-;W Ex £ibris Jdaiislao JZeti £ r«p?i» } ivBg] 'Jr? ^'UPa ^W : ^Ifciifc ffllRni THE CARPENTER AND JOINER’S ASSISTANT. BEING A COMPREHENSIVE TREATISE ON THE SELECTION, PREPARATION, AND STRENGTH OF MATERIALS, AND THE MECHANICAL PRINCIPLES OF FRAMING, WITH THEIR APPLICATION IN CARPENTRY, JOINERY, AND HAND-RAILING; ALSO, A COURSE OF INSTRUCTION IN PRACTICAL GEOMETRY, GEOMETRICAL LINES, DRAWING, PROJECTION, AND PERSPECTIVE; AND AN ILLUSTRATED GLOSSARY OF TERMS USED IN ARCHITECTURE AND BUILDING. BY JAMES XL \Y LANDS. BOROUGH ENGINEER OF LIVERPOOL. flhtsiratRr bn an (feteistfce Series nf |)(afis anb nunur Unntab (Engrabings on Moob, LONDON: BLACKIE & SON, PATERNOSTER BUILDINGS, E.C.; GLASGOW AND EDINBURGH. PREFACE The Framing of Timber for structural purposes may be regarded both as a mechanical and as a liberal art. As a mechanical art, it embraces the knowledge of the various ways of executing different works, of the processes of fashioning timber, of the tools which have to be used, and the manner of handling them. As a liberal art, it includes a knowledge of geometry, of the principles of mechanics, of the nature and strength of the material, and its behaviour under the strains to which it is subjected. On all these branches of knowledge there exist justly esteemed Treatises in our own and other languages. The labours of Barlow, Emy, Jousse, Robison, Rondelet, Nicholson, Tredgold, and many others, are devoted to elucidating the principles of the arts of Carpentry and Joinery; and there are also many useful compilations, foremost among them that of Krafft, the object of which is to present practical examples of the application of these principles. The works of these authors, however, are either too costly to be within the reach of the workman, or the subject is treated in a manner which presupposes greater knowledge of mathematical science than he is likely to possess, or they are written in a foreign language. Further, the information which he seeks is scattered through many separate treatises, none of which singly contains all that he requires to know. The object of the present Publication is to provide, in a compendious form and in plain language, a Complete and Practical Course of Instruction in the Principles of Carpentry and Joinery, with descrip¬ tions and representations of a selection of works actually executed, to illustrate the state of these arts at the present time, and to serve as guides in preparing new designs. The Carpenter and Joiner’s Assistant was projected by Mr. John White, author of Rural Archi¬ tecture, who prepared the greater number of the Drawings for the Plates, but died before he could supply any portion of the Text. The task, therefore, of completing the series of Drawings and preparing the Text devolved on the present Editor, who, while availing himself of Mr. White’s labours, has endeavoured to expand the work into a systematic and comprehensive Treatise. With this view the Work is divided into eight parts. The Fu-st Part is devoted to Practical Geometry, teaching various methods of constructing the angles and the rectilineal and curvilinear figures required in the daily practice of the draughtsman. The Second Part teaches the nature and use of the various kinds of Drawing Instruments. The Third Part is devoted to Stereography, comprehending the projection of lines, surfaces, and solids, and the application of this projection to the problems of Descriptive Carpentry in groins, pendentives, domes, niches, angle-brackets, roofs, hip-roofs, &c. These three parts thus form a complete Treatise on Lines, a knowledge of which is an essential preliminary to the study of Carpentry and Joinery. The Fourth Part treats of the physiology, growth, develop¬ ment, and diseases of Timber trees; of the mode of felling, squaring, and preparing timber for use, and of increasing its durability. It includes a description of the nature, properties, and uses of the various PREFACE. timber trees which in this country are employed by the Carpenter and Joiner; and it elucidates so much of the principles of the composition and resolution of forces, and of the strength and strain of materials, as belongs to Theoretical Carpentry. In the Fifth Part are presented examples of the construction of timber roofs, domes, and spires; of the framing of timber, the formation of joints, straps, truss girders, floors, partitions, timber houses, bridges, centres, and field, park, and dock gates. The Sixth Part is devoted to the illustration of Joinery; comprehending the mouldings used, the formation of joints, gluing up of columns, &c., framing and finishing of doors, windows, and skylights, and the various methods of hinging. The Seventh Part treats of Stairs, Staircases, and Handrailing, and in the latter, which is contributed by Mr. David Mayer, of Cheltenham, the author develops simple methods of getting out the wreath by one bevel and squared ordinates, the advantages of which he has tested in a long course of practice. The Eighth Part advances the student in his knowledge of Drawing, by instruction in the Projections of Shadows, in the method of making Finished Drawings, and in Perspective and Isometrical Projection. To these is added an Index and Illustrated Glossary of the Terms used in Architecture and Building. The number and character of the Illustrations form a prominent feature of the present Work. They consist of above Eight Hundred Geometric, Constructive, and Descriptive Figures interspersed throughout the text, and One Hundred and Fifteen Plates, containing upwards of One Thousand Figures. The Cuts and Plates combined, comprise, it is believed, a larger number of Illustrations than has hitherto been embodied in any similar treatise published in this country; and by incorporating so large a proportion of them in the text, in place of greatly increasing the number of separate Plates, the double advantage to the purchaser has been gained of ready and convenient reference from the text to the figures, and of a considerable modification in his favour in the total cost of the Work. It is impossible in a work like this to quote all the sources of information. Frequent references to authorities are given in the text; but in addition to these, it ought to be stated that the sections on Projection and Perspective are based on J. B. Cloquet’s Nouveau Traite EUmentaire cle Perspective. The Editor has endeavoured to render the Work throughout essentially practical in its character, elucidating the principles and rules not by lengthy demonstrations, but by showing their application in frequent examples. ■ «. * >• PART FIRST. PRACTICAL GEOMETRY. Different Branches of Geometry .... General Terms Used in Geometry. D ef i nition s—Point—Lines—Angles—Plane Figures— Solids, &c. ......... Construction of Angles, Rectilineal Figures, &c.— Problems I. to LIV. Construction of Circles, Circular Figures, &c.— Problems LY. to LXXXIY. Of the Ellipse, Parabola, and Hyperbola— The Ellipse— Problems LXXXY. to XCYI. The Parabola— Problems XCVII. to XCIX. . The Hyperbola— Problem C . Construction of Gothic Arches— The Equilateral Arch—The Lancet Arch—The Drop Arch—The Four-centred Arch—The Ogee Arch PART SECOND. CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. Compasses— Dividers—Hair Dividers—Spring Compasses . Compasses with Moveable Legs. Bow Compasses—Directors, or Triangular Compasses— Proportional Compasses. Beam Compasses—Tubular Compasses—Portable or Turn- in Compasses. Plain Scales — Simply divided Scales — Diagonal Scale— Line of Chords—The Plain Protractor .... Double Scales— The Sector — Plain Scales on the Sector — Sectoral Double Scales—The Line of Lines—The Line of PART THIRD. STEREOGRAPIIY—DESCRIPTIVE CARPENTRY. Projection— Definition of Projection of Points, Lines, and Planes Projection of Solids. The Tetrahedron—The Cube. The Octahedron—The Dodecahedron .... The Icosahedron—Manner of inscribing these five Solids in the same sphere. The Three Curved Bodies—The Cylinder, the Cone, the Sphere— Projections of the Cylinder. Sections of the Cylinder by a Plane .... Projections of the Cone. Sections of the Cone by a Plane. Section of the Sphere by a Plane ..... Tangent Planes to Curved Surfaces— Tangent Plane to a Cylinder . Tangent Plane to a Cone . ...... Tangent Plane to a Sphere. TABLE OF CONTENTS. viii Intersection of Curved Surfaces— Intersections of Cylinders .... Intersection of a Sphere and a Cylinder Intersections of Cones, &c. Intersection of a Cylinder by a Scalene Cone . Of Helices . .... Manner of Taking Dimensions Sections of Solids. Sections of a Cone—of a Cuneoid—of a Cylinder—of a Sphere ......... Sections of an Ellipsoid—of a Cyliudric Ring—of a Pyra¬ mid, &c. .......... Coverings of Solids — Regular Polyhedrons.. Development of the Coverings of Prisms .... Development of Cylinders ...... Development of Right and Oblique Cones Development of the Oblique Cone. Development of Solids whose Surface is of Double Cur¬ vature .. DESCRIPTIVE CARPENTRY. Groins—D efinition of, and Terms Used .... Rectangular Groined Vault—Gothic Groin Welsh, or Under-pitched Groin—Groins on a Circular Plan —Fan Tracery, &c.. Pendentives—D efinition of ..... Ceiling of a Square Room Coved with Spherical Penden¬ tives, &c. ......... An Elliptical Domical Pendentive Roof, &c. . . . Domes—D efinition of ...... Rectangular Oblong Surbased Dome—Octagonal Surbased Dome, &c. ....... Gothic Vault—Spherical Vault—Cylindrical Vault, &c. Niches. Spherical Niches on Different Plans. Elliptical Niches—Octagonal and Semicircular Niches . Angle Brackets ...... Forms of Roofs. Hip Roofs. PART FOURTH. KNOWLEDGE OF WOODS —THEORETICAL CARPENTRY. Physiological Notions of Woods .... Cultivation of Trees .... Diseases of Trees. Timbers Fit for the Carpenter Felling of Timber. Squaring of Timber . Management of Timber after it is Cut . Bending of Timber.102 Seasoning of Timber, and Means Employed to Increase its Durability.104 Insects Injurious to, and Destructive of Timber . . 105 Preservation of Wood by Impregnating it with Chemical Solutions ......... 106 Kyanizing—Margary’s Process — Sir William Burnett’s Process—Payne’s Process, &c.106 Protection of Timber against Fire .108 DESCRIPTIONS OF WOODS. Hard and Soft Woods—T he Oak, various kinds of . . 109 The Chestnut, the Elm, different species of' . . .110 The Walnut—The Beech—The Ash . . ill The Teak—The Green-heart—The Poplar . . . .112 The Alder—The Birch—The Hornbeam—The Maple . 113 The Sycamore—The Lime Tree—The Oriental Plane- — The American or Western Plane. 113 The Willow—The Acacia—The Horse Chestnut . . 114 The Pear Tree—The Apple Tree—The Hawthorn . .114 The Box—Mahogany—Sabicu . . . . . 114 Resinous Woods.115 The Great Pine Shoot of Alnpach.116 The Pine, varieties of, &c. . . .116 The Cedar—The Yew.119 THEORETICAL CARPENTRY. Resolution and Composition of Forces .... 120 Strength and Strain of Materials .... 123 Resistance of Timber to Tension . . . . .124 Experiments by Musehenbroek, Buffou, Barlow, Bevan, and others.124-130 Resistance of Timber to Compression, in the Direction of the Length of its Fibres . . . . . 124 Resistance of Timber to Transverse Strain . . . 126 Summary of Rules ....... 130 Table of the Properties of Timber.133 PART FIFTH. PRACTICAL CARPENTRY. Roofs, Classification of . . . . , . .134 Examples of the Construction of Roofs— Couple Roofs—Roofs with Framed Principals—Hammer Beam Roofs—Roofs with Curved Principals . . 136 Mr. Tredgold’s Rules for Strengths and Proportions of Parts— In a King-post Roof of Pine Timber .... 137 In a Queen-post Roof.137 Descriptions of Various Roofs . . .138 Mansard Roofs.140 Colonel Emy^ System of Construction . . . 14} Pace 63 64 64 66 67 67 68 68 69 69 70 71 71 72 76 77 78 so' 80 81 82 82 83 83 83 84 85 85 91 93 96 96 97 98 99 100 TABLE OF Page De Lorme’s Mode of Constructing Eoofs . . . 144 Gothic Eoofs.145 Conical Eoofs . 145 Domical Eoofs ..145 Timber Steeples and Spires.145 Framino—Joints—Straps 146 Mortises—Joggles, &c.—Scarfing—Fishing . . . 147 Lengthening Beams, &c. . . . . ■ • .148 Dovetailing, Halving, &c..149 Trussed Girders or Beams.149 Experiments of Messieurs Lasnier and Albony, and others 149 Girder of M. Laves,.149 Floors—B ridging-joist or Single-joisted Floors . . 150 Double Floors, or Floors with Bridging-joists . . . 151 Framed Floors.151 Variations in the Modes of Constructing Floors . . 151 French Floors.152 Combination of Timbers of Small Scantling, to Form Floors of Large Span without Intermediate Support . . 153 Wall Plates—Trimmers and Trimming-joists—Binding- joists—Girdei’S—Ceiling-joists.154 Eules for Calculating the Strength of the Component Parts of Floors and Ceilings, . . . . . . 154 Timber Partitions.155 Timber House?.156 Bridges—T he Principles of their Construction . . . 158 Essential Component Parts of a Timber Bridge . . 159 Method of determining the Strain on and the Dimensions of the Chords.160 Diagonal Braces and Knee-braces — Floor Beams and Timber Arches. .161 Classification of Bridges.16L American Bridges . . . ... . . . 165 Haupt’s Analysis of Strains upon Sherman’s Creek Bridge 166 C entres— Their different Species and Modes of Construct)'on . 171 On removing or striking Centres. 175 Gates—E lementary Form, and varieties of . . . 176 Park and Entrance Gates. .177 Dock Gates. 177 PART SIXTH. JOINERY. Mouldings— Grecian and Eoman versions of Classic Mouldings . . 178 Methods of describing Grecian and Eoman Mouldings . 179 Gothic Mouldings.180 Eaking Mouldings. 181 Methods of Enlarging and Diminishing Mouldings . . 181 Joinery— Definition of, &c..1S2 Joints— Various Forms of.182 Gluing up Columns—Diminution of Columns . . 184 Moulding . . 184 CONTENTS. i x Page Framing.185 Floors— Skirting.. . 185 The Operation of Scribing Described .... 186 Doors— Framed or Bound Doors.186 Double Margined Doors ...... 186 Jib and Pew Doors.187 Architraves ......... 187 Windows and Finishings of Windows— Hinged or French Sashes—Suspended Sashes, &c. . .187 Shutters.188 Circular Window .189 Skylights— Skylight with Curved Bars—Irregular Octagonal Skylight 189 Elliptical Domical Skylight—Octagonal Skylight, &c. . 190 Pulpit, with Acoustical Canopy.190 Hinging .190 Varieties of Hinges, and modes of applying them . . 191 Labour-Saving Machines— Sketch of the History of Labour-Saving Machines . . 191 American Saw-bench.192 Furness’ Planing Machine.193 Furness’ Tenoning Machine.193 Furness’ Mortising Machine.194 PART SEVENTH. STAIES AND HAND E AILING. . Stairs —Introductory—Elementary Forms of ... 195 Definitions of Terms used in Stair-building . . . 196 Method of Setting out Stairs ...... 197 Eelative Proportion of Treads and Eisers . . . 197 Plans of Stairs.198 Newel Stairs—Geometrical Stairs—Wellhole Stairs . 198 Formation of Carriages for Elliptical Stairs, &c. . .199 Method of Scribing the Skirting.200 Method of Gluing up Strings.200 Diminishing and Enlarging Brackets .... 201 Handrailing— Definitions of Terms Used in Handrailing . . . 201 Construction of the Falling Mould .... 201 The Section of a Cylinder.202 Mr. Nicholson’s Method contrasted with the Method of the Author.203 Face Moulds and Falling Moulds for Eails . . . 204 Falling Moulds and Face Moulds for Scrolls . . . 205 Sections of Handrails.207 To Form the Section of the Mitre Cap .... 207 To Draw the Swan Neck at the Top of a Rail . . 207 To Form the Knee at the Bottom. Newel .... 207 Methods of Describing Scrolls.207 . * ■ ■ »; , V. . M , .V /- r V" • - ■ sac -** TABLE OF CONTENTS. The Line of Shade in a Cone, and its Shadow cast upon both Planes of Projection ...... 222 The Shadow cast by a Cone upon a Sphere . . . 223 To determine the Shadow of a Concave surface of Re¬ volution ......... 223 Methods of Shading— Surfaces in the Light and Surfaces in the Shade . 224 Shadiug by Flat Tints.224 Shading by Softened Tints ...... 225 Elaboration of Shading and Shadows .... 225 Perspective—I ntroduction—Elementary Illustrations . 227 Perspective Planes ....... 228 Point of View, Station Point, &c, ..... 229 To Find the Perspective of Points and Lines . . 229 To Draw a Square in Perspective.231 To Draw a Cube in Perspective ..... 232 The Principles of Perspective applied .... 233 To Draw a Pavement of Squares in Perspective . . 235 To Draw a Circle in Perspective.236 Perspective of Solids—Tetrahedrons—Cubes—Cylinders 237 Parallel and Oblique Perspective considered . . . 239 To Draw a Sphere in Perspective .... 239 Practical Examples of Perspective Drawing applied to Architecture, &c. ....... 240 Cross, Tower, and Spire.241 A Series of Arches in Perspective, &c.242 A Tuscan Gateway in Perspective.242 A Turkish Bath in Perspective .243 Gothic Spire in Perspective ...... 243 Isometric Projection— Projection of a Cube ........ 243 Projection of the Combination of the Cube and Parallelo- pipedon ......... 244 Projections of a Pyramid, Octagon, Hexagon, Pentagon, &c. 245 Mr. Nicholson’s Table of the Isometric Radius, Semi-axis Minor, and Semi-axis Major of Ellipses . . . 246 PAIIT EIGHTH. PROJECTION OF SHADOWS—PERSPECTIVE- JSOMETRICAL PROJECTION. Projection of Shadows— Properties of Light—The Cone of Rays, &c. . . . Light and Shade—Forms of Cast Shadows, &c. Projections of the Shadow of a Straight Line on Horizontal and Inclined Planes ....... Shadows Projected by Rays of Light which are Parallel among themselves. Projection of the Solar Ray considered .... Shadows cast by a Straight Line upon a Vertical Plane and Curved Surfaces ........ Shadow cast by a Circle upon Horizontal and Vertical Planes . Shadow of a Circle cast upon Two Planes, &c. . Shadow cast by a Cylinder under various conditions Shadow cast in the Interior of a Concave Cylindrical Surface ......... Shadow of a Cone cast upon a Horizontal Plane Shadow cast in the Concave Interior of a Cone The Boundaries of Shade on a Sphere, and its Shadow cast on a Horizontal Plane ...... The Shadow cast in the Concave Interior of a Hemisphere . The Shadow cast in a Niche ...... Shadow cast by a Ring, the Exterior Form of which is a Torus. The Shadow of a Regular Hexagonal Pyramid cast upon both Planes of Projection . The Shadow of a Hexagonal Prism cast upon both Planes of Projection ........ The Limit of Shade on a Cylinder, and its Shadow cast upon the Two Planes of Projection .... 209 210 212 213 215 215 216 218 218 219 220 220 221 LIST OF PLATES, WITH REFERENCES TO THE PAGES WHERE THEY ARE SEVERALLY DESCRIBED. I. . II. . III. IY. V. . YI. YII. VIII. IX. X. XI. XII. XIII. XIV. XV. XVI. XVII. XVIII. XIX. XX. XXI. XXII. XXIII. XXIV. XXV. XXVI. XXVII. XXVIII XXIX. XXX. . XXXI. . XXXII. XXXIII. XXXIV. Centering op Ballochmyle Viaduct, Glasgow and So.-Western Railway. Sections of Solids. Sections of a Cone, Cylinder, Sphere, &c. Coverings of Solids. Covering of a Right Cylinder, &c. Coverings of Solids. Covering of the Frustum of a Cone, required between the parallels. m _-„ In ab take any two points A and c _ n b, and from these points as cen¬ tres, with a radius equal to the line m n, describe the arcs g and —j-S- h; draw the line c D so as to touch these arcs; that is, so as to form their common tangent; and C D will be parallel to A B, as required. Note .—This method of drawing parallels, however current in books on practical geometry, is, to say the least, objectionable, inasmuch as the learner has not been previously informed how to draw tangents to circles or arcs of circles. This objection might be obviated in the following manner: Let ab (Fig. 36) be the given straight line, and the line m n the given distance at which the parallel to ab is to be drawn. From b set off b a=mn, and from a as a c_ centre, with the radius m n, describe the semicircle b e b. Again, from b with mn or b a in the compasses, set off the arc be, and having joined these points, produce b c till the line cd be equal to b c; join da, cutting the circumference in e, and produce d a till af be equal to a e ; then draw f g at any angle to df cutting ab in A, and make hg=fh. Lastly, through the points e and g draw the straight line c d : it will be parallel to a b, sis required. Problem II. — Through a given point c, to draw a straight line parallel to a given straight line A b. In A B (Fig. 37) take any Fi? . S7 . point d, and from d as a centre with the radius d c, describe an arc C e, cutting A B in e, and from c as a centre, with the same radius, describe the arc d D, • Pig. 36. make it equal to C e, join c D, and it will be parallel to A b. Another Method. — Let A b (Fig. 38) be the given line, and c the given point, as before, through which a parallel is required. From c, with a radius sufficient to reach the nearest part of A B, describe an Fig 38 arc, so that A B may form its tangent, as at e; then from any point d in AB, with the radius ce, describe the arc /: A - through c draw C D, touching this arc : the line C D is the parallel required. This (II.) problem may be solved without having re¬ course to arcs, thus:—Let AB (Fig. 39) be the given line, and E the point through which the parallel to Ab is required to be drawn. In A B take Fig , S9 any point F; join E F, join also A E, and pro¬ duce it till E G be equal to A e. Likewise make F K = A F, join G K, and make G H or kh = ef; then, through the points E and H, draw the line CD: it will be parallel to the given line A B. Problem III.— To make an angle equal to a given rectilineal angle. From a given point E (Fig. 40), upon the straight line ef, to make an angle equal to the given Fi „ 40 angle ABC. From the angular point B, with any radius, describe the arc ef, cutting b c and B A in the points e and /. From the point Eon ef with the same radius, describe the arc h g, and make it equal to the arc e f ; then from E, through g, draw the line E d : the angle def will be equal to the angle abc. Problem IV.— To bisect a given angle. Let A b c (Fig. 41) be the given angle. From the angular point b, with any radius, describe an arc cutting B A and B c in the points d and e ; also, from the points d and e as centres, with any radius Fi „ 41 greater than half the dis¬ tance between them, de¬ scribe arcs cutting each other in /; through the point of intersection f, draw b/d : the angle ABC is bisected by the straight line b d ; that is, it is divided in to two equal angles, ABD and cbd. fiR 42 Or thus. —Let ABC (Fig. 42) be the given angle, as before. In A B take any two points D and E. On B c set off B F equal to B d, and B G equal to BE; join Ef and D G, intersecting each other in II; join also B H, and pro¬ duce it to any point K: the angle ABC is bisected by the line B K. Problem V. — To trisect or divide a right angle into three equal angles. Let abc (Fig. 43) be the given right angle. From the an¬ gular point B, with any radius, describe an arc cutting B A and bc in the points d and g\ from the points d and g, with the radius b d or B g, describe arcs cutting the arc G PRACTICAL CARPENTRY AND JOINERY. f d g in e and f: join B e and b /: these lines will trisect tlie angle ABC, or divide it into three F lg , J3 . equal angles. Note. —The trisection of an angle can he effected by means of elementary geometry only in a very few cases ; such, for instance, as those where the arc which measures the proposed angle is a whole circle, or a half, a fourth, or a fifth part of the circumference. Any angle of a pentagon is trisected by dia¬ gonals, drawn to its opposite angles. TO ERECT OR LET FALL PERPENDICULAR LINES. Problem YI.— From a given point c, in a given straight line a b, to erect a perpendicular. From the point C (Fig. 41), with any radius less than CA or CB, de¬ scribe arcs cutting the given line A B in d and e ; from these points as cen¬ tres, with a radius greater than c d or c e, describe arcs intersecting each other in f: join C/, and this line will be the perpendicular required. Problem YII.— From the point b, at the extremity of the line A b, to erect a perpendicular. Above the given line AB (Fig. 45), take any point c, and with the radius or distance c B, de¬ scribe the portion of the circle tZBe; join d c, and extend it to meet the opposite circumference in e: draw the line Be, which will be the perpendicular required. Another Method. —From the point B (Fig. 46), with any radius, describe the arc cde\ and hom the point c, where the arc meets the line A B, with the radius B c, cut the arc in d ; and from d, with the same ra¬ dius, cut it also in e. Again, from the points d and e, with equal radii — greater than half the distance from d to e —describe arcs intersecting each other in /: a. line joining b/ will be the perpendicular required. Or thus .—In A B (Fig. 47) take any point c; from B as a centre, with the radius B c, describe the portion of a circle ce; again, from c, with the same radius, draw an arc cutting the former in d ; also from d as a centre, with the same radius, describe the arc g h \ join c d, and extend it to meet gh in /: a line drawn from /, the point of intersec¬ tion, to B, will be the perpendicular to a b as before. Another Method. —To draw a right angle or erect a perpendicular by means of any scale of equal parts, or standard measure of inches, feet, yards, &c., by setting off distances in pro¬ portion to the numbers 3, 4, and 5, or 6, 8, and 10, or any num¬ bers whose squares correspond to the sides and hypotenuse of a_ right-angled triangle. From any scale of equal parts, as that represented by the line D (Fig. 48), which contains 5; set off from b, on the line A b, the distance B e, equal to 3 of these parts; then from B, with a radius equal to 4 of the same parts, describe the arc a 6; also from e as a centre, with a radius equal to 5 parts, describe another arc intersecting the former in c; lastly join b c ; the line B c will be perpendicular to A b. Note .—This mode of drawing right angles is more troublesome upon paper than the methods previously given; but in laying out grounds or foundations of buildings it is often useful, since only with a measuring rod, line, or chain, perpendiculars may be set out very accurately. The method is demonstrated thus:—The square of the hypotenuse, or longest side of a right-angled triangle, being equal to the sum of the squares of the other two sides, the same property must always be inherent in any three numbers, of which the squares of the two lesser numbers, added together, are equal to the square of the greater. For example, take the numbers 3, 4, and 5; the square of 3 is 9, and the square of 4 is 16; 16 and-9 added together make 25, which is 5 times 5, or the square of the greater number. Although these numbers, or any multiple of them, such as 6, 8, 10, or 12, 16, 20, &c., are the most simple, and most easily retained in tlie memory, yet there are other numbers, very different in propor¬ tion, which can be made to serve the same purpose. Let n denote any number; then »i*-f-l, n*—1, and 2 n, will represent the hypo¬ tenuse, base, and perpendicular of a right-angled triangle. Suppose ?i=6, then « s -f-l=37 , n 3 —1=35, and 2n=12: hence, 37, 35, and 12, are the sides of a right-angled triangle. D Problem YII I.— From a given point c, to let fall a perpendicular to a given straight line a b. From the point c (Fig. 49), with any radius greater than its distance from the line Mg.49. A B, describe an arc cutting A B in the points d and e ; also, from d and e, with equal radii—greater than half the distance between these points—describe arcs on the opposite side of A B, intersecting each other at f- join f c, cutting A B in D; CD will be a perpendicular let fall upon AB, as required. Another Method.— -When the given point whence the perpendicular is to be drawn is nearly opposite the end of the line. Let cbe the given point (Fig. 50), and A Bthe given line, as formerly. In A B, take any con¬ venient point e, and from it, with a radius equal to e c, describe an arc c D : between b and e, take any other point /; and from /, with fc as a radius, describe arcs cutting the former arc in c and D: a straight line drawn through these points of intersection will be perpendicular to the given line A B. Note. —Perpendiculars may also be erected or let fall upon straight lines, without having recourse 11 to the arcs of circles, by means Fig. 61 . of straight lines only. This \ may be shown as follows:— Let a B (Fig. 51) be a given \ G straight line, and c a given point in it; it is required to draw a line from the point c at right angles to a b. Be- K ,,_ _j._n tween c and a take any point d ; , c F draw d e at any angle to D B, and equal to D c; join e c, and \ / produce it too, making cq=cd. E On c b set off cf equal to c E ; join f g, and produce it till GH be equal GEOMETRY—CONSTRUCTION OF RECTILINEAL FIGURES, ETC. 7 c Fig. 52. M X E\ B II F to o F or ac; draw cn: it will be perpendicular to ab. —Again, let a b (Fig. 52) be tlie given straight line, and c tbe given point from which it is proposed to let hill a perpen¬ dicular upon a B. In A B, take any point D, and join c d ; then towards a set off de=cd; also towards b set off D f = d e or d c ; join e c and c f. Again, pro¬ duce E c till e o be equal to E f, and from ef cut off eu = ec; join a n, and extend it on the other side of a b till n k be equal to a n. Then join k e, and cut off k l = c a; join cl, cutting ab in m: cm is the perpendicular let fall upon A b. Problem IX.— To bisect a given straight line. Let A B (Fig. 53) be the given Fig. bs. straight line. From the extreme points A and B as centres, with any equal radii greater than half the length of A b, describe arcs cutting each other in C and D : a straight line drawn through the points of intersection c and D, will bisect the line A B in e. Problem X.— To divide a given straight line into any number of equal parts. Let A B (Fig. 5-1) be the given line to be divided into five equal parts. From the point A draw the straight line A c, forming any angle with A B. On 54 B the line A c, with any convenient opening of the compasses, set off five equal parts towards C; join the extreme points c B ; through the remaining points, 1,2, 3, and 4, draw lines parallel to BO, cutting AB in the corresponding points, 1,2, 3, and 4: A B will be divided into five equal parts, as required. Another Method. —Let ab (Fig. 55) be the given straight line, which we shall suppose,in this instance, is to bo divided into four equal parts. Draw the straight line CD of any conveni¬ ent length, and Rom c set off four equal parts. Then from C, with a radius equal to the distance from c to the last division, or number 4, on the line c n, de¬ scribe an arc; and from the point marked 4, with the same radius, c~ describe another arc cutting the former in g. From the point of intersection G, draw G c, G 1, G 2, G 3, and G 4. Again, from G, with a radius equal to the given line A B, describe an arc cutting G c and G 4 in the points E and F ; join these points : the line V, F will be equal to the given line AB, and it will also Fig. 56. be divided into four equal parts by the lines G 1, g 2. and G 3. A line may be divided into any number of equal parts very simply, by means of a ruler, or scale of equal parts, without the help either of arcs or compasses. Thus :— Let A B (Fig. 56) be the given straight line, and let it be required to divide it into any number—say five equal parts. Draw A c, making any angle with ab; and from A towards c, set off any five equal parts; join 5 B, and produce it indefinitely towards D. Again, from 5, on the line 5 D, with the same or any other scale, set off five equal parts, as before, marked r, li, ill, iv, v; then join A v, 1 iv, 2 hi, 3 II, and 41. From A sei off Act = 4 r, a6 = 3ii,Ac = 2 hi, and A d = 1 rv. Join also al, b 2, c 3, and d 4, cutting AB in 1', 2'. 3', 4': AB is divided into five equal parts by these lines. The bisection of a line by this method is exceedingly simple, as is shown by Fig 57. By either of the two pre¬ ceding methods, scales or drawings may be reduced or enlarged proportionably, so that each part of a given scale or drawing shall bear the same proportion to simi¬ lar parts of another scale or drawing of a different size. A fourth proportional to three given lines, may bo found, in like manner, by this problem. Assume A e, e b, and A D (Fig. 58), to be three given lines, the two first, A E, E B, being placed in the same straight line, and A D, the third line, making Fi-. 58. any angle with A b : having joined D E, through B draw B c, parallel to D e, meeting A d produced in c ; D c is a fourth proportional to these three lines. For, by the first method of this problem, DC and eb are similar portions of the lines AC and AB: wherefore the part DC has the same ratio to the remaining part AD that the part eb of thy line ab has to the remainder AE. As regards scales and drawings. Let A B (Fig. 59), represent the length of one scale or drawing, divided into the given parts Ad, dc, ef f g, gh , and hv>; and D E the length of another scale Fig. 59 . or drawing required to be divided • ** ■ ? ^ }t into similar parts. From the point B draw a line BC = DE, and forming any angle with A B ; join A c, and through the points e > f, g, and h, draAv d Jc, e l } d 1 ->e / m, g n, h 0 , parallel to A c ; and the parts c Jc, lc l, l m, &c., will be to each other, or to the whole line B c, as the lines Ad, d e, ef, &c., are to each other, or to the given line or scale AB. By the second method, as will be evident from the figure, similar divisions can be obtained in lines of any given length. Problem XI. — To describe an equilateral triangle upon a given straight line. Fig. r,o. Let A b (Fig. 60) be the given straight line. From the points A and B, with a radius equal to A b, describe arcs intersecting each other in the point c. Join CA and cb, and ABC will be the equilateral triangle required. Aote. An eminent mathematician haa made the following obser¬ vation regarding this problem:—“It is remarkable that it is not perhaps possible to resolve, without employing the arc of a circle, the very simple problem, and one of the first in the elements of geo¬ metry, viz., to describe an equilateral triangle.” “We have often attempted it,” continues the same author, “but without success, while trying how far we could proceed in geometry by means of straight lines only.” He did right to put in perhaps, as the thing happens to be possible after all, but it shows by what trifles the greatest men will sometimes be baffled. We submit the following PRACTICAL CARPENTRY AND JOINERY. Another Method .—Let ABC be the given triangle (Fig. (56). Produce one of its sides BC till fie - the extension CD be equal to the corresponding side of the proposed triangle. Through the point c draw a line CE parallel to B A; also, through the extreme point d, draw D E parallel to c A, meeting C e in E, and the triangle ecd will be similar to the triangle ABC. Note .—This example illustrates some of the most important pro¬ perties of triangles ; as, for instance, that the alternate angles formed by a straight line, cutting two or more parallel lines, are equal; like¬ wise, that the angles formed by one line falling xipon another, will either be two right angles, or will be together equal to two right angles; wherefore, the angles acb and acd are together equal to two right angles. Problem XYI.— To change a given triangle into another of equal area, having either its base or altitude greater or less than the base or altitude of the given, triangle. Let ABC be the given tri¬ angle (Fig. 67), and e f the altitude of the proposed triangle. From the given height or altitude at f draw the dotted line / B ; and through the point C, draw another dotted line, parallel to /b, and meeting the base line produced in D. Join / D; then the triangle A T>f will be equal in area to the given triangle ABC. Or coro- versely : let A d/ be the given triangle, and let c, or any point in A / produced, be the vertex or altitude of the proposed triangle : draw a dotted line Rom c to D, as be¬ fore ; also, through /, draw / B, parallel to C D ; join C R, which will complete the required triangle ABC. It must be evident from Fig. 67, that the same rule is applicable to any given difference, either in the base or altitude of triangles of equal area Problem XVII. — Two dissimilar triangles being given, to construct a third, which will be similar to the one, and equal in area to the other. Let ABC (Fig. 68) be one of the given triangles, to which the proposed triangle is to be similar, and dee the other given triangle to which it is required to be equal in area. By Problem XVI., change the triangle def into another, D H G, having its altitude equal to that of the triangle ABC. Take any indefinite straight line KL, limited at the one extremity Fip.cs. K ; and from K set off K m equal to A B, and m L equal to D H : bisect the whole line K L in o ; then from o as a cen¬ tre, with a radius equal to ok or o L describe the semi¬ circle K n L : also, from the point m, draw the line m n perpendicular to K l, meeting the circumference in n : mn is a mean proportional between Km and m l, or their equals A B and D H. Lastly, draw the straight line method as remarkably simple and easy:—Let a b (Fig. 61) be the given straight line. It is required to describe an equilateral triangle upon it without making use of the rig. 61 . compasses or arcs of a circle. Bisect a b in d (as shown in a former note), draw A e perpendicular and equal to a d ; join De, and extend da to r, making A F = D E; join also E F ; then from n erect the perpendicular iic=ef, and join a o and c b: abc will be an equilateral triangle. It is easy to see that a c 3 must be 4 ad 3 ; but Acf=A d’ + cd 5 (47 Prop. Buclid), and c d 3 =e f“=f a*-}- a e 3 = a E a -f- d e 3 ; but d e ! =a d -f- a e’= 2 a d 3 . •. c d s =3 a d 3 , and A c*= ab’=4ad ! . Q. E. D. Problem XII.— To construct a triangle whose sides shall be equal to three given ^ 62 lines. r Draw AB (Fig. 62) equal to the given line F. From A as a centre, with a radius equal to the line E, describe an arc; then from B as a centie, with a radius equal to the line D, describe another arc intersecting the former in C ; join c A and c B, and ABC will be the triangle required. Proelem XIII.— To find the length of the hypotenuse, or longest side of a right-angled Fig. 63 . triangle, whose other two sides are equal to two given lines D and e. Draw A b (Fig. 63) equal to the line e, and from the point B draw B c perpendicular and equal to the line D. Join A c, which will be the hypotenuse required. E — Problem XI Y. — The hypotenuse A b, and one side CD, of a right-angled triangle being given, to find the other side. Bisect the hypotenuse AB in e (Fig. 64), and from c as a centre, with a radius equal to e B or e e A, describe an arc ; also from A as a centre, with a radius equal to the given line CD, describe another arc, intersecting the former in E; join EA and E B ; and EB will be the r. -n side required of the right-angled triangle ABE. Problem XV. —On a given line ef, to construct a triangle similar to a given triangle a b c. From the angular point A Flg ®°' of the given triangle (Fig. 65), with any radius, describe the arc de, cutting A B and A c in the points d and e. Also, from E, the one extremity of the given line E F, with the same ra¬ dius, describe the arc m n: take the arc d e in the compasses, and apply it from m to n, so as to make it equal to d e. Again, from the angular point B, with the same or any other radius, describe the arc f g : likewise, from f, the other extremity of ef, with the same radius, draw the arc o p, making it equal to fg. Through the points n and p draw lines from E and F, meeting in D : the triangle efd will be similar to the triangle ABC, as required. GEOMETRY—CONSTRUCTION OF RECTILINEAL FIGURES, ETC. 9 R s equal to m n, and upon this line (by Problem XV.) construct a triangle rst similar to ABC, and it will likewise be equal in area to the triangle dug, or its equal D E f, as required. Note .—The truth of this method may be proved shortly thus:— Put AB = a, DK=i, and the perpendicular height of the triangles ofrwhiek a b and d h are the bases, equal to d. Also let the base of the triangle sought be represented by x, and its height by y. Then by similar triangles a : x :: d : y, and y = ^ Again, because the triangle sought must be equal to dgh, xy=db, and y= — • x — = —■ or dx l = adb and x 1 =a b, or x= r s, is a mean propor¬ tional between a and b, that is, between a b and d h. Problem XVIII. — To inscribe a circle in a given triangle. Let ABC (Fig. 69) be the given triangle. Bisect any two of its angles, as those at A a ’ Pi» M B and c, by the straight lines A d and c D. From the point D, where the bisecting lines meet, let fall the perpendicular d e upon the line A c ; then from D as a centre, with the radius D E, describe a circle. This circle will be inscribed in the triangle ABC, as required. Problem XIX. — To inscribe a circle within three given oblique lines, which, if produced, would form a triangle, but whose angular points are supposed to be inaccessible. From any point g in the line CD (Fig. 70), let fall gib perpendicular to A B ; and from the same point g erect a perpendicular to c D, meeting A B in 1c; bisect the angle k g h by the line g l; Fig. 70 . bisect also the line g l by the perpendi¬ cular line m n. In like manner, find the line op; bisect it also, and through the point of bisection draw the perpendicu¬ lar line r n, meeting m n in the point n. Lastly, from n, the point of intersection, let fall upon A b the perpen¬ dicular ns: ns will be the radius of the required circle. Problem XX. — To construct a triangle equal in mag¬ nitude or area to a given trapezium. Let abdc (Fig. 71) F> g . 71 . D be the given trapezium. Draw the diagonal A D ; then through C draw C E parallel to A d, and meet- ingthe base AB produced in E ; join E D, and BDE will E be a triangle equal in area to the given trapezium abdc. Problem XXI .—To construct a triangle equal in area to a given pentagon. Let A B c D e (Fig. 72) be the pentagon to which the triangle is to be equal. Draw (with dotted lines) the diagonals A d and D B ; draw also E f parallel to d a, and c G parallel to D B, meeting the base extended both ways in F and G. Join Df and dg, and dfg will f~ be a triangle equal in area to the given pentagon, abcde. In like manner, any other figure formed by straight lines may be reduced to a triangle. Should the given figure be a polygon of more than five sides, it will be necessary to change it into another of one side less suc¬ cessively, until it be reduced to five sides, by the method employed in the preceding examples. Problem XXII. — To reduce a hexagon, or six-sided figure, to a pentagon, or five-sided figure. Let A B c D E f (Fig. 73) be the given hexagon. Draw a diagonal between any two of Fig. 73. its alternate angles, as C F ; then through the intermediate angular point D draw the line D G parallel to c F, meeting the base E f ex- ‘ tended in G. Join c G, and ABCGE will be the pentagon required. Problem XXIII.— To construct a rectangle, or parallelogram, equal to a given triangle. Let ABC (Fig. 74) be the given triangle, and the dotted line AD its altitude or perpendicu¬ lar height. Bisect AD in E, and through the point E draw FEO parallel to B D c. Again, through B and c, the extremities of the base, draw B f and c G, each pa¬ rallel to A d, and meeting the line F E g in the points F and G : then bcgf is the rectangle or parallelogram required. Problem XXIV. — To describe a square, or equilateral rectangle, the sides of which shall be equal to a given straight line. Let A B (Fig. 75) be the straight line to which the sides of the square are to be equal. Draw Fig. 75. c D equal to A B, and from c and D as centres, with a radius equal to CD, describe the arcs D F and c E, inter¬ secting each other in g: bisect the arc eg in h: from g as a centre, with radius g h, draw arcs cutting c E and D F in E and F. Join d E, e f, and F C: c D E F is the square re¬ quired. A B Problem XXV.— To construct a rectangle whose sides shall be equal to two given lines. Let AB and CD (Fig. 76) be the given lines. Draw the straight line E F equal to Fig. ? c - A B, and from E draw E h perpen¬ dicular to E F, and equal to CD; then from H and f as centres, with radii equal to A B and c D, describe arcs intersecting in g. Join f G and G H, and efgh will A be the paral lelogram or rectangle c D required. Parallelograms of any form may be drawn in a similar manner. Problem XXVI.— To describe a square equal to a given rectangle. Fig. 77. Let A b c d (Fig. 77) be the given rectangle. Pro¬ duce A b, one side of the rectangle, to E, and make b e equal to B c. Bisect A E in K, and from K as a centre, with the radius K A or k e, describe the semicircle A h e. Produce c B to meet 10 PRACTICAL CARPENTRY AND JOINERY. : the circumference in ii; extend be to F, and make bf equal to BH ; then complete the square BFGH, and it will be equal to the given rectangle. Thus, by means of this, and Problem XXIII., a triangle can be successively changed, first into a rectangle, then from a rectangle into a square, in such a manner that all the three figures shall still be equal in area. Problem XXVII.—To describe a rectangle or 'paral¬ lelogram leaving one of its sides equal to a given line, and its area equal to that of a Fig. 7«. given rectangle. Let A B (Fig. 78) be the given line, and cdef the given rect¬ angle. Produce c E to 0, making E G equal to A B ; from G draw G K parallel to e f, and meeting D F produced in it. Draw the diagonal G f, extending it to meet c D produced in L ; also draw L K parallel to D H, and produce E F till it meet L K in M; then F M k h a a is the rectangle required. Note .—Equal anil similar rhomboids or parallelograms of any dimensions may be drawn after the same manner, seeing the comple¬ ments of the parallelograms which are described on or about the diagonal of any parallelogram, are always equal to each other; while the°parallelograms themselves are always similar to each other, and to the original parallelogram about the diagonal of which they are constructed. Thus, in the parallelogram cokl the complements cefd and f m k n are always equal, while the parallelograms e f n a and dfml about the diagonal ou, are always similar to each other, and to the whole parallelogram c a k l. Another Method.— Let cdef (Fig. 79, No. i) be the given rectangle, and A B (no. 2) a side of the proposed rectangle. Find a fourth proportional to the three fol¬ lowing straight lines, viz., Fi >-' 79 A b the given line, c D and D F sides of the given rectangle. Thus, from any point G (no. 3), draw two diverging lines G u and G K, equal to A B and C D, making any angle at the point G. Join K 11 , and produce c, n till n L be equal to u D F ; then, through L, draw L M parallel to ti K : K M will be a fourth proportional to A B, c D, and D F. Upon the given line A B describe the parallelogram AEON, having each of its sides A N and B o equal to K M, and it will be the rectangle required. "When two sides of one rectangle are reciprocally pro¬ portional to two sides of another, the rectangles must necessarily be equal; because, when four straight lines are geometrically proportional, the product of the first and fourth, or of the extremes, is always equal to the product of the second and third, or of the mean propor¬ tionals. Problem XXVIII. — Upon a given straight line to construct a rectangle equal to a given square. Let A b (Fig. SO, No. 1 ) be the given straight line, and CDEF (no. 2 ) the given square. Draw any two straight lines G n and G K (no. 3), forming any angle at G. Make G H equal to the given line A B, and G K equal to C D, a Produce gh to L, so that Fi". 80 . side of the given square HL may be equal to KG; join K H; then draw M L parallel to KH; join also K M, and it will be a third proportional to A B and c D. Lastly, upon A B as a base, describe a rectangle having its al¬ titude equal to K M, and it will be equal in area to the given square, as required. Problem XXIX.—To describe a square equal to two given squares. Let A and B (Fig. 81) be the given squares. Place them so that a side of each may form the right angle dce; join D E, and upon this hypotenuse describe the square D E G F, and it will be equal to the sum of the squares A and B, which are constructed upon the legs of the right-angled triangle DCE. In the same manner, any other recti¬ lineal figure, or even circle, may be found equal to the sum of other two similar figures or circles. Sup¬ pose the lines C D and c E to be the diameters of two circles, then D E will be the diameter of a third, equal in area to the other two circles. Or suppose c D and C E to be the like sides of any two similar figures, then D E will be the corresponding side of another similar figure, equal to both the former. °Problem XXX.— To describe a square equal to any number of given squares. Let it be required to construct a square equal to the three given squares A, b, and c (Fig. 82). Take the line D E, equal to the side of the square c. From the extremity D erect D F perpendicular to D E, and equal to the side of the square B ; join E F ; then a square described upon this line will be equal to the sum of the two given squares c and B. Again, upon the straight line EF erect the perpendicular fg, equal to the side of the third given square A ; and join G E, which will bo the side of the square G E 11 K, equal in area to A, B, and C. Proceed in the same way for any number of given squares. Problem XXXI.— To describe a square equal to the difference of two unequal squares. Let A and B (Fig. 83) be the given squares. Describe a right- angled triangle,having its base CD equal to the side of the square A, and its hypotenuse CE equal to the side of the square B; then E D, the third side of the right-angled triangle d e c, is the side of a square, the area of which GEOMETRY-CONSTRUCTION OF RECTILINEAL FIGURES. ETC will be equal to the difference of the areas of the two given squares A and B. Pboblem XXXII.— To describe a square which shall be equal to any portion of a given square. Let A (Fig. 84, No. i) be the given square, and let it be required to construct another square, whose area shall be one-third of A. Draw the straight line BC (xo.2) equal to the side of the given square A : produce this line to D, making c D equal to one-third of BC. Upon the whole line B d describe a semicircle, and from c erect C E perpendicular to B D. C E being a mean propor¬ tional between the two segments BC and CD of the line BD, will, consequently, be a side of the square required. In like manner, a square may be described, having any given ratio to a given square, or which may be any given multiple of another square. The first case of the problem is effected (as has been shown) by making the extension or part added to the given line equal to the required ratio; the second, by making the part produced equal to the required multiple of the given square. Remark. —Although, for the sake of brevity and simplicity, the four preceding problems have been restricted to the construction of squares, the same methods are equally applicable to all similar rec- i tilineal, curvilineal, or mixilineal plane figures. For circles, as ! already stated in Problem XXIX., we have only to substitute their i diameters for the sides of squares; whereas, in other cases, the lines forming a right-angled triangle can be supposed the homologous, or like sides of the similar figures to which they belong. Problem XXXIII.— To inscribe a parallelogram in a given quadrilateral figure. Let ABC D (Fig. 85) be the given quadrilateral in which the parallelogram is to be inscrib- Fj „ 85 ed. Bisect each of the sides in the points E, F, G, and n. Join ef, fg, G H, and H E, and the rectilineal figure E F G ti, thus formed, will be the parallelogram required. Problem XXXIY.— To describe a rectangle equal to a given rhomboid. Let A B c D (Fig. 86) be the given rhomboid or parallelo¬ gram to which the rectangle is re¬ quired to be equal. From each of the angular points B and C, upon the same side, let fall perpendiculars b E and c F upon A D, or upon A D produced to F, and the rectangle J B c F e will be equal in area to the given rhomboid A B c D. Problem XXXY.— To describe a rectangle equal to a given irregular quadrilateral figure. Let A B c D (Fig. 87, No. i) be the given quadrilateral. Between any two of its opposite angles, as B and p, draw the diagonal B D; then from the other two opposite angles, at A and C, let fall the perpendicu¬ lars a/ and eg upon the diagonal b d, or upon b d produced if necessary. Again, bisect B D in e, and draw the straight line 11 K (no. 2 ) equal to Be or De, half the diagonal. Upon the line 11 K as a base, construct a rect¬ angle, having its height equal to the sum of the perpen¬ diculars A / and C g: the rectangle thus described will be equal to the given quadrilateral abcd. Problem XXXYI. — To describe a quadrilateral figure equal to a given pentagon. Let abode (Fig. S8) be the given pentagon. Join any two of its alternate angles, as for rig 88. instance those at c and E, by the diagonal line CE; then through the intermediate angular point D draw the line DF parallel to C E, meeting A E produced in f : join C F, and the quadrilateral figure A b c F will be equal to the given pentagon A B c D E, as required. Upon the same principle, any recti¬ lineal figure may be reduced into another having one side less, but still equal in area to the original given figure, as has been already illustrated in Problems XX., XXI., and XXII. Problem XXXVII.— Upon a given straight line to describe any regular polygon. Example I. Upon a given line a b (Fig. 89) to describe a regular pentagon. —Produce A B to c, so that B C may be equal to AB: from B as a centre, with the radius BA or B c, describe the semicircle ADC: *'>g- divide the semi-circumference ADC into as many equal parts as there are sides in the re¬ quired polygon, which in the case before us will be five: through the second division from c draw the straight line B D, which will form another side of the figure. Bisect A B at e and B D at f and draw e G and / G perpendiculars to A B and B D ; then G, the point of intersection, is the centre of a circle, of which A B and D are points in the circumference. From G, with a radius equal to its dis¬ tance from any of these points, describe the circumference A B D II K; then by producing the dotted lines from the centre B, through the remaining divisions in the semi¬ circle A D C, so as to meet the circumference of which G is the centre, in n and K, these points will divide the circle abdhk into the number of parts required, each part being equal to the given side of the pentagon. Example II. Upon a given straight line to describe a regular heptagon. —Let ab (Fig. 90) be the given straight line. As in the former example, from B, with a radius equal to A B, describe the semicircle a d c, and produce A B to meet it in c. Divide the semi-circumference ADC into seven equal parts—the number of sides in a heptagon. Draw B D, as before, through the second division of the semicircle from C: bisect also Fig. 90 . A B in e, and B D in f and draw e G and / G respec¬ tively perpendicular to A B and B D. G, as formerly, is the centre of a circle, whose circum¬ ference passes through the points A B and D. Complete the circle abdii, and it will contain the given side A B se¬ ven times, which is the number of sides required. PRACTICAL CARPENTRY AND JOINERY. c D and c G each equal to CE or CF. Join B D, D E, E F, F G, and G A; then abdefg is a regular hexagon, de¬ scribed upon A b, as required. Problem XL. — To describe a regular octagon upon a given straight line. Let ab (Fig. 95) be the given line. From the extremities A and B erect the perpendicu¬ lars A E and B F: extend the given line both ways to h and l, forming external right angles with the lines A E and b f. Bi¬ sect these external right angles, making each of the bisecting lines A H and B c equal to the given line A B. Draw H G and c D parallel to A e or B F, and each equal in length to A B. From G draw GE parallel to BC, and intersecting A E in E, and from D draw D F parallel to A H, intersecting B F in F. Join E F, and ABCDFEGH is the octagon re¬ quired. Or from D and G as centres, with the given line ab as radius, describe arcs cutting the perpendiculars A e and BF in E and F, and join ge,ef,fd, to complete the octagon. Otherwise, thus. —Let ab (Fig. 96) be the given straight line on which the octagon is to be described. Bisect it in. a, and draw the perpendicular a b equal to A a or b a. Join A b, and produce ab to c, making be equal to a6: join also A c and B c, extending them so as to make c e and c F each equal to Ac or B c. Through c draw cog at right angles to A E. Again, through the same point c, draw D II at right angles to B F, making each of the lines c c, c D, c o, and c H equal to A c or c B, and consequently equal to one another. Lastly, join BC, CD, D E, EF, fg, g H, H a: A BCD efgh will be a regular octagon, described upon A B, as required. Problem XLI.— In a given square to inscribe a given octagon. Let abcd (Fig. 97) be the given square. Draw the dia¬ gonals A c and b d, intersecting each other in e\ then from the angular A points ABC and D as centres, with a radius equal to half the diagonal, viz., ke or ce, describe arcs cutting the sides of the square in the points /, g, n h, h, l, m, n, o , and the straight lines o f, g h, k l, and m ??-, joining D these points will complete the octagon, and be inscribed in the square ABCD, as required. Problem XLI I.— To inscribe any regular polygon in a given circle. Let A ED (Fig. 98) be the given circle, in which it is re- i quired to inscribe a regular pentagon. Draw the diameter ! AB of the given circle, and divide it into the same number | of equal parts as there are sides in the required polygon, i viz., five Bisect AB in «, and erect ec perpendicular to A B, cutting the circumference in F; and make F c, the I part without the circle, equal to three-fourths of the radius I A e or e b. From c, the extremity of the extended radius Remark. — From the preceding examples it is evident that polygons of any number of sides may be constructed upon the same principles, because the circumferences of all circles, when divided into the same number of equal parts, produce equal .angles; and, consequently, by dividing the semi-circumference of any given circle into the number of parts required, two of these parts will form an angle, which will be subtended by its corresponding part of the whole circumference. And as all regular polygons can be insci'ibed in a circle, it must necessarily follow, that if a circle be described through three given angles of that polygon, it will contain the number of sides or angles required. The above is a general rule, by which all regular polygons may be described upon a given straight liue; but there are other methods bv which many of them may be more expeditiously constructed, as shown in the following examples : - Problem XXXVIII .— Upon a given straight line to describe a regular pentagon. Let AB (Fig. 9i) be the given straight line; from its extremity B erect Be perpendicular ^ Fig. n. to A B, and equal to its half. Join A c, and produce it till c d be equal to B c, or half the given line A B. 1 From A and B as centres, with a radius equal to B d, describe arcs intersecting each other in e, which will be the centre of the circum¬ scribing circle abfg h. The side A B applied successively to this circumference, will give the angular points of the pentagon; and these being connected by straight lines, will complete the figure. Another Method .—Let ab (Fig. 92) be the given line, upon which the pentagon is to be described. Erect B d per¬ pendicular and equal to A B. Bisect A b in c, and join c d: produce A b, making c e equal to c d. Then from A and B as centres, with the radius A e, describe the arcs G n and G F, intersecting each other in G. A gain, from the same points A and B, with the radius A B, describe arcs intersecting the former in n and F. Join b f, f G, Gil, and H A, and the rectilineal figure abfgh will be a regular pentagon, having a b as one of its sides, ns required. Problem XXXIX. — Upon a given straight line to describe a regular hexagon. Fig. 93 . Let A B (Fig. 93) be the given straight line. From the extremi¬ ties A and B as centres, with the radius A B, describe arcs cutting each other in g. Again from g, the point of intersection, with the same radius, describe the circle ABC, which will contain the given side ab six times when applied to its circumference, and will be the hexagon required. Another Method .—Upon the given F P line A B (Fig. 94) describe (Problem XI.) the equilateral triangle ABC. Extend the sides A c and BC to E o and F, making CE and CF each equal to a side of the triangle. Bisect the angles A c F and BCE by the straight line a CD, drawn through the common vertex C, and make GEOMETRY—CONSTRUCTION OF RECTILINEAL FIGURES, ETC. draw the straight- line CD through the second division from A of the diameter A B, producing it to meet the opposite circumference at D. Join D A ; then the line or dis¬ tance between the point D thus found, and the adjacent extremity A of the diameter A B, will be a side of the required polygon; and if t successively applied to the circum¬ ference A D B will form the penta¬ gon, as proposed. Again, by the second method, when the polygon to be inscribed is a hexagon, the diameter is divided into six equal parts; and if hues be drawn from the extremity of a perpendicular, whose position and height is determined as before, so as to pass through the first division on each side of the centre, and continued to cut the opposite circumference, the chord which is formed by joining the points of intersection will subtend twice 30, or GO degrees, which is a sixth part of the cir¬ cumference. and therefore a side of the hexagon. These are general rules for the inscription of polygons; but there are other methods of inscribing plane figures in circles, as will be shown in the succeeding examples. Problem XLIII.— In a given circle to describe an equilateral triangle, a hexagon , or a dodecagon. Let A D G E, &c. (Fig. 99), be the given circle. From any point A in the circumference of the oircle, with the radius A B, equal to that of the given circle, describe the arc cbd, and join c d. From C as a centre, with the radius C D, cut the circumference at E, also join DE and EC, then cde will be the equilateral triangle required. For the hexagon, apply the radius A B six times round the circumference of the given circle, and the figure acfegd will be the hexagon sought. Bisect the arc A C in h, and join A h, h C, then either of these lines applied twelve times successively to the circumference, will form the dodecagon, and be contained in the circle. Another Method of inscribing an equilateral triangle. — Let Fig. 100 . ABE (Fig. 100) be the given circle, and C its centre. Draw the diame¬ ter ab, upon which describe the equilateral triangle ADB: join CD, cutting the circumference in E; then through E draw EF parallel to D A, and e G parallel to D B, and meeting the opposite circumfer¬ ence in F and G ; join F G. The tri¬ angle E F G is equilateral, and inscribed in the circle A B E. Problem XL1V.— In a given Fig. 101 . circle to inscribe a square or an octagon. Let abc (Fig. 101) be the given circle. Draw the diameters a c and bd at right angles to each other. Join AB, BC, CD, and DA: these lines will form the square a B c D. Bisect the arcs AB, b c, CD, and D A, in the points e, f g, and h. Join A e, e b, b/, 13 &c., and the octagon will be completed and inscribed, as required. Problem XLY.— To inscribe a regular pentagon or a regular decagon in a given circle. Let A B c D (Fig. 102) be the given circle, of which o is the centre. Draw the diameters A c B D at right angles to each other: bisect the radius A o in e, and from E, with the distance e b, describe the arc B F, cutting A c in F; also from B as a centre, a with the distance bf, describe the arc F G, cutting the circum¬ ference in G. Join GB, and four such chords applied from G round the circumference will terminate in B, and form the pen¬ tagon. Bisect the arc bii in k: join B k and li k. If the same process be repeated with each of the arcs, or if either of the chords B k or k H be carried round the cir¬ cumference, a decagon will be inscribed in the circle, as required. Problem XLYI.— To inscribe a regular polygon in a given circle, by finding the angle at the centre. Divide 3G0 degrees, or the whole circumference of the circle, by the number of sides in Fig. ios. the given polygon, and the quotient will be the number of degrees con¬ tained in the angle at the centre. Suppose, for example, that the polygon to be inscribed in the given circle A is a regular hexagon. By a scale of chords, or any other instrument for measuring angles, make an angle at B the centre of the circle, equal to GO degrees, the legs of which when produced meet the circumference at c and D. Draw the chord C D; this line applied six times successively to the circumference of the given circle, will constitute the required hexagon. To find the angle of any polygon, we have only to subtract the angle at its centre from 180 degrees. For instance, the angle at the centre of a hexagon being 60 degrees, subtract 60 from ISO, and the remainder is 120, the in¬ terior angle of the hexagon, or the angle formed by any two of its adjacent sides. Suppose the required potygon to be an octagon, the angle at the centre of this figure is found, as directed above, to be 45 degrees, which being subtracted from 180, gives for the remainder 135 degrees, the angle formed by the adjoining sides of the octagon. Or, more simply, for the hexagon draw any radius CD (Fig. 103), and upon CD describe the equilateral triangle BCD, which, being repeated round the circle, will complete the hexagon. Again, to find the angle at the base of the elementary- triangle of any regular polygon. Find the interior angle of the polygon, by the rules already given, and one-hall of that angle will be the angle at the base of its elemen¬ tary triangle. As an example, the interior angle of a hexagon is 120 degrees, one half of which is 60 degrees; this is the angle at the base of the elementary triangle, upon which the hexagon is constructed. Also the angle of the octagon is 135 degrees, one-half of which is 6 1\, or 67 degrees 30 minutes—the angle at the base of its elementary triangle. PRACTICAL CARPENTRY AND JOINERY. / Vs V/ Problem XLVII.— To describe a square , as also an octagon, about a given circle. Let A BCD (Fig. 104) be the given circle. Draw the diameters A b and C D, intersecting rig. ku. each other at right angles: through the extremities of these diameters draw the lines ecf, fbg.gdii, and H A E, at right angles to A b and CD, and intersecting at the angular points E, F, g, and H: the figure E F G II is the circumscribing square. Join E G and FTI, cutting the circle in the points a, b, c, and d ; also through these points draw the lines e f, —, g h, —, k l, —, m n, and —, at right angles to their respective radii, and they will complete the octagon, which is also described about the circle A D B C. Proble m XLVIII.— About a given circle to describe a regular polygon. Let it be required to describe a regular pentagon about a given circle (Fig. 105), whose Fig. 105 . centre iso. Divide the circum- ? _ ?■ 1 ference into five equal parts— /;nA the number of sides contained / •, / \ in the given polygon—and from the points ABCD and E, j\ thus found, draw to the centre / V / . \ ' - J \ 0 the radii A 0, Bo, Co, Do, ^ / j \ // and Eo; also through these ' Ad same points draw the lines / A g, g B h, he k, k D l, and 1 IeJ perpendicular to their respective radii, and intersect¬ ing one another in the angular points /, g, h, k, and l: a regular pentagon will be formed, and be described about the circle. Upon the same principle, regular polygons of any num¬ ber of sides may be described about given circles. Problem XLIX.— Any regular polygon being given, to describe another having the same perimeter, but twice the number of sides. Let the regular pentagon (Fig. 106, No. 1 ) represent the Fig. 106. JY given polygon: bisect any two of its adjacent angles, as at A and B, by the straight lines A c and B c, intersecting each other in C; then A b c is the elementary triangle of the pentagon, and c its centre. Draw the straight line D E (no. 2) equal to half A B (no. i), and upon D E describe the triangle def (Problem XY.), similar to ABC. Bisect the base line D E by a perpendicular drawn through the vertex f, and produce this perpendicular upwards to G, making F G equal to F E or F D. Join also G D and G E; then G D E will be the elementary triangle of the decagon, and G the centre. From G, with a radius equal to o D or G E, describe a circle and ten chords, each equal to d e: the base of the elementary triangle, applied successively to the circumference, will produce a decagon having the same perimeter, and, of course, twice the number of sides as the given pentagon. That the decagon thus constructed will have the same perimeter or contour as the pentagon, is evident, seeing that each side of the ten-sided figure is made equal to half a side of the five-sided figure 01 pentagon. Problem L.— Any regular polygon being given, to construct another having the same perimeter, but con¬ taining any different number of sides. For example, let it be required to construct a regular octagon, having its perimeter equal to that of a given hexagon. Divide A b, a side of the given polygon (Fig. 107, No. 1 ), into eight equal parts, the number of sides in the required figure. Let A c be six of these equal parts: draw D E (no. 2 ) equal to A C, or to six-eighths of the given line A B; and upon D F., by Problem XXXVII., describe a regular octagon; its perimeter will be equal to that of the given hexagon. In the preceding example the figure required has a greater number of sides than that which is given ; but to reverse the process, let D E (no. 2) be the side of a given octagon, and let a hexagon be the polygon required. Divide the line D E into six equal parts, and extend it to F, so as to make E F equal to two of these parts, or the whole fine D f equal to eight of these same parts, and con¬ sequently equal to A B, whicli also contains eight parts, each equal to the corresponding divisions on D E. Thus, whatever may be the difference between the number of sides in the given and proposed polygons, it is only neces¬ sary to divide a side of the given figure into the same number of equal parts as there are sides in the one re¬ quired— extending or contracting the given side by the number of equal parts, indicating the excess of the one figure above the other in the number of sides. Problem LI.— Any regular polygon being given, to describe another having the same area, but a different number of sides. Suppose the given polygon to be a regular pentagon: let it be required to describe a regular nonagon, the area of which shall be equal to that of the pentagon (Fig 108, No. 1 ). Find, by Problem XLIX., the elementary triangle A B c of the given pentagon. Divide the base A B into Fig. 10&. nine equal parts, and make D E (no. 2 ) equal to five of these parts; then upon D E, as a base, construct the triangle D E ir, having its altitude or perpendicular height equal GEOMETRY—CONSTRUCTION OF RECTILINEAL FIGURES, ETC. 15 to that of the triangle ABC. The triangle thus found will be five-ninths of ABC. Describe another triangle FGK (so. s). by Problem XLVI., having its altitude equal to that of D E H, and its vertical angle 40 degrees, which is the angle at the centre of a regular nonagon. Again, by Problem X., find a mean proportional between the bases D e and FG of the triangles DEH and FGK, and it will be a side of the nonagon required. Draw a line N 0 (n'o. 4 ), equal to the mean proportional thus found, and upon it describe, by Problem XXXVII., a regular nonagon (no. 4), and its area will be equal to that of the given pen¬ tagon (No. 1 ). Problem LII .—To find the area of any regular polygon. Let the given figure be a hexagon : it is required to find its area. Bisect any two adjacent angles, as those at A and B (Fig. 109), by the straight lines AC and B c, intersecting in c, which will be the centre of the polygon. Mark the altitude of this ele- : mentary triangle, by a dotted line drawn from c perpen¬ dicular to the base AB; then multiply together the base and altitude thus found, and this product by the number of sides: half gives the area of the whole figure. Or otherwise , thus .—Draw the straight line D E, equal to six times, i. e., as many times A B, the base of the ele¬ mentary triangle, as there are sides in the given polygon. Upon D E describe an isosceles triangle, having the same altitude as ABC, the elementary triangle of the given polygon: the triangle thus constructed is equal in area to the given hexagon ; consequently, by multiplying the base and altitude of this triangle together, half the product will be the area required. The rule may be expressed in other words, as follows:—The area of a regular polygon is equal to its perimeter, multiplied by half the radius of its inscribed circle, to which the sides of the polygon are tangents. Problem LIII .—To describe any figure similar and equal to a given rectilineal figure. Let A B c D e f (Fig. 110, No. 1 ) be the given rectilineal figure: it is required to construct another that shall be equal and similar to it. Divide the figure into triangles, by the diagonals A c, c F, and D F : draw a straight line G N (no. 2 ) equal to A F, and upon G N construct the triangle G k N, the three sides of which shall be respectively equal to those of the triangle A C F. Also upon G K, which is by construction equal to A c, describe the triangle G n K, having its sides gh and hk respectively equal to ab and B c. Again, upon it N, which is equal to c F, describe the triangle K L N, having its sides k l and L N respectively equal to CD and D F. And lastly, upon l n, which is equal to D r, construct the triangle L M N, having its sides lm, mn respectively equal to de and ef; then ghklmn will be the figure required. Any rectilineal figure may thus be described equal and similar to a given rectilineal figure, i. e., by dividing the figure or polygon into triangles, and upon a line equal to one of the given sides, constructing a succession of triangles equal and similar to the corresponding triangles into which the original figure is divided. The simplest method of constructing a triangle equal and similar to another, is the following:—Let A c F (Fig. 110, No. 1 ) be the given triangle. Take G n equal to AF; from G as a centre, with a radius equal to AC, de¬ scribe an arc, and from N, with a radius equal to c F, describe another arc cutting the former in K; then, by joining k G and K N, the triangle K G N is formed equal and similar to A c F. The same process may be repeated, till all the triangles in no. 1 are exhausted. Problem LIV .—On a given line to describe a figure similar to a given rectilineal figure. Let ab CD E (Fig. Ill, No. 1 ), be the given rectilineal figure, and FL (no. 2) the given straight line. Divide abcde into triangles by the diagonals B d and B E. From the angular point A, with any convenient radius, describe the arc f g \ and from F (no. 2), one extremity of the given line FL, with the same ra¬ dius, describe the arc mn, making it equal to / g ; likewise from the angular point E (no. 1 ), with any ra¬ dius, describe the arc h k, and from the point L, with the same radius, draw the arc 0 p equal to h Jc; then through the points F and m, as also through L and 0 , draw the straight lines F G and L G, intersecting in G, and the triangle fgl thus found will be similar to ABE. In like manner, upon GL construct the triangle G K L, and upon G K construct the triangle G 11 K, similar to the corresponding triangles bde and BCD (no. 1 ); then FGHEL will be the figure required. Another mode of solution. — Let abcde (Fig. 112, No. 1) be the given figure, as before, and F L (no. 2) the given line, upon which a figure similar to A B c D E is to be con¬ structed. Divide the given figure into the triangles Abe, deb, and B c D. From 0 as a centre (ko. 3), with any moderate radius, describe the circle ab c d, &c. From the point a, round the circumference draw the chords 1C PRACTICAL CARPENTRY AND JOINERY. a b, be, cd, d e, ef, f g, and g h, respectively equal to the lines A E, A B, B E, D e, D B, C D, and B c (No. l). Draw also the radii a o, bo, co, do, &c. (no. s). On the line a o find ok, a fourth proportional to the lines a b, ao, and the given line f L; then through the point lc draw 1c l parallel to the chord a b, and it will be equal by construction to F L (no. From the point l , where the line hi meets the radius b o, draw l m parallel to be; draw also the parallels 7 fi 71 , n o, o p, p r, and r t, meeting the different radii in the points m, n, o, p, r, and t. Then upon F L as a base construct the triangle FGL, having its sides F G and G L equal to the lines l m and m n. Also upon G L describe the triangle G K L, having its sides K L and G K equal to n o and o p. Again, upon G K as a base construct the triangle G K H, having its sides G H and n K respectively equal to the lines p r and r t, and the rectilineal figure fghkl thus formed will be similar to the given recti¬ lineal figure ABODE. CONSTRUCTION OF CIRCLES, CIRCULAR FIGURES, &c. Problem LY.— To find the centre of a given circle. Let A c B D (Fig. 113) be the given circle. Draw the chord line A b between any two points A and B in the circum¬ ference : bisect the line AB by a perpen¬ dicular line CD, produced both ways to meet the circumference in C and D. Again, bisect the perpendicular CD in c, and e is the centre of the circle. Problem LYI.— To draw a tangent to a given circle , that shall pass through a given Fig. m. point in the circumference. ( Let A (Fig. 114) be a given point in the circumference of the circle, whose centre is B. Draw the radius AB,and through the point A draw the line c D perpendicular to A b, and it will be the tangent required. Problem LVII.— To draw a tangent to a circle, or any segment of a circle, through a *>8- os. given point, without having re¬ course to the centime. Let A (Fig. 115) be a given point in the circumference of a circle. Take any other point in the circum¬ ference, as B: join ab, and bisect the arc A B in e: join also A e : then from A as a centre, with a radius equal to A e, the chord of half the arc, describe the arc fe g, making e g equal to e f; then through the points A and g draw the straight line cad, and it will be the tan- Fig 116 gent sought. Another Method. —Let A (Fig. 116) be a given point in the circumference of a cir¬ cle. Take any two other points, as B and D in the cir¬ cumference, equidistant from A : join B D, and through A draw AC parallel to bd- AC is a tangent to the circle. Other methods of drawing a tangent to a circle, from a given point in the circumference, without finding the centre. —Let abc (Fig. 117) be Fig . 117- the given circle, and b the given point in the circumference, from which the tangent is to be drawn, without finding the centre. Take any other two points A and c, in the circumference, one on each side of B, and join them so as to form the triangle A E c. Produce B c to D, making B D equal to A c, and from B, with a radius equal to A B, de¬ scribe an arc; and from d, with the radius B c, describe another arc intersecting the former in E. Through E draw B E f, which is the tangent sought. For, join d e, and as the triangle deb is by construction equal and similar to the triangle A b c, of which the angle dbe is equal to the angle B A c, the angle in the alternate seg¬ ment of the circle, b f must consequently be a tangent to the circle at the point B. Othervjise, thus .—Let abc (Fig. 118) be the given circle, and B the given point, as before. Take any two points A and c in the circumference, equidistant from B; join A c, and bisect it in D. From B and D as centres, with the same radius of any conve¬ nient length, describe arcs in¬ tersecting in E. From E, with . the distance EBorED, describe the semicircle DBF, and join DE, and produce it to meet the semi-circumference in f. Join B F, and it will be a tangent to the circle. For, join B D, and as D B F is an angle in a semicircle, it must be a right angle, and as one of its sides, B D produced, would pass through the centre of the circle, b f must necessarily be a tangent to that circle. Problem LYIII.— To draw a tangent to a circle from a given point without the cii'cum- Fj ference. Let A (Fig. 119) be a given point without a given circle, of which B is the centre; join AB, and upon this line, as a diameter, describe a semicircle cutting the given cir¬ cumference in C; join AC, and it will be the tangent required. Another method, without finding the centre of the given circle. —Take any point c (Fig. 120) in that part of the given circumference which is Fig. 120 . concave towards A; join AC, inter¬ secting the opposite part of the cir¬ cumference in B; produce C A to D, so as to make AD equal to AB. Upon C D, as a diameter, describe a semi¬ circle ; draw A E at right angles to C D, meeting the semi-circumference in E. From A as a centre, with the radius A E, cut the given circle in F ; join A F, and it will be the tangent sought. Problem LIX.— A circle and a tangent being given , to find the point of contact. Let abc (Fig. 121) be a given circle, of which the centre GEOMETRY—CONSTRUCTION OF CIRCLES, CIRCULAR FIGURES, ETC. 17 is E, and CDa tangent to that circle it is required to find the point c. In CD take any point /: join E/, and bisect it in g; then from g, with the radius g E or gf, describe a semicircle in¬ tersecting the tangent and circum¬ ference in c, which is the point of contact sought. Problem LX.— To describe the circumference of a circle through three given points. Let A, B, and c (Fig. 122), be the given points not in ; straight line. Join A Band bc: bisect K _ each of the straight lines AB and BC by perpendiculars meeting in D; then A, b, and c are all equidistant from D; there¬ fore a circle described from D, with the radius DA, db, or DC, will pass through all the three points as re¬ quired. Problem LX I .-Given the span or chord Line, and height or versed sine of the segment of a circle , to find the radius. Let A E (Fig. 123) be the given span or chord line: bisect it in d by the perpendicular line c e, -a Fig. 123. and make DC equal to the given height or versed sine. Join AC, and also bisect it by a line drawn perpendicular to it, and meeting d e in F: join FA, then from f as a centre, with the ra¬ dius F A or f c, describe the arc A c B, which will be the segment required. Another Method. —Let ab (Fig. 124) be the given chord. Bisect it (as before) in D by the line ce, drawn at right angles to it. Make dc equal to the c Fig. 124. given height or versed sine: join Ac, and from the point c, with any radius less than A c, describe the arc f g\ and from A, with the same radius, draw the arc h k, making it equal to fg. Through the point k draw the line A E, which will meet CE in E; then the angle CAE will be equal to the angle A c E. Also from e, with the radius E A or E c, describe the arc ACB, which will form the segment of a circle subtended by the given chord a b,- and having the given height or versed sine D c. Another Method, —Let ab (Fig. 125) be the given chord; and let the line L represent the given height or versed sine of the arc. Bisect ab in c, and draw c D perpendicular to A b, and equal to the line L or given height of the arc. Draw the line A E, equal to CD, and making any angle with ab: join CE, and produce AE to F, so as to make e f equal to A c. Through F draw F G parallel to c E, and meeting AB produced in G. Produce Dcton, making c H equal to CG; then bisect D H in 0 , and it will be the centre of the arc; that is, an arc described from 0 , with the radius D 0 , will pass through the points A and B, and form the segment adb. Note .—The radius of an arc or arch, of which the span, or chord, and height are given, may be obtained by calculation. Thus, divide the square of half the chord by the height of the arc: to the quotient add the height of the arc, and the sum will be the diameter of the circle, or double the radius sought. For example, suppose the chord a b (Fig. 123) 22 feet, and the versed sine CD 7 feet; then half of 22 is 11, the length of the half chord daoi-db; and 11 squared is 121, which, divided by 7, gives 17? for the quotient then 7 added to 17?; is 24?, the diameter of the circle c e, one half of which, 12? feet, is the length of the radius f a or r c. Problem LXII.— The chord of an arc, and also the radius of the circle , being given; to find the height or versed sine of that arc. Let AC (Fig. 126) be the given chord: from the ex¬ tremities A and c, with the given ***• l26 * radius, describe arcs intersecting each other in D : join D A and D c. Bisect A c in E: join also DE, and extend it till D B be equal to D A or D C; then B E will be the height of the arc required. Or otherwise, thus. —Having found d as above; from this centre D, with the given radius, describe the arc ABC; and having bisected A c in E, join D E, and produce it to meet the circumference in B; be is the versed sine, or height of the arc as before. Or by calculation, subtract the square root of the differ¬ ence between the square of the radius and that of half the chord from the radius, and the remainder is the height of the arc. As an example, suppose the half chord E A or E c to be 16 feet, and the radius 20 feet; then the square of 20 is 400, and the square of 16 is 256, which subtracted from 400, gives 144 of a remainder, the square root of which is 12 : this subtracted from 20 leaves a remainder of 8, the height or versed sine E B. Another Method. —Let ab (Fig. 127) be the given chord; and let the straight line L represent the given radius. Produce a b to c, so as to make A c double the length of the line L; and on Fi>. 127. ac, as a diameter, describe a semicircle. Bisect A B in D, and draw c E at right angles to A c, and equal to AD or D B ; then through E draw E F parallel to B c, cutting the semicircle in the point F. From F let fall upon a c the perpendicular F G. Again, from D draw D H at right angles to A b, and equal to C G; then D n is the height or versed sine of the arc. Let H D be extended to o, so as to make H 0 equal to the given radius L; an arc described from 0 as a centre, with the radius 0 H, will also pass through A and B, the ex¬ tremities of the given chord. Note. —Though this method may appear somewhat round about, it is at least novel, and can be demonstrated on strictly geometrical principles. Problem LXII I. — To describe a circle that shall touch, two straight lines given in position , and one of them at a given point. Let A B and A c (Fig. 128) be the given straight lines, and D a given point in AB. If the lines are not parallel, they will meet if produced. Let them meet in the point A. From A C cut off A E, equal to AD; and- in D B take any point F, and make e G equal to D F. J oin E F and d g, C 18 PRACTICAL CARPENTRY AND JOINERY. intersecting in the point o. From o, with the distance o I) or o E, which are equal, describe a circle cutting the given line A B in the points D and H, and A o in e and L. Draw the radius ho, and exteud it to meet the cir¬ cumference in K; join also L o, and produce it to meet the op- positecircumference in m Again, join E m and D K, intersecting in s. From the point of inter¬ section s, as a centre, with the radius se or s d, which are equal, describe the circle END, n j and it will touch both the given lines ab and a c, and the former in the given point d. ■ - 7 '' 1 T 1 - - J ■ ” be the Another Method. —Let ab and AC (Fig. 129) given lines, and D the given point in A c, as before. ae equal to AD: draw d G at right «*■ 129 angles to A b, and meeting c a in G. Cut off A f equal to A G, and join ef, intersecting DG in H. From Ji, with the radius HE or HD, describe the circle E K D, and it will touch the given lines, and A B in D the given point. The plans here given of drawing a tangent from a given point in the circumference, without having Make r.lu» ppnr. nr. f f r\ Kp found in any book of practical geometry that we are ac¬ quainted with. They avoid the somewhat clumsy resource of gauging arcs with the compasses in order to obtain equal angles, which detracts from the elegance of the solution of a problem. The exhibition of arcs ought, if possible, to be avoided, except when they intersect, for the purpose of obtaining certain points. In other cases they mar the effect of the handsomest figures. This considera¬ tion induced having recourse to the circle in the con¬ struction of similar figures. Problem LXIV.— Two circles touching each other being given, to find the point of contact. First, let the circles touch each other internally, as in Fig 130 (n’o. i). Find c and b, the centres of these circles, by Pro¬ blem LV.: join OB, and produce it to meet both the circumferences in A. This will be the point of contact between the two circles. Second, let the circles touch each other externally, as in No then the point B, where the line A c intersects the common boundary of the two circles, is the point of contact; A and C being theircentres. Problem LXV.— To describe with given radii two contiguous circles , which shall also touch a given line. Let ab (Fig. 131) be the given line, and CD and EF the given radii. From any point y in AB, erect a perpendicular g h equal to c D, the greater radius, and set off’ g k equal toEF, the lesser. Through k draw km parallel to ab; and from h, the extremity of the perpendicular g h, with a distance equal to the sum of C d and E f, describe an arc intersecting the parallel km in the point m. Let fall the line m l perpendicular to A B, and it will be equal to E f, one of the given radii. From m, with the radius m l, de¬ scribe a circle. Again, from h, with the radius g h, de¬ scribe also a circle, and it will touch the former in n, while g and l are the respective points of contact with the given line A B. Problem LXVI.—From a given circle to cut ofi a segment, that shall contain an angle equal to a given angle. Let AB D (Fig. 132) represent the given circle. From any point B in the circumference, draw a. tangent BC; also from the point of contact B draw a chord line B A, so as to form an angle equal to the given angle; then the chord A B will divide the circle into two segments. In that segment having its portion of the circumference concave to¬ wards the tangent B c, take any point D, and join AD and D B: the angle adb in this alternate segment is equal to ABC, which was made equal to the given angle; therefore A D B is the segment required. Aote.—If a tangent and chord he drawn from any point in the cir¬ cumference of a circle, the angle formed by these lines will be equal to the vertical angle of any triangle having the chord for its base, and its vertex in any part of the circumference which bounds the alternate segment of the circle. Problem LX VII. — To divide a given circle into any number of equal or proportional parts by concentric divisions. Let ABC (Fig. 133) be the given circle, to be divided into five equal parts. Draw the radius A d, and divide it into the same number of parts as those re¬ quired in the circle; and upon the radius thus divided, describe a A semicircle: then from each point of division on A D, erect perpendi¬ culars to meet the semi-circumfer¬ ence in e,f, g, and h. From D, the centre of the given circle, with radii extending to each of the different points of intersection on the semicircle, describe successive circles, and they will divide the given circle into five parts of equal area as required; the centre part being also a circle, while the other four will be in the form ot rings. Problem LXVIII. - To divide a circle into three con¬ centric parts, bearing to each other the proportion of one , tiuo, three, from the centre. Draw the radius A d (Fig. 134), and divide it into six equal parts. Upon the radius thus divided, describe a semicircle: from the first and third points of division, draw perpen¬ diculars to meet the semi-circum¬ ference in e and /. From D, the centre of the given circle, with radii extending to e and /, describe circles which will divide the given circle into three parts, bearing to each other the same proportion as the divisions on A D, GEOMETRY -CONSTRUCTION OF CIRCLES, CIRCULAR FIGURES, ETC. 19 which areas I, 2, and 3. In like manner, circles may be divided in any given ratio by concentric divisions. Problem LXIN.— To divide a given circle into any number of parts, equal to each other both in area and perimeter. Let abcd (Fig. 135) be the given circle, which we shall suppose is to be divided into five equal and isoperi- metrical areas. Draw the diameter ac, and divide it into five equal parts, at the points e, /, g, and h. Upon Ac, A/, A g, and A h, as diameters, describe a suc¬ cession of semicircles, all upon the same side of the diameter AC. Then reversing the operation, by com¬ mencing at c, describe upon C h. c g, c/, and c e the same number of semicircles, on the contrary side of AC: these opposite semicircular lines will meet in the points e,f g, and h, forming the five equal and isoperime- trical figures into which the circle was to be divided. Note .—It ought to he understood that the diameter AC in the last example, and the directing lines in the two preceding, form no part of the boundary lines by which the respective circles are divided into equal or proportional parts. Problem LXX.— An arc of a circle being given, to raise perpendiculars from any given points in that arc without finding the centre. Let AB (Fig. 136) be the given arc, and A, c , d, and e the given points from which the perpendiculars are to be erected. In the arc e B take any point /, so as to make e f equal to e d : from d and / as centres, with any equal radii greater than half the distance between them, describe arcs inter¬ secting each other in g: join eg, and it will be one of the perpendiculars required: d h and c Jc are found in the same manner. In order to raise a perpendicular from A, the extremity of the arc, suppose the perpendicular ch to be erected: from c, with the distance c A, describe the arc Am; and from A, with the same radius, describe c l , intersecting Am in o: make o l equal to o m, and join A l, which will be the perpen¬ dicular sought. Note. —The perpendicular to any curve, means a line perpendicular to the tangent or chord of that curve. Problem LXXI.— To draw a straight line equal to the circumference of a given circle. Let ABC (Fig. 137) be the given circle. Draw the diameter ab, and divide it into seven equal parts: then draw the straight line D E, equal to three times the length of A B, and one-seventh part more; and it will be a very near approxima¬ tion to the length of the circumfer- i ence. J) The diameter of the circle is to the circumference in the ratio of 1 to 3-1415926, &c. As the decimals might be continued to infinity, it will be seen that the exact pro¬ portion cannot be obtained. The simplest approximation to this ratio is that of 7 to 22, or of 1 to 3-f, and the pre¬ ceding line is drawn according to this last proportion, Fig. 136. which is sufficiently near the truth for most practical purposes. Problem LXXII.— To draw a straight line equal to any given arc of a circle. Let A B (Fig. 138) be the given arc. Find c the centre of the arc, and complete the circle a d b. Draw the diameter B D, and produce it to E, until D E be equal to c D. Join A E, and extend it so as to meet a tangent drawn from B in the point F; then B F will be nearly equal to the arc a b. The following method of finding the length of an arc is equally simple and practical, and not less accurate than the k one given above. Let A B (Fig. 139) be the given arc. Find the centre c, and join ab, B C, and C A. Bi¬ sect the arc AB in D, and join also CD; then through the point D draw the straight lineEDF, at right angles to c D, and meeting c a and C b produced in E and F. Again, bisect the lines A E and B F in the points G and H. A straight line G H, joining these points, will be a very near approach to the length of the arc A B. Note .— Seeing that in very email arcs the ratio of the chord to the double tangent, or, which is the same thing, that of a side of the in¬ scribed to a side of the circumscribing polygon, approaches to a ratio of equality, an arc may be taken so small, that its length shall differ from cither of these sides by less than any assignable quantity; there¬ fore, the arithmetical mean between the two must differ from the length of the arc itself, by a quantity less than any that can be assigned. Consequently the smaller the given arc, the more nearly will the line found by the last method approximate to the exact leDgtli of the arc. If the given arc is above 60 degrees, or two-thirds of a quadrant, it ought to be bisected, and the length of the semi-arc thus found being doubled, will give the length of the whole arc. Since the two preceding problems cannot be exactly solved by any rule founded upon geometrical principles, the two following methods may also be used, which will give the length of a circular arc, or indeed of any curved line whatever, as accurately perhaps as it can possibly be ob¬ tained. The first is to bend a thin slip of wood or any other elastic substance round the curve, then this slip extended out at length will be a very near approach to the length required. The second is to take a small distance between the compasses, and suppose the curve to contain this dis¬ tance any number of times with a remainder. Upon a straight line repeat this distance, or chord, the same number of times, and transfer also the remainder from the curve to the straight line: the straight line thus ex¬ tended will be very nearly equal to the given curve. It is obvious that if a given curve be divided into any number of equal parts or arcs, and if the chords of these arcs be transferred to a straight line, the line thus formed must be somewhat less than the curved line, as the chord of an arc, however small, can never be exactly equal to the arc itself. It is also evident, however, that the smaller the distance between the points, and the more numerous the parts taken on the curve, the more nearly will the straight line to which these parts, or rather their chords, 20 PRACTICAL CARPENTRY AND JOINERY. are transferred, approximate to the length of the curved i line itself. Problem LXXIII.— To construct a triangle equal to I a given circle. Let Fig. 140 (no. i) be the given circle, of which the radius is A B. Draw by Problem LXXI. the straight line CD (no. 2), equal to the circumference of the circle: bisect it in E, and erect E f perpendicular to c D, and equal to the radius A B: ioin c F and D F, and c F D is the - triangle required. Or, as shown by the dotted lines, make c E, which is equal to the semi-circumference, the base of the triangle, and make its altitude g h equal to twice A B, or to the diameter of the circle: join c h and E h, then the triangle c E 7t is also equal to the given circle. Hence the area of a circle is equal to the product of its circumference by half the radius; or to the product of its semi-circumference by half the diameter or radius. Problem LXXIV.— To describe a 'rectangle equal to a given circle. Let Fig. 141 (no. 1 ) be the given circle, of which ab is the radius. By Problem LXXI. draw thestraight line CD (no. 2), equal to the semi¬ circumference of the circle, and upon this line as a base construct the rectangle cdef, having its altitude d e equal to a b, the radius of the given circle, and it will be such as is required. Problem LXXV.— To describe a square equal to a given circle. Let Fig. 142 (xo. 1) be the given circle, which has AB for its radius. Draw the line C D (no. 2), equal to the semi¬ circumference of the given circle. Produce CD to E, making d e equal to the radius a b. Upon the whole line de describe a semicircle, and draw D F perpendicular to de, and meeting the semi¬ circumference in F. Again, draw the straight line G h (no. a), equal to D F, and upon this line as a base describe the square G111 k, and its area will be equal to that of the given circle. Problem LXXYI.— To describe a square or a rectangle that shall be equal to a given circle. Let ABC (Fig. 143, no. i) be the given circle, of which D is the centre. Draw the diameter adc, and draw the double tangent G f n parallel and equal to a c : join d g and D H, cutting the circumference in k and l. Bisect G K and 11 l in m and n, and join m n. Draw the line P R (no. 2 ), equal to M N, and upon it describe the square p R s t, which will be very nearly equal to the area of the circle acc, Again, draw any indefinite line v w (xo. 3). In it take any point X: draw X Y at right angles to V W, and equal to M n or p R. In x w take any point z. and 1. Fig. 143. from z, with the radius Y z, describe the semicircle Q Y w Cut off x a, equal to x q, and describe the rectangle ab w x: it will be equal to the area of the circle. Problem LXXVII.— To describe an arc of a circle through three given points, without finding the centre. Let A, B, and c (Fig. 144), be three given points, not in a straight line. Connect these points, so as to form the triangle ABC, the base of which, A c, will represent the chord of the proposed arc. From A and c, with equal radii, describe the arcs d e and / g, cutting a b and B c in the points h and k. Make the arc k f equal to the arc h e, and d h equal g k. Divide each of these four arcs into the same number of equal parts. Then from A A and c, the extremities of the chord, through the first and second divisions on the arcs d h and k g, draw straight lines, intersecting each other respectively in the points l and m. In like manner, from A and c, through the first and second divisions on the arcs h e and kf draw lines intersecting respectively in the points n and o. A curve line traced through the vertical angular points of the triangles thus formed, will be the arc required. The vertical angles formed as above, are all equal by construction, and as they are upon the same base, they must (according to Euclid) be in the segment of a circle. Note. — The most expeditious, as also the most accurate method for tracing lines of this description, is the following. Having obtained a sufficient number of true points in the proposed curve, and having placed small nails in the several points, bend a thin slip of wood, or some other elastic substance, round these nails; then by drawing the pen along this slip on the side of the nails, the required curve line will be described. Another Construction. — Let A, B, and c (Fig. 145) be thegiven points. Join them so as to form the triangle ABO. Upon AC, as a diameter, de¬ scribe a semicircle, and produce A B and c B to meet the semi-circumference in the points D and E. From A set off on the semicircle the arc A f, Fig Ml. F 2 - E Fig. M3. GEOMETRY-CONSTRUCTION OF CIRCLES, CIRCULAR FIGURES, ETC. equal to the arc CD: join c F, and from A, with a radius equal to CB, cut the line cf inG: join ag, and produce it to meet the semicircle in H. Bisect the arcs d n and ef in the points K and L. Join a k and c L, intersecting each other in M. From M, the point of intersection, let fall upon AC the perpendicular m n, and it will be the altitude of the arc proposed. Another Method. — Let A, b, and C (Fig. 146) be the given points, through which the arc is to be drawn. Join these points, so as to form the triangle ABC. Upon AC describe a semicircle, and extend the lines A B and c B to meet the semi-circumference in D and E. In the semicircle adec insert the chords c F and a g, equal to the chords A E and C D; then the point X, where c F and A G intersect, is a point in the proposed arc. Bisect each of the arcs d f and E G in the points H and k, and join A K and CH by lines intersecting in w, which is also a point in the curve. Fig. 146 . J oin A f and c E, and draw the chord E F. From A F cut off f m, equal to B E, and through m draw m n parallel to F E, and cut¬ ting AEin n. Again,from A F cutoff FV, equal to E n\ also from c E cut off C y, equal to a v ; then a curve traced through A,v,B, w, x,Y, and c, will describe the arc required. Problem LXXY111 .—Three points, neither equidis¬ tant nor in the same straight tine, being given, through which the arc of a circle is to be described, to find the altitude of the proposed arc. Let A, B, and C (Fig. 147) be the given points. Con¬ nect them by the straight lines A B, B c, and A c, forming a triangle, the base of which, viz., a c, will be the chord of the arc whose height is to Fig. 147 . be determined. Bisect the vertical angle ABC by the line J} D, meeting the base in D : bisect also the base A c in e, and from E draw the line E g perpendicular to A c. From the vertex B, with any radius, describe the arc h lc, cutting the sides of the triangle abd, in the points h and 7o; and from any point g in the per¬ pendicular, with the same radius, describe the arc mn, making it equal to the arc hlc: join g m, producing it to meet the base A c in o, or A c produced if necessary ; then draw A f parallel to 0 g, and meeting it produced in f : E f is the extreme height or altitude of the arc proposed. Problem LXXIX.— A segment of a circle being given, to produce the corresponding segment, or to complete the circle without finding the centre. Let abc (Fig. 148) be the given segment; and let B be situated anywhere be¬ tween A and C: join A b and BC: bisect the vertical angle at B by the straight line B D, meeting A c in the point D. From the extrem¬ ities A and C, according to the method previously shown, de¬ scribe angles on the contrary side of the chord A c, equal to the angle abd or cbd, and produce the lines forming Fig. U8 these angles from A and c to meet in the point e; then the vertex e of the triangle A c E is a point in the arc, which is to form the opposite segment. Fig 149. Find by Problem LXXVII any other number of points in the proposed arc, as/, g, h, and lc, and join the extremities of the chord a c by a curved line passing through these points, and it will complete the circle as proposed. Another Method. —Let abc (F ig. 149) be the given segment. From b draw the line bd, cutting AC at any angle, and pro¬ duce it until D e be a fourth proportional to B D, d a, and D c. Take any other point F, and join F D, producing it to G, so as to make dg a fourth projiortional to fd, da, and DC. Find the points K, N, R, &c., in the same manner, and a curve line traced through these points will complete the circle. Problem LXXX.— To describe a semicircle by means of a carpenter’s square, or a right angle, without having recourse to its centre. At the extremities of the diameter AC (Fig. 150), fix two pins, then by slid¬ ing the sides of the square, or other right-angled in¬ strument, D B, B e, in con¬ tact with the pins, a pencil held in contact with the point B will describe the semicircle ABC Problem LXXXI.— To describe the segment of a circle by means of two rods or straight laths, the chord and versed sine being given. Take two rods, eb,ef (Fig. 151), each of which must be at least equal in length to the chord of the proposed segment AC: join them together at B, and expand them, Fig. 151. •B A so that their edges shall pass through the extremities of the chord, and the angle where they join shall be on the extremity b of the versed sine d b, or height of the segment. Fix the rods in that position by the cross piece g li, then by guiding the edges against pins in the extremities of the chord line A c, the curve abc will be described by the point B. Problem LXXXII.— To describe a segment at twice by rods or laths, forming a triangle like the last, or by a triangular mould; the chord and versed sine being given. Let Ac (Fig. 152) be the chord of the segment, and db its height or versed sine: join cB, and draw be parallel to AC, and make Fi* 152 . it equal to B c. Fix a pin in c and- another in B, and with the triangle E B c describe the arc c b. Then remove the pin c to a, and by guiding the sides of the triangle against A and b, describe the other half of the curve A b. Problem LXXXIII.— Having the chord and versed PRACTICAL CARPENTRY AND JOINERY. sine of the segment of a circle of large radius given , to find any number of ‘points in ike curve by means of in¬ tersecting lines. Let A c be the chord and D B the versed sine. Through b (Fig. 153) draw EF indefinitely and parallel to AC: join A B, and draw ae at right angles to A B. Draw also A G at right angles to A C, or divide A D and E B into the same number of equal parts, and number the divisions from A and E respectively, and join the corres¬ ponding numbers by the lines 1 1, 2 2, 3 3. Divide also A G into the same number of equal parts as A d or e b, num¬ bering the divi- s ions from A up¬ wards, 1,2,3, &e.; and from the points 1,2, and 3, draw lines to b; and the points of intersection of these, with the other lines at //, A h, l , will be points in the curve required. Same with b c. A nother Method. —Let A c (Fig. 154) be the chord and u B the versed sine. Join A B, B c, and through B draw E f parallel to A C. From the centre B, with the radius ba or bc, describe the arcs AE, c F. and divide them into any number of equal parts, as 1, 2, 3: from the divisions 1, 2, 3, draw radii to the centre B, and divide each radius into the same number of equal parts as the arcs A E and c F; and the points g, h, l, m, n, o, thus obtained, are points in the required curve. These methods, though not absolutely correct, are suffi¬ ciently accurate when the segment is less than the quadrant of a circle. Problem LXXXIY.— The chord and versed sine of an arc being given, to find the curve , without having recourse to the centre. Let A B (Fig. 155) be the chord, and "D c the versed sine. From c draw the tangent eg parallel to ab; and join c b, and bisect it in/. Make rig. iss. eg equal to cf and from / and g raise per¬ pendiculars to the lines c /, c g, intersecting in e, and e will be a point in the curve. Or, [ which isthe same thing, bisect the angle f eg by the straight line ce, and to this draw the perpendicular from f or g, meeting it in c, which is the point required. In the same way the point h is found by bisecting the angle ecg, then bisecting the line e c by a perpendicular cutting the bisecting line of the angle in h. As the seg¬ ments c c, e B, are equal, another point may be found by joining e B, bisecting it by a perpendicular in h, and making the perpendicular or versed sine equal to that of the segment already found. Proceed thus until a suffi¬ cient number of points is obtained. OF THE ELLIPSE, THE PARABOLA, AND THE HYPERBOLA. Tjif, Ellipse. —I his curve is produced by the section of a cone through both ot its sides, but not parallel to its base. If we expose to the sun a circle of wire, inscribed in a square traversed by two diameters which cross its centre at right angles, and so dispose of it that the rays of the sun may be perpendicular to the plane of the circle (Fig. 156), the shadow projected on a plane parallel to the plane of the circle, will produce a figure in all respects similar to the Fig. 153. A 7 r f i ^ \ \ 1 J \ \ . original figure of wire. But if the circle is turned on one of its diameters A B, with- Fig. us. out changing the situation of the plane on which the shadow is projected, then the shadow of the square E F G ii shall be changed to a parallelogram c f g h, 1 and the shadow of the circle ADBCto an ellipse A cl B C. 2. The shadow of the axis A B is the major axis, ii and the shadow cd of c J) is the minor axis of the ellipse. 3. As the circle may be so turned in regard to the sun and the plane on which the shadow is projected, that its shadow will be only a right line, it follows that, in turning, it can produce all the ellipses possible between a circle and a straight line. 4. If in the interior of the circle ol wire a regular polygon of any number of sides be inscribed, such as a decagon or dodecagon, it is evident that when the shadow of the circle becomes an ellipse, the shadows of the sides of the polygon will form a corresponding polygon, of which the angles, by reason of the parallelism oi the rays of light, will always be at an equal distance from that diameter which is perpendicular to the axis of rotation. Consequently, if we trace on the plane which receives the shadows the parallels e /, k a, m n, r s, g h, which pass through the angles of the polygon ot twelve sides, it will be found that when the frame is turned, the angles follow exactly the path of those lines. This illustration is fertile in suggesting methods for the graphic production of the ellipse, and ot figures resembling the ellipse, composed of arcs of circles. In regard to the first, let us suppose lines parallel to A B, drawn through ; the angles of the smaller inscribed polygon, intersecting the lines drawn through the angles of the larger polygon perpendicular to ab, and their intersections will give points in the elliptic curve. Hence— Problem LXXXV.— To draw an ellipse when the major and minor axes are given. GEOMETRY—CONSTRUCTION OF THE ELLIPSE, PARABOLA, AND HYPERBOLA.. 23 Let ac (Fig. 157) be the axis major, and D b the semi¬ axis minor. On a C describe the semicircle A E c, and from the same centre D, and with the length of the semi-axis minor as radius, describe the semicircle / By. Divide both semicircles into the same number of equal parts, 1, 2, 3, 4, &c.: through the points of division of the greater semicircle draw lines perpendicular to A C, and through the corresponding points of division of the lesser semi¬ circle draw lines parallel to A c, and the intersections of the two sets of lines hkl on n o, &c., will be points in the curve required. In regard to figures resembling the ellipse, composed of arcs of circles, the illustration suggests the following method of producing them graphically. Intersect each side of the polygon by a line perpen¬ dicular to it. Continue the perpendicular from the side of the polygon nearest to the minor axis, until it inter¬ sects the continuation of the axis. Continue the next perpendicular to intersect the last, and so on, and the points of intersection so obtained become the centres from which the flat arcs are described. The intersections of the perpendiculars of the sides nearest the major axis, with the major axis, give the centres of the quicker curves. Problem LXXXVT.—Let A b (Fig. 158) be the axis major, and C D the axis minor. On the semi-axis major, rig. us. from the centre E, describe the quadrant A F, and on the semi-axis minor, from the same centre, the quadrant G C: divide each of these into the same number of equal parts, and through the divisions draw lines parallel to the two axes respectively: the intersections of these lines, 1', 2', 3', 4', indicate the angles of the polygon. N ow, through the centre of the side 4' c, draw a perpendicular cutting the minor axis produced in l, and t is the centre of the arc 4'c. Through the centre of 3' 4' draw a perpendicular cutting rt in v, and v is the centre of the arc 3' 4'; and so on until the last arc A 1', the centre for -which is obtained at the inter¬ section of the perpendicular with the majur axis at p. As the ellipse is a symmetrical curve divided into four equal and similar parts by its axes, the remaining three quar¬ ters can readily be drawn. The ellipse may also be considered as the section of a cylinder. Let a BCD (Fig. 159) be the projection of a cylinder, of which the circle eh fk represents the base divided into twenty equal parts: through each division draw a line parallel to the axis of the cylinder, dividing the moiety of the surface of the cylinder A BCD into ten equal parts. Now if we imagine abcd to be a plane corresponding to the diameter H K, each line will be distant from it by the length of the corresponding perpendicular, 16, 2 c, 3 cZ, 4 e. Now, suppose the diagonal A c to indicate a section of the cylinder oblique to its axis, but perpendi¬ cular to the plane abcd, the ellipse which results from that section will be traced by raising from the points where the parallels meet the line A c, the indefinite per¬ pendiculars, and setting off upon these the distances 1 6, 2 c, 3 d, 4 e. From this is derived the most commonly used method of describing an ellipse by ordinates. Problem LXXXVII. —To draw an ellipse with the trammel. The trammel is an instrument consisting of two prin¬ cipal parts, the fixed part in the form of a cross E ¥ G II (Fig. 160), and the moveable piece or tracer k l m. The fixed piece is made of two rectangular bars or pieces of wood, of equal thickness, joined together so as to be in the same plane. On one side of the frame so formed, a groove is made, forming a right-angled cross. In the groove two studs, k and l, are fitted to slide freely, and carry attached to them the tracer Jc l on. The tracer is generally made to slide through a socket fixed to each stud, and provided with a screw or wedge, by which the distance apart of the studs may be regulated. The tracer has another slider m, also adjustable, which carries a pencil or point. The instrument is used as follows:—Let AC be the major, and HB the minor axis of an ellipse: lay the cross of the trammel on these lines, so that the centre lines of it may coincide with them; then adjust the sliders of the tracer, so that the distance between k and m may be equal to the semi-axis major, and the distance between l and m equal to the semi-axis minor; then by moving the bar round, the pencil in the slider will describe the ellipse. - • PRACTICAL CARPENTRY AND JOINERY 8 yrTy! **• :? r In Fig. 1G1 a modification of the instrument is shown. Here a square e d f is used to form the elliptical quadrant A B instead of the cross, and the studs h l k may be simply pencil in the bight or doubling of the string at n and carrying it round, the curve may be traced. This is based on the well- Fig. ici. known property of the ellipse, that the sum of any two lines drawn from the foci to any points in the circumfer¬ ence is the same. Problem XO. —The axes of an ellipse being giv- n en, to draw the , , . , a\ -t6 curve by inter- e sections. c ' “ Let A c (Fig. 165) be the major, and D b the semi-axis minor. On the major axis construct the parallelogram aefc, and make E / 'J 3 ■* .5 B E its height equal to the semi-axis minor. Divide A E and E B each into the same number of equal parts, and A number the divisions from A and E respectively; then join A 1, 1 2, 2 3, &c., and their intersections will give points through which the curve may be drawn. Problem XCI .—To describe an ellipse by another method of intersecting lines. Let A c (Fig. 166) be the major and EB the minor axis: i / Fig. 1C5 L _ _ _ pins, which can be kept pressed against the sides of the square while the tracer is moved. In this case the adjust¬ ment is obtained by making the distance h l equal to the semi-axis minor, and the distance l k equal to the semi¬ axis major. Problem LXXXYIII.—Fig. 162 shows an ellipse con¬ structed on the rig- principle of the trammel, without using that instru¬ ment. From any points,as/Z inthe A semi - conjugate axis B E, draw Hues so intersect • ing the axis AC in h and n, as that flc and l n may be equal to the difference between the semi transverse and semi¬ conjugate axes: produce Fig. iei. these lines to g, ni, and from the points l, f, &c., on the minor axis, and with the radius DC, strike the small arcs, cutting the lines in g and m. These intersec¬ tions are true points in the curve. This method is obviously the _f same as by the trammel; and in practice it is very useful; a thin straight¬ edge or a piece of stiff paper being used to transfer the points at once. Thus, on the edge of the slip of paper mark off the length of the semi-axis major, a b (Fig. 163), and then from b set off the distance b c, equal to the semi¬ axis minor; then by applying this to the drawing and carrying it round, keeping the points a and c one on each diameter, any number of points in the curve may be obtained. Problem LXXXIX .—An ellipse may cdso be de¬ scribed by means of a string. Let A B (Fig. 164) be the major axis, and D c the minor axis of the ellipse, and F G its two foci. Take a string E G F and pass it over the pins, and tie the ends together, so that when doubled it may be equal to the distance from the focus F to the end of the axis, B; then putting a draw A f and c G each perpendicular to AC, and equal to the semi-axis minor. Divide A D, the semi-axis major, and the lines A F and c G each into the same number of equal parts, in 1,2, 3, and 4; then from e, through the divi¬ sions 1, 2, 3 and 4, on the semi-axis major A D, draw the lines E h, E k, E l, and Em; and from B, through the divisions 1, 2, 3, and 4 on the line A F, draw the lines I, 2, 3, and 4 b; and the intersection of these with the GEOMETRY—CONSTRUCTION OF THE ELLIPSE, PARABOLA, AND HYPERBOLA 25 lines 15 1, 2, 3, and 4, in the points h k l m, will be points in the curve. In the same manner are drawn the Fy. 103. rampant ellipse, Fig. 167, and the segment of the ellipse, Fig- 168, and the rampant segment in Fig. 169, the point F in the two latter figures being the intersection of the major and minor axes. Problem XCII. — To describe with a compass a figure resembling the ellipse. Let A B (Fig. 170) be the given axis, which divide into three equal parts at the points/#. From these points as centres with the radius / a, describe circles which in¬ tersect each other, and from the points of intersection through / and g, draw the diameters c g E, c/d. From c as a centre, with the radius CD, describe the arc D E, which completes the semi-ellipse. The other half of the ellipse may be completed in the same manner, as shown by the dotted lines. Problem XCIII. — Another method of describing a figure approaching the ellipse with a compass. The proportions of the ellipse may be varied by altering the ratio of the divisions of the diameter, as thus:—Divide the major axis of the ellipse a b (Fig. 171), into four equal parts, in the points/# h. On f h construct an equilateral triangle / c h, and produce the sides of the triangle cf, c h indefinitely, as to d and E. Then from the centres / and h, with the radius A / describe the circles ad#,be#; and from the centre c, with the radius CD, describe the arc D E to complete the semi-ellipse. The other half may be completed in the same manner. By this method of construction the minor axis is to the major axis, as 14 Problem XCIV. — To describe an ellipse with the compass , the transverse and conjugate diameters being given. Let A c (Fig. 172) be the transverse diameter, and D B the conjugate semi-diameter. Divide D B into three equal parts in/and #, and make A li, c Jc each equal to two cf these parts: join hf then from h and k, with the radius E A h, describe the circles A m, c n. Bisect the line hf by the perpendicular I, meeting the axis B D produced in E. From E, through h, draw the line E h m, meeting the circle A m, and from e, with the radius E m, describe the arc mBn, completing the semi-ellipse. Another Method.—The two axes of an ellipse being given, to describe the ellipse with a compass. Let A CB E (Fig. 173) be the axes of the ellipse: draw A f parallel and equal to DB: bisect it in 1, and join 1 b. Divide AD also into two equal parts in 1, and from e, -< Fix. 173. through 1, draw the line e 1 k , meeting 1 b in k. Bisect k B by the perpendicular h, meeting the axis b e produced in G: join G k, cutting the transverse axis in m. Then from m, with the radius m A, describe the arc A k, and from G, with the radius G k, describe the arc k B. Problem XCY .—The two axes a c, be, being given, to describe with a compass a figure still more closely approximating to the ellipse. Draw a F parallel and equal to D B (Fig 174): divide it into three equal parts, and draw 1 b, 2 b. Then divide A d also into three equal parts in 1 2 d, and from e, d - ■ , i . .. 2 through the points 1 2 draw lines Em, E Jc, cutting the lines 1 B, 2 b, in m and Jc. Bisect Jc B by a perpendicular h, meeting e b produced in G: join Jc G. Bisect m Jc by a perpendicular /, meeting Jc G in n: join mn. Then G is the centre for tlie arc B Jc, n is the centre for the arc Jc m, cular. The perpendicular g G raised on /d will cut the prolongation of the minor axis in G, which will be. the centre of the arc / D. The perpendicular h H raised on ef meets the line gG in h, which will be the centre of the arc ef. The perpendicular Jc K raised on Ac, meets PRACTICAL CARPENTRY AND JOINERY. I and o is the centre for the arc m A. From the centre D measure the points of intersection of the lines G Jc, n m, with the axis AC, and transfer the measurements to the other side of D ; and set off and make G r equal to G n. Through r draw r s, and the centres G r i, for the other half of the curve, " ill be obtained. Problem: XCVI.— The two axes a c, e b being given, to describe a still 'nearer approximation to the ellipse with the compass. Draw af (Fig. 175) parallel and equal to DB: divide it into four equal parts: divide also ad into four equal parts; and from e and B, through the divisions in D A and A f, draw lines intersecting each other in r, n, Jc. Bisect 1c B by a perpendicular h, meeting b e produced in G: join G h. Bisect n Jc by the perpendicular Z, meeting ^ G in m: join mn. Bisect rn by the perpendicular o, meeting mn in p : join pr. Then the points G, m, 2 >, and s will be the centres for the arcs B Jc, k n, n r, and r a respectively. The lines G t, u y, w x, and the point v are transferred by measurement, as before. Note— By dividing the lines af and a d into a greater number of parts, a still nearer approximation to the elliptic curve will be obtained. The ellipse being a symmetrical curve divided into four equal and similar parts by its axis, it is only necessary to obtain one of these. Thus, on the demi-axis major describe the quadrant ab (Fig. 176), and on the demi-axis minor the quadrant c D. Divide each of these into three equal parts. Through the divisions in A b draw lines parallel to the minor axis, and through the divisions in c D draw lines parallel to the major axis; and the intersections of these in ef indicate the angles of a dodecagon of one quadrant, of which Ac, ef, f D are the sides. On the middle of each of these sides raise an indefinite perpendi¬ the major axis in m, which will be the centre of the arc A e. The three remaining quadrants of the ellipse are easily constructed from the data thus obtained. Tiie Parabola.— This curve is produced by the section of a cone parallel to one of its sides. n The arcli* (Fig. 199) is still flatter than the last. Tli line A B is divided into four equal parts in e, g, f; then Fig- 199. m from the centres A and b, with the radius A / the arcs A Aa \ '* A through the point of their intersection the lines e h, f h are drawn and produced until they meet perpendiculars let fill from e and f. The arcs A Jc, B l are described from e and /, with the radius A e, and the arcs km, l m from d c, with the radius D k. The height is \d of the span. Divide the line a b (Fig. 200) into six equal parts, in the points e, g, h, k, f From Hg . SO n. e and f let fall the perpen¬ diculars 6C,/D: from the points A and B, with the A radius A B, describe the arcs B c, A D cutting the perpen¬ diculars e c, / D in c and D. Draw D el, c fm. Then e and / are the centres of the arcs Al,Bm, and c and D the centres of the arcs m n, In. The height of the arch is, like the last, 3 d of the span. To make the crown of the arch flatter than in the last figure, proceed as before for the centres of the haunch arches, by dividing A B (Fig. Fig. 201. 201) into six parts, in e, f, T\ /A g, h, k: draw A m, B n ; then * 7 it from the centres of these arches e k, with the distance between them as radius, de¬ scribe the arcs e l, k l, and through l draw the lines e l, k l produced to meet the perpendiculars let fall from e and k, in c D. Then the points c and D are the cen- A A \ \ r; \\ // / \ tres of the arcs m 0 , n 0 , ( / \ To draw a four-centred arch when the height, width, or span are given.— Let A c (Fig. 202) be the span of the arch, and D B its height. 'ftp PRACTICAL CARPENTRY AND JOINERY. Divide T) b into five equal parts, in 1, 2, 3, g, B, and set off on the line A c, from A , m g . 202 . and c, three of those parts to A h, c Jc. Then from the point g, with the radius g h, describe the arc n li Jc o, and from the points h Jc, with the radius A Jo or c Jc, describe the arcs A n, c 0 . From the intersections of these arcs with the arc n Ji Jc o, and through thecentres Ji Jc, draw n Ji F, ok E. Then bisect n B, o B in l and m, and produce the lines until they meet 71 h F and ok E in f and E, which two last points are the centres of the arcs n B, o B. Another Method. —Bisect the width of the arch AC (Fig. 203) in D, draw the perpendicular D b, and make it equal to the height of the arch. Divide it into three equal parts: through the second division draw 2 E parallel to A c, intersecting the line c E drawn from c perpendicular to A C in e. Join E B, and draw from B the line bg f at right angles to it. On c A set oft c H equal to D 2 ; and on b f set off B G equal also to D 2: join G H, and bisect it at n. From the point F, where the bisect¬ ing line meets BGF, draw F 11 Jc. Then ll will be the cen¬ tre of the arc c Jc, and F the centre of the arc k B. For the other side of the arch, draw Fm parallel to AC; and from the centre line bd produced, set off m equal to F: draw m l. Another Method .—Divide the height d b (Fig. 201) into two equal parts, and draw 1 E parallel to A c, and meeting the perpen¬ dicular ce in e. Join B E, and draw B F at right angles to it: set off from c and B the points H and G, equal to D 1. Join H G, and bisect the line in Jc. The point F, in which the bisecting line of G H cuts BF, is the centre of the larger arc l b, and H is the centre of the smaller arc C l. To describe a GotJiic arcJi Joy the intersection f of straight lines, wJien tlie span and height are given .— Bisect A c (Fig. 205) in D, and from the point D and the extremities of the line draw A E, D B, C F at right angles to A C, and each equal to the height of the arch : join E B, b F Divide the line D B into any number of equal parts, 1, 2. 3, B, and through the divisions draw lines parallel to A c. Divide the line e b, 205t B F into the same number of equal parts, and from A and C draw lines A 1, A 2, A 3; and their intersection with the horizontal lines in f, h, will be points in the curve required. To draw the arcJies of GotJiic groins, to nnitre truly with a given arcJi of any form. —Let A c (Fig. 206) be the width of body range, and B D its height. Join c B, and divide it into any num¬ ber of equal parts: from the centre D, through the points of division, draw straight lines Di, d2,d3,d4, meeting the circumference of the arch in l, m, n, 0 . From B, through these points in the circumference, draw bo, b n, B m, B l, and produce them to meet a perpendicular raised from c. Let A c (Fig. 207) be the width of the groin arch, and db its height. Join A B, and divide it into the same number of parts as CBin Fig. 206; and draw through the points 1, 2, 3, 4 the lines D 1, D 2, D 3, D 4>. Then from A draw a line perpendicu¬ lar to A e, and transfer to it the divisions from the cor¬ responding line in Fig. 206; and from these divisions draw lines to b. The intersection of these lines with the lines d 1, D 2, &c., will give points through which the curve may be traced. To draw an ogee arcJi. —Divide the width ab (Fig. 208) into four equal parts, *'«; 208 - in d, c, e; and on d, e erect the square d,, f, g, e. The points d, e, f, g, are the centres of the four quad¬ rants A Jc, k l, B Ji, h l, com- J posing the arch. Another Method. —Let ao (Fig d B the height of the arch. Join AB, B C, and bisect the lines in e,f; then from the centres A, e, b, /, c, with the radius A e or e B, describe the arcs intersecting in the points g, Ji, k, l, which are the centres of the four arcs composing the ogee arch. Another Method, when the arch is equilateral. A B (Fig. 210) in C, join A Ji, r, - B Jt. From C, with the • radius A or B, describe the \ arcs A d, B e\ then, to find the centres of the other Fig- 2 arcs, from the points d, e, and Ji as centres, and with the same radius as before, describe arcs intersecting A [ .. X -|b each other in the points / . and g, which are the centres of the arcs h d, h c. 209) be the width and CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. 31 PART SECOND. CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. Before proceeding to introduce the student to stereo¬ graphy, it is proper to describe the instruments used by the architectural draughtsman, and to explain their con¬ struction and application. Cases of drawing instruments generally contain com¬ passes of various kinds, scales, protractors, parallel rulers, and drawing pens. Each of these shall be described in order. Compasses. These are of different kinds, viz :—dividers, compasses with moveable legs, bow-compasses, directors, propor¬ tional-compasses, and beam compasses. Dividers .—These are used for taking off and transfer¬ ring measurements. The common dividers are moderately- sized compasses, without moveable legs, working somewhat stiffly in the joint, and having fine, well-tempered points of equal length, lying fairly upon each other when closed. When dimensions are to be taken with extreme accuracy, these dividers will not, under the most skilful manipu¬ lation, work with the delicacy and certainty that are re¬ quired. In such cases, the draughtsman resorts to his hair dividers: these are compasses in which one leg is acted upon by a spring- and a finely-threaded screw, pressing upon this spring, changes the direction of the point to the nicety of a hairs-breadth. The distance to be measured is first taken as accurately as possible between the points of the compasses, and the screw is then turned until the dimension is obtained with positive exactness. This instru¬ ment is useless unless it be of the very best quality: it must work firmly and steadily; and the points need to be exquisitely adjusted, exceedingly fine, and well-tempered. Instruction for using dividers, which are applied only to measure and transfer distances and dimensions, may appear superfluous; but there are a few simple directions which may save the young draughtsman much perplexity and loss of time, it is, of course, desirable to work the compasses in such a manner that, when the dimension is taken, it may suffer no disturbance in its transfer from the scale to the drawing. In order to this, the instrument is to be held by the head or joint, the forefinger resting on the top of the joint, and the thumb and second finger on either side. When held in this way, there is no pressure except on the head and centre, and the dimension between the points cannot be altered ; but, if the instrument be clumsily seized by a thumb on one leg, and two fingers on the other, the pressure, in the act of transference, must inevitably contract, in some small degree, the opening of the compasses; and if the dimension has to be set off several times, the probability is, that no two transfers will be exactly the same. And, whilst it is all-important to keep the dimension exact, it is also desirable to manipulate in such a way, when setting off the same dimension a number of times, that the point of position be never lost. Persons unaccustomed to the use of compasses, are very apt to turn them over and over in the same direction, when laying down a number of equal measures, and this necessitates a frequent change of the finger and thumb, which direct the movement of the instrument: the con¬ sequence is, either that the fixed leg is driven deep into the drawing, or it loses position. Now, if the movement be alternately above and below the line on which the dis¬ tances are being set off, the compasses can be worked with great freedom and delicacy, and without any liability to shifting. If a straight line is drawn, and semicircles be described alternately above and below the line, it will show the path of the traversing foot. If the two move¬ ments are tried, the superiority of the one recommended will at once be discovered. The forefinger rests gently on the head; and the thumb and second finger, without changing from side to side, direct the movement for setting off any number of times that may be required. Before applying the dividers to the paper, they should be opened wider than the required distance: the point of the near leg is then to be put gently down on the paper, the leg resting against the thumb, and the other leg gently brought to the required distance. The pressure is thus resisted by the thumb, and there is no risk of making a hole in the paper. This remark applies to the use of com¬ passes of all kinds. There is a third sort of dividers, named the Spring Compasses, in which steadiness is combined with the delicacy of adjustment of the hair compasses. The last- named are liable to error, in consequence of the weakness of the spring leg; and without very careful handling, the dimension, though taken with extreme exactness, cannot be laid down correctly. Now, the spring compasses, of which we annex a figure (Fig. 211), have, from their principle of construction, a steadiness and firmness which cannot be surpassed. The legs are fixed to a steel-spring D, whose elasticity keeps the points extended: the screw A B is fastened by a pivot-joint, and passes through a slot at B, and the opening of the instru¬ ment is adjusted by a nut working upon the fine thread of the screw. The legs are jointed below the screw ; and the required dimension can therefore be taken between the points nearly , and afterwards more ac¬ curately determined by a gentle turn of the nut. The instrument is worked by the forefinger and thumb on the head; and, in setting off, the alternate motion before- mentioned is to be observed. The figure gives the exact size of an instrument suit¬ able for small dimensions; but the draughts¬ man ought to provide himself with a variety of sizes, which will take in all the dimen¬ sions he may ordinarily require. And the advantage of having several of these instruments is, that dimensions which occur frequently in a drawing, can be left in one or more of them undisturbed; and thus much PRACTICAL CARPENTRY AND JOINERY. of ilie time saved tliat would otherwise be occupied in re-adjustment. When purchasing spring compasses, the young draughtsman must select only those in which the screw works on a pivot, since, if it be fixed immoveably at A, it cannot adapt itself to the various extensions of the legs, and the fine thread is then much injured by the unequal pressure of the nut. Compasses with Moveable Legs .—Every case of instru¬ ments is provided with a pair of compasses, of which one leg is moveable, and may be substituted by others carry¬ ing a pen or pencil. This instrument serves, in the first instance, as a divider ; and the additional legs enable the draughtsman to describe arcs and circles temporarily in pencil, or permanently in ink. As it is an object to effect the change of leg with little loss of time, some attention must be paid, when selecting the drawing-case, to the contrivance for removing and securing the legs with despatch. The worst construction is that wherein the leg is secured by a screw, since it involves a tedious process of fixing and unfixing; and the best is, perhaps, the bayonet mode of inserting the leg, which is effected in an instant, and makes a firm junction. In working with the pencil and pen legs, it is desirable to keep them vertical to the drawing; and indeed, with the last, it is absolutely neces¬ sary, as otherwise the arc or circle would be described with the side of the pen, and either it would not mark at all, or would produce a ragged, unsightly line. These legs are therefore jointed, so that, in proportion as the com¬ passes are extended, they may be bent inward, and brought to a vertical position. But this adaptation unfits the instrument to describe arcs and circles of very small radii; for the moveable leg has usually a little additional length to compensate for the bending of the joint, and this pre¬ vents a steady adjustment when the points of the com¬ passes are brought near together. In return for this restriction, however, we have a contrivance for describing arcs and circles of larger radii than fall within the usual range of the instrument. It is found, on trial, that if we attempt to describe an arc of more than a certain radius with the pen-leg, we require to throw the other leg into a very oblique position, with the almost certainty of losing its place, and making a false permanent line. To meet this difficulty, a brass lengthening bar is provided, which receives the pen-leg in the one end, and joins to the compasses, by a bayonet-fixing, at the other. When thus lengthened, the instrument will command a radius of six or eight inches with ease and security. The pencil-leg consists of a tube split through half its length, with a ring to move up and down, by which a 6inall short pencil is fixed much on the same principle as the chalk in a portcrayon. The pen-leg is formed of two blades of steel, terminating in thin, rounded, and well- adjusted points. A spring is inserted between the blades, to separate them; and they are brought together by a screw which passes through them, and which is capable of adjusting the pen for a strong line, or for one as fine as a hair. In using this leg, the screw is slackened, and ink inserted between the blades with a quill-pen, or a camel- hair pencil, according to the nature of the colouring fluid used; and the blades are then brought gradually together, i until they will produce a line of the desired quality. The draughtsman will, of course, try the line on his waste-paper before he ventures to describe it on his drawing. A third moveable leg, named the Dotting Pen, is sometimes in¬ cluded in the drawing-case, and though it is an instru¬ ment rather uncertain in its performance, some draughts- Fi g . s* 12. men manage to employ it with very good effect v ,yi in the drawing of dotted lines It is jointed in the same manner as the pen and pencil legs, and consists of two blades terminating in a small revolving wheel E (Fig. 212), which is retained in its position by the screw D. The one wheel might, of course, be permanently fixed, but usu¬ ally there are several given with the pen, to produce dots of greater or less strength; the con¬ trivance of the screw, therefore, admits the ready substitution of one wheel for another. When this pen is used, ink must be inserted between the blades over the wheel; and the latter should be run several times over the waste-paper, until, by its revolution, it takes the ink freely, and leaves a regu¬ larly dotted line in its course. It must be admitted that, with every care, it sometimes fails in its duty, and leaves blank spaces; but where much straight or curved dotted line is required, it will very much abridge the draughts¬ man’s labour; and, if it performs well, will dot with far greater regularity than the steadiest hand. The compasses with moveable legs have frequently to describe an entire circle, and an inexperienced hand finds some difficulty in carrying the traversing leg neatly round the circumference without the other leg losing position. Some persons have recommended the movement of the dividers, that is, to form half the circle in one direction, and half in a reverse direction; and this may answer very well with the pencil-leg, but not with the pen leg; since it is almost impossible, in the latter case, to unite the two semicircles without leaving marks of junction, which very much injure the continuity of the line that forms the circle. This being the case, it is preferable to adopt a method that shall answer equally well with either leg, and which, by one continued sweep, shall complete the figure. It is very desirable to use compasses for circles that have a due rela¬ tion to the radii of the circles to be described; that is to say, such as will allow both the. revolving and fixed leg to be nearly vertical to the paper; for if the fixed leg is in¬ clined obliquely, it is very apt to lose position, or to work a large unsightly hole in the drawing. When the com¬ passes are so adjusted that both legs are vertical, or very nearly so, it is at once a simple and elegant movement that carries the traversing point round the circle. Let the fore¬ finger rest on the head, and the thumb and second finger on the sides; commencing the sweep at the top, and to wards the right hand: the second finger becomes dis engaged when a quadrant is described, and the forefingei then winds the head along the inner part of the thumb, until the point has performed the entire circuit. It is not always desirable to commence the circle at the top, but more frequently from a point which it is to touch accu¬ rately: this, however, presents no difficulty that the method of sweeping does not meet. The only thing ne¬ cessary is to place the fingers and instrument in position in the first instance, with reference to the starting-point and this is readily done by a slight bending of the wrist To one familiar with the use of instruments, these in¬ structions for manipulation may appear unnecessarily minute; but if he will place his compasses for the first CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. ‘33 time in the hands of a youth, and observe his lack of in¬ tuitive dexterity, he will admit that they are in no degree too minute for a tyro. Bovj Compasses .—In every case of instruments, making any pretensions to completeness, there is one pair of Bow Compasses; but we shall advert to several kinds, each recommended by a peculiar excellence or adaptation to the draughtsman’s purposes. This instrument may be described, generally, as small compasses suited for de¬ scribing arcs and circles of short radii, and which can be worked with great facility by the finger and thumb. The most ordinary construction, and that usually found in the drawing-case, has the legs, one of which is a pen, moving freely on a joint, and terminating at the top in a small handle. The pen-blades are a little longer than the other leg, in order that the latter may keep its vertical position throughout a sweep, and not lose its centre. The performance of this small instrument is very satisfac¬ tory ; a succession of small arcs and circles may be de¬ scribed rapidly and delicately, without leaving the centres strongly marked by the fixed point; and this contributes much to the beauty of a drawing, since nothing is more offensive than to see the paper studded with small holes exposing every insertion of the compasses. The annexed engraving (Fig. 213) shows an improvement of the instrument. The vertical position of the pen-leg is secured by the joint C; the blades B are closed by the screw f, which, according as it is tightened or relaxed, renders the line finer or stronger at pleasure; and the box-screw A unites the legs and handle firmly. The leg A d has a socket at its extremity, to admit a steel needle, which is fastened by a clamp e. This last contrivance is simple but valuable. The fine point of the compasses is soon destroyed by continued use, and to renew it by grinding reduces the length of the leg, and in course of time renders the instrument worthless; whereas, a fresh needle can be introduced into the socket as often as is necessary, and a constant delicacy of point maintained. Bow compasses can also be had carrying a pencil-leg. They differ from those previously described only in having a holder for the reception of a thin short pencil, which is held tight by the screw f (Fig. £14). These have also the joint c, the needle-point E, and box-screw joint A. Bows to carry a pencil are seldom included in the drawing- case ; but they, and indeed all, of the other instruments, can be purchased separately. The pencil-leg is certainly a very useful aid to the draughtsman, since there are many occasions where it is desirable to get all the parts of a drawing inserted with the lead, before making them per¬ manent ; and arcs and circles of small radii are not readily described with the larger compasses, supposing them to be properly adapted for sweeping curves of greater magnitude. Another sort of Bows, named Spring Bow-Compasses (Fig. 215), though limited in their application to small curves and circles, are very delicate and exact instru¬ ments, so far as their range extends. They are in principle identical with the Spring Dividers, which we have already described, and one leg is provided with a holder for a pencil or pen. The advantages of this construction can be appreciated only by those who know the difficulty of securing a small radius, with perfect exactness, by compasses that are extended and closed in the ordinary manner; and who have experienced the mortification of seeing an otherwise fine drawing marred and disfigured by small curves or circles, described with a radius de¬ viating from truth in an error of perhaps not more than a hair-breadth, yet failing in one instance to reach the point of junction, and in another passing beyond it. Directors, or Triangular Compasses. —This instrument is used for taking three angular points at once, or for laying down correctly a third point with relation to other two- One form of construction is that of an ordin¬ ary pair of compasses, with an additional leg attached by a universal joint; and another contrivance, much recom¬ mended for simplicity and facility in its use, is a solid plate of three arms, each arm carrying a moveable limb, into which a short pointed needle is inserted at right angles. In using the first, the compasses are opened, and two points taken, and the additional leg is extended in any direction to take up the third point; the manage¬ ment of the second is equally easy, the needle-points are successively adjusted to the angles by the flexure of the moveable limbs. With either instrument, the draughtsman is saved the tedious process of constructing triangles, and determining the relative position of neigh¬ bouring points in his drawing. Proportional Compasses .—These are used for the enlargement or reduction of drawings. The simplest form is that named wholes and halves, which may be described as two bars pointed at each extremity, and working transversely on a box-screw joint, and form¬ ing, as it were, two compasses, the legs of the one being twice the length of those of the other. If any distance be taken between the points of the longer legs, half that distance will be contained at the other end. The application of the instrument to the reducing or enlarging any drawing one-half, is suffi¬ ciently obvious. The proportional compasses, properly so called, is a more complicated contrivance, and admits of more varied application. Its form and general construction are seen in the annexed engraving (Fig. 216). It is in principle the same as the wholes-and-halves, with this difference, that the screw-joint c passes through slides E Jig. 216 . T> E A B 34 PRACTICAL CARPENTRY AND JOINERY. moving in the slots of the bars, and admits of the centre being adjusted for various relative proportions between the openings A B and d e. Different sets of numbers are engraved on the outer faces of the bars, and by these the required proportions are obtained. The instrument must be closed for adjustment, and the nut c loosened; the slide is then moved in the groove, until a mark across it, named the index, coincides with the number required; which done, the nut is tightened again. The scales usually engraved on these compasses are named Lines, Circles, Planes, and Solids. The scale of lines is numbered from 1 to 10, and the index of the slide being brought to any one of these divi¬ sions, the distance D e will measure A B in that propor¬ tion. Thus, if the index be set to 4, d e will be contained four times in A B. The line of circles extends from 1 to 20; and if the index be set to 10 , d e will be the tenth part of the circumference of the circle, whose radius is A B. The line of planes, or squares, determines the proportion of similar areas. Thus, if the index is placed at 3, and the side of any one square be taken by A b from a scale of equal parts, D e will be the side of another square of one-third the area. And if any number be brought to the index, and the same number be taken by A B from a scale of equal parts, d e will be the square root of that number. And in this latter case, D e will also be a mean propor¬ tional between any two numbers, whose product is equal to A B. The line of solids expresses the proportion between cubes and spheres. Thus, if the index be set at 2, and the diameter of a sphere, or the.side of a cube, be taken from a scale of equal parts by a b, then will d e be a diameter of a sphere or side of a cube of half the solidity. And if the slide be set to S, and the same number be taken from a scale of equal parts, then will d e measure 2 on the same scale, or the cube root of 8. The scale of lines and that of circles are those of most value to the draughtsman. The first enables him to reduce oi- enlarge in any required proportion ; and the second gives him the side of the square or polygon, that can be inscribed in a given circle. The instrument needs to be used carefully, since its accuracy depends on the preser¬ vation of the points. If both or either of these are broken, or diminished in length, the proportions cease to be true. In place of using the proportional compasses in setting off a number of times, which would soon wear the points, rather take the distance in the Dividers. Beam Compasses .—The draughtsman has frequently to measure and lay down distances, and to sweep with radii, which the ordinary instruments cannot reach. In these cases, and when extreme accuracy is necessary, he resorts to the Beam Compasses, which are usually made of well- seasoned mahogany, with a slip of holly or box-wood on the face, to carry the scale. Two brass boxes with points are fitted to the beam, one of which moves freely to take in any required distance, and the other is connected with a slow-motion screw working in the end of the beam, and can thus be adjusted with extreme delicacy to any measure or radius. Reading-plates, on the Vernier principle, sub¬ divide the divisions of the scale on the beam, and by'them any measure to three places of figures is taken with extreme truth. Referring to Fig. 217, we proceed to describe it more particularly, c c is the mahogany beam, whose length may be taken at pleasure, although it is not advisable to extend it beyond four or five feet, lest it bend by its own weight: a a is the strip of holly or box-wood on which the scale is engraved: b is the brass box, which moves freely along the beam, and is secured in position by the clamp-screw F : A is the other brass box, made fast to the slow-motion screw d, which works in the end of the beam, and winds it into or out of the box a, to obtain perfect adjustment; and d b are the Vernier scales, or reading- plates. The mahogany beam is sometimes substituted by a brass tube. Before describing the method of setting the instrument, we must explain, in few words, the nature of a Vernier scale. Take any primary division of a scale, and divide it into ten parts, then take eleven such parts and divide the line which they form into tenths likewise; this last then becomes a Vernier or reading-scale. The primary division is 100, its subdivision 10, and the excess of the Vernier division 1; so that if the scale and Vernier are placed parallel and close to each other, a distance or mea¬ sure may be read accurately to the unit of three places of figures. We illustrate by a diagram (Fig. 218), which shows the Vernier attached to the scale a 6 of the ordinary ba¬ ng. 218 . rometer. Here a b is divided into inches and tenths of inches; and c d is the Vernier, con¬ sisting of eleven subdivisions of a b, divided into tenths. Now the zero, or commencement of notation, on the Vernier is, in this case, ad¬ justed to 30 inches on the scale; and its division 10 coincides with 28 inches 9 tenths; hence every division of the Vernier is seen to be one and one-tenth of the scale divisions. To read ofi, therefore, the hundredths of an inch that the zero of the Vernier may be in advance of a tenth, observe what divi¬ sion of the Vernier coincides most nearly with any division of the scale, and that will indicate the hundredths. Thus, taking the adjustment of the figure, the zero corresponds exactly with 30 on the scale, and its division 10, with 28 and 9 tenths ; and we therefore read 30 inches. But if the zero were so posited between 29 and 9 tenths and 30, that the 8 of the Vernier should correspond exactly with a tenth of the scale, we should read 29 inches, 9 tenths, and 8 hundredths. And this is evident, for it zero be 8 hundredths in excess of a tenth, it is only the eighth division of the Vernier that will be found to coin¬ cide exactly with a. tenth of the scale. To adjust the beam compasses for a distance or radius of 13 inches, 5 tenths, 3 hundredths, the box A is to be moved -by the screw D until the zero of the Vernier corresponds with the zero of the beam, and is then to be secured in position by the clamp e : this done, the box b is slid along the beam until the zero of its Vernier coin¬ cides with 13 inches 5 tenths: lastly, the box B is moved CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. by the slow-motion screw, and the third division of the Vernier brought to correspond with the third tenth of the scale, which consequently adds 3 hundredths to the dis¬ tance or radius previously taken. The point of the slide or box f can be removed, and a pen or pencil substituted with accurate adjustment. The beam compasses are sel¬ dom employed, except when extreme accuracy is neces¬ sary. On many occasions, curves of long radius are drawn by means of slips of wood, one edge of which is cut to the required circle. Having described the various sorts of compasses in ordinary use, it is unnecessary to do more than advert briefly to some modifications and improvements in form and detail. It has been thought an advantage to joint both legs of the instrument, in order to bring them to a vertical position at any extension ; but this is a doubtful advantage, especially in compasses designed for drawing arcs and circles, since each leg being equally removed from the centre of motion, there must be a tendency on the part of the one fixed, to tear away from its position. Mr. Brunei has introduced what are called Tubular Com¬ passes, in which the upper part of the legs lengthens out like the slide of a telescope, thus giving greater extent of radius when required. The moveable legs are double, having points at one end, and a pencil or pen at the other; and they move on pivots, so that the pen or pencil can be instantly substituted for the points, or vice versa, and that with the certainty of a perfect adjustment. The design is veiy ingenious, and offers many conveniences, but the instrument is too delicate for ordinary hands. Without extreme care, it must soon be disarranged and rendered useless. The Portable or Turn-in Compasses, is a con¬ trivance which combines dividers, compasses with move- able legs, and bows, in a pocket instrument, folding up to a length of not more than three inches. The upper legs are hollow, and admit either leg of the pen and pencil bows, which can therefore be substituted for each other. When closed for the pocket, one leg of each bow slides into the upper legs, and the other is turned inward towards the head. As a concluding remark, we recommend the draughts¬ man to choose compasses in which the joint is formed by a box-sciew, that can be tightened or relaxed at pleasure. The cheaper kinds have merely a common screw, and these are usually too stiff when first purchased, and in¬ conveniently loose' after being some time in wear. A slight turn of the box-screw, by means of the key, keeps the com¬ passes in good working order, neither so stiff as to spring, nor so loose as to render them uncertain and unsteady in use. Plain and Double Scales. Simply-divided Scales. —Scales are measures and subdi¬ visions of measures laid down with such accuracy, that any diawing constructed by them, shall be in exact proportion in all its details. The plain scale is a series of measures laid down on the face of one small flat ruler, and is thus distinguished from the sector, or double scales, in which two similarly-divided rulers move on a joint, and open to a greater or less angle. In the construction of scales, the subdivision must be carried to as low a denomination as is likely to be required. Thus, for a drawing of limited divisions tenths, or one foot each (Figs. 210 and 220). In the case of large surveys, the primaries become miles, and the lesser divisions furlongs. Indeed the natural size or extent of the object or area, and the surface to be occupied by the delineation, must determine the gradua¬ tion of the scale. But passing from these general remarks, we proceed to the plain scales contained in the drawing case, and laid down on the two sides of a flat ivory ruler^ six inches in length. On one side of the plain scale there is usually a series of simple scales, in which the inch is variously divided, and the primaries subdivided into tenths and twelfths! These may be applied to measurements as inches and tenths, or twelfths; or as feet and tenths, or inches, according to the nature of the drawing. It may be re¬ marked, however, that these small lines of measures are of only limited use, and that the draughtsman must usually lay down a scale with special reference to the work before him; and in all cases it is desirable to have the scale of construction on the margin of the drawing itself, since the paper contracts or expands with every atmospheric change, and the measurements wili therefore not agree at all times with a detached scale; and, more¬ over, a drawing laid down from such detached scale, of wood or ivory, will not be uniform throughout, for on a damp day the measurements will be too short, and on a dry day too long. Mr. Holtzapffel has sought to remedy this inconvenience by the introduction of paper scales; but all kinds of paper do not contract and expand equally, and the error is therefore only partially corrected by his ingenious substitution of one material for another. Diagonal Scale. —The lines to which we have referred give only two denominations, primaries and tenths, or twelfths; but more minute subdivision is frequently re¬ quired, and this is attained by the diagonal scale, which consists of a number of primary divisions, one of which is divided into tenths, and subdivided into hundredths by diagonal lines (Fig. 221). This scale is constructed in the following manner:—Eleven parallel lines are ruled, inclos¬ ing ten equal spaces: the length is set off into ten equal primary divisions, as a e, b c, c 2 , &c.; and diagonals are Fig .221. then drawn from the subdivisions between A and B, to those between d and E, as shown in the diagram. Hence it is evident that at every parallel we get an additional tenth of the subdivisions, or a hundredth of the primaries, and can therefore obtain a measurement with great exactness to three places of figures. To take a measurement of 168, we place one foot of the dividers on the primary 1 11 ) on ol) PRACTICAL CARPENTRY AND JOINERY. (Fig. 22*2), and carry it down to the eighth parallel, and then extend the other foot to the intersection of the dia¬ gonal, which falls from the subdivision 6, with the parallel that measures the eight-hundredth part. More examples or further explanation would only be tedious. The primaries may of course be considered as yards, feet, or inches; and the subdivisions as tenths and hundredths of these respec¬ tive denominations. The diagonal scale is very useful and satisfactory if accurately constructed; but there can be no Fig. 222. -H 1 "i n+c - r i ' ■ » 1 ■nr ff 1 M ' - A Ll rrnx : Tit / question, that one with a Vernier applied to the first sub¬ divisions, would give minute measures with much greater certainty; and no case of instruments ought now to leave the maker without having this addition on one face of the plain scale. The diagonals may be safely applied to a scale where only one subdivision is required. Thus, if seven lines be ruled, inclosing six equal spaces, and the length be divided into primaries, as A B, B c, &c., the first primary A B may be subdivided into twelfths, by two diagonals running from 6, the middle of A B, to 12 and 0. We have here a A 6' TJ C. Fi S- 223 - - 7F3 —ff -\5 wf - AT 7 7 1 yj o 2 > very convenient scale of feet and inches. From c to G, is 1 foot 6 inches; and from c, on the several parallels, to the various intersections of the diagonals, we obtain 1 foot and any number of inches from 1 to 12. All of which is evident from the figure. On the face of the plain scale that carries the diagonal one, there is usually a line of inches. and tenths, and underneath it a decimal scale. These can be used sepa¬ rately, and in conjunction; and in the latter case the primaries of the decimal scale being taken as feet, the subdivisions of the upper line are inches. Line of Chords .—This is usually introduced on the plain scale. It is an unequally divided scale, giving the length of the chord of an arc, from 1 degree to 90 degrees. The quadrant, or quarter of a circle, A c, contained be¬ tween the two radii at right an- on sc Fig . 254 . gles, B A and B c, has its extremi¬ ties joined by the line A c, to which the measures of the chords are to be transferred. The quadrant is divided accurately into nine equal parts; then from c as a centre, each division is transferred by an arc to the line AC, and the chords of every 10 degrees obtained. These primary divisions can be subdivided^into tenths,of 1 degree each, by division of the corresponding arcs. This is rather an illustration of the construction, than a true method of performing it. A line of chords can be laid down accurately only from the tabular sines, delicately set off by the beam compasses. In using this scale, it is to be remembered that the chord of 60 degrees is equal to radius. Therefore, to lay down an angle of any number of degrees, draw an indefinite straight line; take in the compasses the chord of 60 degrees, and from one termina¬ tion of the line, as a centre, describe an arc of sufficient extent; then take from the scale the chord of the required angle, and set it off on the arc; lastly, draw another line from the centre cutting the arc in the measure of the chord. To ascertain the degrees of an angle, extend the angular lines if necessary, that they may be at least equal to the chord of 60 degrees; with this chord in the com¬ passes describe an arc from the angular point; then take the extent of the arc and apply it to the scale, which will show the number of degrees contained in the angle. The Plain Protractor .—The plain scale is sometimes made of greater width, in order to contain all the preced¬ ing lines, and also a protractor for setting off and measur¬ ing angles. The most eligible form for this instrument is the circle or half circle, which construction will presently come before us. It will suffice for the present to say, that the plain scale protractor is a portion of a semicircle, hav¬ ing radii drawn from its centre to every degree of its cir¬ cumference. If, therefore, the centre on the lower side is made to coincide with a given point, an angle of any number of degrees may be measured or set off around its edges. A small roller is sometimes inserted in a slot to make the plain scale serve the purpose of a parallel ruler, but considerable care is necessary in thus applying it, lest the roller slide or shift at either extremity. Double Scales. Each of the scales we have described has a fixed mea¬ sure that cannot be varied; but we come now to speak of those double scales in which we can assume a measure at convenience, and subdivide lines of any length, measure chords and angles to any radius, &c. The Sector .—This instrument consists of two flat rulers, united by a central joint, and opening like a pair of com¬ passes. It carries several plain scales on its faces, but its most important lines are in pairs, running accurately to the central joint, and making various angles according to the opening of the sector. The principle on which the double scales are constructed, is contained in the 4th Prop, of the 6th Book of Euclid, which demonstrates that “the sides about the equal angles of equiangular triangles are proportionals, ’ ’ &c. Now let A c i (Fig. 225) be a sector, or, in other words, an arc of a circle contained between two radii; and let C A, c I, be a pair of sectoral lines, or a double scale. Draw the chord A I, and also the lines e h, d G, e f, parallel to A i. Then shall CE, CD, CB, CA, be proportional to E F, D G, B II, and a i respectively. That is, as c A : A I : : c B : B H, &c. Hence at every opening of the sector, the transverse Fig. 225. CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. 37 distances from one ruler to another, are proportional to the lateral distances, measured on the lines CA, ci; and thus we may apply any radius transversely to the line of chords to measure or lay down any given or required angle; and apply any line transversely to the line of lines, to divide it in any required proportions. The sector is therefore seen to be of universal application, whilst the use of plain scales is limited and special. Plain Scales on the Sector— On the outer edge of the sector is usually given a decimal scale from 1 to 100; and in connection with it, on one of the sides, a scale of inches and tenths. These are identical with the lines on the plain scale, previously mentioned, but the latter are more commodiously placed for use. On the other side we have logarithmic lines of numbers, sines, and tangents; but as these are more complicated than the ordinary plain scales, we defer the consideration of them until we have discussed the double scales. Sectoral Double Scales. —These are respectively named the lines of lines, chords, secants, sines, and tangents. These scales have one line on each ruler, and the two lines converge accurately in the central joint of the sector. The Line of Lines. —This is a line of 10 primaries, each subdivided into tenths, thus making 100 divisions. Its use is to divide a given line into any number of equal parts; to give accurate scale measures for the con¬ struction of a drawing; to form any required scale; to divide a given line in any assigned proportion; and to find third, fourth, and middle proportionals to given right lines. The scale can be applied to other purposes; but, if we take up those mentioned, they will be sufficient illustrations of its uses. Before en¬ tering upon these propositions, we would remark thata^aiemidistance is one taken from the centre down either half of the scale; and a transverse distance is one measured across from scale to scale. Thus (Fig. 226), a 1, a 2, a.3, &c., are lateral distances; and 1.1, 2.2, 3.3, &c., transverse distances. 1. To divide a given line into 8 equal parts. Take the line in the compasses, and open the sector so as to apply it transversely to 8 and S, then the transverse from 1 to 1 will be the eighth part of the line. If the line is to be divided into 5 equal parts, apply it transversely by the compasses to 10 and 10, and the transverse of 2 and 2 is the fifth part. When the line is too long to fall within the opening of the sector, take the half or the third of it. Thus, if a line of too great length is to be divided into 10 parts, take the half and divide into 5 parts; or if into 9 parts, take the third and divide into 3 parts. And in other cases it may be necessary to divide the portion of the line into the original number of parts, and set off twice or thrice to obtain the required division of the whole. 2. To use the line of lines as a scale of equal measures. Open the sector to a right angle, or nearly so, and obtain dimensions by transverse measures from scale to scale, taking care that the points of the compasses are directed to the same division on both rulers. Thus, the transverse measures to the primaries 1.1, 2.2, &c., will give any de¬ nomination, as feet or inches, and similar measures to the same subdivisions on both sides will give tenths. 3. To form any required scale—say, one in which 285 yards shall be expressed by 18 inches. Now, as IS inches cannot be made a transverse, take in the compasses 6 inches, the third part, and make it a transverse to the lateral distance 95, which is the third of 285. The re¬ quired scale is then made; the transverse measures to the primaries being 10 yards, and to the subdivisions so many additional yards. 4. To divide a given line in any assigned proportion— say, a line of 5 inches in the proportion of 2 to 6. Take 5 inches in the compasses, and apply it to the transverse ot 8.8, the sum of the proportions; then will the trans¬ verse distances 2.2, 6.6, divide the given line as required. 5. To find a third proportional to the numbers 9 and 3, or to lines 9 inches and 3 inches in length. Make 3 inches a transverse distance to 9.9; then take the trans¬ verse of 3.3, and this measured laterally on the scale of inches will give 1 inch. For 9 : 3 : : 3 : 1. 6. To find a fourth proportional to the numbers 10, 7, 3, or to lines measuring 10, 7, and 3 inches respectively. Make 7 inches a transverse from 10 to 10, then the trans¬ verse 3.3 will measure on the scale of inches 2^. For 10:7::3:2 T V 7. To find a middle proportional between the numbers 4 and 9, or between 2 lines measuring 4 and 9 inches respectively. To perform this operation the line of lines on the one leg of the sector must first be set exactly at right angles to the one on the other leg. This is done by taking 5 of the primary divisions in the compasses, and making this extent a transverse from 4 on one side to 3 on the other. For 3, 4, and 5, or any of their multiples, form a right-angled triangle. The sector being thus ad¬ justed, take in the compasses a lateral distance of 6 pri¬ maries and 5 tenths, half the sum of the two lines or num¬ bers, and apply this measure transversely from 2 primaries and 5 tenths, half the difference, when the other point of the compasses will reach the primary 6 on the opposite leg of the sector. For 4 : 6 :: 6 : 9. The line of lines is marked L on each leg of the sec tor; and it is to be observed that all measures are to be taken from the inner lines, since these only run accu¬ rately to the centre. This remark will apply to all the double sectoral lines. With reference to some of the pre¬ ceding operations by the line of lines, we may admit that they are suggestive rather than practically useful. They familiarize the young draughtsman with the capabilities of scales, and offer him useful hints for the general con¬ struction and management of lineal measures. The Line of Chords. —The scale of chords on the sector has the same advantage over that on the plain scale that the line of lines has over the simply-divided single scales. With the line of lines we operate on any given line that will come within the opening of the sector; and with the line of chords we can work with any radius of similar ex¬ tent. This last is constructed by making the lateral dis¬ tance of the chord of 60 degrees, which is radius, equal in length to the line of lines. All the intermediate degrees between 1 and 60, are then set off laterally from the centre, on both rulers, by taking on the line of lines a measure equal to twice the natural sine of half the angle. Thus, for 40 PRACTICAL CARPENTRY AND JOINERY. right angles to B c. A B is a line of secants, formed by transfer with the compasses of the radial lines from the centre b. It is therefore seen that the radius, tangent, and secant, are the base, perpendicular, and hypo- thenuse of a right-angled triangle. A line of sines would be formed by graduating the radius 10 b, with 7(J lines drawn through the degrees of the quadrant, and parallel to b c. The lines of sines and taugents are frequently of use to the draughtsman ^ in the determination of a number of points through which an eccentric curve can be drawn. We here give two examples of the use of the sec- 40 toral lines in the solution of ques- so tions in Trigonometry. 1. A right-angled triangle has base 12, perpendicular 16; required the hypothenuse.—Set the sector at right angles, by making the lateral distance of 5 on the line of lines a transverse to 3 and 4. Then take the transverse of 12 on one leg to 16 on the other, and this, measured on the line of lines, will give 20 for the hypothenuse. 2. A right-angled triangle has perpendicular 30, and the angle opposite thereto 37 degrees; required the hypo¬ thenuse.—Take 30 from the line of lines, and make it a transverse to 37 degrees on the line of sines, then the transverse of 90.90, will measure 50 on the line of lines, the length of the hypothenuse. Logarithmic Lines. The Line of Numbers. — This line, commonly called Gunter’s Line, and marked N on the sector, is divided into spaces forming a geometrical series, and is simply a table of logarithms expressed b} r relative measures of length. It is constructed in the following manner:— The entire line is divided into two equal parts, and each of these parts into nine unequal primary divisions, cor¬ responding to the logarithms they are to represent. These primaries are to be the measures of the numbers 2 to 9, whose. logarithms may be considered 30, 47, 60, 70, 78, 84, 90, and 95, as will be seen on reference to the ordinary tables. To make a scale of equal parts for set¬ ting off these quantities, take one-half of the line in the compasses and make it a transverse to 10.10 on the line of lines. Then take successively the transverse distances of 30.30, 47.47, &c., and set them off from the commence¬ ment on the first half of the line of numbers for the pri¬ mary divisions 2, 3, &e. These same spaces may next be transferred to the second half of the line for its primary divisions. Thus we have obtained the logarithms of 20, 30, 40, 50, 60, 70, 80, 90,100 on the first half; and those of 100, 200, &c., on the second half. Now, for the subdivision of the space between 1 and 2, we must set off from the commencement of the line, in succession, the logarithms of 11, 12, &c., to 19; and for that between 2 and 3, the logs, of 21, 22, &c., to 29 ; and thus proceed till we come to the space between 6 and 7, which is too short to admit the decimal divisions. Graduate therefore this last space, and all onward to 10, into two-tenths; and consequently take the logs, of 62, 64, 66, 6S; 72, 74, 76, 78; &c. .All these subdivisions, set off from the commencement on the first half of the line, may also be set off from 1 in the middle, in the second half. Thus, we have found on the one half, the logs, of tens and units, and on the second half those of hundreds and tens. But there is a farther sub¬ division of the space between 1 and 2 on the second half, which is graduated to twenty places; and this halving of the first subdivisions is effected by setting off successively from 1, the logs, of 105, 115, 125, &c. This done, the line i$ constructed. In using this line any value may be attached to the primary divisions, merely observing their relative propor¬ tion to each other. Thus, if the primaries on the first half are units, and their subdivisions tenths, those on the second half will be tens and their subdivisions units. Whatever is the value of a primary or subdivision on the first half, the corresponding primary and subdivision on the second half will have ten times that value. We illustrate the use of the line by a few examples. Take oft the measures of the numbers 896, 1150, 2050. For the first, place one foot of the compasses at the beginning of the line on the left hand, and extend the other over eight primaries and four subdivisions, and nearly to the end of the fifth subdivision. For the second, extend over the ten primaries of the first half and three subdivisions of the second half. For the third, extend over the first ten primaries, and one primary of the second half, and half a subdivision beyond. To multiply, say 135 by 48: take the extent from 1 on the left hand to 48 in the first interval, and apply it to 135 in the second interval, when it will reach to 648, or 6480. To divide 6480 by 135 : extend backwards from 135 to 1 on the left hand, and this will measure back from 6480 to 48. To find a fourth proportional to the numbers 3, 8, and 15: take the extent from 3 to 8 in the first interval, and this will reach from 15 to 40 in the second interval; for 3 : 8:: 15:40. The Line of Sines .—This line gives the sines of an¬ gles to 90 degrees in a geometric series; their logarithms being expressed by relative spaces. It is constructed by laying down the logarithms of the sines from the same scale of equal parts by which the line of numbers was measured. Its two intervals are not, however, of equal length; and hence we cannot set oft' the primaries in both from the same measure. We therefore require a scale of equal parts of twice the length, or one the whole length of the line of numbers, to enable us to set off all the sines from the commencement of the scale on the left hand. The simplest way is to make the length of the line of numbers, a transverse to 10.10 on the sectoral line of lines, and take the transverse measures of half the loga¬ rithms. Now the logs, of the primaries, 1, 2, 3, &c., to 10, in the first interval, and of 20, 30, &c., to 90, in the second interval, are these: 242, 543, 719, 843, 940, 1019, 1085, 1143, 1194, 1239; 1534, 1699, 1808, 1SS4, 1937, 1972, 1993, 2000. Take, therefore, the halves of these logs, tranversely from the line of lines, and lay them down suc¬ cessively on the line of sines, from the beginning of the scale. The subdivision of the primaries into minutes and degrees is proceeded with in the same manner. The degrees iu the first interval are divided into six spaces, each being 10 minutes; but between 10 and 20, in the second interval, there are 20 subdivisions, each represent- CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. 41 ing 50 minutes, or half a degree: between 20 and 30, and 30 and 40, there are 10 subdivisions, each being 1 degree: between 40 and 50, 50 and 60, 60 and 70, there are 5 graduations, each 2 degrees; and the space between 70 and 90 admits only of one subdivision to divide it for 80 and 90 degrees. The commencement of the scale is inter¬ fered with by the sectoral line of sines, so that the measure of 50 minutes is the least sine that can be laid down on the logarithmic line. With this sine, whose log. is 162, we commence the subdivisions, and then proceed to the sines of 1° 10', 1° 20', &c.; 2° 10', 2° 20', &c., until the graduation is completed. There is another method of construction, by which the sines are measured off from the termination of the line on the right hand. For this purpose the logs, of the arith¬ metical complements of the sines are taken, that is to say, the difference between them and radius. Thus, the log. of the sine of 30 degrees, taking all the places of figures, is 9-6989700, and this deducted from radius,or 10 0000000, leaves a remainder of -3010300. If, therefore, 301, or its half by the proposed scale, be laid from the end of the line of sines at the right hand, it will reach the graduation of 30 degrees. There is yet another method: in place of the arithmetical complement of'the sine, take the secant of the complementary angle, viz., 60 degrees, and set oft’from the right hand as in the former case. We mention these vari¬ ous modes of construction to call the young draughtsman's attention to the relation between different angles, and as suggestions for more scientific inquiry concerning them. The manner of taking off a logarithmic sine from the scale is obvious: one foot of the compasses is placed at the commencement, and the other extended to the required degree or minutes. The use of the line in conjunction with the line of numbers may be illustrated by one ex¬ ample. The base of a right-angled triangle is 30, and the angle opposite to it 30 degrees; what is the hypothenuse? Now, Sine of Angle : 30 : : Radius : Hypothenuse. Set one foot of the compasses on 30 degrees, and extend the other to 30 on the line of numbers; and with this opening, set one foot on 90 degrees of the line of sines, and the other foot will reach to 60 on the line of numbers — the hypothenuse required. The Line of Tangents. —This line gives the logarithmic measures of the tangents to 45 degrees, and thence back¬ wards to 88° 30'. The tangent of 45 degrees being equal to radius, or the line of numbers, the graduation cannot be extended beyond this angle; but the upper tangents are obtained by reckoning backwards, 40 for 50, 30 for 60, 20 for 70, &c.; and this method of obtaining the longer tangents is compensated by a peculiarity of operation when the line is wrought in conjunction with the line of numbers. This scale is constructed by measuring off, successively, from the commencement at the left hand, the logarithms of the primaries and subdivisions as required. Thus, the first interval has for its primaries the degrees from 1 to 10; and, in the second, every primary is 10 degrees, except the last, which is only 5. The subdivisions in the first interval are ten minutes; and in the second, between 10 and 20 and 20 and 30, they are 30 minutes, or half a degree: and from 30 to 45, 1 degree each. Make a scale of equal parts, as for the sines, by applying the length of the line of numbers to 10 10 on the line of lines; then take the transverses of half the logarithmic tangents found in the tables. Thus the logs, of 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 20, 30, 40, 45 degrees, are 242, 543, 719, 844, 941, 1021, 1089, 1148, 1199, 1246; 1561, 1761, 1924, 2000. Take therefore the halves of these numbers from the scale, and transfer them to the line for the primaries. The subdivisions are commenced at 1° 30', whose log. is 418; and then continued 1° 40’, 1° 50', 2° 10', &c., until completed. As in the case of the sines, the line of tan¬ gents may likewise be constructed by laying down the arithmetical complements of the tangents backwards, from 45 degrees to the commencement of the scale. The length of a logarithmic tangent is measured off from the commencement of the line at the left hand, by extending the compasses to the degree or minute required. We give two examples of the application of the scale to the solution of questions in trigonometry. 1. The base of a right-angled triangle is 25, and the perpendicular 15; what is the angle opposite to the perpendicular? Here, it the base is considered radius, the perpendicular will be the tangent of the angle opposite to it; therefore, As 25:15 :: Radius : Tangent. Extend the compasses from 15 to 25 on the line of num¬ bers, and this opening will reach backwards from 45 degrees on the line of tangents to 31 degrees, the angle required. 