, or the receptacle 
 of hay and fodder, the carriage-house, and the stable proper, or lodging-house 
 for horses and cows. The first two may be included under one roof, the car- 
 riages on the first floor, and hay in the loft ; but the lodging-place should be 
 distinct, in a wing attached to the barn, that the odours from the animals may 
 not impregnate their food, or the cloth-work of the carriages, or the ammonia 
 tarnish their mountings. 
 
 Hay in bulk, in the mow, occupies about 340 cubic feet per ton ; bales aver- 
 age 2' 4" X 2' 6" X 4', and weigh from 220 to 320 pounds. The door-space for 
 a load of hay in the bulk should be from 12 to 13 feet high and 12 feet wide. 
 The floor beneath the hay should be tight, so that dust and seed may not drop 
 on the carriage. A door for carriages should be 10 feet 6 inches high by 9 feet 
 wide. 
 
 The horse is to be treated with greater care than any other domestic animal. 
 His stable is to be carefully ventilated, that he may have fresh air without 
 being subject to cross-draughts. Preferably the floor should be on the ground, 
 that there may be no cold from beneath. He should stand as near as possible 
 level ; and for this purpose a grated removable floor, with small interstices, 
 should be laid over a concrete bottom, with a drip toward the rear of the stall, 
 and the urine should be collected in a drain and discharged into a trapped 
 manure-tank outside the stable. In Fig. 1524 the pitch of bottom of stalls is 
 to the centre and outward. The manure should never be deposited beneath 
 the stable, but should be wheeled out and deposited in a manure-yard or tank 
 daily. It is as essential that all excrements should be removed entirely from 
 the stable as that the privy should be placed outside the house. 
 
 The breadth of stalls should be from 4 feet 6 inches to 5 feet in the clear; 
 the length, 7 feet 6 inches to 8 feet ; the rack and feed-box require two feet in 
 addition, to which access is given in the best stables by a passage in front. 
 Rack and feed-boxes are often made of iron, and the upper part of stalls fitted 
 with wrought-iron guards. Box-stalls, in which horses are shut up but not 
 tied in cases of sickness or foaling, are about 10 feet square. 
 
 In large stables in cities the first floors are often occupied by the carriages, 
 while the horse-stalls are in the basement or upper stories, with inclined ways 
 of access. In the basement provision must be made for light and ventilation. 
 In the upper stories these may be secured more readily, but the floors must be 
 made tight and deafened, that the urine may not leak through, nor the cold 
 come through from below to make too cool a bed for the horse. 
 
 Fig. 1524 is an elevation in perspective of two first-class stalls, a box shown 
 
632 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 152*. 
 GROUND PLAN. 
 
 iCALC OF FE.E7 
 
 FIG. 1523. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 633 
 
 with the door open, and a single stall. The lower part of the inclosures is of 
 plank, with wrought-iron guards and ramp above. The posts are of oak, and 
 
 FIG. 1524. 
 
 the hay-boxes or mangers of cast-iron ; the hay-rack in the box-stall is of 
 wrought-iron. These are of common manufacture, and are of varied patterns ; 
 but in the country they are usually made of wood, and connected with 
 the stall. 
 
 Fig. 1525 is the plan of a small country stable, showing the desirable pas- 
 
 Open Shed 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 Box Sfall. 
 
 Carriage House 
 
 FIG. 1525. 
 
 sages around the stalls and exterior windows in front of each stall, that the 
 horses may not only have light and air, but can see out. 
 
 Cow-houses, for cows giving milk, should be constructed with care for ven- 
 tilation, light, and cleanliness. Other cattle are usually left out, with sheds 
 under which they can go for shelter. For those housed, the spaces occupied 
 should be about the same per head as the single horse-stall. The manger 
 should be on the floor, 12" to 18" high, and about 18" wide. It is not usual to 
 
634 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 have partitions, but there ought to be between every pair, reaching from the 
 manger half-way to the gutter behind. The floor should be level, grated, with 
 a drip beneath, and cleansed by washing out. The partition and 
 mangers are often of cast-iron, and on sale, but for large stables and 
 in the country they are commonly of wood. 
 
 Greenhouses. Fig. 1526 is the section of a greenhouse, 
 with shelves for plants. The floor is of concrete and 
 the walls are of masonry ; the northern exposure is 
 a blank wall. 
 
 Fig. 1527 is the details of windows. 
 The sides are box-sash, hung with 
 weights (w, w, Fig. 1528). 
 The lower roof sash is 
 firmly fixed, but 
 the upper 
 
 FIG. 1526. 
 
 FEET 
 
 one can be slid down ; it is usually retained in place by a cord attached to the 
 lower part of the sash, passing over a pulley on the upper bar of the frame, 
 with the loose end within reach of the gardener, who can fasten it to a cleat. 
 
 Ventilation and Warming. The purposes of ventilation are not changes of 
 air merely, but the removal of foul and vitiated air, and the substitution there- 
 for of pure air ; and this air may be warm or cool according to the necessities 
 of the season and personal requirements. Open space is not necessarily well 
 ventilated; there must be circulation, outward and inward the latter from 
 
ARCHITECTURAL CONSTRUCTION. 
 
 635 
 
 purer sources than the former. With an equal discharge and sup- 
 ply of pure air, the smaller the room, the more frequent the change 
 of air, the better its distribution and the better the ventilation. 
 But if the means of removal, supply, and distribution of 
 air be proportioned to the size of the room, then 
 the larger the room the better. Apertures 
 do not necessarily mean circulation 
 flue may draw or it may not 
 draw, it may be inert, or 
 the air may come 
 
 j FEET. 
 
 FIG. 1527. 
 
636 ARCHITECTURAL CONSTRUCTION. 
 
 down ; a window may be open, with little or no inward or outward movement 
 of air. In a house exposed to a fresh breeze, on the windward side there is an 
 air-pressure, on the leeward side there is an eddy or vacuum. Air is forced in 
 on the first through every crack of door and window often down chimney- 
 flues and drawn out on the other side. This often happens even with fires in 
 the chimneys, and with insufficient heat in ventilating educts. If one will 
 make an experiment in cold weather, when the windows are closed and there 
 are fires in some rooms, he will often find that there is cold air coming down 
 the unused flues, and will feel the cold current flowing down the stairs and 
 along the floors to the fires. Architects have placed kitchens in the basement 
 and in the attic, and the smell of cooking, rises through the house from the 
 former, and usually descends from the latter when the air is light and muggy. 
 
 Every room should have its distinct flue ; if the current is not upward it 
 will probably be downward, affording a fresh supply of air for ventilation if 
 there is an escape elsewhere. A chimney-flue may be too large for the purposes 
 of a fire; for most fires a flue 8" X 8" is amply sufficient, and will serve for 
 ventilation in the common occupation of a house. If the throat of the chim- 
 ney be made with rounded corners and a diverging sectional area, or a 
 damper hinged at the bottom, for like effect, it should increase upward, and 
 prevent back draught. 
 
 Small circular flues of from 6" to 8" diameter, or of equivalent rectangular 
 section, are now made in concrete or stoneware, which, as they are smoother 
 and with less joints than brickwork, give greater velocities of current with less 
 section, and, laid with care, changes in direction afford but little obstruction. 
 In an ordinary chimney with natural draught the velocity of ascending current 
 is about six feet per second. 
 
 It is usual to depend largely on windows for ventilation, but the space on 
 which they open may be too circumscribed to afford the requisite change of air, 
 or the outer air itself may be too hot, or too cold, or too malarial or offensive, 
 to make the change of air sanitary or pleasant. In tenement or apartment 
 houses care should especially be taken that the inner windows on different flats 
 open into as large air-shafts as possible, and that these shafts should have free 
 opening to the outer air below and at the top, without skylights ; and that the 
 floors should be tight, so that the smells may not pass from one flat to another. 
 Nothing more surely shows faults in ventilation than the diffusion of kitchen 
 smells or tobacco smoke. Distinct flues should be constructed for each room, 
 extending independently well above the roof; and not into an attic with a 
 ventilating louvre, as the air may ascend one flue and descend another, and 
 not out of the louvre. Pipe flues may lead into a single stack if each branch is 
 given the direction of the main current at its connection, without obstructing 
 its flow, as in sewer branches. 
 
 The quantity of air taken into and expired from the lungs by a single indi- 
 vidual is quite small, probably about 14 cubic feet on an average per hour. 
 The usual gas-burner delivers from 4 to 6 cubic feet per hour, under a pressure 
 of 1" and 2" of water. It will be seen, therefore, how small apertures are neces- 
 sary to supply the lungs of a person, if it could be provided directly to him and 
 taken away without vitiating other air. But, in addition, air is vitiated by 
 personal emanations and consumed by lights. These last can readily be ar- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 637 
 
 ranged in connection with flues, not only to remove all their products of com- 
 bustion, but also improve the ventilation of the room. 
 
 All systems of ventilation are based on the idea that so many individuals 
 within a room and so many lights burning vitiate so much air, and that conse- 
 quently a very large quantity of outer air must be introduced to reduce the 
 percentage of vitiation, and generally with very little consideration as to the 
 distribution of this air, although it is in every one's experience that the air in 
 some portions may be fresh, in others stifling ; that in hospital wards there are 
 often dead ends where the air does not circulate, and where patients do not as 
 a rule recover. The system is to provide, somewhere in a room, air enough and 
 trust to chance for its distribution. 
 
 Some architects make the educts at the ceiling, some at the floor, some at 
 both, with registers to control the openings. For sleeping apartments, if there 
 is a fireplace this is all that will be necessary ; if the air goes up or comes down 
 it does not make draughts about the heads of the occupants. 
 
 To make flues draw, various forms of chimney-tops or cowls are adopted. 
 The best and simplest are the Emerson (Fig. 1529) and a modification of the 
 same (Fig. 1530) ; there are also various forms . 
 
 of self-acting flaps, turn-cowls, etc., the prin- 
 ciple being to take advantage of the wind to 
 make a draught. With the wind blowing 
 across the top of a chimney, a bit of square- 
 ended iron pipe extending above the chimney 
 will answer as an expirator, but without a 
 wind the draught must depend on circum- 
 stances within the dwelling and artificial 
 
 draught. When sufficient circulation can not be obtained from natural differ- 
 ences of temperature in the atmosphere, or from winds, it is usual to have 
 recourse to fans to force air into or draw it from a building, or by heat applied 
 to the air in flues, ducts, or chambers in the hot-air furnaces. Both the air 
 and the heat are necessary. 
 
 " No systematic ventilation, however well devised and constructed, however 
 extensive its supply of fresh air, however regularly or judiciously operated, can 
 afford to dispense with the repeated displacement of the air of rooms and sub- 
 stitution of entirely fresh air through open windows and doors, at times, during 
 all seasons of the year" (Briggs). 
 
 Methods of Heating. The open fireplace grate heats by radiation, com- 
 municating heat to objects, which by contact transfer it to the air. Persons 
 coming in contact with rays are themselves heated, while the air around them 
 is cool and invigorating for breathing; the bright glow has a cheering and ani- 
 mating effect upon the system, somewhat like that of sunlight. As a ventilator, 
 an open fire is one of the most important, drawing in air not only for the sup- 
 port of combustion, but also, by the heat of the fire and flue, making a very 
 considerable current through the throat of the chimney above the fire. From 
 this cause, although there is a constant change of air, yet there arises one great 
 inconvenience of disagreeable draughts, especially along the floor, if the air- 
 supply be drawn directly from the outer cold air; but in connection with prop- 
 erly regulated furnaces or stoves, the open fireplace becomes the most perfect 
 
 FIG. 1529. 
 
 Fio. 1530. 
 
638 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 1531. 
 
 means of heating and ventilation. As a heater merely, the open grate in very 
 cold weather is not satisfactory ; its influence is only felt in its immediate 
 vicinity, and but from 10 to 15 per cent, of the heat of the fuel is rendered 
 available. 
 
 Fig. 1531 represents an old form of open fire used in a tavern bar-room and 
 office, which answered admirably for heating and ventilation, and admitted of 
 
 access to many persons. It con- 
 sisted of a circular grate at the 
 level of the floor in the centre of 
 the room. In the cellar beneath 
 was an ash-pit, a, in brickwork, 
 with an opening, o, to supply air 
 for the combustion of the fuel. 
 Above the grate was a counter- 
 weighted sheet-iron hood, 7i, con- 
 nected by a pipe with the chimney, 
 which could be raised or lowered 
 to suit the required draught. 
 Around the grate was a ring-guard 
 to rest the feet on, and the cus- 
 tomers ranged themselves in a circle 
 round the fire. 
 
 Stoves. Open stoves heat by direct radiation, and by heating the air in con- 
 tact with them, and close stoves by the latter way only ; as economical means 
 of heating the latter are the best, and, when properly arranged, give both a com- 
 fortable and wholesome atmosphere. There should be some dish of water upon 
 them to supply a constant evaporation, sufficient to compensate for increased 
 capacity of the air for moisture due to its increased heat. In the hall there will 
 be no objection to a close stove, letting it draw its supply of air as it best can ; 
 but in close rooms the open stove is best, on the plan of the old Franklin stove, 
 or, if a close stove, somewhat on the plan of a furnace, with an outer air-supply 
 for combustion and ventilation. 
 
 Stoves are made of sizes adapted to large and small rooms, in every style and 
 with all possible appliances for comfort, convenience, and economy : self-feed- 
 ers, in which the coal is furnished to the fire in a close chute from the top 
 downward and in proportion to the coal consumption below ; base-burners, in 
 which the draught is reversed so that the base becomes a portion of the heating 
 surface ; doors with mica panels around the circumference, by which the fire 
 is seen with the advantage of radiant heat and easy 'access to the fire-pot. 
 Stoves are usually coal-burners, but plain box-stoves in cast or sheet iron are 
 well adapted for wood-burners and for holding fire and retaining heat. They 
 have appliances for controlling draught by dampers or by opening the smoke- 
 pipe or flue to the room, thereby reducing the velocity of draught, but not 
 throwing the products of combustion outward into the room, as is done by 
 dampers. 
 
 Hot-air furnaces are close cast-iron stoves, inclosed in air-chambers of brick 
 or metal, into which external air is introduced, heated, and distributed by metal 
 pipes to the different rooms of a house. Furnaces have been of late very much 
 
ARCHITECTURAL CONSTRUCTION. 639 
 
 decried, but under proper regulation they are a very cheap, economical, and 
 even healthful means of ventilation, and warming. The heating surface should 
 be very large, the pot thick, or even incased with fire-brick, that it may not be- 
 come too hot ; there should be a plentiful supply of water in the chamber for 
 evaporation, perhaps also beneath the opening of each register ; the air supply 
 should always be drawn from the outer air and unobjectionable sources, through 
 ample and tight ducts, without any chance of draught from the cellar ; the pot 
 and all joints in the radiator should be perfectly gas-tight so that nothing may 
 escape from the combustion into the air-chamber. With these provisions on a 
 sufficient scale, and proper means for distribution of the heated air and escape 
 of foul air, almost any edifice may be very well heated and ventilated. The air 
 should be delivered through the floor or the base-board of the room, and at the 
 opposite side from the flue for the escape of foul air, making as thorough a 
 current as possible across the room, and putting the whole air in motion. In 
 dwelling-houses the fireplace will serve the best means of exit ; in public rooms 
 distinct flues will have to be made for this purpose, and they should be of 
 ample dimensions and well distributed, with openings at the floor and ceiling 
 with registers, and means should be provided for heating the flues. An archi- 
 tect, in laying out flues for heating and ventilation, should, both in plan and 
 elevation, fix the position of hot and foul air flues, and trace in the current of 
 air, always keeping in mind that the tendency of hot air is to rise ; he will then 
 see that, if the exit-opening be directly above the entrance-flue, the hot air will 
 pass out, warming the room but little ; if the exit-opening be across the room 
 and near the ceiling, the current will be diagonal, with a cold corner beneath, 
 where there will be very little circulation or warmth. To heat the exit-flue, a 
 very simple way is to make the furnace-flue of iron, and let it pass up centrally 
 through the exit-flue ; but the current may be obstructed by a high wind. 
 
 Fig. 1532 is one of the many forms of furnaces which consist of the most 
 approved stoves, with a large heating-chamber above in the figure of cast-iron, 
 and composed of numerous flues, but very often a drum of wrought-iron. The 
 whole is inclosed in a brick chamber ; in those denominated portable furnaces 
 the case is of galvanized iron. The figure contains the usual appliances for 
 feeding and clearing fires, with a check draught opening from outside into the 
 smoke flue, and dust damper and flue so arranged that when the grate is shaken 
 no ashes or dust comes into the hot-air chamber. The air is introduced at the 
 bottom of the case, passes up and around the stove, and out through the ducts 
 to different parts of the building. The water-pan is indispensable to the hot- 
 air furnace, and should be of capacity enough for a day's supply, or have auto- 
 matic means of keeping up the supply. 
 
 Air in winter is very dry, but as its volume is enlarged by heat it draws a 
 supply of moisture from everything with which it comes in contact from the 
 skin and lungs, creating that parched and feverish condition experienced in 
 many furnace-heated houses ; from furniture and woodwork, snapping joints 
 and making unseemly cracks. Thus, taking the air at 10 and heating it to 
 70, the ordinary temperature of our rooms requires about nine times the mois- 
 ture contained in the original external atmosphere, and, if heated to 100, as 
 most of our hot-air furnaces heat the air, it would require about twenty-three 
 times. 
 
640 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 The portable furnace is not so economical as the furnace set in brickwork, 
 as more heat escapes through the metallic case. The former are usually made 
 
 Fio 1532. 
 
 from 12" to 36" diameter of pot, from 2' to 6' outside diameter, and 5' to 6' 
 height of case. The brick-set furnaces are from 20" to 32" pot, outside brick- 
 work from 5' to 6' square, walls 4" thick, height 6' to 7'. It is difficult to give 
 any rule for the heating capacity. A 22" pot should be adequate for the heat- 
 ing of a common 25' X 60' city house, and the higher the air-duct the less its 
 diameter. 
 
 The total sectional area of the hot-air educts should be equal to that of the 
 fire-pot ; that to the first floor should be larger than to the other floors, since 
 the column of hot air is shorter it will have less velocity. Air ducts that have 
 outlets at the same level under the same conditions should have greater area if 
 the horizontal pipes are longer. The cold-air duct should have about the same 
 area as the grate, and the inlet should be above the level of the street or back 
 area to avoid dust. If air ducts lead from both sides of the building to the 
 furnace-chamber, the current can be controlled according to the wind, and the 
 hot air distributed more equably through the building. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 641 
 
 The Baltimore heater (Fig. 1533) was the 
 earliest union of the stove with tire furnace. 
 A stove is set in the fireplace of a room in a 
 lower story, of which the exterior or orna- 
 mental half is exposed for the heating of 
 this room, while the inner half acts as a fur- 
 nace for the upper rooms. The smoke-pipe 
 passes up into a chimney above, and is in- 
 closed by an air pipe or jacket to which heat 
 is communicated and distributed by dram 
 pipes and registers to upper stories. 
 
 Steam and hot-water circulation are ap- 
 plied to the heating of buildings by means 
 of wrought- or cast-iron pipes connected 
 with boilers. In the simplest form, as com- 
 mon in workshops and factories, steam is 
 made to give warmth without ventilation by 
 direct radiation from wrought-iron pipes. 
 The general arrangement is by rows of 1" to 
 1" pipe hung against the walls of the room, 
 or suspended from the ceilings, 3' of 1" pipe 
 being considered adequate to heat 200 cubic 
 feet of space ; if there are many windows in 
 the room, or the building is very much ex- 
 posed, more length should be allowed. 
 
 Steam, as a means of heating, is the most 
 convenient and surest in its application to 
 extensive buildings and works. From boilers, 
 located at some central point, steam can be 
 conveyed to points so remote that in many 
 cities it is a matter of sale both for heating 
 and power purposes. The limits of the ex- 
 tension of steam-pipes economically have not 
 yet been determined, but within the range of 
 the buildings occupied by any single textile 
 manufacturing industry steam-heating has 
 proved satisfactory, and is almost universally 
 adopted. For stores, warehouses, large build- 
 ings of all sorts, where there are extensive or 
 numerous rooms to be heated, steam has been 
 long used, and the appliances for its use can 
 be as readily obtained in all our cities and 
 large towns as stoves or grates. Steam is 
 used for heating at either high or low pres- 
 sures ; tinder 5 or 6 pounds would be consid- 
 ered low pressure. A low-pressure apparatus 
 may draw direct from a boiler, or be supplied 
 from the exhaust of a steam-engine ; if from 
 42 
 
 FIG. 1533. 
 
642 ARCHITECTURAL CONSTRUCTION. 
 
 the latter, a certain amount of back pressure must be put on the engine to es- 
 tablish a circulation in the steam-heating pipes. But the loss in power is more 
 than repaid by the utilization of the heat in the steam. If the heating-pipes 
 are ample, they may be arranged to act as condensers, reducing the back 
 pressure below atmosphere and supplying low steam for heating. 
 
 In the operation of heating by steam, the steam, in giving off its latent 
 heat through the pipes to the air of the room, returns to water ; the apparatus 
 would then be nothing but pipes to convey the steam to radiators to condense 
 it, and pipes to return the water to the boiler, were it not for air invariably in 
 water and steam. This necessitates a more complicated circulation; there 
 should be a regular flow outward of steam from the boiler, and inward of water 
 and steam to it, which must be provided for in the design and by care in con- 
 struction that there are no corners or angles forming eddies where air can 
 lodge, and with provision for its continuous movement. 
 
 When hot water is used for heating, there must be circulation throughout 
 the system ; the water flows out from the top of the boiler, gives out its heat, 
 and returns, practically of the same bulk, cold to the bottom of the boiler, and 
 any radiator out of the line of this current is of no use. 
 
 Both steam and water are used for heating rooms either directly or indirectly. 
 Direct heating is like that of common stoves, without any considerations for 
 ventilation ; indirect heating, like that of hot-air furnaces. Radiators are in- 
 closed in a box or chamber, into which air is drawn or forced, and the.n dis- 
 tributed by ducts to the rooms to be warmed and ventilated. With steam or 
 hot-water heating, the metallic surfaces brought in contact with the air usually 
 range from 212 to 250, while the pot of the air-furnace is often from 900 
 to 1000. In a sanitary point of view hot-water or low-steam coils in air- 
 chambers are a more surely healthy means of warming and ventilation ; the 
 greatest objection is their expense, the care requisite in attending them, and 
 the danger of freezing and bursting the pipes if worked intermittently in win- 
 ter. In the arrangement it is usual in dwelling-houses to place the coils at 
 different points in the cellar, as near as possible beneath the rooms to be heated. 
 In public buildings frequently a very large space in the cellar is occupied by 
 the coils, into which the air is forced by a fan, and then distributed by flues or 
 ducts throughout the building. 
 
 All inlet or outlet ventilating flues should be provided with dampers or 
 registers to control the supply or discharge of air, cutting it off when suffi- 
 cient heat is secured, or retaining the warmth when ventilation is not re- 
 quired. 
 
 Fig. 1534 is an elevation showing the usual arrangement of mains, s s, and 
 returns, r r, when the horizontal distance from the boiler is small and the 
 risers few. The inclination of the mains is toward the boiler, and their con- 
 densed water returns by them to the boiler. 
 
 Fig. 1535 is the better practice, and necessary if the steam is higli pressure, 
 the mains extended, and the branches numerous. The inclination of the mains, 
 s s, is from the boiler, and the condensed water flows down to the lowest angle, 
 where it is connected with the return, r, and is by this brought back to the 
 boiler. 
 
 The size of the boiler for a steam-heating apparatus is based on the amount 
 
ARCHITECTURAL CONSTRUCTION. 
 
 643 
 
 of radiating surface, which must include that of the steam- mains and of the 
 returns. 
 
 In Fig. 1535 the steam riser, s, descends from the boiler to the last riser, 
 which is connected at this point with the return, and this should obtain in all 
 forms of steam-heating, keeping the flow of condensed water as far as possible 
 in the direction of the flow of 
 the steam, and removing it 
 from the steam-pipes. 
 
 Figs. 1536 to 1538 are com- 
 mon forms of distributing steam 
 and return pipes of different 
 systems of heating. 
 
 In Fig. 1536 the riser pro- 
 ceeds from the boiler and leads 
 directly to the highest point of 
 
 service. Provision is made for the condensation in the pipe by a direct con- 
 nection. 
 
 In Fig. 1537 the risers are as in Fig. 1536, but there are separate returns 
 for each radiator. 
 
 In Fig. 1538 the main riser is carried directly to the highest story to be 
 warmed, and the distributing mains are led from it with a pitch from the riser, 
 and the descending pipes conveying the steam to the different radiators and 
 the condensed water to the hot well and boiler. This should be the quietest cir- 
 
 Fio. 1534. 
 
 FIG. 1535. 
 
 culation, as the steam, except in the main riser, does not interfere with the flow 
 of the condensed water. 
 
 Valves are introduced in the mains or returns where necessity or conven- 
 ience may require the shutting off of any of the radiators or mains, and air- 
 cocks to relieve stagnation in angles or pockets where circulation must be 
 established before the whole of the heating surface can be utilized. 
 
 The amount of radiating surface depends on the cubic feet of air to be 
 heated and the number of degrees to which it is to be heated. These are de- 
 termined from the outer exposure of the building or room, its plan, material, 
 and construction, whether there is more or less window surface, and its occu- 
 pancy ; whether for living rooms or for business, sedentary or active ; in the 
 store the bookkeeper needs much more heat than the salesman. 
 
 On an average, 100 to 150 cubic feet of room space can be heated from 
 to 70 by 1 square foot of radiating surface say 3' of I" pipe. This covers an 
 average glass exposure which may be taken, according to Mr. Briggs, at 100 
 cubic feet of space to each square foot of glass. 
 
 The effect of glass under air exposure is to be noticed in the different con- 
 
644 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 
 FIG. 1536. 
 
 FIG. 1538. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 645 
 
 ditions experienced in cars while in motion and when stopped in cold weather, 
 and the advantages of double windows in these conveyances. 
 
 Proportions of mains to radiating surface : one of 1" diameter will serve for 
 75 feet of radiating surface, including that of the mains. One and a half inch 
 diameter for 250 square feet, 2" diameter for 500 square feet, 3" diameter for 
 1,250 square feet, 4" diameter for 2,500 square feet. 
 
 For the returns, one size less than that of the steam mains is the rule ; thus, 
 a f " return for a 1" pipe, but no pipe of less diameter than f " is used ; for a 
 2" steam a 2" return, and a larger than 2" is seldom used. It may not be 
 always practicable to return the condensed water, as shown in the figures 
 above, by gravitation, but there are various forms of receivers or traps in which 
 the water is collected and returned by hand or automatically pumping to the 
 boiler. 
 
 Fig. 1539 is a float trap, in which p is the pot with a tight cover, /an open 
 float sliding on the stem, s, at the foot of which is a valve, v ; the condensed 
 water flows in through the inlet, i, raises 
 the float, /, and closes the valve, v ; event- 
 ually the condensed water overflows into/ 
 till it sinks and opens the valve, v, and the 
 condensed water flows out through the 
 valve, v, under the pressure of steam at the 
 inlet ; when blown out, / rises and v closes 
 for another charge. The condensed water 
 is either wasted or returned to the boiler by 
 a pump. The hand valve, A, is an inde- 
 pendent relief to the trap. 
 
 There are traps which return the con- 
 densed water directly to the boiler, in 
 which the condensed water is forced into 
 a chamber above the level of the water in the boiler and the steam pressure 
 then brought upon the chamber, and the water flows down from it into the 
 boiler. 
 
 Heating by indirect radiation is like that by hot-air heaters ; the heaters 
 are inclosed in chambers to which cold air is introduced and the heated air 
 conveyed into different rooms by pipes. It is usual to place the hot-air cham- 
 bers in the cellar and basement, as directly beneath the room to be heated as 
 possible, and extending the cold-air duct to the hot chamber. The chamber is 
 usually made of galvanized iron in a wooden box, and often suspended from 
 the ceiling of the cellar. Where the heating is indirect, as there are more cubic 
 feet of air to be heated, the radiating surface is to be increased, usually to about 
 three times that of the direct heating. 
 
 Heating by Hot Water. The principle of it is based on the fact that water 
 when heated becomes of greater volume and less density, and rises; coming in 
 contact with cooler surfaces, it loses its heat and descends ; the water circulates 
 by the addition and reduction of heat. The possibility of a rapid and thorough 
 circulation gives efficiency to the apparatus. One of its greatest advantages is, 
 that when necessary the circulation will continue with a water temperature at 
 an extremely low point, say 110, and a proportionate consumption of coal. 
 
 FIG. 1539. 
 
646 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Fig. 1540 exhibits the application of hot water to the heating of a building. 
 The boiler shown will serve as an illustration of the water circulation ; it is of 
 
 cast-iron, of which there are numerous 
 forms in this material, as well as wrought- 
 iron. 
 
 The main rises directly to the top, 
 where there must be an expansion cham- 
 ber, C, in connection with it, to provide 
 for the increase of volume in the water 
 due to the heat. It is usual to have this 
 chamber open at the top, and water may 
 be poured down through the funnel noz- 
 zle. A glass at the side shows the level 
 of the water. 
 
 The mains are of a little larger diame- 
 ter than in steam, and reduced in size as 
 the current is distributed into the radia- 
 tors on which valves are placed to control 
 the current ; these valves should be gates, 
 to provide for full water-way. The re- 
 turns are to be of the same diameter as 
 the rising mains. The surface of the ra- 
 diator should be greater than for steam, 
 as the temperature of the water is less. 
 When the radiators are vertical, as shown 
 in the first and second story, there must 
 be air-cocks, a, as shown, for air effectu- 
 ally cuts off circulation. Hot-water cir- 
 culation can be used for direct or indi- 
 rect heating with like appliances as for 
 steam. 
 
 Boilers are of such varied forms and 
 proportions to the area of grate that it is 
 impossible to determine the value of the 
 heating surface except by actual test. As 
 a unit it is better to refer to the area of 
 grate as a measure of the capacity of the 
 boiler, the evaporation of water by the 
 combustion of pounds of coal being the 
 standard of efficiency. 
 
 Under the head of chimneys it is 
 stated that two square inches of flue is 
 sufficient for the combustion of each 
 pound of coal per hour. This has been 
 
 FIG. 1540. 
 
 found in excess for a 5-foot diam. chimney, 
 
 where provision has been made for avoiding eddies by the rounding of corners, 
 and where there is a good natural draught, but there must be a factor of safety 
 when care has not been taken in the location and construction. For the small 
 
ARCHITECTURAL CONSTRUCTION. 
 
 647 
 
 flues connected with the heating of common buildings this rule will not obtain ; 
 flues for this purpose should be at least 8" X 12". But as a large area of chim- 
 ney flue does not interfere with the draught, and as the necessities of chimneys usu- 
 ally increase by the extension 
 of works, it is safer to make 
 the flue larger, but without 
 omitting care in construc- 
 tion. 
 
 The combustion of coal 
 on small grates may be taken 
 at 4 to 6 pounds of anthra- 
 cite coal per square foot of 
 surface ; on grates 4' X 4', 8 
 pounds ; on grates of larger area, 10 pounds, which would be a fair average of 
 this class, and the vents of large chimneys may be calculated on this data, 
 although the draught under equal conditions is in favour of the chimneys of 
 large diameter. 
 
 With forced draught by fans, ejectors, or, as in locomotives, by the exhaust, 
 very large quantities of coal can be consumed on grates. 
 
 The air-ducts from boilerand heaters for smaller and private dwellings have 
 a natural draught ; but for schoolhouses, churches, and other public edifices a 
 forced draught' is usual, is under surer control, and with adequate indirect 
 heaters the quantity of air can be readily furnished. It is common to intro- 
 duce local fans driven by electric motors to supply air at varied points and in 
 quantities suited to the needs of the position. By the use of dampers or valves 
 air of any temperature, from that of the outside to that of the radiator, may be 
 distributed. 
 
 In Figs. 1536, 1537, 1538, and 1540 illustrations are given of the usual radi- 
 ators : the wall coil with return bends, which may be more or less open (with 
 branch Ts and multiple coils they are called box coils), wall coils with branch 
 
 FIG. 1542. 
 
 FIG. 1543. 
 
648 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Ts, vertical tube radiators with box bases of effective heating surface, cast-iron 
 
 radiators of similar design, ornamented and of great variety. Indirect radia- 
 tors, Fig. 1541, in cast-iron with 
 pin or iron projections, and 
 wrought-iron pipes covered with 
 compound coils of wrought-iron 
 ribbons, increase heating surface. 
 In measuring the surface of cir- 
 culating coils include the lengths 
 of angles and all fittings ; in the 
 vertical radiators include the base. 
 Figs. 1542 and 1543 are the 
 end and side elevation of a radia- 
 tor for live or exhaust steam, in- 
 closed in a case for indirect heat- 
 ing. Air is forced through the 
 case into the building by a fan. 
 
 By the location of the radia- 
 
 FIQ 1544 tors in an independent chamber 
 
 and ft valve, Fig. 1544, the air may 
 
 be forced through it hot, or mixed with cold air or without connection with the 
 
 chamber cold. 
 
 FIG. 1545. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 649 
 
 Fig. 1545 is the plan of a portion of a large building heated by steam. B B 
 are two boilers, either of which would be sufficient for the purpose ; the steam 
 mains are shown by black lines following those of the building, with the sizes 
 marked upon them ; the risers by inclined lines, with the square foot of radiat- 
 ing surface on each story marked. This is a very convenient form of drawing, 
 explanatory of the system. It is usual to draw the steam mains and risers in 
 red and the returns in blue, with the diameters on each. 
 
 Plumbing. The conveniences for comfort in modern buildings require the 
 introduction of water and its removal. Most cities have water-supplies and a 
 system of sewers, and the plumber makes the connections with both. In the 
 country, where there are not these public conveniences, their places are largely 
 supplied by pumps and elevated tanks and by cesspools. The quantity used in 
 each household varies with the wants and habits of the occupants. An average 
 bath will take 25 gallons ; each use of a water-closet from 1 to 3 gallons. A 
 wash-tub will hold from 10 to 20 gallons. If the water is to be pumped by 
 hand, from 7 to 10 gallons may be reckoned as the daily use by each person ; if 
 from aqueduct, 30 to 50 gallons is ample. With the popular style of water- 
 closets the use of water has been largely increased, and by carelessness in the 
 selection and use of fixtures the waste has become greater. 
 
 The regulation size of taps for city mains is from " to 1", and the pipes 
 
 FIG. 1546. 
 
650 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 leading into the house from f " to 1" diameter. The pipes are usually of lead, 
 as most waters are not affected sensibly by lead, if the pipes are always kept 
 full, but water which has stood for some time in the pipe should not be used 
 for drinking, and lead-lined tanks should be coated with asphalt varnish. In 
 some cases block-tin pipes are used ; or iron, galvanized, or coated with some 
 preparation of asphalt, or glass-lined. 
 
 The soil or house-sewer pipe connections with the main sewer or cesspool 
 are usually vitrified stoneware pipe, from 4" to 6" diameter, the large size is 
 for the discharge of the sewage, and the rainfall from the roof. Within the 
 house the pipe is either of stoneware or cast-iron ; invariably of the latter if 
 the pipe is exposed. The rising pipe to the roof is here, also, usually of cast- 
 iron, and 4" diameter may be considered ample for a common house ; branches 
 as small as 2" are usually of lead. 
 
 Fig. 1546 is the perspective of a kitchen-range boiler and sink : c is the 
 cold-water pipe leading to the sink and to the boiler ; it enters the top of the 
 boiler, and is led down nearly to the bottom. The hot water is drawn from 
 the top, through the pipe A, is led down to the sink and up 
 for distribution through the house. The water is heated in* 
 the boiler by the water-back, which consists of a closed-box 
 casting, forming the back of the range, r, with two connec- 
 tions with the boiler, the one at the bottom introducing cold 
 water, and \the one at the extreme top discharging it heated 
 into the boiler above, the circulation taking place as in hot- 
 water heating ; the water flows through the pipe, I, is con- 
 nected with the lower part of the water-back, and returns by 
 the pipe, w, from the top of the water-back to a higher point 
 in the boiler; b is the blow-off pipe. Stoves are arranged 
 with water-backs. 
 
 The pipes, a a, are carried above the draw-cocks over the 
 sink, forming air-chambers, to cushion the blow of the water- 
 hammer when the cocks are shut quickly. Beneath the sink 
 there is a trapped connection with the sewer-pipe. 
 
 Fig. 1547 is the elevation of a galvanized-iron boiler, but 
 those in general use here are of copper. 
 
 Fig. 1548 is the perspective drawing of a cast-iron sink, 
 
 B; 
 
 Fiu. 1547. 
 
 FIG. 1548. 
 
 of the usual form and material. They are to be obtained of all suitable dimen- 
 sions, rectangular, from 16" X 12" X 5" deep, to 96" X 24" X 10" deep; also, 
 half-circle and corner sinks, and deep and slop sinks. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 651 
 
 In the kitchen, or a laundry^room adjacent, tubs are set for washing, with 
 hot and cold water service. The water-pipe connections are usually ", the 
 waste connections 2''. The tubs themselves are mostly of wood, but there are 
 many of cast-iron (Fig. 1549), galvanized or enamelled, of slate, of earthen- 
 ware, and of soapstone. 
 
 In the butler's pantry there is usually a sink of planished tinned-copper, 
 with hot and cold water connections. In the chambers and dressing-rooms 
 
 FIG. 1549. 
 
 wash-basins, usually of porcelain or porcelain-lined cast-iron, are set with like 
 connections. The sizes of basins vary from 12" to 18" outside diameters. 
 
 Fig. looO shows the usual form of setting of a wash-basin in a countersunk 
 marble slab, with a back of the same material ; swing faucets for the supply of 
 hot and cold water; self-closing faucets prevent waste, and compression 
 cocks are best suited for high pressures. The waste is closed by a metal or 
 rubber plug, attached to a 
 chain, with the other end 
 fastened to a pin in the 
 marble slab. The sides 
 are inclosed with wood, 
 forming a closet beneath 
 the basin, with usually 
 small drawers for towels 
 at each side of the closet. 
 It is cleaner and neater 
 to support the slabs and 
 
 basins by a metal frame and posts or brackets, with the traps and pipes ex- 
 posed. A later form of wash-basin is an oval (largest size 19" X 15", outside 
 measure), which admits of the washing of the head and shoulders, with a 
 standard waste or hollow pipe overflow, preventing offence in the oxidation of 
 the soapy waste which obtains from pipes, formed on the basin, and which can 
 
 FIG. 1550. 
 
652 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 not be readily cleaned. Fig. 1551 is a plan, and Fig. 1552 a section, of a cast- 
 iron bath-tub porcelain-lined ; there are cocks for hot and cold water ; for the 
 discharge there is a standard waste, which can be taken out entirely, but there 
 is often a common plug and an overflow by an independent pipe. 
 
 The dimensions of tubs are varied to suit the rooms in which they are to 
 be placed ; the largest are 6' X 24", 18" and 19" deep. They may be reduced 
 to 3' 6" in length, but should then be made deeper. Tubs of porcelain are set 
 up on blocks ; porcelain-lined, on cast-iron legs, about 6" in height. Bath- 
 tubs are more generally made of planished tinned-copper in a wooden box for 
 support, and inclosed by wooden panels. In most bath-rooms there are basin 
 
 FIG. 1552. 
 
 FIG. 1551. 
 
 and water-closet often a foot-bath and bidet-pan but it is preferable to make 
 the water-closet a separate room, distinct, with its own water and sewer-service 
 and means of ventilation. 
 
 The construction of one form of water-closet, with all the modern appli- 
 ances for the removal of soil and for ventilation, will be understood from the 
 section (Fig. 1553). The seat is not shown, but is just above the basin, B, 
 which contains some water to receive the defecations, to prevent the soil attach- 
 ing to the side of the basin, and in a measure to check its offensive smell. T 
 is the trap or water-seal which prevents the smell from the soil-pipe S passing 
 up through the basin. The water-discharge from the pipe W is through a rim- 
 flush around the edge of the basin. The sudden discharge washes out the basin 
 B into the trap T, which is also cleaned by the rush of water. The soil-pipe S 
 extends up through the roof, and may or may not also serve as a rain-leader. 
 A sudden flow of water down the soil-pipe often acts as an ejector to draw the 
 water out of the trap T, and break the water-seal ; to prevent this there is a 
 back-air connection, A, leading also to the top of the house. But as the offence 
 of a water-closet is largely due to its recent use, and as smell once getting into 
 the room is with difficulty removed, but generally diffused, there is a ventilat- 
 ing-pipe, V, connecting the basin B with a ventilating-flue ; this is the most 
 important part of the apparatus ; connected with a chamber commode, it would 
 remove all smell, and if there were no trap to the soil-pipe, or were the water- 
 seal broken, it would still prevent any offensive smell from penetrating the 
 house. If the soil-pipe be made also a ventilating-pipe, as is frequently done 
 by its connection with the hot-air flue, then the trap and pipes A and V are 
 unnecessary. 
 
 Many sanitary engineers object to the back-air pipe as imperfect in its 
 
ARCHITECTURAL CONSTRUCTION. 
 
 653 
 
 action (a sudden suction will draw the water out of the trap before the suction 
 can be relieved through a back-air pipe), that it is expensive, and that there are 
 many better ways of protecting the seats by anti-siphon traps or diverging Y- 
 
 FIG. 1553. 
 
 FIG. 1554. 
 
 W 
 
 branches. In Fig. 1554 is shown a branch by which the flow from the up- 
 per pipe dripping into the .lower trap preserves the water seal. As the water- 
 closet must be ventilated, it is better to have a regular flue in the wall with 
 an induced ventilation by heat in some form, or electric fans, and air supplied 
 from outer rooms or windows. 
 
 Fig. 1555 is an elevation of the simplest 
 form of closet the hopper-closet and in many 
 respects the best. A standard waste in the cis- 
 tern c will serve both for the disk- valve and the 
 overflow. A rim-flush is supplied through the 
 pipe W, controlled by a plain cock or by a han- 
 dle A, as in Fig. 1556, lifts the disk valve, clos- 
 ing automatically ; or by cock connecting with 
 the seat, which, when down for occupancy, 
 opens the flush, and, as the sitter rises, the 
 counterbalanced lid rises and closes the cock. 
 
 The service box beneath the cistern continues a temporary flow after the valve 
 is dropped. V is a ventilator branch. 
 
 Fig. 1557 is the section of a pan-closet, for many years the most popular 
 
 FIG. 1555. 
 
654: 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 closet. The copper pan, when shut, cuts off the view of the trap below and 
 any odour from it ; with a small flow of water the basin is readily kept clean, 
 
 but soil is apt to lodge in the iron re- 
 ceiver, and the odour to arise from it 
 when the pan is down. There is an 
 annular ventilating-tube beneath the 
 seat, with an air-shaft attached, but of 
 
 FIG. 1556. 
 
 FIG. 1558. 
 
 SIPHON JET CLOSE.T. 
 
 altogether inadequate dimension for the purpose, as may be said of all such 
 vents attached to water-closets. There is also the air- vent to prevent the water 
 being drawn from the trap. No water connections are shown in the figure. 
 
 If the 2" vent be removed, and the' air- 
 draught pipe be enlarged and connected 
 with a positive draught-flue, all offence 
 from either recent or former use of the 
 closet will be cut off. 
 
 Fig. 1558 is the section of a flap-closet, 
 in which a flap-valve supplies the place of 
 a pan. 
 
 Fig. 1559 is the section of a siphon-jet 
 closet. In addition to the fan flush, /, 
 into the basin, it has a jet-pipe, /, at its 
 
 bottom, inducing a current in the direction of the inclined leg of the trap, and 
 by flush and jet the water is siphoned from the basin. 
 
 FlG. 1559. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 655 
 
 Traps are varied in their form, but all to cut off the air-connection of the 
 soil-pipe with the room in whicli the appliance is placed. The smaller traps 
 are invariably lead, the larger cast-iron. 
 
 Figs. 1560 to 1567 represent the usual forms of lead traps. There are 
 screw-plugs at the bottom of the traps, which can be taken out to remove 
 
 s. 
 
 SHORT BEND. LONG BEND. 
 
 Fia. 1560. 
 
 Fia. 1561. FIG. 1562. 
 
 FIG. 1563. 
 
 FIG. 1564. 
 
 FIG. 1565. 
 
 FIG. 156 
 
 FIG. 1567. 
 
 any obstruction. As the water may be drawn out of any trap by the passage 
 of water down the pipe with which it is connected, air-vents, as already de- 
 scribed, in the water-closet trap, are put on these small traps. But if upper 
 waste pipes be inserted, as in Figs. 1560 and 1561, at a a, and shown in sec- 
 tion (Fig. 1554, p. 653), loss of seal is cut off. 
 
 Figs. 1568 and 1569 are cast-iron traps, with a cap that may be removed to 
 clean the trap, or the aperture may be used for air- vent connection. 
 
 Fig. 1570 is the section of a bell-trap, used on sinks, with a strainer, S, 
 above it. 
 
 Fig. 1571 is a plate with plug, for the bottom of basins and bath-tubs. 
 
 S-TRAP. 
 
 TRAP WITH SIDE OUTLET. 
 
 FIG. 1568. 
 
 FIG. 1569. 
 
 FIG. ISI'O. 
 
 FIG 1571. 
 
 Figs. 1572 to 1577 are common cast-iron bends or angles. 
 
 Figs. 1578 to 1583 are cast-iron branches. The T-branch and cross 
 are objectionable, as the flows from the branches and mains are at right angles, 
 and mutually obstructive ; whereas in the Y, especially in the full Y, the flows 
 are at acute angles with each other, and the currents converge. Similar fittings 
 are used for water, but they are much heavier. 
 
 QUARTER 
 BEND. 
 
 DOUBLE HUB, 
 QUARTER BEND. 
 
 FIG. 1572. 
 
 SIXTH 
 BEND. 
 
 FIG. 1573. 
 
 FIG. 1574. 
 
 FIG. 1575. 
 
 FIG. 1576. 
 
 FIG. 1577. 
 
 Most water-closet basins are inclosed by a lidded seat and riser, but the less 
 wood-work about a basin the better. The seat is generally hung with hinges 
 of brass or composition, so that it can be raised, and the basin makes the 
 
656 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 T-BBANCH. 
 
 CROSS-HEAD. 
 
 HALF 
 Y-BRANCH. Y-BRANOH. 
 
 DOUBLE DOUBLE HALF 
 
 Y-BRANCH. Y-BRANCU. 
 
 FIG. 1578. 
 
 FIG. 1579. 
 
 FIG. 1580. 
 
 FIG. 1581. 
 
 FIG. 1582. 
 
 FIG. 1583. 
 
 FIG. 1584. 
 
 best urinal for men, the upper edge of the basin being covered with an earthen- 
 ware tray, sloping toward the basin. Instead of the tray, if the rim of the 
 
 basin be made square, of sufficient width to 
 support the seat, sloping to the basin, and 
 overhanging, it will make the better urinal, 
 and afford space for an adequate ventilating 
 pipe. 
 
 Urinals, of which one form for males is 
 shown (Fig. 1584), are often used in public 
 buildings, and in open stalls. Although they 
 have water connections, w, and a rim flush, 
 it is almost impossible to keep them sweet ; a 
 cake of carbolic soap is often put in the ba- 
 sin, but the most effectual means adopted on 
 many railway-cars is a piece of ice. 
 
 As direct supply from the service is uncer- 
 tain if there is a draught in another quarter, 
 it is now common to have small cisterns for 
 closets, shown in Fig. 1556. The water from 
 the service pipe is discharged well below the 
 
 surface overflow, generally through a pipe attached to the goose-neck on the 
 ball-valve pipe, to avoid the noise of running water. 
 
 Lighting is one of the present necessities of civilization, and for a great 
 many years gas has been used for lighting in domestic and industrial buildings. 
 Gas fittings are in all forms brackets and pendants, wide branches with 
 fixed, swing, and slide joints ; and burners in great variety bat wings, fish- 
 tail tips, and Argand burners, and many patents for increased light and econ- 
 omy of consumption. 
 
 Service mains are seldom placed in buildings less than f " diameter for 10 
 burners and 100 feet of pipe, to If" and 200 feet length. 
 
 Electric lighting, although not superseding gas for common use, is now 
 largely employed in industrial operations, and for all shows and brilliant illu- 
 minations. 
 
 The three accompanying drawings (Fig. 1584, A, B, C) illustrate three 
 methods of wiring for electric lighting. 
 
 Fig. A is a series installation. The lamps are in series and dynamo series 
 wound. The wires may be led in any direction, care being taken to insulate 
 them properly and avoid proximity to inflammable substances. The current 
 leaves the dynamo at +, passes through each lamp, and returns to the dynamo 
 
ARCHITECTURAL CONSTRUCTION. 
 
 657 
 
 at . This current is maintained at a constant strength required by the con- 
 struction of each style of lamp, and determined by the E. M. F. of the dynamo, 
 and the full resistance of the entire circuit, including the dynamo, the lamps, 
 and the conducting wires. The strength of the current required varies greatly, 
 ranging from ten to twenty amperes for series installations in the various lead- 
 ing arc-light systems for which this method is used. 
 
 Fig. B. One method of wiring is shown for incandescent electric lighting, 
 and is known as the multiple-series installation. Between two main conductors 
 extending from the dynamo are placed, in parallel, several short lines connect- 
 
 rn r 
 
 FIG. 1584. 
 
 ing with the lamps, the current being thus divided among the lamps in pro- 
 portion to their number and resistance, while in the series system the entire 
 current passes through every lamp. This method is common to both the 
 direct and alternating current system, and is convenient for lighting a large 
 room or hall where many lamps are required at one time. A simpler and 
 better arrangement for most purposes is the parallel system, in which two 
 mains are led from the dynamo, and each lamp is placed on a separate branch 
 between the mains, so that it can be lighted or extinguished without inter- 
 fering with the remaining lamps. 
 
 Fig. C shows the Edison three-wire system for incandescent lighting. Two 
 dynamos are joined in series as shown. Each lamp has the same advantage of 
 independent connection with the dynamo as in the two- wire system, shown at 
 B. If an equal number of lamps are burning simultaneously in each row the 
 current will flow through the parallel branches from one dynamo to the other, 
 the central wire remaining neutral ; but if the number is varied by the extin- 
 guishing of lamps or otherwise, the third wire furnishes a path for the surplus 
 current required by the row having the greater number lighted, which will flow 
 to that side in consequence of the reduced resistance resulting from the greater 
 number of branches open through the lighted lamps. 
 
 The chief advantage of this system is in the reduced amount of copper used 
 for conductors, three wires instead of four being used, and each only one half 
 the area, a saving of five eighths being thereby effected. 
 
 A diagram in the Appendix illustrates a graphic method for obtaining the 
 size of wire to be used in electric lighting. 
 43 
 
ARCHITECTURAL CONSTRUCTION. 
 
 GREEK AND ROMAN ORDERS OF ARCHITECTURE, 
 
 as examples of proportions of graceful curves and outlines, are useful as 
 studies and manual practice for the draughtsman. 
 
 The Tuscan, Doric, Ionic, Corinthian, and Composite orders are systems or 
 assemblages of parts subject to certain uniform established proportions, regu- 
 lated by the office each part has to perform, consisting of two essential parts, 
 a column and entablature, subdivided into three parts each : the first into the 
 base, the shaft, and the capital; the second into the architrave, or chief beam, 
 C (Fig. 1585), which stands immediately on the column ; the frieze, B, which 
 lies on the architrave ; and the cornice, A, which is the crowning or uppermost 
 member of an order. In the subdivisions certain horizontal members or mould- 
 ings are used : thus, the ogee (a), the corona (i), the ovolo (c), the cavetto (d), 
 with the fillets, compose the cornice; the fasciae (//), the architrave; the 
 abacus (#), the ovolo (c), the astragal (ii), and the neck (7i), are the capital of 
 the column ; the torus (&) and the plinth (/) (Fig. 1587) are the base. The 
 character of an order is displayed not only in its column, but in its general 
 forms and details, whereof the column is, as it were, the regulator ; the expres- 
 sion being of strength, grace, elegance, lightness, or richness. Though a build- 
 ing be without columns, it is nevertheless said to be of an order, if its details be 
 regulated according to the method prescribed for such order. 
 
 In all the orders a similar unit of reference is adopted for the construction 
 of their various parts. Thus, the lower diameter of the column is taken as 
 the proportional measure for all other parts .and members, for which purpose 
 it is subdivided into sixty parts, called minutes, or into two modules of thirty 
 minutes each. Being proportional measures, modules 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 instance, if the diameter be just five feet, a minute, being 
 one sixtieth, will be exactly one inch. To draw an elevation of any one of the 
 orders, determine the diameter of the column, and from that form a scale of 
 equal parts by sixty divisions, and then lay off the widths and heights of the 
 different members according to the proportions of the required order, as marked 
 in the body or on the sides of the figures. 
 
 Figs. 1585 to 1589 are illustrations of the Tuscan order : <?, in the frieze 
 corresponding to the Doric triglyph, may or may not be introduced. Fig. 1585 
 is an elevation of the capital and entablature ; Fig. 1587 of the base ; and Fig. 
 1586 of another capital. 
 
 A slightly convex curvature, or entasis, is given in execution to the outline 
 of the shaft of a column, by classic architects, to counteract a fancied appear- 
 ance of concave curvature, which might cause the middle of the shaft to ap- 
 pear thinner than it really is. 
 
 Fig. 1588 represents the form of a half-column from the Pantheon at 
 Home. In Fig. 1589, another example, the lower third of the shaft is uni- 
 formly cylindrical. The entasis of the two thirds is constructed by dividing 
 the arc, a i, into equal parts, and the columns into the same number, and pro- 
 jecting the divisions of the arc on to those of the column. The upper diame- 
 ter of column or chord at b is 52 minutes. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 659 
 
 o 
 
 feft 
 
 S 
 
 J/ V 
 
 FIG. 1585. 
 
 ci 
 
 FIG. 1586. 
 
 f 
 
 .g.... 
 
 7 
 
 30 ..... 
 
 rt 
 
 1JT 
 
 at- - 
 
 4 
 
 9 
 -0 
 
 
 <%..... 
 
 .......... 
 
 FIG. 1588. 
 
 FIG. 1589. 
 
 -i 
 
 
 iM 
 
ARCHITECTURAL CONSTRUCTION. 
 
 Figs. 1590 to 1594 exhibit an example of the Doric order, from the Temple 
 of Minerva, in the Island of Egina. Fig. 1590 is an elevation of the capital 
 and the entablature; Fig. 1591 of the base; Fig. 1592 shows the forms of 
 the flutes at the top of the shaft, and Fig. 1593 at the base ; Fig. 1594 the 
 outline of the capital on an enlarged scale. 
 
 The mutules, a a, the triglyphs, b b, the guttse or drops, d d, of the entabla- 
 ture, the echinus,/, and the annulets, g g, of the capital, may be considered 
 characteristic of the Doric. The triglyph is placed over every column, and 
 one or more intermediately over every intercolumn (or span between two col- 
 umns), at such a distance from each other that the metopes, c, or spaces between 
 the triglyphs, are square. 
 
 In the best Greek examples of the order there is only a single triglyph over 
 each intercolumn. The end triglyphs are placed quite up to the edge or outer 
 angle of the frieze. The mutules are thin plates attached to the under side or 
 soffit of the corona, over each triglyph and each metope, with the former of 
 which they correspond in breadth, and their soffits or under surfaces are 
 wrought into three rows of guttae or drops, conical or otherwise shaped, each 
 row consisting of six guttae, or the same number as those beneath each triglyph. 
 The shaft of the Doric column was generally fluted ; the number of channels 
 is either sixteen or twenty, afterward increased in the other orders to twenty- 
 four, a centre flute on each side of the column. 
 
 Figs. 1595 to 1598 exhibit an example of the Ionic order, taken from the 
 Temple of Minerva Polias, at Athens. Fig. 1595 is an elevation of the capital 
 and entablature ; Fig. 1596, of the base ; Fig. 1597 is a sectional half of the 
 plan of the column at the base and the top; Fig. 1598 an elevation of the bal- 
 uster side of the capital. It differs from the Doric in the more slender pro- 
 portions of its shaft, and the addition of a base ; but the capital is the indi- 
 cial mark of the order. 
 
 When a colonnade was continued in front and along the flanks of the 
 building, this form of capital in the end column occasioned an offensive 
 irregularity; for while all the other columns on the flanks showed the volutes, 
 the end one showed the baluster side. It was necessary that the end column 
 should, therefore, have two adjoining volute faces, which was effected by plac- 
 ing the volute at the angle diagonally. 
 
 Figs. 1599 and 1600 represent an example of the Corinthian order, from 
 the Arch of Hadrian, at Athens. This order is distinguished from the Ionic 
 more by its deep and foliaged capital than by its proportions. The capital is 
 considerably more than a diameter in height, varying in different examples 
 from one to one and a half diameter, upon the average about a diameter and a 
 quarter, and has two rows of leaves, eight in each row, so disposed that of the 
 taller ones, composing the upper row, one comes in the middle, beneath each 
 face of the abacus, and the lower leaves alternate with the upper ones, coming 
 between the stems of the latter ; so that in the first or lower tier of leaves there 
 is in the middle of each face a space between two leaves occupied by the stem 
 of the central leaf above them. Over these two rows is a third series of eight 
 leaves, turned so as to support the small volutes which, in turn, support the 
 angles of the abacus. Besides these outer volutes, invariably turned diagonally, 
 there are two other smaller ones, termed catilicoli, which meet each other be- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 661 
 
 M 
 I 11 \ 
 
662 
 
 ARCHITECTURAL CONSTRUCTION. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 663 
 
 I 
 
664 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 1001. 
 
 neath a flower on the face of the abacus. The sides of the abacus are concave 
 in plan, being curved outward so as to produce a sharp point at each corner, 
 which is usually cut off. 
 
 Fig. 1601 represents one of the capitals of the Tower of the "Winds, showing 
 the earliest formation of the Corinthian capital. In this example the abacus 
 
 is square, and the upper row of leaves, of the kind 
 
 i ~7 called water-leaves, are broad and flat, and merely 
 
 carved upon the vase or body of the capital. 
 
 The shaft is, in general, fluted, similarly to that 
 of the Ionic column, but sometimes the flutes are 
 called ; that is, the channels are hollowed out for 
 only about two thirds of the upper part of the 
 shaft, and the remainder cut so that each channel 
 has the appearance of being partly filled up by a 
 round staff or piece of rope. 
 
 The cornice is very much larger than in the 
 other orders, in height and in projection, consisting 
 of a greater number of mouldings beneath the corona, for that and the cymati- 
 um over it are invariably the crowning members. In Fig. 1599 square blocks 
 or dentels are introduced, but often to the dentels is added a row of modillions 
 (Fig. 1719), immediately beneath, and supporting the corona ; and between 
 them and the dentels, and also below the latter, are other mouldings,' some- 
 times cut, at others left plain. 
 
 The Composite Order is a union of the Ionic and Corinthian orders. Its 
 capital consists of a Roman Ionic one, superimposed upon a Corinthian foli- 
 aged base, in which the leaves are without stalks, placed directly upon the body 
 of the base. 
 
 The spacing between the columns, or intercolumn, is from one to one and 
 one half diameters, but modern architects have coupled the columns, making 
 a wide intercolumn between every pair of columns, so that as regards the 
 average proportion between solids and voids, that disposition does not differ 
 from what it would be were the columns placed singly. Supercolumniation, 
 
 or the system of piling up orders, or different 
 
 stages of columns one above another, was em- 
 ployed for such structures merely as were upon 
 too large a scale to admit of the application of 
 columns at all as their decoration, otherwise 
 than by disposing them in tiers. 
 
 The Greeks seldom employed human figures 
 to support entablatures or beams ; the female 
 figures, or Caryatides, are almost uniformly rep- 
 resented in an erect attitude, without any ap- 
 parent effort to sustain any load ; while the 
 male figures, Telamones or Atlantes, display 
 strength and muscular action. Besides entire 
 figures, either Hermes pillars or Termini are 
 occasionally used as substitutes for columns of the usual form, on a moderate 
 scale. The first mentioned consist of a square shaft with a bust or human head 
 
ARCHITECTURAL CONSTRUCTION. 
 
 665 
 
 for its capital ; the latter of a -half-length figure rising out of, or terminating 
 in, a square shaft tapering downward. Hermes pillars are frequently employed 
 by modern architects for the decoration of window architraves. 
 
 The Romans introduced circular forms and curves, not only in elevation 
 and section, but in plan. The true Roman order consists, not in any of the 
 columnar ordinances, but in an arrangement of two pillars (Fig. 1602) placed 
 at a distance from one an- 
 other nearly equal to their 
 own height, and having a 
 very long entablature, 
 which, in consequence, re- 
 quired to be supported in 
 the centre by an arch 
 springing from piers. 
 
 Figs. 1603, 1604, and 
 1605, from the Palace of 
 Diocletian at Spalatro, are 
 illustrations of the differ- 
 ent modes of treatment of 
 the arch and entablature. 
 
 Perhaps the most sat- 
 isfactory works of the Ro- 
 mans are those which we 
 consider as belonging to 
 civil engineering rather 
 than to architecture 
 their aqueducts and via- 
 ducts, all of which, admi- 
 rably conceived and exe- 
 cuted, have furnished 
 practical examples for 
 modern constructions, of 
 which High Bridge across 
 Harlem River may be tak- 
 en as an illustration. 
 
 The history of Roman 
 architecture is that of a 
 style in course of transi- 
 tion, beginning with purely pagan or Grecian, and passing into a style almost 
 wholly Christian. The first form of Christian art was the Romanesque, which 
 afterward branched off into the Byzantine and the Gothic. 
 
 The Romanesque and Byzantine, as far as regards the architectural features, 
 are almost synonymous ; in the earlier centuries there is an ornamental distinc- 
 tion. In its widest signification, the Romanesque is applied to all the earlier 
 round-arch developments, in contradistinction to the Gothic or later pointed 
 arch varieties of the North. In this view the Norman is included in the Ro- 
 manesque. 
 
 The general characteristics of the Gothic are its essentially pointed or ver- 
 
 MiVMyvAwrv Y^ 
 
 FIG. 1604. 
 
666 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 1606. 
 
 FIG. 1607. 
 
 FIG. 1608. 
 
 FIG. 1609. 
 
 FIG. 1611. 
 
 FIG. 1612. 
 
 FIG. 1613. 
 
 FIG. 1615. 
 
 FIG. 1617. 
 
 FIG. 
 
 1618. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 667 
 
 tical tendency, its geometrical 'details, its window-tracery, its openings, its 
 cluster of shafts and bases, its suits of mouldings, the universal absence of the 
 dome, and the substitution of the pointed for the round arch. 
 
 The Romanesque pillars are mostly round or square, and, if square, gener- 
 ally set evenly, while the Gothic square pillar is set diagonally. 
 
 Figs. 1606 to 1610 represent sections of Gothic pillars. Fig. 1611 is half 
 of one of the great western piers of the Cathedral of Bourges, measuring 8 feet 
 on each side. Figs. 1612 and 1613 are elevations of capitals and bases, and sec- 
 tions of Gothic pillars ; one from Salisbury, the other from Lincoln Cathedral. 
 
 Figs. 1614, 1615, and 1616 are examples of Byzantine capitals; Fig. 1617 
 a Norman one, from Winchester Cathedral ; and Fig. 1618 a Gothic capital 
 and base, from Lincoln Cathedral. 
 
 Arches are generally divided into the triangular-headed arch, the round- 
 headed arch, and the pointed arch. Of the round-headed arch, there are 
 semicircular, segmental, stilted (Fig. 1619), and horseshoe (Fig. 1620). Of 
 
 FIG. 1619. 
 
 FIG. 1620. 
 
 FIG 1621. 
 
 the two-centred pointed, the equilateral (Fig. 1621), the lancet, and the ob- 
 tuse. Of the first, the radii of the segments forming the arch are equal to the 
 breadth of the arch, of those of the lancet longer, and of the obtuse shorter. 
 
 Of the complex arches, there are the ogee (Fig. 1622) and the Tudor (Fig. 
 1623). The Tudor arch is described from four centres, two on a level with the 
 spring and two below it. 
 
 Of foiled arches, there are the round-headed trefoil (Fig. 1624), the pointed 
 
 FIG. 1622. 
 
 FIG. 1623. 
 
 FIG. 1624. 
 
 FIG. 1625. 
 
 FIG. 1626. 
 
 trefoil (Fig. 1625), and the square-headed trefoil (Fig. 1626). The points c are 
 termed cusps. 
 
 The semicircular arch is the Roman Byzantine and Norman arch ; the ogee 
 and horseshoe are the profiles of many Turkish and Moorish domes ; the pointed 
 and foliated arches are Gothic. 
 
 Domes and Vaults. The Greek vaulting consisted wholly of spherical sur- 
 faces, the Roman of cylindrical ones. Figs. 1627 and 1628 illustrate this dis- 
 tinction, Fig. 1627 being the elevation of a Roman cylindrical cross-vault, and 
 Fig. 1G28 the elevation of the roof of the church of St. Sophia at Constantino- 
 ple ; and the sprouting of domes out of domes continues to characterize the 
 
668 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 Byzantine style. As a constructive expedient the cross- vault is to be preferred, 
 as the whole pressure and thrust are collected in four definite resultants, ap- 
 
 FIG. 1627. 
 
 KIG. 1C28. 
 
 plied at the angles only, so that it might be supported by four flying buttresses, 
 placed in the direction of the resultants, and strong enough to resist the 
 pressure. 
 
 Fig. 1629 represents a compartment of the simplest Gothic vaulting a, a, 
 groin ribs ; #, 5, #, side ribs. 
 
 The Romans introduced side ribs, appearing on the inside as flat bands, and 
 harmonizing with the similar form of pilasters in the walls, but they never used 
 groin ribs ; the Gothic builders introduced these, and deepened the Roman 
 ribs. The impenetration of vaults, either round or pointed, produces elliptical 
 groin lines, or else lines of double curvature ; yet the early Gothic architects 
 made their groin ribs usually simple pointed arches of circular curvature, 
 thrown diagonally across the space to be groined, and 
 the four side arches were equally simple, the only care 
 being that all the arches should have their vertices at 
 the same level. The strength depended on the ribs, 
 and the shell was made quite light, often not more 
 than six inches, while Roman vaults of the same 
 span would have been three or four feet. The Ro- 
 mans made their vault surfaces geometrically regu- 
 lar, and left the groins to take their chance ; while the 
 early Gothic architects made their groins geometri- 
 cally regular, and let the intermediate surfaces take their chance. 
 
 In the next step the groin ribs were elliptical, and when intermediate ribs 
 or tiercerons were inserted, these ribs had also elliptical or cylindrical curva- 
 tures, different from the groins, and the ribs were placed near each other, in 
 order that the portion of the vault between each pair might practically be 
 almost cylindrical. In the formation of the compound circular ribs three con- 
 ditions were to be observed : 1. That the two arcs should have a common tan- 
 gent at the point of meeting. 2. That the feet of all the ribs should have the 
 same radius, up to the level at which they completely separate from each other. 
 3. That from this point upward their curvature should be so adjusted as to 
 make them all meet their fellows on the same horizontal plane, so that all the 
 ridges of the vaults may be on one level. 
 
 The geometrical difficulty of such works led to what is called fan-tracery 
 vaulting. If similar arches spring from each side of the pillars (Fig. 1629), 
 the portion of vault springing from each pillar would have the form of an in- 
 verted concave-sided pyramid, its horizontal section at every level being square. 
 
 Flo. 1629. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 669 
 
 FIG. 1630. 
 
 The later architects, by converting this section into a circle, the four-sided 
 
 pyramid became a conoid, and all the ribs forming the conoidal surface became 
 
 alike in curvature, so that they all might be made simple circular arcs ; these 
 
 ribs are continued with unaltered curvature till they meet and form the ridge ; 
 
 but in this case the ridges are not level, but 
 
 gradually descend every way from the centre 
 
 point (Fig. 1630). 
 
 In the figure this is not fully carried out, 
 
 for no rib is continued higher than those over 
 
 the longer sides of the compartment, so that a 
 
 small lozenge is still left, with a boss at its 
 
 centre. When the span of the main arch ba 
 
 was large in proportion to that of b ( c, the arch 
 
 b c became a very acute lancet arch, scarcely 
 
 admitting windows of an elegant or sufficient 
 
 size. To obviate this, the compound curve 
 
 was again introduced. 
 
 The four-centred arch is not necessarily flat or depressed ; it can be made 
 
 of any proportion, high or low, and always with a decided angle at the vertex. 
 
 In general, the angular extent of the lower curve is not more than 65, nor less 
 
 than 45. The radius of the upper curve varies from twice to more than six 
 
 times the radius of the lower. The projecting points of the trefoil arch, or 
 
 cusps, are often introduced for ornament merely, but serve constructively, both 
 in vaults and arches, as a load for the sides, to prevent 
 them rising from the pressure on the crown. 
 
 As vaultings, in general, were contrived to collect the 
 whole pressure of each compartment into four single re- 
 sultants, at the points of springing, leaving the walls so 
 completely unloaded that they are required only as in- 
 closures or screens, they might be entirely omitted or re- 
 placed by windows. Indeed, the real supporting walls 
 are broken into narrow strips, placed at right angles to 
 the outline of the building, and called buttresses, and 
 the inclosing walls may be placed either at the outer or 
 inner edge of the buttresses. The first, that adopted by 
 the French architects, gave deep recesses to the interiors, 
 while the other, or English method, served to produce 
 external play of light and shade. 
 
 The Norman buttress (Fig. 1631) resembles a flat 
 pilaster, being a mass of masonry with a broad face, 
 slightly projecting from the wall. They are, generally, 
 of but one stage, rising no higher than the cornice, under 
 which they often, but not always, finished with a slope. 
 Sometimes they are carried up to, and terminate in, the 
 corbel table. 
 
 Fig. 1632 represents a buttress in two stages, with slopes as set-offs. 
 Fig. 1633 is a buttress of the early English style, having a plain triangular 
 
 or pedimental head. The angles were sometimes chamfered off, and sometimes 
 
670 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 ornamented with slender shafts. In buttresses of different 
 stages, the triangular head or gable is used as a finish for the 
 intermediate stages. 
 
 In the Decorated style, the outer surfaces of the buttresses 
 are ornamented with niches, as in Fig. 1634. In the Perpen- 
 dicular style the outer surface is often partially or wholly cov- 
 ered with panel-work tracery (Fig. 1635). 
 
 The buttress was a constructive expedient to resist the 
 thrust of vaulting; but to resist the thrust of the principal 
 vault, or that over the nave or central part of the church, but- 
 tresses of the requisite depth would have filled up the side 
 aisles entirely. To obviate this, the system of flying but- 
 
 FIQ. 1632. 
 
 Fio. 1633. 
 
 FIG. 1634. 
 
 FIG. 1635. 
 
 FIG. 1636. 
 
 Fio. 1637. 
 
 tresses was adopted ; that is, the connection of the interior with 
 the outer buttress, by an arch or system of arches, as shown in 
 Fig. 1636. The outer piers were surmounted by pinnacles, to 
 render them a sufficiently steady abutment to the flying arches. 
 The earlier towers of the Romanesque style were con- 
 structed without spires. All are square in plan, and extremely 
 similar in design. Fig. 1637 is an elevation of the tower at- 
 tached to the church of Sta. Maria, in Cosmedin, and is one of 
 the best and most complete examples of this style. It is 15 feet 
 broad and 110 feet high. These towers are the types of the 
 later Italian campaniles, generally attached to some angle of 
 churches ; if detached, so placed that they still form a part of 
 the church design. Sometimes they are but civic constructions, 
 _as belfries, or towers of defence. The campanile is square, 
 carried up without break or offset to two thirds, at least, of its 
 
ARCHITECTURAL CONSTRUCTION. 
 
 671 
 
 B 
 
 intended height ; it is generally solid to a consid- 
 erable height, or with only such openings as serve 
 to admit light to the staircases. Above this solid 
 part one round window is introduced in each 
 face ; in the next story, two ; in the one above 
 this, three ; then four, and lastly five ; the lights 
 being separated by slight piers, so that the upper 
 story is virtually an open loggia. 
 
 The Gothic towers have projecting buttresses, 
 frequent offsets, lofty spires, and a general pyra- 
 midal form. Fig. 1638 is the front elevation of 
 a simple English Gothic tower; here the plain 
 pyramidal roof, rising at an equal slope on each 
 of the four sides, is intersected by an octagonal 
 spire of steep pitch. The first spires were simple 
 quadrangular pyramids ; afterward the angles 
 were cut off, and they became octagonal, and this 
 
 FIG. 1C38. 
 
 Fio. 1639. 
 
 is the general Gothic form of spire. Often, in- 
 stead of intersecting the square roof, as in the 
 figure, the octagonal spire rests upon a square 
 base, and the angles of the tower are carried up 
 by pinnacles, or the sides by battlements, or by 
 both, as in Fig. 1639, to soften the transition be- 
 tween the perpendicular and sloping part. 
 
 In general the spires of English churches are 
 more lofty than those on the Continent,* the an- 
 gle at the apex in the former being about 10, 
 and in the latter about 15. The apex angle of 
 the spires of Chichester and Lichfield are from 
 12 to 13, or a mean between the two propor- 
 tions, and, according to Ferguson, more pleasing 
 than either. Although having more lofty spires, 
 yet the English construction is much more mas- 
 sive in appearance than the Continental ; the 
 apertures are less numerous, and the surfaces are 
 
ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 1642. FIG. 1640. FIG. 1643. FIG. 1641. 
 
 Fio. 1645. 
 
 Fio. 1646. FIG. 1647. 
 
 FIG. 1644. 
 
 FIG. 1648. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 673 
 
 less cut up and covered with ornaments. The spires of Freiberg Church, and 
 many others on the Continent, are open work. 
 
 Figs. 1640 and 1641 are bell-cots. Figs. 1642 to 1648 are spires. Fig. 1649 
 is an apse, or circular end of a church, from German Gothic examples. 
 
 FIG. 1653. 
 
 FIG. 1654. 
 
 Figs. 1650 and 1651 are examples of spire finials, with weather-cocks. 
 Figs. 1652 and 1653 are examples of towers not connected with church 
 edifices. 
 
 Fig. 1654 is a tower of very recent construction, and is applied to the utili- 
 
 44 
 
674 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 tarian purpose of sustaining a water-tank for the highest service of the Croton 
 in New York city. 
 
 Fig. 1655 represents the upper portion of the tower of Ivan Veliki, at Mos- 
 cow. The Eussian towers are generally constructed independent of their 
 churches, and are intended for the reception of their massive bells. 
 
 Windows. Before the use of painted glass, as very 
 small apertures sufficed for the introduction of the re- 
 quired quantity of light into a church, the windows of 
 the Eomanesque churches were generally small, and de- 
 void of tracery ; and as the Byzantine architects, adorn- 
 ing the walls with paintings, could not use stained glass, 
 they followed in general form the Eomanesque window, 
 apertures with circular heads, either single or in groups 
 (Fig. 1656 or Fig. 1657). The Norman windows were 
 also small, each consisting of a single light, semicircular 
 
 FIG. 1655. 
 
 FIG. 1656. 
 
 FIG. 1657. 
 
 in the head, and placed as high as possible above the ground ; at first splayed 
 on the inside only, afterward the windows began to be recessed with mould- 
 ings and jamb-shafts in the angles, as in Fig. 1657. 
 
 The Lancet, in general use in the early Gothic period, was of the simplest 
 arrangement : in these windows the glass was brought within three or four 
 inches of the outside of the wall, and the openings were widely splayed in the 
 interior. The proportions of these windows vary considerably, in some the 
 height being but five times the width, in others as much as eleven; eight or 
 nine times may be taken as the average. Lancet windows occur singly (Fig. 
 1658), or in groups of two, three, five, and seven, rarely of four and six. The 
 triplet (Fig. 1659) is the most beautiful arrangement of lancet windows. It 
 was customary to mark with greater importance the central light, by giving it 
 additional height, and in most cases increased width also. In some examples 
 the windows of a lancet triplet are placed within one drip-stone forming a 
 single arch, thus bearing a strong resemblance to a single three-light window. 
 The first approximation to tracery appears to have been the piercing of the 
 space over a double lancet window comprised within a single drip-stone (Fig. 
 1660). 
 
 A traceried window is a distinctive characteristic of Gothic architecture; 
 with the establishment of the principle of window tracery the mullions were 
 recessed from the face of the wall in which the window arch was pierced, and 
 
ARCHITECTURAL CONSTRUCTION. 
 
 675 
 
 the fine effect thus produced was speedily enhanced by the introduction of dis- 
 tinct orders of mullions, and by recessing certain portions of the tracery from 
 the face of the primary mullions and their corresponding tracery bars. 
 
 FIG. 1659. 
 
 FIG. 1658. 
 
 FIG. 1660. 
 
 Examples of window tracery, showing its constructive centres and lines, are 
 given on pages 81 and 82, illustrating the chief varieties. The following figures 
 are more complete in position 
 and with architraves, Geomet- 
 
 
 
 FIG. 1601. 
 
 FIG. 1662. 
 
 rical and Flowing ; the former consisting of geometrical figures, as circles, 
 trefoils, quatrefoils, curvilinear triangles, lozenges, etc. ; while in flowing tracery 
 these figures, though still existing, are gracefully blended together in one de- 
 sign. 
 
 Fig. 1661 represents an example of the earlier decorated tracery window- 
 head, consisting of two foiled lancets, with a pointed quatrefoil in the spandrel 
 between them. One half of the windows in this, as in some of the following 
 figures, is drawn in skeleton, to explain their construction. Fig. 1662 is an- 
 other example of Decorated tracery. 
 
 Fig. 1663 is an example of the English leaf tracery; Fig. 1664, of the 
 .French flamboyant. The difference between the two styles is, that while the 
 upper ends of the English loops or leaves are round, or simply pointed, the 
 upper ends of the latter terminate, like their lower ones, in angles of contact, 
 giving a flamelike form to the tracery bars and form pieces. 
 
676 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 In England the Perpendicular style succeeded the Decorated ; the mullions, 
 instead of diverging in flowing or curvilinear lines, are carried up straight 
 through the head of the windows; smaller 
 mullions spring from the head of the prin- 
 
 FIG. 1663. 
 
 FIG. 1664. 
 
 FIG. 1665. 
 
 cipal lights, and thus the upper portion of the window is filled with panel-like 
 compartments. The principal as well as the subordinate lights are foliated in 
 their heads, and large windows are often divided horizontally by transoms. 
 The forms of the window arches vary from simple pointed to the complex 
 four-centred, more or less depressed. 
 
 Fig. 1665 ,is an example of a Perpendicular window. 
 
 Fig. 1666 is a square-headed window, such as were usual in the clear-stories 
 of Perpendicular architecture. 
 
 Figs. 1667 and 1668 are quadrants of circular windows, used more espe- 
 
 FIG. 1666. 
 
 FIG. 1667. 
 
 FIG. 1668. 
 
 FIG. 1669. 
 
 cially in France for the adornment of the west 
 ends and transepts of the cathedrals. 
 
 Besides the tracery characteristic of Gothic 
 architecture, there is a tracery peculiar to the 
 Saracenic and Moorish style, of which Fig. 
 1669 may be taken as an example it being a 
 window of one of the earliest mosques. The 
 general form of the window and door-heads of 
 this style is that of the horse-shoe, either cir- 
 cular or pointed. 
 
 Doorways. Fig. 1670 is the elevation of a 
 circular-headed doorway, which may be con- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 sidered the type of many entrances both in Romanesque, Gothic, and later 
 styles. It consists of two or more recessed arches, with shafts or mouldings 
 in the jambs. In the earlier styles the 
 arches were circular, in the later Gothic, 
 
 FIG. 1671. 
 
 h 
 
 FIG. 1670. 
 
 FIG. 1672. 
 
 generally pointed, but sometimes circular ; in the earlier, the angles in which 
 the shafts are placed are rectangular ; in the latter, the shaft is often moulded 
 on a chamfer plane that is, a plane inclined to the face of the wall, generally 
 at an angle of 45 ; often the chamfer and rectangular plane are used in con- 
 nection. 
 
 Fig. 1671 is a simple head of a depressed four-centred or Tudor-arched 
 doorway, with a hood moulding. 
 
 Fig. 1672 represents the incorporation of a window and doorway. Some- 
 times the doorway pierces a buttress; in that case, the buttress expands on 
 either side, forming a sort of porch. The Gothic architects placed doors 
 where they were necessary, and made them subservient to the beauty of the 
 design. 
 
 Fig. 1673 is an example of a gabled doorway with crockets and finial. 
 
 Fig. 1674 is an example of a perpendicular doorway, with a label or hood 
 moulding above, and ornamented spandrels. 
 
 Fig, 1675 is an example of a Byzantine, and Fig. 1676 of a Saracenic 
 doorway. 
 
 The Renaissance style was, originally, but the revival or a fair rendering of 
 the classical orders of architecture, with ornaments from the Byzantine and 
 Saracenic styles. 
 
 Garbett divides this style into three Italian schools, the Florentine, Vene- 
 tian, and Roman. The, Florentine admits of little apparent ornament, but 
 any degree of real richness, preserving in its principal forms severe contrast; 
 powerful masses self-poised without corbeling, without arching ; breadth of 
 
678 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 everything, of light, of shade, of ornament, of plain wall ; depth of recess in 
 the openings, of perspective in the whole mass, of projection in the cornice. 
 
 FIG. 1(573. 
 
 TIG. 1674. 
 
 Absence of features useless to convenience or stability, admitting of great 
 plainness, or of very florid enrichment. 
 
 The aim of the Venetian school was splendour, variety, show, and orna- 
 ment-; not so much real as effective ornament. Thus, it rarely contains as 
 
 Fio. 1675. 
 
 icre. 
 
 much carving or minute enrichment as the Florentine admits ; but it has 
 larger ornaments, constructed (or built) ornaments, great features useless 
 except for ornament, such as inaccessible porticoes, detached columns, and 
 architraves supporting no ceiling, towers built only for breaking an outline. 
 
 The Roman school is intermediate in every respect between the two other 
 schools. It is better adapted to churches than to any other class of buildings. 
 This fitness arises from the grand, simple, and unitary effect of one tall order, 
 
ARCHITECTURAL CONSTRUCTION. 
 
 679 
 
 generally commencing at or near the ground, obliterating the distinction of 
 two or three stories, making a high building appear a single story. 
 
 Mouldings. " All classical and Romanesque architecture is composed of 
 bold independent shafts, plain or fluted, with bold detached capitals forming 
 arcades or colonnades where they are needed, and of walls whose apertures are 
 surrounded by courses of parallel lines called mouldings, and have neither 
 shafts nor capitals. The shaft system and moulding system are entirely sepa- 
 rate ; the Gothic architects confounded the two ; they clustered the shafts till 
 they looked like a group of mouldings, they shod and capitalled the mouldings 
 till they looked like a group of shafts." The mouldings appear in almost every 
 conceivable position ; from the bases of piers and piers themselves, to the ribs 
 of the fretted vaults which they sustain. 
 
 In the earliest examples of Norman doorways the jambs are mostly simply 
 squared back from the walls ; recessed jambs succeeded, and are common in 
 both Norman and Gothic architecture ; and when thus re- 
 cessed, detached shafts were placed in each angle (Fig. 
 1677). In the later styles the shafts were almost invariably 
 attached to the structure. The angles themselves were 
 often cut or chamfered off, and the mouldings attached to 
 the chamfer-plane. The arrangement of window jambs, 
 during the successive periods, was in close accordance with 
 that of doorways. 
 
 In the richer examples small shafts were introduced, 
 which, rising up to the springing of the window, carried FIG. 1677. . 
 
 one or several of the arch mouldings. Yet mouldings are 
 nevertheless not essential accessories ; many windows of the richest tracery 
 have their mullions and jambs composed of simple chamfers. 
 
 Figs. 1678 to 1686 are examples of arch and architrave mouldings, which, 
 
 FIG. 1680 
 
 FIG. 1682. 
 
680 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 FIG. 1683. 
 
 FIG. 1684. 
 
 FIG. 1685. 
 
 FIG. 1686. 
 
 even when not continuous, partook of the same general 
 arrangement as those in the jambs, with greater rich- 
 ness of detail. When shafts were employed, they car- 
 ried groups of mouldings more elaborate than those 
 of the jambs, though still falling on the same planes. 
 
 Capitals were either moulded or carved with foliage, 
 animals, etc. ; they always consisted of three distinct 
 parts (Fig. 1687) the head mould (A), the bell (B), and the neck mould (C). 
 In Norman examples the head mould was almost invariably square; in the 
 later styles it is circular, or corresponding to the form of the 
 pillar. 
 
 Bases consist of the plinth and the base mouldings. The 
 plinth was square in the Norman style, afterward octagonal ; 
 then, assuming the form of the base mouldings, it bent in 
 and out with the outline of the pier. Base mouldings were 
 also extensively used round the buttresses, towers, and walls 
 of churches. 
 
 String Courses, of which Figs. 1688 to 1693 are exam- 
 ples, were horizontal courses in the face of a wall, the most 
 usual position being under the windows. In the Norman styles they were usu- 
 ally heavy in the outline ; in the later styles they were remarkably light and 
 elegant ; free from restraint or horizontality, they now rose close under the sill 
 
 FIG. 1691. 
 
 FIG. 1692. 
 
 FIG. 1693. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 681 
 
 of the window, and then suddenly dropping to accommodate themselves to the 
 arch of a low doorway, and again .rising to run immediately under the adjoin- 
 ing window. In this way the string courses frequently served the purpose of a 
 drip-stone or hood moulding over doors ; occasionally the hood mould was con- 
 tinued from one window to the other. 
 
 Cornices are not an essential feature in Gothic architecture. In the Nor- 
 man and early English styles the cornice was a sort of enlarged, 
 projecting string course, forming a drip-stone beneath the roof, 
 which, if supported on brackets or corbels, was termed the corbel 
 table. 
 
 The earliest moulding in Norman work is a circular bead 
 strip, worked out of the edges of a recessed arch, called a circu- 
 lar boiotel (Fig. 1694). From a circular form the bowtel soon be- 
 came pointed, and, by an easy transition, into the bowtel of one, two, or three 
 fillets. 
 
 Figs. 1695 to 1700 are sections of Komanesque drip- or cap-stones, adapted 
 to different pitches of roof. 
 
 Fig. 1701 is the scroll moulding ; a simple filleted bowtel, with the fillet 
 
 FIG. 1694. 
 
 FIG. 1C99. 
 
 FIG. 1700. 
 
682 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 undeveloped on one side, as shown by the dotted lines. If this moulding be 
 cut in half, through the centre of the fillet, we have on the developed side the 
 moulding now termed by carpenters the 
 rule joint, which, by rounding off the 
 corners by reverse curves, becomes the 
 wave moulding. 
 
 FIG. 1702. 
 
 FIG. 1703. 
 
 FIG. 1701. 
 
 Fig. 1702 is a Gothic example of the 
 filleted bowtel with prominent alternate 
 hollows. 
 
 Fig. 1703 is an example of the perpendicular style, an insignificant hollow 
 separating groups of mouldings. 
 
 Figs. 1704 to 1709 are examples of moulded timbers, used largely in open- 
 timbered roofs and for exposed beams. It is still the custom, when the fram- 
 ing is not covered in with plastering or ceiling, to corner the edges of the joists 
 and beams, at an angle of 45, for about 1* on each face, but not extending 
 close to the joint or wall ; this is called stop-chamfering. 
 
 FIG. 1704. 
 
 FIG. 1705. 
 
 FIG. 1706. 
 
 FIG. 1707. 
 
 FIG. 1708. 
 
 FIG. 1709. 
 
 Ornament. Architectural ornament is of two kinds, constructive and deco- 
 rative. By the former is meant all those contrivances, such as capitals, brack- 
 ets, vaulting-shafts, and the like, which serve to explain or give expression to 
 the construction ; by the latter, such as mouldings, frets, foliage, etc., which 
 give grace and life, either to the actual constructive form, or to the construe- 
 
ARCHITECTURAL CONSTRUCTION. 
 
 683 
 
 tive decoration. Mouldings of the different styles have been already treated of ; 
 it is proposed to give now what are even more purely decorations of a style. 
 
 In the Grecian orders the Doric (Fig. 1590) has the triglyph mutules and 
 guttae; the Ionic (Fig. 1596) has various mouldings of the cornice, frieze, 
 abacus, and neck of the column enriched. The principal ornament of the 
 neck of the column is the anthemion, commonly known, in its most simple 
 form, as the honeysuckle or palmetto ; in the anthemion, as represented in the 
 figure, the palmetto alternates with the lily or some analogous form. The 
 ornament of the abacus is the egg and dart (Fig. 1710) ; the ornament of the 
 
 FIG. 1710. 
 
 FIG. 1711. 
 
 frieze and cornice (Fig. 1711). The fret (Fig. 1712) and the guilloche (Fig. 
 1713) are also common Greek ornaments, used to adorn the soffits of beams 
 and ceilings. The acanthus is the distinctive ornament of the Corinthian, of 
 which a leaf is represented in front and side view (Figs. 1714 and 1715). 
 
 
 Fio. 1712. 
 
 FIG. 1713. 
 
 FIG. 1714. 
 
 FIG. 1715. 
 
 Figs. 1716, 1717, and 1718 are the side elevation, front elevation, and sec- 
 tion of a Greek bracket, the principal ornaments of which are taken from the 
 anthemion and acanthus. 
 
 Fig. 1719 is an elevation of a portion of an enriched cornice from the 
 temple of Jupiter Stator, at Rome, of the Corinthian order of architecture. 
 Fig. 1720 is the under side of the modillion, on a larger scale. 
 
 The chief characteristic of Roman ornament is its uniform magnificence, an 
 enrichment of the Greek. The most used elements of the Roman decorations 
 are the scroll and the acanthus. The acanthus of the Greeks is the narrow 
 prickly acanthus ; that of the Roman, the soft acanthus. For capitals the 
 Roman acanthus is commonly composed of conventional clusters of olive-leaves. 
 Fig. 1721 represents a Roman acanthus scroll. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 The free introduction of monsters and animals is likewise a characteristic 
 of Greek and Roman ornament, as the sphinx, the triton, the griffin, and 
 others ; they occur, however, more abundantly in the Roman. 
 
 FIG. 17 
 
 FIG. 1717. 
 
 Symbols are the foundation of decorations in the Byzantine and Romanesque. 
 The early symbols were the monogram of Christ, the lily, the cross, the ser- 
 pent, the fish, the aureole, or vesica piscis, and the circle or nimbus, the trefoil 
 and quatrefoil, the first having reference to the Trinity, the second to the 
 
 four Evangelists. Occasionally the 
 symbolic images of the Evangelists, the 
 angel, the lion, the ox, and the eagle, 
 
 FIG. 1719. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 685 
 
 are represented within these circles. The hand in the attitude of benediction, 
 and the lily (the fleur-de-lis), the emblem of the virgin and purity, are com- 
 
 FIG. 1721. 
 
 mon ; also a peculiarly formed leaf, somewhat resembling the leaf of the ordi- 
 nary thistle. The serpent figures largely in Byzantine art as the instrument of 
 the fall, and one type of the redemption. 
 
 FIG. 1722. 
 
 Pagan ornaments, under certain symbolic modifications, were admitted into 
 Christian decorations. Thus the foliations of the scroll were terminated by 
 lilies, or by leaves of three, four, and five blades, the number of blades being 
 significant; and in a similar way the anthemion and every other ancient orna- 
 ment. In the Byzantine, all their imitations of natural forms were invariably 
 conventional ; it is the same even with animals and the human figure ; every 
 saint had his prescribed colours, proportions, and symbols. 
 
 The Saracenic was the period of gorgeous diapers (Figs. 1722 and 1723), for 
 their habit of decorating the entire surfaces of their apartments was highly 
 favourable to the development of this class of design. The Alhambra displays 
 almost endless specimens, and all are in relief and enriched with gold and colour, 
 chiefly blue and red. The religious cycles and symbolic figures of the By- 
 zantine are excluded. Mere curves and angles or interlacings were now to bear 
 the chief burden of a design, but distinguished by a variety of colour. The 
 curves, however, very naturally fell into standard forms and floral shapes, and 
 
686 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 the lines and angles were soon developed into a very characteristic species of 
 tracery, or interlaid strap- work, very agreeably diversified by the ornamental 
 
 FIG. 1733. 
 
 introduction of the inscriptions, which last custom of elaborating inscriptions 
 with their designs was peculiarly Saracenic. Although flowers were not palpa- 
 bly admitted, yet the great mass of the minor details of Saracenic designs are 
 composed of flower forms disguised the very inscriptions are sometimes thus 
 grouped as flowers ; still, no actual flower ever occurs, as the exclusion of all 
 natural images is fundamental to the style in its purity. 
 
 All the symbolic elements of the Byzantine are continued in the Gothic. 
 Ornamentally, the Gothic is the geometrical and pointed element elaborated to 
 the utmost ; its only peculiarities are its combinations of details ; at first the 
 conventional and geometrical prevailing, and afterward these combined with 
 the elaboration of natural objects in its decoration. The most striking feature 
 of all Gothic work is the wonderful elaboration of its geometric tracery ; vesi- 
 cas, trefoils, quatrefoils, cinquefoils, and an infinity of geometric varieties be- 
 sides. The tracery is so paramount a characteristic that the three English 
 varieties, the early English, the decorated, and the perpendicular, and the 
 French flamboyant, are distinguished almost exclusively by this feature. (See 
 Figs. 1661 to 1665.) 
 
 The ornamental mouldings used in the decorative details are numerous, 
 among which the more common is the chevron or zigzag (Fig. 1724), simple 
 as the indented, or duplicated, triplicated, or quadrupled ; the billet, the pris- 
 matic billet, the square billet, and the alternate billet (Fig. 1725) ; the star 
 (Fig. 1726), the fir-cone; the cable (Fig. 1727); the embattled (Fig. 1728); 
 the nail-head (Fig. 1729) ; the dog-tooth (Fig. 1730) ; the ball-flower (Fig. 
 1731) ; and the serpentine vine-scroll. 
 
 Fio. 1724. 
 
 FIG. 1725. 
 
 FIG. 17^6. 
 
 The crocket, in its earliest form, was the simple arrow-head of the episco- 
 pal pastoral staff ; subsequently finished with a trefoil, and afterward still fur- 
 ther enriched. Figs. 1732 and 1733 are early English crockets; Fig. 1734 a 
 decorated one. Fig. 1735 is a finial of the same style. Both finials and crock- 
 ets in detail display a variety of forms. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 687 
 
 The parapets of the early English style are often a simple horizontal course, 
 supported by a corbel table, sometimes relieved by a series of sunk blank trefoil- 
 
 FIG. 1728. 
 
 FIG. 1727. 
 
 FIG. 1729. 
 
 headed panels ; sometimes 
 a low embattled parapet 
 crowns the wall. In the 
 decorated style the hori- 
 zontal parapet is some- 
 times pierced with trefoils, sometimes with wavy, flowing tracery (Fig. 1736). 
 Grotesque spouts or gargoyles discharge the water from the gutters. The para- 
 
 FIG. 1730. 
 
 FIG. 1731. 
 
 FIG. 1732. 
 
 FIG. 1734. 
 
 FIG. 1733. 
 
 FIG. 1735. 
 
 pets of the perpendicular style are frequently embattled (Fig. 1737), covered 
 with sunk or pierced panelling, and ornamented with quatrefoil, or small tre- 
 
 FIG. 1737. 
 
 FIG. 1736. 
 
 foil-headed arches ; sometimes not em- 
 battled but covered with sunk or 
 pierced quatrefoils in circles, or with 
 trefoils in triangular spaces, as in Fig. 
 
 m 
 
 FlG - ir38> 
 
 Among the varieties of ornamental 
 work, the mode of covering small plain surfaces with diapering (Fig. 1739) 
 was sometimes used, the design being in exact accordance with the architec- 
 
688 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 tural features and details of the style. The rose (Fig. 1740), the badge of the 
 houses of York and Lancaster, is often met with in the perpendicular style ; 
 
 FIG. 1740. 
 
 FIG. 1739. 
 
 and tendrils, leaves, and fruit of the vine are carved in great profusion in the 
 hollows of rich cornice mouldings, especially on screen-work in the interior of 
 a church. Fig. 1741, in its original type a Byzantine ornament, an alternate 
 lily and cross, is a common finish to the cornice of rich screen-work in the lat- 
 est Gothic, and is known under the name of the Tudor flower. 
 
 Sculptured foliage (Figs. 1742 to 1747) 
 is much used in capitals, brackets, corbels, 
 bosses, and crockets. Among the forms of 
 foliage the trefoil is most predominant. 
 
 The Ornaments of the Renaissance. The 
 term Eeuaissance is used in a double sense; 
 Flo 1741 in a general sense implying the revival of 
 
 art, and specially signifying a peculiar style 
 
 of ornament. It is also sometimes, in a very confined sense, applied in refer- 
 ence to ornament of the style of Benvenuto Cellini ; or, as it is sometimes des- 
 ignated, the Henry II (of France) style. 
 
 The mixture of various elements is one of the essentials of this style. These 
 
 Fio. 1742. 
 
 FIG. 1743. 
 
 FIG. 1744. 
 
 FIG. 1745. 
 
 FIG. 1? 
 
 FIG. 1747. 
 
ARCHITECTURAL CONSTRUCTION. 
 
 689 
 
 elements are the classical ornaments ; unnatural and natural flowers and foil- 
 asre; men and animals, natural and grotesque; cartouches, or pierced and 
 
 shields, in great promi- 
 
 scrolled 
 
 nence ; tracery independent, 
 
 and 
 
 developed from the scrolls of the 
 cartouches , and jewel forms (Figs. 
 1748 and 1749). 
 
 The Elizabethan is a partial 
 elaboration of the same style; the 
 present Elizabethan exhibits a very 
 striking preponderance of strap and 
 shield work ; but the earlier is 
 much nearer allied to the Continen- 
 tal styles of the time, classical orna- 
 ments but rude in detail, occasional 
 scroll and arabesque work, and strap- 
 work, holding a much more promi- 
 nent place than the pierced or 
 scrolled shields. Fig. 1750 is an ex- 
 ample of the style from the old 
 guard chamber, Westminster. 
 
 Of the earliest and transition 
 
 styles of Renaissance ornament are the Tricento and the Quatrecento. The 
 great features of the first are its intricate tracery and delicate scroll-work of 
 conventional foliage, the style being but a slight remove from the Byzantine 
 and Saracenic ; of the second, elaborate natural imitations of fruit, flowers, 
 birds, or animals (Fig. 1751), all disposed simply with a view to the ornamen- 
 tal ; also occasional cartouches, or scrolled shield-work. 
 
 FIG. 1743. 
 
 FIG. 1749. 
 
 The Renaissance is something more approximative to a combination of pre- 
 vious styles than a revival of any in particular, developed solely on aesthetic 
 principles, from a love of the forms and harmonies themselves, as varieties of 
 effect and arrangements of beauty, not because they had any particular signi- 
 fication, or from any superstitious attachment to them as heirlooms. 
 
 Fig. 1752 is an example of ornament in the Cinquecento style. The ara- 
 besque scroll-work is the most prominent feature of the Cinquecento, and with 
 this in its elements, it combines every other feature of classical art, with the 
 unlimited choice of natural and conventional imitations from the entire animal 
 45 
 
690 
 
 ARCHITECTURAL CONSTRUCTION. 
 
 and vegetable kingdom, both arbitrarily disposed and combined. Absolute 
 works of art, such as vases and implements, and instruments of all kinds, are 
 prominent elements of 
 the Cinquecento ara- 
 besque, but cartouches 
 and strap-work wholly 
 
 FIG. 1750. 
 
 FIQ. 1751. 
 
 FIG. 1752. 
 
 disappear from the best examples. Another chief feature of the Cinquecento 
 is the admirable play of colour in its arabesques and scrolls ; and it is worthy 
 
 of note that the three secondary colours 
 orange, green, and purple perform the chief 
 parts in all the coloured decorations. 
 
 Fig. 1753 is an example of the Louis 
 Quatorze style of ornament. The great medi- 
 um of this style was gilt stucco-work, and 
 this absence of colour seems to have led to its 
 most striking characteristic, infinite play of 
 light, of shade ; colour, or mere beauty of form 
 in detail, having no part in it whatever. Flat 
 surfaces are not admitted ; all are concave or 
 convex : this constant varying of the surface 
 gives every point of view its high lights and 
 brilliant contrasts. 
 
 The Louis Quinze style differs from that 
 of Louis Quatorze chiefly in its absence of 
 symmetry ; in many of its examples it is an 
 almost random dispersion of the scroll and 
 shell, mixed only with that peculiar crimping of shell-work, the coquillage. 
 
 FIG. 1753. 
 
ARCHITECTURAL CONSTRUCTION. 691 
 
 The ornaments of which we have thus given examples are, in general, 
 applied to interior decorations, to friezes, pilasters, panels, architraves, the 
 faces and soffits of arches, ceilings, etc., to furniture, and to art-manufactures 
 in general. For exteriors these ornaments are sparingly applied ; shield and 
 scroll-work, of the later Elizabethan or Eenaissance style, is sometimes used, 
 but very seldom tracery. 
 
 Principles of Design. Professedly treating of architecture only in its most 
 mechanical phase of drawing, the history of it as an art, and the distinctions 
 of styles, have been but briefly treated. To one anxious to acquire knowledge 
 in this department we refer, as the very best compendium within our knowl- 
 edge, to Ferguson's " Hand-Book of Architecture." The study of this work 
 will give direction to a person's observation, but, without referring to actual 
 examples, mere reading will be of little use. Drawings give general ideas of 
 the character of buildings, but no idea of size or of the surroundings of a 
 building. Many a weak design, especially in cast-iron buildings, acquires a 
 sort of strength by the number of its repetitions, giving an idea of extent ; and 
 many a beautiful design on paper has failed in its execution, being dwarfed by 
 its surroundings. With regard to the style of a building, there are none of the 
 ancient styles in their purity adapted to present requirements ; our churches 
 and theatres are more for the gratification of the ear than the eye, and the 
 comforts of our domestic architecture, and the requirements of our stores and 
 warehouses, are almost the growth of the present century. For a design, look 
 first to the requirements of the structure, the purposes to which it is to be 
 applied ; sketch the plan first, arrange the divisions of rooms, the openings for 
 doors and windows, construct the sections, and then the elevations, first in 
 plain outline ; modify each by the exigencies of construction. 
 
 " Construction, including in the term the disposition of a building in ref- 
 erence to its uses, is by some supposed to be the common part of the art of 
 architecture, but it is really the bone, muscle, and nerve of architecture, and 
 the arts of construction are those to which the true architect will look, rather 
 than to rules and examples, for the means of producing two at least of the 
 three essential conditions of building well, commodity, firmness, and delight, 
 which conditions have been aptly said to be the end of architecture as of all 
 creative arts. 
 
 " The two great principles of the art are : First, that there should be no 
 features about a building which are not necessary for convenience, construc- 
 tion, or propriety ; second, that all ornament should consist of enrichment of 
 the essential construction of the building. 
 
 " The neglect of these two rules is the cause of all the bad architecture of 
 the present time. Architectural features are continually tacked on buildings 
 with which they have no connection, merely for the sake of what is termed 
 effect, and ornaments are continually constructed instead of forming the deco- 
 ration of construction to which in good taste they should always be subservient. 
 The taste of the artist ought to be held merely ancillary to truthful disposition 
 for structure and service. The soundest construction is the most apt in the 
 production, or the reproduction, it may be, of real art. The Eddystone Light- 
 house is well adapted to its uses ; it is commodious, firm and stable almost to a 
 miracle, and its form is as beautiful in outline to the delight of the eye as it is 
 
(592 ARCHITECTURAL CONSTRUCTION. 
 
 well adapted to break and mitigate the force of the sea in defence of its own 
 structure. The English Exhibition Building of 1851 was most commodious for 
 the purposes of an exhibition, firm enough for the temporary purpose required 
 of it, and there was delight in the simplicity and truth of its combinations ; and 
 all this may be said to have grown out of propriety of construction, as applied 
 to the material, cast-iron. The use of unfitting material, or fitting material 
 inappropriately, leads almost entirely to incommodiousness, infirmity, and 
 offence, or some of them. 
 
 " Out of truth in structure, and that structure of a very inartificial sort, 
 grow the beautiful forms of the admirable proportions found in the works of 
 the Greeks ; and out of truth in structure, with the strictest regard to the ne- 
 cessities of the composition and of the material employed, and that structure 
 as full of artifice as the artifice employed is of truth and simplicity, grew the 
 classical works vulgarly called Gothic, but now characteristically designated as 
 Pointed, from the arch which is the basis of the style. Structural untruth is 
 not to be justified by authority ; neither Sir Christopher Wren, nor the Athe- 
 nian exemplars of Doric or Ionic in the Propylaeum and in the Minerva Polias, 
 with their irregular and inordinately wide intercolumniation, can persuade 
 even the untutored eye to accept weakness for strength, -or what is false for 
 truth. 
 
 " The Greek examples offer the most beautiful forms for mouldings, and 
 the Grecian mode of enriching them is unsurpassed. It should be borne in 
 mind that the object in architectural enrichment is not to show ornament, but 
 to enrich the surface by producing an effective and pleasing variety of light and 
 shade; but still, although ornament should be a secondary consideration, it 
 will develop itself, and therefore should be of elegant form and composition." 
 
 We have quoted thus at some length from the article "Architecture," 
 " Encyclopaedia Britanuica," because with many authority is necessary, and 
 they distrust their own powers of observation and analysis ; all must feel the 
 truth of the above, but in practice it is very little appreciated or carried out. 
 The present taste in architecture, as in the theatre, is for the spectacular; 
 breadth or dignity of effect is not popular ; edifices are not only covered with, 
 but built up in ornament ; and construction is but secondary. The French, 
 having a building-stone that is very easily worked, cut merely the joints, leav- 
 ing the rough outer surface to be worked after it is laid ; chopping out mould- 
 ings and ornaments almost as readily as though it were in plaster, and the sur- 
 face when finished is covered with enrichments in low relief. The fashion thus 
 set is imitated in this country at immense cost, in the most unfitting materials, 
 marble and granite. Our architectural buildings express fitly our condition 
 a rich country, recent and easily acquired wealth, and a desire and rivalry to 
 exhibit it, or a display as a means of advertising, and in this truth of expression 
 will have an archaeological interest; although it does not contribute much to 
 present excellence in construction, it still has this value : that the architect or 
 constructor need be governed by no rules or principles he can make experi- 
 ments on a pretty extensive scale, and out of much bad construction even forms 
 and ornament may spring up which will stand the test of time, and form a 
 nucleus of a new style adapted to the present wants. 
 
 Cast-iron as a building material, with the exception of exhibition buildings, 
 
ARCHITECTURAL CONSTRUCTION. (593 
 
 has seldom been treated distinctively ; buildings erected with it have been 
 copies of those in stone, and have been even imitated in colour. For the first 
 story of stores, where space is necessary for light and the exhibition of wares, 
 cast-iron columns are almost invariably used, but are objected to architecturally, 
 that they look too weak for the support of the piles of brick and stone above 
 them. The objection should not be to the use, but that the truth of the ade- 
 quate strength of the cast-iron is not conveyed by the form or col^- 1 
 one objects that the ankles of Atlas look too light to support the massive ngure 
 and globe, or wishes him seated to give the idea of stability ; so if the columns 
 and lintels were some other form than Greek or Roman with immense inter- 
 columniations, and coloured fitly, the appearance of weakness would be entirely 
 lost sight of. 
 
 Improvements in the manufacture of iron and steel have led up to the 
 skeleton construction (page 565) frames of cast or rolled iron and steel framed 
 and set before any of the nr onry except that of the foundations is laid. All 
 the columns, girders, and Learns are bolted together, built into, and covered 
 with masonry to add to the rigidity of the structure and for protection against 
 fire. The framing is square, but, for variety and design, arches, soffits, and orna- 
 mental clothing is made in masonry. Little in exterior form can be considered 
 a necessity of construction, and there is as yet no standard of finish. The 
 function of the metal, like the bone in the animal structure, is to give strength 
 to sustain it ; the masonry is the muscle to stiffen, protect, and ornament it. 
 Constructive expedients, like roofs, reduce the appearance of height, and are 
 objectionable from the necessity of gutters and leaders. 
 
 In conclusion, the draughtsman should be conversant with classic and later 
 styles ; still, as he must design to suit the necessities of the times, and the 
 requirements of present tastes and fashions of buildings, he should keep him- 
 self posted on what is being done, and he will find it very convenient to have a 
 scrap-book of cuts from which to draw parts of a design, and afford him ready 
 means of combinations. He will find much in illustrated magazines and news- 
 papers, many cuts unpromising as a whole, yet fruitful in suggestions of parts ; 
 many an agreeable outline unsatisfactorily filled up ; many that are only valuable 
 as showing dimensions requisite for certain uses. But the larger the collection 
 the better for the draughtsman ; it will save time to know, as far as possible, 
 what has been done, that he may judge what forms and proportions it will be 
 best for him to use, and what to avoid. 
 
 It has been our practice to select, from papers and magazines, cuts which 
 we considered of value, and arrange them in scrap-books with appropriate 
 headings. In the Appendix a few pages of " scraps " are given as illustrations. 
 
ISOMETRICAL DRAWING. 
 
 PROFESSOR FARISH, of Cambridge, has given the term Isometrical Per- 
 spective to a particular projection which represents a cube, as in Fig. 1754. 
 The words imply that the measure of the representations of the lines forming 
 the sides of each face are equal. 
 
 The principle of isometric representation consists in selecting, for the plane 
 of the projection, one equally inclined to three principal axes, at right angles 
 to each other, so that all straight lines 
 coincident with or parallel to these 
 axes are drawn in projection to the 
 
 FIG. 1754. 
 
 same scale. The axes are called iso- 
 metric axes, and all lines parallel to 
 them are called isometric lines. The 
 
 planes containing the isometric axes are isometric planes ; the point in the 
 object projected, assumed as the origin of the axes, is called the regulating- 
 point. 
 
 To draw the isometrical projection of a cube (Fig. 1755), draw the horizontal 
 line A B indefinitely ; at the point D erect the perpendicular D F, equal to one 
 side of the cube required; through D draw the lines D b and D/to the right 
 and left, making/D B and b D A each equal an angle of 30. Consequently, 
 the angles F D/ and FD b are each equal to 60. Make D I and D/each 
 equal to the side of the cube, and at b and /erect perpendiculars, making ba 
 and/e each equal to the side of the cube ; connect F a and F e, and draw eg 
 parallel to a F, and a g parallel to F e, and we obtain the projection of the 
 cube. 
 
 If from the point F, with a radius F D, a circle be described, and com- 
 mencing at the point D, radii be laid off around the circumference, forming a 
 
 694 
 
ISOMETRICAL DRAWING. 
 
 695 
 
 regular inscribed hexagon, and the points D a e be connected with the centre 
 of the circle F, we have an isometrical representation of a cube. The point D 
 is called the regulating -point. 
 
 On page 123, Fig. 255, is shown the 
 orthographic projection of a parallelo- 
 pipedon on the several planes, and Fig. 
 254 the revolution of these planes and 
 their cubical representation. Fig. 1756 is 
 the representation of the same solid in iso- 
 metric perspective, producing nearly the 
 same effect. Measures are transferred di- 
 rectly from plans and elevations in or- 
 thographic projections to those in isom- 
 etry. The isometric scale adopted ap- 
 plies only to isometric lines, as F D, F a, 
 and F e (Fig. 1755), or lines parallel 
 thereto ; the diagonals which are equal to 
 each other, and longer than the sides of 
 the cube, are the one less, the other greater. 
 
 Understanding the isometrical projec- 
 tion of a cube, any surface or solid may be 
 similarly constructed, since it is easy to 
 suppose a cube sufficiently large to con- 
 tain within it the whole of the model in- 
 tended to be represented, and, as hereafter 
 will be further illustrated, the position of 
 
 any point on or within the cube, the direction of any line, or the inclination of 
 any plane to which it may be cut, can be easily ascertained and represented. 
 
 In Figs. 1754 and 1755 one face of the cube appears horizontal, and the 
 other two faces appear vertical. If now the figures be inverted, that which 
 
 Fio. 1756. 
 
 FIG. 1757. 
 
 Fio. 1759. 
 
696 
 
 ISOMETRICAL DRAWING. 
 
 before appeared to be the top of the object will now appear to be its under 
 side. 
 
 The angle of the cube formed by the three radii meeting in the centre of 
 the hexagon may be made to appear either an internal or external angle, in 
 the one case the faces representing the interior, and in the other the exterior of 
 a cube. 
 
 Figs. 1757, 1758, and 1759 illustrate the application of isometrical drawing 
 to simple combinations of the cube and parallelopipedon. The mode of con- 
 struction of these figures will be easily understood by inspection, as they con- 
 tain no lines except isometrical ones. 
 
 To draw Angles to the Boundary Lines of an Isometrical Cube. Draw a 
 square (Fig. 1760) whose sides are equal to those of the isometrical cube A 
 
 (Fig. 1761), and from any of its angles describe a quadrant, which divide 
 into 90, and draw radii through the divisions meeting the sides of the square. 
 These will then form a scale to be applied to the faces of the cube ; thus, on 
 D E, or any other, by making the same divisions along their respective edges. 
 
 As the figure is bounded by twelve isometrical lines, and the scale of tan- 
 gents may be applied two ways to each, it can be applied therefore twenty-four 
 ways in all, affording a simple means of drawing, on the isometrical faces of 
 the cube, lines at any angles with their boundaries. 
 
 Figs. 1762 to 1767 show the section of the cube by single planes, at various 
 inclinations to the faces of the cubes. Figs. 1768 and 1769 are the same cube, 
 but turned round, with pieces cut out of it. Fig. 1770 is a cube cut by two 
 planes forming the projection of a roof. Fig. 1771 is a cube with all of the 
 angles cut off by planes, so as to leave each face an octagon. Fig. 1772 repre- 
 sents the angles cut off by planes perpendicular to the base of the cube, form- 
 ing thereby a regular octagonal prism. By drawing lines from each of the 
 angles of an octagonal base to the centre point of the upper face of the cube, 
 we have the isometrical representation of an octagonal pyramid. 
 
 As the lines of construction have all been retained in these figures, they will 
 be easily understood and copied, and are sufficient illustrations of the method 
 of representing any solid by inclosing it in a cube. 
 
ISOMETRICAL DRAWING. 
 
 697 
 
 FIG. 1763. 
 
 FIG. 1763. 
 
 FIG. 1761. 
 
 FIG. 1765. 
 
 FIG. 17C6. 
 
 FIG. 1767. 
 
 FIG. 1768. 
 
 FIG. 1769. 
 
 FIH. 1770. 
 
 FIG. 1772. 
 
698 
 
 ISOMETRICAL DRAWING. 
 
 In the application of this species of projection to curved lines, let A B 
 (Fig. 1773) be the side of a cube with a circle inscribed ; and all the faces of a 
 cube are to have similarly inscribed circles. Draw the diagonals A B, D, and 
 at their intersection with the circumference, lines parallel to A C, B D. Now 
 
 FIG. 1773. 
 
 FIG. 1774. 
 
 draw the isometrical projection of the cube (Fig. 1774), and lay out on the 
 several faces the diagonals and the parallels ; the projection of the circle will 
 be an ellipse, of which the diagonals being the axes, their extremities are de- 
 fined by their intersections /6, e 5, a 2, b 1, d 3, c 4, with the parallels ; having 
 thus the major and minor axes, construct the ellipse by the trammel, or, since 
 the curve is tangent at the centre of the sides, we have eight points in the 
 curve ; it may be put in by curves or by the hand. 
 
 To divide the Circumference of a Circle. First method: On the centre of 
 the line A B (Fig. 1775) erect a perpendicular, C D, making it equal to C A or 
 C B ; then from D, with any radius, describe an arc and divide it in the ratio 
 
 FIG. 1775. 
 
ISOMETRICAL DRAWING. 
 
 699 
 
 required, and draw through the divisions radii from D meeting A B ; then 
 from the isometric centre of the~ circle draw radii from the divisions on A B, 
 cutting the circumference in the points required. 
 
 Second method : On the major axis of the ellipse describe a semicircle, and 
 divide it in the manner required. Through the points of division draw lines 
 
 FIG. 1777. 
 
 perpendicular to A E, which will divide the circumference of the ellipse in the 
 same ratio. On the right hand of the figure both methods are shown in com- 
 bination, and the intersections of the lines give the points in the ellipse. 
 
 Fig. 1776 is an isometrical projection of a bevel-wheel, with a half-plan 
 (Fig. 1777) beneath, and projected lines explanatory of the method to be 
 adopted in drawing the teeth, and of which only half are shown as cut. It 
 will be seen, by reference to the second method given above for the division of 
 the circumference of a circle, that the semicircle is described directly on the 
 major axis of the ellipse. In practice it will be found more convenient, when 
 a full drawing is to be made, to draw the semicircle on a line parallel to the 
 major axis, and entirely without the lines of the main drawing ; and also, as in 
 the example of the bevel-gear, complete on the semicircle, or half-plan, the 
 drawings of all lines, the intersections of which with circles it will be necessary 
 to project on the isometrical drawing. 
 
 Fig. 1778 is an isometrical projection of a complete pillow-block, with its 
 hold-down bolts. By reference to Fig. 700 and Figs. 563 and 564, it will be 
 
700 
 
 ISOMETRIC A L DRAWING. 
 
 FIG. 1778. 
 
 seen how graphically these forms of gearing are given by isometry. Fig. 1779 
 is an isometrical projection of a water-closet cistern with a standard waste. 
 
 FIG. 1 
 
ISOMETRICAL DRAWING. 
 
 701 
 
702 
 
 ISOMETRICAL DRAWING. 
 
 Fig. 1780 is an isometrical projection of a culvert, such as were built be- 
 neath the Croton Aqueduct. 
 
 Fig. 1016 is an isometrical view of the overflow and outlet of the Victoria 
 and Eegent Street sewers in the Thames embankment. 
 
 Fig. 1781 is an isometrical elevation of the roof-truss (Fig. 1151). 
 
 Figs. 1782 and 1783 are the elevation and section in isometry of the district 
 school-house given in Figs. 1495 and 1496. To bring the drawing within the 
 limits of the page, the scale has been necessarily reduced, but it is given in 
 the figure as it should always be, either drawn or written, on all drawings 
 to a scale, not intended for mere pictures or illustrations. The section is 
 drawn at the height of 8 feet above the base course, and higher than is usual 
 in such sections, but it was necessary on account of the extra height of the 
 window-sill above the floor, desirable in all school-rooms. Fig. 1783 is more 
 
 FIG. 1781. 
 
 graphic than the plan (Fig. 1496), especially when more detail is to be shown, 
 but there is nothing in the present drawing that can not be nearly as well 
 shown by the plan ; and to a mechanic, for the purposes of construction, the 
 plan is the simpler. 
 
 By comparing the elevation (Fig. 1782) with the perspective (Fig. 1799), 
 the former appears distorted and out of drawing, but it is much more readily 
 
ISOMETRICAL DRAWING. 
 
 703 
 
 FIG. 1783. 
 
704 
 
 ISOMETRICAL DRAWING. 
 
 drawn, and has this great convenience, that it is drawn to and can be measured 
 by a scale on the isometric lines. 
 
 Fig. 1784 is the isometrical projection, of the wave-line principle, of ship 
 
 \\\\\\\\\\ 
 \\\\\\\\X\ 
 
 FIG. 1784. 
 
 construction, from Russell's "Naval Architecture" as explained and illus- 
 trated on pages 545, 546, and 54.7. 
 
ISOMETRICAL DRAWING. 
 
 705 
 
PERSPECTIVE DRAWING. 
 
 THE science of Perspective is the representation by geometrical rules, upon 
 a plane surface, of objects as they appear to the eye, from any point of view. 
 
 All the points of the surface of a body are visible by means of luminous 
 rays proceeding from these points to the eye. Thus, let the line A B .(Fig. 
 1785) be placed before the eye, C, the lines drawn from the different points 1, 
 2, 3, 4, etc., represent the visual rays emanating from each of these points. If 
 in the place of a line a surface is substituted, the result will be a pyramid of 
 rays. 
 
 The greatest angle under which one or more objects can be distinctly seen 
 is one of 90. If between the object and the eye there be interposed a trans- 
 parent plane, the intersections of this plane with the visual rays are termed 
 
 perspectives of the points from which the 
 rays emanate. In the operations of projec- 
 tion, several important planes are employed 
 in perspective, as : 
 
 1. The horizontal plane A B (Fig. 1786), 
 on which the spectator and the object viewed 
 are supposed to stand, for convenience sup- 
 posed perfectly level, is termed the ground 
 plane. 
 
 2. The plane G N, which has been con- 
 sidered as a transparent plane placed in front 
 of the spectator, on which the objects are 
 
 delineated, is called the plane of perspective or the plane of the picture. The 
 intersection G L of the first and second planes is called the line of projection, 
 the ground, or base line of the picture. 
 
 3. The plane E F passing horizontally through the eye of the spectator, and 
 cutting the plane of the picture at right angles, is called the horizontal plane, 
 and its intersection at D D with the plane of the picture is called the horizon 
 line, the horizon of the picture, or simply the horizon. 
 
 706 
 
 FlO. 1785. 
 
PERSPECTIVE DRAWING. 
 
 707 
 
 4. The plane S T passing vertically through the eye of the spectator, and 
 cutting each of the other planes at a right angle, is called the central plane. 
 
 Point of view, or point of sight, is the point where the eye is supposed to 
 be placed to view the object, as at C, and is the vertex of the optical pyramid. 
 Its projection on the ground plane at S is termed the station point. 
 
 FIG. 1786. 
 
 The projection of any point on the ground plane is called the seat of that 
 point. 
 
 Centre of view, or centre of picture (commonly, though erroneously, called 
 the point of sight), is the point V where the central vertical line intersects the 
 horizon line ; a line drawn from this point to the eye would be in every way 
 perpendicular to the plane of the picture. 
 
 Points of distance are points on the horizon as remote from the centre of 
 view as the eye. 
 
 Vanishing points are points in a picture to which the perspective of all 
 lines converge that in the original object are parallel to each other. 
 
 The vanishing point of any line or system of parallel lines is found by pass- 
 ing a line through the point of sight parallel to the given line, or system of 
 lines. Its intersection with the plane of the picture will be the vanishing 
 point desired. Therefore the vanishing points of all horizontal lines lie on the 
 horizon, and the vanishing points of horizontal lines making an angle of 45 
 with the ground line are at the points of distance. 
 
 Parallel Perspective. An object is said to be seen in parallel perspective 
 when one of its sides is parallel to the plane of the picture. 
 
 Angular Perspective. An object is said to be seen in angular perspective 
 when none of its sides are parallel to the picture. 
 
 The vanishing points of all lines parallel to the plane of perspective are at 
 infinity, or, in other words, such lines have no vanishing points. 
 
 The perspectives of lines parallel to the perspective plane are parallel to 
 their projections on that plane. 
 
 The process of finding the perspective of lines, plane figures, and solids 
 consists merely in finding the perspectives of established points and connecting 
 them. 
 
 To find the Perspective of Two Squares in the Ground Plane whose Sides 
 
708 
 
 PERSPECTIVE DRAWING, 
 
 are Parallel and Perpendicular to the Perspective Plane (Fig. 1787). Let G L 
 be the ground line, a b d c and e f h g the horizontal projections of the two 
 squares; S' is the horizontal, and S the vertical projection of the point of 
 sight ; a b in the ground line is the vertical projection of the two squares. 
 
 FIG. 1787. 
 
 FIG. 1788. 
 
 Draw a S and b S ; these will be the indefinite perspectives of the sides of 
 the squares perpendicular to the perspective plane. Draw h S' ; where this 
 line intersects the ground line, project it vertically to h^ on b S, which is the 
 perspective of h ; the line g h, being parallel to the perspective plane, is parallel 
 to the ground line, and is shown at g l h v In the same way e^ /i and c v d l are 
 found. The perspectives of the diagonals are the lines connecting the corners. 
 
 The line a b is in the perspective plane, and is its own perspective. All 
 lines or plane figures lying in the perspective plane appear in perspective in 
 their true form and size. 
 
 To find the Perspectives of Two Cubes whose Sides are Parallel and Perpen- 
 dicular to the Perspective Plane (Fig. 1788). a b c d is the vertical, and a' b' 
 o' ri and/' e' h' g' the horizontal, projections of the cubes. Draw a S, b S, c S, 
 and d S. These will be the indefinite perspectives of the edges of the cubes 
 perpendicular to the perspective plane. Draw S' g' ; where this strikes the 
 ground line, project to r t and g l on S a and S d, which will be the perspectives 
 of the two corners horizontally projected at g' and vertically projected at a and 
 d; g-i h^ drawn parallel to the ground line and limited by S c and S a, is the 
 perspective of g' h'. In a similar manner the perspectives of all other points 
 and edges are found. 
 
 From the drawing of a square in parallel perspective, we deduce rules for 
 the construction of a scale in perspective. Let D G L D (Fig. 1789) be the 
 
PERSPECTIVE DRAWING. 
 
 709 
 
 plane of the picture. From S'~lay off the distance o S' equal to some unit of 
 measure, as may be most convenient ; from o draw the diagonal to D the point 
 
 FIG. 1789. 
 
 of distance ; now draw 1 1' parallel to the ground line G L, again draw from 1' 
 the diagonal 1' D, and lay off the parallel 2 2', proceed in the same way with 
 the diagonal 2' D and the parallel 3 3', and extend the construction as far as 
 may be necessary. It is evident o S' 1 1', 1' 1 2 2', 2' 2 3 3' are the perspective 
 projections of equal squares, and therefore o S', 1 1', 2 2' 3 3', etc., and S' 1, 
 1 2, 2 3, etc., are equal to each other, and that if o S' is set off to represent any 
 unit of measure, as one foot, one yard, or ten feet, etc., each of these lines 
 represents the same distance, the one being measured parallel to the base line, 
 the others perpendicular to it. In making a perspective drawing a scale thus 
 placed will be found convenient ; which in the centre of the picture might 
 interfere with the construction lines of the object to be put in perspective, is 
 better transferred to the side of the picture a G o, the diagonals to be laid off 
 to a point to the right of D equal to the point of distance. 
 
 The scales thus projected are for lines in the base or ground plane ; for lines 
 perpendicular to this plane the following construction is to be adopted : Upon 
 any point of the base line removed from S', as a, for instance, erect a perpen- 
 dicular, a d ; on this line, lay off as many of the units o S' as may be necessary ; 
 in this example three have been laid off, that is, a d = 3 o S'. From a and d 
 draw lines to the centre of view, and extend the parallels 1 1', 2 2', 33'; at the 
 intersection of these lines with a V erect perpendiculars. The portions com- 
 prehended between the lines a V and d V will be the perspective representa- 
 tions of the line a d, in planes at distances of 1, 2, 3, o S' from the base line, 
 and as #, c, d are laid off at intervals equal to o S', by drawing the lines c V 
 and b"V nine equal squares are constructed, of which the sides correspond to 
 the unit of measure o S'. 
 
