THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF John S.Prell THE MECHANICS AKCHITECTURE WORKS BY E. WYNDHAM TARN. THE SCIENCE OF BUILDING: Ax ELEMENTARY TREATISE ON THK PRINCIPLES OF CONSTRUCTION. By E. WYNDHAM TARN, M.A., Architect. Third Edition, Revised and Enlarged, with 50 Engravings. Fcap. 8vo, 4s. cloth. PRACTICAL GEOMETRY FOR THE ARCHITECT, EX- GINEER, AND MECHANIC. Giving Rules for the Delineation am! Application of various Geometrical Lines, Figures, and Curves. By E. W. TARN, M.A., Architect. Second Edition. With 172 Illus- trations, demy 8vo, 9s. cloth. THE CONSTRUCTION OF ROOFS OF WOOD AND IRON : Deduced chiefly from the Works of Robison, Tredgold, and Humber. By E. WYNDHAM TARN, M.A., Architect. Second Edition, revised. 12mo, Is. 6vo-<.o\oyia dicitur philosophia explicat, QU.iiu necesse est stinliosius novisse, quod habet multas et varias naturales quas- tiones." VITBUVIUS on the Education of the Architect. "Architect! est scientia pluribus disciiiliuis et variis emditionibus ornata .... ea nascitur e fabrica et ratiocinatione .... at qui utrumque perdidicerunt, uti omnibus annis ornati citius cum auctoritate quod fuit propositum simt adsecuti." YITRUVIUS, Book I. Engineering Library TA PEEFACE. THE numerous works that have been written upon the subject of Applied Mechanics seem to be generally intended more for the use of the Engineer than the Architect. The training of the Engineer being essentially a scientific one, he is compelled to devote much time to the study of mechanics ; while the Architect, whose chief aim is to become an Artist, has to give most of his hours of study to the arts of drawing and designing ; so that he has but little time, and often less inclination, to make acquaintance with the science of his profession, being generally contented to take his formulae from some "pocket-book," without caring to enquire into the principles upon which those formulas are based. In the following pages the Author has endeavoured to supply a want which is felt by many Architects, by bringing together, in a small compass, all that it is essential for the Architect to know upon this subject, and to give in as simple a form as possible an outline of the principles upon which all good construction should be based. In doing so it has been assumed 733252 VI PFxEFACE. that the reader possesses a moderate acquaintance with algebra and trigonometry ; and although it has been necessary to the solution of some problems to invoke the powerful aid of the Calculus, yet by far the greater part of the work can be understood and the examples worked out by those who have no knowledge of the higher mathematics. The Author would call particular attention to the theories and examples of Koofs, Arches, Vaulting, and Domes, to which he has given great prominence in this work, as being those subjects in which the Architect is especially interested. He has also worked out numerous examples showing the practical application of every theory and formula, in order that the reader may never be at a loss to understand how to use them. Those who wish to pursue the various subjects more deeply are referred for information to the following Authors, to all of whom the present writer desires to acknowledge his deep obligations : Bow. Economics of Construction, by E. II. Bow. Clarke. Graphic Statics, by G. S. Clarke. Fenwick. Mechanics of Construction, by S. Eenwick. Graham. Graphic and Analytical Statics, by E. II. Graham. Moseley. Mechanical Principles of Engineering and Archi- tecture, by Henry Moseley. Rankine. Manual of Applied Mechanics, by W. J. M. Eankine. Stoney. Theory of Strains, by B. B. Stoney. Weisbach. Mechanics of Engineering, by Julius Weisbach. Wray. Application of Theory to Practice, by H. Wray. CONTENTS. CHAPTER I. FORCES IN EQUILIBRIUM. PAGE 1. Measure of Force 1 2. Resultant of Forces 3 3. Resolution of Forces ........ 4 4. Triangle of Forces . . . 5. Triangular Frame 11 6. Polygon of Forces 12 7. Polygonal Frame . . . . . . . .13 CHAPTER II. MOMENTS OF FORCES. 8. Definition of a Moment 16 9. The Lever 18 10. Couples .- . . . 20 11. Transposition of Couples 21 12. Stress on a rod fixed at one End 22 13. Do. do. supported at each End .... 25 14. Distributed Load 27 15. Beam loaded at two or more Points 29 CHAPTER III. CENTRE OF GRAVITY. 16. Definition of Centre of Gravity 31 17. Centre of Gravity of Triangular Plate . . ; . 32 18. Do. do. of any Trapezium 35 yiii CONTENTS. PAGE 19. Centre of Gravity of any Polygon ..... 36 20. Do. do. of Girder Sections 36 21. Do. do. of Circular Arch 38 CHAPTER IV. RESISTANCE OF MATERIALS TO STRESS. 22. Modules of Elasticity 41 23. Do. of Tensile Elasticity 43 21 Do. of Compressive Elasticity 44 26. Do. of Transverse Elasticity 44 23. Coefficient of Strength 45 27. Resistance to Crushing 46 28. Coefficient of Safety 47 29. Transverse Stress 48 30. Rectangular Beam 52 31. Beam supported at each End 54 82. Safe load on a Beam 55 33. Examples of Application ....... 56 31 Beam fixed firmly at Ends 58 35. Do. supported at three Points 59 36. Beams not Rectangular 62 37. Iron Beams ......... 62 38. Cast-iron Beams 64 39. Wrought-iron Beams 67 40. Angle-iron 69 41. Girders with Angle-irons 71 42. Steel Beams 74 43. Beams of Uniform Strength 75 44. Shearing Stress 77 CHAPTER V. DEFLEXION OF BEAMS. 45. Resistance to Bending ... 79 46. Radius of Curvature . CQ 47. Beam supported at each End . 81 48. Do. with distributed Load c<> 49. Do. fixed at one End ..'.'.'.' 34 CONTENTS. IX PAGE 50. Tredgold's Rule 86 51. Practical Examples 87 52. Scantlings of Floor Timbers . . , . . . . 89 53. Flanged Beams 93 54. Beam of Uniform Strength . , 94 CHAPTER VI. STRENGTH OF PILLARS. 55. Long Pillars 98 56. Hodgkinson's Experimental Formulas 100 57. Shorter Pillars 105 58. Table of Strength of Pillars 108 59. Gordon's Formulae 109 60. Short Pillars . . . Ill CHAPTER VII. ROOFS, TRUSSES. 61. Roofs 112 62. Lean-to Roof 113 63. Span-Roof 117 64. Collar-Roof 120 65. Hammer-Beam Roof ... .... 127 66. Trussed Roofs 131 67. King-post Roof 135 68. Stress Diagram 143 69. Queen-post Roof 147 70. Scantlings of Roof Timbers 151 71. Iron Roofs 151 72. Roofs of large Span 159 73. Bowstring Truss 164 74. Warren Girder . . . . . . . . 168 75. Lattice Girder 171 CHAPTER VIII. ARCHES. 76. Principle of the Arch 174 77. The Inclined Plane 175 & X CONTENTS. PAGE 78. The Wedge ... *.. 178 79. Application to the Arch 1 80. Joint of Rupture . . . 3 81. Stability of the Arch 187 82. Loaded Arch 19 83. Line of Pressures 202 84. Arcades 205 85. Segmental Arch 207 86. Pointed Arch 210 87. Surcharged Pointed Arch 215 88. Tudor Arch 221 89. Surcharged Tudor Arch 227 90. Elliptic Arch 231 91. Surcharged Elliptic Arch 234 92. Vaulted Roofs, Cylindrical . . 237 93. Gothic Vaulting . .243 94. Arched Iron Ribs 246 CHAPTER IX. DOMES, SPIKES. 95. Hemispherical Dome 255 96. Semi-Domes 264 97. Dome of Sta. Sophia at Constantinople . . . .267 98. Surcharged Dome . . . 271 99. Dome of Unequal Thickness 275 100. Gothic Dome 280 101. Do. do. with Lantern 287 102. Conical Dome 290 103. Octagonal Spire .... ... 293 CHAPTER X. BUTTRESSES, SHORING, RETAINING WALLS, FOUNDATIONS. 104. Buttresses . 296 105. Flying Buttress . . 300 106. Shoring . . . . . , . . . 304 107. Retaining- Walls, Pressure of Water 310 CONTENTS. XI PAGE 108. Retaining Walls, Earth Pressure, Surface Level . . . 318 109. Do. Pressure of Earth, Surface Sloping . . 324 110. Do. Buttress of Earth 330 111. Foundations . . '. . . . . . .333 112. Piling 335 CHAPTER XL EFFECT OF WIND ON BUILDINGS. 113. Pressure of Wind on a Plane - . . . . . .340 114. Do. on a Cylinder . . ." . . . 343 115. Do. on Eoofs . . 346 CHAPTER XII. MISCELLANEOUS EXAMPLES AND SOLUTIONS . 352 INDEX . . 365 S. PRELL Civil & Mechanical Engineer SAN FRANCISCO, CAL. MECHANICS OF ARCHITECTURE. CHAPTER I. FORCES IN EQUILIBRIUM. 1. MEASURE OF FORCE. The term " Force " is ap- plied to a certain action of one body upon another which causes a change in the circumstances of one or other, or both of the bodies. It is the object of the science called " Mechanics " to examine into and determine the laws of force and the results which different forces produce. By the term " Applied Mechanics " (as far as the present treatise is con- cerned) we mean the application of the laws of force to Architecture. It is usual to represent mechanical forces by means of straight lines whose directions indicate the direc- tions in which the forces act upon a body ; and a length measured to scale upon any of these lines is taken to represent the magnitude of the force in tons, hundredweights or pounds avoirdupois. When a force acts at any point of a rigid body, that is, of a body which does not yield or change its shape under the action of the force, an exactly similar and equal force is set up in the body, acting however in a diametrically opposite direction to the impressed force ; 2 FORCES IN EQUILIBRIUM. this opposing force is called the "reaction" of the body, and is always equal in magnitude and opposite in direction to the impressed force. Thus, if a load is placed on hard ground, a force exactly equal to the weight of that load presses upwards upon it and prevents it from sinking into the ground. A familiar example of reaction is seen when a man in a boat pushes himself away from the bank of a river by means of a pole ; the reaction of the bank being equal to the pressure exerted by him upon the pole. It is by reaction that the paddles or screw of a steam- vessel, or the oars of a boat, propel the vessel through the water. The only impressed force that we have to take ac- count of in building construction is the force of " gravity," which is the term applied to the attraction of the earth upon every particle of matter which comes within its influence, and which is commonly called weight, being proportional to the " mass " of the body attracted, and measured as before stated in tons, hundredweights and pounds. The force of gravity is always considered, as far as buildings are concerned, to act in parallel lines, and in a direction perpendicular to the surface of still water. When we speak of the " load" which any structure has to sustain, we mean the force which the earth exerts upon every particle of the mass which is placed upon that structure, by pull- ing it towards its surface. There is another kind of force which we shall have to consider, and which is altogether independent of "gravity," namely, the "resistance" which all solid bodies offer to any impressed force tending to crush RESULTANT OF FORCES. 3 them or tear them asunder. This resistance is called the " force of cohesion," and varies greatly in different bodies, as we shall see hereafter. When the force pushes against the body it is called a " compressive " .or " crushing stress ; " when the force tends to pull it asunder, it is called a " tensile " or "extending stress." When it tends to cause the body to bend or break across in the direction of the force, it is termed a " transverse stress ; " and when it acts transversely as .a cutting force tending to cause the body to separate into slices, and the slices to slide off each other, it is termed a " shearing stress." The term " stress " is applied to the action of an .external force upon the fibres or particles of a body when applied to its extremities either as a pushing or pulling force ; while the term " strain " is applied to rthe effect of the stress in changing the shape of the Ibody. Thus the change of length arising from a tensile or compressive force is termed " longitudinal strain." By " unit-stress " is meant the stress upon u omit of sectional area ; " inch-stress " being the stress represents R: and they will act in the directions shown by the arrows. If then any one of the three forces, as W, is known, the other two can be deter- mined, as follows : W AB x sin. ABE, or AB = , AB x cos. ABE W = DE T = BE = = | W x cotan. ABE R = BD = The angle DBE which R makes with the horizontal r is determined from the equation R . sin. DBE = DE = W ; or, sin. DBE = ^ . . . . - (3) A familiar example .---/& O f the triangle of forces will be found in the bracket or crane 7 where an inclined strut BC (fig.8) presses against a wall at B, and also against the end of another beam AC at C where a load W is supported. Let the vertical line C represent the load W, and draw ad parallel to BC, ab parallel to AC, producing AC to d. Then the three lines Ca, ab, bC represent the force W, the stress, TRIANGULAR FRAME. 11 F, in AC, and the stress in OB ; and these forces are in equilibrium if taken in order. Therefore the force F, in AC which balances the stress produced by W at C, must act from C towards A, and is a pulling one, so that AC might be a cord or chain ; while the force acting in BC tp^ balance the stress produced by W must act from U towards B, or the beam BC is sub- jected to a pushing force. 5. TRIANGULAR FRAME. -Suppose a triangular frame to be formed of three rigid straight bars jointed together at their ends, as ABC (fig. 9) ; and to be acted Fig 0. .10. - Ir upon at the vertices A, B, C, by the forces P 1? P*, and P 3 , which keep the frame in equilibrium. From what has been just stated (4) the direction of these three forces must meet in a point D, and the forces them- selves will be proportional to the sides of the triangle abc (fig. 10) which are respectively parallel to their directions. Suppose these forces to produce in the bars of the frame the stresses S 1? S 2 and S 3 . From the vertices of the triangle abc draw lines ao, bo, co, parallel respectively to the sides of the triangle ABC. Then ao being parallel to AB will represent S 1? the FORCES IN EQUILIBRIUM. Ficf.1i stress in the bar AB; bo being parallel to AC will represent S.,, the stress in the bar AC ; co being parallel to BC will represent S 3 , the stress in the bar BC. The first triangle ABC (fig. 9) is called the " frame- diagram," and the second triangle abc (fig. 10) is called the "stress-diagram." The lines which in the former make a closed figure are represented in the latter by lines meeting in a point; and the lines which in the latter make a closed figure are represented in the former by lines meeting in a point. Consequently the two triangles are termed " reciprocal figures." 6. POLYGON OF FORCES. The principle of the " triangle of forces " can be extended to any number of forces which are in equili- brium at a point ; the forces being repre- sented in magnitude and direction by the sides of a polygon re- spectively parallel to their directions, the polygon having as many sides as there are forces. Suppose the five forces P, Q, "R, S and T, to be in equilibrium at the point A (fig. 1 ] ), and to be represented in magnitude and direction by the lines AB, AC, AD, AE, and AK. First, describe a parallelogram ABGC on the lines AB and AC ; then the diagonal AG represents the resultant of P and Q. POLYGONAL FRAME. 13 Next, draw a parallelogram AGHD on the lines AG and AD ; then the diagonal AH represents the resultant of AG and R, or of P, Q and R. Thirdly, draw the parallelogram AHIE, upon the lines AH and AE ; then the diagonal AI represents the re- sultant of AH and S, or of P, Q, R and S. The force T, represented by AK, must be equal in magnitude and opposite in direction to the last resultant AI, in order that the forces may be in equilibrium at the point A. The resultant AI closes the polygon ABGHIA, the sides of which represent the magnitude and direction of the five forces, P, Q, R, S and T. 7. POLYGONAL FRAME. As the principle of the u Triangle of forces " can be extended to ^7 ^2 A a any number of forces acting at a point, so the principle of the "triangular frame" E< (5) can be extended to a frame having any number of sides. Suppose a polygonal frame formed of more than three rigid straight rods jointed at their ends, as ABCDEA (fig. 12) to be kept in equilibrium by the forces P 15 P.,, &c. acting at the joints A, B, &c. and to produce in the bars the stresses S^ S 2 &c. Then each joint must be in equilibrium under the forces- winch act upon it. The forces P 1? B! and S 5 , acting at A must be represented by the three sides of a 14 FORCES IN EQUILIBRIUM. triangle which are parallel respectively to their directions (4). First, let the triangle Oae (fig. 13) have the side Oa parallel to AB representing S, (fig. 12) ; the side Oe parallel to AE representing S 5 ; and the side ae parallel to the direction of P r Next, let the triangle Oab have the side Oa representing S 1? as before ; the side Ob parallel to BC representing S 3 ; the side ab parallel to the direction of P,. Thirdly, let the triangle Obc have the side Ob represent- ing S 2 , as before ; the side be parallel to CD representing S :{ ; the side be parallel to the direction of P.,. Fourthly, let the tri- angle Ocd have the side Oc representing S s , as above ; the side Od parallel to DE representing S 4 ; the side cd parallel to the direction of P 4 . Lastly, we have left the triangle Ode, of which the side Od represents S 4 , as above ; the side Oe parallel to EA represents S 5 ; then the side fie gives the direction and magnitude of P 5 , in order that the forces may balance. Thus each joint of -the frame (fig. 12) furnishes a triangle (fig. 13), and each triangle has one side com- mon to only two other triangles ; and all these triangles put together make up the closed polygon abcdea (fig. 13), the outer lines of which represent the magnitude and direction of the several exterior forces P 15 P., . . . P 3 ; while the inner lines Oa, 0^, ... Of, represent in POLYGONAL FEAME. 15 magnitude and direction the several stresses S 15 S,, . . . S 5 , in the bars of the frame (fig. 1 2). Thus ea represents P 15 ab represents P.,, be represents P 3 , cd represents P 4 , mid de represents P 5 . Oa represents S 1? Ob stands for S,, Oc for 83, Od for S 4 , Oe for S 5 . It will be seen from the above that the fifth force P 5 can only be determined both in magnitude and direction by the line de which closes the stress- diagram, so that if P x , P 2 , P 3 and P 4 are given, or known, beforehand ; the force P 5 which balances them, remains to be determined, as in the theorem of the polygon of forces previously given (6). The line de representing P 5 is said to close the polygon. CHAPTER IT. MOMENTS OF FORCES. 8. DEFINITION OF A MOMENT. Whenever there is a fixed point in a body which is acted upon by external forces, those forces will tend to turn the body round, or cause it to rotate, about the fixed point, provided that the directions of the external forces do not pass through the fixed point. Let be a fixed point in a body which is acted on at A by the forces P, Q and R (fig. 14). Draw OB perpendicular to the direction of the force P ; OC perpendicular to that of force Q; and OD perpendicular to that of the force R. Then P x OB P is called the moment of P about ; Q x OC the moment of Q about ; and R x OD the moment of R about 0. The moment of a force about a point measures the tendency of that force to cause the body to rotate about an axis through the fixed point, such axis being at right-angles to the plane in which the force acts, or to the plane of rotation. If two forces, as P and Q, acting at a point A of a body in which there is a fixed point 0, keep it in equilibrium: then their moments about must be DEFINITION OF A MOMENT. 17 equal in magnitude, but tending to turn the body in opposite directions ; or we must have, P x OB = Q x OC; OB and OC being respectively perpendicular to the directions of P and Q, as in (fig. 14). If R is the resultant (3) of the forces P and Q, then it is demonstrated, in elementary treatises on Mechanics, that the moment of R about is equal to the sum of the moments of P about and of Q about ; or, drawing OD perpendicular to the direction of R, we have R x OD - P x OB + Q x OC . (4) , the moments being taken about an axis through at right-angles to the plane in which the directions of the forces lie. In the diagram (fig. 14) the moment of Q tends to rotate the body in an opposite direction to that which P does, consequently one of the moments must be considered negative and the other positive. When therefore we speak of the sum of the moments of the components, we mean their algebraic sum, each moment being taken with its algebraic sign of plus or minus. In any case where the components have opposite signs, the moment of the resultant is equal to the difference of their two moments. The example previously given (4) of the bracket or crane (fig. 8) will also serve to illustrate the applica- tion of the theory of moments. For if we call F the force acting in AC, and "W the load acting vertically at C, we can determine their relations by taking their 18 MOMENTS OF FORCES. moments about the fixed point B, at the foot of the strut. Draw Be perpendicular to CA, By horizontal or perpendicular to Ca ; then in equilibrium we must have F x Be = TV x Kg, or F:TV = By : Be. Draw B/ parallel to AC , meeting Ca in/; fh parallel to Be ; then the triangles B Ae, fCh, are equal, and are similar to the triangle By/". Also the triangle Cab is similar to the triangle C/B. Therefore we have F : TV = By : Be = B/ : BA = B/:C/ = 4 : C as shown to be the case by the "triangle of forces " (4). 9. THE LEVER. Suppose ACB to be a rigid straight rod having a fixed Fi 9 o 15 point or fulcrum" /TX at C (fig. 15) and ety/ I ''; to be acted on by C X . i / V^ the forces P and ^r ''-' c / -^B Q at its extremities \^ A and B. Draw v'y R Q Ca perpendicular to the direction of P, Cb to that of Q. Then P x Ca and Q x Cb are the moments of P and Q about C. If the rod is kept in equilibrium by these forces, then we have (8) P X Ca = Q x Cb . . . (5) Suppose that the directions of P and Q meet in the THE LEVER. 19 point ; then since their moments balance about C, the moment of their resultant R must be zero, or its direction passes through the point C. In order to find the value of E, or the pressure which the forces P and Q produce upon the fulcrum C, take Oc to represent P, Od to represent Q, and draw the parallelogram Qced, making de parallel to Oc, and ce parallel to Od ; then the diagonal Oe (3) represents the resultant R. If the point is at a considerable distance from C, the length of the diagonal Oe is nearly equal to the sum of the lines Oc and Od, and the further it is off the more nearly do they approach to equality. Consequently, when the directions of P and Q are parallel to one another, or the point is at an infinite distance, we must have R = P + Q and also, P x AC = Q x BC . (6) We have here the principle of the "lever; " and by moving the fulcrum, C, very near to the end, B, of the rod, we see that a very small weight or force acting at A will balance a large one at B ; for we get If Q is 100 Ibs., and BC is the hundredth part of AC, then we find from this equation that P = 1 Ib. will balance at A the weight Q = 100 Ibs. at B. Also since BC = AB - AC . BC _ AB - AC _ AB _ 1 ' 'AC AC AC And we can put the above equation (7) into the form C 2 20 MOMENTS OF FORCES. There are three distinct kinds of lever, which are classed according to the relative positions of the three points A, B, C. In the first kind of lever the fulcrum, C, lies between A and B, as in fig. 15. In levers of the second kind the point B at which the load Q acts lies between A and C (fig. 16). And as the above equation (7) holds good in every case, P must be always less than Q in this kind of lever, since AC is always greater than AB. P Fig. 16. Fig. IT In the third kind of lever (fig. 1 7) the point A at which P acts lies between B and C, the weight Q acting at one end, B, of the lever, the fulcrum C being at the opposite end. Then since AC is always less than BC, P must always be greater than Q. 10. COUPLES. In the third kind of lever (fig. 17) the resultant, B, of the forces P and Q, acting at C, will be the difference of P and Q, since they are acting in opposite directions (8) ; or, II = P Q. Then we have Q x BC = P x AC = P (BC - BA). Therefore, (P - Q) x BC = P x AB or BC = ? -?- ^ X AB := I x AB . . (0) Now the quantity, or ratio, v , gets greater and Jr VI TRANSPOSITION OF COUPLES. 21 greater as P Q diminishes, or as P and Q approach to equality ; so that we may say that when P is equal to Q the point C goes off to an infinite distance, and therefore there is no point of application of the resul- tant K. The two equal and opposite forces, P, Q, acting at A and B, constitute what is called a " couple ; " and the moment P x AB, or Q x BA, is termed the "moment of the couple." The effect of a couple is to communicate an angular motion about an axis perpendicular to the plane in which the forces act ; and two equal and opposite couples acting in the same plane will produce equilibrium in the body on which they act, The effect of a couple upon a body is not altered by any change in the forces P, Q, provided that the moment remains the same, or that the length of the arm of the couple is changed in the inverse ratio ta the change in the forces. 11. TKANSPOSITION OF COUPLES. Suppose a body to have a fixed point, C (fig. 18), and to be acted upon at A and B by two forces, P and Q, whose directions are in the same plane; and that the moments of these two forces about the fixed point C are equal, so that P x AC = BC. At the point C apply the forces P! and P a acting in opposite directions to one another, but in a direction parallel to that of P at A ; and let P t = P 2 = P. 22 MOMENTS OF FORCES. Then since Pj and P. 2 are equal and opposite, the equilibrium of the body remains undisturbed. Also ' apply at C the forces Q t and Q 2 acting in opposite directions to one another, but in a direction parallel to that of Q at B ; and let Q t = Q 2 = Q. Then the equilibrium will remain undisturbed since Qj and Q 2 balance each other. f We have now two equal and opposite couples, namely, f P x AC x P 2 , and Q x BC x Q 2 , which balance each other and can therefore be removed without affecting the equilibrium of the body. When these couples are removed there remain the two forces Pj and Q x acting at C in direction parallel to the original forces P at A and Q at B. Also, by hypothesis, \>e have Pj P and Q! = Q. Hence it follows that when two forces act in the same plane upon a body, and their moments about a fixed point therein are equal to one another, we can transpose those forces unaltered to that fixed point about which they balance; and they may be considered as acting at that fixed point in directions parallel to their original directions respectively, and producing the same effect upon the body as the original forces did. This theorem is of importance in determining the stabilily of arches and domes (Chapters VIII. and IX.) 12. STRESS ON A ROD FIXED AT ONE END. Suppose a rigid rod AB (fig. 19) to be fixed firmly into a wall at A, and to be loaded at the other end B by a weight W. Then W x BA is called the " moment of stress " about the point A ; and the " moment of stress " about any other point D on the rod is W x BD. Using STRESS ON A ROD FIXED AT ONE END. 23 the letter " M" to represent the " moment of stress " at any point, D, we have M = W x BD . . . (10) or, M is proportional to the distance of D from B. If then we take the vertical line AC to represent on any scale the moment, W X BA, and draw the hypo- thenuse BC, the vertical line, or ordinate, DE will represent on the same scale the moment W X BD ; and similarly for any other point between A and B. c Fig . 19. - -E I Fig 20. A. C B \ W 2 i A D \ 4% \ n v | Suppose tlie load W to be divided into two equal parts, half being placed at the end B of the rod, and half at the middle point C (fig. 20). Then the moment of stress at A is, M = ^ x BA + ^ x CA = |WxBA . . (11) or the stress is one-fourth less than when the whole of "NV is placed at the end B. Now let W be divided into four equal parts, one- fourth being placed at each of four equidistant points dividing the rod AB into four equal parts ; then we have 24 MOMENTS OF FORCES. 4 4 = f W x AB . . (12) If we divide AB into eight equal parts, and place one-eighth of W at each point, we obtain in the same way M = - r 'VWxAB . . . (13) It therefore appears that as we increase the number of points over which W is distributed, the value of M approaches nearer and nearer to M = i W x AB . . . (14) and this is its value when the number of points is infinite, or when the load is uniformly distributed over the whole length of the rod ; so that the stress at A when the load is distributed is one-half that produced by the same load placed at the further end B. If we put / for the length of the rod, w for the load per unit of length, then W = n x /, and we have M = i m . / 2 . . . (15) P If x is the distance of \,E Fi( * 21 an y point D from the end :\ B (fig. 21), then putting M \ T""---, f r moment of stress at D, we nave M = i re x A- 2 . . (16) Putting y = i w x x\ we have the equation of the curve called the parabola, the vertex of which is at B. Let the vertical line AC represent on any scale the STRESS ON A ROD SUPPORTED AT EACH END. 25 value of y, or M, when x = BD, as in equation (16) ; then the vertical DE, measured on the same scale, will represent y when x = BD : and similarly for any other point on AB. If then we draw a curve through the points (J, E . . . B, it will be a parabola. If on the other hand a parabola is drawn through C,E . . . B, its " ordinates " DE, etc., will represent the values of J n>. x*. If m is the distance from B of the " focus " of the curve, then by the principle of the parabola, we have x 2 = 4 m . y or, y = x 2 4 m consequently, m = Q . For example, if we put m = I Ib. per inch, AB = 12 inches, then AC = J x 12 2 = 72, and m = J, or the distance of the " focus " from B is T l T th of the height AC. 13. STRESS ON A HOD Fig.M. SUPPORTED AT EACH END. Let the rod A B (fig. 22) be supported at A and B and strained at any intermediate point C by a load W. Suppose P at A and Q at B to represent the vertical reactions of the ends of the "rod upon the supports. Then the stress upon the rod will be the same as if we suppose C to be a fixed fulcrum and then take the 26 MOMENTS OF FORCES. moments of P and Q about C. Since the forces are supposed to keep the rod in equilibrium, their moments (8) about C must be equal, or Q x BC = P x AC, either of which is the moment of stress about C ; and the rod is subjected to the same straining force as if it were fixed in a wall at A, in which case we found (12) M = Q x BC by equation (10). Then since the moments of W at C and Q at B may be supposed to balance when taken about the point A, we have Q x AB = W x AC. Therefore, Q = W ^j. xix5 Substituting this value of Q in the former equation r we get for the moment of stress at C, M = Qx BC = W~ C -*, . . (17). AB The value of M is greatest when AC = BC = AB, or when C is the centre of the rod ; in which case equation (17) becomes, M = I W x AB . . . . (18) The moment of stress at any other point, D, on the rod, is the moment of Q with respect to D, namely, M = Q x BD = W.^ C x _?? . . (19) A. J.5 If we take the vertical CE to represent the quantity w * > and draw the straight lines AE, BE ; DISTRIBUTED LOAD. 27 then the ordinate DF will represent the moment of stress at the point D. Also if ordinates are drawn between any other points on AB to the lines BE, AE, they will represent the moments of stress at those points. Putting AB = /, and AC = x, we have for stress at C, by equation (17), --* . . (20 ) and when x = M = . I . . . . (21) 4 14. DISTRIBUTED LOAD. Suppose that in the last case the load W is uniformly distributed over the whole length of the rod, n being the load per unit Fig. 23 of length; then, W= A *> c E B n- . L To find the moment || \ of stress at any point m ^ C between A and B (fig. 23), bisect AC and BC in D and E ; and let W = P+ Q, where P acts along AC, and Q acts along BC. Then we may consider P and Q to act at D and E. Let M! be the moment of stress at C produced by P, nnd M 2 that produced by Q ; M the total moment of stress at C. Then M = Mj + M 2 ; and we have by equation (19) ,, -p AD x BC P AC x BC Ml=sP AB " = 2'- AB 28 Also, MOMENTS OF FOECES. Therefore, M = M x + M 2 = =- = W AC x BC ~2 ' AB . (22) If C is the centre of the rod, then AC = BO = tf?, and M = - . = . AB (93^ O A 8 \ / Putting \Y = re . AB, the stress at any point C will be found by equation (22) to be, M = ~ AC x_BC = nx AB AC x BC 2 ' AB ~2~ AB = | . AC (AB - AC) . . (24) and when C is at the centre, AC = | AB = %l ; in which case, -* (25) Suppose C is the centre of the rod AB (fig. 24), and D is any other point at distance a- from C, then BEAM LOADED AT TWO OR MORE POINTS. 29 therefore by equation (24) M = ?ADxBD = *(L f _ 2 2 \ 4 is the stress upon the point D. If then we draw a vertical line CF to represent re . / 2 , and draw a parabola AFB, the ordinate DH will represent on the same scale the moment of stress at D ; or the moments of stress at any points along AB are represented by the ordinates of a parabola whose vertex is at F, and whose focus, S, is found by measured FS = --, on the same scale that CF re- presents I . I*. If CD = -. , then DH = f FC. 15. BEAM LOADED AT TWO OR MORE POINTS. Suppose a rod AB, supported at each end, to be loaded at D and E (fig. 25) with the weights Wj and W 2 ; then (13) we have for the stress at the middle point 6, caused by the load W x at D, from equation (10), The moment of stress Fig. 25. Q at C caused by the load Jf D c E \B AV 2 at E is, by equa- HJj ^ J tion (19), , T w BC x BE W, p.,, M. = W t ._ jg- =-BE and M = M! + M 2 is the total moment of stress at the middle point C. 30 MOMENTS OF FORCES. If we suppose D and E to divide the rod into three equal parts, then AD = BE = l\ and supposing also that Wj = W 2 = W, we have for moment of stress at C, = Wx . . . (26) If the rod is divided into four equal parts, and loaded with a weight, W, at each point, then the moment of stress at C will be found in the same manner to be, If there are four such loads placed at equal distances, we find in the same way, CHAPTER III. CENTRE OF GRAVITY. 16. DEFINITION OF CENTRE OF GRAVITY. The several particles of which any body is composed may be considered as so many weights or forces acting in parallel directions ; these forces or weights resulting from the action of " gravity," or the earth's attraction upon the several particles. Such a system of parallel forces must have a " resultant " (2), and the point through which this resultant passes is termed the " centre of gravity " of the body. The centre of gravity may therefore be considered as that point at which the whole weight of the body acts, and where it produces the same effect as the weight of the body would produce. The centre of gravity of a body may also be con- sidered as that point about which the algebraic sum of the moments (8) of all the weights of the particles being taken, is equal to zero ; or the moments of the weights of those which are positive exactly balance the moments of the weights of those which are negative, and tend to turn the body in the opposite direction to the former. Since the force of gravity acts equally on all the particles, the position of the centre of gravity is quite independent of the amount of the force of gravity, and therefore does not vary with any change in the position of the body. 32 CENTRE OF GRAVITY. In order that a heavy body may be supported, the direction of the supporting force must pass through its centre of gravity. Hence it follows that we may consider the weight of a rigid body to be collected at this point, and that if the centre of gravity is in equilibrium under the forces acting upon it, then the body itself is kept in equilibrium. From a knowledge of this fact we are enabled readily to find the position of the centre of gravity of any thin flat substance by means of a simple experiment. Let the body be sus- pended freely at any point on its perimeter, and a plumb-line dropped from the same point, the direction of the plumb-line being marked upon its flat surface. Do the same at some other point on the perimeter, and the intersection of the two plumb-lines is the centre of gravity, since each of them passes through that point. The position of this point in a straight rod of uniform density and dimensions throughout, will evidently be at its middle point; and the same will be the case with a rectangular thin plank of uniform thickness, which may be considered as made up of a number of parallel rods. 17. CENTRE OF GRAVITY OF A TRIANGULAR PLATE. We may consider that a triangular plate as ABC (fig. 26), is made up of a number of straight rods Fiy.26. lying parallel to one side or base, BC or AB, and diminish- ing in length from the base of the vertex. Bisect BC in D, and draw AD ; then the centres of CENTRE OF GRAVITY OF A TRIANGULAR PLATE. 33 gravity of the rods lying parallel to BC must be at their intersection with the line AD, and consequently that of the whole triangle must be on that line. Bisect AB in E, and draw CE ; then the centres of gravity of the rods lying parallel to AB .must be at their intersections with the line CE, and that of the whole triangle must be on that line ; and therefore it is at the intersection Gr of AD and CE. By the geometry of the figure DG- is one-third of AD, and EG one-third of CE. Suppose it is required to find the centre of gravity of a trapezium having two parallel sides, as ABCD (fig. 27), where AD is parallel to BC, but BA and CD will meet if produced in the point E, forming two triangles AED and BEG. Bisect BC in F, and draw EHF. Take F^ equal to one-third of EF, and %' equal one-third of EH ; g and g' will be the centres of gravity of the two triangles. Let P represent the area of the triangle AED, Q that of the trapezium ABCD; then we may consider P and Q as forces acting at the centre of gravity of each figure. Let G be the centre of gravity of the trapezium ABCD, the position of which we have to find. Then the moment of P at/ must balance that of Q at G, both being taken about the centre of gravity g of the whole triangle BEC ; or we have, Fig. 27 P x gtf = Q x G//. 34 CENTRE OF GRAVITY. Then, % = ? x fj ( f = Q (Ey - Ey') = P x (EF - EH). Now as P and Q represent the areas of the two figures AED and ABCD, we have P _ EH x AD _ . Q " Ef x BO - EH x AD' also, 55 = 55 si nce tne triangles are similar. AD D(J therefore, EH x ^ _ AD Q EF -D/> 2 EH XB ~ AD 2 ~ BC 2 - AD* ; therefore, Gy = f x HF We can determine geometrically the position of the point y by taking B& equal to one-third of BC, and drawing ag parallel to BA and intersecting HF in g. So that the position of G can be found without draw- ing the triangle EAD, by simply bisecting AD in H, and BC in F, then joining HF and drawing ag in the manner above described. Then the distance Gy can be calculated by the equation (29). CENTRE OF GRAVITY OF ANY TRAPEZIUM. 35 18. CENTEE OF GRAVITY OF ANY TRAPEZIUM. Let ABCD (fig. 28) be the given trapezium ; draw the diagonals AC and BD intersecting in b. A Fig. 28 Bisect BD in and draw OA, 00. Take Oa = OA, and Oc = 1 00. Then a and e are the centres of gravity of the triangles BAD and BCD re- spectively (17) ; and the centre of gravity, O, of the trapezium will be found on the line ac which is parallel to the diagonal AC. Since the triangles have a common base BD, their areas must be propor- tional to the perpendiculars Ad, Ge, dropped from the vertices A and upon the diagonal BD. And since x Gtf. In working out this problem it is not necessary to draw the perpendiculars Ad and C0, which are only introduced for the purpose of demonstration. Since either diagonal of a parallelogram divides it into two equal triangles, the centre of gravity will evidently be at the intersection of the diagonals. D 2 3G CENTRE OF GRAVITY. 19. CENTRE OF GRAVITY OF ANY POLYGON. Since a polygon can be divided into triangles, we have first to find the centre of gravity of each triangle (17), and then that of two triangles as above (18), and combine this with the centre of gravity of a third triangle, and so on for as many triangles as the figure is divided into, which is two less than the number of sides. Thus if we suppose the triangle BEG to be added on to the trapezium ABCD (fig. 28), we form a five-sided figure divided into three triangles. Having deter- mined G the centre of gravity of the two triangles ABD, BCD, or of the trapezium ABCD in the manner above described (18), we determine , and taking hg = I E^. Join Gy, and divide Gg at K in the proportion of the areas of the trapezium ABCD and of the triangle BEC ; taking* GK to represent the latter and K^7 the former. Then K is the centre of gravity of the whole pentagon. The easiest method, however, in such a case is to cut the figure out in card, and hang it up at two- of its vertices with a plumb-line, as before described (16); then the intersection of the two plumb- lines will determine the centre of gravity of the figure. 20. CENTRE OF GRAVITY OF GIRDER SECTIONS. The figure BAC (fig. 29) represents a section of " angle- iron," consisting of two parallelograms at right angles to each other, the centres of gravity of which, a and , can be determined as above (18). Join ad, and divide d> in the inverse proportion of the areas of the two- CENTRE OF GRAVITY OF GIRDER SECTIONS. 37 arms by the point G, Gb being proportional to the arm AC, and G to that of AB ; or G Area of AC Area of AB* Then G is the centre of gravity of the section. The centre of gravity of a section of " tee-iron," DABC (fig. 30), can be found by taking a the centre of gravity of the vertical parallelogram AD, and A that of the horizontal one BC. Then divide Aa at G in the inverse proportion of the area of BC to that of AD, AG being to Ga in the proportion of the area of AD to that of BC. Then G is the required centre of gravity of the section. Fig 29. Fig. 30. D To find the centre of gravity of the section of a " double-flanged " beam, as DABC (fig. 31), bisect the " web" AD in a, and let A be the centre of gravity of the bottom flange BC, D that of the top flange. By the last method find the centre of gravity b of the web and top-flange. Then divide hJb at G in the inverse proportion of the area of the bottom flange to that of the web and top-flange ; AG being proportional to the CENTRE OF GRAVITY. latter, and Gl> to I the former. Then G is the required centre of gravity. If the top and bottom flanges are equal, it will be evident that G is at the middle point a of the web. 21. CENTRE OF GRAVITY OF CIRCULAR ARCH. The circular arch ABCD (fig. 32) consists of the difference of the two sectors AOB and DOC; being the centre of curva- ture. Proceeding as in the case of a part of a triangle (fig. 29) we have to find in the first place the centres of gravity, a and 6, "of the two sectors. Let the radius OE bisect the angle ^ AOB; and when this angle is less than 60, we may take E as very nearly * OE, or Oa = f OE ; and Fb very nearly one-third of OF, or Ob = f OF. We find by calculation, if we wish to be more exact, that when AOB = 40, then Oa = -654 . OE, and Ob = -654 . OF ; when AOB = 50, Oa = "646 . OE, and Ob = -646 . OF. When AOB = 60, Oa = -636 . OE, and Ob = -636 . OF. When AOB = 70, Oa = -626 . OE, and Ob = -626 . OF. For AOB = 80, Oa = -614 . OE, Ob = -614 . OF. When AOB = 90, Oa = -6 . OE ; Ob = -6 . OF. Then since the areas of the sectors are proportional to the squares of their radii, we find G, the centre of gravity of the arch ABCD, from the equation CENTRE OF GRAVITY OF CIRCULAR ARCH. 39 Also we have, OG = Oa + Gff. As an example, let the angle AOB = 60, OE = 12, OF - 10, then Oa = -636 x OE, Ob = -636 x OF, Oa - Ob = -636 (12 - 10) = 1-272, Oa = 7-632, OE 2 - OF = 144 - 100 = 44. Gte = y> r o x 1-272 = 2-9, OG = 7-632 + 2-0 = 10-532. When the position of the point G is required with great accuracy, it is found from the equation (31) Where the angle 0, or AOC, is expressed in " circular measure,' 1 that is to say, the angle 57-3 is the unit of measurement ; R and r are put for the radii OE and /\ OF. The following Table gives the values of 0, sin. , and for angles from 5 to 90, which will 6 facilitate the calculation ; li arc Q " is the length of an arc to a radius unit}'. The numbers in the second column of the following table give the length of an arc of a circle whose radius is unity, corresponding to the angle given in degrees in the first column. To find the area of any sector of a circle, multiply the length of arc in the second column by half the square of the radius. 40 CENTRE OF GRAVITY a Degrees. Arc 6. Bta.{ Sin. r, " 6 6 Degrees. Arc 0. i Sin.* Sin. j 5 08727 04362 49983 48 83776 40674 48550 6 10472 05234 49981 49 85521 1 -41469 48489 7 12217 06105 '49972 50 87266 42262 48430 8 13963 06976 -49956 51 89012 43051 48365 9 . 15708 07846 '49948 52 90757 43837 48301 10 17453 08716 '49938 53 '92502 44620 48237 11 19199 09585 49922 54 | -94248 45399 48170 12 20944 10453 49908 55 95993 46175 48102 13 22689 11320 49893 56 97738 46947 48035 14 24435 12187 49874 57 99484 47716 47962 1 r 26180 13053 49858 58 1-01229 48481 47892 16 27925 13917 49838 59 1-02974 49242 47820 17" 29671 14781 49816 60 i 1-04 720 50000 47747 18 31416 15643 49794 61 1-06465 50754 -47670 19 33161 16505 49773 62 1-08210 51504 47596 20 34907 17365 49746 63 1-09956 52250 47517 21 36652 18224 49721 64 1-11701 52992 47442 22 38397 19081 49693 65 1-13446 53730 47362 23 40143 19937 49665 66 1-15192 54464 47281 24 41888 20791 49636 67 1-16937 55194 47200 25 43633 21644 49604 68 1-18682 55919 47117 26 45379 22495 49572 69 1-20428 56641 47032 27 47124 23345 49527 70 1-22173 57358 46948 28 48869 24192 49505 71 1-23918 58070 46861 29 50615 25038 49468 72 1-25664 58779 46776 30 52360 25882 49431 73 1-27409 59482 46687 31 54105 26724 49392 74 1-29154 60182 46596 32 55851 27564 49352 75 1-30900 60876 46506 33 57596 28402 -49311 76 1-32645 61566 46414 34 59341 29237 -49270 77 1-34390 62251 46252 35 61087 30071 ! -49226 78 1-36136 62932 46321 36 62832 30902 -49181 79 1-37881 63608 46137 37 64577 31730 '49136 80 1-39626 64279 46036 38 66323 32557 "49089 81 1-41372 64945 45939 39 68068 33381 49040 82 1-43117 65606 4584] 40 69813 34202 -48990 83 1-44862 66262 45740 41 71559 35021 ; -48941 84 '1-46608 66913 45652 42 73304 35837 '48889 85 i 1-48353 67559 45540 43 75049 36650 i -48836 86 1-50098 68200 45436 44 76794 37461 : -48782 87 1-51844 68835 45332 45 78540 38268 '48725 88 1-53589 69466 45239 46 80285 39073 -48668 89 1-55334 70091 45112 47 82030 39875 -48610 90 1-57080 70711 45016 CHAPTER IV. RESISTANCE OF MATERIALS TO STRESS. 22. MODULUS OF ELASTICITY. If a rod of any material, having one square inch of cross section, is subjected to a force S acting in the direction of its length, it will be either shortened or elongated to an amount which will depend on the strength and direc- tion of the force that is applied to it. If the force tends to elongate the rod, it is termed a " tensile stress," and if the force tends to shorten the rod, it is a " compressive stress." Suppose we put L for the original length of the rod, / the amount of increase or decrease in the length pro- duced by the force S ; also that the elasticity of the material remains uninjured during the action of the force, so that when the force is removed the rod returns exactly to its original length; then the change of length, /, is to the original length, L, in the proportion of S to some constant number which depends on the nature of the material, and is determined by experi- ment. For this constant the letter E is used, and it is called the " modulus of elasticity." Then the above proportion can be expressed thus : /: L :: S : E. 42 RESISTANCE OF MATERIALS TO STRESS. When the elongation or shortening I, due to the force S, has been ascertained, we have (32) Owing to the imperfect elasticity which exists in all building materials, the modulus obtained when the ex- periments are made with a tensile force, differs from that obtained when a compressive force is applied. It is also found that the modulus which results from a body subjected to a transverse stress, generally differs from either of the foregoing. We may have therefore three different values of the " modulus of elasticity " for the same material, namely, the modulus of tensile elasticity obtained by employing a stretching force ; the modulus of compressive elasticity obtained by a com- pressing force ; and the modulus of transverse elasticity which is obtained by subjecting a rod to a transverse stress. There is often a great difference in the figures given by different experimenters for the values of the modulus, which sometimes arises from the experiments being performed in different ways ; some persons using a tensile, others a compressive, and others a transverse force. If we put S for the force which produces the elonga- tion or contraction I in a rod of one inch sectional area? and A for the number of square inches of section in another rod, then S x A = F, is the force which will produce the same elongation or contraction / in a rod having A inches of section ; therefore MODULUS OF TENSILE ELASTICITY. 43 T71 T :;nd since = from equation (32), we have, 1 = ^ = F^L > (33) Also, F== E. A./ _ ^ ^ (34) The equation (33) gives the alteration of length for a given force F ; and the equation (34) the force F which will produce the above alteration in the length of the rod. Putting these equations into the form J : 1 L ::E:I . . . . (35) where represents the tensile or compressive force A per unit of area, and =- the proportional change in the Li length ; it follows that the " modulus of elasticity " may be defined as " the constant relation between the tensile or compressive force per unit of area, and the corresponding proportional change in the length which it produces in the rod." 23. MODULUS OF TENSILE ELASTICITY. The following- values of E for different materials expressed in Ibs. avoirdupois, have been found by direct experiment with a tensile force. Those for timbers are taken from a work on " Timber and Timber Trees," by T. Laslett, formerly Inspector of Timber to the Admiralty. If we put / = L, in equation (32) we get E = S ; so that E represents the tensile force per square inch of section 44 RESISTANCE OF MATERIALS TO STRESS. that would stretch a rod to double its length, supposing such a thing to be possible without injuring the elasticity of the material. Kind of Material. English Elm Canadian Elm Dantzic Fir Riga Fir English Oak . French Oak . Pitch Pino Red Canada Pine . Cast Iron . Wrought Iron, bars ,, plates Steel, soft cast ,, hammered Bessemer ., rolled cast . E in Lbs. Av. E in Tons. 1,003,280 448 j 2,474,200 1,105 1.737,570 776 3,009,680 1,344 1,545,600 690 2,480,880 1,107 3,020,940 1,349 2,355,600 1,051 12,000,000 5,357 28,672,000 12,800 26,163,200 11,680 30,240,000 13,500 32,480,000 14,500 31,360,000 14,000 24. MODULUS OF COMPRESSIVE ELASTICITY. As long as the compressive force applied to a rod is not sufficient to injure the elasticity of the material, the modulus of compressicc elasticity may be considered as equal to that of tensile elasticity in timbers. In cast- iron it was found by Hodgkinson that the two moduli were very nearly the same, as given above, so long as the force was less than six tons per square inch ; but with a force of six tons the modulus of com- pressive elasticity was found to be one-fourth more than the modulus of tensile elasticity. With wrought- iron and steel the two moduli may be practically con- sidered as equal. 25. MODULUS OF TRANSVERSE ELASTICITY. When a rod is subjected to a transverse stress at right angles to its length, the resistance to bending is proportional to COEFFICIENT OF STRENGTH. 45 the modulus of elasticity as long as the elasticity of the material remains unimpaired, as will be shown in the chapter on " Deflexion " (47). The value of this modulus is, however, different in some kinds of material to that of the modulus of tensile or of com- pressive elasticity, and can be found experimentally by observing the deflexion caused by a known weight. The modulus of transverse elasticity for cast-iron is found to vary from 15,000,000 Ibs. to 19,000,000 Ibs. ; or taking the mean of these two, we put E = 17,000,000 Ibs = 7,600 tons, for the modulus of transverse elasticity in cast-iron beams. That for timber, wrought-iron, and steel may be taken as equal to the modulus of tensile elasticity given above (23). 26. COEFFICIENT OF STKENGTH. When a rod is sub- jected to a longitudinal stress, F, we find from equa- tion (33) that the change of length, /, varies with the force F. If the force F is sufficient to produce fracture, and / is the alteration of length at the moment of fracture, then F represents the ultimate strength of the material. The quantity S which is the force per unit T71 of section, and is equal to -- , is called the coefficient of A ultimate strength. The values of S are determined by experiment, and vary considerably in the same material for tensile and compressive stress. The following- table gives the ultimate tensile strength of different materials per square inch of section : the coefficient for timbers being taken from Laslett's work mentioned above. 46 RESISTANCE OF MATERIALS TO STRESS. Coefficient of Ultimate Kind of Material. Tensile Strength. S in Lbs. Av. Brick 300 Slate 9,600 to 12,800 Cast Iron .... 11,200 to 44,800 ., Malleable .... 29, 120 to 44,800 Plate-Iron .... 51,000 Bar- 1 ron 60,000 to 70,000 Hoop-Iron .... 64,000 Steel 100,000 to 130,000 English Elm .... Canadian Elm .... 5,460 9,180 Dautzic Fir .... 3,230 Riga Fir 4,050 Spanish Mahogany . American White Oak . 3,790 7,020 English Oak .... French Oak .... 7,570 8,100 Pitch Pine .... 4,670 Canada Spruce .... 3,930 27. RESISTANCE TO CRUSHING. The resistance of a rod of any material to a compressice force follows a very different law to that of the resistance to a tensile force; for whereas the latter is independent of the length of the rod, the former varies greatly with its length, being much greater in short than in long pieces. Since long rods are liable to break by bending when subjected to a compressive force in the direction of their length, it is necessary to make our experi- ments upon very short pieces, as cubes or double cubes, in order to ascertain the actual resisting power of the material to crushing. The value of the coefficient of ultimate strength, S, per square inch of section, for the principal materials used in construction, when subjected to a crushing COEFFICIENT OF SAFETY. 47 force, is given below. The numbers for timbers bein< taken from Laslett's experiments. Kind of Material. Common Brick Fire Brick . Cornish Granite Peterheacl Granite Yorkshire Stone Bolsover Stone . Portland Stone Bath Stone Cast Iron ,, Malleable . Wrought Iron . Steel .... English Elm . Canadian Elm . Dantzic Fir . Riga Fir . English Oak, seasoned American White Oak . Dantzic Oak . French Oak Pitch Pine Canadian Red Pine Canada Spruce Spanish Mahogany Coefficient of Ultimate Crushing Strength. S in Lbs. Av. 500 to 800 1,700 6,400 8,300 to 10,900 7,600 8,300 3,900 1,500 44,800 to 215,040 107,520 to 159,040 36,000 to 40,000 110,000 to 150,000 5,786 8,586 6,948 5,248 7,475 6,451 7,558 7,945 6,462 5,600 4,852 6,415 28. COEFFICIENT OF SAFETY. None of the mate- rials used in construction must ever be subjected to a stress which is anything near the " ultimate-strength," or the stress must never approach to that which would produce fracture. A certain limit has been fixed for the amount of stress to which the materials can be sub- jected without injury to their elasticity, and this limit should never be passed in practice. This limiting stress is called the coefficient of safety, and its ratio to the coefficient of ultimate 'strength (26, 27) varies for 48 RESISTANCE OF MATERIALS TO STRESS. different materials, being fixed arbitrarily according to our experience of the effect of stress in producing strain in those materials. The proportion is also varied according as the stress arises from a moving- load or from a stationary one. In the latter case, the ratio of the coefficient of safety to that of ultimate, strength in steel and wrought-iron is 1 to 3 ; for cast- iron it is 1 to 4 ; in timber 1 to 6 ; and in stone or brick 1 to 8. 29. TRANSVERSE STRESS. When a long beam or rod of any elastic ma- terial is fixed or sup- B ported in a horizontal position, and is sub- jected to forces acting \ B< vertically, they produce a transverse strain on the fibres, and cause the beam to bend in the direction of the forces. Thus, suppose the beam AB (fig. 33) to be fixed firmly and horizontally at one end AD ? and to be loaded at the other end B with a weight W. The end B will then be bent and will take the position B', the upper surface AB' of the beam becoming convex, while the under- side DC becomes concave. It will be seen that the fibres in the upper part of the beam become stretched or lengthened in consequence of this bending, and are therefore subjected to tensile stress. On the TRANSVERSE STRESS. 