2. The base of a right-angled triangle is 20, and the angle opposite to the perpendicular 50 degrees; what is the perpendicular? As Radius : Tan. 50°: : 20 : Perpendicular sought. Extend the compasses from 45 degrees to 50 on the line of tangents, and apply them, thus opened, from 20 towards the right hand, to 23f, the perpendicular. This example shows the method of working when the angle exceeds 45 degrees. The extent taken from the tangents is only from 45 to 40, the complement of 50 degrees; and we therefore apply it from 20 towards the right hand to obtain the length of the perpendicular; but had the angle been 40 degrees, the extent would have been applied from 20 to¬ wards the left hand, to 16f, which would, in that case, have been the perpendicular. We have now gone systematically through the sector, which contains a great deal of what may be termed me¬ chanical mathematics, and offers much that is valuable to the draughtsman in the way of suggestion for the con¬ struction and management of scales. Protractors. We have already referred to the protractor on the plain scale. The semicircle (Fig. 231), though different in form, ^g. 23i. js the same in prin- A ciple. It is a half circle of brass, or other metal, having a double graduation on its circular edge. The degrees run both ways to 180; so that any angle, from 1 to 90 degrees, may be set off on either side. Each graduation marks an angle and its supplement; thus, 10, 20, 30, coincide with 170, 160, 150; and are the PRACTICAL CARPENTRY AND JOINERY. supplements of each other. An angle is protracted or measured by this instrument with great facility. To protract an angle, draw a line, and lay the straight edge of the protractor upon it, with its centre on the point where the angle is to be formed: the required number of degrees is next marked off close to the circular edge: the instrument is then laid aside, and a line drawn from the angular point, to the one which measures the extent of the angle. Thus in the figure, B is the centre, or angular point, D the measure of the angle, and b d the line by which it is formed. The converse operation of measuring an angle is equally simple: the angular point and the centre of the protractor are made to coincide, and the straight edge of the instrument is laid exactly upon one line of the angle, when the other will intersect the cir¬ cular edge, and indicate the number of degrees. The plain scale protractor is used in the same manner; but it is by no means so convenient an instrument as the semi¬ circle. Either of them may be employed occasionally to raise short perpendiculars. For this purpose, make the centre and the graduation of 90 degrees coincide with the line upon which the perpendicular is to be raised. Parallel Ruler. This is a well-known instrument, consisting of two rulers connected by slides, moving on pivots, and so ad¬ justed, that at every opening of the instrument, the rulers and the slides form a parallelogram. In use, its edge is made to coincide exactly with the line to which others are to be drawn parallel: the lower ruler is then held firmly down, and the upper one raised to any required distance, when a line drawn along its edge will be parallel to that Fig. 232. Z from which it. started (Fig. 232). There are methods of uniting the rulers; but we are not f aware that any one has very decided advan¬ tages over the others. The ordinary form, as shown in the figure, is perhaps the simplest, and, therefore, the best. The right-angled straight-edges of the drawing-board and the T-square, are the surest means of all for drawing parallels and perpendiculars; and the parallel ruler will never be used when these can be employed. Drawing Pens. The drawing pen differs from the pen-leg of the compasses only in its having a long straight handle, the top of which is sometimes made to unscrew and form a tracer or pin, to set off angles by the edge of the protractor (Fig. 233). The dotting pen is a similar modification of the dotting leg of the moveable compasses. The use of both instruments is to draw straight, contin¬ uous, or dotted lines in ink. A place is usually provided in the drawing case for a thin pencil, to rule in straight lines, that may afterwards either be obliterated or made permanent by the ink Pricker. This is a simple instrument, consisting of a fine needle¬ point firmly fixed into the end of a wooden or ivory holder, for pricking off distances, the positions of lines, &c., upon the paper. It is so used in conjunction with portable scales, the edges of which, being graduated, are applied to the sheet, and measure off the required dis¬ tance. The pricker to this extent supersedes the dividers, and may be so employed with facility and accuracy. It is also used in copying drawings, by placing the drawing on the top of the sheet upon which the copy is to be made, and pricking through upon the vacant sheet the positions of the lines, angles, and centres of the drawing; thus the copying process is expedited. The ladies’ crochet needle- holder makes a neat handle for the pricking needle. Drawing Paper. Drawing paper, properly so called, is made to certain standard sizes, as follow:— several ig. 233. pen. Demy,. ... 20 inches by 15.1 inches. Medium, 221 „ 171 „ Royal, . ... 24 „ 10.1 „ Super-Royal, 27.1 19-1 Imperial, ... 30 „ 22 „ Elephant, 28 „ 23 „ Columbier, ... 35 „ 231 „ Atlas, . 34 „ 26 ,, Double Elephant, ... 40 „ 27 „ Antiquarian, 53 „ 31 „ Empei'or, ... 68 48 » Of these, Double Elephant is the most generally useful size of sheet. Demy and Imperial are the other useful sizes. Whatman's white paper is the quality most usu¬ ally employed for finished drawings: it will bear wet¬ ting and stretching without injury, and, when so treated, receives shading and colouring easily and freely. For ordinary sketching or working drawings, where damp¬ stretching is dispensed with, cartridge paper, of a coarser, harder, and tougher quality, is to be preferred. Jt bears the use of india-rubber well, receives ink on the original undamped surface freely, shows a good line, and, as it does not absorb very rapidly, tinting lies evenly upon it. For delicate small-scale line-drawing, the thick blue pa¬ per, such as is made by Harris for ledgers, &c., imperial size, answers exceedingly well; but it does not bear damp stretching without injury, and should be merely pinned or waxed down to the board. With good management, there is no ground to fear the shifting of the paper. Good letter-paper receives light drawing very well: of course it does not bear much fatigue. Large sheets, destined for rough usage and frequent reference, should be mounted on linen, previously damped, with a free application of paste. Tracing paper is a preparation of tissue paper, ren¬ dered transparent and qualified to receive ink lines and tinting without spreading. When placed over a draw¬ ing already executed, the drawing is distinctly visible through the paper, and may be copied or traced directly by the ink-instruments: thus an accurate copy may be made with great expedition. Tracings may be folded and stowed away very conveniently; but, for good service, they should be mounted on cloth, or on paper and cloth, with paste. Tracing paper may be prepared from double-double- CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. 4.3 crown tissue paper by lightly and evenly sponging over one surface with a mixture of one part of raw linseed oil or nut-oil. and five parts of turpentine. Five gills of tur¬ pentine, and one of oil, will go over from H to 2 quires of twenty-four sheets. Tracing cloth is a similar preparation of linen, and has the advantage of toughness and durability. Drawing Boards. Drawing boards are made truly rectangular, and for common use may be of two sizes—41 by 30 inches, to carry double-elephant paper, with a margin; and 31 by 24 inches, for imperial and all smaller sizes. Boards much smaller than this are unsuited for ordinary work, but may be necessary for particular purposes. Drawing boards may be of mahogany, oak, or yellow pine, well seasoned; i inch or inch thick for mahogany, and 1 inch for pine, or say 1£- inch to allow for dressing up. They should be barred and dowelled at the ends, to stiffen them, and en¬ able them to resist any tendency to twist, as well as to af¬ ford a suitable edge for the working of the drawing square. It would be an improvement to line the working end of the pine board with a strip of mahogany or other hard wood, as it is liable to wear slightly round at the corners. Boards are occasionally made as loose panels placed in a frame, all flush on the drawing surface, and bound together by bars on the other side. Drawing paper may be fixed down upon the ordinary board, either by damping, and gluing its edges, or by simply fixing it at the corners, and at intermediate points, if necessary, with pins or with sealing wax or wafers. The latter fixing is sufficient where no shading or colour¬ ing is to be applied, and if the sheet is not too long a time upon the board. It has the advantage, too, of preserving to the paper its natural quality of surface. With mounted paper, indeed, there is no other proper way of fixing. For large, coloured, or elaborate drawings, however, a damp- stretched sheet is preferable: with colouring or flat tinting, indeed, damp-stretching is indispensable, as the partial wetting of loose paper by water-colour causes the surface to buckle. Damp-stretching is done in the following way: lay the sheet flat on the board, with that side under¬ most which is to be drawn upon, and pare the thick edges from the paper; draw a wet sponge freely and rapidly over the upper side beginning at the centre, damping the entire surface, and allow the sheet to rest for a few minutes, till it be damped through, and the surface-water disappears. Those parts which appear to revive sooner than others, should be retouched with the sponge. The damping should be done as lightly as possible, as the sponge always deprives the paper of more or less of its sizing. The sheet is now turned over and placed fair with the edges of the board —sufficiently clear of the working edges to permit the free action of the drawing square. The square, or an ordinary straight-edge is next applied to the paper, and set a little within one edge, say about j 3 ^ of an inch, which is then turned up over the square, and smeared all along with melted glue. The paper is then folded back and pressed down by the square, after which the end of a paper-folder, or other smooth article, is rubbed along the ‘Tap,” with a piece of stiff paper interposed, to press out the superfluous glue and bring the paper into intimate contact with the board. The same operation being rapidly applied in succession to the other edges, the sheet is left to dry, and ultimately, by the contraction, turns out perfectly flat and tense. When melted glue is not to be had conveniently, a cake of glue may be dipped in water and rubbed on the margins of the board at the proper places. Lip glue, or artists’ glue, which dissolves very readily, may be used in this case. With loose panelled boards, as described, the panel is taken out, and the frame inverted; the paper being first damped on the back with a sponge, slightly charged with water, is applied equally over the opening to leave equal margins, and is pressed and secured into its seat by the panel and bars. This is a ready enough way of laying a sheet, and for damp sheets is more expeditious than the gluing system. But the large margin required dimi¬ nishes the size of the sheet, and for general use plain boards are sufficient. T-Square. The T-square (Fig. 234) is a blade or “straight-edge ” a, usually of mahogany, fitted at one end with a stock b, 1 Fig. 234. . Details of X-Square. applied transversely at right angles. The stock being so formed as to fit and slide against one edge of the board, the blade reaches over the surface, and presents an edge of its own at right angles to that of the board, by which parallel straight lines may be drawn upon the paper. To suit a 41-inch board, the blade should measure 40 inches long clear of the stock, or one inch shorter than the board, to remove risk of injury by overhanging at the end: it should be 2| inches broad by -fa inch thick, as this section makes it sufficiently stiff laterally and vertically. If thinner, the blade is too slight and too easily damaged by falls and other accidents, and is liable to warp; if thicker, it is too heavy and cumbersome; if broader, it is heavier without being stiffer. The tip of the blade may be secured from splitting by binding it with a thin strip inserted in a saw-cut as shown. The stock should be 14 inches long, to give sufficient bearing on the edge of the board, 2 inches broad, and § inch thick, in two equal thicknesses glued together. With a blade and stock of these sizes, a well proportioned T-square may be made, and the stock will be heavy enough to act as a balance to the blade, and to relieve the operation of handling the square. The blade should be sunk flush into the upper half of the stock on the inside, and very exactly fitted. It should be inserted full breadth, as shown in the figure; notching and dove¬ tailing is a mistake, as it weakens the blade and adds nothing to the security. The lower half of the stock should be only If inches broad, to leave a ^-inch check or lap, by which the upper half rests firmly on the board, and secures the blade lying flatly on the paper. For the second size of board, 31 inches long, the blade should be not mere than 30^ inches, of the same scan- 44 PRACTICAL CARPENTRY AND JOINERY. tling as above, or rather thinner; and the stock a little shorter. One-half of the stock c (Fig. 235), is in some cases made loose, to turn upon a brass pin to any angle Fig. 235. with the blade a, and to be clenched by a n screwed nut and washer. The turning stock *- is useful for drawing parallel lines obliquely to the edges of the board. In most cases, how¬ ever, the sector, and the other appendages to be afterwards described, answer the purpose, and do so more conveniently. A square of this sort should be rather as an addition to the fixed square, and used only when the with Swivelling bevil edge is required, as it is not so handy stock " as the other. The edges of the blade should be very slightly rounded, as the pen will thereby work the more freely. It is a mistake to chamfer the edges-that is, to plane them down to a very thin edge, as is sometimes done, with the object of insuring the correct position of the lines; for the edge is easily damaged, and the pen is liable to catch or ride upon the edge, and to leave ink upon it. A small hole should be made in the blade near the end, by which the square may be hung up out of the way when not in use. No varnish of any description should be applied to the T -square, or indeed to any of the wood instruments em¬ ployed in drawing. The best and brightest varnish will soil the paper long after it has been applied and fur¬ bished up. The natural surface of the wood cleaned and polished occasionally with a dry cloth, is the best and cleanest for working with. Straight edges and Triangles. These appendages to the T-square greatly facilitate the operations of the draughtsman. They should be of close- grained hardwood, as mahogany, well-seasoned; straight¬ edges, when 5 feet long and upwards, may be of ribbon- steel. Wood is more easily kept clean, and is less likely to soil the paper. Straight-edges should, like square blades, be just broad and thick enough for the necessary stiffness, and bevelled a little at one edge. The smallest (as in Fig. 236) may be 9 or 10 inches long, | inch broad, and | inch thick. Triangles, or set-squares, as they are sometimes called, should be barely -jjV inch thick, and flat on the edges, to wear well. They should be right angled, one of them a (Fig. 236), being made with equal sides, and angles of 45 degrees each; the other b, with angles of 60 and 30 degrees. The former, by means of its slant side, is very useful in laying off square figures: the vertical side, too, saves a deal of shifting of the T-square, as, when the hori¬ zontal edge is applied to that of the square, short perpendi¬ cular lines may be drawn by the upright edge. The most convenient size for general use measures from 3| to 4 inches on the side. A larger size, 8 or 9 inches long on the side, is convenient for use in making large scale drawings. Ap¬ plying one or other edge of the triangle b, to the square, the slant side gives at once the boundaries of all hex¬ agonal and triangular figures. This triangle may be of two sizes, 5 inches high and upwards. Of the two set- squares, the second is the more convenient for general use in drawing perpendiculars, as it is larger and has a shorter base, and is more easily handled. Still sharper set- squares are sometimes used; also compound triangles, having the slant-edge broken into two lines of different slopes. The latter is not to be recommended. Circular openings are sometimes made in the body of the triangle for facility in handling. They are of no great use in that respect, but they allow of the triangles being hung up. Triangles are further useful in connection with each other, or with the straight-edge, for drawing short paral¬ lels and perpendiculars without the use of the T-square, as shall be exemplified in the proper place. Sweeps and Variable Curves. For drawing circular arcs of large radius, beyond the range of the ordinary compasses, thin slips of wood, termed Fig. 237. Variable Curve, one-fourth full size. sweeps, are usefully employed, of which one or both edges are cut to the required circle. For curves which are not circular, but variously elliptic or otherwise, “ universal sweeps,” made of thin wood, of variable curvature, are very serviceable. The two examples (Figs. 237, 238) have been found from ex. perience to meet almost all the requirements of ordinary draw¬ ing practice. Whatever be the nature of the curve, some portion of the universal sweep will be found to coincide with its commencement, and it can be continued through¬ out its extent by applying successively such parts of the sweep as are suitable, taking care, however, that the con¬ tinuity is not injured by unskilful junction. Pencils. Pencils are of various qualities, distinguished by letter- marks. The H B (hard and black) quality is usually recommended; but it is too soft to retain long the firm point required for the correct execution of mechanical drawings; and, besides, the softer pencils are the more unctuous, and therefore the less ready in taking on CONSTRUCTION AND USE OF DRAWING INSTRUMENTS. 45 ink lines than the harder. F pencils work pretty well upon smooth paper; but for drawing paper of a thick and rougher quality, especially after having been damp- stretched, H H, and still better, H H H pencils (of two or three degrees of hardness), are better suited to retain their sharpness. They are further recommended by the lightness and delicacy of the lines that may be thrown off by them; for when a pencil drawing is made with the view of being done over with ink lines, the excellence of these lines, as well as the readiness with which they are produced, depends much upon the quality of the pencilling. Pencil lines, intended to be made permanent in ink, ought to be entered very delicately, and made just so dark as to render them distinct, for the more lightly they are executed, the titter they are to receive the ink. A little practice, and a steady hand, will secure the end proposed. The pencil need not be held tightly; a slight hold, without slackness, is what is wanted, inclined a little to the side toward which the line is drawn. Besides a drawing pencil for straight lines, it is well to have one a little softer for sketching in small circles, not requiring the regular application of the pencil bows, as the rounding and filling up of corners, ends of bolts, and the like. Many good draughtsmen consider the following mode of cutting as the one best calculated to prepare the pencil for straight line drawing:—In the first place, it should be cut down to the flat side of the lead, in a plane nearly parallel to the axis; then cut away on the opposite side to a bevel considerabty inclined, and cut, likewise, trans¬ versely, at equal angles. The lead being thus laid bare, should be pared down gradually on the three inclined sides, till brought to a fine edge viewed laterally, and a flat round point in the other aspect (as in Fig. 239). The less inclined side, when applied to a square, admits of the point being brought close to the edge, by which the line is more certainly drawn; and the roundness of the point keeps the pencil longer in working order. The sharpening of a sketching pencil is simply conical, and brings it to a fine point, and many prefer the lining pencil cut also in this manner. To produce a good working pencil, a sharp knife is in¬ dispensable ; it the knife be blunt, the point will invariably break away be¬ fore it is properly brought up—a very fine flat file, or a pumicerstone, or two files set in a stock, so that the sec¬ tion of the blades shall be like the letter V, are sometimes used to bring up the point of the pencil. Amongst the minor things requiring special attention, the cutting and pointing of pencils is one of some consequence both in point of economy and pleasant working. A carelessly cut pencil is constantly requiring a knife; and, at the same time, it works with much uncertainty along the straight edge of the square. Pins. Pins for holding down sheets not fixed by glue or otherwise, are indispensable. These should be made with a broad flat head, of brass, and rounded so as to permit the squares to slide easily over it, and Fi" 240 . the stem, of steel, rivetted into the head. Fig. 240 shows a good form of pin. The stem is in some cases screwed in, but is then liable to wear loose: the taper of the stem should be moderate, so as not to work out when fixed into the board. General Remarks on Drawing.—Management of the Instruments. In constructing preparatory pencil-drawings, it is advis¬ able, as a rule of general application, to make no more lines upon the paper than are necessary to the completion of the drawing in ink; and also to make these lines just so dark as is consistent with the distinctness of the work. And here we may remark the inconvenience of that arbitrary rule, by which it is by some insisted that the pupils should lay down in pencil every line that is to be drawn, before finishing it in ink It is often beneficial to ink in one part of a drawing, before touching other parts at all: it prevents confusion, makes the first part of easy reference, and allows of its being better done, as the sur¬ face of the paper inevitably contracts dust, and becomes otherwise soiled in the course of time, and therefore the sooner it is done with the better. Circles and circular arcs should, in general, be inked in before straight lines, as the latter may be more readily drawn to join the former, than the former the latter. W hen a number of circles are to be described from one centre, the smaller should be inked first, while the centre is in better condition. When a centre is required to bear some fatigue, it should be protected with a thickness of stout card glued or pasted over it, to receive the compass- leg, or a piece of transparent horn should be used, as before remarked when treating of compasses. India-rubber is the ordinary medium for cleaning a drawing, and for correcting errors made in pencilling. For slight work it is quite suitable; but its repeated ap¬ plication raises the surface of the paper, and imparts a greasiness to it, which spoil it for fine drawing, especially if ink-shading or colouring is to be applied. It is much better to leave trivial errors alone, if corrections by the pencil may be made alongside without confusion; as it is, in such a case, time enough to clear away superfluous lines when the inking is finished. For cleaning a drawing, a piece of bread two days old is preferable to india rubber, as it cleans the surface well and does not injure it. When ink lines to any considerable extent have to be erased, a small piece of damped soft sponge may be rubbed over them till they disappear. As, however, this process is apt to discolour the paper, the sponge must be passed through clean water, and applied again to take up the straggling ink. For small erasures of ink lines, a sharp rounded pen-blade applied lightly and rapidly does well, and the surface may be smoothed down by the thumb-nail or a paper-knife handle. In ordinary working drawings a line may readily be taken out by damping it with a hair pencil and quickly applying the india-rubber; and, to smooth the surface so roughened, a 4G PRACTICAL CARPENTRY AND JOINERY. light application of the knife is expedient In drawings intended to be highly finished, particular pains should be taken to avoid the necessity for corrections, as everything of this kind detracts from the appearance. In using the square, the more convenient way is to draw the °lines off the left edge, with the right hand, holding the stock steadily but not very tightly, against the edge of the board with the left hand. I he conveni¬ ence of the left edge for drawing by, is obvious, as we are able to use the arms more freely, and we see exactly what we are doing. To draw lines in ink with the least amount of trouble to himself, the draughtsman ought to take the greater amount of trouble with his tools. If they be well made, and of good stuff originally, they ought to last through three generations of draughtsmen; their working parts should be carefully preserved from injury; they should be kept well set, and above all, scrupulously clean. 1 he set¬ ting of instruments is a matter of some nicety, for which purpose a small oil-stone is convenient, lo dress up the tips of the blades of the pen, or of the bows, as they are usually worn unequally by the customary usage, they may be screwed up into contact, in the first place, and passed along the stone, turning upon the point in a directly perpendicular plane, till they acquire an identical profile. Being next unscrewed, and examined to ascertain the parts of unequal thickness round the nib, the blades are laid separately upon their backs on the stone, and rubbed down at the points, till they be brought up to an edge of j uniform fineness. It is well to screw them together again, and to pass them over the stone once or twice more, to j bring up any fault; to retouch them also on the outer and inner side of each blade, to remove barbs or frasing; and finally to draw them across the palm of the hand. The cliina-ink, which is commonly used for line-draw¬ ing, ought to be rubbed down in water to a certain degree: —avoiding the sloppy aspect of light lining in drawings; and making the ink just so thick as to run freely from the pen. This medium degree may be judged of after a little practice by the appearance of the ink on the pallet. The best quality of ink has a soft feel, free from grit or sedi¬ ment when wetted and rubbed against the teeth, and it has a musky smell. The rubbing of china-ink in water tends to crack and break away the surface at the point: this may be prevented by shifting at intervals the position cf the stick in the hand while being rubbed, and thus round¬ ing the surface. Nor is it advisable, for the same rea¬ son, to bear very hard, as the mixture is otherwise more evenly made, and the enamel of the pallet is less rapidly worn off. When the ink, on being rubbed down, is likely to be for some time required, a considerable quantity of it should be prepared, as the water continually vaporizes: it will thus continue for a longer time in a condition fit for application. The pen should be levelled in the ink, to take up a sufficient charge; and to induce the ink to enter the pen freely, the blades should be lightly breathed upon or wetted before immersion. After each application of ink, the outsides of the blades should be cleaned, to pre¬ vent any deposit of ink upon the edge of the squares. To keep the blades of his inkers clean, is the first duty of a draughtsman who is to make a good piece of work. Pieces of blotting or unsized paper, and cotton velvet, washleather, or even the sleeve of a coat, should always be at hand while a drawing is being inked. When a small piece of blotting paper is folded twice so as to present a corner, it may usefully be passed between the blades of the pen, now and then, as the ink is liable to deposit at the point and obstruct the passage, particularly in tine¬ lining ; and for this purpose the pen must be unscrewed to , admit the paper. But this process may be delayed by I drawing the point of the pen over a piece of velvet, or even over the surface of thick blotting paper; either method clears the point for a time. As soon as any obstruction takes place, the pen should be immediately cleaned, as the trouble thus taken will always improve and expedite the work. If the pen should be laid down for a short time with the ink in it, it should be unscrewed to keep the points apart and so prevent deposit; and when done with altogether for the occasion, it ought to be tho¬ roughly cleaned at the nibs. This will preserve its edges and prevent rusting. For useful reference, to assist the judgment in the pre¬ paration of drawings on paper, the drawing office should be fitted with a vertical scale of full size feet and inches, 6 or 8 feet long, fixed against the wall; and with a hori¬ zontal scale the full length of the office, fixed to the wall at 7 or 8 feet above the floor. The scales should be painted conspicuously in white, with black lining and figures. PART THIRD. STEREOGRAPHY—DESCRIPTIVE CARPENTRY. PROJECTION. Projection has for its object the representing on a given surface the forms of such solid bodies as can have their boundaries properly defined. Since the surfaces of all bodies may be supposed to con¬ sist of points, it is obvious that if the means of determin- | ing the position of any one point be possessed, the means ! of determining the position of all the points are equally possessed, and these will produce the surfaces to which I they belong. As space is unlimited, the position of a point can only be defined by referring it to some other object or objects j whose positions are known. Therefore, if it is required STEREOGRAPHY—PROJECTION. 47 to convey the knowledge of the position of a point, it is necessary to assume some objects of correlation, the posi¬ tions of which are known or may be imagined, and planes are the objects generally selected. Now, suppose a line to pass through the point to be determined, and to be somewhere intersected at a given angle by a plane whose position is known. This inter¬ section will be a point. If the plane and the point of in¬ tersection be given, it is clear that the line which passes through the point sought may be drawn. The point sought must be somewhere in that line. But to fix its locality another element is required, and is obtained by supposing another plane intersecting another straight line passing through the point. The positions of the two planes and the directions of the lines being known, the position of the point is defined by the intersection of the lines. Let A b c D (Fig- 241) and d e r c be the two planes, and ooho the lines, their in¬ tersection at O establishes the position of the point. Although the planes maybe at anjr angle in re¬ spect of each other, yet in practice, for the sake of simplicity, they are supposed to -be at right angles to each other, and the lines passing through the point to be perpendicular to the planes. The intersection of the lines with the planes at G and n are called the projections of the point 0 . In order to determine the -position of a straight line, it is obviously only necessary to determine any two points in it. Let G n (Fig. 242) be a straight line, whose position it is required to determine. Let A B c D be a vertical, and E F D c a horizontal plane, then the position of the point G will be determined by its projections g G g G, and the position of the point II by its projections h H g H, and, conse¬ quently, the position of the line II G is determined; and if the points H G are at the extremities of the line, its length also is determined. The planes are here shown in perspective, but in prac¬ tice they are drawn geometricall}' on the paper, as if they were supposed to be F j g . 2i3< hinged at c D and laid b _ n f flat, and in this posi¬ tion they are repre¬ sented in Fig. 243. It is obvious that all that is required in working the problem is the line of intersection c D. As, however, it will facilitate the comprehension of the subject, the projections are shown both in perspective and geometrically. In the last figure the position of a line was defined by the projections on two planes of two points contained in it. These planes are not the only ones on which the projec- projection may be made. In Fig. 244, op are the tions of the points on a plane BKFD, and in this figure it is seen that h g is the projection of the line on the plane abdc, o p is its projection on bkfd, and g is its projection onCDF E. In Fig. 245 it is shown with the planes laid flat. Omitting the boundary lines, the projection in practice would B D 0 P r / h 9 °\ P\ h 9 be as in Fig. 2 16. By these figures it is made apparent that the drawing of projections is less laborious when the planes are laid flat than if they were vertical; for the vertical and horizontal projections of the line being in¬ tersections of the horizontal and vertical planes, by a plane, h h g' r, perpendicular to them both, which passes through the points n G, and which is therefore perpendi¬ cular to their intersection c D, the straight lines g ' r, r h will also be perpendicular to c D. Hence, if the projection of a point g' on the horizontal plane be known, its pro. jection on the vertical plane laid flat will be in the straight line produced, drawn through cf\ perpendicular to c D. The line n G has been supposed parallel to one of the planes of projection, and its projection on that plane is equal to its length. If the line be oblique to both planes, its length will be greater than either of its projections. Let g li g li (Fig. 247) be the vertical and horizontal projections of a straight Fig . 047 . line, its actual length will be greater than either; but the follow¬ ing considerations show that it may be easily found from them. Let a line G K, lying in the plane passing through G H, be drawn through to meet the perpendicular H h' in K, then G K H is a right- angled triangle, of which G H is the hypothenuse: its side K JI is equal to the vertical projection k h, its base to the projection g Fig . 2 48 . li. Construct this d r triangle, therefore, as shown in Fig. 248, and g r is the length of the line required. The length may be also found from the vertical projection, thus:—Draw kg (Fig. 248) parallel to C D and produce it to s, make k s equal to the horizon¬ tal projection h' g, and join h s, which will be the length of the line required. The projections of a right line being given, and a length taken on one of them, to find the original line 48 PRACTICAL CARPENTRY AND JOINERY. which that length represents, and the angle which it makes with each of its planes of projection. Let ah a' h' (Fig. 249) be the given projections, and c cl the length, Fi S . 249. taken on the horizontal pro¬ jection. As all the points of the line neces¬ sarily corre¬ spond to all the points of its projection, if on c d are raised the indefinite perpendicu¬ i-~* ' V \ i ~— D lars to the common section, these lines will cut the vertical projection a' h' in the points c' and d', and c cl will be the length of the vertical projection of c cl. To find the length of the original line: on cd (Fig. 2-50), raise indefinite perpendiculars, upon which from c carry the length 1 c to c, and from cl the length 2 cl' to D, and draw through C and D the line A r., which is the original line sought; then through c draw Ce parallel to c d, and the angle D c e is the angle which the original line makes with the horizontal plane. In the same manner the angle with the vertical plane is found on the ver¬ tical projection by carrying the length 1 c 2 d from c and cl on the perpendiculars c d' to C'd', and through c' and d', a' b', and then from c', drawing c' e', parallel to c' cl, and the angle d' c e is the angle sought. The projections a’h’ ah" (Fig. 251) of a right line A B, being given, to find the points wherein the prolon¬ gation of that line would meet the planes of projec¬ tion. In the perspective representation of the problem, it is seen that A b, if prolonged, cuts the horizontal plane in c, and the ver¬ tical plane in d, and the projections of the prolongation become c e f cl. Hence, in the following figure, if a h a’ U (Fig. 252) be the projections of A B, the solution of the problem is obtained by producing these lines to meet the common intersection of the planes in f and e, and on these points to raise the perpendiculars feed, when c and cl‘ are the points sought. To draw through a given point a line parallel to the projections of a given Vne. Let a h a h‘ (Fig. 253) be the projections of the ori¬ ginal line A B. From the perspective representation it is evident that the lines a b ci b' of projections of the planes which pass through the original line A B, and the lines of projection of the planes which pass through any line c D, and Fig. 253. parallel to the line A B, are parallel each to each; therefore, if the given point lie in such line C D, the solution of the problem is easy. Let ah a' b’ (Fig. 251) be the given pro¬ jections, and e the given point, then through e draw c d parallel to a b, and through e' draw c d’ parallel to a' b', and c d c'd' are the projections of the line sought. If tlie two lines intersect each other in space, to find from their given projections the angles vjhich they make r with each other. Let A b, c D (Fig. 255) be the given lines intersecting at E. In the perspective representation, if these lines be supposed to lie in a plane which intersects the hori¬ zontal plane in the line A c, this line will be the base of a triangle, AEC. If the plane is perpendicular to the hori¬ zontal plane, the angle A E c is at once known; but, sup¬ pose it inclined to the horizontal plane, then, to find the angle, it is necessary to imagine the plane turned down horizontally on the line A c. as at A e" c. To do this, from E let fall a perpendicular to the horizontal plane, cutting it in e, which is the horizontal projection of E, and the height e E is the height of the vertex of the triangle above the horizontal plane, and e f is the projection of the line E /. There is thus obtained the triangle e E f which suppose laid horizontally, by turning on its base e f then from / as a centre describe the arc E' g E", cutting the line e f produced in e", and join a e" c e", and A e" c is the angle sought. In applying this to the solution of the problem, let STEREOGRAPHY—PROJECTION. 49 a b c d, a' u c d' (Fig. 256) be the projections of the lines: from e draw indefinitely, e E' perpendicular to the line e' f and make e E' equal to e" e', and draw / e': from / as a cen¬ tre, describe the arc E' g e", meeting e f produced in e", and join a e", c e": the angle a E" c is the angle sought. It will be observed that the projections of the point of in¬ tersection of the two lines are in a right line perpendicular to the line of intersection of the planes of projection. Hence this corollary. The projections of the point of intersection of two lines which cut each other in space , are in the same right- line perpendicular to the com¬ mon intersection of the planes of projection. This is further illustrated by the nest problem. To determine, from the projection of two lines which intersect each other in the projections, whether the lines cut each other in space or not. Let ab, c d, a b', c d' (Fig. 257) be the projections of the lines. It might be supposed that as their traces or pro¬ jections intersect each other, that the lines themselves in¬ tersect each other in space, but, on ap¬ plying the corollary of the preceding problem, it is found that the intersec¬ tions are not in the same perpendicular to the line of intersection of the planes of projection a c. This is re¬ presented in perspective in Fig. 258. We there see that the original lines a B c D do not cut each other, although their projections ab,cd,a b', c d', do so. From the point of intersection e raise a perpendicular to the horizontal plane, and it will cut the original line c D in E, and this point therefore belongs to the line c D, but e belongs equally to a B. As the perpendicular raised on e passes through E on the line c D, and through e' on the line a B, these points E e' cannot be the intersection of the two lines, since they do not touch ; and it is also the same in regard to //'. Hence, when two right lines do not cut each other in space, the intersections of their pro¬ jections are not, in the same right line, perpendicular to the common intersection of the planes of projection. 1 he projections of a plane and of a point being given , to draw through the point a plane parallel to the given plane. in the perspective representation, suppose the problem solved, and let B c (Fig. 259) be the given plane, and A c, A B its projections, and E F a plane parallel to the given plane, and g F, G e, its projections. Through any point D, taken at pleasure on the plane e f, draw the vertical plane H j, the horizontal projection of which, I H, is parallel to G E. The plane H j cuts the plane E f in the line k l', and its vertical projection g f in V. The horizontal projec¬ tion of k l' is H I, and its vertical projection k l ’; and as the point D is in k' l', its horizontal and vertical projec¬ tions will be d and d'. Therefore, if through d be traced a line d I, parallel to A B, that line will be the horizontal projection of a vertical plane passing through the original point D; and if on I be drawn the indefinite perpendicular 1 1’, and through d\ the vertical projection of the given point, be drawn the horizontal line d' l', cutting the per¬ pendicular in V, then the line F G drawn through V, parallel to A C, will be the vertical projection of the plane required; and the line G E drawn parallel to A B, its horizontal pro¬ jection. Hence, all planes parallel to each other have theii projections parallel and reciprocally. In solving the pro¬ blem, let A B, A c (Fig. 260) be the projections of the given plane, and d d’ the projections of the given point. Through d draw d I parallel to A B, and from I draw I V perpen¬ dicular to A H : join d d', and through d' draw cl' V parallel to A n. Then F V G drawn parallel to A c, and G e parallel to A B, are the projections required. The projections ab, bc, and ad,dc, of two planes which cut each other being given, to find the projection of their intersections. The planes intersect each other in the straight line A c (Figs. 261 and 262). of which the points A and 0 are the a 50 PRACTICAL CARPENTRY AND JOINERY. projections, since in these points this line intersects the planes of projection. Now, to find these projections, it is only necessary to let fall on the line of intersection the perpendiculars A a, c c (Fig. 261), from the points A and c, and join Ac, ca: Ac will be the horizontal projection, and c a the vertical projection of C a (Fig. 261), or CA (Fig. 232), the line of intersection or arris of the planes. The 'projections of two intersecting planes being given, to find the angle which they make between them. The angle formed by two planes is measured by that of two perpendiculars drawn through the same point of their intersections in each of the two faces. These lines determine a third plane perpendicular to the arris. If, therefore, the two planes are cut by a third plane, the solution of the problem is obtained. On the arris A c (Fig. 263), take at pleasure any point E, and suppose a plane passing through that point, cut¬ ting the two given planes perpendicular to the arris. There results from the section a triangle def, inclined to the horizontal plane, and the angle of which, D E F, is the measure of the inclination of the two planes. The horizontal projection of that triangle is the triangle D e F, the base of which, F D, is perpendicular to A c, and cuts it in the point g, and the line Eg is perpendicular to D F. The line g e is necessarily perpendicular to the arris A C, as it is in the plane def, and its horizontal projection is g e. Now, suppose the triangle def turned on D F as an axis, and laid horizontally, its summit will then be at e", and D e" f is the angle sought. The perpendicular g e i j also in the vertical triangle A c c, of which the arris is the hypothenuse, and the sides A c, c c, are the projec- |. l tions. This description introduces the solution of the pro¬ blem, which applies also to Fig. 264. Through any point g (Fig. 264) taken at pleasure, on the line A c, the horizontal projection of the arris a c, draw FD perpendicular to Ac; from g draw g E' perpendi¬ cular to the arris A c'; and from g as a centre, with the radius g E', describe the arc e' e": join F e", d e". The angle e" f d is the angle sought. Through the projection of a given point a a!, to draw a perpendicular to a plane, b c, CD, also given. Let A E (Fig. Fig. 265 . 265) be the perpendicu¬ lar drawn through the point A to the plane B D, and its intersec¬ tion with the plane is the point E. Suppose a vertical plane a F to pass through A E, this plane would cut BD in the line g D, and its projection a h would be perpendicular to the projection B C. In the same way a'e', the vertical projection of A E, would be perpendicular to c D, the vertical projection of the plane BD. Then, if a line, perpendicular to a h, is drawn through a, it will be the horizontal projection of the plane passing in the line of the perpendicular A E, and h f will be its vertical projection. From a', draw upon c D an in¬ definite perpendicular, and that line will contain the ver¬ tical projection of AE, a a ah contains its horizontal projec¬ tion. Now, to find the point of intersection of the line A E with the plane, construct the vertical projection of the line of intersection of the two planes gv, and the point of intersection of that line with the right line drawn through a , will be the point sought If from that point a perpen¬ dicular is let fall on a h, the point e will be the horizon¬ tal projection of the point of intersection E. In Fig. 266, let b c, c d be the projections of the given plane, and a a of the Fig. 266 . given point. From the given point a, draw a g perpendi¬ cular to B C, which will be the hori¬ zontal projection of a plane passing vertically through ~ 1 a, and cutting the given plane. From a, also draw a A perpendicular to a g, and make it equal to a a! (Fig. 265). From h draw h D perpendicular to h a, and make h D equal to h D. Draw g d , which will be the section of the given plane, and the angle h g d' will be the measure of the inclination of the given plane with the horizontal plane. There is now to be drawn, perpendicular to this line, a line A E STEREOGRAPHY—PROJECTION. 51 through A, which will be the line required. From the point of intersection E let fall upon ah a perpendicular, which will give e as the horizontal projection of E. Therefore:— Where a right line in space is perpendicular to a plane , the projections of that line are respectively per¬ pendicular to the projections of the plane. Through a given point a a', to draw a plane perpen¬ dicular to a right line b c, b 1 c also given. The foregoing problem has shown that the projections of the plane sought must be perpendicular to the projec¬ tions of the line. The plane D E (Fig. 267), is, by construction, perpen¬ dicular to the line B c. Take at pleasure the point A in the plane D E, and through it draw the horizontal line A / , which will be necessarily parallel to the projection D G. and will cut the vertical projection G E in /'. The hori¬ zontal projection of the intersecting point will be f, that of A will be a, and that of a/' will consequently be a f, which, being parallel to D G, will be perpendicular to b c. The solution of the problem consists in making to pass through A a vertical plane A f, the horizontal projection of which will be perpendicular to b c. Through a (Fig. 268), draw the projection a f perpen¬ dicular to b c: from f raise upon k l the indefinite perpendi¬ cular f f', which will be the vertical pro¬ jection of the plane a f f, perpendicu¬ lar to the horizon¬ tal plane, and pass¬ ing through the ori¬ ginal point A (Fig. 267). Then draw through a in the vertical projection a horizontal line, cut¬ ting if in/', which point should be in the projection of the plane sought; and as that plane must be perpendicu¬ lar to the vertical projection of the given right line, draw through /' a perpendicular to b' c, and produce it to cut K l in G. This point G is in the horizontal projection of the plane sought. All that remains, therefore, is from G to draw G D perpendicular to be. If the projections of the straight line are required, proceed as in the previous problem, and as shown by the dotted lines. A right line , a b and a’ b', being given in projection, and also the projection of a given plane , to find the angle which the line makes with the plane. Let A B (Fig. 269), be the original right line intersect¬ ing the plane ce in the fig. 2G9. point B. If a. vertical plane pass through the right line, it will cut the plane c E in the line f b, and the hori¬ zontal plane in the line a b. As the plane aB is in this case parallel to the vertical plane of projection, its projection on that plane will be a quadrilateral figure a b', of the same dimensions; and /B contained in the rectangle will have for its vertical projection a right line D b', which will be equal and similar to / B. Hence the two angles, a' b' D, A Bf\ being equal, will equally be the measure of the angle of inclination Fig. 270. / of the right line A E to the a - 1 / plane c E. Thus the angle a b' T) (Fig. 270) is the angle / sought. D , b This case presents no difficulty; but when the line is in a plane which is not parallel to the plane of pro- « ./ A jection, the problem is more difficult. c In Fig. 271, the right line A B is oblique to the plane may at first sight be imagined to be the angle sought— that is, the measure of the inclination of A B with the plane C E. But it is not so; for the vertical plane which passes through A B, and contains that angle, is oblique to the plane c E; and its projection a b is consequently not perpendicular to the given plane. It is to be recollected that the inclination of a right line to a plane is measured by an angle situated in a plane which shall pass through the right line, and be perpendicular to the plane, which is always possible. If, therefore, a plane perpendicular to c E pass through the right line A B, the projections of the two planes will be perpendicular to each other. Thus a f will be perpendicular to c D, and a' D to D e. These planes intersect in the line/B. This line, as well as the original line A E, will, therefore, be in a plane which is perpendicular to the plane C E, and A b/ will be the angle sought. Through any point A in the given line, draw a perpen- 52 PRACTICAL CARPENTRY AND JOINERY. dicular a / upon the plane c e. This line will also be perpendicular to / B, and will be the third side of a tri¬ angle A / b, rectangular at /, perpendicular to the plane c E, and inclined to the horizontal plane. The projections of the triangle will be a f b on the horizontal, and a r D B in the vertical plane. The projections of the triangle being obtained, it is only necessary to develope it on the horizontal plane by turning it down as on a hinge. To do this, observe that the side / B of the triangle A / B rests on the hypothenuse of a right-angled triangle f b B, which is vertical or perpendicular to the horizontal plane. Lay¬ ing down this triangle flat, by making it turn on its base b /, as on a hinge, it will then appear as / b B', and its hypothenuse will be the side or base of the triangle sought, b/a. It has been seen that A/was perpendicular to /b: raise on / therefore, perpendicular to /b', the side a/, which will be / a' ; draw the line a' b\ which will be the hypothenuse of the triangle sought; and the angle /b' a' will be the measure of the inclination of the line A B with the plane C e. From the above description, the operation may be performed in Fig. 272, in which c D, D E are the projections of the Fig. 272 . plane, and a b, a B the projec¬ tions of the line. Through a in the horizontal projec¬ tion draw a f per¬ pendicular to c D, and join / b. To obtain the de¬ velopment of the triangle D B b, which is in the vertical projection of a right-angled triangle whose base is/ b, imagine this triangle turned down on its base as on a hinge, that is, by construction; make b e' equal to b b, and perpendicular to / b, and join / b'. Then to ob¬ tain the development of the triangle a'n b, on /draw /a' perpendicular to/B', and make it equal to a' D, and join a' b'. Then the angle / b' a, is the measure of the inclina¬ tion of the line A b, on the plane e c. PROJECTIONS OF SOLIDS. There is no general rule for the projections of solids. Their constructions are more or less easy, dependent on the nature of the question; and it is possible always to accom¬ plish them more or less directly by means of the princi¬ ples about to be stated. Given the horizontal projection of a regular tetrahe¬ dron, to Jincl its vertical projection. Let ABcd (Fig. 273) be the given projection of the tetrahedron, which has one of its feces coincident with the horizontal plane. It is evident that the vertical pro¬ jection of that face will be the line acb. If the heio-ht of d above the horizontal plane be known, it is set°otf from c to d\ and by joining a d’, b d', the problem is solved. In proceeding to find the height of d' } let us consider that a perpendicular, let fall from the summit of the tetrahedron on the horizontal plane, is also jDerpen- dicular to the right lines cl a, cl b, d c, and forms with each of the arrises a right-angled triangle, of which two of the sides, the right angle, and the direction of the third side, are known. It is easy, therefore, to con¬ struct one of those tri¬ angles. On cl, draw an indefinite line perpen¬ dicular to cl c, and make c D equal to c B, c A, or A b ; d d will be the height sought, which is carried to the vertical projection from c to cl'. in other ways. A point being given in one of the projections of a tetrahedron, to find the point on the other projection. Let e be the uoint given in the horizontal projection (Fig. 273). It may first be considered as situated in the plane CB cl, inclined to the horizontal plane, and of which the vertical projection is the triangle c B cl'. According to the general method, the vertical projection of the given point is to be found somewhere in a perpendicular raised on its horizontal projection e. If through cl and the point e be drawn a line produced to the base of the triangle in /, the point e will be on that line, and its vertical pro¬ jection will be on the vertical projection of that line/' e d', at the intersection of it with the perpendicular raised on e. If through e be drawn a straight line g h, parallel to c B, this will be a horizontal line, whose extremity h will be on B d ■ The vertical projection of d B is d' B ; therefore, by raising on h a perpendicular to A B, there will be ob¬ tained h', the extremity of a horizontal line represented by hg in the horizontal plane. If through h' is drawn a horizontal line K g, this line will cut the vertical line raised on e in e, the point sought. If the point had been given in g on the arris c d, the projection could not be found in the first manner; but it could be found in the second manner, by drawing through g a line parallel to c B, and prolonging the horizontal line drawn through h to the arris c cl', which it would cut in g\ the point sought. The point can also be found by laying down the rectangular triangle c d d, which is the de¬ velopment of the triangle formed by the projection of the arris c d, the height of the solid and the length of the arris as a hypothenuse, and by drawing through g the line g G perpendicular to c cl, to intersect the hypo¬ thenuse in G, and carrying the height g G from c to g> in the vertical projection. Thus, one or other of these means can be employed according to circumstances. If the point had been given in the vertical, instead of the horizontal projection, the same operations inverted would require to be used Given a tetrahedron, and the projection of a plane cutting it, byzohich it is truncated, to find the projection of the section. First, when the intersecting plane is perpendicular to the base (Fig. 274), the plane cuts the base in two points c /, of which the vertical projections are e and f ; and the STEREOGRAPHY-PROJECTIONS OF SOLIDS. 53 arris B cl is cut in g, the vertical projection of which can readily he found in any of the ways detailed in the last problem. Having found g, join e g', f g', and the triangle e' g f is the projection of the inter¬ section sought. When the intersect¬ ing plane is given in the vertical projection, as e f in Fig. 275, the horizontal projections of the three points e g' f have to be found. The point g in this case may be obtained in several ways. First, by draw¬ ing G' g h through g\ then through h' draw¬ ing a perpendicular to the base, produced to the arris at h, in the horizontal projection, and then drawing h g parallel to c B, cut¬ ting the arris b d in g, which is the point re¬ quired. Second, take d b as the base of a triangle formed by the perpendicu¬ lar, and the arris of which, d B, is the horizontal pro¬ jection, and carry this base, cl b, upon the common line of intersection of the planes from d to B', and draw the arris b' d!. From g draw the horizontal line cutting b' d' in g'; carry g g', which is the distance of the per¬ pendicular of the arris in the horizontal projection, from d to g ; and g is the point sought. The 'projections of a tetrahedron being given , to find its projections when inclined to the horizontal plane in any degree. Let ABC d (Fig. 276) be the projections of a tetrahed¬ ron, with one of its sides coincident with the hori¬ zontal plane, and c' d' b' its vertical projection; it is required to find its pro jections when turned round the arris A B as an axis. The base of the pyramid being a triangle, its verti¬ cal projection is the right line c' B. If this line is raised to c", by turning round B, the horizontal projection will be A c 2 B. When the point c, by the A raising of B c', describes the arc c’ c", the point cl' will have moved to d", and the perpendicular let fall from that point on the horizontal plane will give d 3 , the horizon¬ tal projection of the extremity of the arris c cl; for as the summit d moves in the same plane as c, parallel to the vertical plane of projection, the projection of the sum¬ mit will evidently be in the prolongation of the arris c d, which is the horizontal projection of that plane. The process, therefore, is very simple, and is as follows:—Con¬ struct at the point B the angle required, c B c", and make the triangle c" B d" equal to c'd b' ; from d' let fall a per¬ pendicular cutting the prolongation of the arris cd in d and from c", a perpendicular cutting the same line in c 2 ; join B c 2 , A c 2 , B d 3 , A d 3 . The following is a more general solution of the prob¬ lem :—Let ABC d (Fig. 277) be a pyramid resting with one of its sides on the horizontal plane, and let it be required to raise, by its angle c, the pyramid, by turn¬ ing round the arris A B, until its base makes with the horizontal plane any required angle, as 50°. Conceive the right line c e turning round e, and still continuing to be perpendicular to A B, until it is raised to the required angle. If a perpendicular be now let fall from o, it will give the point c" as the horizontal projection of the angle c in its new position. Conceive a vertical plane to pass through the line c e. This plane will necessarily contain the required angle. Suppose, now, we lay this plane down in the hori¬ zontal projection, thus:— Draw from e the line e c', making with c c an angle of 50°, and from e with the radius e c describe an arc cutting it in o'. From c' let fall on c e a perpendicular on the point c", which will then be the horizontal projection of c in its raised position. On c' e draw the profile of the tetrahedron c' D e inclined to the horizontal plane. From D let fall a perpendicular on c e produced, and it will give d 3 as the horizontal projection of the summit of the pyra¬ mid in its inclined position. J oin A d 3 , B d 3 , A c", B c" to complete the figure. The vertical projection of the tetrahedron in its ori¬ ginal position is shown by a d b, and in its raised position by a, c 2 , d 2 , b. To construct -the vertical and horizontal projections of a cube, the axis of which is perpendicular to the hori¬ zontal plane. The* axis of a cube is the straight line which joins two of its opposite solid angles. If an arris of the cube is given, it is easy to find its axis; as it is the hypothenuse of a right-angled triangle, the shortest side of which is the length of an arris, and the longest the diagonal of a side. Conceive the cube cut by a vertical plane passing through its diagonals EG, AC (Fig. 278), the section will be the rectan¬ gle A E G c. Divide this into two equal right-angled triangles, by the diagonal E c. If, in the upper and lower faces of the cube, we draw the diagonals F H, B D, they will cut the former diagonals in the points / b. Now, as the lines 5 b, bn, f f. fxi, are 54 PRACTICAL CARPENTRY AND JOINERY. perpendicular to the rectangular plane A E G c , f b may be considered as the vertical projection of B F, D H, and from tliis consideration we may solve the problem. Let A E (Fig. 279), be the arris of any cube (the letters here refer to the same parts as those of the preceding diagram, Fig. 278). Through A draw an indefinite line, A c, perpendicular to A E. Set off on this line, from A to c, the diagonal of the square of A e, and join E c, which is then the diagonal of the cube. Draw then the lines E G, c G, parallel respectively to A c and A E, and the resulting rectangle, A E G c, is the section of a cube on the line of the diagonal of one of its faces. Divide the rect¬ angle into two equal parts by the line b /, which is the vertical projection of the lines BF, d II (Fig. 278), and we obtain, in the figure thus completed, the vertical pro¬ jection of the cube, as a c b d (Fig. 280). Through c (Fig. 279), the extremity of the diagonal, draw y z perpendicular to it, and let this line represent the common section of the two planes of projection. Then let us find the horizontal projection of a cube, of which A e G c is the vertical projection. In the vertical projection the axis E c is perpendicular to y z, and, consequently, to the horizontal plane of projection, and we have the height above this plane of each of the points which ter¬ minate the angles. Let fall from each of these points per¬ pendiculars to the horizontal plane, the projections of the points will be found on these perpendiculars. To find, for example, the horizontal projection of the axis E c, take at pleasure, on its prolongation, any point, c' (or e ), which is the projection of both tile extremities of the axis c and e. If we suppose the rectangle aegc turned on the line y z until it is vertical, its projection will be a g. Through o' (or e) draw a line parallel to y*z, and find on it the projection of the rectangle cl c g', by con¬ tinuing the perpendiculars A a, G g. We have now to find the projections of the points b f (which represent d b f H, Fig. 278), which will be somewhere on the per¬ pendiculars b b', f /", let fall from them. We have seen in the preceding Fig. (No. 278), that b f, d h are dis¬ tant from b f by an extent equal to half the diagonal of the square face of the cube. Set off, therefore, on the perpendiculars b b' and f /", from o and m, the distance A b in d b' and and join d a, a b', b' f, f" g‘, g' f', to complete the hexagon which is the horizontal projec¬ tion of the cube. The dotted lines, d o', V c', g cf, show the arrises of the lower side. Knowing the heights of the points in these vertical projections, it is easy to construct i a vertical projection on any line whatever, as that on R s below. In these figures all the points are indicated by the same letters as in the preceding figures. To construct the projections of ct regular octahedron.—The octahedron is formed by the union of eight equilateral triangles; or, more correctly, by the union of two pyramids with square bases, opposed base to base, and of which all the solid angles touch a sphere in which they may be inscribed. Describe a circle (Fig. 281), and divide it into four equal parts by the diameters,and draw the lines ad, db, be, ca; a figure is produced which serves for either the vertical or the horizontal projection of the octahedron, when one of its axes is perpendicular to either plane. One of the faces of an octahedron being given, coinci¬ dent luith the horizontal plane of projection, to construct the projections of the solid. Let the triangle ABC (Fig. 282) be the given face. If A be considered to be the summit of one of the two pyra¬ mids which compose the solid, B c will be one of the sides of the base. This base makes, with the horizontal plane, an angle, which is easily found. Let fall from A a perpen¬ dicular on B c, cutting it in d. With the length B c as a radius, and from d as a centre, describe the indefinite arc e f. The perpendicular A cl will be the height of each of the faces, and, consequently, of that which, turning on A, should meet the side of the Fig> 2 82 . base which has already turned on d. Make this height turn on A, describing from that point as a centre, with the radius A cl, an indefinite arc, cutting the first arc in G, the point of meeting of one of the faces with the square base: draw the lines g a, g d: the first is the profile or inclina¬ tion of one of the faces on the given face ABC, according to the angle d A g ; the second, d G, is the inclination of the square base, which separates the two pyramids in the angle kd,dG. The face adjacent to the side B c is found in the same manner. Through G, draw the hori¬ zontal line G H equal to the perpendicular A cl. This line will be the profile of the superior face. Draw cl h, which is the profile of the face adjacent to b c. From H let fall a perpendicular on A d produced, which gives the point h for the horizontal projection of H, or the summit of the supe¬ rior triangle parallel to the first: draw i Jc, k c, C h, h B, h k, h i, and the projection is obtained. From the heights we have thus obtained, we can now draw the vertical projection M, in which the parts have the same letters of reference. The finding the horizontal projection may be abridged by constructing a hexagon in which may be inscribed the two triangles acb, h i k (Fig. 282 n), and the projection is completed. Given in the horizontal plane the projection of one of the faces of a dodecahedron, to construct its projections. STEREOGRAPHY—PROJECTIONS OF SOLIDS. 55 the assemblage of It is necessary, in The dodecahedron is formed by twelve regular and equal pentagons, order to construct the projection, to dis¬ cover the inclina¬ tion of the faces among themselves. Let the pentagon ABODE (Fig. 283) be one of the faces on which the body is supposed to be seated on the plane. Conceive two other faces, E F G H D, diklc also in the horizontal plane, and then raised by being turned on their bases, E D, D c. By their movement they will describe in space arcs of circles, which will ter¬ minate by the meeting of the sides D n, D I. To find the inclinations of these two faces. —From the points I and H let fall perpendiculars on their bases produced. If each of these pentagons were raised vertically on its base, the horizontal projections of H and I Avould be respectively in 0 2 ; but as both are raised together, the angles H and I would meet in space above h, where the perpendiculars intersect, therefore, h will be the hori¬ zontal projection of the point of meeting of the angles. To find the horizontal projection of G 0 K, prolong indefi¬ nitely z I, and set off from z on zi the length x K in h' ; then from z as a centre, with the radius z k, describe an arc cutting z T produced in the point K', from which let fall on z k' a perpendicular K k", and produce it to x K. If, now, the right-angled triangle, z 1c k', were raised on its base, 1c would be the projection of K\ Con¬ ceive now the pentagon cdikl turned round on cD, until it makes an angle equal to 1c z k' with the horizontal plane, the summit K will then be raised above k by the height k" K, and will have for its projection the point k. In completing the figure practically;—from the centre 0 , describe two concentric circles passing through the points h D. Draw the lines h D, h k, and carry the last round the circumference in mnoprstv: through each of these points draw radially the lines m c, 0 b, r A, t E, and these lines will be the arrises analogous to h D. This being done, the moiety or inferior half of the solid is projected. By reason of the regularity of the figure, it is easy to see that the six other faces will be similar to those already drawn, only that although the superior pentagon will have its angles on the same circumference as the inferior pentagon, the angles of the one will be in the middle of the faces of the other. Therefore, to describe the superior half;— through the angles np sv k, draw the radial lines n 1, p 2, 8 3, v 4, k 5, and join them by the straight lines 1 2, 2 3, 3 4, 4 5, &c. To obtain the length of the axis of the solid, observe that the point k is elevated above the horizontal plane by the height k" K': carry that height to k k”: the point r, analogous to h t is raised by the same height as that point, that is to say h 1 ', which is to be carried from r to it; and the line bk" is the length sought. As this axis should pass through the centre of the body, if a vertical line O o' is drawn, it will cut the vertical projection of the axis in O', and therefore 0 o' is the half of the height of the solid vertically. By doubling this height, and drawing a hori¬ zontal line to cut the vertical lines of the angles of the superior pentagon, the vertical projection of the superior face is obtained, as in the upper portion of Fig. 283, in which the same letters refer to the same parts. One of the faces of a dodecahedron, abode (Fig. 284), being given, to construct the projections of the solid, so that its axis may be perpendicular to the horizontal plane. The solid angles of the dodecahedron are each formed by the meeting of three pentagonal planes. If there be conceived a plane passing through the extremities of the arris of the solid angle, the result of the section would be a triangular pyramid, the sides of whose base would be equal to one of the diagonals of the face, such as B c (Fig. 284). An equilateral triangle b cf (Fig. 285), will repre¬ sent the base of that pyramid inverted, that is, with its summit resting on the horizontal plane. In constructing the projection, it is required to find the height of that pyramid, or, which is the same thing, that of the three points of its base b cf, for as they are all equally ele¬ vated, the height of one of them gives the others. There is necessarily a proportion between the triangle A be (Fig. 285) and ABC (Fig. 284), since the first is the horizontal projection of the second. A g is the pro- Fig. 28(5. jection of AG; but A G is a part of A H, and the projec¬ tion of that line is required for one of the faces of the solid; therefore, as A G: A g :: A H: x, which may be found, by seeking a fourth proportional, to be equal to A li ; or graphically thus:—Raise on A g at. (7 an indefinite perpendi¬ cular, take the length A G (Fig. 284) and carry it from A to g' (Fig. 285), g being one of the points of the base, elevated above the horizontal plane by the height g G'; its height giving also the heights of b cf. Since AG is a portion of A it, a g' will be so also. Produce A g', therefore, and carry on it A H (Fig. 284) from a to H , and from H let fall a perpendicular on A g produced, which gives h the point sought. Produce H h, and carry on it the length H D or H E from h to d and h to e ; draw the lines c d, be, 5G PRACTICAL CARPENTRY AND JOINERY. and the projection of one of the faces is obtained inclined to the horizontal plane, in the angle h' a h. As the two other inferior faces are similar to the one found, the three faces should be found on the circumference of a circle traced from A as a centre, and with A d or A e as a radius. Pro¬ long A n, A o, perpendiculars to the sides of the triangle, and make them equal to A Jc, and through their extremi¬ ties draw perpendiculars, cutting the circumference in the points i Jc, l m. Through these points draw the lines i b, Jcf If m c, and the projections of the three inferior faces are obtained. The superior pyramid is similar and equal to the inferior, and solely opposed by its angles. Describe a circle passing through the three points of the first triangle, and draw within it a second equilateral triangle n op, of which the summits correspond to the middle of the faces of the former one. Each of these points will be the summit of a pentagon, as the points b cf These pentagons have all their sides common, and it is only necessary therefore to determine one of these superior pentagons to have all the others. To obtain the vertical projection (Fig. 286), begin with the three inferior faces. The point A in the horizontal pro¬ jection being the summit of the inferior solid angle, will have its vertical projection in a: the points bcf when raised to the height g G', will be in b' c'/', or simply b' /'. The points bgc being in a plane perpendicular to the vertical plane, will be necessarily confounded with each other. The line a f will be the projection of the arris a/, and a b' will be that of the arrises A b, Ac, and line A g, or rather that of the triangle kb c, which is in a plane perpendicular to the vertical plane. But this triangle is only a portion of the pentagonal face of the solid ori¬ ginally given, and of which A H is the perpendicular let fall from A on the side E D (Fig. 281). This side is com¬ mon to the inferior pentagon, and to the superior penta gon edqpr, which is also perpendicular to the vertical plane; and, consequently, its vertical projection will be e p , equal to a e'. This projection can be now obtained by raising a vertical line through p, the summit of the superior pentagon, and from e as a centre, and with the radius A H' or A H, describing an arc cutting this line in p', the point sought. But pno belong to the base of the superior pyramid; therefore, if the height p is transferred to n', by drawing through p' a line parallel to y x, n will be the projection of the points n and o. Through n' draw s' n a' parallel to a e', cutting perpendiculars drawn through s and A in the horizontal projection. Through s draw s'/' parallel to p e, and join a f, cl p; set off on the perpendicular from r the height of s' above y z at r', and draw r't' parallel to y z, cutting the perpendicular from t, and join ri t'. Make Jc and % the same height as e', and draw Jc i', and join i' b', i' t'\ and we obtain the pro¬ jection sought. One of tlic faces of an icosahedron being given, to construct the projections of the solid when the given face coincides with the horizontal plane of projection. Let the triangle ABC (Fig. 287) be the given face. It may be observed that B c is one of the sides of the base of a pentagonal pyramid. Conceive this base turned on B c and laid in the horizontal plane, it will then be the pentagon b c clef To the centre g, draw the lines B g, c g, d g, e g, f g, and the horizontal projection of the pyramid is obtained. Conceive now the pyramid raised by turning round B c, and its summit e stopped when it has Fif. 587. met the point h, one of the summits of the superior tri¬ angles of the solid, and it will by construction be confound¬ ed with that point. Then, if on h is raised an indefinite perpendicular, and from the centre l, with the radius l e, an arc is described cutting the perpendicular in the point Jc', the line Jc Jc will be the height of the summit of the pentagon, the base of the pyramid; and the line l Jc will be the vertical projection or profile of l e. When l e, in being raised, described the arc e Jc, lg has described the arc g g', and the last point g will, therefore, be the verti¬ cal projection of the centre of the pentagonal base, or the extremity of the perpendicular let fall from the summit of the pyramid on that base. Now, this perpendicular is a portion of the axis of the icosahedron ; therefore, ii through g an indefinite line perpendicular to l Jc be drawn, that line will be the direction of the axis of the solid, and its length has now to be determined to define the summits of the two opposite pyramids. As the pen¬ tagonal pyramid is placed with its summit in the hori¬ zontal plane, its base must be necessarily raised above that plane by the height of the perpendicular let fall from its summit upon that base in g, as g G, and this length, g G, is now to be set off in the direction of the axis from g' to G' ; then drawing G' l, G Jc, a portion of the pro¬ jection is obtained. From m as a centre, and with m g as a radius, describe a circle, in the circumference of which will be found the projections of the summits of five other pentagonal pyramids, equal and similar to the first. To find these:—Through/and d draw lines parallel to l e, cutting the circumference in n o r q, or rather through the summits of the opposite triangles; and to the centre m draw portions of radii B n, Jc o, kp,iq, C ; and through each of the points draw perpendiculars cutting the axis in p, and the lines A Jc', l Jc, in o' n\ and through these points draw the lines shown in the figure, and the projection i3 complete. Fig. 288 shows this projection freed from the lines of construc¬ tion. Having found the heights of the different points, the STEREOGRAPHY—CYLINDER, CONE, AND SPHERE. 57 projection in Fig. 289, which is on a line parallel to a b, can be easily made. A side or an arris of an icosahedron be¬ ing given, to construct the 'projections of the solid, so that one of its axes may be perpendi¬ cular to the horizon¬ tal plane. As in the preceding example of the penta¬ gonal pyramid abode (Fig. 290), placed on its summit F on the horizontal plane, let the side be considered as the given side of the solid. Observe that the superior part of the solid is also a pyra¬ mid ghikl, equal and similar to the first, but having its angles in the centres of the sides thereof. Thus, there will be for the horizontal projections of these pyra¬ mids two pentagons, the arrises of which will be repre¬ sented by lines drawn from each angle to the centre. Observe further, that the two pyramids are separated by ten triangles, which have a certain inclination to each other, and alternately in an inverse order; and that the height of these triangles added to the height of the two pyramids is the length of the axis. As all the triangles have their sides common, the side a b may be considered as the side or base of the triangle a H b, laid on the hori¬ zontal plane. Conceive this triangle raised by turning it round a b until its summit meet h, one of the angles of the base of the superior pyramid. To find its inclination, from H let fall on the base the perpendicular Hro, and draw through h the line h H', perpendicular to H A; then from m as a centre, with the radius m H, describe an arc cutting h H' in n', and join m H' : the line m H' will then be the profile or inclination of the face of the triangle a b H, ac¬ cording to the angle n m h' ; and the projection of the summit is h. By drawing the lines h a, h b, the projection of the triangle a H 6, inclined to the horizontal plane in a h b, is obtained, and the line h h' will be the length of the portion of the axis comprised between the bases of the two pyramids. To find the height of the base of the lower pyramid above the horizontal plane:—As all its angles or points are equally elevated, any of them maybe taken indif¬ ferently, as a. Whatever be the height of this point, such height will always be the side of a right-angled triangle, of which F a is the other side; and the arris, of which F a is the projection, is the hypothenuse. Consequently, if from a is raised an indefinite line a A, perpendicular to F a, and upon it is set off the length of any of the arrises, such as a b, from F to A, the height a A will be the height sought. Thus, having obtained the data for construc¬ ting the vertical projec¬ tion, it may be proceed- Fi s- 29L ed with as follows:— Through / (Fig. 291) draw the line / d g’ n corresponding to the axis, and on it set up the heights fc equal to a A, d g equal to h h', and g n equal to a a. Through d and g' draw lines parallel to the horizontal plane, and on these find the points d', d, a, b', and l, Jc, l', K, by drawing perpendiculars from the points in the horizontal projection, and join these by lines, as in the figure. To inscribe these five solids in the same sphere, pro¬ ceed as follows:—Let A B (Fig. 292) be the diameter of the given sphere: divide it in three equal parts, and make D B equal to one of them: draw D E perpen¬ dicular to A B, and draw the chords A E, E B. A e is the arris of a tetrahedron, and E B the arris of a hexa¬ hedron or cube. From the centre c draw the perpen¬ dicular radius C F, and the chord F B is the arris of the octahedron. Divide B E in extreme and mean proportion in G, and B G is the arris of the dodecahedron. Lastly, make the tangent A n equal to A B, draw c n, and the chord A I is the arris of the icosahedron. THE THREE CURVED BODIES—THE CYLIN¬ DER, THE CONE, THE SPHERE. 1 . —The horizontal projection of a cylinder, the axis of which is perpendicular to the horizontal plane, being given, to find the vertical pro¬ jection. Let the circle abcd (Fig. 293), be the base of the cylin¬ der, and also its horizontal pro¬ jection : from the points A and c raise perpendiculars to the hori¬ zontal plane a c, and produce them to the height of the cylinder —say, for example, a e, cf: draw ef and the rectangle a efc is the projection required. 2. — The horizontal projection of a cylinder, whose axis is par¬ allel to the horizontal plane, being given, to construct its vertical projection. Let the rectangle a c (Fig. 294) be the given projection. From each of the points a, b, c, d, draw perpendiculars to H PRACTICAL CARPENTRY AND JOINERY. x y: on x y set off the height, which of course equals the diameter; and through the points obtained draw a line parallel to x y. Conceive a b, the pro¬ jection of one of the oases of the cylinder, to be turned down on the horizontal plane on the point E, and to be a circle eafb; then the original of the point, of which a is the projection, will be A, which will be elevated above the horizontal plane the height a A. To obtain the vertical projec¬ tion of a or A, therefore, it is H only necessary to carry from x to a', on the perpendicular passing through a, the height a A. In the same way is found the vertical projection of any point in the base, as g : from g draw perpendicular to a b a line cutting the circle in G and H. Draw also from g to the vertical projection the line g g ti, and set on it the height g G, g H, in g' and h', which are the vertical pro¬ jections of g. Thus any number of points may be found, and a curve traced through them. It is evident that as the base cl c is similar and equal to a b, its projec¬ tions will also be similar and equal. These circular bases being in planes which are not pa¬ rallel to the vertical plane, their projections are ellipses, the two axes of which can always be readily found, and the operation of projecting them may thus be shortened. In surfaces of revolution, any point on the surface be¬ longs equally to the generating line, and to the gene¬ rating circle; consequently, if it be required to find the projection of a point on the surface of a cylinder, it is only necessary to draw a line through the point parallel to the sides of the cylinder, cutting the line of projection of one or both of its bases; to draw, from these intersec¬ tions, lines cutting the ellipses in the vertical projection; and from these the projection of the line passing through the point, and consequently the projection of the point itself, is easily found. Let i be the point; through it draw k i l, cutting the base or horizontal projection of the generating circle in k and l; through k draw k 7c', and through l draw l V, and join k' l', and the intersection of the lines k' V, i i", in the vertical projection, defines the point. But it can also be found without referring to the intersections, thus:—Through the point i, draw i k ] j, parallel to b c, and through it draw also i i' perpen¬ dicular to xy; then on the last line set up the height k i, which will give the plane of the point in i" i. The horizontal •projection of the base of a cylinder being given , and also the angle which the base makes with the horizontal plane, to construct the projections of the cylinder. Let the circle agbh (Fig. 295) be the given base, and PROJECTIONS OF THE CONE. A point in one of the projections of a cone being given, to find it in the other projection. let the given angle be 45°. Draw the line AB', making with A B the given angle; and from A as a centre, with A B and a c as radii, describe arcs cutting A b' in b' and c'. Then draw a d, b' e, perpendicular to A b', and equal to the length of the cylinder; and the rectangle A E is the profile of the cy¬ linder inclined to the horizontal plane in an angle of 45°. Now pro¬ long indefinitely the diameter b a, and this line will represent the projection on the verti¬ cal plane of the line in which the generating circle moves, to produce the cylinder. If from B' and G perpendiculars be let fall on A B, k will be the horizontal pro¬ jection of b' a k of the diameter a b, and c of the centre C. Through c draw h g perpendicu¬ lar to A B, and make c h, c g. equal to C H, c G; and the two diameters of the ellipse, which is the projection of the base of the cylinder, will be obtained. In like manner, draw from dfe the lines D d, f/, e e, perpendicular to the diameter A B produced, and their intersections with the diameter and the sides of the cylinder will give the means of drawing the ellipse which forms the projection of the further end of the cylinder. The ellipses may also be found by taking any number of points in the generating circle as I J 0 m, and obtaining their projections i, j, r, q. The method of doing this will be seen by the figure without further explanation. Of the Sections of the Cylinder by a Plane. A cylinder may be cut by a plane in three different ways—1st, the plane may be parallel to the axis—2d, it may be parallel to the base—3d, it may be oblique to the axis or the base. In the first case, the section is a parallelogram, whose length will be equal to the length of the cylinder, and whose width will be equal to the chord of the circle of the base in the line of section. Whence it follows, that the largest section of this kind will be that made by a plane passing through the axis; and the smallest will be when the section plane is a tangent—the section in that case will be a straight line. When the section plane is parallel to the base, the sec¬ tion will be a circle equal to the base. When the section plane is oblique to the axis or the base, the section will be an ellipse. As the manner of constructing the ellipse produced by the section of the cylinder has been already treated of, and it will again come under consideration when treating of the sections of solids, it is not necessary here to dilate further on the subject. STEREOGRAPHY-CYLINDER, CONE, AND SPHERE. 50 Let a (Fig. 296) be the given point. This point belongs equally to the circle which is the section of the cone by a plane parallel to the base, and to a straight line form- ing one of the sides of a triangle which is Fig. 295 . / the section of the cone / by a plane perpendicu- / lar to its base and pass- / ing through its vertex, / and of which fag is ,/ \ the horizontal, and / /' a'g' the vertical pro- / jection. To find the _ / \ \ vertical projection ofu, 17 3' therefore, when th e ho- / d rizontal projection is / given, through a draw / / \ a a perpendicular to / J e | b c, and its intersec- 1 1 f tion with /' g' is the \ \ point required; and \ reciprocally, a in the \ — y.<7 horizontal projection may be found from a in the vertical projection, in the same manner. Otherwise, through a, in the horizontal projection, de¬ scribe the circle a d c, and draw e e' or c c', cutting the sides of the cone in e 1 and c ; draw c e’ parallel to the base, and draw a a', cutting it in a, the point required. Of the Sections of a Gone by a Plane. A cone may be cut by a plane in five different wavs, producing what are called the conic sections:—1st, If it is cut by a plane passing through its axis, the section is a triangle, having the axis of the cone as its height, the diameter of the base for its base, and the sides for its sides. If the plane passes through the vertex, without passing through the axis, as c e" (Fig. 297), the section will still be a triangle, having for its base the chord c’ o, for its height the line E e", and for its sides the sides of the cone, of which the lines c' e, o e, are the horizontal, and the line c e" the vertical projections. 2d. If the cone is cut parallel to the base, as in g’ ti, the sec¬ tion will be a circle, of which g’ ti will be the diameter. 3d. When the section plane is oblique to the axis, and passes through the opposite sides of the cone, as m' p ti, the section will be an ellipse, m n h. 4tli. When the plane is parallel to one of the sides of the cone, as r ti, the resulting section is a parabola r s htu, which may be considered as an ellipse, infinitely elongated. 5th. When the section plane is such as to pass through the sides of another cone formed by producing the sides of the first, the resulting curve in each cone is a hyperbola. Several methods of drawing the curves of the conic sections have already been given in the section on prac¬ tical geometry. Here their projections, as resulting from the sections of the solid by planes, are to be considered. If the mode of finding the projections of a point on the surface of a given cone be understood, the projections of the curves of the conic sections will offer no difficulty. Let the problem be:—First, to find the projections of the section made by the plane m' ti. Take at pleasure upon the plane several points, as p', &c. Let fall from these points perpendiculars to the horizontal plane, and on these will be found the horizontal projection of the points: thus, in regard to the point p’ —Draw through p' a line parallel to AB: this line will be the vertical projection of a hori¬ zontal plane cutting the cone, and its horizontal projec¬ tion will be a circle, with s' n' for its radius. With this radius, therefore, and from the centre c, describe a circle cutting, twice, the perpendicular let fill from p, which will be the projections sought of certain points in the circum¬ ference of the ellipse. In the same manner, any other points may be obtained in its circumference. The opera¬ tion may often be abridged by taking the point p in the middle of the line rti ti ; for then mh will be the major, and n c the minor axis of the ellipse. To obtain the projections of the parabola, more points are required, such as r', 2, s', 3, h! The projections of the section plane which produces the hyperbola, are straight lines, q ti, z h. The development of these curves, that is, their projec¬ tions on planes parallel to the section planes which produce them, may be here illustrated. First, The ellipse. The development of this curve is found by making its major axis equal to A G, and its minor axis g d, equal to Jc l, as has been explained. Second, The parabola (Fig. 297 a). Draw the line u u' parallel to e f, and the line n ti" perpendicular to it and bisecting it. From the horizontal projection take the length g u, and carry it from n to u and u'. Take also g x, and carry it from n to 2; and in the same way transfer the lengths g c, g v, &c., to n t, n 3, &c., and through each of these points draw perpendiculars to E f, and set up on them from the line u n the heights of the corresponding points 2 s’ 3, from the line rti b of the vertical projection: the points through which to trace the curve will thus be obtained. Third, the hyperbola. Draw the line c c (Fig. 299) per- 60 PRACTICAL CARPENTRY AND JOINERY. pendicular to d d', and make d d' equal to the base, and c c equal to the height of the cone. From c as a centre, with the radius c d, describe a semicircle equal to half the base of the cone, and draw r q' the section plane, at the distance from the centre of e q, or c h, in Fig. 2.97. Divide the line r q into any number of equal parts in 1, 2, 3, h, &c., and through them draw lines perpendicular to d d. From c as a centre, with the radii cl, c 2, &c., describe arcs cutting d d' ; and from the points of intersection draw per¬ pendiculars cutting the sides of the cone in 1, 2, 3; and these heights trans¬ ferred to the cor¬ responding perpen¬ diculars drawn di¬ rectly from the points 1, 2, 3, &c., in r q, will give points in the curve. Understanding clearly the principles of construction here developed, no diffi¬ culty will be ex¬ perienced in appre¬ hending the methods of construction employed in de¬ veloping these curves under the head of Sections of Solids. Of the Section of the Sphere by a Plane. A point in one of the projections of the sphere being given , to find it in the other projection. Let a be the given point in the horizontal projection of the sphere h xiv. Any point whatsoever on the surface of a sphere be¬ longs to a circle of that sphere : therefore, if a (Fig. 300) be the point, and a plane b c is made to pass through that point parallel to A b, the section of the sphere by this plane will be a circle, whose diameter will be b c } and the radius, con¬ sequently, d b or d c; and the point a will necessarily be in the circumference of this circle. Since the cen¬ tre d of this circle is situ- a.- ated on the horizontal axis of the sphere, d will also be the centre of the sphere; and as this axis is perpen- h dicular to the vertical plane, its vertical projection will be the point d'. It is evi¬ dent that the vertical pro¬ jection of the given point a, will be found in the cir¬ cumference of the circle described from cl' with the radius, and at that point of it where it is intersected by the line drawn through a, perpendicular to A b. Its vertical projec¬ tion will therefore be either a' or a", according as the point a is on the superior or inferior semi-surface of the sphere. The projection of the point may also be found by an inverse operation, thus:—Conceive the sphere cut by a plane parallel to the horizontal plane of projection pass¬ ing through the given point a. The resulting section will be the horizontal circle described from Jc, with the radius 1c a ; and the vertical projection of this section will be the straight line g' e, or cj e ; and the intersections of these lines with the perpendicular drawn through a, will be the projection of a, as before. The traces of a plane cutting a sphere being given , to find the projections of the action. Let a b (Fig. 301) be the horizontal trace of the section plane. On the line of section take any num¬ ber of points, as a, c, b, and through each of them draw a line per- pendicular to y x. As the point a is situated on the circumference of the great circle of the sphere, its vertical pro¬ jection will be on the vertical projection of the circle at a!. The point b being the extremity of the axis of the sphere, will have its vertical projection b' in the pro¬ jection of the great cir¬ cle e f. The projection of c, or of any other point in the line a b, is found in either of the ways detailed in the preceding problem. Practically, in this case, it is found thus:—Through c draw g c h parallel to y x, and also c c' perpendicular to y x; then with the radius i g, or i h, and from the cen¬ tre l) on the vertical projection, describe arcs of a circle, cutting the perpendicular c c" d in c and c". Then c being in this case the middle of the line of section a b, the vertical projection will be an ellipse, whose major axis will be c c", and minor axis a b'. TANGENT PLANES TO CURVED SURFACES. Tangent Plane to a Cylinder. Let the lines a b, c d (Fig. 302), be tangents to the generating circles e f g h, i lcl m, of the cylinder g l. As the circles are parallel, the tangents will also be paral¬ lel and perpendicular to the radii n i, o e. If through these two tangents a plane, a d, pass, it will be perpen¬ dicular to the rectangle i o ; and on the generating line e i will, consequently, be found the tangent points of all circles which can be con¬ ceived to be drawn between the base and summit of the cylinder. And as, in the formation of the cylinder by the generating circle, the radius n i has been supposed W* 2'J9. STEREOGRAPHY—TANGENT PLANES TO CURVED SURFACES. 61 to pass through all the points of the line e i, the plane a d will contain all the tangents of all the circles sup- posable in the cylinder, such, for example, as the circle p q r s, the tangent of which is t u : the plane a d is therefore a tangent to the cylinder in the right line e i. The axis of the cylinder, o n, is named the directrix; because, in conceiving the cylinder formed by the motion of the generating circle, the centre of the circle will move in the direction on. In considering the circle as formed by the rotation of the rectangle o i round its side o n, it is seen that the generatrix e i is necessarily parallel to the directrix o n, and that i is the point of contact of the generatrix i e and the generating circle i k l m, and through this the tangent passes. Consequently, if through any tangent point on the circum¬ ference of the generating circle of a cylinder, a line is drawn parallel to the directrix, it will be the line in which a tangent plane will touch the cylinder. Tangent Plane to a Gone. The cone differs from the cylinder in that the genera¬ trix a b (Fig. 303) is not paral¬ lel to the directrix c b, and that it passes always through the summit b. Tangent Plane to a Sphere. Let a b (Fig. 304) be a plane perpendicular to the ex¬ tremity of the radius c e. If through any point / in the plane there be drawn the right lines f c, f e, there will be formed the right-angled triangle c ef. As the side c e of the tri¬ angle c e f is equal to the radius of the sphere, the point / will be a point in space without the sphere; and it will obviously be the same in regard to any point taken in a b, except alone the point e. Therefore, every plane perpendicular to a radius of a sphere, and at its extremity, will be a tangent to the sphere. It may also be thus demonstrated:—If the line ef (Fig. 304) is perpendicular to the radius of the circle d i ek, it will be a tangent to that circle; and if the line e n is perpendicular also to the radius of the circle d m el, it will be a tangent to that circle (the radius c e being common to both circles). Now, two straight lines which intersect each other are in the same plane, since the three points nef are not in the same straight line; con¬ sequently, if a plane pass through these lines it will be per¬ pendicular to e c, and tangential to the two circles, which are both generating circles of the same sphere. Hence, if through the point of contact of two generating circles of a sphere two tangents be drawn, these tangents will deter¬ mine the tangent plane of the sphere at that point. Having thus illustrated, generally, the subject of tan¬ gent planes to curved surfaces, it will now be proper to show the practical application of the principles. Through a given point in the circumference of the base of a right cylinder , to draw a tangent plane. Let e (Fig. 305) be the given point on the horizontal pro¬ jection: draw the ra- Fi 305 _ dius o e, and through its extremity e draw perpendicular to it the line a b, which is a tangent to the circle, and is the trace of the tan¬ gent plane sought. 1 hrough e draw the line ei, perpendicu¬ lar to y z } and it gives on the vertical projection the tangent line e'i of the plane, which is parallel to the directrix o n, as has been seen. Through a given point on the surface of a right cylin¬ der to draw a tangent plane. Let a (Fig. 306) be the horizontal projection of the point: through a draw b c parallel to the axis d e, and b c will be the horizontal projec¬ tion of the tan¬ gent line of the plane. Find, then, the vertical pro¬ jection of a by the rules already given. Let this be a': through a' draw b' c parallel to the axis d' e', and this will be the vertical projec¬ tion of the tan¬ gent line. N ow, to find the tan¬ gent plane, draw, in the horizontal projection, the cir¬ cle of the base of the cylinder f g, produce c b to B, the tangent point of the plane with the base, draw the radius D B, and the tangent to it I H, cutting g f produced in H. Then i H will be the profile of the plane, and i H its horizontal projection; and I h i will be the angle which the plane makes with the horizontal plane. From I draw lie parallel to H j, and the horizontal pro¬ jection of the tangent plane is obtained in H j h i, and from this the vertical projection will be easily con¬ structed. To find the tangent plane when the cylinder is oblique. Let a (Fig. 307) be the given point on the surface of an oblique cylinder with a circular base: on its hori¬ zontal projection draw an indefinite line parallel to the axis b c, cutting the two bases of the cylinder, each in two points d e, f g, which will be the extremities of two tangent lines df eg, one on the upper, and the other on the lower part of the surface of the cylinder. To proceed first with the line of the upper surface, d f :— The vertical projection of d will be d' , and that of /will be/'. Since these two points are the extremities of the PRACTICAL CARPENTRY AND JOINERY. 62 tangent line, draw the line cl’ // which will be the tan¬ gent. Through a draw an indefinite line- perpendicular to h j, cutting d'f in a', the vertical projection of a. To obtain the traces of the tangent plane, through d draw indefinitely the tangent H d b, which will be the hori¬ zontal projection sought; and b' f in the plane of the axis b c, will be the vertical projection of the same plane. If it is desired to limit this plane, take, on the horizontal projection b d produced, any point desired, such as H, whose vertical projection will be h. Through this point draw h % parallel to the vertical rig. 308. projection of the cylinder, and this line will be one of the limits sought. In the same way will be obtained the boundaries b j, b' f , ij, % f, &c. To obtain now the tangent plane to the under part of the surface:—Through e draw d g, and the tangent b e k, and operate as directed for the first plane. In Fig. 308 the same pro¬ blem, but with the cone in place of the cylinder, is shown; and as nearly the same letters are used, no other description is required. Through a given 'point on the surface of a sphere , to draw a tangent plane to that surface. Let a (Fig. 309) be the given point in the horizontal Fig. 309. projection. As in the case of the cylinder, it is to be considered whether the point is on the upper or lower part of the surface. Let it be on the lower surface:— Conceive the point turned round horizontally until it has arrived at a', on the diameter b c, which may be considered as the horizontal projection of a great circle of the sphere, parallel to the vertical plane; consequently, the vertical projection of a will be a. If through this point the tangent d e be drawn, this line will be the vertical projection of a tangent plane to the sphere, per¬ pendicular to the vertical plane; consequently, the line- d f', perpendicular to y z, will be the horizontal projec¬ tion of the same plane, which will be inclined to the horizontal plane in the angle e dz: through a" draw a horizontal line a g , cutting the vertical line drawn upon a in the point a' , which will be the vertical projec¬ tion of a, or the tangent point. Conceive now the point a turned back to its first position, and that the plane e’d F, which contains a' , had turned with it, it is clear that when a< is in a, the point K will be in h, and will not have left the horizontal plane; consequently, the trace d f, which was perpendicular to the radius i h', will be in D f, and will continue perpendicular to i h. There re¬ mains only to find the vertical trace of the plane. Draw the line a a'" parallel to D F: it will be perpendicular to the radius i a, as a' a" was to i a' before being turned. This line a a” being horizontal at the height of the point of the tangent, and situated in the tangent plane, its ex¬ tremity will necessarily be in the vertical projection sought. Raise on a"', therefore, the vertical line, cutting a g in k, the point sought. Now, D being also one of the points of the projection, the line D k produced will be the vertical projection sought. If the plane be limited, its vertical and horizontal projections will be the rhom¬ buses D l, D V. The same end can be arrived at in a way more direct, -simple, and expeditious, but which could not be so easily understood without a knowledge of the previous mode. Let a (Fig. 310) be the point given. Through it, and the centre i, draw the diameter a i , which consider to be the pro¬ jection of a section plane. It this is now laid down in the hori¬ zontal plane, there will be obtained a great circle of the sphere, as the vertical projection of the section through the diameter. Through a draw a line perpendicular to a i, cutting the circle in A: through this point draw the tangent x A h, which will be the profile of the tangent plane, meeting the horizontal plane in h. Through h draw d/if perpendicu¬ lar to a i , and D h F will be the horizontal pro¬ jection of the tangent plane. Through a draw a vertical line, and upon it set oft the height a A from d to a'. The remain¬ der of the operation is the same as in the former example STEREOGRAPHY—INTERSECTIONS OF CURVED SURFACES. 63 The next figure (Fig. 311) shows the process when the plane is a tangent to the upper surface. INTERSECTIONS OF CURVED SURFACES. When two solids having curved surfaces penetrate or intersect each other, the intersections of their surfaces form curved lines of various kinds. Some of these, as the circle, the ellipse, &c., can be contained in a plane; but the others cannot, and are named curves of double curva¬ ture. The solution of the following problems depends chiefly on the knowledge of how to obtain, in the most advantageous manner, the projections of a point on a curved surface; and is in fact the application of the principles elucidated in the preceding problems. The manner of constructing the intersections of these curved surfaces which is the most simple and most general in its application, consists in conceiving the solids to which they belong as cut by planes, according to certain con¬ ditions, more or less dependent on the nature of the sur¬ faces. These section planes may be drawn parallel to one of the planes of projection; and as all the points of inter¬ section of the surfaces are found in the section planes, or on one of their projections, it is always easy to construct the curves by transferring these points to the other pro¬ jection of the planes. The projection of two cylinders which intersect at right angles being given, to find the projections of their in¬ tersections. Concei ve, in the horizontal projection (Fig. 312), a series of vertical planes cutting the cylinders parallel to their axes. The vertical projections of all the sec¬ tions will be so many right- angled parallelograms, si¬ milar to e' f, or e" which is the result of the section of the cylinder by the vertical plane e f for this plane cuts the cylinder from surface to surface. The cir¬ cumference of the second cylinder, whose n.vi a is vertical is also cut by the same plane, which meets its upper sur¬ face at the two points g, h, and its under surface at two corresponding points. The vertical projections of these points are on the lines perpendicular to a b, raised on each of them, so that upon the lines e'/', e f, will be situated the intersections of these lines at the points g’ hj g" h", and the same with the other points i, k,l,m, It is not necessary to draw a plan to find these projections. All that is actually required, is to draw the circle repre¬ senting one of the bases, as n o, of the cylinder laid flat on the horizontal plane. Then to produce g h till it cuts the circle at the superior and inferior points a' g, and to take the heights ec!, e G, and carry them, upon a b, Rom g to g', g", and from h to h', li. Fig. 313 is the projection made on the line xz. To construct the projections of tivo cylinders whose axes intersect each other obliquely. Let A (Fig. 314) be the vertical projection of the two Fig. 314. axes. Conceive, in the vertical projections, the cylinders cut by any number of horizontal planes: the horizontal pro¬ jections of these planes will be rectangles, as in the pre¬ ceding example, and their sides will be parallel to the axes of the cylinders. The points of intersection of these lines will be the points sought. Without any previous operation, six of those points of intersection can be ob¬ tained. For example, the point 6 is situated on d" e", the highest point of the cylinder; consequently, the hori¬ zontal projection of c' is on d e, the horizontal projection of d" e", and it is also on the perpendicular let fall from c', that is to say, on the line cf parallel to the axis of the cylinder s h. The point sought will, therefore, be the intersection of those lines at c. In the same way i is obtained. The point j is on the line k l, which is in the horizontal plane passing through the axis d' e': the hori¬ zontal projections of Id l' are k l, and its opposite m n\ therefore, in letting fall perpendiculars from / p , the in¬ tersections of these with k l, m n, give the points j j', p p'. Thus six points are obtained. Take at pleasure an inter¬ mediate point q , through this point draw a line rs paral¬ lel to a b, which will be the vertical projection of a hori¬ zontal plane cutting the cylinder in q. The horizontal projection of this section will be, as in the preceding ex¬ amples, a rectangle which is obtained by taking, in the vertical projection, the height of the section plane above the axis d' e', and carrying it on the base in the horizontal projection from G to T. Through t is then to be drawn PRACTICAL CARPENTRY AND JOINERY. the line Q U perpendicular to G T; and through Q and U lines parallel to the axis; and the points in which these lines are intersected by the perpendiculars let fall from cf u' are the intermediate points required. Any number of intermediate points may be thus obtained; and the curve being drawn through them, the operation is completed. To find the intersections of a sphere and a cylinder. Draw, in the horizontal projection, and parallel to A B (Fig. 315), as many vertical section Fig. 315 . planes as are considered necessary, as efic d. These planes cut at the same time both the sphere and the cylinder, and the result of each sec¬ tion will be a circle in the case of the sphere, and a rectangle in the case of the cylinder. Through each of the points of intersection g, h, i, Jc, draw indefinite lines perpendicular to A B. Take the radius of the circles of the sphere proper to each of these sec¬ tions, and with them cut the cor¬ respondent perpendiculars in g g', h h\ i i', &c., and draw through these points the curves of intersec¬ tion. This operation should be so obvious from the preceding problems, that it is not neces¬ sary to enter more particularly into the description. To construct the intersection of two right cones with circular bases. The solution of this problem is founded on the know¬ ledge of the means of obtaining on one of the projections of a cone a point given on the other. Let ab (Fig. 316) be the common section of the two planes of projection, the circles g d e f ghik, the hori¬ zontal projections of the given cones, and the triangles d'i'f , Ji lie , their vertical projec¬ tions. Suppose these cones cut by a series of hori¬ zontal planes: each section will consist of two circles, which cut¬ ting each other, and the points of their intersection, will be points of intersection of the conical surfaces. For example, the section made by a plane m'ri will have for its hori¬ zontal projections two circles of different diameters, the radius of the one being i m, and of the other l o. The intersecting points of these are p and q, and these points are common to the two circumferences; and their vertical projection on the plane m' n\ will be in p' q. Thus, as many points may be found as is necessary to complete the curve. But there are certain points of intersection which cannot be rigor¬ ously established by this method without a great deal of manipulation, and it is therefore advisable to point out another method of procedure for such cases. The point r in the figure is one of those; for it will be seen that at that point the two circles must be tangents to each other, and it would be difficult to fix the place of the section plane s' 1 so exactly by trial, that it would just pass through that point. It will be seen that the point r must be situated in the horizontal projection of the line g i, which passes through the summits of both cones. This line g i is the projec¬ tion of a vertical plane, which contains^— 1 st, the side of the large cone, passing through the summit G, and terminating at the base in i\ and, 2 d, the side of the smaller cone passing through the summit L, and also ter¬ minating at the base in i. These two lines must inter¬ sect each other at the surface of the cones, and the point r will be the point of intersection. Hence, to find r:— Through i in the horizontal projection raise on g i a perpendicular equal to the height of the cone g g", and draw G g, which will be the side of that cone. Through l raise a perpendicular, and make it equal to the height of the second cone, and draw its side L i; and from the point of intersection let fall a perpendicular on g i, meeting it in r; and through r draw an indefinite line perpendicular to a B, and set up on it from the horizontal projection the height of the point of intersection r. There is still another method by which the operation is abridged. Find the two points i and r, and consider i r as a diameter: from u as a centre, with the radius u r, describe a circle, the circumference of which will be the horizontal projection of the intersection of the two cones. It now remains to find the vertical projection of this circle, which can be done by the methods pointed out in preceding problems. To construct the intersections of a sphere penetrated by an oblique scalene cone (Fig. 317). This problem i-i g . 317 . is not very different from the preceding one; but yet the method of solution given for that could not be advan¬ tageously ap¬ plied to this, and would be quite inapplica¬ ble if the base of the cone were irregular, a circumstance which proves the necessity of knowing several methods of solution for each case. Conceive the cone in this case cut by a number of ver¬ tical planes, all passing through its summit and its base: the sections made by them will all be triangles, easy to determine; and the sections of the sphere by the same planes will be so many circles, quite as easily constructed. Whence it results, that the operation is resolved into find¬ ing the intersections of a straight line and a circle. Not to overcrowd the figure, the operation is shown only in part. STEREOGRAPHY—INTERSECTIONS OF CURVED SURFACES. G5 Lee cd, c'd' (Fig. 318), be the projections of the line given: this line will be analogous to the side cd of the cone in Fig. 317. Conceive cd to be the horizontal projec¬ tion of a vertical plane cutting the sphere. The section resulting from this plane will be a circle contained iu the plane, and of which the radius will be f h, or f g. If this section be turned down on the horizontal plane, there will result the right-angled triangle dec, whose hypo- thenuse d c will cut the circle of the spherical section in I and in j. From these two points let perpendiculars fall on d c, meeting it in ij, which will be the horizontal pro¬ jections of the points of the entry and exit of the cone into the sphere, and the vertical projections of the same will be i' j\ In repeating this operation for every one of the lines in Fig. 318, points will be obtained through which to draw the curves of intersection; but this may be abridged, as now to be shown. Suppose the plane containing the triangle and the circle which was turned down on the horizontal plane, to be raised up by turning on the point c, in describing the arc d d'\ then Fig. 313. this plane will apply to the vertical plane without any alteration; con¬ sequently, the points 1 ' and J' will be eleva¬ ted above a d to the same ex¬ tent as are the projections ij, and so will likewise be the centres f' e of the two circles. From c, as a centre, with the radius c d, describe the arc dd'; and from the same centre, with the radius c f, describe the arc / f. From the last poilit raise a perpendicular, on which set off the height of the radius of the sphere from f to F'; and from f', as a centre, with the radius F G or F H, describe a circle, or rather an arc, cutting d' c in 1 ' j' : from each of these points draw lines parallel to a b, which will cut d c' in i and j, the points of intersection sought. To construct the intersections of two right cones with circular bases. To commence by a very simple example (Fig. 319). Conceive, in the horizontal projection, a vertical plane cutting both cones through their axes: the sections will be two triangles, having the diameters of the bases of the cones as their bases, and the height of the cones as their height. And as in the example the cones are equal, the triangles will also be equal, as the triangles c e' f, g f d, in the vertical projection. Conceive now this same vertical plane passing through the different points of the base, but still passing through the summits of the cones: the sections which result will still be tri¬ angles (as has already been demonstrated), whose bases diminish in proportion as the plane recedes from the centres of the bases, until at length the plane becomes a tangent to both cones, and the result is a tangent line whose projections are h, g, h, f, g e, f It will be observed that the circumferences of the bases cut each other ao m and i, which are the first points of their inter¬ sections, and whose vertical projections are the point m merely. If the projections of the other points of inter¬ section on the lines of the section planes are found (an operation presenting no difficulty, and easily understood by the inspection of the figure), it will be seen that the horizontal triangles ncm, me o, Jc cp, qcr, &c., have for their vertical projections the triangles n e m, m f 0 , Jc e p, &c., and that the intersections of the cone are in a plane perpendicular to both planes of projection, and its projec¬ tions are the right lines i m, m 3. From the known properties of the conic sections, the curve produced by this plane will be a hyperbola. Fig. 320 is the projection of the cones on the line 0 x. The next example (b ig. 321) differs from the first in the inequality of the size of the cones. Suppose an in definite line c D, to be the horizontal pro- A. jection of the vertical section plane, cutting c the two cones through their axes e f Con¬ ceive in tiiis plane an indefinite line ef D, passing through the sum¬ mits of the cones, the vertical projection of this line will be e f d': from d, let fall on C D a perpendicular meetiug it in D: this will be the point in which the line passing through the summits of the cones will meet the horizontal plane; and it is through this point, and through the summits e and /, that the section plane? should be made to pass, as in the preceding example. The horizontal projections of these planes are od, g d, cd, &c.: 0 D is then the projection of a tangent plane to the two conical surfaces 0 e, p/; and the plane passing through the projection G D, and the line e D, cuts the greater cone, and forms by the section the triangles Gen in the horizontal, and g e li in the vertical projection; and it cuts the lesser cone, and forms the triangles i/j, ifj. In the horizontal projection it is seen that the sides n e, 1 / of the triangle intersect in Jc, which is therefore the horizontal projection of one of the points PRACTICAL CARPENTRY AND JOINERY. GG of intersection; and its vertical projection is k. In the same manner, the other points required may be found. It is seen at once that mn, 11', are points in the inter¬ section, and the curve formed by it is traced through the points M k l N in the horizontal, and m k l in the vertical projection. The next example (Fig. 322) is in some respects ana¬ logous to the first, the two cones being of the same height; but the section plane through their axes is not parallel to the plane of projection, and the right cone is pene¬ trated by a scalene cone. There is, how¬ ever, no difference in the operation, which will be un¬ derstood by inspec¬ tion. Construction of the intersections formed by the 'penetration of a cylinder by a sca¬ lene cone. Conceive a line o c c (Fig. 323) parallel to the generatrix of the cylinder, passing through the sum¬ mit of the cone; then all the vertical planes which, passing through this line and the base of the cone, cut the latter and M the cylinder, follow lines which must be in these planes; for the cone will be cut in triangles, and the cylinder in rect¬ angles; and the intersections of these lines from the cone and cylinder will give points in the line sought. Thus, as in this case, the generatrix of the cylinder is vertical, the line d c c drawn parallel to it through the summit of the cone will also be vertical, and its vertical projection will be c c, and its horizontal projection c. Through this point c, therefore, and through the base of the cone, must be made to pass the vertical planes required. For in stance, the vertical plane d c cuts the cone through its axis, and the section is the triangle eld e; and it cuts the cylinder, and the section is the rectangle ef. These two figures intersect each other in ee', g g ; and these inter¬ sections give eg, two of the points sought. The plane n c gives also a triangle and rectangle, intersecting in h h. The plane M c, which is a tangent to the cone, gives the line m c on the surface of the cone in the vertical projec¬ tion, and the rectangle np in the cylinder, and the points of intersection are n and o. The problem thus presents no difficulty. Example 2, Fig. 324.— Conceive both the cone and cylinder cut by a vertical plane, of which c d is the hori¬ zontal trace. Let this section be turned down on the horizontal plane on c d as a hinge, and the triangle cde is obtained as the section of the cone, and the circle/G ii as the section of the cylinder. In this section the line c D is tangent to the circle in G; and a plane parallel to the axis or generatrix of the cylinder, passing through the summit D or d of the cone, and through the point c, would be a tangent to the surfaces of both the cone and the cylinder. The point g, therefore, which is common to both those surfaces, and whose projections are g li, is one of the points of intersection. The line e d, which is the lower side of the cone, enters the circle of the cylinder at I, and leaves it at H. These two points are also, therefore, points in the intersection, and their projections are i h, % li. It is now necessary to find some other point, and to ob¬ tain it, proceed as follows:—Through j, taken at pleasure on the line c d, draw parallel to the generatrix of the cylinder the line K L, which will be the projection of a plane passing through the summit and base of the cone: the triangle K d L will be the horizontal projection of the section formed by that plane. Since the plane of this triangle is parallel to the axis of the cylinder, the sides K d, L d will enter the cylinder at the same moment— each by its own point of intersection; and these points of intersection—the points at which they enter and leave the cylinder—will be on generatrices of the cylinder o p, qr: it is necessary, therefore, to find those generatrices. Through J, a point taken in the middle of the base of the triangle, draw J D, which will enter the circle at M, and leave it at N: through these points draw the generatrices o p, q r, cutting the sides of the triangle in m m, n n, which are the points sought. STEREOGRAPHY—MANNER OF TAKING DIMENSIONS. 07 OF HELICES. Let ah c cl, &c. (Fig. 325), be any curve whatever, traced on a horizontal plane. (In this example it is a circle.) Take on this curve a series of points abed, &c., and through each of them draw a vertical line. Then conceive a curve cutting all these verticals in the points a' b' c'd', in such a manner that the height of the point above the horizontal plane may be in constant relation to the arcs a b, be, c cl ; for ex- tig. 326 . ample, that a may be the zero of height, that b b' may be 1, c c 2, cl cl 3, &c.; then this curve is named a he¬ lix. To construct this curve, carry on the vertical projec¬ tion on each vertical line such a height as has been deter¬ mined, as 1 on b, 2 on c, 3 on d] and through these points a will pass the curve sought. It is easy to see that the curve so traced is independent of the cylinder on which it has been supposed to be traced; a and that if it be isolated, its horizontal projection will be a circle. The helix is named after the curve which is its horizontal projection : thus the helix in the example, is a helix with a circular base. The vertical line fn is the axis of the helix, and j the height b b', comprised between two consecutive in¬ tersections of the curve with a vertical, is the pitch of the .helix. The points abed, &c., being in the circumference of a circle, are, of course, situate at the same distance from its centre. Conceive now that each of these points ap¬ proaches nearer to the centre in a constant ratio, such, for example, as 1, 2, 3, 4, 5 (Fig. 326). The curve then drawn through these points, when supposed to be in the same plane, is called a spiral. If these points, in addition to approaching the centre in a constant ratio, are supposed also to rise above each other by a constant increase of height, a curve will be obtained, which is also called a spiral. This spiral may be conceived to be traced on the surface of a cone (Fig. 326). It may also be traced on the surface of a sphere (Fig. 327). These figures do not require detailed description. MANNER OF TAKING DIMENSIONS. In taking the dimensions of any triangular figure, make a sketch of it as in Fig. 32S, No. 1, and on each line of the sketch mark the dimensions of the side of the figure it represents. Then, in describing the figure, either to its full dimensions, or to an} 1- proportionate scale, draw any straight line os A B, No. 2, and make it equal to the dimension marked on the corresponding line A B of the sketch No. 1. From the centre A, and with the radius ac, describe an arc at C; then from the centre B, with the radius B c, describe an arc intersecting the former: join A c, B C, and the triangle A C B is the figure required. The dimensions of any figure are taken on the principle above illustrated. If the figure is not triangular, it is divided into triangles, in the manner shown by Fig. 32!), Nos. 1 and 2. In Fig. 330, Nos. 1 and 2, the manner of taking dimen¬ sions, when one or more sides of the figure are bounded No. 1. Fig. 3 (0. by curved lines, is illustrated. When, as at A B (No. 1), the side is a circular arc, its centre is obtained as fol¬ lows:—The extreme points A u, and the point of junction c of the intermediate line E c with A c and b c, give three points in the curve. From A and B, therefore, with any radius, describe arcs above and below the curve; from c, with the same radius, intersect these arcs; through the intersections draw straight lines meeting in D; and D is the centre of the curve, and D A, D B, or u c its radius. 68 PRACTICAL CARPENTRY AND JOINERY. SECTIONS OF SOLIDS. Plate I. Plate 1. Fig. 1 .—To draw the sections of a cone made by a line cutting both its sides. Let A D B be the vertical projection of the cone, A c B the horizontal projection of half its base, and E F the line of section. From the points E and F, let fall on A B the perpendiculars eg, fh; and on g h describe the semicircle G 4 h, which is the horizontal projection of half of the section. To find the vertical section—Divide the semi¬ circle G 4 n into any number of equal parts, 1 2 3 4, Szc.] and through these divisions draw lines 1 5 K, 2 6 l, 3 7 m, 4 8 n, perpendicular to the line AB, and meeting the section line E F in the points k l m, Szc. Through him, Szc., draw k t, l u, m v, n w, perpendicular to ef, and make them respectively equal to the corresponding ordinates, 51, 6 2, 7 3, &c.. of the semicircle G 4 H, and points will be obtained through which the ellipse E w F may be traced. It is obvious that, practically, it is necessary only to find the minor axis of the ellipse, the major axis E F being given. If through the points E him n, Szc., lines be drawn parallel to A B, Szc., meeting the side of the cone, as in o p qr s, and from these perpendiculars be let fall on A B, in xyz a b, then arcs described from the centre of the base of the cone I, with the radii I G, I 1, I 2, will meet these perpendiculars. This is applied in the two following figures, to finding the projections of other sections of the cone. Figs. 2 and 3 .—To draw the sections of a cone made by a line parallel to one of its sides. Let A D B be the vertical projection of a right cone, and A c B half the plan of its base; and let E f be the line of section. In ef take any number of points, E a b c d e F, and through them draw lines eh, a 6 1, 5 7 2, &c., per¬ pendicular to A B. Through abode, draw also lines parallel to A B, meeting the side of the cone in fghkl: from these let fall perpendiculars on A B, meeting it in m n o p q. From the centre of the base I, with the radii I to, I n, I o, &c., describe arcs cutting the perpendi¬ culars let fall from the section line in the points 1 2 3 4 5; and through the points of intersection trace the line h 1 2 3 4 5 G, which is the horizontal projection of the section. To find the vertical section—On Y.ab cde, raise perpen¬ diculars to E F, and make them respectively equal to the ordinates in the horizontal projection, as E r equal to E H, a s equal to 6 1, Szc., and the points rstuvw in the curve will be obtained. Fig. 4, Nos. 1-4 .—To draw the section of a cuneoid made by a line cutting both its sides. A cuneoid is a solid ending in a straight line, in which, if any point be taken, a perpendicular from that point may be made to coincide with the surface. The end of the cuneoid may be of any form; but in architecture it is usually semicircular or semi-elliptical, and parallel .to the straight line forming the other end. Let A c B (No. 1) be the vertical projection of the cune¬ oid, and a 5 B the plan of its base, and AB (No. 4) the length of the arris at C, and let D E be the line of section. Divide the semicircle of the base into any number of parts 1 2 3 4 5, and through them draw perpendiculars to A B, cutting it in l m n o p, and join cl, cm, c n, Szc., b} r lines cutting the section line in 6 7 8 9, Szc. From these points draw lines perpendicular to D e, and make them equal to the corresponding ordinates of the semicircle, either by transferring the lengths by the compasses, or by proceeding as shown in the figure. The section on the line D K is shown in No. 2, in which ab equals DK; and the divisions efghk in DK, &c., are transferred to the corresponding points on AB; and the ordinates e l, fm, g n, &c., are made equal to the cor¬ responding ordinates l\, m 2, n 3, of the semicircle of the base. In like manner, the section on the line G ri, shown at No. 3, is drawn. Fig. 5.— To describe cl cylindric section through a line given in position. Let A B G F be a section of a right cylinder passing through its axis; and let CD be the line of the required section. On A B describe a semicircle, and in the arc take any number of points, 1 2 3 4 5, from which draw lines perpendicular to A B, cutting it in o p q r s, and pro¬ duced to meet the line of section c D, in the points 6 7 8 9 10, &c. From these points draw the lines 6 t, 7 u, 8 v, 9 w, 10 x, Szc., perpendicular to c D, and make these ordi¬ nates respectively equal to the ordinates o 1, p 2, q 3, r 4, s 5; then through the points c tuv w, Szc., draw the curve, which will be the section required. The heights of the ordinates may be simply transferred by the compass, or thus:—Produce the line of section CD to E, to meet the diameter A B produced: draw E n perpendicular to E D, and E n perpendicular to E B. From the points in the arc 1 2 3 4 5, draw lines 1 h, 2 k, 3 l, 4 to, 5 n, meeting the line E n ; then with the centre E and radii E h, E k, E l, E m, E n, describe the arcs h h, k k, Szc., and from the points h k l m n, where these arcs meet the line E n, draw the lines n x, m a, l b, k c, h d, cutting the ordinates 6 7 8 9 10, &c., in the points t u v w x a b c cl, through which draw the curve of the required section. Fig. 6. —To describe the cylindric section made by a curved line cutting the cylinder. Let abd E be the section of the cylinder, and CD the line of the section required. On A B describe a semi¬ circle, and divide it into any number of parts as be¬ fore. From the points of division draw ordinates 1 h, 2 k, 3 l, 4 m, Szc., and produce them to meet the line of the section in o p qr s tuv w. Bend a rule or slip of paper to the line c D, and prick off on it the points c opq, Szc.] then draw any straight line F G, and unbending the rule, transfer the points c o p q, Szc., to F abed, Szc. Draw the ordinates a 1, 5 2, c 3, and make them respec¬ tively equal to the ordinates h 1, k 2, l 3, &c., and through the points found trace the curve. ‘Fig. 7.—To describe the section of a sphere. Let A B D C be the great circle of a sphere, and fg the line of the section required. Then, since, as we have seen, all the sections of a globe or sphere are circles, on F G describe a semicircle F 4 G, which will be the section re¬ quired. Or, in F G take any number of points in ml k H, and from the centre of the great circle E, describe the arcs H n, k o, l p , m q, and draw the ordinates H 4, k 3, 12, ml. and n 4, o 3, p 2, q 1 ; then make the ordinates on FG equal to those on B c, and the points so obtained will give the section required. Fig. 8 .—To describe the section of an ellipsoid, when SfECTOOMS ®C f § x with the second hypothenuse / f c- 2 (Fig. 342), as radius; anu L \ proceed thus till the whole ' -r^*\ is completed. ; ——- >1 The lines of the ellipse, pa- f-' —l-■] rabola, and hyperbola are found in the development by first obtaining them on the lines of the triangles, in Fig. 342, and then transferring the lengths to the development. perbola, and by the aid of the intersections, we obtain the horizontal projection of these, figures in No. 2—the para¬ bola passing through M E' F, the hyperbola through Q" L, and the ellipse being represented by the circle d' l\ To obtain points in the circumference of the ellipse upon the development, through the points of intersection o p q r s, draw lines parallel to K H, carrying the heights to the side of the cone g' h, in the points 1 2 3 4 5 6 7, and transfer the lengths g' 1, G' 2, g' 3, Ac., to G 1, o 2, g 3, G 4, Ac., on the radii of the development in No. 3; and through the points thus obtained draw the curve z D I X, To obtain the parabola and hyperbola, proceed in the same manner, by drawing parallels to the base K H, through the points of intersection; and transferring the lengths thus obtained on the sides of the cone G' K, G' ii, to the radii in the development. The projections in Nos. 4 and 5 do not require explanation. Development of the Oblique Gone. In the oblique cone, the position of the summit in the horizontal projection not being coincident with the centre of the circle forming the base, the lines drawn from it to the divisions in the circumference are not radii, and are of unequal lengths. To obtain, therefore, the proper points of intersection, it is necessaiy to construct a right-angled triangle on each of these lines as a base; then the vertical height of the cone is the other side of the right angle, and its hypothenuse is the side of the cone corresponding to the division. Thus, in Fig. 342 the bases of the right-angled triangles c A, c 1, c 2, Ac., are equal to the lines c A, c 1, c2, Ac., in the horizontal projec¬ tion (Fig. 343); the height of all the triangles is equal to the vertical height of the cone cc’; and the hypothenuse of each triangle is thus easily obtained. Development of Solids whose Surface is of Double Curvature. The development of the sphere, and of other surfaces of double curvature, is impossible- except on mppos] ©®V!£[KO Rl©$ ©IF S®1D ®8„ PLATE H. ■f HC..Lowry fc LA.C63E & SOU. GLASGOW.EDrNBE’B.GH S: LOUDON STEREOGRAPHY—COVERINGS OF SOLIDS. 73 of their being composed of a great number of small faces, either plane, or of a simple curvature, as the cylinder and the cone. Thus a sphere or spheroid may be considered as a polyhedron, termi¬ nated, 1st, by a great number ot plane faces, formed by truncated P} 7 ramids, of which the base is a polygon, as in Fig. 34-5 ; 2d, by parts of truncated cones form¬ ing zones, as in Fig. 346; 3d, by parts of cylinders cut in gores, forming flat sides, which diminish in width, as in Fig. 347. In reducing the sphere, or spheroid, to a polyhedron with flat sides, two me¬ thods may be adopted, which differ only in the manner of arranging the developed faces. The most simple me¬ thod is by parallel circles, and others perpendicular to them, which cut them in two opposite points, as in the lines on a terres¬ trial globe. If we suppose that these divisions, in place of being circles, are polygons of the same number of sides, there will result a polyhedron, like that represented in Fig. 345, of which the half, A D B, shows the geometrical elevation, and the other half, A E B, the plan. To find the development, first obtain the summits P, q, r, s, of the truncated pyramids, which form the demi- polyhedron adb, by producing the sides a 1, 12, 23, until they meet the axis E D produced; then from the points P, q, r, and with the radii P A, P1, q 1, q 2, r 2, r 3, and s 3, s 4, describe the indefinite arcs A B, 1 b, 1 b', 2/, 2 /', 3 g, 3 g, 4 h, and from D 4 li, upon which set off the divisions of the demi-polygons A E B, and draw the lines to the summits P, q , r. s, and D, from all the points so set out, as A 1 2 3 4 5 B, for each truncated pyramid. These lines will represent for every band or zone the faces of the truncated pyramids of which they constitute a part. The same development can be made by drawing through the centre of each side of the polygon aeb, indefinite perpendiculars, and setting out upon them the heights of the faces in the elevation, 1 2 3 4 D, and through the points thus obtained drawing parallels to the base. On each of these parallels then set out the widths h, i, k, l, cl, of the corresponding faces in the plan, and there will be thus formed trapeziums and triangles, as in the first development, but arranged differently. This method is used in constructing geographical globes, the other is more convenient in finding the stones of a spherical vault. The development of the sphere by reducing it to conical zones (Fig. 346) is accomplished in the same manner as the reduction to truncated pyramids, with this difference, that the development of the arrises, indicated by A 1 2 3 4, are arcs of circles described from the summits of cones, in place of being polygons. The development of the sphere reduced into parts of cylinders, cut in gores, is produced by the second method described, but in place of joining by lines, the points E, h , i, k, l, cl (Fig. 345), we unite them by a curve, as in Fig, 347. This last method is used in tracing the develop¬ ment of caissons in spherical or spheroidal vaults. The application of these principles to the cases of cover¬ ings which occur most frequently in carpentry, is illus¬ trated in Plates II., III., and IV. Plates II., III., IV. To find the covering of a right cylinder. Plate II — Let abcd {Fig. 1) be the seat or generat¬ ing section. On a d describe the semicircle A 5 D, repre¬ senting the vertical section of half the cylinder, and divide its circumference into any number of equal parts, 1 2 3 4 5, &c., and transfer those divisions to the lines A D and B c produced; then the parallelogram D c, G F will be the covering required. To find the edge of the covering when it is oblique in regard to the sides of the cylinder. Let abcd {Fig. 2) be the seat of the generating section, the edge B c being oblique to the sides A B, DC: draw the semicircle A 5 D, and divide it into any number of parts, as before; and through the divisions draw lines at right angles to A D, producing them to meet B c in rstuv, &c. Produce A D, and transfer to it the divisions of the cir¬ cumference, 1 2 3 4 5 6, ©NQ II S o Fiq.Z. PLATE XIV. *?????> } i f J ^ J. ? i£— zpIFeet. J White del': J. W. Lowry J'c ■BLACK,IE &SON, GLASGOW EDINBURGH So LONDON" - N OGH d@ PLATE XI T «3. /V // fri- 1 /■Li l N* 3 . Fig 7 /' f> 7* 7/ STEREOGRAPHY—NICHES. 83 the width of the panel, and b b the width of the field between the panels. From the centre cl, the circles e e, ff are drawn, touching the lines c a, cb, cb, ca, pro¬ duced; and on the line drawn through d, perpendicular to cl c, the centres of the other circles are found. Having determined the commencement of the divisions, as at c, describe from that point, as a centre, the circle a d equal to the circle //, in No. 1; and through C draw a line, c b, to the centre of the arc forming the side of the vault, and cutting the axis in c. Then, through c draw • from the circumference of the circle a d the tangents c a, c cl, and also produce them beyond the centre b; and be¬ tween them, from b, as a centre, describe the little circle gg. Through the point/, where deb cuts the arc of the vault, and which gives the height of the first field, draw gfc\ and from d, as a centre, describe the circle e e, equal to e e in No. 1, and touching the line g f e\ and through its centre draw D b to the centre of the arc of the vault, cut¬ ting the axis in c. Through c draw the tangents ec, e c, and produce them to beyond b; and from b, as a centre, describe the circle hh between them. It will be seen, by inspection of the figure, how the tangents determine the position of the circles D d'd", c c' c" on the perpendicu¬ lars d c "; and how, by their intersections with the profile of the vault, the heights of the divisions are obtained. To determine the heights of the divisions of a spherical vault. Fig. 3, No. 1, is part of the plafond of a spherical vault, one-half showing the divisions, and the other the mode of framing. To find the height of the divisions:—Produce the meridian lines d b, cl b, representing the width of the panels, and draw the circle b cb, touching the generating circle in G, and the lines in b 6; draw also the meridians d c, dc, representing the width of the field between the panels, and describe the circle H. Through the centre of the circles, a, draw a b indefinitely, and perpendicular to d a. Then having fixed the first horizontal division on the profile of the vault, No. 2, draw through it the line E m n, and draw the circle H, touching it in m, and the larger circle G, touching the lesser one in n\ H and G being respectively equal to the lesser and greater circle in No. 1. Then draw the second line Emw tangential to the circle G, and describe the circle h', touching this line in n; and so proceed, drawing the tangents and the circles H and G alternately; and the intersections of the tangents with the profile of the vault determines the heights of the divisions. To determine the horizontal divisions of the radial panels of a cylindrical vault. Trace the angle ACD {Fig. 4) representing one of the vertical divisions of panels, and by means of the arc c el¬ and the arcs intersecting ate, draw d/ bisecting the angle adc: / is the centre of the circle, which gives the size of the first division. Through the points of intersection of the circumference of the circle, with the centre line E c, draw a g parallel to AD; bisect age, as before, to find /, the centre of the second circle, and so on. Or, from c de¬ scribe the arc F B, and draw B /; then describe the arc a b, and draw b f parallel to B/; the centres / being found equally well by either method. To determine the caissons of an ellipsoidal vault. Plate XV. Fig. 1, No. 1, is a portion of the plan, and No. 2 is the profile of an ellipsoidal vault. The circles M M have their centres on the vertical line as before, and their diameters are determined by the angles made be¬ tween the meridians c D, c D, b D, b D, indicating the divi¬ sions of the plafond. The points //,//, on the profile, are determined by the intersections of the tangents to these circles, which are also tangents to the curves G L, H L, K l. The first of these, G L, is determined in the manner we shall presently describe; the third, K L, is such as to coincide with all the tangents that can be drawn from the lower circumference of the larger circle, in all its positions on the vertical line; and the second, H L, coin¬ cides with all the lower tangents of the smaller curve. To describe the curve G L:— On the major axis of the ellipse adc {Fig. 2), let a and b be the foci; divide the profile A B into any number of equal parts in the lines 12 3 4 5, and join a 5, b 5, a 4, b 4, &c.; then bisect the angle a 5 b in c, by the line 5 c h, cutting the minor axis pro¬ duced in h\ and in the same manner bisect all the angles formed by the lines from the foci, meeting in the divisions 1 2 3 4, &c.; and draw lines through d e fg, cutting the minor axis in lc l m n, then draw the curve o h, coinciding with their intersections. In Fig. 3, No. 1 is the plan of a dome, No. 2 its trans¬ verse, and No. 3 its longitudinal vertical section. The posi¬ tion of the circles opo'p' o" p" o'", on the vertical line, are found as in the former case, and the divisions on the trans¬ verse profile of the dome are obtained by the intersection of the tangents. Then, to find the divisions on the longi¬ tudinal section, draw the circumscribing parallelogram rsut, No. 1, and the diagonals R u, ST. Draw the di visions from the profile No. 2 to the transverse axis A c, No. 1, cutting the diagonals in Imnoprst, and through these points draw lines parallel to A C, and the points wherein these intersect the major axis B D, give the divi¬ sions on the longitudinal profile, Fig. 3. NICHES. Plates XVI., XVII., XVIII. Plate XVI. Fig. 1, Nos. 1 and 2. — Spherical niche on a semicircular plan. The construction of this is precisely like that of a sphe¬ rical dome. The ribs stand in planes, which would pass through the axis if produced. They are all of similar curvature. No. 2 shows an elevation of the niche, with the manner of finding the projections of the ribs from the plan. No. 3 shows the bevelling of the back ribs, a b, against the front rib, at fg h on the plan; a b is the bevel of a, and b c of b. Fig. 2 .—A spherical niche on a segmental plan. No. 1 is the plan. The dotted lines at fg, hi show the manner of finding the representation of the ribs in the elevation: a b and c d are the bevels of the back ribs, where they abut on the front rib. In No. 3, the quadrant F G is drawn with the same radius as the plan of the niche D A, and the lengths and bevels of the back ribs are found by taking the distances fa, g b, from the plan, and setting them on the line F H. Fig. 3 .—A niche, the plan of which is a semicircle, and its elevation a circular segment. The plan No. 1, and elevation No. 2, will be understood on inspection. No. 3 shows the manner of drawing the 84 PRACTICAL CARPENTRY AND JOINERY. back ribs. With the radius hf equal to DA, No. 1, or E a , No. 2, describe the segment F G. Draw H G, and make G lc equal to the height of the segmental head of the niche; and draw k F at right angles to G H. Then F c will be the centre back rib i d; and the lengths and bevels of the others will be found in the same manner as before. Fig. 4. —A niche, of which both the plan and eleva¬ tion are segments of a circle. No. 2 is the elevation of the niche, being the segment of a circle whose centre is at E. No. 1, ABC, is the plan, which is a segment of a circle whose centre is D. It may be made of any depth: the manner of finding the ribs is the same. Having drawn on the plan as many ribs as are required, radiating to the centre D, and cutting the plan of the front rib in a, b c, d e; then through the centre D draw the line G H parallel to AC; and from D describe the curves m l, AG, c H, cutting the line G H; and make D f equal to E 0, No. 2. Then from F as a centre, describe the curves l l and G I li, for the depth of the ribs; and this is the true curve for all the back ribs. To find the lengths and bevel of the ribs:—From the centre d describe the quadrant and arcs a fbg, d h, &c., and draw f f g g, h h perpendicular to D H, cutting the curve 11, and the lines of intersection will give the lengths and bevels of the several ribs. Fig. 5.—A niche whose plan is the segment of a circle and its elevation a semi-ellipse. Let D in the plan (No. 1) be the centre of the segment. Through D draw E F parallel to A c, and continue the curve of the segment to E F. Then to find the curve of the back ribs:—From him n, any points in the curve of the front rib (No. 2), let fall perpendiculars to the line A B, cutting it in ab c d. Then from D as a centre, de¬ scribe the curves a e, bf eg, e h, d h, and from the points where they meet the line E F, draw the perpendiculars e lc, f l, g m, h n, li o, and set up on e k the height e k of the elevation, and the corresponding heights on the other ordinates, when E klmno will be points, through which the curve of the back rib may be traced. The manner of finding the lengths and bevels of the ribs is shown on the other side of the figure, and does not require description. To draw the ribs of a niche elliptic in plan and elevation. Plate XVII.— Fig. 1 : Let No. 1 be the plan, and No. 2 the elevation of the niche. The ribs being all portions of ellipses, may be drawn by the trammel d f e g, as shown at No. 4. The rib c, in the elevation, is seen at a D in the plan No. 1. The bevel of the end li i is seen at A a in No. 3, and that of the end ef at b c. It is not neces¬ sary to describe it more minutely. To draw the ribs of a niche elliptic in plan and ele¬ vation, when the ribs are at right angles to the curve at their points of junction. Fig. 2: Let ABC (No. 1) be the plan of the niche, and No. 2 its elevation. Set off the places of the ribs on ABC. From B as a centre, with any sufficient radius, de¬ scribe a circular segment H I; join H s, I s, n t, 1 1. Bisect the angles H s I, H t I, by the lines swE,hF, meeting the centre line bd of the niche produced in E and F. Complete the parallelogram agbd, and draw its dia¬ gonal G d. In B D take any points o p r, and through them draw the lines o l, p m, r n, parallel to D A, and meeting GD. Draw then Ig, mh, ni parallel to G A; and in the parallelograms thus formed, draw the elliptic quadrants shown by the dotted lines, all parallel to the original curve A B. The intersections of these curves with the seats of the ribs will give points on which the heights of the front rib at a 1, b 2, c 3, d 4, ef, are to be set up, as shown in Nos. 3, 4, and 5. To draw the ribs of an octagonal niche. Fig. 3: Let No. 1 be the plan, and No. 2 the elevation of the niche. It is obvious that the curve of the centre rib H g will be the same as that of either half of the front rib AG, FG. In No. 3, therefore, draw abcde, the half plan of the niche, equal to AB C n G, No 1, and make D G E equal to half the front rib. Divide D G into any number of equal parts 1 2 3 4, &c.; and through the points of division draw lines parallel to A G, meeting the seat of the centre of the angle rib c E in iklmn o. On these points raise indefinite perpendiculars, and set up on them the heights a 1 in i 1, b 2 in k 2, and so on. The shaded parts show the bevel at the meeting of the ribs at G in No. 1. To draw the ribs of an irregular octagonal niche. Fig. 4: Let No. 1 be the plan, and No. 2 the elevation of the niche. Draw the outline of the plan of the niche at A B c D E F (No. 3), and draw the centre lines of the seats of the ribs bo,hi,cg,kg,do, eg; draw also glf equal to the half of the front rib, as given in the eleva¬ tion No. 2, and divide it into any number of equal parts 1 2 3 4. Through the points of division draw d 1, c 2, b 3, a 4, perpendicular to G F, and produced to the seat of the first angle rib G E. Through the points of inter¬ section draw lines parallel to the side ed of the niche, meeting the second angle rib D G; through the points of intersection, again, draw parallels to D c, and so on. The curve of the centre rib is found by setting up from nop q G the heights d 1, c 2, &c., of G F, on the parallel lines which are perpendicular to K G. The curve of the rib B G or e G is found by drawing through the points of intersection of the parallels, perpendiculars to the seat of the rib, and setting up on them, as at h m r I G, the heights d 1, c 2, &c. No. 4 shows the rib C G, and No. 5 the intermediate rib H I. The plan of a semicircular niche in a concave cir- cidar wall being given, to find the ribs. Plate XVIII.— Fig. I: Let abc (No. 1) be the plan of the niche, and Ap c the line of the wall. Join AC, and bisect it at D; and draw the plan of the ribs, l s, n u, p h, &c., and their elevation, as in No. 2, finding their inter¬ sections on the plan at r s, t u, h, Im, no, p. The ribs being in this case segments of a sphere, will all have the same curvature; and their lengths will be ob¬ tained by describing the quadrant ABC (No. 3), in which the radius c A is equal to A d, No. 1 ; and their lengths and bevels at their intersection with the front rib l m, no,p (No. 1), or efk, No. 2, will be obtained by transfer¬ ring the lengths r l, sm,t n, u o, &c., from the plan No. 1 to the points abcde, &c., on the line AC, No. 3, and draw¬ ing the perpendiculars a f, b g, c lc, d i, e k. The back rib v rv x, No. 1, A B c, No. 2, and D, No. 3, is a circular segment, its outer edge described from the centre D, No. 1, with the radius D v, or with the radius CD, No. 3, and the curve of its inner edge with the radius D y, No. 1. The front rib standing over Ap C in the plan is a semi- ellipse, found as shown in Nos. 4, 5, 6. In No. 4 make Fi't. 1. N 0' (D ffl H a < PLATE xm. BLACKIE & SON; GLASGOW, EDINBURGH Sr LONDON. PLATE XVm. Fuj /. Titj.l. M° 4. Fitt 2 Fu/.l..!V? (>. w n © h 11 s. F//.2.JF r //« : / a 1 STEREOGRAPHY—FORMS OF ROOFS. S5 adc equal to A D c, No. 1; describe the semicircle ABC, draw deb perpendicular to A c, and describe the curve of the wall A e c. The figure is thus a plan of the niche ; and in like manner, in No. 5, A B c is an elevation of half the niche on the line A c of the plan, or it is a section on line bd of the plan. Divide the curve EAEC, No. 4, into a number of equal parts 12 3 4 5, and draw 1 a, 2 b, 3 c, 4 cZ, 5 e, perpendicular to A C, and transfer the lengths E1, 1 2, 2 3, &c., from D towards A and c, on the line A D c, No. 6, and draw 1 a, 2 b, 3 c, &c., perpendicular to A c. To find the heights 1 a, 2 b, &c., transfer the divisions abode of the line D c, No. 4, to g e d c b on the line A c, No. 5, and make C/ equal to D p, No. 4. Then draw the perpendiculars g o e m, &c., and transfer the heights to the corresponding or¬ dinates in No. 6, as go to DB, f n to D i, e m to 1 a, dl to 2 b, &c.; and to complete the curve more exactly, divide the last space into two in the point g, No. 4; and draw#/, and transfer the points, in the same manner, to Nos. 5 and 6, for the ordinates ah, gf. To find the mould for the front rib:—On the line A B, No. 7, make the divisions A opqrstn respectively equal to those of the curve abode f c, No. 6; draw A g, o h, p i, q k, r l, s to, t n, perpendicular to A B, and make them equal respectively to D E, a 1, b 2, c 3, d 4, e 5, fg, No. 4; and through ghilclm n B, draw the curve g B, which is the edge of the mould of the front rib. The plan of a semicircidar niche in a convex circu¬ lar wall being given, to find the ribs. Fig. 2 : Let ebf (No. 1) be the plan of the niche, and E D f the curve of the wall. Draw the ribs i h, l to, n, &c., as in the last figure, and draw A c perpendicular to D B. In No. 2 the lines of the elevation are found in the man¬ ner indicated by the dotted lines. In No. 3 the lengths and bevel of the ribs are found as in the last problem; and Nos. 4, 5, 6, and 7 show the manner of describing the front rib and its mould, which must be so easily understood, if the construction of the foregoing figures has been com¬ prehended, as not to require detailed description. ANGLE BRACKETS. Plate XIX. Plate XIX.— Let cab (Fig. 1) be the elevation of the bracket of a cove, to find the angle bracket. First, when it is a mitre bracket in an interior angle, the angle being 45°, divide the curve c B into any number of equal parts 1 2 3 4 5, and draw through the divisions the lines 1 d, 2 e, 3 /, 4#, 5 c, perpendicular to A B, and cutting it in defgc; and produce them to meet the line D E, re¬ presenting the centre of the seat of the angle bracket; and from the points of intersection hi hi c, draw lines h 1, i 2, k 3, l 4, at right angles to D E, and make them equal— li 1 to d 1, i 2 to e 2, &c.; and through F 1 2 3 4 5 draw the curve of the edge of the bracket. The dotted lines on each side of D E on the plan show the thickness of the bracket, and the dotted lines ur,v s,wt, show the manner of finding the bevel of the face. In the same figure is shown the manner of finding the bracket for an obtuse exterior angle. Let DIE be the exterior angle: bisect it by the line I G, which will represent the seat of the centre of the bracket. The lines i H, to 1, n 2, o 3, p 4, c 5, are drawn perpendicular to I G, and their lengths are found as in the former case. To find the angle bracket of a cornice for interior and exterior, otherwise re entrant and salient, angles. Let AAA (Fig. 2) be the elevation of the cornice bracket, E B the seat of the mitre bracket of the interior angle, and H G that of the mitre bracket of the exterior angle. From the points Akab cd A, or wherever a change in the form of the contour of the bracket occurs, draw lines perpendi¬ cular to A i or D c, cutting A i in efg h i, and cutting the line E B in E l m n o b. Draw the lines EG, GL and B n, H K, representing the plan of the bracketing, and the parallel lines from the intersections l m n o, as shown dotted in the engraving; then make b f and rr I each equal to i A, ou to h d, nt to g c, ms to/ b, lr to e a, Ip to ek, and join the points so found to give the contour of the brackets required. The bevels of the face are found as shown by the dotted lines xvyw. To find the angle bracket at the meeting of a concave curved wall with a straight wall. Let AD BE (Fig. 3) be the plan of the bracketing on the straight wall, and D M, E G the plan on the circular wall; cab the elevation on the straight wall, and gmh on the circular wall. Divide the curves cb, gh into the same number of equal parts; through the divisions of c b draw the lines CD, 1 dll, 2 ei, &c., perpendicular to AB, and through those of GH draw the parallel lines, part straight and part curved, 1 m h, 2 n i, 3 o k, &c. Then through the intersections hi kl of the straight and curved lines, draw the curve D E, which will give the line from which to measure the ordinates h 1, i 2, lc 3, &c. To find the angle bracket when the wall is a convex curve. Let B E D C be the plan of the bracketing on the straight wall, and begh the plan on the curved wall. From the points Ak ab c d A of the bracket AAA, where its contour changes, draw perpendiculars as before. Draw H G a radius to the curve of the wall H B, and set on it the divi¬ sions on ml, equal and corresponding to hg f e of the elevation AAA; and draw H I, o u, nt, ms, l r, Ip, per¬ pendicular to G H, and make them equal to i A, h d, g c,fb, e a, e k, of the elevation; then join the points by the lines I u, ut, ts,s r, r p,p G, to obtain the contour of the bracket equal and corresponding to A A A. Through the points on ml draw concentric curves, meeting the per¬ pendiculars from the corresponding points of AAA; from the intersections of the straight and curved lines, o n m l, draw the lines B F, ou, nt, ms, l r, perpendicular to e b, and make them equal to the corresponding lines of the elevation, as before; then join the points ’$utsrp'E,to obtain the contour of the angle bracket. The examples in Figs. 5 and 6 do not require further elucidation. FORMS OF ROOFS. The most simple form of' a building is one erected on a rectangular plan, with two long and two short sides. Such a building is roofed generally, either with a roof of a single slope, called a shed roof, as in Fig. 351— the wall of one of the long sides of the building being carried so much higher than the wall parallel with it, as to give the 80 PRACTICAL CARPENTRY AND JOINERY. required slope to the roof—or with a roof of double slope, as in Fig. 352. In the latter, the planes forming the slopes are equally inclined to the horizon; the meeting of their highest sides makes an arris, which is called the ridge of the roof; and the triangular spaces in the end walls are called gables. When a building is erected on a rectangular plan, of which the four sides are equal, it may be covered with a roof of two slopes. But it may happen that no neces¬ sity may exist for making any of the opposite pairs of sides gables; or there may be reason why all the sides should be gables. In the latter case, two roofs of equal slope intersect each other. This roof, then (Fig. 353), has two ridges a b, c d, and four hollow arrises f e, g e, h e, k e, made by the intersections of the planes of the slopes, and lying over the diagonals of the square. The arrises are form, by their meeting with the former, the arrises or hips a /> c f, be, de, and the form which results at each end n Fi>. 850. b Fin. 857. is called a hip. The roof is called a hipped roof, and the rafters on the lines of the arrises are called hip-rafters. When the end of such a roof is at right angles to its side, as a 6 e (Fig. 357), it is called a right hip; when the angles are unequal, as at d c o, it is an oblique or skewed hip. When the plan of a building is composed of two equal parallelograms crossing each other at right angles, each of the parallelograms may be covered with a roof of two slopes and two gables, as in Fig. 358, or they may each i’ig. 338. Fig. 859. be covered with a hipped roof, as in Fig. 359; in each case forming four valleys at their intersections. When the intersecting parallelograms are unequal in length, the termed valleys or flanks. In the former case, a mode much more simple, and often preferable, because simpler in construction, is to make each of the sides of the roof spring from the sides of the square with an equal slope. The result (Fig. 354) is a pyramid more or less obtuse or acute, and the intersections of the sloping planes form salient angles, or arrises. This kind of roof is called a pavilion roof. If its base be formed by a potygon, of which the sides are equal, the pyramid will be composed of as many triangles as the polygon has sides. The ar¬ rises are also called hips. When the sides of a parallelogram on which a build¬ ing is raised are not very unequal, it Fig. 355 . may be roofed with a pavilion roof, as in Fig. 355; the slopes on cor¬ responding and opposite sides being equal, but those on contiguous sides different. The pavilion roof is applied also to buildings erected on oblong plans. Thus, in the roof (Fig. 356) the sides a b, cd , are truncated at their higher extremities by sides of the same slope rising from the ends ca, db. These fig. 3CO. Fig. 361. shortest may be roofed pyramidally, the slopes of its con¬ tiguous sides being unequal, and the longest with a ridge, as in Fig. 360; or there may be a short ridge common to both, as in Fig. 361. Figs. 362 and 363 are plans of roofs having valleys, hips, and ridges,—the slopes in both being unequal. Fig. 862. Fig. 368. Great edifices are often composed of several masses of building, which form diverse angles with each other; and their extremities may also abut upon streets and roads at various angles. Ordinarily, the different ranges of the building have their crowning cornice on the same level, and their roof of the same height. In Fig. 364 is represented the horizontal projection of the roofs of a building composed of three ranges crossing each other. Their six extremities, square or skewed, are hipped, according to the form of the plan. The lines formed by the intersections of the roofs are valleys. The valleys ca, ce, co, cu are equal, because the build¬ ings A and D are equal in width, and cross each other at STEREOGRAPHY—FORMS OF ROOFS. 87 right angles. The valleys be, b cl, bf, bg are unequal, because the angles formed by the intersecting buildings are not 90°. In this combination of roofs the hips occur together in pairs, and the valleys four and four. In the next figure (Fig. 36-5) the ranges of building inclose a court. Their meeting forms hips on the exterior and valleys in the interior. All the hips divide the angles to which they correspond into two equal parts In the «ame way the valleys divide the interior angle into equal parts; and in general the hips and valleys are equal when they result from the meeting of two ranges of build¬ ings of the same width, with roofs of equal height; but they become irregular when the buildings are of unequal width, though of the same height, or when the opposite sides of the roofs are of different slopes. At the range of building M in the figure, which meets the range A at right angles, the slopes, and consequently the valleys, are un¬ equal, the inequality being proportionate to the deviation from the dotted lines; and the gable is also irregular, as shown by the section N. When the greater width of a building causes its roof to rise higher than the roof which it meets, as the roof of the wide range F meeting D, the connection is completed by extending the slope of D so as to truncate the summit of F, and form the hips b a, c a. When two buildings of unequal width meet each other, and the ridges of the roofs are not kept of equal in the vertical projection a, where the line of the ridge a c of the lower roof meets the slope of the higher roof, to the seat of the ridge * on the p'an below, and joining b cl, b cl for the valleys formed by the intersections. When the slopes are equal, as in the figure, cl b d will be a right angle. In Fig. 367 is shown a pavilion roof, truncated by a plane parallel to its base. Fig. 868, A is the horizontal, and B the vertical projection of a square building, presenting four equal gables and as many equal slopes; but the intersections of the slopes, in place of lying over the diagonals of the square, connect together the summits of the gables. This roof, it If height, as in Fig. 366, the horizontal projection of the lower roof is found by drawing a line a b from the point will be seen by the dotted lines, is the section of a pyrami¬ dal roof cl b c, made by four planes parallel to its diagonal. The combination of hips and gables may be used for figures of any number of sides. If the number of sides is even, the sides may be alternately gabled and horizontal, as in Fig. 369, the plan of which is a hexagon, the result of the trun¬ cation of a triangular pyramid by planes par¬ allel to its opposite sides. Where each side has a gable, as in Fig. 370, the resulting figure is a hexagonal pyramid truncated by six planes, forming gabled sides. 88 PRACTICAL CARPENTRY AND JOINERY. In Eig. 371, A is the horizontal projection of a roof with hips or pavilion ends, truncated; B is the vertical projec¬ tion of the side, and c that of the end. When it is desirable to keep the roofing over a wide building of a rectangular plan low, it may be effected by dividing the span into two, with four principal slopes, two external and two internal. This, which produces a section somewhat resem¬ bling the letter M (Fig. 372), is, from its form, called an IVl-i’oof. At the meeting of the interior slopes is formed a gutter for the water. Fig. 373 is the horizontal projection of such a roof with gables, and Fig. 374 the same with hips. Fig. 373. rig. 374. Fig. 375. ll * In order to avoid the long gutter, another roof is some¬ times introduced, as in Fig. 375, crossing between the two ridges at right angles, and forming valleys by the inter¬ section of its slopes with the interior slopes of the longi¬ tudinal roofs ; and for the sake of Fig 37 G. external appearance, or to collect the water in the centre, two cross roofs are sometimes used, inclosing a cen¬ tral space. When the interior slopes meet in a point, as in Fig. 376, the roof is called a hopper roof. Sometimes the plan of a building is irregular, and its sides are not parallel. If the roof be constructed so that the ridge slopes to the narrow end, its sides will be planes; but if the ridge be made horizontal throughout, the sides of the roof will become twisted or winding. Let add' a (Fig. 377) be the plan of an irregular building. The roof is hipped, having sloped ends forming the two triangles a p a', dg d, which may be isosceles; and the ridge is projected on the line p g, which is not parallel to either of the sides a d, d d\ Two cases here present themselves: the ridge projected on p g is either horizon¬ tal, and its extremities are determined by its meeting with the planes of the sloping ends, which may have the same slope—the larger sides in this case being twisted; or it has an inclination determined by the intersection of the planes apg d, a r pg d\ springing with equal slopes from the wall head. It is easy to observe that the ridge will not in either case appear parallel to the faces of the building. Of these two methods, the most agreeable to the sight is the first, in which the ridge is horizontal: it is also the most economical in construction. The two great¬ est sides of the roof are surfaces generated by a line which, moving from a d or a'd', to meet the ridge p g, is kept in contact with a vertical line passing through core', where the arrises of the hips meet when produced. The twist or wind of the sides being disagreeable, vari¬ ous methods are used to Fig.: get rid of it where the ridge is horizontal. In Fig. 378, two equally inclined planes spring from a d, a’d', and their intersection produces an 6 inclined ridge p f The points q q are taken at the level of p, and the arris/ q is thus symmetrical with fp. The same plan may be covered by a roof with hori¬ zontal ridges, as in Fig. 379. The four sides of the roof are planes, springing at the same in¬ clination from the walls a| abed. The ridges ef e g, are continued at the height determined by the intersection of the hip at e; and a pyramidal construction, efg, is added, at so low a pitch as not to be visible from the ground. In place of these pyramidal roofs, the three sides of the building c b d may have roofs of two slopes intersecting at the lines e h, h f hg, forming what is called an irregular hop¬ per roof. In Fig. 380, another method of preserving the ridge horizontal on one side without twisting the sides, is shown. In this case one of the sides is roofed in two slopes form¬ ing an arris, a b. In Fig. 381, another method of roofing the same space is shown. DESCRIPTIVE CARPENTRY—FORMS OF ROOFS. 89 The Figs. 382 to 3S6 show various methods of roofing a Fig. 382. Fig. 383. trapezium without twisting the sides. In Fig. 382, the sides are planes, and form an irregu¬ lar pavilion roof. In Fig. 383, planes of the same slope rise to the same height, and are united by a platform. In Fig. 384, one side, a, is twisted. In Fig. 385, two irregular pavilion roofs cross each other, forming valleys at their intersec¬ tions. In Fig. 386 there are four gables and four valleys, as in the roof over a square plan, Fig. 353. In the roofs over an oval plan, Fig. 387 shows one with a straight and horizontal ridge, and, consequently, with twisted sides; but in this case the appearance of the side is not disagreeable. In Fig. 388, the sides are not twisted, but slope every¬ where alike; and the roof is truncated and terminated by a platform. Fig. 389 is a conical roof. Fig. 390 is a roof over a rectangular plan, with a semicircular end. The end of the roof is consequently a semi-cone. Fig. 391 is an annular roof. Fig. 392 is the roof of a crescent building, with one end gabled and the other hipped. Fig 393. Fig. 393 is the vertical, and Fig. 394 the horizontal projection of two united conical roofs. Fig. 395 is the junction of a large and a small conical roof. Fig. 396. Fig. 396. Fig. 396 is the junction of a conical and a pavilion roof. Fig. 397 is the junction of a span roof with a large conical roof. Fig. 398 shows the junction of a conical and a cre¬ scent roof. Fig. 399. shows the junction of a conical with an annular roof. In Fig. 400, A is the horizontal, and B the vertical projec¬ tion of a round pavilion roof, formed by a conical roof cut Fig. 397. Fig. 398 by two span roofs, which give two horizontal ridges, four right-lined valleys, and eight hips, which are elliptic curves. In Fig. 401, C is the horizontal, and D the vertical pro- M 90 PRACTICAL CARPENTRY AND JOINERY. jection of a round pavilion roof, formed by a hemispheri¬ cal cupola cut by two span roofs, making two horizontal ridges, four right-lined valleys, and eight hips, which are arcs of circles. Fig. 401. Fie. 402. In Fig. 402, E and f are the horizontal and vertical pro¬ jections of an imperial pavilion made in the same manner. In Fig. 403, G and H are the horizontal and vertical sections of a roof in the form of a hemisphere, truncated by four inclined planes, forming a pavilion roof with four right-lined arrises, and eight arrises of portions of circles. In Fig. 404, I and K are the horizontal and vertical projections of a round pavilion, formed by the truncat¬ ing of a cone b} 7 four inclined planes. This is of the same Fig. 405. kind as the last, only the curves of the arrises are por¬ tions of ellipses. In Fig. 405, A is a horizontal, and B a vertical projection of a conical roof, whose summit is at c, truncated by two sloping planes, forming a horizontal ridge a b, and four hips having elliptic arrises a d, a c, b d, b c. Fig. c is a second vertical projection, on a plane parallel to c d. In Fig. 406, A and B are the horizontal and vertical projections of a conical roof truncated interiorly by two inclined planes. Fig 406. In Fig. 407, A is a horizontal projection of a roof, formed by the setting of a square pyramidal roof diagonally on a pavilion roof of lower elevation. This is the broach or spire of Gothic architecture. B is the vertical projection on a plane parallel to one of the faces of the lower pyra¬ mid, and c the vertical projection on a plane parallel to its diagonal. In Fig. 408, A is the horizontal, and B the vertical pro¬ jection of a pyramidal roof with a square base, set on an octagonal pavilion roof of lower elevation. In Fig. 409, A is the horizontal, and B the vertical pro¬ jection of a pyramidal roof, with an octagonal base, set on a pavilion roof of lower pitch with a square base. In Fig. 410, A and B are the horizontal and vertical projections of a conical roof set on a square pyramid. In Fig. 411, A and B are the horizontal and vertical projections of an acute conical roof set on an octagonal pyramid. In Fig. 412, A is the horizontal, and B the vertical pro- STEREOGRAPH Y —HIP-ROO PS. 91 jection of a roof formed by an octagonal pyramid set on a cone. In Fig. 413, A is the horizontal, and B the vertical projec¬ tion of a roof formed by an acute cone set on an obtuse one. In Fig. 414, A and b are two projections of two conical spires set on a conical roof. Fig. 413. Fig. 414. We shall now proceed to describe the methods of deter¬ mining the places and forms of the rafters in such hip¬ roofs as are of most common occurrence in practice. HIP-ROOFS. Plates XX., XXI. In its most simple form the hip-roof is a quadrilateral pyramid, each triangular side of which is a hip , and the rafter in each angle is a hip-rafter. The common rafters which lie between the hip-rafters in the planes of the sides of the roof, and which, by abutting on the hip rafters, are necessarily shorter than the length of the sloping side, are called jack-rafters. The things required to be determined in a hip-roof are these, viz.:— 1. The angle which a common rafter makes with the plane of the wall-head; that is, the angle of the slope of the roof. 2. The angle which the hip-rafters make with the wall head. 3. The angles which the hip-rafters make with the ad¬ joining planes of the roofs. This is called the backing of the hip. 4. The height of the roof. 5. The lengths of the common rafters. 6. The lengths of the hip-rafters. 7. The length of the wall-plate contained between the hip-rafter and next adjacent entire common rafter. The first, fourth, fifth, and seventh of these are gene¬ rally given, and then all the others can be found from them by construction, as is about to be shown. The plan of a building and the pitch of the roof being given, to find the lengths of the rafters, the backing of the hips, and the shoulders of the jack rafters and purlins. Plate XX.—Let abcd {Fig. 1) be the plan of the roof. Draw G H parallel to the sides AD, bc, and in the middle of the distance between them. From the points ABCD, with any radius, describe the curves ab, ab, cut¬ ting the sides of the plan in a b. From these points, with any radius, bisect the four angles of the plan in rrr r, and from abcd, through the points rrr r, draw the lines of the hip-rafters AG, bg, ch, d h, cutting the ridge line G H in G and n, and produce them indefinitely. The dotted lines ce, df are the seats of the last entire common rafters. Through any point in the ridge line I, draw EIF at right angles to G H. Make I K equal to the height of the roof, and join E K, F K : then E K is the length of a common rafter. Make Go, Ho equal to I K, the height of the roof; and join ao, bo, c o, d o, for the lengths of the hip-rafters. If the triangles aog, and B o G, be turned round their seats, AG, BG, until their planes are perpen¬ dicular to the plane of the plan, the points o o, and the lines Go, Go, will coincide, and the rafters Ao, bo be in their true positions. Let abcd {Fig. 2) be the plan of an irregular roof, in which it is required to keep the ridge level. Bisect the angles of two ends by the lines A b, B b, c G, D G, in the same manner as before; and through G draw the lines ge,gf parallel to the sides c B, D A respectively cutting A b, B b in e and F; join EF: then the triangle E G F is a flat, and the remaining triangle and trapeziums are the inclined sides. Join G b, and draw H I perpendi¬ cular to it: at the points M and N, where H I cuts the lines G E, G F, draw Ml, N L perpendicular to n I, and make them equal to the height of the roof: then draw H K, i L for the lengths of the common rafters. At E, set up E m perpendicular to BE; make it equal to M K or N L, and join B m for the length of the hip-rafter; and proceed in the same manner to obtain A m, c m, D m. To find the hip and valley rafters of a compound ir¬ regular roof {Fig. 3). In the compound roof shown by the plan, in which the ridge is level throughout, although the buildings are of different widths, the method of proceeding to find the hip and valley rafters of the right-lined parts of the roof is the same as in the two former cases, and will be evident on inspection. In the circular part, proceed as follows:—Draw c d a radius to the curve, as the seat of one pair of the common rafters c b, d b, and bisect it in a : through a describe the curve k a w n a, which is the seat of the circular ridge: produce the lines of the other ridges to meet this curved line in a W k, and connect the angles of the meeting roofs with these points, as in the drawing: divide the seat of one pair of the common raf¬ ters in each roof, as xy,pq,t u, and ef into the same number of equal parts; and through the points of division 92 PRACTICAL CARPENTRY AND JOINERY. draw lines parallel to the sides of their respective roofs, intersecting the curved lines drawn through the points of the curved roof; and through the points of intersection draw the curves e, l, to, a, &c., which give the lines of the hips and valleys. On c a, the meeting of the left-hand roof with the circular roof, erect a b at a, and make it equal to the height of roof; and join c b, for length ol valley rafter: proceed in the same manner for the hip- rafter z 6; and for the other hip and valley rafters. To find the valley rafters at the intersection of the roof b with the conical roof e {Fig. 4). Let D II, F H be the common rafters of the conical roof, and K L, I l, the common rafters of the smaller roof, both of the same pitch. On G H set up G e equal to M L, the height of the lesser roof, and draw e d parallel to D f, and from d draw c d perpendicular to D f. The tri¬ angle D d c, will then by construction be equal to the triangle K L M, and will give the seat and the length and pitch of the common rafter of the smaller roof B. Divide the lines of the seats in both figures, D c, K M, into the same number of equal parts; and through the points of division in e, from G as a centre, describe the curves ca, 2g, 1 /' and through those in B, draw the lines 3 f 4 g, M a, parallel to the sides of the roof, and intersect¬ ing the curves in f g a. Through these points trace the curves cfga, Afga , which give the lines of intersec¬ tion of the two roofs. Then to find the valley rafters, join c a, A a; and on a erect the lines a b, a b perpen¬ dicular to c a and A a, and make them respectively equal to ML; then c b, A b is the length of the valley rafter, very nearly. Plate XXI. Fig. 1 —Shows the method of finding the backing of the hip-rafter. Let B b, b c be the common rafters, a d the width of the roof, and A B equal to one-half the width. Bisect B c in a, and join a a, Da. From a set off a c, ad equal to the height of the roof a b, and join Ad, D c; then A d, D c are the hip-rafters. To find the backing : from any point h in A d, draw the perpendicular h g, cutting A a in g\ and through g draw perpendicular to A a the line e /, cutting A B, A D in e and /. Make g k equal to g h, and join k e, k /; the angle e k f is the angle of the back¬ ing of the hip-rafter c. Fig. 2.—Where the angles of the roof are not right angles. Bisect ad in a , and from a describe the semi¬ circle A b D ; draw a b parallel to the sides A B, D c, and join A b, d b, for the seat of the hip-rafters. From b set off on b A, b D, the lengths b d, b e, equal to the height of the roof b c, and join A e, D d, for the lengths of the hip-raf¬ ters. To find the backing of the rafter:—In Ac, take any point k, and draw k h perpendicular to A e. Through h draw f h g perpendicular to A b, meeting a b, ad in / and g. Make h l equal to h k, and join f l, gl\ the angle fig is the backing of the hip. To find the bevel of the shoulder of the purlins. Fig. 3.—First, where the purlin has one of its faces in the plane of the roof, as at E. From c as a centre, with any radius, describe the arc d g ; and from the opposite extremities of the diameter, draw d It, g m perpendicular to B c. From e and / where the upper adjacent sides of the purlin produced cut the curve, draw ei,fl parallel to to d h, g m; also draw c k parallel to d h. From l and i draw l m and i h parallel to B c, and join k h, k m. Then ckm is the down bevel of the purlin, and ckh is its side bevel. When the purlin has two of its sides parallel to the horizon. This simple case is shown worked out at F. It requires no explanation. When the sides of the purlin make various angles with the horizon. Fig. 4 shows the application of the method described in Fig. 3, to these cases. Fig. 5 shows the method of finding the bevel of the jack-rafters. Let A B c D be the plan. Draw the hipped end of the roof A E D in any of the manners already described. Bisect B c in E; and from E as a centre, with the radius E c, describe the semicircle cid Through E draw d e parallel to A D. Bisect the semicircle d e in I, join E I, and produce the line to G; El represents the seat of the common rafter, extending from the wall plate to the junction of the hip-rafters, and the length of the rafter over the seat will of course be equal to b a or C a. Make IG, therefore, equal to b a or C a, and join AG, DG. The triangle A G D will then show the extent of the covering of the hip A E D, and A G, D G give the lengths of the hip-rafters. Produce b c to F and H, and make BF, CH each equal to the length of the common rafter ca or B a, and join A F, dh. The triangle A F B is the covering of the tri¬ angle A E B, and D H C of DEC. To find the bevel of any jack-rafter on the hip A ED: —From its seat on the plan, shown in dotted lines g o, It p, draw the parallel lines g k, h l to the line of the hip- rafter A G in the development, and k l shows the bevel of the jack-rafter on the hip-rafter. Any of the other jack- rafters are found in the same way, as shown by the dotted and shaded lines. The down bevel of all the jack-rafters is the angle c a e. Fig. 6 shows the method of finding the length and bevel of any single jack-rafter. Let A B c be the plan of one angle of a hip-roof, and B E the seat of the hip-rafter. It is required to find the length and bevel of the jack-rafter over the seat D k e. At the point b, where the longest side of the jack-rafter meets the hip-rafter, draw b d at right angles to side of jack- rafter b e, and make the angle bed equal to the slope of roof; draw also b c at right angles to side of hip-rafter a b, and make it equal to b d. Produce b e, h k indefinitely, and make b g equal to de\ and h b, fg is the length of the jack-rafter. To obtain the bevel at h b, draw a l at right angles to B C and make it equal to kf or e g, and join b l ; then the angle lb g is the bevel of the jack-rafter. To find the bevel of a hip-rafter made by a plane par¬ allel to the planes of the common rafters. Let A {Fig. 7) be the seat of the foot of the rafter, b the horizontal projection of its upper end, l m the projec¬ tion of the bevel there, and c the vertical projection of the rafter on a plane parallel to its plane. The backing of the rafter b a d at A, and its cross section r u q v s at f, will be evident without description. The method of finding the bevel at g k E h is as follows:—From g draw the horizontal line g i, and from g as a centre, with the radius g h, describe the arc h i. From i let fall a perpendicular in to the horizontal projection meet¬ ing bto produced in n, then draw l no, and p i o is the proper bevel. KNOWLEDGE OF WOODS. 