. . " . * P 13. .J ** . !! We ! " . * * . TY. A S'. komme 20 . V * V 7 ST WWW . * 1 772 in * . is 7 ZA . , ... 2- . . 3 - . LE Mit t . 1. ' '' . . . . . . . MY .! . A . .. A . . 2 YUMRE VO XX. IN . 11 ; TRAR ! .. . . . B 479990 . WWW * - ** . . W --- -- - - -- * ?! ! .. " I L . . .. . 1 . .. . " . · * at 11 A - - - - . . . a . . A . . . . • - ' ...; t. . * W0 . 7 . i . RM3+ - Toy17 2017 .11 . 21 ".. 27 .. . . A .. ': 2 4 - . - i + . . .. . X1 X2 ** . . *.. - 1 V . .. , w MU . . . . - - - - - - L st is STEIN * . 17.10 sota LUX DEE S From the Library of XS Walter L. Stebbings Class 1883 --- U. of M. Xile w.fo. Stebbings. TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT XTYTTTTTTTTTTTTTTTTTTTT . . - . . . .-. . ... .. . .. i Y . .. . .. ka""ALHA . Architectural Library NA 2520 N63 1853 E lill IBITI . US !! . CARTES SCIENTIA . LIBRARY VERITAS OF THE NIVERSITY OF MICH SON SILTI 571 innin minim MINUUTTITOTTEETTINEN nininilliiniiniitiliniil villiiiiiiiiinilliti tilill . TCEBOR MICU W .. TRIS PENINSULAMA LAMAMOKAMAI CIRCUMSPLU 1.11.17 A O , ts. W l rchitecture IK mill llillllllllllllUNST I KITILTRANEHTIRIRI1111111TL ILILLLER1120011T MIL IN JUNINNI MOBIILINT IIMUSTUS PUMILINI THE GIFT OF Mrs. W. L. Stebbings bullimit Hiilid THE BUILDER'S AND WORKMAN'S NEW DIRECTOR. Glossary P.185 NA 2520 .N63 1853 . un I i he em 1. --- wer -- -- - TI - 7 11 i * - EARLY ENGLISH. . - - NAT Ar 13 . - - -- 11 . . TRANSITION NORMAN. - 0 INC 1 1 17E AN PERPENDICULAR TRACERY. TA Sy 41 - V LI / 1 . * WA 4 .. 2. VY/ 1 .. LAUNDRE . $ w 3 ny 141 N . be - H PER -- - I YER -:- -::-- 1 PERPENDICOLAR . w . . wy ON AT . . D 2 . *- - - - 2 - - . LAS 1:1) EAL. EL 1 T en ECCLESIASTICAL ARCHITECTURE. ... Vio WINDOWS. Pri... i TRANSITION EARLY ENGLISH. naming FLOWING TRACERY. PILLARS. . . Mu www 1 . - - - - - -- . www OM *** . www . 41 A * . - LAS .- www. * n A.Y ármaron . AU . DECORATED. . 1 - - - - - - KT ORAS RE RE + WA SRECE ... 4 05 WWW 17 ... any RA OS . TI . 2 * T OS IN All W- HVIS . - . . * - . . .. . - F. 31 TO US - - - - 1 GEOMETRICAL TRACERY. Irt -- -- NORMAN. SA * SA 1 . 77 NORMAN. ETA 5 Hill 1 +-+- - - - - - - SA - * - _ * - L . + 11 Put TU II I CH THE BUILDER’S AND WORKMAN'S NE Ꮃ Ꭰ Ꭲ Ꭱ Ꭼ Ꮯ Ꭲ 0 Ꭱ : COMPRISING EXPLANATIONS OF THE GENERAL PRINCIPLES OF ARCHITECTURE, OF THE PRACTICE OF BUILDING, AND OF THE SEVERAL MECHANICAL ARTS CONNECTED THEREWITH; ALSO THE ELEMENTS AND PRACTICE OF GEOMETRY IT .ԱՆ. IN ITS APPLICATION TO THE BUILDING ART. A NEW EDITION, REVISED AND MUCH ENLARGED, FROM THE ORIGINAL WORK BY PETER NICHOLSON, ARCHITECT. WITH UPWARDS OF ONE HUNDRED AND FIFTY COPPERPLATES AND NUMEROUS DIAGRAMS. LONDON AND EDINBURGH: A. FULLARTON AND CO. 1853. EDINBURGH: FULLARTOX AND MACNAB, PRINTLRS, LEITA WALI Architecture het. GIFT OF MRS. W. L. STEBBINGS OCT 1. 1942 CONTENTS. . . . . Pp. 1, 2 PUBLISHERS' ADVERTISEMENT TO THIS EDITION, . . PART FIRST. PRELIMINARY REMARKS ON THE PRINCIPLES OF PRACTICAL ARCHITECTURE. . . . . . & 10 . • . . . SECTION I. . . . . . . . Importance of a knowledge of the principles of Foundations, Construction and Proportion, . . $ 145 Piling, Qualities to be studied in every Building, 6 . Walls, Of Sites in general, . . . 7-9 Drains, Character, and Internal and External arrange- Roofs, . ments of Buildings, Floors, Staircases, . . . . 16-22 Apertures, Rooms, . . . . 23_26 Doors, Ceilings, . . . . 27–31 Windows, Requisites to durability, . . 32, 33 | Chimneys, . © 11-17-142 Pp. 5-31 $ 34_44 45–52 53–74 75_84 85_105 106_118 119, 120 121-129 130_-142 143_154 10-15 . . . . . . . . . . . . SECTION II. . . . . . . . Importance of a knowledge of the nature and Sand, . . qualities of Building-materials, . . $ 1-3 Cements, . Classes of Materials, . Concrete, I. Stones, . . . . . 5 Brick, . Natural Stones, . . . 6, 7 III. Wood, . Component parts of Stones, . 8-12 Oak, . Structure and Mechanical properties of Stones, 13_17 Fir, . . Causes of the decomposition of Stones, . 18_25 Beech, &c. . . Siliceous Stones, . , . 26-_32 Strength of Timber, Argillaceous Stones, . . . 33_-37 | IV. Iron, Cast-iron, Calcareous Stones, 38_44 Forged iron, . II. Artificial Stones and Cements, . 45–63 | V. Glass, . . Pp. 32-70 $ 64—66 67_87 88_92 93_113 114_129 130–132 133-139 140-153 . 154 . . . 155_164 165_171 172–178 - . 2_71 PART SECOND. PRACTICAL ARCHITECTURE, OR THE APPLICATION OF GEOMETRY TO THE BUILDING ARTS. SECTION I. MASONRY, . . . . . . . . Pp. 73–87 Definitions, $ 11 Gothic groin, . § 62 I. Walls, Ribbed-groins, .. II. Vaults, Domes, and Groins, . . 8—55 Construction of spherical domes, III. Stone-cutting, . Niches in straight walls, Raking-mouldings, Niches in circular walls, . Arches,. . . Architraves over columns, . . 72 Oblique arches, . Description of the Lewis, . . . 73 An arch in a circular wall, . Stairs, . . . . . 74, 75 Construction of Groins, . . . . 61 | Steps over an area, . . . 76 CONTENTS. . . . § 11 Groin vaults, . . Pp. 88–90 $ 9 . . . Pp. 91–116 • § 31 . 35–42 43__45 . 46–48 49 . 50—53 54–56 60_62 . 63_66 67-81 Pp. 116–131 $ 24 . 25–31 32 . 33, 34 SECTION II. BRICKLAYING, Definition, . . . Brick bond. . . . . SECTION III. CARPENTRY, .. Definitions, . . . Of Frame-work in general, . . Expressions for strength of materials, . Principles of arranging Frame-work, . Joints, . . . . Built Beams, . . . . Scarf Joints, . . . , Scarfing Beams, . . . . Methods of joining timbers, . Naked Flooring, . . . . Truss Girders, . - Roofs. Timbers in a Roof defined, Construction of Hipped Roofs, Trusses Executed, . . . SECTION IV. JOINERY, Definitions, . . . . . Scribing and Mitreing, . . Formation of Curved bodies, . . Formation of Bodies in Parts by joining them with Glue, . . . . Bending Machine, . . . . Hinged Joints, . . . . Door Joints, . . . , SECTION V. PLASTERING, Value of the Plasterer's Art, Internal Plastering, Cements, Lime, .. Plaster of Paris, . . Lathing, Cornices, . . . . . Ornamented Cornices and Enrichments, SECTION VI. PLUMBERY, Definition, . Lead, . . . . . Lead-mines, . . . . Building lead, . . . . Smelting Operations, . Casting Sheet lead, Casting Cisterns, Laying Sheet lead, SECTION VII. HOUSE PAINTING, Definition, . . Nature and object of Painting, . Different kinds of House painting, . White lead, Red lead, . . . . . Litharge, Linseed oil, . . . . Drying oils, . . . $ 1, 2, 1,21 Circular Roofs or Domes, 3, 4 Conic Roofs, 5 Trussed Partitions, . 68 Construction of Niches, . . 9, 10 Pendentive Ceilings, . . . 11-14 Timber Bridges, . . . 15, 16 Frames, . . . . 17 First Class, . . . . Second Class, . . . . 20 Roadway, . Notices of different Timber Bridges, . Moveable Bridges, . . . Centring, . . . . 30 . . . . $ 1-4 | Folding Doors, . . . . 5_91 Window Frames, . . . 10 Sashes, . . . . . Skylights, . . . . 11, 12 Elliptic Archivolt, 13 Mouldings, . . . . 14-21 Stairs, . . . . . 22, 23 | Hand-Railing, . . . . . External Plastering, . . . Roman Cement, . Bayley's Composition, 6-12 Mastic Cement, . . . 13 Keene's Marble Cement, . . 14 Explanation of Terms, . . . . . . Solder, . . 2 Pipes, . . . . . Pumps, . . . . Tools, . . . . . Table of Weights, . Sheet Copper, . . Dotting, . . . . 8-15 Tin plates, . . . . . . . . § 11 Turpentine, . . . List of Colours, . . Painter's Tools, . . . General directions in preparing colours, General directions in laying on paint, Graining, . . . . Ornamental painting, . . Inscription painting, . . 36–58 59-68 . 69__83 Pp. 132—141 § 15, 16 . 17 18 con or . · Stucco, . . 20 . 21 22 Pp. 142–147 . $ 16, 17 18-21 . 22 23 . 25 . 26 27 Pp. 148–155 . $ 9 10 . 11-16 17, 18 19, 20 21-26 . 27-29 CONTENTS. PART THIRD. OF THE GRECIAN ORDERS OF ARCHITECTURE. SECTION I. What constitutes an Order, . . $ ! | Capital, . . . . Egyptian and Gresk Orders, Shaft, . . . . Columns, . 3 | Entablature, . . . Base, . . . . . 4 General remarks on the Orders, . . Pp. 159, 160 $5 . 6 7 . 8-11 . . SECTION II. MOULDINGS, Definitions, Names and shapes of Mouldings, . . . Rules for describing Mouldings, describing Mouldings. Pp. 161–164 $ 13–28 § 1 2-12 . . . Section III. OF DESIGNING COLUMNS, To diminish the shaft of a column, . To give less swell to a column, . To describe the flutes of a column without fillets, . . . . Pp. 165, 166 To describe flutes with fillets on the shaft of a column, 3 . . . SECTION IV. Of The Doric ORDER, . Shaft and capital, . . Architrave, . . . Frieze, . . . . Cornice, . i Tables of comparative proportions, . . $ 1 | Observations thereon, . 2 Table of columnar proportions, 3 Proportions established, 4 4 Roman Doric, . . Pp. 166-170 $ 6-10 . 11-14 15, 16 . 17 . . . SECTION V. OF THE IONIC ORDER, General remarks, Origin of capital, . • . . . Column, Base, i . Volute, . . Flutes, . . Description of plates, General proportions, . . Pp. 171–175 . $ 8 9 . 10-13 14 . 15 16 . . . . . Temple of Minerva Polias, Cornice, . . . Frieze, . Asiatic Ionic cornice and frieze, . . . . . SECTION VI. OF THE CORINTHIAN ORDER, By whom introduced, . . . Has an Attic base, . . . Shaft compared with the Ionic, Capital, . . . . General remarks, . Pp. 176-179 $1 Notices of different examples of this order, $ 9-25 2 Notices of Modern practice in this order, General observations, , . . 4 Table of Proportions, . . · 5-8 / Projection of the capital, . . . SECTION VII. RoMAN ORDER, General observations, . The Tuscan order, P. 180 $ 3 . § 1 Composite, . . SECTION VIII. GENERAL REMARKS, A sixth order, . . General remarks on columns, As engaged or insulated, Intercolumniations, . Pp. 181—184 $ 9-15 . 16-18 19, 20 § 11 Laws of superposition of orders, 2, 3 Of the pedestal, . Of Pilasters, . 5281 . . . . SECTION IX. GLOSSARY OF TERMS, , . . . Pp. 185–191 viii CONTENTS. Pp. 195226 195 PART FOURTH. ELEMENTS AND PRACTICE OF GEOMETRY. SECTION I. ELEMENTS OF GEOMETRY, Signification of Signs, . . . . Axioms, Postulates, Definitions, . . . . . Theorems, . . . . 196 . . 196 196206 206–226 226 · · · · · . . . General Definitions, Curves of the First Order, Of the Ellipse, . . Of the Hyperbola,. . . . . Of the Parabola, . . . Various Orders of Lines, . . SECTION II. PRACTICAL GEOMETRY, . CHAP. I. GEOMETRICAL PROBLEMS AS REGARDS PLANE FIGURES, . CHAP. II. OF RATIO AND PROPORTION, . CHAP. III. ARITHMETICAL OPERATIONS RESPECTING THE CIROLE, CHAP. IV. CURVE LINES, . . . . Of the Ellipse, . . . . Of the Hyperbola, . . . Of the Parabola,. Of Curve Lines of the Higher Orders, . . CHAP. V. PLANE TRIGONOMETRY, . . . . 226 227 227–239 240-244 244–248 249-252 253_339 253—277 278—295 295 301 301—323 301—310 310–315 315—320 321-323 323–339 • . . . . . PART FIFTH THE GEOMETRICAL PRINCIPLES OF ARCEITECTURE AND THE BUILDING ARTS. · · · · . . SECTION I. STEREOTOMY, Planes, . . Solid Angles, . Solids, Cylinder, Cylindroid, . . Cylinoid, . Pp. 343–361 343 344 344 345 345 346 . . Cone, · 348 . · . · . 349–351 351 352 . . ลละ . · . 353 Section of a Cone, Conoid, . . Cuneoid, . Sphere, Solids of Revolution, Ellipsoid, . . Donoid, . Ungulus, . . Architectural Solids, . · . · . . . . . · . .:. . 354 356 357 358 359—361 · . . . . . . . . . . . . . . . . . . . CONTENTS. . SECTION II. ORTHOPROJECTION, . . . . . . SECTION III. DEVELOPMENT OF THE SURFACES OF SOLIDS, . . . SECTION IV. PRINCIPLES OF THE FORMATION OF ARCHES OF DOUBLE CURVATURE, . Pp. 362—379 380—391 392397 PART SIXTH. PERSPECTIVE. Principles of Perspective, Drawing Instruments, . . . . . . Pp. 401–409 . 409_411 . . . . . . . PART SEVENTH. CO NO HISTORICAL NOTICES OF THE RISE AND PROGRESS OF ARCHITECTURE IN GREAT BRITAIN, . . . . . . Pp. 415—426 Roman Architecture in Britain, . § 1, General review of the three orders of Gothic Early Christian Churches, architecture, . . . . . $ 14 Bishop Wilifred's Churches, . . 3 Comparison of Gothic with Classic architecture, 15 Alcuin and Alfred, . . . . 41 Revival of Roman architecture on the Continent, 16 Rise and Progress of Saxon architecture, . 5 | In England, . . . . 17 Introduction of Norman architecture, . . 6 | Inigo Jones, . . . . . . 18 General character of Norman architecture, Sir Christopher Wren, . . . 19 Norman edifices and architects, . . 8, 91 Gibbs and Vanburgh, . . . . 20 Lancet style, Sir W. Chambers, . Progress of architectural taste, , 11, 12 Robert Adam, . . . . . 22 Adulterations of Style, . . 13 Concluding Remarks, . . . 23 PART EIGHTI. APPENDIX. A. On the Principles of Architecture, . . . Pp. 429—443 B. On Concrete, its Qualities, and the Mode of Preparing it, . . . . 444–446 C. Method of Determining whether a Stone will Resist the Action of Frost, . 47, 448 D. Table of the Resistance of Stones to a Crushing force, 449 E. Abstract of the Principal Sandstone, Slate, and Greenstone Quarries in Scotland, with their Analysis, . . . . . . . . . 450, 451 F. Nature and Proportion of Different Mortars, . . 452 G. On the Strength of Materials in general, 453460 H. On the Laws of Rupture in Frame-work, 461-471 YMI DIRECTIONS TO THE BINDER. O It is recommended that the whole of the Plates be bound up, in the order of their numbering, at the end of the Letter-press ; but if it be preferred to scatter them throughout the volume, they may be arranged as follows: 115 131 MASONRY, Plates 1. to XVII. immcdiately following p. 90. CARPENTRY, XVIII. to LII. JOINERY, LIII. to LXXXIII. GRECIAN ORDERS, GEOMETRY, CIX. to CXXI. GEOMETRICAL PRINCIPLES OF ARCHITECTURE, CXXII. to CXLIII, PERSPECTIVE, CXLIV. to CLII. 411 HISTORY OF ARCHITECTURE, CLIII. to CLVI. 426 3410 ADVERTISEMENT. In presenting to the notice of the Public another edition of THE BUILDER'S AND WORKMAN'S New DIRECTOR, it is proper to premise, that it has been deemed advisable to extend very considerably the original plan of the Work. This has been done from a persuasion that it would, by such enlargement, be rendered far more generally useful; and because it has been a frequent subject of complaint and regret, that notwithstanding the numerous and able works on Architecture and the Building art in general, already existing, no single book of moderate compass and expense, uniting in an adequate degree the Theoretical and the Practical principles of those Arts, has hitherto been produced It is presumed that the Work in its present form will be found serviceable, not only to the numerous class of individuals for whose instruction it was originally composed, and also to those young workmen whose efforts are directed to qualify themselves to act as “ Clerks of the works” under Architects, and to the proper performance of whose duties an acquaintance to a certain degree with the Theo- retical principles of the Art is indispensable, but likewise in assisting the studies of young men about, to engage in the practice of Architecture as a profession, and who may-as from the method of instruction generally adopted is too frequently the case_find themselves deficient as respects the Practical branches of Building. With these objects in view, in anew remodelling the plan of THE BUILDER'S AND WORKMAN'S NEW DIRECTOR, it has been the earnest endeavour of the editor to combine as much useful information as possible on the Theory and Practice of Architecture,—the science of Geometry as connected therewith,—and on the several branches of Artificers' works of primary importance in Building; and at the same time to condense the Work within the smallest limits compatible with these objects. Great care has likewise been taken to supply such defects and rectify such errors as have been discovered in the original Work; a considerable body of Notes, several useful Tables, and a number of Plates have been added; and, indeed, no labour or expense has been spared in order to render the present volume as perfect as a Work of this limited extent, on such a comprehensive subject, can be made. 政gt vgt, PRELIMINARY REMARKS ON THE PRINCIPLES OF PRACTICAL ARCHITECTURE. THE BUILDER'S AND WORKMAN'S NEW DIRECTOR. . PRELIMINARY REMARKS ON THE PRINCIPLES OF PRACTICAL ARCHITECTURE. SECTION I. Importance of a knowledge of the principles of Construction and Proportion, 1-5.- Qualities to be studied in every Building, 6. Of Sites in general, 7–9.- Character, and Internal and External arrangements of Buildings, 10–15. - Staircases, 16–22._ Rooms, 23—26.--Ceilings, 27–31.- Requisites to durability, 32, 33. Foundations, 34–44. — Piling, 45–52._ Walls, 53-74.- Drains, 75–84.— Roofs, 85—105.-- Floors, 106-118.- Apertures, 119, 120.- Doors, 121–129. Windows, 130-142.- Chimneys, 143–154. 1. It being intended that the following pages should contain a general summary of the theory and practice of Building, it will be useful, before we proceed to a consideration of the Geometrical prin- ciples of the art, and of the practical details of the several descriptions of Artificers' work which form general principles connected with the subject, the due knowledge and practice of which are of indis- pensable importance to the Builder. 2. The art of Building is, however, so comprehensive, so multifarious in its details, and embraces such a variety of widely differing objects in its practice, that volumes would be necessary to do adequate justice to the subject ; and after all, so many of the most important and interesting practical opera- tions in Building depend on peculiar incidents and local circumstances—which can only be efficiently provided for, or their occurrence foreseen, by an intelligent and experienced superintendent on the spot, as the work proceeds and as the various difficulties present themselves—that little more can be done with effect, in a work of this nature, than to point out the General Principles that are at all times applicable to describe some of those difficulties in execution which are of more frequent occur- rence, together with the most approved and certain means of guarding against or of overcoming them, most correct and useful information may be obtained on the subject. 3. The reader, therefore, must not expect to find, in the following pages, an elaborate treatise on the art of Building, but rather a general outline of the subject; his attention being, moreover, chiefly directed to those objects that are of primary and paramount importance in practice. Such observa- tions will, it is presumed, contribute materially to guide the operations of the less-instructed and less- experienced practitioners of the Building art, for whose use this publication is more especially de- signed ; and tend, it is hoped, to lessen the number and extent of those deplorable failures, now of PRELIMINARY REMARKS ON THE [PART 1 TYN such frequent occurrence in a large class of buildings, in the erection of which, though often of con- siderable magnitude and expense, it is not unusual—from mistaken ideas of economy—to dispense altogether with the assistance of an able and experienced Architect: such failures for the most part arising either from ignorance of or inattention to the primary principles of construction. 4. By a more intimate acquaintance with the essential principles of the art, the disappointment and inconvenience occasioned to the proprietor from deficiency in the accommodation, or awkwardness in the arrangement of the interior of the structure, would be avoided, as well as the disgrace to the taste of the country at large, from the absurdity, incongruity, and inelegance so frequently exem- plified in the preposterous and abortive attempts at decoration in buildings of the class alluded to. Indeed, it may be confidently asserted, that by a very little more attention being given to the subject, even our most ordinary buildings might be vastly improved, both as to arrangement and taste ; for there is no reason that convenience and beauty may not be united, in a certain degree, in buildings of very contracted dimensions, and constructed with the most common description of materials, if skill and judgment are exercised in the designing of them.* 5. Few things are more displeasing to a person observant of Building-operations, and who possesses even a slight knowledge of Architecture, than to see in so many of the extensive, and, in some respects, magnificent rows of dwellings that are daily rising around us, that often, even before they are com- pleted, cracks, settlements, and other defects appear in various parts, arising most commonly from the slight, hasty, and unscientific manner in which they are constructed, and by which they are per- manently disfigured. They consequently remain, as long as they last, monuments of the ignorance, or rapacity, or both, of their projectors; and objects of regret and offence to the scientific Builder, who knows with what a trifling portion of more knowledge or expense these defects might have been prevented ; and likewise to the architectural amateur, who laments that, instead of the splendour that would have been given to the metropolis by the erection of well-proportioned substantial buildings, an appearance of misery, wretchedness, and insecurity is entailed upon it so long as they remain, constant outlay moreover being requisite, on the part of the unfortunate proprietor or occupier, to prevent the dis- figured masses from falling into ruin. The evils which arise from slight and unskilful building are, however, far from being confined to the metropolis; the environs of the cities and large towns through- out England and Scotland very generally present such spectacles as those already alluded to, and they are daily increasing in number ; whereas a proper degree of attention to the principles of con- struction and proportion, and a trifling addition to the original outlay, would ensure the production of elegant and substantial structures, which would become permanent ornaments to the place,—a source of just pride to the inhabitants,--and the admiration of strangers, who might be induced from busi- ness or pleasure to visit it. 6. Qualities to be studied in every Building. 1-In the designing of every building, three things should be particularly considered : viz. Utility, Strength, and Beauty. Utility arises from a just and suitable division and arrangement of the different parts of the structure, by which it is adapted to answer in the most satisfactory manner the purposes for which it is erected. Strength depends upon the nature and quality of the materials employed in the work, and on their appropriate adjustment as to quantity in the different parts. Beauty is produced by the symmetry and true proportion of the several parts of the work, and by their harmony of the whole, as well as with each other.f Sir Henry Wotton, in his Elements of Architecture,' I considers the whole subject under two heads,—the Seat or Situa- tion, and the Work. * See Appendix A. of Ibid. The Elemepts of Architecture collected from the Best Autbors and Examples.' London, 1624 $ Sect. I. PRINCIPLES OF PRACTICAL ARCHITECTURE. 7. Of Sites in General. 1-As to situation, where those who intend to build possess the power of choosing the situation, regard should always be had to the quality, temperature, and salubrity of the air ; proximity to marshes, fens, boggy ground, or stagnant water, should be avoided, particularly if 1 those quarters ill-effects would be frequently experienced. The place should not want the sweet influ- ence of the sun's beams; nor be wholly destitute of breezes of wind, the want of which would cause a stagnation of the air, and render it similar in effect to that of a stagnant pool of water. The ap- proach to it should not be so steep as to be inconvenient either to the occupant or to visiters; and, for the ready transport of the various products of the district, and for facilitating the introduction of the various necessary articles of consumption, the produce of more distant places, it is desirable that it be contiguous to a good road, a canal, a navigable river, or an arm of the sea. It is moreover ad- vantageous to have a house sheltered rather by trees than mountains, because the trees yield a cool. ing and refreshing air, which, during the summer-months, is particularly agreeable as well as salu- brious; and in winter, they serve to break, in some degree, the keenness of the wind in tempests. Mountains, from their position, only protect from certain winds, and, if the situation be either directly east or south, they will be found particularly unpleasant at some seasons of the year. 8. The most desirable situation for a house is upon the slope of a moderate hill, where the ground rises gently from the plain, and continues rising a little behind the house, and where the height is suffi. cient to command a view of the plain below. A situation that has a fine air, plenty of good spring- water, an extensive prospect, with a good soil, and the shelter and defence of trees, may be said to be perfect. 9. In order to form an opinion as to the healthfulness of a situation, notice should be taken of the buildings in the vicinity. If they appear to be clean and fresh on the surface, though so old that the materials are beginning to decay, it is a proof that the air is pure; on the other hand, if the walls be tinged with green or other colours, and moss and herbs grow upon them in abundance, it may be considered as a proof that the air is damp and bad. 10. Character and Arrangement of Buildings.]—In designing a dwelling-house, consideration must always be given to the quality and condition of the person who is to inhabit it, and to the situation in which it is erected; the first, more particularly, when building for an individual by commission; the latter, when building in a town or city, as a speculation for selling or letting. In the latter case, although the Builder may not know the individual who shall occupy the house, he will always be able to guess pretty accurately as to his rank. In either case, it is necessary that the house should be so arranged, and such conveniences provided, as are most adapted to the wants of the occupant; as many things that are desirable, and indeed indispensable, for the accommodation of parties moving in one class of society, are inappropriate and even superfluous to those of another, from the difference in their habits, and in the nature and extent of their establishments. 11. Nothing is more obvious than that the form and arrangement of a dwelling-house should be adapt- ed to the climate of the country, and to the customs and habits of the inhabitants; yet it is no uncom- mon practice to copy in Great Britain the form and arrangements of an Italian house --not omitting even those parts that are especially contrived to collect air and exclude sunshine! We likewise fre- quently see the temples of Greece, with their deep and shady porticoes, and small apertures, con- trived to exclude as much as practicable the excessive heat and glare of sunshine common in that climate, imitated as closely as contracted dimensions and mean materials will admit in this country; and sometimes even converted into habitations ; the portico being moreover placed so as to face the crease in a tenfold degree the chill and gloom of our cloudy and ever-varying atmosphere I PRELIMINARY REMARKS ON THE [PART I. 12. In the arrangement of the several apartments of dwelling-houses in the country, it is to be observed, that studies, libraries, and other rooms principally used in the morning, should be placed facing the east; and those requiring a cool fresh air, or a steady light, towards the north. Something, however, as regards internal arrangement in houses of moderate dimensions, must always depend on local circumstances, of which the views to be obtained of the surrounding country from the principal rooms may be considered as among the most prominent, and ought never to be lost sight of. 13. The several rooms should be so arranged as to be accessible from a general corridor or passage; and care should be taken that the doorways thereto are conveniently placed for ingress and egress, and at the same time so situate that the rooms shall be exposed as little as possible when the doors are opened ; for which purpose they should be made to open towards the fire-place. When practicable, the doors of communication between the several rooms should be placed centrically; by which means a suite of apartments may be formed when required, and the accommodation and general appearance of the whole at such times be much increased. 14. The most convenient and economical form for small insulated dwelling-houses is the square, as it admits of subdivisions with the greatest facility; but for those of greater magnitude, an oblong figure is preferable, it being not only more favourable for obtaining light to the several apartments, but likewise affording greater facility for roofing. In those of a more extensive nature, the triangle, circle, ellipse, or a combination of these figures, is occasionally employed. When the building is of such extent as to require the offices for domestic purposes to be placed in wings connected with the mansion, they should be so arranged as to appear to form a subordinate part of the whole design,- not entirely detached, yet in such order that the more offensive ones shall be removed as far as practicable from the principal apartments, while at the same time ready means of communication with those parts to which access may be necessary are provided. The consideration of the designing of such ex. tensive buildings is, however, foreign to the subject of this work, it being presumed that, in all such cases, an experienced Architect would be called upon to furnish the design. 15. In order to insure the possession of a convenient and commodious habitation, whatever may be its magnitude, it is necessary that a well-digested plan should be formed of each floor,--an elevation of each front,—and two or more sections through different parts of the building. When the struc- ture is of considerable extent or complexity of design, a model should be made of the entire build- ing; indeed, on most occasions, a model will be of such advantage as amply to repay the cost of forming it. 16. Staircases.] In modern buildings, the staircase forms a very important feature. Due attention should, therefore, be given to the proper placing of it, that it may not interfere with the arrangement of the different apartments, and be at the same time easy of access, and afford the means of commu- nicating with the several stories of the building in the most ready and convenient manner. 17. The number, size, form, and disposition of the staircases must be determined by the extent and use to which the structure is to be applied. In a large house, the principal staircase should be spa- 1 The ingenious author of OIKIAIA or Nutshells,' in which work he has treated particularly of houses of moderate dimensions and expense, after having given much useful advice to persons about to build, on the subject of plans, &c., ex. presses himself as follows with respect to models: “I would next advise that he do cause a complete though plain model of the design be has fixed upon to be made very accurately, to a scale of at least a quarter of an inch to a foot; the sev. eral stories to be contrived so as to lift on and off at pleasure, that every part may be easily and minutely scrutinized and measured. Gentlemen who have not been so far conversant in plans as to judge therefrom with certainty, ought not to grudge the trifling charge of three, four, or five guineas for a toy of this kind : the information and advantage they will derive from it may prevent much of the opprobrious work of alteration, and save a great deal of trouble and a consider- able sum of money.' SECT, I.) PRINCIPLES OF PRACTICAL ARCHITECTURE, cious, light, and easy of access and ascent; while the subordinate ones should be so placed as to afford the domestics the easiest and most convenient communication with the various parts of it. 18. In the staircases of ordinary houses, the steps should be from three and a half, to four feet in length, and each step from six to seven inches in height. When the rise is greater than seven inches, the ascent and descent is rendered inconvenient and unpleasant. The width of internal steps should not be less than ten inches and a half ; nor is it desirable that they be more than twelve inches. Landing-places, called half-paces or quarter-paces, should be formed at convenient distances ; where practicable, at about every twelve or thirteen steps; and it is to be observed, that the use of windows should as much as possible be avoided. 19. When the height of the story is considerable, the steps may be arranged so as to form two revolu- ITUL . cannot, however, be adopted where the entire story is less than fourteen feet in height. At the pre- sent time, in ordinary houses, the arrangement and construction of the staircase is too frequently neglected ; consequently, the several members of the family are daily, and some almost hourly, seri- ously inconvenienced thereby; and sometimes spacious and elegant apartments are approached by steps that it would scarcely be pardonable to affix to lead to a hay-loft. 20. We seem in this particular to have fallen into the opposite extreme to our forefathers: they often made their staircases unnecessarily spacious, to the injury of the rooms ; we, on the contrary, in our desire to increase the space in the rooms, unreasonably circumscribe the staircase, thereby rendering it not only incommodious but frequently even dangerous. 21. Staircases may be advantageously lighted from the top by means of sky-lights, or lantern-lights, as the light will be more equally diffused by such means than by windows. 22. In every species of public building, the steps of staircases should be formed of stone. This is desirable likewise in private dwellings, more especially in those erected in large towns and crowded cities, where, from the great value of the ground, the buildings usually consist of several stories, and where, likewise from a variety of causes, danger from fire is more particularly to be apprehended. Economy, however, too often forbids this ; but if persons about to build were duly impressed with a consideration of the security afforded thereby, in the event of fire, few it may be presumed would, under such circumstances, object to incur the comparatively small additional expense of a stone staircase.? 23. Proportions of Rooms. ]—In large edifices, the forms of the several apartments are occasionally varied in order to produce a greater degree of convenience and beauty; in which case, instead of the ordinary rectangular shape, some of the rooms as they rise from the floor, are made either circular, or elliptical; or are compounded of a rectangular portion in the middle, and symmetrical, circular, or elliptical segments at the ends, having the chord or diameter extending the whole breadth of the end to which it is attached, or to a part of such breadth, leaving an equal portion of it at each extremity of the chord. In other instances, hexagonal or octagonal forms are employed; or those compounded of the rectangle, and the last-mentioned figures at one or both ends. 24. Sir William Chambers, speaking of the proportions of rooms, says: “The heights of rooms depend upon their figure. Flat ceiled ones may be lower than those that are coved. If the plan be a square, the height should not exceed fiye-sixths of the side, nor be less than four-fifths; and when it is oblong, the height may be equal to the width. Coved rooms, if square, must be as high as broad; and when oblong, their height may be equal to their width, more one-fifth, one-fourth, or one-third of the differ- ence between the length and width.” It is not practicable, however, always to observe exactly these L 2 Stairs may be advantageously constructed of cast-iron in many situations. B 10 (Part 1. PRELIMINARY REMARKS ON THE proportions. In dwelling-houses, the heights of all the rooms on the same floor are usually similar to each other, though their extent be different ; which renders it extremely difficult where there are several different sized rooms, to proportion all of them well.* 25. The method usually adopted, in buildings where beauty and magnificence are preferred to economy, is to raise the halls, saloons, and galleries higher than the other rooms, by making them oc- cupy two stories; to make the largest sized rooms with flat ceilings; to cove the middle-sized ones one- third, one-fourth, or one-fifth of their height; and in the smaller rooms, where even the highest coves are not sufficient to render the proportion tolerable, it is usual to contrive mezzanines or intersoles above them; these are very convenient as thev afford servants' rooms, bath-rooms, wardrobes, store- rooms, &c. 26. In ordinary dwelling-houses, the rooms are too frequently made less lofty than is desirable, both as regards appearance and salubrity : this arises chiefly from two causes, the one being the great ex- pense of building in this country, particularly in or near the metropolis, and which induces us to con- fine the structure within the smallest possible dimensions compatible with the accommodation abso- lutely necessary; the other, is the coldness and dampness of the climate, which renders it necessary for us to resort, during the greater portion of the year, to artificial means for the warming of our apartments, the expense of fuel at the same time making it important to restrict the consumption of it as much as practicable. From the great improvements that have lately been made in the method of warming buildings on economical principles, we may, however, hope shortly to be enabled to enjoy well-warmed, spacious, and lofty apartments, at a comparatively small expense. 27. Ceilings.]—When the ground-plan of an apartment is rectangular, the ceiling is usually of the same form, and either quite flat and plain, or formed into compartments and pannelled, or enriched with foliage and other ornaments ; but in rooms of a superior description, the forms of the ceilings are frequently much diversified both as respects the plan and section. 28. The most simple form of ceiling, next to that of a flat horizontal surface, is what is termed a waggon-head, which is formed of any portion of a cylinder having its axis parallel to the horizon; the chord of the cylindric section being extended the breadth of the room, so that in every section parallel to one of the ends of the apartment, the upper part of that section will be a segment of a circle. 29. Another form of ceiling frequently used consists of a quadrantal portion of a cylinder, rising from each vertical side of the apartment and meeting the horizontal part in the middle, the curved portion intersecting at each angle; this is denominated a coved ceiling, and admits of a great variety of dispositions and decoration. 30. A ceiling may likewise be formed of intersecting cross arches, so that each arch coincides with the surface of a cylinder having its axis parallel to the horizon. When the summit of each arch rises to the same height, this species of ceiling is termed a groined ceiling. 31. The rectangular plan likewise admits of the formation of a concave ceiling in an ellipsoidal, or spherical form ; such are called domical ceilings. 32. General Requisites to Durability. In order to give strength and durability to a building two things are essentially necessary: viz. the use of good and appropriate materials, and the application of such materials in a judicious and efficient manner. With respect to the former, although the possession of considerable skill and judgment is requisite, to be enabled on all occasions to select the * See Appendix A, $ 34. 3 The ancient practice of forming the ceiling into compartments with pannels, and other plaster enrichments which had become almost obsolete_has happily of late been revived, and is now becoming common even in the better descrip- tion of houses built on speculation. Sect. I.] PRINCIPLES OF PRACTICAL ARCHITECTURE. best sorts of the several species of materials and which indeed can only be acquired by attentive ob- servation and considerable experience-yet it is hoped that the hints given on that subject in the course of the second section of these Preliminary Remarks will be sufficient to prevent the most in- experienced Builder from falling into any important error in the choice or application of those materials which are of chief importance in the construction of strong and durable buildings. What- ever caution however may be exercised in the selection of proper materials, it will be of little avail unless at the same time au adequate degree of attention is given to the application of them in tho structure ; this can only be insured by an attentive and judicious consideration of the nature of the building, and the purposes to which it is to be applied, and a consequent just and accurate appor- tioning of the strength of the several parts of it, so as to be best adapted to the ends proposed. 33. Due attention must likewise be given to the forming of the foundations,—to determine the thick- ness of the walls,—the figure and weight of the roof,—the strength of the floors,—the various com- partitions,—and the number and sizes of the several apertures, -all which should be so arranged and adjusted as to combine, in the greatest practicable degree, convenience, durability, and economy. 34. Foundations. ]— With respect to the foundation, too much attention cannot be given to ascertain the nature of the ground on which the building is intended to be raised. In the event of the exis- tence of any inequality in the texture or substance of the natural soil, it is indispensable that pre- cautions should be taken to prevent, by artificial means, the defects likely to arise therefrom, so that a uniform substratum may be made; otherwise the stability of the structure will be endangered. 35. The ancient Architects, and indeed most writers on Architecture, have attached great impor- tance to this part of the Builder's duty; and though some of their precepts as regards foundations would perhaps, in this age of economy, be generally considered as superfluous and extravagant, yet it should be recollected that it is in no small degree owing to the care, attention, and expense bestowed by the ancients on the foundations of their buildings, that some of their structures have endured up- wards of 2,000 years,4 whilst many even of the most extensive and costly of ours scarcely last the ruinous or dilapidated appearance from cracks and settlements occasioned most frequently from defects in the foundation. Indeed, in the foundations of every description of buildings no precautions should at any time be neglected that it becomes a skilful and diligent builder to take, for in no part is an error of so much consequence,--so difficult to rectify if committed, or so likely to be attended with fatal results. 36. The forming adequate foundations to an extensive building on a bad soil, with a due regard to stability and to economy, is one of the most arduous tasks imposed on the Architect, or Builder ; and in the due performance of it he can rarely gain much positive assistance from even the best theo- retical works on the art, as so much depends on the peculiar circumstances of the individual case, and the precise nature of the difficulties to be overcome. This observation perhaps applies more particularly to extensive buildings of an ordinary description, such as warehouses, manufactories, and the like, where a great difficulty often arises in determining the extent of the precautions abso. lutely necessary to be taken to insure the requisite stability, and at the same time to avoid any un- due increase of expense. 4 Battista Alberti, in the fifth chapter of his third book, 'De Re Ædificatoria,' mentions several ancient buildings in which particular care was bestowed on the forming of the foundations : viz. The Temple of Diana at Ephesus, the Se- pulchre of Antoninus, the Forum Argentarium, the Comitia, and the Tarpeia. The modern excavations in the ancient Roman forum serve to demonstrate the extreme attention and great expense incurred by the ancients in forming founda- tions to their buildings. This is particularly apparent likewise in the remains of the temples of Jupiter Stator, Jupiter Tonans, and the Column of Phocas, 12 [PART 1 PRELIMINARY REMARKS ON THE 37. The marks of a good soil for building on, according to Alberti, are the following: If it does not produce any kind of herb that usually grows in moist places ; if it does not produce any tree at all, or only such as flourish in very hard close earth ; if the place is stony, with large sharp stones, espe- cially flints. The best soil is reckoned to be that which is the hardest to the pickaxe, and which when wetted does not easily dissolve. Hard gravel or stone is the most sure' and firm foundation where all is sound beneath, but there is none which may prove more deceptive, such ground some- times containing cavities beneath ; nor is a foundation of rock itself free from danger of the same kind, caverns being frequent in rocky places, over which, should a heavy building be erected, it might suddenly fall down. The utmost attention should therefore be given to guard against the possibility of an accident of this kind. Palladio recommends throwing down great weights forcibly on the ground, and observing whether it sounds hollow or shakes. He says likewise that a judgment of the firmness of the earth may be formed from the sound of a drum placed on the suspected ground: if on being lightly touched it does not sound again, or if water put in a vessel does not shake, it may be presumed to be solid ; if it be hollow, the effects produced will clearly show it. The usual way adopted to ascertain the solidity of the ground on which a building is about to be erected is by the use of the rammer. If when the ground is struck with this tool it shakes, it should be pierced with a borer, such as is used by well-diggers; and on its being ascertained how far the firm ground is below the surface, the loose or soft ground should be removed. 38. To prepare the bed for a foundation on rock, the thickness of the stratum of rock should first be ascertained, if there are any doubts respecting it; and if there be any reason to suppose that the stratum will not offer sufficient resistance to the weight of the structure, it should be tested by a trial weight at least twice as great as the one it will have to bear permanently. The rock is next properly prepared to receive the foundation-courses by levelling its surface, breaking down all pro- jecting points, filling up cavities either with rubble masonry or with concrete, and carefully removing any portions of the upper stratum which presents indications of having been injured by the weather, The surface prepared in this manner should, moreover, be perpendicular to the direction of the pres- sure ; if this be vertical, the surface should be horizontal, and so for any other direction of the pres- sure. If, owing to a great declivity of the surface, the whole cannot be brought to the same level when the pressure is vertical, it must be broken into steps, in order that the bottom-courses of the foundation throughout may rest on a horizontal surface. If fissures or cavities are met with of so great an extent as to render the filling them with masonry too expensive, an arch must then be formed, resting on the two sides of the fissure, upon which the walls of the structure will be raised. The slaty rocks require most care in preparing them to receive a foundation, as their upper stratum will generally be found injured to a greater or less depth by the action of frost. 