49 other hand those in the lower part are shortened and are subjected to a compressue or crushing stress. The lengthening or shortening of the fibres is also greatest at the top or bottom of the beam, on its outer edges, AB and DC, and get less and less as we proceed inwards. Consequently there must be some part or surface near the middle of the beam, as Nw, where the iibres are neither shortened nor lengthened, the length Nn being equal to the original length AB of the beam, before the action of the load W. This part of the beam is called its " neutral surface," and is parallel to the surfaces AB, DC. The neutral surface may be defined as that surface along which the resultant of the horizontal components of all the diagonal forces is equal to zero. The line Nw is that in which the neutral surface cuts the longitudinal section AB'CD of the beam. Let bRc be the line perpendicular to Nw in which a transverse section of the beam intersects the longitudinal section ; then a horizontal line through H, where the two planes intersect (perpendicular to the plane of the paper) is called the " neutral axis " of that transverse section. Assuming that the elasticity of the beam is uninjured by the stress to which it is subjected, so that when the load "VV is removed the beam will return exactly to its original horizontal position ; and also that the resistances to extension and compression are equal ; we may assume that the neutral axis will pass through the centre of gravity (16) of the transverse section H>. Let be the centre of curvature of a small part, HF, of the line N/z, and assume that HF is very nearly an arc of a circle, so that OF is equal to OH. The forces 50 RESISTANCE OF MATERIALS TO STEESS. of resistance to extension and compression will act parallel to FH, and perpendicular to a transverse section at H ; the sum of the forces of extension acting above H being equal to the sum of the forces of com- pression acting in the opposite direction below H ; or the resultant of these forces must be zero, as above stated. Produce OH to b and OLF to K, on the line AB' ; and draw JiEa parallel to LK. Take any point c on Ha and draw ccl parallel to ab ; then cd represents the extension of one of the fibres above H. Let f repre- sent the force producing this extension cd in the fibre at c, and call a the area of section of the fibre. Then (22) the extension ccl is to the original length FH of the fibre as /divided by a to E, the modulus of elas- ticity ; or by equation (34) we have Also since the triangles HOF and clEc are similar, we have cd : FH = Ed : OH ; so that /= E x a x ~_'f. Ort The moment (8) ofy taken about the point H, isy' x Hrf; and /x EW=E x ax . OH A similar result is obtained if we take the fibres below H, only they will be subjected to compression. If then we put M for the moment of resistance of all TRANSVEHSE STRESS. 51 the fibres in the section, and put x, x^ x t , &c. for H ,&\?., and p for OH, we obtain the value of M, namely, -p M =(aa? + a X{ + a x.? + . . .) P Now the sum of all the quantities, ax 1 + ax? + &e, is called the " moment of inertia " of the transverse section taken about the neutral axis, for which we write the letter " I," its value depending on the geo- metrical form of the section and being determined in different cases by help of the integral calculus. We can therefore put M, the moment of resistance of the section, into the form M = E - X - 1 (36) P Also, since by geometry, ab : H = FH : OH we have 1 _ 1 ._ <& x 1 . OH " p ~ FH fta ' (therefore equation (3Gj becomes If then we put S for the force per square inch pro- ducing the elongation (or compression) ab, and Ha = z the distance of the most extended (or compressed) .fibres from the neutral axis ; we have by equation (32) p v _ Q L x m - S. E 2 52 RESISTANCE OF MATERIALS TO STRESS. Therefore by substituting S for E ^ in the last equation, we have M = S x l . . . (37) When the section of the beam is rectangular, putting- b for the breadth and d for the depth, z being equal to I o Fifth case : let the beam be loaded uniformly over its entire length; then by equation (44), W . . S *^C = *. x . s 3720 = 4060 Ibs. As another example, let us take a beam of English 58 EESISTANCE OF MATERIALS TO STRESS. elm, in which S = 937 for safe-load ; and let b and tf be each 10 inches, and / = 18 feet or 216 inches ; then we have Q b. d- 937 x 1' -7 210 For the first case : W = - = 723 Ibs. In the second case : W = - - = 1446 Ibs. o In the fourth case : W = ?_*_*??? = 2892 Ibs, o In the fifth case : W = 4 X - 4338 = 5784 Ibs, If the length / is expressed in feet instead of inches in the above formulas, the values of S given in the table (32) must be divided by 12. 34. BEAM FIXED FIEMLY AT ENDS. When a beam, AB (fig. 35) is held firmly down at each end, by being built into a wall, or loaded with weights at A and B, it assumes when loaded at any point C, the form of a curve of contrary flexure.* The parts between A and a, B and b, are convex to the direction in which the load W acts, while those between C and a, and C and b, are concave to the direction of W. Consequently the upper parts from A io a and from B to b are in a state of extension, and the lower parts in a state of * Tarn's Practical Geometry, p. 108. BEAM SUPPOKTED AT THKEE POINTS. 59 compression ; while from a to C and b to C their con- dition is reversed. At the points a and b where the flexure changes from concavity to convexity, or the points of contrary-flexure, there will be neither extension nor compression in the fibres, and the stress \vili increase from zero at a and b to a maximum at A, C and B. The theoretical investigation of this problem is rather complicated, and the result obtained is that the resistance of the beam in this case to the stresses produced by a given load is double that of a beam which is only supported at its two ends, as in the previous cases (31). Experimental research is not however in accord with this result, as it is found that the resistance cannot practically be considered as more than half as much again as in the other case, or that the strength of a beam fixed at each end is to that which is only supported in the proportion of 3 to 2. 35. BEAM SUPPOKTED AT THREE POINTS. This problem has been treated differently by the various ;iuthorities upon the subject, who have arrived at some- Fig 36 what varying conclusions. Suppose that ACB (fig. 36) represents a beam supported at its ends A and B, and also by a prop at the middle C, and loaded between these points. Suppose the intermediate prop at C to be removed, then the reactions at A and B are deter- mined in the manner previously given (31), and there is CO RESISTANCE OF MATERIALS TO STRESS. DOW HO reaction at the point C. The equilibrium will not be affected if the middle support is placed so as just to touch the beam, and the pressure of the support may be gradually increased without disturbing- the equilibrium, but the variation of the pressure alters the reactions at A and 13. One of the data of the problem is that all three points of support shall be in one straight line. The reactions therefore of three supports to a beam having a distributed load are in general indeterminate. An attempt has been made to solve this problem theoretically on the supposition that the material is perfectly clastic; the result obtained in the case of beams of uniform depth and section, and having a distributed load W over their whole length, is as follows : For a beam supported at each end and at one point C in the middle, if R t is the reaction at A and at B, R; the reaction at C ; then R! = fV W, It = S W . . (45) If 10 is the load per unit of length, and a is the point of contrary flexure, then 2R, If there are four supports at points dividing the length into three equal parts, ll x the reaction at each end, R 2 the reaction at each of the two intermediate supports, it is found that R! = ,V W, R a = 5* AV . . (46) If however we look at this question from a practical BEAM SUPPORTED AT THREE POINTS. 61 and experimental point of view, and suppose the beam (fig. 36) to be severed at 0, we then have (14) And comparing this with the equations (45) it would appear that the pressure on the intermediate support is greater in a continuous beam than in one that is not continuous; whereas the contrary ought to be, and is practically, the oase, owing to the resistance which the material of the beam offers to its being bent from a straight line. Most of the modern writers who have treated on this subject have adopted these theoretical results in cases of beams with distributed loads sup- ported at three or more points. Some however of these writers * admit that there is no material error in con- sidering each point of support as carrying the load on a part of the beam from centre to centre of two adjoin- ing bays ; which is the result given by the equation (47). This latter distibution of load appears to be the simplest as well as the most practically correct ; and we shall therefore adopt it in all cases where such beams come under our investigations. In the case of one intermediate support at the centre of the beam, we shall take ] W as the reaction, R w at each end, and i W as that, Rq, at the middle, as in equation (47). If the beam is divided into three equal parts, and has two intermediate supports, then we take II, = -J- W, R 2 = % TV . . . (48). * Col. Wray's Application of Theory to Practice, p. 201 ; Femvick's Mechanics of Construction, p. 120. 6' 2 RESISTANCE OF MATERIALS TO STRESS. 36. BEAMS NOT RECTANGULAR. From the reasoning used in obtaining the resistance of a rectangular beam (29, 30), it will be seen that much of the material is wasted and does not offer its full resistance to the stresses. For the moment of stress is greatest at the outer edge and diminishes to nothing at the centre of gravity, consequently the inner fibres have less and less work to do according as they are situated nearer and nearer to the centre, although their actual power of resistance is the same as in the outermost fibres. Theoretically then the beam of section shown by fig. 37 would be as strong as one of rec- tangular section, while half the weight and material of the beam would be saved. But practically the nature of timber prevents us from adopting this section, as the parts cut out would be useless and the beam itself would break across at the centre G. When however we come to use such a mate- rial as iron for our beam, a great saving can be effected by having the top and bottom much wider than the middle part. The form commonly used is that given by fig. 38 and fig. 39, where there are rectangular " flanges " at top and bottom, either equal or unequal as the case may be, which are con- nected by a thin piece of rectangular metal placed vertically between them called the " web." The angles at which these pieces meet are strengthened by tri- angular pieces as shown by the dotted lines ; so that the section is a modified form of fig. 37. 37. IRON BEAMS. The rectangular form of section IRON BEAMS. 63 is only adapted to beams made of timber, and the form of section used for beams of iron depends on the kind of metal used, whether cast-iron, wrought iron or steel. The three sections given (20) in the chapter on " Centre of Gravity," figs. 29, 30 and 31, are those most com- monly in use for this material. Hav- ing first determined the position of j the centre of gravity, G, of the section, we may suppose the neutral axis Nw (fig. 38) to pass through G. N __ It has been shown by equation (37) that if I is the moment of inertia of I . - 1 the sectional area taken about the Fig. .3 8. neutral axis ; z the distance of the most extended or compressed fibres from the neutral axis, then the moment of resistance in a beam subjected to a load acting at right-angles to its length, is M= S-. To find the moment of inertia of the section (fig. 38) we must find that of its three parts, namely, the web or vertical part, and the two horizontal flanges ; and their sum will be the moment of inertia of the whole section. Let I t be the moment of inertia of the top flange about Nw; I 2 that of the web; I 3 that of the bottom flange. Now it is shown in treatises on Mechanics that " the moment of inertia of an area about an axis not passing through its centre of gravity, is equal to its moment of inertia about a parallel line through its own centre of gravity plus the area of its section multiplied (U .RESISTANCE OF MATERIALS TO STRESS. by the square of the distance of its centre of gravity from the given line." Thus, if D is the centre of gravity of the top flange, b and t l its breadth and thickness ; B and t. } the breadth and thickness of the bottom flange whose centre of gravity is A ; d and A, the depth and thickness of the web whose centre of gravity is at a ; then by the above rule, since T V <5 . & is the moment of inertia (29) of a rectangle about an axis through its own centre of gravity, we have I = ii_^L + /, . fl x DGP . . (40) f,.dx G,7 2 . . (.50) L = .:- + B . f, x GA* . . i Then we have, I = I, + I s + I 3 . . (52) The distance z of the fibres in the top flange from G, is z = Gl) + . . . (53) and the distance z of the fibres in the bottom flange from G, is z = GA + - . . . (54) It is only in cast-iron beams that the top and bottom flanges are made of different size; in Drought-iron and steel they are generally equal. 38. CAST-IRON BEAMS. In beams of cast-iron the upper flange is made smaller than the lower one, when the beam is supported at each end ; because the upper CAST-IKON BEAMS. G5 fibres are in compression, and the lower in tension, and the ultimate resistances of this material to compression and extension are in the ratio of 6 to 1. As, however, we are supposed to apply a load which only strains the material within the elastic -limit , or the limit of safety (28), we may consider that these resistances bear a much less ratio to one another ; in fact they ought theoretically to be considered as nearly equal. For an example we take the case of a cast-iron beam 20 feet long between the supports, and 1 3 inches deep, the upper flange being 1 inch thick and 5 inches wide, the lower one 2 inches thick and 10 inches wide, and the web 1 inch thick and 10 inches deep. Let the point D be the centre of gravity of the top flange (fig. 38), the point A that of the lower flange, and the point a that of the web; then by the method pre- viously given (20) we can find the position of G, the centre of gravity of the whole section. rn 114 p . 47 r , 37 = ' GA = ' G = - Then by equation (49) I ^' By equation (50) J3y equation (51) 1. = 1^ + 10X2 x()'= 233, G6 RESISTANCE OF MATERIALS TO STRESS. By equation (52) I Ij + L+ I. = 718. For the distance, z, from the centre of gravity G of the fibres under greatest compression we have, by equation (53) j.iy+l.ffi.Wi, 14 ^ 14 14 For the distance, r, of the fibres under greatest exten- sion we have, by equation (54) * = 4 Z + U = G1 = 4-36. 14 T 14 14 If we suppose that 8 tons per square inch is the safe resistance to compression (S), and put z = 8 - G4, then by equation (37) And by equation (18) M = i W . / = W x 60 = 8 x 718 ; 8'64 therefore, W = C'7 tons. If we suppose that 2 tons per square inch is the safe resistance to extension (S), and put z = 4'36, then by equation (37) And by equation (18) M = lW./=Wx6 therefore, W = 5?> tons. WROUGHT-1RON BEAMS. G7 Hence it appears that we may consider 6 tons as the safe-load to be borne by this beam at the centre. In the kind of beam known as the Hodgkinson girder, which has the area of the bottom flange six times that of the top flange, or in the ratio of the ultimate resistance to compression to the ultimate resistance to extension ; the rule for the breaking- weight at the centre is, W = 2-2 A .' (l in tons . . (55) where A is the area in square inches of the bottom flange, d the total depth in inches of the beam at the middle, / its length infect. Applying this formula to the above example, although in this case the sectional area of the lower flange is only four times that of the upper one, we find W = 2-2 2 - 9 * = 28-6 tons for the breaking -weight at the centre ; so that the safe- load found above, namely, 6 tons, is nearly one-fifth of the breaking-weight as derived from Hodgkinson's formula. If the beam has no top flange, as shown by fig. 30, we have only to put I t = in equation (52) and we get the value of I as before. 39. WROUGUT-IRON BEAMS. Beams of this material are either rolled in one piece, and of the section shown either by fig. 38 or by fig. 39; or they are made of separate plates riveted together by means of angle-irons, and formed into a hollow box as shown by fig. 40. F 2 C8 RESISTANCE OF MATERIALS TO STRESS. Since the ultimate resistances of this material to compression and extension are in the ratio of 2 to 3, we ought theoretically to make the flange which is in tension two-thirds of the area of the flange which is under compression. It seems probable, however, that when the material is subjected to stresses which do not exceed the limit of safety, these resistances are nearly equal, and may be put at 5 tons per square inch of section. Ftg.39. Flg.4O. When the two flanges are unequal, we must proceed as in the case of beams of cast-iron (38), to find the moments of inertia of the several parts about the axis through the centre of gravity of the section, and adding nil these moments together we get the moment of inertia of the whole section. When, however, as is usually the case, both flanges are alike, the moment of inertia of the section is the difference of that of the box taken as solid, and that of the hollow part of the box. If D is the external depth, d the internal depth (fig. 40) between the two flanges, / the total thickness of the plates forming the web, b the breadth of the flanges ; then we have for the moment of inertia of the section, T = h 1 2!_- C* - r/ " 12 ~ , and ~ = i- D. By equation (37) AXGLE-IROX. 69 When the load W is at the centre of the beam, we have by equation (21) M = 1 W . / = 5 b ly ~ (b ~ f) d \ 61) Therefore, W = 5 x -?, x ^5L__^__5 fr . (57) This is the safe-load in tons, when all the dimensions are in inches ; the maximum stress on the fibres at top or bottom of the beam being 5 tons per square inch of section. The following Table gives the safe- load for various beams of wrought-iron having equal top and bottom flanges, the sections being either as fig. 39 or fig. 40 ; W being the load at the middle, and the maximum strain being 5 tons per square inch. ft. j D. *J , 7 = 120". ; 7 = 150". ; 7 = 180". 7 = 240". 7 = 300". Ins. Ins. Ins Ins. W in Tons. Win Tons. W in Tons. W in Tons. W in Tons. 4 8 6| 1 3^ 2'8 2J 1| 5 10 8| i 54 4-6 3-83 2-87 6 12 10J ? 10 8-0 61 5 4 7 15 13 i 19$ 15-5 12 ? 9 9-67 ?T 8 | 16 18J H 29-12 23'3 19-4 14-56 Us i 1 j 40. ANGLE-IKON. To determine the resistance of a beam of this section (fig. 41), we first find its centre of gravity G by the method previously explained (20), making g and y' the centres of gravity of the two arms 70 RESISTANCE OF MATERIALS TO STRESS. DB and BC. Supposing the neutral axis Nrc to pass through G, we determine the moment of inertia, I, of the section by the method previously given (37) ; Ga and GA being the vertical distances of the centre of gravity of the two arms from the neutral axis. Putting BD = (/, BC = b, t the thickness of the metal in each arm ; we have by equations (49, 50, 51, 52, 53), T!= 0, I = -' (l * 4- t d (Ga)* 1 )w if 9 b.t Fig. 41. ** = ^ +!>( (GAV, I = I, + I ;i , ~ = GA + * . Then if "VV is the load on such a beam at its centre, we have from equations (21 and 37), 1 W . / = S \ . As an example, let t = f", d = 4|", b = 5 inches. In order to find the position of G, let g and g e be the centres of the arms ; join yy. Then we have G/ = GA = I x 4] = 17 Gy G 5"x'2 2u' GA + Ga = 37 = Aa = -i (rf -f = ,j G i!0 G Ga G* Therefore, G = ^ x 5 = '^ = 1J, very nearly, o i "^ 61 GA - An - Ga = :> - 4 = 7 ^ 3 G GIRDERS WITH ANGLE- [RONS. 71 GA + * - = 7 + | = JJ7 = 3., very nearly. Then, L = * + J X 4-1 x = 10J, Ti = -*iiP + 5 x 2 x Q 2= 5 *' nearly * Therefore, I - L + I. = 10J + 5-1 = lof Por example, let the length be 120 inches; then -1 W . / = W x 30 = S l = 5 1^1 = 52J. *524 Jlence, W = -~ = 1 f tons, for the safe-load at the oO middle of the beam. 41. GIRDERS WITH ANGLE-IRONS. Large beams of wrought-iron, whether made in the form of fig. 39 or of fig. 40, are composed of plates held together by means of angle-irons riveted at the meeting of the several pieces ; as shown on fig. 40, where A, B, C, D are the .angle-irons riveted to the vertical and horizontal plates. In order to calculate the strength of such a beam with .accuracy, we must add the resistance of the angle-irons to that of the plates, and therefore must find their moment of inertia about N passing through the centre of gravity, G-, of the section, by the method given above (40). Let Ga and Gb be the distances of the centres of gravity of the angle-irons from Nw, A the area of section of each angle-iron, 1' its moment of 72 RESISTANCE OF MATERIALS TO STRESS. inertia about an axis through its own centre of gravity. Then if we put I t for the total of the moments of inertia of all the angle-irons about Nw, we have I t = 41' + 2A (G s + G/r) . . (58) If the top and bottom plates are equal, then Ga -- G/;, and I, = 41' + 4 A x (G) s . . (59) Then the total moment of inertia, I, of the whole section of the beam is, I = T! + L + I, + 1, . . . (60) And when the top and bottom flanges are equal, we have from (39) T _ b . D" - (6 - rf-' , 12" As an example of application let us take the last of the beams in the Table (39), having D = 16", d = 13|", b = 8", t = H", I = 300 ins.; and suppose the plates to be held together by angle-irons of the size given in the above example (40). Then we have, 1' = 15| ; A = ? 7 x A = I 1 ^ = 7, very nearly ; G = 5|, very nearly. Then equation (50) becomes I 4 = 4 x 15| + 4 x 7 x (5-]) 2 = 833. Then we have 1398 + 833 = 2231. GIRDERS WITH ANGLE-IRONS. 73 Then as before, when "W is the load at the centre, ] W . I = S J , and ~ = ^ = 8 which is 7 tons more than when the resistance of the angle-irons was omitted. A deduction ought however to be made from the area of section of the angle-irons for the rivet-holes. In this example, however, the angle-irons have been taken much larger than would be adopted in practice for a girder of these dimensions. Suppose the angle-irons for the above girder to be 3" x 3" x i", then we find by the formulfe given above (40)- j _ 1 3 1 9 28 7 42 12 8 2 10 32 8 4' I:J = 12 X 2 X 8 + 2^ X 2~ X 10 = 48' 1' = I., + T 3 = -3 rr = 2 1, verv nearlv, 4o I t = 4 T + 4 x !1 x ( 2 4 3 Y= 9 + 364 - 372. Therefore, I = 1398 + 372 = 1770, w - 5 x x " 14i tons - This value of W is 3 tons more than was found for the strength of the same girder when the angle-irons were left out of consideration ; we may therefore in general add one-fourth to the strength obtained by 74 RESISTANCE OF MATERIALS TO STRESS. means of equation (57) for the resistance offered by the angle-irons. 42. STEEL BEAMS. The mode of calculation which has been used (39, 41) for finding the safe-load on beams of wrought-iron is equally applicable to those made of steel, whether rolled in one piece or formed of plates riveted together with angle-irons. The coeffi- cient of safety, S, varies very much according to the kind of metal used. A very hard steel, although having a high coefficient, is unfit for the purposes of the architect ; and a comparatively soft or mild steel is the best to use, in which the ultimate tensile and comprcssive strengths are nearly equal and about 40 tons per square inch. Steel has an advantage over ordinary iron in the fact that its elastic limit is nearer to the ultimate strength, so that the coefficient of safety may be put at one-third of the ultimate strength. The value of S may therefore be taken as about 12 tons per square inch, or more than double the coefficient used for wrought-iron. Then the equa- tion from which to determine the safe-load, W, in tons, in a beam loaded at the centre, is found by combining equations (21) and (37) I W. 1= 12 T . If the top and bottom flanges are equal, then z = | D ; 2 x '>.!>-(/, -O^ 3 3 x n.r "' where b, D, t, and I. are the dimensions as previously given (39), all being in inches, and W in tons. If the length is in feet, then BEAM OF UNIFORM STRENGTH. W ^_nLL . (61) 3 D.I As an example we will take a beam of the dimen- sions given above (41) for a beam of wrought-iron : where D = 16", d = 13 ins., b = 8", t = H", / = 25 ft. If the beam is rolled in one piece, and has no angle-irons, then I = 1398, and we have W = 8 x - 1 ----. 16 x 2u If it is made in the box-form with riveted angle- irons, whose section is 3" x 3" x i", then I = 1770, and we have = 8 x = 35-4 tons. 43. BEAM OF UNIFORM STRENGTH. In the beams which we have been considering the depth and breadth have been assumed to be the same throughout the entire length, so that the moment of resistance is the same at all cross sections, whatever their distance from the points of support. But it has been shown by equa- tion (17), that if A and B are the extremities of a beam, and a load is uni- formly distributed over its length, then the mo- ment of stress at the sec- tion at any point, E, is proportional to AE X BE. Consequent!}', the stress is nothing at A and B, and is a maximum at the centre of the beam. Since the moment of resistance at any section is proportional to the breadth, then Fig 42 7G RESISTANCE OF MATERIALS TO STRESS. if we vary the breadth DE (fig. 42) in proportion to AE x BE, keeping the depth uniform, we have a beam whose moment of resistance at any section is propor- tional to the moment of stress at that section. It is a property of the parabola that the ordinate, or half- breadth, DE, varies as AE x BE ; so that to render the strength of the beam uniform throughout, or pro- portional to the stress, we must make the flanges ADB, AFB (when the beam is of iron) in the form of parabo- las ; or arcs of circles may be used instead without material error. This form may be used with advantage in cast-iron beams, and a great reduction in weight of metal is thereby obtained ; but for wrought-iron it is not practicable to vary the width of the flanges. Fig. 4-3 When it is necessary to have the width uniform throughout, we can get a beam of uniform strength by reducing the depth. For as the moment of resistance is proportional to the square of the depth, if DE (fig. 43) is the depth at any point E of the beam, and we make DE 2 proportional to AE x BE, we have a beam in which the resistance at any section is proportional to the stress at that section ; and by making the curve ADB take the form of a semi-ellipse, we obtain a beam of uniform strength, since in the ellipse the square of the ordinate DE is proportional to AE x BE. When the load W is placed at the centre C of the SHEARING-STRESS. 77 beam, we find by equation (19) that the stress at any other point E is proportional to its distance from the nearer end of the beam ; so that in this case the breadth ought to be in the same proportion in order that the resistance of the beam should be the same at all points ; the plan of the flanges should be as shown by fig. 44, being in the form of a parallelogram, and widest at the centre C. The depth is here supposed to be uniform throughout. If the breadth remains the same throughout, we can get a beam of uniform strength, when loaded at the centre, by making the square of the depth DE (fig. 43) proportional to AE, the distance of DE from the nearer end of the beam. The curve ADF which fulfils this condition is the parabola whose vertex is at A; and the curve from B to F will be an equal and similar parabola. This will differ so little from the ellipse, that we can use that curve in either case of load at centre or uniformly distributed. 44. SHEARING - STRESS. There is another kind of stress to which beams are subjected by the action of a load, which we will briefly allude to. Sup- pose the beam AB (fig. 45) to be fixed at A and loaded at B with a weight W : then a vertical action is set up in the beam which we may suppose to be divided into a number of slices parallel to the direction of the force W. The load W tends to cause the end slice to slide down or to shear from the next one, but is pre- Fig.45, 78 RESISTANCE OF MATERIALS TO STRESS. vented from doing so by the lateral resistance of the material. The shearing-force acting on the next slice is equal to that which tends to shear it from the next one ; so that a vertical force equal to W is transmitted from one slice to another through the whole length of the beam, and the load W represents the shearing- stress at each point or slice along it. If the load W, or vc . I, is uniformly distributed over the beam, the shea ring-force at any point whose dis- tance from B we call .r, is represented by n . z, which is nothing at B, and increases up to n~ . I, or W, at A. In a beam AB loaded with W at the middle, C, and supported at each end, a shearing-force equal to ^W is produced at every point. If W, or w . I, is uniformly distributed, the shearing-stress at any point whose distance is x from either end, will be represented by re (| I #), which is | w . I at A and B, and is nothing at the centre C. It is evident from the nature of the force, that shearing-stresses can only exist in pairs ; every shear- ing-stress on a given plane being necessarily accom- panied by a shearing-stress of equal intensity on another plane. The bolts or rivets used for fastening together the plates of iron girders are especially subjected to shear- ing-stress ; and also the plates which form the web of the girder. The shearing-strength of wrought-iron, or its power of resisting a shearing-stress ^ is about 18 tons per square inch of section, or about three-fourths of its tensile strength. That of mild steel is about 29 tons per inch, or about four-fifths of its tensile strength. CHAPTER V. DEFLEXION OF BEAMS. 45. RESISTANCE TO BENDING. When a beam is fixed at one end and loaded with a weight W at the outer end, it assumes a curved form, as wa have previously seen (29) ; the beam AB (fig. S3) which was horizontal before the application of the load, is now deflected from the straight line by the amount BB', which is called the " deflexion " due to the weight W. The amount of deflexion under a given load varies considerably in beams of different material ; thus, in a beam of stone the deflexion will be almost imper- ceptible as long as the load is within the limit of safety (28), or even as the load approaches the ultimate strength or breaking-weight, fracture generally taking place suddenly and without any warning. Almost the same may be said of beams made of cast-iron, but with them there is a measurable deflexion before fracture takes place. With beams of timber or wrought-iron, a very considerable amount of deflexion may be pro- duced long before the load approaches the breaking- weight, and without any serious injury to the material. As however it is essential to the stability of a building that there should be as little deviation as possible from the straight line in the beams used in its construction, it becomes a matter of much importance to determine 80 DEFLEXION OP BEAMS. the load that such beams will bear without being deflected to any great amount. Consequently, we have not merely to calculate the safe-load that may be laid on a beam, but also the amount of deflexion which that load w r ill produce, so that we may regulate the size of the beam accordingly. The resistance which a beam offers to fracture by a load acting transversely, is called its "strength;" while its resistance to deflexion, or to any deviation from a straight line, is called its " stiffness." These two properties of a beam are, as we shall proceed to show, governed by very different laws. 46. RADIUS OF CURVATURE. When considering the case of a beam fixed at one end and loaded at the other (29), we saw that the neutral surface Nw (fig. 33) assumed a curved form, and that any small portion as FH might be taken as very nearly an arc of a circle whose centre was at 0, OF or OH being the radius, p y of this arc. The circle of radius p which most nearly approaches to the curvature of ~Nn at any point, is called its circle of curvature at that point, and its radius, p, is called the radius of curvature of a small arc. This radius varies in length when the whole line Nw differs from a circular curve, and can be found for a point at any distance from N by help of the differen- tial calculus. From equation (36) we have for M the moment of resistance to flexure at any point of the beam M E -I -T' where E is the modulus of elasticity, I the moment of BEAM SUPPORTED AT EACH END. 81 inertia of the section about the neutral axis, p the radius of curvature of the neutral surface at a given point. The above equation can be put into the form __ E.I (62) 47. BEAM SUPPORTED AT EACH END. Suppose a straight rod or beam whose length between the sup- ports is AB, or I, to be supported at A and B (fig. 40) A Fig. 46. D and to be loaded with a weight W at the centre C. If ADB is its position before the load is applied, then the effect of the load is to cause it to bend downwards, as ACB, producing the deflexion DC at the middle. Let R be the reaction at each of the supports A and B, then E = | W ; and the deflexion will be the same if we suppose the beam to be fixed at C and drawn upwards by a force R at each end. Draw CPH parallel to ADB (P being any point on the line CH) and ~Pab perpendicular thereto. Let CP = x, P = y ; then x and y are called the co-ordinates of the curve ACB, .and vary in length with the position of the point P. In order to determine the deflexion at any point, we must find the relation between x and y, or the equation Jto the curve. Since the curvature is but small, we are 82 DEFLEXION OF BEAMS. able to put for the quantity - the second differential coefficient of y with respect to x* From equation (62) we have Since CH = -|-, CP = x, Bfl = PH = A - x; there- fore we get 1 W 1 After two integrations of this equation, we obtain TV , 2 ^ (63 > which is the equation to the curve ACB. TVhen the point whose coordinates are x and y is at the middle of the beam, we have x CH = $ I, and y = HB = CD = 5 = the deflexion of the beam at the middle. Substituting these values for x and y in equation (63) we find 8- W ( l * 13 \- l W ^ ~ 2~ETT Vl6 48/~ 48 * E ' T In the case of a beam of rectangular section of which b is the breadth and d the depth, we have (29), I = T V b . d 3 , and 1 W I 3 /A ~s 8 = 4 E FT* ' ' (60) * See the author's edition of TredgolcVs Carpentry, p. 40. BEAM WITH DISTRIBUTED LOAD. 83 Since the stiffness of a beam, or its resistance to deflexion, is inversely as the amount of deflexion for a given load, it will vary as , or directly as the breadth o and the cube of the depth and inversely as the cube of the length. Comparing this with equation (39) which gives the strength of a beam, we see that the strength varies as the breadth and square of the depth and inversely as the length ; so that we have then the following proportion : Stiffness : strength = ~ : I The deflexion ab at any point P on the beam, where CP = x, Pa = y, is evidently CD - Pa, or 8 - y ; and we have by subtracting equation (63) from equation (64) i-d-y'-J. y equation (65) 1 3173 12 3 x 18 3 _ . s . = 4 X 1003280 X unTTo 3 ~ But by Tredgold's rule (50) the maximum deflexion should be 1$ or '45 inch, so that the load should be reduced in the proportion of 45 to 80, which gives \V r 4. m = 1,7^5 Ibs., or W = 1,504 Ibs. instead of 2,892 Ibs. We will apply equation (72) to the following example : a beam of Dantzic fir whose weight per cubic foot is 30 Ibs., and where E = 1,737,570 lb., has b = 10 inches, d = 15 inches, L = 20 feet; re- quired to find the load at the centre, that will produce a deflexion of |-$ or half an inch. Here n = 625 Ibs. is the weight of the beam, and n = 391 Ibs. From equation (72) 1737J370 x MX. 15' = 8484 Ibs. 17280 20 2 Deducting w from this value of W, we have for the required load at the centre, 8,484 - 391 = 8,093 Ibs. And if the load is uniformly distributed, the weight would be f x 8,093 = 12,949 Ibs. Apply the equation (73) to a similar beam whose depth is 12 inches ; to find the breadth when the load, including the weight of the beam, is 5,000 Ibs. By equation (73) and n = 575 Ibs., f n- = 358 Ibs. So that the load SCANTLINGS OF FLOOll TIMBERS. 89 at the centre producing a deflection of | inch, is 5,000 - 358 = 4,642 Ibs. If the load is uniformly distributed, it will be f x 4,642 = 7,427 Ibs. To find by equation (74) the depth of the beam of Dantzic fir whose breadth is 10 inches, and length 20 feet ; W being 8,484 Ibs. Then by equation (74) we have "9 -' 15 ins. 52. SCANTLINGS OF FLOOR TIMBERS. The following Table gives the scantlings (breadth and depth) of floor joists of fellow fir (where E = 1,737,570 Ibs.) which shall carry a load of 120 Ibs. per foot of length uniformly distributed ; the deflexion not to exceed one-fortieth of the length of bearing, according to Tredgold's rule (50). The joists are supposed to be placed 1 2 inches apart from middle to middle. Single- joisted floors should never exceed a bearing of 18 feet. The above may be considered as the ordinary load which house floors have to support, but in the case of warehouses, workshops, or rooms where very heavy loads are to be sustained, the strength must be pro- portionally greater. Thus, if the joists are required to carry 240 Ibs. per foot, we must either double the breadth of the joist, or else multiply the depth by the cube root of 2, or 1'26. If the load is 360 Ibs. we must either treble the breadth, or else multiply the depth by the cube root of 3, or 1'44; and so on. If the joists are placed further apart we must either 90 DEFLEXION OF BEAMS. diminish the load or increase their scantling propor- tionally ; if placed at 18 inches from middle to middle, they would have to bear half as much again, so that the breadth must be increased in that proportion if the depth is unaltered ; or else the depth increased by multiplying it by the cube-root of 1*5, or 1'145. Bearing. Breadth 2Ius. Breadth 2J Ins. Breadth 3 Ins. Breadth 3J Ins. Breadth 4 Ins. Depth. Depth. Depth. Depth. Depth. 5ft. 4 ins. 4 ills. 4 iiis. 4 ins. 4 ins. 6 4i 4 4 4 4 7 5 4 4 l 4i 4 4 8 5 5 3 4i i 9 6i 6 5 5 5 10 7 64 6 6 5L 11 8 7 6^ 6, 1 6 12 84 8 7 7 6i 13 9 84 8 7i 7 14 10 9 i 9 8 8 15 104 9| 9i 9 8i Ifi 11 io n i4 9 17 11* 10 10i 10 9i 13 12 "I 10i- 10i 4 . When the span exceeds 18 feet it is usual to form a framed-Jloor, consisting of large beams of wood or iron called " Girders," which are placed across from wall to wall at distances of about 10 feet apart ; on these rest some smaller beams called " binders," placed about 6 feet apart ; and on these again rest the " bridging " joists on which the floor is laid ; these latter being about 12 inches apart. The scantling of the bridging joists is determined by the above Table ; that of the Binders is found in the same way by supposing the load on each of them to be six times as great per linear foot as on a bridging-joist. The following Table gives SCANTLINGS OF FLOOR TIMBERS. 91 scantling for binders of jr, on the supposition that the load per foot is 720 Ibs. Length of Bearing. 6" Deep. V" Deep. 8" Deep. 9" Deep. 10" Deep. 11" Deep. Breadth. Breadth. Breadth. Breadth. Breadth. Breadth. 6ft. 4^ ins. 4 ins. Sins. 2 ins. 2 ins. i 2 ins. 7 7 5 4 3 2 2 8 10| 7 5 4 2i 2 9 9 61 5 31 2i 10 12 8| 6 41 Si 11 1U 8 6 44 12 10J 8 6 When the load per foot is greater than 720 Ibs., the scantling must be increased in the proportions men- tioned above for common joists ; if doubled, then either l>y doubling the breadth, or multiplying the depth by the cube root of 2, or 1-26 ; and so on as before. The girders support the ends of the binders at points about 6 feet apart, consequently a girder 10 to 12 feet long will support the end of one binder at the middle, and will therefore carry half the weight of the floor, the other half resting on the two walls ; so that the scantling is that of a beam loaded at the centre by half that part of the weight of the floor which lies between two girders ; and by equation (73) we obtain b when L and d are given b = When the length of the girder is from 15 to 18 feet, we have a binder supported at two points, and one- third of the weight of floor resting at each point ; then 92 DEFLEXION OF BEAMS. by combining equation (26) with equation (73) we get b = 10 L3 For girders whose length is from 20 to 24 feet the binders rest upon three points, at each of which one- fourth of the weight is sustained ; then by combining equation (27) with equation (73) we have When the span exceeds 24 feet, iron girders should be used in place of timber where practicable, the section of which can be determined for a given deflexion by the equation (64). The following Table gives the scantling of fir girders when the load is as given above, namely 120 Ibs. on every square foot of flooring. If the load is 240 Ibs. per square foot, then either the breadth of the girder must be doubled or its depth increased by one-fourth. Span. 12" Deep. 14" Doep. 15" Drep. j 10" Deep. 18" Deep. Breadth. IJivniltli. Breadth. Breadth. Breadth. 10 ft. 4 ins. 3 ins. i 3 ins. 3 ins. 3 ins. 11 5 3 3 3 3 12 6 4 84 3 3 13 7 g 4 4 3 14 9 6 5 44 34 15 11 7 6 4 16 13 9i 7 64 H 17 15 10 8 6^ 5 18 16 11 9 7 64 19 18 13 10i 8 6 20 21 15 12 9 7 21 ... 17 14 m 8 22 19 16 12 9 23 21 18 14 10 24 20 16 11 FLANGED BEAM?. 93 53. FLANGED BEAMS. Beams with * ; flanges " connected by a vertical " web " are made of cast-iron, wrouglit-iron, or steel ; and of section shown by figs. 38, 39, 40. By equation (64) we get the deflexion (5) of such a beam at the middle, where all the dimensions are expressed in inches, W and E in Ibs. or tons. Let us apply this formula to find the deflexion of the cast-iron beam (38) whose depth is 13 inches, top flange 5" x 1", bottom flange 10" x 2", web 10" x 1", the length of bearing being 20 feet, the weight will be just 1 ton. Let W at the centre be 6 tons including f of the beam's own weight, or the actual load on the centre will be of tons. Taking E = 7600 tons, (25) and the value of I being 718, we have By Tredgold's rule (50) the deflexion should not exceed -H or i mcn m tm ' s case - Take the example of the wrought-iron box-girder (41), whose safe- load was found to be 14f tons ; the weight of this beam is 2 tons, five-eighths of which is 1 tons; therefore W = 16 tons. Taking I = 1770, I = 25 feet, E = 10,714 tons, we have for the deflexion in the middle, 1 1 ft O S.3 v 1 '^ 3 6 = 1 x J^-j- X ^^ = '4743 in. 48 10714 1770 By Tredgold's rule (50) the deflexion should not exceed 94 DEFLEXION OF BEAMS. -f-g- or -625 inch, so that the deflexion in this case is well within the rule. As an example of a rolled iron beam, we will take the first beam in the table (39), where the length is 10 feet or 120 inches, and the load at the centre 3| tons. Here the weight of the beam is ^ ton, five- eighths of which is T V ton, so that W = 3 T g- ton, the value of I is found to be 84. Then the deflexion at the middle is, By Tredgold's rule (50) the maximum deflexion is \^ or '25 of an inch. Let us apply the formula to the case of the steel box- girder (42), which has the same dimensions as the wrought-iron girder given above; the load at the middle being 35'4 tons, and five-eighths of its weight being 1*2 tons, we have TV = 36'6 tons ; taking E = 13,500 tons, I = 300 inches, we find the deflexion at the middle, -. . 48 13500 1/70 By Tredgold's rule (50) the maximum deflexion is 625 inch, so that here the deflexion is considerably in excess, and the load ought to be reduced in the proportion of 860 to 625, or nearly as 3 to 2. 54. BEAM OF UNIFORM STRENGTH. It has been shown in the last chapter that a beam may be made of uniform strength throughout, or have its resistance the same at every section, by making the plan of the BEAM OF UNIFORM STRENGTH. 95 flanges in the form of a parabola (43), being widest at the centre, and the depth being the same throughout. In such a beam the contraction or shortening of the fibres in the upper flange, and their extension in the lower one, will be uniform throughout, and consequently the curve which the neutral surface takes when the beam is loaded will be an arc of a circle. Let ABCD (fig. 4?) be such a beam, loaded in the middle, and let be the centre of curvature of the // neutral surface, AKBH the circle of curvature of the top flange. Then if AB (= t) is a horizontal line, / EK is the deflexion, 8, at [ the middle, p = OA = OK \ is the radius of curvature, AC (= cl) the depth of the beam. Draw AF perpen- dicular to CFD, then OF is half the difference in length between the most compressed and most ex- tended fibres. Let CF = \ A. Then HK and AB are two chords of the same circle intersecting in E ; and by Euc. Bk. 3. 35, the rectangle under the segments HE and EK equals the rectangle under the segments AE and EB. And since AE = EB, we have, HE x EK = AE 2 , Fig. 47. or, (2 p- = p . - B;it since 8 is very simll as compared with p, we 96 DEFLEXION OF BEAMS. may put 5 2 = 0. Then we have for the deflexion at the middle, 8 = . i ? very nearly. Also, since the triangle ACF is similar to triangle OAE, therefore OA : AE = AC : CF ; '- Hence we have 8 = ' = ' . (75) 8 (I . I >> d The value of A depends on the modulus of elasticity, E, of the material. If we put c for the conipressive stress per square inch in the extreme upper fibres, and t for the tensile stress per square inch in the extreme lower fibres, we have from equation (32), putting c + t for S, A for /, I for L, *-+4 X /> . . (TO) Substituting this value of A in equation (75) we obtain : The resistance of the web is not taken into account in this formula. If I is expressed in feet, then d must also be in feet, or if d is in inches, I must also be in inches. Take for example the cast-iron beam (38) of which the length was 20 feet or 240 inches, the depth 13 inches, c = 8 tons, t = 2 tons, E = 7,600 tons. Then, if we suppose the flanges to diminish from the centre BEAM OF UNIFORM STRENGTH. 97 towards the ends in the form of a parabola, we have by equation (77) for the deflexion in the middle, 10 400 X 144 = . 73i 7600 8 x 13 which is nearly half as much again as the deflexion allowed by Tredgold's rule (50), and more than double of that found previously (53) for a beam having flanges of equal width throughout. CHAPTER VI. STRENGTH OF PILLARS. 55. LONG PILLARS. By this term we mean pillars whose length exceeds 30 times the diameter or least width. When such a pillar, having a uniform section throughout, is placed in a vertical position with the lower end resting on a hard or impenetrable surface, and a load W is placed on the upper end, it will assume a slightly curved form as ACB (fig. v^r 48), being deflected from the straight line ADB by the amount CD which we call 8 ; and we may assume, without material error, that for a small value of 8 the curve is very nearly an arc of a circle. We can therefore find the amount of deflexion in the same way as in the case of the " Beam of uniform strength " (54), taking A to represent the difference of the lengths of the extreme fibres on the concave and convex sides. Let Fig. 48 d be the least diameter of the pillar, I its length, f the longitudinal unit of stress in the extreme fibres of either side in a horizontal section across its middle. Then the forces acting on the pillar are, W at A, the longitudinal tensile stress on the convex side at C, and the longitudinal compressive stress on the concave side at C. LONG PILLARS. 99 Taking moments about the point C, we have for the moment of resistance M, M = W . 8 ; and by equation (75) a *.'*. 8= 8r7' Therefore, W= ~ ' x M; A . i and by equation (76) Therefore, W = 4 i-^ x M = 4 d . x *j x M (78) For a square section, we have by equation (38) M-/^ 2 - cj ' x/- -J 6 -, - ( . x d , In which case, from equation (78) W = -| x E . . (79) And for a circular section whose radius is r = - , we have from equation (37) M -fl where I is the moment of inertia of the section about its centre, and z = r. For a circle, I = \ 77 r 4 , consequently M - ^ r ' v / w v d * v / 1V1 -- X 'f = - - X - X - , . 4 ^ 2 16 d H 2 100 STRENGTH OF PILLARS. Therefore, W = * X f ,' X E . o L" Hence it appears by comparison of equations (79) and (80), that the strength of a long solid pillar whose section is square, is to that of one whose section is the inscribed circle of the square, in the proportion of 2 x 8 to 3 x TT, or as 17 to 10. For long hollow pillars of circular section, of which d is the internal diameter, Experiment shows that this theoretical result only applies in the case of cast-iron to pillars whose length exceeds 50 diameters ; and in the case of wrought-iron where it exceeds 80 diameters. For pillars of steel or timber the above formula gives correct results for those whose length is not less than 30 diameters. The foregoing investigation has however but little interest for the architect, as he seldom, if ever, uses pillars in which the length exceeds 30 diameters. 56. HODGKINSON'S EXPERIMENTAL FORMULA. Taking the theoretical result above obtained in equa- tions (80) and (81) as the basis on which to work, Mr. Eaton Hodgkinson* proceeded to determine by numerous experiments the laws which govern the resistance of pillars. In the case of long round pillars of cast-iron the strength was found to vary in a rather less degree than the fourth porcer of the diameter, and inversely in a less degree than the square of the length. Comparing the strength of two pillars 10 feet long, * Phil. Trans., 1840 and 1357. HODGKINSONS EXPERIMENTAL FORMULAE. 101 their diameters being 1 J inches and 2| inches respec- tively, it was found that the ratio of their breaking- weights was as 6 .* 1. Since the ratio of their diameters is as 1'6667 : 1, if we put n for the power of the diameter by which the strength varies when the length is the same, we have 1" .* 1-6667" = 1:6, therefore, n = 1 - 1 -^~ = 3-5. log. 1-666; Patting x for the inverse power of the length by which the strength varies when the diameter is the same, it is found that two pillars which are each 2 inches in diameter, and whose lengths are 10 feet and 7i feet respectively, have their breaking- weights in the ratio of 1 to 1-598. And as the diameters are in the ratio of 1 to 1 -3333, we have to determine the value of a; from the equation F : 1-3333* = 1 : 1-598, x = k*^5?? = 1-63. log. 1-3333 The formula for the breaking-weight of a round pillar of cast-iron, whose length is at least thirty times its diameter, is As the result of numerous experiments it was found that the average value of S was 42, when W is in tons, d in inches, I in feet, so that the formula for the 102 STRENGTH OF PILLARS. breaking-weight in tons of a round solid pillar of cast- iron is W = 42 ^- ' ( 82 > a. n l'o3 3 ' 5 . n 3 ' 35 . 4 . 3 6-0 9-0 46-8 49-4 81 3-25 6-8 10-6 60-5 65-6 111-6 3-5 77 12-3 80-2 85-4 150-1 375 8-6 14-1 102-1 109-1 197'S 4 9-6 16-0 128-0 137-2 256-0 4-25 10-6 18-1 158-3 170-1 326-3 4'5 11-6 20-3 193-3 213-3 410-1 4-75 12-7 22-6 233-6 252-5 509-1 5 13'8 25-0 279-0 303-6 625 5-25 14-9 27'6 331-6 360-2 759-7 5-5 16-1 30-3 390-2 424-9 915 575 17-3 33-1 455-9 497-5 1093 6 18-6 36-0 529-0 578-8 1296 6-25 19-8 39-1 610-4 668-9 1526 6-5 21-1 42-3 700-2 768-9 1785 6-75 22-5 45-6 799-0 879-1 2076 7 23-9 490 907-5 1000 2401 7-25 25-3 526 10261 1133 2763 7-5 26-7 56-3 1155-3 1278 3164 775 28-2 60-1 1295-8 1436 3607 8 29-7 64-0 1448-2 1607 4096 8-25 31-2 68-1 | 1612-8 1792 4632 8-5 327 72-3 1790-5 1993 5220 8-75 34-3 76-6 19817 2209 5862 9 35-9 81-0 2237-9 2441 6561 9-25 37'6 85-6 2407 2690 7321 9-5 39-2 90-3 2643 2957 8150 975 40-9 95-1 2894 3243 9037 10 42-7 i 100-0 3162 3548 10000 10-5 46-2 110-3 3751 4219 12161 11 49-8 121 4414 4977 14641 11-5 53-6 132 5158 5828 17497 12 57-4 144 5986 6778 20736 12-5 61-4 156 6905 7835 24422 13 65-4 169 7921 9005 28561 13-5 69-6 182 9040 10296 33225 14 73-8 196 10267 11716 38416 14-5 78-2 210 11610 13270 44216 15 82-6 225 13071 14967 50625 HODGKINSON'S EXPERIMENTAL FORMULAE. 103 where the length I is expressed in feet, and the diameter d in inches. For hollow pillars, in which di is the internal diameter, . . (83) or, the strength of the hollow pillar is the difference between that of a pillar whose diameter is <-/and one whose diameter is d^ As these powers of d and I can only be found by means of logarithms, the preceding table of the values of d 3 ' 5 and I 1 ' 63 will help to facilitate calculation. As an example of the application of equation (82) let the diameter of a round column be 6 ins. and its length 15 feet. Then looking in the table in the column under a" we find 6 35 = 529; and looking in the column under a 1 " 01 we find 15 1T6> = tons ' which is considerably less than the strength of the pillar 15 feet long, and also than that of the cast-iron pillar 12 feet long. This difference is due to the smaller resistance to crushing that wrought-iron offers as compared with cast-iron. Apply the rule to find the strength of a fir pillar 6" square and 12 feet long. Here we have by equation = 8 = 73 tons; and by equation (87) w 72 x 3 x 36 ^ = 7, -+ 3 - x - 3 x -30 = which is greater by 5 tons than that obtained for the same pillar 15 feet long. 58. TABLE OF STKEXGTH OF PILLARS. The follow- ing table gives the breaking- weight in tons of solid GORDON S FORMULA. 109 round cast-iron pillars of various lengths and diameters, from 10 diameters in length up to 40 diameters, cal- culated from the .equations (82, 87). The crushing strength of cast-iron per square inch of section is taken at 50 tons in the shorter pillars. LENGTH IN FF.KT. Diaiu. 6 Ft. 7 Ft. 8 Ft. 9 Ft. 10 Ft. 12 Ft 15 Ft. 18 Ft. 20 Ft. 24 Ft. Inches. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. Tons. 3 98i 84 65 55 46 Si 161 135 115 94 79 59 4 239 203 175 152 126 94 a 337 290 250 219 192 141 98 5 455 394 343 302 267 213 142 105 6 752 660 584 520 465 392 269 200 168 7 1131 1011 907 817 737 607 461 343 289 8 1444 1270 1190 1083 906 707 570 461 342 9 1813 1664 1525 1300 1056 831 712 538 10 2193 2015 1735 1387 1130 1000 747 59. GORDON'S FORMULA. The formulae given by Hodgkinson being rather inconvenient for calculation, a simpler form has been adopted which has the advan- tage of being applicable to all pillars whose length exceeds 10 diameters. Suppose f to represent the breaking-weight per unit of the section, or the " unit- strength " of the pillar, and let r be the ratio of the length to its diameter, or least breadth ; a and b are constants depending on the material and form of section of the pillar, a being a close approximation to the crushing strength per unit of section, or the crushing " unit-strength " of the material. Then Gordon's formula for pillars whose ends are flat and carefully bedded is, /-- (88) 110 STPvENGTH OF PILLARS. For round columns of cast iron, whether solid or hollow, a, = 36 tons, b = T ^. For square cast-iron pillars and stanchions, b = -j^. The following table gives the value of/ for round and square pillars of cast-iron ; then to find the actual breaking-weight of the pillar, multiply the value of / by the area of section in inches. For solid wrought-iron square pillars, a 16 tons, b sinnr- - / 10 12 15 20 Tons. 18 25 30 35 40 Tons. 29 30 Tons. 27 Tons. 23 25 Tons. 14 Tons. 11 Tons. 9 Tons. 7 9 Round cast-iron. / / 23 20 14 16 13 12 10 Square cast-iron. 154 15! 15 13 11 10 Wrought-iron, square. For pillars whose section is angle, tee, channel or cross section, in wrought-iron, a = 19 tons, b = -g-j^-. For example, take the case of a round cast-iron pillar 4 inches diameter and 10 feet long. Here we have r = 30 ; A, the area of section is -n x 2 2 = 12*5604, and/ = 11 ; the breaking-weight is / x A = 138 tons. By Hodgkinson's formula, equation (82), the breaking-weight is only 126 tons. Let the diameter be 6 inches and the length 10 feet; then r = 20,/ = 18, A = T: x 3 2 = 28'2744; the break- ing-weight is therefore / x A = 509 tons ; and by Hodgkinson's formula it is only 465 tons. Take 6" for the diameter of pillar 15 feet long ; then r = 30, / = 11 ; and the breaking weight is /x A = 311 tons ; the breaking-weight of the same pillar by Hodgkinson's formula is only 269 tons. SHORT PILLARS. Ill Take the case of a wrought-iron pillar 6 inches square and 15 feet long; here r = 30, f = 12, A = 36 ; therefore the breaking-weight is / x A = 432 tons; the Hodgkinson formula, equation (84), gives the breaking-weight of such a pillar as only 344-7 tons. The same column 12 feet long, has r = 20, f = 14, andy x A = 504 tons for the breaking-weight, which is half as much again as that found for the same pillar by Hodgkinson's formula. It appears then that Gordon's formula gives a much higher result both for cast and for wrought-iron pillars than is obtained by Hodgkinson's formulas ; and we are disposed to think that Hodgkinson's method of calculating the strength of pillars is the more accurate one. 60. SHORT PILLARS. Pillars whose length is less than 10 diameters are liable to have their material crushed before any bending can take place, and there- fore their breaking-weight is found by multiplying A, the area of section in inches, by the quantity S for that particular material as given by the table (27), or the formula for very short pillars is W = S x A . . . . (89) Thus, a very short square wrought-iron pillar whose section is 6 2 or 36 inches, will have a breaking-weight of 16 x 36 = 576 tons. And a cast-iron round pillar 0> inches diameter, will have a breaking-weight of 50 x TT x 3 2 = 1414 tons. A short pillar of fir 6 inches square has a breaking-weight of 3 x 6 2 108 tons ; arid one of oak has a breaking-weight of 3 X 6 2 = 126 tons. .CHAPTER VII. EOOrS AND TRUSSES. 