93 PART FOURTH. KNOWLEDGE OF WOODS—THEORETICAL CARPENTRY. PHYSIOLOGICAL NOTIONS. The substance named wood is, for the most part, elas¬ tic, tenacious, durable, and easily fashioned—qualities which cause it to be in general request for articles of use and of luxury. It is used as fuel, either in its natural state or as reduced to charcoal: it also affords tar, to pro¬ duce which, much of it is annually consumed. Wood* has always been indispensable in the arts. Iron, from its power of resisting cutting instruments and the attacks of fire, has, in some measure, rivalled it; but wood must still be regarded as the substance which con¬ tributes most to the preservation of man, to his defence, his civilization, and the development of his power. It en¬ ters largely into the fabrication of tools and machines, of arms of warfare, of furniture and utensils, and of all kinds of constructions, from the modest hut, the small foot¬ bridge and frail skiff, to the grandest edifices, bridges of daring span, and mighty ships which extend man’s com¬ merce and his power, and bring together those whom nature would seem to have separated for ever. In the Art of Carpentry, above all, timber is employed in the manner most remarkable, and in the largest masses. The natural form of the tree admits of the timber being obtained in long parallelopipedons, termed beams, girders, &c., the combination and framing of which constitute the means of raising great structures in the most rapid manner. It is not necessary, for the purposes of this work, to enter minutely into the discussion of the subject of vege¬ table physiology: all that is here requisite, is to state some of the more important facts, the knowledge of which is of consequence to the carpenter who desires to possess the power of discernment in the choice of timber. The part that is characterized as timber is obtained from the body of trees, or that part of those which grow with a thick stem, rising high, and little encumbered with branches or leaves, which is called the trunk. The head of the tree consists of the branches, which are adorned with leaves: these attain their full development in the summer, and then, in the great majority of species, fall in the autumn. In carpentry, the wood of the trunk and largest branches alone is used; and only that of the commoner species of trees; leaving to other arts to employ that of the rarer and smaller kinds, for which their beauty, rather than strength, is the recommendation. Some of the timber trees attain an immense size when they are allowed to come to full maturity of growth. Oaks and beeches are found to attain the height of 120 feet; the larch, the pine, the fir grow to the height of 135 feet. Other kinds, as the elm, the aspen, the maple, the alder, and even the walnut, the poplar, and the cypress, reach sometimes a great elevation. In warm regions, the palm grows as large as the oak. The diameter of the trunk varies with the kind and the climate. Oaks and elms may be seen of the enormous size of 36 feet in cir¬ cumference. Earner mentions an elm at Hasfield, in Massachusetts, considered the largest in America, which measured, at the surface of the ground, 34 feet in diameter; and at 5 feet high, 24 feet. Condamine speaks of canoes on the Ama¬ zon river made of single trees, and measuring 90 palms in length, and 10 in width. Other authors mention trees of equally great dimensions, such as the firs of North America, which shoot up to 250 feet in height; and cypresses, which grow to 7 and 8 feet in diameter. The Carolina pine is said to attain a size still more astonishing, the trunk growing to 60 feet in circumfer¬ ence, and rising 300 feet before the head is formed. The diameter which trees attain in our climate, how¬ ever, rarely exceeds 5 feet, and ordinarily they are found of half that size. Ship-building consumes a great quantity of timber. For this service, the curved pieces are more useful; but it seizes also upon the finest trees; which renders it diffi¬ cult to satisfy the wants of the other branches of carpen¬ try, and creates occasion for divers expedients to supply the want of size in the timber, both in length and sec¬ tional area. The difficulty of procuring oaks of sufficient magnitude has directed attention to pines and firs. Botanists classify vegetables, and consequently trees, according to their physiological and structural peculi¬ arities ; and in this way trees are divided into two great classes,—Monocotyledonous, or Endogenous, and Dicotyle¬ donous, or Exogenous trees. The terms Monocotyledonous and Dicotyledonous, be¬ long to the Jussieuan system of nomenclature, and are descriptive of the organization of the seeds. Endogenous and Exogenous are the terms used by modern botanists, and are descriptive of the manner of growth or develop ment of the woody matter of the tree, which is, in the endogens, from the outside inwards towards the interior, and in the exogens, outwards to the exterior. The monocotyledonous or endogenous trees are only used in this country in the formation of articles of luxury. Trees of this class have no branches: their stems, nearly cylindrical, rise to a surprising height, and are crowned by a vast bunch of leaves, in the midst of which grow their flowers and fruits. In this class are the palm trees, orowino- only, in their native luxuriance, in tropical climes, where they are of paramount importance, yielding to the people of those countries meat, drink, and raiment, and timber for the construction of their habitations. The palm tree will serve as a type of the endogenous structure. Dr. Lindleyt says of it—“In the beginning, the embryo of the palm consists of a cellular mass, of a cylindrical form, very small, and not at all divided. As * Colonel Emy, abridged and slightly altered. t Lindley’s Vegetable Kingdom , p. 95. 94 PRACTICAL CARPENTRY AND JOINERY. soon as germination commences, a certain number of cords, of ligneous fibre, begin to appear in the radicle,* deriving their origin from the plumule, t Shortly afterwards, as soon as the rudimentary leaves of the plumule begin to lengthen, spiral and dotted vessels appear in the tissue in connection with the ligneous cords; the latter increase in quantity as the plant advances in growth, shooting through the cellular tissue, and keeping parallel with the outside of the root. At the same time, the cellular tissue in¬ creases in diameter, to make room for the ligneous cords (or woody bundles, as they are called). At last a young leaf is developed, with a considerable number of such cords in connection with its base; and as its base passes all round the plumule, those cords are, consequently, con¬ nected equally with the centre which that base surrounds. Within this a second leaf gradually unfolds, the cellular tissue increasing horizontally at the same time; the lig¬ neous cords, however, soon cease to maintain anything like a parallel direction, but form arcs, whose extremities pass upwards and downwards, losing their extremities in the leaf on the one hand, and in the roots on the other, or in the cellular integument on the outside of the first circle of cords; at the same time, the second leaf pushes the first leaf a little from the centre towards the circum¬ ference of the cone of growth. In this manner, leaf after leaf is developed, the horizontal cellular system enlarging all the time, and every successive leaf, as it forms at the growing point, emitting more woody bun¬ dles curving downwards and outwards, and, consequently, intersecting the older arcs at some place or other; the result of which is, that the first formed leaf will have the upper end of the arcs which belong to it longest and much stretched outwardly, while the youngest will have the arcs the straightest; and the appearaiice produced in the stem will be that of a confused entanglement of woody bundles in the midst of a quantity of cellular tissue. As the stem extends its cellular tissue longitudi¬ nally while this is going on, the woody arcs are, conse¬ quently, in proportion, long, and, in fact, usually appear to the eye as if almost parallel, excepting here and there where two arcs intersect each other. As, in all cases, the greater number of arcs curve outwards as they descend, and eventually break up their ends into a multitude of fine divisions next the circumference, where they assist in forming a cortical integument, it will follow, that the greater part of the woody matter of the stem will be collected near the circumference; while the centre, which is comparatively open, will consist chiefly of cellu¬ lar tissue; and when, as in many palms, the stem has a limited circumference, beyond which it is not its specific nature to distend, the density of its circumference must, it is obvious, be proportionably augmented. “ Never is there any distinct column of pith or medul¬ lary rays, or concentric arrangement of the woody arcs; nor does the cortical integument of the surface of endo¬ genous stems assume the character of bark separating * Radicle, the conical body which forms one extremity of the embryo, and which, when germination takes place, becomes the des¬ cending axis or root of the plant. f Plumule, the growing point of the embryo, situated at the apex of the radicle, and at the base of the cotyledons, by which it is protected when young. It is the rudiment of the future stem of a plant. from the wood below it: on the contrary, as the cortical integument consists very much of the finely divided extremities of the woody arcs, they necessarily hold it fast to the wood, of which they are themselves prolonga¬ tions; and the cortical integument can only be stripped off by tearing it away from the whole surface of the wood, from which it does not separate without leaving myriads of little broken threads behind." Such is the general structure of the most perfect among the endogens, a family of plants which, in this country, are little known as trees; but which, in the cereal and other grasses, enter largely into cultivation as articles of food for men and animals. In other countries, how¬ ever, the endogenous trees fill an important place in structural economy, as well as contribute largely to the food of man. One of them, the bamboo, plays so many parts, as absolutely to make it difficult to say what it is not used for. A recent writer enumerates some of its most common uses in China, in the following amusing manner:— “ Bamboo is used in making soldiers’ hats and shields, umbrellas, soles of shoes, scaffolding poles, measures, baskets, ropes, paper, pencil-holders, brooms, sedan chairs, pipes, flower-stalks, and trellis-work in gardens: pillows are made of the shavings: a kind of cloak for wet weather is made of the leaves. It is used for making sails and covers for boats, fishing rods and fishing baskets, fishing stakes and buoys, aqueducts for water, water wheels, ploughs, harrows, and other implements of hus¬ bandry. Its roots are cut into grotesque figures, and its stem carved into ornaments for the curious, or as incense- burners for the gods. The young shoots are boiled and eaten; and sweetmeats are also made of them. A sub¬ stance found in the joints, called tabasheer, is used in medicine. In the manufacture of tea, it forms the rolling tables, drying baskets, and sieves. The all-important chop-sticks are made of it. It is in universal demand in the house, on the water, in the field: the Chinaman is cradled in it at his birth, through life it is his constant companion in one shape or another, and in it he is carried to his last resting-place to repose even there under the shade of its long oval leaves.” With this catalogue of the uses of the bamboo, we may dismiss the endogens. Dicotyledonous or exogenous trees, which form the second class, are in much greater variety, and much more widely spread over the globe, than the trees of the first class. The form of their trunks is generally conical, tapering from the root to the summit: the summit or head of the tree is formed by the prolongation of the trunk, which divides into sundry primary branches; these again ramify into innumerable secondary branches; and these throw out small twigs, to which the leaves are attached by foot-stalks, larger or smaller. At first sight it appears as if the leaves grew by chance, but an order, regular and constant in each species, presides in their distribution. On making a transverse section of a dicotyledonous tree, we see that it is composed of three parts, easily dis¬ tinguished—the bark which envelops, the pith which forms the core or centre, and the woody substance which lies between the bark and the pith. In the woody substance we distinguish two thicknesses: the one which envelops the pith is the greatest; and is KNOWLEDGE OF WOODS. 95 of a harder nature than that which adjoins the hark. The former is termed perfect wood, the latter alburnum. The inner layer of bark next the alburnum is called the liber, a name given from its being used to form the books ( libri ) of the ancients. Between the liber and the alburnum there is a substance partaking of the qualities of both, and called cambium. This is developed in the spring and autumn: when its internal portion changes insensibly into alburnum, and the exterior into liber. The liber never becomes wood: it is expanded continually by the process of growth in the tree, and forms the bark, which rends and exfoliates externally, because of its drying; and the layer of liber, in growing old, cannot extend in proportion to the augmentation in the circum¬ ference of the tree. Duhamel and Buffon long since proved that alburnum, in process of time, became perfect wood; and there is now no doubt in regard to the manner in which the tree grows and produces its wood. “Exogens, or outwood growers,” says Dr. Lindley,* “are so called because, as long as they continue to grow they add new wood to the outside of that formed in the pre¬ vious year; in which respect they differ essentially from endogens. “In an exogen of ordinary structure, the embryo con¬ sists of a cellular mass, in which there is usually no trace of woody or vascular tissue; but as soon as germination commences, fine ligneous cords are seen proceeding from the cotyledons towards the radicle, meeting in the centre of the embryo, and forming a thread-like axis for the root. As the parts grow, the ligneous fibres are increased in thickness and number, and having been introduced among the cellular mass of the embryo, are separated from each other by a portion of the cellular substance, which con¬ tinues to augment both in length and breadth as the woody cords extend. By degrees, the plumule or rudi¬ mentary stem becomes organized; and having lengthened a little, forms upon its surface one, two, or more true leaves, which gradually expand into thin plates of cellular substance, traversed by ligneous cords or veins, converg¬ ing at the point of origin of the leaves. If at that time the interior of the young plant is again examined, it will be found that more ligneous cords have been added from the bases of the new leaves down to the cotyledons, where they have formed a junction with the first wood, and have served to thicken the woody matter developed upon the first growth. Those ligneous cords which proceed from the base of the leaves, do not unite in the centre of the new stem, there forming a solid axis, but pass down parallel with the outside, and leave a small space of cellu¬ lar tissue in the middle; they themselves being collected into a hollow cylinder, and not uniting in the middle until they reach that point where the woody cords of the coty¬ ledons meet in order to form the solid centre of the root. Subsequently, the stem goes on lengthening and forming new leaves: from each leaf may be again traced a forma¬ tion of woody matter disposed concentrically as before, and uniting with that previously formed, a cylinder of cellular substance being always left in the middle. The solid woody centre of the root proceeds in its growth in a corresponding ratio, lengthening as the stem lengthens, * Vegetable Kingdom, p. 235. and enlarging in diameter as the leaves unfold, and new woody matter is produced. The result of this is, that when the young exogen has arrived at the end of its first year’s growth, it has a root with a solid woody axis, and a stem with a hollow woody axis, surrounding cellular tissue; the whole being covered with a cellular integu¬ ment. But as the woody cords are merely plunged in a cellular basis, the latter passes between them in a radiat¬ ing manner, connecting the centre with the circumference by straight passages, often imperceptible to the naked eye, but always present. “ Here we have the origin of pith in the central cellu¬ lar tissue of the stem of wood in the woody axis, of bark in the cellular integument, and of medullary processes in the radiating passages of cellular tissue, connecting the centre with the circumference.” The woody axis is not, however, quite homogeneous at this time. That part which is near the centre contains vessels of different kinds, particularly dotted vessels (both- renchyma): the part next the circumference is usually des¬ titute of vessels, and consists of woody tissue exclusively: of these two parts, that with the vessels belongs to the wood properly so called, and serves as a mould about which future wood is added: the other belongs to the bark, separates under the form of liber, and in like manner serves as a mould within which future liber is disposed. At the commencement of the second year’s growth the liber separates spontaneously from the true wood; the viscid substance, cambium, is secreted between them; and the stem again lengthens, forming new leaves over its surface. The ligneous cords of the leaves are prolonged into the stem, passing down among the cambium, and adhering in part to the wood, and part to the liber of the previous year; the former again having vessels inter¬ mingled with them, the latter having none. The cellular tissue that connected the wood and the liber is softened by the cambium, and grows between them horizontally, while they grow perpendicularly, extending to make room for them; and, consequently, interposed between the woody cords of which they each consist; forming, in fact, a new set of medullary processes, terminating on the one hand in those of the first year’s wood, and on the other in those of the first year’s liber. The addition of new matter takes place equally in the stem and in the root, the latter extending and dividing at its points, and receiving the ends of the woody cords as they diverge from the main body. The only respects in which the growth of exogens cor¬ responds with that of endogens are, that in both classes the woody matter is connected with the leaves, and in both, a cellular substance is the foundation of the whole structure. As new layers of alburnum are produced, they form concentric circles, which can be easily seen on cutting through the tree; and by the number of these circles one can determine the age of the tree. Some authors assert that this is not so, since a tree may produce in one year several concentric layers of alburnum, and in another year only one. Nevertheless, the commonly received opinion is, that the number of concentric circles in the cross sec¬ tion of the wood, called annual layers, indicates the time it has taken to reach its size. Although a layer of albur¬ num is deposited each year, the process of transformation 90 PRACTICAL CARPENTRY AND JOINERY. of it into perfect wood, otherwise heart-wood, is slow, and, consequently, the alburnum, or sap-wood, compre¬ hends many annual layers. The annual layers become more dense as the tree grows aged; and when there is a great number in a tree of small diameter, the wood is heavy, and generally hard also. In wood which is either remarkably hard or re¬ markably soft, the annual layers can scarcely be distin¬ guished. They cannot, for example, be distinguished in ebony, and other tropical woods, nor in the poplar, and other soft white woods of our climate. In the case of the softer woods in our climate, the layers are frequently thinner and more dense on the northern side than on the opposite. In a transverse section of a box tree, about 7 inches diameter, we reckoned 140 annual layers. The roots of a tree, although buried in the soil, have, as we have seen, an organization resembling that of the trunk and branches. The roots of several trees are em¬ ployed in the arts, but as none of them are used in car¬ pentry we need not dilate on the subject: we shall only remark, that as the branches of a tree divide into smaller branches and twigs, expanding to form a head, so the roots divide also into branches, which expand in every direction in the ground, and these branches again divide, their ultimate division being into filaments, commonly called fibres, which appear to be to the roots what the leaves are to the branches. It has been remarked that there is a sympathy between the branches and the roots in their development. Thus, when several considerable branches of a tree are lopped off, the corresponding roots suffer, and often perish. CULTIVATION OF TREES. Trees are the produce of forests, planted spontaneously, and consequently very ancient, or of forests and planta¬ tions created by man since he has engaged in this kind of culture. The reproduction of trees, their culture, and the felling of timber, belong more to the management of forests than to the art of carpentry; but we shall remark briefly on some qualities which are derived from growth. The size and fine growth of a tree is not an infallible sign of goodness of quality in the wood. The connection of the age of a tree with its development, and the nature of the soil in which it grew, ought to be inquired into to enable a judgment to be formed of the quality of the wood. In general, boggy or swampy grounds bear only trees of which the wood is free and spongy, compared with the wood of trees of the same species grown in good soil at greater elevations. The water, too abundant in low lying argillaceous land, where the roots are nearly always drowned, does not give to the natural juices of the tree the qualities essential to the production of good wood. Trees grown in such places are better adapted for other works than those of the carpenter. The oak, for example, raised in a humid soil, is more proper for the works of the cabinet-maker than for those of the carpenter; because it is less strong and stiff, and is softer and more easy to work than the same wood raised in a dry soil and elevated situation: it is also less liable to cleave and split when only employed for small works. Its strength, compared with that raised in a drier soil is about as 4 to 5, and its specific gravity as 5 to 7. Wet lands are only proper for alders, poplars, and wil¬ lows. Several other species incline to land which is moist or wholly wet; but the oak, the chestnut, the elm, thrive only in dry situations, where the soil is good, and where the water does not stagnate after rain, but is retained only in sufficient quantity to enable the ground to furnish aliment for the vegetation. Resinous trees, too, do not always thrive in the soils and situations proper to the other kinds of timber, and especially in marshy soils: sandy soils are in general the best for their production; and several species affect the neighbourhood of the sea, such as the maritime pine, not less useful for its resin than for its timber. In fine, trees which grow in poor and stony soils, and generally in all such soils as oppose the spreading of their roots, and do not furnish a supply of their proper sap, are slow and stunted in their growth, and produce wood often knotty and difficult to work, and which is mostly used as veneers for ornamenting furniture. The surest tokens of good wood are the beauty, clear¬ ness, and firmness of the bark, and the small quantity of alburnum. It has been remarked that timber on the margin of a wood is larger, more healthy, and of better quality than that which grows in the interior, the effect of the action of the sun and air being less obstructed. DISEASES OF TREES. Trees, like animals, are subject to disease. When, by the effect of old age or disease, a tree dies at the bottom, even before it has arrived at the ordinary limit of exist¬ ence of individuals of the same species, its wood loses the qualities not only essential for timber construction, but also for combustion. It loses flexibility, strength, and durability; it becomes dry and soft; when it falls it ra¬ pidly rots, or becomes the prey of worms; and it burns without flame, and with little heat. If, in place of this, it is felled in its vigour, it ceases to live, it is true; it no more vegetates, it becomes dry, but it preserves all its useful qualities, and is fit for any of the purposes to which timber can be applied. Diseases of plants are, for the most part, dependent on chemical changes in their component parts. The maladies which arise from outward causes are sores, mutilations, and fractures, which may be the result of the action on the bark of the teeth of large animals, of strokes given by accident or design, the effects of wind, or of lightning. This latter agent is the most destruc¬ tive: the trees are shivered partly by mechanical agency, partly by sudden expansion of gas. Dr. Colin observed that the electric fluid first takes the course of the alburnum as the best conductor, splitting off the bark by the sudden expansion of the fluid; a part enters the older portions of the wood, which are compara¬ tively bad conductors, taking the. course of the medullary rays, or points of conduct of the annual layers, or both; and thus splits the tree in various directions, occasionally threading it to the extremity of the roots. KNOWLEDGE OF WOODS. 97 The maladies which can arise from accidents, and from the customary regimen of vegetation, or from the state of the atmosphere, and from meteors, are:—Ulcers, cankers, rottenness, chaps, clefts, and other diseases caused by frost and cold, exfoliations, tumours, knobs, warts, excrescences, plethora, return. Ulcers and Cankers in trees resemble the same diseases in animals. The origin is generally in the roots. A too great abundance of sap in some part of a tree manifests itself in a kind of external suppuration, which is accom¬ panied by a corruption of the fluids, and speedily of the wood adjoining the ulcerated part. The disease sometimes spreads, causes the bark to peel off, and then the tree perishes. Ulcers in elms are due to a collection of corrupt fluid from the decomposition of water pr sap percolated through decayed tissue. The fluid blackens whatever it touches, and is extremely fcetid: when a tree so infected is felled, the odour is very disagreeable while the infected parts are being lopped off. Rottenness proves the existence of some disease in the sap. In this disease, the woody fibre is reduced to powder. Ulcers, cankers, and rottenness also proceed from water obtaining access to the interior of a tree at the points where the branches leave the trunk,—through clefts, which are generally produced by unwonted straining of the branches by the wind. Chaps in the bark are occasioned by scorching winds, drought, sudden changes of season from cold to heat, or vice versa; or from the too violent action of the sun. These expose the wood of the tree to the action of the weather; and their existence is the sign of deterioration in the wood. Circular Chaps, surrounded by other chaps in rays, are supposed to be caused by some insect. Frost Cracks commence in the bark, penetrate the new wood, and sometimes extend deeply into the heart-wood. They are caused by the freezing of the water of the sap, which splits, first the bark, and then the wood of the tree. Rigorous frosts sometimes act upon the external layers ot alburnum, and hinder them from passing into perfect wood, although they continue to exercise their functions in the transmission of the materials which form the new layers, but in such manner that these do not adhere to the layers of the year previous. When these arrive at the state of perfect wood, a void, extending over a consi¬ derable portion of the trunk, and sometimes throughout its circumference, is left between them and the layers in¬ jured by the frost. Two concentric cylinders are thus formed, detached, and separated sometimes by an inter¬ val of a finger s breadth. Twisted Fibres are not the result of disease, but of deformity caused by the prevalent action of the wind in one direction on the head of the tree. The stem, when young and tender, is thus twisted, and its fibres retain their screw form when they pass into perfect wood. Twisted wood is not proper for the use of the carpenter, as in squaring it many of its fibres would be cut through. Exfoliation. —This is a disease of the bark, in which it detaches itself in layers. There results from this an alter¬ ation in the alburnum, and in the wood which it fur¬ nishes. It is believed that exfoliation is caused by some insect, which as yet, however, has escaped microscopical research. Tumours, Warts, Excrescences , and Abscesses, all pro¬ ceed from local disease, which produces deterioration of the alburnum, and an excessive affluence of sap at certain points; resulting in extravasation and accumulation of vegetable substance, forming excrescences and confused contexture. They are often, too, caused by wounds, by the attacks of insects, and by parasitical plants. They injure the wood by disturbing the uniformity of the lig¬ neous fibre. Plethora. 1 his is the result of an over abundance of nutritive juices, drawn irregularly to different parts of the tree, and causing deformity. The quality of the wood is consequently injured by the impairing of its homogeneousness; and its strength could not be trusted, especially if exposed to transverse strains. Return. —This is the last disease of growing trees which have passed the term of their maturity. It commences at the top of the head; and whether it is owing to the obstruction ot the channels which convey nourishment, a deficiency of the nourishment itself, or weakened vital energy, its symptoms are a drying and decay of the top shoots, then of the branches, and lastly of the trunk itself. On the first appearance of the symptoms the tree should be felled, in order to save the timber. TIMBERS FIT FOR THE CARPENTER. In general, regularity in the roundness, and the taper, of a tree, and a fine or uniform texture in the bark, indi¬ cate it to.be of good quality. All appearance of knots, wens, swellings, old sores, al¬ though cicatrized, all traces of canker, or of water having reached the heart of the tree, are infallible signs of dis¬ eased wood. Fresh mosses and lichens on a tree which has been some time felled, are symptomatic of its having lain in a wet place. These may also indicate the locality of some internal disease. It requires a practised eye to judge of the qualities of timber while yet in the unbarked tree; but, as the car¬ penter generally receives the wood on which he operates, squared, this knowledge is not of so much importance to him as it is to the timber-merchant. The qualities which fit wood for works of carpentry are durability, uniformity of substance, straightness of fibre, and elasticity. When wood is squared, its good quality, especially in the case of the oak, the chestnut, and the elm, is known by a fresh and agreeable odour which it exhales, and which is very different from the smell of wood, however freshly cut, which has begun to decay. When timber has been felled for a long time, and has become dry, this peculiar odour is not so perceptible, but the resinous trees retain the smell of the resin for a very long time; the odour being again made perceptible by cutting a slice from the surface. Dry and healthy timber is solid, tenacious, sonorous, and elastic; when it is dead or diseased, it is soft, emits a dull sound when struck, and acquires a disagreeable smell. The good quality of wood is known also by the unifor¬ mity and depth of the colour peculiar to its species. When the colour varies much from the heart to the circumference, and, above all, when it lightens suddenly or too rapidly towards the limit of the alburnum, we may be assured that the tree is affected by some disease. N PRACTICAL CARPENTRY AND JOINERY- 98 The white wood or alburnum of trees should be rejected; and where there is a double layer of white wood, sepa¬ rated by a layer of perfect wood, as is sometimes though rarely the case, the wood is unfit for use. Knotty and cross-grained wood is difficult to work. Cross-grained wood is rarely of great dimensions, and is employed chiefly in the construction of machines, and for purposes in which the tenacity of its fibres is its recom¬ mendation. It is rejected for ordinary work, because it is difficult, and, consequently, expensive to work, and weighs heavy in proportion to its strength. Such timber is very often, however, employed in hydraulic works, es¬ pecially when it is to be wholly under water. It is apt to become shaky diagonally in drying. A great defect is when the fibres do not approach to equality of size. Perfect equality is impossible in tapering timber, but* such an equality as shall not render one part of the piece of timber much less strong than another is obtainable by proper selection. In knotty wood, the knots interrupt that straightness of the fibres which gives strength. The knots are the prolongations of the branches across the perfect wood of the tree from the points where the branches have commenced. Such knots augment in size in the degree that the trunk of the tree increases. If the branches have grown with the tree to the time of its being felled, the knots will be perfect wood, the fibres of the trunk will only be turned slightly from their straightness, and if the knots are few they will not be very hurtful. But if the branch forming the knot has been suppressed or destroyed, or has by any cause ceased growing while the tree grew, the knot formed by it will be inclosed in the new layers of wood, and may become a cause of destruction by the decaying of its substance from contained moisture, and thus a nidus of rottenness is formed within the tree. It is therefore prudent to probe such knots, and, if they are decayed, to cut off all the wood which is traversed by them. In general, the prevalence of knots in a piece of wood indicates that it has been cut from a branch and not from the trunk of a tree. Wood which in growing has been blighted by frost is not fit for the carpenter. The lateral cohesion of its fibres is destroyed, and it contains numerous little chaps which absorb moisture and cause it to rot. When timber in growing has been subjected to strong frosts and thaws, the wood is often alternately alive and dead, and filled with small clefts. It is recognized by an appearance of marbling which it presents on being cut. Krafi't, in the introduction to his Carpentry, has the following remarks:—"It is important,” he says, “in the employment of timber, in pieces used vertically, to place them with the butt on high, and the top downwards. To know if a piece of timber is sound in the middle, saw its two ends, then cause blows with a hammer to be struck at one end, while the ear is placed against the other; and if the sound is dull the timber is bad ; but if, on the con¬ trary, it is clear, the timber is good." Generally when the tree is sound the density decreases from the butt upwards, and from the centre to the circum¬ ference. The greatest strength is found between the centre of the tree and the sap-wood; and the heaviest wood is the strongest. Sound wood under the saw cuts clean, is bright in colour, and, when planed, has a silky lustre: unsound timber wants this lustre, and the saw leaves a woolly surface. FELLING OF TIMBER. In the clearing of a forest, where all the trees without distinction are felled, the work of destruction commences at the borders nearest the roads which bound or traverse it, and proceeds with order and regularity. Care is taken not to embarrass the outlets, or obstruct the roads. The manner of procedure is according to the end proposed. If that be the destruction of the forest and the freeing of the soil from the remains of trees, in order that it may be appropriated to some other species of culture, the trees are removed by the roots, care being taken, in the case of certain species tenacious of life, not to leave any shoot, however small. If, again, the object be to produce copse, the trunks of the trees are cut close to the ground, that the stumps left in the soil may throw out new shoots. In thinning a forest, such trees only are felled as have attained the limit of their development of growth, or at least as have acquired the qualities which fit them for the purposes sought. The felling of timber is performed in either of the three following ways:— 1. The tree is felled with its trunk and stump separated from its roots. 2. It is torn up with all its roots attached. 3. It is cut above the soil, cither by the saw or by the axe, with the intention of removing the stump afterwards, or leaving it for the production of new wood. In practising the first method, the earth is removed from around the tree to the depth which will admit of the roots being cut through; but is retained to fill up the excavation. The roots are then hewn or sawn through, and by means of ropes the tree is pulled over to the side where its fall will least injure the neighbouring trees, or obstruct the operations on them. The roots are then dug out, and the earth is thrown back into the hole made by the removal of the stump. In the case of such trees as have a stump penetrating deeply into the ground, the earth may be removed from around the stump to such a depth only as will admit of the lateral roots being cut through. The tree is then retained in its vertical position by a pivot merely; and by means of chains and levers properly applied, the resistance of this is overcome, and the' tree thrown down as before. This mode cannot be practised in the case of trees which throw their roots vertically downwards, such as the chestnut, the elm, &c. To remove a tree with all its roots attached, the method is as follows:—The earth is removed from the principal roots, so as to admit of cords being passed under them. Then, by means of levers the roots are raised one after the other; cords or chains are next passed under the stool, which is raised in the same manner; and the tree is guided in the direction in which it is wished it should fall Sometimes the tree is overturned by the aid of gun¬ powder. To effect this, it is necessary to remove the earth from around the roots, so as to diminish the resistance of the soil. Then, under the trunk a small metal mortar with a large base is posited upon a piece of wood which KNOWLEDGE OF WOODS. 90 is still larger. The loaded mortar is covered with a cast- iron plate having portions of its corners cut off, so that the force of the explosive gases may he directed against the tree. This method is very little used. The third mode of felling is the most generally adopted. It is the only available method in the case of such trees as do not throw out branches at the foot, when it is wished to leave a stool for the reproduction of wood. The wood¬ cutter makes at the foot of the tree an incision with his axe, cutting deeper at the side on which he wishes the tree to fall. The cuts should nearly meet at the centre; for if too large a solid pivot be left, the falling of the tree would tear it out of the stool, and make a hole there, which Avould cause decay, and unfit the stool for reproduction. When a tree is felled by the axe, it should be cut as close to the ground as possible, to get the largest amount of trunk timber. This is, moreover, serviceable, as in¬ creasing the quality and beauty of the new wood which the stool throws out. The top of the stool, for obvious reasons, should be slightly convex or pyramidal. But it is more economical to cut the tree by the saw than by the axe. Ordinarily, facilities are afforded to the workmen in applying the instrument, by sinking a pit on each side, in which they can stand. As the saw makes way, the weight of the tree would render it impossible to work it: this is remedied by inserting small wedges in the saw-kerf, so that the blade may work freely. Many modifications of saws for the cutting of growing timber have been from time to time invented, and also machinery for giving motion to both circular and recipro¬ cating saws. It has been observed that the costs of the three me¬ thods described bear the following relation. Tearing up by the roots costs twelve times as much as felling by the axe; while the method in which the horizontal roots are cut off and the tree pulled over, costs only twice as much. But by both these methods, the available timber is increased to an extent which more than compensates for the additional cost. When, for culture of the soil, the removal of the stump is desired, either of the two first methods of felling is more advantageous than the last, from the effective leverage which the falling tree exerts. The proper season for the felling of timber has often been debated. In Italy and in Spain, timber is felled in summer, and is found, nevertheless, to be very durable. But it would be improper to draw a rule from the practice in countries where the heat and the dryness of the atmosphere favour the rapid dissipation of the natural juices of the tree. Some authors argue that the tree should be felled at the time of the year when the development of the vegetation is at its height. Others contend that the proper season is when vegetation has ceased. Tradition, which, in mat¬ ters of this kind, submitted to the test of daily experience, is seldom wrong, favours altogether the latter view; and the rule has been to cut the tree between the fall of the leaf in autumn and the rise of the sap in spring—vegeta¬ tion being then inert. Buffon and Duhamel advocated the practice of strip¬ ping the bark from the tree a year before felling, on the supposition that the alburnum by this exposure became perfect wood. The supposition, it need scarcely be said, is wholly erroneous; and a grave evil attending the practice is, that this depriving the tree of its bark while it is yet growing, causes it to die, and injures the elas¬ ticity of the timber. To diminish the quantity of sap in the tree before fell¬ ing it, some have recommended an incision to be made all round the tree immediately above the ground, and so deep as to leave a solid pivot merely large enough to sus¬ tain it. This is termed girdling. The connection between the roots and the trunk and its leaves being thus dis¬ severed, it is supposed that the sap escapes more freely. The practice does not obtain—for these reasons; it is expen¬ sive, it is not known to be beneficial, and it is dangerous in the event of high winds. The rapid exhaustion of the sap could be obtained by placing the tree vertically after it is cut; but the cost of doing so would be very great. SQUARING OF TIMBER. In order to fit the cylindrical stem of a tree for the or¬ dinary purposes of the carpenter, it is reduced to the form of a rectangular prism—a form the most convenient, whether the purpose be the connection of the pieces of timber by uniting them together either in length or in thickness, or in any of the diverse ways which ordinary carpentering requires. It may be said that all the wood delivered to the carpenter to be operated on, with the exception of what is intended for pillars, piles, or posts, requires to be squared. In squaring a tree, the object to be aimed at is to ob¬ tain either as large a parallelopipedon as possible, or one which has its dimensions suited to the purpose for which it is to be employed. The operation of squaring is generally performed by the woodmen who fell the timber; and who, by long experience, can judge at a glance what kind of squared timber a tree, however deformed, will yield. The carpenters thus, in general, have little to do with the squaring of the timber, especially that of foreign growth; but in works in which it is necessary to employ timber the produce of our own country, the tree is often selected and purchased while growing, and is not reduced to the square form until it comes to the hands of the car¬ penter. The operation may be thus shortly described:— The tree being elevated on tressels, transverse sections are made with the saw at both ends perpendicular to its axis, and on the perfect wood thus exposed, the rectangle forming the base of the parallelopipedon is inscribed * * It may be shown that the largest rectangle which can be in¬ scribed in the circle is a square. For if, in Fig. 415, we divide TOO PRACTICAL CARPENTRY AND JOINERY. Let the outer circle in Fig. 417 represent the smaller end of the tree; and let the inner circle distinguish the separation between the heart-wood and sap-wood. The rectangle of the greatest square will be obtained by in¬ scribing the square abed, in a circle which passes this a little. The centre o of this latter circle is easily found geometrically. Through this centre, two lines are drawn at right angles to each other, and from it are set off along these lines the four equal distances o a, o b, o c, o d, extending a little way beyond the line of the true wood; and abed give the arrises of the squared timber. When the tree is elliptical, however, a square is not the greatest rectangle that can be obtained, and it is necessary to proceed as follows:—On the minor axis a b of the ellipse (Fig. 418) describe a circle, and inscribe therein a square edef, and produce the sides c d, fe to g h , h to; then will g Ji h to be the largest rectangle that can be inscribed in the el¬ lipse. In practice, having drawn the two axes of the ellipse, draw the chord line b o, and through the centre draw g to parallel therewith; then from g and to, where this line intersects the line of separation between the alburnum and perfect wood, draw g h, Jem parallel to the major axis, and g Jc, Jim parallel to the minor axis, and the rectangle will be obtained. But it rarely happens that trees are so regular in their form as to admit of such rules being applied. In ordinary cases where the deformity is not great, the application of the callipers to the end will enable the operator to discover the greatest squaring dimensions of the tree. In some cases the deformity is so great that the only guide is practice and a good judgment. MANAGEMENT OF TIMBER AFTER IT IS CUT. The management of cut timber, so as to preserve its qualities from deterioration, and gradually to fit it for the purpose to which it is to be applied, is a subject of great importance. If timber be exposed to great changes of temperature, to alternations of wetness and drought, to a humid and hot atmosphere, it will inevitably suffer a deterioration of those qualities which render it serviceable for the car¬ penter. Timber, when too suddenly dried, is liable to split: these triangles is greater than any other triangle aec, cifc, moie¬ ties of any rectaugle a ecf whatever, inscribed in the same circle; because both sets of triangles having the same base, a c, the alti¬ tudes h b, h d of the first triangles are greater than the altitudes g e, kf of the second triangles; and therefore the square abed, the sum of the two triangles ab c,cda, is greater than the rectangle a e cf, the sum of the two triangles aec, cf a. In the same manner it may be demonstrated that the square is the greatest rectangle that can be formed in the quadrant a b (Fig. 416). when exposed to too high a temperature in a close atmo¬ sphere, its juices are liable to fermentation, followed by a loss of tenacity and a tendency to rot and become worm-eaten. The greater the quantity of timber thus kept together, the more rapidly is it impaired, which is made sensible to the smell by a peculiar odour emitted from it. When timber is exposed to injury from the weather, and lying long exposed on a damp soil, it is attacked by wet rot. The alternations, too, of drought and rain, of frosts and of heat, disorganize the woody fibre, which breaks, and a species of rottenness ensues resembling the decay of growing timber. The means of defending the timber from these various causes of waste, and preserving it in a state fit and proper to be used in construction, we now propose to describe. Although the carpenter has not often to undertake the charge of the timber in the place of its growth, yet as there are cases in which he is called upon to do so, we shall begin by describing the means of preserving the tree from the time it is felled to the period when it is to be used in construction. When the tree is felled, it should be preserved from con¬ tact with the soil, by being elevated on short pieces; and to prevent its too rapid desiccation, and the consequent formation of clefts and shakes, it should be sheltered from the sun, but so as to permit a free circulation of air around it. Where the felling of wood is performed on a large scale, sheds, which may be opened on any side at pleasure, should be constructed, in which the trees may be deposited. In these, the trees, defended from the sun and rain, may be exposed to the air, but with the power of controlling and modifying its action. It has been ob¬ served, that in sheds open on all sides the timber decays and splits more rapidly than in the open air. In piling the timber in these sheds, the trees should not be allowed to be in contact, but should be separated by pieces formed of a quarter section of a tree. They should also be carefully classed according to diameter, in order that the different trunks may be kept level, and each have a solid bearing, to prevent its sagging, and be¬ coming curved. The longest trees should of course be placed undermost. When the timber is squared and cut up, even greater care must be bestowed on it; not alone on the ground that it is then so much the more valuable by the labour which it has cost, but because, by its vessels being divided, it is more easily affected by deteriorating causes; and by its surface being augmented, these causes have also a larger field to operate on. Timber of the same scantling, felled and cut up at the same time, should be piled together; and there should not be mingled in one pile wood of different species. The first layer of the pile (Fig. 419) should be eleva¬ ted above the soil on sleepers, the higher the better, as securing a freer circulation of air, and preventing the growth of fungi. The most perfect security, however, is obtained by paving the site of the pile, and building dwarf walls or piers, with strong girders, to form the foundation for the first tier. Where the space will admit of it, and the timbers are square, they should be laid in tiers crossing each other alternately at right angles, and at least their own width are better when piled so that in the alternate tiers there are only planks sufficient to keep the other tiers from bending. Where space can be afforded, it is well to pile square timber in this way. The diagram (Fig. 420) will best explain this mode. Where space for this cross piling up against a wall or formed into a hollow pile, as shown cannot be afforded, pieces must be inserted between the tiers, as shown in Fig. 421. in Fig. 422. Another way is by crossing the planks, as seen in Fig. 423; and another, which admits the pile 11 Fig. 424. Sometimes it is convenient to pile the wood vertically against a wall, selecting a northern aspect, and shelter¬ ing the timber by a pent-house roof. It is profitable to move the wood from the piles and replace it, turning the sides, altering the relative posi¬ tions of the different pieces in the pile, and changing the points of support; and at the same time picking out and excluding any damaged or deteriorated piece. The pieces between the different tiers should at the same time be in¬ terchanged: and before being used again, they should be carefully inspected, and, if decayed or diseased, rejected. Every vestige of bark should be removed from the wood before it is piled; for the bark often contains the germs of disease, or is infested with insects, and in either case injury is done to the pile. being carried to a greater height, is by erecting four posts and building the timber about them, as seen in Fig. 424 KNOWLEDGE OF WOODS. apart. This method will not do for thin planks, because it would not allow a sufficient circulation of air. These When the timber has been accidentally wetted, or when it is necessary to hasten its desiccation, it should be set 102 PRACTICAL CARPENTRY AND JOINERY. OF THE BENDING OF TIMBER. Curved forms, either essential to the stability of a structure, or necessary for its decoration, require that the carpenter should obtain the timber naturally cur¬ ved, or should possess the power of bending it. Trees which yield timber naturally curved, are generally used for the constructions of the naval architect. If, in ci¬ vil constructions, where curved timber is required, it should be attempted to be formed by hewing it out of straight timber, two evils would ensue: the first, a loss of wood; the second, and greater, the destruction of its strength by the necessary cross-cutting of its fibres. Hence, to maintain the fibres parallel among them¬ selves, and to the curve, recourse is had to curving or bending the timber artificially. This process may be performed on the trees while yet growing, or on the tim¬ ber after it is squared or cut up. The first process is rarely performed, and need not be here described in de¬ tail. We shall therefore proceed to the consideration of the second. The process of bending timber artificially is founded on the property which water and heat have of penetrat¬ ing into the woody substance, rendering it supple and soft, and fitting it to receive forms which it retains after cooling. This process is extremely ancient. A familiar illustra¬ tion of the power of the simultaneous action of heat and moisture in altering the form and dimensions of wood, is the well-known little puzzle of a key sliding in a mor¬ tise: the key is in one piece, and its two ends have pro¬ jections which make their thickness twice that of the part which slides in the mortise; and the wonder is, how, under these circumstances, it was possible to insert the key; and being inserted, how to remove it. The difficulty is overcome by steeping one of the ends in boiling water till thoroughly penetrated, and then squeezing it in a vice till its dimensions are reduced, so that it passes through the mortise: when the wood dries and cools, it resumes its original bulk and form. The softening pre¬ paration for bending timber is effected in the five follow¬ ing ways :— 1. By using the heat of a naked fire. 2. By the softening influence of boiling water. 3. By softening it by vapour. 4. By softening it in heated sand. 5. By vapour under high-pressure. Of the first method of operation, a familiar example is afforded by the cooper, who bends the staves of his casks by kindling within the vessel formed of straight staves, a fire of straw or shavings. It is also used in bending the planks used in ship-building; but it is, on the whole, only applicable to timbers of small scantling; and in such cases as occur seldom, and where one or two pieces only are required to be bent. In the second method, the timber is immersed in water, which is heated until it boils, and is kept boiling until the timber is wholly saturated and softened. The timber being then withdrawn, is immediately forced to assume the required curvature, and is secured by nails or bolts. This proceeding has the defect of weakening the tim¬ ber, and lessening its durability. It should, therefore, be used only in such cases as do not require the qualities of strength and durability. In the third process, the timber is submitted to the ac¬ tion of the steam of boiling water. For this purpose it is inclosed in a box made perfectly air-tight. The box has a series of grated horizontal partitions or shelves on which the timbers are laid. From a steam boiler conveniently situated, a pipe is carried to the box. The steam acts on the timber, and in time softens it and renders it pliant. The time allowed for the action of the steam to produce this effect, is generally one hour for every inch of thick¬ ness in the planks. The fourth method of preparing the wood for bending, is by applying heat and moisture to it through the medium of the sand bath. The apparatus for this pur¬ pose is a furnace with flues, traversing the stone on which the sand is laid, in the manner of hot-house flues. There is also provided a boiler in which water is heated. On the stone a couch of sand is laid: in this the tim¬ bers are immersed, being set edgeways on a bed of sand about 6 inches thick, and having a layer of sand of the same thickness separating them, and being also co¬ vered over with sand. The fire is then lighted in the furnace, and after a time, the sand is thoroughly mois¬ tened with boiling water from the boiler before men¬ tioned. This watering is kept up all the time that the timber is in the stove. Thin deals require, as in the preceding case, an hour for each inch of thickness; but for thick scantlings the time requires to be increased; for instance, a 6-inch timber should remain in the stove eight hours. The fifth mode, by means of high-pressure steam, onty differs from the third process described in this, that the apparatus requires to be more perfect. The box, there¬ fore, is generally made of cast-iron, and all its parts are strengthened to resist the pressure to be employed. When the steam has a pressure of several atmospheres, the soften¬ ing of the wood is very rapid; and it is very effectually done by this method. After the timber is properly softened and rendered pliable, it is bent on a mould having a contour of the form which the timber is required to assume. The simplest method of doing this is shown in outline in Fig. 425. A series of stout posts, a a a, are driven into the ground, on a line representing the desired curve. The piece of wood m n, when softened, is inserted between two posts at the point where the curvature is to beffin, as at a b, and by means of a tackle, applied near that point, it is brought up to the next post, a, where it is fixed by driving a picket, c, on the opposite side. The tackle is shifted successively from point to point; and the pickets, c, d, e, are driven in as the timber is brought up to the posts. It is left in this condition until it is cold and dried; and then it is removed to make KNOWLEDGE OF WOODS. 103 way for another piece. The timbers of the roof, Fig. 3, Plate XXIX., were bent somewhat in this manner. But if the balk is required to be more accurately bent, and out of winding in its breadth, squared sleepers, a a a (Fig. 426, Nos. 1 and 2) are laid truly level across the line of curvature, and the posts b b are also accurately squared on the side next the balk. An iron strap c, which is made to slide freely, is used for attaching the tackle, and as the balk is brought up to the curve, it is secured to the posts 6, b by two iron straps, e e, e e (seen better in the vertical section, No. 2), which embrace the pieces /, on the opposite side, and are wedged up tight by the wedges h h. In operating in either of the ways described, only one piece of timber can be bent at a time. By the following method several pieces may be bent together:— Fig. 427 is a vertical projection, and Fig. 428 a trans¬ verse vertical section, of the apparatus. It consists of the horizontal pieces a a, arranged with their upper surface in the contour of the curve. They are sustained by strong framing b b, c c, cl cl. The timber is laid with its centre on the middle of the frame, and by means of pur¬ chases applied at both sides of the centre, and carried successively along to different points towards each end, it is curved, and secured by iron straps and wedges as before. The frame may be made wide enough to serve for the bending of other pieces, as m, n ; or for a greater number, by increasing the length of the pieces a a, and supporting them properly. The two apparatuses last de¬ scribed are taken from Colonel Emy’s work. These methods are not quite perfect; for in place of the timber assuming a regular curvature, it will obviously be rather a portion of a polygonal contour. To insure per¬ fect regularity in the curve, it is necessary to make a continuous template, in place of the several pieces aaaa. The substitution of the template for these, we need not, however, illustrate or describe, as its construction will be suggested by what has already been said. It is necessary to remark, that in all the cases, the tim¬ ber should be preserved from being injured by the iron strap, by a piece of wood inserted between them, as shown in the figures. Care must be taken that the curvature given to the timber is such as will not too greatly extend, and, perhaps, rupture the fibres of the convex side, and so render it useless. The process of bending timber which we have described, is, as will be seen, restricted to very narrow limits. The effect, when the curve is small, is to cripple the fibres of the inner circumference, and to extend those of the ex¬ terior, and the result is, of course, a weakening of the timber. Kecently, however, a process has been patented by an American gentleman, Mr. T. Blanchard, in which the bending, effected by end pressure, is not only not at¬ tended with injurious effects, but on the contrary, gives to the timber qualities which it did not before possess. In an able article in Household Words , the advantages of this new process are set forth as follows:—“ The principle of bending, as employed in this new application, is based on end-pressure, which, in condensing and turning at the same time, destroys the capillary tubes by forcing them into each other. These tubes are only of use when the tree is growing, and their amalgamation increases the density of the timber, the pressure being so nicely ad¬ justed that the wood is neither flattened nor spread; nor is the outer circumference of the wood expanded, though the inner is contracted. Now, the error of the former process, as expounded by competent judges, has arisen from the disintegrating of the fibre of the wood by ex¬ panding the whole mass over a rigid mould. Wood can be more easily compressed than expanded; therefore, it is plain that a process which in¬ duces a greater closeness in the component parts of the piece under operation—which, as it were, locks up the whole mass by knitting the fibres together—must augment the degree of hardness and power of resistance. The wood thus be¬ comes almost impervious to damp, and to the depredations of insects, while its increased den¬ sity renders it less liable to take fire; and the present method of cutting and shaping timber being super¬ seded, a saving of from two to three fourths of the mate¬ rial is brought about. The action of the machine throws the cross grains into right angles, the knots are compelled to follow the impulse of the bending, the juices are forced out of the cells of the wood, and the cavities are filled up by the interlacing fibres. In the same way, you may sometimes see in the iron of which the barrels of muskets are made, a kind of dark grain, which indicates that the particles of the metal, either in the natural formation or in welding, have been strongly clenched in one another. These specimens are always valued for their extraordinary toughness, as well as for a certain fantastical and mottled beauty. “ Another of the good results of this method is, that the wood is seasoned by the same process as affects the bending. The seasoning of wood is simply the drying of the juices and the reduction of the mass to the mini¬ mum size before it is employed, so that there should be no future warping. But, as we have already shown, the compression resorted to in the American system at once expels the sap, and a few hours are sufficient to convert green timber into thoroughly seasoned wood. Here is an obvious saving of time, and also of money; for the ordi¬ nary mode of seasoning by causing the wood to lie waste for a considerable period, locks up the capital of the trader, and of course enhances the price to the purchaser. Time also will be saved in another way, in searching for pieces of wood of the proper curves for carrying out certain 104 PRACTICAL CARPENTRY AND JOINERY. designs. £ How delighted/ says Mr. Jervis, the United States inspector of timber, ‘ will the shipwright be to get clear of the necessity of searching for crooked pieces of timber; there need be no longer any breaking of hats in the frame, as we have been wont to break them. We shall see Nos. 1, 2, and 3 futtocks, at least, all in one piece.’ An English architect, Mr. Mayhew, remarks that ‘ one of the advantages of this method is, that in its application to all circular, wreathed, or twisted work, it I not only preserves the continuous grain of the wood, which | varies according to the quickness of the sweep, and will give the artist greater freedom in his designs, by allowing him to introduce lines which are now cautiously avoided, in order to prevent the cost of their execution.' ” Mr. Mayhew further observes, that the process has the capability of bending into a permanently set form any Avood up to 16 inches square, hoAvever hard, not only Avithout injuring its fibres, but positively rendering the Avood more rigid, and, at the same time, increasing its strength to such an extent, that in a structural point of view, in many cases, it will supersede the necessity of using iron. Fig. 429 shows the form of the machine for timbers under 6 inches square. Figs. 430 and 431 show the machine for heavy timbers above that scantling. The principle, as lias been stated, is the application of end- pressure; but another characteristic feature is, that the timber, during the process, is subjected to pressure on all sides, by which its fibres are pre¬ vented from bursting or from being crip¬ pled; and, in short, the timber is prevented from altering its form in any other than the desired manner. The set imparted to it becomes permanent after a few hours, during which time it is kept to its form by an enveloping band and a holding bolt, as shown in Fig. 432. SEASONING OF TIMBER, AND THE MEANS EMPLOYED TO INCREASE ITS DURABILITY. The perfect desiccation of timber appears to be one of the best means of insuring its preservation; and this has been sought to be accomplished by applying heat to the pieces of wood. This must be done gradually, as the sudden application of heat with the view of drying the timber has the effect of rending or cracking the exterior before the interior has time to dry. Burying the timber in dry sand, so that the sun may evaporate the moisture gradually, and covering it with quicklime, to produce a gradual heat, have both been resorted to—the former with success in warm countries—the latter, although making the timber dry, compact, and hard, often producing rents by the difficulty of regulating the heat. Stoves or ovens were also resorted to; but these were injudiciously con¬ structed, and it Avas found that timber, when of large di¬ mensions, could not be completely dried in them. There Avas no pro\'ision in them for conveying away the vapour generated by the application of the heat. But recently the plan of drying by subjecting the timber to the action of a current of air highly heated, so as to have its capa¬ city for moisture greatly increased, has been adopted with the happiest results. But mere desiccation does not secure the end aimed at; for it does not exhaust the vegetable matter from out is noAV usually and laboriously done by narrow slips of veneers glued on cores cut across the grain, with many unsightly joints, ill concealed at best, but it will mate¬ rially reduce the cost of all carved work, Avhich now KNOWLEDGE OF WOODS. 105 the pores of the wood: it only dries it there; and when it is exposed to humidity it becomes fluid, and resumes its tendency to fermentation. Immersion in water for such a time as shall permit this matter to be dissolved and washed out of the wood pre¬ vious to the desiccating process being applied, will secure it from the tendency to corruption when again exposed to humidity. For this, running water is obviously pre¬ ferable to stagnant water; and it may fairly be inferred that to the immersion for a long time in the rivers in which they are floated down to the ports for embarkation, is to be attributed the greater durability of the pines of the Baltic, when they are properly treated by thorough drying before being used. But to render this immersion effectual, it is requisite that it be total and complete, and that it be not too long continued. It is considered that the limit of duration is from three to four months. Immersion in hot water effects the same purpose much more rapidly; but as the wood has to be submitted to the action of the water for ten or twelve days, the expense is prohibitory of the process, unless in cases where the con¬ densing water of a steam engine in constant operation can be made available. As we have before remarked, when speaking of the bending of timber, the action of the hot water impairs its strength, and should not be used where strength is an object. Immersion in salt water is a means of adding to the durability of timber. It increases its weight, and adds greatly to its hardness. It is attended, however, by the grave inconvenience of increasing its capacity for mois¬ ture, which renders this kind of seasoning inapplicable for timber to be employed in the ordinary practice of the carpenter. The water seasoning of which we have been speaking, has many objectors; and their strongest arguments are founded on the facts that there are examples of roofs which have existed for ages, the timbers of which have not been subjected to this water seasoning. But numerous experi¬ ments prove, beyond contradiction, that timber immersed in water immediately after being felled and squared, is less subject to cleave and to decay, and that it dries more quickly and more completely; which proves that the water evaporates more readily than the sap, of which it has taken the place. The immersion, however, impairs, to some extent, the strength of the timber; and this consi¬ deration indicates the applicability or non-applicability of the process. When the timber is required for purposes for which dryness and easiness of working are essential, then the water seasoning may be employed with advan¬ tage; but when for purposes in which strength alone is the great requisite, it should not be used. Sir Samuel Bentham found, that large timber, when left with its sap-wood on, in the course of a few years had become dry, compact, and hard in the heart; but where the sap-wood had been taken off, as in sided timber, the exterior became more or less crooked and damaged before the interior was properly seasoned. The greatest objection to this mode of seasoning is its costliness, aris¬ ing from the loss of interest on the capital invested. As the condensation produced by heat increases tne hardness of timber, it has been imagined that charring its surface, by increasing its hardness, would also increase its durability. In this supposition it is probable that the custom of charring the ends of piles and posts which are to be buried in the earth, has originated. The carbon¬ ized portion of the wood may, indeed, hinder the imme¬ diate contact of the humid earth with the non-carbonized wood; but it is to be questioned whether the sound tim¬ ber destroyed in the charring would not have been as good an envelope as the charred surface; and taken quite as long to be destroyed by its contact with the earth as the other would act as a protection. In place of charring the ends of posts or piles, therefore, it would seem better to coat them with some substance impervious to air. But timbers buried in the earth begin as often to rot from within as from without, by the fermentation of their natural juices, as they are too often employed without being submitted to any kind of seasoning process what¬ ever, while, in timbers so placed, the protection of tho¬ rough seasoning is especially requisite. The gradual combination of the combustible elements of a bod} r with the oxygen of the atmosphere, produces a slow combustion or oxidation, to which Liebig applies the term ercmacausis. The ercmacausis of an organic matter is retarded or completely arrested by all those substances which prevent fermentation or putrefaction. Mineral acids, salts of mer¬ cury, aromatic substances, empyreumatic oil, and oil of turpentine, possess a similar action in this respect.* Timber, after being framed, is subject to the same dis¬ eases and causes of decay as before. Often, indeed, the latent diseases only develope themselves when the timber has been worked and framed, and when the replacing of the affected by a sound piece may be very difficult, or altogether impossible. Besides the diseases proper to the species of tree, to the soil, or to the climate, or those caused by any of the acci¬ dents which have been described, timber is liable to the attacks of insects, which are often detrimental to it, and not seldom altogether destructive of it. Among the in¬ sects whose attacks are most fatally injurious to the wood, are the white ant, the Teredo navalis, a kind of Pholas, and the Limnoria terebrans. The white ant devours the heart of the timber, reducing it to powder, while the surface remains unbroken, and affords no indication of the ravages beneath. The teredo and pholas attack wood when submerged in the sea. The teredo, its head armed with a casque or shell in the shape of an auger, insinuates itself into the wood through an almost imperceptible hole; it then in its boring operations follows the line of the fibre of the wood, the hole enlarging as the worm increases in size. It forms thus a tube, extending from the lowest part of the timber to the level of the surface of the water, which it lines with a calcareous secretion. A piece of timber, such as a pile in a marine structure, may be perforated from the ground to the water level by a multitude of these creatures, and yet no indications of their destructive work appear on the exterior. The pholas does not attack timber so frequently as the teredo; and its ravages are more slowly carried on. Its presence in the wood, therefore, though very dangerous, is not so pernicious as the other. For the protection of timber from disease, decay, and * Liebig’s Chemistry of Agriculture and Physiology. O 106 PRACTICAL CARPENTRY AND JOINERY. the ravages of insects, various means are employed. These may be classed as internal and external applications. I. Preservation of Wood by impregnating it with Chemical Solutions. The chemicals usually employed in solution are the dcutochloride of mercury (corrosive sublimate), the prot¬ oxide of iron, the chloride of zinc, the pyrolignite of iron, arsenic, muriate of lime, and creosote. They are either used as baths, in which the timber is steeped, or they are injected into the wood by mechanical means; or the air is exhausted from the cells of the wood, and the solutions being then admitted, fill completely every vacuum. r l he saturation with corrosive sublimate is called Kyan- izing, from the name of the inventor, Mr. Kyan. When this is performed by steeping, the time required is gene¬ rally estimated as follows: — Scantlings of 14 inches square, fourteen days; of 7 inches square, ten da} r s; and for pieces 3 inches square, seven days are sufficient. The saturation with the solution of the chloride of zinc is the patent process of Sir William Burnett. The in¬ jecting sulphate of iron and muriate of lime is Payne’s patent process. The creosote is patented by Mr. Bethell. All of these processes are advantageous under certain circumstances; but it cannot be said that any of them is infallible. It is not easy, however, to discover whether, iu cases of failure, there may not have been some defect in the process; and therefore, in important work, the additional security against the ravages of disease and decay which the impregnation gives, when properly per¬ formed, should not be neglected. But it is to be feared that against the attacks of the marine pests—the teredo, the pholas , and the Limnoria terebrans —the protection these processes afford is at the best doubtful. An exception to this may probably be taken in favour of Mr. BethelPs creosote process. The soluble salts are supposed to act as preservatives of the timber, b} r coagulating its albumen; thus the very quality of combin¬ ing with the albumen destroys the activity of the salts as poisons, and hence although preservatives against decay, they may, when thus combined, be eaten b} 7 an insect with impunity. With creosote, however, the case is different. It fills the vessels of the wood, and its smell is so nauseous that no animal or insect can bear it. It is also insoluble in water, and cannot be washed out. It is thus a protec¬ tion to the wood against the ravages of insects, and also a preservative from decay. But there is great difficulty in injecting it into the heart of the wood; and into hard woods it cannot be perfectly injected. Mr. Rendell con¬ siders that for marine purposes the creosote should be used in the proportion of 10 lbs. to the cubic foot. For ordi¬ nary purposes much less is required. Previous to the application of any of these substances, however, and as a preparative for it, it is essential that the timber be thoroughly deprived of its moisture. In regard to this, Mr. Davidson says, that— 1. Different woods and different thicknesses of wood require different degrees of heat. 2. Hard woods, and thick pieces of wood, require a moderate degree of heat, from 90° to 100". 3. The softer woods, as pine, may be safel} 7 exposed to 120° or even to a higher temperature. When cut exceed- ingly thin, and well clamped, 182° or 200° have been found to harden the fibre and increase its strength. 4. Honduras mahogany boards, of 1 inch thick, may be exposed with advantage as regards colour, beauty, and strength, to even 2S0' or 300’. A piece of this wood 14 inch thick, cut fresh from the log, was deprived wholly of its moisture, amounting to 3G per cent., by exposure to a temperature of 300° for fifty hours. But in practice, from 115° to 120" of temperature arc the best calculated to secure perfect desiccation in slabs of moderate thickness. Supposing the current of heated air to be kept up during twelve hours every day with this temperature, one week may be allowed for every inch thick of the timber, up to 4 inches; but the time must be increased when the thickness exceeds 4 inches, to seven weeks for G inches, and ten weeks for 8 inches. If the tem¬ perature is increased, and the blast of air made continuous, the desiccation may be effected in forty eight hours. An exception must be made in regard to English oak, which should never be exposed to a higher temperature than 105°. The velocit} 7 of the heated current should be 100 feet per second, and the area of outlet for the moisture and used air should be greater than the area of inlet. When the timber is perfectly deprived of its moisture, it is in a condition for the application of the preservative ! agent. The different agents and processes in use may be briefly described. Kyanizing. —In 1S32, Mr. Ivyan took out a patent for soaking timber in chloride of mercury or corrosive subli¬ mate. In cases where this was properly applied, it seems to have been effective; but as it is expensive to apply the solution of sufficient strength, the process came to be im¬ perfectly carried out, and consequently failed. Margary’s Process. —This was patented in 1837- It consisted in soaking the timber in a solution of acetate or sulphate of copper. It has been extensively used, and when the solution is of proper strength, and a sufficient quantity is absorbed, it is also efficient to a certain extent. Sir William Burnett’s Process. —In 1838, this process, which consists in impregnating the timber with a solution of chloride of zinc, was patented. The principle assumed by the patentee is, that the chloride forms an insoluble compound with the albumen of the wood; and this is the theory of action of the chemical compounds already named. It appears, however, that all such agents lose, in time, their efficacy, apparently because the aqueous portion evapo¬ rates, and the timber again absorbs the humidity of the atmosphere. The constant alternations of wet and dry so weaken the solution as to render it inoperative. In Sir William Burnett’s process, the hot solution of the chloride is forced into the timber under pressure in cylinders hermetically sealed. In heating the solution, a horse-shoe boiler on the circulating principle is used, and is found to answer well for this and for Margary’s process-a sufficiently high temperature being maintained at a moderate cost. Payne’s Process. —This was patented in 1811. In this, two solutions are used in succession; the first, an earthy or metallic solution, is forced into the timber under pres¬ sure; and the second, a decomposing fluid, is then forced in, and forms with the former an insoluble compound in the pores of the wood. Thus, sulphate of iron and car¬ bonate of soda will form oxide of iron in the cells of the timber. When this operation is well performed, as in KNOWLEDGE OF WOODS. 107 France, the results have been satisfactory. According to experiments made under the direction of Captain Moor- soin, in 1839, it would appear that the chemical preser¬ vatives injure, to some extent, the transverse strength of the timber. The ratio of strength in Archangel deal and American pine, in their prepared and natural states, appears to be as 976 to 1000. The value of tars and essential oils as preventives of the decay of timber has been long known, and so early as 1737, a patent was granted to Alexander Emerson for the application of hot boiled oil mixed with poisonous substances. In 1754, a patent was granted to John Lewis for the application of a varnish made from the juice of the pitch pine; and also for a process for distilling plantation tar, to be applied for the preservation of wood. None of the processes came into extensive use, chiefly on account of want of skill in their application. Mr. Beth¬ el 1, in 1838, took a patent for impregnating timber with creosote; and this process is so effectual, that it is in con¬ stant use in cases where the odour of the creosote is not an obstacle to its employment. The preservative properties of creosote are said to be owing to its coagulating the albumen, preventing the absorption of moisture, and to its being fatal to animal and vegetable life, thereby arresting the vegetation of the tree, preventing the growth of fungi, and repelling the attacks of insects. M. Boutigny, in conjunction with M. Hutin—proceed¬ ing on the acknowledged theory, that the moisture and oxygen of the air penetrating into the heart of the wood by absorption and filtration, produce eremacausis, and that these elements of destruction appear to act chiefly at the ends of the timber—conceived that if, after the timber was completely deprived of moisture, the ends of its pores were hermetically sealed, absorption, and consequently decay, would be prevented. They accordingly introduced the system of desiccating the timber, partially charring its ends, and then immersing them in oil of scliistus, or some analogous substance. This penetrates with rapidity, the ends are then blazed off, and plunged to the length of a few inches into heated pitch, tar, or gum-lac, which completely seals the pores. Dr. Boucherie, arguing that all the changes in woods are attributable to the soluble parts they contain, which either give rise to fermentation or decay, or serve as food for the worms; and that, as the result of analysis, sound timbers contain from three to seven per cent, of soluble matters, and the decayed and worm-eaten rarely two— commonly, indeed, less than one per cent.—concludes, that since the causes of the changes it undergoes originate in the soluble matters of the wood, it is necessary, for its preservation, either to extract the soluble parts, or to make them unchangeable by introducing substances which should render them unfermentable or inalimentary. This he considers may be effected by many of the metallic salts and earthy chlorides. He shows, by experiments on vege¬ table matters very susceptible of decomposition, such as the pulps of carrot and beet-root, the melon, &c., which differ from wood only in the greater proportion of soluble matter they contain, that in their natural states they rapidly alter, but are preserved by the pyrolignite of iron. Dr. Boucherie conceived that if solutions of sulphate of copper, pyrolignite of iron, or other salts, could be made to take the place of the natural juices of the plant while it yet lived, the vessels of the tree would become filled with the fluid by the process which he calls aspiration. He supposed that by using proper solutions he should be able to protect the wood from dry or wet rot, to augment its hardness, to preserve and clevelope its flexibility and elasticity, to render change of form impossible, to prevent warping and cleaving, and to render it incombustible, or at least to reduce its inflammability, and, lastly, to give to it various colours and odours. The method of proceeding first adopted was to employ the vital energy of the tree to draw the liquid into its vessels by means of the circulation of the sap. This was effected either by sawing the tree above the root, and im¬ mersing it vertically in a bath of the fluid, or by girdling the tree, that is, cutting it all round with a saw, so deep as to leave only a pin in the centre sufficient for its sup¬ port, and surrounding the cut with a trough, into which the fluid was poured. When pyrolignite of iron was the fluid used, the hard¬ ness of the timber was more than doubled. The quality of flexibility was increased by the chloride of lime and other deliquescent salts. Warping and splitting were stayed by a weak infusion of the chloride of lime. Inflam¬ mability was diminished by earthy chlorides. Mineral succeeded better than vegetable colours in the process of dyeing the wood. Resins dissolved in essential oils, by their being absorbed, rendered the "wood impervious to water. It is right to add, that Mr. Bethell, already mentioned, patented a similar process in 1838, two years before Dr. Bouclierie’s method was made known in France. Mr. Beth ell’s specification says: “Trees just cut down may be rapidly impregnated with the solutions, by merely placing the butt ends in tanks containing them. They will thus circulate with the sap throughout the whole tree; or it may be done by bags of water-proof cloth affixed to the butt ends of the trees and filled with the liquid.” Pyro¬ lignite of iron is especially mentioned as circulating freely with the sap. But the process now adopted is that of forcing the liquid through the timber, and is carried on as follows:— After the tree is felled, a saw cut is made across its centre, and nearly through its diameter. By slightly raising the tree under the centre by a wedge, the cut is opened a little, and a piece of string is inserted in it a little within the lip or edge all round. On lowering the centre, the cut closes on the string, which forms a water-tight joint; a hole is then bored obliquely into the cut, and a hollow plug driven into it. A flexible tube is then fitted to the plug, and its other end carried to a cistern containing the solution, and placed high enough to give the requisite pres¬ sure. When the communication is completed, the liquid flows into the cells of the wood from the centre towards the ends, driving out the sap before it. When the solu¬ tion appears at the ends, the impregnation is complete. When the timber cannot be divided in the middle, one of the ends is capped by a piece of board about an inch thick. This is attached by screws, or by screwed dogs. The joint is made as before, the cap is tightened up, and the liquid injected in the same manner. To make certain that the sap has been entirely replaced by the solutions, a chemical test is applied. For example, PRACTICAL CARPENTRY AND JOINERY. I OS when the solution is sulphate of copper, a piece of prus- siate of potash is rubbed on the end of the timber, when, if the solution has reached the end, a deep red-brown stain is produced. The solution now preferred for use is formed of one part b} r weight of sulphate of copper, dissolved in 100 parts of water. All woods do not, of course, absorb the same amount of solution, and the sap-wood absorbs more in propor¬ tion than the heart-wood. From this it may be inferred, which is the fact, that the process is attended with the best results when applied to the commonest and cheapest kinds of timber. The general estimate is, that the quan¬ tity absorbed is equal in cubic extent to one-half the cubic dimensions of the timber. The longer the injecting pro¬ cess is delayed after the felling of the timber, the slower is its progress. In newly felled timber, a log 9 feet long occupies two days, when the head pressure is oh feet. Three months after felling, the process would occupy three days; and four months after felling, four days. One great advantage of this process is, that the timber requires no drying or previous preparation of any kind. II. Preservation by Paints and other Surface Appli¬ cations. Timber, when wrought, and either before it is framed, or when in its place, is coated with various preparations, the object of which is to prevent the access of humidity to its pores. In the application of such surface coatings, it is essential that the timber be thoroughly dry; for if it is not, the coating, in place of preserving it, will hasten its destruction, as any moisture contained in it will be prevented from being evaporated, and will engender in¬ ternal decay. This result will be more speedily developed as the colour of the coating is more or less absorbent of heat. One of the most common applications to timber con¬ structions of large size, is a mixture of tar, pitch, and tallow. The mixture is made in a pot over a fire, and applied boiling hot. In the use of this too great caution cannot be employed to prevent danger from fire. The bridge of Dax, on the Ad our, was entirely burned imme¬ diately after its construction, and when the tarring had just been completed. The pontoon on which the mixture was prepared was carefully kept to the leeward of the bridge; but the mixture in one of the pots having taken fire, and the wind changing suddenly, the flames were driven against one of the piles, which instantly ignited. The fire spread with a prodigious rapidity, enveloping in flames the -whole structure, and in a short time entirely consumed it. Another preservative for large works is painting with sand. It is thus performed:—When the wood is perfectly dry, a coating of some cheap pigment, ground in drying oil, is given to it; and while it is wet, it is dusted over with sand, either by means of a box with a perforated lid, or by a sieve, when the surface is horizontal, or simply by the hand. The sand should be purely silicious, well washed, and perfectly dry. When the first coat is quite dry, it is brushed over with a stiff brush, to detach the loose particles of sand, and then a second coat of paint is applied, and sanded over, like the first. When dry, this is brushed, and a third coat is applied and sanded in like manner. The number of coats depends on the circum¬ stances of the case. The finishing coat should be of good oil paint of a proper colour. When this kind of coating is executed with care and attention, it has great solidity. It fills the cracks of the timber and the joints of the framing, and is a good pre¬ servative. Its surface, however, is necessarily rough and granular; and it, therefore, is not adapted for work in which neatness is desired. But the most universally applicable protective, coating is good oil paint. To render the paint fit for works of carpentry, it is necessary that the oil should be good, the paint insoluble in water, and thoroughly ground with the oil, and that in its application it should be well brushed with the end, and not with the side of the brush. Such a coating has not the disadvantage of weight, like the painting with sand; nor does it, like it, alter the form of the object to which it is applied. The timber to be painted in oil should be planed smooth; and it is essentially requisite that it be dry. It is usual to submit it to the action of the air for some time before painting, and then to take advantage of a dry season to apply the paint. To render effectual any of the surface coatings we have mentioned, it is necessary to take care that the joints of framing are also coated before the work is put together. If this be neglected, it will happen that although any water which may fall on the work will evaporate from the surface, some small portions may insinuate themselves into the joints, and these remaining, will be absorbed by the pores of the wood, and become the cause of rot. The joints of all exposed work should, therefore, be well coated with the protective covering before it is put together. Besides these fluid compositions, timber exposed to the action of marine insects is often covered with a sheathing of metal, usually copper. This metal is, however, very rapidly destroyed by the action of sea-water, and does not afford a protection against the ravages of these crea¬ tures. Broad-headed scupper nails are sometimes used, and the corrosion which ensues by the action of the salt-water indurates the wood so as often effectually to protect it. PROTECTION AGAINST FIRE. To render wood incombustible has frequently been attempted, but with no great success. When we consider the number of structures which are composed of timber, and which, by a slight accident, may become a prey to fire, we cannot wonder at the many attempts which have been made to prevent such a disaster. The means proposed have been —1st. To impregnate the wood with saline solutions; 2d. To cover it with some incombustible coating or cement; and Sd. To sheathe it with metal. All these means are attended with great expense, and incompletely fulfil the purpose. Some of the saline solutions have the effect of rendering the wood more susceptible of atmospheric influences. They enable it to resist only the first attack; as the heat augments, the water of the salts evaporates, and the salts themselves decrepitate, and leave the wood a prey to the flames. It is said, however, that Sir William Burnett’s process for the prevention of decay insures also the incombustibility KNOWLEDGE OF WOODS. 100 of tlie wood; and that in the most intense tire, timber so prepared, would only be charred, and would never burst into llame. The external coatings of non-conducting substances serve also to resist only the first attack of the dames. They are soon either detached from the timber, or they become so heated as to reduce the wood to charcoal. Metallic envelopes, infusible at the first, become soon highly heated, and more speedily reduce the wood to charcoal than the non-conducting coverings. It is, there¬ fore, useless to reckon on the efficacy of any of these means of rendering wood incombustible. DESCRIPTION OF WOODS. HARD WOODS. The Oar.— The oak is the greatest, the strongest, and the most durable of all the forest trees of this country. It is a native of temperate climates, and is not found in either the torrid or frigid zones. Neither is it found, even in temperate climates, at elevations where, the tem¬ perature is very low. It grows naturally in the middle and south of Europe, in the north of Africa, in Asia, in Natolia, the Himalayas, Cochin-China, and Japan. In America it is abundant, especially in the United States. Of the oak there are many species; the most common of which, as the subjects of forest culture in this country, are the Quercus robur or pedunculata (common oak), Quercus sessiliflora (the sessile-fruited oak). The former has its fruit on a long foot-stalk or peduncle; the latter has its fruit sessile, or on a very short stalk. The common oak is of slower growth than the other, which, moreover, tends to grow with a more erect stem and less tortuous branches. The common oak is believed to be more dur¬ able than the sessile-fruited oak; but the cause of the difference in their durability is by some assigned to the modifications produced by soil and climate. The oak, although growing in a wide range of soils, prefers the clayey' - , and it is in the alluvial deposits of Eng¬ land and Scotland where the noblest specimens of this tree are to be found. The oak is the most solid and durable of European woods. It is certain that carpentry structures of oak timber have remained in perfect preservation for more than 600 years. When immersed in water it becomes excessively hard, and is nearly imperishable. Although the oak does not reach the height of some of the pines and palms, and its trunk never attains the enor¬ mous magnitude of those varieties of which we have spoken, it is nevertheless found of very large dimensions. The trunks of trees of this species have been known to grow to a height of 140 feet, and to measure more than 30 feet in circumference. An enormous oak was disco¬ vered in Hatfield Bog, Yorkshire; it was 18 feet in circum¬ ference at the upper end, and 36 feet at the lower end, and although but a fragment, measured 120 feet in length. The general height of British oaks, however, is from 60 to 80 feet; and of American oaks, from 70 to 90 feet. The oak grows very slowly. It has been known at 100 years old to be only 1 foot in diameter. Until the age of forty years it grows pretty fast, but after that its increase becomes less and less sensible. At 200 or 300 ears old, these trees are at their best. Vancouver, from observations on the growth of timber in Hampshire, arrived at the conclusion that the relative growth of wood in that county, taking the trees at ten years’ growth, and the oak as a standard is—Oak 10, elm 16, ash 18, beech 20, white poplar 30. That is to say, in any given time, if the growth of oak be 1, the growth of white poplar will be 3. In 1792, an oak at Vimbush, in Essex, measured 8 feet 5f inches in girth, at 5 feet from the ground; while a larch at the same height measured only 2 feet 4 inches. Thir¬ teen years afterwards, the girth of the oak was 8 feet 10£ inches, and of the larch 5 feet 1 inch. Of the species in commonest use, the following are the general characteristics:— Quercus pedunculata. —The Quercus pedunculata, or common oak, attains the greatest height of any of the oak species, and appears to be the most valuable, in re¬ spect of the durability of its timber. The wood is more stiff, and yet more easily split and broken, than that of the sessiliflora. Its colour is lighter, and its specific gravity not so great. Tredgold gives the following sum¬ mary of results of his experiments on the two kinds:— Quercus Quercus pedunculata. scssilijtora. Specific gravity,. •807 •870 Weight of a cubic foot iu lbs., ... Comparative stiffness, or weight that 00-47 54-97 bent the piece 2 7 yths of au inch, Comparative strength, or weight that 1C7 149 broke the piece, . 322 350 Cohesive force of a square inch in lbs. ... Weight of the modulus of elasticity in 11,502 12,600 lbs. for a square inch, ... 1,648,058 1,4 ; 7l,256 Comparative toughness, ... 81 108 In the Dictionnaire des Eaux et Forets , the following results of experiments made by Hartig, are cited:— Quercus Quercus pedunculata. sessilijtom. The wood, when green, weighs ... 7613 lbs. 80 5 lbs. When half dry, ... ... ... 65"9 „ 07'12 „ When pei’fectly dry, .52-13 „ 51-10 ., The discrepancy in the experiments may be caused by the different circumstances of soil and climate under which the trees were produced. The wood of the pedunculata contains more of the silver grain than the other, and is, on that account, preferred, as more showy, for ornamental work. It also splits clean, which renders it suitable for split paling, laths, barrel staves, and dowels. Its stiffness recommends it for beams, and its quality of resisting alternations of wetness and dryness renders it invaluable for piling. Quercus Ilex .—This is a deciduous tree, and is on this account called by the French chene vert. It grows in the meridional parts of Europe. It is ordinarily tor¬ tuous, which unfits it for general use in carpentry; but as its wood is hard, compact, heavy, and durable, it is em¬ ployed in the construction of machines. Quercus suber. — This species is valuable, chiefly as affording in its bark the material of which corks are made. Its wood rots rapidly when exposed to alterna¬ tions of wetness and dryness. Quercus Pyrenaica (the Pyrenean oak), called also Black Oak.—This species has more alburnum than the others. Emy says—“Its wood is very cross-grained, and no PRACTICAL CARPENTRY AND JOINERY. requires to be left to dry in the bark for five or six years. It is liable to the attack of worms, and grows so tough that it is difficult to work, and breaks the workman’s tools. It is also knotty, and is not a good wood for carpentry." Quercus Ccrris (the Turkey, or Mossy-capped Oak).— This fine species attains a great size. Its wood is of ex¬ cellent quality, and beautifully mottled. Of this kind is the oak of Holland and the Sardinian oak. The former, like all trees of humid soils and climates, grows with a straight fibre, arid has soft wood, easily worked. It is not so strong or durable as the common oak; but under the name of wainscot, it is extensively used for interior finishings and for cabinet work. The Quercus virens is one of the more than seventy species of oaks which are indigenous to America. Its timber in quality and in appearance approaches to that of the British oak, and in the United States it is pre¬ ferred for the purposes of the ship carpenter. The Quer¬ cus virens is confined to the southern states, and is not imported into this country as timber. Stevenson says, that “the sea air seems essential to its existence, as it is rarely found in forests on the mainland, and never more than fifteen or twenty miles from the shore. It is com¬ monly 40 to 50 feet in height, and 1 to 2 feet in diame¬ ter; but it is sometimes much larger.” The Quercus alba , or White Oak, a native of the north¬ ern states of America, is the species from which the sup¬ ply is obtained for the British market. It is not equal to the British oak in strength or durability, and it is in¬ ferior to wainscot in the beauty of its markings or champ; it is also of coarser grain. But its comparative cheap¬ ness causes it to be extensively used in carpentry, joinery, and cabinet-making. The best quality, where durability is required, is the second growth of New Hampshire, which is extensively used in America in ship-framing. The better the quality of this oak, the more it shrinks in drying, and it is liable to split in the sun. The Yellow Oak is much used in ships for hatch-coam¬ ings, windlasses, &c. ; and by agricultural implement makers for axles, as it does not split in the sun. The Chestnut (Castanea vesca).— The leaf is 5 to 7 inches long, and IS to 24 lines broad, bordered with large sharp teeth. The flowers are in bunches as long as the leaves, and the fruit is within a spherical envelope, stud¬ ded with spines. There are two varieties of the chestnut known in Europe. The one produces as fruit the common chestnut, which is slightly flattened by two or three grow¬ ing together in the same envelope; the other produces the large chestnut, which is nearly entirely round, and each nut has a separate cover. The chestnut sometimes grows to a prodigious size. The largest tree in Europe, the Chestnut of an Hundred Horses, on Mount Etna, is of this species. Brydone, in his Tour in Sicily, describes it as appearing at first sight like “a bunch of five large trees growing together;” but a short examination convinced him that it really was a single tree split into five parts. Careful measurement gave the enormous circumference of 204 feet. This tree is supposed to be more than 3000 years old. The Totworth chestnut measured, in 1830, 50 feet in circumference. The mean diameter of this tree, however, is about 37 inches, and it grows to an average height of 44 feet. The wood is very like that of the oak, and is liable to be con¬ founded with it. This resemblance led to the supposition that several old constructions in carpentry were formed of chestnut, but which better examinations have shown to be a variety of the sessile-fruited oak. The chestnut was formerly much used for house car¬ pentry and for furniture. The old wood is rather brittle and shaky, and is liable to internal decay; but the young wood is elastic and durable, and is much used for the rings of ships’ masts, hoops for tubs, churns, &c. In the Transactions of the Society of Arts for 1789, there is an account of the comparative durability of oak and chestnut when used for posts. “Posts of chestnut and others of oak had been put down at Wellington, in Somersetshire, previous to 1745. About 1763, when they had to undergo repair, the oak posts were found to be unserviceable, but the chestnut were little worn. Accordingly, the oak ones were replaced by new, and the chestnut allowed to re¬ main. In twenty-five years (1788) the chestnut posts, which had stood twice as long as the oak, were found in much better condition than those. In 1772, a form was made partly of oak and partly of chestnut, the trees used being of the same age, and were what may be termed young trees. In nineteen years the oak posts had so decayed at the surface as to need to be strengthened by spars, while the chestnut ones required no support. A gate-post of chestnut, on which the gate had swung fifty- two years, was found sound when taken up; and a barn constructed of chestnut in 1743, was sound in every part in 1792. It should seem, therefore, that young chestnut is superior to young oak for all manner of wood-work that has to be partly under ground.” Tredgold states the weight of a cubic foot of chestnut at from 43 to 54 8 lbs., and the specific gravity of the tim¬ ber at -535. Rondelet gives ‘657 as the specific gravity, and 41 lbs. as the weight of a cubic foot. The specimens shown in the Exhibition of 1851, weighed from 27'5 to 36 6 lbs. per cubic foot, and their specific gravity was re¬ spectively, ’438 and ’583. According to Tredgold, the cohesive force is from 9570 to 12,000; Rondelet says 13,300. Its stiffness to that of oak is as 54 to 100 Its strength “ “ 48 “ 100 Its toughness “ “ 85 “ 100 The Elm ( Ulmus ).—The elm is a large tree, common in Europe. Its mean height is 41 feet, and its mean diameter 32 inches. There are fifteen species. Its bark is rough and dark coloured. Its leaves arc oval and toothed, and their colour is a deep rich green. Its flowers appear be¬ fore its leaves, they are disposed in close bundles, and are very numerous along the branches. Its wood is ruddy brown, very fibrous, hard, flexible, and of a dense appear¬ ance, subject to warp, and tough and difficult to work. It is subject to the attacks of worms, and in carpentry it is only used in default of better for works above ground. It is not liable to split, and bears the driving of nails or bolts better than any other wood. "When constantly wet it is exceedingly durable, and is therefore much used for the keels of vessels and in wet foundations, in water¬ works, for piles, pumps, and water-pipes. Its toughness fits it for the naves of wheels, shells for tackle-blocks, and for many uses in turnery, as it bears rough usage without splitting. KNOWLEDGE OF WOODS. m Wycli Elm grows sometimes to the height of 70 feet, and attains a diameter of 3| feet. The stem is less en¬ cumbered with brandies; the wood is lighter, yellower, straighter and finer in the grain than the other. It is tough, and is fitted for works in which it requires to be bent; hence it is much used by coach makers for the naves, poles, and shafts of gigs and carriages, and by shipwrights for jolly boats; it is used, too, for dyers’ and printers’ rollers. The Scotch elm, which is much superior to the English elm, appears to be of this species. It is much finer, harder, closer in the grain, and handsomer in its appearance than the other, and is used in making articles of furniture. In days of old the wood of this species was held in esteem for the making of long bows. Rock Elm is very like the last, and is used by boat builders. Dutch Elm is the worst of all the species. The Twisted Elm yields an ornamental wood, used for furniture. The most profitable age for elms, both for quantity and quality of timber, is probably about fifty years. The wood of elm is sometimes boiled to extract its sap, then washed in aqua-fortis, and stained with a tincture of dragon’s blood and alkanet root, to imitate mahogany. The weight of a cubic foot when green is about 70 lbs., when dry about 48 lbs ; and — Its strength to that of oak is as 82 to 100 Its stiffness “ v 78 “ 100 Its toughness “ “ 80 “ 100 Its absolute cohesive strength, according to Muschenbrock, is 13,200 lbs. It is said to shrink ^th part of its width in seasoning. The elm has been from early times much esteemed as an avenue tree; and Mr. Loudon attributes this to the following qualities—rapidity of growth, straightness of trunk, facility for topping, denseness of foliage, hardness, longevity, and the little care that it requires. Strutt, in his Sylva Britannica , enumerates many elm trees of prodigious size. Among these, the Chipstead elm, GO feet high, and 20 feet circumference at the base: the Crawley elm, on the high road from London to Brighton, the stem of which is 90 feet high, and perforated to the top; it measures G1 feet in circumference at the ground, and 35 feet round the inside at 2 feet from its base: the elms at Mongewell, in Oxfordshire, a group of giants, the principal tree being 70 feet high, 14 feet in circumference, at 3 feet from the ground: the Tutbury Wych elm, and another of the same species at Bagot’s Mile, of immense size, are also figured and described in the same work. Independently of the timber it produces, the elm tree has many economical uses. As fuel it is little inferior to the beech; the charcoal produced is, however, inferior. Its ashes are rich in alkali, the elm in this respect occupying the tenth place in a list of seventy-three trees. Its leaves and young shoots are sometimes used to feed cattle; and the leaves have been used in some places as a substitute for those of the mulberry in feeding silk-worms. They are in parts of Russia used as tea. The inner bark is used for making nets and ropes, and the bark of the Ameri¬ can elm is soaked in water, made supple by pounding, and in the form of ribbands, used for weaving seats as rushes are. The Walnut (JugIans regia). —The walnut tree is a native of Persia, and is of great size. Its branches form a noble head, and its foliage is ample, and of a fine green colour. Its trunk, in the young tree, is smooth, and of a gray colour, but as it grows old the bark becomes chapped and cleft. The walnut is grown in this country chiefly as an ornamental tree. The flexibility of its timber ren¬ ders it unsuitable for beams, although it appears to have been thus used by the ancients. There are many varieties of the walnut. Those chiefly used are two, the Juglans alba and the Juglans nigra, which are procured from America. Of these the nigra, or brown walnut, is the most esteemed. The British walnut tree timber is white in the young tree, and in that state is liable to the attacks of worms. As the tree grows old, the timber darkens in colour, increases in strength and solidity, and becomes easily worked. The timber of the walnut tree is seldom used in this country for works of carpentry, but is highly esteemed for many purposes by the cabinet-maker; aud before the introduction of mahogany it took the place which that timber now occupies. On the Continent it is still prized. The makers of gun stocks consume a large quantity of it; and it is used also in making knife-handles, and in the construction of boxes and drawers to hold articles of polished steel, as it has the advantage of not acting chemically^ on iron or steel. The Juglans alba, the white walnut, or hickory, is, as we have said, produced in North America. It is a large tree, and its timber, when young, is very tough and flexible. The heart-wood of the black walnut, when properly seasoned, is strong, tough, and not liable to warp. It re¬ mains sound for a long time when exposed to heat and moisture. It is never attacked by worms; and it has a grain sufficiently fine and compact to admit of a high polish. The sap-wood, however, speedily decays. In America the black walnut is very extensively used. It is split for shingles. It makes excellent naves for wheels. It is well adapted for naval purposes, as it is not liable to be attacked by sea worms in warm latitudes. On the river Wabash, canoes are made of a single trunk of this tree, sometimes 40 feet long, and 3 feet wide, and are greatly esteemed for their strength and durability. The timber is heavier, stronger, and more durable than the wood of the European walnut. It is fine grained and beautifully veined, and is susceptible of a higher polish. The wood of the walnut, according to Loudon, weighs when green 58 lbs. 6 oz., and when dry 46 lbs. 8 oz. But according to other authorities this is much too high. Musclienbroek gives the specific gravity as -671, the weight of a cubic foot as 41’93 lbs., and its cohesive strength at 8130 lbs. Its strength to that of oak is as 74 to 100 Its stiffness “ “ 49 “ 100 Its toughness “ “ 111 “ 100 The Beech. —Only one variety of this tree (the Fagus sylvatica ) grows in Europe. This tree grows to a great size; and its beautiful, clear, and lustrous foliage, and its shining gray bark, variegated with dark green and yellow 112 PRACTICAL CARPENTRY AND JOINERY. mosses, makes the beech much prized as a forest tree, and an especial favourite with the painter. Its wood varies in colour from white to pale brown; its fibres are compact, but not very hard. It is easily distinguished b} 7 means of the fine and elongated papilla! which cover the surface on which the bark lies, and Or which the impressions are seen in the bark. When the wood is split transversely it presents brilliant satiny facets, like those of the oak, but very much smaller, and not so numerous. The use of the beech has been long abandoned in car¬ pentry works above ground, on account of its tendency to cleave, and its liability to be attacked by worms. The former of these defects is probably owing to injudicious felling, and, it is said, may be avoided by felling the tree at the commencement of summer, when it is full of sap, and leaving it to dry for a year after it is felled; and then, after it is squared, steeping it in soft water for six months. It will not even by this treatment be equal to oak, but will be quite suitable for second-class structures, such as piles, weirs, sluices, floodgates, and the timbering of embankments. In this country at the present day it is used chiefly for making chairs, bedsteads, and panels for carriages, wooden screws, shovels, bakers’ peels, sieve rims, and herring barrels. When used for cabinet-making, it is sometimes stained to imitate mahogany; and when for small articles, such as the handles of jugs and tea pot knobs, it is stained black in imitation of ebony. In France and Germany its uses are manifold. It is employed largely in the fabrication of furniture, and is used for the frames of saddles and horses’ collars, cases for drums, felloes of wheels, bowls, porringers, salt boxes, screws, spinning wheels, pestles, presses and bellows, packing boxes, and scabbards for swords. As fuel, beech wood is highly esteemed; it burns rapidly, with a clear bright flame, and gives much heat. The leaves of the beech are used in place of straw for stuffing mattresses; its bark is used by the tanner, and its fruit affords abundance of an excellent oil, used for burning and also for cooking. For works which are constantly under water it is pccu liarly adapted. It is also much used in the formation of tools, and in cabinet-making, and in the fabrication of a crowd of small objects. It is greatly used for sabots, into which it is converted while yet green, and then to give them durability it is exposed to a flame led by chips of the same wood. It was formerly reduced to very thin leaves, and used for writing upon. The weight of a cubic foot of the timber is 65 lbs. 13 oz. when green, 56 lbs. G oz. when half dry, and 50 lbs. 3 oz. when quite dry; or, according to Barlow, from 43*12 to 53 37; and its absolute cohesive strength 11,500 lbs. Its strength to that of oak is as 103 to 100 Its stiffness “ “ 77 “ 1U0 Its toughness “ “ 138 “ 1U0 Tiie Ash (Fvaxinus excelsi or), Mr. Loudon saj^s, “is excellent for oars, blocks, and pulleys. Few other trees become useful so soon, the wood being fit for walking- sticks at four or five years’ growth, and for handles of spades, and other instruments, at nine or ten years’ growth. An ash pole, 3 inches diameter, is indeed one of the most valuable pieces of timber, for its hulk, that any tree can furnish. For hop poles, hoops, crates, basket handles, rods for training plants, or for forming bowers, light hurdles, fence wallings, the branches of ash, in vari¬ ous stages of growth, are particularly valuable. In the neighbourhood of the Staffordshire potteries, the ash is cultivated to a great extent, and cut every five or six years for crate-wood.” The ash, being hard and heavy, is little used in carpen¬ try ; but these qualities, combined with its toughness and elasticity, render it very serviceable in other arts. Its wood is white, veined longitudinally with yellow¬ ish streaks. Its annual layers are each composed of a zone of compact wood, and another zone in which are many small pores, which show themselves as little holes when a section is made perpendicular to the fibres, and as little interrupted canals in a section parallel to the fibres. This tree is veiy liable to the attacks of worms; and rots rapidly when exposed either to dampness or to alter¬ nations of dryness and moisture. Its toughness and elasticity fit it for resisting sudden and heavy shocks. It is used in making wheel-carriages, implements of husbandry, tools, and the like, but it is too flexible and not sufficiently durable for the carpenter. The weight of a cubic foot of the green wood is 64 lbs. 0 oz., and of the dry wood 49 lbs. 8 oz.; or, according to Barlow, from 43*12 to 53*81; and its cohesive strength 17,000 lbs. Its strength to that of oak is as 119 to 100 I Is stiffness “ “ 89 “ 100 Its toughness “ “ 100 41 100 Tiie Teak ( Tectona grandis). —This wood is in colour light brown; it is porous, and grows quickly. In its fresh state it is more or less impregnated with an aromatic oily substance, and to this it owes much of its value. It is largely used in carpentry and in ship-building. The best kind is from Malabar. Its specific gravity varies from *583 to 1*056. Couch states it at *657, and the weight of a cubic foot at 41*06 lbs.; and Barlow gives 15,000 lbs. as its tenacity per square inch. In thirty-six specimens in the Exhibition of 1851, the specific gravity was—Maximum, 1*056; aver¬ age, *711 ; minimum, *583. The Greeniieaiit (Nectandra rodicei ).—This wood is a native of Guiana, where it is in great abundance. The trees square from IS to 24 inches, and can be procured from 60 to 70 feet long. It is a fine but not even-grained wood. Its heart-wood is deep brown in colour, and the alburnum pale yellow. It is adapted for all purposes where great strength and durability is required, such as house frames, wharfs, bridges, &c. The weight of a cubic foot is from 51*15 to 61*13, and its specific gravity from *831 to 989. The Poplar (Populus).— The wood of the poplar is soft, light, and generally white, or of a pale yellow. It has the property of being only indented and not splin¬ tered by a blow; and hence, and from its lightness, it was used for making bucklers, and this quality fits it also for the sides of carts and barrows used for conveying stones, &c. The principal use of it in construction is for flooring- boards; but it requires to be seasoned for at least two KNOWLEDGE OF WOODS 113 years before it is fit for use in this way. Its whiteness and closeness of grain render it easily kept clean by scouring. It is adapted for all purposes which require light¬ ness and moderate strength, such as for making the large folding doors of barns; and when kept dry it is toler¬ ably durable. The old distich says— “ Though heart of oak be ne’er so stout, Keep me dry, and I’ll see him out.” In Scotland it is sometimes used for mill-work. It is made into dishes and casks by the cooper, and is also used by the cabinet-maker and turner. It is sometimes em¬ ployed as a substitute for lime tree, by musical instru¬ ment makers. It weighs when green 58 lbs. 3 oz. per cubic foot, and from 24 to 38 lbs. 7 oz. when dry. It shrinks and cracks in drying, and loses about a quarter of its bulk. When seasoned it does not warp, and takes fire with difficulty. According to Bevan its tenacity is 7200, and to Muschenbroek 5500 lbs. Alder ( [Alnus ).—The wood of the alder is white when the tree is newly cut down; but the surface of the wound soon becomes of a deep red, this again fades into a pale flesh colour, which is retained by the wood in its dry state. The wood is tender and homogeneous. It has not much tenacity. It is very durable in water. Alder wood is used for all the purposes to which the soft homo¬ geneous woods are generally applied. It is made into wooden vessels, chairs, and tables. When used for the latter purpose, the timber of the old trees, full of knots, is sought after, as it has nearly the beauty of curled maple, with the advantage of a fine deep red colour. When used in constructions above ground, it must be kept perfectly dry. Its most important applications on a large scale are for the purposes of the hydraulic engineer, such as piles for the foundations of bridges, water-pipes, and pump- barrels. Like its congeners, the willow and the poplar, the alder is serviceable to the Cartwright for the sides and bottoms of stone-carts and barrows. It weighs when green 62 lbs. 6 oz., and when dry 39 lbs. 4 oz. It shrinks nearly one-twelfth part of its bulk. Muschenbroek states its tenacity at 13,900 lbs. Birch (Betula alba and Betula nigra). —The wood of the white birch ( Betula alba ) is white, shaded with red; its grain intermediate between coarse and fine. It is easily worked when green, but chips under the tool when dry. The timber of trees grown in temperate cli¬ mates is moderately durable; but that of trees grown in the extreme north is of very great durability. The wood of the birch is used in Bussia in making small rustic carriages; in France, for the felloes of wheels. Chairs and other articles of furniture are also made of it; and it is used by the cooper and the turner. The bark of the tree is used in many ways as a defence against humidity. It is laid as a coping on walls, and it is wrapped round posts and sills inserted in the ground; it is placed over the masonry of vaults, and interposed above the foundation courses of walls—a very objectionable prac¬ tice. But the most familiar application of it as a pro¬ tector against dampness, is the thin layer of it used as an inner sole for shoes, or a lining for hats. Its weight when green is 65 lbs. 6 oz., when dry 45 lbs. 1 oz. Its tenacity is 15,000 lbs. per square inch. Hornbeam (Carpinus betulus). —The wood of this tree is white, and of a fine grain. In drying it shrinks much, which closes its pores and makes it very hard. It is of great use in framing heavy carriages and machines, and in making screws, pulleys, and the wooden teeth of wheels. Its weight, according to Rondelet, is 475 lbs. per cubic foot; its specific gravity, - 760; and Bevan gives 20,240 lbs. as its tenacity. The Maple (Acer campestris). —The wood of the maple is moderately hard, compact, and more or less veined. It is used in various departments of architecture. It is durable when kept dry, but is liable to be attacked by worms. The wood of some of the species takes a fine polish, and is valued by the cabinet-maker. When green it weighs 61 lbs. 9 oz. per cubic foot, and when dry 51 lbs. 15 oz. Its tenacity is 10,584 lbs. The Sycamore (Acer pseudo-platanus). —The wood of the sycamore when young is white, but becomes yellow as the tree grows older, and sometimes even brown to¬ wards the heart. It is compact and firm, without being hard; of a fine grain, and susceptible of a high polish. It does not warp, but is liable to be attacked by worms. It is used in joinery, turning, cabinet-making, and also by musical instrument makers. Cider-presses are made of it, and sometimes also gun-stocks. It used to be greatly in demand for making wooden dishes and spoons, when such articles were used. Its strength to that of oak is a3 81 to 100 Its stiffness “ “ 59 “ 100 Its toughness “ “ 111 “ 100 It weighs when green 64 lbs. per cubic foot, and when dry 48 lbs. It loses one-twelfth part of its bulk in drying. Its tenacity, according to Bevan, is 13,000 lbs. per square inch. Lime Tree (Tilia). —The wood of the lime tree is pale yellow or white, close-grained, soft, light, and smooth. It cuts equally well with or across the grain, and hence is used greatly by carvers. It is used by piano-forte makers for sounding-boards. It is too soft to be employed for works of carpentry, and its use is confined to the car¬ ver, the cabinet-maker, the musical instrument maker, the turner, and the maker of toys. The weight of the cubic foot when dry, according to M. Morin, is 46 lbs.; and its tenacity, according to Bevan, is 23,500 lbs. The Oriental Plane (Platanus orientalis). —Mar¬ shall classes the timber of this tree with that of the sycamore; the French writers class it with the beech and hornbeam. The natives of the East use it for carpenter work, cabinet work, and for boat-building. M. Hassinfratz says that the wood of the plane tree weighs when dry 49 lbs. 3 oz. per cubic foot. It is of a yellowish white colour till the tree attains considerable age, when it becomes brown, mixed with jasper-like veins. In this state it takes a high polish. Bevan states its weight as 40 lbs. per cubic foot, and its tenacity 11,700 lbs. The American, or Western Plane (Platanus occi- dentalis). —A noble tree, of very rapid growth. It bears a general resemblance to the Oriental plane, but it is larger, and more rapid in growth. It is a native of America, where it is called button-wood, and sometimes, from its habitat, the water-beech. It is also called cotton¬ wood, from the thick down which covers the under sur¬ face of the leaves when they first expand. p PRACTICAL CARPENTRY AND JOINERY. 1 14 The timber, when seasoned, is of a dull red, with a fine and close grain. It shrinks much in drying, and is apt to split. Its concentric circles are divided into numerous sections by medullary rays, extending from the centre to the circumference. When the trunk is sawn in a slanting direction, these rays have a remarkable appearance. The cabinet-makers of America do not like the wood on ac¬ count of its tendency to warp; but as it is easily cut in any direction, and takes a fine polish, it is well adapted for cabinet work. The Willow ( Salix ).—The timber of the willow has a wide range of uses. It is sawn into boards for flooring, and into scantling for rafters, and in the latter capacity, when kept dry and ventilated, has been known to last for 100 years. But the purposes more peculiarly its own, are such as require lightness, pliancy, elasticity, and toughness, all of which qualities it possesses in an eminent degree. It also endures long in water, and therefore is in request for paddle-wheel floats, and for the shrouding of water-wheels. It is used in lining carts for conveying stones, or other heavy materials, as it does not splinter; and the same quality renders it fit for guard posts, or fenders. It is also made into cutting boards; it is in demand by the turners and toymakers, and the makers of shoe-lasts. Being susceptible of a fine polish, it is dyed black, and forms an imitation ebony. Young trees, when split in two, are made into styles for ladders. The young and branch timber is made into handles for hay rakes, and other light implements, and into hop poles and props for vines. It is split and made into crates, hurdles, and hampers. The smaller rods and twigs are worked into baskets. Of the strips or shavings the bodies of hats are made. Of all the varieties of the willow, the timber of the Salix Russeliana is the best. This is distin¬ guished from all other willow timber by being of a sal¬ mon colour when dry. When recently cut, the sap-wood is white, and the matured wood slightly reddish; but they both become salmon-coloured when dry. The other varieties of willows cultivated for their tim¬ ber are— Salix alba , which will attain the height of 60 to 80 feet in twenty years; the Salix fragilis, which is often confounded with the Russeliana —this tree grows as rapidly as the alba , but does not attain so great a height; the Salix caprea grows as fast as the fragi¬ lis, and will attain a height of 30 to 40 feet in twenty years. This latter tree, according to Bose, is the most valuable of all the tree willows grown in France. The remarks on the properties of the timber apply, with slight modifications, to all these four species. The specific gravity of willow is *390; the weight of a cubic foot 24 - 37 lbs. Its tenacity, according to Bevan, is 14,000 lbs.; Muschenbroek says 12,500 lbs. Acacia ( Robinia pseudacacia). — The timber called, commonly, Acacia, is the locust-wood of America. Its colour is yellow, with brown veins. M. Hartig places it, in regard to durability, next to the oak, and before the larch. In England, experiments have shown that it is heavier, harder, stronger, more rigid, and more elastic than the best oak; and it is, consequently, fitter than oak for tree-nails. When used for posts, its endurance is next to the yew. Michaux states that it lasts forty years; and on that account its great consumption in America is for sills of doors, and for the posts of the framing of half-tim¬ bered houses that are nearest the ground. It is difficult to procure the timber of large size; as even in districts where the tree thrives best, nine-tenths of the trunks do not exceed 1 foot in diameter, and 40 feet in height. Its strength to that of oak is as 135 to 100. English- grown acacia weighs 44 - 37 lbs. per cubic foot; its specific gravity is ‘710; and, according to Bevan, its tenacity is 16,000 lbs. The Horse Chestnut (/.Esculus Ilippocastanum ).— The wood of the horse chestnut is white, soft, and unfit for purposes requiring strength and durability. It is, nevertheless, applicable to such purposes as the lining of stone carts, and it is said by Boutcher and Duliamel to be well adapted for water-pipes which are to be kept constantly under ground. It is sometimes used for flooring. When green it weighs 60 lbs. 4 oz. per cubic foot, and when dry 37 lbs. 3 oz., or, according to Loudon, 35 lbs. 7 oz.; and loses one-sixteenth part of its bulk. The Service Tree (Sorbus). —The service tree, in foli¬ age and general appearance, closely resembles the moun¬ tain ash. It attains a larger size, and bears larger fruit. In France, trees of this kind are found of the height of 50 or 60 feet. It takes two centuries to attain its full growth, and it is believed that trees exist which are up¬ wards of 1000 years old. The wood of the service tree has a fine and compact grain, and is of a reddish tinge. It is very hard, and takes a high polish. It is in high estimation for the framing of machinery, cogs of wheels, pulleys, screws, and for all such constructions as require great strength and the power of enduring friction. It might, in many cases, be substituted for box. It weighs when dry 72 lbs. 2 oz. per cubic foot. The Pear Tree (Pyrus). —The pear tree yields a wood which is heavy, strong, compact, and of a fine grain. It is slightly tinged with red. Like the service tree, it is of great value for the parts of machines which require to endure much friction, such as screws and the teeth of cogs, and it is used largely in making handles for tools. It is easily stained black, and then so closely resembles ebony as to be with difficulty distinguished from it. It requires to be perfectly dry before it is used. The wood of the wild pear is harder than that of the cultivated pear. The weight of a cubic foot of the wood when green is 70 lbs. 5 oz., and when dry 41 to 53 lbs. Its tenacity, according to Barlow, is 0800 lbs. It shrinks about one-twelfth part of its bulk in drying. The Apple Tree {Modus). —The wood of the apple tree, in its wild state, is fine-grained, hard, and of a brownish colour. It requires to be thoroughly dry before being used, and then it is easily wrought. The wood of the cultivated tree, contrary to what is usually found, has a finer grain than that of the wild tree. The uses to which it is put are nearly the same as in the case of the pear tree; but it possesses the distinguish¬ ing qualities of the latter in a greatly inferior degree. The wild apple tree weighs from 48 to 66 lbs. per cubic foot in a green state, and loses from one-eighth to one- twelfth part of its bulk, and about one-tenth part of its weight, in drying. The cultivated timber is heavier than the other in the proportion of about 66 to 45. Its tena¬ city, as given by Bevan, is 19,500 lbs. KNOWLEDGE OF WOODS. 11 5 The Hawthorn (Crataegus oxyacantha ).—The wood of the hawthorn is white, hard, and difficult to work. Its grain is fine, and it takes a high polish. It is used for the smaller parts of machines, such as cogs and staves for mill-work. It is also made into hammer shafts, flails, and mallets. It weighs 68 lbs. 12 oz. green, and 37 lbs. 5 oz. dry per cubic foot; it contracts one-eighth of its volume in drying. Its tenacity is given by Bevan as 10,500 lbs. The Box (. Buxus ).—This tree, which seldom exceeds the height of 12 feet in Britain, grows in Turkey as high as 25 or 30 feet, with a diameter large in proportion to its height. The wood is remarkably heavy, and is the only Euro¬ pean wood that sinks in water. It is yellow in colour, with a fine uniform grain. It works sweetly, and is very useful for small works exposed to great strain and fric¬ tion, such as screws, and the parts for transmitting mo¬ tion in machinery. It is too valuable to be used in great quantity. It is the only wood employed by the wood engravers, except for large and coarse works. It weighs 80 lbs. 7 oz. per cubic foot when newly cut, and from 60 to 68 lbs. 12 oz. when dry. Its tenacity, according to Bevan, is 19,891 lbs.; Barlow states it at 20,000 lbs. It is sold by weight. Mahogany. —The mahogany tree (. MahoganiSwietenia ) is one of the most beautiful and majestic of trees. Its trunk is often 50 feet high, and 12 feet diameter; and it throws the shelter of its huge arms and beautiful green leaves over a vast extent of surface. It takes probably not less than 200 years to arrive at maturity. The mahogany tree abounds the most and is in greatest perfection between latitudes 11° and 23° 10' N., including within these limits the islands of the Caribbean Sea, Cuba, St. Domingo, and Porto Rico, and in these the timber is superior in quality to that of the adjacent continent of America, owing, it is to be supposed, in some measure, to its growing at greater elevations and on poorer soils. Mahogany timber was used at an early period by the Spaniards in ship-building. In 1597 it was used in the repairs of Sir Walter Raleigh’s ships in the West Indies. It was first imported into England in an unmanufactured state in 1721. The finest mahogany is obtained from St. Domingo, the next in quality from Cuba, and the next from Honduras. In the island of Cuba the tree is felled at the wane of the moon from October to June. The trunks are dragged by oxen to the river, and then, tied together in threes, they are floated down to the rapids. At the rapids they are separated and passed singly, then, collected in rafts, they are floated down to the wharves for shipment. It is considered essential to the preservation of the colour and texture of the wood, that it should be felled when the moon is in the wane. The Honduras mahogany is commonly called Bay wood, and is that most used for the purposes of carpentry. It recommends itself for these purposes by its possessing, in an eminent degree, most of the good and few of the bad qualities of other timber. It works freely; it does not shrink; it is free from acids which act on metals; it is nearly if not altogether exempt from dry rot; and it re¬ sists changes of temperature without alteration. It holds glue well; and it does not require paint to disguise its appearance. It is less combustible than most woods. The weight of a cubic foot is 50 lbs., and its tenacity is given by Barlow at 8000 lbs. Representing the strength of oak by 100, that of Bay wood is 96 “ stiffness of oak by 100, “ “ 93 “ toughness of oak by 100, a “ 99 Sabicu. —The wood of a beautiful tree which grows in Cuba. It is used in the government yards for beams and planking. The weight of a cubic foot is from 57 5 lbs. to 65 lbs. It has been recently used by Sir William Cubitt for the deck floor of the great landing stage at Liverpool. RESINOUS WOODS. Of the timber of the resin-producing trees, belonging to the natural order Conifers, many varieties are used by the carpenter. The yellow deal of Europe, the pro¬ duce of the Pinus sylvestris; the white deal of Norway, the timber of the Abies excclsa; the white pine of Amer¬ ica, which is the Pinus Strobus; the yellow pine of America, Pinus variabilis; the pitch pine, Pinus resin - osa; the silver fir, Pinus Piceci; and the various white firs, or deals, the produce of the Pinus Abies, or spruce fir; and also the larch; are all used in almost every kind of construction for shelter or for ornament. No other kind of tree produces timber at once so long and straight, so light, and yet so strong and stiff; and no other timber is so much in demand for all the purposes of civil architecture and engineering. Log-houses are more conveniently made of the timber of the pine than of any other, because it can be obtained of great length with little taper. In Russia and America roads are made of the trunks of pines. They are rough, it is true, and are very significantly called corduroy roads; but still by their use access is obtained to places which, but for the facilities these trees afford, would be inaccessible. From the growing trees are obtained turpentine, liquid balsam, and the common yellow and black rosin of the shops. Tar is obtained by cutting the wood and roots into small pieces, and charring them, or distilling them in a close oven, or in a heap covered with turf. The lamp¬ black of commerce is the soot collected during this pro¬ cess. Fortunately, the trees of the pine and fir tribe, so useful to man, are found in great abundance in America and Europe. The European pine and fir timber is obtained from the extensive forests of Sweden, Norway, Prussia, Russia, Poland, Germany, Austria, and Switzerland. In many of these places in the Alpine districts, the forests are in¬ accessible; and in others, the timber cannot be made available from the difficulty of conveying it to the streams or rivers which would bear it down to the ports for ship¬ ment. In Sweden the principal river by which the tim¬ ber of that country is floated to the sea is the Gota. It is conveyed by it to Gottenburg. It is also shipped from Stockholm and Gefle. The timber of Norway is floated down the Glommen to Christiana, whence it is called Christiana deal; and down the Drammen to Dram or Drontlieim, whence it is called Dram timber. From the immense forests of Prussia, Russia, and Po¬ land, the timber is brought down the rivers into the ports on the southern shores of the Baltic, whence it is 11G PRACTICAL CARPENTRY AND JOINERY. called Baltic timber. The chief ports are Memel, Dan¬ zig, Riga, Petersburg, Archangel, and Onega. The river Memel being the principal channel through which the pines grown in the north of Prussia reach the sea at the town of that name, the timber they produce is known as Memel timber. The forests of West Prussia and Poland yield timber of a better quality, which, floated down the Vistula and the Bug to Danzig, is known as Danzig timber. The best of the Baltic timber is that which, grown on the banks of the Dnieper, is trans¬ ported to the Dwina, and then, being rafted down to Riga, comes into the market as Riga timber. In the transport of the pines from the Alpine forests, advantage is taken of the slope, and shoots are made, in which the large trees hurry with astonishing velocity to the plains below. When the nature of the ground will not admit of the slioot being formed on the surface with a uniform slope, constructions of great magnitude, made of timber, are carried over gorges and ravines, and even across valleys. These shoots are troughs, the bottoms and sides of which are formed of the trunks of trees. They have such a slope as causes the trees to descend by the force of gravitation alone. The slope required, it is found, need not exceed 20°; and to diminish the friction, a stream of water is made to flow along the shoot. Sometimes, to preserve the timber from injury, it is attached to a species of rude sledge. Of these shoots, the most remarkable for extent, and for boldness of design and construction, was the inclined plane of Alnpach. The pines of the forests on Mount Pilat—pines of the largest and finest quality—rotted where they grew, from the dif¬ ficulty of transporting them to the rivers. The proprie¬ tors of these forests were fully aware of the value of their timber if it could be transported to the rivers; but the boldest and most skilful shrunk from encountering the difficulties that lay in the way of such an enterprise as constructing a shoot in such wild regions. In 1816, how¬ ever, M. Rapp found three proprietors bold enough, along with himselfj to make the attempt. They commenced to make an inclined plane of three leagues in length, and with an uninterrupted slope, to the Lake of Lucerne. The channel of the shoot was 6 feet wide, and 3 feet deep; its bottom was formed of three trunks of trees, placed in juxta-position. In the centre one of the three was formed a channel, into which was turned a stream of water, ali¬ mented at frequent intervals along the line. The inclined plane in its course had 2000 points of support. In several places it was attached to the wall¬ like sides of the rocks along which it had to be carried. It bridged over ravines sometimes at an elevation of 120 feet; and at one point, in order to maintain the slope, it had to be carried through the earth in a tunnel. Wherever practicable, its direction was in right lines, but these could not always be maintained; and where a bend had to be introduced, it was formed on a wide curve. Great as this work was, it was completed in eighteen months by 160 workmen. It was constructed without a single piece of iron being used in fastening it. It con¬ sumed 25,000 trees, and cost £4167 sterling. Other authorities state the cost at £9000, and the date of con¬ struction 1812. From point to point along the line a chain of workmen was established to watch the progress of the trees, and ascertain the time that tree might succeed tree in their passage without danger. By this living telegraph com¬ munication was established between the extremities in three minutes. Pines of 100 feet long, and 10 inches diameter at their smallest end, flashed with such velocity past the watchers, that they appeared only a foot or two in length. They passed from the summit to the lowest extremity in two minutes and a half. To learn the effects produced by such a velocity, ob¬ stacles were placed in the groove so as to throw the trees out in their passage. They were thrown from the chan¬ nel with such force as to enter the ground to the depth of 18 to 24 feet, and one of them striking a tree growing near, split it as if it had been burst by gunpowder. The credit of this magnificent work is entirely due to M. Rapp, who fought his way against a host of preju¬ dices, and overcame difficulties innumerable. The following is a summary of the purposes for which the woods of the various European firs and pines are best adapted. “Memel is the most convenient for size; Riga, the best in qualit} r ; Danzig, when free from large knots, the strongest; Swedish, the toughest. For fram¬ ing, the best deals to be depended on are the Norway, particularly the Christiana battens; and for panelling, the white Christiana; yellow Christiana deals have much sap, and, consequently, cause waste. The best for upper floors are Dram and Christiana white battens; and for ground floors, Stockholm and Gefle yellows. For stair¬ cases, Archangel and Onega planks. Swedish deals are not to be depended on for framing, on account of their warping."* Finns sylvestris .—Red or yellow pine is the produce of the Pinus sylvestris, the wild pine, or Scotch fir. The timber grown in Britain, especially in the southern parts of it, is not so valuable as that produced in the Alpine countries. There are, indeed, exceptions, but this appears to be the rule. It is not so sound; it is coarser in the fibre, it contains more sap-wood, and is not so strong nor so durable. Dr. Smith, however, in his essay on the production of timber, says, that he has seen some Scotch fir grown in the North Highlands, which formed the roof of an old castle, and after 300 years it was as fresh and full of resin as newly-imported Memel. But although the home-grown timber is, in point of fact, less strong and not so durable as that which is im¬ ported, it is worth while to inquire whether much of this difference of quality is not occasioned by the treatment the tree receives. Making every allowance for the in¬ ferior timber produced by planting the Scotch fir on a soil and in a climate not suited to the habits of the tree, it is still hard to believe that, when the soil and climate are judiciously chosen, the timber produced must be so very greatly inferior in quality to the foreign timber. In the Alpine forests the tree is felled at its maturity; it is squared, weather-seasoned, and then water-seasoned in the course of its progress to the port where it is ship¬ ped. It is received in this country after a long interval, and in the hands of the user it is again submitted to a more perfect seasoning and drying before it is * Las ton’s Builders' Price-Book. KNOWLEDGE OF WOODS. 117 finally wrought up. Here, on the contrary, the tree is felled before it attains maturity, the whole process of felling, barking, seasoning, and working, are very quickly gone through, and the timber is generally in its place in the building, whose construction is the immediate cause of the tree’s destruction, in the widest sense, within six months after the time the tree was marked for the axe. Is it a thing to be wondered at, then, that the foreign timber should possess such a superiority over that grown in this country ? Home-grown timber may never rival that from abroad in strength and durability; but a proper attention paid to the selection of the trees, and to the subsequent pro¬ cesses of felling, storing, and seasoning, would render it available for many purposes to which it cannot now be applied; would give durability to such of it as is used in the timbering of farm buildings, its present most frequent application, in place of the decay which renders recon¬ struction a necessity at every renewal of a lease. As a proof that these remarks are not uncalled for, it is only necessary to refer to the excellence of the timber grown in Mar Forest, where due attention has been paid to its selection, cutting, and seasoning. The best wild pine timber is that from the northern parts of Europe, whence it comes in the shape of logs, deals, and spars. The wild pine timber is the most durable of the pine species. Brindley, the celebrated canal engineer, was of opinion that it is as durable as oak. Mr. Semple, the engineer, in his treatise on building in water, expresses a similar opinion. Duhamel states, that on the piles of an old church, which had existed many centuries, being taken up, they were found to be perfectly sound at the centre, with a resinous smell, although the outside was a little decayed. The lightness and stiffness of the Scotch pine timber renders it superior to every other kind of timber for beams, girders, joists, rafters, and, indeed, for framing in general. For joiners’ work, too, it is well adapted, as it is easily worked, and stands better than the harder woods, and if not so durable, which is questionable, it is certainly very much cheaper than they. In the best timber, the saw should leave a clean sur¬ face, not covered with woolly fibres; the annual layers should be thin, never exceeding one-tenth of an inch in thickness. The Riga and Norway timber, as we have said, is the best, and Memel is very little inferior, and being stiffen, it is better adapted for some purposes. In the inferior sorts, the annual layers are thick and soft, the dark part of the ring of a honey yellow. The wood feels clammy; it is heavy, and chokes the saw in cutting. Such timber should not be used where dura¬ bility is required, or where it will be exposed to great strains. In other kinds, the wood, although not heavy, is spongy, and in cutting it the saw leaves a woolly surface. The Swedish timber has often these peculi¬ arities, and is, in such cases, deficient in strength and stiffness. Of the timber of the Pinus sylvestris, Memel supplies three qualities, viz.:— Crown, in baulk, 13x13 inches, and from 28 to 50 feet long. Longer timber is apt to be knotty at the small end. Best Middling. Second Middling , or Brack These are of about the same dimensions as crown, but as they contain large knots, they are not so fit to be cut into small scantlings. Danzig common baulks are from 14 to 16 inches square. Crown baulks are sometimes so large as 26 and 30 inches, and so long as 70 feet; but 40 feet is nearer their average length. As this timber is very sound, it should be used where whole timbers are required; the crown is especially useful for bearing timbers. Riga baulks are 13 to 14 inches square, and average about 40 feet long. The heart of the baulk is often shaky; it should, therefore, be divided longitudinally, and the flitches reversed. It is very hard to tell the difference between Memel and Riga timber when in the log. Norway timber is of smaller dimension than that from Prussia and Russia. It is very durable, and suitable for exposed work, and should be used where the beams do not require to exceed 11 inches square. The timber is also supplied in planks, deals, and battens; and their cha¬ racteristics may be briefly stated as follows, viz.:— Prussian , Memel, and Danzig. —Very durable, adap¬ ted for bridge flooring and external work. Russian, Archangel, and Onega. —Not fit for work exposed to damp. The knots are often surrounded by dead bark, and drop out when the timber is worked. Clean specimens are suitable for joiners’ work. Petersburg and Narva. —Easily takes dry rot in damp unventilated situations. Biornburg. —The planks are 12 feet long, and are like those of Archangel, but more knotty. Finland and Nyland. —These are 14 feet long. They are fit for the carpenter only, and are very durable. Norwegian, Christiana, and Dram. —Deals and bat¬ tens. The Christiana deals have generally much sap wood, and, consequently, cause loss and waste in working. The wood is mellow, and works well under the plane. Of the Dram timber, the upland is the best, the lowland the worst. Frederichstadt. —Durable and mellow, works easily under the plane. Sivedish, Stockholm, Gefle.— Full-sized, and free from sap; but liable to warp, and to be full of large coarse knots. It is useful for ordinary carcass work where cost is an object, as it is cheaper than Norway timber. Gottenburg. —Durable, and fit for the carpenter, but not for the joiner. Hernosand and Sundsvall. — Same characteristics as the last, with all the faults to a greater extent. Tredgold gives the following as the relative strength of foreign pine and of that grown in England, and also in Mar forest, and of oak:— Foreign Pine. English Pine. Mar Forest Pine. Oak. Strength,. ... 80 .. ,. 60 . ... 61 100 Stiffness,. ... 114 .. .. 55 . ... 49 ... 100 Toughness,. ... 56 .. . 65 . .. 76 ... 100 The best foreign timber shrinks about -^th part of its width in seasoning from the log. White Fir, or Deal (European).—This is the produce of the Pinus Abies, or Norway spruce. It is light, elas¬ tic, but varies in durability with the conditions of soil and climate. It is much less resinous than the Scotch fir, PRACTICAL CARPENTRY AND JOINERY. and its colour is a reddisli or yellowish white. This tree affords the Burgundy pitch of commerce, and its bark is used for tanning. The timber being fine-grained, takes a fine polish, and is easily worked, either with or across the grain. It holds glue remarkably well. The spars are from 30 to GO feet long, and from G to 8 inches thick, and are used for scaffold poles, ladders, oars, and masts to small vessels. The American spruce is of two kinds—the White Spruce (Pinus alba), and the Black Spruce ( Pinus nigra). The white American spruce timber is not so resinous as the Norway spruce, nor so heavy; but it is tougher, and is more liable to twist and warp in drying. It decays soon. It is imported in deals and planks. The black spruce is said to produce the best wood; but as it is the bark and not the timber which gives it the distinctive appellation, it is not possible to tell the dif¬ ference, when the wood comes mixed in its cut state. Its distinguishing characteristics, according to Michaux, are strength, lightness, and elasticity. In Maine and Boston it is much used for the rafters of houses. The peculiar characteristics of the European spruce timber may be briefly stated as follows:— Norway Timber. —White deals from Christiana, Fre- derickstadt, and Dram. The two first are of the best description; but in those from Frederickstadt the knots are often surrounded by adhering bark, and are apt to drop out when they are sawn into boards. The lowland Dram timber is apt to shake and warp in drying; the upland deals have not this tendency, and are, therefore, to be preferred. Swedish Timber. — Gottenburg white deals are hard and stringy, and fit only for temporary work. The same remark, in a greater degree, applies to those from Herno- sand and Sundsvall. Russian Timber. —Narva white deals are nearly equal to the Norwegian, and so also are those from Riga, when properly seasoned. Petersburg white deals, however carefully seasoned, expand and contract with every change of weather. American white spruce deals warp and twist very much, and soon decay. They are fit for temporary purposes only. White deal shrinks -^th in becoming perfectly dry, and what are termed dry deals will shrink -g^th. Regarding oak as 100, the strength, &c., of spruce, are— American Norway British Spruce. Spruce. Spruce. Strength,. 86 ... 104 ... 70 Stiffness,. 72 ... 104 ... 81 Toughness,. 102 ... 104 ... 60 Pinus Strobus, the Weymouth pine, or yellow pine, the timber of which, called American white pine, is im¬ ported in large logs. Its wood is light and soft, straight¬ grained, and free from knots, "which fits it for joiners’ work, especially for mouldings. Its colour is a brownish yellow, and the colour and texture are very uniform. It has a peculiar odour. Michaux, in his North American Sylva (1819), says, “ Seven-tenths of the houses, except in the larger capitals, are built of wood, and about three-quarters of these are built almost entirely of white pine; and even in the cities, the beams and principal wood work of the houses are of that wood. The ornamental work of the outer doors, the cornices and friezes of apartments, and the mouldings of fire-places, all of which, in America, are elegantly wrought, are of this wood. It receives gilding well, and is, therefore, selected for looking-glass and picture frames. Sculptors employ it exclusively for the images that adorn the bows of vessels, for which they prefer the kind called the pumpkin pine. At Boston, and in other towns of the northern states, the inside of mahogany furniture and of trunks, the bottoms of Windsor chairs of inferior quality, water-pails, a great part of the boxes used for packing goods, the shelves of shops, and an endless variety of other objects, are made of white pine. In the district of Maine it is employed for barrels to contain salted fish, especially the variety of the timber called the sapling pine, which is of stronger consistence.” But the most important infor¬ mation given by Michaux to the carpenter, is, that in the construction of the magnificent wooden bridges over the Schuylkill at Philadelphia, and the Delaware at Tren¬ ton, and also in the bridges which unite Cambridge and Charlestown with Boston, of which the first is 1500 feet, and the second 3000 feet in length, the white pine has been chosen for its durability. The white pine is also used for sliingles and clapboards. The shingles are commonly 18 inches long, from 3 to G inches wide, £-inch thick at one end, and 1 line at the other. They are made only of the perfect wood, and should be free from knots. In America they last from twelve to fifteen years. They are exported in great quan¬ tities to the West Indies. But however high this pine may rank in America as a timber for the carpenter, it is not esteemed in this country. It is inferior to the Baltic timber in strength and hard¬ ness, and is not to be compared to it in durability. It is liable to dry rot, and is, therefore, and for the other reasons given, never employed in the carpentry of the best buildings, but is exclusively used by the joiner. Its strength to that of oak is as 99 to 100 Its stiffness “ “ 95 “ 100 Its toughness “ “ 103 “ 100 Pitch Pine ( Pinus resinosa). — This pine is highly esteemed in Canada for its strength and durability. Its timber is close-grained, and the concentric circles small. It is exceedingly resinous, and consequently heavy. Its elasticity is remarkable, it may be bent round the bow of a vessel, and after some years it will recover its straight¬ ness. The long-leaved Florida pine yields the best quality; and the fine-grained timber is alone used by the Ameri¬ can government. The pitch pine grown in the northern states in Virginia is not so good, as there the trees are tapped for the pitch, which injures the durability of the wood. It is employed in furnishing planks and deck planks for ships, both in this country and in America. Sometimes planks are obtained 60 feet long without a knot. Stripped of its sap-wood, it makes excellent pumps and troughs for mills, and may be used in situations where it is exposed to damp and dryness, but will not last under ground so long as white pine. It is of a redder colour than Scotch pine, and is sticky and difficult to plane. Its strength compared with oak at 100 is 82; toughness, 92. The Silver Fir (Pinus Picea). — This fir, from which the Strasburg turpentine is obtained, produces a timber which is elastic, light, and stiff. Its grain is irregular, KNOWLEDGE OF WOODS. 119 as the fibres which compose it are partly white and ten¬ der, and partly yellow, or fawn-coloured, and hard. The narrower the white lines are, the more beautiful and solid is the wood. It is used in carpentry works of all kinds. In England it has been used chiefly for floors; and its stiffness gives it an advantage in the case of any slight sinking. Like all of the pine tribe it shrinks considerably in drying; yet Arthur Young and Mitchell affirm that floors of the timber of a full-grown tree may be laid imme¬ diately on its being sawn up without risk of shrinkage. Larch ( Larix europcca.) — The larch is a deciduous tree, frequently attaining great size. It is a native of the mountainous regions of Europe, the west of Asia, and North America. It is found in abundance in the Alpine districts of the south of Germany, Switzerland, Sardinia, and Italy, but not on the Pyrenees, nor in Spain. Of late years it has been extensively cultivated in Great Britain. Among the Homans the larch was highly esteemed for its strength and durability. Its timber was used in the con¬ struction of the forum of Augustus, and several bridges in Home. Vitruvius mentions it, and attributes the decay of buildings in his time to the fact of larch not being used in their construction. The first account of larch trees grow¬ ing in Britain is in 1629. They are at that time spoken of by Parkinson in his Paridisus, as being “rare, and nursed with but a few who are lovers of variety.” Evelyn mentions a large one of “ goodly stature,” as growing in 1664, at Chelmsford, in Essex. Miller, in 1731, says, “ This tree is now pretty common in English gardens.” In the account of the Duke of Athole’s larch plantations, published in the Highland Society’s Transactions, it is stated that Goodwood, the seat of the Duke of Richmond, near Chichester, was probably the first place where the larch was cultivated as a forest tree, and that only on a limited scale. In 1782, an extensive plantation was formed at Halford; and shortly afterwards the Society of Arts of¬ fered premiums for planting the larch, and making known the useful properties of the timber. Public attention was thus drawn to the value of the larch, and the result was that it was extensively planted throughout Britain. In Scotland it is said that the first larch planted is the one known as the crooked larch at Dalwick, in 1725. The popular account, however, is that the Duke of Argyle in- j troduced the larch into Scotland in 1727. Having received them among some exotics from Italy, he treated them all in the same manner, and placed them in a hot-house; when very soon the larches withered, and being supposed dead, were thrown out on a heap of rubbish in a garden. Here they revived, and sending forth shoots, became vigorous trees. In the account in the Highland Society’s Transac¬ tions before alluded to, it is said that “in 1738 Mr. Men- zies, of Migeny, in Glen Lyon, brought a few plants of the larch in his portmanteau from London, five of which he left at Dunkeld, and eleven at Blair in Athole for Duke James.” But whether this be the correct account of its introduction or no, it is indisputable that it was first extensively planted in Blair and at Dunkeld by the Duke of Athole. Between 1740 and 1750 he planted 350 larches at Dunkeld, at an elevation of 180 feet above the sea, and 873 at Blair. In 1759 he planted 700 larches, with the view of experimenting on its value as a timber tree. This plantation was on the face of a hill, from 300 to 400 feet above the sea, and on very poor land. His successor, John, Duke of Athole, between 1764 and 1774 planted 410 acres Scotch, and his plans embraced the planting of 225 acres more. His successor completed this, and during his life continued planting, until, in 1799, he ! had planted, at Logierait, Inver, and Dunkeld, altogether, 800 acres Scotch. In 1800 he continued his operations, and from that date to 1815, completed the planting of 2409 Scotch acres. Some of the land thus planted was at an elevation flu* exceeding the range of growth of the Scotch fir, being from 900 to 1200 feet above the sea. The success which attended the experiment induced him to continue planting till 1826, and at the close of that year he had completed the planting of 8071 Scotch acres with larch alone, or with larch mixed with other trees 8604 Scotch, or 10,324 statute acres. In France attention was early directed to the value of larch timber; and in 1798 a commission was appointed to examine into its suitableness for the construction ol ships. The result, as reported, was:—1. That the wood was more resinous than that of Pinus Laricio , though, at the same time, lighter in the proportion of 25 or 26 to 29. 2. That its fibres were very strong and able to resist twisting. 3. That branches, clear from knots, might be used as topmasts. The wood of the larch, according to Hartig, weighs 60 lbs. 13 oz. per foot when green, 36 lbs. 6 oz. when dry. The wood of trees produced in a good soil is of a yellow¬ ish white; but that of trees grown in a poor soil, and at great elevations, is reddish brown, and very hard. The timber is said to arrive at perfection in forty years; while the Scotch pine takes eighty years to mature its timber. From the nature of its growth, the larch is free from large knots; and although it produces dead knots, yet these are generally sound, and found fast-wedged, as it were, in the timber. It is exceedingly durable, excelling, in this respect, even the oak itself. In timber construc¬ tions it is applicable, in large baulks, or scantlings, as beams and lintels; but when cut into deals, or smaller I scantlings, its tendency to warp or twist is so great, as to i render it much less valuable in this condition. It is said, : however, that if the tree is barked two years before it is cut, the timber loses this tendency. It is difficult to work. As post-piles, or sleepers, or in circumstances in which it is alternately exposed to wet and dry, its durability is very great. Hence, also, it is suitable for mill-work, for the steps of quays, &c. Its strength to that of oak is as 103 to 100 Its stiffness “ “ 79 “ 100 Its toughness “ a 134 “ 100 The Cedar. —Under the general name of cedar are known to us, the red cedar {Juniperus Virginiana), a native of North America; the white cedar (Gupressus thyoides), also an American tree; the cedar of Lebanon {Gedrus Lebani). Of these, the first-named is probably the most familiarly known, by its wood being used for blacklead pencils. The name red cedar has reference to the wood of the heart of the tree merely, for the sap- wood is perfectly white. It is so strong and durable that it would be preferred to every other kind of wood for rural purposes in America, were it not so scarce and high- priced. It is admirably adapted for subterranean water pipes. It is used for the upper parts of the frames of vessels. In this country it is much used for drawers, 120 PRACTICAL CARPENTRY AND JOINERY. wardrobes, and other articles of furniture, as it is not liable to be attacked by insects; and on account of its power of resisting heat, Mr. Brunei has used it for covering locomotive boilers. The timber of the white cedar, from its lightness and its power of resisting alternations of dryness and mois¬ ture, is in common use in Baltimore and Philadelphia for shingles. These are cut transversely to the concentric circles; they are from 2 to 2 feet 3 inches long, 4 to 6 inches broad, and 3 lines thick at the larger end. At Baltimore they are called juniper shingles. These shingles are much more durable than those of the white pine; lasting from thirty to thirty-five years. It is made into pails, buckets, wash tubs, churns, &c.; and the coopers in Philadelphia who make these articles are called cedar coopers. The cedar of Lebanon has timber of a reddish white colour, light, spongy, and easily worked, but very apt to shrink and warp, and by no means durable. In its appearance it bears a close resemblance to the timber of the silver fir. The weight of a cubic foot of red cedar is 260 lbs.; its specific gravity is ’426. Couch gives - 753 as the specific gravity of Canadian cedar, and 47 lbs. as the weight of a cubic foot; and Bevan states its tenacity at 11,400 lbs. per square inch. The Yew ( Taxus ).—The yew is a very slow-growing tree; but if it live a long time it becomes colossal in its dimensions. At Foullebec in France, there is a yew tree 21 feet in circumference. In England there are many noble trees of this kind; and at Fortingall, in Scotland, there is one, or rather the wreck of one, which was found ! by Mr. Tennant to be 56 feet in circumference. As the I tree grows very slowly (so many as 150 annual layers ! have been counted in a tree of 13 inches diameter), the Fortingall yew must have been a flourishing tree at the commencement of the Christian era. The wood of the yew is hard, compact, and of a very fine grain. It is flexible, elastic, and incorruptible. It splits easily. The sap-wood, which is white, and dees not extend to a great depth, is also very hard. The heart-wood is of a fine orange or deep brown colour. It requires a long time to dry, but shrinks very little in drying. It is the finest wood for cabinet-making pur¬ poses, and is generally employed in the form of veneers. Where it is found in sufficient quantity to be used for large works, the yew may be considered to be indestructi¬ ble, even where the most durable of other woods perish. ! In France the timber makes the strongest of all wooden axle-trees. The weight of the specimen in the Exhibition of 1851 was 41 7 lbs. per cubic foot, and its specific gra¬ vity '665. Musclienbroek gives 50 - 43 lbs. as the weight of a cubic foot, and ’807 as the specific gravity of Spanish i yew; and Bevan gives 8000 lbs. as its tenacity.* THEORETICAL RESOLUTION AND COMPOSITION OF FORCES. Carpentry is the art of combining pieces of timber to support a weight, to sustain pressure, or to resist force. It is broadly distinguished from joinery by this, that while the work of the carpenter is essential to the stability of a structure, the work of the joiner is applied more to its completion, its decoration, and rendering it fit for use, 1 and may, in general, be removed without affecting its stability. The principles of carpentry are founded on the doc¬ trine of the composition and resolution of forces—a know¬ ledge of the relative strength of the materials; and it is through a knowledge of these alone that skilful designs are made. The effects of the different forces which act on a piece of timber at rest are these—extension and compression in the direction of its length, lateral compression, and tor¬ sion. To the first, is opposed cohesion; to the second, stiffness; to the third, transverse strength; and to the fourth, the elasticity of torsion. On these resistances of materials, direct experiments have been made, and prac¬ tical formulae for calculating them have been deduced. It is essential, on entering on the subject, that an accurate idea should be formed of the manner in which several forces act when united in their effect, and we shall therefore proceed to lay before our readers the princi¬ ples of the composition and resolution of forces when ac¬ commodated to the chief purposes of the carpenter. If a (Fig. 433) be a force acting on a body in the di- CARPENTRY. rection of the line a b, and c another force acting on the same point in the direction of the line c b, with pressures in the pro¬ portion of the length of the lines a b and c b respectively, then the body will be affected precisely in the same manner as if acted on by a single force d, acting in the di¬ rection d b, with a pressure pro¬ portioned to the line d b, which is the diagonal of a parallelogram formed on a b, c b, and which is called the resultant of the two forces a c. In like manner, if F j g . 43l . F the forces a, c, d (Fig. 434), act on a body b, in the direction of the lines a b, d b, c b, and with inten¬ sities proportioned to the length of these lines, then the resultant of the two forces, a and c, is ex¬ pressed by the diagonal e b of the parallelogram, formed on the lines a b, c, b, and the resultant of this new force e, and the third force d, is / acting in the direction / b, the diagonal formed on eb,db\ there¬ fore, / b expresses, in direction and intensity, the resultant of the three forces a, d, c. In like * Loudon; Strutt; Tredgold; Barlow; Bevan; Rondelet; Emy; Michaux; The Mahogany Tree; Report of the Juries, Exhibition , 1851; Low on Landed, Property; Morton's Cyclopedia; Ponts et Chaussees; E Ingeneur Civile, &c. RESOLUTION AND COMPOSITION OF FORCES. 121 manner, the resultant of the forces abode (Fig. 435) will be found to be h, acting in the direction h o. A sim¬ ple. experiment may be made to prove this. Let the threads a b, a cd, a e f (Fig. 436) have the weights b d f appended to them, and let the two threads a c d, a ef be passed over the pulleys c and e\ then if the weight b be greater than the sum of d /, the assem¬ blage will settle it¬ self in a determinate form, dependent on the weights. If the three weights are equal, the lines a c, a e of the threads will make equal angles with a b ; if the weights d f and b be respec¬ tively 6, S, and 10, then the angle cae will be a right angle, and the lines c a, e a will be of the respective lengths of 6 and 8; and if we produce c a, e a to n and m, and complete the parallelogram a n o m; a n, a m will also be 6 and 8, and the diagonal a o will be 10. The action of the weight b in the direction a o is thus in direct opposition to the combined action of the two weights d f, in the direc¬ tions c a, e a; and if we produce o a to some point k, making a r, a s equal to those weights, we shall manifestly have a k equal to a o. Now, since it is evident that the weight b, represented by a o, would just balance another weight l, pulling directly upwards by means of the pulley k, and as it just balances the two weights d /, acting in the directions a c, a e, we infer that the point a is acted on in the same manner by these weights as by the single weight, and that two pressures acting in ' the directions and with the intensities a c, a e are equal to the single pressure acting in the direction and with the intensity a Jc. In like manner, the pressures a, s a are equivalent to n a , which is equal and opposite to r a; also, o a, r a are equivalent to m a, which is equal and opposite to s a. In the case of a load w (Fig. 437) pressing on the two inclined beams b c,b d, which abut respectively on the points c and d, it is obvious that the pressures will be in the directions b c,b d. To find the amounts of these pres¬ sures, draw the vertical line b e through the centre of | the load, and give it, by a scale of equal parts, as many j units of length as there are units of weight in the load w: draw of, eg parallel to c b,db\ then b g, measured on the same scale, will give the amount of the pressure sus¬ tained by b c, and b f the amount sus¬ tained by b d. A slight consi¬ deration will serve to show that the amount of thrust, or pressure, is not influenced by the lengths of the pieces be, bd. But it must be borne in mind, that although the pressure is not modified in its amount by the length, it is very much modified in its effects, these being greatest in the longest piece. Hence, great attention must be given to this in designing, lest by unequal yield¬ ing of the parts, the whole form of the assemblage be changed, and strains introduced which had not been con¬ templated. If the direction of the beam b d be changed to that shown by the dotted line b i, it will be seen that the pressures on both beams are very much increased, and the more obtuse the angle ibh the greater the strain. By this proposition we can compare the strength of roofs of different pitches.—Let ab,ag (Fig. 438) be rafters of roofs of the respective Fis , 438 heights of F B, F G. Then, because the load on the rafter will increase in the same proportion as its length, the load on the rafter A B of the roof will ff 1 be to the load of a similar covering on A F as A B to A F; but the action of the load on A B, by which it tends to break it, is to that on A F as A F to A B, consequently increased load on A B is diminished by its oblique action; and the diminished load on A F is increased by its direct action; and the transverse strain is the same in both. But the strength of beams, we have seen, is inversely as their length; therefore the power of A B to resist its strain is to the power of A F as A F to A B. If, therefore, a rafter A G is of a scantling just sufficient to carry its load, a rafter A B of a greater pitch would require to be made of a greater scantling, to enable it to carry the same load per foot of length. Hence, steep roofs must have stronger rafters than flat roofs to carry the same weight of covering per square yard of surface, or the rafters must be increased in number so as to reduce the load on each. The increased size of scantling may be found geome¬ trically as follows:—Let the line a f (Fig. 439) be the depth of a beam that would carry the weight required if placed hori¬ zontally. Draw fb perpendicular to a f and make a b equal to the slope or pitch of the rafter: pro¬ duce b a, making a g equal to a f, and draw the semicircle g d b. Then draw a d perpendicular to a b, and a d will be the depth required. When the rafters of a roof are uniformly loaded. Q 12*2 PRACTICAL CARPENTRY AND JOINERY. there is an angle which renders the oblique strain the least possible: this is when the tangent of the angle of inclination is ’7071, or the angle 3516. If, in place of the weight being super¬ imposed on the beams, we suppose it sus¬ pended from their point of junction (Figs. 440 and 441), we ob¬ tain the strains in the same manner; and it is obvious that although the weight is suspended, the strains are still compressing the beams in the direction a b, a c ; or, in other words, rig. mi. the beams act as struts. Substitute, in Fig. 441. for the string the piece of timber ad, and we have now the piece a d stretched in the direction of its length and acting as a tie, and the other tim¬ bers compressed in the direction of their length. It is very important to be able, in all cases, to discrimi¬ nate between struts and ties; and we shall therefore illus¬ trate this more at length. In the jib of the crane ba c (Fig. 442), formed by the pieces b a, c a, we wish to find the measure and description of the strains exerted on these pieces when loaded by the weight w suspended from the point a. Produce b a and a c to d and e ; make a w equal to the weight in equal parts; draw w d parallel to ca, and w e parallel to b a ; then shall a e be the strain on a c, and a d the strain on a b. On attentively considering the figure, it will be seen that while a c is sustaining a com¬ pressing strain, b a is pulled in the direction of its length, and its place might be supplied by a rope. Now, change the directions of the pieces, as shown in the following figure (Fig. 443), and both pieces will appear to be in a state of tension; but it is not so; the place of d a could not here be supplied by a rope; and it is therefore a strut, and not a tie. The following is an invariable rule by which to dis¬ criminate between a strut and a tie in any system of framing; — From the point where the weight or pressure acts, draw a line in the direction of the action of the pressure, and let the length of that line, measured on a scale of equal parts, denote the pressure in lbs., cwts., or tons. From the extremity opposite to the point on which the pressure is exerted, draw lines parallel to the pieces which sustain the strain. The line parallel to the one piece will necessarily cut the other piece, or its direction produced, within or without the framing. If the line cut the piece itself, or its direc¬ tion produced within the fram¬ ing, that piece is compressed — it is a strut; if it cut its direction produced beyond the framing, or point of pressure, the piece is in a state of tension—it is a tie. In the crane- jib b a c, Fig. 442, the line w e parallel to a b, cuts the direction of a c produced within the framing; therefore a c is, by the rule, a strut. The line w d, parallel to c a, cuts the direction of b a, produced in the point d without the framing—and consequently, by the rule, b a is a tie. In like manner, in Fig. 443, w d cuts the piece d a which is a strut, and w c cuts the direction of b a pro¬ duced in c without the framing; b a is therefore a tie. The following is another mode of doing the same thing:—Through the point a, in the two last figures, draw a straight line m n, parallel to the transverse dia¬ gonal of the parallelograms e d, or d c. Then the parts of the flaming on that side of this line, to which the straining force would move if left at liberty, are in a state of com¬ pression, and the parts on the opposite side are in a state of tension. In Fig. 444 the direc¬ tion of the pressure is in the line a b : we therefore set off from a to b, on a scale of equal parts, a b equal to w,the weight, and draw from the point b, op¬ posite to the point of pres¬ sure, the lines b d, be parallel to e a, f a, which, cutting the pieces / a, -j- e a in d c, show that they both '\ | y act as struts. y In Fig. 445, let w be the weight acting on the point a\ —from a draw a cj in the direction of the strain, and STRENGTH AND STRAIN OE MATERIALS. 123 make ag equal to the weight: then, proceeding as before, we find that the parallels cut the directions of the pieces ea,fa produced without the framing; and these pieces are therefore ties. We have shown that the angles the pieces make with each other, influence materially the amount and propor¬ tion of the strains upon them. Generally, the strain on any piece is proportional to the sine of the angle which the straining force makes with the other piece directly, and to the sine of the angle which the pieces make with each other inversely. For, it is plain that the three pres¬ sures, A E, A F, and A G (Fig. 446), which are exerted at the point A, are in the proportions of the lines A e, A f, and F E (f E being equal to a g). But, because the sides of a triangle are pro¬ portional to the sines of the opposite angles, the strains are proportional to the sines of the angles A F E, A E F, and F A E. Therefore, to ascertain the strain on A b arising from any load A E acting in the direction A E ; multiply A E by the sine of the angle E A G, and divide the product by the sine of B A c. It is not necessary in practice to resort to calculation by sines. In designing framing, the measures of the strains can be obtained with equal accuracy by drawing the parallelogram of forces, and measuring from a scale of equal parts. We have hitherto considered the pieces of timber them¬ selves as being subjected to the strains; but it is obvious that they act also as transmitters of the pressures; and it is necessary to consider these as propagated to the points of support, which will be pressed or pulled by the same forces that act on the pieces serving as struts or ties. Thus we learn what supports must be provided for these points. In the truss, Fig. 447, if A B represent'the pressure on A in the direction A B, A c will be the strain on A E, and the magnitude of the pressure on E in the direction A E, and A D the pressure on the point F. The divellent force on the tie-beam e f will be equal to the sum of k c, m D, and the horizontal thrusts at E and F will be as k c, m D, respectively. Further, the pressure on the walls E and F will be unequal when the load A is not in the centre between them. In the figure, the pressure on E will be — A k, and on F = A m or k b. We thus discover, also, what forces are exerted on the joints of the timber at E aud F. For let eb (Fig. 448) be the end of the tie-beam e f, and A c the end of the rafter, the force in the direction of the rafter being repre¬ sented by A C. then the vertical pressure on c will be equal to A B, and the thrust in the direction B E will be equal to B c; and this is the measure of the force acting to splinter off the part d. In a queen-post roof, if the centre of the load A (Fig. 449) corresponds to the centre of the opening e f, the pressure on the points B and c should be alike, and may be found by making B d equal to the load on B, and draw ing d k and m d parallel to B c and B E. Where the weight is not in the centre of the opening, but is in the vertical line o c; from a convenient point o in that verti¬ cal line, draw lines to the supporting points, and B g will be the length of the collar-beam, and g It the place of the queen-post. To find the strains, make B r equal to the whole load, and draw r p parallel to FO; the line rp then represents the strain on g F, the other forces being as before. STRENGTH AND STRAIN OF MATERIALS. The materials employed in construction are exposed to certain forces, which tend to alter their molecular consti¬ tution, and to destroy that attraction which exists between their molecules, named cohesion. It is therefore necessary, in designing constructions, to be able to determine the relation which subsists between these destructive forces and the resistance which the various materials are susceptible of opposing to them. The destructive forces in timber may operate in the manners following:— I. By tension in the direction of the fibres of the wood, producing rupture by tearing it asunder. II. By compression in the direction of the fibres, pro¬ ducing rupture by crushing. III. By pressure at right angles to the direction of the fibres, or transverse strain, which breaks it across, and which, as will be seen, is a combination of the two former strains. IV. By torsion or wrenching. V. By tearing the fibres asunder. It is to the three first of these only that attention need be directed, so far as concerns the work of the carpenter. Every material resists with more or less energy, and for a longer or shorter period, these causes of destruction. The resistance to the first-named force is called the resist¬ ance to extension, or simply, cohesion; to the second, the resistance to compression; and the third, the resistance to transverse force. The measures of these resistances are PRACTICAL CARPENTRY AND JOINERY. I‘24 the efforts necessary to produce rupture by extension, compression, or transverse strain. Those materials which, when they have been submitted to a certain force less than the amount of their resistance, return to their normal condition when that force is with¬ drawn, are termed elastic. The knowledge of the elas¬ ticity proper to any body gives the means of calculating the amount of extension, compression, or flexure, which the body will sustain under a given force. For the purposes of calculation, it is convenient to have a measure of the resilience or elastic power of a body ex¬ pressed either in terms of its own substance, or in weight. This measure is termed the modulus of elasticity of the body. If we suppose the body to have a square unit of surface, and to be by any force compressed to one-half or extended to double its original dimensions, this force is the modulus of the body’s elasticity. No solid substance, it may at once be conceived, will admit of such an extent of com¬ pression or extension; but the expression for the modulus may nevertheless be obtained by calculation on the data afforded by experiment. The moduli for various kinds of woods will be found in the tables. I. Resistance to Tension. Although, mechanically considered, this is the simplest strain to which a body can be subjected, it is yet the one in regard to which fewest experiments have been made, in consequence of the great force required to tear asunder lengthways pieces of timber of even small dimensions. There is, too, a want of agreement sufficiently baffling in the results obtained by different operators. The results of the experiments made by Muschenbroek, Buffon, Barlow, Bevan, and others, are given in the following table, reduced to a section of one inch square:— Table I. —Tenacity of a square inch of different Woods, expressed in the weiyht in lbs., that will produce Rupture. Names of Expeuimentalists. 1 o a 1 5 K -2 •3 S a g Oak, English,. 17,300 13,950 17,000 9,18-7 12,000 8,889 19,800 7,850 13,316 14.7S8 11,376 6,070 Beech,. 17,300 11.461 11,467 22,000 14,720 12,000 17,064 12,406 17,207 14,670 Chestnut, Spanish,. 13,300 10,500 11,900 Sycamore,. 13.00) 5,000 13,000 Poplar. 5,(US 5,260 7,200 6,016 Alder,. 14,186 4,290 14,186 Acacia.. 20,582 16,000 18.291 V alnut,. 8,130 7,800 5.360 8,465 Mahogany,. 8,912 8,000 21,000 Old. 18,950 15,090 S,200 12,460 Aspin, .... 69,90 6,990 Lance-wood,.. .. ... ... 23,400 23,400 Box,. 19,908 19,830 19,890 19.S92 Pear,. . 9,900 9,822 9.861 10,220 ... 8,900 9,560 Riga fir, . 12,253 13,300 12,776 Petersburg,.. 13,300 13,300 Kir,. 8,oOG 5,000 7,818 Pitch pine,. 7.858 7,818 ( VHnr 4,973 11.400 8,186 Norway pine.. 7,237 14,300 10,293 Birch,. ... Hawthorn, . 10,500 10,500 Hazel, . 18,000 18,000 Hollv,. 16,000 16,000 Hornbeam. 20,240 20,240 Laburnum. 10,500 10,500 1 11,800 11,800 Lime tree. 23,500 23,500 Maple, _ . 10 584 10,584 Plane,.. .. .. Willow. ... 11,700 ... 11,700 14.000 14,000 Yew, Spanish... ... 8,000j ... 8,000 Although, on many considerations, it is not to be expected that experiments made on such small scantlings of timber as were used by the experimentalists above cited (some of these being only a square line in section), would agree with each other; yet the discrepancy is so great as to give little confidence in the results. Mr. Bar- low, on mature consideration, has given the following table of the tenacity of wood usually employed by the carpenter; aDd subjoined is the mean result from the fore¬ going table, for the sake of contrast:— Table II .—Tenacity of a square inch of Timber in lbs. Barlow's Mean Results. Mean Results of Table No. I. Oak,. 10,000 13,316 Ash,. 17,000 14,670 Beech,. 11,500 14,720 Teak,. 15,000 12,460 Mahogany, .. 8,000 18,950 Riga fir. 12,000 12,776 As Mr. Barlow’s experiments were very carefully con¬ ducted, his table may be assumed as a safe guide. 1. As the strength of cohesion must be proportional to the number of fibres of the wood, or, in other words, to the area of the section, it follows that the tenacity of any piece of timber, or the weight which will tear it asunder lengthways, will be found by multiptying the number of square inches in its section by the tabular number corres¬ ponding to the kind of timber. Example. —Suppose it is required to find the tenacity of a tie-beam of fir, of 8 X 6 inches scantling. 8x6= 48, which, multiplied by 12,000, the tabular number for fir, gives 576,000 lbs. This is the absolute tenacity. Practically it is not con¬ sidered safe to use more than one-fourth of this weight, or 144,000 lbs. By the rule inverted, the section of the timber may be found when the weight is given, as follows:— 2. Rale. —’Divide the given weight by the tabular number, and the quotient multiplied by 4 is the area of section required for the safe load. Example. —Required the area of section of a piece of fir to resist safely a tensile strain of 144,000 lbs. 144000 _ ]2 x 4 = 4S, the section required. 12000 5 1 II. Resistance of timber to compression in the direction of the length of its fibres. Experiments on this kind of resistance are not nume¬ rous. Mr. Rennie found that to crush a cube of 1 inch on the side, the weights in lbs. required were, for— African oak,. 6,720 lbs. English oak, ... ... ... 3,860 „ White deal, . 1,928 „ American pine, ... ... ... 1,606 „ Elm, ... ... ... ... ... 1,284 „ M. Rondelet obtained higher results ; that is, the tim¬ ber experimented on by him presented a greater re¬ sistance. According to this author, a piece of timber diminishes in strength as it begins to bend; so that the mean strength of the wood of the oak, which is 44 lbs. French for every line superficial in the case of a cube, is i reduced to 2 lbs. for a piece of the same wood, when its STRENGTH AND STRAIN OF MATERIALS. 