39. In stony earths and hard clay, the bed is prepared by digging a trench wide enough to receive the foundation, and deep enough to reach the compact soil which has not been injured by the action of frost; a trench from four to six feet will generally be deep enough for this purpose. The bottom of the trench must be perpendicular to the direction of the pressure. If the ground prove variable, being in some places hard and some soft, it will be necessary to turn arches from one hard spot to another.--If the soil is found to be gravel, it will be proper to examine the thickness of the stratum, and the quality of the strata under it. If the bed of gravel is thick, and the under strata sound and firm, or if it con- sists of a stiff clay, with a mixture of gravel,--no assistance is required ; if, on the contrary, the ground O Some writers on Architecture recommend the digging of wells in order to ascertain the nature of the soil ; and when they are likely to be wanted eventually, it is doubtless a good plan, though much more expensive than boring, SECT, I.) PRINCIPLES OF PRACTICAL ARCHITECTURE is loose, boggy, variable in its texture, and mixed with sand, recourse must be had to artificial means to remedy the defects therein. 40. When the ground is not very soft, and the superincumbent wall is intended to be supported upon narrow piers, a piece of timber or balk is sometimes slit in halves, and then laid so as to be imme- diately under the wall, or at the height of two or three courses from the bottom. This will often prevent settlements, and in many cases will be quite sufficient for every purpose ; care should be taken, however, when this method is adopted, that the timber is not subjected to be wet and dry alternately, as, under such circumstances, it is liable to decay very rapidly. Either oak, fir, elm, or beech, are proper for this purpose. 41. Sometimes large stones, known by the name of Yorkshire-landings or ledgers, are laid in the foundations, and the walls built thereon. They are usually, in such cases, from nine to twelve inches wider than the bottom-footing of the wall ; and vary in thickness from three to six inches according to the weight of the structure intended to be raised on them. In those parts of the coun- try where Yorkshire-landings are not easily obtained, smaller stones of the sort most easily procured on the spot are substituted for them. These, being first chopped, or hammer-dressed, are laid in the trench, prepared of a breadth proportioned to the weight of wall intended to be sustained, and considered necessary; care being taken that the joints of each succeeding course fall as nearly as possible on the middle of the stones in the course immediately beneath. Each of the courses may be diminished in width till the upper course of the stone-work is reduced to a proper width to re- ceive the main wall, the thickness of which must be regulated, first, by a reference to the nature of the materials employed, and, secondly, to the magnitude of the fabric to be erected. Sometimes chalk is used in blocks of four or five cubic feet each, and where the chalk can be procured of such a size, and lies wholly out of the reach of air, it forms an excellent foundation for common houses, but cannot be trusted to bear extraordinary weights. 42. Another method, occasionally employed when the ground is soft and variable, is to form an artificial substratum of broken stone or gravel, mixed with a strong grout composed of stone, lime, and sand. The common way of applying this description of material in foundations is to dig trenches from two to three feet in depth, according to circumstances, about a foot wider than the bottom footing of the respective walls. Into these trenches a quantity of the material is thrown ; the granite, or other hard stone used, -being first broken into pieces of the size of an ordinary hen's egg and spread over the surface, is then rammed to a thickness of six inches; over this a quantity of the grout is thrown, which soon finds its way into the interstices of the stone. A second course or layer is then A 6 Mr. Smeaton, in his Report to the Commissioners of his Majesty's navy upon the defective works in Plymouth Yard, speaking of the clerk of the rope-yard's office-in which a settlement had taken place-observes, “ The north end it seems was built upon a wall of a reservoir which is upon a rock; the rest is supposed to have been built upon planking, which baving decayed about eight years ago, the above settlement took place amounting to about two inches and a half per- pendicular; and as this subsidence is nearly regular, it is most naturally accounted for as above. On this head I must observe that it is generally accounted the most difficult of all foundations, that of being partially upon a rock and partly upon a softer matter, because here being a manifest inequality, it is one of the most difficult problems in Architecture to form a judgment of what may be sufficient as an artificial strengthening, to make it equal with that of a rock which can suffer no compressure. Planking is the common expedient, and where it lies under water and so buried in the ground as nor to be subject to drying, it appears from the works of former ages to be sufficiently durable; but where laid in loose or made ground, so as to get a partial dryness, as may be presumed in the present situation, it appears subject to the rot in a moderate course of years, and therefore to compressure. A much better expedient in such a case is to pave the foundation and build upon the pavement (laying a course of fat stones upon it) rather than upon planking." 14 [PART I. PRELIMINARY REMARKS ON THE UR formed in a similar manner, and the grout again applied ; and so on till the whole mass is brought to the required thickness. Some Architects adopt a different method : they direct the lime and sand to be mixed with the hard material; the whole is then to be wetted, and thrown from a height of from fifteen to twenty feet into the foundation. In cases where the foundations do not go down to that depth, a temporary stage is raised, from which the material is thrown out of wheel-barrows.* 43. Sometimes to save expense, and to avoid an intermediate piece of bad ground, instead of the foundation-walls being continued throughout, piers are brought up, and then arches turned from pier to pier, on which the walls are built. In such cases it is of the greatest importance that the ground on which the piers are built should be rendered equally firm—if not so naturally—otherwise the building will be liable to sustain injury from its settling unequally ; which is likely to be more preju- dicial to it than where the whole of the building settles equally, even in a greater degree, from the ground being uniformly soft; as sometimes happens without the building sustaining much damage. 44. Whenever any inequality in the texture of the soil exists, and it consequently becomes neces- sary to make a difference in the depth of the foundation-walls, and indeed in all cases where it is practicable, it is a good plan to turn inverted arches under the several apertures, by which means the pressure of the entire wall is more equally distributed; whereas if the inverted arches are omitted, the weight being greater where the depth of the wall is greatest, an unequal portion of it is thrown on such part, and a fracture of the brickwork is probably the consequence. Inverted arches should be turned with great exactness, and should never-if it can by possibility be avoided—be less than semicircles. Indeed the parabolic curve is generally recommended, being found the most effective in resisting the reaction of the ground. 45. Piling. 1-In case of boggy earth, or loose sand and gravel, piling is one of the most common means resorted to to form a foundation; and, when properly executed, piles form a very secure and firm foundation. 46. The piles may be formed either of oak, beech, or fir timber. Their dimensions must of course be regulated by the nature of the ground, and the magnitude of the structure intended to be raised thereon. Alberti recommends that they should form a surface twice the breadth of the intended wall; that they should not be less in length than the eighth part of the height of the wall; and that their thickness should be one-twelfth part of their length. He likewise adds, that they should be driven so close together that there shall not be room for one more; this however is now seldom prac- tised, and indeed its necessity is questioned by other writers on Architecture, who recommend that the piles be placed at moderate intervals, for when excessively crowded they are often found to force each other up as they are successively driven. They should however be driven with many repeated strokes, in preference to very heavy ones, which have a tendency to split them; they should likewise in all cases be driven till they reach the hard soil.? Some writers recommend their being charred previous to being used. They are usually pointed, and each pile has a piece of wrought iron fixed * See Appendix B. 7 The measure of resistance may be estimated by taking the absolute stoppage, or the refusal of the pile to penetrate, farther than two-tenths of an inch, from the effect of 30 strokes of a ram weighing 800 pounds, raised to the height of 5 feet at each stroke. 8 Much useful information on the subject of foundations is to be obtained from a work published by Mr. George Semple, of Dublin, in the year 1776, entitled A Treatise on Building in Water,' in which the author gives a very interesting account, in the form of a diary, of the rebuilding of Essex bridge in Dublin, together with some excellent observations on bridge-building in general, and on erecting substantial stone buildings in fresh and salt water, quaking-bogs or morasses, &c. Although Mr. Semple's observations relate chiefly to bridges, yet many judicious remarks are made and directions given by him, that are applicable to other descriptions of buildings. With respect to piles he observes: “There are Sect. I.] 15 PRINCIPLES OF PRACTICAL ARCHITECTURE. to the point, which is termed a shoe, to facilitate its passage through the loose soil, and its entry into the solid earth. The head is also frequently encircled by a strong iron hoop to prevent the ram from splitting it. 47. When the driving of the piles is completed, pieces of timber about six inches in thickness,- termed sleepers—are to be laid along the trenches, on the head of the piles, in such positions that the outer edges of the sleepers shall be severally in a line with the perpendicular face of the superincum- bent wall. These must be firmly fastened to the pile-heads with oak pins or spikes. The space between the sleepers should then be filled with brickwork laid dry without mortar ; and planking three inches thick must then be laid across the sleepers, of such a width as to project about three inches on each side beyond the bottom-footing of the intended wall. 48. In cases of swampy and boggy ground, some Architects, in lieu of piling and planking, have had recourse to cradles of oak or fir timber, strongly framed and braced together in bays of from five to ten feet in length, and of a width proportionable to the superstructure. Over these frames or cradles are laid cross pieces or joists, the whole being firmly bedded in the earth, and the interstices filled with chalk or rubble. This method has by some been considered much superior to the ordinary method of planking already noticed; as in the event of the timbers forming the cradles decaying, the rubble may still remain united, and the sinking of the building be thereby prevented; whereas, in case of the decay of the planking, the sinking of the building is inevitable. 49. In ordinary cases, where the soil is adapted for building on, the best way to proceed is to sink the basement-story of the intended edifice to the proposed level, and then dig trenches for the walls, from two to three feet in depth, and as little wider than is necessary for the footings as is practicable. This is a preferable method to excavating the whole surface to the depth of the footings; inasmuch as it not only saves the labour and expense of removing a large quantity of earth and filling it in again, but it likewise obviates the bad effects of thrusting newly made earth against walls recently built, which should always be carefully avoided. 50. In building on an inclined plane or rising ground, as the ground rises, the foundation should rise accordingly in a series of steps; the distances between, and the height of the several rises, be- ing regulated according to circumstances, but the footings being in every case parallel to the horizon, and not to the surface of the ground. This method will ensure a firm bed for the several courses, and prevent them from sliding, as they would be apt to do if built on the inclined plane, especially in wet seasons, when the moisture in the foundations would have a tendency to allow the inclined part to de- scend towards the lowest point, to the manifest danger of fracturing the walls, and possibly involving the total destruction of the building. 51. In foundations near the edge of water, care should always be taken to examine the substra- tum most minutely and to a considerable depth, as many fatal accidents have occurred from foun- dations having been undermined by rivers. A similar degree of attention should likewise be given several methods that have been made use of to preserve timber. Sir Hugh Platt informs us that the Venetians make use of one which seems very rational, viz. to burn and scorch their timber in a flaming fire, continually turning it round with an engine, till it has got a hard black crusty coal upon it. Others inform us that the Dutch preserve their gates, port- cullises, drawbridges, sluices, &c. by coating them over with a mixture of pitch and tar whereon they strew small pieces of cockle and other shells, beaten almost to powder, and mixed with sea-sand, which incrusts and arms it wonderfully against all assaults of wind and weather. But for my own part, I conclude that the Venetian method is preferable, be- cause I believe it is the sap that is in either oak or fir that is the principal cause of their decaying so soon." 9 On this subject the student will derive much useful information from a perusal of Smeaton's Reports, published by the Society of Engineers in 1812, in three vols. quarto; more particularly that on Hexham bridge, in Northumberland; and likewise from 'A Treatise on Building in Water,' by George Semple, already noticed in Note 8. 16 [PART I. PRELIMINARY REMARKS ON THE where ground has been wrought or dug before. In such cases it ought never to be trusted to, in the condition in which it is left, but should be dug through into the solid earth. 52. In laying foundations in water, two difficulties have to be overcome, both of which require great resources and care on the part of the Architect. The first consist in the means to be used in preparing the bed of the foundation; and the second in securing the bed from the action of the water, to insure the safety of the foundations. The last is generally the more difficult problem of the two ; for a current of water will gradually wear away, not only every variety of loose soils, but also the more tender rocks, or those of a loose texture belonging to the calcareous and argillaceous classes, particularly if stratified as well as most varieties of sandstone. To prepare the bed of a founda- tion in stagnant water, the only difficulty that presents itself is to remove the water from the area on which the structure is to rest. If the depth of water is not more than four feet, this is done by sur- rounding the area with an ordinary water-tight dam of clay, or of some other binding earth. For this purpose, a shallow trench is formed around the area by removing the soft or loose stratum on the bottom; the foundation of the dam is commenced by filling this trench with the clay; and the dam is completed by spreading successive layers of clay about one foot thick, and pressing each layer, as it is spread, to render it more compact. When the dam is completed, the water is pumped out from the enclosed area, and the bed for the foundation is prepared as on dry land. When the depth of stagnant water is more than four feet, and in running water of any depth, the ordinary dam must be replaced by the coffer-dam. This construction consists of two rows of plank, termed sheeting- piles, driven into the soil vertically, forming thus a coffer-work, between which, clay or binding earth is filled in to form a water-tight dam, so as to exclude the water from the area enclosed. 53. Walls. 1-The requisite precautions having been taken, according to the circumstances of the case, in order to insure a safe and solid foundation for the structure, the next subject to be con- sidered is the nature and substance of the several walls. This must be regulated, in some degree, by the quality of the materials used, as well as by the magnitude of the building. 54. A wall should be able to resist a given force, acting upon it either with uniform pressure over the whole surface, or partially upon certain portions. It should be capable of sustaining the pressure of vaults or a roof, acting along a continued abutment, or upon points, as in groin-vaulting; also the power of the wind acting uniformly upon the whole surface. The thickness of walls must be re- gulated by their height, as well as the weight they have to support; consequently the pressure of the roof, the floors, and the additional burdens they may each occasionally be required to sustain, must be taken into consideration, when determining the thickness of the walls of a building, in order to avoid on the one hand making them unnecessarily thick, or falling into the opposite and more dangerous error of forming them of inadequate strength to support the superincumbent weight. The latter is certainly the more common error with modern Builders. 55. The great laws of walling are: First, that the foundations be sound, and the footings of sufficient strength. Second, that the walls stand perpendicularly to the ground-work: the right angle being the ground of all stability.10 Third, that the more massive and heavy materials be the lowest, as fitter to bear than to be borne. Fourth, that the work diminish in thickness as it rises, both for the ease of weight and of expense. Fifth, that certain courses or lodges of more strength than the rest be interlaid like bones to sustain the fabric from total ruin, should some of the under parts chance 10 Of course this principle applies more particularly to the walls of buildings standing detached and free from the pres. sure of any extraneous substance, such as a bank of earth or other matter constantly pressing against it. Where such pressure exists, it is usual to construct the wall on a totally different principle, giving the external surface of the wall án inclination outwards or inwards according to circumstances. PRINCIPLES OF PRACTICAL ARCHITECTURE. 17 to decay. Sixth, that the angles be firmly bound: they being as it were the nerves of the whole fabric. 56. It is generally recommended that the lowest footings of external walls be made twice the width of the superincumbent walls; and those for the partition-walls once and a half their width ; and that the whole be commenced on the same level.11 57. The footings to brick walls should be carried up in thicknesses of two courses ; each footing being diminished half-a-brick in width, till they are reduced to the intended thickness of the wall. The number and thickness of the footings of walls, in large buildings, must be regulated by the weight the different parts have to carry. 58. In executing the brickwork of foundation-walls, it is a very important point that what is termed by the workmen the bond, should be properly observed : as every previous precaution may be rendered nugatory by the adoption of a bad or negligent method of laying and bonding the bricks. i 59. It is desirable that in every two perpendicular courses of the footings, the lower course should | be laid stretching or longitudinally, and the upper one heading or transversely. By this method two inches of the brick is uncovered, and six inches and a half inserted in the solidity of the next two courses and of the superincumbent wall. But if this practice be reversed, and the stretching course placed uppermost, then two inches will be uncovered, and only two inches and a quarter covered by the next courses, and no part whatever will be compressed by the solidity of the wall. In this latter case the insistent weight has a tendency to raise the uninserted part of the brick upwards, and re- moves the intended weight from the extremity of the footing to the perpendicular line of the wall, thereby rendering the spreading of the footing of no avail. 60. Instead of laying the bricks in the interior of the wall, after the external courses are laid directly across the wall, they should be laid alternately, diagonally, transversely, and diagonally the contrary way; the interstices next the outside courses being filled in solid with pieces of brick; and the whole of the bricks being thoroughly bedded in mortar, drawn up and Aushed so that no vacuity whatever shall be left. 61. The more evenly the bricks are laid in a foundation the better, as the strength of the wall is materially increased thereby. The smaller the quantity of mortar used the better also. 62. The strength of brick walls is materially increased by adding occasionally courses of hard stone, which serve like sinews to keep all the rest firmly together, and are of great use when a wall happens to sink more on one side than another. The strength is likewise further increased by fortifying the angles with stone, and at the same time the appearance of the building is improved thereby. 63. In the most perfect way of forming the diminution of walls, the middle of the thinnest part be- the wall must of necessity be perpendicular and plain, it must be the inner, on account of the walls and cross walls. The diminished part of the outside may be covered in this case with a fascia and cornice, which will at once give strength and be ornamental. 64. It is a common method of building to insert a considerable quantity of timber in walls as bond. It is, however, a dangerous practice to construct walls in such a manner that their principal support depends on timber, as timber inserted in new walls is very liable to decay, the lime and damp specific gravityple, suppose a wall fondation be 11 Where the foundation is equal and firm, and the material of similar specific gravity, the breadth ought to be as the area of the vertical section passing through the line in which it is measured. For example, suppose a wall 40 feet high, and 2 feet thick, to have a sufficient foundation at the breadth of 3 feet, what should the breadth of the foundation be when the wall is 60 feet high and 24 feet thick? By taking the proportion 40 X 2:3 :: 60 X 27, we shall have 59 feet. 18 [Part I. PRELIMINARY REMARKS ON THE of the brickwork being active agents in producing putrefaction, particularly where the scrapings of roads is used instead of sand for mortar. It is from this cause that bond-timbers, wall-plates, and the ends of girders, joists, and lintels, are frequently found in a state of decomposition. 65. It was a custom with Builders, in former times, to bed the ends of girders and joists in loam in. stead of mortar: as is directed in the act of parliament for rebuilding the city of London. The usual method of putting bond-timber in walls is to lay it next the inside ; but this bond often decays, and of course leaves the wall resting on the external course or courses of bricks; and fractures, bulges, or total failure, are the natural consequence. These evils may be in some degree avoided by placing the bond-timber in the middle of the wall, so that there shall be brickwork on each side of it, and by not putting continued bond to receive the skirtings, battens, grounds, &c. 66. The bond inserted in walls should always be in considerable lengths, and be continued through the several apertures, in which it ought to be allowed to remain as long as possible. 67. It is of the utmost importance in brick buildings that the bond should be as perfect as possible. There are two kinds of bond in brickwork. The one is termed Old English bond, which consists in laying the bricks throughout the wall in alternate courses of headers and stretchers. The other termed Flemish bond, consists in placing in the same course, headers and stretchers alternately, This latter kind of work was introduced into England during the reign of William III. and has al- most entirely superseded the old English method. 68. Mr. G. Saunders, in his Observations on Brick Bond,'* remarks, with reference to the in- troduction of the Flemish bond, “ Strength was then sacrificed to a minute difference in the outside appearance, and bricks of two qualities were fabricated for the purpose ; a fine brick, often to be rubbed and laid in what is called a close putty joint for the exterior, and an inferior coarse brick for the interior substance of the wall. As these did not correspond in thickness, the exterior and interior of the wall could not be otherwise connected together than by an outside heading-brick, that was here and there continued of its whole length where the exterior and interior courses happened to admit of it, which might not occur for a considerable space. The evil increased so far, that walls are found to con- sist of two separate outside faces, of four inches each in thickness, and the interior substance of little bet- ter than rubbish. Not aware from whence the evil proceeds, considerate bricklayers have projected va- rious schemes for obviating the defects in working with Flemish bond. These defects are, in one or both faces bulging away from the interior substance ; or the failure of the wall, by its separating into two thicknesses along the middle, which sometimes takes place when there is a great superincumbent weight on it, and is called splitting. This latter failure is the great terror of a bricklayer. To prevent this evil, some lay laths or hoop-iron occasionally in the horizontal joints between two courses; others lay diagonal courses of bricks at certain heights from each other ; but the good effect of this last mode is much doubted, as in the diagonal course, by not being continued to the outside, the bricks are much mangled where strength is wanted. Others lay all heading-courses within the outside Flem- i ish bond, a practice in great repute ; making the face-work alternately of nine, and of four inches in thickness. This, as far as relates to the splitting of the wall, is an effectual preventive ; but in curing one evil another is increased, for here there is no stretching-bond, the little that occurs in Flemish bond face-work being too trifling to be of any avail ; so that the least inequality of settlement or weight in the longitudinal direction of the wall, occasions a separation at the vertical joints, as may be often seen in the fronts of buildings. The outside appearance is all that can be advanced in favour of Flemish bond. Of this, however, opinions are far from being in accord. Of those who have considered the subject, some allege that if the courses were alternately of stretchers and of * A pamphlet published by Taylor, 1805. Sect. I.] 19 PRINCIPLES OF PRACTICAL ARCHITECTURE. headers which is the old English bond-executed with the same neatness usually shown in Flemish bond, it would be equally or more pleasing. It is of great importance that all concerned in directing the construction of brick walls, should urge the rejection of the Flemish fashion.” 1 69. In carrying up brick buildings, it is desirable that the walls should be worked in heights of from four to five feet at a time, all round the building simultaneously ; otherwise, as all walls imme- diately shrink after they are built, if the parts are brought up irregularly, those which are connected with others that have already settled will, in drying, shrink and cause a cracking of the joints, which will increase in extent, and become more and more injurious as the work proceeds. Nothing but absolute necessity can justify the work being carried up higher in any particular part, than the height of one scaffold, and when such necessity exists, the work should be sloped-off to receive the bond when the rest is completed. 70. The mortar-bed of brick may be either of ordinary, or thin-tempered mortar ; the latter, how- ever, is the best, as it makes closer joints, and, containing more water, does not dry so rapidly as the other. As brick has great avidity for water, it would always be well not to moisten it before laying it, but to allow it to soak in water several hours before it is used. By taking this precaution, the mortar between the joints will set more firmly than when it imparts its water to the dry brick, which it frequently does so rapidly as to render the mortar pulverulent when it has dried. 71. When brickwork is executed during the winter, care should be taken to preserve the walls as much as possible from the effects of alternate rain and frost ; as from the water penetrating into the walls, and the frost converting such water into ice, its bulk is increased, and consequently it expands, 72. In London and its vicinity, walls are now most commonly constructed of brick ; and, although in buildings of a superior description the exterual walls are frequently faced with wrought stone, they are seldom constructed of stone throughout, as is the case in such parts of England where stone is more abundant, particularly about Bath, nearly the whole of which city is built entirely of stone. The stone used principally in London is brought from the Isle of Portland, from Bath, from Worth in Sussex, and from Bromley-fall and Whitby in Yorkshire. Of late years, considerable quantities of stone bave been sent from Scotland to London, particularly from Cragleith and from Dundee, both of which are of good quality. Various kinds of granite are likewise supplied from Scotland, amongst which the Peterhead from the vicinity of Aberdeen is perhaps the best. Granite is like- wise supplied from Devonshire and from Cornwall, and from the islands of Guernsey and Jersey. The walls of some of the most ancient buildings in London are, however, constructed of Kentish minster, parts of the Tower, and of St. Saviour's Church, Southwark, may be particularly instanced. In several parts of the kingdom, particularly in Norfolk and Suffolk, flints are much used for build- ing with ; and there are some very fine ancient walls now standing in the city of Norwich, formed of that material. 73. Walls built wholly of wrought stone may be less in thickness than those constructed of brick, by one-sixth at the least. In wrought stone walls it is of the greatest consequence to make the joints between the stones as small as possible ; and it is a good practice to lay thin sheet-lead between them, as is often done in constructing the shafts of stone columns. 74. The decay of buildings commonly attributed to time, is in reality principally occasioned by the repeated alternate effects of rain and frost ; but, as in finished edifices the vertical surfaces of the walls only are exposed to such action, the injury produced thereby is not so rapid as in those that are unfinished, where there is likewise a horizontal surface exposed, so that the rain and frost bolli 20 [PART 1. PRELIMINARY REMARKS ON THE penetrate into the body of the work. Whilst a building is in progress, when stormy or frosty weather sets in, precautions should be immediately taken to cover the walls, so as effectually to exclude both wet and frost. Straw and weather-boarding are both useful for this purpose: the best way to pro- ceed is to cover the whole of the tops of the walls with straw, and then lay weather-boarding over it, projecting on each side similar to coping. 75. Drainage.]—Every building, whatever may be its size or the object for which it is constructed, should be provided with adequate means for conveying away the refuse-water and other useless and offensive matter, otherwise it will ever be damp and uncomfortable. For this purpose drains should be constructed of such forms and dimensions that they will not be liable to be choked up by the filth of various kinds that will make its way into them with the water. Mr. Smith, of Deanston, who has introduced the greatest improvements in land drainage, advances the general principle, that the size of sewers should be so adjusted as to have them always as full as possible, with a quick flow; and he contends that the drainage of a city might and should be so constructed as to give rise to as little occasion for men to go through the main drains as there is for men to go through the main pipes for conveying supplies of water. He would make the drains narrow. “Their transverse section should exhibit an oval or egg-shape, having the vertical diameter at least double the length of the horizon- tal. The bricks used should be made on purpose, with radiating sides.” “Care should be taken to make the building water-tight and air-tight, and to prevent the foul water and effluvia passing into the contiguous soil. Where land drainage is to be received, special openings can be made at intervals to receive it. All private drainage should pass into the sewers under ground by well-secured channels or pipes. Strong clay pipes, of an oval section, hard burned, and with a good arrangement for secure jointing, might be cheaply procured for the purpose.” 76. In large buildings it is necessary, and in smaller ones it is desirable, that there should be a main or principal drain of adequate size to receive the contents of the inferior ones from the different parts of the building ; and care should be taken that each drain has a sufficient degree of inclination to form a current. A prevalent practice is to join sewers at angles, frequently at right angles. This occasions eddies and deposits of sediment that would otherwise pass off with the water. 77. In determining the dimension of the drains in any given case, it will be necessary to consider the quantity of water that may at any time be required to be conveyed away, from the roof and other parts, during heavy showers of rain, sudden thaws of snow, and on other extraordinary occasions. Segment-bottomed are much superior to flat-bottomed sewers, which always have a larger amount of deposit with the same flow of water. A good invert is the segment of a circle whose chord, being three feet, the versed sine is six inches. Flat sided sewers are also greatly inferior in strength to those with curved sides; and ought not to be used in clayey or slippery ground where there is much pressure on the sides. 78. As the drains are intended to convey away, not only the refuse-water from the building, but the filth of various descriptions that may accompany it, it is necessary to have, in proper situations, receptacles for such sediment, from whence it may be removed without occasioning inconvenience to the inmates of the structure. These receptacles are termed cesspools. The larger they are made, the better, as they will consequently require cleaning out less frequently. 79. In situations where the soil admits of it, the lower portion of the cesspool should be constructed without the use of mortar, by which means the water collected in the cesspool will filter through the crevices, and thereby render any further drainage unnecessary. Where practicable, it is desirable that the main drain should discharge itself into a running stream, and the use of the cesspool be dis- pensed with. 80. In London, the drains for the most part discharge themselves into the common sewers, which Sect. I.] 21 PRINCIPLES OF PRACTICAL ARCHITECTURE. happily are daily becoming more extended throughout the metropolis and its vicinity, and are doubt- less one of the principal causes of its increased salubrity. The healthiness of a whole town is often essentially improved by the formation of a single sewer or drain. The city of Stuttgard has, by the adoption of a good system of drainage, entirely got rid of a peculiar endemic fever; and Pavia, by the filling up of the city-ditches, has had its average duration of life much raised. 81. The providing of adequate means of drainage is in all cases most important, and cannot be too strongly pressed on the attention of the Builder, as on its completeness and efficiency depends, in a great degree, not only the comfort but likewise the health of the inhabitants ; this is of course more particularly essential in cities and populous towns, but where, nevertheless, it is from various causes too often neglected. Even in London we may sometimes find whole streets of dwelling-houses erected without any adequate means of drainage ; the refuse-water and filth of various descriptions being consequently continually accụmulating and corrupting the air of the place. Indeed, some instances might be pointed out, where the drainage of extensive districts is collected and conducted in open channels for considerable distances through populous neighbourhoods ; and thus torrents of filth and corruption are constantly rushing along and contaminating the surrounding air, and at the same time injuring, to a deplorable extent, the health of the unfortunate people who are constrained to live in the vicinity of these noxious and pestilential currents. 82. When building in a low damp situation is unavoidable, it is advisable to raise the level of the principal floor considerably more above the ground than is usual in ordinary cases. By this means the house will be kept more dry and wholesome, and the labour of excavating for the cellars, &c. will be lessened. The cellars should on such occasions be vaulted, or precautions taken, in forming the floors of the rooms above, to prevent the damp rising through them. 83. It is a good practice in such cases, and more especially where an open area round the house would be inconvenient, to form what is termed a dry drain round the whole of the exterior, in order that the damp soil may be kept from the main walls. This may be effected by excavating the ground round the exterior of the structure to the width of eighteen inches or two feet, building a wall adequate to resist the pressure of the earth, and turning a quadrant arch from the top of the same against the walls of the house; the crown of the arch being kept a few inches below the finished line of the ground, with apertures therein sufficient to ensure a free circulation of the air. 12 84. When bricks abound as a building material, they are particularly convenient for the construc- tion of deep sewers and drains, from the facility of handling in confined spaces ; but it is important their quality should be of the best, since if they scale and decay, great expense must be involved in the repair of the drain. The Tipton, or blue brick, is the best for the face-work of drains. In parts of the country where stone abounds, bricks are often little known, and the resources of the district must be made use of. Where the blue lias limestone occurs, Captain Vetch found it a cheap and ex- cellent material for forming culverts and drains of all sizes. It was used largely for that purpose on the Birmingham and Gloucester railway. 85. Roofs. In determining the method to be adopted for covering a building, two extremes are to be avoided, the making it too heavy, or too light. If too heavy, the pressure on the walls will be excessive ; if too light, the defect will be equally prejudicial to the building ; for the covering is 1100 merely to be considered as a defence from the weather, but it should act as a bond or ligature to 12 The best means of introducing an effective system of drainage-as one principal means of improving the sanitary con- dition of the people, especially of the lower classes, whose dwellings throughout England and Scotland have hitherto been very ill provided with proper drains is now [1843] occupying the attention of a Parliamentary commission. They report that wherever an efficient system of drainage has been introduced, the number of febrile diseases annually occur. ring in a given number of the population has been greatly reduced. 22 [Part 1. PRELIMINARY REMARKS ON THE the structure; when, therefore, the roof is too light, the building is deficient in stability. Of the two extremes, a house top-heavy is certainly the worst. 86. In constructing a roof, care should be taken that the pressure be equally distributed on the several walls, as in that case it becomes one of the principal ties to a building. 87. The forms of roofs are various, and should be adapted to the climate and the material with which they are to be covered. 88. In warm and dry climates, roofs may be made flat, as is the case generally in the East, from which countries it is supposed the ancient Greeks copied the forms of those of their earliest build- ings. The inconvenience of flat roofs being probably soon felt, the roofs of their temples we find were constructed of two inclined rectangular planes, sloping from the middle to the sides, and inclining equally to the horizon ; the wall at each extremity being terminated in the form of an isosceles tri- angle, with an horizontal base. The proportion of the height of this triangle to its base was as one to eight, or one to nine. 89. The Romans adopted the same form of roof, but varied the proportion considerably, making the height from one-fifth to two-ninths of the span. After the decline of the Roman empire, higli pitched roofs were very generally adopted, probably from such proportions being commonly in use | amongst the Northern nations by whom Italy was overrun. 90. In most of the old buildings in Great Britain, whether public or private, the equilateral trangle seems to have been long considered as the standard-proportion for the roof. This pitch continued to be used for several centuries; after which the ridge was somewhat lowered, the rafters being made three-fourths the breadth of the building, which proportion was termed the true pitch. Subsequently the pitch or angle of the ridge formed a right angle; but the height of the roof was gradually dimin- ished from the last noticed proportion to one-third of the span or width, and from that to one- quarter, which is now a very general standard, though some roofs have been executed of a much I lower proportion.13 91. The chief advantage arising from the adoption of a high-pitched roof is that the rain is dis- charged therefrom with greater facility, and the snow continues for a shorter period on the sides of it; such a roof is likewise less liable to be stripped by high winds. But it should at the same time be observed, that, from the increase in the height of the roof, the effect of violent gusts of wind on the body of the structure is much increased. Low roofs require large slates, and the utmost care in the execution ; they have the advantage, however, of being much cheaper, since they admit of the timbers being shorter, and of smaller scantling. 92. The most simple form of a roof is that which has only one row of timbers, arranged in an in- clined plane, which throws the rain wholly to one side. This form of roof is called a shed roof or lean to, and is of course only applicable to inferior buildings of small dimensions. 93. A more elegant form of roof for a rectangular building consists of two rectangular planes of 13 The general beight of roofs varies between one-third and one-sixth of the span. For slates the usual height is one. fourth, which makes the angle of inclination to the borizon 2020. Mr. Tredgold indicates the inclination and height which may be given to other materials as follows: Height of roof in proportion to space. . . Copper or lead . . . Large slates . . Ordinary sized slates . . Stone slate or Plain tiles . Stone tiles . Thatch of straw, reeds, or heath . lucination 3° 50' . 220 26° 33' . 29° 41' 240 : 45° . . Secr. I] 23 PRINCIPLES OF PRACTICAL ARCHITECTURE. equal breadth, having the same degree of inclination to the plane of the wall-head. This is some- times called a pent roof. The triangular piece of wall at each extremity, enclosing the roof, is termed a gable. 94. When all the four sides of a roof are formed by inclined planes, the roof is said to be hipped, The inclined ridges, springing from the angles of the walls, are called the hips. Hipped roofs are frequently truncated. See article 97. 95. There is likewise another species of roof, termed the curb roof, which is formed by four con- tiguous planes, the two forming each side of the roof having an external inclination ; the ridge being the line of concourse of the two uppermost planes, and the highest of the three lines of concourse. This construction is frequently denominated the Mansarde roof, from its inventor Francis Mansart, an eminent French architect. The chief advantage of this form of roof is tliat, as the lower rafters pitch almost perpendicularly to their bases, and consequently form nearly a continuation of the wall, a much larger portion of the space within the roof can be made use of than is practicable in the com- mon roofs, in consequence of the great inclination of their sides. 96. Roofs upon circular bases, with all their horizontal sections circular, the centres of the circles being in a straight line drawn from the centre of the base perpendicular to the horizon, are called gevolved roofs, or roofs of revolution. 97. When the plan of the building is a trapezium, and the wall-leads are properly levelled, the roof cannot be executed in plane surfaces so as to determine it in a level ridge : the sides therefore, instead of being planes, are sometimes made to wind, in order that the summit may be parallel to the horizon; but the preferable method is to make the sides of the roof planes, enclosing a level space or flat in the form of a triangle or trapezium, at the summit of the roof. Roofs flat on the top are said to be truncated ; these are chiefly employed with a view to diminish the height. 98. Sometimes in roofs of rectangular buildings, instead of a flat at the top, a valley is made, which makes the vertical section somewhat in the form of the letter M, or rather in that of an in-` verted W. This species of roof is termed an M roof. 99. When the plan of the roof is a regular polygon, a circle, or an ellipse, the horizontal sections being all similar to the base, and the vertical sections a portion of any curve, convex on the outside, the roof is said to be domical. 100. The ordinary covering for roofs is either thatch, tiles, slates, stone, ironi, lead, or copper. A composition called tessera has of late been occasionally used for covering flat roofs ; and still more recently, malleable zinc has been introduced for covering roofs. Wrought iron, bent in the form of pantiles, has likewise been employed for covering the roofs of extensive sheds and warehouses. Iu some parts of the country shingles, which are small boards similar in form to slates, made of oak, either sawed or cleft, from eight to twelve inches long, and four inches broad, are likewise used for covering roofs. 101. Thatched roofs are formed either of straw or reeds. When well-executed, either will last many years, and form a very good covering to a building. The chief objection to their use is that they form a harbour for vermin. 