 To find the Scale for the Perspective of any Line Oblique to the Perspective 
 Plane (Fig. 1790). Let A B be the perspective of any line oblique to the 
 perspective plane and V its vanishing point ; through the extremity A draw any 
 line A 7 ; divide it into as many parts as is desired to divide the perspective 
 line A B, here seven. Draw 7 B. Connect the remaining points 6, 5, 4, etc., 
 withV ; where these lines intersect the line 7 B they are projected on A B, par- 
 allel to A 7. The divisions A 6", 6* 5", etc., are the perspectives of the seven 
 
T10 
 
 PERSPECTIVE DRAWING. 
 
 equal divisions of A B. If the line A B is fourteen feet long, the divisions 
 A 6", 6" 5", 5" 4* are each equal to two feet. 
 
 To find the Perspective of a Hexagonal Prism with a Pyramidal Top (Fig. 
 
 1791). The perspective plane 
 in the original position is a pro- 
 file plane; abfie is the verti- 
 cal and c' b' c ' d ' e' d' the hori- 
 zontal projection of the figure. 
 M N is the profile perspective 
 plane in which the edge of the 
 prism e i lies. S is the vertical 
 and S' the horizontal projection 
 of the point of sight. Draw rays 
 . of light at S a and S' a', S c, 
 S' c', S d, S' d\ et* These rays 
 pierce the profile perspective plane at points a t GI d^ etc. 
 
 All the points in sight being thus fixed, the profile perspective plane is then 
 transferred to the position M! N t and revolved about its vertical trace into the 
 plane of the paper. Thus, the point projected at a" and a', the perspective of the 
 point originally projected at a and a' is transferred to a'" and revolved to a", which 
 
 FIG. 1790. 
 
 Fio. 1791. 
 
 is its final position. All other points are treated in the same way. The edge e i 
 lying in the perspective plane is the same length in perspective as in projection. 
 
 The transposition of the profile plane from M N to M t N, is only necessary 
 to avoid complicating the perspective with the projections. 
 
 Method of Perpendiculars and Diagonals, to find the Perspective of a Pave- 
 ment in Parallel Perspective (Fig. 1792). a b d c is the pavement, the square 
 blocks running diagonally across it. 
 
PERSPECTIVE DRAWING. 
 
 m 
 
 The projection of the pavement being revolved so as to appear below the 
 ground line, diagonals drawn through points of it to the right will vanish in 
 the point of distance to the left and vice versa. 
 
 FIG. 1792. 
 
 FIG. 1793. 
 
 The sides of the pavement, being perpendicular to the perspective plane, are 
 their own perpendiculars. Thus S a and S b are their perspectives. The lines of 
 division of the blocks, being at 45, are their own diagonals and vanish atD and D,. 
 
 To find the Perspective of a Cube in Parallel Perspective (Fig. 1793), a b d c 
 is the horizontal projection of the cube. The face a bf e being in the plane 
 of perspective, appears in full size. From the point c the diagonal c x disap- 
 pears at D,, and the perpendicular at S. Their intersection c z is the perspective 
 of the lower corner ; d z is found in the same way ; c and d are found by pro- 
 jecting up vertically from c 2 and d 2 and intersecting S e and S /, or they could 
 be found by constructing diagonals in the plane Y Y of the top. 
 
 To find the Perspective of the Frustrum of a Right Square Pyramid by Per- 
 
712 
 
 PERSPECTIVE DRAWING. 
 
 pendiculars and Diagonals (Fig. 1794). a b c d is the horizontal projection of 
 the base and efg h of the top of the pyramid. The point of sight is projected 
 at S' and S ; D and Dj are the points of distance. 
 
 The perspective of each point is found by drawing its perpendicular and 
 diagonal and fixing the intersection of their perspectives. The perpendiculars 
 and diagonals must lie in their proper plane. Thus those at the top of the 
 pyramid must be projected up to the plane s t. 
 
 To find the Perspective of a Horizontal Circle (Fig. 1795). To find the per- 
 spective of any curve it is merely necessary to find the perspectives of enough 
 
 Fro. 1795. Fm. 1796. 
 
 points on it to enable the perspective curve to be traced through them \agec 
 is the horizontal circle. Take any number of equidistant points on it, as a b c, 
 etc. Their perspectives are found by the usual method of perpendiculars and 
 diagonals at a^ b^ c l5 etc. The curve traced through these perspective points is 
 the perspective of the circle. This curve is an ellipse. 
 
 To find the Perspective of a Circle in a Profile Plane (Fig. 1796). a 1 h' is the 
 horizontal and e d the vertical projection of the circle. As before, a number of 
 points are taken on the circle, the location of these points on the horizontal 
 and vertical projections of the circumference are found by revolving the cir- 
 cle about its horizontal diameter parallel to the horizontal plane of projection. 
 The vertical distance of all points on the circumference from the diameter a' h r 
 are thus ascertained and set off on the vertical projection of the circle. 
 
 M M, N N, 0, P P, and Q Q are the vertical traces of the horizontal 
 planes in which the points taken lie. Perpendiculars and diagonals drawn 
 from these points and carried up to get the points in the correct planes give 
 0i c\ 0*1 </n etc., as the perspective of the circle. 
 
 The projection of the circle is only necessary to give the height above the 
 ground line of the planes M M, N N, etc., in which lie the points taken on the 
 
PERSPECTIVE DRAWING. 
 
 713 
 
 circle. Had the circle been in any vertical plane other than a profile plane, the 
 mode of proceeding would have been the same. 
 
 To find the Perspective of a Cylinder whose Axis is Horizontal and at an 
 Angle of 45 'with the Perspective Plane (Fig. 1797). a ejfis the horizontal 
 
 FlG. 1797. 
 
 FIG. 1798. 
 
 projection of the cylinder. The base/y is revolved parallel to the horizontal 
 plane and shown at s f h" j and equidistant points taken on it. Set off above 
 the ground line the heights s t,s h,s u, and s h' and draw horizontal lines 
 through them. These lines will be traces on the perspective plane of horizontal 
 planes passing through the assumed points and corresponding points on the other 
 base. Then finding the perspectives of diagonal and perpendicular lines passing 
 through these points, the perspectives of corresponding points on the bases 
 through which the perspective curves can be drawn. 
 
 To find the Perspective of an Octagonal Prism with its Axis Horizontal and 
 making an Angle of 45 with the Perspective Plane (Fig. 1798). a d h e is the 
 horizontal projection of the prism. M M, N N, and are the traces on the 
 perspective plane of the horizontal planes containing the edges of the prism. 
 The perspectives of the points abed, etc., from the perspectives of the edges 
 vanishing at D, and of the perpendiculars vanishing at S. The intersections 
 of these perspectives give the points desired. 
 
 To draw the Elevation of a Building in Angular Perspective (Fig. 1799). 
 The plan of the two sides which are to appear in perspective are drawn, all 
 openings, projections, and roof plan being indicated. This plan, c, a, #, is 
 placed at the top and about the centre of the drawing board in any desired 
 position ; a line P P', known as the picture plane or plane of measures, is drawn 
 of indefinite length. For convenience in measuring heights it is usual, though 
 not necessary, to draw P P' through the point a of the building. 
 
PERSPECTIVE DRAWING. 
 
 The station point S is selected, largely a matter of judgment, and lines 
 drawn through S parallel to the sides of the building and intersecting the pic- 
 ture plane at P P' ; from P and P' perpendiculars are dropped of indefinite 
 length ; the ground line G L, parallel to P P', is drawn at any convenient place, 
 and the horizon drawn parallel and about six feet above, and intersecting the 
 perpendiculars at V V, the vanishing points, all lines parallel to a c vanishing 
 at V and those parallel to a b at V ; lines are drawn from all points in the 
 plan which are to appear in the perspective to S intersecting P P' and then 
 transferred by vertical lines to the portion of the paper reserved for the per- 
 spective drawing ; the horizontal measures are thus obtained. For the vertical 
 ones the sides of the plan a c and a I are extended till they cut P P' ; at d and e 
 perpendiculars are dropped from these points which are lines of heights. Ele- 
 vations drawn to the same scales as the plan are placed to the right and left 
 of space reserved for the perspective on G- L ; heights are transferred from the 
 elevation on the left to line of heights e e' and vanish at V, heights on the 
 right to d d' and vanish at V. The remaining lines and filling in of details 
 will be understood from the drawing. 
 
 Fig. 1800, the building sketched on page 592, is shown in angular perspec- 
 tive. The general construction will be understood from the description of Fig. 
 1799, the same letters being used to describe similar lines. Additional lines of 
 measures must be taken where there are recesses, projections, chimneys, etc., to 
 locate. Thus, take the chimney/; the line of heights for this is obtained by 
 continuing the line of chimney till it intersects P P' at /', a perpendicular is 
 dropped from this point, and the height transferred to this line from the eleva- 
 tion ; a vanishing point drawn from this intersection to V intersects the hori- 
 zontal limits of the chimney and gives the height ; the bay window similarly. 
 
 At page 906 is a general diagram showing the principal constructive lines 
 necessary in rendering a building in perspective. 
 
 Fig. 1800 A illustrates a method whereby the orthographic plan is dispensed 
 with and a perspective plan substituted, from which the vertical lines and hori- 
 zontal measures may be taken directly. 
 
 The horizon, vanishing points, and point of sight are determined as in the 
 previous examples. Describe arcs from the vanishing points through the point 
 of sight. The intersection of these arcs with the horizon are the points of dis- 
 tance. A line of measures is taken parallel to and at any convenient distance 
 above or below the horizon, and a point, indicating the corner of the building, 
 located where desired on it ; to the right and left of this point the horizontal 
 measures of the two sides of the building are laid off to scale ; from the corner 
 of the building to each vanishing point lines are drawn representing the sides 
 of the plan. The extreme limits of the building laid off on the line of meas^ 
 ures vanish at the points of distance ; that to the left of the corner to the point 
 of distance to the right, and vice versa. Where these lines intersect the sides 
 of the building are the limits of the perspective plan. These lines may then be 
 transferred by perpendiculars to the perspective elevation. 
 
 The perspective plan described above is shown on a larger scale in Fig. 
 1800 B, the dimensions of the doors and windows of the plan being laid off on 
 the line of measures and carried to their respective points of distance as de^ 
 scribed in obtaining the perspective limits of the building. 
 
PERSPECTIVE DRAWING. 
 
 715 
 
 I ' H 
 
 E 
 E 
 
T16 
 
 PERSPECTIVE DRAWING. 
 
 To draw in Parallel Perspective the Interior of a Room, (Fig. 1801). Let a 
 b c d be the plan of the room. A line P P', known as the picture plane of meas- 
 
 * 1 1 * 
 
 '-i "^ 
 
 $ 
 ^ j- 
 
 :$ $; ffor/zon 
 
 1 
 
 5 
 
 1 
 
 i^^X--^ ^ 
 
 ^ i 
 
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 -"" "^^ ^ ' 
 
 \ ^ ^ ^ ""* ^ 
 
 ^'' -- L 
 
 
 \ --^ \ 
 
 
 / / 
 
 
 ^** **" [^ 
 
 / ; 
 
 \ NV - ^""^ ~~^~-V ^" 
 
 ^-"" ^-' i 
 
 x X / 
 
 * ^^ \^~~~^^^"'* 
 
 V ^ 1 
 
 / / 
 
 \ ^^^~^f^^ """*" 
 
 ^ '"' 1 
 
 / / 
 
 
 S?SK^ / 
 
 / / 
 
 \ ^-^ v^x^ ^ 
 
 ^^ X N 1 
 
 / / 
 
 \ ^^" line \of^^>^^ 
 
 'ensures NI y 
 
 / / 
 
 \ X \ 
 
 
 / / 
 
 \ x \ 
 
 / / 
 
 / 
 
 K / 
 
 int of S iff/it 
 
 FIG. 1800. A. 
 
 FIG. 1800. B. 
 
 ures, is drawn far enough forward to include all that is desired to be shown in 
 perspective and usually parallel to the rear of the room. S is taken outside the 
 room and as far away from the plan as the size of the board will admit of. In 
 this example S has been taken to the right of the centre of the room ; if taken 
 in the centre, the perspective would be symmetrical. The points a and d of the 
 plan subtend the greatest angle, which should never be over 60. The same 
 may be said of the vertical angle, a greater angle causing distortion. Dotted 
 lines show rays from various points intersecting P P' and establishing the posi- 
 tion of the vertical lines. The ground line G L is drawn parallel to P P', and 
 the horizon drawn at a suitable elevation. A perpendicular erected at S and 
 intersecting the horizon at V gives the vanishing point for all lines parallel to 
 a b and d c. Vertical heights are laid off on either a a' or d d r , and transferred 
 to their proper position in the perspective. 
 
 To draw an Arched Bridge in Angular Perspective. Let A and B (Fig. 
 1802) be the plans of the piers ; on the line a p, one of the sides of the bridge, 
 lay down the curve of the arch as it would appear in elevation, in this example 
 an ellipse. Divide the width of the arch as at b c d e f g h, carry up lines per- 
 pendicular to b h until they intersect the curve of the arch, and through these 
 points draw lines parallel to b h as k I and m ; let o r be the height of the para- 
 
PERSPECTIVE DRAWING. 
 
 717 
 
718 
 
 PERSPECTIVE DRAWING. 
 
 pet of the bridge above the spring of the arch. Through the station point 
 draw lines parallel to the side a h and end a a of the bridge, till they intersect 
 
 FIG. 1801. 
 
 the assumed base line M M ; project these intersections to the horizon line of 
 the picture for the vanishing points D, D' of perspective lines parallel to a h 
 and a $. Let fall a perpendicular from a to a', and on this perpendicular set 
 off from a' the heights s &, s I, s m, and s r ; from a' and r' draw lines to D and 
 D', and from the points m', l\ Tc 1 to D'. Draw rays from the points a b c d efg 
 h to the station point S, and project their intersection with the base lines to the 
 perspective line a' D' as in previous examples ; the intersection of the lines Tc' 
 D', /' D', m' D' by the perpendiculars thus projected will establish the points 
 
PERSPECTIVE DRAWING. 
 
 719 
 
 of the curve of the arch on the side nearest the spectator. To determine the 
 position of the opposite side of the arch, from a", the perspective of the corner 
 of the pier a" draw a" D', and from h' draw lines to D ; the line /*' p' will be 
 
 FIG. 1802. 
 
 the perspective width of the pier ; draw k' D ; and from &", k" D' ; from g" 
 the intersection of the curve of the arch by the perpendicular to g 1 , draw g* D, 
 the intersection with k" D' will be one point in the curve of the arch on the 
 opposite side of the bridge ; in the same way, from any point in the nearer 
 arc draw lines to D, and the intersection with lines in the same planes on the 
 opposite side of the bridge will furnish points for the further arch ; all below 
 the first only will be visible to the spectator. 
 
 To draw in Perspective a Flight of Stairs (Fig. 1803). Lay off the base 
 line, horizon, centre of view, and point of distance of the picture ; construct 
 the solid a b c d, efg h, containing the stairs, and in the required position in 
 the plane of the picture ; divide the rise a c into equal parts according to the 
 number of stairs, nine, for instance ; divide perspectively the line a b into one 
 less (eight) number of parts ; at the points of division of this latter erect per- 
 pendiculars, and through the former draw lines to the centre of view ; one will 
 form the rise and the other the tread of the steps. From the top of the first 
 step to the top of the upper continue a line a d, till it meets the perpendicular 
 S' V prolonged in v ; this line will be the inclination or pitch of the stairs ; if 
 through the top of the step at the other extremity a similar line be drawn, it 
 
720 
 
 PERSPECTIVE DRAWING. 
 
 will meet the central perpendicular at the same point #, and will define the 
 length of the lines of nosing of the steps, and the other lines may be completed. 
 
 D 
 
 -tfr 
 
 
 
 FIG. 180S. 
 
 As the pitch lines of both sides of the stairs meet the central vertical in the 
 same point, in like manner v will be the vanishing point of all lines having a 
 similar inclination to the plane of the picture. The projection of the other 
 flight of stairs will be easily understood from the lines of construction perpen- 
 dicular to the base line or parallel thereto, lying in planes. 
 
 To find the Reflection of Objects in the Water. Let B (Fig. 1804) be a cube 
 suspended above the water ; find the reflection of the point a by letting fall 
 a perpendicular from it, and setting off the distance a' w below the plane of 
 the water equal to the line a w above this line ; the line w f will also be equal 
 
 D 
 
 Fid. 1804. 
 
PERSPECTIVE DRAWING. 721 
 
 to the line w /; find in the same way the points V and e', through these points 
 construct perspectively a cube in this lower plane, for the reflection of the cube 
 above. 
 
 To find the reflection of the square pillar D removed from the shore : sup- 
 pose the plane of, the water extended beneath the pillar, and proceed as in the 
 previous example. 
 
 The lines of an object which meet in the centre of view V, in the original, 
 have their corresponding reflected lines converging to the same point. If the 
 originals converge to the points of distance, the reflected ones will do the same. 
 To find the reflection of any inclined line, find the reflection of the rectangle 
 of which it is the diagonal, if the plane of the rectangle is perpendicular to 
 the plane of the picture. If the line is inclined in both directions inclose it in 
 a parallelepiped and project the reflection of the solid. 
 
 To find the Perspective Projection of Shadows (Fig. 1805). Let the con- 
 struction points and lines of the picture be plotted. Let A be the perspective 
 projection of a cube placed against another block, of which the face is parallel 
 to the plane of the picture ; to find the shadow upon the block and upon the 
 ground plane, supposing the light to come at such an angle as to cause the 
 projections of it (both vertical and horizontal) to make an angle of 45 with 
 the ground line. Since the angle of light is the diagonal of a cube, construct 
 another cube similar to A, and adjacent to the face bcg\ draw the diagonal 
 b k, it will be the direction of the ray of light, and k will be the shadow of b ; 
 connect fk and c k,fk must be the shadow of the line J/, and c k of b c ; the 
 one upon the horizontal plane and the other in a vertical one : the former will 
 have its direction, being a diagonal, toward the point of distance D', the other 
 being a diagonal in a plane parallel to that of the picture, will be always pro- 
 jected upon this plane in a parallel direction. 
 
 Let B be a cube similar to A ; to find its projection upon a horizontal plane, 
 the shadow of the point b' may be determined as in the preceding example, but 
 the shadow of the point ^', instead of falling upon a plane parallel to the pic- 
 ture, falls upon a horizontal one; its position must be determined as before 
 by b. Construct the cube and draw the diagonal c' I ; in the same way deter- 
 mine the point m the shadow of d' ; connect c k' I m n, and for the shadow of 
 the cube in perspective on a horizontal plane. 
 
 On examination of these projected shadows, it will be found that as the 
 rays of light fall in a parallel direction to the diagonal of the cube, the vanish- 
 ing point of these rays will be in one point V on the line D' L, prolonged, at 
 a distance below D' equal V D' ; and since the shadows of vertical lines upon a 
 horizontal plane are always directed toward the point of distance, the extent of 
 the shadow of a vertical line may be determined by the intersection of the 
 shadow of the ground point of the line by the line of light, from the other ex- 
 tremity. Thus, the point &, cube A, is the intersection of /D' by b V ; the 
 points &', I, m are the intersections of c D', o D', M D' by V V, c'V d'\'. 
 Similarly on planes parallel to that of the picture, k, cube A is intersection of 
 the diagonal c k, by the ray of light b V. 
 
 Applying this rule to the frame C, from r, s, p, draw lines to D' ; from r', 
 s', ;/, draw rays to V ; their intersections define the outline of the shadow of 
 the post. To draw the shadow of the projection, the shadow upon the post 
 47 
 
722 
 
 PERSPECTIVE DRAWING. 
 
PERSPECTIVE DRAWING. 
 
 723 
 
 \ 
 
 UNIVERSITY 
 
724 
 
 PERSPECTIVE DRAWING. 
 
 from t will follow the direction of the diagonal c k. Project u and v upon the 
 ground plane at u' and v' ; from t u' v' and p draw lines to D' ; from t', u, v, w, 
 and x draw rays to V, and the intersection of these lines with their corre- 
 sponding lines from their bases will give the outline required ; as v and w are 
 on the same perpendicular, their rays will intersect the same line v' V. 
 
 With reference to the intensity of " shade and shadow," and the necessary 
 manipulation to produce the required effect, the reader is referred to the article 
 on this subject. 
 
 In treating of Perspective it has been considered not from an artistic point, 
 as enabling a person to draw from Nature, but rather as a useful art to assist 
 the architect or engineer to complete his designs, by exhibiting them in a 
 view such as they would have to the eye of a spectator when constructed. Our 
 examples, owing to size of the page, have been limited in the scale of the 
 figures, and in the distance of the point of view, or distance of the eye from 
 the plane of the picture, unimportant to the mathematical demonstration. It 
 is unnecessary in these particular points that the examples should be copied. 
 The most agreeable perspective representations are generally considered to be 
 produced by fixing the angle of vision at from 45 to 50, and the distance of 
 the horizon above the ground-line at about one third the height of the picture. 
 
 In the early edition of this work there were illustrations of machinery on 
 
 FIG. 1807. 
 
 sale in a kind of perspective, of which two, Figs. 1806 and 1807, specimens of 
 ship work, a windlass and centreboard winch, are reproduced. They are 
 graphic and natural in appearance, and similar illustrations will be found in 
 the collection of " Scraps." 
 
 With the introduction of photography and the ready transfer of the nega- 
 tives to the positive and permanent condition of prints and plates, one is 
 enabled to judge of sizes and dimensions from their surroundings, or by the 
 introduction into the view of appropriate marks or bounds of known distance 
 
PERSPECTIVE DRAWING. 
 
 725 
 
 affording lines for measures. In fact, photography has been applied to surveys 
 with records of angles and topographical views of lines. Views of buildings, 
 circulars of machinery, and objects on sale or of interest are illustrated by the 
 aid of photography. The figure below represents the office of the publishers 
 of this work taken by photography, printed on plain salted paper. The pen- 
 work is done in water-proof ink on the photograph ; the print is then washed 
 in a chemical solution, removing the photographed lines and leaving those in 
 ink. (See Free-Hand Drawing.) 
 
FREE-HAND DRAWING. 
 
 A DRAUGHTSMAN, who has made himself conversant with the rules of pro-, 
 jection as laid down in the preceding pages, and has applied these rules to prac- 
 tice, will be capable of representing correctly such objects as have been illus- 
 trated, or make up similar combinations of his own invention and design for 
 the comprehension of others. But natural objects, as animals, trees, rocks, 
 clouds, etc., can not be imitated on paper with the aid of drawing instruments ; 
 outlines so varied can not be copied in this mechanical way ; it can only be done 
 by hand drawing, an educated eye that can recognise proportion and position, and 
 an educated hand that can execute and portray naturally things recognised by 
 the eye, with the aid of pencil, pen, crayon, brush, or the various tools that now 
 obtain with draughtsmen. But it is by education that one acquires the .facili- 
 ties of such an eye and hand. As the writer acquires facilities by the copying 
 of pothooks, letters, and nourishes which may develop into a valuable distinct- 
 ive hand, so the draughtsman by the study of examples, first of drawings and 
 copying, the learning of proportions, comparison of the works of different art- 
 ists, observations of effect in drawing and nature, will commence an education 
 which will produce pleasant and successful pictures distinctive of the educated 
 artist, appreciated by the public and of mercantile value. 
 
 An educated free hand adds largely to the effect on most drawings, where 
 close measures are not requisite, giving grace and beauty to mechanical designs, 
 and is especially applicable to architectural ornaments and accessories. It will 
 be found impossible to draw many of these in any other way, and there are few 
 drawings that do not require some patching by hand short curves, which can 
 be thus done much more readily, and connections of lines, which can not be 
 done by drawing instruments. 
 
 The pencil or pen should be held by the thumb and first finger, and sup- 
 ported and guided by the second. The two fingers touching the pencil should 
 be placed firmly on it, and be perfectly straight, the end of the middle finger 
 at least one inch above the point of the pencil. In drawing, it is well to com- 
 mence, as in writing, with straight lines. Lines vertical, horizontal, and in- 
 clined, parallel to each other and at angles, light and strong short and long 
 lines, straight and curved, with pen, pencil, or crayon on paper, or chalk on 
 a board. Dot points, and draw lines between them, at a single movement, 
 without going over them a second time, and without patching. Besides 
 direction, lines have a definite length, and the draughtsman must practise 
 himself in drawing lines of equal lengths, or in certain proportions to each 
 other. 
 
 726 
 
FREE-HAND DRAWING. Y27 
 
 Lines equal to each other : 
 
 Lines twice another line : 
 
 Divide a line into any number of equal parts : 
 
 The accuracy of these divisions may be tested by a strip of paper applied 
 along the line, marking off the divisions upon it, and then slipping it along 
 one division, and noting if the divisions on the paper and line still agree. By 
 practice, the eye will be able to make these divisions almost accurately. Having 
 acquired this skill, apply it to the construction of the Geometrical Problems, in 
 the earlier part of the book, in their proper proportions, both in straight and 
 curved lines. The construction should be dependent entirely on eye and hand ; 
 but it will be found, whether the draughtsman draws from copy or nature, that it 
 is almost impossible to get along well without defining positions by some points 
 in the pictures, and sketching in some defined lines which may serve as guides. 
 
 Following this practice of guide lines, it will be well to copy the outlines of 
 architectural mouldings, of which most of the ornaments are conventional rep- 
 resentations of natural objects. 
 
 At page 60 will be found an application to the drawing of acanthus leaves 
 within the guiding lines of squares and the designing for calicoes and woven 
 goods, oilcloths, ceiling, and wall ornamentation based on geometrical figures. 
 
 In such designs " a true artistic end has been accomplished when well-ob- 
 served features of natural objects have been chronicled within the convention- 
 alized limits of a few geometric rules that include proportion, symmetry, and 
 a proper subordination of one part to another." 
 
 To acquire still further readiness in free hand, extend the practice to " let- 
 tering." 
 
 MATERIAL. 
 
 Paper. The different papers manufactured by Whatman are excellent for 
 sketching purposes, and can be purchased either in sheets or in pads of various 
 sizes. Any toothed paper answers the purpose very well, such as common news- 
 paper ; a sketching paper with either a rough grain or a canvas grain may be 
 made by pinning a sheet of thin typewriting paper over a piece of sandpaper 
 or a canvas book in the same manner. 
 
 For pen-and-ink work a hard, unyielding surface is needed ; nothing an- 
 swers the purpose as well as the best quality of Bristol board, although excel- 
 lent results are obtained on Whatman's H. P. (Hot Pressed) paper. 
 
 Pencils. Faber's or Dixon's pencils of medium hardness are best for sketch- 
 ing, but for drawing on Bristol board the harder grades are better. 
 
 Lithographic chalks are now coming into use ; they are much superior to 
 the ordinary chalks, crayons, and charcoal in their not being readily smeared. 
 They find their best medium in Whatman's paper, either H. P. or " not," and 
 in grained scratch-out cardboards they give greater intensity than lead pencil, 
 and reproduce with more certainty. Of course they can not be used where 
 much detail is required, but for generalities they are excellent. 
 
 Pens. The use and selection of pens must be left largely to the draughts- 
 man's own judgment. The ordinary school pen is excellent ; Gillott's 303, or 
 
728 
 
 PEEE-HAND DRAWING. 
 
 the same maker's mapping pens, are also good. The amateur is cautioned 
 against the diminutive crow quill and lithographic pens; the pen is merely a 
 secondary matter, some illustrators doing excellent work with a brush used as 
 a pen, a reed pen, a toothpick, or, indeed, with anything that happens to be 
 at hand. 
 
 Ink. The best ink for reproductive purposes is India ink ; that which has 
 a dull appearance when dry is the best for this purpose ; India ink can be pur- 
 chased in the stick and ground down with water or ready prepared in bottles, 
 either waterproof or otherwise. 
 
 The first step in free-hand drawing is its application to simple objects, of 
 which one must learn to determine the relative proportion of their parts and to 
 lay them down on paper in their proper position and this entirely by eye. Hav- 
 ing thus drawn one object, one proceeds to increase the group of objects. As 
 has been shown in " Perspective," objects appear smaller as they are more re- 
 mote from the spectator, who must know how much the relative scale is changed, 
 not only in objects remote from each other, but also as to what parts of objects 
 can be seen and how they are seen. The rules of perspective give an idea of 
 what can be seen and the proportions, and serve to make the eye intelligent in 
 its observations to be confirmed and strengthened by practice. 
 
 In drawings of the human frame there are numerous charts and rules which 
 may be said to be established which may assist the learner in fixing the form 
 and proportions of parts within certain classical or normal limits. Outlines 
 from these charts may be traced for a brief time by the learner to acquire ideas 
 of the forms, proportions, and positions when at rest ; but when in action limbs 
 are moved, muscles increase under action, and the parts present different lines 
 of sight which must be studied by themselves. Although the length of limbs 
 is not increased, it may be more or less foreshortened. 
 
 " Proportions of the Human Frame" By Joseph Bonomi. 
 
 The following, with the illustrations, are taken from the above work : 
 
 " The human frame is (Figs. 1808 and 1809) divided into four equal 
 measures, by very distinctly marked divisions on its structure and outward 
 form : 
 
 " 1. From the crown of the head to a line drawn across the nipples. 
 
 " 2. From the nipples to the pubes. 
 
 " 3. From the pubes to the bottom of the patella (knee-pan). 
 
 " 4. From the bottom of the patella to the sole of the foot. 
 
 " Again, four measures, equal in themselves, and equal to those just de- 
 scribed, and as well marked in the structure of the human body, are seen when 
 the arms are extended horizontally. They are the following : 
 
 " From the tip of the middle or longest finger to the bend of the arm is 
 one fourth of the height of the person. 
 
 " From the bend of the arm to the pit of the neck is another fourth. 
 
 " These two measures, taken together, make the half of the man's height, 
 and with those of the opposite side equal the entire height. 
 
 " In the figures, the differences in width between the male and female figures 
 are given from the tables of the Count de Clarac of the Apollino and the Venus 
 de Medici. The male figure is in thicker line than the female, and the measure- 
 
FREE-HAND DRAWING. 
 
 729 
 
 ments referring to it are on your right hand, and those referring to the female 
 on your left. 
 
 " The measurements of length, according to Vitruvius and Leonardo da 
 Vinci, are the same in both sexes, and expressed in long horizontal lines run- 
 ning through both the front and profile figures. 
 
 " Almost innumerable are the varieties of character to be obtained by the 
 alterations of widths, without making any change in the measurements of 
 length ; nevertheless, some ancient statues differ slightly in these measurements 
 of length. 
 
 " No measurement is given in the figure of the width of the foot ; its normal 
 
730 
 
 FREE-HAND DRAWING. 
 
 proportion should be one sixteenth of the height. The views of the foot are 
 those of the female. 
 
 " The scale, V, used is 8 heads to the height ; parts, | of a head ; and min- 
 utes, jV of a part. 
 
 " The whole height is usually taken at 8 heads, but there are slight differ- 
 ences in the classic statues ; the height of the Venus de Medici is equal to 7 
 
FREE-HAND DRAWING. 7-3 1 
 
 heads, 3 parts, 10 minutes, that, of the Apollino of Florence, 7 heads, 3 parts, 6 
 minutes. 
 
 " When the student is acquainted with the forms of the body and limbs in 
 two aspects viz., the front and side views and the normal proportions they 
 bear to each other, then will follow the study of the characteristic features of 
 childhood, youth, and mature age, and those niceties of character that the 
 ancients invariably observed in the statues of their divinities, so that in most 
 cases a mere fragment of a statue could be identified as belonging to this or 
 that divinity as, for instance, the almost feminine roundness of the limbs of 
 the youthful Bacchus, the less round and distinctly marked muscles of the 
 Mercury, and of the statues of the Athletae." 
 
 Fig. 1811 is a half-tone reproduction of a photograph showing three views 
 of a plaster model, ecorche i. e., the body with the skin removed, showing the 
 muscles. In the half-tone process a ruled screen of glass is interposed between 
 the drawing or object to be photographed and the negative. The screen of 
 glass is closely ruled with lines crossing at right angles and etched with hy- 
 drofluoric acid ; into the grooves thus produced printing ink is rubbed. It is 
 these lines which produce the crisscross appearance seen in the resulting pic- 
 ture. This process is commonly used in reproducing wash drawings and pho- 
 tographs. 
 
 Figs. 1812, 1813, and 1814 are three views of the above figure drawn in line. 
 A large negative was taken and the print made on " plain salted paper " that 
 is, paper prepared without albumen, which gives to the ordinary print its glossy 
 appearance. This paper is made by being soaked in a solution of 
 
 Chlorate of ammonia 100 grains ; 
 
 Gelatin 10 " 
 
 Water 10 ounces. 
 
 The figures thus made may now be drawn in with pen and water-proof India 
 ink. The pen work should not attempt the fulness of detail given in the pho- 
 tograph. When the drawing has been finished it may be immersed in a solution 
 composed of 1 ounce of bichloride of mercury dissolved in 8 ounces of water, 
 which removes all trace of the photograph, leaving the drawing uninjured on 
 white paper. Omissions may now be supplied, but if there are any conspicuous, 
 the photograph may again be brought out by immersing in a solution of hypo- 
 sulphite of soda in water. 
 
 A readier way is to draw with water-proof ink upon photographs printed on 
 ferro-prussiate paper or blue print paper, the directions for making which will 
 be found on page 52, or it can be purchased. It can then be sent for reproduc- 
 tion as it is, as the blue will not photograph, or the blue may be bleached by 
 immersing the print in a dish of water in which a small piece of washing soda 
 has been dissolved ; then wash the print in clean water. 
 
 Both pen drawing and opaque water colour can be used on the ordinary pho- 
 tograph by mixing a small piece of ox gall with the liquid. 
 
 Figs. 1815 and 1816 are two views of Sandow taken from his photographs, 
 the pen work being executed as above and the photograph washed out. 
 
 In the " Dictionnaire Eaisonne de PArchitecture " of M. Viollet-le-Duc 
 
732 
 
 FREE-HAND DRAWING. 
 
 are given drawings from an album of the middle of the thirteenth century. Cer- 
 tain mechanical processes are given to facilitate the composition and design of 
 figures. According to these sketches, geometry is the generator of movements 
 
 of the human body,, and that of animals, and serves to establish certain relative 
 proportions of the figures. The pen sketch (Fig. 1817) is an example of this 
 practical process. In comparing this mode of drawing with figures in the vi- 
 gnettes of manuscripts, with designs on glass, and even with statues and bas-reliefs, 
 we must recognise the general employment in the thirteenth and fourteenth 
 centuries of these geometrical means, suited to give figures not only their pro- 
 portions but also the justness of their movement and bearing. Rectifying 
 
FREE-HAND DRAWING. 
 
 733 
 
 the canon of Yillard in its proportions by comparison with the best statues, 
 notably those in the interior of the western facade of the Cathedral of Reims, 
 we obtain the Fig. 1818. The line A B, the height of the human figure, is di- 
 
 FIG. 1815. 
 
734: 
 
 FREE-HAND DRAWING. 
 
 FIG. 1816. 
 
 vided into seven equal parts. The upper division is from the top of the head 
 to the shoulders. Let C D be the axis of the figure, the line at the breadth of 
 the shoulders is f of the whole height A B. The point E is the centre of the 
 
FREE-HAND DRAWING. 
 
 735 
 
 line C D ; draw through this point two lines, af and b e, and from the point g 
 two other lines, g e and gf. The line b h is the length of the humerus, and 
 
 FIG. 1817. 
 
 FIG. 1818. 
 
 the line of the knee-pan is on i k. The length of the foot is f of a division, A 1. 
 Having established these proportions, it will be seen by the following cuts how 
 the artisan gave movements to these figures when the movements were not in 
 absolute profile. 
 
 Suppose the weight of the figure to be borne upon one leg (Fig. 1819), the 
 line g e becomes perpendicular, and the axis op of the figure is inclined. The 
 movement of the shoulders and trunk follow this inflection ; the axis of the 
 head and the right heel are in the same vertical line. 
 
 In stepping up (Fig. 1820), the axis of the figure is vertical, and the right 
 heel raised is on the inclined line s , while the line of the neck is on the line 
 Im, and the trunk is vertical, 
 
 In Fig. 1821 it will be seen how a figure can be submitted to a violent move- 
 ment and yet preserve the same geometrical trace. The figure is fallen, sup- 
 ported on one knee and one arm, while the other wards off a blow ; the head 
 is vertical. 
 
 In Fig. 1822, the left thigh being in the line a/, to determine the position 
 of the heel c on the ground, supposed to be level, an arc is to be described from 
 the knee-pan ; the line ef is horizontal. 
 
 It is clear that, in adopting these practical methods, all the limbs can be 
 developed geometrically without shortening. The above will supply to many a 
 
736 
 
 FREE-HAND DRAWING. 
 
 ready means of sketching the human figure in various attitudes, naked, or in 
 the close-fitting dresses of the present fashion ; but in the arrangement of 
 
 FIG. 1819. 
 
 FIG. 1820. 
 
 drapery upon a figure, care must be taken that the drapery should fall in grace- 
 ful folds. " It is necessary to give the body certain inflections which would be 
 ridiculous in a person walking naked. The walk should be from the hips, with 
 
 FIG. 1821. 
 
FREE-HAND DRAWING. 737 
 
 wide-spread legs, and, by the movements of the trunk, make the drapery cling 
 on certain parts and float on others." 
 
 In figures in repose, their centres of gravity must fall within the points of 
 support, but the body can be sustained by muscular exertion, and this should 
 be expressed in such cases by the tension of the muscles on which the position 
 depends. In the act of running, the body inclines forward, its weight assists 
 the movement, and the motions prevent its falling. 
 
 In drawing figures it will be understood that the part that lies behind an- 
 other can not be seen, and that one side of a limb or of the body can not be 
 turned toward you without turning the other from you, and that the length of 
 a body or limb can only be shown in its full length and proportion when it is 
 
 e 
 
 FIG. 1822. 
 
 perpendicular to the line of sight that is, if the arm, for instance, is directed 
 toward the eye, the hand will be the prominent object in view, that the arm will 
 only be shown by the portions prominent beyond the outlines of the hand, that 
 limbs or portions more or less inclined to the line of sight will be more or less 
 foreshortened or show less than their natural length. 
 
 It is very common in the drawing of figures to indicate merely the centre 
 lines of the various portions and then clothe them as shown in several of the 
 figures. 
 
 It is very common with modern draughtsmen to adopt a somewhat similar 
 form of sketching as given in the " Dictionnaire Raisonne de 1'Architecture "- 
 a framework of bones in positions of action and then clothing them with flesh 
 or drapery ; or manikins and lay figures in which the limbs can be set or fixed 
 as desired, and then drawn from as models. Figs. 1823 and 1824 are illustra- 
 tions from Dr. Eimmer's " Elements of Design " of skeleton lines and of manikins. 
 
 Fig. 1825 is taken from photographs of a manikin in our office, which, al- 
 though maimed as to its hands, is one of the best forms of these jointed figures, 
 of which the limbs and body can be set in any required position ; but the sug- 
 gestion is obvious that by the salted-paper or blue-print process, nude figures 
 can be used instead of manikins, and photographs promptly secured without 
 fatigue to the model and in positions of motion impossible in sketching. As a 
 further illustration of this process, a pen drawing is given of the Venus de Milo 
 (Figs. 1826-1827) from two points of view, taken from a plaster model, and a 
 portrait of Alexandre Dumas (Fig. 1858) from a photograph. 
 48 
 
738 
 
 FREE-HAND DRAWING. 
 
 FIG. 1823. 
 
 FIG. 1824. 
 
 FIG. 1825. 
 
FREE-HAND DRAWING. 
 
 Y39 
 
 Artists object that photography is too exact a reproduction, but it is well 
 that they should understand what is an exact reproduction. Tint and colour 
 may produce pleasant impressions without conformity to laws of perspective, but 
 if the picture is to be taken as whole it should be natural, and with the present 
 processes of photography it is well to throw the mechanical drudgery on it. 
 
 FIG. 1826. 
 
 FIG. 1827 
 
 Figs. 1828-1831 are drawings of male hands, Figs. 1832-1838 of legs and 
 feet, with guide-lines to assist the draughtsman. 
 
 Figs. 1839-1841 are drawings of female hands and arms. 
 
 Figs. 1842-1845 are hands and feet of children. 
 
 Figs. 1846 and 1847 are illustrations of the human head and face. 
 
 FIG. 1828. 
 
 FIG. 1829. 
 
 FIG. 1830. 
 
 FIG. 1831. 
 
740 
 
 FREE-HAND DRAWING. 
 
 FIQ. 1837. 
 
 FIG. 1838. 
 
 FIG. 1839. 
 
 Fia. 1840. 
 
 FIG. 1841. 
 
FREE-HAND DRAWING. 
 
 FIG. 1846. 
 
 FIG. 1847. 
 
742 
 
 FREE-HAND DRAWING. 
 
 ELECTIONEER. 
 
 The Forms of Animals. The bodies of most quadrupeds can be included 
 in rectangles as guide-lines, which may be drawn around the illustration of the 
 cow and horse (Figs. 1848 and 1849). The action of the limbs of quadrupeds is 
 chiefly directly forward or directly backward, the -power of lateral motion being 
 limited. The hinder limbs always commence progressive motion, as in the first 
 position of the walk, the fore foot of the same side advances next, then the 
 hind foot of the opposite side, and lastly the fore foot on that side, and so on. 
 In the trot, the hinder leg of one side and the fore leg of the other are raised 
 together. In the canter or gallop, both fore legs and one hind leg are raised 
 together ; when rapidly moving, the two fore legs and two hind legs appear to 
 advance together. In fact, all the movements are rather resultants, as they 
 appear to us, but when instantaneously photographed the legs are wonderfully 
 mixed. 
 
 FIG. 1848. 
 
FREE-HAND DRAWING. 
 
 743 
 
 FIG. 1849. 
 
 Fig. 1850 is one of Landseer's sketches. 
 
 The forms of feet range under two great divisions hoofs (Fig. 1851) and 
 paws (Fig. 1852). All hoofs, whether whole or cloven, approximate to a right- 
 angled triangle, and all paws to a rhomboid. 
 
744 
 
 FREE-HAND DRAWING. 
 
 The J\ ! oses of Animals. Fig. 1853 represents that of the horse ; Fig. 1854, 
 that of the ox and deer tribe; Fig. 1855, those of the carnivori; Fig. 1856, 
 those of the camel, sheep, and goat tribes ; and Fig. 1857, those of the hog 
 tribes. The muzzles of nearly all quadrupeds will be found to range under 
 one or other of these classes, with minute variations to characterize the differ- 
 ent species and individuals. 
 
 FIG. 1853. 
 
 FIG. 1854. 
 
 FIG. 1857. 
 
 In looking over the many sketches and engravings of Landseer which have 
 been published, it will be noticed in how varied a manner they were executed. 
 Sometimes in mere outline with lead-pencil, sometimes with a camel's-hair 
 pencil charged with India ink or sepia for the outlines, giving effect to the 
 subject by slight tints or washes of the same colour ; in others, pen and ink 
 have been alone employed ; some are in oils, others in water-colours ; fre- 
 quently chalks, both black and coloured. " As we look at some of these, we are 
 tempted to believe that, of all the instruments that can be used by the artist, 
 there is none quite so wonderful as the pen. A simple sketch with a pen or 
 lead-pencil is naked, unadorned truth, bearing witness to the skill or its oppo- 
 site of the hand which produced it." 
 
 Strength and boldness in outline are acquired by large scale in drawing ; 
 and in copying, if suitable originals can be obtained, copy them, but if you are 
 confined to the illustration of books and periodicals, recollect that they have 
 usually been reproduced on a reduced scale, and make your drawings two or 
 three times their lineal dimensions. 
 
 The directions and illustrations already given may be considered copying, 
 which is absolutely necessary as an introduction to free-hand drawing, and ex- 
 amples have been selected from figure drawing to give the draughtsman a 
 strong bold hand and an education in proportions. 
 
FREE-HAND DRAWING. 
 
 745 
 
 Sketching from Nature. It is useless to give detailed rules for sketching, 
 as each has his own way and perhaps equally good though dissimilar. If one is 
 inside the house, what one sees through a window is a picture, and it may be 
 transferred to paper if the eye is kept in a single position by a sight fastened 
 to the sash at a convenient distance ; and if the pane of glass be prepared in 
 squares, as described on page 60, the location of different points can be readily 
 established. A simple plan to begin with is to set out carefully the most con- 
 spicuous outline and draw the others with reference to it. If doubtful as to 
 the distances and length of lines, stretch out your arm, holding your pencil 
 vertically, horizontally, or aslant, as the case may be, and shutting one eye, 
 mark off the measure from the end of the pencil by placing your thumb on the 
 spot ; then compare that line or space with any other needed. By noting the 
 relative position of one object with reference to another, all will fall into place 
 almost without thought. Thus you have sketched a house, notice the next ob- 
 ject and observe where it is projected against the building, as at such a window 
 or door, and draw it in. Every artist has his own method of handling the 
 sketch, of using his pen or pencil, charcoal or brush ; the aim of each should 
 be to express what he sees by cross-lines and hatching if the pen is used, or by 
 scribbling if the work is in pencil or charcoal. A general rule is to adapt your 
 strokes as far as possible to the modelling of the subjects you are drawing. 
 Thus if you are representing water, it is natural to do it with horizontal lines. 
 If the water is in motion, the lines, though still horizontal, will be broken and 
 irregular. The reflections which in still water will be represented by vertical 
 lines, in running water are indicated by horizontal lines with closer shading to 
 give depth of tone. 
 
 In drawing the trunk of a tree the characteristics of the bark must be ob- 
 served. The trunk of an oak is rough and broken, while that of the birch 
 requires curved lines across the thickness of the tree. Nature sufficiently indi- 
 cates the treatment. In many cases one must not be content with an exact 
 reproduction. The scene must be interpreted. 
 
746 
 
 FREE-HAND DRAWING. 
 
FREE-HAND DRAWING. 
 
 747 
 
 In all sketching from Nature- the lines must be crisply and unhesitatingly 
 drawn, the forms clearly denned, and the masses decisively indicated. In rapid 
 work a mere outline must suffice, but it must be clearly and cleanly pencilled 
 in. When an elaborate drawing is required, begin by indicating the general 
 arrangement, and then fill in as much detail as may be necessary ; but it is to 
 be remembered that the simplest expression is the best and at the same time 
 the most difficult, the effort being to concentrate the effect on the object to be 
 emphasized, all others being subordinate to it. 
 
 When a sketch has been faithfully made from Nature, as a general rule it is 
 better not to try to improve it afterward, as one is apt to lose the crispness and 
 vigour of the original. It is better to modify or amplify the first notes on an- 
 other sheet of paper. Neither should there be many erasures, as they impart 
 to the paper a dingy appearance and the sketch loses its sharpness. 
 
 The depth of tone of shaded portions, as clouds, in some hasty sketches is 
 indicated by numerals or letters, 1 or A representing the lightest; but it is 
 better if time is afforded to put the shading in the sketch. In sketches of 
 landscapes not intended for reproduction the addition of colour often adds very 
 much to the effect and value of the sketch, but the colour must be light and 
 transparent. 
 
 The foregoing applies chiefly to landscape sketching. If you desire to 
 
 FIG. 1861. 
 
748 
 
 FREE-HAND DRAWING. 
 
 sketch figures or animals, the character of the individual is what you must try 
 to seize ; if an animal in motion, one must work quickly and await the repeti- 
 tion of the action or particular movement you are sketching. Meanwhile em- 
 
 ploy your pencil on those portions that remain longer in one position, except- 
 ing to work on the moving limb the instant it has regained the required 
 position. The sketching of animals in rapid motion is still more difficult and 
 is to a great extent a matter of careful observation and memory, for the limbs 
 are not the only parts that change their position ; the whole attitude of the 
 body is changed. In addition to memorizing all possible of the movement, 
 
FREE-HAND DRAWING. 
 
 T49 
 
 considerable aid may be obtained in observing the animal at rest, which will 
 enable you to understand the details of the structure. 
 
 All sketches made should be preserved for future use, as they furnish the 
 best material for original work that one can have. Sketches made in pencil or 
 other materials which smear may be fixed with a solution composed of one part 
 of gum mastic and seven parts of methylated spirits of wine, and is best applied 
 with an atomizer. 
 
 In transferring your sketches to the paper on which the pen drawing is to 
 be made, commence by making a careful drawing with a hard pencil in outlines, 
 confining yourself to those of shadows as much as possible, to save the surface 
 of the paper, as rubbing spoils it and grays the ink ; or make the drawing on 
 another sheet of paper and transfer it by means of graphite paper to the fair 
 sheet, then ink in. 
 
 If you want a clean, sharp line, the ink must be perfectly black and must 
 stand out alone on the paper. If you want a gray line, it will not be obtained 
 by using light ink, but by making thin lines. A single thin line will come out 
 in the reproduction much blacker than is the intention, for though a number 
 of lines will stand together, a single one will have to be thickened in the type 
 metal by the photo-engraver. 
 
 Fm. 1851). 
 
 FIG. 1S58. 
 
 Fig. 1858, portrait of Alexander Dumas. For description of process see 
 page 738. 
 
 Fig. 1859 is a portrait of Erik Werenskiold, the artist, drawn by himself on 
 Whatman's paper. 
 
 Wash drawings are made with a brush on either Bristol board or water-colour 
 paper, the wash consisting of India ink and water, of various degrees of inten- 
 sity. A number of illustrations are given of this method on page 750, by Paul 
 de Longpre, and reproduced in half tone. 
 
 Fig. 1860 is a design for a small pumping station, drawn Avith India ink 
 and a toothpick, after W. R. Emerson, in the " Technology Architectural Re- 
 view." 
 
750 
 
 FREE-HAND DRAWING. 
 
FREE-HAND DRAWING. 75} 
 
 There are many devices which may be used with effect in pen-and-ink draw- 
 ing, such as splatter work, which" is done by using a small stiff bristle brush (a 
 toothbrush answers the purpose very well), inking it, and holding the bristles 
 downward and inclining toward the drawing and stroking the bristles toward 
 one with a match stick. All parts not intended to be splattered should be cov- 
 ered with paper masks, and even then a waterproof ink must be used in case it 
 is necessary to paint out some portions with Chinese white. The lower portion 
 of Fig. 1860 is in splatter. An inked thumb is also a novel and effective means 
 of representing a background or an imitation of velvet, the lines on the skin 
 being marked on the paper and reproduced excellently. 
 
 An invention of recent years is known among artists as stipple paper or 
 clay board. The surfaces of these cardboards are of various kinds, but are all 
 prepared with a surface of china clay. The simplest variety is that prepared 
 with a plain white surface, upon w.hich the drawing is executed with pen and 
 ink or brush ; the lights are taken out with a sharp ink eraser. It is usual to 
 work upon these boards with a pigmental ink, such as lampblack, ivory black, 
 or India ink. More liquid inks have a tendency to soak through the prepared 
 surface rather than rest upon it, rendering the board useless for scratch-out 
 purposes ; other kinds of boards are impressed with a grain or with plain in- 
 dented lines, which are used similar to the above. Scratch-out boards are diffi- 
 cult of manipulation and are not to be recommended to the amateur. 
 
 Fig. 1861, a portrait of Salvini, is an example of the use of stipple paper. 
 The background or middle tone shows the board in its natural state. The 
 high lights and shadows are obtained respectively by erasing and adding India 
 ink. 
 
 Fig. 1862, " A Venetian Fete on the Seine," is another specimen of work on 
 clay board. In this case the board was entirely black, the various tones being 
 obtained by various degrees of erasing. 
 
 The appearance a drawing will present when reduced may be approximate- 
 ly judged by the use of a " diminishing glass " that is, a concave glass. 
 
 To remove blots or make erasures, use an ink eraser or simply paste a piece 
 of paper over and join the lines at the edges ; or a neater way, cut out the 
 blotted part and paste a piece of paper underneath. 
 
 To clean pen-and-ink drawings use bread one or two days old and not rubber. 
 
 Aerial perspective, or the tones of lights and shadows according to their 
 distance from the observer and the sources of the light, will be acquired by 
 studies of pictures and observations of Nature. The rule in drawing from 
 Nature is to draw only what you see and express it in the most truthful form. 
 
"52 
 
 FREE-HAND DRAWING. 
 
 After a Pen-and-ink Design, by FORTUNT. 
 
FREE-BAND DRAWING. 
 
 753 
 
 G. L. SEYMOUR. 
 
 49 
 
754 
 
 FREE-HAND DRAWING. 
 
FREE-HAND DRAWING. 
 
 755 
 
756 
 
 FREE-HAND DRAWING. 
 
 Study of Oak- Trees', R. LANDSEER. 
 
FREE-HAXD DRAWING. 
 
 757 
 
 Morniny. II. W. ROBBINS. 
 
T58 
 
 FREE-HAND DRAWING. 
 
 Cattle going Home. JAMES M. HART. 
 
FREE-HAND DRAWING. 
 
 759 
 
760 
 
 FREE-HAND DRAWING. 
 
 The Lady of the Woods, 
 
FREE-HAND DRAWING. 
 
 761 
 
762 
 
 FREE-HAND DRAWING. 
 
 rKsV, -.: ;-;. J i f ', 
 
 !W,- VtvX^'-i!'- 1 -''/': ; I ' .1 
 
 '""*Jt<;>'V\ ' , V'V si\ i k'\ J . r:*- 1 '*-" 
 
 C*S?^^%B ' '4' ^ ' -^"H^^ 
 
 ^^apK' ^JKiH'V.i'lS 
 
 : 
 
 
FREE-HAND DRAWING. 
 
 763 
 
764 
 
 FREE-HAND DRAWING. 
 
PL. I. 
 
 Rg.12. 
 
 Fig.ll. I 
 
 Fig.W. 
 
 Fig. 9 
 
 Fig.8. 
 
 Fig. 7. 
 
 Fig. 6. 
 
 a c 
 
 T/ V 
 
PL. I. 
 
 Fig. 4-. 
 
 Rg.3. 
 
 Fig 2. 
 
 Fig.l. 
 
 a _ C 
 
 Fig.5. 
 
 0,2 
 
pL.nr. 
 
PL.IV: 
 
 FIG.4. 
 
 FIG.8 
 
 FIG.7. 
 
PL.Y 
 
Ill 
 
 * 
 I * 
 
 r~--r -_---T: J 
 
 !'i!i!ii ' ' ! 'ilnfj ' 'i 'i'l ! 
 I'i'i'i 1 i ' 'n" ' 'i i'ii 
 
 ijljijij i i [{ijjjj i v j|i] ; 
 
 I'lil'ii ! i 'ill'l ' ill 'I' 1 '! 
 ii 
 
 > i) 
 
 
 ...;ft< 
 
 * C* 
 
 i c 
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 IIJ ' 
 
 j 
 
 A 
 
STATE N ISLAND 
 
 CONTOURS 20 FT. APART. 
 
IX. 
 
PLX. 
 GEOLOGICAL MAP OF 
 
 NEW JERSEY 
 
 GEORGE . H.C OOK, STATE GE O LO GIST . 
 
PL. XII. 
 
 "Topographical Map of Massachusetts" 
 
PL.XIII. 
 
PLXY 
 
PL.XVI. 
 
PLXVII. 
 
APPENDIX. 
 
 PATENT-OFFICE DRAWINGS 
 
 must be made upon pure white paper, of a thickness corresponding to three-sheet Bristol 
 board, with surface calendered and smooth. India ink alone must be used. 
 
 The size of the sheet must be exactly 10 by 15 inches. 1" from its edges single mar- 
 ginal lines are to be drawn, leaving the "sight " precisely 8" by 13". Within this mar- 
 gin all work must be included. Measuring downward from the marginal line of one of 
 the shorter sides, a space of not less than 1J inch is to be left blank for the heading of 
 title, name, number, and date. 
 
 All drawings must be made with the pen only. All lines and letters must be abso- 
 lutely black, clean, sharp, and solid, and not too fine or crowded. Surface shading 
 should be open, and used only on convex and concave surfaces sparingly. Sectional 
 shading should be made by oblique parallel lines, which may be about -" apart. 
 
 Drawings should be made with the fewest lines possible consistent with clearness. 
 The plane upon sectional views should be indicated on the general view by broken or 
 dotted lines. Heavy lines on the shade sides of objects should be used, except where 
 they tend to thicken the work and obscure letters of reference : light to come from the 
 upper left-hand corner, at an angle of 45. 
 
 The scale of the drawing to be large enough to show the mechanism without crowd- 
 ing; but the number of sheets must never be increased unless it is absolutely neaessary. 
 
 Letters and figures of reference must be carefully formed, and, if possible, measure at 
 least \" in height, and so placed as not to interfere with a thorough comprehension of the 
 drawing, and therefore should rarely cross the lines. Upon shaded surfaces a blank space 
 must be left in the shading for the letter. The same part of an invention must always 
 be represented by the same character, and the same character must never be used to 
 designate different parts. 
 
 The signature of the inventor, by himself or by his attorney, is to be placed at the 
 lower right-hand corner of the sheet, and the signature of two witnesses at the lower left- 
 hand corner, all within the marginal line. The title is to be written with pencil on the 
 back of the sheet. The permanent names and title will be supplied subsequently by the 
 office in uniform style. 
 
 Drawings should not be folded for transmission to the office. 
 
 REGISTRATION OF PRINTS AND LABELS. 
 
 A label is a device or representation borne by an article of manufacture or vendible 
 commodity. A print is a device or representation not borne by an article of manufac- 
 ture or vendible commodity, but in some fashion pertaining thereto. A label can not 
 be registered if it bear a device capable of sequestration as a trade-mark until after such 
 device is registered as a trade-mark. Both labels and prints, in order to be entitled to 
 registry, must be intellectual productions in the degree required by the copyright law. 
 
 765 
 
766 APPENDIX. 
 
 MENSURATION. 
 
 The principles of measurement have been quite fully explained under the heads of the 
 Construction of Geometrical Problems and Plotting, but for ready reference there are 
 many rules which are of general application and very necessary in designing and calcula- 
 tion, and are given briefly as follows : 
 To find 
 
 The area of a parallelogram. Multiply the length by the height or perpendicular by 
 the breadth. Multiply the product of two contiguous sides by the natural sine of the 
 included angle. (See Appendix, Table of Natural Sines.) 
 
 The area of a triangle. Multiply the base by the perpendicular height and take half 
 the product. Multiply half the product of two contiguous sides by the natural sine of 
 the included angle. 
 
 The area of a trapezoid. Multiply half the sum of the parallel sides by the perpen- 
 dicular distance between them. 
 
 The area of any quadrilateral figure. Divide the quadrilateral into two triangles ; the 
 sum of the areas of the triangles is the area. 
 
 The area of any polygon. Divide the polygon into triangles and take the sum of their 
 areas. 
 
 The circumference of a circle. Multiply the diameter by 3'1416 = TT. 
 
 The diameter of a circle. Multiply the circumference by the reciprocal of TT. 
 
 The area of a circle. Multiply the square of the diameter by '7854, or the circumfer- 
 ence by one fourth of the diameter. 
 
 The length of an arc of a circle. Multiply the number of degrees in the arc by the 
 radius, and by '01745. 
 
 NOTE. The length of an arc of one degree = radius x '017453. 
 
 " " " " " " minute = " x '000291. 
 
 " " " " " " " second = " x '000005. 
 
 The area of a sector of a circle. Multiply the length of the arc of the sector by half 
 the radius. 
 
 The area of a segment of a circle. From the area of a sector subtract the area of the 
 triangle formed by the radial sides of the sector and its chord. The area of this triangle 
 is the product of the natural sine and cosine by the square of the radial side. 
 
 The area of regular polygons. Find the area of one triangle, and multiply by the 
 number of triangles composing the polygon : 
 
 Or, multiply the total of cosines for the periphery by one half the sine, by the square 
 of the radius for the area. 
 
 No. of Verti- 
 4 Chord. Sides. cal. Product. 
 
 Pentagon -5878 x 5 x -8090 - 2-3776 
 
 Hexagon -5000 x (i x '8660 = 2-5980 
 
 Heptagon -4357 x 7 x -9032 = 2-7478 
 
 Octagon -3827 x 8 x -9238 = 2-8284 
 
 Nonagon -3420 x 9 x '9396 = 2-8925 
 
 Decagon -3090 x 10 x -9510 = 2-9389 
 
 Undecagon -2817 x 11 x -9594 = 2-9739 
 
 Dodecagon -2588 x 12 x -9660 = 3-0000 
 
 Circle = TT = 3-1416 
 
 To find 
 
 The area of the cycloid. Multiply the area of the generating circle by 3. 
 To find 
 
 The area of the parabola. Multiply the base by the height; two thirds of the product 
 is the area. 
 
APPENDIX. 767 
 
 The circumference of an ellipse. Multiply the square root of half the sum of the 
 squares of the two axes by 3 - 1416. 
 
 The area of an ellipse. Multiply the product of the two axes by -7854 = J TT. 
 
 NOTE. The area of an ellipse is equal to the area of a circle of which the diameter is a 
 mean proportional between the two axes. 
 
 To find the area of any curvilineal figure, bounded at the ends by parallel straight 
 lines (Fig. 198). Divide the length of the figure into any number of equal parts and draw 
 ordinates through the points of division, to touch the boundary lines. Or, add together 
 the first and last ordinates, making the sum A ; and add together all the intermediate 
 ordinates, making the sum B. Let L = the length of the figure, and n = the number of 
 divisions, then 
 
 A + 2B 
 
 = x L = area of figure. 
 
 That is to say, twice the sum of the intermediate ordinates, plus the first and last ordi- 
 nates, divided by twice the number of divisions, and multiplied by the length, is equal 
 to the area of the figure. This method is sufficiently exact for most purposes. 
 To find 
 
 The surface of a prism or cylinder. To the product of the perimeter of the end by 
 the height, add twice the area of an end. 
 
 The cubic contents of a prism or a cylinder. Multiply the area of the base by the 
 height. 
 
 The surface of a pyramid or cone. Multiply the perimeter of the base by half the 
 slant height, and add the area of the base. 
 
 The cubic contents of a pyramid or a cone. Multiply the area of the base by one third 
 of the perpendicular height. 
 
 The surface of a frustum of a pyramid or a cone. Multiply the sum of the perime- 
 ters of the ends by half the slant height, and add the areas of the ends. 
 
 The cubic contents of the frustum of a pyramid or a cone. Add together the areas 
 of the two ends and the mean proportional between them (that is, the square root of their 
 product) and multiply the sum by one third of the perpendicular height. 
 
 The cubic contents of a wedge. To twice the length of the base add the length of 
 the edge ; multiply the sum by the breadth of the base, and by one sixth of the height. 
 
 The cubic contents of a prismoid (a solid of which the two ends are unequal but par- 
 allel plane figures of the same number of sides). To the sum of the area of the two ends 
 add four times the area of a section parallel to and equally distant from both ends ; and 
 multiply the sum by one sixth of the length. 
 
 The surface of a sphere. Multiply the square of the diameter by 3 '1416. 
 
 The surface of a sphere is equal to four times the area of one of its great circles. 
 
 The surface of a sphere is equal to the convex surface of its circumscribing cylinder. 
 
 The surfaces of spheres are to one another as the squares of their diameters. 
 
 The curve surface of any segment or zone of a sphere. Multiply the diameter of the 
 sphere by the height of the zone or segment, and by 3 '1416. 
 
 The cubic contents of a sphere. Multiply the cube of the diameter by -5236 = IT. 
 
 The cubic contents of the segment of a sphere. From three times the diameter of the 
 sphere subtract twice the height of the segment ; multiply the difference by the square of 
 the height, and by -5236. 
 
 The cubic contents of a frustum or zone of a sphere. To the sum of the squares of 
 the radii of the ends add of the square of the height; multiply the sum by the height 
 and by 1'5708 = \ TT. 
 
 The cubic contents of a spheroid. (A solid body generated by the revolution of an 
 ellipse around one of its axes.) Multiply the square of the revolving axis by the fixed 
 axis and by '5236. 
 
708 
 
 APPENDIX. 
 
 LINEAL MEASUEE. 
 
 Inches. 
 
 1 - 
 
 12 = 
 
 36 = 
 
 72 = 
 
 7-92 = 
 
 198 = 
 
 792 = 
 
 7920 = 
 
 63360 = 
 
 39-3685 = 
 
 Feet. 
 
 08333 
 1 
 3 
 6 
 
 0-66 
 
 16 J 
 
 66 
 
 660 
 
 5280 
 
 6086-07 
 
 3-2807 
 
 Yards. 
 
 Fath- 
 oms. 
 
 Links. 
 
 Rods. 
 
 Chains. 
 
 Furlongs 
 
 Statute 
 miles. 
 
 Nautical 
 miles. 
 
 Metres. 
 
 02778 
 
 0139 
 
 126 
 
 005 
 
 00126 
 
 000126 
 
 000016 
 
 
 0254 
 
 333 
 
 1667 
 
 1-515 
 
 0606 
 
 0151 
 
 00151 
 
 00019 
 
 .... 
 
 0-3048 
 
 1 
 
 5 
 
 4-545 
 
 182 
 
 0454 
 
 00454 
 
 00057 
 
 
 0-9144 
 
 2 
 
 1 
 
 9-1 
 
 364 
 
 091 
 
 0091 
 
 00114 
 
 
 1-8289 
 
 22 
 
 11 
 
 1 
 
 04 
 
 01 
 
 001 
 
 000125 
 
 
 2012 
 
 5^ 
 
 2f 
 
 25 
 
 1 
 
 25 
 
 025 
 
 003125 
 
 
 5-0294 
 
 22\ 
 
 11 
 
 100 
 
 4 
 
 1 
 
 10 
 
 0125 
 
 
 20-118 
 
 220 
 
 \ 110 
 
 1000 
 
 40 
 
 10 
 
 1 
 
 125 
 
 
 201-18 
 
 1760 
 
 E80 
 
 8000 
 
 320 
 
 80 
 
 8 
 
 ] 
 
 0-86755 
 
 1609-41 
 
 2028-69 
 
 
 
 
 
 
 1-1527 
 
 1 
 
 1855-11 
 
 1-0936 
 
 5468 
 
 
 
 
 
 0-000621 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 Latin prefixes, as milli-, centi-, deci-, to the French units of length (metre), surface (are), weight 
 (gramme), or volume (litre), signify YIOOO, l /ioo, or '/io of the unit ; as, millimetre, '/iooo of a metre, 
 decigramme, '/ID of a gramme. Greek prefixes, as kilo, hekto, deka, multiples of the unit by 1,000. 
 100, or 10, as kilometre = 1000 metres. 
 
 r 
 
 MEASUEES OF SURFACE. 
 
 j. inches. 
 
 8q. feet. 
 
 Sq. yards. 
 
 Sq. rods. 
 
 Roods. 
 
 Acres. 
 
 Sq. miles. 
 
 Sq. metres. 
 
 Ares. 
 
 1 = 
 
 00894 
 
 
 
 
 
 
 
 
 144 = 
 
 1 
 
 Ill 
 
 0037 
 
 
 
 
 0929 
 
 0009 
 
 1296 = 
 
 9 
 
 1 
 
 033 
 
 
 
 
 8361 
 
 0084 
 
 .... 
 
 272i 
 
 30J 
 
 1 
 
 025 
 
 00625 
 
 
 25-293 
 
 0-253 
 
 
 10890 
 
 1210 
 
 40 
 
 1 
 
 25 
 
 
 
 
 
 43560 
 
 4840 
 
 160 
 
 4 
 
 1 
 
 00156 
 
 4046-86 
 
 40-47 
 
 
 27878400 
 
 3097600 
 
 
 .... 
 
 640 
 
 1 
 
 
 25899 
 
 549-8 = 
 
 10-763 
 
 1-196 
 
 0395 
 
 0009 
 
 000247 
 
 
 1 
 
 01 
 
 .... 
 
 1076-31 
 
 119-60 
 
 
 
 02471 
 
 
 100 
 
 1 
 
 BOARD AND TIMBER MEASURE. 
 
 In board measure boards are assumed at one inch in thickness. To obtain the num- 
 ber of feet, board measure (B. M.), of a board or stick of square timber, multiply to- 
 gether the length in feet, the breadth in feet, and the thickness in inches. 
 
 To Compute the Volume of Round Timber. When all dimensions are in feet, multiply 
 the length by one quarter of the product of the mean girth and diameter, and the prod- 
 uct will give the measurement in cubic feet. 
 
 For Square Timber. When all dimensions are in feet, multiply together the length, 
 breadth, and depth : the product will be the volume in cubic feet. When one dimen- 
 sion is given in inches, divide by 12; when two dimensions are given in inches, divide 
 by 144; when all three dimensions are given in inches, divide by 1728. 
 
APPENDIX. 
 
 MEASURES OF CAPACITY. 
 LIQUID MEASURE. 
 
 769 
 
 Gills. 
 
 Pints. 
 
 Quarts. 
 
 Gallons. 
 
 Imp. gallons. 
 
 Litres. 
 
 Cubic feet 
 
 Cubic in. 
 
 Lbs. water 
 at 62. 
 
 I 
 
 0-25 
 
 0-125 
 
 03125 
 
 026 
 
 1183 
 
 0042 
 
 7-219 
 
 26 
 
 4 = 
 
 1 
 
 0-5 
 
 0-125 
 
 1041 
 
 4731 
 
 01671 
 
 28-875 
 
 1-0412 
 
 Bss 
 
 2 
 
 1 
 
 0-25 
 
 2083 
 
 0-9463 
 
 03342 
 
 57-75 
 
 2-0825 
 
 32 = 
 
 8 
 
 4 
 
 1 
 
 0-8331 
 
 3-7852 
 
 0-1337 
 
 231 
 
 8-33 
 
 38-4096 = 
 
 9-6024 
 
 4-8012 
 
 1-2003 
 
 1 
 
 4-5435 
 
 0-1605 
 
 277-27 
 
 10-00 
 
 8-4534 = 
 
 2-1133 
 
 1-0567 
 
 0-26417 
 
 0-2201 
 
 1 
 
 0-0353 
 
 61-0279 
 
 2-2007 
 
 239-36 = 
 
 59-84 
 
 29-92 
 
 7-48 
 
 6-232 
 
 28-320 
 
 1 
 
 1728 
 
 62-321 
 
 138528 = 
 
 034632 
 
 017316 
 
 004329 
 
 0036 
 
 0-01639 
 
 0-000579 
 
 1 
 
 03606 
 
 
 
 
 
 
 
 01604 
 
 27-727 
 
 1 
 
 
 
 
 
 
 
 
 
 
 1 barrel = 31J gallons. Reciprocal = -03175. 
 
 DRY MEASURE. 
 
 Pints. 
 
 Quarts. 
 
 Gallons. 
 
 Pecks. 
 
 Bushels. 
 
 1 = 
 
 0-50 
 
 0-125 
 
 0625 
 
 0-01562 
 
 2 = 
 
 1 
 
 0-25 
 
 0-125 
 
 0-0312 
 
 8 = 
 
 4 
 
 1 
 
 0-50 
 
 0-125 
 
 16 = 
 
 8 
 
 2 
 
 1 
 
 0-^5 
 
 64 = 
 
 32 
 
 8 
 
 4 
 
 i 
 
 The standard bushel contains 2150-42 cubic inches. 
 
 WEIGHTS. 
 
 APOTHECARIES'. 
 
 Grains. 
 
 Scruples. 
 
 Drachms. 
 
 Ounces. 
 
 Pounds. 
 
 1 = 
 
 05 
 
 0167 
 
 0021 
 
 00018 
 
 20 = 
 
 1 
 
 333 
 
 042 
 
 0035 
 
 60 = 
 
 3 
 
 1 
 
 125 
 
 0104 
 
 480 = 
 
 24 
 
 8 
 
 1 
 
 083 
 
 5760 = 
 
 288 
 
 96 
 
 12 
 
 1 
 
 TROY. 
 
 Grains. 
 
 Pennyweights. 
 
 Ounces. 
 
 Pounds. 
 
 1 = 
 
 042 
 
 0021 
 
 00018 
 
 24 = 
 
 1 
 
 05 
 
 0042 
 
 480 = 
 
 20 
 
 1 
 
 083 
 
 5760 = 
 
 240 
 
 12 
 
 1 
 
 AVOIRDUPOIS. 
 
 Drachms. 
 
 Ounces. 
 
 Pounds. 
 
 Hundred-weights. 
 
 Tons. 
 
 French grammes. 
 
 1 = 
 
 0625 
 
 0039 
 
 000035 
 
 00000174 
 
 1-771836 
 
 16 = 
 
 1 
 
 0625 
 
 000558 
 
 000028 
 
 28-34938 
 
 256 = 
 
 16 
 
 1 
 
 00893 
 
 000446 
 
 453-59 
 
 28672 = 
 
 1792 
 
 112 
 
 1 
 
 05 
 
 50802- 
 
 673440 = 
 
 35840 
 
 2240 
 
 20 
 
 1 
 
 1016041-6 
 
 It is common usage here to omit hundred-weights (cwt.) and rate tons at 2,000 pounds as 
 net, and 2240 Ibs. as gross ; 7,000 troy or apothecaries' weight equal 1 pound avoirdupois. 
 50 
 
770 
 
 APPENDIX. 
 
 COMPAKISON OF WEIGHT. 
 
 DYNAMIC TABLE. 
 
 Pounds 
 apothecaries 1 . 
 
 Pounds 
 Troy. 
 
 Pounds 
 avoirdupois. 
 
 Kilo- 
 gramme. 
 
 1 = 
 
 1 
 
 0-8229 
 
 0-37324 
 
 1 = 
 
 1 
 
 0-8229 
 
 0-37324 
 
 1-2153 = 
 
 1-2153 
 
 1 
 
 0-4536 
 
 2-6792 = 
 
 2-6792 
 
 2-2046 
 
 1 
 
 Pounds, 
 feet. 
 
 Kilogramme- 
 metre. 
 
 Horse- 
 power. 
 
 French 
 horse- power. 
 
 1 = 
 
 0-13825 
 
 00003 
 
 000031 
 
 7-2331 = 
 
 1 
 
 000219 
 
 000222 
 
 Per min. 
 
 
 
 
 33-000 = 
 
 4562-3 
 
 1 
 
 1-01386 
 
 32548-9 = 
 
 4500 
 
 0-98633 
 
 1 
 
 CUBIC OK SOLID MEASURE. 
 
 Cubic inches. 
 
 Cubic feet 
 
 Cubic yards. 
 
 Cubic metres. 
 
 United States gallon. 
 
 1 = 
 
 00058 
 
 000021 
 
 000016 
 
 004329 
 
 1728 = 
 
 1 
 
 0-037 
 
 0-0283 
 
 7-48 
 
 46656 = 
 
 27 
 
 1 
 
 0-7646 
 
 201-97 
 
 61016 = 
 
 35-31 
 
 1-3078 
 
 1 
 
 264-141 
 
 231 = 
 
 0-1337 
 
 00495 
 
 00379 
 
 1 
 
 SHIPPING MEASURE. 
 
 Register Ton. For register tonnage or for measurement of the entire internal capacity 
 
 of a vessel : 
 
 100 cubic feet = 1 register ton. 
 Shipping Ton. For the measurement of cargo : 
 
 | 1 U. S. shipping ton. 
 
 40 cubic feet = -( 31 '16 Imp. bushels. 42 cubic feet = 
 
 32-143 U. S. " 
 
 1 British shipping ton. 
 32-719 Imp. bushels. 
 33-75 U. S. " 
 
 Carpenter 's Rule. Weight a vessel will carry = length of keel x breadth at main 
 beam x depth of hold in feet -j- 95 (the cu. ft. per ton). The result will be the tonnage. 
 For a double-decker, instead of the depth of the hold take half the breadth of the beam. 
 
 TABLE OF INCHES AND SIXTEENTHS IN DECIMALS OF A FOOT. 
 
 Inches. 
 
 
 iV 
 
 A 
 
 A 
 
 iV 
 
 A 
 
 A 
 
 A 
 
 A 
 
 A. 
 
 tt 
 
 H 
 
 H 
 
 H 
 
 H 
 
 H 
 
 
 
 000 
 
 005 
 
 010 
 
 016 
 
 021 
 
 026 
 
 031 
 
 036 
 
 042 
 
 047 
 
 062 
 
 057 
 
 062 
 
 068 
 
 073 
 
 078 
 
 1 
 
 083 
 
 089 
 
 094 
 
 099 
 
 104 
 
 109 
 
 115 
 
 120 
 
 125 
 
 130 
 
 135 
 
 141 
 
 146 
 
 151 
 
 156 
 
 161 
 
 2 
 
 167 
 
 172 
 
 177 
 
 182 
 
 187 
 
 193 
 
 198 
 
 203 
 
 208 
 
 214 
 
 219 
 
 224 
 
 229 
 
 234 
 
 240 
 
 245 
 
 3 
 
 250 
 
 255 
 
 260 
 
 266 
 
 271 
 
 276 
 
 281 
 
 286 
 
 292 
 
 297 
 
 302 
 
 307 
 
 312 
 
 318 
 
 323 
 
 328 
 
 4 
 
 333 
 
 339 
 
 344 
 
 349 
 
 354 
 
 359 
 
 365 
 
 370 
 
 375 
 
 380 
 
 385 
 
 391 
 
 396 
 
 401 
 
 406 
 
 411 
 
 5 
 
 417 
 
 422 
 
 427 
 
 432 
 
 437 
 
 443 
 
 448 
 
 453 
 
 458 
 
 464 
 
 469 
 
 474 
 
 479 
 
 484 
 
 490 
 
 495 
 
 6 
 
 500 
 
 505 
 
 510 
 
 516 
 
 521 
 
 526 
 
 531 
 
 536 
 
 542 
 
 547 
 
 552 
 
 557 
 
 562 
 
 568 
 
 573 
 
 578 
 
 7.... 
 
 583 
 
 589 
 
 594 
 
 599 
 
 604 
 
 609 
 
 615 
 
 620 
 
 625 
 
 630 
 
 635 
 
 641 
 
 646 
 
 651 
 
 656 
 
 661 
 
 8 
 
 667 
 
 672 
 
 677 
 
 682 
 
 687 
 
 693 
 
 698 
 
 703 
 
 708 
 
 714 
 
 719 
 
 724 
 
 729 
 
 734 
 
 740 
 
 745 
 
 9 
 
 750 
 
 756 
 
 760 
 
 766 
 
 771 
 
 776 
 
 781 
 
 786 
 
 792 
 
 797 
 
 802 
 
 807 
 
 812 
 
 818 
 
 823 
 
 828 
 
 10 
 
 833 
 
 839 
 
 844 
 
 849 
 
 854 
 
 859 
 
 865 
 
 870 
 
 875 
 
 880 
 
 885 
 
 891 
 
 896 
 
 901 
 
 906 
 
 911 
 
 11 
 
 917 
 
 922 
 
 927 
 
 932 
 
 937 
 
 943 
 
 948 
 
 953 
 
 958 
 
 964 
 
 969 
 
 974 
 
 979 
 
 984 
 
 990 
 
 995 
 
APPENDIX. 771 
 
 ELECTRICAL UNITS. 
 
 C. G. S. 
 
 Unit of space, 1 centimetre, C. ; of mass, 1 gramme, G. ; of time, 1 second, S. 
 
 The definitions of units as adopted at the International Electrical Congress at Chicago 
 in 1893, established by Act of Congress of the United States, July 12, 1894, are as follows: 
 
 The ohm (the unit of resistance, represented by R) is equal to 10* (or 1,000,000,000) 
 units of resistance of the C. G. S. system, and is represented by the resistance offered to 
 an unvarying electrical current by a column of mercury at 32 Fahr. (14-4521 grammes 
 in mass) of a constant cross-sectional area, and of the length of 106 '3 centimetres. 
 
 The ampere (the unit of current strength, or rate of flow, represented by C) is one tenth 
 of the unit of current of the C. G. S. system, and is the equivalent of the unvarying cur- 
 rent which, when passed through a solution of nitrate of silver in water in accordance 
 with standard specifications, deposits silver at the rate of -001118 gramme per second. 
 
 The volt (the unit of electro-motive force, or difference of potential, represented by E) 
 is the electro-motive force that, steadily applied to a conductor whose resistance is 1 
 ohm, will produce a current of one ampere, and is equivalent to -fj (or -6974) of the 
 electro-motive force between the poles or electrodes of a Clark's cell at a temperature 
 of 15 C., and prepared in the manner described in the standard specifications. 
 
 The coulomb (or ampere-second, the unit of quantity, Q) is the quantity of electricity 
 transferred by a current of one ampere in one second. 
 
 The farad (the unit of capacity represented by K) is the capacity of a condenser 
 charged to a potential of one volt by one coulomb of electricity. 
 
 The joule (volt-coulomb, the unit of energy or work, W) is equal to 10,000,000 units 
 of work in the C. G. S. system, and is practically equivalent to the energy expended in 
 one second by an ampere in one ohm. 
 
 The watt, or ampere-volt (the unit of power, P), is equal to 10,000,000 units of power 
 in the C. G. S. system, and is practically equivalent to the work done at the rate of one 
 joule per second ; 746 watts = 1 H. P. 
 
 The henry (the unit of induction) is the induction in a circuit when the electro-motive 
 force induced in this circuit is one volt, while the inducing current varies at the rate of 
 one ampere per second. 
 
 The ohm. volt, etc., as above defined, are called the " international " ohm, volt, etc., 
 to distinguish them from the " legal " ohm, B. A. unit, etc. 
 
 The value of the ohm, determined by a committee of the British Association in 1863, 
 called the B. A. unit, was the resistance of a certain piece of copper wire preserved in 
 London. The so-called " legal " ohm as adopted by the International Congress of Elec- 
 tricians in Paris in 1884, was a correction of the B. A. unit, and was defined as the re- 
 sistance of a column of mercury 1 square millimetre in section and 106 centimetres long, 
 at a temperature of 32 F. 
 
 1 legal ohm = 1-0112 B. A. units, 1 B. A. unit = 0-9889 legal ohm. 
 
 1 international ohm = 1-0136 " " 1 " " = 0-9866 int. " 
 1 " " = 1 -0023 legal ohm, 1 legal ohm = 0-9977 " " 
 
 UNIT OF HEAT. 
 
 The British Thermal Unit (B. T. U.) is that quantity of heat required to raise the 
 temp, of 1 pound of pure water 1 F. at or near 39-1 F., the maximum density- of water. 
 
 The French thermal unit, or calorie, is that quantity of heat which is required to raise 
 the temperature of 1 kilogramme of pure water 1 C., at or about 4 C., which is 
 equivalent to 39'1 F. 
 
 1 French calorie = 3'968 British thermal units, 1 B. T. U. = '252 calorie. 
 
772 
 
 APPENDIX. 
 
 FIFTH POWERS, TABLE OF. 
 
 
 Fifth Power. 
 
 
 Fifth Power. 
 
 
 Fifth Power. 
 
 
 Fifth Power. 
 
 11 
 
 1 61051 
 
 33 
 
 391 35393 
 
 55 
 
 5032 84375 
 
 77 
 
 27067 84157 
 
 12 
 
 248832 
 
 34 
 
 454 35424 
 
 56 
 
 5507 31776 
 
 78 
 
 28871 74368 
 
 13 
 
 3 71293 
 
 35 
 
 525 21875 
 
 57 
 
 6016 92057 
 
 79 
 
 30770 56399 
 
 14 
 
 5 37824 
 
 36 
 
 604 66176 
 
 58 
 
 6563 56768 
 
 80 
 
 32768 00000 
 
 15 
 
 7 59375 
 
 37 
 
 693 43957 
 
 59 
 
 7149 24299 
 
 81 
 
 34867 84401 
 
 16 
 
 10 48576 
 
 38 
 
 792 35168 
 
 60 
 
 7776 00000 
 
 82 
 
 37073 98439 
 
 17 
 
 14 19857 
 
 39 
 
 902 24199 
 
 61 
 
 8445 96301 
 
 83 
 
 39390 40643 
 
 18 
 
 18 89568 
 
 40 
 
 1024 00000 
 
 62 
 
 9161 32832 
 
 : 84 
 
 41821 19424 
 
 19 
 
 24 76099 
 
 41 
 
 1158 56201 
 
 63 
 
 9924 36543 
 
 1 85 
 
 44370 53125 
 
 20 
 
 32 00000 
 
 42 
 
 1306 91232 
 
 64 
 
 10737 41824 
 
 86 
 
 47042 70176 
 
 21 
 
 40 84101 
 
 43 
 
 1470 08443 
 
 65 
 
 11602 90625 
 
 87 
 
 49842 09207 
 
 22 
 
 51 53632 
 
 44 
 
 1649 16224 
 
 66 
 
 12523 32576 
 
 88 
 
 52773 19168 
 
 23 
 
 64 36343 
 
 45 
 
 1845 28125 
 
 67 
 
 13501 25107 
 
 89 
 
 55840 59449 
 
 24 
 
 79 62624 
 
 46 
 
 2059 62976 
 
 68 
 
 14539 33568 
 
 90 
 
 59049 00000 
 
 25 
 
 97 65625 
 
 47 
 
 2293 45007 
 
 69 
 
 15640 31349 
 
 91 
 
 62403 21451 
 
 26 
 
 11881376 
 
 48 
 
 2548 03968 
 
 70 
 
 16807 00000 
 
 92 
 
 65908 15232 
 
 27 
 
 143 48907 
 
 49 
 
 2824 75249 
 
 71 
 
 18042 29351 
 
 93 
 
 69568 83693 
 
 28 
 
 172 10368 
 
 50 
 
 3125 00000 
 
 72 
 
 19349 17632 
 
 94 
 
 73390 40224 
 
 29 
 
 205 11149 
 
 51 
 
 3450 25251 
 
 73 
 
 20730 71593 
 
 95 
 
 77378 09375 
 
 30 
 
 243 00000 
 
 52 
 
 3802 04032 
 
 74 
 
 22190 06624 
 
 96 
 
 81537 26976 
 
 31 
 
 286 29151 
 
 53 
 
 4181 95493 
 
 75 
 
 23730 46875 
 
 97 
 
 85873 40257 
 
 32 
 
 335 54432 
 
 54 
 
 4591 65024 
 
 76 
 
 25355 25376 
 
 99 
 
 95099 00499 
 
 CAST-IRON BALLS, VOLUME AND WEIGHT OF. 
 
 Diameter. 
 
 Volume. 
 
 Weight. 
 
 Diameter. 
 
 Volume. 
 
 Weight. 
 
 Diameter. 
 
 Volume. 
 
 Weight. 
 
 Inches. 
 
 Cu. inches. 
 
 Pounds. 
 
 Inches. 
 
 Cu. inches. 
 
 Pounds. 
 
 Inches. 
 
 Cu. inches. 
 
 Pounds. 
 
 2 
 
 4-19 
 
 1-09 
 
 H 
 
 87-1 
 
 22-7 
 
 9 
 
 381-7 
 
 99-4 
 
 2i 
 
 8-18 
 
 2-13 
 
 6 
 
 113-1 
 
 29-5 
 
 9| 
 
 448 9 
 
 116-9 
 
 3 
 
 14-1 
 
 3-68 
 
 6i 
 
 143-8 
 
 37-5 
 
 10 
 
 523-6 
 
 136-4 
 
 3* 
 
 22-5 
 
 5-85 
 
 7 
 
 179-6 
 
 46-8 
 
 11 
 
 696-9 
 
 181-8 
 
 4 
 
 33-5 
 
 8-73 
 
 7* 
 
 220-9 
 
 57-5 
 
 12 
 
 904-8 
 
 235-9 
 
 4| 
 
 47-7 
 
 12-4 
 
 8 
 
 268-1 
 
 69-8 
 
 13 
 
 1150-3 
 
 299-6 
 
 5 
 
 65-5 
 
 17-0 
 
 8* 
 
 321-5 
 
 83-7 
 
 14 
 
 1436-8 
 
 374-6 
 
 Weight for other metals vary as their specific gravities and for diameters as their cubes. 
 CAST-IRON PIPES, STANDARD WEIGHTS OF. 
 
 
 
 a.S 
 
 d 
 
 2 a 
 
 g . . 
 
 2S 
 
 
 
 
 
 "t 
 
 
 
 5S.& 
 
 
 
 o 
 
 i3 
 
 
 0,^-42 
 
 gg 
 
 15 
 
 
 D' 
 
 Id 
 
 1 
 
 rt j2ji 
 
 ssl 
 
 H O 
 
 - ' CM 
 
 ^ o 
 
 
 E 
 
 5-5 
 
 H D 
 
 *o 
 
 *0 <D M 
 
 *<N M 
 
 S^l 
 
 ^5 
 
 ^ 
 
 s 
 
 S 
 
 S to 
 
 3 
 
 i,wi 
 
 1 E^ j 
 
 i2 p 
 
 3 
 
 S 
 
 A 
 
 SP a5 H 
 
 .Sf s 
 
 -^5 
 
 .S<w<M 
 
 11 
 
 a. 
 
 
 
 H 
 
 
 >> oP " 
 
 
 o o.ts 
 
 a co 
 
 
 09 
 
 
 ^ 
 
 * 
 
 ^~ 
 
 
 O 
 
 O 
 
 Inches. 
 
 Inches. 
 
 Pounds. 
 
 Pounds. 
 
 Pounds. 
 
 Pounds. 
 
 Pounds. 
 
 U. S. gal. 
 
 4 
 
 0-40 
 
 17-27 
 
 18-75 
 
 225 
 
 3600 
 
 720 
 
 6528 
 
 6 
 
 0-42 
 
 26-46 
 
 28-92 
 
 347 
 
 2515 
 
 503 
 
 1-469 
 
 8 
 
 0-45 
 
 37-33 
 
 40-50 
 
 486 
 
 2025 
 
 405 
 
 2-611 
 
 10 
 
 0-50 
 
 51-54 
 
 56-17 
 
 673 
 
 1800 
 
 360 
 
 4-081 
 
 12 
 
 0-55 
 
 67-76 
 
 73-75 
 
 885 
 
 1650 
 
 330 
 
 5-876 
 
 14 
 
 0-58 
 
 83-02 
 
 90-67 
 
 1088 
 
 1490 
 
 298 
 
 7-997 
 
 16 
 
 0-60 
 
 97-78 
 
 106-75 
 
 1281 
 
 1350 
 
 270 
 
 10-440 
 
 18 
 
 0-64 
 
 117-11 
 
 126-67 
 
 1520 
 
 1280 
 
 256 
 
 13-22 
 
 20 
 
 0-70 
 
 142-25 
 
 153-43 
 
 1841 
 
 1260 
 
 252 
 
 16-32 
 
 24 
 
 0-80 
 
 194-77 
 
 210-33 
 
 2524 
 
 1200 
 
 240 
 
 23-50 
 
 30 
 
 0-90 
 
 273-00 
 
 285-33 
 
 3524 
 
 1080 
 
 216 
 
 36-72 
 
 36 
 
 1-00 
 
 363-22 
 
 390-50 
 
 4686 
 
 1000 
 
 200 
 
 52-88 
 
 From " Water Works," by Howland and Ellis. 
 
APPENDIX. 
 
 773 
 
 rH 
 
 
 ft 
 
 CM 
 
 8 
 
 a 
 
 go- 
 o-is 
 
 Tf * 
 
 
 
 oot 
 
 GO-* rH 
 
 CO-* 
 COb-0 
 COCO * 
 
 OS 
 
 
 
 
 
 
 i 
 
 CM 
 CM 
 
 (M OtO 
 
 rH-*b- 
 
 o CO te 
 
 00 
 
 
 
 
 
 
 Sg 
 
 (MCM 
 
 *O-* 
 TT CO CO 
 
 Si co 
 O5 CM 
 CMCMCO 
 
 t- 
 
 
 
 
 
 
 OtOO 
 
 tO rH CM 
 
 t-coos 
 
 O GO CM 
 to Jj ^ 
 
 O OS CM 
 
 coooo 
 
 to 
 
 
 
 
 
 f, 
 
 OOrH CO 
 TT t GO 
 
 TI<OOO 
 
 oot o 
 
 OCM T)< 
 CM CM CM 
 
 f 
 
 
 
 
 
 
 TH 
 
 OrH 
 
 o * >i 
 
 tOCMt- 
 
 OS 00 tO 
 
 CM -*tO 
 
 (Mb- CM 
 
 OCOCM 
 
 00 O CM 
 
 
 
 
 
 
 
 ocoo 
 
 CM tOO 
 
 to to to 
 OOb-O 
 
 OCO t- 
 
 COOSO 
 
 t--*l-H 
 
 rHCOO 
 
 OOOCM 
 
 tooo o 
 
 $ 
 
 
 
 
 i 
 
 b-COOS 
 
 t 00 O 
 OCM^ 
 
 tooto 
 
 O b GO 
 
 o joo 
 b- 65 os 
 
 tO r- tO 
 
 OSMCO 
 
 rH tO rH 
 
 O to GO 
 
 * 
 
 
 
 
 
 v-5 
 
 too-* 
 
 COrHOO 
 
 00 OO 
 CM 00 CM 
 
 
 
 s 
 
 SbJg 
 
 CJSOCM 
 
 CO * tO 
 
 CO 
 
 
 
 
 523 
 
 S5;t 
 
 OS t-t 
 
 coo to 
 
 CZ lO 
 COOSO 
 
 5jg 
 
 COOSO 
 
 to tot 
 
 88S2 
 
 tOOOrH 
 CM COO 
 
 if 
 
 
 
 Q 
 
 Sr?0 
 
 * TfCO 
 
 OS GO t 
 
 SS2 
 
 GOOO 
 00 to-* 
 
 5i?S 
 
 co-*o 
 
 OtOb- 
 
 Sct| 
 
 b-OSrH 
 
 r* 
 
 co 
 
 
 
 oco 
 
 oos to 
 
 COrHGO 
 b-CMtO 
 
 So 
 
 SCO 
 
 c? 
 
 55? 9 
 
 38S 
 
 O r OO 
 
 b- CO OS 
 
 us 
 
 CO 
 
 
 
 rH-*b- 
 OSGOCO 
 
 OSCMt- 
 
 00 * * 
 
 G^ OO CO 
 
 fc^ OO O> 
 
 S2cM 
 
 O CM O 
 
 b-OO 
 
 sss 
 
 ooo 
 
 t- 00 OS 
 
 OrH CM 
 
 CM 
 
 
 5 
 
 o 
 
 CM 
 
 b-OCM 
 
 CM CO CO 
 
 * tO GO 
 
 t-oto 
 
 TOt- 
 COOSO 
 
 T-l C^CO 
 
 WOO- 
 
 * b rH 
 
 coco-* 
 
 tooo 
 
 *OO 
 
 s^s 
 
 ls 
 
 * 
 
 
 2 
 
 CMCO * 
 
 SOS OS 
 tooo 
 
 OrH CM 
 
 THCOO 
 
 CM OOOO 
 OS COb- 
 
 So 
 
 S3 
 
 OI-OS 
 (M CM (M 
 
 CO COCO 
 
 CM tOO 
 
 gt o 
 tot- 
 
 TflCOrH 
 COOSO 
 
 i? 
 
 
 cf-t, 
 
 8 
 
 $coS 
 
 coS,; 
 
 o3 
 
 r -* -+ 
 
 t-COOS 
 
 t-OOO 
 rHrHCM 
 
 CMcUcM 
 
 ct 
 
 53 9 
 
 SSo 
 
 ocoo 
 1- >f ~; 
 
 CM 
 
 CO 
 
 2cSS 
 
 THOOO 
 
 CM OS CO 
 
 toSSS 
 
 *0 CO 
 
 tO GO CO 
 
 coooo 
 
 OtOGO 
 
 CMCMoS 
 
 OtOO 
 
 cScoS 
 
 b-CO O 
 Tj< O to 
 
 5SS 
 
 rH 
 
 OS 
 
 CO b- 
 
 to * -- 
 
 r^S 
 
 1H05CM 
 r-OO 
 
 GO CM b- 
 *-*CO 
 
 cQi^g 
 
 OS 00 t- 
 
 OrH 
 
 Sr-S 
 
 2 
 
 (MCOtO 
 
 CM O 
 COCO 
 
 5^S 
 
 gs? 
 
 * 
 
 CO^fO 
 OOOrH 
 
 t CO OS 
 
 cotcoo 
 
 r?4?g 
 
 ooo 
 
 OSCMb- 
 
 tO OOrH 
 CM t CO 
 
 to CC' t- 
 CO-* * 
 
 oSS 
 
 t-OOO 
 
 rHCMCO 
 
 OtOb- 
 
 00 O CM 
 
 CMCMCO 
 
 ooo 
 
 gss 
 
 * 
 
 tO CM t CM 
 
 5SS 
 
 00 OS-* 
 
 S-ot- 
 
 t 00 OS 
 
 Or? CM 
 
 ijSc? 
 
 -coS 
 
 o to t oo 
 
 =o- 
 
 CMCO-* 
 
 O<000 
 
 CM CM CM 
 
 Scoco 
 
 =Jg 
 
 - 
 
 CO CM O GO t 
 
 GO CM tO 
 
 O OOS 
 
 cot- to 
 
 S2?S 
 
 ggco 
 
 sgsss 
 
 cotK ooto 
 
 t- oo o 
 
 OOrH 
 
 CM COO 
 
 to ooo 
 
 rHrHCM 
 
 co too 
 
 CMCMCO 
 
 OOl-O 
 
 coco-* 
 
 X 
 
 O OS tO * rH GO 
 
 853 
 
 T* 00 rM 
 COOCO 
 
 O OS to 
 
 SSS 
 
 *f 00 CM 
 
 <ooo 
 
 COCMb- 
 
 CMCMCO rJlOO 
 
 tOt- GO 
 
 COOSO 
 
 rHrH CO 
 
 *2S 
 
 g3c 
 
 OstMO 
 
 CMCCCO 
 
 * 
 
 rn CO tO O CMO 
 CM O rH b * O 
 
 tScoos 
 
 S1-I-* 
 CM GO 
 
 I Ol 
 
 TflrHCO 
 
 SOS tO 
 GO rH 
 
 gse 
 
 tooo 
 
 CM b-CO 
 
 rH CMCMCO CO * O 
 
 ototo 
 
 t-oooo 
 
 OSOr- 
 
 CM CO O 
 
 b O CM 
 rHCMCM 
 
 ot o 
 
 CMCMCO 
 
 * 
 
 rHCO -*-HCO tOOOrH 
 COO OOrHtO rHtOSM 
 
 b-SM t 
 
 CO 00 CO 
 
 OOT(lT)< 
 
 sss 
 
 SS OS 
 
 StOtO 
 CM 
 
 rHrH CMC* COCO-* 
 
 *00 
 
 WtOb- 
 
 b- 00 OS 
 
 OrH CM 
 
 ^KtOOO 
 
 sss 
 
 * 
 
 SOtO b-COrH COOt- 
 OCM TtltO OCOCO 
 
 b2cM?8 
 
 S^c? 
 
 CM * -r> 
 
 COIO 
 
 CM to O 
 
 5,52 
 
 SoS 
 
 rHrH rHrHCM CMCMCO 
 
 CO-*-* 
 
 OOO 
 
 to tot 
 
 COOS 
 
 COO 
 
 -2 
 
 X 
 
 t- COOSO OtOOO O CO 
 
 * tot-os rnojo cncMO 
 
 *tOb- 
 
 S-S3 
 
 TfOOO 
 
 t-oto 
 
 CM 1C '/) 
 CO OSO 
 
 T+l O tO 
 
 -T OtO 
 
 tO OS- 
 
 rHTHrH rH CM CM 
 
 CMCOCO 
 
 eOT)<Tj< 
 
 -*00 
 
 to tot 
 
 00 O rH 
 
 CM COO 
 
 X 
 
 rHrH CM CO CO CO * O tO b- GO 
 CMCO -* O tO t GO O CM * tO 
 
 OSrH 03 
 
 S-* o 
 t OS 
 
 tot- OS 
 rHCO t- 
 
 SSo 
 
 o-*oo 
 
 OS b-O 
 
 CM tOO 
 * CM rH 
 
 rH --- 
 
 rHCMCM 
 
 CM CM CM 
 
 cococo 
 
 *-*0 
 
 otot- 
 
 OOOSO 
 
 X. 
 
 OOO rH tO rH tO CM CO CO * * 
 OrHrH CMCMCO CO 7*" O to b- 
 
 oo to 
 
 OSOrH 
 
 tOb-b- 
 
 CMCO-* 
 
 coooo 
 otoos 
 
 T CM CO 
 T-HCOO 
 
 *toos 
 
 OS CO b 
 
 7,-7'r? 
 
 rHrH rnrHrH rH rH CM CM CM CM CO CO * * O 
 
 uoij panoy 
 
 -* t b tO CO OS CM O CO OOOO 
 
 coSo 
 
 OOOS-* 
 
 SCO b- 
 coo 
 
 2CM 
 
 to co co 
 
 2?2 
 
 OOO OrHrH CM CM T* OCOO 
 
 SrScM 
 
 CM CM" CO 
 
 I CM CO 
 00 * O 
 
 S 
 
 rHrHCM 
 
 SS 
 
 CM CO CO 
 
 j.";^ 
 
 ~~ -*"* "-:? T? 
 
 cfiirsr 
 
 CO CO CO 
 
 * 
 
 00*tO 
 
 (00 OS 
 
 o en 
 
774 APPENDIX. 
 
 WEIGHTS OF WROUGHT-IRON AND BRASS PLATES AND WIRE, SOFT ROLLED. 
 
 BIRMINGHAM UAUGE. 
 
 No. of 
 
 gauge. 
 
 AMERICAN GAUGE (BROWN A 8HAEPE). 
 
 Plate iron. 
 
 Thickness of 
 each number. 
 
 Thickness 
 of each 
 number. 
 
 PLATES PER SQUARE FOOT. 
 
 WIRE PER LINEAL TOOT. 
 
 Wrought 
 iron. 
 
 Brass. 
 
 Wrought 
 iron. 
 
 Brass. 
 
 Lbs. 
 17-025 
 
 Inch. 
 454 
 
 0000 
 
 Inch. 
 46 
 
 Lbs. 
 17-25 
 
 Lbs. 
 19-68 
 
 Lbs. 
 5607 
 
 Lbs. 
 6051 
 
 15-9375 
 
 425 
 
 000 
 
 4096 
 
 15-361 
 
 17-53 
 
 4447 
 
 4799 
 
 14-25 
 
 38 
 
 00 
 
 3648 
 
 13-68 
 
 15-61 
 
 3527 
 
 3806 
 
 12-75 
 
 34 
 
 
 
 3248 
 
 12-182 
 
 13-90 
 
 2797 
 
 3018 
 
 11-25 
 
 3 
 
 1 
 
 2893 
 
 10-848 
 
 12-38 
 
 2218 
 
 2393 
 
 10-65 
 
 284 
 
 2 
 
 2576 
 
 9-661 
 
 11-02 
 
 1759 
 
 1898 
 
 9-7125 
 
 269 
 
 3 
 
 2294 
 
 8-603 
 
 9-81 
 
 1395 
 
 1505 
 
 8-925 
 
 238 
 
 4 
 
 2043 
 
 7-661 
 
 8-74 
 
 1106 
 
 1193 
 
 8-25 
 7-6125 
 
 22 
 203 
 
 5 
 6 
 
 1819 
 1620 
 
 6-822 
 6-075 
 
 7-78 
 6-93 
 
 0877 
 0695 
 
 0946 
 0750 
 
 6-75 
 
 18 
 
 7 
 
 1442 
 
 5-410 
 
 6-17 
 
 0551 
 
 0595 
 
 6-1875 
 
 165 
 
 8 
 
 1284 
 
 4-818 
 
 5-49 
 
 0437 
 
 0472 
 
 5-55 
 
 148 
 
 9 
 
 1144 
 
 4-291 
 
 4-89 
 
 0347 
 
 0374 
 
 5-025 
 
 134 
 
 10 
 
 1018 
 
 3-820 
 
 4-36 
 
 0275 
 
 0296 
 
 4-5 
 
 12 
 
 11 
 
 0907 
 
 3-402 
 
 3-88 
 
 0218 
 
 0235 
 
 4-0875 
 
 109 
 
 12 
 
 0808 
 
 3-030 
 
 3-45 
 
 0173 
 
 0186 . 
 
 3-5625 
 
 095 
 
 13 
 
 0719 
 
 2-698 
 
 3-07 
 
 0137 
 
 0148 
 
 3-1125 
 
 083 
 
 14 
 
 0640 
 
 2-403 
 
 2-74 
 
 0109 
 
 0117 
 
 2-7 
 
 072 
 
 15 
 
 0570 
 
 2-140 
 
 2-44 
 
 00863 
 
 00931 
 
 2-4375 
 
 065 
 
 16 
 
 0508 
 
 1-905 
 
 2-17 
 
 00684 
 
 00758 
 
 2-175 
 
 058 
 
 17 
 
 0452 
 
 1-697 
 
 1-93 
 
 00542 
 
 00585 
 
 1-8375 
 
 049 
 
 18 
 
 0403 
 
 1-511 
 
 1-72 
 
 00430 
 
 00464 
 
 1-575 
 
 042 
 
 19 
 
 0358 
 
 1-345 
 
 1-53 
 
 00341 
 
 00368 
 
 1-3125 
 
 035 
 
 20 
 
 (319 
 
 1-198 
 
 1-36 
 
 00271 
 
 00292 
 
 1-2 
 
 032 
 
 21 
 
 0284 
 
 1-067 
 
 1-21 
 
 00215 
 
 00231 
 
 1-05 
 
 028 
 
 22 
 
 0253 
 
 9505 
 
 1-08 
 
 00170 
 
 00183 
 
 9375 
 
 025 
 
 23 
 
 0225 
 
 8464 
 
 9660 
 
 00135 
 
 00145 
 
 826 
 
 022 
 
 24 
 
 0201 
 
 7537 
 
 8602 
 
 00107 
 
 00115 
 
 75 
 
 02 
 
 25 
 
 0179 
 
 6712 
 
 7661 
 
 00085 
 
 000916 
 
 675 
 
 018 
 
 26 
 
 0159 
 
 5977 
 
 6822 
 
 000673 
 
 000726 
 
 6 
 
 016 
 
 27 
 
 0141 
 
 5323 
 
 6075 
 
 000534 
 
 000576 
 
 525 
 
 014 
 
 28 
 
 0126 
 
 4740 
 
 5410 
 
 000423 
 
 000457 
 
 4875 
 
 013 
 
 29 
 
 0112 
 
 4221 
 
 4818 
 
 000336 
 
 000362 
 
 45 
 
 012 
 
 30 
 
 0100 
 
 3759 
 
 4290 
 
 000266 
 
 000287 
 
 375 
 
 01 
 
 31 
 
 0089 
 
 3348 
 
 3821 
 
 000211 
 
 000228 
 
 3375 
 
 009 
 
 32 
 
 0079 
 
 2981 
 
 3402 
 
 000167 
 
 000180 
 
 3 
 
 008 
 
 33 
 
 00708 
 
 2655 
 
 3030 
 
 000132 
 
 000143 
 
 2625 
 
 007 
 
 34 
 
 00630 
 
 2364 
 
 2698 
 
 000105 
 
 000113 
 
 1875 
 
 005 
 
 35 
 
 00561 
 
 2105 
 
 2402 
 
 0000836 
 
 00009015 
 
 15 
 
 004 
 
 36 
 
 005 
 
 1875 
 
 214 
 
 0000662 
 
 0000715 
 
 
 
 37 
 
 00445 
 
 1669 
 
 1905 
 
 0000525 
 
 00005671 
 
 
 
 38 
 
 00396 
 
 1486 
 
 1697 
 
 0000416 
 
 00004496 
 
 
 
 39 
 
 00353 
 
 1324 
 
 1511 
 
 0000330 
 
 00003566 
 
 
 
 40 
 
 00314 
 
 1179 
 
 1345 
 
 0000262 
 
 00002827 
 
 Copper is about 5 per cent heavier than brass. Lead is about 47 per cent heavier than wrought 
 iron. Zinc is about 7 per cent lighter than wrought iron. Sheet copper is rated by weight at so 
 many ounces per square foot, and sheet lead at so many pounds per square foot. 
 
APPENDIX. 
 
 775 
 
 TABLE OF DIMENSIONS AND WEIGHT OF WEOUGHT-1RON WELDED TUBES. 
 
 Nominal 
 diameter. 
 
 External 
 diameter. 
 
 Thick- 
 ness. 
 
 Internal 
 diameter. 
 
 Internal 
 circum- 
 ference. 
 
 External 
 circum- 
 ference. 
 
 Length of 
 pipe per 
 square 
 toot of 
 internal 
 surface. 
 
 Length of 
 pipe per 
 square 
 foot of 
 external 
 surface. 
 
 Internal 
 area. 
 
 Weight 
 per foot. 
 
 No. of 
 
 threads 
 per 
 inch of 
 screw. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Feet. 
 
 Feet. 
 
 Inches. 
 
 Lbs. 
 
 
 Yi 
 
 40 
 
 068 
 
 27 
 
 85 
 
 1-27 
 
 14-15 
 
 944 
 
 057 
 
 24 
 
 27 
 
 / 
 
 54 
 
 088 
 
 36 
 
 ri4 
 
 1-7 
 
 10-5 
 
 7-075 
 
 104 
 
 42 
 
 18 
 
 3 /8 
 
 67 
 
 091 
 
 49 
 
 1-55 
 
 2-12 
 
 7-67 
 
 5-657 
 
 192 
 
 56 
 
 18 
 
 V. 
 
 84 
 
 .109 
 
 62 
 
 1-96 
 
 2-65 
 
 6-13 
 
 4-502 
 
 305 
 
 84 
 
 14 
 
 3 /4 
 
 1-05 
 
 .113 
 
 82 
 
 2-59 
 
 3'3 
 
 4-64 
 
 3-637 
 
 533 
 
 1-13 
 
 14 
 
 1 
 
 1-31 
 
 134 
 
 1-05 
 
 3-29 
 
 4-13 
 
 3-66 
 
 2-903 
 
 863 
 
 1-67 
 
 ll 1 /* 
 
 l'/4 
 
 1-66 
 
 14 
 
 1-38 
 
 4-33 
 
 5-21 
 
 2-77 
 
 2-301 
 
 1-496 
 
 2-26 
 
 117* 
 
 !/ 
 
 1-9 
 
 145 
 
 1-61 
 
 5-06 
 
 5-97 
 
 2-37 
 
 2-01 
 
 2-038 
 
 2-69 
 
 nv 
 
 2 
 
 2-37 
 
 154 
 
 2-07 
 
 6-49 
 
 7-46 
 
 1-85 
 
 1-611 
 
 3-355 
 
 3-67 
 
 "V. 
 
 2/ 2 
 
 2-87 
 
 204 
 
 2-47 
 
 7-75 
 
 9-03 
 
 1-55 
 
 1-328 
 
 4-783 
 
 5-77 
 
 8 
 
 3 
 
 3-5 
 
 217 
 
 3-07 
 
 9-64 
 
 11- 
 
 1-24 
 
 1-091 
 
 7-388 
 
 7-55 
 
 8 
 
 / 
 
 4- 
 
 226 
 
 3-55 
 
 11-15 
 
 12-57 
 
 1-08 
 
 0-955 
 
 9-887 
 
 9-05 
 
 8 
 
 4 
 
 4-5 
 
 237 
 
 4-07 
 
 12-69 
 
 14-14 
 
 95 
 
 0-849 
 
 12-73 
 
 10-73 
 
 8 
 
 */ 
 
 5- 
 
 247 
 
 4-51 
 
 14-15 
 
 15-71 
 
 85 
 
 0-765 
 
 15-939 
 
 12-49 
 
 8 
 
 5 
 
 5-56 
 
 259 
 
 5-04 
 
 15-85 
 
 17-47 
 
 78 
 
 0-629 
 
 19-99 
 
 14-56 
 
 8 
 
 6 
 
 6-62 
 
 28 
 
 6-06 
 
 19-Q5 
 
 20-81 
 
 63 
 
 0-577 
 
 28-889 
 
 18-77 
 
 8 
 
 7 
 
 7-62 
 
 301 
 
 7-02 
 
 22-06 
 
 23-95 
 
 54 
 
 0-505 
 
 38-737 
 
 23-41 
 
 8 
 
 8 
 
 8-62 
 
 322 
 
 7-98 
 
 25-08 
 
 27-1 
 
 48 
 
 0-444 
 
 50-039 
 
 28-35 
 
 8 
 
 9 
 
 9-69 
 
 344 
 
 9- 
 
 28-28 
 
 30-43 
 
 42 
 
 0-394 
 
 63-633 
 
 34-08 
 
 8 
 
 10 
 
 10-75 
 
 366 
 
 10-02 
 
 31-47 
 
 33-77 
 
 38 
 
 0-355 
 
 78-838 
 
 40-64 
 
 8 
 
 Nominal 
 diameter. 
 
 Thickness, 
 extra strong. 
 
 Thickness, double 
 extra strong. 
 
 Actual inside diameter. 
 Extra strong. 
 
 Actual inside diameter. 
 Double extra strong. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 Inches. 
 
 '/ 
 
 o-ioo 
 
 
 
 0-205 
 
 . . 
 
 l /4 
 
 0-123 
 
 
 0-294 
 
 ..... 
 
 3 /8 
 
 0-127 
 
 
 
 0-421 
 
 ... . 
 
 '/ 
 
 0-149 
 
 0-298 
 
 0-542 
 
 0-244 
 
 3 /4 
 
 0-157 
 
 0-314 
 
 0-736 
 
 0-422 
 
 1 
 
 0-182 
 
 0-364 
 
 0-951 
 
 0-587 
 
 W4 
 
 0-194 
 
 0-388 
 
 1-272 
 
 0-884 
 
 l'/2 
 
 0-203 
 
 0-406 * 
 
 1-494 
 
 1-088 
 
 2 
 
 0-221 
 
 0-442 
 
 1-933 
 
 1-491 
 
 f/ 
 
 0-280 
 
 0-560 
 
 2-315 
 
 . 1-755 
 
 3 
 
 0-304 
 
 0-608 
 
 2-892 
 
 2-284 
 
 S'/ 
 
 0-321 
 
 0-642 
 
 3-358 
 
 2-716 
 
 4 
 
 0-341 
 
 0-682 
 
 3-818 
 
 3-136 
 
 BOILER TUBES. 
 
 External 
 
 Thickness, 
 
 Average 
 
 External 
 
 Thickness, 
 
 Average 
 
 diameter. 
 
 wire gauge. 
 
 weight. 
 
 diameter. 
 
 wire gauge. 
 
 Weight. 
 
 Inches. 
 
 No. 
 
 Lbs. per foot. 
 
 Inches. 
 
 No. 
 
 Lbs. per foot 
 
 W4 
 
 16 
 
 1- 
 
 3 
 
 11 
 
 3-5 
 
 !*/ 
 
 15 ' 
 
 1-16 
 
 8'/4 
 
 11 
 
 4' 
 
 W4 
 
 14 
 
 1-63 
 
 4 
 
 8 
 
 6'4 
 
 2 
 
 13 
 
 2- 
 
 5 
 
 7 
 
 9-1 
 
 2Y4 
 
 12 
 
 2-16 
 
 6 
 
 6 
 
 12'3 
 
 272 
 
 12 
 
 2-56 
 
 7 
 
 6 
 
 15-2 
 
 *VH 
 
 11 
 
 2-2 
 
 8 
 
 6 
 
 16- 
 
776 
 
 APPENDIX. 
 
 HEAVY PIPE FOE DRIVEN WELLS. 
 Tested at 1200 pounds hydraulic pressure. Furnished in five-foot lengths. 
 
 Size (inches) 
 
 li 
 
 1| 
 
 2 
 
 2 
 
 3 
 
 3* 
 
 4 
 
 
 
 
 
 
 
 
 
 Weight per foot, Ibs 
 
 3'62 
 
 2-75 
 
 3-75 
 
 6-00 
 
 7'75 
 
 9-25 
 
 11-00 
 
 
 
 
 
 
 
 
 
 HEAVY WROUGHT GALVANIZED IKON SPIRAL K1VETED PIPES, 
 
 WITH FLANGED CONNECTIONS. 
 Tested at 150 pounds hydraulic pressure. Regalvanized after riveting. 
 
 Inside diameter (inches ) . . . 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 19, 
 
 
 
 
 
 
 
 
 
 
 
 
 Wire gauge, Nos 
 
 20 
 
 20 
 
 20 
 
 18 
 
 18 
 
 18 
 
 18 
 
 16 
 
 16 
 
 16 
 
 Nominal weight per foot, Ibs. . . 
 
 H 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 12 
 
 13 
 
 14 
 
 Manufactured lengths, 20 feet or less. Elbows and other fittings, cast iron. 
 LIGHT PIPE, SUITABLE FOB HOUSE LEADERS, VENTILATING, AIR, AND BLOWER PIPES, ETC. 
 
 
 2 
 
 21 
 
 3 
 
 3* 
 
 4 
 
 4* 
 
 5 
 
 5$ 
 
 6 
 
 
 
 
 
 
 
 
 
 
 
 Nominal weight per foot, Ibs 
 
 1 
 
 i 
 
 i 
 
 1 
 
 H 
 
 If 
 
 1* 
 
 If 
 
 1* 
 
 TABLE OF COPPER AND BRASS RODS ONE FOOT IN LENGTH. 
 
 To find the weight of copper or brass pipe, take the weight of the exterior diameter from the 
 table, and subtract from it the weight of a rod equal to that of the interior diameter, or bore. . 
 
 Diamet'r 
 in 
 inches. 
 
 Copper. 
 
 Brass. 
 
 Diamet'r 
 in 
 inches. 
 
 Copper. 
 
 Brass. 
 
 Diamet'r 
 in 
 inches. 
 
 Copper. 
 
 Brass. 
 
 v 
 
 047 
 
 045 
 
 I'/* 
 
 7-993 
 
 7-593 
 
 4'A 
 
 55-62 
 
 52-27 
 
 '/. 
 
 106 
 
 101 
 
 I 11 /!. 
 
 8-630 
 
 8-198 
 
 4 3 /8 
 
 58-94 
 
 5539 
 
 'A 
 
 189 
 
 179 
 
 ! 3 /4 
 
 9-270 
 
 8-806 
 
 4'A 
 
 62-36 
 
 58-60 
 
 'A- 
 
 296 
 
 281 
 
 l ls / 
 
 9-950 
 
 9-452 
 
 4'/8 
 
 65-87 
 
 61-90 
 
 '/ 
 
 426 
 
 405 
 
 i'/i 
 
 10-642 
 
 10-110 
 
 4% 
 
 69-48 
 
 6 
 
 v 
 
 579 
 
 550 
 
 i/i. 
 
 11-370 
 
 10-801 
 
 *'/ 
 
 73-19 
 
 68-77 
 
 '/ 
 
 757 
 
 719 
 
 2 
 
 12-108 
 
 11-503 
 
 5 
 
 77-43 
 
 72-76 
 
 A. 
 
 958 
 
 910 
 
 i'/a 
 
 13-668 
 
 12-985 
 
 5'/8 
 
 80-89 
 
 76-00 
 
 5 /8 
 
 1-182 
 
 1-123 
 
 2'A 
 
 15-325 
 
 14-559 
 
 5'/4 
 
 84-88 
 
 79-76 
 
 "A. 
 
 1-431 
 
 1-360 
 
 2 3 /8 
 
 17-075 
 
 16-221 
 
 5 8 /8 
 
 88-97 
 
 83-60 
 
 % 
 
 1-703 
 
 1-618 
 
 2'A 
 
 18-916 
 
 17-970 
 
 5'A 
 
 93-15 
 
 87-53 
 
 '/. 
 
 1-998 
 
 1-898 
 
 2 5 A 
 
 20-856 
 
 19-808 
 
 5 6 / 8 
 
 97-44 
 
 91-56 
 
 V 
 
 2-318 
 
 2-202 
 
 2 3 A 
 
 22-891 
 
 21-746 
 
 5 3 /4 
 
 101-81 
 
 95-68 
 
 16 Ae 
 
 2-660 
 
 2-527 
 
 2 7 /8 
 
 25-019 
 
 23-768 
 
 5'/8 
 
 106-29 
 
 99-88 
 
 1 
 
 3-027 
 
 2-876 
 
 3 
 
 27-243 
 
 25-881 
 
 6 
 
 110-85 
 
 104-15 
 
 I 1 /" 
 
 3-417 
 
 3-246 
 
 3'/8 
 
 29-559 
 
 28-081 
 
 6>/4 
 
 12030 
 
 113-04 
 
 1'A 
 
 3-831 
 
 3-639 
 
 3'A 
 
 31-972 
 
 30-373 
 
 6'A 
 
 130-10 
 
 122-26 
 
 l/>e 
 
 4-269 
 
 4-056 
 
 3 3 / 8 
 
 34-481 
 
 32-757 
 
 6 3 /4 
 
 140-32 
 
 131-85 
 
 *v 
 
 4-723 
 
 4-487 
 
 3 1 /* 
 
 37-081 
 
 35-227 
 
 7 
 
 150-86 
 
 141-76 
 
 !/! 
 
 5-214 
 
 4-953 
 
 3 6 / 8 
 
 39-777 
 
 37-788 
 
 VV4 
 
 161-87 
 
 152-10 
 
 ! 3 /8 
 
 5-723 
 
 5-437 
 
 3/4 
 
 42-568 
 
 40-440 
 
 '/ 
 
 173-22 
 
 162-77 
 
 l'/M 
 
 6-255 
 
 5-943 
 
 3'/8 
 
 45-455 
 
 43-182 
 
 7 3 /4 
 
 184-97 
 
 173-81 
 
 W. 
 
 6-811 
 
 6-470 
 
 4 
 
 48-433 
 
 46-000 
 
 8 
 
 197-03 
 
 185-14 
 
 ! 9 /ie 
 
 7-390 
 
 7-020 
 
 4V 8 
 
 52-40 
 
 49-24 
 
 
 
 
APPENDIX. 
 
 NUMBER OF BURDEN'S RIVETS IN ONE HUNDRED POUNDS. 
 
 Lengths. 
 
 DIAMETER. 
 
 Lengths. 
 
 3S.B. 
 
 i 
 
 1 
 
 H 
 
 i 
 
 1 
 
 1,092 
 
 665 
 
 .... 
 
 
 5 
 
 90 
 
 i 
 
 1,027 
 
 597 
 
 
 
 * 
 
 85 
 
 1 
 
 940 
 
 538 
 
 450 
 
 .... 
 
 6 
 
 80 
 
 H 
 
 840 
 
 512 
 
 415 
 
 
 6} 
 
 75 
 
 H 
 
 797 
 
 487 
 
 389 
 
 356 
 
 7 
 
 70 
 
 if 
 
 760 
 
 460 
 
 370 
 
 329 
 
 n 
 
 67 
 
 1* 
 
 730 
 
 440 
 
 357 
 
 280 
 
 8 
 
 65 
 
 if 
 
 711 
 
 420 
 
 340 
 
 271 
 
 H 
 
 61 
 
 if 
 
 693 
 
 390 
 
 325 
 
 262 
 
 9 
 
 67 
 
 *i 
 
 648 
 
 375 
 
 312 
 
 257 
 
 H 
 
 54 
 
 2 
 
 608 
 
 360 
 
 297 
 
 243 
 
 10 
 
 61 
 
 2i 
 
 573 
 
 354 
 
 .... 
 
 
 10$ 
 
 47 
 
 H 
 
 555 
 
 347 
 
 280 
 
 232 
 
 
 
 H 
 
 525 
 
 335 
 
 260 
 
 220 
 
 
 
 2f 
 
 500 
 
 312 
 
 242 
 
 208 
 
 
 
 3 
 
 460 
 
 290 
 
 224 
 
 197 
 
 
 
 3* 
 
 433 
 
 267 
 
 212 
 
 180 
 
 
 
 i 
 
 413 
 
 248 
 
 201 
 
 169 
 
 
 
 Si 
 
 395 
 
 241 
 
 192 
 
 160 
 
 
 
 4 
 
 
 230 
 
 184 
 
 158 
 
 
 
 4i 
 
 .... 
 
 220 
 
 177 
 
 150 
 
 
 
 H 
 
 
 
 210 
 
 171 
 
 146 
 
 
 
 4f 
 
 
 200 
 
 166 
 
 138 
 
 
 
 5 
 
 
 190 
 
 161 
 
 135 
 
 
 
 6* 
 
 
 180 
 
 156 
 
 130 
 
 
 
 H 
 
 .... 
 
 172 
 
 151 
 
 124 
 
 
 
 5f 
 
 
 164 
 
 145 
 
 120 
 
 
 
 6 
 
 .... 
 
 157 
 
 140 
 
 115 
 
 
 
 6* 
 
 .... 
 
 150 
 
 138 
 
 111 
 
 
 
 64 
 
 .... 
 
 146 
 
 134 
 
 107 
 
 
 
 H 
 
 .... 
 
 143 
 
 129 
 
 104 
 
 
 
 7 
 
 .... 
 
 140 
 
 125 
 
 100 
 
 
 
 WEOUGHT SPIKES NUMBER TO A KEG OF ONE HUNDRED AND FIFTY POUNDS. 
 
 LENGTH. 
 
 i" 
 
 A" 
 
 1" 
 
 TV 
 
 i" 
 
 Inches 
 3 
 
 2 250 
 
 
 
 
 
 3| 
 
 1,890 
 
 1,208 
 
 
 
 
 4 
 
 1,650 
 
 1,135 
 
 
 
 
 44 . 
 
 1,464 
 
 1,064 
 
 
 
 
 5 
 
 1,380 
 
 930 
 
 742 
 
 
 
 6 
 
 1,292 
 
 868 
 
 570 
 
 
 
 7 . 
 
 1,161 
 
 662 
 
 482 
 
 445 
 
 806 
 
 8 
 
 
 635 
 
 455 
 
 384 
 
 256 
 
 9 
 
 
 573 
 
 424 
 
 300 
 
 240 
 
 10 
 
 
 
 391 
 
 270 
 
 222 
 
 11 
 
 
 
 
 249 
 
 203 
 
 12 
 
 
 
 
 236 
 
 180 
 
 
 
 
 
 
 
778 APPENDIX. 
 
 LENGTHS OF CUT NAILS AND SPIKES, AND NUMBER IN A POUND. 
 
 Size. 
 
 Length. 
 
 No. 
 
 Size. 
 
 Length. . 
 
 No. 
 
 Size. 
 
 Length. 
 
 No. 
 
 
 Inches. 
 
 
 
 Inches. 
 
 
 
 Inches. 
 
 
 3d. 
 
 1 
 
 420 
 
 8d. 
 
 2i 
 
 100 
 
 SOd. 
 
 4 
 
 24 
 
 4 
 
 1* 
 
 270 
 
 10 
 
 3 
 
 65 
 
 40 
 
 *i 
 
 20 
 
 5 
 
 If 
 
 220 
 
 12 
 
 H 
 
 52 
 
 60 
 
 
 
 6 
 
 2 
 
 175 
 
 20 
 
 H 
 
 28 
 
 
 
 
 APPROXIMATE NUMBER OF WIRE NAILS PER POUND. 
 
 LENGTH, INCHES. 
 