125 height is 72 times its base. From a great number of experiments, he compiled the following progression:— Tn a cube, the height of which is 1, the strength is 1' la a piece, the height of which is 12, „ •833 „ „ 24, •5 „ „ „ 36, •33 „ „ „ 48, „ T66 „ „ „ 60, „ •083 „ » » 72, n •041 Thus, in a cube of oak of 1 inch super¬ ficial of base, submitted to the action of a force pressing vertically, the mean force is expressed by 144 X 44 = 6336, In a bar of the same wood of the same base, and 12 inches high, the strength by the progression would be 144 X 44 X -833 = 5278, . In a bar 24 inches high, the strength by the formula, 144 X 44 X '5 = 3168, In a bar 36 inches, the strength by the formula, 144 X 44 X '333 = 2110, In a bar 48 = 144 X 44 X T34 = 849. In a bar 60 = 144 X 44 X '083 = 526. In a bar 72 = 144 X 44 X '042 = 266. In fir bars the following results were obtained:— By Calculation. By Experiment. In the cube of 1 inch, 144 X 52 X 1' = 7,488 7,490 Iu the bar of 12 inches, „ X '833 = 6,238 6,355 „ 24 „ „ X -5 = 3,744 3,429 „ 36 „ „ X '33 = 2,471 2,575 These results, M. Rondelet observes, accord with the experiments of MM. Perronnet, Lamblardie, and Girard. In an experiment described by the latter, a piece of wood 2 metres 273 millemetres long, and 155 by 104 millemetres in section, broke under a weight of 33,120 kilogrammes. Reducing these to English measures, the length is 7'35 feet, the sides of the base 6 05 and 4 - 06 inches respectively. The area of the section is therefore 24'5 inches, and as the weight is 72,864 lbs., the weight on the square inch is 2974 lbs. The height of the piece is about twenty-two times the size of its base, and the progression would give a reduc¬ tion of half the strength. If, therefore, 2974 is doubled, 5948 lbs. are obtained as the absolute resistance per square inch—a result which agrees very closely with the experiments of M. Rondelet. The same author arrived at the following conclusions :— 1. That the resistance does not sensibly diminish in a prism, the height of which does not exceed eight times its base. 2. That when the height of the prism is ten times its base, it begins to yield by bending. 3. That when the height is 16 times the base, the piece of wood is incapable of resistance. Tredgold says that when the length of the wood is less than eight times its diameter, the force causes it to expand in the middle of its length, and split into several pieces, but when it exceeds this length it yields by bending. As in practice the last case is almost the only one which occurs, we shall confine our observations to it alone. As the first degree of flexure would prove fatal to any piece of framing, the strength necessary to resist this is what is required. According to Tredgold, the strain is directly as the weight, and inversely as the strength, which is inversely as the cube of the diameter in the case of a column. The strain is also directly as the square of the 1 While the mean of ) ^ experiments gave, ) *’ 5,310 2,911 2,163 length, and inversely as the diameter, which is directly as the deflection. Therefore, for cylindrical posts he gives the formula T7 e X L 2 x W = D 4 , in which e is a con¬ stant number for the kind of timber (see p. 126). When the post is rectangular, and D is its least side, e x L s x W = B D 3 . When the post is square, 4 e X L 2 x W = D 4 , where D is the diagonal of the square. The stiffest rectangular post is that in which the greater is to the less side as 10 to 6. The equation is then 0‘6 e X L- x W = D 4 . When D is the least side, divide by 0 6 to find the greater. These rules, expressed in words, are as follows:— To find the diameter of a post that will sustain a given weight when the length exceeds ten times the diameter. Rule. —Multiply the weight in lbs. by 17 times the value of e, then multiply the product by the square of the length in feet, and the fourth root of the last product will be the diameter in inches. Examples. —1. The height of a cylindrical oak post being 10 feet, and the weight to be supported by it 10,000 lbs., required its diameter. The tabular value of e for oak is '0015— therefore, 17 X '0015 X 100 X 10000 = 2550, the fourth root of which is 7'106, the diameter required. By inverting the operation, we find the weight, when the dimensions are given, thus—The height of a cylin¬ drical oak post being 10 feet, and its diameter 7106 inches, required the weight it will support. The fourth power of 7 106 inches, as we have seen, is 2550, therefore— 17 X 2 -g—- - = = 10,000. 0015 X 100 - 2o5 To find the scantling of a rectangular post to support a given weight. Rule. —Multiply the weight in lbs. by the square of the length in feet, and the product by the tabular value of e. Divide this product by the breadth in inches, and the cube root of the quotient will be the thickness in inches. 2. Let the height of the post, as before, be 10 feet, and the weight to be suppor ted 10,000 lbs., required the thick¬ ness of the post when its breadth is 5 inches. • 0015 X 10- X 10000 _ 3()0 5 the cube root of which is 6 69, therefore the section of the post is 6 69 X 5 inches. To find the dimensions of a square post that will sus¬ tain a given weight. Rule. —Multiply the weight in lbs. by the square of the length in feet, and the product by four times the value of e, and the fourth root of the product will be the diagonal of the post in inches. 3. Let a square oak post be 10 feet long, and let the weight to be supported be 10,000 lbs., required the dimen¬ sions of its sides. The value of e is -0015 — therefore, '0015 x 4 x 100 X 10000 = 6000, the fourth power of the diagonal of the square, therefore the diagonal of the square is 8’8 inches, and its side 6'22. To find the stiffest rectangular post that will support a given weight. Rule. —Multiply the weight in lbs. by 0'6 times the tabular value of e; multiply the product by the square of PRACTICAL CARPENTRY AND JOINERY. IPG the length in feet; the fourth root of this product will be the least side in inches. Divide this by 06 for the greatest side. 4. Let the length of the stiffest rectangular oak post be 10 feet, and the weight to be supported 10,000 lbs., required the side of the post. 0-6 x 0015 x 10x10 x 10000 = 9000, the fourth root of which is 5-477, the least side, which, divided by O'G, gives 9T3 as the greatest side. From experiments made by Lamande and Girard, by loading posts till a small amount of deflection was visible, and thus ascertaining the values of l W D, Mr. Tredgold calculated the value of the constant e, so as to give the load to which the post might be safely exposed. These constants for different woods are found in the following table:— Table 1IT .—Constant lVumhers, to be used in calculating the Dimen¬ sions of Posts, Pillars, dec., of Timber, pressed in the direction of their Lengths. Kind of Wood. Value of the Constant c. English oak. .... | -0015 Beech, . •00195 Ash,. .... -001G8 Elm,. .... 1 -00184 Spanish mahogany, . •00205 Honduras do. . -00161 Teak, . .... : '00118 Riga fir, . .... 1 -00152 Memel ditto, . .... , -00133 Norway spruce,. .... -00142 Larch,. .... -0019 III. Resistance of timbers to transverse strain. When a piece of timber is fixed horizontally at its two ends, then, either by its own proper weight, or by the addition of a load, it bends, and its fibres become curved. If the curvature do not exceed a certain limit, the tim¬ ber may recover its straightness when the weight is re¬ moved; but if it exceed that limit, although the curvature diminishes on the removal of the load, the timber never recovers its straightness, its elasticity is lessened, and its strength is partly lost. On the load being augmented by successive additions of weights, the curvature in¬ creases until rupture is produced. Some woods, how¬ ever, break without previously exhibiting any sensible curvature. It may be supposed that, in the case of the timber being exactly prismatic in form, and homogeneous in structure, the rupture of its fibres would take place in the middle of its length, in the vertical line, where the curves of the fibres attain their maxima. In the rupture by transverse strain of elastic bodies in general, and consequently in wood, all the fibres are not affected in the same manner. Suppose a piece of timber, composed of a great number of horizontal ligneous layers, subjected to such a load as will bend it, then it will be seen that the layers in the upper part aie contracted, and those in the lower part extended, while between these there is a layer which suffers neither compression nor tension; this is called the neutral plane or axis. If the position of the neutral axis could be determined with precision, it would render more exact the means of calculating transverse strains; but as the knowledge of the ratios of extensibility and compressibility is not exact, the position of the neutral axis can only be vaguely deduced from experiment. Were the ratios of compres¬ sion and extension equal, the neutral axis would be in the centre of the beam; but experiments show that this equality does not exist. Barlow found that in a rectangu¬ lar fir beam the neutral axis was at about five-eighths of the depth;* and Duhamel cut beams one-third, and one- half, and two-thirds through, inserting in the cuts slips of harder wood, and found the weights borne by the uncut and cut beams to be as follows:— Uncut Beam. One-third Cut. One-half Cut. Two-thirds Cut. 45 lbs. 51 lbs. 4S lbs. 42. Results which clearly show, that less than half the fibres were engaged in resisting extension; and it has been long known that a beam of soft wood, supported at its extremities, may have a saw-cut made in the centre, half-way through its thickness, and a hard wood piece inserted in the cut, without its strength being materially impaired. It is not here necessary to enter into an investigation of the theory of transverse strain. The results of it, cor¬ rected by experimental evidence on which rules of practical utility may be founded, are all that need be sought for. The transverse strength of beams is— Directly as the breadth, Directly as the square of the depth, ami Inversely as the length; or substituting the letter b for the breadth, d for the depth, and l for the length, and placing the ratios together, the general expression of the relation of strength to the dimensions of a beam is obtained as follows:— b X cl 1 L But this forms no rule for application, since beams of different materials do not break by the application of the same load; and it is therefore necessary to find by experi¬ ment a quantity to express the specific strength of each material. Let this quantity be represented by S, and the formula becomes— bxddx S . . . - I - - = breaking weight. By this formula experiments can be reduced so as to give the value of S. It is only necessary to find the breaking weight of a beam whose dimensions are known, and then by transposition of the equation— l X breaking weight _ 0 YxTP “ “ b - S thus becomes a constant for all beams of the same material as the experimental beam. Although this ratio of strength to the dimensions of the beam is very nearly correct, it is not absolutely so; and the French writers modify it in the manner which will be stated subsequently. The length of a beam appears to * Mr. Barlow’s method of operating was as follows:—He ran a saw cut to 5-8ths of the depth of the beam, and inserted a thin slip of pear tree, sufficiently tight to preserve the stiffness of the beam, but not so tight as to cripple it. The beam was then loaded till it broke. On examining the slip of pear-tree after the fracture of the beam, the impression of the fibres was found distinctly marked on it, strongest at top, and weakening gradually to the bottom, where compression ceased. STRENGTH AND STRAIN OF MATERIALS. 1*27 influence tlie strength to a greater extent than the theory allows; for similar beams are rather more than twice as strong when half as long, and the strength does not increase quite so rapidly as the square of the depth. When the value of S for various kinds of wood is deter¬ mined, the formula may be used for computing the strength of a given beam, or the size of a beam to carry a given load. For any three of the quantities, l, b, d, W, being given, we can find the fourth thus:— I. When the beam is fixed at one end and loaded at the other, and when l W 6 d 2 S. The length, breadth, and depth being given, to find the weight— _ S b dr The weight, breadth, and depth being given, to find the length— l = S b di W The weight, length, and depth being given, to find the breadth— b = l W S d 2 The weight, length, and breadth being given, to find the depth- d = / ^y l_ w When the section of a beam is square, that is, when b = d; then b or d = ^ . II. When the beam is supported at one end, and loaded l W in the middle former cases— then 4 6 d 2 S, and we have for the W = d = y / i b d- S l l W l = 4 b d" S W ~ b - l W and where b = d. b or d 4 6 S ] II. When the beam is fixed at both ends— 4d ! S ’ l W . 4 S When loaded in the Middle. 6 6d 2 S W = b = l 6 b d- S . W ’ IW . 6 d 3 S ’ l w . 6 b S ’ t i 3/ nv d=b= When loaded at an Intermediate roint. 1 = b = d =>J 2 m n 2 m n W 3 b d 3 S * 2 m nW 3 Id* S ’ 2 m n W , , o, 2 m n W d = b = & U6 ■ Where to n is the rectangle of the segments into which the load divides the beam. IY. When the beam is supported at both ends, but not fixed— When load is uniformly diffused. 2 b d 3 S w = b = l 2 b d 3 S _ W~~ 5 IW . 2 b d 3 ’ . / lW ■ d ~ V 2b S ’ b= d= — When loaded at an intermediate point. „ r lb d* S . m n W 1 ~ b d 2 S ’ m n W _ b ~ Id* S ; , , inn W d=i = & ~ When the beam is inclined, the horizontal distance be¬ tween its supports is the length or l. The following table contains the results of experiments made by Mr. Barlow on the transverse strength of various kinds of wood, with the value of S, calculated according to the formula S = :— 4 6 ci- Table IY. 1. 2. 3. s. 6. 7. 8. 9. 10. 1 11. 1 -s 8 ■j?3 !> ■S-3 Name of Woods. 3 1 3 1 .3 0-5 ua ll § ol 2^ S f 8, s= 4 -i£. c, c _iW C ~bd‘- |1 1 i-i W o f £ Ai-n-i" 2 1195 621 ! 4-16 2 727 1304 ... Middle. 702 1304 2037 679 Outside. 7 2 760 225 772 2026 675 3 GOG 593 1556 4-16 ? 'J 702 1104 1820 607 Middle. 030 1304 2037 679 Outside 4-06 o G4S 1253 17S5 595 Middle. 051 1174 1334 G05 2 Outside. 4 16 2 2 1020 1000 2046 882 2 416 2 2 GS0 906 1562 521 2 590 110S 1731 577 Elm. 416 2 2 543 714 1115 372 2 Fir, New England, ... 7 2 2 150 420 1102 307 3 ,, Riga. 2 2 753 125 422 1108 369 3 2 738 150 407 1051 350 3 GOG 125 430 1144 3S0 3 2 G08 150 561 1202 420 4 16 2 2 1000 1752 584 2 32o S53 284 445 142 445 1036 345 9 577 200 655 1474 401 3 Oak, English,. 7 2 900 150 450 1181 394 0 934 200 037 1072 557 3 2 2 872 225 073 1700 OSS 3 756 200 500 1457 4S5 3 2 903 150 520 13S3 448 3 English. 416 2 2 903 999 15611 >{ Fa8t grown. 856 677 1058 >{ Slow grown. 972 999 1561 it East grown. 835 943 1473 § M grown. 74S 1447 2261 In store 756 1304 2037 Mean 553 In "store 10 years. Bitch pine,.. 7 2 2 660 150 622 1632 544 3 Hod pine,. 7 2 657 150 511 1341 447 3 745 In using Mr. Barlow’s formulae for transverse strength, the eighth column of the table gives the constants S for the various kinds of timber; and all the dimensions of the timber must be in inches. The rules for two of the cases, expressed in words, are as follows: — To find the strength of a rectangular beam, fixed at one end and loaded at the other. Rule .—Multiply the value of S by the area of the sec¬ tion, and by the depth of the beam, and divide the pro¬ duct by the length in inches. The quotient will be the breaking weight in lbs. Example .—A beam of Riga fir projects 10 feet beyond its point of support, and its section is 8 X 6 inches, what is its breaking weight? Area 8 X 6 = 48, multiplied by the depth 8 = 384. Multiply this by the constant 1108, and divide the product by the length, = 3545 lbs. The fourth part of this is the safe weight to impose in practice, therefore— 354 jL = 886 lbs. 4 To find the strength of a rectangular beam, when it is supported at the ends and loaded at the middle. Rule .—Multiply S by four times the depth, and by the area of the section in inches, and divide the product by the length between the supports in inches, and the quotient will be the greatest weight the beam will bear in lbs. Example .—A beam of Riga fir is 20 feet long between 128 PRACTICAL CARPENTRY AND JOINERY. its supports, and its section is 8 X 6 inches; required its breaking weight. 1108x8x4x8x6 240 = 7091-2. The fourth part of this is the safe load, therefore — 7091 —-— = ii(2 lbs. 4 The constant C, in the same table, column 9, and in column 11 of Table VIII., is for the formulae as given by Mr. Tredgold, in which it is not necessary to convert the length of the bearing into inches. When the beam is supported at one end, and loaded at the other, the equation is — breaking weight. Or, in words, as follows:—Multiply the breadth in inches by the square of the depth in inches, and by the constant C for the kind of timber; and divide the product by four times the length in feet, for the breaking weight. Example .—A beam of Riga fir projects 10 feet beyond its point of support, and is 8 X 6 inches in section; re¬ quired its breaking weight. The value of C for Riga fir is 369, therefore— 369 X 6 x 64 — = 3o42'4 = the breaking 40 light. 3542 4 = 88 5= safe weight. When the beam is supported at both ends, and loaded in the middle. Rule .—Multiply the breadth by the square of the depth, and by C, and divide by twice the length ; the quotient is the breaking weight. Example.—he t the beam be 8 x 6 in section, and 20 feet long between its supports; required the breaking weight. 369 X 6 X 64 20 - = 7084. In the summary of formulae in the sequel (p. 130), will be found all the rules written in words. It has been stated that some French writers use for¬ mulae different from those here given, and intended to unite more closely the results of theory and experiment; but they are difficult of application in practice, and the results vary so little from those of the formulae given, that the correction is not worth making. Mr. Gwilt, in his Encyclopcedia of Architecture, not only ignores all the laborious experiments that have been made in this country, but also speaks disparagingly of the labours of the able men who have endeavoured to benefit the architect and engineer by bringing the aid of mathe¬ matical investigation to found upon those experiments safe and general rules for practice. Mr. Gwilt cites Buffon’s experiments, as given by Rondelet, and in his introduc¬ tory notice of them says, “ They are worth more than all which has hitherto been done in this country. The treatises on mechanical carpentry seem to have been written more with the view of perplexing than of assisting the student/' And this, too, notwithstanding the labours of Robison, Young, Barlow, Tredgold, Hodgkinson, Bevan, Rennie, and a host of others. But a little investigation would have made it apparent that the constants derived by these writers from their ex¬ periments give results most singularly in accordance with those obtained by the able French philosopher; and, which is of greater importance, perfectly safe when applied in practice. The discrepancy between the results of the English for¬ mulae here given and the experiments of Buffon occurs chiefly in short bearings; and as the strength obtained by the formulae is sufficient for long bearings, it follows that it must be sufficient also for the short lengths, erring only in being slightly in excess, or “a little stronger than strong enough/’ The rule given by Rondelet is expressed in terms of only two of the three dimensions of the timber, and requires a constant which is empirical, like that of the formulae already given. It is as follows:— Subtract from the primitive strength one-third of the quantity which expresses the number of times the depth is contained in the length of the beam. Multiply the remainder by the square of the length. Divide the product by the number that expresses the relation of the depth to the length. In the following table, the four first columns give the result of certain of Buffon’s experiments, reduced to Eng¬ lish measures: the two other columns contain constants for Barlow’s and Tredgold’s formula), calculated from the other columns:— Table Y.— Results of certain of M. Buffon's Experiments on Transverse Strength. No. Length. Side of Square. Mean Weight which brake the pieces. Breaking Weight by Rondelet's Formula. Value of S, Value of c, 1 . 7-462 4-264 5,768 5,768 1670 546 8'526 4,869 4,943 1612 538 9-51)4 4.387 4,301 1635 545 10-660 3,946 3.799 1644 548 12-792 3,279 3,018 1623 541 7-462 5 "33 12,496 12,496 1847 616 8"52S 10,626 10,750 1800 600 9 594 8,635 9,429 1440 480 ln-660 7,765 S,357 1441 480 12-792 6,644 6,748 167S5 559 14-924 5,819 4,600 1730 577 17-096 4,810 4,738 1620 540 19-188 4,120 4,066 1500 500 21-320 3,624 3,530 1531 510 23-452 3,364 3,092 1563 521 25-584 2,502 2,726 1260 420 29-848 2,112 2,151 1247 412 7 462 6-396 20,635 19,196 1654 554 8-52S 16,804 16,562 1637 545 9-594 14,292 14,547 2011 670 10-660 12,197 12,877 1500 500 12-792 9,938 9,420 1446 482 14-924 8,210 8,666 1400 466 17-056 7,030 7,348 1365 442 19-188 6,187 6,319 1300 434 21-320 5,495 5,506 1337 446 10-660 S'528 30,148 30,363 1828 609 12-792 25,540 24,883 1804 601 14-924 21,605 20,854 1843 614 17-056 17.968 17,833 1740 5S0 19-188 14,577 15,482 1587 529 21-320 13,303 13,593 1570 523 Average... ...1621... ... 540 The constants S and C agree very nearly with those derived from the experiments of Mr. Barlow and Mr. Tredgold: the latter assumes 550 as the value of S. In any beam exposed to transverse strain, it is mani¬ fest that there must be some certain proportion between the breadth and depth which will afford the best results. It is found that this is obtained when the breadth is to the depth as 6 to 10. Therefore, when it is required to find the least breadth that a beam for a given bearing should have, the formula is as follows:— 0-6 = 6 ; or, expressed in words- Rule .—Divide the length in feet by the square root of the depth in inches, and the quotient, multiplied by the decimal 0-6, will give the least breadth the beam ought to have. STRENGTH AND STRAIN OF MATERIALS. 129 The nearer a beam approaches to tlie section given by this rule, the stronger it will be; and from this rule is derived the next. To find the strongest form of a beam so as to use only a given quantity of timber. Rule .—Multiply the length in feet by the decimal 0 6, and divide the given area in inches by the product, and the square of the quotient will be the depth in inches. j Example .—Let the given length be 20 feet, and the given area of section 60 inches. Then ^ — 5 00, 4\) X D b the square of which is 25 inches, the depth required, and the breadth is consequently 2-4 inches. The stiffest beam is that in which the breadth is to the depth as *58 to 1. The stiffest beam which can be cut out of a round tree may be found Fig. 450. graphically as follows:—Let a b (Fig. 450) be the diameter of the circle, then with a radius equal to the radius of the circle, and from the extremities of the diameter, draw the arcs ode, f dg; join a c, c b, b d, d a, and the parallelogram thus formed is the section of the stiffest beam which can be cut from the tree. The French writers make the ratio of breadth to depth as 5 : 7. It will be seen by the formulae that there is an un¬ questionable advantage obtained by fixing the ends of bearing timbers, as the increase of strength is then as 3 to 2. This, although not easily accomplished, nor indeed proper when the ends of the timbers are built into walls, as the stability of these may be injured, yet may be done in certain circumstances; and it further leads to important practical rules. The chief of these is, that girders and joists, and all bearing timbers whatever, when laid over several points of support, should be made as long as pos¬ sible; and that purlins, rafters, and joists should, when¬ ever the space will admit of it, be notched on the sup¬ ports, in place of being framed between them. But the breaking weight of a beam has seldom, for practical purposes, to be ascertained. For when a beam is loaded, and even by the effort of its own weight, when its length is great compared with its other dimensions, it sinks down or bends in the middle. If the deflection produced by a load does not exceed a certain limit, the timber will recover its straightness when the load is removed; but beyond that limit, the elasticity of the fibres becomes diminished; the timber loses part of its strength, and the addition of weight at length causes rupture of the fibres. Mr. Barlow gives one-fourth of the breaking weight as the greatest weight that should be used; but experiment shows that one-fifth of the breaking weight produces a permanent set in a beam, and consequently a diminution of its strength; and the French authors invariably give only one-tentli of the breaking weight as the safe load when the load is fixed; but if a moving load has to be provided for, they say that it should not exceed one- twelfth or one-thirteenth of the breaking weight. But the load which a beam may sustain without per¬ manent injury, may cause such an amount of deflection as to unfit it for the purposes of the carpenter; for a beam, forming part of a system of framing, cannot be deflected without a sensible alteration of its length, and thus the framing becomes deranged. Moreover, in girders, beams, and joists which sustain floors and ceilings, where the work should not only be true, but appear true, a very small amount of curvature would mar its beauty. Mr. Tredgold assumes that this curvature is not sensible when it does not exceed the -Tj-J-yth part of the length of the beam; and in forming his rules, he has taken the fortieth part of an inch in every foot as the allowable deflection. The term stiffness is opposed to flexibility: thus, when a piece of timber bends under a weight in a very small degree, it is said to be stiff; and when it bends consider¬ ably, it is said to be flexible. Now, the stiffness of beams is proportional to the space they are bent through by a given weight when the lengths are the same, but that two pieces of different lengths may be equally stiff, the deflection should be proportional to their lengths. For a deflection which would not be injurious in a beam 40 feet long, would be highly detrimental in one 10 feet long. The extension of any part of a beam is directly as the force that produces it, and as it is known by experiment that the deflection is as the weight, all other things being the same, the deflection is therefore as the extension. The extension is as the weight and the cube of the length directly, and as the breadth and cube of the depth inversely: the deflection will consequent^ be in the same proportion. If L be the length in feet; W, the weight in lbs.; B, the breadth in inches; and D, the depth in inches; L 3 x W . , , ,. B " x ~ D 3 1S as ™ ie deflection. In order that a beam may be equally stiff, as we have seen above, the deflection should be inversely as the length: consequently, the weight that a beam will sus¬ tain will be ^ X — \y where d is the deflection in d x L- inches; and ~ X . ^, — a constant number for the same ’ BxD"c2 kind of timber. But, before these rules can be applied, it is necessary to obtain, experimentally, the value of d. This being done, we should have ^ — a constant quantity; which being given, the deflection in any other case may be found. Experiments, with the view of determining this con¬ stant, have been made by various writers. Of these, probably, the most satisfactory were by Duhamel; for this reason, that the scantlings operated on were of the size generally used in building. From these and other experi¬ ments, many of which were made by himself, Mr. Tred¬ gold computed the constants for the various kinds of tim¬ ber given in the following table. In computing the value of a, Mr. Tredgold assumed the deflection to be ^th °f an inch per foot; and the formula B X D 3 became x ~W = a ‘ ^ ien ^ ie deflection is required to be less than this, say I of ^ T , then multiply the constant a by 2; if a of T V> multiply a by 3; or if required greater than -jV’ multiply a by any number of times that the deflection may exceed ^Vth of an inch per foot. R mo PRACTICAL CARPENTRY AND JOINERY. Table VI.— Table oj the Values of “a" Oak, old, ... ... -0099 „ young, ... ■0105 „ „ . ... -0164 „ „ •0197 „ Riga, ... -0107 „ English,. •0119 „ Canadian. ... ... *0090 » Adriatic, •0193 Dauzic ... -0105 9 ) T119 Mean, ... -0124 Mean of 10 examples of English oak, •0124 White spruce, ... -00957 •0138 To find the scantling of a piece of timber which, when laid in a horizontal position, and supported at both ends, will resist a given transverse strain, witli a deflection not exceeding f a th of an inch per foot. 1. When the breadth and length are given, to find the depth. Rule. —Multiply the square of the length in feet by the weight to be sustained in lbs., and the product by the tabular number a (Table VI.; and column 10, Table VIII.) ; divide the product by the breadth in inches, and the cube root of the quotient will be the depth in inches. Example. —Required the depth of a pitch pine-beam, having a bearing of 20 feet, and a breadth of 6 inches, to sustain a weight of 1000 lbs. The square of the length, 20 feet = 400 Multiplied by the weight . 1000 And the product .... 400,000 By the decimal . . . . . 016 Divide the product by the breadth, 6 inches.= 6400 000 Gives ...... 1066*666 The cube root of which is 102 inches, the depth re¬ quired. 2. When the depth is given. Rule. —Multiply the square of the length in feet by the weight in lbs., and multiply this product by the tabu¬ lar value of a: divide the last product by the cube of the depth in inches, and the quotient will be the breadth required. Example. —Length of pitch-pine beam 20 feet; depth, 10-2 inches; weight, 1000 lbs. Then 20 x 20 x 1000 x -016 6100 = 6, the breadth 10-2 x 102 x 10 2 “ 1061 required. 3. When neither the breadth nor the depth is given, but they are to be determined by the proportion before given, that is, breadth to depth as 0 6 to 1. Rule .—Multiply the weight in lbs. by the tabular num¬ ber a, and divide the product by 0*6, and extract the square root: multiply the root by the length in feet, and extract the square root of this product, which will be the depth in inches, and the breadth will be equal to the depth multiplied by 0 6. Example. —Weight, 1000 lbs.; value of a, "016; length, 20 feet. v /lOOOX J 0 L 6 xao = m V 106 = 10 2, the depth required ; then 10'2 X 0 6 = 6 0, the breadth required. The following are the formulce given by Mr. Barlow:— B = the breadth, D = the depth, L = the length, W = the weight, and d — the deflection. When the beam is supported at one end. 1. Where the weight is at the extremity, the breadth, length, and amount of deflection being given, to find the depth. / W V EBcT {The length in feet, the breadth, depth, and deflection in inches.) 2. Where the weight is uniformly spread, the breadth, length, and deflection being given, to find the depth. ,/S*78W V EBi {The length in feet, the breadth, depth, and deflection in inches .) And when the beam is a cylinder, and the weight at its end, the deflection will be 1*7 times that of a square beam. When the beam is supported at both ends, and the length, weight, and deflection given. ,xL = D. x L = d. The weight being in the middle. w P , = B D". The weight uniformly spread. ■625 W L 3 -=bd\ 16 E d ’ 16 ed {The length in feet, the breadth, depth, and deflection in inches.) And in a square beam— W L 3 _ _ -625 W L s 16Ed • 16Ed When the beam is a cylinder. Multiply the quotient by 17, and the fourth root of the product is the diameter. = B 1 = D 4 Table VII.— Table of the Value of E in the above Formu CALCULATED. Ash,. E = -244 Beech, . ... „ T95 Birch, . „ -240 American ditto,. ... „ ’256 Deal, Christiana, „ -230,- Deal, Memel, ... „ T90 Elm, . „ T01 Fir, New England, •317 Fir, Riga, . „ -167 Fir, Mar Forest, „ -94 Greenheart, ... „ -384 Larch,. . . -91 \ Do., . .. ., -152 J Do.,. •119) Norway Spar, ... ,, -211 Oak, English,... „ -210 Oak, Canadiau, ... ... „ -310 Oak, Danzig, ... „ T49 Oak, Adriatic, ... „ T42 Pitch pine, •177 Red pine, ... ... „ ‘272 Teak, ... „ -349 ( Mean, T20. Summary of Rules. I. Resistance to Tension or Tenacity. To find the tenacity of a piece of timber. 1. Rule —Multiply the number of square inches in its section by the tabular number corresponding to the kind of timber (Table I.; or column 4, Table VIII.). STRENGTH AND STRAIN OF MATERIALS. 131 To find the area of section when the weight is given. 2. Rule. —Divide the given weight by the tabular number, and multiply the quotient by 4 for the area of section required for the safe load. II. Resistance to Compression. It is not necessary to give rules for the absolute crush¬ ing force of timber. Those that follow are applicable to the cases of posts whose length exceeds ten times their diameter, and which yield by bending. To find the diameter of a post that will sustain a given weight. 3. Rule— Multiply the weight in lbs. by l - 7 times the value of e (Table III.; or column 9, Table VIII.); then multiply the product by the length in feet, and the fourth root of the last product is the diameter in inches required. To find the scantling of a rectangular post to sustain a given weight. 4. Rule. —Multiply the weight in lbs. by the square of the length in feet, and the product by the value of e: divide this product by the breadth in inches, and the cube root of the quotient will be the depth in inches. To find the dimensions of a square post that will sustain a given weight. 5. Rule. —Multiply the weight in lbs. by the square of the length in feet, and the product by 4 times the value of e\ and the fourth root of this product will be the diagonal of the post in inches. To find the stiffest rectangular post to sustain a given weight. 6. Rule. —Multiply the weight in lbs. by 06 times the tabular value of e, and the product by the square of the length in feet; and the fourth root of this product will be the least side in inches: divide the least side by 0 6 to I obtain the greatest side. III. Resistance to Transverse Strain. 1st. When the beam is fixed at One end and loaded at the other. To find the breaking weight, when the length, breadth, and depth are given. 7. Rule. —M ultiply the square of the depth in inches by the breadth in inches, and the product by the tabular value of S (Table V.; or column 12, Table VIII.), and divide by the length in inches: the quotient is the breaking weight. To find the length, when the breadth, depth, and break¬ ing weight are given. 8. Rule. —Multiply the square of the depth by the breadth, and by the value of S, and divide by the weight: the quotient is the length. To find the breadth, when the depth , length, and break¬ ing weight are given. 9. Rule. —Multiply the weight by the length in inches, and divide by the square of the depth in inches multi¬ plied by the value of S: the quotient is the breadth. To find the depth, when the breadth, length, and weight are given. 10. Ride. —Multiply the length in inches by the weight, divide the product by the breadth in inches multiplied by S, and the square root of the quotient is the depth. To find the side of a square beam, when the length and weight are given. 11. Rule. —Multiply the length in inches by the weight, divide the product by S, and the cube root of the quo¬ tient is the side of the square section. 2d. When the beam is supported at one end and loaded in the middle. The length, breadth, and depth, all in inches, being given, to find the iveight. 12. Rule. —Multiply the square of the depth by 4 times the breadth, and by S, and divide the product by the length for the breaking weight. The weight, breadth, and depth being given, to find the length. 13. Rule. — Multiply 4 times the breadth by the square of the depth, and by S, and the product divided by the weight is the length. The weight, length, and depth being given, to find the breadth. 14. Rule. —Multiply the length by the weight, and the product divided by 4 times the square of the depth multiplied by S, is the breadth. The weight, length, and breadth being given, to find the depth. 15. Rule. —Multiply the length by the weight, and divide the product by 4 times the breadth multiplied by S. When the section of the beam is square , and the weight and length are given, to find the side of the square. 16. Rule. —Multiply the length by the weight, and divide the product by 4 times S: the cube root of the quotient is the breadth or the depth. 3d. When the beam is fixed at both ends and loaded in the middle. The breadth, depth, and length being given, to find the weight. 17. Rule. —Multiply 6 times the breadth by the square of the depth, and by S, and divide the product by the length for the weight. It is not necessary to repeat all the transpositions of the equation. 4th. When the beam is fixed at both ends and loaded at an intermediate point. 18. Rule. —Multiply 3 times the length by the breadth, and by the square of the depth, and by S; and divide the product by twice the rectangle formed by the segments into which the weight divides the beam. For example, if the beam is 20 feet long and the weight is placed at 5 feet from one end, then the segments are respectively 5 feet and 15 feet, or, in inches, 60 and 180; and the rectangle is 60 X 180 = 10800; and twice this amount, or 21600, is the divisor. Suppose the beam of Riga fir, fixed at both ends, and its section 8x6 inches, and the weight placed at 5 feet from one end, required its breaking weight: then, three times the length = 720, multiplied by the product of the breadth into the square of the depth, and by the tabular value of S = 306339840; which divided by 21600, as above, gives 14,182 lbs. as the breaking weight. 5th. When the beam is supported at both ends, but not fixed, and when the load is in the middle. To find the weight, when the length, breadth, and depth are given. 19. Rule. — Multiply 4 times the breadth by the square of the depth and by S, and divide the product by the length: the product is the breaking weight. 6th. When the weight is uniformly diffused. 20. Ride. —Multiply twice the breadth by the square 132 PRACTICAL CARPENTRY” AND JOINERY. of the depth and by S, and divide the product b}' the length: the quotient is the breaking weight. Note. —The beam bears twice as much when the load is uniformly diffused, as when it is applied in the middle of its length. 7tli. When the load is at an intermediate point. 21. Rule. —Multiply the length by the breadth, by the square of the depth, and by S, and divide the product by the rectangle of the segment, that is, by the product of the shorter and longer divisions multiplied together. IV. Rules for the Dimensions of Beams to resist a Transverse Strain with a deflection of not more than ■fijth of 1 inch per foot. The following are Mr. Tredgold's formulae:— 8th. When the beam is supported at both ends and loaded in the middle. When the weight, and the length and breadth are given, to find the depth. 22. Rule. —Multiply the square of the length in feet by the weight to be supported in lbs., and the product by the tabular value of a (Table VIII., column 10): divide the product by the breadth in inches, and the cube root of the quotient is the depth in inches. When the weight, and the length and depth are given, to find the breadth. 23. Rule. —Multiply the square of the length in feet by the weight in lbs., and the product by the tabular value of a: divide by the cube of the depth in inches, and the quotient will be the breadth in inches. When the weight and length arc given, and the ratio of the breadth to the depth is to be as 06 to 1. 2k Rule. —Multiply the weight in lbs. by the tabular number a: divide the product by 0 - 6, and extract the square root: multiply the root by the length in feet, and extract the square root of the product, which will be the depth in inches. To find the breadth ,—multiply the depth by 0 6. The following are the rules given by Mr. Barlow, for cases in which the amount of deflection is given. 9th. When the beam is fixed at one end, and loaded at the other, the weight in lbs., length in feet, and breadth and deflection in inches, being given, to find the depth. 25. Rule. —Divide the weight in lbs. by the tabular value of E (Table VII.; and column 13, Table VIII.) multiplied by the breadth and by the deflection; and the cube root of the quotient, multiplied by the length, will be the depth required. 10th. When the load is uniformly distributed. 26. Rule. —Take fths of the actual weight, or, which is the same, multiply the weight by '375, and then pro¬ ceed as above. 11th. When the beam is supported at both ends, and loaded in the middle. Given the weight in lbs., the length in feet, and the deflection in inches, to find the other dimensions. 27. Rule. —Multiply the weight by the cube of the length: divide the product by 16 times E, multiplied by the deflection, and the quotient is the breadth multiplied by the cube of the depth. When the beam is intended to be square, the fourth root of the above quotient is the depth or breadth. When it is a cylinder, multiply the quotient by 17, and the fourth root of the product is the diameter. 12th. When the load is uniformly distributed. 29. Rule. —Multiply the weight by -625, and by the cube of the length, and divide the product by 16 times E, multiplied by the deflection: the quotient is the breadth multiplied by the cube of the depth. If the reader has considered attentively the formulae, he will have been led to the conclusion that in a parallel¬ sided beam exposed to transverse strain, much of the material does not aid in resisting the force of the weight; that, in point of fact, there must be some other form than that of the parallelopipedon, which shall give the greatest results with the least quantity of timber. This, which is called the form of equal strength, has been thoroughly investigated, and we shall here present the results. In a beam fixed at one end, and loaded at the other, the form of equal strength is produced when the under side is made a parabola, and the breadth uniform (Fio\ 451). When the depth is uniform, the figure of the beam is of a wedge form. When the breadth and depth both vary, the form is a cubic parabola. In a beam fixed at one end, and having its load uni¬ formly distributed, the form ol equal strength is a triangle (Fig. 452). When a beam is supported at both ends, and loaded at the middle or any intermediate point, and the breadth is the same throughout, its upper side should be composed of Fig. 454. two parabolas whose vertices are at the points of support, and its lower side should be a straight line (Fig. 453). STRENGTH AND STRAIN OF MATERIALS. 133 When the depth is constant, the horizontal section or When it is loaded uniformly over its length, the upper side should be elliptical (Fig. 455). Fig. 456. When it is uniformly loaded, and the depth constant, the outline of the breadth should be two parabolas (Fig. 456). Table VTII .—Table of the Properties of Timber. 1. 2. - 3. 4. 5. 6. 7. 8. 9. Tredgold’s Formula). Barlow' Formulas. Specific Gravity, Weight of a foot, Weight of a Bur, Absolute Tenacity of a sq. inch. Tenacity of a sq. inch Modulus Modulus Crushing Constants for 10. 11. 12. 13. 1 foot long. without of Elasticity, of Elasticity, ’osts, and being in lbs. 1 inch sq., in feet. in lbs. value of Value of Value of Valuo of Value of 1-0. in lbs. in lbs. in lbs. f - a. C. S. E. Acacia,'. •710 44-37 •30 18,290 1,152,000 373,900 •621 •1867 Alder,. •soo 50- •347 14,186 6,895 1 ’ } •690 43-12 Ash,. \ •845 53-81 17,207 3,540 1,644,800 4,970,000 8,683 } •00168 •0105 •677 •2036 •244 \ •760 47-5 •33 1,640,000 9,363 ) Bay tree,. •822 5137 •35 12,396 7,158 Beech, . j •690 to 'S54 43- 5337 ‘315 14,720 2,3*60 1,353,600 4,600,000 7,733 9,363 •00195 •0*127 ’552 •1556 •195 Birch,. •792 49-5 •34 15,000 5,406,000 6,402 •0141 •643 •1881 •240 „ American, . •648 40-5 •28 11,663 1,257,600 3,388,000 11,663 •605 •1834 •256 Box,. •960 60- •41 19,891 Bullet tree, . 1-029 64-31 •446 2,601,600 5,878,000 •882 •2646 Cane,. •400 25- •174 6,300 Cedar, new,. ,, ,, seasoned,. Chestnut,. •909 10,293 5,674 •753 47-06 •32 4,912 •657 41-06 •285 11,900 ■0187 •765 47-81 •33 j 7,1*48 Deal, Christiana,. •698 43-62 •30 12,400 1,672,000 5,378,000 6,586 •0095 •521 •1562 •230-115 „ Metnel,. •590 36-87 •25 1,536,200 6,26S,000 •0089 •577 •1731 *190 „ Norway spruce,. •340 21-25 •147 17,600 7,293 •00142 „ English spruce,. •470 29-37 •20 7,000 •0124 Elder,. •695 43-43 •30 10,230 ( 9,973 ( 8,467 Elm,. | •544 •588 34- 3675 •236 13,489 3,240 1,340,000 699,840 5,680,000 10,3*31 ! •00184 •017 •372 •1115 •101 Fir, Riga, . •753 47-06 •32 | 11,549 12,776 1,328,800 869,600 4,080,000 5,748 j 6,819 J •00152 •00115 •369 •1108 •167 „ Red,. „ Mar,. •30 2,797,000 ■0233 •380 •1144 •94 Hawthorn,. •91 38-12 •26 10,500 ■86 •36 •32 18,000 16,000 Holly,. •76 47-5 •76 47-5 •32 20,240 7,289 •92 57-50 •40 io'soo Lance,. 1-022 63-87 •44 23,400 Larch, . j •522 32-62 •243 10,220 2^0*65 10,740,000 4,415,000 5,568 ) •0019 •0128 •284 •853 ■120 •560 35' 8,900 1,052,800 ... j Lignum-vitas,. . 1-22 76-25 •53 11,800 Lime tree, . Mahogany, Spanish,. •760 47-50 •32 23,500 •0152 •800 50- •34 16,500 8,198 •00205 0137 ,, Honduras,. Maple,. •560 35- •243 18,950 3,800 1,596,300 6,570,000 •00161 •0109 •793 49-56 10,584 •0197 Oak, English, . j •830 •934 52- 58-37 •36 13,316 3,960 1,700,000 1,451,220 4,730,000 ( 4,684 { 9,509 ( 10,058 | -0015 •0124 •553 •1658 •210 „ Canadian,. •872 54-50 •378 10,253 2,148,800 5,674,000 ( 4,231 j- •009 •588 •1766 •310 „ Danzic. •756 47-24 •327 12,780 1,191,200 974,400 3,607,000 7,731 •0087 •560 •1457 •149 ,, Adriatic, . •993 6206 •43 2,257,000 •526 •1383 „ African,. •972 60-75 •42 ' ... 2,282,300 5,583,000 •661 41-31 •283 9,861 7,518 ( 6,790 ) ( 5,445 5 ■0215 Pine, Pitch, . •660 41-25 •283 7,818 1,225,600 4,364,000 •0166 •544 ■1632 •177 „ Red,. •607 41-06 •26 10,000 1,840,000 6,423,000 5,375 ) •0109 •447 •1341 •272 ,, American yellow,... •461 •640 28-81 40- •20 •28 11,700 3,900 1,600,000 8,700,000 5,445 •0112 •0128 10,493 Plum tree,. •786 49-06 •338 11,351 ... j 9,367 3,657, wet •383 23-93 •164 6,016 13,000 ( 3,107 ) { 5,124 ] •0224 •69 43-1 •296 •0168 7,417,000 •00118 •0076 •820 •2462 •349 Walnut,. 8,465 14,000 8,000 7,227 •020 Willow,'. •390 24-37 •167 6,128 •031 •807 50-43 •347 134 PRACTICAL CARPENTRY AND JOINERY. PART FIFTH. PRACTICAL CARPENTRY. ROOFS. Roofs may be variously classed, according to their forms, and the combinations of their surfaces. The sim¬ plest are those which have either plane surfaces, or cylin- dric surfaces having their generatrix horizontal. The slope given to a roof is for the purpose of throwing off rapidly the water of rain or of snow, in order that the materials of the roof may be quickly dry. Many authors have occupied themselves in investigating the different slopes which should be given to roofs, according to the climate and the materials used as covering. The flat roofs or terraces of the East, and the high- pitched roofs of the countries of the North, the extremes of the scale, might lead to the supposition that climate alone had determined the proper slope to be adopted. Accordingly, M. Quatrembre de Quincy proposed to regu¬ late the slopes of roofs rigorously in accordance with the latitude; thus, commencing at zero at the equator, he elevates the roof 3° for each geographical climate* when the covering materials is pan tiles, and adds 3° more when the covering is of Roman tiles, 6° when it is of slates, and 8° when formed of plain tiles. This proposi¬ tion of M. de Quincy has been converted into a simple formula by M. Belmas, thus expressed:—Make the slope of the roof equal to the excess of the latitude of the place where it is constructed over that of the tropics. The latitude of the tropics being 23° 28’, the rule would give, for a roof in latitude 25°, a slope of only 1° 32'. But on so small an inclination as this, the necessary overlap of the tiles or slates would cause them to slope the reverse way, and the water would consequently penetrate under them. But climate does not appear in practice to be the regu¬ lator of the slopes of roofs; for in the same place are found roofs of various slopes, and neither the original proposi¬ tion of M. de Quincy, nor the form of it proposed by M. Belmas, are verified by experience. Indeed, the rule would seem to be based on a too limited series of observa¬ tion, and gives slopes too low for moderate climates, and too high for those in warm regions where terraces are invariably used. The following table shows the slopes actually in use in certain places, and the slopes according to M. de Quincy’s rifle:— * The space comprised between the equator and each polar circle is divided into 24 zones or climates, the limits of which are deter¬ mined by a difference of half an hour in the length of the day in the summer solstice. Thus, leaving the equator where the latitude is zero, and the length of the longest day 12 hours, the circles of separation are fixed as in Hie following table, where the Roman numerals are the climates, and the opposite corresponding numbei’s are the lati¬ tudes at which the climates finish, up to the 24th, which terminates at the polar circle, where the length of the longest day is 24 hours:_ I. II. S° 25' VII. 45" 29' XIII. 59° 58' 16 25 VIII. 49 1 XIV. 61 18 23 50 IX. 51 58 XV. 62 25 30 20 X. 54 27 XVI. 63 22 36 28 XL 56 37 XVI r. 64 6 41 22 XII. 58 29 XVIII. 64 49 XIX. 65° 24' XX. 65 47 XXI. 66 6 XXII. 66 20 XXIIT. 66 28 XXIV. 66 30 Places. Climates. Slopes according to M. de Quincy’s Rule. Covering used. Actual Slopes. St. Petersburg,.., 14° 40° 24' Iron. 18° to 20° Copenhagen,. 11 32 48 Slates. Hamburg,. 10 37 48 Plain tiles. 45 to 60 Brussels,. 9 34 30 60° ( 32 36 Slates. 45 to 60 Paris,. 9 < 30 36 33 to 45 ( 24 36 Pan tiles. 18 to 25 Colmar,. 9 32 0 Plain tiles. 60° The empiric rule, then, giving results so widely diffe¬ rent from what experience has taught, is not to be relied on, and some other data on which to base our practice must be found. M. Rondelet, indeed, regards the slope to be given to roofs as altogether arbitrary, and depen¬ dent on taste alone, with such restrictions only as the more or less perfect nature of the materials impose. It is, indeed, extremely probable that the slopes of roofs were regulated originally according to the materials used as a covering. The inclination is generally uniform in all places where the same kind of materials is used. A thatch of leaves, bark, straw, or reeds, probably the first kind of covering employed, required a very steep slope that the water might be speedily thrown off. And when in course of time the more perfect covering of tiles and slates came to be applied, the habit of imitation would for a while prevent any change in the accustomed slope. However this may be, it is evident that the variety of slopes in the same localities shows that no precise rule can now be drawn from existing examples. In roofs covered with slates, the height, or pitch, of the roof is made from one-fourth of the width of the span to the whole width of the span, that is, the slope varies from an angle of 26° 30' with the horizon to 60°; and writers assuming 26° 30' as the smallest angle for common slates, have given the following rates of inclination for other materials:— Kind of Covering. Inclination Height of Roof to the Horizon. in parts of the Span. a Square of Roofing. Copper,. i 3° 50' A ( 100 ( 700 1120 Large slates,. 22 0 i Common slates,... 26 30 i 500 to 900 Stone slates,. 29 41 2380 29 41 24 0 i 1780 650 Pan tiles,. Thatch,. 45 0 * Colonel Emy says, that the inclination of roofs covered with plain tiles varies in France within the limits of 40° and 60°. As the tiles are not nailed like slates, they are made to resist the wind by the pressure of their weight. The angle of 45", he observes, is the best for roofs covered with slates or plain tiles, a slope that permits the use of the interior of the roof as garrets. When the slope is more gentle, the scantling of the roof timbers require to be increased; and when more steep, the increase of roof surface and augmentation of the EXAMPLES OF THE CONSTRUCTION OF ROOFS. 135 length of the timbers increases the cost, without a cor¬ responding benefit, as the height gained cannot be use¬ fully occupied internally, unless by making two stories of apartments in the roof, a practice which, in the pre¬ sent day, is very properly abandoned. The result of an extended consideration of the subject is given by Colonel Emy as follows:— In roofs covered with tiles hung on laths, the slope should not be greater than that at which the materials would slide naturally. It should, therefore, not exceed an angle of 27° with the horizon. Its lowest limit should be such that the tiles should never Lave so small a slope by their overlap that the water would stagnate. In roofs with metallic coverings, the slope requires to be only sufficient to cause the flow of the water; and, therefore, need not exceed -j^th of the span. The following table is extracted from his work:— On the subject of the pitch of roofs, Professor Robison remarks as follows:—“ A high-pitched roof will undoubt¬ edly shoot off the rains and snows better than one of a lower pitch. The wind will not so easily blow the drop¬ ping rain in between the slates, nor will it have so much power to strip them off. A high-pitched roof will exert a smaller thrust on the walls, both because its strain is less horizontal, and because it will admit of higher cover¬ ing. But it is more expensive, because there is more of it. It requires a greater size of timbers to make it equally strong, and it exposes a greater surface to the wind. There have been great changes in the pitch of roofs: our forefathers made them very high, and we make them very low. It does not, however, appear that this change has been altogether the effect of principle. In the simple, unadorned habitations of private persons, everything comes to be adjusted by our experiences, which have re¬ sulted from too low-pitched roofs; and their pitch will always be such as suits the climate and covering. Our architects, however, go to work on different principles. Their professed aim is to make a beautiful object. The sources of the pleasure arising from what we call taste are so various, so complicated, and even so whimsical, that it is almost in vain to look for principle in the rules adopted by our professed architects.,” — ‘‘The Greeks, after making a roof a chief feature of a house, went no further, and contented themselves with giving it a slope suited to their climate. This we have followed, because in the milder parts of Europe we have no cogent reasons for deviating from it. And if any architect should deviate greatl}’ - in a building where the outline is exhi¬ bited as beautiful, we should be disgusted; but the dis¬ gust, though felt by almost every spectator, has its origin in nothing but habit. In the professed architect or man of education, the disgust arises from pedantry; for there is not such a close connection between the form and uses of a roof as shall give precise determinations; and the mere form is a matter of indifference. We should not, therefore, reprobate the high-pitched roofs of our ancestors, particularly on the Continent. It is there where we see them in all the extremity of the fashion; and the taste is by no means exploded as it is with us.” The conclusion to be arrived at is then, as expressed by Professor Robison, “that there is not such a close con¬ nection between the form and uses of a roof as shall give precise determinations, and the mere form is a matter of indifference.” EXAMPLES OF THE CONSTRUCTION OF ROOFS. Roofs of two slopes in narrow buildings are composed of rafters alone, with a cross piece, forming each pair of opposite rafters into what is termed a couple. The rafters without the cross piece, or tie-beam , would tend to thrust out the walls on which they rest; and this cross piece is intended therefore to act as a tie to counteract this thrust. Its position is consequently of importance; and from a false economy, or from ignorance of its function, it is generally, in buildings of an inferior class, placed so high as to be of little use in counteracting the thrust. This kind of roof, the couple-roo£ is only practicable in buildings of very moderate width. In wide buildings, the rafters would bend by their own weight, unless made of a preposterous size or supported in some manner. When the width of the building, therefore, exceeds these moder¬ ate limits, the rafters are kept from bending by a piece of timber parallel to the tie-beam, and called a collar-beam. But it will be obvious that couple-roofs so formed, inde¬ pendently of consuming a great quantity of timber, can only, after all, be used for small spans; and hence it is necessary to have recourse to the framed roof. In framed roofs, the rafters are sustained by pieces of timber, which lie under them horizontally, and divide their length into spaces within the limit of their flexure under the weight of the covering. These horizontal pieces are called 'pur¬ lins , and are sustained by trussed frames of carpentry, distributed transversely at equal distances in the length of the building, the distances being calculated with rela¬ tion to the strength of the purlins. Fig. 457 illustrates this kind of roofs, in which A A, B c is the trussed frame of carpentry, called a principal , d d Fig. 457- are the purlins, and e e the rafters. It will be useful here to consider tlie principles of trussing. Kinds of Roofs and of Coverings. Proportion of the Span. Inclination to the Horizon. Vs and £ | J and I t 1 !* 9° 28' to 14° 2' 18 26 to 26 33 33° 46' 36 52 45 0 60 0 Hollow tiles, Roman tiles, Slates, lowest inclination,.. Thatch, common slates, PRACTICAL CARPENTRY AND JOINERY. 13G Let A B, c B (Fig. 458) be two rafters, placed on walls at A and C, and meeting in a ridge B. Even by their own weight, and much more when loaded, these rafters would have a tendency to spread outwards at A and c, and to ing is the same. The rafters are compressed, the strain¬ ing beam is compressed, and the tie-beam and posts, the latter now called queen posts, are in a state of tension. In some roofs, for the sake of effect, the tie-beam does not stretch across between the feet of the principals, but is interrupted. In point of fact, although occupying the sink at b. If this tendency be restrained by a tie estab¬ lished betwixt A and C, and if A B, B C be perfectly rigid, and the tie A c incapable of extension, b will become a fixed point. This, then, is the ordinary couple roof, in which the tie A c is a third piece of timber; and which may be used for spans of limited extent; but when the span is so great that the tie A c tends to bend down¬ wards or sag, by reason of its length, then the condi¬ tions of stability obviously become impaired. Now, if from the point B a string or tie be let down and attached to the middle d, of A c, it will evidently be impossible for A c to bend downwards so long as A B, B c remain of the same length: D, therefore, like B, will become a fixed point, if the tie bd be incapable of extension. But the span may be increased, or the size of the rafters ab,cb be diminished, until the latter also have a tendency to sag; and to prevent this, pieces D E, D F are introduced, extending from the fixed point D to the middle of each rafter, and establishing F and E as fixed points also, so long as D e, d F remain unaltered in length. Adopting the ordinary meaning of the verb “to truss," as expressing to tie up (and there seems to be no reason why we should seek further for the etymolog}'), we truss or tie up the point D, and the frame A b c is a trussed frame. In like manner, F being established as a fixed point, G is trussed to it. In every trussed frame there must obviously be one series of the component parts in a state of compression, and the other in a state of extension. The functions of the former can only be filled by pieces which are rigid, while the place of the latter may be supplied by strings. In the diagram, the pieces A B, c B are compressed, and A c, D B are extended; yet in general the tie D b is called a ldng-post , a term which conveys an altogether errone¬ ous idea of its duties. Thus we see how the two princi¬ pal rafters, by their being incapable of compression, and the tie-beam by its being incapable of extension, serve, through the means of the king-post, to establish a fixed point in the centre of the void spanned by the roof, which again becomes the point d’appui of the struts, which at the same time prevent the rafters from bending, and serve in the establishing of other fixed points; and the combina¬ tion of these pieces is called a king-post roof. It is sometimes, however,- inconvenient to have the centre of the space occupied by the king-post, especially where it is necessary to have apartments in the roof. In such a case recourse is had to a different manner of truss¬ ing. Two suspending posts are used, and a fourth ele¬ ment is introduced, namely, the straining beam a b (Fig. 459), extending between the posts. The principle of truss- place of, it does not fill the office of a tie-beam, but acts merely as a bracket attached to the wall (Fig. 460). It is then called a hammer-beam. It is a general rule that wood should be used as struts and iron as ties; and in many modern trusses this rule has been admirably exemplified by the combination of both materials in the frames. There is another class of principals in which tie-beams are not used. Such are the curved principals of De Lorme and Emy. In the system of Philibert de Lorme, arcs formed of small scantlings of timber are substituted for the framed principals; and in that of Colonel Emy, lami¬ nated arcs are used. The principals of roofs may therefore, in respect of their construction, be divided broadly into two classes— First, Those with tie-beams; and, Second, Those without tie-beams. The first class, those with tie-beams, may be further classified as king-post roofs and queen-post roofs. The second class may be arranged as follows:— 1st. Hammer-beam roofs. 2d. Curved principal roofs. And sometimes the classes are combined. Examples of the varieties of all these will be found in the plates; and we shall now proceed to describe these examples. Our explanations shall, however, be limited to such parts of the construction for which an explana¬ tion is indispensable, and which cannot be readily under¬ stood by an inspection of the engravings. As preliminary to the explanation of the examples, it may however be well here to give Mr. Tredgold’s rules for proportioning the strength of the various pieces com¬ posing the roof; but while we do so, it is necessary to caution the student that these rules are empirical and too general to be relied on, except in simple cases. It is far better, however, although it is commonly attended with more labour, to trust to the formulae given in the article on the strength of timber, applying the rules specially to each case. In estimating the pressure on a roof, for the purpose of apportioning the proper scantlings of timber to be used, not onty the weight of the timber and the slates, or other covering, must be taken, but also the weight of snow which in severe climates may be on its surface, and also the force of the wind, which we may calculate at 40 lbs. per superficial foot. The weight of the covering materials, and the slope of roof, which is usually given, are contained in the following table:— EXAMPLES OF THE CONSTRUCTION OF ROOFS. 137 Material. Inclination. Weight on a Square Foot. Till. Rise 1 inch to a foot. | to lj lbs. Copper. „ i „ 1 u Lead. „ 2 „ 4 „ 7 „ Zinc. „ 3 „ U „ 2 Short pine shingles.... , Lon" cypress shingles „ 6 „ 4 „ 5 Slate. „ 6 „ 5 „ 9 With the aid of this table, and taking into account the pressure of the wind and the weight of snow, the strength of the different parts may be calculated, as we have said, by the rules given under the head—“ Strength of Materials; ” but the following empirical rules, deduced by Mr. Tredgold from these, and from experience, will be found of easy application, and useful for simple cases. Mr. Tredgold assumes 66£ lbs. as the weight on each square foot. It is customary to make the rafters, tie-beams, posts, and struts all of the same thickness. Mr. Tredgold’s Rules. IN A KING POST ROOF OF PINE TIMBER. To find the dimensions of the principal rafters. Rule. —Multiply the square of the length in feet by the span in feet, and divide the product by the cube of the thickness in inches; then multiply the quotient by 0 96 to obtain the depth in inches. Mr. Tredgold gives also the following rule for the rafters, as more general and reliable:— Multiply the square of the span in feet by the distance between the principals in feet, and divide the product by 60 times the rise in feet: the quotient will be the area of the section of the rafter in inches. If the rise is one-fourth of the span, multiply the span by the distance between the principals, and divide by 15 for the area of section. When the distance between the principals is 10 feet, the area of section is two-thirds of the span. To find the dimensions of the tie-beam, when it has to support a ceiling only. Rule. —Divide the length of the longest unsupported part by the cube root of the breadth, and the quotient multiplied by 147 will give the depth in inches. To find the dimensions of the king-post. Rule. —Multiply the length of the post in feet by the span in feet: multiply the product by O'12, which will give the area of the section of the post in inches. Divide this by the breadth for the thickness, or by the thick¬ ness for the breadth. To find the dimensions of struts. Rule. —Multiply the square root of the length sup¬ ported in feet by the length of the strut in feet, and the square root of the product multiplied by 0‘8 will give the depth, which multiplied by 06 will give the thickness. IN A QUEEN-POST ROOF. To find the dimensions of the principal rafters. Rule. —Multiply the square of the length in feet by the span in feet, and divide the product by the cube of the thickness in inches: the quotient multiplied by 0155 will give the depth. To find the dimensions of the tie-beam. Rule. —Divide the length of the longest unsupported part by the cube root of the breadth, and the quotient multiplied by P47 will give the depth. To find the dimensions of the queen-ptosts. Rule. —Multiply the length in feet of the post by the length in feet of that part of the tie-beam it supports: the product multiplied by 027 will give the area of the post in inches; and the breadth and thickness can be found as in the king-post. The dimensions of the struts are found as before. To find the dimensions of a straining-beam. Rule. —Multiply the square root of the span in feet by the length of the straining-beam in feet, and extract the square root of the product: multiply the result by 09, which will give the depth in inches. The beam, to have the greatest strength, should have its depth to its breadth in the ratio of 10 to 7; therefore, to find the breadth, multiply the depth by 0 7. To find the dimensions of purlins. Rule. —Multiply the cube of the length of the purlin in feet by the distance the purlins are apart in feet, and the fourth root of the product will give the depth in inches, and the depth multiplied by 06 will give the thickness. To find the dimensions of the common rafters , when they are placed 12 inches apart. Rule. —Divide the length of bearing in feet by the cube root of the breadth in inches, and the quotient mul¬ tiplied by 072 will give the depth in inches. We shall, by way of practice, test the scantlings ot some of the examples of executed roofs by these rules, which is preferable to working out supposititious examples in this place. It may be well here, also, before describing the exam¬ ples, to note some practical memoranda of construction which cannot be too closely kept in mind in designing roofs. Beams acting as struts should not be cut into or mortised on one side, so as to cause lateral yielding. Purlins should never be framed into the principal rafters, but should be notched. When notched, they will carry nearly twice as much as when framed. Purlins should be in as long pieces as possible. Rafters laid horizontally are very good in construction, and cost less than purlins and common rafters. The ends of tie-beams should be kept with a free space round them, to prevent decay. It is said that girders of oak in the Chateau Roque d’Ondres, and girders of fir in the ancient Benedictine monastery at Bayonne, which had their ends in the wall wrapped round with plates of cork, were found sound, while those not so protected were rotten and worm-eaten. It is an injudicious practice to give an excessive cam¬ ber to the tie-beam: it should only be drawn up when deflected, as the parts come to their bearings. The struts should always be immediately underneath that part of the rafter whereon the purlin lies. The diagonal joints of struts should be left a little open at the inner part, to allow for the shrinkage of the heads and feet of the king and queen posts. It should be specially observed that all cranks or bends in iron ties are avoided. And, as an important final maxim— Every construction should be a little stronger than strong enough. S 138 PRACTICAL CARPENTRY ANI) JOINERY. Description of the Plates. Plate XXII.— Fig. \ is the elevation of a king-post roof, designed by Mr. White, for a span of 30 feet. By the rules given for calculating the scantling, it will 13 X 5 inches, 8|X5 „ 4 X2i „ 7£ X 5 „ found to be as follows:— A, Tie-beam,. b. Principal rafters, ... c, Struts, D, King-post, ... Fig. 2 is the design for a king-post roof, for a span of 33 feet 6 inches. The purlins here are shown framed into the principals, a mode of construction to be avoided, unless rendered absolutely necessary by particular circumstances. The scantling, as determined by the rules, is as follows:_ Principal rafters, . 10 X 5 inches. Tie-beam, . 11 J x 6 King-post,. 7j X 6 „ Struts, . ... 4 x 2 J Purlins, . 10 X 6 " The principals being 10 feet apart. Fig. 3.—A compound roof for a span of SO feet. It is composed of a curved rib c c, formed of two thicknesses of 2-inch plank bolted together. Its ends are let into the tie-beam; and it is also firmly braced to the tie-beam by the king-post and suspending pieces B B, which are each in two thicknesses, one on each side of the rib and tie- beam, and by the straps a a. A is the rafter; d, the gutter-bearer; c and b , the straps of the king-post. The second purlins, it will be observed, are carried by the upper end of the suspending pieces B B. Full details of the straps and bolts of this and the succeeding examples will be found in Plates XXXVIII. and XXXIX., “Joints and Straps.” Fig. 4.—A queen-post roof, with an iron kinc-bolt, intended for a span of 32 feet. a, Principal rafter, ... b, Straining-piece, . . c, Queen-post, . . Struts, . 11X5 inches. 11X5 9X5 .. 5X4 „ c, King-bolt. The common rafters are 8x3 inches, and project over the walls to form a projecting cornice: a is the short ceiling-joist of the cornice; b, an ornamental bracket. Fig. 5.—A queen-post roof for a span of 60 feet. The scantlings are as follows: — Principal rafters, 11 X 6 inches. Tie-beam, 12£ X 6 Queen-post B, ... ... . 8X6 Suspending post a. H X 3 Struts (large), ... . ..H 41 X 3i „ ,, (small),. 31 X 2J „ Fig. 6.—Nos. 1 , 2, 3, and 4 are the elevation and details of the queen-post roof of the railway workshops at Wor- cester. The principals are placed 1 5 feet apart, and the purlins are trussed. The details are as follows: — Principal rafters, 8X8 inches. Tie-beam, 12 X 8 Queen-post 8X6 Struts ... ... ... . 41 X 41 „ Straining-beam, 9X8 Common rafters, 4i X 2 Purlins, in two flitches each (trussed with 7 stirrup pieces and iron ties), j y o „ The tie-beams are carried on iron shoes. No. 1 is the elevation of the roof. No. 2 is a section of one hay of tlie roof. No. 3 shows the under side, and No. 4 the side of a purlin drawn to a larger scale. Fig. 7.—Elevation of the principal of a platform roof for a span of 70 feet. The tie-beam in this example is scarfed at a and b, and the centre part of the roof has counter-braces c c. The longitudinal pieces e e, secured to the heads of the queen-posts, and the piece cl, carry the platform joists A. The details of the scarfing and strengthening the tie-beam will be found described in the section on “ Scarfing and Lengthening Beams,” and illustrated in Plate XXXIX. Plate XXIII.— Fig. 1 is a queen-post M-roof, for a span of 47 feet; or rather, having a king-bolt in the centre, it is a compound roof:— A, Tie-beam, 13 X 6 inches. b, Principal rafter, ... 11 X 6 c, Straining-beam, ... 11 X 6 d, Queen-post, ... 7X6 e, Strut, . 7X6 f, Counter-brace, 6X6 o, Common rafter, 6 X 21 „ a, Wall-plate for common rafters, 7X9 b b, Purlins, ... 11 X 6 „ eg, Kiug-bolt, ... 2 inches diameter. d, Ridge-rafter, . 12 X 21 inches. h, Gutter-bearer, 3 X 21 „ In this roof, the purlins are shown framed into the principals, a practice which has already been censured. Fig. 2.—A simple queen-post roof for a span of 40 feet:— a, Tie-beam, 12 X 6 inches. b, Principal rafter, 10 X 6 c, Straining beam, .. 9X6 D, Queen-post, 8X6 e, Strut, 6X6 f, Common rafter, ... 6 X 21 „ a, Wall-plate, 9X6 b, Purlin, 12 X 9 Ridge-rafter; d, strap at foot of common rafter; e, ditto at foot of queen-post; /, ditto at head of ditto; k, straining-cill. Fig. 3. — King-post roof for a span of 38 feet 9 inches :— As Executed As Determined by the Rule. Tie-beam, . 11X9 ins. lli X 6 ins. King-post, 10 X 8 „ 8 X 6 „ Suspending posts or queen-posts, 10 X 7 „ 5 X 3 „ Principal rafters, 10 X 7 „ 8 X 6 „ Principal struts, . 6X6 „ 6 X 3 „ Secondary struts, 5X5 „ 3 X 2 „ Purlins, 7 X 9 „ Cleets at back of purlins, 8X6 „ Ridge piece, ... 8 X li„ On comparing the scantlings of this roof as executed, with those derived from the application of Mr. Tredgold’s formulas, there will be found an excess of strength. The scantlings, in point of fact, are nearly as large as those of other examples, of much greater span, which we have given; and this roof, therefore, suffers much in com¬ parison with the next example, where the scantlings, as executed, are, generally speaking, of smaller dimensions than those resulting from the application of the formulas. The tie-beam is strapped to the king and queen posts, and the principal rafters are secured by screwed bolts and nuts. Fig. 4.—No. 1 is the elevation of ODe of the trusses of George Heriot’s Schools, Edinburgh, designed by Alex¬ ander Black, Esq., architect, and executed under his superintendence. The dimensions of the scantlings by the rules are here J. Whits del . J.W.JLawy, fc ■ PLATE XXIV. Fig,J. Fu/. 2. Fig. 2. Fig. 4. A.White, dd. BLAU E lie SOX . GLASGOW. KD1NBUR.CVH 8e LONDON. IV. A.Beever, Sc. EXAMPLES OF THE CONSTRUCTION OF ROOFS. 139 placed in juxtaposition with the dimensions of the scant¬ lings as executed, as follows:— By the Rule. As Executed. a, Tie-beam, ... ... Ilfx4inches. 12x4inclies. B, Principal rafters, ... 8X4 „ 8x4 „ c, Common rafters, ... ... 5 X 2$g „ D, Purlins, ... ... 9 X5| inches. 9x6 „ e, King-post (oak), ... 8X4 „ 5X4 „ f, Queen-posts (oak), ... 5X5 „ 4x4 „ G, H, Inner struts, ... ... 4| X 3^ „ 4X4 „ i, Outer struts, . 3 X 2$j „ 4 X 3£ ,, k, Cleets to do., . ... 6X4 „ No. 2 shows the details of the king-post to a larger scale. The heads of the rafters and the feet of the struts are received by cast-iron sockets bolted to the king-post. The tie-beam is suspended to the post by an iron strap with wedges. A section through the tie-beam, king-post, and strap, is given at No. 3. Nos. 4* and 5 show the scarfing of the purlins; and Nos. 6 and 7 the end of the tie-beam, with the iron shoe which receives the foot of the rafter, and the strap which secures it. Plate XXIV. Fig. 1.—No. 1, elevation of one of the principals of the roof of Wellington Street Church, Glas¬ gow, designed by John Baird, Esq., architect, and erected in 1823. No. 2, part of the upper side of the tie-beam:— a, Tie-beam, 12X9 iuches. B, Principal rafter, 13 ins. at bottom, 11 at top, and 9 inches thick. c, Straining-beam, forming support of platform, and cambered, 13 ins. deep at centre, 11 at ends, and 9 inches thick. d, Principal queen-posts, 13X9 ins. at top and bottom, and 9X9 in smallest part. e, Second queen-posts, 10 X 9 ins. at top and bottom, and 7 by 9 in smallest part. f, Principal strut, 9X9 inches. G g, Secondary struts, 7X9 inches. H, Straining piece between principal struts, 6X9 inches. ni m, Platform joists, 10 X 24 inches, and 15 ins. apart from centre to centre, covered with boarding If inch thick for lead. n n, Common rafters, laid horizontally, 6 X 2| inches, covered with slate-boarding § inch thick. The principals are placed 9 feet 4 inches apart. All the timbers are joined by mortise-and-tenon joints. The platform joists and horizontal rafters are notched on the straining-beams and principals. The tie-beams are in two lengths of timber, scarfed, as shown in No. 2. The scarf is secured by iron straps, each 3 inches wide and ■§ inch thick, and bolted. The iron work is of the follow¬ ing dimensions:— Straps at feet of principal rafters, 2$ X § inches. Three-tailed straps connecting principal rafters, queen-posts, and straining-beam, 2f X £ inches; and each tail 2 feet 6 inches long. Straps at feet of queen-posts, 2| X 4 inch, bolted and keyed. King-bolt If inch diameter, screwed up hard. Fig. 2.—Elevation of a roof-principal of the parish church, Elgin, by Simpson of Aberdeen. There are twelve principals similar to the elevation in Fig. 2. They are placed 6 feet 6 inches apart between cen¬ tres; and the scantlings are as follows:— A, Tie-beam, in two flitches, each 13 x 5| inches. b c, Principal rafters, 11 inches deep at lower end, 8 inches at top, and 6 inches thick. D, Collar-beam, 7 X 5f inches. E, Struts, 5 X 5i inches. F, Struts, 5 X 4f inches. m in, Horizontal rafters, 4f X 2$ inches, 13 inches apart, and covered with groove-and-tongued deal 1 inch thick, and lead weigh¬ ing 7 lbs. to the superficial foot. The tie-beams have cast-iron shoes, h, at each end, with abutments formed for the rafters, and secured with t-inch diameter bolts, with nuts and washers. The suspending rods are 1 inch square, and have abut¬ ment pieces for the rafters and struts. Fig. 3.—One of the roof-principals of the City Hall, Glasgow. The following are the dimensions of its timbers:— A, Tie-beam, 14 X 12 iuches. b, Cill piece, 12 X 12 inches. c, Principal rafter, at the end, 9X7 inches, and at s. 11 itches deep. d, Ditto, where doubled at lower end, 8X7 inches. e, King-post, in two thicknesses, each 10 X 6 inches. f, Queen-posts, at top and bottom, 13X7 inches, and in middle, 10 X 7 inches. g, Straining-beam, 10 X 7 inches. H, Struts, 6X6 inches. K, Common rafters, 6X2| iuches. m, Ridge board, 10 X 2 inches; batten over it, rounded for lead, 3| X 3 inches. n, Wall-plates, under common rafters, 12 X 1$ inches, with pole- plate, 2X2 inches. o o, Purlins, 8X5 inches. p, Outer wall-plates, 14 X 3 inches. r, Inner wall-plates, resting on corbels, 11X5 inches. The iron straps are 4 inches broad by § inch thick. Their bolts are £ inch square. The bolts securing the ends of the rafters, and the beams, are 1 inch square; and their washers are the full breadth of the beams. The principals are placed 12 feet apart from centre to centre. Fig. 4.—One of the principals of the roof of the East parish church, Aberdeen. The following are the dimen¬ sions of the timbers:— There are five principal trusses, placed 14 feet apart. A, Tie-beam, in two thicknesses, 14 X 10 inches. Principal rafters, 13 inches deep at bottom, Ilf inches at top, and 10$ inches thick. The rafters bear on oak abutment pieces 11 X 7$ inches, bolted between the ties and to each other. d, Collar-beam, in two thicknesses, one on each side of the rafter, and notched and bolted, 12X5$ inches each. E, Purlins. The two lower, 13 X 6$ inches; the upper, 11$ X 8$ iuches; notched on the rafters and bolted. F, Common rafters, 5$ X 2$ inches, and 13 inches apart. The discharging posts between the bracket pieces and the stone corbel are of oak, 6 inches square. Binding pieces, 9$ X 3f inches, extend between the tie- beams, and are mortised into them; and into these binding pieces the ceiling joists, which are 13 inches apart, and 6 X If inches, are mortised. The dimensions of the iron work are as follows:— King-rod, If in. square, with a cast-iron key piece at top. Queen-rods, 1$ in. square, having solid heads at rafters, and secured at foot by being passed through solid oak pieces k, placed between flitches of tie-beams, and securely bolted, and there fastened with cast-iron washers and nuts. Four bolts at abutment end of ties, . £ inch square. Two do. at each oak piece, for suspending rods, £ „ ., Two do. at each eud of collar-beam, ... ... § „ „ Purlin bolts,. f „ „ The abutments of the rafters at both ends, and the bearings of the bolts, have pieces of milled lead interposed; and all the joints of the framing were coated with white lead and oil before being put together. Plate XXV.— Figs. 1 and 2 show a wider use of iron in the parts of the framing acting as ties. They are the 140 PRACTICAL CARPENTRY AND JOINERY. •aa*''! roofs of sheds at the Liverpool Docks. The scantlings are as follows:— Fig. 1. Principal rafters, ... 12 X 8 inches. Struts, . 8X8 „ Purlins.10 X 4 „ Common rafters, ... 4J X 2 „ Tie-rod and suspending-rod, 1^ inch diameter. Fig. 2. Principals,. 14 X 8 inches. Collar-pieces. ... 11 X 3, one on each side of rafter. Purlins, ... ... 16 X 4 inches. Tie-rods and suspending-rod, If inch diameter. The details are similar to those of the roof shown in Figs. 5, 6, 7, 8, 9, and 10 of the same plate. Fig. 3.—A roof adapted to a hall or church with nave and aisles. The framing is simple and good:— A, Principal tie. B, Tie of aisle roof, c, Gilder supported by the iron column d. e, Story-post. Fig. 4 is a queen-post roof, adapted to the same use as the last. Fig. 5.—Roof of the East Quay shed of the Salthouse Dock, Liverpool. Jesse Hartley, Esq., engineer. The dimensions are all marked on the detailed draw¬ ings, which are made to a larger scale, and are contained in Figs. 6 to 10. The scantlings are as follows:— Principal rafters, . 16 X 9 inches. Common rafters, ... .. 41 X 2 „ Purlins, ... .. . 15 X 5 „ Collar-beam, ... ... ... ... 15 X 9 „ 1 ie and suspending rods, 2 inches diameter. Plate XXVI.— Fig. 1 shows the principal of a roof of 44 feet 8 inches span. In this, wrought-iron is used for the suspension rods, and cast-iron shoes as abutments for the timbers acting as struts. At C, on the wall-head, is a cast-iron shoe, to receive the tie-beam and the foot of the principal rafter. The sole-plate of the shoe is prolonged, to admit of its being secured by bolts to the tie-beam. The head of the principal rafter, and the end of the straining beam, are inserted into a cast-iron socket, an elevation of which is seen, enlarged, at No. 1. The sus¬ pension rod A D, it wall be seen, passes through the solid part of the socket. It has a head at its upper end, and at its lower end it is screwed, and secured by a nut. On the side of the socket is cast a rest for the end of the purlin a b. To avoid cutting the principal rafters, the other purlin at B is also carried in a cast-iron rest bolted to the rafter. The centre suspending rod at E passes through a cast-iron socket, which serves as an abutment to the two main struts. Similar abutments are provided for the lower end of the struts. Fig. 2.—This principal, for a roof of 45 feet span, has details of the same kind as those described above. The detail No. 2 is a section of the shoe at head of king-bolt, into which upper ends of principals are inserted. Fig. 3.—This is a principal also with wrought-iron suspension rods. The tie-beams, principals, and struts are first framed together; the suspending rods are then introduced, and screwed up by the nuts at their lower end until the framing is firmly united. A roof of this construction, 54 feet span and 212 feet long, is erected at the passengers’ shed of the Croydon railway station. Fij- 4 shows a roof, the principal rafters of which are constructed of timber and iron. They are in three thick¬ nesses, the centre flitch being timber, and the side plates wrought-iron, bolted together through the timber, as shown more at large in the section No. 7. No. 3 shows the foot of one of the rafte”s, with the iron girder on which it is supported; the mode of attaching the tension- rod, and the manner of constructing the gutter. No. 4 shows the cast-iron strut at B. No. 8 is a section of the rib of the strut on the line A. Nos. 5 and 6 show, in plan and elevation, the manner of connecting the tension-rods at the apex; the letters refer to the same parts in both. Fig. 5 is a roof-principal, formed with iron rafters, struts, straining-beam, queen-posts, purlins, and tension- rod. The iron parts are connected together by hinge- joints, as at c. The purlins are supported by sockets on the principals, as at B. The common rafters are of timber. Plate XXVII.—To diminish the excessive height of roofs, their sharp summit is suppressed, and replaced by a roof of a lower slope. Francis Mansard, who died in 1666, brought this sort of roof into fashion in France, and was for a long time re¬ garded as its inventor.* The roof is known, indeed, as the Mansard roof; and the garrets formed in such roofs were called Mansards. The Mansard roof may be de¬ scribed in several manners:— 1st. In Fig. 461, the triangle a cl b represents the pro¬ file of a high-pitched roof, the height being equal to the base. At the point e, in the middle of the height c d, draw a line horizontally li e i, parallel to the base a b, to represent the upper side of the tie-beam, and make ef equal to the half of ed\ then a h f i b will be the profile of the Mansard roof. 2d. In Fig. 462, make c e the height of the true roof, equal to half the width a, b, and construct the two squares a d e c, c e gb\ also make cl h, e /, and g i each equal to one-third of the side of a square; then will a hf i b be the profile required. Fi*. 41. 3d. In Fig. 463, on the base a b, draw the semicircle a d b, and divide it into four equal parts, a e, e cl, df,fb; join the points of division, and the resulting demi-octagon is the profile required. 4th. Fig. 464.—Whatever be the height of the Man- * Bullet says that Mansard truncated his roofs after the example of oue at Chilly, by Metezeau. Mesanges asserts that he took the idea from a frame composed by Segallo, and that Michael Angelo employed it in the construction of the dome of St. Peter's; but Krafft, in his work on Carpentry , seeks a more remote origin for this kind of roof. He remarks that it existed in the Louvre, built by Pierre Lescot for Henri II., in 1570. He adds, that the houses in Lower Brittany were covered with these roofs in the end of the fifteenth century. EXAMPLES OF THE CONSTRUCTION OF ROOFS. Ml sard, c e or b g, D’Aviler makes g i equal to the half of that height; and the height, e /, of the false roof, equal to the half of c i. 5th. Fig. 465. — Describe the semicircle a f b, and Fig. 464. rig. 465. divide each half of the base a b into three equal parts. From the last divisions p p, the perpendiculars p r, p r are erected-, cutting the semicircle in r r ; then a r f rb is the profile. 6th. Fig. 466. — This method, which is described by JBeledor in La Science Fie. 4'-:6. des Engine VjTs, litis been generally adopted on account of its simplicity, and the good effect which it produces. Describe a semicircle on the base a b, and divide its a circumference into five equal parts, in 1 2 3 4; then the chords a 1, b 4 are the sides of the true roof, and 1 /, 4 / those of the false roof. It is by this method that the lines in Fig. 1, Plate XXVII., are set out. The forms of the Mansard roof, it will be seen, may be infinitely varied, by choosing, in the diameter or in the semi-circumference, other points of division, or by alter¬ ing the relation between the height and width of the roof, according to the uses to which it is to be converted. Plate XXVII., Fig. 1.—A Mansard roof for an arched ceiling, selected from Ivrafft. Fig. 2.—A king-post Mansard roof. In this example there is a wide space available as an apartment. The construction would be improved by adding straps to the feet of the story-posts. Fig. 3.— A roof with two stories of apartments in its height. This is an example also taken from Krafft. It is the roof of the Chateau de Florimont, in Alsace, de¬ signed by General Kleber. It is difficult to conceive what good is obtained by the introduction of the story- post in the centre. Fig. 4.—A queen-post Mansard roof. This is regularly trussed, and is erected over the riding-house at Copenha¬ gen. Colonel Einy remarks that there should be struts a under the purlins, and a small collar-piece b added; that the tie B is too heavy, and that the cross-piece above D serves no useful purpose, and might be dispensed with. Figs. 5 to 10, on the same plate, are examples of the modes of forming and framing dormers, such as at A and B, in Fig. 3. When Colonel Emy was called upon, in 1819, to con¬ struct the roof of a building upwards of 60 feet wide, at the barracks of Libourne, it was proposed to him to follow the method of De Lorme, a notice of whose system is prefixed to the description of Plate XXX. But, while acknowledging the merit of De Lorme, and the beauty of the results obtained in constructing roofs in accord¬ ance with his method, it had yet appeared to M. Emy, that where timber of tolerable length could be obtained, results equally good might be produced, without cutting it into the short scantlings required by De Lorme’s system. Accordingly, as the country afforded pines of from 36 to 40 feet long, and fir still longer could be easily obtained from the Pyrenees, he sought to compose a roof in which the timber might be used in its whole length, and which should combine the necessary solidity with the lightness, elegance, and economy of the system of De Lorme. He succeeded in designing a roof which, in his judgment, satisfied all those conditions; but did not obtain authority to carry out his designs. He was authorized at length to make a trial at Marac, near Bayonne, in 1825, in roof¬ ing a building of nearly the same dimensions. The success of this trial was such as, in 1826, determined the authori¬ ties to roof, in the same ihanner, the manege at Libourne, for which the system was originally designed. The execution of M. Einy’s system is within the power of the ordinary carpenter; and the workmanship is less than in the roofs of De Lorme, as the wood is all in straight pieces. There are neither mortises nor tenons, except at the ridge; and the process of construction and setting the principals in their places is so simple, that, as M. Emy says, twelve workmen, two-thirds of whom were common labourers, were able to put together, and raise and set in their places, two principals each week at Marac. Principals have often been constructed of great arcs, or centres, formed of several pieces of timber superimposed on each other. But such pieces have been of considerable scantling. They have been very short; their connection has been by iron; and their curvature produced by the adze or by heat. Now, the construction invented by M. Emy is a timber arch composed of a series of long and thin planks applied on the flat, the flexibility of which permits them to be easily and quickly bent without the aid of heat; and their rigidity, properly regulated, has the property of maintaining the form given, and destroy¬ ing the thrust. It would be impossible to bend, even with the aid of fire, timbers of the same scantling as these composite arcs. Even supposing that pieces of half the length could be so bent, then, to form the whole arch, butt-joints, occupy¬ ing the whole section of the timber, would have to be introduced; while, by building these arch-beams of thin planks, the joints can be properly broken without weaken¬ ing the strength of the beam. The combination of this system may be varied infinitely by the number, the span, and form of the arcs; and the strength of the arcs may be increased, according to the necessity of the case, without changing the system, or injuring the elegance of its appearance, by simply adding more planks either to the whole length of the arc, or to such part as trial, always indispensable in large con¬ structions, shows to be necessary. Since the invention of M. Emy has been made public by his own publications, and by the report of the Society for the Encouragement of National Industry, in March, 1831, roofs have been constructed on his principle with great success both for large and small spans. The ex¬ amples we have engraved embrace the roof of a shed at Marac, and the roof of the riding-house at Libourne, constructed by himself (Plate XXVIII.); and the appli¬ cation of his system to the roof of a Gothic church erected PRACTICAL CARPENTRY AND JOINERY. 142 at Grassendale, near Liverpool, by Mr. Arthur Hill Holme, architect (Plate XXIX.) Plate XXVIII.—Roof of a shed at Marac, near Bayonne, France. Each principal of this roof {Fig. 1) is composed of a semicircular arch, two principal vertical pieces, two prin¬ cipal rafters, two struts, a king-post, and a collar-beam, the whole tied together by pieces which are at right angles to the curve. These radial pieces, as well as the sides of the arch, are notched upon each other. The vertical pieces are distant from the face of the wall about 4 inches. The three first radial pieces on each side are prolonged beyond the uprights, and enter recesses made in the wall to receive them, as seen in Fig. 12. The object of this is merely to steady the frames, and keep them vertical. Between the radial pieces, the plates composing the arc are bolted together with cylindrical bolts, which are driven tightly into accurately made holes by a heavy mallet. These keep the plates from sliding on each other. The plates are further firmly tied together by iron straps. The bolts are -fo inch diameter, and about 2 feet 6 inches apart. The plates of wood forming the arc are 1| inch thick, inches broad, and about 40 feet long. Two and a half plates of this length, joined end to end, make up the whole length of the curve. The joints are so distributed that those of one row do not correspond to those of another row, and that each joint is carried by one of the radial pieces. All the plates cannot, of course, have only three joints; and several of them have only two. Thus there may be only from ten to twelve joints in the whole arc. The vertical pieces are 7f inches thick; the principal rafters 5^ inches thick. The principals are placed 9 feet 10 inches apart; and maintained in this position by the braces seen in Fig. 2, and on a larger scale in Fig. 6, by the purlins, and by a line of double ties stretching between the fourth radial pieces. When this roof was proposed, it was alleged that it would alter its form, and exercise a thrust upon the walls, especially when loaded with its covering. Colonel Emy, therefore, judged it proper to make several experimental principals on this construction, which could be submitted to proof, and enable him to determine what weight they could support without alteration of form, and also the number of plates of which the arcs should be composed. The experimental arcs were composed of five plates. They were sustained by thick plates of oak laid on the ground, which had first been truly levelled and beaten solid. When the arc was raised up and left to itself it drooped a little. Then, by long cords, there were suspended to the points of the arc, which represented fairly the points of pressure, platforms of wood at about 20 inches above the ground. These platforms were then loaded gradually with cast- iron, until the weight on each reached 2200 lbs., making the total load 10'8 tons — a weight which was one quarter more than the principal would have to sustain. The plates in these experimental arcs were fastened to¬ gether solely by the radial pieces and iron straps, as Colonel Emy wished to reserve a means of increasing the strength by inserting the bolts after the experiment, and when the principals were set in the places they were finally to occupy. As the Aveights were added, the arc appeared to flatten. At the end of twenty-four hours, its curvature was tested by a radial rod of wood of 32 - 8 feet long mounted with iron at each end, and centred truly on an iron axis, and established with precision on the head of a pile driven in to the level of the springing of the arch. It was found that the king-post had fallen down, but that the curvature of the arc comprehended between the seventh radial pieces had not sensibly altered. There was an augmentation of the curvature below these points, the maximum being at radial No. 4; and the disposition of the principal rafters and uprights was, of course, also slightly affected. The diameter of the arc, however, did not vary; and there¬ fore the plates must have slid on each other to the extent of not quite an eighth of an inch each. The conclusion derived from the experiments was, that the stiffness of the arc should not be the same throughout, and that it was necessary to reinforce it in the places that had yielded the most, by supplementary plates. The proper result was obtained by adding, on the two sides of each arch, one supplementary plate to a part of the extrados, and two plates to a part of the intrados. The following is the proportion of the number of plates, and their width, which Colonel Emy adopted as a rule:— Width. Ft. In. From the springing to radial No. 1, ... 7 plates, ... 1 3 From radial No. 1 to the tie placed ) g ^ y between radials Nos. 6 and 7, 1 From the above tie to radial No. 9, ... 6 „ ... 1 0 From radial No. 9 to king-post,. 5 „ (nearly) Oil The supplementary plates were of oak, and of the same thickness as the others. The principals thus strengthened were again submitted to proof without change of form. The manner of constructing the principals was as fol¬ lows :—The ground having been dressed and beaten to a level, a semicircle of 65 feet 7 inches diameter was described, representing the intrados of the arc of five plates; and the chief lines of the principal were then traced, and strong sleepers were laid down and fastened by pickets. The sleepers, twenty-four in number, and 10 inches square in section, were all laid radially, and dis¬ tributed so as to fall between the radial pieces of the arc and the iron straps; two only were on the outside of the springing. On these sleepers a floor was laid large enough to hold the draught of the principal; and the centre of the arc was formed by an iron axis fixed on the head of a pile. The floor being laid, and the draught made on it, pieces of wood 8 inches square were fastened through it by long spikes to the sleepers, and to these the template for curving the plates forming the arc was fixed. The process of bend¬ ing timber for this and similar purposes has already been described in detail at pages 102, 103, substituting a con¬ tinuous template for the polygon. To this the reader is referred. The arc being formed by the process there described, the other parts of the principal were properly fitted, but not fastened; and on being completed, the parts were taken asunder, numbered, and put aside, to be raised to their places on the completion of the whole. This was necessary, Fu/ ■ #©ffilFS • ROOF OF A SHED AT MARAC NEAR BAYONNE. FRANCE. Di'su/h cd and eaxouhsct 7/y <-o7..£jh\ Fit/. 6. S<-,lie tv Fins. 3 atui, 4. Scale to-Fias. I and. 1‘. ' o . locuss. del. US © ©f S o eL.rrn XXJX ROOF OF THE SALLE DES CATECHI SMES. CATHEDRALOF AMIENS, tigs. L itad cUtails. ROOF OF GRASSENDALE CHURCH Kigs. S.-t.ami tUtaUs. 1-3 EXAMPLES OF THE CONSTRUCTION OF ROOFS. 143 as Colonel Emy found himself unable to raise the princi- | pal entire. The erecting the principals in their places on the walls i was thus accomplished:—A moveable scaffold was pre¬ pared, easily erected and removed, and provided with a template similar to the one on which the arc was formed. ; This is shown in Figs. 3 and 4. When the scaffold was brought exactly to the place where the principal was to be erected, all the pieces ol the latter were raised as numbered, and put in their places; and then, when completely fastened together, the template or centre was detached, and the arc allowed to rest on its wall-plates. The bolts were then added to the arc before it received the weight of the roof timbers. Colonel Emy considered, after completing his work in the way described, that he might have constructed the arcs, and fitted all the pieces on the vertical template at once, and thus have saved time, and made the work more perfect; but the idea came too late. The principal being thus placed, was maintained in its position by wedges, the ends of the radials, in the recesses in the walls before mentioned, and by stays nailed to the principals temporarily; and the scaffold was removed to the place of the next principal. We shall now describe particularly the figures on the plate. Fig. 1.—Transverse section of the building, and eleva¬ tion of one of the principals. Fig. 2.—Longitudinal section. Fig. 3.—Transverse section of the building before the placing of the principals, and section of the scaffold. Fig. 4.—Section of the scaffold, at right angles to the preceding. Fig. 5 .—Elevation of the summit of the principal in Fig. 1 drawn to a larger scale. Fig. 6.—Section through E F of Fig. 5, showing the counter-bracing and ties between the principals. Fig. 7.—Section on the line G it of Fig. 5. Fig. 8.—Section on the line I J of Fig. 5. Fig. 9.—Section of one of the radials on the line K L, Fig. 5, corresponding to the longitudinal point of the arc. Fia. 10.—End elevation on the plane M N of the radial No. 9. Fig. 11.—Section through the arc on the line O P, showing one of the iron straps. Fig. 12.—Side elevation of the springing of one of the arcs, showing two of the radials Q R, ST, and the recesses, Q and s, in the wall, to receive their prolongations. Fig. 13.—Front elevation of the same. Fig. 14.—Portion of the plan of the wall-plates, show¬ ing the manner in which the parts of the principal are framed into them. Fig. 15.—Section of the principal on the line of the upper face of the radial No. 2, Q R, Fig. 12. Fig. 16.—Section of the principal on the line of the upper face of the radial No. 1, s t, Fig. 12. Fig. 17.—Junction of the upright and principal rafter: I Q is the elevation; P is the profile on the line u v. For the sake of distinctness, the radial is not shown. Fig. 18.—Iron strap and screw used in bending the arc plates. Fig. 19.—One of the straps used in securing the plates to the template. , Fig. 20.—One of the ties of timber serving the same purpose. Figs. 21 to 24 are parts of the roof of the riding-house at Libourne. This roof, on the same principle, was erected in 1826. The principals differ from those of Marac. just described, in this: that as the walls at Libourne were of great thickness, and strengthened by great counterforts, it was not necessary so carefully to guard against lateral thrust, and therefore the arcs were composed of fewer plates each throughout. The diameter of the intrados of the arcs is 68 feet 8 inches; the principals are placed 10 feet 6 inches apart, from centre to centre. In constructing these principals, a working floor was erected at about 2 inches below the wall-plate, and the draught was laid down on it, and a polygonal mould was formed in the manner described at pages 102, 103. This, as remarked at that place, is not so perfect as the con¬ tinuous template, such as was used at Marac, and is more apt to rupture the wood at the points of contact, unless it is freshly cut and very flexible. The floor on which the principals were constructed extended only half the length of the building; therefore, although the principals were easily raised to the vertical position, by the application of shears and windlasses, each had to be moved to its proper place. This was effected by placing under each springing a little carriage running on the wall-plates. The frames were kept upright by proper stays, and the carriages dragged along by tackling, and forced by levers till they reached their places. Figs. 21 and 22.—Front and side elevations of the springing of one of the principals. The centre of the arc is on the line a b. The principal is mounted on one of the carriages mentioned above. Fig. 23.—Section through the principal on the line c D. Fig. 24.—Horizontal section on the line A b. Plate XXIX.— Fig. 1. Roof of the Salle des Cate- chismes, Amiens Cathedral. Fig. 1 is a transverse, and Fig. 2 a longitudinal section of this roof. It is composed of principals formed with principal rafters, curved ribs, king-post, tie-beam, and collar-beam. The ends of the tie-beam, in addition to their wall-hold, are supported by framed brackets, resting on stone corbels in the wall. The brackets, tie-beam, king-post, and curved ribs are all exposed to view, and are moulded in a very simple and effective manner. The veiling is vaulted, and formed of boarding, ornamented with vertical moulded ribs, placed about 18 inches apart. Fig. 1.—No. 1 is an elevation of the lower portion of the king-post, showing the mode of finishing the chamfering. No. 2 is a section through the octagonal portion of the post. No. 3 is a vertical section of the tie-beam. No. 4 is a section of the cornice from which the arched ceiling springs. Fig. 3 is the elevation, and Fig. 4 the longitudinal sec¬ tion of a Gothic roof on Col. Emy’s principle, designed by Mr. Arthur Hill Holme, of Liverpool, and erected under his direction at Grassendale Church, Aigburth; and Nos. 1, 2, and 3, Fig. 3, are the details of the same roof drawn to a larger scale. The mode of construction was in every respect similar to what has already been de¬ scribed, and need not therefore be repeated. PRACTICAL CARPENTRY AND JOINERY. 5Pj ' 144 In 1561, Philibert de Lorme published his book, entitled New Inventions for Building Well at Little Expense. In his address to the reader, he says, among other tilings, that as it is difficult to find trees large enough to serve for beams, and the other timbers of mansions, he has long sought for some invention which would enable him to use all kinds of wood, and even the small pieces, and so to dispense with the great trees hitherto used. The result of his researches was the system of framing to which the name of the inventor is given, and which is here illustrated and described in detail. The system of Philibert de Lorme is composed of arcs or hemicycles formed of planks, used as substitutes for the framed principal. The planks forming one layer or thickness are placed end to end, and their joints are cut radially to the centre. The joints of one layer or thickness of plank correspond to the middle of the planks of the second layer, and for small spans each plank is only about 4 feet long, by about 8 inches wide and 1 inch thick. The feet of the hemi¬ cycles are tenoned into the wall-plates. The shoulders of the tenons are about 1 inch. The hemicycles are all traversed in the joints by ties 1 inch thick and 4 inches wide. Keys of 1 inch thick and 1| inch wide, and of a length nearly equal to the width of the planks, traverse the ties. They serve to maintain the hemicycles in their vertical planes at their proper distance apart, which is about 2 feet, and, at the same time, to tie, in each hemicycle, the planks together. The mortices iu the ties are a little less apart than twice the thickness of the rib-planks. Some make only one mortice, with the view of saving labour. The mode of construction illustrated by all the figures in Plate XXX., was first employed by De Lorme in roofing the pavilions of the Chateau de la Muette, at St. Germains-en-Laye. The walls of these pavilions were in a defective state, and would not bear the weight either of stone vaulting, or of heavy carpentry, even if trees large enough to make the roof of the ordinary construc¬ tion in use at that time could have been obtained, which, we learn from the work of De Lorme, published in 1561, was not the case. The advantage of the system, according to its author, is the saving of expense, because very light and short timbers are proper for the work, and the walls need not be so thick as for heavier carpentry; great vehicles for the transportation of the wood, and ropes and engines for the raising it, are not required; and in countries where only small scantlings of timber are obtainable, it permits of roofs of greater span to be made than would otherwise be possible. In Plate XXX., fig. 5 shows a portion of one of the hemicycles, as he called his frames, for spans of from 21 to SO feet. Each hemicycle, A B, in this case, is built of two thicknesses of wood, each of which, e f is in pieces of 3 or 4 feet long, 8 inches wide, and 1 inch thick. The joints of the one series are made to fall on the middle of the length of the other; each piece has a mortice cut in the middle of its length, and a half-mor¬ tice at each end. The mortices are 4 inches long, and a little more than 1 inch wide. They serve to receive ties g g, which may be of any length, and otherwise of the same dimensions as the mortice. The ties are secured in their places by keys h h, driven through mortices made in the ties, one on each side of every hemicycle. The mortices in the ties are made with a little draw. The keys are best when made of split wood. The two thicknesses of timber in each hemicycle are first framed together by small pins, to prevent their sliding, and then the hemicycles are united by their ties, and the two fas¬ tened by the keys. They are then placed on wall-plates, 10 or 12 inches wide, and 8 or 9 inches thick, having mortices sunk at 2 feet apart to receive the ends of the hemicycles. The mortices are 2 inches wide, 3 inches deep, and 6 inches long. In the roof of the pavilion of the Chateau de la Muette, where the span was 64 feet, the scantling was increased to 13 inches wide, and 1| inch thick. The ties were al¬ ternately double and single, and were 3 inches by 1 4 inch. Each hemicycle was double tenoned into the wall-plate. The general elevation of the roof is shown in Fig. 1, and parts of the hemicycles to a larger scale in Fig. 6. The same letters refer to the same parts in both figures— A A, one of the hemicycles; B, a terrace or gallery, used as a belvidere; c c, double ties; d d, single ties; e e, wall-plates; / f eaves-rafters (coyaux). The notches for double ties are just so deep, that the outside surface of the tie is flush with the edge of the hemicycle. When the span is small, and the curve of the roof is so quick that it becomes impossible to cover it with slates or tiles, De Lorme adopts the expedient shown in Figs. 2 and 3. Fig. 4 shows the application of the principle in the construction of a groined vault, with a pendant in the centre. The dimensions of the pieces of which these arcs or hemicycles are composed, increase of course with the increase of the span of the arch; and, as has been men¬ tioned above, the single ties give place to double and single ties placed alternately. In the roofs of ordinary buildings, where the span does not exceed 24 feet, the author directs the pieces which compose the hemicycles to be made 1 inch thick and 4 feet long; for roofs of 36 feet span, the thickness to be 14 inch; for roofs of 60 feet, the thickness to be 2 inches; for roofs of 90 feet, the thickness to be 24 inches; and for roofs of greater dimensions, the thickness to be 3 inches. Plate XXXI.—Ptoof of the great hall, Hampton Court. Fig. 1 is a longitudinal, and Fig. 2 a transverse section of the roof. The great hall at Hampton Court is 106 feet long, 40 feet wide, and 45 feet high in the walls. It was completed in 1536 or 1537. The roof consists of seven bays in length, one of which is the subject of Fig. 1, and by referring to the transverse section, Fig. 2, the con¬ struction, which is similar to that of Westminster Hall, will be clearly comprehended. Each principal consists of a centre arcli and two half-arches, and the principals are connected by three tiers of arches, as seen in Fig. 1. These, with their enriched panels and pendants, produce an exquisite richness of effect. Fig. 4 is one of the large pendants. Fig. 3.—One of the pendants of the second tier of arches. Fig. 6.—One of the pendants of the third tier of arches. Figs. 7 and 8 are two of the wall-corbels from which the roof springs. The exterior of the roof has a double pitch like the Mansard roof. Wsf.Beever.Sc. I J LATE XXXI. R © (0) W S o ROOF OF THE GREAT HALL. HAMPTON COURT M Of 5 , ROOF OF WESTMINSTER HALL. 'PLATE XmZl Fit/. 4'. Fi/j. 7 . J White, del? Fit/. 2 /•'/ of upper part of Scaffold inn. Fig 5. if. A. Beever. Sc. i of base of Spire, and part of Roof. Scale /hr Figs:2. 2.3.4. ■■ Scale for Figs: Plan of base of Scaffolding. so 56 so Feel. Fig. o'. Fig. 2. Fi g. 7. Elevation of Spire and part of Roof. A. F. Orridge.Dell BLACK1E & S'. GLASGOW.EDINBURGH & LONDON. 1\aV"\\nI FRAMING—JOINTS—STRAPS. 147 When two pieces of wood are joined by the simple contact of the end of the one piece with its bed on the other, we say that they abut, or are joined by a plain joint. This mode of joining does not prevent the one piece slid¬ ing on the other, unless it is fastened with nails or bolts. The contrivances by means of which one piece is pre¬ vented from sliding on the other are called mortises , joggles, &c. The putting together of two pieces of wood may be done in three ways:— 1. They may meet and form an angle; and this mode has three cases— 1. The end of one piece may bear upon a point in the length of the other. This case is the most frequent, and gives rise to the mortise-and-tenon joint, the joggle-joint, and to all those which are modifications of these two. 2. The two pieces can be joined mutually by their extremities under any angle whatever. This forms the angle-joint. 3. They may cross each other; and this result is the notch-joint. II. Two pieces of wood may be joined in a right line by lapping and indenting the meeting ends on each other. This is called scarfing. III. Two pieces of wood may be joined longitudinally end to end, the joint being secured by covering it on opposite sides by pieces of wood bolted to both beams. This process is termed fishing. It is requisite to consider the joint as formed by two pieces only, because joints formed by more than two pieces can always be resolved into this. The mortise-and-tenon joint is the principle of the greatest number of the other joints. It is necessary, therefore, to describe it first at length. In the simplest case of a tenon-and-mortise joint, the two pieces of wood meet at right angles (Fig. 467). The tenon a is formed at the extremity of the piece A, in the direction of its fibres and parallel to its axis m n, by two notches, which take from each side a parallelopipedon. The planes of the sides / g of the tenon are always par¬ allel to the face b of the timber, and the other planes of the notch at right angles to it. The mortise is hollowed in the face of the piece B, and is of exactly the same size and form as the tenon, which therefore perfectly fills it. The two sides of the mortise which correspond to the breadth of the tenon should be parallel to the direc¬ tion of the fibres of the wood. The sides of the mortice are called its cheelcs . and the square parts of the timber A from which the tenon projects, and which rest on the cheeks of the mortise, are called the shoulders of the tenon, and its springing from these is called its root. As the cheeks of the mortise and the tenon are exposed to the same amount of strain in a system of framing, it follows that each should be equal to one-third of the thickness of the timbers in which they are made. The length of the tenon should be equal to the depth of the mortise, so that its end should press home on the bottom of the mortise when its shoulders bear upon the cheeks; but as perfection in execution is unattainable, the tenon in practice is always made a very little shorter than the depth of the mortise, that its shoulders may come close. When the mortise-and-tenon joint is cut, adjusted, and put together, the pieces are united by a key or trenail. The key is generally round, with a square head, and in diameter is about equal to a fourth part of the thickness of the tenon. It is generally inserted at the distance of one-third of the length of the tenon from the shoulder. But a key should never be depended on as a means of securing the joint; for the immobility of a system of fram¬ ing should result from the balancing of the forces and the precision of the execution. A frame fixed definitely in its place should be stable and solid without the aid of keys, which are to be regarded as mere auxiliaries, useful during its construction. If the endeavour is made to apply the same manner of forming the mortise and tenon when the timbers are not at right angles, but oblique, several disadvantages arise. Such a joint is represented in the subjoined Fig. 46S (No. 1) by abed. If there were no other inconveniences, the impossibility of inserting the tenon in the mortise when the pieces form a portion of a system, would ob¬ viously preclude its adoption, as it would require to be thrust into the mortise in the direction of the arrow; but added to this, there is the difficulty of working the mortise, and the tendency of the thrust of the tenon to rend the piece B in the line b c. All these inconveniences are remedied in a very simple manner, by truncating the tenon on the line a /, as shown in No. 2, by a plane perpendicular to the axis of the mortise-piece B. The execution is thus rendered easy and exact, the evil from the thrust of the tenon is ob¬ viated, and the pieces can be put together by dropping the tenon-piece vertically into the mortise. This is the simplest form of the mortise-and-tenon joint for oblique thrusts. But, obviously the only resistance offered to the sliding of the tenon-piece along the mortise- piece is offered by the strength of the tenon, which is quite insufficient in large carpentry works; and it is therefore necessary to modify the form so as to bring new bearing surfaces into action. Plate XXXVII. Fig. 1.—No. 1 shows the joint formed by the meeting of a principal rafter and tie- beam, c being the tenon. The cheeks of the mortise are cut down to the line d fi so that an abutment e d is formed of the whole width of the cheeks, in addition to that of the tenon; and the notch so formed is called a joggle. No. 2 shows the parts detached and in perspec- 1-13 PRACTICAL CARPENTRY AND JOINERY. tive. It will be seen that a much larger bearing surface is thus obtained. Fig. 2. —No. 1 is a geometrical elevation of a joint, differing from the last by having the anterior part of the rafter truncated, and the shoulder of the tenon returned in front. It is represented in perspective in No. 2. Fig. 3.—Nos. 1 and 2 show the geometrical elevation and perspective representation of an oblicpie joint, in which a double abutment or joggle is obtained. In all these ioints, the abutment, as d e, Fig. 1, should be perpendi¬ cular to the line d /; and in execution, the joint should be a little free at /, in order that it may not be thrown out at d by the settling of the framing. The double abutment is a questionable advantage; it increases the difficulty of execution, and, of course, the evils resulting from bad fitting. It is properly allowable only where the angle of meeting of the timbers is very acute, and the bearing surfaces are consequently very long. Fig. 4.—Nos. 1 and 2 show a means of obtaining resis¬ tance to sliding by inserting the piece c in notches formed in the rafter and the tie-beam: d e shows the mode of securing the joint by a bolt. Fig. 5.—Nos. 1 and 2 show a very good form of joint, in which the place of the mortise is supplied by a groove c in the rafter, and the place of the tenon by a tongue d in the tie-beam. As the parts can be all seen, they can be more accurately fitted, which is an advantage in heavy work. In No. 1 the mode of securing the joint by a strap a b and bolts is shown. Fig. 6.—Nos. 1 and 2 is another mortise joint, secured by a strap a b and cotter or wedge a. Fig. 7 shows the several joints which occur in framing the king-post into the tie-beam, and the struts into the king-post. A is the tie-beam; B, the king-post; and c and D, struts. The joint at the bottom of the king-post has merely a short tenon e let into a mortice in the tie- beam. The abutment of the strut D is made square to the back of the strut, as far as the width of the king-post admits, and a short tenon / is inserted into a mortice in the king-post. The abutment of the joint of c is formed as nearly square to the strut as possible. The term king-post , as has been already stated, gives quite an erroneous notion of its functions, which are those of a suspension tie. Hence the necessity for the long strap b a bolted at d d, and secured by wedges at c, in the manner more distinctly shown by the section, Fig. 8, No. 2. The old name king-piece is better than king-post. Fig. 8.—No. 1 shows the equally inappropriately named queen-post. A is the tie-beam; B, the post tenoned at e ; c, the strut; and D, the straining-piece. The strap ba, and bolts d d. Fig. 9.—In this figure, the superior construction is shown, in which a king-bolt of iron C D is substituted for the king-post. On the tie-beam A, is bolted by the bolts a e, df the cast-iron plate and sockets abed, the inner parts of which, h g, hg, form solid abutments to the ends of the struts B B. The king-bolt passes through a hole in the middle of the cast-iron socket-plate, and is secured below by the nut D. A bottom-plate ef prevents the crushing of the fibres by the bolts. Plate XXXVIII.— Figs. 1 to 5 show various,methods of framing the head of the rafters and king-posts by the aid of straps and bolts. Fig. 6 shows the heads of the j rafters halved and bolted at their junction, and a plate I laid over the apex to sustain the bolts which are substi- i tuted for the king-post. One bolt necessarily has a link I formed in it for the other to pass through. Fig. 7 shows at d what may be considered the upper part of the same king-bolt as is shown in Plate XXXVII. Fig. 9, with the mode of connecting the rafters. A cast- iron socket-piece C receives the tenons a a of the rafters A A, and has a hole through it for the bolt, the head of which, b, is countersunk, b is the ridge-piece set in a shallow groove in the iron socket-piece. An elevation of the side is given, in which G is the bolt, F the socket- piece, and E the ridge-piece. Figs. 8, 9, 10, and II illustrate the mode of framing [ together the principal rafter, queen-post, and straining- piece. In the first three examples the joints are secured by straps and bolts; and in the last example the queen-bolt D passes through a cast-iron socket piece c, which receives the ends of the straining-piece and rafter, as those of the two rafters are received in Fig. 7. Figs. 12 and 13 show modes of securing the junction of the collar-beam and rafter by straps; and Figs. 14 and 15, modes of securing the junction of the strut and the rafter by straps. Lengthening Beams, &c.— In large works in carpentry it is often necessary to join timbers in the direction of their length, in order to procure scantlings of sufficient longi¬ tudinal dimensions. When it is necessary to maintain the same depth and width in the lengthened beam, the mode of joining called scarfing is employed. Scarfing is per¬ formed in a variety of ways, dependent upon whether the lengthened beam is to be subjected to a longitudinal or transverse strain. This method of joining is illustrated in Plate XXXIX., Figs. 1 to 13. In Fig. 1 a part of the thickness of the timber is cut obliquely from the end of each piece, and being lapped over each other, the joint is secured by bolts. In this case the joint depends entirely on the bolts. Iron plates are interposed between the nuts and the timber, to pre¬ vent the screwing up of the nuts injuring the beam. In Fig. 2 a key is added in the middle of the joint, notched equally into both beams. In Fig. 3 the joint is improved by its surface being indented on each joint, and the key driven between. In this example continuous plates of iron are placed to prevent injury from the bolts. Figs. 4, 5, 6, 7, 8, 9 Are all variations on the last figure. In Fig. 10 the beams are halved together vertically, as shown by the plan No. 2 and section No 3. They are keyed at the centre and secured by iron straps. In Fig. 11 the joint is made much larger and halved, the end of each beam is scarfed and keyed, as in Fig. 3, and the joint is secured by two straps and seven bolts. No. I is the side, and No. 2 the top of the beam. Figs. 12 and 13 are examples of scarfs formed by the interposition of a third piece b. When the beam does not require to be of the same dimensions throughout, it is sometimes lengthened by the process termed fishing. The ends of the beams a a. Fig. 14, are abutted together, and a piece of timber b b is placed on each side, and secured by bolts and keys. Fig. 15 is an example of a fished beam, in which the fishing-pieces b b and timbers a a are tabled, and indented and keyed together. J W.Lottry fc. BXACEIE fc SON: GLASGOW. EDINBUHGH fc LOUDON. JOflMTTS &W© SYW&PS. Tig 3.If?. . ' PI ATE XKXVJIf- TE XL. TRUSSED GIRDERS OR BEAMS. 149 Dovetailing, Halving, &c. — Fig. 16 shows two | pieces of timber joined together at right angles by a dove- I tailed notch. As to dovetails in general, it is necessary to remark that they should never be depended upon in car¬ pentry for joints exposed to a strain, as a very small de¬ gree of shrinkage will allow the joint to draw considerably. Figs. 17 and 18 show modes of mortising wherein the tenon has one side dovetailed or notched, and the corres¬ ponding side of the mortise also dovetailed or notched. The mortise is made of sufficient width to admit the tenon, and tlie dovetailed or notched faces are brought in contact by driving home a wedge C. Of these, Fig. 1 8 is the best. Fig. 19.—Nos. 1 and 2 show the halving of the timbers crossing each other. Fig. 20.—Nos. 1 and 2 show a joint similar to those in Nos. 17 and IS, but where the one timber b is oblique to the other a. Fig. 21.—Nos. 1 and 2 show the mode of notching a collar-beam tie into the side of a rafter by a dovetailed joint. The general remark as to dovetailed joints applies with especial force to this example. TRUSSED GIRDERS OR BEAMS. Plates XL., XLI. The general principle of trussing, and the object sought to be gained by its use, have been spoken of in the intro¬ duction to the article on roofs. Plate XL. Fig. 1 .-^No. 1 is an elevation of a trussed girder, with one of the flitches removed to show the trussing. No. 2 is a plan of the beam, and No. 3 a section through the line ah. The trussing-bars c, No. 1, are of iron, and are shown in section enlarged at cl, No. 3. An iron tension-plate D extends along the bottom of the beam, and connects the abutment bolts A A. These bolts pass between the flitches, and are screwed down upon an iron plate b. The central bolt B fulfils the functions of the king-post of a trussed roof. The beam is generally sawn in two, and the ends reversed, when put together in a truss. Fig. 2.—Nos. 1 and 2 are the plan and elevation of what may be called a queen-trussed beam. The construc¬ tion is the same as the preceding, with the substitution of the queen-bolts B B for the king-bolt. Fig. 3.—Nos. 1 and 2 show another example of a queen- bolt truss, where greater depth, and consequently greater stability, is obtained for the truss by the use of binding and bridging joists, A being the trussed beam, c the binding joists, and B the bridging joists. The tension- strap is joined together at the points c c by cotter-wedges, which have what is technically called a draw, so that the driving home of the wedge may bring together the parts Fig. 4 is an example of a girder trussed with a stirrup- piece B, end-plates A A, and a tension-rod A b A. No. 1 is an elevation of the beam; No. 2 is a plan; and No. 3 is an enlarged vertical section through the line a b. It is diffi¬ cult to balance the tensile and compressive resistances in a beam of this kind, so that they may be in action to the same extent and at the same time; and this application of iron in trussing is now considered by many practical men to be nearly useless. The beam is considered to be crippled before the iron begins to be strained, and there¬ fore this mode of trussing is not now in much favour. Fig. 5. —Nos. 1 and 2 illustrate the application of the tension-rod on what may be considered the queen-post principle, there being two stirrups at a a. Fig. 6,—Nos. 1 and 2 show a combination of timber and wrought-iron. The beam is composed of three flitches, the two outer being of timber, and the central of boiler¬ plate. The flitches are bolted together. In the elevation it is the iron flitch that is shown. Fig. 7 is an enlarged drawing of the connection of the trussing-piece d with the abutment bolt. The portion shown is the end of the girder, Fig. 2; b, the cog on the tension-plate notched into the bottom of the beam; d, the trussing-piece ; a, a hole through the beam for the trans¬ verse bolt, against which the abutment bolt is pressed; c, the cross-plate on the top of the beam, on which the nut is screwed down. Fig. 8 shows the connection of the trussing-pieces a b with the abutment bolt in same girder. Fig. 9 shows the links which connect the tension-rods of girder, Fig. 5. Fig. 10.—An enlarged drawing of the joints of the ten¬ sion-rods of girder, Fig. 3. Fig. 11.—A section through a 6 of trussing-piece in Fig. 3. Plate XLI. — Example from Krafft. Fig. 1.—Nos. 1, 2, and 3 represent what is usually called a truss, but what is property a built beam in three flitches. The three flitches are indented, as shown by the plan, No. 2, and the parts are brought home by the keys c c. They are then bolted together with tiers of bolts. Krafft remarks that this and similar trusses are only suit¬ able for situations in which they are exposed to tensile strains. Fig. 2.—Nos. 1 and 2 are representations of a com¬ pound beam, a modification by Mr. White of the system of M. Laves, architect to the King of Hanover. The system of M. Laves has for its object the reducing the weight of frames of carpentry, and economizing the timber which enters into their composition. He effects this by making a saw-cut horizontally along the centre of the piece of timber, and extending nearly to its ends. At the ends of the saw-cut he introduces bolts to prevent its extending further, and then forces the halves asunder in the middle of their length to a distance equal to one or one and a half times the total thickness of the beam by inserting pieces of wood, as shown in the figure. Several experiments to test his system were made by the inventor and others; also by Messieurs Lasnier and Albony, two skilful carpenters of Paris, assisted by M. Emmery, inspector-general of roads and bridges, and M. Biet, architectural inspector of buildings. In the experiments of M. Laves, four pieces of pine, each 40 feet long, 9 in. thick, and 7^ in. wide, were taken. Three of the pieces were prepared according to his system, the halves being forced apart to the distances respectively of the half of the thickness of the piece, the thickness of the piece, and one and a half times its thickness. The other beam was used in its natural state for comparison. Each of the four pieces was loaded with weights, be¬ ginning at 100 lbs. and increasing to 1700 lbs. Their deflections under their loads were as follows:— The piece in its natural state, The first prepared beam, ... The second „ „ The third „ „ 5i inches. 3 ^ „ H .. PRACTICAL CARPENTRY AND JOINERY. 150 In the experiment of M. Albony, made in 1840, two pieces of pine were taken from the same tree. Each piece was 51 feet 3§ inches long, 7 jV inches thick, and nearly 1 1 inches deep. The distance between the supports was 49 feet 24 inches. The saw-kerf in the prepared piece was as thin as could be made, and the two halves were separated 5 to inches, by a piece of timber. Before commencing to load the beams, the deflection due to their own weight was measured. In the unpre¬ pared beam it was found to be rather more than 14 inch, but in the prepared beam nothing. The beams were loaded with weights very gradually increased until they reached 793g lbs., when the deflection was—in the un¬ prepared beam 19 to inches, and in the prepared beam 2r 8 o inches. Under this load the pieces broke. M. Lasnier’s experiment was made with two beams of very dry pine, each 19to by 3^ inches, and the distance between the supports was 41 feet 10 inches. The halves were forced asunder 9 5 inches. The beams were loaded by weights of 272 lbs., applied successively, and distributed at three points of suspension. When the load reached 272 x 3 = 816 lbs. the prepared beam was deflected 2to inches, and then, after sustaining it a few minutes, it broke. The unprepared beam broke before the weight reached 800 lbs. M. Emy, who records these experiments, says—“It does not appear that the strength of a beam prepared according to the system of M. Laves much exceeds that of a beam in its natural state; but its stiffness is much augmented; which may render the preparation useful in several cases.” The beam represented in Fig. 2, No. 1, has been sawn in two horizontally, the two pieces again put together, secured by the bolts and straps drawn to a larger scale in No. 2, and then the two halves forced apart by the pieces b b inserted between them. M. Laves applied beams prepared on his system in the construction of floors, roofs, and bridges. Fig. 3 is a truss on the principle of the queen-post roof: A is the tie-beam, C one of the principal braces, E one of the queen-posts, D the straining-piece, and E and G are struts. Fig. 4 is a truss formed by the beams A B, straining- pieces b b, and braces and counter-braces cl d. When the braces, straining-pieces, and punchions c are inserted, the whole frame is made rigid by screwing the nuts of the three bolts. Fig. 5 is a combination of iron and timber. B is a cast- iron beam, and A a timber beam: on the top of the latter is the tension-rod cl, and on the upper side of this, and under side of the iron beam, are sockets formed for the punchions / and braces and counter-braces e e. Wrought iron straps embrace the framing at each punchion, and are tightened by cotters at c c. Fig. 6 is the trussed framing for the gallery of a church, where the ceiling underneath is curved. The principal B is notched on the wall-plate G, and also on the beam E: the tie A is secured on the wall-plate B, and bolted to the principal. F is a beam serving the office of a purlin, to carry the gallery joists; D is a strut; b b are the floors of the pews; and 0 c c the partitions, c is a hammer-piece resting on the beam E, and bolted to the principal B: its outer extremity carries the piece I, which supports the gallery front. Fig. 7 is another example of the trussed frame for a gallery. Here a framed principal adce, resting on the wall-plate H, and front beam E, supports the beam k, which carries the gallery joists B: a a and b b are the floors and partitions of the pews. Fig. 8 , Nos. 1, 2, and 3, shows the curb and ribs of a circular opening c B A, No. 2, cutting in on a sloping ceil¬ ing: as, for example, a circular-headed window occurring between two principals, such as that shown in Fig. 7. No. 1 is a section through the centre B D, No. 2, and efi, No. 3. The height L K is divided into equal parts in efg h i , and the same heights are transferred to the main rib in A 1 2 3 4 5 B. Through the points A 1 2 3 4 5, in No. 2, lines are drawn parallel to the axis bei; and through the points efghi in No. 1, lines are drawn parallel to the slope K H. The places of the ribs 1 2 3 4 5 in the latter, and their site on the plan No. 3, and also the curve of the curb, are found by intersecting lines in the manner the student is already acquainted with. FLOORS. Plates XLIL—XLIY. Floors are the horizontal partitions which divide a building vertically into stages or stories. The timbers which enter into the composition of floors are bridging-joists, binding-joists, girders, ceiling joists, and the boards which form the platform. All these, except the last-mentioned, are comprehended under the term “naked flooring,” and are strictly within the pro¬ vince of the carpenter. When the bearing between the points of support is not great, bridging-joists alone are used to support the floor¬ ing-boards, and, it may be, ceiling-joists. They are laid across the opening or void, and rest on the wall at each end. A piece of timber, called a wall-plate, is interposed between the ends of the joists and the wall, to equalize their bearing. A floor of bridging-joists, called a single-joisted floor, is the strongest that can be made with a given quantity of timber; but when the bearing is long, the joists, from their elasticity, bend under a moving weight, and thus disturb the ceiling below. When, therefore, the bearing is of such length as to cause the joists to bend, their elasticity is considerably diminished by placing underneath them stronger timbers, called binding-joists. This construction is called a double floor. When it is calculated that the bearing will exceed the limit of strength of the binding- joists, a third mode of construction is adopted, in which larger timbers, called girders, are introduced to support the binding-joists. This construction is called a framed floor. These three kinds of construction shall now be described and illustrated in order. Plate XLIL Fig. 1. Bridging-joist or Single-joisted Floors. —No. 1 is the plan of an apartment: act a a are the walls, b b the wall-plates, cccc, &c., the bridging-joists, d cl part of the flooring-boards. The bridging-joists are usually placed from 10 to 12 inches apart: their scantling is dependent on their length, their distance apart, and the weight they have to carry. Rules for calculating their size from these data will be found in the sequel. No. 2 shows a section through the joists at right angles ’• •; _• • •- FLOORS. 151 to their direction: c c are the bridging-joists, d the edge I of one of the flooring-boards, e e the side of a ceiiing-joist. The ceiling-joists cross the bridging-joists at right angles, as seen at(U, No. 1, and are notched up to them and fastened with nails. Sometimes every third or fourth bridging-joist is made deeper than its fellows, and the ceiling-joists are then fixed to them only. This has the advantages of preventing sound passing so readily, and making the ceiling stand better. When the bearing of single joists exceeds 8 feet, they should be strutted between, to prevent their twisting, and to give them stiffness. When the bearing exceeds 12 feet, two rows of struts are necessary; and so on, adding a row of struts for every increase of 4 feet in the bearing. There are three modes of strutting employed. The first and most simple is to insert a piece of board, nearly of the depth of the joists, between every two joists, so as to form a continuous line across. The struts should fit rather tightly, and are simply nailed to keep them in position. The second mode is to mortise a line of stout pieces into the joists in a continuous line across, but the mortises materially weaken the joists. The third mode is represented in the section, No. 2: //are double struts, of pieces from 3 to 4 inches wide, and li inch thick, crossing each other, and nailed at the crossing to each other, and at their ends to the joists. The struts should be cut at then- ends to the bevel proper for their inclination. To save the trouble of boring holes for the nails, two slight cuts are made at each end with a wide-set saw, and the strut is nailed through these with clasp-nails. Of the three modes, the last is the best. In No. 1,/// show three lines of struts. When some joists would, from their position, run into a fire-place or flues in a wall, it is improper to give them a bearing there. In the case of the floor, Fig. 1, two short timbers, called “trimmers,” are introduced—one on each side of the place to be cleared, with one end resting in the wall, and the other framed into the third joist from it: into the outer side of these, respectively, the end portions of the two first joists are framed, the intermediate portion being dispensed with. The joist into which the trimmers are framed is called the “trimming-joist,” and is made thicker than the others, according to the number of joists dependent on it for support. The hearth rests on a brick arch turned between trimmer and wall. Trimming is also resorted to for stair and other openings. In order to effectually prevent the passage of sound from one story to another, a second floor, of rough board¬ ing, is sometimes inserted between the joists, and covered with some non-conductor of sound,—the usual composition, which is called in England pugging, and in Scotlanddeci/eir- ing, being a mixture of lime mortar, earth, and the light ashes from a smithy. This sound-boarding, or deafening¬ boarding, as the secondary floor is called, is supported on fillets nailed to the sides of the joists. It is shown in the section, Fig. 1 , No. 2, where g g is the boarding, and h h the fillets. Along the joist next the wall a fillet is nailed, so as to fill up the space between the joist and the wall, and admit of the pugging being used there to effectually stop communication. Fig. 2. Double Floor, or Floor with Binding-joists .— No. 1, is a plan of this kind of floor: a a are the binding- joists, having their ends resting in the walls, but with templates b b, which are short pieces of timber or stone, interposed to lengthen their bearing; c c are the bridg¬ ing-joists, d d the wall-plates, e e is a trimmer opposite the fire-place, and / part of the flooring. The section, No. 2, shows the connection of the parts; a one of the binding-joists, c e the bridging-joists, notched over the binding-joist, and g g the ceiling-joists under it. Where the saving of depth in the framing is an object, the ceiling-joists are framed into the binding joists by a chase-mortise, as at h in the same figure. No. 3 is a sec¬ tion of the same floor parallel to the direction of the bridging and ceiling joists. The same letters of refeience apply to the same parts in both figures. Fig. 3. Framed Floor; the third mode of construction. _No. 1 represents the plan: A A are the girders, with their ends bearing on templates in the walls; ebb are the binding-joists, and c c the bridging-joists; d d the wall- plates ; e e the trimmer at fire-place. As the wall h con¬ tains flues along nearly all its length, the binding-joists do not rest in it, but are framed into an additional girder A. In this case, the tenon passes through the mortise, and is keyed on the other side, as shown in section in No. 4, in which A is the wall-girder, and B the binding-joist. No. 2 is a section through the girder, showing the man¬ ner of framing the binding-joists into it; A the giidei, BB the binding-joists, is the bridging-joists, d d the ceiling-joists. No. 3 is a section through the floor at right angles to the last section: in it the same letters refer to the same parts. In framing the binding-joists into the girders, it is necessary to effect a compromise between two evils; for the tenon is stronger the nearer it is to the lower side of the joist or binder on which it is formed, and the mortise weakens the beam or girder the least when it is near the upper side thereof; that is, when it is above the neutral axis. A contrivance, therefore, called a “ tusk tenon,, ^ is used, which is seen in the sections Nos. 2 and 4. The tenon a is a little above the middle of the joist; but its efficiency is increased by the tusk b, which relieves it of its bearing, and the shoulder above the tenon is cut back obliquely; and thus, without unduly weakening the girder, a great depth of bearing is obtained for the joist. It is necessary to take great care in fitting the bearing parts to the corresponding parts of the mortise. The tenon a should be equal to one-sixtli of the depth of the girder; and, according to the best practice, it should be inserted at one-third the depth of the girder from the lower side. It is a good practice to saw girders down the middle, and to reverse the ends and bolt the halves together with the sawn side outwards, with slips between to admit a circulation of air. By this means the heart of the timber can be examined, and the beam be rejected if unsound: the timber being reduced to a smaller scantling also dries more readily, and is rendered less liable to decay; and as the butt and top of a tree are rarely of the same strength, the girder must be improved by the process, which tends to equalize its strength throughout. When the bearing of a girder exceeds 22 feet, it is often difficult to get timber of a sufficient size. In this case the process of trussing the girder is resorted to. Various modes of trussing are figured in Plates XL. and XLI., and described in p. 149. Variations in the Modes of Constructing Floors .— In framed floors, especially in Scotland, binders are PRACTICAL CARPENTRY AND JOINERY. 152 frequently omitted, the girders are more numerous, and the bridging-joists are either notched down on them if the space will admit, or tenoned into them if otherwise. The ceiling-joists, too, in place of being notched or tenoned, are suspended to the bridging-joists by small straps of wood. Thus, the separation between the floor and the ceil¬ ing is more complete, and sound is less readily transmitted. Plate XLIII. Fig. 1 is the plan, Fig. 2 a transverse section across the direction of the girders, and Fig. 3 a longitudinal section, at right angles to the last, of such a floor. The same letters refer to the same parts in all three figures. a a, girder; b b, bridging-joists dovetailed into the girders; c c, ceiling-joists hung to the bridging-joists by the straps cl cl ; e e, fillets for the support of the sound¬ boarding or deafening-boards, nailed to the sides of the bridging-joists; //, sound-boarding lying loosely on the fillets e e; g g, flooring-boards grooved and tongued; h h, ceiling-laths; m m, plaster ceiling; n n, pugging or deafening, composed generally of lime, earth, and forge ashes in equal proportions. Fig. 4 is the plan, and Fig. 5 a section of part of a warehouse floor composed of girders a a a, supported by cast-iron columns b b b, and supporting the floor of planks c c c. Figs. 6 and 7 are' plan and section of part of a ware¬ house floor composed of trussed girders a a, supported on iron columns b b, bridging-joists c c , and flooring- boards d d. Fig. 8 is a vertical section through the floor at right angles to that shown in Fig. 7; and Figs. 9 and 10 are plan and section of the head of the cast-iron pillar, drawn to double the scale. Plate XLIV. French Floors. — Fig. 1 is the plan of a portion of a floor composed of joists sustaining floor¬ ing-boards; and various modes of disposing the latter are shown. Fig. 2 is a longitudinal section on the line A b of Fig. 1; and Fig. 3 a transverse section on the line c D of the same figure. Fig. 4 is a transverse section on the broken line E f. a a are the joists on which the flooring-boards are nailed; b b boards, the full width of the deals or other timber out of which they are cut: these are gauged to a width, and jointed together by a groove-and-tongue joint: they are generally 1 inch to 1^ inch thick, according as the joists are nearer or further apart. Each board is attached-to each joist by two or three nails, according to its width; and when all the boards are laid and nailed, the joints are dressed off with a plane. c c shows the floor composed of narrow deals, jointed with groove and tongue, and each deal fastened by two nails to each joist. When the flooring-boards are of narrow deal, they are generally planed on both sides, to reduce them to uniform thickness; and in this case the upper surface of each joist is also planed, and all the joists are carefully adjusted in the same perfectly level plane. The end joints of the boards are arranged so that the joints of two contiguous boards shall never fall on the same joist; and care is taken, for the sake of appearance, to make the joints of alternate boards fall on the same straight line across the apartment, and at the middle of the length of the intermediate boards. But when it is possible to obtain boards the whole length of the apart- [[ ment, the preference is given to a floor without end joints. The end joints of the boards are, in many cases, 1 made also with groove-and-tongue; but as the joints [ occur only on the middle of a joist, and can be well nailed, it is by many considered superfluous. To render a floor still more solid, and prevent the passage of sound, a second course of boarding is laid above the first, with a space between. This is shown on the extreme right in Fig. 1. //are fillets nailed on the first laid boarding, conformably with the joists; g g are the boards j of the second floor; and to deafen the floor, the intervals ' between the fillets are filled with lime-mortar, or with lime and ashes, or with dry moss. When lime-mortar is used, the upper boarding must be laid before it is quite dry, lest the hammering required in fixing it should break up the deafening. When dry moss is used, it is driven in as the upper boards are laid, and rammed hard. The second floor-boards do not require their joints to be grooved and tongued, as the penetration of dust, &c., is prevented by the grooving and tongueing of the first floor. k h shows another method of laying the flooring-boards, where the joints meet in a straight line on a joist; and i i shows the manner of laying, called in this country j herring-boning. In either of these two last methods, the width of the board should not be less than a twelfth, ! nor more than a sixth of its length; and the best mode of j ointing is by grooving and tongueing. Where it is customary to wash floors with water, M. | Emy considers a plain joint preferable to a groove and- tongue joint for the boarding; for when the board grows 1 old, the surface rots or decays, and the edge of the board, in the case of the groove-and-tongue joint, having little ! solidity, the fibres splinter off. In nailing the boards in common floors, what are called floor-nails are used. These have the shank square in section, the head large and round, and its top shaped like a very flat diamond point. For better work, nails called pointes de Paris are used. The shank of this kind of nail is cylindrical, and the head small, so that it may be driven under the surface of the board by a punch. But still j better are the clous a parquet, which correspond to the English flooring-brads. Sometimes, too, screws are used; in which case the upper surface of the boards is countersunk by a cylin¬ drical hole, so as to receive entirely the head of the nail, and admit of the surface of the floor being planed off. 1 The cylindrical holes are filled in with pieces of wood of the same kind as the boards, with their fibres in the same direction, and strongly glued, and driven in with a mallet. This method is used chiefly for oaken floors. As it is not usual in France to cover the floors with carpets, more attention is paid to the appearance of their surface than with us. Sometimes, boards of different I kinds of wood are used, and combined so as to produce contrast in colour, and in the direction of the fibres; and even with one kind of wood agreeable combinations are produced by merely contrasting the latter. Floors of parquetiy are not here touched upon, as be¬ longing, in France, more to the cabinet-maker than the carpenter and joiner. The flooring-boards cover only the upper surfaces of the joists. Sometimes their other three faces are left FLOORS. visible; but more often their under side is covered to form a ceiling to the apartment below. At e, Fig. 3 is a section of the platfond or ceiling, composed of thin planks, jointed longitudinally with a groove-and-tongue joint, and nailed on the under side of the joists. The deal used for this purpose being preferably lime-tree, which in French is called tilleul, the covering itself came to be called tillis. The joints of the platfond boards are sometimes beaded, and the whole platfond is generally painted like the rest of the wood-work of the apartment. Sometimes the platfond boarding does not stretch across the under side of the joists, but is framed in between them, as in Fig. 5, which is a section perpendicular to the direc¬ tion of the joists a a. When the fibres of the boards are perpendicular to the direction of the joists, the work is more solid. When the platfond boarding is flush with the under side of the joist, as in Fig. 5, the tongue has to be worked on the upper edge of the board, in which case auy shrinkage of the timber makes a visible opening; but when the boarding is a little recessed, as in Figs. ( and 8, the tongue can be made on the under side, and the shrinkage is not observed. The platfonds are frequently decorated by being divided into compartments by the joists, and these compartments are enriched with paintings and sculptures. This sort of platfond is much in use where plaster is not easily ob¬ tained. In Figs. 6, 7, and 8 is represented a platfond of this kind. Fig. 6 is a plan of the ceiling looking up. Fig. 7 is a section of the floor by a vertical plane passing through A B in Fig. 6, and Fig. 8 another vertical section by a plane perpendicular to the former on the line c D. In these are shown the joists, the boarding of the floor, and the cross pieces framed between the joists with mor¬ tise and tenon, to form the compartments. In Fig. 7, the platfond boards are cut in the direction of their fibres, which is perpendicular to the fibres of the joists, and in Fig. 8 they are cut across the direction of their fibres. If such a platfond were ornamented by painting, the shrink¬ age of the wood would obviously mar the work by making the joints visible; the practice is therefore to prepare frames to fit the panels or compartments, and on these to stretch cloth, on which the ornamental painting is made. At Paris, and in other places where plaster is abundant, flooring of stone or tile is often substituted for the timber floor. This mode of construction is shown in Figs. 9 and 10— Fig. 9 being a section on a line crossing the direction of the joists, and Fig. 10 a section passing through the middle of the interval between two joists; a a the joists, b b laths of oak crossing the joists and nailed to them, c c composition of plaster on which the stones or tiles are laid, cl cl the stone or tile floor, e e laths to support the ceiling, /pugging or deafening between the joists, g plaster ceil¬ ing united to the deafening through the interstices between the laths. The pugging, f not only prevents the passage of sound, but also of disagreeable odours. It is therefore especially used over kitchens and stables. The pugging is formed of a coarse mortar, composed of Jime and pieces of stone or of old plaster. It is from 3 to 4 inches thick at the middle of the interval between the joists, but at the sides it is carried up to the under side of the laths which support the floor, thus forming a sort of 153 trough; and to make it better adhere to the wood, the sides of the joists are studded with nails or wooden pins. When this extent of pugging is not required, the under side of the floor laths merely is plastered, as seen in the sections, Figs. 11 and 12. In floors of great span, the elasticity of the joists would break the plaster ceilings attached to them. To prevent this, one series of joists is used to carry the floor, and another series of slenderer joists to carry the ceiling. A vertical section of this arrangement is shown in Fig. 13, in which a a are the flooring joists, bb the flooring boards, and o o the ceiling joists. When strong split laths of oak are used, the ceiling joists are placed farther apart, as in Fig. 14, in which a a are the floor joists, bb the flooring boards, and o o the ceiling joists. Combination of timbers of small scantling to form floors of large span without intermediate support. The first variety of such floors to be described is that invented by Sebastien Serlio, a celebrated architect, who was born in Bologna in 1518, and died at Paris in 1552. On the principle of construction adopted, by Serlio, the principal timbers form great rectangular divisions, each timber having one end supported by the wall and its other end supported by the adjacent timber. Four great joists, a d a a (Fig. 15), have each one of their ends cc' a! a' a', resting in the wall of a square apart¬ ment, aud they are arranged perpendicularly to each other, so that the outer end of each beam is supported by the middle of the next adjoining. This principle will be familiar to most of our readers from the amusing illustration of it shown in books of philosophical recreations, which consists in placing three or four knives resting on the ends of their handles and interlacing their blades by crossing them alternately, by which means a considerable weight may be supported at the meeting of their points. The spaces between the main joists and the walls are filled in with ordinary joists resting in the wall and framed into the main joists, which serve as trimmers, and the central square is filled in also with joists placed diagonally, so that the weight may be borne equalty by the four main joists. Fig. 16 is the plan of part of a floor of the Palace in the Wood, at the Hague. It is an extension of the system of Serlio. The hall, of which this is the floor, is 60 feet on the side; and the figure represents one of the four angles. The floor is constructed of small girders of oak, forming 300 square panels. Any one of the girders, such for example as c, is tenoned at each end into two other girders, as b and /, and carries the ends of other two girders e and cl, which are tenoned into it at the middle of its length. Those girders which run on the walls are tenoned into wall-plates C D, imbedded in, and fixed to the masonry. The floor is composed of a double thickness of boards, crossing each other at right angles, grooved and tongued, and nailed to the girders. Fig. 17 is a section on the line A B on the plan, showing the two thicknesses of boards. The girders are cut below so as to form a slightly con¬ cave surface, with the object of compensating for any sag¬ ging; which would have had a disagreeable effect. The result, also, is to diminish the weight at the centre of the floor. In constructing this kind of floor, the sides should u 154 PRACTICAL CARPENTRY AND JOINERY. be divided into an unequal number of spaces, that there may be a square compartment in the centre. Fire-proof Floors. —Of late, much attention has been given in this country, and still more in France, to the construction of fire-proof floors. In Plate XLIII., illus¬ trations are given of both the English and the French j methods of construction. Figs. 11 and 12 are longitudi¬ nal and transverse sections of Messrs. Fox and Barrett’s system, a a are the main girders, consisting of a pair of rolled joists; bb, cc, the ordinary joists, which are alter- i nately 6 inches and 4i inches deep; d d, wooden bat¬ tens, about 1 inch square, resting on the bottom flanges of the ordinary joists and main girders, and placed like laths with a narrow space between them. On the top of 1 these is spread a layer of coarse mortar, which being i pressed down between the battens, forms a good surface of attachment for the ceiling. The space between the | main girders and the spaces between the joists are then j filled in with concrete, e e, in which are imbedded small ! quarterings or joists, //, to which the flooring-boards are nailed. The concrete sets and consolidates the whole floor into one slab. Fig. 13 is a French fire-proof floor on the system of Thrasnd. It consists of rolled double-flanged girders, sup¬ porting common bridging-joists of timber, b b, which sup¬ port the flooring-boards. The iron girders are connected together by transverse tie-rods bent round the lower flanges. These support hollow bricks, c c, which are cemented together with plaster of Paris. In Fig. 14 is a section of a lighter floor. The gir¬ ders are connected by light iron ties bent over them; and with these ties are interlaced cross bars of light iron, so as to form together a complete trellis work, on whicli is run plaster of Paris; flat boards being placed provision¬ ally below the trellis as a mould, till the plaster is set. It is now necessary to put the reader in possession of the means of calculating the strengths of the various timbers which enter into the composition of the three varieties of naked flooring already described, and to give, as guides in construction, such practical rules as experi¬ ence has developed. This will best be done by taking each description of timber in its order. Primarily, it may be remarked, that flooring timbers may be of such scantling as to be sufficiently strong for safety, and yet be deficient in stiffness. A slight de¬ flection in the flooring would injure the ceilings of the i apartment below; and for that and other obvious reasons, 1 the rules for calculating the scantling of timbers in naked flooring are based on stiffness—or resistance to deflection, i and not on absolute strength—or resistance to transverse j breaking-strain. Wall plcdes. —These, like all bearing timbers, should be increased in dimensions as the span becomes longer. It is not possible to give a rule by which to calculate this increase; nor is it necessary. Tredgold has laid down the following proportions as a guide; and they are practically j safe :— Rule .—Bearing, 20 feet. Wall plates, 4lt by 3 inches. „ " 30 „ „ 6 by 4 „ „ 40 „ „ 7| by 5 „ Where wall-plates extend over openings, and have to sustain the ends of timbers, their scantling must be calcu¬ lated by the general rules for transverse strain. Wall- plates should be carefully notched together at every angle and return, and scarfed at every longitudinal joint. The notch joint is shown in Plate XXXIX., Figs. 16 and 19, and the scarf at Plate XXXIX., Figs. 1 to 15. Single-joisted Floor. —The timber in the joists should be so disposed that these may be as deep as is consistent with the thickness requisite to prevent splitting in nail¬ ing the boards. The least thickness which it is safe to give is 2 inches. To find the depth of the joist, when its thickness and the length of the bearing are given, and when the distance apart from centre to centre is 12 inches. Rule. —Divide the square of the length in feet by the breadth in inches, and the cube root of the quotient, mul¬ tiplied by 2'2 for fir, and 2 3 for oak, will give the depth in inches. Example. — Required the depth of a joist when its length is 15 feet and its breadth 3 inches. — * — 75, the cube root of which is 4 21. Therefore, 4'21 x 22 = 9 - 26 for fir, and 4 21 X 2 3 = 9 68 for oak. But when joists are so thick as 3 inches, they injuriously affect the keying of the ceiling; and in the example above a better relation between the depth and thickness would have been obtained by making the latter 2 inches. In this case— 15x15 ,, , , » —— = 112 5, the cube root of which is 4 - 4; which multiplied by 2 2 and 2 3, gives 1056 for fir, and 11 for oak. All single-joisted floors, it has been already said, should be strutted; and, accorling to the rule laid down, the rows of struts should not be more than 5 or 6 feet apart. Sometimes, to afford a better key to the plaster of the ceiling, the under sides of the joists are crossed with battens l.y by 1 inch, and 12 inches apart; and to these the laths are nailed. This process is in Scotland called brandering. Trimmers and Trimming-joists. —The thickness of the trimmer is found by the rule given for binding-joists. The trimming-joists are made -J- of an inch thicker for every joist carried by the trimmer. In trimming, tusk tenons should be used. The tongue of the tenon should run at least 2 inches through, and be draw-pinned; and if it does not completely fill the length of the mortise, it should be wedged also. A proper fillet requires to be nailed to the trimmer to form a skew-back for the brick arch; and each trimmer should have two or more bolts, according to its length, to tie it to the wall. Binding-joists. —Binding-joists should not exceed 6 feet apart. Their depth is often, but not necessarily, regulated by the depth of the floor; and therefore it is necessary to know how to find the breadth when the length and depth are given, as well as to find the depth when the length and breadth are given. To find the breadth, when the length and depth are given. Rule. —Divide the square of the length in feet by the cube of the depth in inches, and the quotient, multiplied by 40 for fir, and 44 for oak, will give the breadth in inches. To find the depth, when the length and breadth arc aiven. Rule. —Divide the square of the length in feet by the breadth in inches, and the cube root of the quotient, mul- - PARTITIONS. 155 tiplied by 842 for fir, and by 353 for oak, will give the J depth in inches. Girders.—To find the depth of a girder, when the length and breadth are given. Rule .—Divide the square of the length in feet by the breadth in inches, and the cube root of the quotient mul¬ tiplied by 4-2 for fir, and 4 34 for oak, will give the depth in inches. Example .—Let the length be 20 feet clear, and the breadth 10 inches. Then - = 40, the cube root of which, 341, multiplied by 4'2 for fir, gives 14 32 inches as the depth, say 14^. To find the breadth, when the length and depth are given. Rule .—Divide the square of the length in feet by the cube of the depth in inches, and the quotient, multiplied by 74 for fir, or 82 for oak, will give the breadth in inches. Take the same data as before— -i — = 136, which multiplied by 74, gives as the breadth 2936 10 inches. Ceiling-joists .—Divide the length in feet by the cube root of the breadth in inches, and multiply the quotient by 0-64 for fir, or 0 67 for oak, to obtain the depth. Let the length be 6 feet and the breadth 2 inches. Then - - =. 47, which multiplied by 0 64 = 3 inches. 126 The foregoing rules apply to ordinary cases of floors, such as those of dwelling-houses. The greatest load on such floors is when they are covered with people, equal to about 120 lbs. per superficial foot, and allowing for the weight of the floor, altogether about 150 lbs. per foot. In warehouse floors, however, the absolute weight must be taken at about 24 cwts. to the yard superficial, or 300 lbs. per foot, and to allow for the shock caused by throwing down heavy goods, 3S0 lbs. to the foot should be assumed as the dead weight. PARTITIONS. Plate XLY. Timber ‘partitions are internal vertical divisions used in the upper stories of a building, to make the separations required in forming the apartments. When such apart¬ ments are more numerous than in the lower stories, the partitions should be so constructed as in no way to in¬ fluence, by their weight, the integrity of the ceilings of the rooms beneath; and their weight therefore should be transferred, by the system of framing, to the immove¬ able points of the structure. To accomplish this, trussed or quartered partitions are used. These are framed on the same principle as a king or queen post roof; and are equally capable of bearing a strain proportionate to the scantling of the timbers of which they are composed. Timber partitions should not be used in dividing the ground floors into apartments, because of their liability to be affected by damp. Stones or bricks are the proper materials to use in such places. Plate XLV.—The figures in this plate are all examples of trussed or quartered partitions. Fig. 1 is a partition trussed on the principle of the queen-post roof. The object aimed at in this case is to resolve all the pressures or weights of the partition into vertical or downward pressure on the walls, which in the example before us is rendered easy by the symmetrical arrangement of the openings. For it will be readily seen that the pieces D D, with the intertie A A, the straining- piece h, and the struts cl, acting in the same manner as roof principals, form a queen-post truss; the intertie A A being rendered continuous as a tie-beam by the straps at a a. The strut c serves to discharge the downward pres¬ sure at m to the wall; and the counter-struts, the pressure at n to the foot of the queen-post D. The actual stability of the partition, however, depends on the upper trussing; that is, on the framing composed of the tie A A, the posts D D, the principals d d, and the straining-piece h. c is the headpiece of the partition, B its cill, l the door¬ posts, and i i the door-frame; b b, b b are the joists of the floors above and below. The counter-braces, such as g, prevent the sagging of the main struts, and give addi¬ tional stiffness and firmness to the framing. This partition is at l ight angles to the direction of the joists b b, and therefore when the door-posts do not fall upon a joist, it is necessary to support them by pieces, as k. The door-casing is shown, also the headpiece, and the joists of the floor above at h k. Fig. 2, No. 1.—In this example, the intertie b, the post D, and the struts g, form a king-post truss. The door¬ posts l, are secured by the straps at o and p, the intertie is continuous, and the king-post is rendered so by the strap at m. The cill is sustained by the strap at n, and thus the whole system of the framing is dependent on the upper portion of the truss: e and f are the struts, and k k the doorcase; A the headpiece, and h It joists of the floor above. In No. 2, the upper portion of the truss is on the queen- post principle, d is the inter tie, which, as before, is the tie-beam ; e is the strut forming the principal, and/ the straining-piece. The door-posts A B are suspended by the straps c c, E is the cill, d a strut or brace, g a counter brace, and c the headpiece. Fig. 3, No. 1 .—The partition in this case runs in the direction of the joists A and B, and the truss serves to give strength and stiffness to the floor. The upper por¬ tion of the truss, consisting of the intertie c, struts e, straining-pieces /, and queen-posts D D, is still in this example the main support of the framing. The straps l l tie up the joist to the queen-posts, which form one door¬ post of each door: the other door-post, i, is framed into the joist and intertie, and has its strut or brace g. Be¬ tween the queen-posts are counter-braces h. The straps k k render the intertie continuous; m m is the door¬ case. No. 2.—In this partition there is only one door at the side, and the framing is therefore not symmetrical. The I intertie T>, the post C, and the strut h, are the parts essen¬ tial to secure the stability of the framing. The braces fgi are as before. The joist B is suspended by the strap m, and the strap l ties together the intertie D, the post c, and the brace h. A row of struts or dwangs (Scot) k k, is introduced to give stiffness to the quarterings. This par¬ tition may be regarded as half of a queen-post truss, C being one of the posts, and the wall serving the purpose of the other. 156 PRACTICAL CARPENTRY AND JOINERY. The imperfections in joints, and the tendency in all timber to shrink, frequently cause settlements in framed partitions, and consequent cracks in the plaster. To diminish the risk of this damage, it is essential that all the timbers be well seasoned, and the joints made with the greatest exactness. It is advisable to have all parti¬ tions put up some time before they are plastered, so that any imperfection occasioned by warping or shrinking may be seen in time, and remedied. The arrises of all timbers of, or exceeding, 3 inches in thickness should be taken off, to admit of a better key for the plastering. The distance apart of the quarterings, or filling-in timbers, should be adapted to the length of the laths, which is generally about 4 feet; and, therefore, when this is the case, the timbers should be about 1 foot apart from centre to centre. TIMBER-HOUSES. Plates XLVI. and XLVII. Houses and other edifices constructed of timber, and raised on plans of which the perimeters are right lines, are composed— First, Of vertical walls of carpentry forming the facade and returns. Second, Of interior partitions of carpentry, dividing the interior space horizontally into apartments. Third, Of floors or horizontal partitions, dividing the interior space vertically into stages, stories, or floors. Fourth, Of roofs which cover and defend the inclosure. Fifth, Of stairs which afford access to the different stories. The use of timber walls, doubtless, preceded the use of walls of masonry. Now, however, that the means of construction are multiplied, wooden structures are only erected in this country where other building material is scarce and timber plenty, when cheapness without regard to durability is aimed at, or when expedition in construc¬ tion is the object. Walls constructed wholly of carpentry would consume an immense quantity of timber, and would be more ex¬ pensive than if built of brick or stone: the timber, there¬ fore, in thick walls is used only in sufficient quantity to form a frame-work which shall insure the stability of the structure; and the interspaces or panels of the frames are filled in with masonry of small stones, with thin brick work, or with lath and plaster. This mode of construction is in the north called post and pan or post and petrail, and the square of framing is called a pan; but such erec¬ tions are more universally called half-timbered houses. The combination of principles in these timber erections varies very little. The general type of the compositions is presented in Fig. 470, and it may be traced in all the figures. Ordinarily, to preserve the apartments and the timber from damp, the level of the first floor is raised consider¬ ably above the soil, on a wall of masonry. Frequently, the walls of the first story are formed entirely of masonry, and the carpentry work commences above this, carried up sometimes in the same plane, and sometimes projecting on corbels. Every pan is composed of a ground-cill, which receives the tenons of the principal posts, and these posts receive the tenons of the horizontal beams, which are termed bressummers, and which serve to carry the floors, and are crowned at the top by beams corresponding to the ground-cill, and which, like it, are mortised to receive the tenons of the posts. These crowning beams are sometimes also named bressummers: when they support the feet of the rafters they are called raising-plates; but a general name for them is capping-pieces, which better describes their place and function. The inclined timbers in the framing, termed braces, divide the parallelograms formed by the vertical and hori¬ zontal timbers into triangles, and thus preserve the in¬ tegrity of their form. They, as well as all the other pieces, should be very exactly fitted at their abutments, to diminish as much as possible the creaking by the play of the joints, caused by the flexibility of the long timbers. Between the beams or bressummers, the door-posts, called jamb-posts, and the window-posts, called by old writers prick-posts, are framed. The horizontal pieces framed into these to form the heads of the openings are termed transoms and lintels, and those introduced between the principal horizontal timbers are called interduces or inter ties. The posts which have no other function to perform than to sustain the edifice, are called posts simply. The panels or spaces between the doors and windows, and between the different posts, are filled in with vertical pieces called quarters. Pans which are of great length should be divided into bays, either equal or symmetrical, by principal posts, which, like those at the angles, should rise in one piece from the ground-cill to the capping-piece. As these interrupt the continuity of the bressummers, the ends of the bressum¬ mers which are framed into them should be tied together by iron straps within and without. All the beams, too, which are framed into the corner posts, and have return beams, should be tied in the same manner by a right- angled strap embracing the post. When the timber framing of the carcase is completed, all the intervals between the posts are, in the case of post and petrail construction, filled in with small stones or bricks set in good mortar. When the timbering is to re¬ main visible, as is generally the case in such houses, the masonry or brick-work filling is done with great neatness, and the timbers are dressed on their exposed faces before they are framed. When this mode of construction is not adopted, and the timbers are to be hid, the exterior and interior sur¬ faces are lathed, and the space between is either left void or, which is better, filled with some non-conductor of heat, and then the lathing is covered with plaster and deco¬ rated with the usual architectural decorations of strings, cornices, architraves, &c. In place of lathing and plastering the exterior surface, it may be covered with boarding, and wooden mouldings may be applied in decoration, as in Plate XLVI., Fig. 3. To insure better connection between the masonry fill¬ ing, the plaster, and the timber, it was formerly the cus¬ tom to groove the latter, and to plant it all over with pins of wood: now this is seldom done. In Sweden, where timber-houses are almost universally employed, the mode of construction is the same now as TIMBER-HOUSES. 157 ifc has been for centuries, and is also of the type here described. A foundation plinth of rough granite is first laid to a height of 2 feet above the ground. On this are laid the ground-cills, with mortise holes for the uprights. The up¬ rights are from 6 to 8 inches square, and are mortised to receive the bressummers, and are otherwise tied together by the interties, window-cills, lintels, and braces. When this framing is completed, it is covered on the inside with f-inch deal boarding, and on the outer side with two thicknesses of f-inch deal, the first laid horizontally and the second vertically. The vertical joints are again covered by slips nailed to them. The space between the outer and inner lining is Fig. 4G9. n t , \ / s. -. 1 ” >-iA % •:. ^ ':v . .i . • -v- -V • -• filled in with shavings, moss, or some non-conducting sub¬ stance (see Fig. 469). The lowest floor is double lined, and filled iu between the linings in the same manner; and houses thus con¬ structed are impervious to the colds of winter. A house of this kind can be erected in a few days, and where timber is abundant, costs very little. But its liability to be destroyed by fire renders it a very hazardous kind of building. Fig. 470 is the elevation of part of one bay of a building constructed of wood on stone pillars. The ground-cill a is pillars. These pieces are called in Frencli poitrail, evidently the same as the petrail of the north country, and which gives to this style of building the name of post and petrail. In order to throw the weight as much as possible over the stone pillars, discharging struts D D are introduced at each story. These sustain the principal posts P, and are framed into them, and into the ground-cill, in the same way that a principal rafter is framed into its king-post and tie- beam, and is like it secured by iron straps d. The braces in the panels F K are halved on each other, and form the St. Andrew’s cross, for the sake both of effect and stiffness. The horizontal pieces OOO are the interties. The figures given in Plate XLVI., and about to be described, are the kind of constructions in timber which obtain in the present day. The more ancient edifices were constructed much in the same manner, the framing applicable to all such constructions not being susceptible of great variety. Fig. 3, Plate XLVII., is the elevation of the gable of a modern imitation of an ancient timber gable, in which the arrangement of the timber forms the principal decoration. In designing pans of wood, the greatest care should be taken to make all the principal timbers coincide verti¬ cally, so that from the ground-cill to the headpiece the principal posts, story-posts, door-posts, and even the quar- terings respectively of each story should be in the same vertical line, so that they may not have the effect of twist¬ ing or bending the horizontal timbers. For the same reason the openings of the doors and windows should be com¬ prised within the same vertical lines. This sound rule of construction, void over void and solid over solid, is appli¬ cable to timber constructions as well as to those of stone or brick; and produces, by the symmetry and correspondence of parts which arise from its being adhered to, an effect which is always agreeable. When it is necessary to make one or more of the open¬ ings of a greater width than the others, as a gateway for example, the panel in which it is made should, if possible, be carried up; and the weight of the intermediate framing above should be discharged by struts to the posts which form the panel, so that its lintel should have nothing to carry but the short quarterings between it and the cills of the windows above. As the charge to be sustained by the timbers diminishes story by story as they ascend, it is customaiy and proper to diminish their scantling. The batter which ensues is confined to the exterior, the interior surface being kept in the same plane throughout. The effect of this is to increase the stability of the edifice by extending its base. Plate XLVI., Figs. 1 and 2.—In these the principles of construction described are exemplified, the ground-cill A, principal posts b', bressummer E c, and diagonal struts B B and D D, form a truss which sustains the structure. The parts are connected by mortise-and-tenon joints, and se¬ cured by straps ctb c deg, &c. The principal posts, story- posts, and quarterings of both stories are in the same ver¬ tical lines. Fig. 3 shows timber construction adapted to modern street architecture: A, the ground-cill; B F G, one of the principal posts; B, the angle principal; C D e, bressummers scarfed at the points d, b, and e, and secured by straps ci b c df g\ K, an additional story-post at the angle. Plate XLVII.— Figs. 1 and 2 show the framing of the Townhall of Milford, Massachusetts. Fig. 1 is an eleva¬ tion of the side of the structure: A A, the ground-cill; B B, bressummer; D D, intertie; CC, capping-piece. The princi¬ pal vertical posts K L, K L, K L, correspond in number and position with the principals of the roof, and all the other principal timbers are in the same vertical lines. The same principle of construction is developed in the end elevation, Fig. 2, where A A is the ground-cill; A C, F F, principal posts, which are continued to form the tower shown in Plate XXXV ®; B B, bressummer ; ee,e e, interties, forming trusses with B B by means of queen-bolts and struts h h ; g kmn are braces, and d d, d d, interties above the win¬ dows and doors. Fig. 3 is the elevation of a gable at Chester, a recent work, in which the ancient style is elegantly imitated, the arrangement of the timbers forming the principal decoration: A A is a lintel supported by the corbels, a a; 158 PRACTICAL CARPENTRY AND JOINERY. DD, principal posts; B B, intertie; c c, capping-piece; E e, story-posts; e e e e, quarterings, the panels between which are filled in with ornamental ribs. Fig. 4 is a barge-board or gable board from a house at Droitwich, and Fig. 5 is an example of a gable-board from a house at Worcester. BRIDGES. A bridge is a platform supported at intervals to form a roadway over a river, valley, or other depression. Bridges are as various in kind as the circumstances that necessitate their construction. They are formed of various materials, as timber, iron, stone, and brick, or of the combination of all these. In the timber bridge, for example, to the consideration of which this introduction properly refers, there may be piers or abutments of stone or brickwork, and the carpenter can only combine his timbers by the aid of the straps, bolts, hoops, screws, nuts, washers, and other ironwork supplied by the smith. In bridges constructed with arches of masonry or brick¬ work, the aid of the carpenter is sometimes required in preparing the foundation by a cofferdam, or in making platforms of timber on which the stones or bricks are placed, and his utmost skill is, in large constructions, always called forth in designing and constructing the centres or mould on which the arch is formed. Of all the materials used in bridge construction, timber, connected together by iron, is the most extensively avail¬ able for the platform, and although liable to serious objec¬ tions on account of its destructibility by natural decay or by accident, and its liability to change under hygro- metric influence, yet the readiness with which it can be made available, its cheapness, and the ease with which it can be renewed, added to the improvements introduced by science to obviate its defects, render it in new coun¬ tries, and especially in those where the material is abun¬ dant, such as America, one of the most important auxili¬ aries in civilization. It is to these countries accordingly, and preferably to America, that it is necessary to look for the boldest ex¬ amples of timber-bridge construction. The most simple support for a roadway is a series of longitudinal timbers laid between two piers or abutments. When the span becomes considerable, single beams are insufficient and framed trusses become necessary. The consideration of the case of a single beam involves the principle of a framed truss. The same forces exist in both, the manner of resisting them alone is different. The forces which act on a single beam, when loaded either uniformly or at certain points of its length, have already been investigated (page 123 et seq.)-, but it maybe well to reproduce in this place a summary of the results, and to extend somewhat the consideration of the subject. In any loaded beam, as we have seen, the fibres on the upper side are compressed, while those on the lower side are extended; and within the elastic limits those forces are equal. The intensity of the strain, also, varies directly as the distance of any fibre from the neutral axis. But there is another series of forces which has now to be considered. Whatever be the form of a beam, it is always necessary that the area of cross section at the points of support be sufficient to resist the force tending to crush the fibres in a direction perpendicular to their length. This resistance is proportioned directly to the area, and therefore the dimensions at the point of support must never be less than is obtained by making the resistance per square inch, multiplied by the breadth and depth, equal to the weight, or, in other words, by dividing the weight by the resistance per square inch, to find the area of section. This vertical strain in a loaded beam occurs at all points between the middle and the ends. In the middle point it is almost nothing; at each end it is equal to half the weight of the beam and its load; and at intermediate points, it is proportional to the distance from the middle of the span—a consideration of great importance in bridge construction. In what follows, an endeavour is made to place before the reader succinctly, and in a manner suited to the cha¬ racter of this work, the reasoning of Mr. Haupt on the principles of bridge construction. If the parts of a beam near the neutral axis, which, we have seen, are little strained and oppose but little resist¬ ance, could be removed; and if the same amount of material could be disposed at a greater distance from the axis; the strength and stiffness would be increased in exact propor¬ tion to the distance at which it could be made to act. Hence, in designing a bridge truss, the material, to resist the hori¬ zontal strain, must be placed as far from the neutral axis as the nature of the structure will allow. Suppose to the single beam A B (Fig. 471) we add another CD, and unite them by vertical connections, then it might be supposed that we were doing as above suggested; that is, making a compound beam by disposing the material ad¬ vantageously at the greatest distance from the neutral axis. But it is not so. There are only two beams resisting with their individual strength and stiffness the load, which is increased by the weight of the vertical connections, and they would sink under the pressure into the curve shown by the dotted lines. It is necessary, therefore, to use some means whereby the two beams will act as one, and their flexure under pressure be prevented. This is found in the use of braces, as in the next figure (472); and we shall Fig 472. c a proceed to consider what effect a load would produce on a truss so formed. The load being uniformly distributed, the depression in the case of flexure will be greatest in the middle, and the diagonals of the rectangles ab, c d, will have a tendency to shorten. But, as the braces are incapable of yielding BRIDGES. 