102. The materials used for covering roofs vary in different parts of the country. The natural production of the district being of course the easiest obtained, is most frequently adopted for ordinary buildings; thus, in Gloucestershire and Oxfordshire, we find them very generally covered with thin slabs of stone ; whilst in Sussex, Surrey, and Norfolk, tiles are more common. The effects of the daily increasing facilities for transmission are, however, very apparent in this particular, as we now find the use of slate becoming very general in almost every county of England and Scotland, par- ticularly in the superior description of buildings. 24 [PART 1 PRELIMINARY REMARKS ON THE 103. In London and its vicinity, slates, either from Wales or Westmoreland, are very generally adopted; though tiles are likewise still very common. Tiles, when they are of a good description, and glazed—as is the common practice in some parts of the country-form a very excellent and durable covering; though, from their great weight, they require the timbers of the roof to be made much stronger than is necessary for slates. 104. A composition invented by. Lord Stanhope, and used by the late Mr. Nash, for covering the nearly flat fire-proof roofs of Buckingham Palace, is described as being composed of Stockholm tar, dried chalk in powder, and sifted sand, in the proportion of three gallons of tar to two bushels of chalk and one bushel of sand, the whole being well boiled and mixed together in an iron pot. It is laid on in a fluid state, in two separate coats, each about three-eighths of an inch in thickness, squared slates being imbedded in the upper coat, allowing the mixture to flush up between the joints the whole thickness of the two coats, and the slates being about an inch. The object in imbedding the slates in the composition, is to prevent its becoming softened by the heat of the sun, and sliding down to the lower part of the roof, an inclination being given of only one inch and a half in ten feet, which is sufficient to carry off the water, when the work is carefully executed. One gutter, or water- course, is made as near to the centre as possible, in order to prevent any tendency to shrink from the walls, and also that the repairs, when required, may be the more readily effected. It is stated, that after a fall of snow, it is not necessary to throw it from the roof, but merely to open a channel along the water-course, and that no overflowing has ever occurred; whereas with metal roofs it is necessary to throw off the whole of the snow on the first indication of a thaw. These roofs have been found to prevent the spreading of fires. Another advantage is stated to be the facility of repair which the composition offers, as, if a leak occurs, it can be seared and rendered perfectly water-tight by passing a hot iron over it; and when taken up, the mixture can be re-melted and used again. The author proposes to obviate the disadvantage of the present weight of these roofs by building single brick walls at given distances, to carry slates, upon which the composition should be laid, instead of filling the spandrels of the arches with solid materials, as has been hitherto the custom. . 105. A square of 100 feet of the undermentioned materials used in roofing weighs as follows:-- Cwt. qrs. lbs. Sheet copper, 16 ounces per square, including seams, weighs about . 1 0 0 Lead, 7 lbs. to the foot, including soldering and laps, weighs . . . 7 0 0 Fine slate . . . . . . . . . . . Coarse slate . . . . . . . . . . . . 10 0 0 Pantiles and mortar . . . . . . . . . . 9 2 0 Plain tiles and mortar . . . . . . . . .' . 18 0 0 106. Floors.)-Floors are of various kinds. Those formed of brick or stone are termed pavements; some are formed of earth, and are consequently called earthen-floors ; others of plaster, and are thence deno- minated plaster-floors ; whilst such as are constructed of timber and boarding, are called timber floors. 107. Under the term floor is included not only the boarding, but likewise the whole of the timber-work necessary to its support. The timbers which support the boarding are termed the naked flooring. .Floors are said to be single, or double, or framed, according to the manner in which they are constructed. 108. Single floors consist of a series of timbers of equal depth termed joists, generally laid parallel to each other, breadthways of the apartment, and about twelve inches asunder; the ends being inserted into and supported by the opposite walls. On the upper edges of the joists, boards are fastened which form the surface of the floor. This description of floor is only adapted for the smaller and inferior apartments. A floor of this kind may, however, be much strengthened by nailing a series of small pieces of quartering, termed struts, extending in a line from one end of the floor to the other, in an angu. lar position against the sides of the several joists, so as to cross each other in the form of the letter & Sect. I.] 25 PRINCIPLES OF PRACTICAL ARCHITECTURE, 109. Sometimes every third or fourth joist is made deeper than the other, and the ceiling joists are fixed to the deep joists at right angles. 110. Double floors consist of three tiers of joists, viz.: the binding joists, the bridging joists, which are laid on them, and the ceiling joists, which are notched and fixed to the under side of the binding- joists. 111, When the bearing is considerable, and it is required that the ceilings should be good, framed floors become necessary. In this case the binding-joists are framed into large pieces of timber, ex- tending transversely across the building, and termed girders. The distance between the girders should never exceed twelve feet. The ends of the girders, which are inserted into the opposite walls from nine to twelve inches, should be laid on pieces of stone or timber, and a small vacuity should be left on each side and at top to admit of a circulation of air around the timber, as it is otherwise liable to decay. 112. The binding-joists are framed into the girders, at distances of about five feet apart. Across them, other series of timbers, termed bridging joists, are laid at distances of about twelve inches apart, and to these the floor-boards are attached. The ceiling-joists are nailed to the binding-joists as in other floors. 113. When the span of the floor exceeds twenty feet (if required to sustain considerable weight it is somewhat less) the girders should be trussed. The excellence of every truss is to dispose the timbers in positions as direct to each other as possible. Trusses are variously constructed, according to the width of the building, the contour of the roof, and the circumstances of walling below. Ex- amples of different species of trusses are given in this volume under the head of CARPENTRY. 114. Particular attention should always be given to the trussing of beams; as unless it is done with judgment, and in an effectual manner, it is rather prejudicial than useful. An iron tie should always be introduced as an abutment for the truss ; because the failure of trusses is often occasioned by the enormous compression applied upon a small surface of timber. 115. When beams are trussed, they should be cambered about one inch in a length of fifteen or sixteen feet, as all timber has a tendency to sag. 116. In determining the situation of the principal timbers of a floor, the placing of girders over openings or near to flues should always be avoided. If it can be done in no other way, the girders should be laid in an inclined direction. 117. Floors when first framed should be about three-quarters of an inch higher in the middle than at the sides of a room. The ceiling-joists likewise should be fixed about three-quarters of an inch in twenty feet, higher in the middle than at the sides of a room; for all floors, however well-constructed, will settle in some degree. 118. It may be here observed, that in laying floors the boards should be made to rise a little under the doorways, in order that the doors may shut close without dragging. This likewise assists in caus- ing them to clear the carpet. It has also been recommended that when floors are laid, the joints of the boards should be shot and fitted, and the boards tacked down only the first year; the nailing them down being deferred till the next. By this means they will lie stanch, close, and without shrinking in the least, as if they were all of one piece. · 119. Apertures.]—The apertures in buildings are the doors, windows, and chimneys. The number, position, and dimensions of the several apertures in a building must of course be regulated by the uses to which they are to be applied, either for the purposes of ingress or egress, as doors,-or for the admission of light and air, as windows. They should, moreover, be justly proportioned to the size of the building. 120. The doors and windows should always, as far as practicable, be placed centrically, both as to PKN 11 26 [Part I. PRELIMINARY REMARKS ON THE 2 arrangement of plan, and the vertical situation in the walls, particularly the latter. This principle is, however, in modern buildings too frequently disregarded, to the detriment of the stability as well as the appearance of the structure. 121. Doors.1-Doorways are either external or internal, and vary both in size and style of decora- tion, accordingly. 122. External doorways are usually much larger than internal ones, and should be proportioned to the size and intended use of the building. Those to public edifices should be of larger dimensions than those to private dwellings. 123. Entrance doors to private houses should not be less than three feet six inches, nor more than four feet wide. The height will vary according to circumstances, but should never be less than seven feet, so that a man may easily pass through them. It is a good proportion, in small doors, to have their dimensions in the ratio of three to seven ; and in large doors, of one to two. 124. The bottom of the door-frame is termed the sill ; the vertical sides, the jambs; the part oppo- site the sill, the head or soffite. When an aperture is left above the door to admit light, the horizontal rail between the head of the door-frame and the door itself, is called a transom. 125. The decorations of doorways consist usually of an architrave running up each side and along the top. When a superior degree of enrichment is required, a frieze and cornice is added, the ends of which are occasionally supported by carved trusses or consoles. Sometimes the doorway is decorated with pilasters, or with semi, or three-quarter columns, with a regular entablature ; and in buildings of superior importance the entrance is adorned with a portico of detached columns, the door itself being enriched in a corresponding degree. 126. As respects internal doors, it is to be observed, that, in our climate, the fewer doors a room has the more comfortable it will be ; and if those necessary for communication between the different rooms are placed opposite to each other, such an arrangement not only contributes towards the regu- larity of the decoration, but when required to be set open, produces a freer circulation of the air, and likewise adds to the splendour of the appearance when the rooms are lighted up. 127. When more than one door is placed on the same side of the room, care should be taken that they are equidistant from the centre,-a point too often disregarded, in consequence of which a needless and offensive irregularity is produced. Doors should be hung so as to open inwards and „towards the fire-place, in order that persons sitting round the fire may be protected as much as pos- sible from the current of cold air admitted by the opening of the door. A door should never be placed close by the side of a fire-place. In bed-chambers, care should be taken to avoid as much as possible the placing of doors by the sides of the bed, both on account of the draughts of air from them and the noise communicated through them, as well as the inconvenience attending the opening and shutting them in such situations. 2-ion 128. Internal doors should never be less than two feet nine inches wide ; nor need they generally be more than three feet three, or three feet six inches. Should it, however, be necessary to make them of a greater width, they should be made to open in two flaps or leaves, by which means the door will not only be lighter, but it will not project so far into the apartment, and may in many cases be made to fold entirely into the thickness of the wall. 129. Doors are usually made either of deal or wainscot. The better sort are often made of ma- hogany, and inlaid with different kinds of rare woods: in such cases they are most commonly veneered. The door itself is either framed, battened, or ledged. 130. Windows.]-Windows are of various forms, being sometimes squares or parallelograms ; and having sometimes semi-circular, semi-elliptical, or segment heads, in other cases being perfect circles, Sect. I.] 27 PRINCIPLES OF PRACTICAL ARCHITECTURE. or ellipses. They are, however, most commonly rectangular ; the external jambs being either quite plain, or ornamented with stone or plaster dressings. 131. The vertical sides of the aperture are termed the jambs ; the top, the soffite ; and the bottom, the sill. When the soffite of the aperture is curved, the points where the curve unites with the vertical jambs are termed the springing points. When the section of the soffite is less than a semicircle, the arch is usually called by workmen a scheme-arch. 132. The dressings consist of the sill, and the insisting architrave surrounding the upper part; some- times a frieze and cornice is added. The breadth of the architrave varies, according to circumstances, from one-fifth to one-seventh of the opening ; and the number and size of the mouldings forming the cornice likewise vary according to the effect, as to lightness or solidity, that is required to be given to the building. Sometimes the frieze is altogether suppressed, and the lower moulding of the cornice rests on the architrave. In edifices of importance windows are usually ornamented in the same manner as doors. 133. Windows should be arranged, as to number and size, so as to admit an adequate quantity of light and air to the apartment in which they are placed. The facility of obtaining this must depend on the situation and aspect of the building; and by these circumstances therefore, in a greater or lesser degree, the number and forms of the windows will be regulated. In this climate but little incon- venience is experienced from a predominance of sunshine at any season of the year, or in any situa- tion ; it is desirable therefore on all occasions to admit a sufficiency of light to counteract the gloom which more commonly prevails in wet and cloudy weather, as recourse can always be had to blinds or shutters whenever an excess of light and heat is admitted by the windows. 134. Great changes have taken place at different periods in this country with respect to the num- ber and arrangement of the windows in our dwellings. Formerly the windows were very numerous, the piers between them being so slender as even to appear inadequate to support themselves: as may still be seen in many old houses in different parts of the country. Latterly we have fallen into the opposite extreme, and we now often make the windows too small and not sufficiently numerous. This has perhaps been caused partly from motives of economy, since a tax has been imposed on light and air, and partly by a desire to introduce the Italian and Grecian taste into our dwellings, without due consideration as to the difference of climate. 135. It is scarcely possible to give a general rule of universal application, by which to determine the number and size of windows necessary for different apartments, as so much must always depend on the situation of the building, and the particular use to which the room is to be applied. Sir William Chambers, in treating on this part of the art, after objecting to the rules given by Palladio and others with respect to windows, as inapplicable to this climate, observes : “ In the course of my own practice I have generally added the depth and the height of the rooms on the principal floors to- gether, and taken one-eighth part thereof for the width of the window, a rule to which there are but few objections.”* Even this rule, however, will not always produce a satisfactory result. A large lofty apartment requires more light than one circumscribed in its dimensions ; and the quantity admitted should be regulated so as to produce in the minds of the spectators either gay, cheerful, or solemn sensations, according to the nature and purpose for which the structure is designed. 136. When series of windows occur in the exterior of a building, the sills should all range; and the intermediate spaces or piers between them should be of equal width, and never, when it is possible to avoid it, less than the width of the window. The most general practice limits the breadth of piers Treatise on the Decorative Part of Civil Architecture,' (page 134, Fourth Edition, * See Sir William Chambers's Taylor, London, 1826. 28 [PART 1. PRELIMINARY REMARKS ON THE to the same as that of the aperture, or to a half more. When any fanciful arrangement is attempted, both the apertures and piers should be disposed symmetrically; and the extreme piers should on all occasions be made wider than the intermediate ones, and should correspond with each other. An odd number of windows in the same front is preferable to an even number: for as it is desirable to have the entrance-door in the middle of the length, an even number regularly disposed, would oc- casion a pier to be over the doorway, which is by no means graceful. 137: The dimensions of windows should be adapted to the size of the principal apartments. A good proportion for a rectangular window is for the height to be two and a quarter times the width, which, in ordinary dwellings, may vary from three and a half to four feet. In large buildings the width must be increased according to circumstances.15 138. In some instances, where the space will not admit of the introduction of two windows, and when one of the ordinary size is not sufficient, what is termed a Venetian window is employed. This consists of a window divided into three compartments in width, the centre-space being equal to the width of an ordinary window, and the spaces on each side being considerably less. Occasionally the centre space is filled by a sash-door. . 139. French casements, as they are termed, are likewise sometimes substituted for hanging-sashes ; and, in houses in the country, when it is desirable to have the means of walking out of an apartment on to a lawn or terrace, they are doubtless very convenient, but they always render a room very cold from the difficulty of effectually fastening them so as to exclude the air and damp, which indeed can only be accomplished at a considerable expense. As these casements are often applied in the ordi- nary houses about London, they are not only exceedingly inconvenient from the causes already noticed, but their effect is perfectly ridiculous. 140. The sashes of windows in ordinary dwelling-houses are most commonly made of deal; in those of a superior description, of wainscot, or mahogany; occasionally the bars are made of copper, brass, or iron.16 Sometimes sash-frames and sashes are made altogether of cast-iron, particularly those for churches, warehouses, and manufactories. Some years since a composition of lead and tin was much in use for sashes, and more particularly for fan-lights; but from the increased facility of manufac- turing iron and copper into sash-bar, the use of this species of metal has decreased. 141. In some cases, the jambs of apertures are not perpendicular to the sills, but stand at equal angles therewith, so as to approach nearer to each other at top than at bottom. Such examples are frequently to be met with both in ancient and modern works. Vitruvius, in his fourth book, has laid down rules for Doric, Ionic, and Attic doors, all of which have their apertures narrower at the top than the bottom. 142. Sometimes the jambs of apertures are formed to stand obliquely to the walls in which they occur, the area of the inner surface of the aperture being larger than the outer: they are in such cases said to be splayed. 143. Chimneys.—The subject of chimneys is one of great importance in building. A chimney con- sists of an aperture in the wall to receive a grate or stove ; and is sometimes named a fire-place. From it a vacant space, named a vent, shaft, or fue is carried, within the thickness of the wall, to the level of the top of the roof, to convey away the smoke. SIL Na 15 The dimensions of windows are restricted by act of parliament to twelve feet in height, and four feet nine inches in width; if exceeding these dimensions the window, if not belonging to a shop, warehouse, or licensed public-house, is charged as two, unless it is not more than three feet six inches high. 16 Of late years the use of wrought iron rolled sash-bar has become very common, not only for greenhouses and con. servatories, but for skylights and sashes generally. Srct. I.] PRINCIPLES OF PRACTICAL ARCHITECTURE. 29 144. The bottom of the fire-place should be laid with square tiles, stone, marble, or an iron plate, in order to receive the cinders and ashes; this is termed the inner hearth. Upon a level with this, or a little above it, and immediately before the fire-place, a space equal in length to the breadth of the chimney, and about two feet in breadth, should be laid with the same sort of materials as the inner hearth; it is termed the slab, and is laid either in a wooden boxing, containing sand and mortar, or upon a flat brick arch, which has been turned between the wall and a trimming-joist in the floor. The front and vertical sides of the chimney apertures are termed Jambs, and are composed of stone, marble, wood, or iron. The part which reaches across the top of the aperture is called the mantel, and is of the same sort of materials as the jambs. As in thin walls, the part which the chimney occupies is projected into the room, in this, what is over the chimney, is termed the breast. Within the fire-place, the parts which reach between the jambs and back, are named covings. Where the space is contracted from the size of the fire-place to that of the flue, it is called the gathering wings, or throat. The portion of the chimney which rises above the roof, is named the chimney top. Where several flues, either in the wall or at the top, approach very near each other, the partitions between them are named withs; and the whole is termed a stack. 145. In rooms of ordinary dimensions, the flues, in rough stone walls, are from 12 to 14 inches square ; in hewn stone or brick work, about 10 by 14 inches ; but the section must be enlarged in rooms of large dimensions, kitchens, &c. As soot is apt to gather in the angles of square flues, the circular form is preferable. They should be made quite smooth within, and free of quick bendings. It is of advantage to have the flues a great height; but if raised much above the level of the roof, it is difficult to render them ornamental. 146. As much of the comfort of an apartment depends upon its being free of smoke, and there is much economy in causing the heat to be reflected into the room, instead of suffering it to be unnecessarily absorbed by the materials, or dissipated in the flue, &c., great attention is necessary in fitting up the fire-place. For economy, stone which will stand the fire is preferable to metal. The side-covings should be levelled, or wholly circular ; and the opening at the back of the fire-place into the throat should not exceed four inches in breadth ; frequently one inch and a half is sufficient. With regis- ter-stoves this can be regulated to great advantage. The throat part should fall back from this aper- ture, and, in general, no air should be admitted into the throat, but what passes over the fire ; it is therefore advisable to fit up the fire-place very accurately, and place the front of the grate forward to the line of the face of the wall of the apartment. In countries where stoves are used, it is custom- ary to place them altogether before the face of the wall. This admits of much decoration, and also throws the greatest part of the heat into the room. 147. Chimneys have always been considered important features, but the style of decoration has varied greatly in different ages. In the Norman castles, they were frequently large, and accom- panied by rude pillars, sculptures, and ornamented mouldings. After the revival of Roman architec- ture, the whole space between the fire-place and the ceilings, called the chimney brace, was covered with architectural decorations of great labour and expense. Wood was succeeded by stucco work and ornamented pannels, to receive paintings; but of late these have been abandoned, and the chimney has been reduced into the smallest possible bounds, making it for elegance to depend upon the marble-dressings, and highly-polished and engraved steel register-stoves. 148. The difficulty of constructing open fires, as is our custom, so as to secure the greatest possi- ble effect therefrom, and at the same time to avoid, under all circumstances, annoyance from smoke, is one of the most formidable that a Builder has to encounter. Consideration should therefore be given, when designing a building, to arrange and construct the chimneys in such a manner as to pre- clude as far as possible the danger of their smoking. 30 [PART 1 PRELIMINARY REMARKS ON THE ! 149. The principal causes of the smoking of chimneys, according to Dr. Franklin, * who gave great attention to and published a treatise on the subject, are the following :- “First. Smoky chimneys in a new house are such frequently for mere want of air. The workmanship of the rooms being good, the joints of the boards of the floors and the pannels of the wainscotting are all true and tight, the sashes and doors shut with exactness, added to which perhaps the walls not being thoroughly dry, cause a dampness in the air of the rooms which keeps the woodwork swelled and close. “Secondly. The openings of the chimneys in the rooms being too large; they being often regulated rather with regard to symmetry and beauty than to utility. “Thirdly. The chimney being constructed with too short a funnel. “ The fourth cause, is their overpowering one another; if, for instance, there be two chimneys in one large room, and a fire is made in each of them, if the doors and windows shut close, it will soon be found that the greater and stronger fire will overpower the weaker one, and draw air down its funnel to supply its own demand. If, instead of two chimneys being in one room, the two chimneys are in different rooms which communicate by a door, the effect is the same when- ever the door is opened. “The fifth cause is when the tops of the chimneys are commanded by higher buildings, or by a hill, so that the wind blowing over such eminences falls like water over a dam, and sometimes almost perpendicularly on the tops of the chim. neys that lie in its way, and beats down the smoke contained in them. - The sixth cause of smoky chimneys is the reverse of the last mentioned, namely, where the top of the chimney is so situated as to be below the roof or parapet wall of a building, so that when the high part happens to be exposed to the wind it forms a kind of dam against its progress, and is then forced down the funnel of the chimney, driving the smoke contained therein down into the room. “Seventhly. Chimneys often smoke in consequence of the improper situation of a door; this happens more particularly when the door is situated by the side of the fire-place, and is bung so as to open against the adjoining wall, which often occurs, as being, when open, more out of the way; consequently, when the door is opened in part, a current of air rushes in, passes along the wall and across the opening of the fire-place, and drives some of the smoke out into the room; like- wise when the door is shutting the current is augmented, and a similar effect is produced. “Eighthly. A room that has no fire in its chimney is sometimes filled with smoke, which is received at the top of its funnel and descends into the room; the smoke being forced down by the pressure of the descending current, occasioned by the difference between the external air and that of the flue. “Ninthly. Chimneys that usually draw well will occasionally give smoke into the room, in consequence of its being driven down by strong winds passing over the tops of their funnels; this is more particularly the case where the funnels are too short.” 150. Having enumerated these several causes, the Doctor gives at considerable length the remedies which have been found to be efficacious in removing the inconvenience, and for which the reader is referred to the treatise already alluded to. 151. A new invention, as respects the construction of fiues, has been brought before the public, and a patent obtained by Mr. J. W. Hiort, of his Majesty's Office-works. The invention consists in building circular smoke-flues, within the usual thickness of walls, and incorporated with the common brickwork. Each flue is surrounded in every direction, from bottom to top, by cavities commencing at the back of the fire-place and connected with each other. The air confined within these cavities is by the heat of any one fire rendered sufficiently warm to prevent condensation within all the flues contained in the same stack of chimneys. The patentee states that these flues are erected without difficulty, and can be adapted with equal ease to every possible bend, turn, or direc- tion, without the smallest deviation from their original form and capacity. The circular flue com- mences at the throat of the chimney, below the usual line of the chimney bar, and immediately over the fire, and the half-circle continues thence down to the hearth, forming the centre of the back of the fire-place. The usual filling in brickwork by this means becomes unnecessary, and the angles within the fire-place may be altogether avoided. Thus the throat of the chimney is made to contain * See a Tract entitled · Observations on Smoky Chimneys,' hy B. Franklin, LL.D. London, 1793. SECT. I ] 31 LA PRINCIPLES OF PRACTICAL ARCHITECTURE. no more air than can be heated by the fuel ordinarily consumed, nor can the air of the room connect with that of the chimney without passing through, or coming in contact with, the fire ; and should the upper part of the flue admit of a counter current of descending colder air, it must at a certain point become rarified, and return to the centre spiral column of ascending smoke and heated air. 152. In building these flues, no other material is used than the patentee's newly invented bricks, and the cement by which they are united. These bricks require no cutting, being made on systematic principles, and when applied to the purposes intended, the joints, both horizontally and vertically, are as those of an arch, and therefore rendered capable of resisting great external pressure ; and the rim of the flue being in two thicknesses, the interior is essentially protected from any injury to which the outside facing of the wall may be liable, by plugs driven into the mortar-joints. The patentee like- wise states, that, from the construction of these chimneys, and the nature of the materials of which they consist, no danger need be apprehended should the soot ignite, for such an accumulation of soot as the common chimneys are liable to cannot take place within these tunnels, there being no angles in which the soot can lodge, the draft of air through them being much stronger, and the necessity of cleansing rendered less frequent by the inside of the bricks being vitrified to prevent adhesion. 153. These chimneys do not require to be carried up to a great height above the roof or parapet, as chimneys constructed according to the common method usually do. The tops or terminations of the newly invented chimneys may in some situations be entirely concealed; if not, by their formation, and their not being required to be lofty, they are perfectly secure in the most tempestuous weather. Each flue terminates with a small brick or stone shaft, with a stone cap and base, so contrived as to preserve to the summit a continuation of the hot air cavities, and so as to render unnecessary any conducted in an ascending direction, so that the smoke issues out unimpeded, and protected under that line of ascension. The wind in its transit upwards, producing a quick exhaustion of the air within the chimney, causes the smoke to be emitted with a velocity in proportion to the force of the wind from whatever quarter it may blow. 154. These flues have been introduced into many public and private buildings, and the result has been highly satisfactory. The expense it is said is not on the whole materially greater than that of ordinary flues, when the cost of the requisite shafts for them and the constant repairs which are neces- sary thereto is taken into the account. SECTION II. OF MATERIALS. Importance of a knowledge of the nature and qualities of Building-materials, 1-3:- Classes of Materials, 4.- I. Stones, 5. Natural Stones, 6, 7.- Component parts of Stones, 8–12. Structure and Mechanical properties of Stones, 13–17.---Causes of the decomposition of Stones, 18-25.___Siliceous Stones, 26–32. Argillacous Stones, · 33-37.-_ Calcareous Stones, 38—44.- II. Artificial Stones and Cements, 45–63.- Sand, 64-66.- Cements, 67–87. Concrete, 88–92.- Brick, 93—113.- III. Wood, 114—129,-_ Oak, 130-132,_ Fir, 133—139. -Beech, &c. 140—153.- Strength of Timber, 154. IV. Iron, Cast-iron, 155—164.- Forged iron, 165-171. -- V. Glass, 172-178. 1. The attention of the reader having, in the preceding pages, been directed to such General Prin- ciples as are most important to be attended to in the designing and constructing of edifices, a few observations relative to the nature and qualities of the different kinds of materials, and likewise as to their selection for use in a building, will now be introduced, with some valuable additional matter from Mahan's • Elementary Course of Civil Engineering.'* 2. An accurate knowledge of the physical and chemical properties of materials is of essential im- portance to the Builder or Engineer, to enable him to form a correct estimate of the advantages to be derived from their proper application to the purposes of constructions, so as to satisfy the condi. tions of judicious economy and skilful workmanship. He should therefore be acquainted with their absolute and relative strength, the resistance which they offer to friction and shocks,—the changes which they undergo from exposure to the atmosphere, and to the more ordinary chemical agents, as fire, salt-water, &c.—and, finally, with the time and labour required in preparing them for the pur- poses of buildings. 3. The strength and durability of buildings depend chiefly upon the nature of the materials of which they are constructed; yet so little regard has been paid to the selection of durable materials for architectural buildings in this country, that in a thousand years almost every specimen of British genius which now adorns this Island, will be mouldered into dust! The Egyptians, the Greeks, and the Romans, have rendered themselves famous by the magnitude and solidity of their buildings; and so judicious have they been in the choice of materials, that their Temples, their Palaces, their Thea- tres, and their Aqueducts are still the objects of admiration, though two thousand years have elapsed since some of them were erected. With such examples before us, and with the aid of sciences un- known to the ancients, it is little to our credit that we have not buildings more worthy of the age, and better capable of perpetuating the memorable times in which we live.t 4. The materials used in building may be arranged in four classes: 1. Natural stones; 2. Artifcius stone and Cements ; 3. Wood ; 4. The Metals. I.-STONES. 5. The difference in the qualities of building-stones is chiefly occasioned by a difference either of chemical composition or of mechanical structure. Those stones are considered to be of the best quality that are compact and of an uniform texture,-capable of resisting moisture, and the effects of frost or of heat without splitting, and the effects of the weather without perceptible decay. Every • Professor Barlow's edition. Glasgow, 4to. A. Fullarton & Co. † See Appendix Co SECT. II.] PRINCIPLES OF PRACTICAL ARCHITECTUREL 33 . one must be sensible, that, in order to form an opinion respecting the qualities of stone as a building- material, we ought to be in some degree acquainted with its chemical and mechanical properties ; and when the properties of those from a new quarry are the same or similar to those that are known to be strong, durable, and suited for building-purposes, we may then conclude that they also will be strong, durable, and fit for building. A knowledge of the chemical properties only of stone, however, is not sufficient; because from experience we find that the structure or mechanical arrangement of the stone has an equal, if not a greater, degree of influence on its strength and durability. 6. Natural stones or Rocks. ]-Natural stones, or rocks, are composed of an aggregation of several simple mineral substances. They are variously classified by naturalists, either according to their chemical constituents, or to their external appearances and physical properties. These classifications, although essential for scientific arrangement, are of minor importance to the Builder ; as the princi- pal points requiring his attention are those which render stone a suitable material for building. 7. The most essential properties of stone as a building-material, are strength, or the resistance which is offered by it to rupture, caused either by compression, extension, or a cross strain ; hardness, or its capability of resisting shocks, and attrition ; and durability, or its unchangeable character when exposed to the extremes of temperature, to the atmosphere and to chemical agents. These properties can be readily ascertained by a few simple experiments which will be noticed under their proper heads. 8. Component parts of Stones.)-Stones are compound bodies. They are chiefly composed of what chemists have called earths. Of these earths there are four kinds : 17 some of which are always found to be mixed or combined in different proportions in different stones.18 The names of these four earths are, 1. Silex ; 2. Alumina ; 3. Lime ; and, 4. Magnesia. . 9. Silex, or the earth of flints, when perfectly pure, appears in the form of a white powder, which is incombustible, infusible, nearly insoluble in water, and not acted upon by common acids. It is found in almost all stones, particularly in gravel, sand, quartz, and flint, of which it forms nearly the whole substance. Almost all stones which strike fire with steel contain silex. Stones which contain a large portion of silex, if compact, are extremely durable, but they are very difficult to work. Those stones which derive their peculiar qualities from silex, are called siliceous stones. See article 26 of this section. 10. Aluinina, or pure clay, in its perfect state, is white like silex ; it adheres strongly to the tongue, is incombustible, insoluble in water, but soluble in acids and in alkaline lixivia. It enters into the composition of many building-stones, but there are few in which it forms the principal part; never- theless it appears to be the cause of their peculiar qualities. Alumina has a considerable attraction for the metallic oxides, and the other earths. Common clay is a mixture of alumina and silex, but it also frequently contains metallic oxides and other earths. Those stones which derive their peculiar qualities from alumina are called aluminous stones; but as they generally contain a mixture of silex also, they are described under the general head of argillaceous stones. See article 33 of this section, 11. Lime, or calcareous earth, is the substance well-known, in its pure state, under the name of quicklime. It exists in stones, in combination with fixed air or carbonic acid, when it is called car- bonate of lime; and is the substance which constitutes the principal part of chalk, limestone, and mar- ble. Those stones which are chiefly composed of lime are called calcareous or limestones. See article 38 of this section. . 12. Magnesia, when pure, appears in a lighter and whiter powder than any of the other earths ; it is soluble in acids, but not in alkaline lixivia. It is a component part of several minerals. The 17 There are several other earths, viz. Barytes, Strontian, Zircon, &c. ; but they very rarely occur in building-stones. 18 Any of these earths may be obtained in a separate state by the methods described in Henry's ' Epitome of Chemistry Park's Clieinical Catechism ;' Thomson's 'System of Chemistry;' or Brande's Manual of Chemistry.' 34 [Part 1 PRELIMINARY REMARKS ON THE stones which contain a considerable portion of magnesia, have generally a smooth and greasy feel, a greenish colour, a fibrous texture, and a silky lustre. 13. Structure and Mechanical Properties of Stones. ]-Stones are either simple or compound. Lime- stone is an instance of the first kind, it being uniformly of the same substance throughout; but a com- pound stone is an aggregation of parts of simple minerals. Compound stones are either cemented or aggregated. This distinction is founded on the mode of their formation. The grains or masses, in the case of cemented stones, appear to have been formed previous to their being cemented together. The strength and durability of these stones depend on the nature of the cementing material. When it is a pure siliceous cement, the stones are always very durable ; but, when the cementing matter is alumina with a mixture of iron, the stones are very subject to be disintegrated. In calcareous sandstones, where the cementing material is of a soft kind, it is liable to be washed out by rain, and the stones fall in pieces or moulder away. 14. In aggregated stones, the present structure appears to have been the original one; and the simple minerals which constitute them are immediately connected together, without the interposition of a cementing material. When the simple minerals which constitute an aggregated stone, measure nearly the same in length, breadth, and thickness, the structure is called granular ; as in granite. When the constituent parts are thin, and lie flat, the structure is said to be slaty or beddy. Sometimes stones are granular slaty ; part of the constituents being granular and the other part slaty. In some aggregated stones one of the constituent parts forms a basis in which the other parts are imbedded, either in the form of grains or crystals. The structure is then said to be porphyritic. 15. Stones which are of a fine and uniform grain, compact texture, and deep colour, are the strongest; and when the grain, colour, and texture are the same, those are the strongest which are the heaviest ; but otherwise the strength does not increase with the specific gravity. There are certain defects that can be ascertained by the eye, which should cause stone to be rejected as a building- material, though belonging to a good class: these are cracks, cavities, and foreign minerals, particu- larly the various forms under which iron is found. The effect of these is to render the stone weak, brittle, and liable to disintegration, from the decomposition of the metallic ores. Great hardness is likewise objectionable when the stone is to be prepared by the chisel, owing to the labour required to work it; and as the stones of this character generally wear smooth, and become polished by attrition, they are unsuitable for stairs, pavements, &c., where accidents might happen from slipping. Brittle- ness is a defect which frequently accompanies hardness, particularly in coarse-grained stones; it pre- vents the stone from being wrought to a true surface, and from receiving a smooth edge at the angles. The coarse-grained stones are, moreover, more liable to rapid disintegration than those of a fine texture. 16. Experiments on an extensive scale have been made both in France and England, to ascertain the comparative and absolute strength of the most common building-stones.* Small cubical blocks, the sides of which measured about four superficial inches, were selected for the experiments, and the following general results were obtained : Ist. The strongest and hardest stones were found to be those having a dark colour, a compact uniform texture, and the greatest specific gravity. These last qualities generally accompanied each other; and the strength and hardness increased in proportion as they predominated, without however exhibiting any uniform law of variation. 20. That, for the same stone, the strength varied nearly in the proportion of the area of the base. For equal bases, the circle gave the most favourable results for strength ; and, after it, those polygons which most nearly approximated to it. * See an extensive series of experiments on this subject in piso Appendices D. E. and F. The Philosophical Transactions' for 1818, Part 1. See Secr. II.] PRINCIPLES OF PRACTICAL ARCHITECTURE. For the same volumes, those were strongest in which the altitudes were equal to the diameters of the bases; and the strength decreased, either as the altitude or the area of the base was increased, the volume remaining the same. 3d. In the blocks submitted to a compressive force, it was observed that a slight yielding took place under a weight equal to about one-half of that which was necessary to crush the stone; and when rupture ensued, those of a crystalline texture generally split into needle-formed fragments, parallel to the direction of the force; wbilst the amorphous stones broke into small pyramidal fragments, having a common vertex near the centre of the block. 4th. From a comparison of the weight borne by the stone in some of the boldest structures in Europe, with the results of these experiments, it appears that it would not be safe to submit any stone to a permanent strain which would be greater than one-tenth of the weight required to crush a small block of it, of the size of those used for the experiments. 5th. That the following order of strength and hardness was observed in the more common building-stones: basalt, granite, limestone, and sandstone. 17. The terms hard and soft, as applied to stones, are not very definite ; but workmen call those stones hard, which can be sawed into slabs only by the agency of the grit-saw ; and those soft, which can be separated by a common-saw. A knowledge of these qualities is essential to the Builder, to enable him to fix a price on the preparation of the material, and also to select such as are most suit. able to resist attrition and shocks. 18. Of the Causes of the Decomposition of Building-stones.]