61. ROOFS. The covering of a building, whether of slate, tile, lead, zinc, or other material, has to be carried upon beams of wood or iron laid across from wall to wall, and generally placed at a considerable angle with the horizontal, called the " pitch " of the roof, so as to allow the rain- water to run oif freely. The angle of pitch depends upon the kind of covering used, and for slates or tiles should not be less than 27, which is the pitch adopted by Tredgold ; but where lead or zinc is used the pitch may be as low as 4. The weight of tlio covering per square foot also varies with the material employed, being about \\ Ib. for zinc, exclusive of the boarding on which it is laid; for lead it is about 7 Ibs., for slates from 7 'to 11 Ibs., and for tiles from \^ to 24 Ibs. In addition however to the constant load of the covering there is the occasional load of snow to be sustained, and above all. that of the wind pressure, which is greater for roofs of high than of low pitch. Tredgold puts down 40 Ibs. per foot for the wind pressure and other occasional forces, including snow, for roofs of 27 pitch, and the scantlings of timbers given in his " Carpentry " are calculated for an allow- ance of 66 Ibs. on the square foot, including the weight of timbers, boarding, &c. LEAN-TO ROOF. 113 Some authors however prefer to take the pressure of the wind separately from the dead load, as the wind can only act on one side of a roof at a time, and consequently subjects it to a racking force tending to strain the timbers unequally. We shall however in the following pages adopt Tredgold's method and take 66 Ibs. as the load per foot which the timbers of a roof must be prepared to sustain with safety, when the pitch is about 27, or the value of the cotangent of the angle of pitch is 2. Roofs may be divided into two distinct classes, namely, untrussed roofs and trussed roofs. In the former we have simple rafters placed either from wall to wall, as in the " lean-to " roof (62), or the " span roof" (63) where the rafters are in pairs meeting in the middle and forming a ridge. These rafters are covered with battens or boards nailed upon them, on which the material of the covering is fixed. Such roofs are only adapted to small spans, or where the walls have a considerable thickness. Trussed roofs on the other hand may be used for spans of almost unlimited amount, and are formed of a number of beams fitted together in such a manner as to be a mutual help to one another, and to prevent any outward thrust being caused by the weight of the covering. In these roofs the " trusses " are placed about 10 or 12 feet apart, and the load of the roof presses on them by means of horizontal beams called purlins. 62. LEAN-TO ROOF. The simplest form of untrussed roof is that in which the rafters are supported by two walls, one at a higher level than the other, as AB 114 ROOFS AND TRUSSES. (fig. 49), the lower end B being laid upon a piece of timber called a " wall-plate," which lies horizontally along the wall and dis- tributes the load uni- formly over its entire length. The upper end A is notched out and bedded upon a similar piece of timber. If W is the load upon the . 4 9 rafterj then Lalf W ig supported by the plate at A and at B. Putting Rj for the reaction of the plate at B, R 2 for that at A, we have Rj = I W = B, . . . ' . (90) Tredgold says* of a rafter placed in this manner, "cut so as to rest upon two level plates at A and B, the beam would have no tendency whatever to slide, notwithstanding its inclined position, and con- sequently it would have no horizontal thrust." " By cutting the rafters of a shed-roof, so that they may rest level upon the plates, the roof will have no tendency to push out the lower wall." Let G be the centre of the rafter AB, % the load per foot length upon the rafter whose length is I; then we have "W = w . I, and the transverse stress P at G is, P=w.l.cos.0 . . . (91) where B is the angle of pitch ; and the load being uniformly distributed, the deflexion will be that produced by a load at the centre (48) equal to x * TrcdgolcVs Carpentry, art. 42, 7th edition, by E. W. Tarn. LEAN-TO ROOF. 115 rc . I . cos. 0. Putting = 27, we have, w . I . cos. 6 = 89 m . I; and the deflexion is -55 w . L Supposing the rafters to be placed one foot apart from middle to middle, and w to be 66 Ibs., we have for the transverse stress, 36-3 I in Ibs. Then from equation (73), if we put ' ~ . = , we have by Tredgold's rule (50), for the scantling of the rafter when the deflexion is T V th of the length " 100 (92) where b and d are its breadth and depth in inches, I its length in feet. The following table of scantlings of rafters for a pitch of 27 has been calculated from this rule. Length of Rafter. j Breadth 2". Breadth 2J". Breadth 3". 5ft. Depth. 3 ins. Depth. 3 ins. Depth. 3 ins. 6 3.V i 3i 3 7 4 8| 8J 8 4| 4 10 6 5J 5 11 t;i 6 51 12 6| 6Jr 6 13 7 6i 14 8 yj 7 15 *i 8 7i If instead of notching the rafter AB on the plate at A, we cause it simply to lean against the face of the wall, we have a similar case to that shown by fig. 7, where a beam AB is supported at B and rests against a vertical wall at A, which supports it by i 2 116 ROOFS AND TRUSSES. means of the horizontal reaction T at A. The load W acting vertically at C meets the direction of the reaction T and of the reaction R of the wall B, in the point D ; since, as has been already demonstrated (4), the direc- tions of three forces in equilibrium must meet in one point. If the line DE (fig. 7) represents W, then BE will represent T, which will be the horizontal thrust of the rafter at B ; and we have (putting 6 for the angle of pitch) T : W = BE : DE = BE : 2 CE, or, T = W j^J. = i W . cotan. (93) L/hi We can obtain the same result by taking moments (8) of T at A and W at C about B (fig. 7) ; and in equilibrium we have T x DE = W x BE : or since DE = 2 CE, T>Tj1 T = W -==, = \ W . cotan. 0, as in equation (93). When = 30, we find T = -866 W ; for B = 45% T = | W ; for = 60% T = -29 W. If we put as before, W = w . /, and s = span of roof, we have s = I . cos. 0, or, I = ; and equa- tion (93) becomes m ?vl cos. w s 2 sin. 2 sin. 6 Then since sin. diminishes with the angle, 0, we see SPAN ROOF. 117 that the thrust T increases with decrease of 0. When B = 30, T = w . s ; when Q = 20, T = 1-46 ro.s; for = 10, T = 2-88 w . S-, and when 6 = 5, T = 5*75 w . s. 63. SPAN ROOF. This is the common form of roof, where the rafters are in pairs, and meet together in a " ridge " as AB, AC (fig. 50), the lower ends resting Fig. 50. on walls at B and C. If we suppose the load, W, to be uniformly distributed over each rafter, w the load per square foot, and I the length of the rafter in feet ; then "W = w . I is the load on each, and | TO . I the vertical pressure at B and C, m . I or W the vertical pressure at A. Let the vertical line A/i represent W at A ; draw he parallel to AC, and kg parallel to AB ; and draw the horizontal line efg cutting Ah iny. Then by the parallelogram of forces, Ae represents the com- pression P down the rafter AB, ef the horizontal thrust, T, which it produces at B. Then we have (B being the angle of pitch) Af= 2 Ah = Ae x sin. 0, 118 KOOFS AND TRUSSES. or, \ W = P . sin. ; therefore, P = t> ^ . . (95) Also since ef = Ae x cos. 0, therefore, T = P x cos. = ^ . c _^ = ? x cotan. (96) 2 sin. 2 w . I cos. 2 ' sin. 0' or, putting s for the "span" of the roof from B to C r we have | s = I . cos. 0, s = 2 I . cos. ; therefore, T = J^ . . . (07) The same result can be obtained by taking T for the mutual pressure of the rafters at A, W for the load on the rafter acting vertically at G ; the directions of these two forces will meet in the point D, consequently the direction of the reaction at B must meet them in D. Then if DE represents W, EB will be the horizontal thrust T at B, BD the reaction R at B ; or, by taking moments of T and W about B, we have T x DE = W x BE ; or, T x 2 EG = W x BE ; as obtained before, in equation (96) ; also we have SPAN ROOF. 119 R = VIF+T? . . (98) Produce DB to b, take the point a on tlie vertical line through the middle of the wall BH, and ab to represent on any convenient scale the value of R from equation (98) : and the vertical line ac the value of Q the weight of 1 foot length of the wall BH, on the same scale. Complete the parallelogram abed, and the diagonal ad will represent the resultant of II and Q both in direction and magnitude (2). Produce ad to cut the base of the wall ; then if the point where it cuts the base lies within the thickness of the wall it will be in a condition of stability under the action of the forces ; but if it lies outside the point H the wall will be overturned. We can determine the conditions of stability of the wall, or the thickness necessary to be given to it to insure its stability, by taking moments of the forces T, W, and Q, about the point H. Let h be the height and t the thickness of the wall. Suppose W and Q to act vertically down the centre of the wail. Then we have for the moments in equi- librium about H T xh = Q x L + W x |. rW the compression down DB. Now suppose the collar DE to be placed in position, then the pressure at D is supported by the two beams DE and DB ; we must therefore resolve in the direction of the beams as was done at A in fig. 50, by drawing a F parallel to BD, and Ba parallel to DF ; then DF re- presents the compression V in the collar when it acts as a strut. Then, V : W = DF : Da = Ea : Da ; and since V in DE, and since ~ = cotan. 0, we have for the compression W Y = _!!. cotan. . . (104) If we suppose the collar to have a square section, d being its breadth and depth in inches, I its length in feet ; then by equation (86) we have cl 1 = J x I* . . (105) b where V is expressed in tons, and the value of S is 1 ton for fir, and !- ton for oak. Putting V = ^ ton, S = 1, Z = 9 feet, we find by equation (105), cl = 2-28- 126 ROOFS AND TRUSSES. inches ; so that a collar 3" x 3" will suffice for the purpose. The following method for finding the horizontal thrust on the tie-beam when there is a collar acting as a strut, as in fig. 52, is employed by some writers on this subject. Suppose the collar to be placed any- where between A and B, and that AD = x ; then m . x is the load on AD, m (I x) that on BD ; and the vertical pressure at A from the load on AD and AE is n- . x. If P is the pressure down AD, we have from equation (95) p _ VO . X 2 sin. 0' The vertical pressure at D by the load on BD is u X (I x), and by the load on AD it is m . x ; there- fore the whole load at D is the sum of these two quantities, or \ m . I. Let Q be the pressure down DB from the load at D, then by equation (95) Q= 2 sin. The whole compression down the rafter is P + Q, and -the horizontal thrust, T, at B and C is therefore, T = (P + Q) cos. = -J- m (I + x) cotan. 6 . (106) which is the tensile stress produced upon the tie-beam. The value of T varies with that of x, or the distance of the collar from the ridge; being least when x 0, when there is no collar, in which case T = | W . cotau. 6, as in equation (96). It appears therefore from equa- tion (106) that the effect of the collar-strut is to in- crease the tension in the tie-beam, For when x = HAMMER-BEAM ROOF. 127 /, being 27, \ve have T = W ; but when there is no strut we get T = W. In this latter case, however, the bending of the rafters will tend to increase the thrust on the tie-beam. So that we have the alterna- tive of either increasing the scantling of the rafters, or of introducing a collar-strut to prevent them from bending. Another method of determining the horizontal thrust at the rafter feet when the " collar " acts as a " strut/' is as follows : Take the moments of the forces about D, namely, R = w . I acting vertically upwards at B, the weight on BD, w (I x), acting vertically down- wards at a point half-way between B and D, and the thrust T acting horizontally at B. The moments of these forces in equilibrium give the equation, m .1(1 #) cos. m (I x) x -J- (I x) cos. = T x (/ - -r) sin. 6 ' whence we get T = ~ (I + #) cotan. 0, which is the same as given by equation (106). 65. HAMMER-BEAM ROOF. This is a form of roof frequently found over buildings of the late Gothic period, the largest in this country being that which covers Westminster Hall, and having a span of 68 feet. The hammer-beam roof is generally a modification of the " collar " roof (64) the upper part usually having a collar, as DH (fig. 53) placed across horizontally between- the rafters and at about one-third or one-half of the distance from the vertex to the springing. The 128 ROOFS AND TRUSSES. feet of the rafters are framed into a horizontal piece called the "hammer-beam" which rests at one end on the wall and is supported at its outer end by a strut. Thus, in fig. 53, BF is the hammer-beam, and GF the inclined strut supporting its outer end F and resting against the wall at Gr. The foot of the rafter AB is framed into the hammer-beam at B, and is further stiffened by a vertical strut EF resting on the outer end of the hammer-beam. In order to investigate the action of the load in a roof of this kind, we will suppose that the weight of the roof covering is supported by purlins at A, D, E W and B ; and that - is supported by each rafter at A, 5- at B, at D and E. We will assume as before o o HAMMER-BEAM ROOF. 129 (64) that the collar takes all the horizontal thrust arising from the pressure down AD ; also that the strut EF takes the load W at E ; so that the only horizontal thrust, T, on the hammer-beam at B will arise from the action of the load W at D. Then if 9 is the angle of pitch, we have, as in fig. 51, W T = Jl x cotan. . . (107) o Since the beams BF and FG form a bracket (as shown by fig. 8), where the load W acts on the outer end F by means of the strut EF, there must be a ten- sile stress F, produced in the beam BF (4) acting from B towards F. Taking moments about G of the load at F and of the tension F in BF, we have in eqnili- 3 brium, F x BG = ^ x FB, o F = |x||. .(108) In order that there may be no outward thrust at B, we must have F in equation (108) equal to T in equa- tion (107), or it is requisite that = cotan.., or that the strut FG shall be parallel to BE. If BG is increased then F becomes less than T, and if BG is diminished, F becomes greater than T, so that the greater the inclination of FG to the wall BG the greater 130 ROOFS AND TRUSSES. will be the inward stress upon BF. By means of the strut FG the horizontal thrust is taken lower down the wall, and therefore the tendency to overturn it about its base is diminished. If BG- represents the load ^ W acting at F. then BF will represent the horizontal thrust F, on the wall at G, or we have F : W = BF : BG, > Take, for example, a roof having a pitch of 45, and let AB = 18ft., the span being 25| ft. Taking the load on the purlins at 660 Ibs. per foot of rafter, the principals being supposed 10 ft. apart, we have W = 11,800 Ibs., | W = 3960 Ibs., tan. =cotan.0= 1,BF = BG when FG is parallel to BE, so that there is no thrust at B. The horizontal thrust F at G will be J W or 3960 Ibs. Suppose the wall to be 20 feet high, BG being equal to BF, or to one-sixth of the span, or 4| ft. ; so that the horizontal thrust, F or T, will act at a distance of 15f feet from the base. Let t be the thickness of the wall, which it is required to find ; and suppose that the load W of the roof acts vertically down the middle of the wall. Let Q be the weight of 10 feet length of wall, w its weight per cubic foot, say 120 Ibs. ; then if h is the given height of the wall, we have Q = w . h . t . 10 = 120 x 20 x 10* = 24,000 . Taking moments about the outer edge of the wall, of the forces W, Q, and T, we have in equilibrium TKUSSED ROOFS. 131 |1 + 11880 | = 3960 x , or, t* + -5 - 5-2 = 0, from which we find t = 2*04ffc. as the minimum thick- ness of the wall that will keep it from being overturned. In order that the wall may have sufficient stability its thickness should be 3 feet. The tension in the hammer-beam in this case is 3,9601bs., which is also the compression down EF. The compression down the strut FQ is |^ = ~^ = 5600 Ibs. The tension in the collar DH is | W or 1980 Ibs. and the compression down AD is ^ = 2800 Ibs. ; that down DEB will be 8400 Ibs. ; the load J- W at E being taken by the strut EF. The transverse stress at right angles to the W rafter at D is -g cos. = 2800 Ibs. ; then by Tredgold's rule (50) we can determine the scantling of the rafter from the equation (72), namely, 2800 = 100 -^ . Putting the breadth b = 5 inches, we find d = 9^ inches, in order that the rafter may not deflect more than three-tenths of an inch at D. 66. TRUSSED ROOFS. This term is applied to those roofs in which a number of beams are connected together in such a manner as to form what is called a " truss." These pieces of framing are placed on the walls at intervals of 10 or 12 feet, and support the covering of the roof by means of longitudinal K.2 182 ROOFS AND TRUSSES. "purlins," as explained before (65). The whole weight of the roof is therefore borne by the trusses at the points where the purlins rest upon them. For roofs of moderate span it will be sufficient to make the truss of a pair of " principal rafters," as AB and AC 3 Fio.54. (fig. 54), which are prevented from thrusting out the walls by having their feet B and C framed into a tie- beam BC, to which they are secured by iron straps and bolts. The purlins are fixed on the "principals" at A, B, C, D and E ; and if AV is the load on each side, we have ^ W sustained at each, of the points A, D and E, and ^ W at B and at C. To find the stress on the beams, take the vertical line Aa to represent | W, draw ab parallel to AE, etc parallel to AD, and be horizontal; then the compression P down the rafter AD is represented by the line A; and we have W P . sin. 6 = j- , where B is the angle of " pitch ; " W Ps= 4isrfl- Let the load |\Y at D be represented by the vertical line Del, draw cl/ parallel to the rafter, Df at right TRUSSED ROOFS. 133 angles to the rafter, ed parallel to D/. Then the line De represents the compression Q down the rafter from the load at D, and the line I)/ or ed the transverse stress on the rafter at D. Then we have w Q = f sin. e, and the total compression down DB is represented by A.b + D, or by P + Q ; and we have compression in rafter DB in -' (109) The tension, T, in the tie-beam is represented by bh + e g^ or by (Kb + De) cos. 6 ; therefore T = (P + Q) cos. 6 = ~ cotan. e + ^ sin. 6. cos. . (110) 4 6 The transverse stress, V, at D is represented by D/ or Del . cos. 0, or we have V - ^.cos.0 . . . (Ill) For example, taking Q = 27, let AB = 10 ft., W = 6600 Ibs., then cotan. = 2, cos. 9 = -89, sin. 6 = '45, J W = 3300 Ibs. Then from equation (111) we find V = 2937 Ibs.; and by Tredgold's rule for the deflexion (50) we determine the scantling of the rafter from the equation, 2937 = 100 b -^, or, b . d 3 = 2937. 134 KOOFS AND TRUSSES. If we assume the breadth, b, to be 5 inches, we get 8 J ins. for the depth ; or if b = 4 ins., then d 9 ins. Such a scantling will be found more than sufficient to- resist the compression given by equation (109). The tension, T, in the tie-beam is found by equa- tion (110), T = ^ + ^x-4 = -7x 6600 = 4620 Ibs. which is about the safe tensile stress on 9 square inches of fir, so that a tie-beam 3" x 3" would be sufficient for this purpose. In practice, however, the breadth of the tie-beam must equal that of the rafter which is framed into it. It must also be of sufficient depth so- as not to bend by its own weight more than -fa inch for each foot of length. Taking the scantling as 4" x 4",. the weight per foot will be about 4 Ibs., and the length being 18 feet, the total weight will be 72 Ibs., and we must take f of this as the load at the centre causing deflection, or 45 Ibs. ; we have then by Tredgold's rule (50), K = 10oi^i 3 = 79 Ibs. lo" Therefore it is evident that a scantling of 4" x 4" is more than sufficient to resist any stress that the tie-beam has to bear. In such a case, where the tie-beam has no load to carry, an iron rod or bar may be used in its place. When the tie-beam has to carry the weight of a ceiling the load per foot must be taken at 120 Ibs., if the trusses are 10 feet apart ; in which case we have w = 120 x 18 = 2160 Ibs. Then we find the scantling from the equation KING-POST ROOF. 135 X 2160 _ 100 which gives b . d 3 = 4374 ; and if we put 5, then we find d = 9 J ins. When there is a floor to be carried by the tie-beam, as well as a ceiling, the beam must be made strong enough to bear a distributed load of 12001bs. per foot length ; or b . cl? will be ten times as great as when there is only a ceiling. 67. KING-POST ROOF. When the tie-beam of a trussed roof has a considerable load to carry, it is advisable to support it in the middle by means of a piece of timber called a " king-post," as AF (fig. 55). The heads of the principal rafters are framed into the head of the post at A, so as to hold up the post, and the tie-beam is held up in the middle by means of a strap passed round it and bolted to the post. By this means half the load on the tie-beam is carried by the rafters, which causes an increase in the compression down the rafters, and a corresponding increase in the tension in the tie-beam. This beam will then be divided into two equal parts, each of which has half the load to carry, and is half the length of the whole 136 ROOFS AND TRUSSES. beam ; consequently the moment of stress is reduced to one-fourth, and the strength of the tie-beam to bear a transverse strain need only be one-fourth of what it was when there was no king-post. A light iron rod may be used for a king-post instead of wood, when it has only a tensile stress to bear. If w is the load on the whole length of the tie-beam, then we can deter- mine the scantling necessary according to Tredgold's rule (50) by the equation, or, fl.rf' = i |L xL' . . (112) If TV = 2160 Ibs. as in the last example (66), where L = 18 ft., then we find from equation (112), b . d? = 547. Putting b = 4, we find d = 5J ins. ; so that the effect of the king-post is to reduce the quantity of timber in the tie-beam by one-half. The tensile stress in the king-post will be , and the sectional area can be found from the equation = S.fl.rf, ord.d= . (113) where S is the coefficient of safety. Taking S = 500 Ibs. per square inch for fir, we have l).d= = 2-16 inch 1000 in the foregoing example. KING -POST ROOF. 137 As however the rafters are framed into the head of the king-post, its breadth must be the same as that of the rafters. If an iron rod is used for a king-post, we can put S = 4 tons, or about 16 times as much as when fir is used ; so that a rod of iron J inch in diameter will serve the purpose. The compression R down each rafter from the stress on the king-post is ~ 4~sTn7e>' which must be added to P + Q in equation (109) to get the total compression down the rafter DB, namely, P + Q + J--. . _ 4 sm. 02 4 sin. -V + *+%**. . (114) 4 sm. 2 The tensile stress in the tie-beam is now increased by B. cos. = - cotan. ; so that the total tension, T, in the tie-beam is, T = ^L+J"- cotan. + ^ sin. . cos. . (115) which in the foregoing example is 5700 Ibs., or the safe tensile stress on 12 square inches of fir ; so that the scantling given above is amply sufficient. The final step to be taken in completing the king- post truss is to insert a strut between the foot of the king-post and the middle points D and E (fig. 56) of the rafters, by which means the greater part of the 138 ROOFS AND TRUSSES. load at those points is taken off the rafters, and we are enabled to reduce their scantling in the same way as we reduced that of the tie-beam by inserting the king- post. There will then be no transverse stress on the rafter to be taken into consideration, but only the compression down it, which will be increased by the Fig. 56 strut conveying the load at D and E to the king-post, and thence to the head of the rafters at A. When there was no strut at D, as in fig. 54, the \V load at D was resolved parallel and perpendicular to the rafter ; but the insertion of the strut alters the directions of the forces, and we must treat the beams DB and DF as if they were a pair of rafters, and resolve ^ down each, as we did in fig. 54. Taking Del to represent | TV, and drawing da parallel to DF, db parallel to BD, we have Da representing the com- pression clown DB, Db that down the strut. Drawing the horizontal diagonal ab, we have | db or ac repre- senting the horizontal thrust on the foot of the rafter and consequently the additional tension in the tie- beam. The line DC will represent the additional stress KING-POST ROOF. 139 in the king-post arising from the pressure down one strut, or Dd will be the additional stress in the king- post arising from the two struts ; so that an addition of | W must be made to the stress in the king-post. The total stress in the king-post is therefore - + ^ Taking Q for the compression down the strut, we have This is also the corresponding compression down the rafter DB due to the load at D. If W = 6600 Ibs., 6 = 27, sin. = '454, we find Q = 3633 ibs. Apply- ing the equation (105), and putting Q = 1| tons, we have for fir, d 4 = il x 5 2 = 42, when the length of the strut is 5 ft., or, d = 2*546 inches. The additional tension in the king-post caused by W the two struts, is 2 Q sin. = -^ . Putting F for the total tensile stress in the king-post, we have In the foregoing example W = 6600 Ibs., w = 2160 Ibs., therefore F = 4380 Ibs. To find the minimum scantling of the king-post we must put, as in equa- tion (113) F = S . b . d. 140 EOOFS AND TRUSSES. Therefore, J . rf = !?_+ _^ . . (118) Taking S = 500 for fir, we get b . d = 8*8 square inches, and as the breadth must be 4 inches, the scantling will be 4" x 2J". Putting R for the compression down each rafter caused by the tensile stress in the king-post, we have R.sin.0 = |; B = W + 4 sin. The total compression down the rafter AD is therefore 4 sin. d 4 sin. 6 2 W + m 4 sin. 6 (119) In the above example we find P + R = 8458 Ibs. To find the compression down DB, we have to add Q as found by equation (116) to P + R, so that the pressure down D B is In the above example we find P+Q + R=12,100 Ibs. The tension, T, in the tie-beam is T = (P + Q + R) cos. 6 KING-POST EOOF. 141 In the foregoing example we have, since cotan. 6=2 nearly, T = 10,781 Ibs. To find the scantling of the tie-beam to resist this tensile stress, we must put T - S . b . d, as in equation (113); and in the above example this will give b.d= - = 2H square ins. 500 so that a scantling of 5" x 4J" will be sufficient. To find whether the scantling 5" x 4J" is sufficient for the rafter, we must consider the part BD as a pillar 5 feet long with a pressure of 12,100 Ibs. or 5'4 tons; and that it will bear one-fourth more than a pillar 4 ins. square. The breaking weight of a fir pillar 4 ins. square and 5 feet long can be found by equations (86, 87) to be equal to 39 tons ; and the strength of one 5" x 4'' will be nearly 49 tons, which is about 9 times the actual pressure ; so that this scantling may be con- sidered as sufficient. "We have now seen how by the addition of each piece of the truss the stress on the other parts is changed, and also the kind of stress to which each part is sub- jected. There are other stresses which have to be borne besides those we have been considering, such as the compression of the head of the king-post by the heads of the principal rafters being framed into it, and the compression on the foot of the king-post by the heels of the struts. There is also the compression on the tie-beam by the feet of the rafters. Summing up all the results obtained in the foregoing 142 ROOFS AND TRUSSES. example, where Q = 27, and the rafter is 10 ft. long, the span being 18 feet, and the tie-beam carrying the weight of a ceiling only, we have Compression down rafter AD = P + R = 8458 Ibs. do. do. DB = P + Q + R = 12,100 Ibs. do. do. strut DF = Q = 3633 Ibs. Tension in the tie-beam BO = T = 10,781 Ibs. do. king-post AF = F = 4380 Ibs. From these we find the minimum scantling of the beams to be as follows : Tie-beam . . . . 5" x 4J" Principal rafters . . . 4" x 5" do. struts . . . 3" x 2|" do. king-post . . . 4" x 2" The scantling of the purlin at D, which is placed square with the rafter, can be determined by Tredgold's W rule (50) the load - being uniformly distributed along iC it, and its bearing being 10 feet. The actual transverse W stress is therefore - x cos. 0, and for deflexion under A a distributed load we take five-eighths of this for the load at the middle. Consequently the equation from which to determine its scantlin is 82 ID 2 from which we get b . d 3 = 1856 ; and if we put b = 4 inches we find d = 8 inches, or the scantling of tlie purlin is 4" x 8". STRESS-DIAGRAM. For the purlin at A, we have 143 82 10* or, b . d 3 = 2063, . and the same scantling, 4" x 8", will do in this case. 68. STRESS-DIAGRAM. It has been previously shown (6) & (7) that when a jointed polygonal frame is in equilibrium from forces acting at the joints, the relative magnitude of the stresses they produce in the direc- tions of the several bars can be found by drawing a " stress-diagram " whose sides are respectively parallel to the directions of the impressed forces, and from the vertices of the new polygon drawing lines parallel to the sides of the frame. This method is now generally Fig. 57 employed for determining the stress in every bar of a truss, and we shall proceed to show its application to the king-post roof which we have just been considering. First draw an outline of the truss, as fig. 57, and mark the loads supposed to act at each of the joints. Thus at A we have - "W supported by the end of each rafter, so that | W is the total pressure at A ; at D and E we have | W ; and at B and C we have ^ W. All these forces act vertically downwards. At B and C there is 144 ROOFS AND TRUSSES. also the reaction R acting vertically upwards, and if m is the load on the tie-beam, we have R = W + -J. In the example given above we have W = 6600, re = 2160, |W = 3300, W = 1650, \w = 540, R = 7140. To form the stress-diagram, draw a vertical line b X y (fig. 58) to represent on any convenient scale the value of 2 W, and bisect this line in X. Take Xf/, Xe, gf and ba each to represent ^ "W ; gk and each to represent R. Draw horizontal lines through k and I, as and /ew. Then draw an parallel to the rafter BD ; fm parallel to the rafter CE, meeting In and km in the points n and m. Draw dp parallel to the rafter AD, meeting np parallel to the strut DF. Draw eq parallel to the rafter AE, meeting mq parallel to the strut EF. Then, if the figure is correctly drawn, the line joining p and parallel to KL. Draw dn parallel to EC, meeting mn parallel to EK; also co parallel to CF, meeting op parallel to FK ; and draw the vertical line no. Beginning at the point A, the polygon of forces in equilibrium at A is, ah, X, X^, ia, in which ia acting towards A, gives the compression in the rafter AD, and Xz acting from A is the tension in the tie AH. Proceeding to the joint H, we have the polygon formed by the lines XJ, //, z'X, all acting from H, BOW-STRING TRUSS. 167 which gives li the tension in the brace DH, and X/ that in HI. At D, the polygon of forces is, ca, ai, il, Ik, he, which gives kc acting towards D for the compression in DE, and Ik acting from D for the tension in the brace DL At I, we have the polygon, Xw, mk, Id, /X, all Acting from I, which gives mk the tension in El, and Xm that in IK. At E, the polygon of forces is, dc, ck, km, mn, nd, giving nd acting towards E for the compression in CE, and mn acting from E for the tension in EK. At the joint C, the forces in equilibrium are re- presented by X for the angle which OK makes with the horizontal OD, we have P = p . sin. . Take the line K = P . (134) Let Rj, E, be the reactions of the two sides, meeting in the point a ; p . R the friction ; then we can deter- mine E, in the manner previously shown in the case of the wedge (78) ; and the resistance of friction is found as in equation (133); putting for the angle KOE, and Q for the resistance of the two sides, we have Q = ji . P . cotan. Q = /x . p . sin. $ . cotan. . . (135) And since the force Q acts with P to prevent the re- action 2R from pushing back the voussoir, conse- quently the voussoir will remain in its place so long as 2R is less than P + Q. 182 ARCHES. If p and are the same at each voussoir it will be seen from equation (134) that P and Q decrease as the angle < decreases, or as the voussoirs approach the springing; being greatest when $ = 90, or at the " key - stone," and least, when = 0, in a semi- circular arch. From this we see the necessity of in- creasing the value of p as we get further away from the crown of the arch, otherwise it will have a tendency to break up by the rising, or pushing back, of the voussoirs at the haunches. The reaction R, and conse- quently the friction F, is however greater in the lower voussoirs than in the higher ones, owing to the weight of the upper ones pressing upon the lower. If there is a heavy load on the crown of the arch and an insufficient load on the haunches, the arch will break up by the falling in of the crown, the joints AB and El opening at B and I, while those at LJ and MN open at M and L ; thus throwing all the pressure on the outer edges of the voussoirs, and causing them to crumble to pieces. It will be seen from equation (132) that the reaction, R, of the sides of the voussoirs increases as sin. 6 diminishes, or as the angle which the joints make with each other is lessened. Also from equation (133) we see that the force Q increases with the increase of the cotangent of Q or with the decrease of the angle 0. Hence it will be evidently an advantage to make the arch consist of a large number of narrow voussoirs, rather than a small number of wide ones, as the former will be able to carry a heavier load. Since the reaction, R, measures the pressure which the surface of the bed of the stone has to sustain, we APPLICATION TO THE ARCH. 183 must take care that it does not exceed the limit of safety for the resistance of the material to a crushing force. If b . d is the area in inches of the bed, and S is the load per square inch that may be safely borne ? which is about one-tenth of the crushing- weight, we must have R not greater than S . b . d. If Yorkshire stone is used the value of S is about 760 Ibs. ; if Port- land stone, S is 390 Ibs., and for Bath stone S is 150 Ibs. It will be evident that by increasing the area of the bed of each voussoir, or its depth d } we increase the strength of the arch itself, or its usefulness for supporting a heavy load. For example, we will suppose an arch of 1 foot breadth of soffit has to support a wall 20 feet high, the back of each voussoir being 12 inches ; to find the necessary depth d of the voussoirs, b being 1 foot. Take the keystone as bearing a load of 20 cubic feet of brickwork weighing 1 cwt. per cubic foot ; then we have P = 20 cwt. = 2240 Ibs. Let = 6, the arch being semi-circular ; then we have sin. Q -10453; and from equation (132) p = 2 sin. 112() - = 11,000 Ibs. nearly. 10453 Also we have R = S . b . d = 12 S . d, since b = 12 ins. ; Al , , 11000 therefore, a = - -ia 184 AKCHES. If we put S = 150, for Bath stone, we find 11000 , . The radius of the arch in this case is 4 feet, or the span 8 feet. 80. JOINT OF RUPTURE. Suppose we have a semi- circular arch, as GAL (fig. 83), resting on two supports at CD and KL. Then, from what has been previously shown (79), if the haunches at F and H are insuffi- ciently loaded, the voussoirs will have a tendency to rise at F and H, being pushed back by the pressure of the upper part of the arch. The arch in this case will give way by the falling in of the crown, the joint at AB opening at B, so that the whole pressure comes upon the edge at A. The haunches will rise at F and H by the opening of the joints EF and GH at F and H, whereby the whole of the pressure is thrown upon JOINT OF RUPTURE. 185 the edges at E and Gr. Let N represent the mutual reaction of the two halves of the arch meeting at AB ; then at the instant of rupture the force N acts horizon- tally at the edge A, and in order to ascertain the con- ditions of equilibrium in the arch, we must find the position of the joint EF, where the thrust, N, of the other half of the arch will have its greatest effect ; that joint being called the "joint of rupture." Let be the centre of curvature from which all the joints of the arch radiate ; let P be the weight of any portion of the arch between AB and the joint EF ; then P will act vertically at the centre of gravity, y, of the arch, and its moment about E will be, P x EJW. The moment of N about E is, N x Am ; and in order that there may be equilibrium in the arch, we must have N x Am = P x E/> ; N = P| . . (136) In order to find the "joint of rupture," or that at which the value of N is greatest, we must express these quan- tities in terms of the angle, 0, which OEF makes with the vertical OB ; and by calculating N for several values of 6, we find what value of makes N greatest. It is necessary in the first place to ascertain the position, ffj of the centre of gravity of the arch ABEF. which we do by means of equation (31), namely, where OA = R, OB = r, and is the angle BOE. 186 ARCHES. Then we have E/? = Ew mp Q = r . sin. Q Ov . sin. 2 4 R 3 -?- 1 (137) Aw = OA - Om = R - r . cos. . . (138) Taking one foot for the thickness of the arch, and making 8 represent the weight per cubic foot of the material ; we have P = (R + r) (R - r) | . 8 - 8 1 (R - r) . . . (139) Then by combining the equations (137, 138, 139), we obtain the value of N for any given values of 0, R and r, by means of equation (136). In order to find the value of which makes N greatest, we take the case of R = 12, r = 10, and calculate N for various values of 0. Take = 56 ; or, arc = -9774, see table (21). Then, sin. = -829, cos. = -5592, sin. 2 -f- = -2204 ; and we find N = 11-127 x 8. Now take = 58; or, arc = 1-0123, sin. = -848, STABILITY OF THE AKCH. 187 cos. e = -5299, sin. 2 -^ = -2354; from which we find N = 11-161 x 6. Let = 60 ; or, arc e = 1-0472, sin. 6 = -866, cos. f\ = '5, sin. 2 _ = -25, from which N = 11-17 x 6. Let e = 62 ; or, arc = 1-0821, sin. = -883, cos. a = -4695, sin. 2 -- -2653, and we get N = 11-155 x . Let = 04; or, arc = 1-117, sin. = -8988, cos. = -4384, sin. 2 1 = -2808, N = 11-106 x 8. Hence it appears that the value of N is greatest when = 60, its values diminishing as either in- creases or decreases. The joint EF which makes 60 with the vertical may therefore be considered to be the " angle of rupture." 81. STABILITY OF THE ARCH. Having found the " joint of rupture/' we can now proceed to determine the necessary strength of the pier or abutment of given height to resist the thrust of the arch. By means of equation (136) we obtain the value of the horizontal thrust N, and since the moments of N and P balance about the point E, we can by the " transposition of couples," which has been previously demonstrated (11), consider N and P as acting at the point E, without 188 AECHES. alteration of direction or value. Let ZX be the base of the pier which supports the arch, h its height DX, t its required thickness ZX ; then we have to equate the p moments of P and N acting at E, of the weight - of the arch between CD and EF acting at /, and of the weight Q of the pier acting at its centre of gravity q ; all these moments to be taken about the outer edge Z of the pier. From equation (139) we have (when Q = 60) P = 8 x -5236 (R 2 - r 2 ) . . (140) EJ = -866 r - '3183 r^- - (141) Am = H- L . . (142) N = P x %L. Am The moment of P, acting at E, about Z, is P x El = P (/ + t - r . sin. 0) = P (t + -134 r). The moment of N, acting at E, about Z, is N x IZ = The moment of | P, at ^', about Z, is i p x J/ = i P (t + r - O/ . cos. DO/) STABILITY OF THE ARCH. 189 The moment of Q, acting at q, about Z, is Equating the moment of N with all the other moments, we obtain the " Equation of Equilibrium," namely, = P( + -134 r)+^ (143) The value of N is found from the equations (140, 141, 142) ; and if the arch and abutment are of similar material we can omit 8 from the equations, or consider 8 = 1. For example, let R = 12 ft., r = 10 ft., h = 10 ft, then we find from equation (140), P = 23-038, P = 11-519 ; from equation (141) EJW = 3-394; from equa- tion (142) Am = 12 - 5 = 7; N _ 23-038 x 3-394 = n . 1? The moment of N at E, about Z, is 11-17 x 15 = 167-55; the moment of P at E, about Z, is 23-038 (t + 1-34) = 23-038 t + 30-87 ; the moment of J P at g', about Z, is 1 1-519 (*- -54) = 11-519* -6-22; 190 ARCHES. the moment of Q at = 8-66 - 5-58 = 3'08, from equation (145) Km = 12 5 = 7, from equation (140). Then by equation (148) we have N = 0338_x3^8 x _ H . m ^ t In order to determine the thickness^ of the pier of given height h, we take the moments of N and P, as acting at E, about the point Z ; and the pier can be taken as consisting of the two rectangles HVXZ, called Q, and o 194 ARCHES. HFLK, called F, whose moments about Z are Q x and F x gh. Let h = 10 - DX. KL - MK - ML = t + r - -866 R = t - '392, F = KL x FL = 11 t - 4-312, gh = KL = i (t - -392). Then the moment of F about Z is F x <7/i = 5-5 (t* - -784* + -154) = 5-5 t' 2 - 4-31 1 + -847 HZ = h + ~ = 16, Q . L = HZ x = 8 2 . The moment of N, acting at E, and taken about Z, is N x Z = 4-33 - mp = 1-56. From equation (146), Am = 6 - 2'5 = 3-5. mi f AT T> Ew 25-98 x 1-56- n - * Therefore, N = P -^- = , o = 11-58 . 8: Am 3-o F = KL x FL = 6 (* - -196); F x ah = 3 (f - -392* + -04) ; Q x [ = HZ x-J= 6-5 f; Ze = 10 + 2-5 = 12-5; Ee = t + 5 x -134 = t + -67. Then the Equation of Stability when pier and arch are of same material, is, 2 x 11-58 x 12'5 = 25-98 (t + -67) + 3 (f - '392 t + -04) + 6'5 f ; which reduces to, f + 2 -61* - 28'63 = ; from which we find, t = 4*2. Example 2. R = 10, r = 8, k =4, k = .10 ; then R - r = 2, BM = 6, FL = 9, ML = 8-66, KL = t - -66, HZ = 15, Ze = 14, Ee = t + 1-072. P = ?JL? X 8-66.5 = 65.5; 8-66 24 ' LOADED ARCH. 197 Eju = 6-928 - 4-62 = 2-308; Km = 10 - 4 = 6 ; 65 x 2-308 F x gh = ? (t - -66)" = 4-5 (f - 1-32 + -44) Q x ~ = 15 x|- = 7-5f Then the equation of stability becomes 50 x 14 = 65 (t + MJ72) + 4-5 (f- - 1-32 t + -44) + 7-5 f or, f + 5 t - 52-36 = t = 5'15. Example 3. Let R = 12, r = 10, k = 5, A = 10; then R - r = 2 ; BM = 7, FL = 11, ML = 10-392, KL = t - -392, HZ = 16, Zg = 15, Ee = t + I'M. P = 93-528 . 8 nip = 5*58, E/> = 3 - 08, Am = 7 F x ffh = 5-5 (f - -784 + -154) Q . I - = 8 f Then the equation of stability becomes 1234-56 = 89- 218 t + 13-5 f + 126-177 or, t 2 + 6-6 t - 82-1 = t = 6-35. Example 4. Let R = 16, r = 14, k = 7, h = 15 ; 198 ARCHES. then R - r = 2, BM = 9, FL = 15, ML = 13-856, KL = t + '144, HZ = 23, r Le = 22, EC = t + 1-876 ; P = 166-27 . 5 tnp = 7-5, Ep = 12-124 - 7-5 = 4'624, Az = 16 - 7 = 9. N= 166-27 x 4-624 F x yh = (t + -144) 2 = 7-5 (f + -288 1 + -021) The equation of stability becomes 3759 = 19 f + 168^+312, or, f+ 9^-181=0 t = 9-65. Example 5. Let R = 20, r = 17, k = 10, /< =20; then R - r = 3, BM = 13, FL = 20, ML = 17-32, KL = t - -32, HZ = 30, T Le = 28-5, Ve = t + 2-278 P = 285-8 mp = 9-27, E^ = 14-72 - 9-27 = 5-45, Km =11-5 N= 285 '^ 5 ' 45 .5 = 135-44. 6 F x gh = 10 (t - -32) 2 = 10 (f - -64^ + -102) Q x^-isr. The equation of stability is 135-44 x 57 = 285-8 (* + 2-278) + 10 (f 3 - -64 1 + '102) + 15 t\ LOADED ARCH. 199 or, f + 11-176 - 282-7 = 0, t= 12-1. Example 6. Let R - 24, r = 20, 7; = 12, = 25 ; then R - r = 4, BM = 16, FL = 24, ML = 20-78, KL = t - -784, HZ = 37, r Lc = 35, Ee = t + 2^8 ; P = 415-6. 5 zj = 11-09, E;J = 17-32 - 11-09 = 6-23, Km = 24 - 10 = 14 __ B 14 F x gh = 12 (t - -784) 2 = 12 (f - 1-568 1 + '61 5) Q . 4- = 18-5 e. The equation of stability becomes 185 x 70 = 30-5 f + 396-8 t + 1106-4, or, f+ 13 #-388 = 0, t = 14-25. The following Table gives the results obtained in the foregoing examples, the material of the arch and pier being supposed to be the same in all cases, so that the value of 5 can be omitted. R. r. fc, h. t. 6 5 3 10 4-20 10 8 4 10 5-15 12 10 5 10 6-35 16 14 i 15 9-65 20 17 10 20 12-10 24 20 12 25 14-25 200 ARCHES. Fig. 86. The following method of approximating to the calculation of the thrust of a loaded arch, will be found more simple in practice, although perhaps not quite so accurate as the foregoing ; the result obtained will however be sufficiently exact for the purposes of the architect. Let AB (fig. 85) be the intrados of the semi-arch, DKL B the load line, AX the height of the pier, XZ its thick- ness. Draw the line AD, and sup- pose the triangle ADK to represent (very nearly) the area of the figure EBDKF, and let G be the centre of gravity of the tri- angle (17). The radius OE is drawn at an angle of 30 with the horizontal line OA, being the centre of the arch. Through E draw the horizontal line FEmn ; the point F being on the vertical KAX. We will consider F to be the point about which the moments of P at Gr and N at C balance one another ; LOADED ARCH. 201 N being the horizontal thrust at C, P the weight of the triangle ADK acting at G. Then, as before, we have P x Fm = N x Gti N = P ^ Gn which determines the value of N. We can now take N arid P as acting horizontally and vertically at the point F, and equate their moments about Z with the moment of the weight (Q) of the pier whose sectional area is LKXZ, the force Q acting down the centre of the pier, as ek. Putting t for the thick- ness XZ of the pier, we have for equilibrium And for stability ^ For example, let OA = 10, BC = 1 ; CD x 2, AX = 10, FX = 15, Q = 24 t, P = *l2Li? = 70, Fw = CM = 13 O = M ; then the diagonal ch represents the resultant R of the forces N, P, and Q, and will cut the base ZX at a point I within the base. In this way we can determine the amount of stability possessed by the structure for any given thickness of the pier. If the point I lies outside Z, the structure will be overthrown ; if it falls at Z, there will only just be equilibrium ; and for stability the point I should be at least one-fourth of ZX within the base, or U should be not less than \ ZX. 83. LINE OF PKESSUKES. The stability of an arch and its abutments can be determined by geometrical LINE OF PRESSURES. 203 methods in the following manner. Let ABCD (fig. 86) represent the half of a semi-circular arch having a surcharge MK, and a supporting pier DZ. From O the centre of curvature draw OEF, making the angle EOD equal to 30 with the horizontal line ODC ; then EF is the " joint of rupture" as previously determined 204 ARCHES. (80). Divide ABEF into two equal voussoirs with a common joint RI ; draw the verticals IJ and FL. Find G, the centre of gravity of the arch ABEF and its surcharge, and also the weight P which will act at G, by the methods previously given (82). Also find the centre of gravity g of the voussoir ABEI and its sur- charge, and the centre g' of IREF and its surcharge ; and call w and ro' the weights of these two parts respectively, acting at g and /. Draw the horizontal line Ew, and the vertical G/> ; then find the value of the horizontal thrust N by the method given above, namely N = P - E '' . Am When the arch is in a condition of stability the hori- zontal pressure N will act at the centre n of the joint AB. Suppose the line na to represent on any convenient scale the force N, and the line ad the weight of t]ie part JMBRT, acting through the centre g; take nc, equal to ad and draw the diagonal ca, which will repre- sent the resultant of the two forces. Produce ea to meet the joint RI in s, and also to meet a vertical be through (j in the point b. The points n and s are called "centres of resistance." Take be to represent the weight of the second part acting through y', and make bf equal to ae ; draw the vertical fk equal to be, and draw the diagonal kb, producing it to meet the joint EF in , and also to meet the vertical through /, the centre of gravity of the abutment, in the point r ; then t is another " centre of resistance," and kb repre- sents the resultant of the forces on the second voussoir. ARCADES. 205 A curved line drawn through the " centres of resis- tance," n, s, t, &c., is called the " line of pressures," and in order to secure the stability of the arch it is essential that this curve should lie entirely within the depth of the arch, and should in no place come nearer to the extrados or intrados than one-fourth of the depth of the arch. To determine the stability of the abutment, draw the vertical ry, through /, the centre of gravity, of the abutment, and let ry represent Q on the same scale that an represents N; produce tr to x, making rx equal to lib ; draw the vertical xu equal to ry ; then the diagonal ru represents the resultant of all the forces acting on the pier. If the point 2, where ru cuts the base of the pier, lies within the base ZX, the structure will be in a condition of stability, but if it falls either at Z or outside of it, the pier will be pushed over by the arch. The distance Zi should be at least one- fourth of ZX in order to secure stability to the structure. When applying this method in practice, the arch should be drawn on a large scale and divided into several voussoirs, so as to get as many " centres of re- sistance " as possible. An example of the application of the geometrical method will be found in a paper on " Vaulting " by T. H. Eagles, read before the Royal Institute of British Architects on June 1, 1874. 84. ARCADES. It is a common occurrence in Archi- tecture to have a number of arches arranged in a row, two adjoining arches being made to spring from the same pier, so as to form an "Arcade." When the 206 ARCHES. arches are equal in span and carry an equal load, their horizontal thrusts will counterbalance one another, so that the strength of the piers need not be more than sufficient to resist the vertical load or crushing weight of the superstructure ; except in the case of the abut- ments at the two extremities of the arcade, which must be made sufficiently strong to resist the hori- zontal thrust N of the end arches, as calculated in the foregoing examples (82). When however it happens, as is frequently the case, that the span of the arches varies, then the thrust of the larger arches will only be partially counteracted by that of the smaller ones; and as we have seen (81) that the horizontal thrust in- creases as the square of the span in arches of similar construction, it is evident that if one arch is double the span of that next to it, the pier which supports them both must be made strong enough to resist three- fourths of the thrust of the larger arch, as the thrust of the smaller one is only one-fourth that of the larger. If one arch is 10 feet span and the next one is 14 feet span, or in like proportion, the larger will have just double the thrust of the former, so that only one-half of its thrust is balanced by the thrust of the smaller arch, and the other half of the thrust must therefore be sus- tained by the supporting pier. The necessity of making the end piers of an arcade etronger than the intermediate ones was pointed out long ago in the 6th book of Vitruvius, showing that the thrust of an arch formed of wedge-shaped voussoirs was well understood in his days. He says " itemque qute pilatim aguntur aedificia, cum cuneorum divisioui- bus coagmentis ad centrum respondentibus fornices SEGMENT A L ARCH. 207 concluduntur, extremes pilae in his latiores spatio sunt faciundte, uti vires eaj habentes resistere possint, cum cunei ab oneribus parietum pressi per coagmenta ad centrum se prenienter extrudunt incumbas. Itaque si angulares pilas erunt spatiosis magnitudinibus, contiuendo cuneos firmitatem operibus prasstabunt." 85. SEGMENTAL ARCH. An arcli which is formed by an arc of a circle less than a semi-circle is called " segmental," as EFABE'F (fig. 87), which subtends at an angle less than two right angles. If the angle EODj or 0, which the joint EF makes with the hori- zontal line OD, is less than or equal to 30, then the oint which makes 30 with OD is the "joint of 208 AECHES. rupture," as in the case of the semi-circular arcli (82). When 6 is greater than 30 the springing joint or " skewback " EF will be the "joint of rupture," and we proceed to find P and N in the same way as before (82). We can get their values approximately by drawing BF, FL, and considering P to be the area of the trapezium FLMB, as in the case of a semi-circle. Bisect FL in ? and BM in /, draw B/ and Ly, and take af = - Lyj li = J Bz ; join a and b ; then take G I aG = BM : FL. Then G is the centre of gravity of the figure BFLM. Draw the verticals be, ad, Gq, meeting the line of surcharge MK. Draw the horizontal line ETW, and the vertical Gp ; then mp = ]%, and as before, equa- tion (144), ML /, FL \ t" I 1 + BM + FIJ ML = R . cos. 0, FL = MA + R - R . sin. 6 = k + R (1 - sin. 0), BM = k + R r, E/> = E/w - mp = r . cos. w/>. Aw = OA - OM = R - r . sin. N P ^ J Aw For example, let 6 = 45, R = 12 ft., r = 10 ft., / = 5 ft., 7* = EX = 10 ft. Then, sin. = cos. = -707, ML = 8*484, FL = 8-52, BM = 7. SEG MENTAL ARCH. 209 mp = 2-828 x 24 ' 04 = 4-446 J. 0*0^ E/? = 7-07 - 4-45 = 2-62, Km = 12 x 7-07 = 4'93 P = 4-242 x 15-52 . 8 = 65-83 . 5 65-83 x 2-62 . N = 4-03 To find the requisite thickness t of the pier, we have, as before, to suppose P and N to act at E, and to take their moments about Z. We have also the moment of the rectangle FHKL, which we call F, acting at y, and of the rectangle HVXZ, which we call Q, acting at /. F = KL x FL = FL x (t + r . cos. - ML) = 8-52 (t - 1-414) ; and gh = KL; so that, F x gh = 4-26 (t - 1-414)' = 4-26 (f -2-828 t + 2) 6627x 1>128 . 8 = 2-33 . 5. . sin. Let = 48, "arc 0" = -83776, ^ = -4855; a then P = 18-4307 . b, Am = 8-891, E^ = Ml 71, N - 18-4307 x M171 = 2 . 316 g> 8*891 Hence it appears that the maximum value of N is obtained when = 45, and we can take the joint EF which makes 15 with the horizontal as the "joint of rupture." If we draw HE from H the middle point of the span, we find that the angle EHD is nearly 30, 214 ARCHES. when EOD is 15. In any other form of pointed arch the position of the "joint of rupture" can be found approximately by drawing HE at an angle of 30 with the horizontal. For an equilateral arch of any other dimensions, we can find P and N from the following equations : P = -3927 (R 2 - **) . 6 . (150) . . (151) E/> = -96593 r - -79335 x OG . (152) Am = M54 R - -547 r . (153) N = pJ- . . (154) Am In order to determine the thickness, t, of the abut- ment necessary to resist the thrust of such an arch, we proceed as before, in the semi-circular arch (81), to take N and P as acting horizontally and vertically at E ; and equate their moments about Z with the moments of the lower part CDEF of the arch and of the pier DZ. Put W for the area of CDEF, then in this arch we have W = J P, Also, to determine the position of g the centre of CDEF, we have, from equation (31) = ~ x ' 49858 = ' 6Q5 w (155) ag = r + t - Oy . cos. 71 = t + r - -99144 Off. SURCHARGED POINTED ARCH. 215 In the present example where R = 12, r = 10, we find Oy = 11, ag = t- -906, W = ? = i^|^ 5 o o = 5-7597 8. Let //., the height DX of the pier, be taken as ten feet ; then the moment of N about Z is N x IZ = N (h + r . sin. 15) = N (h + '2588 r} ; and in this case, N x IZ = 2-33346 x 12-588 . 8 = 29-373 . 8. The moment of P, at E, about Z, is P x El, or P (t + r - r cos. 15) = P (t + '03407 r). Also, P x El = 17-279 (t + -3407) = (17-279 t + 5-887) 8 The moment of "W, acting at g, about Z, is W x (i(j = (t -906) = 5-7597 (t - -906) 8 3 = (5-7597 t - 5-217) 8. The moment of Q, the area of the pier, is Q x fy' = /4 = 5^.8. The equation of stability is therefore (18), 58-746 = 17-279 t + 5-887 + 5-7597 I - 5-217 + 5 & or, f + 4-61 t - 58-078 = ; whence, t = 5 feet. 87. SURCHARGED POINTED ARCH. We have now to consider the Gothic arch as loaded with a wall level 216 ARCHES. at the top, as MK (fig. 89). Let the arch be equi- lateral as before, having for its centre, OD = r the span, 00 = R. Draw OEF making the angle EOD = 15 , and draw the vertical FL. Then, as in the case of the semi-circular arch, we can approximate Fig.89. to the area of FLMBE, or P, by drawing BF and taking P to represent the area of the trapezium BFLM, of which G is the centre of gravity. Draw the vertical Gp meeting the horizontal line Em in p \ then, as before, we find SURCHARGED POINTED ARCH. 217 BM = MA + AB = k + 1-155 (R - r) FL = BM + r . sin. 60 - R . sin. 15 = BM + -866r - -259 R; ML = R . cos. 15 - r . cos. 60 = -96593 R - -or; Ew == r . cos. 15 - r . cos. 60 = -46593r; E/? = E?w mp. By equation (153) Am = AB + B? = 1-154R - -547r By equation (154) For example, let R = 12, r = 10, /<: = 5 ; then, BM = 7-31, FL = 12-862, ML = 6-59116, P = 20-172 x 3-2963 = 66-486 . 5, mp = 3-603, Em = 4-66, ftp = 1-057, Am =8'38 N - 66 ' 486 t x ^ 8 = 8-386 . 6. 8*38 To find the thickness t of the pier DZ whose height is ^, we proceed as before to take moments of the forces about Z. Put F for the area of the rectangle KLFJ, acting at g its centre of gravity ; then F x cjn is its moment about Z. Let Q be the area or weight of the rectangle JVXZ, whose centre is g' ; and the moment of Q is Q x ; * . 218 ARCHES. F x gn = FL x 5^ = ~ (MK - ML) 2 = ?(t + r- -965S3 R) 2 , = 6-431 (t - 1-592)* . 8 = (e x 6-431 - 20-476 t + 16-297) 8, Q x ~= (h + E sin. 15) - .8 = (/& + '259R)|.8 <*w fv *w = 6-554 ? 2 . 8. The moment of N, at E, taken about Z, is N x ZI, or, N (h + r sin. 15) = N (h + '259 r) = 105*57 . 8. The moment of P, at E, taken about Z, is P x El, or P( r 4. t r . cos. 15) = P (t + '034 r) = (66-486 * + 22-605) 8. Then the equation of stability (81) is 211-14 = 66-486 t + 22-605 + 6-431 t s - 20-476 + 16-297 + 6-554 t or, f + 3-54 t - 13-25 = 0. Whence we find, t = 2'3 feet; when the arch and pier are built of similar materials. Example 2. Let R = 16, r = 14, k = 7, h = 12 ; then BM = 9-31, ML = 8-46, FL = 17-27, KL = t - 1-456 . P = 112-44 . 8, mp = 4-6523, E/> = 1-871, Km = 10-8 N= 112-44 xl-87 l5 = 19 . 5 10-8 SURCHARGED POINTED ARCH. 219 F x gn = 8-63 (t - 1-456)' . 8 = 8-63 (* 2 - 2-012* + 2-12) . 8 N x ZI = 19-5 x 15-626 -8 = 304-7 . 5 P x El = 112-44 t . 8 + 53-52 . 5 Q X / = 7-134, E;J = 2-184, Km = 16-76; 306 x 2-184 N = 39-87 16-76 F x gn = 13-86 (t - 3-182)' . 8 = 13-86 (f - 6-364(5 + 10-12) . N x ZI = 39-87 x 25-18 . b = 1004 . 8, P X El = 306 t . 8 + 208 . 8, Q x Ig = 26-2 = 13-11 f . 