159 in the direction of their length, the shortening cannot take place, neither can the flexure. A truss of this de¬ scription, therefore, when properly proportioned, is capa¬ ble of resisting the action of a v/niform load, as in the case of an aqueduct. If the load is not uniformly distributed, the pressures will be found thus:—Let the weight be applied at some point C (Fig. 473), and represented by c p. Now resolve this into its components in the direction CA,CB, and construct the parallelogram pm,co, then c m will represent the strain Fig. 473. a c _ on C B and C o the strain in the direction c A. By trans¬ ferring the force C m to the point b, and resolving it into vertical and horizontal components, the vertical pressure on B will be found equal to C n and that on A equal to n P. That is, the pressures on A and B are directly pro¬ portional to their distance from the place of the applica¬ tion of the load. In the same manner, if the load were at R, it would be discharged by direct lines to A and B. The effect of the oblique force C A acting on R is to force it upwards, and the direction and magnitude of the strain would be the diagonal of a parallelogram constructed on AC, C R. The consequence of this is, that in a truss a weight at one side produces a tendency to rise at the other side, and, therefore, while the diagonals of the loaded side are compressed those of the unloaded side are extended. Hence, while the simple truss shown in the last two figures is perfectly sufficient for a structure uniformly loaded, because the weight on one side is balanced by the weight on the other, it is not sufficient for one subjected to a variable load. For available load, it is therefore necessary either that the braces should be made to resist extension by having iron ties added to them, or that other braces to resist com¬ pression in the opposite direction should be introduced; and thus we obtain a truss composed of four elements, namely, chords A B and c D (Fig. 474), vertical ties ef,g h, Jc m, braces e c, g f g m, Jc D, and counter-braces A /, e h, Jc Jc, B m, or, in place of the latter, tie-rods added to the braces. It has been shown that in any of the parallelograms of such a truss as has been described, the action of a load is to compress the braces a d, a b, and to extend the counter-braces a b, a c. Suppose, (Fig. 475) that the counter-braces have been extended to the length a m, and the braces compressed to an equal extent; then if a wedge be closely fitted into the interval a m, it will neither have any effect on the framing, nor will itself be affected in any way so long as the weight which has pro- \ duced the flexure continues. But on the removal of the weight, the wedge becomes compressed by the effort of the truss to return to its normal condition. This effort is re¬ sisted by the wedge, and there is, consequently, a strain on the counter-brace equal to that which was pro¬ duced by the action of the weight. The effect of the addition of a similar weight, therefore, would be to relieve the strain on the counter¬ brace, without adding anything to the strain on the brace a d. As the vibration of a bridge is caused by its effort to regain its normal form after the change of form produced by a passing load, it is evident that it will be much diminished by counterbracing in the way described. There is thus required for the proper construction of a bridge at least four sets of timbers. First, The horizontal main timbers, called chords. Second, The vertical pieces uniting those, called ties. Third, The main braces. Fourth, The counter-braces; and to these may be added arch-braces, which will be noticed hereafter. In proportioning these several parts, regard must be had to the following considerations :— The chords being unsupported in the intervals between, the ties must be so strong that no sagging or deflection can take place. The braces must be incapable of yielding by lateral bending. Now, the proper proportions of these depend on the distance apart between the ties, and in arranging this, care must be taken to avoid extremes of number on the one hand, and weakness on the other. To trace the effects of a weight through the system of timbers which compose the truss of a bridge, would be a very complex problem. It would, moreover, be beyond the scope of this work, and would exceed what has been assumed as the limit of mathematical knowledge of the majority of its readers. Such investigations, however, have been made, and their results assume the form of maxims of construction. Some of the most useful of these maxims, gleaned from the work of Mr. Haupt, are now presented to the reader; and his investigation of the weights and strains on the timbers and ironwork of the Sherman’s Creek Bridge is given at length in p. 165—168. As the parts of the frame act only in distributing the forces which are applied to it, and, whatever be the incli¬ nation of the braces, the pressure on the abutment and the strain upon the centre of the chords must remain the same, it might be inferred that the degree of inclination of the braces was of little consequence, but such is not the case. For the braces must not be so long as to yield by lateral flexure, and the chords must be supported at such intervals that no injurious flexure shall be produced by the passage of a load. Again, as the ties approach each other the angle of the brace increases; and the number of ties and braces, and consequently the weight of the structure, is increased. When the maximum load and the size of the chords are known, the limit of the intervals can be determined, by considering the portion of the chord between every two ties, as in the condition of a beam supported at both ends and loaded in the middle. 160 PRACTICAL CARPENTRY AND JOINERY. It has been shown that the strain upon the counter- | braces is not increased by the passing of a load; and that ' by driving a wedge at the joint of the counter-brace, a | strain can be permanently thrown on the brace, equal to that which would result from the passage of the maxi¬ mum load. It will therefore be safe to calculate the strength of the counter-brace by the condition that it shall produce the required compression on the brace. This strain upon the counter-brace is equal to the pres¬ sure on the brace. Hence, as any accidental load that can I ever act at a single point is small when compared with | the uniform load, and it is to give the required immo- j bility under the accidental load that the counterbracing is used, it follows that the counter-braces may be very small as compared with the other timbers. It has also been shown that in a single beam supported at the ends and uniformly loaded, there exist horizontal and vertical forces at every point except the middle and the extremities. At the middle, the strains are altogether horizontal; but at other points, the distances to the extre¬ mities being unequal, the horizontal strains no longer balance each other, and the difference must be compen¬ sated by a cross strain on the fibres. This vertical force, at zero at the centre, increases to¬ wards the extremities, where it is equal to one-half the whole uniform weight, and in this case the increase is proportional to the distance from the centre. Hence it follows that the ties and braces which resist the vertical forces may be smaller at the centre than at the abut¬ ments. Each successive brace, therefore, as it recedes from the centre, should theoretically be increased in size; but as this adds greatly to the trouble and expense of framing, it is better in practice to make them uniform in size, and to compensate for the additional strains at the ends by adding other braces called arch-braces. In the construction of a bridge with arch-braces, the simplest plan is to depend upon the latter to sustain the weight of the structure, having only a light truss with counter-braces or diagonal ties to give connection and stiff¬ ness to the various parts, and to resist the action of vari¬ able loads. Instead of arch-braces, arches are sometimes used, which are beneficial, and produce somewhat the same effects as the arch-braces. An arch of timber cannot alone be depended on to sus¬ tain a variable load; it requires always to be connected with a system of trussing to give it the necessary stiffness, j The importance of having a beam, continuous over its supports has been pointed out; and equally great advan¬ tages also accrue from having a bridge in which there are several spans in succession connected over the supports so as to be made continuous. The strength of the chords in the central span of a series would be double that of the same span disconnected; and the extreme spans would be stronger, in the ratio of three to two, than if discon¬ nected. In applying these results to practice, it is necessary first to determine the weight of the bridge and its load. The weight of the bridge is found by preparing a bill of timbers of assumed dimensions, and multiplying the num¬ ber of cubic feet by the weight of a cubic foot of the material, which may be taken on an average at 35 lbs. The average quantity of material in the Howe bridges on the Philadelphia railroad is about 30 cubic feet per foot lineal; and therefore this may be assumed as a guide in calculations. The greatest load a bridge can sustain would consist of locomotive engines, which would give 1 ton per foot lineal of the bridge. Hence 1 ton per foot for the load, and half a ton per foot for the weight of the structure, may be assumed as a maxi¬ mum load when the span does not exceed 200 feet. The safe strain for timber will be considered as 1000 lbs. per square inch, and for iron ten times as much. To find the strain on the chords. 1. The strain of compression on the upper chord .— Multiply half the weight of the bridge by the distance of the centre of gravity from the abutmeut (which is nearly a quarter of the span), and divide the product by the height of the truss, measured from the centres of the upper and lower chords. Example. — Let the span be 160 feet, and the height 17 feet; required the cross section of the upper chord in the centre. The weight at 1^ ton per foot is 480,000 lbs.; and as¬ suming the depth of the chords at 12 inches, the distance from centre to centre will be 16 feet. mi r , n , 480,000 x 160 _... then, according to the rule, -—— ?v> --= 600,000, as the maximum strain at the centre; which, divided by 1000 lbs., as the resistance per square inch, gives 600 square inches of section; which, divided by 12 inches, the depth of the chord, gives a total breadth of 50 inches, or 25 inches to each truss, if there are two trusses. 2. The strain of tension on the lower chord at the centre. —This is equal to the compressive strain on the upper chord; but from the occurrence of joints, the power of resistance is diminished. To compensate this, the quan¬ tity of material is increased to such an extent that the resisting area shall be obtained exclusive of the timber in which the joint occurs. A good practical way of doing this is to make the upper chord of three, and the lower chord of four timbers to each truss; and if a joint then should occur in each panel, each piece of timber should be equal to the leugth of four panels; and three of the four timbers should therefore contain sufficient resisting area for the whole strain. 3. Strain at the ends of the chords. —In a bridge of a single span, the horizontal strain at the end of the brace nearest the abutment will equal the weight on the brace, multiplied by the co-tangent of its inclination. If the inclination be 45°, the horizontal strain will be equal to the vertical weight. If the angle with the horizontal line is greater than 45°, the strain will be less than the weight. As this is generally the case, it is safe to assume the hori¬ zontal strain at the end of the chord as equal to the ver¬ tical force acting on the first brace. This vertical force is half of the whole weight of 12 feet wide; and it is proper to take the strain on the whole panel as the minimum strain. The weight on one panel is 36,000 pounds; requiring a cross section of 36 inches, or 9 inches to a tie, and 1J | to a brace. This cross sec¬ tion, for a brace of 20 feet long, is so small that it would yield with lateral flexure; and recourse must therefore be had to the formula for long posts, unless the braces are supported at the middle of their length. BRIDGES. 161 The dimensions at the ends and at the centre having been obtained, the intermediate timbers should increase from the former towards the latter by regular additions. 4. The strain on the Counter-braces .—It has been shown that the strain on any counter-brace is equal to that produced by the action of a variable load on the corre¬ sponding brace. It will consequently be equal to the strain on the braces of the middle panel; and if each panel contains two braces and one counter-brace, the size of the latter should be uniform, and equal to 30 square inches, in the truss of the dimensions assumed. Hence, if supported in the middle, the counter-braces should be 6x5 inches. 5. Horizontal Ties and Braces at top and bottom .—The use of these is to give lateral stiffness to the bridge, and guard against the effects of the wind, which is the greatest disturbing cause. Assuming the force of the wind to be 15 lbs. per square foot, and the truss to be close boarded, and its height 18 feet, the total force over the surface would be 43,200 lbs. or 21,600 lbs. to each series of braces at the top and bottom of the bridge. If the calculation, therefore, is continued with the same dimensions as before, the half weight will be 240,000 lbs., and the cross section to resist it 240 square inches, or little more than one-third of the dimensions at the centre. The dimensions at the ends and at the centre having been obtained, a uniform increase between the points can be made. The size of the chords might be deduced from the for¬ mula applicable to a beam supported at both ends and loaded in the middle, in respect of that portion of them which lies between any two posts; but as the dimensions determined in this way are smaller, the rule already given is safer; and the excess of size is amply sufficient to resist the additional cross strain from any passing load. 6 . The strain on the Ties and Braces .—The end braces which project from the abutments bear the whole of the load, and there is a decrease of strain to the eentre. The weight of the bridge, as before, is 480,000 lbs., or 240,000 lbs. at each end; which, at 1000 lbs. per square inch, is 240 square inches of section for the ties if of wood, and 24 inches if of iron. If the panels be 12 feet wide, and the height, as before, 16 feet, the length of the diagonal or brace will be 20 feet, and the strain on it will be 240,000 X 20 _ oqo 000 lbs. The section, therefore, will 16 require to contain 300 square inches, which, divided among four braces, gives 75 square inches to each. The strain at the middle is theoretically nothing, but in practice it is the same as that on the panel, on the above assumption. This would be estimated as the strain on a bridge produced by a uniform load; and if the bracing is in squares, the diagonals will be to the sides as 1 *4 :1 nearty. The strain on the end braces will be —x 1 4 = 15,120, which, at 1000 lbs. per inch, gives only 1512 square inches to resist the strain. Practically, the end braces in this case might be 5 X 4 inches, and those at the middle of the span very much lighter. In the middle panel, they might even be omitted without injury. Diagonal Braces and Knee-braces .—When the road¬ way is on the top of the truss, braces occupying the direc¬ tion of diagonals to the cross section of the truss can be used to prevent side motion; but where the roadway is on the bottom, knee-braces must be used. Experience has shown that scantlings 7x5 are large enough for diagonal braces, and 6x5 for knee-braces. Floor Beams .—The formula for these is the same as in the case of other floors. The case assumed by Mr. Haupt for the same truss is as follows:—Length of floor- beam between supports 14 feet, depth of same 14 inches, greatest load equal to 6 tons, applied at the centre; re¬ quired the breadth, so that the deflection shall not exceed one-fortieth of an inch to a foot. w l' 2 x ‘0125 : W 5 or, substituting the figures, 13,440 x 14 2 x'0125 . — = 12 inches. 14 Timber Arches.— Mr. Haupt considers that the usual course of making a truss sufficiently strong to resist the weight, and then adding arches as greater security, should be reversed; and that the arches should be made the main dependence, and a light truss be used in combination with them, to prevent change of form, and to give the proper support to the roadway. He assumes, for the sake of illus¬ tration, the same data as before, namely, the span at 160 feet, the rise of the arches, four in number, 20 feet, and the weight on the bridge 1 ^ ton per foot. The weight is then 268,800 lbs. to each half of the bridge, and the strain on the arches in the centre is 268,800X40 not? cr\n it •• ko.-/.- i e —-- = 537,600 lbs., requiring 537 o inches of cross section. Four arches 16 inches deep and 8'4 inches wide could supply the amount of material. The compression at the ends will be to that at the centre as 40 2 + 20* : 40, or as 2' 2 + 1 : 2; hence it will be 537,600 x linearly = 601,055 lbs., and will require 601 square inches of section; therefore, if the arch is 8’4 inches wide as before, its depth must be 18 inches nearly. As in this case the whole of the weight is sustained by the arch, and the truss is used only to stiffen it and carry the roadway, the braces have no more strain at the ends than at the centre; and the principle of proportioning them in arithmetical progression from the centre to the end is no longer applicable. With scarcely an exception, the examples of bridges contained in Plates XLVIII. and LYI. maybe resolved into the following element¬ ary figures:— 1. Trusses which are below the roadway, and which depend for their stability on the abut¬ ments. Of these, Fig. 476 is the type, and the illustrations will be found in Figs. 3, 4, 6 , and 7, Plate XLVIII.; Figs. 1 and 9, Plate XLIX.; Fig. 1, Plate LII.; Fig. 1, Plate LIII., and Fig. 1, Plate LV. 2. Trusses which are above the roadway, and have only vertical pres¬ sure. Of these, Fig. 477 is the type, and the illustration will be found in Plate L. 3. Trusses below the roadway, composed of timber x Fig. 476. PRACTICAL CARPENTRY AND JOINERY. i 012 arches, and ties and braces, but dependent on the abut¬ ments for resistance to lateral thrust, the type of which is Fig. 4 ( 8. irig 478 _ 4. Trusses above or below the roadway, com¬ posed of timber arches, and ties and braces, and which have only vertical pressure, the type of which is Fig. 479, and the illus¬ trations Figs. 1, 5, and 12, Plate LIY. 5. Lattice trusses above yj g . 479. the roadway, the illus¬ tration of which is Fig. 8, Plate LIY. The foregoing analysis will enable the reader to find at a glance the illustration he requires, and with this key, the plates will now be described in their order. Plate XLVIII. — Fig. 1 is the elevation of a timber draw-bridge on the Gotha Canal, Sweden. Fig. 2. Plan of the draw-bridge. In this, part of the tloor timbers is removed, in order to show the framing, and the position of the rack for moving the bridge. Fig. 3 is the elevation of the simplest form of trussing for a bridge, when the roadway is above the truss, and the abutments are sufficiently strong to resist the lateral thrust. A chord, B strut, c straining-piece. Examples of the same kind of truss, applied in works recently executed, will be found in Plate XL1X., Figs. 9, 10, and 11, and Plate LV., Fig. 1. Fig. 4. Elevation of a bridge truss, in which the chord- piece, A A, is in two lengths, joined in the middle of the span, and the fence of the roadway is used as an auxiliary truss, with a king-bolt, B. This would be improved in sustaining a variable load by introducing counter-bracing. Fig. 5. The elevation of a bridge truss, composed of chord piece A, laminated arch B, struts and straining- piece c c, and cross-pieces D, connecting the framing. Further illustrations of the same principle in detail will be found in Plate LI., Figs. 1 and 6, in Plate LIII, Fig. 3, and in Plate LVI., Fig. I. Fig. G shows the elevation of a timber bridge of 34 feet span, with chord-piece A, two sets of struts c c, strain¬ ing-piece B, and suspending-pieces D. Illustrations of the extended application of this prin¬ ciple of framing will be found in the next figure in this .plate, in Plate XLIX., Fig. 1, Plate LIT., Plate LIII., Fig. 1, and Plate LV., Fig. 1. Fig. 7. In this truss there are chord-pieces c, straining- pieces D d, struts E F G, counter-struts, or radial posts, ir, transverse connecting pieces M M, radial straps a b, a b, cushion and straining-pieces to the under side of the truss K L, and cast-iron shoes, c, fixed to the abutment to receive the ends of the struts. The joints of the struts, and between the struts and straining-pieces, are secured by straps e, d. Although from the contour of the under side of this truss, and the radiating posts and straps, it assumes the form of an arch, it has none of its characteristics: it is simply a truss of the same nature as those belonging to the class illustrated in Fig. G; and the material is not so well dis¬ posed as in the examples of that class above referred to, and especially as in the following example. Plate XLIX.— Figs. 1 to 8 showthe elevation, plan, and J details of construction, of a timber bridge erected over the Spey at Laggan Kirk, by Telford. In this example, the material is employed most judiciously to obtain the greatest result by the smallest means; and the details, as in all the works of this excellent engineer, are worthy of careful study. The work belongs to Class No. 2 of the generalization. As the dimensions of the timbers are figured on the drawings, and the plan, elevation, and sec¬ tion explain themselves, it is only necessary to note that Figs. 4, 5, and G represent the details of the cast-iron sockets which serve to unite the struts to the straining- pieces, and Figs. 7 and 8, the details of the cast-iron shoe attached to the abutments to receive the ends of the struts. Fig. 9 is a side elevation, Fig. 10 a transverse section, and Fig. 11 the plan, of part of a timber bridge, belonging to Class No. 1. In this, there are the usual chord-pieces A, straining- pieces B, struts e. The piers are piles D d d, and are connected by two rows of waling timbers—one, g g, under the chord-pieces, and the other under the abutment of the struts. The chord-pieces are further secured to the heads of the piles by straps c d, seen in Fig. 11. Fig. 10 is a section on the line L K of Figs. 9 and 11. Plate L —Timber bridge over the river Don, at Iu- verury, on the Great North of Scotland Railway. This bridge belongs to the second class. It consists of 10 bays, or spans, four of which span the ordinary bed of the river Don, one spans a mill-lead, and five are land or flood openings. It is situated about 200 } r ards above the confluence of the rivers Don and Ury, crossing the former at a consider¬ able angle, adjacent to a great bend immediately above in the course of the river, and standing at the south end of a considerable flat, or haugh, which is liable to be flooded. The bays are spanned by ordinary queen post trusses: this form being adopted in order to obtain the greatest height and clear water-way at the least expense. In 1829 happened the greatest flood on record in Aber¬ deenshire: both the above rivers were swollen to an enor¬ mous extent; the level of the water, throughout the haugh referred to and on the site of the bridge, being within 2 feet 3 inches of the level of the under side of the tie-beams ot the trusses. It was therefore necessary for the engineers to provide sufficient water-way for a similar flood; and after the most careful investigations of the history of that flood, the design of construction, and extent of openings of the viaduct, as now executed, were determined on. It is scarcely necessary to give a detailed description of the parts, as the accompanying drawings fully explain them¬ selves ; but a short general statement may not be outof place. All the timber used in the structure is of the best Memel. The piles and braces of piers are 12" x 12" scantling. The head-pieces of the piers 12" X 9". The tie-beams of the trusses 12" x 9". The struts, straining-beams, and queen-posts of the trusses 12" x 12", and the diagonals 12" x 8". All the wrought-iron work of the bolts and straps is of B.B. crown Staffordshire iron. The toes of the pier tim¬ bers are shod with Staffordshire boiler-plate; and the caps over the queen-posts, the shoes under the tie-beams, and the strap bars at the springing of the trusses, are of ordi¬ nary gray cast iron. Preparations for the erection of the viaduct were com¬ menced in the autumn of 1853, by the construction of a FIATE I. sbg&decdes ato ©pijmESo .Inis: J in ,<. Si-nr Bridge over Sue River FortG-lJVorth.crf Scotland Ttathruy BY B. & El. B LYTH C. E$ 6 told. Centring ot BaUnrJanyU Viaduct. /das qow and South. Western Mcdbray. BY JOHN MILLER C. E. Fig. 1 WATeerer. Sc, IV. ('. fonss Dd. 'cole to Figs. 1.2.6. ■ Fig. 10. Fig. 6 BRIDGES. temporary bridge across the Don, parallel to, and above the site of the viaduct, with side gangways at each pier for carrying the piling engines. The piles were driven to varying depths, as shown in Fig. 1. They were shod with ordinary four-armed shoes weighing 20 lbs. each, and driven in till a perfectly firm rest was obtained. The bed of the river consisted of large loose gravel and stones, lying on an under stratum of strong dark-coloured clay, into which the piles were driven several feet. The water piers consist of seven piles, with longitudinal braces, diagonals, and cap-cills, arranged in such manner as to insure rigidity and strength. The distance between the extreme piles of each pier is 45 feet, tnis distance be¬ ing considered necessary to resist the shocks given on the breaking up of frost in winter, when the river, swelling to a great height, hurls along with it enormous quantities of ice. Lumps of ice, weighing from 5 to 8 or 10 tons, have been seen shooting through this bridge with the velocity of seven miles an hour; and, in severe winters, sheets of ice half a mile in length frequently pass through. The piers are all covered to a foot below the summer level of the. water with 3-inch planks, spiked at the crossing of each timber. Filling-in pieces, 1 2" x 6 ", are introduced at the centre pile to receive this covering. Toe-pieces, 12" X 6", triangular in form, are faced on the outer piles; and these, and the ends of the covering, are further protected by sheets of boiler plate, 5 feet long, firmly bolted and spiked. Below the planking, the inter¬ vals between the piles are filled in with rough pitching of large rubble stones, and their exterior is surrounded with the same material. All joinings of the timbers of the piers are secured by straps on each side 3" wide by f" thick, bolted toge¬ ther with f-inch diameter round bolts: where timbers cross each other, round bolts 1 ^-inch diameter are used. The ends of the timbers are tenoned into mortises formed in adjoining beams. The diagonals of the piers are placed so as to form supports under the ends of the struts of the trusses. Each truss consists of two parallel tie-beams, queen- posts, struts, straining-beam, and diagonals. The tie- beams are placed 9" apart, and the queen-posts and struts pass through between them, having shoulders I 3 inches deep on each side. The straining-beams, struts, queen- posts, and diagonals are all mortised and tenoned together, and fastened with straps, 3" X with £" bolts. A round bolt 1 f" diameter is passed through each queen-post and its cast-iron cap and shoe, and being firmly screwed up with nut and jam-nut, the whole truss is braced together, so as to give a slight camber to the tie-beam. The struts of the adjoining trusses abut against each other, the abutment being about 8 inches deep. They are secured by straps, 3" X passed round the ties and screwed at the ends, to receive the cross cast-iron bars. The straps are increased in size in the centre to receive bolts which pass through the ties and strut. The tie-beams are scarfed over the piers, and secured by bolts. The roadway consists of transverse beams, laid 4 feet apart from centre to centre. The beams are 18 feet long, and 12 x 10 inches scantling: they are supported by the trusses, which are 14 feet apart, from centre to centre. 1 03 Each beam is fastened by bolts, 1 inch diameter, to one of the tie-beams at each end; and the whole surface be¬ tween the trusses is covered with 3" planking, on which the railway chairs are laid, and spiked through to the transverse beams. The piles, the covering of the piers, and the roadway, are payed over with two coats of Archangel tar and pitch at a boiling heat; and the remainder of the timber work is painted three coats in white lead and oil. All the timber is cleaned except the planking, which is rough from the saw. The iron work was painted first one coat in red lead, and then two coats in black lead and oil. During the winter of 1853-54, the temporary bridge was carried away by the breaking up of the ice; and two of the permanent piles, which were not connected with the others, and therefore unsupported, were broken over: beyond this no casualty occurred, nor was any special difficulty incurred in the execution of the viaduct. The railway was opened in September, 1854; and though it has been severely tested by both summer and winter floods, the whole structure still continues in excellent re¬ pair. The engineers, however, thought it advisable to pro¬ vide against extraordinary cases, and erected ice-fenders in front of each water-pier. These ice-fenders consist of three piles, braced, covered wiU> planking, and faced with boiler-plate, similar to the end part of the water-piers. They are placed three feet clear of the piers, so as to receive any shock without allowing the main structure to participate in it. These ice-fenders are not shown on the drawing. Fig. 1 is a general elevation, and Fig. 2 a general plan. Fig. 3, one of the bays drawn to a larger scale, with the dimensions of the timbers figured. Fig. 4, transverse section through the bridge. Fig. 5, plan of the bay shown in elevation in Fig. 3. Figs. 6 , 7, 8 , 9 and 10 on this plate will be described in the section on “ Centres.” Plate LI.—The figures on this plate illustrate Class 3. They are designs by Mr. White for railway and road bridges Fig. 1 is an elevation of the framing of a railway bridge. It is a combination of a laminated arch with light masonry. C C is the laminated arch, abutting at the ends on iron plates, e e. F, the chord. E, the straining-piece. K N o, struts. r s, R s, radial posts or braces, with double iron straps, and cross-pieces at R and s. M L, M l, braces and counter-braces. G, a continuation in timber of the torus-moulding of the piers. H, the railing, or fence, which is braced and counter- braced, and so connected to the other framing by bolts (seen in Fig. 4) as to add strength and stiffness to the structure. Fig. 2, a plan of part of the structure, which shows that it is composed of six parallel trusses, 000 , like that shown in Fig. 1, united by transverse bracing, q q, at each line of radial posts, and stiffened by horizontal braces, w w. At R R are seen the plates and cross-pieces connect¬ ing the straps of the radial posts. 164 PRACTICAL CARPENTRY AND JOINERY. z shows a portion of the planking of the floor. Fir/. 3 is a transverse section of the bridge, on the line A B of Fig. 1, in which c c c are the arches, e e e e the abutment-plate, and cl the cross-piece and transverse- bracing. f, the chords. e, the straining-pieces. G, the torus. ii, the fence, p the floor, / the railway sleeper, and g the guide or guard piece. Fig. 4 is an enlarged section, showing in greater detail c the arch, EFOG the horizontal timbers above it, and H the railing, q the cross timbers, p the floor, cl x y the transverse ties and braces, and t a transverse iron tie at every radial post, bolted to the chords. Fig. 5, an enlarged section, showing f o the chords, and the manner in which the}' are embraced by the straps and cross-pieces of the radial posts, p the floor, / the sleeper and rail, and g the guide-piece. Fig. G is the elevation of a road bridge belonging to the same class as last example. It consists of two lami¬ nated arches in combination with straight trussing. c c', C c' are the laminated arches, with radial posts, K s, and braces and counter-braces, c and cl. F is the chord. E, the straining-piece. . K, the principal struts. OOO, vertical posts, with struts or braces, N N, and counter-braces. G is the timber continuation of the moulding of the piers. H, the fence, consisting, as in the last example, of light framing of posts, rails, braces, and counter-braces. Fig. 7 is a portion of the plan of the structure, show¬ ing the upper side of the trusses o o o o, the floor-timbers q q, the planking z, the horizontal braces iu w, and the cross-bracing at each line of radial posts. Fig. 8, a transverse section of the bridge on the line A B, Fig. 6, in which c C are the laminated arches, E the straining-piece, f the chord, h the fence, e e the abutment- plate, of cast-iron, s the carriage-way, and r r the foot¬ paths of the road. Fig. 9 is a section of one of the frames to a larger scale. c, the lower arch. c', the upper arch, c d the brace and counter-brace, halved at their intersection. E, the straining-piece. f, the chord. II, the fence. x y, the cross-bracing. t, an iron tie connecting the upper part of all the trusses. q, the floor-timbers. z, the planking. G, wooden moulding. Fig. 10 is the plan of the top of the straps which unite the arches to the truss, and Fig. 11, the side elevation of the same. Plate LI I.— Elevation, Plan, Section, and Details of the Timber Bridge over the River Tyne, at Linton, North British Railway. —This bridge was originally constructed entirely of freestone. Between the abutments, from which the struts spring, there were two stone arches, with a pier in the centre of the river; but, owing to the insufficiency of the foundation of this pier, it was swept away by a flood of unprecedented magnitude in September, 1816, after the opening of the North British Railway: the abutments, however, remained uninjured. It became a matter of importance to repair this accident as soon as possible, to admit of the passage 'of trains; and with this view it was determined to erect, between the abutments of the arches which had fallen, the timber bridge shown on the draw¬ ings. This erection was put up very rapidly, owing to the simplicity of its construction; and it has been found to answer its purposes perfectly. The drawings, in them¬ selves, are so complete as to render any detailed account of the structure unnecessary. Fig. 1. Elevation cf the bridge, the opening of which is 90 feet wide. The dimensions of all the parts are figured on the drawings, and the construction is shown in detail in Fig 4, Nos. 1 and 2. The bridge belongs to the third class in our enumera¬ tion. It consists of a built or laminated chord consisting of five planks, each 12x3, as seen in section, Fig. 4, No. 2. Immediately beneath this is a straining-piece, 12 X 6 inches, and 72 feet long; and struts c, 12 x 9. Under this a series of eight transverse timbers serve to unite all the frames. The next straining-piece is 12 X 6, and 54 feet 5 inches long; and the struts c are 12x9 inches. Under this is another series of six transverse pieces. The next straining-piece is 12 x 6, and 36 feet 7 inches long. This has a series of four transverse pieces separating it from the next and lowest straining-piece A, which is 12 x 12, its struts also being 12 x 12. The posts, E E, are 12X12 inches, and their straps are 3;| x | inch. The manner of securing the straps is shown in detail in Figs. 5 and 6. The lower ends of the struts are housed in radiating cast-iron shoe pieces, attached to a cast-iron abutment-plate, seen at F in Figs. 1 and 3, and at D in Fig*. 2. Fig. 2 is a plan of the structure, showing A, the truss frames; C, the transverse pieces; B, the horizontal braces; D, the abutment-plate; E, an upper plate of iron connect¬ ing all the trusses, seen also at G in Fig. 3, and in Fig. 4, No. 2; and F F, the rails. Fig. 3 is a transverse section on the line A b, Fig, 1. G is the upper connecting plate, and F the abutment- plate. Fig. 4. No. 1 is an enlarged elevation, and No. 2 an enlarged section of the truss and fence rail. A, the lowest straining-piece; BB, straining-pieces; CC, transverse pieces; f the upper straining-piece; g, the built or laminated chord; E E, straps. The fence railing is framed with rails K, and capping piece i, posts G, and braces and counter - braces, H. A bolt, eg f, No. 2, passes through the upper rail K, the post G, and the chord and uppermost straining-piece, and thus unites the truss and the fence railing firmly together. Figs. 5 and 6 show details of the head of the strap e, already referred to. Plate LIII.— Figs. 1 and 2 show the elevation and plan of a skew bridge, of Class No. 2, designed by Mr. White. The scantlings may be nearly the same as those in the last example. Fig. 1 is the elevation. Each frame is composed of a BRIDGES. 1(35 chord N, and straining-pieces L H F, and struts A B C, D E, G, k, and M; posts and straps ab,a 6 , a6; and transverse pieces opr connect all the frames. Fig. 2 is the plan showing the trusses A A, six in number, the connecting transverse pieces o p R s, and the horizontal braces, 1 2 3 4 5 6 7. The ends of the chords rest on a cast- iron wall-plate w w. Figs. 3 and 4 show another design, by Mr. White, for a bridge composed of the combination of a laminated arch with the ordinary straight truss. In the plan, it will be seen, there are no floor beams, but in their place planking close jointed, on which the floor¬ ing planks are laid, so as to cross them at right angles. In the elevation, Fig. 3, there is shown a double chord F E, a straining-piece D, struts BCG, and posts and straps ab, ab, a b. Transverse pieces at c and / connect all the frames A A A to the laminated arches. In Fig. 4, the plan, A A are the trusses, six in number; cj ‘ the transverse pieces; R, the iron wall-plate; 1 2 3 4 5, the horizontal braces. Plate LIY.— American Bridges: Fig. 1 is the elevation of a bridge belonging to Class 4. Fig. 2, part of the plan of the same. Figs. 3 and 4, the detail to a larger scale of the tension-posts, braces, and counter-braces, upper and lower chords, and their iron fastenings. In this mode of combining an arch with a trussed frame, the arches are connected with the tension-posts, and the posts with the chords, by screw fastenings, as seen in Fig. 4; and all is so arranged as to admit of changing the position of the arches relatively to the chords, or of draw¬ ing together the chords without changing the position of the arches, and thus regulating and distributing the strain over the different parts of the bridge at pleasure. The following are the directions of Mr. Steele, the patentee of this mode of construction, to be attended to in the erection of one of his bridges, extracted from Haupt's Treatise on Bridges :— The truss must first be erected, provided with suitable cast-iron skew-backs to receive the braces and tension- posts; and the several parts of the chords should be con¬ nected with cast-iron gibs. Wedging under the counter¬ braces must be avoided, by extending the distance between the top skew-backs sufficiently to bring the tension-posts on the radii of the curve of camber of the bridge. The tension-posts must be about eight inches shorter than the distance between the chords; and, in screwing up the truss, care must be taken not to bring their ends in contact with the chords; but they must be equidistant, and about four inches from them. When the truss is thus finished, it must be thrown on its final bearings; and it is then ready to receive the arches, which should be constructed on the curve of the parabola, with the ordinates so calculated as to be measured along the central line of the tension-posts. They must be firmly fastened to the posts and bottom chords by means of strong screw-bolts and connecting plates, as shown at d d, and should foot on the masonry some distance below the truss, which can be done with safety, as the attachment to the posts and chords will re¬ lieve the masonry of much of their horizontal thrust. When a bridge so constructed is put into use, it will be found, as the timber becomes seasoned, that the weight will be gradually thrown upon the arches, which will ultimately bear an undue portion of the load. To avoid this, the camber must be restored, and the posts moved up, so as ao-ain to divide the strain between the truss and the arches. This adjustment must take place once or twice in each year, until the timber becomes perfectly seasoned; after which, in a well-constructed bridge, but little attention will be required.. Plates of iron should in all cases be in¬ troduced between the abutting surfaces of the top chords and arches; and all possible care taken to prevent two pieces of timber from coming in contact, by which decay is hastened : care should also be taken to obtain the curve of the parabola for the arches; as it is the curve of equi¬ librium, and of greatest strength, as has been shown by experiment.* Bridges constructed on this plan, will be found to pos¬ sess an unusual amount of strength for the quantity ot material contained in them; and, if well built and protected, great durability. Fig. 5. Elevation of a truss of Class 4. Fig. 6. Plan of the same. Fig. 7. Vertical section through the centre of the bridge. The bridge is in two spans, each 148 feet 3 inches from skew-back to skew-back, or 154 feet 6 inches from the middle of the pier to the end of the truss. The pier is 3 feet 2 inches wide on the top, and 6 feet at the skew-backs. The truss consists of three rows of top and bottom chords, and two sets of posts and braces. It is counter- braced by rods of inch iron between the braces. The panels increase in width from the end towards the middle of the span. The first are 9 feet If inches from centre to centre of posts, and the middle ones 12 feet 1^ inches. The quantity of materials in this bridge is given by Mr. Haupt as lofiows:— Timber for one Span. 3 wall-plates. 8 X 16 18 feet long B.M. 576 20 chords . 6 X 13 36 „ „ 4,680 10 „ . 8 X 13 36 „ „ 3,120 10 „ . 8 X 10 36 „ „ 2,400 20 * . 6 X 10 36 „ 3,600 56 posts, yellow pine 9 X 12 23 „ „ 11,592 4 king-posts. 9 X 16 23 „ „ 1,104 15 floor-beams ... 8 X 14 18 „ „ 2,520 14 „ . 7 X 14 18 2,058 56 lateral braces... 4£ X 7 8* » 1,213 3 „ „ 4i X 7 13 „ 103 30 roof-braces ... 4 X 5 17 „ 850 56 check-bi'aces 9 X 20 3 „ „ 2,520 56 „ 9 X 23 3 „ 2,898 60 main-braces 6 X 9 19 >, 5,130 15 tie-beams 8 X 10 19 „ „ 1,900 8 purlins . 4 X 6 20 „ „ 320 135 rafters . 3 X 5 10 \ „ „ 1,772 15 roof-posts . 4 X 6 3 „ „ 75 30 knee-braces ... 5 X 5 6 „ „ 312 16 track stringers ... 8 X 10 20 „ „ 2,133 3300 feet B.M. £-inch sheeting for roof 3,300 56 arch-pieces ... 9 X 11 25 „ 11,550 7000 feet B.M. inch boards, 20 feet long . for weather-boarding, 7,000 "Weight per lineal foot, 1416 pounds. 72,726 No. of cubic feet per foot lineal, 40. * The parabola is the curve of equilibrium when no load is upon the bridge, and also when the load is uniform; but there can be no curve of equilibrium for the variable load of a passing train. Stiffness can be secured in this case, only by an efficient system of counter¬ bracing. The plan proposed fulfils every condition of a good bridge. 106 PRACTICAL CARPENTRY AND JOINERY. Counter-brace Rods for one Span. 4 rods for 1st panels, each 24 't. 3 in. long 1 in. diam. 97'0 ft. 4 „ 2d „ „ 24 2 5) 1 „ 97-0 4 3d ,, „ 24 „ 8 1 „ 987 4 4th „ „ 25 ,. 0 , 1 „ 100-0 4 5 th „ 25 „ 2 1 „ 100-7 4 6th „ 25 „ 8 1 „ 102-7 4 7th or mid pan. 26 „ 0 ,, 1 1 „ 104-0 Total lineal feet 700T Weight in pounds at 2‘65 per foot = 1855 pounds. Arch Suspension Rods for one Span. 4 rods, each 6 feet 8 inches') 8 8 8 8 10 „ 13 „ 15 feet 6 inches 17 » 2 „ 'r inch diameter. 8 „ 18 „ 2 Total length 322 feet. Weight at 4 I ° 0 ' 1 o lbs. per foot, 1590 pounds. Lateral Brace Rods for one Span. 15 rods, each 16 feet 9 inches long, 1 inch diameter, 655 pounds. Small Bolts for one Span. 60 bolts, through arches, 47 inches long, 1 inch diam., 622 lbs. 60 bolts, through chords and posts, 34 inches long, | inch diam. 255 30 roof-bolts, 36 inches long, | inch diam. 135 224 spikes for braces, f pound each. 168 Mr. Haupt lias given ail analysis of tile strains upon this bridge; and as it affords an example of his process of working out the problem of the stability of a timber bridge of this kind, the whole is here extracted:— Dimensions and Data for Calculation of Brida e at Sherman'. Creek. Span at skew-backs .^ 148 ft. 3 in. Whole length of truss for one span. 154 ,, Out to out of chords . 20 „ Middle to middle of chords . 19 „ Resisting cross-section of upper chords 400 so. in. Resisting cross-section of 6 lower chords, de¬ ductions for splice,, check-brace, and bolt, and allowing for scarT-key. 280 sq. in. Versed sine of lower arch. 20 feet. Cross-section of 8 arches . 800 sq. in. Span 148b and rise 20, will give radius... 172 25 feet. And 172-25, 152 - 25, and 74’125, express the proportion of the hypotheuuse, perpendicu¬ lar, and base of skew-back. Hypothenuse of skew-back covered by arches 18 inches. Perpendicular „ „ „ 16 „ Base „ „ „ 7"6 „ Distance from skew-back to bottom of arch 4i „ „ „ middle of skew-back to middle of chord . 4 ft. 5 in. Width from out to out of chords . 16 „ 2 „ „ between chords in the clear . 11 >, distance from centre to centre of door-beams 51 feet. Weight of one-half span complete (77 feet) 120,000 lbs. Distance of centre of gravity from point of support. 37 feet. Weight of one-half span with load. 275,000 lbs. Distance between shoulder of post . 151 feet. Calculation of Truss ivilhout the Arches* Let x = distance of neutral axis from top chord. 19 — x — distance of ditto from bottom chord. P = pressure per square inch on top chord. * The reader will readily discover that Mr. Haupt, in general, URes the approximative round numbers in his calculations, for the sake of simplicity. -- (19 — .t) = strain per square inch on bottom chord. x — 83 = distance of neutral axis from top chord. 19 — x = 10 - 7 = distance of do. from bottom chord. P= 1532lbs. = pressure per square inch on top chord. (1532-i-8'3) X (19—83) = 1975 lbs.=strain per square inch on bottom chord. P 400 P x = 280 —(19 — x)'\ 400 P X 8-3 + 280 P x x 10 7 = 275,000 x 37. The bottom chords derive some assistance from the ma¬ sonry; but as the roadway is on the bottom of the truss, little opportunity is given for wedging the lower chords; and for this reason the assistance to be derived from this service is not estimated. Ties and Braces. The weight upon the middle panel (12^ lineal feet) is 45,000 lbs. To resist this there are four posts, the cross- section of each being 72 square inches, or the united cross- section 288, equivalent to 156 25 lbs. per square inch. The distance between the shoulders of the posts being 15i feet, and the width of the middle panel, exclusive of posts, 11^ feet, the diagonal will be 19 3. 193 The strain upon the diagonal will be 45,000 X j >-;> = 56,000 lbs., which divided by the cross-section of the four , , , . , 56,000 braces, aviII make the pressure per square inch -g—~ = 260 lbs. The expression for the limit of the resistance to flexure, , 9000 BD* . ,, , 9000x9x6' { w = 4 x- j7> -, gives the present case w= - pyip - = 46,000 pounds, or for The four braces. 184,000 pounds. The actual pressure . 56,000 „ Difference in favour of stability . 128,000 „ The strain upon the end ties, which sustain the weight of half the bridge, will be 275,000 pounds, the cross-section being as before 288 square inches: the strain per square inch will thus be 955 pounds. The width of the end panel being 8^ feet exclusive of posts, and the distance between the shoulders of the posts being as before 15.^ feet, the diagonal will be 17'7 feet, and . , , , 275,000x177 the pressure in the direction ot the braces--= 314,000 pounds = 1451 pounds per square inch. The limit of the resistance to flexure for the 4 braces is , , 90 00 x 9 x 6 3 , 0S) . AAA 3 expressed by w =- YpV* - X = 223,000 pounds. As the pressure is 314,000 pounds, it appears that with the assumed weight of a train of locomotives, or one ton per lineal foot besides the weight of the structure, the end braces would yield by lateral flexure in the direction of the plane of the truss, if not supported in the middle. If an intermediate support be used, the resistance will be quadrupled, and will be amply sufficient. It is also necessary to examine whether the braces, if supported in the middle in the direction of the plane of truss, could yield laterally in the direction of the perpen¬ dicular to this plane: the relative resistances in the two BRIDGES. 1G7 cases arc as G x !) :i : 9 X 6 3 , or as 9 : 4. The limit in this M „ , 223,000 x 9 case would therefore be -^-= 502,000 pounds, which is more than the pressure (314,000 pounds). It appears therefore from this calculation, that if the arches are omitted, the end braces should be supported in the middle by diagonals in the opposite direction. As an additional security, the depth should be increased to nine inches. In the other panels, they should diminish gra¬ dually to the middle of the span, where the original dimensions are sufficient. Floor Beams. The floor beams are 7x14 inches, width in clear between supports 11 feet, distances from centre to centre 5h feet. The weight on the drivers of a locomotive, 18 tons, may be considered as distributed nearly equally over 3 floor beams, which will give 6 tons for each beam. 6x3-t-5 o = 33 tons = the equivalent weight in the middle of the beam R = 18 wl 18 x 6600x 11 b d- 7 X 14- strain per square inch. = 952 pounds = maximum Lateral Braces. The lateral braces are x 7 ins., and 8 feet long. The prevalent winds are in a direction nearly parallel to the axes of the bridge, so that its exposure is not great. Assume as the basis of a calculation that the sides are closely boarded 20 feet high, and that the perpendicular force of wind may be 15 pounds per square foot: the whole pressure upon one span will be say 45,000 pounds. As there is lateral bracing both above and below, this pres¬ sure would be resisted by 4 lateral rods I inch diameter = 314 square inches, or 14,330 pounds per square inch. The proportional strain upon the lateral braces would 45,000x8 be - — 72,000, to resist which, are four braces 4| x 7 = 126 square inches = 571 pounds per square inch. Ihe bearing surface at the joints does not much exceed one-half the area of the cross-section, consequently the actual pressure at the joints will be about 1000 pounds. Ihe limit of flexure of the four braces is expressed by 9000x7x4i 3 w — g! — X 4 = 360,000 pounds nearly. The maximum pressure is 72,000 pounds. Difference in favour of stability, 2SS,000 pounds. The lateral braces cannot yield either by crushing or bending, and are, therefore, amply sufficient. Gould the bridge, if not loaded , be blown away ? The weight of one span has been found to be 240,000 pounds. The resistance to eliding would be. 120,000 pounds. The pressure of wind. 45,000 „ Difference in favour of stability ... 75,000 „ Could the bridge yield to the force of the wind by ro¬ tation around the outer edge of the chord ? The effect of the wind, 45,000 pounds, actmg with a leverage of 10 feet, would .give for the disturbing force . 450,000 pounds. The resistance, = weight of bridge X half¬ width from out to out = 240,000 X8= 1,920,000 „ Difference in favour of stability... 1,470,000 Strain wpon the Knee-braces. Let ACBD (Fig. 480) represent the cross-section. The Fig. 480 . effect of the pressure of wind on AC is equivalent to half that pressure applied at the point A. A force at A tends to produce rotation around u and c, which may be resisted by a brace in the direction of the dia¬ gonal A B. The pressure upon the brace will bear to the force at A the proportion of the diagonal to the side A D. If the brace be removed, the pressure must, nevertheless, still continue; and if it be resisted by a brace ef the pressure upon e f will be greater than that upon A b, in the proportion of A D to e D ; because D is a fulcrum, and A D and e D the leverages of the acting and resisting forces. If e f is parallel to A b, which is generally a very favourable direction, the lengths e f and A B will be in proportion to the distances D e and D A, and may be substituted for them. In the pre¬ sent case, the force of wind, 45,000 pounds, acting with a leverage of ten feet, will give its moment 450,000, or 225,000 pounds acting at a distance of 20 feet. The length of the diagonal is V 20- + 16- = 25-6 feet, and the strain in the direction of the diagonal2^,500 X15 6 _ gg qoo 16 The length of the knee-braces being 5 feet, the strain 256 upon them will be 36,000 x—= 184,000 pounds. This is resisted by 15 braces (one to each post). The cross- section of each is 25 square inches; but, as the bearing surface of the joint does not extend over the whole surface of the section, the resisting portion will be reduced to 15 square inches. The strain per square inch will there- , ! 184,000 010 lore be ——-— = 818 pounds. It) X lo 1 For the resistance to flexure of the fifteen braces, w = 9000x5 x5 3 -- X 15 = 3,375,000, or about 20 times the pres¬ sure. The strain upon the bolts at D, will be to the vertical component at A, in the proportion of D e to e a, or as 5 : (25 6 — 5). The vertical component at A, = 22,500 = ad _ 16 ~Fc = ”2,500 X 9 Q= say 17,000. Hence the strain upon the 15 bolts will be 17,000 X 4 = 68,000. or 4533 pounds to each bolt, or 10,000 pounds per square inch if the bolts are f inch diameter. Pressure wpon the Arch. For this calculation we have, from the table of data, Span, 148 feet. Distance of centre of gravity from abutment, 37 feet. Rise of arch, 20 feet. Proportion of l^pothenuse, perpendicular, and base of skew-back = 18, 16, and 7 6 . Cross-section of 8 arches, 800 square inches. 800 x Ys =711 proportion to resist horizontal thrust at skew-back. 800 X Yg = 338 square inches to resist vertical pressure at skew-back. 1C8 PRACTICAL CARPENTRY AND JOINERY. The weight for one half span loaded is 275,000. 800 x 20 xP = 275,000 x 37. P= 448=pressure, per square inch, on arches, in middle. The resisting cross-section at the skew-backs is the same as at the crown. The pressure is greater in the proportion of the hypothenuse 18 to the perpendicular: it will therefore be 4-18 X jg = 504lbs. The arches are therefore more than sufficient to sustain the whole weight. When both systems act as one, The data required to determine the strains upon the chords and arches are, Distance from middle of upper to middle of lower chord .. 19 feet. Distance from middle of skew-hack to middle of lower chord. 4’2 „ Distance from middle of top chord to middle of arch . 3 - 5 „ Cross-section of upper chords. 400 square in. „ lower „ . 2S0 „ „ arch at crown . 800 „ „ „ skew-backs. 711 „ =dist.of top chord from neutral axis. = „ arch at crown „ = ,, bottom chord „ = ,, arch at skew-back ,, = pressure per sq. in. on top chord. ,, arch at crown „ bottom chord. „ arch at skew- (19-*) = ready been found to be 223,000 pounds, which will be pro- 223,000 X 15'5 duced by a vertical pressure of— ~ ’ , _ _—— = 195,282 j l 17'7 pounds: this is the extreme limit of the power of resist¬ ance of the end braces. The proportion of surface at the skew-back which resists the vertical pressure is 388 square inches. If we suppose the vertical pressure on the base of the skew-back to be the same per square inch as the horizontal pressure upon the perpendicular, it will be capable of resisting 180,830 lbs.: this, deducted from the whole pressure, . . 275,000 ,, will leave for the portion to be sustained by the braces. 94,170 lbs. which is below the resisting power. The actual limit of the resisting power of the arch is very great; but assuming that in practice it is not safe to exceed 1000 pounds per square inch, the proportions of the weight sustained by the truss and arch would be, 108,663 For tlie truss 275,000 X 33 p u0 + ](W(663 = CG.8S0. And for the arch 275,000 x 4 , 4 . 0’5 53 = 208,000. These numbers will give for the strain per square inch 206,100 on the arch, „ —(23'5 *)= back, horizontally. The equations in this case are, P P 400 P a -f — 800 (x — 3 - 5 ) 2 + - 280 (19—.r)' — x) 2 = 275,000 x 37, P P and 400 Px + 800— (.r — 3 5)“ = 280—(19 - x)- + 711 338 For the end braces = 615 lbs. 66,880 x 17-7 : 353 lbs. - (23-5 — x)\ From the second of these we find x = IDS. Consequently the distance of the neutral axis will be, Below top chord . 11 ‘8 feet. 15-5 X 216 It has been stated that the bridge at the western end is sustained by an abutment-pier: it is proper to examine whether the resistance which that is capable of opposing is sufficient to counterbalance the thrust of the arch, on the supposition that it should bear the whole of the load. The dimensions of the abutment-pier are given in Fig. 481, except the length, which may be taken at 16 feet. We will examine the conditions of equili¬ brium on the supposi¬ tion that rotation takes place around the point B. The disturbing force is the horizontal com¬ ponent of the thrust of the arch = 358,750 lbs. acting with a leverage of 164 feet: its moment will there¬ fore be 358,750 x 1 64 = 5,919,375. The resistances are:— 1. The weight of the masonry above c B = 110 perches, of 3750 lbs. = 412,500 lbs. The distance of centre of gra¬ vity from B is 5 feet; the moment will be 2,062,500. 2. The adhesion of the mortar, estimating it at 50 lbs. per square inch, or one-half the tabular strength of hydraulic cement, will be, on a surface of 160 sq. ft., = 1,152,000 lbs., and its moment, with a leverage of 5 feet, = 5,760,000 lbs. 3. The vertical pressure of the arch itself, 275,000 lbs., acting with a leverage of 9 feet, will give a moment, 275,000 x 9 = 2,475,000. j 2,062,500 The sum of the moments of the resisting forces will be { 5,760,000 ... = 4,378,125 BRIDGES. 106 As this difference is less than the adhesion of the mortar, it appears that an abutment pier of dry masonry of the same dimensions would be overturned. It has been supposed, in this calculation, that the arch bears the whole weight, and that the abutment resists the whole thrust. The actual horizontal thrust, with the twosys- tems acting together, was found to be 377 X 711 =268,01-7. The moment will be 268,047 X 16^ = 4,422,775. The re¬ sistance, omitting the strength of the mortar,= 4,537,500. From which it appears that if we disregard the adhesion of the mortar, the system as a whole would be very nearly in a state of equilibrium, the difference being in favour of stability. The practice of the writer in proportioning abutments on rock foundations is, to disregard the adhe¬ sion of the mortar, throwing this, whatever it may be, in favour of stability: there is so little uniformity in the strength of mortar, and so much liability to cracks occa¬ sioned by jars, when partially set, that it is not safe to an abutment do not prevent it from overturning, without takin"- the strength of the mortar into consideration, it is too weak. ■When the base is to any extent compressible, it is not sufficient that the disturbing and resisting forces should be in a state of equilibrium—a condition which requires the resultant of all the forces to pass through the point of rotation; but it is proper that the resultant should pass through the middle of the base.* Summary. Span . 148 ft. 3 in. Width of pier on top . 3 „ 2 „ „ „ skew-back . 6 „ Timber in one span . .. 72,726 „ No. of cubic feet per foot lineal. 40 „ Width from out to out of chords 20 „ „ middle to middle of chords 19 „ V ersed sine of lower arch . 20 „ Radius . ... 17,255 „ Weight of timber per lineal foot 1,416 pounds. Weight of iron in one span . ... 5,280 Weight of half-span loaded. ..275,000 Strain upon floor beams per square inch 952 „ lateral brace-rods per square nch 3,444 „ „ lateral braces . 571 } „ „ knee-braces per square inch 818 „ Pressure per square inch on top chord 381 „ „ „ arch at crown 268 „ „ „ „ lower chord 232 „ „ ., „ arch at skew-back 615 „ „ „ ., end-braces ... 353 „ ,. „ „ middle braces 260 „ Fig. 8 is the elevation of the common lattice bridge; Fig. 9, a section of the same when the roadway is above the latticed sides; and Fig. 10, a section when the road¬ way is supported on the under side of the lattice. Fig. 11, plan of one of the latticed sides. Although when first introduced the lattice construction at once obtained great favour from its simplicity, economy, and elegant lightness of appearance, yet experience has shown that it is only adapted for small spans and light loads, unless fortified by arches or arch braces. When * This calculation was maile before the completion of the bridge: the correctness of the conclusions was soon confirmed: the pier began to crack after the opening of the road, and au increase of thickness by the addition of buttresses was found necessary. well constructed, however, it is useful for ordinary road bridges where the transport is not heavy. On the subject of lattice bridges, Mr. Haupt makes the following perti¬ nent remarks introductory to his notice and description of the kind of construction called the improved lattice, shown in Figs. 12, 13, and 14:— “ One of the first defects apparent in some old lattice bridges, is the warped condition of the side-trusses. The cause which produces this effect cannot, perhaps, be more simply explained than by comparing them to a thin and deep board placed edgeways on two supports, and loaded with a heavy weight: so long as a proper lateral support is furnished, the strength may be found sufficient; but when the lateral support is removed, the board twists and falls. A lattice-truss is composed of thin plank, and its con¬ struction is in every respect such as to render this illustra¬ tion appropriate. Torsion is the direct effect of the action of any weight, however small, upon the single lattice. A second defect may be found in the inclined position of the tie. All bridge-trusses, whatever may be their par¬ ticular construction, are composed of three series of tim¬ bers; those which resist and transmit the vertical forces are called ties and braces, and those which resist the hori¬ zontal force are known by the names of chords, caps, &c. In every plan, except the common lattice, these ties are either vertical, or perpendicular to the lower chords or arches; and, as the force transmitted by any brace is naturally resolved into two components, one in the direc¬ tion of, and the other at right angles to the chord or arch, it would seem that this latter force could be best resisted by a tie whose direction was also perpendicular. The short ties and braces at the extremities, furnishing but an insecure support, render these points, which require the greatest strength, weaker than all others; this defect is generally removed by extending the truss over the edge of the abutment, a distance about equal to its height, or to such a distance that the short ties will not be re¬ quired to sustain any portion of the weight, the effect of which is to provide a remedy at the expense of economy, by the introduction of from 15 to 30 feet of additional truss. A bridge whose corresponding timbers in all its parts are of the same size, is badly proportioned; some parts must be unnecessarily strong or others too weak, and a useless profusion of material must be allowed, or the structure will be insufficient. If, for example, the forces acting on the chords in¬ crease constantly from the ends to the centre, the most- scientific mode of compensation would appear to be, to increase gradually the thickness of the chords; and, for similar reasons, the ties and braces should increase in an inverse order from the centre to the ends. In accordance with this, it is found that in bridges that have settled to a considerable extent, the greatest deflection is always near the abutment; that is, the chords are bent more at this point than in the centre, and the joints of the braces are much more compressed. It is also found that the weakest point of a lattice bridge is near the centre of the lower chord; this might be expected, since, from the nature of the force and the mode of con¬ nection, the joints of the lower chords are only half as strong as the corresponding ones of the upper chord, it Y 170 PRACTICAL CARPENTRY AND JOINERY. being assumed that the resistances to compression and extension are equal. This defect may be in a great degree removed by inserting wedges behind the ends of the lower chords. A variation in the size of every timber, according to the pressure it is to sustain, would, of course, be inconvenient and expensive; but, as the principle of proportioning the parts to the forces acting upon them is of great importance, such other arrangements should be adopted as will secure its advantages, and at the same time possess sufficient simplicity for practice; this is effected by the introduction of arch-braces or arches, than which a more simple, scientific, and efficacious mode of strengthening a bridge could not perhaps be devised, as they not only serve, with the addition of straining-beams, to relieve the chords, and give them that increase of thick¬ ness at the points of maximum pressure which is essential to strength, but they also relieve the ties and braces by transmitting directly to the abutments or other fixed sup¬ ports, a great part of the weight that they would other¬ wise be required to sustain. It may, perhaps, be objected that the pressure of the arch-braces or arches would injure the abutments: in answer to this it may be remarked, that a certain degree of pressure is very proper; the embankment behind an abutment exerts a very great force upon it, the tendency of which is to push it forward. If, then, a counter-pres¬ sure can be produced by the thrust of arch-braces or by wedging behind the ends of the lower chords, two impor¬ tant advantages are gained; the abutment is not only increased in stability, but the tension on the lower chord of the bridge is diminished by an amount equal to the degree of pressure thus produced. It is, however, proper to observe, that when the situa¬ tion of the embankment exposes it to the danger of being washed away from the back of an abutment, the pressure on its face must not be sufficient to destroy its equili¬ brium; should this effect be apprehended, the horizon¬ tal ties must be sufficient to sustain the thrust of the bridge. An essential condition in every good bridge is, that it shall not only be sufficient to resist the greatest dead weight that it can ever be required to sustain in the or¬ dinary course of service, but it must also be secure against the effects of variable loads. This is generally effected by the addition of counter-braces; but the lattice truss possesses this peculiarity, that it is counterbraced without the addition of pieces designed exclusively for this purpose: to prove this, invert the truss, when it will be apparent that the braces become ties, and the ties braces, possessing the same strength in both positions. Fig. 12 is the elevation, and Figs. 13 and 1-4 details of the improved lattice. The difference between this and the common lattice is— 1st. The braces, instead of being single, as in the common lattice, are in pairs, one on each side of the truss, between which a vertical tie passes; this arrangement increases the stiffness, upon the same principle that a hollow cylinder is more stiff than a solid one with the same quantity of material, and of the same length, and obviates the defect of warping. 2d. The tie is vertical, or perpendicular to the lower chord, a position which is more natural, and in which it is more efficacious than when inclined. | 3d. The end-braces all rest on and radiate from the abutment, by which means a firm support is given to the structure, and the truss is not required of greater length i than is sufficient to give the braces room. 4th. The truss is effectually counterbraced, the braces becoming ties, and the ties braces, when called into action by a variable load, and are capable of opposing a resistance on the principle of the inclined tie of the ordinary lattice bridge. It is readily admitted that the strength in the inverted is less than in the erect position, but it must be remem¬ bered that the unloaded bridge is always in equilibrium; that the action of the parts which renders counterbracing necessary, results entirely from the variable load, and that, therefore, a combination of timbers to resist its effects should not be as strong as that which sustains both the permanent and the variable loads. Behind the ends of the lower chords at the abutments, and between them over the piers, double wedges are driven, the object of which is, by the compression which they produce, to relieve the tension of the lower chord. For ordinary spans, the dimensions of the timbers may be— Braces . . . . 2 in. ly 10 in. in pairs. Ties . . . 3 „ 12 „ Arches or arcli-br.ioos G „ 12 „ Chords . . . . 3 „ 14 „ .apped. Pius ... 2j in. in diameter. In conclusion, it is proper to remark that the proposed plan is not recommended as the best under all circum¬ stances, but it is as economical in first cost as any other that can be used. The arrangement will bo found even more simple than the ordinary lattice, and it is equally applicable for bridges on common roads or railroads, and for roof or deck bridges. The braces, in consequence of being placed in pairs, require a slight increase of timber over the common plan, in the proportion of -40 to 36, but the diminished lengths of the ties and of the truss more than counterbalance this increase. The cost of workmanship on the truss is very trifling, and less than on the common lattice; if the timbers are cut to the proper lengths, the auger will be the only tool required in putting it together. Plate LY., Figs. 1 to 6, illustrate the construction of a skew bridge erected over the Leith Branch Railway, Portobello, on the North British Railway. This viaduct was formed of timber, principally on ac¬ count of the ground being of a nature unfavourable to the construction of a stone bridge, and also owing to the very great angle at which the public road and Leith Branch Railway is crossed. To allow sufficient headway for the Leith Branch, it is spanned by cast-iron girders of an elliptical form, resting on a timber sole, so as to render struts unnecessary. The spans beyond the crossing of the road and railway for the remainder of the viaduct, are thrown on the square by a simple method which is shown on the plan. The details of the structure are to a great extent deline¬ ated in the plate; it is therefore unnecessary to enter into them here. Fig. 1, the elevation of the bridge. Fig. 2, the plan. Fig. 3, a vertical transverse section. PLATE IV Mil [©(DIES h§§B DiE^TTMES BY JOHN MII.LER, C E ation. Plan ,ond Details, erf Shew Bridae over the Leith Braneh Railway See. at 1‘artohello. art. the North British Railway. X 3 JLp entii/ui of Viaduct over the VrcLem Canal near FaVdrlc on the EJinburah and Olasr/ow Railway -'.nr.; Dvnula.ix Bum on die North British Railwa,v Via dn it teirlrni. 200 Teet j.bo Teet J. IV.Linvn/c. BLACK IE & SON -, GLASGOW, EDLNBUUGII & LONDON CENTRES. 171 Fig. 4, a section to a larger scale, showing one of the trasses and rail Fig. 5, details to a larger scale of piers, chords, struts, waling pieces, and the iron work serving to unite them. Fig. 6, the cast-iron capping of the pier piles. Figs. 7 to 11 on this plate are described under the section Centres. Plate LYI.— Figs. 1 and 2 illustrate a bridge of the third class; which is erected across the river Tweed at Mertoun. This bridge consists of five arches, each 70 feet span, erected on stone abutments and piers. It was intended originally to be built entirely of stone, and although tim¬ ber was adopted for the arches at the time of its erection, it was considered probable that stone arches might at some future time be substituted for the wooden framing. The piers and abutments were therefore made of such dimen¬ sions and strength as to be sufficient for stone arches. The work was commenced in 1839 and finished in 1841, by Mr. William Smith, contractor, Montrose. The whole of the piers and abutments were built on rock, which was reached by means of cofferdams, and which was penetrated to the extent of from 1 to 3 feet, on purpose to get a solid and level foundation. The depth of the different foundations varies from G feet 3 inches to 11 feet 7 inches below the summer level of the river, and the height from the summer level to the springing of the arches is 18 feet. It was intended originally to make this only 16 feet, but in consequence of an unusually high flood occurring during the building pf the piers, the height was increased 2 feet. The extreme length of the body of the piers in the plan is 29 feet 6 inches, and their thickness is 10 feet. The foundation extends 1 foot on every side beyond these dimensions, and is diminished to them by two footings of 6 inches each. The whole of the mason work is of freestone, obtained in the neighbour¬ hood of the bridge. Each arch consists of three laminated beams or ribs; each rib is formed of 5 thicknesses of half logs 12x6 inches, which were bent on a frame or centre, and fixed together, previous to their erection. The depth of the rib at the ends is 2 feet 6 inches, but in the centre it is diminished to 2 feet, the upper layer being cut horizontally to con¬ nect it with the longitudinal beam of the roadway. Each curved rib is bound together by eleven clasp hoops of iron, three of them embracing also the roadway beam; and the different layers of bent timber are kept from sliding on each other by oak keys, to the number of 60 in each rib, inserted horizontally, and sunk half their thickness into the con¬ tiguous layers. The longitudinal roadway beams are each 12 inches square, and connected with the curved rib by the iron straps above mentioned; also by the insertion be¬ tween them of six upright braces 10x6 inches, and further by clamps 6 inches square, one on each side of the rib, halved upon the beams, so as to fit close on both beams and braces. The outward clamps of the external trusses extend upwards and form the posts of the parapet railing, and all the parts are securely fixed together by | -inch bolts. Between the upright ties, braces 12x6 inches are fitted, reaching from the curved rib to the horizontal beam. In each arch, the. curved beams are connected by six cross ties, notched upon them, and stiffened by four pairs of diago¬ nals, 4^ inches square, all fixed by strong spikes. The roadway is supported by cross-beams, 9x4 inches, placed 4 feet apart between centres, supported by the lon¬ gitudinal beams, notched 1 inch upon them, and bolted down. (In these, 3-incli planking is laid, and covered by wood sheathing inch thick, crossing the planking obliquely, and having tarred paper interposed. Over this sheathing lies a coat of gravel, bedded in Archangel and coal-tar pitch to the thickness of 1 inch, forming a com¬ pletely water-tight sole; covering this is a layer of stiff clay puddle 2 inches in depth, and over all a layer of common road metal, blinded with fine gravel and sand. The width of the roadway within the parapets is 18 feet. The parapet railing is formed by posts 6 inches square, being the continuation of the ties of the arch trusses, secured to the roadway planking by iron stirrup straps, and tenoned at the top into a rail, 6 inches by 4 inches; the I top rail is surmounted by a coping 7 inches broad, l£ inch i thick in the middle, and 1 inch at the sides, and it is fur- I ther secured by an iron strap at each post, passing over the top and down each side of the post, fixed with nails. Each compartment is fitted with braces and counter-braces, and a panel rail at bottom, 4 inches broad by 2h in depth, the former halved at their crossing, and tongued into the posts at head and foot, and the latter tenoned into the posts. Two lines of bars, 5x2 inches, run along the inner side of the railing, notched into and upon the posts each 4 inch, and securely nailed both to the posts and to the diagonals. For descriptions of Figs. 3 and 4 on this plate, see Centres. CENTRES. Centres are works in carpentry which serve to sustain the masonry of vaults or arches during their construction, and until the insertion of the keystone gives them the power of sustaining themselves. Under this view centres are true scaffolds. Centres are of different species, according to the nature of the curvature of the vault. The disposition of the tim¬ bers of which they are composed is analogous to that ot | timber bridges and of roofs, and the different elementary j pieces are known by the same names. Each centre is com- ] posed of a series of frames placed parallel to each other, | and perpendicular to the axis of the vault or arch. They i are tied together horizontally, and covered with boarding, technically termed logging , which forms the cradle or mould to support the masonry. Centi-es are distinguished also by the mode in which they are constructed. Thus there are flexible centres, which may undergo a change of form during the con- j struction of the vault, by the varying nature of the load, j Of this kind was the centre of the bridge of Neuilly, j constructed by Peronnet. There are also fixed centres, I which maintain their form under the varying load, and j these flexible and fixed centres may be either in one span j supported only at the springing of the vaults, or they may | be sustained by intermediate supports. The flexible centre is composed of a series of triangular trusses, arranged so as to form concentric polygons, the angles of the one corresponding to the sides of the other. ITardouin Mansard has the reputation of being the inven¬ tor of this principle. It was applied by him in construct- 172 PRACTICAL CARPENTRY AND JOINERY. ing the centres for building the bridge of Moulins in 1706, but there is proof that it had been previously proposed by Claude Perrault for a wooden bridge designed to cross the Seine at Sevres. In Plate L1X., Fig. 2, is shown Peronnets centre for the bridge over the Seine at Neuilly, an example on the largest scale of this kind of centre. The annexed diagram (Fig. 482) is a representation of the framing at A in the plate. The flexibility of this system arises from the smallness of the angle of inclination between the timbers, and the number of articulations or joints on which the pieces can turn and change their inclination relatively to each other. The result is, that as soon as the centre is charged with the weight of the masonry on its haunches, it sinks there and rises at the crown. To remedy this grave inconveni¬ ence, which is attended by a complete change in the form of the vault, and may result in seri¬ ous accidents, it is necessary to load the crown of the centre with such a weight as shall equilibriate the load on the haunch, and to vary this as the work pro¬ ceeds, so as to main¬ tain the integral form. In practice this equilibriating load is really the voussoirs which are laid temporarily in their place before being properly set, a process involving a great expense of labour and a continual manipulation, attended by considerable risk. The sinking of the haunches and rising of the crown were so great in the centres of the bridge at Neuilly, that to restore them to their primitive form, and to enable them to retain it during the construction of the arch, it was found necessary to load their summits successively with weights of 120, 420, and 448 tons. When the arches were keyed, the general sinking of the centres amounted to from 275 inches to 315 inches in twenty-four hours. When the centres were struck the arch descended. It is not necessary to enlarge further the notice of the species of centre which has been classed as the flexible centre, as in this country it is entirely unknown; and although it was used in some magnificent works, and its use was sanctioned by the high authority of Peronnet, yet it is now abandoned in France also. The second class, which has been termed the fixed or inflexible centre, is that which has alone been used in this country, which has produced examples exhibiting a happy combination of science and constructive skill. It may well be supposed that the art of disposing the pieces of tim¬ bers, which enter into the composition of a centre, in such a manner that they may sustain, without change of form, all the efforts of the voussoirs. varying with the progress of the work until the key-stone is placed, and to determine the dimensions of the timber, is not of easy attainment. The theories which the learned have given to the world on the subject of centres are so general, as to be totally inapplicable to particular cases; and they are founded besides on abstract reasoning, without reference to the materials, and those means and appliances of the work¬ men which sometimes totally change the state of the question and always modify the results. Unless, indeed, the design of the framing of a centre be of a very simple nature, it would be very difficult to attempt to estimate the forces and the strengths required to sustain them. The best that can be done is only approximative, and no more is attempted in the rules which are subsequently given. The result of investigations on the pressure of arch¬ stones on a centre is, that the centre should be combined in such a manner as to withstand as advantageously as possible the effort of the stones to slide upon their beds. Experiment has shown that hard stones have not any tendency to slide on the bed until it is elevated to about 30°; and it has also shown that when the stone is set in fresh mortar it does not begin to slide until the bed is ele¬ vated to an angle of from 34° to 36°. Voussoirs of soft stone, absorbent of moisture, have been raised to an angle of 45° without sliding, when the centre of gravity did not fall without their base. Reasoning from these experiments, and assuming 32° as the limiting angle of resistance, the conclusion would be arrived at, that the centre did not require to com¬ mence until the arch stones had reached that angle; and in the Pont du Gard and the arch of Cestius at Rome, the corbels on which the centres were supported remain at from 25° to 2S° above the springing. Beyond 32° the weight on the centre goes on increas¬ ing as it approaches the key-stone; but in practice it is safe to consider the whole weight of the stone as resting on the centre, when a vertical line drawn through its centre of gravity falls without the lower bed of the stone; and the amount of error is not great, and is on the safe side, if this is taken to be the case when the bed of the stone exceeds 60°. But to make this observation more accurate, we quote Mr. Tredgold’s w'ords. lie says, “When the depth of the arch stone is nearly double its thickness, the whole of its weight may be considered to rest upon the centre, at the joint which makes an angle of about 60° with the horizon. If the length be less than twice the thickness, it may be considered to rest wholly upon the centre when the angle is below 60°, and if the length exceed twice the thickness, the angle will be considerably above 60° before the whole weight will press on the centre.” To find the pressure of the arch stones on the centre, in a direction perpendicular to its curve, Mr. Tredgold has given the following formula:—VV (sin a —/cos. a) = P. Where W is the weight of the stone, P the pressure on the centre, a = the angle which the lower bed makes with the horizon, and / = the fraction; and by apply this formula he obtains the following result:— When the bed makes, with the the pressure = the weight, horizon, an anglo of multiplied by 34° . . -04 3G . . -08 38 . . -12 40 . . -17 42 . . -21 44 . . -25 46 . . -29 4S . . -33 50 . . *37 52 . . -40 54 . . -44 56 . . -48 58 . '52 60 . . -51 F4. 482. B A t'ttntrifu/ nr Ctrrtl'emd /'raiqs Br? df.fr I'rntri/m of (h'er Jirtih/r Irh’iniwirr. ( rntmu) 'rDeati Bvrth/r, KJinhimih . J IV l.own- /c Tio.l EEPJTB1ES. CENTRES. The application of this may he illustrated by an example. Suppose it was required to find the pressure of the arch¬ stones on the centre, in a space comprehended between the angles of 32° and 42°, that is in 10°, take out of the preceding table the decimal opposite every second degree, for the first 10°, and add them together as follows:— •12 = -G2 •17 •21 J Multiply this number by the weight of a portion of the arch stones comprehended between two degrees, and the product will be the pressure on the centre. Suppose the centres to be 6 feet apart from outside to outside, and the depth of the arch stones to be 8 feet, as in Waterloo Bridge, and the space comprehended between two degrees, measured at the middle of the depth of the stone, is 2 feet, there is obtained— 6 feet = the distance betwixt the centres. 8 feet = depth of the arch stones. 2 feet = width of the portion sought. 6 x 8 X 2 = 96 cubic feet, which, multiplied by 160 lbs. the weight of a cubic foot of stone, = 15,360 lbs., the weight of two degrees; and this, multiplied by '62, gives 9523 lbs. for the pressure of 10° of the arch stones on one centre frame. The pressure on the centre, it has been said, may be considered equal to the whole weight of the stone, from 60° to 90°, and from 60° to the angle of repose the pres¬ sure rapidly diminishes. It would he manifestly errone¬ ous, therefore, to make the frame of the centre equally strong throughout; and the endeavour in designing it should be to apply the strength only where it is really required. How this has been managed by different de¬ signers will be seen by examining the drawings of examples of centres in Plates L., LV., LVf., LYII., LVIII.,and LIX. In designing centres, the observation which Mr. Smea- ton makes on his own design for the centering of Cold¬ stream Bridge may be held to be a sound practical maxim. “ What I had therefore in view,” he says, “ was to distri¬ bute the supporters equally under the burden, preserving at the same time such a geometrical connection through¬ out the whole, that if any one pile or row of piles should settle, the incumbent weight would be supported by the rest. With respect to the scantlings, I did not so much contrive how to do with the least quantity of timber, as how to cut it with the least waste; for as I took it for granted the centre would be constructed with east coun¬ try fir, I have set down the scantlings, such as they usually are, in whole baulks, or cut in two lengthways; and as I think the pieces will suffer less by notchings in the middle intersection, and being cut into small pieces, it will re¬ main of value for common building after it has been done with as a centre.” And he adds in favour of simplicity, “ As the construction is more obvious, and less exactness required in the handling, I should expect to get a good centre made by some in this way that would make but bad work of the other.” Fig. 1, Plate LVIL, shows a centre for a small span. It consists of a trussed frame, of which A is the tie, B the principal, or, as its outer edge is curved to the contour of the arch, it is called by Mr. Smeaton the felloe, c the post 173 or puncheon, and f a strut. The centre is carried by the piles D, on the top of which is a capping piece E, extending across the opening; and the wedge blocks a are interposed betwixt it and the tie-beam. Figs. 2 and 3 are centres, also for small spans; and of this Fig. 2 is the best in its arrangement. In Fig. 4 the weight of the centre of the arch is carried directly by the struts to the ends of the tie-beam, the tie- ; beam struts and king-post A making a simple king-post ! truss. Two other trusses support the arch above the i haunches, and have a collar-piece between them at half the height of the arch. The ends of the cross-braces arc seen at a. Fig. 5 shows a centre with intermediate supports and simple framing, consisting of two trusses formed on the puncheons over the intermediate supports as king-posts, and subsidiary trusses for the haunches, with struts from their centres parallel to the main struts. In Fig. 6 the weight is fairly distributed between the three points of support. The ends of transverse braces connecting the trusses are seen at a. Fig. 7 shows a system of supporting the arch rib from the intermediate supports by radiating struts, which, with modifications to suit the circumstances of the cases, has been very extensively adopted in many large works, and of which other examples are here presented. The struts abut at their upper end on straining pieces, or apron pieces, as they are sometimes termed, which are bolted to the rib, and serve to strengthen it. The ends of the trans¬ verse braces are seen at a a. Fig. 8 is the centre of a bridge over the river Don, in Aberdeenshire. The bridge consists of five arches, each of 75 feet span. It was erected from designs of Thomas Telford, Esq., by Messrs. John Gibb & Son, of Aberdeen, who designed the centres. In this centre the weight is in a very simple and inge¬ nious manner discharged to the points of support. The piles are cross-braced; the sides of the braces are seen at d/C; and the puncheons above are also connected by a system of cross-bracing, b b. Plate LVJII., Fig. 1.— Centering of Gloucester Over- bridge. —This centre, designed and constructed by Mr. Cargill, the contractor for building the bridge, consists of a series of trusses supported on piles, which being in some cases 16 feet apart, allowed the navigation to be carried on. There are six parallel rows of piles fixed in the current of the river, each row connected with cross-braces and caps, and each supporting a rib, which forms the actual centering. The piles and ribs are further steadied by diagonal braces. Between the pile caps and the ribs are placed the wedges or slack-blocks by which the centering is lowered after the keying of the masonry. Mr. Cargill, in a letter to Mr. Telford, dated March 26, 1832, thus describes the construction of the centre:— “ In constructing the centering for this bridge, I first laid a platform perfectly level, and a little larger than the centering which was to be made; I then struck it out full size upon this platform, firmly fixing centres to the dif¬ ferent radii. The timber was Dantzic, being much harder and of larger dimensions than Memel, and mostly 15 inches square. The iron straps were of the best iron. “ The piles upon which the centre was to stand were ! then driven. They were of Memel timber, with wrought 174 PRACTICAL CARPENTRY AND JOINERY. iron shoes, and caps framed upon the tops to the proper height. Upon these caps was laid another tier of beams lengthways of the centre one, under each rib; upon these beams were fixed the wedges, which were of three thick¬ nesses, the bottom one being bolted down to these beams, the tongue or driving piece in the middle being of oak, and well hooped at the driving end; the top side of the upper piece was laid perfectly level and straight, both transversely and longitudinally. The wedges were rubbed with soft soap and black lead before they were laid on each other. “ Each rib of the centre was then brought and put to¬ gether upon a scaffold made on the tops of these wedge- pieces, and lifted up whole by means of two barges in the river and two cranes on the shore. The scaffold was ex¬ tended thirty feet beyond the striking end of the wedges, to lay the last ribs upon previous to raising, also to stand upon for finally striking. After the ribs were properly braced, they were covered with 4-inch sheeting piles, which had been used in the cofferdams. “ That this centre was well suited to its purpose is known by its not sinking more than one inch when we keyed the arch. My greatest dread was the coal-boats which trade on the Ledbury Canal, forced adrift by floods in the Severn, and striking against the centre before we could close the arch. To prevent mischief of this kind, I drove the piles for extending the up-stream side of the scaffold (or rather the platform on which it was originally constructed) very firmly into the clay, so that they might resist the stroke of a boat befoi’e she could touch any of the supports of the centering. “ In the month of December, when within twenty feet of closing the arch, a very high flood being in the Severn, two of these boats loaded with coal came adrift, struck the outside piles, which were of Memel logs, broke two of them, and then sank against the main bearing piles which supported the centre, one boat on the top of the other. These boats being seventy feet Jong, raised a considerable head of water over them, and lay there until the flood sub¬ sided, which was many weeks; had not these upper guard piles weakened the shock, I believe the whole centre and arch would have been destroyed. “When the spandrel walls were built up two courses below the crown of the arch, and the internal brick walls to the same height, we struck the centre, which was done by placing beams upon the top of the work directly over the ends of the wedges. To these beams successively was fixed a tackle, to which, at the lower end, was slung the heavy ram with which we drove the piles, with tail ropes to it, and swung exactly so as to strike, in its swinging, when pulled back, the driving end of the tongue piece of the wedge. This ram, of 12 cwt., when pulled back by eight men, and two men to pull it forward, gave a most tremendous blow, yet twenty or thirty blows were re¬ quisite before we could perceive the wedges to move; but after they once moved, they slid themselves, and we put in pieces to stop them going further than was required. The whole time of striking did not, I think, exceed three hours, although we had the ram to remove and the tackle to refix at every set of wedges. I was afraid that no force we could bring against these wedges would move them under such a weight as the entire arch, they themselves being a heavy body, and it was no small joy to see this effected so easily. I am persuaded no wedges placed in the usual way could have been disengaged, as no force could be brought to act upon them sufficient for the purpose. “Wethen disengaged the covering (which, it will be re¬ membered, was composed of.sheeting piles from the coffer¬ dam), and let down the ribs as they were put up; took them to pieces and carried them ashore. The whole of the bearing piles were then drawn by two levers, each made of two forty-feet logs and strong chains. Every pile was drawn, and although the expense was consider¬ able, they paid well for the labour.” Fig. 2 .—The Centering of Dean Bridge, Edinburgh, constructed by Telford, in 1831.—The height from the bed of the river to the roadway of this bridge is 106 feet, and the bridge consists of four arches, each of 90 feet span. The carriage way is carried on the inner arch, and the larger wing arch, which projects 5 feet beyond the other, supports the footpath. No. 1 shows the centering for this latter arch, and No. 2 the centering for the main arch. Fig. 3.— TIlc Centering of Cartland Craigs Bridge is here represented .—Cartland Craigs Bridge, built by Tel¬ ford, in 1821, spans the precipitous banks of the Mouse River. It consists of three arches, each of 52 feet span, and the centre arch is 122 feet in height. Plate L.— Figs. 6, 7, 8, 9, and 10, in this plate, illus¬ trate the construction of the centering used in construct¬ ing the Ballochmyle Viaduct on the Glasgow and South- Western Railway. The river Ayr, which the viaduct spans by an arch ol 180 feet, offers natural facilities for crossing it by one span, owing to its high and rocky banks. This viaduct is erected on private grounds of great beauty, rendered classic by the song of Burns, and the railway company were bound not to interfere further than was actually necessary with the natural beauties of the river and its banks. With the view of carrying out that arrangement, the viaduct, as represented in the plate, was designed and constructed. The main arch of this viaduct, which is semicircular, has a span of 180 feet, and is in height, from the bed of the river to the crown of the arch, about 160 feet. It was therefore necessary to construct scaffolding and cen¬ tering of no ordinary description. The principle on which this was done is new and simple. It was executed as much as possible with whole timbers, of which the uprights consisted, and where the distance of the points of thrust of the frame was not great, the struts and walings were of half timbers; the lagging consisted of battens. It will be seen from the representation in Plate L., and more distinctly in the picturesque view taken during the opera¬ tions, and which forms the engraved title to this work, that the materials for constructing the viaduct were con¬ veyed along by large waggons on a stage projected from the centering, and from them removed by travelling cranes moving longitudinally and transversely. The details of construction of this stage, and the general arrangement of the centering, do not call for special re¬ mark. It may, however, be necessary to add, that the whole timber work in connection with the construction of this viaduct perfectly answered the purpose. Plate LV.— Fig. 7, in this plate, is the centering of viaduct over the Union Canal, near Falkirk, on the Edin¬ burgh and Glasgow Railway. CENTRES. The reason for constructing this large arch over the canal, and the description of centering shown in the plate, was to meet the requirements of a clause in the railway company’s act, by which certain heights and widths were to be preserved over the footway, carriage-way, towing- path, and canal, and no piers were allowed to be placed in the space between the abutments of the large arch, even for temporary purposes. The figure in the plate shows the principle on which the centering was con¬ structed and kept in its position. The centering shown on the smaller arch of this viaduct does not call for any par¬ ticular remark. Both were constructed so as to be simply and easily put together, using whole timber, so as to prevent waste. Figs. 8 to 11, also in Plate LV., illustrate the con¬ struction of centres for the viaduct of the North British Railway, carried over Dunglass Burn. The span of the large arch is 132 feet, and its height 110 feet. The principle of construction is nearly that of Balloch- myle Viaduct, already described. The smaller land arches, however, show a variation in the construction. Plate LVI., Figs. 3 and 4.— Fig. 3 is a longitudinal, and Fig. 4 a transverse section of one of the arches of a via¬ duct, illustrating the mode of constructing and supporting the centering. The viaduct was erected over the Lugar Water, near Old Cumnock, in 1849, by J. Miller, Esq, C.E. The supporting framing consists of four sets of horizon¬ tal timbers, sustained at their ends either by corbels in the piers, or by vertical timbers carried up from the ground, and resting at their centre on the capping-pieces of the series of seven vertical posts, seen in Fig. 4. These verti¬ cal posts rest on a sleeper below, and are firmly braced, counter-braced, and shored to give lateral stiffness. The two exterior posts carry the scaffolding used in the con¬ struction of the arch, and each of the five interior posts is placed under the middle of a centre. The upper horizontal timber supported by the posts carries the centre striking wedges, and the two extreme wedges are carried by the impost of the piers. On these wedges the tie-beams of the centres rest. Plate LIX. — Fig. 1 is the centre used in the construc¬ tion of Waterloo Bridge. Fig. 2.—The centre used in the construction of the elegant bridge over the river Seine at Neuilly, and already noticed in the introductory portion of this section. Fig. 3.—A design by Mr. White. Centre for a seg¬ mental arch of 120 feet span, somewhat after the manner of the centre for Waterloo Bridge. A c B c A are the felloes or arch of the centre, on which the lagging rests. To these are bolted the abutment-pieces d d of the struts a a a a , and the lower ends of the struts rest on iron sole- plates on the tops of the upper striking wedges F F. The radial posts fff are in pairs, one on each side of the struts a a; they are bolted together, and secured to the felloes by iron straps: c c are transverse ties in pairs, bolted together through the radial posts: h h additional transverse ties, h principal tie, and b b straining pieces. r> D f F are striking wedges; e e e f struts resting on sleepers laid on the footings of the piers, and having capping-pieces g g, on which the striking wedges rest. On Removing Centkes. The removal of the centres of a bridge when loaded 177 with the weight of the arch stones is always a delicate operation, demanding prudence and patience. It is neces¬ sary to allow sufficient time for the setting of the mortar | used in the construction of the arch. It may be conceived J that unless this is done, and if the centres are struck, as it I is termed, before the mortar acquires proper consistency to | resist the pressure thrown on it, there would be a sudden sinking of the whole arch, which might pass the limits of I safety. In the bridge of Nemours, in France, the arches of I which were surbased, the pressure on the centres was I found to be so great that the usual mode adopted in | France for the striking of the centres could not be fol¬ lowed. The mode of supporting the arch stones on the centres there adopted was somewhat different from the j practice of this country. In place of a continuous cover- rig. 483. ing of boards, which is here called lagging, the bridge in question had the following contrivance (Fig. 483): a is part of the centre; on this is placed the wedge-piece b, support¬ ing the plank c, which continues across the arch, there being one for every course of voussoirs. The voussolr c is supported on c by the wedge-piece d. The striking the centres should have been performed by withdrawing the wedge-pieces b and d of every alter¬ nate plank, and then the plank c, and then repeating the operation, leaving a fourth-part remaining, and so on till all were withdrawn; but, as has been said, this could not be done, and recourse was had to the expedient of cutting away gradually the feet of the principals, which rested on the corbels of the abutments—a mode clumsy in the ex¬ treme, and also dangerous. The method of striking the centres practised in this country is preferable in every respect to the French mode. That usually adopted is seen in Plate LIX. Figs. 1 and 3. In Fig. 3 it will be seen that each centre frame is sup¬ ported on posts e e e, springing from the footings of the pier. These posts have capping-pieces, g g, extending across the whole width of the arch. Between the capping- pieces and the frame are interposed three pieces of timber, the two outer of which, f F, have their inner faces stepped in wedge-shaped surfaces, and the intermediate piece, D D, is doubly stepped in the same way to correspond. If the position which these pieces relatively hold is remarked, it will be seen that when the piece D is driven back towards the arch, the two pieces F will approach each other, and the centre will thus be gently lowered. A great advan¬ tage is the power that this mode gives of merely easing the centre first, and then lowering it by degrees, so that the voussoirs come gently to their bearing. A method of striking the centre, closely resembling the French method described above, was practised at the Chester Bridge. Each centre frame there had a rim of two thicknesses of 4-incli plauk bent round it; and on this the lagging, 4,j inch thick, was supported by a pair of folding wedges, 15 or 16 inches long, 10 or 12 inches broad, and tapering about 1^ inch. As there were six centre frames in the width of the bridge there were neces¬ sarily six pairs of striking wedges for each course of vous¬ soirs. Hi is arrangement gave the power of easing any portion of the arch, or of tightening one part and slacken- 176 PRACTICAL CARPENTRY AND JOINERY. iug another, as the symptoms exhibited by the stone work as it came to its bearing required. The able constructor of this centre, with the power which this mode cf con¬ struction gave him, preferred striking the centre while the mortar of the arch-stones was yet green and pasty, easincr it a little at first to permit the joints to accommodate themselves to each other, and so proceeding gradually till they obtained their perfect bearing. A different method was employed by the constructor of the centre for Gloucester Bridge. In this centre the wedges were placed under all the supports, and in this way a very great control over its movements could be exercised. A similar arrangement was adopted by Mr. Telford in many of his larger works, as in the Dean Bridge and Cartland Craigs Bridge, Plate LVIII. GATES. A gate, in its ordinary acceptation, may be regarded as a moveable portion of a fence or inclosure; and under this aspect its most obvious representative is a rectangular plane of wood or iron, a door, in point of fact, and this is one of the forms it usually assumes. But in many cases such a gate would be objectionable on account of its weight and costliness; and there is, therefore, substituted for the rectangular plane of wood a rectangular frame of wood or iron, sparred or barred in such a manner as to prevent the passage of animals. In ordinary field-gates the width of the opening be¬ tween the posts, to which the gate is hinged, is usually 9 feet 8 inches, which gives a clear roadway of 9 feet when the gate is opened. The height of the gate de¬ pends, of course, on the height of the fence, of which it forms a part. In ordinary cases this may be five feet, and allowing for the height of the lower bar of the frame from the ground, there are obtained as the average dimen¬ sions of a field-gate—length 9 feet, and height -1 feet. This rectangular frame, then, is the elementary form of the gate. Its vertical sides are called styles; that to which the hinges are attached being called the hanging-style, and that to which the fastening is attached the falling- style. The horizontal sides of the frame, and all the bars parallel to them, are called rails. Such a frame suspended by one of its shorter sides would not maintain the rectangular form; it would be¬ come rhomboidal by the falling down of the other sides by their own weight. To enable it to maintain the rectangular form it is necessary to add an angle brace, which may be applied either as a tie or a strut, as the material used is iron or wood. But the gate may be resolved into a simple elementary form thus:—Let the diagram (Fig. 4S4) represent —a the hanging-style, b the top-rail, and c the . brace or strut of a gate, all firmly united. a This is evidently a t simple truss, like the jib of a crane; and if a weight w be hung to its outer end, the rail b will obviously be in a state of tension, and the brace c in a state of compression; that is, b is a tie, and c is a strut. Again, let a (Fig. 485) be the hang¬ ing-style, b the lower rail, and c the brace, and it is now obvious that c is a tie and b is a strut. Therefore, keep, ing in mind the constructing maxim that iron should be used as a tie and wood as a strut, when the brace is placed as in Fig. 484 it should be of timber, and when as in Fig. 485 it should be of iron. But it may be objected that, if this rule were adhered to, it would not be possi¬ ble to construct a timber gate, as it requires both ties and struts. Now, the pieces of wood of which a gate is formed are framed together, and held in their places by bolts, nails, or wooden pins, and the frame is hung to the posts by iron hinges. The straps by which these hinges are attached to the frame are generally so long as to embrace a considerable portion of the length of the rail, and may, therefore, be made subservient to rendering the timber rail b (Fig. 484) competent as a tie; whereas if timber be used for the brace c, as in Fig. 485, it is evident that the strength of the brace has very little to do with the stability of the framing; that, in point of fact, the stability is due entirely to the strength of the nails, or to the slight resistance to tearing that the fibres of the timber between where the nails are driven and the end of the brace offer; or it must be insured by adding an iron strap to each end of the brace. But this extra iron is expensive; and as by simply making the brace a strut in place of a tie, the iron strap of the upper hinge can be made to supplement the deficiency of the upper wooden rail as a tie. Fig. 486 is the ordinary field gate, constructed on the principles above described. The top rail a becomes a tie, and is secured to the hanging-post by the strap of the upper hinge embracing it, and being bolted through it. The elementary frame is thus rendered perfectly rigid, and the addition of the front or falling-post b, and the bars e, f, g, h, completes the fence. By this mode of con¬ struction the tensile strain is thrown on the bolts and strap of the upper hinge. Having thus pointed out the principle of stability in the framing of the gate, we shall proceed to give some practical details, showing its application to frames of wood and iron. In a timber gate, then, the diagonal bar should form a strut, as in Fig. 485, and not a tie. Were we merely to consider, in the application of the diagonal bar, the angle which should be the best fitted to insure the frame maintaining its form, we should adopt the angle oi 45°. But the bar placed at this angle would not extend half-way along the top rail, and the result would be the introduction of a new element of destruction in the cross¬ strain, to which the top bar would then be exposed; for the point of the strut would be a rigid fulcrum, over which the top bar would be liable to be broken, by a GATES. 177 weight hung to its outer end. Practically, therefore, it is better that the strut should make a smaller angle with the horizon; that in fact it should be attached to the top bar, at just such a distance from the end of the latter as shall not so much weaken it, as to prevent it forming a perfect abutment to the thrust of the strut; and, in ordi¬ nary cases, ten inches or a foot is sufficient. The junction of the strut and top rail should be by the same kind of joint as that by which the toe of a rafter is let into the tie-beam. Fig. 486 shows the general appearance; and Fig. 487 some of the constructive details of a gate, of the construction here advocated. The hanging-style of the gate is 4£ inches square in section; the falling-style is 8 inches square. The top rail a is 4| inches square at the hanging style; and 3 inches by 4 1 1 at the falling-style. It is tenoned into the styles. The diagonal bar m is 4^ inches deep, and 2 inches thick; tapering to the upper end for the sake of light¬ ness. It is tenoned into the hanging-style, and notched into the top rail. The other rails e, f, g, h, are 4^ inches deep at the hanging-style, and taper to 3^ inches deep at the falling-style. They are 1§ inch thick, and are tenoned into the styles, the tenons having only one shoulder on the outside, so as to allow of larger cheeks to the mortises in the styles. In Fig. 487, b is a vertical section through the rails and diagonal, looking towards the hanging-style. The upper hinge has its straps pro¬ longed, as seen in Fig. 486, so as to embrace a considerable portion of the top rail, that the bolts and nails which fasten it may have a secure hold in the solid wood. The under hinge does not require this prolongation of the straps, as the force upon it is a thrusting, and not a draw¬ ing force. As the bottom rail is so much thinner than the hang¬ ing-style, a small piece of wood, of the depth of the rail, is generally added at e, Fig. 487, as a rest for the strap of the hinge. The tenons of the rails are secured in the mortises by pins, and the diagonal is securely nailed to the rails at its intersection with them. Before putting the parts together, the tenons and the intersecting parts of the rails and diagonals should be coated with white lead in oil. The great destroyer of the gate is rain, which falling on the thin top bar, as usually constructed, soaks into the joints and induces rot. The wide top rail in the gate described, affords protection against this; and the only parts exposed to it are the intersections of the dia¬ gonal and rails; but, by giving the upper edges of these a slight bevel, so as to throw the water from the joint, the risk of injury from this cause is destroyed. The top rail should be saddle-backed, or rounded on the top. Some¬ times vertical bars are added to the gate; but these, as we have already said, add to its weight, and not to its strength; and, moreover, introduce new joints, exposed to the action of the rain, and they should therefore be dis¬ pensed with. Plate LX. —Park and Entrance Gates. — Figs. 1, 2, and 3 are examples of park gates, of open frames. Figs. 4, 5, 6 and 7 are elevations of entrance gates in various styles. In Fig. 5, No. 1, it will be observed that two small rollers are inserted in the bottom rail of the gate, and run on iron rails, laid on stone sleepers fixed in the ground. The liinge-pin, too, is continuous between the top and bottom hinge, and serves merely as an axis on which the gate rotates, the whole of the weight being sustained by the rollers. It has sufficient play to allow the gate to rise as it opens. Fig. 5, No. 2, is the strap of the top hinge, and the same construction of strap is applicable to all the previous examples, where the object is to extend the hold of the strap on the top rail. The strap of the bottom hinge may in all cases be very short, and not as in Fig. 1, where for uniformity’s sake it is extended to the same length as the upper strap. Fig. 6, No. 2, is a section through the upper part of the lower rail of Fig. 6, No. 1; and Fig. 7, No. 2, a section through the rail of Fig. 7, No. 1. Both of them show that at the lower side of the panel the mouldiug is sub¬ stituted by a splay or weathering to throw off the water. Plate LXI. —Dock Gates. — Fig. 1 is an elevation of the convex side of one of the gates of the Coburg Dock, Liver¬ pool. Fig. 2 is an elevation of its concave side. Fig. 3 a horizontal section through the gate, immediately above the sill. Fig. 4, a horizontal section under the top rail. The framing of each gate or leaf consists of a heel-post, on a pivot fixed to which the gate turns; of a head-post, or, as it is sometimes called, a mitre-post, and thirteen curved horizontal rails, called bars or ribs, tenoned iuto the head and heel posts. The whole are fastened by iron straps on each side securely bolted. The vertical straps, it will be seen, have horizontal branches at each rib, and the diagonal strap, serving as a tie, is united to it at the top. The ribs are rebated to receive the close planking which extends to the first rib under the top, and which is flush with their face. There are also three vertical posts on each side of the gate, at equal distances apart, securely bolted, extending from the bottom of the gate to the top of the rib where the planking ends. The back of the heel-post is formed truly circular in section to fit closely the segmental groove in the stonework, called the hollow quoin. Each gate is hung at the top with a wrought-iron collar in a cast-iron anchor block let into the stone-work, and at the bottom it turns on a pivot pin of hard brass, moving in a cast-iron cup let into the masonry (see the detailed drawing of this, Fig. 7, Plate LXII.) The top of the pivot is let into a brass socket fixed in the bottom of the heel-post by wedging, and between them there is interposed a ball of hard steel. All these parts are truly turned and fitted. The outer end of each gate is sup ported by a brass roller with a lever adj usting apparatus. The roller traverses the cast-iron segments let into the masonry, as seen in Fig. 3, and detailed in section in Fig. 6, Plate LXII. The two central divisions, formed by the vertical bars, contain the paddles or sluices, which are formed of cast- iron faced with brass, and work in cast-iron framing let into the gate. Part of the cast-iron work of the sluices is shown in detail in Fig. 5, Plate LXII., and the whole of their moving parts will be understood by an inspection of Fig. 1 of the same plate. On the top of the gate is a gangway, supported on brackets, and provided with iron stanchions, and chains 178 PRACTICAL CARPENTRY AND JOINERY. or rails as a fence. The outside stanchions are fitted in sockets, and are made to unship. Plate LXII.— Fig. 1 is the elevation, Fig. 2 a ver¬ tical section, and Figs. 3 and 4 horizontal sections of the top and bottom of one of the gates of the Victoria Dock, Hull; and Figs. 5, 6, and 7 are details of the parts which have been already referred to. The description of the construction of the Coburg Dock gates applies equally to this, and need not be repeated. The following is an ex¬ tract from a specification of the engineer of these works, Mr. J. B. Hartley, for a gate of similar dimensions:— There are to be two pairs of gates for a clear opening of GO feet from the outer face of the hollow quoins. The ribs, heads, and heels of these gates are to be either of English oak of the very best and quickest grown timber; or of African oak, but no mixture of these woods in the framing of any gate will be allowed. The planking to be of greenlieart timber. Each rib, head, and heel of each gate is to be formed of a single and self-contained piece of timber squared through to the full dimensions figured upon the drawings. The whole of the bolts, spikes, nuts, washers, and all other wrought-iron work connected with the gates, excepting the up and down cross and diagonal straps, the gangway-stanchions, and anchor-bolts, are to be galvanized in the best possible manner. The sluices in the gates are to be faced with brass, and fitted up in every respect in the best style of workmanship, and in every respect in accordance with the drawings. A cast-iron cup must be prepared and let into the masonry to receive the pivot pin in the heel of each gate. These pivot pins are to be of hard brass, and to be truly turned, so as to fit the cup, and also a steel ball working between the top of the pin and the heel casting as shown, &c., which must be bored to receive them, as also must the heel castings of the gates. The heel cast¬ ings are to be of hard brass, firmly wedged into the foot of the heel-post. When the gates have been framed and fitted together the heels are to be dressed off truly round, and are to be completely covered for 15 feet above the heel castings, with broad, flat-headed copper nails, 226 to the pound, to fit the hollow quoins, which must be rubbed and polished perfectly smooth, and truly vertical from top to bottom, in the hollow between the water lines, to receive them. The segment plates, on which the truck wheels of the gate are to travel, are to be of cast iron, in every respect in accordance with the drawings; they are to be carefully bedded down, very truly, upon the masonry prepared to receive them, and when bedded are to be fastened down by lewis bolts, as shown upon the drawings. Crab-boxes and gearing are to be provided on the gang¬ way of each gate to work the sluices in the gates. Wrought-iron stanchions are to be fixed on the inside of the gangway of each gate of a permanent character, and are to be provided with two rails of 1^ inch round iron, screwed up to each stanchion, on each side, as shown in the drawings. The stanchions on the outside of the gangways are to be moveable, and made to ship and unship into sockets provided for them; these moveable stanchions are to be provided with chains of galvanized quarter-inch round iron. Anchor-blocks are to be provided of cast iron, and let into the top of each set of hollow quoins, from which the anchor-bolts are to be carried into the masonry pro¬ vided for the purpose. These bolts are to be of 2|-inch round iron, and are to bo provided at each end with sufficient screws and washers, and are, when let into masonry, to be run round with lead and securely fixed. PAR T SIXTH. JOINERY. MOULDINGS. Plates LXIII.—LXIX. Before entering on the consideration of the subject of Joinery, it may be well, as introductory to it, to illustrate and describe the various ornamental mouldings which may have to be formed by the joiner. Plate LXIII.—Grecian and Roman versions of the same mouldings are shown on this plate. Fillet or Listel right-angled mouldings require no de¬ scription. The Astragal or Bead .—To describe this moulding, divide its height into two equal parts, and from the point of division as a centre, describe a semicircle, which is the contour of the astragal. Boric Annulets .—The left-hand figure shows the Roman, and the right-hand figure the Grecian form of I this moulding. To describe the latter proceed thus:— Divide the height b a into four equal parts, and make the projection equal to three of them. The vertical divi¬ sions give the lines of the under side of the annulets, and the height of each annulet, c c, is equal to one-fifth of the projection; the upper surface of c is at right angles to the line of slope. Listel andFascia. —(Roman.)—Divide the whole height into seven equal parts, make the listel equal to two of these, and its projection equal to two. With the third vertical division as a centre, describe a quadrant. (Gre¬ cian.)—Divide the height into four equal parts, make the fillet equal to one of them, and its projection equal to three-fourths of its height. Gavetto or Hollow .—In Roman architecture this mould¬ ing is a circular quadrant; in Grecian architecture it is an elliptical quadrant, which may be described by any of the methods given in the first part of the work. PLA TE LX1U. &q ® cm o= m i.m © s . Roman. Fillet or Lisi el. Grecubn. Ash an a Z or Be a <./. Doric, Annule/s. y_ L is tel and Facia . On'die or Ho U.o w. Ovolo or (Jnarlo Round. _ Cymcu Recta . Oymcb Reversa . _J _ Troc/dlns or Scolia , _* Iorus. J. White,, deb. BLACKIE , GLASGOW, EDINBURGH & LONDON. W.E.Bceva- Sc. Eli !■ =.] ih X Fin 1. jV] © OJ L 0 0 M S 3 o PLATE T2XJV. t h a Fiji. 2 . i > ■" ) Fi q . 2. d y : // j _ / n 1 ^ ^ d a /> Fi panel. Plate LXXIV., Fig. 1, is the elevation of a double- margined door of large dimensions, such as a folding-door used to close the communication between two drawing¬ rooms. But the door here illustrated, in place of folding, is made to slide into recesses in the wall or partition. Fig. 2 shows the plan of the doors and recesses, and Fig. 3 is a vertical section through the recessed part of the parti¬ tion and the door. Fig. 4 is a section to an enlarged scale showing the top rail of the door, the recess in the parti¬ tion, one of the pullies and the straps by which the door is hung to it, and the iron bar on which the pullies run. Figs. 5 to 13 are examples of French doors, showing many varieties of panelling and ornamentation. Plate LXXIII., Fig. 1, is the elevation of a jib door in the side of an apartment which has a base b, dado and dado moulding a; and Fig. 3 is a vertical section through the door. In Fig. 2 is drawn to a larger scale a horizon¬ tal section through part of the hanging style of the jib door and the frame in which it is hung. The dotted lines show the line of the hinging through the base mouldings, and the extent to which the door can be opened. Fig. 4 is a section through the dado moulding, and Fig. 5 a sec¬ tion through the skirting and base moulding, both to the same enlarged scale as Fig. 2. Fig. 6 is an elevation of a pew door; and Fig. 7 a section through its hanging style and the style of the framing to which it is hung. It will- be seen that it is treated precisely as a jib door. The jambs of doorways are finished with wooden linings which are either plain or framed, and moulded to cor¬ respond with the door; they are in either case fixed to the grounds, and if the jamb be wide the lining may re¬ quire backings or cross pieces to stiffen them. The doors are hung in their places in two ways. In the first a door frame is set in the wall or partition; to this the door is hinged, and the linings being kept back from the edge of the frame to a distance equal to the thickness of the door, thus form the rebate to receive the door and the stop against which it shuts. In the second mode the lining covers the whole of the jamb, and is rebated to form the recess to receive the door, and a corresponding rebate is formed on the other edge of the lining for appearance sake. Doorways are in general surrounded by an ornamental wooden margin or border, not merely, however, for orna¬ ment, but especially to cover the junction between the plaster and wooden ground in the case of a wall, or be¬ tween the plaster and door frame in the case of a partition. 1 These margins are sometimes plates of wood ornamented with mouldings, and are termed architraves, examples of which are given in Plate LXIX., or they may consist of pillars or pilasters with proper entablature. The pilasters are set on solid blocks of the same height as the skirting, and so also are the architraves in good work; but mother cases they run down and are scribed to the floor. It is said above that the chief use of the architrave is to cover the joint between the wooden ground or the door frame and the plaster. The ground being fixed and the door frame set in its place, serve as guides for floating the plaster by, and when the plaster is dry the architrave should be applied so as to lap over the joint and effectually cover it; but in ignorance of good construction it is common to fix | the architraves before the plastering is complete, a practice which cannot be too severely reprehended. WINDOWS, AND FINISHINGS OF WINDOWS. Plates LXXVL—LXXYIII. Windows consist of the glazed frames called sashes , and of the frames or cases of various kinds which contain these. The sashes may be either fixed, or hinged to open like a door, or suspended by lines over pullies and balanced by weights. The frame for the fixed sash consists of solid sides or styles, a head piece or lintel, and a sill, which is made wider than the other pieces and weathered. This frame is rebated to receive the sash, and the latter is retained in its place by a slip of wood nailed round the inside of the frame. The hinged or French sashes, as they are termed, ha ve rebated solid frames, and in their construction every care is required to make them weather tight at the sills and where they meet in the middle. In exposed places they should always be made to open outwards, as the effect of wind is then to close them and make their joints tighter. Suspended sashes are hung on frames provided with boxes or cases to contain the balancing weights. In order that the reader may become familiarized with the several parts of the sash-window, its frame, its shutters, and finish¬ ings, sketches are here presented of the horizontal and vertical sections through a window, and these being de¬ scribed and a notion of them acquired, it will not be necessary to embarrass him with repetitions of the descrip¬ tion of the same details in the plates. Fig. 507 shows a horizontal section, and Fig. 508 a vertical section of the window frame. The frame consists of sides or breasts of about H inch thick, grooved down the middle for the reception of a beaded piece o o (Fig. 508), called a parting bead, from its serving to part the sashes. The sides or breasts are called the pulley styles, and the frame is completed by the sill below and the lintel above. To the outside edge of the pulley style the beaded pieces// forming the sides of the casing or boxing are attached, and the beaded edge projects so far beyond the face of the style as, with the parting bead, to form the outer path or channel in which the sash slides. On the inner edge of the pulley style is fixed the piece b b (Fig. 507), called the inside lining, to which the shutters are hinged; a back piece extending between the inside lining and outside piece f parallel to the pulley style, is added, to complete the case or boxing, and the box has sometimes, and should always 188 PRACTICAL CARPENTRY AND JOINERY. have, a division in the centre to separate the weights of the upper and lower sashes. The path for the inner sash is formed by a slip or stop bead, fixed to the styles by nails or, preferably, by screws. In the lower end of the path of the outer sash a hole is cut in the pulley style suffi¬ ciently large to admit the weights, so that the sashes may be hung after the frames are fixed, and the lines repaired at any time. This is called the pocket, and it is covered by a piece of wood attached by screws. The sashes themselves consist of an outer frame, which is composed of styles and rails. The bottom rail of the lower sash is deeper than the others, and is throated to prevent the water from driving under it. The meeting rails of the upper and lower sashes are made wider than the others and fit together in the manner shown by A, Fig. 508. The horizontal and vertical bars, which divide the sashes into panes, are termed sash bars. The vertical bars, like the styles in framing, extend, in single pieces, between the rails ot the sash, the horizontal bars being cut to fit their places, and dowelled together through the vertical bars. The fittings of a window consist of the boxings for the shutters, if there be any, the linings, the shutters, with their back flaps, and the architraves, or other finishings of the opening in the apartment. All these parts are exhibited in Fig. 507 in horizontal section. The boxings are formed in the space between the inside lining ol the sash frame and the framed ground. The back of the recess is sometimes plastered; but, in better works, it is covered with a framed lining, as at a in the Fig. 507, called a-back lining. This has generally bead and flush panels, and is fitted in between the inside lining b, and the framed ground c, and is gene¬ rally tongued into both. The shutters cl, are framed as doors, and panelled and moulded in the same manner. They are hinged to the inside lining. The back flaps are generally lighter than the shutters, and are sometimes framed and moulded, so that the whole exposed surface shall present the same appearance when the shutters are closed; but they more frequently consist of bead and flush framing. The shutters in place of being hinged, are some¬ times suspended and balanced by weights, like the sashes. The wood work M M, Fig. 508, which extends from the window sill to the skirting, is called the breast lining; and that on the side of the recess, extending from the bottom of the shutters to the skirting, the elboiu lining. The ceiling of the window recess P P is also formed of wood, and is termed the soffit lining. These linings are all framed and moulded to correspond with the doors and other framed work of the room. The margin of the window opening is finished with architraves or other ornamental appliances, in the same manner as the doors. Plate LXXVI.— Fig. 1, No 1, is the elevation of a sashed window with its finishings. Immediately under No. 1 is an enlarged section of one of the jambs of the window, showing in detail the lower sash, the pulley piece, boxing, weights, shutter boxing, back lining, archi¬ trave and shutters. Fig. 1, No. 3, is a vertical section to the same enlarged scale, through the lower part of the window and the window breast, showing the rail of the lower sash, the sill of the window frame, the breast lining and skirting. The upper part of the window is shown in Fig. 1, No. 3, at the bottom of the plate. It shows the rail of the upper sash, the lintel of the window frame, the soffit lining, and the architrave. Fig. 1, No. 4, is a section through the meeting rails of the upper and lower sashes, showing the nature of the rebate, or check, as it is called in Scotland. Fig. 2, No. 1, is a sashed window double-margined, in imitation of a Fi'encli window. Fig. 2, No. 2, shows the details, on a larger scale, of the shutters, which are so arranged as to form the pilasters at the sides of the opening in the apartment. Fig. 2, No. 3, is a section through the top of the window, showing the details of the entablature over the pilasters; and Fig. 2, No. 4, is a section through the centre style ot the sash. A sash bar, full size, is shown in section on the left- hand side of the plate. Plate LXXYII.— Fig. 1 shows part of the elevation, and Fig. 2 part of the plan of a window on an irregular octagonal plan. In this the shutter of the widest part of the window sinks into the breast and is suspended over pullies, and balanced by weights like the sashes. Fig. 3 shows the manner of attaching the suspending lines by means of a bracket carried below the bottom of the shutter, so that the pulley and weights may be entirely contained in the window breast. Fig. 4 is a plan or horizontal sec¬ tion through the breast; and Fig. 5 is a vertical section of the same. The shutters of the side windows are hinged to the frame in the usual way, as seen in Fig. 2, which repre¬ sents a horizontal section of the window above the sill, with the shutters closed. Fig. 6 is a section of a sash bar, full size. ' Plate LXXVIII. —Fig. 1 shows the finishing of a window, looking upwards towards the soffit. The dotted lines A B show the shutters and back flap when closed ; ff[] NOSIK]3MSS :0iF WONDOWS PLATE LXXVI. IBLACK1E & SON. C.LASGff#,EDINBURGH. 8c LONDON. i T.W. Loyvry fc. PI, A Tti LXXIX. J While, del Fill ■ © J H S ULAK W ] i x J © © W o Fig. 1. Fig 3 i Feet Fig 2 - hbbbbhb SKYLIGHTS. 180 and the shaded parts, A b, the shutter and back flap when folded into its boxing; C H D E is the boxing and back lining; G the ground; and f the architrave. Fig. 2 shows the soffit, the shutters, and finishings of a window in a wall thicker at one side than the other; the drawings are so detailed as to be self-explanatory. Fig. 3, the soffit, shutters, and finishings of a window in a circular wall. CIRCULAR WINDOW. Plates LXXIX, LXXX. Plate LXXIX., Fig. 1 , No. 1 , is the plan, and No. 2, part of the elevation of a circular window, with diamond formed panes. In No. 1, a are the jambs, b the sash, c the pulley style, e f the window sill, d the inside lining. The centre lines of the divisions for the panes are set out on the plan of the lower rail of the sash, at c d e /; the thickness of the bars at a b, k l, o, &e.; and the sites of the crossings or intersections of the bars at h i, n, &c. The heights are set out in a similar manner on No. 2, and from these data the elevation can be drawn, as in the manner already familiar to the reader. Fig. 2 is an eleva¬ tion of the lower sash on the stretch-out; and Fig. 3 the stretch-out of the diagonal bar A D, showing the twist occasioned by the curvature. To find tlce mould for the curvature of the diagonal sash-bar.— In Fig. 4, let A b e d be the plan of the bottom rail, and let it be divided into any number of equal parts —1, 2, 3, 4, &c. Through these draw ordinates at right angles to the chord line A B, meeting it in a b c d, and produced beyond it to meet the chord line A c of the dia¬ gonal bar in l m n o p. Through these points of inter¬ section draw ordinates perpendicular to A c, and make them equal to the corresponding ordinates of A B, as l qv to a f 1, m r w to b g 2, &c. Fig. 5 shows the section of a sash bar, full size, with a dowel hole A; and underneath is shown the mitring of the bars BDCE, and the dowel A A, all full size. Plate LXXX.—The figures in this plate illustrate the framing of a circular headed sash in a circular wall. Fig. 1, No. 1, is a plan of the window, and No. 2 an elevation of the circular headed sash. In No. 1, A A are the jambs, s the outside lining, b c the upper sash, g g the pulley styles, H the inside lining, e the parting bead, F the stop bead, or batten rod, as it is sometimes called, M the sill. To find the veneer for the arch-bar k l m called the cot-bar, or chord-bar .—Set out the stretch-out of the arc K L M, Fig. 1, on the line AB, Fig. 4, and draw lines from the divisions in the arc to any chord line, as no, No. 1; then make the ordinate c a D, Fig. 4, equal to o z p, No. 1, 3 be equal to r t y, and so on; then gedfh, Fig. 4, will be the veneer for the arch-bar. To find the mould for the radial bars. —From p in No. 1 draw P R a tangent to the curve; and on it draw lines from division l in the radial bars fheg, -and pro¬ duce them to cross the plan of the lower sash; then transfer the ordinates R h i, j k l, &c. to H r s, 3 t u, aud the ordi¬ nates of the bars in No. 2, and the moulds lg Fs of the bars E g F H will be obtained. To find the face mould for the circular outside lining. —The dotted line aklmno p. Fig. 1, No. 2. shows the lower edge of the lining; and lines drawn through these points perpendicular to A c, cut the line s g (No. 1), in ab c d ef g. Transfer these on the stretch-out to the line A B, Fig. 5, and draw ordinates perpendicular to A B, on which set up the corresponding heights from No. 2, as b k to b g, c l to c h, dm to d k, &c. To obtain the moulds for the head of the sash frame apply the stretch-out of the outside of the arch in No. 2 to the base line A B in Fig. 2, and set out on the ordinates drawn through the divisions, the corresponding ordinates from the chord I K in No. 1. To obtain the mould for the underside of the sash, Fig. 3, set out the divisions of the underside of the arch in Fig. 1, No. 2, along the base line A B in Fig. 3, and proceed in the same manner as above, but setting out the ordinates from the chord line L M. Fig. 1, No. 3, shows the first division of the sash frame A n in No. 2, and the plan No. 1; the thickness of stuff' required to work it out of the solid is shown at E N. The joint at N, No. 2, and k h, No. 3, is shown at b c in Fig. 2, and c F in Fig. 3. Figs. 6, 7, and 8, are sections through the sash frame and sash at three points: 1st, above the springing; 2d, at /; and 3d, at the centre. The part corresponding to the pulley style is now divided into two pieces, B and c, and the parting bead b is inserted between them, a is the outer, and c the inner lining. The latter beaded and grooved for the recejition of the soffit lining. A is the sash. SKYLIGHTS. Plates LXXXL—LXXXIII. Plate LXXXI., Fig. ,1.—In the skylight, of which No. 1 is the plan, and No. 2 the elevation, it is required to find the length and backing of the hip. Let A B be the seat of the hip; erect the perpendicular A c, and make it equal to the vertical height of the sky¬ light, and draw B c, which is the line of the underside of the hip; the dotted line g h shows its upper side. To find the backing, from any point in B c, as n: draw perpendicular to b c, a line n F meeting a b in f, and through F draw a line at right angles to A B, meeting the sides of the skylight in D and E. Then from F as a centre, and with F n as radius, cut the line A B in m, and join D m, E m. The angle D m e is the backing of the hip, and the bevel k ml will give the angle of backing when applied to the perpendicular side of the hip bar. In Fig. 2, in which No. 1 is the. plan, and No. 2 the elevation of a skylight with curved bars, to find the hip: let A B be the seat of the centre bar, and d e the seat of the hip. Through any divisions 1 2 3 4 c of the rib, over a b draw lines at right angles to A b, and produce them to meet ED in p o n m D. From these points draw lines perpendicular to E D, and set up on them the correspond¬ ing heights from A B, as l 1 in p 1, k 2 in o 2, &c. In the irregular octagonal skylight, Fig. 3, Nos. 1 and 2, the length and backing of the hips is formed as in Fig. 1, No. 1, by drawing a c perpendicular to A B, and set- ting up on it the height of the skylight in A c, then PRACTICAL CARPENTRY AND JOINERY. 190 drawing b c. The point F is found by drawing g F per¬ pendicular to B c, from any part of B c, and through F drawing d E at right angles to A B to meet the adjacent sides of the skylight in D e, then making f h equal to F g, and joining d h E h. In the octagonal skylight, Fig, I, No. 1 and 2, which has curved ribs, the process of finding the hips is exactly similar to that employed in Fig 2, and need not be again described. Plate LXXXIL, Fig. 1.—Nos. 1, 2 and 3 are the plan, side elevation, and end elevation of an irregular octagonal skylight, and Fig. 2, Nos. 1, 2, and 3, the plan, side eleva¬ tion, and end elevation of an elliptical skylight, neither of which requires detailed description. Fig. 3, No. 1, is the plan, No. 2 the end elevation, and No. 3 the side elevation, of an elliptical domical skylight. The section of the skylight on the minor axis is a circular segment, as seen in No. 2. To find the ribs in Fig. 4, describe the quadrant A B, and in c B make the height, D B, equal to the height of the segment in No. 2; draw E D, and make E L equal to the length of the rib over the minor axis, and draw c L to find the bevil, L w, of the end. Divide the arc E L into any number of parts, and through them draw lines per¬ pendicular to A c, and produce them indefinitely; draw also through, in the lower end of the rib, the line m K per¬ pendicular to A c. Then from D as a centre, with the length of the longest semidiameter of the ellipse as radius, cut the line m K n K, and draw D K, and produce it to n to meet the perpendicular A n from A. Then the line D n will be the semi-axis major of an ellipse, as A c in Fig. 5, and the segment of it formed by the lines L u m K in Fig. 4 will be the rib standing over the semi-axis major. But all the ribs may be drawn by ordinates thus: From D, Fig. 4, as a centre, and with the lengths of the several ribs as radii, cut the line m K in H G F, and through these draw lines from D, meeting A n. Then the points where these lines are crossed by the perpendiculars to A c, pass¬ ing through the divisions 1 2 3 4 L in the arc E L are the places of corresponding ordinates, by which the curves may be drawn, as D u t K in Fig. 5, Du(H in Fig. 6, D u t G Fig. 7, and d u t f in Fig. 8. Plate LXXXIIL, Fig. 1.—No. 1 is the plan, and No. 2 the elevation, of an octagonal skylight. No. 3 is one of the sides laid over on the horizontal plane of projection. Fig. 2, No. 1, is the projection of a portion of the inside of the skylight looking up, and No. 2 is an elevation of a portion of the interior corresponding to the last. The mode of finding the lengths and backings of the hips and ribs is developed in the lower half of the plan. First, the hip B D. Make D E equal to the vertical height, and join B E. From any point b in b E let fall a perpendicular meeting B D in c; make c a equal to c b, and join c a and A a, and produce the latter to T. Then c a T is the bevil for the backing of the hip to be applied to the vertical side of the rib. It will be seen that the rib K I is found in a similar manner from the right angled triangle K I L, of which the hypotenuse K L is the length of the rib as before. In obtaining the ribs on the hither side of the octagon a compendious method is adopted. Let P Q be the seat of the hip, and N t), G f the seats of any other ribs; on f G construct the right angled triangle, F G H, as before, and from any puint, R, in the hypotenuse draw R s, parallel to H g, and R e at right angles to f n. From R as a centre with any radius describe a circle as cl e f g, and through e and / draw lines parallel to H G. At the points where these cut the seats of the ribs erect perpendiculars, as at n m Jc Lp o*i h, and iutersect them by tangents from the circle parallel to H g, as dl m, g h o\ then join m s, and o s, and we obtain rso and Rsmas the bevils for the backing of the rib P Q, and in like manner the backing of any other rib is obtained. Fig. 3, No. 1, shows the rib at f g in Fig. 1, No. 1, to a larger scale, and No. 2 shows a hip rib, the angle of backing, Q D s, being the same as A a c in Fig. 1, No. 1. No. 3' shows the common bar corresponding to the line i K in Fig. 1, No. 1, the angle pfo, being the same as vv r x in the latter Fig. No. 4 is the hip as seen in Fig. 2, Nos. 1 and 2. The manner of finding the mouldings of the angle bars and ribs, as exemplified in this Fig., has been already^ described in detail, and on examination it will be seen that the same letters refer to the same parts in all the mouldings, by which their correspondence can be readily traced. Fig. 4 is the window bar. Plate LXXXIII?— Pulpit with Acoustical Canopy .— The drawings sufficiently Explain the construction of the pulpit; and of the canopy, which is novel in design; we insert the following explanation, kindly supplied by its author, Mr. Wylson. The concave interior surface is generated from a point which appears in the flank eleva¬ tion, a little above the level, and in advance, of the desk. In the first place, a parabola was drawn, planways and sectionally, having its focus in what was considered as the average position of the speaker’s mouth; and a point was then found from which, as a centre, could be drawn, on plan and section, a circular arc coinciding as closely as possible with the parabolic curve, which point is the one already mentioned. The adopting a spherical surface in lieu of a paraboloidal one was for the purpose of simplify¬ ing the construction. The curves appear, in dotted lines, in the plan and flank elevation. The exterior, or back, of the canopy, bears no affinity to the interior it being straight horizontally. The canopy stands independently of the pulpit, and could be removed without interfering therewith. HINGING. Plates LXXXIV.—LXXXVI. The art of hanging two pieces of wood together, such as a door to its frame, a shutter to the lining, or a back flap to a shutter by certain ligaments that permit one or other of them to revolve. The ligament is termed a hinge. Hinges are of many sorts, among which may be enumerated, butts, rising hinges, casement hinges, chest hinges, coach hinges, folding hinges, garnets, screw hinges, scuttle hinges, shutter hinges, desk hinges, back fold hinges, esses, and centre-pin or centre-point hinges. As there are many varieties of hinges, there are also many modes of applying even the simplest of them, and much dexterity and delicacy is frequently required. In some cases the hinge is visible, in others it is necessary that it should be concealed. In some it is required not [POJILIPU To PLAT]-: IPOSIkP IT AND A©® tm ST 3 S A 1L Si&SiSIPY, PSEE STLBKIES flMffltRSIH (BLASSJW. JAMES WYLSON, ARCHT: HaJ / Thin of I Plllpil Half Plan of part below Pulpit. 1. XXXIII <1 © J. Hy l.ion, del. B1.AOKJE V, ElTlNBURC.il Sc 1. ONI)OX . H'.J.Pt SKY [L 0 © [HI Y S o PLATE LXXX11. tr~ fit/. 7. y}. uv i ni r i LidjjJ- 5 - g n Feat. BLACKIE & SON , GLASGOW. EDINBURGH & LONDON.. //’. / Bearer. Sc. KUl.NUr LABOUR SAVING MACHINES. 191 only that the one hinged part shall revolve on the other, but it shall be thrown back to a greater or lesser distance from the joint. In Plates LXXXIV.—LXXXVI. are figured a great variety of modes of hinging. Plate LXXXIV. — Fig. 1, No. 1, shows the hinging of a door to open to a right angle, as in No. 2. Fig. 2, Nos. 1 and 2, and Fig. 3, Nos. 1 and 2. These figures show other modes of hinging doors to open to 90°. Fig. 4, Nos. 1 and 2. These figures show a manner of hinging a door to open to 90°, and in which the hinge is concealed. The segments are described from the centre of the hinge g, and the dark shaded portion requires to be cut out to permit it to pass the leaf of the hinge g f. Fig. 5, Nos. 1 and 2, show an example of centre-pin hinge permitting the door to open either way, and to fold back against the wall in either direction. Draw a b at right angles to the door, and just clearing the line of the wall, or rather representing the plane in which the inner face of the door will lie when folded back against the wall; bisect it in /, and draw f d the perpendicular to a b, which make equal to a f or / 6, and d is the place of the centre of the hinge. Fig. 6, Nos. J and 2, another variety of centre-pin hinging opening to 90°. The distance of b from a c is equal to half of a c. In this, as in the former case, there is a space between the door and the wall when the former is folded back. In the succeeding figures this is obviated. Fig. 7. No. 1. Bisect the angle at a by the line a b ; draw d e and make e g equal to once and a half times a d; draw f g at right angle to e d, and bisect the angle f g e by the line c g, meeting a b in b, which is the centre of the hinge. No. 2 shows the door folded back when the point e falls on the continuation of the line/gr. Fig. 8, Nos. 1 and 2. To find the centre draw a b, making an angle of 45° with the inner edge of the door, and draw c b parallel to the jamb, meeting it in b, which is the centre of the hinge. The door revolves to the extent of the quadrant d c. Plate LXXXV. — Fig. 1, Nos. I and 2; Fig. 2, Nos. 1 and 2; and Fig. 3, Nos. 1 and 2, examples of centre-pin joints, and Fig. 4, Nos. 1 and 2, do not require detailed description. Fig. 5, Nos. 1, 2, and 3, show the flap with a bead a closing into a corresponding hollow, so that the joint can¬ not be seen through. Fig. 6, Nos. 1, 2, and 3, show the hinge a b equally let into the styles, and its knuckle forming a part of the bead on the edge of the style B. The beads on each side are equal and opposite to each other, and the joint pin is in the centre. Fig. 7, Nos. 1, 2, and 3. In this example, the knuckle of the hinge forms portion of the bead on the style B, which is equal and opposite to the bead on the style A. In Fig. 8, Nos. 1, 2, and 3, the beads are not opposite. Plate LXXXVI. — Fig. 1, shows the hinging of a back flap when the centre of the hinge is in the middle of the joint. Fig. 2, Nos. 1 and 2, shows the manner of hinging a back flap when it is necessary to throw the flap back from the joint. Fig. 3, Nos. 1 and 2, is an example of a rule-joint, such as is required for the shutter in Fig. 2, No. 2, Plate LXXVI. The further the hinge is imbedded in the wood, the greater will be the cover of the joint when opened to a right angle. Fig. 4, Nos. 1 and 2, shows the manner of finding the rebate when the hinge is placed on the contrary side. Let / be the centre of the hinge, a b the line of joint on the same side, h c the line of joint on the opposite side, and b c the total depth of the rebate. Bisect be hi e and join e /; on e f describe a semicircle cutting a b in g, and through g and e draw g h cutting d c in h, and join d h, h g, and g a to form the joint. Fig. 5, Nos. 1 and 2, is a method of hinging employed when the flap on being opened has to be at a distance from the style. It is used in doors of pews to throw the opened flap or door clear of the mouldings of the coping. Fig. 6, Nos. 1 and 2, is the ordinary mode of hinging the shutter to the sash frame. LABOUR SAVING MACHINES. In many of the operations of the joiner, in which numerous copies of the same thing have to be produced, accuracy is insured by introducing the principle of the guide, either to direct the tool over the work or the work over the tool. Examples of this are found in the mitre- box, the shoot blocks, and in the various kinds of fences and stops. These appliances are obviously the first step towards seeking the aid of machinery in performing operations requiring frequent repetition; and, accord¬ ingly, we find the principle of the guide applied first to simple sawing, then to planing, and subsequently to grooving, tonguing, mortising, tenoning, and shaping. At first, as is usually the case, the applications of machinery to these works were in direct imitations of the actions of the workman. Thus, in the first planing machines the work was fixed, and the plane made to pass over it with a reciprocating motion ; but, eventually the same effect came to be better produced by means entirely different. It was not till near the end of the last cen¬ tury that circular saws were introduced into England, although they were previously used on the Continent for small work in all kind of materials. The first attempt to construct a planing machine, is stated by Mr. Moles- wortli to have been made by a Mr. Hatton in 1776,* and the next attempt was made by Sir Samuel Bentham in 1791. When Sir Samuel Bentham was in Russia, previous to the date mentioned, he had made considerable progress in contriving machinery for shaping wood, so as to insure accuracy and save manual labour. Besides the general operations of planing, rebating, mortising, dovetailing, grooving, bevelling, and sawing in curved, winding, and transverse directions, he had completed, in the way of example, an apparatus for preparing all the parts of a highly finished sash window; another for preparing every part of an ornamental carriage-wheel, and nothing re¬ mained for finishing the work of the joiner or wheel¬ wright, but the putting the several component parts * On the Conversion of Wood by Machinery, a paper read before the Institute of Civil Engineers, Nov. 17, 1857, by Guilford Lindsay Molesworth. 102 PRACTICAL CARPENTRY AND JOINERY. together. In 1793, Sir Samuel Bentham patented* several inventions, and as Mr. Molesworth well says, “had his ideas been carried out and the appliances of engineer¬ ing been more perfect, he would doubtless have introduced many of those inventions which were years afterwards brought out and patented as new.” In 1802, Mr. Bramah patented machinery for producing straight, smooth, parallel, and curvilinear surfaces on wood. In his planing machine the cutting tools were fixed in apertures near the circumference of a horizontal wheel, moving with great velocity on a vertical axis, and the timber laid on a horizontal carriage was moved for¬ ward under their action. It was in principle somewhat the same as the machine figured in Plate LXXXVII. In 1807, Mr. Brunei's famous block machinery was set to work in Portsmouth dockyard. It is not necessary here to trace the progress of conversion of timber by machinery; suffice it to say that Thomson’s machinery for sawing, gauging, and grooving and tonguing flooring-boards was in operation in 1826, and in 1827, Mr. Muir, of Glasgow, patented a machine for working flooring-boards, which has since served as a model for those subsequently intro¬ duced in Britain. In this machine the bottom of the board was roughly planed by a rotatory adze, while another similar adze operated on the upper surface. The board then passed between two fixed cutters set obliquely, which removed a shaving of the length and width of the deal, while two revolving cutters or saws made the sides parallel, and two other cutters grooved or tongued the edges. In the hands of Mr. M‘Dowall, of Johnstone, the flooring machine has in design and workmanship ap¬ proached perfection. But it is not to these machines which we here desire to direct the attention of the reader. Machinery so ex¬ tensive and capable of producing so much, will not supply the requirements of the ordinary workshop. It is ex¬ pensive in its first cost, and requires a great amount of the same kind of work to keep it in remunerative opera¬ tion. For the ordinary workshop, where the trade is limited and much varied, the simpler though less perfect machines, which are used in America in aid of the work¬ man, are more suitable; and it is to the description and illustration of one or two of the more generally useful of these machines that we propose to devote these pages and plates. Plate LXXXYII. — Figs. 1, 2, 3. The small saw bench which is here figured, is suitable for jobbing work. It occupies little space, and can be applied in plain and bevel sawing, mitring, tenoning, rebating, &c. The bench is supported on four stout legs, firmly united by means of the top rails, and the middle rails which carry the plutnmer blocks of the driving pulley. It is only 3 feet 2 inches long, 2 feet 2 inches wide, and 3 feet 64 inches high. Its dimensions are confined within these narrow limits, by the mode adopted of banding the pulleys, about to be described, and which is the subject of a patent. Fig. 1 is a plan of the top; Fig. 2 is an elevation of one end; and Fig. 3 is an elevation of the front of the * In referring to the specification of this patent of 23d April, 1793, the editor of the Mechanic's Magazine , a competent authority, says it is a perfect treatise on the subject, indeed the only one worth quoting from which has to this day been written on the subject. Dec. 16, 1848. j bench. The same letters refer to the same parts in all the figures, a a a a are the supports, b b rails on which the plummer blocks of the driving pulley are fixed, and which carry also the braced upright c c, to which the cast-iron ; bracket, supporting the axis of the saw d d, is attached; e e is the driving pulley, moved by a handle attached to its axis; ff are radial iron bars moving freely on the axis of the driving pulley, and carrying at their outer end the tension pulley g g. The diameter of the driving pulley is 234 inches, the diameter of the tension pulley is 7f inches, the axle of the saw is 1 ^ inch diameter; it is suspended on conical steel centres, and serves as a third pulley. The band passes over the tension pulley and over the saw axle, and its lower web is pressed against the periphery of the driving pulley by the weight of the | tension pulley. The saw is 8 inches in diameter, and ! makes 154 revolutions for each turn of the handle. Fig. 1 is the plan of the top of the machine. It is com- I posed of a front board h h, hinged to the frame at h, so that its other end can be elevated by means of a wooden screw m, as seen by the dotted lines in Fig. 3. Through a slot in the top of this the saw d d works; n n is the back board, which is capable of being slidden along parallel to the blade of the saw, being guided by a fillet on its under side, sliding in a groove in the fixed top- board of the machine below it, seen at 0 0 , Fig. 2. This sliding board has various holes, some screwed and others plain, for fixing the fences, guides, and stops to be here¬ after mentioned; and as the pins and screws which secure j these project below the bottom of the board, there are longitudinal channels grooved out in the path of these holes in the fixed top p p, underneath. When the machine is used for ripping or cross-cutting, j a parallel motion fence q q is screwed by a wooden liand- i screw to the front board h h, Fig. 1. The distance between : the fence and blade of the saw regulates the scantling to j be cut off*; a stop is fixed to one of the holes in the sliding ! board, against which the stuff is held, and the board being slidden along in the direction of the arrow, the 1 stuff is acted on by the saw. By raising the front board by means of the screw m, the machine can be adjusted for tenoning. The board is raised until just so much of the saw is exposed as is equal to the depth of the shoulder of the tenon, and the fence is set to the proper length of the tenon. The stuff is passed along, and the shoulder on one side is cut; it is then reversed and the operation repeated for the other side. When the shoulders of all the pieces have been cut, the front board is dropped until the saw projects through to the length of the tenon; the fence is then moved towards the saw, till the space between them is just equal to the depth of the shoulder, and the stuff is then passed through, on end, twice; and if the adjustment has been correctly made the tenon is formed perfectly square and clean. For bevel cutting and mortising, the guide shown at ft, Fig. 1, is used. It con¬ sists of a stock, which, by means of the handscrew r, can be fixed to the sliding top. To the end of the stock are hinged the arms 2 2, and to a block sliding freely in a slot in the stock are hinged the short arms 3 3, hinged also at their other end to 2 2. By means of the screw 4 the block is made to traverse the slot, and the arms 2 2 are moved so as to make a greater or less angle with the stock. : B m 8 m ( I II LAI0I® 3 A'yiHS M ASM OKIE So Furness’s patent mortising and tenoninc machines. PLATE LAWXI 7i - LABOUR SAYING MACHINES. When mitring is to be performed, the arms 2 2 are adjusted, to make an angle of 45° with the plane of the saw, and the stuff being held against one or other of them, the top board is shot so as to bring it under the operation of the saw. On the inner end of the shaft of the driving pulley, there is a crank which may be acted on by a treadle like a turning lathe when the stuff is thin; but, in ordinary cases, the machine requires two operators. It is not necessary to describe further the many operations which . this little handy bench may aid in performing with accu¬ racy and despatch. Figs. 4, 5, 6, illustrate Furness’ patent planing machine. In this machine the stuff is operated upon by cutters, held by horizontal arms fixed to a vertical shaft, and it is in this respect similar to the machine patented by Mr. Bramah in 1S02, but it is much simpler and less expen¬ sive. The same letters refer to the same parts in all the figures. A A ft a is the fixed framing of the machine. B B a travelling bed piece, on which the stuff to be ope¬ rated on is fixed. Its upper surface, on which the stuff rests, is itself planed true by the revolving cutters. It travels on cast-iron ways b' b', and has a rack z z fixed to its under side, into which a pinion D on the hand- wheel 1, and also a pinion E on the bevel-wheel 2, gear. The cutter frame c C is of cast-iron; it slides between the vertical pieces of the main framing a a, and can be raised and lowered by means of the toothed rack b, which is acted upon by the winch handle c, by means of the screw and bevel pinion d and e. This frame carries a vertical spindle /, with a long pulley of timber g, fitted to it. The lower end of the spindle has attached to it the arms h h, which carry the cutters m m. On the lower part of the ! frame c is fixed the cast-iron disk plate n n, which presses on the timber while the cutters operate upon it close to its periphery. The main driving belt gives motion to the shaft o o, by means of the fast pulley p, and is shifted to the loose pulley p, when the machine is to be stopped. The pulley q on the shaft o drives the long pulley g on the cutter shaft, and another pulley on the top of o drives the pulley r on the shaft s. A third pulley t, fixed on the shaft o, gives motion to a pulley v, carried by an inter¬ mediate shaft, shown by dotted lines, and through this in- j termediate shaft and its pulleys, to the pulleys w w ' iv”, moving also on the shaft s. xx is a clutch, by means of which either the pulley r or the pulleys w can be keyed to the shaft, and while the other remains loose, and i thus it may be made to revolve in either direction. The ! shaft s has two sliding bevel pinions y y, at its lower end, either of which can be thrown into gear with the bevel wheel 2, which, as has been said, has a pulley on its shaft gearing into the rack z on the under side of the travelling bed B B. When by means of the upper handle and clutch x x, the pulley r is made to revolve with the shaft, the travelling bed brings the stuff forward in the direction of the arrows, under the operation of the cutters. When the timber has all been passed through, r is thrown loose, and w is fixed, and the carriage moves back with rapidity. The carriage can also be moved back¬ wards or forwards, by throwing the pinions y y out of or into gear, and it can also be moved by hand through the wheel s. Plate LXXXVIl."— Fig. 1, Nos. 1 to 10 are plans, 193 elevations, and details of Furness’ patent tenoning ma¬ chine. Fig. 1, No. 1 is the plan, Fig. 1, No. 2, a side elevation, and No. 3 an end elevation. The same letters refer to the same parts in all these. This machine, by means of revolving cutters a a, forms both sides of the tenon simul¬ taneously. The stuff to be operated upon is laid on a travelling bed M M, which moves transversely across the machine, so as to pass the stuff between the cutters. The lower cutter b is supported by the main frame B B, and the upper cutter a by the moveable frame c c, which can be raised or lowered, so that it can be adjusted to the thick¬ ness of stuff or depth of shoulder required; and the bed M M can also be raised or lowered, so as to complete the adjustment. The moveable stop p q regulates the length of the tenon. The main driving pulley / has another pulley 2 r, keyed on the same shaft, a belt from which passes over pulley h of the upper cutter, and then under the pulley h of the lower cutter, and over a tension pulley L, and then returns. By this arrangement the cutters move in opposite directions; and to allow of the adjust¬ ment for the various thicknesses of wood, and keep the belt tight on the pulleys, a simple contrivance is used. A leather belt s s is attached to the bottom of the stirrup spar 11, which is attached to the hinged cross-head K, carry¬ ing the tension pulley L, and the belt then passes through under an eccentric paul at 2 2, which holds it tight at any point required. We shall now describe the parts of the machine in detail. A A upright supports, which with the rails B b form the main framing of the machine, car¬ rying the driving pulley and the lower cutter b ; c c upright framing, which moves round c as a centre, and its opposite end can be lowered or raised by adjusting the radial slot at E, and for finer adjustment by the vertical screw f. It is further supported by the wrought- iron stay-rod g. This frame carries the upper cutter a, and, by means of the uprights H h, and jointed cross piece K, the tension pulley L. M M the bed frame on which the stuff to be operated on is laid. It is supported on the slotted upright board m m, the under edge of which rests on the wedge pieces n n. By means of a screw, the handle of which is seen at o, the wedges have a back¬ ward and forward motion, and raise and lower the frame as occasion may require; and it is fixed when adjusted by the screw mm, working through the slots in the up¬ right supports. The moveable fence p q consists of two parts, the distance between which can be regulated, and indicates the length of the tenon. The stuff is placed between the fence q and the cutter, and the fence p is set to the proper length of the tenon; the stuff* is then advanced so as to rest against p, and the other end con¬ sequently projects through the cutters to the required distance. The fences are kept in their position by springs, which can be depressed by the pressure of the wood. The wood is also pressed against the stops rr, and is kept fast on the frame by the lever s s, the outer end of which is grasped in the left hand of the workman, together with a little iron handle t, fixed to the frame, which he thus uses like pincers, and the bed frame is at the same time pushed from him by his right hand, so as to bring the stuff under the action of the cutters. The operation, which takes long to describe, is performed very rapidly. Fig. 1, No. 4 is the side view of one of the cutters. 104 PRACTICAL CARPENTRY AND JOINERY. No. 5 is the inside view of the disk of the same, and No. 6 the front or outside of the disk, all drawn to a larger scale. We have described the machine as adapted for tenon¬ ing, but it can be used also for sawing and boring. Nos. 7 and 8 are a profile and front view of the saw, which when used takes the place of the lower cutter, and is driven by reversing the banding, so as to cause the saw to revolve In a direction opposite to the motion of the sliding frame. The boring tool is seen in profile in No. 9. It is sus¬ pended between the piece u, fixed to the outside of the frame c c, Fig. 1, No. 3, and the slotted piece v, No. 10, which is attached to the inner side of the frame by a clamp screw working through the slot. It is driven bv a belt from the small pulley w, Fig. 1, No. 2, passing over the pulley x, carried by its own spindle. Fig. 2, Nos. 1 and 2 show Furness’ patent mortising machine. Fig. 2 is a side elevation, and Fig. 2, Nos. 1 and 2, details of the chisel socket. The machine consists of a sill A, bolted to the floor or otherwise steadied, two uprights B b, and two cross rails c c. D is a lever moving on a centre at d, and having the mortising tool attached to its other end at e, a treadle e, acting on the lever d, by the two iron straps f f, and the interposed ratchet lever f. The chisel socket, besides being attached to the lever D, is suspended to a sjjring pole G fixed to the ceiling of the apartment, whose elas¬ ticity raises it after every cut. The chisel socket H slides in the two eyes h h, formed on the ends of the bars of the iron frame Jc 1c. These bars slide through the uprights B B, and by means of the screw L can be moved forwards or backwards, so as to move the cutting chisel further from or nearer to the frame, according to the thickness of stuff. I is the bench or rest on which the stuff is laid. It is carried by the piece m, which has a bolt passing through a slot in the upright, so that by means of the clamping screw n it can be raised or lowered. To pre¬ vent the stuff rising with the chisel, there is a stop s s on each side. These work through eyes in the slotted pieces r r, and by means of screws can be adjusted along with the chisel. The depth of the cut of the chisel is regulated by moving the straps f f to various notches in the levers D F and e. The Figs. 2, Nos. 1 and 2, show in detail the manner of suspending the chisel socket. 1 is a rope attached to the spring pole G, 2 is a triangular iron link, to the top of which is attached the rope, and to the bottom a leather belt 4, clamped by the screw apparatus at 3, so that it can be lengthened or shortened at will. 5 is the end of the lever D, showing the manner in which it is attached to the socket; 8 is the socket turning freely round its axis in a head 6. This head has two wings 6 6 diametri¬ cally opposite, witli square notches in them, into which the detent 7 falls. When half the length of the mortise has been cut, the detent is withdrawn from the notch it Ki g . 509 may happen to be in, and the chisel is turned £ JL round till the detent falls into the opposite notch, and the remaining half of the mortise is completed, working from its extremity again P towards the centre. j' I Fig. 509, c d is a side, and b a a front view d b of the chisel used in mortising. This machine is also used for making dowels or wooden pins, by substituting a cylindrical cutter for the chisel. In conclusion, we note a few practical points to be ob¬ served in the construction of these machines. In machiues witli revolving cutters the general opinion is, that the greater the speed of the cutting-tool the better will be the quality of the work. The practical limit, however, appears to be between 2500 and 3500 revolu¬ tions per minute. A higher velocity heats the bearings, destroys the balance, and causes injurious vibrations. To produce a good result the travel of the work should be very slow relatively to the travel of the cutters. In some of the planing machines the cutters revolve with a velocity of 7000 feet per minute, while the work advances at the rate of only 30 feet, but as a general rule the work travels about of an inch for each stroke of the cutters. In order to withstand this high velocity, the framing ot the machine requires to be perfectly constructed. It should be of hard wood; and Mr. Molesworth directs— “ that the joints be not made so as to depend simply on mortise and tenon; they should be shouldered in. “ The bearings are sometimes made of an alloy, composed of 100 parts tin, 10 parts antimony, and 2 parts copper. In forming them the spindle is accurately fitted in its place, and the alloy is cast round it into an iron shell, which forms the plummer block. The parting of the base and cap is made by inserting a thin sheet of iron in the proper position.” “ Another peculiarity is the method of securing steadiness in high-speeded shafts; this consists in cutting in the journal a succession of angular threads, with angular grooves between. The alloy is cast round the journal as before, and great steadiness of action is secured; whilst the oil remains in the bearings without difficulty. An adjusting axle box is also much used; it is centred on two set screws, so as to allow it to turn slightly in the event of the opposite bearing being un- . evenly worn, and thus the extra wear and chatter which II would ensue at high speeds are obviated. STAIRS. 195 PART SEVENTH STAIRS AND HAND-RAILING* Stairs are constructions composed of horizontal planes elevated above each other, forming steps; affording the means of communication between the different stories of a building. In the distribution of a house of several stories, the stairs occupy an important place. In new constructions their form may be regular, but in the reparation or re¬ modelling of old buildings, the first consideration is gen¬ erally to make the distribution suitable for the living and lodging rooms, and then to convert to the use of the stairs the spaces which may remain; giving to them such forms in plan as will render them agreeable to the sight, and commodious in the use. A great variety of form in the plans of stairs is thus in a measure forced on the designer, leading to many ingenious contrivances for overcoming difficulties, disguis¬ ing defects, and enhancing accidental beauties, which he might not have adopted if unfettered in his choice. These inventions, originated by necessity, are again applied in cases where the necessity may not exist, recommended by their intrinsic beauty, or by the desire for variety in design. As introductory to the construction of stairs, a selec¬ tion of some of the more simple contrivances are here presented. That kind of stair which, after the common lad¬ der, is the most simple, is formed of a thick plank placed at a con¬ venient angle to form the ascent, and upon it are nailed pieces of wood to give a. firm foot¬ ing. This (Fig. 510) is often used in scaffolding. The stair next in degree is composed of horizontal planks forming steps, just sufficiently wide to give a footing; the planks are tenoned on the ends and let into mortises in two raking planks; the mortises are sometimes rectangular, as at d (Fig. 511), and sometimes they follow the inclination of the sides, as b and c. In the better sort the outer edge of the step has a nosing, as at c. The tenons of the steps are sometimes made so long as to pass entirely through the sides, and are secured by keys on the outside:—to preserve the planks which form the steps from splitting, the sides of the raking pieces are grooved to receive their ends. The opposite side pieces, too, are often bound together by iron rods; one end of each rod having a rivet head, and the other end being screwed with a nut to embrace the side Fig. 511. pieces. Such rods should be placed near the middle of a step, and close to its under side. Another method of forming a stair expeditiously, is to notch out the side pieces on their upper edge sufficiently Fig. sis. to receive the steps and risers, thus; a a the side pieces, b b the risers, and c c the step boards or treads (Fig. 512). The risers are nailed at the ends to the sides or strings, and the steps are nailed to the risers and also to the strings. Such methods as have been described are often used in ware¬ houses, factories, and agricultural buildings. There is a contrivance for economizing space sometimes used, which, perhaps, it may be well to mention, as the ascent is thereby made in about one half the space other¬ wise required. The width of this kind of stair is divided into two sets of steps, both of equal length and width, but the risers, except the first and last, are made twice the usual height; thus, if the line a n (Fig. 513) be 72 inches, and the width CD 33, and it is necessary to rise 80 in., divide the line a B in nine equal parts, and make the step equal to two of these parts; also, di¬ vide the width in two equal parts, and the height into ten equal parts, which gives 8 inches for the tread, 8 inches for the bottom riser, and 16 inches for the intermediate risers a a, &c., and 8 for the top riser b. Arrange the risers in such order that the face line of one riser shall be in the midway betwixt the face of the one next below and the one Fig. 514. Fig. 516. next above, as will better be seen by reference to Fig. 514. The height of the risers is so disposed that the bot¬ tom riser shall have the face of the first step 8 inches from * Contributed by Mr. Mayer, Montpelier Villas. Cheltenham. PRACTICAL CARPENTRY AND JOINERY. 196 the floor, whilst the first step on b shall be 16 inches from the floor, and the succeeding risers 16 inches each. In using this stair, one foot is placed on a step of one flight, as at a (Fig. 513), and the other on a step of the other flight, as at b, and so on alternately. Such stairs will only admit the passage of one person at a time. When it is required to admit of two persons passing each other, three flights are neces- Fig. sie. sary, the centre flight being ■ -- made wider than the exterior flights (Figs. 515 and 516). y-l This contrivance may be used in places not sufficiently spa- cious to admit of stairs of __ the usual construction. When houses began to be built in stories, the stairs were placed from story to story in straight flights like ladders. They were erected on the exterior of the build¬ ing, and to shelter them when so placed, great projection was given to the roofs. To save the extent of space re¬ quired by straight flights, the stairs were made to turn upon themselves in a spiral form, and were inclosed in turrets. A newel, either square or round, reaching from the ground to the roof, served to support the inner ends of the steps, and the outer ends were let into the walls, or supported on notched boards attached to the walls. At a later period the stairs came to be inclosed within the building itself, and for a long time preserved the spiral form, which the former situation had necessitated. Definitions. —The apartment in which the stair is placed, is called the staircase. The horizontal part of a step is called the tread, the vertical part the riser, the breadth or distance from riser to riser the going, the distance from the first to the last riser in a flight the going of the flight. When the risers are parallel with each other, the stairs are of course straight. When the steps are narrower at one end than the other, they are termed winders. When the bottom step has a circular end, it is called a round-ended step; when the end is formed into a spiral, it is called a curtail step. The wide step introduced as a resting-place in the ascent is a landing, and the top of a stair is also so called. When the landing at a resting place is square, it is designated a quarter space. When the landing occupies the whole width of the stair¬ case it is called a half space. So much of a stair as is included between two landings is called a flight, especially if the risers are parallel with each other: the steps in this case are fliers. The outward edge of a step is named the nosing; if it project beyond the riser, so as to receive a hollow mould¬ ing glued under it, it is a moulded nosing. A straight-edge laid on the nosings represents the angle of the stairs, and is denominated the line of nosings. The raking pieces which support the ends of the steps are called strings. The inner one, placed against the wall, is the wall string; the other the outer string. If the outer string be cut to mitre with the end of the riser, it is a cut and mitred string; but when the strings are grooved to receive the ends of the treads and risers, they are said to be housed, and the grooves are termed housings. Stairs in which the outer string of the upper flight stands perpendicularly over that of the lower flight are called dog-legged stairs, otherwise newel stairs, from the fact of a piece of stuff called a newel, being used as the axis of the spiral of the stair; the newel is generally or¬ namented by turning, or in some other way. The outer strings in such stairs are tenoned into the newel, as also are the first and last risers of the flight. When the upper and lower strings are separated by an interval, the space is called the well-hole. If the front string is mitred or bracketed, it is called an open string; if grooved, a close string. Where there is a well-hole and no newel, and the string is continued in a curve, the curved part of the string is said to be wreathed, and the stair is then a geometrical stair. Besides the support afforded by the strings the stair is sustained by pieces placed below the fliers; these are called carriages; they are composed of longitudinal and transverse pieces; the former are called rough strings, the latter pitching pieces; and the rough strings have trian¬ gular pieces called rough brackets, fitted to the underside of the tread and riser. The winders are supported by rough pieces called bearers, wedged into the wall, and secured to the strings. When the front string is ornamented with brackets, it is called a bracketed stair. Where communication between the stories is frequent, the qualities necessary in the stairs are ease and conveni¬ ence in using, combined with sufficient strength and dura¬ bility. Economy of space in the construction of stairs is an important consideration. To obtain this, the stairs are made to turn upon themselves, one flight being carried above another at such a height as will admit of head room to a full-grown person. Method of setting out stairs where the building is al¬ ready erected, or the general plan of the building is understood. The first objects to be ascertained are the situation of the first and last risers, and the height of the story wherein the stair is to be placed. A sketch is made of the plan of the hall to the extent of 10 or 12 feet from the supposed place of the foot of the stair, and all the doorways, branch¬ ing passages, or windows which can possibly come in con¬ tact with the stair from its commencement to its expected termination or landing are noted^ This sketch necessarily includes a portion of the entrance-hall in one part, and of the lobby or landing in the other, and on it have to be laid down the expected lines of the first and last risers. The height of the story is next to be exactly determined and taken on a rod; then, assuming a height of riser suitable to the place, a trial is made, by division, how often this height is contained in the height of the story, and the quotient, if there be no remainder, will be the number of risers in the story. Should there be a remain¬ der on the first division, the operation is reversed, the number of inches in the height being made the dividend, and the before-found quotient the divisor, and the opera¬ tion of division by reduction is carried on, till the height of the riser is obtained to the thirty-second part of an inch. These heights are then set off on the story rod as exactly as possible. The next operation is to show the risers on the plan, but for this no arbitrary rule can be given; the designer must exercise his ingenuity. STAIRS. 197 When two flights are necessary for the story, it is desir¬ able that each flight should consist of an equal number of risers; but this will depend on the form of the staircase, the situation and height of the doors, and other obstacles to be passed over or under, as the case may be. Try what the width of the tread will be by setting off, upon the line n a in Fig. 519, the width of the landing from the wall ab; and dividing the length of the flight into as many equal spaces as it is intended there should be steps in each flight. The landing covers one riser, and therefore the number of steps in a flight will be always one fewer than the number of risers. The width of tread which can be obtained for each flight will thus be found, and consistent with the situation, the plan will be so far decided. A pitch-board should now be formed to the angle of inclina¬ tion : this is done by making a piece of thin board in the shape of a right-angled triangle, the base of which is the exact going of the step, and its perpendicular the height of the riser. If the stair be a newel stair, its width will be found by setting out the plan and section of the newel on the landing; (if one newel, it should, of course, stand c in the middle of the width;) then, in connection with the newel, mark the place of the outer or front string, and also the place of the back or wall string, according to the intended thickness of each. This should be done not onty to a scale on the plan, but likewise to the full size on the rod. Set off on the rod, in the thickness of each string, the depth of the grooving of the steps into the string; mark also on the plan the place and section of the bottom newel; the same figure answers for the place of the top newel of the second flight, the flights being supposed of equal length. The front string is usually framed into the middle of the newel, and thus the centres of the rail, the newels, the balusters, and the front string range with each other; the width of the flights will thus be shown on the rod. It is a general maxim that the greater the breadth of a step the less should be the height of the riser; and con¬ versely, the less the breadth of step, the greater should be the height of the riser. Experience shows that a step of 12 inches width and 5^ inches rise, may be taken as a standard; and if from this it is attempted to deduce a rule of proportion, substituting, for the sake of whole numbers, the dimensions in half-inches, namely 21 and 11, then, in order to find any other width corresponding in inverse proportion, these values for t and r in the formula, we have - ^ ^ = 8^ inches as the breadth of tread. Suppose, again, the given breadth to be 13 inches, we 12 x 51- have-—— = 5,^- inches as the height of riser. This process of inverse proportion may be performed graphically as follows:— Let the tread and riser of a step of approved proportion be represented by the sides c b, 6 a, of the triangle ab c, Fig. 517. Through the pointa, draw a line daf parallel to the step line c b. Then, to find the riser for any other step, set off on the line c b, from the point c to d , the required y Fig. 617 Say as 24 : 24 : 24 : 22 13-8 13-2 Thus, it will be seen that a step of 6 inches in width will require the riser to be 11 inches, a step of 9| inches will need the riser to be nearly 7 inches, and that a step of 10 inches requires a riser of about 6f inches. The same thing is thus otherwise expressed. Let T be the tread, and R the riser of any step which is found to have proper proportion, then to find the proportion of any other tread t, and riser r,-= t , or —-— y. r t Take, for example, a step with a tread of 12 and a riser of 5^ inches as the standard, then to find the breadth of the tread when the given riser is 8 inches, and substituting width of a step, say 10 inches, and draw d d; draw also c d, and continue it to the line b a, and the point of in¬ tersection there will show the height of riser corresponding to the tread c d. In like manner, if the width given be 18 inches, set it off in the point 6; draw 6 e and ce, and the intersection at h will be obtained, giving 3-| inches for the height of the riser. A width of 20 inches will show a height of 3 3 inches. On the right side of the figure is shown each step we have mentioned, connected with its proper riser, thus exhibiting the angle of pitch. The same end nearly is arrived at thus:—In the right-angled triangle a b c, Fig. 518, make a b equal to 2d inches, and b c equal to 11 inches, according to the previous stan¬ dard propor¬ tion ; then to find the riser corresponding toagiven tread, from b set oft' on a b the length of the tread, as bd, and through d draw the perpendicular d e, meeting the hypothenuse in e ; then d e is the height of the riser, and if we join b e, the angle d b e is the slope of the ascent. In like manner, where b f is the width of the tread, / g is the riser, and b g the slope of the stair. A width of tread, b h, gives a riser of the height of h Jc, and a width of tread equal to 6 l , gives a riser equal to l m. It is conceived, however, that a better ? r ® lds - . Ris ° rs - proportion for steps and risers may be ob¬ tained by the annexed method:— Set down two sets of numbers, each in arithmetical progression; the first set show¬ ing the width of the steps, ascending by inches, the other showing the height of the riser, descending by half inches. It will readily be seen that each of these steps and risers are such as may suitably pair together. 