—The principal causes of the decay of building-stones, are, the dissolving power of moisture exerted upon their alkaline or calcareous matter ; the expansion of water in the pores and fissures of the stones ; changes of temperature, and the ac- tion of the external air. 19. Stones of a slaty structure are very subject to be shivered into pieces by the frost. This is oc- casioned by the cracks and fissures getting filled with water in wet weather, and when this is followed by frost, the expansion of the water in freezing splits the stones. Indeed the force of the confined fluid is so great that, in some places, large blocks of stone are split into slates by the expansion of freezing water. The stones being quarried in the autumn, are frequently placed in a position con- trary to that they had in the quarry; the rain insinuates itself between the layers of which the stones are composed, and the water, expanding as it freezes during the winter, splits the stones into slates. For this reason, stones of a slaty structure should always be laid in the same position they had in the quarry. Stones that are regularly porous, are not so much affected by the frost, because the water is less confined; but almost all stones that have stripes or veins in them, and are of an equal texture, are more or less subject to be injured by the frost. The veins are generally softer than the other parts of the stone. covered with different kinds of lichens, and afterwards with moss. As soon as the surface of a stone begins to be softened, the seeds of lichenswhich are constantly floating in the air-find a resting. place upon it, and it becomes covered with vegetation ; but stones which decay rapidly change their surfaces so often that the seeds have not time to take root upon them. Hence, when stones become covered with lichens, it is not a bad criterion of their durability. But even lichens tend to destroy stones---particularly those which present an inclined surface and prepare for mosses; and these again for larger plants, which slowly decompose the surface for sustenance. In London we may fre- quently observe some stones nearly black, and others presenting a clear white surface. The black stones are those which are more compact and durable, and preserve their coating of smoke, while the white stones have had their surfaces decomposed, and, presenting a fresh surface, appear as if they had been recently scraped. 21. The qualities of stone from quarries which have been long wrought, may easily be known by inspecting old buildings that have been erected with it. But, * in order to form a judgment of the durability of any building-stone, which has not had the test of experience, it is desirable to examine S 1 G 36 [Part I. PRELIMINARY REMARKS ON THE it in its native bed, particularly those parts of the bed which have been long exposed to the air. This may not unfrequently be done in some parts of the country where the stone is quarried; for as each stratum rises in a certain direction, it will come to the surface some where, if not covered by soils. The stone, in such situations, offers certain indications of the effect which atmospheric agency pro- duces upon it. Where this examination cannot be made, all stones that are not calcareous may in some degree be proved by observing what effect is produced by immerging them in water for a given time,—by exposing them to a red heat and to frost,---or by covering them with dilute nitric acid for by the action of heat, frost, or acids, may be fairly considered as most capable of resisting the decom- posing or disintegrating effects of moisture and change of temperature.”* A method of ascertaining the comparative durability of marbles, and all stones purely calcareous, is described by the writer just quoted, which is nearly as follows. Let specimens of different marbles be cut into cubes, or any other regular figures of a given magnitude, and immersed in dilute muriatic acid of the same degree of strength : those which dissolve most slowly, will be least liable to decay. Soft stones, says Pal- ladio, t and such as we are not well-acquainted with the nature of, should be quarried in the summer, and exposed to the effects of air and frost two years before they are used. This practice would be the means of preventing those stones from being used for external walls which, after such a trial, ex- hibited symptoms of decay. 22. Strength and durability are the most essential qualities of building-stones for the purpose of public monuments, bridges, quays, docks, &c. ; but for dwelling-houses, comfort, convenience, and eco- nomy are of still greater importance than extreme durability. Stones which attract moisture, and are affected by every variation of the weather, are very unfit for the walls of dwelling-houses. These are chiefly of the argillaceous class, such as basalt, whinstone, &c. Walls constructed of these stones are always damp, and the timber which comes in contact with them is generally soon affected by the dry rot. 23. In this climate it is also desirable that the walls of dwelling-houses should be constructed of materials which are slow or imperfect conductors of heat, in order that the houses may be warm in winter, and cool in summer. Stones which are light, dry, and porous, are generally imperfect con- ductors of heat; those that are compact, and of a glassy or vitreous nature, are in general good con- ductors; and where it is necessary, on account of the expense, to use the latter kind, the walls should be lined with brick.19 24. In the choice of stone, for ornamental buildings, the colour becomes an object of importance. It ought to agree with the colours of the surrounding objects. A glaring white, or a gloomy dark coloured stone, is equally unfit for ornamental purposes ; yet a little variety of colour adds much to the beauty of a building. 25. As the stones commonly used for building may be arranged in three classes, it is usual, for greater convenience, to adopt this classification, viz. 1. The Siliceous class, or the one of which silex is the base, or principal constituent; 2. The Argillaceous, of which argile—which is a mixture of silex and alumine is the base ; and, 3. The Calcareous, of which lime is the base. 26. Siliceous stones. ]—Granite, gneiss, and sienite are commonly known to Builders by the general appellation of granite, owing to the great resemblance of their external characters and of their physical properties. Granite and gneiss differ, in fact, rather in the aggregation of their constituent * Rees' Cyclopædia, Article Stone. † Leoni's Palladio, book I. chap. 3. 19 Count Rumford has made some interesting experiments and observations on the conducting powers of bodies in his Essays, vol. ii. Secr. 11. ] 37 TI PRINCIPLES OF PRACTICAL ARCHITECTURE. LU elements than in any other essential particular. In the former, the aggregation of the particles is mostly homogeneous, giving the stone a uniform appearance, and the property of splitting readily in all directions; whereas in gneiss, the particles are disposed in layers, which give the stone a laminated appearance, and cause it to yield to the chisel and wedge more readily in the direction of the layers than in any other. 27. The constituent elements of these stones are quartz, feldspar, and mica. These elements aro easily distinguishable on an examination of a sample of the stone. The quartz presents a transparent or semi-transparent appearance, somewhat approaching to that of glass of a milky hue ; the feldspar is usually either whitish or reddish, presenting more of an opaque appearance than the quartz, and where the surface has been for some time exposed to the action of the weather, having a dull white character; the mica is found in scales of greater or less size, of a dark colour when seen in the mass, but transparent, and resembling the small scales of a fish when detached from the block. When uni. formly distributed throughout the mass, these constituents give the stone a uniform colour, generally some shade of gray, but occasionally of a slight reddish hue, owing mostly to the colour of the feldspar. 28. The quality of the stone depends on the aggregation of the particles, their size, and the pro- portion of each: the best is usually that in which the particles are fine and uniformly disseminated V be hard and brittle, and will, in consequence, present great difficulties to being wrought and dressed to a uniform surface. The feldspar decomposes when exposed to the atmosphere for a long period ; and, if in excess, will be injurious to the quality. The mica is also subject to decomposition ; and, when in excess, gives the stone a character of weakness, which causes it to be known by the appella- tion of tender or soft granite. Besides their peculiar elements, foreign minerals are almost always to be met with in these two stones. The most deleterious are schorl, iron, and its ores, more especially the sulphurets. The iron becoming oxidized, when the stone is exposed to the air, destroys it very rapidly, particularly if in large quantities; the sulphurets, by decomposition, yield sulphuric acid, which soon destroys the texture of the stone by acting on the mica and feldspar; and schorl, when in abundance, gives the stone a character of great brittleness, which, in some cases, renders it entirely unfit for a building material. Sienite, though frequently mistaken for granite, is composed principally of particles of hornblende, and feldspar. The general appearance of the mass is either gray, or has a reddish tint; and particles of the hornblende are readily distinguishable by their greenish tint when the stone is moistened. Quartz and mica are likewise generally found in sienite, and give it more the appearance of granite. The structure of this stone usually resembles that of granite ; but it is sometimes found in layers. 29. Granite, gneiss, and sienite, for strength, hardness, and durability, occupy the first rank as building-materials, but they will not resist very high temperatures ; although gneiss, when the mica in it is very abundant, has in some cases been used with success as a facing for fire-places and fur- naces subjected to a strong heat. Granite and sienite are the most suitable for the purposes of cut in which they can be procured from the quarry, and the perfect accuracy with which the surfaces can be wrought.20 Gneiss seldom splits evenly, and is therefore more suitable for rubble and hammered 20 The granite of Aberdeenshire is of fine quality, and is understood to be almost inexhaustible. It is found of a variety of shades, and of course presents considerable varieties of texture and component parts. It is, perhaps, not generally known that it is capable of taking on a high and very permanent polish, as may be seen in two beautiful pillars wrought of the red Aberdeenshire granite, in the British museum. The process by which this exceedingly hard and dur- able material is wrought into a variety of beautiful forms, and made to receive a high and perfect polish, is the invention of Mr. Alexander Macdonald of Aberdeen.- The ancient Egyptians appear to have attained great dexterity in the working 38 [PART 1. PRELIMINARY REMARKS ON THE stone. It is also an excellent material for flagging-stone ; and all three of these stones are in very common use for structures requiring great solidity and permanency; as for revetment-walls of forti- fications,* quay-walls, sea-walls, light-houses, &c. 30. There are two principal varieties of sandstone, the red and the gray, which is generally known to builders under the name of free-stone. It is composed of small particles of quartz, united by an argillaceous or calcareous cement. Both varieties are very extensively used in building. They are generally strong and durable ; and, though they yield readily to the chisel and other tools, are suffi- ciently hard to resist the wear and tear to which any part of an edifice is ordinarily exposed. Sand- stone is frequently so porous as to absorb a large quantity of moisture, which, when acted upon by the frost, causes the surface of the stone to disintegrate, or to split off in scales. The gray sand- stone is more liable to this defect than the red, and requires a thin coating of mortar, paint, or a white-wash of hydraulic lime, to protect it from the action of the atmosphere. But both varieties are used in the construction of public works : in some cases as the principal material, but mostly for the cut stone of the angles,-for the coping, for the water-tables, &c. Its inferiority to granite, and its liability to disintegrate, render it more suitable to ordinary structures. It should, moreover, only be used as ashlar, or cut stone ; because it adheres very badly to mortar, and is, therefore, not suitable for rubble work, the principal strength of which depends on the adhesion between the stones and mortar. 31. The principal supply of sandstone for architectural and engineering purposes in England, is derived from the quarries of Bromleyfall near Leeds, in Yorkshire --Whitby in the same county, - Portland in Dorsetshire, and Purbeck,--and from Dundee and Craigleith in Scotland. The most extensive freestone quarry in Scotland, is that of Craigleith, out of which nearly the whole New Town of Edinburgh has been built. There are also valuable quarries at Humbie and at Hailes in Edinburghshire; at Pencaitland in East Lothian ; at Binnie in Linlitligowshire; at Cullelo in Fife ; and at Garscube, Woodside, and Balgray in Lanarkshire. 32. All the stones belonging to the siliceous class of which there is a great variety--are emi- of this obdurate material, and it is still extensively employed in the East. In cutting the hardest granite, the Hindoo workmen employ only a small steel chisel, and an iron mallet. The chisel is short, and Dr. Kennedy thinks it probable that it is formed of the hard steel known as Berar wootz. It tapers to a round point like that of a drawing pencil. The mallet, weighing a few pounds, is somewhat longer than the chisel. The head, set on at right angles to the handle, may be from two to three inches long. It has only one striking face, formed into a pretty deep hollow, which is lined with lead to deaden the blow. With two such simple tools to have detached the most massy granite column from its native bed, to have formed, fashioned, and scarped the granite rock which forms the tremendous fortress of Dowlatabad, and to have excavated the wonderful caverns at Ellora, are instances of the incredible patience of the lindoo, and of the simple and apparently inadequate means by which he accomplishes the most difficult undertaking. It seems probable that the Hindoo stone-cutters never worked with any other tools. When the stone is brought to a smooth surface, it next under- goes the dressing with water, in the manner usual with masons. The fine black shining polish is given in the following manner :--A block of granite of considerable size is rudely fashioned into the shape of the end of a large pestle. The lower face of this is hollowed out into a cavity, and this is filled with a mass composed of pounded corundum stone, mixed with melted lac. This block is moved by means of two sticks, and pieces of bamboo, placed one on each side of its neck, and bound together by cords twisted and tightened by sticks. The weight of the whole is as much as two workmen can easily manage. They seat themselves on, or close to the stone they are to polish, and by moving the block backwards and forwards between them, the polish is given by the friction of the mass of wax and corundum. Granite finished in this way is the most common material of which the tombstones of princes and great men in India are con- structed. As a beautiful glossy black, it is scarcely inferior to the finest black marble. A granite gateway, supposed to be five hundred years old, in the ancient city of Warankal, has lost nothing of its original lustre. * A revetment consists of a facing of stone, wood, sods, or any other material, to sustain an embankment, when it rercives a slope steeper than the natural slope. Sect. II ] 39 PRINCIPLES OF PRACTICAL ARCHITECTURE. nently suitable for the purposes of building, either as cut stone or rubble. Among those less in use are the buhr or millstone, which, from its great hardness, durability, and porosity, forms an excellent rubble stone,—the soap-stone, which is principally used as a fire-stone for the facing of fire-places and furnaces, and mica-slate, which is also a good fire-stone, and fornis a good material both for rubble work and for flagging. 33. Argillaceous stones. ]-Nearly all the stones known to builders as slate-stone belong to this class. The most remarkable varieties are those denominated trap rocks by mineralogists, which consist either of basalt, or green stone. Basalt is very remarkable for its great strength and hardness; though it is less durable than many varieties of the siliceous class. 34. Green stone, so called from the greenish tint it exhibits when wet, is found very abundantly in Britain. It is a good building material, when it does not contain any large quantity of iron ; for this metal, by becoming oxidized, very soon entirely destroys the texture of the stone, causing it to break up into small fragments or scales. It is only suitable for rubble work, owing to its being found chiefly in small tabular prismatic masses ; but from the facility with which it is quarried, and its unchange- ableness in salt-water, it has been sometimes used for break-water stone. 35. Gray wacke, and gray wacke slate, properly belong to the sandstones. They are composed of the fragments of several other minerals, in a granular state, united by an argillaceous cement. Both of these stones make a good building-material for rubble work ; and the gray wacke slate is in very common use as a flagging and coping-stone. 36. Common roof slate requires no particular description. There are many varieties of this stone which are very suitable for rubble work. The best for roof-covering is that which splits into thin the sulphurets, which are more deleterious to it. Slate, to be of a quality for roofing, should have the property of splitting into thin laminæ, and should resist the absorption of water, and the natu- ral process of decomposition by air and moisture ; this depends much on its chemical composition S decay will be the sooner covered with lichens and mosses. The strength of slate is very great in com- parison with any kind of freestone. It has been ascertained that a slate one inch in thickness will support in a horizontal position, as much weight as a piece of Portland stone, of equal superficies, five inches in thickness. As the hardness of slate arises principally from the silex it contains, which is of all earths the least favourable to vegetation, those slates which are the hardest when first taken from the quarry, and which have the least specific gravity, are to be preferred; for the increase of weight in slate is owing to the presence of iron, either in pyrites, or a state of oxide ; the pyrites being decomposed by moisture, and the iron admitting a still higher degree of oxygenation, the sur. face of the stone peels off, or falls into an ochrey powder. In order to judge correctly as to the quality of the slate, it must be struck. If it yield a sonorous, bell-like sound, it is a sign of its good- ness. Some judgment may likewise be formed from its colour; the dark blue sort being more liable to imbibe and retain moisture than that which is of a light blue colour: the touch likewise affords some criterion as to the quality of the slate ; a good firm slate feels hard and rough, whereas an open, porous one feels smooth and greasy. Another mode of trying slate is recommended by some, viz. then immerse them in water for twelve hours; take them out and wipe them as dry as possible with a linen cloth, and if they weigh more than they did previously to immersion, by more than one dram in twelve pounds, it denotes a species of slate that has a tendency to imbibe water, and is consequently objectionable. 37. Quarries of slate and slate-stones are abundantly distributed in various districts in England; 40 [PART I. i PRELIMINARY REMARKS ON THE but the best and principal supply of roofing.slate is obtained from Caernarvonshire in Wales. The quarries of Green-moor in Yorkshire supply excellent paving-flags,-as do also those of Valentia in Ireland, some of which are remarkably fine and durable. The principal slate-quarries in Scotland occur at Eisdale and at Ballahulish in Argyleshire, at both which places a dark blue slate is wrought, of a very hard quality ; at Huntly in Aberdeenshire, where a blue and green slate is wrought; at Aberfoyle in Perthshire, where a harder blue and green slate occurs ; at Foudland in Aberdeenshire, where the slate is of a light blue colour, and moderately hard ; and at Luss in Dumbartonshire, and in the isle of Bute, where a pale blue slate is quarried. 38. Calcareous stones. ]—This very abundant class, composed of innumerable varieties, is the most useful building material known to the Builder and Architect, both for common and ornamental pur- poses : arising from the strength, hardness, durability, and beauty both of colour and polish, which it is known to possess. It also furnishes the principal ingredient in the composition of every variety of cement used for uniting stones artificially. 39. Calcareous stones-distinguished by the more common appellation of limestone and marble- are composed principally of carbonate of lime, combined with the metallic oxides, and several other foreign minerals. They seldom occur in a pure state. When found so, the colour of the stone is a pure white, and it is shown by analysis, to be composed of lime, carbonic acid, and a small quan- tity of water. The general properties of this class of stone, both physical and chemical, are so well kuown as hardly to require any description. Its effervescence with acids, and the effects of heat on it, both of which disengage the carbonic acid, are facts with which almost every person must be acquainted. 40. Mineralogists distinguish two general divisions of this class : 1. Granular limestone ; 2. Com- pact limestone. The granular presents the distinct appearance of an aggregation of grains of vari- able size, from very fine to coarse, apparently the result of an irregular crystallization. The compact has a fine uniform texture, without any appearance of grains, some of the varieties being quite loose and earthy in their texture. 41. As a building-material, the calcareous stones are classed in two divisions: 1. The common line- stone ; 2. The marbles. Each of these divisions furnishes an equally good stone for building ; but the marbles are mostly reserved for ornamental purposes, owing to the fine polish which the stones from which they are procured are susceptible of receiving. The term marble is frequently applied by Builders to all stones which receive a high polish ; and it is the proper signification of the term ; but it is now usually applied only to those varieties of limestone which are polished. The compact lime- stone furnishes a great variety of variegated marbles; but generally they are not so highly estimated as those furnished by the granular, owing to their inferiority in hardness and polish. 42. The colours of variegated marble are owing to the metallic oxides; and the names of the differ- ent varieties are taken from some peculiarity either in the appearance or colour of the stone. Some of the best known are : the veined; the bird's-eye ; the conglomerate, of which there are two kinds ; the breccia, composed of broken angular fragments united by a natural calcareous cement; and the pudding-stone, composed of round pebbles similarly united; the lumachella, which exhibits a variety of shells united by a calcareous cement; the Florence-stone, or ruin-marble, so called from some fancied resemblance to ruins exhibited by the figures on the polished surface; the verd antique, which is of a green colour, and is named after a stone much esteemed by the ancients, &c. &c. 43. Limestone is found in a great many parts of England; but the best kind for building-purposes is derived from Devonshire, near Mary-church. The Isle of Anglesea also produces good building marble, as do many parts of Ireland, and several parts of Scotland; though no part of the United Kingdom is known to produce marble fit for statuary purposes. The price of Italian marble is from SECT. II.) PRINCIPLES OF PRACTICAL ARCHITECTURE. 40s. to 60s. per cubit foot; of Irish, from 10s. to 128. The best English lime for mortar is obtained from the neighbourhood of Dorking and Marstram in Kent. 44. Plaster of Paris, or gypsum, is a sulphate of lime ; its principal use is for stucco-work in the interior of edifices; for which purpose it is prepared by calcination over a strong fire. It is too soft for a building-stone ; and although forming, when properly prepared, a cement which hardens very rapidly, it is not suitable for the purposes of mortar, because, having a strong affinity for water, it absorbs it from the atmosphere, and increases so much in volume as to occasion cracks in walls built with it; and, in exposed situations, it very soon commences to exfoliate. TE II.-ARTIFICIAL STONES, AND CEMENTS. 45. Artificial stones, and Cements.)-The term artificial stone is applied to any composition to which, by an artificial process, the general properties of natural stone are imparted; such, for example, are mortar and brick, both of which, when properly prepared, possess in an eminent degree the qualities of good stone. The term cement is applied to certain mineral substances, found either in a natural state, or prepared artificially, which, being mixed with common lime, impart to it the property of hardening under water. 46. The ingredients which usually enter into the composition of mortar are slaked lime, and sand; to which sufficient water is added to bring the mixture to a proper consistence or temper before using it for the purposes to which it is to be applied. When the mortar is to be used for hydraulic works, a certain proportion of cement is to be added to the other ingredients. 47. Lime. When limestone is submitted to a high temperature for some length of time, the water, and nearly all the carbonic acid, which enter into its composition, are driven off, and the result ob- tained is known by the name of quick lime. The stone in this new state shows a strong avidity for water, which it absorbs even from the atmosphere ; and when water is poured over the stone it swells and cracks, evolving a very considerable degree of heat, and finally, falls into a fine white powder, in which state it is denominated slaked lime, and belongs to that class of chemical substances denominated hydrates. If the limestone is a perfectly pure carbonate, it will absorb about three-and-a-half times its bulk in the process of slaking, and the hydrate will be found increased in the same proportion. 48. As a building-material, lime is divided by engineers into two classes : 1. Common lime ; 2. Hydraulic, or Water lime. Common lime is also sometimes termed fat lime, from the appearance and feeling of the paste made from it with water; whilst hydraulic lime, with the same quantity of water, yielding a thin paste, is denominated meagre lime. This difference of appearance in the paste of the two, it must however be observed, does not serve in all cases to distinguish the two classes ; for some varieties are very meagre, without possessing the slightest hydraulic properties. The distinction between the two classes consists in the uses to which they are applicable. The mortar of common lime will never harden under water, or in very moist places, as in the foundations of edifices, or in the interior of very thick walls ; and therefore is only suitable for dry positions and thin walls; whereas hydraulic lime yields a mortar which sets readily, and soon becomes nearly as hard as stone in all moist situations. 49. Very few limestones or chalks consist entirely of carbonate of lime. If they did, they would be uniformly the same in their nature, properties, and effects ; and as the different qualities of lime- stones for cements depend on the nature of the substances which are mixed with the carbonate of lime, it is desirable that we should be able to ascertain when those substances are present that im- prove the quality of the lime. When a limestone does not copiously effervesce in acids, and is suf- ficiently hard to scratch glass, it contains silex, and probably clay. When it is deep brown, or red, or strongly coloured of any of the shades of brown or yellow, it contains oxide of iron, and burns into PRELIMINARY REMARKS ON THE [PART I . S a buff-coloured lime. When a limestone is not sufficiently hard to scratch glass, but effervesces slowly, and makes the acid in which it effervesces milky, it contains magnesia. Magnesian limestones are usually of a brown or pale yellow colour. They are found in Somersetshire, Leicestershire, Derbyshire, Shropshire, Durham, Yorkshire, and Nottinghamshire. 50. To ascertain the properties of a limestone, direct experiment should always be resorted to; for the external appearance of the stone does not present any indications which can be relied on with certainty. The simplest method consists in calcining a small portion of the stone to be tried over a common fire on a plate of iron, slaking it and kneading it into a thick paste ; this paste being placed at the bottom of a glass-vessel, and carefully covered with water, will, in a short time, give evidence of the quality of the stone. If, after several days, it is found not to have set, the stone may be pronounced as affording common lime ; but if it has become firmer, or hard, it may be safely classed among the hydraulic varieties, and its excellence will be shown by the quickness with which it hardens. 51. It is not a difficult matter to analyze limestone with sufficient accuracy for all the purposes of the Builder or Architect. A few directions for ascertaining the constituent parts of this material cannot fail of being both interesting and useful : The instruments required for the analysis of limestones are few, and not very expensive. They are an accurate balance and a series of weights, a pestle and mortar, some filtering paper, and a few reagents and tests. The principal reagents required are muriatic acid, sulphuric acid, nitric acid, solution of prussiate of potash, solution of neutral carbonate of pot- ash, carbonate of ammonia, and spirit of wine. (1.) Having determined the quantity to be analyzed, as 100 grains, let it be reduced to a fine powder, and weigh it carefully. Next expose it to the action of muriatic acid. The acid should be poured upon the earth in an evaporating bason, in a quantity equal to about twice the weight of the earthy matter, but diluted with double its volume of water. The mixture should be often stirred, and suffered to remain an hour or more before it be examined. If the limestone contains any carbonate of lime or magnesia, they will have been dissolved in this time by the acid, which sometimes takes up a little oxide of iron, but very seldom any alumine. The fluid should be passed through a filter; and the solid matter collected, washed with rain-water, dried at a moderate heat, and weighed. Its loss will denote the quantity of solid matter dissolved by the acid. (2.) The washings must be added to the solution, which if not sour to the taste, must be made so by adding fresh acid; then a little solution of prussiate of potash must be mixed with the whole, and if a blue precipitate occurs, it indicates the presence of oxide of iron; in which case the solution of the prussiate must be dropped in till no further effect is produced. Filter the solution and collect the precipitate; heat it to redness; the result is oxide of iron. (3.) Into the fuid, freed from oxide of iron, a solution of carbonate of potash must be poured till all effervescence ceases in it, and till its taste and smell indicate a considerable excess of alkaline salt. The precipitate that falls down is carbonate of lime. It must be collected on a filter, and dried at a heat below that of redness. The remaining fluid must be boiled for a quarter of an hour, when the magnesia, if there be any, will be precipitated from it, combined with car- bonic acid, and its quantity, is to be ascertained in the same manner as that of the carbonate of lime. (4.) The insoluble part separated by the first process (1.) may contain silex, alumine, and oxide of iron. To separate these from each other, the solid matter should be boiled for two or three hours with sulphuric acid, diluted with four times its weight of water. The quantity of the acid should be regulated by the quantity of solid matter to be acted upon, allowing for every 100 grains 120 grains of acid. The substance remaining undissolved by the acid may be con- sidered as silex; and it must be separated and weighed, after washing and drying it in the same manner as the other precipitates. (5.) The alumina and oxide of iron, if there be any, will both have been dissolved by the sulphuric acid, and may be separated by adding carbonate of ammonia to excess, which throws down the alumine, and leaves the oxide of iron in solution, and this substance may be separated from the fluid by boiling it. (6.) When the examination of the limestone is completed, the quantities of the different substances, arranged in the order of the experiments, should be added together, and if they nearly equal the original quantity of limestone, the analysis may be considered accurate. Thus, suppose a hundred grains of limestone to have been analyzed; they may be found to contain- Of carbonate of lime,. . . . . . . 51:3 Carbonate of magnesia, . . 27 Sect. II.) 43 PRINCIPLES OF PRACTICAL ARCHITECTURE. . . Silex, . Alumine, : Oxide of iron, . . . . . . . . . . 12 5. . . . . . . . . Water and loss, . . . . . . . : 98.3 1.7 100.0 When the experimentalist has become acquainted with the relation between the external and chemical characters of lime- stones, he will seldom find it necessary to perform all the processes that have been described; for instance, few limestones contain magnesia, therefore the process (4.) may be omitted in such cases. 21 52. It is only within a few years back that scientific men have come to any certain conclusions as regards the causes of the peculiar property of hydraulic lime. For a long time it was ascribed to the presence of metallic oxides; then to the manner of slaking the lime and mixing the ingredients of mortar; but careful analysis and experience have finally settled the question, and it is now fully ascertained that this property is owing to the presence of argile or common clay in the stone, which, after the latter is calcined, forms a compound possessing this highly important quality. It still, how- ever, remains to be determined, whether the presence of both the elements of argile—silex and alu- mine-are necessary to impart this property. Alumine alone, it is known, does not; and an hydraulic lime has been found in France, in which it is stated that silex is alone present. Whatever may be the solidifying principle, a most important fact is put beyond a doubt, that an artificial hydraulic lime, equal in quality to the best natural varieties, can at any time be made by mixing common lime, in a slaked state, with any mineral substance of which argile is the predominant constituent, by simply exposing it to a suitable degree of heat, and afterwards converting it into a fine powder, before mixing it with the lime. 53. One of the most celebrated limes in France is that of Metz. Its excellent qualities are highly extolled by Belidor* and Rondelet.t The stone from which it is made, is of a grey colour, very hard, and is composed of Carbonic acid, • . 39.00 Lime, . . . . 44.5 Silex, . . 5.25 Alumine, · · · · · · 1.25 Oxide of iron, . . . . . . . 3.2 Oxide of manganese, ganese, . · · · · · · · 3.5 Water and loss, . . . . . 3:3 in orid . 100-1 The oxide of manganese, it appears, has the same effect in cement as the oxide of iron has. Indeed Bergman supposed that the property of setting in water proceeded from the manganese contained in poor limes; but later inquiries have shown that it only possesses this property in common with iron and silica. 54. “ Pennarth limestone,” says Bp. Watson, § “is washed in large cobbles from the cliffs on the 21 The reader who wishes to pursue this interesting subject may consult Henry's “Epitome of Chemistry,' part ü. sect. ii.; Sir H. Davy's 'Agricultural Chemistry,' sect. iv. p. 164; Kirwan's · Elements of Mineralogy,' vol. i. p. 459, 28 ed. 1794. * Science des Ingenieurs,' liv. iii. chap. 3. ť •L'Art de Bàtir,' tome i. p. 249. Gauthey's • Construction des Ponts,' tome ii. p. 279; or Nicholson's Journal, vol. v. p. 111, 4to series. $ Chemical Essays, vol. iv. p. 346. 44 (PART . PRELIMINARY REMARKS ON THE Welsh side of Bristol Channel. The lime made from it is highly esteemed in that country from its setting under water. It is called lion lime (perhaps lien) from its binding quality. The stone is of a grey colour, and besides the proper earth of lime, contains a large proportion of clay and iron." 55. The stone of Lena, in Upland, mentioned by Bergman, is grey, mixed with greenish white particles, and very hard; generally invested with the brown oxide of manganese. It loses 0.39 of its weight by burning, and affords a brown lime. The presence of manganese may be discovered by melting the stone with double its weight of nitre; the manganese, if it contain any, will leave a green trace on the sides of the crucible. * 56. The limestone of Senonches, which furnishes the best water-lime to the environs of Paris, according to Descostils, contains, besides carbonate of lime, a considerable quantity--perhaps a fourth—of extremely fine silica, with a very small portion of magnesia, alumina, and oxide of iron. 57. When Smeaton undertook the erection of the much celebrated lighthouse on the Eddystone rocks, he made a great number of experiments on the nature and properties of water-limes. Experi- ments made by Smeaton could not fail of being useful and instructive; they are calculated to remove many prejudices, and establish many important facts.22 He found the Aberthaw and Watchet limes to be superior to any other kind he tried. Aberthaw lime is made from a stone found at Aberthaw, on the coast of Glamorganshire. “This stone, before burning, was of a very even, but dead sky-blue colour, with a very few shining particles; but when burnt and sifted it was of a bright buff colour;" it contains about 13 per cent. of bluish clay.I The Watchet lime is made from the blue lias lime- stone of Watchet in Somersetshire. It “is of a dead sky-blue, of a very fine frosted grain when broken, with a few shining particles;" it gives a buff coloured lime. 58. The Barrow lime, from Barrow in Leicestershire, is made from a stone which has the appear- ance of blue lias, only rather of a more yellow tinge, and slaty structure. It burns to a buff coloured lime like that of Watchet and Aberthaw, and on dissolution affords about 21 per cent. of blue clay, and a minute quantity of dirty grey sand. “ It does not appear to acquire quite so firm and stony a hardness as the blue lias of Somersetshire,''though it occurs on the same deposit: the blue lias stone strata stretching across England, in a direction from N. E. to S. W., from the sea-coast at Whitby to Lyme-Regis in Dorsetshire. | Hitherto the lias used by the London builders has been brought from Lyme-Regis; but it is little used in the metropolis, being about 25 per cent. dearer than the Dorking lime. 59. The Clunch line, from near Lewis in Sussex, Smeaton says, is in great repute for water- works," and indeed deservedly so." This stone is found in thick masses, as chalk generally is; it is harder than common chalk, but of the lowest degree of what may be denominated a stony hardness, and inclining towards a yellowish ash colour. 60. The Sutton lime (Lancashire) is made from a stone of a deep brown colour, with a white clayey coat on the outside. Like the Aberthaw and other good limes, it is of a buff colour when burnt, it contains nearly 19 per cent. of brown or red clay, and about 21 per cent. of fine brown sand. 61. The following table shows the result of Smeaton's experiments on nine different English water limestones. I The last two columns are added, -the one to show the proportion of clay in a hundred parts of the stone,—the other, the proportion of fine sand. CD * Kirwan's Mineralogy, vol. i. p. 117. t Journal des Mines,' tome xxxiv. p. 309. 22 Smeaton's experiments are detailed in the 4th chapter of his · Description of the Construction of Eddystone Light- house,' a work which cannot be too earnestly recommended to the attention of the reader. | Eddystone Lighthouse, p. 105. § Idem. p. 114. | An attempt is now [1843] making to supply the London market with this lime from a valuable working at Southam in Warwickshire. It contains from 87 to 95 per cent. of carbonate of lime. Eddystone Lighthouse, p. 117. Sect. II.] PRINCIPLES OF PRACTICAL ARCHITECTURE. In 100 parts. Kind of limestone. Proportion of clay. Colour of the clay. Reduction of weight by burn- ing. Colour of the brick. Proportion of clay. Proportion of sand. 13 1 2 3 4 to 3 43 3_2 12 Sminute Abertbaw . Watchet Barrow Long Bennington Sussex Clunch Dorking (Surrey) Berryton grey lime VOOR A CO | Gray stock brick S Lightish colour reddish hue Gray stock brick Dirty blue Ash colour quantity, Lead colour Ditto Ditto Ditto Ash colour Ditto Ditto Ditto Brown 3-2 21:4 136 18.7 5.8 8.3 10:5 18.7 Gilford . . Sutton i 2.5 62. Smeaton's experiments show that the hard Plymouth stone is not any better for the purpose of making water-cements than common chalk-lime. Smeaton also tried shell-lime, which has been sup- posed to be superior to any other kind. He found that it would set hard and readily without admix- ture of sand, tarras, or other matter; on being put into water it did not fall in pieces immediately, but gradually macerated and dissolved from the surface inwards.* Hence it appears that neither shell-lime nor any other pure lime can be used for a water-cement; indeed Smeaton lays it down as a fundamental position, that no composition can be made with marble or chalk-lime and sand that will ever acquire a stony hardness under water, or where it can be perpetually supplied with moisture from the tides.f The Agnes lime, made near Ashbourn, Derbyshire, is stated to be one of those that made good water-mortar. I 63. In the fifth volume of the Prize Essays of the Highland Society of Scotland,'s there is a good account of the principal lime-quarries in Scotland, by Mr. James Carmichael, to which we are in- debted for the following comparative Table of the Scottish quarries : . . Price per Bushel 690. 310. 34d. 340. 2 d. Ro 3d. 31d. . No. NAME. 1. Burdiehouse, . 2. Mount Lothian, 3. Fullarton, . . 4. Side, .. 5. Bents, . . 6. Whitefield, . 7. Carlops, . 8. Hemperston, 9. Middleton, . 10. Blinkbonny, . 11. Chrigtondean, 12. Cousland, . 13. Salton, . . 14. Jerusalem, . 15. Sunnyside, . 16. East Barns, . 17. Skateraw, . 18. Harelaw, . 19. East Camps, 20. Raw Camps, . . Annual Produce. 15,000 15,000 8,000 12,000 6,000 8,000 7,000 5,500 12,000 5,000 24,000 16,000 24,000 8,000 7,000 7,000 18,000 8,000 5,500 3,500 Carbonate of Lime in 100 parts. 99.7 98.9 99.0 98.5 99.0 98.9 99.0 99.0 94.5 99.5 96.8 98.8 97.5 98.0 96.0 99.0 99.0 Sale Price. 2s. 3d. P. B. Is. 8d. B. B. 13. 8d. Is. 8d. - Is. 4d.com Is. 6d. - 18. 6d. ls. 9d. 1s. 9d. - Is. 6d. Is. 9d. ls. 100. - ls. 100. ls. 10d. - 28. - 1s. 6d. P. B. Is. 7d. - 18. 10d. B. B. 6d. cwt. 10s. P, ton. Imperial Bushels. 59,640 87,120 46,464 69,696 34,848 46,464 40,656 31,944 69,696 29,040 139,392 92,928 139,392 46,464 40,656 27,832 71,568 46,464 106,484 93,336 3}d. 3d. . . 31d. . . 30. . . 3fd. 38d. . . 4d. . . . . 4£d. 44d. 31d. 41d. 41d. 97.7 99.7 99.7 . . * Eddystone Lighthouse, p. 106. Edinburgh, 1837, 8vo. + Smeatou's Reports, vol. iii. p. 381. Darwin's Phytologia, p. 220. 46 [PART 1. PRELIMINARY REMARKS ON THE . Annual Produce. 12,000 7,000 8,000 20,000 Sale Price. 23. 6d. B. B. 2s. 6d. - 2s. 6d. - Price per Bushel. 5d. 5d. . . Carbonate of Lime in 100 parts. 99.0 98.5 99.2 99.7 99.0 99.0 50 . 23. - 4d. 7,500 Imperial Bushels. 69,696 40,656 46,464 116,160 29,820 17,892 245,440 122,720 55,998 51,200 44,800 No. NAME. 21. Leven Seat, . . 22. Gateshiell, . . 23. North Silvermine, . 24. Duddingston, . 25. Murrayshall, .. 26. Craigend, . . 27. Cumbernauld, . 28. Netherwood, 29. Campsie So., 30. Hurlet, . . 31. Househill, . . 32. Arden, . . 33. Thornton, 34. Braehead, . . 99.5 99.3 48d. 410. 4d. 4d. 41d. 98.5 99.8 99.5 9d. 5d. 2s. 4d. P. B. 2s. 4d. - ls. 20. B. } 18s. 8d. Ch. 13s. 4d. - 12s. Q. 4 12s. P. Ch. 9d. P. bus. 13s. 4d. Ch. 13s. 4d. – 1s. 20. Im. 8d. 2 Bus. 4d. P. B. 8d. 2 B. 8d. - 9d. - 11d. - Is. - 3d. to 4d. 4d. 5d. 31d. 4d. 4d. 4d. 4d. . 4,500 . 80,000 2,500 1,200 . 1,600 . 1,400 . ...... 1,800 . 3,000 5,000 30,000 . 24,000 . 35,000 . 20,000 . 14,000 . 11,000 12,000 . 100,000 . 20,000 , 160,000 . 24,000 18,000 . 400,000 . 13,000 8,000 . 16,000 . 18,000 . 12,000 . 25,000 8,000 . 3,000 . 3,400 54d. 6d. 3d. to. 41d. 4d. 55.0 99.0 99.5 99.7 99.5 99.4 99.8 99.5 98.3 95.5 99.0 99.2 97.5 99.9 99.0 99.3 57,600 96,000 20,000 60,000 48,000 70,000 40,000 28,000 22,000 48,000 100,000 20,000 320,000 48,000 36,000 1,600,000 52,000 46,848 93,676 105,408 70,272 146,470 32,000 18,024 20,424 36. Balgreggan, 37. Halton, . . . 38. Craighead, . . 39. Tarmitchell, . . 40. Aldoon, . . 41. Craigniell, . . 42. Gaswater, .. 43. Benston, . . . 44. Craigdullet, . 45. Closeburn, . . 46. Porteston, . . 47. Bangry, . . . 48. Charleston, . . 49. Douloch, . . 50. Invertiel, . 51. Chapel, . . . 52. Forther, . . 53. Pitlessie, . . 54. Roscobbie, . . 55. Hedderwick, .. 56. Limefield, . . 57. West Pittendriech,. 6d. 51d. 6d. 96.5. 414. 41d. 4}a. 11d. - ls. 1s. 7d. 4 B. Is. 6d. - 25. 2d. B. M. 2s. ld. - 35. - 29. 90.- 2s. - 2s. 11d. 4 B. 38. B. M. 38. 4d. 6d. 54d. 99.5 97.0 99.5 99.7 99.5 98.5 99.0 99.0 99.7 4d. 8 d. 6d. 6d. 64. Sand. The term sand is applied to any mineral substance in a granular state, when the grain is of an appreciable size, and insoluble in water. 65. Sand is classified, either from its principal constituent element, as siliceous sand, argillaceous, &c.; or from the size of the grain, as coarse, fine, and middling sand; the latter classification being chiefly in use among builders. As this material is either procured from pits, or from shores or rivers, or the sea, it is denominated pit-sand, sea-sand, or river-sand, from the locality where it is obtained. 66. Pit-sand is superior to river-sand for all purposes where mortar is required to possess great strength. Its grain is more angular and porous than that of river-sand: the latter becoming smooth and polished from the constant attrition between the grains ; this roughness gives the lime a better hold on the grains ; besides, it is generally freer from the impurities, as salts, &c., which are found in the sand taken from the shores of the sea and tide-water rivers. River-sand, owing to its superior white- ness and the small size of its grain, is well-suited for plastering in the interior of edifices. Fine pit- 1 . SECT. [I.] PRINCIPLES OF PRACTICAL ARCHITECTURE. 