8 The equation of stability becomes 2008 = 306 t + 208 + 13-86 1 - 88'3 t + 140-3 + 13-11 f which can be reduced to f + 8 t - 65 = ; therefore we have, t = 5 feet. The following Table gives the results obtained in the foregoing examples, the arch and pier being supposed to be built of similar materials. R. r. /,-. h. t. 12 10 5 10 2-30 16 14 7 12 3-65 20 17 10 15 3-92 2-1 20 12 20 5*00 TUDOR ARCH. 221 88. TUDOR ARCH. We now propose to investigate the thrust of an arch which is usually drawn by the architect by means of two different lengths of radius and from four different centres. The height MB (fig. 90) of this kind of arch is generally less than half Fiq 9O the span, and the mathematical curve which most nearly approaches it in form is that known as the Parabola, for modes of drawing which the reader is referred to ihe author's " Practical Geometry." Let DEB be a part of a parabola which nearly represents the half of a Tudor arch, S being the focus of the curve, and the ordinate HS = 2DS. Let DM, the half span of the arch, be 9 times DS, MB = 6 times DS, according to the property of the curve, namely, that the square of the ordinate MB equals 4 times DS 222 ARCHES. multiplied by DM the abscissa. Or, if we put DS = m, sc any abscissa measured from D, y any ordinate belonging to the abscissa x, we have, y z =. 4 m . x. There will be a similar half-arch turned the opposite way abutting against the joint AB, and producing the thrust N at A, the two half-arches together making an obtuse angle at the crown A. The normal at any point, as E', can be easily drawn by letting fall the vertical E'K' and measuring K'O' = 2DS = 2m ; then E'O' is the normal, or perpendicular to the tangent at the point E'. The radius of curvature at any point of the curve lies in the normal, and its length, R'E' or p, is found from the equation 2 m P ~ sinT0' where is the angle which the normal makes with the vertical. The line K'O' is called the " subnormal,'" and its length is the same for all points on the curve. Also O'E' . sin. 6 = O'K' = 2m, or, O'E' = -? sin. e ' E'K' = O'E' x cos. = 2m . cotan. e ; DK' = f- = m . cotan. 2 0; 4 m O'R' = p O'E' = --. - - - = TUDOR ARCH. 225 07427, S^ = 1-3721, = -89805. Then, sin. 1 e sin. 2 6 P = 20-72652 . 8, mp = 5-0437, E> = 3'42844, Am = 6-65492 ; and we get ^ 20-72652x3-42844, 1ft _ ft ., 6-65492 l6 ' 8 ' 8 ' Let = 56; then we find P = 21-05 . 8, Wp = 3-4344, Am = 6-759 ; giving N = 10-697 . 8 Let = 58 ; then P = 21-357 . 8, E> = 3-436, Am = 6-8,")8 ; N = 10-700 . 8 Let 6 = 60; then P = 21-649 . 8, E> = 3-4342, Am = 6-9533 ; N = 10-692 . 8 Let = 62; then P - 21-9272 . 8, E> = 3*4289, AM = 7-0446 ; N = 10-673 . 8 From which it appears that as we approach to or recede from 6 = 58, the value of N diminishes ; hence we may assume that for this case the point E, where N is a maximum, is that at which the normal makes an angle of 58 with the vertical, and this will therefore be the "joint of rupture." To find the point E where the normal EO makes 58 with the vertical, take DK = -39046, KO = 2. Putting F for the area or weight of the lower part of the arch between EF and CD, we have to integrate the 226 ARCHES. equations for P and mp, given above, between the limits = 58 and 6 = 90 ; from which we obtain F = 3-7707 . 8 and the distance qg' is obtained by the integral for mp, taken between the same limits, giving qg' = 9-907, therefore D/ = qtf - MD = 9-907 - 9 = -907 ; and y f x = t-T)f= t -907. The moment of F about Z is F x y'x = F (*-DF) = 3-7707 t.b - 3-42 . 5. Putting Q for the area or weight of the pier whose height is h and thickness t, we have for the moment of Q about Z Q x *- = %h . f .. The moment of N, acting horizontally at E, taken about Z, is N (h + KE) = N (h + 2 cotan. 58) = 10-7 (h+ 1-24974)3; The moment of P, acting vertically at E, taken about Z, is P (UK + t) = P (-39 + = 21-357 t . 6 + 8'33 . 6 Taking k = 10, we have for the equation of stability (81), 241 = 21-357* + 8-33 + 3-7707 - 3'42 + 32* which reduces to F + Zt- 47-2 = ; from which we obtain, t = 4'8. SURCHARGED TUDOR ARCH. 227 It appears therefore that for a Tudor arch without surcharge having a span of 18 feet, and a height at the centre of 6 feet or one-third of the span, the depth of the voussoirs being 2 feet, the thickness of a pier 10 feet high should be 4*8 feet in order to ensure stability. Fig 91 . 89. SURCHARGED TUDOR ARCH. Let us now sup- pose that the parabolic arch, whose "joint of rupture " we found above (88), has a surcharge whose height AT (fig. 91) above the crown we call/fc, the top of the sur- charge being taken as level. Let EF be the "joint of rupture " as above determined, making 58 with the vertical; DM = 9, BM = 6, CD = EF = 2, DK = 2 228 AUCHES. -39, AT =3 = 7*, BT-= 5-108, OE = 2-358, EK = 1-24974. If P represents the area or weight of the arch and its surcharge EFUTB, acting at G its centre of gravity, we can approximately find P and mp by assuming as in the former cases (82) (87) that the trapezium FUTB represents P; then dividing it into two triangles by the line BU, we find a and b the centres of these triangles on the lines Uf and Be; and G is determined from the proportion aG:6G = FU : BT. Then, P = BT t FU xTU; = TU/ FU \ 3 \ r BT + FU/ ' ~Ep = m E - nip = MK mp, MK = 9 - DK = 8-6, BT = 5-108, FU = MT - (OE + 2) . sin. 32 - 8-799, TU = MK + 2 . cos. 32 = 10-296, VU = 9 + t - TU = t - 1-296. From which we find, P = 71-593, mp = 5-93, ~E>p = 8-6 - 5-93 = 2-67, Km =6-858, N = P-^-= 27-873. A.m Let the height h of the pier be 10, and F the area or weight of the rectangle VUFI, Q that of the rectangle- SURCHARGED TUDOR ARCH. 229 IJXZ ; then the moment of F, acting at y, taken about Z, is F x = 140. The moment of Q, the area or weight of the pier, is Then the equation of stability becomes 280 = 11-5 * + 15-4 + 5-75 t - 3'11 +5 C, or, f + 3-45 # - 53-6 =0, = 5-78. Comparing this with the thickness of pier requisite for the " equivalent circular arch" of the above dimen- sions (81), we find that t = 5 in the circular arch, so that is nearly one-sixth more in the elliptic arch of same span. In applying this method to an elliptic arch of any form and dimensions, there is no necessity to draw the circular arch, as the point E can be determined on the ellipse by taking OK = -866 x OD, and EK = OB ; and the horizontal dimensions are the same as in a circular arch in which OD = r, and OC = R. 91. SURCHARGED ELLIPTIC ARCH. When the elliptic- arch has a surcharge MR (fig. 93), we can approximate to the value of JST in the same manner as we have done for other forms of arch. Let OD be the half span, 01> the height of the intrados, 00 the half major-axis, and OA the half minor-axis of the extrados. With as a centre and OD for a radius, describe the arc DE'; and draw OE' making an angle of 30 with OD, or take KE' equal to half OD. Draw the vertical KE' cutting the intrados of the elliptic arch at E ; then, as- SURCHARGED ELLIPTIC AECH. 235 before shown (90), E is the point at which the thrust N has its greatest effect. Then OK = OE' x cos. 30 = '866 x OD. If HEF is the normal to the curve at E, then we find Fig 93 Take e the middle point between A and B, and draw E; consequently P = . ? x MI, 236 AKCHES. MI MI/ E \ a \ ^ EI + mr Let F be the area or weight of the rectangle EJLI acting at# its centre; then F = El x EJ; El = OM - KB = OM- 5?j EJ = -l:U x 01) = DK; M < = k + ^; MI = OK = -86(5 x OD; *> = Em - mp = OK - mp ; Aw - OA - KE = OA - ^ B ; bj = # + ?J = # + -067 x 01); K/ = DX + OM = // + /< + OA; EK = | 013. Then the moment of N acting horizontally at A, and taken about E, will be in equilibrium with the moment of P acting at Gr, together with the moment of F acting at (j. The moment of F about E bowever acts in the opposite direction to that of P, consequently we have P x % - F x ^ Am We can then transpose N, P, and F, to the point E and equate their moments about / with that of Q the weight of pier RLXZ, as before. For example, let OD = 10, OB = 5, 00 = lV>, AO = 6, k = MA = 2, h = DX = 10, AB = 1. Then we find, El = 8 - 2-5 = 5-5, Me = 2-5, MI - 8 66, EJ = 1-34, P = 34-64, mp = 2'89 --'- = 4'87o, EJJ = 8-lJG 8 _ 4-875 = 3-785, Am = 6 2'5 = 3 -5, P x E/ = 131-11 .8, F x ?? = 4-94 . b, VAULTED ROOFS CYLINDRICAL. 237 > o'O The moment of N (omitting 5), supposed to act at E, and taken about Z is, N 1 x (h + EK) = 36 x (10 +2-5) = 450. The moment of P, acting at E, and taken about Z, is, P (t + DK) = 34-64 (t + 1-34) = 34-64 + 46-4. The moment of F, acting at E, and taken about Z, is F (t + DK) - 5-5 x 1-34 (t + 1-34) = 7-37^ + 9-88. The moment of Q is, Q x f = (/* + OM) = 9 f Then the equation of stability (81) becomes 000 = (34-64 +7-37) t + 46-4 + 9-88 + 9f or, +4-67 - 93-75 = 0, and t = 7*625. 92. VAULTED ROOFS. CY- Fig. 94 . LINDIUCAL. When a roof is formed of a stone or brick vaulting, the thrust is often concentrated at certain points on the wall or abutment, and is not distributed uniformly along the wall. This kind of vaulting consists of a main rib as shown on plan by AB, ED, CF, (fig. 94) placed at right angles to the walls EF and CD, and also two (or more) diagonal ribs, as AC, AD, BF, BE, springing from the same points on the wall, so that we have three ribs all pressing on the same point. The diagram 238 ARCHES. shows the ribs on plan, the diagonal ribs being hero drawn as making 30 with the main transverse ribs ; but in practice we find them placed at different angles, usually varying from 30 to 45. The diagonal arches will be of greater span than the transverse arch, and consequently if the latter is semicircular, the former must be elliptical if the height is the same. Since the angle CAB = 30, we have AC = 1-1.54 X AB, and if a and b are the semi-major and semi- minor axes of the elliptic arches AC, &c., then we have b ; a = 1 : 1-154, or, - = -860. Let ABOD (fig. U5) represent the half sec- tion of one of the transverse ribs, rest- ing at its springing on a corbel let securely into the wall, and filled in at the back up to the level of E, EO \j making 30 with the horizontal OD, and EF being the ''joint of rupture " as pre- viously determined Fig.95 (80). Pllt K for OF the radius of the ex- trados, r for OE the radius of the intrados, and let I be the lateral breadth of the rib. Then if P is the weight of the arch EFAB, g its centre of gravity, 6 the VAULTED ROOFS CYLINDRICAL. 239 weight of a cubic foot of the material, we have from equation (140) P = -5236 .b. I . (R 2 - ;*) ; By equation (141) E/? = -866 r - -3183 (^-3) ; And by equation (142) Km = R - r - N = P ^~ Am and N is the horizontal thrust of the arch acting at E. For the diagonal elliptical ribs we have (90) to take N from a semi-circular arch of same span whose depth iit the crown is to that of the elliptic arch in the proportion of a to #, or as 1*154 ; 1. If then II r is the depth of the elliptic arch, that of the " equivalent circular arch" is, 1-154 (R r). Putting R' and r' for the radii of the u equivalent circular arch/' we have / = 1-154 r, R' = r' + 1-154 (11 - r) = 1-154 R P' = -5236 . I . b (R' 2 - r") = R' - r ' = 1-154 R - . A m The thrust produced by each diagonal rib upon the 240 ARCHES. wall is the resolved part of W perpendicular to the wall, namely, N' cos. 30 = '806 N', and for the two diagonal ribs the thrust will be twice this, or 1-732 N'. Then the total thrust, T, upon the wall at A or B (fig. 93) is . T = N + 1-T32 N' whicli ma}' be considered as acting at the level of the point E (fig. 94). For example, let us suppose the span to be 20 feet, the depth and width of the ribs 1 foot ; then It = 11 . r = 10, I = 1 ; P = -5236 . 5 x 21 = 11 . 8 ; E/? = 8 -60 - 5 = 3-66, Am = 6. N= n *l^. 8 = 0-71. 8 P' = 14-7 . 5, E>' - 4-2, A'm' = 6-02, W = iL>l 2 . 8 = 8-916 . 5 6*92 T = (6-71 + 1-732 x 8-916) 8 = 22-15 b. To determine the thickness, t, of the wall, we have to take the moments about Z of T, P and 2P' acting at E ; T acting horizontally, P and 2P' acting vertically. Also we have the moment of F the weight of the lower part JEDC of the transverse arch, and 2F' the same for the two diagonal ribs. We can find approximately the value of F, by con- sidering it to be the area or weight of the triangle JEC, or F = A x JE x CJ = A(ll - 8-66) x 5 - 5'85 . 8. VAULTED ROOFS CYLINDRICAL. 241 Also, F' = F, very nearly ; and if y' is the centre of gravity, then ag' = t + ^ JE = t + -78. Let Q be the weight of the wall IJXZ, whose thick- ness XZ = t, and whose length will be AF (fig. 94) or AC X sin. 30 = ~ = R' = 12-7. Let h = CX - 20, then IZ = 25 ; and the moment of Q about Z is Q x 4 = 12-7 x 25 x ~ . 8 = 158-8 . 8 . f . The moment of T is T x IZ = 25 x 22-15 . 8 = 553-75 . 8. The moment of P + 2 P' is (P + 2 P') (t + JE) = 40-4 (t + 2-34) 8 - (40'4 1 + 94-5) 8. The moment of F + 2 F' or 3 F is 3 F (t + -78) = 17-55 (t + -78) 8 = (1 7'55 t + 13-60) 8. Then the equation of stability is 1107-5 = 40-4 1 + 94-5 + 17-55 1 + 13-69 + 158-8 f t or, f + -365 - 6-3 = ; t = 2-3 ft. Suppose that instead of a solid wall from E to F, we had only piers 3 feet wide at E, A, and F ; to determine the thickness, t, of the pier? The value of Q x 4> will then be 3 x 25 x | . 8 = 37-5 f . 8 ; all the other moments remaining as before. Then the equation of stability is reduced to K 242 ARCHES. f+ 1-545?- 26-65 = 0; 2 = 4-5 ft. In the latter case, however, the quantity of walling is less than one-half what it is in the former, the pro- portion being 13"5 to 29-2; consequently it is more economical to have piers than a continuous wall. In the foregoing investigation we have taken no account of the panelling or filling in between the ribs. This is usually made as light as possible, and we will assume that its thickness decreases from E towards B (fig. 94), in such a manner that the pressure on every part of the ribs is uniform, and also that the effect of the panelling is the same as if we doubled the width I of the ribs, keeping the depth B, r the same as before. In this case we have to double the values of P and P', and therefore also of N and N', while F may be considered as unaltered. Applying this to the fore- going example, we have 1 = 2, and all dimensions the same as before. Then we have P = 22 . 8, F = 29-4 . 8, T = 44-3 8. Then for a continuous wall 12- 7 feet in length the- equation of stability becomes, when reduced, f + -62 t - 12-T, or t = 3-5 feet. For a pier 3 feet wide we have to determine t from the- equation f +2-6?- 53-7 =0; t = 6-2 feet. So that the strength of the abutment must be increased in the proportion of 3 to 2, to allow for the panelling of the vault. GOTHIC VAULTING. 243 Instead of making the diagonal ribs elliptical, the plan is sometimes adopted of making them semicircular, and "stilting" the transverse rib so as to bring the crowns on the same level. In the foregoing case the semi-circular transverse arch will have to be stilted by 154 r, or about a foot and a half when r = 10; the springing of the semicircle in the transverse rib being so much above that of the diagonal ribs. 93. GOTHIC VAUL- TING. In Gothic vaulting the ribs are formed of pointed arches, as previously described (86); and we shall here assume that the ribs, both transverse and diago- nal, are equilateral arches. Let ABCD (fig. 96) represent in section one of the transverse ribs, the "joint of rupture" z_ EFO making 15 with the horizontal OD, as before determined (86). We will also assume that the masonry is solid up to the level of El, I being the top of the wall or pier. Let the plan (fig. 94) of the ribs be the same as in the case of the semicircular ribs, the diagonals making 30 with the transverse. Let R and r be the radii of the transverse rib, R' and r those of the R 2 Fig 96 . 244 ARCHES. diagonals; then as before, R' = 1-154 R, ?*' = 1*154 r. From equation (150) we have P = -3927 (R 2 -r) 8 . /, where I is the width of the arch. From equations (151, 152) we have - E; = -960 r- -516 STL And from equation (153) Am = M54R- -547 r N = P x -%. A;# For example, Jet R = 21, r = 20, / = 1 ; then we find, P = 16-1 . 8, Ep = 3-32, Am = 13'3, N = 4 . 8 ; F = (1-154)' x P = 1-331 x P, Ey - 1-154 x Ej>, AW = 1-154 x Am; W = (M54) 8 x N = 1-331 x N = 5-324 .8; P' = 21-438. The moment of N acting at E, and taken about Z, is N (h + r . sin. 15) = 100-72 . 8. The moment of N' at E, taken about Z, is 866 . N (h + r' sin. 15) = 120 . 8. The moment of P at E, taken about Z, is P (t + R - r . cos. 15) = (16-1 t + 27-1) 8. The moment of P' about Z, is (for the diagonals) P" (t + R' - r' . cos. 15) = (21-43 t + 36) 8. Taking F for the weight of the part of the arch below GOTHIC VAULTING. 245 E, and represented by the triangle CJE, whose centre is Jt ?4^----^- 7 rX, v ri -ft f\ Ns y 1 r yp ^ 2 c/> X" Fta.97. thrust, N, will be greatest. Let ABCD (fig. 97) be the half of a rib standing on a pier or support at CD, let O be the centre of the circle, and draw OEF making 30 with OD ; bisect EF in e, then e will be the point at which N, supposed to act at , the middle of the joint AB, will have its greatest effect (80). The covering is generally carried by means of purlins resting upon the rib at intervals of a few feet, but it will be sufficient to consider the load as uniformly distributed over the whole circumference of the rib, and, as in the Chapter on Roofs, we take this load, including the effects of wind, snow, &c., as 66 Ibs. to 248 ARCHES. the square foot of roofing. Then, if the ribs are 10 feet apart, we have 060 Ibs. for the load on every foot of their circumference. If we put AY for the weight of each half-rib and its load, R and r for the external and internal radii of the rib, \ve have AY = 7 R x CGO = 1037 R. Let P be the weight of the part between AB and EF, F that of the part between EF and CD ; then P = | AY = 691 . R F = L AY = | P = 346 . R. Draw the vertical OB, and let g be the centre of gravity of the arch between AB and EF ; then P acts at g in the vertical line gq, eg and An are determined by the equations (141, 142) namely eg = -866 :+" r - -3183 . j? '~-~ Then we have, N = P which gives the horizontal thrust at e. To determine the strength of the rib at EF to resist the stress upon it from the moment of P about c, we have to compare the moment P x eg with the moment of resistance of a beam of I section about the point e as given by equation (56) cr, P X eg with - -' (b . D : ' - JT^t . rl"), ARCHED IRON RIBS. 249 where D and d are the depths, b the breadth and t the thickness of the rib ; and the fibres at the top and bottom are subjected to a stress of 5 tons per square inch of section ; all the dimensions being expressed in inches. The resistance of the pier or support is determined as before by taking moments about Z, of P and N acting at e, F acting at y', and Q, the weight of the pier itself, acting at its centre. As an example, let E = 11, r - 10, R -+ = 10-5, DX = h = 20, B; - r 2 = 21, R 3 - r 3 = 331, P = 3-4 tons, F = 1-7 tons, eq = 9-09 5-07 = 4, an = 5-25. N = 3 ' 4 x 4 = 2-6 tons. Transposing N and P to e, and taking their moments about Z, we have, for the moment of N about Z N ?i + t^ = 2-6 x 25-25 = 65-65. The moment of P about Z is P x eb = P (t + I)/) = P (t + r - -433 (R + r) ) = 3-4 + 3-1. The moment of F acting at /, about Z, is F x d t) . d 3 ) = 267, or resistance is to stress as 267 to 356, which shows that the strength of the rib is insufficient for safet}'. Let the thickness of the metal be f inch, then 1) = 12, d = 10'5, t = '75, b = 8 ; and the moment of resistance is found to be 377, or rather more than the moment of stress. Since the moment of stress diminishes from EF towards the crown, it will be evident that a saving" of material may be effected by diminishing the depth of 252 ARCHES. the rib towards the crown, where it need only be strong enough to resist the direct pressure of the force N. A very considerable saving can be effected in the quantity of walling to sustain the thrust of the rib, by making the ribs rest on piers with an open space between them. In the last example suppose the pier to be 2*5 ft. long instead of 10 ft., then the moment of Q is 1*56 t' ; and the equation of stability is t 2 + 4-7 t - 155 = 0; t = 10'3ft. Here the amount of walling is less than half what it was in the former case, or as 26 to 56. In the foregoing investigation no account has been taken of the stiffness of the iron arch, or its resistance to change of form, which is a very important element, and constitutes the chief advantage obtained by the employment of iron for this purpose; for until the arch begins to bend at the crown there will be little or no thrust upon the supporting piers. If we supposed the half-arch from EF to AB, together with the cor- responding half-arch on the opposite side, to be super- seded by a straight beam, we can find easily the deflexion in the middle caused by the load P at y, by the rules given in Chapter V. Let W be the load at A whose moment about e will be equal to the moment of P ; then we have ^ X ne = P x ye, or, W = 2 P x g -. Then W represents the vertical pressure at A, caused by the load P at g in the two sides of the arch. For a horizontal beam whose length is I = 2 tie, the deflexion (53) will be- ARCHED IRON RIBS. 253 1 W I 3 ~ 48 ' E ' I > where I = g_-_t> J ~ _g_-_0_g. Taking the foregoing example in which D = 12, d = 11, b = 4i, = |, we = 9-09, we find TTT f>'8 x 4 Taking E = 10,000 tons, we find 6 = -315 for the deflexion at the middle of a beam 18'18 ft. long. The deflexion allowed by Tredgold for such a beam would be '455 inch. The deflexion of less than |- inch would not produce a perceptible change of form in an arch 12 inches deep, and the actual deflexion at the crown would certainly not exceed that of a horizontal beam, but would rather be less. Consequently, it would appear that the resistance of the continuous arch to bending will counteract the greater part of the hori- zontal thrust as calculated above, as long as the moment of stress does not exceed the safe moment of resistance. The strength of the piers will then be little more than is necessary to resist the vertical load upon the rib. Since the resistance to deflexion increases as the cube of the depth, when other dimensions remain unaltered, it follows that by increasing the depth of the rib we can ensure any amount of stiffness in the .arch, and thereby overcome all horizontal thrust upon the supports. A nearer approximation to the resistance to bending can be obtained by supposing the arch to be divided at AB, and a straight beam (as shown by the dotted lines 254 ARCHES. AF and BE) to take the place of the arch between EF and AB, the length of which beam will be R ~j~~ The fV deflexion (8) of such a beam loaded with a distributed weight W is, by equation (71) 1 W I 3 = 8 E ' T' where I is the moment of inertia of the section. W is here represented by the resolved part of P at right angles to the length of the beam ; or W = -866 P. Putting D = 12", d = 11", I = 4J", t = ", we have Let E = 10,000, R = 11 ft., r = 10 ft., I = ?L+ r . rv = 10| ft. = 126 ins., P = 3-4 tons, W = -866 P = 2-944 tons ; 1 2-944 (10-5) 3 x 1728 ., . , L - = ' 361 mch - 8 By Tredgold's rule (50) the maximum deflexion is *25 inch, so that the above is half as much again. Taking the same dimensions, but making b = 6", we have I = 252, 5 = -292 inch. Let D = 13", d = 12", then iii the last example we find I = 307, 8 = -24 inch. When D = 14", and d = 13", we find I = 305, 5 = -2 inch. CHAPTER IX. DOMES, SPIRES. 95. HEMISPHERICAL DOME. A Dome or Cupola is an arched roof covering a circular chamber, and consists of a series of arches both vertical ajid hori- zontal. The vertical arches have all the same radius of rig 98 . curvature but dimmish in width from the springing towards the vertex, and may be considered as " lunes " cut out of the hemisphere. The horizontal arches diminish in diameter from the springing towards the 256 DOMES, SPIEES. vertex. In order to investigate the stability of a dome, we shall consider it as made up of a number of vertical " limes " or ribs, as ABCD (fig. 98), whose base Dd subtends a very small angle, <, at the centre 0. The horizontal arches are shown by the dotted lines as EE', VdW. Let N be the horizontal pressure acting at the vertex, A, when the dome is about to break up by opening at B and F; and let P be the weight of the "lime" between EF and AB, acting at G its centre of gravity. Draw the horizontal line Ew, and the vertical G/>; then, supposing the moments of P and N to balance about E, we have P x E/> = N x Am, , - Am In order to find the "joint of rupture," as in the arch (80), we must express N in terms of the angle 6 which any joint, as OEF, makes with the vertical ; or, = BOE ; and then find by calculation what value of gives the greatest value of N. We will suppose for convenience that the angle DO/, or <&, subtends 2 at O, or the arc T)d to be T |-^th part of the circle ; then, "arc (/> " = "0349, expressed in circular measure, as in the Table previously given (21). Let OE = r, OF = R ? 6 = weight of 1 cubic foot of the arch and pier or " drum " which supports it. Then we find, by means of the Integral Calculus, that P = | (1 _ cos. 6) (R 3 - r 3 ) . 6 . (162) o - -01163 (1 - cos. 0) (R 3 - r 3 ) . 6. HEMISPHERICAL DOME. 257 Also we find $ sin. 2 sin. -J since may be taken as very nearly equal to unity. j 2 E/J = E?ra ^ = r . sin. mp . (164) Aw = R - r . cos. . . (165) In order to find the value of which gives N a maximum value, we take, as in the arch, R = 11, r = 10, and calculate N by means of the foregoing equations for different values of 0. We have in this case R 2 - r 2 = 21, R 3 - t* = 331, R 4 - r 1 = 4641. When is 69, we find N = -73577 . 5 When is 70, we find N - -73607 . S When is 71, we find N = -73575 . 8. We therefore conclude that the maximum value of N is obtained when the angle BOE, or 0, is 70, the angle EOD being 20; and that EF is the "joint of 258 DOMES, SPIRES. rupture" when EOD = 20. In this case we have "arc 0"= 1-22173, sin. = -93969, cos. 6 = -34202, sin. 20 = -64279 ; consequently the value of N, or the thrust of the 180th part of the dome, from equation (161) becomes TJ _ -007192 (R 3 -r>*)r- -Q039273 (R 4 - r') n(t , R - -34202 r From equation (162) we have for the value of P, P = -007656 (R 3 - r 5 ) . 8 . (167) Putting F for the weight of the part of the arch between CD and EF, we find from equation (162) F = -00398 (R 3 - r 5 ) . 8 . . (168) If Q is the weight of the portion of the "drum" which supports the " lune " ABCD, we find Q = -017453 (2r + f) . t . h . (169) where t is the thickness and h the height of the "drum" from the surface of the ground. In order to determine the thickness, t, of the walls of the " drum," we equate twice the moment of N acting at E, and taken about the outer edge Z of the wall, with the moment of P acting at E, the moment of F acting at the centre of gravity of the lower part of the " lune," and the moment of Q. And if the wall and dome are built of similar material, we may omit the quantity 8 from our equations. The moment of 1ST about Z, is N (h + r . cos. 0) = N (h + '34202 r) . (170) HEMISPHERICAL DOME. 259 The moment of P about Z is P (t+ -06031 r) . . (171) The moment of F about Z is r-' 7351 1^) . (172) The moment of Q about Z is Q fj$j = '0058177 (3 r f + f ) A . (173) For example, let It = 11, r = 10, h = 50 j then the moment of N is, from equation (170) 39-32. The moment of P is, from equation (171) 2-534 t + 1-53. The moment of F is, from equation (172) 1-32 t - -404. The moment of Q is, from equation (173) 8-7265 1 2 + -29 f . Then the "equation of stability" becomes 78-64 = 2-534 t + 1-53 + 1'32 t - '404 + 8'73 f + '29 t 3 which reduces to t 3 + 30 2 + 13-3 - 267 = 0; from which we find, t = 2-66. [The solution of this equation is obtained by means of " Homer's process," which will be found described in the Appendix to De Morgan's Arithmetic.] s 2 260 DOMES, SPIRES. If we put 6 = 120 Ibs., then in the above example we find N = 88'33 Ibs., P = 304*08 Ibs. Suppose W to represent the pressure on the joint EF, then W = N. sin. 20 + P . cos. 20 = 88-33 x -34204 + -93969 x 304-08 = 316 Ibs. If the dome is of Bath stone, the safe load is 150 Ibs. per square inch, and if of brick it is 50 Ibs. per square inch. Now the area, A, of the joint EF in inches is A = -|(R 2 -r 2 ). cos. 20 x 144 = 2-3612 (R - r 2 ) - 49 inches. From which it appears that the load is only ^ih of the safe load in the case of Bath stone, and |th of the safe load for brick ; so that the dome might be safely loaded to a considerable extent by a " lantern " or other heavy weight without affecting its stability. When the dome has only its own weight to carry, and has no - f lantern," the values of N and P diminish with the width as long as the internal radius is unaltered. Thus, if we make E = 10-3, r = 10, we have R 2 - r 2 - 6, R 3 - r 5 = 92, R 4 - r 4 = 1248 ; so that A is reduced by f- or -286, P is reduced by -278, and N is reduced by -339, or nearly one third. Hence it appears that the thickness of a dome may be reduced almost indefinitely when it has only its own weight to carry. The thrust of the dome at the level of EF may be counter-balanced by means of a ring of wrought iron HEMISPHERICAL DOME. 261 fixed tightly round it. Now, according to Rankine (Applied Mechanics, p. 184), "The thrust round a circular ring under an uniform normal pressure, is the product of the pressure on an unit of circumference by the radius." Consequently, if T is the total normal pressure on the ring at the level of EF, where the radius is R . cos. 20 ; we have, the pressure on an unit T of circumference = 2 TT . R . cos. 20 and the thrust round the ring is - T 2 TT . R . cos. 20 C x R. cos. 20 = ~ = -16 T (174) and this represents the tension in a ring of iron placed round the dome at the level of the joint EF. In the case of a dome in which R = 11, r = 10, we have, T = 180 N = 88-33 X 180 = 15900 Ibs. = 7'1 tons. The tension in a ring of iron is therefore, from equation (174), = -16 T = 1-136 tons. Taking the safe tensile strength of wrought iron as 5 tons per inch of section, we find that the sectional area 1*1 ^.f* of a belt for this dome, is i^ = '227 inches, or about o | of a square inch, so that 1" x \" will suffice. If this belt is fixed round the dome, the strength of the wall of the "drum" need only be sufficient to support the vertical weight of the dome, which is (P + F) 180 = 462-48 x 180 = 83246 Ibs. = 37-3 tons. 262 DOMES, SPIRES. If the thickness of the drum is the same as that of the dome, or 1 foot, the area will be TT (R 2 r) = 66 feet, and the pressure will be 1261 Ibs. per foot of walling. A thickness of 1 foot would suffice for a wall only 10 feet high, but for a height of 50 feet it should be at least 2 feet. As another example, let R = 31*5, r = 30, h = 60 ; then we have, R 2 - r 2 = 93, R 3 r 3 = 4280, R 1 - r* = 176049; from equation (167), P = 32-767 . 5; from equation (166), N = 10-924 . 8 ; from equation (168), F = 16-03 . 5. The moment of ~N, at E, taken about Z, from equa- tion (170), is, 767-5. The moment of P, at E, about Z, from equation (171), is, 32-767 . t + 59-285. The moment of F about Z, from equation (172), is, 16-03 . t - 3-767. The moment of Q about Z, from equation (173), is, 31-4154 . t* + -34906 . t\ The equation of stability is therefore (putting 2N for N) 1535 = 32-767 t + 59-285 + 16-03 t- 3-767 + 31-4154 f + -34906 t 3 ; which can be reduced to f + 90 + 140 t - 4241 - 0, from which we find, t = 6, nearly. The total normal thrust of the whole dome at the level of the joint EF in this example, is T = 10-924 x 120 x 180 = 234,078 Ibs. = 105 tons. HEMISPHERICAL DOME. 263 The tensile strain on an iron belt to counteract this thrust, by equation (174), is, -16 x 105 = 17 tons, so that the belt should have a sectional area of = o 3 "4 square inches ; so that an iron belt 7" x |" will suffice. In this example the area of the joint EF is 220 inches, and the pressure upon it is W = 120 (10-924 x -34204 + 32-767 x -93969) - 4143 Ibs., which is nearly 20 Ibs. on the square inch, or two- fifths of the safe pressure when the material of the dome is brick. Suppose that in the last example the thickness is reduced to -5, or R = 30-5 ; then R 2 - r 2 = 30, R 3 - 3* = 1,365, R 4 r l = 55,133; P = 10-45 . 5, N = 3-86 . 5, F = 5-433 . 8 ; h = 60. The moment of N about Z is 271-2 ; That of P about Z is 10-45 t + 18-9 ; That of F about Z is 5-433 1 + 1-684 ; That of Q is 31-4154 f + -34906 t\ The equation of stability is therefore - 542-4 - 15-883 1 + 31-4154 t 2 + -349 f + 20-584, which can be reduced to f + 90 f + 45-5 1 - 1496 = ; from which we find, t = 3-8, nearly. In this case the area of the joint EF is 71 inches, and the pressure upon it is 1,337 Ibs., or 19 Ibs. on the 264 DOMES, SPIRES. square inch. The total normal pressure round tlie whole of the dome at the level of the joint EF is, T = 37*2 tons, and the tensile strain on an iron belt by equation (174) would be -16 x 37'2 = 6 toes. It will be seen from equation (166) that for similar domes, or those in which the radii R and r are in the same proportion to one another, the thrust N varies as the cube of the radii. Thus, a dome whose radii are R 33 and r 30, will have 27 times the thrust of a dome in which R = 11, and r = 10. When the quantity R r is small as compared with r, we can simplify the formulae given above, by putting in which case the equation (163) becomes - 3 1 cos. 6 and the equation (166) becomes 96. SEMI-DOMES. We find in many ancient build- ings, especially in the churches of the Byzantine period, that there are side-chapels, built out from the main structure, which are semicircular on plan and are covered with a domical roof. This roof is similar to that shown on fig. 97, only instead of the half-dome abutting against another half-dome, a vertical wall or arch is placed as at OA (fig. 99), against which the half-dome abuts. SEMI-DOMES. 265 The thrust of this half-dome upon the wall on which it stands, DZ, can be calculated by the formula given above, the value of N for a lime subtending an angle of 2 at 0, being calculated either by equation (166) or equation (175); and the strength of pier or wall supporting it is found by equations(170, 171, 172, 173). Equating twice the moment of N to the moments of P, F, and Q, taken about Z, we obtain a cubic equation in terms of t, the thickness of the wall. ' For example, let us take the case of the " exhedra " of Sta. Sophia at Constanti- nople, where R = 52 feet, r = 50 feet, and h, the height of the sustaining wall, is 75 feet. We will suppose that 8, the weight of a cubic foot of the material, is 1 cwt. throughout, so that it can be omitted from the calculation. From equation (167) we have, P = -007656 (R 3 - r 3 ) 5 = 119-58. From equation (175), we have, N = 43-7 8 = 2-185 tons. Fig. 99 266 DOMES, SPIEES. From equation (168), we have, F = -00398 x 15608 8 = 62-12 8. The moment of N acting horizontally at EF, taken about Z, is by equation (170) 43-7 (75 + 17-1 ) 8 = 4024-77 5. The moment of P acting vertically at E, is by equa- tion (171) 119-5 (t + 3-015) 8 = (119-5 t + 360*3) 8. The moment of F acting vertically at the centre of gravity of the part CDEF, is by equation (172) 62-12 (t + 1) 8 = (62-12 t + 62-12) 8. The moment of Q, the weight of the pier DZ, acting vertically at its centre of gravity, is by equation (173) 00582 (150 f + f) 75 = (65-5 f + -4365 f) 8. Then the equation of stability becomes 8049-54 = -4365 f + 65'5 f + 181-62 t + 422-42 which reduces to f + 150 f + 416 t - 17473 = ; from which we find t = 9 feet very nearly. We have next to consider the thrust of the crown AB of the semi-dome against the wall or arch OA, as the vertices of all the lunes into which we suppose the dome to be divided meet at AB, and therefore concentrate the thrust of the half-dome on that point. The value of N obtained by equation (175) will be DOME OF STA. SOPHIA AT CONSTANTINOPLE. 267 the horizontal pressure at the apex of the half-dome of each lune, but acting obliquely to the plane of the wall or arch OA, according to the position of the lune in the dome. Therefore to obtain the pressure, R, at right angles to that plane, we must multiply N by the sine of the angle which it makes therewith ; or the pressure of each half of the semi-dome at the vertex is .1 R = N (sin. 1 + sin. 3 + + sin. 89) = N x 28-64938 R = 2 N x 28-64938 = 57-29876 x N . (176) which is the pressure of the dome at its apex at right angles to the wall or arch OA. For example, we will apply this to the semi-dome or exhedra of Sta. Sophia, as given above. Here we have N = 43-7 8 Therefore, R = 25048 = 125-2 tons. This pressure R serves the purpose of resisting the outward thrust of the central dome as calculated below (97). 97. DOME OF STA. SOPHIA AT CONSTANTINOPLE. We will now apply the formula found above (95) to the investigation (approximately) of the thrust of the central Dome of Sta. Sophia. This dome is circular on plan and covers a square chamber by means of "pendentives" at the four angles. The dome presents on the inside the appearance of being nearly hemispherical, having a radius of about 55 feet ; but constructional!*/ it must 268 DOMES, SPIRES. be considered a segmental dome, the springing of which is at the joint EF (fig. 100), the radius OE making an angle of 60 with the vertical OA, where is the centre of curvature. The base of the dome from F to D is thickened out so as to form an abutment, the inner surface only being worked to the curve of the dome. Referring to article (95) and taking the 180th part of the circumference, we have P = weight of the lime ABEF ; 8 = weight of a cubic foot of material ; N = the horizontal thrust at A of the opposite lune ; G, the centre of gravity of P ; 6 = the angle BOB ; R = outer radius of the dome, as OA or OF ; r = inner do. do. as OB or OE. Then, by equation (161) we have Am' By equation (162) P = -01163 (1 - cos. 0) (R 3 - r 3 ) 8, and when 6 = 60, cos. = J. Also R = 57, ; = 55. Therefore, P = -01163 x 5 x 18818 . 8 = 109-27 . 8. Also, by equations (163,164) Ej0 = 55 x -866 - | x 74-68 1 ' 047 7 ' 433 = 13-24. By equation (165) Am = 57 55 x *5 = 29-5. /a/* FLOOR LINE Fig . IOO 270 DOMES, SPIRES. Consequently, we have 109-27 x 13-24 = thrust of 180th part of the dome, at EF. Putting 8 = 1 cvvt., we have the thrust of the 180th part of the dome equal to 49 cwts. or nearly 2| tons. So that the total thrust of the dome at the level of EF is 180 x 2i or 450 tons. The radius Em is 46 feet, therefore the circumference at E is 289 feet, arid the thrust of the dome at the level of EF is or 1*56 tons per foot length of circum~ ference. If a ring or belt of iron is placed round the dome at F to counteract the thrust, and T is the total normal pressure on the ring at that level, we have T = 180 N = 450 tons. And by equation (174) the tensile stress in the belt is 16 T = 72 tons. Taking 5 tons per square inch of section as the safe- strain, we find that the belt must have a sectional area of = 14'4 square inches, or be 14^ inches wide 5 by 1 inch thick. The weight, W, of the portion of the "drum " or base of the dome, FH, corresponding to the 180th part of the circumference, may be roughly estimated at 12| tons, acting vertically through its centre of gravity, g. Compounding this with the resultant, Q, of the forces. SURCHARGED DOME. 271 P and N at E, we get IR as the direction of the resul- tant R of these forces, making an angle of about 8 with the vertical at I. The forces N, P, Q, W, and R, are very nearly in the proportions of 5, 11, 12, 25, and 36. For the resistance offered by the abutments to this oblique thrust, we have on each of two sides of the chamber, a semi-dome of 50 feet radius ; the vertex of which abuts at H, and acts as a flying buttress. We have also the arch HU, 100 feet span, whose weight will also tend to counter-balance the thrust. On the two other sides we have four solid piers 18 feet x 25 feet carrying arches 70 feet span. These ought to be sufficient to resist outward thrust and also to bear the crushing weight of the whole of the super- structure. It has been shown (96) that the thrust of the semi- dome against the arch forming one side of the chamber carrying the large dome, amounts to 125 tons ; and we have seen above that the total thrust of the large dome is 450 tons. If we suppose one-fourth of this to press, against each of the four walls, we have 112 tons for the pressure against each ; so that the above thrust of the half-dome, namely, 125 tons, will counterbalance this thrust on two sides. 98. SURCHARGED DOME. Domical vaults are frequently used to carry a heavy surcharge, in cases where great strength is required in the roof of a circular chamber, as in bomb-proof powder magazines. Suppose the horizontal line MK (fig. 101) to represent the top of the surcharge, ABCD the section of the dome whose thickness is R r : and let AM = k. 272 DOMES, SPIRES. EF is the " angle of rupture " as before determined (95), the radius OE making 20 with the horizontal OD. Draw the horizon- L M tal lines, Vn and E/#, and let G be the centre of gravity of a piece, whose weight is P, cut out of the dome by two vertical planes making an angle $ ( = 2) with each other at the verti- "* cal OM ; drop a vertical from G cutting Ew in p. Then P is the mass MLFEB, or we may assume, very nearly, P = MLFw - BE/rcB : Fig. 101. Qr P = * R* (cos. 20) 2 x FL - EB>ra = -0154 R 2 (k + '658 R) . 8 - EB/w EBwz = BEOB - OEwzO = ^(l _ cos, 70) 8 3 v * r 2 (sin. 70) 2 X r . cos. 70 x 8 = (-00765 r 3 -001756 r>) 8 = -0059 r 3 . 8, Therefore, P = (-0154 R 2 (k + '658 R) - -0059 ?>) 8 . (177) Let a be the centre of gravity of the part MLFy*, at SURCHARGED DOME. 273 its distance from the axis OM ; I that of the part BE OB, bh its distance from OM ; d that of the part OEwO, de its distance from OM. Then ai = ~ ML = ~' R . sin. 70 = -626 R ; o o when d = 70. P x mp = {-0154 R 3 (k + '058 R) x -625 R + 001756 . r 1 x -47 r - -00765 r 1 x -513 r} x6 = { -00964 R 3 (k + -658 R) - -0031 . r< } . 6 . (178) E/> = r . sin. 70 - mp = -03969 > - mp, Km = R - r . cos. 70 = R - -34^ r, N = P|. Am Since N and P balance about E, we can consider them as acting at E parallel to their original directions, and take their moments from that point about Z, the outer edge of the supporting wall or u drum." Let F be the mass of the part KLFJ, acting at y its centre of gravity ; let Q be that of JIXZ acting at y' its centre of gravity. Then to obtain the equation of stability we have to equate twice the moment of N about Z, with the sum of the moments of P, F, and Q. Let DX = //. The moment of N is N (h + r . sin. 20) = N (h + -342 r). 274 DOMES, SPIRES. The moment of P is P (t + -0603 r). F = f- { (r + O 2 - R 2 . (cos.20 ) 2 ]- (k + R-R . sin.20)8 J _ -883 R 2 ) (k + '658 R) 5, 2 O + O 3 - '7796 R 3 - -3 (,-"+ O f --883"K'~ 1 (^-M) 3 j^2-649 R 2 (r + f) + 1-559 R 3 3" r + t' - "883 R 2 F x ^ = -0058 (/f + -658 R) {(; + f) 3 - 2-65 R 2 (; + t) + 1-56R S }8, which is the moment of F about Z. The moment of Q is the same as in equation (173), namely 0058177 (3rf + f)k . 8. As an example of application of these formulas, let R = 11, ,- = 10, h = 20, k = 4. Then we find from equation (177), P = 16'8 .8; and, from equation (178), P x mp = 113-228, mp = 6-74, E/ = 2-55, Am = 7'58; Therefore, N = !!^* ~L?- 5 8 = 5-65 8. Z'Oo The moment of N, at E, about Z (omitting 8), is 5-G5 x 23-42 = 132-32. The moment of P, at E, about Z, is 16-8 + 10-13. The moment of F, at g, about Z, is 0652 f + 1-965 2 s - 1-37* 15-20. DOME OF UNEQUAL THICKNESS. 275 The moment of Q, at /, about Z, is 11635f + 3-51 1\ The equation of stability becomes 264-64 = -18155 f + 5-466 t 2 + 15-53 1 - 5-13, which can be reduced to f + 30 f + 85-5 1 - 1485-5 = 0, from which we find t = 5*4 ft., very nearly. The pressure on the joint EF in this example is W = -342 N + -94 P = 2127 Ibs. = 150 x 14-18; so that if Bath stone is used, the area of the joint must be 14'18 square inches. If brick is used, in which 50 Ibs. per inch is the safe load, the area must be 42-54 inches. The actual area in this case is 49 inches, so that the resistance will be ample when brick is used. 99. DOME OF UNEQUAL THICKNESS. Since the heaviest pressure is borne by the lower part of the dome, a saving in weight and material can be effected by reducing the thickness from the springing towards the crown. We will here ^J Uf Fiy.W2. take an example in which the depth at the crown is half that at the springing. Let AC (fig. 102) be the outer curve of the dome T 2 276 DOMES, SPIRES. struck from as a centre, BD the inner curve struck from 0' as a centre ; call OA = R, O'B = r, then CD = 11- r, AB = ll ~ r Draw O'E making an angle of 70 with the vertical OA, then the "joint of rupture" will be at E. Draw the horizontal line FEw, and let P be the weight of the part of a " lune " of the dome between EF and AB. Let P! be the weight of the wedge AF/w, P, that of BEw ; ~j and -., the horizontal distances of their centres of gravity from AO. Let Gr be the centre of gravity of P, mp or ~, its distance from AO. Then 2-2 ~~" Let be a small angle which a small luxe of the dome subtends at 0', which, as before, we take as 2% so that in circular measure we have " arc " = '0349. From the geometry of the figure we find t{ R2 (R- "3- j * ^ov} 8 O'm = r . cos. 20 = -342 r Om = O'm + -*- (R - r) = -5 R - '158 r r - O'w = '658 r r 1 - O'm 3 = -96 r' R _ Om = '5R+ -158 r DOME OF UNEQUAL THICKNESS. 277 E/> = EM - mp = -9397 r - z Am = AO - Ow = R - 0/w = -5 R + '158 r l\ = -0175 R 2 (-5 R + -158 r) 3 j ' P, = -0175 (-658 r' - -32 r<) . 6 = -00592 r 3 . 8. If we put A 1 for any measurement along OA, eitlior from or 0', then we have the limits of integration being from x = Om to x = R ; when IV, = -01103 3 l-5708 - sin.- . - the limits of integration being from x = O'm to * = r. P = -01163 3 1-5708 - sin.- > . . 5 / ) = -003101 r 4 . 5 278 DOMES, SPIRES. For example, let B = 11, r = 10 ; then O'm = 3-42 ; Om = 5-5 - 1-58 = 3-02 ; P l = 7'58 . 8, P 2 = 5-92 . 8, P = 1\ - P 2 = 1-66 . 8. P^i = 44-34 . 8, P z z.,= 31-01 . 8, P^ - P 2 ^ 2 = 13*33 . 5 13-33 V-'-TST 1 "* E^ = 9-307 -8 = 1-397 Am = 5-5 + 1-58 = 7-08 ^ 1-66 x 1-397 , , , -7-08 ----- 8= 3 ~' - 8 ' Patting F for the weight of the lower portion of the lune between EF and CD, we can approximate to its value by taking it to equal the area of the section at EF plus that of the section at CD, multiplied by half OM ; or we have F = -t(R 2 - r 2 ) (1 + cos. 2 20) 9^ . 8 = -016475 (R 2 - ?*) (-5 R - -158 ;) . 5. The lever arm of F about Z may be taken as very nearly or, the moment of F about Z, is If Q is the weight of the part of the wall DZ, Q . q its moment about Z is found by equation (173), and is Q.0 = -0058177 (3rf + fth. 8. DOME OF UNEQUAL THICKNESS. 279 The moment of N acting at E, about Z, is N (h + Om') = N (h + -5 II - -158 r). Patting h = 50, we have for the mommt of N in the above example N (h + Om) = 19-27 . 5. The mo:nsnt of P acting at E, about Z, is P + t - r x -9397) = P (t + -0603 r) = (1-GG t + 1) . 6, in this example. The moment of F about Z is F Y t - R " r \ = A -356 * - -678) . 6. The moment of Q is Q . q = (8-7265 ji 2 + -29 f) . 8. Then the equation of stability becomes 38-54 = 1-66 t + I + 1-35C t -'678 + 8-7265 f + -29 ^ which reduces to f + 30 f + 10-4 - 131-8 = 0; from which we find, t = 1*88. Comparing this dome with the dome of uniform thickness and same diameter (95), we see that the horizontal thrusts are in the proportion of 39 : 88 ; or the thrust of the dome of varying thickness is less than half what it is in the dome of uniform thickness ; also that the thickness required for the walls supporting them is nearly in the proportion of 2 to 3, where the height is the same. 280 DOMES, SPIRES. 100. GOTHIC DOME. This term is applied to a dome whose section is a Gothic or pointed arch, as ABCD (fig. 103), where is the centre of curvature for one half of the section, AH being the axis or centre line of the dome, and DH the half- span. Let n be the middle point of the vertical joint AB ; then the form of arch varies according to the angle, a, which On makes with the vertical OK. If the section is an equi- lateral arch (86), then a = 30 ; and in the great dome of the Cathedral at Florence we find a = 22|. We will call OC or OF = R, OD or OE = r ; then On = H R + ?')> CD = EF = U - r. We will take as before a small lune cut out of the dome and subtend- ing an angle = 2 at the centre 0; in circular measure "arc c/>" = -0349. Let N, the horizontal thrust, act at the point n, and we have in the first place, as before, to find the joint EF at which N is greatest. Let G be the centre of gravity of the part of the lune between AB arid EF whose weight is P, and let OE make the ariiile with OH. Then we have to Fig 1O3. GOTHIC DOME. 281 express N in terms of 0, and find what value of makes N a maximum. The value of P is determined from the Integral P = . 6 . // r (sin. (a + 0) - sin. a)clr . d) (179) where the limits of integration are r and K, and = to = ; so that P = -01 1 03 (H 3 - r 1 ) (cos. a - cos. (a + 0) - . sin. a) 5 sin. -- // )'' (sin. (a + 0) sin. a) 2 dr . d$ mp = $- JJ > :i (sin. (a + 0) - sin. a) dr . (10 and since $ is small, we can put -r = 1 ; and we have mp = 3 ^^ x | |- (1 + 2 sin. 2 a) + 1 (sin. 2 a - sin. 2 (a + 0) ) - 2 sin. a (cos. a cos. (a + 0) ) j- -T- -j cos. a cos. ('a + 0) . sin. a |- 2 (1 + 2 sin. 2 a) + sin. 2 a - sin. 2 (a + 0) - 8 sin. a (cos. a cos. (a + 0) ) > -r ! COS. a COS. (a + 0) . sin. a >- 282 DOMES, SPIRES. E/> = r (sin. (a + 6) sin. a) mp mn ~ - R + r cos - a r cos - a N - P . E /'. 7W# We will apply these formulae to the case where a = 22|, or "arc a" = -3927, sin. a = -38268, cos. a = 92388, sin. 2 a = cos. 2 a = -70711, sin. 2 a = -14645, 1 +2 sin. 2 a = 1-2929. P = 1)1163(R 3 -O{-9239(l-cos.0)--3827(0-sin.0)}2 1 2-5858 + -707 (1 - cos. 2 - sin. 2 0) - 2-828 (1 - cos. 0) - 1-172 sin. j -r | '9239 (1 - cos. 0) - -3827 (0 - sin. 0) j Ep = r {-9239 sin. - -3827 (1 - cos. 0)} - mp mn = -4619 (R + r) - (-9239 cos. - -3827 sin. 0) r. In order to find the greatest value of N, we put R = 11, r = 10, and calculate the above formulae for different values of ; then we find that N is greatest when = 54, or, a + = 76J, so that OE makes 13 with the horizontal line OC. In this case we have = -94248, cos. = -5878, sin. = -80902, sin. 20 = 95106, cos. 20=- -30902. P = -003837 (R' - r 1 ) 5 mp = -32756 |^ GOTHIC DOME. 283 N = -002262 (R 3 - r 1 ) ; - -001258 (R 4 - ?") 46194 K + -&J4Ur If we put F for the weight of the part of the lune between EF and CD acting at its centre of gravity ; we find from the equation (179), by integrating from = 54 to e = 90 - a, that which in the present case becomes F = -001667 (R 3 - r') . 8. The lever arm of F about Z may be taken as where t is the thickness of the wall of the " drum ; " or the moment of F about Z is The forces P and N being supposed to act at E, the moment of N about Z, (putting h for the height of the drum) is N (h + r . cos. (a + 0) ) = N (k + -23345 r). The moment of P about Z, is P (t + r - r . sin. (a + 0) ) = P (t + '02763 r). If Q is the weight of the portion of the wall of the "drum" corresponding to the lune ABCD, acting at its centre of gravity, q its lever arm about Z ; then we have for the moment of Q, from equation (173), putting s for the half-span, Q . q = -0058177 (3sf + f) h . 5. 284 DOMES, SPIRES. The half span HU of the dome is s = HD = r - I (R + r) sin. a P*I = r- -ID 134 ( It + r), when a = 22|. Suppose, for example, that the span is 20 feet and the thickness, CD or EF, is 1 foot, then R = r + 1, HD = 10 = 8,3= 10 = r - -19134 (2r + 1) 10-19134 ,, . r = = 16*5 : 01732 therefore R = 17-5. Let/* = 60 feet. Then we find T = 3-327 . 8, N = -64334 . 8, so that the thrust at E is less in this dome than in the hemispherical one of equal span and thickness, in the proportion of about 7 to 8. The moment of N, at E, about Z, is 35 ; that of P is 3-328 t + 1-517. The moment of F is 1-446 t - 723, and that of Q is -291 (' + 30 f). The equation of stability becomes 70 = 3-328* + 1-517 + 1-446*- 723 + -291 (f + 30 f) or, f + 30 e + 16-5 t - 239 = ; therefore, t = 2-416 feet. If we put 8 = 120 Ibs., then the value of N is 77-2 Ibs. and multiplying this by 180 we have 13,876 Ibs. or 6*2 tons, for the horizontal thrust of the entire dome at the level of the joint EF. If an iron belt is placed round the dome at the level of F, we find by equation (174) that its sectional area should be '16 X 6*2 -f- 5, or one-fifth of an inch, if we allow 5 tons per inch as the safe stress on wrought iron ; so that a belt 1" x \" GOTHIC DOME. 285 would entirely counteract the outward thrust of the dome at this point. We will now apply these formula? to a circular dome of the same diameter as the octagonal dome of Florence, the diameter of which is 130 feet; and will take the thickness at C feet. We find in this case that r = 107 feet, and R = 113 feet; R + r = 220, R 2 = 12,769, r = 11,449, R* - r< = 217,854, R' - /' = 31,967,760. We will suppose that the dome stands upon a solid circular wall 175 feet high. We have then P = -003837 x 217854 . 8 = 836 8 -002262 xl 07 x 217854 - -00 1258< 31967760 , x 1 1 3 r+ -22849 ~xl07~ ~ = 162-5 . 5. The moment of N about Z is N (h + -23345 r) = 162'5 + 200 . S = 32500 6. The moment of P about Z is p (t + -02763 r) = (836 t + 2462) 8. The moment of F about Z is F (t - 5_Z1^ = /363< - 1080) 5. The moment of Q about Z is Q. f j = 1-018 (321 f + O 8. Tlic equation of stability becomes 65000 = 836 t + 2462 + 363 t - 1089 + 1-018 (321 f + f) 286 DOMES, SPIRES. which reduces to f + 321 f + 1119* - 62500 = 0; the solution of which is t = 12*12 feet. Putting 8 = 120 Ibs., we have N = 19,500 Ibs. as the horizontal thrust at F for the 180th part of the dome; and the total thrust of the dome is, T = 3,510,000 Ibs. or 1,567 tons. Multiplying this by *16, equation (174), and dividing by o, we get 50 square inches for the sectional area of an iron belt placed round the dome about the level of F, that will counteract the thrust ; or a belt 2 inches thick and 25 inches wide would be required for this purpose. The radius of this dome is 6| times that of the first example where the span was 20 feet, and the cube of 6| is 275. We find from the formulas that the thrusts of the two domes are in the ratio of '64334 to 162*5, or as 1 to 252, which is nearly as the cube of their diameters. It will be evident from the foregoing formulae that the position of the "joint of rupture'' EF must vary with the value of the angle a, which is nothing in a hemispherical dome, and is 30 in a dome whose section is an equilateral pointed arch, in which latter case we find OE makes an angle of 11 with the horizontal line OC. By substituting the angle 30 for a and 49 for in the expressions for P, 1S T , and F, we can calculate their values for a dome whose section is an equilateral arch. In this case we find the following equations, P = -00288 (R 3 - r 3 ) b N = -001387 (R 3 - r 3 ) r - -000407 (R 4 - r<) g 43302 li + '24222 r GOTHIC DOME WITH LANTERN. 287 The lever arm of P about Z is t + -01837 r. The lever arm of N about Z is h + -1908 r. The moment of F about Z is 001103 (R 3 - r 1 ) The moment of Q is the same as before. 101. GOTHIC DOME IVITH LANTERN. When a dome is used to form the roof of such an edi- fice as the Cathe- dral of Florence, it is usually sur- mounted by an or- namental lantern, the weight of which must be taken into consideration when investigating the thrust of the dome upon its supporting walls. Let IKL (fig. 104) represent the section of such -000407 (R 4 - r 4 ). rig 104- a lantern, and S the weight of 180th part of it acting in the direction Iq cutting Ew in q. Then in finding 288 DOMES, SPIKES. the outward pressure, N, at E, we must equate the moment of N about E with the sum of the moments of P and of S ; P being the weight of the 180th part of the dome between AB and EF ; so that we have N x mn = P x Ey> + S x Ey, or N = p * E /> + S x Ey mn Taking the case of the Florentine dome, we will suppose the total weight of the lantern to be 100 tons, and put S = 10 . 8; also mq = 10-5 ; It = 113, r = 107 ; ~&m = -58969 r = 63-1 ; E? = Em - mq = 52-6. mn = -02388 5-tf- -23345 r = 76-65; P = 836.8. mp = -32756 ^ ~ ''] = 48-07. Ej = Era /; = 03-1 - 48-07 = 15-03. N = 836 x 15-03 + 10 x 52-6 g = j-j g This is only -oVth more than the thrust of the same dome without a lantern, which we found above (99) to be 162-5 . 5. In this case the equation of stability becomes 08400 = 1-018 O 3 + 321 t~) + 1209 t + 2501 - 1089, or, f + 321 1~ + 1188 t - 65803 = 0; t = 12-38ft. Without a lantern we found that t = 12-12 ft., so that its weight has comparatively very little effect upon the thrust of the dome. If we reduce the thickness of the dome one-half, we find that the thrust caused GOTHIC DOME WITH LANTERN. 289 by the weight of the lantern is increased in proportion to that caused by the weight of the dome itself. Put- ting R - r = 3ft. instead of 6ft, we have P = -003837 (HO 3 - 107 3 ) 5 = 406-6 . 6. mp = -32756 ^^ = 47-4. E/> - 63-1 - 47-4 = 15-7. mn = 100-24 - 25 = 75-24. Without the weight of a lantern, we have N = 406-6 xl5-7 5=j 75-24 With a lantern of the above weight, we have N = 406-6 x 15-7 + 10 x 52-6 _ = 01 . 8 g . 75*24 or the thrust with a lantern is in this case -^th more than without it. The total outward thrust of this dome with the lantern amounts to 885 tons, so that an iron belt placed round it to counteract the thrust would require to be by equation (174), '16 x 885 -f- 5, or 28-32 square inches in section ; and if it were 2 inches thick its width must be 14^ inches. The area of the joint EF in this example is A = | (R 2 - r 2 ) cos. 13i = ll'036ft. = 1589 square inches. The pressure on the joint is W = N . sin. 13i + (P + S) . cos. 13| = 426-51 8. Putting 5 = 120lbs., we find W = 51,1811bs., or 32"2lbs. on every square inch of the joint. Since the DOMES, SPIRES. safe load for brick is oOlbs. per inch, we see that we might safely reduce the thickness of the dome still further. 