5 9 6 8* 7 8 8 7£ 9 7 10 6* 11 6 12 13 5 14 4i 15 4 16 3* 17 3 18 24 198 PRACTICAL CARPENTRY AND JOINERY It is seldom, however, that the proportion of the step and riser is exactly a matter of choice—the room allotted to the stairs usually determines this proportion; but the above will be found a useful standard, to which it is desirable to approximate. In better class buildings the number of steps is consid¬ ered in the plan, which it is the business of the architect to arrange, and in such cases the height of the story rod is simply divided into the number required. Plans of Stairs .—Before giving examples of the vari¬ ous forms of stairs ordinarily occurring in practice, we shall with some minuteness illustrate the mode of laying down the plan of a stair, where the height of the story, the number of the steps, and the space which they are to occupy are all given. The first example shall be of the simplest kind, or dog- legged stairs. Let the height (Fig. 519) be 10 feet, the number of risers 17, the height of each riser consequently TyV, an d the breadth of tread 9§; the width of the staircase 5 feet 8 inches. Proceed first to lay down on the plan the width of the landing, then the size of the newel a in its proper position, rig. 519. Fig. 620. the centre of the newel being on the riser line of the landing, which should be drawn at a distance from the back wall equal to the semi-width of the staircase, and at right angles to the side wall. Bisect the last riser a b at o, and describe an arc from the centre of the newel, as o n, on which set out the breadth of the winders; then to the centre of the newel, draw the lines indicating the face of each riser. If there be not space to get in the whole of the steps, winders may be also introduced on the left hand side, instead of the quarter space, as shown. The next example is a geometrical staircase. Let abcd (Fig. 520) be the plan of the walls where a geometrical stair is to be erected, and the line c be the line of the face of the first riser; let the whole height of the story be 11 feet 6 inches, and the height of riser 6 inches, the number of risers will consequently be twenty-three. The number of steps in each flight will be one fewer than the number of risers, and according to the preceding rule the tread should be 11 inches, so if there are two flights there will be twenty-one steps; or if winders are necessary, there will be twenty-two steps in all, from the first to the last riser. Having first set out the opening of the well-hole, or the line of balusters, divide the width of the stairs into two equal parts, and continue the line of division with a semicircle round the circular part, as shown by the dotted line in the figure; then divide this line from the first to the last riser into twenty-two equal parts, and if a proper width for each step can thus be obtained, draw the lines for the risers. This would, however, give a greater width of step than is required; take therefore 11 inches for the width of step, and this, repeated twenty times, will reach to the line cl, which is the last riser. There is in this case eight winders in the half space, but four winders might be placed in one quarter space, the other quarter space might be made a landing, and the rest of the steps being fliers, would bring the last riser to the line A c. The usual place for the en¬ trance to the cellar stairs is at D, but allowing for the thickness of the carriages, the height obtainable there will be only about 6 feet, which is not sufficient. At E, in this example, would be a better situation for the entrance to the cellar steps. Plates LXXXVIIL—XCIII., XCY. Plate LXXXVIIL, Fig. 1.—Nos. 1 and 2, show a plan and elevation of a newel stair. The first quarter space con¬ tains three winders, the next quarter space is a landing; the lower flight is shown partly in section, exposing the rough string D D, and its connection with the bearers c c. The front string a A should be tenoned into the newels below and above. Fig. 3.—No. 1, shows the plan of a geometrical stair with winders. In the first quarter space, or lower half of the figure, the lines of the steps are drawn to the centre of the well-hole, and this is the usual way of placing the risers; but drawn thus as radii of the circle, they are, obviously, too narrow at the inner end next the well hole, and too wide next the wall, and if two persons were passing each other they would both be forced to use parts of the tread, most inconvenient to walk upon. Further, as the risers of the steps are all of equal height, it follows that the slope or ramp of the string board along the ends of the fliers, from the first to the seventh step, will be much less steep than that which subtends the narrow ends of the winders, and the result will be a very ungraceful knee at their junction. Both of these inconveniences can be overcome by adjusting the steps in such a way as to distribute the inequality amongst them, or as the French term it, by making the steps dance, as is shown in the upper half of the figure. This may be accomplished either by calculation or graphically. By the first method, the step which is in the centre of the circular arc is regarded as a fixed line, and the divergence from parallelism has to be made between it and the extremes either way. But it is not necessary to begin the diver¬ gence at the first step, nor indeed is it advisable, and in general the first and last three or four steps are. left unal¬ tered, so that they may be perfectly parallel to the landing. Suppose then that the divergence is fixed to commence at the fourth step, it becomes necessary to distribute eight spaces along the centre of the string, commencing at the centre line of the stairs, which, from the centre line to the fourth riser, shall follow some law of uniform progres¬ sion, say that of arithmetical progression, as being the most simple. The progression then will consist of eight terms, the sum of which shall be equal to the length from the centre to the fourth step. Suppose that its develop- STAIRS. 1.0.9 ment is 66 inches, a length composed of the breadth of three fliers, 4 5 6, namely 36 inches, and the sum of the widths of the ends of the five winding steps, 7 8 9 10 11, namely 30 inches, Subtracting from . . . . 66 inches. The width of eight steps of the same width as the winders, . . . . 48 ,, There is obtained the difference . 18 from which is to be furnished the progressive increase to the steps as they proceed from the centre to riser No. 4. Suppose these increments to follow the law of the natural numbers 1234567 8, the sum of which is 36, divide the difference 18 by 36, and the quotient, 0 - 5 inches, is the first line of the progression, and the steps will increase as follows:— The end of step No. 11 = 6 - 5 „ 10 = 7 „ 9 = 7o „ 8 = 8 „ 5 = 9‘5 „ 4 = 10 The sum of which is 6G These widths, taken from a scale, are to be set off on the line of balusters, and from the points so obtained, lines are to be drawn through the divisions of the centre line. It is easy to perceive that by this method, and by vary¬ ing the progression, any form may be given to the curve of the string. The graphic method, however, now to be described, is preferable to the method by calculation, seeing that it is important to give a graceful curve to the development of the string. Let the dotted line s m p , Fig. 3, No. 2, represent the kneed line made by the first division of the stairs in the lower part, corresponding to the nosing of the fliers, and the upper part m n to that of the winders. Bisect the line of the winders m n in p, and raise a perpendicular p i. Then set off m s equal to m p , and make s r perpendicu¬ lar to s m. The intersection of these two perpendiculars, s r and p i, gives the centre of the arc of a circle, tangen¬ tial in s and p to the sides of the angle s m p. In like manner is found the arc to which pn,no are tangents, and a species of cyma is formed by the two arcs, which is a graceful double curve line without knees. This line is met by the horizontal lines, which indicate the surface of the treads, the point p being always the fixed point of the centre step, the twelfth in this example. Therefore, the heights of the risers are drawn from the story rod to meet the curved line of development, s p o, and are thence transferred to the baluster line on the plan. Fig. 2.—Nos 1 and 2 show the plan and elevation of a well-hole stairs, with a landing in the half space. The well-hole is here composed of two circular quadrants con¬ nected by a small portion of straight line; this figure is not so graceful as the perfect semicircle in Fig. 3, No. 1, but it allows more room on the landing.* Plate XC. Fig. 1.—Nos. 1 and 2 are the plan and elevation of a geometrical stair, composed of straight flights, with quarter-space landings, and rising 15 feet 9 inches. * For description of Plate LXXXIX., see pages 200, 201, and 207. The first flight is shown in Fig. 1, No. 2, partly in sec¬ tion, exhibiting the carriage c c, T the trimmer joists for quarter space, and v the trimmer joists of the floor below, with the lower end of the iron baluster fastened by a screw and nut d, at the under side of the trimmer joist v. Fig. 2.—No. 1, exhibits the plan, and No. 2, the eleva¬ tion of a geometrical stair, with straight flights connected by winders on the quarter spaces. Plate XCI.— Fig. 1 is a plan, and Fig. 3 an elevation of a geometrical stair, with a half space of winders. The positions of the rough strings or carriages are shown on the plan by dotted lines, e g, e /, h i,fk. This is a simpler mode of forming the carriages of stairs than that gene¬ rally practised; having fewer joints it is also stronger. It is fully illustrated and described as applied to the more intricate example of elliptical stairs in Plate XCII. Fig. 2 shows the plan, and Fig. 4 the elevation of a geometrical stair with part winders, and part landing, well adapted for a situation where a door has to be entered from the landing. The line A B on the plan shows the situa¬ tion where the principal carriage should be introduced. Plate XCII. exhibits a plan (Fig. 1, No. 1) and eleva¬ tion (Figs. 1 and 2) of an elliptical stair with winders throughout. On the plan is shown the position of the carriages for such a stair, and we shall now describe the formation of such carriages. Plate XCIII. represents the formation of carriages for the elliptical stairs in Plate XCII. Fig. 1 is the longest carriage, or rough-string, and is formed of one deal, 11 inches wide by 3 or 4 in thickness; its length of bear¬ ing betwixt the walls is about 15 feet. To find the best position for the carriages, lay a straight edge on the plan, and by its application find where a right line will be divided into nearly equal parts by the intersection of the risers. The object of this will readily be understood, if it is considered that in a series of steps of equal width and risers of equal height, the angles will be in a straight line, whereas in a series of unequal steps and equal risers, the angles will deviate from a straight line in proportion to the inequality in the width of steps. Notwithstanding the inequality in the width of steps, which thus often occurs, it seldom happens that carriages may not be applied to stairs, if their situation be carefully selected by the means above mentioned. The double line a b is taken from the plan (Fig. 1, No. 1, Plate XCII.), with the lines of risers crossing at various angles of inclination. These lines re¬ present the back surface of each riser, according to the num¬ ber on each. The double line A B will therefore be under¬ stood as representing the thickness of the piece. Lines drawn from the intersections of each of the risers perpen¬ dicularly on a B (Fig. 1, Plate XCIII.), will present the width of bevel which each notching will require in the car¬ riage at the junction of the wall. No. 8 crosses very ob¬ liquely; No. 9 with somewhat less obliquity; No. 10 with still less, and the obliquity continually diminishes, till at 13 the crossiug is at right angles, presenting only one line. The remaining numbers are bevelled in the reverse direc¬ tion, gradually increasing to No. 19, where the carriage enters the wall. The complete lines show the side of the carriage next the well-hole, whilst the dotted lines repre¬ sent the side next the wall. The most expeditious method of setting out such carriages is to draw them out at full size on a floor. Having first set out the plan of the stairs 200 PRACTICAL CARPENTRY AND JOINERY. at full size, take off the width of every step, in the order in which it occurs, marking that width, and at right angles thereto draw the connecting riser, thus proceeding step by step, till the whole length of the carriage is com¬ pleted ; next set out one side of the carriage as a face side, and square over to the back, allowing the bevel as found on the plan; then, witli a pair of compasses, prick off to the under edge at each angle, for the strength; this will define the curvature for the underside with its proper wind, to suit the ceiling surface of the stairs. The bearer, c D, Fig. 1, No. 1, Plate XCII., is a level piece wedged in the wall, with its square end abutting against the side of the carriage, A B. The dotted line on the upper side of the carriage, Fig. I. Plate XCIII., and the straight dotted line on its under side, are intended to show the edges of an 11-inch deal previous to its being cut; the shaded part at each end shows its bearing in the wall; at the riser 18 is shown a corpsing, to receive the lower end of the carriage, Fig. 3, C L; and at the riser 16, a similar corpsing to receive the carriage, Fig. 4, G H; Fig. 2 is the carriage, E F Fig. 1, No. 1, Plate XCII., par¬ allel with A B, Fig. 1, against which the front string is nailed. Each of the last mentioned is formed in the same manner as the one already described. This method of forming the carriages of stairs is not yet much practised. It was introduced by the author more than thirty years since, and has given greater satisfaction than the more laborious process of framing for every step, which is not only weaker from the greater number of joints, but is also more expensive. It is now gradually coining into use. Plate XCY. exhibits a stair winding round a large cylindrical newel. Fig. 1 is the plan, and Fig. 2 the elevation. The lower part of the newel is composed of cylindrical staving, of 2-inch plank, into which the risers and bearers to each step are fixed, the detail of which is better seen in Fig. 4, drawn to a larger scale. The manner in which the steps and risers are put together is shown in Fig. 3; the risers are grooved, and the steps tongued into them. This figure shows the ends of the steps before the last thin casing of string board is fixed. They are united by a band of iron screwed to the bearers throughout the entire length. This kind of construction has a near resemblance to the method for carriages that has been much in use, and called framed carriages for well-hole stairs, the objections to which are stated above. Method of Scribing the Skirting.—Pl. LXXXIX. Fig. 4 shows the method of scribing down skirting on stairs. The instrument used for this purpose is shown in two positions, A and B. It is something like a bevel in form, but has a slider with a steel point at the end; this slider moves steadily in collars, so that while the steel point rests on any point on the stairs, another point on the slide denotes on the skirting board the corresponding point, thus remedying a defect of the common compasses by maintaining always a parallel motion. Fig. 5 is another view of the same instrument, showing the mortise in which the slide works. Strings.— Fig. 6 shows a portion of a string board for the steps {Fig. 8); the middle part being a flexible veneer intended to be bent on a cylinder of a suitable curva¬ ture, and blocked on the back by pieces in a perpen¬ dicular position. Fig. 9 is the string board in development for the smaller end of the winders. Fig. 7 a more enlarged view of the same, showing the mode of easing the angle by intersecting lines. In circular strings, the string board for the circulai part is prepared in several different ways. Each of these will now be described, the first being that adopted in veneered strings. One indispensable requisite in forming a veneered string, is called by joiners a cylinder; it is, however, in fact, a semicylinder joined to two parallel sides. An apparatus of this kind must first be formed of a diameter equal to the distance betwixt the faces of the strings in the stairs. Take some flexible material, as a slip of paper, and measure the exact stretch-out of the circular part of the cylinder, from the springing line on one side to the spring¬ ing line on the other. Lay this out as a straight line, on a drawing board; then examine the plan of the stairs, and measure therefrom the precise place of each riser coming in contact with or near to the circular part of the well-hole as it intersects on the line of the face of the string, and also the distance of such riser from the spring- ing-lines. These distances should all be carefully marked on the slip of paper, and transferred to the drawing-board; then, with the pitch board, set out the development of the line of steps, by making each step equal to the width found, and connecting with it at right angles, its proper height of riser. When the whole development has been set out on the drawing-board, mark from the angles of the steps downwards, the dimension for the strength of car¬ riage; by this means it will be seen what shape and size of veneer will be required. The whole of the setting out must now be transferred to the face of the veneer; then with the point of an awl prick through the angles of the steps and risers, and trace the lines on the back, as well as on the front. The veneer must now be bent down on the cylinder, bringing the springing lines and centre lines of the string to coincide as exactly as possible with those of the cylinder; the whole string must then be carefully backed by staving pieces glued on it, with the joints and grain parallel to the axis of the cylinder. The lines on the back of the string will serve to indicate the quan¬ tity of the veneer to be covered by the staving. The whole must be allowed to remain on the cylinder, till suffi¬ ciently dry and firm. It is next fitted to the work, by cutting away all the superfluous wood as directed by the lines on the face of the veneer, and then being perfectly fitted to the steps, risers, and connecting string; it must be firmly nailed both to the steps and risers, and also to the carriages. Each heading joint in the string should be grooved and tongued with a glued tongue. There is another method of gluing up the strings some¬ times practised. In this the string is set out as before described, but instead of using a thin veneer, an inch board is taken, on the face of which the development of steps, risers, springing, and centre lines must be carefully set out as before. The edge of the board must be gauged from the face, equal to the thickness of a veneer, which would bind round the cylinder; the string must then be confined down on the work-bench, and grooves made by a dado grooving plane on its back in the direction of the riser, and at about half an inch distant from each other, till the whole width of the cylindric surface is formed into a HANDBAILING. 201 regular succession of grooves and projections; the string must then be bent on the cylinder, and the grooves filled with small bars of wood, carefully glued in. When diy, this is to be fitted to the stairs, as in the former method. Another method is making staves hollowed on the face to the curvature of the well-hole, and setting out as much of the string on each piece as will cover its width, then gluing the staves, edge to edge, without any veneer. This method, though expeditious , is not safe. A fourth method is sometimes practised, when the curved surface is of great length and large sweep, as in the back strings of circular stairs. In this a portion of cylindric surface is formed on a solid piece of plank, about three or four feet in length; and the string, being set out on a veneer of board sufficiently thin to bend easity, is laid down round the curve, with such a number of pieces of like thickness as will make the required thickness of the string-board. In working this method, the glue is intro¬ duced between the veneers with a thin piece of wood, and the veneers quickly strained down to the curved piece with hand-screws. A string can be formed in this way to almost any length by gluing a few feet at a time, and when that dries, removing the cylindric curve and gluing down more, till the whole is completed. The manner of jointing the staves is shown in Plate XCIII., Fig. 5, where a bevel is set with the tongue in the line of the radius, whilst the stock coincides with the back of the stave piece. Fig 6 also shows how a back string is formed for the stair in Plate XCII., and a base moulding formed for the same in thicknesses, and applied to the string. Fig. 7, No. 1, Plate XCIII., shows a portion of front string with bracket, and the mitred end of a riser at a. No. 2 shows the back of the same riser and how it is shouldered and mitred to receive the front string and bracket; B shows the thickness of the front string, A the carriage, c the thickness of the tread, d the hollow, and e the end of the bracket. Diminishing and Enlarging Brackets. — Plate LXXXIX. Fig. 1. To diminish the bracket of the fliers to suit the winders, make one of the fliers marked B, the base of a right-angled triangle, and settiug off any con¬ venient distance, B C, for the perpendicular, draw a line from the extreme point of the bracket, to form the hypo- thenuse of the triangle; set A, the length of the shorter bracket required to be drawn, as a perpendicular, under the hypothenuse; draw ordinates through each raeipber of the original bracket, and through the points of their intersection with B, draw lines converging to the point c. The intersection of these with the line A will divide it for the corresponding set of ordinate lines, which draw, and make equal respectively to those on the line B; trace the contour through the various points thus obtained, and the bracket A will be produced. To enlarge a bracket, it is only necessary to reverse the process by making the shorter bracket, as A, the base of the triangle, producing from it the perpendicular and hypothenuse. This procedure is so obvious, that no detailed description is necessary. HANDRAILING. Plates XCIV., XCYL—C. Although many authors had written on the subject of handrailing before the time of Mr. Peter Nicholson, the methods described by them for producing the face mould were erroneous in principle, and attended with great waste of material. The merit of introducing a better system is due to Mr. Nicholson, who taught the true theory of cylin¬ drical sections, and illustrated it by practical solutions of the problem of producing the section of a cylinder through any three points on its surface. In the following treatise, the author, in illustrating the same theory, has introduced methods of solving the problems, which will be found less intricate and easier understood than those of Mr. Nicholson, and, what is not less important, more readily applied in practice. The difference between the two methods will be described in the sequel. Definitions.— In the following article, there will be frequent occasion to make use of certain terms which it is of importance to have fully understood. The horizontal , or ground plane , is that plane on which the plan is drawn. The vertical plane is any plane considered as standing perpendicular on the ground plane. The oblique plane, cutting plane, or plane of the plank, is that plane on which the mould of the rail is produced. The trace of any plane is a line forming the termination of one plane and its junction with another; thus the angle of a block of marble is the trace of the plane of an}’ - one of its sides on another side which it meets. The trace therefore is a line, and the only line which can be drawn common to either of two planes, meeting each other at an angle. A cylinder is a solid, described by geometricians as generated by the rotation of a rectangle about one of its sides, supposed to be at rest; this quiescent side is called the axis of the cylinder, therefore the base and top of the cylinder are equal or similar circles. A prism is a solid, whose base and top are similar right line figures, with sides formed in planes, and rising per¬ pendicularly from the base to the top. The cylinder, so called by joiners, is a solid figure, compounded of the two last-mentioned figures; its base is composed of a semicircle joined to a right-angled paral¬ lelogram. This last compound figure is intended when¬ ever the word cylinder occurs in the following article, unless the word geometrical be prefixed. Of the Construction of the Falling Mould.— The height of the handrail of a stair, as the following considera¬ tions will show, need not be uniform throughout, but may be varied within the limits of a few inches, so as to secure a graceful line at the changes of direction. In ascending a stair the body is naturally thrown forward, and in descending it is thrown back, and it is only when stand¬ ing or walking on the level that it maintains an upright position. Hence the rail may be with propriety made higher where it is level at the landings, the position of the body being then erect, than at the sloping part, where the body is naturally more or less bent. The height of the rail on the nosings of the straight part of the stairs should be 2 feet 71 inches, measuring from the tread to its upper side; to this there should be added at the landings the height of half a riser. In winding stairs, regard should be had, in adjusting the height of the rail, to the position of a person using it, who may be thrown further from it at some points than 2 C 202 PRACTICAL CARPENTRY AND JOINERY. at others, not only by the narrowing of the treads, but by the oblique position of the risers. Take, for example, the elliptical stairs (Fig. 1, No. 1, Plate XCII.), and sup¬ pose the rails of uniform height. A person in ascending, with the foot on the nosing of steps 6 or 7, will find the rail lower to the hand than when standing on the nosings of 19 or 20. The risers of steps 3, 13, and 23 are square with the rail, while those of the other steps are more or less oblique. In such a case it is advisable to make the rail of the average height over 3, 13, and 23, to raise it several inches higher at 7, and to depress it to an equal extent over 19 and 20; to raise it, also, at the top of the stairs, the more especially as the easing of the rail will tend to lower it there. It is seldom that the rail will require to be lowered below the assumed standard more than 3 inches, or raised above it more than 4 inches, and unless these variations in the height are adjusted in accordance with the foregoing considerations, the effect will be very disagreeable. It is necessary to guard the reader against the common error of raising the rail over winders, especially such as are of steep pitch. The height should be uniform, except in the instances adduced above. The falling mould (Fig. 1, Plate XCIY.) is given in strict agreement with Mr. Nicholsons method, but it is quite at variance with the rule just named. The Section of a Cylinder. Plate XCVI.—If any cylindric body, as A B 0 (Fig. 1), standing on a horizontal plane, be cut by an oblique plane, V op, it is obvious that a third or vertical plane, VP Bp, maybe so applied to the cylinder, that it shall not only be at right angles to the ground plane, but also to the plane of section. It can also be easily shown that on such a vertical plane the oblique plane would be projected, according to the rules of projection, as a right line; for, if the position of the oblique or cutting plane can in any way be defined, then the trace of the oblique plane, on its line of intersection o p on the ground plane, can be known, and any vertical plane standing at right angles to the trace of the oblique plane 0 p, will be one on which the trace of the oblique plane will be pro¬ jected as a right line; that is, the representation of the plane 0 2 ? on v p will be simply a geometrical line; conse¬ quently the vertical plane, by construction, is at right angles to the ground plane, and also to the oblique plane. It is evident, then, that if any figure whatever be de¬ scribed on the ground plane, and it be required to describe such a figure that its various parts in every point shall be immediately over the figure on h p, nothing more is necessary than to drqw lines through the various parts of the figure on h p, perpendicular from P B. Continue those lines perpendicularly upon the vertical plane V p, and return them on the oblique plane; and then measure on those lines from the line vp, the same each to each on the plane v o p, as the corresponding lines on the ground plane; thus, the line v 6 will be made equal to P D, the line 5 equal to C, and so of any other line. Those lines are called ordinate lines, and the method here described is called tracing by ordinates. It is thus particularly de¬ scribed, because unless the process be perfectly understood by the learner, he cannot know anything of the way of producing the section of a cylinder or the face-moulds for handrails geometrically. The manner of obtaining such face-moulds will now be described. To produce the section of a cylinder through any three points on its convex surface. Figs. 2, 3.—First draw the plan of the cylinder, or part of the cylinder, as ABC. Let A be the lowest point in the section, B the seat of the intermediate height, and c the seat of the greatest height. The height on A is nothing, and is therefore a mere point on the plan; the height on the point B is equal to h h, and the height on C is equal to C p. These heights are sometimes called the resting points. Draw a right line from A, the seat of the lowest point of the section, to c, the point on the plan agreeing to the highest point of the section; draw cp and make it equal to the greatest height of the sec¬ tion ; complete the triangle A C p by drawing the line A h p. Take the intermediate height and apply it to the triangle wherever it can be applied, as a perpendicular under the line A p; in other words make h h parallel with C p, and equal in length to the third or middle height; then draw the line from h to B, the seat of the middle height, and it will be the leading ordinate; then at right angles with b b, and touching the convex line of the plan of the cylinder, draw the line a B D—this line is the trace of the vertical plane. Draw the line c p at right angles to a B, and passing through the point c on the plan, make D P equal to C p; complete the triangle D P a by drawing the line P a\ continue the line b B until it intersect the line a P at 2. This triangle completes the representation of the vertical plane. Nothing more now remains to be done but to draw the ordinates c d and e parallel to D P; to square out from the line a P the corresponding ordinates 1 2 3 4 5 6, and to make them respectively equal to the corresponding ordinates on the plan; thus 6 P is equal to d C, 5 is equal to e, 4 equal to d, 3 equal to c, 2 is equal to B, and 1 a is equal to a A, and the mould for the oblique plane will be completed by tracing a line through the points 1 2 3 4 5 6. The angle a P D is what is usually called the pitch of the plank, the use of which will be explained hereafter. In order further to demon¬ strate this subject, consider the figure of the ground plan to be drawn on a level plane, and a P D to stand vertically on its trace a D; then suppose the oblique plane to be turned on its trace a P, so as to form a right angle with the vertical plane, and it will follow of necessity that the point 1 will coincide with the point A; and the point P being thereby elevated to the full height of the section P 6 will be brought into a position parallel with D c; 6 P being equal to D c, the point 6 will of necessity be in the exact situation of the highest point of the section; so also the ordinate 2, being equal to the ordinate B and parallel to B, must bring that point equal to the intermediate height; so also of all the other ordinates: therefore the mould 1 2 3 4 5 6 is the section of the cylinder to the three heights and points required.* * The author, when a lad, in 1824, heard a fellow-workman read a passage from Mr. Nicholson’s work, The Builder , and was struck with the words: Section of a cylinder through any three given points.” He was at that time familiar with the method of produc¬ ing groins, angle ribs, brackets, &c., by ordinates, and at once pro¬ ceeded to solve the problem indicated in Mr. Nicholson’s words. HANDRAILING 203 Figs. 4 and o differ nothing in principle from that we have already described. The centre line of the plan of the rail is here substituted for the line A B c in Figs. 2 and 3. Draw the line A C, connecting the seats of the highest and lowest points; make C p equal to the greatest height, and at right angles with A C; make b h equal to the in¬ termediate height, and draw the line b B, the leading ordinate, as in the preceding figures; square from it for the base of the section, and make all the ordinates on the plan parallel with the leading ordinate b B. Make D p equal to c p, the height of the section; draw P a, con¬ tinuing the ordinates to ftP; square them out from a P, and make each ordinate on the section equal to its rela¬ tive ordinate on the plan; draw the figure through the various points 12 3, &c. ; and the moulds will be com¬ pleted. In drawing the first triangle AC p the line C p may be made equal to the whole height, or to any fraction of the height, provided the intermediate height be drawn in the same manner; for it is evident that if the perpen¬ dicular c p be lengthened or shortened, the perpendicular b h will be lengthened or shortened in the same propor¬ tion; and it will sometimes be found more convenient to use a part than the whole of the height, as will be seen in constructing the moulds, Plate XCYII. Other means might also be adopted of finding the trace of the oblique plane on the ground plane, one of which we have shown in Plate XCIV, Fig. 2. Before proceeding further, it may be well, in a few words, to describe the leading points of difference between Mr. Nicholson’s method and that here taught. In Mr. Nicholson’s method two bevels are required in cutting the plank, and the ordinates are bevelled both on the plan and on the oblique plane. The seat points of the heights used are: — one at each corner of the mould and one on the convex part of the rail. The heights are measured to the top side of the falling mould for the lower wreath, and to the under side down¬ wards for the upper wreath. The vertical plane, from which the ordinates are traced, is generally, if not always, made to pass through the inner angles of the joint lines of the plan. The oblique plane, on which the face mould is pro¬ duced, is sometimes at a right angle, sometimes at an acute angle, and sometimes at an obtuse angle to the vertical plane. This is what he calls the spring bevel, and it is necessary to bevel the plank in accordance with this before the pitch bevel can be applied. In the method here taught the pitch bevel only is used, and the ordinates are squared on the plan and on the oblique plane. The pitch bevel of this method is there¬ fore equivalent to the two bevels of the other. The seat points of the heights are taken on the centre line of the rail, and the heights are taken to the centre line of the falling mould; the trace of the vertical plane He succeeded in liis endeavours, and practised his own system for a year without being aware what amount of affinity existed between it and that of Mr. Nicholson. The general method here explained was first taught by the author in Cheltenham in the year 1826, and in other parts of England previous to 1830, in which year he visited the United States of America, and practised and taught this method in the city of Phila¬ delphia, where he resided for more than seven years. is always square with the trace of the oblique plane, con¬ sequently the use of the spring bevel is not required, the piece having only to be bevelled at once to the pitch bevel. Fig. 2, Plate XCIV., is intended to illustrate the dif¬ ference between the two methods, No. 1 showing the method of the author, and No. 2 that of Mr. Nicholson; and in order that the illustration may be complete, the heights or resting points in both figures are taken in accordance with the latter method. Plate XCIV., Fig. 2, No. 1.— A base line is drawn from 1 to 3, connecting two of the resting points, and parallel therewith a line is drawn through 6, which is the intermediate resting point, indefinitely towards the point 8; the line 1 2 is drawn perpendicular to 1 3, and equal in height to the height of the section or a fraction of the same—in this case one-fourth—it being taken, according to Mr. Nicholson’s method, to the top of the rail. The perpendicular 3 4 is made equal to the intermediate height, or the same fraction of that height (one-fourth); the line 2 4 is drawn through the point 4 indefinitely, the line 1 3 is continued to inter¬ sect 2 4 in 5, and the line 7 8 is drawn parallel with 2 5, intersecting 6 8 at the point S, giving the line 8 5 as the line of the trace of the plank on the ground plane. This gives a trace or leading ordinate precisely parallel with that we have adopted; but we think it is far from being so direct in its application to the subject as that we have generally used. Mr. Nicholson’s most usual method of finding the trace of the oblique plane is shown in Fig. 2, No. 2, where the seats and heights are the same as in No. 1; p 2 being the greatest height, t 6 the inter¬ mediate height, and 9 v the least height. The line P o is drawn connecting the corners of the plan mould ; P 2 is drawn perpendicular to P s; 6 T is parallel to P 2, and equal to the intermediate height; V s is parallel to P 2, and equal to the lowest height; 2 v is drawn indefi¬ nitely till it meets a line drawn through the points P and 9 in Q, 6 L is drawn parallel to p Q, and T l parallel to 2 Q; this gives the trace of the plane L Q, which is con¬ tinued to o, where it meets the line P s; then draw the ordi¬ nate 9 on the plan, meeting p o where the ordinate g on the vertical plane is made equal to 9 v; g 2 is then drawn, meeting p s in 0. This is Mr. Nicholson’s most usual and roundabout method of finding the trace of the oblique plane; but besides this it is also necessary to find the leading ordinate on the oblique plane. To do this, draw S M perpendicular to 2 o, and from the point 0 describe the arc L M, and draw 0 M, the leading ordinate. This method requires a spring bevel, which is shown at 0 , and is found by setting one foot of the compass in s, and ex¬ tending the other to the line 0 2, making the portion of the arc there seen meeting the line po; from the point of intersection in p o, draw a line to L, and the spring bevel will thereby be produced. The leading ordinates being found, all other ordinates are drawn as parallels, and the mould is traced according to the figures. We have thus contrasted the method of Mr. Nicholson by bevelled ordinates with our own method by squared ordinates, to satisfy the reader that, if the same heights and seats of heights are used, the same mould is pro¬ duced by squared as by bevelled ordinates. The plan mould on the left is exactly the same as that on the right; 204 PRACTICAL CARPENTRY AND JOINERY. the heights are the same, both being taken from the kill¬ ing mould Fig. 1*, each height being taken from the line A to the top of the falling mould in each case. It will at once be seen that the mould is identical, the tracing in one case being from the line on the concave side, and in the other from that on the convex side of the mould. Plate XCVII., Fig. 1, shows the face mould, and Fig. 2 the falling mould of a rail suited to the stairs, Fig. 1, Plate XCI. For the falling mould describe the quadrant B 3 {Fig. 2) to the radius of the concave side of the rail on the plan, and make D B equal to its development; then set out the lines of steps and risers in the order they occur in the stairs, placing all the risers at their pro¬ per situation as to the springing and centre lines; make 1 equal to the last flier, 2 equal to the first winder, 3, 4, 5, and 6 equal to the succeeding winders, taken on the concave curve of the rail; draw the bottom lines of the falling mould, making them touch the angles of the steps excepting where the curved part necessarily leaves them; draw a line for the centre line of the falling mould at a distance from the bottom equal to the half of the depth of the rail, also a line at the distance answering to the top of the rail, and draw lines at right angles through the thickness of the rail for the butt-joint, as at p and A; draw a line through the centre of the lower butt-joint parallel to D b, meeting pe in C; make the line b h per¬ pendicular to c b, at or near the centre of the length of the killing mould. We shall then have P c for the greatest height, and h b for the intermediate height of the section; the lowest point of the section being the point A in the plan, the seats of those points on the plan must ever be in the centre line of the rail. The method of producing the face mould differs nothing from the general method. In Fig. 1 draw A C; make C p perpendicular thereto, and equal to the greatest height, or to some fraction of the same—in this case it is one-third; then draw the hypothenuse p h A, and take the intermediate height, or its corresponding fraction one-third, and apply it to form the perpendicular b h. From b to the seat of the middle height on the plan, draw the leading ordinate b B, and square the base of the vertical plane as a tangent to the plan mould; draw all the ordinates on the plan parallel to the leading ordinate, and through as many points as may be needed for the tracing; make P f equal to the greatest height, and draw the hypothenuse Pet, from which square out all the ordinates, making them respectively equal to the corresponding ones on the plan; carry one ordinate through the centre of the plan, and take off the distance of that point, applying it as at n. Draw the dotted line n 4, and make the butt-joint at 6 by squaring from n 4 through the centre of the bevel joint. Make the butt-joint at the straight part by simply squaring it from the side of the mould through the centre of the bevel joint; then trace through the points 1 2 3 4, &c., and the mould will be complete. The Figs. 3 and 4 of this plate are drawn on precisely the same principle, as before described. They are here intro¬ duced as a specimen of a wreath of a small well-hole, with a very sharp ascent, the radius of the inner curve being only 3 inches with three winders in the quarter-space. The risers are here supposed to be drawn to the centre, the riser between the last flier and the first winder is conse¬ quently identical with the springing line; the same line is also here made the line of the middle height. The falling mould is constructed, as before described, with the under¬ sides touching the angles of the steps, excepting only where made conformable to a fair curve; the height of the section is found, as in the former case, by drawing a horizontal line through the joint line at A, and taking the height from this line to p for the greatest height, both it and the intermediate height being taken to the centre line of the falling mould. A line is drawn in Fig. 3, through two of the seat points, namely, A and c; one third of the greatest height is used as a perpendicular, and the same fraction is used for the middle height, b h; the leading ordinate is drawn through the seat of the middle height, that is, through the centre of the springing line; the base of the vertical plane is squared from the leading ordinate, and the entire height of the section set up, as P 1, the perpen¬ dicular; and having drawn p a, and continued and squared out the ordinates, the mould is pricked off by making each ordinate 1 2 3 4, &c., equal to its respective ordinate on the plan. The butt-joint is drawn by squaring it from the line, ng, through the centre of the bevel joint. In this case the sharp pitch of the mould produces it a great width at each end; if this were to be cut out of the plank to its proper bevel, C P a, it would take at least twice the amount of material that would be absolutely needed by the means we shall now point out. Let the centre line be pricked out on the mould, parallel with which draw the dotted line, shown on the same. Now if the mould be laid on the plank in this form it will appear as at Fig. 5; this may be cut quite square out of the plank, and will be quite sufficient to produce the rail in the most perfect shape; c D shows the edge of the plank, the oblique line c D being the proper bevel just mentioned, which is to be drawn on the edge of the plank; A B is drawn on the mould when in its position in Fig. 3, as a parallel to P a, and is called the backing line; this line should be drawn on the plank on both sides, and perfectly opposite on each. When the piece has been cut out square, as shown at Fig. 5, the point e of the mould should be slid to the point c, where the mould is a second time to be marked on the material, keeping the line A B on the mould to agree with the line A B on the plank; then let the mould be applied to the other side of the plank, by bringing the point e on the mould to coincide with the point D on the plank, and the line A B on the mould to coincide with the line AB on the piece; mark the piece again in this position of the mould; this is what is called backing the mould, and the piece is now pro¬ perly lined for wreathing. This is done by placing the piece in the vice with the concave edge upwards, and tak¬ ing off the superfluous wood down to the lines just de¬ scribed on the surface of the plank. It will, however, be found requisite sometimes to place the mould on the piece, and fix both in the vice together, in order to supply that portion of the line which will be deficient by reason of cutting square through the plank, instead of the old method of bevelling. When the concave cylindric surface is thus produced, the falling mould may be applied. This is done by making the bevel joint line of the falling mould to correspond with the bevel joint line of the piece, while the butt-joint lines of the falling moulds also coincide with the butt-joints of the piece; the butt-joints of the falling mould thus applied will now show the position of the joint M A !K1 [0 ifi A 0 iL n N © o PLATE xcvm. n. Mavcr drJ\ Indies p 9 6 5 0 3 Feel J. W. lowy fc B-LA.CKIF, & SON, C.J.ASGOW, EDINBURGH fcLONDON !H AIM'D R k DU KG U LACK IF. SON OI.ASOI KDINIM HOU X. LONDON HANDRAILING. one way, whilst; those of the face mould will show it the other, thus rendering the joint complete; mark on the piece on each side of the falling mould for the under side and top side of the rail; square from the concave surface, both for the top and bottom of the rail, and take off the superfluous wood, using a pair of callipers to gauge off the back side of the piece. What has here been stated will hold good as a general rule; but if the face mould be not much wider than the rail itself, it would be absurd to be at any undue trouble to save material. It will, however, generally be best to cut the piece out square from the plank, and bevel it after in the manner de¬ scribed. Plate XCYIII.— Fig. 1 is the falling mould, and Fig. 2 the face mould of a wreath suitable to the stairs shown at Plate XCI., Fig. 2.* Fig. 1 shows the stretch-out of A b, the internal curve of the plan. D L is the width of the last flier, 2 3 and 4 are widths of winders, 5 is the half width of the landing; the line L D, Fig. 1, is equal to A g, Fig. 2, or the length of the straight part on the plan; the lines c d e f and P, Fig. 1, are all at equal distance from each other, and represented at Fig. 2, by the portions of radii, g k l, &c, drawn across the rail. The position of the oblique plane is found here (Fig. 2) by our usual method as shown in the former examples, Cp being a third of the greater height, and b h a third of the intermediate height; and as B is the seat of that height, b B becomes the leading ordinate, and a B the base of the vertical plane; P a, is the pitch of the plank; the lines t r and u s, show the upper and lower surface of the plank; E f is drawn parallel with a P at any convenient distance; p and g are drawn through each end of the joint line, and con¬ tinued to the height from the line B a, equal to the dis¬ tance from E to the section of the falling mould at P Fig. 1. This section, or rather the section of the square rail, is thus shown by the small parallelogram at P; in like manner the section of the rail is shown at /from the section of falling mould a /, Fig. 1 ; the parallelograms e d c and a are produced in the same manner from e d c, and L, in Fig. ]. It will now be needful to draw lines through the angles of each parallelogram, and square with the line E F; the mould is then produced by making the ordinate 19 equal to q on the plan, and measured from the line E F 18, the same one answering for the top, and the other for the bottom, of the representation of the piece; 20 and 21 are both equal to p, on the plan; 14 and 15 are equal to o on the plan, 16 and 17 to n, 10 and 11 equal to m, 12 and 13 are equal to l, and the other points are traced in a similar way, and parallelograms are then drawn through the points, as seen in the plate; each of these parallelograms will represent a section of the solid square rail as it would appear on the plane of the plank drawn by orthographic projection; if lines be drawn through these angles, they will represent the square rail; such a mould may be laid on the plank, and the piece cut square out; the piece is then set out on the edge, as shown in the elevation, by the vertical sections of the square rail; the distances of the angles may be measured and set oft, and each angle found with great precision, * This plate exhibits the method of cutting the wreath out square, as first taught by Mr. Nicholson, but is not contained in his Car¬ penter's Qiiidc. 205 first finding the outside angle of the piece, and then gaug¬ ing the other angles therefrom. Fig. 3 of this plate shows the manner of setting out a wreath for such a situation as that exhibited by the lauding stairs, Fig. 1, on Plate XC. A c is the length of the straight parts at the upper end of the piece, c d the length of the circular, and D B the length of the straight part at the lower end of the piece; make L the landing, equal to the development of c D, and the half steps r and g h, equal respectively to AC and D b, bisect c D and draw E f; make the perpendicular at g equal to half a riser, also the same at c; make the perpendicular at / equal to half a riser from L the landing line, to the centre of the rail from this point; set off the half width of the falling mould above h, and draw the bottom line of the falling mould by making the hypothenuse lines at r and A; connect these by a fair curve passing through the proper height at F; draw also the centre liue and top of the falling mould. It will be seen that if our usual method of finding the position of the oblique plane were here applied, it would necessarily produce the line g as the leading ordinate, for the height F is half the height k G, and the base k i is bisected by the line F; consequently the l^pothenuse should in like manner be bisected by placing the intermediate height; this fact is mentioned to show that there are many cases occurring where a moment s reflection will serve to convince the practical man what the position of the plank should be, without drawing one line. Therefore for the face mould (see Fig. 4) bisect the quadrant in g , and draw the ordinate g, from which square the ordinates from D A, and also from the hypothenuse, and make the ordinates 1 2 3 4, &c., equal to the corresponding ordinates A bed, Ac.; and draw the face mould through the respective points to pro¬ duce the butt-joint square from the side of the rail through the centre of the bevel joint. The application of the falling mould is shown at Fig. 5, the line F of the falling mould (Fig. 3) is placed in the posi¬ tion g, answering to g on the plan, and in the centre of the thickness of the plank; s and s show the springing lines, A and B the bevel joints; a butt-joint is shown at the lower end of the piece: its application will be easily compre¬ hended, and will generally serve for the performance of this work. No example of a perfectly straight falling mould has been given, as it would be superfluous; it will be easily seen, from what has been said, that the leading ordinate in such cases must always fall in the same man¬ ner as in the last instance. Plate XCIX.—This plate exhibits the manner of pro¬ ducing the falling moulds, and face moulds for scrolls. Fig. 1, No. 1 is the face mould, and No. 2 the falling mould, for a small scroll. In tracing moulds of this de¬ scription, there is no need of any process to find the position of the plank; no better position can be found than that in which a plank would be if laid flat on the nosings of the stairs, and the pitch board gives this angle of inclination. Take the pitch board, and lay the step side of it against the side of the straight part of the level mould A B, and by means of the upper edge of the pitch board, mark off the line c d ; draw any number of ordi¬ nates on the plan mould square with the straight rail, continue them to c d, square them out, and prick off the mould by making the ordinates/ g h i k l , respectively, PRACTICAL CARPENTRY AND JOINERY. 200 equal to tlie ordinates 1 2 3 4 5, on the plan mould, and draw the face mould through the points. This mould may be drawn without using more ordinates than are needed to find the joint, and to show the width of the straight rail. The mould itself is merely a quarter of an ellipsis, both for the inner and outer curve, and its trans¬ verse diameter is equal to the diameter of that circle from which it is generated on the plan; therefore if the ordi¬ nate f be continued till equal to the length of radius, it will represent so much of the transverse diameter, and a line drawn through its extremity parallel to c d, will re¬ present the conjugate diameter of the ellipsis, and the mould may be produced by the trammel, or by any of the means explained at pages 23 and 24. The shaded part a b shows the piece wrought and in position. The falling mould, No. 2, is produced by making the line e, 12 3, &c., equal to the line e, 1 2 3, &c., in No. i, on the convex side of the rail. From the top of the ordinate line 1, draw the line b to the pitch of the rail, and con¬ nect this line to the line 9 e, by a fair curve; this forms the bottom line of the falling mould; the top line is drawn parallel to it at a distance equal to the depth of the rail. The ordinates 12 3, &c., will now show the height of the rail, in as many points, from the bottom of the scroll. The falling mould, No. 3, is produced by making the line, 12 3, &c., equal to the internal curve of the scroll, and the ordinates 12 3, &c., on No. 3, equal to the ordinates 12 3, &c., on No. 2; the piece is jointed at A on the plan (No. 1) to the level portion of the scroll, the line 5 on the falling mould (No. 3) showing the same as a perpendicular joint, which in this part differs but little from a butt-joint, which might of course be used if pre¬ ferred. Fig. 2 of this plate exhibits the side view of two pieces of handrail of similar character to the scroll pieces just described, but are here shown as applied to a landing, m being the landing, o the riser, and n the step below; whilst L shows the riser, and Jc, the step above. Such a landing may be seen in plan at c, Fig. 1, Plate XCI., being the top of the first flight of stairs. The risers in this case do not pass through the centre of the well-hole; they are so arranged that the centre B of the rail on the return shall be precisely half a step from the line of risers, l o. By this arrangement the piece of rail in this part is of the simple kind just mentioned, less indent of the well- hole into tjie landing is made, and the rail itself has a better appearance than when a greater amount of it is thrown on the level at the landing. The pitch board is applied with its step side against the side of the plan mould, as at A, and the pitch line produces the line A a. Draw all the ordinates a b c, &c., parallel to the riser line of the pitch board, and return them on the plane of the mould or plank at right angles to A a; make the ordinates a b c, &c., equal to the ordinates 12 3, &c., on the plan, and draw the mould. This mould, as also the preceding one for the scroll, must be bevelled by the pitch board. Lay, therefore, the hypothenuse of the pitch board to coincide with the surface of the plank, and the riser line of the board will give the bevel on the edge of the plank for backing the mould; this will be better comprehended by the position the piece will have when placed in the work, as shown by C D and E F. In Fig. 3, is shown another and somewhat more expeditious method of working. Let B c be the plan mould, with its centre line as shown, and h e d the pitch; square out the line d/and make it equal to e c; then square out from the ordinate h, making it also equal to e c. The semiconjugate and semi transverse diameters of the ellip¬ tical mould will thereby be obtained, and it may be drawn by the trammel, or by any of the methods already men¬ tioned. It is possible, by thus working to the centre line only, to make the rail without cutting it out one-eighth of an inch wider in any place than its exact width. Fig. 4 shows the same piece cut out from the square plank; the end is then, by the use of the pitch board, to have the line a drawn through the centre of its thick¬ ness; which enables us to draw the centre line of the mould / (No. 1) on the piece, as shown by the curve line at a. By using a pair of callipers the vertical sides of the rail may be readily produced, and the bottom and top squared from them. Fig. 5 shows, at No. 1, the scroll adapted for such a stair as the elliptical stair, Plate XCII., or wherever a scroll comes immediately in connection with winders. First decide at what point on the plan mould the scroll shall come to its level position; and as the plank is usually about |-inch more in thickness than the rail is in depth, it will be possible to obtain that extent of rising in the first part or level portion of the scroll; this is sup¬ posed to be at the point g, on the plan mould, which is therefore made the place of the first joint. The point c is that point of the rail where the third riser occurs, and may with propriety be made the place of the second joint. Take the stretch-out of the exterior curve, d to g, and make the line c g, in No. 2, equal thereto, and set out the bottom line A B of the hilling mould two inches below c g. On A B, set off the height of the rail at c, equal to two risers; draw the line l to the inclination of the rail, and the intermediate portion of the falling mould, as a fair curve, connecting the straight lines; make the centre and top lines of the falling mould parallel to the bottom line, to suit the depth of the rail. This completes the falling mould. For the face mould, in No. 1, draw through the centre of the joint lines the lines eg ; make cd perpendi¬ cular thereto, and draw d g. Take the distance, c e, from No. 2, and set it out from a to b, No. 1, and make the point b in the centre of the width of the rail; this will be the seat of the intermediate height, and c and g the seats ot the highest and lowest heights. Take the length of ef, No. 2, and apply it at e /, No. 1, parallel to d c; draw the line e b for the leading ordinate and square from it, as a tangent to the curve of the scroll; draw b t, the base line for the wreath, and draw t s and g n perpendi¬ cular to b t, then draw the hypothenuse, and continue the ordinates a i hi vi, &c., to meet it. From the points or intersection draw the ordinates 1 2 3 4 5 6 7, &c., making them respectively equal to the ordinates a a t i h b l m, &c ; and through the points thus obtained, prick off the mould. We have thus endeavoured to give not only one general method of producing a face mould, and by one demonstra¬ tion sought to make it apply to any number of cases what¬ ever, but have also given a variety of instances of applying the same in actual practice. We shall now proceed to describe certain details, which could not, without embarrassing the subject, be noticed before. PLATE r. HANDRAILING. 207 Sections of Handrails. —In Plate XCIV. some of the usual forms of the sections of handrails are given. To describe Fig. 3, divide the width 6 6 in twelve parts, bisect it by the line A B, at right angles to 6 6; make C B equal to seven, A C equal to three such parts, and B i also equal to three parts; set off one part from 6 to 7, draw the lines 7 i on each side of the figure; set the compasses in 4 4, extend them to 6 6, and describe the arcs at 6 6 to form the sides of the figure; also.set the compasses in b, extending them to A, and describe the arc at A to form the top; make l B equal to two parts, and draw the line Jclk; take four parts in the compasses, and from the points 4 4 describe the arcs e f, then with two parts in the compasses, one foot being placed in Jc, draw the inter¬ secting arcs g A; from these intersections as centres, de¬ scribe the remaining portions of the curves, and by join¬ ing h i, k i, complete the figure. Fig. 5 is another similar section of handrail. The width 6 G is divided into twelve equal parts as before; the point 4 is the centre for the side of the figure, which is described with a radius of two parts; Am is made equal to three parts, and B m to eight parts, and m n equal to seven parts; then will A B be the radius, and B the centre for the top of the rail. Take seven parts in the compasses, and from the centre 6 in the vertical line A B, describe the arcs g h, g Jr, take six parts in the compasses, and from the centre 4, describe the arcs ef,ef; draw the line d d through the point n\ from the intersections at e f g It, as a centre, with the radius of four parts, and from 4, as a centre, with the radius of two parts, describe the curve of contrary flexure forming the side of the rail; then from d, with the radius of one part, describe the arc at d, form¬ ing the astragal for the bottom of the rail. In Fig. 4 divide the width c D into twelve equal parts; make 6 m equal to 6 parts; 6 B and m h respectively, equal to two parts, and m 1 equal to three parts; make e li and h f respectively, equal to two parts; then in/ and e set one foot of the compasses, and with a radius equal to one and a half parts, describe the arcs g g ; from the point m, with the radius m A, describe the arc at A meeting the arcs g g, to form the top reed of the figure; from 2 with a radius equal to two parts, describe the side reeds c and D; draw 1 d parallel to A B; and with a radius of one part from the points d d describe the reed d for the bottom of the rail, which completes the figure. Fig. 6.—To describe this figure, let the width G 6 be divided into 12 parts; make m 4 equal to four parts, m 6 equal to 6 parts, and 6 8 equal to 2 parts; make 6 d equal to 5 parts, and draw the dotted lines d 4; also the lines 4 g. On these lines make l 4 equal to two parts, l o equal to half a part, and o g equal to four parts; also make m k equal to one part, and draw the lines g k; from k, as a centre, describe the arc at A for the top of the rail; from g describe the arcs li o. At 4 and 4, with the radius of two parts, describe the arcs at 6 for the sides of the rail; then from d set off the distance of two parts on the line d 4, and from this point as a centre, with a radius of two parts, describe the curves of contrary flexure terminating in d d, which will complete the curved parts of the figure. Continue the line 6 6 the distance of four parts on each side to the points 4': from these points, and through the points d d, draw the lines d d for the chamfer at the bottom of the rail, thus completing the entire figure. To FORM THE SECTION OF THE MlTRE CAP. — Fig. 3, Plate LXXXIX, exhibits the method of producing the section of the mitre cap from the section of the handrail. Let A b c D, &c., be the section of the handrail. Draw the line G G in the centre of the section, and draw across it, at right angles, the line A B; describe a circle 0, 11, j, h, having its centre on the line G G, and its diameter equal to the size of the cap. From the outsides of the rail A B, draw lines A h, B j, parallel to the line G G, and meeting the circle of the cap at Jij. From the points of intersec¬ tion h j, draw lines meeting on the line G G at a point i, as far into the mitre cap as it is proposed to carry the mitre. Then draw lines parallel to G G, through as many points in the rail as may be required, as B C D E F, con¬ tinuing them till they meet the mitre lines h i, j i; set one foot of the compasses in the centre of the circle o 11 j h, and extending the other to each of the points in succession, describe circular arcs meeting the diameter 0, 11. From the points of meeting draw the ordinates, 1 2 3 4 5, &c., making them respectively equal to the cor¬ responding ordinates, B C D E F, &c., and draw the figure through those points. TO DRAW THE SWAN-NECK AT THE TOP OF A RAIL, as in Plate LXXXVTII. Fig. 1, No. 1. — Continue the bottom line of the rail upwards till it intersects the line of the back of the last baluster; draw a horizontal line through the top of the newel, measure from this line down the back of the baluster to the intersection, and set off the same distance downwards on the under side of the rail, from which square out a line to intersect the horizontal line above; this will give the centre point of the curve. A slight variation from this will be seen in Fig. 2, Plate LXXXIX., the rail being there brought nearer to the newel. This variation will be easily understood, and needs no description. To FORM THE KNEE AT THE Bottom Newel.— Draw out the width of one step, as at A b (Fig. 521), and the risers connected with it above and below b c, o A, and join A c; continue the line of the first riser o A upwards to the height of half a riser at D; and through D draw a hori¬ zontal line meeting the hypo- thenuse A G in G. From D set off towards G, half the width of the mitre cap D E, make G F equal to GE; draw F m square from the under side of the rail, and make E M perpendicular to D G, and the point M will be the centre of the curve. Scrolls.—Plate C. — In Fig. 1 is shown a very simple manner of describing a scroll. Take the width of the rail in the dividers, and repeat it three times on the line 12 3 which gives the first or greatest radius for the quadrant A. Refer now to Fig. 1, No. 2, where the scheme of centres is drawn out at full size. Draw 1 2 at right angles to the first line, and make it equal to two-thirds the width of the rail. Draw 2 3 at right angles to 1 2; make it equal to three-fourths of 1 2, and join 3 1; through 2, and at right angles to 3 1 draw the line 2 4; then draw the line 208 PRACTICAL CARPENTRY AND JOINERY. S 4 at right angles to 3 2, the line 4 5 at right angles to 3 4, or parallel to 3 2, and so on with the other lines, always squaring from the one last drawn, and thus the centres are obtained, from which the quadrants B, c, D, E, F in Fig. 1 are drawn. Fig. 2, No. 1, is shown as a rail of 2f inches in width; the square is here made half of that width, or inch, the first radius is made 7^ times the square, and as the side of the square is once lost by the half revolution, the width of the scroll will be equal to 14 times the side of the square, or 19^ inches; the construction of the square will easily be understood by a reference to Fig. 2, No. 2, where it is shown full size, the numbers showing the points for the centres in succession, beginning with the least radius, and ending with the greatest. Fig. 3 is perhaps more simple than the preceding, and is adapted for a large rail, where only a small scroll can be used. The first radius is made equal to 8 inches. This distance is divided into five equal parts, and the square is made equal to one of the parts. The angles of the square are the first four centres, the middle of the side is the fifth, I' and the centre of the square the sixth centre. Fig. 4 shows a ready method of producing a converging series in geometrical progression, as such a series is often found useful in setting off the radii of scrolls. The lengths of any two lines being known—to form a series from the same: take the longest line, as A b, and make it the per¬ pendicular of a right-angled triangle, the base of which B c may be made of any convenient length; let d be the length of the second line in the series; from A B draw the perpendicular b d meeting A c in d and join b C; draw the line d e at right angles to B c, continuing it to meet B c, draw e f perpendicular to a b, then draw f g per¬ pendicular to B c, and g h perpendicular to A B ; this pro¬ cess may be continued to any extent, and the lines A B, d, f K & c -> and also b, e, g, &c., and b d, ef g h, &c., will be a series in geometrical progression. Fig. 6 is a method of producing a scroll by sixths of a circle. Describe a circle as A B, and divide its circumfer¬ ence into six equal parts, and draw the diameters shown by the darker lines on the drawing. Divide one of the divisions of the circle into six equal parts, and set off one of the divisions, equal to 10 degrees, from each diameter; then draw the second series of diameters shown by fainter lines; or the ten degrees may be set off at once by a pro¬ tractor. At the distance of two inches from the centre draw the first radius a parallel to the faint diameter and intersecting a B; from the point of intersection draw the next radius parallel to the next faint diameter, inter¬ secting the next succeeding darker lined diameter, and continue drawing the radii parallel with the faint-lined diameter, and their points of intersection with the first series of diameters from the centres of the curve of the scroll. The lines so drawn form a converging series, and their lengths are to each other in geometrical progression. In the figure the series is continued from a inwards, through one revolution and a half. Figs. 5, 7, and 8 are methods of drawing scrolls by eighths of a circle. As they differ only in the quickness of their convergency a description of one will suffice for alL In Fig. 7 proceed first to make the double cross by drawing right angles and bisecting the same, as shown on the figure. The centre of the largest arc of this scroll is situated at a distance of two inches from the centre of the scroll to the right, on the line b, and the next centre on the increasing side, 2£ inches from the centre; the most ready method of producing the converging radii is by cutting a small piece of paper to the angle which the radius of the curve makes with the diameter of the scroll, and using this as a bevel to the next diameter, and so on in succession, either converging or diverging; thus the angle of radius and diameter, taken at c, may be ap¬ plied at b, at a, and so on in succession, producing each centre by its intersection with the next diameter. The lines of the radii cb a are continued out in the open space of the scroll in this figure, beyond where their use occurs, that the manner of obtaining one from the other may be the better seen. Fig. 9.—This shows a vertical scroll, sometimes used for terminating a hand-rail when space cannot be afforded for a horizontal scroll. The method of drawing it is so obvious as to need no description. Fig. 10 is a scroll step suitable for the scroll of the rail shown at Fig 7. The centres of the various arcs are found as in Fig. 7, the same centres being used for the line of balusters and the line of nosings; then, to describe the block and step, take the length of radius of one of the arcs in the rail mould from its centre to the centre of the rail, and from its corresponding centre in the block, which will extend to the centre of the baluster, as from c to e; draw out the section of the baluster e to the intended size; then extend the compasses from the centre of the curve to the inside of the baluster, and describe from each centre in succession, to produce the interior curve of the block. The width of the block at its neck, that is, at g, should always be commensurate with the size of the baluster, as there shown; from this place the outer or convex line of the block is determined, and is struck round from the same centres as before, which are also used for the nosing line, thereby showing the size of the stepboard. At a is shown what is usually called the tail of the block. It is secured by a screw to the thick part of the riser. At i is shown the shoulder of the riser; from this point the riser is reduced to a veneer, which is carried round the convex portion of the block as far as the point h, where it is secured by a pair of counter-wedges, there shown in section. On the back part of the step is a line indicating the position of the second riser of the stairs, and the sec¬ tion of the baluster on the second step is shown; from this baluster to the next at e, should be equal to half the going of one step; and, in spacing the balusters round the scroll, it is desirable that their distances from each other should gradually diminish as they approach nearer to the centre of the scroll, and that the balusters of the inner revolu¬ tion should be as near as possible in the centres betwixt those of the outer revolution; otherwise they will look crowded and irregular. PROJECTION OF SHADOWS. 209 PART EIGHTH. PROJECTION OF SHADOWS, PERSPECTIVE, ISOMETKTCAL PROJECTION. PROJECTION OF SHADOWS. As a luminous point can be seen on all sides at the same instant, it follows that there must proceed from it an in¬ finite number of rays diverging in every direction, and extending indefinitely in straight lines. The point may thus be considered as the centre of a luminous sphere, and that sphere itself may be conceived to be composed of an assemblage of pyramids or cones, whose summits are in its centre, and whose sides are indefinitely extended. Thus an eye placed at any distance from a luminous point, C, (Fig. 522), will receive a certain number of its rays, which may be considered as forming a cone, whose base is the pupil of the eye, and whose summit is the luminous point. That one of the rays which passes through the centre of the pupil of the eye, and through the point, will be the axis of the cone. If the eye, in place of being round, were square or triangular, there would be a pyramid in place of a cone of rays. If at the distance 1, from a luminous point c, (Fig. 522), there is placed a square plane, A b, it will intercept a cer¬ tain number of the rays from c. These rays will form a that the intensity of light increases or diminishes in the ratio of the square of the distance. Again, suppose a (Fig. 523) to be a luminous point, and b, c, a, a pyramid of rays, and let b c be a plane cutting it perpendicularly. This plane receives the sum of the rays measured by the angle, b a c. Suppose the plane now turned round/as an axis into the position d e, draw the rays d a, e a, and the plane will receive now only the Vig. 523. 7 ' pyramid, whose base will be A b, and whose apex will be c. Conceive the sides of the pyramid prolonged indefinitely, and on its . aX ^ C ° 5 se ^ ^ ie e( l ua l lengths c M, M N, V > N 0, or, which is the same thing, divide the axis c 0 into equal parts in M N 0; and at each division draw a square plane perpendicular to the axis, and cutting the sides of the pyramid in 2 D, 3 E; the side of the square, 2 d, will be double the side of the square A b, and the square itself will consequently, in area, be quad¬ ruple the extent of A b. But the surface of the second square receives only the same number of rays as the sur¬ face of the first, consequently it will be only one-fourth part as light. In the same way a square at 3 will have its side three times as great as the side of the first square, and its surface thus being nine times greater will receive only a ninth part of the light. The light then evidently diminishes in respect of the distance of the illuminated object from the luminous point, in the ratio of 1 , a, for the distances, 1, 2, 3, 4; or, as it is ordinarily ex¬ pressed -. — The intensity of the light is in the inverse ratio of the square of the distance. It also follows, by a parity of reasoning, that if it be re¬ garded as converging towards any point, its intensity will inciease in the same manner; and hence the general rule_- sum of the rays, measured by the angle d a e. Conceive it further turned until it is horizontal, as d" e", then the plane, it is evident, receives no light, and is therefore in shade. Hence the rule: when a surface receives the light perpendi¬ cular to its plane it will be lighted to its maximum inten¬ sity, and that intensity will diminish in the ratio of the obliquity of the surface to the direction of the light. When a luminous body has extent, it is to be regarded as composed of an infinite number of light- , 0 points like a, sending out rays in all directions, which cross without confounding each other. The following simple experiment will prove this. Insert two tubes a b, c. d, in one side of a box e f g h (Fig. 524), and let them be in the same plane, and inclined to each other. Let there be an aperture in one side of the box so that the interior can be seen when the eye is ap- Fig. 524. - ^ plied to it. Then furnish one tube with a blue coloured glass, and the other with a red coloured glass. If the tube with the blue glass be stopped, and the light be ad¬ mitted by the other, there will be an oval of red light thrown on the side of the box opposite to the tube at lc m, and if the red glass tube be stopped, and the blue glass tube be opened, there will be an oval of blue light at n o. If both tubes be opened together, there will be an oval of led and an oval of blue, although the rays cross each other in their passage. Hence, the rays of light cross each other in every direction, without obstructing or confound¬ ing each -other. These preliminary notions of the properties of light are necessary to the proper understanding of what follows on tlie subject of shadows. Let a (Fig. 525) be a luminous point from which rays diverge in every direction, and let b be an opaque body, a cube for example, and c another opaque body, say a 210 PRACTICAL CARPENTRY AND JOINERY. sphere. The parts of these bodies which receive the rays will be more or less illuminated, and the parts which do not receive the rays will be more or less deprived of light, and will be, as it is termed, in shade. Thus, the face b d, 0 Fig. 526. of the cube will alone be illuminated, and its other five sides will be in shade. In the same way the rays a e, af, a g, &c., tangents to the sphere, determine a seg- j ment which will be illuminated, and the other segment ; will be in shade. But the rays which touch the boun- j daries of the face of the cube, if prolonged, form an indefi- I nite pyramid, truncated by the plane b d, the part a b d will be a luminous pyramid formed by the unintercepted rays, and the indefinite portion beyond will be the shadow thrown by the opaque body. Thus, the shadow thrown by a body may be considered as a solid, the form of which is dependent on the luminous body emitting the light, the opaque body intercepting the light, and on the positions of these bodies relatively to each other. The shadow there¬ fore thrown by the cube in the figure will be a quad¬ rangular pyramid, and that by the sphere, a cone. The shadow thrown by a body will appear to increase in intensity in proportion as the light which illuminates the body increases in intensity, but this is simply the effect of contrast. If the rays of the pyramid and of the cone be cut by planes m n, o p, perpendicular to their axes, the projections will be a square and a circle. If the planes be oblique to the axes as m r, o s, the projections will be a lozenge and an ellipse. The luminous body, in respect of that which is illumi¬ nated by it may have three different dimensions. 1st. It may be smaller, and in this case it may be compared to the light of a candle, and the portion of the opaque body illuminated will be smaller than the portion in shade. 2nd. It may be of the same size, in which case the illu¬ minated part of the body will be equal to the shaded part. But here another kind of shadow makes its appearance. Let a (Fig. 52G) be the luminous, and c the opaque body, both spheres of the same size. Now, from every point of the surface of a luminous rays emanate, but those alone situated on the hemisphere 12 3 are projected on c. These will be all tangents to the opaque sphere at the points : be 4, and will determine the extent of the illuminated seg- I ment, which will manifestly be the hemisphere bo 4, I and the other hemisphere will be in shade; the primary shadow thrown by the body will be a cylinder, and its intersection by the plane r s, perpendicular to the axis, will be a circle. But as the luminous rays emanating from fg, and the other infinity of points on the surface of the body a, are intercepted by the opaque body b at the tangent J points 6 7, &c., the shadow of each of these points will also | be thrown on the plane r s. Therefore, and confining the | illustration for simplicity to these points in the meanwhile, j it will appear that a second shadow will be thrown on the plane, considerably augmenting the size of the first. On attentive consideration, however, the two shadows will be seen to be very different, for, from the point 2 j emanate rays 2 o, 2 p, which are not intercepted by the I opaque body, and which, therefore, illuminate the plane within the space occupied by the secondary shadow, and which consequently diminish its intensity. This will be seen to be the case with the rays emanating from all the ; points of the surface of the luminous body contained be- ' tween/and g, and by drawing the figure to a large scale, I and projecting rays from a great number of points, it will be satisfactorily seen that the second shadow, in propor¬ tion as it extends beyond the first, will continue to decrease in intensity. The second shadow is called a penumbra. 3d. The luminous body may be greater than the body receiving light from it. In this case, as is made evident by the Fig. 527, the segment of the body illuminated, shadow b' 4' thrown on the plane c D will be less than the intercepting body. The penumbra f g is analogous to the preceding one. It ma}’’ therefore be concluded that the shadow thrown I by any body diminishes always in intensity from the first limits of the penumbra to the last; and, further, that the parts of the same surface which are nearest the luminous j point will be more highly illuminated than those that are more distant. If a luminous body at a (Fig. 528) illuminate a plane b c, the rays ab, a c, together with the plane, will make the triangle b a c. If it be removed to a' a", &c., the triangle becomes more and more acute, and its sides ap¬ proach nearer to parallel lines, and as the lines approach parallelism, the rays emanating from a, and forming the pyramid, approach equality in length; and hence the plane b c will be more equally lighted the more distant the luminous point is from it, as if, in point of fact, the luminous body were equal to the surface illuminated. This supposition gives great simplicity in the projection of PROJECTION OF SHADOWS. 211 shadows, where we suppose the light of the sun or of the moon as the illuminating medium, and from the immense distance of these bodies consider the rays as parallel; and hence, surfaces illuminated by their light are regarded as illuminated in equal intensity in all their extent. Before entering on the consideration of the modification of shadows by reflection, and their modification also by colour, it will be necessary to carry this investigation into optics somewhat further; but in the meantime, enough has been furnished as an introduction to what follows:— ON THE PROJECTION OF SHADOWS. The preliminary matter has unfolded the general idea of shadows, and prepared for the consideration of the con¬ struction of shadows produced by bodies exposed to different kinds of lights. Problem I .—The 'projection of a luminous point being given , and also of a straight line, to find the length and direction of the shadoiv of the line on the horizontal plane. Let a b (Fig. 529) be the common section of the two planes, c the luminous point, c c its projections, and cl, d' d' those of the straight line. Then draw the line C D representing the ray from c, passing through D, and continue it to meet the horizontal plane in e. This Fig. 5?9. point will be the shadow of the point d. Other rays Cl, c 2, will be intercepted by d d, and there will be behind d D, therefore, the right angled triangle due deprived of light. This triangle may be considered as a vertical plane cutting the horizontal plane, and its inter¬ section will therefore be the straight line cl e. It may also be considered as the projection of the hypothenuse D e of the triangle doe. Suppose the triangle pro¬ longed indefinitely to the left, and it is evident that its plane will pass through the light c. The light, therefore, the given straight line, and the projection of the latter, are manifestly in the same plane; and since d e is the projection of d e, c d will be that of c D, or rather c e will be the projection of the ray c e, or of the hypothenuse of the triangle c c e. Consequently, the shadow thrown by the line d d will be found in the trace of a vertical plane, passing through the horizontal projections of the light, and of the given line. Let there be given the line D cl (Fig. 530), the height of the luminous point c c , and let it be required to find the shadow of the line on the horizontal plane. Through c' and d' the horizontal projections of the light and of the b line, draw the indefinite line c d' e', which will be the trace of a vertical plane, passing through c and D; then through D, the extremity of the straight line in the vertical projection, draw an indefinite ray from C, meeting the common section of the two planes in e, which gives the place of the shadow of the point D. Transfer this to the trace in the horizontal plane in e', and the shadow sought will be the line d’ e'. Let c', o', d’, cl D, be the horizontal and vertical pro¬ jections of the light, and of the straight line. Through c and cl' draw an indefinite straight line, which consider as the trace of the vertical plane. Suppose this plane turned over on c' e' as an axis until it lies horizontally, which is done in drawing an indefinite perpendicular to the trace c' e', aud carrying on it the height c C of the luminous point from c' to c'; and in the same manner drawing the perpendicular to cl', and setting off on it the height cl' d' of the straight line cl D, and through c' and d', drawing a line meeting the trace in e', which determines the length of the shadow d' e'. To determine the shadow of a straight line, of which a part is intercepted by a vertical plane. Let / f (Fig. 530) be the given line. Through it, and through the light c, suppose to pass a vertical plane, of which the horizontal trace is c' g', then draw the ray cf prolonged to meet the common section, and cutting it in ff> f g would be the length of the shadow of / f if there were no obstacle. But suppose a vertical plane m lc interposed so as to receive a portion of the shadow, and that it is required to find that portion. If, as before, the triangle c c g is folded down on c' g as an axis, it will produce the triangle c c' g' on the horizontal plane, and there will also be found the line/ F in/' f', the indefinite intersection of the plane m h in m' ¥, and it is only necessary to draw o' f' meeting the vertical plane in l and the line m' l will be the shadow. Problem II. — Given a luminous point, and a straight line inclined to the horizontal plane, to find the shadow of the line on the plane. Let A b, a b (Fig. 531, No. 1) be the projections of the line, and d, d d’ of the point. If the former problem has been understood this will offer no difficulty. Through b M 212 PRACTICAL CARPENTRY AND JOINERY. from d" draw a straight line, meeting the common inter- I section in e; draw through e an indefinite line, perpen- j dicular to ft s, the common intersection, and through B Fig 531. No. 1. draw from the horizontal projection of the luminous point d a straight line cutting the last perpendicular in e'; join A e', which will be the shadow sought. This problem may also be solved by laying the light and the straight line in the horizontal plane, as shown in the figure. It is so simple that it requires no description. When the shadow is in part intercepted by a vertical plane. Problem III. — The projections of a straight line in¬ clined to tiuo planes being given, to find its shadow on the two planes. Let f G ,fg (Fig. 531, No. 2) be the projections of the line, and cl, d'd" those of the light. Now, a luminous ray passing through G will project the shadow of that point on the horizontal plane at h if no obstacle intervenes. Through F draw the straight line F h, which will be the But in solving this problem there is a didiculty fre¬ quently occurring as follows:—If the extremity G, or g of the line were more elevated than in this example, the luminous ray would meet the horizontal plane at a dis¬ tance too great to be within the limits of the paper, and if G were as high as the luminous point, it is evident that the ray would be parallel to the horizontal plane. In such cases take an}^ point in the straight line F a, as l , and through it, on the horizontal projection, draw l L parallel to d D; draw through l and through d an indefinite straight line, and then through d and L draw a ray which cuts d l in to, which is one of the points of the shadow sought. Through this point and F draw a straight line, prolonged to meet the vertical plane J K, and the point i of intersection will be obtained as in the preceding opera¬ tion. If it is required to operate by the projections of the straight line and luminous point, draw the ray d" l' to meet the common intersection of the two planes in to'; and from to' let fall the perpendicular to'to, cutting the prolongation of d l in to, which is the point sought. These last two pro¬ blems may be ren¬ dered very easy of comprehension, by the study of the perspective diagram 531, No. 3, wherein the same letters are used to refer i -. "X to the same -_^. /I parts. Problem IV.— The projections of a straight line, and of a plane inclined to the planes of projection being given, to find the portion of the shadoiv of the line inter¬ cepted by the plane. Let A B (Fig. 532) be the common section of the two planes; c, c c' the projections of the light; d, d d' those of Fig. 532. PROJECTION OF SHADOWS. 213 fche horizontal plane; that the ray c h is intercepted by the plane f g at the point K; that this point is at the intersection of the ray, and the line I J; that this line, as well as the given right line cl cl and the luminous point, are all in the piano of the right angled triangle h c c, and therefore, all that is required is to turn over this tri¬ angle, and its contained lines, so that it shall lie in the horizontal plane, which is thus accomplished:— Draw an indefinite line c cl, which will be the trace of a vertical plane passing through the luminous point and the given straight line. Then from c and cl let fall per¬ pendiculars, and carry on them the heights c c, and d cl From j draw also j J perpendicular to h c, and set off on it j J, equal to the height of G above g. From j draw J I which will be the intersection of the inclined plane. Through c and cl draw the ray, cutting I j in K, which is the point sought, and from K let fall onka perpendicular K lc, the extremity k of which, is the horizontal projection of K. The length of the shadow on the inclined plane will then be I K, and its horizontal projection I lc, conse¬ quently the entire shadow will be d I K, or d F lc,'. The problem may also be solved thus: through c and d in the vertical projection draw an indefinite straight line, and through c and d' draw the ray, cutting F G in lc') from this point let fall a perpendicular on c d produced, and this gives, as in the former case, the point k. In the same way the operation is performed in the case of the plane p q or p m', on the right hand, which is inclined to the straight line in the contrary way to the former. Pkoblem V.— Of shadoivs projected by rays of light which are parallel among themselves. In the problems already given, the distance of the object illuminated from the light has been supposed to be known, as it is more easy thus to conceive the construc¬ tion of shadow. For, if a luminous body, such as the sun, had been taken as the source of the light, it would have been impossible to have determined its place on the planes of projection relatively to the places of the bodies illumi¬ nated by it, and the question would have appeared more abstract. It is necessary now to see how, in the absence of the place of the light in the planes of projection, the shadows of bodies can be constructed. Let c, c c' (Fig. 533) be the projections of a straight line, and e, e e' those of the light. If the light is supposed laid over on the horizontal plane its projection will be E; E / will be the ray, e f the projection of the ray, and cythe shadow of the given line. Conceive the ray / E produced indefinitely towards z, its projection will also be indefinitely produced towards 0 . Conceive also the luminous point occupying the places successively of G and H on the prolongation of the ray, the projections of these points will necessarily be also on the prolongation of the projections of the ray in g h, cj h', &c. But it is evident that in whichever of these points, G n, the luminous point, is placed, the direction and dimensions of the shadow cf remain the same. Hence the actual position of the light¬ giving body, when its rays are parallel, is not required; it is sufficient to know the projections of a single ray. Problem VI.— The projections of a solar ray , and of a straight line being given, to determine the shadow of the line on the horizontal plane. Let 0 /, z. j (Fig. 533), be the projections of the ray, and i, 1 i those of the line. If through i is di ’awn a line lc l parallel to / z, and through i another line parallel to / z\ these two lines will be tire traces of a plane passing through the light, whatever may be its distance. These lines may be also considered as the projections of an indefinite ray, making certain angles with the planes of projections. As the right line given is equal to the right line c 0, the shadow i lc will be equal to the shadow cf. If it is re¬ quired to operate directly by the ray the problem is very easy. L lc is the direction of the ray, and the angle L lc l is the measure of its inclination with the horizontal plane. The projections of a solar ray then suffices to determine the shadows of different bodies. As the bodies can be illumined in an infinity of ways, the artist can choose that which is best adapted for his purpose. But architects, and engineers, and designers generally, have long agreed to adopt a certain angle, at which the luminous rays are supposed to fall upon the planes of projection, as possessing many advantages. This convention is the position that all the rays of light are such, that every one of their pro¬ jections makes an angle of 45°, with the common inter¬ section of the planes of projection, which can only have place in the case where the ray is in the direction of the diagonal of a cube, one of the faces of which is parallel to the vertical plane. The great advantage of this con¬ vention is expressed in this, that when the projections of the light make an angle of 45°, with the common section of the planes of projection, the length of the shadow of a straight line perpendicular to one of the planes is equal to the diagonal of the square of the same line. Let this be illustrated by an example. Since the rays of the sun are parallel, and their projec- PRACTICAL CARPENTRY AND JOINERY. 214 / g , d 7i, e i, parallel to c d , and through the points FDc' draw the rays F g, D h, c' i, parallel to E d, and their in¬ tersections will give the points of the shadow g h i on the horizontal plane. Join these by the lines f g, g h, h i, parallel to c d, F D , c d, and there is obtained the shadow of the cube on the horizontal plane. It is very important that the ray should not be con¬ founded with its projections, and it may be well to demonstrate this by taking the points of the cube successively. Let c h, h e, (Fig. 5 35) be the projections of the light, making A - an angle of 45° with the common section of the planes. From any point whatever, such as c, raise c C perpendicular to c h, and make it equal to c c. Carry the height c c from c to X, and continue the line c X, which is the diagonal of the square, from c to d ; draw the indefi¬ nite line c d, which will be the diagonal of the cube, or the ray sought; and the angle c d c, which is about 35° 16', will be the measure of the ray upon the horizontal plane, and not 45°. For if the ray were inclined 45° to the plane, as c X, its length would equal the height of the right line, or the side of the square of that line, in place of its diagonal. It is important, therefore, not to confound the ray with its projections. Following out the construction of the problem, the points F D E, and their projections g h d. are found in the same manner as c and x. Plate CI.—The projections of the diagonals oi the ima¬ ginary cube which denote the direction of the rays of light, being equal in both planes, it follows, that in all cases and whatever be the form of the surface upon which the shadow is cast, the oblique lines joining the projections of the point which throws the shadow,and that which denotes it, are also equal. In illustration of this, let R r' Fig. 1, Plate CI., be the projections of a ray of light, and A a' those of a point, the shadow of which is required to be projected on the vertical plane X Y. Draw the straight lines A a, a' a, parallel to R r', and from a', where A' a' meets X Y, the trace of the vertical plane, draw the perpen¬ dicular a' a to meet the oblique line A a, and the inter¬ section a is the position of the shadow of the point A. It will be at once seen that the line A a in the vertical is equal to the line a' a' in the horizontal projection, and the point a might have been obtained by the compasses, in setting off on A a a length equal to a' a'. Problem YII.— To find the shadow cast on a vertical wall by a straight line ab (PI. CI., Fig. 1). As we have already seen, the shadow will be a straight line, and all that is required is to find two points in that line. The shadow of A is already found at a , and we have only to find that of B at b in the same manner, and to join a and b. Suppose, now, that in place of a mere line we have a rectangular slip A B c D, then the shadow cast by this will be a similar and equal rectangle abed. Hence we have the general proposition that when a surface is 'parallel to a plane , its shacloiu thrown upon that plane is a figure equal and similar to it. When the object is not parallel to the plane, the shadow is no longer an equal and similar figure, but the method of determining it is the same. In Fig. 2, Plate CI., let A B c D be the vertical projection, and a' b' the horizontal projection of the rectangular slip, and x Y the trace of the vertical plane, which is oblique to a' b'. Draw the lines A a , b' b', meeting the trace of the plane in a' and b'; draw the vertical lines a' a, b' b, meeting the oblique lines A a, C c, and b b, D d, in ab c d, and join a b c cl to form the figure of the shadow. The mode of construction is the same when the given plane remains parallel to the vertical plane of projection, and the rectangular object is oblique to it. When there are mouldings or projections from the face of the vertical plane, the boundaries of the shadow will be an exact reproduction of the contour or section of such mouldings or projections. On the vertical plane x Y (Fig. 3) there is a moulding, across which the shadow of the rectangle is thrown. The fillet of the moulding is here regarded as a vertical plane in advance of x Y, and the plane of the shadow is found by drawing from its axis a line meeting a' b' in e'. The oblique lines drawn from E F in the vertical projection give, by their intersection with the axis of the moulding, the situation of the shadow across the fillet. The points abed are found as before, and the shadow across the curved part of the moulding is a reproduction, in the horizontal projection, of its section or contour. To find the shadow of the rectangular slip cast upon two vertical planes meeting in any angle. Fig. 4. Let X Y, Y z, be the traces of the vertical planes, a' b' the trace of the slip, and a' a, b' b', the projections of the rays. From Y, the meeting of the planes, draw Y e' parallel to a' a, and a' e' will then indicate the portion of the slip whose shadow will fall on the plane X Y, and e' b' the portion which will fall on Y z. The shadow in the vertical projection will consist of two parallelograms, having a common side, e f, in the intersections of the planes. The method of drawing these is obvious, and need not be described. Figs. 5 and 6.— To find the shadow cast by a straight line A B upon a curved surface , either convex or eoncawe , whose horizontal projection is represented by the line x e' y. We have already explained that the shadow of a point upon any surface whatever is found by drawing a straight line through that point, parallel to the direction of the light, and marking its intersection with the given surface. Therefore, through the projections A and a! of one of the points in the given straight line, draw the lines a a, A V, at an angle of 45°; and through the point a, where the latter meets the projection of the given surface, raise a perpendicular to the ground-line; its intersection with the line A a , is the position of the shadow of the first point taken; and so for all the remaining points in the line. If it be required to delineate the entire shadow east by a slip abcd. as before, upon the surfaces under consi¬ deration, we shall be enabled, by the construction above explained, to trace two equal and parallel curves a e 6, ,1. W. Lowry Jr. PROJECTION OF SHADOWS. 215 c f cl, representing the shadows of the sides AB and CD; while those of the remaining sides will be found denoted by the vertical straight lines a c and b d, also equal and parallel to each other, and to the corresponding sides of the figure, seeing that these are themselves vertical and parallel to the given surfaces. Fig. 7.—When the slip is placed perpendicularly to a given plane X-Y, on which a projecting moulding, of any form whatever, is situated, the shadow of the upper side A' b' which is projected vertically in A, will be simply a line A a, at an angle of 45°, traversing the entire surface of the moulding, and prolonged unbroken beyond it. This may easily be demonstrated by finding the position of the shadow of any number of points such as d', taken at plea¬ sure upon the straight line A' b'. The shadow of the opposite side, projected in c, will follow the same rule, and be denoted by the line c c, parallel to the former. From this example we are led to state as a useful general rule: that in all cases where a straight line is 'perpen¬ dicular to a plane of projection, it throws a shadow upon that plane, in a straight line, forming an angle of 45° with the ground-line. Fig. 8 represents still another example of the shadow cast by the slip in a new position; here it is supposed to be set horizontally in reference to its own surface, and perpendicularly to the given plane X Y. Here we see that the shadow commences from the side D B, which is in contact with this plane, and terminates in the horizon¬ tal line a c, which corresponds to the opposite side A c of the slip. Problem YIII.— The projections of a circle and of the light being given, to find the shadow of the circle on the horizontal plane. Let c cl', d c' (Fig. 536) be the projections of the light, and e fg hi jh l and g' Jc those of the circle. Take in the deemed necessary, and through each of them di*aw right lines parallel to c d. These lines will be the horizontal projections of as many luminous rays. Through each cor¬ responding point in the vertical projection, draw lines parallel to c' cl, which will be the vertical projections of the same rays. Through the intersections of these lines with a B, draw perpendiculars cutting the horizontal pro¬ jections of the light, and each point of intersection will be the shadow of the corresponding point. Thus the shadow of h will be m, that of l will be n, and so on; and the circular shadow np m s, will be the shadow of the given circle. In this simple example, the shadow being a circle, the lines given are merely for the sake of illustration. In practice it would be sufficient to find the shadow of the centre, as in the vertical projection, Fig. 1, Plate CII. To find the shadow of the same circle on the vertical plane (Fig. 537), points are taken arbitrarily in the cir- plane obliquely to its base.- and the horizontal projections of the rays are drawn to meet A b, and from each point of inter- -B section perpendicu- larsareraised cutting the vertical projec¬ tions of the rays. This shadow will be an ellipse, because the cylinder of rays is cut by the vertical -See also illustration Fig. 2, Plate CII. Figs. 538 and 539 present no difficulty, but will be under¬ stood by inspection. The subject is further illustrated by Figs. 3 and 4, Plate CII. In Fig. 540, the shadow is thrown equally on the two planes of projection. The construction of this also will be ob¬ vious on inspection, as also that of Fig. 5, Plate CII., where the planes form a salient angle, and Fig. 6, where the shadow is thrown on a circular wall. The next figure (Fig. 541) is an ex¬ ample of the shadow of a circle situated in the plane of the lumi¬ nous rays. Let c d, c'd' be the projection of the circle, and e f, f e those of the light. Through c d draw the indefinite line eg, which will be the trace of a plane passing through PRACTICAL CARPENTRY AND JOINERY. 210 tlie ray. To the ellipse, which is the vertical projection of the circle, draw two tangents paral¬ lel to e £ and pro¬ duce them to A B; from J k, the points of their intersection with A B, draw per¬ pendiculars, cutting c # in l g. These pointsdetermine the length l g of the shadow sought. But as in what follows it is necessary to determine exactly the tangent points h' i', they are found thus on the shadow. Lay the circle on the hori¬ zontal plane, and draw in the direction of the ray the tangents to it, M # N l, which give H I the points sought; and by letting fall perpendi¬ culars from these on c g, their horizontal projections h i, and consequently the projections of 1l i', are exactly determined. It is proper to remark, that by this second operation the shadow l g might have been easily found without employing the vertical projection of the circle, and this knowledge affords a ready means of solving the next pro¬ blem. Problem IX .—To find the shadow of a circle whose horizontal projection a h (Fig. 542) is perpendicular to the trace c d of a plane passing through the ray. It has been seen that when the ray is in the direction of the diagonal of a cube, the length of the shadow on the horizontal plane of a right line is equal to the diagonal of the square of its height. Take therefore e a ox eh, which carry upon any perpendicular whatever, as e to/; and then take / a or / h and carry it from e to g and g to h, c' and e h will then be the length of the shadow of the vertical diameter of the circle; then through g draw j i perpendicular to e h, and make g j, g i, each equal to the radius of the circle, and the ellipse e j h i will be the shadow required. Problem X. — To find on the circumference of a circle the tangent points of planes passing through the light when the circle is not on the plane of the light. Ler c d', d c (Fig. 543) be the projections of the light, e f, e f those of the circle, and let the shadow of the circle be found by the means already known. Produce indefinitely the plane e f of the circle towards g, and con¬ sider/# as the common section of the plane of projection. | In c d take any point whatever, as h, and from it raise a : perpendicular h H; and having determined the ray H i, the right angled triangle i h H is obtained, and its hori¬ zontal projection is i h. Find on the line/# the projec¬ tion of this triangle, which is done by letting fall upon that line perpendiculars from h! i. It is evident that the j point i will be its own projection, and the projection of h! will be h; It i then will be the horizontal projection of the triangle i h n. The original point of which lo is the projection, will be elevated above the horizontal plane by the height h n. Consequently this height will have to be carried from It to h', and the hypothenuse h' i drawn, which will be the ray n i brought back to the plane of the circle/#; for the triangle i h n being conceived to be raised on its base, li i will be absolutely in the plane / #, which is that of the circle. There remains, therefore, only to draw to the circle two tangents parallel to h', which give J k, the points sought, the projections of which will be j k, and their Vi* 644. shadow on the circum¬ ference of the ellipse will be l and m. Problem XI .—The projections of the light and those of a cylin¬ der being given, to find the sliadovj of the cylinder. This operation is re¬ duced to finding the shadow of the centre o of the circular top of the cylinder, which will be o" (Fig. 544). From this point with the radius o" g de¬ scribe the circle # k h, and draw to it the tangents f g e h, and the shadow is determined. Fig. 515 represents a cylinder, the position of which is analogous to that of the circle in Fig. 537, and the con¬ struction is the same. Fig. 546.—This is analogous to Fig. 538, and is con- upon which take again any point A; take the height a A and set it from a to c; draw the line A c, the diagona of the square, and carry it upon a b from a to d'\ thei draw A cV, which will be the diagonal of the cube, the ray or the hypothenuse of the right-angled triangle A a cZ'; pro ject this triangle on ef, the plane of the circle produced by letting fall on e f from the points a d the perpendi culars a a, d d, and the line a d will be the projection o a d', the base of the triangle. Carry the height a A fron a to A , and diaw a d, which will be the projection sough of the ray brought into the plane of the circle. Drav parallel to this ray the tangents m /, g h, and there wil be obtained the tangent points sought. From these point: Fig. 548. z PROJECTION structed in the same manner. The tangent j' z' should be considered as the trace of a plane perpendicular to the vertical plane, passing through that luminous ray which is a tangent to the cylinder in the line J j. The learner should repeat this projection with the cylinder removed further from the ver¬ tical plane, as in Fig. 2, Plate CIII. Fig. 547.—In this figure we require to find the projec¬ tions of the shadows of the two bases of a cylinder, in the same manner as the shadow of the circle is found in Fig. 539. These formed, we draw tangents to the ellipses, and wo shall have the sha¬ dow of the cylinder on the horizontal plane. But it is ne¬ cessary, moreover, to find the tangent lines of the planes of the light, as has been done in Fig. 543, and which we shall here repeat. Through any point a, taken at pleasure in the projec¬ tion of the direction of the light, draw a perpendicular, | let fall on the horizontal projection of the cylinder the lines i Jc, l n, which will be the tangent lines or limits of the shadow of the cylinder above and below. Lastly, carry the heights i A, l g, on the vertical projection, and we obtain i h' g' n' } as the limits of the shadow before and behind in the vertical projection. Fig. 548.—When we have made the two portions of the ellipses, as in Fig. 540, let us lay down the circle x y in the horizontal plane. Then project the ray in the plane of the circle, and draw tangents parallel to the ray. From a and b draw the lines a c, b d, which are the limits of the shadows, and from e the extremity of the tan- OF SHADOWS. 217 gent b e draw the line e f tangent to the ellipse, and this line will be the shadow carried from the tangent line of the cylinder to the horizontal plane. Lastly, lay the cir¬ cle y z in the ver¬ tical plane, and it will be in the same plane as the ver¬ tical projection of the light; draw to it two tangents gh, i h parallel to the projection of the ray. From g and i draw the lines l m, n o, which are the limits of the shadow on the vertical projection. From k draw Jc p, and this line is the shadow carried on the ver¬ tical plane by the tangent Tio; and thus the whole shadow of the cylin¬ der is found in the two planes. The next figure, No. 549, is analogous to Fig. 541, and does not require description. Fig. 550 is constructed like Fig. 542. It is only necessary to observe that the cylinder will receive more light in the line / e than on any other part, k l will be the touching points of the lumin¬ ous planes tangents to the surface of the cylinder, and the 2 E PRACTICAL CARPENTRY AND JOINERY. 218 projections of these lines will he k a, l b in the horizontal plane. The shadows of these lines in the horizontal planes will be m L K. Fig. 551 is the application to a cylinder of the prin¬ ciple of construction exemplified in Fig. 543. Problem XII.— To find the shadow of the interior of a concave cylindrical surface. Let c D E, c h (Fig. 552) be the projections of a concave semicylinder. Through c draw the horizontal projection of the light, cutting the curve in D; from this point raise an indefinite perpendicular D d. Through /', the vertical projection of the higher point of the cylinder (raised ver¬ tically over c), draw the vertical projection of the light, which will cut the line raised on the point D in i'. This point of intersection will be the shadow of /’ in the in¬ terior of the cylinder. To obtain a second point of the shadow. Through any part of the curve, as J, draw a line parallel to c D, cutting the curve in the point k\ from this point raise a perpen¬ dicular, and from /, the vertical projection of J, draw a line parallel to/' i', cutting the perpendicular in the point K, which will be the shadow of /. Lastly, draw the horizontal projection of the light in such manner as that it may be a tangent to the curve at L. The vertical pro¬ jection of this point will be l', and it will be the commence¬ ment of the shadow in the cylindrical cavity. Through these points (or any number similarly obtained) draw a curve V k' i', which will be the shadow of the circular part L, c. Then draw the straight line i'd, which will be the shadow of/' m', a portion of the line/'A, and the shadow of the other portion m' A will be in c D in the horizontal projection. To find this shadow directly from the luminous rays. If we consider c D in the horizontal projection as the trace of a plane cutting the cylinder, the resulting sec¬ tion will be the rectangle D F. Draw through F a ray cutting D G in i, and the point of intersection will be the shadow of F. Carry the height D i from d to i' in the vertical projection, and we have the point sought. Repeat this for the section J k , and we shall obtain the point K, and the height k K will be equal to k k'\ the tan¬ gent point L will give as its section only the line L l', and consequently the point L will be itself the point in the shadow. Fig. 553.—This is the reverse of Fig. 552, and presents no difficulty. Fig. 554.—The principles illustrated in Figs. 543 and r "“ lu ! U as the plane of the light, are ap¬ plied in the solution of this pro¬ blem. With these in view, the solution of this is not difficult. Figs. 555, 556. — Problem XIII .—To find the shadow of a cone on the horizontal plane. We have already seen that the shadow on the surface of a cylinder, and also the shadow thrown by a cylinder on the horizontal plane, are deter¬ mined by the line wherein a plane tangent to the surface Fig. 653. PROJECTION OF SHADOWS. 219 of the cylinder touches that surface. It is the same in re¬ gard to the cone. Suppose the problem solved as in Fig. 555. Conceive two planes c D E, c F E, passing through the light through the limits of the projected shadow, and by the sha¬ dow of the cone, these planes will be tangents to the surface of the cone, and conse- — quently also to the circle of its base, according to the lines D c, F c, and the line of their intersection, or the axis c E, will be in the di¬ rection of a lumin¬ ous ray passing through the sum¬ mit C, and projec¬ ting the shadow of that point on the horizontal plane in E. The projections of these planes will evidently be the triangles cde,cfe. Hence, to solve this problem, it is sufficient that we have the shadow of the summit (Fig. 556), or the point E, which we can obtain by the ray c E, or by the vertical projection c' e. From this point we draw tangents to the circle of the base ED, ef, and the radii dcfc, which will be the limits of the shadow on the surface of the cone, and the tangents will be the limits of the shadow thrown on the horizontal plane. Suppose the cone placed on its summit. The projection of the shadow in this case is very easy, if we are content with a mere mechanical solution. First find cl (Fig. 557), the shadow of the centre c of the base. From cl as a centre, with a radius equal to the ra¬ dius of the base, describe a cir¬ cle; and from the summit C draw the tangents C e, C /, and the sha¬ dow projected on the horizontal plane will be de¬ termined. To find the boundaries be¬ tween light and shade on the sur¬ face of the cone— 1st. Draw through the tan¬ gent point e a line parallel to c cl, cutting the circum¬ ference of the base in g, and from g draw g c, and we ob¬ tain the horizontal projection of the tangent sought. 2 d. From the centre c raise a perpendicular upon c e, and the radius o g will be the line sought. S cl. On d c produced, take the point h, distant from the centre by a space equal to c d\ from h draw to the base of the cone the tangents h g , h i, which will give us the points g i. If two planes e H, / H (Fig. 558), be conceived to pass through the rectilineal boundaries of the projected sha¬ dows, they will be tangents to the cone in the lines c G, c I, and will cut each other in the line c H. This line of intersection makes with the horizontal plane an angle H c h equal to that made by the lumin¬ ous rays; consequent¬ ly, the point h being the projection of H (which is raised above the horizontal plane by the height c c ) will be distant from c the extent cd\ for a ray passing through c, and projecting a shadow from that point on the horizontal plane, will be equal and parallel to H c. Problem XIV. — To draw the shadow on the concave interior of a cone. Conceive a vertical plane passing through the horizontal projection of the light, and cutting the cone (Fig. 559). Turn down this section on the horizontal plane, and we have the triangle acb. To find now the shadow of the point a : through A, its vertical projection, draw a ray which will project the shadow of the point in D on the side B c of the triangle ABC. This side has for its horizontal projec¬ tion the line c b, and as all the points in B c correspond to those in c b, we have only to let fall a perpendicular from D on b c to give us d for the point sought as the shadow of a. Take any other point, e, and suppose a vertical plane passing through it parallel to the first; it is evident that the shadow of e will be found in ef, and so on, and the result of the operation will be a hyperbola. This method of finding the shadow, point by point, how¬ ever, is very tedious, and we shall therefore describe a more ready solution. Produce the plane A B indefinitely towards I ; through the summit c draw the radius c I; from I let fall upon the prolongation of a b a perpendicular, which will give the point i; from that point draw the tangents i J, i K, and the points J K will be the commencement of the shadow projected on the interior of the cone by the arc J a K; and J, k, and d will be three points in the shadow sought. Take another point, as e : then through i and the given 220 PRACTICAL CARPENTRY AND JOINERY. point e draw i e l, which will be the trace of a plane which we suppose to pass through the summit c. Now, every section of a cone by a plane passing through its summit is a triangle; draw, therefore, l c, e c, and we have the triangle E c d', as the vertical projection of this section. Draw now through E a ray carrying this point to H on the side d' c of the triangle, and this gives us h on the horizontal projection l c. From this last operation is deduced a short method of finding the point. The point h is situated at the inter¬ section of l c and e f which contain or form the hyper¬ bola. Through the given point e draw e f parallel to a b, and the point sought will be found on that line. Through i and through e draw i e l, from l draw l c, and the point sought is also to be found in that line, and must necessarily be at h its intersection with e f. Problem XV. — To determine the boundaries of the shadow on the surface of a sphere , and the projection of the shadow from the sphere on the horizontal plane. Consider the line c d (Fig. 560), the horizontal projec¬ tion of the light, as the trace of a vertical plane passing through the centre of the sphere. The resulting section will be a great circle, which suppose turned over on to the horizontal plane, or, to avoid confusion of lines, pro¬ jected vertically on the line a b. Consider this line a b to be the common section of the planes of projection, and project on it also the luminous ray at the angle of the diagonal of a cube. This done, draw rays tangentially to the circle, and from the points e' f' let fall on c d the perpendiculars e' e, f'f and the new points e f will be the projections of e' f. The shadow of e will be pro¬ jected on a b in g, and that of / in d‘ ’, consequently the length of the shadow projected by the great circle e' n' f of the sphere will be cf d' on a b, or g d on c d. Now consider this vertical circle as a sphere, and it is evident the line e' f will be the boundary between the light and dark portions, as well as the vertical projection of a great circle inclined to the horizontal plane. The point p , the extremity of a horizontal diameter, as well as its opposite or antipodes, will have its horizontal pro¬ jection in h i, and the line h i will be the horizontal projection of the diameter of which p' is one extremity. The rays which pass through p' and its opposite project the shadows of these points in;, the middle of g'd'; and the horizontal projection of the shadows of these points will be; k, and we thus have—1st, the two axes, ef h i, of an ellipse, which will be the horizontal projection of the inclined circle (of which e' p f is the vertical projection), and therefore the horizontal projection of the boundaries of the light and shade on the surface of the sphere; 2d, the two axes, g d, j k, of another ellipse, which will be the section of the cylinder formed by the rays which are tangents to the surface of the sphere by the horizontal plane, and therefore the boundaries of the shadow thrown by the sphere on the horizontal plane. These ellipses may be then traced by the aid of a slip of paper, as described ante, p. 24, Fig. 163; or they may be traced by finding the horizontal projection of the section of the sphere on the line n! p o’, and thus obtaining points in its circumference, as at q, n. Problem XVI.— To find the shadow in the concave interior of a hemisphere. Find the tangent points D B (Fig. 561), and through C' the centre draw the line A o C. This line will be the hori¬ zontal projection of the light, and the tangent points D B will be the commencement of the shadow sought. To find another point in the shadow on the line A c, conceive, as in the preceding case, the hemisphere to be cut by a vertical plane, whose trace is the line A c. Turn down this section on the horizontal plane in a'd c'. Through a', the vertical projection of A, draw a ray, which will give the shadow of that point in a' in the concavity of the curve. From that point let fall a perpendicular upon A C, which will cut the line in a, the point sought. We have now three points in the shadow sought. To find others, take any point on the arc BAD, as E, draw through E the horizontal projec¬ tion of a ray, which will cut the hemisphere in e f, and the vertical projec¬ tion of that section will be the semicircle E' f' ; draw through e' a ray carrying the shadow of that point on the curve, and let fall from the intersection a perpendicular on E F: and e is the point PROJECTION OF SHADOWS. 221 sought. Proceed in the same manner to obtain other points, and draw through them the curve DeaB, the boundary of the shadow. If the hemisphere is in the vertical projection, as in Fig. 562, we take e g, the vertical projection of the light, for the section plane, and proceed precisely as in the fore¬ going case. Problem XVII .—To determine the shadow in a niche. The inferior part of the niche being cylindrical and its superior part spherical, we have to solve this problem by the combination of methods we have just learned. Through A and a' (Fig. 563), draw the projections of the light, and at the extremity of a' a', raise a perpendicular, which gives the point a. The line a' a will be the boun¬ dary of the shadow in the cylindrical part. Through o, the centre of the spherical portion, draw B h, the vertical projection of the light, cutting the hemisphere, of which project the section in B 2 i' E. Draw the ray B 2 i', and let fall a perpendicular on B b, and the intersection i will be the shadow of b in the interior of the hemisphere. (The niche is in fact only a quarter of a sphere, but for the purpose of the problem we regard it as a hemisphere.) We have then the two semi-axes of an ellipse b o, i o, consequently we can construct the quadrant d e i of that ellipse. Find now the horizontal projection of B in b', and draw through it the plane b' b'. Through b' draw a per¬ pendicular, cutting B i in b, the point sought, and we can, by drawing the curve e b a, complete the boundary of the shadow. If the drawing is large, it is necessary to find more points, as shown by the dotted lines. Problem XVIII .—To determine the shadows of a cylinder of which the axis is circular (such as a ring ) and the exterior form of which is a torus. Let c d be the horizontal projection of the light. The result of the section by a plane on this line will be two equal circles, having for their diameters ef g h ; and, as || these circles are in the same condition as respects the light, the results will be equal; hence we require only to operate for one of them. Let this be g h, and all the points we find for this car be transferred to e f Lay this circle in the horizontal plane in G E, and draw the rays I d, K l tan¬ gents to the circle at I and K. These points are the boun daries between the light and shade of the circle, and the shadows of these points on the hori¬ zontal plane will be at the intersections of the rays with the line c d at the points d and l\ and if we let fall on c d perpendiculars from the points I K, we shall have their horizontal pro¬ jections in i 1c, i being on the upper and 1c on the lower side. If we now draw a ray to M, the centre of the circle, we shall have N as the point most highly illuminated, and n for its horizontal projection. By these operations we have obtained the points n, lc, l , i, d, which we transfer upon c d in o, p, q, r, s. If we now make a section on t u parallel to the common section A B, we have two circles equal to thepreceding, and the projection of the light on this plane will be the same as in the vertical plane; that is, the line V u will make an angle of 45° with t u. Consequently, the tangent points will have v x as their horizontal pro¬ jections, and y u as their shadows projected on the hori¬ zontal plane. These two points are not in their place, and, as we require only the one point u , we may, if we think fit, reject y. Through V, the horizontal projection of V, draw an indefinite line parallel to c d; the shadow of V will be found in that line at z, as follows:—1st, by raising from u a perpendicular, cutting the line in z. 2d, in carrying from v to z the diagonal of the square of the height v v. 3d, in raising from V the line v' V' equal to V V and perpendicular to V z, then drawing through Y a ray, Y z, parallel to I cl, since v z is parallel to c d. Make still another section by the line 1 2, and the pro¬ jection of the light on this plane will be an angle of 90°; the tangent points will consequently be at 3 and 4, and their horizontal projections in 5 and 2. The shadow thrown on the horizontal plane by the point 4 will be 6. We shall find this point by carrying from 2 to 6 the dia¬ gonal of the square of the height 2 4. We have now a sufficient number of points for tracing the curve i V 2, the quarter of the shadow of the body, and the curve dz 6 2, the quarter of the projected shadow. The other portions can be found from this. Problem XIX — To find the outline of the shadow cast upon both planes of projection by a regular hexa¬ gonal pyramid. Plate CIII., Fig. 1. In this figure it is at once obvious that the three sides A ' B' f', a' b' o', and A' c'd' alone receive the light; con¬ sequently the edges A.' F' and A' D' are the lines of shade. To solve this problem, then, we have only to determine the shadow cast by these two lines, which is accomplished by drawing, from the projections of the vertex of the 222 PRACTICAL CARPENTRY AND JOINERY. pyramid, the lines A b and A' ct', parallel to the ray of light; then raising from the point b a perpendicular to the ground line, which gives at a' the shadow of the ver¬ tex on the horizontal plane, and finally by joining this last point a’ with the points d' and F'; the lines D a' and F a ' are the outlines of the required shadow on the hori¬ zontal plane. But as the pyramid happens to be situated sufficiently near the vertical plane to throw a portion of its shadow, towards the vertex, upon it, this portion may be found by raising from the point c where the line a' ct' cuts the ground line, a perpendicular c a, intersecting the line A b in a; the lines a d aud a e, joining this point with those where the horizontal part of the shadow meets the ground line, will be its outline upon the vertical plane. Problem XX.— To find the shadow cast by a hexa¬ gonal 'prism upon both planes of projection. Fig. 4, Plate CIII. The shadows cast upon the two planes of projection are delineated in the figures, and the lines of construc¬ tion which are also given are sufficient to indicate the mode of operation without the help of further explana¬ tions. Problem XXL— Required to determine the limit of shade in a cylinder , and likewise its shadow cad upon the two planes of projection. Fig. 2, Plate CIII. When the cylinder is placed vertically, the lines of shade are at once found by drawing two tangents to its base, parallel to the ray of light; and projecting, through the points of contact, lines parallel to the axis of the cylinder. Draw the tangents I)' d' and c' c', parallel to the ray f'; these are the outlines of the shadow cast upon the horizontal plane. Through the point of contact c draw the vertical line C E; this line denotes the line of shade upon the surface of the cylinder. It is obviously unnecessary to draw the perpendicular from the opposite point d'; because it is altogether concealed in the vertical elevation of the solid. In order to ascertain the points c' and D' with greater accuracy, it is proper to draw, through the centre o', a diameter perpendicular to the ray of light f\ Had this cylinder been placed at a somewhat greater distance from the vertical plane of projection, its shadow would have been entirely cast upon the horizontal plane, in which case it would have terminated in a semicircle drawn from the centre o', with a radius equal to that of the base. But, as in our example, a portion of the shadow of the upper part is thrown upon the vertical plane, its outline will be defined by an ellipse drawn in the manner indicated in Fig. 2 of the preceding Plate. Fig. 5. —W 7 hen the cylinder is placed horizontally, and at the same time at an angle with the vertical plane, the construction is the same as that explained above; namely, lines are to be drawn parallel to the ray of light, and touching the opposite points of either base of the cylinder; and, through the points of contact A and c, the horizontal lines A B and C D are to be drawn, denoting the limits of the shade on the figure. The latter of these lines only is visible in the elevation; while, on the other hand, the former, A B alone, is seen in the plan, where it may be found by drawing a perpendicular from A meeting the base f' g' in a'. The line a' e' drawn parallel to the axis of the cylinder is the line of shade required. The example here given presents the particular case in which the base of the cylinder is parallel to the direction of the rays of light in the horizontal projection. This case admits of a simpler solution than the preceding, in which the necessity for drawing the vertical projection of the figure is dispensed with. All that is required in order to determine the line A' e' is to ascertain the angle which the ray of light makes with the projection of the figure. Draw a tangent to the circle f' A 2 o' (which represents the base of the cylinder laid down on the horizontal plane), in such a manner as to make with f' g' an angle of 35° 16', and through the point of contact A 2 draw a line parallel to the axis of the cylinder; this line e' a' will be the line of shade as before. Problem XXII .—To find the line of shade in a cone , and its shadoiu cast upon the two planes of project ion. Fig. 3, Plate CIII. By a construction similar to that of Fig. 1, we find the point a'; from this point draw tangents to the opposite sides of the base; these two lines will denote the outlines of the shadow cast upon the horizontal plane. Their points of contact b' and c', joined to the centre a', will give the lines a' b' and a' c' for the required lines of shade in the plan; of these, the first only will be visible at A B in the elevation. If the cone be situated in the reverse position, as in Fig. 6, the shade is determined in the following manner: —From the centre a' of the base, draw a line parallel to the light; from the point a', where it intersects the perpendicular, describe a circle equal to the base, and from the point A' draw the lines A' b' and a' c, touching this circle; these are the outlines of the shadow cast upon the horizontal plane. Then, from the centre A', draw the radii a' b' and a' c', parallel to a' U and a' c'; these radii are the horizontal projections of the lines of shade, the former of which, transferred to B D, is alone visible in the elevation. But in order to trace the outline of that portion of the shadow which is thrown upon the vertical plane, it is necessary to project the point c' to c, from which, by a construction which will be manifest from inspection of the figures, we derive the point c, and the line c d as part of the cast shadow of the line c' A'. The rest of the outline of the vertical portion of the cast shadow, is derived from the circumference of the base, as in Fig. 2. Problem XXIII .—The projections of a cone and a sphere being given, to determine the shadoiu thrown by the first body on the second. Suppose, in the first instance, that the shadows belong¬ ing to both bodies have been found. 1st. Let c d (Fig. 565), be the horizontal projection of the light; e d the ray; e e the height of the cone; f d h the shadow thrown by the cone on the horizontal plane; (the horizontal projection of the sphere being supposed removed); efi eh the boundaries of shade on the surface of the cone ; i E the vertical projection of the shadow of the cone or of the lines efi eh; cEg the vertical projection of the cone. The lines f d, h cl are the traces of two planes inclined to the horizontal plane, tangents to the surface of the cone in the lines efi e h, and intersecting each other in the arris E d or its horizontal projection e d. Suppose now the sphere by which portion of the shadow is to be intercepted, cut by the vertical plane c cl in the diameter PROJECTION OF SHADOWS. 223 g q\ make the vertical projection of this section, and we have the circle K on 7c, which will intercept the ray e d in the point N, which has n as its horizontal projection. This point then is the projection of the shadow of the summit of the cone upon the sphere. We could in the same man¬ ner obtain other points, but this method is ope- rose, and wants precision, in conse¬ quence of the obli¬ quity of the inter¬ secting lines. We proceed to consider a method more direct and involving less labour. 2d. In this method we regard the sphere as being cut by plane inclined to the horizontal plane, and whose trace is f d. We find easily the inclination of the plane, because it is a tangent to the surface of the cone in the line f e, which is the horizontal projection of i E. We have thus a rectangular triangle, and f e as one of its sides. Its second side is the axis of the cone, perpendicular to e f and of which the height e e' is known, and the hypothenuse of the triangle is f E 7 . Conceive this triangle raised on its base / e, and we shall have an idea of the inclination of the plane dfe. Now from 7c, the centre of the horizontal projec¬ tion of the sphere, let fall upon d f a perpendicular a b. This line will be parallel to / e, and will serve as the vertical plane for the projection of the sphere, as well as the incliued section plane. Raise upon a b, from the point e, a perpendicular, on which set off the height e e' from e to e". Then project/in f; draw f E", and we have the first right-angled triangle ef E 1 ', similar and equal to the first, and consequently we have also the inclination of the section plane in the angle e f e". Now, with the radius 7c 7c' perpen¬ dicular to a b, describe a circle as the vertical projec¬ tion of the sphere. We now see that this is cut by the inclined plane f e", or by the shadow of the cone in the line o' p', one of the diameters of the circle of that section. Divide the horizontal projection o p of that diameter in two equal parts at q\ let fall from this a perpendicular on a b, cutting the circumference / \ Ki e- ° 66 - at the points r s, and the line r s will be the major, and o p the minor axis of the ellipse, which is the horizontal projec¬ tion of the circle produced by the section of the sphere. This ellipse, or so much of it as we require, we can trace by means of a slip of paper. Having obtained this, we pro¬ ceed in the same way in regard to the trace h d, and, the operations being completed, we transfer these shadows to the vertical projection, Fig. 556. vertical projection is very different from the others; and if we were to continue to operate in the same manner we should have the sections irregular, and the lines cutting too obliquely to give a precise result. Conceive, there¬ fore, that the circle q r is an indefinite horizontal plane, on which the shadow from a b is to be thrown. To obtain this shadow we have only to draw through oj (the vertical projection of the centre y) a ray meeting the plane q r in z, and from z to let fall a perpendicular cut¬ ting the horizontal projection of the line of light passing through the centre y, and it will give 0 as the shadow of Problem XXIV. —To determine the shadow of a coo\- cave surface of revolution. Let A b be the projections of the body, and 1 2 3 4, &c., traces of planes cutting it in the direction of the rays of light. Let us obtain first the vertical projection of the section 2, which will serve as a model for the others. Divide the concave part of the body B by horizontal sec¬ tions cd,efg h, i Jc, which will be expressed by so many circles in figure A. In doing this we should avoid multi¬ plicity of lines, by making as many of the circles as pos¬ sible of the same diameter. Find now the vertical section on 2. It is evident that the first point 2 will be on th« section q r in 2; the second point l, will be on the section i 7c in l , and also in a b in p; the third m, will be on g 7i, c d in m and 0 ; and the fourth n, on e f in n. Through 2 Im n 0 p draw a curve, which will be the vertical projection of 2 l m n. Through p draw the pro¬ jection of the ray, which will meet the curve in s, the first point of the shadow sought. Find in the same manner the points w t v u, &c., and through them draw the curve wtv u, which will be the portion of the shadow projected by the arc a y. The point u belongs to the horizontal section 4, whoso Fi(f. 5.17. 224 PRACTICAL CARPENTRY AND JOINERY. the centre of the circle a b on the horizontal plane of q r. As the circle is parallel to this plane its shadow will also be a circle of the same radius; therefore from the centre z, with the radius a y or b y, describe the circumference cutting the horizontal circle q o r in 6, which will be a point in the shadow sought, but as 6 is very near r, we can take without sensible error r as its vertical projection. Find then a point in the horizontal section i k in the same manner. The ray y z cuts the section i k in the point 7. By letting fall from 7 a perpendicular, we obtain the centre 7, from which we describe a circle equal to the first, or simply an arc of it on the circumference l k of the section i k , which gives a second point S. Proceed thus with all the sections, which will give the centres 9, 10, &c., and the arcs 11, 12, of which the projections are 11, 12. Through the intersections of the arcs 12, II, 8, 6, with the circles in A, draw a curve, which will be the horizontal projection of the shadow thrown on the concave portion of the figure, and the corresponding vertical projections are obtained by drawing perpendiculars from the same points to cut the sections qr, i k, g h, e f in B. It is not necessary to describe the method of obtaining the re¬ mainder of the shadow on the horizontal projection, as an inspection of the figure, coupled with the previous pro¬ blems, should be sufficient to enable the learner to do it. METHODS OF SHADING. Plates CIY.—CVI. The intensity of a shade or shadow is modified by the various peculiarities in the forms of bodies, by the inten¬ sity of the light, and by the position which objects may occupy in reference to it. Flat surfaces wholly exposed to the light, and at all points equidistant from the eye, should receive a uni¬ form tint or tone. In geometrical drawings, where the visual rays are imagined parallel to the plane of projection, every surface parallel to this plane is supposed to have all its parts at the same distance from the eye. When two surfaces thus situated are parallel, the one nearer the eye should receive a lighter tint than the other. Every surface exposed to the light, but not parallel to the plane of projection, and, therefore, having no two points equally distant from the eye, should receive an un¬ equal tint. The tint should, therefore, gradually increase in depth as the parts of such a surface recede from the eye. If two surfaces are unequally exposed to the light, the one which is more directly opposed to its rays should receive the fainter tint. When a surface entirely in the shade is parallel to the plane of projection, it should receive a tint uniformly dark. When two objects parallel to each other are in the shade, the one nearer the eye should receive the darker tint. When a surface in the shade is inclined to the plane of projection, the part which is nearest the eye should receive the deepest tint. If two surfaces exposed to the light, but unequally inclined to its rays, have a shadow cast upon them, the shadow upon the lighter surface will be more intense than that on the darker surface. We shall now proceed to give some directions for using the brush, or hair-pencil, and explain the usual methods em¬ ployed in producing this conventional tinting and shading. The methods of shading most generally adopted are either by the superposition of any number of flat tints, or of tints softened off at their edges. The former method is the more simple of the two, and should be the first attempted. Shading by Flat Tints .—Let it be proposed to shade the prism, Fig. 4, Plate CIY., or Fig. 3, Plate CVI., by means of flat tints. According to the position of the prism, as shown by its plan, Fig. 1, Plate CIV., one face, a b, is parallel to the plane of projection, and, therefore, entirely in the light. This face should receive a uniform tint, either of Indian ink or sepia. When the surface to be tinted happens to be very large, it is advisable to put on a very light tint first, and then to go over the surface a second time with a tint sufficiently dark to give the desired tone to the surface. The right-hand face b g being inclined to the plane of projection, should receive a graduated tint as it recedes. This gradation is obtained by laying on a succession of flat tints in the following manner:—First, divide the side into equal parts by vertical lines. These lines should be drawn very lightly in pencil, as they merely serve to cir¬ cumscribe the tints. A grajdsh tint is then spread over the first division, cb 11, Fig. 2. When this is dry, a similar tint is laid on, extending over the first and second divisions, and so on, till, lastly, a tint covering the whole surface imparts the desired graduated shade to that side of the prism, as in Fig. 3. The number of tints designed to express such a graduated shade depends upon the extent of the surface to be shaded; and the depth of tint must vary according to the number. As the number of washes is increased, the whole shade gradually presents a softer appearance, and the lines which border the different tints become less harsh and perceptible. For this reason the foregoing method of representing a shade or graduated tint by washes succes¬ sively passing over each other, is preferable to that some¬ times employed, of first covering the whole surface, and then gradually narrowing the tint at each successive wash, because in this way the outline of each wash remains un¬ touched, and presents, unavoidably, a harshness, which, by the former method, is in a great measure subdued. The left-hand face a is also inclined to the plane of pro¬ jection ; but, as it is entirely in the light, it should be covered by a series of much fainter tints than the last surface, which is in the shade. It should darken as it recedes. The gradation of tint is effected in the same way as before, as shown in Figs. 3 and 4. Let it be proposed to shade a cylinder by means of flat tints, Figs. 5 to 12, Plate CIV. In shading a cylinder it will be necessary to consider the difference in the tone proper to be maintained between the part in the light and that in the shade. It should be remembered that the line of separation between the light and shade, a b, is determined by the radius, o a', Fig. 5, drawn at an angle of 45°, and perpendicular to the rays of light. That part, therefore, of the cylinder, which is in the shade, is comprised between the lines a b and c d. This portion, then, should be shaded conformably to the rule previously laid down for treating surfaces in the shade MHETIH1©©® a IF @IUABil$, time civ. METHODS OF SHADING. 225 inclined to the plane of projection. All the remaining part of the cylinder which is visible presents itself to the light; but, in consequence of its circular figure, the rays of light form angles varying at every part of its surface. In order to represent with effect the rotundity, it will be necessary to determine with precision the part of the surface which is most directly affected by the light. This part is situated about the line e i, Fig. 12, in the vertical plane of the ray of light, R o, Fig. 5. As the visual rays, however, are perpendicular to the vertical plane, and therefore parallel to V o, it follows that the part which appears clearest to the eye will be near this line V o, and may be limited by the line T o, which bisects the angle v 0 R and the line R 0 . By projecting the points e' and m', and drawing the lines e i and m n, Fig. 12, the surface comprised between these lines will represent the lightest part of the cylinder. This part should have no tint upon it whatever, if the cylinder happen to be polished—a turned iron shaft, or a marble column for instance; but if the surface of the cylinder be rough, as in the case of a cast-iron pipe, then a very light tint—considerably lighter than on any other part—may be given it. Again, let us suppose the half-plan of the cylinder to be divided into any number of equal parts. Indicate these divisions upon the surface of the cylinder by faint pencil lines, and begin the shading by laying a tint over all that part of the cylinder in shade. This will at once render evident the light and dark parts. When this is dry put on a second tint, extending over that division which is to be deepest in colour, then spread a third tint over this division, and one on each side of it. Proceed in this way until the whole of that part of the cylinder which is in the shade is covered. Treat in a similar manner the left-hand side, and com¬ plete the operation by covering the whole surface of the cylinder—excepting only the division in full light—with a very light tint. Shading by Softened Tints. —The advantage which this method possesses over the one just described, consists in imparting to the shade a softer appearance; the limita¬ tions of the different tints being imperceptible. It is, how¬ ever, more difficult. Let it be proposed to shade by this method the former example of a prism. Apply a narrow strip of tint to the nearest division of the shaded side, and then, qualifying the tint in the brush with a little water, join another lighter strip to this, and finally, by means of another clean brush moistened with water, soften off the edge of this second strip, which may be taken as the limit of the first tint. When the first tint is dry, cover it with a second, which must be similarly treated, and should extend beyond the first. Proceed in this manner with other tints, until the whole face is shaded, as presented in Fig. 3, Plate CVI. In the same way the left-hand face is to be covered, though with a tint considerably lighter, for the rays of light fall upon it almost perpendicularly. Let it now be proposed to shade the cylinder, Fig. 7, Plate CVI., by means of softened tints. The boundary of each tint being indicated as before, the first strip of tint must cover the line of extreme shade, and then be softened off on each side. Other and suc¬ cessively wider strips of tint are to follow, and receive the same treatment as the one first put on. As this method requires considerable practice before it can be performed with nicety, the learner need not be discouraged at the failure of his first attempts, but per¬ severe in practising on simple figures of different sizes. If, after shading a figure by the foregoing method, any inequalities in the shade present themselves, such defects may be remedied, in some measure, by washing off redun¬ dancies of tint with the brush or a damp sponge, and by supplying a little colour to those parts which are too light. Dexterity in shading figures by softened tints is best acquired in practising upon large surfaces; this is the surest way of overcoming timidity and hesitation. Elaboration of Shading and Shadoivs.— Having thus laid down the simplest primary rules for shading isolated objects, and explained the easiest methods of carrying them into operation; it is now proposed to illustrate their application to more complex forms, to show where the shading may be modified or exaggerated, to introduce additional rules, and to offer some observations and direc¬ tions for shading architectural and mechanical drawings. Whatman’s best rough-grained drawing-paper is better adapted for receiving colour than any other. Of this paper, the Double Elephant size is preferable, as it pos¬ sesses a peculiar consistency and grain. A larger paper is seldom required, and even for a small drawing, a por¬ tion of a Double Elephant sheet should be used. The paper for a coloured drawing ought always to be strained upon a board with glue, or by means of a straining frame. Before proceeding to lay on colour, the face of the paper should be washed with a sponge well charged with water, to remove any impurities from its surface, and to prepare it for the better reception of the colour. The whole of the surface is to be damped, that the paper may be subjected to a uniform degree of ex¬ pansion. It should be only lightly touched by the sponge, and not rubbed. Submitted to this treatment, the sheet of paper will present, when thoroughly dry, a clean smooth surface, not only agreeable to work upon, but also in the best possible condition to take the colour. The size of the brushes to be used will, of course, de¬ pend upon the scale to which the drawing is made. Long thin brushes, however, should be avoided. Those possess¬ ing corpulent bodies and fine points are to be preferred, as they retain a greater quantity of colour, and are more manageable. During the process of laying on a flat tint, if the sur¬ face be large, the drawing may be slightly inclined, and the brush well charged with colour, so that the edge of the tint may be kept in a moist state until the whole surface is covered. If in tinting a small surface the brush should be too fully charged with colour, the surface will unavoidably present rugged edges, and an uneven appear¬ ance throughout. A moderate quantity of colour in the brush, well and expeditiously spread on the paper, is the only method of giving an even, close-grained aspect to the surface. As an invariable rule let it be remembered, that no tint, shade, or shadow, is to be passed over or touched again until it is quite dry, and that the brush is not to be moved backwards and forwards through the colour. In the examples of shading which are given in this 2 f 226 PRACTICAL CARPENTRY AND JOINERY. work, it may be observed that all objects with curved outlines have a certain amount of reflected light imparted to them. It is true that all bodies, whatever may be their form, are affected by reflected light; but, with a few ex¬ ceptions, this light is only appreciable on curved surfaces. The judicious degree and treatment of this light is of con¬ siderable importance. All bodies in the light reflect on the objects near them some of the rays they receive. The shaded side of an isolated object is lighted by rays reflected from the ground on which it rests, or from the air which surrounds it. In proportion to the degree of polish, or brightness in the colour of a body, is the amount of reflected light which it communicates to adjacent objects, and also its own sus¬ ceptibility of illumination under the reflection from other bodies. A polished column, or a white porcelain vase, receives and imparts more reflected light than a rough casting or a stone pitcher. Shade, even the most inconsiderable, ought never to extend to the outline of any smooth circular body. On a polished sphere, for instance, the shade should be deli¬ cately softened off just before it meets the circumference, and when the shading is completed, the tint intended for the local colour may be carried on to its outline. This will give transparency to that part of the sphere influ¬ enced by reflected light. Very little shade should reach the outlines even of rough circular bodies, lest the colour¬ ing look harsh and coarse. Shadows also become lighter as they recede from the bodies which cast them, owing to the increasing amount of reflection which falls on them from surrounding objects. Shadows, too, are modified in intensity by the air, as they recede from the spectator; they thus appear to increase in depth as their distance from the spectator diminishes. In nature this difference in intensity is only appreciable at considerable distances. Even on exten¬ sive buildings inequalities in the depth of the shadows are hardly perceptible; it is most important, however, for the effective representation of architectural subjects drawn in plan and elevation, that the variation in the distance of each part of an object from the spectator should at once strike the eye; and therefore a conventional exaggeration is practised. The shadows on the nearest and most pro¬ minent parts are made very dark, to give scope for the due modification in intensity in those parts which recede. The same direction is applicable to shades. The shade on a cylinder, for instance, situated near the spectator, ought to be darker than on one more remote. As a general rule, the colour on an object, no matter what it may be intended to represent, should become lighter as the parts on which it is placed recede from the eye. Plate CV. presents some examples of finished shad- mg. The remarks which we now propose to offer upon each of these figures are applicable alike to all forms of a similar character. Fig. 1 represents a hexagonal prism surmounted by a fillet. The most noticeable part of this figure is the shadow of the prism in the plan view. It presents an example of the graduated expression which should be given to all shadows cast upon plain surfaces. Its two extremities are distinctly different in their tone. This difference is an exaggeration of the natural appearance necessary for the effect aimed at. Figs. 2, 3, and 6 exemplify the complex appearance of shade and shadow presented on concave surfaces. It is worthy of notice that the shadow on a concave surface is darkest towards its outline, and becomes lighter as it nears the edge of the object. Reflection from that part of the surface on which the light falls, causes this gradual dimi¬ nution in the depth of the shadow; the part most strongly illuminated by reflected light beiug opposite to that most strongly illuminated by direct light. No brilliant or extreme lights should be left on con¬ cave surfaces, as they tend to render doubtful, at first sight, whether the objects represented are concave or con¬ vex. After the local colour has been put on, a faint wash should be passed very lightly over the whole concavity. This will modify and subdue the light, and tend to soften the tinting. The lightest part of a sphere {Fig. 4) is confined to a mere point, around which the shade commences and gradually increases as it recedes. This point is not in¬ dicated on the figure, because the actual shade tint on a sphere ought not to be spread over a greater portion of its surface than is shown there. The very delicate and hardly perceptible progression of the shade in the immediate vicinity of the light point, should be effected by means of the local colour of the sphere. In like manner, all polished or light-coloured curved surfaces should be treated; the part bordering upon the extreme light should be covered with a tint of local colour some¬ what fainter than that used for the flat surfaces. In curved unpolished surfaces, the local colour should be gradually deepened as it recedes from that part of the surface most exposed to the light. In shading a sphere, the best way is to put on two or three softened-off tints in the form of crescents converging towards the light point, the first one being carried over the point of deepest shade. A ring {Fig. 5) is a difficult object to shade. To change with accurate and effective gradation the shade from the inside to the outside of the ring, to leave with regularity a line of light upon its surface, and to project its shadow with precision, require a degree of attention and care in their execution, greater perhaps than the shade and shadow of any other simple figure. The learner, there¬ fore, should practise the shading of this figure, as he will seldom meet with one presenting greater difficulties. Figs. 7 and 8 show the peculiarities of the shadows cast by a cone on a sphere or cylinder. The rule that the depth of a shadow on any object is in proportion to the degree of light which it encounters on the surface of the object, is. in these figures very aptly illustrated. It will be seen, by referring to the plan {Fig. 7), that the shadow of the apex of the cone falls upon the lightest point of the sphere, and this is therefore the darkest part of the shadow. So also the deepest portion of the shadow of the cone on the cylinder, in the plan {Fig. 8), is where it comes in contact with the line of extreme light. Flat surfaces are similarly affected; the shadows thrown on them being less darkly expressed, according as their in¬ clination to the plane of projection increases. The local colour on a flat surface should, on the contrary, increase in depth as the surface becomes more inclined to this plane. These figures also show that shadows as well as shades are affected by reflected light. This is very observable where the shadow of the cone falls upon the cylinder. HTraptfi©? rosniwi®' i'5M©TH©= PERSPECTIVE. 227 Notwithstanding the most careful exertions of the colourist to keep every feature of a drawing clear and distinct, some amount of uncertainty, resulting from the proximity and natural blending of the different parts, will pervade the lines which separate its component members. For practical working purposes, therefore, a completely coloured drawing is unsuitable. On the other hand, a mere outline, although perhaps intelligible enough to those who are familiarly acquainted with the object de¬ lineated, has an undecided appearance. As complete colouring renders it difficult for the eye to separate the various parts, owing to an apparently too intimate re¬ lationship between them; a line drawing, on the contrary, perplexes the eye to discover any relation between them at all, or to settle promptly their configuration. The eye involuntarily asks the question, Is that part round or square, is it in the plane of the contiguous parts or more remote? As a means of avoiding the indefiniteness pre¬ sented by the outline in the coloured drawing, and the want of adequate coherence and doubtfulness in the mere line drawing, recourse is not unfrequently had to a kind of semi-colouring, or rather shading and tinting the parts. In this kind of drawing, it is advisable to follow a di¬ rection previously given, viz., to modify the colour on every part according to its distance from the eye. It may be as well also, for the purpose of maintaining harmony in the colouring, and of equalizing its appearance, to colour less darkly large shades than small ones, although they may be situated at an equal distance from the eye. The tinting should be very considerably lighter than on finished coloured drawings; and, indeed, no very dark shading should be employed. Besides presenting too violent a contrast between the parts coloured and those without any colour at all, dark shading would produce, in some measure, the indistinctness which is objectionable in completely tinted drawings. When, however, any architectural or other object is represented in perspective, the aim of the artist should be to avoid all the conventional exaggerations of which we have spoken, and to imitate to the best of his ability the appearance the object would have in nature. PERSPECTIVE. As an introduction to this study, it is necessary to ob¬ serve, that a luminous point emits rays in all directions, and that all the points of the surface of a body are ren¬ dered visible by means of rays, which represent the axes of different cones formed by the emanation of bundles of rays from these points. Let the line A B be placed before the eye c. It is evi¬ dent that the sum of the visual rays which 1 emanate from each of 3 the points of that 3 line to the eye, as t 1 C, 2 c, 3 C, &c., forms a triangle 1 c 7, of which the base is I 7 and the summit C. It 7 is easy to see that if in place of the line a plane or curved surface is sub¬ stituted, the result will be a pyramid of rays in place of a triangle. Let A b (Fig. 569) be a straight line, and let the globe of the eye be represented by a circle, and its pupil by the point C. The ray emanating from A, entering through c, will proceed to the retina of the eye, and be depicted at a. And as it follows that all the points of A b will send rays, entering the eye through c, the whole image of A B will be depicted on the retina of the eye in a curved line a 3 b. Conceive the line A B moved to a greater distance from the eye, and placed at A' b', then the optic angle will be reduced, and the image a' 3 b' will be less than before; and as our visual sensations are in proportion to the magni¬ tude of the image painted on the retina, it may be con¬ cluded that the more distant an object is from the eye, the smaller the angle under which it is seen becomes, and consequently, the farther the same object is removed from the eye the less it appears. Observation has rendered it evident, that the greatest angle under which one or more objects can be dis¬ tinctly seen is one of 90°. If between the object and the eye there be interposed a transparent plane (such as one of glass m n), the intersection of this plane with the visual rays are termed perspectives of the points from which the rays emanate. Thus a is the perspective of A, b of B, and so on of all the intermediate points; but, as two points determine the length of a straight line, it follows that ab is the perspective of A B, and a ' b' the perspective of a' b'. It is evident from the figure that objects appear more or less great according to the angle under which they are viewed; and further, that objects of unequal size may appear equal if seen under the same angle. For draw fg, and its perspective will be found to be the same as that of a' b' . It follows, also, that a line near the eye may be viewed under an angle much greater than a line of greater di¬ mensions but more distant, and hence a little object may appear to be much greater than a similar object of larger dimensions. Since, therefore, unequally-sized objects may appear equal in size, and equally-sized objects unequal, and since objects are not seen as they are in effect, but as they appear under certain conditions, perspective may be defined to be a science which affords the means of representing, on any surface whatever, objects such as they appear when seen from a given point of view. It is divided into two branches, the one, called linear perspec¬ tive, occupying itself with the delineation of the contours of bodies; the other, called aerial perspective, with the 228 PRACTICAL CARPENTRY AND JOINERY. gradations of shade and colour produced by distance. The former of these only is proposed to be discussed here. The perspective of objects, then, is obtained by the in¬ tersection of the rays which emanate from them to the eye by a plane or other surface (which is called the picture), situated between the eye and the objects. From the explanation and definition we have just given, it is easy to conceive that linear perspective is in reality the problem of constructing the section, by a surface of some kind, of a pyramid of rays of which the summit and the base are given. The eye is the summit, the base may be re¬ garded as the whole visible ex¬ tent of the object or objects to be represented, and the inter¬ secting surface is the picture. A good idea of this will be obtained by supposing the pic¬ ture to be a transparent plane, through which the object may be viewed, and on which it may be depicted. Let us suppose any object, as the pyramid ab (Fig. 570), to be viewed by a spectator at c through a transparent plane D e. From the points of the pyramid visual rays will pass to the eye of the spectator, and if the points where they intersect the transparent plane be joined by lines on it, a representation of the object, as seen by the spectator, will be obtained. The transparent plane represents the picture, and the problem in perspec¬ tive, is, as we have said, to make a section of the pyramid or cone of rays, as the case may be, by a plane, curved, or other surface. The figure illustrates the mode of doing this. A horizontal projection of the visual rays is made, that is obtained from the following figures, in which the process is repeated geometrically. Let o and o' be the projection of the eye, e f ef those of the picture, and abg , a b c cl g, of a pyramid with a square base. Now, if from the eye a line is drawn to the points A a of the object, we shall have for the projections of that line, the lines AO, a o'. The points a' a", where these pro¬ jections cut the projections of the picture, are evidently the piojections of the points in which the visual rays meet to say, from the plane or horizontal projection of the point required to be found in perspective, a line is drawn to the position or station point of the spectator, as A a c, and another line from its vertical projection to the eye of the spectator, as A a' o. At the points of intersection of the first set of lines with the horizontal projection of the picture, a perpendicular a a' is drawn, and the intersection of this with the corresponding line from the vertical pro¬ jection, gives the point a' required. All the other points are obtained in the same manner. A much better idea of the mode of operation will be the picture, and all that is required is to find the position of that point on the picture itself. Conceive e' f' to be the elevation of the face of the picture. To its base E 1 D transfer the points a" b", b"g", g" c", c" d", in which the rays in the horizontal projection cut the picture, and from these points draw indefinite lines perpendicular to e'd. On the line a" a set up from the base E' D the height Ea', in the vertical projection of the picture, and a! will be the perspective of the point required. Proceed in the same manner to obtain the other points. As on the problem of finding the perspec¬ tive of any point the whole science of per¬ spective rests, the student should make himself thoroughly master of it, and although he may not be able to perceive the direct utility of what immediately follows, he is recommended to study it with care and attention, so as to understand the principles. The applica¬ tion of these will be developed by and by, and methods of abridging the labour will be pointed out; the student will also be enabled to devise others for himself. In addition to the vertical and horizontal planes with which we are familiar in the ope¬ rations of projection, several auxiliary planes are employed in perspective, and particularly the four following:— 1. The horizontal plane A B (Fig. 572), on which the spectator and the objects viewed are supposed to stand; this is therefore generally termed the ground plane or geometrical plane. 2. The plane c R, which has been considered as a transparent plane placed in front of the spectator, on which the objects are delineated. It is called the plane of projection or the plane of the picture. The intersec¬ tion c D of the first and second planes is called the line of projection , the ground line, or base of the picture. PERSPECTIVE. 229 3. The plane E f passing horizontally through the eye of the spectator, and cutting the plane of the picture at Fi?. 672. N _ F I II D ./M _/ right angles in the line H I, is called the horizontal 'plane, and its intersection with the plane of the picture is called the horizon line, the horizon of the picture , or simply the horizon. 4. The plane M N passing vertically through the eye of the spectator, and cutting each of the other planes in a right angle, is called the vertical plane, and sometimes the central plane; hut as the term vertical plane is applied to any plane that is perpendicular to the ground plane, we shall use-the term central plane for the sake of avoid¬ ing confusion. Point of view, or point of sight, is the point where the eye is supposed to be placed to view the object, as at o (Fig. 571), and is the vertex of the optic cone. Its pro¬ jection on the ground plane in Fig. 572 is M, and is termed the station point. The projection of any point on the ground plane is called the seat of that point. Problem I.— To find the perspective of a given point. Let Jc (Fig. 573) be the given point, draw the visual ray k o, which will meet the picture in k’, the perspective of k, and it is only necessary now to know how to determine the position of k'. Since o is the horizontal projection of the eye, if we draw k o' it will be the horizontal projection of the visual consequently be perpendicular to the ground plane A B. We have already seen that k o, the hypothenuse of this triangle, cuts the picture in Jc , and we perceive that the side k o' of the triangle cuts the base of the picture in P; and as the two points P k' are in the plane of the triangle, and in the plane of the picture, the intersection of the picture and triangle will be the line P Jc' —whence it fol¬ lows, that to determine in the picture the perspective of k } we draw from that point a line k o', cutting the base of the picture in P ; from P we elevate a perpendicular indefinitely; we draw then the visual ray k o, cutting this vertical line in Jc' , which will be the point sought. Observe that the triangle k o o' is intersected in p Jc' parallel to its side o o', and consequently the points of the triangle will be proportional among themselves; thus Jc o' : o o' :: Jc P : P Jc', and the height of Jc', may be ob¬ tained by seeking a fourth proportional to the three lines Jc o, o o', Jc P, which will be P Jc. These three lines may always be known, for Jc o’ is the distance of the object from the position of the spectator o', called the station point; o o' is the vertical height of the eye above the ground plane, and Jc P is the distance of the object from the pic¬ ture. Thus the distance of the object from the station point, is to the height of the eye as the distance of the object from the picture is to the height of the perspective point in the picture. The triangles Jc o' o, L o' o, are similar, since they are the same height, and are comprised between parallels. These triangles will therefore be proportional; thus V, the perspective of L, will have the same height in the picture as k', the perspective of Jc. To obtain the perspective of L, therefore, we can use the triangle Jc o' o. To do this, project L upon the horizontal trace of the central plane, M N in the point Jc, which may then be considered as the vertical projection of L. The remainder of the operation need not be described. Problem II.— To find tJie perspective of a given right line. Let A b (Fig. 574) be the given line. From its extre¬ mities A B, draw to the eye of the spectator the rays A o, B o. It is evident that, as the picture E f is not cut by these rays, each of the points will be at the same time the original and perspective points. Hence we have:— Rule I.— When a straight line lies in tJie plane of the picture, it suffers no change, but its perspective repre¬ sentation is the same as its original. Let the line D K be situated in the ground plane, draw from its extremities d 0 , Jc o, the projections of the rays D o, k o, and we have the triangle doe, the base of which will be parallel to the line of projection; and con¬ sequently every point in its base, as d k, will be equi¬ distant from the line of projection, and the height of all the points in the picture will be the same; therefore, the straight line between the points dk, which is the perspec¬ tive of d K, will be parallel to the line of projection. Further, the triangle doe will be cut by the picture parallel to its base D e, and the intersection d Jc will there¬ fore be parallel to that base and to the line of projection. From this we obtain-- 230 PRACTICAL CARPENTRY AND JOINERY. Rule II. — When an original line is parallel to the base of the picture, the perspective of that line will also he parallel to it. The line A D, it will be seen, is in the central plane; the triangle d A o is therefore also in'that plane, and consequently vertical. The triangle, therefore, will cut the picture in the line A d perpendicular to its base. A d produced would be the trace of a plane perpendicular to the original plane, which would necessarily pass through the point of sight. The line A o' will also contain the projection of the vertical triangle D o' 0, and the inter¬ section of the triangle d A o with the picture will be the line A d, which is the perspective of A D, and it will tend to the projection of the point of sight. The straight line B K, which is the base of the triangle K B o, is not in a plane perpendicular to the original plane, but inclined to it; for kb is beyond the central plane, while o is in that plane. Consequently, the pro¬ jection of the inclined plane in which this triangle is situated, will be the triangle K B o' on the original plane, and its intersection with the plane of projection in the picture will be bo'; and, as the intersection B k is part of B o', the perspective b k is also directed towards the point of sight. It can be shown that this would also be the case with all other lines perpendicular to the picture; and therefore it can be concluded— Rule III.— The perspectives of all lines perpendicular to the picture pass through the point of sight. Let A b (Fig. 5(5) be a straight line, making with B c, the base of the picture, an angle of 45°. The perspective ol the line will be a' B, which, being produced, would meet the horizon in the point d', and this will be the point of convergence of the perspectives of all lines parallel to A b. It is easy to perceive that the original line A b is the base of a scalene triangle ABE, formed by that line and the rays A E, be, and which triangle has its base in the original plane, and its summit in the eye of the spectator. It will be inclined to both planes of projection, and will cut the picture in the line a' B. The vertical projection of this triangle in the pic¬ ture will be the triangle c e b. Now, any triangle may be regarded as the moiety of a quadrilateral figure; therefore, if through e we draw a line E F parallel to B A, and another A f parallel to the ray b e, we shall obtain a quadrilateral figure A b e f, double the first triangle A b e, divided into two equal parts by the diago¬ nal or ray A E. Since, then, the side e f is of the height of the eye, it will necessarily meet the plane of the pic¬ ture at a point d in the horizon, and the line cV B will be the intersection of the plane A b with the picture. There¬ fore the point cl will be the vanishing point of A b, and consequently of all the original lines parallel to it. Suppose perpendiculars let fall from E f upon the ori¬ ginal plane, and we obtain the points ef as the projections of E and F, draw e b, ef, f a, and we have the quadrila¬ teral ab, ef, for the horizontal projection of the inclined plane A E. The lines f e, f e, with the perpendiculars E e, F /, form a rectangle e F, or / e, which passes through the perpendiculars E e, F/. Consequently, this plane is vertical, and cuts the plane of the picture in the line cl d 1 . Observe that the lines E d 1 , e cl, being parallel to A B, make with the picture an angle of 45°, and therefore that E e is equal to e d\ But E e’ is the principal ray, or the distance of the eye from the picture, and therefore dl may be regarded as the point of distance carried upon the horizon from e’ to d\ Hence we obtain— Rule IV. —The perspectives of all original lines mabing an angle of 45° with the picture, vanish in the point of distance. If the original line make a greater angle than 45° with the picture, its vanishing point will be found‘between the point of sight and the point of distance; and if a less angle, its vanishing point will be beyond the point of distance; and the general rule is thus expressed:— Rule V. — The perspective of an original line, making any angle ivhatever with the picture, will have its vanishing point on the horizon at the intersection with the picture of a plane parallel to the original line passing through the point of sight. We have seen that the principal ray is necessarily parallel to the lines which are perpendicular to the pic¬ ture, and that its intersection with the picture, or the point of sight, is the vanishing point for all such lines. We have seen, also, that the vanishing point of lines making an angle of 45° with the picture, is at the inter¬ section with the picture of a line drawn through the point of sight parallel to the original line. And as in this, so in the case of any line making any angle whatever with the picture. Whence follows the general rule. Rule VI.—The vanishing point for horizontal straight lines, foi'ming any angle whatever with the picture, is at the intersection with the picture of a parallel to these lines, drawn through the point of sight. To show this geometrically, let ABCDEFG (Fig. 576) be the horizontal pro¬ jection, or plan of an original object, n K the picture line, and o the station point; then the vanishing b point of A G, and all its parallels, will be found by drawing 0 a parallel to it, to intersect the picture line produced, when a is the vanishing point. The vanishing point of is and its parallels will be b. The vanishing point of F E and all other lines PERSPECTIVE. 231 perpendicular to the picture will be the point of sight e ; and the lines G F, B c, D E, being parallel to the picture, their perspectives will also be parallel to it. The rules thus established, enable us to abridge, in many instances, the operations of drawing perspectives, as may be thus illustrated. Let ABCD (Fig. 577) be the plan of a square, 0 the place of the spec¬ tator, e f the line Fig. 677 . of projection, and o c the central plane. Draw ef to re¬ present the ver¬ tical projection of the base of the picture, G K the horizontal line, 0 the point of sight, G and o 0 the ver¬ tical plane. Then to draw the per¬ spective of the square, transfer the side c D to 0 b (see Prob. I., Rule I.), and draw b a , which is the perspective of the side B A of the square produced to its vanishing point (see Prob. I., Rule III.) Then, as the diagonals of the square form an angle of 45° with the picture, from 0 , set off on the horizontal line o a, 0 K, each equal to the distance 0 c (see Prob. I., Rule IV.), and draw b G, 0 K, the perspectives of the diagonal produced to their vanishing points, and join the points a cl where these intersect the perspectives of the sides by the line d a, parallel to the base of the picture (see Prob. I., Rule II.), and od ab will be the perspective of the square. If the perspective of the point A alone had to be sought, the operation would be simply to draw b 0, and to inter¬ sect it by 0 K. Let it be required, for example, to find the perspective of an original point A, situated 35 feet from the central plane of the picture, and 63 feet from the base. Set off from any convenient scale from 0 to f (Fig. 578), 35 feet, and draw f c, c being the point of sight; set off from / to b, 63 feet, and draw to E, the distance of the spectator or point of distance, the line b E, and the inter¬ section of the lines at A will be the perspective of the point required. Fig . 578 _ Let it be required to draw __e_ e _ the perspective of a square, ' situated in the original % plane, at any angle what- ever to the plane of the picture. To solve this problem it is necessary only / to find the perspective of ~T o a single point, which may be done in various ways. Let A B (Fig. 579) be the plan of the square, o the place of the spectator or station point, E K the line of projection, and 0 c the central plane. We see that the point A is on the horizontal trace of the central plane, and at the distance A c from the picture, and that, there¬ fore, the perspective of A is to be found on the central line, in the vertical projection o c. Set off the distance A c from c to g , and draw from g to the point of distance h the line g h, and its intersection with the central line at a is the perspective of A. Thus two points in the perspec¬ tive are obtained, as B, being on the base of the picture, will have for its perspective b. The vanishing points of the sides of the square are found to be K and m, by drawing parallels to these sides through 0 to meet the picture. Transfer the distances, therefore, K and m, to the horizon line e f in n p, and to these, from b and a, draw the perspective representations of the sides of the square. The operation is shown re¬ peated above the horizon, the same parts being indicated by the same letters accented. In the examples hitherto, we have operated by methods more or less indirect; it is now necessary to show the direct method of solving the same problems. Let it be required to draw in perspective a square the same as the last. From the station point o (Fig. 580), draw to the angles of the square abed visual rays O a, o b, &o., and find the vanishing points m! 1c as before. Draw in the vertical projection the ground line m k, and the horizon n p, and transfer to the ground line the in¬ tersection of the visual rays with the picture; draw from a the lines a m', a k\ and their intersections with the per¬ pendiculars 1, 2, 3, will give the limits of the sides; then draw a k', b k', intersecting at c, which completes the square. If, as in the last figure, another square be drawn at a height above the given plane, equal to the side of the original square, we shall obtain the representation of a cube. Let ABCD (Fig. 581) be the horizontal projections of four straight lines perpendicular to the picture B D, b d. The perspective of these lines 6 a, d c, f e, h g, will con¬ verge to the point of sight i', and if the original lines | were infinite, they would appear to meet in the point 232 PRACTICAL CARPENTRY AND JOINERY. of sight. These lines may be regarded as the boundaries of four planes, two horizontal, one / g above, and the other b c below the horizontal plane, and two vertical, be, cl g; and as the boundary lines of these planes may be considered as of infinite extent, so may also the planes them¬ selves. It is obvious from the figure, that the perspectives of any line situated in a horizontal plane, can never in any case pass the hori¬ zon ; that the horizon is in like manner the van¬ ishing line of all hori¬ zontal planes; and that the trace of the central plane is the vanishing line for all vertical planes parallel to it. In the example the vertical planes are shown to be squares, and their diagonals are conse¬ quently inclined to the picture in an angle of 45°. The perspectives of these diagonals will have their vanishing points in the vanishing lines of the planes which contain them, or in the trace of the central plane. The distance i h, i l of these points h l, will be equal to the distance I i of the eye from the picture. The dia¬ gonals of the horizontal planes will have their vanishing points in the point of distance, as we have already seen. The plane A c being parallel to the picture, will have for its perspective the si¬ milar square a e g c. The line b c is evi¬ dently inclined to the ground plane, and it is also situated in a vertical plane per¬ pendicular to the pic¬ ture. Its perspective will therefore have its vanishing point in the vertical of the picture, and in a point m which will be more or less high, as the inclination of the line is greater or less. The perspec¬ tives of the diagonals of the side d g will have their vanishing points in n m. From this we deduce— Rule VII .