47 sand should not leave any stain on the fingers when rubbed between them. If it is found to contain earthly impurities, or salts, it must be passed through several waters in flat vats ; the water being renewed and poured off until it no longer appears turbid. The siliceous class is superior to every other, owing to the great strength and hardness of its grains. 67. Cements. ]—These substances are obtained sometimes in a natural state ; or they can be pre- pared artificially, by calcining pure argile, and most of the argillaceous stones. Their uses, as has been already stated, (Art. 45,) are to form an artificial hydraulic lime, by mixing them with common lime. 68. All the substances which serve as cements contain nearly the same constituent elements, and in about the same proportions. They are argile, in which the alumine varies between one-fourth and one-half of the silex, with a small portion of the oxides of iron and manganese, and the carbonate of lime, potash, and soda. 69. The argile, or clay, constitutes the essential ingredient of cements. The metallic oxides seem to play a neutral part when not in excess; but the oxide of iron, when very abundant in clay, appears, from some experiments, to act injuriously on the quality of the cement. The carbonates of soda and potash, and the muriate of soda, or common salt, produce, it is said, a favourable effect, when the heat is by accident carried beyond the degree for suitable calcination. The other foreign ingredients found in clay, as the carbonate of magnesia, &c., do not appear to affect the quality of the cement. 70. Puzzolano,23 and trass or terras, are the most celebrated natural cements. They are both of yol- canic origin, the former being found in a pulverulent state near Mount Vesuvius; the latter near Andernach on the Rhine, where it occurs in fragments, and is ground fine, and exported for hydraulic works. The constituent elements of both these natural products are nearly the same, and as follows, in one hundred parts :- . . . . . . Silex, . Alumina, Iron, . Lime, . . . . . . . . . . 55 to 60 parts. 20- 19 - 20— 15 -- 5- 6 . . . . . . 71. Puzzolano appears to have been used by the Romans in the construction of their ordinary buildings in the neighbourhood of the place where it is obtained.* Its colours are reddish or reddish brown, and grey or greyish black. That of Naples is generally grey, that of Civita Vecchia more generally reddish or reddish brown. Its surface is rough and uneven, and of a baked appearance ; when broken and examined with a magnifying glass, it appears to be a spongy substance, with innumerable little cavities like a cinder, and not much harder. It comes to us in pieces of the size of a nut to that of an egg. It is very brittle and has an earthy smell ; its specific gravity varies from 2.570 to 2.785. It does not effervesce with acids, and is not diffusible in cold water; but in hot water it gradually deposits a fine earth. When heated it assumes a darker colour, and easily melts into a black slag. 72. According to Bergman, puzzolano contains from 19 to 20 parts of iron. The iron is not oxidated, and being finely divided and dispersed through the whole mass, it offers a large surface which quickly decomposes the water with which it is mixed when made into mortar. The uninn of 23 For a method of making artificial puzzolano, and on the nature of cements and mortars generally, the reader should consult · A Practical Treatise on Calcareous Mortars, Cements,' &c. By T. L. Vicat, translated by Capt. Smith, Madras Engineers. * Vitruvius, book ii. chap. vi. . . . . . . . . . 48 | Part I. PRELIMINARY REMARKS ON THE the oxygen of the water with the iron is supposed to be the principal cause of the hardening of the mortar; and the heat given out by the lime seems to be of use in promoting this union.* Smeaton found puzzolano to be equal, if not superior, to tarras, when mixed with Aberthaw lime, as a water-cement; and in wet and dry work superior to any he had ever seen; it also exceeded in hard- ness any of the compositions commonly used in dry work.t 73. Tarras is found near Andernach in the department of the Rhine and Moselle, and is called by the Dutch Tras of Andernach. Its colour is light grey, ash colour, or brownish grey. Its surface is rough and porous ; fracture, earthy and sometimes lamellar. It is more frequently mixed with heterogeneous particles than puzzolano is; and is sometimes so hard that it with difficulty yields to the knife. It feels dry and harsh, and scarcely effervesces with acids. It is not diffusible in cold water, but in hot water it gives an earthy smell, and deposits a fine earth. 74. According to Bergman, its constituents are nearly the same as those of puzzolano, except that it contains a little more lime. But M. Sganzin, in his Cours de Construction,' gives the constitu- ents of tarras as follows:8– Silica, . . . . . . . . 57 Alumina, . . . . . . . . 28 Lime, . . . . . . . 6:5 Oxide of iron, . . . . . . . 8:5 100 The quantity of iron is much less than that given by Bergman's analysis, and the different pro- perties of the substances when made into mortar renders it more likely to be correct. 75. Mortar made with tarras does not stand the frost. When the work is wet and dry, it must be continually wet: and in that case it throws out a kind of stony concrescence (called stalactites) which becomes very hard, and deforms the face of the work. || 76. There are few places that can be supplied with puzzolano or tarras at so low a price as to per- mit of its being used, with freedom, whenever it is necessary to make use of a water-cement; it is desirable, therefore, that some cheap substitutes should be generally known, as some of them may be obtained in almost any situation. The Dutch were the first that attempted to supply the place of these natural productions by an artificial combination of substances that were to be had in their own country; and, in this they have succeeded extremely well. An artificial tarras is made at Amster- dam by calcining a kind of clay that is got out of the sea; it is afterwards reduced to powder by mills for that purpose. This artificial tarras has been analyzed, and consists of Silica, · · · · · · · · 57 Alumina, . . . . . . . . 20 Lime, . . . . . . . 5:5 Oxide of iron, . . . . . . . 17'5 1000 77. In the neighbourhood of Tournay is found a very hard blue stone which produces an excellent lime. During the burning of this stone into lime, part of it, in a half-calcined state, falls through the grating of the furnace and mixes with the ashes of the coals. This mixture of coal-ashes and refuse * Kirwan's Mineralogy, vol. i. p. 412. Kirwan's Mineralogy, vol. i. p. 413. | Smeaton's Eddystone, pp. 108 and 109. + Smeaton's Eddystone, p. 109. $ Gauthey, tome ii. p. 281. | Gauthey, tome ii. p. 281. SECT. II.] 49 PRINCIPLES OF PRACTICAL ARCHITECTURE. of stone, forms what is called the cinder of Tournay.* sists of The best kind has been analyzed, † and con- 0 . . . . . . . . Silica, Alumina, Lime, Iron, . . . . . . . . . 44 40 705 8.5 . . . . . . . . . . . . . . . 100 Though not equal in goodness to the kinds before described, it answers very well, and is much used. 78. Artificial tarras may be made by burning schistus or slates that abound in iron, and grinding them to a fine powder. A hard black slate has been used for this purpose in Sweden, after being twice strongly calcined in a limekiln. The experiments of M. Gratien le Pere, show that the schistus of Cherbourgh may be used with much advantage as a substitute for puzzolano or tarras. It is black, hard, ferruginous, and falls off in scales of various thicknesses. Subsequent experiments, however, prove that the slaty schistus of Roule, in the environs of Cherbourgh, is better; and that good mortar may be made with the ferruginous schist of Haineville, but inferior to the two former ones. The schist of Haineville has been analyzed by Descostils, I and contains Silica, . . . . . . . . 46 Alumina, . . . . . . . . 26 Magnesia, . . . . . . . 8 Lime, Oxide of iron, . . . . . . . 100 79. Decayed basalt may be obtained in many parts of the British Islands, and when calcined and pulverised, will make an excellent substitute for puzzolano or tarras. M. Guyton, in 1787, sent some to Cherbourgh for trial, which was found to answer nearly as well as puzzolano; and some experi- ments that have been made since, with basalt from the department of the Haut-Loire, have given still more advantageous results. Of basalt the following table shows the constituent parts of several varieties. Localities. Silica. Alumina. | Magnesia. | Lime. | Oxide of Oxide of 1 Soda iron. manganese. Authority. Basalt of Staffa Rowley Rag or Amorphous basalt Toadstone of Derbyshire . Basalt of Hassenburg. Basalt of Hunneberg - 48 47.5 63 44.5 50 16 32.5 14 16.75 | 15 III la Kennedy Withering Ditto. Klaproth Bergman 0.12 2.6 8 Basalt is found in many parts of England, and in vast abundance in Scotland. At Edinburgh a stratum of basalt forms the Calton and Castle Hills, and Salisbury Crags. It is also very plentiful in Ireland ; Sir H. Davy says, “an excellent red tarras may be procured in any quantities from the Giant's Causeway" in Ireland. 80. The argillaceous iron ores are capable of forming a very hard cement that will set in water. They should be calcined till they assume a deep brown colour. The siftings of the iron stone, after calcination at the iron furnaces, was used by Smoaton; he calls it minion, and says it was “supposed by Mr. Michel to be what chiefly falls from the outside of the lumps of iron stone, and therefore con- tains more clay.” | * Belidor's Science des Ingénieurs. f Gauthey's Construction des Ponts, tome ii. p. 283. | Philosophical Magazine, vol. xxxiv, p. 180. Gauthey's Construction des Ponts, tome ii. p. 283. § Agricultural Chemistry, p. 327. |Smeaton's Eddystone Lighthouse, p. 117. 50 [PART I. PRELIMINARY REMARKS ON THE NA 81. Black oxide of iron is a combination of iron and oxygen, in which there is a less quantity of oxygen than there is in the red oxide. The scales that are detached from iron in the operation of forging, are a black oxide of iron. We cannot have a better proof of the use of the black oxide of iron in water-cements than Smeaton's experiments. He found that scales from the smith's forge were equal to puzzolano, when mixed with lime in the same proportion. 82. Cement-stones are earthy compounds of iron and lime, with silex and alumina. They are fre- quently found in rounded nodules, intersected by thin septa of carbonate of lime, and are called sep- taria, clay-balls, ludas helmontii, &c.; but the stone is also found stratified along with the alum'shale. The market is at present supplied from the coast of the Isle of Sheppey, from Harwich, and from the Yorkshire coast near Whitby. It is not however confined to these places, but is abundantly distri- buted. We have seen it both in Scotland and within the district of the great Northumberland coal- field. It is also found on the coast of France, near Boulogne. Its colour is brownish grey of various degrees of intensity, compact, hard, and fine grained; specific gravity 2:59. Its fractured surfaces become brown or red brown by exposure with moisture,-evidently owing to the oxidation of the iron the stone contains. On being calcined it loses about one-third of its weight, the loss consisting of water and carbonic acid ; its colour becomes reddish brown. The calcination is effected in kilns or ovens, and requires care to prevent vitrification taking place. The stones after being properly calcined, are ground to a very fine powder, which is called Roman cement, and should be kept in a very dry place. The more closely it is packed, and the less it is exposed to the air the better; for it has a strong attraction for moisture, and acquires it rapidly from the air, particularly in a damp atmosphere ; but being close packed and dry it may be kept a considerable length of time. It is sent out from the cement-manufactories in close casks, usually lined with paper, containing five bushels. But two-bushel casks are sometimes used ; and for immediate use it is frequently sent out in three-bushel casks. Six casks of cement, including the casks, weigh one ton. · The component parts of the cement-stones differ considerably; and so does the quality and colour of the cement; the colour is of much importance for stucco-work, and is besides a good criterion of the quality. We give the analysis of the principal varieties used, and their qualities as far as our experience extends : CD Constituent parts. Sheppey cement-stone. Harwich cement. Platre cement-stone from Boulogne. 65.7 52 69.5 733 70.25 18 13 66 Carbonate of lime. . Carbonate of magnesia. Silex . . . . . Aluinina Protoxide of iron . . Carbonate of iron Carbonate of manganese Alkaline and earthy salts .. Water and loss 9.375 9.5 17.75 9:9 · 44 3.5 11:3 1.9 1.3 7.5 3.875 1.1 2.75 1000 Berthier 100.000 100 Hatchett Good 100.0 Vanquelin 100.0 Guyton 100.00 Drapier Analyst . Quality of ditto Inferior The stone does not appear to be a regular chemical compound; bụt when it is good the composi- tion does not materially vary from Hatchett’s analysis. 83. The Roman cement is the most valuable of water-cements, and its use most extensive, besides it has the advantage of being easily applied ; hence, we begin with it. On a clean mortar-board put a quantity of cement, sufficient to serve the time the cement requires to set in, (about 15 minutes,) and add to it not less than an equal quantity by measure of dry and clean river-sand composed of SECT. II.) 51 PRINCIPLES OF PRACTICAL ARCHITECTURE. 4. V . unequal sized and angular grains. Mix the cement-powder and the sand well together, and then form the mixture rapidly into a paste by adding-at once if possible-as much clean water as will render it of the proper consistence for use. It should be quickly used, and not disturbed in the slightest de- gree after it has commenced setting. During the time of setting, its surface becomes dry and sensi- bly warm ; and, if it be not exposed to water, it is useful to apply water frequently, which causes the cement to become more hard, compact, and durable. The work to be cemented or joined with cement should be clean from mortar and dust; and should be well-wetted before applying the cement; whether the material be stone or brick, new surfaces, cleared well of dust, always unite the best. For building, good cement will bear one and a half part of sand to one of cement, and sometimes two parts of sand to one of cement may be used. Forty bushels of cement with a proper quantity of sand will build a rod of brick-work. In coating or lining with cement, it should always be applied in one coat, on clean surfaces, which have been well-wetted, and not less than about three-fourths of an inch in thickness. For it is well known that cement does not adhere so well to cement, as to stone or brick; besides, when cement is applied in two coats, the finishing one is usually done with finer sand, and the two coats being of unequal texture partially separate in drying, and often fail with the frost of the first winter. When it' is applied in one coat, the less the surface is worked the better. It should be left with the appearance of the grain of sandstone, it will then stand the frost, and be free from small cracks. There should not be less sand than equal parts of sand and cement-powder, and more if the mixture will bear it so as not to be too crumbling to work. A bushel of cement, with a proper quantity of sand, will cover two square yards of brick wall three-fourths of an inch thick ; which is the usual thickness of cement- stucco. 84. The use of cement, or of cement and sand, in casting ornaments, is very considerable; it casts exceedingly well, and when sand can be used it forms very durable works. It is much used in Gothic buildings. It has been sometimes improperly applied in the restoration of ancient works ; but no one will regret the change its use has produced in the aspect of London, where it has been formed into tops for chimneys of every variety of form, and completely driven the red earthen pots out of use for new works of a respectable kind. All the precautions we have noticed apply to casting in cement; the neglect of them causes those frequent failures which happen in works of this kind, and prevent it becoming a successful rival of carving in stone. 85. Before the time when Parker discovered the properties of the Sheppey cement stone, and still in places where it cannot be procured at a reasonable cost, it is necessary to produce similar proper- ties by mixture of suitable materials. Good hydraulic lime is usually chosen as the basis of the composition ; but common lime by peculiar treatment in burning becomes nearly as good ; and to give either of these the property of hardening in water, either puzzolano, tarras, basalt, or some of the other substances we have described, are mixed with the water or hydraulic lime, and such a pro- portion of sand as is adapted to the purpose. The nature of these compositions was carefully studied by Smeaton, both for Eddystone Lighthouse, and his later works.* The following table shows the various mixtures and proportions he used. Table of the Composition of Twenty Kinds of Mortar for Hydraulic Works. No. Water lime with puzzolano. der bushels. bushels. 1 Eddystone mortar . 2:32 2 Stone ... ditto .. 2 . L . 1 . 2.68 3 Ditto ... second sort 2 1 . 2 . 5.57 4 Face mortar . . 2 . 1 . 3 . 4:67 * See Hist. Account, chap. iv. and Reports. Lime pow. Puzzolano Common sand bushels. No. of cubic feet. aaaa 4.67 PRELIMINARY REMARKS ON THE [Part L No. Water lime with puzzolano. Lime pow- der bushels. Puzzolano bushels. Common sand bushels. No. of cubic feet. 0 4.17 2 2 . . . . 04 , 3 . 4.04 Minion. 5 Face mortar second sort 6 Backing mortar . Water lime with minion.* 7 Face mortar . . 8 Ditto calder composition 9 Backing mortar . 10 Ditto ... second sort cara · · 322 3:57 4.17 4:04 . . . · . . . . 2 2 04 Of . . 3 3 . . Common lime with tarras. Tarras. O · · - araraaraa · · Co Co Co N · · Minion. · · aa a a Co Co Co Ng 4.05 · Tarras mortar . . 1 1.67 increased . 2:50 13 - further ditto . . . . 2 . 3.45 14 - still further ditto . 4.35 15 Tarras backing mortar . . 3.50 second sort . . 3:37 Common lime with minion. 17 Ordinary face mortar . . . . 2 . 2 . 2 2.75 18 second sort . . 4.34 19 Ordinary backing mortar . . 20 - - second sort . . . . 2 . 03 . 3 . 3.92 86. The following observations ought to be carefully attended to: First, The materials are all supposed to be in a dry state when measured. Second, The lime is supposed to be put into the measure with a shovel with some degree of force, but not pressed ; the same may be said of the puzzolano, minion, and tarras. Third, Sand measures more when moist than dry; so that in moist sand something must be allowed in the measure. Fourth, If the sand is not a mixture of coarse and fine, it must be rendered so by admixture. Fifth, “ The due beating of the mortar is of great consequence, and without going into that repeti- tion of beating which has always been looked upon as essential to tarras mortar, such as No. 11, a degree of beating sufficient to give it all the possible consistence and toughness before it is used, is in reality indispensable.” 1 Sixth, " The customary allowance for tarras mortar beating, is a day's work of a man for every | bushel of tarras.” No. 5. Smeaton supposed equal in firmness and validity to No. 11; and No. 10 to 15. The mea- sure is the Winchester bushel striked. The lime chiefly used by Smeaton was the Aberthaw for the best compositions. 87. The peculiar advantages of Roman cement had early drawn attention to forming an artificial compound to possess the same properties, and these attempts have been in some degree success- ful. Of the ingredients alumine seems to be the least active, nevertheless its presence is an advan- tage in getting the other ingredients to the proper state, by means of calcination. The difficulty of the case consists in getting the ingredients intimately mixed before they be calcined. The setting is a consequence of a compound hydrate of silex, lime, and oxide of iron being formed; and hence, the more completely these bodies have been deprived of water, the greater will be the energy with which they will combine. 74* Minion is the siftings of the iron stones after calcination at the iron furnaces. Sect. II.] 53 PRINCIPLES OF PRACTICAL ARCHITECTURE. 88. Concrete.]—In preparing concrete, the following proportions have been found to succeed per- fectly in some recent structures. . . . . . . . . . . . . . Hydraulic lime, (unslaked,) Sand, (middling,) . Cement, (common clay,) Gravel, (coarse,) . Chippings of stone, . . . . . . . . 0.30 parts. 0.30 - 0.30 - 0.20 - 0.40 - . . . . . . . . . . . . . of a hard temper ; this mass is suffered to rest in a heap about twelve hours; it is then spread out into a layer about six inches thick, and the gravel and stone are evenly spread over it, and the whole well mixed up. The mass, before it is used, is suffered to remain until it has partially set, which will re- 2 improve the quality of the concrete. This material depends on the quality of the mortar for its excellence. It is not stronger than simple hydraulic mortar, but it is far more economical. The gravel, which enters into its composition, is used to fill up the voids between the fragments of stone, which would otherwise be filled by the mortar alone. Broken brick may be used instead of frag- ments of stone when the latter cannot be had; or gravel alone may be used. 89. There is no subject connected with the art of the Engineer or Builder upon which more in- genuity has been uselessly expended than upon that of mortar. Misled by erroneous or forced in- terpretations of some passages of the ancients, particularly of Vitruvius, various hypotheses have been formed to explain the superior properties of the mortar found in the remains of ancient edifices, over that of a modern date ; and almost universal failure, for a long period, attended all the experiments made in conformity with these hypotheses, as they were not conducted according to the only sure method of investigation in such cases, a careful analysis. The fallacy, both of the hypotheses adopted, and the results obtained, led scientific engineers to treat the subject in a more rational manner, and with a success which has fully repaid the care bestowed on it. The true nature both of lime and mortar-thanks to the labours of Vicat, Raucourt, and Treussart,-men who stand at the head of the professions of civil and military engineers in France-is now perfectly understood; and the re- sults, owing to the light that they have thrown on the subject, may with certainty be predicted. It ing the ingredients, nor the age, are the causes of the great strength and hardness of some kinds of mortar, although they doubtless exercise some influence ; but that these qualities are attributable, almost solely, to the nature of the lime; secondly, that with common lime and sand a mortar is ob- tained which is suitable only for dry exposures; and that no age nor preparation will cause it to harden in moist situations, such as foundations, the interior of heavy walls, and constructions under water ; thirdly, that there are natural varieties of limestone which possess this peculiar property of hardening under water and in moist situations, and are, therefore, alone suitable for hydraulic mor- tar; and that wherever this natural hydraulic lime cannot be procured, an artificial mortar can be prepared, fully equal to that made of the natural lime, by adding some natural or artificial cement to common lime and sand. With regard to the action of the lime on the sand, the most careful analysis, thus far, has not been able to detect any appearance of a chemical combination between the two; and it is the received opinion that the union between them is simply of a mechanical character: the lime entering the pores of the sand, and thus connecting the particles much in the same way as the particles of granular stones are connected by a natural cement. The sand itself serves the im- portant purposes of causing the mass to shrink uniformly, whilst the hardening or setting of the mor- tar is still in progress, and thus prevents any cracking, which must always be the result of irregularity 54 (Part I. PRELIMINARY REMARKS ON THE in the shrinking; it promotes the rapil desiccation of the mass ; and is conducive both to solidity and economy, from its superior strength, hardness, and cheapness, to lime. No perfectly satisfactory solution has yet been given for the hardening of either common or hydraulic mortar. That the former acquires strength and hardness with age, experience has very conclusively shown; and it was, for some time, supposed that this arose from a gradual conversion of the lime into a carbonate by the slow absorption of carbonic gas from the air; but, from experiments conducted with great care, it seems that only a very thin coating on the surface undergoes this change, and that no more gas can be detected in the interior of the mass than is usually retained by lime which has been submitted to the greatest heat of ordinary kilns. As to the action which takes place in hydraulic lime, it is ac- counted for on the supposition that a chemical combination takes place between the lime and argile when mixed in a moist state, being a compound formed with new properties distinct from those of the constituent elements: this combination requiring a longer or shorter time, depending on the energy of the ingredients, to become complete. 90. From the above views of the nature of mortar, it appears that its good qualities, as a building material, essentially depend, l. on the kind of lime ; 2. on the strength and hardness of the sand; 3. on the adhesion between the lime and sand, which will depend on the roughness and porosity of the particles of sand, and the care bestowed in thoroughly incorporating the ingredients. 91. The experiments made on the strength of mortar have led to no satisfactory conclusions, except so far as to institute a comparison of the effects produced by using various proportions of the same or different ingredients, and from the more or less care taken in mixing them. As to the absolute strength, no definite results, of course, can be arrived at, owing to the variable proportions of the ingredients, until some greater uniformity shall be adopted in the practice of Engineers and Builders generally, for determining the proportions, and the method of mixing them. 92. The most interesting experiments on the strength of mortar are those detailed by General Treussart in his work on this subject. He chose for his experiments small rectangular parallelopipeds, six inches long, and two inches square. These were placed on props at each end, leaving a bearing of four inches between the props. A common weighing scale was attached by a hook or stirrup to the middle point of the parallelopiped, and weights added until it broke: He found that mortar, formed of equal parts of common lime, sand, and cement, bore in this way before rupture took place, from 220 to 440 pounds; and he recommends, that for heavy masonry exposed to the air, the mortar used should bear from 220 to 230 pounds when submitted to this test, and that for hydraulic works, from 330 to 440 pounds. With regard to mortar of common lime and sand, its strength was found, by his experiments, to be very inferior to that in which cement entered. The best samples were those made with one part of lime and two parts of sand ; some of these bore, at the moment of rupture, from 60 to 100 pounds.24 93. Brick.]—This material is properly an artificial stone, formed by submitting common clay, which has undergone suitable preparation, to a temperature sufficient to convert it into a semi- vitrified state. 94. Brick may be used for nearly all the purposes to which stone is applicable: for when carefully made, its strength, hardness, and durability, are but little inferior to the more ordinary kinds of 24 Experiments on this very interesting subject to Builders have been made by Colonel Totten, of the United States' Corps of Engineers at Fort Adams. So far as they have been prosecuted, they agree with the results given in the pre- ceding remarks. Mr. Brunel also, previous to commencing the Tunnel under the Thames, made various experiments on the strength of different kinds of mortars and cements. These are described in Barlow's 'Treatise on the Strength of Timber, Iron, and other Building Materials.' SECT. II.] 55 PRINCIPLES OF PRACTICAL ARCHITECTURE. building-stone. It remains unchanged under the extremes of temperature; resists the action of water; sets firmly and promptly with mortar; and being both cheaper and lighter than stone, is preferable to it for many kinds of structures, as arches, the walls of houses, &c. . . 95. The art of brick-making is a distinct branch of the useful arts, and does not properly belong to that of the Builder ; but as he is frequently obliged to prepare this material himself, the following outline of the process may prove of service :- The best brick-earth is composed of a mixture of pure clay and sand, deprived of pebbles of every kind, but particu- larly of those which contain lime, and pyritous or other metallic substances; as these, when in large quantities, and in the form of pebbles, act as fluxes, destroy the shape of the brick, and weaken it by causing cavities and cracks, but when present in small quantities, and equally diffused throughout the earth, they assist the vitrification, and give it a more uniform character. Good brick-earth is frequently found in a natural state, and requires no other preparation for the purpose of the brick- maker. When he is obliged to prepare the earth by mixing the pure clay and sand, direct experiments should in all cases be made, to ascertain the proper proportions of the two. If the clay is in excess, the temperature required to semi-vitrify it will cause it to warp, shrink, and crack; and if there is an excess of sand, complete vitrification will ensue under similar circumstances. The quality of the brick depends as much on the care bestowed on its manufacture, as on the quality of the earth. The first stage of the process is to free the earth from pebbles: which is most effectually done by digging it out early in the autumn, and exposing it in small heaps to the weather during the winter. In the spring the heaps are carefully riddled if necessary, and the earth is then in a proper state to be kneaded or tempered. The quantity of water required in tem- plastic as to admit of its being easily moulded by the workman. About half a cubic foot of water to one of the earth, is, in most cases, a good proportion. If too much water be used, the brick will not only be very slow in drying, but it will, in most cases, crack, owing to the surface becoming completely dry before the moisture of the interior has had time to escape; the consequence of which will be, that the brick when burnt will be either entirely unfit for use, or very weak. Machinery is now coming into very general use in moulding brick; it is superior to manual labour, not only from the labour saved, but from its yielding a better quality of brick, by giving it great density, which adds to its strength. Mr. Bakewell of Manchester's clay-tempering machine, and also bis brick-making machine, are described in The Mechanic's Magazine' of May 14, 1831. Great attention is requisite in drying the brick before it is burned. It should be placed for this purpose in a dry ex- posure, and be sheltered from the direct action of the wind and sun, in order that the moisture may be carried off slowly and uniformly from the entire surface. When this precaution is not taken, the brick will generally crack from the un- equal shrinking, arising from one part drying more rapidly than the rest. Too large a proportion of sand will render the brick brittle under this process; while too large a proportion of clayey matter will be indicated by the brick shrinking and cracking. The burning and cooling should be done with equal care. A very moderate fire should be applied under the arches of the kiln for about twenty-four hours, to expel any remaining moisture from the raw brick; this is known to be com- pletely effected when the smoke from the kiln is no longer black. The fire is then increased until the bricks of the arches attain a white heat; it is then allowed to abate in some degree, in order to prevent complete vitrification; and is alternately raised and lowered in this way until the burning is complete, which may be ascertained by examining the bricks at the top of the kiln. The cooling should be slowly effected, otherwise the bricks will not withstand the effects of the weather. This is done by closing the mouths of the arches, and the top and sides of the kiln, in the most effec- tual manner, with moist clay and burnt brick, and allowing the kiln to remain in this state until the warmth has perfectly subsided.- A kiln 13 feet long, 10 feet 6 inches wide, and 12 feet high, will burn 18,000 bricks. 96. Brick of a good quality exhibits a fine, compact, uniform texture when broken across --gives found in the kiln ;-those which form the arches, denominated arch-brick, are always vitrified in part, and present a grayish, glassy appearance at one end ; they are very hard but brittle, and of inferior strength, and set badly with mortar ;--those from the interior of the kiln, usually denomi- nated body, hard, or cherry-brick, are of the best quality ;—those from near the top and sides are generally under burnt, and are denominated soft, pale, or salmon-brick, they have neither sufficient 56 [Part I. PRELIMINARY REMARKS ON THE strength nor durability for heavy masonry, nor the outside courses of walls which are exposed to the weather. 97. The bricks in general use are distinguished-amongst English bricklayers—as malms, gray stocks, red stocks, and place bricks. The variety in their qualities and colours is caused by the differ- ence in the nature of the materials of which they are formed, and the care and attention with which they have been manufactured and burnt. 98. Malms or malm stocks, are of two sorts. Those of the first quality are termed cutters; and, on account of the facility with which they are cut or rubbed down, are used for forming what are called gauged arches, over doors and windows, and other similar purposes. Those of an inferior quality are denominated seconds, and are most commonly selected for facing the fronts of buildings. Mr. Lees, by mixing together certain proportions of chalk and loam, discovered that a good substitute for malm stocks might be produced ; and his practice is now generally adopted in the vicinity of London. 99. The next in quality are the gray stocks. These are formed of good earth, well-wrought with little mixture ; and when thoroughly burnt, are sound and durable. 100. Red stocks are the common sort of bricks made in many parts of the country, and owe their colour principally to the earth of which they are composed, though partly to the manner in which they are burnt. The difference between the colour of the bricks commonly made in the vicinity of London, and those made in the country generally, is ascribed in a great degree to the effect pro- duced in burning, by the ashes of sea-coal which is usually mixed with the brick earth in preparing the former. Red cutting bricks of a very superior quality are made within a few miles of London, particularly at the village of Hedgerley, near Windsor. They make excellent fire-bricks as well as cutters. A variety of facing bricks of superior quality differing somewhat in colour are brought to London from different parts of the country, more particularly from Southampton, Ipswich, Ware, and Rochford in Essex. 101. Place bricks, or as they are sometimes termed, sandel or samel bricks, are those which are left in the clamp after the malms and stocks have been selected ; and consist of those bricks that are not uniformly burnt. They are consequently of a very inferior quality, and should never be used either for external work, or where any weight is to be sustained. 102. There is likewise another sort of brick called burs or clinkers. They are such as have been over-burnt and partly vitrified ; two or three being sometimes run together. . 103. The size of bricks is regulated by Act of Parliament on account of the heavy duty that is laid on them. The standard-size of bricks made for sale must, when burnt, be ten inches in length, five inches in width, and three inches in thickness. Bricks may indeed be made of any size, but all above the standard-size pay a higher duty. The duty when first levied in 1784, was 2s. 6d. per 1000; it is now 5s. 10d. The quantity charged with duty in England in 1839, was 1,569,020,952 ; in Scotland, 42,267,633. Ireland is exempt from the duty on bricks. 104. In selecting bricks, care should be taken to choose those that when struck ring well, yielding a metallic sound, and such as will bear a smart blow without breaking. They should likewise be of an uniform and regular shape, and not readily imbibe water; as those that are most porous are liable in a greater degree to be affected by frost, and will in a few years probably crumble to powder. But the quality of good brick may be improved by soaking it for some days in water, and re-burning it. This process increases both the strength and durability, and renders the brick more suitable for hydraulic constructions, as it is found not to imbibe water so readily after having undergone it. 105. Brick presents great diversity in its strength, arising principally from its greater or less density; the densest made of the same earth, being uniformly the strongest. It was found on ex- Sect. II.) PRINCIPLES OF PRACTICAL ARCHITECTURE. are periment, that good brick, having the specific gravity of 2.168, required 1,200 pounds on a square inch to crush it. 106. Fire-brick. –This material is used for the facing of ovens, furnaces, fire-places, &c., where a very high degree of temperature is to be sustained. It is composed of a very refractory species of clay, that will remain unimpaired by a degree of heat which would vitrify and completely destroy ordinary brick. Bricks of this description are made in England, in the vicinity of Windsor ; at Stourbridge, near Cambridge ; in some of the iron mining counties; and in Wales. There is a peculiar large sort of fire-brick of excellent quality termed Welch lumps. A very remarkable brick of this nature has been made of agaric mineral; it remains unchanged by the highest temperature, is one of the worst con- ductors of heat known, and is so light as to float on water. 107. There are several sorts of bricks known by the name of paving bricks. Those made in Eng- land are formed of a peculiar sort of earth, and are burnt so as to become very hard. With respect to size, they are similar to other bricks, excepting as to thickness, which generally does not exceed one inch and a half. Considerable quantities of this sort of brick are imported from Holland and Flanders. They are very hard, and of a dirty brimstone colour. The Flemish bricks are more yellow than the Dutch, but the latter are usually better baked. The hardest kind of all are termed clinkers, and are chiefly used for paying yards, stables, &c., and in constructing ovens, lining soap- boilers, cisterns, &c. 108. Compass bricks are circular on the plan, and are chiefly used for steyning-wells and cess-pools. Concave bricks are made like common bricks, but are hollowed on one side in the direction of the length, and are sometimes employed in the construction of drains and water-courses. There is like- wise a species of hollow brick made of the same material as plain tiles, and called by the workmer cones or pots. They are of a cylindrical form, but have one end brought into a square shape, the other being filled in to the circular form. They vary from five to nine inches in length, and are much used for turning domes and arches, especially where lightness of construction is requisite. 109. Tiles. — As a roof covering, tiles are, in many cases, superior to slate and metallic coverings, . both for strength and durability. They are, therefore, very suitable for the roofing of arches, as their great weight is not so objectionable as in the case of common roofs of frame-work. Tiles are made of the best potter's clay; and, in order to make them both thin and strong, are moulded with great care to give them the greatest density. They are of very variable form and size, the worst being the flat square form, which, owing to the warping of the clay in burning, seldom makes a per- fectly water-tight covering. 110. Plain tiles are a species of tiles used for covering roofs. They are of a rectangular form, and perfectly flat; and are usually about ten inches long, six inches broad, and five-eighths of an inch thick. 111. Pan tiles have a rectangular outline, and are bent so that the surface is both concave and convex. They are thirteen inches long, eight inches broad, and half an inch thick. They are some- times glazed, which adds greatly to their value. 112. Ridge-tiles and Hip-tiles are of a semi-cylindric form, and used-as their names import—for covering the ridges and hips of roofs ; as are likewise another kind denominated gutter tiles, which are however but seldom used now, their place being supplied by lead. 113. Paving tiles are a long flat kind of tile, about nine inches long, four inches and a-half broad, and one inch and a-half thick. They are principally in use for the floors of dairies, cheese-stores, &c. There are other kinds of square tiles used for paving, and known by the name of Ten-inch tiles, or Twelve-inch tiles. 58 [PART 1. PRELIMINARY REMARKS ON THE III.-WOOD. 114. This material holds the next rank to stone, owing to its durability and strength, and the very general use made of it in constructions. To suit it to the purposes of the Builder, the tree is felled after having attained its mature growth, and the trunk, the larger branches that spring from the trunk, and the main parts of the root, are cut into suitable dimensions and seasoned: in which state the term timber is applied to it. The crooked or compass timber of the branches and roots, is mostly applied to the purposes of ship-building, for the knees and other parts of the frame-work of vessels. The trunk furnishes all the straight timber. 115. The trunk of a full grown tree presents three distinct parts: the bark, which forms the ex- terior coating,—the sap-wood, which is next to the bark,--the heart, or inner part, which is easily distinguishable from the sap-wood by its greater firmness and darker colour. The heart forms the essential part of the trunk as a building-material. The sap-wood possesses but little strength, and is subject to rapid decay, owing to the great quantity of fermentable matter contained in it; and the bark is not only without strength, but if suffered to remain on the tree after it is felled, it hastens the decay of the sap-wood and heart. Trees should not be felled for timber until they have attained their mature growth, nor after they exhibit symptoms of decline ; otherwise the timber will be less strong, and far less durable. Most forest-trees arrive at maturity in between fifty and one hundred years ; and commence to decline after one hundred and fifty, or two hundred years. The age of the tree can, in all cases, be ascertained by cutting into the centre of the trunk, and counting the rings or lay- ers of the sap and heart, as a new ring is formed each year in the process of vegetation. When the tree commences to decline, the extremities of the old branches, and particularly the top, exhibit signs of decay. 116. Trees should not be felled whilst the sap is in circulation; for this substance is of a pecu- liarly fermentable nature, and, therefore, very productive of destruction to the wood. The proper seasons for felling are, in winter, during the months of December, January, and February; and, in midsummer, during July. All other seasons are bad; but the spring is peculiarly so, for the tree then contains the greatest quantity of sap. As the sap-wood, in most trees, forms a large portion of the trunk, experiments have been made for the purpose of improving its strength and durability. These experiments have been mostly directed towards the manner of preparing the tree before felling it. One method consists in girdling, or making an incision with an axe around the trunk completely through the sap-wood, and suffering the tree to stand in this state until it is dead; the other consists in barking, or stripping the entire trunk of its bark, without wounding the sap-wood, early in the spring, and allowing the tree to stand until the new leaves have put forth and fallen before it is felled. The sap-wood of trees, treated by both these methods, was found very much improved in hardness, strength, and durability; the results from girdling were, however, inferior to those from barking. 117. The seasoning of timber is of the greatest importance, not only to its durability, but to the solidity of the structure for which it may be used; as a very slight shrinking of some of the pieces, arising from the seasoning of the wood, if used in a green state, might, in many cases, cause material injury, if not complete destruction to the structure. Timber is considered as sufficiently seasoned, for the purposes of frame-work, when it has lost about one-fifth of the weight which it has in a green state 25 Several methods are in use for seasoning timber ;-they consist either in an exposure to the air for a certain period in a sheltered position, which is termed natural-seasoning ; in immersion in water, termed water-seasoning; or in boiling or steaming. On the 25 Oak, when completely seasoned, has been found to have lost one-third of its original weight.-Barlow Strength of Timber.' Sect. II.] 59 PRINCIPLES OF PRACTICAL ARCHITECTURE. 118. For natural seasoning, the trunk, so soon as the tree is felled, should be deprived of its bark, and removed to some dry position until it can be sawed into suitable scantling. It should then be piled in a perfectly dry situation, and be exposed to a free circulation of the air, but sheltered from the direct action of the wind, rain, and sun. By using these precautions, an equable evaporation of the moisture will take place over the entire surface, which will prevent either warping or splitting, which necessarily ensues when one part dries more rapidly than another. It is farther recommended, in- stead of piling the pieces on each other in a horizontal position, that they be laid on cast-iron sup- ports properly prepared, and with a sufficient inclination to facilitate the dripping of the sap from one end; and that heavy round timber be bored through the centre to expose a greater surface to the air, as it has been found that it cracks more in seasoning than square timber. Natural seasoning is pre- ferable to any other, as timber seasoned in this way is both stronger and more durable than when pre- pared by any artificial process. Most timber will require, on an average, about two years to become fully seasoned in the natural way. 119. Water-seasoning may be resorted to when despatch is necessary; the trunk is immersed in water about a fortnight, and then taken out and dried in a sheltered position before using it. The sap-wood is rendered less liable to decay by this process, as a large proportion of the fermentable matter is dis- solved by the water ; but the general strength of the timber is impaired by this loss. Fresh running water is considered the best for timber which is to be used in the frame-work of houses, as the salt which is taken up by timber, immersed in salt-water, keeps it always in a moist state, by attracting moisture from the atmosphere. 