102. CONICAL DOME. In this, which is the strongest form of dome, the maximum value of N is at the base CD (fig. 105), or CD may be considered as the "joint of rupture," and the moments of P and N balance about the outer edge C. Let a be the angle which the slant side BD makes with the vertical OB ; let dn, whose length is /, be the centre line of the thick- ness of the dome, OA the axis or central line of the dome. Then we take, as before, a small slice of the dome equal to 180th of the whole, and put P for its weight acting at G its centre of gravity ; N the hori- zontal thrust acting at n the middle point between A and B. We have then On = I . cos. a, Qd = I . sin. a, 00 = R, OD = r, 0*KR+r). Let the vertical from G cut the line OD in p, then Op = Od = (R + r), C> = R - Op = J (2 R - r) Fig. 10 5 CONICAL DOME. 291 P _ *_/ ff A -,'P I H " 180 I 3 3 j = -00582 (R 3 - r 1 ) b . cot. a On = H R + >') cot, a H-pflK OK = -00388 ?-~ -- (2 R - >) 5, which is independent of the angle a, and depends only on the values of R and r. For example, let R = 11, r = 10, a = 5, \ (R + r) = 10-5, cot. a = 11-43, I = 10 ' 5 = J^?-,= 120-47. sin. a -08/16 P = 22 . 5, N = -734 8. To determine the thickness t of the wall which we suppose to be circular on plan, we take P and N as acting at C parallel to their original directions, and equate the moment of twice N, taken about Z, with that of P and of Q the weight of the wall. The moment of Q is given by equation (173), and if we put h = 100 for the height of the wall, we have Q . q = -582 (f + 30 f) 5. The moment of P is P(*-(R-r)) = (22*- 22)6. The moment of N is N x h = 73-4 . 5. "The equation of stability is 146-8 = -582 .(f + 3W J ) + 22 t -22, u 2 292 DOMES, SPIRES. which reduces to f + 30 f + 38 t - 290 = ; from which we get, t = 2'46. The outer dome of St. Paul's Cathedral in London is formed of a framework of timber,* supported upon a conical brick dome 18 inches thickj in which a = 23, and the radius R is 50ft,, r = 48'37ft., E - r = l-63ft., R + r = 98-37, 2R - r = 51-63, R J - r 3 = 11912, cot. 23 = 2-35585, P = -00582 x 11912 x 2-355855 = 163-338. ' 0388 x H912 x 51-63 98'37 Putting 8 = 120lbs., we have N = 2847-61bs., and the total outward thrust of the dome is 228 tons. To find the thickness t of a solid wall 200 feet high that will support this dome, we have for the moment of N, N x h = 4746 . 8. The moment of P is P (t - 1-63) = (163-33 * - 266-2) 8 Q.q= -0058177 (3 r f + f) h . 8 = 1-16354 (f + 145f).8. The equation of stability is 9492 - 1-16354 (f + 145 f) + 163-33 - 266-2, which reduces to f + 145 e + 140*- 8387 = 0; t = 7ft., nearly. * See the author's edition of Trcdgold's Caiyeutry, p. 144. OCTAGONAL SPIHE. 293 Fig. 106 103. OCTAGONAL SPIRE. This form of spire may be considered as a case of the " Conical Dome " (102), only having eight flat sides instead of being circular on plan. The plan of the spire is shown by fig. 106, where IKLM is one c side of the base, OD the radius of the inscribed circle of the inner octagon, 00 that of the outer octagon. Let 00 = R, OU = r, Od = i(R + r) ; IK = S, LM =r s ; then a we have, S = -8286 n, s - -8.286 r. If we call A the area of IKLM, we have A = R.S-|r* = -4143 (R;- r). We can, with very slight error, consider d to be the centre of gravity of the trapezium IKLM. Let a be the angle which two opposite sides make with each other at the vertex of the spire ; ml = I (fig. 105); then the centre of gravity, Gr, of one side may be considered to be in the line nd (fig. 105). Ofl = I . sin. a; Oti = I . cos. a = \ (R + r) . cot. a ; sin. a 2 sin. a Let P be the weight of one side acting at G, N the horizontal thrust at the vertex ; then we can take 294 DOMES, SPIKES. P = | A . 1 . 8 = (R 2 - r 2 ) (R + >') 8, dp = i Or/, CW = B - Orf, 0/> - i (R + r), Qp = B-Q ? , = 2jy-r o Taking moments about C of P and N, we liave N x Ow = P x Gp ; , N = P JP = : 06 i (R 2 _ r) (2 R - r) 5. On v cos. a To determine the thickness of the supporting wall, we proceed as before (102) to equate twice the moment of N, acting at CD, about Z the base of the wall, with the moment of P at C, and the moment of Q at ^, Q being the weight of the wall forming one side of the octagonal tower. Q = -4143 ( (/ + O 2 - r") h . 5, where h = L>X. = -4143 (2rt + f) k . 5. The moment of N about Z, is N x h. The moment of P about Z is P (t - (R - r) ). The moment of Q is Q x -^ , very nearly. Therefore the equation of stability becomes For example, let R = 11, r = 10, k = 100, a = 5, cos. a = -99619, sin. a = -08716, /= 120'47,R + r= 21, R - r = 1, R 2 - r 2 = 21 ; 2 R - r = 12 ; OCTAGONAL SPIRE. 295 Q = 41-43(^ + 200.5. The equation of stability is therefore 3512 = 524 - 524 + 20-715 (f + 20f) which reduces to ^ + 20f + 25-3(5 -195 = 0; t = 2-44. [The foregoing investigations are based upon papers by the author, which were read before the " Royal " Society in 1866, and published in the " Proceedings" No. 85, 1866; and subsequently in the "Civil En- gineer and Architect's Journal " for February and March, 1868, "On the Stability of Domes."] CHAPTER X. BUTTRESSES. SHORING. RETAINING WALLS. FOUNDATIONS. 104. BUTTRESSES. It often happens in the construc- tion of buildings that the horizontal thrust, as of a roof or vault, is concentrated on a few points of the outer walls, while the intermediate parts have little if any thrust to sustain. We have seen this to be the case in vaulted roofs (92) (93), where the ribs of the vault- ing cause the whole thrust to be borne at the points from which they spring. This is also the case with roof-trusses where there is no tie-beam, as in the Hammer-beam roof (65). In such cases it is evidently more advantageous and economical to in- crease the strength of the wall at the points where the thrust is greatest, rather than making the whole wall of a sufficient thickness to resist the thrust. The method usually adopted where there is suffi- cient space outside the walls, is to build out a mass of masonry called a " buttress." v > Thus, let ABXY (fig. 107) be the vertical section of the wall having a horizontal BUTTRESSES. 297 Fig.WQ. thrust, T, as its summit, and let BCYZ be a " buttress" built out from the wall and of the same height. Tho plan of the wall and buttress is shown by abed (fig. 108). Suppose P to represent the weight of the portion of the wall from centre to centre of the buttresses, and to act at its centre Gr. We will first suppose the but- tress to have a uniform projection x for its whole height ; t being the thickness and h the height of the wall, I its length from centre to centre of the buttresses. Let b be the breadth ab (fig. 108) of the buttress, and Q its weight acting at its centre g. Then, in order that there may be equilibrium between the forces T, P and Q, their moments about the outer edge Z of the buttress must balance ; or the equation of equilibrium is . (180) giving a quadratic equation in terms of #, by which we can determine the projection x of the buttress, for a given thickness b and thrust T. Or, if x and b are given, we can find the amount of thrust that can safely be put on the wall and buttress. Take, for example, the case of a wall which has to sustain the thrust of a vaulted roof, as in a previous example (92) where T = 44-3 5 = 5,316 Ibs., when 5 = 120 Ibs. ; let h = 20 feet, I = 10 feet, t = 1'5 feet, b = 2 feet. Then we have T x k = pl + And the equation of stability is 298 BUTTRESSES, SHORING, RETAINING WALLS, ETC. P = h X I X t X 8 = 300 8 = 40 x . 8 Then the equation of stability (180) becomes 40 x 44-3 = 300 (f + x) + 20 x* or, 88-6 = 15 ar + 11-25 + x z , or, x" + 15 x 77*35 = ; x = 4 feet. Now suppose that x is given as 3 feet, all the other dimensions being as above ; to find what thrust, T, can be safely put upon the top of the wall. The equation (180) becomes 40 x T = 300 x 3| + 20 x 9 T = 32-65 8 = 3,918 Ibs. Let us now reduce the projection i y of the upper half of the buttress by one-half, (fig. 109) making BC = | x, CE = i h ; then the weight of the part BCED is \ Q, and its moment about Z is \ Q X \ - x ; the weight of the part FDYZ is | Q, and its moment about Z is | Q x far; or the moment of the weight of the buttress is ri which is one-eighth less than in the former case (fig. 107), while the weight is reduced by one-fourth. BUTTRESSES. 299 The equation of stability (180) becomes in this case Fig.lW. which in the example above given is 88-6 = lfo+ 11-25 + -875^, or, x"- + \lx - 88-4 = 0; x = 4-18. Hence it appears that very little stability is lost by the reduction of the buttress, but a considerable saving in material is effected. Now suppose the buttress to be triangular in section, as BYZ (fig. 110); then its weight is Q, and its moment about Z is |Q x %x, the weight acting at g the position of which is found as previously described (17) ; so that the moment of Q about Z is reduced by one-third, while the weight Q is reduced by one-half. The equation of stability (180) in this case becomes 0; which in the above example gives x* + 22-Dx - 116 x = 4-32. From this it appears that the triangular form of buttress is the most economical, being the most 300 BUTTRESSES, SHORING, RETAINING WALLS, ETC. effective for a given weight of material. Sucli a buttress is generally built in steps as shown by the dotted lines. The equation of stability (180) being a quadratic, there musl, by the rules of algebra, always be two " roots " or values of x that will satisfy it ; if one root is positive and the other negative, we take the positive root as the value of x. If one root is zero, and the other is negative, it shows that a buttress is not re- quired, the wall itself being strong enough to resist the thrust. In the first example (fig. 107) suppose T = 11-25 . 8; then the equation (180) becomes 11-25 = 15* + 11-25 + * 2 , or, x* + 15* = 0, the " roots " of which are x = 0, and x = 15 ; so that in this case no buttress is required. Again, let T = 9-25 8, then the equation (180) is 9-25 = 15* + 11-25 + x\ or, x* + 15* + 2 =0, in which case both the " roots " are negative, being * = '135 and * = 29-73 ; showing that the strength of the wall is more than sufficient to resist the thrust. In order that the equation may have a real positive "root," the third term which is independent of *, must be negative. 105. FLYING BUTTRESS. A "flying-buttress" is an arched rib of masomy, the base of which rests on the top of one"wall, while the vertex presses against another wall at a distance from the former. This form of buttress is adopted where a horizontal thrust has to FLYING BUTTRESS. 301 Fig. Ill <- T be sustained by the higher wall, the weight of which is greatly reduced by openings, so that it is of insuffi- cient strength to bear the thrust, which is conveyed by the flying buttress to a lower and outer wall against which an ordinary buttress can be built. This ar- B A rangement is shown by fig. Ill where BCE is the flying-buttress resting on the wall CEXY, the arch form being adopted to prevent it from breaking up by its own weight. The thrust T is received at the top, AB, of the higher wall, whose weight however we will neglect in the consideration of the problem. We will suppose that the thrust on the top of the outer or lower wall is counteracted by the weight, Q, of a triangular but- tress, and the weight, F, of the wall itself. The flying buttress acts as an inclined beam, and produces a thrust N upon the top of the wall independently of the thrust T. Draw the line HGL passing through Gr the centre of gravity of the flying buttress and meeting the 302 BUTTRESSES. SHORING, RETAINING WALLS, ETC. top of the wall at its middle point L. Let N be the reaction against the wall at H acting horizontally in HI ; draw the vertical IG/>, meeting the horizontal line CED in p ; and let P be the weight of the flying- buttress acting vertically at G. Then, as we have previously shown (4), the resultant of N and P must act in the direction of the line IL, and we have N : P = LJJ : ip :-* , - : The forces N and P can be considered as acting horizontally and vertically at L. Let a be the angle which HL makes with DC ; and let DE = s, EC = t, CY = /<, YZ = z\ then DL - s + \ ; = 2s -+- t ; DC = s + t; BD = (s + tan. a; Ip = HD = ?--+J tan. a, ., DL 2s + t T ! m 2s + t HL = = - - -r- ; Lw = A DL = . cos. a 2 cos. a 4 Let b be the breadth and d the average depth of the flying buttress ; then 2 s + t , P = b . d x HL x 6 = b . d . 2 cos. a _ tan. a 4 sm. a If F is the weight of the lower wall from centre to FLYING BUTTRESS. 303 centre of the buttresses, I the length, k the height, and t the thickness ; we have F = h . t . I . 5 ; and the moment of F about Z is The moment of P, acting at L, about Z, is P^4 The moment of the buttress Q about Z is Qxfaa}A.d;axf#..a = U . A . x 2 . 8. The equation of stability by which to determine x is, 2 (N x A + T (h + BD)) fQ-* . (181) For example, let a = 30, h = 20, t = 1-5, b = 2, d = 2, s = 10, / = 10 ; then sin. a = '5, cos. a = -866, tan. a = -5774 ; BD = 11'5 x -;>774 = 6'64. F = 20 x 1-5 x 10 .5 = 3008 P = 2 x 2 . 2 J-2^ 8 = 50 . 8 N = 2 x 2 2 | > ' 5 d = 43. 8, Q = x 20 x 2 ar . d = 20 # . d. Suppose T to be the thrust from the ribs of a vaulted roof (92), and to be equal to 44-3 . 8, as in the 304 BUTTRESSES, SHORING, RETAINING WALLS, ETC. example given. Then the equation of stability (181) becomes 2 (43 x 20 + 44-3 x 26-64) = 350 (f + x) + 13-3^, which reduces to or - 287 = ; x = 8-35 ft. Now, suppose that a = 45, all the other dimensions remaining as above. Then cos. a = sin. a = '707, tan. a = 1, BD = s + t = 1 1-5 ; P = 61 . 8, N = 30'4 . 8 ; and the equation (181) is 2 (30-4 x 20 + 44-3 x 31-5) = 361 (f + x) + 13'3a- 2 , which reduces to ~" T a* + 27^-281 = 0; x = 8 ft. 106. SHOEING. When v the wall of a building has a tendency to fall out- wards from any thrust that may act upon it from within, a slanting piece of timber called a u raking-shore" is placed against it to act as a kind of temporary buttress until it can be secured by a more permanent structure. The shore CZ (fig. 112) rests firmly on a template laid on the ground at Z and the Fig. 112 SHORING. 305 upper end C is secured to a nailing-piece CE and prevented from moving upwards by a needle driven through the wall at C. In this case the weight of the shore itself has very little to do with resisting the thrust at C, being kept in its place by the weight of the part above C which presses upon the head. Sup- pose that a horizontal thrust, T, acting at AB tends to push the wall over about its base Y, and that when the shore is fixed at G the wall is just upon the point of falling. Then the pressure of the head of the shore produces a reaction N at C; and when the forces N and T are in equilibrium about Y, we have T x BY - N x CY. Let CY = h, BC - y ; W the weight of wall below C, P the weight of wall above C, t its thickness. Then the above equation becomes T ( h + y) = N x L Taking moments of T, W, and P, about Y, we have in equilibrium -N.J Therefore, N = (W + P) ~ . . (182) Now in order that the shore may not slip upward* from the pressure N, we must have the weight, P of the part above C, sufficient to keep it down ; and we have to determine the value of y which will suffice for this purpose, as this will give the highest point at which the head of the shore can be fixed against the wall that 306 BUTTRESSES, SHORING, RETAINING WALLS, ETC. will render it effective. Let s be the length of wall to be supported by the shore, 8 its weight per cubic foot. Then we have Let a be the angle which the shore makes with the horizontal, then taking moments of P and N about Z, the shore foot, we have N x h = P x h . cot. a, or, P = N x tan. a = (W + P) * ' ?f"' a , from equation (182), from which we obtain P (2/i - t . tan. a) = W . t . tan. a . . (183) Substituting for P and W, their values as given above, in this equation, we have an equation from which to determine the minimum value of y that will preserve equilibrium ; the angle a being given, as well as the dimensions t, /t, and s. For example, suppose a = 60, or, tan. a 1-732 : let h = 20 feet, t = H feet, s = 10 feet ; then W = 300 8, P = 15 y 8 ; and the equation (183) becomes 15 y (40 - 2-C) = 300 x 2'G y= 300x^6 =1 . 3gft lo x 3r4 which is the least height of wall above C, supposing the whole length of the wall to act as a solid mass ; but if this is not the case, or the wall is liable to fracture from the upward pressure of the shore, a much SHORING. 307 greater height will be necessary to secure the stability of the shore. As another example, let a = 75, or tan. a = 3'732 ; h = 30 feet, t = 1-5 feet, s = 10 feet. Then t . tan. a = 5-6, W = 450 8, P = 15y . 5; and equation (183) becomes 15 y X 54-4 = 450 X 5'6 y = 3-1 feet. If there is a roof on the top of the wall, its weight must be taken into consideration, as it will have an important effect in keeping down the head of the shore. Let R be the weight of roof distributed over the length, s, of the wall, then we must add R to P in equation (182), so that we have N = (W + P + B) l k and taking moments about Z of N and P + R, we get N x h = (P + R) h . cot. a, or, P + R = N . tan. a = (W + P + R) ^ll a |^ a , therefore, P (2/i - t . tan. a) = (W + B) * . tan. a -R x 2k . . (184) Let us apply this to the examples given above, and suppose R to represent the weight of one side of a roof, 10 feet long, whose rafters are 20 feet long, and the load is 20 Ibs. per square foot of roofing. Then in the first example R = 20 x 10 X 20 = 4,000 Ibs. x 2 308 BUTTRESSES, SHORING, RETAINING WALLS, ETC. Let the weight of the wall be 120 Ibs. per cubic foot; then W = 300 x 120 = 36000, P = 15 y x 120 = 1800 y. Equation (184) becomes therefore 1800y x 37-4 = 40000 x 2-6 - 4000 x 40 280 The height BC or y being negative in this case, shows that the pressure, R, of the roof alone is more than sufficient to keep down the head of the shore, so that the point C may be placed at B, the top of the wall, or immediately below the wall-plate of the roof. In the second example we have R = 4000, W = 450 x 120 = 54000, P = 15 y x 120 = 1800 y. Then we have from equation (184) y x 54-4 x 1800 = 58000 x 5-6 - 4000 x 60 Here we find y is positive, and the head of the shore must be at least lOf inches below the top of the wall. We have now to determine the strength of the shore necessary to resist the pressure placed upon its head. The forces acting down the shore are the resolved parts of P, R, and N ; and if we put F for the compression down the shore, we have F = (P + R) sin. a + N. cos. a . (185) Apply this to the examples given above, and first SHORING. 309 suppose that R = ; then in the first example, we have, cot. a = '5774, sin. a = -366, cos. a = -5, P = 21 . 6, N = P x cot. a - 12 . 5; then equation (185) becomes F = (21 x -866 + 12 x -5) 6 = 2,904 Ibs. Let I be the length of the shore, then I = -JL- = 24 ft. sin. a From equation (86) for long square pillars of fir we have, for safe load F = 2240 ~ i or, the " angle of repose." Then the only portion which presses against the wall is the wedge AED of earth filled in between the wall and the natural slope DE. Now supposing the wall was Fig . 117 just on the point of giving way under the pressure P of the earth, the whole mass AED would not yield at once, but a portion as ADF would be the first to fall, breaking away from the general mass in some line DF between DE and DA, making an angle Q with the horizontal. This wedge of earth is supported by P, the resistance of the wall, and by the resistance of the surface DF on which it tends to slide. Putting Q for the weight of this wedge 1 foot wide, _/? the weight of a cubic foot of earth, d the depth AD ; then we have Q = lp . d\ cot. . . (192) The weight Q may be considered as acting vertically at 320 BUTTRESSES, SHORING, RETAINING WALLS, ETC. G the centre of gravity of the wedge, and will meet the horizontal force P at the point S on the inclined plane DF. If SN is the normal to the plane at S, then R the resultant of the resistances of the different points on the plane DF is inclined to SN at the angle <, which is the " limiting angle of resistance " between any two contiguous surfaces of earth.* We have then the forces P, Q, and R in equilibrium with each other at the point S ; consequently, by the principle of the "inclined plane" (77) they will be proportional respectively to the sides of the triangle SP, PR, RS ; and we have P : Q = SP : PR = sin. PRS : sin. RSP = sin. GSR I cos. GSR = sin. (0 - 0) : cos. (6 0) since GSN = 0, RSN = , GSR = 0-0. Therefore P - O v ' X sin " ( e ~ cos. (0 -"ft = Q x tan. (0 - 0) or putting for Q its value from equation (192), P = \p . d* . cot. . tan. (6 - 0) . .(193) We have now to determine the value of which makes P a maximum, since the pressure on the wall will vary according to the values given to the angle ; and if the wall is made strong enough to support the wedge of earth whose inclination corresponds with the * Moseley's Engineering. RETAINING WALLS, EARTH-PRESSURE, ETC. 321 maximum value of P, and which thus requires the greatest resistance to support it; then will the earth be prevented from slipping at any inclination whatever. If the wall supplies a resistance which is equal to the maximum value of P in respect to the variable angle 0, it will not be pushed over by the pressure of the earth against it ; but if its resistance is less than the maxi- mum value of P, it will be overthrown. We have then to determine when the quantity involving the angle 0, (equation 193) cot. 6 x tan. (0 $) is greatest, with the variation of 0. Now, by the rules of Trigonometry, cot. B . tan. -- _ sm. 6 . cos. (0 - 0) sin. (0 + ) = sin. (2 d ) = sin. . cos. (0 ) + cos. . sin. (0 - 0) sin. (0 0) = sin. = sin. . cos. (0 - $] cos. . sin. (0 0). By subtraction sin. (20 0) sin. = 2 cos. . sin. 0-0. By addition sin. (20 0) + sin. = 2 sin. . cos. 0-0. Therefore we have / . v sin. (2 ) - sin. cot. . tan. (0 0) = -, ^ -^ -. sin. (20 0) + sm. _, 2 sin. ~ sin7(2~0^q" Y 322 BUTTRESSES, SHORING, RETAINING WALLS, ETC. It will be evident that this quantity is greatest when the fractional part is least, or when sin. (2 ) = 1 = sin. 90, or, when 6 = 45 + J , or < = 45 i Hence the equation (193) becomes P = p . d 2 . cotan. ^45 + $\ . tan. ^45 - \ = | p.d* tan. 2 (45 - |) . . (194) = |^.^ 1 -^- S ] rK l = i^x A. 1 + sin. 4> Comparing this with equation (186) we see that the pressure of earth against a wall is the same as that of a fluid whose weight per cubic foot is *- Bn -* A = p . tan.' 1 + sin. Also since SD = J DF, the point K where P acts is given by DK = J DA, since the pressure against the wall increases from the top downwards, as in the case of the pressure of water. The value of A is given in the foregoing Table for each kind of earth, and has only to be multiplied by half the square of the height of the wall in feet to obtain the pressure P on the wall at K. Suppose, as before, that ro is the weight per cubic foot of the wall, t its thickness, when the wall is of rectangular section ; the equation of stability is RETAINING WALLS, EARTH-PRESSURE, ETC. 323 i*.rf.fi2Px4 o = | d 3 x A t = -82 I . d (195) \/ m Putting, m = 125 Ibs., we have, when A = 20'6, or A = -165, t = '82 x -406 . d = r/, When A = 26-1, as for dry clay, A = -21. w * = '82 x -458 d = -38 d; when A = 03-74, for wet clay, . = -51, = 82 x -714f/=-586f/. Putting d = 20 in each of these cases, we have for the thickness t, 6 feet, 7 feet, and llf feet, respec- tively. Let us now suppose that the thickness of the wall at the base is double that at the top, the outer face of the wall being battered as in fig. 115, then if. a is the angle of slope which BC makes with the horizontal, and t is the thickness of the wall at the summit, we have- cot, a = , or, d . cot. a = t. Then from (107) we have for the moment of resistance of the weight of the wall, taken about C, K . d . t (d . cot, a + ~\ + lw . d? . cot. 2 a = -- m . d . f + w . ( I . f Y 2 324 BUTTRESSES, SHORING, RETAINING WALLS, ETC. = ~. n . d . f- (196) So that the eqitation of stability in this case is "w.rf.f- * *.A o # = 427 l.d (197) V W If = '165, we have w t = -427 x -406 x d= -173f/. If d = 20 feet, then t = 3J feet, and the base is 7 feet ; so that the quantity of material in the wall is the same as that in a rectangular wall 5] feet thick, or nearly one-fourth of the material is saved by using this form of section instead of the rectangular one, where the thickness was found above to be 6f feet. If A = -21, then t = '427 x '458 . d = -196 d\ m and when d = 20 ft., t = 3-92 ft. If A = -51, then t = "427 x -714 d = '305 d; m and when d = 20 ft, t = 7-1 ft. 109. RETAINING WALLS. PEESSUKE OF EARTH: SURFACE SLOPING. Instead of having the surface of the earth level with the top of the retaining wall, as in the last article, we now suppose it to be rilled up to a height h or AL (fig. 118) above the top of the wall, and sloped back therefrom at an angle $ with the horizontal, the summit of the earth being level. Suppose that when the wall is about to yield to the pressure,. PRESSURE OF EARTH, SURFACE- SLOPING. 325 the tendency of the earth is to break off in the line DF, making the angle 6 with the horizontal DX. Let Q be the weight of the mass AKFD ; then if AD = d, AL = hj p = the weight of a cubic foot of earth ; we have Ftg-JW If P is the horizontal pressure against the wall, we have, as before, P = Q . tan. (0 - ) J cot. e - cot. * } tan. (6 - 0). In order to find the pressure against the wall, we have to determine the value of Q which makes P a maximum. This however involves a too complicated 326 BUTTRESSES, SHORING, RETAINING WALLS, ETC. process to give here, but the reader will find it worked oat in Moseley's " Mechanical Principles of Engineer- ing." We shall here proceed to ascertain by calcula- tion the value of for different values of < the angle of repose, by taking a particular case : and the same method can easily be applied to any other case. Suppose that h = (L so that A+ d . = 2 ; then we li have to take different values of 6 and find which one makes P greatest for a given value of . Putting P = \p1f x P', we have to calculate the values of P' = (4 cot. 6 - cot, <) tan. (0 - ) Let < = 45, cot. = !; for B = 55, P' = -3173, for 6 = 57| , F = -3432, for 6 = 60, P' = -3508, for0=62, P' = '3417, Hence we may conclude that in this case the maximum value of P is obtained when = 60% or (f> = 15 ; then since cot, = -5774, tan. (0 -(/>) = -268; we have P = lp . (P x -3508. Equating twice the moment of P, acting at |- d above D, with that of the weight of the wall, taken about C, we have \ m . d . t- = | p . d 3 x -3508, or t = '484 L . cL PRESSURE OF EARTH, SURFACE-SLOPING. 327 If we put p = TV, then t = -484 d ; and when d = 20, t = 9-68. Now take = 20, cot. < = 2-7475. f for = 30, P' = -7358, for = 32f, F = -7829, for = 35, P' = -7943, for = 371, P' = -7772. We may therefore conclude that when = 20, = 35 gives the maximum value to P ; so that in both cases we find

+ 15 ; which we may, without material error, consider as the value of d which makes P a maximum for any value of ; so that the general equation for P will be P = I P K I ( 7 ^J~- )' cot. (0 + 15) - cot. * | x -268 . . . (198) Taking = 20, we have, when h = d P = i p . r/ 2 x '7943. Equating twice the moment of P with the moment of the weight of the wall, we have \ro.d. t=\p.d*K '7943 t = '728 I p . d. s/ n lip = w, then t = '728 . d\ and when d = 20, t = 14-56. When the face of the wall is made to batter, and the thickness at the base is twice that at the summit, or 2 t, we have, as before, by equation (196), when = 45 328 BUTTRESSES, SHORING, RETAINING WALLS, ETC. - 11 w . d . t 2 = l -p . d 3 X "3508 6 3 ^ t = -253 1 2- .d. ^/ w If p = re, then we have t = '253 r/; and when d = 20 feet, t = 5-06 feet, the width of the base being 10-12 feet. When = 20, we have - o = p o x -7943 = 38 /.rf. V w If p = m, then t = -38 d\ and when d = 20 feet, t 7-6 feet, the width at base being 15-2 feet. Fig W) We will now suppose that the earth slopes away to an indefinite distance, as AF (fig. 119), or that the height h is indefinite. Supposing that when the wall is about to PRESSURE OF EARTH, SURFACE-SLOPING. 329 give wav, the tendency of the earth is to break off in the line DF, making an unknown angle 9 with the horizontal ; we have Q = | p . d . AF . cos. , and, AF ; d = cos. : sin. (9 ). T^fore, Q^.^r^ P = Q . tan. (-#)->. rf- . COS . C ^ T) cos. #. Now since can never be less than , the greatest value of P must be when 9 = , and cos. (0 <) = 1, or DF coincides with the natural slope DE. Hence it appears that in this case the maximum value of P is P = \p . d 2 . cos. 2 4> . . (199) For a rectangular wall, the equation of stability is \ w . d . f = % p . d 3 . cos. 2 (j) ^ = 816 I P- .d.cos.Q. V m Ip = w, t = -816 . cos. . d. When = 45, cos. = '707, t = "577 d; if d = 20, * = 11-54. When = 20, cos. = -9397, * = -767 d; if rf = 20, t = 15J. When the thickness of the wall at the base is double that at the top, or is equal to 2 1, we have from equation (196) 11 n>.d.f = -J- p . d* . cos. 2 4> 6 o 330 BUTTRESSES, SHOEING, RETAINING WALLS, ETC. t = '426 / COS. (/> . (I. V m Ifp = w, we have, t = '426 . cos. $ . d. When (/> = 45, cos. = -707, t = '301 . d; d = 20 feet, t = 6'02 feet, for the thickness of the wall at top ; the thickness at base being 12'04 feet. When $ = 20, cos. = '9397 ; t = -4 d; d = 20 feet, t = 8 feet, for the thickness at top of wall, the base being 1C feet. 110. RETAINING WALLS. EARTH BUTTRESS. We now suppose that the earth is filled in against a wall and left to form its natural slope, as AED (fig. 120), and Fig . 120 we have to determine the pressure which it exerts against the wall. We will suppose, as before, that when the wall is about to give way under the pressure, the earth has a tendency to yield in the line DF making some angle d with the horizontal. Then if Q is the weight of the wedge AFD, p that of a cubic foot of earth, we have Q = | p . AD x AF x cos. <. RETAINING WALLS, EARTH BUTTRESS. 331 Now, AF : AD = cos. 6 I sin. ( + 0} therefore, If P is the horizontal pressure against the wall, we have from equation (103) ..__, tan. 6 + tan. $ and we have to determine when P is a maximum for the variation of 6. Since tan. (0 - *) = tan a- to. j. 1 + tan. . tan. $ P = i If we put = 45, or tan. (/> = 1, then we find P to be a maximum when = 72, or 6 = 27 ; and tan. 27 is very nearly equal to '5 or i ; tan. 72 = 3 ; so that generally we can put = 27 + ; therefore we have P # p = i 4 tan. (27 + ) + tan. $ When <\> = 45, equation (200) becomes p = s^JL Equating the moment of the wall about C with that 332 BUTTRESSES, SHORING, RETAINING WALLS, ETC. of twice P, which is supposed, as before, to act at a point one-third off/ above D ; we have for stability 7 , pd' 2 d pel* If j = w;, then = -3 r/ ; and when d = 20 feet, = 6 feet; the section of the wall being rectangular. If there is a horizontal force, T, acting at the top of the wall and tending to push the wall over about its base D, then the embankment of earth will act as a buttress ; and by equating the moment of T, about D, with the sum of the moments of W and P about D, we can ascertain what is the amount of the force, T, that would just be on the point of overturnin the wall. We have then Let n> = p = 120 ; then T = 60 (f + -^ \ = 60 (t' 2 + -0417 = 20, then 6 = 27, 6 = 47 ; tan. = -364, tan. (27 + ) = tan. 47 = 1-0724; therefore P =p . d 2 x -174. Then the equation of stability for the wall becomes FOUNDATIONS. 333 \m . d . f = p . cl 3 x '116 t = . 48 /P . d. V m If p w, then t = '48 d ; and when d = 20 ft., t = 9-6 T . rf = i w . rf . t 2 + J j . cl 3 x -174. Putting/? = TK = 120; we have T = 60 (f + -116O- If = 9-6, d = 20, we have T - 60 (92-2 + 46-4) = 8,316 Ibs., and this is the pressure upon one foot length of wall that will just, balance the resistances of the weight of wall and weight of the earth buttress. 111. FOUNDATIONS. When a heavy building is erected upon a yielding soil, the load must be distributed over a sufficiently wide area in order that the footings may not sink into the ground. It is also essential that the earth should be excavated to a considerable depth, so as to reach solid ground, and the trenches made perfectly level at the bottom before commencing to lay the foundations. [Speaking of foundations, Vitruvius (Book 3) says, " Fundationes eorum operum fodiautur si queat inveniri ad solidum et in solido quantum ex amplitudine operis pro ratione videbitur, extruaturque structura totum solum quam solidissima."] The amount of pressure per square foot that the earth will bear must depend on the nature of the soil, as it will be less with loose soils or those in which (the angle of repose) is small, than with those in which is large; so that the resistance of the soil must evidently be some 334 BUTTRESSES, SHORING, RETAINING WALLS, ETC. function of the angle <. The following Theorem is given by Rankine as the basis of his investigations upon this subject ; " It is necessary to the stability of a granular mass, that the direction of the pressure between the portions into which it is divided by any plane should not at any point make with the normal to that plane an angle exceeding the angle of repose." If then p Y is the greatest and p., the least pressure, we must have * not greater than sin. rf, Pi+P* or putting we have 1 + sin. <}> = PjP* Pi + p-l Pi + Pt 1 sin. (i> = ' 1 ^ } ' 2 Pi * J_* __o_ . Pi+P* Pi+P* Therefore, ^ = l+_sirM>_ p., I sin. If we put/*' for the greatest horizontal pressure at a depth 1 sin. $' And if P is the greatest vertical pressure per square foot . (201) 1 sin. (/>/ \1 sin. (/>/ Putting A for the area of the footings or foundation PILING. 335 of the wall, in feet ; and Q for the weight of earth displaced by it, we have Q = m . d . A. If TV is the weight of the wall, then TV = P x A. Then, the limit of the ratio in which the weight of the building exceeds the weight of the earth displaced by it, when the pressure is uniformly distributed, is TV = P. -A P w . d . A w . d .*\ from equation (201) 1 - sin. / When = 45, we find, ^ = 34; When = 20, we find, ~ = 4'2. ' Q Suppose, for example, that m = 120 Ibs. d = 5 feet; then for = 45, P = 120 x 5 x 34 = 20,400 Ibs. per square foot. TVhen = 20, P = 120 x 5 x 4-2 = 2,520 Ibs. per square foot. Multiplying the value of P thus obtained by the area A of the foundations in feet, we obtain the utmost load that the earth will bear. To ensure stability we must either increase the area A at least threefold, or else divide the weight W by three or more. 112. PILING. TVhen the soil is too soft to allow of ordinary foundations being laid upon it, it becomes 336 BUTTRESSES, SHORING, RETAINING WALLS, ETC. necessary to reach the firmer substrata by means of long baulks of timber, called " piles," shod with iron having a point or knife-edge, which are driven through the soft earth by means of a heavy " ram," or " monkey," let fall from a height upon the head of the pile. [The use of piling for the foundations of buildings to be erected on very loose soil, has been known from the earliest period, as is shown by the allusion made by Vitruvius (Book 3), in the following words, to this method of construction : "^sin autem solidum non invenietur, sed locus erit congesticius ad imum aut paluster, tune is locus fodiatur exinamiaturque et palis alneis aut oleagineis aut robusteis ustilatis configatur, sublicaeque machinis adigantur quam creberrimae, carbonibusque expleantur intervalla palorum et tune structures solidissimis fundamenta impleantur."] On the top of these piles the foundations of the walls are laid, and it is essential that we should be able to determine beforehand what load the piles will safely bear after they have been driven in as far as they will go. The investigation of a formula which approxi- mately gives the maximum pressure which a pile will bear when the weight of the ram, the weight of the pile, the height of fall, and the distance through which it is driven at the last blow, are known, will be found in Weisbach's " Mechanics of Engineering," and is based on the following laws which govern the impact of bodies : first Lam. The momentum, or mass x velocity r , of two bodies which impinge, at the end of the impact, is equal to the momentum at the beginning of the impact. Second Law. The vis-viva, or mass x velocity- PILING. 337 squared) lost by the inelastic impact is equal to the sum of the products of the masses and the squares of their gain or loss of velocity. Third Law. If a body falling from a height k, acquires the velocity v in feet per second, h is called the height due, to the velocity v ; and if g represents the force of gravity, or the velocity that a body acquires in falling from rest in one second of time, we have the value of fj being generally taken as 32'2 feet. In this investigation we suppose the pile to be in- elastic and not to be perceptibly compressed by the blows, so as not to complicate the formula. If then the ram falls through a height of h feet, v being the velocity with which it strikes the pile, we have by the 3rd law, tf = 2 cj h. Let W be the weight of the ram, P that of the pile ; the pile will have moved through a certain space during the time of impact, and will have a velocity V at the end of that time ; and if H is the height due to the velocity V, we have by the 3rd law, V 2 = 2 g H. Now by the first law the momentum at the end of the time of impact equals the momentum at the beginning of the impact, or V (W + P) = v . W ; W therefore, V - - - v 338 BUTTRESSES, SHORING, RETAINING WALLS, ETC. Now suppose that at the last blow the pile sinks a distance of s feet, x being the velocity at the end of the last blow ; then by the 3rd law, v? = 2 g s. Let L be the resistance of the earth, or. the load which the pile can just support without sinking further into the ground, then by the 2nd law, we have L x v? = (W + P) V 2 , or, L = -X' OV + P) = ,_ W \^ (W + P) ?-^ W + P ' s neglecting altogether the action of W and P in opposi- tion to the resistance of the earth. In using the equation (202) to find the greatest load a pile can bear, we must express both h and s in feet or both in inches, and each of the quantities, W, P, and L, either in tons, cwts. or Ibs. For example, let W = 1 ton, P = f ton, h = 10 feet, and suppose s is found by measurement to be -36 inch or *03 feet ; then by equation (202) we have L = J_ x -1? = 190-5 tons. 1'75 '03 If W and P are expressed in cwts. or Ibs., L will also be in cwts. or Ibs. PILING. 339 According to Eankine, piles are generally driven till L amounts to between 2000 and 3000 Ibs. per square inch of the area of section of the pile, and are loaded permanently with from 200 to 1000 Ibs. per square inch, so that the /actor of safety is from 10 to 3. Thus, if we suppose in the foregoing example that the pile is 13 inches square, its area of section will be 169 square inches ; turning 190'5 tons into Ibs. and dividing by 169 we get 2525 Ibs. per square inch for the maximum load ; taking 5 for the factor of safety when the pile has been driven down to firm ground, we find that it can be safely loaded with 505 Ibs. per square inch ; if however it has failed to reach a firm bottom, we must take 10 as the factor of safety and not load it with more than 253 Ibs. per square inch. The momentum M, with which the ram strikes the pile, is, M = mass x velocity = W X \X^4'4 x h ; which is the force in tons with which a ram weighing W tons strikes the pile. Thus when h = I foot, and W = I ton, then M = 8 tons; or if the ram weighs 12 cwt., then M = 8 x 12 = 96 cwts. If k = 5 feet, M = 18 tons, when the ram is 1 ton ; or M = 219 cwts. when the ram is 12 cwts. If h 10 feet, M = 25'4 tons, when the ram is 1 ton, or M = 305 cwts. when the ram is 12 cwts. The pressure M, which the ram of weight "W pro- duces on-thehead of the pile at the moment of striking, with a fall of h feet, can therefore be approximately calculated by the formula M = W v'64'4 . h z 2 CHAPTER XL EFFECT OF WIND ON BUILDINGS. 113. PRESSURE OF WIND ON A PLANE. The action of wind upon lofty buildings or those erected in exposed situations is of too powerful a character to allow of its being neglected by the architect. Tredgold estimated that the pressure of wind upon roofs amounted to more than all the other forces combined to which they were subjected ; and since the wind acts directly upon one side only of the roof at a time, it produces a far greater strain on its timbers than an equal pressure placed uniformly over both sides would do. It has been found as the result of experiment, that the normal pressure of wind against a plane surface is very nearly proportional to the square of the velocity with which the wind is moving ; arid it has been ascer- tained that wind moving with a velocity of 35 miles an hour, or 51 feet a second, produces a normal pressure of 6 Ibs. on every square foot of a plane surface ; and by assuming that the pressure varies as the square of the velocity, we can calculate the pressure for any given velocity- When the velocity is 2 x 35 or 70 miles an hour, the pressure will be 2 2 x 6 or 24 Ibs. per square foot, and when it is 3 x 35 or 105 miles an hour, the pressure will be 3 2 x 6 or 54 Ibs. If p is the pressure on a square foot due to a velocity v, A PRESSURE OF WIND ON A PLANE. 341 the given area, in feet, we have for the pressure P on the area A, P = p x A. And since the pressure p is 6 Ibs. when v is 35 miles, we have 6 V35 or, p - Also we can find v when p is given, from the equation v = V~204]tf . . (204) The following Table gives the value of p for different values of v. v, in Miles per Hour. 1>, in Ibs. per Foot. r, in Miles per Hour. p, in Ibs. per Foot. 20 1-96 70 24-00 30 4-41 80 31-36 35 6-00 90 39-69 40 7-84 105 54-00 50 12-25 128 80-00 60 17-64 The highest pressure that has been observed is about 80 Ibs. per square foot, but this is a very exceptional amount and is rarely attained in the British Isles. In considering the pressure of the wind upon a vertical wall against which it is blowing horizontally, it maybe assumed to be uniform from top to bottom, so that the resultant pressure will act at the middle of the wall. If h is the height of the wall, the pressure, P, upon 1 foot length of wall will be, p ->:* 342 EFFECT OF WIND ON BUILDINGS. and the moment of this pressure about the outer edge of the wall, will be Putting W for the weight of a wall one foot long and h feet in height, t for its thickness supposed uniform throughout ; n> the weight of a cubic foot of walling ; we have W = re . h . t and its moment about the outer edge, is w .4. = 1 . h . f . fV fW Equating this to twice the moment of P, we have for stability, For example, let jy = 24, w = 144, /* = 36, then by equation (205) t = -^-_ = 2 V 3 = 3-464 ft. V3 If the thickness at the base is double that at the summit, we have from equation (196), putting for the least thickness, *;-*>^8|..* > PRESSURE OF WIND ON A CYLINDER. 343 from which we find, t - 6 - i.fii and the thickness at base is 3-62 feet. Example 2. Let the pressure/* be 80 Ibs. ; then for a wall of uniform thickness and 36 feet high, we have from equation (205) ' = /}x c -^ft. If t is thickness at top and 2 t that at base, we have by equation (206) , _ / 6 80 fi _ f ~ V TT 144 > and the base of the wall is 2 t or 6'6 feet. 114. PRESSURE OF WIND ON A A CYLINDER. To determine the pres- sure of wind on the surface of a cylindrical tower, we must assume that the direction of the pressure is everywhere parallel to one axis of the cylinder. Let ABC (fig. 121) represent a quarter plan of the " fig. 121. tower, C the centre and AB the surface pressed upon by the wind which we will suppose to be acting parallel to BC. Take any two points, D and E, very near together on the arc AB, and let the radius EC = r, making the angle 6 with BC; and call the angle ECD the differential of 6, or d6 ; then the area dA, of a small part of the surface of the tower whose height is %, is rfA = h . r . dd. 344 EFFECT OF WIND ON BUILDINGS. If v is the velocity of wind parallel to BC, then the velocity normal to the surface at E is v . cos. Q, and the pressure on clA. is proportional to ?' . cos. 2 6. Resolving this pressure parallel to BO, we have the pressure in that direction proportional to V"' . COS. 3 0. Consequently, the pressure on the curved surface AB. is to that on a flat surface equal to AB, as r . cos. 3 . dd - r sin. r J cos. 3 . (10 (cos. 2 + 2). Taking the limits of integration from = to = 90, this ratio becomes equal to 2 ; 3, or the pressure on the cylindrical tower is two-thirds of that on a square one of same diameter.* If we put 2 R for the diameter, k for the height, P for the pressure on the surface, P' for that on a square tower, we have P' = p . 2 R . h ...... (207) Let it be required to find the necessary thickness for stability of the walls of a cylindrical tower for a given * Rankine says, ' ' the total pressure of the wind against the side of a cylinder is about one-half of the total pressure against a diametral plane of that cylinder." PRESSURE OF WIND ON A CYLINDER. 345 value of P) t Jbeing the thickness, and r the internal radius ; then t = R r, or, r = R t ; W being the weight of the tower, and m the weight of 1 cubic foot of its material. Then, W = m . TT . (R 2 - r 1 ) . h = TV . -TT (2R - f) t . /t, and the moment of W about the outer edge is . The moment of the pressure is, from equation (207) r *...,. B.*. Equating the moment of W with twice that of P, we have for stability W . E = 2 P A, or, Ap . A = w . 77 (2 R; - f), f-2R.*+ = 0; o TT ?y or, f - 2 R . t + -424 ^..7^ = (208) w For example, let 2 R = 10 ft., P- = J?i = i, ^ = 100 ft. Then equation (208) becomes e - 10*+ 7-1 = 0; = -769 ft. = 9 ins. If E- = J^. = |., the equation (208) becomes TV 144 o 346 EFFECT OF WIND ON BCILDINGS. f - 10 + 16 = ; t = 2 ft. 115. WIND PRESSUKE ON ROOFS. The experiments which have been made to determine the pressure on surfaces inclined at different angles to the direction of the wind have produced rather anomalous results, arising from the use of small planes moving rapidly through the air; but these offer a very different resist- ance to that offered by a large roof completely covered in ; since with a plane moving through the air a partial vacuum is formed behind it which greatly affects the resistance; so that little reliance can be placed on such experiments as far as roofs are concerned. In the case of a roof we have the wind impinging directly on a large surface enclosed all round, so that no wind can get behind it; and we may therefore consider its effect as following the ordinary laws of djoiamics. Let AB (fig. 122) represent the sloping side of a roof inclined at an angle with the horizontal AC, the angle being the "pitch" of the roof. We will suppose the wind to be. blowing horizontally with a Fig. 122. velocity v\ then as before (114), we have the velocity normal to the plane at B represented by the quantity v . sin. 6 and the pressure/* on a square foot of inclined surface WIND PRESSURE ON ROOFS. 347 is proportional to # 2 . sin. 2 0; therefore from equa- tion (203) = ?/.sin.*0 (2Q9) 204 If Q = 30, sin. 2 Q = , and the pressure per square foot en the sloping roof is one-fourth of that on a vertical plane ; if 6 = 60, sin. 2 6 = f , and the pressure per square foot is three-fourths -of that on a vertical plane, or three times as much as when the angle is 30. In the following Table the pressure per square foot has been calculated for roofs of different pitch, and for various velocities of wind. PITCH OF ROOF. 20. 30". 40. 50. CO . 70. Velocity of Wind. Miles per Hour. inlbs. per ft. inlbs. per ft. infbs. per ft. inVbs. per ft. iJb, per ft, inlbs. per ft. 20 23 49 81 115 1-47 1-73 30 52 1-10 1-82 2-59 3-31 3-89 35 70 1-50 2-48 3-52 4-50 5-30 40 92 1-96 3-24 4-60 5-88 6-92 50 1-43 3-06 5-06 7-19 9-19 10-82 60 2-06 4-41 7-29 10-35 13-23 15-58 70 2-81 6-00 9-S8 14-08 18-00 21-19 80 3-67 7-84 12-96 18-40 23-52 27-70 90 4-64 9-92 16-40 23-29 29-76 35-05 105 6-32 13-50 22-31 31-69 40-50 47-68 128 9'36 20-00 33-06 46-94 60-00 70-64 We will now show how the stress on the timbers of a king-post roof can be calculated, which is caused by the wind blowing upon one side of the roof. In calcu- lating the stress upon a roof (68) we included the pressure of wind as part of the dead load of 66 Ibs. per foot superficial, as adopted by Tredgokl. A more 348 EFFECT OF WIND BUILDINGS. accurate method, however, is to take the dead load of covering, timbers and snow, separately from the pressure of wind, and form the stress diagram as before (68) ; and then to take the pressure of the wind as a dead load on one side only of the roof. Let ABC (fig. 123) represent the truss of a king- post roof in which the rafters AC and AB make the angle with the tie-beam BC. Suppose the wind to be blowing on the side AC, and P to be the total pressure resolved perpendicularly to AC. Then if A is the area of roof supported by AC, we have where I is the length of rafter, and b the breadth between two trusses. Draw AK and EL perpendicular to AC, and cutting BC in K and L. We wiil consider that half the pressure, P, is borne at E and one-fourth at each point A and C. The pressure P at A produces a reaction E : at B, and a reaction E 2 at C; taking moments of E : and -- about C, and of E., and B, we have P about WIND PRESSURE OX ROOFS. 349 The pressure J- P at E produces a reaction R 3 at B, and a reaction R 4 at C ; taking moments of R 3 and \ P p about C, and of R A and t> about B, we have ^ttP P vlUnr-TJ P V BL P 4COS. 2 0-1 4 xB =^xBL,or,E t =--x B - = -. .-, Q The total reaction R' at B is The total reaction R at C is P 4 cos. 2 ^ - 4 cos. 2 e For example, Iet0 = 30, then cos. 2 <9 = 3 , R' = 1 . P We now proceed to draw the stress diagram, as before (68), by first taking a line Ih (fig. 124) parallel to the direction of the pressure, P; that is, perpendicular to AC (fig. 123). Take ab on this line to represent 350 EFFECT OF WIND ON BUILDINGS. i P, or the pressure at C, and be to represent R the reaction at C. Draw cd horizontally at c meeting ad drawn at right angles to bac. Take ae to represent | P, the pressure at E, and draw ef parallel to AC and at right angles to ae, meeting df drawn parallel to EF. Draw a vertical line jfg meeting cd in y, and draw gh parallel to AB. Then be represents the reaction R at C, he the reaction R' at B. For the forces in equilibrium at C, we have ab representing \ P, be representing R, cd acting from the centre C, representing the tension in CF, da acting towards C, and representing the compression in EC. For the forces in equilibrium at E, we have ea repre- senting | P, ad acting towards E, the compression in CE ; df acting towards E, the compression in FE ; fe- acting towards E the compression in EA. For the forces at A, we have he the force P, ef the pressure in E A, y// acting y)w% A the tension in AF ? gh acting towards A the compression in AD. For the forces at B, ch is the reaction R' ; hg acting towards B the compression in BA ; gc acting from B> the tension in BF. The strong lines in fig. 124 show the parts in com- pression and the dotted lines those in tension. We WIND PRESSURE ON ROOFS. 351 have, by calculation, when = 30, the following values of the different forces represented by the lines in the stress diagram (fig. 124) : ab = IP, ae = 2 E-, ac = | . -?-, he = -J, he - 7 4 P ,/.. 7 4 V 3 P / 1 > ^^ J l"./~> >M - 1-fO * 30 r ~ 4 r63 x 7.51-03 -Q.Q x 26-7 ,73-5 = 42 256 ' therefore, d s ' 5 = ~ x 256 = 457. By the Table (56) we find that (5'75) 3 ' 5 = 455-9, so that we may take of- inches as the required diameter of a solid iron column sufficiently strong to carry this load in safety. Now suppose the column to be hollow and 10 inches, in external diameter, required to find d^ its internal diameter, or the necessary thickness of metal. Using the equation (83) we have MISCELLANEOUS EXAMPLES. 357 "I A3'5 ,1 .T5 O1 /2O -7 *t x 8 or 22| cwts. 9. What weight at the middle will break a beam of Riga fir, 20 feet bearing, 12" deep and 9" wide? From equation (42) we have From the Table (32) we have S = 6 x 775 Ibs. for breaking stress ; therefore, TV = 3100 x 9 X 12 * = 16,740 Ibs. = 7-47 tons. 12 x 20 The safe-load will be one-sixth of this. 10. A beam AB having a span of 32 feet, is supported at A and B, and loaded uniformly through- out a part of its length, AC, equal to 24 feet, with 1 cwt. per foot run. It is required to find the reactions, TV! and W 2 , of the supports at A and B. Let TV be the load on AC, equal to 24 x 1 j or 30 cwts. which acts at a point halfway between A and C or 12 feet from A and 20 feet from B. Taking moments of the forces TV t and TV about B, we have TVj x 32 = TV x 20 - 30 x 20; therefore, \\\ = 80 ^ 20 = 18f cwts. MISCELLANEOUS EXAMPLES. 3G1 Again, taking moments of W 2 and W about A, we have W 2 x 32 = W x 12 = 30 x 12; therefore, W 2 = 3Q * 12 = 11 cwts. OrW 11. A girder AB having a bearing of 50 feet 6 inches is supported at A and B, its weight, w, being -1-th ton per foot run. It has also a load, W, of 64'8 tons distributed over a length, BC, of 23 feet. Required to find the point in the beam where the stress is greatest. The greatest stress is at the point where the Re- sultant (2) of all the forces acts on the beam. Now the resultant of the load W = 64-8 tons distributed over BC acts at a point half-way between B and C, or at 11 feet 6 inches from B. The resultant of the weight w = 1O1 tons acts at the centre of the beam. We have then to find the position of the resultant of the two forces W and w } whose distance from B we call x. Taking moments of the forces about B, we have = W x 11-5 + w x 25-25; therefore, _ 64-8 x 11-5 + 10-1 X 25-85 74-9 12. A beam is supported at A and B and has a span of 18 feet ; it is loaded at C, 5 feet from A, with 3 tons, and at D, 7 feet from B, with 2 tons. It is required to 362 MISCELLANEOUS EXAMPLES. find the reactions P and Q at A and B, or the propor- tion of load which is carried at each point. Taking moments about B of all the forces, we have P x AB = 3 x BC + 2 x BD = 3x13 + 2x7 = 53 Therefore, P = 53 tons. Taking moments about A, we have Q x BA = 2 x AD + 3 x AC = 2x11 + 3x5 = 37 Therefore, Q = ?! tons - lo P + Q = ^ = 5 tons, lo 13. A horizontal beam, AB (fig. 125) of which C is the centre, is supported f . ^ 6 at each end and loaded A | E | F B uniformly over its middle D O D portion EF only, the parts AE and BF being unloaded; to find the moment of stress at the middle point C. Let m be the weight per foot of load on EF, AB = I, BF = AE = a-, M the moment of stress at C. First, suppose the beam to be loaded uniformly throughout its length with w per foot, and M' the moment of stress at C. Let Mj represent the moment MISCELLANEOUS EXAMPLES. 363 of stress at C produced by the load on AE or BF, the resultant of which acts at D, where AD or BD = f . By equation (25) we have ir -}*.*, The load on AE or BF is m . r, and by equation (19) Therefore, M = M' - 2 M x = rcl x - - 4 m . x x - x ~ 8 8 For example, let x = - ; w = I ton, I = 20 feet, x = 5 ft., j = |. Then we have M =|( a0 - 2 0xl) = | Also, M' = 20 x 2| = 50 = -J So that, M ; M' = 3 : 4 Therefore M = f M', or the moment of stress at C with the load wper foot over the middle half only, is three- fourths of the moment of stress produced by w per foot over the whole length of the beam. It will be seen that the value of M x is independent of the length / of the beam, and depends only on the length x of AE and BF. In the above example, M t = 364 MISCELLANEOUS EXAMPLES. G'25, as the value of the moment of stress about C of the load on AE or BF. Let us apply the above equations to determine the safe-load per foot in the beam of Riga fir described at (33), where I = 15 ft. = 180 inches, b = G", d = 12". Here the moment of resistance is 4 S . . d* = 6200 x 144 = 892800, when S = 775 for safe-load. Let TV' be the load per foot when the beam is loaded uniformly throughout its entire length ; then we have M' = w . V = 892800, 892800 x 8 180* = 221 Ibs. Let n\ be the load per foot when the beam is loaded only on AE and BF ; then if x = l } we have 2 M! = | n\ . ^ = 892800, \ - 452 Let re be the load per foot when the beam is loaded only on EF ; then M = i w . I (l -4x* } = 892800. \ v _ 892800 x 8 _ OT ' 1 x 180* INDEX. ABUTMENTS, 187 Angle-iron, 69 Angle of repose, 318 Arcades, 205 Arch, centre of gravity of, 38 elliptical, 231 iron, 246 parabolic, 221 pointed, 210 principle of, 174 segmented, 207 semi-circular, 184 Tudor, 221 BEAMS, CAST IRON, 64 deflexion of, 79 flanged, 62 rectangular, 52 steel, 74 - strength of, 52 wrought iron, 67 of uniform strength, 75 Bowstring truss, 164 Bracket, 10 Buttresses, 296 earth, 330 flying, 300 CAST IRON BEAMS, 64 pillars, 98 Cast iron, strength of, 109 Centre of gravity, 31 pressure, 312 Circular arch, centre of gravity of, 38 stability of, 187 Coefficient of strength, 45 safety, 47 Collar roof, 120 Conical dome, 290 Constantinople, Sta. Sophia at, 267 Couples, 20 Crushing strength, 46 Curvature, radius of, 80 DEFLEXION OF BEAMS, 79 Domes, conical, 290 Gothic, 280 hemispherical, 255 semi, 264 Dome of Sta. Sophia, Constanti- nople, 267 unequal thickness, 275 EARTH BUTTRESS, 330 pressure of, 318 Elasticity, modulus of, 41 Elliptic arch, 231 Equilibrium, 4 366 INDEX. FlR JOISTS, SCANTLING OF, strength of, 56 Flanged beam, 93 Fluid pressure, 310 Flying buttress, 300 Forces, measure of, 1 polygon of, 12 resultant of, 3 triangle of, 9 Foundations, 333 Friction, 176 GIRDERS, STRENGTH OF, 71 Gordon's formula, 109 Gothic arch, 210 dome, 280 vaulting, 243 Gravity, centre of. 31 HAMMER-BEAM ROOF, 127 Hemispherical dome, 255 Hodgkinson's cast-iron beams, 67 pillars, 100 INCLINED PLAKE, 175 Inertia, moment of, 51 Iron beams, 62 pillars, 109 roofs, 151 strength of, 46 JOINT OF RUPTURE OF ARCHES, 184 Joists, scantling of, 89 KING-POST HOOF, 135 LATTICE BEAM, 171 Xean-to roof, 62 Lever, principle of, 18 Line of pressures, 202 Loaded arch, 190 Loads on beams, 48 MEASURE OF FORCE, 1 Modulus of elasticity, 41 Moment of a force, 16 inertia, 51 NATURAL SLOPE OF EARTH, 318 Neutral axis in beams, 49 OCTAGONAL SPIRE, 293 PARABOLIC ARCH, 221 Parallelogram of forces, 4 Piling, 335 Pillars, strength of, 98 Pitch of a roof, 112 Pointed arch, 210 Polygon of forces, 12 stress, 14 Polygonal frame, 13 Pressure, centre of, 312 of earth, 318 of water, 310 QUEEN-POST ROOF, 147 RADIUS OF CURVATURE, 80 Raking shore, 304 Rectangular beam, 52 Resistance of materials, 46 Resolution of forces, 4 Resultant of forces, 3 Retaining walls, 310 Roofs, 112 Rupture, joint of, 184 INDEX. 