—All lines inclined to the horizon , parallel among themselves, and inclined in ver¬ tical planes perpen¬ dicular to the pic¬ ture , have their vanishing points in the vanishing line of the planes which contain them. Rule VIII. — The perspectives of planes parallel to the picture cct/nnot have vanishing points, but will always be of the same figures as their originals. Further to illustrate the vanishing points of inclined lines, let abc D (Fig. 582) be two original parallel lines; their perspectives will beaBCD; the direction of which will be to the vanishing point e, situated at the inter¬ section of the picture with a plane E /, passing through j the eye f, parallel to the planes of the triangles A a b, I CCD, passing through the given original lines. PERSPECTIVE. 233 The line F E is a line parallel to the given lines drawn through the eye to meet the picture. As it is in the plane E /, its intersection with the picture determines the vanishing points of the lines a B, c D. This problem may be considered in a different manner. Let A B C D (Fig. 583) be the original lines as before: H but in place of supposing them situated in two triangles, let us suppose them situated in a plane b c, inclined to the original plane. Let the plane G e, passing through the eye F, be parallel to B c, and let it cut the picture in the line H E, which, as we know, will then be the vanishing line for all planes parallel to B c. The line F e is drawn through the eye parallel to the original lines given, it lies in the plane G E, and cuts the picture in E. It lies also, however, in the plane /e, and therefore E is the vanishing point sought. Let us consider the pi*actical application of this problem, | with the view to its more perfect elucidation:— Let A B (Fig. 584) be the plan of a cube, and c D the elevation of one of its sides, with diagonal lines drawn on it. Draw the visual rays 0 A, o B, the central plane 0 M, the picture line H L, and the lines o H, o L, to determine the vanishing points of the sides. Then to find the vanishing points of the oblique lines; from O in the ground plan, on the line o L, construct a right-angled tri¬ angle o l N, of which the angle N o L is equal to G c D, and set up the height L n, from l to n, in Fig. 585, which gives n as the vanishing point for C G and all lines parallel to it. Set the same length off downwards from l to m, and m is the vanishing point for F D, and all lines parallel to it. In the same way find the vanishing points on the left-hand side, by drawing the triangle 0 H K, and set off the length H K in h and r above and below the vanish¬ ing point s of the horizontal lines of the cube. The reason of this process is obvious, for we have only to imagine the triangles O L N, o H K, revolved round 0 L and o H until their plane is at right angles to the paper, and we then perceive that n and k are the heights over the horizontal vanishing points, that a plane passing through o at the height of the eye of the spectator, and at 45° with the horizontal plane, would intersect the plane of the picture. The drawing of the figure is explained by the dotted lines. In what we have hitherto advanced are comprehended all the principles of perspective, and we shall now pro¬ ceed to apply these principles in the solution of various problems. Problem III. — The distance of the 'picture and the perspective of the side of a square being given, to com¬ plete the square, without having recourse to a pla/n. 1st. When the given side is parallel to the base of the picture, let a b (Fig. 586) be the side of the square, o Fig. 686. the point of sight, d the d point of distance. Draw a 0 , b o, for the indefinite perspectives of the side, and a d for the perspec¬ tive of the diagonal, and where it intersects b o in c, draw c e parallel to a b, and the perspec¬ tive of the square is completed. 2d. When the diagonal of the square is perpendicular to the base of the picture, let a (Fig. 587) be the hither or nearest angle of the square, o the point of sight, d d' the points of distance. The diagonal of the square being perpendicular to the picture, will have the point of sight for its vanishing point. Draw, therefore, a 0 as its inde¬ finite perspective. Set off from a to b the length of the diagonal, and draw b d intersecting a o in c, which is the perspective of the farther extremity of the diagonal; then draw the sides (which make angles of 45° with the pic¬ ture) to the distance points d d'. 3d. When one side of a square is given, making any angle with the picture; let a b (Fig. 588) be the given side; produce it to the horizon in c. Set off the distance 2 G ' 234 PRACTICAL CARPENTRY AND JOINERY of the eye on the central line, from E to F, and draw c F, which will be a parallel to the given side a b, and c is consequently the vanishing point for all lines parallel to a 6. The point a belongs equally to two sides of the square, one the given side a b , the other the hither side of the square. Let us see how this latter is to be found. As this side makes a right angle with a b , from F draw F g at right angles to F c, and continue it till it meets the horizon. In this case the lines would extend beyond the limits of the paper, and to avoid that inconvenience we may adopt another method of working, viz., by the diagonal. Therefore, from F draw F d, making an angle of 45° with F c, and its intersection d with the horizon will be the vanishing point of one diagonal of the square. From f also draw F h, at an angle of 45°, with F c for the vanishing point h of the other diagonal. Now to complete the square, draw from the extremities of the giveu side the perspectives of the diagonals a d,b h, and produce the latter indefinitely; and to find the length of the sides, take the following means, founded on what has previously been learned. Through the intersection of the diagonals at o draw a straight line m n, parallel to the base of the picture, and make o n equal to o m, and n will be a point in the side of the square, through which draw r s, intersecting the diagonals, and join b s, a r, and the square is com¬ pleted. Problem IY. — To divide a line given in perspective in any proportion. Let A B or B c (Fig. 589) be the given line, and let it be required to divide it into two equal parts. Now, A B being parallel to the base of the picture, will have its perspective exactly the same as the original, and if divided into equal parts, its per¬ spective will also be di¬ vided into equal parts. But B c being oblique to the base of the picture will, in reality, be divided into two equal parts in the point e, although these parts appear unequal. If through A and c we draw a line produced to the horizon, the point 0 will be the vanishing point of all lines parallel to A o. Hence, if we draw d o, b o, these Fig. 589. three lines will be parallel among themselves. And as the lines A B, B c, are comprised within the parallels A o, b o, and are cut by the third parallel D o, it follows that their parts are proportional among themselves. And, therefore, if A D is the half of a b, c e will be the half of B c. The same will follow if the lines are in the vertical plane, as 1c g divided into two equal parts in o. From what has been stated it will be seen that the problem reduces itself to these cases:—1. When the line is situated in a plane parallel to the picture, the divisions are made as in any other straight line. 2. When it makes any angle whatever with the picture, we draw through one of its extremities a line, either parallel to the base, or to the side of the picture, accordingly as the given line is situated in a horizontal or vertical plane, and we divide this line into the number of equal parts demanded; then through the last division, and through the other extremity of the given line, we draw a line produced to the horizon, and from all the points of division draw parallels to this line, which will, by intersection, give the perspective divisions of the line required. For example, let it be required to divide the line m n (Fig. 590)into three equal parts; through n draw the horizontal line n 3 indefinite¬ ly, and set along it three equal divi¬ sions of the com¬ passes, opened at pleasure in 1 2 3, through the last divi¬ sion, and through m draw a line to the horizon, inter¬ secting it in 0, which will be the vanishing point of all parallels to 3 o, and of course of the lines 2 o, 1 o, and by drawing these the perspective line m n is divided into three equal parts. When the given line is in¬ clined both to the picture and the ground plane, as A B (Fig. 591), it is necessary to find the divisions, first on its hori¬ zontal projection A b, and through them to draw ver¬ tical lines which will cut the perspective line in the points required. Problem V.— Through a given point in a picture to draw a line parallel to the base or side of the picture, and perspectively equal to another given line A b. Let a (Fig. 592) be the given point. From it draw an indefinite line parallel to the base of the picture, or to its side, and then to determine on either of these lines a length perspectively equal to the given line, draw through a any line at pleasure, cutting the base of the picture, say at 0 , and the horizon at e. Set off from o a length, 0 6, equal to the given line, and draw b e, which will cut the indefinite horizontal line in g, and a g will be equal to A b. In the same way proceed to obtain the length of the line a /parallel to the side of the picture. It is easy by this means to find the perspective height of a figure at any distance from the base of the picture. Let r, for example, be the point on the original plane at which the feet of a figure are supposed to be situated, and it is required to know its height, and suppose A B the real II from a or b draw a 0 , 6 o, which will be the perspective height of the figure, then through r draw a horizontal ;| direction of the diagonals of the square, and their inter¬ sections with the perspectives of the sides produced will give their points. Problem VIII .—To draw the‘perspective of a pavement of squares, the sides of which form any angle with the picture. Let A B (Fig. 595) be the horizontal projection of the Fig. 695 235 duced, and the perspective lengths of the side of the squares se.t out on it, and again as at e f if necessary. Problem VII .—To draw a pavement of squares in perspective, when the sides are inclined at angle of 45° to the base of the picture. In this case let a b (Fig. 594) be the side of the square, and its diagonal will consequently be a 1. Lay off along the base of the picture the divisions 12 3, fee., each equal to the length of the diagonal. Now, as the sides of the square make angles of 45° with the picture, the distance points will be their vanishing points, and nothing more is required to be done than to draw from 12 3, and to D D the lines 1 D, 2 D. If it is required to fill the space * This is most conveniently done by transferring them first to the edge of a strip of paper, from which they can be transferred to the picture-line. PERSPECTIVE. line cutting the parallels 0 e, b e, in s and t, and t s will be the height sought, which has to be carried vertically from r to H. Or otherwise, from the point of intersection t raise a perpendicular, which gives the point h, and t h will equally be the height sought. Problem VI .—To draw the perspective of a pavement of squares, two of the sides of the squares being parallel to the base of the picture. The squares having two of their sides parallel to the case of the picture, their other two sides will have for their vanishing point the point of sight, and the point of distance will be the vanishing point of the diagonals. No plan is required. Set off on the base of the picture at 1 2 3, &c. (Fig. 593), divisions equal to the sides of the square, draw through these to the point of sight 0 , lines which are the perspectives of the sides of the squares, perpendicular to the picture, and intersect these by a diagonal drawn from any point, as 1 to D, the point of distance, and through the points of intersection draw lines parallel to the base of the picture. The squares can be extended so as to fill the picture by extending the base line and setting off more divisions. But if there is not room to do this, any of its parallels, as A B, may be pro¬ Fig. 593. base of the picture, cutting the squares (of which only d two rows are here shown) at any angle. Transfer to the vertical projection a b of the base of the picture the divisions 1, 2, 3. 4, 5, 6, 7.* Through the seat of the eye o draw o p parallel to the sides of the squares, and p will be the vanishing point for them. Transfer the distance of p from the central 12 plane op to the horizon at o' p', and draw 1 p' 2 p', &lc., which gives the perspectives of the sides of the squares parallel to m n, and we have now to find those which are perpendicular to them. Thi^we might do by finding their vanishing points; but this mode would be inconvenient, as it would extend the vanishing point far beyond the limits of the paper, and we shall rather operate by the diagonals. Draw therefore through o, the seat of the eye, the lines o s, O t, parallel to the diagonals, to meet the picture-line produced in s t, and transfer o s, o t 236 PRACTICAL CARPENTRY AND JOINERY. to the horizon-line in o' s', o' t'. Then transfer the points 4 or v , where the diagonal meets the picture-line A b, to the vertical projection of the picture-line a b, and from these points draw lines to the vanishing points, as shown in the figure. Problem IX. — To draw a hexagonal 'pavement in perspective, when one of the sides of the hexagon is parallel to the base of the picture. Let A b (Fig. 596) be the given hexagon. Set off along the base of the picture, divisions equal to the side of the hexagon, then draw its diagonals, which will divide it into six equilateral triangles, and find the vanishing points of the diagonals. This may be done in the follow¬ ing simple manner. Let o be the intersection of the central plane with the horizon, and 0 c be equal to the distance of the spectator; then from c draw the lines C D, c D, parallel to the diagonals of the hexagon, and their intersection with the horizon in the points D D will be the vanishing points of the diagonals. The remainder of the operation requires no description. Problem X. —To draw the perspective of a circle. Let A b a b (Fig. 597) be the projections of the picture, c c those of the eye, o the point of distance, ijkl the given circle. ^ The most expeditious method of operating is to cir¬ cumscribe the circle by a square. The circle touches the square at four points; and if the diagonals of the square are drawn, they intersect the circle at four other points, which gives eight points, the perspectives of which are easily found. Thus, then, we draw the perspectives of the square and of its diameters and diagonals, and then project on the base of the picture a b the points I J K l in k' and l' and k" and l", and from these points draw the lines k" c, l" c , J which cut the diagonals in i j k l, and through these and the four others E f g h the circle is to be traced j by hand. As the circle is a figure which has very frequently to be drawn in perspective, we shall consider it under an¬ other aspect. Let there be any number of points taken in the cir¬ cumference of the original circle (Fig. 597), and suppose lines drawn to them from the eye c, as the tangents M c, N c. Now it is evident that the collection of all these lines forms the projection of a scalene cone, having its i base circular, and its summit in the eye of the spectator. ! Let us conceive this cone cut by the plane of the picture, I and its section in the picture will be the perspective of its base or of the given circle. This operation can be performed by the rules of descriptive geometry, and the result will be the same as by the problem above. The perspective of this circle is, then, necessarily an ellipse, since it is the result of the section of a cone by a plane which passes through both sides, and is not parallel to its | base. It is further to be observed, that in the ellipse the | principal axis m n does not pass through the point o, the . perspective of the original centre of the circle, but is the perspective of a chord M N, determined by the tangents | M C N C. Problem XI. —To inscribe a circle in a square given j in perspective, and of which one side is parallel to the j base of the picture. We know by the preceding problem that the ellipse should touch the four sides of the square in their apparent or perspective centres at the points e, f g, h (Fig. 598). We can, therefore, consider the line e f as the minor axis of the ellipse which we seek; and as we know that the major axis should cross it perpendicularly in its centre, we divide e f into two equal parts, and through i, the centre of the ellipse, draw perpendicularly to e f an inde¬ finite line, which will be the direction of the major axis, of which we have to determine the length. Take / i, or i e, and carry it from h to j, and draw through h and j a straight line to the minor axis at k, and h k will be equal to half the major axis which we set off from i to l and m. Having now the major and minor axes of the ellipse, it is easy to draw it with the aid of a slip of paper, or in any other way. In the next figure the same method may be thus ap¬ plied:— Let a c (Fig. 599) be the given square; through the intersection of its diagonals draw / e, and divide it into two equal parts in i, and through i draw an indefinite line parallel to g li. Through q, the perspective centre of the circle, draw a line perpendicular to g h, and produce it both ways, when it will cut the side d c in s, and the line l m in r. Carry the length s r from g or h upon l m to p or y, and draw g p or h y to cut s q in the point o, PERSPECTIVE. 237 and either of the lines g p o or h y o will be the rule with which to operate in describing the ellipse as before. If it is required to divide the periphery of the circle into any number of parts, equal or otherwise, it may be thus performed:— Though the points of division, on the geometrical plan of the circle (Fig. 600), draw radii, and produce them to intersect the sides of the cir- Fig. 600 . cumscribing square. Then from the intersections visual rays may be drawn, and the corresponding points obtained in the perspective square, Rom which radii drawn to the perspective centre will cut the perspective circle in the points required. But we may in most cases dispense with the visual rays, and obtain the perspective divisions of the square by Problem IV., thus:— From the hither angle A of the square (Fig. 601) draw any line, as A B, A G, equal to the side of the square on the plan, and on it set off the intersections of the radii. Then from B draw through c a line cutting the horizon in D, and from G through F a line cutting the horizon in E, and from the divisions of the line A B, A G, draw lines to these points, which will divide A C, A F perspectively in Through the divisions in the plan of the circle (Fig. 602), draw lines parallel to one of the sides of the square, and pro¬ duce them to intersect any of its sides, and from the per¬ spectives of these intersections draw lines to the vanish¬ ing point of the side, to which the lines drawn through the divisions of the circle are parallel. Perspective of Solids. Problem XII. — The horizontal 'projections of two tetrahedrons being given , to draw the perspective of the solid. Fig. 603. d c r> Let A b, a b (Fig. 603) be the projections of the picture, There is yet another method, which is of very great applicability. its angles, and the other on one of its arrises, to draw them in perspective. 238 PRACTICAL CARPENTRY AND JOINERY. Here the operation is so simple that no explanation is required. Problem XIV. — To draw three equal given cylinders in perspective (Fig. 605). It is not necessary to repeat the method of putting a circle in perspective, but it may be well to observe, that the upper surface of the cylinder may be so near the hori¬ zon that it is physically impossible to inscribe an ellipse. In such a case an approximate solution is all that can be arrived at. A few principal points should be obtained, and the ellipse traced by approximation. This example presents a singularity, which at first sight appears a paradox, and yet is nothing less than that. The cylinder A, although evidently further from the eye than b, and seen consequently under a less angle, appears in the picture to have a greater diameter. The reason is, that its optical cone is cut more obliquely by the picture than that of b, and hence the intersection of A is longer. This, which is held to be a proof of the incorrectness of per¬ spective, is, on the contrary, a proof of its correctness; for, if the subject is attentively considered, and we proceed to view the picture under the same conditions as to distance and height of the eye, as we have supposed to exist in viewing the object itself, we shall perceive that the repre¬ sentations of the objects must be seen under the same angles as the objects themselves, and therefore, although the diameter of the further column measures more in the representation than that of the nearer one, yet, from the proper point of view, the angle under which it is seen is less, and therefore it will appear to be smaller. The result, although geometrically correct, is yet a distorted representation, when viewed in the way we usually look at a picture. It is a correct section of the cone of rays, but not made by a plane so situated relatively to the object and spectator, as we should place a picture or a transparent plane through which to view the object. Before proceeding further, we shall take the opportunity of making some remarks on the conditions under which objects may be properly represented. The greatest angle under which objects can be viewed with distinctness is one of 90°. But when viewed under this angle, it is with such an effort as to produce an un¬ comfortable sensation. Let us, however, suppose that we view an object under an angle of 90°, and we may consider the sum of the rays which can enter the eye under that angle as forming a cone, having the eye at its summit. Now, according to our proposition, the only objects which can be distinctly seen are those contained within the base of the cone. A picture is in this con¬ dition. Let ah, ab (Fig. 606), be the projections of the base of a square picture. Now, in order that this may be inscribed within the base of a right-angled cone, it is necessary that the axis of the cone be equal to the radius of the circle of its base. Thus from c\ the centre of the picture, with c a or c b as radius, we describe a circle which contains the whole of the picture. We carry the radius in the horizontal projection from c to c', and the line c c will of course be the shortest distance which we should take in order to see distinctly the picture A B. If we make a right angle at c, we shall have an isosceles triangle, of which the base o o will be equal to the diameter of the base of the cone. Now this distance c c', which is sufficient for the picture A B or a b, is too little for the picture a G, which has the same base as AB, but a greater height, or for a II, which has the same height, but a greater width; for neither of these are contained entirely within the optic cone o c' o, but require the base of the cone to be increased to p p. The distance, therefore, ought to vary with the height or width of the picture. So much for the principle; but we have said that a lesser angle than 90° is better to be adopted, and it is not easy to give any rule for this. In general, however, the distance should never be less than the diagonal of the picture, which would give o r o as the angle under which the square picture AB or ah should be viewed, and p s p for the others. The distance, too, should depend in some measure on the height of horizon assumed, and in general the higher the horizon the greater should be the distance. We now have to speak on a point to the misconception of which we attribute much that has been said against the art of perspective. The central plane, it will be remembered, was defined to be “ a plane passing vertically through the eye of the spectator, and cutting the ground plane, the horizontal plane, and the plane of the picture at right angles.” Now let us take the horizontal projections of an object A A (Fig. 607), the picture B b, the eye of the spectator c, and the central plane c D a. The picture is parallel to the longest side of the object, and the central plane bisects the angle formed by the visual rays which proceed from the extre¬ mities of the object to the eye. This, then, we assume to be the correct conditions of a picture; but suppose it were required to introduce into the same picture other objects B B, as seen from the same point of view, we should no longer place the picture parallel to the long sides of the object, with the face of the spectator directed towards a, but we would again bisect the angle formed by the PERSPECTIVE. 239 visual rays proceeding from the extremities of the object to the eye of the spectator by the central plane c e, and make the line of the picture perpendicular to c E, as at F F. Perspective has been divided into parallel and oblique perspective, and this division has introduced the miscon¬ ception animadverted upon. Parallel perspective, so called, can only exist when the central plane, bisecting the angle formed by the visual rays, bisects also the object to be represented. N evertheless, in views of interiors, of streets, and in architectural representations generally, it is un¬ scrupulously used, although the intersection of the central plane with the picture is at one-third of the width of the latter. In the figures CAB (Fig. 608), three different repre¬ sentations of the same interior are given to illustrate these remarks. In fig. C the central plane bisects the visual angle; and the lines of the further side of the apartment, being parallel to the picture, are also parallel in the repre¬ sentation. In A the central plane does not bisect the visual angle, but is at one-third of the width of the pic¬ ture ; and this is the condition animadverted on. If it be desired to have the point of sight not in the centre of the apartment, like C, but at one side, then the representa¬ tion is only correct when it is like B, in which the cen¬ tral plane bisects the picture, and the lines of the further side of the apartment being no longer parallel to the picture, have their proper vanishing points. To revert to the example which has led to this digression, we observe by Fig. 605, that the distortion arises from assuming a position for the picture in which the central plane does not bisect the visual angle. But let us place the picture in the position E f, in which it is perpendicular to the central plane 0 c, bisecting the visual angle, and we have the cylinders in what is called oblique perspective (a distinctive term which should not exist), and there is no longer any distortion. Problem XV. — To drazu a sphere in perspective. Let ah, a b (Fig. 609) be the projections of the pic¬ ture, c c' those of the eye, the circle d that of the given sphere. . If the centre of the sphere is at the height of the horizon, the vertical projection of its centre will be the point of sight c'. Draw from the eye c the tangents E c F c. Draw also the chord e f, which may be re¬ garded as the base of a cone formed by the visual rays, tangents to the sphere, and of which the eye c is the summit. The section of this cone by the picture will be a circle, since the cone is cut parallel to its base. This circle will have for its diameter e /; and consequently cf or c e as its radius. Then, if with this radius from the centre c'we describe a circle, it will be the perspective sought. When the sphere is below the horizon, let D (Fig. 610) be the hori¬ zontal projection of a sphere, c that of the eye, and L M the picture-line. Draw visual rays tangents to the sphere, and we obtain on the picture-line ef as the per¬ spective horizontal diameter of the sphere. At k set off the diameter ef so obtained, and from the points e f raise indefinite perpendiculars. Now, suppose D to be the vertical projection of the sphere, and c the vertical pro¬ jection of the eye. From c' draw visual rays tangents to the sphere, and we obtain r s as the intersection of the cone of rays by the picture, and the length r s as the per¬ spective vertical diameter of the sphere. But the diameter r s is greater than the diameter e f and therefore the perspective representation of a sphere viewed under the conditions premised, must be an ellipse. When we obtain the two diameters, we obtain all the measurements neces¬ sary for the representation of the sphere when below the horizon, and in the central plane, viz., the major and minor axes of the ellipse, and the curve may be tram¬ melled by the aid of a slip of paper. The point g in which the sphere touches the picture is the projection of its centre, and of its axis D g, and its perspective representation is k In this there is another of those apparent contradictions, for it is certain that a sphere always appears to us to be round on whichever side we regard it; while in perspective, in every case except that in which its centre is in the point of sight, it must be drawn an ellipse with its major axis directed towards the point of sight. An attentive consideration of the figures and description will render this evident, and the reader may also advert to the explanation of this apparent contradiction, which is given in the text treat¬ ing of Figs. 605—607. We shall now proceed to give some other examples of spheres in perspective. PRACTICAL CARPENTRY AND JOINERY. 240 Let A B , ab (Fig. 611), be the projections of the picture, C c' those of the eye, D that of the given sphere in contact with the ground plane. Draw from the eye the tangents E c, F c, through E and F, draw parallel to the picture the lines E G, F h, and draw also as many parallels to them, IK, op, LM, &c., as may be considered necessary. These parallels, then, are traces of vertical planes cutting the sphere parallel to the picture, and all the sections made by them will be so many circles, which will comprehend all the visible portion of the sphere, as determined by the tangents EC, F c. The perspectives of the circles h f, I k, &c., which are parallel to the plane of the picture, will also be circles. If we envelope all these perspective circles by a curved line, this line will be the perspective of the sphere. It is easy to find the perspectives of the circles. First draw the diameter r s, which will be the horizontal projection of one of the axes of the sphere, pass¬ ing through the centres of the circles h f, i k, l m, &c. Find then the perspective direction of that axis; observe that the point r is raised above the ground plane by the height of the radius of the sphere, and that it touches the picture. Its perspective consequently will be in r, and as the axis is perpendicular to the picture, its vanishing point will be the point of sight (f. Having drawn R c 7 , the perspective of the axis, find on it the centres of the circles. Find first the diameter of the circle EG; through l, the centre, draw l c, cutting the picture in l, and from l raise a vertical line cutting the axis R c' in l', the centre sought. The radius l E has for its perspective the radius V e, with which, from the perspective centre Z, we describe the circle e e, and so on for all the other circles, and finally we circumscribe them by the elliptic curve. This method is a little long, and has not all the pre¬ cision which could be desired, for the two axes- of the ellipse have not first of all been determined, which would simplify the operation; but this figure was necessary to show that the union of all the circles contains all the visible parts of the sphere, and forms an ellipse in the picture; and, moreover, it was absolutely required for the understanding of the following method, which is a conse¬ quence of it, but which is much more simple and precise. After having drawn the tangents EC.FC (Fig. 612), the lines EG, F H, and found the point R, we draw the line R c, and produce it indefinitely towards R. We then seek on A B the intersections of the centres 1 6 of the circles EG, F H, and from 1' 6' raise to the picture ver¬ ticals which cut the direction of the major axis in 1" 6". We take the perspective of the radius E 1, which is e 1 or 6/, and in the picture from 1" and 6” as centres, and with el or / 6 as radius, describe an arc to the right and left on R c' to the points 7 and 8, and the line 7 8 will be the major axis of the ellipse. We divide this into two equal parts, and through the middle 9" draw a line vertical to the picture-line a b, producing it to the hori¬ zontal projection of the picture-line A B, which it will cut in 9'. From the point of sight c we draw through this point a line produced to meet the diameter r s, and from the point of intersection 9' we draw parallel to the picture c a line I M; from I we draw a line I c which cuts A B in l. We then take Z 9', the perspective of the radius L 9', and from 9" as a centre cut the direction of the minor axis in 10 11, and the minor axes will be determined. Having the two axes, the ellipse is easily traced. Practical Examples of Perspective Drawing applied to Architecture, &c.—Plates CYII.—CXI. Having thus described the principles of perspective, and shown their application to the drawing of elementary figures, we propose now, with the view to their more PERSPECTIVE. perfect illustration, to show their application to the drawing of architectural and other objects. Plate CVII.— Fig. 3 is the plan of a cross, and A w, Fig. 4, is a vertical section of the same. Having selected the position of the spectator, or station point s, we draw from it visual rays, including as much of the object as we wish to represent. The angle included by these visual rays is then bisected, and the central plane s 0 drawn. The picture-line X Y is next drawn at right angles to the central plane, and then the vanishing poiuts are found by drawing from the station point lines, sw, sv, parallel to the sides of the object, to intersect the picture¬ line. The point of intersection, or vanishing point, for the left-hand side of the figure, is at x; the vanishing point lor the right-hand side, and the lines parallel to it, are beyond the limits of the paper. We have advisedly chosen in this case such a point of view for the picture, and such a distance as would throw one of the vanishing points beyond the limits of the paper, as such conditions of both station point and vanishing points are of constant occur¬ rence in practice, and it is right that the learner should at the outset be made acquainted with the difficulties of the ait and the means of overcoming them. These means in such cases may be two. The first and most ready is to fix slips of lath to the drawing-board, on which the horizon¬ line may be extended to the requisite distance. The other is to draw the converging lines by the centrolinead in¬ vented by Mr. Nicholson. This instrument, however, is not much used, probably because it requires frequent alte¬ ration to adjust it to the various vanishing points. We prefer the simple plan of fixing laths to the board, and inserting at each vanishing point a needle, against which j the straight-edge may work as a centre. We shall now | proceed to describe the process in detail. We have already described the manner of drawing the 1 extreme visual rays, picture-line, and central plane; these I being drawn, we proceed to draw the visual rays from the j various points of the plan EDCABCDE, and/ fff gg, gg } ! and by producing to meet the picture-line x y, the sides | of the cross and the steps B g 1, c 4, Do. From the points 1 2 3 4 5, where these intersect the picture-line, we let fall indefinite perpendiculars. These lines bein<* in the plane of the picture will have the original heights of the object. Intersecting, therefore, these by horizontal lines from the different heights in the section of the cross a w, we obtain the heights of the different parts of the cross represented by the lines 1 2 3 4, brought into the plane of the picture. Now let us trace the drawing of any point in the plane, as A or b, to its perspective repre¬ sentation. From B, which is one of the angles of the shaft of the cross, draw the visual ray b b cutting the picture- 1 line in 6; and from b draw towards the ground plane of the perspective representation in Fig. 4 the line b b, which is an indefinite representation of the angle or arris of the shaft. Then, from the figure B n on the left, draw from the heights of the different horizontal divisions of the shaft, lines to the right-hand vanishing point, and we obtain their perspective heights on b b. In like manner we obtain the perspective representation of the points//, // and the lines which join them, representing on the plan one of the limbs of the cross, by first drawing the visual rays, and then letting fall perpendiculars from the intersec¬ tions of these with the picture-line, and intersecting these 211 perpendiculars by lines drawn from the left-hand section B H to the vanishing point. It is not necessary to parti¬ cularize further the drawings of this figure; the letters will enable the reader to follow the lines, and from his previous knowledge he will be able to reproduce a per¬ spective-so simple. It will have been observed, that the labour of drawing is abridged and simplified by producing the lines of the object on the plan to the picture-line, a method of proce¬ dure which we shall adopt as much as possible in all the illustrations. It may be as well here to show how an object may be drawn under any angle of view, that is to say, how the extreme visual rays may be made to include such an angle as may be determined upon. Let ABCD, Fig. 1, be any object, which it is required to delineate as seen from a point of sight a, under the angle of view A G c, &c. Let e f be the pictpre-line. In Fig. 2, construct on the base B c the triangle bac, the included angle B A c being the angle required. Then, from the point c, Fig. 1, draw c G parallel to A B, and from A draw A G parallel to A c, meeting in the point G; bisect the angle AGCby the line G o, and at any required distance from the object draw E F perpendi¬ cular to G o, and the object will then be seen under the angle agc equal to bac, and E F will be the picture-line. The next illustration is a pavilion, the plan of which is seen in Fig. 5, and the half elevation in Fig. 6. Let s be the station point. From s draw the extreme visual rays, including all parts of the plan, and bisect the angle by the central plane S 0; then proceed to find the van¬ ishing points by drawing from s the lines S w, s v parallel to the sides of the plan to meet the picture-line x y, and from the various points of the building draw visual rays Ee > E /> d cl, &c., to meet the picture-line. Then, having drawn the ground line Q z, Fig. 7, and the horizon-lino H h, and the central plane s o, and set off on the horizon¬ line the right and left hand vanishing points: at the in¬ tersection of the visual ray from f, the hither angle of the building, let fall a perpendicular to the ground plane, Fig. 7. Produce F F, Fig. 5, to meet the picture in f', and draw the perpendicular f' f". Then from f", where this perpendicular meets the ground line, Fig. 7, draw a line to the left-hand vanishing point, to intersect the per¬ pendicular from the visual ray of the point f, and we obtain the hither angle of the base of the building on the ground plane. In like manner produce D A to the picture in a', and draw a' a", on which the heights for the hither angle of the body of the building are to be set up. The perspective height of the apex of the roof is obtained by producing the centre line of the building to the line of the picture, and on the perpendicular drawn from its in¬ tersection setting up the height of the apex. The hither angle of the balcony e E touches the picture-line, or comes into the plane of the picture, consequently is of the same height in the perspective as in the original. The divi¬ sions of the balcony railing are found by the applica¬ tion of Problem IV. as follows:—At any convenient place draw the horizontal line 6 1 6, and set off on it from 1 to 6 the divisions required; then from 1 draw to the vanishing points the lines 1 P, 1 p, intersecting the perpendiculars let fall from the visual rays of the further angles of the balcony. From 6 6 through these points of intersection p R draw lines as 6 P p, intersecting the horizon in p\ 2 h 242 PRACTICAL CARPENTRY AND JOINERY. then to p as a centre draw radii from the points of divi¬ sion 2 3 4 5, and these will divide the line 1 P perspec- tively in the same ratio, and the divisions can be trans¬ ferred to the balcony by drawing perpendiculars, as in the example. In the third example, which is that of a broach, or tower and spire, seen in plan in Fig. 8, and in half elevation in Fig. 9, the same method of procedure is adopted. On the plan, Fig. 8, the square tower abcd, and its porch g e f h, are represented, and also the spring¬ ing of the octagonal spire fghklmnr. The planes passing through the centre of the plan are produced in P p' and P G to meet the picture-line, and the heights of the apices of the gables and the spire are set up on the perpendiculars let fall from p' and G. Plate CVIII. — To draw a series of arches in per¬ spective. Let Fig. 1, No. 1, be the plan of the piers supporting the arches, and let Fig. 1, No. 2, be the elevation of an arch. From s, the station point, draw the visual rays, and find the vanishing points and the centre of the picture in the usual manner. Draw the ground line Q z, and the horizon H H, and set ofF on the latter the centre of the pic¬ ture and vanishing points. The perspectives of the semi¬ circles, forming the arches, are shown obtained at M N in the plan, and m' n' in Fig. I, No. 2, in the different methods illustrated in Problem IX., Figs. 600, 601, and 602. At M in the plan the intersections of the radii 1 2 3 4, No. 3, with the curve, and also with B F, are laid down at o 12 3 and o'r2'3', and from these points visual rays are drawn to Q z, and from the points of intersection of these with the latter the perpendiculars 12 3 are drawn to meet the radial line drawn to the perspective centre 0 . The arch stones may be found in the same way, but in the figure they are shown as obtained by Problem II. The length of the line dEF, No. 2, is set off on any line afe, and through e and f', the perspective representation of f, a line is drawn to meet the horizon in D. The divisions of the arch stones on the line ef, No. 2, being then carried to the line/ e, the perspective division of a f' is obtained by drawing lines from the divisions in / e to D. Another method of producing the perspective curve of the arch is shown at N on the plan, and at n' in No. 2. Through any points in the arch a 1 2 3, lines are drawn to the plan in a 1 2 o, and a' 1'2' 3', and corresponding lines a 1 2 o 1' 2', for the elevation; visual rays are drawn from the points in the plan, and perpendiculars raised on QZ, at the points of intersection. To obtain the heights the line R s is produced to the plane of the picture in T, and T b is drawn perpendicular to Q z, and on it are set up the heights of the pier T c, and of the divisions of the arch a 1' 2' 3'. Lines are drawn from these to the right-hand vanishing point, to intersect the perspective representation of the angle of the pier, and through these intersections cal 2 b, are drawn lines converging to the left-hand vanish¬ ing point, intersecting the corresponding perpendiculars. To draw a circular vault pierced by a circular headed window. Fig. 2, No. 1, ACB is the elevation of the vault, and Fig. 2, No. 2, an elevation of its side, with fghkl the circular headed window. Let s be the point of sight, and S D the distance of the spectator, that is D, the point of distance. The positions of the divisions goo, o o k, are found by drawing from them lines to the point of distance, intersecting m n, the perspective representation of L M, and the perspective of the semicircle in the perpendiculars outside of the vault is found by the intersection of d D, c D, &c., with the vertical lines drawn through o o o. Through these intersections horizontal lines are drawn, and the points in which they are intersected by lines drawn to the point of sight s, through the divisions in the vault d' c' H, are points through which to trace the curve d" c" h" of the circular headed opening. Plate CIX.— To draw a Tuscan gateway in perspec¬ tive. Having drawn the plan mm {Fig. I), and fixed the station point s, draw the extreme visual ra} r s s B-, SB 3 , and with an}’ - radius, from s as a centre, describe an arc R R cutting the rays, and from r r describe the intersect¬ ing arcs t t; draw the central line s o through the inter¬ sections, and the picture-line Y z at right angles to s o. Then from s draw the necessary visual rays intersecting the picture-line, and produce also the leading lines of the object to intersect the picture. Draw the vanishing lines s v, s w. In Fig. 3 draw the ground-line G G, the horizon¬ line H H, and the central line o o. Proceed now to trans¬ fer the divisions on the picture-line Y z to the line F f, by describing arcs from N as a centre; and to draw the perspective, by carrying the heights from Fig. 2 to meet the lines let fall from the points where the corresponding lines of the members of the cornice intersect the picture¬ line, as shown by the faint lines in the plate; and from the points thus obtained draw lines to the right-hand vanishing point. Thus the height B {Fig. 2) is carried to B 4 , to meet the line let fall from N, the intersection of b' B 3 , Fig. 1, with the picture-line, and from b 4 a line is drawn to the right-hand vanishing point to meet the line b b\ let fall from the intersection of the visual ray s B 2 with the picture-line; thus the heights b ] b ] are obtained. To obtain b 2 a line is drawn from b' to the left-hand vanishing point v^, to meet the line let fall from the inter¬ section of the visual ray s B 1 with the picture-line. The pediment may be drawn by finding vanishing points for the inclined lines, as described at page 232, or in either of the following manners. Produce the centre line A A 1 {Fig. I), to meet the picture-line in a 1 . Transfer the point A 1 to ff in A 2 , and let fall the perpendicular A 2 A 3 . Draw horizontal lines from the heights of the pediment cornice intersecting this last line, and also the other intersecting lines of the different mouldings. From the heights on a 3 draw lines to the left-hand vanishing point v 2 , and intersect them by lines drawn from the cor¬ responding heights. For example, the height of the apex of the pediment A is carried to a* and a 3 {Fig. 3). From A ? - a line is drawn to the left-hand vanishing point, and from a) a line is drawn to the right-hand vanishing point, and their intersection at a 2 is the perspective height of the apex. The other method of drawing the pediment is by visual rays from the seats of the various members where they intersect the line A A 1 {Fig. 1), and then draw¬ ing the heights from the line A 2 A 3 , to intersect the per¬ pendiculars let fall from the points where these visual rays cut the picture-line. It is not necessary to give further details of the steps of the process, which must now be familiar to the student. In making a complicated drawing, there is a multipli- PLATE CVII1. E ©TT D V [£ J. W Lowry fcidp' BLACK!E & SON ; GLASGOW, EDINBURGH & LONDON (P E IS § [p © t ] y e BLACKIE R- SOU. GLASGOW,EDINBURGH &• LOUDON. H:J.£ee- ISOMETRIC PROJECTION. 243 city of lines required, which tend to embarrass the learner. The lines of construction are of course first drawn in pen¬ cil, and only such portions of them as are eventually to appear in the work are drawn in ink; and it is well in practice, after the lines which are to appear in the draw¬ ing are obtained, to ink them in, and obliterate such lines used in obtaining them as are no longer required. Plate CX.—This plate is an example of the method of drawing a building in perspective. Fig. 1 is so much of the plan of the building, which is a Turkish bath, as is required for the operation, and Figs. 2 and 3 are eleva¬ tions of the two sides under view. The station s is first selected, and the extreme visual rays drawn and bisected by drawing the arc a o b, and thus obtaining the central line s 0 . The picture line x y is drawn at right angles to s o, and in contact with the hither angle of the building and the vanishing lines s v, s v are drawn parallel to the sides B c, B A respectively. The heights in the perspective are obtained, as in the former case, by producing the lines of the plan to intersect the picture, and then transferring the heights of the elevation (Fig. 3) to the corresponding perpendiculars let fall from these intersections. The steps and terrace D are in advance of the picture, and their heights are obtained by setting them up on a perpendicular let fall from the intersection of the original line D with the picture, and drawing lines through them from the right-hand vanishing point, to intersect the per¬ pendiculars let fall from the intersection of the visual rays. The plate shows all the lines of operation, and no further description is therefore necessary. Plate CXI. is an example of the method of drawing a Gothic broach or spire in perspective. Fig. 1 shows horizontal sections of the spire at six different heights. The first is above the moulding of the tower, and shows the windows there; the second at the springing of the spire, the third at the first spire light, the fourth at the point where the lower pyramid termi¬ nates, the fifth at the second tier of lights, and the sixth at the third tier. Fig. 2 is the elevation of one side. Although apparently complicated, this figure is by no means difficult, and the student is recommended to repro¬ duce this figure, or one of a similar kind, on a larger scale. The lines of operation, if carefully studied, will render description unnecessary. ISOMETRIC PROJECTION. This is a conventional manner of representing an object, in which it has somewhat the appearance of a perspective drawing, with the advantage of the lines situated in the three visible planes at right angles to each other, retain¬ ing their exact' dimensions. For the representation of such objects, therefore, as have their principal parts in planes at right angles to each other, this kind of projection is particularly well adapted. The name isomdrical was given to this projection by Professor Farish, of Cambridge. The principle of isometric representation consists in selecting for the plane of the projection, one equally in¬ clined to three principal axes, at right angles to each other, so that all straight lines coincident with or parallel to these axes, are drawn in projection to the same scale. ' The axes are called isometric axes, and all lines parallel to them are called isometric lines. The planes containing j the isometric axes are isometric planes; the point in the ! object projected, assumed as the origin of the axes, is called the regulating point. If any of the solid angles of a cube (Fig. 613) be made the regulat¬ ing point, and the three lines which meet in it the isometric axes, then it may be demonstrated that the plane of projection, to be such that these axes will make equal angles with it, must be at right angles to that diagonal of the cube which passes through the regulating point. The projection of the cube will therefore beasABCDEFG in the figure. Let r B, s D (No. 1, Fig. 614) be the side of a cube, and r D, the diagonal of the side, produce s D, and make C D equal to the diagonal ?’D, complete the parallelogram CA, B D, and draw its diagonal A d, which is then the diago¬ nal of a cube, of which teds is the side, and which is represented in plan in No. 2, a' b' c' f'. Through D, draw K L at right angles to the diagonal A D, and K L is the trace of the plane of projection. If to this line we draw through the points ecfb of the cube, lines parallel to the diagonal a d, and therefore perpendicular to K L, we find that the projection c D of the edge of the cube A c, is equal to the projections of the diagonals A B, CD, of the top and bottom surfaces of the cube, and we know that the projection of the other diagonal c' f' of the top (No. 2), and the projection of one diagonal on each side of the cube, will be equal to the original line, as they lie in planes parallel to the plane of projection. Produce a d indefinitely, and at any point of it a" (No. 3), with the radius D c or nb describe a circle. Draw its other diameter b" c", and produce the lines E e F f through its circumference; join the points c" e' f' b"/ e' c", to complete the hexagon, then join a" f', a"/', and we have the isometrical projection of a cube, one of the sides of which is r B s D. The lines drawn from the plan above (No. 4), show that the projection f'/ of the diagonal c D is of the same size as the original, and the triangle c" f which 244 PRACTICAL CARPENTRY AND JOINERY. is the projection of the section of the cube by a plane cf' (No. 1), parallel to KL, the plane of projection, shows that the projection of one diagonal on each of the sides of the cube must also be of the same size as the original. The relations of the lines of the projection to the ori¬ ginal lines, are as follows:—The lines c" f', f and f c", and all lines parallel to them, are equal to their original lines. The isometrical axes and isometrical lines are to the original lines as -8164 to 1. The diagonals a" b", a" e', and A cf are to their originals as - 5773 to 1; or otherwise, calling the minor axis unity, then the isometrical lines are 1‘41421, and the major axis equal to the original, is 1 73205, their ratio being as yC, V 2, V 3. But in practice it is not necessary to find these lengths by computation; it is much more simple and easy to con¬ struct scales having the proper relation to each other, as we shall proceed to show in Fig. 615. Let it be required, for example, to draw the iso¬ metrical projection of a cube of 7 feet on the side. Now, we have seen that the projection of one of the diagonals of the upper surface of the cube, is of the same size as the ori¬ ginal. Draw, therefore, an indefinite line A b, and at any point of it A draw A c, making an angle of 45° with b a, and make A c from a scale of equal parts equal to the length of the side of the cube, which is here 7 feet. Also from A draw A E indefinitely, making an angle of 30° with B A. From c draw c o perpendicular to BA, and indefinitely produced, and cutting A E in E, then A E is the isometrical length of the side of the cube A c. From A, with A E as radius, cut C o produced in f, and from F as a centre, with the same radius, describe a circle A E, bb; produce c F to D, draw ac'bg parallel to C d, and join D o', D G, and B E, and we shall then have a hexagon inscribed in the circle. Then draw the radii F a, f b, and we obtain the projection of the cube. Now, divide AC into seven equal parts, and through the divisions draw perpendiculars to B A, cutting A E, and A E will be the scale for the isometric axes and lines, and their parallels. The scale for E f, and the other minor diagonals, is made by drawing lines at an angle of 30 3 from the divisions of the original scale, set off on A B, intersecting the perpen¬ dicular e f. The divisions on e f form the scale for e f, Fig 616. . I | | Natural Scale -(■— J - 1 -- 1 —f- 1 - 1 - 1 -1- 1 - 1 - 1 - 1 -1— , Dragon ala F o', F G, and all lines parallel to them. As these ratios are invariable, scales may be constructed for permanent use, as in Fig. 616. Although it is quite essential that the mode of forming scales for the isometric and other lines should be clearly understood, it is seldom that these scales require to be used in practice. For as we can adopt any original scale at pleasure, it is generally more convenient to adopt such a scale as we can apply at once to the isometrical lines. And in place of constructing the hexagon each time, it is convenient to have a set square with its ano-les 90° 60° and 30°, by meaus of which and a T-square or parallel ruler, all the lines of construction may be drawn. Having assumed a scale, therefore, and fixed on the regulating point, suppose we wish to draw a cube, we proceed as follows:—From the regulating point F (Fig. 615), by means of the set square of 30°, draw the right and left hand isometricals FA, fb, and make them by the scale equal to the side of the cube. Then with the same set square, draw a E, be, to complete the upper surface. Draw, by means of the set square of 90°, A c', F D, B G, make them equal to fa, or f b, and, by the aid of the set square of 30°, join D cf, D G, and the cube is completed. The following figures illustrate the application of iso¬ metrical drawing to simple combinations of the cube and parallelopipedon. In Fig. 617, one mode of construction Fig. 618. is shown by dotted lines, but we may proceed directly as in drawing the cube in the manner above described, be¬ ginning at the hither angle of the largest block on the figure, and adding the minor parts. Fig. 618 shows the interior of what may be considered a box or a building. Fig. 619 requires no description. The following figures show how lines which are not isometric may be obtained by the aid of those which are. In Fig. 620, A is the half-plan of a pyramid with a Fig. 615. ISOMETRIC PROJECTION. 245 square base. By including it in an isoraetrical square, its projection is readily obtained. Fig. 621 is an octagon, Fig. 622 a hexagon, and Fig. 623 a pentagon. The projections are obtained by the inter¬ sections of their lines, or their lines produced with the sides of the circumscribing square; and it may be observed by these examples, that the projection of any line making any angle whatever with the isometric lines, can be very easily obtained. In the figures, all the lines of construc¬ tion arising from the various intersections are shown for the sake of illustration. But in practice it is easily to be seen, that it is only necessary to obtain a few of the inter¬ sections. As for example, in Fig. 621, the points abed ef, and in Fig. 622 the points ab c d ef g h. We shall now proceed, in Fig. 624-, to show how the projection of lines at any angle may be obtained directly. Let ab, No. 1, be the isometrical projection of a cube, on any of the sides of which it is required to draw lines at various angles. Draw a square, No. 2, and from any of its angles describe a quadrant, which divide into 90°, and draw radii through the divisions meeting the sides of the square. These will then form a scale to be applied to the isometric faces of the cube, No. 1; thus, from e, or any other angle of the cube, draw a line E F at any angle; make it equal to the side of the square, No. 2, and transfer the divisions of that side to it. Join G F, and draw parallels to G F through the other divisions of ef, meeting eg, which they will divide in the same proportion, and repeat the operation to find the divisions of the remaining sides; or from the angle c of the square, No. 2, draw a line CD, Tig. 621. and make it equal to E G, and draw parallels in the same manner. As the figure has twelve isometrical sides, and the scale of tangents may be applied two ways to each, it can be applied, therefore, twenty-four ways in all. We thus have a simple means of drawing, on the isometrical faces of the cube, lines forming any angles with their boundaries. We have now to consider the application of this species of projection to curved lines. Let A b, No. 1, Fig. 625, be the side of a cube with a circle inscribed; and suppose all the faces of the cube to have similarly inscribed circles. Let us draw the isome¬ trical projection of the cube, No. 2. Then, as the pro- and consequently one of the diameters of the circle, is of the same size as the original, we have at once the major axis of the ellipse which the projection of the circle forms, and as the circle touches each side of the square, we have also four points in the circumference of the ellipse, and we have only to find the isometric projection of its minor axis. From the intersections of the diagonals of the faces of the cube, set off on the major axis the radius of the circle at a b c def y and through the points thus obtained draw isometric lines cutting the minor axis in 1 2 3 4 5 6, and we thus obtain the length of the minor axis. The ellipse can then be sketched by hand, or trammelled by a slip of paper. We may divide the circumference of a circle in two ways, as shown in Fig. 626. First, on the centre of the line A B erect a perpendicular c D, and make it equal to C A or c B. Then from D with any radius, describe an Fia. 626. arc, and divide it in the ratio required, and draw through the divisions, radii from D, meeting A B. Then from the isometric centre of the circle draw radii from the divisions on A B, cutting the circumference of the circle in the points required. Second, on the major axis of the ellipse describe a semi¬ circle, and divide it in the manner required. Through the points of division thus obtained draw lines perpen¬ dicular to A E, which will divide the circumference of the ellipse in the same ratio. On the right hand of the figure both methods are shown in combination, and the intersection of the lines gives the points in the ellipse. 24 G practical carpentry and joinery Mr. Nicholson, in his Treatise on Projection, lias given II minor, and semi-axis major of ellipses, from 1 inch to the following table of the isometric radius, semi-axis II 9 feet; which, in certain cases, may be found useful: Example of tiie Use of the Table. —Let it be required to find the semi-axis of an ellipse u-hich is the isometrical projection of a circle, the isometrical radius being Stject 8 inches. Tu one of the columns under isometrical radius, will be found 2 feet 8 inches: and in the same line, in the next column, on the right hand, will be found 1 foot right, under semi-axis major, will be found 3 feet 3 inches. Examples might be introduced to show the applica¬ bility of this mode of projection to buildings and the parts of buildings, but its principles are so obvious, and its practice, when these are mastered, so easy, that to multiply examples would be mere surplusage. We shall therefore close this subject with the remark and caution, that although isometrical projection is a valuable addition to the ordinary plan, section, and elevation of the >§ inches, under semi-axis minor; and in the same line further to the draughtsman, and may be most advantageously used as explanatory of these, it does not give so truthful or pleasing a representation of an object as a proper per¬ spective drawing. It should only be used, therefore, when the object in view is the elucidation or explanation of a subject, and never when pictorial representation alone is intended. Within the limits which we have indicated it is of extended utility, beyond them it is caricature. INDEX TO THE CARPENTER AND JOINER’S ASSISTANT; AND GLOSSARY OF TERMS USED IN ARCHITECTURE AND BUILDING. ABACISCUS ABACISCUS.—1. Any flat member.—2. The square compartment of a mosaic pavement. ABACUS. —A table constituting the upper or crowning member of a column and its capital. It is rectangular in the Tuscan, Doric, and Ionic orders; but in the Corinthian and Composite orders its sides are curved inwards. These curves are called the arches of the abacus, and the meeting of the curves its horns. The arches are generally de¬ corated with an ornament in the centre of the curve, called the rose of the abacus. The type of the abacus is found in the Grecian Doric order, in which it is a square member. In the Tuscan and Roman Doric it has a moulding and fillet round its upper edge, called the cymatium. In the Grecian Ionic the profile of its side is an ovolo or ogee, and in the Roman Ionic an ovolo or ogee with a fillet over; in the Corinthian and Com¬ posite orders its mouldings are a cavetto, a fillet, and an ovolo. Grecian Doric Abacus. Roman Doric Abacus. In mediaeval architecture the abacus is a strongly marked feature in the earlier styles; but loses this character in the later styles, in which there is no real line of separation between it and the rest of the capital. ABELE or Abel Tree. —The white poplar, Populus alba. See p. 112. ABIES. See Fir. ABSCESSES in Trees detrimental, p. 97. ABSCISSA.—A part of the diameter, ortrans- verse axis of a conic section, in¬ tercepted between the vertex, or some other fixed point, and a semi-ordinate. Thus in the para¬ bolic figure B c A, the part of the axis D c intercepted between the semi-ordinate B D, and the ver¬ tex c, is an abscissa. ABUT, to. —To adjoin at the end, to be con¬ tiguous to ; generally contracted to But. ABUTMENT.—1. The solid pier or mound of earth from which an arch springs.—2. Abutments of a bridge, the solid extremities on, or against which, the arches rest. ACACIA.—Properties and uses of, p. 114. ACANTHUS.—The plant bear’s-breech, the leaves of which are imitated in the foliage of the Corinthian and Composite capitals. ACCIDENTAL POINT.—In perspective that point in which a right line drawn from the eye of the spectator parallel to another given right line, cuts the plane of the picture. Thus: let a B be the given line, c F E the plane of the picture, D the eye, c D the line parallel to A B, then is c the accidental point. ACER. See Maple. ACOUSTICAL PULPIT, Plate LXXXIII., description of, p. 190. ACROTER, Acroteriom, Acroteria. —A small pedestal, placed on the apex or angles of a A Pediment with Acroteria, AAA pediment, for the support of a statue or other or¬ nament. The term is also used to denote the pin¬ nacles or other ornaments on the horizontal copings or parapets of buildings, and which are sometimes called Acrotcral Ornaments. ADHESION of Surfaces glued together. From Mr. Bevan’s experiments it appears that the surfaces of dry ash-wood, cemented by glue newly made, in the dry weather of summer would, after twenty- four hours’ standing, adhere with a force of 715 lbs. to the square inch. But when the glue has been frequently melted and the cementing done in wet weather, the adhesive force is reduced to from 300 to 500 lbs. to the square inch. When Scotch fir cut in autumn was tried, the force of adhesion was found to be 562 lbs. to the square inch. Mr. Bevan found the force of cohesion in solid glue to be equal to 4000 lbs. to the square inch, and hence concludes that the application of this substance as a cement is capable of improvement. ADHESIVE FORCE of Nails and Screws in different kinds of Wood. Mr. Bevan’s experiments were attended with the following results:—Small sprigs, 4560 in the pound, and the length of each T 4 0 4 jj of an inch, forced into dry Christiania deal to the depth of 0‘4 inch, in a direction at right angles to the grain, required 22 lbs. to extract them. Sprigs half an inch long, 3200 in the pound, driven in the same deal to 0 4 in. depth, required 37 lbs. to extract them. Nails 618 in the pound, each nail inch long, driven 0'5 in. deep, required 58 to extract them. Nails 2 inches long, 130 in the pound, driven 1 inch deep, took 320 lbs. Cast- iron nails, 1 inch long, 3S0 in the pound, driven 0 - 5 in., took 72 lbs. Nails 2 inches long, 73 in the pound, driven 1 inch, took 170 lbs. ; when driven 1A inch they took 327 lbs., and when driven 2 inches 530 lbs. The adhesion of nails driven at right angles to the grain was to force of adhesion when driven with the grain, in Christiania deal, 2 to 1, and in green elm as 4 to 3. If the force of adhesion of a nail and Christiania deal be 170, then in similar circumstances the force for green sycamore will be 312, for dry oak 507, for dry beech 667. A common screw j; of an inch diameter was found to hold with a force three times greater than a nail 2i inches long, 73 of which weighed a pound, when both en¬ tered the same length into the wood. ADZE(formerly written Addice). —A cuttingtool i used for chopping a surface of timber. It consists 217 AMPHIPROSTYLE of a blade of iron, forming a portion of a cylindrical surface, ground to an edge from the concave side outwards at one end, and having a socket at the other end for the handle, which is set radially. The handle is from 24 to 30 inches long. The weight of the blade is from 2 to 4 lbs. The w ork is ger.e- rally laid in a horizontal position, and the instru¬ ment held in both hands. The operator, standing in a stooping posture, swings the instrument in a circular path, nearly of the same curvature as the ! blade. His arms, from the shoulder joint, which I forms the centre of motion with the tool, make nearly an inflexible radius, and he thus makes his strokes in a succession of short arcs. The extent I of the stroke is gauged by his right thigh, with ; which his arm comes in contact at each stroke. Standing on his work, the operator, in preparatory work, generally directs the strokes between his feet; but in finer work he directs them under his toe, penetrating the wood with uneiring piecision, and perfect safety to himself.— Holtzapfdl. AERIAL PERSPECTIVE.—That branch of perspective which treats of the representation of the effects of air and atmosphere ; and the several gradations, depths, ami intensities of light, colour, and shadow, produced by intervening air on ob¬ jects, as they recede from the eye of the spectator. AISLE (pronounced lie). —The wing of a build¬ ing; usually applied to the lateral divisions of a church, which are separated from the central part, called the nave and choir, by pillars and piers. The nave is frequently, though incorrectly, termed the middle aisle, and the lateral divisions side aisles. See woodcut, Cathedral. A LA GREC^UE, A la Grec.—O ne of the varieties of the fret ornament. il nr rF ru r-k I — 1 i— 1 I — 1 A la Grocque.—Greek Border Ornament. ALBURNUM.—The white and softer part of the wood of exogenous plants; sap-wood. Seep. 95. ALDER, Aldus—F or description and qualities of, see p. 113. ALTERNATE ANGLES.-The angles formed by two straight lines, A----7B united at their extremi¬ ties by a third straight line. Thus, the lines _j, a B, 0 D, united by the / line B c, give the alter- c — nate angles ABC and BcD. When the angles are equal, the lines AB, CD are parallel. AMBIT. — 1. The perimeter of a figure.— 2. The periphery or circumference of a circular body. AMBO.—A pulpit or reading-desk AMBRY.—1. A cupboard or closet.—2. In ancient churches a cupboard formed in a recess in the wall, with a door to it, placed by the side of the altar, to contain the sacred utensils. AMERICAN Bench Circular Saw, p. 192. AMERICAN or Western Plane, p 113. AMERICAN SPRUCE ( Pin-us alba, and Pinus nigra). —Properties and uses of, p. 118. AMPHIPROSTYLE.—Structures having the ! Tjl f «!! m f) 1 1 ® <§) <§ > 1 f ® Plan of n Ampliiprostyle Building. form of an ancient Greek or Roman rectangular temple, with a prostyle or portico on each of its INDEX AND GLOSSARY. ANAMORPHOSIS ends or fronts, but with no columns on its sides or | ANNULET.—A small moulding, whose hori- | zontal section i3 circular. It isuscu indiscriminately Anaruorphi Hank: ANAMORPHOSIS.—A term in perspective, denoting a drawing executed in such a manner as to present a distorted image of the object repre¬ sented; but which, when viewed from a certain point, or reflected by a curved mirror, shows the object in its true proportions. ANCHOR.—An ornament shaped like an anchor or arrow¬ head, used in all the orders of architecture. It is applied as an enrichment to the ovolo-cchinus or quarter-round, anil as it invariably alternates with the egg ornament, the combination is popularly called egg-and - anchor, egg-and-dart, or egg-and- tongue. ANCON.—An elbow or an¬ gle, whi-nce the French term coin , a corner; also the English quoins, or corner-stones. The corners of walls and beams are sometimes Ancones. ANCON E S. —- Orna¬ ments cut on the key-stones of arches, or on the sides of door-cases. Called also Consoles and Trusses. ANGLE.—To make an angle equal to a given angle. Prob. Ill, p. 5.—To bisect an angle, Prob. IV. p. 5; Prob. V. p. 5. ANGLE op Repose. —That angle at which one body will just rest on another without slipping. It is called also the limiting angle of resistance. ANGLE-BAR.—The vertical bar at each angle of windows constructed on a polygonal plan. ANGLE-BEAD, Angle-Staff. —A piece of wood fixed vertically upon the exterior or salient angle of an apartment, to preserve it from injury, and also to serve as a guide by which to float the plaster. It is called also stuff-bead. ANGLE-BRACE.—1. A piece of timber fast¬ ened at each end to one of the pieces forming the adjacent sides of a system of framing, and subtending the angle formed by their junction. When it is fixed between the opposite angles of a quadrangular frame it is called a diagonal brace. It is also called angle-lie and diagonal tie. —2. A boring tool for working in corners and other places where there is not room to swing round „ the cranked handle of the ordinary brace. It is made of metal, with a pair closet, cabinet, and ot bevel pinions, and a winch handle, which revolves at right angles to the axis of the hole to be bored. ANGLE-BRACKET.—A bracket placed in an interior or exterior angle, and uotat right angles with as a synonym for list, cincture, fillet, tenia, &c. CoiTectly, annulets are the fillets or bands which encircle the lower part of the Doric capital, above its neck or trachelium. ANT.E.—The pier-formed ends of the ptero- mata or side walls of temples, when they are pro¬ longed beyond the face of the end walls. A term er. See columns Portico iu Antis. A A, AnUo. applied to pilasters when they stand opposite a column. A portico in antis, is one in which columns stand between antte. ANTEFIX*-E.—Upright blocks ornamented on the face, placed at regular intervals on the crown- the planes which form it. See Angle-Bkackets, p. 85, and PI. XIX. ANGLE-BRACKETS for coves in straight, concave, and convex walls, p. 85, PI. XIX. ANGLE-CAPITAL.—An Ionic capital on the flank column of a portico, having volutes on three sides, the exterior volute being placed at an angle of 135° with the plane of the frieze, on front and flank. ANGLE-IRON. p. 4. ANGLE-RAFTER. — A rafter placed in the line of meeting of the inclined planes of a hip-roof. It is called also kip-rafter, and in Scotland piend- rafter. See Hip-Roofs, p. 91, and Roof, p. 135, and Plates XX. and XXI. ANGL E- RIB,—A curvedpieceof timber placed m the angle between two adjacent sides of a coved or arched ceiling, so as to range with the common ribs. p. 80. ANGLES.—Construction of. Geometry, p. 5. ANGULAR CAPITAL.—A term applied to a comparatively modern variety of the Ionic capital, which has its four sides alike, and all its volutes placed at an angle of 135° with the plane of the frieze. ANGULAR PERSPECTIVE.—A term ap¬ plied to that kind of perspective in which neither of the sides of the principal object is parallel to the plane of the picture, so that in the representation the horizontal lines of the original object converge to vanishing points. It is called also oblique per¬ spective. ANNULAR VAULT.—A vault springing from two walls, both circular on the plan, the one being concentric to the other. ANNULATED COLUMNS.—Columns clus¬ tered together or joined by rings or bands. They are much used in early English architecture. ing member of a cornice. These ornaments were originally used to terminate the ends of the cover¬ ing tdes of the roof. ANTIUM.—In ancient architecture, aporchto a southern door; that to the north was called Por- tiuni. APARTMENT.—1. Used in the singular, is synonymous with room or chamber.—2. The term was formerly used to denote a suite of rooms compris¬ ing, at the least, a hall, ante-chamber, chamber, ardrobe, with the necessary j conveniences for cooking and the accommodation of domestics. APERTURES.—The openings in the walls of a building, such as doors and windows. APOPHYGE, Apollusis, Apophysis. —The parts at the top and bottom of the shaft of a column which spring out to meet the edges of the fillets. The apophyge is usually moulded into a concave sweep or cavetto. and it is often called the spring or the scape. In French it is termed conge. APPLE TREE, The.—Description of the pro¬ perties and uses of, p. 114. APPLICATE-ORDINATE.—A right line at right angles applied to the axis of any conic section, and bounded by the curve. Thus in the figure, the right line B A, at right angles to c D, the axis of the parabola b c a, is termed an appiicate- ordinate. APPLICATION oftheprin- ciples of the resolution of forces to determine the strains on pieces of framing, and the strains trans¬ mitted by them, p. 123. APRON.—1. A platform or flooring of plank at the entrance of a lock, on which the gates are shut.— 2. A term used by plumbers in the north of England and iu Scotland as synonymous with flash ing. APRON-LINING-—The facing of the apron- piece. APRON-PIECE.—A piece of timber fixed into the walls of a staircase, and projecting horizontally, to support the carriage pieces and joisting in the half spaces or landings. It is called also pitching- piece, p. 196. APSIS.—A term applied to that part of any building which has a circular or polygonal termina¬ tion and a vaulted roof. The eastern portion of the church, where the clergy sat and where the altar was placed. It generally had a circular or poly- 24- 8 ARCH ! gonal termination, and was vaulted ; woodcut, Cathedral. APTERAL.—A temple having along its flanks or sides. | ARABESQUE.—Arabesques or Moresques, style of ornament com¬ posed of representations of a mixture of fruit and flowers, buildings, and other objects. In pure an¬ cient arabesques, such as are found in the Alhambra, no animal representations are used. ARJEO STYLE.—A term applied to a columnar arrangement when the colurn ns are far apart. The interval assigned is four diameters, and it is pro¬ perly applicable to the Tuscan order only. AIL'EOSY STY LE — An arrangement of coupled columns, in which four columns are placed in a space equal to eight diame¬ ters and a half. The cen- Arabesqne. tral iutercolumniation is equal to three diameters and a half, and the others on each side to half a diameter. ARCADE.—A series of arches supported on piers or pillars, used generally as the screen and roof support of an ambulatory or walk; but in the \ Arcade, Romcey Church, Hampshire. architecture of the middle ages also applied as an ornamental dressing to a wall, as in the figure. ARCPI.—A structure composed of separate inelastic bodies, having the shape of truncated wedges, arranged on a curved line, so as to retain their position by mutual pressure. Arches are usually constructed of stones or of bricks. The separate stones which compose the arch, are called voussoirs, or arch-stones; the extreme, or lowest voussoirs, are termed springers, and the uppermost, or central one, is called the keystone. The under, or concave side of the voussoirs. is called the intra- dos, and the upper, or convex side, the extrados of the arch. When the curves of the intrados and extrados are concentric, or parallel, the arch is said to be extradosed. The supports which afford rest¬ ing and resisting points to the arch, are called piers and abutments. The upper part of the pier or abutment where the arch rests—technically, where it springs from—is the impost. The s]>an of an arch, is in circular arches the length of its chord, and generally, the width between the points of its opposite imposts whence it springs. The rise' of an arch, is the height of the highest point of its intra¬ dos above the line of the impost; this point is some¬ times called the underside of the crown, the highest point of the extrados being the crown. Arches are designated in two ways; first, in a general manner, according to their properties, their uses, their posi¬ tion in a building, or their exclusive employment in a particular style of architecture. Thus, there are arches of equilibration, equipollent arches, arches of discharge, askew and reversed arches, and Roman, pointed, and .Saracenic arches. Second, they are named specifically, according to the curve their intrados assume, when that curve is the section of any of the geometrical solids, as circular, seg¬ mental, cycloidal, elliptical, parabolical, hyperboli¬ cal, or catenarian arches; or from the resemblance ARCH INDEX AND GLOSSARY. BALUSTRADE of the whole contour of the curve to some familiar object, as lancet-arch, and horse-shoe arch; or from Equilatoral Arch. Sogniontal Arc.li. the method used in describing the curve, as three- centred arches, four-centred arches, and the like. Horsc-shoo Arch. Four-centred or Tudor Arch. When any arch has one of its imposts higher than the other, it is said to be rampant , ARCH, Equilateral, to draw, p. 28. ARCH, the Ogee, to draw, p. 30. ARCH, the Lancet, to draw, p. 29. ARCH, the Drop, to draw, p. 29. ARCH, the Four-centred, to draw, p. 29, 30. ARCH, GoTHrc. —To describe by the intersec¬ tion of straight, lines, p. 30. ARCH-BRACES in bridge construction, p. 159. ARCHANGEL TIMBER. See Pinus syl- VESTRIS, p. 116, 117. ARCHITECTURE.—The .art of building ; but in a more limited and appropriate sense, the art of constructing houses, bridges, and other buildings for the purposes of civil life. A rchilectwre is usually divided into three classes, civil, military, and naval, but when the term architecture is used without a qualifying adjective, civil architecture is always understood.— Civil architecture is the art of design¬ ing and constructing palaces, houses, churches, bridges, and other edifices for the purposes of civil life; but in a more limited and appropriate sense, it is restricted to such edifices as display symmetri¬ cal disposition and fitting proportions of their parts, and are adorned by pillars, entablatures, arches, and other contrivances for their embellishment.— Military architecture is the art of fortification.— Naval architecture is the art of building ships. ARCHITRAVE.—1. The lower division of an entablature, or that part which rests immediately on the capital of the column. It is sometimes called the cpistylium. See woodcut, Column. — 2. The moulded lining on the faces of the jambs and lin¬ tels of a door or window opening, or niche. ARCHITRAVE-CORNICE.—An entablature consisting of an architrave and cornice only, the frieze being omitted. ARCHITRAVES.—Illustrations of, PI.LXIX. p. 182. ARCHIVOLT.—The architrave or ornamental band of mouldings on the face of an arch followin'* the contour of the intrados. AREA.—The superficial content of any figure. The method of ascertaining the area of the various geometrical figures will be found under the name of the figure. ARRIERE-VOUSSURE.—A rear-vault; an arch placed within the opening of a window or door, and of a different form, to increase the light- Arriere-voussure. way of the window,, or to admit of the better opening of the door; it seems also to have served the purpose of an arch of discharge. ARRIS.—The line in which two straight or curved surfaces of any body, forming an exterior angle, meet each other; an edge. ARRIS-FILLE P.—A triangular piece of wood used to raise the slates of a roof when they abut against the shaft of a chimney or a wall, so as to throw off more effectually the rain from the join¬ ing. It is called also a tilting-fillet. ARRIS-GUTTER.— A wooden gutter of the form of a V in section, fixed to the eaves of a building. ARRIS-WISE.—Tiles or bricks laid diagonally are said to be laid arris-wise. A SH TREE.—Properties and uses of, p. 112. ASHLERING.—Timber quarterings in garrets for affixing lath to, in forming partitions, to cut off the acute angle made by the meeting of the sloping roof with the floor. They are usually two or three feet high, perpendicular to the floor, and fixed at top to the rafters. AST EL.—A board or plank used for partition¬ ing overhead in tunnelling. ASTRAGAL. — A small moulding, semicircular in its profile. It is frequently ornamented by being carved into the representation of beads or berries. ATLANTES.—A term applied to figures or half figures of men used in the place of columns or pilasters, to support an entablature. They are called also Telamones. ATTACHED COLUMNS.—Those which pro¬ ject three-fourths of their diameter from the wall. ATTIC BASE.—A peculiar base used by the ancients in the Ionic order or column, and by Pal¬ ladio and others in the Doric. It consists of an upper torus, a scotia, and lower torus, with fillets between them. ATTIC ORDER.—An order of small square pillars at the uppermost extremity of a building, above the main cornice. The pillars are never less than a quarter, nor more than one-third of the height of the order over which they are placed. ATTIC STORY.—The uppermost story of a house when the ceiling is square with the sides; distinguished from garret, in which the ceiling, or part of the ceiling, is inclined. Rooms in the attic story are called attics. AUGER.—A tool used by carpenters and other artificers in wood, for boring large holes. It con¬ sists of an iron blade ending in a steel bit, and having a handle placed at right angles to the blade. Modern augers have a small pointed screw at the extremity, for better entering the wood, and a spiral groove formed in the blade. The lower ex¬ tremities of the threads of this screw, formed by this groove, are sharpened, to form the bit or cutter; these are called screw-augers. Those made with a straight groove or channel are sometimes called pocl-augers. An ingenious improvement has been recently made by an American. The cutting edges of the lower end of the screw, in place of being parallel to the axis, are rounded off, in imitation of the boring apparatus of the Teredo navalis. AWL.—An iron instrument for piercing small holes. See Brad and Brad-awl. AXIS.—1. In geometry, the straight line in a plane figure, round which it revolves to generate a solid.—2. Generally, a supposed right line drawn from the centre of one end to the centre of the other, in any figure.— Axis of a sphere, a cylindei•, cone, &c., is the straight line round which the gene¬ rating semicircle, rectangle, triangle, &c., revolves. — Axis-minor, conjugate axis, or second axis of a hyperbola and ellipse, a straight line drawn through the centre perpendicularly to the axis-major or trans¬ verse axis.— Axis-major, or transverse axis in the 249 ellipse and hyperbola, a straight line passing through the two foci and the two principal vertices of the figure. In the ellipse the axis-major is the longest diameter; in the hyperbola it is the shortest. B. BACK.—The side op When a piece of timber i an inclined position, the Quoins with Back-filleted Margins. called a bach filet, and backflletted. posite the face or Ireast. 3 laid in a horizontal or under side is called the breast, and the upper side the bach. Thus, we have the back of a hand¬ rail, the back of a rafter, &c., meaning the upper side. BACK-FILLET.— When the margins of a quoin, or those of the jambs of a door or win¬ dow, project beyond the face of the wall, the re¬ turn of the projection is tie margin is said to be BACK-FLA PS. p. 188. BACK-LINING.—The piece of a sash-frame parallel to the pulley piece and next to the jamb. See p. 187. BACKER.—A term used to denote a narrow slate laid on the back of a broad, square-headed slate, where the slates begin to diminish in width. BACKING of the hip, p. 91. BADIGEON.—A mixture of plaster and free¬ stone ground together and sifted, used by statuaries to fill the small holes and repair the defects of the stones of which they make their statues. BAGUETTE.—An astragal or bead. BALANCE-BEAM.—A long beam attached to the gate of a lock, serving to open and shut it. BALCONY.—A projection in front of a house; a frame of wood, iron, or stone, supported by pil¬ lars, columns, or consoles, and encompassed with a balustrade railing or parapet. Balconies are com¬ mon before windows. BALECTION-MOULDINGS. See Bolec- TION. BALK.—Apiece of timber from 4 to 10 inches square. BALL-FLOWER,- Ball-flower. BALTIC TIMBER.- p. 115-118. -An ornament resembling a ball inclosed in a circu¬ lar flower, the three petals of which form a cup round it. The ball- flower ornament is usu¬ ally found inserted in a hollow moulding, and may be considered as one of the characteristics of the Decorated style. -Description and uses of, BALUSTER. — A small column having a swelling in the middle and mouldings to form a base and capital, used in balustrades. The lateral part of the Ionic capital is also called the baluster. BALUSTRADE.—A row of balusters set on a Balustrade. continuous plinth, and surmounted by a cap or rail, serving as a fence for altars, balconies, terraces, 2 l INDEX AND GLOSSARY. BAMBOO steps, staircases, tops of buildings, &c. Balus¬ trades are sometimes used solely as ornaments. BAMBOO.—Multifarious uses of, p. 94. BAND.—In classic architecture, any flat mem¬ ber with small projection. In mediaeval architec¬ ture, the round mouldings or suite of mouldings which girds the middle of the shafts in the early English style. BANDELET, Bandlet.—A narrow band. BANDING PLANE.—A plane intended for cutting out grooves and inlaying strings and bands in straight and circular work. BANISTER. — A corruption of baluster. BAN K ER.—A bench upon which masons place the stones about to be hewn. In Scotland termed a siege. BAR or Barred Door. —The Scottish synonym for ledged door; a door formed of narrow deals joined by grooving and tongueing or by rebating, and secured by bars or ledges nailed across the back. BAR-POSTS.—Posts driven into the ground to form the two sides of a field gateway. They have holes corresponding to each other, into which bars are inserted to form the fence. BARGE-BOARDS, called also Gable-Boards. —The raking-boards at the gable of a building, placed to cover the ends of the roof timbers when they project beyond the walls. They are some¬ times called verge-boards, and are variously orna¬ mented. See illustration, Plate XLVII., Timber- houses; Figs. 3 and 4, Gable-boards. BARGE-COUPLES.—The exterior couples of a roof which project beyond the gable. BARGE-COURSE.—The course of tiles which covers and overhangs the gable-wall of a building, and is made up below with mortar; also, a coping to a wall formed of a course of bricks set on edge. BASE.—The bottom of anything, considered as its support or that whereon it stands or rests. The base of a pillar or column is that part which lies between the top of the pedestal and the bottom of the shaft; but where there is no pedestal it is the moulding or series of mouldings between the bot¬ tom of the shaft and the plinth; and in the Grecian Doric, the steps on which the column stands form its base. The lowest part of a pedestal, and the plain or moulded fittings which surround the bot¬ tom of a wall next the floor, are also termed the base of the pedestal and apartment respectively. BASE-LINE.—In perspective, the common sec¬ tion of the picture and the horizontal plane, p. 228. BASE-MOULDINGS.—The mouldings imme¬ diately above the plinth of a wall, pillar, or pedestal. BASEMENT.—1. The ground floor on which the order or columns which decorate the principal story are placed.—2. A story below the level of the ' street. BASIL.—The slope or angle of the cutting I part of a tool or instrument, such as a chisel or plane. All edge tools may be regarded as wedges | formed by the meeting of two straight or curved j surfaces, or of a straight and curvilinear surface, at angles varying from 20° to 12°. Occasionally the tool is ground with two basils, as in the case of the j hatchet, the turner's chisel, and some others.—The j angle of the basil in cutting tools depends on the hardness or softness of the material to be operated upon, and on the direction of its fibres. Mr. Holt- zapfell classifies cutting tools in the three following groups: — 1 st. Paring tools, with their edges the angles of which do not exceed 60°; one plane form¬ ing the edge being nearly c&incident with the work produced. These tools remove the fibres princi¬ pally in the direction of their length. 2d. Scrap¬ ing tools with thick edges, varying from 60° to 120 °, the planes of the edges forming nearly equal angles with the surfaces produced. Such tools re¬ move the fibres in all directions with nearly equal facility, producing fine dust like shavings, by act¬ ing superficially. 3d. Shearing or separating tools, with edges from 60° to 90°, generally duplex, and then applied on opposite sides of the substance to be operated upon. One plane of each tool, or of the single tool, is coincident with the plane pro¬ duced.— Holtzapfdl chiefly. See Cutting Tools, and particular description under the name of each separate tool. BASKET-HANDLE ARCH (Fr. ansc dc punter).—Any arch whose vertical height is less than half its horizontal diameter; consequently the term includes all surbased and semi-elliptic arches. BASS-RELIEF. See Relief. BASSOOLAH.—The Indian adze. In place of being circular, like the European adze, this is formed at a direct angle of 45 J to 5U\ Its handle is very short, and it is used with great precision by the nearly exclusive motion of the elbow joint. In different districts the instrument varies in weight, and in the angle which the cutting face forms with the line of the handle. The average weight, how¬ ever, may be stated at 11 lbs. 12 oz., and the length of the handle 12 or 13 inches. In using, it is grasped so near the blade that the fore finger rests on the metal, the thumb nearly on the back of the handle, the other fingers grasping the front of it, with their nails approaching the ball of the thumb. When the head of the instrument is made about 2 lbs. weight, it is a very handy tool for blocking out hard or soft woods.— Holtzapfdl. BASTON, Baton, Batoon.— Another name for the torus, or round moulding in the base of a column, or otherwise applied. See woodcut, Column. BATTEN.—-A piece of timber from 11, inch to 7 inches broad, and ^ inch to 2^ inches thick. The battens of commerce are 7 inches by 2i inches. BATTEN DOOR.—A ledged door or barred door. BATTENING. — Narrow battens fixed to a wall, to which the laths for plastering are nailed. They are attached to the wall, either by nailing to bond-timbers built in for the purpose, or fixed directly to the wall by holdfasts of wrought iron; and they should always be so fixed when crossing flues. BATTER.—To incline from the perpendicular. Thus a wall is said to batter when it recedes as it rises. BATTLEMENTi—A parapet of a building provided with openings or embrasures, or the em¬ brasures themselves. The portions of wall which separate the embrasures are called merlons. BAULK.—A piece of whole timber, being the squared trunk of any of the trees usually employed in buildings.—The tie-beam of a common couple roof is called a baulk in Scotland. BAY. — A term applied in architecture without much preci¬ sion.— 1. Any opening in a wall left for the insertion of a door or window.—2. Any dis¬ tinct recess in a build¬ ing.—3. The quadran¬ gular space between the principal divisions of a groined roof, over which a pair of dia¬ gonal ribs extend, and rest on the four angles. — 4. The horizontal space between two principals. — 5. The division of a build¬ ing comprised between two buttresses. — 6 . The part of a window included between two mullions, called also day or light. —7. In a barn, a low inclosed space for depositing straw or hay ; or the space between the thrashing-floor and the end of the barn. If a barn consists of a floor and two heads, where corn is laid, it is called a barn of two bays. BAY-WINDOW.—A projecting window, ris¬ ing from the ground or base¬ ment on a semi-octagonal or some other polygonal plan, but generally understood to be straight-sided. When a projecting window is circu¬ lar in its plan, it is a bow- window; when it is support¬ ed on a bracket or corbel, and is circular or polygonal, it is an oriel. These distinc¬ tions are too little attended to in practice, the terms being often used synony¬ mously. BEAD. — A round moulding of very frequent use, called also baguette. — A series of beads parallel to and in contact with each other is called a reed .—In joinery the bead is of con¬ stant occurrence, and is formed, or run, as the term is. on the edges of boards which have to be jointed together, and thus they serve to admit of and yet to disguise any shrinkage 250 BENCH which the wood may undergo. The bead is also much used in framed work. When it is flush with the face of the work it is called a ^itfri-bead; when it is raised, a cocfc-bead. See p. 183, 184, Joinery. BEAD-BUTT. Joinery, p. 185. BEAD-FLUSH. Joinery, p. 185. BEAK.—Synonymous with bird's mouth (which see). BEAM. See Girder. BEAM-COMPASS.—An instrument used in describing large circles. It consists of a wooden or brass beam, having sliding sockets, with steel and pencil or ink points. See description and use of. p. 34. BEAM-FILLING.—Filling in between tim¬ bers with masonry or brick-work. BEARERS, in staircases, p; 19G. BEARING.—The space between the two fixed extremes of a piece of timber; the unsupported part of a piece of timber; also, the length of the part that rests on the supports. BED-MOULDING.—Properly those members of a cornice which lie below the corona. BEECH TREE.—For description of properties and uses, see p. Ill, 112. BEETLE.—A heavy wooden maul or hammer. BE LECTION.—SeeBALECTiON and Boleotion. BELFRY.—That part of a steeple or other building in which a bell or bells are hung, and more particularly the timber work for sustaining the bell. BELL —The body of a Corinthian or Composite capital, supposing the foliage stripped off. BELL-GABLE. — In small Gothic churches and chapels, a kind of turret placed on the apex of a gable, at the west end, and carrying a bell or sometimes two l ells. BELL-ROOF.—A roof shaped like a bell, its vertical section being a curve of contrary flexure. Plate XXXVI Fig. 1, No. 1. BENCH.—A strong table on which carpenters, Cabinetmaker’s Bench. joiners, cabinetmakers, and other artisans prepare their work. In respect of those required by the cabinetmaker, joiner, and carpenter, a few remarks may be made. These benches are made in various ways, from a few rough boards nailed together, to very complete structures, with various means and appliances for holding and fastening the work while being operated on. A cabinetmaker’s bench, of the most complete kind, is shown in the figure. The framing is connected by screw-bolts and nuts. The top surface is a thick plank planed very true. It has a trough at a to receive small tools, and a drawer at z. Two side-screws c d, which, with the chop e, constitute a vice for fixing work. An end-screw g, and sliding-piece h, form another vice for thin works which require to be held at right angles to the position of the other chop c ; but its chief use is to hold work by the two ends. Work, when laid on the top of the bench, is steadied by the iron bench- hook &, which slides in a mortise in the top, and has teeth at the end which catches the wood. When work would be injured by the bench-hook, the stop m, sliding stiffly in a square mortise in the bench-top, serves to stay it. The stop and bench- hook are shown separately above, drawn to a much Slop. BENCH-HOOK INDEX AND GLOSSARY. BRACE larger scale. There are several square holes along the front of the top, also, at distances apart from eacli other equal to the motion of the sliding-piece h, which has a similar hole. In these bench-holes the iron stop n is inserted, and a similar stop is also inserted in the hole in h. Thus, any piece of wood whose length does not exceed the distance between the end-hole of the bench and the stop in h when it is drawn out to the full extent of its range, may be secured. The face of the stop n is slightly roughened. A holdfast o, sliding loosely in a mortise, is used in holding square pieces of work on the bench. It is fixed by driving on the top, and released by driving on the back. At p is a pin, which is placed in any of the holes shown in the piece in which it is fixed, to support the end of long pieces, which are held by the screws c d, at their other extremity. Various improvements in the bench-hooks, stops, and hold¬ fasts have been from time to time suggested, such as making them work by screws j but being in their simple form sufficiently manageable, and the im¬ provements being more expensive, they have not obtained general use.—The carpenter’s bench is composed of a platform or top, supported on strong framing. It is furnished with a bench-hook at the left-hand end; at which end also the side-board has the screw and screw-cheek, together forming the vice or bench-screw. The side-board and right-hand leg of the bench are pierced with holes, into any one of which a pin is inserted, to hold up the end of any long piece of work clamped in the bench-screw. The length of the bench may be 10 to 12 feet, the breadth 2 feet 6 inches, the height about 2 feet 8 inches. The legs should be 3' inches square, well braced; front top - board should be 2 inches thick; the further boards may be 1 j inch. These two benches may be regarded as the opposite extremes of the scale, and between them may be many varieties both in size and in the number of the fittings, as inclina¬ tion or the necessities of the workman may dictate. BENCH-1IOOK, Bench-Holdfast. See pre¬ vious word. BENCH-PLANES.—The following planes, used for surfaces by the joiner, are usually called bench-planes:— Length. Width. Jack Plano, . . 12 to 17 ins. 2J to 3 ins. Trying Plane, . 20 to 22 ,, 31 to 3J ,, Long Plane, . . 24 to 26 „ 3jf „ Jointer,. . . . 28 to 30 ,, 3f „ Smoothing Plane, 6J to 8 ,, 2$ to 3J ,, Pluck Plano, . . 12 Compass Plane, . 6J to S ,, 2| to 3J „ Width of If to 2| „ If to 2| ,, BENDING TIMBER.—Various methods de¬ scribed, p. 102; Colonel Emy's method, p. 103; Mr. T. Blanchard’s process, p. 103. BEVEL.—An instrument for drawing angles. It consists of two limbs jointed together, one called the stock, and the other the blade, which is move- able on a pivot at the joint, so that it may be adjusted to include any angle between it and the BEVEL-TOOLS.—Tools used in turning hard woods. They are in pairs, and their cutting edges are bevelled off right and left. BILLET.—An ornament much used in Norman architecture. It consists of small rounded billets, Billet-moulding. like an imitation of small pieces of stick, placed in a hollow moulding at intervals apart generally equal to their length. BILLS.—1. The ends of compasses.—2. Knee- timbers. BINDING-JOISTS.—Beams in framed floors which support the bridging-joists above and the ceiling-joists below. See Floors, Plate XLII., and description, p. 150. — Binding-joists. Rule for find¬ ing the strength of, p. 154. BINDING-R AFTERS.—The same as Purlins. BIORNBURG TIMBER. See Pinus SYL- VESTRIS, p. 116, 117. BIRCH.—The common birch, Bctula alba; the mahogany birch of America, Betula lenta; tall or yellow birch, Bctula excelsa ; black birch, Betula nigra. For description of qualities and uses of the varieties generally used, see p. 113. BIRD’S MOUTH.—An interior angle or notch cut across the grain at the extremity of a piece of timber, for its reception on the edge of another piece. BIT.—1. The cutting part of a plane.—2. A name common to all those exchangeable boring tools forwoodappliedbymeansof the crank-formed handle known as the carpenter’s brace. The similar tools used for metal, and applied by the drill-bow, ratchet, brace, lathe, or drilling-machine, are termed drills or drill-bits. The distinction, however, is not uni¬ formly maintained: very frequently all those small revolving borers which admit of being exchanged in their holders or stocks, .are included under the name of bits. The variety is, therefore, very great, and the particular names used to designate them are derived, in most cases, from their forms and the purposes for which they are employed. For wood, the typical form is the shell-bit (fig. a), which is shaped like a gouge, with the piercing end sharpened to a semicircular edge for shearing the fibres round the circumference of the hole. When large, it is termed a gouge-bit, and when small, a quill-bit. Sometimes the piercing end is drawn to a radial j point, and it is then known as the spoon-bib — of which the cooper's dowel-bit and the table or furni- \ ture bit are examples. Occasionally the end is bent into a semicircular form horizontally, and it then becomes the duck-nose bit. The centre-bit (fig. b), is another typical form, of which there are many modifications. The end is flat, and provided with a centre-point or pin, filed triangularly, and which serves as a guide for position; a shearing edge or nicker serving to cut the fibres round the margin of the hole, and a broad chisel-edge or cutter to pare away and remove the wood within the circle defined by the nicker. The plug centre-bit, used chiefly for making countersinks for cylinder-headed screws; the button-tool, which retains only the centre-pin and nicker, and is used for cutting out discs of leather and the like; th e flute-drill, the cup-key tool, the wine-cooper's bit, are all modifications of this borer, suited to special kinds of work. The half- round bit (fig. c), is employed for enlarging holes in metal, and is usually fixed in the lathe or vertically. The cutting end is ground with an incline to the right angle, both horizontally and vertically, three to six degrees, according to the hardness of the material to be bored. The rose-bit (fig. d), is cylin¬ drical, and terminates in a truncated cone, the oblique surface of which is cut into teeth like the rose-countersink, of which it is a modification. It is also used for enlarging holes of considerable depth in metals and hard woods. BLACK WALNUT TREE.—Properties and uses of, p. 111. BLANK DOOR, Blank Window.— A recess in a wall, made to appear like a door or window as the case may be. BLOCKING COURSE.—The course of stones whatever be its horizontal Burface.— Body range of a groin. The larger of the two vaults, by the inter¬ section of which the groin is formed, p. 78.— Body of a room. The main part of an apartment, inde¬ pendent of any recesses. - 1 BOLECTION-MOULD- ] < INGS.—Mouldings in framed work which project beyond the surface of the framing. Called also balection and bclection. See p. 185. BOLSTERS of the Ionic Capital. —1. The lateral part which joins two volutes.—2. Same as balusters (which see). BOLT.—A cylindrical piece of wrought iron for fastening together the parts of framing or ma¬ chinery. Mr. Farey gives the following rule for proportioning the size of the bolt to the strain to which it is to be exposed, viz.: Divide the given strain in lbs. by 2200, and the square root of the quotient is the proper diameter of the pin of the bolt. BOND, in masonry and brickwork, signifies that disposition of the materials by which the joints of one course are covered by the stones or bricks of the next course horizontally and vertically, so as to make the whole aggregate act together, and be mutually dependent on each other. See also Eng¬ lish Bond and Flemish Bond. BOND-TIMBER.—Timbers placed in horizon¬ tal tiers at certain intervals in the walls of buildings for attaching battens, laths, and other finishings of wood. BONING ; in Scotland termed Borning.— The act of judging of a plane surface, or of setting ob¬ jects in the same plane or line by the eye. BORING TOOLS. See Auger, Awl, Bit, BnoAcn, &c. BOSS.—An ornament placed at the intersection It is or bricks erected on the upper part of a corniee to make a termination. BLOCKINGS.—Small pieces of wood fitted to the interior angle of two meeting boards, and glued there to strengthen the joint, as m m m, Fig. 1, Nos. 3 and 4, Plate LXXII. BOARD.—A piece of timber sawed thin, and of considerable length and breadth as compared with its thickness. BO A RDIN G - J OISTS.—The bridging - joists to which the floor boarding is nailed. BOASTING or Scarbung. —In stone cutting, an operation performed with a chisel and mallet. The chisel is about 2 inches broad, and its cutting edge ground quite sharp. With this tool, impelled by the mallet, the ridges left between the grooves formed in brolching are worked off, till the whole surface is reduced to the plane of the draughts. The workman commences at the angle of the stone most remote from him on his right hand, and runs the chisel draughts diagonally towards his left hand, or he commences at the angle nearest to him on his right hand. BODY of a Niche. — The vertical surface, 251 Boss, York (Jathodr.il. of the ribs of groined or cross-vaulted roofs, frequently richly sculptured. BOTTOM PANEL. — The lowest panel in framed work. See p. 186. BOTTOM RAIL.—A term used to denote the lowest rail in a piece of framed work. See p. 186. BOULTINE.—A convex moulding, the con¬ tour of whose section is a quadrant. It is gener¬ ally used below the abacus of the Tuscan and Doric capitals. It is called more commonly ovolo or quarter-round. See Plate LXIII., Mouldings. BOW-COMPASSES.—Different kinds of, and instructions how to use, see p. 33. BOW - SAW ; called also Frame - Saw and Sweep-Saw. —It is used for cutting curves. The frame of this saw consists of a central rod or stretch¬ er, to which are mUrth-ed two end pieces that have a slight motion of rbmtion on the stretcher. These end pieces are each adapted at one extremity to receive the saw-blade, and the other ends are con¬ nected by a coil of string, in the middle of which is a short lever. On turning round the lever, the string is twisted, and thereby shortened. It thus draws together those ends of the cross pieces to which it is attached, and separates the opposite ends, by which means the saw is stretched. In using the bow-saw, the work is usually fixed verti¬ cally, and the saw worked horizontally; but the frame is placed at all angles, so as to clear the work. BOWTEL.—The shaft of a clustered pillar, or any plain round moulding. BOX TREE. — For properties and uses, see p. 115. BOXED SHUTTERS. — Those which fold back into a box or case. BOXINGS of a Window.— The cases, one on each side of the window, and opposite to each other, into which the shutters are folded. The shutters in this case are termed boxed-sh utters. See illustra¬ tion, p. 188, and Plate LXXVIII., Window- Finishings. BRACE.—A piece of timber in any system of framing extending across the angle between two other pieces at right angles. See action of braces BRACE AND BIT anti counterbraces, Bridges, p. 159, 166; and illustration of braces in framing timber houses, Fig. 470, p. 157. BRACE and BIT.—The brace is an instru¬ ment made of wood or iron. It consists of a cranked shaft, having at its one end a socket, called the pad, to receive the bits or boring tools, and at the other a swivelled head or shield, which, when the instrument is used horizontally, is pressed forward by the workman's breast, and when vertically, by his left hand, which is commonly placed against his forehead. See Bit. BRACKET.—A small support against a wall for a figure, clock, &c. Brackets, in joinery, are either cut out of deal or framed with three pieces of timber, viz., a vertical piece attached to the wall, a horizontal piece attached to the shelf to be sup¬ ported, and an angle brace framed between the horizontal and vertical pieces. BRACKETED STAIRS, p. 196. BRACKETS, Diminishing and Enlarging, p. 201. BRAD. — A particular kind of nail, used in floors or other work where it is deemed proper to drive nails entirely into the wood. To this end it is made without a broad head or shoulder on the shank. BRAD-AWL.—An awl used to make holes for brads. BRANCHED WORK.—The carved and sculp¬ tured ornaments in panels, friezes, &c,, composed of leaves and branches. BRAN DERING.—Covering the under side of joists with battens about 1 inch square in the sec¬ tion, and 12 to 14 inches apart, to nail the laths to, in order to secure a better key for the plaster of a ceiling. See p. 154. BRANDISHING or Brattishing. —A crest, battlement, or other parapet. BRANDRETH.—-A fence or rail round the opening of a well. BREAK.—A recess; also, any projection from the general surface of a wall or building. BREAKING-JOINT. — That disposition of joints by which the occurrence of two contiguous joints in the same straight line is avoided. BREAST-LINING, p. 1S8. BRESSUMMER or Breastsummer. — A summer or beam used in the face or breast of a wall, as a lintel to support a superincumbent wall. Its use is generally restricted to a beam used as a lintel in an external wall, such as over the wide openings of shop fronts. See Timber Houses, p. 154. BRICK-NOGGING.—Brick-work carried up and filled in between timber framing. BRICK-TRIMMER.—A brick arch abutting against the wood trimming joist in front of a fire¬ place, and used to support the hearth. See Floors p. 151. BRICK-WORK is valued by the cubic yard, and also by the rod. A rod of brick-work is a quantity of 272£ superficial feet, of the thickness of a brick and a half, or 13& inches. The quarter of a footjs generally disregarded, and the round number 272 feet is reckoned a rod. Hence a rod of brick-work is 306 cubic feet, and contains 4500 bricks, allowing for waste. BRIDGE.—Any structure of wood, stone, or iron raised over a river, poud, lake, or hollow of any kind, to support a roadway for the passage of men, vehicles, &c. Among rude nations, bridges - are sometimes formed of other materials than those enumerated; and sometimes they are formed of boats, or logs of wood lying on the water, fastened together, covered with planks, and called floating bridges. A bridge over a marsh is made of logs or other materials laid upon the surface of the earth.— In suspension or chain bridges, the flooring, or main body of the bridge, is supported on strong iron chains or rods, hanging in the form of an inverted arch from one point of support to another. The points of support are the tops of strong pillars or towers, erected for the purpose at each ex¬ tremity of the bridge. Over these pillars the chains pass, and are attached beyond them to rocks or massive frames of iron firmly secured under ground. J lie flooring is connected with the chains by means of strong upright iron rods.—A draw bridge is one which is made with hinges, and may be raised or opened. Such bridges are constructed in fortifica. turns, to hinder the passage of a ditch or moat; and over rivers, dock-entrances, and canals, that the passage of vessels may not be interrupted. — A fl'Jtag-bridge is made of pontoons, light boats hol¬ low beams, empty casks, or the like. It is made as occasion requires, for the passage of armies. A flying-bridge is also constructed in such a manner as to move from one side of a ' INDEX AND GLOSSARY. in the middle of the river by an anchor, and by the action of the helm it can be made to swing from one bank to the other. BRIDGE.—Classification of the usual forms of bridge trusses, p. 161. BRIDGE-BOARD or Notch-Board. — A board into which the ends of wooden steps are fas¬ tened t-'ee p. 195, Fig. 512. BRIDGE GUTTER. — Gutters formed of boards covered with lead, supported on wooden bearers. BRIDGE OVER.—A piece of timber which is laid over parallel lines of support, crossing them presents this mode. In the first method the prac- transversely, is said to bridge over them. Thus, in I tical rule to find the size of the keys:—To the flooring, the upper joists to which the boards Fig. l. BUTTRESS frieze of the entablature, in the Ionic and Corin¬ thian orders of architecture. BUILDING.—A fabric or edifice constructed for use or convenience; as a house, a church, a shop. BUILDING a Beam is accomplished in the simplest manner, by laying the flitches above each other, and bolting or hooping them together. The sliding of the pieces is prevented by the insertion of keys. This mode is shown in fig. 1. Another method is to table or indent the surfaces of the pieces together, and secure with hoops. Fig. 1 are attached bridge over the binding-joists which extend transversely beneath them, and they are therefore called bridging joists. BRIDGES.—Consideration of the forces which act on framed trusses as applied in the construction of, p. 158. BRIDGES. — Mr. Uaupt's rules for cal¬ culating the strains on the different pieces com¬ posing the truss. See p.160. ! Building a Beam. struetion, of, p. 158. BRIDGES illustrated BRIDGES, TIMBER.—Theory of the con- depth of the beam in inches add three-eighths of the j depth, and divide the sum by the number of keys I to be used; the quotient will be the thickness of i each key, and their breadth should be twice their i of the depth of the beam. The indents should be made to form abutments to the pressure. See p. 148. BUILT BEAM.—One composed of several pieces. ' BULL-NOSED BRICKS.—Those with one I of their vertical angles rounded. BULLER NAILS.—Round-headed nails with ort shanks, turned and lackered; used chiefly for the hangings of rooms. BULLET MOOD. — A wood of a greenish hazel colour, close and hard, the produce of the Virgin Isles, West Indies. It resembles green- I heart. BULL’S-EYE.— A small circular or elliptical j window. BULL'S-NOSE.—The external angle of a polygon, or any obtuse angle. BUTMENTS. Abutments. BUTT END of Timber. —That which is near¬ est the root of the tree. BUTT-HINGES. —Those which are placed on the edges of doors, &c., with their knuckle on the side on which the door opens. See Hinging p 180, and Plates LXXXIV.-LXXXVI. BUTT-JOINT.—That formed by two pieces of timber united endways. BUTTRESS. — 1. A prop. — 2. A projection from a wall to impart additional strength and sup- Timber draw-bridge, Gotha Canal Plate. Tage. XLVIII. 162 Simple bridge truss, XL VIII. 162 Simple bridge truss, XLVIII. 162 ' Laminated arch truss, . XLVIII. 162 Elevation of a bridge 34 feet span, XLVIII. 162 Elevation of a bridge truss, . XLVIII. 162 Bridge over the Spey at Laggan- kirk,. XLTX. 162 Elevation of a timber bridge, XLIX. 162 Bridge over the Don at Inverury, L. 162 Railway-bridge, designed by Mr. White,. LI. 163 Road-bridge, designed by Mr. White,. LI. 164 Timber bridge over the Tyne at Linton, ..... LII. 164 Skew-bridge, desigued by Mr. White, ..... LIII. 164 Trussed bridge, with laminated arch braces, designed by Mr. White,. Lin. 165 American timber bridge, LIV. 165 American timber bridge, L1V. 165 Common lattice bridge, LIV. 169 Improved lattice bridge, LIV. 170 Skew-bridge at Portobello, on the North British Railway, LV. 170 Bridge over the Tweed at Mertoun, LVI. 171 as to move from one side of a river to the other Tf- , o T ,, , Ui P l,ureu orn aments repre- consist, uf a boat s ec„ re d to a W £& ,™*«» BRIDGING - FLOORS. — Those in which bridging-joists are used. See p. 150. BRIDGING-JOISTS.—The upper joists in a framed floor, to which the flooring boards are nailed. See Bridge over Floors, p. 150; and illustration. Plate XLII. BRIDGING-JOISTS. —Rules for calculating the strength of, p. 154. BRIDGINGS.—Pieces of wood placed between two beams or other timbers, to prevent their ap¬ proaching each other. More generally termed strain¬ ing or strutting pieces. BRINGING UP, or Carrying Up, signifies the same as building up. BROACH. — A general name for all tapered boring bits or drills. Those for wood are fluted like the shell-bit, but tapered towards the point; but those for metal are solid, and usually three, four, or six sided. Their usual forms are shown in the annexed figures. Broaches are also known as wideners and rimers. Fig. a is an example of the broach or rimer for wood, and fig. b of those for metal. BROACH. An old English term for a spire; still in use m the north of England, as Hesslebroach, &c.; and in some other parts of the country, as in Leicestershire, it is used to denote a spire springing 1,0 ™the tower without any intermediate parapet. BROAD. An edge tool for turning soft wood. 1 he edge of the broad is at right angles to the handle, and the blade is either square or triangular. I he triangular broad is used principally for turning large pieces the plank way of the grain. BUCRANIA. — Sculptured ornaments repre- tinp- OY-aU-nllo c^n.T. n ,l _,i ■ made tt\i ri., . , , . •' '. , " 1 “'-“ ‘t-quuou some counter support to resist the Y r,Lu l' , Sty o m, y ertlCal /Y t 0utw:lnI p™ 8ure of the vaulting. The roof, in the a cupola oi dome—2. The sole port of the Early Engli.hstyle, appears always to hove been high hian and Composite capitals: n.'i h-i n 1 <,,t„ i , pitched. In the interior the arches are usually lan¬ cet-shaped, and the pillars often reduced to very slender proportions. As if to give still greater light¬ ness of appearance, they are frequently made up of a centre pillar, surrounded by slight detached shafts, only connected with the pillar by their capi¬ tals and bases, and bands of metal placed at inter¬ vals. Hiese shafts arc generally of Purbeck mar¬ ble, the pillar itself being of stone, and, from their extreme slenderness, they appear as if quite inade¬ quate to support the weight above them. Some of the best examples are to lie seen in Salisbury Ca¬ thedral. The architects of this style carried their E. EARLY ENGLISH ARCHITECTURE.— The first of the Pointed or Gothic styles of archi¬ tecture that prevailed in this country. It suc¬ ceeded the Norman towards the end of‘the twelfth century, and gradually merged into the Decorated at the end of the thirteenth. Oue of the leading peculiarities in this style is the form of the win° dows, which are narrow in proportion to their height, and terminate in a pointed arch, resem¬ bling the blade of a lancet. Throughout the early period of the style, they are very plain, particu- arly in small churches, but in cathedrals and other arge buildings, the windows, frequently combined two or more together, are carried to a great height, are richly and deeply moulded, and the jambs orna- rnentod with slender shafts. On the eastern and western fronts of small churches the windows are 260 Early English Capital, Salisbury Cathedral ideas of lightness to the utmost limits of prudence, and their successors have been afraid to imitate their example. The abacus of the capitals is gen¬ erally made up of two bold round mouldings, with a deep hollow between. The foliage is peculiar, generally very gracefully drawn, and thrown into elegant curves; it is usually termed stiff-leaved, from the circumstance of its rising with a stiff stem from the neck-mould of the capital. The trefoil is commonly imitated, and is very characteristic of the style. The mouldings of this style have great boldness, and produce a striking effect of light and shade. They consist chiefly of rounds separated by deep hollows, in which a peculiar ornament, called the dog's-tooth, is used, whenever ornament can be introduced. This ornament is as characteristic of the Early English as the zigzag is of the Norman.' INDEX AND GLOSSARY. ENTASIS EARTH-TABLE EARTH-TABLE.—The course of stones in a wall seen immediately above the surface of the ground, now called the plinth. It is also termed tjrass-tablc and ground-table. EAVES.—That part of a roof which projects beyond the face of a wall. EAVES-BOARD; called also Eaves catch and Eaves-LATH. —An arris-fillet nailed across the rafters at the eaves of a roof, to raise the slates a little. EAVES-GUTTER.—A gutter attached to the eaves. ECHINUS.—An ornament in the form of an egg, peculiar to the ovolo or quarter-round mould¬ distance, and the doorway was flanked by two towers of a peculiar shape, broad in front and ing; whence this moulding is sometimes called echinus. EGG-and-ANCHOR, Egg-and-Daut, Egg- and-Tongue. See Anchor and Echinus. EGYPTIAN ARCHITECTURE.—The style of building which prevailed in ancient Egypt, and the remains of which, as exhibited in the pyramids, Interior of tlic Temple of Esneh. tombs, and ancient temples scattered over that country, alike excite our wonder, and attest the magnificence and advanced civilization of the pe¬ riod to which they belong. Its remains are pro¬ bably the most ancient of any in the world, the only buildings that may possibly vie with them in point of antiquity being the rock-cut temples and other monuments of India. The characteristics of Egyp¬ tian architecture are solidity, boldness, and ori¬ ginality ; the object being to fill the mind of the spectator with astonishment and awe. The col¬ umns are numerous, close, and very large, being sometimes ten or twelve feet in diameter. They are generally without bases, aud had a great va¬ riety of capitals, from a simple square block orna¬ mented with hieroglyphics or faces, to an elaborate composition of palm-leaves, bearing a distant re¬ semblance to the Corinthian capital. The shafts are either plain or worked into reeds or flutes, and frequently are constructed to represent various plants, such as the lotus, the date-palm, or the papyrus. The entablature is very simple, consist¬ ing of a plain architrave, surmounted by a large torus, and a large overhanging concave moulding, which serves as a cornice. The roofs of the tem¬ ples were formed by large blocks of stone, extend¬ ing from wall to wall or from column to column, and not unfrequently the roof was wholly absent. In the construction of the portico of the temples the greatest magnificence was displayed. It was frequently approached by an avenue of sphinxes and other sculptures, extending to a considerable Egyptian Capital, Denderah. narrow at the sides, with the walls sloping back at a slight angle from the perpendicular. The walls, pillars, and interior of the temple generally, were covered with a profusion of hieroglyphics, pictures, and symbolical figures. Many of the pictures still display great brilliancy of colouring, the dryness of the climate having preserved them nearly intact after so great a lapse of time. No use appears to have been made of wood, and the stones employed in the construction of the columns were of the most gigantic di¬ mensions. Another prominent character¬ istic of Egyptian architecture is the absence of the arch, no specimen of which occurs among their ancient monuments. EIDOGRAPH.—A form of the para¬ graph, for copying, enlarging, or reducing drawings, invented by Professor Wallace of Edinburgh. ELBOW-LINING.—The lining of the elbows of a window^ See p. 188. ELBOWS. — The upright sides which flank any panelled work, as in windows below the shutters. ELEVATION.—A geometrical delinea¬ tion of any object according to its vertical and horizontal dimensions, without regard to its thickness or projections ; the front or any other extended face of a building. ELIZABETHAN ARCHITECTURE. —A name given to the impure architecture which prevailed in the reigns of Elizabeth and James I., when the worst forms of Gothic and debased Italian were combined together, producing singular heterogeneous¬ ness in detail, but wonderful picturesque¬ ness in general effect. Its chief character¬ istics are windows of great size, both in the plane of the wall and deeply embayed, and galleries of great length, combined with a profuse use of ornamental strap-work in the para¬ pets, window-heads. &c. Elizabethan Window, Rushton nail, cir. 1590. ELLIPSE, Description of the, p. 22. ELLIPSE, to draw, when the major and minor axes are given, Prob. LXXXV. p. 22, and Prob. LXXXVI. p. 23. ELLIPSE, the, to draw, with the trammel, Prob. LXXXVII. p. 23. ELLIPSE, the, to draw, on the method of the trammel, without using the instrument, Prob. LXXXVIII. p. 24. ELLIPSE, to describe, by means of a string, Prob. LXXXIX. p. 24. ELLIPSE, to draw with the compass a figure approaching the, Prob. XCIII. and XCIV. p. 25, and Prob. XCVI. p. 26. . ELLIPSE.—To find the circumference of an ellipse.—Rule: Multiply half the sum of the two diameters by 3T416.—To find the area of an ellipse. Rule: Multiply the largest diameter by the short¬ est, and the product by '7854. ELLIPSES, to draw, by intersecting lines, Prob. XC. and XCI. p. 24. ELLIPSOID, an, to describe the section of, p. 68, Plate I. Fig. 8. ELLIPSOIDAL VAULTS. — To determine the caissons of, p. S3. ELLIPTICAL-DOMICAL PENDENTIVE, p. 81. ELM, The.—Properties and uses of, p 110. ELM, Wych, p. 111. ELM, Rock, p. 111. ELM, Dutch, p. 111. ELM, Twisted, p. 111. EMBATTLEMENT, or Battlement. — An indented parapet, belonging originally to military works, the indents, crenelles, or embrasures being used for the discharge of missiles. It was after¬ wards adopted extensively aR a decoration in me¬ diaeval architecture. EMBOSS, v .—To form bosses or protuberances; to cut or form with prominent figures. EMBRASURE.—An opening in a wall, splay¬ ing or spreading inwards. The term is usually ap¬ plied to the indent or crenelle of an embattled parapet. EMY’S Laminated Roofs, p. 141. ENCARPA, Encarpus.— A festoon of fruit or flowers on a frieze or capital. See Festoon. ENDOGENOUS PLANTS. See Endogens. ENDOGENS.—Plants whose stems are increased by the devel¬ opment of woody matter towards the centre, instead of at the cir¬ cumference, as in exogens. To this class belong palms, grasses, rushes, &c. Stems of this sort have no distinct concentric layers or medullary rays. ENDOGEN S.—G eneral char¬ acteristics of, p. 93. ENGAGED COLUMN.—A column attached to a wall, so that part of it is concealed. Engaged columns have seldom less than a quarter, or more than a half of their diameter in the solid of the wall. ENGLISH BOND.—That disposition of bricks in brick work which consists of courses of headers and stretchers alternately. The figures show the first and second courses of a 14-inch wall. A is a Engaged Column. course consisting of a row of stretchers a a, and headers b b ; and B, the next succeeding course, shows the disposition of these reversed. ENLARGING AND Diminishing Mouldings. —Method of, p. 181. ENNEAGON.—A polygon, with nine sides or nine angles. ENRICH.—To adorn with carving or sculp¬ ture. ENSTYLE.—An intercolunmiation of two and a quarter diameters. ENTABLATURE.—That part of an order which lies upon the abaci of the columns. It con¬ sists of three principal divisions—the architrave, the frieze, and the cornice. See woodcut, COLUMN. ENTAIL.—The more delicate and elaborate parts of carved work. ENTASIS.—A swelling; the curved line in 261 INDEX AND GLOSSARY. column diminishes - baluster. the I ENTERCLOSE which the shaft of . swelling in the middle of 2, 3, and 4. Plate LXXIi. ENTERC LOSE. — A rooms. ENTRESOL.—A low story between two other stories. EPIST\ LIUM, Epistyle. —An ancient name for the architrave. EQUILATERAL Arch, to draw. p. 2S. EQUILATERAL Triangle, to describe, Prob. XI. p. 7- EREMACAUSIS, or slow combustion or oxidation, the cause of decay in timber. See p. 105. EREMACAUSIS, remedies for, p. 105. ESCAPE.—That part of a column where it springs out of the base; the apophyge ; the conge. ESCUTCHEON.—1. A shield for armorial bearings.—2. A plate for protecting the keyhole of a door, or to which the handle is attached. ESTRADE.—An elevated part of the floor of a room ; a public room. EXCRESCENCES in Trees, p. 97. EXEDRA, Exhedra.— In ancient architec¬ ture, tlie name given to vestibules or apartments in public buildings where the philosophers disputed, and also to apartments or vestibules in private houses used for conversation. In medueval archi¬ tecture. the term is sometimes applied to the porch of a church, especially to the galilee or western porch. The apsis, too, was sometimes termed the exedra. EXFOLIATION of the bark of trees, p. 97. EXOGEN.—A plant whose stem increases by development of woody matter towards the outside. To this class belong all our timber trees. EXOGENS.—General characteristics of p. 94 EXPANDING CENTRE-PIT.—A hand- instrument, chiefly used for cutting out discs of leather and other thin material, and for making the margins of circular recesses. It consists of a central stem a, and point b, mount¬ ed on a transverse bar c, which carries a cutter d at one end, and is adjust¬ able for rai lius. The arm c being carried FALLING MOULD in Hand-bailing, to oee J’igs. construct, p. 201. I FALLING STYLE.—That style of a gate or passage between two door in which the lock, latch, or other fastening, is ' placed. See Gates, p. 176. FALSE ATTIC. — An architectural finish, bearing some resemblance to the Attic order, but without pilasters or balustrade. It is used to crown a building and to receive a bas-relief or inscription. FALSE ROOF.—The open space between the ceiling of an upper apartment and the rafters of the roof. FAN-LIGHT.—Properly a semicircular win¬ dow over the opening of a door, with radiating bars in the form of an open fan, but now used for any window over a door. FAN-TRACERY VAULTING.—The very complicated mode of roofing, much used in the Per¬ pendicular style, in which the FISHING dually to the points of suspension, from which the ends generally hang down. The festoon, in archi¬ tecture, is sometimes composed of an imitation of drapery similarly disposed, and frequently of an assemblage of musical instruments, implements of war or of the chase, and the like, according to the purpose to which the building it ornaments is ap¬ propriated. , FILLET.—A small moulding, generally rect- mult is covered by | angular in section, and having the appearance of a narrow band. It has many synonymes.—In carpentry and joinery, any small scantling less than a batten. FILLISTER.—A kind of plane used for grooving timber or for forming rebates. FINE STUFF.—Plaster used in common ceilings and walls for the reception of paper or colour. EINIAL.—The ornamental termination to a Fan-tracery, North Aisle, St. George’s Chapel, Windsor. round the fixed points a and b, the cutter d describes a circle of which the radius is the distance b d. EXTRADOS.—The exterior curve of an arch. See Arch, F. front of an edifice, of the patterns for FAQADE.—The face oi FACE-MOULD.—One marking the board or plank out of which the hand¬ rails for stairs and other works are to be cut. See Staircases and Hand-bailing, in text. FACETS, Facettes.— Small projections be¬ tween the flutings of columns. FACIA. See Fascia. FACING.—1. The thin covering of polished stone, or of plaster or cement, on a rough stone or brick wall.—2. The wood-work which is put as a border round apertures, either for ornament or to cover and protect the junction between the frames of the apertures and the plaster.—3. Sometimes in joinery used synonymously with lining. FACTABLE.— The same as coping. FAG US SYLVATICA.—The beech tree. For description, seep. 111. FALDSTOOL.—A kind of stool placed at ti e south side of the altar, at which the Kings of Eng¬ land kneel at their coronation.—2. A small desk, at which the Litany is enjoined to be sung or said; sometimes called a Litany-stool.— 3.°The chair of a bishop, inclosed by the railing of the altar.—4. An arm-chair; a folding-chair. FALLING MOULDS.—The two moulds which, in forming a hand-rail, are applied, the one to its convex, and the other to its concave vertical side, in order to form the back and under-surface, and finish the squaring. ribs and veins of tracery, of which all the principal j lines have the same curve, and diverge equally in i every direction from the springing of the vault, as j in Henry VII.’s Chapel, Westminster, and St. George’s Chapel, Windsor. ttvi.’ a _ i i.i „ FASCIA.—1. A band or fillet.—2. Anv flat an T aJn»'7^h?p.Tif erm ihecentreof member with a little projection, as the band of an anything, as the eye of a volute, of a dome. | architrave.-!!. In brick buildings, the jutting of the bricks beyond the windows in the several stories except the highest. FASTIGIUM.—The summit, apex, or ridge of a house or pediment. FEATHER-BOARDING. —A covering of boards, in which the edge of one board overlaps a part of the one next it. It is also called weather- boarding. FEATHER-EDGED BOARDS. - Boards made thin on one edge. FEATHERINGS, or Foliations. —The cusps or arcs of circles with which the divisions of a Gothic window are ornamented. FELLOE.—The outer rim of the frame of a centre or mould under the lagging or covering- boards. See p. 173. FELLING of Timber.—D ifferent modes of procedure, p. 98. FELLING of Timber. —Comparative cost of various methods, p. 99.- FELLING of Timber. —Proper season for %- the operation, p. 99. FELT GRAIN.—Timber split in a direction crossing the annular layers towards the centre. ^ hen split conformably with the layers it is called the quarter grain. FEMUR.—In architecture, the interstitial 4 space between the channels in the- triglyph of a- the Doric order. FENDER-PILES.—Piles driven to protect work, either on land or water, from the concus¬ sion of moving bodies. FENESTRAL. — A small window. Used also to designate the framed blinds of cloth or c canvas that supplied the place of glass previous <: to the introduction of that material. < " FESTOON.—A sculptured ornament in imi¬ tation of a garland of fruits, leaves, or flowers, suspended between two points. The garland is of I the greatest size in the middle, and diminishes gra- 262 pinnacle, consisting usually of a knot or assemblage of foliage. By old writers finial is used to denote not only the leafy termination, but the whole pyramidal mass. FINISHING COAT.—In plastering, the last coat of stucco-work where three coats are used. FINLAND TIMBER. See Binds sylves- tris, p. 116, 117. FIR TREE. See description and uses, p. 116. FIRE-PLACE.—The lower part of a chimney which opens into the room or apartment, and in which the fuel is burned. FIRMER.—A paring chisel. See Chisel. FIRRINGS.—Pieces of wood nailed to any range of scantlings to bring them to one plane, applied generally to the pieces added to joists which are under the proper level for laying the floor. Galled also fur rings. See p. 185. FIRST COAT.—The first plaster coat laid on laths is so called when only two coats are used. When three coats are used, it is called the prickiny- up coat. In brick-work, the first of two-coat work is called rendering, and the first of thi ee-coat work roughing up. FISHING, Fished Beam.—A built beam, composed of two beams placed end to end, and Fig. i secured by pieces of wood covering the joint on opposite sides. Fishing is performed in three diffe- INDEX AND GLOSSARY. FISTUCA rent ways. In the first the ends of the beam are abutted together, and a piece of wood is placed on each side and secured by bolts, fig. 1. Secondly, the parts may be indented together, so as better to resist a tensile strain, as in fig-2. Thirdly, pieces, termed lceys, may be notched equally into the beams and the side-pieces, as at A B, fig. 3. Bee p. 147, 148. FISTUCA.—An instrument for driving piles, with two handles, raised by pulleys, and guided in its descent so as that it may fall upon the head of the pile, and drive it into the ground. It is called by the workmen a monkey. FIXED or In¬ flexible Centre, in bridge-building, p. 172. FLAMBOYANT Style of Archi¬ tecture. — A term applied by French writers to that style of Gothic architec¬ ture in France which was coeval with the Perpendicular style in Britain. Its chief characteristic is a wavy, flame - like tracery in the win¬ dows, panels, &c. ; whence the name. FLANK.— 1. The side of a build¬ ing.—2. The Scotch term for a valley in Flamboyant Window, a roof. Church of St. Ouen at Rouen. PLANNING— The splaying of a door or window-jamb internally. FLAPS.—Folds or leaves attached to window- shutters. FLASHINGS.—In plumbing, pieces of lead, zinc, or other metal, used to protect the joinings where a roof comes in contact with a wall, or where a chimney, shaft, or other object comes through a roof. The metal is let into a joint or groove cut in the wall, and then folded down so as to cover and protect the joining-s. When the flashing is folded down over the upturned edge of the lead of a gutter, it i3 termed in Scotland an apron. FLAT PANELS, p. 185. FLATTING.—A coat of paint which, from its mixture with turpentine, leaves the work flat or without gloss. FLECHE.—A name for a spire when the alti¬ tude is great compared with the base. FLEMISH BOND.—That disposition ofbricks which exhibits externally alternate headers and stretchers in each course, whereas in English bond the headers and stretchers are in alternate rows. The figure shows two courses of a 14-inch wall in Flemish bond. FLEURON.—Foliage such as that in the cen¬ tre of the abacus of the Corinthian capital. It has been defined to be such foliage as is not in direct imitation of nature. FLEXIBLE CENTRES, p. 171. FLIERS.—Steps of a stair which are parallel¬ sided ; such as do not wind. See p. 196. FLIGHT.—A series of fliers from one level to another. See p. 196. FLOATING. —Reducing the surface of plaster- work to a plane. It is thus performed:—The whole surface of the work is divided into bays or compartments, by ledges of lime and hair, from 6 to 8 inches wide, extending from the top to the bottom of the walls, and across the whole width of the ceiling. These are termed screeds, and are formed at 4, 5, or 6 feet apart, by the plumb-rule and straight-edge, so as to be accurately in the same plane. They thus become gauges or guides for the rest of the work. When the screeds are thus prepared, the panels or interspaces are filled in flush with plaster, and a long float being made to traverse them, all the plaster which projects beyond is struck off, and the whole surface reduced to one plane. FLOATS. — Plasterers' tools, consisting of straight rules, which are moved over the surface of plaster while soft, to reduce it to a plane. They are of three sorts: the hand flout, used by one man; the quick-float, used in angles; and the derby or two- handed float, which is so long, that two men arc required to work it. FLOOR.—1. That part of a building or room on which we walk. See p. 150.—2. A platform of boards, planks, or other material, laid on timbers. FLOORING. — The whole structure of the floor-platform of a building, including the support¬ ing timbers. The weight of flooring is estimated at from 30 to 80 lbs. per foot superficial, and floors of dwelling-houses are generally calculated to carry 150 lbs. per foot superficial, including their own weight. FLOORING-MACHINE — A machine for preparing complete flooring-boards with great despatch; the several operations of sawing, plan¬ ing, grooving and tonguing being all carried on at the same time by a series of saws, planes, and re¬ volving chisels. FLOOR.—Constructed by Serlio at Bologna, in 1518, p- 153. FLOOR of the Palace in the Wood at the 1 Hague, p. 153. FLOOR-TIMBERS.—The timbers on which the floor-boards are laid. FLOORS.—Variation in the mode of construc¬ tion common in Scotland, Plate XLIII. Figs. 1 and 2, p. 151. FLOORS.—Mode of construction of, used in France, Plate XLIV. p. 152. FLOORS.—Construction of warehouse - floors, Plate XLIII. Figs. 4-10, p. 152. FLOORS.—Formed of a combination of small timbers, Plate XLIV. Figs. 15-17, p. 153. FLOORS, Fire-proof, Plate XLIII. Figs. 12- 14, p. 154. FLOORS.—Rules for calculating the strength of timbers which enter into the composition of, p. 154. FLORIATED.—Having florid ornaments; as, the floriated capitals of early Gothic pillars. FLORID STYLE. —A term employed by some writers on Gothic architecture to designate that highly enriched and decorated architecture which prevailed in the fifteenth and beginning of the sixteenth century. It is often called the Tudor style, as it prevailed chiefly in the Tudor era. FLUE.—A passage for smoke in a chimney, leading from the fireplace to the top of the chim¬ ney, or into another passage; as, a chimney with iour flues. Also, a pipe or tube for conveying heat to water, in certain kinds of steam-boilers. The same name is given to passages in walls for the pur¬ pose of conducting heat from one part of a building to the others. FLU ING. — Expanding or splaying, as the jambs of a window. FLUSH.—A term applied to surfaces which are in the same plane.— To flush a joint, is to fill it until the filling material is in the plane of the sur¬ faces of the bodies joined. FLUSH PANEL, p. 185. FLUTINGS, or Flutes, are the hollows or channels cut perpendicularly in columns, &c. When the flutes are partially filled by a smaller round moulding, they are said to be cabled. See Cable. FLYERS.—Steps in a flight of stairs which are parallel to each other. See Fliers. FLYING-BUTTRESS.—In Gothic architec¬ ture, a buttress in the form of an arch springing from a solid mass of masonry and abutting against 263 FORMERETS another building, to resist the thrust of an arch or of a roof. It is seen in many cathedrals; and there its office is to act as a counterpoise against the vault¬ ing of the central pile. FOCUS.—1. A point in which any number of rays of light meet, after being reflected or refract¬ ed ; as, the focus of a lens.— Virtual focus or point of divergence, the point from which rays tend after refraction or reflection. — Geometrical focus, the point in which rays of light ought to be concen¬ trated when reflected from a concave mirror, or re¬ fracted through a lens, the point in which they are actually found being termed the refracted focus. These foci are separated from one another, in pro¬ portion to the degree of spherical aberration.—2. A certain point in the parabola, ellipsis, and hyper¬ bola, where rays reflected from all parts of these curves concur or meet. The focus of an ellipse is a point toward each end of the longer axis, from which two right lines, drawn to any point in the circumference, shall together be equal to the longer axis. The focus of a parabola is a point in the axis within the figure, and distant from the vertex by the fourth part of the parameter. The focus of a hyperbola is a point in the principal axis, within the opposite hyperbolas, from which, if any two lines are drawn, meeting in either of the opposite hyperbolas, the difference will be equal to the principal axis. FOILS. — The small arcs in the tracery of Gothic windows, panels, &c., which are said to be Trefoil. Quatrefoil. Cinquefoil. trefoiled, quatrefoiled, cinquefoiled, and multifoiled, according to the number of arcs they contain. FOLDED FLOORING, p. 185. FOLIATION.—The use of small arcs or foils in forming tracery. FOMERELL.—A lantern-dome or cover. FONT. — A vessel employed in Protestant churches to hold water for the purpose of baptism, and in Catholic churches used also for holy water. Font, Colobrooke. There are a great many fonts in England, curious both for their antiquity and their architectural designs. In the Decorated style, their form is usually octagonal, sometimes hexagonal; and in the Perpendicular style, the octagonal form is al¬ most invariably used. FOOT-PACE.—A landing or resting place at the end of a flight of steps. If it occurs at the angle where the stairs turn, it is called a quarter- pace. FOOT-STALL.—The plinth or base of a pillar. FOOTING.—A spreading course at the founda¬ tion of a wall. The footings appear like steps, as in the figure. FOOTING-BEAM.— The tie-beam of a roof. FORE-PLANE.—The first plane used after the saw or axe. FORE-SHORTEN¬ ING—In perspective, the diminution which in repre¬ sentation a body suffers in one of its dimensions, as compared with the others, owing to the obliquity of the diminished part to the plane of the picture. Footing of a Wall. FORMERETS. — The arches which in Gothic groins lie next the wall, and are consequently only half the thickness of those which divide the wall into compartments. FOUNDATIONS FOUNDATIONS.—The solid ground on which the walls of a building rest; also that part of the building or wall which is under the surface of the ground. FOUR-CENTERED ARCH, to draw, p. 29. FOUR-LEAVED FLOWER.—An ornamen¬ tal member, much used in hollow mouldings, very Four-Ioaved Flowers, W characteristic of the Decorated period of Gothic architecture. FOX-TAIL WEDGING.—A method of fix¬ ing a tenon in a mortise, by splitting the end of the tenon beyond the mortise, and inserting a wedge. The wedge being driven forcibly in, enlarges the tenon, and renders the joint firm. When the mor¬ tise is not cut through and through, the wedge is inserted in the end of the tenon, and the tenon en¬ tered into the mortise, and then driven home. The bottom of the mortise resists the wedge, and forces it further into the tenon, which is thus made to expand, and press firmly against the sides of the mortise. FRAM E.—A term applied to any assemblage of pieces of timber firmly connected together. F RAMED-FLOQR.—One with girders, bind¬ ing-joists, and bridging-joists. See p. 151, Plate XLII. Fig. 3. FRAMED ou Bound Doors, p. 186. FRAMING. — 1. Fitting and joining in con¬ struction. See p. 146.—2. The rough timbers of a house, including beams, flooring, roofing, and partitions. FRAXINUS.—The ash-tree. Seep. 112. FREDERICKSTADT TIMBER. Seep. 116, 117, PlNUS SYLVESTRIS. I'RLNCH DOORS. — Examples of, Plate LXXIV. Figs. 5-15, p. 187. FRESCO.—A method of painting on walls, j performed with water-colours on fresh plaster, or on a wall coated with mortar not yet dry. The ! colours, incorporating with the mortar, and drying ^ a witli it, become very durable. Only as much of ! purposes. INDEX AND GLOSSARY. FRIEZE-RAIL.—The rail next below the top rail of a door of six or more panels. .See p. 186. FRONTISPIECE.—The face of a building. FRONTON.—A pediment. FROST CRACKS in trees, p. 97. FRUSTUM.—The part of a solid uext the base, left by cutting olf the top or segment by a plane parallel to the base; as the frustum of a cone, of a pyramid, of a conoid, of a spheroid, or of a sphere, which latter is any part comprised between two parallel circular sections; and the middle frustum of a sphere is that whose ends are equal circles, having the centre of the sphere in the middle of it, and equally distant from both ends. FURNESS’ Mortising Machine, p. 194. FURNESS’ Planing Machine, p. 193. FURNESS’ Tenoning Machine, p. 193. FUST or a Column.—T he shaft. scriptions of, Plates LXI., LXII., andp. GATES.—Park and Entrance Gates, G. GABLE. — The triangular end of a house or other building, from the cornice or eaves to the top. GABLE-ROOF.—A roof open to the sloping rafters or spars, finishing against gable-walls. GABLE- WIN DOW.— A window in a gable, or a window shaped like a gable. GABLET.—-A small gable or gable-shaped i ,. .. • , : = ----- -1--" .. "J umouuo ill UWUIIUU, UU Slgllll V decoration, frequently introduced on buttresses, i a rebate made round the opening of a doorway screens, ft. o. .. __■ .. V , ° . GOTHIC GATES.—Dock Gates, illustrations and de- 177, 17S. . ates, illustra¬ tions and description of, Hate LX. and p. 177. GATEWAY.— 1. A way through the gate of some inclosure.—2. A building to be passed at the entrance of the area before a mansion. GATHERING of the Wings. — The lower part of the funnel of a chimney. GAUGE.—Measure; dimension. GAUGED-ARCHES are those in which the bricks are cut and rubbed to a gauge or mould, so as to make exactly fitting joints. GAUGED-PILES.—Large piles placed at re¬ gular distances apart, and connected by horizontal beams, called runners or wale pieces, fitted to each side of them by notching, and firmly bolted. A gauge or guide is thus formed for the sheet\inrj or filling piles, which are drawn between the wale- pieces, and fill up the spaces between the gauged- piles. Gauged-piles are called also standard piles. GAUGE D-STUFF.—In plastering, stuff com¬ posed of three parts of putty-lime and one part of plaster of Paris. GAVEL. See Gable. GEFLE TIMBER. See Pinus sylvestris p. 116, 117. GEMMELS.—An old name for hinges. GEOMETRICAL ELEVATION. —A draw¬ ing of the front or side of a building or any object. —The projection, on a vertical plane, of the front or side of a building or other object. GLOME 1RICAL STAIRS.—Those stairs the steps of which are supported at one end only by being built into the wall. See p. 196. GIRLET-CHECK, or Jiblet-Cheek. — A term used by stone-masons in Scotland, to signify screens, &c. GAGE, or Gauge.— 1. The length of a slate or tile below the lap ; also the measure to which any j substance is confined.—2. The quantity of plaster °f Paris used with common plaster to hasten its ( setting. GAIN.—1. A bevelling shoulder —2. A lapping of timbers.—3. The cut that is made to receive a ; timber. GALILEE.—A small gallery or balcony, open , to the nave of a conventual church, from which j visitors might view processions; also a porch or I portico annexed to a church, and used for various the wall is coated with plaster as the artist expects GALLERY.—1. Anapartmentof much mater to finish during a day s work, and at the end of j length than breadth, serving as the Hnv the iinrfinn r. /at .... _ .... • .. „ .. I.... ° the day the portion not painted on is carefully cut off by the outline of a figure, or other well-defined form, so as to conceal the joining in commcnt-ino- next day’s work. From difficulty of alteration when the colour is once absorbed, the greatest pre¬ cision of design is necessary before commencing the work. With this view, a drawing on paper, the exact size of the work to be executed, called a cartoon, is prepared beforehand, and from it the outlines of the design are carefully tracod on the wet plaster. This method of painting is very ancient. It was used by the Greeks, and in later times was much employed by the great masters of cornice hp5-.nir>" s , chools - Jt 7 is calle d fresco, either roof-water beyond the walls. because it is done on fresh plaster, or because it is used on walls and buildings in the open air. FRET. — Work raised in protuberances. A kind of knot or meander, consisting of two fillets nterlaced, used as an ornament in architecture. All the embattled, crenelled, and lozenge mouldings ire frets. An u la Grccque. FRETTED ROOFS. — Groined roofs, much ntersecteil by arches. FRIEZE.—That part of an entablature which _. . , -ge of c„... munication for the different rooms of a building._ 2. Any apartment whose length greatly exceeds its breadth, used for the reception of pictures, statues, &c., its use being denoted by a qualifying word, as picture-gallery, sculpture-gallery, &c. — 3. A plat¬ form projecting from the walls of a building, sup¬ ported on piers, pillars, brackets, or consoles, as the gallery of a church or of a theatre.—4. A long portico, with columns on one side. GALLERY of a Church.— Framing for the, p. 150. GARGOYLE, or Gurgoyle.— A spout in the parapet of a building, for throwing the Gargoyles, in classic and Sculptured Friozo, Temple on the IIIjsh is between the architrave and cornice. It is a flat member, usually enriched with sculptures. FRIEZE-PANEL.—Theupper panelof a of six or more panels. See p. 1S6. Gargoyle, Stony-Stratford. architecture, were ornamented with masks, carved into the representations of heads of animals, especially of the head of the lion. In the architec¬ ture of the middle ages the gargoyle became longer, and assumed a vast variety of forms. GARNET-HINGE.—A species of binge, re¬ sembling the letter T laid horizontally, thus f-; called in Scotland a cross-tailed hinge. GARRET.—1. That part of a house which is on the upper floor, immediately under the roof.— 9 Rotten wood. forming a recess to receive 'a docTr or gate which I opens outwards. GIBS.—Pieces of iron employed to clasp to¬ gether such pieces of a framing as are to be keyed, previous to inserting the keys. GIRDER.—A main beam to support the joists . of n fioor. It may be in one or two pieces, plain or trussed. See Trussed Girders, p. 149 and Plate XL. GIRDERS.—Rules to determine the strength ; of, p. 155. GIRDLING Trees, to exhaust their sap before felling, p. 99. GIRTH.—In practice, the square of the quar¬ ter girth multiplied by the length, is taken as the . solid content of a tree. GLACIS.—An easy, insensible slope. GLASS-PLATE.— Specific gravity, 2-453; weight of a cubic foot, 153 lbs.; expansion by 180° J of beat, from 32° to 212°, -00086 inch. GLAZING, as is now practised, embraces the ; cutting of all kinds of-glass manufactured for win- ] dows, together with the fixing of it in sashes by 1 means of brads and putty ; also the formation of I casements, and seaming the glass by bands of lead J fastened to outside frames of iron. GLUING-UP COLUMNS, p. 184. GLYPHS.—In sculpture and architecture, a j notch, channel, or cavity, intended as an orna- | ment. Used in combination chiefly, as tryglyph I (which see). j GO BBETS.—Blocks of stone; squared blocks of stone. GODROON, Gadroon. — An ornament, con¬ sisting of headings or cablings. GORE.—A wedge-shaped or triangular piece. GOING of a Step, and Going of the Flight, p. 196. GOTHIC.—In architecture, a term at first ap¬ plied opprobriously to the architecture of the middle ages, but now in general use as its distinctive ap¬ pellation. By some the term Gothic is considered to include the Romanesque, Saxon, and Norman styles, which have circular arches, but it is strictly applicable only to the styles which are distinguished by the pointed arch. Gothic architecture so re¬ stricted has been divided into three distinct styles or periods, which have been variously named by different authors, but the terms most generally used are those bestowed by Mr. Rickman. By him the first period is named the Early English, it pre¬ vailed in the thirteenth century; the second, the Decorated, prevailed in the fourteenth centuiy; and the third, the Perpendicular style, commenced in the end of the fourteenth, and continued in use’ till the middle of the sixteenth century. The chief characteristics of Gothic architecture are:-"" The I predominance of the arch and the subserviency and PARPFTTTwn T „ I subordination of all the other parts to this chief upper panelof a door ; ston P in tuJ - - * e Insertm g snia11 splinters of ! feature ; the tendency of the whole composition to i Hi 1 ° r St0n . e ‘ n thu I 01nt8 of coarse masonry, after the walls vertical lines; the absence of the column and en- e built. Flint walls are frequently garretted 264 | tablature of classic architecture, of square edges GOTHIC ARCHES INDEX AND GLOSSARY. .and rectangular surfaces, and the substitution of clustered shafts, contrasted surfaces, and members multiplied in rich variety. GOTHIC ARCHES.-Construction of, p.2S-30. GOTHIC GROINS, to draw the arches of, to mitre truly with a given arch of any form, p 30. GOTHIC MOULDINGS —Characteristics of the various periods or styles of, p. 130. GOTHIC MOULDINGS. —Illustrations of, Plate LXVa p. ISO. GOTHIC VAULT. —Manner of dividing it into compartments, p. 82. GOTTEN BURG TIMHER. See Pinus stl- vestris, p. 116, 117. GRADE.—A step or degree. GRADE, v. —To reduce to a certain degree of ascent or descent, as a road or way. GRADIENT.— ; The degree of slope or inclina¬ tion of a road. GRAINING. — Painting in imitation of the grain of wood. GRASS TABLE.—In Gothic buildings, the first horizontal or slightly inclined surface above tlie ground ; the top of the plinth. GRATING. —A framework of timber, com¬ posed of beams crossing each other at right angles, used to sustain the foundations of heavy buildings iu loose soils. See G hi I,lag e. GRECIAN ARCHITECTURE.—This term is used to distinguish the architecture which flour¬ ished in Greece from about 500 years before the Christian era, or perhaps a little earlier, until the Roman conquest. It comprehends the Doric, Ionic, and Corinthian orders, to which may probably be added the Caryatic order. Of these the Doric is the most distinctive, and may be regarded as the national style. The architecture of the Greeks is known to us only through the remains of their sacred edifices and monuments, and we have no means of ascertaining in what manner it was ap¬ plied to their houses. .Simple and grand in their general composition, perfect in proportion, en¬ riched, yet not encumbered with ornament of con¬ summate beauty, these remains cannot be sur¬ passed in harmony of proportion and beauty of detail. GREEN-HEART TIMBER.—Description of, and properties and uses of, p. 112. GREES, Gue.se, -Gryse.—T his word, which is variously spelled, signifies a step or degree. GREY-STOCK BRICKS. — The hardest of the malm bricks. They are of a pale brown colour. GRILLAGE. — A framework composed of beams laid longitudinally, and crossed by similar beams notched upon them, used to sustain walls, and prevent their irregular settling in soils of un¬ equal compressibility. The grillage is firmly bedded and the earth packed in the interstices between the beams, a flooring of thick planks, termed a plat¬ form, is then laid, and on this the foundation courses of the wall rest. GROIN.—The line made by the intersection of simple vaults crossing each other at any angle. See p. 76. GROINED ARCn. —An arch formed by the intersection of two semicylinders or arches. Groined Arch. GROINED ROOF, or Celling. — A ceiling formed by three or more intersecting vaults, every two of which form a groin at the intersection, and all the groins meet in a common point, called the apex or summit. The curved surface between two adjacent groins is termed the sectroid. Groined roofs are common to classic and mediaeval archi¬ tecture, but it is in the latter style that they are seen in their greatest perfection, 'in this style, by increasing the number of intersecting vaults, vary¬ ing their plans, and covering their surface with rib3 and veins, great variety and richness were ob¬ tained, and at length the utmost limit of com¬ plexity was reached in the fan groin tracery vaulting. See Fan Tracery. GROINS in rectangular vaults, p. 77, 78. I GROINS on a circular plan, p. 79. GROINS on an octagonal plan, p. 79, 80. GROOVING and Tonguing, Grooving and ; Feathering, Ploughing and Tonguing. — In 1 joinery, a mode of joining boards, which consist's I in forming a groove or channel along the edge of H A Grooving and Tonguing. one board, and a continuous projection or tongue on the edge of another board. When a series of boards is to be joined, each board has a groove on its one edge and a tongue on the other. See p. 182. GROTESQUE-— 1. Applied to artificial grotto- work. decorated with rock-work, shells, &c.—2. That style of ornament which, as a whole, has no type in nature; the parts of animals, plants, and other incongruous elements being combined to¬ gether. GROUND-FLOOR.—Properly, that floor of a house which is at the base, but usually that which is on a level with ora little above the ground with¬ out. GROUND - JOISTS.—Joists which rest on dwarf-walls, prop-stones, or bricks laid on the ground; sleepers. GROUND-LINE.—In perspective, the inter¬ section of the plane of the picture with the ground- plane. See p. 228. GROUND-MOULD.—An invert mould, used in tunnelling operations, or any mould by which I the surface of ground is formed. GROUND-PLAN.—The plan of that story of ( a house which is on the level of the surface of the ground, or a little above it. GROUND - PLANE. — In perspective, the plane on which the objects to be represented are supposed to be situated. Sea p. 228. GROUND-PLATE, or Ground-Sill. —The lowest horizontal timber into which the principal and other timbers of a wooden erection arc in- i serted. GROUND-TABLE STONES.—The top of the plinth of a Gothic building. See Earth-Table, Grass-Table. GROUNDS.—In joinery, pieces of wood at¬ tached to a wall for nailing the finishings to. They have their outer surface flush with the plastering. See p. 185. GROUPED COLUMNS, or Pilasters.— A term used to denote three, four, or more columns or pilasters assembled on the same pedestal. When two only are placed together, they are said to be coupled. GROUT.—Mortar in a fluid state, used to fill in the joints in brick-work, or the cavities in rub¬ ble building. GUILLOCHE. — An interlaced ornament,formed ! by two or more intertwin- j ing bands, frequently used ! in classical architecture to enrich the torus and other mouldings. G UTTaE.— Ornaments resembling drops, used in the Doric entablature, ! immediately under the triglyph and mutule. LILLI H. HACKING.—In walling, a manner of building in which a course of stones, begun with single stones in height, is interrupted and carried on in two stones in height, but so as to make the two courses at the | one end equal in height to the one course at the other. HACKING-OUT TOOL.—A knife for remov¬ ing old putty out of the rebates of a sash, prepara- I tory to inserting a new pane of glass. HACKMATACK.—The popular name of the red larch, Pinus microcar pa. but more commonly j applied to the Pinus pendula. HACKS.—The rows in which bricks are laid to dry after they are moulded. HAF FIT.—The fixed part of a lid or cover, to which the moveable part is hinged. HALF-HEADER.—In bricklaying, a brick j either cut longitudinally into two equal parts, or cut into four parts by these halves being cut across 265 HANDRAILING | transversely, used to close the work at the end of a | course. HALF-LONG (Scotch, JIa/jlin ).—One of the , bench-pianes. HALF-ROUND.-—A moulding whose profile is I a semicircle; a bead; a torus. HALF-SPACE, or Foot-Pace.— The resting place of a staircase; the broad space or interval be- I tween two flights of steps. When it occurs at the I angle turns of a stair, it is called a quat ler-space. See p. 196. HALF-TIMBERED HOUSES. See descrip- ! tion and illustration, p. 156, Plates XLVI. and | XL V II. HALVING.—A mode of joining two timbers by letting them into each other. See p. 1-19. HAMMER-BEAM.—A short beam attached to the foot of a principal rafter in a roof, in the place of the tie-beam. Hammer-beams are used in pairs, and project from the wall, but do not extend half way across the apartments. The hammer- beain is generally supported by a rib rising up from a corbel below; and in its turn forms the support Hainmor-benn Roof, Westminster Hall. of another rib, constituting with that springing from the opposite hammer-beam an arch. Al¬ though occupying the place of a tie in the roofing, it does not act as a tie; it is essentially a lever, as will be obvious on an examination of the figure, Here the inner end of the hammer-beam a receives the weight of the upper portion of the roof, which is balanced by the pressure of the principal at its outer end. See also p. 115, description of the roof of Westminster Ilall, and illustration, Plate XXXII. IIANCE.—A term in mediaeval architecture, and that which immediately succeeded it, which seems to have been limited in its application to the small arches, at the springing of three and four centred arches, and to the small arches by which a straight lintel is sometimes united to its jamb or impost. HANDRAIL.—A rail to hold by. It is used in staircases to assist in ascending and descending. When it is next to the open newel, it forms a cop¬ ing to the stair balusters. HANDRAILING. — Definition of terms, p. 201. HANDRAILING, Elucidation of the princi¬ ples of.—Section of a cylinder, p. 202. HANDRAILING.—To produce the section of a cylinder, through any three points on its convex surface, p. 202. HANDRAILING.—Summary of the leading points of difference between the method of Mr. Nicholson and that here taught, p. 203. HANDRAILING.—Method of producing the face-mould and falling-mould for the stairs, Fig. 1, Plate XCI. p. 204. HANDRAILING.—Method of producing the falling and face moulds for the stairs, b ig. 2, Plate XCI. p. 205. HANDRAILING.—Method of producing the falling and face moulds for, Fig. 1, Plate XC. p. 205. HANDRAILING.—Method of producing the face and falling moulds for scrolls, p. 205, 206. HANDRAILING. — Sections of handiails, how to draw, p. 207. HANDRAILING.—Mitre cap, how to form the section of the, p. 207. HANDRAILING.—To form the swan-neck at the top of a rail, p. 207. HANDRAILING.—To form the knee at the bottom newel, p. 207. HANDRAILING.—Scrolls, how to draw, p. 207. HANDRAILING.—The scroll step, how to form, p. 20S. HANDRAILING. — Vertical scrolls, how to draw, p. 208. 2 L 'is' Hanging Buttress. i a plas- HANGING BUTTRESS HANGING BUTTRESS. — A buttress not rising from the ground, but supported on a corbel. Applied chiefly as a decoration, and used only in the Decorated & T and Perpendicular styles. 1 ik| HANGING STYLE of a Vf Door or Gatr —That to which & tlie hinges are fixed. cj» HANGINGS.— Linings for Jfe 'vSl, rooms, consisting of tapestry. || leather, paper, and the like. They were originally invented to hide the rudeness of the car¬ pentry or the harsh appearance of the bare ■wall. Paper-hang¬ ings were introduced early in the seventeenth century. HATCHET. —A small axe with a short handle, used with one haml for reducing the edges of boards, Ac HAUNCH of an Arch.— The middle part between the ver¬ tex or crown and the springing. HAUNCIIING, ].. 182. HAWK.—A small quadran¬ gular board, with a handle un¬ derneath, used by plasterers to hold their plaster. HAWK-BOY.—A boy who attends terer, and supplies his hawk with stuff. HAWTHORN, The.—Properties and uses of, P- 05. HEADPIECE.—The capping-piece of a quar¬ tered partition, or of any series of upright timbers. IIEAD-POST.—The post in the stall-partition of a stable which is nearest to the manger. HEAD-WALL.—The wall in the same plane j as the face of the arch which forms the exterior of a bridge. HEADER.—1. In masonry, stones extending over the thickness of the wall; through stones.— 1 2. In brick-work, bricks which are laid lengthways across the thickness of the wall. HEADING-COURSE.—A course of stones or bricks laid lengthways across the thickncssof a wall. 1 HEADING-JOINT.—The joint of two or more boards at right angles to their fibres. HEART-BON D —!n masonry, a kind of bond in which two stones forming the breadth of a wall have one stone of the whole breadth placed over them. HEART-WOOD.—The central part of the , trunk of a tree : the duramen. See description of Exogens, p. 94-96. HELICAL LINE of a Handrail. —The spiral line twisting round the cylinder, representing the squared handrail before it is moulded. ( HELICES.—Projection of, p. 67. HELIX. — A scroll or volute; in the plural, helices- The small vol¬ ute or caulicule under the abacus of the Cor¬ inthian capital. In every perfect capital there are sixteen heli¬ ces, two at each angle and two meeting under the middle of each face of the abacus. HEMICYCLE.— A semicircle. See de¬ scription of the liemicy Lorme, p, 144. HEMISPHERE, A.- the concave INDEX AND GLOSSARY. are placed obliquely to the right and left alter¬ nately. It receives its name from the resemblance which the courses have to the bones of herring. See Plate XLI.V., French Floors, Fig. 1, for illus¬ tration of Flooring-boards laid herring-bone fashion. IIE X JE D RON.—A cube. HEX/ED RON. — One of the five regular solids. It is bounded by six squares. To find the Surface. Multiply the square of its linear side by 6 0000000. To find the solid content. Multiply the cube of its linear side by 1-000000. HEXAGON. — A figure of six sides and six angles. To find its area. Multiply the square of the side by 2-5930762. HEXAGON.—To reduce a hexagon to a pen¬ tagon, Prob. XXH. p. 9. HEXAGON.— Upon a given straight line, to describe a regular hexagon, Prob. XXXIX. p. 12 . HEXAGONAL PYRAMID, A.—To find the shadow of, p. 221. HEXASTYLE, Hexastylos. — A building with six columns in front. HICKORY WOOD (Juglans alba). p. 111. HILING, Heling.—T he covering of the roof of a building; slating; tiling. HINGED or French Sashes, p. 187. HINGES.—The hook or joint on which a door or gate turns. Hinges are the joints on which doors, lids, gates, shutters, and an infinite number of articles, are made to swing, fold, open, or shut up. They are made in a great variety of forms, to adapt them to particular purposes. ?See p. 191. HINGING.- Various modes of, described, p. 191. HIP.—A piece of timber placed in the line of meeting of the two inclined sides of a hipped roof, to receive the jack-rafters. It is also called a hip- rafter, and in Scotland a piend-rafter. IIIP-KNOB.—A finial or other similar orna¬ ment placed on the top of the hip of a roof, or on of M. Philibert de lie shadow on To find rface of, p. 220. HENDECAGON, or Endkcagon.— A figure of eleven sides and eleven angles. To find its area. Multiply the square of the side by 9-3656411. HEPTAGON. — A figure having seven sides and angles. To find its area. Multiply the sauare HEKNOSAND TIMBER. See Prxcs SYL- VESTRIS, p. 116, 117. H ERRING-BONE WORK.—Courses of stone Hip-knob, Colon try, the point of a gable. When used upon timber gables, or on gables with barge-boards, the hip- knob generally terminates with a pendant. HiP-MOULDING, or IIip- Mould.— Any moulding on the hip-rafter. More commonly used to denote the backing of a hip-rafter. HIP-RAFTER.—The rafter which forms the hip of a roof; a piend-rafter. See p. 91. HIP-ROOF.—A roof, the ends of which rise from the wall-plates, with the same inclination as ICOSAHEDRON I It contains 1134 cul )ic inches, or S duodecimal 1 inches; and two hods of mortar are equal to a bushel nearly. Four hods of mortar will lay 100 bricks. A hod contains 20 bricks. HOGGING.—The drooping of the extremities and consequent com ex appearance of any timber , supported in the middle. I HOLDFAST.—A hook or long nail, with a ' flat short head, for securing objects to a wall; a bench-hook. HOLING.—Piercing the holes for the rails of a stair. j HOLLOW.—A concave moulding. Sometimes I called a casement. HOLLOW NEWEL. — In architecture, an opening in the middle of a staircase. It is used in | contradistinction to a solid newel, which has the end of the steps built into it. In the hollow newel the ends of the steps next the hollow are unsup¬ ported, the other ends being oidy supported by the i surrounding wall of the staircase. HOLLOW QUOINS.—The part of the piers of a lock-gate, in which the heel-post or hanging- post of the gate turns. HOLLOW WALL.—A wall built in two thicknesses, with a cavity between, either for the purpose of saving materials or of preserving a uni- iormity of temperature in the apartments. IIGOD - MOULDING, IIood - M ou ld.— 'The upper and projecting moulding over a Gothic door or window, &c.; called also a label, drip, or weathcr- moulding. See woodcut, Dripstone. HORIZONTAL LINE.—In perspective, the line of intersection of the horizontal plane with the . plane of the picture. See p. 229. HORIZONTAL PLANE.—In perspective, a plane parallel to the horizon, passing through the j eye and cutting the plane of the picture at right angles. See p. 228. HORNBEAM.—For description of properties and uses, see p. 113. IIORSE-CHESTNUT. — For description of J properties and uses, see p. 114. HOUSE, v .—To excavate a space in one tim- I her for the insertion of another. See p. 196. HOUSING.— 1. The space taken out of one , solid to admit of the insertion of the extremity of another, for the purpose of connecting them. See j p. 196.—2. A niche for a statue. HOVELLING.—A inode of preventing chim- I neys from smoking, by carrying up the two sides i which are liable to receive strong currents, to a greater height than the others, or by leaving aper¬ tures in the sides, so that when the wind blows i over the top the smoke may escape below. HYRffiTHRAI,.—A building or temple un¬ covered by a roof, as the famous temple of Neptune I at Pass turn. HYPERBOLA.— Construction of, p. 27, 28. HYPERBOLA.—Description of the, p. 27. HYPERBOLA.—To find the focus of a liyper- I bola, p. 28. HYPERBOLA.—Mode of describing graphi¬ cally, p. 2S. HYPERBOLA.—To draw-, by means of a rule and a string, p. 28. HYPERBOLA.—To draw tangents and per¬ pendiculars to the curve of a hyperbola, p. 28. HYPERBOLA.—To find points in the curve of a hyperbola, the axis, vertex, and ordinate being given, p. 28. HYPOTKACHELIUM. — The neck of the capital of a column ; the part which forms the junction of the shaft with its capital. See Neck. Hip-roof. I the other two sides. Called in Scotland a piend- ! roof. 8-ee p. 91. HIP-ROOFS.—Preliminary notions, p. 91. HIP-ROOFS. — Construction of, for regular and irregular plans, and methods of finding the , lengths of the rafters, the backing of the hips, and the bevels of the shoulders of the jack-rafters and the purlins, p. 91, 92. HOARDING. — A timber inclosure round a building, to store materials when the building is in course of erection or undergoing repair. HOD.—A kind of tray used in bricklaying, for carrying mortar and bricks. It is fitted with a handle, and borne on tlie shoulder. A hod for nortar is 9 inches by 9 inches, and 14 inches long. 266 ICHNOGRAPHY.—In architecture and per¬ spective, the horizontal section of a building or other object; apian. ICOSAHEDRAL. — Having twenty equal sides. ICOSAHEDRON.—Tlie projections of an, to construct, p. 56, 57. ICOSAHEDRON.—A solid of twenty equal sides. The regular icosahedron is a solid, consist¬ ing of twenty triangular pyramids, whose vertices meet in the centre of a sphere supposed to circum¬ scribe it; and therefore they have their bases and heights equal.—To find the surface of an icosahe¬ dron. Multiply the square of its linear side by 8-6602540.—To find the solidity of an icosahedron. Multiply the cube of its linear side by 2TS16950- IMPAGES INDEX AND GLOSSARY. ISODOMON IMPAGES.—A term used by Vitruvius, and i ancients, varied almost in every building. Vitru- posed to signify the rails of a door. ' vius enumerates five varieties of intercolumniation. and assigns to them definite proportions, expressed in measures of the inferior diameter of the cflumn. These are—The pycnostyle, of one diameter and a half; the systyle, of two diameters; tl.e diastyle, of three diameters; the areostyle, of four diameters; and the eustyle, of two and a quarter diameters. It is found, however, on examining the remains of ancient architecture, that they rarely or never agi ee with Vitruvian dimensions, which must therefore be regarded as arbitrary; and indeed the words them¬ selves, as will be found on referring to them, con¬ vey no idea of an exactly defined space, but are in their very vagueness more applicable to the re¬ mains of ancient art. INTERDUCE.—An inteitie. See that term. INTER JOIST.—The space between two joists. INTERLACING ARCHES.—Circular arches which intersect each other, as in the figure. They supposed to signify IMPERIAL.—A roof or dome in the form of an imperial crown. IMPERIALS. — In slating, slates measuring 2 feet G inches by 2 feet. IMPOST.—The congeries of mouldings forming a cap or cor¬ nice to a pier, abutment, or pilas¬ ter, from which an arch springs. INCERTUM. — A mode of building used by the Romans, in which the stones were not squa 1 ed nor the joints placed regularly. It corre-iponds to the modern rubble-work. INCISE.—To cut in; to carve. &. impost. INDENTED.— Cut in the edge or margin into points like teeth, as an in¬ dented moulding. Indented mouldings are much used in the transition from the Norman to the Early English style, and sometimes in the Early English style itself. INDIAN ARCHITECTURE. — The archi¬ tecture of Hindoostan, in its details, bears a strik¬ ing resemblance to the architecture of Persia and Egypt, and they are considered to have a common origin. Its monuments may be divided into two classes, the excavated, which is either in the form of a cavern, or in which a solid rock is sculptured into the resemblance of a complete building: and the constructed, in which it is actually a building, or formed by the aggregation of different materi¬ als. The first class is exemplified in the caves of Elephanta and Ellora, and the sculptured temples j the Pointed style to have had its origin INTERMOD I L- LION.—The space be¬ twixt two modillions. j INTERPILASTER. , —The space between two pilasters. INTERQUARTER. —The space between two ' quarters. INTERTIE, I.ntek- I DUCE. —A short piece of timber introduced hori¬ zontally between uprights, to bind them together or to stiffen them. See p. 155, Partitions, and illus¬ tration, Plate XLV. Fig. 1, A A; Fig. 2, No. 2, D; Fig. 3, No. 1, C; andp. 156, Timber-houses, and illustration, Fig. 470, p. 157, OOO. INTRADOS.— 1. The interior or under concave curve of an arch. The exterior or convex curve is called the extrados. See woodcut, Arch. —2. i vault. It is called also Interlacing Arcade, Norwich Cathedral. are frequent in arcades of the Norman style of the twelfth century, and in them Dr. Milner supposed Jain Temple, Mount Aboo, Gujerat.—Forgussou’s Hindoo Architecture. of Mavalipooram, and the second class in the pa¬ godas of Chillimbaram, Tanjore, and others. The architecture of India resembles, in its details, that of Egypt, but its differences are also very striking. In the architecture of Egypt massiveness and soli¬ dity are carried to the extreme ; in Indian archi¬ tecture these have no place. In the former the ornaments are subordinate to the leading forms, and enrich without hiding them. In the latter the principal forms are overwhelmed and decomposed by the accessories. In the one grandeur of effect is the result, while littleness is the characteristic of the other. Besides the various styles of Hindoo architecture, properly so called, there is in Hindoo¬ stan a distinct series of buildings belonging to tin Mahometan conquerors of that country, and con sisting of palaces, mosques, and tombs. These par take strongly of the characteristic features of Sara cenic architecture. INJURY to Timber, from being exposed to sudden or rapid changes of temperature, p. 100. INNER PLATE.—The innermost of the two wall-plates in a double wall-plated roof. INSERTED COLUMN.—The same as en¬ gaged column (which see). INSERTUM. See Incertum. INSULATED COLUMNS. — Those which stand clear from the walls, as opposed to attached or engaged columns. INTAGLIO.—Literally, a cutting or engrav¬ ing ; hence, anything engraved, or a precious stone with arms or an inscription engraved on it, such as we see in rings, seals. &c. INTERCOLUMNIATION.—The space be¬ tween two columns, " The concave surface of douellc. INVERTED ARCH.—An arch with its in- trados below the axis or springing line, and of which, therefore, the lowest stone is the keystone. Inverted arches are used in foundations to connect particular points, and distribute their weight or \ pressure over a greater extent of surface, as in ! piers and the like. IONIC ORDER'—The second of the "three Grecian and third of the five Roman orders. The distinguishing characteristic of this order is the voluted capital of the column. In the Grecian Ionic ! capital the volutes appear the same on the rear as ! on the front, and are connected on the flanks by a I peculiar roll-moulding, called the baluster or bolster. In the exterior columns of a portico, however, the volutes are repeated on the outer Hank, and are thus necessarily angular. The Grecian Ionic may be j considered in three parts—the stylobate, column, and entablature. The stylobate is from four-fifths j to a whole diameter in height, and is in three re- | ceding steps. The column is rather more than nine , j diameters in height. Of this two-fifths of a dia- This, in the practice of the j meter are given to the base, and from three-fourths 267 to seven-eighths to the capital, including the hypo- trachelium. The base is divided into three nearly equal parts in height, with two equal fillets sepa¬ rating them. The lowest is a torus, which rests on Grecian Ionic Capital. the stylobate; this is separated from a scotia above by a fillet, and another fillet intervenes between the scotia and another torus, and from a third fillet the scape or apopliyge of the shaft springs. The shaft diminishes in a curved line to its upper diameter, which is five-sixths of its lower diameter, or some¬ times more. It is fluted with twenty-four flutes, with fillets one-fourth of their width between. The hypotrachelium or necking of the capital is some¬ times separated from the shaft by a plain fillet, and sometimes by a carved bead, and is generally orna¬ mented. Above this the mouldings of the capital spring out. They consist generally of a bead, an ovolo, and a torus, all richly caived, and on these rest the square mass, on the faces of which are the volutes, and on this rests the abacus, whose edges are moulded into the form of an ovolo, and sometimes ornamented with the egg-and-tongue. The entablature is rather more than two diameters in height. If this be divided into five parts, two of them may be given to the architrave, two to the frieze, and the remaining part to the cornice. If the architrave, again, be divided into nine parts, seven of them may be given to three equal fascias, and the remainder to the band of architrave mouldings. The frieze may be plain or enriched with sculpture in low-relief. The cornice consists of bed-mouldings, corona, and crowning mouldings. [ The bed-mouldings are composed of a bead and a \ cyma reversa, both carved. The cyma reversa is contained in the depth of the corona, whose bed is hollowed out for that purpose. The crown mouldings are rather more than one-fourth of the height of the cornice, and consist of a carved bead, an ovolo, also carved, and a crowning fillet. This is a general description of the composition of the Ionic order of the Erechtheium on the Acropolis at Athens, the most perfect example left to us, pro¬ duced probably about 420 years R.C., the great epoch of Athenian art.—The Roman Ionic. This order in the hands of the Romans suffered great debase¬ ment. The angular volute of the Greeks was a clumsy enough expedient to get rid of a difficulty which need never have arisen if the order had been used either in antis or attached ; but the Romans, not content with this, made in many cases volutes * at all the four angles of the column, and then curved j all the sides of the abacus. The Ionic, as used by I the Italian architects, varied much in proportions ; J but the following are those given by Sir William j Chambers, in diameters and minutes, or sixtieths I of a diameter:—The column is nine diameters high, ! of which the base occupies thirty, and the capital I twenty-one minutes. The architrave, divided into I two fascias, is forty and a half minutes, the frieze l also forty and a half minutes, and the cornice fifty-four minutes. The separation between the I fascias of the architrave is made by a fillet and ' carved ovolo. The architrave mouldings consist of a carved ogee and a fillet. The frieze is plain. I It is surmounted by an ogee serving as a bed¬ moulding to a dentil band equal in depth to rather i more than a fifth of the cornice. Over this is an ovolo, serving qg the bed-moulding of the corona; then the corona, in height equal to nearly a fifth of the cornice ; and then the crowning mouldings, consisting of an ogee, a fillet, and a cyma recta. When a pedestal is used, the same authority makes it two diameters and six-tenths in height. IRON.—To find the weight of.—1. Wrought iron close hammered: find the number of cubic inches contained in the mass, multiply this by 28, cut off two figures to the right hand, and the re¬ mainder is lbs. 2. Cast iron : proceed as above, but multiply by 26 instead of 28. IRON TIE-RODS.—Dimensions of, for various spans. See Tie-rod. ISLE, Ile, a spelling formerly incorrectly used, instead of aile or aisle (which see). ISODOMON, Isodomdm.—I n Grecian archi¬ tecture, a species of walling in which the courses were of equal thickness and equal lengths. ISOMETRICAL PROJECTION INDEX AND GLOSSARY. KYANIZE ISOMETRICAL PROJECTION, P . 242. ITALIAN ARCHITECTURE.—Under this term are Comprehended the three great architec¬ tural schools of Italy—the Florentine, the Roman, and the Venetian. The architecture of Florence is best displayed in its palaces. In the facades of these, columns are used only as ornamental acces¬ sories, and however many the horizontal divisions or stories, the reigning cornice is proportioned to the whole height of the building, considered as an order, and is in general boldly pronounced and richly decorated. This is the severest of the Italian schools, and the exteriors of these palaces have a solidity, monotony, and solemnity which would make them appear as fortified places, if it were not for their richly-ornamented cornices. In the Roman school the architecture is less massive; columns are introduced freely, and grandeur of effect without severity is studied. The Venetian school is char¬ acterized by lightness and elegance, and the free use of columns, pilasters, and arcades. J. JACK-ARCH.—An arch of a brick in thick¬ ness. JACK-PLANE.—One of the bench-planes. It is about eighteen inches long, and is used in reduc¬ ing inequalities in the timber preparatory to the use of the trying-plane. JACK-RAFTER. See Jack timbers, andp. 91, voce Hip-roof. JACK-RIB. See Jack-timbers. JACK-TIMBER'S.—Those timbers in a series which, being intercepted by some other piece, are shorter than the rest. Thus, in a hipped-roof, each rafter which is shorter than the side-rafters is a jack-rafter. 1 AM B-LINING S.— The linings of the vertical sides of a doorway. JAMB-POSTS.