120. Steaming is mostly in use for ship-building, where it is necessary to soften the fibres for the purpose of bending large pieces of timber. It impairs the strength of the timber, but renders it less subject to decay, and to warp and crack. About four hours is said to be sufficient for steaming the largest sized pieces.26 121. When timber is used for posts partly imbedded in the ground, it is usual to char the part thus imbedded, to preserve it from decay. This method is only serviceable when the timber has been previously well-seasoned, as it then acts as a preventative both of worms and the rot; but for green timber it is highly injurious, as by closing the pores it prevents the evaporation from the surface, and thus causes fermentation and rapid decay within. 122. The most durable timber is procured from trees of a close compact texture, which, on analy- sis, yields the largest quantity of carbon. And those which grow in moist and shady localities furnish timber which is weaker and less durable than that from trees growing in a dry open exposure. 123. Timber is subject to defects, arising either from some peculiarity in the growth of the tree, or from the effects of the weather. Straight-grained timber, free from knots, is superior in strength and quality, as a building material, to that which is the reverse. 124. The action of high winds and severe frosts injures the tree whilst standing: the former sepa- rating the layers from each other, forming what is denominated rolled timber ; the latter cracking the timber in several places, from the surface to the centre. These defects, as well as those arising from worms or age, are easily seen by examining a cross section of the trunk. 125. The wet and dry rot are the most serious causes of the decay of timber, and all the remedies thus far proposed to prevent them are too expensive to admit of a very general application. Both of these causes have the same origin,-fermentation, and consequent putrefaction. The wet rot takes place in wood exposed alternately to moisture and dryness : and the dry rot is occasioned by want of a free circulation of air, as in confined warm localities like cellars, and the more confined parts of ves- 26 The rule in our dock-yards is an hour to every inch in the thickness. The detect in strength is very inconsiderable 60 [PART I. PRELIMINARY REMARKS ON THE sels. Trees of rapid growth, which contain a large portion of sap-wood, and timber of every descrip- tion, when used green, where there is a want of a free circulation of air, decay very rapidly with the rot. 126. Of the various remedies proposed to prevent the rot, the application of salt around the timber is said to succeed in ship-building; and boiling the timber for some hours in a solution of copperas, or in one of corrosive sublimate, 27 is said to answer the purpose for house-carpentry. The best means is to use only well-seasoned timber, and to procure a free circulation of fresh air around it. 127. The durability of timber varies greatly under different circumstances of exposure. If placed in a sheltered position, and exposed to a free circulation of fresh air, it will last for centuries without any very sensible changes in its physical properties; and the same is known to take place when it is entirely immersed in water, or imbedded in the ground, or in thick walls, so as not to be affected by the atmosphere. In salt-water, however, particularly in warm climates, timber is rapidly destroyed by several kinds of worms, which soon reduce it to a honey-comb state. 128. The best-seasoned timber will not withstand the effects of exposure to the weather for a much greater period than twenty-five years, unless it is protected by a coating of paint or pitch ; or of oil laid on hot when the timber is partly charred over a light blaze. These substances themselves, being of a perishable nature, require to be renewed from time to time, and will, therefore, be service- able only in situations which admit of their renewal. They are, moreover, more hurtful than service- able to unseasoned timber, as by stopping up the pores of the exterior surface, they prevent the moisture from escaping from within, and, therefore, promote one of the chief causes of decay.28 129. Britain produces at present but a small proportion of the timber employed for building and other purposes. There are said to be about thirty species of forest-trees indigenous to Great Britain, which attain the height of 30 feet. But, according to the best authorities, there are no less than 140 species which attain a similar height, indigenous to the United States. We are supplied principally from the Baltic, from Africa, and from America. 130. Oak.]—Of the several kinds of timber applicable for building, Oak is universally allowed to be the most valuable, it being the strongest and likewise the most durable. It is very lasting in water, and in a dry state it has been known to remain uninjured a thousand years. 131. Of the oak it is said there are nearly fifty different species. The Quercus robur, or common English oak, however, far excels all other kinds in the known world. That from Sussex is considered to be the best that England produces; the next in quality is that which is grown in the south-west parts of Kent, and the north-east parts of Hampshire. Its excellence, which consists in the compact- ness and closeness of its pores, is supposed to be owing partly to the soil, and partly to the method of management adopted as respects the wood. Excellent oak is likewise grown in some parts of Scotland. The open, porous, foxy-coloured oak which is grown in Lincolnshire, and some other parts of the country, is far from being durable. The heart of such oak is considered to be scarcely superior to the sap of the better kinds. The oak from the north of Europe is generally superior in quality to that brought from North America ; but the American live oak, as it is called, (Quercus virens,) is greatly superior to that usually imported under the name of white oak. A species of oak is brought from Norway, denominated clapboard. Another termed Dutch wainscot, is imported from Holland, though it is grown in Germany and floated down the Rhine. Clapboard is distinguished from wainscot by the light-coloured streaks which cross it in various directions. Wainscot is softer than the com- mon oak, but it is not so liable to warp and split; in this respect clapboard is far inferior to wainscot. 27 This is known in England under the denomination of Kyanized timber. Another method of more recent invention consists in impregnating the timber with sulphate of iron and sulphate of lime. The process renders timber as hard and nearly as heavy as stone; and the expense is only about 8 per cent. 28 For various means of preserving timber, see Tredgold. On the Principles of Carpentry.' SECT. II.] 61 PRINCIPLES OF PRACTICAL ARCHITECTURE. S 132. Oak is best adapted for building-purposes when it has attained its full growth, which it rarely does in less than from eighty to one hundred years ; but that under its maturity is rather to be pre- ferred to that which has much passed it. Oak is useful for most purposes of the Carpenter, particu- larly in situations where it is exposed to the weather. It is likewise much esteemed for wall-plates, sleepers, ties, templets, king-posts, and indeed for all purposes where its warping in drying and its flexibility do not render it objectionable. For general purposes those pieces should be chosen that have the straightest grain, though for knees and similar uses a curvilinear direction of the fibres of the timber may be occasionally desirable. It may be taken as a general rule, that of two pieces of oak equally dry and of equal dimensions, that the heavier is the better piece. It is to be observed likewise, that in the oak, as in all other trees, the wood of the boughs and branches is weaker by far than that of the body of the tree ; that of the great limbs stronger than that of the small limbs; and the wood from the heart of a sound tree the strongest of all.-- The weight of a cubic foot of dry oak is about fifty pounds. 133. Fir. ]— The species of Fir that are principally used as timber for building are," first, the Pinus picea, or silver fir tree, which is common in the mountainous parts of Scotland, in Switzerland, in Norway, and on the shores of the Baltic. Its timber is of a yellow colour 134. The second species is the Pinus abies, or spruce fir, which produces the white sort of timber. It is a native of Norway and Denmark, where it grows spontaneously; it is likewise plentiful in the Highlands of Scotland. 135. The third species is the Pinus strobus, commonly known as the American pitch pine. This tree grows to an amazing height. It is a white sort of wood, and is imported into this country chiefly from South Carolina. 136. The best sorts of fir that are imported—for comparatively but little of our own growth is used -are from Norway, Riga, Memel and Dantzic. An inferior sort, which is imported in smaller logs, is called Dranton or dram timber. Of late years, a considerable quantity of fir timber has been im- ported into this country from North America, particularly from Canada, and, being cheaper, in con- sequence of the import duty being considerably less on it than on the Baltic timber, it has been much used in the inferior description of buildings, especially in such as have been erected on specu- lation. The ordinary sort is, however, certainly very inferior in quality to the Baltic timber, and is said to be much more liable to be affected with the dry rot than that is ; it is seldom, therefore, em- ployed in buildings of a superior class ; never indeed in the form of timber, though occasionally it is introduced in some parts of the joiner's work, for which, from the facility with which it is worked, it is peculiarly applicable.29 Some excellent fir-wood is obtained from the forest of Mar in Aberdeenshire. 137. Deals are the wood of the fir tree cut up into certain thicknesses in the countries from whence they are imported. The deals imported into England are principally from Christiana and other parts of Norway, from Dantzic, and from St. Petersburg. They are usually three inches in thickness and nine inches in width ; when wider they are termed planks. Deals vary in length from eight feet to upwards of twenty feet, but in purchasing them they are usually calculated at twelve feet; 120 twelve- feet deals, nine inches wide, making what is called a hundred. They are denominated white or yel- low, from the species of timber out of which they are formed. 138. The greatest care should be taken that deals are properly seasoned before they are used in a 29 By the new tariff of 1842, the duty on colonial, that is, North American timber is reduced to 1s. the load; and to 2s. on deals, and 6d. on lath-wood. The duty on foreign timber is 30s. the load; until October 10, 1843, when it is to be reduced to 25s. On foreign deals the duty is 35., but will be reduced to 30s. after that date. The duty on foreign lath-wood is 10s. the load. Between 1829 and 1833, the average quantity of colonial timber imported into Great Britain was 412,682 loads ; of foreign or Baltic, 122,783 loads. on foreigh wood. The un colonial, that foreign timber is 3 dis 105. the load. " deals the duty is 62 [Part I PRELIMINARY REMARKS ON THE building, as the work will otherwise shrink and fly. In order to season deals, they should be piled in stacks so as to allow as free a circulation of air around them as possible. Yellow deals are the most proper to be used in external works, as they stand the weather better than white deals, which from being softer, and consequently more easily wrought, are more proper to be employed in internal fit- tings and finishings. 139. Fir wood is peculiarly applicable for joiners' work, both from the facility with which it is wrought, its lightness, and the readiness with which it takes the glue ; but as it is easily compressed by a force acting at right angles to its fibres, wherever compression of that kind is likely to arise, oak, or one of the hard woods, should be employed in preference to deal: it will, however, it is said, bear a weight in the direction of its fibres better than oak. The darker part of the annual ring is the hardest ; dry wood is harder than green, consequently more difficult to work.—The weight of a cubic foot of dry Fir is from twenty-five to thirty pounds. 140. Of the Beech, the wood of different trees varies considerably in its quality, caused doubtless by the difference of soil and situation in which it is grown. The colour of beech is a whitish brown, of different shades; the darker kind is called brown, and sometimes black beech. The lighter, which is much the harder of the two, is called white beech : the black beech is, however, the toughest, and is said to be the most durable of the two. Beech is not very difficult to work, and may be brought to a very smooth surface. It is grown in considerable quantities in the southern parts of Buckingham- shire and in Sussex. It decays rapidly when used in damp situations, and is liable to be affected by worms whether in a dry or damp state : it is, however, durable when constantly immersed in water, and is consequently useful for piles, in situations where it is kept constantly wet. It is likewise adapted for making various tools, and for furniture of an ordinary description.—The weight of a cubic foot of beech when dry varies from forty-three to fifty-three pounds. 141. The Chestnut (Castanea sativa) which is indigenous in this country, is amongst the most valuable species of trees for building ; for which purpose it was formerly much used, though latterly its cultivation has been rather neglected. In substance, quality, and colour, it so much resembles oak, that it is difficult to distinguish the one from the other. It is said, however, that a certain opinion may be formed from observing whether a black substance is formed round the part into which a nail is driven : this is always the case with oak, but the same effect is not produced in chestnut, Chestnut does not shrink or swell so much as other woods, it is easier to work than British oak, and is likewise tougher. The wood of young trees is found to be superior even to oak in durability. The weight of a cubic foot of dry Chestnut is from thirty-five to forty pounds. 142. Of the Elm there are several species. The common rough leaved Elm (Ulmus campestris) is very generally met with in scattered woods and hedges in the southern parts of England; it is harder and more durable than the other species, and as it resists moisture particularly well, it is commonly used for coffins. Elm is much esteemed on account of its durability in situations where it is kept constantly wet, and is therefore frequently used for piles and planking in wet foundations, and for pipes, pumps, and other kinds of water works; it is likewise much used for dressers, chopping-blocks, and other domestic purposes. The colour of the heart-wood of elm is generally darker than that of oak, and of a redder brown; the sap-wood is of a yellowish, or brownish white, with pores inclining to red. Elm is in general porous and cross-grained, and frequently it is very coarse grained. It twists and warps much in drying, and shrinks considerably both in length and breadth. It is difficult to work, but it is not liable to split, and bears the driving of bolts and nails better than any other species of timber.—The weight of a cubic foot of dry elm is from thirty-four to thirty-seven pounds ; of seasoned, from thirty-six to fifty pounds. 143. Mahogany is a wood too generally known to require particular description here. It is a native SECT. II.) 63 PRINCIPLES OF PRACTICAL ARCHITECTURE. of the warmest parts of America, is found in the islands of Cuba, Jamaica, and Hispaniola, and like- wise in the Bahama islands. The Jamaica mahogany is much harder and more durable than the Honduras or American, and may be easily distinguished by its pores having a white appearance as if filled with chalk, whilst the pores of the latter appear quite dark. The Hispaniola—or as it is generally termed, Spanish mahogany—is extremely hard, and twists or warps less than any other kind of wood; it takes a fine polish, and is admirably adapted for superior kinds of finishings, such as doors, window-shutters, sashes, and handrails of staircases; it is likewise much used for various descriptions of furniture. In Jamaica it has been frequently used for joists, floors, rafters, shingles, &c. Ships have likewise been built of mahogany, for which purpose it appears to be well-adapted, inasmuch as it is found that shot will bury itself in this wood without causing it to splinter. The finest kinds of the Spanish wood are extremely valuable for cabinet-work, and for the purpose of being cut into veneers.—A cubic foot of Jamaica mahogany weighs about fifty-four pounds. A cubic foot of Honduras mahogany rarely weighs more than about thirty-eight pounds. 144. The Alder tree is a native of Europe and Asia, and grows in wet grounds, and on the banks of rivers. Its wood is extremely durable in water, or in damp ground. Vitruvius has remarked that, in a wet state, alder will sustain the weight of very heavy piles of building without risk of accident; and that the whole of the buildings at Ravenna, which is situated in a marsh, were founded upon piles of this wood. Evelyn says he found piles of alder were used under the Rialto at Venice, which was built in 1591. From the durability of alder in water, it is much esteemed for piles, planking, sluices, pumps, and in general for any purpose where it is kept constantly wet. It soon rots, however, when exposed to the weather, and in a dry state is very subject to worms. It is much used for turners, wares, and other light purposes. The alder is of a reddish yellow colour of different shades, and its texture is very uniform. It is soft and works easily, and is well-adapted for carving, and for making models for casting from.--The weight of a cubic foot of alder in a dry state is from thirty-four to fifty pounds. 145. Of the Plane tree there are several species; the most common are the Oriental and the Occi- dental plane. The Oriental plane is a native of the Levant, and is considered one of the finest of trees. Its wood is much like beech, but more figured, and is used for furniture as well as for different kinds of joiners' work. The Occidental plane is a native of North America. On the banks of the Ohio and Mississippi it is said to attain to an enormous size, sometimes exceeding twelve feet in diameter: The wood of the Occidental plane is harder than that of the Oriental kind ; it is very durable in water, and is therefore used by the Americans for wooden quays, in preference to any other descrip- tion of wood. The colour of the wood of the plane is nearly the same as that of beech, which it likewise closely resembles in its structure. It works easily and stands well.–A cubic foot when dry weighs from forty to forty-six pounds. 146. The Sycamore, or great maple ( Acer pseudo-platinus) is a native of the mountains of Ger- many, and is very common in Great Britain. In the north of England it is generally called the plane tree. The wood is very durable in a dry state, when it can be protected from worms, by which it is as liable to be affected as the beech. The colour of the sycamore is generally of a brownish white, sometimes of a yellowish white ; in young wood it approaches very nearly to white, with a silky lustre. The wood is sometimes beautifully curled. It is generally easy to work, being some- what softer than beech.—A cubic foot of sycamore, when dry, varies in weight from thirty-four to forty-two pounds. 147. The Ash is a tree that grows in most parts of Britain ; it is not however much employed in building, being in its nature too flexible and difficult to work. But it is much used by millwrights and wheelwrights, and for making implements of husbandry, as it is very elastic and tough, and generally 64 [PART I. PRELIMINARY REMARKS ON THE of a straight and even grain, well-calculated to sustain sudden shocks. The timber varies consider. ably in quality, according to the soil and situation in which it is grown. It is said that if ash is felled when full of sap, it is very liable to be affected by the worm; it soon rots when exposed to damp, or to alternate dryness and moisture, but in dry situations it is very durable. The colour of the wood of old trees is oak-brown, with a more veined appearance, the veins being darker than in oak; the wood is sometimes very beautifully figured; the wood of young trees is a brownish white with a shade of green ; the young wood is by much the more valuable.—A cubic foot of ash weighs fifty pounds. 148. Walnut is of several sorts. The walnut-tree was formerly much cultivated in England, as well for its timber as its fruit ; but since the importation of mahogany has increased so greatly, the cultivation of the walnut tree has been less attended to. It is now used chiefly for cabinet-work, for which pur- pose it is much esteemed ; it is likewise used for gunstocks, &c. It is less liable to be affected by worms than any other timber excepting cedar: but from its brittle and cross-grained texture is not well-adapted for the main timbers of a building, though formerly it was much used for floors, roofs, &c.—The weight of a cubic foot of walnut-tree, in a dry state, varies from forty to forty-eight pounds. 149. The tree from which the Teak wood or Indian oak is produced, is a native of the mountain- ous parts of the Malabar and Coromandel coasts, as well as of Java, Ceylon, and other parts of the East Indies. The teak tree is of rapid growth; the trunk grows erect to a vast height. The wood is light, easily worked, and though porous it is strong and durable ; it requires but little seasoning, and shrinks very little. Malabar teak is esteemed superior to any other in India. From some ex- periments that have been made, it would appear that it is superior in strength and stiffness to oak, though not quite equal to it in toughness. -The weight of a cubic foot of teak, when dry, varies from forty-one to fifty-three pounds. 150. The common Acacia, or locust-tree, is a native of the mountains of America, from Canada to Carolina. It is a tree of quick growth, and attains a considerable size. The wood is much val- ued for its durability ; it is adapted for all those purposes for which oak is applicable, and is an ex- cellent material for posts, stakes, and pales. It requires about as much labour to work it as ash does. - When seasoned, a cubic foot of it weighs from forty-nine to fifty-six pounds. 151. Of the Poplar tree there are five species common in England. The common white pop- lar ; the black poplar; the aspen, or trembling poplar ; the abele, or great white poplar; and the Lombardy poplar. The wood of most of these species makes very good flooring for bed-rooms, and places where there is not much wear: it is sufficiently strong for sustaining light weights, but is not fit for large timbers. This wood does not inflame readily, but rather moulders away than maintains any solid heat; it is likewise recommended for farm-buildings, cheese-rooms, &c., because neither mice nor mites will attack it. The weight of a cubic foot of the several species, when dry, is as follows: common white poplar thirty-three pounds; aspen and black poplar twenty-six pounds; abele thirty-two pounds ; Lombardy poplar twenty-four pounds. 152. Of the Larch tree there are three species: one European and two American. The European larch is a native of the Alps, of Switerland, Italy, Germany, and Siberia. It has been introduced, within the last sixty years, into Great Britain with much success, especially on the estates of the Duke of Athole in Perthshire. It is extremely durable in all situations, failing only where any other kind of wood would fail. It is not liable to be attacked by worms, nor does it inflame readily. For these valuable properties, larch has been celebrated from the time of Vitruvius, who regrets that it could not be easily transported to Rome, where such a species of timber would have been so valuable. Scamozzi likewise extols the larch for every purpose of building, and its value has been fully proved when grown in pro- per soils and situations in Great Britain. In countries where larch abounds it is frequently used to Sect. II.] 65 PRINCIPLES OF PRACTICAL ARCHITECTURE. cover buildings. In two or three years it becomes covered with resin, and as black as charcoal; this resin forms a kind of impenetrable varnish which effectually resists the weather. It is peculiarly adapted for flooring-boards in situations where there is much wear, and for staircases; when rubbed with oil, its colour is superior to that of the black oak staircases to be seen in some old mansions. It is likewise well-adapted for doors, shutters, and other finishings of the like description : when varnished it is of a beautiful colour, so that painting is not necessary. It is more difficult to work than Riga, or Memel timber, but the surface is better when once obtained. When perfectly dry it stands well, but it warps much in seasoning. The wood of the European larch is generally of a honey-yellow colour: the hard part of the annual rings of a redder cast; sometimes of a brownish white.—The weight of a cubic foot, when dry, varies from twenty-nine to forty pounds. 153. The Cedar, or Pinus cedrus of botanists, is a native of Mount Libanus, whence it has its name. The finest cedars in the time of Vitruvius grew in the Isle of Crete, the modern Candia, in Africa, Apollo at Utica, cedar was found 1,200 years old. According to Vitruvius the statue of Diana, in the famous temple at Ephesus, was of cedar, as well as the timber-work of the floor and of the ceil- ing of that edifice ; and he adds that the timber-work of the most celebrated temples of antiquity was in general executed in cedar, on account of its extreme durability. The cedar is a resinous wood, has a powerful odour, with a slightly bitter taste, and is not subject to the worm. It is straight grained and is easily worked, but readily splits. The weight of a cubic foot of seasoned cedar is from thirty and a half to thirty-eight pounds. 154. Strength of Timber. ]—From a variety of experiments made to ascertain the strength of differ- ent kinds of wood, it appears that there is but little difference in the strength of those varieties which are in most common use as building-materials. The resistance which timber of oak or pine offers to a force of extension, acting parallel to the direction of the fibres, is very nearly the same in each ; and on average, may be stated, according to the results of experiments, at 10,000 pounds on the square inch before rupture ensues. The resistance to rupture by compression is about one-half the resistance to rupture by a force of extension ; and may be taken, on an average for the two kinds of timber, at 4,000 pounds on the square inch. In practice, timber should not be exposed to a perma- nent strain greater than one-fifth of that which will cause rupture, when the force acts parallel to the direction of the fibres. To ascertain the limits of the resistance to a force acting in a perpendicular direction to the fibres, it will be necessary to examine the analytical expressions given by writers on the strength of materials ; and as these expressions are equally applicable to all materials, they are given, with their applications, in our Appendix, Article F. IV.-IRON. 155. Cast-iron.]—Within these few years, in consequence of the improvements that have been made in the manufacture of cast-iron, that material has been much used in the construction of the floors and roofs of buildings as well as in bridges. The safety iron affords against fire, and the circumstance of its not being subject, like timber, to sudden or rapid decay, render it, when judiciously employed, a most valuable material to the Builder, as well as to the Machinist and Engineer. A certain degree of prejudice has however hitherto been entertained, both by Architects and Builders, against an ex- tensive use of cast-iron in the construction of buildings, from the circumstance of failures having in some instances occurred where it has been employed. It may be observed, however, that these fail- ures have seldom if ever arisen from any defect in the material itself; but rather from injudicious attempts at extreme lightness of construction or from an over-strained and ill-judged economy. 66 [Part 6 PRELIMINARY REMARKS ON THE Numerous examples might be adduced of buildings—particularly in the warehouses and manufac- tories which have been of late years constructed in the manufacturing districts both of England and Scotland—wherein a judicious application of cast-iron, both in the floors and roofs, has been attended with the most satisfactory results. 156. There are several different modes of employing cast-iron in the construction of floors. Some- times it is applied partially, in conjunction with wood; the girders, or principal beams only being of cast-iron, in which case they are formed with proper sockets to receive the ends of the timber-joists. In other instances both the girders and binding-joists are of iron, the bridging and ceiling-joists being of wood. Occasionally the floor is formed by means of cast-iron beams, between which arches are turned, extending from beam to beam, and formed either with common bricks, paving bricks, or with the hollow cones already described. When the arches are turned with bricks, they are capable of sustaining either stone paving, a plaster floor, or slight joists and a wood floor. In some cases the brick arches are altogether dispensed with, and the floor is formed of thick Yorkshire landings låid on the iron beams, both faces of which being rubbed, 'they answer for the floor, and likewise for the ceiling of the apartment beneath. 157. Another advantageous application of cast-iron, and which appears to be daily increasing, is in lieu of timber bressummers and story posts, for supporting the fronts of houses wherein shops and large show-windows are required on the ground-floor. For this purpose beams of cast-iron are now fro- quently employed ; but a still preferable method is that of a framed cradle of wrought-iron, supported, if necessary, by cast-iron pillars. This method of construction adds much to the stability and security of the building; and in the event of fire, the injury sustained therefrom is much diminished; indeed, the advantage is so apparent that it is matter of surprise that, in any moderately well-built houses, the old plan of a timber bressummer and story posts should be adhered to. The chief objection to the use of cast-iron beams in such situations is, that in the event of fire and the beam becoming heated, it would be liable to crack, especially if subjected to the action of a stream of cold water. 158. In using cast-iron, attention must however be given to proportion the scantlings of the beams to the weight they are required to sustain, in the same manner as is necessary when timber is employed ; is of indefinite and almost infinite strength. It is likewise to be observed that a beam of uniform thickness is not equally strained in every part, it may therefore be reduced in size, so as to lessen the strain, and at the same time the expense of the material. 159. The following rules, necessary to be observed in using cast-iron in buildings, are given in the late Mr. Tredgold's • Practical Essay on the Strength of Cast-Iron.' In determining the scantlings of cast-iron beams, they should always be such that the beam will be capable of sustain- ing, if equally diffused over its entire surface, six times the weight that would, if applied in the centre of the beam, be sufficient to break it, and in calculating the weight of the load to be supported, the weight of the beam itself must always be included. To find the weight of a beam of cast-iron, multiply the area of the section in inches, by the length in feet and by 3-2, which will give the weight in pounds. If the weight a cast-iron bar will support be multiplied by 1:12 the product will be the weight a wrought-iron bar of the same size will support: the flexure of the wrought-iron bar will be found by multiplying the flexure of the cast-iron one by 0·86. If the depth of a cast-iron bar be multiplied by 0-937, the product will be the depth of a square bar of wrought-iron of equal stiffness. If the depth of a cast-iron beam be multiplied by 1.83, the product will be the depth of a square beam of oak of equal stiffness, and if by 1.71, the product will be the depth of a beam of yellow fir of equal stiffness. Sect. II.] 67 PRINCIPLES OF PRACTICAL ARCHITECTURE, load of a cast-iron one of the same size : the flexure of the oak beam will be found by multiplying the flexure of the cast- iron one by 2:8. To find the weight a beam of yellow fir will bear, multiply the weight a cast-iron one of the same size will sustain by 0.3, and to find its flexure multiply the flexure of a cast-iron one by 2.6. 160. When cast-iron is employed in buildings, the utmost care should be taken to render the iron in each casting of a uniform quality, because in iron of different qualities the shrinkage varies con- siderably, by which an unequal tension is caused amongst the parts of the metal which impairs its strength, and renders it liable to sudden and unexpected failure. To ensure the attainment of this most desirable object, it is necessary, especially in the lighter, description of castings, that they should be formed from the second fusion of the metal, and not from the blast-furnace, as is frequently done with the heavier kind of work. When the texture is not uniform, the surface of the casting is usually uneven where it ought to have been even. This unevenness, or the irregular swellings and hollows on the surface of a casting, may however arise either from the unequal shrinkage of the metal in cooling, or from the defective manner in which the moulding has been performed. 161. Cast-iron is divided into two principal varieties: the gray cast-iron, and white cast-iron. There exists a very marked difference between the properties of these two varieties. There are besides many intermediate varieties, which partake more or less of the properties of these two, as they approach, in their external appearances, nearer to the one or the other. 162. Gray cast-iron, when of a good quality, is slightly malleable in a cold state, and will yield readily to the action of the file, when the hard outside coating is removed. This variety is also sometimes termed soft gray cast-iron; it is softer and tougher than the white iron, and when broken, the surface of the fracture presents a granular structure; the colour is gray, and the lustre is what is termed metallic, resembling small brilliant particles of lead strewed over the surface. White cast- iron is very hard and brittle ; when recently broken, the surface of the fracture presents a distinctly marked crystalline structure, the colour is white, and lustre vitreous, or bearing a resemblance to the reflected light from an aggregation of small crystals. The gray iron is most suitable where strength is required, and the white where hardness is the principal requisite. 163. The colour and lustre presented by the surface of a recent fracture, are the best indications of the quality of iron. A uniform dark-gray colour, and high metallic lustre, are indications of the best and strongest. With the same colour, but less lustre, the iron will be found to be softer and weaker, and to crumble readily. Iron without lustre, of a dark and mottled colour, is the softest and weakest of the gray varieties. Iron of a light gray colour, and high metallic lustre, is usually very hard and tenacious. As the colour approaches to white, and the metallic lustre changes to vitreous, hardness and brittleness become more marked, until the extremes of a dull or grayish white colour, and a very high vitreous lustre are attained, which are the indications of the hardest and most brittle of the white variety. The quality of cast-iron may also be tested by striking a smart stroke with a hammer on the edge of a casting. If the blow produces a slight indentation, without any appearance of fracture, it shows that the iron is slightly malleable, and therefore of a good quality; if, on the contrary, the edge is broken, it indicates brittleness in the material, and a consequent want of strength. 164. The strength of cast-iron will depend not only on the quality of the melted metal, but also upon its temperature at the moment it is thrown into the mould, the position of the mould itself, and the manner in which the cooling is performed. All of these circumstances render it very difficult to judge of the quality of a casting from a bare inspection of its external characters: but, in general, if the exterior presents a uniform appearance, without any inequalities on the surface, it will be an in- dication of uniform strength throughout. Gray cast-iron offers a greater resistance to a force of ex- 68 [PART I. PRELIMINARY REMARKS ON THE tension than the white cast, in a ratio of nearly eight to five; but the white cast offers the greatest resistance to a compressive force. The strength of the gray cast-iron is very variable, depending on the quantity of carbon that is combined with it. Its resistance to rupture by a force of extension, in the best varieties, does not exceed 20,000 pounds on the square inch.30 It is found, moreover, that the strength of bars, cast in vertical moulds, is superior to those which are cast horizontally; and that large bars are stronger than small ones, in a ratio which is greater than the areas of their sections. The resistance of cast-iron to compression is very great. From experiments, it appears that it will bear a weight varying between 90,000 and 140,000 pounds on the square inch, before rupture takes place by compression. 165. Forged iron.)-The colour, lustre, and texture of a recent fracture, present also the most certain indications of the quality of forged iron. The fracture submitted to examination should be of bars at least one inch square; or if flat bars, they should be at least half an inch thick, otherwise the texture will be so greatly changed—arising from the greater elongation of the fibres in bars of smaller dimensions-as to present none of those distinctive differences observable in the fracture of large bars. The surface of a recent fracture of good iron presents a clear gray colour and high metallic lustre ; the texture is granular, and the grains have an elongated shape, and are pointed and slightly crooked at their ends, giving the idea of a powerful force having been employed to produce the fracture. When a bar, presenting these appearances, is hammered or drawn out into small bars, the surface of fracture of these bars will have a very marked fibrous appearance, the filaments being of a white colour, and very elongated. When the texture is either laminated or crystalline, it is an indication of some defect in the metal, arising either from the mixture of foreign ingredients, or from some neglect in the process of forging. 166. Burnt iron is of a clear gray colour, with a slight shade of blue, and of a slaty texture. It is soft and brittle. Cold short iron, or iron that cannot be hammered when cold without breaking, pre- sents nearly the same appearance as burnt iron, but its colour inclines to white. It is very hard and brittle. Hot short iron, or that which breaks under the hammer when heated, is of a dark colour without lustre. 167. The fibrous texture, which is only developed in small bars by hammering, is an inherent quality of good iron ; those varieties which are not susceptible of receiving this peculiar texture, are of an inferior quality, and should never be used for purposes requiring great strength : the fila- ments of these varieties are short, and the fracture is of a deep colour, between lead-gray and dark-gray. 168. The best forged iron presents two varieties; the hard and soft. The hard variety is very strong and ductile, but does not yield to the hammer so readily as the soft. It preserves its granular texture a long time under the action of the hammer, and only developes the fibrous texture when beaten or drawn out into small rods : its filaments then present a silver-white appearance. The soft variety is weaker than the hard; it yields easily to the hammer, and it commences to exhibit, under its action, the fibrous texture in tolerably large bars. The colour of the fibres is between a silver- white and lead-gray. 169. Iron may be naturally of a good quality, and still, from being badly refined, not present the appearances which are regarded as sure indications of its excellence. Generally, however, if the surface of fracture presents a texture partly crystalline and partly fibrous, or a fine granular texture, in which some of the grains seem pointed and crooked at the points, together with a light-gray 30 A mean of several experiments by Mr. G. Rennie, and others by Captain Brown, on cast-iron bars of various sizes, announts only to 18.000 pounds per square inch. See Barlow On the Strength of Timber and Iron.' Sect. II.) PRINCIPLES OF PRACTICAL ARCHITECTURE. colour without lustre, it will indicate natural good qualities, which require only careful refining to be fully developed. 170. The strength of forged iron is very variable, as it depends not only on the natural qualities of the metal, but also upon the care bestowed in forging, and the greater or less compression of its fibres when drawn or hammered into bars of different sizes. The resistance offered by the best kinds of forged iron to rupture by a force of extension may be stated, on an average, at 60,000 pounds on the square inch, for bars whose cross section is greater than one square inch. It has been found, that in comparing the relative strength of bars of different sizes, small bars are the strongest. Bars having a cross section of half a square inch, will require a force of extension equivalent to 70,000 pounds the square inch, to produce rupture; and bars having a cross section of a quarter of a square inch, will bear from 80,000 to 95,000 pounds on the square inch before rupture ensues.81 With equal areas, flat bars are stronger than square ones, and round bars are stronger than either the flat or square bars. There are no satisfactory experiments on the resistance of forged iron to rupture from a compressive force : indeed a knowledge of this resistance could be of little practical use, as forged iron is never used for vertical supports, cast-iron and wood being much superior for such purposes. 171. Wrought-iron is principally used in buildings for chimney-bars, ties, chain-bars, cramps, bolts and nuts, straps, cradle-bars, window-guards, light fences, staircase railing, shutter-bars, bolts, locks, and other light fittings. The subjoined table of the weights of the several descriptions of iron will be found useful in forming calculations or estimates. A cubic foot of cast-iron weighs 450 lbs. A cubic foot of wrought-iron weighs 481 lbs. A piece of iron twelve inches long of the respective qualities and sizes stated will weigh as follows. Wrought flat and bar iron. Rod iron Ibs. in. in. who feel 1} Cast-iron. in. by 1 ... 1 ... 17 ... 1 ... 1 ... 1 ... 1 ... 1 ... 2 in. diameter. inch ......... 1... ... ... in. 1 1 1 I 1 1 1 1 1 i 1 1 1 1 1 1 3.125 3.515 3.906 4.297 4.687 5.078 6.151 6.593 6.250 by ... ... ... ... ... ... cakes lbs. 3.340 3.757 4.175 4.592 5.010 5427 5.845 6.262 6:680 lbs. 1.473 2.623 3.320 4.099 4.959 5.902 6.926 8:033 10.492 Al lf ... ..... cole ......... 1 ... 2 V.-GLASS. 172. The kinds of glass in most general use in buildings, are the following: Crown, or Table-glas. ; German sheet, and Plate-glass. 173. Crown, or table-glass, which is the common sort of window-glass, is manufactured in differ- ent parts of the kingdom, particularly at Newcastle-upon-Tyne, Shields, and Bristol. The best English crown-glass is composed of 120 parts (by weight) white sand, 60 fine pearl-ash, 30 saltpetre, 2 borax, 1 arsenic. As it varies in quality, it is distinguished as best, second, and third. The best crown-glass is that which is made from the purest silex and alkali ; and should be clear, bright, and free from blemishes, specks, and wreaths. The second glass is not 31 These numbers are certainly overrated. The best iron from the rolls scarcely exceeds 25 tons per inch; and the larger bars are proportionally as strong as the smaller. A large bar, however, reduced by hammer to less section, gains strength. See Barlow 'On the Strength of Timber anu Irun.' 70 [Part I. PRELIMINARY REMARKS ON THE PRINCIPLES, &c. so free from defects, and the third is still inferior in quality. The colour is one of the most important considerations in estimating the quality of glass; and on this account chiefly, the glass manufactured in the vicinity of Newcastle is most esteemed in the market. Crown-glass is manufactured in pieces of a circular form, termed tables, each table having a thick knob in the centre, called the knot. These tables vary from four feet to five feet in diameter, but the most common size is from forty-eight to forty-nine inches in diameter. Crown glass, of a proper quality and substance, will weigh from ten to eleven ounces the square foot: varying somewhat according to the materials of which it is composed. The tables are packed in crates: each crate of best glass containing twelve tables; a crate of seconds, fifteen tables; and a crate of thirds eighteen tables. There is an inferior kind of glass, termed Green glass, which is used for ordinary purposes, such as glazing the windows of cottages, garden-lights, hot houses, &c., and which is much cheaper than the descriptions before noticed. Its green colour is owing to the presence of iron in the imper- fectly purified alkaline substances used in its manufacture. 174. German Sheet-glass is of an excellent quality, particularly as respects colour ; but from the manner in which it is manufactured, one side-which of course is placed outermost in the sash-has an uneven, and consequently a very unpleasant appearance. It was formerly, however, much in use ; 80 mm flatting of crown glass, together with the reduction that has been made in the price of plate-glass, it is not much in request now in this country. 175. Plate-glass far excels all other sorts in quality and beauty. It is nearly colourless, and is cast in plates of sufficient thickness to admit of its being polished with the greatest accuracy and delicacy. Plates of almost any size, from 144 to 11,000 square inches, may be obtained. The principal manu- factory of this beautiful substance, in England, is near Prescot in Lancashire. The following pro- portions of materials are used in the manufacture of plate-glass : Lynn sand, . . . . . . 