 B. W. G. 
 
 X 
 
 % 
 
 X 
 
 K 
 
 % 
 
 1 
 
 1* 
 
 IX 
 
 IX 
 
 2 
 
 2* 
 
 3 
 
 3X 
 
 4 
 
 4# 
 
 5 
 
 6 
 
 7 
 
 00... 
 
 
 
 
 
 
 
 33 
 34 
 
 45 
 52 
 60 
 72 
 85 
 99 
 120 
 137 
 165 
 198 
 251 
 329 
 429 
 568 
 701 
 913 
 1246 
 1655 
 2133 
 3000 
 
 27 
 29 
 
 38 
 44 
 50 
 60 
 71 
 82 
 100 
 115 
 138 
 165 
 209 
 274 
 357 
 473 
 584 
 761 
 1038 
 1379 
 1778 
 
 23 
 25 
 32 
 37 
 43 
 51 
 60 
 71 
 85 
 98 
 118 
 142 
 179 
 235 
 306 
 406 
 500 
 653 
 890 
 1182 
 
 20 
 21 
 28 
 32 
 38 
 45 
 53 
 62 
 75 
 86 
 103 
 124 
 157 
 204 
 268 
 350 
 438 
 571 
 779 
 
 16 
 17 
 23 
 26 
 30 
 36 
 42 
 50 
 60 
 69 
 82 
 99 
 125 
 164 
 214 
 284 
 350 
 
 14 
 
 15 
 19 
 22 
 25 
 30 
 35 
 41 
 50 
 57 
 69 
 83 
 105 
 137 
 178 
 236 
 
 12 
 13 
 16 
 19 
 22 
 26 
 30 
 35 
 43 
 49 
 59 
 71 
 90 
 117 
 153 
 
 10 
 11 
 14 
 16 
 19 
 23 
 26 
 31 
 37 
 43 
 52 
 62 
 79 
 103 
 
 9 
 10 
 13 
 14 
 17 
 20 
 24 
 28 
 33 
 39 
 46 
 55 
 70 
 
 8 
 9 
 11 
 13 
 15 
 18 
 21 
 25 
 30 
 35 
 41 
 50 
 
 7 
 8 
 10 
 11 
 13 
 15 
 18 
 21 
 25 
 29 
 
 6 
 
 8 
 9 
 11 
 13 
 15 
 18 
 
 o 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 57 
 65 
 
 76 
 90 
 106 
 123 
 149 
 172 
 207 
 248 
 314 
 411 
 536 
 710 
 876 
 1143 
 1558 
 2069 
 2667 
 3750 
 4444 
 
 2 
 
 
 
 
 
 
 3 
 
 
 
 
 
 100 
 120 
 141 
 164 
 200 
 229 
 276 
 333 
 418 
 548 
 714 
 947 
 1168 
 1523 
 2077 
 2758 
 .3556 
 5000 
 5926 
 7618 
 
 4 
 
 
 
 
 
 5 
 
 
 
 211 
 
 247 
 299 
 345 
 414 
 496 
 628 
 822 
 1072 
 1420 
 1752 
 2280 
 3116 
 4138 
 5334 
 7500 
 8888 
 11428 
 
 169 
 197 
 239 
 275 
 331 
 397 
 502 
 658 
 857 
 1136 
 1402 
 1828 
 2495 
 3310 
 4267 
 6000 
 7111 
 9143 
 
 6 
 
 
 
 7 
 
 
 
 8 
 
 
 
 9 
 
 
 
 10 . 
 
 
 663 
 837 
 1096 
 1429 
 1893 
 2336 
 3048 
 4156 
 5517 
 7112 
 10000 
 11850 
 15237 
 
 11 
 
 
 13 
 
 
 13 
 
 
 14 
 
 2840 
 3504 
 4571 
 6233 
 8276 
 10668 
 15000 
 17777 
 ;>v>H5(i 
 
 15 
 16 
 
 B. W. G. 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 ~3K 
 3% 
 W 
 5# 
 
 17... 
 
 
 
 5 
 5^ 
 7 
 8 
 10 
 11 
 
 W 
 5 
 
 6 
 
 7 
 8 
 10 
 
 4 
 4K 
 5% 
 ?,% 
 T# 
 9 
 
 3% 
 4 
 5 
 6 
 
 18 
 
 1 
 
 19... 
 
 2 
 
 20 
 21 .. 
 
 3 
 
 4 
 
 22... 
 
 5... 
 
 TESTS OF TELEGRAPH WIRE. 
 
 The following data are taken from a table given by Mr. Prescott relating to tests of E. B. 
 B. galvanized wire furnished the Western Union Telegraph Company : 
 
 Size 
 of 
 Wire. 
 
 Diameter 
 Parts of 
 One Inch. 
 
 WEIGHT. 
 
 Length, 
 Feet per 
 Pound. 
 
 RESISTANCE. 
 TEMP., 75-8 FAHB. 
 
 Ratio of 
 Breaking 
 Weight to 
 Weight per 
 Mile. 
 
 Grains, per 
 Foot. 
 
 Pounds per 
 Mile. 
 
 Feet per 
 Ohm. 
 
 Ohms per 
 Mile. 
 
 4 
 
 238 
 
 1043-2 
 
 886-6 
 
 6-00 
 
 958 
 
 5-51 
 
 
 5 
 
 220 
 
 891-3 
 
 673-0 
 
 7-85 
 
 727 
 
 7-26 
 
 
 6 
 
 203 
 
 758-9 
 
 572-2 
 
 9-20 
 
 618 
 
 8-54 
 
 3-05 
 
 7 
 
 180 
 
 596-7 
 
 449-9 
 
 11-70 
 
 578 
 
 10-86 
 
 3-40 
 
 8 
 
 165 
 
 501-4 
 
 378-1 
 
 14-00 
 
 409 
 
 12-92 
 
 3-07 
 
 9 
 
 148 
 
 403-4 
 
 304-2 
 
 17-4 
 
 328 
 
 16-10 
 
 3-38 
 
 10 
 
 134 
 
 330-7 
 
 249-4 
 
 21-2 
 
 269 
 
 19-00 
 
 3-37 
 
 11 
 
 120 
 
 265-2 
 
 200-0 
 
 26-4 
 
 216 
 
 24-42 
 
 2-97 
 
 12 
 
 109 
 
 218-8 
 
 165-0 
 
 32-0 
 
 179 
 
 29-60 
 
 3-43 
 
 14 
 
 083 
 
 126-9 
 
 95-7 
 
 55-2 
 
 104 
 
 51-00 
 
 3-05 
 
 SIZES OF GALVANIZED WIRE USED IN TELEGRAPH AND TELEPHONE 
 
 LINES. 
 
 No. 4. Now used on important lines where the multiplex systems are applied. No. 
 5. Little used. No. 6. Used for important circuits between cities. No. 8. Medium size 
 for circuits of 400 miles or less. No. 9. For similar locations to No. 8, but on somewhat 
 
APPENDIX. Y79 
 
 shorter circuits ; until lately was the size most largely used in this country. Nos. 10, 11. 
 For shorter circuits, railway telegraphs, private lines, police and fire-alarm lines, etc. 
 No. 12. For telephone lines, police and fire-alarm lines, etc. Nos. 13, 14. For telephone 
 lines and short private lines : steel wire is used most generally in these sizes. 
 
 The grades of line wire are generally known to the trade as "Extra Best Best " (E. 
 B. B.), "Best Best" (B. B.), and "Steel." 
 
 STANDARD I BEAMS. 
 
 Depth 
 
 Min. wt. 
 
 
 
 
 
 
 
 WEIGHT PER FOOT. 
 
 Increase 
 of b and t 
 
 
 of 
 beam. 
 
 per 
 foot. 
 
 Min. 
 b. 
 
 Min. 
 t. 
 
 z. 
 
 s. 
 
 P- 
 
 y- 
 
 Intermediate. 
 
 for each 
 Ib. inc. of 
 weight. 
 
 Max. 
 
 24" 
 
 80-00 
 
 7-00 
 
 50 
 
 3-250 
 
 60 
 
 542 
 
 1-142 
 
 Vary by 5 Ibs. 
 
 0123 
 
 100-00 
 
 20" 
 
 64-00 
 
 6-25 
 
 50 
 
 2-875 
 
 55 
 
 479 
 
 1-029 
 
 65 Ibs. then by 5 Ibs. 
 
 015 
 
 75-00 
 
 15" 
 
 42-00 
 
 5-50 
 
 41 
 
 2-545 
 
 41 
 
 424 
 
 0-834 
 
 45 " " 5 " 
 
 020 
 
 55-00 
 
 12" 
 
 31-50 
 
 5-00 
 
 35 
 
 2-325 
 
 35 
 
 388 
 
 0-738 
 
 
 025 
 
 35-00 
 
 10" 
 
 25-00 
 
 4-66 
 
 31 
 
 2-175 
 
 31 
 
 363 
 
 0-673 
 
 Vary by 5 Ibs. 
 
 029 
 
 40-00 
 
 9" 
 
 21-00 
 
 4-33 
 
 29 
 
 2-020 
 
 29 
 
 337 
 
 0-627 
 
 25 Ibs. then by 5 Ibs. 
 
 033 
 
 35-00 
 
 8" 
 
 18-00 
 
 4-00 
 
 27 
 
 1865 
 
 27 
 
 311 
 
 0-581 
 
 Vary by 2 Ibs. 
 
 037 
 
 25-50 
 
 7" 
 
 15-00 
 
 3-66 
 
 25 
 
 1-705 
 
 25 
 
 284 
 
 0-534 
 
 " " 2| " 
 
 042 
 
 20-00 
 
 6" 
 
 12-25 
 
 3-33 
 
 23 
 
 1-550 
 
 23 
 
 258 
 
 0-488 
 
 2i " 
 
 049 
 
 17-25 
 
 5" 
 
 9-75 
 
 3-00 
 
 21 
 
 1-395 
 
 21 
 
 233 
 
 0-443 
 
 " 2i " 
 
 059 
 
 14-75 
 
 4" 
 
 7-50 
 
 2-66 
 
 19 
 
 1-235 
 
 19 
 
 206 
 
 0-396 
 
 " " 1 Ib. 
 
 074 
 
 10-50 
 
 3" 
 
 5-50 
 
 2-33 
 
 17 
 
 1-080 
 
 17 
 
 180 
 
 0-350 
 
 " " 1 " 
 
 098 
 
 7-50 
 
 STANDARD CHANNELS. 
 
 
 
 
 
 
 
 
 
 WEIGHT PER FOOT. 
 
 Increase 
 
 
 Depth 
 
 Min wt 
 
 
 
 
 
 
 
 
 of b and t 
 
 
 of 
 chan- 
 
 per 
 foot. 
 
 Min. 
 b. 
 
 Min. 
 
 t. 
 
 z. 
 
 s. 
 
 P- 
 
 y- 
 
 Intermediate. 
 
 for each 
 Ib. inc. of 
 
 Max. 
 
 nel. 
 
 
 
 
 
 
 
 
 
 weight. 
 
 
 15" 
 
 33-00 
 
 3-40 
 
 40 
 
 3-00 
 
 40 
 
 500 
 
 900 
 
 35 Ibs. then by 5 Ibs. 
 
 020 
 
 55-00 
 
 12" 
 
 20-50 
 
 2-94 
 
 28 
 
 2-66 
 
 28 
 
 443 
 
 723 
 
 25 " " 5 " 
 
 025 
 
 40-00 
 
 10" 
 
 15-00 
 
 2-60 
 
 24 
 
 2-36 
 
 24 
 
 393 
 
 633 
 
 Vary by 5 Ibs. 
 
 029 
 
 35-00 
 
 9" 
 
 13-25 
 
 2-43 
 
 23 
 
 2-20 
 
 23 
 
 367 
 
 597 
 
 15 Ibs. then bv 5 Ibs. 
 
 033 
 
 25-00 
 
 8" 
 
 11-25 
 
 2-26 
 
 22 
 
 2-04 
 
 22 
 
 340 
 
 560 
 
 Vary by 2i Ibs. 
 
 037 
 
 21-25 
 
 7" 
 
 9-75 
 
 2-09 
 
 21 
 
 1-88 
 
 21 
 
 313 
 
 523 
 
 2 " 
 
 042 
 
 19-75 
 
 6" 
 
 8-00 
 
 1-92 
 
 20 
 
 1-72 
 
 20 
 
 287 
 
 487 
 
 2| ' 
 
 049 
 
 15-50 
 
 5" 
 
 6-50 
 
 1-75 
 
 19 
 
 1-56 
 
 19 
 
 260 
 
 450 
 
 2i " 
 
 059 
 
 11-50 
 
 4" 
 
 5-25 
 
 1-58 
 
 18 
 
 1-40 
 
 18 
 
 233 
 
 413 
 
 1 Ib. 
 
 074 
 
 7-25 
 
 3" 
 
 4-00 
 
 1-41 
 
 17 
 
 1-24 
 
 17 
 
 207 
 
 377 
 
 1 " 
 
 098 
 
 6-00 
 
 During the preparation of this work the Association of American Steel Manufacturers 
 has been formed, from whose circular the above table, has been extracted. 
 
 GRADES OF STEEL. 
 
 Rivet Steel. Ultimate strength, 48,000 to 58,000 pounds per square inch. 
 Elongation, 26 per cent. 
 
780 
 
 APPENDIX. 
 
 Soft Steel. Ultimate strength, 52,000 to 62,000 pounds per square inch. 
 Elongation, 25 per cent. 
 
 Medium Steel. Ultimate strength, 60,000 to 70,000 pounds per square inch. 
 Elongation, 22 per cent. 
 
 Elastic limit, not less than one half the ultimate strength. 
 
 Bending test, 180 degrees flat on itself, or equal to thickness of piece tested, without 
 fracture on outside of bent portion. 
 
 WEIGHTS OF LEAD PIPE PEE FOOT IN LENGTH. 
 
 Caliber. 
 
 i 
 
 I 
 
 i 
 
 f 
 I 
 
 1 
 
 u 
 
 if 
 
 2 
 
 MARK. 
 
 AAA 
 
 AA 
 
 A 
 
 B 
 
 C 
 
 D 
 
 E 
 
 
 Lbs. oz. 
 
 Lbs. oz. 
 
 Lbs. oz. 
 
 Lbs oz. 
 
 Lbs. oz. 
 
 Lbs. oz. 
 
 Lbs. oz. 
 
 Lbs. oz. 
 2 
 
 10 
 
 9i 
 
 
 
 
 
 
 
 2 
 
 1 12 
 
 1 5 
 
 1 2 
 
 1 
 
 14 
 
 7 
 
 3 .. 
 2 8 
 3 8 
 
 2 
 
 1 10 
 
 1 3 
 
 1 
 
 10 
 12 
 1 4 
 
 12 
 
 2 12 
 
 2 8 
 
 2 
 
 1 7 
 
 4 14 
 
 3 3 
 
 3 
 
 2 3 
 
 1 12 
 
 1 3 
 
 1 
 
 
 6 .. 
 
 4 8 
 
 4 
 
 3 4 
 
 2 8 
 
 2 4 
 
 2 
 
 1 8 
 
 6 12 
 
 5 12 
 
 4 11 
 
 3 11 
 
 3 
 
 2 8 
 
 2 
 
 . . . . 
 
 8 
 
 7 
 
 6 4 
 
 5 
 
 4 4 
 
 3 8 
 3 
 3 10 
 
 2 
 
 . . . . 
 
 
 8 8 
 
 6 7 
 
 5 
 
 4 
 
 10 11 
 
 8 14 
 
 7 
 
 6 
 
 5 
 
 4 
 
 . . 
 
 . . . . 
 
 
 . . 
 
 . . 
 
 
 . . 
 
 3 
 
 
 
 THICKNESS. 
 
 WASTE. 
 
 1 
 
 A 
 
 i 
 
 A 
 
 16 11 
 
 13 10 
 
 10 10 
 
 7 3 
 
 6 
 
 4 
 
 19 9 
 22 8 
 25 6 
 
 16 
 18 7 
 20 14 
 
 12 9 
 14 8 
 16 7 
 
 9 4 
 10 12 
 12 2 
 
 5 
 
 3 8 
 
 
 .. .. 
 
 8 
 
 6 
 
 
 . . . . 
 
 . . . . 
 
 .. 
 
 18 6 
 
 13 9 
 
 
 
 10 8 
 
 7 6 
 
 31 3 
 
 .. .. 
 
 20 5 
 
 15 
 
 .. .. 
 
 
 10 8 
 12 
 
 . . . . 
 
 
 
 
 
 
 
 TABLE OF THE WEIGHT OF A CUBIC FOOT OF WATER AT DIFFERENT TEM- 
 PERATURES. 
 
 Fahren- 
 heit. 
 
 Centi- 
 grade. 
 
 Weight in 
 pounds. 
 
 Fahren- 
 heit. 
 
 Centi- 
 grade. 
 
 Weight in 
 pounds. 
 
 Fahren- 
 heit. 
 
 Centi- 
 grade. 
 
 Weight in 
 pounds. 
 
 Degrees. 
 32 
 
 Degrees. 
 
 
 62-42 
 
 Degrees. 
 95 
 
 Degrees. 
 35 
 
 62-06 
 
 Degrees. 
 167 
 
 Degrees. 
 
 75 
 
 60-87 
 
 39 
 
 4 
 
 62-42 
 
 104 
 
 40 
 
 61-95 
 
 176 
 
 80 
 
 60-68 
 
 41 
 
 5 
 
 62-42 
 
 113 
 
 45 
 
 61-83 
 
 185 
 
 85 
 
 60-48 
 
 50 
 
 10 
 
 62-41 
 
 122 
 
 50 
 
 61-69 
 
 194 
 
 90 
 
 60-27 
 
 59 
 
 15 
 
 62-37 
 
 131 
 
 55 
 
 61-55 
 
 203 
 
 95 
 
 60-04 
 
 68 
 
 20 
 
 62-32 
 
 140 
 
 60 
 
 61-39 
 
 212 
 
 100 
 
 59-83 
 
 77 
 
 26 
 
 62-25 
 
 149 
 
 65 
 
 61-23 
 
 
 
 
 86 
 
 30 
 
 62-16 
 
 158 
 
 70 
 
 61-06 
 
 
 
 
APPENDIX. 
 
 T81 
 
 FIG. 1863. 
 
 THE FLOW OF WATER. 
 
 With the increased consumption of water by the mills at Lowell, Mass., there was 
 found a necessity for the accurate gauging of the quantities severally used, and, as at 
 times the total was beyond the flow of the stream, that it should be properly distributed 
 and that none should be wasted. At this time the late James B. Francis was the engi- 
 neer of the Locks and Canals Company, to whom the charge of the canals and water dis- 
 tribution was committed. He decided that the weir was 
 the form in which the water from the several wheels 
 should be measured as most economical in application and 
 accurate in results. 
 
 Figs. 1863 and 1864 are the sectional elevation and 
 plan of the common form of weir, in which the lower 
 edges, bottom, and sides are chamfered off, with edges 
 about i" wide, making a perfect rectangle. In his ex- 
 periments Mr. Francis made the bottom plate of cast iron 
 with the upper edge planed and set accurately horizontal 
 and the sides planed for the vertical edges and for the 
 joints with the bottom plate. The iron rectangle could 
 be accurately measured, but it was necessary to deter- 
 mine the wetted rectangle from the height of the water 
 above the bottom edge. This was done by the hook 
 gauge, in which the hook was stbmerged in the water 
 and gradually raised by a micrometer screw till it showed 
 the sign of a rising at the surface of the water (Fig. 1865). 
 The of the scale of the gauge was accurately referred 
 to the edge of the weir. It was necessary that the surface 
 of the water at the hook be kept perfectly still. It was 
 therefore submerged in a tight box which had communi- 
 cation with the water in the flume by a very small hole or 
 by a small pipe on the bottom of the flume, across it par- 
 allel to the weir and at a slight distance from it. In this 
 pipe at equal intervals in the width of the weir small holes 
 were drilled. The effect of this pipe was to give an aver- 
 age water level in the gauge box. 
 
 The general formula established by the experiments, 
 on which the following table is calculated, is Q = 3'33 
 (I - 2A) AJ, in which Qis the discharge in cubic feet per 
 second, I the length of the weir, and h the height of water 
 above the crest of the weir, both in feet. 
 
 For complete end contractions the side of the weir 
 should be at least equal to the depth of the water on it 
 from the side of the canal. The bottom contraction, also 
 complete, is shown by a body of air below the crest of 
 the weir. Where there is no end contraction, provision 
 should be made for introducing and maintaining free 
 communications of the air beneath the water sheet. For 
 large velocities of approach to the weir, divide the area of a 
 
 water section above it by its discharge from the table for , in the equation h' = , and 
 add h' to previous depth for corrected discharge. 
 
 In the table, the discharge is given for one foot in length ; but as in weirs there are 
 usually two end contractions, virtually reducing the length, and met in the formula above 
 by -%h, a column of correction has been added, which is to be subtracted from the 
 product of discharge, as shown in example. 
 
 FIG. 1864. 
 
 . I 
 
 ~ C-t-J-C ~ ~ +. 1 
 
 FICJ. 1805. 
 
782 
 
 APPENDIX. 
 
 DISCHAKGE, IN CUBIC FEET PEE SECOND, OF A WETE ONE FOOT LONG, WITH- 
 OUT CONTRACTION AT THE ENDS ; FOE DEPTHS FEOM 0-200 TO 0-999 FEET. 
 
 Correction 
 for con- 
 tractions. 
 
 Depth. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 012 
 
 0-20 
 
 0-298 
 
 0-300 
 
 0-302 
 
 0-305 
 
 0-307 
 
 0-309 
 
 0-311 
 
 0-314 
 
 0-316 
 
 0-318 
 
 013 
 
 21 
 
 0-320 
 
 0-323 
 
 0-325 
 
 0-327 
 
 0-330 
 
 0-332 
 
 0-334 
 
 0-337 
 
 0-339 
 
 0-341 
 
 015 
 
 22 
 
 0-344 
 
 0-346 
 
 0-348 
 
 0-351 
 
 0-353 
 
 0-355 
 
 0-358 
 
 0-360 
 
 0-362 
 
 0-365 
 
 017 
 
 23 
 
 0-367 
 
 0-370 
 
 0-372 
 
 0-374 
 
 0-377 
 
 0-379 
 
 0-382 
 
 0-384 
 
 0-387 
 
 0-389 
 
 019 
 
 24 
 
 0-391 
 
 0-394 
 
 0-396 
 
 0-399 
 
 0-401 
 
 0-404 
 
 0-406 
 
 0-409 
 
 0-411 
 
 0-414 
 
 021 
 
 25 
 
 0-416 
 
 0-419 
 
 0-421 
 
 0-424 
 
 0-426 
 
 0-429 
 
 0-431 
 
 0-434 
 
 0-436 
 
 0-439 
 
 023 
 
 26 
 
 0-441 
 
 0-444 
 
 0-447 
 
 0-449 
 
 0-452 
 
 0-454 
 
 0-457 
 
 0-459 
 
 0-462 
 
 0-465 
 
 025 
 
 27 
 
 0-467 
 
 0-470 
 
 0-472 
 
 0-475 
 
 0-478 
 
 0-480 
 
 0-483 
 
 0-485 
 
 0-488 
 
 0-491 
 
 028 
 
 28 
 
 0-493 
 
 0-496 
 
 0-499 
 
 0-501 
 
 0-504 
 
 0-507 
 
 0-509 
 
 0-512 
 
 0-515 
 
 0-517 
 
 030 
 
 29 
 
 0-520 
 
 0-523 
 
 0-525 
 
 0-528 
 
 0-531 
 
 0-534 
 
 0-536 
 
 0-539 
 
 0-542 
 
 0-544 
 
 033 
 
 0-30 
 
 0-547 
 
 0-550 
 
 0-553 
 
 0-555 
 
 0-558 
 
 0-561 
 
 0-564 
 
 0-566 
 
 0-569 
 
 0-572 
 
 036 
 
 31 
 
 0-575 
 
 0-577 
 
 0-580 
 
 0-583 
 
 0-586 
 
 0-589 
 
 0-591 
 
 0-594 
 
 0-597 
 
 0-600 
 
 039 
 
 32 
 
 0-603 
 
 0-606 
 
 0-608 
 
 0-611 
 
 0-614 
 
 0-617 
 
 0-620 
 
 0-623 
 
 0-625 
 
 0-628 
 
 042 
 
 33 
 
 0-631 
 
 0-634 
 
 0-637 
 
 0-640 
 
 0-643 
 
 0-646 
 
 0-649 
 
 0-651 
 
 0-654 
 
 0-657 
 
 045 
 
 34 
 
 0-660 
 
 0-663 
 
 0-666 
 
 0-669 
 
 0-672 
 
 0-675 
 
 0-678 
 
 0-681 
 
 0-684 
 
 0-687 
 
 048 
 
 35 
 
 0-689 
 
 0-692 
 
 0-695 
 
 0-698 
 
 0-701 
 
 0-704 
 
 0-707 
 
 0-710 
 
 0-713 
 
 0-716 
 
 052 
 
 36 
 
 0-719 
 
 0-722 
 
 0-725 
 
 0-728 
 
 0-731 
 
 0-734 
 
 0-737 
 
 0-740 
 
 0-743 
 
 0-746 
 
 056 
 
 37 
 
 0-749 
 
 0-752 
 
 0-755 
 
 0-759 
 
 0-762 
 
 0-765 
 
 0-768 
 
 0-771 
 
 0-774 
 
 0-777 
 
 059 
 
 38 
 
 0-780 
 
 0-783 
 
 0-786 
 
 0-789 
 
 0-792 
 
 0-795 
 
 0-799 
 
 0-802 
 
 0-805 
 
 0-808 
 
 063 
 
 39 
 
 0-811 
 
 0-814 
 
 0-817 
 
 0-820 
 
 0-823 
 
 -'0-827 
 
 0-830 
 
 0-833 
 
 0-836 
 
 0-839 
 
 067 
 
 0-40 
 
 0-842 
 
 0-846 
 
 0-849 
 
 0-852 
 
 0-855 
 
 0-858 
 
 0-861 
 
 0-865 
 
 0-868 
 
 0-871 
 
 072 
 
 41 
 
 0-874 
 
 0-877 
 
 0-881 
 
 0-884 
 
 0-887 
 
 0-890 
 
 0-893 
 
 0-897 
 
 0-900 
 
 o'gos 
 
 076 
 
 42 
 
 0-906 
 
 0-910 
 
 0-913 
 
 0-916 
 
 0-919 
 
 0-923 
 
 0-926 
 
 0-929 
 
 0-932 
 
 0-936 
 
 081 
 
 43 
 
 0-939 
 
 0-942 
 
 0-945 
 
 0-949 
 
 0-952 
 
 0-955 
 
 0-959 
 
 0-962 
 
 0-965 
 
 0-969 
 
 085 
 
 44 
 
 0-972 
 
 0-975 
 
 0-978 
 
 0-982 
 
 0-985 
 
 0-988 
 
 0-992 
 
 0-995 
 
 0-998 
 
 1-002 
 
 090 
 
 45 
 
 1-005 
 
 1-009 
 
 1-012 
 
 1-015 
 
 1-019 
 
 1-022 
 
 1-025 
 
 1-029 
 
 1-032 
 
 1-035 
 
 095 
 
 46 
 
 1-039 
 
 1-042 
 
 1-046 
 
 1-049 
 
 1-052 
 
 1-056 
 
 1-059 
 
 1-063 
 
 1-066 
 
 1-070 
 
 100 
 
 47 
 
 1-073 
 
 1-076 
 
 1-080 
 
 1-083 
 
 1-087 
 
 1-090 
 
 1-094 
 
 1-097 
 
 1-100 
 
 1-104 
 
 106 
 
 48 
 
 1-107 
 
 1-111 
 
 1-114 
 
 1-118 
 
 1-121 
 
 1-125 
 
 1-128 
 
 1-132 
 
 1-135 
 
 1-139 
 
 111 
 
 49 
 
 1-142 
 
 1-146 
 
 1-149 
 
 1-153 
 
 1-156 
 
 1-160 
 
 1-163 
 
 1-167 
 
 1-170 
 
 1-174 
 
 118 
 
 0-50 
 
 1-177 
 
 1-181 
 
 1-184 
 
 1-188 
 
 1-191 
 
 1-195 
 
 1-199 
 
 1-202 
 
 1-206 
 
 1-209 
 
 124 
 
 51 
 
 1-213 
 
 1-216 
 
 1-220 
 
 1-223 
 
 1-227 
 
 1-231 
 
 1-234 
 
 1-238 
 
 1-241 
 
 1-245 
 
 130 
 
 52 
 
 1-249 
 
 1-252 
 
 1-256 
 
 1-259 
 
 1-263 
 
 1-267 
 
 1-270 
 
 1-274 
 
 1-278 
 
 1-281 
 
 136 
 
 53 
 
 1-285 
 
 1-288 
 
 1-292 
 
 1-296 
 
 1-299 
 
 1-303 
 
 1-307 
 
 1-310 
 
 1-314 
 
 1-318 
 
 143 
 
 54 
 
 1-321 
 
 1-325 
 
 1-329 
 
 1-332 
 
 1-336 
 
 1-340 
 
 1-343 
 
 1-347 
 
 1-351 
 
 1-355 
 
 150 
 
 55 
 
 1-358 
 
 1-362 
 
 1-366 
 
 1-369 
 
 1-373 
 
 1-377 
 
 1-381 
 
 1-384 
 
 1-388 
 
 1-392 
 
 157 
 
 56 
 
 1-395 
 
 1-399 
 
 1-403 
 
 1-407 
 
 1-410 
 
 1-414 
 
 1-418 
 
 1-422 
 
 1-425 
 
 1-429 
 
 164 
 
 57 
 
 1-433 
 
 1-437 
 
 1-441 
 
 1-444 
 
 1-448 
 
 1-452 
 
 1-456 
 
 1-459 
 
 1-463 
 
 1-467 
 
 171 
 
 58 
 
 1-471 
 
 1-475 
 
 1-478 
 
 1-482 
 
 1-486 
 
 1-490 
 
 1-494 
 
 1-498 
 
 1-501 
 
 1-505 
 
 178 
 
 59 
 
 1-509 
 
 1-513 
 
 1-517 
 
 1-521 
 
 1-524 
 
 1-528 
 
 1-532 
 
 1-536 
 
 1-540 
 
 1-544 
 
 186 
 
 0-60 
 
 1-548 
 
 1-551 
 
 1-555 
 
 1-559 
 
 1-563 
 
 1-567 
 
 1-571 
 
 1-575 
 
 1-579 
 
 1-583 
 
 194 
 
 61 
 
 1-586 
 
 1-590 
 
 1-594 
 
 1-598 
 
 1-602 
 
 1-606 
 
 1-610 
 
 1-614 
 
 1-618 
 
 1-622 
 
 202 
 
 62 
 
 1-626 
 
 1-630 
 
 1-633 
 
 1-637 
 
 1-641 
 
 1-645 
 
 1-649 
 
 1-653 
 
 1-657 
 
 1-661 
 
 210 
 
 63 
 
 1-665 
 
 1-669 
 
 1-673 
 
 1-677 
 
 1-681 
 
 1-685 
 
 1-689 
 
 1-693 
 
 1-697 
 
 1-701 
 
 218 
 
 64 
 
 1-705 
 
 1-709 
 
 1-713 
 
 1-717 
 
 1-721 
 
 1-725 
 
 1-729 
 
 1-733 
 
 1-737 
 
 1-741 
 
 227 
 
 65 
 
 1-745 
 
 1-749 
 
 1-753 
 
 1-757 
 
 1-761 
 
 1-765 
 
 1-769 
 
 1-773 
 
 1-777 
 
 1-781 
 
 236 
 
 66 
 
 1-785 
 
 1-790 
 
 1-794 
 
 1-798 
 
 1-802 
 
 1-806 
 
 1-810 
 
 1-814 
 
 1-818 
 
 1-822 
 
 245 
 
 67 
 
 1-826 
 
 1-830 
 
 1-834 
 
 1-838 
 
 1-843 
 
 1-847 
 
 1-851 
 
 1-855 
 
 1-859 
 
 1-863 
 
 254 
 
 68 
 
 1-867 
 
 1-871 
 
 1-875 
 
 1-880 
 
 1-884 
 
 1-888 
 
 1-892 
 
 1-896 
 
 1-900 
 
 1-904 
 
 263 
 
 69 
 
 1-909 
 
 1-913 
 
 1-917 
 
 1-921 
 
 1-925 
 
 1-929 
 
 1-934 
 
 1-938 
 
 1-942 
 
 1-946 
 
 273 
 
 0-70 
 
 1-950 
 
 1-954 
 
 1-959 
 
 1-963 
 
 1-967 
 
 1-971 
 
 1-975 
 
 1-980 
 
 1-984 
 
 1-988 
 
 283 
 
 71 
 
 1-992 
 
 1-996 
 
 2-001 
 
 2-005 
 
 2-009 
 
 2-013 
 
 2-017 
 
 2-022 
 
 2-026 
 
 2-030 
 
 293 
 
 72 
 
 2-034 
 
 2-039 
 
 2-043 
 
 2-047 
 
 2-051 
 
 2-056 
 
 2-060 
 
 2-064 
 
 2-068 
 
 2-073 
 
 303 
 
 73 
 
 2-077 
 
 2-081 
 
 2-085 
 
 2-090 
 
 2-094 
 
 2-098 
 
 2-103 
 
 2-107 
 
 2-111 
 
 2-115 
 
 314 
 
 74 
 
 2-120 
 
 2-124 
 
 2-128 
 
 2-133 
 
 2-137 
 
 2-141 
 
 2-146 
 
 2-150 
 
 2-154 
 
 2-159 
 
 324 
 
 75 
 
 2-163 
 
 2-167 
 
 2-172 
 
 2-176 
 
 2-180 
 
 2-185 
 
 2-189 
 
 2-193 
 
 2-198 
 
 2-202 
 
APPENDIX. 
 
 783 
 
 DISCHAKGE, IN CUBIC FEET PEE SECOND, OF A WE1K ONE FOOT LONG, WITH- 
 OUT CONTKACTION AT THE ENDS;FOK DEPTHS FKOM 0-200 TO 0-999 FEET. 
 
 (Continued.) 
 
 Correction 
 for con- 
 tractions. 
 
 Depth. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 335 
 
 76 
 
 2-206 
 
 2-211 
 
 2-215 
 
 2-219 
 
 2-224 
 
 2-228 
 
 2-232 
 
 2-237 
 
 2-241 
 
 2-246 
 
 346 
 
 77 
 
 2-250 
 
 2-254 
 
 2-259 
 
 2-263 
 
 3-267 
 
 2-272 
 
 2-276 
 
 2-281 
 
 2-285 
 
 2-290 
 
 358 
 
 78 
 
 2-294 
 
 2-298 
 
 2-303 
 
 2-307 
 
 2-312 
 
 2-316 
 
 2-320 
 
 2-325 
 
 2-329 
 
 2-334 
 
 369 
 
 79 
 
 2-238 
 
 2-343 
 
 2-347 
 
 2-351 
 
 2-356 
 
 2-360 
 
 2-365 
 
 2-369 
 
 2-374 
 
 2-378 
 
 381 
 
 0-80 
 
 2-383 
 
 2-387 
 
 2-392 
 
 2-396 
 
 2-401 
 
 2-405 
 
 2-410 
 
 2-414 
 
 2-419 
 
 2-423 
 
 393 
 
 81 
 
 2-428 
 
 2-432 
 
 2-437 
 
 2-441 
 
 2-446 
 
 2-450 
 
 2-455 
 
 2-459 
 
 2-464 
 
 2-468 
 
 406 
 
 82 
 
 2-473 
 
 2-477 
 
 2-482 
 
 2-486 
 
 2-491 
 
 2-495 
 
 2-500 
 
 2-504 
 
 2-509 
 
 2-513 
 
 413 
 
 83 
 
 2-518 
 
 2-523 
 
 4-527 
 
 2-532 
 
 2-536 
 
 2-541 
 
 2-545 
 
 2-550 
 
 2-554 
 
 2-559 
 
 431 
 
 84 
 
 2-564 
 
 2-568 
 
 2-573 
 
 2-577 
 
 2-582 
 
 2-587 
 
 2-591 
 
 2-596 
 
 2-600 
 
 2-605 
 
 444 
 
 85 
 
 2-610 
 
 2-614 
 
 2-619 
 
 2-623 
 
 2-628 
 
 2-633 
 
 2-637 
 
 2-642 
 
 2-646 
 
 2-651 
 
 457 
 
 86 
 
 2-656 
 
 2-660 
 
 2-665 
 
 2-670 
 
 2-674 
 
 2-679 
 
 2-684 
 
 2-688 
 
 2-693 
 
 2-698 
 
 470 
 
 87 
 
 7-702 
 
 2-707 
 
 2-712 
 
 2-716 
 
 2-721 
 
 2-726 
 
 2-730 
 
 2-735 
 
 2-740 
 
 2-744 
 
 484 
 
 88 
 
 2-749 
 
 2-754 
 
 2-758 
 
 2-763 
 
 2-768 
 
 2-772 
 
 2-777 
 
 2-782 
 
 2-786 
 
 2-791 
 
 498 
 
 89 
 
 2-796 
 
 2-801 
 
 2-805 
 
 2-810 
 
 2-815 
 
 2-819 
 
 2-824 
 
 2-829 
 
 2-834 
 
 2-838 
 
 512 
 
 0-90 
 
 2-843 
 
 2-848 
 
 2-853 
 
 2-857 
 
 2-862 
 
 2-867 
 
 2-872 
 
 2-876 
 
 2-881 
 
 2-886 
 
 526 
 
 91 
 
 2-891 
 
 2-895 
 
 2-900 
 
 2-905 
 
 2-910 
 
 2-915 
 
 2-919 
 
 2-924 
 
 2-929 
 
 2-934 
 
 541 
 
 92 
 
 2-938 
 
 2-943 
 
 2-948 
 
 2-953 
 
 2-958 
 
 2-963 
 
 2-967 
 
 2-972 
 
 2-977 
 
 2-982 
 
 555 
 
 93 
 
 2-986 
 
 2-991 
 
 2-996 
 
 3-001 
 
 3-006 
 
 3-011 
 
 3-015 
 
 3-020 
 
 3-025 
 
 3-030 
 
 570 
 
 94 
 
 3-035 
 
 3-040 
 
 3-044 
 
 3-049 
 
 3-054 
 
 3-059 
 
 3-064 
 
 3-069 
 
 3-074 
 
 3-078 
 
 586 
 
 95 
 
 3-083 
 
 3-088 
 
 3-093 
 
 3-098 
 
 3-103 
 
 3-108 
 
 3-113 
 
 3-117 
 
 3-122 
 
 3-127 
 
 601 
 
 96 
 
 3-132 
 
 3-137 
 
 3-142 
 
 3-147 
 
 3-152 
 
 3-157 
 
 3-162 
 
 3-166 
 
 3-171 
 
 3-176 
 
 677 
 
 97 
 
 3-181 
 
 3-186 
 
 3-191 
 
 3-196 
 
 3-201 
 
 3-206 
 
 3-211 
 
 3-216 
 
 3-221 
 
 3-226 
 
 632 
 
 98 
 
 3-231 
 
 3-235 
 
 3-240 
 
 3-245 
 
 3-250 
 
 3-255 
 
 3-260 
 
 3-265 
 
 3-270 
 
 3-275 
 
 648 
 
 99 
 
 3-280 
 
 3-285 
 
 3-290 
 
 3-295 
 
 3-300 
 
 3-305 
 
 3-310 
 
 3-315 
 
 3-320 
 
 3-325 
 
 Example. Let the weir, with end contractions, be 5 '3 feet long, and depth of water, 
 or h = 0-612. 
 
 By table the discharge for one foot in length is 1 -594 
 
 5-3 
 
 Correction 
 
 8-4482 
 196 
 
 Discharge in cubic feet per second 8'2522 
 
 The measures of the discharge of the different wheels was accurately determined; it 
 was equally important to determine the quantities used by the different corporations. 
 Mr. Francis's arrangements for this purpose were the construction of rectangular plank 
 flumes in the different canals or feeders, of which the widths were divided into feet 
 marked on the upstream faces of the timbers, extending above and across the ends of the 
 flume, which enabled the average thread of the float in its passage through the flume to 
 be ascertained. The depth of the water in the flume was measured by a gauge placed 
 centrally on one side of the flume. The floats measuring the velocity of the currents 
 were adjusted to the depths of water, having some two inches clearance from the bottom. 
 The tubes were of two to three inches diameter, loaded at the bottom, and marked with 
 their water line. They were brought to the flume and those of length appropriate to 
 its depth were selected. From the notes thus taken the average path of the float was 
 determined, on which was plotted its velocity, as represented by the different circles 
 (Fig. 182). Through the average of these circles on the lines of flow is drawn the full 
 line of averages from which the average of the total flow was determined. 
 
784 
 
 APPENDIX. 
 
 In a similar way gaugings may be made of natural streams where the sections are ap- 
 proximately smaller. Soundings are to be made across the stream and sections drawn. 
 The average velocities may be taken through these sections, which multiplied by the 
 products of the sections and nine tenths of their sum will give approximately the flow 
 of the streams. For this kind of measure I have used two rubber balls connected by an 
 adjustable soft cotton cord, the lower ball being filled with water. A light strong thread 
 is attached to the upper ball with the ends in the hands of the observers on each side of 
 the stream, by which the ball may be guided in the thread of its appropriate section. 
 
 It was customary to find the average flow of a stream from the surface velocities in 
 different threads of it according to the sections by means of apples loaded with shingle 
 nails to nearly the specific gravity of the water, casting them into the stream, and taking 
 the time of transit between two cords stretched across it, and taking as an average about 
 eighty per cent that of the whole observations by the area of the entire section. Surface 
 velocities only to be considered as loose approximations, as they are very much influenced 
 by the direction and strength of the wind and the uniformities of flow in the different 
 sections and diversions or eddies. 
 
 The miner's inch is designated as the flow through one square inch, but under vari- 
 ous heads in different states. P. J. Flynn in "Irrigation Canals" gives one cubic foot 
 per second as the equivalent flow through fifty miner's inches under a mean head of four 
 inches. 
 
 Fro. 1866. 
 
 FLOW OF WATER IN PIPES AND CONDUITS. 
 
 The general formula for the flow of water in all channels is v = c \/fis, in which is 
 the velocity, c a coefficient determined for the particular form of channel, It the 
 hydraulic mean depth that is, the area of water cross-section divided by its wetted 
 perimeter of channel ; s is the slope of difference of level or pressure in feet between two 
 points divided by the length in feet. 
 
 Fig. 1866 is a section of a water pipe in which the capacities are shown of its differ- 
 ent sections from the bottom to a full section without any head. The friction of the 
 sides of the full pipe reduces the velocity, and the maximum is at about eighty per cent 
 of full, and the maximum of the curve of discharge or velocity by area of section a little 
 above ninety per cent. It must be observed that the velocity at half section is the same 
 as when the pipe is full. 
 
 Messrs. Ganguillet and Kutter have established formulae, of which there is a modifi- 
 cation according to the character of the surface of the wetted perimetre. 
 
 The Kutter formula, as it is called, is very complicated, but it has been simplified 
 graphically by Mr. Rudolph Hering and published in the " Transactions of the A. S. C. 
 E.," January, 1879, from which the figures have been redrawn. 
 
APPENDIX. 
 
 785 
 
 IRON ; CEMENT ; TERRA COTTA PIPES, WELL-JOINTED AND IN BEST ORDER ; CAREFULLY 
 
 PLASTERED SURFACE. 
 
 Grades in feef per hundred n..o// 
 
 n -/ .? .4 .6 .8 / J. 4. 5. 6. 
 
 51 
 
786 
 
 APPENDIX. 
 
 OLD IRON ; CEMENT AND TERRA COTTA PIPES NOT WELL-JOINTED NOR IN PERFECT 
 ORDER ; WELL LAID BRICKWORK. 
 
 Grades in feel per hundred n .0/3 
 
 .1 .2 .4 .6 .81- t 3. 4. 5. 6. 
 
 60 
 
 36D/am 
 
APPENDIX. 
 
 787 
 
 FOUL AND SLIGHTLY TUBEKCTJLATED IRON ; CEMENT AND TERRA COTTA PIPES WITH 
 IMPERFECT JOINTS, AND IN BAD ORDER ; WELL-DRESSED STONEWORK AND SECOND 
 CLASS BRICKWORK. 
 
 Grades fn feet per hundred n= .0/6 
 
 .1 .? A .6 .8 /. 2. 3. 4 6. 6. 
 
788 
 
 APPENDIX. 
 
 IRON ; CEMENT ; TEKRA COTTA PIPES, WELL-JOINTED AND IN BEST ORDER. 
 
 .OZ.04.06* .1 
 
 Grades in feet per hundred] /?= -Oil 
 
 .2 .3 .4 .6 .7 .$.3 /.O /.6 2.0 26 3.O 
 
 56X83 5X76 46X6'9 
 
 1'6XZ3 
 
APPENDIX. 
 
 789 
 
 OLD IRON ; CEMENT 
 
 AND TERRA COTTA PIPES NOT WELL-JOINTED NOR IN PERFECT 
 ORDER ; WELL LAID BRICKWORK. 
 
 Grades m feef per hundred /? .0/3 
 
 02.04O6./ .2 .3 .4 .J 6 .7 .8 .3/.O X.v5 2.O Z5 3.O 
 
 0X9 66X83 6X76 46"X63 
 
790 
 
 APPENDIX. 
 
 FOTJL AND SLIGHTLY TUBERCULATED IRON ; CEMENT AND TERRA COTTA PIPES WITH 
 IMPERFECT JOINTS, AND IN BAD ORDER ; WELL-DRESSED STONEWORK AND SECOND 
 CLASS BRICKWORK. 
 
 Grades //? feet per hundred 
 
 -3 * -5 .6 .7.8.9*0 '-5 
 
 n~.OI5 
 
 46X69 
 
APPENDIX. 
 
 T91 
 
 For tables of the Kutter formula see "Irrigation Canals," by P. J. Flynn, C. E. 
 
 It will be observed that n depends on the condition of the wetted surface, and that 
 with iron surfaces it must vary largely with the time exposure of these surfaces and the 
 characteristics of the water passing through. 
 
 Where pipes of iron can not be cleaned, it is impossible for the engineer to form re- 
 liable judgment of their condition after years of use. They can be tested 
 by measuring flows through a hydrant which discharges from a single 
 length of pipe and determining the s in that length. 
 
 For a main with no available gate the following is suggested : 
 
 Let a d be the main ; establish s between a & and cdby reliable gauges. 
 At g put in a branch controlled by a gate. After s s are established as 
 above, open the gate and measure its discharge accurately by a weir. With 
 this factor see what is between a and & and check the quantities by the 
 changes between c and d and compare with flows at different values of n. 
 
 Should the values of ?&, as thus established by experiment, be outside 
 of the limits of n = 'Oil, '013, '015, as given by the diagrams, curves 
 can be established from these values and extended to include those of 
 the experiments, which may answer as approximations. 
 
 FLOW OF AIR. 
 
 The flow of air is subject to the same laws as the flow of water. The 
 preceding diagrams may therefore be used in finding the discharge of air in cubic feet 
 per second, under the same heads of water in feet, by multiplying by 27'6, the square 
 root of 761, the difference in density of a cubic foot of the two fluids. 
 
 The theoretical discharge of a pipe is as the square root of the fifth power of the 
 diameter, from which the following table, derived from the circular of the B. F. Sturte- 
 vant Company, is based : 
 
 $ 
 
 TABLE FOE EQUALIZING THE DIAMETER 
 
 OF PIPES. 
 
 1 
 
 i 
 
 
 
 P 
 
 irties putting up blast pipes 
 four 6-inch pipes is the same 
 
 are very liable to think, because the combined area of 
 as one 12-inch pipe, that the four pipes will convey the 
 
 2 
 
 5.7 
 
 2 ] 
 
 3 
 
 1C 
 
 2.7| 3 
 
 4 
 
 32 
 
 5.7 2. 
 
 4 
 
 same quantity of air with the same ease and freedom that the 12-inch will, where- 
 
 5 
 
 5C 
 
 9.8 
 
 I.I 
 
 1.8| 5 
 
 as it actually does take 5-7 almost six 6-inch 
 
 pipes. Again, 
 
 16 3-inch pipes 
 
 6 
 
 88 
 
 1C 
 
 .] 
 
 2.3| 1.6| 6 
 
 have the combined area of one 12-inch pipe, but in actual practice it takes 
 
 7 
 
 129 
 
 23j 8.3 
 
 4.1 
 
 2.3| 1.5 
 
 7 
 
 just 32 3-inch pipes to do the work of one 12-inch. 
 
 8 
 
 180 
 
 32 | 12 
 
 .'..7 
 
 3.2| 2.1| 1.4 
 
 8 
 1.1 
 
 T" 
 
 
 Thi 
 
 s is 
 the 
 
 lue 
 sin 
 Th 
 
 to t 
 
 lllp 
 
 el* 
 
 an 
 
 he < 
 ipos 
 K c 
 icte 
 Tl 
 
 xee 
 
 OV( 
 
 Iffn 
 
 rs ii 
 le fi 
 til 
 
 SS 
 
 r th 
 -es f 
 inc 
 _nm 
 le \ 
 
 
 "fri 
 at i 
 t, tl 
 lies 
 s a 
 ith 
 pes 
 O 
 
 ctio 
 n th 
 
 e to 
 
 oft 
 the 
 the 
 |0f 
 
 f til 
 
 4 
 
 1 
 
 1 fc 
 o la 
 
 po 
 he I 
 in 
 ve 
 the 
 100 
 :ipa 
 
 r every cubic foot of ai. 
 rge. 
 f each column give the 
 run c'li pipes, 
 ersection of the horizo 
 rtical give the numbe 
 diameter given at the 
 umn, that will be equa 
 city for conveying aii 
 one given opposite in 
 first column. 
 
 1 
 
 r in 
 di- 
 
 ntal 
 r of 
 top 
 1 in 
 to 
 the 
 
 9 
 
 244 
 
 42 
 
 1C 
 
 7 f 
 
 4.3| 2.8 
 
 1.9 
 
 10 
 
 317 
 
 56 
 
 20 
 
 I.I 
 
 5.7[ S:6| 2.4| 1.7| 1.3 
 
 10 
 
 11 
 
 402 
 
 71 
 
 26 
 
 12 
 
 7.0| 4.5 3.1 
 
 I.I 
 
 1.7 
 
 t.l 
 
 11 
 
 12 
 
 501 
 
 88 
 
 32 
 
 16 
 
 9.0[ 5.7) 3.S 
 
 2.8[ 2.0) 1.6 
 
 1.1 
 
 12 
 
 13 
 
 613 
 
 107 
 
 .11) 
 
 l 
 
 11 | 6.9 
 
 4.7 
 
 3.4 
 
 2.5 
 
 I.I 
 
 U 
 
 1.1 
 
 13 
 
 14 
 IS 
 
 737 
 
 129 
 
 47 
 
 2.1 
 
 13 | 8.3| 5.7| 4.1 
 
 3.0| 2.3 
 
 I.I 
 
 1..- 
 
 l.i 
 
 14 
 
 16 
 
 026 
 
 180 
 
 6, 
 
 M 
 
 18 
 
 n 
 
 7.9| 5.7 
 
 4.2 
 
 1.9 
 
 2, 
 
 t.l 
 
 1.7 
 
 1.4 
 
 1.2 
 
 16 
 
 17 
 
 
 
 
 
 
 
 
 
 4.9[ 8.8) 2.9 
 
 
 
 
 
 1.2(17 
 
 18 
 
 375 
 
 239 
 
 88 
 
 43 
 
 M 
 
 16 
 
 10 
 
 1.1 
 
 5.7 
 
 4.8| 3.4 
 
 2.8| 2.3| 1.9[ 1.6 
 
 1.3 
 
 1.2| 18 
 
 19 
 20 
 
 580 
 797 
 
 275 
 
 no 
 
 411 
 
 23 
 
 18 
 
 12 
 
 M 
 
 8 8 
 
 t.S 
 
 5, 
 
 3.9 
 
 3.2 
 
 I.I 
 
 |J.2| 1.8 
 
 ,.:, 
 l.T 
 
 1.3 
 
 \.: 
 
 1.2! 19 | 
 1.3) 1.1J20 
 
 22 
 24 
 
 284 
 834 
 
 493 
 
 80 
 
 88 
 
 M 
 
 .12 
 
 18 
 22 
 
 1C 
 
 12 
 
 |.| 
 
 7.1 
 
 4.1 
 
 5.7 
 
 3.7|3.1 
 4.6| 3.8 
 
 3.2 
 
 2.9 
 
 2.4 
 
 I.I 
 
 1.6 
 
 i.< 
 
 22 | 
 | 1.2|24] 
 
 aal 
 
 26 
 
 474 
 
 C05 
 
 19 
 
 1M 
 
 62 
 
 n 
 
 IT 
 
 1* 
 
 14 
 
 11 
 
 8.6 
 
 I 1 ."* 
 
 s,r 
 
 6.8 .. 
 
 4.1 
 
 4.0|3.4 
 
 2.9 2.5 
 
 2.1 
 
 1.9| 1.5| 1.2|26 
 
 30 
 
 963 
 
 8C4 
 
 11 
 
 IH 
 
 8.8 
 
 56 
 
 .18 
 
 2.8 
 
 20 
 
 1C 
 
 12 
 
 9.9 
 
 8.0| 6.7 
 
 4.8| 4.1 
 5.7J 4.7 
 
 4.1 
 
 I.I 
 
 3. 
 
 2.6| 2.2( 1.7| 1.4 
 
 1.2|3O| 
 
 36 
 
 818 
 
 1361 
 
 n 
 
 243 
 
 139 
 
 M 
 
 w 
 
 43 
 
 32 
 
 2.1 
 
 19 | 16 
 
 13 
 
 11 
 
 8.9 
 
 7.C 
 
 c.;, 
 
 .'..7 
 
 >'' 
 
 4.3 
 
 3.4| 2.7[ 2.2|1.9|l.C|38| 
 
 48 
 
 989 
 
 2792 
 
 81 |492 |282 
 
 180 
 
 123 
 
 88 
 
 CO 
 
 50 
 
 39 
 
 .12 
 
 26 
 
 22 
 
 18 
 
 16 
 
 13 
 
 12 
 
 10 
 
 8.1. 
 
 7.0| 5.7i 4.7|3.8J3.2|2.1|1.4|48| 
 
 54 | 
 
 i 560 
 
 3753 
 
 08 671 384 
 
 244 
 
 160 
 
 11 'J 
 
 88 
 
 C8 
 
 53 | 43 
 
 .1J 
 
 29 
 
 24 
 
 21 
 
 18 
 
 1C 
 
 15 
 
 12 
 
 9.4| 7.6| 6.25. 2 
 
 4.3;2.8|1.9 
 
 1.3|54 
 
 6O ]27913 
 
 4879 ] 
 
 81 
 
 872 
 
 499 
 
 .114 
 
 215 
 
 1M 
 
 111 
 
 H 
 
 69 
 
 M 
 
 40 
 
 3f- 32 
 
 27 
 
 23 
 
 20 
 
 18 
 
 10 
 
 12 | 9.9| 8.1 
 
 0.', 
 
 5.7|3.6|2.4|l.8|l.S 
 
792 
 
 APPENDIX. 
 
APPENDIX. 
 
 793 
 
 : 
 
 : : 
 
 tr- 
 
 U5I 
 
794 
 
 APPENDIX. 
 
 The foregoing diagrams of the flow of gas have been made from the formula con- 
 tained in "Practical Treatise on Heat," by Thomas Box, and is as follows : 
 Q= (H x (3-7D) 5 -^L)i, 
 
 in which Q = discharge cubic feet per minute; H = pressure (grade) in inches of water; 
 D = diameter of pipe in inches ; L = length in yards. The specific gravity (G) of the 
 gas in this formula is -42. 
 
 Formula in previous editions of this work was Q = 1350 D 2 4/ , and agrees very 
 
 closely with the results from the diagrams given below. 
 
 The diagrams for the flows of water (pages 785-790) in pipes are equally applicable 
 to the movement of gaseous fluids, the velocities being directly as the square roots of 
 their specific gravities. Thus, the water discharge of a conduit 2 feet in diameter under 
 a grade of .0833' per 100 feet in the diagram n = -015 is 5 '5 cubic feet per second. By 
 Tables of the Weight of Water (page 780) compared with that of air below it will be 
 seen that at 62 F. the weight of a cubic foot of water is to that of air as 1 to 820. 
 5-5 x 1/820 = 5-5 x 28-636 = 157 cubic feet per second. 
 
 As the diagrams for the flow of water are in feet per hundred and those of air or gas 
 in inches, to convert the grades of the former into those of the latter, multiply the dis- 
 charge by the square root of '0833' (1") or '2889'. Taking above example, diagram 
 under 1: 100 gives a discharge of 19 cubic feet per second, which x '2889 x 28'636 = 157 
 cubic feet per second ; but as the flow of air is usually given in cubic feet per minute, 
 multiply by 60 and we obtain 9,420 cubic feet of air per minute. 
 
 In the movement of compressed air by the action of the pump the air is heated, and in 
 its passage through the pipe this heat is gradually dissipated with a change of its specific 
 gravity, for which allowance is to be made in velocity of movement. With illuminating 
 gas the specific gravity of '42 adopted in the diagram is a common one. 
 
 The products of combustion escaping into a large chimney may be taken as of the 
 same specific gravity as illuminating gas at '42, but the velocity in the general rules is 
 much less than that given by the diagrams. There is no objection to large chimneys ex- 
 cept in their cost and back draughts, which last may be met by uniformity of section 
 without eddies or by a Venturi converging and diverging tube, at the bottom. 
 
 Insufficient draught in short chimneys, induced by positions or necessities of con- 
 struction, is met by fans discharging directly into a fire-room or ash-pit, or by steam-jet 
 blowers, as illustrated on page 368. 
 
 VOLUME AND WEIGHT OF DRY AIR 
 
 At Different Temperatures under a Constant Atmospheric Pressure of 29-92 Inches of the 
 Barometer, the Volume at 32 F. being the Unit. 
 
 Temp. F. 
 
 Volume. 
 
 Weight per 
 cub. ft., 
 Pounds. 
 
 Temp. F. 
 
 Volume. 
 
 Weight per 
 cub. ft., 
 Pounds. 
 
 Temp. F. 
 
 Volume. 
 
 Weight per 
 cub. ft., 
 Pounds. 
 
 
 
 935 
 
 0864 
 
 132 
 
 1-204 
 
 0671 
 
 325 
 
 1-597 
 
 0506 
 
 12 
 
 960 
 
 0842 
 
 142 
 
 1-224 
 
 0659 
 
 350 
 
 1-648 
 
 0490 
 
 22 
 
 980 
 
 0824 
 
 152 
 
 1-245 
 
 0649 
 
 375 
 
 1-689 
 
 0477 
 
 32 
 
 1-000 
 
 0807 
 
 162 
 
 1-265 
 
 0638 
 
 400 
 
 1-750 
 
 0461 
 
 42 
 
 1-020 
 
 0791 
 
 172 
 
 1-285 
 
 0628 
 
 450 
 
 1-852 
 
 0436 
 
 52 
 
 1-041 
 
 0776 
 
 182 
 
 1-306 
 
 0618 
 
 500 
 
 1-954 
 
 0413 
 
 62 
 
 1-061 
 
 0761 
 
 192 
 
 1-326 
 
 0609 
 
 550 
 
 2-056 
 
 0384 
 
 72 
 
 1-082 
 
 0747 
 
 202 
 
 1-347 
 
 0600 
 
 600 
 
 2-150 
 
 0376 
 
 82 
 
 1-102 
 
 0733 
 
 212 
 
 1-367 
 
 0591 
 
 650 
 
 2-260 
 
 0357 
 
 92 
 
 1-122 
 
 0720 
 
 230 
 
 1-404 
 
 0575 
 
 700 
 
 2-362 
 
 0338 
 
 102 
 
 1-143 
 
 0707 
 
 250 
 
 1-444 
 
 0559 
 
 800 
 
 2-566 
 
 0315 
 
 112 
 
 1-163 
 
 0694 
 
 275 
 
 1-495 
 
 0540 
 
 900 
 
 2-770 
 
 0292 
 
 122 
 
 1-184 
 
 0682 
 
 300 
 
 1-546 
 
 0522 
 
 1000 
 
 2-974 
 
 0268 
 
APPENDIX. 
 
796 
 
 APPENDIX. 
 
 A Babcock and Wilcox water tube boiler (Fig. 1867) is composed of lap-welded 
 wrought-iron tubes placed in an inclined position and connected with each other and 
 with a horizontal steam and water drum, by vertical passages at each end, while a mud 
 drum is connected to the rear and lowest point in the boiler. 
 
 The end connections are in one piece for each vertical row of tubes, and are of such 
 form that the tubes are staggered. The sections thus formed are connected with the 
 drum, and with the mud drum also, by short tubes expanded into bored holes. The 
 openings for cleaning, opposite the end of each tube, are closed by hand-hole plates. 
 
 To utilize waste heat heaters are set in a chamber in connection with the flues leading 
 to the chimney. Fig. 1868 is an elevation of one of these forms of apparatus, the Green 
 economizer, consisting of ranges of vertical pipes, connected at the top and bottom 
 with horizontal pipes, into which the feed water is introduced at the bottom and leaves 
 at the top. The whole is inclosed in a brick chamber with the products of combustion 
 passing among the pipes. The outsides of the pipes are ckaned by automatic scrapers. 
 Where the heat is necessary to insure draft in the smoke flue there can be no economy in 
 the apparatus, but an obstruction to the draft. 
 
 In operation, the fire is made under the front and higher ends of the tubes, and the 
 products of combustion pass up between the tubes into a combustion chamber under the 
 steam and water drum ; from thence they pass down between the tubes, then once more 
 up through the spaces between the tubes and off to the chimney. The water inside the 
 tubes, as it is heated, tends to rise toward the higher end, and as it is converted into 
 steam, the mingled column of steam and water rises through the vertical passages into 
 the drum above the tubes, where the steam separates from the water, and the latter flows 
 back to the rear and down again through the tubes in a continuous circulation. 
 
 The steam is taken out at the top of the steam drum, near the back end of the boiler, 
 after it has thoroughly separated from the water. 
 
 Scale..l in.=13 ft. 
 
 FIG. 1868. 
 
APPENDIX. 
 
 797 
 
 L J 
 
 The Heine boiler (Fig. 1869) is composed of wrought-iron tubes extending between 
 the inside of two "water legs," or end connections, between the tubes and a steam and 
 water drum placed above them. These end chambers are of approximately rectangular 
 shape, drawn in at the top to fill the curvature of the shells. Each is composed of a 
 head plate and a tube sheet, flanged all around and joined at bottom and sides by a butt 
 strap of the same material, strongly riveted to both. They are further stayed by hollow- 
 stay bolts of hydraulic tubing of large diameter, so placed that two stays support each 
 tube and hand hole. The water legs are joined to the shell by flanged and riveted joints, 
 and the shells are cylinders with heads dished. The steam space in front is about two 
 thirds the diameter of the shell, while at the rear the water occupies two thirds of the 
 shell, the whole contents being equally divided between steam and water. On the top 
 of the shell, near the front end, is riveted a nozzle for a steam and safety valve. A flue, 
 or breeching, connects the furnace to the chimney. 
 
798 
 
 APPENDIX. 
 
 TABLE OF SATURATED STEAM. 
 
 ENGLISH UNITS. 
 
 CECIL H. PEABODY, B. S. 
 
 to \ 
 
 II 
 
 5 
 
 2 
 
 p 
 
 
 a 
 .0 
 
 OJ 
 
 S 
 
 DENSITY. 
 
 1-8 
 
 ji 
 
 "2 
 '3 
 
 
 j 
 
 6 
 
 S 
 
 DENSITY. 
 
 
 
 IS 
 
 - CO 
 
 o 1 
 
 
 g 
 
 
 
 S'o-" 
 
 M 
 
 jffi 
 
 .S 
 
 
 "S 
 
 3 
 
 .S"o- 
 
 O 
 
 P-4 ^ 
 
 5$ 
 
 3 
 
 g 
 
 i 
 
 o 
 
 
 
 PH d 1 
 
 3 ca 
 
 ~ 
 
 ji-j 
 
 jjj 
 
 9 
 
 I 
 
 f* 
 
 ri 
 
 M ^ 
 
 Mf 
 
 -*j 
 
 "o 
 
 4) 
 03 
 
 B 
 
 g 
 
 o| 
 
 o 
 
 i 
 
 o 
 
 .? 
 
 i * 
 
 V QC 
 
 CMP 
 
 ? 
 
 44 
 
 1 
 
 "3 
 
 !> 
 
 
 1 
 
 rff . 
 
 1 
 
 1 
 
 B 
 B 
 
 I 
 
 2 
 
 B 
 
 1 
 
 ^ffO 
 
 PH 
 
 r 
 
 B 
 
 i 
 
 B 
 
 a 
 
 02 
 
 ^PnOEn 
 
 P 
 
 < 
 
 Q 
 
 A 
 
 r 
 
 S 
 
 y 
 
 P 
 
 t 
 
 q 
 
 A 
 
 r 
 
 S 
 
 y 
 
 1 
 
 101-99 
 
 70-0 
 
 1113-1 
 
 1043-0 
 
 334-6 
 
 0-00299 
 
 51 
 
 282-10 
 
 251-5 
 
 1168-0 
 
 916-5 
 
 8-259 
 
 0-1211 
 
 2 
 
 126-27 
 
 94-4 
 
 1120-5 
 
 1026-1 
 
 173-6 
 
 0-00576 
 
 52 
 
 283-32 
 
 252-7 
 
 1168-4 
 
 915-7 
 
 8-110 
 
 0-1233 
 
 3 
 
 141-62 
 
 109-8 
 
 1125-1 
 
 1015-3 
 
 118-4 
 
 0-00844 
 
 53 
 
 284-53 
 
 253-9 
 
 1168-7 
 
 914-8 
 
 7-968 
 
 0-1255 
 
 4 
 
 153-09 
 
 121-4 
 
 1128-6 
 
 1007-2 
 
 90-31 
 
 0-01107 
 
 54 
 
 285-72 
 
 255-1 
 
 1169-1 
 
 914-0 
 
 7-829 
 
 0-1277 
 
 5 
 
 162-34 
 
 130-7 
 
 1131-5 
 
 1000-8 
 
 73-22 
 
 0-01366 
 
 55 
 
 286-89 
 
 256-3 
 
 1169-4 
 
 913-1 
 
 7-696 
 
 0-1299 
 
 6 
 
 170-14 
 
 138-6 
 
 1133-8 
 
 995-2 
 
 61-67 
 
 0-01622 
 
 56 
 
 288-05 
 
 257-5 
 
 1169-8 
 
 912-3 
 
 7-568 
 
 0-1321 
 
 7 
 
 176-90 
 
 145-4 
 
 1135-9 
 
 990-5 
 
 53-37 
 
 0-01874 
 
 57 
 
 289-19 
 
 258-6 
 
 1170-1 
 
 911-5 
 
 7-443 
 
 0-1344 
 
 8 
 
 182-92 
 
 151-5 
 
 1137-7 
 
 986-2 
 
 47-07 
 
 0-02125 
 
 58 
 
 290-31 
 
 259-7 
 
 1170-5 
 
 910-8 
 
 7-323 
 
 0-1366 
 
 9 
 
 188-33 
 
 156-9 
 
 1139-4 
 
 982-5 
 
 42-13 
 
 0-02374 
 
 59 
 
 291-42 
 
 260-8 
 
 1170-8 
 
 910-0 
 
 7-208 
 
 0-1387 
 
 10 
 
 193-25 
 
 161-9 
 
 1140-9 
 
 979-0 
 
 38-16 
 
 0-02621 
 
 60 
 
 292-51 
 
 261-9 
 
 1171-2 
 
 909-3 
 
 7-096 
 
 0-1409 
 
 11 
 
 197-78 
 
 166-5 
 
 1142-3 
 
 975-8 
 
 34-88 
 
 0-02866 
 
 61 
 
 293-59 
 
 263-0 
 
 1171-5 
 
 908-5 
 
 6-987 
 
 0-1431 
 
 12 
 
 201-98 
 
 170-7 
 
 1143-6 
 
 972-9 
 
 32-14 
 
 0-03111 
 
 62 
 
 294-65 
 
 264-1 
 
 1171-8 
 
 907-7 
 
 6-882 
 
 0-1453 
 
 13 
 
 205-89 
 
 174-6 
 
 1144-7 
 
 970-1 
 
 29-82 
 
 0-03355 
 
 63 
 
 295-70 
 
 265-2 
 
 1172 1 
 
 906-9 
 
 6-779 
 
 0-1475 
 
 14 
 
 209-57 
 
 178-3 
 
 1145-8 
 
 967-5 
 
 27-79 
 
 0-03600 
 
 64 
 
 296-74 
 
 266-2 
 
 1172-4 
 
 906-2 
 
 6-680 
 
 0-1497 
 
 15 
 
 213-03 
 
 181-8 
 
 1146-9 
 
 965-1 
 
 26-15 
 
 0-03826 
 
 65 
 
 297-77 
 
 267-2 
 
 1172-7 
 
 905-5 
 
 6-583 
 
 0-1519 
 
 16 
 
 216-32 
 
 185-1 
 
 1147-9 
 
 962-8 
 
 24-59 
 
 0-04067 
 
 66 
 
 298-78 
 
 268-3 
 
 1173-0 
 
 904-7 
 
 6-490 
 
 0-1541 
 
 17 
 
 219-44 
 
 188-3 
 
 1148-9 
 
 960-6 
 
 23-22 
 
 0-04307 
 
 67 
 
 299-77 
 
 269-3 
 
 1173-3 
 
 904-0 
 
 6-401 
 
 0-1562 
 
 18 
 
 222-40 
 
 191-3 
 
 1149-8 
 
 958-5 
 
 22-00 
 
 0-04547 
 
 68 
 
 300-76 
 
 270-3 
 
 1173-6 
 
 903-3 
 
 6-314 
 
 0-1584 
 
 19 
 
 225-24 
 
 194-1 
 
 1150-7 
 
 956-6 
 
 20-90 
 
 0-04786 
 
 69 
 
 301-74 
 
 271-2 
 
 1173-9 
 
 902-7 
 
 6-228 
 
 0-1606 
 
 20 
 
 227-95 
 
 196-9 
 
 1151-5 
 
 954-6 
 
 19-91 
 
 0-05023 
 
 70 
 
 302-71 
 
 272-2 
 
 1174-3 
 
 902-1 
 
 6-144 
 
 0-1628 
 
 21 
 
 230-55 
 
 199-5 
 
 1152-3 
 
 952-8 
 
 19-01 
 
 0-05259 
 
 71 
 
 303-66 
 
 273-2 
 
 1174-6 
 
 901-4 
 
 6-063 
 
 0-1649 
 
 22 
 
 233-06 
 
 202-0 
 
 1153-0 
 
 951-0 
 
 18-20 
 
 0-05495 
 
 72 
 
 304-61 
 
 274-1 
 
 1174-9 
 
 900-8 
 
 5-984 
 
 0-1671 
 
 23235-47 
 
 204-5 
 
 1153-7 
 
 949-2 
 
 17-45 
 
 0-05731 
 
 73 
 
 305-54 
 
 275-1 
 
 1175-2 
 
 900-1 
 
 5-908 
 
 0-1693 
 
 24 
 
 237-79 
 
 206-8 
 
 1154-4 
 
 947-6 
 
 16-76 
 
 0-05966 
 
 74 
 
 306-46 
 
 276-0 
 
 1175-4 
 
 899 -i 
 
 5-834 
 
 0-1714 
 
 25 
 
 240 04 
 
 209-1 
 
 1155-1 
 
 946-0 
 
 16-13 
 
 0-06199 
 
 75 
 
 307-38 
 
 276-9 
 
 1175-7 
 
 898-8 
 
 5-762 
 
 0-1736 
 
 26 
 
 242-21 
 
 211-2 
 
 1155-8 
 
 944-6 
 
 15-55 
 
 0-06432 
 
 76 
 
 308-28 
 
 277-8 
 
 1176-0 
 
 898-2 
 
 5-691 
 
 0-1757 
 
 27 
 
 244-32 
 
 213-4 
 
 1156-5 
 
 943-1 
 
 15-00 
 
 0-06666 
 
 77 
 
 309-18 
 
 278-7 
 
 1176-2 
 
 897-5 
 
 5-621 
 
 0-1779 
 
 28 
 
 246-36 
 
 215-4 
 
 1157-1 
 
 941-7 
 
 14-49 
 
 0-06899 
 
 78 
 
 310-06 
 
 279-6 
 
 1176-5 
 
 896-9 
 
 5-554 
 
 0-1801 
 
 29 
 
 248-34 
 
 217-4 
 
 1157-7 
 
 940-3 
 
 14-03 
 
 0-07130 
 
 79 
 
 310-94 
 
 280-5 
 
 1176-8 
 
 896-3 
 
 5-488 
 
 0-1822 
 
 30 
 
 250-27 
 
 219-4 
 
 1158-3 
 
 938-9 
 
 13-59 
 
 0-07360 
 
 80 
 
 311-80 
 
 281-4 
 
 1177-0 
 
 895-6 
 
 5-425 
 
 0-1843 
 
 31 
 
 252-15 
 
 221-3 
 
 1158-8 
 
 937-5 
 
 13-18 
 
 0-07590 
 
 81 
 
 312-66 
 
 282-3 
 
 1177-3 
 
 895-0 
 
 5-362 
 
 0-1865 
 
 32 
 
 253-98 
 
 223-1 
 
 1159-4 
 
 936-3 
 
 12-78 
 
 0-07821 
 
 82 
 
 313-51 
 
 283-2 
 
 1177-6 
 
 894-4 
 
 5-3010-1886 
 
 33 
 
 255-76 
 
 224-9 
 
 1159-9 
 
 935-0 
 
 12-41 
 
 0-08051 
 
 83 
 
 314-36 
 
 284-1 
 
 1177-8 
 
 893-7 
 
 5-2400-1908 
 
 34 
 
 257-50 
 
 226-7 
 
 1160-4 
 
 933-7 
 
 12-07 
 
 0-08280 
 
 84 
 
 315-19 
 
 285-0 
 
 1178-1 
 
 893-1 
 
 5-1820-1930 
 
 35 
 
 259-19 
 
 228-4 
 
 1161-0 
 
 932-6 
 
 11-75 
 
 0-08508 
 
 85 
 
 316-02 
 
 285-8 
 
 1178-3 
 
 892-5 
 
 5-125 
 
 0-1951 
 
 36 
 
 260-85 
 
 230-0 
 
 1161-5 
 
 931-5 
 
 11-45 
 
 0-08736 
 
 86 
 
 316-84 
 
 286-7 
 
 1178-6 
 
 891-9 
 
 5-069 
 
 0-1973 
 
 37 
 
 262-47 
 
 231-7 
 
 1162-0 
 
 930-3 
 
 11-16 
 
 0-08964 
 
 87 
 
 317-65 
 
 287-5 
 
 1178-8 
 
 891-3 
 
 5-014 
 
 0-1994 
 
 38264-06 
 
 233-3 
 
 1162-5 
 
 929-2 
 
 10-88 
 
 0-09191 
 
 88 
 
 318-45 
 
 288-4 
 
 1179-1 
 
 890-7 
 
 4-961 
 
 0-2016 
 
 39265-61 
 
 234-8 
 
 1163-0 
 
 928-2 
 
 10-62 
 
 0-09417 
 
 89 
 
 319-25 
 
 289-2 
 
 1179-3 
 
 890-1 
 
 4-909 
 
 0-2037 
 
 40 
 
 267-13 
 
 236-4 
 
 1163-4 
 
 927-0 
 
 10-37 
 
 0-09644 
 
 90 
 
 320-04 
 
 290-0 
 
 1179-6 
 
 889-6 
 
 4-858 
 
 0-2058 
 
 41 
 
 268-62 
 
 237-9 
 
 1163-9 
 
 926-0 
 
 10-13 
 
 0-09869 
 
 91 
 
 320-83 
 
 290-8 
 
 1179-8 
 
 889-0 
 
 4-808 
 
 0-2080 
 
 42270-08 
 
 239-3 
 
 1164-3 
 
 925-0 
 
 9-906 
 
 0-10090 
 
 92 
 
 321-60 
 
 291-6 
 
 1180-0 
 
 888-4 
 
 4-760 
 
 0-2101 
 
 43271-51 
 
 240-8 
 
 1164-8 
 
 924-0 
 
 9-690 
 
 0-10320 
 
 93 
 
 322-37 
 
 292-4 
 
 1180-3 
 
 887-9 
 
 4-712 
 
 0-2122 
 
 44272-91 
 
 242-2 
 
 1165-2 
 
 923-0 
 
 9-484 
 
 0-10540 
 
 94 
 
 323-14 
 
 293-2 
 
 1180-5 
 
 887-3 
 
 4-665 
 
 0-2144 
 
 45274-29 
 
 243-6 
 
 1165-6 
 
 922-0 
 
 9-287 
 
 0-10770 
 
 95 
 
 323-89 
 
 294-0 
 
 1180-7 
 
 886-7 
 
 4-619 
 
 0-2165 
 
 46275-65 
 
 245-0 
 
 1166-0 
 
 921-0 
 
 9-097 
 
 0-10990 
 
 96 
 
 324-64 
 
 294-8 
 
 1181-0 
 
 886-2 
 
 4-574 
 
 0-2186 
 
 47276-99 
 
 246-3 
 
 1166-4 
 
 920-1 
 
 8-914 
 
 0-11220 
 
 97 
 
 325-38 
 
 295-6 
 
 1181-2 
 
 885-6 
 
 4-5300-2208 
 
 48 
 
 278-30 
 
 247-6 
 
 1166-8 
 
 919-2 
 
 8-740 
 
 0-11440 
 
 98 
 
 326-12 
 
 296-4 
 
 1181-4 
 
 885-0 
 
 4-4860-2229 
 
 49 
 
 279-58 
 
 248-9 
 
 1167-2 
 
 918-3 
 
 8-573 
 
 0-11660 
 
 99 
 
 326-86297-1 
 
 1181-6 
 
 884-5 
 
 4-4440-2250 
 
 50 
 
 280-85 
 
 250-2 
 
 1167-6 
 
 917-4 
 
 8-414 
 
 0-11880 
 
 100 
 
 327-58 
 
 297-9 
 
 1181-9 
 
 884-0 
 
 4-4030-2271 
 
APPENDIX. 
 
 TABLE OP SATURATED STEAK Continued. 
 
 f-g 
 
 J3 
 
 2 
 
 
 g 
 
 S 
 
 DENSITY. 
 
 94 
 
 a 
 
 2 
 
 
 g 
 
 S 
 
 DENSITY. 
 
 o M 
 
 fM 
 
 .2* 
 
 
 l 
 
 a 
 
 .3*8-2 
 
 O M 
 
 iT 83 
 
 a 1 
 
 
 ^O 
 
 i 
 
 S'6-2 
 
 8'? 
 
 3 r. 
 
 J 
 
 | 
 
 J 
 
 1 
 
 J3 
 ,3 
 
 (Eg, 
 
 5 8 
 
 j3 
 
 4 
 
 i 
 
 3 
 
 . 
 
 
 
 1i Jyn 
 I! 
 
 o 
 1 
 
 
 
 1 
 
 || 
 
 o 
 
 9 
 
 "3 
 
 
 
 Hsl 
 
 II 
 
 ao? 
 
 o 
 
 1 
 3 
 
 ll 
 
 o 
 
 <c 
 "3 
 
 ! 
 
 .2f o c o 
 
 
 
 4) 
 
 B 
 
 H 
 
 
 
 1 
 
 K 
 
 02 
 
 
 
 H 
 
 H 
 
 K 
 
 1 
 
 4> 
 
 I 
 
 02 
 
 SftOfa 
 
 p 
 
 t 
 
 Q 
 
 A 
 
 r 
 
 S 
 
 y 
 
 p 
 
 t 
 
 Q 
 
 * 
 
 r 
 
 8 
 
 V 
 
 101 
 
 328-30 
 
 298-6 
 
 1182-1 
 
 883-5 
 
 4-362 
 
 0-2293 
 
 151 
 
 358-78 
 
 330-5 
 
 1191-4 
 
 860-9 
 
 2-992 
 
 0-3342 
 
 102 
 
 329-02 
 
 299-4 
 
 1182-3 
 
 882-9 
 
 4-322 
 
 0-2314 
 
 152 
 
 359-30 
 
 331-1 
 
 1191-5 
 
 860-4 
 
 2-973 
 
 0-3363 
 
 103 
 
 329-73 
 
 300-1 
 
 1182-5 
 
 882-4 
 
 4-282 
 
 0-2335 
 
 153 
 
 359-82 
 
 331-6 
 
 1191-7 
 
 860-1 
 
 2-955 
 
 0-3384 
 
 104 
 
 330-43 
 
 300-9 
 
 1182-7 
 
 881-8 
 
 4-244 
 
 0-2356 
 
 154 
 
 360-34 
 
 332-2 
 
 1191-8 
 
 859-6 
 
 2-937 
 
 0-3405 
 
 105 
 
 331-13 
 
 301-6 
 
 1182-9 
 
 881-3 
 
 4-206 
 
 0-2378 
 
 155 
 
 360-86 
 
 332-7 
 
 1192-0 
 
 859-3 
 
 2-919 
 
 0-3426 
 
 106 
 
 331-83 
 
 302-3 
 
 1183-1 
 
 880-8 
 
 4-169 
 
 0-2399 
 
 156 
 
 361-37 
 
 333-3 
 
 1192-2 
 
 858-9 
 
 2-901 
 
 0-3447 
 
 107 
 
 332-52 
 
 302-0 
 
 1183-4 
 
 880-4 
 
 4-132 
 
 0-2420 
 
 157 
 
 361-88 
 
 333-8 
 
 1192-3 
 
 858-5 
 
 2-884 
 
 0-3467 
 
 108 
 
 333-20 
 
 303-8 
 
 1183-6 
 
 879-8 
 
 4-096 
 
 0-2441 
 
 158 
 
 362-39 
 
 334-3 
 
 1192-5 
 
 858-2 
 
 2-867 
 
 0-3488 
 
 109 
 
 333-88 
 
 304-5 
 
 1183-8 
 
 879-3 
 
 4-061 
 
 0-2462 
 
 159 
 
 362-90 
 
 334-9 
 
 1192-7 
 
 857-8 
 
 2-850 
 
 0-3509 
 
 110 
 
 334-56 
 
 305-2 
 
 1184-0 
 
 878-8 
 
 4-026 
 
 0-2484 
 
 160 
 
 363-40 
 
 335-4 
 
 1192-8 
 
 857-4 
 
 2-833 
 
 0-3530 
 
 111 
 
 335-23 
 
 305-9 
 
 1184-2 
 
 878-3 
 
 3-992 
 
 0-2505 
 
 161 
 
 363-90 
 
 335-9 
 
 1193-0 
 
 857-1 
 
 2-816 
 
 0-3551 
 
 112 
 
 335-89 
 
 306-6 
 
 1184-4 
 
 877-8 
 
 3-959 
 
 0-2526 
 
 162 
 
 364-40 
 
 336-4 
 
 1193-1 
 
 856-7 
 
 2-799 
 
 0-3572 
 
 113 
 
 336-55 
 
 307-3 
 
 1184-6 
 
 877-3 
 
 3-926 
 
 2547 
 
 163 
 
 364-90 
 
 337-0 
 
 1193-3 
 
 856-3 
 
 2-783 
 
 0-3593 
 
 114 
 
 337-20 
 
 308-0 
 
 1184-8 
 
 876-8 
 
 3-894 
 
 0-2568 
 
 164 
 
 365-39 
 
 337-5 
 
 1193-4 
 
 855-9 
 
 2-767 
 
 0-3614 
 
 115 
 
 337-86 
 
 308-7 
 
 1185-0 
 
 876-3 
 
 3-862 
 
 0-2589 
 
 165 
 
 365-88 
 
 338-0 
 
 1193-6 
 
 855-6 
 
 2-751 
 
 0-3635 
 
 116 
 
 338-50 
 
 309-4 
 
 1185-2 
 
 875-8 
 
 3-831 
 
 0-2610 
 
 166 
 
 366-37 
 
 338-5 
 
 1193-7 
 
 855-2 
 
 2-736 
 
 0-3655 
 
 117 
 
 339-14 
 
 310-0 
 
 1185-4 
 
 875-4 
 
 3-801 
 
 0-2631 
 
 167 
 
 366-85 
 
 339-0 
 
 1193-9 
 
 854-9 
 
 2-721 
 
 0-3675 
 
 118 
 
 339-78 
 
 310-7 
 
 1185-6 
 
 874-9 
 
 3-770 
 
 0-2653 
 
 168 
 
 367-33 
 
 339-5 
 
 1194-0 
 
 854-5 
 
 2-706 
 
 0-3695 
 
 119 
 
 340-42 
 
 311-4 
 
 1185-8 
 
 874-4 
 
 3-740 
 
 0-2674 
 
 169 
 
 367-81 
 
 340-0 
 
 1194-2 
 
 854-2 
 
 2-691 
 
 0-3716 
 
 120 
 
 341-05 
 
 312-0 
 
 1186-0 
 
 874-0 
 
 3-711 
 
 0-2695 
 
 170 
 
 368-29 
 
 340-5 
 
 1194-3 
 
 853-8 
 
 2-676 
 
 0-3737 
 
 121 
 
 341-67 
 
 312-7 
 
 1186-2 
 
 873-5 
 
 3-683 
 
 0-2715 
 
 171 
 
 368-77 
 
 341-0 
 
 1194-4 
 
 853-4 
 
 2-661 
 
 0-3758 
 
 122 
 
 342-29 
 
 313-3 
 
 1186-3 
 
 873-0 
 
 3-655 
 
 0-2736 
 
 172 
 
 369-24 
 
 341-5 
 
 1194-6 
 
 853-1 
 
 2-647 
 
 0-3778 
 
 123 
 
 342-91 
 
 314-0 
 
 1186-5 
 
 872-5 
 
 3-627 
 
 0-2757 
 
 173 
 
 369-71 
 
 342-0 
 
 1194-7 
 
 852-7 
 
 2-632 
 
 0-3799 
 
 124 
 
 343-52 
 
 314-6 
 
 1186-7 
 
 872-1 
 
 3-599 
 
 0-2779 
 
 174 
 
 370-18 
 
 342-5 
 
 1194-8 
 
 852-3 
 
 2-618 
 
 0-3820 
 
 125 
 
 344-13 
 
 315-2 
 
 1186-9 
 
 871-7 
 
 3-572 
 
 0-2800 
 
 175 
 
 370-65 
 
 343-0 
 
 1195-0 
 
 852-0 
 
 2-603 
 
 0-3841 
 
 126 
 
 344-73 
 
 315-9 
 
 1187-1 
 
 871-2 
 
 3-546 
 
 0-2820 
 
 176 
 
 371-12 
 
 343-5 
 
 1195-1 
 
 851-6 
 
 2-589 
 
 0-3862 
 
 127 
 
 345-33 
 
 316-5 
 
 1187-3 
 
 870-8 
 
 3-520 
 
 0-2841 
 
 177 
 
 371-59 
 
 344-0 
 
 1195-3 
 
 851-3 
 
 2-575 
 
 0-3883 
 
 128 
 
 345-93 
 
 317-1 
 
 1187-4 
 
 870-3 
 
 3-494 
 
 0-2862 
 
 178 
 
 372-05 
 
 344-4 
 
 1195-4 
 
 851-0 
 
 2-561 
 
 0-3904 
 
 129 
 
 346-53 
 
 318-7 
 
 1187-6 
 
 869-9 
 
 3-469 
 
 0-2883 
 
 179 
 
 372-51 
 
 344-9 
 
 1195-6 
 
 850-7 
 
 2-548 
 
 0-3925 
 
 130 
 
 347-12 
 
 318-4 
 
 1187-8 
 
 869-4 
 
 3-444 
 
 0-2904 
 
 180 
 
 372-97 
 
 345-4 
 
 1195-7 
 
 850-3 
 
 2-535 
 
 0-3945 
 
 131 
 
 347-71 
 
 319-0 
 
 1188-0 
 
 869-0 
 
 3-419 
 
 0-2925 
 
 181 
 
 373-43 
 
 345-9 
 
 1195-9 
 
 850-0 
 
 2-522 
 
 0-3966 
 
 132 
 
 348-29 
 
 319-6 
 
 1188-2 
 
 868-6 
 
 3-395 
 
 0-2946 
 
 182 
 
 373-88 
 
 346-4 
 
 1196-0 
 
 849-6 
 
 2-508 
 
 0-3987 
 
 133 
 
 348-87 
 
 320-2 
 
 1188-4 
 
 868-2 
 
 3-371 
 
 0-2967 
 
 183 
 
 374-33 
 
 346-8 
 
 1196-1 
 
 849-3 
 
 2-495 
 
 0-4008 
 
 134 
 
 349-45 
 
 320-8 
 
 1188-5 
 
 867-7 
 
 3-347 
 
 0-2988 
 
 184 
 
 374-78 
 
 347-3 
 
 1196-2 
 
 848-9 
 
 2-482 
 
 0-4029 
 
 135 
 
 350-03 
 
 321-4 
 
 1188-7 
 
 867-3 
 
 3-3230-3009 
 
 185375-23 
 
 347-8 
 
 1196-4 
 
 848-6 
 
 2-470 
 
 0-4049 
 
 136 
 
 350-60 
 
 322-0 
 
 1188-9 
 
 866-9 
 
 3-3000-3030 
 
 186 
 
 375-68 
 
 348-2 
 
 1196-5 
 
 848-3 
 
 2-457 
 
 0-4070 
 
 137 
 
 351-17 
 
 322-6 
 
 1189-0 
 
 866-4 
 
 3-277 
 
 0-3051 
 
 187 
 
 376-12 
 
 348-7 
 
 1196-6 
 
 847-9 
 
 2-445 
 
 0-4090 
 
 138 
 
 351 73 
 
 323-2 
 
 1189-2 
 
 866-0 
 
 3-255 
 
 3072 
 
 188 
 
 376-56 
 
 349-2 
 
 1196-8 
 
 847-6 
 
 2-432 
 
 0-4111 
 
 139 
 
 352-29 
 
 323-8 
 
 1189-4 
 
 865-6 
 
 3-234 
 
 0-3092 
 
 189 
 
 377-00 
 
 349-6 
 
 1196-9 
 
 847-3 
 
 2-420 
 
 0-4132 
 
 140 
 
 352-85 
 
 324-4 
 
 1189-5 
 
 865-1 
 
 3-212 
 
 0-3113 
 
 190 
 
 377-44 
 
 350-1 
 
 1197-1 
 
 847-0 
 
 2-408 
 
 0-4153 
 
 141 
 
 353-40 
 
 325-0 
 
 1189-7 
 
 864-7 
 
 3-191 
 
 0-3134 
 
 191 
 
 377-88 
 
 350-5 
 
 1197-2 
 
 846-7 
 
 2-396 
 
 0-4174 
 
 142 
 
 353-95 
 
 325-6 
 
 1189-9 
 
 864-3 
 
 3-170 
 
 0-3155 
 
 192378-32 
 
 351-0 
 
 1197-3 
 
 846-3 
 
 2-385 
 
 0-4194 
 
 143 
 
 354-50 
 
 326-1 
 
 1190-1 
 
 864-0 
 
 3-149 
 
 0-3176 
 
 1931378-75 
 
 351-4 
 
 1197-4 
 
 846-0 
 
 2-373 
 
 0-4215 
 
 144 
 
 355-05 
 
 326-7 
 
 1190-2 
 
 863-5 
 
 3-128 
 
 0-3197 
 
 194379-18 
 
 351-9 
 
 1197-6 
 
 845-7 
 
 2-361 
 
 0-4236 
 
 145 
 
 355-59 
 
 327-2 
 
 1190-4 
 
 863-2 
 
 3-107 
 
 0-3218 
 
 195 
 
 379-61 
 
 352-4 
 
 1197-7 
 
 845-3 
 
 2-349 
 
 0-4257 
 
 146 
 
 356-13 
 
 327-8 
 
 1190-6 
 
 862-8 
 
 3-087 
 
 0-3239 
 
 196 
 
 380-04 
 
 352-8 
 
 1197-8 
 
 845-0 
 
 2-337 
 
 0-4278 
 
 147 
 
 356-67 
 
 328-3 
 
 1190-7 
 
 862-4 
 
 3-068 
 
 0-3259 
 
 197 
 
 380-47 
 
 353-3 
 
 1198-0 
 
 844-7 
 
 2-325 
 
 0-4298 
 
 148 
 
 357-20 
 
 328-9 
 
 1190-9 
 
 862-0 
 
 3-049 
 
 0-3280 
 
 198 
 
 380-89 
 
 353-7 
 
 1198-1 
 
 844-4 
 
 2-314 
 
 0-4318 
 
 149 
 
 357-73 
 
 329-4 
 
 1191-0 
 
 861-6 
 
 3-030 
 
 0-3300 
 
 199 
 
 381-31 
 
 354-1 
 
 1198-2 
 
 844-1 
 
 2-304 
 
 0-4338 
 
 150 
 
 358-26 
 
 330-0 
 
 1191-2 
 
 861-2 
 
 3-011 
 
 0-3321 
 
 200 
 
 381-73 
 
 354-6 
 
 1198-4 
 
 843-8 
 
 2-294 
 
 0-4359 
 
800 
 
 APPENDIX. 
 
 EXPANSIVE WORKING OP STEAM. 
 
 Cut-off in 
 percentage 
 of stroke. 
 
 Number 
 of expan- 
 sions. 
 
 MEAN ABSOLUTE PRESSURE, 
 IN DECIMAL PARTS, OF 
 ABSOLUTE PRESSURE OP 
 ADMISSION. 
 
 Cut-off in 
 percentage 
 of stroke. 
 
 Number 
 of expan- 
 sions. 
 
 MEAN ABSOLUTE PRESSURE, 
 IN DECIMAL PARTS, OF 
 ABSOLUTE PRESSURE OF 
 ADMISSION. 
 
 Dry satu- 
 rated steam. 
 
 Moderately 
 moist steam. 
 
 Dry satu- 
 rated steam. 
 
 Moderately 
 moist steam. 
 
 800 
 
 ii 
 
 9760 
 
 9784 
 
 1 
 
 10 
 
 3145 
 
 3303 
 
 667 
 
 14 
 
 9403 
 
 9366 
 
 095 
 
 10J 
 
 3035 
 
 3189 
 
 571 
 
 11 
 
 8857 
 
 8910 
 
 091 
 
 11 
 
 2933 
 
 3089 
 
 500 
 
 2 
 
 8394 
 
 8465 
 
 087 
 
 114 
 
 2839 
 
 2991 
 
 444 
 
 2* 
 
 7960 
 
 8050 
 
 083 
 
 12 
 
 2751 
 
 2904 
 
 400 
 
 24 
 
 7560 
 
 7664 
 
 080 
 
 1S4 
 
 2669 
 
 2818 
 
 364 
 
 2f 
 
 7200 
 
 7316 
 
 077 
 
 13 
 
 2592 
 
 2742 
 
 333 
 
 3 
 
 6870 
 
 6997 
 
 074 
 
 184 
 
 2520 
 
 2666 
 
 308 
 
 3i 
 
 6570 
 
 6705 
 
 071 
 
 14 
 
 2452 
 
 2599 
 
 286 
 
 84 
 
 6300 
 
 6437 
 
 069 
 
 144 
 
 2388 
 
 2532 
 
 267 
 
 8i 
 
 6051 
 
 6192 
 
 067 
 
 15 
 
 2327 
 
 2472 
 
 250 
 
 4 
 
 5820 
 
 5965 
 
 065 
 
 15i 
 
 2270 
 
 2411 
 
 235 
 
 4* 
 
 5608 
 
 5757 
 
 063 
 
 16 
 
 2216 
 
 2358 
 
 222 
 
 4* 
 
 5410 
 
 5564 
 
 061 
 
 14 
 
 2164 
 
 2303 
 
 211 
 
 4f 
 
 5230 
 
 5386 
 
 059 
 
 17 
 
 2115 
 
 2255 
 
 200 
 
 5 
 
 5060 
 
 5218 
 
 057 
 
 m 
 
 2069 
 
 2205 
 
 191 
 
 5k 
 
 4904 
 
 5063 
 
 056 
 
 18 
 
 2024 
 
 2161 
 
 182 
 
 6J 
 
 4760 
 
 4918 
 
 054 
 
 184 
 
 1982 
 
 2116 
 
 174 
 
 54 
 
 4620 
 
 4781 
 
 053 
 
 19 
 
 1942 
 
 2181 
 
 167 
 
 6 
 
 4490 
 
 4653 
 
 051 
 
 19| 
 
 1903 
 
 2034 
 
 160 
 
 ej 
 
 4370 
 
 4533 
 
 050 
 
 20 
 
 1866 
 
 1998 
 
 154 
 
 64 
 
 4250 
 
 4418 
 
 048 
 
 21 
 
 1796 
 
 1926 
 
 148 
 
 6f 
 
 4150 
 
 4311 
 
 045 
 
 22 
 
 1736 
 
 I860 
 
 143 
 
 7 
 
 4046 
 
 4208 
 
 043 
 
 23 
 
 1672 
 
 1790' 
 
 138 
 
 71 
 
 3950 
 
 4112 
 
 042 
 
 24 
 
 1610 
 
 1741 
 
 133 
 
 7* 
 
 3857 
 
 4020 
 
 040 
 
 25 
 
 1566 
 
 1688 
 
 129 
 
 7* 
 
 3770 
 
 3933 
 
 038 
 
 26 
 
 1518 
 
 1638 
 
 125 
 
 8 
 
 3688 
 
 3849 
 
 037 
 
 27 
 
 1474 
 
 1591 
 
 121 
 
 8J 
 
 3608 
 
 3769 
 
 036 
 
 28 
 
 1431 
 
 1547 
 
 117 
 
 84 
 
 3533 
 
 3694 
 
 034 
 
 29 
 
 1392 
 
 1506 
 
 114 
 
 8f 
 
 3461 
 
 3622 
 
 033 
 
 30 
 
 1355 
 
 1467 
 
 111 
 
 9 
 
 3392 
 
 3552 
 
 
 
 
 
 108 
 
 9* 
 
 3326 
 
 3494 
 
 
 
 
 
 105 
 
 94 
 
 3274 
 
 3422 
 
 
 
 
 
 103 
 
 9f 
 
 3203 
 
 3359 
 
 
 
 
 
 TABLE OF FACTORS OF EVAPORATION. 
 
 Pressure 
 Pounds 
 per 
 Sq. Inch. 
 
 Boiling 
 Point 
 T,. 
 Fahr. 
 
 INITIAL TEMPERATURE OF FEED WATER, T 2 . 
 
 32 
 
 50 
 
 68 
 
 86 
 
 104 
 
 122 
 
 140 
 
 158 
 
 176 
 
 194 
 
 212 
 
 14-70 
 
 212 
 
 1-19 
 
 1-17 
 
 1-15 
 
 1-13 
 
 1-11 
 
 1-10 
 
 1-08 
 
 1-06 
 
 1-04 
 
 1-02 
 
 1-00 
 
 20-78 
 
 230 
 
 1-20 
 
 1-18 
 
 1-16 
 
 1-14 
 
 1-12 
 
 1-10 
 
 1-08 
 
 1-06 
 
 1-04 
 
 1-02 
 
 1-01 
 
 28-82 
 
 248 
 
 1-20 
 
 1-18 
 
 1-16 
 
 1-14 
 
 1-13 
 
 1-11 
 
 1-09 
 
 1-07 
 
 1-05 
 
 1-03 
 
 1-01 
 
 39-26 
 
 266 
 
 1-21 
 
 1-19 
 
 1-17 
 
 1-15 
 
 1-13 
 
 1-11 
 
 1-09 
 
 1-07 
 
 1-06 
 
 1-04 
 
 1-02 
 
 52-56 
 
 284 
 
 1-21 
 
 1-20 
 
 1-18 
 
 1-16 
 
 1-14 
 
 1-12 
 
 1-10 
 
 1-08 
 
 1-06 
 
 1-04 
 
 1-02 
 
 69-27 
 
 302 
 
 1-22 
 
 1-20 
 
 1-18 
 
 1-16 
 
 1-14 
 
 1-12 
 
 1-11 
 
 1-09 
 
 1-07 
 
 1-05 
 
 1-03 
 
 89-95 
 
 320 
 
 1-22 
 
 1-21 
 
 1-19 
 
 1-17 
 
 1-15 
 
 1-13 
 
 1-11 
 
 1-09 
 
 1-07 
 
 1-05 
 
 1-03 
 
 115-22 
 
 338 
 
 1-23 
 
 1-21 
 
 1-19 
 
 1-17 
 
 1-15 
 
 1-14 
 
 1-12 
 
 1-10 
 
 1-08 
 
 1-06 
 
 1-04 
 
 145-75 
 
 356 
 
 1-23 
 
 1-22 
 
 1-20 
 
 1-18 
 
 1-16 
 
 1-14 
 
 1-12 
 
 1-10 
 
 1-08 
 
 1-06 
 
 1-04 
 
 182-27 
 
 374 
 
 1-24 
 
 1-22 
 
 1-20 
 
 1-18 
 
 1-17 
 
 1-15 
 
 1-13 
 
 1-11 
 
 1-09 
 
 1-07 
 
 1-05 
 
 225-56 
 
 392 
 
 1-24 
 
 1-23 
 
 1-21 
 
 1-19 
 
 1-17 
 
 1-15 
 
 1-13 
 
 1-11 
 
 1-09 
 
 1-07 
 
 1-06 
 
 276-54 
 
 410 
 
 1-25 
 
 1-23 
 
 1-22 
 
 1-20 
 
 1-18 
 
 1-16 
 
 1-14 
 
 1-12 
 
 1-10 
 
 1-08 
 
 1-06 
 
 336-26 
 
 428 
 
 1-25 
 
 1-24 
 
 1-22 
 
 1-20 
 
 1-18 
 
 1-16 
 
 1-14 
 
 1-12 
 
 1-11 
 
 1-09 
 
 1-07 
 
APPENDIX. 
 
 801 
 
 The Twenty-eighth Street Central Station of the United Electric Light and Power 
 Company. By H. W. York, "Trans. A. S. C. E M " March 18, 1896. 
 
 In this station 20,000 H. P. of engines, together with the boilers, condensing appa- 
 ratus, dynamos, and switchboard, and storage for 6,000 tons of coal, are all on a plot of 
 ground 160 feet 11 inches by 197 feet 6 inches. All machinery, including the boilers, is 
 on the ground floor, and yet there is plenty of light, air, and ample space for working 
 around all the apparatus. Fig. 1870 is a plan of the foundation walls and piers of the 
 
802 
 
 APPENDIX. 
 
 building. The entire front wall is hollow and carried up above the roof to prevent the 
 noise of the machinery annoying the patients in Bellevue Hospital, which is directly 
 across the street. Fig. 1871 is a cross-section of the entire structure. 
 
 Loop System. The main steam and exhaust piping is shown in plan on Fig. 1872. A 
 16-inch header is run the length of the boiler-room, and a similar header is run the length 
 of the engine room between the two rows of foundations and parallel to the boiler-room 
 header. Each of these headers are divided into five sections by means of four gate- 
 
APPENDIX. 
 
 803 
 
 valves, and each section of the boiler-room header is connected to the corresponding sec- 
 tion of the engine-room header by a 14-inch branch rising from the top of one and dis- 
 charging into the top of the other, a valve being placed on each end where a connection 
 is made to the header. Each boiler lias an independent connection to the boiler-room 
 header, supplied with two stop-valves, one in the customary position just beyond the 
 safety-valves, and the other at the point where the pipe enters the header. This second 
 valve has its stem extended through the wall into the repair shop, so that in case of 
 trouble any boiler may be cut out of a room having no communication with the boiler 
 house. 
 
 Each engine on the east side of the engine room is connected to one of the 14-inch 
 
804 
 
 APPENDIX. 
 
 branches previously mentioned, while outlets are left on the engine-room header for con- 
 nections to the west row of the engines as soon as they are placed in position. In case 
 any section of the engine-room header is cut out, one engine connected to this section 
 can be fed directly from the 14-inch branch, leaving only one which can not be run, and 
 in case any section of the boiler-room header is cut out no engine need be shut down, as 
 
 the one connected to the 14-inch branch can be 
 fed back from the engine-room header. 
 
 The boilers are of the upright water-tube 
 type of the Clonbrock pattern, and are in 600 
 H.-P. units, occupying little ground room 'per 
 unit of capacity (Fig. 1873). 
 
 The conveyer for handling coal and ashes 
 consists of an endless chain of gravity buckets, 
 which are loaded by means of a filler and can be 
 dumped at any desired point. The driver is in 
 the north end of the ventilator over the coal 
 bunker. The coal filler is in a vault under the 
 sidewalk, and the coal is dumped into this ap- 
 paratus through a grating situated about the 
 street level. After being deposited in the buck- 
 ets the coal is carried up into the ventilator over 
 the coal bunker and dumped into any portion 
 of the bunker desired. From the hoppers in 
 the bottom of the bunker the coal is spouted 
 to the different boilers. The arrangement is 
 such that the coal trims itself and will con- 
 tinue running down the spouts as required and 
 without assistance so long as any remains in the 
 bunker. 
 
 Under each boiler is an ash hopper deliver- 
 ing the ashes to a second movable filler, which 
 deposits them in the buckets of the conveyer 
 when it is not used for coal. The conveyer 
 dumps the ashes at a point from which they are 
 spouted over to a tank in the southeast corner 
 of the coal bunker. 
 
 The engines are Westinghouse double acting 
 "Columbian steepled compounds." Fig. 1874 
 shows one of these engines in section. The low 
 pressure is placed over the high pressure, and 
 both pistons are connected to the same rod. 
 The crank is inclosed in the same manner as 
 the Westinghouse engines. The low-pressure 
 valve is operated by a fixed eccentric placed in- 
 side the crank case, while the high-pressure 
 valve receives its motion from a shifting eccen- 
 tric outside the crank case, operated automati- 
 cally by the governor, which is placed on the 
 shaft outside of the eccentric. The low-pressure valve is of the slide-valve type, while 
 that for the high-pressure cylinder is a hollow piston valve, being constructed in this 
 manner to allow the exhaust from the lower end of the high-pressure cylinder to 
 pass up through it. Diameter high-pressure cylinder, 21J inches ; diameter low- 
 pressure cylinder, 37 inches ; stroke, 22 inches. The speed is 200 revolutions -per 
 
 Elevation. 
 
APPENDIX. 
 
 805 
 
 minute and the rated horse power 1,200 when operating condensing, with 150 pounds 
 initial steam pressure. 
 
 Each main engine is directly connected to a 600- kilowatt Westinghouse ^alternator by 
 a rigid coupling, both engine and generator being set on a firm cast-iron bedplate. The 
 generator has but one bearing, the armature being swung between the engine and this 
 single support. 
 
 For exciting the fields of the alternators, 75-kilowatt direct-current Westinghouse 
 dynamos of the railway generator type are used. 
 
 FIG. 1874. 
 
 FOUNDATION FOB VERTICAL ENGINES. 
 
806 
 
 APPENDIX. 
 
 Zoss j/i Yo/fs. 
 
 FIG. 1875. 
 
 In the " Universal "Wiring Com- 
 puter, '' by Carl Hering, charts are giv- 
 en which give directly, and without 
 calculation or the use of formulae, 'the 
 gauge number or cross section in cir- 
 cular mils of lead for any number of 
 lamps of any make at any distance or 
 for any loss. One of these tables is 
 given above as an illustration on 
 "graphics." 
 
 Follow the general direction of 
 the broken line and the arrows from 
 one set of diagonals to the next. 
 
 Example : What size wire is re- 
 quired for 10 lamps of '775 amperes 
 each, at 50 feet, for a loss of one volt ? 
 
 Solution : Starting with the cur- 
 rent for one lamp, -775 amperes (see 
 scale below centre), follow it to the 
 left until it intersects the diagonal 
 representing one volt loss, thence up 
 to the diagonal representing 10 lamps, 
 thence to the right to the diagonal 
 representing 50 feet, and from here 
 down to the scale of the circular mils 
 or gauge numbers, on which the read- 
 ing is found to be about 8,200 circu- 
 lar mils, or a No. 11 B. S. wire. 
 
 Fig. 1875 is an incandescent elec- 
 tric lamp socket of the Bryant Elec- 
 tric Company. 
 
 Fig. 1876 is a switch of the Hart 
 and Hegeman Company. 
 
APPENDIX. 
 
 sor 
 
 All of the above drawings are made in a style in which the exterior shell is trans- 
 parent, showing the interior mechanism, and known as "ghost cuts." 
 
 FIG. 1876. 
 
 
 
 Area. 
 
 Chms pr 1000 ft. 
 
 No. 
 
 
 
 at 70 F. 
 
 B. & S. 
 
 Diameter, 
 Mils. 
 
 Circular 
 
 Current 
 
 Guage. 
 
 
 Mils (d2) 
 
 allowed by 
 
 
 
 1 mil = -001 in. 
 
 Underwriter's 
 
 
 
 
 Code. 
 
 0000 
 
 460-000 
 
 211600-00 
 
 175 
 
 .000 
 
 409-640 
 
 167805-00 
 
 145 
 
 00 
 
 364-800 
 
 133079 '04 
 
 120 
 
 
 
 324-860 
 
 105534-03 
 
 100 
 
 1 
 
 5289-300 
 
 83694-20 
 
 95 
 
 2 
 
 257-630 
 
 66373-00 
 
 70 
 
 3 
 
 229-420 
 
 52633-40 
 
 60 
 
 4 
 
 204-310 
 
 41742-57 
 
 50 
 
 5 
 
 181-940 
 
 33102-00 
 
 45 
 
 6 
 
 162-020 
 
 26250-50 
 
 35 
 
 7 
 
 144-280 
 
 20816-70 
 
 30 
 
 8 
 
 128-490 
 
 16509-00 
 
 25 
 
 9 
 
 114-430 
 
 13094-00 
 
 
 10 
 
 101 -890 
 
 10381-00 
 
 20 
 
 11 
 
 90-742 
 
 8234-10 
 
 
 12 
 
 80-808 
 
 6529-90 
 
 15 
 
 13 
 
 71-901 
 
 5178-40 
 
 
 14 
 
 64-084 
 
 4106-80 
 
 10 
 
 
 LUNDELL MOTOR. 
 
808 
 
 APPENDIX 
 
 TABLE OP DENSITY OP GASES AND VAPOURS, AIR AT THE SAME TEM- 
 PERATURE AND PRESSURE BEING 1-0 ; ALSO THE WEIGHT OF A 
 CUBIC FOOT AT 62 FAHR., UNDER AN ATMOSPHERIC PRESSURE 
 OF 29-92 INCHES OF MERCURY. 
 
 
 Density of 
 air. 
 
 Specific 
 gravity. 
 
 Weight of 
 cubic foot 
 in pound. 
 
 Cubic feet. 
 
 Air (atmospheric) 
 
 1-00000 
 
 001221 
 
 07610 
 
 13-14 
 
 
 06926 
 
 000085 
 
 00527 
 
 189-70 
 
 Oxygen ga^ . 
 
 1-10563 
 
 001350 
 
 08414 
 
 11-88 
 
 Nitrogen gas 
 
 97137 
 
 001185 
 
 07383 
 
 13-54 
 
 Carbonic-acid gas 
 
 1-52901 
 
 001870 
 
 11636 
 
 8-59 
 
 Carbonic-oxide gas 
 
 9674 
 
 00118 
 
 07364 
 
 13-60 
 
 Vapour of water 
 
 6235 
 
 000761 
 
 04745 
 
 21-07 
 
 ' " alcohol 
 
 1-589 
 
 00194 
 
 12092 
 
 8-27 
 
 " " sulphuric ether 
 
 2-586 
 
 00316 
 
 19680 
 
 5-08 
 
 " " oil of turpentine 
 
 4-760 
 
 00581 
 
 36224 
 
 2-76 
 
 
 
 
 
 
 SPECIFIC GRAVITY OP LIQUIDS AT 60 FAHR. 
 
 Acid, muriatic 1-200 
 
 Acid, nitric 1-217 
 
 Acid, sulphuric 1*849 
 
 Alcohol, pure -794 
 
 Ammonia, 27-9 per cent -891 
 
 Carbon disulphide 1-260 
 
 Ether, sulphuric . -720 
 
 Oil, linseed '940 
 
 Oil, olive -920 
 
 Oil, petroleum '780- -880 
 
 Oil, turpentine 870 
 
 Oil, whale -920 
 
 Tar 1-000 
 
 Vinegar 1-080 
 
 Water 1-000 
 
 Water, sea. . . , . . 1-026-1 -030 
 
 SPECIFIC GRAVITIES AND WEIGHTS OF EARTHS, ETC. 
 
 SUBSTANCE. 
 
 Common 
 specific 
 gravity. 
 
 Average 
 weight 
 per cubic 
 foot. 
 
 SUBSTANCE. 
 
 Common 
 specific 
 gravity. 
 
 Average 
 weight 
 per cubic 
 foot. 
 
 Alabaster 
 
 2-73 
 3-57 
 1-13 
 1-40 
 4-0-4-86 
 2-74 
 1-71 
 1-6-1-8 
 1-6-2-16 
 2-3 
 1-28 
 96 
 1-93 
 1-40 
 1-24-1-30 
 1-0 
 2-08 
 
 171 
 192 
 70 
 87 
 250-304 
 171 
 107 
 100-115 
 100-125 
 140-150 
 80 
 60 
 121 
 80-95 
 77-81 
 62 
 120-140 
 80-110 
 250 
 162 
 150-200 
 160-170 
 138 
 110 
 135 
 61 
 
 Gypsum . 
 
 2-24 
 
 80 
 2-88 
 2-4 
 2-72 
 2-08 
 2-08 
 2-72 
 2-72 
 2-24 
 2-80 
 2-80 
 96 
 2-64 
 8-96 
 1-09 
 95 
 2-16 
 1-60 
 2-32 
 2-80 
 2-72 
 2-03 
 1-95 
 2-88 
 
 140 
 50 
 180 
 150 
 170 
 130 
 130 
 170 
 170 
 140 
 175 
 175 
 60 
 165 
 560 
 68 
 60 
 135 
 100 
 145 
 175 
 170 
 127 
 122 
 180 
 
 Asbestos 
 
 ,7f- -" 
 
 Lime quick 
 
 Asphalt, California 
 " Trinidad 
 
 Limestone 
 
 Magnesia carbonate .... 
 Marble . 
 
 Barvtes 
 
 Basalt 
 
 Masonry, concrete 
 
 " rubble 
 
 Borax 
 
 Brick, masonry 
 
 " granite. . . . 
 
 Bricks 
 
 " limestone 
 " sandstone .... 
 Mica 
 
 " fire 
 
 Cement, Portland 
 
 " Rosendale 
 Clay 
 
 Porphyry 
 
 
 Coal 
 
 Quartz . 
 
 Coal tar 
 
 Red lead . 
 
 Coke 
 
 
 Concrete 
 
 Rubber pure 
 
 Earth 
 
 Salt, common and rock . 
 Sund 
 
 Emerv 
 
 4-0 
 
 2-60 
 
 Feldspar 
 
 
 Glass 
 
 Slate 
 
 Gneiss and granite 
 Graphite 
 
 2 : 20 
 
 Soapstone 
 
 Sulphur 
 
 Gravel 
 
 
 Grindstone 
 
 2-14 
 98 
 
 Trap 
 
 Gutta-percha 
 
 
 
APPENDIX. 
 
 809 
 
 SPECIFIC GRAVITY AND WEIGHT OF WOOD. 
 
 WOOD. 
 
 Specific 
 gravity. 
 
 Average 
 weight 
 per cubic 
 foot. 
 
 WOOD. 
 
 Specific 
 gravity. 
 
 Average 
 weight 
 per cubic 
 foot. 
 
 Apple . . 
 
 73- -80 
 
 47 
 
 Lignum-vitas 
 
 65-1-33 
 
 62 
 
 Ash 
 
 60- -84 
 
 45 
 
 Lime 
 
 80 
 
 50 
 
 Bamboo 
 
 31- -40 
 
 22 
 
 Linden. 
 
 60 
 
 37 
 
 Beech 
 
 62- -85 
 
 46 
 
 Locust 
 
 73 
 
 46 
 
 Birch 
 
 56- -74 
 
 41 
 
 Mahogany 
 
 56-1-06 
 
 51 
 
 Box 
 
 91-1-33 
 
 70 
 
 Maple 
 
 57- -79 
 
 42 
 
 Butternut 
 
 38 
 
 24 
 
 Oak, live 
 
 96-1-10 
 
 64 
 
 Cedar 
 
 49- -75 
 
 39 
 
 " white 
 
 69- -86 
 
 48 
 
 Cherry 
 
 61- -72 
 
 41 
 
 " red 
 
 73- -75 
 
 46 
 
 Chestnut 
 
 46- -66 
 
 35 
 
 Palmetto 
 
 
 
 Cork 
 
 24 
 
 15 
 
 Pine, long leaf 
 
 70 
 
 44 
 
 Cypress 
 
 41- -66 
 
 33 
 
 " white 
 
 35- -55 
 
 25 
 
 Ebony . . 
 
 1-13-1-33 
 
 76 
 
 " yellow 
 
 46- -76 
 
 38 
 
 Elm. 
 
 - 55 * 78 
 
 38 
 
 Poplar 
 
 38- -58 
 
 30 
 
 Fir 
 
 48- -70 
 
 37 
 
 Spruce 
 
 40- -50 
 
 28 
 
 Gum 
 
 84-1-00 
 
 57 
 
 Sycamore 
 
 59- -62 
 
 37 
 
 Hackmatack 
 
 59 
 
 37 
 
 Tamarack 
 
 40 
 
 25 
 
 Hemlock 
 
 36- -41 
 
 24 
 
 Teak 
 
 66- -98 
 
 51 
 
 Hickory ... .... 
 
 69- -94 
 
 48 
 
 Walnut 
 
 50- -67 
 
 36 
 
 Hornbeam . . 
 
 76 
 
 47 
 
 " black. . 
 
 50 
 
 31 
 
 PROPERTIES OF METALS. 
 
 METAL. 
 
 Specific 
 gravity. 
 
 Weight per 
 cubic foot. 
 
 Melting point, 
 degrees Fahr. 
 
 Tensile strength, 
 pounds per square 
 inch. 
 
 Expansion 
 in 100 ft. 
 from temp. 
 32 to 212 F. 
 
 Aluminum 
 
 2-63 
 
 166 
 
 1,400 
 
 26,000 
 
 
 Antimony 
 
 6'76 
 
 421 
 
 842 
 
 
 1083 
 
 Bismuth 
 
 9-82 
 
 612 
 
 510 
 
 6,400 
 
 1392 
 
 Cadmium 
 
 8-65 
 
 539 
 
 
 
 
 Calcium 
 
 1-58 
 
 
 
 
 
 Chromium 
 
 5-00 
 
 
 
 
 
 Cobalt 
 
 8-60 
 
 
 
 
 
 Copper 
 
 8-7to 8-9 
 
 450 
 
 1,930 
 
 20,000 to 30,000 
 
 1722 
 
 Gold pure 
 
 19-30 
 
 1215 
 
 1,915 
 
 
 1552 
 
 " 22 carats 
 " 20 " 
 Iron cast 
 
 17-49 
 15-71 
 7-20 
 
 450 
 
 2 000 to 2,200 
 
 
 1109 
 
 " wrought 
 
 7-68 
 
 480 
 
 2,700 to 2,900 
 
 46,000 to 48,000 
 
 1220 
 
 Lead 
 
 11-36 
 
 709 
 
 625 
 
 
 2848 
 
 Magnesium 
 
 1-70 
 
 106 
 
 1,200 
 
 
 
 Manganese 
 
 8-00 
 
 499 
 
 
 
 
 Mercury, 60 
 
 13-58 
 
 847 
 
 662 
 
 
 
 Nickel 
 
 8'3to 8-9 
 
 537 
 
 3,000 
 
 
 
 
 21-50 
 
 1341 
 
 
 
 0857 
 
 Potassium 
 
 0-86 
 
 
 144 
 
 
 
 Selinium 
 
 4-30 
 
 
 220 
 
 
 
 Silver 
 
 10-50 
 
 655 
 
 1,750 
 
 
 1909 
 
 Sodium 
 
 0-97 
 
 
 
 
 
 Steel 
 
 7-84 
 
 490 
 
 2,500 
 
 IHigh grade, 70,- 
 000 to 80,000. 
 Medium grade, 
 (50 000 to 70 000 
 
 1-1079 
 
 Strontium 
 
 2-54 
 
 
 
 Soft grade, 52,000 
 to 60,000. 
 
 
 Tellium 
 
 6-11 
 
 
 
 
 
 Thallium 
 
 11-85 
 
 
 
 
 
 Tin. 
 
 7*35 
 
 458 
 
 442 
 
 3,500 
 
 2173 
 
 Tungsten 
 
 17-00 
 
 
 
 
 
 
 18-33 
 
 
 
 
 
 Zinc 
 
 7-14 
 
 445 
 
 780 
 
 5,000 to 6,000 
 
 2942 
 
810 
 
 APPENDIX. 
 
 SOLDERS. 
 
 
 Copper. 
 
 Tin. 
 
 Lead. 
 
 Zinc. 
 
 Antimony. 
 
 Silver. 
 
 Nickel. 
 
 Lead, melts at 441 
 Tin melts at 340 
 
 
 
 33 
 67 
 
 67 
 33 
 
 
 
 
 
 Spelter soft . 
 
 50 
 
 
 
 50 
 
 
 
 
 " hard 
 
 65 
 
 
 
 35 
 
 
 
 
 Steel 
 
 13 
 
 
 
 5 
 
 
 82 
 
 
 Brass or copper 
 
 50 
 
 
 
 50 
 
 
 
 
 
 47 
 
 
 
 47 
 
 
 6 
 
 
 
 66 
 
 
 
 33 
 
 1 
 
 
 
 Copper ... 
 
 53 
 
 47 
 
 
 
 
 
 
 German silver . . 
 
 38 
 
 
 
 54 
 
 , . 
 
 
 8 
 
 Solders of lead and tin are termed soft solders, in plumbers' work the joints are wiped 
 and the solder is in a mass. In soldering cornices, where there is a strain, the metal 
 should be riveted as well. Solders containing copper are used for brazing. 
 
 Sheet lead for sulphuric-acid chambers is joined by burning the sheets by a blowpipe. 
 The welding of various metals is now effected by electricity. In joining one metal to 
 another, one must be fluid; and mercury, being usually fluid, combines with most metals 
 except iron and platinum. An amalgam of tin or cold solder can be applied by friction 
 to polished iron and form a surface which admits of being soldered. 
 
 Fusible Alloys. A fusible alloy composed of 50 bismuth, 25 tin, and 25 lead melts at 
 212 , the melting point may be still further reduced by a larger percentage of bismuth, 
 and it is used either as a solder or a fusible plug. In automatic fire extinguishers the 
 cap of the sprinkler is soldered with this alloy, and melts at about 170. For fusible 
 plugs in boilers, the United States supervising inspectors specify Banca tin, which melts 
 at about 445 F. The plug must be at least |" smallest diameter. 
 
 ALLOYS AND COMPOSITIONS. 
 
 
 Cop- 
 per. 
 
 Zinc. 
 
 Tin. 
 
 Nickel. 
 
 Lead. 
 
 Alu- 
 mi- 
 num. 
 
 Steel. 
 
 
 Anti- 
 mony. 
 
 Tensile 
 strength 
 per sq. 
 inch. 
 
 Aluminum bronze 
 
 90-0 
 33 
 
 
 
 
 
 10-0 
 
 lumi 
 90-0 
 
 
 
 
 75.000- 
 90,000 
 
 78,327 
 
 " brass 
 
 33i 
 
 Add 
 10-0 
 89-3 
 
 88-9 
 81-0 
 
 33i per 
 
 cent t 
 
 num bi 
 
 onze . . 
 
 
 " composition . 
 Babbitt's metal, light duty 
 " " for bear- 
 ing's 
 
 
 
 1-8 
 3-7 
 
 
 
 
 
 
 
 8-9 
 
 7.4 
 16-0 
 
 
 
 
 
 
 
 
 
 2-0 
 
 1-0 
 
 
 
 
 
 
 
 
 
 
 
 98-0 
 
 2-0* 
 
 
 
 51-1 
 
 31-9 
 
 3-2 
 
 13-8 
 
 
 
 
 
 
 
 
 
 
 
 42.560 
 49,280 
 57,000 
 
 Muntz metal 
 
 6-25 
 67-3 
 
 93-75 
 13-0 
 
 
 
 
 
 
 
 
 Manganese alloy \ 
 
 
 
 
 1-2 
 
 96 : 75 
 
 18-0^ 
 
 5|| 
 
 
 
 3-25 
 
 
 Sheathing metal 
 
 56-0 
 66-0 
 50-0 - 
 
 61-0 
 
 44-0 
 
 2i : 6' 
 
 35-0 
 
 
 
 
 
 
 
 
 22-0 
 29-0 
 
 2-0 
 
 
 
 12-0 
 
 
 
 
 
 it 
 
 
 
 
 
 
 
 Sterrometal 
 
 
 
 
 Iron. 
 2-0 
 
 
 Anti- 
 
 monv. 
 25 -0 
 12-5 
 
 56-8 
 
 85,120 
 
 Type metal 
 
 
 75-0 
 
 87-5 
 
 
 
 
 
 
 
 
 
 
 
 White " 
 
 7-4 
 69-8 
 
 7.4 
 
 25-8 
 
 28-4 
 4-4 
 
 
 
 
 
 
 " " hard.., 
 
 
 
 
 
 
 
 * Chromium, 2'0. f Twenty-five per cent manganese to copper doubles its tensile strength 
 without diminishing its ductility. \ Manganese, 18'0. || Silicon, '5. 
 
APPENDIX. 
 
 811 
 
 TABLES OF THE CIRCUMFEEENCES OF CIRCLES TO THE NEAREST FRACTION OF 
 PRACTICAL MEASUREMENT ;' ALSO, THE AREAS OF CIRCLES, IN INCHES AND 
 DECIMAL PARTS, LIKEWISE OF FEET AND DECIMAL PARTS. 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 ence in feet 
 
 in 
 
 in square 
 
 in square 
 
 ence in feet 
 
 in 
 
 in square 
 
 in square 
 
 and inches. 
 
 inches. 
 
 inches. 
 
 feet. 
 
 and inches. 
 
 inches. 
 
 inches. 
 
 feet. 
 