367 STA. SOPHIA, DOME OF, 267 Safe load ou a Learn, 55 Scantling of joists, 89 roof timbers, 151 Segmented arch, 207 Semi-domes, 264 Shearing, 77 Shoring, 304 Span-roof, 117 Spire, 293 Stability of arches, 187 domes, 258 Stancheons, 110 Steel beams, 74 Steel, strength of, 46 Strength of beams, 52 pillars, 109 Stress, 3 diagram, 143 Surcharged arch, 215 dome, 271 'TENSILE STRENGTH, 43 Thrust of arches, 184 domes, 260 Thrust of roofs, 117 Tie-beam, 135 Transposition of couples, 21 Transverse strength of beams, 56 stress, 48 Trapezium, centre of gravity of, 35 Triangle of forces, 9 Triangular frame, 11 Tredgold's rule, 86 Trussed roofs, 131 Tudor arch, 221 VAULTING, 237 Velocity of wind, 341 pile driver, 337 Voussoirs, 174 WATER, PRESSURE OF, 310 Warren girder, 168 Wedge, 178 Wind, pressure of, 340 velocity of, 341 Wrought iron beams, 67 pillars, 104 BRADBURY, AONEW, & CO. LD., PRINTERS, WHITEFRIARS. LIST OF USEFUL BOOKS FOE BUILDERS, CONTRACTORS, DECORATORS, &e. LOCKWOOD'S BUILDER'S PRICE-BOOK: a Comprehen- sive Handbook of the Latest Prices of every kind of Material and Labour f Trades connected with Building ; including a great variety of useful In- formation in all matters concerning these Trades. With many useful Memoranda and Tables. New Edition, Re-written, Re-modelled, and greatly Enlarged. Edited by FRANCIS T. W. MILLER, A.R.I.B.A. 700 pages, Crown Svo, 4s. cloth. ARCHITECTURAL PERSPECTIVE : The Whole Course and Operations of the Draughtsman in Drawing a Large House in Linear Perspective. Illustrated by numerous Diagrams. By F. O. FERGUSON. Demy Svo, 3s. M. boards. THE HANDBOOK OF SPECIFICATIONS. By Professor T. L. DONALDSON. Svo, 1 11s. W. cloth. SPECIFICATIONS FOR PRACTICAL ARCHITEC- TURE. By F. ROGERS, Architect. Svo, 15s. cloth. THE HOUSE-OWNER'S ESTIMATOR; or, What will it Cost to Build, Alter, or Repair? By JAMES D. SIMON and FRANCIS T. W. MILLER. Crown Svo, 3s. (x?. cloth. THE POCKET TECHNICAL GUIDE, MEASURER, AND ESTIMATOR FOR BUILDERS AND SURVEYORS. By A. C. BEATON. Is. W. QUANTITIES AND MEASUREMENTS, How to Calculate and Take. By A. C. BEATON. Is. 6(i. cloth. PLUMBING : a Text-book to the Practice of the Art or Craft of the Plumber. By W. P. BUCHAN. Sixth Edition, Enlarged. With 330 Illustrations. 4s. cloth. VENTILATION": a Text-book to the Practice of the Art of Ventilating Buildings. By W. P. BUCHAN. With 170 Illustrations. 4s. cloth. SANITARY WORK IN THE SMALLER TOWNS AND IN VILLAGES. By CHARLES SLAGG, A.M. Inst. C.E. 12mo, 3s. 6d. cloth. HOUSE-PAINTING, GRAINING, MARBLING, AND SIGN-WRITING. By ELLIS A. DAVIDSON. Fifth Edition. Os. cloth. A GRAMMAR OF COLOURING, applied to DECORA- TIVE PAINTING AND THE ARTS. By GEORGE FIELD. Revised by E. A. DAVIDSON. 3s. Gd. cloth. SCHOOL OF PAINTING FOR THE IMITATION OF WOODS AND MARBLES. By A. R. VAN DER BURG and P. VAN DER BURG. Royal folio, 18J by 12 in. Illustrated with 24 full-size Coloured Plates. Also 12 Plain Plates, comprising 154 Figures. 1 11s. 6<7. ELEMENTARY DECORATION, as applied to the Interior and Exterior Decoration of Dwelling Houses, &c. By JAMES WILLIAM FACEY. 2s. cloth. PRACTICAL HOUSE DECORATION. By JAMES WILLIAM FACET. 2s. M. cloth. LONDON : CROSBY LOCKWOOD shelf of all practical engineers. "I 2 CROSBY LOCK WOOD < SOWS CATALOGUE. Handbook for Works' Managers. THE WORKS' MANAGER'S HANDBOOK OF MODERN RULES, TABLES, AND DATA. For Engineers, Millwrights, and Boiler Makers; Tool Makers, Machinists, and Metal Workers; Iron and Brass Founders, &c. By W. S. HUTTON, C.E., Author of " The Practical Engineer's Handbook." Fourth Edition, carefully Revised, and partly Re-written. In One handsome Volume, medium 8vo, 15$. strongly bound. [Just published. tS" The Author having compiled Rules and Data for his own use in a great variety of modern engineering work, and having found his notes extremely useful, decided to publish them revised to date believing that a practical work, suited to the DAILY REQUIREMENTS OF MODERN ENGINEERS, would be fav ourably rece ive d. In the Third Edition, the following among other additions have been made, viz.: Rules for the Proportions of Riveted Joints in Soft Steel Plates, the Results of Experi- ments by PROFESSOR KENNEDY for the Institution of Mechanical Engineers Rules for the Proportions of Turbines Rules for the Strength of Hollow Shafts of Whit- worth's Compressed Steel, &c. ** OPINIONS OF THE PRESS. " The author treats every subject from the point of view of one who has collected workshop notes for application in workshop practice, rather than from the theoretical or literary aspect. The volume contains a great deal of that kind of information which is grained only by practical experi- ence, and is seldom written in books."--^r. "The volume is an exceedingly useful one, brimful with engineers' notes, memoranda, 'and rules, and well worthy of being on every mechanical engineer's bookshelf." Mechanical World. " The information is precisely that likely to be required in practice. . . . The work forms a desirable addition to the library not only of the works manager, but of anyone connected with engineering." Mining Jour 'ormidable mass of facts and figures, rei . Such a volume will be found absolutely necessary as a book of reference in all sorts A formidable mass of facts and figures, readily accessible through an elaborate index of 'works 'connected with the metal trades." Ryland 's iron Trades Circular. " Brimful of useful information, stated in a concise form, Mr. Hutton's books have met a press- Ing want among engineers. The book must prove extremely useful to every practical man possessing a copy." Practical Engineer. Practical Treatise on Modern Steam-Boilers. STEAM-BOILER CONSTRUCTION. A Practical Handbook for Engineers, Boiler-Makers, and Steam Users. Containing a large Col- lection of Rules and Data relating to the Design, Construction, and Working- of Modern Stationary, Locomotive, and Marme Steam-Boilers. By WALTER S. HUTTON, C.E., Author of "The Works' Manager's Handbook," &r, With upwards of 300 Illustrations. Medium 8vo, i8s. cloth. \_Justpublished. "Every detail, both in boiler design and management, is clearly laid before the reader. The volume shows that boiler construction has been reduced to the condition of one of the most exact sciences; and such a book is of the utmost value to the fin de siecle Engineer and Work-/ Manager." Marine Engineer. " There has long been room for a modern handbook on steam boilers ; there is not that room now, because Mr. Mutton has filled it. It is a thoroughly practical book for those who are occu- pied in the construction, design, se'ection, or use of boilers." Engineer. "The Modernised Templeton." THE PRACTICAL MECHANIC'S WORKSHOP COM- PA NION. Comprising a great variety of the most useful Rules and Formulae in Mechanical Science, with numerous Tables of Practical Data and Calcu- lated Results for Facilitating Mechanical Operations. By WILLIAM TEMPLE- TON, Author of "The Engineer's Practical Assistant," &c. &c. Sixteenth Edition, Revised, Modernised, and considerably Enlarged by WALTER S. HUTTON, C.E., Author of "The Works' Manager's Handbook," "The Practical Engineer's Handbook," &c. Fcap. 8vo, nearly 500 pp., with Eight Plates and upwards of 250 Illustrative Diagrams, 6s., strongly bound for workshop or pocket wear and tear. [Just published. OPINIONS OF THE PRESS. valuab In Its modernised form Hutton's ' Templeton ' should hav 3le information which the mechanic will often find of u he might look for in vain in other works. This modernised edition will be appreciated by all who have learned to value the original editions of ' Templeton.' "Knglish Mechanic. *' It has met with great success in the engineering workshop, as we can testify ; and there are a great many men who, in a great measure, owe their rise in life to this little book." Building News. This familiar text-book-well known to all mechanics and engineers-is of essential service to the every-day requirements of engineers, millwrights, and the various trades connected with engineering and building. The new modernised edition is worth its weight in gold."- Builditif "'"Vhis'well-known and largely used book contains information, brought up to date, of the sort so useful to the foreman and draughtsman. So much fresh information has been introdu as to constitute M:chanical W MECHANICAL ENGINEERING, etc. Stone-working Machinery. STONE-WORKING MACHINERY, and the Rapid and Economi. cat Conversion of Stone. With Hints on the Arrangement and Management of Stone Works. By M. Powis BALE, M.I.M.E. With Illusts. Crown b vo 7 "Should be in the hands of every mason or student of stone- work. "-Coltery Guardian Pindb 0k ' r *" Wh manipuUte stone for building or ornamental purposes." Pump Construction and Management. PUMPS AND PUMPING : A Hani' Notes on Selection, M.I.M.E., Author of " The matter is set forth as concisely as possible. In fact, condensation rather than diffusere ;t>*vn M//M. jaLwntiyememfm PUMPING : A Handbook for 'Pump Users. Being i, Construction and Management. By M. Powis BALE" >f " Woodworking Machinery," &c. Crown 8vo, 2S. 6d. Milling Machines, etc. MILLING : A Treatise on Machines, Appliances, and Processes em- ployed in the Shaping of Metals by Rotary Cutters, including Information on Making and Grinding the Cutters. By PAUL N. HASLUCK, Author of" Lathe- work." With upwards of 300 Engravings. Large crown 8vo, 125. 6d. cloth. Turning. V" st Plushes. LATHE-WORK : A Practical Treatise on the Tools, Appliances, and Processes employed in the Art of Turning. By PAUL N. HASLUCK.. Fourth Edition, Revised and Enlarged. Cr. 8vo, 5$. cloth. "Written by a man who knows, not only how work ought to be done, but who also knows how to do it, and how to convey his knowledge to others. To all turners this book would be valuable. " Screw-Cutting. SCREW THREADS : And Methods of Producing Them. With Numerous Tables, and complete directions for using Screw-Cutting Lathes. By PAUL N. HASLUCK, Author of " Lathe- Work," &c. With Fifty Illustra- tions. Third Edition, Enlarged. Waistcoat-pocket size, is. 6d. cloth. " Full of useful information, hints and practical criticism. Taps, dies and screwing-tools gene- rallyare illustrated and their action described."- Mechanical World. " It is a complete compendium of all the details of the screw cutting lathe ; in fact a tnulltim-- Smith's Tables for Mechanics, etc. t TABLES, MEMORANDA, AMD CALCULATED RESULTS', FOR MECHANICS, ENGINEERSr\ARCHITECTS, GUILDERS, etc. Selected and Arrariged by FRANCIS Smirk Fifth Edition, thoroughly Revised and Enlarged, witlV^LNew Section of ELECTRICAL TABLES, F\RMUL,E, and MEMORANDA. Waistqpat- pocket size, is. 6d\ limp leather. [Jmt published, " It would, perhaps, be as dWSf ult to make a small poc\et-book selection of notes ind formulae to suit ALL engineers as it would ke to make a universal rtl^dicine ; but Mr. SmithV^ " pocket collection may be looked upcrtras a successful attempt. '^Engineer. "The best example we have ever Veen of 250 pages of useful\matter packed into the dimen- sions of a card-case." Building News. "A veritable pocket treasury of knowledge." Iron. Engineer's and Machinist's Assistant. THE ENGINEER'S, MILLWRIGHT'S, and MACHINIST'S PRACTICAL ASSISTANT. A collection of UsefulTables, Rules and Data. By WILLIAM TEMPLETON. 7th Edition, with Additions. i8mo, zs. 6d. clotb. " Occupies a foremost place among booksofthis kind. A more suitable present to an appren- tice to any of the mechanical trades could not possibly be made." Building News. "A deservedly popular.work.it should be in the 'drawer' of every mechanic." English Mechanic. Iron ana Steel. " IRON AND STEEL " : A Work for the Forge, Foundry, Factory, and Office. Containing ready, useful, and trustworthy Information for Iron- masters ; Managers of Bar, Rail, Plate, and Sheet Rolling Mills ; Iron and Metal Founders ; Iron Ship and Bridge Builders ; Mecnanical, Mining, and Consulting Engineers ; Contractors, Builders, &c. By CHARLES HOARK. Eighth Edition, Revised and considerably Enlarged. 32mo, 6s. leather. "One of the best of the pocket books." English Mechanic. pocket books." English end this boo " Naval Science. . dially recommend this book to those engaged in considering the details of all kinds of l works." 4 CROSBY LOCK WOOD & SON'S CATALOGUE. Engineering Construction. PATTERN -MAKING : A Practical Treatise, embracing the Main Types of Engineering Construction, and including Gearing, both Hand and Machine made, Engine Work, Sheaves and Pulleys, Pipes and Columns, Screws, Machine Parts, Pumps and Cocks, the Moulding of Patterns in Loam and Greensand, &c., together with the methods of Estimating the weight of Castings; to which is added an Appendix of Tables for Workshop Reference. By a FOREMAN PATTERN MAKER. With upwards of Ihree Hundred and Seventy Illustrations. Crown Svo, 75. 6d. cloth. " A well-written technical guide, evidently written by a man who understands and has prac- tised what he has written about. . . . We cordially recommend it to engineering students, ycui'g journeymen, and others desirous of being initiated into the mysteries of pattern-making." Builder. " We can confidently recommend this comprehensive treatise.' Building- A'nvs. " Likely to prove a welcome guide to many workmen, especially to draughtsmen who have lacked a training in the shops, pupils pursuing their practical studies m our factories, and to em- ployers and managers in engineering works. Hard-ware Trade Journal. "More than 370 illustrations help to explain the text, which is, however, always clear and ex- plicit, thus rendering the work an excellent trade mecum for the apprentice who desires to become master of his trade." English Mechanic, Dictionary of Mechanical Engineering Terms. LOCKWOOD1S DICTIONARY OF TERMS USED IN THE PRACTICE OF fyECHANICAL ENGINEERING, embracing those current in the Drawing Office, Pattern Shop, Foundry, Fitting, Turning, Smith's and Boiler Shops, &c. &c\ Comprising upwards of 6,000 Definitions. Edited by A FOREMAN PATTERN-MAKER, Author of " Pattern Making." Crown 8vo, 7$. 6d. cloth. \k "Just the sort of handy dictHnary required by the various trades engaged in mechanical en- gineering. The practical engineermg pupil will find thevbook of great value in his studies, and every foreman engineer and mechahic should have a copy^Buildinf Xtics. "Alter a careful examination ofV^he book, and trying alMnanner of words, we think that the engineer will here find all he is likely-ha require. It will be lateely used." Practical Engineer. "One of the most useful books which can be presented toV mechanic or student." English Mechanic. \ " Not merely a dictionary, but, to a ce\tain extent, also a mo* valuable guide. It a happy idea to combine with a definition df the phra it treats." Machinery Market. " Nc word having connection with anV branch of constructW engineering seems to be omitted. No more comprehensive work hasbeen, so far, issued. Knowledge. "We strongly commend this useful and reliable adviser to our friends in the workshop, and ( students everywhere." Colliery Guardian. \ Steam Boilers. A TREATISE ON S\TEAM BOILERS: Their Strength, Con- struction, and Economical WUfking. By ROBERT"WILSON, C.E. Fifth Edition, izmo, 6s. cloth. "The best treatise that has ever beeHjDublished on steam bonVs." Engineer. "The author shows himself perfect maker of his subject, and *ie heartily recommend all em- ploying steam power to possess themselves o{ the work." Ry land's iron Trade Circular. Boiler Chimneys. BOILER AND FACTORY CHIMNEYS; Their Draught-Power and Stability. With a [Chapter on ^Lightning Conductors. By ROBERT WILSON, A. T.C.E., Authdvpf "A Treatise on Steam Boilers," &c. Second Edition. Crown 8vo, 3$. 64, cloth. \ " Full of useful information, definite in statement, Vid thoroughly practical in treatment. The Local Gmemmcnt Chronicle. \ "A valuable contribution to the literature of scientific building." The Builder. Boiler Making. THE BOILER-MAKER'S READY RECKONER &> ASSIST- A NT. With Examples of Practical Geometry and Templating, for the Use of Platers, Smiths and Riveters. By JOHN COURTNEY, Edited by D. K. CLARF, M.I.C.E. Third Edition, 480 pp., with i4olllusts. Fcap. Svo, 7$. half-bound. " No workman or apprentice should be without this book." Iron Trade Circular. " Boiler-makers will readily recognise the value of this volume. . . . The tables are clearly Dinted, and so arranged that they can be referred to with the greatest facility, so that it cannot be doubted that they wifl be generally appreciated and much used." Mining- Journal. Warming. HEATING BY HOT WATER; with Information and Sug- gestions on the best Methods of Heating Public, Private and Horticultural Buildings. By WALTER JONES. With Illu.'trations, crown Svo, 25. cloth. We confidently recommend all interested in heating by hot water to secure a copy of this valuable little treatise. "-The Plumber and Decorator. MECHANICAL ENGINEERING, etc. Steam Engine. TEXT-BOOK ON THE STEAM ENGINE. With a Sup- plement on Gas Engines, and PART II. ON HEAT ENGINES. By T. M. GOODEVE, M.A., Barrister-at-Law, Professor ot Mechanics at the Normal School of Science and the Royal School of Mines; Author of "The Princi- ples of Mechanics," "The Elements ot Mechanism," &c. Eleventh Edition, Enlarged. With numerous Illustrations. Crown 8vo, 6s. cloth. . "Professor Goodeve has given us a treatise on the steam engine which will bear comparison with anything written by Huxley or Maxwell, and we can award it no higher praise." Enfinttr. " Mr. Goodeve 's text-book is a work of which every young engineer should possess himself." Gas Engines. ON GAS-ENGINES. Being a Reprint, with some Additions, of the Supplement to the Text-book on the Steam Engine, by T. M. GOODEVE, M.A. Crown 8vo, zs. 6d. cloth. Mr. Go : little v Steam. THE SAFE USE OF STEAM. Containing Rules for Un- professional Steam-users. By an ENGINEER. Sixth Edition. Sewed, 6d. ' If steam-users would but learn this little book by heart boiler explosions would become sensations by their rarity." English Mechanic. Reference Book for Mechanical Engineers. THE MECHANICAL ENGINEER'S REFERENCE BOOK, for Machine and Boiler Construction. In Two Paris. Part I. GENERAL ENGINEERING DATA. Part II. BOILER CONSTRUCTION. With 51 Plates and numerous Illustrations. By NELSON FOLEY, M.I.N.A. Folio, 5 5.'. half- bound. [Just published. Coal and Speed Tables. A POCKET BOOK OF COAL AND SPEED TABLES, for Engineers and Steam-users. By NELSON FOLEY, Author of " Boiler Con- struction." Pocket-size, 35. 6d. cloth ; 4$. leather. "These tables are designed to meet the requirements of every-day use j and maybe com- mended to engineers and users of steam."- Iron. " This pocket-book well merits the attention of the practical engineer. Mr. Foley has com- piled a very useful set of tables, the information contained in which is frequently required by engineers, coal consumers and users of steam." Iron and Coal Trades Xevinu. Fire Engineering. FIRES, FIRE-ENGINES, AND FIRE-BRIGADES. With a History of Fire-Engines, their Construction, Use, and Management; Re- marks on Fire-Proof Buildings, and the Preservation of Life from Fire ; Foreign Fire Systems, &c. By C. F. T. YOUNG, C E. With numerous Illustrations, 544 pp., demy 8vo, i 45. cloth. " To such of our readers as are interested in the subject of fires and fire apparatus, we can most heartily commend this book." Bngijucring. " It displays much evidence of careful together. It is evident enough that his acquaintance steam fire engines is accurate and luU." Engineer. Estimating for Engineering Worlc, C.E. Crown 8vo, zs. 67. cloth. 6 CROSBY LOCKWOOD & SON '5 CATALOGUE. THE POPULAR WORKS OF MICHAEL REYNOLDS (" THE ENGINE DRIVER'S FRIEND "). Locomotive-Engine Driving. LOCOMOTIVE-ENGINE DRIVING : A Practical Manual for Engineers in charge of Locomotive Engines. By MICHAEL REYNOLDS, Member of the Society of Engineers, formerly Locomotive Inspector L. B.and S.C. R. Eighth Edition. Including a KEY TO THE LOCOMOTIVE ENGINE. With Illus- trations and Portrait of Author. Crown 8vo. 45. 6rf. cloth. "Mr. Reynolds has supplied a want, and has supplied it well. We can confidently i the book, not only to the practical driver, but to everyone who takes an interest in the performance of locomotive engines." The Engineer. " Mr. Reynolds has opened a new chapter in the literature of the day. This admirable practical treatise, of the practical utility of which we have to speak in terms of warm commendation." Athentztim. " Evidently the work of one who knows his subject thoroughly."- Rail-way Service Gazette. "Were the cautions and rules given in the book to become part of the every-day working of o ir engine-drivers, we might have fewer distressing accidents to deplore." Scotsman. Stationary Engine Driving. STATIONARY ENGINE DRIVING : A Practical Manual for Engineers in charge of Stationary Engines. By MICHAEL REYNOLDS. Fourth Edition, Enlarged. With Plates and Woodcuts. Crown 8vo, 45. 6d. cloth. "The author is thoroughly acquainted with his subjects, and his advice on the various points treated is clear and practical. ... He has produced a manual which is an exceedingly useful one for the class for whom it is specially intended." Engineering; " Our author leaves no stone unturned. He is determined that his readers shall not only know something about the stationary engine, but all about it." Engineer. "An engineman who has mastered the contents of Mr.Reynolds's bookwill require but little actual experience with boilers and engines before he can be trusted to look after them."ErigJiskMecha>iit. The Engineer t Fireman, and Engine-Boy. THE MODEL LOCOMOTIVE ENGINEER, FIREMAN, and ENGINE-BOY. Comprising a Historical Notice of the Pioneer Locomotive Engines and their Inventors. By MICHAEL REYNOLDS. With numerous Illus- trations and a fine Portrait of George Stephenson. Crown 8vo, 45. 64. cloth. "From the technical knowledge of the author it will appeal to the railway man of to-day more forcibly than anything written by Dr. Smiles. ... The volume contains information of a tech- nical kind, and facts that every driver should be familiar with." -lin^li.h .Mechanic. "We should be glad to see this book in the possession of everyone in the kingdom who has ever laid, or is to lay, hands on a locomotive engine." Iron. Continuous Hallway Brakes. CONTINUOUS RAILWAY BRAKES: A Practical Treatise on the several Systems in Use in the United Kingdom ; their Construction and Performance. With copious Illustrations and numerous Tables. By MICHAEL REYNOLDS. Large crown 8vo, gs. cloth. " A popular explanation of the different brakes. It will be of great assistance in forming public opinion, and will be studied with benefit by those who take an interest in the brake." English Mechanic. "Written with sufficient technical detail to enable the principle and relative connection ot the various parts of each particular brake to be readily grasped." Mechanical World. Engine-Driving Life. ENGINE-DRIVING LIFE : Stirring Adventures and Incidents in the Lives of Locomotive-Engine Drivers. By MICHAEL REYNOLDS. Second Edition, with Additional Chapters. Crown 8vo. 2s. cloth. "From first to last perfectly fascinating. Wilkie Collins's most thrilling conceptions are thrown Into the shade by true incidents, endless in their variety, related in every page." North British Mail. "Anyone who wishes to get a real insight into railway life cannot do better than read ' Engine- Driving Life' for himself ; and if he once take it up he will find that the author's enthusiasm and real love of the engine-driving profession will carry him on till he has read every page." Saturday Review. Pocket Companion for Enginemen. THE ENGINEMAN'S POCKET COMPANION AND PR AC- TICAL EDUCATOR FOR ENGINEMEN, BOILER ATTENDANTS, AND MECHANICS. By MICHAEL REYNOLDS. With Forty-five Illustra- tions and numerous Diagrams. Second Edition, Revised. Royal i8mo, 35. t>d., strongly bound for pocket wear. "This admirable work is well suited to accomplish its object, being the honest workmanship of a competent engineer." Glasgow Herald: " A most meritorious work, giving in a succinct and practical form all the information an engine- minder desirous of mastering the scientific principles of his daily calling would require." Miller " A boon to those who are striving to become efficient mechanics." Daily Chronicle. CIVIL ENGINEERING, SURVEYING, etc. 7 French-English Glossary for Engineers, etc. A POCKET GLOSSARY of TECHNICAL TERMS ENGLISH- FRENCH, FRENCH-ENGLISH ; with Tables saitable for the Architectural, Engineering, Manufacturing and Nautical Professions. By JOHN JAMES FLETCHER, Engineer and Surveyor, zco pp. Waistcoat-pocket size, is. 6d., limp leather. " It ought certainly to be in the waistcoat-pocket of every professional man." Iron. " It is a very great advantage for readers and correspondents in France and England to have to large a number of the words relating to engineering and manufacturers collected In a liliputian volume. The little book will be useful both to students and travellers.' Architect. " The glossary of terms is very complete, and many of the tables are new and well arranged. We cordially commend the book.' Mechanical World Portable Engines. THE PORTABLE ENGINE; ITS CONSTRUCTION AND MANAGEMENT. A Practical Manual for Owners and Users of Steam Engines generally. By WILLIAM DYSON WANSBROUGH. With 90 Illustra- tions. Crown 8vo, 35. 6rf.-cloth. " This is a work of value to those who use steam machinery. . . . Should be read by every- one who has a steam engine, on a farm or elsewhere." Mark Lane Express. " We cordially commend this work to buyers and owners of steam engines, and to those who have to do with their construction or use." Timber Trades Journal. " Such a general knowledge of the steam engine as Mr. Wansbrough furnishes to the reader should be acquired by all intelligent owners and others who use the steam engine. "Building Nevis. which describes with all necessary Purchasers contain a good deal of CIVIL ENGINEERING, SURVEYING, etc. MR. HUMBER'S IMPORTANT ENGINEERING BOOKS. The fFater Supply^of Cities and Towns. A CQMPREHENSIVE^TREATISE on the WATER-SUPPLY OF CITIES AND TOWNS. \Zy WILLIAM HUMBER, A-M.Inst.C.E., and M. InstA^l.E., Author of " CasKand Wrought Iron Bridge Construction," ated with 50 DWble Plates, I Single Plate, Coloured Frontispiec\ and upwards of 250 Woodcuts, and containing 400 pages of Text. Imp. $tp, 6 6s, elegantly and substantially half-bound in morocco. Ltst of Contente. I. Historical Sketch 6 that have been adopted for to Cities and Towns. II. reign Matter usually Rainfall and Evapor the water-bearing formations tricts. V. Measurement and E flow of Water VI. On the Se Source of Supply. -VI I. Wells. VIII. voirs. IX. The Purification of Wat Pumps. XI. Pumping Machinery "The most systematic and valuable work Conduits.-XIII. Distribution of Water.-XIV. Works iilus exhaustiveness much more distinc ~ E "f\v"can congratulate Mr. H tion on a subject so to[ ~" nber, are mostly drawing ogether with Specifications < ted, 3si n K which wiU be found : anterbury, Dundee. . . n, bublin. and other: ^s characterised aim German than ( to give of Infor- of every engineer whose pfactice may He in thi"s"branch"of the profession. "Builder. Cast and Wrought Iron Bridge Construction. A COMPLETE AND PRACTICAL TREATISE ON CAST AND WROUGHT IRON BRIDGE CONSTRUCTION, including Iron Foundations. In Three 'Parts Theoretical* Practical, and Descriptive. By WILLIAM HUMBER, A.M^Inst.C.E., and MUnst.M.E. Third Edition, Re- vised and much improved\with "5 Doublfe Plates (20 of whichnow first appear in this edition), and numerous Additions to the Text. In Two Vols., ' ip. ito, 6 i6s. 6d. half-bound in morocco. \ Jery valuable contribution to the^tandard literature > civil engineering-. In addition to '. . ---. - i i- j-:i- ^ , T ;, f r. rhi\h very much enhance the nistruc- ; eminent enjfineers, are CROSBY LOCK WOOD &- SON'S CATALOGUE. MR. H UMBER'S GREAT WORK ON MODERN ENGINEERING. Complete in Four Volumes, imperial 410, price Volume sold separately as i 12 i2s., half-morocco. Each Hows : A RECORD OF THE PROGRESS OF MODERN ENGINEER- ING. FIRST SERIES. Comprising Civil, Mechanical, Marine, Hydraulic, Railway, Bridge, and other Engineering Works, &c. By WILLIAM HUMBER, A-M.Inst.C.E., &c. Imp. 4to, with 36 Double Plates, drawn to a large scale, Photographic Portrait of John Hawkshaw, C.E., F.R.S., &c., and copious descriptive Letterpress, Specifications, &c., 3 35. half-morocco. List of the Plates and Diagrams. Thames, West London Extension Railway (5 plates) ; Armour Plates : Suspension Bridge, Thames (4 plates); The Allen Engine; Sus- pension Bridge, Avon (3 plates) ; Underground G. N. Railway ; Roof of Station, Dutch Railway (3 plates). Rhenish Rail (2 plates); Bridge over th. " Handsomely lithographed and printed. In a permanent form copies of the plans and specifica tractors for many important engineering works.' Engineer. HUMBER'S RECORD OF MODERN ENGINEERING. SECOND SERIES. Imp. 410, with 36 Double Plates, Photographic Portrait of Robert Stephenson, C.E., M.P., F.R.S., &c., and copious descriptive Letterpress, Specifications, &c., 3 35. half-morocco. List of the Plates and Diagrams. Victoria Station and Roof, L. B. & S. C. R. (8 plates) ; Southport Pier (2 plates) ; Victoria Station and Roo-, L. C. & D. and G. W. R. (6 plates); Roof of Cremorne Music Hall; Bridge ----- - - -- -jutch Birkenhead Docks. Low Water Basin (15 pUtes); Charing Cross Station Roof, C. C. R.-ilway (3 plates) ; Digswell Viaduct, Great Northern Railway ; Robbery Wood Viaduct, Great Northern Railway; Iron Permanent Way; Clydach Viaduct, Merthyr, Tredegar, and Abergavenny Railway; Ebbw Viaduct, thyr, erga , Tr ' degar, and Abergavenny Rail- . , -_J Wood Viaduct. Cornwall Rail- way ; Dublin Winter Palace Roof (3 plates) ; Bridge over the Thames, L. C. & D. Railway (6 plates) ; Albert Harbour, Greenock (4 plates/. " Mr. Humber has done the profession good and true service, by the fine collection of examples he has here brought before the profession and the public." Practical Mechanics Journal. HUMBER'S RECORD OF MODERN ENGINEERING. THIRD SERIES. Imp. 4to, with 40 Double Plates, Photographic Portrait of J. R. M'Clean, late Pres. Inst. C.E., and copious descriptive Letterpress, Speci- fications, &c., 3 3$. half-morocco. List of the Plates and Diagrams. MAIN DRAINAGE, METROPOLIS. North Side. Map showing interception of Sewers ; Middle Level Sewe'r (2 plates) ; Outfall Sewer, Bridge over River Lea (3 plates) ; Outfall Sewer, Bridge over Marsh Lane, North Woolwich Sewer, Reservoir and Outlet (4 plates) ; Outfall Sewer, Filth Hoist; Sections of Sewers (North and South Sides). THAMES EMBANKMENT. Section of River Wall ; Steamboat Pier, Westminster J2 plates); tion Po'utfaU Sewer, Brfdge"^ver" Bow and' Barking Railway (3 plates); Outfall Sewer, Bridge over East London Waterworks 1 Feeder (2 plates) ; Outfall Sewer, Reservoir (2 plates) ; Outfall Sewer, Tumbling Bay and Outlet ; Out- fall Sewer, Penstocks. South Side. Outfall Waterloo Bridges ; York Gate (2 plates) ; Over, flow and Outlet at Savoy Street Sewer (3 plates) ; Steamboat Pier, Waterloo Bridge (3 plates) ; Junction of Sewers, Plans and Sections ; Gullies, Plans and Sections ; Rolling Stock ; Granite and Iron Forts. Sewer, Bermondsey Branch (2 plates) ; Outfall " The drawings have a constantly increasing value, and whoever desires to possess clear repre- sentations of the two great works carried out by our Metropolitan Board will obtain Mr. Humber's volume." Engineer. HUMBER'S RECORD OF MODERN ENGINEERING. FOURTH SERIES. Imp. 410, with 36 Double Plates, Photographic Portrait of John Fowler, late Pres. Inst. C.E., and copious descriptive Letterpress, Speci- fications, &c., 3 35. half-morocco. List of the Plates and Diagrams. Abbey Mills Pumping Station, Main Drain- Mesopotamia ; Viaduct over the River Wye, Midland Railway (3 plates) ; St. Germans Via- duct, Cornwall Railway plates) ; Wrought- Iron Cylinder for Diving Bell; Millwall Docks (6 plates) ; Milroy's Patent Excavator ; Metro- politan District Railway (6 plates) ; Harbours, Ports, and Breakwaters (3 plates). 'We gladly welcome another year's issue of this valuable publication from the able pen of Mr. Humber. The accuracy and general excellence of this work are well known, while its useful- n *, in giving the measurements and details of some of the latest examples of engineering , as carried out by the most eminent men in the profession, cannot be too highly prized." Artixan, nge. Metropolis (4 plates) ; Barrow Docks (5 plates); Manquis Viaduct, Santiago and Val- paraiso Railway (2 plates) ; Adam's Locomo- tive, St. Helen's Canal Railway (2 plates) ; Cannon Street Station Roof, Charing Cross CIVIL ENGINEERING, SURVEYING, etc. g MR. HUMBER'S ENGINEERING BOOKS continued. Strains, Calculation of. A HANDY BOOK FOR THE CALCULATION OF STRAINS IN GIRDERS A ND SIM ILA R STR UCTURES, A ND THEIR STRENGTH. Consisting of Formulae and Corresponding Diagrams, with numerous details for Practical Application, &c. By WILLIAM HUMBER. A-M.Inst.C.E., &c. Fourth Edition. Crown 8vo, nearly 100 Woodcuts and 3 Plates, 75. 6d. cloth. " The formula; are neatly expressed, and the diagrams good."Athen SON'S CATALOGUE. Statics, Graphic and Analytic. GRAPHIC AND ANALYTIC STATICS, in their Practical Appli. cation to the Treatment of Stresses in Roofs, Selid Girders, Lattice, Bowstring and Suspension Bridges, Braced Iron Arches and Piers, and other Frameworks. By R. HUDSON GRAHAM, C.E. Containing Diagrams and Plates to Scale. With numerous Examples, many taken from existing Structures. Specially arranged for Class-work in Colleges and Universities. Second Edition, Re- vised and Enlarged. 8vo, i6s. cloth. * ' Mr. Graham's book will find a place wherever graphic and analytic statics are used or studied. ' ' Engineer. " The work is excellent from a practical point of view, and has evidently been prepared with much care. The directions for working are ample, and are illustrated by an abundance of well- selected examples. It is an excellent text-book for the practical draughtsman." Athenaum. Student's Text-Book on Surveying. PRACTICAL SURVEYING: A Text-Book for Students pre- paring for Examination or for Survey-work in the Colonies. By GEORGE W. USILL, A.M.I. C.E., Author of "The Statistics of the Water Supply of Great Britain." With Four Lithographic Plates and upwards of 330 Illustra- tions. Second Edition, Revised. Crown 8vo, 75. 6d. cloth. " The best forms of instruments are described as to their construction, uses and modes of employment, and there are innumerable hints on work and equipment such as the author, in his experience as surveyor, draughtsman and teacher, has found necessary, and which the student in his inexperience will find most serviceable." Engineer. " The latest treatise in the English language on surveying, and we have no hesitation in say- ing that the student will find it a belter guide than any of its predecessors .... Deserves to be recognised as th first book which should be put in the hands of a pupil of Civil Engineering, and every gentleman of education who sets out for the Colonies would rind it well to have a copy." Architect. " A very useful, practical handbook on field practice. Clear, accurate and not too con- densed." Journal of Education. Survey Practice. AID TO SURVEY PRACTICE, for Reference in Surveying, Level- ling, and Setting-out ; and in Route Surveys of Travellers by Land and Sea. With Tables, Illustrations, and Records. By Lowis D'A. JACKSON, A.M.I.C.E., Author of " Hydraulic Manual," "Modern Metrology," &c. Second Edition, Enlarged. Large crown 8vo, I2S. 6d. cloth. "Mr. Jackson has produced a valuable vade-mecum for the surveyor. We can recommend this book as containing an admirable supplement to the teaching of the accomplished surveyor." " As a text-book we should advise all surveyors to place It in their libraries, and study well the matured instructions afforded in its pages." Colliery Guardian. " The author brings to his work a fortunate union of theory and practical experience which, aided by a dear and lucid style of writing, renders the book a very useful one." Builder. Surveying, Land and Marine. LAND AND MARINE S URVEYING, in Reference to the Pre- paration of Plans for Roads and Railways ; Canals, Rivers, Towns' Water Supplies; Docks and Harbours. With Description and Use of Surveying Instruments. By W. D. HASKOLL, C.E., Author of " Bridge and Viaduct Con- struction,' 1 &c. Second Edition, Revised, with Additions. Large cr. 8vo, gs. cl. " This book must prove of great value to the student. We have no hesitation in rr commend- ing it, feeling assured that it will more than repay a careful study. "-Mechanical H'orl*. ' A most useful and well arranged book for the aid of a student. We can strongly recommend it as a carefully-written and valuable text-book. It enjoys a well-deserved repute among surveyors." Builder. " This volume cannot fail to prove of the utmost practical utility. It may be safely recommended to all students who aspire to become clean and expert surveyors." Mining Journal. Tunnelling. PRACTICAL TUNNELLING. \Explaining in detail the Setting, out of the works, Shaft-sinkingjand Heading-driving, Ranging the Lines and Levelling underground, Sub-B*cavating\Timbering, and the Construction of the Brickwork of Tunnels, with] the amooat of Labour required for, and the Cost of, the various portions of thi work. B\FREDERICK W. SIMMS, F.G.S., M.Inst.C.E. Third Edition, Revised and Extended by D. KINNEAR CLARK, M.Inst.C.E. Imperial 8vo, with 21 FoldingNPlates and numerous Wood Engravings, 30$. cloth. "The estimation in which Mr. Slmms's book on tunnelling has been held for over thirty years cannot be more truly expressed than in the words of the late Prof. Rankine : ' The best source of in- formation on the subject of tunnels is Mr.F.W.Simms'swork on Practical Tunnelling."' Architect. " It has been regarded from the first as a text book of the subject. . . . Mr. Clarke has added immensely to the value of the book." Engineer. CIVIL ENGINEERING, SURVEYING, etc. n Levelling. A TREATISE ON THE PRINCIPLES AND PRACTICE OF LEVELLING. Showing its Appttaation to ^purposes of Railway and Civil Engineering, in the Construction of Roads; witKMr.TELFORD's Rules for the same. By FREDERICK W. SIMMS, F.G.S\, M.Inst.Q.E. Seventh Edition, with the addition of LAW'S Practical Examplekfor Setting-out Railway Curves, and IRAUTWINE'S Field Practice of Laymg-dut Circular Curves. With 7 Plates and numerous Woodcuts, 8vo, 8s. 6d. clpth. *ATRAUIWINE on Curves may be had separate, 55. " The text-book on levelling in most of our engineering schools and colleges. "-Engineer. " The publishers have rendered a substantial service to the profession, especially to the younger nbers, by bringing out the present edition of Mr. Sii Heat, Expansion by. EXPANSION OF STRUCTURES BY HEAT. By JOHN KEILY, C.E., late of the Indian Public Wor& and Victorian Railway Depart- ments. CroWn 8vo, 35. 6d. cloth. SUMMARY OF CONTENT Section I. FORMITC.AS AND DATA. Section I VI. MECHANICAL FORCE OF Section II. METAL BIARS. V HEAT. Section III. SIMPLE FIRAMES. Section VKL WORK OF EXPANSION Section IV. COMPLEX\ FRAMES AND \ AND CONTRACTION. PLATESX^ Section VIIIA SUSPENSION BRIDGES. Section V. THERMAL CONDUCTIVITY. Section IX. ''MASONRY STRUCTURES. " The aim the author has sei before him, viz., to show the effects of heat upon metallic and other structures, is a laudable onp, for this is a branch of physics upon which the engineer or archi- tect can find but little reliable and comprehensive data in books." Jiuilrftr. " Whoever is concerned to know the effect of changes of temperature on such structures as suspension bridges and the like, could not do better than consult Mr. Keily's valuable and handy exposition of the geometrical principles involved in these changes." Scotsman. Practical Mathematics. MATHEMATICS FOR PRACTICAL MEN; Being a Common- place Book of Pure and. Mixed Mathematics. Designed chiefly for the use of Civil Engineers, Architects and Surveyors. By OLINTHUS GREGORY, LL.D., F.R.A.S., Enlarged by HENRY LAW, C.E. 4th Edition, carefully Revised by J. R. YOUNG, formerly Professor of Mathematics, Belfast College. With 13 Plates, 8vo, i is. cloth. " The engineer or architect will here find ready to his hand rules for solving nearly every mathe- ' ' i his practice The rules are in all cases explained by means of the process is clearly worked out." Builder. " One of the most serviceable books for practical mechanics. . . It is an instructive book for the student, and a text-book for mm who, having once mastered the subjects it treats of, needs occasionally to refresh his memory upon them." Building News. Hydraulic Tables. HYDRA ULIC .TABLES, CO-EFFICIENTS, and FORMULM for finding the Discharge of Water from Orifices, Notches, Weirs, Pipes, and Rivers. With New\Formulae, Tablets, and General Information on Rainfall, Catchment- Basins, Drainage, Sewertfg*. Water Supply for Towns and Mill Power. By JOHN NEVILLE, Civil Engineer, M.R.I.A. Third Ed., carefully Revised, with considerable Additions. Numerous Illusts. Cr. 8vo, 14$. cloth. " Alike valuable to students arid engineers in practii avoidable failures, and assist them V> select the readie given work connected with hydraulicyngineering." Minikg Journal. "It is, of all English books on theVubject, the one near&t to completeness. . . . From th (tood arrangement of the matter, the clear explanations, an* abundance of formulae, the carefully calculated tables, and, above all, the thorough acquaintance with both theory and construction, which is displayed from first to last, the book will be found to be an acquisition. 'Architect, Hydraulics. HYDRA ULIC MANUAL, Consisting of Working Tables and Explanatory Text. ^Intended as a Guide in Hydraulic Calculations and Field Operations. By Lowis^'A. JACKSO K Author of "Aid to Survey Practice," " Modern Metrology," &c\ Fourth Edition, Enlarged. Large cr. 8vo, i6s. cl. " The author has had a wide experience in hyclrauVjc engineering and has been a careful ob- server of the facts which have comeAmder his notice, attd from the great mass of material at his command he has constructed a maiuial which mvy beWcepted as a trustworthy guide to this branch of the engineer's profession. Ve can heartily reJbmmend this volume to all who desire to be acquainted with the latest development of this important subject." Engineering. ^^S^SS^^^^^^^^t^SS-^ superannuated, and Its thorough adoption of recent experiments ; the text is, in fact, in great part a short account of the great modern experiments." Naturt, 12 CROSBY LOCK WOOD < SON '5 CATALOGUE. Drainage. ON THE DRAINAGE OF LANDS, TOWNS AND BUILD- INGS. By G. D. DEMPSEY, C.E., Author of " The Practical Railway En- gineer," &c. Revised, with large Additions on RECENT PRACTICE IN DRAINAGE ENGINEERING, by D. KINNEAR CLARK, M.Inst.C.E. Author of " Tramways : Their Construction and Working," " A Manual of Rules, Tables, and Data for Mechanical Engineers." &c. &c. Crown 8vo, 75. 6d. cloth. " As a work on recent practice in drainage engineering, the book is to be commended to all who are making that branch of engineering science their special study." Iron. "A comprehensive manual on drainage engineering, and a useful introduction to the student." Building News. Tramways and their Working. TRAMWAYS: THEIR CONSTRUCTION AND WORKING. Embracing a Comprehensive History of the System ; with an exhaustive Analysis of the various Modes of Traction, including Horse-Power, Steam, Heated Water, and Compressed Air ; a Description of the Varieties of Rolling Stock : and ample Details of Cost and Working Expenses : the Progress recently made in Tramway Construction, &c. &c. By D. KINNEAR CLARK, M.Inst.C.E. With over 200 Wood Engravings, and 13 Folding Plates. Two Vols., large crown 8vo, 305. cloth. " All interested in tramways must refer to it, as all railway engineers have turned to the author's work ' Railway Machinery."' Engineer. " An exhaustive and practical work on tramways, in which the history of this kind of locomo- tion, and a description and cost of the various modes of laying tramways, are to be found." Building News. " The best form of rails, the best mode of construction, and the best mechanical appliances are so fairly indicated in the work under review, that any engineer about to construct a tramway will be enabled at once to obtain the practical information which will be of most service to him." Athenaum. Oblique Arches. A PRACTICAL TREATISE ON THE CONSTRUCTION OP OBLIQUE ARCHES. By JOHN HART. Third Edition, with Plates. Im- perial ovo, 8s. cloth. Curves, Tables for Setting-out. TABLES OF TANGENTIAL ANGLES AND MULTIPLES for Setting-out Curves from 5 to 200 Radius. By ALEXANDER BEAZELEY, M.Inst.C.E. Third Edition. Printed on 48 Cards, and sold in a cloth box, waistcoat-pocket size, 3$. 6d. " Each table is printed on a small card, which, being placed on the theodolite, leaves the hands free to manipulate the instrument no small advantage as regards the rapidity of work." E ngineer. "Very handy ; a man may know that all his day's work must fall on two of these cards, which he puts into his own card-case, and leaves the rest behind." Athinaum. Earthwork. EARTHWORK TABLES. Showing the Contents in Cubic Yards of Embankments, Cuttings, &c.,of Heights or Depths up to an average of 80 feet. By JOSEPH BROADBENT, C.E., and FRANCIS CAMPIN, C.E. Crown 8vo, 55. cloth. " The way in which accuracy is attained, by a simple division ot each cross section into three elements, two in which are constant and one variable, is ingenious." Athtnaum. Tunnel Shafts. THE CONSTRUCTION OF LARGE TUNNEL SHAFTS: A Practical and Theoretical Essay. By I. H. WATSON BUCK, M.Inst.C.E., Resident Engineer, London and North- Western Railway. Illustrated with Folding Plates, royal 8vo, 125. cloth. " Many of the methods given are of extreme practical value to the mason ; and the observations on the form of arch, the rules for ordering the stone, and the construction of the templates will be found of considerable use. We commend the book to the engineering profession." Building News. " Will be regarded by civil engineers as of the utmost value, and calculated to save much time and obviate many mistakes." Colliery Guardian. Girders. Strength of. GRAPHIC TABLE FOR FACILITATING THE COMPUTA- TION OF THE WEIGHTS OF WROUGHT IRON AND STEEL GIRDERS, etc., for Parliamentary and other Estimates. By J. H. WATSON BUCK, M.Inst.C.E. On a Sheet, zs.Gd. CIVIL ENGINEERING, SURVEYING, etc. 13 River Engineering. RIVER BARS: The Causes of their Formation, and their Treat- .- ment by " Induced Tidal Scour; " with a Description of the Successful Re- J duction by this Method of the Bar at Dublin. By I. J. MANN, Assist. Eng. w to the Dublin Port and Docks Board. Royal 8vo, 75. bd. cloth. " We recommend all interested in harbour works and, indeed, those concerned in the im- provements of rivers generaUy-to read Mr. Mann's interesting work on the treatment of river Trusses. TRUSSES OF WOOD AND IRON. Practical Applications of Science in Determining the Stresses, Breaking Weights, Safe Loads, Scantlings, and Details of Construction, with Complete Working Drawings. By WILLIAM GRIFFITHS, Surveyor, Assistant Master, Tranmere School of Science and Art. Oblong 8vo, 45. 6d. cloth. " This handy little book enters so minutely into every detail connected with the construction of toof trusses, that no student need be ignorant of these matters." Practical Engineer. Hallway Working. SAFE RAILWAY WORKING. A Treatise on Railway Acci- dents : Their Cause and Prevention ; with a Description of Modern Appliances and Systems. By CLEMENT E..STRETTON, C.E\Vice-President and Con- sulting Engineer, Amalgamated ^ociety of Railway Servants. With Illus- trations and Coloured Plates. Second Edition, Enlarged. Crown 8vo, 3$. 6d. cloth. \ \ [Just published. " A book for the engineer, the directors, the managers ; and, in shoX all who wish for informa- tion on railway matters will find a perfect encycldpzdia in Safe Railw% Working. 1 "Rail-way Review, " We commend the remarks on railway signalling to all railway managers, especially where a uniform code and practice is advocated." Herepath's Rail-way Journal. "The author maybe congratulated on having collected, in a very convenient form, much valuable information on the principal questions affecting the safe working of railways." Rail. JField-Boofc for Engineers. THE ENGINEER'S, MINING SURVEYOR'S, AND CON- TRA CTOR 'S FIELD-BOOK. Consisting of a Series of Tables, with Rules, Explanations of Systems, and use of Theodolite for Traverse Surveying and Plotting the Work with minute accuracy by means of Straight Edge and Set Square only ; Levelling with the Theodolite, Casting-out and Reducing Levels to Datum, and Plotting Sections in the ordinary manner ; setting-out Curves with the Theodolite by Tangential Angles and Multiples, with Right and Left-hand Readings of the Instrument : Setting-out Curves without Theodolite, on the System of Tangential Angles by sets of Tangents and Off- sets ; and Earthwork Tables to 80 feet deep, calculated for every 6 inches in depth. By W. DAVIS HASKOLL, C.E. With numerous Woodcuts. Fourth Edition, Enlarged. Crown 8vo, 12S. cloth. ' ' The book is very handy ; the separate tables of sines and tangents to every minute will make it useful for many other purposes, the genuine traverse tables existing all the :i]:i:;' u 20, 21. Mahogany: Earlier Stages and Finished Specimen 22, 2,,% 4 . Sienna Marble: Varieties of Grain, Preliminary Stages and Finished Specimen 25, 26, 27. Juniper Wood : Methods of producing Grain, &c. : Preliminary Stages and Finished Specimen-28, 29, 30. Vert de Mer Marble : Varieties of Grain and Methods of Working Unfinished and Finished Speci- mens 31. 32. 31. Oak: Varieties of Grain, Tools Employed, and Methods of Manipulation, Pre- liminary Stages and Finished Specimen 34, 35, 36. Waulsort Marble: Varieties of Grain, Un- finished and Finished Specimens. "Those who desire to attain skill in the art of painting woods and marbles will find advantage In consulting this book. . . . Some of the Working Men's Clubs should give their young men tke opportunity to study it." B-uildfr. " A comprehensive guide to the art. The explanations of the processes, the manipulation and management of the colours, and the beautifully executed plates will not be the least valuable to the rtudent who aims at making his work a faithful transcript nf nature." Building A'tns. DECORATIVE ARTS, etc. 17 House Decoration. ELEMENTARY DECORATION. A Guide to the Simpler Forms of Everyday Art, as applied to the Interior and Exterior Decoration of Dwelling Houses, &c. By JAMES W. FACEY, Jun. With 68 Cuts, izmo, zs. cloth limp. PRACTICAL HOUSE DECORATION : A Guide to the Art of Ornamental Painting, the Arrangement of Colours in Apartments, and the principles of Decorative Design. With some Remarks upon the Nature and Properties of Pigments. By JAMES WILLIAM FACEY, Author of " Elementary Decoration," &c. With numerous Illustrations. I2mo, 2s. 6d. cloth limp. N.B.The above Two Works together in One Vol., strongly half-bound, 5*. Colour* A GRAMMAR OF COLOURING. Applied to Decorative Painting and the Arts. By GEORGE FIELD. New Edition, Revised, Enlarged, and adapted to the use of the Ornamental Painter and Designer. By ELLIS A. DAVIDSON. With New Coloured Diagrams and Engravings, izmo, 35. 64. cloth boards. "The book is a most useful resume of the properties of pigments." Builder. House Painting, Graining, etc. HOUSE PAINTING, GRAINING, MARBLING, AND SIGN WRITING, A Practical Manual of. By ELLIS A. DAVIDSON. Sixth Edition. With Coloured Plates and Wood Engravings. i2mo, 6s. clolh boards. " A mass of information, oi use to the amateur and of value to the practical man." English " Simply invaluable to the youngster entering upon this particular calling, and highly service- able to the man who is practising \t." Furniture Gazette. Decorators, Receipts for. THE DECORATOR'S ASSISTANT: A Modern Guide to De- corative Artists and Amateurs, Painters, Writers, Gilders, &c. Containing upwards of 600 Receipts, Rules and Instructions ; with a variety of Informa- tion for General Work connected with every Class of Interior and Exterior Decorations, &c. Fourth Edition, Revised. 152 pp., crown 8vo, is. in wrapper. " Full of receipts of value to decorators, painters, gilders, &c. The book contains the gist of larger treatises on colour and technical processes. It would be difficult to meet with a work so full of varied information on the painter's art."- Huilding Ne-ws. " We recommend the work to all who, whether for pleasure or profit, require a guide to decora- tion." Plumber and Decorator, Moyr Smith on Interior Decoration. ORNAMENTAL INTERIORS, ANCIENT AND MODERN. By J. MOYR SMITH. Super-royal 8vo, with 32 full-page Plates and numerous smaller Illustrations, handsomely bound in cloth, gilt top, price i8s. "The book is well illustrated and handsomely got up, and contains some true criticism and a good many good examples of decorative treatment. ''The Builder. ' This is the most elaborate and beautiful work on the artistic decoration of interiors that we have seen. . . . The scrolls, panels and other designs from the author's own pen are very beautiful and chaste ; but he takes care that the designs of other men shall figure even more thar, his own." Liverpool Albion. " To all who take an interest in elaborate domestic ornament this handsome volume will be welcome. "Graph ic. British and Foreign Marbles. MARBLE DECORATION and the Terminology of British and Foreign Marbles. A Handbook for Students. By GEORGE H. BLAGROVE, Author of " Shoring and its Application," &c. With 28 Illustrations. Crown &vo. 35. 6d. cloth. This most useful and much wanted handbook should be in the hands of every architect and ,ders generally." Saturday rltten treatise ; the work is essentially practical."-5^>o. Marble Working, etc. MARBLE AND MARBLE WORKERS: A Handbook for Architects, Artists, Masons and Students. By ARTHUR LEE, Author of " A Visit to Carrara," " The Working of Marble," &c. Small crown 8vo, 2$. cloth. " A really valuable addition to the technical literature of archi ects and ' ffews. 1 8 CROSBY LOCK WOOD & SON'S CATALOGUE. DELAMOTTE'S WORKS ON ILLUMINATION AND ALPHABETS. A PRIMER OF THE ART OF ILLUMINATION, for the Use of Beginners : with a Rudimentary Treatise on the Art, Practical Directions for its exercise, and Examples taken from Illuminated MSS., printed in Gold and Colours. By F. DELAMOTTE. New and Cheaper Edition. Small 4to, 6s. orna- mental boards. "The examples of ancient MSS. recommended to the student, which, with much good sense, the author chooses from collections accessible to all, are selected with judgment and knowledge, as well as taste." AtheiHZitm. ORNAMENTAL ALPHABETS, Ancient and Medieval, from the Eighth Century, with Numerals ; including Gothic, Church-Text, large and small, German, Italian, Arabesque, Initials for Illumination, Monograms, Crosses, &c. &c., for the use of Architectural and Engineering Draughtsmen, Missal Painters, Masons, Decorative Painters, Lithographers, Engravers, Carvers, &c. &c. Collected and Engraved by F. DELAMOTTE, and printed in Colours. New and Cheaper Edition. Royal 8vo, oblong, zs. 6d. ornamental boards. 