—The upright timbers on each side of a doorway, called also priclc-posts. See p. JAMB-STONES. — Those employed in con¬ structing the vertical sides of an opening. JAMBS.—The vertical sides of any aperture, such as a door, a window, or chimney. JERKIN-HEAD.—The end of a roof not hip- j ped down to the level of the side walls, the gable j being carried up higher than those walls. A trun- I cated hipped roof. JESTING-BEAM. — A beam introduced for j the sake of appearance and not for use. J ETTY. — A projecting portion of a building. JIB-DOOR. — A door with its surface in the i plane of the wall in which it is set. Jib-doors are intended to be concealed, and therefore they have no architraves or finishings round them, and the plinth and dado are carried across them. See p. 1S7 and Plate LXXIII. JOGGLES, or Joggle-joints. — In architec¬ ture, the joints of stones or other bodies, so con- \ structed and fitted together, as to prevent them from sliding past each other by any foree acting in a direction perpendicular to tlic pressure by which | mm E [3J3 : a a a, Jojiglejoints. u The but JogRle. they are thus held together. In masonry, this term is applied to almost every sort of jointing in which J one piece of stone is let or fitted into another, so as ! to prevent all sliding on the joints. In carpentry, the struts of a roof are said to be joggled into the J truss-posts and into the rafters. JOINER.—-Materials used by the, p. 182. JOINERY.—The art or practice of dressing, framing, joining, and fixing wood-work, for the in- | ternal and external finishings of houses. See De finition. p. 182. JOINT.—1. In architecture, the surface of se¬ paration between two bodies that are brought into contact, and held firmly together by means of ce- ■ ment, mortar, &c., or by a superincumbent weight. ; I The nearer the surfaces of separation approach each other, the more perfect the joint, but in masonry the contact cannot be made very close on account of the coarseness of the cement.— 2. In carpentry and joinery, the place where one board or member is connected with another. Joints receive various names, according to their fonns and uses. Pieces of timber are framed and joined to one another most generally by mortises and tenons, of which there are several kinds, and by iron straps and bolts. When it is required to join two pieces of timber so as to make a beam of a given length, and equal in strength to one whole piece of the same dimensions and length, this is done by scarfing. — A longitudinal joint, one in which the common seam runs parallel with the fibres of both.— Abutting or butt joint, one in which the plane of the joint is at right angles to the fibres, and the fibres of both pieces in the same straight line.— Square joint, one in which the plane j of the joint is at right angles to the fibres of one j piece, and parallel to those of the other.— Bcvel- ! joint, one in which the plane of the joint is parallel I to the fibres of one piece, and oblique to those of ' the other.— Mitre joint, one in which the plane of | the joint makes oblique angles with both pieces. ! — Dovetail-joint. Bee Dovetail, also Mortise, Tenon, and Scarfing, and p. 146, 182. JOINTER. —The largest jilane used in straight¬ ening the edges of boards to be jointed together. In bricklaying, a crooked piece of iron bent in two opposite directions, and used for drawing, by the aid "I the jointing-rule, thehorizontaland vertical joints of the work. JOIMTNG-RULE.—A straight-edge used by bricklayers for guiding the jointer in drawing in the joints of brick-work. JOINTS AND STRAPS, p. 146, 182. JOISTS.—The pieces of timber to which the boards ol a floor, or the laths of a ceiling are nailed, and which rest on the walls or on girders, and some¬ times on both. They are laid horizontally, in pa¬ rallel equidistant rows. They are of a rectangular ! form, and placed with their edges uppermost, as the lateral strength of a horizontal rectangular beam t° resist a force acting upon it is proportional to the breadth of the transverse section multiplied into the square of the depth. Flooring with only one i tier of joists is termed single-flooring, and when two tiers are used, it is termed double-flooring. — Trim.- [ ming-jnists, two joists, into which each end of a small beam, called a trimmer, is framed. See Trimmer. — Binding joists, or binders, in a double flnor, are those which form the principal support of the floor, and run from wall to wall.— Bridging- joists, those which are bridged on to the binding- : joists, and carry the floor: the}- are laid across the binding-joists.— Ceiling joists, cross-pieces fixed to the binding-joists underneath, to sustain the lath and plaster. See Floors, p. 150. JUBE.—The rood-loft, or gallery into the choir. JUFFERS.—Pieces of timber fouror five inches square in section. JUGLANS ALBA.—White walnut or hick¬ ory, p. 111. JUGLANS NIGRA.— Black or brown wal¬ nut. Seep. 111. JUMP.—An abrupt rise from a level course. JUMPER.— A name given by masons ami miners to a long iron chisel used in boring shot- holes, for blasting large masses of stone, by which they may be split into smaller ones. JUTTY. — A projection in a building. The same as jetty. K. KEEP.—The stronghold of an ancient castle. KEEPING the Perpendicular. — Causing the vertical joints in brick-work to recur in the same straight line in each alternate course. KERB or Kirb plate. See Curb-plate. IvERF.—The channel or way made through wood by a saw. KERNEL.—The same as crenelle (which see). EE\ .—A name given to all fixing wedges. KEY-PILE.—The centre pile plank of one of the divisions of sheeting piles contained between two gauge piles of a cofferdam, or similar work. It is made of a wedge form, narrowest at the bottom, and when driven, keys or wedges the whole to¬ gether. 268 KEYED DADO.—In architecture, a dado that is secured from warping, by having bars of wood grooved into it across the grain at the back. KEYHOLE SAW.—A saw used for cutting , out sharp curves, such as keyholes require, whence its name. It consists of a narrow blade, thickest on the cutting or serrated edge, its teeth having no twist or set, and a long handle perforated from end to end, into which the blade is thrust to a greater or lesser extent, according to the nature of the work to be performed. The handle is provided with a pad and screw for fastening the blade when it is adjusted. It is also called a turning-saw. KEYING a mortise joint, p. 147. KEYS.—-In naked flooring, pieces of timber fixed in between the joists by mortise and tenon. Y hen tbese are fastened with their ends projecting against the sides of the joists, they are called ztrut- ting-picces. KEY STONE.—The highest central stone of an arch ; that placed on the top or vertex, to bind the two sweeps together. In some cases the keystone projects from the face, and is moulded and enriched. In vaulted Gothic roofs, the keystones are usually ornamented with a boss or pendant. See first woodcut under Arch. KILLESSE, Cullis, Coulisse. — A gutter, groove, or channel. The term is corruptly .applied in some districts to a hipped roof; as a kil/cssed or cullidged roof. A dormer-window, too, is some¬ times called a killcsscd or cullidged window. KING-PIECE.—Another and more appropriate name for king-post. KING-POST. — The post which, in a truss, extends between the apex of two inclined pieces and the tie-beam, which unites their lower ends as in a king-post roof. See p. 148, and woodcut, Roof. KING-TABLE. — In medireval architecture, conjectured to be the string-course, with ball and flower ornaments, usual under parapets. KIOSK.—A Turkish word signifying an open pavilion or summer-house supported by pillars. KIRB-PLATE. See Curb-plate. KIRB-ROOF. Sec Curb roof. KNEE.—1. A piece of timber somewhat in the form of the human knee when bent.—2. A part of the back of a hand-rail, which is of a convex form, the reverse of a ramp, which is concave.— Knee- piece or knee-rafter, an angular piece of timber used to strengthen the joining of two pieces of timber in a roof. KNOCKER.—A kind of hammer fastened to a door, to lie used in seeking for admittance. The knockers of medimval buildings present many Knocker, Village of Street, Somersetshire. beautiful specimens of forged iron-work. They are often highly decorated, and sometimes assume very quaint and fantastic forms. KNOT, OR Knob.—A bunch of leaves, flowers, or similar ornament, as the bosses at the ends of labels, the intersections of ribs, and the bunches of foliage in capitals. KNOTS in Wood.— Some kinds render wood unfit for the carpenter; some kinds are not preju¬ dicial. See p. 98. KNOTTING.—A process to prevent the knots of wood from appearing, by laying on a size com¬ posed of red led, white lead, and oil, or a coat of gold size, as the preliminary process of painting. KNOTTY and Cross-grained Wood.— Unfit for ordinary carpentry works. See p. 98. KNOWLEDGE OF WOODS.—Physiological notions, p. 93. KNUCKLE.—A joint of a cylindrical form, with a pin as axis, sucli as that by which the straps of a hinge are held together. See p. 190, Hinging; and Plates LXXXIV. —LXXXYI. KYANIZE, v .—To steep in a solution of cor¬ rosive sublimate, as timber, to preserve it from the dry-rot. LABEL INDEX AND GLOSS All Y. LONG PLANE L. LABEL.—A projecting moulding over a door, window, or other opening; called also dripstone, weather-moulding, and, when in the interior, hood- moulding. LABOUR - SAYING MACHINES.—Sketch of the introduction of, p. 191- LABOUR - SAVING MACHINES. — Sir Samuel Bentham’s inventions in. p. 191. LABOUR-SAVING M A CHINES.—Ameri¬ can circular saw bench, and Furness’ planing- machine, Plate LXXXVII. and p. 192; Fur¬ ness’ patent mortising and tenoning machines, Plate LXXXVIIand p. 193. LABYRINTH FRET.—A fret with many in¬ volved turnings. LACUNA RIA on Lacunars. — Panels or coffers in a ceiling. LADIES.—In slating, small slates measuring about 15 inches long and 8 inches wide. LADY-CHAPEL.—A chapel dedicated to the Virgin Mary, frequently attached to large churches. It was variously placed', but generally to the east¬ ward of the high altar. In churches of an earlier date than the thirteenth century, the lady-chapel is generally an additional building. The term is of modern application. See woodcut, Cathe- -M: !: DUAL. LAGGINS, Lagging. —The planking laid on the ribs of the centering of a tunnel or bridge, to carry the brick or stone work. See p. 171. LAMINATED ARCHES.—Arches composed of thin plates of wood fastened together. See p. 141. LANCET ARCH.—One whose head is shaped like the point of a lancet. See p. 29. LANCET WINDOW. — A window with a lancet arch. This kind of window is characteristic. i ’ of the Early English style of architecture. Lancet windows have no tracery, and were often double or triple, and sometimes five were placed together. Though separated on the outside, lancet windows which are placed together are in the interior com¬ bined into one design by a wide splaying of the openings, and thus form, to a certain extent, a com¬ pound window. LANDING. — The Lancot Window,’ Coinberton. first part of a floor at the end of a flight of steps. Also, a resting-place be¬ tween flights. See p. 196. LANTERN.— 1. A drum-shaped erection, on the top of a dome, or the roof of an apartment, to give light, and serve as a sort of crowning to the fabric. It may be either circular, square, elliptical, or polygonal (see Louvre). Also, the lower part of a tower placed at the junction of the cross in a cathe¬ dral or large church, having windows on all sides.— 2. A square cage of carpentry placed over the ridge of a corridor or gallery, between two rows of shops, to illuminate them, as in many public arcades. LAP, v. —To lap boards is to lay one partly over the other. L AQU EAR.—The same as lacunaria or lacunar. LARCH.—Description and uses of, see p. 119. LARMIER.—The corona or drip of a cornice; corruptly lorimcr. LATH.—1. A thin narrow board or slip of wood nailed to the rafters of a building, to support the tiles or covering.—2. A thin narrow slip of wood nailed to the studs, to support the plastering; also, a thin cleft piece of wood used in slating, tiling, and plastering. There are two sorts of laths, single and double ; the former being barely a quarter of an inch, while the latter are three-eighths of an inch thick. Pantile laths are long square pieces of fir, on which the pantiles hang.— Lath floated and set fair, three-coat plasterer’s work, in which the first is called pricking up, the second floating, the third or finishing is done with fine stuff.— Lath laid and set, two-coated plasterer’s work ; except that the first is called laying, and is executed without scratching, unless with a broom.— Lath plastered, set, and coloured, the same as lath laid, set, and coloured.— Lath pricked up, floated, and set for paper; the same as lath floated and set fair. LATTICE.—Any work of wood or iron, made by crossing laths, rods, or bars, and forming open chequered or reticulated work. — Lattice, or lattice window, a window made of laths or strips c>f iron which cross one another like net-work, so as to leave open interstices. It is only used when air rather LatticeWork, Cairo. than light is to be admitted. Such windows are common in hot countries, and in these the lattice work is frequently arranged in handsome devices. LATTICE BRIDGE, p. 109. LAYER BOARDS.—The boards for sustain¬ ing the lead of gutters. LEAD, in excavator’s work, is the distance to which the materials have to be removed. LEAD NAILS.—Nails used to fasten lead, leather, canvas, &c., to wood. They are of the same form as clout nails, but are covered with lead or solder. LEAF.—The side of a double door. LEAF BRIDGE.—A bridge consisting of two opening leaves. LEAN TO.— A building whose rafters pitch against or lean on to another building, or against a wall. LEAR BOARD. — The same as layer hoard. LECTERN OR Lf.ttern.— The reading desk in the choir of ancient churches and chapels. It was generally of brass, and sometimes elaborately carved. Its use has been almost entirely superseded in Eng¬ land by the modern reading desk, or rather reading pew. LEDGE.—A surface projecting horizontally, or slightly inclined to the horizon; a string course; also, the side of a rebate, against which a door or shutter is stopped, or a projecting fillet serving the same purpose as a door stop, or the fillet which confines a window frame in its place. LEDGE I > DOORS.—Doors .formed of deals, with cross pieces on the back to strengthen them. See p. 1S6. LEDGERS.—The horizontal timbers used in scaffolding. LEDGMENT.—1. A laying out; the develop¬ ment of the surface of any solid on a plane, so that its dimensions may be readily obtained.—2. The same as ledge; a string course or horizontal mould¬ ing. LEDGMENT TABLE.—In mediaeval archi¬ tecture, a name given to any of the tables of the base, except the ground table. LENGTHENING BEAMS.—By scarfing and by fishing, p. 148. LEWIS, Lewisson (Fr. louve, louveceau).- An instrument of iron, used in raising large stones to the upper part of a building, which operates by the dovetailing of one of its ends into an open¬ ing in the stone. It con¬ sists of two moveable parts a a, perforated at their heads to admit the pin or bolt c d. These are inserted, by hand, into the cavity formed in the stone; and between them the part b is introduced, which pushes their points out to the sides of the stone, thus filling the ca¬ vity; e a half-ring bolt, with a perforation at each Lewis, end; to this the tackle I above is attached by a hook. The fastening-pin passes horizontally through all the holes, entering at the right side d, and forelocking on the other end c. LICH - GATE — A shed over the gate of a i churchyard to rest the corpse under; called also a corpse-gate. LIME TREE, Linden tree. —Description and uses of. p. 113. LINE op Lines on the Sector, p. 37. LINE op Chords on the Sector. —Construc¬ tion and use of, p. 37. LINE of Polygons on tpe Sector. — Con¬ struction and use of, p. 38. LINE op Projection. —In perspective, the intersection of the plane of the picture with the ground plane. LINE of Nosings, p. 196. LINE op Secants on the Sector. —Construc¬ tion and use of, p. 39. LINE of Sines on the Sector. —Construc¬ tion and use of, p. 39. 2G9 LINE of Tangents on the Sector.— Con¬ struction and use of, p. 39. LINEAL MEASURES. See Weights and Measures. LINEAR PERSPECTIVE.—That branch of perspective which regards only the positions, mag¬ nitudes, and forms of the objects delineated. LINING.—In architecture, the covering of the surface of any body with a thinner substance. The term is only applied to coverings in the interior of a building, coverings on the exterior being properly termed casings. Lining of boxings for window shutters, are the pieces of framework into which the shutters are folded back. Linings of a door are the coverings of the jambs and soffit of the aperture.— Lining out stuff, drawing lines on a piece of board or plank, so as to cut it into thinner pieces. LINTEL.—A horizontal piece of timber, iron, or stone placed over an opening. See p. 156. LIST. Listed.— A fillet moulding. LOBBY.—1. A small hall or waiting-room ; also, an inclosed space surrounding or communi¬ cating with one or more apartments; such as the boxes of a theatre. When the entrance to a prin¬ cipal app.rtment is through another apartment, the dimensions of which, especially in width, do not entitle it to be called a vestibule or antechamber, it is called a lobby.—2. A small apartment taken from a hall or entry. LOCK.—1. Lock, in its primary sense, is any¬ thing that fastens; but in the art of construction the word is appropriated to an instrument composed of springs, wards, and bolts of iron or steel, used to fasten doors, drawers, chests, &c. Locks on outer doors are called stock locks; those on chamber doors, spring locks; and such as are hidden in the thick¬ ness of the doors to which they are applied, are called mortise locks. —2. A basin or chamber in a canal, or at the entrance to a dock. It has gates at each end, which may be opened or shut at plea¬ sure. By means of such locks vessels are trans¬ ferred from a higher to a lower level, or from a lower to a higher. Whenever a canal changes its level on account of an ascent or descent of the ground through which it passes, the place where the change takes place is commanded by a lock. LOCK-CHAMBER.—In canals, the area of a lock inclosed by the side walls and gates. LOCK-GATE.—The gate of a lock provided with paddles, &c. See Dock-gates, p. 177, and Plates LXI. and LXII. LOCK-PADDLE.—The sluice in a lock which serves to fill or empty it. LOCK-PIT.—The excavated area of a lock. LOCK-RAIL.—The middle rail of a door, to which the lock or fastening is fixed. See p. 186. LOCK-SILL.—An angular piece of timber at the bottom of a lock, against which the gates shut. LOCK-WEAR.—A paddle-wear, in canals, an over-fall behind the upper gates, by which the waste water of the upper pond is let down through the paddle-holes into the chamber of the lock. LOCKER.—1. A small cupboard.—2. A small closet or recess, frequently observed near an altar in Catholic churches, and intended as a depository for the sacred vessels, water, oil, &c. LOCUTORY.—An apartment in a monastery, in which the monks were allowed to converse when silence was enjoined elsewhere. LODGE.—1. A small house in a park, forest, or domain, subordinate to the mansion ; a tempo¬ rary habitation; a hut.—2. A small house or cottage appended to a mansion, and situated at the gate of the avenue leading to the mansion; as a porter’s lodge. LOFT.—In modem usage thiR term is restricted to the place immediately under the roof of a build¬ ing, when not used as an abode; as liay-lofl. The gallery of a church is sometimes termed the loft in Scotland. LOGARITHMIC LINES on the Sector.— Construction and use of, p. 40. LOMBARD ARCHITECTURE. —A name given to the round-arched Gothic of Italy, intro¬ duced by the conquering Goths and Ostrogoths, which superseded the Romanesque, and reigned from the eighth to the twelfth century. “At first,” says Mr. Fergusson, “when the barbarians were few, and the Roman influence still strong, they of course were forced to adopt the style of their pre¬ decessors, and to employ Italian builders to execute for them works they themselves were incapable of producing; but as they became stronger they threw off the trammels of an art with which they had no sympathy, to adopt one which expressed their own feelings; and although the old influence still lay beneath, and occasionally even came to the surface, the art was Gothic in all essentials, and remained so during nearly the whole of the middle ages. LONG PLANE, or Jointer. See Jointer. LOOP-HOLE INDEX AND GLOSSARY. METRE LOOP-HOLE.—A narrow opening in a wall. LOUIS-QUATORZE, Style of. — A meretri¬ cious style of ornament anil ornamental decoration developed in France during the reign of Louis XIV. The great medium of this style was gilt stucco¬ work, and its most striking characteristics are an infinite play of light and shade, and a certain dis¬ regard of symmetry of parts, and of symmetrical Louis-Quatorze Ornament. arrangement. The characteristic details are the scroll and shell. The classical ornaments, and all the elements of the Cinque-cento, from which the Louis-Quatorze or Louis XIV. proceeded, are ad¬ mitted, under peculiar treatment, or as accessories; the panels are formed by chains of scrolls, the con¬ cave and convex alternately; some clothed with an acanthus foliation, others plain. LOUIS-QUINZE, Style of.—A variety of the Louis-Quatorze style of ornament, which pre¬ vailed in France during the reign of Louis XV., in which the want of symmetry in the details, and of Louis-Quinzo Ornament. symmetrical arrangement, which characterize the Louis XIV. style, are carried to an extreme. An utter disregard of symmetry, a want of attention to masses, and an elongated treatment of the foliations of the scroll, together with a species of crimped conventional shell-work, are characteristics of this style. LOUVRE, Loover, Lover, or Lantern.— i A dome or turret rising out of the roof of the hall I in our ancient domestic edifices ; formerly open at I the sides, but now generally glazed. They were originally intended to allow the smoke to escape, Louvre, Abbot's Kitchen, Glastonbury. when the fire was kindled on dogs in the middle of the room. The open windows in church-towers are I called louvre-windows, and the boards or bars which j are placed across them to exclude the rain, are called ' louvre-boards, corruptly luf,r-boards. LOZENGE MOULDING, or Lozenge Fret. An ornament used in Norman architecture, pre¬ senting the appearance of diagonal ribs inclosing j diamond-shaped panels. LUFFER - BOARDING. See Louvre. LUMBER.—In America, timber sawed or split ' for use. 1 I LUNETTE.—An aperture in a concave ceiling for the admission of light. LYING PANELS.—Those in which the fibres | of the wood lie in a horizontal direction. LYSIS.—A plinth or step above the cornice of the podium which surrounds the stylobate. M. M ROOF.—A kind of roof formed by the junc¬ tion of two common roofs, with a valley between them. MACHICOLATIONS.—In castellated archi¬ tecture, openings made through the roofs of portals to the floor above, but more generally openings , made in the floor of projecting galleries for the pur¬ pose of defence, by pouring through I them boiling pitch, molten lead, &c., upon the besiegers. ~ In the j latter case they are formed by the I parapet or breast-work B being set I out on corbels D, beyond the face of | the wall c. The spaces E between ' the corbels, which are open through¬ out, are the machicolations. From | its striking appearance, the cor- ; belled gallery with its parapet, was i frequently used when machicola- I tions were not required for the pur- : poses of defence, and the so-called apertures were Machicolations, Herstmonceux Cootie. omitted. Machicolations do not appear to have been used earlier than the twelfth century. MACHINES with Revolving Cutters. — General remarks on, p. 1D4. MAHOGANY, p. 115. MAHOGANY TREE.—Description and uses of, p. 115. MAIN BRACES, p. 159. MAIN COUPLE. -A name given to the ; trussed principal of a roof. MALM BRICKS.—Those composed of clay, I sand, and comminuted chalk. They burn to a pale i brown colour, more or less inclined to yellow, which . is an indication of magnesia. MANAGEMENT of Timber after it is cut, p. 100. MANNER of Dividing a Gothic vault into ! compartments, p. 82. MANNER of Dividing conical, spherical, i and other vaults, into compartments and caissons, p. 82. MANSARD ROOF.— A roof formed with an ' upper and under set of rafters on each side, the under set less and the upper set mure inclined to the horizon. It is called a mansard roof from the J name of the architect, Francois Mansard, who re- ! vived its use in France. It is called also a curb- 1 roof, from the French courber, to bend, descriptive of the double inclination of its sides. See Man¬ sard Roof, p. 140, and Plate XXVII. MANTELPIECE, Mantlepiece.— The orna- ! mental dressing or front to the mantle-tree. MANTLE - SHELF.—The woi •k over a fire¬ place in front of the chimney. MANTLE - TREE. —- The lintel of a fire- I place. MAPLE.—Description and uses of, p. 113. MARBLE.—The popular name of any species of calcareous stone or mineral, of a compact tex¬ ture, and of a beautiful appearance, susceptible of a good polish. Marble is limestone, or a stone which may be calcined to lime, a carbonate of lime; but limestone is a more general name, comprehend¬ ing the calcareous stones of an inferior texture, as well as those which admit a fine polish. The term j is limited by mineralogists and geologists to the ' 270 several varieties of carbonate of lime, which have more or less of a granular and crystalline texture. | In sculpture, the term is applied to several compact or granular kinds of stone, susceptible of a very fine polish. The varieties of marble are exceedingly numerous, and greatly diversified in colour. In modern times, the quarries of Carrara, in Italy, almost supply the world with white marble. Of variegated marbles, there are many sorts found in this country of singular beauty. Marble is much used for statues, busts, pillars, chimney-pieces, monuments, &c. MARGIN DRAl GHT.—In stone cutting, a line of chiselling along the edge of a stone. MARGIN of a Course. —In slating, that part of a course of slating which is not covered by the next superior course. MARGINS, OR Margents.— The flat parts of the styles and rails of framed or panelled work. Doors which are made in two leaves are called double- margined doors, in consequence of the styles bein'-- repeated in the centre, as are also those in one lea£ made in imitation of a two-leaved door. MARIGOLD WINDOW.—The same as Catherine wheel window and rose window. See Rose Window. MARKET- CROSS.-—A cross set up where a market is held. Most market towns in England and Scotland had, in early times, one of these. The primitive form was that of simple shaft and cross stone, but they afterwards were constructed in a much more elaborate manner, so as frequently to lose the cruciform structure as a distinguishing char¬ acteristic. MARL BRICKS.—Fine bricks used for gauged arches and the fronts of buildings. MARQUETRY- — Inlaid work consisting of thin pieces of wood of different colours, arranged on a ground so as to form various figures. Used in cabinet work. The term is also used as synonymous with mosaic. MASK. — A piece of sculpture representing some grotesque form, to fill and adorn vacant places, as in friezes, panels of doors, keys of arches, &c. MASON.—One who prepares and sets stones ; a builder in stone. MASONRY.—-The art of shaping, arranging, and uniting stones together to form walls and other parts of buildings. MASTIC.—A kind of cement made by mixing litharge, or the red protoxide of lead, with pulver¬ ized calcareous stones, sand, and linseed oil. The proportions of the ingredients vary. MATCH PLANES.—Planes in pairs, used in joining boards by grooving and tonguing, one plane being used to form the groove, and the other to form the tongue. MAUSOLEUM—In modern times, a sepul¬ chral chapel, or edifice erected for the reception of a monument, or to contain tombs. MEANDER.—An ornament composed of two =QWSJrSJMf^ Meander. or more fillet mouldings intertwined in various ways; a fret. MEDALLION.—A circular, oval, or some¬ times square tablet, bearing on it objects represented in relief, or an inscription. MEDIAEVAL ARCHITECTURE.—A term properly applied to denote the architecture which prevailed throughout the middle ages, or from the fifth to the fifteenth century. It thus comprises the Romanesque, the Byzantine, the Saracenic, the Lombard, and other styles, besides the Norman and the Gothic. In popular language, however, it is restricted to the Norman and early Gothic styles, which prevailed in Great Britain and on the Con¬ tinent from the eleventh to the fourteenth century. MEMBER.—Any subordinate part of a build¬ ing, order, or composition, as a frieze or cornice; and any subordinate part of these, as a corona, a cymatium, a fillet. MEM EL TIMBER, Crown Memel, Best Middling, Second Middling, or Brack. Purus SYLVESTR1S, p. 116, 117. MERLON.—The plain parts of an embattled parapet, between the crenelles or embrasures. See woodcut. Battlement. METHODS of Piling newly-felled timber, p. 100. METOPE.—The space between the triglyphs of the Doric frieze. METRE.—A French measure equal to 39'37 English inches. INDEX AND GLOSSARY. MEZZANINE M EZZ AN IX K — A story of small height intro- | duced between two higher ones. MEZZO - RELIEVO. — Middle relief. See Demi-Relievo. MIDDLE PANEL, p. 18G. MIDDLE POST.—The same as king-post. The rail of a door level with the hand, and on which the j lock is generally fixed; whence it is usually termed j the lock-rail. MILE.—A measure of distance. The English i mile = 5280 feet; the geographical or nautical, GO 75-6 feet; ratio of geographical to English, | 1*15068 to l. MILLED LEAD.—Lead rolled out into sheets by machinery. MILLIAItY COLUMN.—A column set up to mark distances ; a milestone. See Column. MINARET.—A slender, lofty turret rising by different stages or stories, surrounded by one or more projecting balconies, common in mosques in Mahometan countries. The priests from the bal¬ conies summon the people to prayers at stated times of the day ; so that they answer the purpose of bel¬ fries in Christian churches. MINSTER.—A monastery ; an ecclesiastical convent or fraternity; but it is said originally to have been the church of a monastery; a cathedral church. Both in Germany and England this title is given to several large cathedrals; as York minster, the minster of Strasburg, &c. It is also found in the names of several places which owe their origin to a monastery ; as, Westminster, Leominster, &c. MITRE.—The line fonned by the meeting of surfaces or solids at an angle. It is commonly applied, however, when the objects meet in a right angle, and the mitre-line bisecting this makes an angle of 45° with both. MITliE-BOX.—A box or trough with three sides, used for forming mitre-joints. It has cuts in its vertical sides, the plane passing through which crosses the box at an angle of 45°. The piece of wood to be mitred is laid in the box, and the saw being worked through the guide-cuts, forms the mitre-joint in the wood. MITRE - SQUARE. — A bevel with a fixed blade, for striking an angle of 45° on a piece of stuff, in order to its being mitred. MODILLION.—A block carved into the form of an enriched bracket, used under the corona of the Corinthian and Composite entablatures. Mo- dillions less ornate are occasionally used in the Ionic Modillion. entablature. The derivation of the word is pro¬ bably from modulus (a measure of proportion), ex¬ pressive of the arrangement of the brackets at regu¬ lated distances. MODULAR PROPORTION.—That which is j regulated by the module. MODULE, Modulus. —A measure which may be taken at pleasure to regulate the proportions of an order, or the disposition of the whole building. The diameter or semi-diameter of the column at the bottom of the shaft has usually been selected by architects as their module, and this they subdivide into parts or minutes, the diameter generally into sixty, and the semi-diameter into thirty minutes. Some architects make no certain or stated divisions of the module, but divide it into as many parts as may be deemed requisite. MONASTERY.—A house of religious retire¬ ment or seclusion from ordinary temporal concerns, whether an abbey, a priory, a nunnery, or a con¬ vent. The word is usually applied to the houses of monks. MONIAL, or Montcale. See Mullion. MONKEY. — The ram or weight of a pile¬ driving engine. See Fistuca. MONOCOTYLEDON! JUS PLANTS, p. 93. MONOLITHIC.—Formed of a single stone ; as a monolithic obelisk. MONOPTEROS. —A term used by Vitruvius to denote a temple composed of columns arranged in a circle, and supporting a conical roof or a tholus, but having no cella. Such a temple, however, would be more correctly denominated cyclostylar. MONOTRIGLYPH.—The intercolumniation ! of the Grecian Doric most usually followed. It is that in which space is left for the insertion of only one triglypli between those immediately over two contiguous columns. MONT ANTS, Mountings, Muntins. —The intermediate styles in a piece of framing, which are tenoned into the rails. Seep. 186. MOORISH, or Moresque Architecture. See Saracenic Architecture. MORTAR.—A mixture of lime and sand with water, used as a cement for uniting stones and bricks in walls. The proportions vary from 1^ part of sand to 1 part of lime, to 4 and 5 parts of sand to 1 of lime. When limestones contain consider¬ able portions of silica and alumina, they form what is termed hydraulic lime, and the mortars made with them are called hydraulic mortars, which are used for building piers or walls under water, or ex¬ posed to it, because they soon harden in such situa¬ tions, and resist the action of the water. MORTISE, Mortice. —A cavity cut in a piece I of wood or other material, to receive a correspond- I ing projecting piece called a tenon, formed on another | piece of wood, &c., in order to fix the two together ! at a given angle. The sides of the mortise are four ; planes, generally at right angles to each other and ' to the surface where the cavity is made. The junc- ! lion of two pieces in this manner is termed a mor¬ tise joint. See p. 147, Joints, and p. 182. MORTISE LOCK.—A lock made to fit into a i mortise cut in the style and rail of a door to re¬ ceive it. MOSAIC WORK is an assemblage of little pieces of glass, marble, precious stones, &c., of va¬ rious colours, cut square, and cemented on a ground j of stucco, in such a manner as to imitate the colours I and gradations of painting. This kind of work was used in ancient times both for pavements and orna- I menting walls. In recent times, two kinds of mosaic 1 are particularly famous—the Roman .and the Flo¬ rentine. In the former, the pictures are formed by ; joining very small pieces of stone. In the Floren- | tine style larger pieces are used. MOUCIIARABY.—A balcony with a parapet ' and machicolations projected over a gate to defend the entrance. The parapet may be either embattled or plain. MOULD-STONE.—The jamb stone of an aperture. MOULDED NOSING, p. 196. 271 NATURAL BEDS OF STONE MOULDING, or Forming the Surface of wood into various square and curved contours,p. 184. MOULDING PLANES.—Joiners’ planes used in forming the contours of mouldings. MOULDINGS, Gothtc. —Examples of, p. 180. MOULDINGS, Greek and Roman. —Mode of describing the various, p. 178. MOULDINGS. Planted or Laid in, p. 185. MOULDINGS' Stuck, p. 184. MULLION, Munnion, Monycat.e, Monial. —A vertical division between the lights of win¬ dows, screens, See., in Gothic architecture. Mul- lions are rarely found earlier than the early English style. Their mouldings are very various. Some¬ times the styles in wainscotting are called mullions. MUTULE.—An ornament in the Doric cor¬ nice, answering to the modillion in the Corinthian, but differing from it in form, being a square block, from which the guttse depend. NAIL.—A small pointed piece of metal, usu¬ ally with a head, to be driven into a board or other piece of timber, and serving to fasten it to other timber. The larger kinds of instruments of this } sort are called spikes; and a long, thin kind, with a fiattish head, is called a brad. Nails are exten- I sively used in building, and generally in the con- I structive arts. There are three leading distinctions . of iron nails, as respects the state of the metal from which they are prepared, namely, wrought or forged | iron nails, cut or pressed iron nails, and cast iron nails. Of the wrought or forged nails there are | about 300 sorts, which receive different names, ex¬ pressing for the most part the uses to which they are applied, as hurdle, pail, deck, scupper, mop, &c. , Some are distinguished by names expressive of their I form: thus, rose, clasp, diamond, Ac., indicate the ! form of their heads, and flat, sharp, spear, &c., ! their points. The thickness of any specified form is expressed by the terms fine, bastard, strong. Nails are made both by hand and by machinery. ' NAIL-HEAD MOULDING.—An ornament • common in Norman architecture. It is so named from being formed by a series of diamond-pointed knobs, resembling the heads of nails. NAILS, Adhesion of, in Wood. See Adhe- : sive Force of Nails and Screws. NAKED.—Any continuous surface, as opposed i to the ornaments and projections which arise from it. Thus the naked of a wall is the continuous sur¬ face of the wall, as opposed to its projections or | ornamented parts. NAKED FLOORING.—The supporting tim- ! bers on which the floor-boards are laid. See p. 150. NAOS.—The body of an ancient temple, some¬ times, but erroneously, applied to the cella or inte¬ rior. The space in front of the temple was called I pronaos. NARTHEX.—The name of an inclosed space J in the ancient basilicas when used as Christian I churches, and also of an ante-temple or vestibule i without the church. To the narthex the cate- I chumens and penitents were admitted; and there ‘ appears to have been several such apartments in I each church, but nothing certain is known of their ' position. Narthex is frequently used as synonym¬ ous with porch and portico. I NARVA TIMBER. — Pinus sylvestris, p. 116, 117. NATTES.— A name given to an ornament used ! in the decoration of surfaces in the architecture of the twelfth century, from its resemblance to the interlaced withs of matting. NATURAL BEDS of Stone are the surfaces NAVE INDEX AND GLOSSARY. NOTCH in stratified rocks, in which the Inminse separate. As all stones of this kind exfoliate rapidly when these surfaces are exposed, it becomes necessary, in using them in building, to lay them on their natu¬ ral beds, or, in other words, so to place them in the wall that their exfoliating surfaces shall be horizon¬ tal, or at right angles to its face. The contrary use of the stone is described as setting on edge. NAVE,—The central avenue or middle part of a church, extending from the western porch to the transept, or to the choir or chancel, according to the nature and extent of the church. In the larger structures it has generally one or more aisles on each side, and sometimes a series of small chapels beyond these. In smaller buildings it is commonly without aisles. See woodcut, Cathedbal. NEBULE MOULDING. — A moulding whose edge takes the form of an undulating line. It is used in corbel-tables and archivolts. NECK, Necking, ok Hypotrachelium. —In architecture, the part which selves to connect a capital or head with its body or shaft; thus the neck of a capital is that part which lies between the j lowest moulding of the capital and the highest | moulding of the shaft. In the Grecian Doric it is 1 the space between the annulets and the channel, and in the Roman Doric it is the space between the annulets and the astragal. In the same way the I neck of a finial is the part in which the finial joins I the obelisk, and the channels, astragals, or other members which terminate the shaft or body, arc called the luck-mouldings. NECK-MOULDINGS See preceding word. NEEDLE.—A beam of timber supported on upright posts, used to carry a wall temporarily during alterations or repairs. NEEDLEWOH K.—A"term somethin ;s applied to the framework of timber, of which old houses are constructed. NERVURES, Nerves, or Branches.— The ribs which bound the sides of a groined compart¬ ment in a vaulted roof, as distinguished from the diagonal ribs. NEUTRAL AXIS.— That plane in abeam in which theoretically the tensile and compressive forces terminate, and in which the stress is there¬ fore nothing. See Strength and Strain op Ma¬ terials —Transverse Strain, p. ]26. NEWEL.—The upright cylinder or pillar, round which, in a winding staircase, the steps turn, and are supported from the bottom to the top. In stairs where the steps are pinned into the wall and there is no central pillar, the staircase is said to have an open newel. The newel is sometimes con¬ tinued through to the roof, and serves as a vaulting- shaft from which the ribs branch off in all directions. NEWEL STAIRS, p. 196. NICHE —A recess in a wall for the reception of a statue, a vase, or of some other ornament. In classic architecture, niches were generally semicir¬ cular in the plan, and terminated in a semi-dome at the top. They were sometimes, however, square in the plan, and sometimes also square-headed. I They were ornamented with pillars, architraves, consoles, and in other ways. In the architecture Niche, All-Souls' College, Oxford. of the middle ages niches were extensively used as decorations, and for the reception of statues. In the Norman style they were so shallow as to he little more than panels, and the figures were fre¬ quently carved on the back in alto-rilievo. In the early English style they become more deeply re¬ cessed, and are highly enriched, and in the Deco¬ rated style they become infinitely varied. They were chiefly semi-octagonal or semi-hexagonal in plan, and their heads were formed into groined vaults, with ribs, and bosses, anil pendants. They were projected on corbels, and adorned with pillars, buttresses, and mouldings of various kinds, and bad canopies added to them, sometimes flat and sometimes projecting in every variety of plan, and elaborately carved and enriched. In the Perpen¬ dicular style this variety and elaboration were con¬ tinued. NICHE, Spherical, on a circular plan, p. S3. N ICHE, Spherical, on a segmental plan, p.83. NICHE, on a semicircular plan, with a seg¬ mental elevation, p. S3. NICHE, segmental in plan and elevation, p. S4 NICHE, segmental in plan and elliptical in ele¬ vation, p. 84. NICHE, a, elliptical in plan and elevation, p. 84. NICHE, a, octagonal in plan, p. 84. NICHE, a semicircular, in a concave wall, p. 84. NICHE, a semicircular, in a convex wall, p. 85. NICHE, A.—To determine the shadow in the interior of, p. 22. NIDGING.—In masonry, nidging is a mode -if dressing used chiefly for granite, but sometimes also applied to other stones. It is thus performed. The face of the stone being prepared as for rubbing or tooling, the beds and joints are squared up, and a margin draught, about jfths of an inch wide, run round the face. Lines are then drawn around the margin, parallel with the beds and joints, and cut in by a sharp boaster, with light blows of the mal¬ let. The nidging then begins. The tool used for ; this is a hammer about 3 lbs. weight, having its ends formed like an axe. The axe-face of the hammer is j held crossing the stone at right angles to its length; i each blow makes a slight indentation by abrasion, and, by a succession of blows, the stone is furrowed all over with shallow furrows, and reduced to a ! uniform surface. No portion of stone should be left between the successive series of furrows. Nidg¬ ing, when well done, is a characteristic mode of working hard stones; but it is not proper to be used I for soft stones, as it makes their surface more liable to be acted on by the weather. NOGGING.—Brick-work carried up in panels between timber quarters. NOGGING PIECES. — Horizontal pieces | fitted in between and nailed to the quarters for | strengthening the briok-nogging. NOGS.—A north of England term for wood bricks or timber bricks. NONAGON.—A figure having nine sides and nine angles. To find the area of a nonagon. Mul¬ tiply the square of its side by GT818242. NONIUS. See Vernier. NORMAN ARCHITECTURE.—A stylo of architecture imported into England imroedfoely from Normandy, at the time of the Conquest,TTn. 1066. It continued in use till towards the end of the twelfth century, when it was superseded by the first of the Pointed or Gothic styles, the Early Eng¬ lish. The Norman is readily distinguished from the styles which succeeded to it by its general mas¬ sive character, round-headed doors and windows, and low square central tower. The doorways are I generally very highly enriched by a profusion of decorated mouldings, for the most part peculiar to I the style. The windows have no mullions, and in the early examples are quite plain. In later speci¬ mens they have frequently small shafts in the jambs, or are enriched with the zigzag moulding peculiar to the style. The piers which support the arches are in the earlier examples strikingly solid and massive, being merely plain square or circular masses of ! masonry, sometimes having capitals and bases, and j j sometimes merely an impost to relieve the outline, i The square piers were frequently recessed at the ! angles,, and in some cases had half pillars attached j to their sides; and the circular ones in some in- 272 stances had the plain surface relieved by lines cut in a lozenge or spiral form. As the style advanced, these solid piers were reduced to more moderate pro¬ portions of round or octagonal pillars, and in the time of the transition were frequently very tall and slender. The capitals of these piers and pillars are among the most important features of this style. Tim upper member or abacus is in general square, and its profile is also square, having its lower edge sloped or chamfered off. One of the earliest forms of the capital, and which, with various modifications, is found in all periods of the style, is what is called the cushion capital. (See woodcut under that term.) It is frequently divided into two or more parts, and is also sometimes enriched with sculpture of foliage and figures ; but under all these modifications it may still he taken as the primary form of the Nor¬ man capital. The arches were almost universally round-headed until the period of tlie transition, when the pointed form was used along with or fre- quently instead of it. The pointed arch must not be taken as a certain criterion of transition date, as we have examples of it combined with solid early Norman piers, as at Malmesbury Abbey; but these examples are rare, and the mixture of the two f rms may generally be taken as evidence of transition. The windows were universally round-headed, until the transition period. NORWAY SPRUCE.—Description and uses of, p. 117. NOBWAY TIMBER. See Linus sylves- TRIS, p. 116, 117. NOSING.—The projecting edge of a moulding or drip. NOSING of Steps. —The projecting moulding on the edge of a step, consisting generally of a torus, with a fillet below, joined by a sweep or cavetto to the face of the riser. See p. 196. NOTCH, v .—To cut a hollow on the face of a piece of timber, for the reception of another piece. The piece in which the hollow is cut is said to he notched upon the other piece, and if the notched piece is superimposed, it is said to be notched down upon the inserted piece ; as the bridging-joists are notched clown on the binding-joists in naked floor¬ ing. (See p. 151.) The figures show the varieties of notching in common use. a is the method termed halving; that is, when a notch equal in depth to half the thickness of the stuff is made in both pieces ; h is a dovetail notch; in c the notch is formed a little way from the end of each piece, so that the joint cannot be drawn asunder in either direction ; NOTCH-BOA KD INDEX AND GLOSSARY. OVER-STORY in d the width of the notch is not so great as the j width of the piece on which it is to be let down, J which is also partially notched to receive it. This last, however, belongs rather to caulking or cogging than to notching. NOTCH-BOARD.—A board which is notched j or grooved, to receive the ends of the boards which form the steps of a wooden stair. NUMBERS. The line of, on the sector, p. 40. NYLAND TIMBER. See Pin us syi.vks- j IBIS, p. 116, 117. 0 . Obelisk at Luxor. OAK.— Description and uses of, p. 109. OBELISK.—A lofty, quadrangular, monolithic column of a pyramidal form ; not, however, termi¬ nating iri a point, nor truncated, but crowned by a flatter pyramid. The proportion of the thickness to the height is nearly the same in all obelisks ; that is, between one-ninth and one-tenth; and the thickness at the top is never less than half, nor greater than three- fourths of the thickness at the bottom. Egypt abounded with obelisks, which were always of a single block of stone; and many have been removed thence to Rome and other places. It is generally believed that obelisks weie ori¬ ginally erected as mo¬ numental structures, serving as ornaments to the open squares in which they were usu¬ ally placed, or intended to celebrate some im¬ portant event, and to perpetuate its remem¬ brance. They were usu¬ ally adorned with hiero¬ glyphics. The two larg¬ est obelisks were erected by Sesostris, in Helio¬ polis; the height of these was 180 feet. They were removed to Rome by Augustus. OBJECTIVE LINE. — In perspective, any line drawn on the geometrical plane, the represen- . tation of which is sought on the draught or picture, i OBJECTIVE PLANE.—Any plane situated on the horizontal plane, whose perspective repre- i sentation is required. OBLIQUE ARCHES, or Oblique Bridges. ' —Those arches or bridges whose direction is not at 1 right angles to*their axes. See Skew Bridges. OCCULT LINES.—Such lines as are required ■ in the construction of a drawing, but which do not j appear in the finished work. Dotted lines are also so termed. OCTAGON.—A figure of eight sides and eight angles.—To find the superficies of an octagon. Mul¬ tiply the square of its side by 4‘8284272. OCTAGON.—Upon a given straight line, to ; describe a regular octagon, Prob. X L. p. 12. OCTAGON.—In a given square, to inscribe a | regular octagon, Prob. XLI. p. 12. OCTAGON.—About a given circle, to describe ' an octagon, Prob. XLVII. p. 14. OCTAHEDRON, Octaedron —One of the five regular solids. It is contained by eight equal equi- ; lateral triangles.—To find the surface of an octa- | hedron. Multiply the square of the linear side by 1 3'4G4101(J.—To find the solidity. Multiply the ! cube of the linear side by 0'4714045. OCTAHEDRON, Projections of, to con¬ struct, p. 54. OCTANGULAR PYRAMID.—To find the section of an, p. 69, Plate I. Fig. 12. OCTASTYLE, Octosty-le. —A temple or other building having eight columns in front. OCULUS.—A round window. It was some¬ times simply termed an 0. ODEUM, Odeon. — A kind of theatre, in which poets and musicians sulmitted their works for the judgment of the public, and contended for prizes. (ECUS.—In ancient architecture, the banquet- ing-room of a Roman house; an apartment adjoin¬ ing the drawing-room. OFF-SET, or Set-off. —A horizontal break in a wall at a diminution in its thickness. In Scot¬ land termed a scarcement. OGEE.—In classic architecture, a moulding consisting of two members, one concave and the other convex, it is called also cyma revei'sa. See FI?. 3. Fig. 2. Fig. 1. Ogoo Mouldings. Mouldings, and also Plate LXIII. In medkeval architecture, the ogee moulding assumed different forms at different periods. 1’ ig. 1 is Early English, Fig. 2 is Decorated, Fig. 3 is late Perpendicular. OGEE ARCH.—A pointed arch, the sides of which are each formed with a double curve. It is used in the Decorated style, and less frequently throughout the Perpen¬ dicular style, and is gen¬ erally introduced over doors, niches, tombs, and windows, its inflected curves weakening it too • much to permit of its application for the support of a great weight. OGEE ARCH.—Methods of drawing, p. 30. OGEE PYRAMID, with a hexangular base, to find the section of an, p. 69, Plate I. Fig. 13. OGIVE.—The French term for the ogee arch, but it is also applied to the diagonal ribs of a groined vault. The Pointed stylo of a’chitecture is termed by the French Le style Ogival. OIL LETS, or Oylets. —In the walls of build¬ ings of the middle ages, small openings or eyelet- holes, through which missiles were discharged. ONEGA TIMBER. See Pinus sylvestris, p. 116, 117. OPEN NEWELLED STAIRS. — Winding stairs which have no solid pillar or newel in the centre. OPEN STRING, p. 196. OPISTHODOMUS.—A term applied to the hinder part of a temple, when there is a regular i entrance, and a facade of columns, as in front. The same as the Roman posticum. ORATORY. — A small private chapel, or a closet near a bed-chamber, furnished with an altar, a crucifix, &c., and set apart for the purposes of private devotion, such as commonly existed in the better class of dwellings previous to the Reforma¬ tion, and is still often used by Roman Catholics. The small chapels attached to churches were also often called by the same name. ORB.—-A plain circular boss. The inedueval name for the tracery of blank windows or stone panels. ORDERS of ARCHITECTURE.-The term order, in architecture, signifies a system or assem¬ blage of parts subject to certain uniform established proportions, which are regulated by tlio office each part has to perform. An order may be said to be the genus, of which the species are five, viz., Tuscan, Doric, Ionic, Corinthian, and Composite (see these terms), but it is usual to give to these five the name of orders. Each order consists of two essential parts, a column and an entablature; the column is divided into three parts, the base, the shaft, and the capital; and the entablature is divided into three parts also, the architrave, the frieze, arid the cornice. In the subdivisions certain horizontal members are used, which, from the curved forms of their edges, are called mouldings; as the ovolo, cyma, cavetto, torus, &c. The character of an order is displayed, not only in its column, but in its general forms and detail, of which the column is, as it were, the regulator, the expression being of strength, grace, elegance, lightness, or richness. The scale for the proportions—that is, not the ac¬ tual but the relative dimensions of the different parts compared with each other—is taken from the lower diameter of the shaft of the column, which is divided into two modules or sixty minutes. See Column. ORDONNANCE.—The right assignment, for convenience and propriety, of the measure of the several apartments, that they be neither too large nor too small for the purposes of the building, and that they be conveniently distributed and lighted. ORIEL WINDOW.—A large bay or recessed window in a hall, chapel, or other apartment. It usually projects from the outer face of the wall, either in a semi-octagonal or semi-square plan, and 273 is of various kinds and sizes. When not on the ground floor it is supported on brackets or corbels. Some writers restrict the term oriel window to such Oriel Wiudovv, Ballol College, Oxford. as project from the outer face of the wall and arc supported on corbels, and apply the term bay- window to such as rise from the ground. ORIENTAL PLANE. See Platanus ori- ENTALIS, p. 113. ORIENTATION. — An eastern direction or aspect; the art of placing a church so as to have its chancel pointing to the east. ORIGINAL LINE.—Any line belonging to an original object. ORIGINAL OBJECT.—In perspective, any object whatever. ORIGINAL PLANE.—In perspective, any plane on which an original object is situated, or any plane of the object itself. ORNAMENTS. — In architecture, are the smaller and detailed parts of the main work, not essential to it, but serving to adorn and enrich it. ORTHOGRAPHY.—1. In geometry, the art of delineating the fore right plane or side of any object, and of expressing the elevations of each part; so called because it determines things by perpen¬ dicular lines falling on the geometrical plane.—2. In architecture, the elevation of a building, show¬ ing all the parts in their true proportion. It is either external or internal. The first is the repre¬ sentation of the external part or front of a building, as seen by the eye of the spectator, placed at an infinite distance from it. The second, commonly called the section, exhibits the building as if the external wall were removed and separated from it. —3. In perspective, the fore right side of any plane, that is, the side or plane that lies parallel to a straight line that may be imagined to pass through the outward convex points of the eyes, continued to a convenient length. ORTHOSTYLE.—A columnar arrangement, in which the columns are placed in a straight line. OSTIUM.—In ancient architecture, the door of a chamber. OUNDY, or Undy Moulding. —A moulding with a wave-like outline. OUT and IN BOND.—A Scotch term for al- Fig. 2. Tiro Courses of Door and of Window Jamb or Rebates. ternate header and stretcher in quoins, and window and door jambs. OUTER DOORS.—Those which are common to the interior and exterior sides of the walls of a building. OUTER STRING, p. 196. OVA.—Ornaments i'M*e form of eggs, into which the ovolo moulding is often carved. OVER-STORY.—The clere story. 2 M OVOLO INDEX AND GLOSSARY. PEAR TREE OVOLO.—A moulding, the vertical section of which is, in Roman architecture, a quarter of a circle; it is thence called the quarter-round. In Grecian architecture the section of the ovolo is elliptical, or rather egg-shaped. OVOLO.—To describe an ovolo, its projection and a tangent to it being given, p. 179. OVOLO, The Hyperbolic.—To describe, its projection and a tangent to it being given, p. 1/9. P. PACE.—A portion of a floor slightly raised above the general level; a dais. PACKING.—In masonry, small stones im¬ bedded in mortar, used to fill up the interstices of the larger stones in rubble walls. PAD.—A handle. PADDLE.—A small sluice.— Paddle-holes are the passages which conduct the water from a dock or the upper pond of a canal, into the lock-chamber, and out of the lock-chamber into the lower pond. PAGODA.—A temple in the East Indies, in which idols are worshipped. The pagoda is gene¬ rally of three subdivisions. First, an apartment whose ceiling is a dome, resting on columns of stone or marble; this part is open to all persons. Second, an apartment forbidden to all but Brahmins. Third and last, the cell of the deity or idol inclosed with a massy gate. The idol itself is sometimes called a pagoda. The most remarkable pagodas are those ! of Benares, Siam, Pegu, and particularly that of j Juggernaut in Orissa. Pagodas are also common | in China, where they are called tads. PAL/ESTRA. —Among the Greeks, a place for athletic exercises. PALLET MOULDING. — In brick-making, I that kind of moulding in which sand is used to pre- ' vent the clay from adhering to the mould, one | mould only being used; and the brick when moulded turned out on a flat board called a pallet, on which it is carried by the assistant to the hack-barrow or the hack. PAMPRES. — Ornaments consisting of vine | leaves and grapes, with which the hollows of the ; circumvolutions of twisted columns are sometimes decorated. PAN of Wood, or Pas of Framing. See Timber Houses, p. 150. PANACHE. The French name for a species of pendenfcive, formed by a portion of a domical vault intercepted between one horizontal and two vertical surfaces. It occurs when a round tower or dome is carried over a square substructure, as when j Panache. a dome is raised on the square formed by the cross- , ing of the nave and transept of a church. In this ■ case the panache p becomes a spherical triangle, bounded by three arcs, viz., the arch of the nave a, i the arch of the transept b, and the circle c, which serves as the springing of the dome or tower. PANEL.—In architecture, an area sunk from the general face of the surrounding work; also a ' compartment of a wainscot or ceiling, or of the sur- ! face of a wall, &c.; sometimes inclosing sculptured ! ornaments. In joinery, it is a tympanum or thin j piece of wood, framed or received in a groove by two upright pieces or styles, and two transverse I pieces or rails : as the panels of doors, window- j shutters, &c. In masonry, a term sometimes ap¬ plied to one of the faces of a hewn stone. PANEL SAW.—A saw used for cutting very thin wood in the direction of the fibres or across them. Its blade is about 2G inches long, and it has about six teeth to the inch. PANELLING.—In architecture, the operation of covering or ornamenting with panels; panelled work. PANELS in Joinery, p. 185. PANTAGRAPII, Pantograph. —An instru¬ ment for copying, enlarging, or reducing drawings. See Eidograph. PANTHEON.—A temple dedicated to all the gods. The term is also applied to places of public exhibition, in which every variety of amusement is to be found. PANTILE, OR Pentile.—A tile in the form of a parallelogram, straight in the direction of its length, but with a waved surface transversely. Each tile is about 134 inches long and 7 inches wide, hut the development of its surface is of course greater; it is about half an inch thick. It has a small tongue or projection from its under side at its upper end, which serves to hook it to the lath. Pantiles are set cither dry or in mortar. They over¬ lap laterally, the down bent edge of the one tile covering the upturned edge of the other. Having only 3 or 4 inches of longitudinal overlap, pan¬ tiling is little more than half the weight of plain tiling, but it is not so warm a covering, and is more apt to be injured by storms. The ridges and hips of roofs covered either with pan or plain tiles, are finished with large concave tiles, called hip or ridge tiles, and sometimes crown tiles; these are not over¬ lapped, but aie set in mortar and fastened with nails or pins. To find the number of pantiles re¬ quired to cover a roof, the gauge being 10 inches. Rule; Multiply the area in superficial feet by 1-80. And to find the weight in tons. Multiply the area in superficial feet by '00377- PARABOLA.— Description of the, p. 2G. PARABOLA.—To draw, p. 2G. PARABOLA.—To draw by means of intersec¬ ting lines, Prob. XCVII. and XCVIII. p. 27. PARABOLA.—To draw, bymeans of a straight rule and a square, Prob. XCIX. p. 27 PARABOLA. — To find the parameter of a parabola, p. 2G. PARABOLA.—To draw perpendiculars to the curve of a parabola, p. 2G. PARABOLA.—To find the area of a parabola or its segment. Multiply the base by the perpen¬ dicular height, and two-thirds of the product is the area. PARADISE. — In mediaeval architecture, a small private apartment or study. PARALLEL.—To draw a straight line par¬ allel to a given straight line, Prob. I. and II. p. 5. PARALLEL COPING. — Coping of equal thickness throughout. PARALLEL PERSPECTIVE. — That in which the picture is supposed to be situated, so as to be parallel to tire side of the principal object to be represented. See Perspective, p. 227. PARALLELOGRAM.— To inscribe in a given quadrilateral figure, Prob. XXXIII. p. 11. PARALLELOGRAM of Forces, p. 120. PARALLELOGRAM of Forces.— Applica¬ tion of, to discover the stress on parts of framing, p. 121. PARAPET.—Literally, a wall or rampart to the breast, or breast high.—In military structures, the parapet is a wall intended for defence, and is either plain or battlemented, and pierced with loop¬ holes and oillets for the discharge of missiles.—In civil and ecclesiastical buildings, the parapet, like the balustrade, is to be regarded chiefly in the light of an ornament. The plain and simple embattled parapet, indeed, is to be found in buildings of the middle ages, from the early Norman to the latest Perpendicular; but, in general, the parapet assumes the character of the various styles, proceeding from comparative plainness in the earlier styles, to being ornamented with panelled and pierced work in those which succeeded it.—In common language, a parapet is abreast-wall raised on the sides of bridges, quays, &c., for protection. PARGE BOARD.—See Barge Board. PARGET.— 1. Gypsum; plaster stone.— 2. Plaster laid on roofs or walls.—3. A plaster formed of lime, hair, and cow-dung, used for plastering flues. PARGETTING, Pergetting, Pergeuring, Pabge-WORK. —Plastering; as a noun, plaster or stucco. Also, a term used for plaster-work of va¬ rious kinds, but commonly applied to a particular sort of plaster-work, with patterns and ornaments 274 raised or indented upon it, much used in the inte¬ rior, and often in the exterior of houses in the time of Queen Eliza¬ beth. The term is now seldom used, except for the plastering of chimney flues. PARING CHISEL. — A broad, flat chisel used by joiners; it is worked by the impulsion of the hand alone, and not by the blows of a mallet, like the socket-chisel, firmer, &c. PARK and Entrance Gates, p. 177. PARPEND, or Perpend. —A stone reaching through the thickness of a wall so as to he visible on both sides, and therefore worked on both ends. PARPEND WALL.—A wall built of parpends or stones which reach through its entire thickness. PARQUETRY.—A species of joinery or cabinet-work, which consists in making an inlaid floor composed of small pieces of wood, either square or tri¬ angular, which, by the manner of their disposition, are capa¬ ble of forming vari¬ ous combinations of figures. Such floors are much used in France. PARREL.—A chimney-piece; the dressings and orna¬ ments of a fireplace. PARTHENON.—A celebrated Grecian temple of Minerva in the Acropolis of Athens. It was built of marble, and was a peripteral octostyle, with 17 columns on the sides; its length 223 feet, breadtli 102, and height from the base to the pediment 65 feet. It was almost reduced to ruins in 1G87 by the explosion of a quantity of gunpowder which the Turks had placed in it; but dilapidated as it now is, it still retains an air of inexpressible gran¬ deur and sublimity. PARTING BEAD.—The beaded slip inserted into the centre of the pulley-style of a window, to keep apart the upper and lower sashes. PARTITION.—A wall of stone, brick, or tim¬ ber, which serves to divide one apartment from another in a building. PARTITIONS, Timber, p. 158. PARTY-WALLS.—A wall formed between houses to separate them from each other, and pre¬ vent the spreading of fire. PATAND, Paten.— 1. A piece of timber laid on the ground to receive and sustain the ends of vertical pieces.—2. A bottom plate; a sill. PATERA.—1. An open vessel in the form of a cup, used by the Greeks and Romans in their sacri¬ fices and libations.—2. The representation of a cup or round dish in flat relief, used as an ornament in friezes ; but many flat ornaments are called pateras which have no resemblance to cups or dishes. PATTEN.—The base of a column or pillar. PAVILION.—1. A turret or small building, usually isolated, and having a tent-formed roof, Pavilion of Flora, Tuilcries, Paris. whence its name.—2. A projecting part of a building, when it is carried higher than the general structure, and provided with a tent-formed roof. PEAR TREE.—Description of the properties and uses of, p. 114. PECKINGS INDEX AND GLOSSARY. PERSPECTIVE PECKINGS. See Place-Bricks. PECKY.—A term in America applied to tim¬ ber in which the first symptoms of decay appear. PEDESTAL.—An insulated basement or sup¬ port for a column, a statue, or a vase. It usually consists of a base, a die or dado, and a cornice, called also a surbase or cap. When a range of columns is supported on a continuous pedestal the latter is called a podium or stylobate. PEDIMENT.—In classic architecture, the tri- | angular finishing above the entablature at the end of buildings or over porticoes. The mouldings of j the entablature bound the inclined sides of the pedi¬ ment. Also the triangular finishing over doors and windows. In the debased Roman style the same name is given to these same parts, though not tri¬ angular in their form, but circular, elliptical, or interrupted. In the architecture of the middle ages, small gables and triangular decorations over open¬ ings, niches, &c., are called pediments. These have the angle at the apex more acute than the corre¬ sponding decoration of classic architecture. TEEN.—The same as piend (which see). PENCILS.—Qualities and uses of, p. 45. PENDANT, Pendent. —A hanging ornament used in the vaults and timber roofs of Gothic archi¬ tecture. In the former, pendants are formed of stone and generally richly sculptured, and in tim¬ ber-work they are of wood, variously decorated with carving. See Plate XXXI. PENDANT POST.—1. In a mediaeval prin¬ cipal roof-truss, a short post placed against the wall, its lower end supported on a corbel or capital, and its upper end carrying the tie-beam or hammer- beam.—2. The support of an arch across the angles of a square. PENDENTIVE.—-The portion of a dome- shaped vault, which descends into a corner of an angular building, when a ceiling of this kind is Peadenlive Roof. placed over a straight-sided area. Thus, when a portion of a sphere, as the hemisphere in the figure, is intersected by cylindrical or cylindroidal arches, as a a a, the vaults b b are formed, which are pen- dentives. In Gothic architecture, the portion of a groined ceiling springing from one pillar or impost, and bounded by the apices of the longitudinal and transverse vaults, is called a pendentive. PENDENTIVE of an irregular octagonal plan over an apartment, the plan of which is a parallelo¬ gram, p. 82. PENDENTIVE formed by the intersection of an octagonal domical vault with a square, p. 81. PENDENTIVE BRACKETING.—The coved bracketing springing from the wall of a rectangular area in an upward direction, so as to form the horizontal plane into a complete circle or ellipse. See Pendentive. PENDENTIVE CRADLING.—The tim¬ ber work for sustaining the lath and plaster in pendentives. PENT-HOUSE.—A shed with a roof of a single slope. PENT-ROOF. — A roof formed like an inclined plane, the slope being all on one side ; called also a shed-roof. PENTADORON.—In ancient architecture, a brick of five palms in length, used by the Greeks in the construction of their sacred edifices. PENTAGON.—A figure with five equal T sides and angles is a regular pentagon ; if the l sides be unequal, it is an irregular pentagon.— To find the area of a regular pentagon. Multi¬ ply the square of its side by P7204774. PENTAGON.—On a given straight line, to describe a regular pentagon, Prob. XXXVIII. p. 12. PENTAGRAPH. See Pantagraph. PENTASTYLE. — An edifice having five column in front. PERCH.—-An old name sometimes applied to a bracket or corbel. PERCLOSE, Parclose. —The raised carved I timber back to a bench or seat; the parapet round I a gallery; a screen or partition. PERGETTING, or Pergeuring. See Par- j GETTING. PERI BOLUS.—In ancient architecture, a| court surrounding a temple, and itself surrounded i by a wall inclosing the whole of the sacred ground. I It was commonly adorned with statues, altars, and i monuments, and sometimes contained other smaller | temples or a sacred grove. A perfect example of | the peribolus exists in the temple of Isis at Pom- j peii, and remains of others are found at Palmyra i and elsewhere. PERIDROMUS.—The space in a peripteral { temple, between the walls of the cella or body and | the surrounding columns. PERIMETER.—The circuit or boundary of any plane figure. In round figures it is equivalent to circumference or periphery, but the term is more J frequently applied to figures composed of straight j lines. PERIPHERY.—The circumference of a circle or ellipse, or of any curvilinear figure. PERIPTERAL.—-A temple, the cella of which | is surrounded with columns, those on the flanks being at a distance from the wall equal to their [ intercolumniation. PERISTYLE, Peristylium. — A range of | columns surrounding anything, as the cella of a temple, or any place, as a court or cloister. It is 1 frequently but incorrectly limited in signification 1 to a range of columns round the interior of a place. | PERPEND, Perpyn, Perpent. SeePARPEND. ! PERPENDICULAR LINES.—To erect or 1 let fall, p. 6-18. PERPENDICULAR STYLE. —The third* and last of the pointed or Gothic styles used in this | country, called also the florid style of Gothic. It was developed from the Decorated during the latter part of the fourteenth century, and continued in use till the middle of the sixteenth. The broad distinc- i tion between this and the preceding styles lies in ! the preponderance of perpendicular lines, particu¬ larly observable in the tracery of windows, the ! panelling of flat surfaces within and without, and the multiplicity of small shafts with which the piers, &c., are overlaid. The vertical line every¬ where predominates, catching the eye at first sight, so that when once this characteristic has been pointed out, it is impossible to mistake a building in this style. Another peculiarity is the increased width of the windows and the lowness of the roofs, which are frequently so low as not to rise above the parapet. This is owing to the use of the four- centred depressed arch, which gave an opportunity of employing greater width, without increasing the height of the windows. To such an extent is this peculiarity carried, that the chancel of a church in this style is almost as light as a conservatory, the whole space between the buttresses be'ng occupied with the windows. The upper tier of windows, or clear-story, offers another peculiarity. Tn the pre¬ ceding styles these windows were generally small; but in the Perpendicular, when that style became fully developed, they are often so large, and placed so closely together, that the whole clear-story be- panelling, parapets, buttresses, and turrets. It will be seen that the principal mullions, instead of run¬ ning into flowing tracery, are here carried straight through to the head of the window, and that the subordinate tracery is likewise converted into straight lines. In this consists the essential dif¬ ference between the Decorated and Perpendicu'ar styles. The Perpendicular style of Gothic archi¬ tecture was peculiar to England. PERPENT-STONE. See Parpend. PERRON.—A term denoting a staircase lying open or outside the building ; or more properly the steps in the front of a building which lead into the first story, where it is raised a little above the level o f the ground. PERSIAN.— A figure in place of a column, used to support an entablature. See Caryatides. PERSPECTIVE.-—1. The science which teaches the representation of an object or objects on a definite surface, so as to affect the eye when viewed from a given point, in the same manner as the object or objects themselves. Correctly defined, a perspective delineation is a section, by the plane or other surface, on which the delineation is made, of the cone of rays proceeding from every part of the object to the eye of the spectator. It is inti¬ mately connected with the arts of design, and is indispensable in architecture, engineering, fortifi¬ cation, sculpture, and generally all the mechanical arts; but it is particularly necessary in the art of painting, as without a correct observance of the rules of perspective, no picture can have truth and life. Perspective alone enables us to represent fore¬ shortenings with accuracy, and it is requisite in delineating even the simplest positions of objects. Perspective is divided into two branches, linear and aerial. Linear perspective has reference to the posi¬ tion, form, magnitude, &c., of the several lines or contours of objects, &c. 'J he outlines of such ob¬ jects as buildings, machinery, and most works of human labour which consist of geometrical forms, or which can be reduced to them, may be most accurately obtained by the rules of linear perspec¬ tive, since the intersection with an interposed plane of the rays of light proceeding from every point of such objects, may be obtained by the principles of geometry. Linear perspective includes the various kinds of projections; as scenoyraphic, orthographic, ichnoyraphic, stei'cographic projections, &c. Aerial perspective teaches liow to give due diminution to the strength of light, shade, and colours of objects according to their distances, and the quantity of light falling on them, and to the medium through which they are seen.— Perspective plane, the surface on which the object or picture is delineated, or it is the transparent surface or plane through which we suppose objects to be viewed ; it is also termed the plane of projection, and the plane of the picture. —Parallel perspective is where the picture is sup¬ posed to be so situated, as to be parallel to the side comes one large window, merely divided by the mullions. The annexed view of Beauchamp Chapel presents a very perfect instance of Perpendicular architecture, both in the windows, and also in the 273 of the principal object in the pictme; as a build¬ ing, for instance.— Oblique perspective is when the plane of the picture is supposed to stand oblique to the sides of the object represented; in which case the representations of the lines upon those sides will not be parallel among themselves, but will tend towards their vanishing point.—2. A kind of painting, often seen in gardens and at the end of a gallery, designed expressly to deceive the sight by representing the continuation of an alley, a build¬ ing, a landscape, or the like.— Jsometrical perspec¬ tive, or more correctly isometrical projection, a kind of orthographic projection, so named and brought prominently into notice by Pro¬ fessor Parish, of Cambridge, by which solids, of the form of rect¬ angular parallelopipeds, or such as are reducible to this form, or can be contained in it, can be represented with three of their contiguous planes in one figure, which gives a more intel¬ ligible idea of their form than can be done by a separate plan and elevation. At the same time, this method admits of their dimensions being PERSPECTIVE INDEX AND GLOSSARY. measured by a scale as directly as in the usual mode of delineation. As applied to buildings and machinery, it gives the elevation and ground plan in one view, and is therefore considered more use¬ ful, as explanatory of the ordinary geometrical drawings, than linear perspective, it is also easier and simpler in its application. PERSPECTIVE.—1 ntroductory obs ervations, p. 227. PERSPECTIVE. — Definitions of terms, p. 228, 229. PERSPECTIVE.—To find the perspective of a given point, Prob. I. p. 229. PERSPECTIVE.—To find the perspective of a given right line, Prob. II. p 229. PERSPECTIVE.—Rules in, p. 229-232. PERSPECTIV E.—The distance of the picture and the perspective of the side of a square being given, to complete the square without having re¬ course to a plan, Prob. III. p. 233. PERSPECTIVE.—To divide a line given in perspective in any proportion, Prob. IV. p. 234. PERSPECTIVE—Through a given point in a picture, to draw a line parallel to the base or side of the picture, and perspective^ equal to another given line. Prob. V. p. 234. PERSPECTIVE.—To draw the perspective of a pavement of squares, Prob. VI.-VIII. p. 235. PERSPECTIV E.—To draw a hexagonal pave¬ ment in perspective, Prob. IX. p. 23(5. PERSPECTIV E.—To draw the perspective of a circle, Prob. X. p. 236. PIEDOUCHE. — A small pedestal or base, serving to support a bust, candelabrum, or other ornament. PIEDIIOTT.—The jamb of an opening, includ¬ ing the face and retiring side. It is moie specifi¬ cally applied to the jamb of an arched opening, when crowned with an impost moulding. PIEND.—An arris; a salient angle; a hip. PIEND CHECK.—A term applied in Scot- 1 land to the rebate formed on the piend or anHe at I the bottom of the riser of the stone step of a stair, ns at a a a in the figure. PIER. — 1. The support of the arches of a bridge; the solid parts between openings in a wall, such as the door, windows, &c. (See woodcut, Abch.) —2. A mole or jetty carried out into the sea, whether intended to serve as an embankment to protect vessels from the open sea, or merely as a landing place. For this latter purpose suspension : chain-piers are sometimes employed.—3. The pillars i in Norman and Gothic architecture are generally. ] though not very correctly, termed piers. PIER-ARCHES. — In Gothic architecture, arches supported on piers (or pillars) between the central parts and aisles of a church. PILASTER.—A debased pillar: a square pillar projecting from a pier or from a wall, to the extent of from one-fourth to one third of its breadth. Pil¬ asters originated in the Grecian antce. In Roman architecture they were sometimes tapered like columns, ami finished with capitals modelled after the order with which they were used. PILE-DRIVER, or Pjle-Engixe.—A n engine for driving down piles. It consists of a large ram or block of iron, termed the monhey, which slides between two guide-posts. Being drawn up to the - . ... ... n circle m ; square given in perspective, Prob. XI. p. 236. PERSPECTIVE.—To draw a tetrahedron ii perspective, Pmb. XII. p. 237. PERSPECTIV E.—To draw a cross in perspcc tive, p. 241. PERSPECTIVE.—'To draw a pavilion in per spective, p. 241. PERSPECTIV E.—To draw a broach or spin in perspective, p. 242, 213, Plate CVII. and CXI PERSPECTIVE.—To draw a Tuscan gate Win in perspective, p. 242, Plate CIX. PERSPECTIVE.—To draw a Turkish bath ii perspective, p. 243. Plate CX. PERSPECTIVE.—To draw a series of arches in perspective, p. 242, Plate CVJI1. PERSPECTIVE.—To draw a circular vaul pierced by a circular-headed window in perspective p. 242, Plate CVIII. PERSPECTIVE.—Isometric projection, defi mtion and illustration of, p. 242. PERSPECTIVE. — Isometric scales, how t< construct, p. 243. PERSPECTIVE.—Tsometrical projection, ap plication of, to curved lines p 245 PERSPECTIVE.-To draw a cube in per spective. Trob. XIII. p. 237. PERSPECTIVE.—To draw cylinders in per spective, Trob. XIV. p. 238 PERSPECTIVE. Proper „„»Ie at which oh jects should be viewed, p. 238. PERSPECTIVE.—Argument against the use ot the distinctive terms parallel and oblique pro¬ spective, p. 239. 1 PERSPECTIV E.—To draw a sphere in per spective, Prob. XV. p. 239. PERSPECTIVE.—Practical examples of per spective drawing applied to architecture, &e p -'40 PETERSBURG TIMBER. S„ P.aA av t VESTRIS, p. 116, 117. PEW DOOR, Plate LXXIII. Fig. 6. p. 187. PHOLAS. —A marine animal./injurious tc timber, p. 105. i .,?. IAZZA —A Sf l uare open space surrounded b> buildings or colonnades. The term is frequently but improperly, used to signify an arcaded or col¬ onnaded walk. PIECE-WORK.—Work done and paid for by the measure of quantity, in contradistinction tc work done and paid for by the measure of time. top, and then let fall from a considerable height, it comes down on the head of the pile with a violent blow. It may be worked by men or horses, or a steam-engine. The most improved pile-driver is that constructed by Mr. James Nasmyth, being an ingenious application of the principle of bis cele¬ brated steam-hammer. PILE - PLANKS. -—Planks about 9 inches broad, and from 2 to 4 inches thick, sharpened at their lower end, and driven with their edges close together into the ground in hydraulic works. Two rows of pile-planks thus driven, with a space be¬ tween them filled with puddle, is the means used to form water-tight coffer-dams and similar erections. PILES.—Beams of timber, pointed at the end, driven into the soil for the support of some super¬ structure. They are either driven through a com¬ pressible stratum, till they meet with one that is I incompressible, and thus transmit the weight of the : structure erected on the softer to the more solid { material, or they are driven into a soft or compres¬ sible structure in such numbers as to solidify it. In the first instance, the piles.are from 9 to IS inches I in diameter, and about twenty times their diameter in length. They are pointed with iron at their lower end, and their head is encircled with an iron 276 PISE ring, to prevent its being split by the blows of the pile-driver. In the second case, the piles are from 6 to 12 feet long, and from 6 to 9 inches in dia¬ meter. In constructing coffer-dams and other hy¬ draulic works, other kinds of piles besides those described are used, such as gauge-piles, sheeting- piles, pile-planks, key-piles. These will he found under their proper heads. PILLAR.—1. A pile, or columnar mass com¬ posed of several pieces, and the form and propor¬ tions of which are arbitrary, that is, not subject to the rules of classic architecture. A square pillar is a massive work, called also a pie)- or piedroit, serv¬ ing to support arches, &c.—2. A supporter; that which sustains or upholds; that on which some superstructure rests. PIN.—A piece of wood or metal, square or cylindrical in section, and sharpened or pointed, used to fasten timbers together. Large metal pins are termed bolts, and the wooden pins used in ship¬ building treenails. PINACOTHECA.—A picture gallery. PINES AND Firs.—D escriptions and uses of, p. 115, (t seq. PINNACLE.—]. A turret, or part of a build¬ ing elevated above the main building.—2. In medi- reval architecture, a term applied to any lesser Early English Pinnacle, Perpendicular Pinnacle, Beverley Minster. T.inlly Ch., Cambridge. euucuiie in urniiiuciiii, unii.sisung hi a uociy or snail I terminated by a pyramid or spire, used either ex- ' teriorly or interiorly. | PINNING. — Fastening tiles or slates with pins ; inserting small pieces of stone to fill vacuities. PINNING IJP. Driving in wedges in the \ process of underpinning, so as to bring the upper i work to bear fully on the work below. PINS-DRAWING, p. 45. PINUS STROBUS. - The Weymouth pine, 1 p. 118. PINUS SYL VESTRIS. p. 116, 117. PISCINA.—A niche on the south side of the ! altar in Roman Catholic churches, containing a small basin and water-drain, through which the Piscina, Fiefield, Essex. priest emptied the water in which he had washed his hands, and also that with which the clialicc had been rinsed. PISE.—A species of wall constructed of stiff earth or clay, rammed into moulds, which arc carried up as the work is carried up. It has been used in INDEX AND GLOSSARY. POLYGON' PITCH OF A ROOF France of late years, but it is as old as the days of Pliny. PITCH op A Roof. —The inclination of the sloping sides of a roof to the horizon, or the vertical ano-le formed by the sloping side. The pitch is usually designated by the ratio of the height to the span. PITCH OB’ A Roof. — Opinions of various authors as to the pitch which should be given to a roof, to suit the climate and the material used for covering, p. 134, 135. PITCH PINE.—Properties and uses of, p. 118. PITCH PINE. p. 118. PITCHING PIECE, p. 196. PITCHING PIECE.—A piece of timber pro¬ jecting horizontally from a wall, to support the rough strings in staircasing. See Apron-Piece. PLACE BRICKS.—Those bricks which, having been outermost or farthest from the fire in the clamp or kiln, have not received sufficient heat to burn them thoroughly. They arc consequently soft, uneven in texture, and of a red colour. They are also termed pcckings, and sometimes sandcl or samel bricks. PLAFOND, Platfond. —1. The ceiling of a room, whether flat or arched.—2. The under side of a cornice.—3. Generally, any soffit. PLAIN TILES are simple parallelograms, generally about 10), inches long, lij inches wide, and f-inch thick, and weighs 2 lbs. 5 oz. Each tile has a hole at one end to receive the wooden pin by which it is secured to the lath. Plain tiles are laid on laths on mortar, with an overlap of 6 to 8 inches. At 6 inches gauge, it takes to cover a square of roofing 768 plain tiles ; at 7 inches gauge, it takes 655 tiles; and at S inches gauge, 576 tiles. The average square of plain tiling is 700, and weighs 14 cwts. 2 qrs. PLAN.—A draught or form; properly, any¬ thing drawn or represented on a plane, as a map or chart, but the word is usually applied to the hori¬ zontal geometrical section of anything, as a build¬ ing, for example. The term is applied also to the draught or representation on paper of any projected work, as the plan of a house, of a city, of a har¬ bour, &c. PLANCEER, 1’LANCHER. —A ceiling, or the soffit of a cornice. PLANE.—The plane is a cutting instrument on the guide principle. It is, in fact, a chisel guided by the stock or wooden handle in which it is set. The guide or sole of the stock is, in general, an exact counterpart of the form it is intended to produce. Planes are of various kinds, as the jack Plough Moulding. and the second for sinking the opposite edge. The plough is a plane for sinking a channel or groove in a surface, not close to the edge of it. Mould- ing planes are for forming mouldings, and must vary according to the design. The bead plane is used for mouldings whose section is semicircular. Planes are also used for smoothing metal, and are wrought by ma¬ chinery. See Planing Machine. PLANE, The Oriental. —Description of pro¬ perties and uses, p. 113. PLANE, American or Western, p. 113. PLANE of Projection, Plane of Delinea¬ tion, Transparent Plane. —In perspective, the same as plane of the picture. See p. 228. PLANING MACHINE. —A tool or instru¬ ment wrought by steam power, for saving manual labour in producing a perfectly plane surface upon wood or metal. This is usually accomplished, in j metal-planing machines, by such an arrangement of mechanism, as will cause the object which is to be ' operated upon to traverse backwards and forwards | upon a perfectly smooth and level bed, while the 1 cutting tool is fixed to a cross slide above it, and | slightly penetrates the surface as it is carried along. | The tool is acted upon by screws, so as to enable j the attendant to adjust the depth of the cut, and to j move it with unerring precision over every part of ! the surface which it is required to plane. Planing I machines for wood are described in the text, under Labour-saving Machines, p. 191-193. PLANING MACHINE. — Furness’ patent, p. 193. PLANK. — A broad piece of sawed timber, differing from a board only in being thicker. Broad pieces of sawed timber which are not more than 1 inch, or lj inch thick, are called boards ; like pieces, from 1A to 3 or 4 inches thick, are called planks. Sometimes pieces more than 4 inches thick are called planks. PLANS of Stairs.—T he mode of setting out, p. 198. PLANTED.—In joinery, a projecting member wrought on a separate piece of stuff, and afterwards fixed in its place, is said £ to be planted; as a planted moulding. PLASTER.—A composition of lime, water, and sand, well mixed into a kind of paste, and used for coating walls and partitions of houses. This composition when dry becomes hard, but still re¬ tains the name of plaster. Plaster is sometimes made of different materials, as chalk, gypsum, &c., and is sometimes used to parget the whole surface of a building. Plaster is also the material of which ornaments are cast in architecture, and also that with which the fine stuff or gauge for mouldings and other parts is mixed, when quick setting is re¬ quired. PLAT, or Plot. —A word used by old authors for plan. l’L AT-BAND. —1. A ny flat rectangular mould¬ ing, the projection of which is much less than its width; a fascia.—2. A lintel formed with voussoirs in the manner of an arch, but with the intrados horizontal.—3. The fillets between the flutes of the Ionic and Coiinthian pillars. PLATE.—A general name for all timber laid horizontally in a wall to receive the ends of other timbers, such as a wall plate. PLATFORM.—A flat covering or roof of a building, suited for walking on ; a terrace or open walk on the top of a building. PLETHORA .—A disease of trees. See p. 97- the utmost degree of smoothness to the surface of i PLINTH.—A square member serving as the the wood; the compass plane, which is similar to . base of a column, the base of a pedestal, or of a r - a. & plane (about 17 inches long), used for taking off the roughest and most prominent parts of the stuff; the trying plane, which is used after the jack plane; the long plane (26 inches long), used when a piece of stuff is to be planed very straight; the jointer, still longer than the former, which is used for ob¬ taining very straight edges; the smoothing plane (7i inches long), and block plane (12 inches long), chiefly used for cleaning off finished work, and giving the smoothing plane, but has its under surface vex, its use being to form a concave cylindrical surface. The foregoing are technically called bench- planes. There is also a species of planes called rebate planes, the first of which is simply called the rebate plane, being chiefly used for making rebates. Of the sinking rebating planes there are two sorts, -, sido and end. the moving fillister and the sash fillister, the first for sinking the edge of the stuff next the workman, wall. The square member, for instance, under the torus of the Tuscan base is the plinth. See wood- cut, Column. PLOT.—A plan. PLOT, v .—To make a plan of anything. PLOTTING.—In surveying, the describing or laying down upon paper the several angles and lines of a tract of land which has been surveyed and measured. It is usually performed by means of a protractor ; sometimes by the plotting scale (which see). PLOTTING-SCALE.—A scale of equal parts, with its divisions along its edge, so that measure¬ ments may be made by application. A particular kind of plotting-scale is sometimes used in setting off the lengths of lines in surveying. It consists of two graduated ivory scales, one of which is perfo¬ rated nearly its whole length by a dovetail-shaped slit, for the reception of a sliding-picce. The second 277 scale is attached to this sliding-piece, and moves along with it, the edge of the second scale being always at right angles to the edge of the first. By this means the rectangular co-ordinates of a point are measured at once on the scales, or the position of the point laid down on the plan. PLOUGH.—A joiners grooving-plane. See Plane. PLUG CENTRE-BIT.—A modified form of the ordinary centre-bit, in which the centre-point or pin is enlarged into a stout cylindrical plug, which may exactly fill a hole previously bored, and guide the tool in the process of cutting out a cylindrical countersink around the same, as, for example, to receive the head of a screw-bolt. PLUMB. — Perpendicular, that is, standing according to a plumb-line. The post of the house or the wall is said to be plumb. PLUMB-LINE.—A line perpendicular to the plane of the horizon; or a line directed to the centre of gravity in the earth. See Plummet, Plumb-Rule. PLUMB-RULE.—A simple instrument for the same purpose as the plumb-line or plummet, used by masons, bricklayers, and carpenters It con¬ sists of a board with parallel edges; a line is drawn down the middle of the board, and to the upper end of this line the end of a string is attached, carrying a piece of lead at its lower end. When the edge of the board is applied to a wall or other upright ob¬ ject, the exact coincidence of the plumb-line with the line marked on the board indicates that the wall or other object is vertical, while the deviation of the plumb-line from that on the board, shows how far the object is from the perpendicular. Some¬ times another board is fixed across the lower end of the plumb-rule, having its lower edge at right angles to the line drawn on the other. In this case it becomes a level. PLUMM ET.—1. A long piece of lead attached to a line, used in sounding the depth of water.—2. An instrument used by carpenters, masons, &c., in adjusting erections to a perpendicular line. The terms plummet, plumb-line, and plumb-rule are often used synonymously. See Plumb-Ltne. POCKET. — A hole in the pulley style of a sashed window. See p. 188. POD-AUGER.—A name given in some loca¬ lities to an auger formed with a straight channel or groove. See Auger. PODIUM. — In architecture, a continuous pedestal; a stylobate; also, a projection which sur¬ rounded the arena of the ancient amphitheatre, where sat persons of distinction. POINTED, or Christian Architecture. See Gothic. POITRAIL, Poitrel. See Timber Houses, p. 156. POLE-PLATE.—A sort of smaller wall-plate laid on the top of the wall, and on the ends of the tie-beams of a roof, to receive the rafters. POLINGS.—Boards used to line the inside of a tunnel during its construction, to prevent the fall¬ ing of the earth or other loose material. POLYCHROMY. — A modern term used to express the ancient practice of colouring statues, and the exteriors and interiors of buildings. This practice dates from the highest antiquity, but pro¬ bably reached its greatest perfection in the twelfth and thirteenth centuries. POLYFOIL.—An ornament formed by a moulding disposed in a number of segments of circles. POLYGON. — To find the area of a regular polygon. Multiply half the perimeter by the per¬ pendicular let fall from the centre upon one of the sides; or multiply the square of the side by the multiplier corresponding to the figure in the follow¬ ing table:— Table of Polygons. Name. No. of 42SS S3& Multipliers. Equilateral Triangle, 3 120° 0-2SS6751 0-4330127 Square. 4 90° 0-500000 1-ooooooi 0-6S81910 1-7204774 g 0-S660254 2"5980762 7 5i;-° 1-0382607 3-6339124 Octagon, .... 8 45° 1-2071068 4 8284272 9 1 -3737387 6-1818242 Decagon, .... 10 30° 1-5388418 7 0942088 Undecagon, . . . i 1 32 A” 1-7028439 9-3056411 Dodecagon, . . . 12 30° 1-8600254 11 1961524 POLYGON.—To find the area of any regular polygon, Prob. LII. p. 15. POLYGON of Forces, p. 120. POLYGON, Regular. —About a given circle, to describe any regular polygon, Prob. XLVIII. p. 14. POLYGON, Regular.—T o describe a regular polygon with the same perimeter as another given polygon, but with twice the number of sides, Prob. XLIX. p. 14. POLYGON INDEX AND GLOSSARY. PUTTY POLYGON, Regular. —To construct with the same perimeter as a given polygon, but with any different number of sides, Prob. L. p. 14. POLYGON, Regular.— To describe with the same area as a given polygon, but a different num¬ ber of sides, Prob. LI. p. 34. POLYGON, Regular. —Jn a given circle, to inscribe any regular polygon, Prob. XLII.p. 12, Probs. X LI 11.-XL VI. p. 13. POLYGONS, Regular.— On a given straight line, to describe any of the regular polygons, Prob. XXXVII. Examples 1 and 2, p. 11. POLYGONS, Line of, on the Sector.—Con¬ struction and use of, p. 3S. POLYSTYLE.—An edifice in which there are many columns. POMEL, Pommel.—A knob or ball used as a finial to the conical or dome shaped roof of a turret, pavilion, &c. POPLAR, The.—D escription of the properties and uses of, p. 112. POPPY HEAD.—An ornament carved on the raised ends of seats, benches, and pews in old churches. PORCH.—An exterior appendage to a building, forming a covered approach or vestibule to a door¬ way. The porches in some of the older churches are of two stories, having an upper apartment, to which the name parvise is sometimes applied. PORTAL.—1. The lesser gate when there are two of different dimensions at the entrance of a building.—2. A term formerly applied to a little square corner of a room, separated from the rest by a wainscot, and forming a short passage into a 1 room.—3. A kind of arch over a door or gate, or the frame-work of the gate. PORTCULLIS —A strong grating of timber ' or iron, resembling a harrow, made to slide in ver- , tical grooves in the jambs of the entrance gate of a j fortified place, to protect the gate in case of assault. | The vertical bars, when of wood, were pointed witli iron at the bottom, for the purpose of striking into the ground when the grating was dropped, or of injuring whatever it might hill upon. In general | FortcullU. there were a succession of portcullises in the same gateway. It is sometimes called a portcluse. The I portcullis, with the chains by which it was moved | attached to its upper angles, formed an armorial [ bearing of the house of Lancaster; and is of fre¬ quent occurrence as a sculptured ornament on buildings erected by the monarchs of the Lancas- [ ter family, as on Henry VII.'s Chapel and King’s I College Chapel, Cambridge. As an architectural device, it is usually surmounted by a crown, and placed alternately with a rose, also surmounted by I a crown. PORTICO. — An open space before the en- I trance of a building, fronted with columns. Por¬ ticoes are distinguished as prostyle or in antis, as | they project before or recede within the building. They are further distinguished by the number of I their columus ; as a tetrastyle, hexastyle, and octa- style portico. .See Amphiprosttle and Ant.e. POST.— A piece of timber set upright, and in- ■ tended to support something else; as the posts of a house, the posts of a door, the posts of a gate, the posts of a fence. It also denotes any vertical piece j of timber, whose office is to support or sustain in a vertical direction ; as a kingpost, queen-post, truss- post, door-post, &c. — Post and pailing, a close wooilcn fence, constructed with posts fixed in the ground and pales nailed between them. —Post and railing, a kind of open wooden fence for the pro¬ tection of young quickset hedges, consisting of posts and rails, &c. These terms are sometimes con¬ founded. POST AND PANE, Post and Pan, Post and Pbtrail —Another name for half-timbered houses, or those in which the walls are composed of timber framing, with panels of brick, stone, or lath and plaster. See p. 156, Plates XLVI. and XLVII. POST AND PETRA IL. See Post and Pane. POSTERN.—Primarily, a back door or gate; a private entrance. Hence, any small door or gate. POSTICUM. — The part of a temple at the I rear of the cella or body. See Proxaos. POSTIQUE. — Superadded ; done after the main work is finished. Applied to a superadded ornament of sculpture or architecture. POYNTELL.—Paving set in squares or lozenge forms. | PRACTICAL CARPENTRY.—Introduction to, p. 134. PRaECINCTIONES.—T he passages between the rows of seats in the Roman theatres, called also bailee or belts. PRESERVATION OF TIMBER by Sur¬ face Applications, as tar, pitch, tallow, paints, I paint with sand, and sheathing with copper or j copper nails, p. 108. PRESERVATION OF WOOD by impreg¬ nating it with Chemical Substances.—K yan’s process with corrosive sublimate, called kganizing, p. 106; Margary’s process, acetate or sulphate of copper, p. 106 ; Burnett’s process, chloride of zinc, p. 106; Payne’s process, two solutions, p. 106, 107; Bethell’s process, creosote, p. 107; Boutigny’s pro¬ cess, oil of schistus, p. 107; Bouclierie’s process with various chemical solutions, absorbed by as¬ piration, p. 107 ; mode of impregnating the wood with solutions for its preservation, p. 107, 108. PRICK-POST.—The same as queen-post (which see), p. 156. PRICKER, p. 42. PRICKING-UP.—In plastering, the first coats of plaster in three coat work upon lath. PRINCIPALS, or Principal Rafters.— Those which are larger than the common rafters, and which are framed at their lower ends into the tie-beam, and at their upper ends are either united , j at the king-post, or made to bear against the ends of the straining-beams when queen-posts are used. | The principals support the purlins, which again | carry the common rafters, and thus the whole weight of the roof is sustained by the principals. | The struts, braces, &c., used in framing with the principal rafters, are sometimes called principal • struts, principal braces, &c. PRINT.—A plaster cast of a flat ornament, or ! an ornament of this kind formed of plaster from a i mould. PRISM.— A solid, of which the ends are equal, ' similar, and parallel rectilineals; and the other sides j are parallelograms.—To find the surface of a prism. Rule: Find the area of one of its ends, and to its i double add the sum of the areas of the parallelo¬ grams.—To find the solidity of a prism. Rule: Find the area of one of its ends, and multiply it by , the length or perpendicular height. PRISM, or Pyramid. — Development of a, j p. 70. PRISM. — Development of the covering of a prism, p. 71. PRISMOID.—A body that approaches to the form of a prism; a solid having for its two ends any | dissimilar parallel plane figures of the same number of sides, and its upright sides trapezoids. PROFILE.—The outline or contour of any¬ thing. such as a building, a figure, a moulding. PROJECTION, Projecture. — The jutting out of certain parts of a building beyond the naked wall, or the jutting out of anything in advance of a normal line or surface. PROJECTION. — Definition of projection, p. 46; general principles of projection, p. 47; planes of projection illustrated, p. 47. PROJECTION, — Intersections of lines and planes under various conditions, p. 47-52. PROJECTION OF SOLIDS, p. 52. PROJECTIONS.—To find the projections of the intersections of two cylinders under various conditions, p. 63; to find the projections of the in¬ tersections of a sphere and c^lincfer, p. 64; to find the projections of the 'intersections of two right cones, p. 64. PRONAOS.—The space in front of the naos or cella of a Greek temple. The term is sometimes used for portico. See Naos. PROPERTIES OF TIMBER.— 1 Table of the, p. 133. PROPORTION. — In architecture, the just magnitude of each part, and of each part compared with another in relation to the end or object in PROPORTION of Tread and Riser of a Step, p. 196. PROPORTIONAL COMPASSES. — Differ¬ ent kinds of, and their use, p. 33. PROPYLARUM, Propylon. — The porch, vestibule, or entrance of an edifice. PROSCENIUM.—That part of a theatre from the curtain or drop-scene to the orchestra. In the ancient theatre it comprised the whole of the stage. PROSTYLE.—A range of columns standing detached from the building to which they belong. See Portico. PROTECTION OF WOOD against Fire, p. 108; means adopted, p. 108; not efficacious, p. 3 09. PROTRACTOR. — An instrument for laying down and measuring angles on paper. See p. 41. PSEUDISIDOMON. — A manner of building among the Greeks, in which the height, length, and thickness of the courses differed. PSEUDO - DIPTERAL. — Falsely or imper¬ fectly dipteral, the inner range of columns being omitted. A term denoting a building or temple, wherein the distance from each side of the cella to the columns on the flanks is equal to two inter- colunmiations, the inner range of columns necessary to a dipteral edifice being omitted. As a noun, an imperfect peripteral, in which the columns at the wings were set within the walls. See Peripteral. PSEUDO - PERIPTERAL.—A term applied to a temple having the columns on its sides attached to the walls, instead of being arranged as in a per¬ ipteral . PSEUDO-PROSTYLE. — A term suggested I by Professor Hosking, to denote a portico, the pro¬ jection of which from the wall is less than the width of its intercolumniation. PTEROMA.—The space between the wall of the cella of a temple and the columns of the peri¬ style; called also ambulatio. PUG-PILES.—Piles mortised into each other by a dove-tail joint. 'They are also called do re¬ tailed piles. PUG-PILING.—A mode of fixing piles by mortising them into each other by a dove tail joint. Also termed dove-tailed piling. PUGGING. — Any composition, generally a coarse kind of mortar, laid on the sound boarding under the hoards of a floor, to prevent the trans¬ mission of sound. In Scotland it is termed deaf¬ ening. See p. 151. PULLEY-MORTISE.— The same as chase- mortise (which see). PULLEY-STYLE.—The style of a window- case, in which the pulleys are fixed. See p. 187. PULPIT, Pulpitum. — An elevated place or inclosed stage in a church, in which the preacher stands. It is called also a desk. Pulpits in mo¬ dern churches are of wood, hut in ancient times some were made of stone, others of marble, and richly carved. Pulpits were also sometimes erected on the outside of churches as well as within. PULPIT, with Acoustical Canopy. — Illustra¬ tion and description of, Plate LXXXIIJa p. 190. PULVINATED.— A term used to express a swelling in any portion of an order, such for in¬ stance as that of the frieze in the modern Ionic order. PUMP-BTT.—A species of large auger with removable shank, such as is commonly used for boring wooden pump-barrels. PUNCHEON. — A post (see Fig. 3, Plate XXXII., Roof of Westminster Hall, and descrip¬ tion, p. 145); a small upright piece of timber in a partition, now called a quarter. PURLIN.—A piece of timber laid horizontally, resting on the principals of a roof to support the common rafters. Purlins are in some places called ribs. PURLINS.—To find the dimensions of, p. 137. PUTLOGS.—Short pieces of timber used in scaffolds to carry the floor. They are placed at right angles to the wall, one end resting on the ledgers of the scaffold and the other in holes left in the wall, called putlog-holes. See Ledger. PUTTY.—1. A very fine cement made of lime only. The lime is mixed with water until it is of such consistence that it will just drop from the end of a stick. It is then ran through a hair sieve, to remove the gross parts. Putty differs from fine - INDEX AND GLOSSARY. PYCNOSTYLE Stuff in the mode of its preparation, and in having no hair mixed with it.—2. A cement compounded of whiting and linseed oil, kneaded together to the consistence of dough, used for stopping small cavi¬ ties in wood-work and for fixing the glass in window frames. PYCNOSTYLE. — In ancient architecture, a building where the columns stand very close to each other. To this intercolumniation one diame- ^ ter and a half is assigned. . PYRAMID.—A solid which has a rectilinear ( figure for its base, and for its sides triangles with a ; common vertex.—To find the surface of a pyramid. Rule: Find separately the area of the base and the areas of the triangles which constitute its sides, and j add them; their sum will be the whole surface. - To find the solid content of a pyramid. Rule : Find the area of the base, and multiply it by the perpen¬ dicular height, and one-third of the product will be ( the solid content. QUIRKED OVOLO, The.—To describe, P " VjUOIN.— 'The external angle of a building, and joined, generally the stones of which that angle is formed, j When the quoin stones project beyond the general surface of the wall, and have their arrises chain- f er ed, they are called rustic quoins. Quoins are sometimes in Scotland called external corners. | REINS OF A VAULT a joint formed by making rebates or longitudinal recesses in the opposite edges of the boards to be See p. 182, 183. 1 V \ Rohate Joints. R. Q. RABBET. See Rebate. RAD AND Dab.—A substitute for brick-nogging partitions, consisting of cob or a mixture of clay j l . , r II -.1 ■ _1.. 11,., nnlil REBATE.—A kind of hard freestone used in pavements; also, a piece of wood fastened to a long stick, for beating mortar. REBATE (more commonly spelled rybat ).— The rebated reveal of a door or window. See illus¬ tration, Out and In Bond. REBATE PLANES.—Planes used in forming and finishing rebates in joiner-work ; or, as it is technically termed, sinking rebates. Of these there are the moving fillister, used, in sinking rebates, on the edge of the board next to the workman, and the sash fillister in sinking the rebate on the edge fur- and chopped straw, filled in between laths of split j t j iest f rom him; and the guillaumes, skewed and oak or hazel; called also wattle and dab. ; S q Uare the former for finishing the rebate across RADIUS OF A Circle.— The span or chord , direction of the fibre, and the latter for finish- —-. > . c , the direction of the fibre, — v..v and versed sine of a circle being given, to find the ing ifc in the direction of the fibre, radius, Prob. LXI. p. 17- | ° RECESS.—A small cavity o: RAFTERS.—Pieces of timber winch form the QUADRA.—A square frame or border inclos- I ing a bas-relief, but sumetimes used to signify any i frame or border; the plinth of a podium. ( QUADKJE.—'The fillets above and below the scotia of the Attic base. QUADRANGLE.—A square surrounded with buildings, as a cloister or the buildings of a college. QUADRILATERAL figure equal to a penta¬ gon.—To describe, Prob. XXXVI. p. 11. QUARREL.—A lozenge-shaped pane of glass used in leaded casements, or any pane of glass so used; also, the opening in which the glass is set; a small square or lozenge-shaped paving tile or stone. QUARTER-GRAIN.—When timber is split in the direction of its annular plates or rings. When it is split across these, towards the centre, it is called the felt-grain. QUARTER-ROUND.—The echinus moulding. QUARTER-SPACE.—The name given to the resting-place or foot-pace of a stair, when it occurs at the angle turns of the stair. See p. 196. QUARTERED PARTITION. —A partition formed with quarters. See Plate XLV. p. 155. QUARTERING.—Forming partitions with quarters. . QUARTERS.—The vertical timber-framing of a partition to which the laths are nailed; called also studs, and in Scotland standards. See p. 156 QUATREFOIL.—A piercing or p; by cusps or foliations into four leaves, or more cor¬ rectly the leaf-shaped figure formed by the cusps. An ornament similar to the four leaves of a cruci- I the wall of a building. ity or niche formed in Exhcdne, tribunes, alcoves framework of the slopes of a roof. Common rafters j come un( j er this denomination, and afford consider- those to which the slate boarding or lathin attached. See Roof. RAFTERS.—To calculate the dimensions of, p. 137- RAG-WORK.—A kind of rubble masonry, formed with stones about the thickness of a brick. RAG LET, or Raglin (corruption of regula). — A rectangular groove cut in stone or brick work. „ „ RAGLINS.—A term used in the north of Eng¬ land for the slender ceiling-joists of a building. RAILS.—The horizontal timbers in any piece of framing. See Framing and Door, and also p. 186. RAISED PANELS, p. 185. RAISING PLATE, Reson Plate.— The wall- plate; or more generally, any horizontal timber which, laid on walls or borne by posts or puncheons, sustains other timbers. See p. 156. RAKE.—-A slope or inclination. RAKER.—An iron tool used by bricklayers to rake out decayed mortar from the joints of brick¬ work, preparatory to repointing them. RAKING COURSES.—Diagonal courses of brick, laid in the heart of a thick wall, between the outside courses. AU(J RAKING MOULDINGS.—Those which are ianel formed | inclined from the horizontal line, as in the sides of a pediment. RAKING MOULDINGS.—To describe, RAMP.—Literally, a spring < bound ; any form flower, frequently used as a decoration in hol¬ low mouldings in the early English and Decorated styles, but which it has been proposed to distinguish by the term quatrelobe. QUEEN-POST.—The suspending posts in the framed principal of a roof, or in a trussed partition, or other truss where there are two. See woodcut, under Roof. When there is only one post, it is called a king-post or crown-post. See p. 136. QUEENS.—In slating, slates measuring 3 feet by 2 feet. QUERCUS ROBUR, p. 109; Q. pcdunculata, p. 109 ; Q. sessilijlora, p. 109 ; Q. ilex, p. 109 ; Q. suber, p. 109; Q. pyrenaica, p. 109 ; Q. cerris, p. 110; Q. virens, p. 110; Q. alba, p. 110. QUINDECAGON.—A plane figure with fifteen sides and fifteen angles. QUIRK.—A deep indentation; the hollow under the abacus of a column. QUIRK BEAD. p. 184. QUIRK MOULDINGS.—Mouldings whose sudden rising interrupting the continuity of a slop¬ ing line; a concave sweep connecting a higher and i lower part of some work, as the coping of a wall, or the higher and lower parts of a stair-railing at a half or quarter pace; a flight of steps, or the line tangential to the steps. RAMP and Twist.—A line which rises and winds at the same time, as the section of a vertical I cylinder by an inclined plane. RAMPANT ARCH.—One whose imposts are not of the same height. See Rampant Ellipses, ! Figs. 167, 169, p. 24, 25. | RANCE.—A shore or prop. [Scotch.] RAND.—A border or margin, or a fillet cut from a border or margin in the process of straight- apparent projection is increased by the addition of a quicker curve. ening it. RANDOM TOOLING. —In Scotland called droving, is a mode of hewing the face of a stone, either as preparatory to some other process, or as a finishing operation. The chisel used is from 2 to 4 inches broad, and is called variously a broad tool, a looler, or a drove. Having ground it to a fine and even cutting edge, the workman holding it in his left hand with its edge upon the stone, strikes its head with a wooden mallet, so as to advance it about g inch, producing a cut or indentation of that length, and of the whole width of the tool. By successive strokes the tool is advanced to the extre¬ mity of the stone, and the result is a series of in¬ dentations or flutings at right angles to its .path, or to the draught, as it is termed. The same opera¬ tion is repeated till the whole surface of the stone is worked over. The excellence of the work depends [ on the regularity of the minute flutings, and the j absence of ridges between the draughts. REBATE.—A rectangular longitudinal recess made in the edge of any substance. Thus the rect- angular recess made an a- door-frame, into which j the door shuts, is a rebate.—Rebate joint, in joinery, 279 able additional space. They add to the commodious¬ ness of dining-rooms, drawing-rooms, libraries, &e. RECESSED ARCH. —One arch within an¬ other. Such arches are sometimes called double, triple, See., and sometimes compound arches. RECTANGLE.—To construct a rectangle equal to a given triangle, Prob. XXIII. p. 9.— To construct a rectangle whose sides shall be equal to two given lines, Prob. XXV. p. 9.— To describe a rectangle or parallelogram having one of its sides equal to a given line, and its area equal to a given rectangle, Prob. XXVII. p. 10.—To construct a rectangle upon a given line, and equal to a given square, Prob. XXVIII. p. 10.—To describe a rect¬ angle equal to a given rhomboid, Prob. XXXIV. p. 11 .—To describe a rectangle equal to a given irregular quadrilateral figure, Prob. XXXV. p. 11. —To describe a rectangle equal to a given circle, Prob. LXXIV. p. 20. RECTILINEAR FIGURE.—To describe any figure equal and similar to a given rectilinear figure, Prob. LIII. p. 15.—To describe, on a given line, a figure similar to a given rectilinear figure, Prob. LIV. p. 15. RED or Yellow Pine. p. 116. RED or Yellow Pink. —Causes to which the rapid decay of home-grown timber may be attri¬ buted. See p. 117- RED or Yellow Pine. — Characteristics of Memel, Norway, and Swedish timber. See p. 117- REEDS.—A moulding consisting of several beads side by side. REGLET, Regula. —A small moulding, rect¬ angular in its section; a fillet or listel. Also, a rectangular groove. REGRATING. — Rehewing or otherwise re¬ newing the surface of a hewn stone. REGULAR BODIES.—Those which have all their sides, faces, and angles similar and equal. They are only five in number—namely, the tetra¬ hedron, hextedron, octahedron, dodecahedron, and icosahedron. These are also called the Platonic bodies.— To find the surface of any of the regular bodies. Rule: Multiply the square of its linear side by the proper number in the annexed table under “surface;” and to find the solid content, multiply the cube of its linear side by the proper number under “solidity.” Tetrahedron. . Hextedron . . Octahedron . Dodecahedron . Icosahedron . . 17320508 6-0000000 3-4641016 20-6457788 8-6602540 S&T 04714045 7-6631180 2-1816950 REGULAR FIGURES.—Those whose sides and angles are equal, as the square and equilateral triangle, pentagon, hexagon, &c. Circles can be inscribed within and about all the regular figures, and the area of any of them may be found by the particular rule given under the proper word, or by j this general rule : Multiply half the perimeter of the figure by the perpendicular let fall from the cir- ! cumscribed or inscribed circle, on any of its sides. REGULAR POLYHEDRONS, The.—De¬ scription of, p. 69. „ REGULAR POLYHEDRONS, The. — De¬ velopment of, p. 70. REINS of A Vault. —The sides or walls which ' sustain it. INDEX AND GLOSSARY. RELIEF RELIEF, Rilievo.— The projectnre or promi¬ nence of a figure above or beyond the ground or plane on which it is formed. Relief is of three kinds: high relief {alto rilievo ), low relief {basso High Heliel'. rilierO), and half relief {mezzo rilievo). The differ¬ ence is in the degTce of projection. High relief is formed from nature, as when a figure projects as Low Relief. much as the life. Low relief is when the figure pro¬ jects but little, as in medals, festoons, foliages, and other ornaments. Half relief is when one-half of the figure rises from the plane. REMOVING CENTRES, or Striking Centres, p. 175. RENAISSANCE.—A term applied to the style of building and decoration which came into vogue in the early part of the sixteenth century, profes¬ sedly a return to the classic architecture of Greece and Rome. It was, nevertheless, a tasteless ad- herence to the dogmas and rules of Vitruvius, in 1 which everything was to be designed by rule and ! line, and nothing left to the invention of the archi¬ tect. Chateaubriand characterizes the French ex¬ amples of the Renaissance as ‘‘bastard Roman, cold and servile, neither in harmony with the climate, i nor suited to the wants of the people.” RENDER.—To plaster on walls, slates, or tiles, directly and without the intervention of laths. 1 RENI)ER and Set. —Two-coat plasteron walls. RENDER, Float, and Set. — Three-coat ! plaster on walls. REREDOS, Rerdos, Reredosse.—T he back of a fire-place ; an altar-piece ; a screen or partition wall separating the chancel fromthebodyofachureh. RESINOUS WOODS.—General description of, P. 1IA RESISTANCE OfTimbeu.—T o tension, p.124. —To compression, p. 124.—To transverse strain p. 126.—To torsion, p. 124. RESOLUTION and Composition of Forces. --The term resolution of forces or of motion, in j dynamics, signifies the dividing of any single force or motion into two or more others, which, acting in different directions, shall produce the same effect as ^ the given motion or force This is the reverse of composition of forces or of motion. Thus let a is, in the annexed diagram, represent the quantity and direction of some given force; draw any lines A c, a d ; and join c B, D u, and complete the parallelo¬ grams a d r k, a c B f. Then by composition of forces the force a r is equivalent to A D and A E, or to a c and A F. Hence it is evident that a given 1 D force, as a b, maybe resolved into as many pairs of I forces as there can be triangles described upon a i given straight line a B, or parallelograms about it. 1 And as the forces represented by ad, db, or a c c B, may also be resolved into other pairs of forces,’ it appears that by proceeding in the same manner with the successive pairs of forces, a given force j may he resolved into an unlimited number of others, acting in all possible directions. Sec p. 120. RESPOND. A pilaster or half pillar respond¬ ing to another similar, or to a whole pillar opposite j to it. RESTING PLACE.—A half or quarter pace in a stair. RESTING POINTS.—In handrailing, the heights set up to obtain the section of a cylinder in forming the wreath. See p. 202. ROMANESQUE The workman then, with a pitching tool and mal¬ let, or with a hammer similar to a nidging ham¬ mer, breaks or splits away pieces from the face and arrises of the stone, striving to avoid the appear¬ ance of formality, and taking care not to leave tool marks. It is an especial object, in taking out the pieces from the edges, that those in the two con¬ tiguous stones shall correspond as nearly as may be in size and depth, so that the whole surface of the wall, when completed, may look as inartificial i RESULTANT.—In dynamics, the force which possible. It is only’in dressing such stones as i results from the composition of two or more for 1 acting upon a body. When the two forces act upon i » body in the same line of direction, the resultant : is equivalent to the sum of both ; when they act in opposite directions, the resultant is equal to their | difference, and acts in the direction of the greater. If the lines of direction of the two forces are inclined to each other, then on taking in each direction, from 1 the point where they intersect, a straight line to represent each of the forces respectively, and con¬ structing a parallelogram of which these lines are the adjacent sides, the resultant is represented in intensity and direction by the diagonal of the paral¬ lelogram passing through the point of intersection. I By combining this resultant with a third force, a new resultant will be obtained; and in this manner | the resultant of any number of forces may be deter- I mined. See p. 120. RETICULATED MOULDING—In arclii- ! tecture, a member composed of a fillet interlaced in various ways like network. It is seen chiefly in buildings in the Norman style. RETICULATED WORK.—In architecture, that wherein the stones are square and laid lozenge- admit of a piece being struck out of their face by a blow, without leaving a hammer mark, that rock- work is admissible. Rock-work, formed by the chisel and mallet, is insipid in the extreme, and its :-k of judg- i well as a la< lnsipu use evinces bad taste, rnent. ROCOCO. — A debased variety of the Louis- Quatorze style of ornament, proceeding from it through the degeneracy of the Louis-Quinze. 1 1 is Reticulated Work. wise, resembling the meshes of a net. This species of masonry was very common among the ancients RETURN.—Adis ease of trees. Sec p. 97. RETURN, in building, denotes a side or part that falls away from the front of any straight work. RETURN BEAD.—One which shows the same appearance on the face and edge of a piece of stuff, forming a double quirk. REVEALS, or Revels.— The sides of an opening for a door or window, between the frame¬ work and the face of the wall. In Scotland termed frequently rybat-liead, or, probably from the way in which it is cut rebate-head. See woodcut, under Back-Fillet. RIBBING.—An assemblage of ribs. RIBS, -in carpentry and joinery, are curved pieces of timber to which the laths are fastened, in forming domes, vaults, niches, &c. In architecture, projecting bands or mouldings used in ornamented ceilings, both flat and curved, but more commonly in the latter, especially when groined. RIDGE.—'1 be highest part of the roof of a building. But in architecture the term is more par¬ ticularly applied to the meeting of the upper end of the rafters. When the upper end of the rafters abut against a horizontal piece of timber, it is called a ridge-piece or ridge-plate. Ridge is also used to signify the internal angle or nook of a vault, liidgc- tile, a convex tile made for covering the ridge of a roof. RIDGE-PIECE, Ridge-Plate. — A piece of timber at the ridge ot a roof, against which the common rafters abut. Called also pole-plate. RIDGE-ROLL, or Ridge-Batten—A roundel piece of timber, over which the lead is turned in the ridges and hips of a roof. It is gene¬ rally about 2 inches diameter, and fixed to the ridge of the roof by spike3 about 4 feet apart. RIGA TIMBER. Pinos sylvestris, p. 116. RILIEVO. See Relief. RING-COURSE.—The outer course of &tone or brick in an arch. RIPPING SAW.—One used for cutting wood ! m the direction of the fibres. RISERS OF STEPS, p. 196. RISING HINGE.—One so constructed as to ! raise the door to which it is attached, as it opens. )RK. ou Rocking, in masonry, a generally a meaningless assemblage of scrolls and crimped conventional shell-work, wrought into all sorts of irregular and indescribable forms, without individuality and without expression. 'I bis term is sometimes applied in contempt to anything bad or tasteless in ornamental decoration. ROD.—A measure of length equal to 16.J feet. A square roil is the usual measure of brick-work, and is equal to 272j square feet. ROLL - MOULDING. — A round moulding divided longitudinally along the middle, the upper half of which projects over the lower. It occurs often in the later period of the Early English and Roll Moulding. in the Decorated style, where it is profusely used for drip-stones, string-courses, abacuses, &c. —lloll and fillet moulding, around moulding with a square Roll and Fillet Moulding. fillet on the face of it. It is most usual in the early Decorated style, and appears to have passed by various gradations into the ogee. ROMAN ARCHITECTURE.—The style of architecture used by the Romans. Founded on the Grecian architecture, the Roman is, though less chaste and simple, more varied, richer, and in some respects bolder and more imposing. It embraces two additional orders of columns, the Tuscan and the Composite. All its curved mouldings are more circular and have greater projection, and its pedi¬ ments are steeper. Ornaments, too, are more fre¬ quently introduced. It is further characterized by the use of the arch, which in its late periods was one of its leading features, and was unknown in the architecture of the Greeks. ROMANESQUE.—A general term for all those styles of architecture which, commencing with the Christian era, sprung from the Roman, and flour¬ ished in Europe till the introduction of Gothic ar¬ chitecture. In all these there is an evident imita¬ tion of the features of classical Roman architecture, altered and debased. There is still a prevalence of JOCK-WORK, or Rocking, in masonry, as horizontal lines, of rectangular faces, and square- us name implies, is that mode in which the stone 1 edged projections, and of arches supported on pillars us an ai tihcial roughness given to it to imitate retaining traces of classical proportion?. The open- 11 j 1 l . lni ;ice °* :i r °ck. It is thus performed: ings in the walls are small, and subordinate to the a . re ?ut r " un .d the face of the stone, j surfaces in which they occur; the members of the id joints are squared up. i architecture massive and heavy. The styles art SO from which the beds ; TNDEX AND GLOSSARY. known in their various modifications by the names of Byzantine, Lombard, Saxon, &c. ROOD.—A measure equal to 36 square yards, by which rubble masonry is valued in Scotland. Bubble walls at and below 18 inches thick are reduced to 1 foot, and above 18 inches thick to 2 feet. ROOD.—A measure of land, the fourth part of an acre, and equal to 40 square poles or J210 square yards. ROOD.—A cross, crucifix, or figure of Christ on the cross, placed in a church. The holy rood was a cross with an effigy of our Saviour, generally as large as life, elevated at the junction of the nave and choir, and facing the western entrance to the church. Sometimes images of the Virgin Mary and St. John were placed, the one on the one side, and the other on the other side, of the image of Christ. ROOD-LOFT, Rood-Tower.— The gallery in a church where the rood and its appendages were placed. This loft or gallery was commonly placed over the chancel-screen in parish churches, or be¬ tween the nave and chancel; but in cathedral churches it was placed in other situations. The rood tower or steeple was that which stood over the intersection of the nave with the transepts. ROOF.—The cover of a building, irrespective of the materials of which it is composed. Roofs are distinguished: 1st. By the materials of which they are formed, as stone, brick, wood, slate roofs, &c. 2d. By their form and mode of construction, of which there is great variety, as shed, curb, hip, gable, pavilion, and ogee roofs. 3d. They are fur¬ ther divided into high-pitched or low-pitched roofs, as their inclined sides make a greater or lesser angle with the horizon. In carpentry, roof signifies the timber frame-work by which the roofing or cover¬ ing materials of the building are supported. This consists in general of the principal ratters, the pur- two varieties of principals which are in common use; the first, the king-post principal, and the second, the queen-post principal, with the purlins and common rafters in situ. The mode of framing here exhibited is termed a truss. Sometimes, when the width of the building is not great, common raf¬ ters are used alone to support the roof. They are in that case joined together in pairs, nailed wheie they meet at top, and connected with a tie at the bottom. They are then termed couples or couple close. See p. 85. In Asia, the roofs of houses are flat or horizontal. The same name, roof, is given to the sloping covers of huts and cabins, to the arches of oven-furnaces, &c. ROOF COVERING.—Weight of the various kinds of, p. 137. ROOFS.—Various forms of, as arising from variety in the forms of buildings, p. 85-91. ROOFS.—Practical memoranda of construction in designing, p. 137. ROOFS.—Classified according to their forms and the combination of their surfaces, p. 134. ROOFS.—Slope of, according to various au¬ thors, p. 134; as settled by climate, p. 134; by M. Rondelet considered arbitrary, p. 134 ; as settled by material used as the covering, p. 134; Colonel Emy's remarks on, p. 134; Professor Robison’s remarks on, p. 135. ROOFS.—Examples of constructions of, p. 135. ROOFS.—Illustration of the principles of truss¬ ing of, p. 136. ROOFS. — Table of the weight of the usual coverings of, p. 137. ROOFS.—Mr. Tredgold’s rules for the dimen¬ sions of the timbers in, p. 137. ROOFS illustrated and described:— ROSE-WINDOW.—A circular window divid¬ ed into compartments by mullions or tracery radi- lins, and the common rafters. The principal rafters, or principals, as they are more commonly termed, are set across the building at about 10 or 12 feet apart; the purlins lie horizontally upon these, and sustain the common rafters, which carry the cover¬ ing of the roof. The preceding figures show the King-post roif, span 30 feet, King-post roof, span 33 feet 6 in., Compound roof, span 30 feet, Queen-post roof span 32 feet, Queen-post roof, span 60 feet. Queen-post roof, Railway Work¬ shops at Worcester, . Platform roof, span 70 feet, Queen-post M roof, span 47 feet, . Queen-post roof, span 40 feet, King-post roof, span 38 feet 9 in., Roof of George lleriot’s Schools, Edinburgh. Roof of Wellington Street Church, Glasgow,. Roof of the Parish Church, Elgin. Roof of the City Hall, Glasgow, . Roof of the East Parish Church, Aberdeen, .... Roof of Sheds, Liverpool Docks, Roof of Sheds, Liverpool Docks, Roof for a hall or church with nave and aisles, Queen-post roof for a hall or church with nave and aisles, Roof of the East Quay Shed, Salt- house Dock, Liverpool, Roof of 44 feet 8 inches span, Roof of 45 feet span, . Roof of 54 feet span, . Roof with principals constructed of timber and iron, . Roof with iron rafters, r iron on a wall with plaster, mortar, cement, lead, or other binding. SEAM OF GLASS.—The quantity of 120 lbs., of 24 stones of 5 lbs. each. SEASONING OF TIMBER, and the means employed to increase its durability, p. 104. SEASONING OF TIMBER.—By stoving, p. 104 ; by burying in dry sand, 104 ; by immersion in cold water, 105; by immersion in hot water, 105 ; by immersion in salt water, 105 ; by charring the surface, 105 ; by’ coating the surface with sub¬ stances impervious to the air, 105; Sir Samuel Bentham's observations on, 105. SECANTS, Line of, on the Sector. —Con¬ struction and use of, p. 39. SECOND BRICKS.—Bricks of a quality next to the finest mail stocks or cutters. They are used in the principal fronts of buildings. SECOND COAT.—In architecture, either the finishing coat as in laid and set plaster, or in ren¬ dered and set plaster; or it is the floating when the plaster is roughed in, floated, and set for paper. SECTION. —In architecture, the projection or geometrical representation of a building supposed to be cut by a vertical plane for the purpose of exhibiting the interior, and describing the height, breadth, thickness, and manner of construction of the walls, arches, domes, &c. SECTOR.—Construction and use of, p. 36. SECTROID.—The curved surface between two adjacent groins. See p. 77. SEDILIA.—The Latin name for a seat, which has come to be pretty generally applied by way of distinction to the seats for the priests in the south wall of the choir or chancel of many churches and j Scdilia, Bolton Percy, Yorkshire. cathedrals. In this country they are usually re- i cessed in the wall like niches, and three in number, for the use of the priest, the deacon, and sub-deacon, i during part of the service of high-mass. SELF-FACED.—A term used to denote the natural face or surface of a flag-stone, in contra¬ distinction to dressed or hewn. SEPTARIA.—A name given to nodules or 1 spheroidal masses of calcareous marl, whose inte- I rior presents numerous fissures or seams of some I crystallized substance, which divide the mass, i When calcined and reduced to powder, these sep- I taria furnish the valuable mortar called Roman or ' Parkers cement, which has the property of harden¬ ing under water. SERVICE TREE. — Properties and uses of p. 114. SET SQUARES, p. 44. SET-OFF, or Offset— The part of a wall, &c., which is exposed horizontally when the portion above it is reduced in thickness.—Also, the sloped mouldings which divide Gothic buttresses into stages. See Soakcement. SETTING.—The quality of hardening in plaster or cement; also, the fixing of stones in Avails or vaults.— Setting coat, the best sort of plastering on ceilings or walls. SETTING-OUT ROD.—A rod used by joiners for setting-out frames, as of windows, doors, &c. SETTLEMENTS.—Failures in a building oc- | casioned by sinking. SEVEREY, Seyery, Seberee, Sibary.—A I compartment in a vaulted roof; also, a compart- , ment or division of scaffolding. SHADING—Methods of, p. 22 4. SHADING by Flat Tints, p. 224. SHADING by Softened Tints, p. 225. SHADOWof a straight line.—Tofirnl the length j and direction of, the projections of the straight line and of the luminous point being given, Prob. I. P . 211. SHADOW.—To find the shadoAv of a straight i line inclined to the horizontal plane, the projections of the luminous point and of the straight line being given, Prob. II. p. 211. SHADOW.—To find the shadow of a straight i line inclined to two planes. Prob. III. p. 212. SHADOW.—To find the portion of the sha- 1 dow of a straight line, interrupted by a plane inclined to tho planes of projection, Prob. IV. j P- 212 - SHADOW.—To determine the shadow of a I straight line on the horizontal plane, the projec- tions of a solar ray and of the straight line being | given, Prob. VI. p. 213. SHADOW.—To determine the shadow cast by ' a straight line on a vertical wall, Prob. VII. p. 214. SHADOW.—To find the shadow cast by a I straight line upon a curved surface, p. 214. SHADOW.—To find the shadow of a circle upon tlie horizontal plane, p. 215. SHADOW.—To find the shadow of a circle on the vertical plane, p. 215. SHADOW.—To find the shadow of a circle on two planes, p. 215. SHADOW of a circle on a circular wall, p. 215. SHADOW of a circle situated in the plane of I the luminous rays, p. 215. SHADOW of a circle, wltose horizontal projec- | tion is perpendicular to a trace of a plane passing through the luminous ray, Prob. IX. p. 216. SHADOW.—To find the shadow of a cylinder under various conditions, Prob. XI. p. 216-218. SHADOW.—To find the shadow of the inte¬ rior of a concave cylindrical surface, Prob. XII. p. 21S. SHADOW.—To find the shadow of a cone on the horizontal plane, Prob. XIII. p. 218. SHADOW.—To find the sliadoAv on the con- ca\ - e interior of a cone, Prob. XIV p. 219. SHADOW. — To determine the shadow of a sphere on the horizontal plane, and the boundaries of shade on the sphere, Prob. XV. p. 220. SHADOW.—To find the shadow on the con¬ cave interior of a hemisphere, Prob. XVI. p. 220. SHADOW. — To determine the shadow in a niche, Prob. XVII. p. 221. SHADOW.—To find the shadow of a regular hexagonal pyramid on both planes of projection, Prob. XIX. p. 221. SHADOW. — To find the shadow cast by a hexagonal prism upon both planes of projection, Prob. XX. p. 222. SHADOW.—To determine the limit of shade in cylinders placed vertically, and likewise its shadow on both planes of projection, Prob. XXI. p. 222. [ SHADOW.—To determine the limit of shade in a cylinder placed horizontally, and its shadow on both planes of projection, p. 222. SHADOW.—To find the limit of shade in a cone, and its shadow on the two planes of projec- | tion, Prob. XXII. p. 222. SHADOW.—To find the shadow thrown by a cone upon a sphere, Prob. XXIII. p. 222. SHADOW.—To determine the shadow of a concave surface of revolution, Prob. XXIV. p. 223. SHADOWS, Projection of.—I ntroductory remarks, p. 209. SHADOWS projected by rays of light which are parallel among themselves Prob. V. p. 213. SHADOWS.—To find, on the circumference of a circle, the tangent points of planes passing through the light, when the circle is not in the plane of the light, Prob. X. p. 216. SHADOWS.—To determine the shadows of a cylinder whose axis is circular (such as a ring), Prob. XVIII. p. 221. SHAFT.—The shaft of a column is the body of it, between the base and the capital. It is also called the/iw< or trunk of the column. It always diminishes in diameter, sometimes from the bot¬ tom, sometimes from a quarter, and sometimes from a third of its height, and sometimes its out¬ line is a convex curve, called the entasis. In the Ionic and Corinthian columns, the difference of the upper and lower diameters of the shaft, varies from a fifth to a twelfth of the lower diameter. (See 28U | Column.) — Vaulting shafts, those which support ribs, or other parts of a vault —Shaft of a king-post, the part between the joggles .—Shaft of a chimney, the part which r ses above the roof for discharging the smoke into the air. SHAFTED IMPOST. — In mediaeval archi¬ tecture, an impost with horizontal mouldings, the section of the mouldings of tho arch above the im¬ post being different from that of the shaft below it. In a banded impost the sections are alike. SHAKE.—A fissure or rent in timber, occa¬ sioned by its being dried too suddenly, or exposed to too great heat. Shakes frequently occur in grow¬ ing timber from various causes. SHANK—Another name for the shaft of a column.— Shanks , or legs, names given to the plain j space between the channels of the triglyph of a | Doric frieze. SHANTY.—A hut or mean dwelling. SHED ROOF.—The simplest kind of roof, 1 formed by rafters sloping between a high aud a low I wall. SHEERS —Two masts or spars lashed or ! bolted together at or near the head, provided with a pulley, and raised to nearly a vertical position, used in lifting stones arid other building materials. SHEET PILES, Sheeting-Piles. —Piles formed of thick plank, shot or jointed on the edges, and sometimes grooved and tongued, driven closely together between the main or gauge piles of a coffer¬ dam or other hydraulic work, to inclose the space so as either to retain or exclude water, as the case may be. Sheeting-piles have of late been formed of iron. SHELL-BIT.—A boring tool used with the brace in boring wood; it is shaped like a gouge, that is, its sectiou is the segment of a circle, and when used it shears the fibres round the margin of the hole, and removes the wood almost as a solid core. SHINGLE.—A small piece of thin woed, used like a slate for covering a roof or building. Shingles are from 8 to 12 inches long, and about 4 inches broad, thicker on one edge than the other. In America they arc extensively used, and are there manufactured by machinery of a very ingenious and simple description. SHINGLE-ROOFED.—Having a roof covered witli shingles. SHINLOG.—The brick building by which the mouth of a brick kiln is closed. SHOE.—1. The inclined piece at the bottom of a water-trunk or lead pipe, for turning the course of tho water, and discharging it from the wall of a building.—2. An iron socket used in timber framing to receive the foot of a rafter or the end of a strut. SHOOT.—To plane straight, or fit by planing. SHOOTING. — In joinery, the operation of planing the edge of a board straight, and out of winding. SHOOTING BOARD.—An external fence or 1 guide used in shooting or planing the edges of boards, in which the piece to be planed is narrower than the face of the plane. The annexed figures Fig. J. Fig. 2. j are sections of shooting boards, fig. 1 being used ■ for a rectangular joint, and fig. 2 for a mortise I joint. In both figures, a is a piece of board on which the plane e lies on its side, and b, another piece on which the board to be planed, cl, is laid, c is a stop against which the edge of the wood is pressed. There are many other forms of shooting boards. SHORE.—A piece of timber or other material placed in such a manner as to prop up a wall or other heavy body.— Dead-shore, an upright piece fixed in a wall that has been cut or broken through for the purpose of making some alterations in the building. SHOULDER.—Among artificers, a horizontal or rect angular projection from the body of a thing. —Shoulder of a tenon, the plane transverse to the length of a piece of timber from which the tenon projects. It does not. however, always lie in the plane here defined, but sometimes lies iu different planes. See p. 147. SHOULDERING.—In slating, a fillet of haired lime laid under the upper edge of the smaller and I thicker kind of slates, such as those of Argyleshire, | to raiso them there and prevent their being open INDEX AND GLOSSARY. SHREDDINGS at the overlap, and also to make the joint weather- tight. SHREDDINGS.—In old buildings, short, light pieces of timber, fixed as bearers below the roof, forming a straight line with the upper side of the rafters. SHRINE.—1. A reliquary, or box for holding the bones or other remains ot departed saints. The primitive form of the shrine was that of a small church with a high-ridged roof, and similar to the j m h o ; i D i? SPAN-ROOF Squares. 1 ton of Westmoreland Slates will cover 2 I ton of Welsh Rags . . ,, 1J to 2 10U0 Duchess Slates . . 9 1000 Countess Slates . . 5 1000 Ladies Slates . . ,, 3.j 1000 Tavistock Slates ,, 2£ SLEEPERS.—Pieces of timber on which are laid the ground joists of a floor, and also, and more usually, the ground joists themselves. Formerly the term was used to denote the valley-rafters of a roof.—In railways, sleepers are beams of wood or blocks of stone firmly imbedded in the ground to I SITE.—The position or seat of a building; the | place whereon it stands. , SKETCH.—An outline or general delineation of anything; a first rough or incomplete draught of ! a plan or any design; as the sketch of a building. j SKEW.—A term used in Scotland for a gable- i coping or factable. SKEW, on A skew. —Oblique; as a stew-bridge. SKEW-ARCH.—An arch whose direction is not at right angles to its axis ; it is also frequently termed an oblique arch. i SKEW - BACK. — The sloping abutment in I | brick-work or masonry for the ends of the arched head of an aperture. In bridges, it is the course of ! sustain the rails, which are usually fixed to the ! masonry forming the abutment for the voussoirs of sleepers by means of cast-iron supports called chairs. j a segmental arch, and in iron bridges it is the abut- j SLIDING-RULE.—A mathematical instru- rortable Shrine, Malmesbury Abbey. hog-backed tombs of the ancient Greeks, still seen in Anatolia. Hence,—2. A tomb, of shrine-like configuration ; and,—3. A mausoleum of a saint, of any form; as the shrine of St. Thomas h, Becket at Canterbury. SHUTTERS. — The boards which close the aperture of a window. The shutters of principal windows are usually in two divisions or halves, each subdivided into others, so that they may be received within the boxings into which tire shutters are folded or fall back. The front shutter is of the exact breadth of the boxing, and also flush with it; the others, which are hidden in the boxing, are somewhat less in breadth, and are termed backfolds or backjlaps. Shutters, as above defined, may be considered as the doors of window openings, and arc formed upon the same principles as doors, but sometimes in place of being hinged to fold back, they are suspended and counterbalanced like win¬ dow-sashes, so as to slide ; and they are also made of laths jointed together and wound round a roller placed either horizontally above the soffit, or verti¬ cally at the side of the opening. See p. 188. SHUTTERS.—Linings for. p. 188. SIB ARY. See Sevebey. SICAMORE. See Sycamore. SIDE-HOOK.—In joinery, a rectangular pris¬ matic piece of wood, with a projecting knob at the euds of its opposite sides. The use of the side-hook is to hold a board fast, its fibres being in the direc¬ tion of the length of the bench-, while the workman is cutting across the fibres with a saw or grooving- plane, or in traversin'/ the wood, which is planing it in a direction perpendicular to the fibres. SIDE-POSTS.—In architecture, a kind of truss-posts placed in pairs, each disposed at the same distance from the middle of the truss, for the purpose of hanging the tie-beam below. In ex¬ tended roofs, two or three pairs of side-posts are used. Throughout the text they are called primary and secondary queen-posts. SIDE-TIMBERS, Side-Wavers.— 1 The former is the Somersetshire, and the latter the Lincoln¬ shire local name for purlins. SIEGE.—The name given in Scotland to the bench or other support on which a mason places his stone to be hewn, a term derived from the French. In England it is termed a banker. SILL.—The horizontal piece of timber or stone at the bottom of a framed case; such as that of a door or window.— Ground sills are the timbers on the ground which support the posts and superstruc¬ ture of a timber building.—The word sill is also | ment formed for the ribs. SKEW-BRIDGE.—A bridge in which the | passages over and under the arch intersect each I other°obliquely. In conducting a road or railway I through a district in which there are many natural i or artificial watercourses, or in making a canal I through a country in which roads are frequent, such j intersections very often occur. Before the introduc- I tion of railways skew-bridges were seldom erected, it being more usual to build the bridge at right i engineer, gauger, &c. , ° , , i- , _ ..r a. _i SiT.T-p.K'P’ATWfi'. ment or scale, consisting of two parts, one of which slides along the other, and each having certain sets of numbers engraved on it, so arranged that when a given number on the one scale is brought to coin¬ cide with a given number on the other, the pro¬ duct or some other function of the two numbers is obtained by inspection. The numbers may be adapted to auswer various purposes, and slide rules made to suit the necessities of the carpenter, angles, and to divert the course of the road or the stream to accommodate it. But in a railway, and sometimes in a canal, such a deviation from the straight line of direction is often inadmissible, and it therefore becomes necessary to build the bridge obliquely. SKEW-BRIDGES illustrated and described:— Skew-bridge over the river Don, Skew-bridge on the system of M. Somet, Skew - bridge over the Leith Branch Railway, near Portobello, . . Plate. Page. L. 1G2 LIII. 164 LV. 170 SKEW-CORBEL, Skew-Put, Skew-Table.— A stone built into the bottom of a gable to form an abutment for the coping. SKEW-FILLET.—A fillet nailed on a roof along the gable coping, to raise the slates there and throw the water away from the joining. SKIRTING, Skirting-Board. —The narrow vertical board placed round the margin of a floor. Where there is a dado this board forms a plinth fol¬ ks base; otherwise it is a plinth for the room itself. See p. 186. SKIRTING.—Method of scribing, p. 200. SKY-DRAIN.—A cavity formed round the walls of a building, to prevent the earth from lying SLIP-FEATHER, p. 1S2. SLIT-DEAL.—Fir boards a full half-inch thick. SLOP-MOULDING.—In brick-making, that kind of moulding in which water is used to free the clay from the mould, in place of the sand used in pallet-moulding. SMOOTHING-PLANE. See Plane. SNECKING.—A peculiar method of building in rubble-work. See Rubble. SNIPE’S-BILL PLANE.—In joinery, a plane with a sharp arris for forming the quirks of mould¬ ings. SOCKET-CHISEL. — A chisel made with a socket; a stronger sort of chisel, used by carpenters for mortising, and worked with a mallet. SOCLE.—A flat square member of less height than its horizontal dimension, serving to raise pedestals, or to support vases, or other ornaments. It differs from a pedestal in being without base or capital. A continued socle is one continued round a building. SOFFIT.—The under side of the lintel or ceil¬ ing of an opening; the lower surface of a vault or arch. It also denotes the under horizontal surface of an architrave between columns, and the under surface of the corona of a cornice. SOFFIT-LINING, p. 188. SOLAR, Sollar.—A loft; an upper chamber. See Sollar. SOLID OF Revolution generated by an ogee curve.—To describe the section of a, p. 69, Plate I. Fig. 10. SOLID OF Revolution generated by a lancet- formed curve.—To describe the section of a, p. 69, Plate I. Fig. 11. SOLIDS.—Sections of, p. 68. SOLIDS.—Coverings of, p. 69. SOLIDUM.—The die of a pedestal. SOLIVE.—A joist, rafter, or piece of wood, .igainst them and causing dampness, called also ! either slit or sawed. The word is i'rench, and is air-drain and dry-drain. ! sometimes, though rarely used by English writers. SKY-LIGHT.—A window placed in the top ; SOLLAR. — Originally an open gallery or of a house, or a frame consisting of one or more | balcony at the top of a house, exposed to the sun ; nclined planes of glass placed in a roof to light pas¬ sages or rooms below. " SKY-LIGHTS.—To find the length and back¬ ing of an hip, p. 189. SKY-LIGHTS, Octagonal, Domical, &c.—To find the ribs, window-bars, &c., p. 190. SLABS.—The outside planks or boards, mainly of sap-wood, sawn from the sides of round timber. SLACK-BLOCKS.—The wedges on which the used to denote the bottom pieces which support i centres used in the construction of bridges are sup- quarter and truss partitions, and the flat stones used to form the bottom of a drain are also called sills. SILVER FIR.—Properties and uses of, p. 118. SINES, Line of, on the Sector.—Construction and use of. p. 39. SINGLE FLOOR, Single Flooring, Single Joists, Single-joist Floor.—A pplied to naked flooring, consisting of bridging-joists only, p. 150. SINGLE HUNG.—Applied to a window with two sashes, when one only is moveable. SISSOO.—A tree well known throughout the Bengal presidency, and highly valued on account of its timber, which furnishes the Bengal ship¬ builders with their crooked timbers and knees. It is universally employed both by Europeans and natives of the north-west provinces of India, where strength is required. It is the Dalbcryia sissoo of botanists, and belongs to the papilionaceous divi¬ sion of the natural order Legumiuoste. ported, p. 173, 17. SLAP-DASH.—A provincial term for rough¬ casting. SLATE BOARDING.—Close boarding cover¬ ing the rafters of a roof, on which the slates are laid. In Scotland, called sarkiny. SLATES.—The various sizes of slates are thus named :— Doubles, Ladies, Countesses, Duchesses,. Imperials, . Queens, Welsh Longs, or Rags, 11x06 13x08 1 8 x 0 10 2 0 A square of slating i3100 superficial feet. A square of Westmoreland or Welsh Rag slating will weigh 10 cwts., and of Duchesses, Countesses, or Ladies slating 6 cwts. but latterly used to signify any upper room, loft, or | garret. I SOMMER. See Summer. SOUND TIMBER.—Krafft's mode of judging of, p. 98. SOUND-BOARDING. —The sound-boarding of floors consists of short boards generally, and preferably, split, not sawn, which are disposed transversely between the joists, and supported by fillets fixed to the sides of the joists, for holding the substance called pugging, intended to preventsound from being transmitted from one story to another. See Pugging. In Scotland, sound-boarding is termed deafening-boarding. SOUNDING-BOARD, or Sound Board.— A board or structure placed over a pulpit or other place occupied by a public speaker, to reflect the sound of his voice, and thereby render it more au¬ dible. Sounding-boards are generally flat, and placed horizontally, but concave parabolic sounding- boards have been tried, and found to answer better. See p. 190, and Plate LXXXIII?' SOURCE, Souse. —A support or under prop. SPAN.—In architecture, an imaginary line across the opening of an arch or roof, by which its extent is estimated. See Arch. SPAN-PIECE.—A name given in some places to the collar-beam of a roof. SPAN-ROOF.—A name sometimes given to the common roofing, which is formed by two inclined 28-5 INDEX AND GLOSSARY. SPANDREL i'n contradistinction to ; STALLS planes or sides, in contradistinction to a shed or lean-to. SPANDREL.—The irregular triangular space comprehended between the outer curve or ex trad os of an arch, a horizontal line drawn from its apex, and a perpendicular line from its springing. In Gothic architecture, spandrels are usually orna¬ mented with tracery, foliage, the. See Plate XXXII. Fig. 5, and Door-head with Spandrels, under Dripstone. gradually elongated, and more and more acute, SPAN D RE L-BRACKETING.—A cradling of suited at length in the elegant tapering spire; among brackets which is placed between curves, each of the many existing examples of which, probably, which is in a vertical plane, and in tho circumfer- that of Salisbury is the finest. In mediaeval archi- nce of a circle whose plane is horizontal. j lecture, to which alone they are appropriate, A wall built on the spires are generally square, octagonal, or circular in plan;"they are sometimes solid, more frequently charging water from a building. .' SQUARE STAFF.—Asqui SPANDREL-WALL, back of an arch filling in thespandr SPAR.—A small beam or rafter. In arehitec- hollow, and are variously ornamented with band: ture, spars are the common rafters of a roof, as dis- encircling them, with panels more or less enriched, tinguished from the principal rafters. and with spire lights, which are of infinite variety. SPECIFICATION.—A statement of particu- Their angles are sometimes cvocketted, and they lars, describing the manner of executing any work are almost invariably terminated by a finial. In about to be undertaken, and the quality, dimen- the later styles the general pyramidal outline is ob- sions, and peculiarities of the materials to be used, tained by diminishing the diameter of the building SPERE.—An old term for the screen across ; in successive stages, and this has been imitated in the lower end of a dining-hall, to shelter the en- 1 modern spires, in which the forms and details of trance. classic architecture have been applied to structures SPERVER.— An old term for the wooden | essentially-mediaeval. The term spire is sometimes frame at the top of a bed or canopy. .Sometimes restricted to signify such tapering buildings, crown- the term includes the tester, or head piece. It signi- ing towers or turrets, as have parapets at their base, fied originally a tent. When the spire rises from the exterior of the wall SPHERE, ok Globe, is a solid bounded by a of the tower without the intervention of a parapet, curved surface, every point of which is equidistant I it is called a broach. from a point within it called the centre.—To find , SPIRE-LIGHTS.—The windows of a spire, the surface of a sphere, or of any segment or zone SPIRIT-LEVEL.—An instrument employed ol it. Rule : Multiply the circumference of a great ! for determining a line or plane par alle l to the hori- circle of the sphere by the axis, or by the part of it zon, and also the relative heights of ground at two corresponding to the segment or the zone required: or more stations. It consists of a tube of glass the product will be the surface.—To find the solid nearly filled with spirit of wine or distilled water, coutentof asphere. Rule: Multiply the cube of the and hermetically sealed at both ends; so that when axis by '5230.—To find the solid content of a seg- held with its axis in a horizontal position, the bubble ment of a sphere. 1. When the axis and height of <>f air which occupies the part not filled with the segment are given. Rule: From three times the liquid rises to the upper surface, and stands exactly lance, and subtract the product from one-third of ! To describe a square equal to any number of given the cube of half the length of the spindle, then four j squares, Prob. XXX. p. 10.—To describe asquaie times the remainder multiplied by 3T410 will give equal to the difference between two unequal squares, the content. j Prob. XXXI. p. 10.—To describe a square equal •SPIRE.—The pyramidal or conical termination to any portion of a given square, Prob. XXXII. of a tower or turret. The earliest spires were merely | p. 10.—To describe a square about a given circle pyramidal or conical roofs, specimens of which still Prob. XLVII. p. 14.—To describe a square equal exist in Norman buildings, as that of the tower of j to a given circle, Prob. LXXV. p. 20, and Prob. Than Church in Normandy. These roofs, becoming ! LXXVI p. 20. SQUARE FRAMED.—In joinery, a work is said to be square framed or framed square, when the framing has all IJie angles of its styles, rails, and mountings square without being moulded. SQUARE FRAMING, p. 185. SQUARE SHOOT.—A wooden trough for dis- x fillet used as an igle staff in place of a bead-moulding, in rooms that are prepared for papering. SQUARING A Handrail.— The method of cutting a plank for a rail to a staircase, so that all the vertical sections may be rectangular. SQUARING of Timber. — Methods usually adopted, p. 99. S Q U I N C FI.—The small pendentive arch formed across the angle of a square tower, to sup¬ port the side of a superimposed octagon. The axis subtract twice the height, and multiply the re¬ mainder by the square of the height and by '523(5. 2. When the height and radius of the base are givei Rule: To three times the square of the radius add in the middle of the tube. The tube is placed within a brass or wooden case, having a long •pening on ! the side which is to be uppermost, so that the posi¬ tion of tho air-bubble may be readily seen. When , the square of the height, multiply the sum by the i the instrument thus prepared is laid on a horizontal height, and by *5236. The product is the content. r .. .. To find the contentof the middle zoneof a sphere. Rule: From the square of the axis or greatest dia¬ meter subtract one-third of the square of the height, then multiply the remainder by the height and by | '7354. —To find the content of any zone of a sphere. Rule : Add the square of the radii of the two ends to . the sum by twice the height and’by '7854. SPHERE.—To find the limits of shade on, and the shadow thrown by a sphere, p. 220. SPHERE penetrated by a cylinder.—To find the projections of, p. 64. SPHERE.—To find the projections of a sphere penetrated by an oblique scalene cone. p. 64. SPHERE.—Development of the surface of a sphere, p. 73. SPHERE.—To find the projections of a scalene cone penetrating a sphere, p. 64. SPHERE.—Tangent plane to a sphere, p. 61. SPHERE.—To construct the sections of a sphere by a plane, p. 00. SPH E R E.—To describe the section of a sphere, p. G8, Plate I. Fig. 7. SPHERE, The — Development of, p. 72. SPHERICAL BRACKETING.—llrackets sc Flan Section of Gottiic 'Window. A A. Thu internal Splay. opening through : vail for urface, the air-bubble stands in the very middle of , application of the term to these pendentives may the tube; when the surla.ee slope*, the bubble rises have been suggested by their resemblance to a corner to the higher end. It is used by carpenters and j cupboard, which was also called a squinch or sconce. joiners for ascertaining whether the upper surface , SQUINT.—In medueval architecture, a name of any work be horizontal. When employed in sur- given to an oblique opening in the wall of a church, veymg, it is attached to a telescope, the telescope 1 Squints were generally so placed as to afford a view . , , . , , a, 1 "! beil, g fifc ted to a frame or cradle of brass, I of the high altar from the transept or aisles. :-th.rd of the square of the height, then multiply "'Inch is supported on three legs. j STACK of Wood.—A pile containing 10S cubic SPLAY.—A sloped surface, or a surface which i feet, makes an oblique angle with another; as when the ' STAFF-ANGLE. See Square Staff. I STAFF-BEAD See Angle-Bead. STAGE.—The part between one sloping projec- | tion and another, in a Gothic buttress. Also, the ' horizontal division of a window separated by tran¬ soms. Sometimes the term is used to signify a floor, a story. STAIR.—A step, but generally used in the plural to signify a succession of steps, arranged as a way between two points at different heights in a building, &c. A succession of steps in a continuous 3oor, window, &c, [ line is called a flight of stairs; the termination of widens inwards. A large chamfer is called a splay. ; the flight is called a landing. Stairs are further SPOKE-SHAVE.—A sort of small plane used 1 distinguished by the various epithets, dog-legged, for dressing the spokes of wheels and other curved i newelled. open newelled, &c. work, where the common plane cannot be applied. STAIRCASE.—The building or apartment , ,, ,. „ . , ., ;.- SPOON-BIT.—A hollow bit with a taper point which contains the staiis, see p. 196. L . f C °/ the ’^h-and-plaster work < for boring wood. STAIRS and Handrailing.-I ntroductory ,11 2^pL 8 > u PP°_ rt “‘jk^apbencal.surface. SPRING BEVEL of a Rail. —The angle remarks, p. 195. which the top of the plank makes with a vertical | STAIRS.—Simple contrivances as substitutes plane which has its termination in the concave side, ! for stairs, p. 195.—Contrivances for economizing and touches the ends of the rail-piece. See p. 203. space in stairs, p. 195. SPRING COMPASSES.—Instructions in the I STAIRS.—Definitions of terms, p. 196. use of, p. 31. I STAIRS.—Method of laying down the plan of: SPRINGER.—The point where the vertical As applied to dog-legged stairs, . . 198 See support of an arch terminates, and the curve begins. As applied to newel stairs, . . . 198 SPHPRfiTn . . Thu lowest of the series of voussoirs of which an As applied to geometrical stairs, . . 199 * _r 3 ’ r fi f? ur ? approaching > arch is formed, being the stone which rests imme- As applied to elliptical stairs, ... 199 TTiftrv .spherical. In gee- diately upon the impost. See woodcut, AncH. The STAIRS. —Method of setting out when the I..';,.' ’ <• by ■ lie ivvu- but! b ilding neral plan underst.I, lution of an ellipse about one of its axes. When the | . I --- ’ b 1 ’ generating ellipse revolves about its Ioi axis, the spheroid is oblong or prolate: course of stones from v the SPHERICAL PENDENTIVES.—To the ceiling of a room with, p. 80. SPHERICAL \ AULT.—To determini heights of the divisions of a, p. S3. SPHERO - CYLINDRIC GROIN. — One formed by the intersection of a cylindrical vault with a spherical vault of greater dimensions, p- 7" its less o iger or major , when about • minor axis, the spheroid is oblate. The earth is an oblate spheroid, that is, flattened at the pules, so that its polar diameter is shorter than its equatorial diameter. SPHEROIDAL BRACKETING.—Bracket¬ ing which lias a spheroidal surface. SPINDLE, Circular.— A circular spindle is the solid generated by the revolution of a segment of a circle about its chord.—To find the solid tent of a circular spindlf called a springer. I p. J 96 , SPRINGING.—The point from which an arch i STAIRS.—Methods of lessening the inequality springs or rises .—Springing course, the horizontal of width between the ends of the winding steps by ines from which an arch springs orrises, i calculation, and also graphically, p. 199.—Forma- In carpentry, in boarding a roof, the setting the tion of carriages for various kinds of stairs, p. boards together with bevel joints, for the purpose of 199, 200. keeping out. the rain. STAKE-FALD HOLES.—A local name for SPRUCE.—Description and uses of, seep. 117. I putlog holes. SPUR.—Often used as a synonyme for strut. 1 STALLS.—Fixed seats, inclosed either wholly SQUARE.—To construct a square, the sides of ■ or partially at the back and sides.—The choir or which shall be equal to a given straight line, Prob. j chancel of a cathedral, collegiate church, and even XX \ . p. 9. To describe a square equal to a given of most small churches, bad, previous to the Eefor- if tlie upnmiln , ‘ , „ ' ?J ' dll I'L v the area rectangle. Prob. XXVI. p. 9.—To describe a square I mation, one or more ranges of wooden stalls at its ® ‘ ° 6 n 'J a the central dis- equal to two given squares, Prob. XXIX. p. 10.— I west end, the seats of which were separated from 286 INDEX AND GLOSSARY SUBSELLIA of securing them together. See Plates XXVII.— XXIX., and p. 146. STRENGTH and Strain of Materials. p. 123. STRETCHED OUT.— In architecture, a term applied to a surface that will just cover a body so extended that all its parts are in a plane, or may be made to coincide with a plane. STRETCHER.—A brick or stone laid hori¬ zontally with its length in the direction of the face of the wall. It is thus distinguished from a header, which is laid lengthwise across the thickness of the wall, so that its head or end is seen in the external face of the wall. STRETCHING COURSE. — A course of stretchers ; that is, of stones or bricks laid hori¬ zontally with their lengths in the direction of the face of the wall. See Heading Course. STRIKE-BLOCK.—A plane shorter than a jointer, used for shooting a short joint. STRIKING.—In architecture, the drawing of lines on the surface of a body; the drawing of lines on the face of a piece of stuff for mortises, and cut¬ ting the shoulders of tenons. In joinery, the act of running a moulding with a plane. —The striking of a centre is the removal of the timber framing, upon which an arch is built after its completion. STRIKING CENTRES, p. 175. STRING - BO ARD, String-Piece, or Stringer.— A board placed next to the well- hole in wooden stairs, and terminating the ends of the steps. STRING-COURSE. — A narrow moulding or projecting course continued horizontally along the face of a building, frequently under windows. It is sometimes merely a flat band. STRING S.—Eormation of, by various methods, p. 200, 201. STRINGS in staircases, p. 196. STRUT.—Any piece of timber in a system of framing which is pressed or crushed in the direction of its length. STRUTS.—In flooring, short pieces of timber about 1 i inch thick, and 3 to 4 inches wide, inserted STANCHION each other by large projecting elbow; were often enriched by panelling, am by canopies of tabernacle work, e STOCK and Bits. Sue Brace and Bits. STOCKHOLM TIMBER. See Pin us syl- vestris, p. 110, 117. STOOTHING.—A provincial term for batten¬ ing. STOPS.—In joinery, pieces of wood nailed on the frame of a door to form the recess or rebate into which the door shuts. STORY.—A stage or floor of a building, called in Scotland a flat; a subdivision of the height of a house; or a set of rooms on the same floor or level. A story comprehends the distance from one floor to another ; as a story of nine, ten, twelve, or sixteen feet elevation. Hence each floor terminating the space is called a story; as a house of one story, of two stories, of five stories. In the United States, the floor next the ground is the first story; in France and England tins is called the ground-floor, and the second from the ground is called the first floor or story. STORY-POSTS.—Upright posts to support a floor or superincumbent wall, through the medium of a beam placed over them. STORY - ROD.—A rod used in setting up a staircase, equal in length to the height of a story of a house, and divided into as many parts as there are intended to be steps in the stair, so that the steps may be measured, and distributed with ac¬ curacy. See p. 196. S'l’OUP. — A basin for holy water, usually placed in a niche at the entrance of Roman Catholic Stalls, Ilijjliam Ferrers Church, Northamptonshire. crochets, pinnacles, &c. Many beautiful examples , of these yet remain. STANCHION.— A prop or piece of timber j giving support to one of the main parts of a roof; also, one of the upright bars, wood or iron, of a ! window, screen, railing, &c. STANDARD.—In joinery, any upright in a j framing, as the quarters of partitions, the frame of a door, and the like. STANZA. — An apartment or division in a building. STARLINGS, or Sterlings. —An assemblage of piles driven round the piers of a bridge to give them support. They are sometimes called stilfs. STEENING, or Steaning. — The brick or stone wall, or lining of a well or cess-pool, the use of which is to prevent the irruption of the sur¬ rounding soil. STEEPLES, Towers, and Spires of Timber. —Construction of w described p. 145, 146, and illus¬ trated in Plate, XXXV., XXXVa, and XXXVI. STEP. — One of the gradients in a stair; it is composed of two fronts, one horizontal, called the tread, and one vertical, called the riser. STEREO BATE.—The same as stylobate (which see). STEREOGRAPHY. —The act or art of de¬ lineating the forms of solid bodies on a plane; a branch of solid geometry which shows the construc¬ tion of all solids which are regularly defined. See p. 46. STEREOTOMY.—A branch of stereography, which teaches the manner of making sections of solids under certain specified conditions. STICKING.—The operation of forming mould¬ ings by means of a plane, in distinction from the operation of forming them by the hand. STTLTED ARCH.—A term applied to a form of the arch used chiefly in the twelfth century. In between flooring joists sometimes diagonally, as in the figure, to stiffen them. See p. 151, and illus¬ trations of various modes of strutting. Plate XLII. STRUTS and Ties. — To find whether a piece of timber in a system of framing is acting as a strut or a tie. p. 122. STRUTTING BEAM, Strut-Beam.— An old name fora collar-beam. STRUTTING-PIECE.—The same as strain¬ ing-piece. STUB-MORTISE.—A mortise which does not pass through the whole thickness of the timber. STUCCO—A word applied as a general term to plaster of any kind, used as a coating for walls, and to give them a finished surface. The third coat of three-coat plaster, consisting of fine lime and sand, is termed stucco ; it is floated and trowelled. There is a species called bastard stucco, in which a small portion of hair is used. It is merely floated and brushed with water. STUCK MOULDINGS.—In joinery, mould¬ ings formed by planes, instead of being wrought by the hand. STUDS.—In carpentry, posts or quarters placed in partitions, about a foot distant from each other. STUDWORK.—A wall of brick-work built between studs. STUMP-TENON, p. 1S2. STYLE. See Stile, and p. 1S6. STYLOBATE.—In architecture, in a general sense, any sort of basement upon which columns are placed to raise them above the level of tho ground or floor; but in its technical sense, it is ap¬ plied only to a continuous unbroken pedestal, upon which an entire range of columns stand, contradis¬ tinguished from pedestals, which are merely de¬ tached fragments of a stylobate placed beneath each column. SUBPLINTH.—A second and lower plinth I placed under the principal one in columns and I pedestals. SUBSELLIA.—The small shelving seats in the stalls of churches or cathedrals, made to turn up upon hinges so as to form either a seat, or a form STRAIGHT-EDGE. - In Slr " 8 '"' A " 1 ' joinery, a slip of wood made perfectly straight on the edge, and used to ascer¬ tain whether other edges are straight, or whether the face of a board is planed straight. It is made of different lengths, according to the required mag- , nitude of the work. Its use is obvious, as its ap¬ plication will show whether there is a coincidence between the straight-edge and the surface or edge to which it is applied. It is also used for drawing straight lines on the surface of wood. See Winding Sticks, p 44. STRAIGHT-JOINT FLOOR, p. 185. STRAIGHT LIN E.—To bisect a straight line, Prob. IX. p. 7-—To divide a straight line into any number of equal parts, Prob. X. p. 7.—To find the shadow of a straight line inclined to the horizontal plane, p. 211.—To find the shadow of a straight line inclined to two planes, p. 212.—To find the shadow of a straight line intercepted by a plane in¬ clined to the plane of projection, p. 212.—To de¬ termine the shadow of a straight line on the hori¬ zontal plane, p. 213.—To determine the shadow of a straight line on the vertical plane, p 214.—Shadow thrown by a straight line on a curved surface, p. 214. STRAIGHT STATRS. p. 196. STRAIN and Strength of Materials. p 123. STRAINING PIECE.—A beam placed be¬ tween two opposite beams to prevent their nearer approach ; as rafters, braces, struts, &c. If such a piece performs also the office of a sill, it is called a straining sill. STRAINING PIECES in a partition. Plate XLV. Fig. 1. h ; Fig. 2, No. 2, /; Fig. 3, No. 1, /, p. 155. STRAP.—In carpentry, an iron plate placed ; across the junction of two timbers for the purpose 287 this form the arch docs not spring immediately from the imposts, but is raised as it were upon stilts for some distance above them. STINK-TRAP.—A contrivance to prevent the passage of noxious vapours from sewers and drains. Section of Drain-trap. It is variously formed. The figure shows one of the forms commonly used. STIRRUP PIECE.—A name given to a piece of wood or iron in framing, by which any part is suspended; a vertical or inclined tie. INDEX AND GLOSSARY. to kneel upon, as occasion required. They are still in constant use on the Continent, though compara- TILE-CREASING TAKING DIMENSIONS.—The manner of, five regular solids. It is hounded by four equilateral triangles.—To find its surface. Rule: Multiply the square of its linear side by 17320508.—To find its solidity. Rule: Multiply the cube of its linear side by 0-1178511. TETR A HEDRON.—The horizontal projection of a tetrahedron being given, to find the vertical projection, p. 52.—A point in one of the projections of a tetrahedron being given, to find the point in the other projection, p. 52. f ■ TETRAHEDRON.—To find the projection of it the courses of the shaft , the section of a tetrahedron cut bv a plane, p. 52. T A NY' r n „ , p i - i i-i I TETRAHEDRON. — To find the projections of A ‘Z 1 ,. . Jo fc f cWs and Slrn,liir tools : a tetrahedron when inclined to the horizontal plane, TALON.—The French term for the ogee moulding. TALUS.—A slope or inclined plane. TAMBOUR.—1. A term applied to the naked part of Corinthian and Composite capitals, which bears some resemblance to a drum. It is also called the vase, and campana, or the Idt. Also, the wall <>f a circular temple surrounded with columns, and I the circular vertical part of a cupola.—2. A cylin- 1 drical stone, such as one " " of a column -To right and oblique a cone, p. 61.—To a 'cd surfaces, the Sector.— tively seldom used in England. They are also called misereres. SUMMARY of rules for calculating the strength of timber, p. 130. SUMMER.—1. A large stone, the first that is laid over columns and pilasters. The first voussoir of an arch above the impost.—2. A large timber supported on two stone piers or posts, serving as a lintel to an opening.— 3. A large timber or beam laid as a bearing beam ; a ! g < Lower Diameter in minutes liO' 00' 60' 00' PiameteratNeek 45' ! 45' 45' 4S' 50' 461/ Height of the Col in diameters 7 i '30 6 7 7 0-54 Height of the En- tabiat.ure . . C4-14 1-5-Ji 1-30 1-40 145 Height of Archi- 354' 32' 30' 25' 314' 31 Height of Frieze -(■' : 39' 35' 314' 33V Height of Cornice 434' 41' 30' 40' 1 42' 394' For an example of the Tuscan Order, see the wood- I cut under Column. TUSK TENON. — Described p. 151, Plate , XLII., Figs. 2 and 4. TYMPAN, Tympanum.— In architecture, the space in a pediment, included between the cornice of the inclined sides and the fillet of the corona. , The term is also used to signify the die of a pedes- I tal, and the panel of a door. The tympan of an arch is the spandrell. TWISTED FIBRES. — In trees, render the wood unfit for the carpenter, p. 97- u. ULCERS in trees, p. 97. ULMUS. — The elm tree. Description and uses of, p. 110. UNDER-CROFT.—A vault under the choir j or chancel of a church. UNDERFOOT.—The same as underpin (which | see). UNDERPIN, v. —1. To support a wall, or a mass of earth or rock, when an excavation is made beneath it, by building up under it from the lower level .—To underset and to underfoot are used in the same sense. UNDERPINNING.—1. The act of bringing up a solid building, to replace soft earth or other material removed from beneath a wall or over¬ hanging bank of earth or rock. In Scotland this process is called goufing. —2. Solid building substi- 200 tuted for soft materials excavated from under a wall, bank of earth, or mass of rock. UNOU LA.—In geometry, a part cut off from a cylinder, cone, ifcc., by a plane passing obliquely through the base and part of the curved surface. Hence it is bounded by a segment of a circle which is part of the base, and by a part of the curved surface of the cone or cylinder, and by the cutting plane.—It is so named from its resemblance to the hoof of a horse. tlngula. y. VALLEY RAFTER.—The rafter in the re¬ entrant angle of a roof. See Hip Roof, p. 91. VANISHING POINT.—In perspective, the point in which an imaginary line passing through the eye of the observer parallel to any original line cuts the horizon. See p. 230, 231. VASE.—The body of the Corinthian and Com¬ posite capitals. See Drum and Tambour. VAULT.—A continued arch, or an arched roof, so constructed that the stones, bricks, or other material of which it is composed, sustain and keep each other in their places. Vaults are of various kinds, cylindrical, elliptical, single, double, cross, diagonal, Gothic, &c. When a vault is of greater height than half its span, it is said to be surmounted, and when of less height, surbased. A rampant vault is one which springs from planes not parallel to the horizon. One vault placed above another constitutes a double vault. A conic vault is formed of part of the surface of a cone, and a spherical vault of part of the surface of a sphere, as fig. 4. A vault is simple, as figs. 1 and 4. when it is formed by the surface of some regular solid, around one axis; and compound, as figs. 2 and 3, when compounded of more than one surface of the same solid, or of two different solids. A groined vault, fig. 3, is a compound vault, rising to the same height in its surfaces as that of two equal cylinders, or a cylinder with a cylindroid. VAULTING SHAFT, Vaulting Pillar.— A pillar sometimes rising from the floor to the spring of the vault of a roof; more frequently, a short pillar attached to the wall, rising from a cor¬ bel, and from the top of which the ribs of the vault spring. The pillars between the triforium windows of Gothic churches rising to and supporting the vaulting maybe cited as examples. VAULTS.—Method of dividing into compart¬ ments, p. 82. VENEER.—A facing of superior wood placed in thin leaves over an inferior sort. Generally, a facing of superior material laid over an inferior material. VENETIAN WINDOW.—A window of large size divided by columns or piers resembling pilas¬ ters into three lights, the middle one of which is usually wider than the others, and is sometimes arched. VERANDA, Verandah. —An oriental word denoting a kind of open portico, or a sort of light external gallery in front of a building with a sloping roof, supported on slender pillars, and frequently partly inclosed in front with lattice-work. In India almost every house is furnished with a ve¬ randa, which serves to keep the inner rooms cool and dark. INDEX AND GLOSSARY. ZOOPHORUS WIND-BEAM.—An old name for collar-beam. WINDERS.—Those steps of a stair which, radiating from a centre, are narrower at one end than at the other. WINDING.—A surface whose parts are twisted so as not to lie in the same plane. When a surface is perfectly plane it is said to be out of winding. WINDING-STICKS. —Two slips of wood, each straightened on one edge, and having the op¬ posite edge parallel. Their use is to ascertain whether the surface of a board, &c., winds or is twisted. For this purpose, one of the slips is placed across one end of the board, and the other across the other end, with one of the straight-edges of each upon the surface. The workman then looks in a longitudinal direction over the upper edges of the two slips, and if he finds that these edges coincide throughout their length, he concludes that the sur¬ face is out of winding; but if the upper edges do not coincide, it is a proof that the surface winds. See Winding. WINDING-STAIRS, p. 196. WINDOW.—An opening in the wall of a build¬ ing for the admission of light, and of air when necessary. This opening has a frame on the sides, in which are set moveable sashes, containing panes of glass. The sashes are generally made to rise and fall, for the admission or exclusion of air, but some¬ times the sashes are made to open and shut verti¬ cally, like the leaves of a folding door. WINDOWS and Window Finishings, p. 187. WINDOW-FRAME.—The frame of a window which receives and holds the sashes. WINDOW-SASH.—The sash or light frame in which panes of glass are set for windows. See Sash. WINDOW-SHUTTERS. — Hung to sink into the breast, p. 188. WINDOW-SILL. See Sill. WING.—A smaller part or building attached to the side of the main edifice. WOOD. See Timber. WOOD blighted by frost unfit for the carpenter, p. 98. WOOD-BRICKS.— Blocks of wood of the shape and size of bricks, inserted in the interior walls of a building as holds for the joinery. WREATHED STRING, p. 196. VERGE-BOARDS VERGE-BOARDS.—See Barge-Boards. VERMICULATED WORK. —In masonry that in which the stones are so dressed as to have the appearance of having been eaten into or tracked by worms. VERNIER —A small moveable scale, p. o4. VERSED SINE, or Height of an Arc.—To . Prob. LXII. WALES or Waling-Pieces.— The horizontal timbers serving to connect a row of main piles together. WALL-STRING, p. 196. WALNUT WOOD.—Properties and uses of, p. 111. WARPING. See Casting. WARTS in trees detrimental, p. 97- WASH-BOARD.—The plinth or skirting of a room. WASTING.—In stone-cutting, splitting off the surplus stone with a wedge-shaped chisel, called a point, or with a pick. By either of these the faces of the stone are reduced to nearly plane surfaces, and it is said to be wasted off; in Scotland called clouring. WEATHER, v .—To slope a surface, so that it may throw off the water. WEATHER-BOARDING. — Boards nailed with a lap on each other, to prevent the penetration of rain and snow. WEDGE.—To find the surface of. Rule: Find the areas of the rectangle, the two parallelograms or trapezoids, and the two triangles of which its surface consists, and add them together.—To find the solidity of a wedge. Rule: To twice the length of the base add the length of the edge, and multiply the same by the breadth of the base and by one- sixth of the perpendicular from the edge upon the base ; the product will be the content. WEIGHT OF ROOF-COVERING:— find, the chord and radius being given, p. 17. VERTICAL PLANE, p. 47, 229. VESICA PISCIS.—A name given to a figure formed usually by the intersection of two equal circles cutting each other in their centres, but often also assuming the form of an ellipse or an oval. It is a common figure given to the aureole, or glory, by which representations of each of the three persons of the Holy Trinity and of the Blessed Virgin are surrounded in the paintings and sculp¬ tures of the middle ages. The form is also found 1 square of pan-tiling will weigh. .. 7 2 0 c«t. qra 1 do. plain tiling.from 14 0 0 to 14 2 1 do. countess or ladies'slating 6 0 0 1 do. Welsh rag or Westmore¬ land do. 10 0 0 1 do. 7 lbs. lead. 6 1 0 1 do. copper, 16 oz. or 1 lb. p. ft. 0 3 16 1 do. zinc cast, inch thick.. 2 0 6 1 do. do. inch thick.. 10 4 WELL-HOLE, Well. —In a flight of stairs, the spade left in the middle beyond the ends of the steps. See p. 196. WELSH-GROIN or Underpitch Groin.— A groin formed by the intersection of two cylin¬ drical vaults, of which one is of less height than the other. WELSH LAYS.—In slating, slates measuring 3 feet by 2 feet. WET-ROT.—Causes of, p. 100. WEYMOUTH PINE or Yellow Pine, called also American White Pine.—P roperties and uses of, p. 118. WHEEL - WINDOW. See Catherine Wheel, and Rose Window. WHIP-SAW.—A saw usually set in a frame for dividing or splitting wood in the direction of the fibres. It is wrought by two persons. WHITE ANT destructive to timber, p. 105. WHITE FIR or White Deal —The produce of the Pinus dbies or Norway Spruce.—Properties and uses of, p. 117- WHITE SPRUCE.—Properties and uses of, p. 118. WHITE WALNUT.—Properties of, p. 111. WHITE WOOD OR Alburnum of trees unfit for carpentry, p. 98. WICKET.—A small door formed in a larger one, to admit of ingress and egress without open¬ ing the whole. WILLOW, The —Properties and uses of, p. 114. WIMBLE.—An instrument used by carpenters and joiners for boring holes; a kind of augur, WIND—To cast or warp; to turn or twist any ! surface, so that all its parts do not lie in the same WAGGON-HEADED CEILING or Vault¬ ing. —The same as Cglindric vaulting (which see). WAINSCOT.—The timber-work that serves to line the walls of a room, being usually made in panels, to serve instead of hangings. The wood originally used for this purpose was a foreign oak, known by the name of roagenschot, and hence the name of the material came by degrees to be cor¬ rupted into wainscot, and applied to the work itself. Hence, also, the name wainscot is often applied to oak deal. GLASGOW: W. G. BLACKIE AND CO., PRINTERS, VILLAFIELD. (A GETTY RESEARCH INSTITUTE 3 3125 01059 3354 1,/Mm jjg|| Iglly r wi , jy ^ ■]>? j - ’*/ ’■“> ) \ HBMwBg^^i^ji M ^piSK^i