 
 
 
 
 1 65 
 
 6 
 
 28-27 
 
 196 
 
 20 
 
 iV 
 
 003 
 
 
 1 74 
 
 64 
 
 29-46 
 
 204 
 
 39 
 
 1 
 
 012 
 
 
 1 7f 
 
 64 
 
 30-68 
 
 212 
 
 59 
 
 A 
 
 028 
 
 
 1 8 
 
 6f 
 
 31-92 
 
 220 
 
 78 
 
 | 
 
 049 
 
 
 1 8f 
 
 64 
 
 33-18 
 
 228 
 
 98 
 
 ft 
 
 077 
 
 
 1 8f 
 
 6| 
 
 34-47 
 
 237 
 
 1-18 
 
 f 
 
 no 
 
 
 1 M 
 
 M 
 
 35-78 
 
 246 
 
 1-37 
 
 TV 
 
 150 
 
 
 1 94 
 
 63 
 
 37-12 
 
 256 
 
 1-57 
 
 | 
 
 196 
 
 
 1 10 
 
 7 
 
 38-48 
 
 267 
 
 1-77 
 
 ft 
 
 248 
 
 
 1 lOf 
 
 74 
 
 39-87 
 
 277 
 
 1-96 
 
 f 
 
 307 
 
 
 1 10 J 
 
 H 
 
 41-28 
 
 287 
 
 2-16 
 
 H 
 
 371 
 
 
 i 114 
 
 71 
 
 42-72 
 
 297 
 
 2-36 
 
 $ 
 
 442 
 
 
 i 11* 
 
 
 
 44-18 
 
 307 
 
 2-55 
 
 
 
 518 
 
 
 i n4 
 
 7f 
 
 45-66 
 
 318 
 
 2-75 
 
 f 
 
 601 
 
 
 2 Of 
 
 7f 
 
 47-17 
 
 328 
 
 2-94 
 
 If 
 
 690 
 
 
 2 03 
 
 74 
 
 48-71 
 
 338 
 
 34 
 
 i 
 
 785 
 
 0054 
 
 2 14 
 
 8 
 
 50-26 
 
 349 
 
 H 
 
 i* 
 
 994 
 
 0069 
 
 2 14 
 
 84 
 
 51-85 
 
 360 
 
 35 
 
 14 
 
 1-23 
 
 0085 
 
 2 15 
 
 84 
 
 53-46 
 
 371 
 
 44 
 
 if 
 
 1-48 
 
 0103 
 
 2 2| 
 
 81 
 
 55-09 
 
 383 
 
 4* 
 
 il 
 
 1-77 
 
 0123 
 
 2 2f 
 
 84 
 
 56-74 
 
 394 
 
 4 
 
 M 
 
 2-07 
 
 0144 
 
 2 3 
 
 8| 
 
 58-43 
 
 406 
 
 & 
 
 i* 
 
 2-40 
 
 0167 
 
 2 3f 
 
 8| 
 
 60-13 
 
 428 
 
 5l 
 
 il 
 
 2-76 
 
 0192 
 
 2 3j 
 
 85 
 
 61-86 
 
 430 
 
 64 
 
 2 
 
 3-14 
 
 0218 
 
 2 44 
 
 9 
 
 63-62 
 
 442 
 
 3 
 
 24 
 
 3-55 
 
 0246 
 
 2 4f 
 
 9 
 
 65-40 
 
 455 
 
 7 
 
 H 
 
 3-98 
 
 0276 
 
 2 5 
 
 94 
 
 67-20 
 
 467 
 
 7f 
 
 21 
 
 4-43 
 
 0307 
 
 2 5| 
 
 9^ 
 
 69-03 
 
 480 
 
 11 
 
 4 
 
 4-91 
 
 0341 
 
 2 5j 
 
 94 
 
 70-88 
 
 493 
 
 if 
 
 25 
 
 5-41 
 
 0376 
 
 2 64 
 
 9f 
 
 72-76 
 
 506 
 
 8| 
 
 8* 
 
 5-94 
 
 0412 
 
 2 6f 
 
 i 
 
 74-66 
 
 519 
 
 9 
 
 24 
 
 6-49 
 
 0450 
 
 2 7 
 
 95 
 
 76-59 
 
 532 
 
 9| 
 
 3 
 
 7-07 
 
 0490 
 
 2 7? 
 
 10 
 
 78-54 
 
 545 
 
 9* 
 
 Si 
 
 7-67 
 
 0532 
 
 2 7 
 
 104 
 
 80-51 
 
 559 
 
 104 
 
 8* 
 
 829 
 
 0576 
 
 2 84 
 
 104 
 
 82-52 
 
 573 
 
 103 
 
 3| 
 
 8-95 
 
 0621 
 
 2 84 
 
 101 
 
 84-54 
 
 587 
 
 11 
 
 84 
 
 9-62 
 
 0668 
 
 2 9 
 
 104 
 
 86-59 
 
 601 
 
 11? 
 
 8f 
 
 10-32 
 
 0716 
 
 2 9| 
 
 lOf 
 
 88-66 
 
 615 
 
 ill 
 
 3j 
 
 11-04 
 
 0766 
 
 2 9J 
 
 lOf 
 
 90-76 
 
 630 
 
 124 
 
 33 
 
 11-79 
 
 0818 
 
 2 104 
 
 io4 
 
 92-88 
 
 645 
 
 1 04 
 
 4 
 
 12-57 
 
 087 
 
 2 104 
 
 11 
 
 95-03 
 
 660 
 
 1 1 
 
 44 
 
 13-36 
 
 093 
 
 2 10? 
 
 ii4 
 
 97-21 
 
 675 
 
 1 11 
 
 *i 
 
 14-19 
 
 099 
 
 2 114 
 
 114 
 
 99-40 
 
 690 
 
 1 If 
 
 4! 
 
 15-03 
 
 105 
 
 2 11 J 
 
 ill 
 
 101-62 
 
 705 
 
 1 24 
 
 4* 
 
 15-90 
 
 111 
 
 3 04 
 
 114 
 
 103-87 
 
 720 
 
 1 24 
 
 43 
 
 16-80 
 
 118 
 
 3 04 
 
 ill 
 
 106-14 
 
 736 
 
 1 25 
 
 4j 
 
 17-72 
 
 124 
 
 3 Oj 
 
 11* 
 
 108-43 
 
 752 
 
 1 34 
 
 i 
 
 18-66 
 
 130 
 
 3 14 
 
 ill 
 
 110-75 
 
 768 
 
 1 3| 
 
 5 
 
 19-63 
 
 136 
 
 3 If 
 
 12 
 
 113-10 
 
 785 
 
 1 44 
 
 54 
 
 20-63 
 
 14S 
 
 3 2 
 
 124 
 
 115-47 
 
 802 
 
 1 44 
 
 54 
 
 21-65 
 
 150 
 
 3 24 
 
 124 
 
 117-86 
 
 819 
 
 1 45 
 
 5 
 
 22-69 
 
 157 
 
 3 23 
 
 12f 
 
 120-28 
 
 836 
 
 1 54 
 
 54 
 
 23-76 
 
 165 
 
 3 3J: 
 
 124 
 
 122-72 
 
 853 
 
 1 5f- 
 
 5| 
 
 24-85 
 
 173 
 
 3 3f 
 
 12* 
 
 125-19 
 
 870 
 
 1 6 
 
 5f 
 
 25-97 
 
 181 
 
 3 4 
 
 iaf 
 
 127-68 
 
 887 
 
 1 6f 
 
 6l 
 
 27-11 
 
 189 
 
 3 4| 
 
 124 
 
 130-19 
 
 904 
 
812 
 
 APPENDIX. 
 
 TABLES OF THE CIKCUMFEKENCES OF CIECLES, ETC. (Continued.) 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 ence in feet 
 
 in 
 
 in square 
 
 in square 
 
 ence in teet 
 
 in feet and 
 
 in square 
 
 in square 
 
 and inches. 
 
 inches. 
 
 inches. 
 
 feet. 
 
 and inches. 
 
 inches. 
 
 inches. 
 
 feet. 
 
 3 4f 
 
 13 
 
 132-73 
 
 922 
 
 5 2J 
 
 20 
 
 314-16 
 
 2-182 
 
 3 51 
 
 131 
 
 135-30 
 
 939 
 
 5 Si- 
 
 201 
 
 318-10 
 
 2-209 
 
 3 5f 
 
 131 
 
 137-89 
 
 956 
 
 5 3| 
 
 20i 
 
 322-06 
 
 2-237 
 
 3 6 
 
 13f 
 
 140-50 
 
 974 
 
 5 4 
 
 20f 
 
 326-05 
 
 2-265 
 
 3 6f 
 
 131 
 
 143-14 
 
 992 
 
 5 4f 
 
 20^ 
 
 330-06 
 
 2-293 
 
 3 6f 
 
 18f 
 
 145-80 
 
 1-011 
 
 5 41 
 
 20| 
 
 334-10 
 
 2-321 
 
 3 71 
 
 18f 
 
 148-49 
 
 1-030 
 
 5 5 
 
 20f 
 
 338-16 
 
 2-349 
 
 3 7f 
 
 181 
 
 151-20 
 
 1-050 
 
 5 5| 
 
 201 
 
 342-25 
 
 2-377 
 
 3D 
 o 
 
 14 
 
 159-94 
 
 1-069 
 
 5 6 
 
 21 
 
 346-36 
 
 2-405 
 
 3 8f- 
 
 141 
 
 156-70 
 
 1-088 
 
 5 6| 
 
 211 
 
 350-50 
 
 2-434 
 
 3 8f 
 
 141 
 
 159-49 
 
 1-107 
 
 5 6J 
 
 ill 
 
 354-66 
 
 2-463 
 
 3 91 
 
 14? 
 
 162-30 
 
 1-126 
 
 5 7| 
 
 21f 
 
 358-84 
 
 2-492 
 
 3 91 
 
 141 
 
 165-13 
 
 1-146 
 
 5 71 
 
 211 
 
 363-05 
 
 2-521 
 
 3 9| 
 
 14* 
 
 167-99 
 
 1-166 
 
 5 7 
 
 21| 
 
 367-28 
 
 2-550 
 
 3 101 
 
 14f 
 
 170-87 
 
 1-186 
 
 5 81 
 
 2li 
 
 871-54 
 
 2-580 
 
 3 lOf 
 
 14i 
 
 173-78 
 
 1-206 
 
 5 8f 
 
 211 
 
 375-83 
 
 2-610 
 
 3 HI 
 
 15 
 
 176-71 
 
 1-227 
 
 5 9J 
 
 22 
 
 380-13 
 
 2-640 
 
 3 111 
 
 151 
 
 179-67 
 
 1-247 
 
 5 9| 
 
 221 
 
 384-46 
 
 2-670 
 
 3 Hi 
 
 1M 
 
 182-65 
 
 1-267 
 
 5 9 
 
 221 
 
 388-82 
 
 2-700 
 
 4 01 
 
 155 
 
 185-66 
 
 1-288 
 
 5 10J 
 
 22} 
 
 393-20 
 
 2-730 
 
 4 Of 
 
 151 
 
 188-69 
 
 1-309 
 
 5 lOf 
 
 221 
 
 397-61 
 
 2-761 
 
 4 1 
 
 Iftf 
 
 191-75 
 
 1-330 
 
 5 11 
 
 22f 
 
 402-04 
 
 2-792 
 
 4 11 
 
 I6f 
 
 194-83 
 
 1-352 
 
 5 11J 
 
 22* 
 
 406-49 
 
 2-823' 
 
 4 11 
 
 16* 
 
 197-93 
 
 1-374 
 
 5 11| 
 
 22i 
 
 410-97 
 
 2-854 
 
 4 21 
 
 16 
 
 201-06 
 
 1-396 
 
 6 01 
 
 23 
 
 415-48 
 
 2-885 
 
 4 2f 
 
 161 
 
 204-22 
 
 1-418 
 
 6 Of 
 
 231 
 
 420-00 
 
 2-917 
 
 4 3 
 
 161 
 
 207-39 
 
 1-440 
 
 6 1 
 
 23ir 
 
 424-56 
 
 2-949 
 
 4 33 
 
 lt 
 
 210-60 
 
 1-462 
 
 6 It 
 
 23| 
 
 429-13 
 
 2-981 
 
 4 S| 
 
 161 
 
 213-82 
 
 1-484 
 
 6 If 
 
 231 
 
 433-74 
 
 3-013 
 
 4 41 
 
 16$ 
 
 217-08 
 
 1-507 
 
 6 2| 
 
 28| 
 
 4S8-36 
 
 3-045 
 
 4 4^ 
 
 16f 
 
 220-35 
 
 1-530 
 
 6 2| 
 
 23| 
 
 443-01 
 
 3-077 
 
 4 5 
 
 16i 
 
 223-65 
 
 1-553 
 
 6 3 
 
 23| 
 
 447-69 
 
 3-109 
 
 4 5| 
 
 17 
 
 226-98 
 
 1-576 
 
 6 8| 
 
 2 
 
 45239 
 
 3-142 
 
 4 6J 
 
 171 
 
 230-33 
 
 1-599 
 
 6 4| 
 
 2 01 
 
 461-86 
 
 3-207 
 
 4 61 
 
 171 
 
 233-70 
 
 1-622 
 
 6 4 
 
 2 OJ 
 
 471-44 
 
 3-273 
 
 4 6l 
 
 171 
 
 237-10 
 
 1-645 
 
 6 5| 
 
 2 Of 
 
 481-11 
 
 3-341 
 
 4 ft? 
 
 171 
 
 240-53 
 
 1-669 
 
 6 6J 
 
 2 1 
 
 490-87 
 
 3-408 
 
 4 7t 
 
 171 
 
 243-98 
 
 1-693 
 
 6 71 
 
 2 11 
 
 500-74 
 
 3-477 
 
 4 7* 
 
 171 
 
 247-45 
 
 1-718 
 
 6 81 
 
 2 1J 
 
 510-71 
 
 3-547 
 
 4 8l 
 
 IT! 
 
 250-95 
 
 1-743 
 
 6 8| 
 
 2 If 
 
 520-77 
 
 3-617 
 
 4 81 
 
 18 
 
 254-47 
 
 1-767 
 
 6 9| 
 
 2 2 
 
 530-93 
 
 3-687 
 
 4 8| 
 
 18* 
 
 258-02 
 
 1-792 
 
 6 lOt 
 
 2 21 
 
 541-19 
 
 3-758 
 
 4 94 
 
 181 
 
 261-59 
 
 1-817 
 
 6 lli 
 
 2 2 
 
 551-55 
 
 3-830 
 
 4 9f 
 
 18f 
 
 265-18 
 
 1-842 
 
 7 
 
 2 2f 
 
 562-00 
 
 3-904 
 
 4 101 
 
 181 
 
 268-80 
 
 1-868 
 
 7 0* 
 
 2 3 
 
 572-56 
 
 3-976 
 
 4 101 
 
 18f 
 
 272-45 
 
 1-893 
 
 7 If 
 
 2 3J 
 
 583-21 
 
 4-050 
 
 4 lOi 
 
 18* 
 
 276-12 
 
 1-918 
 
 7 2| 
 
 2 3i 
 
 59396 
 
 4-124 
 
 4 111 
 
 18i 
 
 279-81 
 
 1-943 
 
 7 3 
 
 2 8| 
 
 604-81 
 
 4-200 
 
 4 Hf 
 
 19 
 
 283-53 
 
 1-969 
 
 7 3 
 
 2 4 
 
 615-75 
 
 4-276 
 
 6 
 
 191 
 
 287-27 
 
 1-995 
 
 7 4f 
 
 2 41 
 
 626-80 
 
 4-352 
 
 5 03 
 
 19* 
 
 291-04 
 
 2-021 
 
 7 51 
 
 2 44- 
 
 637-94 
 
 4-430 
 
 5 OJ 
 
 19| 
 
 294-83 
 
 2-047 
 
 7 61 
 
 2 4| 
 
 649-18 
 
 4-508 
 
 6 It 
 
 19i 
 
 298-65 
 
 2-074 
 
 7 7 
 
 2 5 
 
 660-52 
 
 4-586 
 
 5 If 
 
 191 
 
 302-49 
 
 2-101 
 
 7 7* 
 
 2 51 
 
 671-96 
 
 4-666 
 
 5 2 
 
 19* 
 
 306-36 
 
 2-128 
 
 7 81 
 
 2 5} 
 
 683-49 
 
 4-747 
 
 5 2| 
 
 in 
 
 310-25 
 
 2-155 
 
 7 91 
 
 2 5f 
 
 695-13 
 
 4-827 
 
APPENDIX. 
 
 813 
 
 TABLES OF THE CIECUMFEKENCES OF CIRCLES, ETC. (Continued.) 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 ence in feet 
 
 in feet and 
 
 in square 
 
 in square 
 
 ence in teet 
 
 in feet and 
 
 in square 
 
 in square 
 
 and inches. 
 
 inches. 
 
 inches. 
 
 feet. 
 
 and inches. 
 
 inches. 
 
 inches. 
 
 feet. 
 
 1 10J 
 
 2 6 
 
 706-86 
 
 4-908 
 
 11 6J 
 
 3 8 
 
 1520-5 
 
 10-56 
 
 7 11 
 
 2 6J 
 
 718-69 
 
 4-990 
 
 11 7 
 
 3 8i 
 
 1537-9 
 
 10-68 
 
 7 11J 
 
 2 64 
 
 730-62 
 
 5-073 
 
 11 7f 
 
 3 8i 
 
 1555-3 
 
 10-80 
 
 8 0| 
 
 2 6f 
 
 742-64 
 
 5-157 
 
 11 8J 
 
 3 8| 
 
 1572-8 
 
 10-92 
 
 8 1| 
 
 2 7 
 
 754-77 
 
 5-241 
 
 11 9| 
 
 3 9 
 
 1590-4 
 
 11-04 
 
 8 2k 
 
 2 7i 
 
 766-99 
 
 5-326 
 
 11 lOfr 
 
 3 9i 
 
 1608-1 
 
 11-17 
 
 8 23- 
 
 2 7 
 
 779-31 
 
 5-411 
 
 11 lOJt 
 
 3 9 
 
 1626-0 
 
 11-29 
 
 8 3f 
 
 2 7f 
 
 791-73 
 
 5-498 
 
 11 llf 
 
 3 9f 
 
 1643-9 
 
 11-41 
 
 8 4 
 
 2 8 
 
 804-25 
 
 5-585 
 
 12 OJ 
 
 3 10 
 
 1661-9 
 
 11-54 
 
 8 5| 
 
 2 Si 
 
 816-86 
 
 5-673 
 
 12 1 
 
 3 10J 
 
 1680-0 
 
 11-67 
 
 8 6| 
 
 2 8J 
 
 829-58 
 
 5-761 
 
 12 2 
 
 3 10i 
 
 1698-2 
 
 11-79 
 
 8 6 
 
 2 8| 
 
 842-39 
 
 5-849 
 
 12 21 
 
 3 10* 
 
 1716-5 
 
 11-92 
 
 8 7& 
 
 2 9 
 
 855-30 
 
 5-939 
 
 12 3f 
 
 3 11 
 
 1734-9 
 
 12-05 
 
 8 8i 
 
 2 9| 
 
 868-31 
 
 6-029 
 
 12 4| 
 
 3 lit 
 
 1753-4 
 
 12-18 
 
 8 9i 
 
 2 9| 
 
 881-41 
 
 6-120 
 
 12 5i 
 
 3 1H 
 
 1772-0 
 
 12-30 
 
 8 10 
 
 2 9f 
 
 894-62 
 
 6-212 
 
 12 6 
 
 3 llf 
 
 1790-8 
 
 12-43 
 
 8 lOJ 
 
 2 10 
 
 907-92 
 
 6-305 
 
 12 6f 
 
 4 
 
 1809-6 
 
 12-57 
 
 8 11 
 
 2 10J 
 
 921-32 
 
 6-398 
 
 12 U 
 
 4 OJ 
 
 1828-6 
 
 12-70 
 
 9 Of 
 
 2 10A 
 
 934-82 
 
 6-491 
 
 12 8| 
 
 4 0^ 
 
 1847-4 
 
 12-83 
 
 9 IJ 
 
 2 lOf 
 
 948-42 
 
 6-586 
 
 12 0J- 
 
 4 Of 
 
 1866-5 
 
 12-96 
 
 9 11 
 
 2 11 
 
 962-11 
 
 6-681 
 
 12 91 
 
 4 1 
 
 1885-7 
 
 13-09 
 
 9 2| 
 
 2 11J 
 
 975-91 
 
 6-777 
 
 12 lOf 
 
 4 li 
 
 1905-0 
 
 13-23 
 
 9 3^ 
 
 2 11J 
 
 989-80 
 
 6-874 
 
 12 11J 
 
 4 1* 
 
 1924-4 
 
 13-36 
 
 9 4 
 
 2 llf 
 
 1003-8 
 
 6-970 
 
 13 0| 
 
 4 14 
 
 1943-9 
 
 13-50 
 
 9 5 
 
 3 
 
 1017-9 
 
 7-069 
 
 13 1 
 
 4 2 
 
 1963-5 
 
 13-63 
 
 9 5J 
 
 3 Oi 
 
 1032-1 
 
 7-167 
 
 13 11 
 
 4 2i 
 
 1983-2 
 
 13-77 
 
 9 6| 
 
 3 OJ 
 
 1046-3 
 
 7-266 
 
 13 2| 
 
 4 2* 
 
 2003-0 
 
 13-91 
 
 9 74 
 
 3 Of 
 
 1060-7 
 
 7-366 
 
 13 3f 
 
 4 2f 
 
 2022-8 
 
 14-05 
 
 9 Si 
 
 3 1 
 
 1075-2 
 
 7-466 
 
 13 4| 
 
 4 3 
 
 2042-8 
 
 14-19 
 
 9 9 
 
 3 1J 
 
 1089-8 
 
 7-567 
 
 13 5 
 
 4 8i 
 
 2062-9 
 
 14-32 
 
 9 J 
 
 3 H 
 
 1104-5 
 
 7-669 
 
 13 5f 
 
 4 3^ 
 
 2083-1 
 
 14-46 
 
 9 10$ 
 
 3 If 
 
 1119-2 
 
 7-772 
 
 13 6i 
 
 4 3f 
 
 210S-3 
 
 14-61 
 
 9 11| 
 
 3 2 
 
 1134-1 
 
 7-876 
 
 13 7f 
 
 4 4 
 
 2123-7 
 
 14-75 
 
 10 Oj- 
 
 3 2* 
 
 1149-1 
 
 7-979 
 
 13 8J- 
 
 4 4J 
 
 2144-2 
 
 14-89 
 
 10 OJ 
 
 3 2 
 
 1164-2 
 
 8-085 
 
 18 8| 
 
 4 4^ 
 
 2164-7 
 
 15-03 
 
 10 If 
 
 3 2| 
 
 1179-3 
 
 8-189 
 
 13 9f 
 
 4 4f 
 
 2185-4 
 
 15-18 
 
 10 2* 
 
 3 3 
 
 1194-6 
 
 8-295 
 
 13 10| 
 
 4 5 
 
 2206-2 
 
 15-32 
 
 10 3i 
 
 3 3i 
 
 1:409-9 
 
 8-403 
 
 13 11J 
 
 4 5J 
 
 2227-0 
 
 15-46 
 
 10 4 
 
 3 3k 
 
 1225-4 
 
 8-509 
 
 14 
 
 4 5| 
 
 2248-0 
 
 15-61 
 
 10 4 
 
 3 3f 
 
 1241-0 
 
 8-617 
 
 14 
 
 4 5| 
 
 2269-1 
 
 15-76 
 
 10 5| 
 
 3 4 
 
 1256-6 
 
 8-727 
 
 14 1| 
 
 4 6 
 
 2290-2 
 
 15-90 
 
 10 6| 
 
 3 4J 
 
 1272-4 
 
 8-836 
 
 14 2| 
 
 4 6J 
 
 2311-5 
 
 16-05 
 
 10 7i 
 
 3 44 
 
 1288-2 
 
 8-946 
 
 14 3i 
 
 4 6i 
 
 2332-8 
 
 16-20 
 
 10 8 
 
 3 4f 
 
 1304-2 
 
 9056 
 
 14 4 
 
 4 6f 
 
 2354-3 
 
 16-35 
 
 10 8| 
 
 3 5 
 
 1320-2 
 
 9-169 
 
 14 4f 
 
 4 7 
 
 2375-8 
 
 16-50 
 
 10 9* 
 
 3 5\ 
 
 1336-4 
 
 9-211 
 
 14 5J- 
 
 4 7} 
 
 2397-5 
 
 16-65 
 
 10 lOf 
 
 3 5 
 
 1352-6 
 
 9-394 
 
 14 6| 
 
 4 74 
 
 2419-2 
 
 16-80 
 
 10 11J 
 
 3 5f 
 
 1369-0 
 
 9-506 
 
 14 7 
 
 4 7f 
 
 2441-1 
 
 16-95 
 
 10 111 
 
 3 6 
 
 1385-4 
 
 9-62 
 
 14 7^ 
 
 4 8 
 
 2463-0 
 
 17-10 
 
 11 Of 
 
 3 6i 
 
 1402-0 
 
 9-73 . 
 
 14 8| 
 
 4 8* 
 
 2485-0 
 
 17-26 
 
 11 1J 
 
 3 6J 
 
 1418-6 
 
 9-84 
 
 14 9i 
 
 4 8^ 
 
 2507-2 
 
 17-41 
 
 11 2i 
 
 3 6f 
 
 1435-4 
 
 9-96 
 
 14 lOi 
 
 4 8f 
 
 2529-4 
 
 17-56 
 
 11 3 
 
 3 7 
 
 1452-2 
 
 10-08 
 
 14 11 
 
 4 9 
 
 2551-8 
 
 17-72 
 
 11 8J 
 
 3 7i 
 
 1469-1 
 
 10-20 
 
 14 ll s 7 
 
 4 9J: 
 
 2574-2 
 
 17-88 
 
 11 4| 
 
 3 7 
 
 1486-2 
 
 10-32 
 
 15 Of 
 
 4 9J 
 
 2596-7 
 
 18-03 
 
 11 5fc 
 
 3 7f 
 
 1503-3 
 
 10-44 
 
 15 If 
 
 4 9f 
 
 2619-3 
 
 18-19 
 
814: APPENDIX. 
 
 TABLES OF THE CIKCUMFEKENCES OF CIRCLES, ETC. (Continued.) 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 ence in feet 
 
 in feet and 
 
 in square 
 
 in square 
 
 ence in feet 
 
 in teet and 
 
 in square 
 
 in square 
 
 and inches. 
 
 inches. 
 
 inches. 
 
 feet. 
 
 and inches. 
 
 inches. 
 
 inches. 
 
 feet. 
 
 15 2 
 
 4 10 
 
 2642-1 
 
 18-35 
 
 18 10i 
 
 6 
 
 4071-5 
 
 28-27 
 
 15 3 
 
 4 10J 
 
 2664-9 
 
 18-51 
 
 18 10 
 
 6 OJ 
 
 4099-8 
 
 28-47 
 
 15 3f 
 
 4 10$ 
 
 2687-8 
 
 18-66 
 
 18 llf 
 
 6 Oi 
 
 4128-2 
 
 28-67 
 
 15 4i 
 
 4 lOf 
 
 2710-8 
 
 18-82 
 
 19 
 
 6 Of 
 
 4156-8 
 
 28-87 
 
 15 r>t 
 
 4 11 
 
 2734-0 
 
 18-98 
 
 19 l| 
 
 6 1 
 
 4185-4 
 
 29-07 
 
 15 6i 
 
 4 Hi 
 
 2757-2 
 
 19-15 
 
 19 2i 
 
 6 1J 
 
 4214-1 
 
 29-27 
 
 15 6 
 
 4 iii 
 
 2780-5 
 
 19-31 
 
 19 2} 
 
 6 H 
 
 4242-9 
 
 29-47 
 
 15 7f 
 
 4 llf 
 
 2803-9 
 
 19-47 
 
 19 3 
 
 6 l| 
 
 4271-8 
 
 29-67 
 
 15 8i 
 
 5 
 
 2827-4 
 
 19-63 
 
 19 4$ 
 
 6 2 
 
 4300-8 
 
 29-87 
 
 15 9J 
 
 5 Oi 
 
 2861-0 
 
 19-80 
 
 19 5J 
 
 6 2i 
 
 4329-9 
 
 30-07 
 
 15 10 
 
 5 0^ 
 
 2874-8 
 
 19-96 
 
 19 6 
 
 6 2i 
 
 4359-2 
 
 30-27 
 
 15 lOf 
 
 5 Of 
 
 2898-6 
 
 20-13 
 
 19 6f 
 
 6 2 
 
 4388-5 
 
 30-47 
 
 15 llf. 
 
 5 1 
 
 2922-5 
 
 20-29 
 
 19 7i 
 
 6 3 
 
 4417-9 
 
 30-68 
 
 16 Of 
 
 5 11 
 
 2946-5 
 
 2046 
 
 19 8| 
 
 6 3\ 
 
 4447-4 
 
 30-88 
 
 16 U 
 
 5 if 
 
 2970-6 
 
 20-63 
 
 19 9i 
 
 6 3| 
 
 4477-0 
 
 31-09 
 
 16 2 
 
 5 If 
 
 2994-8 
 
 20-80 
 
 19 9| 
 
 6 3f 
 
 4506-7 
 
 31-30 
 
 16 2f 
 
 5 2 
 
 3019-1 
 
 20-96 
 
 19 lOf 
 
 6 4 
 
 4536-5 
 
 31-50 
 
 16 3* 
 
 6 2* 
 
 3043-5 
 
 21-13 
 
 19 Hi 
 
 6 4J 
 
 4566-4 
 
 31-71 
 
 16 4J 
 
 5 2i 
 
 3068-0 
 
 21-30 
 
 20 01 
 
 6 4^ 
 
 45963 
 
 31-92 
 
 16 &l 
 
 5 2| 
 
 3092-6 
 
 21-48 
 
 20 l| 
 
 6 4f 
 
 4626-4 
 
 32-13 
 
 16 5 
 
 5 3 
 
 3117-2 
 
 21-65 
 
 20 11 
 
 6 5 
 
 4656-6 
 
 32-34 
 
 16 6 
 
 5 3* 
 
 3142-0 
 
 21-82 
 
 20 2| 
 
 6 5J 
 
 4686-9 
 
 32-55 
 
 16 7 
 
 5 3J 
 
 3166-9 
 
 21-99 
 
 20 3J 
 
 6 5J 
 
 4717-3 
 
 82-76' 
 
 16 8i 
 
 5 3f 
 
 3191-9 
 
 22-17 
 
 20 4J 
 
 6 5f 
 
 4747-8 
 
 32-97 
 
 16 9 
 
 5 4 
 
 3217-0 
 
 22-34 
 
 20 5 
 
 6 6 
 
 4778-3 
 
 33-18 
 
 16 9f 
 
 5 4J 
 
 3242-2 
 
 22-51 
 
 20 5| 
 
 6 6 
 
 4809-0 
 
 33-40 
 
 16 10| 
 
 5 4\ 
 
 3267-5 
 
 22-69 
 
 20 6 
 
 6 6 
 
 4839-8 
 
 33-61 
 
 16 llf 
 
 5 4f 
 
 3292-8 
 
 22-87 
 
 ^!0 7f 
 
 6 6| 
 
 4S70-7 
 
 33-82 
 
 17 Oi 
 
 5 5 
 
 3318-3 
 
 23-04 
 
 20 8| 
 
 6 7 
 
 4901-6 
 
 34-04 
 
 17 1 
 
 5 5J. 
 
 3343-9 
 
 23-22 
 
 20 8| 
 
 6 7J 
 
 4932-7 
 
 34-25 
 
 17 If 
 
 5 5| 
 
 3369-6 
 
 23-40 
 
 20 9| 
 
 6 7^ 
 
 4963-9 
 
 34-47 
 
 17 2 
 
 5 5f 
 
 3395-3 
 
 23-58 
 
 20 10| 
 
 6 7t 
 
 4995-1 
 
 34-69 
 
 17 3| 
 
 5 6 
 
 3421-2 
 
 23-76 
 
 20 11 J 
 
 6 8 
 
 5026-5 
 
 34-91 
 
 17 4J 
 
 5 64 
 
 3447-2 
 
 23-94 
 
 21 OJ 
 
 6 8J 
 
 5058-0 
 
 35-12 
 
 17 4| 
 
 5 6 
 
 3473-2 
 
 24-12 
 
 21 0| 
 
 6 81 
 
 5089-5 
 
 35-34 
 
 17 6$ 
 
 5 6f 
 
 3499-4 
 
 24-30 
 
 21 if 
 
 6 8| 
 
 5121-2 
 
 35-56 
 
 17 6i 
 
 5 7 
 
 3525-1 
 
 24-48 
 
 21 2| 
 
 6 9 
 
 5153-0 
 
 35-78 
 
 17 7i 
 
 5 7 
 
 8552-0 
 
 24-67 
 
 21 3J 
 
 6 9J 
 
 5184-8 
 
 36-01 
 
 17 8 
 
 5 7i 
 
 3578-5 
 
 24-85 
 
 21 4 
 
 6 9| 
 
 5216-8 
 
 36-23 
 
 17 8f 
 
 5 7f 
 
 3605-0 
 
 25-03 
 
 21 4| 
 
 6 9| 
 
 5248-8 
 
 36-45 
 
 17 9| 
 
 5 8 
 
 3631-7 
 
 25-22 
 
 21 6| 
 
 6 10 
 
 5281-0 
 
 36-67 
 
 17 10| 
 
 5 8J 
 
 3658-4 
 
 25-40 
 
 21 6| 
 
 6 10 J 
 
 5313-2 
 
 86-89 
 
 17 Hi 
 
 5 8i 
 
 3685-3 
 
 25-59 
 
 21 7i 
 
 6 10i 
 
 5345-6 
 
 37-12 
 
 17 11* 
 
 5 8| 
 
 3712-2 
 
 25-78 
 
 21 7* 
 
 6 10| 
 
 5378-0 
 
 37-35 
 
 18 Of 
 
 5 9 
 
 3739-3 
 
 25-96 
 
 21 8| 
 
 6 H 
 
 5410-6 
 
 37-57 
 
 18 H 
 
 5 9i 
 
 3766-4 
 
 26-15 
 
 21 9* 
 
 6 11J 
 
 5443-2 
 
 37-80 
 
 18 2i 
 
 5 9i 
 
 3793-7 
 
 26-34 
 
 21 10| 
 
 6 III 
 
 5476-0 
 
 38-03 
 
 18 3i 
 
 5 9f 
 
 3821-0 
 
 26-53 
 
 21 ll| 
 
 6 llf 
 
 5508-8 
 
 38-26 
 
 18 8J 
 
 5 10 
 
 3848-5 
 
 26-72 
 
 21 11| 
 
 7 
 
 5541-7 
 
 38-48 
 
 18 4| 
 
 5 10J 
 
 3876-0 
 
 26-92 
 
 22 0$ 
 
 7 0^ 
 
 5574-8 
 
 38-71 
 
 18 5i 
 
 5 10| 
 
 3903-6 
 
 27-11 
 
 22 If 
 
 7 O.V 
 
 5607-9 
 
 38-94 
 
 18 6i 
 
 5 lOf 
 
 3931-4 
 
 27-30 
 
 22 2i 
 
 7 9| 
 
 5641-1 
 
 39-17 
 
 18 7 
 
 5 11 
 
 3959-2 
 
 27-49 
 
 22 3 
 
 7 1 
 
 5674-5 
 
 39-41 
 
 18 7f 
 
 5 1H 
 
 3987-1 
 
 27-69 
 
 22 3t 
 
 7 H 
 
 5707-9 
 
 39-64 
 
 18 8| 
 
 5 Hi 
 
 4015-2 
 
 27-88 
 
 22 4^ 
 
 l H 
 
 5741-4 
 
 39-87 
 
 18 9| 
 
 5 llf 
 
 4043-3 
 
 28-08 
 
 22 5| 
 
 7 If 
 
 5775-0 
 
 40-10 
 
APPENDIX. 815 
 
 TABLES OF THE CIECUMFERENCES OF CIKCLES, ETC. (Continued.) 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 Circumfer- 
 
 Diameter 
 
 Area 
 
 Area 
 
 ence in feet 
 and inches. 
 
 in feet and 
 inches. 
 
 in square 
 inches. 
 
 in square 
 feet. 
 
 ence in feet 
 and inches. 
 
 in feet and 
 inches. 
 
 in square 
 inches. 
 
 in square 
 ieet. 
 
 22 6J 
 
 7 2 
 
 5808-8 
 
 40-34 
 
 26 2 
 
 8 4 
 
 7853-9 
 
 54-54 
 
 22 6 
 
 7 2} 
 
 5842-6 
 
 40-57 
 
 26 5^ 
 
 8 5 
 
 8011-9 
 
 55-64 
 
 22 7* 
 
 7 2J 
 
 5876-5 
 
 40-80 
 
 26 8| 
 
 8 6 
 
 8171-3 
 
 56-75 
 
 22 8fr 
 
 7 2| 
 
 5910-5 
 
 41-04 
 
 26 11 
 
 8 7 
 
 8332-3 
 
 57-86 
 
 22 9 
 
 7 3 
 
 5944-6 
 
 41-28 
 
 27 2| 
 
 8 8 
 
 8494-9 
 
 58-99 
 
 22 10* 
 
 7 3* 
 
 5978-9 
 
 41-52 
 
 27 5* 
 
 8 9 
 
 8659-0 
 
 60-13 
 
 22 10| 
 
 7 3J 
 
 6013-2 
 
 41-76 
 
 27 9 
 
 8 10 
 
 8824-7 
 
 61-28 
 
 22 11| 
 
 7 3| 
 
 6047-6 
 
 42-00 
 
 28 
 
 8 11 
 
 8892-0 
 
 62-44 
 
 23 Of 
 
 7 4 
 
 6082-1 
 
 42-24 
 
 28 3J 
 
 9 
 
 9160-9 
 
 63-62 
 
 23 li 
 
 7 4 
 
 6116-7 
 
 42-48 
 
 28 6| 
 
 9 1 
 
 9331-3 
 
 64-80 
 
 23 2 
 
 7 4i 
 
 6151-4 
 
 42-72 
 
 28 9^ 
 
 9 2 
 
 9503-3 
 
 66-00 
 
 23 2i 
 
 7 4| 
 
 6186-2 
 
 42-96 
 
 29 0| 
 
 9 3 
 
 9676-9 
 
 67-20 
 
 23 3f 
 
 7 5 
 
 6221-1 
 
 43-20 
 
 29 3f 
 
 9 4 
 
 9852-1 
 
 68-42 
 
 23 4 
 
 7 5i 
 
 6256-1 
 
 43-44 
 
 29 7 
 
 9 5 
 
 10028-8 
 
 69-64 
 
 23 5J 
 
 7 5 
 
 6291-2 
 
 43-68 
 
 29 10 
 
 9 6 
 
 10207-1 
 
 70-88 
 
 23 6 
 
 7 51 
 
 6326-4 
 
 43-93 
 
 30 1 
 
 9 7 
 
 10386-9 
 
 72-13 
 
 23 6| 
 
 7 6 
 
 6361-7 
 
 44-18 
 
 30 4f 
 
 9 8 
 
 10568-3 
 
 73-39 
 
 23 1\ 
 
 7 6J- 
 
 6397-1 
 
 44-43 
 
 30 7| 
 
 9 9 
 
 10751-3 
 
 74-66 
 
 23 8 
 
 7 6 
 
 6432-6 
 
 44-67 
 
 30 10f 
 
 9 10 
 
 10935-9 
 
 75-94 
 
 23 9 
 
 7 6| 
 
 6468-2 
 
 44-92 
 
 31 If 
 
 9 11 
 
 11122-0 
 
 77-24 
 
 23 9J 
 
 7 7 
 
 6503-8 
 
 45-17 
 
 
 
 
 
 23 10J 
 
 7 7i 
 
 6539-6 
 
 45-41 
 
 31 5 
 
 10 
 
 11309-8 
 
 78-54 
 
 23 llf 
 
 7 V* 
 
 6575-5 
 
 45-66 
 
 31 8J 
 
 10 1 
 
 11499-0 
 
 79-85 
 
 24 0^ 
 
 7 7f 
 
 66115 
 
 45-91 
 
 31 ll| 
 
 10 2 
 
 11689-9 
 
 81-18 
 
 
 
 
 
 32 2| 
 
 10 3 
 
 11882-3 
 
 82-52 
 
 24 1 
 
 7 8 
 
 6647-6 
 
 46-16 
 
 32 5| 
 
 10 4 
 
 12076-3 
 
 83-86 
 
 24 H 
 
 7 8J- 
 
 6683-8 
 
 46-42 
 
 32 8% 
 
 10 5 
 
 12271-9 
 
 85-22 
 
 24 2J 
 
 7 8i 
 
 6720-0 
 
 46-67 
 
 32 llf 
 
 10 6 
 
 12469-0 
 
 86-59 
 
 24 3J 
 
 7 8J 
 
 6756-4 
 
 46-92 
 
 33 2 
 
 10 7 
 
 12667-7 
 
 87-97 
 
 24 4i 
 
 7 9 
 
 6792-9 
 
 47-17 
 
 33 6i 
 
 10 8 
 
 12868-0 
 
 89-36 
 
 24 4 
 
 7 9i 
 
 6829-4 
 
 47-43 
 
 33 9| 
 
 10 9 
 
 13069-8 
 
 90-76 
 
 24 61 
 
 7 9 
 
 6866-1 
 
 47-68 
 
 34 Of 
 
 10 10 
 
 13273-3 
 
 92-17 
 
 24 6 
 
 7 9* 
 
 6902-9 
 
 47-94 
 
 34 3| 
 
 10 11 
 
 13478-2 
 
 93-60 
 
 24 7 
 
 7 10 
 
 6939-7 
 
 48-19 
 
 34 6f 
 
 11 
 
 13684-8 
 
 95-03 
 
 24 8 
 
 7 10J 
 
 6976-7 
 
 48-45 
 
 34 9f 
 
 11 1 
 
 13892-9 
 
 96-48 
 
 24 8 
 
 7 10 
 
 7013-8 
 
 48-71 
 
 35 0| 
 
 11 2 
 
 14142-6 
 
 97-93 
 
 24 9$ 
 
 7 10| 
 
 7050-9 
 
 48-96 
 
 35 4^ 
 
 11 3 
 
 14313-9 
 
 99-40 
 
 24 10| 
 
 7 11 
 
 7088-2 
 
 49-22 
 
 35 7} 
 
 11 4 
 
 14526-8 
 
 100-88 
 
 24 11| 
 
 7 11} 
 
 7125-5 
 
 49-48 
 
 35 lOf 
 
 11 5 
 
 14741-2 
 
 102-37 
 
 25 
 
 7 ll| 
 
 7163-0 
 
 49-74 
 
 36 H 
 
 11 6 
 
 14S57-2 
 
 103-87 
 
 25 01 
 
 7 llf 
 
 7200-5 
 
 50-00 
 
 36 4 
 
 11 7 
 
 15174-7 
 
 105-38 
 
 
 
 
 
 36 7f 
 
 11 8 
 
 15393-8 
 
 106-90 
 
 25 1 
 
 8 
 
 7238-2 
 
 50-26 
 
 36 10| 
 
 11 9 
 
 15614-5 
 
 108-43 
 
 25 2| 
 
 8 OL 
 
 7275-9 
 
 50-53 
 
 37 2 
 
 11 10 
 
 15836-8 
 
 109-98 
 
 25 3 
 
 8 0* 
 
 7313-8 
 
 50-79 
 
 37 5J 
 
 11 11 
 
 16060-6 
 
 111-53 
 
 25 3 
 
 8 Of 
 
 7351-7 
 
 51-05 
 
 
 
 
 
 25 4| 
 
 8 1 
 
 7389-8 
 
 51-32 
 
 37 8| 
 
 12 
 
 16286-0 
 
 113-10 
 
 25 5J 
 
 8 1} 
 
 7427-9 
 
 51-58 
 
 37 Hi 
 
 12 1 
 
 16513-0 
 
 114-67 
 
 25 6i 
 
 8 1 
 
 7466-2 
 
 51-85 
 
 38 2f 
 
 12 2 
 
 16741-6 
 
 116-26 
 
 25 7 
 
 8 H 
 
 7504-5 
 
 52-11 
 
 38 5f 
 
 12 3 
 
 16971-7 
 
 117-86 
 
 
 
 
 
 38 8| 
 
 12 4 
 
 17203-4 
 
 119-47 
 
 25 7 
 
 8 2 
 
 7542-9 
 
 52-38 
 
 39 
 
 12 5 
 
 17436-7 
 
 121-09 
 
 25 8$ 
 
 8 21- 
 
 7o81'5 
 
 52-65 
 
 39 3J- 
 
 12 6 
 
 17671-5 
 
 122-72 
 
 25 9-$ 
 
 8 2| 
 
 7620-1 
 
 52-92 
 
 39 6| 
 
 12 7 
 
 17907-9 
 
 124-36 
 
 25 10 
 
 8 2| 
 
 7658-8 
 
 53-19 
 
 39 9* 
 
 12 8 
 
 18145-9 
 
 126-01 
 
 25 11 
 
 8 3 
 
 7697-7 
 
 53-46 
 
 40 Of 
 
 12 9 
 
 18385-4 
 
 127-68 
 
 25 11| 
 
 8 3| 
 
 7736-6 
 
 53-73 
 
 40 3f 
 
 12 10 
 
 18626-6 
 
 129-35 
 
 26 
 
 8 3| 
 
 7775-6 
 
 54-00 
 
 40 6| 
 
 12 11 
 
 18869-2 
 
 131-04 
 
 26 H 
 
 8 3f 
 
 7814-7 
 
 54-27 
 
 
 
 
 
816 
 
 APPENDIX. 
 
 TABLE OF DIAMETERS, CIRCUMFERENCES, AND AREAS OF CIRCLES. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 1 
 
 3-1416 
 
 0-7854 
 
 51 
 
 160-22 
 
 2042-82 
 
 101 
 
 317-30 
 
 8011-85 
 
 2 
 
 6-28 
 
 3-14 
 
 52 
 
 163-36 
 
 2123-72 
 
 102 
 
 320-41 
 
 .8171-28 
 
 3 
 
 9-42 
 
 7-07 
 
 53 
 
 166-50 
 
 2206-18 
 
 103 
 
 323-58 
 
 8332-29 
 
 4 
 
 12-57 
 
 12-57 
 
 54 
 
 169-65 
 
 2290-22 
 
 104 
 
 326-73 
 
 8494-87 
 
 5 
 
 15-71 
 
 19-63 
 
 55 
 
 172-79 
 
 2375-83 
 
 105 
 
 329-87 
 
 8659-01 
 
 6 
 
 18-85 
 
 28-27 
 
 56 
 
 175-93 
 
 2463-01 
 
 106 
 
 333-01 
 
 8824-73 
 
 7 
 
 21-99 
 
 38-48 
 
 57 
 
 179-07 
 
 2551-76 
 
 107 
 
 336-15 
 
 8992-02 
 
 8 
 
 25-13 
 
 50-27 
 
 58 
 
 182-21 
 
 2642-08 
 
 108 
 
 339-29 
 
 9160-88 
 
 9 
 
 28-27 
 
 63-62 
 
 59 
 
 185-35 
 
 2733-97 
 
 109 
 
 342-43 
 
 9331-32 
 
 10 
 
 31-42 
 
 78-54 
 
 60 
 
 188-50 
 
 2827-43 
 
 110 
 
 345-57 
 
 9503-32 
 
 11 
 
 34-56 
 
 95-03 
 
 61 
 
 191-64 
 
 2922-47 
 
 111 
 
 348-72 
 
 9676-89 
 
 12 
 
 37-70 
 
 113-10 
 
 62 
 
 194-78 
 
 3019-07 
 
 112 
 
 351-86 
 
 9852-03 
 
 13 
 
 40-84 
 
 132-73 
 
 63 
 
 197-92 
 
 3117-25 
 
 113 
 
 355-00 
 
 10028-75 
 
 14 
 
 43-98 
 
 153-94 
 
 64 
 
 201-06 
 
 3216-99 
 
 114 
 
 358-14 
 
 10207-03 
 
 15 
 
 47-12 
 
 176-71 
 
 65 
 
 204-20 
 
 3318-31 
 
 115 
 
 361-28 
 
 10386-89 
 
 16 
 
 50-26 
 
 201-06 
 
 66 
 
 207-34 
 
 3421-19 
 
 116 
 
 364-42 
 
 10568-32 
 
 17 
 
 53-41 
 
 226-98 
 
 67 
 
 210-49 
 
 3525-65 
 
 117 
 
 367-57 
 
 10751-32 
 
 18 
 
 56-55 
 
 254-47 
 
 68 
 
 213-63 
 
 3631-68 
 
 118 
 
 370-71 
 
 10935-88 
 
 19 
 
 59-69 
 
 283-53 
 
 69 
 
 216-77 
 
 3739-28 
 
 119 
 
 373-85 
 
 11122-02 
 
 20 
 
 62-83 
 
 314-16 
 
 70 
 
 219-91 
 
 3848-45 
 
 120 
 
 376-99 
 
 11309-73 
 
 21 
 
 65-97 
 
 346-36 
 
 71 
 
 223-05 
 
 3959-19 
 
 121 
 
 380-13 
 
 11499-01 
 
 22 
 
 69-11 
 
 380-13 
 
 72 
 
 226-19 
 
 4071-50 
 
 122 
 
 383-27 
 
 11689-87 
 
 23 
 
 72-26 
 
 415-48 
 
 73 
 
 229-34 
 
 4185-39 
 
 123 
 
 386-42 
 
 11882-29 
 
 24 
 
 75-40 
 
 452-39 
 
 74 
 
 232-48 
 
 4300-84 
 
 124 
 
 389-56 
 
 12076-28 
 
 25 
 
 78-54 
 
 490-87 
 
 75 
 
 235-62 
 
 4417-86 
 
 125 
 
 392-70 
 
 12271-85 
 
 26 
 
 81-68 
 
 530-93 
 
 76 
 
 238-76 
 
 4536-46 
 
 126 
 
 395-84 
 
 12468-98 
 
 27 
 
 84-82 
 
 572-56 
 
 77 
 
 241-90 
 
 4656-63 
 
 127 
 
 398-98 
 
 12667-69 
 
 28 
 
 87-96 
 
 615-75 
 
 78 
 
 245-04 
 
 4778-36 
 
 128 
 
 402-12 
 
 12867-96 
 
 29 
 
 91-11 
 
 660-52 
 
 79 
 
 248-19 
 
 4901-67 
 
 129 
 
 405-26 
 
 13069-81 
 
 30 
 
 94-25 
 
 706-86 
 
 80 
 
 251-33 
 
 5026-55 
 
 130 
 
 408-41 
 
 13273-23 
 
 31 
 
 97-39 
 
 754-77 
 
 81 
 
 254-47 
 
 5153-00 
 
 131 
 
 411-55 
 
 13478-22 
 
 32 
 
 100-53 
 
 804-25 
 
 82 
 
 257-61 
 
 5281-02 
 
 132 
 
 414-69 
 
 13684-78 
 
 33 
 
 103-67 
 
 855-30 
 
 83 
 
 260-75 
 
 5410-61 
 
 133 
 
 417-83 
 
 13892-91 
 
 34 
 
 106 -Bl 
 
 907-92 
 
 84 
 
 263-89 
 
 5541-77 
 
 134 
 
 420-97 
 
 14102-61 
 
 35 
 
 109-96 
 
 962-11 
 
 85 
 
 267-03 
 
 5674-50 
 
 135 
 
 424-11 
 
 14313-88 
 
 36 
 
 113-10 
 
 1017-88 
 
 86 
 
 270-18 
 
 5808-80 
 
 136 
 
 427-26 
 
 14526-72 
 
 37 
 
 116-24 
 
 1075-21 
 
 87 
 
 273-32 
 
 5944-68 
 
 137 
 
 430-40 
 
 14741-14 
 
 38 
 
 119-38 
 
 1134-11 
 
 88 
 
 276-46 
 
 6082-12 
 
 , 138 
 
 433-54 
 
 14957-12 
 
 39 
 
 122-52 
 
 1194-59 
 
 89 
 
 279-60 
 
 6221-14 
 
 139 
 
 436-68 
 
 15174-68 
 
 40 
 
 125-66 
 
 1256-64 
 
 90 
 
 282-74 
 
 6361-73 
 
 140 
 
 439-82 
 
 15393-80 
 
 41 
 
 128-80 
 
 1320-25 
 
 91 
 
 285-88 
 
 6503-88 
 
 141 
 
 442-96 
 
 15614-50 
 
 42 
 
 131-95 
 
 1385-44 
 
 92 
 
 289-03 
 
 6647-61 
 
 142 
 
 446-11 
 
 15836-77 
 
 43 
 
 135-09 
 
 1452-20 
 
 93 
 
 292-17 
 
 6792-91 
 
 143 
 
 449-25 
 
 16060-61 
 
 44 
 
 138-23 
 
 1520-53 
 
 94 
 
 295-31 
 
 6939-78 
 
 144 
 
 452-39 
 
 16286-02 
 
 45 
 
 141-37 
 
 1590-43 
 
 95 
 
 298-45 
 
 7088-22 
 
 145 
 
 455-53 
 
 16513-00 
 
 46 
 
 144-51 
 
 1661-90 
 
 96 
 
 301-59 
 
 7238-23 
 
 146 
 
 458-67 
 
 16741-55 
 
 47 
 
 147-65 
 
 1734-94 
 
 97 
 
 304-73 
 
 7389-81 
 
 147 
 
 461-81 
 
 16971-67 
 
 48 
 
 150-80 
 
 1809-56 
 
 98 
 
 307-88 
 
 7542-96 
 
 148 
 
 464-96 
 
 17203-36 
 
 49 
 
 153-94 
 
 1885-74 
 
 99 
 
 311-02 
 
 7697-69 
 
 149 
 
 468-10 
 
 17436-62 
 
 50 
 
 157-08 
 
 19B3-50 
 
 100 
 
 314-16 
 
 7853-98 
 
 150 
 
 471-24 
 
 17671-46 
 
APPENDIX. 817 
 
 TABLE OF DIAMETERS, CIRCUMFERENCES, AND AREAS OF CIRCLES. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 151 
 
 474-38 
 
 17907-86 
 
 201 
 
 631-46 
 
 31730-87 
 
 251 
 
 788-54 
 
 49480-87 
 
 152 
 
 477-52 
 
 18145-84 
 
 202 
 
 634-60 
 
 32047-39 
 
 252 
 
 791-68 
 
 49875-92 
 
 153 
 
 480-66 
 
 18385-39 
 
 203 
 
 637-74 
 
 32365-47 
 
 253 
 
 794-82 
 
 50272-55 
 
 154 
 
 483-80 
 
 18626-50 
 
 204 
 
 640-88 
 
 32685-13 
 
 254 
 
 797-96 
 
 50670-75 
 
 155 
 
 486-95 
 
 18869-19 
 
 205 
 
 644-03 
 
 33006-36 
 
 255 
 
 801-11 
 
 51070-52 
 
 156 
 
 490-09 
 
 19113-45 
 
 206 
 
 647-17 
 
 33329-16 
 
 256 
 
 804-25 
 
 51471-86 
 
 157 
 
 493-23 
 
 19359-28 
 
 207 
 
 650-31 
 
 33653-53 
 
 257 
 
 807-39 
 
 51874-76 
 
 158 
 
 496-37 
 
 19606-68 
 
 208 
 
 653-45 
 
 33979-47 
 
 258 
 
 810-53 
 
 52279-24 
 
 159 
 
 499-51 
 
 19855-65 
 
 209 
 
 656-59 
 
 34306-98 
 
 259 
 
 813-67 
 
 52685-29 
 
 160 
 
 502-65 
 
 20106-19 
 
 210 
 
 659-73 
 
 34636-06 
 
 260 
 
 816-81 
 
 53092-96 
 
 161 
 
 505-80 
 
 20358-34 
 
 211 
 
 662-88 
 
 34966-71 
 
 261 
 
 819-96 
 
 53502-11 
 
 162 
 
 508-94 
 
 20611-99 
 
 212 
 
 666-02 
 
 35298-94 
 
 262 
 
 823-10 
 
 53912-87 
 
 163 
 
 512-08 
 
 20867-24 
 
 213 
 
 669-16 
 
 35632-73 
 
 263 
 
 826-24 
 
 54325-21 
 
 164 
 
 515-22 
 
 21124-07 
 
 214 
 
 672-30 
 
 35968-09 
 
 264 
 
 829-38 
 
 54739-11 
 
 165 
 
 518-36 
 
 21382-46 
 
 215 
 
 675-44 
 
 36305-03 
 
 265 
 
 832-52 
 
 55154-59 
 
 166 
 
 521-50 
 
 21642-43 
 
 216 
 
 678-58 
 
 36643-61 
 
 266 
 
 835-66 
 
 55571-63 
 
 167 
 
 524-65 
 
 21903-97 
 
 217 
 
 681-73 
 
 36983-61 
 
 267 
 
 838-80 
 
 55990-25 
 
 168 
 
 527-79 
 
 22167-08 
 
 218 
 
 684-87 
 
 37325-26 
 
 268 
 
 841-95 
 
 56410-44 
 
 169 
 
 530-93 
 
 22431-76 
 
 219 
 
 688-01 
 
 37668-48 
 
 269 
 
 845-09 
 
 56832-20 
 
 170 
 
 534-07 
 
 22698-01 
 
 220 
 
 691-15 
 
 38013-27 
 
 270 
 
 848-23 
 
 57255-53 
 
 171 
 
 537-21 
 
 22965-83 
 
 221 
 
 694-29 
 
 38359-63 
 
 271 
 
 851-37 
 
 57680-43 
 
 172 
 
 540-35 
 
 23235-22 
 
 222 
 
 697-43 
 
 38707-56 
 
 272 
 
 854-51 
 
 58106-90 
 
 173 
 
 543-50 
 
 23506-18 
 
 223 
 
 700-57 
 
 39057-07 
 
 273 
 
 857-65 
 
 58534-94 
 
 174 
 
 546-64 
 
 23778-71 
 
 224 
 
 703-72 
 
 39408-14 
 
 274 
 
 860-80 
 
 58964-55 
 
 175 
 
 549-78 
 
 24052-82 
 
 225 
 
 706-86 
 
 39760-78 
 
 275 
 
 863-94 
 
 59395-74 
 
 176 
 
 552-92 
 
 24328-49 
 
 226 
 
 710-00 
 
 40115-00 
 
 276 
 
 867-08 
 
 59828-49 
 
 177 
 
 556-06 
 
 24605-79 
 
 227 
 
 713-14 
 
 40470-78 
 
 277 
 
 870-22 
 
 60262-82 
 
 178 
 
 559-20 
 
 24884-56 
 
 228 
 
 716-28 
 
 40828-14 
 
 278 
 
 873-36 
 
 60698-72 
 
 179 
 
 562-34 
 
 25164-94 
 
 229 
 
 719-42 
 
 41187-07 
 
 279 
 
 876-50 
 
 61136-18 
 
 180 
 
 565-49 
 
 25446-90 
 
 230 
 
 722-57 
 
 41547-56 
 
 280 
 
 879-65 
 
 61575-22 
 
 181 
 
 568-63 
 
 25730-43 
 
 231 
 
 725-71 
 
 41909-63 
 
 281 
 
 882-79 
 
 62015-82 
 
 182 
 
 571-77 
 
 26015-53 
 
 232 
 
 728-85 
 
 42273-27 
 
 282 
 
 885-93 
 
 62458-00 
 
 183 
 
 574-91 
 
 26302-20 
 
 233 
 
 731-99 
 
 42638-48 
 
 283 
 
 889-07 
 
 62901-75 
 
 184 
 
 578-05 
 
 26590-44 
 
 234 
 
 735-13 
 
 43005-26 
 
 284 
 
 892-21 
 
 63347-07 
 
 185 
 
 581-19 
 
 26880-25 
 
 235 
 
 738-27 
 
 43373-61 
 
 285 
 
 895-35 
 
 63793-97 
 
 186 
 
 584-34 
 
 27171-63 
 
 236 
 
 741-42 
 
 43743-54 
 
 286 
 
 898-49 
 
 64242-43 
 
 187 
 
 587-48 
 
 27464-59 
 
 237 
 
 744-56 
 
 44115-03 
 
 287 
 
 901-64 
 
 64692-46 
 
 188 
 
 590-62 
 
 27759-11 
 
 238 
 
 747-70 
 
 44488-09 
 
 288 
 
 904-78 
 
 65144-07 
 
 189 
 
 593-76 
 
 28055-21 
 
 239 
 
 750-84 
 
 44862-73 
 
 289 
 
 907-92 
 
 65597-24 
 
 190 
 
 596-90 
 
 28352-87 
 
 240 
 
 753-98 
 
 45238-93 
 
 290 
 
 911-06 
 
 66051-99 
 
 191 
 
 600-04 
 
 28652-11 
 
 241 
 
 757-12 
 
 45616-71 
 
 291 
 
 914-20 
 
 66508-30 
 
 192 
 
 603-19 
 
 28952-92 
 
 242 
 
 760-26 
 
 45996-06 
 
 292 
 
 917-34 
 
 66966-19 
 
 193 
 
 606-33 
 
 29255-30 
 
 243 
 
 763-41 
 
 46376-98 
 
 293 
 
 920-49 
 
 67425-65 
 
 194 
 
 609-47 
 
 29559-26 
 
 244 
 
 766-55 
 
 46759-47 
 
 294 
 
 923-63 
 
 67886-68 
 
 195 
 
 612-61 
 
 29864-77 
 
 245 
 
 769-69 
 
 47143-52 
 
 295 
 
 926-77 
 
 68349-28 
 
 196 
 
 615-75 
 
 30171-86 
 
 246" 
 
 772-83 
 
 47529-16 
 
 296 
 
 929-91 
 
 68813-45 
 
 197 
 
 618-89 
 
 30480-52 
 
 247 
 
 775-97 
 
 47916-36 
 
 297 
 
 933-05 
 
 69279-19 
 
 198 
 
 622-03 
 
 30790-75 
 
 248 
 
 779-11 
 
 48305-13 
 
 298 
 
 936-19 
 
 69746-50 
 
 199 
 
 625-18 
 
 31102-55 
 
 249 
 
 782-26 
 
 48695-47 
 
 299 
 
 939-34 
 
 70215-38 
 
 200 
 
 628-32 
 
 31415-93 
 
 250 
 
 785-40 
 
 49087-39 
 
 300 
 
 942-48 
 
 70685-83 
 
 53 
 
818 APPENDIX. 
 
 TABLE OF DIAMETERS, CIRCUMFERENCES, AND AREAS OF CIRCLES. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 301 
 
 945-62 
 
 71157-86 
 
 351 
 
 1102-70 
 
 96761-84 
 
 401 
 
 1259-78 
 
 126292-81 
 
 302 
 
 948-76 
 
 71631-45 
 
 352 
 
 1105-84 
 
 97314-76 
 
 402 
 
 1262-92 
 
 126923-48 
 
 303 
 
 951-90 
 
 72106-62 
 
 353 
 
 1108-98 
 
 97867-68 
 
 403 
 
 1266-06 
 
 127553-73 
 
 304 
 
 955-04 
 
 72583-36 
 
 354 
 
 1112-12 
 
 98422-96 
 
 404 
 
 1269-20 
 
 128189-55 
 
 305 
 
 958-19 
 
 73061-66 
 
 355 
 
 1115-26 
 
 98979-80 
 
 405 
 
 1272-34 
 
 128824-93 
 
 306 
 
 961-33 
 
 73541-54 
 
 356 
 
 1118-41 
 
 99538-22 
 
 406 
 
 1275-49 
 
 129461-89 
 
 307 
 
 964-47 
 
 74022-99 
 
 357 
 
 1121-55 
 
 100098-21 
 
 407 
 
 1278-63 
 
 130100-42 
 
 308 
 
 967-61 
 
 74506-01 
 
 358 
 
 1124-69 
 
 100659-27 
 
 408 
 
 1281-77 
 
 130740-52 
 
 309 
 
 970-75 
 
 74990-60 
 
 359 
 
 1127-83 
 
 101222-90 
 
 409 
 
 1284-91 
 
 131382-19 
 
 310 
 
 973-89 
 
 75476-76 
 
 360 
 
 1130-97 
 
 101787-60 
 
 410 
 
 1288-05 
 
 132025-43 
 
 311 
 
 977-03 
 
 75964-50 
 
 361 
 
 1134-11 
 
 102353-87 
 
 411 
 
 1291-19 
 
 132670-24 
 
 312 
 
 980-18 
 
 76453-80 
 
 362 
 
 1137-26 
 
 102921-72 
 
 412 
 
 1294-34 
 
 133316-63 
 
 313 
 
 983-32 
 
 76944-67 
 
 363 
 
 1140-40 
 
 103491 13 
 
 413 
 
 1297-48 
 
 133964-58 
 
 314 
 
 986-46 
 
 77437-12 
 
 364 
 
 1143-54 
 
 104062-12 
 
 414 
 
 1300-62 
 
 134614-10 
 
 315 
 
 989-60 
 
 77931-13 
 
 365 
 
 1146-68 
 
 104634-67 
 
 415 
 
 1303-76 
 
 135265-20 
 
 316 
 
 992-74 
 
 78426-72 
 
 366 
 
 1149-82 
 
 105208-80 
 
 416 
 
 1306-90 
 
 135917-86 
 
 317 
 
 995-88 
 
 78923-88 
 
 367 
 
 1152-96 
 
 105784-49 
 
 417 
 
 1310-04 
 
 136572-10 
 
 318 
 
 999-03 
 
 79422-60 
 
 368 
 
 1156-11 
 
 106361-76 
 
 418 
 
 1313-19 
 
 137227-91 
 
 319 
 
 1002-17 
 
 79922-90 
 
 369 
 
 1159-25 
 
 106940-60 
 
 419 
 
 1316-33 
 
 137885-29 
 
 320 
 
 1005-31 
 
 80424-77 
 
 370 
 
 1162-39 
 
 107521-01 
 
 420 
 
 1319-47 
 
 138544-24 
 
 321 
 
 1008-45 
 
 80928-21 
 
 371 
 
 1165-53 
 
 108102-99 
 
 421 
 
 1322-61 
 
 139204-70 
 
 322 
 
 1011-59 
 
 81433-22 
 
 372 
 
 1168-67 
 
 108686-54 
 
 422 
 
 1325-75 
 
 139866-85 
 
 323 
 
 1014-73 
 
 81939-80 
 
 373 
 
 1171-81 
 
 109271-66 
 
 423 
 
 1328-89 
 
 140530-51 
 
 324 
 
 1017-88 
 
 82447-96 
 
 374 
 
 1174-96 
 
 109858-35 
 
 424 
 
 1332-03 
 
 141195-74 
 
 325 
 
 1021-02 
 
 82957-68 
 
 375 
 
 1178-10 
 
 110446-62 
 
 425 
 
 1335-18 
 
 141862-54 
 
 326 
 
 1024-16 
 
 83468-98 
 
 376 
 
 1181-24 
 
 111036-45 
 
 426 
 
 1338-32 
 
 142530-92 
 
 327 
 
 1027-30 
 
 83981-84 
 
 377 
 
 1184-38 
 
 111627-86 
 
 427 
 
 1341-46 
 
 143200-86 
 
 328 
 
 1030-44 
 
 84496-28 
 
 378 
 
 1187-52 
 
 112220-83 
 
 428 
 
 1344-60 
 
 143872-38 
 
 329 
 
 1033-58 
 
 85012-28 
 
 379 
 
 1190-66 
 
 112815-38 
 
 429 
 
 1347-74 
 
 144545-46 
 
 330 
 
 1036-73 
 
 85529-86 
 
 380 
 
 1193-80 
 
 113411-49 
 
 430 
 
 1350-88 
 
 145220-12 
 
 331 
 
 1039-87 
 
 86049-01 
 
 381 
 
 1196-95 
 
 114009-18 
 
 431 
 
 1354-03 
 
 145896-35 
 
 332 
 
 1043-01 
 
 86569-73 
 
 382 
 
 1200-09 
 
 114608-44 
 
 432 
 
 1357-17 
 
 146574-15 
 
 333 
 
 1046-15 
 
 87092-02 
 
 383 
 
 1203-23 
 
 115209-27 
 
 433 
 
 1360-31 
 
 147253-52 
 
 334 
 
 1049-29 
 
 87615-88 
 
 384 
 
 1206-37 
 
 115811-67 
 
 434 
 
 1363-45 
 
 147934-46 
 
 335 
 
 1052-43 
 
 88141-31 
 
 385 
 
 1209-51 
 
 116415-64 
 
 435 
 
 1366-59 
 
 148616-97 
 
 336 
 
 1055-57 
 
 88668-31 
 
 386 
 
 1212-65 
 
 117021-18 
 
 436 
 
 1369-73 
 
 149301-05 
 
 337 
 
 1058-72 
 
 89196-88 
 
 387 
 
 1215-80 
 
 117628-30 
 
 437 
 
 1372-88 
 
 149986-70 
 
 338 
 
 1061-86 
 
 89727-03 
 
 388 
 
 1218-94 
 
 118236-98 
 
 438 
 
 1376-02 
 
 150673-93 
 
 339 
 
 1065-00 
 
 90258-74 
 
 389 
 
 1222-08 
 
 118847-24 
 
 439 
 
 1379-16 
 
 151362-72 
 
 340 
 
 1068-14 
 
 90792-03 
 
 390 
 
 1225-22 
 
 119459-06 
 
 440 
 
 1382-30 
 
 152053-08 
 
 341 
 
 1071-28 
 
 91326-88 
 
 391 
 
 1228-36 
 
 120072-46 
 
 441 
 
 1385-44 
 
 152745-02 
 
 342 
 
 1074-42 
 
 91863-31 
 
 392 
 
 1231-50 
 
 120687-42 
 
 442 
 
 1388-58 
 
 153438-53 
 
 343 
 
 1077-57 
 
 92401-31 
 
 393 
 
 1234-65 
 
 121303-96 
 
 443 
 
 1391-73 
 
 154133-60 
 
 344 
 
 1080-71 
 
 92940-88 
 
 394 
 
 1237-79 
 
 121922-07 
 
 444 
 
 1394-87 
 
 154830-25 
 
 345 
 
 1083-85 
 
 93482-02 
 
 395 
 
 1240-93 
 
 122541-75 
 
 445 
 
 1398-01 
 
 155528-47 
 
 346 
 
 1086-99 
 
 94024-73 
 
 396 
 
 1244-07 
 
 123163-00 
 
 446 
 
 1401-15 
 
 156228-26 
 
 347 
 
 1090-13 
 
 94569-01 
 
 397 
 
 1247-21 
 
 123785-82 
 
 447 
 
 1404-29 
 
 156929-62 
 
 348 
 
 1093-27 
 
 95114-86 
 
 398 
 
 1250-35 
 
 124410-21 
 
 448 
 
 1407-43 
 
 157632-55 
 
 349 
 
 1096-42 
 
 95662-28 
 
 399 
 
 1253-49 
 
 125036-17 
 
 449 
 
 1410 57 
 
 158337-06 
 
 350 
 
 1099-56 
 
 96211-28 
 
 400 
 
 1256-64 
 
 125663-71 
 
 450 
 
 1413-72 
 
 159043-13 
 
 
 
 
 
 
 
 
 
 
APPENDIX. 
 
 TABLE OP DIAMETERS, CIRCUMFERENCES, AND AREAS OF CIRCLES. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 Diam- 
 eter. 
 
 Circum- 
 ference. 
 
 Circular 
 area. 
 
 451 
 
 1416-86 
 
 159750-77 
 
 501 
 
 1573-94 
 
 197135-72 
 
 551 
 
 1731-02 
 
 238447-67 
 
 452 
 
 1420-00 
 
 160459-99 
 
 502 
 
 1577-08 
 
 197923-48 
 
 552 
 
 1734-16 
 
 239313-96 
 
 453 
 
 1423-14 
 
 161170-77 
 
 503 
 
 1580-22 
 
 198712-80 
 
 553 
 
 1737-30 
 
 240181-83 
 
 454 
 
 1426-28 
 
 161883 13 
 
 504 
 
 1583-36 
 
 199503-70 
 
 554 
 
 1740-44 
 
 241051-26 
 
 455 
 
 1429-42 
 
 162597-06 
 
 505 
 
 1586-50 
 
 200296-17 
 
 555 
 
 1743-58 
 
 241922-27 
 
 456 
 
 1432-57 
 
 163312-55 
 
 506 
 
 1589-65 
 
 201090-20 
 
 556 
 
 1746-73 
 
 242794-85 
 
 457 
 
 1435-71 
 
 164029-62 
 
 507 
 
 1592-79 
 
 201885-81 
 
 557 
 
 1749-87 
 
 243668-99 
 
 458 
 
 1438-85 
 
 164748-26 
 
 508 
 
 1595-93 
 
 202682-99 
 
 558 
 
 1753-00 
 
 244544-61 
 
 459 
 
 1441-99 
 
 165468-47 
 
 509 
 
 1599-07 
 
 203481-74 
 
 559 
 
 1756-15 
 
 245422-00 
 
 460 
 
 1445-13 
 
 166190-25 
 
 510 
 
 1602-21 
 
 204282-06 
 
 560 
 
 1759-29 
 
 246800-86 
 
 461 
 
 1448-27 
 
 166913-60 
 
 511 
 
 1605-35 
 
 205083-95 
 
 561 
 
 1762-43 
 
 247181-30 
 
 462 
 
 1451-42 
 
 167638-53 
 
 512 
 
 1608-49 
 
 205887-42 
 
 562 
 
 1765-57 
 
 248062-30 
 
 4(33 
 
 1454-56 
 
 168365-02 
 
 513 
 
 1611-64 
 
 206692-45 
 
 563 
 
 1768-72 
 
 248946-87 
 
 464 
 
 1457-70 
 
 169093-08 
 
 514 
 
 1614-78 
 
 207499-05 
 
 564 
 
 1771-86 
 
 249832-01 
 
 465 
 
 1460-84 
 
 169822-72 
 
 515 
 
 1617-92 
 
 208307-23 
 
 565 
 
 1775-00 
 
 250718-73 
 
 466 
 
 .1463-98 
 
 170553-92 
 
 516 
 
 1621-06 
 
 209116-97 
 
 566 
 
 1778-14 
 
 251607-01 
 
 467 
 
 1467-12 
 
 171286-70 
 
 517 
 
 1624-20 
 
 209928-29 
 
 567 
 
 1781-28 
 
 252496-87 
 
 468 
 
 1470-26 
 
 172021-05 
 
 518 
 
 1627-34 
 
 210741-18 
 
 568 
 
 1784-42 
 
 253388-30 
 
 469 
 
 1473-41 
 
 172756-97 
 
 519 
 
 1630-49 
 
 211555-63 
 
 569 
 
 1787-57 
 
 254281-30 
 
 470 
 
 1476-55 
 
 173494-45 
 
 520 
 
 1633-63 
 
 212371-66 
 
 570 
 
 1790-71 
 
 255175-86 
 
 471 
 
 1479-69 
 
 174233-51 
 
 521 
 
 1636-77 
 
 213189-26 
 
 571 
 
 1793-85 
 
 256072-00 
 
 472 
 
 1482-83 
 
 174974-14 
 
 522 
 
 1639-91 
 
 214008-43 
 
 572 
 
 1796-99 
 
 256969-71 
 
 473 
 
 1485-97 
 
 175716-35 
 
 523 
 
 1643-05 
 
 214829-17 
 
 573 
 
 1800-13 
 
 257868-99 
 
 474 
 
 1489-11 
 
 176460-12 
 
 524 
 
 1646-19 
 
 215651-49 
 
 574 
 
 1803-27 
 
 258769-85 
 
 475 
 
 1492-26 
 
 177205-46 
 
 525 
 
 1649-34 
 
 216475-37 
 
 575 
 
 1806-42 
 
 259672-27 
 
 476 
 
 1495-40 
 
 177952-37 
 
 526 
 
 1652-48 
 
 217300-82 
 
 576 
 
 1809-56 
 
 260576-26 
 
 477 
 
 1498-54 
 
 178700-86 
 
 527 
 
 1655-62 
 
 218127-85 
 
 577 
 
 1812-70 
 
 261481-83 
 
 478 
 
 1501-68 
 
 179450-91 
 
 528 
 
 1658-76 
 
 218956-44 
 
 578 
 
 1815-84 
 
 262388-96 
 
 479 
 
 1504-82 
 
 180202-54 
 
 529 
 
 1661-90 
 
 219786-61 
 
 579 
 
 1818-98 
 
 263297-67 
 
 480 
 
 1507-96 
 
 180955-74 
 
 530 
 
 1665-04 
 
 220618-32 
 
 580 
 
 1822-12 
 
 264207-94 
 
 481 
 
 1511-11 
 
 181710-50 
 
 531 
 
 1668-19 
 
 221451-65 
 
 581 
 
 1825-26 
 
 265119-79 
 
 482 
 
 1514-25 
 
 182466-84 
 
 532 
 
 1671-33 
 
 222286-53 
 
 582 
 
 1828-41 
 
 266033-21 
 
 483 
 
 1517-39 
 
 183224-75 
 
 533 
 
 1674-47 
 
 223122-98 
 
 583 
 
 1831-55 
 
 266948-20 
 
 484 
 
 1520-53 
 
 183984-23 
 
 534 
 
 1677-61 
 
 223961-00 
 
 584 
 
 1834-69 
 
 267864-76 
 
 485 
 
 1523-67 
 
 184745-28 
 
 535 
 
 1680-75 
 
 224800-59 
 
 585 
 
 1837-83 
 
 268782-80 
 
 486 
 
 1526-81 
 
 185507-90 
 
 536 
 
 1683-89 
 
 225641-75 
 
 586 
 
 1840-97 
 
 269702-59 
 
 487 
 
 1529-96 
 
 186272-10 
 
 537 
 
 1687-04 
 
 226484-48 
 
 587 
 
 1844-11 
 
 270623-86 
 
 488 
 
 1533-10 
 
 187037-86 
 
 538 
 
 1690-18 
 
 227328-77 
 
 588 
 
 1847-26 
 
 271546-70 
 
 489 
 
 1536-24 
 
 187805-19 
 
 539 
 
 1693-32 
 
 228174-66 
 
 589 
 
 1850-40 
 
 272471-12 
 
 490 
 
 1539-38 
 
 188574-10 
 
 540 
 
 1696-46 
 
 229022-10 
 
 590 
 
 1853-54 
 
 273397-10 
 
 491 
 
 1542-52 
 
 189344-57 
 
 541 
 
 1699-60 
 
 229871-12 
 
 591 
 
 1856-68 
 
 274324-66 
 
 492 
 
 1545-66 
 
 190116-62 
 
 542 
 
 1702-74 
 
 230721-71 
 
 592 
 
 1859-82 
 
 275253-78 
 
 493 
 
 1548-80 
 
 190890-24 
 
 543 
 
 1705-88 
 
 231573-86 
 
 593 
 
 1862-96 
 
 276184-48 
 
 494 
 
 1551-95 
 
 191665-43 
 
 544 
 
 1709-03 
 
 232427-59 
 
 594 
 
 1866-11 
 
 277116-75 
 
 495 
 
 1555-09 
 
 192442-19 
 
 545 
 
 1712-17 
 
 233282-89 
 
 595 
 
 1869-25 
 
 278050-59 
 
 496 
 
 1558-23 
 
 193220-51 
 
 546 
 
 1715-31 
 
 234139-76 
 
 596 
 
 1872-39 
 
 278985-99 
 
 497 
 
 1561-37 
 
 194000-42 
 
 547 
 
 1718-45 
 
 234998-20 
 
 597 
 
 1875-53 
 
 279922-97 
 
 498 
 
 1564-51 
 
 194781-89 
 
 548 
 
 1721-59 
 
 235858-21 
 
 598 
 
 1878-67 
 
 280861-53 
 
 499 
 
 1567-65 
 
 195564-93 
 
 549 
 
 1724-73 
 
 236719-79 
 
 599 
 
 1881-81 
 
 281801-65 
 
 500 
 
 1570-80 
 
 196349-54 
 
 550 
 
 1727-88 
 
 237582-94 
 
 600 
 
 1884-96 
 
 282743-34 
 
820 
 
 APPENDIX. 
 
 TABLE OF SQUARES, CUBES, SQUAEE AND CUBE ROOTS OF NUMBERS. 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 1 1 
 
 1 
 
 i-ooo 
 
 1-000 
 
 4096 
 
 262144 
 
 64 
 
 8-000 
 
 4-000 
 
 4 
 
 8 
 
 2 
 
 1-414 
 
 1-259 
 
 4225 
 
 274625 
 
 65 
 
 8-062 
 
 4-020 
 
 9 
 
 27 
 
 3 
 
 1-732 
 
 1-442 
 
 4356 
 
 287496 
 
 66 
 
 8-124 
 
 4-041 
 
 16 
 
 64 
 
 4 
 
 2-000 
 
 1-587 
 
 4489 
 
 300763 
 
 67 
 
 8-185 
 
 4-061 
 
 25 
 
 125 
 
 5 
 
 2-236 
 
 1-709 
 
 4624 
 
 314432 
 
 68 
 
 8-246 
 
 4-081 
 
 36 
 
 216 
 
 6 
 
 2-449 
 
 1-817 
 
 4761 
 
 328509 
 
 69 
 
 8-306 
 
 4-101 
 
 49 
 
 343 
 
 7 
 
 2-645 
 
 1-912 
 
 4900 
 
 343000 
 
 70 
 
 8-366 
 
 4-121 
 
 64 
 
 512 
 
 8 
 
 2-828 
 
 2-000 
 
 5041 
 
 357911 
 
 71 
 
 8-426 
 
 4-140 
 
 81 
 
 729 
 
 9 
 
 3-000 
 
 2-080 
 
 5184 
 
 373248 
 
 72 
 
 8-485 
 
 4-160 
 
 100 
 
 1000 
 
 10 
 
 3-162 
 
 2-154 
 
 5329 
 
 389017 
 
 73 
 
 8-544 
 
 4-179 
 
 121 
 
 1331 
 
 11 
 
 3-316 
 
 2-223 
 
 5476 
 
 405224 
 
 74 
 
 8-602 
 
 4-198 
 
 144 
 
 1728 
 
 12 
 
 3-464 
 
 2-289 
 
 5625 
 
 421875 
 
 75 
 
 8-660 
 
 4-217 
 
 169 
 
 2197 
 
 13 
 
 3-605 
 
 2-351 
 
 5776 
 
 438976 
 
 76 
 
 8-717 
 
 4-235 
 
 196 
 
 2744 
 
 14 
 
 3-741 
 
 2-410 
 
 5929 
 
 456533 
 
 77 
 
 8-774 
 
 4-254 
 
 225 
 
 3375 
 
 15 
 
 3-872 
 
 2-466 
 
 6084 
 
 474552 
 
 78 
 
 8-831 
 
 4-272 
 
 256 
 
 4096 
 
 16 
 
 4-000 
 
 2-519 
 
 6241 
 
 493039 
 
 79 
 
 8-888 
 
 4-290 
 
 289 
 
 4913 
 
 17 
 
 4-123 
 
 2-571 
 
 6400 
 
 512000 
 
 80 
 
 8-944 
 
 4-308 
 
 324 
 
 5832 
 
 18 
 
 4-242 
 
 2-620 
 
 6561 
 
 531441 
 
 81 
 
 9-000 
 
 4-326 
 
 361 
 
 6859 
 
 19 
 
 4-358 
 
 2-668 
 
 6724 
 
 551368 
 
 82 
 
 9-055 
 
 4-344 
 
 400 
 
 8000 
 
 20 
 
 4-472 
 
 2-714 
 
 6889 
 
 571787 
 
 83 
 
 9-110 
 
 4-362 
 
 441 
 
 9261 
 
 21 
 
 4-582 
 
 2-758 
 
 7056 
 
 592704 
 
 84 
 
 9-165 
 
 4-379 
 
 484 
 
 10648 
 
 22 
 
 4-690 
 
 2-802 
 
 7225 
 
 614125 
 
 85 
 
 9-219 
 
 4-396 
 
 529 
 
 12167 
 
 23 
 
 4-795 
 
 2-843 
 
 7396 
 
 636056 
 
 86 
 
 9-273 
 
 4-414 
 
 676 
 
 13824 
 
 24 
 
 4-898 
 
 2-884 
 
 7569 
 
 658503 
 
 87 
 
 9-327 
 
 4-431 
 
 625 
 
 15625 
 
 25 
 
 5-000 
 
 2-924 
 
 7744 
 
 681472 
 
 88 
 
 9-380 
 
 4-447 
 
 676 
 
 17576 
 
 26 
 
 5-099 
 
 2-962 
 
 7921 
 
 704969 
 
 89 
 
 9-433 
 
 4-'464 
 
 729 
 
 19683 
 
 27 
 
 5-196 
 
 3-000 
 
 8100 
 
 729000 
 
 90 
 
 9-486 
 
 4-481 
 
 784 
 
 21952 
 
 28 
 
 5-291 
 
 3-036 
 
 8281 
 
 753571 
 
 91 
 
 9-539 
 
 4497 
 
 841 
 
 24389 
 
 29 
 
 5-385 
 
 3-072 
 
 8464 
 
 778688 
 
 92 
 
 9-591 
 
 4-514 
 
 900 
 
 27000 
 
 30 
 
 5-477 
 
 3-107 
 
 8649 
 
 804357 
 
 93 
 
 9-643 
 
 4'530 
 
 961 
 
 29791 
 
 31 
 
 5-567 
 
 3-141 
 
 8836 
 
 830584 
 
 94 
 
 9-695 
 
 4-546 
 
 1024 
 
 32768 
 
 32 
 
 5-656 
 
 3-174 
 
 9025 
 
 857374- 
 
 95 
 
 9-746 
 
 4-562 
 
 1089 
 
 35937 
 
 33 
 
 5-744 
 
 3-207 
 
 9216 
 
 884736 
 
 96 
 
 9-797 
 
 4-578 
 
 1156 
 
 39304 
 
 34 
 
 5-830 
 
 3-239 
 
 9409 
 
 912673 
 
 97 
 
 9-848 
 
 4-594 
 
 1225 
 
 42875 
 
 35 
 
 5-916 
 
 3-271 
 
 9604 
 
 941192 
 
 98 
 
 9-899 
 
 4-610 
 
 1296 
 
 46656 
 
 36 
 
 6-000 
 
 3-301 
 
 9801 
 
 970299 
 
 99 
 
 9-949 
 
 4-626 
 
 1369 
 
 50653 
 
 37 
 
 6-082 
 
 3-332 
 
 10000 
 
 1000000 
 
 100 
 
 10-000 
 
 4-641 
 
 1444 
 
 54872 
 
 38 
 
 6-164 
 
 3-361 
 
 10201 
 
 1030301 
 
 101 
 
 10-049 
 
 4-657 
 
 1521 
 
 59319 
 
 39 
 
 6-244 
 
 3-391 
 
 10404 
 
 1061208 
 
 102 
 
 10-099 
 
 4-672 
 
 1600 
 
 64000 
 
 40 
 
 6-324 
 
 3-419 
 
 10609 
 
 1092727 
 
 103 
 
 10-148 
 
 4-687 
 
 1681 
 
 68921 
 
 41 
 
 6-403 
 
 3-448 
 
 10816 
 
 1124864 
 
 104 
 
 10-198 
 
 4-702 
 
 1764 
 
 74088 
 
 42 
 
 6-480 
 
 3-476 
 
 11025 
 
 1157625 
 
 105 
 
 10-246 
 
 4-717 
 
 1849 
 
 79507 
 
 43 
 
 6-557 
 
 3-503 
 
 11236 
 
 1191016 
 
 106 
 
 10-295 
 
 4-732 
 
 1936 
 
 85184 
 
 44 
 
 6-633 
 
 3-530 
 
 11449 
 
 1225043 
 
 107 
 
 10-344 
 
 4-747 
 
 2025 
 
 91125 
 
 45 
 
 6-708 
 
 3-556 
 
 11664 
 
 1259712 
 
 108 
 
 10-392 
 
 4-762 
 
 2116 
 
 97336 
 
 46 
 
 6-782 
 
 3-583 
 
 11881 
 
 1295029 
 
 109 
 
 10-440 
 
 4-776 
 
 2209 
 
 103823 
 
 47 
 
 6-855 
 
 3-608 
 
 12100 
 
 1331000 
 
 110 
 
 10-488 
 
 4-791 
 
 2304 
 
 110592 
 
 48 
 
 6-928 
 
 3-634 
 
 12321 
 
 1367631 
 
 111 
 
 10-535 
 
 4-805 
 
 2401 
 
 117649 
 
 49 
 
 7-000 
 
 3-659 
 
 12544 
 
 1404928 
 
 112 
 
 10-583 
 
 4-820 
 
 2500 
 
 125000 
 
 50 
 
 7-071 
 
 3-684 
 
 12769 
 
 1442897 
 
 113 
 
 10-630 
 
 4-834 
 
 2601 
 
 132651 
 
 51 
 
 7-141 
 
 3-708 
 
 12996 
 
 1481544 
 
 114 
 
 10-677 
 
 4-848 
 
 2704 
 
 140608 
 
 52 
 
 7-211 
 
 3-732 
 
 13225 
 
 1520875 
 
 115 
 
 10-723 
 
 4-862 
 
 2809 
 
 148877 
 
 53 
 
 7-280 
 
 3-756 
 
 13456 
 
 1560896 
 
 116 
 
 10-770 
 
 4-876 
 
 2916 
 
 157464 
 
 54 
 
 7-348 
 
 3-779 
 
 13689 
 
 1601613 
 
 117 
 
 10-816 
 
 4-890 
 
 3025 
 
 166375 
 
 55 
 
 7-416 
 
 3-802 
 
 13924 
 
 1643032 
 
 118 
 
 10-862 
 
 4-904 
 
 3136 
 
 175616 
 
 56 
 
 7-4S3 
 
 3-825 
 
 14161 
 
 1685159 
 
 119 
 
 10-908 
 
 4-918 
 
 3249 
 
 185193 
 
 57 
 
 7-549 
 
 3-848 
 
 14400 
 
 1728000 
 
 120 
 
 10-954 
 
 4-932 
 
 3364 
 
 195112 
 
 58 
 
 7-615 
 
 3-870 
 
 14641 
 
 1771561 
 
 121 
 
 11-000 
 
 4-946 
 
 3481 
 
 205379 
 
 59 
 
 7-681 
 
 3-892 
 
 14834 
 
 1815848 
 
 122 
 
 11-045 
 
 4-959 
 
 3600 
 
 216000 
 
 60 
 
 7-745 
 
 3-914 
 
 15129 
 
 1860867 
 
 123 
 
 11-090 
 
 4-973 
 
 3721 
 
 226981 
 
 61 
 
 7-810 
 
 3-930 
 
 15376 
 
 1906624 
 
 124 
 
 11-135 
 
 4-986 
 
 3844 
 
 238328 
 
 62 
 
 7-874 
 
 3-957 
 
 15625 
 
 1953125 
 
 125 
 
 11-180 
 
 5-000 
 
 3969 
 
 250047 
 
 63 
 
 7-937 
 
 3-979 
 
 15876 
 
 2000376 
 
 126 
 
 11-224 
 
 5-013 
 
APPENDIX. 
 
 821 
 
 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS ( Continued). 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 
 roots. 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 16129 
 
 2048383 
 
 127 
 
 11-269 
 
 5-026 
 
 36100 
 
 6859000 
 
 190 
 
 13-784 
 
 5-748 
 
 16384 
 
 2097152 
 
 128 
 
 11-313 
 
 5-039 
 
 36481 
 
 6967871 
 
 191 
 
 13-820 
 
 5-758 
 
 16641 
 
 2146689 
 
 129 
 
 11-357 
 
 5-052 
 
 36864 
 
 7077888 
 
 192 
 
 13-856 
 
 5-768 
 
 16900 
 
 2197000 
 
 130 
 
 11-401 
 
 5-065 
 
 37249 
 
 7189517 
 
 193 
 
 13-892 
 
 5-778 
 
 17161 
 
 2248091 
 
 131 
 
 11-445 
 
 5-078 
 
 37636 
 
 7301384 
 
 194 
 
 13-928 
 
 5-788 
 
 17424 
 
 2299968 
 
 132 
 
 11-489 
 
 5-091 
 
 38025 
 
 7414875 
 
 195 
 
 13-964 
 
 6-798 
 
 17689 
 
 2352637 
 
 133 
 
 11-532 
 
 5-104 
 
 38416 
 
 7529536 
 
 196 
 
 14-000 
 
 5<808 
 
 17956 
 
 2406104 
 
 134 
 
 11-575 
 
 5-117 
 
 38809 
 
 7645373 
 
 197 
 
 14-035 
 
 5-818 
 
 18225 
 
 2460375 
 
 135 
 
 11-618 
 
 5-129 
 
 39204 
 
 7762392 
 
 198 
 
 14-071 
 
 6-828 
 
 18496 
 
 2515456 
 
 136 
 
 11-661 
 
 5-142 
 
 39601 
 
 7880599 
 
 199 
 
 14-106 
 
 5-838 
 
 18769 
 
 2571353 
 
 137 
 
 11-704 
 
 5-155 
 
 40000 
 
 8000000 
 
 200 
 
 14-142 
 
 5-848 
 
 19044 
 
 2628072 
 
 138 
 
 11-747 
 
 5-167 
 
 40401 
 
 8120601 
 
 201 
 
 14-177 
 
 5-857 
 
 19321 
 
 2685619 
 
 139 
 
 11-789 
 
 5-180 
 
 40804 
 
 8242408 
 
 202 
 
 14-212 
 
 5-867 
 
 19600 
 
 2744000 
 
 140 
 
 11-832 
 
 5-192 
 
 41209 
 
 8365427 
 
 203 
 
 14-247 
 
 5-877 
 
 19881 
 
 2803221 
 
 141 
 
 11-874 
 
 5-204 
 
 41616 
 
 8489664 
 
 204 
 
 14-282 
 
 5-886 
 
 20164 
 
 2863288 
 
 142 
 
 11-916 
 
 5-217 
 
 42025 
 
 8615125 
 
 205 
 
 14-317 
 
 5-896 
 
 20449 
 
 2924207 
 
 143 
 
 11-958 
 
 5-229 
 
 42436 
 
 8741816 
 
 206 
 
 14-352 
 
 5-905 
 
 20736 
 
 2985984 
 
 144 
 
 12-000 
 
 5-241 
 
 42849 
 
 8869743 
 
 207 
 
 14-387 
 
 5-915 
 
 21025 
 
 3048625 
 
 145 
 
 12-041 
 
 5-253 
 
 43264 
 
 8998912 
 
 208 
 
 14-422 
 
 5-924 
 
 21316 
 
 3112136 
 
 146 
 
 12-083 
 
 5-265 
 
 43681 
 
 9129329 
 
 209 
 
 14-456 
 
 5-934 
 
 21609 
 
 3176523 
 
 147 
 
 12-124 
 
 5-277 
 
 44100 
 
 9261000 
 
 210 
 
 14-491 
 
 5-943 
 
 21904 
 
 3241792 
 
 148 
 
 12-165 
 
 5-289 
 
 44521 
 
 9393931 
 
 211 
 
 14-525 
 
 6-953 
 
 22201 
 
 3307949 
 
 149 
 
 12-206 
 
 5-301 
 
 44944 
 
 9528128 
 
 212 
 
 14-560 
 
 5-962 
 
 22500 
 
 3375000 
 
 150 
 
 12-247 
 
 5-313 
 
 45369 
 
 9663597 
 
 213 
 
 14-594 
 
 5-972 
 
 22801 
 
 3442951 
 
 151 
 
 12-288 
 
 5-325 
 
 45796 
 
 9800344 
 
 214 
 
 14-628 
 
 5-981 
 
 23104 
 
 3511008 
 
 152 
 
 12-328 
 
 5-336 
 
 46225 
 
 9938375 
 
 215 
 
 14-662 
 
 5-990 
 
 23409 
 
 3581577 
 
 153 
 
 12-369 
 
 5-348 
 
 46656 
 
 10077696 
 
 216 
 
 14-696 
 
 6-000 
 
 23716 
 
 3652264 
 
 154 
 
 12-409 
 
 5-360 
 
 47089 
 
 10218312 
 
 217 
 
 14-730 
 
 6-009 
 
 24025 
 
 3723875 
 
 155 
 
 12-449 
 
 5-371 
 
 47524 
 
 10360232 
 
 218 
 
 14-764 
 
 6-018 
 
 24336 
 
 3796416 
 
 156 
 
 12-489 
 
 5-383 
 
 47961 
 
 10503459 
 
 219 
 
 14-798 
 
 6-027 
 
 24649 
 
 3869893 
 
 157 
 
 12-529 
 
 5-394 
 
 48400 
 
 10648000 
 
 220 
 
 14-832 
 
 6-036 
 
 24964 
 
 3944312 
 
 158 
 
 12-569 
 
 5-406 
 
 48841 
 
 10793861 
 
 221 
 
 14-866 
 
 6-045 
 
 25281 
 
 4019679 
 
 159 
 
 12-609 
 
 5-417 
 
 49284 
 
 10941048 
 
 222 
 
 14-899 
 
 6-055 
 
 25600 
 
 4096000 
 
 160 12-649 
 
 5-428 
 
 49729 
 
 11089567 
 
 223 
 
 14-933 
 
 6-064 
 
 25921 
 
 4173281 
 
 161 
 
 12-688 
 
 5-440 
 
 50176 
 
 11239424 
 
 224 
 
 14-966 
 
 6-073 
 
 26244 
 
 4251528 
 
 162 
 
 12-727 
 
 5-451 
 
 50625 
 
 11390625 
 
 225 
 
 15-000 
 
 6-082 
 
 26569 
 
 4330747 
 
 163 
 
 12-767 
 
 5-462 
 
 51076 
 
 11543176 
 
 226 
 
 15-033 
 
 6-099 
 
 26896 
 
 4410944 
 
 164 
 
 12-806 
 
 5-473 
 
 51529 
 
 11697083 
 
 227 
 
 15-066 
 
 6-100 
 
 27225 
 
 4492125 
 
 165 
 
 12-845 
 
 5-484 
 
 51984 
 
 11852352 
 
 228 
 
 15-099 
 
 6-109 
 
 27556 
 
 4574296 
 
 166 
 
 12-884 
 
 5-495 
 
 52441 
 
 12008989 
 
 229 
 
 15-132 
 
 6-118 
 
 27889 
 
 4657463 
 
 167 
 
 12-922 
 
 5-506 
 
 52900 
 
 12167000 
 
 230 
 
 15-165 
 
 6-126 
 
 28224 
 
 4741632 
 
 168 
 
 12-961 
 
 5-517 
 
 53361 
 
 12326391 
 
 231 
 
 15-198 
 
 6-135 
 
 28561 
 
 4826809 
 
 169 
 
 13-000 
 
 5-528 
 
 53824 
 
 12487168 
 
 232 
 
 15-231 
 
 6-144 
 
 28900 
 
 4913000 
 
 170 
 
 13-938 
 
 5-539 
 
 54289 
 
 12649337 
 
 233 
 
 15-264 
 
 6-153 
 
 29241 
 
 5000211 
 
 171 
 
 13-076 
 
 5-550 
 
 54756 
 
 12812904 
 
 234 
 
 15-297 
 
 6-162 
 
 29584 
 
 5088448 
 
 172 
 
 13-114 
 
 5-561 
 
 55225 
 
 12977875 
 
 235 
 
 15-329 
 
 6-171 
 
 29929 
 
 5177717 
 
 173 
 
 13-152 
 
 5-572 
 
 55696 
 
 13144256 
 
 236 
 
 15-362 
 
 6-179 
 
 30276 
 
 5268024 
 
 174 
 
 13-190 
 
 5-582 
 
 56169 
 
 13312053 
 
 237 
 
 15-394 
 
 6-188 
 
 30625 
 
 5359375 
 
 175 
 
 1-8-228 
 
 5-593 
 
 56644 
 
 13481272 
 
 238 
 
 15-427 
 
 6-197 
 
 30976 
 
 5451776 
 
 176 
 
 13-266 
 
 5-604 
 
 57121 
 
 13651919 
 
 239 
 
 15-459 
 
 6-205 
 
 31329 
 
 5545233 
 
 177 
 
 13-304 
 
 5-614 
 
 57600 
 
 13824000 
 
 240 
 
 15-491 
 
 6-214 
 
 31684 
 
 5639752 
 
 178 
 
 13-341 
 
 5-625 
 
 58081 
 
 13997521 
 
 241 
 
 15-524 
 
 6-223 
 
 32041 
 
 5735339 
 
 179 
 
 13-379 
 
 5-635 
 
 58564 
 
 14172488 
 
 242 
 
 15-556 
 
 6-231 
 
 32400 
 
 58S2000 
 
 180 
 
 13-416 
 
 5-646 
 
 59049 
 
 14348907 
 
 243 
 
 15-588 
 
 6-240 
 
 32761 
 
 5929741 
 
 181 
 
 13-453 
 
 5-656 
 
 59536 
 
 14526784 
 
 244 
 
 15-620 
 
 6-248 
 
 33124 
 
 6028568 
 
 182 
 
 13-490 
 
 5-667 
 
 60025 
 
 14706125 
 
 245 
 
 15-652 
 
 6-257 
 
 33489 
 
 6128487 
 
 183 
 
 13-527 
 
 5-677 
 
 60516 
 
 14886936 
 
 246 
 
 15-684 
 
 6-265 
 
 33856 
 
 6229504 
 
 184 
 
 13-664 
 
 5-687 
 
 61009 
 
 15069223 
 
 247 
 
 15-716 
 
 6-274 
 
 34225 
 
 6331625 
 
 185 
 
 13-601 
 
 5-698 
 
 61504 
 
 15252992 
 
 248 
 
 15-748 
 
 6-282 
 
 34596 
 
 6434856 
 
 186 
 
 13-638 
 
 5-708 
 
 62001 
 
 15438249 
 
 249 
 
 15-779 
 
 6-291 
 
 34969 
 
 6539203 
 
 187 
 
 13-674 
 
 5-718 
 
 62500 
 
 15625000 
 
 250 
 
 15-811 
 
 6-299 
 
 35344 
 
 6644672 
 
 188 
 
 13-711 
 
 5-728 
 
 63001 
 
 15813251 
 
 251 
 
 15-842 
 
 6-307 
 
 35721 
 
 6751269 
 
 189 
 
 13-747 
 
 5-738 
 
 63504 
 
 16003008 
 
 252 
 
 15-874 
 
 6-316 
 
822 
 
 APPENDIX. 
 
 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS (Continued) 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 1 
 ; Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 64009 
 
 16194277 
 
 253 
 
 15-905 
 
 6-324 
 
 99856 
 
 31554496 
 
 316 
 
 17-116 
 
 . 6-811 
 
 64516 
 
 16387064 
 
 254 
 
 15-937 
 
 6-333 
 
 100489 
 
 31855013 
 
 317 
 
 11-804 
 
 6-818 
 
 65025 
 
 16581375 
 
 255 
 
 15-968 
 
 6-341 
 
 101124 
 
 32157432 
 
 318 
 
 11-832 
 
 6-825 
 
 65536 
 
 16777216 
 
 256 
 
 16-000 
 
 6-349 
 
 101161 
 
 32461759 
 
 319 
 
 11-860 
 
 6-832 
 
 66049 
 
 16974593 
 
 257 
 
 16-031 
 
 6-351 
 
 102400 
 
 32768000 
 
 320 
 
 11-888 
 
 6-839 
 
 66564 
 
 17173512 
 
 258 
 
 16-062 
 
 6-366 
 
 103041 
 
 33076161 
 
 821 
 
 11-916 
 
 6-841 
 
 67081 
 
 17373979 
 
 259 
 
 16093 
 
 6-314 
 
 103684 
 
 33386248 
 
 322 
 
 17-944 
 
 6-854 
 
 6*7600 
 
 17576000 
 
 260 
 
 16-124 
 
 6-382 
 
 104329 
 
 33698267 
 
 323 
 
 17-972 
 
 6-861 
 
 68121 
 
 17779581 
 
 261 
 
 16-155 
 
 6-390 
 
 104976 
 
 34012224 
 
 324 
 
 18-000 
 
 6-868 
 
 68644 
 
 17984728 
 
 262 
 
 16-186 
 
 6-398 
 
 105625 
 
 34328125 
 
 325 
 
 18-027 
 
 6-815 
 
 69169 
 
 18191447 
 
 263 
 
 16-217 
 
 6-406 
 
 106276 
 
 34645976 
 
 326 
 
 18-055 
 
 6-882 
 
 69696 
 
 18399744 
 
 264 
 
 16-248 
 
 6-415 
 
 106929 
 
 34965783 
 
 327 
 
 18-083 
 
 6-889 
 
 70225 
 
 18609625 
 
 265 
 
 16-278 
 
 6-423 
 
 107584 
 
 35287552 
 
 328 
 
 18-110 
 
 6-896 
 
 70156 
 
 18821096 
 
 266 
 
 16-309 
 
 6-431 
 
 108241 
 
 35611289 
 
 329 
 
 18-138 
 
 6-903 
 
 71289 
 
 19034163 
 
 267 
 
 16-340 
 
 6-439 
 
 108900 
 
 35937000 
 
 330 
 
 18-165 
 
 6-910 
 
 71824 
 
 19248832 
 
 268 
 
 16-370 
 
 6-441 
 
 109561 
 
 36264691 
 
 331 
 
 18-V93 
 
 6-911 
 
 12361 
 
 19465109 
 
 269 
 
 16-401 
 
 6-455 
 
 110224 
 
 36594368 
 
 332 
 
 18-220 
 
 6-924 
 
 72900 
 
 19683000 
 
 270 
 
 16-431 
 
 6-463 
 
 110889 
 
 36926037 
 
 333 
 
 18-248 
 
 6-931 
 
 13441 
 
 19902511 
 
 271 
 
 16-462 
 
 6-411 
 
 111556 
 
 37259104 
 
 334 
 
 18-275 
 
 6-938 
 
 13984 
 
 20123643 
 
 272 
 
 16-492 
 
 6-419 
 
 112225 
 
 31595375 
 
 335 
 
 18-303 
 
 6-945 
 
 14529 
 
 20346417 
 
 273 
 
 16-522 
 
 6-481 
 
 112896 
 
 37933056 
 
 336 
 
 18-330 
 
 6-952 
 
 15076 
 
 20570824 
 
 274 
 
 16-552 
 
 6-495 
 
 113569 
 
 38272753 
 
 337 
 
 18-351 
 
 6'958 
 
 15625 
 
 20796875 
 
 275 
 
 16-583 
 
 6-502 
 
 114244 
 
 38614472 
 
 338 
 
 18-384 
 
 6-965 
 
 76176 
 
 21024576 
 
 276 
 
 16-613 
 
 6-510 
 
 114921 
 
 38958219 
 
 339 
 
 18-411 
 
 6-912 
 
 76729 
 
 21253933 
 
 277 
 
 16-643 
 
 6-518 
 
 115600 
 
 39304000 
 
 340 
 
 18-439 
 
 6-919 
 
 77284 
 
 21484952 
 
 278 
 
 16-678 
 
 6-52$ 
 
 116281 
 
 89651821 
 
 341 
 
 18-466 
 
 6-&S6 
 
 77841 
 
 21717639 
 
 279 
 
 16-703 
 
 6-534 
 
 116964 
 
 40001688 
 
 342 
 
 18-493 
 
 6-993 
 
 78400 
 
 21952000 
 
 280 
 
 16-733 
 
 6-542 
 
 117649 
 
 40353607 
 
 343 
 
 18-520 
 
 l-ooo 
 
 18961 
 
 22188041 
 
 281 
 
 16-163 
 
 6-549 
 
 118336 
 
 40701584 
 
 344 
 
 18-541 
 
 1-006 
 
 19524 
 
 22425768 
 
 282 
 
 16-792 
 
 6-557 
 
 119025 
 
 41063625 
 
 345 
 
 18-574 
 
 1-013 
 
 80039 
 
 22665187 
 
 283 
 
 16-822 
 
 6-565 
 
 119716 
 
 41421736 
 
 346 
 
 18-601 
 
 1-020 
 
 80656 
 
 22906304 
 
 284 
 
 16-852 
 
 6-573 
 
 120409 
 
 41781923 
 
 347 
 
 18-621 
 
 1-021 
 
 81225 
 
 23149125 
 
 285 
 
 16-881 
 
 6-580 
 
 121104 
 
 42144192 
 
 348 
 
 18-654 
 
 1-033 
 
 81196 
 
 23393656 
 
 286 
 
 16-911 
 
 6-588 
 
 121801 
 
 42508549 
 
 349 
 
 18-681 
 
 1-040 
 
 82369 
 
 23639903 
 
 287 
 
 16-941 
 
 6-596 
 
 122500 
 
 42815000 
 
 350 
 
 18-108 
 
 1-041 
 
 82944 
 
 23887872 
 
 288 
 
 16-970 
 
 6-603 
 
 123201 
 
 43243551 
 
 351 
 
 18-134 
 
 1-054 
 
 83521 
 
 24137569 
 
 289 
 
 17-000 
 
 6-611 
 
 123904 
 
 43614208 
 
 352 
 
 18-161 
 
 1-060 
 
 84100 
 
 24389000 
 
 290 
 
 17-029 
 
 6-619 
 
 124609 
 
 43986911 
 
 353 
 
 18-188 
 
 1-067 
 
 84681 
 
 24642171 
 
 291 
 
 17-058 
 
 6-626 
 
 125316 
 
 44361864 
 
 354 
 
 18-814 
 
 7-074 
 
 85264 
 
 24897088 
 
 292 
 
 17-088 
 
 6-634 
 
 126025 
 
 44738875 
 
 355 
 
 18-841 
 
 7-080 
 
 85849 
 
 25153757 
 
 293 
 
 17-117 
 
 6-641 
 
 126736 
 
 45118016 
 
 356 
 
 18-861 
 
 7-081 
 
 86436 
 
 25412184 
 
 294 
 
 17-146 
 
 6-649 
 
 127449 
 
 45499293 
 
 357 
 
 18-894 
 
 1-093 
 
 87025 
 
 25672375 
 
 295 
 
 17-175 
 
 6-656 
 
 128164 
 
 45882712 
 
 358 
 
 18-920 
 
 1-100 
 
 81616 
 
 25934836 
 
 296 
 
 17-204 
 
 6-664 
 
 128881 
 
 46268279 
 
 359 
 
 18-941 
 
 1-101 
 
 88209 
 
 26198073 
 
 297 
 
 11-233 
 
 6-671 
 
 129600 
 
 46656000 
 
 360 
 
 18-913 
 
 1-113 
 
 88804 
 
 26463592 
 
 298 
 
 11-262 
 
 6-679 
 
 130321 
 
 47045831 
 
 361 
 
 19-000 
 
 1-120 
 
 89401 
 
 26730899 
 
 299 
 
 11-291 
 
 6-686 
 
 131044 
 
 47437928 
 
 362 
 
 19-026 
 
 1-126 
 
 90000 
 
 27000000 
 
 300 
 
 11-320 
 
 6-694 
 
 131769 
 
 47832147 
 
 363 
 
 19-052 
 
 1-133 
 
 90601 
 
 27270901 
 
 301 
 
 11-349 
 
 6-701 
 
 132496 
 
 48228544 
 
 864 
 
 19-078 
 
 1-140 
 
 91204 
 
 27543608 
 
 302 
 
 11-318 
 
 6-709 
 
 133225 
 
 48627125 
 
 365 
 
 19-104 
 
 1-146 
 
 91809 
 
 27818127 
 
 303 
 
 11-406 
 
 6-716 
 
 133956 
 
 49027896 
 
 366 
 
 19-131 
 
 1-153 
 
 92416 
 
 28094464 
 
 304 
 
 17-435 
 
 6-723 
 
 134689 
 
 49430863 
 
 367 
 
 19-151 
 
 1-159 
 
 93025 
 
 28372625 
 
 305 
 
 17-464 
 
 6-731 
 
 135424 
 
 49836032 
 
 368 
 
 19-183 
 
 1-166 
 
 93636 
 
 28652616 
 
 306 
 
 17-492 
 
 6-738 
 
 136161 
 
 50243409 
 
 369 
 
 19-209 
 
 1-112 
 
 94249 
 
 28934443 
 
 307 
 
 17-521 
 
 6-745 
 
 136900 
 
 50653000 
 
 370 
 
 19-235 
 
 1-179 
 
 94864 
 
 29218112 
 
 308 
 
 17-549 
 
 6-753 
 
 137641 
 
 51064811 
 
 371 
 
 19-261 
 
 7-185 
 
 95481 
 
 29503609 
 
 309 
 
 11-578 
 
 6-160 
 
 138384 
 
 51478848 
 
 372 
 
 19-281 
 
 7-191 
 
 96100 
 
 29791000 
 
 310 
 
 11-606 
 
 6-161 
 
 139129 
 
 51895117 
 
 373 
 
 19-313 
 
 7-198 
 
 9.8121 
 
 30080231 
 
 311 
 
 11-635 
 
 6-775 
 
 139876 
 
 62313624 
 
 374 
 
 19-339 
 
 7-204 
 
 91344 
 
 30371328 
 
 312 
 
 11-663 
 
 6-782 
 
 140625 
 
 52734375 
 
 375 
 
 19-364 
 
 7-211 
 
 97969 
 
 30664297 
 
 313 
 
 11-691 
 
 6-789 
 
 141376 
 
 53157376 
 
 376 
 
 19-390 
 
 7-217 
 
 98596 
 
 30959144 
 
 314 
 
 11-120 
 
 6-796 
 
 142129 
 
 53582633 
 
 377 
 
 19-416 
 
 7-224 
 
 99225 
 
 31255875 
 
 315 
 
 11-148 
 
 6-804 
 
 : 142884 
 
 54010152 
 
 378 
 
 19-442 
 
 7-230 
 
APPENDIX. 
 
 823 
 
 TABLE OF SQUARES, CUBES, SQUAEE AND CUBE ROOTS OF NUMBERS ( Continued). 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 143641 
 
 54439939 
 
 379 
 
 19-467 
 
 7-236 
 
 195364 
 
 86350888 
 
 442 
 
 21-023 
 
 7-617 
 
 144400 
 
 54872000 
 
 380 
 
 19-493 
 
 7-243 
 
 196249 
 
 86938307 
 
 443 
 
 21-047 
 
 7-623 
 
 145161 
 
 55306341 
 
 381 
 
 19-519 
 
 7-249 
 
 197136 
 
 87528384 
 
 444 
 
 21-071 
 
 V-628 
 
 145924 
 
 55742968 
 
 382 
 
 19-544 
 
 7-255 
 
 198025 
 
 88121125 
 
 445 
 
 21-095 
 
 7-634 
 
 146689 
 
 56181887 
 
 383 
 
 19-570 
 
 7-262 
 
 198916 
 
 88716536 
 
 446 
 
 21-118 
 
 7-640 
 
 147456 
 
 56623104 
 
 384 
 
 19-595 
 
 7-268 
 
 199809 
 
 89314623 
 
 447 
 
 21-142 
 
 7-646 
 
 148225 
 
 57066625 
 
 385 
 
 19621 
 
 7-274 
 
 200704 
 
 89915392 
 
 448 
 
 21-166 
 
 7-651 
 
 148996 
 
 57512456 
 
 386 
 
 19-646 
 
 7-281 
 
 201601 
 
 90518849 
 
 449 
 
 21-189 
 
 7-657 
 
 149769 
 
 57960603 
 
 387 
 
 19-672 
 
 7-287 
 
 202500 
 
 91125000 
 
 450 
 
 21-213 
 
 7-663 
 
 150544 
 
 58411072 
 
 388 
 
 19-697 
 
 7-293 
 
 203401 
 
 91733851 
 
 451 
 
 21-236 
 
 7-668 
 
 151321 
 
 58863869 
 
 389 
 
 19-723 
 
 7-299 
 
 204304 
 
 92345408 
 
 452 
 
 21-260 
 
 7-674 
 
 152100 
 
 59319000 
 
 390 
 
 19-748 
 
 7-306 
 
 205209 
 
 92959677 
 
 453 
 
 21-283 
 
 7-680 
 
 152881 
 
 59776471 
 
 391 
 
 19-773 
 
 7-312 
 
 206116 
 
 93576664 
 
 454 
 
 21-307 
 
 7-685 
 
 153664 
 
 602S6288 
 
 392 
 
 19-798 
 
 7-318 
 
 207025 
 
 94196375 
 
 455 
 
 21-330 
 
 7-691 
 
 154449 
 
 60698457 
 
 393 
 
 19-824 
 
 7-324 
 
 207936 
 
 94818816 
 
 456 
 
 21-354 
 
 7-697 
 
 155236 
 
 61162984 
 
 394 
 
 19-849 
 
 7-331 
 
 208849 
 
 95443993 
 
 457 
 
 21-377 
 
 7-702 
 
 156025 
 
 61629875 
 
 395 
 
 19-874 
 
 7-337 
 
 209764 
 
 96071912 
 
 458 
 
 21-400 
 
 7-708 
 
 156816 
 
 62099136 
 
 396 
 
 19-899 
 
 7-343 
 
 210681 
 
 96702579 
 
 459 
 
 21-424 
 
 7-713 
 
 157609 
 
 62570773 
 
 397 
 
 19-924 
 
 7-349 
 
 211600 
 
 97336000 
 
 460 
 
 21-447 
 
 7-719 
 
 158404 
 
 63044792 
 
 398 
 
 19-949 
 
 7-355 
 
 212521 
 
 97972181 
 
 461 
 
 21-470 
 
 7-725 
 
 159201 
 
 63521199 
 
 399 
 
 19-974 
 
 7-361 
 
 213444 
 
 98611128 
 
 462 
 
 21-494 
 
 7-730 
 
 160000 
 
 64000000 
 
 400 
 
 20-000 
 
 7-368 
 
 214369 
 
 99252847 
 
 463 
 
 21-517 
 
 7-736 
 
 160801 
 
 64481201 
 
 401 
 
 20-024 
 
 7-374 
 
 215296 
 
 99897344 
 
 464 
 
 21-540 
 
 7-741 
 
 161604 
 
 64964808 
 
 402 
 
 20-049 
 
 7-380 
 
 216225 
 
 100544625 
 
 465 
 
 21-563 
 
 7-747 
 
 162409 
 
 65450827 
 
 403 
 
 20-074 
 
 7-386 
 
 217156 
 
 101194696 
 
 466 
 
 21-587 
 
 7-752 
 
 163216 
 
 65939264 
 
 404 
 
 20-099 
 
 7-392 
 
 218089 
 
 101847563 
 
 467 
 
 21-610 
 
 7-758 
 
 164025 
 
 66430125 
 
 405 
 
 20-124 
 
 7-398 
 
 219024 
 
 102503232 
 
 468 
 
 21-633 
 
 7-763 
 
 164836 
 
 66923416 
 
 406 
 
 20-149 
 
 7-404 
 
 219961 
 
 103161709 
 
 469 
 
 21-656 
 
 7-769 
 
 165649 
 
 67419143 
 
 407 
 
 20-174 
 
 7-410 
 
 220900 
 
 103823000 
 
 470 
 
 21-679 
 
 7-774 
 
 166464 
 
 67917312 
 
 408 
 
 20-199 
 
 7-416 
 
 221841 
 
 104487111 
 
 471 
 
 21-702 
 
 7-780 
 
 167281 
 
 68417929 
 
 409 
 
 20-223 
 
 7-422 
 
 222784 
 
 105154048 
 
 472 
 
 21-725 
 
 7-785 
 
 168100 
 
 68921000 
 
 410 
 
 20-248 
 
 7-428 
 
 223729 
 
 105823817 
 
 473 
 
 21-748 
 
 7-791 
 
 168921 
 
 69426531 
 
 411 
 
 20-273 
 
 7-434 
 
 224676 
 
 106496424 
 
 474 
 
 21-771 
 
 7-796 
 
 169744 
 
 69934528 
 
 412 
 
 20-297 
 
 7-441 
 
 225625 
 
 107171875 
 
 475 
 
 21-794 
 
 7-802 
 
 170569 
 
 70444997 
 
 413 
 
 20-322 
 
 7-447 
 
 226576 
 
 107850176 
 
 476 
 
 21-817 
 
 7-807 
 
 171396 
 
 70957944 
 
 414 
 
 20-346 
 
 7-453 
 
 227529 
 
 108531333 
 
 477 
 
 21-840 
 
 7-813 
 
 172225 
 
 71473375 
 
 415 
 
 20-371 
 
 7-459 
 
 228484 
 
 109215352 
 
 478 
 
 21-863 
 
 7-818 
 
 173056 
 
 71991296 
 
 416 
 
 20-396 
 
 7-465 
 
 229441 
 
 109902239 
 
 479 
 
 21-886 
 
 7-824 
 
 173889 
 
 72511713 
 
 417 
 
 20-420 
 
 7-470 
 
 230400 
 
 110592000 
 
 480 
 
 21-908 
 
 7-829 
 
 174724 
 
 73034632 
 
 418 
 
 20-445 
 
 7-476 
 
 231361 
 
 111284641 
 
 481 
 
 21-931 
 
 7-835 
 
 175561 
 
 73560059 
 
 419 
 
 20-469 
 
 7-482 
 
 232324 
 
 111980168 
 
 482 
 
 21-954 
 
 7-840 
 
 176400 
 
 74088000 
 
 420 
 
 20-493 
 
 7-488 
 
 233289 
 
 112678587 
 
 483 
 
 21-977 
 
 7-846 
 
 177241 
 
 74618461 
 
 421 
 
 20-518 
 
 7-494 
 
 234256 
 
 113379904 
 
 484 
 
 22-000 
 
 7-851 
 
 178084 
 
 75151448 
 
 422 
 
 20-542 
 
 7-500 
 
 235225 
 
 114084125 
 
 485 
 
 22-022 
 
 7-856 
 
 178929 
 
 75686967 
 
 423 
 
 20-566 
 
 7-506 
 
 236196 
 
 114791256 
 
 486 
 
 22-045 
 
 7-862 
 
 179776 
 
 76225024 
 
 424 
 
 20-591 
 
 7-512 
 
 237169 
 
 115501303 
 
 487 
 
 22-068 
 
 7-867 
 
 180625 
 
 76765625 
 
 425 
 
 20-615 
 
 7-518 
 
 238144 
 
 116214272 
 
 488 
 
 22-090 
 
 7-872 
 
 181476 
 
 77308776 
 
 426 
 
 20-639 
 
 7-524 
 
 239121 
 
 116930169 
 
 489 
 
 22-113 
 
 7-878 
 
 182329 
 
 77854483 
 
 427 
 
 20-663 
 
 7-530 
 
 240100 
 
 117649000 
 
 490 
 
 22-135 
 
 7-883 
 
 183184 
 
 78402752 
 
 428 
 
 20-688 
 
 7-536 
 
 241081 
 
 118370771 
 
 491 
 
 22-158 
 
 7-889 
 
 184041 
 
 78953589 
 
 429 
 
 20-712 
 
 7-541 
 
 242064 
 
 119095488 
 
 492 
 
 22-181 
 
 7-894 
 
 184900 
 
 79507000 
 
 430 
 
 20-736 
 
 7-547 
 
 243049 
 
 119823157 
 
 493 
 
 22-203 
 
 7-899 
 
 185761 
 
 80062991 
 
 431 
 
 20-760 
 
 7-553 
 
 244036 
 
 120553784 
 
 494 
 
 22-226 
 
 7-905 
 
 186624 
 
 80621568 
 
 432 
 
 20-784 
 
 7-559 
 
 245025 
 
 121287375 
 
 495 
 
 22-248 
 
 7-910 
 
 187489 
 
 81182737 
 
 433 
 
 20-808 
 
 7-565 
 
 246016 
 
 122023936 
 
 496 
 
 22-271 
 
 7-915 
 
 188356 
 
 81746504 
 
 434 
 
 20-832 
 
 7-571 
 
 247009 
 
 122763473 
 
 497 
 
 22-293 
 
 7-921 
 
 189225 
 
 82312875 
 
 435 
 
 20-856 
 
 7-576 
 
 248004 
 
 123505992 
 
 498 
 
 22-315 
 
 7-926 
 
 190096 
 
 82881856 
 
 436 
 
 20-880 
 
 7-582 
 
 249001 
 
 124251499 
 
 499 
 
 22-338 
 
 7-931 
 
 190969 
 
 83453453 
 
 437 
 
 20-904 
 
 7-588 
 
 250000 
 
 125000000 
 
 500 
 
 22-360 
 
 7-937 
 
 191844 
 
 84027672 
 
 438 
 
 20-928 
 
 7-594 
 
 251001 
 
 125751501 
 
 501 
 
 22-383 
 
 7-942 
 
 192721 
 
 84604519 
 
 439 
 
 20-952 
 
 7-600 
 
 252004 
 
 126506008 
 
 502 
 
 22-405 
 
 7-947 
 
 193600 
 
 85184000 
 
 440 
 
 20-976 
 
 7-605 
 
 253009 
 
 127263527 
 
 503 
 
 22-427 
 
 7-952 
 
 194481 
 
 85766121 
 
 441 
 
 21-000 
 
 7-611 
 
 254016 
 
 128024064 
 
 504 
 
 22-449 7-958 
 
824. 
 
 APPENDIX. 
 