11 For those who Insert enamelled sentences round gilded chalices, who blazon shop legends over shop-doors, who letter church walls with pithy sentences from the Decalogue, this book will be use- ful. K A thenau m. EXAMPLES OF MODERN ALPHABETS, Plain and Ornamental; including German, Old English, Saxon, Italic, Perspective, Greek, Hebrew, Court Hand, Engrossing, Tuscan, Riband, Gothic, Rustic, and Arabesque ; with several Original Designs, and an Analysis of the Roman and Old English Alphabets, large and small, and Numerals, for the use of Draughtsmen, Sur- veyors, Masons, Decorative Painters, Lithographers, Engravers, Carvers, &c. Collected and Engraved by F. DELAMOTTE, and printed in Colours. New and Cheaper Edition. Royal 8vo, oblong, zs. fid. ornamental boards. "There is comprised in it ev:ry possible shape into which the letters of the alphabet and numerals can be formed, and the talent which has been expended in the conception of the various plain and ornamental letters is wonderful." Standard. MEDIEVAL ALPHABETS AND INITIALS FOR ILLUMI- NATORS. By F. G. DELAMOTTE. Containing 21 Plates and Illuminated Title, printed in Gold and Colours. With an Introduction by J. WILLIS BROOKS. Fourth and Cheaper Edition. Small 4to, 45. ornamental boards. " A volume In which the letters of the alphabet come forth glorified in gilding and all the colours ol the prism interwoven and intertwined and intermingled." Sun. THE EMBROIDERER'S BOOK OF DESIGN. Containing Initials, Emblems, Cyphers, Monograms. Ornamental Borders, Ecclesiastical Devices, Mediaaval and Modern Alphabets, and National Emblems. Col- lected by F. DELAMOTTE, and printed in Colours. Oblong royal 8vo, is. 6*J. ornamental wrapper. assistance to ladies and yc .... ._ 1 pretty work." East Anglian Times. Wood Carving. INSTRUC7TONS IN WOOD-CARVING, for Amateurs: with Hints on Design. By A LADY. With Ten Plates. New and Cheaper Edition. Crown 8vo, as. in emblematic wrapper. " The handicraft of the wood-carver, so well as a book can Impart It, may be learnt from ' A Lad " ^"direction's given arfplain and easily understood." English Mechanic. Glass Painting. GLASS STAINING AND THE ART OF PAINTING ON GLASS. From the German of Dr. GESSERT and EMANUEL OTTO FROMBERG. With an Appendix on THE ART OF ENAMELLING, izmo, zs. 6d. cloth limp. Letter Painting. THE ART OF LETTER PAINTING MADE EASY. By JAMES GREIG BADENOCH. With 12 full-page Engravings of Examples, is. 6ci. cloth limp. " The system is a simple one, but quite original, and well worth the careful attention of letter painters. It can be easily mastered and remembered." Building News. CARPENTRY, TIMBER, etc. 19 CARPENTRY, TIMBER, etc. Tredgold's Carpentry, Revised & Enlarged by Tarn. THE ELEMENTARY PRINCIPLES OF CARPENTRY. A Treatise on the Pressure and Equilibrium of Timber Framing, the Resist- ance of Timber, and the Construction of Floors, Arches, Bridges, Roofs, Uniting Iron and Stone with Timber, &c. To which is added an Essay on the Nature and Properties of Timber, &c., with Descriptions of the kinds of Wood used in Building ; also numerous Tables of the Scantlings of Tim- ber for different purposes, the Specific Gravities of Materials, &c. By THOMAS TREDGOLD, C.E. With an Appendix of Specimens of Various Roofs of Iron and Stone, Illustrated. Seventh Edition, thoroughly revised and considerably enlarged by E. WYNDHAM TARN, M.A., Author of "The Science of Build- ing," &c. With 61 Plates, Portrait of the Author, and several Woodcuts. In one large vol., 410, price i 55. cloth. "Ought to be in every architect's and every builder's library." Builder. work whose monumental excellence must commend it wherever skilful carpentry Is con- cerned. The author's principles are rather confirmed than unpaired by time. The additional plates are of great intrinsic value." Building News. WoodivorJcing Machinery. WOODWORKING MACHINERY : Its Rise, Progress, and Construction. With Hints on the Management of Saw Mills and the Economi- cal Conversion of Timber. Illustrated 4 with Examples oi Recent Designs by leading English, French, and American Engineers. By M. Powis BALE, A.M.Inst.C.E., M.I.M.E. Large crown 8vo, izs. 6d. cloth. "Mr. Bale is evidently an expert on the subject and he has collected so much information that his book is all-sufficient for builders and others engaged in the conversion of timber." Architect. "The most comprehensive compendium of wood- working machinery we have seen. The author is a thorough master of his subject." Building Ncwi. " The appearance of this book at the present time will, we should think, give a considerable Impetus to the onward march of the machinist engaged in the designing and manufacture of wood-working machines. It should be in the office of every wood-working factory." Eiiflish Saw Mills. SA W MILLS : Their Arrangement and Management, and the Economical Conversion of Timber. (A Companion Volume to " Woodwork- ing Machinery.") By M. Powis BALE. With numerous Illustrations. Crown 8vo, IDS. 6d. cloth. " The administration of a large sawing establishment is discussed, and the subject examined from a financial standpoint. We could not desire a more complete or practical treatise." Builder. " We highly recommend Mr. Bale's work to the attention and perusal of all those who are en- gaged in the art of wood conversion, or who are about building or remodelling saw-mills on im- proved principles." Building News. Carpentering. THE CARPENTER'S NEW GUIDE ; or, Book of Lines for Car- penters ; comprising all the Elementary Principles essential for acquiring a knowledge of Carpentry. Founded on the late PETER NICHOLSON'S Standard Work. A New Edition, Revised by ARTHUR ASHPITKL, F.S.A. Together with Practical Rules on Drawing, by GEORGE PYNK. With 74 Plates, 4to, i is. cloth. Handrailinff and Stairbuildinff. A PRACTICAL TREATISE ON HANDRAILING : Showing New and Simple Methods for Finding the Pitch of the Plank, Drawing the Moulds, Bevelling, Jointincj-up, and Squaring the Wreath. By GEORGE COLLINGS. Second Edition, Revised and Enlarged, to which is added A TREATISE ON STAIRBUILDING. With Plates and Diagrams, izmo, zs. 6d. ' Will befound of practical utility in the execution of this difficult branch of joinery. "-Builaer. " Almost every difficult phase of this somewhat intricate branch of joinery is elucidated by the aid of plates and explanatory letterpress." Furniture Gazette. Circular Work. CIRCULAR WORK IN CARPENTRY AND JOINERY: A Practical Treatise on Circular Work of Single and Double Curvature By GEORGE COLLINGS, Author of " A Practical Treatise on Handrailmg." Illus- trated with numerous Diagrams. Second Edition. I2mo, is. 6d. cloth limp. An excellent example of what a book of this kind should be. Cheap in price, clear in dcfii.i- tion and practical in the examples selected." liiituter. 20 CROSBY LOCKWOOD & SON'S CATALOGUE. Timber Merchant's Companion. THE TIMBER MERCHANT'S AND BUILDER'S COM- PANION. Containing New and Copious Tables of the Reduced Weight and Measurement of Deals and Battens, of all sizes, from One to a Thousand Pieces, and the relative Price that each size bears per Lineal Foot to any given Price per Petersburg Standard Hundred ; the Price per Cube Foot of Square Timber to any given Price per Load of 50 Feet ; the proportionate Value of Deals and Battens by the Standard, to Square Timber by the Load of 50 Feet ; the readiest mode of ascertaining the Price of Scantling per Lineal Foot of any size, to any given Figure per Cube Foot, &c. &c. By WILLIAM DOWSING. Fourth Edition, Revised and Corrected. Cr.8vo.3s. cl. glad to see a fourth edition of these admirable tables, which for correctness and nplicity of arrangement leave nothing to be desired." Timber Trades Journal. "An exceedingly well-arranged, cle sell timber." Journal of Forestry. "An exceedingly well-arranged, clear, and concise manual of tables for the use of all who buy " Practical Timber Merchant. THE PRACTICAL TIMBER MERCHANT. Being a Guide for the use of Building Contractors, Surveyors, Builders, &c., comprising useful Tables for all purposes connected with the Timber Trade, Marks of Wood, Essay on the Strength of Timber, Remarks on the Growth of Timber, &c. By W. RICHARDSON. Fcap. 8vo, 35. 6d. cloth. " This handy manual contains much valuable information for the use of timber merchant?-, builders, foresters, and all others connected with the growth, sale, and manufacture of timber. ' Journal of Forestry. Timber Freight Book. THE TIMBER MERCHANT'S, SAW MILLER'S, AND IMPORTER'S FREIGHT BOOK AND ASSISTANT. Comprising Rules, Tables, and Memoranda relating to the Timber Trade. By WILLIAM RICHARDSON, Timber Broker; together with a Chapter on "SPEEDS OF SAW MILL MACHINERY," by M. Powis BALK, M.I.M.E., &c. I2mo, 35. 6ws in, as z makeweight, a host of material concerning bricks, columns, cisterns. Sic." English MARINE ENGINEERING, NAVIGATION, etc. 21 MARINE ENGINEERING, NAVIGATION, etc. Chain Cables. CHAIN CABLES AND CHAINS. Comprising Sizes and Curves of Links, Studs, &c., Iron for Cables and Chains, Chain Cable and Chain Making, Forming and Welding Links, Strength of Cables and Chains, Certificates for Cables, Marking Cables, Prices of Chain Cables and Chains, Historical Notes, Acts of Parliament, Statutory Tests, Charges for Testing, List of Manufacturers of Cables, &c. &c. By THOMAS W. TRAILL, F.E.R.N., M. Inst. C.E., Engineer Surveyor in Chief, Board of Trade, Inspector of Chain Cable and Anchor Proving Establishments, and General Superin- tendent, Lloyd's Committee on Proving Establishments. With numerous Tables, Illustrations and Lithographic Drawings. Folio, a 2S. cloth, bevelled boaids. "It contains a vast amount of valuable information. Nothing- seems to be wanting to make it I complete and standard work of reference on the subject." Nautical Magazine. Marine Engineering. MARINE ENGINES AND STEAM VESSELS (A Treatise on). By ROBERT MURRAY, C.E. Eighth Edition, thoroughly Revised, with considerable Additions by the Author and by GEORGE CARLISLE, C.E., Senior Surveyor to the Board of Trade at Liverpool, izmo, 55. cloth boards. "Well adapted to give the young steamship engineer or marine engine and boiler maker a general introduction into hU practical work." Mechanical World. " We feel sure that this thoroughly revised edition will continue to be as popular in the future as it has been in the past, as, for its size, it contains more useful information than any similar treatise. " Industries. The information given is both sound and sensible, and well qualified to direct young sea- going hands on the straight road to the extra chiet's certificate. Most useful to survejors, inspectors, draughtsmen, and all \oung engineers who take an interest in their profession." Glasgow Herald. "An indispensable manual for the student of marine engineeiing." Liverpool Mercury. English J Pocket-BooJc for Marine Engineers. A POCKET-BOOK OF USEFUL A POCKET-BOOK OF USEFUL TABLES AND FOP- MULM FOR MARINE ENGINEERS. By FRANK PROCTOR, A.I.N.A. Third Edition. Royal 32mo, leather, gilt edges, with strap, 45. 11 supply a long-felt >- " ' ." United Service 11 We recommend it to our readers as going far to supply a long-felt want." Naval Science. "A most useful companion to all marine engineers." United Service Gazette. Introduction to Marine Engineering. ELEMENTARY ENGINEERING : A Manual for Young Marine Engineers and Apprentices. In the Form of Questions and Answers on Metals, Alloys, Strength of Materials, Construction and Management of Marine Engines and Bjiltrs, Geometry, &c. &c. With an Appendix of Useful Tables. By JOHN SHERREN BREWER, Government Marine Surveyor, Hong- kong. Small crown 8vo, as. cloth. " Contains much valuable information for the class for whim it is intended, especially la the chapters on the management of boilers and eng nes." Nautical Magazine. - A useful introduction to the more elaborate text books. "-S-oM'/m*. ^^^LJo astudi-nt v>-h . t.;uth>: n-.jiii^te desire and resolve to attai.i a thorough knowledge. Mr. nHPfr offers decidedly useful help.' Athtnaum. Navigation. PRACTICAL NAVIGATION. Consisting of THE SAILOR'S SEA-BOOK, by JAMES GKEENWOOD and W. H. ROSSER ; together with the requisite Mathemaiiral and Nautical Tab'esforthe Working of the Problems, by HENRY LAW, C.E. , and Professor J. R. YOUNG. Illustrated, izmo, 71. stroagly half-bound. 22 CROSBY LOCKWOOD &> SON'S CATALOGUE. MINING AND METALLURGY. Metalliferous Mining in the United Kingdom. BRITISH MINING : A Treatise on the History , Discovery , Practical Development, and Future Prospects of Metalliferous Mines in the United King- dom. By ROBERT HUNT, F.R.S., Keeper of Mining Records ; Editor of " Ure's Dictionary of Arts, Manufactures, and Mines, &c. Upwards of 950 pp., with 230 Illustrations. Second Edition, Revised. Super-royal 8vo, 2 2S. cloth. " One of the most valuable works of reference of modern times. Mr. Hunt, as keeper of mining records of the United Kingdom, has had opportunities for such a task not enjoyed by anyone else, and has evidently made the most of them. . . . The language and style adopted are good, and the treatment of the various subjects laborious, conscientious, and scientific. "^Engineering. "The book is, in fact, a treasure-house of statistical information on miningr subjects, and we know of no other work embodying so great a mass of matter of ihis kind. Were this the only merit ef Mr. Hunt s volume, it would be sufficient to render it indispensable in this library of everyone interested in the development of the mining and metallurgical industries of this country." thenaum. "A mass of information not elsewhere available, and of the greatest value to those who may be interested in our great mineral industries." Engin " 'A sound, business-like collection of interesting facts. . . . The amount of Information Mr. Hunt has brought together is enormous. . . . The volume appears likely to convey more instruction upon the subject than any work hitherto published." Mining Journal. Colliery Management. THE COLLIERY MANAGER'S HANDBOOK: A Compre- hensive Treatise on the Laying-out and Working of Collieries, Designed as a Book of Reference for Colliery Managers, and for the Use of Coal-Mining Students preparing for First-class Certificates. By CALEB PAMELY, Mining Engineer and Surveyor; Member of the North of England Institute of Mining and Mechanical Engineers ; and Member of the South Wales Insti- tute of Mining Engineers. With nearly 500 Plans, Diagrams, and other Illustrations. Medium 8vo, about 00 pages. Price i js. strongly bound. [Just published. Coal and Iron. THE COAL AND IRON INDUSTRIES OF THE UNITED KINGDOM. Comprising a Description of the Coal Fields, and of the Principal Seams of Coal, with Returns of their Produce and its Distribu- tion, and Analyses of Special Varieties. Also an Account of the occurrence the Rise and Progress of Pig Iron Manufacture. By RICHARD MEADE, Assistant Keeper of Mining Records. With Maps. 8vo, i 8s. cloth. " The book is one which must rind a place on the shelves of all Interested In coal and iron production, and in the iron, steel, and other metallurgical industries." Engineer. " Of this book we may unreservedly say that it is the best of its class which we have ever met. . . A book of reference which no one engaged in the iron or coal trades should omit from his library." Iron and Coal Trades Review. Prospecting for Gold and other Metals. THE PROSPECTOR'S HANDBOOK: A Guide for the Pro- spector and Traveller in Search of Metal-Bearing or other Valuable Minerals. By J. W. ANDERSON, M.A. (Camb.), F.R.G.S., Author of "Fiji and New Caledonia." Fifth Edition, thoroughly Revised and Enlarged. Small crown 8vo, 35. 6d. cloth. "Will supply a much felt want, especially among Colonists, in whose way are so often thrown many mineralogical specimens the value of which it is difficult to determine. " E ngiiieer. "How to find commercial minerals, and how to identify them when they are found, are the 1 jading points to which attention is directed. The author has managed to pack as much practical detail into his pages as would supply material for a book three times its size." Mining yournal. Mining Notes and Formulce. NOTES AND FORMULA FOR MINING STUDENTS. By JOHN HERMAN MERIVALE, M.A., Certificated Colliery Manager, Professor of Mining in the Durham College of Science, Newcastle-upon-Tyne. Third Edition, Revised and Enlarged. Small crown Svo, zs. dd. cloth. " Invaluable to anyone who is working up for an examination on mining subjects." Coal and Iron Trade; Review. " The author has done his work in an exceedingly creditable manner, and has produced a book that will be of service to students, and those who are practically engaged in mining operations.'' Engineer. " A vast amount of technical matter of the utmost value to mining engineers, and of considei- able interest to students." Schoolmaster. MINING AND METALLURGY. Explosives. A HANDBOOK ON MODERN EXPLOSIVES. Being a Practical Treatise on the Manufacture and Application of Dynamite, Gun- Cotton, Nitro-Glycerine and other Explosive Compounds. Including the Manufacture of Collodion-Cotton. By M. EISSLER, Mining Engineer and Metallurgical Chemist, Author of " The Metallurgy of GolJ," &c. With about 100 Illustrations. Crown 8vo, IDS. 6d. cloth. " Useful not only to the miner, but also to officers of both services to whom blasting and the use of explosives generally may at any time become a necessary auxiliary." Nature. " itable mine of information on the subject of explosives employed for military, mining " and blasting purposes." Army and Navy Ga " The book is clearly written. that should be found of great ly d f explosives." Atfientzu Gold, Metallurgy of. THE METALLURGY OF GOLD : A Practical Treatise on the Metallurgical Treatment of Gold-hearing Ores. Including the Processc s ot Concentration and Chlorination, and the Assaying, Melting and Refining of Gold. By M. EISSLER, Mining Engineer and Metallurgical Chemist, formerly Assistant Assayer of the U.S. Mint, San Francisco. Third Edition, Revised and greatly Enlarged. With 187 Illustrations. Crown 8vo, izs. 6d. cloth. "This book thoroughly deserves its title of a Practical Treatise.' The whole process of gold of the bullion, is described in cle ess of detail." Saturday Review. aluable data, and we strongly recommend it to all professional men engaged in the gold-mining industry." Mining Journal Silver, Metallurgy of. THE METALLURGY OF SILVER : A Practical Treatise on the Amalgamation, Roasting and Lixiviation of Silver Ores, Including the Assaying, Melting and Refining of Silver Bullion. By M. EISSLER, Author of "The Metallurgy of Gold " Second Edition/Enlarged. With 150 Illus- trations. Crown 8vo, los. 6d. cloth. [Jmt published. "A practical treatise, and a technical work which we are convinced will supply a long felt want amongst practical men, and at the same time be of value to students and others indirectly connected with tne industries." Mining Journal. "From first to last the book is thoroughly sound and reliable."-C0/AVry Guardian. " For chemists, practical miners, assayers and investors alike, we do not know of any work on the subject so handy and yet so comprehensive." Glasgow Herald. Silver-Lead, Metallurgy of. THE METALLURGY OF ARGENTIFEROUS LEAD: A Practical Treatise on the Smelting of Silver-Lead Ores and the Refining of Lead Bullion. Including Reports on various Smelting Establishments and Descriptions of Modern Furnaces and Plants in Europe and America. By M. EISSLER, M.E., Author of "The Metallurgy of Gold," &c. Crown 8vo. 400 pp., with numerous Illustrations, its. 6d. cloth. [Just published. Metalliferous Minerals and Mining. TREATISE ON METALLIFEROUS MINERALS AND MINING. By D. C. DAVIES, F.G.S., Mining Engineer, &c., Author of "A Treatise on Slate and Slate Quarrying." Illustrated with numerous Wood Engravings. Fourth Edition, carefully Revised. Crown 8vo, I2J. 6d. cloth. Neither the practical miner nor the general reader interested in mines canhave a better book for his companion and his guide. "-Mining Journal. {Mining World. ' We are doing- our readers a service in calling their attention to this valuable work. " As a history of the present state of mining throughout the world this book has a real value, and it supplies an actual want." Athenaeum. Earthy Minerals and Mining. A TREATISE ON EARTHY & OTHER MINERALS AND MINING. By D. C. DAVIES, F.G.S. Uniform with, and forming a Com- panion Volume to, the same Author's " Metalliferous Minerals and Mining." With 76 Wood Engravings. Second Edition. Crown 8vo, 12S. 6d. cloth. " We do not remember to have met with any English work on mining matters that contains " same amount of information packed in equally convenient form." Academy. " We should be inclined to rank it as among the very best of the handy technical and trades nuals which have recently appeared." British Quarterly Review. 24 CROSBY LOCK WOOD & SON'S CATALOGUE. Mineral Surveying and Valuing. THE MINERAL SURVEYOR AND VALUER'S COMPLETE GUIDE, comprising a Treatise on Improved Mining Surveying and the Valua- tion of Mining Properties, with_ New Traverse Tables. By WM. LINTERN, Mining and Civil Engineer. Third Edition, with an Appendix on " Magnetic and Angular Surveying," with Records of the Peculiarities of Needle Dis- turbances. With Four Plates of Diagrams, Plans, &c. ismo, 45. cloth. " Mr. Lintern's book forms a valuable and thoroughly trustworthy guide." Iron and Coal Trades Review, " This new edition must be of the highest value to colliery surveyors, proprietors and mana- gers." Colliery Guardian. Asbestos and its Uses. ASBESTOS: Its Properties, Occurrence and Uses. With some Account of the Mines of Italy and Canada. By ROBERT H. JONES. With Eight Collotype Plates and other Illustrations. Crown 8vo, I2s. 6d. cloth. "An interesting and invaluable work." Colliery Guardian. " We couns"! our readers to get this exceedingly interesting work for themselves ; they will find in it much that is suggestive, and a great deal that is of immediate and practical usefulness." " A'valuable addition to the architect's and engineer's library." Building News. Underground Pumping MacJiinery. MINE DRAINAGE. Being a Complete and Practical Treatise on Direct-Acting Underground Steam Pumping Machinery, with a Descrip- tion of a large number of the best known Engines, their General Utility and the Special Sphere of their Action, the Mode of their Application, and their merits compared with other forms of Pumping Machinery. By STEPHEN MICHELL. 8vo, 155. cloth. " Will be highly esteemed by c generally who require to be acquainted with the best Is a most valuable work, and stands almost alone in the literature of steam pumping machinery.' Colliery Guardian. " Much valuable information is given, so that the book Is thoroughly worthy of an extensive circulation amongst practical men and purchasers of machinery.' 1 Mining Journal. Mining Tools. A MANUAL OF MINING TOOLS. For the Use of Mine Managers, Agents, Students, &c. By WILLIAM MORGANS, Lecturer on Prac- tical Mining at the Bristol School of Mines, izmo, as. dd. cloth limp. ATLAS OF ENGRAVINGS to Illustrate the above, contain- ing 235 Illustrations of Mining Tools, drawn to scale. 4to, 45. 6d. cloth. " Students in the science of mining, and overmen, captains, managers, and viewers may gain practical knowledge and useful hints by the study of Mr. Morgans' manual." Colliery G-uardian. "A valuable work, which will tend materially to improve our mining literature." Mining Journal. Coal Mining. COAL AND COAL MINING: A Rudimentary Treatise on. By the late Sir WARINGTON W. SMYTH, M.A., F.R.S., &c., Chief Inspector of the Mines of the Crown. Seventh Edition, Revised and Enlarged. With numerous Illustrations. I2mo, 45. cloth boards. "As an outline is given of every known coal-field in this and other countries, as well as of the principal methods of working, the book will doubtless interest a very large number of readers." Mining Journal. Subterraneous Surveying. SUBTERRANEOUS SURVEYING, Elementary and Practical Treatise on, vrith and without the Magnetic Needle. By THOMAS FEN WICK, Surveyor of Mines, and THOMAS BAKER, C.E. Illust. izmo, $s. cloth boards. Granite Quarrying. GRANITES AND OUR GRANITE INDUSTRIES. By GEORGE F. HARRIS, F.G.S., Membre de la Societe Beige de Geologic, Lec- turer on Economic Geology at the Birkbeck Institution, &c. With Illustra- tions. Crown 8vo, zs. 6d. cloth. " A clearly and well-written manual for persons engaged or interested In the granite industry." Scots ELECTRICITY, ELECTRICAL ENGINEERING, etc. 25 ELECTRICITY, ELECTRICAL ENGINEERING, etc. Electrical Engineering. THE ELECTRICAL ENGINEER'S POCKET-BOOK OF MODERN RULES, FORMULA, TABLES AND DATA. By H. R. KEMPE, M.Inst.E.E., A.M.Inst C.E., Technical Officer Postal Telegraphs, Author of "A Handbook of Electrical Testing," &c. With numerous Illus- trations, royal 32010, oblong, 55. leather. [Just published. . " There is very little in the shape of formulae or data which the electrician is likely to want in a hurry which cannot be found in its pages." Practical Engineer. "A very useful book of reference for daily use in practical electrical engineering and its various applications to the industries of the present day." Iron. " It is the best book of its kind." Electrical Engineer. "The Electrical Engineer's Pocket- Book is a good one."- Electrician. _ " Strongly recommended to those engaged in toe various electrical industries." Electrical Electric Lighting. ELECTRIC LIGHT FITTING : A Handbook for Working Electrical Engineers, embodying Practical Notes on Installation Manage- ment. By JOHN W. URQUHART, Electrician, Author of " Electric Light,' 1 &c. With numerous Illustrations, crown 8vo, 55. cloth. \_Just published. " This volume deals with what may be termed the mechanics of electric lighting, and is addressed to men who are already engaged in the work or are training for it. The work traverses a great dtal of ground, and may be read as a sequel to the same author's useful work on Electric Light.' " Electrician. s an attemt to state in the simlest lanua h guidance of those who have to run erusal of the workmen for whom it is written." Electrical Kevin'. " Eminently practical and useful. . . . Ought to be in the hands of everyone in charge of an electric light plant." Electrical Engineer. " A really capital book, which we have no hesitation in recommending to the notice of working electricians and electrical engineers." Mechanical World. Electric lAgUt. ELECTRIC LIGHT : Its Production and Use. Embodying Plain Directions for the Treatment of Dynamo-Electric Machines, Batteries, Accumulators, and Electric Lamps. By J. W. URQUHART, C.E., Author of " Electric Light Fitting," &c. Fourth Edition, Revised, with Large Additions and 145 Illustrations. Crown 8vo, 75. 6J. cloth. [Just published. ' The book is by far the best that we have yet met with on the subject." Atfienaum. " It is the only work at present available which gives, in language intelligible for the most part ordin present 1 ' The book contains a ge not only as obtained from voltaic or galvanic batteries, but treats at length of the dyn machine in several of its forms." Colliery Guardian. Construction of Dynamos. DYNAMO CONSTRUCTION : A Practical Handbook for the Use of Engineer Constructors and. Electricians in Charge. With Examples of leading English, American and Continental Dynamos and Motors. By J. W. URQUHART, Author of " Electric Light," &c. Crown 8vo, 75. t>d. cloth. ' [Just published. existed. " The author has produced a bonk for which a demand has long existed. The subject is treated in a thoroughly practical manner." Mechanical ll'orld. Dynamic Electricity and Magnetism. THE ELEMENTS OF DYNAMIC ELECTRICITY AND MAGNETISM. By PHILIP ATKINSON, A.M., Ph.D. Crown 8vo. 400 pp. With 120 Illustrations, los. 6d. cloth. \_Jtistpublish.d. Text Book of Electricity. THE STUDENT'S TEXT-BOOK OF ELECTRICITY. By HENRY M. NOAD, Ph.D., F.R.S., F.C.S. New Edition, carefully Revised. With an Introduction and Additional Chapters, by W. H. PREECE, M.I.C.E., Vice-President of the Society of Telegraph Engineers, &c. With 470 Illustra- tions. Crown 8vo, izs. 6d. cloth. " We can recommend Dr. Noad's book for clear style, great range of subject, a good Indrx, tnd a plethora of woodcuts. Such collections as the present are indispensable/'-^ much wanted. Th: :u--d. The chapter t e fifty times the price of the book." Scottish Leather Trader. Dentistry. MECHANICAL DENTISTRY : A Practical Treatise on the Construction of the various kinds of Artificial Dentures. Comprising also Use- ful Formulae, Tables and Receipts for Gold Plate, Clasps, Solders, &c. &c. By CHARLES HUNTER. Third Edition, Revised. With upwards of 100 Wood Engravings. Crown 8vo, 3$. 6d. cloth. " The work is very practical." Monthly Review of Denial Surgery. " We can strongly recommend Mr. Hunter's treatise to all students preparing for the profession of dentistry, as well as to every mechanical dentist." Dublin Journal of Medical Science. Wood Engraving. WOOD ENGRA VING : A Practical and Easy Introduction to the Study of the A rt. By WILLIAM NORMAN BROWN. Second Edition. With numerous Illustrations, izino, is. 6rf. cloth limp. " The book is clear and complete, and will be useful to anyone wanting to understand the first elements of the beautiful art of wood engraving." Grathic. 32 CROSBY LOCKWOOD & SON'S CATALOGUE. HANDYBOOKS FOR HANDICRAFTS. By PAUL N. HASLUCK. Metal Turning. THE METAL TURNER'S HANDYBOOK. A Practical Manual for Workers at the Foot-Lathe : Embracing Information on the Tools, Appliances and Processes employed in Metal Turning. By PAUL N. HAS- LUCK, Author of " Lathe- Work." With upwards of One Hundred Illustra- tions. Second Edition, Revised. Crown 8vo, 2s. cloth. " Clearly and concisely written, excellent in every way." Mechanical World. Wood Turning. THE WOOD TURNER'S HANDYBOOK. A Practical Manual for Workers at the Lathe : Embracing Information on the Tools, Appliances and Processes Employed in Wood Turning. By PAUL N. HASLUCK. With upwards of One Hundred Illustrations. Crown 8vo, zs. cloth. hitherto sought in vain f WOOD AND METAL TURNING. By P. N. HASLUCK. (Being the Two preceding Vols. bound together.) 300 pp., with upwards of 200 Illustrations, crown bvo, 35. 6d. cloth. Watch Repairing. THE WATCH JOBBER'S HANDYBOOK. A Practical Manual on Cleaning, Repairing and Adjusting. Embracing Information on the Tools, Materials, Appliances and Processes Employed in Watchwork. By PAUL N. HASLUCK. With upwards of One Hundred Illustrations. Cr. 8vo, zs. cloth, ' All young persons connected with the trade should acquire and study this excellent, and at the same time, inexpensive vioi'k.."Clerkeirwell Cnronicle. Clock Repairing. THE CLOCK JOBBER'S HANDYBOOK: A Practical Manual on Cleaning, Repairing and Adjusting. Embracing Information on the Tools, Materials, Appliances and Processes Employed in Clockwork. By PAUL N. HASLUCK. With upwards of 100 Illustrations. Cr. 8vo, zs. cloth. " Of inestimable service to those commencing the trade." Coventry Standard. WATCH AND CLOCK JOBBING. By P. N. HASLUCK. (Being the Two preceding Vols. bound together.) 320 pp., with upwards of aoo Illustrations, crown 8vo, 35. 6rf. cloth. Pattern MaJcing. THE PATTERN MAKER'S HANDYBOOK. A Practical Manual, embracing Information on the Tools, Materials and Appliances em- ayed in Constructing Patterns for Founders. By PAUL N. HASLUCK. ith One Hundred Illustrations. Crown 8vo, zs. cloth. "This handy volume contains sound information of considerable value to students and artificers." Hardware Trades Journal. Mechanical Manipulation. THE ME CHA NIC'S WORKSHOP HA ND YBOOK. A Practical Manual on Mechanical Manipulation. Embracing Information on various Handicraft Processes, with Useful Notes and Miscellaneous Memoranda. By PAUL N. HASLUCK. Crown 8vo, as. cloth. " It is a book which should be found in every workshop, as it is one which will be continually referred to for a very great amount of standard information." Saturday Revim. Model Engineering. THE MODEL ENGINEER'S HANDYBOOK: A Practical Manual on Model Steam Engines. Embracing Information on the Tools, Materials and Processes Employed in their Construction. By PAUL N. HASLUCK. With upwards of 100 Illustrations. Crown 8vo, zs. cloth. " By carefully going through the work, amateurs may pick up an excellent notion of the con- struction of full-sized sttam engines." Te^rajtAic Journal. Cabinet Making. THE CABINET WORKER'S HANDYBOOK: A Practical Manual, embracing Information on the Tools, Materials, Appliances and Processes employed in Cabinet Work. By PAUL N. HASLUCK, Author of " Lathe Work," &c. With upwards of 100 Illustrations. Crown 8vo, as. cloth. [Glasgow Herald. ' Thoroughly practical throughout. The amateur worker in wood will find it most useful." INDUSTRIAL AND USEFUL ARTS. 33 Electrolysis of Gold, Silver, Copper, etc. ELECTRO-DEPOSITION : A Practical Treatise on the Electrolysis of Gold, Silver, Copper, Nickel, and other Metals and Alloys. With descrip- tions of Voltaic Batteries, Magneto and Dynamo-Electric Machines, Ther- mopiles, and of the Materials and Processes used in every Department of the Art, and several Chapters on Electro-Metallurgy. By ALEXANDER WATT. Third Edition, Revised and Corrected. Crown 8vo, gs. cloth. "Eminently a book for the practical worker in electro-deposition. It contains practical descriptions of methods, processes and materials as actually pursued and used in the workshop." Engineer. Electro-Metallurgy. ELECTRO-MET ALL URG Y ; Practically Treated. By ALEXANDER WATT. Author of " Electro-Deposition," &c. Ninth Edition, Enlarged and Revised, with Additional Illustrations, and including the most recent Processes. i2mo, 45. cloth boards. "From this book both amateur and artisan may learn everything necessary for the successful prosecution of electroplating." Iron. Electroplating. ELECTROPLATING: A Practical Handbook on the Deposi- tion of Copper, Silver, Nickel, Gold, Aluminium, Brass, Platinum, &c. &c. With Descriptions of the Chemicals, Materials, Batteries and Dynamo Machines used in the Art. By J. W. URQUHART, C.E. Second Edition, with Additions. Numerous Illustrations. Crown 8vo. 55. cloth. " An excellent practical manual." -Engineering. " An excellent work, giving the newest information." Horological Journal. Electrotyping. ELECTROTYPING : The Reproduction and Multiplication of Print- ing Surfaces and Works of Art by the Electro-deposition of Metals. By J. W. URQUHART, C.E. Crown 8vo, 55. cloth. " The book is thoroughly practical. The reader is, therefore, conducted through the leading laws of electricity, then through the metals used by electrotypers, the apparatus, and the depositing processes, up to the final preparation of the work." Art Journal. Horology. A TREATISE ON MODERN HOROLOGY, in Theory and Prac- tice. Translated from the French of CLAUDIUS SAUNIER, by JULIEN TRIP- PLIN, F.R.A.S., and EDWARD RIGG, M.A., Assayer in the Royal Mint. With 78 Woodcuts and 22 Coloured Plates. Second Edition. Royal 8vo, 2 2s. cloth ; 2 IDS. half-calf. " There is no horologicat work in the English language at all to be compared to this produc- tion of M. Saunier's for clearness and completeness. It is alike good as a guide for the student and as a reference for the experienced horologist and skilled workman." Horological Journal. " The latest, the most complete, and the most reliable of those literary productions to which continental watchmakers are indebted for the mechanical superiority over their English brethren -in fact, the Book of Books, is M. Saunier's Treatise. '"-Watchmaker, Jeweller and Silversmith. Watchmaking. THE WATCHMAKER'S HANDBOOK. A Workshop Com- panion for those engaged in Watchmaking and the Allied Mechanical Arts. From the French of CLAUDIUS SAUNIER. Enlarged by JULIEN TRIPPLIN, F.R.A.S., and EDWARD RIGG, M.A., Assayer in the Koyal Mint. Woodcuts and Copper Plates. Third Edition, Revised. Crown 8vo, gs. cloth. " Each part is truly a treatise in itself. The arrangement is good and the language is clear and concise. It is an admirable guide for the young watchmaker." I: ngmeering. ' It is impossible to speak too highly of its excellence. It fulfils every requirement in a hand- book intended for the use of a workman."-^*/, and Clocknuike- . . This book contains an immense number of practical details bearing on the daily occupation of a watchmaker." Watchmaker and Metal-worker (Chicago). Goldsmiths' Work. THE GOLDSMITH'S HANDBOOK. By GEORGE E. GEE, Jeweller, &c. Third Edition, considerably Enlarged, ismo, 3$. 6d. cl. bds. "A good, sovnd educator, and will be accepted as an authority." Horological Journal. Silversmiths' Work. THE SILVERSMITH'S HANDBOOK. By GEORGE E. GEE, Jeweller, &c. Second Edition, Revised, with numerous Illustrations, nmo, ".vorke'rs iri the Wade S will speedily discover its merits when they sit down to study it. M - Enz is ec anu^ ^^ ^ W orks together, strongly half-bound, price 75. 34 CROSBY LOCKWOOD & SON'S CATALOGUE. Bread and Biscuit Baking. THE BREAD AND BISCUIT BAKER'S AND SUGAR- BOILER'S ASSISTANT. Including a large variety of Modern Recipes. With Remarks on the Art of Bread-making. By ROBERT WELLS, Practical Baker. Second Edition, with Additional Recipes. Crown 8vo, 2s. cloth. " A large number of wrinkles for the ordinary cook, as well as the baker." Saturday Review. Confectionery. THE PASTRYCOOK AND CONFECTIONER'S GUIDE. For Hotels, Restaurants and the Trade in general, adapted also for Family Use. By ROBERT WELLS, Author of " The Bread and Biscuit Baker's and Sugar Boiler's Assistant." Crown 8vo, 2S. cloth. " We c our reader Ornamental Confectionery. ORNAMENTAL CONFECTIONERY : A Guide for Bakers, Confectioners and Pastrycooks ; including a variety of Modern Recipes, and Remarks on Decorative and Coloured Work. With 129 Original Designs. By ROBERT WELLS. Crown 8vo, 55. cloth. "A valuable work, and should be in the hands of every baker and confectioner. The illus- trative designs are alone worth treble the amount charged for the whole work."-aAcrs' 'Jimcs. Flour Confectionery. THE MODERN FLOUR CONFECTIONER. Wholesale and Retail.. Containing a large Collection of Recipes for Cheap Cakes, Biscuits, &c. With Remarks on the Ingredients used in their Manufacture, &c. By R. WELLS, Author of " Ornamental Confectionery," "The Bread and Biscuit Baker," " The Pastrycook's Guide," &c. Crown 8vo, zs. cloth. Laundry Work. LA UN DRY MANAGEMENT. A Handbook for Use in Private and Public Laundries, Including Descriptive Accounts ot Modern Machinery and Appliances for Laundry Work. By the EDITOR of " The Laundry Journal." With numerous Illustrations. Crown 8vo, as. 6d. cloth. CHEMICAL MANUFACTURES & COMMERCE. New Manual of Engineering Chemistry. ENGINEERING CHEMISTRY: A Practical Treatise for the Use of Analytical Chemists, Engineers, Iron Masters, Iron Founders, Students, and others. Comprising Methods of Analysis and Valuation of the , . Principal Materials used in Engineering Work, with numerous Analyses, Examples, and Suggestions. By H. JOSHUA PHILLIPS, F.I.C., F.C.S., Analytical and Consulting Chemist to the Great Eastern Railway. Crown bvo, cloth. [Just published. small service to a numerous body of practical men. . . . The analytical methods may be pronounced most satisfactory, being as accurate as the despatch required of engineering chemists permits." Chemical News. Analysis and Valuation of Fuels. FUELS: SOLID, LIQUID AND GASEOUS, Their Analysis and Valuation. For the Use of Chemists and Engineers. By H. J. PHILLIPS, F.C.S., Analytical and Consulting Chemist to the Great Eastern Railway. Crown 8vo, 35. 6d. cloth. " Ought to have its place in the laboratory of every metallurgical establishment, and wherever fuel is used on a large sc*le."~CJtemicat News. "Cannot fail to be of wide interest, especially at the present time." Railway A'rrvs. Alkali Trade, Manufacture of Sulphuric Acid, etc. A MANUAL OF THE ALKALI TRADE, including the Manufacture of Sulphuric Acid, Sulphate of Soda, and Bleaching Powder. By JOHN LOMAS. 390 pages. With 232 Illustrations and Working Drawings. Second Edition. Royal 8vo, i ios. cloth. "This book is written by a manufacturer for manufacturers. The working details of the most approved forms of apparatus are given, and these are accompanied by no less than 232 wood en- gravings, all of which may be used for the purposes of construction. 1 ' Athentzum, AGRICULTURE, FARMING, GARDENING, etc. 35 The Blowpipe. THE BLOWPIPE IN CHEMISTRY, MINERALOGY, AND GEOLOGY. Containing all known Methods of Anhydrous Analysis, Work- ing Examples, and Instructions for Making Apparatus. By Lieut.-Col. W. A. Ross, R.A. With 120 Illustrations. New Edition. Crown 8vo, 55. cloth. "The student who goes through the course of experimentation here laid down will gain a better insight into inorganic chemistry and mineralogy than if he had -got up' any of the best text-books ot the day, and passed any number of examinations in their contents. pI -CAto/ News . Commercial Chemical Analysis. THE COMMERCIAL HANDBOOK OF CHEMICAL ANA- LYSIS; or, Practical Instructions tor the determination oi the Intrinsic or Commercial Value of Substances used in Manufactures.Trades, and the Arts. By A. NORMANDY. New Edition by H. M. NOAD, F.R.S. Cr. 8vo, its. 6d. cl. "Essential to the analysts appointed under the new Act. The most recent results are given, and the work is well edited and carefully written." Nature. Brewing. A HANDBOOK FOR YOUNG BREWERS. By HERBERT EDWARDS WRIGHT, B.A. New Edition, much Enlarged. [In the press. Dye-Wares and Colours. THE MANUAL OF COLOURS AND DYE-WARES : Their Properties, Applications, Valuation, Impurities, and Sophistications. For the use of Dyers, Printers, Drysalters, Brokers, &c. By J. W. SLATER. Second Edition, Revised and greatly Enlarged. Crown 8vo, 75. 6d. cloth. "A complete encyclopaedia of the materia tinctoria. The information given respecting each article is full and precise, and the methods of determining the value of articles such as these, so liable to sophistication, are given with clearness, and are practical as well as valuable." Chemist " There is no other work which covers precisely the same ground. To students preparing for examinations in dyeing and printing it will prove exceedingly useful." Chemical News. Pigments. THE ARTIST'S MANUAL OF PIGMENTS. Showing their Composition, Conditions of Permanency, Non- Permanency, and Adul- terations; Effects in Combination with Each Other and with Vehicles ; and the most Reliable Tests of Purity. By H. C. STANDAGE Second Edition. Crown 8vo, as. 6d. cloth. " This work is indeed multum-in-parvo, and we can, with good conscience, recommend it to all who come in contact with pigments, whether as makers, dealers or users." Chemical Review. Gauging. Tables and Rules for Revenue Officers, Brewers, etc. A POCKET BOOK OF MENSURATION AND GAUGING : Containing Tables, Rules and Memoranda for Revenue Officers, Brewers, Spirit Merchants, &c. By J. B. MANT (Inland Revenue). Second Edition Revised. Oblong i8mo, 45. leather, with elastic band. " This handy and useful book is adapted to the requirements of the Inland Revenue Depart- ment, and will be a favourite book of reference." Civilian. " Should be in the hands of every practical brewer." Brewers' Journal. AGRICULTURE, FARMING,. GARDENING, etc. Youatt and Burn's Complete Grazier. THE COMPLETE GRAZIER, and FARMER'S and CATTLE- BREEDER'S ASSISTANT. Including the Breeding, Rearing, and Feeding of Stock; Management of the Dairy, Culture and Management of Grass Land, and of Grain and Root Crops, &c. By W. YOUATT and R. SCOTT BURN. An entirely New Edition, partly Re-written and greatly Enlarged, by W. FREAM, B.Sc.Lond., LL.D. In medium 8vo, about 1,000 pp. [/ the press. Agricultural Facts and Figures. NOTE-BOOK OF AGRICULTURAL FACTS AND FIGURES FOR FARMERS AND FARM STUDENT late Professor of Agriculture, Glasgow Veter .^hrmo 3 st?omp 4 le; e le and h c e omprehensiveNote.bookfor have Jen UHterT teems with information, and we can c with agrcuilture."-M>r/A British Agriculturist. armers and Farm St rdia llyrecommend it t 36 CROSBY LOCKWOOD & SON'S CATALOGUE. Flour Manufacture, Milling, etc. FLOUR MANUFACTURE: A Treatise on Milling Science and Practice. By FRIEDRICH KICK, Imperial Regierungsrath, Professor of Mechanical Technology in the Imperial German Polytechnic Institute, Prague. Translated from the Second Enlarged and Revised Edition with Supplement. By H. H. P. POWLKS, A.M.I.C.E. Nearly 400 pp. Illustrated with 28 Folding Plates, and 167 Woodcuts. Royal 8vo, 255. cloth. " This valuable work is, and will remain, the standard authority on the science of milling. , . The miller who has read and digested this work will have laid the foundation, so to speak, of a suc- cessful career ; he will have acquired a number of general principles which he can proceed to apply. In this handsome volume we at last have the accepted text-book of modern milling in good, sound English, which has little, if any, trace of the German idiom."- Tlie Miller. " The appearance of this celebrated work in English is very opportune, and British millers will, we are sure, not be slow in availing themselves of its pages."-Milters' Gazette. Small Farming. SYSTEMATIC SMALL FARMING; or, The Lessons of my Farm. Being an Introduction to Modern Farm Practice for Small Farmers in the Culture of Crops ; The Feeding of Cattle; The Management of the Dairy, Poultry and Pigs, &c. &c. By ROBERT SCOTT BURN, Author of " Out- lines of Landed Estates' Management." Numerous Illusts., cr. 8vo, 6s. cloth. "This is the completes! book of its class we have seen, and one which every amateur farmer will read with pleasure and accept as a guide." Field. "The volume contains a vast amount of useful information. No branch of farming is le't untouched, from the labour to be done to the results achieved. It may be safely recommended to 11 who think they will be in paradise when they buy or rent a three-acre farm" Glasgow Herald. Modern Farming. OUTLINES OF MODERN FARMING. By R. SCOTT BURN. Soils, Manures, and Crops Farming and Farming Economy Cattle, Sheep, and Horses Management of Dairy, Pigs and Poultry Utilisation of Town-Sewage, Irrigation, &c. Sixth Edition. In One Vol., 1,250 pp., half- bound, profusely Illustrated, 125. " The aim of the author has been to make his work at once comprehensive and trustworthy, he has succeeded to a degree which entitles him to much credit." Morning " " and in th Advertiser. " No farmer should be without this book." Banbury Guardian. Agricultural Engineering. FARM ENGINEERING, THE COMPLETE TEXT-BOOK OF. Comprising Draining and Embanking; Irrigation and Water Supply ; Farm Roads, Fences, and Gates ; Farm Buildings, their Arrangement and Con- struction, with Plans and Estimates; Barn Implements and Machines ; Field Implements and Machines; Agricultural Surveying, Levelling, &c. By Prof. JOHN SCOTT, Editor of the " Farmers' Gazette," late Professor of Agriculture and Rural Economy at the Royal Agricultural College, Cirencester, &c, &c. In One Vol., 1,150 pages, half-bound, with over 600 Illustrations, i2s. "Written with great care, as well as with knowledge and ability. The author has done his work well ; we have found him a very trustworthy guide wherever we have tested his statements. The volume will be of great value to agricultural students. "-Mark Lane Express. " For a youn Bell's Weekly English Agriculture. THE FIELDS OF GREAT BRITAIN: A Text-Book of Agriculture, adapted to the Syllabus of the Science and Art Department. For Elementary and Advanced Students. By HUGH CLEMENTS (Board of Trade). Second Ed., Revised.with Additions. i8mo, 2s. 6d. cl. "A most comprehensive volume, giving a mass of information." Agricultural Economist. " It is a long time since we have seen a book which has pleased us more, or which contains such a vast and useful fund of knowledge." Educational Times. Tables for Farmers, etc. TABLES, MEMORANDA, AND CALCULATED RESULTS for Farmers, Graziers, Agricultural Students, Surveyors, Land Agents Auc- tioneers, etc. With a New System of Farm Book-keeping. Selected and Arranged by SIDNEY FRANCIS. Second Edition, Revised. 272 pp., waist- coat-pocket size, is. 6d. limp leather. " Weighing less than i oz., and occupying no more space than a match box, it contains a mass of facts and calculations which has never before, in such handy form, been obtainable. Every operation on the farm is dealt with. The work may be taken as thoroughly accurate, the whole ot tne tables having been revised by Dr. Fream. We cordially recommend it." ell's Weekly Messenger. " A marvellous little book. . . . The agriculturist who possesses himself of it will not be disappointed with his investment." The Farm. AGRICULTURE, FARMING, GARDENING, ett. 37 Farm and Estate Book-keeping. BOOK-KEEPING FOR FARMERS & ESTATE OWNERS. A Practical Treatise, presenting, in Three Plans, a System adapted for all Classes of Farms. By JOHNSON M. WOODMAN, Chartered Accountant. Second Edition, Revised. Cr. 8vo, 3 s. 6d. cl. bds. ; or zs. 6d. cl. limp. .. i?n V' olume ls a capital study of a most important subject." AgriciMur at Gazette. Will be found of great assistance by those who intend to commence a system of book-keep- ing-, the author's examples being clear and explicit, and his explanations, while full and accurate, being to a large extent free from technicalities."-/:. Stock Journal. Farm Account Book. WOODMAN'S YEARLY FARM ACCOUNT BOOK. Giving a_Weekly Labour Account and Diary, and showing the Income and Expen- diture under each Department of Crops, Live Stock, Dairy, &c. &c. With Valuation, Profit and Loss Account, and Balance Sheet at the end of the Year, and an Appendix of Forms. Ruled and Headed for Entering a Com- plete Record of the Farming Operations. By JOHNSON M. WOODMAN, Chartered Accountant. Folio, ys. 6d. half bound. [.culture. "Contains every requisite form for keeping farm accounts readily and accurately." Agri- Early Fruits, Flowers and Vegetables. THE FORCING GARDEN ; or, How to Grow Early Fruits, Flowers, and Vegetables. With Plans and Estimates for Building Glass- houses, Pits and Frames. By SAMUEL WOOD. Crown 8vo, 35. 6d. cloth. " A good book, and fairly fills a place that was in some degree vacant. The book is written with great care, and contains a great deal of valuable teaching." Gardeners' Magazine. " Mr. Wood's book is an original and exhaustive answer to the question ' How to Grow Early Fruits, Flowers and Vegetables? ' "Land and Water. Good Gardening. A PLAIN GUIDE TO GOOD GARDENING ; or, How to Grow Vegetables, Fruits, and Flowers. With Practical Notes on Soils, Manures, Seeds, Planting, Laying-out of Gardens and Grounds, &c. By S. WOOD. Fourth Edition, with numerous Illustrations. Crown 8vo, 35. 6d. cloth. "A very good book, and one to be highly recommended as a practical guide. The practical directions are excellent." Athcnizum. " May be recommended to young gardeners, cottagers, and specially to amateurs, for the plain, simple, and trustworthy information it gives on common matters too often neglected." Gardener? Chronicle. Gainful Gardening. MULTUM-IN-PARVO GARDENING; cr, How to make One Acre of Land produce 620 a-year by the Cultivation of Fruits and Vegetables ; also, How to Grow Flowers in Three Glass Houses, so as to realise 176 per annum clear Profit. By S. WOOD. Fifth Edition. Crown 8vo, is. sewed. "We are bound to recommend it as not only suited to the case of the amateur and gentleman's gardener, but to the market grower." Gardeners' Magazine. Gardening for Ladies. THE LADIES' MULTUM-IN-PARVO FLOWER GARDEN, nd Amateurs' Complete Guide. By S. WOOD. With Illusts. Cr.Svo, 35. &t. cl. " This volume contains a good deal of sound, common sense instruction." Florist. " Full of shrewd hints and useful instructions, based on a lifetime of experience." Scotsman. Receipts for Gardeners. GARDEN RECEIPTS. By C. W. QUIN. i2mo, is. 6d. cloth. "A useful and handy book, containing a good deal of valuable information." Athencrum. Market Gardening. MARKET AND KITCHEN GARDENING. By Contributors to "The Garden." Compiled by C. W. SHAW, late Editor of "Gardening Illustrated." iamo, 3$. fid. cloth boards. " The most valuable compendium of kitchen and market-garden work published." Farmer. Cottage Gardening. COTTAGE GARDENING; or, Flowers, Fruits, and Vegetables for Small Gardens. By E. HOBDAY, izmo, is. 6d. cloth limp. Potato Culture. POTATOES : How to Grow and Show Them. A Practical Guide to the Cultivation and General Treatment of the Potato. By JAMES PINK. Second Edition. Crown bvo, as. cloth. 38 CROSBY LOCKWOOD & SON'S CATALOGUE. LAND AND ESTATE MANAGEMENT, LAW, etc. Hudson's Land Valuer's Pocket-Book. THE LAND VALUER'S BEST ASSISTANT: Being Tables on a very much Improved Plan, for Calculating the Value of Estates. With Tables for reducing Scotch, Irish, and Provincial Customary Acres to Statute Measure, &c. By R. HUDSON, C.E. New Edition. Royal 32010, leather, elastic band, 45. "This new edition includes tables for ascertaining: the value of leases for any term of years ; and for showing how to lay out plots of ground of certain acres in forms, square, round, &c., with valuable rules for ascertaining the probable worth of standing timber to any amount ; and is of incalculable value to the country gentleman and professional man." Farmers Journal. Ewart's Land Improver's Pocket-Book. THE LAND IMPROVER'S POCKET-BOOK OF FORMULA, TABLES and MEMORANDA required in any Computation relating to the Permanent Improvement of Landed Property. By JOHN EWART, Land Surveyor and Agricultural Engineer. Second Edition, Revised. Royal same, oblong, leather, gilt edges, with elastic band, 45. "A compendious and handy little volume." Spectator. Complete Agricultural Surveyor's Pocket-Book. THE LAND VALUER'S AND LAND IMPROVER'S COM- PLETE POCKET-BOOK. Consisting of the above Two Works bound to- gether. Leather, gilt edges, with strap, 75. 6d. " Hudson's book is the best ready-reckoner on matters relating to the valuation of land and crops, and its combination with Mr. Ewart's work greatly enhances the value and usefulness of the latter-mentioned. ... It is most useful as a manual for reference." North of England Fanner. Auctioneer's Assistant. THE APPRAISER, A UCTIONEER, BROKER, HOUSE AND ESTA TE A GENT A ND VA L UER'S POCKET A SSISTANT, lor the Valua- tion for Purchase, Sale, or Renewal of Leases, Annuities and Reversions, and of property generally; with Prices for Inventories, &c. By JOHN WHEELER, Valuer, &c. Fifth Edition, re-written and greatly extended by C. MORRIS, Surveyor, Valuer, &c. Royal 32010, 55. cloth. " A neat and concise book of reference, containing an admirable and clearly-arranged list of prices for inventories, and a very practical guide to determine the value of furniture, &c." Standard. " Contains a large quantity of varied and useful information as to the valuation for purchase, sale, or renewal of leases, annuities and reversions, and of property generally, with prices for Inventories, and a guide to determine the value of interior fittings and other effects." BuiUer. Auctioneering. AUCTIONEERS: THEIR DUTIES AND LIABILITIES. A Manual of Instruction and Counsel for the Young Auctioneer. By ROBERT SQUIBBS, Auctioneer. Second Edition, Revised and partly Re- written. Demy 8vo, I2S. 6d. cloth. "The position and duties of auctioneers treated compendiously and clearly." Builder. "Every auctioneer ought to possess a copy of this excellent work." Ironmonger. Legal Guide for Pawnbrokers. THE PAWNBROKERS', FACTORS' AND MERCHANTS' GUIDE TO THE LAW OF LOANS AND PLEDGES. With the Statutes and a Digest of Cases on Rights and Liabilities, Civil and Criminal, as to Loans and Pledges of Goods, Debentures, Mercantile and other Se- curities. By H. C. FOLKARD, Esq., Barrister-at-Law, Author of " The Law of Slander and Libel," &c. With Additions and Corrections. Fcap. 8vo, 35. 6d. cloth. " This work contains simply everything that requires to be known concerning the department of the law of which it treats. We can safely commend the book as unique and very nearly perfect.' Iron. " The task undertaken by Mr. Folkard has been very satisfactorily performed. . . . Such ex- planations as are needful have been supplied with great clearness and with due regard to brevity. ' City Press. LAND AND ESTATE MANAGEMENT, LAW, etc. 39 Law of Patents. PATENTS FOR INVENTIONS, AND HO W TO PROCURE THEM. Compiled for the Use of Inventors, Patentees and others. By G. G. M. HARDINGHAM, Assoc.Mem.Inst.C.E., &c. Demy 8vo, cloth, price 2S. 6d. Metropolitan Rating Appeals. REPORTS OF APPEALS HEARD BEFORE THE COURT OF GENERAL ASSESSMENT SESSIONS, from the Year 1871 to 1885. By EDWARD RYDE and ARTHUR LYON RYDE. Fourth Edition, brought down to the Present Date, with an Introduction to the Valuation (Metropolis) Act, 1869, and an Appendix by WALTER C. RYDE, ot the Inner Temple, Barrister- at-Law. 8vo, i6s. cloth. " A useful work, occupying a place mid-way between a handbook for a lawyer and a guide to the surveyor. It is compiled by a gentleman eminent in his profession as a land agent, whose spe- cialty, it is acknowledged, lies i the direction of assessing property for rating purposes." Land Agents' Record. " It is an indispensable work of reference for all engaged in assessment business." Journal of Gas Lighting. House Property. HANDBOOK OF HOUSE PROPERTY. A Popular and Practi- cal Guide to the Purchase, Mortgage, Tenancy, and Compulsory Sale of Houses and Land, including the Law of Dilapidations and Fixtures; with Examples of all kinds of Valuations, Useful Information on Building, and Suggestive Elucidations of Fine Art. By E. L. TARBUCK, Architect and Surveyor. Fourth Edition, Enlarged. 121110, 55. cloth. " The advice is thoroughly practical." Law Journal. " For all who have dealings with house property, this is an indispensable guide." Decoration. "Carefully brought up to date, and much improved by the addition of a division on fine art. " A well-written and thoughtful work." Land Agent's Record. Inwood's Estate Tables. TABLES FOR THE PURCHASING OF ESTATES, Freehold, Copy hold, or Leasehold; Annuities, A dvowsons, etc., and for the Renewing of Leases held under Cathedral Churches, Colleges, or other Corporate bodies, for Terms ol Years certain, and for Lives ; also for Valuing Reversionary Estates, Deferred Annuities, Next Presentations, &c. ; together with SMART'S Five Tables of Compound Interest, and an Extension of the same to Lower and Intermediate Rates. By W. INWOOD. 23rd Edition, with considerable Additions, and new and valuable Tables of Logarithms for the more Difficult Computations of the Interest oi Money, Discount, Annuities, &c., by M. FEDOR THOMAN, oi the Socie^ Credit Mobilier of Paris. Crown 8vo, 8s. cloth. "Those interested in the purchase and sale of estates, and in the adjustment of compensation cases, as well as in transactions in annuities, life insurances, &C H will find the present edition of " 'Inwood's Tables' still maintain a most enviable reputation. The new issue has been enriched by large additional contributions by M. Fedor Thoman, whose carefully arranged Tables cannot lail to be of the utmost utility." Mining Journal. Agricultural and Tenant-Right Valuation. THE AGRICULTURAL AND TENANT-RIGHT-VALUER'S ASSISTANT. A Practical Handbook on Measuring and Estimating the Contents, Weights and Values of Agricultural Produce and Timber, the Values of Estates and Agricultural Labour, Forms of Tenant-Right-Valua- tions, Scales ol Compensation under the Agricultural Holdings Act, 1883, &c. &c. By TOM BRIGHT, Agricultural Surveyor. Crown 8vo, 35. 