 TABLE OF SQUARES, CUBES, SQUAEE AND CUBE EOOTS OF NUMBEES (Continued'). 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 
 roots. 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 255025 
 
 128787625 
 
 505 
 
 22-472 
 
 7-963 
 
 322624 
 
 183250432 
 
 568 
 
 23-832 
 
 8-281 
 
 256036 
 
 129554216 
 
 506 
 
 , 22-494 
 
 7-968 
 
 323761 
 
 184220009 
 
 569 
 
 23-853 
 
 8-286 
 
 257049 
 
 130323843 
 
 507 
 
 22-516 
 
 7-973 
 
 324900 
 
 185193000 
 
 570 
 
 23-874 
 
 8-291 
 
 258064 
 
 131096512 
 
 508 
 
 22-538 
 
 7-979 
 
 326041 
 
 186169411 
 
 571 
 
 23-895 
 
 8-296 
 
 259081 
 
 131872229 
 
 509 
 
 22-561 
 
 7-984 
 
 327184 
 
 187149248 
 
 572 
 
 23-916 
 
 8-301 
 
 260100 
 
 132651000 
 
 510 
 
 22-583 
 
 7-989 
 
 328329 
 
 188132517 
 
 573 
 
 23-937 
 
 8-305 
 
 261121 
 
 133432831 
 
 511 
 
 22-605 
 
 7-994 
 
 329476 
 
 189119224 
 
 574 
 
 23-958 
 
 8-310 
 
 262144 
 
 134217728 
 
 512 
 
 22-627 
 
 8-000 
 
 330625 
 
 190109375 
 
 575 
 
 23-979 
 
 8-315 
 
 263169 
 
 135005697 
 
 513 
 
 22-649 
 
 8-005 
 
 331776 
 
 191102976 
 
 576 
 
 24-000 
 
 8-320 
 
 264196 
 
 135796744 
 
 514 
 
 22-671 
 
 8-010 
 
 332929 
 
 192100033 
 
 577 
 
 24-020 
 
 8-325 
 
 265225 
 
 136590875 
 
 515 
 
 22-693 
 
 8-015 
 
 334084 
 
 193100552 
 
 578 
 
 24-041 
 
 8-329 
 
 266256 
 
 137388096 
 
 516 
 
 22-715 
 
 8-020 
 
 335241 
 
 194104539 
 
 579 
 
 24-062 
 
 8-334 
 
 267289 
 
 138188413 
 
 517 
 
 22-737 
 
 8-025 
 
 336400 
 
 195112000 
 
 580 
 
 24-083 
 
 8-339 
 
 268324 
 
 138991832 
 
 618 
 
 22-759 
 
 8-031 
 
 337561 
 
 196122941 
 
 581 
 
 24-103 
 
 8-344 
 
 269361 
 
 139798359 
 
 519 
 
 22-781 
 
 8-036 
 
 338724 
 
 197137368 
 
 582 
 
 24-124 
 
 8-349 
 
 270400 
 
 140608000 
 
 520 
 
 22-803 
 
 8-041 
 
 339889 
 
 198155287 
 
 583 
 
 24-145 
 
 8-353 
 
 271441 
 
 141420761 
 
 521 
 
 22-825 
 
 8-046 
 
 341056 
 
 199176704 
 
 584 
 
 24-166 
 
 8-358 
 
 272484 
 
 142236648 
 
 522 
 
 22-847 
 
 8-051 
 
 342225 
 
 200201625 
 
 585 
 
 24-186 
 
 -363 
 
 273529 
 
 143055667 
 
 523 
 
 22-869 
 
 8-056 
 
 343396 
 
 201230056 
 
 586 
 
 24-207 
 
 8-368 
 
 274576 
 
 143877824 
 
 524 
 
 22-891 
 
 8-062 
 
 344569 
 
 202262003 
 
 587 
 
 24-228 
 
 8-372 
 
 275625 
 
 144703125 
 
 525 
 
 22-912 
 
 8-067 
 
 345744 
 
 203297472 
 
 588 
 
 24-248 
 
 8-377 
 
 276676 
 
 145531576 
 
 526 
 
 22-934 
 
 8-072 
 
 346921 
 
 204336469 
 
 589 
 
 24-269 
 
 8-382 
 
 277729 
 
 146363183 
 
 527 
 
 22-956 
 
 8-077 
 
 348100 
 
 205379000 
 
 590 
 
 24-289 
 
 8-387 
 
 278784 
 
 147197952 
 
 528 
 
 22-978 
 
 8-082 
 
 349281 
 
 206425071 
 
 591 
 
 24-310 
 
 8-391 
 
 279841 
 
 148035889 
 
 529 
 
 23-000 
 
 8-037 
 
 350464 
 
 207474688 
 
 592 
 
 24-331 
 
 8-396 
 
 280900 
 
 148877000 
 
 530 
 
 23-021 
 
 8-092 
 
 351649 
 
 208527857 
 
 593 
 
 24-351 
 
 8-40*1 
 
 281961 
 
 149721291 
 
 531 
 
 23-043 
 
 8-097 
 
 352836 
 
 209584584 
 
 594 
 
 24-372 
 
 8-406 
 
 283024 
 
 150568768 
 
 532 
 
 23-065 
 
 8-102 
 
 354025 
 
 210644875 
 
 595 
 
 24-392 
 
 8-410 
 
 284089 
 
 151419437 
 
 533 
 
 23-086 
 
 8-107 
 
 355216 
 
 211708736 
 
 596 
 
 24-413 
 
 8-415 
 
 285156 
 
 152273304 
 
 534 
 
 23-108 
 
 8-112 
 
 356409 
 
 212776173 
 
 697 
 
 24-433 
 
 8-420 
 
 286225 
 
 153130375 
 
 535 
 
 23-130 
 
 8-118 
 
 357604 
 
 213847192 
 
 598 
 
 24-454 
 
 8-424 
 
 287296 
 
 153990656 
 
 536 
 
 23-151 
 
 8-123 
 
 358801 
 
 214921799 
 
 599 
 
 24-474 
 
 8-429 
 
 288369 
 
 154854153 
 
 537 
 
 23-173 
 
 8-128 
 
 360000 
 
 216000000 
 
 600 
 
 24-494 
 
 8-434 
 
 289444 
 
 155720872 
 
 538 
 
 23-194 
 
 8-133 
 
 361201 
 
 217081801 
 
 601 
 
 24-515 
 
 8439 
 
 290521 
 
 156590819 
 
 539 
 
 23-216 
 
 8-138 
 
 362404 
 
 218167208 
 
 602 
 
 24-535 
 
 8-443 
 
 291600 
 
 157464000 
 
 540 
 
 23-237 
 
 8-143 
 
 363609 
 
 219256227 
 
 603 
 
 24-556 
 
 8-448 
 
 292681 
 
 158340421 
 
 541 
 
 23-259 
 
 8-148 
 
 364816 
 
 220348864 
 
 604 
 
 24-576 
 
 8-453 
 
 293764 
 
 159220088 
 
 542 
 
 23-280 
 
 8-153 
 
 366025 
 
 221445125 
 
 605 
 
 24-596 
 
 8-457 
 
 294849 
 
 160103007 
 
 543 
 
 23-302 
 
 8-158 
 
 367236 
 
 222545016 
 
 606 
 
 24-617 
 
 8-462 
 
 295936 
 
 160989184 
 
 544 
 
 23-323 
 
 8-163 
 
 368449 
 
 223648543 
 
 607 
 
 24-637 
 
 8-467 
 
 297025 
 
 161878625 
 
 545 
 
 23-345 
 
 8-168 
 
 369664 
 
 224755712 
 
 608 
 
 24-657 
 
 8-471 
 
 298116 
 
 162771336 
 
 546 
 
 23-366 
 
 8-173 
 
 370881 
 
 225866529 
 
 609 
 
 24-677 
 
 8-476 
 
 299209 
 
 163667323 
 
 547 
 
 23-388 
 
 8-178 
 
 372100 
 
 226981000 
 
 610 
 
 24-698 
 
 8-480 
 
 300304 
 
 164566592 
 
 548 
 
 23-409 
 
 8-183 
 
 373321 
 
 228099131 
 
 611 
 
 24-718 
 
 8-485 
 
 301401 
 
 165469149 
 
 549 
 
 23-430 
 
 8-188 
 
 374544 
 
 229220928 
 
 612 
 
 24-738 
 
 8-490 
 
 302500 
 
 166375000 
 
 550 
 
 23-452 
 
 8-193 
 
 375769 
 
 230346397 
 
 613 
 
 24-758 
 
 8-494 
 
 303601 
 
 167284151 
 
 551 
 
 23-473 
 
 8-198 
 
 376996 
 
 231475544 
 
 614 
 
 24-779 
 
 8-499 
 
 304704 
 
 168196608 
 
 552 
 
 23-494 
 
 8-203 
 
 378225 
 
 232608375 
 
 615 
 
 24-799 
 
 8-504 
 
 305809 
 
 169112377 
 
 553 
 
 23-515 
 
 8-208 
 
 379456 
 
 233744896 
 
 616 
 
 24-819 
 
 8-508 
 
 306916 
 
 170031464 
 
 554 
 
 23-537 
 
 8-213 
 
 380689 
 
 234885113 
 
 617 
 
 24-839 
 
 8-513 
 
 308025 
 
 170953875 
 
 556 
 
 23-558 
 
 8-217 
 
 381924 
 
 236029032 
 
 618 
 
 24-859 
 
 8-517 
 
 309136 
 
 171879616 
 
 556 
 
 23-579 
 
 8-222 
 
 383161 
 
 237176659 
 
 619 
 
 24-879 
 
 8-522 
 
 310249 
 
 172808693 
 
 557 
 
 23-600 
 
 8-227 
 
 384400 
 
 238328000 
 
 620 
 
 24-899 
 
 8-527 
 
 311364 
 
 173741112 
 
 558 
 
 23-622 
 
 8-232 
 
 385641 
 
 239483061 
 
 621 
 
 24-919 
 
 8-531 
 
 312481 
 
 174676879 
 
 559 
 
 23-643 
 
 8-237 
 
 386884 
 
 240641848 
 
 622 
 
 24-939 
 
 8-536 
 
 313600 
 
 175616000 
 
 560 
 
 23-664 
 
 8-242 
 
 388129 
 
 241804367 
 
 623 
 
 24-959 
 
 8-540 
 
 314721 
 
 176558481 
 
 561 
 
 23-685 
 
 8-247 
 
 389376 
 
 242970624, 
 
 624 
 
 24-979 
 
 8-545 
 
 315844 
 
 177504328 
 
 562 
 
 23-706 
 
 8-252 
 
 390625 
 
 244140625 
 
 625 
 
 25-000 
 
 8-549 
 
 316969 
 
 178453547 
 
 563 
 
 23-727 
 
 8-257 
 
 391876 
 
 245314376 
 
 626 
 
 25-019 
 
 8-554 
 
 318096 
 
 179406144 
 
 564 
 
 23-748 
 
 8-262 
 
 393129 
 
 246491883 
 
 627 
 
 ?,5'039 
 
 8-558 
 
 319225 
 
 180362125 
 
 565 
 
 23-769 
 
 8-267 
 
 394384 
 
 247673152 
 
 628 
 
 25-059 
 
 8-563 
 
 320356 
 
 181321496 
 
 566 
 
 23-790 
 
 8-271 
 
 395641 
 
 248858189 
 
 629 
 
 25-079 
 
 8-568 
 
 321489 (182284263 
 
 567 
 
 23-811 
 
 8'276 
 
 396900 
 
 250047000 
 
 630 
 
 25-099 
 
 8-572 
 
APPENDIX. 
 
 825 
 
 TABLE OF SQUARES, CUBES, SQUAEE AND CUBE EOOTS OF NUMBERS (Continued). 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 
 roots. 
 
 398161 
 
 251239591 
 
 631 
 
 25-119 
 
 8-577 
 
 481636 
 
 334255384 
 
 694 
 
 26-343 
 
 8-853 
 
 399424 
 
 252435968 
 
 632 
 
 25-139 
 
 8-581 
 
 483025 
 
 335702375 
 
 695 
 
 26-362 
 
 8-857 
 
 400689 
 
 253636137 
 
 633 
 
 25-159 
 
 8-586 
 
 484416 
 
 337153536 
 
 696 
 
 26-381 
 
 8-862 
 
 401956 
 
 254840104 
 
 634 
 
 25-179 
 
 8-590 
 
 485809 
 
 338608873 
 
 697 
 
 26-400 
 
 8-866 
 
 403225 
 
 256047875 
 
 635 
 
 25-199 
 
 8-595 
 
 487204 
 
 340068392 
 
 698 
 
 26-419 
 
 8-870 
 
 404496 
 
 257259456 
 
 636 
 
 25-219 
 
 8-599 
 
 488601 
 
 341532099 
 
 699 
 
 26-438 
 
 8-874 
 
 405769 
 
 258474853 
 
 637 
 
 25-238 
 
 8-604 
 
 490000 
 
 343000000 
 
 700 
 
 26-457 
 
 8-879 
 
 407044 
 
 259694072 
 
 638 
 
 25-258 
 
 8-608 
 
 491401 
 
 344472101 
 
 701 
 
 26-476 
 
 8-883 
 
 408321 
 
 260917119 
 
 639 
 
 25-278 
 
 8-613 
 
 492804 
 
 345948408 
 
 702 
 
 26-495 
 
 6-887 
 
 409600 
 
 262144000 
 
 .640 
 
 25-298 
 
 8-617 
 
 494209 
 
 347428927 
 
 703 
 
 26-514 
 
 8-891 
 
 410881 
 
 263374721 
 
 641 
 
 25-317 
 
 8-622 
 
 495616 
 
 348913664 
 
 704 
 
 26-532 
 
 8-895 
 
 412164 
 
 264609288 
 
 642 
 
 25-337 
 
 8-626 
 
 497025 
 
 350402625 
 
 705 
 
 26-551 
 
 8-900 
 
 413449 
 
 265847707 
 
 643 
 
 25-357 
 
 8-631 
 
 498436 
 
 351895816 
 
 706 
 
 26-570 
 
 8-904 
 
 414736 
 
 267089984 
 
 644 
 
 25-377 
 
 3-635 
 
 499849 
 
 353393243 
 
 707 
 
 26-589 
 
 8-908 
 
 416025 
 
 268336125 
 
 645 
 
 25-396 
 
 8-640 
 
 501264 
 
 354894912 
 
 708 
 
 26-608 
 
 8-912 
 
 417316 
 
 269586136 
 
 646 
 
 25-416 
 
 8-644 
 
 502681 
 
 356400829 
 
 709 
 
 26-627 
 
 8-916 
 
 418609 
 
 270840023 
 
 647 
 
 25-436 
 
 8-649 
 
 504100 
 
 357911000 
 
 710 
 
 26-645 
 
 8-921 
 
 419904 
 
 272097792 
 
 648 
 
 25-455 
 
 8-653 
 
 505521 
 
 359425431 
 
 711 
 
 26-664 
 
 8-925 
 
 421201 
 
 273359449 
 
 649 
 
 25-475 
 
 8-657 
 
 506944 
 
 360944128 
 
 712 
 
 26-683 
 
 8-929 
 
 422500 
 
 274625000 
 
 650 
 
 25-495 
 
 8-662 
 
 508369 
 
 362467097 
 
 713 
 
 26-702 
 
 8-933 
 
 423801 
 
 275894451 
 
 651 
 
 25-514 
 
 8-666 
 
 509796 
 
 363994344 
 
 714 
 
 26-720 
 
 8-937 
 
 425104 
 
 277167808 
 
 652 
 
 25-534 
 
 8-671 
 
 511225 
 
 365525875 
 
 715 
 
 26-739 
 
 8-942 
 
 426409 
 
 278445077 
 
 653 
 
 25-553 
 
 8-675 
 
 512656 
 
 367061696 
 
 716 
 
 26-758 
 
 8'946 
 
 427716 
 
 279726264 
 
 654 
 
 25-573 
 
 8-680 
 
 514089 
 
 368601813 
 
 717 
 
 26-776 
 
 8-950 
 
 429025 
 
 281011375 
 
 655 
 
 25-592 
 
 8-684 
 
 515524 
 
 370146232 
 
 718 
 
 26-795 
 
 8-954 
 
 430336 
 
 282300416 
 
 656 
 
 25-612 
 
 8-688 
 
 516961 
 
 371694959 
 
 719 
 
 26-814 
 
 8-958 
 
 431649 
 
 283593393 
 
 657 
 
 25-632 
 
 8-693 
 
 518400 
 
 373248000 
 
 720 
 
 26-832 
 
 8-962 
 
 432964 
 
 284890312 
 
 658 
 
 25-651 
 
 8-697 
 
 519841 
 
 374805361 
 
 721 
 
 26-851 
 
 8-966 
 
 4S4281 
 
 286191179 
 
 659 
 
 25-670 
 
 8-702 
 
 521284 
 
 376367048 
 
 722 
 
 26-870 
 
 8-971 
 
 435600 
 
 287496000 
 
 660 
 
 25-690 
 
 8-706 
 
 522729 
 
 377933067 
 
 723 
 
 26-888 
 
 8-975 
 
 436921 
 
 288804781 
 
 661 
 
 25-709 
 
 8-710 
 
 524176 
 
 379503424 
 
 724 
 
 26-907 
 
 8-979 
 
 438244 
 
 290117528 
 
 662 
 
 25-729 
 
 8-715 
 
 525625 
 
 381078125 
 
 725 
 
 26-925 
 
 8-983 
 
 439569 
 
 291434247 
 
 663 
 
 25-748 
 
 8-719 
 
 527076 
 
 382657176 
 
 726 
 
 26-944 
 
 8-987 
 
 440896 
 
 292754944 
 
 664 
 
 25-768 
 
 8-724 
 
 528529 
 
 384240583 
 
 727 
 
 26-962 
 
 8991 
 
 442225 
 
 294079625 
 
 665 
 
 25-787 
 
 8-728 
 
 529984 
 
 385828352 
 
 728 
 
 26-981 
 
 8-995 
 
 443556 
 
 295408296 
 
 666 
 
 25-806 
 
 8-732 
 
 531441 
 
 387420489 
 
 729 
 
 27-000 
 
 9-000 
 
 444889 
 
 296740963 
 
 667 
 
 25-826 
 
 8-737 
 
 532900 
 
 389017000 
 
 730 
 
 27-018 
 
 9-004 
 
 446224 
 
 298077632 
 
 668 
 
 25-845 
 
 8-741 
 
 534361 
 
 390617891 
 
 731 
 
 27-037 
 
 9-008 
 
 447561 
 
 299418309 
 
 669 
 
 25-865 
 
 8-745 
 
 535824 
 
 392223168 
 
 732 
 
 27-055 
 
 9-012 
 
 448900 
 
 300763000 
 
 670 
 
 25-884 
 
 8-750 
 
 537289 
 
 393832837 
 
 733 
 
 27-073 
 
 9-016 
 
 450241 
 
 302111711 
 
 671 
 
 25-903 
 
 8-754 
 
 538756 
 
 395446904 
 
 734 
 
 27-092 
 
 9-020 
 
 451584 
 
 303464448 
 
 672 
 
 25-922 
 
 8-759 
 
 540225 
 
 397065375 
 
 735 
 
 27-110 
 
 9-024 
 
 452929 
 
 304821217 
 
 673 
 
 25-942 
 
 8-763 
 
 541696 
 
 398688256 
 
 736 
 
 27-129 
 
 9-028 
 
 454276 
 
 306182024 
 
 674 
 
 25-961 
 
 8-767 
 
 543169 
 
 400315553 
 
 7S7 
 
 27-147 
 
 9-032 
 
 455625 
 
 307546375 
 
 675 
 
 25-980 
 
 8-772 
 
 544644 
 
 401947272 
 
 738 
 
 27-166 
 
 9-036 
 
 456976 
 
 308915776 
 
 676 
 
 26-000 
 
 8-776 
 
 546121 
 
 403583419 
 
 739 
 
 27-184 
 
 9-040 
 
 458329 
 
 310288733 
 
 677 
 
 26-019 
 
 8-780 
 
 547600 
 
 405224000 
 
 740 
 
 27-202 
 
 9-045 
 
 459684 
 
 311665752 
 
 678 
 
 26-038 
 
 8-785 
 
 549081 
 
 406869021 
 
 741 
 
 27-221 
 
 9-049 
 
 461041 
 
 313046839 
 
 679 
 
 26-057 
 
 8-789 
 
 550564 
 
 408518488 
 
 742 
 
 27-239 
 
 9-053 
 
 462400 
 
 314432000 
 
 680 
 
 26-076 
 
 8-793 
 
 552049 
 
 410172407 
 
 743 
 
 27-258 
 
 9-057 
 
 463761 
 
 315821241 
 
 681 
 
 26-095 
 
 8-797 
 
 553536 
 
 411830784 
 
 744 
 
 27-276 
 
 9-061 
 
 465124 
 
 317214568 
 
 682 
 
 26-115 
 
 8-802 
 
 555025 
 
 413493625 
 
 745 
 
 27-294 
 
 9-065 
 
 466489 
 
 318611987 
 
 683 
 
 26-134 
 
 8-806 
 
 556516 
 
 415160936 
 
 746 
 
 27-313 
 
 9-069 
 
 467856 
 
 320013504 
 
 684 
 
 26-153 
 
 8-810 
 
 558009 
 
 416832723 
 
 747 
 
 27-331 
 
 9-073 
 
 469225 
 
 321419125 
 
 685 
 
 26-172 
 
 8-815 
 
 559504 
 
 418508992 
 
 748 
 
 27-349 
 
 9-077 
 
 470596 
 
 322828856 
 
 686 
 
 26-191 
 
 8-819 
 
 561001 
 
 420189749 
 
 749 
 
 27-367 
 
 9-081 
 
 471969 
 
 324242703 
 
 687. 
 
 26-210 
 
 8-823 
 
 562500 
 
 421875000 
 
 750 
 
 27-386 
 
 9-085 
 
 473344 
 
 325660672 
 
 688 
 
 26'22y 
 
 8-828 
 
 564001 
 
 423564751 
 
 751 
 
 27-404 
 
 9-089 
 
 474721 
 
 327082769 
 
 689 
 
 26-248 
 
 8-832 
 
 565504 
 
 425259008 
 
 752 
 
 27-422 
 
 9-093 
 
 476100 
 
 328509000 
 
 690 
 
 26-267 
 
 8-836 
 
 567009 
 
 426957777 
 
 753 
 
 27-440 
 
 9-097 
 
 477481 
 
 329939371 
 
 691 
 
 26-286 
 
 8'840 
 
 568516 
 
 428661064 
 
 754 
 
 27-459 
 
 9-101 
 
 478864 
 
 331373888 
 
 692 
 
 26-305 
 
 8-845 
 
 570025 
 
 430368875 
 
 755 
 
 27-477 
 
 9-105 
 
 480249 
 
 332812557 
 
 693 
 
 26-324 
 
 8-849 
 
 571536 
 
 432081216 
 
 756 
 
 27-495 
 
 9-109 
 
826 
 
 APPENDIX. 
 
 TABLE OF SQUARES, CUBES, SQUARE AND CUBE ROOTS OF NUMBERS (Continued). 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 573049 
 
 433798093 
 
 757 
 
 27-513 
 
 9-113 
 
 672400 
 
 551368000 
 
 820 
 
 28-635 
 
 9-359 
 
 574564 
 
 435519512 
 
 758 
 
 27-531 
 
 9-117 
 
 674041 
 
 553387661 
 
 821 
 
 28-653 
 
 9-363 
 
 576081 
 
 437245479 
 
 759 
 
 27-549 
 
 9-121 
 
 675684 
 
 555412248 
 
 822 
 
 28-670 
 
 9-367 
 
 677600 
 
 438976000 
 
 760 
 
 27-568 
 
 9-125 
 
 677329 
 
 557441767 
 
 823 
 
 28-687 
 
 9-371 
 
 579121 
 
 440711081 
 
 761 
 
 27-586 
 
 9-129 
 
 678976 
 
 559476224 
 
 824 
 
 28-705 
 
 9-375 
 
 580644 
 
 442450728 
 
 762 
 
 27-604 
 
 9-133 
 
 680625 
 
 561515625 
 
 825 
 
 28-722 
 
 9-378 
 
 582169 
 
 444194947 
 
 763 
 
 27-622 
 
 9-137 
 
 682276 
 
 563559976 
 
 826 
 
 28-740 
 
 9-382 
 
 583696 
 
 445943744 
 
 764 
 
 27-640 
 
 9-141 
 
 683929 
 
 565609283 
 
 827 
 
 28-757 
 
 9-386 
 
 685225 
 
 447697125 
 
 765 
 
 27-658 
 
 9-145 
 
 685584 
 
 567663552 
 
 828 
 
 28-774 
 
 9-390 
 
 586756 
 
 449455096 
 
 766 
 
 27-676 
 
 9-149 
 
 687241 
 
 569722789 
 
 829 
 
 28-792 
 
 9-394 
 
 .688289 
 
 451217663 
 
 767 
 
 27-694 
 
 9-153 
 
 688900 
 
 571787000 
 
 830 
 
 28-809 
 
 9-397 
 
 589824 
 
 452984832 
 
 768 
 
 27-712 
 
 9-157 
 
 690561 
 
 573856191 
 
 831 
 
 28-827 
 
 9-401 
 
 591361 
 
 454756609 
 
 769 
 
 27-730 
 
 9-161 
 
 692224 
 
 575930368 
 
 832 
 
 28-844 
 
 9-405 
 
 592900 
 
 456533000 
 
 770 
 
 27-748 
 
 9-165 
 
 693889 
 
 578009537 
 
 833 
 
 28-861 
 
 9-409 
 
 594441 
 
 458314011 
 
 771 
 
 27-766 
 
 9-169 
 
 695556 
 
 580093704 
 
 834 
 
 28-879 
 
 9-412 
 
 695984 
 
 460099648 
 
 772 
 
 27-784 
 
 9-173 
 
 697225 
 
 582182875 
 
 835 
 
 28-896 
 
 9-416 
 
 597529 
 
 461889917 
 
 773 
 
 27-802 
 
 9-177 
 
 698896 
 
 584277056 
 
 836 
 
 28-913 
 
 9-420 
 
 599076 
 
 463684824 
 
 774 
 
 27-820 
 
 9-181 
 
 700569 
 
 586376253 
 
 837 
 
 28-930 
 
 9-424 
 
 600625 
 
 465484375 
 
 775 
 
 27-833 
 
 9-185 
 
 702244 
 
 588480472 
 
 838 
 
 28-948 
 
 9-427 
 
 602176 
 
 467288576 
 
 776 
 
 27-856 
 
 9-189 
 
 703921 
 
 590589719 
 
 839 
 
 28-965 
 
 9-431 
 
 603729 
 
 469097433 
 
 777 
 
 27-874 
 
 9-193 
 
 705600 
 
 592704000 
 
 840 
 
 28-98?, 
 
 9-435 
 
 605284 
 
 470910952 
 
 778 
 
 27-892 
 
 9-197 
 
 707281 
 
 594823321 
 
 841 
 
 29-000 
 
 9-439 
 
 606841 
 
 472729139 
 
 779 
 
 27-910 
 
 9-201 
 
 708964 
 
 596947688 
 
 842 
 
 29-017 
 
 9-442 
 
 608400 
 
 474552000 
 
 780 
 
 27-928 
 
 9-205 
 
 710649 
 
 599077107 
 
 843 
 
 29-034 
 
 9-446 
 
 609961 
 
 476379541 
 
 781 
 
 27-946 
 
 9-209 
 
 712336 
 
 601211584 
 
 844 
 
 29-051 
 
 9-450 
 
 611524 
 
 478211768 
 
 782 
 
 27-964 
 
 9-213 
 
 714025 
 
 603351125 
 
 845 
 
 29-068 
 
 9-454 
 
 613089 
 
 480048687 
 
 783 
 
 27-982 
 
 9-216 
 
 715716 
 
 605495736 
 
 846 
 
 29-086 
 
 9-457 
 
 -614656 
 
 481890304 
 
 784 
 
 28-000 
 
 9-220 
 
 717409 
 
 607645423 
 
 847 
 
 29-103 
 
 9-461 
 
 616225 
 
 483736625 
 
 785 
 
 28-017 
 
 9-224 
 
 719104 
 
 609800192 
 
 848 
 
 29-120 
 
 9-465 
 
 617796 
 
 485587656 
 
 786 
 
 28-035 
 
 9-228 
 
 720801 
 
 611960049 
 
 849 
 
 29-137 
 
 9-468 
 
 619369 
 
 487443403 
 
 787 
 
 28-053 
 
 9-232 
 
 722500 
 
 614125000 
 
 850 
 
 29-154 
 
 9-472 
 
 620944 
 
 489303872 
 
 788 
 
 28-071 
 
 9-236 
 
 724201 
 
 616295051 
 
 851 
 
 29-171 
 
 9-476 
 
 622521 
 
 491169069 
 
 789 
 
 28-089 
 
 9-240 
 
 725904 
 
 618470208 
 
 852 
 
 29-189 
 
 9-480 
 
 624100 
 
 493039000 
 
 790 
 
 28-106 
 
 9-244 
 
 727609 
 
 620650477 
 
 853 
 
 29-206 
 
 9-483 
 
 625681 
 
 494913671 
 
 791 
 
 28-124 
 
 9-248 
 
 729316 
 
 622835864 
 
 854 
 
 29-223 
 
 9-487 
 
 627264 
 
 496793088 
 
 792 
 
 28-142 
 
 9-252 
 
 731025 
 
 625026375 
 
 855 
 
 29-240 
 
 9-491 
 
 628849 
 
 498677257 
 
 793 
 
 28-160 
 
 9-256 
 
 732736 
 
 627222016 
 
 856 
 
 29-257 
 
 9494 
 
 630436 
 
 500566184 
 
 794 
 
 28-178 
 
 9-259 
 
 734449 
 
 629422793 
 
 857 
 
 29-274 
 
 9-498 
 
 632025 
 
 502459875 
 
 795 
 
 28-195 
 
 9-263 
 
 736164 
 
 631628712 
 
 858 
 
 29-291 
 
 9-502 
 
 633616 
 
 504358336 
 
 796 
 
 28-213 
 
 9-267 
 
 737881 
 
 633839779 
 
 859 
 
 29-308 
 
 9-505 
 
 635209 
 
 506261573 
 
 797 
 
 28-231 
 
 9-271 
 
 739600 
 
 636056000 
 
 860 
 
 29-325 
 
 9-509 
 
 63680* 
 
 508169592 
 
 798 
 
 28-248 
 
 9-275 
 
 741321 
 
 638277381 
 
 861 
 
 29-342 
 
 9-513 
 
 638401 
 
 510082399 
 
 799 
 
 28-266 
 
 9-279 
 
 743044 
 
 640503928 
 
 862 
 
 29-359 
 
 9-517 
 
 640000 
 
 512000000 
 
 800 
 
 28-284 
 
 9-283 
 
 744769 
 
 642735647 
 
 863 
 
 29-376 
 
 9-520 
 
 641601 
 
 513922401 
 
 801 
 
 28-301 
 
 9-287 
 
 746496 
 
 644972544 
 
 864 
 
 29-393 
 
 9-524 
 
 643204 
 
 515849608 
 
 802 
 
 28-319 
 
 9-290 
 
 748225 
 
 647214625 
 
 865 
 
 29-410 
 
 9-528 
 
 644809 
 
 517781627 
 
 803 
 
 28-337 
 
 9-294 
 
 749956 
 
 649461896 
 
 866 
 
 29-427 
 
 9-531 
 
 646416 
 
 519718464 
 
 804 
 
 28-354 
 
 .9-298 
 
 751689 
 
 651714363 
 
 867 
 
 29-444 
 
 9-535 
 
 648025 
 
 521660125 
 
 805 
 
 28-372 
 
 9-302 
 
 753424 
 
 653972032 
 
 868 
 
 29-461 
 
 9-539 
 
 649636 
 
 523606616 
 
 806 
 
 28-390 
 
 9-306 
 
 755161 
 
 656234909 
 
 869 
 
 29-478 
 
 9-542 
 
 651249 
 
 525557943 
 
 807 
 
 28-407 
 
 9-310 
 
 756900 
 
 658503000 
 
 870 
 
 29-495 
 
 9-546 
 
 652864 
 
 527514112 
 
 808 
 
 28-425 
 
 9-314 
 
 758641 
 
 660776311 
 
 871 
 
 29-512 
 
 9-550 
 
 654481 
 
 529475129 
 
 809 
 
 28-442 
 
 9-317 
 
 760384 
 
 663054848 
 
 872 
 
 29-529 
 
 9-553 
 
 656100 
 
 531441000 
 
 810 
 
 28-460 
 
 9-321 
 
 762129 
 
 665338617 
 
 873 
 
 29-546 
 
 9-557 
 
 657721 
 
 533411731 
 
 811 
 
 28-478 
 
 9-325 
 
 763876 
 
 667627624 
 
 874 
 
 29-563 
 
 9-561 
 
 659344 
 
 535387328 
 
 812 
 
 28-495 
 
 9-329 
 
 765625 
 
 669921875 
 
 875 
 
 29-580 
 
 9-564 
 
 660969 
 
 537367797 
 
 813 
 
 28-513 
 
 9-333 
 
 767376 
 
 672221376 
 
 876 
 
 29-597 
 
 9-568 
 
 662596 
 
 539353144 
 
 814 
 
 28-530 
 
 9-337 
 
 769129 
 
 674526133 
 
 877 
 
 29-614 
 
 9-571 
 
 664225 
 
 541343375 
 
 815 
 
 28-548 
 
 9-340 
 
 770884 
 
 676836152 
 
 878 
 
 29-631 
 
 9-575 
 
 665856 
 
 543338496 
 
 816 
 
 28-565 
 
 9-344 
 
 772641 
 
 679151439 
 
 879 
 
 29-647 
 
 9-579 
 
 667489 
 
 545338513 
 
 817 
 
 28-583 
 
 9-348 
 
 774400 
 
 681472000 
 
 880 
 
 29-664 
 
 9-582 
 
 669124 
 
 647343432 
 
 818 
 
 28-600 
 
 9-352 
 
 776161 
 
 683797841 
 
 881 
 
 29-681 
 
 9-586 
 
 70761 
 
 549353259 
 
 819 
 
 28-618 
 
 9-356 
 
 777924 
 
 686128968 
 
 882 
 
 29-698 
 
 9-590 
 
APPENDIX. 
 
 82T 
 
 TABLE OF SQUAEES, CUBES, SQUAEE AND CUBE KOOTS OF NUMBERS ( Continued). 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 Squares. 
 
 Cubes. 
 
 No. 
 
 Square 
 roots. 
 
 Cube 
 roots. 
 
 779689 
 
 688465387 
 
 883 
 
 29-715 
 
 9-593 
 
 894916 
 
 846590536 
 
 946 
 
 30-757 
 
 9-816 
 
 781456 
 
 690807104 
 
 884 
 
 29-732 
 
 9-597 
 
 896808 
 
 849278123 
 
 947 
 
 30-773 
 
 9-820 
 
 783225 
 
 693154125 
 
 885 
 
 29-748 
 
 9-600 
 
 898704 
 
 851971392 
 
 948 
 
 30-789 
 
 9-823 
 
 784996 
 
 695506456 
 
 886 
 
 29-765 
 
 9-604 
 
 900601 
 
 854670349 
 
 949 
 
 30-805 
 
 9-827 
 
 786769 
 
 697864103 
 
 887 
 
 29-782 
 
 9-608 
 
 902500 
 
 857375000 
 
 950 
 
 30-822 
 
 9-830 
 
 788544 
 
 700227072 
 
 888 
 
 29-799 
 
 9-611 
 
 904401 
 
 860085351 
 
 951 
 
 30-838 
 
 9-833 
 
 790321 
 
 702595369 
 
 889 
 
 29-816 
 
 9-615 
 
 906304 
 
 862801408 
 
 952 
 
 30-854 
 
 9-837 
 
 792100 
 
 704969000 
 
 890 
 
 29-832 
 
 9-619 
 
 908209 
 
 865523177 
 
 953 
 
 30-870 
 
 9-840 
 
 793881 
 
 707347971 
 
 891 
 
 29-849 
 
 9-622 
 
 910116 
 
 868250664 
 
 954 
 
 80-886 
 
 9-844 
 
 795664 
 
 709732288 
 
 892 
 
 29-866 
 
 9-626 
 
 912025 
 
 870983875 
 
 955 
 
 30-903 
 
 9-847 
 
 797449 
 
 712121957 
 
 893 
 
 29-883 
 
 9-629 
 
 913936 
 
 873722816 
 
 956 
 
 30-919 
 
 9-851 
 
 799236 
 
 714516984 
 
 894 
 
 29-899 
 
 9-633 
 
 915849 
 
 876467493 
 
 957 
 
 80-935 
 
 9-854 
 
 801025 
 
 716917375 
 
 895 
 
 29-916 
 
 9-636 
 
 917764 
 
 879217912 
 
 958 
 
 30-951 
 
 9-857 
 
 802816 
 
 719323136 
 
 896 
 
 29-933 
 
 9-640 
 
 919681 
 
 881974079 
 
 959 
 
 30-967 
 
 9-861 
 
 804609 
 
 721734273 
 
 897 
 
 29-949 
 
 9-644 
 
 921600 
 
 884736000 
 
 960 
 
 30-983 
 
 9-864 
 
 806404 
 
 724150792 
 
 898 
 
 29-966 
 
 9-647 
 
 923521 
 
 887503681 
 
 961 
 
 31-000 
 
 9-868 
 
 808201 
 
 726572699 
 
 899 
 
 29-983 
 
 9-651 
 
 925444 
 
 890277128 
 
 962 
 
 31-016 
 
 9-871 
 
 810000 
 
 729000000 
 
 900 
 
 30-000 
 
 9-654 
 
 927369 
 
 893056347 
 
 963 
 
 31-032 
 
 9-875 
 
 811801 
 
 731432701 
 
 901 
 
 30-016 
 
 9-658 
 
 929296 
 
 895841344 
 
 964 
 
 31-048 
 
 9-878 
 
 813604 
 
 733870808 
 
 902 
 
 30-033 
 
 9-662 
 
 931225 
 
 898632125 
 
 965 
 
 31-064 
 
 9-881 
 
 815409 
 
 736314327 
 
 903 
 
 30-049 
 
 9-665 
 
 933156 
 
 901428696 
 
 966 
 
 31-080 
 
 9-885 
 
 817216 
 
 788763264 
 
 904 
 
 30-066 
 
 9-669 
 
 935089 
 
 904231063 
 
 967 
 
 31-096 
 
 9-888 
 
 819025 
 
 741217625 
 
 905 
 
 30-083 
 
 9-672 
 
 937024 
 
 907039232 
 
 968 
 
 31-112 
 
 9-892 
 
 820836 
 
 743677416 
 
 906 
 
 30-099 
 
 9-676 
 
 938961 
 
 909853209 
 
 969 
 
 31-128 
 
 9-895 
 
 822649 
 
 746142643 
 
 907 
 
 30-116 
 
 9-679 
 
 940900 
 
 912673000 
 
 970 
 
 31-144 
 
 9-898 
 
 824464 
 
 748613312 
 
 908 
 
 30-133 
 
 9-683 
 
 942841 
 
 915498611 
 
 971 
 
 3T160 
 
 9-902 
 
 826281 
 
 751089429 
 
 909 
 
 30-149 
 
 9-686 
 
 944784 
 
 918330048 
 
 972 
 
 31-176 
 
 9-905 
 
 828100 
 
 753571000 
 
 910 
 
 30-166 
 
 9-690 
 
 946729 
 
 921167317 
 
 973 
 
 3T192 
 
 9-909 
 
 829921 
 
 756058031 
 
 911 
 
 30-182 
 
 9-694 
 
 948676 
 
 924010424 
 
 974 
 
 31-208 
 
 9-912 
 
 831744 
 
 758550528 
 
 912 
 
 30-199 
 
 9-697 
 
 950625 
 
 926859375 
 
 975 
 
 31-224 
 
 9-915 
 
 833569 
 
 761048497 
 
 913 
 
 30-215 
 
 9-701 
 
 952576 
 
 929714176 
 
 976 
 
 31-240 
 
 9-919 
 
 835396 
 
 763551944 
 
 914 
 
 30-232 
 
 9-704 
 
 954529 
 
 932574833 
 
 977 
 
 31-256 
 
 9-922 
 
 837225 
 
 766060875 
 
 915 
 
 30-248 
 
 9-708 
 
 956484 
 
 935441352 
 
 978 
 
 31-272 
 
 9-926 
 
 839056 
 
 768575296 
 
 916 
 
 30-265 
 
 9-711 
 
 958441 
 
 938313739 
 
 979 
 
 31-288 
 
 9-929 
 
 840889 
 
 771095213 
 
 917 
 
 30-282 
 
 9-715 
 
 960400 
 
 941192000 
 
 980 
 
 31-304 
 
 9-932 
 
 842724 
 
 773620632 
 
 918 
 
 30-298 
 
 9-718 
 
 962361 
 
 944076141 
 
 981 
 
 31-320 
 
 9-936 
 
 844561 
 
 776151559 
 
 919 
 
 30-315 
 
 9-722 
 
 964324 
 
 946966168 
 
 982 
 
 31-386 
 
 9939 
 
 846400 
 
 778688000 
 
 920 
 
 30-331 
 
 9-725 
 
 966289 
 
 949862087 
 
 983 
 
 31-852 
 
 9-943 
 
 848241 
 
 781229961 
 
 921 
 
 30-347 
 
 9729 
 
 968256 
 
 952763904 
 
 984 
 
 81-368 
 
 9-946 
 
 850084 
 
 783777448 
 
 922 
 
 30-364 
 
 9-732 
 
 970225 
 
 955671625 
 
 985 
 
 31-384 
 
 9-949 
 
 851929 
 
 786330467 
 
 923 
 
 30-380 
 
 9 ; 736 
 
 972196 
 
 958585256 
 
 986 
 
 31-400 
 
 9-953 
 
 853776 
 
 788889024 
 
 924 
 
 30-397 
 
 9-739 
 
 974169 
 
 961504803 
 
 987 
 
 31-416 
 
 9-956 
 
 855625 
 
 791453125 
 
 925 
 
 30-413 
 
 9-743 
 
 976144 
 
 964430272 
 
 988 
 
 31-432 
 
 9-959 
 
 857476 
 
 794022776 
 
 926 
 
 30-430 
 
 9-746 
 
 978121 
 
 967361669 
 
 989 
 
 31-448 
 
 9-963 
 
 859329 
 
 796597983 
 
 927" 
 
 30-446 
 
 9-750 
 
 980100 
 
 970299000 
 
 990 
 
 31-464 
 
 9-966 
 
 861184 
 
 799178752 
 
 928 
 
 30-463 
 
 9-753 
 
 982081 
 
 973242271 
 
 991 
 
 31-480 
 
 9-969 
 
 863041 
 
 801765089 
 
 929 
 
 30-479 
 
 9-757 
 
 984064 
 
 976191488 
 
 992 
 
 31-496 
 
 9-973 
 
 864900 
 
 804357000 
 
 930 
 
 30-495 
 
 9-761 
 
 986049 
 
 979146657 
 
 993 
 
 31-511 
 
 9-976 
 
 866761 
 
 806954491 
 
 931 
 
 30-512 
 
 9-764 
 
 988036 
 
 982107784 
 
 994 
 
 31-527 
 
 9-979 
 
 868624 
 
 809557568 
 
 932 
 
 30-528 
 
 9-767 
 
 990025 
 
 985074875 
 
 995 
 
 31-543 
 
 9-983 
 
 870489 
 
 812166237 
 
 933 
 
 30-545 
 
 9-771 
 
 992016 
 
 988047936 
 
 996 
 
 31-559 
 
 9-986 
 
 872356 
 
 814780504 
 
 934 
 
 30-561 
 
 9-774 
 
 994009 
 
 991026973 
 
 997 
 
 31-575 
 
 9-989 
 
 874225 
 
 817400375 
 
 935 
 
 30-577 
 
 9-778 
 
 996004 
 
 994011992 
 
 998 
 
 31-591 
 
 9-993 
 
 876096 
 
 820025856 
 
 936 
 
 30-594 
 
 9-782 
 
 998001 
 
 997002999 
 
 999 
 
 31-606 
 
 9-996 
 
 877969 
 
 822656953 
 
 937 
 
 30-610 
 
 9-785 
 
 1000000 
 
 1000000000 
 
 1000 
 
 31-622 
 
 10-000 
 
 879844 
 
 825293672 
 
 938 
 
 30-626 
 
 9-788 
 
 1000201 
 
 1003003001 
 
 1001 
 
 31-638 
 
 10-003 
 
 881721 
 
 827936019 
 
 939 
 
 30-643 
 
 9-792 
 
 1004004 
 
 1006012008 
 
 1002 
 
 31-654 
 
 10-006 
 
 883600 
 
 830584000 
 
 940 
 
 30-659 
 
 9-795 
 
 1006009 
 
 1009027027 
 
 1003 
 
 31-670 
 
 10-009 
 
 885481 
 
 833237621 
 
 941 
 
 30-676 
 
 9-799 
 
 1008016 
 
 1012048064 
 
 1004 
 
 31-685 
 
 10-013 
 
 887364 
 
 835896888 
 
 942 
 
 30-692 
 
 9-802 
 
 1010025 
 
 1015075125 
 
 1005 
 
 31-701 
 
 10-016 
 
 889249 
 
 838561807 
 
 943 
 
 30-708 
 
 9-806 
 
 1012036 
 
 1018108216 
 
 1006 
 
 31-717 
 
 10-019 
 
 891136 
 
 841232384 
 
 944 
 
 30-724 
 
 9-809 
 
 1014049 
 
 1021147343 
 
 1007 
 
 31-733 
 
 10-023 
 
 893025 
 
 843908625 
 
 945 
 
 30-740 9-813 
 
 1016064 
 
 1024192512 1008 
 
 31-749 
 
 10-026 
 
828 
 
 APPENDIX. 
 TABLE OP RECIPROCALS. 
 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 oo 
 
 10-000 
 
 5-0000 
 
 3-3333 
 
 2-5000 
 
 2-0000 
 
 1-6667 
 
 1-4286 
 
 1-2500 
 
 1-1111 
 
 1 
 
 1-00000 
 
 90909 
 
 83333 
 
 76923 
 
 71428 
 
 66667 
 
 62500 
 
 58823 
 
 55555 
 
 52631 
 
 2 
 
 50000 
 
 47619 
 
 45454 
 
 43478 
 
 41667 
 
 40000 
 
 38461 
 
 37037 
 
 35714 
 
 34483 
 
 3 
 
 33333 
 
 32258 
 
 31250 
 
 30303 
 
 29412 
 
 28571 
 
 27778 
 
 27027 
 
 26316 
 
 25641 
 
 4 
 
 25000 
 
 24390 
 
 23810 
 
 23256 
 
 22727 
 
 22222 
 
 21739 
 
 21277 
 
 2083.3 
 
 20408 
 
 5 
 
 20000 
 
 19608 
 
 19231 
 
 18868 
 
 18519 
 
 18182 
 
 17857 
 
 17544 
 
 17241 
 
 16949 
 
 6 
 
 16667 
 
 16393 
 
 16129 
 
 15873 
 
 15625 
 
 15385 
 
 15152 
 
 14925 
 
 14706 
 
 14493 
 
 7 
 
 14286 
 
 14085 
 
 13889 
 
 13699 
 
 13514 
 
 13333 
 
 13158 
 
 12987 
 
 12821 
 
 12658 
 
 8 
 
 12500 
 
 12346 
 
 12195 
 
 12048 
 
 11905 
 
 11765 
 
 11628 
 
 11494 
 
 11364 
 
 11236 
 
 9 
 
 11111 
 
 10989 
 
 10870 
 
 10753 
 
 10638 
 
 10526 
 
 10417 
 
 10309 
 
 10204 
 
 10101 
 
 10 
 
 10000 
 
 09901 
 
 09804 
 
 09709 
 
 09615 
 
 09524 
 
 09434 
 
 09346 
 
 09259 
 
 09174 
 
 11 
 
 09091 
 
 09009 
 
 08929 
 
 08850 
 
 08772 
 
 08696 
 
 08621 
 
 08547 
 
 08475 
 
 08403 
 
 12 
 
 08333 
 
 08264 
 
 08197 
 
 08130 
 
 08065 
 
 08000 
 
 07937 
 
 07874 
 
 07813 
 
 07752 
 
 13 
 
 07692 
 
 07634 
 
 07576 
 
 07519 
 
 07463 
 
 07407 
 
 07353 
 
 07299 
 
 07246 
 
 07194 
 
 14 
 
 07143 
 
 07092 
 
 07042 
 
 06993 
 
 06944 
 
 06897 
 
 06849 
 
 06803 
 
 06757 
 
 06711 
 
 15 
 
 06667 
 
 06623 
 
 06579 
 
 06536 
 
 06494 
 
 06452 
 
 06410 
 
 06369 
 
 06329 
 
 06289 
 
 16 
 
 06250 
 
 06211 
 
 06173 
 
 06135 
 
 06098 
 
 06061 
 
 06024 
 
 05988 
 
 05952 
 
 05917 
 
 17 
 
 05882 
 
 05848 
 
 05814 
 
 05780 
 
 05747 
 
 05714 
 
 05682 
 
 05650 
 
 05618 
 
 05587 
 
 18 
 
 05556 
 
 05525 
 
 05495 
 
 05464 
 
 05435 
 
 05405 
 
 05376 
 
 05348 
 
 05319 
 
 05291 
 
 19 
 
 05263 
 
 05236 
 
 05208 
 
 05181 
 
 05155 
 
 05128 
 
 05102 
 
 05076 
 
 05051 
 
 05025 
 
 20 
 
 05000 
 
 04975 
 
 04950 
 
 04926 
 
 04902 
 
 04878 
 
 04854 
 
 04831 
 
 04808 
 
 04785 
 
 21 
 
 04762 
 
 04739 
 
 04717 
 
 04695 
 
 04673 
 
 04651 
 
 04630 
 
 04608 
 
 04587 
 
 04566 
 
 22 
 
 04545 
 
 04525 
 
 04505 
 
 04484 
 
 04464 
 
 04444 
 
 04425 
 
 04405 
 
 04386 
 
 04367 
 
 23 
 
 04348 
 
 04329 
 
 04310 
 
 04292 
 
 04274 
 
 04255 
 
 04237 
 
 04219 
 
 04202 
 
 04184 
 
 24 
 
 04167 
 
 04149 
 
 04132 
 
 04115 
 
 04098 
 
 04082 
 
 04065 
 
 04049 
 
 04032 
 
 04016 
 
 25 
 
 04000 
 
 03984 
 
 03968 
 
 03953 
 
 03937 
 
 03922 
 
 03906 
 
 03891 
 
 03876 
 
 03861 
 
 26 
 
 03846 
 
 03831 
 
 03817 
 
 03802 
 
 03788 
 
 03774 
 
 03759 
 
 03745 
 
 03731 
 
 03717 
 
 27 
 
 03704 
 
 03690 
 
 03676 
 
 03663 
 
 03650 
 
 03636 
 
 03623 
 
 03610 
 
 03597 
 
 03584 
 
 28 
 
 03571 
 
 03559 
 
 03546 
 
 03534 
 
 03521 
 
 03509 
 
 03497 
 
 03484 
 
 03472 
 
 03460 
 
 29 
 
 03448 
 
 03436 
 
 03425 
 
 03413 
 
 03401 
 
 03390 
 
 03378 
 
 03367 
 
 03356 
 
 03344 
 
 30 
 
 03333 
 
 03322 
 
 03311 
 
 03300 
 
 03289 
 
 03279 
 
 03268 
 
 03257 
 
 03247 
 
 03236 
 
 31 
 
 03226 
 
 03215 
 
 03205 
 
 03195 
 
 03185 
 
 03175 
 
 03165 
 
 03155 
 
 03145 
 
 03135 
 
 32 
 
 03125 
 
 03115 
 
 03106 
 
 03096 
 
 03086 
 
 03077 
 
 03067 
 
 03058 
 
 03049 
 
 03040 
 
 33 
 
 03030 
 
 03021 
 
 03012 
 
 03003 
 
 02994 
 
 02985 
 
 02976 
 
 02967 
 
 02959 
 
 02950 
 
 34 
 
 02941 
 
 02933 
 
 02924 
 
 02915 
 
 02907 
 
 02899 
 
 02890 
 
 02882 
 
 02874 
 
 02865 
 
 35 
 
 02857 
 
 02849 
 
 02841 
 
 02833 
 
 02825 
 
 02817 
 
 02809 
 
 02801 
 
 02793 
 
 02786 
 
 36 
 
 02778 
 
 02770 
 
 02762 
 
 02755 
 
 02747 
 
 02740 
 
 02732 
 
 02725 
 
 02717 
 
 02710 
 
 37 
 
 02703 
 
 02695 
 
 02688 
 
 02681 
 
 02674 
 
 02667 
 
 02660 
 
 02653 
 
 02646 
 
 02639 
 
 38 
 
 02632 
 
 02625 
 
 02618 
 
 02611 
 
 02604 
 
 02597 
 
 02591 
 
 02584 
 
 02577 
 
 02571 
 
 39 
 
 02564 
 
 02558 
 
 02551 
 
 02545 
 
 02538 
 
 02532 
 
 02525 
 
 02519 
 
 02513 
 
 02506 
 
 40 
 
 02500 
 
 02494 
 
 02488 
 
 02481 
 
 02475 
 
 02469 
 
 02463 
 
 02457 
 
 02451 
 
 02445 
 
 41 
 
 02439 
 
 02433 
 
 02427 
 
 02421 
 
 02415 
 
 02410 
 
 02404 
 
 02398 
 
 02392 
 
 02387 
 
 42 
 
 02381 
 
 02375 
 
 02370 
 
 02364 
 
 02358 
 
 02353 
 
 02347 
 
 02342 
 
 02336 
 
 02331 
 
 43 
 
 02326 
 
 02320 
 
 02315 
 
 02309 
 
 02304 
 
 02299 
 
 02294 
 
 -02288 
 
 02283 
 
 02278 
 
 44 
 
 02273 
 
 02268 
 
 02262 
 
 02257 
 
 02252 
 
 02247 
 
 02242 
 
 02237 
 
 02232 
 
 08227 
 
 45 
 
 02222 
 
 02217 
 
 02212 
 
 02208 
 
 02203 
 
 02198 
 
 02193 
 
 02188 
 
 02183 
 
 02179 
 
 46 
 
 02174 
 
 02169 
 
 02165 
 
 02160 
 
 02155 
 
 02151 
 
 02146 
 
 02141 
 
 02137 
 
 02132 
 
 47 
 
 02128 
 
 02123 
 
 02119 
 
 02114 
 
 02110 
 
 02105 
 
 02101 
 
 02096 
 
 02092 
 
 02088 
 
 48 
 
 02083 
 
 02079 
 
 02075 
 
 02070 
 
 02066 
 
 02062 
 
 02058 
 
 02053 
 
 02049 
 
 02045 
 
 49 
 
 02041 
 
 02037 
 
 02033 
 
 02028 
 
 02024 
 
 02020 
 
 02016 
 
 02012 
 
 02008 
 
 02004 
 
 
 o 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
APPENDIX. 
 TABLE OF RECIPROCALS Continued. 
 
 829 
 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 50- 
 
 02000 
 
 01996 
 
 01992 
 
 01988 
 
 01984 
 
 01980 
 
 01976 
 
 01972 
 
 01969 
 
 01965 
 
 51 
 
 01961 
 
 01957 
 
 01953 
 
 01949 
 
 01946 
 
 01942 
 
 01938 
 
 01934 
 
 01931 
 
 01927 
 
 52 
 
 01923 
 
 01919 
 
 01916 
 
 01912 
 
 01908 
 
 01905 
 
 01901 
 
 01898 
 
 01894 
 
 01890 
 
 53 
 
 01887 
 
 01883 
 
 01880 
 
 01876 
 
 01873 
 
 01869 
 
 01866 
 
 01862 
 
 01859 
 
 . -01855 
 
 54 
 
 01852 
 
 01848 
 
 01845 
 
 01842 
 
 01838 
 
 01835 
 
 01832 
 
 01828 
 
 01825 
 
 01821 
 
 55 
 
 01818 
 
 01815 
 
 01812 
 
 01808 
 
 01805 
 
 01802 
 
 01799 
 
 01795 
 
 01792 
 
 01789 
 
 56 
 
 01786 
 
 01783 
 
 01779 
 
 01776 
 
 01773 
 
 01770 
 
 01767 
 
 01764 
 
 01761 
 
 01757 
 
 57 
 
 01754 
 
 01751 
 
 01748 
 
 01745 
 
 01742 
 
 01739 
 
 01736 
 
 01733 
 
 01730 
 
 01727 
 
 58 
 
 01724 
 
 01721 
 
 01718 
 
 01715 
 
 01712 
 
 01709 
 
 01706 
 
 01704 
 
 01701 
 
 01698 
 
 59 
 
 01695 
 
 01692 
 
 01689 
 
 01686 
 
 01684 
 
 01681 
 
 01678 
 
 01675 
 
 01672 
 
 01669 
 
 60 
 
 01667 
 
 01664 
 
 01661 
 
 01658 
 
 01656 
 
 01653 
 
 01650 
 
 01647 
 
 01645 
 
 01642 
 
 61 
 
 01639 
 
 01637 
 
 01634 
 
 01631 
 
 01629 
 
 01626 
 
 01623 
 
 01621 
 
 01618 
 
 01616 
 
 62 
 
 01613 
 
 01610 
 
 01608 
 
 01605 
 
 01603 
 
 01600 
 
 01597 
 
 01595 
 
 01592 
 
 01590 
 
 63 
 
 01587 
 
 01585 
 
 01582 
 
 01580 
 
 01577 
 
 01575 
 
 01572 
 
 01570 
 
 01567 
 
 01565 
 
 64 
 
 01563 
 
 01560 
 
 01558 
 
 01555 
 
 01553 
 
 01550 
 
 01548 
 
 01546 
 
 01543 
 
 01541 
 
 65 
 
 01538 
 
 01536 
 
 01534 
 
 01531 
 
 01529 
 
 01527 
 
 01524 
 
 01522 
 
 01520 
 
 01517 
 
 66 
 
 01515 
 
 01513 
 
 01511 
 
 01508 
 
 01506 
 
 01504 
 
 01502 
 
 01499 
 
 01497 
 
 01495 
 
 67 
 
 01493 
 
 01490 
 
 01488 
 
 01486 
 
 01484 
 
 01481 
 
 01479 
 
 01477 
 
 01475 
 
 01473 
 
 68 
 
 01471 
 
 01468 
 
 01466 
 
 01464 
 
 01462 
 
 01460 
 
 01458 
 
 01456 
 
 01453 
 
 01451 
 
 69 
 
 01449 
 
 01447 
 
 01445 
 
 01443 
 
 01441 
 
 01439 
 
 01437 
 
 01435 
 
 01433 
 
 01431 
 
 70 
 
 01429 
 
 01427 
 
 01425 
 
 01422 
 
 01420 
 
 01418 
 
 01416 
 
 01414 
 
 01412 
 
 01410 
 
 71 
 
 01408 
 
 01406 
 
 01404 
 
 01403 
 
 01401 
 
 01399 
 
 01397 
 
 01395 
 
 01393 
 
 01391 
 
 72 
 
 01389 
 
 01387 
 
 01385 
 
 01383 
 
 01381 
 
 01379 
 
 01377 
 
 01376 
 
 01374 
 
 01372 
 
 73 
 
 01370 
 
 01368 
 
 01366 
 
 01364 
 
 01362 
 
 01361 
 
 01359 
 
 01357 
 
 01355 
 
 01353 
 
 74 
 
 01351 
 
 01350 
 
 01348 
 
 01346 
 
 01344 
 
 01342 
 
 01340 
 
 01339 
 
 01337 
 
 01335 
 
 75 
 
 01333 
 
 01332 
 
 01330 
 
 01328 
 
 01326 
 
 01325 
 
 01323 
 
 01321 
 
 01319 
 
 01318 
 
 76 
 
 01316 
 
 01314 
 
 01312 
 
 01311 
 
 01309 
 
 01307 
 
 01305 
 
 01304 
 
 01302 
 
 01300 
 
 77 
 
 01299 
 
 01297 
 
 01295 
 
 01294 
 
 01292 
 
 01290 
 
 01289 
 
 01287 
 
 01285 
 
 01284 
 
 78 
 
 01282 
 
 01280 
 
 01279 
 
 01277 
 
 01276 
 
 01274 
 
 01272 
 
 01271 
 
 01269 
 
 01267 
 
 79 
 
 01266 
 
 01264 
 
 01263 
 
 01261 
 
 01259 
 
 01258 
 
 01256 
 
 01255 
 
 01253 
 
 01252 
 
 80 
 
 01250 
 
 01248 
 
 01247 
 
 01245 
 
 01244 
 
 01242 
 
 01241 
 
 01239 
 
 01238 
 
 01236 
 
 81 
 
 01235 
 
 01233 
 
 01232 
 
 01230 
 
 01229 
 
 01227 
 
 01225 
 
 01224 
 
 01222 
 
 01221 
 
 82 
 
 01220 
 
 01218 
 
 01217 
 
 01215 
 
 01214 
 
 01212 
 
 01211 
 
 01209 
 
 01208 
 
 01206 
 
 83 
 
 01205 
 
 01203 
 
 01202 
 
 01200 
 
 01199 
 
 01198 
 
 01196 
 
 01195 
 
 01193 
 
 01192 
 
 84 
 
 01190 
 
 01189 
 
 01188 
 
 01186 
 
 01185 
 
 01183 
 
 01182 
 
 01181 
 
 01179 
 
 01178 
 
 85 
 
 01176 
 
 01175 
 
 01174 
 
 01172 
 
 01171 
 
 01170 
 
 01168 
 
 01167 
 
 01166 
 
 01164 
 
 86 
 
 01163 
 
 01161 
 
 01160 
 
 01159 
 
 01157 
 
 01156 
 
 01155 
 
 01153 
 
 01152 
 
 01151 
 
 87 
 
 01149 
 
 01148 
 
 01147 
 
 01145 
 
 01144 
 
 01143 
 
 01142 
 
 01140 
 
 01139 
 
 01138 
 
 88 
 
 01136 
 
 01135 
 
 01134 
 
 01133 
 
 01131 
 
 01130 
 
 01129 
 
 01127 
 
 01126 
 
 01125 
 
 89 
 
 01124 
 
 01122 
 
 01121 
 
 01120 
 
 01119 
 
 01117 
 
 01116 
 
 01115 
 
 01114 
 
 01112 
 
 90 
 
 01111 
 
 OHIO 
 
 01109 
 
 01107 
 
 01106 
 
 01105 
 
 01104 
 
 01103 
 
 01101 
 
 01100 
 
 91 
 
 01099 
 
 01098 
 
 01096 
 
 01095 
 
 01094 
 
 01093 
 
 01092 
 
 01091 
 
 01089 
 
 01088 
 
 92 
 
 01087 
 
 01086 
 
 01085 
 
 01083 
 
 01082 
 
 01081 
 
 01080 
 
 01079 
 
 01078 
 
 01076 
 
 93 
 
 01075 
 
 01074 
 
 01073 
 
 01072 
 
 01071 
 
 01070 
 
 01068 
 
 01067 
 
 01066 
 
 01065 
 
 94 
 
 01064 
 
 01063 
 
 01062 
 
 01060 
 
 01059 
 
 01058 
 
 01057 
 
 01056 
 
 01055 
 
 01054 
 
 95 
 
 01053 
 
 01052 
 
 01050 
 
 01049 
 
 01048 
 
 01047 
 
 01046 
 
 01045 
 
 01044 
 
 01043 
 
 96 
 
 01042 
 
 01041 
 
 01040 
 
 01038 
 
 01037 
 
 01036 
 
 01035 
 
 01034 
 
 01033 
 
 01032 
 
 97 
 
 01031 
 
 01030 
 
 01029 
 
 01028 
 
 01027 
 
 01026 
 
 01025 
 
 01024 
 
 01022 
 
 01021 
 
 98 
 
 01020 
 
 01019 
 
 01018 
 
 01017 
 
 01016 
 
 01015 
 
 01014 
 
 01013 
 
 01012 
 
 01011 
 
 99 
 
 01010 
 
 01009 
 
 01008 
 
 01007 
 
 01006 
 
 01005 
 
 01004 
 
 01003 
 
 01002 
 
 01001 
 
 
 o 
 
 1 
 
 2 
 
 3 
 
 4 
 
 6 
 
 6 
 
 7 
 
 8 
 
 9 
 
830 
 
 APPENDIX. 
 
 LATITUDES AND DEPARTURES. 
 
 1 
 
 1 
 
 2 
 
 3 
 
 4 
 
 S 
 
 If 
 
 1 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 1 
 
 0~ 
 
 1-000 
 
 o-ooo 
 
 2-000 
 
 o-ooo 
 
 3-000 
 
 o-ooo 
 
 4-000 
 
 o-ooo 
 
 5-000 
 
 90 
 
 Oir 
 
 1-000 
 
 0-004 
 
 2-000 
 
 0-009 
 
 3-000 
 
 0-013 
 
 4-000 
 
 0-017 
 
 5-000 
 
 89| 
 
 o* 
 
 1-000 
 
 0-009 
 
 2-000 
 
 0-017 
 
 3-000 
 
 0-026 
 
 4-000 
 
 0-035 
 
 5-000 
 
 894 
 
 Of 
 
 rooo 
 
 0-013 
 
 2-000 
 
 0-026 
 
 3-000 
 
 0-039 
 
 4-000 
 
 0-052 
 
 5-000 
 
 89 
 
 1 
 
 1-000 
 
 0-017 
 
 2-000 
 
 0-035 
 
 3-000 
 
 0-052 
 
 3-999 
 
 0-070 
 
 4-999 
 
 89' 
 
 1 
 
 1-000 
 
 0-022 
 
 2-000 
 
 0-044 
 
 2-999 
 
 0-065 
 
 3-999 
 
 0-087 
 
 4-999 
 
 88f 
 
 14 
 
 1-000 
 
 0-026 
 
 1-999 
 
 0-052 
 
 2-999 
 
 0-079 
 
 3-999 
 
 0-105 
 
 4-998 
 
 884 
 
 if 
 
 1-000 
 
 0-031 
 
 1-999 
 
 0-061 
 
 2-999 
 
 0-092 
 
 3-998 
 
 0-122 
 
 4-998 
 
 88| 
 
 2 
 
 0-999 
 
 0-035 
 
 1-999 
 
 0-070 
 
 2-998 
 
 0-105 
 
 3-998 
 
 0-140 
 
 4-997 
 
 88 
 
 aj 
 
 0-999 
 
 0-039 
 
 1-998 
 
 0-079 
 
 2-998 
 
 0-118 
 
 3-997 
 
 0-157 
 
 4-996 
 
 87|- 
 
 a* 
 
 0-999 
 
 0-044 
 
 1-998 
 
 0-087 
 
 2-997 
 
 0-131 
 
 3-996 
 
 0-174 
 
 4-995 
 
 874 
 
 2f 
 
 0-999 
 
 0-048 
 
 1-998 
 
 0-096 
 
 2-997 
 
 0-144 
 
 3-995 
 
 0-192 
 
 4-994 
 
 874- 
 
 3 
 
 0-999 
 
 0-052 
 
 1-997 
 
 0-105 
 
 2-996 
 
 0-157 
 
 3-995 
 
 0-209 
 
 4-993 
 
 87' 
 
 3* 
 
 0-998 
 
 0-057 
 
 1-997 
 
 0-113 
 
 2-995 
 
 0-170 
 
 3-994 
 
 0-227 
 
 4-992 
 
 86f 
 
 3* 
 
 0-998 
 
 0-061 
 
 1-996 
 
 0-122 
 
 2-994 
 
 0-183 
 
 3-993 
 
 0-244 
 
 4-991 
 
 864 
 
 81 
 
 0-998 
 
 0-065 
 
 1-996 
 
 0-131 
 
 2-994 
 
 0-196 
 
 3-991 
 
 0-262 
 
 4-989 
 
 86 
 
 4 
 
 0-998 
 
 0-070 
 
 1995 
 
 0-140 
 
 2-993 
 
 0-209 
 
 3-990 
 
 0-279 
 
 4-988 
 
 86 
 
 *i 
 
 0-997 
 
 0-074 
 
 1-995 
 
 0-148 
 
 2-992 
 
 0-222 
 
 3-989 
 
 0-296 
 
 4-986 
 
 85f- 
 
 *i 
 
 0-997 
 
 0-078 
 
 1-994 
 
 0-157 
 
 2-991 
 
 0-235 
 
 3-988 
 
 0-314 
 
 4-985 
 
 854 
 
 4f 
 
 0-997 
 
 0-083 
 
 1-993 
 
 0-166 
 
 2-990 
 
 0-248 
 
 3-986 
 
 0-331 
 
 4-983 
 
 85 
 
 5 
 
 0-996 
 
 0-087 
 
 1-992 
 
 0-174 
 
 2-989 
 
 0-261 
 
 3-985 
 
 0-349 
 
 4-981 
 
 85 
 
 i 
 
 0-996 
 
 0-092 
 
 1-992 
 
 0-183 
 
 2-987 
 
 0-275 
 
 3-983 
 
 0-366 
 
 4-979 
 
 84f 
 
 5* 
 
 0-995 
 
 0-096 
 
 1-991 
 
 0-192 
 
 2-986 
 
 0-288 
 
 3-982 
 
 0-383 
 
 4-977 
 
 844 
 
 5f 
 
 0-995 
 
 0-100 
 
 1-990 
 
 0-200 
 
 2-985 
 
 0-301 
 
 3-980 
 
 0-401 
 
 4-875 
 
 84i 
 
 6 
 
 0-995 
 
 0-105 
 
 1-989 
 
 0-209 
 
 2-984 
 
 0-314 
 
 3-978 
 
 0-418 
 
 4-973 
 
 84 
 
 6 
 
 0-994 
 
 0-109 
 
 1-988 
 
 0-218 
 
 2-982 
 
 0-327 
 
 3-976 
 
 0-435 
 
 4-970 
 
 83fc 
 
 64 
 
 0-994 
 
 0-113 
 
 1-987 
 
 0-226 
 
 2-981 
 
 0-340 
 
 3-974 
 
 0-453 
 
 4-968 
 
 834 
 
 6| 
 
 0-993 
 
 0-118 
 
 1-986 
 
 0-235 
 
 2-979 
 
 0-353 
 
 3-972 
 
 0-470 
 
 4-965 
 
 83 
 
 7 
 
 0-993 
 
 0-122 
 
 1-985 
 
 0-244 
 
 2-978 
 
 0-366 
 
 3-970 
 
 0-487 
 
 4-963 
 
 83 
 
 *i 
 
 0-992 
 
 0-126 
 
 1-984 
 
 0-252 
 
 2-976 
 
 0-379 
 
 3-968 
 
 0-505 
 
 4-960 
 
 82f- 
 
 n 
 
 0-991 
 
 0-131 
 
 1-983 
 
 0-261 
 
 2-974 
 
 0-392 
 
 3-966 
 
 0-522 
 
 4-957 
 
 824 
 
 7 t 
 
 0-991 
 
 0-135 
 
 1-982 
 
 0-270 
 
 2-973 
 
 0-405 
 
 3-963 
 
 0-539 
 
 4-954 
 
 82 
 
 
 0-990 
 
 0-139 
 
 1-981 
 
 0-278 
 
 2-971 
 
 0-418 
 
 3-961 
 
 0-557 
 
 4-951 
 
 82 
 
 8* 
 
 0-990 
 
 0-143 
 
 1-979 
 
 0-287 
 
 2-969 
 
 0-430 
 
 3-959 
 
 0-574 
 
 4-948 
 
 Slf- 
 
 8* 
 
 0-989 
 
 0-148 
 
 1-978 
 
 0-296 
 
 2-967 
 
 0-443 
 
 3-956 
 
 0-591 
 
 4-945 
 
 814 
 
 8| 
 
 0-988 
 
 0-152 
 
 1-977 
 
 0-304 
 
 2-965 
 
 0-456 
 
 3-953 
 
 0-608 
 
 4-942 
 
 81* 
 
 9 
 
 0-988 
 
 0-156 
 
 1-975 
 
 0-313 
 
 2-963 
 
 0-469 
 
 3-951 
 
 0-626 
 
 4-938 
 
 81 
 
 9 
 
 0-987 
 
 0-161 
 
 1-974 
 
 0-321 
 
 2-961 
 
 0-482 
 
 3-948 
 
 0-643 
 
 4-935 
 
 80f 
 
 9* 
 
 0-986 
 
 0-165 
 
 1-973 
 
 0-330 
 
 2-959 
 
 0-495 
 
 3-945 
 
 0-660 
 
 4-931 
 
 804 
 
 9f 
 
 0-986 
 
 0-169 
 
 1-971 
 
 0-339 
 
 2-957 
 
 0-508 
 
 3-942 
 
 0-677 
 
 4-928 
 
 80| 
 
 10 
 
 0-985 
 
 0-174 
 
 1-970 
 
 0-347 
 
 2-954 
 
 0-521 
 
 3-939 
 
 0-695 
 
 4-924 
 
 80 
 
 10J 
 
 0-984 
 
 0-178 
 
 1-968 
 
 0-356 
 
 2-952 
 
 0-534 
 
 3-936 ' 
 
 0-712 
 
 4-920 
 
 79f 
 
 104 
 
 0-983 
 
 0-182 
 
 1-967 
 
 0-364 
 
 2-950 
 
 0-547 
 
 3-933 
 
 0-729 
 
 4-916 
 
 794 
 
 10f 
 
 0-982 
 
 0-187 
 
 1-965 
 
 0-373 
 
 2-947 
 
 0-560 
 
 3930 
 
 0-746 
 
 4-912 
 
 79^ 
 
 11 
 
 0-982 
 
 0-191 
 
 1-963 
 
 0-382 
 
 2-945 
 
 0-572 
 
 3-927 
 
 0-763 
 
 4-908 
 
 79 
 
 Hi 
 
 0-981 
 
 0-195 
 
 1-962 
 
 0-390 
 
 2-942 
 
 0-585 
 
 8-923 
 
 0-780 
 
 4-904 
 
 78f 
 
 11* 
 
 0-980 
 
 0-199 
 
 1-960 
 
 0-399 
 
 2-940 
 
 0-598 
 
 3-920 
 
 0-797 
 
 4-900 
 
 784- 
 
 llf 
 
 0-979 
 
 0-204 
 
 1958 
 
 0-407 
 
 2-937 
 
 0-611 
 
 3916 
 
 0-815 
 
 4-895 
 
 78^ 
 
 12 
 
 0-978 
 
 0-208 
 
 1-956 
 
 0-416 
 
 2-934 
 
 0-624 
 
 3-913 
 
 0-832 
 
 4-891 
 
 78" 
 
 12J 
 
 0-977 
 
 0-212 
 
 1-954 
 
 0-424 
 
 2-932 
 
 0-637 
 
 3-909 
 
 0-849 
 
 4-886 
 
 77f 
 
 12* 
 
 0-976 
 
 0-216 
 
 1-953 
 
 0-433 
 
 2-929 
 
 0-649 
 
 3-905 
 
 0-866 
 
 4-881 
 
 774 
 
 12f 
 
 0-975 
 
 0-221 
 
 1-951 
 
 0-441 
 
 2-926 
 
 0-662 
 
 3-901 
 
 0-883 
 
 4'877 
 
 77i 
 
 13 
 
 0-974 
 
 0-225 
 
 1-949 
 
 0-450 
 
 2-923 
 
 0-675 
 
 3-897 
 
 0-900 
 
 4-872 
 
 77 
 
 13J 
 
 0-973 
 
 0-229 
 
 1-947 
 
 0-458 
 
 2-920 
 
 0-688 
 
 3-894 
 
 0-917 
 
 4-867 
 
 76 
 
 13* 
 
 0-972 
 
 0-233 
 
 1-945 
 
 0-467 
 
 2-917 
 
 0-700 
 
 3-889 
 
 0-934 
 
 4-862 
 
 764 
 
 13| 
 
 0-971 
 
 0-238 
 
 1-943 
 
 0-475 
 
 2-914 
 
 0-713 
 
 3-885 
 
 0-951 
 
 4857 
 
 76i 
 
 14 
 
 0-970 
 
 0-242 
 
 1-941 
 
 0-484 
 
 2-911 
 
 0-726 
 
 3-881 
 
 0-968 
 
 4-851 
 
 76 
 
 14i 
 
 0-969 
 
 0-246 
 
 1-938 
 
 0-492 
 
 2-908 
 
 0-738 : 
 
 3-877 
 
 0-985 
 
 4-846 
 
 75f 
 
 14^ 
 
 0-968 
 
 0-250 
 
 1-936 
 
 0-501 
 
 2-904 
 
 0-751 
 
 3-873 
 
 1-002 
 
 4-841 
 
 754 
 
 14f 
 
 0-967 
 
 0-255 
 
 1-934 
 
 0-509 
 
 2-901 
 
 0-764 
 
 3-868 
 
 1-018 
 
 4-835 
 
 75i 
 
 15 
 
 0-966 
 
 0-259 
 
 1-932 
 
 0-518 
 
 2-898 
 
 0-776 
 
 3-864 
 
 1-035 
 
 4-830 
 
 75 
 
 Ml 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 bio 
 
 
 
 1 
 
 2 
 
 3 
 
 A 
 
 5 
 
 <$ 
 
APPENDIX. 
 
 831 
 
 LATITUDES AND DEPARTURES. 
 
 b>] 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 t*i 
 
 .a 
 
 
 
 
 
 
 If 
 
 1 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 
 
 
 o-ooo 
 
 6-000 
 
 o-ooo 
 
 7-000 
 
 o-ooo 
 
 8-000 
 
 o-ooo 
 
 9-000 
 
 o-ooo 
 
 90 
 
 oj 
 
 0-022 
 
 6-000 
 
 0-026 
 
 7-000 
 
 0-031 
 
 8-000 
 
 0-035 
 
 9-000 
 
 0-039 
 
 89f 
 
 04 
 
 0-044 
 
 6-000 
 
 0-052 
 
 7-000 
 
 0-061 
 
 8-000 
 
 0-070 
 
 9-000 
 
 0-079 
 
 894 
 
 Of 
 
 0-065 
 
 5-999 
 
 0-079 
 
 6-999 
 
 0-092 
 
 7-999 
 
 0-105 
 
 8-999 
 
 0-118 
 
 89J 
 
 
 0-087 
 
 5-999 
 
 0-105 
 
 6-999 
 
 0-122 
 
 7-999 
 
 0-140 
 
 8-999 
 
 0-157 
 
 89 
 
 li 
 
 0-109 
 
 5-999 
 
 0-131 
 
 6-998 
 
 0-153 
 
 7998 
 
 0-175 
 
 8-998 
 
 0-196 
 
 88|- 
 
 ii 
 
 0-131 
 
 5-998 
 
 0-157 
 
 6-998 
 
 0-183 
 
 7-997 
 
 0-209 
 
 8-997 
 
 0-236 
 
 884 
 
 if 
 
 0-153 
 
 5-997 
 
 0-183 
 
 6-997 
 
 0-214 
 
 7-996 
 
 0-244 
 
 8-996 
 
 0-275 
 
 88 
 
 2 
 
 0-174 
 
 5-996 
 
 0-209 
 
 6-996 
 
 0-244 
 
 7-995 
 
 0-279 
 
 8-995 
 
 0-314 
 
 88 
 
 H 
 
 0-196 
 
 5-995 
 
 0-236 
 
 6-995 
 
 0-275 
 
 7-994 
 
 0-314 
 
 8-993 
 
 0-353 
 
 87|- 
 
 H 
 
 0-218 
 
 5-994 
 
 0-262 
 
 6-993 
 
 0-305 
 
 7-992 
 
 0-349 
 
 8-991 
 
 0-393 
 
 874 
 
 24 
 
 0-240 
 
 5-993 
 
 0-288 
 
 6-992 
 
 0-336 
 
 7-991 
 
 0-384 
 
 8-990 
 
 0-432 
 
 87J 
 
 3 
 
 0-262 
 
 5-992 
 
 0-314 
 
 6-f90 
 
 0-366 
 
 7-989 
 
 0-419 
 
 8-988 
 
 0-471 
 
 87* 
 
 if 
 
 0-283 
 
 5-990 
 
 0-340 
 
 6-989 
 
 0-397 
 
 7-987 
 
 0-454 
 
 8-966 
 
 0-510 
 
 86f 
 
 H 
 
 0-305 
 
 5-989 
 
 0-366 
 
 6-987 
 
 0-427 
 
 7-985 
 
 0-488 
 
 8-983 
 
 0-549 
 
 864 
 
 3| 
 
 0-327 
 
 5-987 
 
 0-392 
 
 6-985 
 
 0-458 
 
 7-983 
 
 0-523 
 
 8-981 
 
 0-589 
 
 sei- 
 
 4 
 
 0-349 
 
 5-985 
 
 0-419 
 
 6-983 
 
 0-488 
 
 7-981 
 
 0-558 
 
 8-978 
 
 0-628 
 
 se 
 
 H 
 
 0-371 
 
 5-984 
 
 0-445 
 
 6-981 
 
 0-519 
 
 7-978 
 
 0-593 
 
 8-975 
 
 0-667 
 
 85 
 
 4 
 
 0-392 
 
 5-982 
 
 0-471 
 
 6-978 
 
 0-549 
 
 7-975 
 
 0-628 
 
 8-972 
 
 0-706 
 
 854 
 
 4| 
 
 0-414 
 
 5-979 
 
 0-497 
 
 6-976 
 
 0-580 
 
 7-973 
 
 0-662 
 
 8-969 
 
 0-745 
 
 85 
 
 5 
 
 0-436 
 
 5-977 
 
 0-523 
 
 6-973 
 
 0-610 
 
 7-970 
 
 0-697 
 
 8-966 
 
 0-784 
 
 85 
 
 i 
 
 0*458 
 
 5-975 
 
 0-549 
 
 6-971 
 
 0-641 
 
 7-966 
 
 0-732 
 
 8-962 
 
 0-824 
 
 84f 
 
 6* 
 
 0-479 
 
 5-972 
 
 0-575 
 
 6-968 
 
 0-671 
 
 7-963 
 
 0-767 
 
 8-959 
 
 0-863 
 
 844 
 
 5f 
 
 0-501 
 
 5-970 
 
 0-601 
 
 6-965 
 
 0-701 
 
 7-960 
 
 0-802 
 
 8-955 
 
 0-902 
 
 84 
 
 6 
 
 0-523 
 
 5-967 
 
 0-627 
 
 6-962 
 
 0-732 
 
 7-956 
 
 0-836 
 
 8-951 
 
 0-941 
 
 84 
 
 6 
 
 0-544 
 
 5-964 
 
 0653 
 
 6-958 
 
 0-762 
 
 7-952 
 
 0-871 
 
 8-947 
 
 0-980 
 
 83f 
 
 64 
 
 0-566 
 
 5-961 
 
 0-679 
 
 6-955 
 
 0-792 
 
 7-949 
 
 0-906 
 
 8-942 
 
 1-019 
 
 834 
 
 6f 
 
 0-588 
 
 5-958 
 
 0-705 
 
 6-951 
 
 0-823 
 
 7-945 
 
 0-940 
 
 8-938 
 
 1-068 
 
 83 
 
 7 
 
 0-609 
 
 5-955 
 
 0-731 
 
 6-948 
 
 0-853 
 
 7-940 
 
 0-975 
 
 8-933 
 
 1-097 
 
 83 
 
 7i 
 
 0-631 
 
 5-952 
 
 0-757 
 
 6-944 
 
 0-883 
 
 7-936 
 
 1-010 
 
 8-928 
 
 1-136 
 
 82f 
 
 n 
 
 0-653 
 
 5-949 
 
 0-783 
 
 6-940 
 
 0-914 
 
 7-932 
 
 1-044 
 
 8-923 
 
 1-175 
 
 824 
 
 7f 
 
 0-674 
 
 5-945 
 
 0-809 
 
 6-936 
 
 0-944 
 
 7-927 
 
 1-079 
 
 8-918 
 
 1-214 
 
 82 
 
 8 
 
 0-696 
 
 5-942 
 
 0-8i>5 
 
 6-932 
 
 0-974 
 
 7922 
 
 1-113 
 
 8-912 
 
 1-253 
 
 82 
 
 8i 
 
 0-717 
 
 5-938 
 
 0-861 
 
 6-928 
 
 1-004 
 
 7-917 
 
 1-148 
 
 8-907 
 
 1-291 
 
 81f 
 
 i 
 
 0-739 
 
 5-934 
 
 0-887 
 
 6-923 
 
 1-035 
 
 7-912 
 
 1-182 
 
 8-901 
 
 1-330 
 
 814 
 
 8f 
 
 0-761 
 
 5-930 
 
 0-913 
 
 6-919 
 
 1-065 
 
 7-907 
 
 1-217 
 
 8-895 
 
 1-369 
 
 81i 
 
 9 
 
 0-782 
 
 5-926 
 
 0-939 
 
 6-914 
 
 1-095 
 
 7-902 
 
 1-251 
 
 8-889 
 
 1-408 
 
 81 
 
 9* 
 
 0-804 
 
 5-922 
 
 0-964 
 
 6-909 
 
 1-125 
 
 7-896 
 
 1-286 
 
 8-883 
 
 1-447 
 
 80f 
 
 94 
 
 0-825 
 
 5-918 
 
 0-990 
 
 6-904 
 
 1-155 
 
 7-890 
 
 1-320 
 
 8-877 
 
 1-485 
 
 80^ 
 
 9| 
 
 0-847 
 
 5-913 
 
 1-016 
 
 6-899 
 
 1-185 
 
 7-884 
 
 1-355 
 
 8-870 
 
 1-524 
 
 80J 
 
 10 
 
 0-868 
 
 5-909 
 
 1-042 
 
 6-894 
 
 1-216 
 
 7-878 
 
 1-389 
 
 8863 
 
 1-563 
 
 80 
 
 10 
 
 0-890 
 
 5-904 
 
 1-068 
 
 6-888 
 
 1-246 
 
 7-872 
 
 1-424 
 
 8-856 
 
 1-601 
 
 79f 
 
 104 
 
 0-911 
 
 5-900 
 
 1-093 
 
 6-883 
 
 1-276 
 
 7-866 
 
 1-458 
 
 8-849 
 
 1-640 
 
 794 
 
 104 
 
 0-933 
 
 5-895 
 
 1-119 
 
 6-877 
 
 1-306 
 
 7-860 
 
 1-492 
 
 8-842 
 
 1-679 
 
 79J 
 
 11 
 
 0-954 
 
 5-890 
 
 1-145 
 
 6-871 
 
 1-336 
 
 7-853 
 
 1-526 
 
 8-835 
 
 1-717 
 
 79 
 
 Hi 
 
 0-975 
 
 5-885 
 
 1-171 
 
 6-866 
 
 1-366 
 
 7-846 
 
 1-561 
 
 8-827 
 
 1-756 
 
 78fc 
 
 11* 
 
 0-997 
 
 5-880 
 
 1-196 
 
 6-859 
 
 1-396 
 
 7-839 
 
 1-595 
 
 8-819 
 
 1-794 
 
 784 
 
 nf 
 
 1-018 
 
 5-874 
 
 1-222 
 
 6-853 
 
 1-425 
 
 7-832 
 
 1-629 
 
 8-811 
 
 1-833 
 
 7S 
 
 12 
 
 1-040 
 
 5-869 
 
 1-247 
 
 6-847 
 
 1-455 
 
 7-825 
 
 1-663 
 
 8-803 
 
 1-871 
 
 T8 
 
 12 
 
 1-061 
 
 5-863 
 
 1-273 
 
 6841 
 
 1-485 
 
 7-818 
 
 1-697 
 
 8-795 
 
 1-910 
 
 77f 
 
 12| 
 
 1-082 
 
 5-858 
 
 1-299 
 
 6-834 
 
 1-515 
 
 7-810 
 
 1-732 
 
 8-787 
 
 1-948 
 
 774 
 
 12| 
 
 1-103 
 
 5-852 
 
 1-324 
 
 6-827 
 
 1-545 
 
 7-803 
 
 1-766 
 
 8-778 
 
 1-986 
 
 77 
 
 13 
 
 1-125 
 
 5-846 
 
 1-350 
 
 6-821 
 
 1-575 
 
 7-795 
 
 1-800 
 
 8-769 
 
 2-025 
 
 77 
 
 13i 
 
 1-146 
 
 5-840 
 
 1-375 
 
 6-814 
 
 1-604 
 
 7-787 
 
 1-834 
 
 8-760 
 
 2-063 
 
 76f 
 
 134 
 
 1-167 
 
 5-834 
 
 1-401 
 
 6807 
 
 1-634 
 
 7-779 
 
 1-868 
 
 8-751 
 
 2-101 
 
 764 
 
 13J 
 
 1-188 
 
 5-828 
 
 1-426 
 
 6-799 
 
 1-664 
 
 7-771 
 
 1-902 
 
 8-742 
 
 2-139 
 
 76 
 
 14 
 
 1-210 
 
 5-822 
 
 1-452 
 
 6-792 
 
 1-693 
 
 7-762 
 
 1-935 
 
 8-733 
 
 2-177 
 
 76 
 
 14J 
 
 1-231 
 
 5-815 
 
 1-477 
 
 6'78i 
 
 1-723 
 
 7-754 
 
 1-969 
 
 8-723 
 
 2-215 
 
 75f 
 
 144 
 
 1-252 
 
 5-809 
 
 1-502 
 
 6-777 
 
 1-753 
 
 7-745 
 
 2-003 
 
 8-713 
 
 2-253 
 
 754 
 
 14f 
 
 1-273 
 
 5-802 
 
 1-528 
 
 6-769 
 
 1-782 
 
 7-736 
 
 2-037 
 
 8-703 
 
 2-291 
 
 76i 
 
 15 
 
 1-294 
 
 5-796 
 
 1-553 
 
 6-761 
 
 1-812 
 
 7-727 
 
 2-071 
 
 8-693 
 
 2-329 
 
 75 
 
 tec 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 i> 
 
 't- 
 
 03 
 
 d 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
832 
 
 APPENDIX. 
 
 LATITUDES AND DEPARTURES. 
 
 i 
 
 1 
 
 9 
 
 3 
 
 41 
 
 & 
 
 ttcs 
 
 _== 
 
 I 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 i 
 
 15 
 
 0-966 
 
 0-259 
 
 1-932 
 
 0-518 
 
 2-898 
 
 0-776 
 
 3-864 
 
 1-035 
 
 ! 4-830 
 
 75 
 
 i-H 
 
 0-965 
 
 0-263 
 
 1-930 
 
 0-526 
 
 2-894 
 
 0-789 
 
 3-859 
 
 1-052 
 
 4-824 
 
 74f 
 
 151 
 
 0-964 
 
 0-267 
 
 1-927 
 
 0-534 
 
 2-891 
 
 0-802 
 
 3-855 
 
 1-069 
 
 4-818 
 
 74* 
 
 15f 
 
 0-962 
 
 0-271 
 
 1-925 
 
 0-543 
 
 2-887 
 
 0-814 
 
 3-850 
 
 1-086 
 
 4-812 
 
 74^ 
 
 16 
 
 0-961 
 
 0-276 
 
 1-923 
 
 0-551 
 
 2-884 
 
 0-827 
 
 3-845 
 
 1-103 
 
 4-806 
 
 74 
 
 16* 
 
 0-960 
 
 0-280 
 
 1-920 
 
 0-560 
 
 28.80 
 
 0-839 
 
 3-840 
 
 1-119 
 
 4-800 
 
 73| 
 
 16! 
 
 0-959 
 
 0-2S4 
 
 1-918 
 
 0-568 
 
 2-876 
 
 0-852 
 
 3-835 
 
 1-136 
 
 4-794 
 
 73| 
 
 16| 
 
 0-958 
 
 0-288 
 
 . 1-915 
 
 0-576 
 
 2-873 
 
 0-865 
 
 3-830 
 
 1-153 
 
 4-788 
 
 73 
 
 IT 3 
 
 0-956 
 
 0-292 
 
 1-913 
 
 0-585 
 
 2-869 
 
 0-877 
 
 3-825 
 
 1-169 
 
 4-782 
 
 73 
 
 17i 
 
 0-955 
 
 0-297 
 
 1-910 
 
 0-593 
 
 2-866 
 
 0-890 
 
 b'820 
 
 1-186 
 
 4-775 
 
 72f 
 
 1V1 
 
 0-954 
 
 0-301 
 
 1-907 
 
 0-601 
 
 2-861 
 
 0-902 
 
 3-815 
 
 1-203 
 
 4-769 
 
 72! 
 
 17f 
 
 0-952 
 
 0-305 
 
 1-905 
 
 0-610 
 
 2-857 
 
 0-915 
 
 3-810 
 
 1-220 
 
 4-762 
 
 72 
 
 18 
 
 0-951 
 
 0-309 
 
 1-902 
 
 0-618 
 
 2-853 
 
 0-927 
 
 3-804 
 
 1-236 
 
 4-755 
 
 72 
 
 18i 
 
 0-950 
 
 0-313 
 
 1-899 
 
 0-626 
 
 2-849 
 
 0-939 
 
 3-799 
 
 1-253 
 
 4-748 
 
 71f 
 
 181 
 
 0-948 
 
 0-317 
 
 1-897 
 
 0-635 
 
 2-845 
 
 0-952 
 
 3-793 
 
 1-269 
 
 4-742 
 
 fll 
 
 18| 
 
 0-947 
 
 0-321 
 
 1-894 
 
 0-643 
 
 2-841 
 
 0-964 
 
 3-788 
 
 1-286 
 
 4-735 
 
 71* 
 
 19 
 
 0-946 
 
 0-326 
 
 1-891 
 
 0-651 
 
 2-837 
 
 0-977 
 
 3-782 
 
 1-302 
 
 4-728 
 
 71 
 
 19i 
 
 0-944 
 
 0-330 
 
 1-888 
 
 0-659 
 
 2-832 
 
 0-9S9 
 
 3-776 
 
 1-319 
 
 4-720 
 
 70f 
 
 191 
 
 0-943 
 
 0-334 
 
 1-885 
 
 0-668 
 
 2-828 
 
 1-001 
 
 3-771 
 
 1-335 
 
 4-713 
 
 70^ 
 
 19| 
 
 0-941 
 
 0-338 
 
 1-882 
 
 0-676 
 
 2-824 
 
 1-014 
 
 3-765 
 
 1-352 
 
 4-706 
 
 70i 
 
 20 
 
 0-940 
 
 0-342 
 
 1-879 
 
 0-684 
 
 2-819 
 
 1 026 
 
 3-759 
 
 1-368 
 
 4-698 
 
 70 
 
 20J 
 
 0-938 
 
 0-346 
 
 1-876 
 
 0-692 
 
 2-815 
 
 1-038 
 
 3-753 
 
 1-384 
 
 4-691 
 
 69f 
 
 20! 
 
 0-937 
 
 0-350 
 
 1-873 
 
 0-700 
 
 2-810 
 
 1-051 
 
 3-747 
 
 1-401 
 
 4-683 
 
 69! 
 
 20f 
 
 0-935 
 
 0-354 
 
 1-870 
 
 0-709 
 
 2-805 
 
 1-063 
 
 3-741 
 
 1-417 
 
 4-676 
 
 69^ 
 
 21 
 
 0-934 
 
 0-358 
 
 1-867 
 
 0-717 
 
 2-801 
 
 1-075 
 
 3-734 
 
 1-433 
 
 4-668 
 
 69 
 
 21i 
 
 0-932 
 
 0-362 
 
 1-864 
 
 0-725 
 
 2-796 
 
 1-087 
 
 3-728 
 
 1-450 
 
 4-660 
 
 68f 
 
 211 
 
 0-930 
 
 0-367 
 
 1-861 
 
 0-733 
 
 2-791 
 
 1-100 
 
 3-722 
 
 1-466 
 
 4-652 
 
 681 
 
 21| 
 
 0-929 
 
 0-371 
 
 1-858 
 
 0-741 
 
 2-786 
 
 1-112 
 
 3-715 
 
 1-482 
 
 4-644 
 
 68J 
 
 22 
 
 0-927 
 
 0-375 
 
 1-854 
 
 0-749 
 
 2-782 
 
 1-124 
 
 3-709 
 
 1-498 
 
 4-636 
 
 68 
 
 22J 
 
 0-926 
 
 0-379 
 
 1-851 
 
 0-757 
 
 2-777 
 
 1-136 
 
 3-702 
 
 1-515 
 
 4-628 
 
 67| 
 
 22! 
 
 0-924 
 
 0-383 
 
 1-848 
 
 0-765 
 
 2-772 
 
 1-148 
 
 3-696 
 
 1-531 
 
 4-619 
 
 ; 67! 
 
 22f 
 
 0-922 
 
 0-387 
 
 1-844 
 
 0-773 
 
 2-767 
 
 1-160 
 
 3-689 
 
 1-547 
 
 4-611 
 
 67i 
 
 23 
 
 0-921 
 
 0-391 
 
 1-841 
 
 0-781 
 
 2-762 
 
 1-172 
 
 3-682 
 
 1-563 
 
 4-603 
 
 67 
 
 23J 
 
 0-919 
 
 0-395 
 
 1-838 
 
 0-789 
 
 2-756 
 
 1-184 
 
 3-675 
 
 1-579 
 
 4-594 
 
 66J 
 
 23! 
 
 0-917 
 
 0-399 
 
 1-834 
 
 0-797 
 
 2-751 
 
 1-196 
 
 3-668 
 
 1-595 
 
 4-585 
 
 661 
 
 23f 
 
 0-915 
 
 0-403 
 
 1-831 
 
 0-805 
 
 2-746 
 
 1-208 
 
 3-661 
 
 1-611 
 
 4-577 
 
 66J 
 
 24 
 
 0-914 
 
 0-407 
 
 1-827 
 
 0-813 
 
 2-741 
 
 1-220 
 
 3-654 
 
 1-627 
 
 4-568 
 
 66 
 
 24* 
 
 0-912 
 
 0-411 
 
 1-824 
 
 0-821 
 
 2-735 
 
 1-232 
 
 3-647 
 
 1-643 
 
 4-559 
 
 65f 
 
 24! 
 
 0-910 
 
 0-415 
 
 1-820 
 
 0-829 
 
 2-730 
 
 1-244 
 
 3-640 
 
 1-659 
 
 4-550 
 
 65! 
 
 24| 
 
 0-908 
 
 0-419 
 
 1-816 
 
 0-837 
 
 2-724 
 
 1-256 
 
 3-633 
 
 1-675 
 
 4-541 
 
 65J 
 
 25 
 
 0-906 
 
 0-423 
 
 1-813 
 
 0-845 
 
 2-719 
 
 1-268 
 
 3-625 
 
 1-690 
 
 4-532 
 
 65 
 
 25 
 
 0-904 
 
 0-427 
 
 1-809 
 
 0-853 
 
 2-713 
 
 1-280 
 
 3-618 
 
 1-706 
 
 4-522 
 
 64f 
 
 25! 
 
 0-903 
 
 0-431 
 
 1-805 
 
 0-861 
 
 2-708 
 
 1-292 
 
 3-610 
 
 1-722 
 
 4-513 
 
 64| 
 
 25f 
 
 0-901 
 
 0-434 
 
 1-801 
 
 0-869 
 
 2-702 
 
 1-303 
 
 3-603 
 
 1-738 
 
 4-503 
 
 64 
 
 26 
 
 0-899 
 
 0-438 
 
 1-798 
 
 0-877 
 
 2-696 
 
 1-315 
 
 3-595 
 
 1-753 
 
 4-494 
 
 64 
 
 26J 
 
 0-897 
 
 0-442 
 
 1-794 
 
 0-885 
 
 2-691 
 
 1-327 
 
 3-587 
 
 1-769 
 
 4-484 
 
 63| 
 
 26! 
 
 0-895 
 
 0-446 
 
 1-790 
 
 0-892 
 
 2-685 
 
 1-339 
 
 3-580 
 
 1-785 
 
 4-475 
 
 63! 
 
 26| 
 
 0-893 
 
 0-450 
 
 1-786 
 
 0-900 
 
 2-679 
 
 1-350 
 
 3-572 
 
 1-800 
 
 4-465 
 
 63 
 
 2T 
 
 0-891 
 
 0-454 
 
 1-782 
 
 0-908 
 
 2-673 
 
 1-362 
 
 3-564 
 
 1-816 
 
 4-455 
 
 63 
 
 27| 
 
 0-889 
 
 0-458 
 
 1-778 
 
 0-916 
 
 2-667 
 
 1-374 
 
 3-556 
 
 1-831 
 
 4-445 
 
 62f 
 
 271 
 
 0-887 
 
 0-462 
 
 1-774 
 
 0-923 
 
 2-661 
 
 1-385 
 
 3-548 
 
 1-847 
 
 4-435 
 
 62! 
 
 27| 
 
 0-885 
 
 0-466 
 
 1-770 
 
 0-931 
 
 2-655 
 
 1-397 
 
 3-, 40 
 
 1-862 
 
 4-425 
 
 62^ 
 
 28 
 
 0-883 
 
 0-469 
 
 1-766 
 
 0-939 
 
 2-649 
 
 1-408 
 
 3-532 
 
 1-878 
 
 4-415 
 
 62 
 
 28^ 
 
 0-881 
 
 0-473 
 
 1-762 
 
 0-947 
 
 2-643 
 
 1-420 
 
 3-524 
 
 1-893 
 
 4-404 
 
 61f 
 
 28! 
 
 0-879 
 
 0-477 
 
 1-758 
 
 0-954 
 
 2-636 
 
 1-431 
 
 3-515 
 
 1-909 
 
 4-394 
 
 611 
 
 28| 
 
 0-877 
 
 0-481 
 
 1-753 
 
 0-962 
 
 2-630 
 
 1-443 
 
 3-507 
 
 1-924 
 
 4-384 
 
 61i 
 
 29 
 
 0-875 
 
 0-485 
 
 1-749 
 
 0-970 
 
 2-624 
 
 1-454 
 
 3-498 
 
 1-939 
 
 4-373 
 
 61 
 
 29 
 
 0-872 
 
 0-489 
 
 1-745 
 
 0-977 
 
 2-617 
 
 1-466 
 
 3490 
 
 1-954 
 
 4-362 
 
 60f 
 
 291 
 
 0-870 
 
 0-492 
 
 1-741 
 
 0-985 
 
 2-611 
 
 1-477 
 
 3-481 
 
 1-970 
 
 4-352 
 
 60! 
 
 Ft 
 
 0-868 
 
 0-496 
 
 1-736 
 
 0-992 
 
 2-605 
 
 1-489 
 
 3-473 
 
 1-985 
 
 4-341 
 
 60i 
 
 30 
 
 0-866 
 
 0-500 
 
 1-732 
 
 rooo 
 
 2-598 
 
 1-500 
 
 3-464 
 
 2-000 
 
 4-330 
 
 60 
 
 Sf 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 _j= 
 
 1 
 
 1 
 
 9 
 
 3 
 
 4 
 
 & 
 
 I 
 
APPENDIX. 
 
 833 
 
 LATITUDES AND DEPARTURES. 
 
 i> 
 
 
 
 6 
 
 7 
 
 8 
 
 9 
 
 
 
 j 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 i 
 
 15 
 
 1294 
 
 5-796 
 
 1-553 
 
 6-761 
 
 1-812 
 
 7-727 
 
 2-071 
 
 8-693 
 
 2-329 
 
 75 
 
 151 
 
 1-315 
 
 5-789 
 
 1-578 
 
 6-754 
 
 1-841 
 
 7-718 
 
 2-104 
 
 8-683 
 
 2367 
 
 74| 
 
 15* 
 
 1-336 
 
 5-782 
 
 1-603 
 
 6-745 
 
 1-871 
 
 7-709 
 
 2-138 
 
 8-673 
 
 2-405 
 
 
 15| 
 
 1-357 
 
 5-775 
 
 1-629 
 
 6737 
 
 1-900 
 
 7-700 
 
 2-172 
 
 8-662 
 
 2-443 
 
 741 
 
 16 
 
 1-378 
 
 5-768 
 
 1-654 
 
 6-729 
 
 1-929 
 
 7-690 
 
 2-205 
 
 8-651 
 
 2-481 
 
 74 
 
 161 
 
 1-399 
 
 5-760 
 
 1679 
 
 .6-720 
 
 1-959 
 
 7-680 
 
 2-239 
 
 8-640 
 
 2-518 
 
 73S- 
 
 16* 
 
 1-420 
 
 5-753 
 
 1-704 
 
 6-712 
 
 1-988 
 
 7-671 
 
 2-272 
 
 8-629 
 
 2-556 
 
 73* 
 
 16f 
 
 1-441 
 
 5-745 
 
 1-729 
 
 6-703 
 
 2-017 
 
 7-661 
 
 2-306 
 
 8-618 
 
 2-594 
 
 731 
 
 17 
 
 1-462 
 
 5-738 
 
 1-754 
 
 6-694 
 
 2-047 
 
 7-650 
 
 2-339 
 
 8-607 
 
 2-631 
 
 73 
 
 171 
 
 1-483 
 
 5-730 
 
 1-779 
 
 6-685 
 
 2-076 
 
 7-640 
 
 2-372 
 
 8-595 
 
 2-669 
 
 72f 
 
 m 
 
 1-504 
 
 5-722 
 
 1-804 
 
 6-676 
 
 2-105 
 
 7-630 
 
 2-406 
 
 8-583 
 
 2-706 
 
 72* 
 
 17f 
 
 1-524 
 
 5-714 
 
 1-829 
 
 6-667 
 
 2-134 
 
 7-619 
 
 2439 
 
 8-572 
 
 2-744 
 
 721 
 
 18 
 
 1 -545 
 
 5-706 
 
 1-854 
 
 6-657 
 
 2-163 
 
 7-608 
 
 2-472 
 
 8-560 
 
 2-781 
 
 72 
 
 181 
 
 1-566 
 
 5-698 
 
 1-879 
 
 6-648 
 
 2-192 
 
 7598 
 
 2-505 
 
 8-647 
 
 2-818 
 
 71f 
 
 18* 
 
 1-587 
 
 5-690 
 
 1-904 
 
 6-638 
 
 2-221 
 
 7-587 
 
 2-538 
 
 8-535 
 
 2-856 
 
 71* 
 
 18J 
 
 1-607 
 
 5-682 
 
 1929 
 
 6-629 
 
 2-250 
 
 7-575 
 
 2-572 
 
 8-522 
 
 2-893 
 
 
 19 
 
 1-628 
 
 5-673 
 
 1-953 
 
 6-619 
 
 2-279 
 
 7-564 
 
 2-605 
 
 8-610 
 
 2-930 
 
 71 
 
 191 
 
 1-648 
 
 5-665 
 
 1-978 
 
 ti-609 
 
 2-308 
 
 i 7-553 
 
 2-638 
 
 8-497 
 
 2-967 
 
 70f 
 
 19* 
 
 1-669 
 
 5-656 
 
 2-003 
 
 6-598 
 
 2-337 
 
 7-541 
 
 2-670 
 
 8-484 
 
 3-004 
 
 70i 
 
 19f 
 
 1-690 
 
 5-647 
 
 2-028 
 
 6-588 
 
 2-365 
 
 7-529 
 
 2-703 
 
 8-471 
 
 3-041 
 
 701 
 
 20 
 
 1-710 
 
 5-638 
 
 2-052 
 
 6-578 
 
 2394 
 
 7-518 
 
 2-736 
 
 8-467 
 
 3-078 
 
 70 
 
 201 
 
 1-731 
 
 5-629 
 
 2-077 
 
 6-567 
 
 2-423 7-506 
 
 2-769 
 
 8-444 
 
 3-115 
 
 
 20* 
 
 1-751 
 
 5-620 
 
 2-101 
 
 6-557 
 
 2-451 7-493 
 
 2-802 
 
 8-430 
 
 3-152 
 
 69-* 
 
 20|- 
 
 1-771 
 
 5-611 
 
 2-126 
 
 6-546 
 
 2-480 
 
 7-481 
 
 2-834 
 
 8-416 3-189 
 
 691 
 
 21 J 
 
 1-792 
 
 5-601 
 
 2-150 
 
 6-535 
 
 2-509 
 
 7-469 
 
 2-867 
 
 8-402 
 
 8-225 
 
 69 
 
 211 
 
 1-812 
 
 5-592 
 
 2-175 
 
 6-524 
 
 2-537 
 
 7-456 
 
 2-900 
 
 8-388 
 
 3-262 
 
 68$ 
 
 21* 
 
 1-833 
 
 5-582 
 
 2-199 
 
 6-513 
 
 2-566 
 
 7-443 
 
 2-932 
 
 8-374 
 
 3-299 
 
 68* 
 
 21J 
 
 1-853 
 
 5-573 
 
 2-223 
 
 6-502 
 
 2-594 
 
 7-430 
 
 2964 
 
 8-359 
 
 3-335 
 
 681 
 
 22 
 
 1-873 
 
 5-563 
 
 2-248 
 
 6-490 
 
 2-622 
 
 7-417 
 
 2-997 
 
 8-345 
 
 3-371 
 
 68 
 
 221 
 
 1-893 
 
 5-553 
 
 2-272 
 
 6-479 
 
 2-651 
 
 7-404 
 
 3-029 
 
 8-330 
 
 3-408 
 
 67J 
 
 22* 
 
 1-913 
 
 5-543 
 
 2-296 
 
 6-467 
 
 2-679 
 
 7-891 
 
 3-061 
 
 8-315 
 
 3-444 
 
 67i 
 
 22| 
 
 1-934 
 
 5-533 
 
 2-320 
 
 6-455 
 
 2-707 
 
 7-378 
 
 3-094 
 
 8-300 
 
 3-480 
 
 671 
 
 23 
 
 1-954 
 
 5-523 
 
 2-344 
 
 6-444 
 
 2-735 
 
 7-364 
 
 3-126 
 
 8-285 
 
 3-517 
 
 67 
 
 231 
 
 1-974 
 
 5-513 
 
 2-368 
 
 6-432 
 
 2763 
 
 7-350 
 
 3-158 
 
 8-269 
 
 3-5e3 
 
 66f 
 
 23* 
 
 1-994 
 
 5-502 
 
 2-392 
 
 6-419 
 
 2-791 
 
 7-336 
 
 3-190 
 
 8-254 
 
 3-5^9 
 
 66* 
 
 23 
 
 2-014 
 
 5-492 
 
 2-416 
 
 6-407 
 
 2-819 
 
 7-322 
 
 3-222 
 
 8-238 
 
 3-625 
 
 661 
 
 24 
 
 2-034 
 
 5-481 
 
 2-440 
 
 6-395 
 
 2-847 
 
 7-308 
 
 3-254 
 
 8-222 
 
 3-661 
 
 6 
 
 241 
 
 2-054 
 
 5-471 
 
 2-464 
 
 6-382 
 
 2-875 
 
 7"294 
 
 3-286 
 
 8-206 
 
 3-696 
 
 664 
 
 24* 
 
 2-073 
 
 5-460 
 
 2-488 
 
 6-370 
 
 2-903 
 
 7-280 
 
 3-318 
 
 8-190 
 
 3-732 
 
 65 
 
 24| 
 
 2-093 
 
 5-449 
 
 2-512 
 
 6-357 
 
 2-931 
 
 7-265 
 
 3-349 
 
 8-173 
 
 3-768 
 
 651 
 
 25 
 
 2-113 
 
 5-438 
 
 2-536 
 
 6-344 
 
 2-958 
 
 7-250 
 
 3-381 
 
 8-157 
 
 3-804 
 
 65 
 
 251 
 
 2-133 
 
 5-427 
 
 2-559 
 
 6-331 
 
 2-986 
 
 7-236 
 
 3-413 
 
 8-140 
 
 8-839 
 
 64f 
 
 25* 
 
 2-153 
 
 5-416 
 
 2-583 
 
 6-318 
 
 3-014 
 
 7-221 
 
 3-444 
 
 8-123 
 
 3-875 
 
 4* 
 
 25| 
 
 2-172 
 
 5-404 
 
 2-607 
 
 6-305 
 
 3-041 
 
 7-206 
 
 3-476 
 
 8-106 
 
 3-910 
 
 641 
 
 26 
 
 2-192 
 
 5-393 
 
 2-630 
 
 6-292 
 
 3-069 
 
 7-190 
 
 3-507 
 
 8-089 
 
 3-946 
 
 64 
 
 261 
 
 2-211 
 
 5-381 
 
 2-654 
 
 6-278 
 
 3-096 
 
 7-175 
 
 3-538 
 
 8-072 
 
 3-981 
 
 63f 
 
 26* 
 
 2-231 
 
 5-370 
 
 2-677 
 
 6-265 
 
 3-123 
 
 7-160 
 
 3-670 
 
 8-054 
 
 4-016 
 
 63* 
 
 26f 
 
 2-250 
 
 5-358 
 
 2-701 
 
 6-251 
 
 3-151 
 
 7-144 
 
 3-601 
 
 8-037 
 
 4-061 
 
 631 
 
 27 
 
 2-270 
 
 5-346 
 
 2-724 
 
 6-237 
 
 3-178 
 
 7-128 
 
 3-682 
 
 8-019 
 
 4-086 
 
 63 
 
 271 
 
 2-289 
 
 5-334 
 
 2-747 
 
 6-223 
 
 3-205 
 
 7-112 
 
 3-663 
 
 8-001 
 
 4-121 
 
 62f 
 
 27* 
 
 2-309 
 
 5-322 
 
 2-770 
 
 6-209 
 
 3-232 
 
 7-096 
 
 3-694 
 
 7-983 
 
 4-156 
 
 62* 
 
 27| 
 
 2-328 
 
 5-310 
 
 2-794 
 
 6-195 
 
 3-259 
 
 7-080 
 
 3-725 
 
 7-965 
 
 4-190 
 
 621 
 
 28 
 
 2-347 
 
 5-298 
 
 2-817 
 
 6-181 
 
 3-286 
 
 7-064 
 
 3-756 
 
 7-947 
 
 4-225 
 
 62 
 
 281 
 
 2-367 
 
 5-285 
 
 2-840 
 
 6-1S6 
 
 3-313 
 
 7-047 
 
 3-787 
 
 7-928 
 
 4-260 
 
 61f 
 
 28* 
 
 2-386 
 
 5-273 
 
 2-863 
 
 6-152 
 
 3-340 
 
 7-031 
 
 3-817 
 
 7-909 
 
 4294 
 
 61-* 
 
 28| 
 
 2-405 
 
 5-260 
 
 2-886 
 
 6-137 
 
 3-367 
 
 7-014 
 
 3-848 
 
 7-891 
 
 4-329 
 
 611 
 
 29 
 
 2-424 
 
 5-248 
 
 2-909 
 
 6-122 
 
 3-394 
 
 6-997 
 
 3-878 
 
 7-872 
 
 4-363 
 
 61 
 
 291 
 
 2-443 
 
 6-235 
 
 2-932 
 
 6-107 
 
 3-420 
 
 6-980 
 
 3-909 
 
 7-852 
 
 4-398 
 
 60f 
 
 29i 
 
 2-462 
 
 5-222 
 
 2-955 
 
 6-093 
 
 3-447 
 
 6-963 
 
 3-939 
 
 7-833 
 
 4-432 
 
 60* 
 
 29| 
 
 2-481 
 
 5-209 
 
 2-977 
 
 6-077 
 
 3-474 
 
 6-946 
 
 3-970 
 
 7-814 
 
 4-466 
 
 601 
 
 30 
 
 2-500 
 
 5-196 
 
 3-000 
 
 6-062 
 
 S-500 
 
 6-928 
 
 4-000 
 
 7-794 
 
 4-600 
 
 60 
 
 3 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 S 
 
 J 
 
 ff 
 
 
 
 
 
 ! 
 
 9 
 
 
 
 54 
 
834 
 
 APPENDIX. 
 
 LATITUDES AND DEPARTURES. 
 
 
 \ 
 
 I 
 
 i i 
 
 i 
 
 J 
 
 1 
 
 4 
 
 I 
 
 5 
 
 bi 
 
 
 
 
 
 
 
 
 
 
 
 .a 
 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 1 
 
 30 
 
 0-866 
 
 0-500 
 
 1-732 
 
 1-000 
 
 2-598 
 
 1-500 
 
 3-464 
 
 2-000 
 
 4-330 
 
 60 
 
 30i 
 
 0-864 
 
 0-504 
 
 1-728 
 
 1-008 
 
 2-592 
 
 1-511 
 
 3-455 
 
 2-015 
 
 4-319 
 
 59f 
 
 30 
 
 0-862 
 
 0-508 
 
 1-723 
 
 1-015 
 
 2-585 
 
 1-523 
 
 3-447 
 
 2-030 
 
 4-308 
 
 59* 
 
 30 
 
 0-859 
 
 0-511 
 
 1-719 
 
 1-023 
 
 2-578 
 
 1-534 
 
 3-438 
 
 2-045 
 
 4-297 
 
 59i 
 
 31 
 
 0-857 
 
 0-515 
 
 1-714 
 
 1-030 
 
 2-572 
 
 1-545 
 
 3-429 
 
 2-060 
 
 4-286 
 
 59 
 
 311 
 
 0-855 
 
 0-519 
 
 1-710 
 
 1-038 
 
 2-565 
 
 1-556 
 
 3-420 
 
 2-075 
 
 4-275 
 
 58f 
 
 31* 
 
 0-853 
 
 0-522 
 
 1-705 
 
 1-045 
 
 2-558 
 
 1-567 
 
 3-411 
 
 2-090 
 
 4-263 
 
 58* 
 
 31f 
 
 0-850 
 
 0-526 
 
 1-701 
 
 1-052 
 
 2-551 
 
 1-579 
 
 3-401 
 
 2-105 
 
 4-252 
 
 58 
 
 32 
 
 0-848 
 
 0-530 
 
 1-696 
 
 1-060 
 
 2-544 
 
 1-590 
 
 3-392 
 
 2-120 
 
 4-240 
 
 58 
 
 32J 
 
 0-846 
 
 0534 
 
 1-691 
 
 1-067 
 
 2-537 
 
 1-601 
 
 3-383 
 
 2-134 
 
 4-229 
 
 57f 
 
 32* 
 
 0-843 
 
 0-537 
 
 1-687 
 
 1-075 
 
 2-530 
 
 1-612 
 
 3-374 
 
 2-149 
 
 4-217 
 
 57* 
 
 32| 
 
 0-841 
 
 0-541 
 
 1-682 
 
 1-082 
 
 2-523 
 
 1-623 
 
 3-364 
 
 2-164 
 
 4-205 
 
 671 
 
 33 
 
 0-839 
 
 0-545 
 
 1-677 
 
 1-089 
 
 2-516 
 
 1-634 
 
 3-355 
 
 2-179 
 
 4-193 
 
 57 
 
 33 
 
 0-836 
 
 0-548 
 
 1-673 
 
 1-097 
 
 2-509 
 
 1-645 
 
 3-345 
 
 2-193 
 
 4-181 
 
 56f 
 
 33* 
 
 0-834 
 
 0-552 
 
 1-668 
 
 1-104 
 
 2-502 
 
 1-656 
 
 3-336 
 
 2-208 
 
 4-169 
 
 56* 
 
 33f 
 
 0-831 
 
 0-556 
 
 1-663 
 
 1-111 
 
 2-494 
 
 1-667 
 
 3-326 
 
 2-222 
 
 4-157 
 
 56 
 
 34 
 
 0-829 
 
 0-559 
 
 1-658 
 
 1-118 
 
 2-487 
 
 1-678 
 
 3-316 
 
 2-237 
 
 4-145 
 
 56 
 
 34J 
 
 0-827 
 
 0-563 
 
 1-653 
 
 1-126 
 
 2-480 
 
 1-688 
 
 3-306 
 
 2-251 
 
 4-133 
 
 56f 
 
 34* 
 
 0-824 
 
 0-566 
 
 1-648 
 
 1-133 
 
 2-472 
 
 1-699 
 
 3-297 
 
 2-266 
 
 4-121 
 
 55* 
 
 34| 
 
 0-822 
 
 0-570 
 
 1-643 
 
 1-140 
 
 2-465 
 
 1-710 
 
 3-287 
 
 2-280 
 
 4-108 
 
 65i 
 
 35 
 
 0-819 
 
 0-574 
 
 1-638 
 
 1-147 
 
 2-457 
 
 1721 
 
 3-277 
 
 2-294 
 
 4-096 
 
 55 
 
 85J 
 
 0-817 
 
 0-577 
 
 1-633 
 
 1-154 
 
 2-450 
 
 1-731 
 
 3-267 
 
 2-309 
 
 4-083 
 
 54J 
 
 35* 
 
 0-814 
 
 0-581 
 
 1-628 
 
 1-161 
 
 2-442 
 
 1-742 
 
 3-257 
 
 2-323 
 
 4-071 
 
 54* 
 
 35f 
 
 0-812 
 
 0-584 
 
 1-623 
 
 1-168 
 
 2-435 
 
 1-753 
 
 3-246 
 
 2-337 
 
 4-058 
 
 54 
 
 36 
 
 0-809 
 
 0-588 
 
 1-618 
 
 1-176 
 
 2-427 
 
 1-763 
 
 3-236 
 
 2-351 
 
 4-045 
 
 54 
 
 36J 
 
 0-806 
 
 0-591 
 
 1-613 
 
 1-183 
 
 2-419 
 
 1-774 
 
 3-226 
 
 2-365 
 
 4-032 
 
 53f 
 
 3<H 
 
 0-804 
 
 0-595 
 
 1-608 
 
 1-190 
 
 2-412 
 
 1-784 
 
 3-215 
 
 2-379 
 
 4-019 
 
 53* 
 
 36| 
 
 0-801 
 
 0-598 
 
 1-603 
 
 1-197 
 
 2-404 
 
 1-795 
 
 3-205 
 
 2-393 
 
 4-006 
 
 53J 
 
 31 
 
 0-799 
 
 0-602 
 
 1-597 
 
 1-204 
 
 2-396 
 
 1-805 
 
 3-195 
 
 2-407 
 
 3-993 
 
 53 
 
 37J 
 
 0-796 
 
 0-605 
 
 1-592 
 
 1-211 
 
 2-388 
 
 1-816 
 
 3-184 
 
 2-421 
 
 3-980 
 
 52f 
 
 37* 
 
 0-793 
 
 0-609 
 
 1-587 
 
 1-218 
 
 2-380 
 
 1-826 
 
 3-173 
 
 2-435 
 
 3-967 
 
 52* 
 
 37| 
 
 0-791 
 
 0-612 
 
 1-581 
 
 1-224 
 
 2-372 
 
 1-837 
 
 3-163 
 
 2-449 
 
 3-953 
 
 52i 
 
 38 
 
 0-788 
 
 0-616 
 
 1-576 
 
 1-231 
 
 2-364 
 
 1-847 
 
 3-152 
 
 2-463 
 
 3-940 
 
 52 
 
 88J 
 
 0-785 
 
 0-619 
 
 1-571 
 
 1-238 
 
 2-356 
 
 1-857 
 
 3-141 
 
 2-476 
 
 3-927 
 
 5 If 
 
 38* 
 
 0-783 
 
 0-623 
 
 1-565 
 
 1-245 
 
 2-348 
 
 1-868 
 
 3-130 
 
 2-490 
 
 3-913 
 
 51* 
 
 38f 
 
 0-780 
 
 0-626 
 
 1-560 
 
 1-252 
 
 2-340 
 
 1-878 
 
 3-120 
 
 2-504 
 
 3-899 
 
 511 
 
 39 
 
 0-777 
 
 0-629 
 
 1-554 
 
 1-259 
 
 2-331 
 
 1-888 
 
 3-109 
 
 2-517 
 
 3-886 
 
 51 
 
 39 
 
 0-774 
 
 0-633 
 
 1-549 
 
 1-265 
 
 2-323 
 
 1-898 
 
 3-098 
 
 2-, 31 
 
 3-872 
 
 50f 
 
 39* 
 
 0-772 
 
 0-636 
 
 1-543 
 
 1-272 
 
 2-315 
 
 1-908 
 
 3-086 
 
 2-544 
 
 3-858 
 
 50* 
 
 39f 
 
 0-769 
 
 0-639 
 
 1-538 
 
 1-279 
 
 2-307 
 
 1-918 
 
 3-075 
 
 2-558 
 
 3-844 
 
 50J 
 
 40 
 
 0-766 
 
 0-643 
 
 1-532 
 
 1-286 
 
 2-298 
 
 1-928 
 
 3-064 
 
 2-571 
 
 3-830 
 
 50 
 
 40J 
 
 0-763 
 
 0-646 
 
 1-526 
 
 1-292 
 
 2-290 
 
 1-938 
 
 3-053 
 
 2-584 
 
 3-816 
 
 49f 
 
 40* 
 
 0-760 
 
 0-649 
 
 1-521 
 
 1-299 
 
 2-281 
 
 1-948 
 
 3042 
 
 2-598 
 
 3-802 
 
 49* 
 
 40f 
 
 0-758 
 
 0-653 
 
 1-515 
 
 1-306 
 
 2-273 
 
 1-958 
 
 3-030 
 
 2-611 
 
 3-788 
 
 491 
 
 41 
 
 0-755 
 
 0-656 
 
 1-509 
 
 1-312 
 
 2-264 
 
 1-968 
 
 3019 
 
 2-624 
 
 3-774 
 
 49 
 
 41J 
 
 0-752 
 
 0-659 
 
 1-504 
 
 1-319 
 
 2-256 
 
 1978 
 
 3-007 
 
 2-637 
 
 3-759 
 
 48f 
 
 41* 
 
 0-749 
 
 0-663 
 
 1-498 
 
 1-325 
 
 2-247 
 
 1-988 
 
 2-996 
 
 2-650 
 
 3-745 
 
 48* 
 
 41f 
 
 0-746 
 
 0-666 
 
 1-492 
 
 1-332 
 
 2-238 
 
 1-998 
 
 2-984 
 
 2-664 
 
 3-730 
 
 48i 
 
 42 
 
 0-743 
 
 0-669 
 
 1-486 
 
 1-338 
 
 2-229 
 
 2-007 
 
 2-973 
 
 2-677 
 
 3-716 
 
 48 
 
 42* 
 
 0-740 
 
 0-672 
 
 1-480 
 
 1-345 
 
 2-221 
 
 2-017 
 
 2-961 
 
 2-689 
 
 3-701 
 
 47f 
 
 42* 
 
 0-737 
 
 0-676 
 
 1-475 
 
 1-351 
 
 2-212 
 
 2-027 
 
 2-949 
 
 2-702 
 
 3-686 
 
 47* 
 
 42f 
 
 0-734 
 
 0-679 
 
 1-469 
 
 1-358 
 
 2-203 
 
 2-036 
 
 2-937 
 
 2-715 
 
 3-672 
 
 47i 
 
 43 
 
 0-731 
 
 0-682 
 
 1-463 
 
 1-364 
 
 2-194 
 
 2-046 
 
 2-925 
 
 2-728 
 
 3-657 
 
 41 
 
 43J 
 
 0-728 
 
 0-685 
 
 1-457 
 
 1-370 
 
 2-185 
 
 2-056 
 
 2-913 
 
 2-741 
 
 3-642 
 
 46f 
 
 43* 
 
 0-725 
 
 0-688 
 
 1-451 
 
 1-377 
 
 2-176 
 
 2-065 
 
 2-901 
 
 2-753 
 
 3-627 
 
 46* 
 
 43| 
 
 0-722 
 
 0-692 
 
 1-445 
 
 1-383 
 
 2-167 
 
 2-075 
 
 2-889 
 
 2-766 
 
 3-612 
 
 46i 
 
 44 
 
 0-719 
 
 0-695 
 
 1-439 
 
 1-389 
 
 2-158- 
 
 2-084 
 
 2-877 
 
 2-779 
 
 3-597 
 
 46 
 
 44i 
 
 0-716 
 
 0-698 
 
 1-433 
 
 1-396 
 
 2-149 
 
 2-093 
 
 2-865 
 
 2-791 
 
 3-582 
 
 45f 
 
 44* 
 
 0-713 
 
 0-701 
 
 1-427 
 
 1-402 
 
 2-140 
 
 2-103 
 
 2-853 
 
 2-804 
 
 3-566 
 
 45* 
 
 44f 
 
 0-710 
 
 0-704 
 
 1-420 
 
 1-408 
 
 2-131 
 
 2-112 
 
 2-841 
 
 .2-816 
 
 3-551 
 
 45J 
 
 45 
 
 0-707 
 
 1707 
 
 1-414 
 
 1-414 
 
 2-121 
 
 2-121 
 
 2828 
 
 2-828 
 
 3-536 
 
 45 
 
 f' 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 1 
 
 
 ] 
 
 I 
 
 i * 
 
 t 
 
 J 
 
 1 
 
 4 
 
 L 
 
 & 
 
 1 
 
APPENDIX. 
 
 835 
 
 LATITUDES AND DEPARTURES. 
 
 t 
 
 s 
 
 6 
 
 7 
 
 S 
 
 
 
 * 
 
 SO 
 
 Dep. 
 
 Lat. Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 1 
 
 30 
 
 2-500 
 
 5-196 3-000 
 
 6-062 
 
 3-500 
 
 ! 6-928 
 
 4-000 
 
 7-794 
 
 4-500 
 
 60 
 
 80$ 
 
 2-519 
 
 5-183 
 
 3-023 
 
 6-047 
 
 3-526 
 
 6-911 
 
 4-030 
 
 7-775 
 
 4534 
 
 69J 
 
 301 
 
 2-538 
 
 5-170 
 
 3-045 
 
 6-031 
 
 3-553 
 
 6-893 
 
 4-060 
 
 7-755 
 
 4-568 
 
 591 
 
 30f 
 
 2-556 
 
 5-156 
 
 3-068 
 
 6-016 
 
 3-579 
 
 6-875 
 
 4-090 
 
 7735 
 
 4-602 
 
 59} 
 
 31 
 
 2-575 
 
 5-143 
 
 3-090 
 
 6-000 
 
 3-605 
 
 6-857 
 
 4-120 
 
 7-715 
 
 4-635 
 
 59 1 
 
 81* 
 
 2-594 
 
 5-129 
 
 3113 
 
 5-984 
 
 3-631 
 
 6-839 
 
 4-150 
 
 7-694 
 
 4-669 
 
 58f 
 
 81* 
 
 2-612 
 
 5-116 
 
 3-135 
 
 5-968 
 
 3-657 
 
 6-821 
 
 4-180 
 
 7-674 
 
 4-702 
 
 581 
 
 31f 
 
 2-631 j 
 
 5-102 
 
 3-157 
 
 5-952 
 
 3-683 
 
 6-803 4-210 
 
 7-653 
 
 4-736 
 
 58} 
 
 32 
 
 2-650 
 
 5-088 
 
 3-180 
 
 5-936 
 
 3-709 
 
 6-784 4-239 
 
 7-632 
 
 4-769 
 
 58 
 
 32} || 2-668 
 
 5-074 
 
 3-202 
 
 5-920 
 
 3-735 
 
 6-766 4-269 
 
 7-612 
 
 4-802 
 
 57f 
 
 321 2-686 
 
 5-060 
 
 3-224 
 
 5-904 
 
 3-761 
 
 6-747 : 4-298 
 
 7-591 
 
 4-836 
 
 571 
 
 32f 
 
 2-705 
 
 5-046 
 
 3-246 
 
 5-887 
 
 8-787 
 
 6-728 ! 4328 
 
 7-569 
 
 4-869 
 
 57} 
 
 33 
 
 2-723 
 
 5-032 
 
 3-268 
 
 5-871 
 
 8-812 
 
 6-709 4-357 
 
 7-548 
 
 4-902 
 
 5T 
 
 33} 
 
 2-741 
 
 5-018 
 
 3-290 
 
 5-854 
 
 8-838 
 
 6-690 : 4-386 
 
 7-627 
 
 4-935 
 
 56f 
 
 331 
 
 2-760 
 
 5-003 
 
 3-312 
 
 5-837 
 
 8-864 
 
 6-671 4-416 
 
 7-505 
 
 4-967 
 
 561 
 
 33f 
 
 2-778 
 
 4'989 
 
 3333 
 
 5-820 
 
 8-889 
 
 6-652 4-445 
 
 7-483 
 
 5-000 
 
 56} 
 
 34 
 
 2-796 
 
 4-974 
 
 3-355 
 
 5-803 
 
 8-914 
 
 6-632 4-474 
 
 7-461 
 
 5-033 
 
 56 
 
 34} 
 
 2-814 
 
 4-960 
 
 3-377 
 
 5-786 
 
 8-940 
 
 6-613 4-502 
 
 7-439 
 
 5-065 
 
 55f 
 
 341 
 
 2-832 
 
 4-945 
 
 3-398 
 
 5-769 
 
 8-965 
 
 6-593 
 
 4-531 
 
 7-417 
 
 5-098 
 
 651 
 
 34f 
 
 2-850 
 
 4-930 
 
 3-420 
 
 5-752 
 
 8-990 
 
 6-573 
 
 4-560 
 
 7-395 
 
 5-130 
 
 66} 
 
 35 
 
 2-868 
 
 4-915 
 
 3-441 
 
 5-734 
 
 4-015 
 
 6-553 
 
 4-589 
 
 7-372 
 
 5-162 
 
 "slF 
 
 35} 
 
 2-886 
 
 4-900 
 
 3-463 
 
 5-716 
 
 4-040 
 
 6-533 
 
 4-617 
 
 7-350 
 
 5-194 
 
 54f 
 
 351 
 
 2-904 
 
 4-885 
 
 3-484 
 
 5-699 
 
 4-065 
 
 6-513 
 
 4-646 
 
 7-827 
 
 5-226 
 
 541 
 
 35| 
 
 2-921 
 
 4-869 
 
 3-505 
 
 5-681 
 
 4-090 
 
 6-493 
 
 4-674 
 
 7-304 
 
 5-258 
 
 64} 
 
 36 
 
 2-939 
 
 4-854 
 
 3-527 
 
 6-663 
 
 4'115 
 
 6-472 
 
 4-702 
 
 7-281 
 
 5-290 
 
 54 
 
 36} 
 
 2-957 
 
 4-839 
 
 3-548 
 
 5-645 
 
 4-139 
 
 6-452 
 
 4-730 
 
 7'28 
 
 5-322 
 
 53f 
 
 361 
 
 2974 
 
 4-823 
 
 3-569 
 
 6-627 
 
 4-164 
 
 6-431 
 
 4-759 
 
 7-235 
 
 5-353 
 
 581 
 
 36f 
 
 2-992 
 
 4-808 
 
 3-590 
 
 6-609 
 
 4-188 
 
 6-410 
 
 4-787 
 
 7-211 
 
 5-385 
 
 63} 
 
 3T 
 
 3-009 
 
 4-792 
 
 3-611 
 
 5-590 
 
 4-213 
 
 6-389 
 
 4-816 
 
 7-188 
 
 5-416 
 
 53 
 
 37} 
 
 3-026 
 
 4-776 
 
 3-632 
 
 5-572 
 
 4-237 
 
 6-368 
 
 4-842 
 
 7-164 
 
 5-448 
 
 52f 
 
 371 
 
 3-044 
 
 4-760 
 
 3-653 
 
 5-554 
 
 4-261 
 
 6-347 
 
 4-870 
 
 7-140 
 
 5-479 
 
 621 
 
 37f 
 
 3-061 
 
 4-744 
 
 3-673 
 
 5-535 
 
 4-286 
 
 6-326 
 
 4-898 
 
 7-116 
 
 5-510 
 
 52} 
 
 38 
 
 3-078 
 
 4-728 
 
 3-694 
 
 5-516 
 
 4-310 
 
 6-304 
 
 4-925 
 
 7-092 
 
 5-541 
 
 52 
 
 38} 
 
 3-095 
 
 4-712 
 
 3-715 
 
 5-497 
 
 4-334 
 
 6-283 
 
 4-953 
 
 7-068 
 
 5-672 
 
 61f 
 
 381 
 
 3-113 
 
 4-696 
 
 3-735 
 
 5-478 
 
 4-358 
 
 6-261 
 
 4-980 
 
 7-043 
 
 5-603 
 
 511 
 
 38| 
 
 3-130 
 
 4-679 
 
 3-756 
 
 5-459 
 
 4-381 
 
 6-239 
 
 6-007 
 
 7-019 
 
 5-633 
 
 51} 
 
 39 
 
 3-147 
 
 4-663 
 
 3-776 
 
 5-440 
 
 4-405 
 
 6-217 
 
 6-035 
 
 6994 
 
 5-664 
 
 51 
 
 39} 1 
 
 3-164 
 
 4-646 
 
 3-796 
 
 5-421 
 
 4-429 
 
 6-195 
 
 6-062 
 
 6-970 
 
 6-694 
 
 50f 
 
 391 
 
 3-180 
 
 4-630 
 
 3-816 
 
 5-401 
 
 4-453 
 
 6-173 
 
 6-089 
 
 6-945 
 
 5-725 
 
 501 
 
 39f 
 
 3-197 
 
 4-613 
 
 3-837 
 
 6-382 
 
 4-476 
 
 6-151 
 
 5-116 
 
 6-920 
 
 5-765 
 
 50} 
 
 40 
 
 3-214 
 
 4-596 
 
 3-857 
 
 6-362 
 
 4-500 
 
 6-128 
 
 5-142 
 
 6-894 
 
 5-785 
 
 50 
 
 40} 
 
 3-231 
 
 4-579 
 
 3-877 
 
 5'343 
 
 4-523 
 
 6-106 
 
 5-169 
 
 6-869 
 
 5-815 
 
 49f 
 
 401 
 
 3-247 
 
 4-562 
 
 3-897 
 
 6-323 
 
 4-546 
 
 6-083 
 
 5-196 
 
 6-844 
 
 5-845 
 
 491 
 
 40f 
 
 3-264 
 
 4-545 
 
 3-917 
 
 5-303 
 
 4-569 
 
 6-061 
 
 6-222 
 
 6-818 
 
 5-875 
 
 49} 
 
 41 
 
 3-280 I 
 
 4-528 
 
 3-936 
 
 5-283 
 
 4-592 
 
 6-038 
 
 5-248 
 
 6-792 
 
 5-905 
 
 49 
 
 41* 
 
 3-297 
 
 4-511 
 
 3-956 
 
 6-263 
 
 4-615 
 
 6-015 
 
 6-275 
 
 6-767 
 
 5-934* 
 
 48f 
 
 411 
 
 3-313 
 
 4-494 
 
 3-976 
 
 6-243 
 
 4-638 
 
 5-992 
 
 5-301 
 
 6-741 
 
 6-964 
 
 481 
 
 41* 
 
 3-329 
 
 4-476 
 
 3-995 
 
 5-222 
 
 4-661 
 
 6-968 
 
 5-827 
 
 6-715 
 
 5-993 
 
 fi 
 
 42 
 
 3-346 
 
 4-459 
 
 4-015 
 
 5-202 
 
 4-684 
 
 5-945 
 
 5-353 
 
 6-688 
 
 6-022 
 
 48 
 
 42} 
 
 3-362 
 
 4-441 
 
 4-034 
 
 5-182 
 
 4-707 
 
 5-922 
 
 5-379 
 
 6-662 
 
 6-051 
 
 47i 
 
 421 
 
 3-378 
 
 4-424 
 
 4-054 
 
 5-161 
 
 4-729 
 
 5-898 
 
 5-405 
 
 6-635 
 
 6'080 
 
 471 
 
 42| 
 
 3-394 
 
 4-406 
 
 4-073 
 
 6-140 
 
 4-752 
 
 5-875 
 
 6-430 
 
 6-609 
 
 6-109 
 
 4 lt 
 
 43 
 
 3-410 
 
 4-388 
 
 4-092 
 
 5-119 
 
 4-774 
 
 5-851 
 
 5-456 
 
 6-582 
 
 6-138 
 
 4T 
 
 43} 
 
 3-426 
 
 4-370 
 
 4-111 
 
 5-099 
 
 4-796 
 
 5-827 
 
 5-481 
 
 6-556 
 
 6-167 
 
 46f 
 
 484 
 
 3-442 
 
 4-352 
 
 4-130 
 
 5-078 
 
 4-818 
 
 5-803 
 
 6-507 
 
 6-528 
 
 6-196 
 
 461 
 
 43| 
 
 3-458 ! 
 
 4-334 
 
 4-149 
 
 5-057 
 
 4-841 
 
 5-779 
 
 5-532 
 
 6-501 
 
 6-224 
 
 46} 
 
 44 
 
 3-473 
 
 4-316 
 
 4-168 
 
 5-035 
 
 4-863 
 
 5-755 
 
 5-657 
 
 6-474 
 
 6-262 
 
 46 
 
 44} 
 
 3-489 
 
 4-298 
 
 4-187 
 
 5-014 
 
 4-885 
 
 5-730 
 
 5-582 
 
 6-447 
 
 6-280 
 
 45| 
 
 441 
 
 3-505 
 
 4-280 
 
 4-206 
 
 4-993 
 
 4-906 
 
 5-706 
 
 5-607 
 
 6-419 
 
 6-308 
 
 451 
 
 44| 
 
 3-520 
 
 4-261 
 
 4-224 
 
 4-971 
 
 4-928 
 
 5-681 
 
 6-632 
 
 6-392 
 
 6-336 
 
 45} 
 
 45 
 
 3-536 
 
 4-243 
 
 4-243 
 
 4.950 
 
 4-950 
 
 5-657 
 
 5-667 
 
 6-364 
 
 6-364 
 
 45 
 
 be 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. Lat. 
 
 Dep. 
 
 Lat. 
 
 Dep. 
 
 Lat. 
 
 t"i 
 ( C3 
 
 J 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 1 
 
836 
 
 APPENDIX. 
 
 NATURAL, SINES AND COSINES. 
 
 
 O 
 
 1 
 
 2" 
 
 3 
 
 4= 
 
 
 / 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 f 
 
 
 
 00000 
 
 Unit. 
 
 01745 
 
 99985 
 
 03490 
 
 99939 
 
 05234 
 
 99863 
 
 06976 
 
 99756 
 
 60 
 
 1 
 
 00029 
 
 Unit. 
 
 01774 
 
 99984 
 
 03519 
 
 99938 
 
 05263 
 
 99861 
 
 07005 
 
 99754' 
 
 59 
 
 2 
 
 00058 
 
 Unit. 
 
 01803 
 
 99984 
 
 03548 
 
 99937 
 
 05292 
 
 99860 
 
 07034 
 
 99752 
 
 58 
 
 3 
 
 00087 
 
 Unit. 
 
 01832 
 
 99983 
 
 03577 
 
 99936 
 
 05321 
 
 99858 
 
 07063 
 
 99750 
 
 57 
 
 4 
 
 00116 
 
 Unit. 
 
 01862 
 
 99983 
 
 03606 
 
 99935 
 
 05350 
 
 99857 
 
 07092 
 
 99748 
 
 56 
 
 5 
 
 00145 
 
 Unit. 
 
 01891 
 
 99982 
 
 03635 
 
 99934 
 
 05379 
 
 99855 
 
 07121 
 
 99746 
 
 55 
 
 6 
 
 00175 
 
 Unit. 
 
 01920 
 
 99982 
 
 03664 
 
 99933 
 
 05408 
 
 99854 
 
 07150 
 
 99744 
 
 54 
 
 7 
 
 00204 
 
 Unit. 
 
 01949 
 
 99981 
 
 03693 
 
 99932 
 
 05437 
 
 99852 
 
 07179 
 
 99742 
 
 53 
 
 8 
 
 00233 
 
 Unit. 
 
 01978 
 
 99980 
 
 03723 
 
 99931 
 
 05466 
 
 99851 
 
 07208 
 
 99740 
 
 52 
 
 9 
 
 00262 
 
 Unit. 
 
 02007 
 
 99980 
 
 03752 
 
 99930 
 
 05495 
 
 99849 
 
 07237 
 
 99738 
 
 51 
 
 10 
 
 00291 
 
 Unit. 
 
 02036 
 
 99979 
 
 03781 
 
 99929 
 
 05524 
 
 99847 
 
 07266 
 
 99736 
 
 50 
 
 11 
 
 00320 
 
 99999 
 
 02065 
 
 99979 
 
 03810 
 
 99927 
 
 05553 
 
 99846 
 
 07295 
 
 99734 
 
 49 
 
 12 
 
 00349 
 
 99999 
 
 02094 
 
 99978 
 
 03839 
 
 99926 
 
 05582 
 
 99844 
 
 07324 
 
 99731 
 
 48 
 
 13 
 
 00378 
 
 99999 
 
 02123 
 
 99977 
 
 03868 
 
 99925 
 
 05611 
 
 99842 
 
 07353 
 
 99729 
 
 47 
 
 14 
 
 00407 
 
 99999 
 
 02152 
 
 99977 
 
 03897 
 
 99924 
 
 05640 
 
 99841 
 
 07382 
 
 99727 
 
 46 
 
 15 
 
 00436 
 
 99999 
 
 02181 
 
 99976 
 
 03926 
 
 99923 
 
 05669 
 
 99839 
 
 07411 
 
 99725 
 
 45 
 
 16 
 
 00465 
 
 99999 
 
 02211 
 
 99976 
 
 03955 
 
 99922 
 
 05698 
 
 99838 
 
 07440 
 
 99723 
 
 44 
 
 17 
 
 00495 
 
 99999 
 
 02240 
 
 99975 
 
 03984 
 
 99921 
 
 05727 
 
 99836 
 
 07469 
 
 99721 
 
 43 
 
 18 
 
 00524 
 
 99999 
 
 02269 
 
 99974 
 
 04013 
 
 99919 
 
 05756 
 
 99834 
 
 07498 
 
 99719 
 
 42 
 
 19 
 
 00553 
 
 99998 
 
 02298 
 
 99974 
 
 04042 
 
 99918 
 
 05785 
 
 99833 
 
 07527 
 
 99716 
 
 41 
 
 20 
 
 00582 
 
 99998 
 
 02327 
 
 99973 
 
 04071 
 
 99917 
 
 05814 
 
 99831 
 
 07556 
 
 99714 
 
 40 
 
 21 
 
 00611 
 
 99998 
 
 02356 
 
 99972 
 
 04100 
 
 99916 
 
 05844 
 
 99829 
 
 07585 
 
 99712 
 
 39 
 
 22 
 
 00640 
 
 99998 
 
 02385 
 
 99972 
 
 04129 
 
 99915 
 
 05873 
 
 99827 
 
 07614 
 
 99710 
 
 38 
 
 23 
 
 00669 
 
 99998 
 
 02414 
 
 99971 
 
 04159 
 
 99913 
 
 05902 
 
 99826 
 
 07643 
 
 99708 
 
 37 
 
 24 
 
 00698 
 
 99998 
 
 02443 
 
 99970 
 
 04188 
 
 99912 
 
 05931 
 
 99824 
 
 07672 
 
 99705 
 
 36 
 
 25 
 
 00727 
 
 99997 
 
 02472 
 
 99969 
 
 04217 
 
 99911 
 
 05960 
 
 99822 
 
 07701 
 
 99703 
 
 35 
 
 26 
 
 00756 
 
 99997 
 
 02501 
 
 99969 
 
 04246 
 
 99910 
 
 05989 
 
 99821 
 
 07730 
 
 99701 
 
 34 
 
 27 
 
 00785 
 
 99997 
 
 02530 
 
 99968 
 
 04275 
 
 99909 
 
 06018 
 
 99819 
 
 07759 
 
 99699 
 
 33 
 
 28 
 
 00814 
 
 99997 
 
 02560 
 
 99967 
 
 04304 
 
 99907 
 
 06047 
 
 99817 
 
 07788 
 
 99696 
 
 32 
 
 29 
 
 00844 
 
 99996 
 
 02589 
 
 99966 
 
 04333 
 
 99906 
 
 06076 
 
 99815 
 
 07817 
 
 99694 
 
 31 
 
 30 
 
 00873 
 
 99996 
 
 02618 
 
 99966 
 
 04362 
 
 99905 
 
 06105 
 
 99813 
 
 07846 
 
 99692 
 
 30 
 
 31 
 
 00902 
 
 99996 
 
 02647 
 
 99P65 
 
 04391 
 
 99904 
 
 06134 
 
 99812 
 
 07875 
 
 99689 
 
 29 
 
 32 
 
 00931 
 
 99996 
 
 02676 
 
 99964 
 
 04420 
 
 99902 
 
 06163 
 
 99810 
 
 07904 
 
 99687 
 
 28 
 
 33 
 
 00960 
 
 99995 
 
 02705 
 
 99963 
 
 04449 
 
 99901 
 
 06192 
 
 99808 
 
 07933 
 
 99685 
 
 27 
 
 34 
 
 00989 
 
 99995 
 
 02734 
 
 99963 
 
 04478 
 
 99900 
 
 06221 
 
 99806 
 
 07962 
 
 99683 
 
 26 
 
 35 
 
 01018 
 
 99995 
 
 02763 
 
 99962 
 
 04507 
 
 99898 
 
 06250 
 
 99804 
 
 07991 
 
 99680 
 
 25 
 
 36 
 
 01047 
 
 99995 
 
 02792 
 
 99961 
 
 04536 
 
 99897 
 
 06279 
 
 99803 
 
 08020 
 
 99678 
 
 24 
 
 37 
 
 01076 
 
 99994 
 
 02821 
 
 99960 i 
 
 04565 
 
 99896 
 
 06308 
 
 99801 
 
 08049 
 
 99676 
 
 23 
 
 38 
 
 01105 
 
 99994 
 
 02850 
 
 99959 
 
 04594 
 
 99894 
 
 06337 
 
 99799 
 
 08078 
 
 99673 
 
 22 
 
 39 
 
 01134 
 
 99994 
 
 02879 
 
 99959 
 
 04623 
 
 99893 
 
 06366 
 
 99797 
 
 08107 
 
 99671 
 
 21 
 
 40 
 
 01164 
 
 99993 
 
 02908 
 
 99958 
 
 04653 
 
 99892 
 
 06395 
 
 99795 
 
 08136 
 
 99668 
 
 20 
 
 41 
 
 01193 
 
 99993 
 
 02938 
 
 99957 
 
 04682 
 
 99890 
 
 06424 
 
 99793 
 
 08165 
 
 99666 
 
 19 
 
 42 
 
 01222 
 
 99993 
 
 02967 
 
 99956 
 
 04711 
 
 99889 
 
 06453 
 
 99792 
 
 08194 
 
 99664 
 
 18 
 
 43 
 
 01251 
 
 99992 
 
 02996 
 
 99955 
 
 04740 
 
 99888 
 
 06482 
 
 99790 
 
 08223 
 
 99661 
 
 17 
 
 44 
 
 01280 
 
 99992 
 
 03025 
 
 99954 
 
 04769 
 
 99886 
 
 06511 
 
 99788 
 
 08252 
 
 99659 
 
 16 
 
 45 
 
 01309 
 
 99991 
 
 03054 
 
 99953 
 
 04798 
 
 99885 
 
 06540 
 
 99786 
 
 08281 
 
 99657 
 
 15 
 
 46 
 
 01338 
 
 99991 
 
 03083 
 
 99952 
 
 04827 
 
 99883 
 
 06569 
 
 99784 
 
 08310 
 
 99654 
 
 14 
 
 47 
 
 01367 
 
 99991 
 
 03112 
 
 99952 
 
 04856 
 
 99882 
 
 06598 
 
 99782 
 
 08339 
 
 99652 
 
 13 
 
 48 
 
 01396 
 
 99990 
 
 03141 
 
 99951 
 
 04885 
 
 99881 ! 
 
 06627 
 
 99780 
 
 08368 
 
 99649 
 
 12 
 
 49 
 
 01425 
 
 99990 
 
 03170 
 
 99950 
 
 04914 
 
 99879 
 
 06656 
 
 99778 
 
 08397 
 
 99647 
 
 11 
 
 50 
 
 01454 
 
 99989 
 
 03199 
 
 99949 
 
 04943 
 
 99878 
 
 06685 
 
 99776 
 
 08426 
 
 99644 
 
 10 
 
 51 
 
 01483 
 
 99989 
 
 03228 
 
 99948 
 
 04972 
 
 99876 
 
 06714 
 
 99774 
 
 08455 
 
 99642 
 
 9 
 
 52 
 
 01513 
 
 99989 
 
 03257 
 
 99947 
 
 05001 
 
 99875 
 
 06743 
 
 99772 
 
 08484 
 
 99639 
 
 8 
 
 53 
 
 01542 
 
 99988 
 
 03286 
 
 99946 
 
 05030 
 
 99873 
 
 06773 
 
 99770 
 
 08513 
 
 99637 
 
 7 
 
 54 
 
 01571 
 
 99988 
 
 03316 
 
 99945 
 
 05059 
 
 99872 
 
 06802 
 
 99768 
 
 08542 
 
 99635 
 
 6 
 
 55 
 
 01600 
 
 99987 
 
 03345 
 
 99944 
 
 05088 
 
 99870 
 
 06831 
 
 99766 
 
 08571 
 
 99632 
 
 5 
 
 56 
 
 01629 
 
 99987 
 
 03374 
 
 99943 
 
 05117 
 
 99869 
 
 06860 
 
 99764 
 
 08600 
 
 99630 
 
 4 
 
 57 
 
 01658 
 
 99986 
 
 03403 
 
 99942 
 
 05146 
 
 99867 
 
 06889 
 
 99762 
 
 08629 
 
 99627 
 
 3 
 
 58 
 
 01687 
 
 99986 
 
 03432 
 
 99941 
 
 05175 
 
 99866 
 
 06918 
 
 99760 
 
 08658 
 
 99625 
 
 2 
 
 59 
 
 01716 
 
 99985 
 
 03461 
 
 99940 
 
 05205 
 
 99864 
 
 06947 
 
 99758 
 
 08687 
 
 99622 
 
 1 
 
 60 
 
 01745 
 
 99985 
 
 03490 
 
 99939 
 
 05234 
 
 99863 
 
 06976 
 
 99756 
 
 08716 
 
 99619 
 
 
 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 
 t 
 
 8O 
 
 88 | 
 
 87" 
 
 8G 
 
 85 
 
 / 
 
APPENDIX. 
 
 837 
 
 NATURAL. SINKS AND COSINES. 
 
 
 5 
 
 G 
 
 7-0 
 
 8 
 
 
 
 
 / 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 / 
 
 
 
 08716 
 
 99619 
 
 10453 
 
 99452 
 
 12187 
 
 99255 
 
 13917 
 
 99027 
 
 15643 
 
 98769 
 
 60 
 
 1 
 
 08745 
 
 99617 
 
 10482 
 
 99449 
 
 12216 
 
 99251 
 
 13946 
 
 99023 
 
 15672 
 
 98764 
 
 59 
 
 2 
 
 08774 
 
 99614 
 
 10511 
 
 99446 
 
 12245 
 
 99248 
 
 13975 
 
 99019 
 
 15701 
 
 98760 
 
 58 
 
 3 
 
 08803 
 
 99612 
 
 10540 
 
 99443 
 
 12274 
 
 99244 
 
 14004 
 
 99015 
 
 15730 
 
 98755 
 
 57 
 
 4 
 
 08831 
 
 99609 
 
 10569 
 
 99440 
 
 12302 
 
 99240 
 
 14033 
 
 99011 
 
 15758 
 
 98751 
 
 56 
 
 5 
 
 08860 
 
 99607 
 
 10597 
 
 99437 
 
 12331 
 
 99237 
 
 14061 
 
 99006 
 
 15787 
 
 98746 
 
 55 
 
 6 
 
 08889 
 
 99604 
 
 10626 
 
 99434 
 
 12360 
 
 99233 
 
 14090 
 
 99002 
 
 15816 
 
 98741 
 
 54 
 
 7 
 
 08918 
 
 99602 
 
 10655 
 
 99431 
 
 12389 
 
 99230 
 
 14119 
 
 98998 
 
 15845 
 
 98737 
 
 53 
 
 8 
 
 08947 
 
 99599 
 
 10684 
 
 99428 
 
 12418 
 
 99226 
 
 14148 
 
 98994 
 
 15873 
 
 98732 
 
 52 
 
 9 
 
 08976 
 
 99596 
 
 10713 
 
 99424 
 
 12447 
 
 99222 
 
 14177 
 
 98990 
 
 15902 
 
 98728 
 
 51 
 
 10 
 
 09005 
 
 99594 
 
 10742 
 
 99421 
 
 12476 
 
 99219 
 
 14205 
 
 98986 
 
 15931 
 
 98723 
 
 50 
 
 11 
 
 09034 
 
 99591 
 
 10771 
 
 99418 
 
 12504 
 
 99215 
 
 14234 
 
 98982 
 
 15959 
 
 98718 
 
 49 
 
 12 
 
 09063 
 
 99588 
 
 10800 
 
 99415 
 
 12533 
 
 99211 
 
 14263 
 
 98978 
 
 15988 
 
 98714 
 
 48 
 
 13 
 
 09092 
 
 99586 
 
 10829 
 
 99412 
 
 12562 
 
 99208 
 
 14292 
 
 98973 
 
 16017 
 
 98709 
 
 47 
 
 14 
 
 09121 
 
 99583 
 
 10858 
 
 99409 
 
 12591 
 
 99204 
 
 14320 
 
 98969 
 
 16046 
 
 98704 
 
 46 
 
 15 
 
 09150 
 
 99580 
 
 10887 
 
 99406 
 
 12620 
 
 99200 
 
 14349 
 
 98965 
 
 16074 
 
 98700 
 
 45 
 
 16 
 
 09179 
 
 99578 
 
 10916 
 
 99402 
 
 12649 
 
 99197 
 
 14378 
 
 98961 
 
 16103 
 
 98695 
 
 44 
 
 17 
 
 09208 
 
 99575 
 
 10945 
 
 99399 
 
 12678 
 
 99193 
 
 14407 
 
 98957 
 
 16132 
 
 98690 
 
 43 
 
 18 
 
 09237 
 
 99572 
 
 10973 
 
 99396 
 
 12706 
 
 99189 
 
 14436 
 
 98953 
 
 16160 
 
 98686 
 
 42 
 
 19 
 
 09266 
 
 99570 
 
 11002 
 
 99393 
 
 12735 
 
 99186 
 
 14464 
 
 98948 
 
 16189 
 
 98681 
 
 41 
 
 20 
 
 09295 
 
 99567 
 
 11031 
 
 99390 
 
 12764 
 
 99182 
 
 14493 
 
 98944 
 
 16218 
 
 98676 
 
 40 
 
 21 
 
 09324 
 
 99564 
 
 11060 
 
 99386 
 
 12793 
 
 99178 
 
 14522 
 
 98940 
 
 16246 
 
 98671 
 
 39 
 
 22 
 
 09353 
 
 99562 
 
 11089 
 
 99383 
 
 12822 
 
 99175 
 
 14551 
 
 98936 
 
 16275 
 
 98667 
 
 38 
 
 23 
 
 09382 
 
 99559 
 
 11118 
 
 99380 
 
 12851 
 
 99171 
 
 14580 
 
 98931 
 
 16304 
 
 98662 
 
 37 
 
 24 
 
 09411 
 
 99556 
 
 11147 
 
 99377 
 
 12880 
 
 99167 
 
 14608 
 
 98927 
 
 16333 
 
 98657 
 
 36 
 
 25 
 
 09440 
 
 99553 
 
 11176 
 
 99374 
 
 12908 
 
 99163 
 
 14637 
 
 98923 
 
 16361 
 
 98652 
 
 35 
 
 26 
 
 09469 
 
 99551 
 
 11205 
 
 99370 
 
 12937 
 
 99160 
 
 14666 
 
 98919- 
 
 16390 
 
 98648 
 
 34 
 
 27 
 
 09498 
 
 99548 
 
 11234 
 
 99367 
 
 12966 
 
 99156 
 
 14695 
 
 98914 
 
 16419 
 
 98643 
 
 33 
 
 28 
 
 09527 
 
 99545 
 
 11263 
 
 99364 
 
 12995 
 
 99152 
 
 14723 
 
 98910 
 
 16447 
 
 98638 
 
 32 
 
 29 
 
 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 
 
 11349 
 
 99354 
 
 13081 
 
 99141 
 
 14810 
 
 98897 
 
 16533 
 
 98624 
 
 29 
 
 32 
 
 09642 
 
 99534 
 
 11378 
 
 99351 
 
 13110 
 
 99137 
 
 14838 
 
 98893 
 
 16562 
 
 98619 
 
 28 
 
 33 
 
 09671 
 
 99531 
 
 11407 
 
 99347 
 
 13139 
 
 99133 
 
 14867 
 
 98889 
 
 16591 
 
 98614 
 
 27 
 
 34 
 
 09700 
 
 99528 
 
 11436 
 
 99344 
 
 13168 
 
 99129 
 
 14896 
 
 98884 
 
 16620 
 
 98609 
 
 26 
 
 35 
 
 09729 
 
 99526 
 
 11465 
 
 99341 
 
 13197 
 
 99125 
 
 14925 
 
 98880 
 
 16648 
 
 98604 
 
 25 
 
 36 
 
 09758 
 
 99523 
 
 11494 
 
 99337 
 
 13226 
 
 99122 
 
 14954 
 
 98876 
 
 16677 
 
 98600 
 
 24 
 
 37 
 
 09787 
 
 99520 
 
 11523 
 
 99334 
 
 13254 
 
 99118 
 
 14982 
 
 98871 
 
 16706 
 
 98595 
 
 23 
 
 38 
 
 09816 
 
 99517 
 
 11552 
 
 99331 
 
 13283 
 
 99114 
 
 15011 
 
 98867 
 
 16734 
 
 98590 
 
 22 
 
 39 
 
 09845 
 
 99514 
 
 11580 
 
 99327 
 
 13312 
 
 99110 
 
 15040 
 
 98863 
 
 16763 
 
 98585 
 
 21 
 
 40 
 
 09874 
 
 99511 
 
 11609 
 
 99324 
 
 13341 
 
 99106 
 
 15069 
 
 98858 
 
 16792 
 
 98580 
 
 20 
 
 41 
 
 09903 
 
 99508 
 
 11638 
 
 99320 
 
 13370 
 
 99102 
 
 15097 
 
 98854 
 
 16820 
 
 98575 
 
 19 
 
 42 
 
 09932 
 
 99506 
 
 11667 
 
 99317 
 
 13399 
 
 99098 
 
 15126 
 
 98849 
 
 16849 
 
 98570 
 
 18 
 
 43 
 
 09961 
 
 99503 
 
 11696 
 
 99314 
 
 13427 
 
 99094 
 
 15155 
 
 98845 
 
 16878 
 
 98565 
 
 17 
 
 44 
 
 09990 
 
 99500 
 
 11725 
 
 99310 
 
 13456 
 
 99091 
 
 15184 
 
 98841 
 
 16906 
 
 98561 
 
 16 
 
 45 
 
 10019 
 
 99497 
 
 11754 
 
 99307 
 
 13485 
 
 99087 
 
 15212 
 
 98836 
 
 16935 
 
 98556 
 
 15 
 
 46 
 
 10048 
 
 99494 
 
 11783 
 
 99303 
 
 13514 
 
 99083 
 
 15241 
 
 98832 
 
 16964 
 
 98551 
 
 14 
 
 47 
 
 10077 
 
 99491 
 
 11812 
 
 99300 
 
 13543 
 
 99079 
 
 15270 
 
 98827 
 
 16992 
 
 98546 
 
 13 
 
 48 
 
 10106 
 
 99488 
 
 11840 
 
 99297 
 
 13572 
 
 99075 
 
 15299 
 
 98823 
 
 17021 
 
 98541 
 
 12 
 
 49 
 
 10135 
 
 99485 
 
 11869 
 
 99293 
 
 13600 
 
 99071 
 
 15327 
 
 98818 
 
 17050 
 
 98536 
 
 11 
 
 50 
 
 10164 
 
 99482 
 
 11898 
 
 99290 
 
 13629 
 
 99067 
 
 15356 
 
 98814 
 
 17078 
 
 98531 
 
 10 
 
 51 
 
 10192 
 
 99479 
 
 11927 
 
 99286 
 
 13658 
 
 99063 
 
 15385 
 
 98809 
 
 17107 
 
 98526 
 
 9 
 
 52 
 
 10221 
 
 99476 
 
 11956 
 
 99283 
 
 13687 
 
 99059 
 
 15414 
 
 98805 
 
 17136 
 
 98521 
 
 8 
 
 53 
 
 10250 
 
 99473 
 
 11985 
 
 99279 
 
 13716 
 
 99055 
 
 15442 
 
 98800 
 
 17164 
 
 98516 
 
 7 
 
 54 
 
 10279 
 
 99470 
 
 12014 
 
 99276 
 
 13744 
 
 99051 
 
 15471 
 
 98796 
 
 17193 
 
 98511 
 
 6 
 
 55 
 
 10308 
 
 99467 
 
 12043 
 
 99272 
 
 13773 
 
 99047 
 
 15500 
 
 98791 
 
 17222 
 
 98506 
 
 5 
 
 56 
 
 10337 
 
 99464 
 
 12071 
 
 99269 
 
 13802 
 
 99043 
 
 15529 
 
 98787 
 
 17250 
 
 98501 
 
 4 
 
 57 
 
 10366 
 
 99461 
 
 12100 
 
 99265 
 
 13831 
 
 99039 
 
 15557 
 
 98782 
 
 17279 
 
 98496 
 
 3 
 
 58 
 
 10395 
 
 99458 
 
 12129 
 
 99262 
 
 13860 
 
 99035 
 
 15586 
 
 98778 
 
 17308 
 
 98491 
 
 2 
 
 59 
 
 10424 
 
 99455 
 
 12158 
 
 99258 
 
 13889 
 
 99031 
 
 15615 
 
 98773 
 
 17336 
 
 98486 
 
 1 
 
 60 
 
 10453 
 
 99452 
 
 12187 
 
 99255 
 
 13917 
 
 99027 
 
 15643 
 
 98769 
 
 17365 
 
 98481 
 
 
 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 
 / 
 
 84 
 
 83 
 
 83 
 
 81" 
 
 8O 
 
 / 
 
838 
 
 APPENDIX. 
 
 NATURAL, SINES AND COSINES. 
 
 
 1O 
 
 11 
 
 13 
 
 13 
 
 14 
 
 
 / 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 f 
 
 
 
 17365 
 
 98481 
 
 19081 
 
 98163 
 
 20791 
 
 97815 
 
 22495 
 
 97437 
 
 24192 
 
 97030 
 
 60 
 
 1 
 
 17393 
 
 98476 
 
 19109 
 
 98157 
 
 20820 
 
 97809 
 
 22523 
 
 97430 ! 
 
 24220 
 
 97023 
 
 59 
 
 2 
 
 17422 
 
 98471 
 
 19138 
 
 98152 
 
 20848 
 
 97803 
 
 22552 
 
 97424 
 
 24249 
 
 97015 
 
 58 
 
 3 
 
 17451 
 
 98466 
 
 19167 
 
 98146 
 
 20877 
 
 97797 
 
 22580 
 
 97417 
 
 24277 
 
 97008 
 
 57 
 
 4 
 
 17479 
 
 98461 
 
 19195 
 
 98140 
 
 20905 
 
 97791 
 
 22608 
 
 97411 
 
 24305 
 
 97001 
 
 56 
 
 5 
 
 17508 
 
 98455 
 
 19224 
 
 98135 
 
 20933 
 
 97784 
 
 22637 
 
 97404 
 
 24333 
 
 96994 55 
 
 6 
 
 17537 
 
 98450 
 
 19252 
 
 98129 
 
 20962 
 
 97778 
 
 22665 
 
 : 97398 
 
 24362 
 
 96987 54 
 
 7 
 
 17565 
 
 98445 
 
 19281 
 
 98124 
 
 20990 
 
 97772 
 
 22693 
 
 97391 
 
 24390 
 
 96980 
 
 53 
 
 8 
 
 17594 
 
 98440 
 
 19309 
 
 98118 
 
 21019 
 
 97766 
 
 22722 
 
 97384 
 
 24418 
 
 96973 
 
 52 
 
 9 
 
 17623 
 
 98435 
 
 19338 
 
 98112 
 
 21047 
 
 97760 
 
 22750 
 
 j 97378 
 
 24446 
 
 96966 
 
 51 
 
 10 
 
 17651 
 
 98430 
 
 19366 
 
 98107 
 
 21076 
 
 97754 
 
 22778 
 
 : 97371 
 
 24474 
 
 96959 
 
 50 
 
 11 
 
 17680 
 
 98425 
 
 19395 
 
 98101 
 
 21104 
 
 97748 
 
 22807 
 
 97365 
 
 24503 
 
 96952 
 
 49 
 
 12 
 
 17708 
 
 98420 
 
 19423 
 
 98096 
 
 21132 
 
 97742 
 
 22835 
 
 97358 
 
 24531 
 
 96945 
 
 48 
 
 13 
 
 17737 
 
 98414 
 
 19452 
 
 98090 
 
 21161 
 
 97735 
 
 22863 
 
 97351 
 
 24559 
 
 96937 
 
 ; 47 
 
 14 
 
 17766 
 
 98409 
 
 19481 
 
 98084 
 
 21189 
 
 97729 
 
 22892 
 
 97345 
 
 24587 
 
 96930 
 
 46: 
 
 15 
 
 17794 
 
 98404 
 
 19509 
 
 98079 
 
 21218 
 
 97723 22920 
 
 97338 
 
 24615 
 
 96923 
 
 45 
 
 16 
 
 17823 
 
 98399 
 
 19538 
 
 98073 
 
 21246 
 
 97717 '' 22948 
 
 97331 
 
 24644 
 
 96916 
 
 44 
 
 17 
 
 17852 
 
 98394 ! 
 
 19566 
 
 98067 
 
 21275 
 
 97711 1 22977 
 
 97325 
 
 24672 
 
 96909 
 
 43 
 
 18 
 
 17880 
 
 98389 
 
 19595 
 
 98061 
 
 21303 
 
 97705 1 23005 
 
 97318 
 
 24700 
 
 96902 
 
 '42 
 
 19 
 
 17909 
 
 98383 
 
 19623 
 
 98056 
 
 21331 
 
 97698 ! 23033 
 
 97311 : 
 
 24728 
 
 96894 
 
 41 
 
 20 
 
 17937 
 
 98378 
 
 19652 
 
 98050 
 
 21360 
 
 97692 23062 
 
 97304 
 
 24756 
 
 1 96887 
 
 40 
 
 21 
 
 17966 
 
 98373 
 
 19680 
 
 98044 
 
 21388 
 
 97686 ; 23090 
 
 97298 
 
 24784 
 
 96880 
 
 39 
 
 22 
 
 17995 
 
 98368 
 
 19709 
 
 98039 
 
 21417 
 
 97680 i 23118 
 
 97291 
 
 24813 
 
 96873 
 
 : 38 
 
 23 
 
 18023 
 
 98362 
 
 19737 
 
 98033 
 
 21445 
 
 97673 ; 23146 
 
 97284 
 
 24841 
 
 96866 
 
 37 
 
 24 
 
 18052 
 
 98357 
 
 19766 
 
 98027 
 
 21474 
 
 97667 
 
 23175 
 
 97278 
 
 24869 
 
 96858 36 
 
 25 
 
 18081 
 
 98352 
 
 19794 
 
 98021 
 
 21502 
 
 97661 
 
 23203 
 
 97271 
 
 24897 
 
 96851 35 
 
 26 
 
 18109 
 
 98347 
 
 19823 
 
 98016 
 
 21530 
 
 97655 
 
 23231 
 
 97264 
 
 24925 
 
 '96844 
 
 34 
 
 27 
 
 18138 
 
 98341 
 
 19851 
 
 98010 
 
 21559 
 
 97648 
 
 23260 
 
 97257 
 
 24954 
 
 96837 
 
 33 
 
 28 
 
 18166 
 
 98336 
 
 19880 
 
 98004 
 
 21587 
 
 97642 
 
 23288 
 
 97251 
 
 24982 
 
 96829 
 
 32 
 
 29 
 
 18195 
 
 98331 
 
 19908 
 
 97998 
 
 21616 
 
 97636 
 
 23316 
 
 97244 
 
 25010 
 
 96822 
 
 31 
 
 30 
 
 18224 
 
 98325 
 
 19937 
 
 97992 
 
 21644 
 
 97630 
 
 23345 
 
 97237 
 
 25038 
 
 ; 96815 
 
 30 
 
 31 
 
 18252 
 
 98320 
 
 19965 
 
 97987 
 
 21672 
 
 97623 
 
 23373 
 
 97230 
 
 25066 
 
 96807 
 
 29 
 
 32 
 
 18281 
 
 98315 
 
 19994 
 
 97981 
 
 21701 
 
 97617 
 
 23401 
 
 97223 
 
 25094 
 
 96800 
 
 28. 
 
 33 
 
 18309 
 
 98310 
 
 20022 
 
 97975 
 
 21729 
 
 97611 
 
 23429 
 
 97217 
 
 25122 
 
 96793 
 
 27 
 
 34 
 
 18338 
 
 98304 
 
 20051 
 
 97969 
 
 21758 
 
 97604 
 
 23458 
 
 97210 
 
 25151 
 
 96786 
 
 26 
 
 35 
 
 18367 
 
 98299 
 
 20079 
 
 97963 
 
 21786 
 
 97&9S 
 
 23486 
 
 ' 97203 
 
 25179 
 
 96778 
 
 25 
 
 36 
 
 18395 
 
 98294 
 
 20108 
 
 97958 
 
 21814 
 
 97592 
 
 23514 
 
 : 97196 
 
 25207 
 
 96771 
 
 24 
 
 37 
 
 18424 
 
 98288 
 
 20136 
 
 97952 
 
 21843 
 
 97585 
 
 23542 
 
 97189 
 
 25235 
 
 96764 
 
 23 
 
 38 
 
 18452 
 
 98283 
 
 20165 
 
 97946 
 
 21871 
 
 97579 
 
 23571 
 
 97182 
 
 25263 
 
 96756 
 
 22 
 
 39 
 
 18481 
 
 98277 
 
 20193 
 
 97940 
 
 21899 
 
 97573 
 
 23599 
 
 97176 
 
 25291 
 
 96749 
 
 21 
 
 40 
 
 18509 
 
 98272 
 
 20222 
 
 97934 
 
 21928 
 
 97566 
 
 23627 
 
 97169 
 
 25320 
 
 96742 
 
 20 
 
 41 
 
 18538 
 
 98267 
 
 20250 
 
 97928 
 
 21956 
 
 97560 
 
 23656 
 
 97162 
 
 25348 
 
 96734 
 
 19 
 
 42 
 
 18567 
 
 98261 
 
 20279 
 
 97922 
 
 21985 
 
 97553 
 
 23684 
 
 97155 
 
 25376 
 
 96727 
 
 18 
 
 43 
 
 18595 
 
 98256 
 
 20307 
 
 97916 
 
 22013 
 
 97547 
 
 23712 
 
 97148 
 
 25404 
 
 96719 
 
 17 
 
 44 
 
 18624 
 
 98250 
 
 20336 
 
 97910 
 
 22041 
 
 97541 
 
 23740 
 
 97141 
 
 25432 
 
 96712 
 
 16 
 
 45 
 
 18652 
 
 98245 
 
 20364 
 
 97905 
 
 22070 
 
 97534 
 
 23769 
 
 97134 
 
 25460 
 
 96705 
 
 15 
 
 46 
 
 18681 
 
 98240 
 
 20393 
 
 97899 
 
 22098 
 
 97528 
 
 23797 
 
 97127 
 
 25488 
 
 96697 
 
 14 
 
 47 
 
 18710 
 
 98234 
 
 20421 
 
 97893 
 
 22126 
 
 97521 
 
 23825 
 
 97120 
 
 25516 
 
 96690 
 
 13 
 
 48. 
 
 18738 
 
 98229 
 
 20450 
 
 97887 
 
 22155 
 
 97515 
 
 23853 
 
 97113 
 
 25545 
 
 96682 
 
 12 
 
 49 
 
 18767 
 
 98223 
 
 20478 
 
 97881 
 
 22183 
 
 97508 
 
 23882 
 
 97106 
 
 25573 
 
 96675 
 
 11 
 
 50 
 
 18795 
 
 98218 
 
 20507 
 
 97875 
 
 22212 
 
 97502 
 
 23910 
 
 97100 
 
 25601 
 
 96667 
 
 10 
 
 51 
 
 18824 
 
 98212 
 
 20535 
 
 97869 
 
 22240 
 
 97496 
 
 23938 
 
 97093 
 
 25629 
 
 96660 
 
 9 
 
 52 
 
 18852 
 
 98207 
 
 20563 
 
 97863 
 
 22268 
 
 97489 
 
 23966 
 
 97086 
 
 25657 
 
 96653 
 
 8 
 
 53 
 
 18881 
 
 98201 
 
 20592 
 
 97857 
 
 22297 
 
 97483 
 
 23995 
 
 97079 
 
 25685 
 
 96645 
 
 7 
 
 54 
 
 18910 
 
 98196 
 
 20620 
 
 97851 
 
 22325 
 
 97476 
 
 24023 
 
 97072 
 
 25713 
 
 96638 
 
 6 
 
 55 
 
 18938 
 
 98190 
 
 20649 
 
 97845 
 
 22353 
 
 97470 
 
 24051 
 
 97065 
 
 25741 
 
 96630 
 
 5 
 
 56 
 
 18967 
 
 98185 
 
 20677 
 
 97839 
 
 22382 
 
 97463 
 
 24079 
 
 97058 
 
 25769 
 
 96623 
 
 4 
 
 57 
 
 18995 
 
 98179 
 
 20706 
 
 97833 
 
 22410 
 
 97457 
 
 24108 
 
 97051 
 
 25798 
 
 96615 
 
 3 
 
 58 
 
 19024 
 
 98174 
 
 20734 
 
 97827 
 
 22438 
 
 97450 
 
 24136 
 
 97044 
 
 25826 
 
 96608 
 
 2 
 
 59 
 
 19052 
 
 98168 
 
 20763 
 
 97821 
 
 22467 
 
 97444 
 
 24164 
 
 97037 
 
 25854 
 
 96600 
 
 1 
 
 60 
 
 19081 
 
 98163 
 
 20791 
 
 97815 
 
 22495 
 
 97437 
 
 24192 
 
 97030 
 
 25882 
 
 96593 
 
 
 
 
 Cosine. 
 
 : Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 
 / 
 
 70 
 
 78 
 
 77 
 
 70" 
 
 75 
 
 f 
 
APPENDIX. 
 
 NATURAL, SINES AND COSINES. 
 
 
 15 
 
 16 
 
 17 
 
 18 
 
 10" 
 
 
 / 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 / 
 
 
 
 25882 
 
 96593 
 
 27564 
 
 96126 
 
 29237 
 
 95630 
 
 30902 
 
 95106 
 
 32557 
 
 94552 
 
 60 
 
 1 
 
 25910 
 
 96585 
 
 27592 
 
 96118 
 
 29265 
 
 95622 
 
 30929 
 
 95097 
 
 32584 
 
 94542 
 
 59 
 
 2 
 
 25938 
 
 96578 
 
 27620 
 
 96110 
 
 29293 
 
 95613 
 
 30957 
 
 95088 
 
 32612 
 
 94533 
 
 58 
 
 3 
 
 25966 
 
 96570 
 
 27648 
 
 96102 
 
 29321 
 
 95605 
 
 30985 
 
 95079 
 
 32639 
 
 94523 
 
 57 
 
 4 
 
 25994 
 
 96562 
 
 27676 
 
 96094 
 
 29348 
 
 : 95596 
 
 31012 
 
 95070 
 
 32667 
 
 94514 
 
 56 
 
 5 
 
 26022 
 
 96555 
 
 27704 
 
 96086 
 
 29376 
 
 95588 
 
 31040 
 
 95061 
 
 32694 
 
 94504 
 
 55 
 
 6 
 
 26050 
 
 96547 
 
 27731 
 
 96078 
 
 29404 
 
 95579 
 
 31068 
 
 95052 
 
 32722 
 
 94495 
 
 54 
 
 7 
 
 26079 
 
 96540 
 
 27759 
 
 96070 
 
 29432 
 
 95571 
 
 31095 
 
 95043 
 
 32749 
 
 94485 
 
 53 
 
 8 
 
 26107 
 
 96532 
 
 27787 
 
 96062 
 
 29460 
 
 95562 
 
 31123 
 
 ! 95033 
 
 32777 
 
 94476 
 
 52 
 
 9 
 
 26135 
 
 96524 
 
 27815 
 
 96054 
 
 29487 
 
 95554 
 
 31151 
 
 J95024 
 
 32804 
 
 :94466 
 
 51 
 
 10 
 
 26163 
 
 96517 
 
 27843 
 
 96046 
 
 29515 
 
 95545 
 
 31178 
 
 J95015 
 
 32832 
 
 94457 
 
 : 50 
 
 11 
 
 26191 
 
 96509 
 
 27871 
 
 96037 
 
 29543 
 
 95536 
 
 31206 
 
 95006 
 
 32859 
 
 94447 
 
 49 
 
 12 
 
 26219 
 
 96502 
 
 27899 
 
 96029 
 
 29571 
 
 95528 
 
 31233 
 
 94997 
 
 32887 
 
 94438 
 
 48 
 
 13 
 
 26247 
 
 96494 
 
 27927 
 
 96021 
 
 29599 
 
 95519 
 
 31261 
 
 94988 
 
 32914 
 
 94428 
 
 47 
 
 14 
 
 26275 
 
 96486 
 
 27955 
 
 96013 
 
 29626 
 
 95511 
 
 31289 
 
 94979 
 
 32942 
 
 94418 
 
 46 
 
 15 
 
 26303 
 
 96479 
 
 27983 
 
 96005 
 
 29654 
 
 95502 
 
 31316 
 
 94970 
 
 32969 
 
 94409 
 
 45 
 
 16 
 
 26331 
 
 96471 
 
 28011 
 
 95997 
 
 29682 
 
 95493 
 
 31344 
 
 94961 
 
 32997 
 
 '94399 
 
 44 
 
 17 
 
 26359 
 
 96463 
 
 28039 
 
 95989 
 
 29710 
 
 ;95485 
 
 31372 
 
 ,94952 
 
 33024 
 
 ^94390 
 
 43 
 
 18 
 
 26387 
 
 96456 
 
 28067 
 
 95981 
 
 29737 
 
 95476 
 
 31399 
 
 i94943 
 
 33051 
 
 94380 
 
 142 
 
 19 
 
 26415 
 
 96448 
 
 28095 
 
 95972 
 
 29765 
 
 95467 
 
 31427 
 
 94933 
 
 33079 
 
 94370 
 
 41 
 
 20 
 
 26443 
 
 96440 
 
 28123 
 
 95964 
 
 29793 
 
 95459 
 
 31454 
 
 94924 
 
 33106 
 
 94361 
 
 40 
 
 21 
 
 26471 
 
 96433 
 
 28150 
 
 95956 
 
 29821 
 
 95450 
 
 31482 
 
 94915 
 
 33134 
 
 94351 
 
 i39 
 
 22 
 
 26500 
 
 96425 
 
 28178 
 
 95948 
 
 29849 
 
 95441 
 
 31510 
 
 94906 
 
 33161 
 
 94342 
 
 38 
 
 23 
 
 26528 
 
 96417 
 
 28206 
 
 95940 
 
 29876 
 
 95433 
 
 31537 
 
 ;94897 
 
 33189 
 
 94332 
 
 37 
 
 24 
 
 26556 
 
 96410 
 
 28234 
 
 95931 
 
 29904 
 
 :95424 
 
 31565 
 
 94888 
 
 33216 
 
 94322 
 
 188 
 
 25 
 
 26584 
 
 96402 
 
 28262 
 
 95923 
 
 29932 
 
 95415 
 
 31593 
 
 94878 
 
 33244 
 
 94313 
 
 ! 35 
 
 26 
 
 26612 
 
 96394 
 
 28290 
 
 95915 
 
 29960 
 
 95407 
 
 31620 
 
 194869 
 
 33271 
 
 94303 
 
 |34 
 
 27 
 
 26640 
 
 96386 
 
 28318 
 
 95907 
 
 29987 
 
 95398 
 
 31648 
 
 :94860 
 
 33298 
 
 94293 
 
 '33 
 
 28 
 
 26668 
 
 96379 
 
 28346 
 
 95898 
 
 30015 
 
 95389 
 
 31675 
 
 '94851 
 
 33326 
 
 94284 
 
 :32 
 
 29 
 
 26696 
 
 96371 
 
 28374 
 
 95890 
 
 30043 
 
 95380 
 
 31703 
 
 94842 
 
 33353 
 
 94274 
 
 31 
 
 30 
 
 26724 
 
 96363 
 
 28402 
 
 95882 
 
 30071 
 
 95372 
 
 31730 
 
 ;94832 
 
 33381 
 
 94264 
 
 30 
 
 31 
 
 26752 
 
 96355 
 
 28429 
 
 95874 
 
 30098 
 
 95363 
 
 31758 
 
 194823 
 
 33408 
 
 94254 
 
 29 
 
 32 
 
 26780 
 
 96347 
 
 28457 
 
 95865 
 
 30126 
 
 95354 
 
 31786 
 
 94814 
 
 33436 
 
 94245 
 
 28 
 
 33 
 
 26808 
 
 96340 
 
 28485 
 
 95857 
 
 30154 
 
 95345 
 
 31813 
 
 94805 
 
 33463 
 
 : 94235 
 
 27 
 
 34 
 
 26836 
 
 96332 
 
 28513 
 
 95849 
 
 30182 
 
 95337 
 
 31841 
 
 94795 
 
 33490 
 
 94225 
 
 26 
 
 35 
 
 26864 
 
 96324 
 
 28541 
 
 95841 
 
 30209 
 
 95328 
 
 31868 
 
 94786 
 
 33518 
 
 94215 
 
 25 
 
 36 
 
 26892 
 
 96316 
 
 28569 
 
 95832 
 
 30237 
 
 95319 
 
 31896 
 
 94777 
 
 33545 
 
 94206 
 
 24 
 
 37 
 
 26920 
 
 96308 
 
 28597 
 
 95824 
 
 30265 
 
 95310 
 
 31923 
 
 94768 
 
 33573 
 
 ;94196 
 
 S3 
 
 38 
 
 26948 
 
 96301 
 
 28625 
 
 95816 
 
 30292 
 
 95301 
 
 31951 
 
 :94758 
 
 ; 33600 
 
 94186 
 
 ;22 
 
 39 
 
 26976 
 
 96293 
 
 28652 
 
 95807 
 
 30320 
 
 95293 
 
 31979 
 
 94749 
 
 '33627 
 
 94176 
 
 21 
 
 40 
 
 27004 
 
 96285 
 
 28680 
 
 95799 
 
 30348 
 
 95284 
 
 32006 
 
 94740 
 
 ' 33655 
 
 94167 
 
 :20 
 
 41 
 
 27032 
 
 96277 
 
 28708 
 
 95791 
 
 30376 
 
 95275 
 
 32034 
 
 ;94730 
 
 33682 
 
 94157 
 
 19 
 
 42 
 
 27060 
 
 96269 
 
 28736 
 
 95782 
 
 30403 
 
 95266 
 
 32061 
 
 94721 
 
 33710 
 
 94147 
 
 18 
 
 43 
 
 27088 
 
 96261 
 
 28764 
 
 95774 
 
 30431 
 
 95257 
 
 32089 
 
 94712 
 
 33737 
 
 94137 
 
 17 
 
 44 
 
 27116 
 
 96253 
 
 28792 
 
 95766 
 
 30459 
 
 95248 
 
 32116 
 
 94702 
 
 33764 
 
 94127 
 
 16 
 
 45 
 
 27144 
 
 96246 
 
 28820 
 
 95757 
 
 30486 
 
 95240 
 
 32144 
 
 94693 
 
 33792 
 
 94118 
 
 15 
 
 46 
 
 27172 
 
 96238 
 
 28847 
 
 95749 
 
 30514 
 
 95231 
 
 32171 
 
 94684 
 
 33819 
 
 : 94108 
 
 14 
 
 47 
 
 27200 
 
 96230 
 
 28875 
 
 95740 
 
 30542 
 
 95222 
 
 32199 
 
 94674 
 
 33846 
 
 94098 
 
 13 
 
 48 
 
 27228 
 
 96222 
 
 28903 
 
 95732 
 
 30570 
 
 95213 
 
 32227 
 
 94665 
 
 33874 
 
 94088 
 
 12 
 
 49 
 
 27256 
 
 96214 
 
 28931 
 
 95724 
 
 30597 
 
 95204 
 
 32254 
 
 94656 
 
 33901 
 
 94078 
 
 11 
 
 50 
 
 27284 
 
 96206 
 
 28959 
 
 95715 
 
 30625 
 
 95195 
 
 32282 
 
 94646 
 
 33929 
 
 94068 
 
 10 
 
 51 
 
 27312 
 
 96198 
 
 28987 
 
 95707 
 
 30653 
 
 95186 
 
 32309 
 
 94637 
 
 33956 
 
 94058 
 
 9 
 
 52 
 
 27340 
 
 96190 
 
 29015 
 
 95698 
 
 30680 
 
 95177 
 
 32337 
 
 94627 
 
 33983 
 
 94049 
 
 8 
 
 53 
 
 27368 
 
 96182 
 
 29042 
 
 95690 
 
 30708 
 
 95168 
 
 32364 
 
 94618 
 
 34011 
 
 94039 
 
 7 
 
 54 
 
 27396 
 
 96174 
 
 29070 
 
 95681 
 
 30736 
 
 95159 
 
 32392 
 
 94609 
 
 34038 
 
 94029 
 
 6 
 
 55 
 
 27424 
 
 96166 
 
 29098 
 
 95673 
 
 30763 
 
 95150 
 
 32419 
 
 94599 
 
 34065 
 
 94019 
 
 5 
 
 56 ! 27452 
 
 96158 
 
 29126 
 
 95664 
 
 30791 
 
 !95142 
 
 32447 
 
 94590 
 
 34093 
 
 94009 
 
 4 
 
 57 
 
 27480 
 
 96150 
 
 29154 
 
 95656 
 
 30819 
 
 :95133 
 
 32474 
 
 94580 
 
 34120 
 
 93999 
 
 3 
 
 58 
 
 27508 
 
 96142 
 
 29182 
 
 95647 
 
 30846 
 
 ; 95124 
 
 32502 
 
 94571 
 
 34147 
 
 93989 
 
 2 
 
 59 
 
 27536 
 
 96134 
 
 29209 
 
 95639 ! 30874 
 
 95115 
 
 32529 
 
 94561 
 
 34175 
 
 93979 
 
 1 
 
 60 
 
 27564 
 
 96126 
 
 29237 
 
 95630 
 
 30902 
 
 95106 
 
 32557 
 
 94552 
 
 34202 
 
 93969 
 
 i 
 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 / 
 
 / 
 
 74 73 
 
 73 
 
 71 
 
 7O 
 
 
840 
 
 APPENDIX. 
 
 NATURAL. SINES AND COSINES. 
 
 
 30 
 
 31 
 
 22" 
 
 23" 
 
 34= 
 
 
 / 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 / 
 
 
 
 34202 
 
 93969 
 
 35837 
 
 93358 
 
 37461 
 
 92718 
 
 39073 
 
 92050 
 
 40674 
 
 91355 
 
 60 
 
 1 
 
 34229 
 
 93959 
 
 35864 
 
 93348 
 
 37488 
 
 92707 
 
 39100 
 
 92039 
 
 40700 
 
 91343 
 
 59 
 
 2 
 
 34257 
 
 93949 
 
 35891 
 
 93337 
 
 37515 
 
 92697 
 
 39127 
 
 92028 
 
 40727 
 
 91331 
 
 58 
 
 3 
 
 34284 
 
 93939 
 
 35918 
 
 93327 
 
 37542 
 
 92686 
 
 39153 
 
 92016 
 
 40753 
 
 91319 
 
 57 
 
 4 
 
 34311 
 
 93929 
 
 35945 
 
 93316 
 
 37569 
 
 92675 
 
 39180 
 
 92005 
 
 40780 
 
 91307 
 
 56 
 
 5 
 
 34339 
 
 93919 
 
 35973 
 
 93306 
 
 37595 
 
 92664 
 
 39207 
 
 91994 
 
 40806 
 
 91295 
 
 55 
 
 6 
 
 34366 
 
 93909 
 
 36000 
 
 93295 
 
 37622 
 
 92653 
 
 39234 
 
 91982 
 
 40833 
 
 91283 
 
 54 
 
 7 
 
 34393 
 
 93899 
 
 36027 
 
 93285 
 
 37649 
 
 92642 
 
 39260 
 
 91971 
 
 40860 
 
 91272 
 
 53 
 
 8 
 
 34421 
 
 93889 
 
 36054 
 
 93274 
 
 37676 
 
 92631 
 
 39287 
 
 91959 
 
 40886 
 
 91260 
 
 52 
 
 9 
 
 34448 
 
 93879 
 
 36081 
 
 93264 
 
 37703 
 
 92620 
 
 39314 
 
 91948 
 
 40913 
 
 91248 
 
 51 
 
 10 
 
 34475 
 
 93869 
 
 36108 
 
 93253 
 
 37730 
 
 92609 
 
 39341 
 
 91936 
 
 40939 
 
 91236 
 
 50 
 
 11 
 
 34503 
 
 93859 
 
 36135 
 
 93243 
 
 37757 
 
 92598 
 
 39367 
 
 91925 
 
 40966 
 
 91224 
 
 49 
 
 12 
 
 34530 
 
 93849 
 
 36162 
 
 93232 
 
 37784 
 
 92587 
 
 39394 
 
 91914 
 
 40992 
 
 91212 
 
 48 
 
 13 
 
 34557 
 
 93839 
 
 36190 
 
 93222 
 
 37811 
 
 92576 
 
 39421 
 
 91902 
 
 41019 
 
 91200 
 
 47 
 
 14 
 
 34584 
 
 93829 
 
 36217 
 
 93211 
 
 37838 
 
 92565 
 
 39448 
 
 91891 
 
 41045 
 
 91188 
 
 46 
 
 15 
 
 34612 
 
 93819 
 
 36244 
 
 93201 
 
 37865 
 
 92554 
 
 39474 
 
 91879 
 
 41072 
 
 91176 
 
 45 
 
 16 
 
 34639 
 
 93809 
 
 36271 
 
 93190 
 
 37892 
 
 92543 
 
 39501 
 
 91868 
 
 41098 
 
 91164 
 
 44 
 
 17 
 
 34666 
 
 93799 
 
 36298 
 
 93180 
 
 37919 
 
 92532 
 
 39528 
 
 91856 
 
 41125 
 
 91152 
 
 43 
 
 18 
 
 34694 
 
 93789 
 
 36325 
 
 93169 
 
 37946 
 
 92521 
 
 39555 
 
 91845 
 
 41151 
 
 91140 
 
 42 
 
 19 
 
 34721 
 
 93779 
 
 36352 
 
 93159 
 
 37973 
 
 92510 
 
 39581 
 
 91833 
 
 41178 
 
 91128 
 
 41 
 
 20 
 
 34748 
 
 93769 
 
 36379 
 
 93148 
 
 37999 
 
 92499 
 
 39608 
 
 91822 
 
 41204 
 
 91116 
 
 40 
 
 21 
 
 34775 
 
 93759 
 
 36406 
 
 93137 
 
 38026 
 
 92488 
 
 39635 
 
 91810 
 
 41231 
 
 91104 
 
 39 
 
 22 
 
 34803 
 
 93748 
 
 36434 
 
 93127 
 
 38053 
 
 92477 
 
 39661 
 
 91799 
 
 41257 
 
 91092 
 
 38 
 
 23 
 
 34830 
 
 93738 
 
 36461 
 
 93116 
 
 38080 
 
 92466 
 
 39688 
 
 91787 
 
 41284 
 
 91080 
 
 37 
 
 24 
 
 34857 
 
 93728 
 
 36488 
 
 93106 
 
 38107 
 
 92455 
 
 39715 
 
 91775 
 
 41310 
 
 91068 
 
 36 
 
 25 
 
 34884 
 
 93718 
 
 36515 
 
 93095 
 
 38134 
 
 92444 
 
 39741 
 
 91764 
 
 41337 
 
 91056 ' 
 
 35. 
 
 26 
 
 34912 
 
 93708 
 
 36542 
 
 93084 
 
 38161 
 
 92432 
 
 39768 
 
 91752 
 
 41363 
 
 91044 
 
 34 
 
 27 
 
 34939 
 
 93698 
 
 36569 
 
 93074 
 
 38188 
 
 92421 
 
 39795 
 
 91741 
 
 41390 
 
 91032 
 
 33 
 
 28 
 
 34966 
 
 93688 
 
 36596 
 
 93063 
 
 38215 
 
 92410 
 
 39822 
 
 91729 
 
 41416 
 
 91020 
 
 32 
 
 29 
 
 34993 
 
 93677 
 
 36623 
 
 93052 
 
 38241 
 
 92399 
 
 39848 
 
 91718 
 
 41443 
 
 91008 
 
 31 
 
 30 
 
 35021 
 
 93667 
 
 36650 
 
 93042 
 
 38268 
 
 92388 
 
 39875 
 
 91706 
 
 41469 
 
 90996 
 
 30 
 
 31 
 
 35048 
 
 93657 
 
 36677 
 
 93031 
 
 38295 
 
 92377 
 
 39902 
 
 91694 
 
 41496 
 
 90984 
 
 29 
 
 32 
 
 35075 
 
 93647 
 
 36704 
 
 93020 
 
 38322 
 
 92366 
 
 39928 
 
 91683 
 
 41522 
 
 90972 
 
 28 
 
 33 
 
 35102 
 
 93637 
 
 36731 
 
 93010 
 
 38349 
 
 92355 
 
 39955 
 
 91671 
 
 41549 
 
 90960 
 
 27 
 
 34 
 
 35130 
 
 93626 
 
 36758 
 
 92999 
 
 38376 
 
 92343 
 
 39982 
 
 91660 
 
 41575 
 
 90948 
 
 26 
 
 35 
 
 35157 
 
 93616 
 
 36785 
 
 92988 
 
 38403 
 
 92332 
 
 40008 
 
 91648 
 
 41602 
 
 90936 
 
 25 
 
 36 
 
 35184 
 
 93606 
 
 36812 
 
 92978 
 
 38430 
 
 92321 
 
 40035 
 
 91636 
 
 41628 
 
 90924 
 
 24 
 
 37 
 
 35211 
 
 93596 
 
 36839 
 
 92967 
 
 38456 
 
 92310 
 
 40062 
 
 91625 
 
 41655 
 
 90911 
 
 23 
 
 38 
 
 35239 
 
 93585 
 
 36867 
 
 92956 
 
 38483 
 
 92299 
 
 40088 
 
 91613 
 
 41681 
 
 90899 
 
 22 
 
 39 
 
 35266 
 
 93575 
 
 36894 
 
 92945 
 
 38510 
 
 92287 
 
 40115 
 
 91601 
 
 41707 
 
 90887 
 
 21 
 
 40 
 
 35293 
 
 93565 
 
 36921 
 
 92935 
 
 38537 
 
 92276 
 
 40141 
 
 91590 
 
 41734 
 
 90875 
 
 20 
 
 41 
 
 35320 
 
 93555 
 
 36948 
 
 92924 
 
 38564 
 
 92265 
 
 40168 
 
 91578 
 
 41760 
 
 90863 
 
 19 
 
 42 
 
 35347 
 
 93544 
 
 36975 
 
 929] 3 
 
 38591 
 
 92254 
 
 40195 
 
 91566 
 
 41787 
 
 90851 
 
 18 
 
 43 
 
 35375 
 
 93534 
 
 37002 
 
 92902 
 
 38617 
 
 92243 
 
 40221 
 
 91555 
 
 41813 
 
 90839 
 
 17 
 
 44 
 
 35402 
 
 93524 
 
 37029 
 
 92892 
 
 38644 
 
 92231 
 
 40248 
 
 91543 
 
 41840 
 
 90826 
 
 16 
 
 45 
 
 35429 
 
 93514 
 
 37056 
 
 92881 
 
 38671 
 
 92220 
 
 40275 
 
 91531 
 
 41866 
 
 90814 
 
 15 
 
 46 
 
 35456 
 
 93503 
 
 37083 
 
 92870 
 
 38698 
 
 92209 
 
 40301 
 
 91519 
 
 41892 
 
 90802 
 
 14 
 
 47 
 
 35484 
 
 93493 
 
 37110 
 
 92859 
 
 38725 
 
 92198 
 
 40328 
 
 91508 
 
 41919 
 
 90790 
 
 13 
 
 48 
 
 35511 
 
 93483 
 
 37137 
 
 92849 
 
 38752 
 
 92186 
 
 40355 
 
 91496 
 
 41945 
 
 90778 
 
 12 
 
 49 
 
 35538 
 
 93472 
 
 37164 
 
 92838 
 
 38778 
 
 92175 
 
 40381 
 
 91484 
 
 41972 
 
 90766 
 
 11 
 
 50 
 
 35565 
 
 93462 
 
 37191 
 
 92827 
 
 38805 
 
 92164 
 
 40408 
 
 91472 
 
 41998 
 
 90753 
 
 10 
 
 51 
 
 35592 
 
 93452 
 
 37218 
 
 92816 
 
 38832 
 
 92152 
 
 40434 
 
 91461 
 
 42024 
 
 90741 
 
 9 
 
 52 
 
 35619 
 
 93441 
 
 37245 
 
 92805 
 
 38859 
 
 92141 
 
 40461 
 
 91449 
 
 42051 
 
 90729 
 
 8 
 
 53 
 
 35647 
 
 93431 
 
 37272 
 
 92794 
 
 38886 
 
 92130 
 
 40488 
 
 91437 
 
 42077 
 
 90717 
 
 7 
 
 54 
 
 35674 
 
 93420 
 
 37299 
 
 92784 
 
 38912 
 
 92119 
 
 40514 
 
 91425 
 
 42104 
 
 90704 
 
 6 
 
 55 
 
 35701 
 
 93410 
 
 37326 
 
 92773 
 
 38939 
 
 92107 
 
 40541 
 
 91414 
 
 42130 
 
 90692 
 
 5 
 
 56 
 
 35728 
 
 93400 
 
 37353 
 
 92762 
 
 38966 
 
 92096 
 
 40567 
 
 91402 
 
 42156 
 
 90680 
 
 4 
 
 57 
 
 35755 
 
 93389 
 
 37380 
 
 92751 
 
 38993 
 
 92085 
 
 40594 
 
 91390 
 
 42183 
 
 90668 
 
 3 
 
 58 
 
 35782 
 
 93379 
 
 37407 
 
 92740 
 
 39020 
 
 92073 
 
 40621 
 
 91378 
 
 42209 
 
 90655 
 
 2 
 
 59 
 
 35810 
 
 93368 
 
 37434 
 
 92729 
 
 39046 
 
 92062 
 
 40647 
 
 91366 
 
 42235 
 
 90643 
 
 1 
 
 00 
 
 35837 
 
 93358 
 
 37461 
 
 92718 
 
 39073 
 
 92050 
 
 40674 
 
 91355 
 
 42262 
 
 90631 
 
 
 
 I 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 
 
 eo 
 
 es 
 
 G7 
 
 GO" 
 
 05 
 
 / 
 
APPENDIX. 
 
 841 
 
 NATURAL. SINES AND COSINES. 
 
 
 35 
 
 30 
 
 37 
 
 3S 
 
 39 
 
 
 
 Siiie. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine 
 
 Sine. 
 
 Cosine 
 
 Sine. 
 
 Cosine 
 
 / 
 
 
 
 42262 
 
 90631 
 
 43837 
 
 89879 
 
 45399 
 
 89101 
 
 46947 
 
 88295 
 
 48481 
 
 87462 
 
 60 
 
 1 
 
 42288 
 
 90618 
 
 43863 
 
 89867 
 
 45425 
 
 89087 
 
 46973 
 
 88281 
 
 48506 
 
 87448 
 
 59 
 
 2 
 
 42315 
 
 90606 
 
 43889 
 
 89854 
 
 45451 
 
 89074 
 
 46999 
 
 88267 
 
 48532 
 
 87434 
 
 58 
 
 3 
 
 42341 
 
 90594 
 
 43916 
 
 89841 
 
 45477 
 
 89061 
 
 47024 
 
 88254 
 
 48557 
 
 87420 
 
 57 
 
 4 
 
 42367 
 
 90582 
 
 43942 
 
 89828 
 
 45503 
 
 89048 
 
 47050 
 
 88240 
 
 48583 
 
 87406 
 
 56 
 
 5 
 
 42394 
 
 90569 
 
 43968 
 
 89816 
 
 45529 
 
 89035 
 
 47076 
 
 88226 
 
 48608 
 
 87391 
 
 55 
 
 6 
 
 42420 
 
 90557 
 
 43994 
 
 89803 
 
 45554 
 
 89021 
 
 47101 
 
 88213 
 
 48634 
 
 87377 
 
 54 
 
 7 
 
 42446 
 
 90545 
 
 44020 
 
 89790 
 
 45580 
 
 89008 
 
 47127 
 
 88199 
 
 48659 
 
 87363 
 
 53 
 
 8 
 
 42473 
 
 90532 
 
 44046 
 
 89777 
 
 45606 
 
 88995 
 
 47153 
 
 88185 
 
 48684 
 
 87349 
 
 52 
 
 9 
 
 42499 
 
 90520 
 
 44072 
 
 89764 
 
 45632 
 
 88981 
 
 47178 
 
 88172 
 
 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 
 
 90483 
 
 44151 
 
 89726 
 
 45710 
 
 88942 
 
 47255 
 
 88130 
 
 48786 
 
 87292 
 
 48 
 
 13 
 
 42604 
 
 90470 
 
 44177 
 
 89713 
 
 45736 
 
 88928 
 
 47281 
 
 88117 
 
 48811 
 
 87278 
 
 47 
 
 14 
 
 42631 
 
 90458 
 
 44203 
 
 89700 
 
 45762 
 
 88915 
 
 47306 
 
 88103 
 
 48837 
 
 87264 
 
 46 
 
 15 
 
 42657 
 
 90446 
 
 44229 
 
 89687 
 
 45787 
 
 88902 
 
 47332 
 
 88089 
 
 48862 
 
 87250 
 
 45 
 
 16 
 
 42683 
 
 90433 
 
 44255 
 
 89674 
 
 45813 
 
 88888 
 
 47358 
 
 88075 
 
 48888 
 
 87235 
 
 44 
 
 17 
 
 42709 
 
 90421 
 
 44281 
 
 89662 
 
 45839 
 
 88875 
 
 47383 
 
 88062 
 
 48913 
 
 87221 
 
 43 
 
 18 
 
 42736 
 
 90408 
 
 44307 
 
 89649 
 
 45865 
 
 88862 
 
 47409 
 
 88048 
 
 48938 
 
 87207 
 
 42 
 
 19 
 
 42762 
 
 90396 
 
 44333 
 
 89636 
 
 45891 
 
 88848 
 
 47434 
 
 88034 
 
 48964 
 
 87193 
 
 41 
 
 20 
 
 42788 
 
 90383 
 
 44359 
 
 89623 
 
 45917 
 
 88835 
 
 47460 
 
 88020 
 
 48989 
 
 87178 
 
 40 
 
 21 
 
 42815 
 
 90371 
 
 44385 
 
 89610 
 
 45942 
 
 88822 
 
 47486 
 
 88006 
 
 49014 
 
 87164 
 
 39 
 
 22 
 
 42841 
 
 90358 
 
 44411 
 
 89597 
 
 45968 
 
 88808 
 
 47511 
 
 87993 
 
 49040 
 
 87150 
 
 38 
 
 23 
 
 42867 
 
 90346 
 
 44437 
 
 89584 
 
 45994 
 
 88795 
 
 47537 
 
 87979 
 
 49065 
 
 87136 
 
 37 
 
 24 
 
 42894 
 
 90334 
 
 44464 
 
 89571 
 
 46020 
 
 88782 
 
 47562 
 
 87965 
 
 49090 
 
 87121 
 
 36 
 
 25 
 
 42920 
 
 90321 
 
 44490 
 
 89558 
 
 46046 
 
 88768 
 
 47588 
 
 87951 
 
 49116 
 
 87107 
 
 35 
 
 26 
 
 42946 
 
 90309 
 
 44516 
 
 89545 
 
 46072 
 
 88755 
 
 47614 
 
 87937 
 
 49141 
 
 87093 
 
 34 
 
 27 
 
 42972 
 
 90296 
 
 44542 
 
 89532 
 
 46097 
 
 88741 
 
 47639 
 
 87923 
 
 49166 
 
 87079 
 
 33 
 
 28 
 
 42999 
 
 90284 
 
 44568 
 
 89519 
 
 46123 
 
 88728 
 
 47665 
 
 87909 
 
 49192 
 
 87064 
 
 32 
 
 29 
 
 43025 
 
 90271 
 
 44594 
 
 89506 
 
 46149 
 
 88715 
 
 47690 
 
 87896 
 
 49217 
 
 87050 
 
 31 
 
 30 
 
 43051 
 
 90259 
 
 44620 
 
 89493 
 
 46175 
 
 88701 
 
 47716 
 
 87882 
 
 49242 
 
 87036 
 
 30 
 
 31 
 
 43077 
 
 90246 
 
 44646 
 
 89480 
 
 46201 
 
 88688 
 
 47741 
 
 87868 
 
 49268 
 
 87021 
 
 29 
 
 32 
 
 43104 
 
 90233 
 
 44672 
 
 89467 
 
 46226 
 
 88674 
 
 47767 
 
 87854 
 
 49293 
 
 87007 
 
 28 
 
 33 
 
 43130 
 
 90221 
 
 44698 
 
 89454 
 
 46252 
 
 88661 
 
 47793 
 
 87840 
 
 49318 
 
 86993 
 
 27 
 
 34 
 
 43156 
 
 90208 
 
 44724 
 
 89441 
 
 46278 
 
 88647 
 
 47818 
 
 87826 
 
 49344 
 
 86978 
 
 26 
 
 35 
 
 43182 
 
 90196 
 
 44750 
 
 89428 
 
 46304 
 
 88634 
 
 47844 
 
 87812 
 
 49369 
 
 86964 
 
 25 
 
 36 
 
 43209 
 
 90183 
 
 44776 
 
 89415 
 
 46330 
 
 88620 
 
 47869 
 
 87798 
 
 49394 
 
 86949 
 
 24 
 
 37 
 
 43235 
 
 90171 
 
 44802 
 
 89402 
 
 46355 
 
 88607 
 
 47895 
 
 87784 
 
 49419 
 
 86935 
 
 23 
 
 38 
 
 43261 
 
 90158 
 
 44828 
 
 89389 
 
 46381 
 
 88593 
 
 47920 
 
 87770 
 
 49445 
 
 86921 
 
 22 
 
 39 
 
 43287 
 
 90146 
 
 44854 
 
 89376 
 
 46407 
 
 88580 
 
 47946 
 
 87756 
 
 49470 
 
 86906 
 
 21 
 
 40 
 
 43313 
 
 90133 
 
 44880 
 
 89363 
 
 46433 
 
 88566 
 
 47971 
 
 87743 
 
 49495 
 
 86892 
 
 20 
 
 41 
 
 43340 
 
 90120 
 
 44906 
 
 89350 
 
 46458 
 
 88553 
 
 47997 
 
 87729 
 
 49521 
 
 86878 
 
 19 
 
 42 
 
 43366 
 
 90108 
 
 44932 
 
 89337 
 
 46484 
 
 88539 
 
 48022 
 
 87715 
 
 49546 
 
 86863 
 
 18 
 
 43 
 
 43392 
 
 90095 
 
 44958 
 
 89324 
 
 46510 
 
 88526 
 
 48048 
 
 87701 
 
 49571 
 
 86849 
 
 17 
 
 44 
 
 43418 
 
 90082 
 
 44984 
 
 89311 
 
 46536 
 
 88512 
 
 48073 
 
 87687 
 
 49596 
 
 86834 
 
 16 
 
 45 
 
 43445 
 
 90070 
 
 45010 
 
 89298 
 
 46561 
 
 88499 
 
 48099 
 
 87673 
 
 49622 
 
 86820 
 
 15 
 
 46 
 
 43471 
 
 90057 
 
 45036 
 
 89285 
 
 46587 
 
 88485 
 
 48124 
 
 87659 
 
 49647 
 
 86805 
 
 14 
 
 47 
 
 43497 
 
 90045 
 
 45062 
 
 89272 
 
 46613 
 
 88472 
 
 48150 
 
 87645 
 
 49672 
 
 86791 
 
 13 
 
 48 
 
 43523 
 
 90032 
 
 45088 
 
 89259 
 
 46639 
 
 88458 
 
 48175 
 
 87631 
 
 49697 
 
 86777 
 
 12 
 
 49 
 
 43549 
 
 90019 
 
 45114 
 
 89245 
 
 46664 
 
 88445 
 
 48201 
 
 87617 
 
 49723 
 
 86762 
 
 11 
 
 50 
 
 43575 
 
 90007 
 
 45140 
 
 89232 
 
 46690 
 
 88431 
 
 48226 
 
 87603 
 
 49748 
 
 86748 
 
 10 
 
 51 
 
 43602 
 
 89994 
 
 45166 
 
 89219 
 
 46716 
 
 88417 
 
 48252 
 
 87589 
 
 49773 
 
 86733 
 
 9 
 
 52 
 
 43628 
 
 89981 
 
 45192 
 
 89206 
 
 46742 
 
 88404 
 
 48277 
 
 87575 
 
 49798 
 
 86719 
 
 8 
 
 53 
 
 43654 
 
 89968 
 
 45218 
 
 89193 
 
 46767 
 
 88390 
 
 48303 
 
 87561 
 
 49824 
 
 86704 
 
 7 
 
 54 
 
 43680 
 
 89956 
 
 45243 
 
 89180 
 
 46793 
 
 88377 
 
 48328 
 
 87546 
 
 49849 
 
 86690 
 
 6 
 
 55 
 
 43706 
 
 89943 
 
 45269 
 
 89167 
 
 46819 
 
 88363 
 
 48354 
 
 87532 
 
 49874 
 
 86675 
 
 5 
 
 56 
 
 43733 
 
 89930 
 
 45295 
 
 89153 
 
 46844 
 
 88349 
 
 48379 
 
 87518 
 
 49899 
 
 86661 
 
 4 
 
 57 
 
 43759 
 
 89918 
 
 45321 
 
 89140 
 
 46870 
 
 88336 
 
 48405 
 
 87504 
 
 49924 
 
 86646 
 
 3 
 
 58 
 
 43785 
 
 89905 
 
 45347 
 
 89127 
 
 46896 
 
 88322 
 
 48430 
 
 87490 
 
 49950 
 
 86632 
 
 2 
 
 59 
 
 43811 
 
 89892 
 
 45373 
 
 89114 
 
 46921 
 
 88308 
 
 48456 
 
 87476 
 
 49975 
 
 86617 
 
 1 
 
 60 
 
 43837 
 
 89879 
 
 45399 
 
 89101 
 
 46947 
 
 88295 
 
 48481 
 
 87462 
 
 50000 
 
 86603 
 
 
 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 
 / 
 
 O4" 
 
 ea 
 
 63 
 
 ei 
 
 eo 
 
 / 
 
842 
 
 APPENDIX. 
 
 NATURAL SINES AND COSINES^ 
 
 
 3O 
 
 31 
 
 33 
 
 33 
 
 34 
 
 / 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 f 
 
 
 
 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 
 
 54537 
 
 83819 
 
 55992 
 
 82855 
 
 57 
 
 4 
 
 50101 
 
 86544 
 
 51604 
 
 85657 
 
 53091 
 
 84743 
 
 54561 
 
 83804 
 
 56016 
 
 82839 
 
 56 
 
 5 
 
 50126 
 
 86530 
 
 51628 
 
 85642 
 
 53115 
 
 84728 
 
 54586 
 
 83788 
 
 56040 
 
 82822 
 
 55 
 
 6 
 
 50151 
 
 86515 
 
 51653 
 
 85627 
 
 53140 
 
 84712 
 
 54610 
 
 83772 
 
 56064 
 
 82806 
 
 54 
 
 7 
 
 50176 
 
 86501 
 
 51678 
 
 85612 
 
 53164 
 
 84697 
 
 54635 
 
 83756 
 
 56088 
 
 82790 
 
 53 
 
 8 
 
 50201 
 
 86486 
 
 51703 
 
 85597 
 
 53189 
 
 84681 
 
 54659 
 
 83740 
 
 56112 
 
 82773 
 
 52 
 
 9 
 
 50227 
 
 86471 
 
 51728 
 
 85582 
 
 53214 
 
 84666 
 
 54683 
 
 83724 
 
 56136 
 
 82757 
 
 51 
 
 10 
 
 50252 
 
 86457 
 
 51753 
 
 85567 
 
 53238 
 
 84650 
 
 54708 
 
 83708 
 
 56160 
 
 82741 
 
 50 
 
 11 
 
 50277 
 
 86442 
 
 51778 
 
 85551 
 
 53263 
 
 84635 
 
 54732 
 
 83692 
 
 56184 
 
 82724 
 
 49 
 
 12 
 
 50302 
 
 86427 
 
 51803 
 
 85536 
 
 53288 
 
 84619 
 
 54756 
 
 83676 
 
 56208 
 
 82708 
 
 48 
 
 13 
 
 50327 
 
 86413 
 
 51828 
 
 85521 
 
 53312 
 
 84604 
 
 54781 
 
 83660 
 
 56232 
 
 82692 
 
 47 
 
 14 
 
 50352 
 
 86398 
 
 51852 
 
 85506 
 
 53337 
 
 84588 
 
 54805 
 
 83645 
 
 56256 
 
 82675 
 
 46 
 
 15 
 
 50377 
 
 86384 
 
 51877 
 
 85491 
 
 53361 
 
 84573 
 
 54829 
 
 83629 
 
 56280 
 
 82659 
 
 45 
 
 16 
 
 50403 
 
 86369 
 
 51902 
 
 85476 
 
 53386 
 
 84557 
 
 54854 
 
 83613 
 
 56305 
 
 82643 
 
 44 
 
 17 
 
 50428 
 
 86354 
 
 51927 
 
 85461 
 
 53411 
 
 84542 
 
 54878 
 
 83597 
 
 56329 
 
 82626 
 
 43 
 
 18 
 
 50453 
 
 86340 
 
 51952 
 
 85446 
 
 53435 
 
 84526 
 
 54902 
 
 83581 
 
 56353 
 
 82610 
 
 42 
 
 19 
 
 50478 
 
 86325 
 
 51977 
 
 85431 
 
 53460 
 
 84511 
 
 54927 
 
 83565 
 
 56377 
 
 82593 
 
 41 
 
 20 
 
 50503 
 
 86310 
 
 52002 
 
 85416 
 
 53484 
 
 84495 
 
 54951 
 
 83549 
 
 56401 
 
 82577 
 
 40 
 
 21 
 
 50528 
 
 86295 
 
 52026 
 
 85401 
 
 53509 
 
 84480 
 
 54975 
 
 83533 
 
 56425 
 
 82561 
 
 39 
 
 22 
 
 50553 
 
 86281 
 
 52051 
 
 85385 
 
 53534 
 
 84464 
 
 54999 
 
 83517 
 
 56449 
 
 82544 
 
 38 
 
 23 
 
 50578 
 
 86266 
 
 52076 
 
 85370 
 
 53558 
 
 84448 
 
 . 55024 
 
 83501 
 
 56473 
 
 82528 
 
 37 
 
 24 
 
 50603 
 
 86251 
 
 52101 
 
 85355 
 
 53583 
 
 84433 
 
 55048 
 
 83485 
 
 56497 
 
 82511 
 
 36 
 
 25 
 
 50628 
 
 86237 
 
 52126 
 
 85340 
 
 53607 
 
 84417 
 
 55072 
 
 83469 
 
 56521 
 
 82495 
 
 35 
 
 26 
 
 50654 
 
 86222 
 
 52151 
 
 85325 
 
 53632 
 
 84402 
 
 55097 
 
 83453 
 
 56545 
 
 82478 
 
 34 
 
 27 
 
 50679 
 
 86207 
 
 52175 
 
 85310 
 
 53656 
 
 84386 
 
 55121 
 
 83437 
 
 56569 
 
 82462 
 
 33 
 
 28 
 
 50704 
 
 86192 
 
 52200 
 
 85294 
 
 53681 
 
 84370 
 
 55145 
 
 83421 
 
 56593 
 
 82446 
 
 32 
 
 29 
 
 50729 
 
 86178 
 
 52225 
 
 85279 
 
 53705 
 
 84355 
 
 55169 
 
 83405 
 
 56617 
 
 82429 
 
 31 
 
 30 
 
 50754 
 
 86163 
 
 52250 
 
 85264 
 
 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 
 
 56689 
 
 82380 
 
 28 
 
 33 
 
 50829 
 
 86119 
 
 52324 
 
 85218 
 
 53804 
 
 84292 
 
 55266 
 
 83340 
 
 56713 
 
 82363 
 
 27 
 
 34 
 
 50854 
 
 86104 
 
 52349 
 
 85203 
 
 53828 
 
 84277 
 
 55291 
 
 : 83324 
 
 56736 
 
 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 
 
 85157 
 
 53902 
 
 84230 
 
 55363 
 
 1 83276 
 
 56808 
 
 82297 
 
 23 
 
 38 
 
 50954 
 
 86045 
 
 52448 
 
 85142 
 
 53926 
 
 84214 
 
 55388 
 
 ! 83260 
 
 1 56832 
 
 82281 
 
 22 
 
 39 
 
 50979 
 
 86030 
 
 52473 
 
 85127 
 
 53951 
 
 84198 
 
 55412 
 
 i 83244 
 
 56856 
 
 82264 
 
 21 
 
 40 
 
 51004 
 
 86015 
 
 52498 
 
 85112 
 
 53975 
 
 84182 
 
 55436 
 
 83228 
 
 56880 
 
 82248 
 
 20 
 
 41 
 
 51029 
 
 86000 
 
 52522 
 
 85096 
 
 54000 
 
 84167 
 
 55460 
 
 83212 
 
 56904 
 
 82231 
 
 19 
 
 42 
 
 51054 
 
 85985 
 
 52547 
 
 85081 
 
 54024 
 
 84151 
 
 55484 
 
 83195 
 
 56928 
 
 82214 
 
 18 
 
 43 
 
 51079 
 
 85970 
 
 52572 
 
 85066 
 
 54049 
 
 84135 
 
 55509 
 
 83179 
 
 56952 
 
 82198 
 
 17 
 
 44 
 
 51104 
 
 85956 
 
 52597 
 
 i 85051 
 
 54073 
 
 84120 
 
 55533 
 
 83163 
 
 56976 
 
 82181 
 
 16 
 
 45 
 
 51129 
 
 85941 
 
 52621 
 
 ! 85035 
 
 54097 
 
 84104 
 
 55557 
 
 83147 
 
 57000 
 
 82165 
 
 15 
 
 46 
 
 51154 
 
 85926 
 
 52646 
 
 . 85020 
 
 ' 54122 
 
 84088 
 
 55581 
 
 ! 83131 
 
 57024 
 
 82148 
 
 14 
 
 47 
 
 51179 
 
 85911 
 
 52671 
 
 ' 85005 
 
 54146 
 
 84072 
 
 55605 
 
 83115 
 
 57047 
 
 82132 
 
 13 
 
 48 
 
 51204 
 
 85896 
 
 52696 
 
 ; 84989 
 
 54171 
 
 84057 
 
 55630 
 
 83098 
 
 57071 
 
 82115 
 
 12 
 
 49 
 
 51229 
 
 85881 
 
 52720 
 
 1 84974 
 
 54195 
 
 84041 
 
 55654 
 
 83082 
 
 57095 
 
 82098 
 
 11 
 
 50 
 
 51254 
 
 85866 
 
 52745 
 
 ' 84959 
 
 54220 
 
 84025 
 
 55678 
 
 83066 
 
 57119 
 
 82082 
 
 10 
 
 51 
 
 51279 
 
 85851 
 
 52770 
 
 84943 
 
 54244 
 
 84009 
 
 55702 
 
 83050 
 
 57143 
 
 82065 
 
 9 
 
 52 
 
 51304 
 
 85836 
 
 52794 
 
 84928 
 
 54269 
 
 83994 
 
 55726 
 
 83034 
 
 57167 
 
 82048 
 
 8 
 
 53 
 
 51329 
 
 85821 
 
 52819 
 
 84913 
 
 54293 
 
 83978 
 
 55750 
 
 83017 
 
 57191 
 
 82032 
 
 7 
 
 54 
 
 51354 
 
 85806 
 
 52844 
 
 84897 
 
 54317 
 
 83962 
 
 55775 
 
 83001 
 
 57215 
 
 82015 
 
 6 
 
 55 
 
 51379 
 
 85792 
 
 52869 
 
 84882 
 
 54342 
 
 83946 
 
 55799 
 
 82985 
 
 57238 
 
 81999 
 
 5 
 
 56 
 
 51404 
 
 85777 
 
 52893 
 
 84866 
 
 54366 
 
 83930 
 
 55823 
 
 82969 
 
 57262 
 
 81982 
 
 4 
 
 57 
 
 51429 
 
 85762 
 
 52918 
 
 84851 
 
 54391 
 
 83915 
 
 55847 
 
 82953 
 
 57286 
 
 81965 
 
 3 
 
 58 
 
 51454 
 
 85747 
 
 52943 
 
 84836 
 
 54415 
 
 83899 
 
 55871 
 
 82936 
 
 57310 
 
 81949 
 
 2 
 
 59 
 
 51479 
 
 85732 
 
 52967 
 
 : 84820 
 
 54440 
 
 83883 
 
 55895 
 
 82920 
 
 57334 
 
 81932 
 
 1 
 
 60 
 
 51504 
 
 85717 
 
 52992 
 
 84805 
 
 54464 
 
 83867 
 
 55919 
 
 82904 
 
 57358 
 
 81915 
 
 
 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 'Cosine. 
 
 Sine. 
 
 Cosine. 
 
 ; Sine. 
 
 Cosine. 
 
 Sine. 
 
 
 / 
 
 5O 
 
 58 
 
 S7 
 
 5O 
 
 ; 55 
 
 : / 
 
APPENDIX. 
 
 843 
 
 NATURAL, SINES AND COSINES. 
 
 
 35 
 
 3O 
 
 37 1 " 
 
 38 
 
 3O 
 
 
 / 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 , /" 
 
 
 
 57358 
 
 81915 
 
 58779 
 
 80902 
 
 60182 
 
 79864 
 
 61566 
 
 78801 
 
 62932 
 
 77715 
 
 60 
 
 1 
 
 57381 
 
 81899 
 
 58802 
 
 80885 
 
 60205 
 
 ' 79846 
 
 61589 
 
 78783 
 
 62955 
 
 77696 
 
 59 
 
 2 
 
 57405 
 
 81882 
 
 58826 
 
 80867 
 
 60228 
 
 79829 
 
 61612 
 
 78765 
 
 62977 
 
 77678 
 
 58 
 
 3 
 
 57429 
 
 81865 
 
 58849 
 
 80850 
 
 60251 
 
 79811 
 
 61635 
 
 78747 
 
 63000 
 
 77660 
 
 57 
 
 4 
 
 57453 
 
 81848 
 
 58873 
 
 ; 80833 
 
 60274 
 
 79793 
 
 61658 
 
 78729 
 
 63022 
 
 77641 
 
 56 
 
 5 
 
 57477 
 
 81832 
 
 58896 
 
 80816 
 
 60298 
 
 79776 
 
 61681 
 
 78711 
 
 63045 
 
 77623 
 
 55 
 
 6 
 
 57501 
 
 81815 
 
 58920 
 
 80799 
 
 60321 
 
 79758 
 
 61704 
 
 78694 
 
 63068 
 
 77605 
 
 54 
 
 7 
 
 57524 
 
 81798 
 
 58943 
 
 80782 
 
 60344 
 
 79741 
 
 61726 
 
 78676 
 
 63090 
 
 77586 
 
 53 
 
 8 
 
 57548 
 
 81782 
 
 58967 
 
 80765 
 
 60367 
 
 79723 
 
 61749 
 
 78658 
 
 63113 
 
 77568 
 
 52 
 
 9 
 
 57572 
 
 81765 
 
 58990 
 
 80748 
 
 60390 
 
 79706 
 
 61772 
 
 78640 
 
 63135 
 
 77550 
 
 51 
 
 10 
 
 57596 
 
 81748 
 
 59014 
 
 80730 
 
 60414 
 
 79688 
 
 61795 
 
 78622 
 
 63158 
 
 77531 
 
 50 
 
 11 
 
 57619 
 
 81731 
 
 59037 
 
 80713 
 
 60437 
 
 79671 
 
 61818 
 
 78604 
 
 63180 
 
 77513 
 
 49 
 
 12 
 
 57643 
 
 81714 
 
 59061 
 
 80696 
 
 60460 
 
 79653 
 
 61841 
 
 78586 
 
 63203 
 
 77494 
 
 48 
 
 13 
 
 57667 
 
 81698 
 
 59084 
 
 80679 
 
 60483 
 
 79635 
 
 61864 
 
 78568 
 
 63225 
 
 77476 
 
 47 
 
 14 
 
 57691 
 
 81681 
 
 59108 
 
 80662 
 
 60506 
 
 79618 
 
 61887 
 
 78550 
 
 63248 
 
 77458 
 
 46 
 
 15 
 
 57715 
 
 81664 
 
 59131 
 
 80644 
 
 60529 
 
 79600 
 
 61909 
 
 78532 
 
 63271 
 
 77439 
 
 45 
 
 16 
 
 57738 
 
 81647 
 
 59154 
 
 ; 80627 
 
 60553 
 
 : 79583 
 
 61932 
 
 78514 
 
 63293 
 
 77421 
 
 44 
 
 17 
 
 57762 
 
 81631 
 
 I 59178 
 
 80610 
 
 60576 
 
 79565 
 
 61955 
 
 78496 
 
 63316 
 
 77402 
 
 43 
 
 18 
 
 57786 
 
 81614 
 
 59201 
 
 80593 
 
 60599 
 
 79547 
 
 61978 
 
 78478 
 
 63338 
 
 77384 
 
 42 
 
 19 
 
 57810 
 
 81597 
 
 59225 
 
 80576 
 
 60622 
 
 79530 
 
 62001 
 
 78460 
 
 63361 
 
 77366 
 
 41 
 
 20 
 
 57833 
 
 81580 
 
 59248 
 
 80558 
 
 60645 
 
 79512 
 
 62024 
 
 78442 
 
 63383 
 
 77347 
 
 40 
 
 21 
 
 57857 
 
 81563 
 
 59272 
 
 80541 
 
 60668 
 
 : 79494 
 
 62046 
 
 78424 
 
 63406 
 
 77329 
 
 39 
 
 22 
 
 57881 
 
 81546 
 
 59295 
 
 80524 
 
 60691 
 
 79477 
 
 62069 
 
 i 78405 
 
 63428 
 
 77310 
 
 38 
 
 23 
 
 57904 
 
 81530 
 
 59318 
 
 80507 
 
 60714 
 
 i 79459 
 
 62092 
 
 78387 
 
 63451 
 
 77292 
 
 37 
 
 24 
 
 57928 
 
 81513 
 
 59342 
 
 80489 
 
 60738 
 
 79441 
 
 62115 
 
 ! 78369 
 
 63473 
 
 77273 
 
 36 
 
 25 
 
 57952 
 
 81496 
 
 59365 
 
 80472 
 
 60761 
 
 79424 
 
 62138 
 
 78351 
 
 63496 
 
 77255 
 
 35 
 
 26 
 
 57976 
 
 81479 
 
 59389 
 
 i 80455 
 
 60784 
 
 79406 
 
 62160 
 
 78333 
 
 63518 
 
 77236 
 
 34 
 
 27 
 
 57999 
 
 81462 
 
 59412 
 
 80438 
 
 60807 
 
 79388 
 
 62183 
 
 78315 
 
 63540 
 
 77218 
 
 33 
 
 28 
 
 58023 
 
 81445 
 
 59436 
 
 80420 
 
 60830 
 
 79371 
 
 62206 
 
 78297 
 
 63563 
 
 77199 
 
 32 
 
 29 
 
 58047 
 
 81428 
 
 59459 
 
 80403 
 
 60853 
 
 79353 
 
 62229 
 
 i 78279 
 
 63585 
 
 77181 
 
 31 
 
 30 
 
 58070 
 
 81412 
 
 59482 
 
 80386 
 
 60876 
 
 79335 
 
 62251 
 
 78261 
 
 63608 
 
 77162 
 
 30 
 
 31 
 
 58094 
 
 81395 
 
 59506 
 
 80368 
 
 60899 
 
 79318 
 
 62274 
 
 78243 
 
 63630 
 
 77144 
 
 29 
 
 32 
 
 58118 
 
 81378 | 59529 
 
 : 80351 
 
 60922 
 
 . 79300 
 
 62297 
 
 78225 
 
 63653 
 
 77125 
 
 28 
 
 33 
 
 58141 
 
 81361 
 
 59552 
 
 80334 
 
 60945 
 
 79282 
 
 62320 
 
 78206 
 
 63675 
 
 77107 
 
 27 
 
 34 
 
 58165 
 
 81344 
 
 59576 
 
 ! 80316 
 
 60968 
 
 ; 79264 
 
 62342 
 
 78188 
 
 63698 
 
 77088 
 
 26 
 
 35 
 
 58189 
 
 81327 59599 
 
 80299 
 
 ; 60991 
 
 ! 79247 
 
 62365 
 
 78170 
 
 63720 
 
 77070 
 
 25 
 
 36 
 
 58212 
 
 81310 59622 
 
 80282 
 
 . 61015 
 
 : 79229 
 
 ! 62388 
 
 ; 78152 
 
 63742 
 
 77051 
 
 24 
 
 37 
 
 58236 
 
 81293 
 
 59646 
 
 80264 
 
 61038 
 
 79211 
 
 62411 
 
 78134 
 
 63765 
 
 77033 
 
 23 
 
 38 
 
 58260 
 
 81276 
 
 59669 
 
 80247 
 
 61061 
 
 79193 
 
 62433 
 
 78116 
 
 63787 
 
 77014 
 
 22 
 
 39 
 
 58283 
 
 81259 59693 
 
 80230 
 
 . 61084 
 
 79176 ' 62456 
 
 78098 
 
 63810 
 
 76996 
 
 21 
 
 40 
 
 58307 
 
 81242 59716 
 
 80212 
 
 61107 
 
 79158 
 
 62479 
 
 78079 
 
 63832 
 
 76977 
 
 20 
 
 41 
 
 58330 
 
 81225 59739 
 
 80195 
 
 61130 
 
 79140 
 
 62502 
 
 78061 
 
 63854 
 
 76959 
 
 19 
 
 42 
 
 58354 
 
 81208 59763 
 
 80178 
 
 61153 
 
 : 79122 
 
 62524 
 
 : 78043 
 
 63877 
 
 76940 
 
 18 
 
 43 
 
 58378 
 
 81191 
 
 59786 
 
 80160 
 
 61176 
 
 79105 
 
 62547 
 
 78025 
 
 63899 
 
 76921 
 
 17 
 
 4*4 
 
 58401 
 
 81174 
 
 59809 
 
 80143 
 
 61199 
 
 79087 
 
 '62570 
 
 78007 
 
 63922 
 
 76903 
 
 16 
 
 45 
 
 58425 
 
 81157 
 
 59832 
 
 80125 
 
 61222 
 
 79069 
 
 62592 
 
 77988 
 
 63944 
 
 76884 
 
 15 
 
 46 
 
 58449 
 
 81140 
 
 59856 
 
 80108 
 
 61245 
 
 79051 
 
 : 62615 
 
 77970 
 
 63966 
 
 76866 
 
 14 
 
 47 
 
 58472 
 
 81123 
 
 59879 
 
 80091 
 
 61268 
 
 79033 
 
 62638 
 
 77952 
 
 63989 
 
 76847 
 
 13 
 
 48 
 
 58496 
 
 81106 
 
 59902 
 
 80073 
 
 61291 
 
 79016 
 
 62660 
 
 77934 
 
 64011 
 
 ; 76828 
 
 12 
 
 49 
 
 58519 
 
 81089 
 
 59926 
 
 80056 
 
 61314 
 
 78998 ! 62683 
 
 77916 
 
 64033 
 
 76810 
 
 11 
 
 50 
 
 58543 
 
 81072 
 
 59949 
 
 ' 80038 
 
 61337 
 
 78980 
 
 62706 
 
 i 77897 
 
 64056 
 
 76791 
 
 10 
 
 51 
 
 58567 
 
 81055 
 
 59972 
 
 80021 
 
 61360 
 
 78962 
 
 62728 
 
 : 77879 
 
 64078 
 
 76772 
 
 9 
 
 52 
 
 58590 
 
 81038 
 
 59995 
 
 80003 
 
 61383 
 
 78944 
 
 62751 
 
 i 77861 
 
 64100 
 
 76754 
 
 8 
 
 53 
 
 58614 
 
 81021 
 
 60019 
 
 79986 
 
 61406 
 
 78926 
 
 62774 
 
 77843 
 
 64123 
 
 76735 
 
 7 
 
 54 
 
 58637 
 
 81004 
 
 60042 
 
 79968 
 
 61429 
 
 78908 
 
 62796 
 
 j 77824 
 
 64145 
 
 76717 
 
 6 
 
 55 
 
 58661 
 
 80987 
 
 60065 
 
 79951 
 
 61451 
 
 78891 
 
 62819 
 
 77806 
 
 64167 
 
 76698 
 
 5 
 
 56 
 
 58684 
 
 80970 
 
 60089 
 
 79934 
 
 61474 
 
 78873 
 
 62842 
 
 77788 
 
 64190 
 
 76679 
 
 4 
 
 57 
 
 58708 
 
 80953 
 
 60112 
 
 79916 
 
 61497 
 
 78855 
 
 62864 
 
 77769 
 
 64212 
 
 76661 
 
 3 
 
 58 
 
 58731 
 
 80936 
 
 60135 
 
 79899 
 
 61520 
 
 78837 
 
 62887 
 
 77751 
 
 64234 
 
 76642 
 
 2 
 
 59 
 
 58755 
 
 80919 
 
 60158 
 
 79881 
 
 61543 
 
 78819 
 
 62909 
 
 77733 
 
 64256 
 
 76623 
 
 1 
 
 60 
 
 58779 
 
 80902 
 
 60182 
 
 79864 
 
 61566 
 
 78801 
 
 62932 
 
 77715 
 
 64279 
 
 76604 
 
 
 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 
 / ' 
 
 54,0 
 
 53 
 
 53 
 
 51" 
 
 SO 
 
 / 
 
844 
 
 APPENDIX. 
 
 NATURAL. SINKS AND COSINES. 
 
 
 4O 
 
 <tl 
 
 43 
 
 4,3 
 
 4=4= 
 
 
 / 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 / 
 
 
 
 64279 
 
 76604 
 
 65606 
 
 75471 
 
 66913 
 
 74314 
 
 68200 
 
 73135 
 
 69466 
 
 71934 
 
 60 
 
 1 
 
 64301 
 
 76586 
 
 65628 
 
 75452 
 
 66935 
 
 74295 
 
 68221 
 
 73116 
 
 69487 
 
 71914 
 
 59 
 
 2 
 
 64323 
 
 76567 
 
 65650 
 
 75433 
 
 66956 
 
 74276 
 
 68242 
 
 73096 
 
 69508 
 
 71894 
 
 58 
 
 3 
 
 64346 
 
 76548 
 
 65672 
 
 75414 
 
 66978 
 
 74256 
 
 68264 
 
 73076 
 
 69529 
 
 71873 
 
 57 
 
 4 
 
 64368 
 
 76530 
 
 65694 
 
 75395 
 
 66999 
 
 74237 
 
 68285 
 
 73056 
 
 69549 
 
 71853 
 
 56 
 
 5 
 
 64390 
 
 76511 
 
 65716 
 
 75375 
 
 67021 
 
 74217 
 
 68306 
 
 73036 
 
 69570 
 
 71833 
 
 55 
 
 6 
 
 64412 
 
 76492 
 
 65738 
 
 75356 
 
 67043 
 
 74198 
 
 68327 
 
 73016 
 
 69591 
 
 71813 
 
 54 
 
 7 
 
 64435 
 
 76473 
 
 65759 
 
 75337 
 
 67064 
 
 74178 
 
 68349 
 
 72996 
 
 69612 
 
 71792 
 
 53 
 
 8 
 
 64457 
 
 76455 
 
 65781 
 
 75318 
 
 67086 
 
 74159 
 
 68370 
 
 72976 
 
 69633 
 
 71772 
 
 52 
 
 9 
 
 64479 
 
 76436 
 
 65803 
 
 75299 
 
 67107 
 
 74139 
 
 68391 
 
 72957 
 
 69654 
 
 71752 
 
 51 
 
 10 
 
 64501 
 
 76417 
 
 65825 
 
 75280 
 
 67129 
 
 74120 
 
 68412 
 
 72937 
 
 69675 
 
 71732 
 
 50 
 
 11 
 
 64524 
 
 76398 
 
 65847 
 
 75261 
 
 67151 
 
 74100 
 
 68434 
 
 72917 
 
 69696 
 
 71711 
 
 49 
 
 12 
 
 64546 
 
 76380 
 
 65869 
 
 75241 
 
 67172 
 
 74080 
 
 68455 
 
 72897 
 
 69717 
 
 71691 
 
 48 
 
 13 
 
 64568 
 
 76361 
 
 65891 
 
 75222 
 
 67194 
 
 74061 
 
 68476 
 
 72877 
 
 69737 
 
 71671 
 
 47 
 
 14 
 
 64590 
 
 76342 
 
 65913 
 
 75203 
 
 67215 
 
 74041 
 
 68497 
 
 72857 
 
 69758 
 
 71650 
 
 46 
 
 15 
 
 64612 
 
 76323 
 
 65935 
 
 75184 
 
 67237 
 
 74022 
 
 68518 
 
 72837 
 
 69779 
 
 71630 
 
 45 
 
 16 
 
 64635 
 
 76304 
 
 65956 
 
 75165 
 
 67258 
 
 74002 
 
 68539 
 
 72817 
 
 69800 
 
 71610 
 
 44 
 
 17 
 
 64657 
 
 76286 
 
 65978 
 
 75146 
 
 67280 
 
 73983 
 
 68561 
 
 72797 
 
 69821 
 
 71590 
 
 43 
 
 18 
 
 64679 
 
 76267 
 
 66000 
 
 75126 
 
 67301 
 
 73963 
 
 68582 
 
 72777 
 
 69842 
 
 71569 
 
 42 
 
 1? 
 
 64701 
 
 76248 
 
 66022 
 
 75107 
 
 67323 
 
 73944 
 
 68603 
 
 72757 
 
 69862 
 
 71549 
 
 41 
 
 20 
 
 64723 
 
 76229 
 
 66044 
 
 75088 
 
 67344 
 
 73924 
 
 68624 
 
 72737 
 
 69883 
 
 71529 
 
 40 
 
 21 
 
 64746 
 
 76210 
 
 66066 
 
 75069 
 
 67366 
 
 73904 
 
 68645 
 
 72717 
 
 69904 
 
 71508 
 
 39 
 
 22 
 
 64768 
 
 76192 
 
 66088 
 
 75050 
 
 67387 
 
 73885 
 
 68666 
 
 72697 
 
 69925 
 
 71488 
 
 38 
 
 23 
 
 64790 
 
 76173 
 
 66109 
 
 75030 
 
 67409 
 
 73865 
 
 68688 
 
 72677 
 
 69946 
 
 71468 
 
 37 
 
 24 
 
 64812 
 
 76154 
 
 66131 
 
 75011 
 
 67430 
 
 73846 
 
 68709 
 
 72657 
 
 69966 
 
 71447 
 
 36 
 
 25 
 
 64834 
 
 76135 
 
 66153 
 
 74992 
 
 67452 
 
 73826 
 
 68730 
 
 72637 
 
 69987 
 
 71427 
 
 3.5 
 
 26 
 
 64856 
 
 76116 
 
 66175 
 
 74973 
 
 67473 
 
 73806 
 
 68751 
 
 72617 
 
 70008 
 
 71407 
 
 34 
 
 27 
 
 64878 
 
 76097 
 
 66197 
 
 74953 
 
 67495 
 
 73787 
 
 68772 
 
 72597 
 
 70029 
 
 71386 
 
 33 
 
 28 
 
 64901 
 
 76078 
 
 66218 
 
 74934 
 
 67516 
 
 73767 
 
 68793 
 
 72577 
 
 70049 
 
 71366 
 
 32 
 
 29 
 
 64923 
 
 76059 
 
 66240 
 
 74915 
 
 67538 
 
 73747 
 
 68814 
 
 72557 
 
 70070 
 
 71345 
 
 31 
 
 30 
 
 64945 
 
 76041 
 
 66262 
 
 74896 
 
 67559 
 
 73728 
 
 68835 
 
 72537 
 
 70091 
 
 71325 
 
 30 
 
 31 
 
 64967 
 
 76022 
 
 66284 
 
 74876 
 
 67580 
 
 73708 
 
 68857 
 
 72517 
 
 70112 
 
 71305 
 
 29 
 
 32 
 
 64989 
 
 76003 
 
 66306 
 
 74857 
 
 67602 
 
 73688 
 
 68878 
 
 72497 
 
 70132 
 
 71284 
 
 28 
 
 33 
 
 65011 
 
 75984 
 
 66327 
 
 74838 
 
 67623 
 
 73669 
 
 68899 
 
 72477 
 
 70153 
 
 71264 
 
 27 
 
 34 
 
 65033 
 
 75965 
 
 66349 
 
 74818 
 
 67645 
 
 73649 
 
 68920 
 
 72457 
 
 70174 
 
 71243 
 
 26 
 
 35 
 
 65055 
 
 75946 
 
 66371 
 
 74799 
 
 67666 
 
 73629 
 
 68941 
 
 72437 
 
 70195 
 
 71223 
 
 25 
 
 36 
 
 65077 
 
 75927 
 
 66393 
 
 74780 
 
 67688 
 
 73610 
 
 68962 
 
 72417 
 
 70215 
 
 71203 
 
 24 
 
 37 
 
 65100 
 
 75908 
 
 66414 
 
 74760 
 
 67709 
 
 73590 
 
 68983 
 
 72397 
 
 70236 
 
 71182 
 
 23 
 
 38 
 
 65122 
 
 75889 
 
 66436 
 
 74741 
 
 67730 
 
 73570 
 
 69004 
 
 72377 
 
 70257 
 
 71162 
 
 22 
 
 39 
 
 65144 
 
 75870 
 
 66458 
 
 74722 
 
 67752 
 
 73551 
 
 69025 
 
 72357 
 
 70277 
 
 71141 
 
 21 
 
 40 
 
 65166 
 
 75851 
 
 66480 
 
 74703 
 
 67773 
 
 73531 
 
 69046 
 
 72337 
 
 70298 
 
 71121 
 
 20 
 
 41 
 
 65188 
 
 75832 
 
 66501 
 
 74683 
 
 67795 
 
 73511 
 
 69067 
 
 72317 
 
 70319 
 
 71100 
 
 19 
 
 42 
 
 65210 
 
 75813 
 
 66523 
 
 74664 
 
 67816 
 
 73491 
 
 69088 
 
 72297 
 
 70339 
 
 71080 
 
 18 
 
 43 
 
 65232 
 
 75794 
 
 66545 
 
 74644 
 
 67837 
 
 73472 
 
 69109 
 
 72277 
 
 70360 
 
 71059 
 
 17 
 
 44 
 
 65254 
 
 75775 
 
 66566 
 
 74625 
 
 67859 
 
 73452 
 
 69130 
 
 72257 
 
 70381 
 
 71039 
 
 18 
 
 45 
 
 65276 
 
 75756 
 
 66588 
 
 74606 
 
 67880 
 
 73432 
 
 69151 
 
 72236 
 
 70401 
 
 71019 
 
 15 
 
 46 
 
 65298 
 
 75738 
 
 66610 
 
 74586 
 
 67901 
 
 73413 
 
 69172 
 
 72216 
 
 70422 
 
 70998 
 
 14 
 
 47 
 
 65320 
 
 75719 
 
 66632 
 
 74567 
 
 67923 
 
 73393 
 
 69193 
 
 72196 
 
 70443 
 
 70978 
 
 13 
 
 48 
 
 65342 
 
 75700 
 
 66653 
 
 74548 
 
 67944 
 
 73373 
 
 69214 
 
 72176 
 
 70463 
 
 70957 
 
 12 
 
 49 
 
 65364 
 
 75680 
 
 66675 
 
 74528 
 
 67965 
 
 73353 
 
 69235 
 
 72156 
 
 70484 
 
 70937 
 
 11 
 
 50 
 
 65386 
 
 75661 
 
 66697 
 
 74509 
 
 67987 
 
 73333 
 
 69256 
 
 72136 
 
 70505 
 
 70916 
 
 10 
 
 51 
 
 65408 
 
 75642 
 
 66718 
 
 74489 
 
 68008 
 
 73314 
 
 69277 
 
 72116 
 
 70525 
 
 70896 
 
 9 
 
 52 
 
 65430 
 
 75623 
 
 66740 
 
 74470 
 
 68029 
 
 73294 
 
 69298 
 
 72095 
 
 70546 
 
 70875 
 
 8 
 
 53 
 
 65452 
 
 75604 
 
 66762 
 
 74451 
 
 68051 
 
 73274 
 
 69319 
 
 72075 
 
 70567 
 
 70855 
 
 7 
 
 54 
 
 65474 
 
 75585 
 
 66783 
 
 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 
 
 68115 
 
 73215 
 
 69382 
 
 72015 
 
 70628 
 
 70793 
 
 4 
 
 57 
 
 65540 
 
 75528 
 
 66848 
 
 74373 
 
 68136 
 
 73195 
 
 69403 
 
 71995 
 
 70649 
 
 70772 
 
 3 
 
 58 
 
 65562 
 
 75509 
 
 66870 
 
 74353 
 
 68157 
 
 73175 
 
 69424 
 
 71974 
 
 70670 
 
 70752 
 
 2 
 
 59 
 
 65584 
 
 75490 
 
 66891 
 
 74334 
 
 68179 
 
 73155 
 
 69445 
 
 71954 
 
 70690 
 
 70731 
 
 1 
 
 60 
 
 65606 
 
 75471 
 
 66913 
 
 74314 
 
 68200 
 
 73135 
 
 69466 
 
 71934 
 
 70711 
 
 70711 
 
 
 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 Cosine. 
 
 Sine. 
 
 
 
 4O 
 
 48 
 
 4:7 
 
 46 
 
 4> 
 
 / 
 
APPENDIX. 
 
 845 
 
 LOGARITHMS OP NUMBERS. 
 
 N. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4L 
 
 5 
 
 6 
 
 7 
 
 S 
 
 O 
 
 I>. 
 