6d. cloth. Full of tables and examples in connection with the valuation of tenant-right, estates, labour, contents, and weights of timber, and farm produce of all kinds." Agricultural Gazette. " An eminently practical handbook, full of practical tables and data of undoubted interest am! value to surveyors and auctioneers in preparing valuations of all kinds." Farmer. Plantations and Underwoods. POLE PLANTATIONS AND UNDERWOODS: A Practical Handbook on Estimating the Cost of Forming, Renovating, Improving and Grubbing Plantations and Underwoods, their Valuation for Purposes ol Transfer, Rental, Sale or Assessment. By TOM BRIGHT, F.S.Sc., Author of " The Agricultural and Tenant-Right-Valuer's Assistant," &c. Crown 8vo, 3 s. 6d. cloth. .[?< published. " Will be found very useful to those who are actually engaged in managing vooA."-Belft s and agents it will be a welco ssist the valuer in the discha , and use both to surveyors and auctioneers in preparing valuations of all kimls."-A>f HeraM. Be wo- "tkc. bv-a.tr 40 CROSBY LOCKWOOD &> SON'S CATALOGUE. A Complete Epitome of the Laws of this Country. EVERY MAN'S OWN LAWYER: A Handy-Book of the Principles of Law and Equity. By A BARRISTER. Twenty-ninth Edition. Revised and Enlarged. Including the Legislation of 1891, and including careful digests of The Tithe Act, 1891 ; the Mortmain and Charitable Uses Act, 1891; the Charitable Trusts (Recovery) Act, 1891 ; the Forged Transfers Act, 1891; the Custody of Children Act, 1891; the Slander of Women Act, 1891; the Public Health (London) Act, 1891; the Stamp Act, 1891; the Savings Bank Act, 1891; the Elementary Education (" Free Education") Act, 1891; the County Councils (Elections) Act, 1891; and the Land Registry (Middlesex Deeds) Aet, 1891; while other new Acts have been duly noted. Crown 8vo, 688 pp., price 6s. 8rf. (saved at every consultation ! ), strongly bound in cloth. [Just published. VTHE BOOK WILL BE FOUND TO COMPRISE (AMONGST OTHER MATTER) THE RIGHTS AND WRONGS OF INDIVIDUALS LANDLORD AND TENANT VENDORS AND PURCHASERS PARTNERS AND AGENTS COMPANIES AND ASSOCIATIONS MASTERS. SERVANTS AND WORKMEN LEASES AND MORTGAGES CHURCH AND CLERGY, RITUAL LIBEL AND SLANDER CONTRACTS AND AGREEMENTS BONDS AND BILLS OF SALE CHEQUES, BILLS AND NOTES RAILWAY AND SHIPPING LAW BANKRUPTCY AND IN- ELECTIONS COUNTY COUNCILS MUNICIPAL CORPORATIONS PARISH LAW, CHURCH- WARDENS, ETC. PUBLIC HEALTH AND NUISANCES FRIENDLY AND BUILDING SOCIETIES COPYRIGHT AND PATENTS TRADE MARKS AND DESIGNS HUSBAND AND WIFE, DIVORCE, ETC. TRUSTEES AND EXECUTORS INTESTACY, LAW OF GUARDIAN NOTE. The object of this work is to enable those who consult it to help them- selves to the law; and thereby to dispense, as far as possible, with professional assistance and advice. There are many wrongs and grievances which persons sub- mit to from time to time through not knowing how or where to apply for redress ; and many persons have as great a dread of a lawyer's office as of a lion's den. With this book at hand it is believed that many a SIX-AND-EIGHTPENCE may be saved; many a wrong redressed ; many a right reclaimed; many a law suit avoided ; and many an evil abated. The work has established itself as the standard legal adviser of all classes, and also made a reputation for itself as a useful book of reference for lawyers residing at a distance from law libraries, who are glad to have at hand a work em- bodying recent decisions, and enactments. *** OPINIONS OF THE PRESS. " It is a complete code of English Law, written in plain language, which all can understand. . . Should be in the hands of every business man, and all who wish to abolish lawyers' bills." Weekly Times. " A useful and concise epitome of the law, compiled with considerable care." Law Magazine. "A complete digest of the most useful facts which constitute English law." Globe. " Admirably done, admirably arranged, and admirably cheap." Leeds Mercury. ' A concise, cheap and complete epitome of the English law So plainly written that he who runs may read, and he who reads may understand." Figaro. ' A dictionary of legal facts well put together. The book is a very useful one." Spectator. "The latest edition of this popular book ought to be in every business establishment, and on every library table." Sheffield Post. Private Bill Legislation and Provisional Orders. HANDBOOK FOR THE USE OF SOLICITORS AND EN- GINEERS Engaged in Promoting Private Acts of Parliament and Provi- sional Orders, for the Authorization of Railways, Tramways, Works for the Supply of Gas and Water, and other undertakings of a like character. By L. LIVINGSTON MACASSEY, of the Middle Temple, Barrister-at-Law, Supply of Gas and Water, and other undertakings of a like character. By L. LIVINGSTON MACASSEY, of the Middle Temple, Barrister-at-Law, M.Inst.C.E. ; Author of " Hints on Water Supply." 8vo, 950 pp., 255. cloth. "The volume is a desideratum on a subject which can be only acquired by practical experi- ence, and the order of procedure in Private Bill Legislation and Provisional Orders is followed. The author's suggestions and notes will be found of great value to engineers and others profession- ally engaged to this class of practice." Building News. The author's double experience as an engineer and barrister has eminently qualified him for the task, and enabled him to approach the subject alike from an engineering and legal point of view. The volume will be found a great help both to engineers and lawyers engaged to promoting Private Acts of Parliament and Provisional Orders." Local Government Chronicle. LONDON, 1862. THE PRIZE MEDAL Was awarded to the Publishers of "WEALE'S SERIES." A NEW LIST OF WEALE'S SERIES RUDIMENTARY SCIENTIFICjEDUCATIONAL, AND CLASSICAL. Comprising nearly Three Hundred and Fifty distinct works in almost every department of Science, A rt, and Education, recommended to the notice of Engineers, Architects, Builders, Artisans, and Students generally, as well as to those interested ibraries, Literary and Scientific Institutions, Colleges, Schools, Science Ct c., &>c. IS" " WEALE'S SERIES includes Text-Books on almost every branch of Science and Industry, comprising such subjects as Agriculture, Architecture and Building, Civil Engineering, Fine Arts, Mechanics and Mechanical Engineering, Physical and Chemical Science, and many miscellaneous Treatises. The whole are constantly undergoing revision, and new editions, brought up to the latest discoveries in scientific research, are constantly issued. The prices at which they are sold are as low as their excellence is assured." American Literary Gazette. " Amongst the literature of technical education, WEALE'S SERIES has ever enjoyed a high reputation, and the additions being made by Messrs. CROSBY LOCKWOOD & SON render the series more complete, and bring the informa- tion upon the several subjects down to the present time." Mining Journal. " It is not too much to say that no books have ever proved more popular with, or more useful to, young engineers and others than the excellent treatises comprised in WEALE'S SERIES." Engineer. " The excellence of WEALE'S SERIES is now so well appreciated, that it would be wasting our space to enlarge upon their general usefulness and value." Builder. "The volumes of WEALE'S SERIES form one of the best collections of elementary technical books in any language." Architect. " WEALE'S SERIES has become a standard as well as an unrivalled collection of treatises in all branches of art and science." Public Opinion. PHILADELPHIA, 1876. THE PRIZE MEDAL Was awarded to the Publishers for Books : Budimentary, Scientific, "WEALE'S SERIES," ETC. CROSBY LOCKWOOD & SON, 7, STATIONERS' HALL COURT, LUDGATE HILL, LONDON, E.G. ' WEALlTS RUDIMENTARY SERIES. WEALE'S RUDIMENTARY SCIENTIFIC SERIES. ** The volumes of this Series are freely Illustrated with Woodcuts, or otherwise, where requisite. Throughout the fol- lowing List it must be understood that the books are bound in limp cloth, unless otherwise stated ; but the -volumes marked with a t may also be had strongly bound in cloth boards for 6d. extra, N.B. In ordering from this List it is recommended, as a means of facilitating business and obviating error, to quote the numbers affixed to the volumes, as -well as the titles and prices. CIVIL ENGINEERING, SURVEYING, ETC. No. 31. WELLS AND WELL-SINKING. By JOHN GEO. SWINDELL, A.R.I.B.A., and G. R. BURNELL, C.E. Revised Edition. With a New Appendix on the Qualities of Water. Illustrated. 2s. 35. THE BLASTING AND QUARRYING OF STONE, for Building and other Purposes. By Gen. Sir J. BURGOYNE, Bart. is. 6d. 43. TUBULAR, AND OTHER IRON GIRDER BRIDGES, par- ticularly describing the Britannia and Conway Tubular Bridges. By G. DRYSDALE DEMPSEY, C.E. Fourth Edition. 2s. 44. FOUNDATIONS AND CONCRETE WORKS, with Practical Remarks) on Footings, Sand, Concrete, Beton, Pile-driving, Caissons, and Cofferdams, &c. By E. DOBSON. Seventh Edition, is. 6d. 60. LAND AND ENGINEERING SURVEYING. By T. BAKER, C.E. Fifteenth Edition, revised by Professor J. R. YOUNG. 2S.t 80*. EMBANKING LANDS FROM THE SEA. With examples and Particulars of actual Embankments, &c. By J. WIGGINS, F.G.S. 8l. WATER WORKS, for the Supply of Cities and Towns. With a Description of the Principal Geological Formations of England as in- fluencing Supplies of Water, &c. By S. HUGHES, C.E. New Edition. 45.$ gland as in- , . . , .. dition. 45.$ 118. CIVIL ENGINEERING IN NORTH AMERICA, a Sketch of. By DAVID STEVENSON, F.R.S.E., &c. Plates and Diagrams. 35. 167. IRON BRIDGES, GIRDERS, ROOFS, AND OTHER WORKS. By FRANCIS CAMPIN, C.E. zs. 6d4 197. ROADS AND STREETS. By H. LAW, C.E., revised and enlarged by D. K. CLARK, C.E., including pavements of Stone, Wood, Asphalte, &c. 4S. 6d.t 203. SANITARY WORK IN THE SMALLER TOWNS AND IN VILLAGES. By C.SLAGG, A.M.I.C.E. Revised Edition. 35.* 212. GAS-WORKS, THEIR CONSTRUCTION AND ARRANGE- MENT; and the Manufacture and Distribution of Coal Gas. Originally written by SAMUEL HUGHES, C.E. Re-written and enlarged by WILLIAM RICHARDS, C.E. Eighth Edition, with important additions. 55. 6d.t 213. PIONEER ENGINEERING. A Treatise on the Engineering Operations connected with the Settlement of Waste Lands in New Coun- tries. By EDWARD DOBSON, Assoc. Inst. C.E. 45. 6d.t 216. MATERIALS AND CONSTRUCTION; A Theoretical and Practical Treatise on the Strains, Designing, and Erection of AVorks of Con- struction. By FRANCIS CAMPIN, C.E. Second Edition, revised. 35.* 219. CIVIL ENGINEERING. By HENRY LAW, M.Inst. C.E. Including HYDRAULIC ENGINEERING by GEO. R. BURNELL, M.Inst. C.E. Seventh Edition, revised, with large additions by D. KINNEAR CLARK, M.Inst. C.E. 6s. 6d., Cloth boards, 75. 6d. 268: THE DRAINAGE OF LANDS, TOWNS, &> BUILDINGS. By G. D. DEMPSEY, C.E. Revised, with large Additions on Recent Practice in Drainage Engineering, by D. KINNEAR CLARK, M.I.C.E. Second Edition, Corrected. 41. 6d.t 8" The t indicates that these vols, may be had strongly bound at 6d. extra. LONDON: CROSBY LOCKWOOD AND SON, WEALE'S RUDIMENTARY SERIES. MECHANICAL ENGINEERING, ETC. 33. CRANES, the Construction of, and other Machinery for Raising Heavy Bodies. By JOSEPH GLYNN, F.R.S. Illustrated, is. 6d. 34. THE STEAM ENGINE. By Dr. LARDNER. Illustrated. is.6d. 59. STEAM BOILERS: their Construction and Management. By R. ARMSTRONG, C.E. Illustrated, is. 6d. 82. THE POWER OF WATER, as applied to drive Flour Mills, and to give motion to Turbines, &c. By JOSEPH GLYNN, F.R.S. 2S.J 98. PRACTICAL MECHANISM, the Elements of; and Machine Tools. By T. BAKER, C.E. With Additions by J. NASMYTH, C.E. as. 6d. 139. THE STEAM ENGINE, a Treatise on the Mathematical Theory of, with Rules and Examples for Practical Men. By T. BAKBR, C.E. is. 6d. 164. MODERN WORKSHOP PRACTICE, as applied to Steam .Engines, Bridges, Ship-building, Cranes, &c. ByJ. G. WINTON. Fourth Edition, much enlarged and carefully revised. 35. bd.t [Just published. 165. IRON AND HEAT, exhibiting the Principles concerned in the Construction of Iron Beams, Pillars, and Girders. By J. ARMOUR, as. 6d.t 166. POWER IN MOTION: Horse-Power, Toothed- Wheel Gearing, Long and Short Driving Bands, and Angular Forces. ByJ. ARMOUR, zs.J 171. THE WORKMAN'S MANUAL OF ENGINEERING DRAWING. ByJ. MAXTON. 6th Edn. With 7 Plates ad 350 Cuts. 35. 6d.t 190. STEAM AND THE STEAM ENGINE, Stationary and Portable. Being an Extension of the Elementary Treatise on the Steam Engine of MR. JOHN SEWELL. By D. K. CLARK, M.I.C.E. 35. 6d.t 200. FUEL, its Combustion and Economy. By C. W. WILLIAMS. With Recent Practice in the Combustion and Economy of Fuel Coal, Coke Wood, Peat, Petroleum, &c. by D. K. CLARK, M.I.C.E. 3s.6d.t 202. LOCOMOTIVE ENGINES. By G. D. DEMPSEY, C.E. ; with large additions by D. KINNEAR CLARK, M.I.C.E. 33.* 211. THE BOILERMAKER'S ASSISTANT in Drawing, Tern- plating, and Calculating Boiler and Tank Work. By JOHN COURTNEY . Practical Boiler Maker. Edited by D.K. CLARK, C.E. too Illustrations. 2s, 217. SEWING MACHINERY: Its Construction, History, &c., with full Technical Directions for Adjusting, &c. By J. W. URQUHART, C.E. zs.* 223. MECHANICAL ENGINEERING. Comprising Metallurgy, Moulding, Casting, Forging, Tools, Workshop Machinery, Manufacture of the Steam Engine, &c. By FRANCIS CAMPIN, C.E. Second Edition, zs. 6d. 236. DETAILS OF MACHINERY. Comprising Instructions for the Execution of various Works in Iron. By FRANCIS CAMPiN.'C.E. 3$.* 237. THE SMITHY AND FORGE; including the Farrier's Art and Coach Smithing. By W. J. E. CRANK. Illustrated, zs. 6d.J 238. THE SHEET-METAL WORKER'S GUIDE; a Practical Hand- book for Tinsmiths, Coppersmiths, Zincworkers, &c. With 94 Diagrams and Working Patterns. By W. J. E. CRANE. Second Edition, revised, is. Sd. 251. STEAM AND MACHINERY MANAGEMENT: with Hints on Construction and Selection. By M. Powis BALE, M.I.M.E. 2S. 6d4 254. THE BOILERMAKER'S READY-RECKONER. By J. COURTNEY. Edited by D. K. CLARK, C.E. 45., limp ; 55., half-bound. 255. LOCOMOTIVE ENGINE-DRIVING. A Practical Manual for Engineers in charge of Locomotive Engines. By MICHAEL REYNOLDS, M.S.E Eighth Edition. 35. 6d., limp ; 45. 6d. cloth boards. 256. STATIONARY ENGINE-DRIVING. A Practical Manual Engineers in charge of Stationary Engines. By MICHAEL REYNOLDS, M.S.E. Third Edition. 35. 6d. limp ; 45. 6d. cloth boards. 260. IRON BRIDGES OF MODERATE SPAN: their Construe- tion and Erection. By HAMILTON W. PBNDRED, C.E. 2s. g~ The \ indicates that these vols. may be had strongly bound at ftd. extra. 7, STATIONERS' HALL COURT, LUDGATE HILL, E.G. WEALE'S RUDIMENTARY SERIES. MINING, METALLURGY, ETC. 4. MINERALOGY, Rudiments of; a concise View of the General Properties of Minerals. By A. RAMSAY, F.G.S., F.R.G.S., &c. Third Edition, revised and enlarged. Illustrated. 35. 6d.t 117. SUBTERRANEOUS SUR VEYING, with and without the Mag- netic Needle. By T. FENWICK and T. BAKER, C.E. Illustrated, as. 6d. t 135. ELECTRO-METALLURGY; Practically Treated. By ALEX- ANDER WATT. Ninth Edition, enlarged and revised, with additional Illus- trations, and including the most recent Processes. 35. 6d.t 172. MINING TOOLS, Manual of. For the Use of Mine Managers, Agents, Students, &c. By WILLIAM MORGANS. 2s. 6d. 172*. MINING TOOLS, ATLAS of Engravings to Illustrate the above, containing 235 Illustrations, drawn to Scale. 4to. 45. 6d. 176. METALLURGY OF IRON. Containing History of Iron Manu- facture, Methods of Assay, and Analyses of Iron Ores, Processes of Manu- facture of Iron and Steel, &c. By H. BAUERMAN, F.G.S. Sixth Edition, revised and enlarged, ss.t [Just published. 180. COAL AND COAL MINING. By the late Sir WARINGTON W. SMYTH, M.A., F.R.S. Seventh Edition, revised. 35. 6d.t [Just published. ioc. THE MINERAL SURVEYOR AND VALUER'S COM- PLETE GUIDE. By W. LINTERN, M.E. Third Edition, including Mag- netic and Angular Surveying. With Four Plates. 35. 6d.t 214. SLATE AND SLATE Q UA RRYING, Scientific, Practical, and Commercial. By D. C. DAVIES, F.G.S., Mining Engineer, &c. 35.! 264. A FIRST BOOK OF MINING AND QUARRYING, with the Sciences connected therewith, for Primary Schools and Self Instruction. By J. H. COLLINS, F.G.S. Second Edition, with additions, is. 6d. ARCHITECTURE, BUILDING, ETC. 16. ARCHITECTURE ORDERS The Orders and their ^Esthetic Principles. By W. H. LEEDS. Illustrated, is. 6d. 17. ARCHITECTURE STYLES The History and Description of the Styles of Architecture of Various Countries, from the Earliest to the Present Period. By T. TALBOT BURY, F.R.I. B.A., &c. Illustrated. 2s. ** ORDERS AND STYLES OF ARCHITECTURE, in One Vol., 3.1. 6d. 1 8. ARCHITECTURE DESIGN -The Principles of Design in Architecture, as deducible from Nature and exemplified in the Works of the Greek and Gothic Architects. ByE.L. GARBETT, Architect. Illustrated. 2s.6d. ' The three preceding Works, in One handsome Vol., half bound, entitled " MODERN ARCHITECTURE," price 6s. 22. THE ART OF BUILDING, Rudiments of. General Principles of Construction, Materials used in Building, Strength and Use of Materials, Working Drawings, Specifications, and Estimates. By E. DOBSON, 2S.J 25. MASONRY AND STONECUTTING : Rudimentary Treatise on the Principles of Masonic Projection and their application to Con- struction. By EDWARD DOBSON, M.R.I.B.A., &c. 2s. 6d.t 42. COTTAGE BUILDING. By C. BRUCE ALLEN, Architect. Tenth Edition, revised and enlarged. With a Chapter on Economic Cottages for Allotments, by EDWARD E. ALLEN, C.E. 2s. 45. LIMES, CEMENTS, MORTARS, CONCRETES, MASTICS, PLASTERING, &c. By G. R. BURNELL, C.E. Thirteenth Edition, is. 6d. 57. WARMING AND VENTILATION. An Exposition of the General Principles as applied to Domestic and Public Buildings, Mines, Lighthouses, Ships, &c. By C. TOMLINSON, F.R.S., &c. Illustrated. 35. ill. ARCHES, PIERS, BUTTRESSES, &c.: Experimental Essays on the Principles of Construction. By W. BLAND. Illustrated, is. 6d. g^p The J indicates that these vols. may be had strongly bound at bd. extra. LONDON : CROSBY LOCKWOOD AND SON, WEALE'S RUDIMENTARY SERIES. Architecture, Building, etc., continued. n6. THE ACOUSTICS OF PUBLIC BUILDINGS; or, The Principles of the Science of Sound applied to the purposes of the^A.rchitect and Builder. By T. ROGER SMITH, M.R.I.B.A., Architect. Illustrated, is 6d. 127. ARCHITECTURAL MODELLING IN PAPER, the Art of. By T. A. RICHARDSON, Architect. Illustrated, is. 6d. 128. VITRUVIUSTHE ARCHITECTURE OF MARCUS VITRUVIUS POLLO. In Ten Books. Translated from the Latin b JOSEPH GWILT, F.S.A., F.R.A.S. With 23 Plates. 55. 130. GRECIAN ARCHITECTURE, An Inquiry into the Principles of Beauty in ; with an Historical View of the Rise and Progress of the Art in Greece. By the EARL OF ABERDEEN, is. V The two preceding Works in One handsome Vol., half bound, entitled "ANCIENT ARCHITECTURE," price ds. 132. THE ERECTION OF DWELLING-HOUSES, Illustrated by a Perspective View, Plans, Elevations, and Sections of a pair of Semi- detached Villas, with the Specification, Quantities, and Estimates, &c. By S. H. BROOKS. New Edition, with Plates, as. 6d.t 175. LOCKWOO&S BUILDERS PRICE BOOK FOR 1892. A Comprehensive Handbook of the Latest Prices and Data for Builders, Architects, Engineers, and Contractors. Re-constructed, Re-written, and further Enlarged. By FRANCIS T. W. MILLER, A.R.I.B.A. 700 pages. 3 s. 6d. ; cloth hoards, 4 s. [Just Published. 182. CARPENTR Y AND JOINERY THE ELEMENTARY PRIN- CIPLFS OF CARPENTRY. Chiefly composed from the Standard Work of THOMAS TREDGOLD, C.E. With a TREATISE ON JOINERY by E. WYNDHAM TARN, M.A. Fifth Edition, Revised. 3s. 6d.f 182*. CARPENTRY AND JOINERY. ATLAS of 35 Plates to accompany the above. With Descriptive Letterpress. 4to. 6s. 185. THE COMPLETE MEASURER ; the Measurement of Boards, Glass, &c.; Unequal-sided, Square-sided, Octagonal-sided, Round Timber and Stone, and Standing Timber, &c. By RICHARD HORTON V Fifth Edition. 45. ; strongly bound in leather, 5s. 187. HINTS TO YOUNG ARCHITECTS. By G. WlGHTWlCK. New Edition. By G. H. GUILLAUMB. Illustrated. 35. 6d.t 188. HOUSE PAINTING, GRAINING, MARBLING, AND SIGN WRITING : with a Course of Elementary Drawing for House-Painters, Sign- Writers, &c., and a Collection of Useful Receipts. By ELLIS A. DAVIDSON. Sixth Edition. With Coloured Plates. 55. cloth limp ; 6s. cloth boards. 189. THE RUDIMENTS OF PRACTICAL BRICKLAYING. In Six Sections : General Principles ; Arch Drawing, Cutting, and Setting : Pointing; Paving, Tiling, Materials; Slating and Plastering; Practical Geometry, Mensuration, &c. By ADAM HAMMOND. Seventh Edition, is. 6d. 191. PL UMBING. A Text-Book to the Practice of the Art or Craft of the Plumber. With Chapters upon House Drainage and Ventilation. Sixth Edition. With 380 Illustrations. By W. P. BUCHAN. 35. 6d.t IQ2. THE TIMBER IMPORTER'S, TIMBER MERCHANTS, and BUILDER'S STANDARD GUIDE. By R. E. GRANDY. as. 206. A BOOK ON BUILDING, Civil and Ecclesiastical, including CHURCH RESTORATION. With the Theory of Domes and the Great Pyramid, &c. By Sir EDMUND BECKETT, Bart., LL.D., Q.C., F.R.A.S. 4 s. 6d.t 226 THE JOINTS MADE AND USED BY BUILDERS in the Construction of various kinds of Engineering and Architectural Works. By WYVILL J.CHRISTY, Architect. With upwardsofi6oEngravtngson Wood. 35.* M 8 THE CONSTR UCTION OF ROOFS OF WOOD AND IRON. By E. WYNDHAM TARN, M.A., Architect. Second Edition, revised, it. 6d. (g- The * indicates that these vols. may be had strongly bound at 6d. extra. 7, STATIONERS' HALL COURT, LUDGATE HILL, B.C. WEALE'S RUDIMENTARY SERIES. Architecture, Building, etc., continued. 229. ELEMENTARY DECORATION: as applied to the Interior and Exterior Decoration of Dwelling- Houses, &c. By J. W. FACKY. 2S. 257. PRACTICAL HOUSE DECORATION. A Guide to the Art of Ornamental Painting. By JAMES W. FACBY. as. 6d. %* The two preceding Works, in One handsome Vol., half -bound, entitled " HOUSE DECORATION, ELEMENTARY AND PRACTICAL," price $s. 230. HANDRAILING. Showing New and Simple Methods for finding the Pitch of the Plank, Drawing the Moulds, Bevelling, Jointing-up, and Squaring the "Wreath. By GEORGE COLLINGS. Second Edition, Revised, including A TREATISE ON STAIRBUILDING. Plates and Diagrams. 2s.6d, 247. BUILDING ESTA TES : a Rudimentary Treatise on the Develop- ment, Sale, Purchase, and General Management of Building Land. By FOWLER MAITLAND, Surveyor. Second Edition, revised. 2s. 248. PORTLAND CEMENT FOR USERS. By HENRY FAIJA, Assoc. M. Inst. C.E. Third Edition, corrected. Illustrated. 2s. 252. BRICKWORK ': a Practical Treatise, embodying the General and Higher Principles of Bricklaying, Cutting and Setting, &c. By F. WALKER. Third Edition, Revised and Enlarged, is. 6d. 23. THE PRACTICAL BRICK AND TILE BOOK. Comprising: 189. BRICK AND TILE MAKING, by E. DOBSON, A.I.C.E.; PRACTICAL BRICKLAY- 165. ING, by A. HAMMOND ; BRICKCUTTING AND SETTING, by A. HAMMOND. 534 pp. with 270 Illustrations. 6s. Strongly half-bound. tft.THE TIMBER MERCHANT'S, SAW-MILLER'S, AND IMPORTER'S FREIGHT-BOOK AND ASSISTANT. By WM. RICH- ARDSON. With a Chapter on Speeds of Saw-Mill Machinery, &c. By M. Powis BALE, A.M.Inst.C.E. 3 s.t 258. CIRCULAR WORK IN CARPENTRY AND JOINERY. A Practical Treatise on Circular Work of Single and Double Curvature. By GEORGE COLLINGS. Second Edition, 2s. 6d. 259. GAS FITTING: A Practical Handbook treating of every Description of Gas Laying and Fitting. By JOHN BLACK. With 122 Illus- trations. 2S. 6d.t 261. SHORING AND ITS APPLICATION: A Handbook for the Use of Students. By GEORGE H. BI.AGROVB. is. 6d. 265. THE AR T OF PR A CTICAL BRICKCUTTING & SETTING. By ADAM HAMMOND. With 90 Engravings, is. 6d. [Just published. 267. THE SCIENCE OF BUILDING : An Elementary Treatise on the Principles of Construction. Adapted to the Requirements of Architec- tural Students. By E. WYNDHAM TARN, M.A. Lond. Third Edition, Revised and Enlarged. With 59 Wood Engravings. 35. 6d.t 271. VENTILATION : a Text-book to the Practice of the Art of Ventilating Buildings, with a Supplementary Chapter upon Air Testing. By WILLIAM PATON BUCHAN, R.P., Sanitary and Ventilating Engineer, Author of " Plumbing," &c. 35. 6d.t [.Just Published. SHIPBUILDING, NAVIGATION, MARINE ENGINEERING, ETC. 51. NA VAL ARCHITECTURE. An Exposition of the Elementary Principles of the Science, and their Practical Application to Naval Construc- tion. By J. PBAKB. Fifth Edition, with Plates and Diagrams. 35. 6d4 53*. SHIPS FOR OCEAN & RIVER SER VICE, Elementary and Practical Principles of the Construction of. By H. A. SOMMERFELDT. is. 6d. 53**. AN ATLAS OFENGRA VINGS to Illustrate the above. Twelve large folding plates. Royal 4to, cloth. 75. 6d. 54. MASTING, MAST-MAKING, AND RIGGING OF SHIPS, Also Tables of Spars, Rigging, Blocks ; Chain, Wire, and Hemp Ropes, &c., relative to every class of vessels. By ROBERT KIPPING, N.A. 2s. f^~ The * indicates that these vols. may be had strongly bound at fxl. extra. LONDON : CROSBY LOCKWOOD AND SON, WEALE'S RUDIMENTARY SERIES. Shipbuilding, Navigation, Marine Engineering, etc., cont. 54*. IRON SHIP-BUILDING. With Practical Examples and Details. By JOHH GRANTHAM. C.E. Fifth Edition. 43. 55. THE SAILOR'S SEA BOOK: a Rudimentary Treatise on Navigation. By JAMBS GREENWOOD, B.A. With numerous Woodcuts and Coloured Plates. New and enlarged edition. By W. H. KOSSHR. 2S. 6d.t 80. MARINE ENGINES AND STEAM VESSELS. By ROBERT MURRAY, C.E. Eighth Edition, thoroughly Revised, with Additions by the Author and by GEORGE CARLISLE, C.E. 45. 6d. limp ; 55. cloth boards. 83 Jw. THE FORMS OF SHIPS AND BOATS. By W. BLAND. Eighth Edition, Revised, with numerous Illustrations and Models, is. 6d, 99. NAVIGATION AND NAUTICAL ASTRONOMY, in Theory and Practice. By Prof. J. R. YOUNG. New Edition. 2s. 6d. 106. SKIPS' ANCHORS, a Treatise on. By G. COTSELL, N.A. is. 6d. 149. SAILS AND SAIL-MAKING. With Draughting, and the Centre of Effort of the Sails ; Weights and Sizes of Ropes : Masting, Rigging, and Sails of Steam Vessels, &c. I2th Edition. By R. KIPPING, N.A., 2s. 6d.t 155. ENGINEER'S GUIDE TO THE ROYAL & MERCANTILE NAVIES. By a PRACTICAL ENGINEER. Revised by D. F. M'CARTHY. 35. 55 PRACTICAL NAVIGATION. Consisting of The Sailor's Sea-Book. By JAMBS GREENWOOD and W. H. ROSSER. Together with the requisite Mathematical and Nautical Tables for the Working of the ' Problems. By H. LAW, C.E., and Prof. J. R. YOUNG. 75. Half-bound. AGRICULTURE, GARDENING, ETC. bi. A COMPLETE READY RECKONER FOR THE ADMEA- SUREMENT OF LAND, &c. By A. ARMAN. Third Edition, revised and extended by C. NORRIS, Surveyor, Valuer, &c. 2s. 131. MILLER'S, CORN MERCHANT'S, AND FARMER'S READY RECKONER. Second Edition, with a Price List of Modern Flour-Mill Machinery, by W. S. HUTTON, C.E. 2s. 140. SOILS, MANURES, AND CROPS. (Vol. I. OUTLINES OF MODERN FARMING.) By R. SCOTT BURN. Woodcuts. 23. 141. FARMING 6- FARMING ECONOMY, Notes, Historical and Practical, on. (Vol. 2. OUTLINES OF MODERN FARMING.) By R. SCOTT BURN. 33. 142. STOCK; CATTLE, SHEEP, AND HORSES. (Vol. 3. OUTLINES OF MODERN FARMING.) By R. SCOTT BURN. Woodcuts, as. 6d. 145. DAIRY, PIGS, AND POULTRY, Management of the. By R. SCOTT BURN. (Vol. 4. OUTLINES OF MODERN FARMING.) 2s. 146. UTILIZATION OF SEWAGE, IRRIGATION, AND RECLAMATION OF WASTE LAND. (Vol. 5. OUTLINES OF MODERN FARMING.) By R. SCOTT BURN. Woodcuts. 25. 6d. % Nos. 140-1-2-5-6, in One Vol., handsomely half-bound, entitled " OUTLINES OF MODERN FARMING." By ROBERT SCOTT BURN. Price ias. 177. FRUIT TREES, The Scientific and Profitable Culture of. From the French of Du BREUIL. Revised by GEO. GLBNNY. 187 Woodcuts. 33. 6d.t 198. SHEEP; THE HISTORY, STRUCTURE, ECONOMY, AND DISEASES OF. By W. C. SPOONBR, M.R.V.C., Sec. Fifth Edition, enlarged, including Specimens ot New and Improved Breeds. 35. 6d.t 201. KITCHEN GARDENING MADE EASY. By GEORGE M. F. GLENNY. Illustrated, is. 6d.$ 207. OUTLINES OF FARM MANAGEMENT, and the Organi- zation of Farm Labour. By R. SCOTT BURN. 2s. 6d.t 208. OUTLINES OF LANDED ESTATES MANAGEMEN2. By R. SCOTT BURN. 2s. 6d. . Nos. 207 & 208 in One Vol., handsomely half-bound, entitled " OUTLINES OF LANDED ESTATES AND FARM MANAGEMENT." By R. SCOTT BURN. Price 6s. (g- The % indicates that these vols. may be had strongly bound at 6J. extra. 7, STATIONERS' HALL COURT, LUDGATE HILL, E.G. 8 WEALE'S RUDIMENTARY SERIES. Agriculture, Gardening, etc., continued. 209. THE TREE PLANTER AND PLANT PROPAGATOR. A Practical Manual on the Propagation of Forest Trees, Fruit Trees, Flowering Shrubs, Flowering Plants, &c. By SAMUEL WOOD. 2S. 210. THE TREE PRUNER. A Practical Manual on the Pruning of Fruit Trees, including also their Training and Renovation ; also the Pruning of Shrubs, Climbers, and Flowering Plants. By SAMUEL WOOD. is. 6d. V Nos. 209 6f 210 in One Vol., handsomely half -bound, entitled "THE TRBB PLANTER, PROPAGATOR, AND PRUNBR." By SAMUEL WOOD. Price 3*. 6d. 218. THE HAY AND STRAW MEASURER : Being New Tables for the Use of Auctioneers, Valuers, Farmers, Hay and Straw Dealers, &c. By JOHN STEELE. Fifth Edition. 2s. 222. SUBURBAN FARMING. The Laying-out and Cultivation of Farms, adapted to the Produce of Milk, Butter, and Cheese, Eggs, Poultry, and Pigs. By Prof. JOHN DONALDSON and R. SCOTT BURN. 33. 6d.t 231. THE ART OF GRAFTING AND BUDDING. By CHARLES BALTBT. With Illustrations. 2s. 6d.t 232. COTTAGE GARDENING; or, Flowers, Fruits, and Vegetables for Small Gardens. By E. HOBDAY, is. 6d. 233. GARDEN RECEIPTS. Edited by CHARLES W. QUIN. is. 6d. 234. MARKET AND KITCHEN GARDENING. By C. W. SHAW, late Editor of " Gardening Illustrated." 33.* 239. DRAINING AND EMBANKING. A Practical Treatise, em- bodying the most recent experience in the Application of Improved Methods. By JOHN SCOTT, late Professor of Agriculture and Rural Economy at the Royal Agricultural College, Cirencester. With 68 Illustrations, is, 6d. 240. IRRIGA TION AND WA TER SUPPL Y. A Treatise on Water Meadows, Sewage Irrigation, and Warping ; the Construction of Wells, Ponds, and Reservoirs, &c. By Prof. JOHN SCOTT. With34lllus. is. 6d. 241. FARM ROADS, FENCES, AND GATES. A Practical Treatise on the Roads, Tramways, and Waterways of the Farm; the Principles of Enclosures ; and the different kinds of Fences, Gates, and Stiles. By Professor JOHN SCOTT. With 75 Illustrations, is. 6d. 242. FARM BUILDINGS. A Practical Treatise on the Buildings necessary for various kinds of Farms, their Arrangement and Construction, with Plans and Estimates. By Prof. JOHN SCOTT. With 105 Illus. 23. 243. BARN IMPLEMENTS AND MACHINES. A Practical Treatise on the Application of Power to the Operations of Agriculture ; and on various Machines used in the Threshing-barn, in the Stock-yard, and in the Dairy, &c. By Prof. J. SCOTT. With 123 Illustrations. 2S. 244. FIELD IMPLEMENTS AND MACHINES. A Practical Treatise on the Varieties now in use, with Principles and Details of Con- struction, their Points of Excellence, and Management. By Professor JOHN SCOTT. With 138 Illustrations. 2s. 245. AGRICULTURAL SURVEYING. A Practical Treatise on Land Surveying, Levelling, and Setting-out ; and on Measuring and Esti- mating Quantities, Weights, and Values of Materials, Pro'duce, Stock, &c. By Prof. JOHN SCOTT. With 62 Illustrations, is. 6d. %* Nos. 239 /024S in One Vol., handsomely half-bound, entitled "THE COMPLETE TEXT-BOOK OF FARM ENGINEERING." By Professor JOHN SCOTT. Price 12*. 250. MEAT PRODUCTION. A Manual for Producers, Distributors, &c. By JOHN EWART. 2s. 6d.t 266. BOOK-KEEPING FOR FARMERS & ESTATE OWNERS. By J. M. WOODMAN, Chartered Accountant. 2s. 6d. cloth limp ; 35. 6d. cloth boards. 8&~ The t indicates that these vols. may be had strongly bound at dd. extra. LONDON: CROSBY LOCKWOOD AND SON, WEALE'S RUDIMENTARY SERIES. MATHEMATICS, ARITHMETIC, ETC. 32. MATHEMATICAL INSTRUMENTS, a Treatise on; Their Construction, Adjustment, Testing, and Use concisely Explained. By J. F. HEATHER, M.A. Fourteenth Edition, revised, with additions, by A.- T. WALMISLEY, M.I.C.E., Fellow of the Surveyors' Institution. Original Edi- tion, in i vol., Illustrated. 2S.t %* In ordering- the above, be careful to say, " Original Edition " (No. 32), to distin- guish it from the Enlarged Edition in 3 vols. (Nos. 168-9-70.) 76. DESCRIPTIVE GEOMETRY, an Elementary Treatise on; with a Theory of Shadows and of Perspective, extracted from the French of G. MONGE. To which is added, a description of the Principles and Practice of Isometrical Projection. By J. F. HEATHER, M.A. With 14 Plates, as. 178. PRACTICAL PLANE GEOMETRY: giving the Simplest Modes of Constructing Figures contained in one Plane and Geometrical Con- struction of the Ground. By J. F. HEATHER, M.A. With 215 Woodcuts. 2*. 83. COMMERCIAL BOOK-KEEPING. With Commercial Phrases and Forms in English, French, Italian, and German. By JAMES HADDON, M.A., Arithmetical Master of King's College School, London, is. 6d. 84. ARITHMETIC, a Rudimentary Treatise on : with full Explana- tions of its Theoretical Principles, and numerous Examples for Practice. By Professor J. R. YOUNG. Eleventh Edition, is. 6d. 84*. A KEY to the above, containing Solutions in full to the Exercises, together with Comments, Explanations, and Improved Processes, for the Use of Teachers and Unassisted Learners. By J. R. YOUNG, is. 6d. 85. EQUATIONAL ARITHMETIC, applied to Questions of Interest, Annuities, Life Assurance, and General Commerce ; with various Tables by which all Calculations may be greatly facilitated. By W. HIPSLBY. zs. 86. ALGEBRA, the Elements of. By JAMES HADDON, M.A. With Appendix, containing miscellaneous Investigations, and a Collection of Problems in various parts of Algebra. 2s.. 86*. A KEY AND COMPANION to the above Book, forming an extensive repository ol Solved Examples and Problems in Illustration of the various Expedient* necessary in Algebraical Operations. By J. R. YOUNG, is. 6d. 88. EUCLID, THE ELEMENTS OF : with many additional Propositions 8q and Explanatory Notes : to which is prefixed, an Introductory Essay ' Logic. By HENRY LAW, C.E. 25. 6d.t Sold also separately, viz. . 88. EUCLID, The First Three Books. By HENRY LAW, C.E. is. 6d. 89. EUCLID, Books 4, 5, 6, 11, M. By HENRY LAW, C.E. is. 6d. 90. ANALYTICAL GEOMETRY AND CONIC SECTIONS, By JAMES HANN. A New Edition, by Professor J. R. YOUNG. 2s.t 91. PLANE TRIGONOMETRY, the Elements of. By JAMES HANN, formerly Mathematical Master of King's College, London, is. 6d. 92. SPHERICAL TRIGONOMETR Y, the Elements of. By JAMES HANN. Revised by CHARLES H. DOWLING, C.E. is. V Or with " The Elements of Plane Trigonometry," in One Volume, 2S. 6d. gg. MENSURATION AND MEASURING. With the Mensuration and Levelling of Land for the Purposes of Modern Engineering. By T. BAKER, C.E. New Edition by E. NUGENT, C.E. Illustrated, is. 6d. 101. DIFFERENTIAL CALCULUS, Elements of the. By W. S. B. WOOLHOUSE, F.R.A.S., &c. is. 6d. 102. INTEGRAL CALCULUS, Rudimentary Treatise on the. By HOMERSHAM Cox, B.A. Illustrated, is. 136. ARITHMETIC, Rudimentary, for the Use of Schools and s Self- Instruction. By JAMBS HADDON, M.A. Revised by A. ARMAN. is. 6d. 137. A KEY TO HADDON'S RUDIMENTARY ARITHMETIC. By A. ARMAN. is. 6d. - The indicates that these vols. may be had strongly bound at 6J. extra. 7, STATIONERS' HALL COURT, LUDGATE HILL, E.C. 10 WEALE'S RUDIMENTARY SERIES. Mathematics, Arithmetic, etc., continued. 168. DRAWING AND MEASURING INSTRUMENTS. Includ- ing I. Instruments employed in Geometrical and Mechanical Drawing, and in the Construction, Copying, and Measurement of Maps and Plans. II. Instruments used for the purposes of Accurate Measurement, and for Arithmetical Computations. By T. F. HEATHER, M.A. Illustrated, is. 6d 169. OPTICAL INSTRUMENTS. Including (more especially) Tele- scopes, Microscopes, and Apparatus for producing copies of Maps and Plans by Photography. By J. F. HEATHER, M.A. Illustrated, is. 6d. 170. SURVEYING AND ASTRONOMICAL INSTRUMENTS. Including I. Instruments Used for Determining the Geometrical Features of a portion of Ground. II. Instruments Employed in Astronomical Observa- tions. By J. F. HEATHER, M.A. Illustrated, is. 6d. ** The above three volumes form an enlargement of the Author's original work "Mathematical Instruments." (See No. 32 in the Series.} ^MATHEMATICAL INSTRUMENTS. By J. F. HEATHER, 169. > M.A. Enlarged Edition, for the most part entirely re-written. The 3 Parts as 170 J above, in One thick Volume. With numerous Illustrations. 45. 6d.t 158. THE SLIDE RULE, AND HOW TO USE IT; containing full, easy, and simple Instructions to perform all Business Calculations with unexampled rapidity and accuracy. By CHARLES HOARE, C.E. Sixth Edition. With a Slide Rule in tuck of cover. 2s. 6d.t 196. THEORY OF COMPOUND INTEREST AND ANNUI- TIES ; with Tables of Logarithms for the more Difficult Computations of Interest, Discount, Annuities, &c. By F^DOR THOMAN. Fourth Edition. 454 199. THE COMPENDIOUS CALCULATOR; or, Easy and Concise Methods of Performing the various Arithmetical Operations required in Commercial and Business Transactions ; together with Useful Tables. By D. O'GORMAN. Twenty-seventh Edition, carefully revised by C. NORRIS. 2S. 6d., cloth limp ; 35. 6d., strongly half-bound in leather. 204. MATHEMATICAL TABLES, for Trigonometrical, Astronomical, and Nautical Calculations ; to which is prefixed a Treatise on Logarithms. By HENRY LAW, C.E. Together with a Series of Tables for Navigation and Nautical Astronomy. By Prof. J. R. YOUNG. New Edition. 45. 204*. LOGARITHMS. With Mathematical Tables for Trigonometrical, Astronomical, andNautical Calculations. By HENRY LAW, M.Inst.C.E. New and Revised Edition. (Forming part of the above Work). 35. 221. MEASURES, WEIGHTS, AND MONEYS OF ALL NA- TIONS, and an Analysis of the Christian, Hebrew, and Mahometan Calen- dars. By W. S. B. WOOLHOUSB, F.R.A.S..F.S.S. Seventh Edition, 2s.6d.t 227. MATHEMATICS AS APPLIED TO THE CONSTRUC- TIVE ARTS. Illustrating the various processes of Mathematical Investi- gation, by means of Arithmetical and Simple Algebraical Equations and Practical Examples. Bv FRANCIS CAMPIN. C.E. Second Edition. 35.* PHYSICAL SCIENCE, NATURAL PHILO- SOPHY, ETC. 1. CHEMISTRY. By Professor GEORGE FOWNES, F.R.S. With an Appendix on the Application of Chemistry to Agriculture, is. 2. NATURAL PHILOSOPHY, Introduction to the Study of. By C. TOMLINSON. Woodcuts, is. 6d. 6. MECHANICS, Rudimentary Treatise on. By CHARLES TOM- LINSON. Illustrated, is. 6d. 7. ELECTRICITY; showing the General Principles of Electrical Science, and the purposes to which it has been applied. By Sir W. SNOW HARRIS, F.R.S., &c. With Additions by R. SABINB, C.E., F.S.A. is. 6d. 7. GALVANISM. By Sir W. SNOW HARRIS. New Edition by ROBERT SABINB, C.E., F.S.A. is. 6d. 8. MAGNETISM; being a concise Exposition of the General Prin- ciples of Magnetical Science. By Sir W. SNOW HARRIS. New Edition, revised by H. M. NOAD, Ph.D. With 165 Woodcuts. 35. 6d4 J^y~ The t indicates that these vols. may be had strongly bound at bd. extra. LONDON : CROSBY LOCKWOOD AND SON, WEALE'S RUDIMENTARY SERIES. II Physical Science, Natural Philosophy, etc., continued. 11. THE ELECTRIC TELEGRAPH; its History and Progress; with Descriptions of some of the Apparatus. ByR. SABINB, C.E., F.S.A. 35. 12. PNEUMATICS, including Acoustics and the Phenomena of Wind Currents, for the Use of Beginners By CHARLES TOMLINSON, F.R.S. Fourth Edition, enlarged. Illustrated, is. 6d. 72. MANUAL OF THE MOLLUSC A; a Treatise on Recent and Fossil Shells. By Dr. S. P. WOODWARD. A.L.S. Fourth Edition. With Plates and 300 Woodcuts, ys. 6d., cloth. 96. ASTRONOMY. By the late Rev. ROBERT MAIN, M.A. Third Edition, by WILLIAM THYNNK LYNN, B.A., F.R.A.S. as. 97. STATICS AND DYNAMICS, the Principles and Practice of; embracing also a clear development of Hydrostatics, Hydrodynamics, and Central Forces. By T. BAKER. C.E. Fourth Edition, is. 6d. 173. PHYSICAL GEOLOGY, partly based on Major-General PORT- LOCK'S "Rudiments of Geology." By RALPH TATu,A.L.S.,&c. Woodcuts, as. 174. HISTORICAL GEOLOGY, partly based on Major-General PORTLOCK'S " Rudiments." By RALPH TATB, A.L.S., &c. Woodcuts, as. 6d. 173 RUDIMENTARY TREATISE ON GEOLOGY, Physical and & Historical. Partly based on Major-General PORTLOCK'S " Rudiments of 174. Geology." By RALPH TATB, A.L.S., F.G.S., &c. In One Volume. 43. 6d4 183 ANIMAL PHYSICS, Handbook of. By Dr. LARDNER, D.C.L., & formerly Professor of Natural Philosophy and Astronomy in University -o. College, Lond. With 520 Illustrations. In One Vol. ys. 6d., cloth boards. V Sold also in Two Parts, as follows . 183. ANIMAL PHYSICS. By Dr. LARDNBR. Part I., Chapters I. VII. 45. 184. ANIMAL PHYSICS. By Dr. LARDNBR. Part II., Chapters VIII. XVIII. 35. 269. LIGHT: an Introduction to the Science of Optics, for the Use of Students of Architecture, Engineering, and other Applied Sciences. By E. WYNDHAM TARN, M.A. is. 6d. [Just published. FINE ARTS. to. PERSPECTIVE FOR BEGINNERS. Adapted to Young Students and Amateurs in Architecture, Painting, &c. By GBORGB PYNB. as. 40 GLASS STAINING, AND THE ART OF PAINTING ON GLASS. From the German of Dr. GBSSERT and EMANUBL OTTO FROM- BBRO. With an Appendix on THE ART OF ENAMELLING, as. 6d. 69. MUSIC, A Rudimentary and Practical Treatise on. With numerous Examples. By CHARLES CHILD SPENCER, as. 6d. 71. PIANOFORTE, The Art of Playing the. With numerous Exer- cises & Lessons from the Best Masters. By CHARLES CHILD SPENCER. is.6d. 69-71. MUSIC & THE PIANOFORTE. In one vol. Half bound, 55. 181. PAINTING POPULARLY EXPLAINED, including Fresco, Oil, Mosaic, Water Colour, Water-Glass, Tempera, Encaustic. Miniature. Painting on Ivory, Vellum. Pottery, Enamel, Glass. &c. With Historical Sketches of the Progress of the Art by THOMAS JOHN GULLICK, assisted JOHN TIMBS, F.S.A. Sixth Edition, revised and enlarged. 55.* 186. A GRAMMAR OF COLOURING, applied tc Decorative Painting and the Arts. By GEORGE FIELD. New Edition, enlarged and adapted to the Use of the Ornamental Painter and Designer. By ELLIS A. DAVIDSON. With two new Coloured Diagrams, &c. 38.* 446. A DICTIONARY OF PAINTERS, AND HANDBOOK FOR PICTURE AMATEURS ; including Methods of Painting, Cleaning, Re- lining and Restoring, Schools of Painting, &c. With Notes on the Copyists and Imitators of each Master. By PHILIPPE DARYL. as. 6d.J fg- The t indicates that these vols. may be had strongly bound at 6d. extra. 7, STATIONERS' HALL COURT, LUDGATE HILL, E.C. 12 WEALE'S RUDIMENTARY SERIES. INDUSTRIAL AND USEFUL ARTS." 23. BRICKS AND TILES, Rudimentary Treatise on the Manufac- ture of. By E. DOBSON, M.R.I.B.A. Illustrated, 354 67. CLOCKS, WATCHES, AND BELLS, a Rudimentary Treatise on. By Sir EDMUND BECKETT, LL.D., Q.C. Seventh Edition, revised and en- larged. 45. 6d. limp ; 55. 6d. cloth boards. 83. CONSTRUCTION OF DOOR LOCKS. Compiled from the Papers of A. C. HOBBS, and Edited by CHARLES TOMLINSON. F.R.S. 2s. 6d. 162. THE BRASS FOUNDER'S MANUAL; Instructions for Modelling, Pattern-Making, Moulding, Turning, Filing, Burnishing, Bronzing, &c. With copious Receipts, &c. By WALTER GRAHAM. 2S4 205. THE ART OF LETTER PAINTING MADE EASY. By T.G.BADBNOCH. Illustrated with 12 full-page Engravings of Examples. is.6d. 215. THE GOLDSMITHS HANDBOOK, containing fufi Instruc. tions for the Alloying and Working of Gold. By GEORGE E. GEB, 3$.* 225. THE SILVERSMITH'S HANDBOOK, containing full In- structions for the Alloying and Working of Silver. By GEORGE E. GEE. js.* ,* The two preceding Works, in One handsome VOL., half-bound, entitled " THB GOLDSMITH'S & SILVERSMITH'S COMPLETE HANDBOOK,'' 7$. 249. THE HALL-MARKING OF JEWELLERY PRACTICALLY CONSIDERED. By GEORGE E. GEE. 354 224. COACH BUILDING, A Practical Treatise, Historical and Descriptive. By J. W. BURGESS, as. 6d.t 235. PRACTICAL ORGAN BUILDING. By W. E. DICKSON, M.A., Precentor of Ely Cathedral. Illustrated. 2s. 6d.I- 262. THE ART OF BOOT AND SHOEMAKING. By JOHN BEDFORD LENO. Numerous Illustrations. Third Edition, as, 263. MECHANICAL DENTISTRY: A Practical Treatise on the Construction of the Various Kinds of Artificial Dentures, with Formulae, Tables, Receipts, &c. By CHARLES HUNTER. Third Edition. 33.$ 270. WOOD ENGRA VING : A Practical and Easy Introduction to the Study of the Art. By W. N. BROWN, is. 6d. MISCELLANEOUS VOLUMES. 36. A DICTIONARY OF TERMS used in ARCHITECTURE, BUILDING, ENGINEERING, MINING, METALLURGY, ARCH&- OLOGY, the FINE ARTS, <5w. By JOHN WEALE. Sixth Edition. Revised by ROBERT HUNT, F.R.S. Illustrated. 55. limp ; 6s. cloth boards. 50. THE LAW OF CONTRACTS FOR WORKS AND SER- VICES. By DAVID GIBBONS. Third Edition, enlarged. 3S. 112. MANUAL OF DOMESTIC MEDICINE. By R. GooniNG, B.A., M.D. A Family Guide in all Cases of Accident and Emergency a*. lit*. MANAGEMENT OF HEALTH. A Manual oi Home and Personal Hygiene. By the Rev. JAMES BAIRD, B.A. is. 150. LOGIC, Pure and Applied. By S. H. EMMENS. is. 6d. 153. SELECTIONS FROM LOCKE'S ESSAYS ON THB HUMAN UNDERSTANDING. With Notes by S. H. EMMENS. 2s. 154. GENERAL HINTS TO EMIGRANTS. 2s. 157. THE EMIGRANTS GUIDE TO NATAL. By ROBERT JAMES MANN, F.R.A.S., F.M.S. Second Edition. Map. 2s. 193. HANDBOOK OF FIELD FORTIFICATION. By Major W. W. KNOLLYS, F.R.G.S. With 163 Woodcuts. 354 194. THE HOUSE MANAGER: Being a Guide to Housekeeping. Practical Cookery, Pickling and Preserving, Household Work, Dairy Management, &c. By AN OLD HOUSEKEEPER. 35. 6d4 194. HOUSE BOOK (The}. Comprising : I. THE Hous^MANAGER. 112 & By an OLD HOUSEKEEPER. II. DOMESTIC MEDICINE. By R. GOODING, M.D. 112*. "* MANAGEMENT OF HEALTH. By J. BAIRD. In One Vol., half-bound, 6. 85^" The t indicates that these vols may be had strongly bound at 6d. extra. LONDON : CROSBY LOCKWOOD AND SON, WEALE'S EDUCATIONAL AND CLASSICAL SERIES. 13 EDUCATIONAL AND CLASSICAL SERIES. HISTORY. i. England, Outlines of the History of; more especially with reference to the Origin and Progress of the English Constitution. By WILLIAM DOUGLAS HAMILTON, F.S.A., of Her Majesty's Public Record Office. 4 th Edition, revised. 55. ; cloth boards, 6s. 5. Greece, Outlines of the History of; in connection with the Rise of the Arts and Civilization in Europe. By W. DOUGLAS HAMILTON of University College, London, and EDWARD LBVIBN, M.A., of Ballioi College, Oxford, as. 6d. ; cloth boards, 3 s. 6d. 7. Rome, Outlines of the History of: from the Earliest Period to the Christian Era and the Commencement of the Decline of the Empire. By EDWARD LBVIEN, of Ballioi College, Oxford. Map, 23. 6d. ; cl. bds. 33. 6d. 9. Chronology of History, Art, Literature, and Progress, from the Creation of the World to the Present Time. The Continuation by W. D. HAMILTON, F.S.A. 3 s. ; cloth boards, 3 s. 6d. 50. Dates and Events in English History, for the use of Candidates in Public and Private Examinations. By the Rev. E. RAND. is. ENGLISH LANGUAGE AND MISCELLANEOUS. 11. Grammar of the English Tongue, Spoken and Written. With an Introduction to the Study of Comparative Philology. By HYDB CLARKE, D.C.L. Fifth Edition, is. 6d. 12. Dictionary of the English Language, as Spoken and Written. Containing above 100,000 Words. By HYDE CLARKE, D.C.L. 35. 6d. ; cloth boards, 45. 6d. ; complete with the GRAMMAR, cloth bds., 55. 6d. 48. Composition and Punctuation, familiarly Explained for those who have neglected the Study of Grammar. By JUSTIN BRBNAN. i8th Edition, is. 6d. 49. Derivative Spelling-Book: Giving the Origin of Every Word from the Greek, Latin, Saxon, German, Teutonic, Dutch, French, Spanish, and other Languages ; with their present Acceptation and Pronunciation. By J. ROWBOTHAM, F.R.A.S. Improved Edition, is. 6d. 51. The Art of Extempore Speaking: Hints for the Pulpit, the Senate, and the Bar. By M. BAUTAIN, Vicar-General and Professor at the Sorbonne. TranslatedfromtheFrench. 8th Edition, carefully corrected. zs.6d. 54. Analytical Chemistry, Qualitative and Quantitative, a Course of. To which is prefixed, a Brief Treatise upon Modern Chemical Nomencla- ture and Notation. By WM. W. PINK and GEORGE E. WEBSTER, as. THE SCHOOL MANAGERS' SERIES OF READING BOOKS, Edited by the Rev. A. R. GRANT, Rector of Hitcham, and Honorary Canon of Ely j formerly H.M. Inspector of Schools. INTRODUCTORY PRIMER, $d. t. d. FIRST STANDARD SECOND THIRD FOURTH STANDARD FIFTH ' ,,....16 SIXTH . . . .16 LESSONS FROM THE BIBLE. Part I. Old Testament, is. LESSONS FROM THE BIBLE. Part II. New Testament, to which is added THE GEOGRAPHY OF THE BIBLE, for very young Children. By Rev. C. THORNTON FORSTBR. is. 2d. *.* Or the Two Parts in One Volume, a. 7, STATIONERS' HALL COURT, LUDGATE HILL, B.C. 4 WKALE'S EDUCATIONAL AND CLASSICAL SERIES. FRENCH. 24. French Grammar. With Complete and Concise Rules on the Genders of French Notms. By G. L. STRAUSS, Ph.D. is 6d 25. French-English Dictionary. Comprising a large number of New Terms used in Engineering, Mining, &c. By ALFRED ELWES. is. 6d. 26. English-French Dictionary. By ALFRED ELWES. 2s. 25,26. French Dictionary (as above). Complete, in One Vol.. 35.; cloth boards, 33. 6d. ** Or with the GRAMMAR, cloth boards 45 6d 47. French and English Phrase Book : containing Intro- ductory Lessons, with Translations, several Vocabularies of Words, a Col- lection of suitable Phrases, and Easy Familiar Dialogues, is. 6d. GERMAN. 39. German Grammar. Adapted for English Students, from Heyse's Theoretical and Practical Grammar, by Dr. G. L. STRAUSS, is. 6d 40. German Reader : A Series of Extracts, carefully culled from the most approved Authors of Germany ; with Notes, Philological and Ex- planatory. By G. L. STRAUSS, Ph.D. is. 41-43. German Triglot Dictionary. By N. E. S. A. HAMILTON. In Three Parts. Part I. German-French-English. Part II. English-Ger- man-French. Part III. French-German-English. 35., or cloth boards, 43. 41-43 German Triglot Dictionary (as above), together with German & 39. Grammar (No. 39), in One Volume, cloth boards, 53. ITALIAN. 27. Italian Grammar, arranged in Twenty Lessons, with a Course of Exercises. By ALFRED ELWBS. is. 6d. 28. Italian Triglot Dictionary, wherein the Genders of all the Italian and French Nouns are carefully noted down. By ALFRED ELWBS. Vol. i. Italian-English-French. 2S . 6d. 30. Italian Triglot Dictionary. By A. ELWES. Vol. 2. English-French-Italian. 2s. 6d. 32. Italian Triglot Dictionary. By ALFRED ELWES. Vol. 3. French-Italian-English. 2s. 6d. 28,30, Italian Triglot Dictionary (as .ibove). In One Vol., 73. 6d. 32. Cloth boards. SPANISH AND PORTUGUESE. 34. Spanish Grammar, in a Simple and Practical Form. With a Course of Exercises. By ALFRED ELWES. is. 6d. 35. Spanish-English and English-Spanish Dictionary. Including a large number of Technical Terms used in Mining, Engineering, &c. with the proper Accents and the Gender of every Noun. By ALFRED ELWES 43. ; cloth boards, 55. ** Or with the GRAMMAR, cloth boards, 6s. 55. Portuguese Grammar, in a Simple and Practical Form. With a Course of Exercises. By ALFRED ELWES. is. 6d. 56. Portuguese-English and English-Portuguese Dic- tionary. Including a large number of Technical Terms used in Mining, Engineering, &c., with the proper Accents and the Gender of every Noun, By ALFRED ELWES. Second Edition, Revised, 53. ; cloth boards, 6s. %* Or with the GRAMMAR, cloth boards, 73. HEBREW. 46*. Hebrew Grammar. By Dr. BRESSLAU. is. 6d. 44. Hebrew and English Dictionary, Biblical and Rabbinical ; containing the Hebrew and Chaldee Roots of the Old Testament Post- Rabbinical Writings. By Dr. BRESSLAU. 6s. 46. English and Hebrew Dictionary. By Dr. BRESSLAU. 35. 44,46. Hebrew Dictionary (as above), in Two Vols., complete, with 46*. the GRAMMAR, cloth boards, 128. LONDON : CROSBY LOCKWOOD AND SON, WEALE'S EDUCATIONAL AND CLASSICAL SERIES. 15 LATIN. 19. Latin Grammar. Containing the Inflections and Elementary Principles of Translation and Construction. By the Rev. THOMAS GOODWIN, M.A., Head Master of the Greenwich Proprietary School, is. 6d. 20. Latin-English Dictionary. By the Rev. THOMAS GOODWIN JVI.A. 2S. 22. English-Latin Dictionary; together with an Appendix of R3Sy^gs^a^*s their origin from the La *' n - By * 20,22. Latin Dictionary (as above). Complete in One Vol., 35. 6d. cloth boards, 45. 6d. % Or with the GRAMMAR, cloth boards, 55. 6d. LATIN CLASSICS. With Explanatory Notes in English. 1. Latin Delectus. Containing Extracts from Classical Authors. with Genealogical Vocabularies and Explanatory Notes, by H. YOUNG, is. 6d: 2. Csesaris Commentarii de Bello Gallico. Notes, and a Geographical Register for the Use of Schools, by H. YOUNG. 25. 3. Cornelius INT epos. With Notes. By H. YOUNG, is. 4. Virgilii Maronis Bucolica et Georgica. With Notes on the Buco- lics by W. RUSHTON, M.A., and on the Georgics by H. YOUNG, is. 6d. 5. Virgilii Maronis ^Eneis. With Notes, Critical and Explanatory, by H. YOUNG. New Edition, revised and improved With copious Addi- tional Notes by Rev. T. H. L. LBARY, D.C.L., formerly Scholar of Brasenose College, Oxford. 3 s. 5* Part i. Books i. vi., is. 6d. 5** Part 2. Books vii. xii., 2s. 6. Horace; Odes, Epode, and Carmen Saeculare. Notes by H. YOUNG, is. 6d. 7. Horace ; Satires, Epistles, and Ars Poetica. Notes by W. BROWN- RIGG SMITH, M.A., F.R.G.S. is. 6d. 8. Sallustii Crispi Catalina et Bellum Jugurthinum. Notes, Critical and Explanatory, by W. M. DONNB, B.A., Trin. Coll., Cam. is. 6d. 9. Terentii Andria et Heautontimorumenos. With Notes, Critical and Explanatory, by the Rev. JAMES DA VIES, M.A. is. 6d. to. Terentii Adelphi, Hecyra, Phormio. Edited, with Notes, Critical and Explanatory, by the Rev. JAMES DAVIES, M.A. as. 11. Terentii Eunuchus, Comcedia. Notes, by Rev. J. DAVIES, M.A. is. 6d. 12. Giceronis Oratio pro Sexto Roscio Amerino. Edited, with an Introduction, Analysis, and Notes, Explanatory and Critical, by the Rev JAMES DAVIBS, M.A. is. 6d. 13. Ciceronis Orationes in Catilinam, Verrem, et pro Archia. With Introduction, Analysis, and Notes, Explanatory and Critical, by Rev. T. H. L. LBARY, D.C.L. formerly Scholar of Brasenose College, Oxford, is. 6d. 14. Ciceronis Cato Major, Lselius, Brutus, sive de Senectute, de Ami- citia, de Claris Oratoribus Dialogi. With Notes by W. BROWNRIGG SMITH M.A., F.R.G.S. 2s. 16. Livy : History of Rome. Notes by H. YOUNG and W. B. SMITH, M.A. Part i. Books i., ii., is. 6d. 16*. Part 2. Books iii., iv., v., is. 6d. 17. Part 3. Books xxi., xxii., is. 6d. 19. Latin Verse Selections, from Catullus, Tibullus, Propertius, and Ovid. Notes by W. B. DONNB, M.A., Trinity College, Cambridge. 2$. 20. Latin Prose Selections, from Varro, Columella, Vitruvius, Seneca, Quintilian, Florus, Velleius Paterculus, Valerius Maximus Sueto- nius, Apuleius, &c. Notes by W. B. DONNB, M.A. 2S. 21. Juvenalis Satirae. With Prolegomena and Notes by T. H. S. ESCOTT, B.A., Lecturer on Logic at King's College, London, as. 7, STATIONERS' HALL COURT, LUDGATE HILL, E.C. 1 6 WEALE'S EDUCATIONAL AND CLASSICAL SERIES. GREEK. 14. Greek Grammar, in accordance with the Principles and Philo- logical Researches of the most eminent Scholars of our own day. By HANS CLAUDB HAMILTON, is. 6d. 15,17. Greek Lexicon. Containing all the Words in General Use, with their Significations, Inflections, and Doubtful Quantities. By HBNRY R. HAMILTON. Vol. i. Greek-English, 2 s. 6d. ; Vol. 2. English-Greek, 2 s. Or the Two Vols. in One, 4 s. 6d. : cloth boards, 55. 14, 15. Greek Lexicon (as above). Complete, with the GRAMMAR, in 17. One Vol., cloth boards, 6s. GREEK CLASSICS. With Explanatory Notes in English. I. Greek Delectus. Containing Extracts from Classical Authors, with Genealogical Vocabularies and Explanatory Notes, by H. YOUNG. New Edition, with an improved and enlarged Supplementary Vocabulary, by JOHN HUTCHISON, M.A., of the High School, Glasgow, is. 6d. *, 3. Xenophon's Anabasis ; or, The Retreat of the Ten Thousand. Notes and a Geographical Register, by H. YOUNG. Part i. Books i. to Hi., is. Part 2. Books iv. to vii., is. 4. Lucian's Select Dialogues. The Text carefully revised, with Grammatical and Explanatory Notes, by H. YOUNG, is. 6d. 5-12. Homer, The Works of. According to the Text of BAEUMLEIN. With Notes, Critical and Explanatory, drawn from the best and latest Authorities, with Preliminary Observations and Appendices, by T. H. L. LEARY, M.A., D.C.L. THB ILIAD: Parti. Books i. to vi., is. 6d. Part 2. Books vii. to xii., is.6d. THB ODYSSEY: Parti. Books i. to vi., is. 6d Parts. Books vii. to xi i., is. od di., is. 6d. Part 3. Books xiii. to xviii., is. 6d Part 4. Books xix. to xxiv., is. 6d. Part 3. Books xiii. to xviii., is. 6d. Part 4. Books xix. to xxiv., and Hymns, 2s. 13. Plato's Dialogues : The Apology of Socrates, the Crito, and the Phaedo. From the Text of C. F. HERMANN. Edited with Notes, Critical and Explanatory, by the Rev. JAMES DAVIBS, M.A. as. 14-17. Herodotus, The History of, chiefly after the Text of GAISFORD. With Preliminary Observations and Appendices, and Notes, Critical and Explanatory, by T. H. L. LBARY, M.A., D.C.L. Part i. Books i., ii. (The Clio and Euterpe), 25. Part 2. Books iii., iv. (The Thalia and Melpomene), 2S. Part 3. Books v.-vii. (The Terpsichore, Erato, and Polymnia), 2s. Part 4. Books viii., ix. (The Urania and Calliope) and Index, is. 6d. 18. Sophocles : CEdipus Tyrannus. Notes by H. YOUNG, is. to. Sophocles: Antigone. From the Text of DINDORF. Notes, Critical and Explanatory, by the Rev. JOHN MILNBR, B.A. 2s. 23. Euripides : Hecuba and Medea. Chiefly from the Text of Dm- DORF. With Notes, Critical and Explanatory, by W. BROWNRIGG SMITH, 26. Euripides : Alcestis. Chiefly from the Text of DINDORF. With Notes, Critical and Explanatory, by JOHN MILNER, B.A. is. 6d. 30. ^Eschylus : Prometheus Vinctus : The Prometheus Bound. From the Text of DINDORF. Edited, with English Notes, Critical and Explanatory, by the Rev. JAMBS DAVIBS, M.A. is. 12 ^Eschylus : Septem Contra Thebes : The Seven against Thebes. From the Text of DINDORF. Edited, with English Notes, Critical and Ex- planatory, by the Rev. JAMBS DAVIBS, M.A. is. 40. Aristophanes : Acharnians. Chiefly from the Text of C. H. WBISB. With Notes, by C. S. T. TOWNSHBND, M.A. is. 6d. 41. Thucydides : History of the Peloponnesian War. Notes by H. YOUNG. Book i. is. 6d. 42. Xenophon's Panegyric on Agesilaus. iNotes and Intro- duction by LL. F. W. JEWITT. is. 6d. 43. Demosthenes. The Oration on the Crown and the Philippics. With English Notes. By Rev. T. H. L. LBARY, D.C.L., formerly Scholar of Brasenose College, Oxford, is. 6d. CROSBY LOCKWOOD AND SON, 7, STATIONERS' HALL COURT, E.G. E. ft F. N. SPON, 125, STRAND, UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the last date stamped below. MAY 2 8 STACK JL72 Fcrm L9-100m-9,'52(A3105)444