 100 
 
 00 0000 
 
 0434 
 
 0868 
 
 1301 
 
 1734 
 
 2166 
 
 2598 
 
 3029 
 
 3461 
 
 3891 
 
 432 
 
 101 
 
 4321 
 
 4751 
 
 5181 
 
 5609 
 
 6038 
 
 6466 
 
 6894 
 
 7321 
 
 7748 
 
 8174 
 
 428 
 
 102 
 
 * 8600 
 
 9026 
 
 9451 
 
 9876 
 
 +300 
 
 0724 
 
 1147 
 
 1570 
 
 1993 
 
 2415 
 
 424 
 
 103 
 
 01 2837 
 
 3259 
 
 3680 
 
 4100 
 
 4521 
 
 4940 
 
 5360 
 
 5779 
 
 6197 
 
 6616 
 
 419 
 
 104 
 
 * 7033 
 
 7451 
 
 7868 
 
 8284 
 
 8700 
 
 9116 
 
 9532 
 
 9947 
 
 +361 
 
 0775 
 
 416 
 
 105 
 
 02 1189 
 
 1603 
 
 2016 
 
 2428 
 
 2841 
 
 3252 
 
 3664 
 
 4075 
 
 4486 
 
 4896 
 
 412 
 
 106 
 
 5306 
 
 5715 
 
 6125 
 
 6533 
 
 6942 
 
 7350 
 
 7757 
 
 8164 
 
 8571 
 
 8978 
 
 408 
 
 107 
 
 * 9384 
 
 9789 
 
 +195 
 
 0600 
 
 1004 
 
 1408 
 
 1812 
 
 2216 
 
 2619 
 
 3021 
 
 404 
 
 108 
 
 03 3424 
 
 3826 
 
 4227 
 
 4628 
 
 5029 
 
 5430 
 
 5830 
 
 6230 
 
 6629 
 
 7028 
 
 400 
 
 109 
 
 * 7426 
 
 7825 
 
 8223 
 
 8620 
 
 9017 
 
 9414 
 
 9811 
 
 +207 
 
 0602 
 
 0998 
 
 396 
 
 110 
 
 04 1393 
 
 1787 
 
 2182 
 
 2576 
 
 2969 
 
 3362 
 
 3755 
 
 4148 
 
 4540 
 
 4932 
 
 393 
 
 111 
 
 5323 
 
 5714 
 
 6105 
 
 6495 
 
 6885 
 
 7275 
 
 7664 
 
 8053 
 
 8442 
 
 8830 
 
 389 
 
 112 
 
 * 9218 
 
 9606 
 
 9993 
 
 +380 
 
 0766 
 
 1153 
 
 1538 
 
 1924 
 
 2309 
 
 2694 
 
 386 
 
 113 
 
 05 3078 
 
 3463 
 
 3846 
 
 4230 
 
 4613 
 
 4996 
 
 5378 
 
 5760 
 
 6142 
 
 6524 
 
 382 
 
 114 
 
 *6905 
 
 7286 
 
 7666 
 
 8046 
 
 8426 
 
 8805 
 
 9185 
 
 9563 
 
 9942 
 
 +320 
 
 379 
 
 115 
 
 06 0698 
 
 1075 
 
 1452 
 
 1829 
 
 2206 
 
 2582 
 
 2958 
 
 3333 
 
 3709 
 
 4083 
 
 376 
 
 116 
 
 4458 
 
 4832 
 
 5206 
 
 5580 
 
 5953 
 
 6326 
 
 6699 
 
 7071 
 
 7443 
 
 7815 
 
 372 
 
 117 
 
 * 8186 
 
 8557 
 
 8928 
 
 9298 
 
 9668 
 
 +038 
 
 0407 
 
 0776 
 
 1145 
 
 1514 
 
 369 
 
 118 
 
 07 1882 
 
 2250 
 
 2617 
 
 2985 
 
 3352 
 
 3718 
 
 4085 
 
 4451 
 
 4816 
 
 5182 
 
 366 
 
 119 
 
 5547 
 
 5912 
 
 6276 
 
 6640 
 
 7004 
 
 7368 
 
 7731 
 
 8094 
 
 8457 
 
 8819 
 
 363 
 
 120 
 
 *9181 
 
 9543 
 
 9904 
 
 +266 
 
 0626 
 
 0987 
 
 1347 
 
 1707 
 
 2067 
 
 2426 
 
 360 
 
 121 
 
 08 2785 
 
 3144 
 
 3503 
 
 3861 
 
 4219 
 
 4576 
 
 4934 
 
 5291 
 
 5647 
 
 6004 
 
 357 
 
 122 
 
 6360 
 
 6716 
 
 7071 
 
 7426 
 
 7781 
 
 8136 
 
 8490 
 
 8845 
 
 9198 
 
 9552 
 
 355 
 
 123 
 
 * 9905 
 
 +258 
 
 0611 
 
 0963 
 
 1315 
 
 1667 
 
 2018 
 
 2370 
 
 2721 
 
 3071 
 
 351 
 
 124 
 
 09 3422 
 
 3772 
 
 4122 
 
 4471 
 
 4820 
 
 5169 
 
 5518 
 
 5866 
 
 6215 
 
 6562 
 
 349 
 
 125 
 
 * 6910 
 
 7257 
 
 7604 
 
 7951 
 
 8298 
 
 8644 
 
 8990 
 
 9335 
 
 9681 
 
 +026 
 
 346 
 
 126 
 
 10 0371 
 
 0715 
 
 1059 
 
 1403 
 
 1747 
 
 2091 
 
 2434 
 
 2777 
 
 3119 
 
 3462 
 
 343 
 
 127 
 
 3804 
 
 4146 
 
 4487 
 
 4828 
 
 5169 
 
 5510 
 
 5851 
 
 6191 
 
 6531 
 
 6871 
 
 340 
 
 128 
 
 * 7210 
 
 7549 
 
 7888 
 
 8227 
 
 8565 
 
 8903 
 
 9241 
 
 9579 
 
 9916 
 
 +253 
 
 338 
 
 129 
 
 11 0590 
 
 0926 
 
 1263 
 
 1599 
 
 1934 
 
 2270 
 
 2605 
 
 2940 
 
 3275 
 
 3609 
 
 335 
 
 130 
 
 3943 
 
 4277 
 
 4611 
 
 4944 
 
 5278 
 
 5611 
 
 5943 
 
 6276 
 
 6608 
 
 6940 
 
 333 
 
 131 
 
 *7271 
 
 7603 
 
 7934 
 
 8265 
 
 8595 
 
 8926 
 
 9256 
 
 9586 
 
 9915 
 
 +245 
 
 330 
 
 132 
 
 12 0574 
 
 0903 
 
 1231 
 
 1560 
 
 1888 
 
 2216 
 
 2544 
 
 2871 
 
 3198 
 
 3525 
 
 328 
 
 133 
 
 3852 
 
 4178 
 
 4504 
 
 4830 
 
 5156 
 
 5481 
 
 5806 
 
 6131 
 
 6456 
 
 6781 
 
 325 
 
 134 
 
 *7105 
 
 7429 
 
 7753 
 
 8076 
 
 8399 
 
 8722 
 
 9045 
 
 9368 
 
 9690 
 
 +012 
 
 323 
 
 135 
 
 13 0334 
 
 0655 
 
 0977 
 
 1298 
 
 1619 
 
 1939 
 
 2260 
 
 2580 
 
 2900 
 
 3219 
 
 321 
 
 136 
 
 3539 
 
 3858 
 
 4177 
 
 4496 
 
 4814 
 
 5133 
 
 5451 
 
 5769 
 
 6086 
 
 6403 
 
 318 
 
 137 
 
 6721 
 
 7037 
 
 7354 
 
 7671 
 
 7987 
 
 8303 
 
 8618 
 
 8934 
 
 9249 
 
 9564 
 
 315 
 
 138 
 
 *9879 
 
 +194 
 
 0508 
 
 0822 
 
 1136 
 
 1450 
 
 1763 
 
 2076 
 
 2389 
 
 2702 
 
 314 
 
 139 
 
 14 3015 
 
 3327 
 
 3639 
 
 3951 
 
 4263 
 
 4574 
 
 4885 
 
 5196 
 
 5507 
 
 5818 
 
 311 
 
 140 
 
 6128 
 
 6438 
 
 6748 
 
 7058 
 
 7367 
 
 7676 
 
 7985 
 
 8294 
 
 8603 
 
 8911 
 
 309 
 
 141 
 
 * 9219 
 
 9527 
 
 9835 
 
 +142 
 
 0449 
 
 0756 
 
 1063 
 
 1370 
 
 1676 
 
 1982 
 
 307 
 
 142 
 
 15 2288 
 
 2594 
 
 2900 
 
 3205 
 
 3510 
 
 3815 
 
 4120 
 
 4424 
 
 4728 
 
 5032 
 
 305 
 
 143 
 
 5336 
 
 5640 
 
 5943 
 
 6246 
 
 6549 
 
 6852 
 
 7154 
 
 7457 
 
 7759 
 
 8061 
 
 303 
 
 144 
 
 * 8362 
 
 8664 
 
 8965 
 
 9266 
 
 9567 
 
 9868 
 
 +168 
 
 0469 
 
 0769 
 
 1068 
 
 301 
 
 145 
 
 16 1368 
 
 1667 
 
 196*7 
 
 2266 
 
 2564 
 
 2863 
 
 3161 
 
 3460 
 
 3758 
 
 4055 
 
 299 
 
 146 
 
 4353 
 
 4650 
 
 4947 
 
 5244 
 
 5541 
 
 5838 
 
 6134 
 
 6430 
 
 6726 
 
 7022 
 
 297 
 
 147 
 
 7317 
 
 7613 
 
 7908 
 
 8203 
 
 8497 
 
 8792 
 
 9086 
 
 9380 
 
 9674 
 
 9968 
 
 295 
 
 148 
 
 17 0262 
 
 0555 
 
 0848 
 
 1141 
 
 1434 
 
 1726 
 
 2019 
 
 2311 
 
 2603 
 
 2895 
 
 293 
 
 149 
 
 3186 
 
 3478 
 
 3769 
 
 4060 
 
 , 4351 
 
 4641 
 
 4932 
 
 5222 
 
 5512 
 
 5802 
 
 291 
 
 150 
 
 6091 
 
 6381 
 
 6670 
 
 6959 
 
 7248 
 
 7536 
 
 7825 
 
 8113 
 
 8401 
 
 8689 
 
 289 
 
 151 
 
 *8977 
 
 9264 
 
 9552 
 
 9839 
 
 126 
 
 0413 
 
 0699 
 
 0985 
 
 1272 
 
 1558 
 
 287 
 
 152 
 
 18 1844 
 
 2129 
 
 2415 
 
 2700 
 
 2985 
 
 3270 
 
 3555 
 
 3839 
 
 4123 
 
 4407 
 
 285 
 
 153 
 
 4691 
 
 4975 
 
 5259 
 
 5542 
 
 5825 
 
 6108 
 
 6391 
 
 6674 
 
 6956 
 
 7239 
 
 283 
 
 154 
 
 *7521 
 
 7803 
 
 8084 
 
 8366 
 
 8647 
 
 8928 
 
 9209 
 
 9490 
 
 9771 
 
 +051 
 
 281 
 
 155 
 
 19 0332 
 
 0612 
 
 0892 
 
 1171 
 
 1451 
 
 1730 
 
 2010 
 
 2289 
 
 2567 
 
 2846 
 
 279 
 
 156 
 
 3125 
 
 3403 
 
 3681 
 
 3959 
 
 4237 
 
 4514 
 
 4792 
 
 5069 
 
 5346 
 
 5623 
 
 278 
 
 157 
 
 5900 
 
 6176 
 
 6453 
 
 6729 
 
 7005 
 
 7281 
 
 7556 
 
 7832 
 
 8107 
 
 8382 
 
 276 
 
 158 
 
 * 8657 
 
 8932 
 
 9206 
 
 9481 
 
 9755 
 
 +029 
 
 0303 
 
 0577 
 
 0850 
 
 1124 
 
 274 
 
 159 
 
 20 1397 
 
 1670 
 
 1943 
 
 2216 
 
 2488 
 
 2761 
 
 3033 
 
 3305 
 
 3577 
 
 3848 
 
 272 
 
 N. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 e 
 
 7 
 
 8 
 
 O 
 
 r>. 
 
846 
 
 APPENDIX. 
 
 LOGARITHMS OF NUMBERS. 
 
 IV. 
 
 O 
 
 1 
 
 9 
 
 3 
 
 4 ' 
 
 5 i 
 
 ; 
 
 7 ' 
 
 8 
 
 O 
 
 r>. 
 
 160 
 
 20 4120 i 
 
 4391 
 
 4663 
 
 4934 
 
 5204 
 
 5475 
 
 5746 
 
 6016 
 
 6286 
 
 6556 
 
 271 
 
 161 
 
 6826 ' 
 
 7096 
 
 7365 
 
 7634 
 
 7904 
 
 8173 
 
 8441 
 
 8710 
 
 8979 
 
 9247 
 
 269 
 
 162 
 
 * 9515 
 
 9783 
 
 *051 
 
 0319 
 
 0586 
 
 0853 
 
 1121 
 
 1388 
 
 1654 
 
 1921 
 
 267 
 
 163 
 
 21 2188 
 
 2454 
 
 2720 
 
 2986 
 
 3252 
 
 3518 
 
 3783 
 
 4049 
 
 4314 
 
 4579 
 
 266 
 
 164 
 
 4844 
 
 5109 
 
 5373 
 
 5638 
 
 5902 
 
 6166 
 
 6430 
 
 6694 
 
 6957 
 
 7221 
 
 264 
 
 165 
 
 7484 
 
 7747 
 
 8010 
 
 8273 
 
 8536 
 
 8798 
 
 9060 
 
 9323 
 
 9585 
 
 9846 
 
 262 
 
 166 
 
 22 0108 
 
 0370 
 
 0631 
 
 0892 
 
 1153 
 
 1414 
 
 1675 
 
 1936 
 
 2196 
 
 2456 
 
 261 
 
 167 
 
 2716 
 
 2976 
 
 3236 
 
 3496 
 
 3755 
 
 4015 
 
 4274 
 
 4533 
 
 4792 
 
 5051 
 
 259 
 
 168 
 
 5309 
 
 5568 
 
 5826 
 
 6084 
 
 6342 
 
 6600; 
 
 6858 , 
 
 7115 
 
 7372 
 
 7630 
 
 258 
 
 169 
 
 *7887 
 
 8144 
 
 8400 
 
 8657 
 
 8913 
 
 9170: 
 
 9426' 
 
 9682 
 
 9938 
 
 *193 
 
 256 
 
 170 
 
 23 0449 
 
 0704 
 
 0960 
 
 1215 
 
 1470 
 
 1724 
 
 1979 
 
 2234 
 
 2488 
 
 2742 
 
 254 
 
 171 
 
 2996 
 
 3250 
 
 3504 
 
 3757 
 
 4011 
 
 4264 
 
 4517 
 
 4770 
 
 5023 
 
 5276 
 
 253 
 
 172 
 
 5528 
 
 5781 
 
 6033 
 
 6285 
 
 6537 
 
 6789 
 
 7041 
 
 7292 
 
 7544 
 
 7795 
 
 252 
 
 173 
 
 *8046 
 
 8297 
 
 8548 
 
 8799 
 
 9049 
 
 9299 
 
 9550 
 
 9800 
 
 050 
 
 0300 
 
 250 
 
 174 
 
 24 0549 
 
 0799 
 
 1048 
 
 1297 
 
 1546 
 
 1795 
 
 2044 
 
 2293 
 
 2541 
 
 2790 
 
 249 
 
 175 
 
 3038 
 
 3286 
 
 3534 
 
 3782 
 
 4030 
 
 4277 
 
 4525 
 
 4772 
 
 5019 
 
 5266 
 
 248 
 
 176 
 
 5513 
 
 5759 
 
 6006 
 
 6252 
 
 6499 
 
 6745 
 
 6991 
 
 7237 . 
 
 7482 
 
 7728 
 
 246 
 
 177 
 
 *7973 
 
 8219 
 
 8464 
 
 8709 
 
 8954 
 
 9198 
 
 9443 
 
 9687 
 
 9932 
 
 *176 
 
 245 
 
 178 
 
 25 0420 
 
 0664 
 
 0908 
 
 1151 
 
 1395 
 
 1638 
 
 1881 
 
 2125 
 
 2368 
 
 2610 
 
 243 
 
 179 
 
 2853 
 
 3096 
 
 3338 
 
 3580 
 
 3822 
 
 4064 
 
 4306 
 
 4548 
 
 4790 
 
 5031 
 
 242 
 
 180 
 
 5273 
 
 5514 
 
 5755 
 
 5996 
 
 6237 
 
 6477 
 
 6718 
 
 6958 
 
 7198 
 
 7439 
 
 241 
 
 181 
 
 7679 
 
 7918 
 
 8158 
 
 8398 
 
 8637 
 
 8877 
 
 9116 
 
 9355 
 
 9594 
 
 9833 
 
 239 
 
 182 
 
 26 0071 
 
 0310 
 
 0548 
 
 0787 
 
 1025 
 
 1263 
 
 1501 
 
 1739 
 
 1976 
 
 2214 
 
 238 
 
 183 
 
 2451 
 
 2688 
 
 2925 
 
 3162 
 
 3399 
 
 3636 
 
 3873 
 
 4109 
 
 4346 
 
 4582 
 
 237 
 
 184 
 
 4818 
 
 5054 
 
 5290 
 
 5525 
 
 5761 
 
 5996 
 
 6232 
 
 6467 
 
 6702 
 
 6937 
 
 235 
 
 185 
 
 7172 
 
 7406 
 
 7641 
 
 7875 
 
 8110 
 
 8344 
 
 8578 
 
 8812 
 
 9046 
 
 9279 
 
 2 % 34 
 
 186 
 
 * 9513 
 
 9746 
 
 9980 
 
 213 
 
 0446 
 
 0679 
 
 0912 
 
 1144 
 
 1377 
 
 1609 
 
 233 
 
 187 
 
 27 1842 
 
 2074 
 
 2306 
 
 2538 
 
 2770 
 
 3001 
 
 3233 
 
 3464 
 
 3696 
 
 3927 
 
 232 
 
 188 
 
 4158 
 
 4389 
 
 4620 
 
 4850 
 
 5081 
 
 5311 
 
 5542 
 
 5772 
 
 6002 
 
 6232 
 
 230 
 
 189 
 
 6462 
 
 6692 
 
 6921 
 
 7151 
 
 7380 
 
 7609 
 
 7838 
 
 8067 
 
 8296 
 
 8525 
 
 229 
 
 190 
 
 *8754 
 
 8982 
 
 9211 
 
 9439 
 
 9667 
 
 9895 
 
 123 
 
 0351 
 
 0578 
 
 0806 
 
 228 
 
 191 
 
 28 1033 
 
 1261 
 
 1488 
 
 1715 
 
 1942 
 
 2169 
 
 2396 
 
 2622 
 
 2849 
 
 3075 
 
 227 
 
 192 
 
 3301 
 
 3527 
 
 3753 
 
 3979 
 
 4205 
 
 4431 
 
 4656 
 
 4882 
 
 5107 
 
 5332 
 
 226 
 
 193 
 
 5557 
 
 5782 
 
 6007 
 
 6232 
 
 6456 
 
 6681 
 
 6905 
 
 7130 
 
 7354 
 
 7578 
 
 225 
 
 194 
 
 7802 
 
 8026 8249 
 
 8473 
 
 8696 
 
 8920 
 
 9143 
 
 9366 
 
 9589 
 
 9812 
 
 223 
 
 195 
 
 29 0035 
 
 0257 
 
 0480 
 
 0702 
 
 0925 
 
 1147 
 
 1369 
 
 1591 
 
 1813 
 
 2034 
 
 222 
 
 196 
 
 2256 
 
 2478 
 
 2699 
 
 2920 
 
 3141 
 
 3363 
 
 3584 
 
 3804 
 
 4025 
 
 4246 
 
 221 
 
 197 
 
 4466 
 
 4687 
 
 4907 
 
 5127 
 
 5347 
 
 5567 
 
 5787 
 
 6007 
 
 6226 
 
 6446 
 
 220 
 
 198 
 
 6665 
 
 6884 
 
 7104 
 
 7323 
 
 7542 
 
 7761 
 
 7979 
 
 8198 
 
 8416 
 
 8635 
 
 219 
 
 199 
 
 * 8853 
 
 9071 
 
 9289 
 
 9507 
 
 9725 
 
 9943 
 
 *161 
 
 0378 
 
 0595 
 
 0813 
 
 218 
 
 200 
 
 30 1030 
 
 1247 
 
 1464 
 
 1681 
 
 1898 
 
 2114 
 
 2331 
 
 2547 
 
 2764 
 
 2980 
 
 217 
 
 201 
 
 3196 
 
 3412 
 
 3628 
 
 3844 
 
 4059 
 
 4275 
 
 4491 
 
 4706 
 
 4921 
 
 5136 
 
 216 
 
 20'2 
 
 5351 
 
 5566 
 
 5781 
 
 5996 
 
 6211 
 
 6425 
 
 6639 
 
 6854 
 
 7068 
 
 7282 
 
 215 
 
 203 
 
 7496 
 
 7710 
 
 7924 
 
 8137 
 
 8351 
 
 8564 
 
 8778 
 
 8991 
 
 9204 
 
 9417 
 
 213 
 
 204 
 
 * 9630 
 
 9843 
 
 056 
 
 0268 
 
 0481 
 
 0693 
 
 0906 
 
 1118 
 
 1330 
 
 1542 
 
 212 
 
 205 
 
 31 1754 
 
 1966 
 
 2177 
 
 2389 
 
 2600 
 
 2812 
 
 3023 
 
 3234 
 
 3445 
 
 3656 
 
 211 
 
 206 
 
 3867 
 
 4078 
 
 4289 
 
 4499 
 
 4710 
 
 4920 
 
 5130 
 
 5340 
 
 5551 
 
 5760 
 
 210 
 
 207 
 
 5970 
 
 6180 
 
 6390 
 
 6599 
 
 6809 
 
 7018 
 
 7227 
 
 7436 
 
 7646 
 
 7854 
 
 209 
 
 208 
 
 8063 
 
 8272 
 
 8481 
 
 8689 
 
 8898 
 
 9106 
 
 9314 
 
 9522 
 
 9730 
 
 9938 
 
 208 
 
 209 
 
 32 0146 
 
 0354 
 
 0562 
 
 0769 
 
 0977 
 
 1184 
 
 1391 
 
 1598 
 
 1805 
 
 2012 
 
 207 
 
 210 
 
 2219 
 
 2426 
 
 2633 
 
 2839 
 
 3046 
 
 3252 
 
 3458 
 
 3665 
 
 3871 
 
 4077 
 
 206 
 
 211 
 
 4282 
 
 4488 
 
 4694 
 
 4899 
 
 5105 
 
 5310 
 
 5516 
 
 5721 
 
 5926 
 
 6131 
 
 205 
 
 212 
 
 6336 
 
 6541 
 
 6745 
 
 6950 
 
 7155 
 
 7359 
 
 7563 
 
 7767 
 
 7972 
 
 8176 
 
 204 
 
 213 
 
 *8380 
 
 8583 
 
 8787 
 
 8991 
 
 9194 
 
 9398 
 
 9601 
 
 9805 
 
 *008 
 
 0211 
 
 203 
 
 214 
 
 33 0414 
 
 0617 
 
 0819 
 
 1022 
 
 1225 
 
 1427 
 
 1630 
 
 1832 
 
 2034 
 
 2236 
 
 202 
 
 215 
 
 2438 
 
 2640 
 
 2842 
 
 3044 
 
 3246 
 
 3447 
 
 3649 
 
 3850 
 
 4051 
 
 4253 
 
 202 
 
 216 
 
 4454 
 
 4655 
 
 4856 
 
 5057 
 
 5257 
 
 5458 
 
 5658 
 
 5859 
 
 6059 
 
 6260 
 
 201 
 
 217 
 
 6460 
 
 6660 
 
 6860 
 
 7060 
 
 7260 
 
 7459 
 
 7659 
 
 7858 
 
 8058 
 
 8257 
 
 200 
 
 218 
 
 * 8456 
 
 8656 
 
 8855 
 
 9054 
 
 9253 
 
 9451 
 
 9650 
 
 9849 
 
 047 
 
 0246 
 
 199 
 
 219 
 
 34 0444 
 
 0642 
 
 0841 
 
 1039 
 
 1237 
 
 1435 
 
 1632 
 
 1830 
 
 2028 
 
 2225 
 
 198 
 
 N. 
 
 1 
 
 2 
 
 3 
 
 4 5 O 7 
 
 8 
 
 O 
 
 J>. 
 
APPENDIX. 
 
 847 
 
 LOGARITHMS OP NUMBERS. 
 
 IV. 
 
 o 
 
 1 3 
 
 . 
 
 4, 
 
 5 
 
 ; r- 
 G 7 ' 
 
 8 
 
 e 
 
 r>. 
 
 220 
 
 34 2423 
 
 2620 
 
 2817 
 
 3014 
 
 3212 
 
 3409 
 
 3606 
 
 3802 
 
 3999 
 
 4196 
 
 197 
 
 221 
 
 4392 
 
 4589 
 
 4785 
 
 4981 
 
 5178 
 
 5374 
 
 5570' 
 
 5766 
 
 5962 
 
 6157 
 
 196 
 
 222 
 
 6353 
 
 6549 
 
 6744 
 
 6939 
 
 7135 
 
 7330 
 
 7525 
 
 7720 
 
 7915 
 
 8110 
 
 195 
 
 223 
 
 * 8305 
 
 8500 8694 
 
 8889 
 
 9083 
 
 9278 
 
 9472 
 
 9666 
 
 9860 
 
 4054 
 
 194 
 
 224 
 
 35 0248 
 
 0442 | 0636 
 
 0829 
 
 1023 
 
 1216 
 
 1410 
 
 1603 
 
 1796 
 
 1989 
 
 193 
 
 225 
 
 2183 
 
 2375 
 
 2568 
 
 2761 
 
 2954 
 
 3147 
 
 3339 
 
 3532 
 
 3724 
 
 3916 
 
 193 
 
 226 
 
 4108 
 
 4301 
 
 4493 
 
 4685 
 
 4876 
 
 5068 
 
 5260 
 
 5452 
 
 5643 
 
 5834 
 
 192 
 
 227 
 
 6026 6217 
 
 6408 
 
 6599 
 
 6790 
 
 6981 
 
 7172 
 
 7363 
 
 7554 
 
 7744 
 
 191 
 
 228 
 
 7935 
 
 8125 
 
 8316 
 
 8506 
 
 8696 
 
 8886 
 
 9076 
 
 9266 
 
 9456 
 
 9646 
 
 190 
 
 229 
 
 * 9835 
 
 *025 
 
 0215 
 
 0404 
 
 0593 
 
 0783 
 
 0972 
 
 1161 
 
 1350 
 
 1539 
 
 189 
 
 230 
 
 36 1728 
 
 1917 
 
 2105 
 
 2294 
 
 2482 
 
 2671 
 
 2859 
 
 3048 
 
 3236 
 
 3424 
 
 188 
 
 231 
 
 3612 
 
 3800 
 
 3988 
 
 4176 
 
 4363 
 
 4551 
 
 4739 
 
 4926 
 
 5113 
 
 5301 
 
 188 
 
 232 
 
 5488 
 
 5675 
 
 5862 
 
 6049 
 
 6236 
 
 6423 
 
 6610 
 
 6796 
 
 6983 
 
 7169 
 
 187 
 
 233 
 
 7356 
 
 7542 
 
 7729 
 
 7915 
 
 8101 
 
 8287 
 
 8473 
 
 8659 
 
 8845 
 
 9030 
 
 186 
 
 234 
 
 * 9216 
 
 9401 
 
 9587 
 
 9772 
 
 9958 
 
 143 
 
 0328 
 
 0513 
 
 0698 
 
 0883 
 
 185 
 
 235 
 
 37 1068 
 
 1253 
 
 1437 
 
 1622 
 
 1806 
 
 1991 
 
 2175 
 
 "2360 
 
 2544 
 
 2728 
 
 184 
 
 236 
 
 2912 
 
 3096 
 
 3280 
 
 3464 
 
 3647 
 
 3831 
 
 4015 
 
 4198 
 
 4382 
 
 4565 
 
 184 
 
 237 
 
 4748 
 
 4932 
 
 5115 
 
 5298 
 
 5481 
 
 5664 
 
 5846 
 
 6029 
 
 6212 
 
 6394 
 
 183 
 
 238 
 
 6577 
 
 6759 
 
 6942 
 
 7124 
 
 7306 
 
 7488 
 
 7670 
 
 7852 
 
 8034 
 
 8216 
 
 182 
 
 239 
 
 * 8398 
 
 8580 
 
 8761 
 
 8943 
 
 9124 
 
 9306 
 
 9487 
 
 9668 
 
 9849 
 
 *030 
 
 181 
 
 240 
 
 38 0211 
 
 0392 
 
 0573 
 
 0754 
 
 0934 
 
 1115 
 
 1296 
 
 1476 
 
 1656 
 
 1837 
 
 181 
 
 241 
 
 2017 
 
 2197 
 
 2377 
 
 2557 
 
 2737 
 
 2917 
 
 3097 
 
 3277 
 
 3456 
 
 3636 
 
 180 
 
 242 
 
 3815 
 
 3995 
 
 4174 
 
 4353 
 
 4533 
 
 4712 
 
 4891 
 
 5070 
 
 5249 
 
 5428 
 
 179 
 
 243 
 
 5606 
 
 5785 
 
 5964 
 
 6142 
 
 6321 
 
 6499 
 
 6677 
 
 6856 
 
 7034 
 
 7212 
 
 178 
 
 244 
 
 7390 
 
 7568 
 
 7746 
 
 7923 
 
 8101 
 
 8279 
 
 8456 
 
 8634 
 
 8811 
 
 8989 
 
 178 
 
 245 
 
 * 9166 
 
 9343 
 
 9520 
 
 9698 
 
 9875 
 
 *051 
 
 0228 
 
 0405 
 
 0582 
 
 0759 
 
 177 
 
 246 
 
 39 0935 
 
 1112 
 
 1288 
 
 1464 
 
 1641 
 
 1817 
 
 1993 
 
 2169 
 
 2345 
 
 2521 
 
 176 
 
 247 
 
 2697 2873 3048 
 
 3224 
 
 3400 
 
 3575 
 
 3751 
 
 3926 
 
 4101 
 
 4277 
 
 176 
 
 248 
 
 4452 4627 
 
 4802 
 
 4977 
 
 5152 
 
 5326 
 
 5501 
 
 5676 
 
 5850 
 
 6025 
 
 175 
 
 249 
 
 6199 
 
 6374 
 
 6548 
 
 6722 
 
 6896 
 
 7071 
 
 7245 
 
 7419 
 
 7592 
 
 7766 
 
 174 
 
 250 
 
 7940 
 
 8114 
 
 8287 
 
 8461 
 
 8634 
 
 8808 
 
 8981 
 
 9154 
 
 9328 
 
 9501 
 
 173 
 
 251 
 
 * 9674 9847 
 
 *020 
 
 0192 
 
 0365 
 
 0538 
 
 0711 
 
 0883 
 
 1056 
 
 1228 
 
 173 
 
 252 
 
 40 1401 
 
 1573 
 
 1745 
 
 1917 
 
 2089 
 
 2261 
 
 2433 
 
 2605 
 
 2777 
 
 2949 
 
 172 
 
 253 
 
 3121 
 
 3292 
 
 3464 
 
 3635 
 
 3807 
 
 3978 
 
 4149 
 
 4320 
 
 4492 
 
 4663 
 
 171 
 
 254 
 
 4834 
 
 5005 
 
 5176 
 
 5346 
 
 5517 
 
 5688 
 
 5858 
 
 6029 
 
 6199 
 
 6370 
 
 171 
 
 255 
 
 6540 
 
 6710 
 
 6881 
 
 7051 
 
 7221 
 
 7391 
 
 7561 
 
 7731 
 
 7901 
 
 8070 
 
 170 
 
 256 
 
 8240 
 
 8410 
 
 8579 
 
 8749 
 
 8918 
 
 9087 
 
 9257 
 
 9426 
 
 9595 
 
 9764 
 
 169 
 
 257 
 
 * 9933 
 
 *102 
 
 0271 
 
 0440 
 
 0609 
 
 0777 
 
 0946 
 
 1114 
 
 1283 
 
 1451 
 
 169 
 
 258 
 
 41 1620 
 
 1788 
 
 1956 
 
 2124 
 
 2293 
 
 2461 
 
 2629 
 
 2796 
 
 2964 
 
 3132 
 
 168 
 
 259 
 
 3300 
 
 3467 
 
 3635 
 
 3803 
 
 3970 
 
 4137 
 
 4305 
 
 4472 
 
 4639 
 
 4806 
 
 167 
 
 260 
 
 4973 
 
 5140 
 
 5307 
 
 5474 
 
 5641 
 
 5808 
 
 5974 
 
 6141 
 
 6308 
 
 6474 
 
 167 
 
 261 
 
 6641 6807 
 
 6973 
 
 7139 
 
 7306 
 
 7472 
 
 7638 
 
 7804 
 
 7970 
 
 8135 
 
 166 
 
 262 
 
 8301 8467 
 
 8633 
 
 8798 
 
 8964 
 
 9129 
 
 9295 
 
 9460 
 
 9625 
 
 9791 
 
 165 
 
 263 
 
 * 9956 121 
 
 0286 
 
 0451 
 
 0616 
 
 0781 
 
 0945 
 
 1110 
 
 1275 
 
 1439 
 
 165 
 
 264 
 
 42 1604 
 
 1768 
 
 1933 
 
 2097 
 
 2261 
 
 2426 
 
 2590 
 
 2754 
 
 2918 
 
 3082 
 
 164 
 
 265 
 
 3246 
 
 3410 
 
 3574 
 
 3737 
 
 3901 
 
 4065 
 
 4228 
 
 4392 
 
 4555 
 
 4718 
 
 164 
 
 266 
 
 4882 5045 
 
 5208 
 
 5371 
 
 5534 
 
 5697 
 
 5860 
 
 6023 
 
 6186 
 
 6349 
 
 163 
 
 267 
 
 6511 6674 
 
 6836 
 
 6999 
 
 7161 
 
 7324 
 
 7486 
 
 7648 
 
 7811 
 
 7973 
 
 162 
 
 268 
 
 8135 8297 
 
 8459 
 
 8621 
 
 8783 
 
 8944 
 
 9106 
 
 9268 
 
 9429 
 
 9591 
 
 162 
 
 269 
 
 * 9752 
 
 9914 
 
 075 
 
 0236 
 
 0398 
 
 0559 
 
 0720 
 
 0881 
 
 1042 
 
 1203 
 
 161 
 
 270 
 
 43 1364 
 
 1525 
 
 1685 
 
 1846 
 
 2007 
 
 2167 
 
 2328 
 
 2488 
 
 2649 
 
 2809 
 
 161 
 
 271 
 
 2969 3130 
 
 3290 
 
 3450 
 
 3610 
 
 3770 
 
 3930 
 
 4090 
 
 4249 
 
 4409 
 
 160 
 
 272 
 
 4569 4729 
 
 4888 
 
 5048 
 
 5207 
 
 5367 
 
 5526 
 
 5685 
 
 5844 
 
 6004 
 
 159 
 
 273 
 
 6163 6322 
 
 6481 
 
 6640 
 
 6799 
 
 6957 
 
 7116 
 
 7275 
 
 7433 
 
 7592 
 
 159 
 
 274 
 
 7751 
 
 7909 
 
 8067 
 
 8226 
 
 8384 
 
 8542 
 
 8701 
 
 8859 
 
 9017 
 
 9175 
 
 158 
 
 275 
 
 * 9333 
 
 9491 
 
 9648 
 
 9806 
 
 9964 
 
 122 
 
 0279 
 
 0437 
 
 0594 
 
 0752 
 
 158 
 
 276 
 
 44 0909 
 
 1066 
 
 1224 
 
 1381 
 
 1538 
 
 1695 
 
 1852 
 
 2009 
 
 2166 
 
 2323 
 
 157 
 
 277 
 
 2480 
 
 2637 
 
 2793 
 
 2950 
 
 3106 
 
 3263 
 
 3419 
 
 3576 
 
 3732 
 
 3889 
 
 157 
 
 278 
 
 4045 
 
 4201 
 
 4357 
 
 4513 
 
 4669 
 
 4825 
 
 4981 
 
 5137 
 
 5293 
 
 5449 
 
 156 
 
 279 5604 
 
 5760 
 
 5915 
 
 6071 
 
 6226 
 
 6382 
 
 6537 
 
 6692 
 
 6848 
 
 7003 
 
 155 
 
 IV. 
 
 13 
 
 *J 
 
 4, 
 
 5 
 
 G 
 
 7 
 
 H 
 
 
 
 r>. 
 
848 
 
 APPENDIX. 
 
 LOGARITHMS OF NUMBERS. 
 
 N. 
 
 
 
 1 
 
 3 
 
 3 
 
 4, 
 
 5 
 
 6 
 
 7 
 
 8 
 
 O 
 
 r>. 
 
 280 
 
 44 7158 
 
 7313 
 
 7468 
 
 7623 
 
 7778 
 
 7933 
 
 8088 
 
 8242 
 
 8397 
 
 8552 
 
 155 
 
 281 
 
 *8706 
 
 8861 
 
 9015 
 
 9170 
 
 9324 
 
 9478 
 
 9633 
 
 9787 
 
 9941 
 
 095 
 
 154 
 
 282 
 
 45 0249 
 
 0403 
 
 0557 
 
 0711 
 
 0865 
 
 1018 
 
 1172 
 
 1326 
 
 1479 
 
 1633 
 
 154 
 
 283 
 
 1786 
 
 1940 
 
 2093 
 
 2247 
 
 2400 
 
 2553 
 
 2706 
 
 2859 
 
 3012 
 
 3165 
 
 153 
 
 284 
 
 3318 
 
 3471 
 
 3624 
 
 3777 
 
 3930 
 
 4082 
 
 4235 
 
 4387 
 
 4540 
 
 4692 
 
 153 
 
 285 
 
 4845 
 
 4997 
 
 5150 
 
 5302 
 
 5454 
 
 5606 
 
 5758 
 
 5910 
 
 6062 
 
 6214 
 
 152 
 
 286 
 
 6366 
 
 6518 
 
 6670 
 
 6821 
 
 6973 
 
 7125 
 
 7276 
 
 7428 
 
 7579 
 
 7731 
 
 152 
 
 287 
 
 7882 
 
 8033 
 
 8184 
 
 8336 
 
 8487 
 
 8638 
 
 8789 
 
 8940 
 
 9091 
 
 9242 
 
 151 
 
 288 
 
 * 9392 
 
 9543 
 
 9694 
 
 9845 
 
 9995 
 
 146 
 
 0296 
 
 0447 
 
 0597 
 
 0748 
 
 151 
 
 289 
 
 46 0898 
 
 1048 
 
 1198 
 
 1348 
 
 1499 
 
 1649 
 
 1799 
 
 1948 
 
 2098 
 
 2248 
 
 150 
 
 290 
 
 2398 
 
 2548 
 
 2697 
 
 2847 
 
 2997 
 
 3146 
 
 3296 
 
 3445 
 
 3594 
 
 3744 
 
 150 
 
 291 
 
 3893 
 
 4042 
 
 4191 
 
 4340 
 
 4490 
 
 4639 
 
 4788 
 
 4936 
 
 5085 
 
 5234 
 
 149 
 
 292 
 
 5383 
 
 5532 
 
 5680 
 
 5829 
 
 5977 
 
 6126 
 
 6274 
 
 6423 
 
 6571 
 
 6719 
 
 149 
 
 293 
 
 6868 
 
 7016 
 
 7164 
 
 7312 
 
 7460 
 
 7608 
 
 7756 
 
 7904 
 
 8052 
 
 8200 
 
 148 
 
 294 
 
 8347 
 
 8495 
 
 8643 
 
 8790 
 
 8938 
 
 9085 
 
 9233 
 
 9380 
 
 9527 
 
 9675 
 
 148 
 
 295 
 
 * 9822 
 
 9969 
 
 *116 
 
 0263 
 
 0410 
 
 0557 
 
 0704 
 
 0851 
 
 0998 
 
 1145 
 
 147 
 
 296 
 
 47 1292 
 
 1438 
 
 1585 
 
 1732 
 
 1878 
 
 2025 
 
 2171 
 
 2318 
 
 2464 
 
 2610 
 
 146 
 
 297 
 
 2756 
 
 2903 
 
 3049 
 
 3195 
 
 3341 
 
 3487 
 
 3633 
 
 3779 
 
 3925 
 
 4071 
 
 146 
 
 298 
 
 4216 
 
 4362 
 
 4508 
 
 4653 
 
 4799 
 
 4944 
 
 5090 
 
 5235 
 
 5381 
 
 5526 
 
 146 
 
 299 
 
 5671 
 
 5816 
 
 5962 
 
 6107 
 
 6252 
 
 6397 
 
 6542 
 
 6687 
 
 6832 
 
 6976 
 
 145 
 
 300 
 
 7121 
 
 7266 
 
 7411 
 
 7555 
 
 7700 
 
 7844 
 
 7989 
 
 8133 
 
 8278 
 
 8422 
 
 145 
 
 301 
 
 8566 
 
 8711 
 
 8855 
 
 8999 
 
 9143 
 
 9287 
 
 9431 
 
 9575 
 
 9719 
 
 9863 
 
 144 
 
 302 
 
 48 0007 
 
 0151 
 
 0294 
 
 0438 
 
 0582 
 
 0725 
 
 0869 
 
 1012 
 
 1156 
 
 1299 
 
 144 
 
 303 
 
 1443 
 
 1586 
 
 1729 
 
 1872 
 
 2016 
 
 2159 
 
 2302 
 
 2445 
 
 2588 
 
 2731 
 
 143 
 
 304 
 
 2874 
 
 3016 
 
 3159 
 
 3302 
 
 3445 
 
 3587 
 
 3730 
 
 3872 
 
 4015 
 
 4157 
 
 143 
 
 305 
 
 4300 
 
 4442 
 
 4585 
 
 4727 
 
 4869 
 
 5011 
 
 5153 
 
 5295 
 
 5437 
 
 5579 
 
 142 
 
 306 
 
 5721 
 
 5863 
 
 6005 
 
 6147 
 
 6289 
 
 6430 
 
 6572 
 
 6714 
 
 6855 
 
 6997 
 
 142 
 
 307 
 
 7138 
 
 7280 
 
 7421 
 
 7563 
 
 7704 
 
 7845 
 
 7986 
 
 8127 
 
 8269 
 
 8410 
 
 141 
 
 308 
 
 8551 
 
 8692 
 
 8833 
 
 8974 
 
 9114 
 
 9255 
 
 9396 
 
 9537 
 
 9677 
 
 9818 
 
 141 
 
 309 
 
 * 9958 
 
 *099 
 
 0239 
 
 0380 
 
 0520 
 
 0661 
 
 0801 
 
 0941 
 
 1081 
 
 1222 
 
 140 
 
 310 
 
 49 1362 
 
 1502 
 
 1642 
 
 1782 
 
 1922 
 
 2062 
 
 2201 
 
 2341 
 
 2481 
 
 2621 
 
 140 
 
 311 
 
 2760 
 
 2900 
 
 3040 
 
 3179 
 
 3319 
 
 3458 
 
 3597 
 
 3737 
 
 3876 
 
 4015 
 
 139 
 
 312 
 
 4155 
 
 4294 
 
 4433 
 
 4572 
 
 4711 
 
 4850 
 
 4989 
 
 5128 
 
 5267 
 
 5406 
 
 139 
 
 313 
 
 5544 
 
 5683 
 
 5822 
 
 5960 
 
 6099 
 
 6238 
 
 6376 
 
 6515 
 
 6653 
 
 6791 
 
 139 
 
 314 
 
 6930 
 
 7068 
 
 7206 
 
 7344 
 
 7483 
 
 7621 
 
 7759 
 
 7897 
 
 8035 
 
 8173 
 
 138 
 
 315 
 
 8311 
 
 8448 
 
 8586 
 
 8724 
 
 8862 
 
 8999 
 
 9137 
 
 9275 
 
 9412 
 
 9550 
 
 138 
 
 316 
 
 *9687 
 
 9824 
 
 9962 
 
 *099 
 
 0236 
 
 0374 
 
 0511 
 
 0648 
 
 0785 
 
 0922 
 
 137 
 
 317 
 
 50 1059 
 
 1196 
 
 1333 
 
 1470 
 
 1607 
 
 1744 
 
 1880 
 
 2017 
 
 2154 
 
 2291 
 
 137 
 
 318 
 
 2427 
 
 2564 
 
 2700 
 
 2837 
 
 2973 
 
 3109 
 
 3246 
 
 3382 
 
 3518 
 
 3655 
 
 136 
 
 319 
 
 3791 
 
 3927 
 
 4063 
 
 4199 
 
 4335 
 
 4471 
 
 4607 
 
 4743 
 
 4878 
 
 5014 
 
 136 
 
 320 
 
 5150 
 
 5286 
 
 5421 
 
 5557 
 
 5693 
 
 5828 
 
 5964 
 
 6099 
 
 6234 
 
 6370 
 
 136 
 
 321 
 
 6505 
 
 6640 
 
 6776 
 
 6911 
 
 7046 
 
 7181 
 
 7316 
 
 7451 
 
 7586 
 
 7721 
 
 135 
 
 322 
 
 7856 
 
 7991 
 
 8126 
 
 8260 
 
 8395 
 
 8530 
 
 8664 
 
 8799 
 
 8934 
 
 9068 
 
 135 
 
 323 
 
 * 9203 
 
 9337 
 
 9471 
 
 9606 
 
 9740 
 
 9874 
 
 009 
 
 0143 
 
 0277 
 
 0411 
 
 134 
 
 324 
 
 51 0545 
 
 0679 
 
 0813 
 
 0947 
 
 1081 
 
 1215 
 
 1349 
 
 1482 
 
 1616 
 
 1750 
 
 134 
 
 325 
 
 1883 
 
 2017 
 
 2151 
 
 2284 
 
 2418 
 
 2551 
 
 2684 
 
 2818 
 
 2951 
 
 3084 
 
 133 
 
 326 
 
 3218 
 
 3351 
 
 3484 
 
 3617 
 
 3750 
 
 3883 
 
 4016 
 
 4149 
 
 4282 
 
 4414 
 
 133 
 
 327 
 
 4548 
 
 4681 
 
 4813 
 
 4946 
 
 5079 
 
 5211 
 
 5344 
 
 5476 
 
 5609 
 
 5741 
 
 133 
 
 328 
 
 5874 
 
 6006 
 
 6139 
 
 6271 
 
 6403 
 
 6535 
 
 6668 
 
 6800 
 
 6932 
 
 7064 
 
 132 
 
 329 
 
 7196 
 
 7328 
 
 7460 
 
 7592 
 
 7724 
 
 7855 
 
 7987 
 
 8119 
 
 8251 
 
 8382 
 
 132 
 
 330 
 
 8514 
 
 8646 
 
 8777 
 
 8909 
 
 9040 
 
 9171 
 
 9303 
 
 9434 
 
 9566 
 
 9697 
 
 131 
 
 331 
 
 *9828 
 
 9959 
 
 090 
 
 0221 
 
 0353 
 
 0484 
 
 0615 
 
 0745 
 
 0876 
 
 1007 
 
 131 
 
 332 
 
 52 1138 
 
 1269 
 
 1400 
 
 1530 
 
 1661 
 
 1792 
 
 1922 
 
 2053 
 
 2183 
 
 2314 
 
 131 
 
 333 
 
 2444 
 
 2575 
 
 2705 
 
 2835 
 
 2966 
 
 3096 
 
 3226 
 
 3356 
 
 3486 
 
 3616 
 
 130 
 
 334 
 
 3746 
 
 3876 
 
 4006 
 
 4136 
 
 4266 
 
 4396 
 
 4526 
 
 4656 
 
 4785 
 
 4915 
 
 130 
 
 335 
 
 5045 
 
 5174 
 
 5304 
 
 5434 
 
 5563 
 
 5693 
 
 5822 
 
 5951 
 
 6081 
 
 6210 
 
 129 
 
 336 
 
 6339 
 
 6469 
 
 6598 
 
 6727 
 
 6856 
 
 6985 
 
 7114 
 
 7243 
 
 7372 
 
 7501 
 
 129 
 
 337 
 
 7630 
 
 7759 
 
 7888 
 
 8016 
 
 8145 
 
 8274 
 
 8402 
 
 8531 
 
 8660 
 
 8788 
 
 129 
 
 338 
 
 * 8917 
 
 9045 
 
 9174 
 
 9302 
 
 9430 
 
 9559 
 
 9687 
 
 9815 
 
 9943 
 
 072 
 
 128 
 
 339 
 
 53 0200 
 
 0328 
 
 0456 
 
 0584 
 
 0712 
 
 0840 
 
 0968 
 
 1096 
 
 1223 
 
 1351 
 
 128 
 
 N. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4= 
 
 5 
 
 O 
 
 7 
 
 & 
 
 O 
 
 T>. 
 
APPENDIX. 
 
 84:9 
 
 LOGARITHMS OP NUMBERS. 
 
 IV. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4, 
 
 5 
 
 G 
 
 7 
 
 a 
 
 O 
 
 r>. 
 
 340 
 
 53 1479 
 
 1607 
 
 1734 
 
 1862 
 
 1990 
 
 2117 
 
 2245 
 
 2372 
 
 2500 
 
 2627 
 
 128 
 
 341 
 
 2754 
 
 2882 
 
 3009 
 
 3136 
 
 3264 
 
 3391 
 
 3518 
 
 3645 
 
 3772 
 
 3899 
 
 127 
 
 342 
 
 4026 
 
 4153 
 
 4280 
 
 4407 
 
 4534 
 
 4661 
 
 4787 
 
 4914 
 
 5041 
 
 5167 
 
 127 
 
 343 
 
 5294 
 
 5421 
 
 5547 
 
 5674 
 
 5800 
 
 5927 
 
 6053 
 
 6180 
 
 6306 
 
 6432 
 
 126 
 
 344 
 
 6558 
 
 6685 
 
 6811 
 
 6937 
 
 7063 
 
 7189 
 
 7315 
 
 7441 
 
 7567 
 
 7693 
 
 126 
 
 345 
 
 7819 
 
 7945 
 
 8071 
 
 8197 
 
 8322 
 
 8448 
 
 8574 
 
 8699 
 
 8825 
 
 8951 
 
 126 
 
 346 
 
 * 9076 
 
 9202 
 
 9327 
 
 9452 
 
 9578 
 
 9703 
 
 9829 
 
 9954 
 
 +079 
 
 0204 
 
 125 
 
 347 
 
 54 0329 
 
 0455 
 
 0580 
 
 0705 
 
 0830 
 
 0955 
 
 1080 
 
 1205 
 
 1330 
 
 1454 
 
 125 
 
 348 
 
 1579 
 
 1704 
 
 1829 
 
 1953 
 
 2078 
 
 2203 
 
 2327 
 
 2452 
 
 2576 
 
 2701 
 
 125 
 
 349 
 
 2825 
 
 2950 
 
 3074 
 
 3199 
 
 3323 
 
 3447 
 
 3571 
 
 3696 
 
 3820 
 
 3944 
 
 124 
 
 350 
 
 4068 
 
 4192 
 
 4316 
 
 4440 
 
 4564 
 
 4688 
 
 4812 
 
 4936 
 
 5060 
 
 5183 
 
 124 
 
 351 
 
 '5307 
 
 5431 
 
 . 5555 
 
 5678 
 
 5802 
 
 5925 
 
 6049 
 
 6172 
 
 6296 
 
 6419 
 
 124 
 
 352 
 
 6543 
 
 6666 
 
 6789 
 
 6913 
 
 7036 
 
 7159 
 
 7282 
 
 7405 
 
 7529 
 
 7652 
 
 123 
 
 353 
 
 7775 
 
 7898 
 
 8021 
 
 8144 
 
 8267 
 
 8389 
 
 8512 
 
 8635 
 
 8758 
 
 8881 
 
 123 
 
 354 
 
 * 9003 
 
 9126 
 
 9249 
 
 9371 
 
 9494 
 
 9616 
 
 9739 
 
 9861 
 
 9984 
 
 +106 
 
 123 
 
 355 
 
 55 0228 
 
 0351 
 
 0473 
 
 0595 
 
 0717 
 
 0840 
 
 0962 
 
 1084 
 
 1206 
 
 1328 
 
 122 
 
 356 
 
 1450 
 
 1572 
 
 1694 
 
 1816 
 
 1938 
 
 2060 
 
 2181 
 
 2303 
 
 2425 
 
 2547 
 
 122 
 
 357 
 
 2668 
 
 2790 
 
 2911 
 
 3033 
 
 3155 
 
 3276 
 
 3398 
 
 3519 
 
 3640 
 
 3762 
 
 121 
 
 358 
 
 3883 
 
 4004 
 
 4126 
 
 4247 
 
 4368 
 
 4489 
 
 4610 
 
 4731 
 
 4852 
 
 4973 
 
 121 
 
 359 
 
 5094 
 
 5215 
 
 5336 
 
 5457 
 
 5578 
 
 5699 
 
 5820 
 
 5940 
 
 6061 
 
 6182 
 
 121 
 
 360 
 
 6303 
 
 6423 
 
 6544 
 
 6664 
 
 6785 
 
 6905 
 
 7026 
 
 7146 
 
 7267 
 
 7387 
 
 120 
 
 361 
 
 7507 
 
 7627 
 
 7748 
 
 7868 
 
 7988 
 
 8108 
 
 8228 
 
 8349 
 
 8469 
 
 8589 
 
 120 
 
 362 
 
 8709 
 
 8829 
 
 8948 
 
 9068 
 
 9188 
 
 9308 
 
 9428 
 
 9548 
 
 9667 
 
 9787 
 
 120 
 
 363 
 
 * 9907 
 
 +026 
 
 0146 
 
 0265 
 
 0385 
 
 0504 
 
 0624 
 
 0743 
 
 0863 
 
 0982 
 
 119 
 
 364 
 
 56 1101 
 
 1221 
 
 1340 
 
 1459 
 
 1578 
 
 1698 
 
 1817 
 
 1936 
 
 2055 
 
 2174 
 
 119 
 
 365 
 
 2293 
 
 2412 
 
 2531 
 
 2650 
 
 2760 
 
 2887 
 
 3006 
 
 3125 
 
 3244 
 
 3362 
 
 119 
 
 366 
 
 3481 
 
 3600 
 
 3718 
 
 3837 
 
 3955 
 
 4074 
 
 4192 
 
 4311 
 
 4429 
 
 4548 
 
 119 
 
 367 
 
 4666 
 
 4784 
 
 4903 
 
 5021 
 
 5139 
 
 5257 
 
 5376 
 
 5494 
 
 5612 
 
 5730 
 
 118 
 
 368 
 
 5848 
 
 5966 
 
 6084 
 
 6202 
 
 6320 
 
 6437 
 
 6555 
 
 6673 
 
 6791 
 
 6909 
 
 118 
 
 369 
 
 7026 
 
 7144 
 
 7262 
 
 7379 
 
 7497 
 
 7614 
 
 7732 
 
 7849 
 
 7967 
 
 8084 
 
 118 
 
 370 
 
 8202 
 
 8319 
 
 8436 
 
 8554 
 
 8671 
 
 8788 
 
 8905 
 
 9023 
 
 9140 
 
 9257 
 
 117 
 
 371 
 
 * 9374 
 
 9491 
 
 9608 
 
 9725 
 
 9842 
 
 9959 
 
 +076 
 
 0193 
 
 0309 
 
 0426 
 
 117 
 
 372 
 
 57 0543 
 
 0660 
 
 0776 
 
 0893 
 
 1010 
 
 1126 
 
 1243 
 
 1359 
 
 1476 
 
 1592 
 
 117 
 
 373 
 
 1709 
 
 1825 
 
 1942 
 
 2058 
 
 2174 
 
 2291 
 
 2407 
 
 2523 
 
 2639 
 
 2755 
 
 116 
 
 374 
 
 2872 
 
 2988 
 
 3104 
 
 3220 
 
 3336 
 
 3452 
 
 3568 
 
 3684 
 
 3800 
 
 3915 
 
 116 
 
 375 
 
 4031 
 
 4147 
 
 4263 
 
 4379 
 
 4494 
 
 4610 
 
 4726 
 
 4841 
 
 4957 
 
 5072 
 
 116 
 
 376 
 
 5188 
 
 5303 
 
 5419 
 
 5534 
 
 5650 
 
 5765 
 
 5880 
 
 5996 
 
 6111 
 
 6226 
 
 115 
 
 377 
 
 6341 
 
 6457 
 
 6572 
 
 6687 
 
 6802 
 
 6917 
 
 7032 
 
 7147 
 
 7262 
 
 7377 
 
 115 
 
 378 
 
 7492 
 
 7607 
 
 7722 
 
 7836 
 
 7951 
 
 8066 
 
 8181 
 
 8295 
 
 8410 
 
 8525 
 
 115 
 
 379 
 
 8639 
 
 8754 
 
 8868 
 
 8983 
 
 9097 
 
 9212 
 
 9326 
 
 9441 
 
 9555 
 
 9669 
 
 114 
 
 380 
 
 *9784 
 
 9898 
 
 +012 
 
 0126 
 
 0241 
 
 0355 
 
 0469 
 
 0583 
 
 0697 
 
 0811 
 
 114 
 
 381 
 
 58 0925 
 
 1039 
 
 1153 
 
 1267 
 
 1381 
 
 1495 
 
 1608 
 
 1722 
 
 1836 
 
 1950 
 
 114 
 
 382 
 
 2063 
 
 2177 
 
 2291 
 
 2404 
 
 2518 
 
 2631 
 
 2745 
 
 2858 
 
 2972 
 
 3085 
 
 114 
 
 383 
 
 3199 
 
 3312 
 
 3426 
 
 3539 
 
 3652 
 
 3765 
 
 3879 
 
 3992 
 
 4105 
 
 4218 
 
 113 
 
 384 
 
 4331 
 
 4444 
 
 4557 
 
 4670 
 
 4783 
 
 4896 
 
 5009 
 
 5122 
 
 5235 
 
 5348 
 
 113 
 
 385 
 
 5461 
 
 5574 
 
 5686 
 
 5799 
 
 5912 
 
 6024 
 
 6137 
 
 6250 
 
 6362 
 
 6475 
 
 113 
 
 386 
 
 6587 
 
 6700 
 
 6812 
 
 6925 
 
 7037 
 
 7149 
 
 7262 
 
 7374 
 
 7486 
 
 7599 
 
 112 
 
 387 
 
 7711 
 
 7823 
 
 7935 
 
 8047 
 
 8160 
 
 8272 
 
 8384 
 
 8496 
 
 8608 
 
 8720 
 
 112 
 
 388 
 
 8832 
 
 8944 
 
 9056 
 
 9167 
 
 9279 
 
 9391 
 
 9503 
 
 9615 
 
 9726 
 
 9838 
 
 112 
 
 389 
 
 * 9950 
 
 +061 
 
 0173 
 
 0284 
 
 0396 
 
 0507 
 
 0619 
 
 0730 
 
 0842 
 
 0953 
 
 112 
 
 390 
 
 59 1065 
 
 1176 
 
 1287 
 
 1399 
 
 1510 
 
 1621 
 
 1732 
 
 1843 
 
 1955 
 
 2066 
 
 111 
 
 391 
 
 2177 
 
 2288 
 
 2399 
 
 2510 
 
 2621 
 
 2732 
 
 2843 
 
 2954 
 
 3064 
 
 3175 
 
 111 
 
 392 
 
 3286 
 
 3397 
 
 3508 
 
 3618 
 
 3729 
 
 3840 
 
 3950 
 
 4061 
 
 4171 
 
 4282 
 
 111 
 
 393 
 
 4393 
 
 4503 
 
 4614 
 
 4724 
 
 4834 
 
 4945 
 
 5055 
 
 5165 
 
 5276 
 
 5386 
 
 110 
 
 394 
 
 5496 
 
 5606 
 
 5717 
 
 5827 
 
 5937 
 
 6047 
 
 6157 
 
 6267 
 
 6377 
 
 6487 
 
 110 
 
 395 
 
 6597 
 
 6707 
 
 6817 
 
 6927 
 
 7037 
 
 7146 
 
 7256 
 
 7366 
 
 7476 
 
 7586 
 
 110 
 
 396 
 
 7695 
 
 7805 
 
 7914 
 
 8024 
 
 8134 
 
 8243 
 
 8353 
 
 8462 
 
 8572 
 
 8681 
 
 110 
 
 397 
 
 8791 
 
 8900 
 
 9009 
 
 9119 
 
 9228 
 
 9337 
 
 9446 
 
 9556 
 
 9665 
 
 9774 
 
 109 
 
 398 
 
 * 9883 
 
 9992 
 
 +101 
 
 0210 
 
 0319 
 
 0428 
 
 0537 
 
 0646 
 
 0755 
 
 0864 
 
 109 
 
 399 
 
 60 0973 
 
 1082 
 
 1191 
 
 1299 
 
 1408 
 
 1517 
 
 1625 ' 
 
 1734 
 
 1843 
 
 1951 
 
 109 
 
 N. 
 
 O 
 
 1 
 
 2 
 
 3 
 
 4= 
 
 5 
 
 e 
 
 r 
 
 8 
 
 9 
 
 r>. 
 
 55 
 
850 
 
 APPENDIX. 
 
 LOGARITHMS OF NUMBERS. 
 
 TV. 
 
 O 
 
 1 
 
 3 3 
 
 4 
 
 5 
 
 7 
 
 8 
 
 9 
 
 r>. 
 
 400 
 
 60 2060 
 
 2169 
 
 2277 2386 
 
 2494 
 
 2603 
 
 2711 
 
 2819 
 
 2928 
 
 3036 
 
 108 
 
 401 
 
 3144 
 
 3253 
 
 3361 
 
 3469 
 
 3577 
 
 3686 
 
 3794 
 
 3902 
 
 4010 
 
 4118 
 
 108 
 
 402 
 
 4226 
 
 4334 
 
 4442 
 
 4550 
 
 4658 
 
 4766 
 
 4874 
 
 4982 
 
 5089 
 
 5197 
 
 108 
 
 403 
 
 5305 
 
 5413 
 
 5521 
 
 5628 
 
 5736 
 
 5844 
 
 5951 
 
 6059 
 
 6166 
 
 6274 
 
 108 
 
 404 
 
 6381 
 
 6489 
 
 6596 
 
 6704 
 
 6811 
 
 6919 
 
 7026 
 
 7133 
 
 7241 
 
 7348 
 
 107 
 
 405 
 
 7455 
 
 7562 
 
 7669 
 
 7777 
 
 7884 
 
 7991 
 
 8098 
 
 8205 
 
 8312 
 
 8419 
 
 107 
 
 406 
 
 8526 
 
 8633 
 
 8740 
 
 8847 
 
 8954 
 
 9061 
 
 9167 
 
 9274 
 
 9381 
 
 9488 
 
 107 
 
 407 
 
 * 9594 
 
 9701 
 
 9808 
 
 9914 
 
 +021 
 
 0128 
 
 0234 
 
 0341 
 
 0447 
 
 0554 
 
 107 
 
 408 
 
 61 0660 
 
 0767 
 
 0873^ 
 
 0979 
 
 1086 
 
 1192 
 
 1298 
 
 1405 
 
 1511 
 
 1617 
 
 106 
 
 409 
 
 1723 
 
 1829 
 
 1936 
 
 2042 
 
 2148 
 
 2254 
 
 2360 
 
 2466 
 
 2572 
 
 2678 
 
 106 
 
 410 
 
 2784 
 
 2890 
 
 2996 
 
 3102 
 
 3207 
 
 3313 
 
 3419 
 
 3525 
 
 3630 
 
 3736 
 
 106 
 
 411 
 
 3842 
 
 3947 
 
 4053 
 
 4159 
 
 4264 
 
 4370 
 
 4475 
 
 4581 
 
 4686 
 
 4792 
 
 106 
 
 412 
 
 4897 
 
 5003 
 
 5108 
 
 5213 
 
 5319 
 
 5424 
 
 5529 
 
 5634 
 
 5740 
 
 5845 
 
 105 
 
 413 
 
 5950 
 
 6055 
 
 6160 
 
 6265 
 
 6370 
 
 6476 
 
 6581 
 
 6686 
 
 6790 
 
 6895 
 
 105 
 
 414 
 
 7000 
 
 7105 
 
 7210 
 
 7315 
 
 7420 
 
 7525 
 
 7629 
 
 7734 
 
 7839 
 
 7943 
 
 105 
 
 415 
 
 8048 
 
 8153 
 
 8257 
 
 8362 
 
 8466 
 
 8571 
 
 8676 
 
 8780 
 
 8884 
 
 8989 
 
 105 
 
 416 
 
 * 9093 
 
 9198 
 
 9302 
 
 9406 
 
 9511 
 
 9615 
 
 9719 
 
 9824 
 
 9928 
 
 +032 
 
 104 
 
 417 
 
 62 0136 
 
 0240 
 
 0344 
 
 0448 
 
 0552 
 
 0656 
 
 0760 
 
 0864 
 
 0968 
 
 1072 
 
 104 
 
 418 
 
 1176 
 
 1280 
 
 1384 
 
 1488 
 
 1592 
 
 1695 
 
 1799 
 
 1903 
 
 2007 
 
 2110 
 
 104 
 
 419 
 
 2214 
 
 2318 
 
 2421 
 
 2525 
 
 2628 
 
 2732 
 
 2835 
 
 2939 
 
 3042 
 
 3146 
 
 104 
 
 420 
 
 3249 
 
 3353 
 
 3456 
 
 3559 
 
 3663 
 
 3766 
 
 3869 
 
 3973 
 
 4076 
 
 4179 
 
 103 
 
 421 
 
 4282 
 
 4385 
 
 4488 
 
 4591 
 
 4695 
 
 4798 
 
 4901 
 
 5004 
 
 5107 
 
 5210 
 
 103 
 
 422 
 
 5312 
 
 5415 
 
 5518 
 
 5621 
 
 5724 
 
 5827 
 
 5929 
 
 6032 
 
 6135 
 
 6238 
 
 103 
 
 423 
 
 6340 
 
 6443 
 
 6546 
 
 6648 
 
 6751 
 
 6853 
 
 6956 
 
 7058 
 
 7161 
 
 7263 
 
 103 
 
 424 
 
 7366 
 
 7468 
 
 7571 
 
 7673 
 
 7775 
 
 7878 
 
 7980 
 
 8082 
 
 8185 
 
 8287 
 
 102 
 
 425 
 
 8389 
 
 8491 
 
 8593 
 
 8695 
 
 8797 
 
 8900 
 
 9002 
 
 9104 
 
 9206 
 
 9308 
 
 102 
 
 426 
 
 * 9410 
 
 9512 
 
 9613 
 
 9715 
 
 9817 
 
 9919 
 
 +021 
 
 0123 
 
 0224 
 
 0326 
 
 102 
 
 427 
 
 63 0428 
 
 0530 
 
 0631 
 
 0733 
 
 0835 
 
 0936 
 
 1038 
 
 1139 
 
 1241 
 
 1342 
 
 102 
 
 428 
 
 1444 
 
 1545 
 
 1647 
 
 1748 
 
 1849 
 
 1951 
 
 2052 
 
 2153 
 
 2255 
 
 2356 
 
 101 
 
 429 
 
 2457 
 
 2559 
 
 2660 ( 
 
 2761 
 
 2862 
 
 2963 
 
 3064 
 
 3165 
 
 3266 
 
 3367 
 
 101 
 
 430 
 
 3468 
 
 3569 
 
 3670 
 
 3771 
 
 3872 
 
 3973 
 
 4074 
 
 4175 
 
 4276 
 
 4376 
 
 100 
 
 431 
 
 4477 
 
 4578 
 
 4679 
 
 4779 
 
 4880 
 
 4981 
 
 5081 
 
 5182 
 
 5283 
 
 5383 
 
 100 
 
 432 
 
 5484 
 
 5584 
 
 5685 
 
 5785 
 
 5886 
 
 5986 
 
 6087 
 
 6187 
 
 6287 
 
 6388 
 
 100 
 
 433 
 
 6488 
 
 6588 
 
 6688 
 
 6789 
 
 6889 
 
 6989 
 
 7089 
 
 7189 
 
 7290 
 
 7390 
 
 100 
 
 434 
 
 7490 
 
 7590 
 
 7690 
 
 7790 
 
 7890 
 
 7990 
 
 8090 
 
 8190 
 
 8290 
 
 8389 
 
 99 
 
 435 
 
 8489 
 
 8589 
 
 8689 
 
 8789 
 
 8888 
 
 8988 
 
 9088 
 
 9188 
 
 9287 
 
 9387 
 
 99 
 
 436 
 
 *9486 
 
 9586 
 
 9686 
 
 9785 
 
 9885 
 
 9984 
 
 +084 
 
 0183 
 
 0283 
 
 0382 
 
 99 
 
 437 
 
 64 0481 
 
 0581 
 
 0680 
 
 0779 
 
 0879 
 
 0978 
 
 1077 
 
 1177 
 
 1276 
 
 1375 
 
 99 
 
 438 
 
 1474 
 
 1573 
 
 1672 
 
 1771 
 
 1871 
 
 1970 
 
 2069 
 
 2168 
 
 2267 
 
 2366 
 
 99 
 
 439 
 
 2465 
 
 2563 
 
 2662 
 
 2761 
 
 2860 
 
 2959 
 
 3058 
 
 3156 
 
 3255 
 
 3354 
 
 99 
 
 440 
 
 3453 
 
 3551 
 
 3650 
 
 3749 
 
 3847 
 
 3946 
 
 4044 
 
 4143 
 
 4242 
 
 4340 
 
 98 
 
 441 
 
 4439 
 
 4537 
 
 4636 
 
 4734 
 
 4832 
 
 4931 
 
 5029 
 
 5127 
 
 5226 
 
 5324 
 
 98 
 
 442 
 
 5422 
 
 5521 
 
 5619 
 
 5717 
 
 5815 
 
 5913 
 
 6011 
 
 6110 
 
 6208 
 
 6306 
 
 98 
 
 443 
 
 6404 
 
 6502 
 
 6600 
 
 6698 
 
 6796 
 
 6894 
 
 6992 
 
 7089 
 
 7187 
 
 7285 
 
 98 
 
 444 
 
 7383 
 
 7481 
 
 7579 
 
 7676 
 
 7774 
 
 7872 
 
 7969 
 
 8067 
 
 8165 
 
 8262 
 
 98 
 
 445 
 
 8360 
 
 8458 
 
 8555 
 
 8653 
 
 8750 
 
 8848 
 
 8945 
 
 9043 
 
 9140 
 
 9237 
 
 97 
 
 446 
 
 * 9335 
 
 9132 
 
 9530 
 
 9627 
 
 9724 
 
 9821 
 
 9919 
 
 +016 
 
 0113 
 
 0210 
 
 97 
 
 447 
 
 65 0308 
 
 0405 
 
 0502 
 
 0599 
 
 0696 
 
 0793 
 
 0890 
 
 0987 
 
 1084 
 
 1181 
 
 97 
 
 448 
 
 1278 
 
 1375 
 
 1472 
 
 1569 
 
 1666 
 
 1762 
 
 1859 
 
 1956 
 
 2053 
 
 2150 
 
 97 
 
 449 
 
 2246 
 
 2343 
 
 2440 
 
 2536 
 
 2633 
 
 2730 
 
 2826 
 
 2923 
 
 3019 
 
 3116 
 
 97 
 
 450 
 
 3213 
 
 3309 
 
 3405 
 
 3502 
 
 3598 
 
 3695 
 
 3791 
 
 3888 
 
 3984 
 
 4080 
 
 96 
 
 451 
 
 4177 
 
 4273 
 
 4369 
 
 4465 
 
 4562 
 
 4658 
 
 4754 
 
 4850 
 
 4946 
 
 5042 
 
 96 
 
 452 
 
 5138 
 
 5235 
 
 5331 
 
 5427 
 
 5523 
 
 5619 
 
 5715 
 
 5810 
 
 5906 
 
 6002 
 
 96 
 
 453 
 
 6098 
 
 6194 
 
 6290 
 
 6386 
 
 6482 
 
 6577 
 
 6673 
 
 6769 
 
 6864 
 
 6960 
 
 96 
 
 454 
 
 7056 
 
 7152 
 
 7247 
 
 7343 
 
 7438 
 
 7534 
 
 7629 
 
 7725 
 
 7820 
 
 7916 
 
 96 
 
 455 
 
 8011 
 
 8107 
 
 8202 
 
 8298 
 
 8393 
 
 8488 
 
 8584 
 
 8679 
 
 8774 
 
 8870 
 
 95 
 
 456 
 
 8965 
 
 9060 
 
 9155 
 
 9250 
 
 9346 
 
 9441 
 
 9536 
 
 9631 
 
 9726 
 
 9821 
 
 95 
 
 457 
 
 * 9916 
 
 +011 
 
 0106 
 
 0201 
 
 0296 
 
 0391 
 
 0486 
 
 0581 
 
 0676 
 
 0771 
 
 95 
 
 458 
 
 66 0865 
 
 0960 
 
 1055 
 
 1150 
 
 1245 
 
 1339 
 
 1434 
 
 1529 
 
 1623 
 
 1718 
 
 95 
 
 459 
 
 1813 
 
 1907 
 
 2002 
 
 2096 
 
 2191 
 
 2286 
 
 2380 
 
 2475 
 
 2569 
 
 2663 
 
 95 
 
 N. 
 
 
 
 1 
 
 3 
 
 3 
 
 4= 
 
 5 
 
 O 
 
 7 
 
 a 
 
 
 
 r>. 
 
APPENDIX. 
 
 851 
 
 LOGARITHMS OP NUMBERS. 
 
 TV. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4 5 
 
 G 
 
 7 
 
 8 
 
 O 
 
 l>. 
 
 460 
 
 66 2758 
 
 2852 
 
 2947 
 
 3041 
 
 3135 
 
 3230 
 
 3324 
 
 3418 
 
 3512 
 
 3607 
 
 94 
 
 461 
 
 3701 
 
 3795 
 
 3889 
 
 3983 
 
 4078 
 
 4172 
 
 4266 
 
 4360 
 
 4454 
 
 4548 
 
 94 
 
 462 
 
 4642 
 
 4736 
 
 4830 
 
 4924 
 
 5018 
 
 5112 
 
 5206 
 
 5299 
 
 5393 
 
 5487 
 
 94 
 
 463 
 
 5581 
 
 5675 
 
 5769 
 
 5862 
 
 5956 
 
 6050 
 
 6143 
 
 6237 
 
 6331 
 
 6424 
 
 94 
 
 464 
 
 6518 
 
 6612 
 
 6705 
 
 6799 
 
 6892 
 
 6986 
 
 7079 
 
 7173 
 
 7266 
 
 7360 
 
 94 
 
 465 
 
 7453 
 
 7546 
 
 7640 
 
 7733 
 
 7826 
 
 7920 
 
 8013 
 
 8106 
 
 8199 
 
 8293 
 
 93 
 
 466 
 
 8386 
 
 8479 
 
 8572 
 
 8665 
 
 8759 
 
 8852 
 
 8945 
 
 9038 
 
 9131 
 
 9224 
 
 93 
 
 467 
 
 * 9317 
 
 9410 
 
 9503 
 
 9596 
 
 9689 
 
 9782 
 
 9875 
 
 9967 
 
 *060 
 
 0153 
 
 93 
 
 468 
 
 67 0246 
 
 0339 
 
 0431 
 
 0524 
 
 0617 
 
 0710 
 
 0802 
 
 0895 
 
 0988 
 
 1080 
 
 93 
 
 469 
 
 1173 
 
 1265 
 
 1358 
 
 1451 
 
 1543 
 
 1636 
 
 1728 
 
 1821 
 
 1913 
 
 2005 
 
 93 
 
 470 
 
 2098 
 
 2190 
 
 2283 
 
 2375 
 
 2467 
 
 2560 
 
 2652 
 
 2744 
 
 2836 
 
 2929 
 
 92 
 
 471 
 
 3021 
 
 3113 
 
 . 3205 
 
 3297 
 
 3390 
 
 3482 
 
 3574 
 
 3666 
 
 3758 
 
 3850 
 
 92 
 
 472 
 
 3942 
 
 4034 
 
 4126 
 
 4218 
 
 4310 
 
 4402 
 
 4494 
 
 4586 
 
 4677 
 
 4769 
 
 92 
 
 473 
 
 4861 
 
 4953 
 
 5045 
 
 5137 
 
 5228 
 
 5320 
 
 5412 
 
 5503 
 
 5595 
 
 5687 
 
 92 
 
 474 
 
 5778 
 
 5870 
 
 5962 
 
 6053 
 
 6145 
 
 6236 
 
 6328 
 
 6419 
 
 6511 
 
 6602 
 
 92 
 
 475 
 
 6694 
 
 6785 
 
 6876 
 
 6968 
 
 7059 
 
 7151 
 
 7242 
 
 7333 
 
 7424 
 
 7516 
 
 91 
 
 476 
 
 7607 
 
 7698 
 
 7789 
 
 7881 
 
 7972 
 
 8063 
 
 8154 
 
 8245 
 
 8336 
 
 8427 
 
 91 
 
 477 
 
 8518 
 
 8609 
 
 8700 
 
 8791 
 
 8882 
 
 8973 
 
 9064 
 
 9155 
 
 9246 
 
 9337 
 
 91 
 
 478 
 
 *9428 
 
 9519 
 
 9610 
 
 9700 
 
 9791 
 
 9882 
 
 9973 
 
 063 
 
 0154 
 
 0245 
 
 91 
 
 479 
 
 68 0336 
 
 0426 
 
 0517 
 
 0607 
 
 0698 
 
 0789 
 
 0879 
 
 0970 
 
 1060 
 
 1151 
 
 91 
 
 480 
 
 1241 
 
 1332 
 
 1422 
 
 1513 
 
 1603 
 
 1693 
 
 1784 
 
 1874 
 
 1964 
 
 2055 
 
 90 
 
 481 
 
 2145 
 
 2235 
 
 2326 
 
 2416 
 
 2506 
 
 2596 
 
 2686 
 
 2777 
 
 2867 
 
 2957 
 
 90 
 
 482 
 
 3047 
 
 3137 
 
 3227 
 
 3317 
 
 3407 
 
 3497 
 
 3587 
 
 3677 
 
 3767 
 
 3857 
 
 90 
 
 483 
 
 3947 
 
 4037 
 
 4127 
 
 4217 
 
 4307 
 
 4396 
 
 4486 
 
 4576 
 
 4666 
 
 4756 
 
 90 
 
 484 
 
 4845 
 
 4935 
 
 5025 
 
 5114 
 
 5204 
 
 5294 
 
 5383 
 
 5473 
 
 5563 
 
 5652 
 
 90 
 
 485 
 
 5742 
 
 5831 
 
 5921 
 
 6010 
 
 6100 
 
 6189 
 
 6279 
 
 6368 
 
 6458 
 
 6547 
 
 89 
 
 486 
 
 6636 
 
 5726 
 
 6815 
 
 6904 
 
 6994 
 
 7083 
 
 7172 
 
 7261 
 
 7351 
 
 7440 
 
 89 
 
 487 
 
 7529 
 
 7618 
 
 7707 
 
 7796 
 
 7886 
 
 7975 
 
 8064 
 
 8153 
 
 8242 
 
 8331 
 
 89 
 
 488 
 
 8420 
 
 8509 
 
 8598 
 
 8687 
 
 8776 
 
 8865 
 
 8953 
 
 9042 
 
 9131 
 
 9220 
 
 89 
 
 489 
 
 * 9309 
 
 9398 
 
 9486 
 
 9575 
 
 9664 
 
 9753 
 
 9841 
 
 9930 
 
 *019 
 
 0107 
 
 89 
 
 490 
 
 69 0196 
 
 0285 
 
 0373 
 
 0462 
 
 0550 
 
 0639 
 
 0728 
 
 0816 
 
 0905 
 
 0993 
 
 89 
 
 491 
 
 1081 
 
 1170 
 
 1258 
 
 1347 
 
 1435 
 
 1524 
 
 1612 
 
 1700 
 
 1789 
 
 1877 
 
 88 
 
 492 
 
 1965 
 
 2053 
 
 2142 
 
 2230 
 
 2318 
 
 2406 
 
 2494 
 
 2583 
 
 2671 
 
 2759 
 
 88 
 
 493 
 
 2847 
 
 2935 
 
 3023 
 
 3111 
 
 3199 
 
 3287 
 
 3375 
 
 3463 
 
 3551 
 
 3639 
 
 83 
 
 494 
 
 3727 
 
 3815 
 
 3903 
 
 3991 
 
 4078 
 
 4166 
 
 4254 
 
 4342 
 
 4430 
 
 4517 
 
 88 
 
 495 
 
 4605 
 
 4693 
 
 4781 
 
 4868 
 
 4956 
 
 5044 
 
 5131 
 
 5219 
 
 5307 
 
 5394 
 
 88 
 
 496 
 
 5482 
 
 5569 
 
 5657 
 
 5744 
 
 5832 
 
 5919 
 
 6007 
 
 6094 
 
 6182 
 
 6269 
 
 87 
 
 497 
 
 6356 
 
 6444 
 
 6531 
 
 6618 
 
 6706 
 
 6793 
 
 6880 
 
 6968 
 
 7055 
 
 7142 
 
 87 
 
 498 
 
 7229 
 
 7317 
 
 7404 
 
 7491 
 
 7578 
 
 7665 
 
 7752 
 
 7839 
 
 7926 
 
 8014 
 
 87 
 
 499 
 
 8101 
 
 8188 
 
 8275 
 
 8362 
 
 8449 
 
 8535 
 
 8622 
 
 8709 
 
 8796 
 
 8883 
 
 87 
 
 500 
 
 8970 
 
 9057 
 
 9144 
 
 9231 
 
 9317 
 
 9404 
 
 9491 
 
 9578 
 
 9664 
 
 9751 
 
 87 
 
 501 
 
 * 9838 
 
 9924 
 
 *011 
 
 0098 
 
 0184 
 
 0271 
 
 0358 
 
 0444 
 
 0531 
 
 0617 
 
 87 
 
 502 
 
 70 0704 
 
 0790 
 
 0877 
 
 0963 
 
 1050 
 
 1136 
 
 1222 
 
 1309 
 
 1395 
 
 1482 
 
 86 
 
 503 
 
 1568 
 
 1654 
 
 1741 
 
 1827 
 
 1913 
 
 1999 
 
 2086 
 
 2172 
 
 2258 
 
 2344 
 
 86 
 
 504 
 
 2431 
 
 2517 
 
 2603 
 
 2689 
 
 2775 
 
 2861 
 
 2947 
 
 3033 
 
 3119 
 
 3205 
 
 86 
 
 505 
 
 3291 
 
 3377 
 
 3463 
 
 3549 
 
 3635 
 
 3721 
 
 3807 
 
 3895 
 
 3979 
 
 4065 
 
 86 
 
 506 
 
 4151 
 
 4236 
 
 4322 
 
 4408 
 
 4494 
 
 4579 
 
 4665 
 
 4751 
 
 4837 
 
 4922 
 
 86 
 
 507 
 
 5008 
 
 5094 
 
 5179 
 
 5265 
 
 5350 
 
 5436 
 
 5522 
 
 5607 
 
 5693 
 
 5778 
 
 86 
 
 508 
 
 5864 
 
 5949 
 
 6035 
 
 6120 
 
 6206 
 
 6291 
 
 6376 
 
 6462 
 
 6547 
 
 6632 
 
 85 
 
 509 
 
 6718 
 
 6803 
 
 6888 
 
 6974 
 
 7059 
 
 7144 
 
 7229 
 
 7315 
 
 7400 
 
 7485 
 
 85 
 
 510 
 
 7570 
 
 7655 
 
 7740 
 
 7826 
 
 7911 
 
 7996 
 
 8081 
 
 8166 
 
 8251 
 
 8336 
 
 85 
 
 511 
 
 8421 
 
 8506 
 
 8591 
 
 8676 
 
 8761 
 
 8846 
 
 8931 
 
 9015 
 
 9100 
 
 9185 
 
 85 
 
 512 
 
 * 9270 
 
 9355 
 
 9440 
 
 9524 
 
 9609 
 
 9694 
 
 9779 
 
 9863 
 
 9948 
 
 033 
 
 85 
 
 513 
 
 71 0117 
 
 0202 
 
 0287 
 
 0371 
 
 0456 
 
 0540 
 
 0625 
 
 0710 
 
 0794 
 
 0879 
 
 85 
 
 514 
 
 0963 
 
 1048 
 
 1132 
 
 1217 
 
 1301 
 
 1385 
 
 1470 
 
 1554 
 
 1639 
 
 1723 
 
 84 
 
 515 
 
 1807 
 
 1892 
 
 1976 
 
 2060 
 
 2144 
 
 2229 
 
 2313 
 
 2397 
 
 2481 
 
 2566 
 
 84 
 
 516 
 
 2650 
 
 2734 
 
 2818 
 
 2902 
 
 2986 
 
 3070 
 
 3154 
 
 3238 
 
 3323 
 
 3407 
 
 84 
 
 517 
 
 3491 
 
 3575 
 
 3650 
 
 3742 
 
 3826 
 
 3910 
 
 3994 
 
 4078 
 
 4162 
 
 4246 
 
 84 
 
 518 
 
 4330 
 
 4414 
 
 4497 
 
 4581 
 
 4665 
 
 4749 
 
 4833 
 
 4916 
 
 5000 
 
 5084 
 
 84 
 
 519 
 
 5167 
 
 5251 
 
 5335 
 
 5418 
 
 5502 
 
 5586 
 
 5669 
 
 5753 
 
 5836 
 
 5920 
 
 84 
 
 TV. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4= 
 
 5 
 
 G 
 
 7 
 
 a 
 
 O 
 
 r>. 
 
852 
 
 APPENDIX. 
 
 LOGARITHMS OF NUMBERS. 
 
 N. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4, 
 
 5 
 
 7 
 
 8 
 
 9 
 
 I>. 
 
 520 
 
 71 6003 
 
 6087 
 
 6170 
 
 6254 
 
 6337 
 
 6421 
 
 6504 
 
 6588 
 
 6671 
 
 6754 
 
 83 
 
 521 
 
 6838 
 
 6921 
 
 7004 
 
 7088 
 
 7171 
 
 7254 
 
 7338 
 
 7421 
 
 7504 
 
 7587 
 
 83 
 
 522 
 
 7671 
 
 7754 
 
 7837 
 
 7920 
 
 8003 
 
 8086 
 
 8169 
 
 8253 
 
 8336 
 
 8419 
 
 83 
 
 523 
 
 8502 
 
 8585 
 
 8668 
 
 8751 
 
 8834 
 
 8917 
 
 9000 
 
 9083 
 
 9165 
 
 9248 
 
 83 
 
 524 
 
 * 9331 
 
 9414 
 
 9497 
 
 9580 
 
 9663 
 
 9745 
 
 9828 
 
 9911 
 
 9994 
 
 *077 
 
 83 
 
 525 
 
 72 0159 
 
 0242 
 
 0325 
 
 0407 
 
 0490 
 
 0573 
 
 0655 
 
 0738 
 
 0821 
 
 0903 
 
 83 
 
 526 
 
 0986 
 
 1068 
 
 1151 
 
 1233 
 
 1316 
 
 1398 
 
 1481 
 
 1563 
 
 1646 
 
 1728 
 
 82 
 
 527 
 
 1811 
 
 1893 
 
 1975 
 
 2058 
 
 2140 
 
 2222 
 
 2305 
 
 2387 
 
 2469 
 
 2552 
 
 82 
 
 528 
 
 2634 
 
 2716 
 
 2798 
 
 2881 
 
 2963 
 
 3045 
 
 3127 
 
 3209 
 
 3291 
 
 3374 
 
 82 
 
 529 
 
 3456 
 
 3538 
 
 3620 
 
 3702 
 
 3784 
 
 3866 
 
 3948 
 
 4030 
 
 4112 
 
 4194 
 
 82 
 
 530 
 
 4276 
 
 4358 
 
 4440 
 
 4522 
 
 4604 
 
 4685 
 
 4767 
 
 4849 
 
 4931 
 
 5013 
 
 82 
 
 531 
 
 5095 
 
 5176 
 
 5258 
 
 5340 
 
 5422 
 
 5503 
 
 ,5585 
 
 5667 
 
 5748 
 
 5830 
 
 82 
 
 532 
 
 5912 
 
 5993 
 
 6075 
 
 6156 
 
 6238 
 
 6320 
 
 6401 
 
 6483 
 
 6564 
 
 6646 
 
 82 
 
 533 
 
 6727 
 
 6809 
 
 6890 
 
 6972 
 
 7053 
 
 7134 
 
 7216 
 
 7297 
 
 7379 
 
 7460 
 
 81 
 
 534 
 
 7541 
 
 7623 
 
 7704 
 
 7785 
 
 7866 
 
 7948 
 
 8029 
 
 8110 
 
 8191 
 
 8273 
 
 81 
 
 535 
 
 8354 
 
 8435 
 
 8516 
 
 8597 
 
 8678 
 
 8759 
 
 8841 
 
 8922 
 
 9003 
 
 9084 
 
 81 
 
 536 
 
 9165 
 
 9246 
 
 9327 
 
 9408 
 
 9489 
 
 9570 
 
 9651 
 
 9732 
 
 9813 
 
 9893 
 
 81 
 
 537 
 
 * 9974 
 
 *055 
 
 0136 
 
 0217 
 
 0298 
 
 0378 
 
 0459 
 
 0540 
 
 0621 
 
 0702 
 
 81 
 
 538 
 
 73 0782 
 
 0863 
 
 0944 
 
 1024 
 
 1105 
 
 1186 
 
 1266 
 
 1347 
 
 1428 
 
 1508 
 
 81 
 
 539 
 
 1589 
 
 1669 
 
 1750 
 
 1830 
 
 1911 
 
 1991 
 
 2072 
 
 2152 
 
 2233 
 
 2313 
 
 81 
 
 540 
 
 2394 
 
 2474 
 
 2555 
 
 2635 
 
 2715 
 
 2796 
 
 2876 
 
 2956 
 
 3037 
 
 3117 
 
 80 
 
 541 
 
 3197 
 
 3278 
 
 3358 
 
 3438 
 
 3518 
 
 3598 
 
 3679 
 
 3759 
 
 3839 
 
 3919 
 
 80 
 
 542 
 
 3999 
 
 4079 
 
 4160 
 
 4240 
 
 4320 
 
 4400 
 
 4480 
 
 4560 
 
 4640 
 
 4720 
 
 80 
 
 543 
 
 4800 
 
 4880 
 
 4960 
 
 5040 
 
 5120 
 
 5200 
 
 5279 
 
 5359 
 
 5439 
 
 5519 
 
 80 
 
 544 
 
 5599 
 
 5679 
 
 5759 
 
 5838 
 
 5918 
 
 5998 
 
 6078 
 
 6157 
 
 6237 
 
 6317 
 
 80 
 
 545 
 
 6397 
 
 6476 
 
 6556 
 
 6635 
 
 6715 
 
 6795 
 
 6874 
 
 6954 
 
 7034 
 
 7113 
 
 80 
 
 546 
 
 7193 
 
 7272 
 
 7352 
 
 7431 
 
 7511 
 
 7590 
 
 7670 
 
 7749 
 
 7829 
 
 7908 
 
 79 
 
 547 
 
 7987 
 
 8067 
 
 8146 
 
 8225 
 
 8305 
 
 8384 
 
 8463 
 
 8543 
 
 8622 
 
 8701 
 
 79 
 
 548 
 
 8781 
 
 8860 
 
 8939 
 
 9018 
 
 9097 
 
 9177 
 
 9256 
 
 9335 
 
 9414 
 
 9493 
 
 79 
 
 549 
 
 * 9572 
 
 9651 
 
 9731 
 
 9810 
 
 9889 
 
 9968 
 
 *047 
 
 0126 
 
 0205 
 
 0284 
 
 79 
 
 550 
 
 74 0363 
 
 0442 
 
 0521 
 
 0600 
 
 0678 
 
 0757 
 
 0836 
 
 0915 
 
 0994 
 
 1073 
 
 79 
 
 551 
 
 1152 
 
 1230 
 
 1309 
 
 1388 
 
 1467 
 
 1546 
 
 1624 
 
 1703 
 
 1782 
 
 1860 
 
 79 
 
 552 
 
 1939 
 
 2018 
 
 2096 
 
 2175 
 
 2254 
 
 2332 
 
 2411 
 
 2489 
 
 2568 
 
 2646 
 
 79 
 
 553 
 
 2725 
 
 2804 
 
 2882 
 
 2961 
 
 3039 
 
 3118 
 
 3196 
 
 3275 
 
 3353 
 
 3431 
 
 78 
 
 554 
 
 3510 
 
 3588 
 
 3667 
 
 3745 
 
 3823 
 
 3902 
 
 3980 
 
 4058 
 
 4136 
 
 4215 
 
 78 
 
 555 
 
 4293 
 
 4371 
 
 4449 
 
 4528 
 
 4606 
 
 4684 
 
 4762 
 
 4840 
 
 4919 
 
 4997 
 
 78 
 
 556 
 
 5075 
 
 5153 
 
 5231 
 
 5309 
 
 5387 
 
 5465 
 
 5543 
 
 5621 
 
 5699 
 
 5777 
 
 78 
 
 557 
 
 5855 
 
 5933 
 
 6011 
 
 6089 
 
 6167 
 
 6245 
 
 6323 
 
 6401 
 
 6479 
 
 6556 
 
 78 
 
 558 
 
 6634 
 
 6712 
 
 6790 
 
 6868 
 
 6945 
 
 7023 
 
 7101 
 
 7179 
 
 7256 
 
 7334 
 
 78 
 
 559 
 
 7412 
 
 7489 
 
 7567 
 
 7645 
 
 7722 
 
 7800 
 
 7878 
 
 7955 
 
 8033 
 
 8110 
 
 78 
 
 560 
 
 8188 
 
 8266 
 
 8343 
 
 8421 
 
 8498 
 
 8576 
 
 8653 
 
 8731 
 
 8808 
 
 8885 
 
 77 
 
 561 
 
 8963 
 
 9040 
 
 9118 
 
 9195 
 
 9272 
 
 9350 
 
 9427 
 
 9504 
 
 9582 
 
 9659 
 
 77 
 
 562 
 
 *9736 
 
 9814 
 
 9891 
 
 9968 
 
 *045 
 
 0123 
 
 0200 
 
 0277 
 
 0354 
 
 0431 
 
 77 
 
 563 
 
 75 0508 
 
 0586 
 
 0663 
 
 0740 
 
 0817 
 
 0894 
 
 0971 
 
 1048 
 
 1125 
 
 1202 
 
 77 
 
 564 
 
 1279 
 
 1356 
 
 1433 
 
 1510 
 
 1587 
 
 1664 
 
 1741 
 
 1818 
 
 1895 
 
 1972 
 
 77 
 
 565 
 
 2048 
 
 2125 
 
 2202 
 
 2279 
 
 2356 
 
 2433 
 
 2509 
 
 2586 
 
 2663 
 
 2740 
 
 77 
 
 566 
 
 2816 
 
 2893 
 
 2970 
 
 3047 
 
 3123 
 
 3200 
 
 3277 
 
 3353 
 
 3430 
 
 3506 
 
 77 
 
 567 
 
 3583 
 
 3660 
 
 3736 
 
 3813 
 
 3889 
 
 3966 
 
 4042 
 
 4119 
 
 4195 
 
 4272 
 
 77 
 
 568 
 
 4348 
 
 4425 
 
 4501 
 
 4578 
 
 4654 
 
 4730 
 
 4807 
 
 4883 
 
 4960 
 
 5036 
 
 76 
 
 569 
 
 5112 
 
 5189 
 
 5265 
 
 5341 
 
 5417 
 
 5494 
 
 5570 
 
 5646 
 
 5722 
 
 5799 
 
 76 
 
 570 
 
 5875 
 
 5951 
 
 6027 
 
 6103 
 
 6180 
 
 6256 
 
 6332 
 
 6408 
 
 6484 
 
 6560 
 
 76 
 
 571 
 
 6636 
 
 6712 
 
 6788 
 
 6864 
 
 6940 
 
 7016 
 
 7092 
 
 7168 
 
 7244 
 
 7320 
 
 76 
 
 572 
 
 7396 
 
 7472 
 
 7548 
 
 7624 
 
 7700 
 
 7775 
 
 7851 
 
 7927 
 
 8003 
 
 8079 
 
 76 
 
 573 
 
 8155 
 
 8230 
 
 8306 
 
 8382 
 
 8458 
 
 8533 
 
 8609 
 
 8685 
 
 8761 
 
 8836 
 
 76 
 
 574 
 
 8912 
 
 8988 
 
 9063 
 
 9139 
 
 9214 
 
 9290 
 
 9366 
 
 9441 
 
 9517 
 
 9592 
 
 76 
 
 575 
 
 * 9668 
 
 9743 
 
 9819 
 
 9894 
 
 9970 
 
 4045 
 
 0121 
 
 0196 
 
 0272 
 
 0347 
 
 75 
 
 576 
 
 76 0422 
 
 0498 
 
 0573 
 
 0649 
 
 0724 
 
 0799 
 
 0875 
 
 0950 
 
 1025 
 
 1101 
 
 75 
 
 577 
 
 1176 
 
 1251 
 
 1326 
 
 1402 
 
 1477 
 
 1552 
 
 1627 
 
 1702 
 
 1778 
 
 1853 
 
 75 
 
 578 
 
 1928 
 
 2003 
 
 2078 
 
 2153 
 
 2228 
 
 2303 
 
 2378 
 
 2453 
 
 2529 
 
 2604 
 
 75 
 
 579 
 
 2679 
 
 2754 
 
 2829 
 
 2904 
 
 2978 
 
 3053 
 
 3128 
 
 3203 
 
 3278 
 
 3353 
 
 75 
 
 3V. 
 
 O 
 
 1 
 
 9 
 
 3 
 
 A 
 
 5 
 
 e 
 
 7 
 
 
 
 O 
 
 r>. 
 
APPENDIX. 
 
 853 
 
 LOGARITHMS OF NUMBERS. 
 
 IV . 
 
 o 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 G 
 
 7 
 
 & 
 
 9 
 
 r>. 
 
 580 
 
 76 3428 
 
 3503 
 
 3578 
 
 3653 
 
 3727 
 
 3802 
 
 3877 
 
 3952 
 
 4027 
 
 4101 
 
 75 
 
 581 
 
 4176 
 
 4251 
 
 4326 
 
 4400 
 
 4475 
 
 4550 
 
 4624 
 
 4699 
 
 4774 
 
 4848 
 
 75 
 
 582 
 
 4923 
 
 4998 
 
 5072 
 
 5147 
 
 5221 
 
 5296 
 
 5370 
 
 5445 
 
 5520 
 
 5594 
 
 75 
 
 583 
 
 5669 
 
 5743 
 
 5818 
 
 5892 
 
 5966 
 
 6041 
 
 6115 
 
 6190 
 
 6264 
 
 6338 
 
 74 
 
 584 
 
 6413 
 
 6487 
 
 6562 
 
 6636 
 
 6710 
 
 6785 
 
 6859 
 
 6933 
 
 7007 
 
 7082 
 
 74 
 
 585 
 
 7156 
 
 7230 
 
 7304 
 
 7379 
 
 7453 
 
 7527 
 
 7601 
 
 7675 
 
 7749 
 
 7823 
 
 74 
 
 586 
 
 7898 
 
 7972 
 
 8046 
 
 8120 
 
 8194 
 
 8268 
 
 8342 
 
 8416 
 
 8490 
 
 8564 
 
 74 
 
 587 
 
 8638 
 
 8712 
 
 8786 
 
 8860 
 
 8934 
 
 9008 
 
 9082 
 
 9156 
 
 9230 
 
 9303 
 
 74 
 
 588 
 
 *9377 
 
 9451 
 
 9525 
 
 9599 
 
 9673 
 
 9746 
 
 9820 
 
 9894 
 
 9968 
 
 *042 
 
 74 
 
 589 
 
 77 0115 
 
 0189 
 
 0263 
 
 0336 
 
 0410 
 
 0484 
 
 0557 
 
 0631 
 
 0705 
 
 0778 
 
 74 
 
 590 
 
 0852 
 
 0926 
 
 0999 
 
 1073 
 
 1146 
 
 1220 
 
 1293 
 
 1367 
 
 1440 
 
 1514 
 
 74 
 
 591 
 
 1587 
 
 1661 
 
 1734 
 
 1808 
 
 1881 
 
 1955 
 
 2028 
 
 2102 
 
 2175 
 
 2248 
 
 73 
 
 592 
 
 2322 
 
 2395 
 
 2468 
 
 2542 
 
 2615 
 
 2688 
 
 2762 
 
 2835 
 
 2908 
 
 2981 
 
 73 
 
 593 
 
 3055 
 
 3128 
 
 3201 
 
 3274 
 
 3348 
 
 3421 
 
 3494 
 
 3567 
 
 3640 
 
 3713 
 
 73 
 
 594 
 
 3786 
 
 3860 
 
 3933 
 
 4006 
 
 4079 
 
 4152 
 
 4225 
 
 4298 
 
 4371 
 
 4444 
 
 73 
 
 595 
 
 4517 
 
 4590 
 
 4663 
 
 4736 
 
 4809 
 
 4882 
 
 4955 
 
 5028 
 
 5100 
 
 5173 
 
 73 
 
 596 
 
 5246 
 
 5319 
 
 5392 
 
 5465 
 
 5538 
 
 5610 
 
 5683 
 
 5756 
 
 5829 
 
 5902 
 
 73 
 
 597 
 
 5974 
 
 6047 
 
 6120 
 
 6193 
 
 6265 
 
 6338 
 
 6411 
 
 6483 
 
 6556 
 
 6629 
 
 73 
 
 598 
 
 6701 
 
 6774 
 
 6846 
 
 6919 
 
 6992 
 
 7064 
 
 7137 
 
 7209 
 
 7282 
 
 7354 
 
 73 
 
 599 
 
 7427 
 
 7499 
 
 7572 
 
 7644 
 
 7717 
 
 7789 
 
 7862 
 
 7934 
 
 8006 
 
 8079 
 
 72 
 
 600 
 
 8151 
 
 8224 
 
 8296 
 
 8368 
 
 8441 
 
 8513 
 
 8585 
 
 8658 
 
 8730 
 
 8802 
 
 72 
 
 601 
 
 8874 
 
 8947 
 
 9019 
 
 9091 
 
 9163 
 
 9236 
 
 9308 
 
 9380 
 
 9452 
 
 9524 
 
 72 
 
 602 
 
 * 9596 
 
 9669 
 
 9741 
 
 9813 
 
 9885 
 
 9957 
 
 029 
 
 0101 
 
 0173 
 
 0245 
 
 72 
 
 603 
 
 78 0317 
 
 0389 
 
 0461 
 
 0533 
 
 0605 
 
 0677 
 
 0749 
 
 0821 
 
 0893 
 
 0965 
 
 72 
 
 604 
 
 1037 
 
 1109 
 
 1181 
 
 1253 
 
 1324 
 
 1396 
 
 1468 
 
 1540 
 
 1612 
 
 1684 
 
 72 
 
 605 
 
 1755 
 
 1827 
 
 1899 
 
 1971 
 
 2042 
 
 2114 
 
 2186 
 
 2258 
 
 2329 
 
 2401 
 
 72 
 
 606 
 
 2473 
 
 2544 
 
 2616 
 
 2688 
 
 2759 
 
 2831 
 
 2902 
 
 2974 
 
 3046 
 
 3117 
 
 72 
 
 607 
 
 3189 
 
 3260 
 
 3332 
 
 3403 
 
 3475 
 
 3546 
 
 3618 
 
 3689 
 
 3761 
 
 3832 
 
 71 
 
 608 
 
 3904 
 
 3975 
 
 4046 
 
 4118 
 
 4189 
 
 4261 
 
 4332 
 
 4403 
 
 4475 
 
 4546 
 
 71 
 
 609 
 
 4617 
 
 4689 
 
 4760 
 
 4831 
 
 4902 
 
 4974 
 
 5045 
 
 5116 
 
 5187 
 
 5259 
 
 71 
 
 610 
 
 5330 
 
 5401 
 
 5472 
 
 5543 
 
 5615 
 
 5686 
 
 5757 
 
 5828 
 
 5899 
 
 5970 
 
 71 
 
 611 
 
 6041 
 
 6112 
 
 6183 
 
 6254 
 
 6325 
 
 6396 
 
 6467 
 
 6538 
 
 6609 
 
 6680 
 
 71 
 
 612 
 
 6751 
 
 6822 
 
 6893 
 
 6964 
 
 7035 
 
 7106 
 
 7177 
 
 7248 
 
 7319 
 
 7390 
 
 71 
 
 613 
 
 7460 
 
 7531 
 
 7602 
 
 7673 
 
 7744 
 
 7815 
 
 7885 
 
 7956 
 
 8027 
 
 8098 
 
 71 
 
 614 
 
 8168 
 
 8239 
 
 8310 
 
 8381 
 
 8451 
 
 8522 
 
 8593 
 
 8663 
 
 8734 
 
 8804 
 
 71 
 
 615 
 
 8875 
 
 8946 
 
 9016 
 
 9087 
 
 9157 
 
 9228 
 
 9299 
 
 9369 
 
 9440 
 
 9510 
 
 71 
 
 616 
 
 *9581 
 
 9651 
 
 9722 
 
 9792 
 
 9863 
 
 9933 
 
 +004 
 
 0074 
 
 0144 
 
 0215 
 
 70 
 
 617 
 
 79 0285 
 
 0356 
 
 0426 
 
 0496 
 
 0567 
 
 0637 
 
 0707 
 
 0778 
 
 0848 
 
 0918 
 
 70 
 
 618 
 
 0988 1059 
 
 1129 
 
 1199 
 
 1269 
 
 1340 
 
 1410 
 
 1480 
 
 1550 
 
 1620 
 
 70 
 
 619 
 
 1691 
 
 1761 
 
 1831 
 
 1901 
 
 1971 
 
 2041 
 
 2111 
 
 2181 
 
 2252 
 
 2322 
 
 70 
 
 620 
 
 2392 
 
 2462 
 
 2532 
 
 2602 
 
 2672 
 
 2742 
 
 2812 
 
 2882 
 
 2952 
 
 3022 
 
 70 
 
 621 
 
 3092 3162 
 
 3231 
 
 3301 
 
 3371 
 
 3441 
 
 3511 
 
 3581 
 
 3651 
 
 3721 
 
 70 
 
 622 
 
 3790 
 
 3860 
 
 3930 
 
 4000 
 
 4070 
 
 4139 
 
 4209 
 
 4279 
 
 4349 
 
 4418 
 
 70 
 
 623 
 
 4488 
 
 4558 
 
 4627 
 
 4697 
 
 4767 
 
 4836 
 
 4906 
 
 4976 
 
 5045 
 
 5115 
 
 70 
 
 624 
 
 5185 
 
 5254 
 
 5324 
 
 5393 
 
 5463 
 
 5532 
 
 5602 
 
 5672 
 
 5741 
 
 5811 
 
 70 
 
 625 
 
 5880 
 
 5949 
 
 6019 
 
 6088 
 
 6158 
 
 6227 
 
 6297 
 
 6366 
 
 6436 
 
 6505 
 
 69 
 
 626 
 
 6574 
 
 6644 
 
 6713 
 
 6782 
 
 6852 
 
 6921 
 
 6990 
 
 7060 
 
 7129 
 
 7198 
 
 69 
 
 627 
 
 7268 
 
 7337 
 
 7406 
 
 7475 
 
 7545 
 
 7614 
 
 7683 
 
 7752 
 
 7821 
 
 7890 
 
 69 
 
 628 
 
 7960 
 
 8029 
 
 8098 
 
 8167 
 
 8236 
 
 8305 
 
 8374 
 
 8443 
 
 8513 
 
 8582 
 
 69 
 
 629 
 
 8651 
 
 8720 
 
 8789 
 
 8858 
 
 8927 
 
 8996 
 
 9065 
 
 9134 
 
 9203 
 
 9272 
 
 69 
 
 630 
 
 9341 
 
 9409 
 
 9478 
 
 9547 
 
 9616 
 
 9685 
 
 9754 
 
 9823 
 
 9892 
 
 9961 
 
 69 
 
 631 
 
 80 0029 
 
 0098 
 
 0167 
 
 0236 
 
 0305 
 
 0373 
 
 0442 
 
 0511 
 
 0580 
 
 0648 
 
 69 
 
 632 
 
 0717 
 
 0786 
 
 0854 
 
 0923 
 
 0992 
 
 1061 
 
 1129 
 
 1198 
 
 1266 
 
 1335 
 
 69 
 
 633 
 
 1404 
 
 1472 
 
 1541 
 
 1609 
 
 1678 
 
 1747 
 
 1815 
 
 1884 
 
 1952 
 
 2021 
 
 69 
 
 634 
 
 2089 
 
 2158 
 
 2226 
 
 2295 
 
 2363 
 
 2432 
 
 2500 
 
 2568 
 
 2637 
 
 2705 
 
 69 
 
 635 
 
 2774 
 
 2842 
 
 2910 
 
 2979 
 
 3047 
 
 3116 
 
 3184 
 
 3252 
 
 3321 
 
 3389 
 
 68 
 
 636 
 
 3457 
 
 3525 
 
 3594 
 
 3662 
 
 3730 
 
 3798 
 
 3867 
 
 3935 
 
 4003 
 
 4071 
 
 68 
 
 637 
 
 4139 
 
 4208 
 
 4276 
 
 4344 
 
 4412 
 
 4480 
 
 4548 
 
 4616 
 
 4685 
 
 4753 
 
 68 
 
 638 
 
 4821 
 
 4889 
 
 4957 i 5025 
 
 5093 
 
 5161 
 
 5229 
 
 5297 
 
 5365 
 
 5433 
 
 68 
 
 639 
 
 5501 
 
 5569 
 
 5637 5705 
 
 5773 
 
 5841 
 
 5908 
 
 5976 
 
 6044 
 
 6112 
 
 68 
 
 IV. 
 
 O 
 
 133 
 
 4 
 
 5 
 
 O 
 
 9 
 
 8 
 
 
 
 r>. 
 
854: 
 
 APPENDIX. 
 
 LOGARITHMS OF NUMBERS. 
 
 N. 
 
 o 
 
 1 
 
 3 
 
 3 
 
 4, 
 
 5 
 
 7 
 
 8 
 
 9 
 
 D. 
 
 640 
 
 80 6180 
 
 6248 
 
 6316 
 
 6384 
 
 6451 
 
 6519 
 
 6587 
 
 6655 
 
 6723 
 
 6790 
 
 68 
 
 641 
 
 6858 
 
 6926 
 
 6994 
 
 7061 
 
 7129 
 
 7197 
 
 7264 
 
 7332 
 
 7400 
 
 7467 
 
 68 
 
 642 
 
 7535 
 
 7603 
 
 7670 
 
 7738 
 
 7806 
 
 7873 
 
 7941 
 
 8008 
 
 8076 
 
 8143 
 
 68 
 
 643 
 
 8211 
 
 8279 
 
 8346 
 
 8414 
 
 8481 
 
 8549 
 
 8616 
 
 8684 
 
 8751 
 
 8818 
 
 67 
 
 644 
 
 8886 
 
 8953 
 
 9021 
 
 9088 
 
 9156 
 
 9223 
 
 9290 
 
 9358 
 
 9425 
 
 9492 
 
 67 
 
 645 
 
 * 9560 
 
 9627 
 
 9694 
 
 9762 
 
 9829 
 
 9896 
 
 9964 
 
 +031 
 
 0098 
 
 0165 
 
 67 
 
 646 
 
 81 0233 
 
 0300 
 
 0367 
 
 0434 
 
 0501 
 
 0569 
 
 0636 
 
 0703 
 
 0770 
 
 0837 
 
 67 
 
 647 
 
 0904 
 
 0971 
 
 1039 
 
 1106 
 
 1173 
 
 1240 
 
 1307 
 
 1374 
 
 1441 
 
 1508 
 
 67 
 
 648 
 
 1575 
 
 1642 
 
 1709 
 
 1776 
 
 1843 
 
 1910 
 
 1977 
 
 2044 
 
 2111 
 
 2178 
 
 67 
 
 649 
 
 2245 
 
 2312 
 
 2379 
 
 2445 
 
 2512 
 
 2579 
 
 2646 
 
 2713 
 
 2780 
 
 2847 
 
 67 
 
 650 
 
 2913 
 
 2980 
 
 3047 
 
 3114 
 
 3181 
 
 3247 
 
 3314 
 
 3381 
 
 3448 
 
 3514 
 
 67 
 
 651 
 
 3581 
 
 3648 
 
 3714 
 
 3781 
 
 3848 
 
 3914 
 
 3981 
 
 4048 
 
 4114 
 
 4181 
 
 67 
 
 652 
 
 4248 
 
 4314 
 
 4381 
 
 4447 
 
 4514 
 
 4581 
 
 4647 
 
 4714 
 
 4780 
 
 4847 
 
 67 
 
 653 
 
 4913 
 
 4980 
 
 5046 
 
 5113 
 
 5179 
 
 5246 
 
 5312 
 
 5378 
 
 5445 
 
 5511 
 
 66 
 
 654 
 
 5578 
 
 5644 
 
 5711 
 
 5777 
 
 5843 
 
 5910 
 
 5976 
 
 6042 
 
 6109 
 
 6175 
 
 66 
 
 655 
 
 6241 
 
 6308 
 
 6374 
 
 6440 
 
 6506 
 
 6573 
 
 6639 
 
 6705 
 
 6771 
 
 6838 
 
 66 
 
 656 
 
 6904 
 
 6970 
 
 7036 
 
 7102 
 
 7169 
 
 7235 
 
 7301 
 
 7367 
 
 7433 
 
 7499 
 
 66 
 
 657 
 
 7565 
 
 7631 
 
 7698 
 
 7764 
 
 7830 
 
 7896 
 
 7962 
 
 8028 
 
 8094 
 
 8160 
 
 66 
 
 658 
 
 8226 
 
 8292 
 
 8358 
 
 8424 
 
 8490 
 
 8556 
 
 8622 
 
 8688 
 
 8754 
 
 8820 
 
 66 
 
 659 
 
 8885 
 
 8951 
 
 9017 
 
 9083 
 
 9149 
 
 9215 
 
 9281 
 
 9346 
 
 9412 
 
 9478 
 
 66 
 
 660 
 
 * 9544 
 
 9610 
 
 9676 
 
 9741 
 
 9807 
 
 9873 
 
 9939 
 
 +004 
 
 0070 
 
 0136 
 
 66 
 
 661 
 
 82 0201 
 
 0267 
 
 0333 
 
 0399 
 
 0464 
 
 0530 
 
 0595 
 
 0661 
 
 0727 
 
 0792 
 
 66 
 
 662 
 
 0858 
 
 0924 
 
 0989 
 
 1055 
 
 1120 
 
 1186 
 
 1251 
 
 1317 
 
 1382 
 
 1448 
 
 66 
 
 663 
 
 1514 
 
 1579 
 
 1645 
 
 1710 
 
 1775 
 
 1841 
 
 1906 
 
 1972 
 
 2037 
 
 2103 
 
 65 
 
 664 
 
 2168 
 
 2233 
 
 2299 
 
 2364 
 
 2430 
 
 2495 
 
 2560 
 
 2626 
 
 2691 
 
 2756 
 
 65 
 
 665 
 
 2822 
 
 2887 
 
 2952 
 
 3018 
 
 3083 
 
 3148 
 
 3213 
 
 3279 
 
 3344 
 
 3409 
 
 65 
 
 666 
 
 3474 
 
 3539 
 
 3605 
 
 3670 
 
 3735 
 
 3800 
 
 3865 
 
 3930 
 
 3996 
 
 4061 
 
 65 
 
 667 
 
 4126 
 
 4191 
 
 4256 
 
 4321 
 
 4386 
 
 4451 
 
 4516 
 
 4581 
 
 4646 
 
 4711 
 
 65 
 
 668 
 
 4776 
 
 4841 
 
 4906 
 
 4971 
 
 5036 
 
 5101 
 
 5166 
 
 5231 
 
 5296 
 
 5361 
 
 65 
 
 669 
 
 5426 
 
 5491 
 
 5556 
 
 5621 
 
 5686 
 
 5751 
 
 5815 
 
 5880 
 
 5945 
 
 6010 
 
 65 
 
 670 
 
 6075 
 
 6140 
 
 6204 
 
 6269 
 
 6334 
 
 6399 
 
 6464 
 
 6528 
 
 6593 
 
 6658 
 
 65 
 
 671 
 
 6723 
 
 6787 
 
 6852 
 
 6917 
 
 6981 
 
 7046 
 
 7111 
 
 7175 
 
 7240 
 
 7305 
 
 65 
 
 672 
 
 7369 
 
 7434 
 
 7499 
 
 7563 
 
 7628 
 
 7692 
 
 7757 
 
 7821 
 
 7886 
 
 7951 
 
 65 
 
 673 
 
 8015 
 
 8080 
 
 8144 
 
 8209 
 
 8273 
 
 8338 
 
 8402 
 
 8467 
 
 8531 
 
 8595 
 
 64 
 
 674 
 
 8660 
 
 8724 
 
 8789 
 
 8853 
 
 8918 
 
 8982 
 
 9046 
 
 9111 
 
 9175 
 
 9239 
 
 64 
 
 675 
 
 9304 
 
 9368 
 
 9432 
 
 9497 
 
 9561 
 
 9625 
 
 9690 
 
 9754 
 
 9818 
 
 9882 
 
 64 
 
 676 
 
 * 9947 
 
 +011 
 
 0075 
 
 0139 
 
 0204 
 
 0268 
 
 0332 
 
 0396 
 
 0460 
 
 0525 
 
 64 
 
 677 
 
 83 0589 
 
 0653 
 
 0717 
 
 0781 
 
 0845 
 
 0909 
 
 0973 
 
 1037 
 
 1102 
 
 1166 
 
 64 
 
 678 
 
 1230 
 
 1294 
 
 1358 
 
 1422 
 
 1486 
 
 1550 
 
 1614 
 
 1678 
 
 1742 
 
 1806 
 
 64 
 
 679 
 
 1870 
 
 1934 
 
 1998 
 
 2062 
 
 2126 
 
 2189 
 
 2253 
 
 2317 
 
 2381 
 
 2445 
 
 64 
 
 680 
 
 2509 
 
 2573 
 
 2637 
 
 2700 
 
 2764 
 
 2828 
 
 2892 
 
 2956 
 
 3020 
 
 3083 
 
 64 
 
 681 
 
 3147 
 
 3211 
 
 3275 
 
 3338 
 
 3402 
 
 3466 
 
 3530 
 
 3593 
 
 3657 
 
 3721 
 
 64 
 
 682 
 
 3784 
 
 3848 
 
 3912 
 
 3975 
 
 4039 
 
 4103 
 
 4166 
 
 4230 
 
 4294 
 
 4357 
 
 64 
 
 683 
 
 4421 
 
 4484 
 
 4548 
 
 4611 
 
 4675 
 
 4739 
 
 4802 
 
 4866 
 
 4929 
 
 4993 
 
 64 
 
 684 
 
 5056 
 
 5120 
 
 5183 
 
 5247 
 
 5310 
 
 5373 
 
 5437 
 
 5500 
 
 5564 
 
 5627 
 
 63 
 
 685 
 
 5691 
 
 5754 
 
 5817 
 
 5881 
 
 5944 
 
 6007 
 
 6071 
 
 6134 
 
 6197 
 
 6261 
 
 63 
 
 686 
 
 6324 
 
 6387 
 
 6451 
 
 6514 
 
 6577 
 
 6641 
 
 6704 
 
 6767 
 
 6830 
 
 6894 
 
 63 
 
 687 
 
 6957 
 
 7020 
 
 7083 
 
 7146 
 
 7210 
 
 7273 
 
 7336 
 
 7399 
 
 7462 
 
 7525 
 
 63 
 
 688 
 
 7588 
 
 7652 
 
 7715 
 
 7778 
 
 7841 
 
 7904 
 
 7967 
 
 8030 
 
 8093 
 
 8156 
 
 63 
 
 689 
 
 8219 
 
 8282 
 
 8345 
 
 8408 
 
 8471 
 
 8534 
 
 8597 
 
 8660 
 
 8723 
 
 8786 
 
 63 
 
 690 
 
 8849 
 
 8912 
 
 8975 
 
 9038 
 
 9101 
 
 9164 
 
 9227 
 
 9289 
 
 9352 
 
 9415 
 
 63 
 
 691 
 
 * 9478 
 
 9541 
 
 9604 
 
 9667 
 
 9729 
 
 9792 
 
 9855 
 
 9918 
 
 9981 
 
 +043 
 
 63 
 
 692 
 
 84 0106 
 
 0169 
 
 0232 
 
 0294 
 
 0357 
 
 0420 
 
 0482 
 
 0545 
 
 0608 
 
 0671 
 
 63 
 
 693 
 
 0733 
 
 0796 
 
 0859 
 
 0921 
 
 0984 
 
 1046 
 
 1109 
 
 1172 
 
 1234 
 
 1297 
 
 63 
 
 694 
 
 1359 
 
 1422 
 
 1485 
 
 1547 
 
 1610 
 
 1672 
 
 1735 
 
 1797 
 
 1860 
 
 1922 
 
 63 
 
 695 
 
 1985 
 
 2047 
 
 2110 
 
 2172 
 
 2235 
 
 2297 
 
 2360 
 
 2422 
 
 2484 
 
 2547 
 
 62 
 
 696 
 
 2609 
 
 2672 
 
 2734 
 
 2796 
 
 2859 
 
 2921 
 
 2983 
 
 3046 
 
 3108 
 
 3170 
 
 62 
 
 697 
 
 3233 
 
 3295 
 
 3357 
 
 3420 
 
 3482 
 
 3544 
 
 3606 
 
 3669 
 
 3731 
 
 3793 
 
 62 
 
 698 
 
 3855 
 
 3918 
 
 3980 
 
 4042 
 
 4104 
 
 4166 
 
 4229 
 
 4291 
 
 4353 
 
 4415 
 
 62 
 
 699 
 
 4477 
 
 4539 
 
 4601 
 
 4664 
 
 4726 
 
 4788 
 
 4850 
 
 4912 
 
 4974 
 
 5036 
 
 62 
 
 IV. 
 
 
 
 1 
 
 3 
 
 3 
 
 4, 
 
 5 
 
 G 
 
 7 
 
 * 
 
 O 
 
 I>. 
 
APPENDIX. 
 
 855 
 
 LOGARITHMS OF NUMBERS.! 
 
 IV. 
 
 <> 
 
 1 
 
 3 
 
 3 
 
 4, 
 
 5 
 
 O 
 
 7 
 
 8 
 
 9 
 
 r>. 
 
 700 
 
 84 5098 
 
 5160 
 
 5222 
 
 5284 
 
 5346 
 
 5408 
 
 5470 
 
 5532 
 
 5594 
 
 5656 
 
 62 
 
 701 
 
 5718 
 
 5780 
 
 5842 
 
 5904 
 
 5966 
 
 6028 
 
 6090 
 
 6151 
 
 6213 
 
 6275 
 
 62 
 
 702 
 
 6337 
 
 6399 
 
 6461 
 
 6523 
 
 6585 
 
 6646 
 
 6708 
 
 6770 
 
 6832 
 
 6894 
 
 62 
 
 703 
 
 6955 
 
 7017 
 
 7079 
 
 7141 
 
 7202 
 
 7264 
 
 7326 
 
 7388 
 
 7449 
 
 7511 
 
 62 
 
 704 
 
 7573 
 
 7634 
 
 7696 
 
 7758 
 
 7819 
 
 7881 
 
 7943 
 
 8004 
 
 8066 
 
 8128 
 
 62 
 
 705 
 
 8189 
 
 8251 
 
 8312 
 
 8374 
 
 8435 
 
 8497 
 
 8559 
 
 8620 
 
 8682 
 
 8743 
 
 62 
 
 706 
 
 8805 
 
 8866 
 
 8928 
 
 8989 
 
 9051 
 
 9112 
 
 9174 
 
 9235 
 
 9297 
 
 9358 
 
 61 
 
 707 
 
 9419 
 
 9481 
 
 9542 
 
 9604 
 
 9665 
 
 9726 
 
 9788 
 
 9849 
 
 9911 
 
 9972 
 
 61 
 
 708 
 
 85 0033 
 
 0095 
 
 0156 
 
 0217 
 
 0279 
 
 0340 
 
 0401 
 
 0462 
 
 0524 
 
 0585 
 
 61 
 
 709 
 
 0646 
 
 0707 
 
 0769 
 
 0830 
 
 0891 
 
 0952 
 
 1014 
 
 1075 
 
 1136 
 
 1197 
 
 61 
 
 710 
 
 1258 
 
 1320 
 
 1381 
 
 1442 
 
 1503 
 
 1564 
 
 1625 
 
 1686 
 
 1747 
 
 1809 
 
 61 
 
 711 
 
 1870 
 
 1931 
 
 1992 
 
 2053 
 
 2114 
 
 2175 
 
 2236 
 
 2297 
 
 2358 
 
 2419 
 
 61 
 
 712 
 
 2480 
 
 2541 
 
 2602 
 
 2663 
 
 2724 
 
 2785 
 
 2846 
 
 2907 
 
 2968 
 
 3029 
 
 61 
 
 713 
 
 3090 
 
 3150 
 
 3211 
 
 3272 
 
 3333 
 
 3394 
 
 3455 
 
 3516 
 
 3577 
 
 3637 
 
 61 
 
 714 
 
 3698 
 
 3759 
 
 3820 
 
 3881 
 
 3941 
 
 4002 
 
 4063 
 
 4124 
 
 4185 
 
 4245 
 
 61 
 
 715 
 
 4306 
 
 4367 
 
 4428 
 
 4488 
 
 4549 
 
 4610 
 
 4670 
 
 4731 
 
 4792 
 
 4852 
 
 61 
 
 716 
 
 4913 
 
 4974 
 
 5034 
 
 5095 
 
 5156 
 
 5216 
 
 5277 
 
 5337 
 
 5398 
 
 5459 
 
 61 
 
 717 
 
 5519 
 
 5580 
 
 5640 
 
 5701 
 
 5761 
 
 5822 
 
 5882 
 
 5943 
 
 6003 
 
 6064 
 
 61 
 
 718 
 
 6124 
 
 6185 
 
 6245 
 
 6306 
 
 6366 
 
 6427 
 
 6487 
 
 6548 
 
 6608 
 
 6668 
 
 60 
 
 719 
 
 6729 
 
 6789 
 
 6850 
 
 6910 
 
 6970 
 
 7031 
 
 7091 
 
 7152 
 
 7212 
 
 7272 
 
 60 
 
 720 
 
 7332 
 
 7393 
 
 7453 
 
 7513 
 
 7574 
 
 7634 
 
 7694 
 
 7755 
 
 7815 
 
 7875 
 
 60 
 
 721 
 
 7935 
 
 7995 
 
 8056 
 
 8116 
 
 8176 
 
 8236 
 
 8297 
 
 8357 
 
 8417 
 
 8477 
 
 60 
 
 722 
 
 8537 
 
 8597 
 
 8657 
 
 8718 
 
 8778 
 
 8838 
 
 8898 
 
 8958 
 
 9018 
 
 9078 
 
 60 
 
 723 
 
 9138 
 
 9198 
 
 9258 
 
 9318 
 
 9379 
 
 9439 
 
 9499 
 
 9559 
 
 9619 
 
 9679 
 
 60 
 
 724 
 
 * 9739 
 
 9799 
 
 9859 
 
 9918 
 
 9978 
 
 +038 
 
 0098 
 
 0158 
 
 0218 
 
 0278 
 
 60 
 
 725 
 
 86 0338 
 
 0398 
 
 0458 
 
 0518 
 
 0578 
 
 0637 
 
 0697 
 
 0757 
 
 0817 
 
 0877 
 
 60 
 
 726 
 
 0937 
 
 0996 
 
 1056 
 
 1116 
 
 1176 
 
 1236 
 
 1295 
 
 1355 
 
 1415 
 
 1475 
 
 60 
 
 727 
 
 1534 
 
 1594 
 
 1654 
 
 1714 
 
 1773 
 
 1833 
 
 1893 
 
 1952 
 
 2012 
 
 2072 
 
 60 
 
 728 
 
 2131 
 
 2191 
 
 2251 
 
 2310 
 
 2370 
 
 2430 
 
 2489 
 
 2549 
 
 2608 
 
 2668 
 
 60 
 
 729 
 
 2728 
 
 2787 
 
 2847 
 
 2906 
 
 2966 
 
 3025 
 
 3085 
 
 3144 
 
 3204 
 
 3263 
 
 60 
 
 730 
 
 3323 
 
 3382 
 
 3442 
 
 3501 
 
 3561 
 
 3620 
 
 3680 
 
 3739 
 
 3799 
 
 3858 
 
 59 
 
 731 
 
 3917 
 
 3977 
 
 4036 
 
 4096 
 
 4155 
 
 4214 
 
 4274 
 
 4333 
 
 4392 
 
 4452 
 
 59 
 
 732 
 
 4511 
 
 4570 
 
 4630 
 
 4689 
 
 4748 
 
 4808 
 
 4867 
 
 4926 
 
 4985 
 
 5045 
 
 59 
 
 733 
 
 5104 
 
 5163 
 
 5222 
 
 5282 
 
 5341 
 
 5400 
 
 5459 
 
 5519 
 
 5578 
 
 5637 
 
 59 
 
 734 
 
 5696 
 
 5755 
 
 5814 
 
 5874 
 
 5933 
 
 5992 
 
 6051 
 
 6110 
 
 6169 
 
 6228 
 
 59 
 
 735 
 
 6287 
 
 6346 
 
 6405 
 
 6465 
 
 6524 
 
 6583 
 
 6642 
 
 6701 
 
 6760 
 
 6819 
 
 59 
 
 736 
 
 6878 
 
 6937 
 
 6996 
 
 7055 
 
 7114 
 
 7173 
 
 7232 
 
 7291 
 
 7350 
 
 7409 
 
 59 
 
 737 
 
 7467 
 
 7526 
 
 7585 
 
 7644 
 
 7703 
 
 7762 
 
 7821 
 
 7880 
 
 7939 
 
 7998 
 
 59 
 
 738 
 
 8056 
 
 8115 
 
 8174 
 
 8233 
 
 8292 
 
 8350 
 
 8409 
 
 8468 
 
 8527 
 
 8586 
 
 59 
 
 739 
 
 8644 
 
 8703 
 
 8762 
 
 8821 
 
 8879 
 
 8938 
 
 8997 
 
 9056 
 
 9114 
 
 9173 
 
 59 
 
 740 
 
 9232 
 
 9290 
 
 9349 
 
 9408 
 
 9466 
 
 9525 
 
 9584 
 
 9642 
 
 9701 
 
 9760 
 
 59 
 
 741 
 
 * 9818 
 
 9877 
 
 9935 
 
 9994 
 
 +053 
 
 0111 
 
 0170 
 
 0228 
 
 0287 
 
 0345 
 
 59 
 
 742 
 
 87 0404 
 
 0462 
 
 0521 
 
 0579 
 
 0638 
 
 0696 
 
 0755 
 
 0813 
 
 0872 
 
 0930 
 
 58 
 
 743 
 
 0989 
 
 1047 
 
 1106 
 
 1164 
 
 1223 
 
 1281 
 
 1339 
 
 1398 
 
 1456 
 
 1515 
 
 58 
 
 744 
 
 1573 
 
 1631 
 
 1690 
 
 1748 
 
 1806 
 
 1865 
 
 1923 
 
 1981 
 
 2040 
 
 2098 
 
 58 
 
 745 
 
 2156 
 
 2215 
 
 2273 
 
 2331 
 
 2389 
 
 2448 
 
 2506 
 
 2564 
 
 2622 
 
 2681 
 
 58 
 
 746 
 
 2739 
 
 2797 
 
 2855 
 
 2913- 
 
 2972 
 
 3030 
 
 3088 
 
 3146 
 
 3204 
 
 3262 
 
 58 
 
 747 
 
 3321 
 
 3379 
 
 3437 
 
 3495 
 
 3553 
 
 3611 
 
 3669 
 
 3727 
 
 3785 
 
 3844 
 
 58 
 
 748 
 
 3902 
 
 3960 
 
 4018 
 
 4076 
 
 4134 
 
 4192 
 
 4250 
 
 4308 
 
 4366 
 
 4424 
 
 58 
 
 749 
 
 4482 
 
 4540 
 
 4598 
 
 .4656 
 
 4714 
 
 4772 
 
 4830 
 
 4888 
 
 4945 
 
 5003 
 
 58 
 
 750 
 
 5061 
 
 5119 
 
 5177 
 
 5235 
 
 5293 
 
 5351 
 
 5409 
 
 5466 
 
 5524 
 
 5582 
 
 58 
 
 751 
 
 5640 
 
 5698 
 
 5756 
 
 5813 
 
 5871 
 
 5929 
 
 5987 
 
 6045 
 
 6102 
 
 6160 
 
 58 
 
 752 
 
 6218 
 
 6276 
 
 6333 
 
 6391 
 
 6449 
 
 6507 
 
 6564 
 
 6622 
 
 6680 
 
 6737 
 
 58 
 
 753 
 
 6795 
 
 6853 
 
 6910 
 
 6968 
 
 7026 
 
 7083 
 
 7141 
 
 7199 
 
 7256 
 
 7314 
 
 58 
 
 754 
 
 7371 
 
 7429 
 
 7487 
 
 7544 
 
 7602 
 
 7659 
 
 7717 
 
 7774 
 
 7832 
 
 7889 
 
 58 
 
 755 
 
 7947 
 
 8004 
 
 8062 
 
 8119 
 
 8177 
 
 8234 
 
 8292 
 
 8349 
 
 8407 
 
 8464 
 
 57 
 
 756 
 
 8522 
 
 8579 
 
 8637 
 
 8694 
 
 8752 
 
 8809 
 
 8866 
 
 8924 
 
 8981 
 
 9039 
 
 57 
 
 757 
 
 9096 
 
 9153 
 
 9211 
 
 9268 
 
 9325 
 
 9383 
 
 9440 
 
 9497 
 
 9555 
 
 9612 
 
 57 
 
 758 
 
 * 9669 
 
 9726 
 
 9784 
 
 9841 
 
 9898 
 
 9956 
 
 +013 
 
 0070 
 
 0127 
 
 0185 
 
 57 
 
 759 
 
 88 0242 
 
 0299 
 
 0356 
 
 0413 
 
 0471 
 
 0528 
 
 0585 
 
 0642 
 
 0699 
 
 0756 
 
 57 
 
 IV. 
 
 
 
 1 
 
 3 
 
 3 
 
 * 
 
 5 
 
 e 
 
 7 
 
 8 
 
 O 
 
 r>. 
 
856 
 
 APPENDIX. 
 
 LOGARITHMS OP NUMBERS. 
 
 IV. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4, 
 
 5 
 
 e 
 
 7 
 
 8 
 
 9 
 
 r>. 
 
 760 
 
 88 0814 
 
 0871 
 
 0928 
 
 0985 
 
 1042 
 
 1099 
 
 1156 
 
 1213 
 
 1271 
 
 1328 
 
 57 
 
 761 
 
 1385 
 
 1442 
 
 1499 
 
 1556 
 
 1613 
 
 1670 
 
 1727 
 
 1784 
 
 1841 
 
 1898 
 
 57 
 
 762 
 
 1955 
 
 2012 
 
 2069 
 
 2126 
 
 2183 
 
 2240 
 
 2297 
 
 2354 
 
 2411 
 
 2468 
 
 57 
 
 763 
 
 2525 
 
 2581 
 
 2638 
 
 2695 
 
 2752 
 
 2809 
 
 2866 
 
 2923 
 
 2980 
 
 3037 
 
 57 
 
 764 
 
 3093 
 
 3150 
 
 3207 
 
 3264 
 
 3321 
 
 3377 
 
 3434 
 
 3491 
 
 3548 
 
 3605 
 
 57 
 
 765 
 
 3661 
 
 3718 
 
 3775 
 
 3832 
 
 3888 
 
 3945 
 
 4002 
 
 4059 
 
 4115 
 
 4172 
 
 57 
 
 766 
 
 4229 
 
 4285 
 
 4342 
 
 4399 
 
 4455 
 
 4512 
 
 4569 
 
 4625 
 
 4682 
 
 4739 
 
 57 
 
 767 
 
 4795 
 
 4852 
 
 4909 
 
 4965 
 
 5022 
 
 5078 
 
 5135 
 
 5192 
 
 5248 
 
 5305 
 
 57 
 
 768 
 
 5361 
 
 5418 
 
 5474 
 
 5531 
 
 5587 
 
 5644 
 
 5700 
 
 5757 
 
 5813 
 
 5870 
 
 57 
 
 769 
 
 5926 
 
 5983 
 
 6039 
 
 6096 
 
 6152 
 
 6209 
 
 6265 
 
 6321 
 
 6378 
 
 6434 
 
 56 
 
 770 
 
 6491 
 
 6547 
 
 6604 
 
 6660 
 
 6716 
 
 6773 
 
 6829 
 
 6885 
 
 6942 
 
 6998 
 
 56 
 
 771 
 
 7054 
 
 7111 
 
 7167 
 
 7223 
 
 7280 
 
 7336 
 
 7392 
 
 7449 
 
 7505 
 
 7561 
 
 56 
 
 772 
 
 7617 
 
 7674 
 
 7730 
 
 7786 
 
 7842 
 
 7898 
 
 7955 
 
 8011 
 
 8067 
 
 8123 
 
 56 
 
 773 
 
 8179 
 
 8236 
 
 8292 
 
 8348 
 
 8404 
 
 8460 
 
 8516 
 
 8573 
 
 8629 
 
 8685 
 
 56 
 
 774 
 
 8741 
 
 8797 
 
 8853 
 
 8909 
 
 8965 
 
 9021 
 
 9077 
 
 9134 
 
 9190 
 
 9246 
 
 56 
 
 775 
 
 9302 
 
 9358 
 
 9414 
 
 9470 
 
 9526 
 
 9582 
 
 9638 
 
 9694 
 
 9750 
 
 9806 
 
 56 
 
 776 
 
 *9862 
 
 9918 
 
 9974 
 
 +030 
 
 0086 
 
 0141 
 
 0197 
 
 0253 
 
 0309 
 
 0365 
 
 56 
 
 777 
 
 89 0421 
 
 0477 
 
 0533 
 
 0589 
 
 0645 
 
 0700 
 
 0756 
 
 0812 
 
 0868 
 
 0924 
 
 56 
 
 778 
 
 0980 
 
 1035 
 
 1091 
 
 1147 
 
 1203 
 
 1259 
 
 1314 
 
 1370 
 
 1426 
 
 1482 
 
 56 
 
 779 
 
 1537 
 
 1593 
 
 1649 
 
 1705 
 
 1760 
 
 1816 
 
 1872 
 
 1928 
 
 1983 
 
 2039 
 
 56 
 
 780 
 
 2095 
 
 2150 
 
 2206 
 
 2262 
 
 2317 
 
 2373 
 
 2429 
 
 2484 
 
 2540 
 
 2595 
 
 56 
 
 781 
 
 2651 
 
 2707 
 
 2762 
 
 2818 
 
 2873 
 
 2929 
 
 2985 
 
 3040 
 
 3096 
 
 3151 
 
 56 
 
 782 
 
 3207 
 
 3262 
 
 3318 
 
 3373 
 
 3429 
 
 3484 
 
 3540 
 
 3595 
 
 3651 
 
 3706 
 
 56 
 
 783 
 
 3762 
 
 3817 
 
 3873 
 
 3928 
 
 3984 
 
 4039 
 
 4094 
 
 4150 
 
 4205 
 
 4261 
 
 55 
 
 784 
 
 4316 
 
 4371 
 
 4427 
 
 4482 
 
 4538 
 
 4593 
 
 4648 
 
 4704 
 
 4759 
 
 4814 
 
 55 
 
 785 
 
 4870 
 
 4925 
 
 4980 
 
 5036 
 
 5091 
 
 5146 
 
 5201 
 
 5257 
 
 5312 
 
 5367 
 
 55 
 
 786 
 
 5423 
 
 5478 
 
 5533 
 
 5588 
 
 5644 
 
 5699 
 
 5754 
 
 5809 
 
 5864 
 
 5920 
 
 55 
 
 787 
 
 5975 
 
 6030 
 
 6085 
 
 6140 
 
 6195 
 
 6251 
 
 6306 
 
 6361 
 
 6416 
 
 6471 
 
 55 
 
 788 
 
 6526 
 
 6581 
 
 6636 
 
 6692 
 
 6747 
 
 6802 
 
 6857 
 
 6912 
 
 6967 
 
 7022 
 
 55 
 
 789 
 
 7077 
 
 7132 
 
 7187 
 
 7242 
 
 7297 
 
 7352 
 
 7407 
 
 7462 
 
 7517 
 
 7572 
 
 55 
 
 790 
 
 7627 
 
 7682 
 
 7737 
 
 7792 
 
 7847 
 
 7902 
 
 7957 
 
 8012 
 
 8067 
 
 8122 
 
 55 
 
 791 
 
 8176 
 
 8231 
 
 8286 
 
 8341 
 
 8396 
 
 8451 
 
 8506 
 
 8561 
 
 8615 
 
 8670 
 
 55 
 
 792 
 
 8725 
 
 8780 
 
 8835 
 
 8890 
 
 8944 
 
 8999 
 
 9054 
 
 9109 
 
 9164 
 
 9218 
 
 55 
 
 793 
 
 9273 
 
 9328 
 
 9383 
 
 9437 
 
 9492 
 
 9547 
 
 9602 
 
 9656 
 
 9711 
 
 9766 
 
 55 
 
 794 
 
 *9821 
 
 9875 
 
 9930 
 
 9985 
 
 +039 
 
 0094 
 
 0149 
 
 0203 
 
 0258 
 
 0312 
 
 55 
 
 795 
 
 90 0367 
 
 0422 
 
 0476 
 
 0531 
 
 0586 
 
 0640 
 
 0695 
 
 0749 
 
 0804 
 
 0859 
 
 55 
 
 796 
 
 0913 
 
 0968 
 
 1022 
 
 1077 
 
 1131 
 
 1186 
 
 1240 
 
 1295 
 
 1349 
 
 1404 
 
 55 
 
 797 
 
 1458 
 
 1513 
 
 1567 
 
 1622 
 
 1676 
 
 1731 
 
 1785 
 
 1840 
 
 1894 
 
 1948 
 
 54 
 
 798 
 
 2003 
 
 2057 
 
 2112 
 
 2166 
 
 2221 
 
 2275 
 
 2329 
 
 2384 
 
 2438 
 
 2492 
 
 54 
 
 799 
 
 2547 
 
 2601 
 
 2655 
 
 2710 
 
 2764 
 
 2818 
 
 2873 
 
 2927 
 
 2981 
 
 3036 
 
 54 
 
 800 
 
 3090 
 
 3144 
 
 3199 
 
 3253 
 
 3307 
 
 3361 
 
 3416 
 
 3470 
 
 3524 
 
 3578 
 
 54 
 
 801 
 
 3633 
 
 3687 
 
 3741 
 
 3795 
 
 3849 
 
 3904 
 
 3958 
 
 4012 
 
 4066 
 
 4120 
 
 54 
 
 802 
 
 4174 
 
 4229 
 
 4283 
 
 4337 
 
 4391 
 
 4445 
 
 4499 
 
 4553 
 
 4607 
 
 4661 
 
 54 
 
 803 
 
 4716 
 
 4770 
 
 4824 
 
 4878 
 
 4932 
 
 4986 
 
 5040 
 
 5094 
 
 5148 
 
 5202 
 
 54 
 
 804 
 
 5256 
 
 5310 
 
 5364 
 
 5418 
 
 5472 
 
 5526 
 
 5580 
 
 5634 
 
 5688 
 
 5742 
 
 54 
 
 805 
 
 5796 
 
 5850 
 
 5904 
 
 5958 
 
 6012 
 
 6066 
 
 6119 
 
 6173 
 
 6227 
 
 6281 
 
 54 
 
 806 
 
 6335 
 
 6389 
 
 6443 
 
 6497 
 
 6551 
 
 6604 
 
 6658 
 
 6712 
 
 6766 
 
 6820 
 
 54 
 
 807 
 
 6874 
 
 6927 
 
 6981 
 
 7035 
 
 7089 
 
 7143 
 
 7196 
 
 7250 
 
 7304 
 
 7358 
 
 54 
 
 808 
 
 7411 
 
 7465 
 
 7519 
 
 7573 
 
 7626 
 
 7680 
 
 7734 
 
 7787 
 
 7841 
 
 7895 
 
 54 
 
 809 
 
 7949 
 
 8002 
 
 8056 
 
 8110 
 
 8163 
 
 8217 
 
 8270 
 
 8324 
 
 8378 
 
 8431 
 
 54 
 
 810 
 
 8485 
 
 8539 
 
 8592 
 
 8646 
 
 8699 
 
 8753 
 
 8807 
 
 8860 
 
 8914 
 
 8967 
 
 54 
 
 811 
 
 9021 
 
 9074 
 
 9128 . 
 
 9181 
 
 9235 
 
 9289 
 
 9342 
 
 9396 
 
 9449 
 
 9503 
 
 54 
 
 812 
 
 * 9556 
 
 9610 
 
 9663 
 
 9716 
 
 9770 
 
 9823 
 
 9877 
 
 9930 
 
 9984 
 
 +037 
 
 53 
 
 813 
 
 91 0091 
 
 0144 
 
 0197 
 
 0251 
 
 0304 
 
 0358 
 
 0411 
 
 0464 
 
 0518 
 
 0571 
 
 53 
 
 814 
 
 0624 
 
 0678 
 
 0731 
 
 0784 
 
 0838 
 
 0891 
 
 0944 
 
 0998 
 
 1051 
 
 1104 
 
 53 
 
 815 
 
 1158 
 
 1211 
 
 1264 
 
 1317 
 
 1371 
 
 1424 
 
 1477 
 
 1530 
 
 1584 
 
 1637 
 
 53 
 
 816 
 
 1690 
 
 1743 
 
 1797 
 
 1850 
 
 1903 
 
 1956 
 
 2009 
 
 2063 
 
 2116 
 
 2169 
 
 53 
 
 817 
 
 2222 
 
 2275 
 
 2328 
 
 2381 
 
 2435 
 
 2488 
 
 2541 
 
 2594 
 
 2647 
 
 2700 
 
 53 
 
 818 
 
 2753 
 
 2806 
 
 2859 
 
 2913 
 
 2966 
 
 3019 
 
 3072 
 
 3125 
 
 3178 
 
 3231 
 
 53 
 
 819 
 
 3284 
 
 3337 
 
 3390 
 
 3443 
 
 3496 
 
 3549 
 
 3602 
 
 3655 
 
 3708 
 
 3761 
 
 53 
 
 3V. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 o [ r>. 
 
APPENDIX. 
 
 S5T 
 
 LOGARITHMS OF NUMBERS. 
 
 ]V. 
 
 o 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 O 
 
 I > 
 
 820 
 
 91 3814 
 
 3867 
 
 3920 
 
 3973 
 
 4026 
 
 4079 
 
 4132 
 
 4184 
 
 4237 
 
 4290 
 
 53 
 
 821 
 
 4343 
 
 4396 
 
 4449 
 
 4502 
 
 4555 
 
 4608 
 
 4660 
 
 4713 
 
 4766 
 
 4819 
 
 53 
 
 822 
 
 4872 
 
 4925 
 
 4977 
 
 5030 
 
 5083 
 
 5136 
 
 5189 
 
 5241 
 
 5294 
 
 5347 
 
 53 
 
 823 
 
 5400 
 
 5453 
 
 5505 
 
 5558 
 
 5611 
 
 5664 
 
 5716 
 
 5769 
 
 5822 
 
 5875 
 
 53 
 
 824 
 
 5927 
 
 5980 
 
 6033 
 
 6085 
 
 6138 
 
 6191 
 
 6243 
 
 6296 
 
 6349 
 
 6401 
 
 53 
 
 825 
 
 6454 
 
 6507 
 
 6559 
 
 6612 
 
 6664 
 
 6717 
 
 6770 
 
 6822 
 
 6875 
 
 6927 
 
 53 
 
 826 
 
 6980 
 
 7033 
 
 7085 
 
 7138 
 
 7190 
 
 7243 
 
 7295 
 
 7348 
 
 7400 
 
 7453 
 
 53 
 
 827 
 
 7506 
 
 7558 
 
 7611 
 
 7663 
 
 7716 
 
 7768 
 
 7820 
 
 7873 
 
 7925 
 
 7978 
 
 52 
 
 828 
 
 8030 
 
 8083 
 
 8135 
 
 8188 
 
 8240 
 
 8293 
 
 8345 
 
 8397 
 
 8450 
 
 8502 
 
 52 
 
 829 
 
 8555 
 
 8607 
 
 8659 
 
 8712 
 
 8764 
 
 8816 
 
 8869 
 
 8921 
 
 8973 
 
 9026 
 
 52 
 
 830 
 
 9078 
 
 9130 
 
 9183 
 
 9235 
 
 9287 
 
 9340 
 
 9392 
 
 9444 
 
 9496 
 
 9549 
 
 52 
 
 831 
 
 * 9601 
 
 9653 
 
 9706 
 
 9758 
 
 9810 
 
 9862 
 
 9914 
 
 9967 
 
 +019 
 
 0071 
 
 52 
 
 832 
 
 92 0123 
 
 0176 
 
 0228 
 
 0280 
 
 0332 
 
 0384 
 
 0436 
 
 0489 
 
 0541 
 
 0593 
 
 52 
 
 833 
 
 0645 
 
 0697 
 
 0749 
 
 0801 
 
 0853 
 
 0906 
 
 0958 
 
 1010 
 
 ]062 
 
 1114 
 
 52 
 
 834 
 
 1166 
 
 1218 
 
 1270 
 
 1322 
 
 1374 
 
 1426 
 
 1478 
 
 1530 
 
 1582 
 
 1634 
 
 52 
 
 835 
 
 1686 
 
 1738 
 
 1790 
 
 1842 
 
 1894 
 
 1946 
 
 1998 
 
 2050 
 
 2102 
 
 2154 
 
 52 
 
 836 
 
 2206 
 
 2258 
 
 2310 
 
 2362 
 
 2414 
 
 2466 
 
 2518 
 
 2570 
 
 2622 
 
 2674 
 
 52 
 
 837 
 
 2725 
 
 2777 
 
 2829 
 
 2881 
 
 2933 
 
 2985 
 
 3037 
 
 3089 
 
 3140 
 
 3192 
 
 52 
 
 838 
 
 3244 
 
 3296 
 
 3348 
 
 3399 
 
 3451 
 
 3503 
 
 3555 
 
 3607 
 
 3658 
 
 3710 
 
 52 
 
 839 
 
 3762 
 
 3814 
 
 3865 
 
 3917 
 
 3969 
 
 4021 
 
 4072 
 
 4124 
 
 4176 
 
 4228 
 
 52 
 
 840 
 
 4279 
 
 4331 
 
 4383 
 
 4434 
 
 4486 
 
 4538 
 
 4589 
 
 4641 
 
 4693 
 
 4744 
 
 52 
 
 841 
 
 4796 
 
 4848 
 
 4899 
 
 4951 
 
 5003 
 
 5054 
 
 5106 
 
 5157 
 
 5209 
 
 5261 
 
 52 
 
 842 
 
 5312 
 
 5364 
 
 5415 
 
 5467 
 
 5518 
 
 5570 
 
 5621 
 
 5673 
 
 5725 
 
 5776 
 
 52 
 
 843 
 
 5828 
 
 5879 
 
 5931 
 
 5982 
 
 6034 
 
 6085 
 
 6137 
 
 6188 
 
 6240 
 
 6291 
 
 51 
 
 844 
 
 6342 
 
 6394 
 
 6445 
 
 6497 
 
 6548 
 
 6600 
 
 6651 
 
 6702 
 
 6754 
 
 6805 
 
 51 
 
 845 
 
 6857 
 
 6908 
 
 6959 
 
 7011 
 
 7062 
 
 7114 
 
 7165 
 
 7216 
 
 7268 
 
 7319 
 
 51 
 
 846 
 
 7370 
 
 7422 
 
 7473 
 
 7524 
 
 7576 
 
 7627 
 
 7678 
 
 7730 
 
 7781 
 
 7832 
 
 51 
 
 847 
 
 7883 
 
 7935 
 
 7986 
 
 8037 
 
 8088 
 
 8140 
 
 8191 
 
 8242 
 
 8293 
 
 8345 
 
 51 
 
 848 
 
 8396 
 
 8447 
 
 8498 
 
 8549 
 
 8601 
 
 8652 
 
 8703 
 
 8754 
 
 8805 
 
 8857 
 
 51 
 
 849 
 
 8908 
 
 8959 
 
 9010 
 
 9061 
 
 9112 
 
 9163 
 
 9215 
 
 9266 
 
 9317 
 
 9368 
 
 51 
 
 850 
 
 9419 
 
 9470 
 
 9521 
 
 9572 
 
 9623 
 
 9674 
 
 9725 
 
 9776 
 
 9827 
 
 9879 
 
 51 
 
 851 
 
 * 9930 
 
 9981 
 
 +032 
 
 0083 
 
 0134 
 
 0185 
 
 0236 
 
 0287 
 
 0338 
 
 0389 
 
 51 
 
 852 
 
 93 0440 
 
 0491 
 
 0542 
 
 0592 
 
 0643 
 
 0694 
 
 0745 
 
 0796 
 
 0847 
 
 0898 
 
 51 
 
 853 
 
 0949 
 
 1000 
 
 1051 
 
 1102 
 
 1153 
 
 1204 
 
 1254 
 
 1305 
 
 1356 
 
 1407 
 
 51 
 
 854 
 
 1458 
 
 1509 
 
 1560 
 
 1610 
 
 1661 
 
 1712 
 
 1763 
 
 1814 
 
 1865 
 
 1915 
 
 51 
 
 855 
 
 1966 
 
 2017 
 
 2068 
 
 2118 
 
 2169 
 
 2220 
 
 2271 
 
 2322 
 
 2372 
 
 2423 
 
 51 
 
 856 
 
 2474 
 
 2524 
 
 2575 
 
 2626 
 
 2677 
 
 2727 
 
 2778 
 
 2829 
 
 2879 
 
 2930 
 
 51 
 
 857 
 
 2981 
 
 3031 
 
 3082 
 
 3133 
 
 3183 
 
 3234 
 
 3285 
 
 3335 
 
 3386 
 
 3437 
 
 51 
 
 858 
 
 3487 
 
 3538 
 
 3589 
 
 3639 
 
 3690 
 
 3740 
 
 3791 
 
 3841 
 
 3892 
 
 3943 
 
 51 
 
 859 
 
 3993 
 
 4044 
 
 4094 
 
 4145 
 
 4195 
 
 4246 
 
 4296 
 
 4347 
 
 4397 
 
 4448 
 
 51 
 
 860 ( 
 
 4498 
 
 4549 
 
 4599 
 
 4650 
 
 4700 
 
 4751 
 
 4801 
 
 4852 
 
 4902 
 
 4953 
 
 50 
 
 861 
 
 5003 
 
 5054 
 
 5104 
 
 5154 
 
 5205 
 
 5255 
 
 5306 
 
 5356 
 
 5406 
 
 5457 
 
 50 
 
 862 
 
 5507 
 
 5558 
 
 5608 
 
 5658 
 
 5709 
 
 5759 
 
 5809 
 
 5860 
 
 5910 
 
 5960 
 
 50 
 
 863 
 
 6011 
 
 6061 
 
 6111 
 
 6162 
 
 6212 
 
 6262 
 
 6313 
 
 6363 
 
 6413 
 
 6463 
 
 50 
 
 864 
 
 6514 
 
 6564 
 
 6614 
 
 6665 
 
 6715 
 
 6765 
 
 6815 
 
 6865 
 
 6916 
 
 6966 
 
 50 
 
 865 
 
 7016 
 
 7066 
 
 7117 
 
 7167 
 
 7217 
 
 7267 
 
 7317 
 
 7367 
 
 7418 
 
 7468 
 
 50 
 
 866 
 
 7518 
 
 7568 
 
 7618 
 
 7668 
 
 7718 
 
 7769 
 
 7819 
 
 7869 
 
 7919 
 
 7969 
 
 50 
 
 867 
 
 8019 
 
 8069 
 
 8119 
 
 8169 
 
 8219 
 
 8269 
 
 8320 
 
 8370 
 
 8420 
 
 8470 
 
 50 
 
 868 
 
 8520 
 
 8570 
 
 8620 
 
 8670 
 
 8720 
 
 8770 
 
 8820 
 
 8870 
 
 8920 
 
 8970 
 
 50 
 
 869 
 
 9020 
 
 9070 
 
 9120 
 
 9170 
 
 9220 
 
 9270 
 
 9320 
 
 9369 
 
 9419 
 
 9469 
 
 50 
 
 870 
 
 9519 
 
 9569 
 
 9619 
 
 9669 
 
 9719 
 
 9769 
 
 9819 
 
 9869 
 
 9918 
 
 9968 
 
 50 
 
 871 
 
 94 0018 
 
 0068 
 
 0118 
 
 0168 
 
 0218 
 
 0267 
 
 0317 
 
 0367 
 
 0417 
 
 0467 
 
 50 
 
 872 
 
 0516 
 
 0566 
 
 0616 
 
 0666 
 
 0716 
 
 0765 
 
 0815 
 
 0865 
 
 0915 
 
 0964 
 
 50 
 
 873 
 
 1014 
 
 1064 
 
 1114 
 
 1163 
 
 1213 
 
 1263 
 
 1313 
 
 1362 
 
 1412 
 
 1462 
 
 50 
 
 874 
 
 1511 
 
 1561 
 
 1611 
 
 1660 
 
 1710 
 
 1760 
 
 1809 
 
 1859 
 
 1909 
 
 1958 
 
 50 
 
 875 
 
 2008 
 
 2058 
 
 2107 
 
 2157 
 
 2207 
 
 2256 
 
 2306 
 
 2355 
 
 2405 
 
 2455 
 
 50 
 
 876 
 
 2504 
 
 2554 
 
 2603 
 
 2653 
 
 2702 
 
 2752 
 
 2801 
 
 2851 
 
 2901 
 
 2950 
 
 50 
 
 877 
 
 3000 
 
 3049 
 
 3099 
 
 3148 
 
 3198 
 
 3247 
 
 3297 
 
 3346 
 
 3396 
 
 3445 
 
 49 
 
 878 
 
 3495 
 
 3544 
 
 3593 
 
 3643 
 
 3692 
 
 3742 
 
 3791 
 
 3841 
 
 3890 
 
 3939 
 
 49 
 
 879 
 
 3989 
 
 4038 
 
 4088 
 
 4137 
 
 4186 
 
 4236 
 
 4285 
 
 4335 
 
 4384 
 
 4433 
 
 49 
 
 IV. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 a 
 
 
 
 D. 
 
858 
 
 APPENDIX. 
 
 LOGARITHMS OF NUMBERS. 
 
 IV. 
 
 o 
 
 1 
 
 a 
 
 3 
 
 4 
 
 5 
 
 O 
 
 7 
 
 8 
 
 9 
 
 r>. 
 
 880 
 
 94 4483 
 
 4532 
 
 4581 
 
 4631 
 
 4680 
 
 4729 
 
 4779 
 
 4828 
 
 4877 
 
 4927 
 
 49 
 
 881 
 
 4976 
 
 5025 
 
 5074 
 
 5124 
 
 5173 
 
 5222 
 
 5272 
 
 5321 
 
 5370 
 
 5419 
 
 49 
 
 882 
 
 5469 
 
 5518 
 
 5567 
 
 5616 
 
 5665 
 
 5715 
 
 5764 
 
 5813 
 
 5862 
 
 5912 
 
 49 
 
 883 
 
 5961 
 
 6010 
 
 6059 
 
 6108 
 
 6157 
 
 6207 
 
 6256 
 
 6305 
 
 6354 
 
 6403 
 
 49 
 
 884 
 
 6452 
 
 6501 
 
 6551 
 
 6600 
 
 6649 
 
 6698 
 
 6747 
 
 6796 
 
 6845 
 
 6894 
 
 49 
 
 885 
 
 6943 
 
 6992 
 
 7041 
 
 7090 
 
 7140 
 
 7189 
 
 7238 
 
 7287 
 
 7336 
 
 7385 
 
 49 
 
 886 
 
 7434 
 
 7483 
 
 7532 
 
 7581 
 
 7630 
 
 7679 
 
 7728 
 
 7777 
 
 7826 
 
 7875 
 
 49 
 
 887 
 
 7924 
 
 7973 
 
 8022 
 
 8070 
 
 8119 
 
 8168 
 
 8217 
 
 8266 
 
 8315 
 
 8364 
 
 49 
 
 888 
 
 8413 
 
 8462 
 
 8511 
 
 8560 
 
 8609 
 
 8657 
 
 8706 
 
 8755 
 
 8804 
 
 8853 
 
 49 
 
 889 
 
 8902 
 
 8951 
 
 8999 
 
 9048 
 
 9097 
 
 9146 
 
 9195 
 
 9244 
 
 9292 
 
 9341 
 
 49 
 
 890 
 
 9390 
 
 9439 
 
 9488 
 
 9536 
 
 9585 
 
 9634 
 
 9683 
 
 9731 
 
 9780 
 
 9829 
 
 49 
 
 891 
 
 * 9878 
 
 9926 
 
 9975 
 
 024 
 
 0073 
 
 0121 
 
 0170 
 
 0219 
 
 0267 
 
 0316 
 
 49 
 
 892 
 
 95 0365 
 
 0414 
 
 0462 
 
 0511 
 
 0560 
 
 0608 
 
 0657 
 
 0706 
 
 0754 
 
 0803 
 
 49 
 
 893 
 
 0851 
 
 0900 
 
 0949 
 
 0997 
 
 1046 
 
 1095 
 
 1143 
 
 1192 
 
 1240 
 
 1289 
 
 49 
 
 894 
 
 1338 
 
 1386 
 
 1435 
 
 1483 
 
 1532 
 
 1580 
 
 1629 
 
 1677 
 
 1726 
 
 1775 
 
 49 
 
 895 
 
 1823 
 
 1872 
 
 1920 
 
 1969 
 
 2017 
 
 2066 
 
 2114 
 
 2163 
 
 2211 
 
 2260 
 
 48 
 
 896 
 
 2308 
 
 2356 
 
 2405 
 
 2453 
 
 2502 
 
 2550 
 
 2599 
 
 2647 
 
 2696 
 
 2744 
 
 48 
 
 897 
 
 2792 
 
 2841 
 
 2889 
 
 2938 
 
 2986 
 
 3034 
 
 3083 
 
 3131 
 
 3180 
 
 3228 
 
 48 
 
 898 
 
 3276 
 
 3325 
 
 3373 
 
 3421 
 
 3470 
 
 3518 
 
 3566 
 
 3615 
 
 3663 
 
 3711 
 
 48 
 
 899 
 
 3760 
 
 3808 
 
 3856 
 
 3905 
 
 3953 
 
 4001 
 
 4049 
 
 4098 
 
 4146 
 
 4194 
 
 48 
 
 
 
 
 
 
 
 n 
 
 
 
 
 
 
 900 
 
 4243 
 
 4291 
 
 4339 
 
 4387 
 
 4435 
 
 4484 
 
 4532 
 
 4580 
 
 4628 
 
 4677 
 
 48 
 
 901 
 
 4725 
 
 4773 
 
 4821 
 
 4869 
 
 4918 
 
 4966 
 
 5014 
 
 5062 
 
 5110 
 
 5158 
 
 48 
 
 902 
 
 5207 
 
 5255 
 
 5303 
 
 5351 
 
 5399 
 
 5447 
 
 5495 
 
 5543 
 
 5592 
 
 5640 
 
 48 
 
 903 
 
 5688 
 
 5736 
 
 5784 
 
 5832 
 
 5880 
 
 5928 
 
 5976 
 
 6024 
 
 6072 
 
 6120 
 
 48 
 
 904 
 
 6168 
 
 6216 
 
 6265 
 
 6313 
 
 6361 
 
 6409 
 
 6457 
 
 6505 
 
 6553 
 
 6601 
 
 48 
 
 905 
 
 6649 
 
 6697 
 
 6745 
 
 6793 
 
 6840 
 
 6888 
 
 6936 
 
 6984 
 
 7032 
 
 7080 
 
 4*8 
 
 906 
 
 7128 
 
 7176 
 
 7224 
 
 7272 
 
 7320 
 
 7368 
 
 7416 
 
 7464 
 
 7512 
 
 7559 
 
 48 
 
 907 
 
 7607 
 
 7655 
 
 7703 
 
 7751 
 
 7799 
 
 7847 
 
 7894 
 
 7942 
 
 7990 
 
 8038 
 
 48 
 
 908 
 
 8086 
 
 8134 
 
 8181 
 
 8229 
 
 8277 
 
 8325 
 
 8373 
 
 8421 
 
 8468 
 
 8516 
 
 48 
 
 909 
 
 8564 
 
 8612 
 
 8659 
 
 8707 
 
 8755 
 
 8803 
 
 8850 
 
 8898 
 
 8946 
 
 8994 
 
 48 
 
 910 
 
 9041 
 
 9089 
 
 9137 
 
 9185 
 
 9232 
 
 9280 
 
 9328 
 
 9375 
 
 9423 
 
 9471 
 
 48 
 
 911 
 
 9518 
 
 9566 
 
 9614 
 
 9661 
 
 9709 
 
 9757 
 
 9804 
 
 9852 
 
 9900 
 
 9947 
 
 48 
 
 912 
 
 * 9995 
 
 042 
 
 0090 
 
 0138 
 
 0185 
 
 0233 
 
 0280 
 
 0328 
 
 0376 
 
 0423 
 
 48 
 
 913 
 
 96 0471 
 
 0518 
 
 0566 
 
 0613 
 
 0661 
 
 0709 
 
 0756 
 
 0804 
 
 0851 
 
 0899 
 
 48 
 
 914 
 
 0946 
 
 0994 
 
 1041 
 
 1089 
 
 1136 
 
 1184 
 
 1231 
 
 1279 
 
 1326 
 
 1374 
 
 47 
 
 915 
 
 1421 
 
 1469 
 
 1516 
 
 1563 
 
 1611 
 
 1658 
 
 1706 
 
 1753 
 
 1801 
 
 1848 
 
 47 
 
 916 
 
 1895 
 
 1943 
 
 1990 
 
 2038 
 
 2085 
 
 2132 
 
 2180 
 
 2227 
 
 2275 
 
 2322 
 
 47 
 
 917 
 
 2369 
 
 2417 
 
 2464 
 
 2511 
 
 2559 
 
 2606 
 
 2653 
 
 2701 
 
 2748 
 
 2795 
 
 47 
 
 918 
 
 2843 
 
 2890 
 
 2937 
 
 2985 
 
 3032 
 
 3079 
 
 3126 
 
 3174 
 
 3221 
 
 3268 
 
 47 
 
 919 
 
 3316 
 
 3363 
 
 3410 
 
 3457 
 
 3504 
 
 3552 
 
 3599 
 
 3646 
 
 3693 
 
 3741 
 
 47 
 
 920 
 
 3788 
 
 3835 
 
 3882 
 
 3929 
 
 3977 
 
 4024 
 
 4071 
 
 4118 
 
 4165 
 
 4212 
 
 47 
 
 921 
 
 4260 
 
 4307 
 
 4354 
 
 4401 
 
 4448 
 
 4495 
 
 4542 
 
 4590 
 
 4637 
 
 4684 
 
 47 
 
 922 
 
 4731 
 
 4778 
 
 4825 
 
 4872 
 
 4919 
 
 4966 
 
 5013 
 
 5061 
 
 5108 
 
 5155 
 
 47 
 
 923 
 
 5202 
 
 5249 
 
 5296 
 
 5343 
 
 5390 
 
 5437 
 
 5484 
 
 5531 
 
 5578 
 
 5625 
 
 47 
 
 924 
 
 5672 
 
 5719 
 
 5766 
 
 5813 
 
 5860 
 
 5907 
 
 5954 
 
 6001 
 
 6048 
 
 6095 
 
 47 
 
 925 
 
 6142 
 
 6189 
 
 6236 
 
 6283 
 
 6329 
 
 6376 
 
 6423 
 
 6470 
 
 6517 
 
 6564 
 
 47 
 
 926 
 
 6611 
 
 6658 
 
 6705 
 
 6752 
 
 6799 
 
 6845 
 
 6892 
 
 6939 
 
 6986 
 
 7033 
 
 47 
 
 927 
 
 7080 
 
 7127 
 
 7173 
 
 7220 
 
 7267 
 
 7314 
 
 7361 
 
 7408 
 
 7454 
 
 7501 
 
 47 
 
 928 
 
 7548 
 
 7595 
 
 7642 
 
 7688' 
 
 7735 
 
 7782 
 
 7829 
 
 7875 
 
 7922 
 
 7969 
 
 47 
 
 929 
 
 8016 
 
 8062 
 
 8109 
 
 8156 
 
 8203 
 
 8249 
 
 8296 
 
 8343 
 
 8390 
 
 8436 
 
 47 
 
 930 
 
 8483 
 
 8530 
 
 8576 
 
 8623 
 
 8670 
 
 8716 
 
 8763 
 
 8810 
 
 8856 
 
 8903 
 
 47 
 
 931 
 
 8950 
 
 8996 
 
 9043 
 
 9090 
 
 9136 
 
 9183 
 
 9229 
 
 9276 
 
 9323 
 
 9369 
 
 47 
 
 932 
 
 9416 
 
 9463 
 
 9509 
 
 9556 
 
 9602 
 
 9649 
 
 9695 
 
 9742 
 
 9789 
 
 9835 
 
 47 
 
 933 
 
 * 9882 
 
 9928 
 
 9975 
 
 021 
 
 0068 
 
 0114 
 
 0161 
 
 0207 
 
 0254 
 
 0300 
 
 47 
 
 934 
 
 97 0347 
 
 0393 
 
 0440 
 
 0486 
 
 0533 
 
 0579 
 
 0626 
 
 0672 
 
 0719 
 
 0765 
 
 46 
 
 935 
 
 0812 
 
 0858 
 
 0904 
 
 0951 
 
 0997 
 
 1044 
 
 1090 
 
 1137 
 
 1183 
 
 1229 
 
 46 
 
 936 
 
 1276 
 
 1322 
 
 1369 
 
 1415 
 
 1461 
 
 1508 
 
 1554 
 
 1601 
 
 1647 
 
 1693 
 
 46 
 
 937 
 
 1740 
 
 1786 
 
 1832 
 
 1879 
 
 1925 
 
 1971 
 
 2018 
 
 2064 
 
 2110 
 
 2157 
 
 46 
 
 938 
 
 2203 
 
 2249 
 
 2295 
 
 2342 
 
 2388 
 
 2434 
 
 2481 
 
 2527 
 
 2573 
 
 2619 
 
 46 
 
 939 
 
 2666 
 
 2712 
 
 2758 
 
 2804 
 
 2851 
 
 2897 
 
 2943 
 
 2989 
 
 3035 
 
 3082 
 
 46 
 
 IV. 
 
 
 
 1 
 
 3 
 
 3 
 
 4, 
 
 5 
 
 
 
 7 
 
 S 
 
 Q 
 
 l>. 
 
APPENDIX. 
 
 859 
 
 LOGARITHMS OF NUMBERS. 
 
 N. 
 
 O 
 
 1 
 
 2 
 
 3 
 
 4= 
 
 5 
 
 e 
 
 7 
 
 8 
 
 & 
 
 l>. 
 
 940 
 
 97 3128 
 
 3174 
 
 3220 
 
 3266 
 
 3313 
 
 3359 
 
 3405 
 
 3451 
 
 3497 
 
 3543 
 
 46 
 
 941 
 
 3590 
 
 3636 
 
 3682 
 
 3728 
 
 3774 
 
 3820 
 
 3866 
 
 3913 
 
 3959 
 
 4005 
 
 46 
 
 942 
 
 4051 
 
 4097 
 
 4143 
 
 4189 
 
 4235 
 
 4281 
 
 4327 
 
 4374 
 
 4420 
 
 4466 
 
 46 
 
 943 
 
 4512 
 
 4558 
 
 4604 
 
 4650 
 
 4696 
 
 4742 
 
 4788 
 
 4834 
 
 4880 
 
 4926 
 
 46 
 
 944 
 
 4972 
 
 5018 
 
 5064 
 
 5110 
 
 5156 
 
 5202 
 
 5248 
 
 5294 
 
 5340 
 
 5386 
 
 46 
 
 945 
 
 5432 
 
 5478 
 
 5524 
 
 5570 
 
 5616 
 
 5662 
 
 5707 
 
 5753 
 
 5799 
 
 5845 
 
 46 
 
 946 
 
 5891 
 
 5937 
 
 5983 
 
 6029 
 
 6075 
 
 6121 
 
 6167 
 
 6212 
 
 6258 
 
 6304 
 
 46 
 
 947 
 
 6350 
 
 6396 
 
 6442 
 
 6488 
 
 6533 
 
 6579 
 
 6625 
 
 6671 
 
 6717 
 
 6763 
 
 46 
 
 948 
 
 6808 
 
 6854 
 
 6900 
 
 6946 
 
 6992 
 
 7037 
 
 7083 
 
 7129 
 
 7175 
 
 7220 
 
 46 
 
 949 
 
 7266 
 
 7312 
 
 7358 
 
 7403 
 
 7449 
 
 7495 
 
 7541 
 
 7586 
 
 7632 
 
 7678 
 
 46 
 
 950 
 
 7724 
 
 7769 
 
 7815 
 
 7861 
 
 7906 
 
 7952 
 
 7998 
 
 8043 
 
 8089 
 
 8135 
 
 46 
 
 951 
 
 8181 
 
 8226 
 
 8272 
 
 8317 
 
 8363 
 
 8409 
 
 8454 
 
 8500 
 
 8546 
 
 8591 
 
 46 
 
 952 
 
 8637 
 
 8683 
 
 8728 
 
 8774 
 
 8819 
 
 8865 
 
 8911 
 
 8956 
 
 9002 
 
 9047 
 
 46 
 
 953 
 
 9093 
 
 9138 
 
 9184 
 
 9230 
 
 9275 
 
 9321 
 
 9366 
 
 9412 
 
 9457 
 
 9503 
 
 46 
 
 954 
 
 9548 
 
 9594 
 
 9639 
 
 9685 
 
 9730 
 
 9776 
 
 9821 
 
 9867 
 
 9912 
 
 9958 
 
 46 
 
 955 
 
 98 0003 
 
 0049 
 
 0094 
 
 0140 
 
 0185 
 
 0231 
 
 0276 
 
 0322 
 
 0367 
 
 0412 
 
 45 
 
 956 
 
 0458 
 
 0503 
 
 0549 
 
 0594 
 
 0640 
 
 0685 
 
 0730 
 
 0776 
 
 0821 
 
 0867 
 
 45 
 
 957 
 
 0912 
 
 0957 
 
 1003 
 
 1048 
 
 1093 
 
 1139 
 
 1184 
 
 1229 
 
 1275 
 
 1320 
 
 45 
 
 958 
 
 1366 
 
 1411 
 
 1456 
 
 1501 
 
 1547 
 
 1592 
 
 1637 
 
 1683 
 
 1728 
 
 1773 
 
 45 
 
 959 
 
 1819 
 
 1864 
 
 1909 
 
 1954 
 
 2000 
 
 2045 
 
 2090 
 
 2135 
 
 2181 
 
 2226 
 
 45 
 
 960 
 
 2271 
 
 2316 
 
 2362 
 
 2407 
 
 2452 
 
 2497 
 
 2543 
 
 2588 
 
 2633 
 
 2678 
 
 45 
 
 961 
 
 2723 
 
 2769 
 
 2814 
 
 2859 
 
 2904 
 
 2949 
 
 2994 
 
 3040 
 
 3085 
 
 3130 
 
 45 
 
 962 
 
 3175 
 
 3220 
 
 3265 
 
 3310 
 
 3356 
 
 3401 
 
 3446 
 
 3491 
 
 3536 
 
 3581 
 
 45 
 
 963 
 
 3626 
 
 3671 
 
 3716 
 
 3762 
 
 3807 
 
 3852 
 
 3897 
 
 3942 
 
 3987 
 
 4032 
 
 45 
 
 964 
 
 4077 
 
 4122 
 
 4167 
 
 4212 
 
 4257 
 
 4302 
 
 4347 
 
 4392 
 
 4437 
 
 4482 
 
 45 
 
 965 
 
 4527 
 
 4572 
 
 4617 
 
 4662 
 
 4707 
 
 4752 
 
 4797 
 
 4842 
 
 4887 
 
 4932 
 
 45 
 
 966 
 
 4977 
 
 5022 
 
 5067 
 
 5112 
 
 5157 
 
 5202 
 
 5247 
 
 5292 
 
 5337 
 
 5382 
 
 45 
 
 967 
 
 5426 
 
 5471 
 
 5516 
 
 5561 
 
 5606 
 
 5651 
 
 5696 
 
 5741 
 
 5786 
 
 5830 
 
 45 
 
 968 
 
 5875 
 
 5920 
 
 5965 
 
 6010 
 
 6055 
 
 6100 
 
 6144 
 
 6189 
 
 6234 
 
 6279 
 
 45 
 
 969 
 
 6324 
 
 6369 
 
 6413 
 
 6458 
 
 6503 
 
 6548 
 
 6593 
 
 6637 
 
 6682 
 
 6727 
 
 45 
 
 970 
 
 6772 
 
 6817 
 
 6861 
 
 6906 
 
 6951 
 
 6996 
 
 7040 
 
 7085 
 
 7130 
 
 7175 
 
 45 
 
 971 
 
 7219 
 
 7264 
 
 7309 
 
 7353 
 
 7398 
 
 7443 
 
 7488 
 
 7532 
 
 7577 
 
 7622 
 
 45 
 
 972 
 
 7666 
 
 7711 
 
 7756 
 
 7800 
 
 7845 
 
 7890 
 
 7934 
 
 7979 
 
 8024 
 
 8068 
 
 45 
 
 973 
 
 8113 
 
 8157 
 
 8202 
 
 8247 
 
 8291 
 
 8336 
 
 8381 
 
 8425 
 
 8470 
 
 8514 
 
 45 
 
 974 
 
 8559 
 
 8604 
 
 8648 
 
 8693 
 
 8737 
 
 8782 
 
 8826 
 
 8871 
 
 8916 
 
 8960 
 
 45 
 
 975 
 
 9005 
 
 9049 
 
 9094 
 
 9138 
 
 9183 
 
 9227 
 
 9272 
 
 9316 
 
 9361 
 
 9405 
 
 45 
 
 976 
 
 9450 
 
 9494 
 
 9539 
 
 9583 
 
 9628 
 
 9672 
 
 9717 
 
 9761 
 
 9806 
 
 9850 
 
 44 
 
 977 
 
 * 9895 
 
 9939 
 
 9983 
 
 +028 
 
 0072 
 
 0117 
 
 0161 
 
 0206 
 
 0250 
 
 0294 
 
 44 
 
 978 
 
 99 0339 
 
 0383 
 
 0428 
 
 0472 
 
 0516 
 
 0561 
 
 0605 
 
 0650 
 
 0694 
 
 0738 
 
 44 
 
 979 
 
 0783 
 
 0827 
 
 0871 
 
 0916 
 
 0960 
 
 1004 
 
 1049 
 
 1093 
 
 1137 
 
 1182 
 
 44 
 
 980 
 
 1226 
 
 1270 
 
 1315 
 
 1359 
 
 1403 
 
 1448 
 
 1492 
 
 1536 
 
 1580 
 
 1625 
 
 44 
 
 981 
 
 1669 
 
 1713 
 
 1758 
 
 1802 
 
 1846 
 
 1890 
 
 1935 
 
 1979 
 
 2023 
 
 2067 
 
 44 
 
 982 
 
 2111 
 
 2156 
 
 2200 
 
 2244 
 
 2288 
 
 2333 
 
 2377 
 
 2421 
 
 2465 
 
 2509 
 
 44 
 
 983 
 
 2554 
 
 2598 
 
 2642 
 
 2686 
 
 2730 
 
 2774 
 
 2819 
 
 2863 
 
 2907 
 
 2951 
 
 44 
 
 984 
 
 2995 
 
 3039 
 
 3083 
 
 3127 
 
 3172 
 
 3216 
 
 3260 
 
 3304 
 
 3348 
 
 3392 
 
 44 
 
 985 
 
 3436 
 
 3480 
 
 3524 
 
 3568 
 
 3613 
 
 3657 
 
 3701 
 
 3745 
 
 3789 
 
 3833 
 
 44 
 
 986 
 
 3877 
 
 3921 
 
 3965 
 
 4009 
 
 4053 
 
 4097 
 
 4141 
 
 4185 
 
 4229 
 
 4273 
 
 44 
 
 987 
 
 4317 
 
 4361 
 
 4405 
 
 4449 
 
 4493 
 
 4537 
 
 4581 
 
 4625 
 
 4669 
 
 4713 
 
 44 
 
 988 
 
 4757 
 
 4801 
 
 4845 
 
 4889 
 
 4933 
 
 4977 
 
 5021 
 
 5065 
 
 5108 
 
 5152 
 
 44 
 
 989 
 
 5196 
 
 5240 
 
 5284 
 
 5328 
 
 5372 
 
 5416 
 
 5460 
 
 5504 
 
 5547 
 
 5591 
 
 44 
 
 990 
 
 5635 
 
 5679 
 
 5723 
 
 5767 
 
 5811 
 
 5854 
 
 5898 
 
 5942 
 
 5986 
 
 6030 
 
 44 
 
 991 
 
 6074 
 
 6117 
 
 6161 
 
 6205 
 
 6249 
 
 6293 
 
 6337 
 
 6380 
 
 6424 
 
 6468 
 
 44 
 
 992 
 
 6512 
 
 6555 
 
 6599 
 
 6643 
 
 6687 
 
 .6731 
 
 6774 
 
 6818 
 
 6862 
 
 6906 
 
 44 
 
 993 
 
 6949 
 
 6993 
 
 7037 
 
 7080 
 
 7124 
 
 7168 
 
 7212 
 
 7255 
 
 7299 
 
 7343 
 
 44 
 
 994 
 
 7386 
 
 7430 
 
 7474 
 
 7517 
 
 7561 
 
 7605 
 
 7648 
 
 7692 
 
 7736 
 
 7779 
 
 44 
 
 995 
 
 7823 
 
 7867 
 
 7910 
 
 7954 
 
 7998 
 
 8041 
 
 8085 
 
 8129 
 
 8172 
 
 8216 
 
 44 
 
 996 
 
 8259 
 
 8303 
 
 8347 
 
 8390 
 
 8434 
 
 8477 
 
 8521 
 
 8564 
 
 8608 
 
 8652 
 
 44 
 
 997 
 
 8695 
 
 8739 
 
 8782 
 
 8826 
 
 8869 
 
 8913 
 
 8956 
 
 9000 
 
 9043 
 
 9087 
 
 44 
 
 998 
 
 9131 
 
 9174 
 
 9218 
 
 9261 
 
 9305 
 
 9348 
 
 9392 
 
 9435 
 
 9479 
 
 9522 
 
 44 
 
 999 
 
 9565 
 
 9609 
 
 9652 
 
 9696 
 
 9739 
 
 9783 
 
 9826 
 
 9870 
 
 9913 
 
 9957 
 
 43 
 
 3V. 
 
 O 
 
 1 
 
 3 
 
 3 
 
 4 
 
 5 
 
 e 
 
 7 
 
 8 
 
 O 
 
 D. 
 
860 
 
 APPENDIX. 
 
 The Application of Logarithms. The logarithm of a number is set down as a decimal, 
 and addition of ciphers to numbers does not change the logarithm ; it is the same for 11, 
 110 1100, but the value of the number is established by figures to the left of the decimal 
 point ; thus, if the number is among the units, the characteristic is ; if in the tens, 1 ; 
 in the hundreds, 2 ; thousands, 3 ; tens of thousands, 4, and so on ; if the number is a 
 decimal fraction and the first figure a tenth, the characteristic is 1, if hundredths 2, thou- 
 sandths 3. 
 
 Multiplication of two numbers is performed by the addition of their logarithms and 
 characteristics, and finding the number corresponding to their sum ; thus, to multiply 119 
 
 by 2760. 
 
 Characteristic of 119 2, logarithm. 2-075547 
 
 " 2760 3, " 3-440909 
 
 5-516456 
 3284 403 
 
 401 I) = 132)53(401 
 
 328440-1 528 
 
 200 
 132 
 ~68 
 
 As the characteristic is 5, the result is 6 figures of whole numbers. 
 Division is performed by subtracting the logarithm of the divisor from that of the divi- 
 dend, and finding the logarithm of the remainder for the quotient. But if the divisor is 
 the larger, then the characteristic of the remainder is . 
 Thus, to divide 500 by 63008. 
 
 Logarithm of 500 2-698970 
 
 Logarithm of 63000 = 4-799341 
 
 D = 69 8 -^ = 65-2 
 
 10 
 
 Logarithm of 63008 4.799396 
 
 Corresponding number -007935 = 3-899574 
 
 Numbers are raised to any power by multiplying their logarithm by the exponents, and 
 roots are extracted by dividing the logarithm. Thus, to get the square of any number, its 
 logarithm is multiplied by 2, for the cube by 3, for the 4th power by 4 ; in like manner, to 
 obtain the square root of the number, divide the logarithm by 2 ; by 3 for \f ; by 4 
 
 for 
 
 . 
 
 The roots of numbers are better expressed by fractional exponents, thus: Va by a~ ; * 
 |/a by a 8 . 
 
 The raising of numbers to different powers is extremelj' simple, by logarithms, when 
 the numbers are whole numbers, but becomes somewhat more complicated when the num- 
 bers are decimals. 
 
 Thus, to find the 4th power of -07. 
 
 Logarithm -07 2-845098 
 
 _ _ 4 
 
 8 3-380392 
 
 Number -00002401 5-380392 
 To extract the 4th root of -07 
 
 Logarithm -07 2-845098 
 
 Add 2 to the characteristic to make it 
 
 divisible by 4, and a positive 2 to the 2-2-845098 
 
 logarithm to balance it. 4)4'2-845098 
 
 Number -5143 1- 711274 
 
APPENDIX. 861 
 
 The exponent of a root is often a decimal ; thus the 1/'07 may be expressed by -07- a8 . 
 
 Logarithm -07 2-845098 
 
 -25 
 
 4225490 
 1690196 
 
 5-21127450 
 5-5 
 
 Number -5143 1-71127450 
 
 NOTE. In this example, -5 is added to the resultant characteristic to bring it to an integer, and 
 an equal positive amount to the logarithm to balance it. 
 
 The same logarithm as by dividing by 4- and corresponding to the number -5143. The 
 rule is to consider the logarithm as a plus quantity, and multiply by the exponent and the 
 characteristic as minus, and. after similar multiplication, subtract it from the first product. 
 When a characteristic has a minus sign (3), and it is to be subtracted, the sign is changed 
 and added. 
 
 Thus, to divide 10- by T V 
 
 Logarithm 10- 1-00000 
 
 rV I 
 
 Logarithm of 100- 2-000 
 
 To divide T V Logarithm TOOOOO 
 
 by T^T 2- 
 
 Logarithm of 10' 1*0000 
 
 To divide y^ 1 ^ Logarithm 3-00000 
 
 by 100 2 
 
 Logarithm of -00001 5 
 
 TABLE OF RECIPROCALS, PAGES 828 AND 829. 
 
 Use of Reciprocals. Reciprocals may be conveniently used to facilitate computations 
 in long divisions. Instead of dividing as usual, multiply the dividend by the reciprocal 
 of the divisor. The method is especially useful when many different dividends are re- 
 quired to be divided by the same divisor. In this case find the reciprocal of the divisor, 
 make a small table of its multiples up to nine times, and use this as a multiplication 
 table, instead of actually performing the multiplication in each case. 
 
SCRAPS. 
 
 IT is good practice to collect, from the circulars of manufacturers, and from 
 illustrated newspapers and magazines, varied illustrations of tools and machines, 
 engineering structures, buildings, etc., and arrange them under their appropri- 
 ate heads in scrap-books. They will be found very useful in designing, not 
 only enabling one the more readily to make drawings, but to convey to the 
 draughtsman the character and proportions of the design which is to be made. 
 And those parts which are of common use and purchasable in the market can 
 be readily arranged in position and executed more economically than from a 
 new design. There is a saving in the matter of drawing, and also in the cost 
 of construction. 
 
 A proper combination and arrangement of parts which have served a pur- 
 pose will afford material for a more practical and satisfactory design than ean 
 usually be made from attempts at originality. Knowledge of what has been 
 done is economy in all labour. If the construction or machine can not be seen, 
 its picture can supply its place, and its details can be studied at leisure ; and 
 as the education of the eye is of essential importance to the draughtsman, let 
 him see as much as he can practically, and at the same time acquire a good 
 collection of scraps from which to design. There are few constructions from 
 which something of education can not be drawn, parts if not a whole. 
 
 In this view a small collection of scraps has been made pertinent to the 
 book. Its page does not admit of the sizes which will be found in the illus- 
 trated papers and magazines the quarto will be found much more generally 
 useful and a library of such scrap-books will furnish material for a draughts- 
 man which can not be found in any encyclopaidia. 
 
 862 
 
SCRAPS. 
 
 863 
 
 Compound Steam Cylinders. H. M. S. Spartan. 
 
 Wrought-Iron Plates and Covers. 
 
 Compressed- Air Locomotive, St. Gothard Tunnel. 
 
864 
 
 SCRAPS. 
 
 Three-Throw Crank. 
 
 Forged weight, 2$. tons 11 cwt. 
 Finished " 15 " 8 " 
 
 Corrugated Boiler-Flues. 
 
 Weight, 25 tons 10 cwt. 
 
SCRAPS. 
 
 865 
 
 56 
 
 Screw Propeller. 
 Vessel, 1400 jrross tons. Engines, 130 nominal English horse-power 
 
866 
 
 SCRAPS. 
 
 Spherical Bearing. 
 
 O QJ83OOO 
 
 1 
 I , 
 
 k? 
 
 l-l 
 
 
 il IPi 
 
 ^ i^ 
 1^ U* 
 
 C) ?5 ^ ^ ^. 
 
 Conventional Signs of Riveting. 
 
 F sfi 
 
 L 
 
 In the Wilkinson stoker the coal is fed mechanically through an inclined pipe on to 
 the dead plate P and slides down upon the bars, which are hollow and set at an angle. 
 The top of the bars is stepped, and tuyere-shaped openings about J x 3 inches are pro- 
 vided in each riser. The bars are carried at their ends on hollow boxes, and are 4-inch 
 centres. For the feeding, adjacent bars move in opposite directions by a system of tog- 
 gles driven by the stoker engine. 
 
SCRAPS. 
 
 867 
 
 The Coxe stoker consists of a travelling grate with fire on its tipper surface. The 
 coal is fed by the motion of the grate at the front. There are four blast compartments, 
 A, B, C, D, under the fire connected by dampers. The sides of the furnace are pro- 
 tected by wrought-iron water backs, through which the water circulates under slight 
 pressure. 
 
 The Stirling Boiler consists of three upper steam-drums and a lower mud-drum. All 
 the steam-drums are connected at the top, but the front and middle drums only 'are con- 
 nected in the water space. Tubes, 3V diameter ; movement of flame shown by arrows 
 around baffle plates. 
 
868 
 
 SCRAPS. 
 
 The Abendroth and Boot Boiler was the earliest of its type on the market, and has 
 been modified in its details since its introduction to meet necessities which were devel- 
 oped by use. The section gives the latest form. The angles of the tubes, with the hori- 
 zontal and the baffle plates beneath and around the tubes making a positive circulation 
 of the flame, are of the original design, but longitudinal drums extend lengthwise over 
 the tubes with a w r ater circulation in the lower half toward the rear and downward 
 by vertical pipes to the lower ends of the tubes. On these vertical pipes there are 
 two cross-drums the intermediate one to receive the feedwater, the lower for a mud- 
 drum. The long drums connect with the steam-drum placed above and crosswise of 
 them. 
 
SCRAPS. 
 
 869 
 
8TO 
 
 SCRAPS. 
 
 Andover, Mass., Steam Pumping Plant, ~by the Deane Steam Pump Co., Holydke, Mass. 
 Diameter H. P. cylinder, 17"; L. P., 30". Pump plunger, 8|". Stroke, 30". Ca- 
 pacity at 205' plunger speed, 1,205 gals, per minute. Average observed steam pressure, 
 90-3; water, 139-79 ; in test trial, pump exceeded contract duty of 125,000,000. 
 
 Reidler Valve, from LeavitVs Thames- Ditton Pump. (See page 365.) 
 
SCRAPS. 
 
 871 
 
 F THR 
 
 UNIVERSITY D 
 
872 
 
 SCRAPS. 
 
SCRAPS. 
 
 873 
 
 - 2 - i w r "i~""".4"""".-f.".~""!| jj 
 
 S 
 
 i 
 
 * Sf 9T $- tS- 
 
 Class "R? 
 
 K--6'9' -->) 
 
 From the "Engineering News." 
 
 Elliptic Spring applied to Car Truck. 
 
 Bolster Spring. 
 
 OF THK 
 
 UNTVEBSITY J . 
 
SCRAPS. 
 
 Third Avenue Elevated Railroad, 
 
 Cable-car Grip. 
 
 The accompanying figures rep- 
 resent a side elevation and an end 
 view respectively of a cable-car 
 grip. 
 
 The gripping apparatus is 
 shown in its open position, and 
 the cable is therefore running in- 
 operatively through it. When it 
 is required to grip the cable it is 
 merely necessary to pull over the 
 lever A, whereupon the lever, its 
 quadrant frame and shank attach- 
 ments C, are raised up bodily 
 about the fixed fulcrum of the 
 link c, and the pair of rollers E, 
 carried at the lower extremities 
 of C, forces the jaws G close to- 
 gether, and tightly grasps the 
 cable between the concave dies or 
 packing pieces h. 
 
SCRAPS. 
 
 875 
 
 o l> 7 8 3 10 11 12 13 14 15 
 
 Derrick. 
 
 OF THT? 
 
 UNIVERSITY 
 
876 
 
 SCRAPS. 
 
 Canvas Dams. (From Trans. A. S. C. E.) 
 
 Earth Portion of the New Croton Dam, showing RuUtle Masonry Care. 
 
SCRAPS. 
 
 877 
 
 Sweetwater Arched Masonry Dam, Southern California. 
 
 The dimensions of this dam are : Thickness at base, 46 ft. ; thickness at top, 12 ft. ; 
 height, 90 ft. ; radius of arch at top, 222 ft. The upper face batter is 1 to 6 to within 
 6 ft. of the top, thence vertical ; on the lower face it is 1 to 3 for 28 ft. ; 1 to 4 for 32 
 ft. ; thence 1 to 6 to the coping. In January, 1895, a freshet discharge flooded the waste 
 weir and rose 22 inches over the parapet wall. The dam was subjected to this cataract 
 action for a period of forty hours without injury. 
 
 Sear Valley Arched Masonry Dam, California. 
 
 This receives a sufficient support from the arch-action, and has stood since 1884. Its 
 length at top is 270 ft., and its radius of curvature is 355 ft. At the centre the deviation 
 of the dam from a straight line is about 27 ft. , this being the versed sine of the sub- 
 tended angle. It is 38 inches in thickness at top and 102 inches at a point 48 ft. 
 below. 
 
878 
 
 SCRAPS. 
 
SCRAPS. 
 BUILDERS' HARDWARE. 
 
 8T9 
 
 Mortise-Lock, cover off. 
 
 Front 
 
 Boxed Strike, front. SHding-door 
 
 Lock. 
 
 Thumb-Piece. 
 
 Knob and Rose. 
 
 Escutcheons. 
 
880 
 
 SCRAPS. 
 
 Sash-Lifts. 
 
 Hook and Eye 
 
 Skitter- 
 Knob. 
 
SCRAPS. 
 
 881 
 
 Examples of Ancient Hinges and Doors. 
 
 w 
 
 s\\\\\y SSS3ESSSS 253 
 
 
 ^i 
 
 /K 
 
 ///xvy/Vy^/A^y/y/y^/vvx/xi 
 
 Se&steeT 
 
 ^3v 
 
 
 > : 
 
 
 hi 
 
 Safe Construction. 
 
 ^' 
 
 
 \;/N/ 
 
 
 ^5 
 
 ^ / / 
 
 
 Cast- Iron Tread. 
 
 57 
 
882 
 
 SCRAPS. 
 
 gaa aag B i i . g a i .gaa ai .aH 
 
 FEET 
 
 , Section, and Elevation of a Wooden Mantel and Fire-Place. 
 
SCRAPS. 
 
 883 
 
 Sectional Plan of Grate and Flue. 
 Details of Fireplace. 
 
884 
 
 SCRAPS. 
 
 Vestibule Doors. 
 
SCRAPS. 
 
 885 
 
 Examples of Inlaid Floors or Marquetry. 
 
886 
 
 SCRAPS. 
 
 Railing. 
 
SCRAPS. 
 
 887 
 
888 
 
 SCRAPS. 
 
 Enameled Tile. 
 
 ri 
 
 Terra Cotta. 
 
SCRAPS. 
 
 889 
 
890 
 
 SCRAPS. 
 
 
SCRAPS. 
 
 891 
 
892 
 
 SCRAPS. 
 
SCRAPS. 
 
 893 
 
894 
 
 SCRAPS. 
 
SCRAPS. 
 
 895 
 
896 
 
 SCRAPS. 
 
SCRAPS. 
 
898 
 
 SCRAPS. 
 
SCRAPS. 
 
 f i 
 
900 
 
 SCRAPS. 
 
SCRAPS. 
 
 901 
 
902 
 
 SCRAPS. 
 
SCRAPS. 
 
 903 
 
904' 
 
 SCRAPS. 
 
 
SCRAPS. 
 
 905 
 
906 
 
 SCRAPS. 
 
 ! 
 
 P/cfure f/grre or 
 
 L/ne of Measures 
 
 I \ 
 
 i 
 
 \ X 
 
 / ' 
 / / 
 
 ' / 
 
 t.. v 
 
 *i\l \ 
 
 ! 
 
 ;K\ 
 
 \ \ 
 
 \ 
 
 Perspective Diagram, (See page 714.) 
 
SCRAPS. 
 
 907 
 
908 
 
 SCRAPS. 
 
SCRAPS. 
 
 909 
 
910 
 
 SCRAPS. 
 
SCRAPS. 
 
 Coneij Island. 
 
912 
 
 SCRAPS. 
 
 Coney Island. 
 
INDEX. 
 
 Abendroth & Root water tube boiler, 868. . 
 
 Acanthus leaf or scroll, 863. 
 
 Accumulator for water pressure, 412. 
 
 Acoustics, general principles of, 630. 
 
 Adiabatic curve, 206. 
 
 ACrial perspective, 751. 
 
 Air-chambers of pumps, 365, 411, 412. 
 
 Air ducts to furnaces, area of, 640. 
 
 Air, flow of, diagrams, 791. 
 
 Air-lock, use of, 431 ; Barr-Moran, 435. 
 
 Air taken into and expired from the lungs of a 
 
 person, 636. 
 Alloys and compositions, table of, 810. 
 
 brass, Muntz metal, Babbit-metal, copper with 
 various metals and proportions, 180. 
 
 chart of strength, Prof. Thurston, 180. 
 Aluminum, properties of, 180. 
 Anchor bolts, kinds and strength of, 253. 
 Anchors for beams and walls, 559. 
 Angle blocks in truss bridges, 499. 
 
 irons, equal and unequal legs, dimensions of, 246. 
 Angles, definition of, 4; sum of, in figures, 16. 
 Angular perspective, example of, 714. 
 Anthemion or honeysuckle, architecture, 683. 
 Antimony, properties and use of, 179. 
 Apartment houses, 609, 610. 
 Apron, for protection of dam, 443. 
 Apse, circular end of a church, 673 ; of basilicas, 
 
 624. 
 Arch and architrave mouldings, 679. 
 
 bridges, parts and proportions of, 518. 
 
 Melan concrete, Stockbridge, Mass., 522. 
 Arched bridge in angular perspective, 716. 
 Arches, complex, ogee, Tudor, trefoil, triangular, 
 roundheaded, and pointed, 667. 
 
 of the Minneapolis Viaduct, 520, 
 
 table of dimensions of, 521. 
 Artificial building material, 174. 
 Ashti reservoir, for irrigation, India, 438. 
 Asphalt lining for reservoirs, pavement with con- 
 crete foundation, 476. 
 Axle and rolling friction, 199. 
 Axles, car, 261. 
 
 Babbit-metal for journal-boxes, 180. 
 Babcock and Wilcox water tube boiler, 796. 
 Ball-and-socket joint for flexible pipe, 405. 
 
 59 913 
 
 Ball valves, varieties of, 374. 
 
 Baltimore Academy of Music, ventilation of, 623. 
 
 heater, 641. 
 
 Baluster and newel post, 579. 
 Base and base mouldings, 680. 
 Batter and offsets to retaining walls, 436. 
 Beams, loading of, transverse stress, 235. 
 Bear Valley arched masonry dam, 877. 
 Beetaloo dam, concrete, South Australia, 444. 
 Bell-cots, designs for, 673. 
 Bell-trap for sink, 655. 
 
 Belts, tight and loose, 287 ; transmission of power 
 by diagram, 292 ; speed of, 292 ; width and 
 thickness of, 293 ; leather, canvas, rubber, 293. 
 Bevel gears, relative sizes of, 310; mortise, 316 ; 
 projection of, 321; skew,>323. 
 
 wheel, isometrical projection of, 699. 
 Bismuth, properties of, in fusible alloys, 179. 
 Blocks for running rigging, 333 ; dimensions of, 
 
 table, 334. 
 
 Blowers to improve chimney draft, 368. 
 Blue print paper for reproduction, 731. 
 Board and timber measure, 768. 
 Body plans of vessels, wave lines, 546. 
 Boiler, locomotive, details of, 394-396. 
 
 setting, horizontal and tubular, 523. 
 
 stays, forms of, 391. 
 
 tubes, 775. 
 
 corrugated flre-boxes, 395-397. 
 
 Shapley upright, 396. 
 
 Boilers, horizontal, tubular, proportions of, number 
 of tubes, 389, 390. 
 
 water tube, 395 ; Babcock and Wilcox, 796 ; 
 Heine, 797; Clonbrock, 804; Stirling, 867; 
 Abendroth and Root, 868. 
 Bolts and nuts, forms of threads, 250. 
 Bolts, strength of, 255. 
 Boston "Water Works conduit, 458. 
 Boulevard, wide avenues, 474. 
 Boundary lines on topographical drawings, 119. 
 Bowtel moulding, simple fillet and rule joint, 681. 
 Box car, elevation and plan of, 539. 
 
 end of a locomotive rod, 351. 
 
 girders, strength and thickness of steel, 245, 246. 
 Braces and counter braces, 484. 
 Bracing truss of wrought iron between wooden, 
 beams, 249. 
 
914 
 
 INDEX. 
 
 Bracket for baluster, 583 ; ornamental, 886. 
 
 Brass, composition of, 180. 
 
 Brick arches, architectural, 557. 
 pavements, laying of, 478. 
 walls, bond of, 557. 
 walls for foundations, 428. 
 
 Bricks arid brickwork, dimensions and varieties 
 of, 174. 
 
 Bridges, general principles of bracing, 483 ; Howe 
 and Pratt trusses, 499 ; Howe truss highway 
 bridge, 499-501 ; combination truss, Northern 
 Pacific R. E., 502, 503 ; iron bridge, N. Y. & 
 N. II. R. R., 503-505 ; Phoenix Bridge Company, 
 505-507 ; Pratt truss from the Lima and Oroya 
 R. R., 509, 510; highway bridge, King Bridge 
 Company, 508-510 ; ferry landing bridge, 
 512; Rivermont bridge, Lynchburg, 514; ele- 
 vation of a pier and bridge over the Rio Galis- 
 teo, N. M. and S. P. R. R., 515; arch bridges, 
 518 ; viaduct at Minneapolis, 520 ; Cabin John 
 Bridge, 521 ; dimensions of arch bridges, 521 ; 
 Melan concrete arch bridge, 522 ; suspension 
 bridge, 523. 
 spaces between the ports of steam cylinders, 217. 
 
 Bridge trusses, rules for, 498. 
 
 Bridging of floor beams, 558. 
 
 Bristol board, 54. 
 
 Broad Street station, Philadelphia, Pa., 899. 
 
 Brooklyn Water Works conduit, 458. 
 
 Brushes for tints, 162. 
 
 Builders' hardware, 879, 880. 
 
 Building heated by steam, plan of, 649. 
 in angular perspective, 713. 
 materials, 168. 
 
 Built columns, sections of, 232. 
 
 Bulkhead wall, New York city, 419. 
 
 Butler's pantry, water connections, 651. 
 
 Butterfly valves, 375. 
 
 Buttress, Norman, English, flying, 669. 
 
 By-pass pipe to valves, 376. 
 
 Byzantine and Saracenic doorways, 677. 
 ornaments, 685. 
 
 Cabin John Bridge, Washington conduit, 521. 
 Cable-car grip, 874. 
 
 Caisson, steel, 428 ; framing of wooden, 432. 
 Caissons for piers, of the Poughkeepsie, of the 
 
 Susquchanna bridge, 429. 
 Cam punch and shear, 414. 
 Cams, eccentrics, wipers, 343-346. 
 Canal, representation of earth bank, 167. 
 Canals, Erie, Delaware and Raritan, Chesapeake, 
 
 and Canadian, 452. 
 
 Cantilever beams for foundations, 416. 
 Canvas dams, 876. 
 Cape Cod Bay, map of, 105. 
 
 Capitals, Byzantine, Norman, Gothic, 667 ; dis- 
 tinct parts, 080. 
 Car-axles, M. C. B. A., 260. 
 Caryatides, Atlantes, Hermes pillars, 664. 
 Casement of French window, 578. 
 Castings, crystallization in cooling, 178. 
 
 Cast-iron balls, volume and weight of, 772 ; beams, 
 forms of, 241 : shafts, 258 ; connecting rods, 
 354 ; girders, 492 ; pintle and joint details, 
 566 ; pipes, standard weights of, 772 ; posts 
 protected from fire, 564 ; stairs and carriages, 
 584 ; treads and platforms, 881. 
 
 Cast- and wrought- iron piles, 427. 
 
 Catch basins for sewers, 471-473. 
 
 Cathedral of Bourges, piers of, 667. 
 
 Cedar-block pavement, 479. 
 
 Ceilings, furring strips for, 560, 587. 
 
 Cement-faced walks, 475. 
 
 Cement, Portland, natural, sand, 176. 
 
 Centennial Exhibition 1876, buildings of, 907. 
 
 Central Park roads, New York city, 473. 
 
 Centre plates of railway truck, 539. 
 
 Centrolinead, 9. 
 
 Chains, cables, couplings, 336. 
 
 Chains, power transmission by, 299. 
 
 Chain wheels with pockets, 335. 
 
 Chamfer plane, moulded, 677. 
 
 Channel beams, section and dimensions of, 245. 
 
 Chimneys, drawing and description of, wrought- 
 iron, 526-530. 
 
 Chimney tops or cowls, 637. 
 
 Chinese anchors, 429 ; capstan, 194. 
 
 Chords, definition of, 3 ; scale of, 25. 
 
 Churches, 900-906. 
 
 Churches, theatres, lecture rooms, music and legis- 
 lative halls, 620-631. 
 
 Circles, 2 ; radius, diameter, chord, segment, sec- 
 tor, quadrant, 3 ; tangent, 10 ; circles inscribed 
 in polygons, 17 ; circle in a profile plane, per- 
 spective of, 712. 
 
 Circumference of a circle, diameter, arc, 766. 
 
 Circumferences and areas of circles, tables of, 811- 
 819. 
 
 Cisterns and tanks of wood and wrought iron, 
 461. 
 
 Clearances of steam cylinder, 209, 369. 
 
 Clevis, standard, 510. 
 
 Clonbrock water tube boiler, 804. 
 
 Clutch, cylinder-friction, 280. 
 
 Coal, fire, and steam, representation of, 185. 
 
 Coaling bins for locomotives, 497. 
 
 Cofi'er-dam,417. 
 
 Cohoes dam, 442. 
 
 Coils, spiral, flat, of wrought-iron pipe, 403. 
 
 Cold rolled wrought-iron shafts, 178. 
 
 Columns, cast-iron, wrought-iron, Phoenix, Key- 
 stone, strength of, 230-232. 
 
 Combination bridge truss, 503. 
 
 Compacting sands for foundations, 417-419. 
 
 Compasses, 44 ; portable, beam, 45 ; use of, 88. 
 
 Composite beams, wood, iron-trussed, 249. 
 
 Composition and design of figures, Dictionnaire 
 Raisonne" de P Architecture, 731. 
 
 Compound steam engines, 209. 
 steam cylinders, 863. 
 
 Concrete base blocks, Department of Docks, New 
 York city, 421. 
 
 Concrete, lime, and bituminous cements, 176 ; 
 
INDEX. 
 
 915 
 
 floors, 566 ; sewer in situ, 468 ; walls for 
 houses, 558. 
 
 Conduit for electric and cable lines, 482. 
 
 Conduits for water supplies, of wood, cast and 
 wrought iron, of masonry for Brooklyn, Bos- 
 ton, and New York city, 457-461. 
 
 Cone pulleys, 286. 
 
 Conic sections, orthographic projection of, 127. 
 
 Connecting and coupling rods, 347-354. 
 
 Connections of angles for I-beams and Z-bars, 
 566-568. 
 
 Contours, representation of topography, 99 ; head 
 of Franklin, 100. 
 
 Co-ordinates of curvature for maps, table of, 115. 
 
 Copper and brass rods, table of weights, 776. 
 
 Copper in alloys, 179. 
 
 Corbels and brackets, 681. 
 
 Corliss steam engine, 219, 869; valve gear, 220. 
 
 Cornice from the temple of Jupiter Stator at Kome, 
 683. 
 
 Cornices in plaster, 587. 
 
 Corrugated boiler Hues and furnaces, 392, 864. 
 
 Cottage, rural style, 606, 902. 
 
 Cottered joints, 347. 
 
 Cotton spindles, friction in driving, 199. 
 
 Country house, plan and elevation of, 599. 
 
 Coupled I-bearns, 242. 
 
 Coupling and pulley combined, 283. 
 
 Coupling rod, stub end of, 354. 
 
 Couplings, of shafts, face, 273 ; sleeve, screw, cone, 
 274; clamp, box, horn, 275; pipe, Oldham's, 
 Hooke's universal, 277 ; clutch, 278 ; friction 
 cone, Weston double friction cone, 279; elas- 
 tic, Weston disk, 280; cylinder friction 
 clutches, 281 ; magnetic coupling, spring hub, 
 282. 
 
 Cow houses, 633. 
 
 Crank, path of, 211. 
 
 Cranks and crank axles, proportions of, hand, 340, 
 864. 
 
 Crib dam in Colorado, 439. 
 
 Crib dock, 422. 
 
 Cross-head and guides of horizontal engine, 357. 
 
 Crossing stones in streets, 475. 
 
 Cross-section paper, 104. 
 
 Croton conduits, old and new, reservoirs, 458. 459. 
 
 Croton dam, earth portion of new, 876. 
 
 Crowfoot to rafter, 488. 
 
 Culvert, isometrical projection of, 702. 
 
 Curbstones, 476. 
 
 Curves, variable, adjustable, 40 ; elliptic, etc., 41. 
 
 Curvilineal figures, area of, 767. 
 
 Cycloid, 303; epicycloid, hypocycloid, 304; area 
 of, 766. 
 
 Cylindrical surfaces, representation of, 58. 
 
 Cylindrical valves, 372. 
 
 Damper valve, 380. 
 
 Damp stretching of drawing papers, 54. 
 
 Dams, Lake McMillan dam across the Pecos Kiver, 
 Colorado, 437; Ashti tank, India, 438; crib 
 dam, Colorado, 439 ; Holyoke crib across the 
 
 Connecticut River, 440; across the Croton 
 River, 441 ; across the Merrimac at Lowell, 
 442 ; across the Mohawk at Cohoes, 442 ; 
 Beetaloo dam, South Australia, 443; canvas 
 dams, 876; Sweetwater dam, Bear Valley 
 dam, 877 ; movable dam, Great Kanawha, 878 ; 
 section of the new Croton dam, 876. 
 Dash-pot of a Corliss engine, 220. 
 Dead points in crank motion, 212. 
 Deafening of floors, 560. 
 Deane steam pump at Holyoke, 870. 
 Deck beams, 242. 
 
 Density of gases and vapours, table of, 808. 
 Derrick, drawings and details of, 875. 
 Designing of a house, 591. 
 
 Designs, enlarging and reduction of, cloth and 
 wall ornamentation, 60; ornamental, in line 
 and tracery, 77-82. 
 
 Diagram of comparison of United States and met- 
 ric units, 71. 
 
 velocity and path of water in a flume, 72. 
 difference between the charge per ton of transit 
 
 on canal and railroad, 73. 
 annual product of pig iron in the CJ. S., 74. 
 railway time-table, 75. 
 mortality record with range of temperature and 
 
 humidity, 76. 
 
 velocity of falling bodies, 196. 
 expansions under pressures, 208. 
 link movement, 223. 
 strength of wrought -iron columns, 234. 
 strength of wrought-iron beams, 248. 
 strength of shafts, 260. 
 horse power transmitted by shafts; 262. 
 pressures on thrust collars, areas to resist, 273. 
 horse power transmitted by belts, 292. 
 horse power transmitted by ropes, 297. 
 distance between pulleys in rope driving, 297. 
 pitches and faces of gears and stress, 311. 
 elbows, tecs, crosses, and branches for wrought- 
 iron pipes, 402. 
 
 flow of water through pipes, 785-790. 
 proportions of the human figure, 729. 
 flow of gas through pipes, 792, 793. 
 wiring computer, 806. 
 Diapering, architectural, 687. 
 Dike breakwater, 426. 
 Dikes of earth across salt marshes, 438. 
 Dimensions of suspension bridges, table of, 523. 
 
 walls, New York city building laws, 569. 
 Diminishing glass, use of, 751. 
 Discharge of weirs, table of, 782. 
 Dining rooms, kitchens, and parlours, sizes of. 
 
 591. 
 Disengagement of large pulley from main shaft, 
 
 277. 
 
 Dished head for wrought-iron cylinders, 412. 
 Distribution of water mains, 462. 
 Dividers, hair, 44 ; three-legged, proportional, 45. 
 Docks, bulkhead of New York Dock Department, 
 420 ; crib dock west bank, New York harbour, 
 423 ; Thames embankment, London, England, 
 
916 
 
 INDEX. 
 
 424 ; iron pier Coney Island, quay at Calais, 
 
 427. 
 
 Domes and vaults, 667. 
 Doors, dimensions of, parts of, stiles, bottom rail, 
 
 lock, parting, top panels, muntin, architrave, 
 
 studs, jambs; sliding and folding; side and 
 
 transom lights, 571-574. 
 
 Doorways, circular-headed, 676 ; pointed, 677. 
 Dormer windows, 578. 
 
 Double-beat valves for steam and water, 372. 
 Drawing-board, 36. 
 Drawing pen, exercises with, 57. 
 Drawing-table, 37. 
 Drip or cap stones, 681. 
 
 Driver or leader, and driven or follower, 302. 
 Drums or wooden pulleys, 284. 
 Dry rot, 169. 
 Dynamic table, 770. 
 
 Earth, shrinkage in refill, clay, glacier till, hard- 
 pan, quicksand, 167. 
 
 Eccentrics, 342 ; curve, drawing of, 344 ; strap with 
 metallic disks, 349. 
 
 Egg and dart, architecture, 683. 
 
 Egg-shaped sewers, equivalent circular areas, table 
 of, 468. 
 
 Electrical units, 771. 
 
 Electric conduit, 481. 
 
 Electric lighting, wiring for, scries, multiple, three- 
 wire systems, 656. 
 
 Electric switch, lamp socket, 806. 
 
 Elevated Kailroad, Third Avenue, New York, 874. 
 
 Elizabethan style, 689. 
 
 Ellipse, construction of, 30 ; circumference of, 767. 
 
 Elliptic spring applied to car truck, 873. 
 
 Enamelled brick, 175 ; tile, 888. 
 
 English basement house, 599. 
 
 Entasis on columns, 658. 
 
 Equilibrium, stable and unstable, 187. 
 
 Erie Canal, locks of, 453. 
 
 Evaporation, factors of, 800. 
 
 Expansion bolts, 254 : expansion coupling, 271. 
 
 Expansion, law of, for gases, Mariotte, 206. 
 
 Expansive working of steam, table of, 800. 
 
 Factor of safety, 229. 
 
 Fan flush to water-closet, 654. 
 
 Fang nut, 254. 
 
 Fan in connection with radiators, 648. 
 
 Fan-tracery vaulting, 668. 
 
 Fifth powers, table of, 772. 
 
 Fire brick, 175. 
 
 Fireplaces and mantel, 584. 
 
 Fireproof buildings, 563. 
 
 Fireproof of old builders, 561. 
 
 Fire-retarding constructions, 569-571. 
 
 Flange connections for steam and water pipes, 
 
 398. 
 
 Flash boards on dams, 444. 
 Flashings for roofs, 587. 
 
 Flexible joints for submerged water mains, 405. 
 Float trap for condensed water, 645. 
 
 Flooring frame, headers, trimmers, tail beams, 558. 
 
 Floor plan of steel girders and beams, 565. 
 
 Floors, load on, 559, 560. 
 
 Flows of air, 791 ; of gas, diagram, 792, 793. 
 of water and air, comparison of, 794 ; of water in 
 pipes and conduits, 784-790. 
 
 Flues, stacks for house, 585 ; for every room, 636. 
 
 Flumes, discharge of, 783 ; penstock to water 
 wheel, 457. 
 
 Fly-wheels, 408-411. 
 
 Foliage, sculptured, 688. 
 
 Foot-pan and bidet-pan, 652. 
 
 Free-hand drawing, illustrations : proportions of 
 the human frame, 729 ; half-tone of e'corche' 
 figures, 730; pen drawing of ecorche, 732; of 
 Sandow, 733, 734 ; drawing of figures geomet- 
 rically, 735-737 ; figures in skeleton lines and 
 manikins, 738 ; pen drawing of Venus de Milo, 
 739; pen drawings of male hands, 739 ; of logs 
 and feet, of female hands and arms, 740 ; of 
 children's hands and arms, of human head and 
 face, 741; of Electioneer, 742 ; of cow, horse, 
 donkey, 743 ; of hoofs and paws of animals, 
 noses, 744; pen drawing of Southern sketch, 
 745; pumping station, drawn with toothpick 
 and splatter, 746 ; Salvini, Venetian fete on 
 the Seine on stipple paper, 747, 748 ; pen draw- 
 ings of Alexandre Dumas, Erik Werenskiold, 
 749; wash drawings of flowers, 750; design m 
 pen and ink by Fortuny, 752 ; and woodcuts 
 of various sketches and paintings, 753-764. 
 
 Foot-walks in cities. 474. 
 
 Force, definition of, 186. 
 
 Formula for the strength of wrought-iron beams, 
 247. 
 
 Foundations for structures, 415. 
 
 Four-centred arch, proportions of, architecture, 669. 
 
 Framing for stairs, headway, 581. 
 of a caisson, 433. 
 
 Freezing process for foundations, 435. 
 
 Freight shed, of wood, for railroad, 494. 
 
 Fret, guilloche, architectural ornaments, 683. 
 
 Frictional gear, 331-333. 
 
 Friction, coefficient of, Morin's tables, 197 ; of rail- 
 way trains. Chanute, 200. 
 
 Friction-wheels and friction-rollers, 199. 
 
 Furring strips, 560, 587. 
 
 Fusible alloys, 179, 810. 
 
 Gases and vapours, density of, 808. 
 
 Gas fittings, service mains, 656. 
 
 Gas, flow of, 792, 793. 
 
 Gates, guard and canal, Cohoes, 444 ; Lowell, 447 i 
 
 Holyoke, 449 ; Cheney, tubular gates at 
 
 Windsor Locks, 449; Sudbury Kiver Conduit, 
 
 451 ; lock gates, 453. 
 
 Gate valves, Peet, Coffin, Pratt and Cady, 380. 
 Gauging of streams, 784. 
 Gearing, spur, bevel, and screw, 301. 
 Geological map of the United States, 108 ; sections 
 
 of the earth's crust, 109. 
 Geometrical and flowing traceries, 675. 
 
INDEX. 
 
 917 
 
 Girders, cast-iron, table of strength of, 242 ; plate 
 
 and lattice, 502 ; beams, floor plan, 565. 
 Glacier till, 167. 
 Glass, representation and varieties of, transparency 
 
 of, 183. 
 
 Globe valves, dimensions of, 377-378. 
 Glue, mouth, 54. 
 Gold, properties of, 182. 
 Gothic architecture, characteristics of, 665 ; roofs 
 
 of churches, technical names, 627 ; towers, 
 
 spires, 671. 
 
 Governors, balls, shifting cams, 407. 
 Grades of roads and highways, 474. 
 
 of steel, 779. 
 
 Granite block pavement, 475. 
 Gravity, centre of, 186 ; velocity due to, 196. 
 Green economizer in chimneys, 796. 
 Greenhouses, designs for, 634. 
 Greenwood Cemetery, contoured map of, 101. 
 Grid, flexible, for indicator cards, 207. 
 Groined arches in concrete, 563. 
 Grooved pulley, shadow on, 157. 
 Groove packing, test of, 365. 
 Guides, cross-head, 357. 
 Guide pulleys for belts, 289. 
 Gutters of buildings, 586. 
 
 Hard-pan, 167. 
 
 Handrails of stairs, 582. 
 
 Hangers, 265-268 : hanger bolts, 255. 
 
 Hearth and supports, 585. 
 
 Heating by hot water, 645 ; direct and indirect 
 radiation, 642-645. 
 
 Heat and electrical unit, 771. 
 
 Heavy bearings, friction of, 200. 
 
 Height of stories of dwelling houses, 593. 
 
 Heine water tube boiler, 797. 
 
 Helix, orthographic projection of, 139. 
 
 High-stoop city house, 599. 
 
 Highway bridge, Pratt truss, 510. 
 
 Hills, representation of, by verticals, 98 ; by con- 
 tours, 99. 
 
 Hinges and doors, ancient, 881. 
 
 Hodgkinson, experiments of, on columns and 
 beams, 98. 
 
 Hoisting apparatus for small water gates, 447. 
 
 Hollow brick, 563. 
 
 Hood moulding, architecture, 681. 
 
 Hooks, proportions of, 337. 
 
 Hoosac Tunnel, construction and completion of, 
 537. 
 
 Horizontal thrust of arch, 519. 
 
 Hospitals, 630. 
 
 Hot-air furnaces, 638. 
 
 House, designing of, 591. 
 
 Housing for journals, 414. 
 
 Howe truss, 499 ; highway bridge, 500. 
 
 Hydrants, 382. 
 
 Hydraulic press, 195; riveting machine, 412; tees 
 and crosses, 404. 
 
 Hydrometrical and marine survey, plots of, 105. 
 
 Hyperbola, construction of, 34. 
 
 I-beams, 241 ; table of dimensions and strength of 
 
 iron and steel, 243, 244, 779. 
 Idlers, or binders for belts, 291. 
 Inches and sixteenths in decimals of a foot, table 
 
 of, 770. 
 Inclined forces, resultant of, 192 ; plane, principle 
 
 of, 190. 
 
 India ink, grinding, 56 ; slabs for, 57. 
 Indicator cards of a steam engine, 206-210. 
 Injector for boiler feed, 336. 
 
 Inked thumb, for representing a background, 751. 
 Inking in of topographical drawings, 118. 
 Instruments, drawing, management of, 55. 
 Internal gearing, 308 ; wheel driven by a pinion 
 
 and driving a pinion, 325. 
 Involute, construction of, 305 ; teeth, rack, and 
 
 pinion, 308. 
 Iron and plank pipes for the conveyance of water, 
 
 457. 
 
 Iron roofs, corrugated and framed, 490. 
 Iron shoes and plates for braces and rafters of 
 
 roofs, 488. 
 
 Iron tank with inverted-dome bottom, 461. 
 Isle of Wight, chart of, 106. 
 Isometrical drawing, illustrations : Principles of 
 
 cubic projections, curved lines, 698; practical 
 
 application to projection of gear wheels, 699 ; 
 
 pillow-block, water-closet cistern, culvert, 701 ; 
 
 roof frame, plan and elevation of a school- 
 house, ship construction, 704 ; elevation of a 
 
 seaside resort, 705. 
 Italian campaniles, 670; schools of architecture, 
 
 677. 
 
 Jack-rafters, dimensions of, 490. 
 
 Jack-screw, 414. 
 
 Janney car coupler, 215. 
 
 Joinings for beams, 490. 
 
 Joints, pipe, under heavy pressure, 399 ; steam 
 
 pipe, 400. 
 
 Journal bearings of box, M. C. B. A., 272. 
 Joy's valve gear, 226. 
 
 Kentucky, geological survey of, 106. 
 Keys, metal strips to secure hubs to shafts, 259. 
 Kinzua Viaduct, 514. 
 Kitchen range, boiler, and sink, 650. 
 Knuckle joint, 347. 
 
 Korting blower to increase draft in flues, 368. 
 Kutter's formula for flow of water in pipes, graph- 
 ically, 784-790. 
 
 Landing bridge for ferry, 513. 
 Lap and lead, slide valve, 217. 
 Latitudes and departures, table of, 830-835. 
 Lattice bars, spacing of, 232. 
 
 deck bridge, bill of material, 505. 
 Lead, properties of, 181. 
 
 pipe, weights of, 780. 
 Leather link belting, 301. 
 
 packing for pumps and cylinders of hydraulic 
 presses, 363. 
 
918 
 
 INDEX. 
 
 Legislative halls, requirements of, 630. 
 
 Lettering, varieties of, triangles for, 62. 
 
 Lever, principle of, 188 ; hand and foot, 338 ; under 
 inclined forces, 195. 
 
 Lewis, for raising stones, 254. 
 
 Lineal measure, table of, 768. 
 
 Line, geometrical, 2 ; horizontal, vertical, parallel, 
 5 ; irregular, plotted, 91 ; orthographic projec- 
 tion of, 122. 
 
 Lines of shafting, laying out of, 262. 
 position and division, guide lines, 727. 
 
 Link belting, 300. 
 motion, 222. 
 
 Lintels, 557. 
 
 Liverpool water-works, Norton Tower, 461. 
 
 Load, dead, live, 229. 
 
 Lock nut, 251 ; washers, 256. 
 
 Locks of canals, 453. 
 
 Locomotive, driving-wheel of, 341 ; boiler for, 395 ; 
 plan and elevation of frame of, 546 ; No. 999, 
 N. Y. C. & H. R. R. R., 871 ; N. Y., O. & W., 
 872 ; distribution of load, 873. 
 
 Logarithms of numbers, table of, 845-859 ; applica- 
 tion of, 860. 
 
 Longitude, table of length of a degree of, 
 116. 
 
 Loop system of piping, 802. 
 
 Louis Quatorze style, Louis Quinze style, 690. 
 
 Lundell electric motor, 807. 
 
 Machine and blacksmith shop, city, 614. 
 
 foundations, 535. 
 Machines, location of, 530. 
 Malleable cast-iron, 178. 
 Man-hole, 390; hand-hole covers, 863. 
 Man-holes, for sewer, 470. 
 Mansard roof, 585-586. 
 Mantel and fireplace, plan, section, and elevation 
 
 of, 882, 883. 
 
 Mantels, flues, jambs, 584. 
 Map projections, orthographic, stereograph ic, 
 
 globular, Mercator's, conic, Bonne's, polyconic, 
 
 110-115. 
 
 Marine boiler of steamer Minneapolis, 395. 
 Mariotte, law of, 206. 
 Marquetry, examples, 885. 
 Masonry, conventional signs of, technical terms 
 
 for, representations of, 171. 
 Masonry curbs sunk by water jets, 427. 
 
 terms of, 171. 
 
 Materials, earth and wood, characteristics and rep- 
 resentation of, 167-171. 
 Measures of surface, 768 ; of capacity, 769 ; cubic 
 
 or solid, 770. 
 Mechanical stokers, Wilkinson, Coxe, 866. 
 
 work or effect, 201. 
 Mensuration, 766. 
 Mercator's projection, 112. 
 Meridians, topographical drawing, 119. 
 Metals, antimony, bismuth, copper, lead, tin, and 
 
 zinc, properties of, 179 ; conventional signs to 
 
 represent, 177 ; table of properties of, 809. 
 
 Metres and United States units, graphic compari- 
 son of, 71. 
 
 Mill constructions, tiro-retarding, 569. 
 
 Miner's inch, 784. * 
 
 Morin, experiments of, on friction, and table of 
 sliding and rolling, 197. 
 
 Mortality and disease by graphics, 76, 77. 
 
 Mortars, lime, cement, sand cement, 175, 176. 
 
 Mortise wheels, proportions of, 316. 
 
 Motion, 211, 228. 
 
 Moulded timbers, 682. 
 
 Mouldings, classical, Romanesque, Gothic, Nor- 
 man, 679. 
 
 Greek and Roman, 588 ; stuck in wood, 589 ; 
 perpendicular style of, 682. 
 
 Mounting paper and drawings, varnishing, 55. 
 
 Movable d>m, Great Kanawha River, 878. 
 
 Muntz metal, 180. 
 
 Mutules and guttse, 683. 
 
 Nails and spikes, weight, table of, 777-778. 
 
 Natural sines and cosines, table of, 836-844. 
 
 Neutral surface under transverse stress, 240. 
 
 New Haven, map of the harbour and city of, 101. 
 
 New York city schoolhouse, a, 619. 
 
 New York State canals, lock specification, 455. 
 
 Nipple, close and shoulder, 403. 
 
 Northern Canal at Lowell, Mass., section of, 452. 
 
 Nuts, various forms of, 251. 
 
 Open fire in a tavern, 638. 
 Orders of architecture, Tuscan, 659. 
 Doric, 660. 
 Ionic, 662. 
 
 Corinthian and Composite, 664. 
 Organs of churches, 627. 
 
 Ornamental mouldings, chevron, billet, star, fir 
 cone, cable, embattled, nail head, dog tooth, 
 ball flower, serpentine, vine scroll, 686. 
 Ornament, architectural, 682. 
 Orthographic projection, 121 ; of a point, of a line, 
 of a solid, of simple bodies, 123; conic sec- 
 'tions, 127 ; intersection of solids, 130 ; the 
 helix, 139. 
 
 Ox gall for drawing on the ordinary photograph, 
 731. 
 
 Packing for water-pumps, 362 ; of stuffing-boxes, 
 
 70. 
 
 Paints, 184. 
 
 Palace of Diocletian, 665. 
 Pan-closet, 653. 
 Panels, ceilings in Italy, 562. 
 Pantagraph, 51. 
 
 Paper, drawing, tracing, transfer, parchment, he- 
 liographic, 52. 
 
 pencils, chalks, pens, ink, 727-728. 
 
 profile and cross-section, 71. 
 Parabola, construction of, 33 ; area of, 766. 
 Parallel motions, 215. 
 Parallelogram, rhombus, rhomboid, 16. 
 
 of forces, 193. 
 
INDEX. 
 
 919 
 
 Parallels, 22 ; parallel ruler, drawing of, 39. 
 
 Parapets, architectural, 687. 
 
 Paris boulevard, 475. 
 
 Partitions, framing of, 556. 
 
 Passenger car, elevations and sections of, 539. 
 
 Patent Office requirements, drawings, registration, 
 765. 
 
 Pavements, granite- block, with and without con- 
 crete foundation, 476 ; asphalt, 477 ; Salt Lake 
 City, 478 ; brick, cedar block, 479. 
 
 Pediments, brackets, railing, 886. 
 
 Pelton water-wheel, 204. 
 
 Pen-and-ink drawings, to clean, 751. 
 
 Pencils, marks of, 1. 
 
 Pen, drawing or right-line, 42; railroad, border, 
 curve, 42 ; dotting, 43. 
 
 Perspective drawing, planes of, 706 ; points of, par- 
 allel and angular, 707 ; of squares, cubes, 
 scales for, prisms, pavement, horizontal circle, 
 in profile, cylinder, octagonal prism, building, 
 interior of room, arch bridge, schoolhouse, 
 cottage, stairs, reflection of objects in water^ 
 projection of shadows ; capstan and winch, 725. 
 
 Pews, length of, 627. 
 
 Pier, iron curb of, with piles driven inside, 431. 
 
 Piers of the Third Avenue Bridge, 431. 
 
 Piers, Poughkeepsie Bridge, 430 ; pile, 497 ; trestle 
 bent, 498; Kinzua Viaduct, 513; over Kio 
 Galisteo, N. M. and S. P. R. R., 513 ; Third 
 Avenue Elevated Suburban, 515, 516 : stone 
 pier of railroad bridge over Susquehanna at 
 Havre de Grace, 516; of bridge across the 
 Missouri at Bismarck, 517. 
 
 Pile-pier, 497. 
 
 Piles for foundations, 417 ; splicing of, 419. 
 
 Pillow-block, isometrical projection of, 699. 
 
 Pillow or plumber-block, standard and hangers, 
 263. 
 
 Pin-nut, standard, 510. 
 
 Pinion driving a rack, 324. 
 
 Pipe coupling, lead, 404. 
 for driven wells, table of, 776. 
 air-chamber, 650. 
 
 Pistons of steam engines, and of pumps, 360. 
 
 Piston-ring and packing, 362. 
 
 Pitch of roof, 486. 
 
 Plan and elevations of a small house, drawings of, 
 548-553. 
 
 Plane table, 84. 
 
 Plan of church, transept, nave, and chancel, 625. 
 
 Plastering, 176: furring of walls, 587. 
 
 Plate girder, bill of material for, 503. 
 
 Plates and covers, wrought-iron, 863. 
 and wire, weights of wrought iron and brass, 
 table of, 774. 
 
 Platforms for foundations, grillages, 415. 
 
 Plots, transferring of, 110. 
 
 Plotting, scales used in, 83 ; traverse table used in, 
 87 ; meridian assumed in, 89 ; irregular lines, 
 91. 
 
 Plough for electric conduit, 482. 
 
 Plugs and caps for pipes, 403. 
 
 Plumbing, 649. 
 
 Pneumatic piles, 431. 
 
 Point, geometrical, 2 ; pricking, 43 ; tracing, 44. 
 
 Polygons, 14; irregular, 17; inscribed, 18; con- 
 struction of, 19; similar, 26; regular, areas of, 
 766. 
 
 Pondage, rule of, for permanent mill powers, 436. 
 
 Pop safety valve, 382. 
 
 Porous brick tiles, 564. 
 
 Ports of steam cylinders, 217 ; exhaust, 217 ; di- 
 mensions of, 371. 
 
 Principles of architectural design, 691. 
 
 Printing frame for heliographic paper, 53. 
 
 Proportions of the members of a roof, 489. 
 
 Protractor, 21, 25, 50. 
 
 Privies, water-closets, and outhouses, 593. 
 
 Pulley, principle of, 190. 
 
 Pulleys, 282 ; cast-iron, 283 ; plate, wrought-iron 
 rim, split, pulley and coupling combined, 
 wooden-plate pulley, drums, cone, fast and 
 loose, 286 ; guide, 289 ; idlers or binders, 291. 
 
 Pumping engine at St. Louis, 345 ; Leavitt pump, 
 365; Worthington, 366; Deane pump, Hoi- 
 yoke, 870 ; Reidler valves, to Leavitt's pump, 
 870. 
 
 Quicksand, 167. 
 
 Quoin and pintal for bedpost of lock, 455. 
 
 Rack gear and pinion, 301 ; involute teeth of, 309 ; 
 
 rack driving a pinion, 325; pinion driving 
 
 rack, 324. 
 
 Radiating surface for heating, 643. 
 Radiators, wall coil, box coil, wrought-iron tube, 
 
 cast-iron loop, cast-iron pin, 647. 
 Rail joints of the West Shore Railroad, 480. 
 Railroads, standard sections of permanent way, 
 
 480. 
 
 Rails, standard, drawing and dimensions of, 481. 
 Railway rolling stock, 539-543. 
 
 surveys, plots of, 103. 
 Ranges, United States survey, 93. 
 Reciprocals, table of, 828 ; use of, 861. 
 Register valve for steam, 381. 
 Reidler water valve, 870. 
 Renaissance style, 677 ; ornaments of. 688 ; Tri- 
 
 cento and the Quatrecento, Cinquecento, 689. 
 Reservoirs for water-works, 459. 
 Retaining walls, 435. 
 
 River wall, Thames embankment, 422-426. 
 Riveted joints, lap, single, double, treble riveted, 
 
 butt and angular connections, 383-386. 
 Riveting, conventional signs of, 866. 
 Rivets for plate girders, 247; forms and dimen- 
 sions of, 383 ; pitch of, 777. 
 Roads and highways, dirt, gravel, oyster shells, 
 
 Macadam, Telford, 472, 473. 
 Rolled I-beams, section of, 242. 
 iron, table of weight of, 773. 
 Rolling friction on roads, 200-201. 
 Romanesque and Byzantine architecture, 665. 
 church or basilicon, 624 ; pillars, 667. 
 
920 
 
 INDEX. 
 
 Roman order, characteristics of, 665; vaulting, 
 
 608 ; school of architecture, 678. 
 Roofs, plans and sections of, 585. 
 Roof truss, isometricul projection of, 702. 
 
 parts of, 485 ; varieties of, 490. 
 Rooms, proportions and distribution of, 589, 594. 
 Ropes, transmission of power by, 293, 299. 
 Rubber, properties and use of, 184. 
 
 ring joint, 405. 
 
 valves, 374. 
 
 Rudder post and screw frame, 864. 
 Rulers and triangles, 36-38. 
 Russian towers, architecture, 674. 
 
 Safety valve, 381. 
 
 Safe-vault construction, 881. 
 
 Sag of rope in driving, 298. 
 
 Salted paper, recipe for, 731. 
 
 Sand cement, 176. 
 
 Saracenic diapers, architecture, 685. 
 
 Saturated steam, table'of, 798. 
 
 Scales, 25 ; application of, 27 ; forms of, plotting, 
 46; diagonal, 47; vernier, 48; oft-set, 92; on 
 drawings for photography, 119; for lines in 
 perspective, 709. 
 
 School-houses, 616-622 ; isometrical view of, 702. 
 
 Screw pile, 427. 
 
 Screw, principle of, 191 ; the differential, 194. 
 
 Screw propeller, 865. 
 
 Screws, shades and shadows on, 157 ; wood and 
 metal, 252 ; drawing of triangular and square 
 threaded, 327. 
 
 Scroll moulding, 681. 
 
 Seasoning of timber, 168. 
 
 Seats, desks, school furniture, 616; space occupied 
 by, 624. 
 
 Sector, drawing instrument, 49 ; area of, 766. 
 
 Sewers, 466 ; of vitrified ware or cement, 467 ; 
 Washington, Brooklyn, 466. 
 
 Shade lines, 144. 
 
 Shading and shadows, manipulation of, and meth- 
 ods of tinting, 159-166. 
 
 Shadows, perspective projection of, 721. 
 
 Shafting, diagram of strength and length between 
 bearings, 260. 
 
 Shafts, cold-rolled, 178 ; wooden, iron, and steel 
 257 ; cast-iron, plan and sections, 259. 
 
 Sheds for wood or coal, 594. 
 
 Sheet-iron arches for concrete floors, 563. 
 
 Sheet-piling, 417. 
 
 Ship construction, wave-line, isometrical projec- 
 tion of, 704. 
 
 Shipping measure, 770. 
 
 Shoe for wooden curb of well, 428. 
 
 Silver, properties of, 182. 
 
 Sine, cosine, versed sine, secant, tangent, 21. 
 
 Sink, cast-iron, 650. 
 
 Skeleton construction of iron and steel, 565, 693, 
 898. 
 
 Skeleton frame of working beam, 354. 
 Sketching from Nature, 745. 
 Skew bevels, plan of, 323. 
 
 Skew bridges, 520. 
 
 Smith's process for coating pipes, 465. 
 
 Sockets for wire ropes, 336. 
 
 Soil or house-sewer pipe, 650 ; extension to roof, 
 652. 
 
 Soldering union, nipple, 403. 
 
 Solders, composition and use of, 810. 
 
 Solids, orthographic projection of, 122; intersec- 
 tions, 130. 
 
 Spaces occupied by check valves, table of, 376 ; 
 globe valves, 378. 
 
 Specials for water mains, 463. 
 
 Specifications of pipe mains, Brooklyn, N. Y., 465. 
 
 Specific gravity of liquids, earths, 808; woods, 
 metals, and gases, 809. 
 
 Speed of belts, 291, 292. 
 
 Sphere, development of the surface of, 144 ; shade 
 on, 155. 
 
 Spherical bearing, 866. 
 
 Spike frames, 555. 
 
 Spiral, construction of, 35. 
 riveted pipes, 776. 
 
 Spire finials, 673. 
 
 Spires, 900, 901 ; of English churches, 671. 
 
 Splatter work in drawing, 751. 
 
 Split pulleys, 283. 
 
 Sponge, means of correcting errors in drawings, 
 166. 
 
 Springs, driving, equalizing bar, elliptic, bolster,' 
 873. 
 
 Sprocket wheels, 300. 
 
 Spur wheel, drawing of, 317; oblique projection 
 of, 319. 
 
 Spur wheels, parts of, 302, 306. 
 
 Square, multiples of, 27 ; on the hypothenuse, 29 ; 
 reduction of areas of, 30. 
 
 Squares, cubes, and roots of numbers, table of, 820- 
 
 827. 
 divisions into triangles and octagons, 59. 
 
 Stables, barn, carriage house, stable proper, floor 
 of, 631. 
 
 Stairs, 578; treads and risers, fliers and wind- 
 ers, landing, headway, nosing, strings, car- 
 riages, newel post, and baluster, 579 ; laying 
 out, framing for, 581 ; hand rail, 582 ; wrought- 
 iron strings and rails, cast-iron treads and 
 risers, 584. 
 
 Stalls, pitch of bottom and breadth of, 631. 
 
 Stamp mill, 347. 
 
 Standard for the support of shafting, 265. 
 
 Standard I-beams, channels, A. A. S. M., 779. 
 
 Standard rails, dimensions of, 481. 
 
 Stanhope levers, 212. 
 
 Stationary boilers, Philadelphia Water- Works, 392. 
 
 Stay bolts, proportions of, 392. 
 
 Steam and hot-water circulation as a means of 
 heating, 641. 
 
 Steam cylinders, 359. 
 
 Steam engine, horizontal frame, 356 ; Corliss, 219, 
 220, 869. 
 
 Steam heating, arrangement of mains and returns 
 for, 642. 
 
INDEX. 
 
 921 
 
 Steam, its application, 205. 
 
 Steam jacket, 346. 
 
 Steam piston packing, 347. 
 
 Steam valve, plan and section of, 379. 
 
 Steel, homogeneous metal, 178 ; from pure wrought- 
 iron, 179. 
 
 Steelyard and platform scales, 194. 
 
 Step for an upright shaft, 269. 
 
 Steps for stairs, breadth of, treads of, and height 
 of risers, 579. 
 
 Stiffeners for the webs of plate girders, 247. 
 
 Stipple paper or clay board, 751. 
 
 Stirling boiler, 867. 
 
 Stirrup irons for wooden beams, 559. 
 
 Stokers, mechanical, Wilkinson and Coxe, 866. 
 
 Stones and masonry, conventional signs of, 171. 
 
 Stones, granitic, argillaceous, sand, lime, charac- 
 teristics of, 173. 
 
 Stop chamfering, 682. 
 
 Stores and warehouses, 612, 614. 
 
 Stoves, open and close, 638. 
 
 Straight edges, 37. 
 
 Strength of men and animals, 202. 
 
 Stress, tensile, compressive, shearing, transverse, 
 tortional, 230, 234. 
 
 String courses, 680. 
 
 St. Sophia, roof of, 667. 
 
 Studs for house framing, 556. 
 
 Stuffing boxes and glands, packings for, 369. 
 
 Stylus, tracing point, 44. 
 
 Suspension bearings, 270. 
 
 Suspension bridges, table of dimensions, 523'. 
 
 Sulphur, characteristics of, 182. 
 
 Summer house, 887. 
 
 Surfaces, development of, cylinders and cones, 141. 
 in shade, tinting, 160. 
 
 Sweetwater arched masonry dam, 877. 
 
 Table of railway curves, 103. 
 co-ordinates of curvature for maps, 115. 
 length of degree of longitude, 116. 
 sliding and rolling friction, Morin, 198. 
 safe loads of cast-iron columns, 231. 
 safe loads of Phoenix columns, 232. 
 safe central load of yellow-pine beams, 239. 
 strength of wrought-iron I-beams, 243. 
 strength of steel beams, 244. 
 strength of steel box girders, 245. 
 dimensions and weights of Z-bars, 247. 
 dimensions of bolts and nuts, United States 
 
 standard, 256. 
 
 proportions of sunk keys, 260. 
 distances between bearings of shafts, 261. 
 horse power transmitted by shafts, 262. 
 horse power transmitted by wire ropes, 298. 
 pitch, diameters, and teeth of gears, 312. 
 relation of diametral to circular pitch, 313. 
 radius of arcs of circles for gear teeth, Adcock, 
 
 313-315. 
 
 sizes of sheaves and blocks, 334. 
 capacities and sizes of hooks. 337. 
 dimensions of eyes and cranks, 339. 
 
 Table of space occupied by check valves, 376. 
 
 dimensions of globe valves, 378. 
 
 dimensions of single-riveted lap joints, 384. 
 
 dimensions of double-riveted lap joints, 385. 
 
 dimensions of treble-riveted lap joints, 386. 
 
 number of tubes in horizontal and tubular boil- 
 ers, 390. 
 
 proportions of stay bolts for flat surfaces, 392. 
 
 dimensions of pipe flanges and cast-iron pipes, 
 399. 
 
 dimensions of wrought-iron tubes and coup- 
 lings, 401. 
 
 diameters and thicknesses of cast-iron pipes, 
 with lead for joints, 463. 
 
 size of egg-shaped sewer and circular equiva- 
 lents, 468. 
 
 dimensions of standard rails, 481. 
 
 dimensions of parts of roofs, 489, 490. 
 
 dimensions of a wrought-iron roof, 492. 
 
 material for plate-girder bridges, 503. 
 
 material for lattice-girder bridges, 505. 
 
 arch bridges, dimensions of, 521. 
 
 suspension bridges, dimensions of, 523. 
 
 loads for floors, 559. 
 
 theatres and their dimensions, 630. 
 
 polygons, chords, verticals, and areas, 766. 
 
 lineal measures and of surfaces, 768. 
 
 capacity, liquid and dry measure, 769. 
 
 weights, apothecaries', Troy, avoirdupois, dy- 
 namic, 770. 
 
 cubic or solid measure, 770. 
 
 inches and sixteenths in decimals of a foot, 770. 
 
 fifth powers, 772. 
 
 weight of cast-iron balls, of cast-iron pipes, 772. 
 
 weight of rolled iron, 773. 
 
 weight of wrought-iron and brass plates, 774. 
 
 wrought-iron welded tubes, of boiler tubes, 775. 
 
 weight of pipes for driven wells, spiral pipe, 
 light pipe for leaders, and air pipes, 776. 
 
 weight of copper and brass rods, 776. 
 
 weight of rivets, spikes, 777 ; of cut nails, ot 
 iron nails, of telegraph wire, 778. 
 
 weight of beams and channels of the A. ot 
 A. S. M., 779. 
 
 weights of lead pipe, of a cubic foot of water, 780. 
 
 discharges of water over weirs, 782, 783. 
 
 equalizing the diameter of pipes, 791. 
 
 volume and. weight of dry air, 794. 
 
 saturated steam, 798, 799. 
 
 expansive working of steam, factors of evapora- 
 tion, 800. 
 
 mils and ohms, 807. 
 
 specific gravities of gases, of liquids, of earths, 
 etc., 808. 
 
 woods and metals and properties of, 809. 
 
 circumferences, diameters, and areas of circles, 
 811-819. 
 
 squares, cubes, and roots of numbers, 820-827. 
 
 reciprocals, 828, 829. 
 
 latitudes and departures, 830-835. 
 
 natural sines and cosines, 836-844. 
 
 logarithms of numbers, 845-859. 
 
922 
 
 INDEX. 
 
 Tanks, lead-lined, coated with asphalt varnish, 
 650. 
 
 Taps for city mains, sizes of, 649. 
 
 Telegraph and telephone lines, table of sizes of 
 wire, 778. 
 
 Teredo and Limnoria, 169. 
 
 Terra cotta, 175, 888. 
 
 Thames-Ditton pump, 365. 
 
 Theatres, dimensions and plans of, 630. 
 
 Thrust bearings for screw-propeller shafts, 272. 
 
 Thumb-nut, 338. 
 
 Timber frames, forms and dimensions of parts, 
 555. 
 
 Timber, sections, conventional signs, seasoning, 168. 
 
 Tin, properties of, 182. 
 
 Tinting and shading, manipulation of, 159; pre- 
 paring colours for, 164. 
 
 Tires of wagons, 201. 
 
 Titles of maps and charts, 69. 
 
 Toggle-joint, 195. 
 
 Topographical drawing, conventional signs of, 95 
 
 Topography, coloured, 116; conventional colours 
 of, 117. 
 
 Torsional stress, 234. 
 
 Tower for water tank in New York city, 674. 
 
 Towers, Eomanesque, 670. 
 
 Traceries, perpendicular, leaf, flamboyant, Sara- 
 cenic, and Moorish, 676. 
 
 Tracing cloth, 52. 
 
 Trains, time-table of, 74. 
 
 Trammel, ellipsograph, 31. 
 
 Transverse stress of beams, 235. 
 
 Traps, antisiphoning, 653 ; for sewer pipes, 655. 
 
 Trestle bent of elevated railroad, 498. 
 
 Trestles for drawing-table, 37. 
 
 Triangles for drawing, 37. 
 
 Triangle and square, use of, 22. 
 
 Triangles, isosceles, equilateral, right-angled, simi- 
 lar, 26. 
 
 Triple compound steam engine, 210. 
 
 Trundle pins or wheels, 301. 
 
 Truss bridges, effect of unequal loading, 484: 
 wooden, Howe, Pratt, 499. 
 
 Trusses for roof and floor of a gymnasium, 493. 
 
 Trussing of a beam by struts and tension rods, 250. 
 
 T-square, 38. 
 
 Tubes, weight of wrought-iron welded, table of 
 775. 
 
 Tubs set for washing, 651. 
 
 Tudor arched doorway with hood mouldings, 677. 
 
 Tunnels and principles of timbering, 535. 
 
 Turbine, Fourneyron, Boyden, and Jonval, 204. 
 
 Turn-buckle or swivels, 255. 
 Turn-table, 510. 
 Tusk tenons, 558. 
 
 United Electric Light and Power Station, 801. 
 
 United States Survey, ranges of, 93. 
 
 Unit of force and space, 201. 
 
 Upright boiler, Shapley's, 396. 
 
 Upset of bolts, 255. 
 
 Urinals, 656. 
 
 Valve diagrams, steam, 218. 
 gear, Corliss, link, Joy's, Walschaert, 219-228. 
 motion of St. Louis pumping engine, 345. 
 motion, slide valves, 216. 
 
 Valves, automatic, double-beat, 372; poppet, disk, 
 rubber, flap, ball, air, pump, loaded flap, but- 
 terfly, check, 377 ; safety, 381 ; controlled by 
 hand. 376; cocks, bibs, plain, hose, compres- 
 sion, air, stop and plug cocks, globe, straight, 
 angle and cross, damper, 380; regulator, 
 steam-hammer, hydrant, 383. 
 Variable speed gear, 333. 
 Vaulting, fan-tracery, 668. 
 Vaults and domes, 667. 
 Venetian school of architecture, 678. 
 Ventilation and warming, 634. 
 
 by compressed air in French exhibition, 368. 
 Vessels in launching, friction of, 198. 
 Vestibule doors, 884. 
 Villa, Italian, 607. 
 
 Wagon tires, 473. 
 Wall girders, position of, 568. 
 Walls, dimensions of, New York city laws, 569. 
 Walls in masonry, 556 ; concrete, 555. 
 Walschaert valve gear, 226. 
 Wash-basins, sizes of, 651. 
 Wash drawings, 749. 
 Washers, table of, 256. 
 Water-back, 650. 
 
 Water-closets, appliances of, 652 ; basins, 655 ; cis- 
 tern, isometrical projection of, 700. 
 
 washout, hopper, pan, flap, siphon-jet, 652-654. 
 Water, diagram of path and velocity in flume, 72. 
 
 flow of, 781. 
 
 jet for the sinking of piles, 426. 
 
 lines of a ship, 546. 
 
 mains, dimensions, and weight of, 462, 463. 
 
 pipe of sheet iron, 460. 
 
 power and its applications, 203. 
 
 weight of a cubic foot of, at different tempera- 
 tures, table of, 780. 
 
 wheels, tub, flutter, breast, overshot, undershot, 
 
 Scotch, turbines, Pelton, 203, 204. 
 Wave-line principle of ship construction, 545. 
 Waves of sound in halls of audience, 623. 
 Weather-cocks, 673. 
 Weaving-room, location of looms, 531. 
 Wedge gearing, 332. 
 Weight of gas mains, 472. 
 Weights, apothecaries', Troy, avoirdupois, 769. 
 
 of material, 177. 
 
 Weirs, table of discharge of, 782. 
 Westinghouse engines, steepled compound, 804. 
 Weston-Capen double-friction clutch, 279. 
 Wheel and axle, principle of, 189. 
 Whitworth's quick return motion, 213. 
 Winch, centreboard, 724. 
 Windlass, 724. 
 
 Window frames, sashes, blinds, 573-578 ; dimen- 
 sions of, 577. 
 Windows and doors, examples of, 889-897. 
 
INDEX. 
 
 923 
 
 Windows and doors, Byzantine, Romanesque, Nor- 
 man, Lancet, traceried, 674. 
 Wire nails, weight of, 778. 
 
 ropes, sockets for, 336. 
 Wiring computer, Carl Hering, 806. 
 Wooden plate pulley, 285. 
 shaft in plan and section, 258. 
 steps for shafts, 271. 
 packing for pump pistons, 363. 
 Woods, characteristics and use of, 169 ; white pine, 
 Southern pine, Canadian red, Norway, and 
 silver pines, spruce, hemlock, ash, chestnut, 
 black walnut, butternut, hickory, beech, live 
 oak, white oak, bass, poplar, white wood, 
 cedar, locust, elm, maples, 170. 
 Working beam, 354. 
 Working strain of one-inch rope, 296. 
 
 World's Fair buildings, 908-910. 
 Worm and worm wheel, 330; the Albro, 3'28. 
 Worthington stearn pump, 3.66. 
 Wrought-iron columns, strength of, 233. 
 
 diagram of, 234. 
 
 rim pulleys, 283. 
 
 crank connections of river-boat engines, 354. 
 
 pipe connections, 400. 
 
 tubes and couplings, dimensions of, 401. 
 
 curb pier with inside piles, 431. 
 
 trestles for bridges, 514. 
 
 chimney stack, 530. 
 
 string and rail for stairs, 583. 
 
 spikes, table of weight of, 777. 
 
 Zinc, properties and uses of, 182. 
 
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 a century in the libraries not only of this country, but of Europe also. The painstaking in- 
 vestigations and rare erudition it embodies, the broad field it has so diligently delved and 
 gleaned, and the reverent Christian spirit it manifests, make it a monumental work, destined 
 to become a classic authority on the subjects it has made its own henceforth. The previous 
 works in this field, such as Dr. Draper's ' Conflict of Religion and Science,' and Professor 
 Shields's ' Final Philosophy,' must yield pre-eminence to President White's admirable history. 
 The new book is far fuller and more accurate in its narrative, clearer in its treatment, and 
 more judicious in its judgments." The New World, London. 
 
 " It is a complete survey of the whole field of battle. . . . The chapters are full of striking 
 interest. The author of these volumes shows that the warfare of science has never been with 
 religion but only with old errors that were confounded with it, and that the Eternal Verities 
 become the more clear and sure as the ancient guesses which have been confounded with 
 them are cleared finally away." London Daily News. 
 
 " Such an honest and thorough treatment of the subject in all its bearings that it will carry 
 weight and be accepted as an authority in tracing the process by which the scientific method 
 has come to be supreme in modern thought and life." Boston Herald. 
 
 " It is graphic, lucid, even-tempered never bitter nor vindictive. No student of human 
 progress should fail to read these volumes. While they have about them the fascination of a 
 well-told tale, they are also crowded with the facts of history that have had a tremendous 
 bearing upon the development of the race. " Brooklyn Eagle. 
 
 " A conscientious summary of the body of learning to which it relates, accumulated during 
 long years of research. . . . A monument of industry." New York Evening Post. 
 
 " So interesting as to enchain the attention at once and keep it enchained. Concise as a 
 history of the universe could be made, tabulated so that instant reference to a particular bit 
 of history, theory, or biography may be had, it will be valuable as a lexicon relating to 
 religious controversy." Chicago Times- Herald. 
 
 " Dr. White knows much of science and he knows much of theology. But his point of 
 view is that of a historian. It is this fact which gives the greatest value to his book. We 
 have whole libraries of controversial works dealing with the relations between science and 
 theology, but they have been written either by scientists or theologians. President White 
 occupies the impartial position of the historic scholar, who has no prejudices against the 
 truth of science and no hostility toward the truth of religion." N. Y. Review of Reviews. 
 
 ' ' The work is a masterpiece of a mind as devoid of wanton iconoclasm as of moral 
 cowardice. It is a definite statement of where the best thinkers of the world now stand in 
 the religio-scientific conflict. It is clear, honest, brave, and must be given a place among the 
 great books of the year." Chicago Tribune. 
 
 "This is and will continue to be one of the great books of the world, like Thucydides's 
 ' History of the Peloponnesian War.' So long as Science and Theology retain their place in 
 human interest, this history of the conflict of ages between them will exert its attraction and 
 read its lesson. It is a great book." The Outlook. 
 
 NEW YORK: D. APPLETON & CO., 72 FIFTH AVENUE. 
 
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