720 parts. Alkaline salt, . . . . . . 450 - Slaked lime, . Nitre, . . . . . . . 25 - Broken plate-glass, . . . . . 425 1700 176. Window glass is frequently stained of various colours, -as red, orange, yellow, blue, purple, and green. The three former colours are produced by the application of chemical solutions to the glass, which is afterwards subjected to great heat in kilns made for the purpose, by which means the colours are burnt in, and become as permanent as the glass itself. Glass of the three latter colours is usually made from the pot. 177. A very beautiful species of glass has been produced, denominated Embossed glass. It is formed from the better desci'ption of crown-glass, first made perfectly flat, and ground, and then orna- mented with a multiplicity of devices, such as scrolls, fretts, flowers, and other decorations, by means of a chemical application which is burnt into the glass, and produces a most rich, brilliant, and beautiful effect, not unlike matted silver. It may moreover be executed of any of the above-mentioned colours. 178. Glaziers have likewise less expensive methods of giving glass the effect of opacity, without lessening materially its power of admitting light. The most common way is by rubbing the polish off the surface with sand and water, or with emery. The square of glass being first bedded in plaster, is rubbed till the polish is entirely removed ; it is afterwards washed and dried, and is then ready for use. It is distinguished in the trade by the appellation of Ground glass. • The present Government duty on crown-glass is 735. 60. per cwt.; on broad-glass, 30s. per cwt.; on plate-glass 603. Part Seconi. PRACTICAL ARCHITECTURE, OR THE APPLICATION OF GEOMETRY TO THE BUILDING ARTS. PART II. PRACTICAL ARCHITECTURE, OR THE APPLICATION OF GEOMETRY TO THE BUILDING ARTS. The construction of Architectural works is divided into several distinct Arts; and of those to which the principles of Geometrical science apply with most advantage we now proceed to treat. We shall begin with MASONRY and BRICKLAYING ; then proceed to CARPENTRY and JOINERY ; and end with an explanation of the ORDERS used in Architecture. SECTION I. MASONRY Definitions, 1.- I. Walls, 2—7.- II. Vaults, Domes, and Groins, 8–55. III. Stone-cutting, 56. Raking- mouldings, 57.__ Arches, 58.-— Oblique arches, 59.- An arch in a circular wall, 60.- Construction of Groins, 61.- Gothic groin, 62.- Ribbed-groins, 63. — Construction of spherical domes, 64.- Niches in straight walls, 65–70.— Niches in circular walls, 71.- Architraves over columns, 72. Description of the Lewis, 73. Stairs, 74, 75. -Steps over an area, 76. 1. Definitions. ]—Masonry is the art of cutting stones, and building them into masses, for the pur- poses of Architecture. Stones prepared from the quarry, for building, are generally of a rectangular form. I.-WALLS. 2. A wall is a mass composed of stones, generally joined with cement, and so arranged that a plumb-line from any point of the surface may not fall without the solid. 3. In modern house-building, the bedding-joints of the squared stones or bricks have always a hori. zontal position. In piers, quays, and bridges, the bedding-joints are generally laid at a right angle with the latter of the outside faces; and these latter are sometimes worked in the reticulated manner, that is, the courses are laid at an angle of 45° with the horizon. 4. The footings in the foundations of stone walls should be composed of large square stones, all of the same course, being of an equal thickness. If the foundations are made to taper much, the super- structure will depend on the back parts of the lowermost stones, and unless these are very truly worked and laid, it will therefore be liable to give way. Where the direction of a wall is up the face of steep ground, cut into steps, the footings must be bedded with great care; and all the upright joints of an upper footing-stone, should break joint, or fall upon the middle of the stones below. In laying the foundations of thin walls, where stones of proper size can be had, each course of the footing should consist of stones reaching the entire thickness of the walls, with the proper projections. But in thicker 74 [Part II. PRACTICAL ARCHITECTURE. walls, when only a part of the stones can be had of sufficient length, then every alternate stone may be laid quite across the wall; the interval consisting of two stones in breadth, after the manner of Flemish bond in brickwork. If even these bond-stones cannot be procured, then every alternate stone may be in length two-thirds of the thickness on one side the wall, and on the other side a stone of one-third of the same breadth may be placed, and the order reversed in the next course, which will form a sufficiently strong bond. In broader foundations, where stones cannot be procured equal to two-thirds of the breadth, they may be built with the joints of the upper course of each footing rest- ing nearly on the middle of the stones in the course below it. 5. When the superstructure of a wall consists of unhewn stone laid in mortar, it is called a Rubble wall, which may be either coursed or uncoursed ; of these the latter is very common in ordinary build- ings. The greater part of the stones are used as they come from the quarry; some have a slight hammer dressing. This sort of wall is very inconvenient for receiving bond-timbers; but if bond- timbers be preferred to plugging, the backing must be levelled in every height where the bond-timbers are required. When the stones are very small, and of irregular shape, it will add to their stability to work in slender laths or old hoop-iron ; care must be taken to dispose the longest sort of the stones so as to create the greatest possible bond; and the work must be carried very regularly up round every wall of the building, and with not too much rapidity, in order that it may have time to indurate or set, to a certain degree of firmness before a great weight comes upon it; and on no account should thin stones be set on edge, which thoughtless and unprincipled workmen are but too much inclined to, either from haste, or for the purpose of making a smooth outside facing. 6. Coursed rubble, where proper stones can be obtained, is much preferable to the other, and is more favourable to the disposition of bond-timbers. The courses are of various thicknesses, adjusted by means of a gauge or sizing-rule; and the stones are either hammered, dressed, or axed. 7. Walls faced with squared stones, and backed with rubble or brick, are called ashler work, and the stones themselves are called ashlers. The average size of each ashler is from 24 to 36 inches in length horizontally, 8 to 12 inches in breadth or depth, and 10 to 16 inches in height. The best figure for the stones of an ashler-facing, is that of a truncated wedge, that is, thinner at one end than the other in the thickness of the wall, so that those in one course may form in their back parts indentations like the teeth of a saw, the next course having its indentations varied from that below it; the whole is therefore toothed or united with the rubble backing, much more effectually than if the backs of the ashlers were parallel with the face. Bond-stones should be introduced in every course of ashler facing: they should be in quantity equal to one-sixth of the face of the wall, and of a length to reach at least one foot into the back, but the more the better. Every bond-stone should, if possible, be placed in the middle between those in the course below. When the jambs of piers are coursed with ashler, or when the jambs are of one entire height, every alternate stone next the aper- ture in the former case, and next to the jambs in the latter, should bond through the wall; and every other stone should be placed lengthwise, in each return of an angle, not less than the average length of an ashler. Bond-stone should have no taper in their beds, nor should their ends, or the ends of the return stones, be ever less than 12 inches. Closers should never be admitted, unless they bond at least two-thirds of the thickness of the wall. All upright joints should be square or at a right angle with the face for about two inches back, after which they may widen a little towards the back. The upper and lower beds of every stone should be quite level or parallel to each other for their whole breadth. All the joints, for the distance of about one inch from the face, should be cemented with fine mortar, or with a mixture of oil-putty and white-lead; the former is practised at Edinburgh, the latter at Glasgow; at the latter place the joints of the polished ashler work are uncommonly fine and accu- rate. The remainder of the ashler, and all the rubble, should be laid in good lime mortar ; that for UU u SECT. I.] MASONRY 75 the rubble, should be made with coarser sand. All the stones should be laid in their natural beds. Wall plates should always be placed on a number of bond-stones, to which they may be either joggled or fixed by iron cramps. II.-VAULTS, DOMES, AND GROINS. 8. A vault in masonry is a mass of stones overtopping an area of a given boundary, and supported by one or more walls, or pillars, placed without the boundary of that area ; the stones being so arranged that they mutually balance and support one another. 9. The surface of the vault which is opposite to the area is called the soffit, or intrados, or vaulted surface; and is generally concave towards the area, or composed of surfaces that are concave, gen- erally consisting of the portion of a cylinder, cylindroid, or sphere, but never exceeding half the solid. The exterior or convex curve of the superior surface of an arch, is called the extrados. 10. As it is the proper adjustment of the weight of any part of a vault to the surrounding mass which prevents that part from falling, it becomes necessary, in the act of construction, to build it upon a mould until the whole be closed. The mould used for this purpose is called a centre. 11. The part of the top-surface of a wall, or pillar, on which the first stones of a vault rest, is called the spring of the vault, or bed of the vault. 12. If the face of a wall, or pillar, and the soffit of the vault, meet together in the bed of that vault, the line of concourse is called the spring line of the vault. 13. A complex vault formed by the intersection of several archoids, whether of the same or of different heights, is denominated a groin vault. 14. The intrados of every vault used in Architecture may be considered the concave surface of some geometrical solid, or the surface of a solid compounded of one or more geometrical solids. 15. When solids of revolution are employed in vaulting, the axis is generally either perpendicular or parallel to the horizon. 16. The surface of a cylinder, or cylindroid, can be employed in vaulting only when the axis is parallel or inclined to the horizon ; for when it is perpendicular, the concave surface of the cylinder, or cylindroid, is only the interior surface of a wall upon a circular or elliptical plan. 17. A conic surface may form the intrados of a vault, whether the axis be, perpendicular or parallel to the horizon. Such surfaces are, however, seldom used, except it be in the heads of apertures of doors and windows, with the axis horizontal; or in kilns where the axis is vertical. 18. When the surface of a cylindroid is employed in vaulting, one of the axes of the elliptic ends is always in a horizontal, and the other in a vertical plane. The position of the axes of a cylinder, or cylindroid, is most generally horizontal; the inclined position occurring very rarely. 19. If the surface of a vault be cylindric, or cylindroidic, a portion of the surface of the cylinder, or cylindroid, not exceeding the half, is generally employed. The edges which terminate the surface, and which are parallel to the axis, are called the springing lines of that surface. 20. When a spheric surface is employed in vaulting only, a segment not exceeding the half is used. 21. The surface of every conoid may be used in vaulting; and, when used, the position of the axis is perpendicular to the horizon. 22. When the surface of a vault is ellipsoidal, the shorter axis has generally a vertical position. 23. A cylindrical, or cradle vault, consists of a plain arch ; the figure of whose extrados is a portion of a cylindrical surface, terminating on the top of the walls which support it, in a horizontal plane parallel to the axis of the cylinder. 24. A cylindroidal vault consists also of a plain arch, the figure of whose extrados springs from a hori- 76. [PART II. PRACTICAL ARCHITECTURE. zontal plane, but its section perpendicular to those lines is everywhere a semi-ellipsis, equal and similar throughout, having its base that of either axis ; otherwise, it is sometimes the segment of an ellipsis less than a semi-ellipsis, having an ordinate parallel to the axis for its base. 25. A vault, rising from a circular, elliptical, or polygonal plan, with a concavity within and a con- vexity without, so that all horizontal sections of the intrados may be of similar figures, having their centres in the same vertical line, or common axis, is called a dome. 26. Various names are given to domes, according to the figure of their plan, as polygonal, circular, or elliptic. Circular domes may be either spherical, spheroidal, ellipsoidal, hyperbolical, parabolical, &c. Such as rise higher than the radius of the base are called sumnounted domes ; and such as are below this altitude are termed diminished, or surbased domes. If a dome be a portion of a sphere, that is, if its base be a circle, and its vertical section through the centre of its base the segment of a circle, it is called a cupola. A spherical dome, or cupola, may be intersected by a cylindric vaulting in any direction; and the intersection will always be circular, provided the axis of the oylinder tend to the centre of the sphere, because every section of a sphere made by a plane is a circle, as is also every section of a right cylinder perpendicular to the axis. Suppose, therefore, the sphere to be cut by a plane forming a section equal to that of the cylinder, and the two sections applied together, the right line drawn from the centre of the circle, which is the section of the sphere, to the centre of such sphere, will be perpendicular to the plane of this section ; and, since the axis of the cylinder is also perpen- dicular to the same plane, it will be in the same right line with the remainder of the radius of the sphere. From this we deduce, that, when the axis of a cylindrical vaulting is horizontal, and tends to that of a spherical vault, their intersection must be in the circumference of a circle, whose plane will be perpendicular to the horizon ; and hence those beautiful sphero-cylindrical groins, so greatly and justly admired in our principal buildings. 27. Upon this principle, any building that has a polygonal base may be made to terminate a circle, and sustain a cupola, or cylindric wall; for, if the tops of the side walls of the polygon be brought to a level, and equal segments of circles, whether semicircles or less portions, be raised on the top, meet- ing in the lines of intersection of the sides of the polygon, and if the angular spaces between the circular-headed walls be made good to the level of the summit of the arches, so as to coincide with the circumference of a great circle of the sphere, they will terminate in a ring at the level of the summit of the arches, and be portions of the sphere, called by our workmen spandrels, and by the French pendentives. On the ring so formed, a cornice is usually laid, on which the cylindric wall or dome is raised. 28. The plans of apartments intended to be covered with cupolas, are, in general, either square or octangular. The pendentives are likewise commonly equal in number to the angles of the walls ; but this is not essential, because, in polygonal plans, arches may be thrown across the angles, to double the number of the sides of the polygon, still preserving the equal sides. Over the middle of the walls, equal and similar arches may be built, that shall touch those across the angles at the bottom, and have their tops in the same level. Or, instead of walls, piers may be carried to an ade- quate height upon each angle of the polygon, and return upon either side of it. Archivolts may then be turned over every two adjacent piers, and the spandrels be filled in to the level of the summits of the arches or archivolts, as before, and the termination will be a circle on the inside, as already stated. 29. There do not appear any instances among the Roman buildings, of pendentives or spandrels being supported by four pillars, or by quadrangular or polygonal walls, and which support themselves on a spherical dome or a cylindrical wall. Pendentives rising from pillars, and surmounted by a dome, were originally introduced in the celebrated church of St. Sophia at Constantinople. St. Paul's, and $t. Stephen's Walbrook, London, are beautiful specimens of this sort. Sect. I.] 77 MASONRY. 30. When two or more plain vaults penetrate or intersect each other, with their summits in the same horizontal plane or level, the figure of the intrados, formed by the several branches of the vaults, is called a groin. In other words, a groin is a vault in which two geometrical solids may be trans- versely applied, one after another, so that a portion of the groin will have been in contact with the first solid, and the remainder with the second when the first is removed, and that the summit of the one may intersect that of the other. This definition will be found almost universal, as it applies not only to plain vaults intersecting each other, but also to such as are annular, or in the form of semi- cylindric rings, intersected by cylindric or cylindroidal plain vaults, whose axes tend to that of the annuli; but it does not include the species used in the chapel of Henry VII. at Westminster, and in King's College chapel at Cambridge, where, instead of the horizontal sections of the curved surfaces presenting exterior right angles, as is generally the case, they present convex arches of circles. 31. A property common to every kind of groins, is, that the several branches intersect and form arches of equal height upon each enclosing wall, the perpendicular surface of which is continued on both sides, till intercepted by the intrados of the arches; consequently, the upright of each wall is equal in height to that of the common apex of the arches. This forms a striking and characteristic difference between domes and groins: the latter is a branched vault, terminating in each branch against the enclosing wall; whereas the former is a vault without branches, having its curves springing from all points of the wall or walls around the bottom of its circumference, whether upon a polygonal, circular, or elliptical plan. Another property of the dome is, that all its horizontal sections are similar figures, whether made by the exterior or interior surface. 32. Groins are variously denominated, according to the surfaces of the geometrical bodies which form the simple vault, viz. When the axes of the simple vaults are in two vertical planes, crossing at right angles to each other, they form a rectangular groin. 33. When three or more simple vaults of one common height pierce each other, and form a com- plex vault, in such a manner that if the surfaces of the several solids of which each is formed were respectively applied, one at a time, to succeeding portions of the surface of the complex vault, each portion of the complex vault would come in contact with certain corresponding portions of the surface of each of the solids : the complex surface thus generated is called a multangular groin. 34. When the several axes of the simple vaults form equal angles around the same point, and when each of the vaults are of the same width, the surface of the solid is called an equiangular groin. 35. When the breadths of the cross vaults, or openings of a groined vault are equal, the groin is said to be equilateral. 36. The species of every groin, formed by the intersection of two vaults of unequal width, is denoted by two preceding words; the first ending in o, indicates the simple vault of the greater width, and the second terminating with ic, denotes the simple vault of the less width. 37. Thus, a cylindro-cylindric groin, is a groin in which the cylindric portion is wider than the cylindroid. In this species of groin, the section of the cylindroidic part has its lesser axis horizontally posited. 38. A cylindroido-cylindric groin is one which has the greater axis of the cylindroidic part horizon- tally posited. 39. When the two portions of a groin are of equal width, the groin is either cylindric or cylindroidic, accordingly as the portions are both cylinders or both cylindroids. 40. When one vault pierces another of less height, the angle formed at the intersection is called an arch, and its species is indicated by the two preceding words, as in groins ; that ending in o, in- dicates the simple vault of the greater height, and the other ending in ic, denominates that of the lesser height. WA . 78 [PART IL PRACTICAL ARCHITECTURE. CD 41. A cylindro-cylindric arch, is an arch made of two cylindric portions ; but the portion indicated by the word ending in o, is higher than the portion ending in ic. 42. A sphero-cylindric arch is that in which the spheric portion surmounts the cylindric, portion, or that the principal or master vault is a sphere, and the other which perforates it a cylinder. 43. A cylindro-spheric arch is one in which the higher arch is a portion of a cylinder, and the other a portion of a sphere, as must be the case with a spherical headed niche in a cylindric wall, or in a cylindric vault. 44. It does not appear that the Greeks made use of vaults or arches prior to the Roman conquest; but from that period they employed not only plain vaults with cylindrical intradosses, but also quadri- lateral equal-pitched groined vaults, with cylindrical or cylindroidal intradosses, or a mixture of both, as may be observed over the passages of the theatres and gymnasia. 45. The dome was invented by the Romans or Etrurians. The Pantheon, which is generally reputed to have been built by Agrippa, son-in-law to Augustus, (though some writers maintain that he only added the portico,) is one of the earliest remaining structures with arches : it consists of a hemispherical cavity enriched with coffers, and terminates upwards in an aperture called the eye of the dome. The exterior side rises from degrees or steps, placed in a sloping direction, and forming nearly a tangent to the several internal groins of the steps, presenting to the eye a truncated segment of a sphere, much less than a hemisphere. This forms the general character of the Roman dome. 46. Domes were of very frequent use among the Romans, as may be deduced from their groins, and the remains of their ancient edifices. But ancient Greece does not furnish a single example of a dome, if we except that which covers the monument of Lysicrates, but which being only of a single stone, may rather be deemed a lintel than a built dome. Vitruvius abserves, (Book iii. chap. 3,) that the floors of temples were frequently supported by vaults, and (Book v. chap. 1,) that the roofs of basilicas were vaulted in the tortoise form, which he distinguished by the term testudo. This mode of vaulting is very flat, and has four curved sides springing from the four walls, approaching nearly to the form of a flat dome upon a rectangular plan. From the remains of Roman buildings we also observe that their ceilings were vaulted over their apartments, as may be seen in the chapels of the temple of Peace, and the side apartments of Dioclesian's baths, which are furnished with vaults having cylin- drical intradosses; while the great rectangular apartments, in both these edifices, are vaulted with groins. Nor is it a little remarkable that these groins are not formed by the intradosses of the vaults of the chapel, whose summits rise but a small distance above the springing of the middle groins. The piers between the chapels also have small arcades, the summits of which are considerably below the cylindrical intradosses of the side vaults; a mode to be discovered in many other buildings. 47. The Romans used annular vaults, as in the temple of Bacchus, where, as in the temple of Peace and the baths of Dioclesian, the summits of the arcades, supporting the cylindric wall and dome of the central apartment, do not intersect the annular intrados, but this convex side of the cylindric wall which supports it, consequently they do not form groius. 48. The intradosses of the Roman domes, as we have already hinted, are of a semicircular section, as in the Pantheon, and the temple of Bacchus at Rome, the temple of Jupiter, and the vestibule of the palace of Dioclesian at Spalatro ; while the vertical section of the extrados through the axis, ex- hibits a much less segment, as may be seen in the first and last of the examples quoted. The latter observation, however, only applies to edifices built prior to the reign of the emperor Justinian; after which period, from the completion of the dome of St. Sophia at Constantinople, to the finishing of St. Paul's cathedral at London, the domes are all of the surmounted kind, and approach in a certain degree to the proportions of spires or towers, so much affected in the middle ages. Since the labours and taste of Mr. Stewart and others have revived the legitimate Grecian architecture, the contour of SECT. I.] , MASONRY 79 the ancient Roman dome has been also restored, especially in cases where the structure is ornamented with any of the orders. 49. In the interior of the large towers of our Gothic cathedrals, over the intersections of the cross, we find domes, rising from a square base, generally pierced with two windows over each wall, and forming beautiful groins, by their intersection with the interior domic ceiling. 50. Though the equilibrium and pressure of domes are very different from those of ordinary arches, yet they have some common properties, as will appear from the following comparison : if, in their cylindrical or cylindroidal vaulting of uniform thickness, the tangent to the arch at the bottom be perpendicular to the horizon, the vault cannot stand ; neither can it be built with a concave contour in the whole, or in any part; and to make the arch in equilibrium, whether its section be circular or elliptical, supposing the intrados to be given, the extremes must be loaded vastly high, between the extrados of the curve, which runs upward, and the tangent to the arch, which is an asymptote, ris- ing vertically from each foot or extreme of the arch. In like manner, in thin domical vaulting of equal thickness, if the curved surface rise perpendicular from the base, the bottom will burst, let the contour be as it may. 51. Notwithstanding this agreement, dome-vaulting differs in other particulars very essentially from the common sort; for instance, to bring the figure of a dome into proper equilibrium, after the convexity has been carried to its full extent of equilibrium around, and equidistant from the summit 2 courses is less than that of the exterior, the stones cannot fall inwardly, whatever be the outward pressure, unless they be squeezed into a less compass, which is supposed impossible ; consequently, they must be crushed to powder before such a vault can give way. For the same reason, a vault may be constructed, that shall be convex within, and concave outwardly, and yet be sufficiently firm. The strongest form, however, of a circular vault, intended to bear a load at the top, is that of a truncated cone, similar to Sir Christopher Wren's contrivance for supporting the stone lanthorn and exterior dome of St. Paul's. In this kind of vault, the pressure is communicated in the sloping right line of the sides of the cone perpendicular to the joints, consequently the conic sides have no tendency to bend to one side more than to the other, unless it be from the gravity of the materials tending towards the axis, which is counteracted by the abutting vertical joints ;-a form so strong, as to be adequate to sustain or repel any force acting on its summit that we can possibly conceive. 52. In dome-vaulting, on the contrary, the contour being convex, there is a certain load, which, if laid on the apex of the dome, must cause it to burst outwardly. The power of this load will be greater or less, according to the approximation of the contour towards, or its recession from, the chords of the arches of the two sides, or to a conic vaulting on the same base, carried up to the same altitude, and ending in the same circular course. In exemplification of this, if we begin at the key- stone, and proceed downwards, from course to course, supposing a horizontal line to be a tangent at the vertex, we shall discover that every successive coursing-joint may be made to slope so much, and consequently, the pressure of the archstones of any course towards the axis may be so great, as to be more than sufficient to resist the weight of all the part above; hence it is evident that a certain degree of curvature may be given to the contour, which will be just sufficient to prevent the stones, in any succeeding course, from being forced outwardly. 53. A circular vault, thus balanced, is what is called an equilibrated dome; but it is the weakest of all vaults, between that of its own contour, and that of a cone upon the same base, rising to the same height, and ending in a key-stone, or finished with an equal circular course. 54. From these data we may conclude, that the equilibrated dome has the boldest contour, but is only the limit of an indefinite number of inscribed circular vaults, all stronger than itself. ik , old at 1pts. I reprehend other. A ENTIDAK T o rino V eri - harau " ' -- * Xorxe L unar 80 [PART II. PRACTICAL ARCHITECTURE. 55. In other respects circular vaulting differs from the straight, in being built with courses in circular rings ; so that the stones in each course being of equal length, and pressing equally towards the axis, cannot slide inwardly. Hence circular vaults may be left open at the top; and even the equilibrated dome may carry a lantern of equal gravity with the part that would have been necessary to complete the whole. But domes of a more flat contour may carry more, according as they approach nearer to a cone, as already remarked ; and those circular vaults that are either straight or concave on the sides, may be loaded without limit, and can never fail till the materials are crushed, provided they be hooped at the bottom. III. STONE-CUTTING. [Plate I.] 56. In stone-cutting, a narrow surface formed by a chisel, or point, on the surface of a stone, so as to coincide with a straight edge, is called a draught. PROBLEM I. To form the face of a stone into a Plane surface. Run a draught first along and close to an edge of the stone ; and then another along and close to the adjacent edge, meeting the former draught. Run a third draught along the diagonal, so as to meet the other two, and form a triangle ; then run a fourth draught along the other diagonal, so as to pass through the meeting of the first two, and through the first diagonal draught. Then reduce the protuberant parts between the draughts so that every part of the surface may coincide with the straight edge, and be in a plane with the former draughts. The reason of this is obvious, since the three sides of a triangle are always in one plane. Plate I. Fig. 1, exhibits the method of forming the face of a stone to a plane surface. No. 1, shows the first two draughts along the edges AD and AB; No. 2, the first two draughts AD, AB, and the first diagonal draught BD, connecting the extremities of the two first draughts ;'and No. 3, shows all the four draughts AD, AB, AC, and BD. It would also be convenient to form two other draughts along the edges CD and CB, in order to reduce the surface to a plane in the easiest manner, and to prevent the edge of the stone from break- ing within the surface. PROBLEM II. To form a Winding surface. Run four draughts along the edges all in one plane, by the preceding Problem; and, having wrought the other sides to a square, set AH, Plate I., Fig. 2, on the perpendicular edge equal to the quantity of winding, and draw HD and HB for the draught-lines. Draw the equidistant lines li, mj, n k, parallel to AH; 1p, m q, nr, parallel to AD; and join ip, j q, k r; then HD, i p, jq, k r are the draught-lines. If the surface be now reduced, so that a straight edge parallel to the plane AFGB may everywhere coincide with the surface, and at the same time with the cross draughts, it will be in the form re- quired: as in Fig. 2, No. 2. n . STONE CUTTING. FORMATION OF PLANE, WINDING & SPHERICAL SURFACES. PLATE I. Fig. 1. N92. Mig L 103, Frig. 1. N.. TÓN Fig.3. N:3. 761,7 Fig. 2. NO 2. -- - ----- Wi -- ----- ------- ***** More *promenament .. Fig.3 NO 1. Fig. 3.V.2. - Fig. 3. N°4. Fig. 3 NO7. Fig. 3. N.6. Fig. 3. 1.5, INSTITUT // /. / / / 21/ /// / 7 / / / / / / SA SC Z Whi / 11/IIIIIIIIII SUZ ZZZZ 12777 III 7: : 2 Tui Will // / 7 / /17 %2 / 2 Pl Milli ! SITE 21 Kui / / / / / / IZ 771 HALAM 1/1171/117/ MZN- CARE . 777V /11 II 2 77. 11171/11 777 2 7// 27 WETUDU / TI IPX/ 11/12 . Will 16. . // . PNicholson. Armstrong A Frillarton&C London & Edinburgh STONE CUTTING. . RAKING MOULDINGS. Fig.1. . Fig.4. Fig. 2. bi Fig. 3. Et Pa -. my -. ---- mitt Fig. 5. NO2. 11. Invented by P.A. -- -- - - Drzwn In P. Nicholsom, A Fullarton&COL mdon&Edinburgh ech SECT. I.] 81 MASONRY PROBLEM III. To reduce the face of a stone to a Spherical surface. The first thing is to curve the edge of a rule to the curvature of the sphere. Before we proceed further, therefore, we may show how this is done ; supposing the stone to be four feet in its longest dimension; and the radius of curvature thirty feet. Find the versed sine of an arc, whose chord is 4 feet, to a circle of which the radius is 30 feet, by Problem lxii. PRACTICAL GEOMETRY. This will be found to be very nearly of an inch ; and the segment of the arc may be described by Problem liii. PRACTICAL GEOMETRY. Suppose Plate I. Fig. 3, Nos. 6 and 7, to be the rule thus formed; No. 5 to be a portion of the ato drawn by the edge of the rule ; and No. 3 to be the upper face of the stone squared. Let the stone of which the surface is to be spherical be first squared as in No. 1, and let the chord a b, No. 5, of the arc a lb, be equal in length to the diagonal AC or BD, No. 3. Bisect the chord a b, No. 5, by the perpendicular c l. In No. 1 let a b c d be a face of a stone ; and on the perpendicular edges make a f, bg, ch, de, each equal to cl, the versed sine of No. 5. In No. 1 bisect the sides a b, 6 c, cd, d a, in the points 1, m, n, k. Join I n, k m, meeting in i, and draw lp, mq, nr, k o, on the perpendicular faces parallel to the edges a f, bg, ch, de. In the chord a b, No. 5, make c g and c f each equal to il, or in. Draw giand f k, perpendicular to a b, meeting the arc in j and k. In No. 1 make lp and n r each equal to the difference between gi or f k, and cl, No. 5. Again, in the chord a b, No. 5, make c e and c d each equal to i k or i m. Draw e h and d i par- allel to cl, meeting the arc in h and i. In No. 1 make h o and m g each equal to the difference between h e or di, and cl, No. 5. Then, sinking the draughts in No. 1 from the lines a c, b d, k m, I n, by the rule No. 6, we shall be enabled to form the convex surface of the stone. No. 4 is the same as No. 5, except that the chords a c, a d, are set off from the end a. From what has now been described of working the convex face of a stone, No. 1, the process of forming a concave face will be easily understood. The rule No. 7 applies in this case; and the mode of finding the draught-lines is shown in No. 2. vi 57. RAKING MOULDINGS. [Plate II.) Fig. 1, Plate II. exhibits the elevation of a pediment with modillions. The construction of a modillion, as B, is shown in Fig. 3. Fig. 2 shows the method of tracing the section which is perpendicular to the raking or inclined direction of the moulding, the horizontal one being given. Fig. 4 shows the same for a Grecian ovolo. Draw AB perpendicular to the horizontal moulding, and AP perpendicular to AB. Draw PM parallel to AB, meeting the curve in M, Mm parallel to the raking-line, and aC perpendicular to it. Make a p equal to AP, and draw pm parallel to aC; then will m be a point in the section of the moulding. In the same manner as we have found the point m, we may find as many more points as we please. This description applies to Figs. 2, 3, and 4. 82 [Part II. PRACTICAL ARCHITECTURE. 27 Fig. 5 shows a method of tracing the angle-munnion of a Gothic window, one of the intermediate munnions in the front being given. Let AB be the horizontal line of front, and ABC the angle; and let AD, perpendicular to AB, be the central line of the section of the given munnion, and BE bisect the angle ABC. In BE, or BE produced, take any point a, and draw a p perpendicular to a E. In the curve No. 1 take any point M, and draw Mm parallel, and MP perpendicular to AB, meeting AB in P. Make a p equal to AP, and draw p m parallel to BE, and m will be a point in the section of the angle-munnion. In the same manner as the point m has been found, we may find as many more points in the section No. 2 as we please. 58. ARCHES. [Plate III.) The construction of an arch for a Gothic window, with a reveal, and label-moulding, is shown in Plate III. No. 1 is an elevation of the inside of the arch; No. 2 a plan with the splay at the jambs, and as the arch rises the splay decreases so as to be level at the crown. No. 3 is the splayed part of the first arch stone as ABC; No. 4, the second stone DEF; No. 5, the third GHI; and so on. No. 10 is a plan of one of the beds with two double plugs; No. 11 is another; and No. 12 a section of the label. S 59. OBLIQUE ARCHES. [Plate IV.] When an arch intersects a wall obliquely, as in Plate IV., and we have given the plan of the jambs HAC, and DEFG ; k k ashlar joining the arch ; PKLQ the elevation of the outer face of the arch, and RMNS the inner face on the elevation; Nos. 1, 2, 3, 4, 5, on the elevation, gives the projection of each joint or bed from the face; and by making a b equal to the breadth of the wall, and transferring b c, e f, &c. to Nos. 1, 2, 3, 4, 5, in b c, b e, bg, the bevels of the beds from the face are found; as d ac, da e, dag, &c., the bevel being applied across the bed from the face. No. 11 is the arch-mould applied on the face ; No. 12, a shifting-stock applied from the face across the bed, which gives the exact bevel from K to M on elevation ; No. 13 is the same as No. 1. If 909, op, mn, the bevel of the joint, be continued down T and V, it will be found to be square from the face at the crown. 60. AN ARCH IN A CIRCULAR WALL. [Plate V.] In Plate V. No. 1 is an elevation of the arch ; and No. 2 a plan of the bottom-bed from A to B. From p to o is what it gains on the circle, from the bottom-bed to the joint f; from n to m is the projection from the bottom-bed to g on the joint; from o to m is equal to the projection of f g at the joint; and a b c d shows what it gains on the circle as it goes round. By dropping a perpendicular line from the bottom-bed to p, and from f to o, from the bottom-bed inside to n; also from g to m; and then squaring from those lines, where they intersect with the plan, we obtain the proper curves for all the different moulds. Nos. 3 and 4 are done in the same way. In No. 3, po is equal to the MASONRY. SHUI GOTHIC ARCH PLATE.III. NO]. N:10. N"). 1 hllad NO2. 휴 ​무 ​유자 ​우 ​No 4 No | 4. No || 5.1 N:72. BĆE I E M R.Johnson. S.Porter. Á Fullarton& C Landon&Edinburgh fullt 3 of 1 STONE CUTTING. OBLIQUE ARCH. PLATE IV 20.5. N4 imgstat .---- K m 11 .. k im . . - - - . - -.- .... - - - - - - - - - - -- - . , . .- - - - - 8 SASA w w enn N°13. N. 12. NU. SS - - - - AVA / het IM -- - - - - - N°1. JIJI-T -JTIN N.2.11 .3. i 1. N4. No 5. .. ad S . ASSE R.Johnson, S. Porter. A Fullarlon & CO. London & Edinburgh STONE CUTTING, ARCH IN A CIRCULAR WALL. PLATE V. NN N° 7 Invented by Johnson Enard HR Raste A Fullerton&C London& Edinburgh STONE CUTTING. CONSTRUCTION OF GROINS. PIATE VZ NO 2. Invented by Johnson, Drom hy/Nicholson. Earned by Armstym, A Fullerton & C onan Edinburgh ارت: ہوا Fraraced bvR:Johnsen: . X SR No 7. 11 11 IP 6 VIIDUAUDIT IN No8. SAS - NO 3 N:9. - NO9. 1 . rite NY No 5. ST 1 LE AK II milli UTI 11111 mit: 18 . 1 A Fullarion & Co.London&Edinburgh SA - CS T 71 WIZI GOTHIC GROIN. STONE CUTTING, JIH - S- TNX SY . - - . . . . . . . . . . . . . Ae VET WW ON C1 SV S- SEL ANNUT . 151 UNITY SIS S 11 TE 4 . - .... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .... .......... I IIIIIIIIII! INI T IIIII XIII III . II HUB No 2. NOZ. . - 3 2 . ++- N. 4. + 111111 UNIMITTI LLORE . . XIII . NA 1 UT RE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 111111111111111 --- SA 5X1111 ha WA - - - V A VIVIDUIN w IN IIIIIIII SER PS LOUI SA KA Engrooed by S Portor PLATE IT INUTI MUNTANCHETTA T IUNTINGTINIMINNA PIIMTINUINO JUNI Sect. I.] 83 MASONRY. projection of f h, and h i is equal to that of om. In No. 4, n ne is equal to that of kl. No. 5 is a mould for bottom-bed of the first arch-stone, g s being equal to AB on the plan, No. 2; r s equal to ac; s t equal to Bn; and t u equal to pn. By laying the plan and elevation down at full size, and taking the same methods, the arch may be worked without any difficulty. No. 6 is a mould for applying on the bed or joint f g. No. 7 is a mould for the bed at hi. No. 8 is a mould for the bed kl, which becomes also almost straight and square ; and No. 9 is an arch or face-mould. Draw two lines on the face-mould square from the base line ; then run two chisel-draughts straight and out of winding ; next draw other two lines across the face, parallel from the base line, and run two draughts with the hollow mould as found from the plan at EF; then the whole face may be worked with a straight edge, taking care to apply it square from the base. There may be as many upright lines as may be thought necessary to work the stone correctly. 61. CONSTRUCTION OF GROINS. [Plate VI.] The first preparation is to lay down the arch at full size, as shown in Plate VI. (or half of it will do) on a straight floor, or on a piece of oil-cloth, which is a very good thing for the purpose. Then divide out the number of courses that is intended to be in the arch, as 1, 2, 3, 4, 5, 6, 7; draw all the joints of half of the arch, and draw a line from each of these joints level or parallel to the base ; make a correct bevel to each of these lines; and having prepared a straight face of an arch-stone, square it; then run a chisel-draught round where the point of the mould cuts it in No. 1 section. At 2 and 3, or the second joint from the keystone, apply the bevel as shown at letter I, No. 2, with the arch-mould applied from the bevel. In No. 2 section, the bevels are found in the same manner; as there are two bevels for each joint of a quoin. ABCD, on section No. 1, shows the size of the quoin-stones; h i j k l m on the plan of the ceiling shows the same quoin when fixed, and opqrstu o w shows its face and beds when prepared for fixing; 3 letter E, shows the size of an arch-stone with the bevel and mould applied; 3 at F, 3 at G, the same as letter E; and 3 at H a single mould. 62. GOTHIC GROIN. [Plate VII.] In Plate VII. ABCD is the plan of the walls of a Gothic groin; and GDEF the plan of the pillars from whence the ribs spring. No. 1 is a section showing the manner of connecting the ribs with the arch-stones, as kg, la, mb, nh, oc, pd, gt, re, and s f. Prepare the rib, the height of two courses of the arch-stones; and, at every third course, work the rib on quoin, as shown on the section from a to b, or from t to U. HL,IK are level and straight ribs intersecting at the crown of the arch, and O is the keystone. No. 2 is a plan of the ceiling ; No. 3, a section showing the manner of connecting the quoins, as y u x tw, &c. ; No. 4, the keystone with the mould applied ; No. 5, a section through the keystone; No. 6, a plan of the keystone, showing the intersections of the angles; No. 7, the plain surface of the stone with lines drawn for the first process. No. 8 is a plan of the course at a on the section ; No. 9, a plan of the course from 6 to m on section. No. 10 is a section of a single rib with plugs. It is much stronger to do the arch in this way than to run the ribs all single, as they are generally done. PRACTICAL ARCHITECTURE. [PART II. 63. RIBBED GROINS. [Plate VIII.] section across AB or GH, in a vertical plane passing through the axis of the pillars. No 2 is the rib, which has a f for its seat. No. 3 is one of the diagonal ribs insisting upon its seat bg; No. 4, the rib which has ch for its seat; No. 5, the rib which has di for its seat; and No. 6, the rib which has Fk for its seat. Suppose the pillar to be in the circumference of a circle of which the centre is t, and let the arc GF be the quadrant of the circumference. Divide the arc GF into as many equal parts as the number of parts between the ribs, and let a, b, c, d, be the points of division. Draw the chord aG, and produce aG to meet the seat of the apex of the ribs in Q. Through P, any point in aG, draw Qp meeting a f in p. Join 6 a, and produce it to meet the seat of the apex line R. Through p draw Rp, meeting a g in p. Join b c, and produce b c to meet the apex line of the transverse vault in S. From p in 6 g, draw ps, meeting c h in p. Join cd, and produce it to meet the seat of the apex line of the transverse vault in T. From p in ch, draw pt meeting di in p. Join dF, and produce dF to meet the seat of the apex of the transverse vault in U. From p in di draw pU meeting Fk in p. In GC, No. 1, make GP equal to GP on the plan, Fig. 1; in a f, No. 2, make a p equal to a p in a f on the plan ; in b g, No. 3, make bp. equal to b p in bg on the plan, Fig. 1, and so on. Then, supposing the rib GML, No. 1, to be a given rib, draw PM perpendicular to BC. In Nos. 2, 3, 4, &c. draw pm perpendicular to the base, and make p m in each equal to PM, No. 1; then m will be a point in the curve of each rib. In the same manner we may find as many points as we please for drawing the curve of each rib. Fig. 2 shows one quarter at equal angles. 64. CONSTRUCTION OF SPHERICAL DOMES. [Plate IX.] In this example, Plate IX., the stones are first supposed to be wrought to a conical surface. The bevels y o t, x 5, vlq, &c.; then apply on the convex surface to the horizontal joints, the circular leg being bent upon the surface. ABC, ABC, &c., Nos. 2, 3, &c. are the bevels that apply on the convex surface to the vertical joints. Nos. 6 and 7 exhibit the bevels for working the conical joints. The versed sines Wa, XB, Yy, &c. of the arcs Nos. 2, 3, 4, &c. are taken from No. 1, between the similar letters and the half-chords wt, xu, yv, &c. Nos. 2, 3, 4, are also taken from the dis- tances between the similar letters No. 1. The straight leg AB of the bevel bisects the chord of the half arc of Nos. 1, 2, 3, &c. perpendicularly. CM - - -- PLATE VIIL - " .. . . ... ... .. ot - .. - - - Fig. 2. * ..... X1.- STONE CUTTING. GROINS. NO2 N°4. Introducut by H:"Tühol.son. È. Bullarind Co I, ondan & Edinburgh - - - S*** i L ............... ................ T 627 Tant ..... ::::::: :::::::: :: ..: . . ..... .... . . .. ........ ... i STONE CUTTING. CONSTRICTION OF SPHERICAL DOMES. PLATE IX ................ No 2. ...... -- -- --- -- W ----- No 2. - . ---- ------ - . N. 3. Auto- NO 4. 1.7 Y --- 10 EĽ V05. P.Vicholson. C.Armstrong A Fullarton&Co Limdon & Edinburgh CH. *PLATEX : STONE **** CUTTINE Mere CUTTING , • NICHES IN STRAIGHT WALLS. Fig. 1. #@. ? | ਰ ਰ ਲਈ " ? ? . 17 1. 1 Iq , 3. Fig. t. 1 3. ਘਾ 1. W? 1. Fig. 5. Fig. 6. ? 2 , | 2 . A 7. .1 1 , Getrmstrong. K.Jomwon w A Fullarton & COLondon&Edinburgh M0/ SECT. I.] MASONRY. 65. NICHES IN STRAIGHT WALLS. [Plate X.] TIT These decorations consist of recesses in a wall, either for the purpose of embellishment, or for re- ceiving statues or other ornaments. They may be formed with spherical heads, and cylindrical backs, or entirely with hemispherical backs, or with spheroidal backs, having the transverse or con- jugate axis of the ellipses vertical, as may best comport with the character of the object to be placed therein. Those with spheroidal backs may have their horizontal sections in circles of different diameters, and consequently, their sections through the vertical axis, all equal semi-ellipses, similar to each other; or all their horizontal sections may be similar ellipses, and the sections through the vertical axis of the niche will be dissimilar ellipses of equal heights, at least for one half of the niche; but spheroidal niches with such sections are difficult of execution, and more pleasing to the eye than those with circular horizontal sections. Among the works of the Romans, niches have either a circular or rectangular plan ; the heads of those of the circular kind are generally spherical. In the middle of the attic of Nerva, at Rome, a niche is seen with a rectangular elevation, and a cylindrical back and head. Those upon elliptic plans were not much used by the ancients; though in Wood's 'Ruins of Palmyra,' there are two niches with elliptical heads, within the entrance portico of the Temple of the Sun; but the author has given no plan of them. Most frequently, those upon rectangular plans have horizontal heads, though a few are to be met with that have cylindrical heads: those upon circular and rectangular plans, are, for the sake of variety, most commonly placed alternately. 66. The plans of niches with cylindrical backs, should be semicircular, when the thickness of the walls will admit of it; and the depth of those upon rectangular plans, should be the half of their breadth, or as deep as may be necessary for the statues they are to contain: their heights depend upon the character of the statues, or on the general form of groups introduced, yet seldom exceeding twice and a half their width, not being less than twice. Those for busts only, should have nearly the same proportion in respect to each other. In some cases their height may rather exceed the measure of their breadth : they may be of any of the forms used by the ancients, or of those mentioned at the beginning of this article. 67. In point of decorations, niches admit of all such as are applicable to windows; and whether their heads be horizontal, cylindrical, or spherical, the enclosure may be rectangular. In antique remains, we frequently meet with tabernacles as ornaments, disposed with alternate and arched pediments; the character of the architecture should be similar to that placed in the same range with them. 68. Niches are sometimes disposed between columns and pilasters, and sometimes ranged alternately, in the same level with windows: in either case they may be ornamented in plain, as the space will admit, but in the latter, they should be of the same dimensions with the aperture of the windows. When the intervals between the columns or pilasters happen to be very narrow, niches had better be omitted, than have a disproportionate figure, or be of a diminutive size. 69. When intended for containing statues, vases, or other works of sculpture, they should be con- trived to exhibit them to the best advantage, and consequently the plainer the niche, the better will it answer the design, as every species of ornament, whether of mouldings or sculpture, has a tendency to confuse the outline. 70. Plate X. exhibits the various modes of construction that may be employed for niches, whether the stones be laid in horizontal courses, or in courses tending to a centre, according to the number of stones. No. 1 is the plan ; and No. 2, the elevation in each of the six figures. 111 86 [PART II PRACTICAL ARCHITECTURE. 71. NICHES IN CIRCULAR WALLS. [Plate XI.] The construction of a niche in a circular wall is slown in Plate XI. No. 1 is an elevation of the niche with the joints of the stones; No. 2, a plan of the niche at the springing; No. 3, a plan of the top of the arch-stones of the niche; No. 4, a plan of the top-bed of the keystone, No. 3 on the plan ; No. 5, a section of keystone; and No. 6, the keystone turned upon the top-bed, showing the projec- tion from A to B, from B to C, and from C to L, of No. 5. The top-bed and rough back of No. 6 is shown in the section of No. 5; and No. 7 is a section of keystone at the back. GI on the elevation is the level bed at the springing; klmnopqrstu are joints of the niche-stones ; ABCDEF the plan at the springing ; IL the centre line, GG ashlar, HH the plan of the springing-stones of the niche. By laying down the plan, elevation, and section at full size, the execution is rendered easy. 72. ARCHITRAVES OVER COLUMNS. When columns are at a considerable distance apart, the difficulty of supporting an architrave of stone, from column to column, is not easily overcome. Where the stone will bear its own weight with safety the following method is one of the best in use. In Plate XII., Fig. 1, No. 1 is a plan, elevation, and section of an architrave over columns ; ABCDE being a plan of the top-bed of the architrave, with a chain bar of wrought iron, and collars let in flush with the top-bed, and run with lead. ABCD is an elevation of the architrave; e f g h, a brick arch over architrave, to relieve it of the weight of the upper parts ; x y z, a section through the middle at f, showing the tail of the cornice on the crown of the brick-arch to take part of the weight off the stone ; x shows the springer for the brick-arch both ways; i, the end of frieze, &c., and k is a piece of iron let into both stones to prevent the springer moving; it has no other abutment. No. 2 is the end-elevation of C the stone over the column; D is another middle stone to join C; lmnop is a dotted line showing the joggle joint, and q r the upright joint, seen at u u u in No. 2. The end ow corresponds to mnop. In No. 1, t is the rope with the lewis let into the stone. 73. Description of the Lewis. ]-In Plate XII., Fig. 2, No. 1, is an elevation of the Lewis; No. 2, a section ; No. 3, the bolt; No. 4, the ring; No. 5, the outside piece; No. 6, the middle piece; and No. 7, the coteral. In No. 5, when the two are put together, they should be three-eighths of an inch wider at the bottom than the three when put together at the neck under the bolt; and the two outside pieces should always be made straight. Some masons make them hollow, but we think this is a bad plan, for when they are made so, they bear too much at the point; whereas when made straight, they bear more regularly all the way up. 74. STAIRS. (Plates XIII, and XIV.] No. 1, Plate XIII. is a plan of the staircase ; No. 2, a development of the well-hole end of the steps laid down at full length. ABCDEF is the line of the soffit, showing the quoin-ends of winders and flyers, with the rabbet drawn square from the soffit at the quoin ends of the winders, as ghi. By laying down the section from B to E at full size, we determine the quantity the stone should be hollow STONE CUTTING. NICHE. PLATE. XI. N. 4. N:1. WIN N. 5. Vell Wh N6. No7 13. R.Johnson, S. Porter. A Fullarton & CO London & Edinburgh MASONRY, ARCHITRAVE OVER COLUMNS. ᏢᏞᎪᎢᎬ XII. NE A . : Y ІНННННННННННЕ Fig. 1. " megang цHHHHHHHHH. V.1. 45 . - VO2. mu w I- UJD Fig. ? N. 2. V? 1. We . 17. . .. mewa www M 10.7. V"6. V.03. IN - SA = M - SL - - M -ULU LAN U . R. Johnson. G. Gladwin. Å Fullarion Co. London & Edinburgh STONE CUTTING. STAIRS. PLATE XII. SA -- - -- . 1. -- - Invented & Drawn by PNicholson. A Fullartan&C'London & Edinburgh OF I) STOVE (IT TING. PLATE XIV STAIRS. Fig. 1. 1 N? 2. Fig. 2. TA Vºl. V Dosyasyon De V: 2. - - - M Introduced hv R.Johnson. Engraved by R.Rorte. of IV. I Farkus 1. Im Eirburg MASONRY, STEPS OVER AREA TO ENTRANCE DOOR. PLATE 11. N. 3. V 1 . 27 ST / . / / / / ** N: 1. S lilli NO 4. GEN N INN VW VVV 1 UUTUU NNN 111 FTS11 1 NIIN SU 11111! 11 . IC 11 FITNET ATT NAPI NU www 11111/ UUTA TATA . 111 A 2 NO 2. VW N V WW W USININ NUNUN WY VVVVV UN NUNUN V WWW VY WWWWWW WWW WWW LUNYAI NNNN NNNNNNNNNNNNNNNN E NWIN . 1 // PILITIN OF 11H11 VIN TV TIL 21/VIII/ // / / . / // // / / ANTI . // . Engraved by E. Turrel. Introduced by Johnson. Aluiarvio.CLoudm3 Edmborgh SECT. I.] 87 MASONRY or round at BCDE; and AB being a section of two flyers and two winders at the wall-line, BC a section across the end, and CD two winders and two flyers, by laying these sections down at full size, it is at once seen where the soffit will be crippled, and how to avoid it. The steps can be made easy to the eye ; and by making all the joints of the winders square from the rake, the stone is stronger, it is not liable to flush, and the back-rabbet will wind the same as the soffit. When the rabbet of the winders is cut in with the same mould as the flyers, it leaves a very acute angle, and liable to be broken off. 75. Plate XIV. shows another staircase. No. 1 is the plan ; A, the bottom or first step, BC, the top-landings. Fig. 1, No. 2, is a section of the staircase. The first flight from A to No. 1 can be done without any support, by using an additional thickness of stone, as shown on the section from E to D shaded dark, and by making all the back-rabbets b c, b c, b c, arch joints. Fig. 2, No. 1, is an elevation of the sixth step, showing the rabbet with two iron plugs, an inch square, let into each stone 21 inches, and by working off the additional thickness about 12 or 15 inches in from the end with an ogee, it does not offend the eye. No. 2 is an elevation of the end of one step, showing the plugs the other way. At F on the section, the soffit should join the landing with a hollow. Wherever landings go into a wall, a course of stout Yorkshire paving, or of thin landings, should be laid; over that, three courses of bricks with sand, nine inches in ; and over that, another course of landings to go through the whole thickness of the wall. When the staircase is ready for fixing, the bricks are taken out with little trouble, and without injuring the wall in the least. The landings in all public buildings should be done so. 76. STEPS OVER AN AREA TO AN ENTRANCE DOOR. [Plate XV.] In Plate XV. No. 1 is an elevation of the steps ; No. 2, a plan ; No. 3, a section, and No. 4, a section of the second step; where a f, b c, are the rabbets made arch-joints, the stones being thicker and each joint plugged. By building a brick pier from A on the section, one at each end of the step, the steps will be sufficient without a brick arch below them. SECTION II. BRICKLAYING. Definition, 1.- Brick bond. 2–8.-Groin vaults, 9. 1 BRICKLAYING is the art of building with bricks; and as bricks are prepared of the forms most commonly required in the construction of edifices, the art is in a great measure limited to the methods of bonding together bricks, so that the joint effect of the adhesion of the mortar, and the over-lapping of the bricks, called bond, may produce the strongest work possible. See articles 57 to 72 of Part I., Sect. I. BRICK BOND. 2. Where the greatest degree of strength is the chief object of the Builder, it is best to dispose all the bricks in every second course, with the length in the direction of the wall. The bricks thus laid are called stretchers. The length of the bricks of all the intermediate courses should be across the wall; the bricks so laid being called headers. The wall is thus composed of alternating courses of headers and stretchers, with the joints crossed ; and the strength is the greatest that can be obtained with rectangular bricks. This mode of construction should be employed in all rough works, and for walls to be covered with stucco. It is called Old English bond. See articles 67 and 68 of Part I., Sect. I. 3. In order to render the apparent joints similar in all the coursesma circumstance which con- tributes much to the appearance of good brickwork-each course is composed of alternate headers and stretchers, crossing the joints of the adjacent courses, and also with the header crossing the middle of the stretcher of the course below it. This disposition is called Flemish bond, and should be used in finished brickwork, and where the work is jointed. It is defective, however, in strength, particularly at the angles; hence, good mortar is essential where this kind of bond is used. 4. The strongest bond is produced by the Dutch method of disposing the bricks diagonally, in respect to the face of the wall, and with the joints crossed. The apparent surface of the work being afterwards dressed smooth and rubbed, it appears very uniform and neat. The waste of material, and the amount of labour necessary, will prevent this method being used, except by the curious. 5. The bed-joints of brickwork should not be more in thickness than the size of the largest grains of sand in the mortar. The usual allowance is 3 inches for the brick and joint; but unless the bricks be thicker than they are in general, this allowance is too much; and for kiln-burnt bricks it is sometimes too little; contractors being in such cases compelled to lay bricks with too little mortar, -a defect more serious than of using too much. By the rule above, the difficulty of the ordinary method is avoided. The distance of the screen wires should correspond with the thickness of the joints. 6. Mortars and cements are prepared in the same manner for brickwork as for masonry ; (see articles 45—92, Part I., Sect. II. ;) but it requires more attention in building with brick to keep their surfaces free from dust. If the surface of a brick be coated with loose sand or dust when it is laid, the mortar will not adhere to it; hence, all such dust or sand should be removed by washing, and …. 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SSSSSSSSSSSSS 以S 11 年11 -- 。 工作​, 以 ​HTT11 * A:….l. * WWW 女人 ​RSS 三中学 ​a. . . . ::: - - + ---. - -- … -- - - - - -- : :-- - .…- .,LT … . ……. .- .- 一 ​::tt ti.…… .) … :………. .….. 自 ​. :: …… ….…… 11.11.13 …… 了 ​一样 ​「 - --- … 是 ​- - # 了 ​, 产 ​是 ​! “ 上一中​, A. 非上 ​ap: 产一子​。 ,在一 ​- , 。 一一一一 ​1 。 , -- , 工作​; 十月 ​. ,, , MI、 ith , - n WWW - 了 ​ht.. 业​。 是一 ​. . ……. 重重重重 ​. ……… La : … ER y 。 11 .it.. : 一一一 ​「 一下​。 。 了 ​, 一一一 ​th 3.11.id... o-pic CARPENTRY PLATE. k Tony Of I AND BRICKLAYING, PLATE TW GROLI VAULTS. Fig. 3. Fig. 2. --- -- LAH # .- 1 * - ЧТ 15-НЫНА 1 19 p1 ..... ii. 1 -. - I - - — - - - - 111.11 - - Fig.2. . y - - - - ... - - - . .. . . . . . . . . 1- . .. AK .. .. .. AH 1 - - - TILI - - - UNHH - - . . . . . . . . . . . - 1 fa. . 1. 2. Fig... Fig.6. Fingrared him.(Armation. 21 Afiliariikli.nader. Famiz: $ ) 3 C* CH Sect. II.] BRICKLAYING. brushing the bricks in water; and it is further desirable (as stated in art. 70, Part I., Sect. II.) to have the bricks well-wetted before laying them in dry weather. 7. The preparation of bricks for arches, niches, &c., is dependent on the same geometrical principles as the like works in masonry; differing only in the size of the masses used, and the methods of cutting them to their proper form. The joints of niches generally radiate from a centre, as in Figs. 3 and 4, Plate X. ; and in No. 1, Plate XI. In vaulting, the chief difficulty is in the preparation of the centring, particularly in the case of intersecting on groined vaults; the manner of doing which may be understood from the following example. 8. Light brick-arches require no other particular care than to have the joints of the string-courses perpendicular to the surface of the soffit. The bricks are usually laid on an end on the centre. Heavy brick-arches, above 2 feet in thickness, if the curvature of the arch is considerable, require great care in the arrangement of the bond in order to procure the utmost solidity, by placing the greatest num- ber of bricks in the arch, and the greatest strength, by the connection of the different courses. Were the bricks so laid that the joints between each string-course were continuous from the soffit to the back of the arch, however close the joints might be at the soffit, they would be very open at the back; and the strength of the arch would therefore depend on that of the mortar in the joints; unless the precaution was taken to fill in the joints with fragments of slate closely packed as the successive courses of brick are laid from the soffit to the back, an operation which would present considerable practical difficulty in preserving the proper direction of the joints. On the other hand, were the bricks laid in successive courses over the soffit, or, as it is termed, in shells, the greatest possible number of brick would then be laid in the arch ; but as there would be no other union between tlie successive shells than the mortar between them, it is to be apprehended that they might easily separate should any motion take place in the arch, and the whole mass would therefore offer but little comparative resistance. Hence, to unite the advantages of the two methods, the entire arch, (Plate XVI., fig. 1,) should be divided into several portions, by joints running entirely through from the soffit to the back, the brick being laid in these successive portions alternately in shells, and in blocks, with joints running entirely through the arch from the sofit to the back. Any bond may be adopted for the portions laid in shells. If the arch is not more than three feet in thickness, a very solid mass can be obtained by dividing the thickness into two equal shells. In the arrangement here explained, the blocks, in which the joints run entirely through, should not consist of more than three or four bricks in thickness estimated along the curve of the soffit. The bricks which form the key of the arch require to be laid with great care. The first course on the soffit may be formed (Plate XVI., fig. 2,) of a thickness of three bricks laid on their ends in very thin tempered mortar, and well wedged in, if necessary, with pieces of strong slate. The next course should be formed of five bricks laid also on an end, and forming continuous joints with those below them. This course can be laid in grout; for, by dividing the length of the course into several compartments separated by a single row of bricks laid in mortar, the grout may be first poured into these compartments, and the bricks be set in it, and the joints be then filled with pieces of slate. The third and following courses are laid in a similar manner; increasing the thickness of each successive course by two bricks. TU GROIN VAULTS. [Plate XVII.] 9. Plate XVII. exhibits the plan and elevation of a series of groin vaults. Fig. I is the plan. The cross lines which extend from the angles are the seats of the groins, US 90 [Part II. PRACTICAL ARCHITECTURE. which are the intersections of the upper surfaces of the boards. The hyperbolic curves within the cross lines are the seats of the intersections of the under sides of the boards of the side arches, with the upper surfaces of the boards of the large arch, and the two side-arches parallel to the Fig. 2 is an elevation or section of the principal vault, and the two side-vaults. Fig. 3 a transverse elevation or section of the crossing vaults. Figs. 4, 5, and 6 exhibit the moulds for bending over the boarding, in order to find the true inter- sections agreeable to the seats on the plan.-See Part VII. ON THE DEVELOPMENT OF SURFACES, AND ORTHOPROJECTION. SECTION III. CARPENTRY. Definitions, 1, 2.__Of Frame-work in general, 3, 4.- Expressions for strength of materials, 5. Principles of arranging Frame-work, 6-8.- Joints, 9, 10.- Built Beams, 11–14. Scarf Joints, 15, 16.- Scarfing Beams, 17.- Methods of joining timbers, 18, 19. Naked Flooring, 20. — Truss Girders, 21.– Roofs. Timbers in a Roof defined, 22. — Construction of Hipped Roofs, 23—29.___ Trusses Executed, 30.- Circular Roofs or Domes, 31.-_ Conic Roofs, 32, 33.- Trussed Partitions, 34.- Construction of Niches, 35-42. Pendentive Ceilings, 43–45. __ Timber Bridges, 46–48.- Frames, 49. First Class, 50—53. Second Class, 51--56.- Road- way, 57-59.-- Notices of different Timber Bridges, 60-62._Moveable Bridges, 63—66. Centring, 67–81. 1. CARPENTRY is the art of arranging beams of timber for the various purposes to which they are applied in structures. 2. The term frame is applied to any combination of beams firmly connected with each other. 3. The frame-work of a structure may be supported either by suspending it from fixed points above it, or by resting it on fixed points below it. In both cases the arrangement must be such as to pre- sent a state of stable equilibrium. When the frame is suspended from fixed points above it, this state of equilibrium will exist, whatever may be the degree of flexibility of the figure of the frame ; but when the frame is supported from beneath, the arrangement of the parts of the frame, in order that this state shall exist, must present a figure of an invariable form. 4. As the frames of structures are generally supported from beneath, the first point to be attended * to in their arrangement is to combine the pieces to obtain a figure presenting an invariable form. This is effected by making such a disposition of the principal beams of the frame, that the pressures, thrown on its different points, shall be transmitted in right lines, parallel to the fibres of these piecos, directly to the points of support. By this disposition the pressure will be thrown directly on the fixed points, and will have no tendency to change the figure of the frame by an action on those beams which do not immediately rest on these points. As an arrangement of this nature is not always practicable, it will be necessary, in some cases, to adopt a disposition in which counteracting pressures may conie into play; by placing the beams in such positions, that a pressure, whose line of direction does not pass through a fixed point, will be destroyed by an equal one in an opposite direction. The pressure on each beam in the frame, as well as the line of its direction, can be determined by the laws of statics; and the size of the beam must be so regulated that the effects of the pressure shall not impair its elastic force. The points of junction of the beams—which are termed the joints—must, moreover, be firmly con- nected, by proper artificial means, to prevent any yielding at those points. The beams composing the frame-work of a structure may be either straight or curved. The positions of the straight beams may be either horizontal, vertical, or inclined ; and they may rest on one or more points of the support. As the directions of the pressures may be either perpendicular, parallel, or oblique, to the fibres of the beams, it will be necessary, in the first place, to determine their nu. merical values in the different cases here assumed. 5. The algebraical expressions, relative to the flexure and rupture of beams for the most simple cases, are given in Appendix G. The expressions which apply to the laws of rupture in the cases that most usually occur in frame-work, are added in Appendix H. 6. In the arrangement of every system of frame-work, as one of the main objects is to procure a 92 (PART II. PRACTICAL ARCHITECTURE. 22 figure of an invariable form, such a disposition of the parts must be made, that any pressure on one part, which may have a tendency to produce a change of figure, shall be counteracted by some other part. The simplest manner of producing this effect is to combine the parts of the frame to form a series of triangular figures; because no change can take place in these figures without the pieces forming their sides becoming shorter or longer by a force either of compression or of extension. If therefore any of the main pieces of a frame intersect each other, forming quadrilateral figures, it will be necessary to introduce other pieces placed in the direction of the diagonals of the quadri- laterals, for the purpose of counteracting any tendency to a change of form in these figures. An arrangement of this character, generally termed diagonal bracing, is used for all frame-work requiring great strength and stiffness. 7. The pieces in a system which resist a compressing force are usually termed struts ; those that counteract a force of extension are termed ties; and those which are used to add stiffness to the frame, by preventing any tendency to flexure in the other parts, are termed braces. 8. It is important that the different pieces of a frame should be as little grain cut as possible, that is, cut in a direction oblique to the natural fibres, otherwise their strength would be greatly impaired. Beams, the cross sections of which are the same throughout, and of a rectangular form, are, on this account, mostly used for frames. In some cases of resistance to a cross strain, it might be preferable to use a beam whose longitudinal section should be of the form of a solid of equal resist- ance; but as these solids present less resistance to flexure than rectangular beams having a uniform depth, equal to the greatest depth of the solid of equal resistance, and as stiffness is, in almost every practical case, of as much importance as ultimate strength, the rectangular beam of uniform depth is to be preferred. * 9. Joints.]—The bearing surface of the joints should be as great as the nature of the case will permit, in order to prevent the joint from being crippled, either by the indentation, or crushing of the fibres of the parts in contact. The bearing surface of the joints should, moreover, be perpen- dicular to the directions of the pressure on them, to prevent any tendency to sliding on either of the surfaces. 10. In arranging the joints, the simplest forms that are most suitable to the object in view, should be adopted, in order that there may be the least inconvenience in obtaining an accurate fit of the parts. The parts should not, in all cases, fit close; but allowance should be made for the settling, arising from the shrinking of the fibres, and the new direction of the pressures which may arise from this cause. If this allowance were not made, the pieces would frequently be liable to split and give way at the joints. In cases of this character in heavy frame-work, it is recommended to make the surfaces of the joints circular, as the surfaces would then continue in contact should any change take place in the position of the pieces from settling, or any other cause. 11. Built beams.]—To procure as great strength as the nature of the case will admit of, the different courses of a built beam must be connected in such a way that they will not slide on each other when submitted to a cross strain. This may be effected either by placing pieces of hard wood in notches cut in the beams, as represented in Plate XVI., fig. 3; or else, by indenting the beams as repre- sented in Plate XVI., fig. 4. The courses are then firmly connected by screw bolts, or by iron hoops, or by a stirrup which will admit of being tightened if the stirrup works loose from the shrinking of the fibres, or from any other cause. 12. The keys of hard wood may be either simple blocks of a rectangular form, or double wedges which will admit of being driven in the notch for the purpose of bringing the surfaces in close II UU WII * On the subject of Flooring, the reader may advantageously consult Tredgold's · Principles of Carpentry.' CARPENTRY, SCARFING BEAMS. PLATE XVIII. Fig. 1. Ant Fig. 2. NOI. N° 2. Fig.3. I O © D Fig.4. NO 2. N°1. Fig. 5. NO2. Fig.6. Nº. . Fig. 7. 19 N. 2 Fig. 8. N!1 Pe Nicholso Hi Lowry OF A Falartor&CLondon&Edinburgh CH CARPENTRY, SCARFING BEAMS. PLATE XX Fig. 1 Fig. 2. o Fig. 3. Introduced by:P Nicholson. Fig. 4. --- ---- un . . - --- . - . 1.-. Fig. 5. Fig.6. anda na ano E Fig. 7. - - - - Fig 8. -- -- - Fig..9. Introduced by A. Morton.. Fig.10. N1. Fig.10. N.° 2. W Lowry. wiarton (Londos & Lainbirgli Tchi CARPENTRY, PLATE XX. Fig.1. METHODS OF JOINING TIMBERS . Fig.IN°1. Fig. LV. 2. Fig. 1.N!3 Fig.1.1.4. ---- - --- - - - --.- wwwmirithm Moyra "Мумин/ Fig. 2 Fig. 2.Nº.1. Fig.3. Mwanan many wym powing Fig. 2.NO %. Fig.3. N° 1. Fig.4.NO 2 Fig4.N°4. Lig4. Fra vist Fig4 N:1. FigaNO3. umyne in numenti MWANAE ANAM kn/min L A FilarWa&. Locial Edinburgh echy Sect. III.] 93 CARPENTRY. contact. This, however, requires care in practice, as the wedges, if driven in with force, might cripple the fibres. 13. The position of the indents should be regulated to prevent the courses from sliding. It has been recommended in built beams formed of two courses, to make the upper course of two separate pieces, abutting against an iron bolt, termed a king bolt: as experiment has shown that a beam sawed across at top, to a depth nearly one-third of the entire depth, having a piece of hard wood inserted into the cut of the saw, offered more resistance to a cross strain than a whole beam. 14. The more perfect the contact between the courses, the stronger will be the beam ; but, as the where the built beam is exposed to the weather, to use keys instead of indents, and to leave sufficient space between each course for the circulation of the air, in order that moisture may not be retained long enough in the joint to cause the rot. 15. Scarf joints. ]—When it is necessary to unite two beams at their ends, the simplest and strongest method consists in placing the two pieces end to end, and confining them in this position by two or four pieces bolted on two or four of the sides, as the case may require. This method is termed fishing a bearn ; it is used only for rough heavy work. The side pieces may be simply con- fined by screw bolts; or else they may be connected with the main pieces by keys, or indents. 16. When the beam is required to be of the same thickness throughout, a joint, termed a scarf, is used in place of fishing. The form of the scarf will depend on the nature of the strains to which the beam is to be submitted. 17. SCARFING BEAMS. [Plates XVIII. and XIX.] 1 Several examples of the forms of joints for lengthening timbers, or scarfing, are shown in Plate XVIII. Fig. 1 is a plain scarf with parallel surfaces bolted together. The parts are shown separated in Fig. 2. Fig. 3 is a scarf with an oblique joint bolted together; and Fig. 4 shows the two parts separated. Fig. 5 is a scarf with parallel surfaces tabled together, so that a pair of wedges may be applied to tighten the joint. The parts are shown asunder in Fig. 6. Fig. 7 is a scarf with oblique surfaces made to tighten, by means of a pair of wedges. Fig. 8 shows the parts separated. Plate XIX., Figs. 1 to 4, show different modes of bolting, and of forming the scarfs described in the most simple form on the last plate. Figs. 5 and 6 show methods of joining where the timbers have to resist a lateral stress; Fig. 6 applying to the case where the timbers are too short to join without addition. Figs. 7 and 8, represent methods of building beams out of smaller timbers. Fig. 9 is a very good kind of scarf, only difficult to execute. Fig. 10 shows the parts separated. 18. METHODS OF JOINING TIMBERS. [Plate XX.] Various methods of joining timbers are shown in Plate XX. Fig 1 is a common joint where the two pieces are halved upon each other. Nos. 1 and 2 are the 94 [PAвт II PRACTICAL ARCHITECTURE. two pieces before the surfaces are brought into contact; Nos. 3 and 4 exhibit another method. In this case, the end of one piece does not pass the outer surface of the other. Fig. 2 shows methods of joining timbers where the end of each piece passes the end of the other to a small distance. Fig. 3 shows how two timbers are joined by mitering them together. In this case the two pieces ought to be fixed together with a bolt at right angles to the mitre joint. Fig. 4 shows how one piece of timber is joined to another, when one of the pieces is extended on both sides of the other piece. fore be taken to prevent the surfaces in contact from being crippled, as well as any displacement of the pieces. This can only be effected by making the bearing surfaces as great as possible, and by securing the pieces by a judicious arrangen:ent of the bolts and straps. In addition to these, it has been proposed to insert pieces of lead, or iron, between the bearing surfaces, to prevent the crippling of the fibres. 20. NAKED FLOORING. [Plates XXI. and XXII.] Plate XXI. exhibits the construction of a double floor. The large piece of timber in the middle is called a girder. The pieces of timber which are supported by the wall and the girder, and which appear to run in a transverse direction to the girder, are called binding-joists. The pieces of timber which are supported by the binding-joists, and fixed near the lower edges of them, and which are parallel to the girder, are called ceiling-joists. The pieces of timber that appear upon the walls, and which support the binding-joists, are called the wall-plates. The pieces of timber that run parallel to the girder and over the binding-joists, are called bridging-joists. The bridging-joists are those next to the wall where the chimney is shown, and prevent the binding- joists and ceiling-joists from being seen. For this reason the bridging-joists are removed in the other half, in order to show the binding and ceiling joists. Fig. 2, No. 1, shows the plan of the floor ; and No. 2, a section of the same, drawn in the manner of working drawings. Plate XXII. contains the parts to a larger scale, to show the mode of making the joints. Fig. 1 exhibits the manner of joining the binding-joists and girders. No. 1, girder with binding-joists on each side of it joined together. No. 2, a binding-joist prepared for being joined to the girder, No. 3. Fig. 2, a section of the girder, with one binding-joist inserted, and the other ready for insertion, as in Fig. 1, Nos. 2 and 3. Fig. 3, a section of the binding-joists, showing the notches for the bridging and ceiling joists. Fig. 4 exhibits in No. 1 the bridging and binding joists joined together. No. 2, the bridging-joist with the lower edge uppermost, in order to show the notch. No. 3, binding-joist, exhibiting the notches. Fig. 5 exhibits the method of tumbling in a joist between two binding-joists or girders, when the joists are to be framed flush at the top. PLATE IN . Onirmstronno -- - - - - - - - - .- .. . = Nev SEN ton - -- - : . . SNS V . -, ANAS 2 1 M . . 1 19 SERI ILE . . PHOTO NON S. R OTI M SHWA ISSN . . SPRING ME TE - - .. .* WY US i . . . - - - - MA Sa TOLE SERIE W NNNN - O - u e XVI EEEEEE w DIIIII WWW . & C London& Edinburan. WWII M W . HVI VO/ 112 - - - VAKED FLOORING, CARPENTRY, WS S .VO RAM WHI NIMI WIZ : A Fullar WW I . . S Om M aa- 1: MA 119.2. - - IT _ WWW 1 IIIMII BA WWE WWII 2 . . .. . . W MUN WE - -- * KER ISINIS FI A int SERA THITIM FIT STE . UT NA . . . HIL WI ow S - t - . - . - . . -- - - . - - - - - - ULLIR LA ALE WILL 1 WW Ul WW VW 1. - 1!1 MAN WWW MINI 12. Dis man HU TT- .. :-. WW WOIIII PIDU RA ALLE ANNO 2 ++ * VE . .. - VINT .-. 11 ti DES - . . . W www un S - WN WHIL lililodisrin? - CARPENTRY, NAKED FLOORING. PLATE XXI. N.° 1. Fig. I. N" 2. N®3. V illi 4490 min Fig.2. Fig. 3. M.2 MANAM NC 10 3. mammor Fig. 4. N° 1. N.° 3. No 2. Fig. 5. PMoholson. R Rotti. NI A Fullarwn&C? Landon&Edinburgh CH ! CARPENTRY, TRUSS GIRDERS Fig.l. P717E XXIII. : RT M . 141 7 . E- 3 SARE Fig. 2. NOL. 1 1 . M ! iN WIN S ..-.. - - 14 17 di!! AS2 . . NO 2. MA . Fig 3. NOT . "RES - - - - - - 13 N° 2. = = = = . * NS- UCH: Es Fig.5. Fig. 7. N°2. Fig.6. - VOZ NO1. NO2. N° No2 Hullige . BE THEW Mill Fig. 4. ---- --- --- ----- ---- --- -- --- --- WE . on the -- WM RADI! S P icholson. H. Lowry. A Fuliarton & Co London&Edinburgh CARPEYTRY, TRISS GIRDERS PLATE XXIV Toil 2. At . 2 WA STAVEB HII INANITISHI N DHWAATITHIRRITO DIE HOR ES mooth Still LIMITINUTURIMBINGKAULA TAHAN! பாபு p H it it MUDHIHIRU 11111!1!1:11 ... H .. 221 E . wieder . L DID roll * RSS 2 IN HE A VO I MW WIL EST ulit WE HINA S AN . 1 ! NU IV. \ . inlitt E 11 . - 1 2 1. . Pool i nie. SER SASS CNN ES SE SA SEL SH 242 105. 16. VA LEN . . introduced by Foster Cruired by 2VUT F:21. i mba се IRPENTRY, TIMBERS OF ROOF DEFINED. L’LITE XXV. - . MIL US W 2 * I! ! . PE==TESSE - : : EEEEEE Tre EN Teac Mind 1 SUNIT TA W . WW li - M numento . Low- re . .. * * .. VIIMI au ****. . ammaren M . wenn : Hmm Ay VIIIIII . . ZA SA . A W } Win Sis Wie XI/1 www. MD 2 . !. WW . . 11 PETER *** Illini WE SEN . - . "IZ. WWW.H LE ill OWN w 19 : BUL :: them Pilt Tsets XS 1 24 SSS MIN TE volal KUND TERTE SUB16M ! ..:) "! S VIII !T!." lilililo :: IRIN WILL 1: 19. 1997 Utsi !! 2 ! ! OUT ITULI XHAUTA AMURUWERTO 113. .. . SEBEBE SARRERAS ISIES - POR GAS SSS T :V ELA Fiy.. N MAGURUDATARARLARLANARINSKA POTILISTA LLEKITARAKTER TIENERALITAT DE LA SELERATORS TARTARIANONIMALTA TAGASINERINGE DONIO SI DELE LH SITIS ::- 2013. SZESE EESEE - TTE ---.* .W. 0 . 1 HC RE - - -- ++ 9 . -- Sri :--23: - -- -- - SU TEL - HI rini.Nazan:- -* L EU ---..:: ... .--.- Se : -=-=- ! 9. GREE . --- SAR ------ - 1 ---------- anne * * SZsas T.. .X1, znansa MUUNDULTUNG W irusinging WHERHADH UMIN WIL!!1210l O RIHIHITIMURIT t ri libutaráter Pumasimamia zinaanza kushinnanbombeyninikud BIX - - -- 2 - - - - u - - P. Nicholson. Cifrm.tr100. A Fullarion&.C!Indons Fairbagi. # ?? s . non 1.0.Ali. Sect. III] 95 CARPENTRY. The method is thus : Lay the joist which is to be framed in, with its top-edge undermost upon the girders or binding-beams. Then, with a sharp pointed instrument, draw a line upon the upper side, now undermost, close to the adjacent faces of the girders or of the binding-joists ; then turn the edge, now uppermost, underneath, and shift the joist in the direction of its length until the line drawn on the top fall in the plane of the surface of the girder, or that of the binding-beam. Place a straight edge so that it will at once coincide with the vertical surface of the girder and that of the joists ; in this position draw a straight line, which is the line for cutting the joists. The other end will be found in the same manner. 21. TRUSS GIRDERŞ. [Plates XXIII. and XXIV.1 Plate XXIII. shows some of the methods of trussing girders. Fig. 1 is an iron truss bolted together at the ends. Fig. 2, a girder divided into two lengths. No. 1, a vertical longitudinal section. No. 2, plan of the girder from the top. This example consists of two braces uniting a king-bolt in the middle. Fig. 3 is a girder divided into three lengths. Fig. 4 is a design for a girder extending between two walls at a great distance from each other. In this design the surface of the floor is raised at a greater height above the ceiling, as the span between the walls is greater. Fig. 5 is a washer seen on the top of Fig. 2. Fig. 6 a bolt at the end of Fig. 2. Fig. 6 the bolt in the middle of Fig. 2, shown in No. 1, also in Fig. 3, No. 1, at the end of the horizontal piece in the middle. Figs. 6 and 7 are the bolts. These trusses are liable to fail when they have not an iron tie as in Fig. 1. The truss described in Plate XXIV. is entirely made of iron. No. 1 is the perspective elevation ; e and f are screws in order to tighten the two iron ties between a and b. It is also tightened by means of wedges shown at A and B. The recesses at i and k are sockets for the ends of the bind. ing-joists. No. 2 is a longitudinal section showing the ends of the binding-joists, ceiling-joists, and bridging. joists longitudinally. No. 3 is a plan of the joisting, seen from the top. No. 4 is a plan of the joisting inverted, showing the tie-rods. Nos. 5 and 6 are transverse views at ends. When trusses of this kind have the proper strengthe they are superior to wooden trusses. ROOFS. 22. TIMBERS IN A ROOF DEFINED. [Plate XXV.] In Plate XXV. the pieces of timber which lie on the tops of walls are called wall-plates. The pieces of timber which extend over the area and rest upon each wall-plate, are called tie-beams. The piece of timber which rises from the middle of each tie-beam perpendicular to it, is called a king-post. The two pieces of timber that branch out from the bottom of each king-post, are called struts. The pieces of timber which rise obliquely from the ends of the tie-beam, and which meet 1 96 [Part II. PRACTICAL ARCHITECTURE. . the king-post at the top, and which are supported in the middle by the strut, are called principal rafters. The piece of timber which is supported by the ends of the tie-beams, and overtopping each wall- plate, is called a pole-plate. The piece of timber on either side which is supported by the principal rafters at the middle, and parallel to the wall and pole-plates, is called a purlin. The piece of timber which runs along the tops of the king-posts, is called the ridge-piece. The pieces of timber on either side which are supported in the middle by the purlin, at the bottom by the pole-plate, and at the top by the ridge-piece, are called common rafters or spars. Fig. 2 is a plan of the roof, and Fig. 3 is a section made in the manner of working drawings. CONSTRUCTION OF HIPPED ROOFS. [Plates XXVI.-XXIX.] " Laler. 23. Plates XXVI. XXVII. XXVIII. and XXIX. exhibit the construction of hipped roofs. By means of the plan we have the seats of all the lines; therefore, if we have the heights of these lines, we can easily find their lengths by drawing a right-angled triangle, of which one of the legs is the seat of the line on the plan, and the other the difference of the heights of each end of the line from its seat.-See ORTHOPROJECTION, Problem vi. We may find all the angles from the principles of projection ; but the method employed throughout all the figures exhibited in the four plates is that used in the construction of trehedrals, Fig. 3, Plate XXIX. Draw Hi perpendicular to the side CD, meeting the seat g e of the edge of the flat in i. In ge, make iI equal to the height of the roof, and draw HI for the length of a common rafter. Let k be the upper angle of the end of a purlin in HI; draw m n parallel to HI; from k, with any convenient radius, describe the semicircle mlon. Draw kl perpendicular to HI, and ms, n u parallel to CD: draw also lr, k q, and o t parallel to the height of the roof, and join AĒ. Draw AB perpendicular to A0, No. 1, and e f perpendicular to DC, Fig. 3, meeting CD in I. In AB No. 1, make Ad equal to DI, Fig. 3, and join de. Let g be the place of the purlin in the rafter AE. Draw gf perpendicular to Ae, meeting Ao in f. From f with the radius f g describe the arc gh meeting Ao in h. Draw f n and h l parallel to AB, and draw g k parallel to AB, meeting de in k, and draw k l parallel to Ao, and join il; then hil is the inward bevel as required. 24. But we may here find the other three inward bevels of the purlin for Fig. 3, in a much more rapid manner than that now described. For this purpose, draw g2 perpendicular to CD, meeting CD in 2. Also draw lines from e and f perpendicular to AB. Proceeding now with No. 1: In AB make Ac equal to C2, Fig. 3. Produce li and Ae to meet each other in 0, and join c o, meeting f n in m; then fi o is the inward bevel of the purlin on the hip represented by its seat De, Fig. 3, to CD, meeting gC in r, q, t. Draw r s and t u parallel to Hi. Join qs and qu; then pgs will be the inward bevel of the purlin, and pou that which is applied in the side of the roof. 25. Fig. 1, Plate XXVI. exhibits the development of a roof upon an irregular quadrangular plan. The principle is shown under the article SOLID TRIGONOMETRY. In such constructions it is much better to terminate the side of the roof in a plane parallel to the horizon, than to make it meet in a SC With regard to the inward bevel of the purlin, the method of finding it is more particularly CAR PENTRY. CONSTRUCTION OF HIP ROOFS. PLATE XXVI, Menu . .- Friq. I. Fig 2. Fig. 3. -- - Fig. 2. 1.1. Invented Klimawm-by P. Nicholson A Fallarton&C'London& Edinburgh ( Of I CARPENTRY. CONSTRUCTION OF HIP ROOFS. PLATE YXYIT Fig. 2. Hiy. 2. 0 ---- Fig.l E -- - - - - - - - - - - - - J!. 12. 19-3. Fiy.f.' lig.3. Invented & Drawn by.P. Nicholson. A Frillzatu 8-Landa dicingh SH. CARPENTRY. CONSTRICTION OF HIP ROOF S. PLATE XXVTIL. 10.7. NO 7. Fig.2. W AR Fig. 1. NO2 - ŽA 729.3. N°/ AK I ----------------- ----- 1 to ! my V3 N: 2 Drawn by P Viholson. A Fullartond (Lordul 3. Edinburg CARPENTRY. CONSTRUCTION OF HIP ROOFS, PLATE. XXIX. Fig.2. NO2 ra. A Vol.le Fig.2. D .-- . . . - - BE 11 2 H Fig.3. Fig.2, NOI. Fig. 2. 1 2 . LI ---...com Fia:3. NOM. I'1o. 3, NO 2. S IM .. . .4.m.. .. ........... SO VB . Drawn by P. Nicholson, UN A Fillarton & CºLondon & Edinburgh Iori inio SECT. III.] 97 CARPENTRY. shown in Fig. 1, No. 1, which, though apparently different, is the same in principle as Fig. 3, No. 1, Plate XXIX. 26. Fig. 2, Plate XXVI. was intended to show the geometrical demonstration of the principle of finding the inward bevel of the purlin. We shall, however, pass over the investigation, leaving it to the learner. 27. It will not be necessary to explain every particular figure in the construction of hipped roofs, as the same principle applies to all, whether the plans be regular or irregular; only observing that, when a plan has all its angles equal, the finding of one backing of hip, or of one bevel of a purlin for any hip rafter, serves equally for all the others. By the development of the inclined sides of a roof, we obtain not only the bevels of the purlins in the sides of the roof, but the bevels at the top and bottom of every rafter and jack rafter in the most natural way. 28. Plate XXVII., Fig. 1, Nos. 1, 2, 3, show how the bevels are applied in order to back the hip rafter. No. 1 is a right section of the hip. In Nos. 2 and 3 the point a is in the middle of the thick- ness at the upper end or back of the rafter. No. 2 shows how the bevel is applied on one side from a; and No. 3 shows how it is applied on the other side. When the plan is irregular, as in Fig. 1, No. 1, Plate XXVIII., the most expeditious method of finding the lengths of all the hips is exhibited at No. 2. The construction is this :-Draw an indefinite straight line At, and draw tТ perpendicular to AT. Make tT equal to the common height of the roof, and make TA equal to the seat of any rafter ; then AT joined will be the length of that rafter. To apply this to the present example, make tA equal to QA, No. 1; t2 equal to QB or Q2, No. 1; t3 equal to Q3, No. 1, &c. Join AT, 2T, 3T, &c. ; then AT, 2T, 3T, &c. will be the lengths of the rafters represented by their seats, QA, QB, Q3, &c. To find the backing of the hip for any one of these rafters, as that whose seat is AQ, No. 1:-In No 1 make the angle QAt equal to tAT, No. 2, and proceed as in the former cases. To find the bevel of a purlin by the principles of projection :-In Fig. 3, Plate XXVIII. let FGH represent the angle of a purlin where FG is in the plane of the roof, and draw Fl, Gm, and Hk par- allel to AB, meeting the seat Bt of the hip in l, m, k, Draw k i perpendicular to Gm, meeting Gm in j, and Fl in i. Any where apart :-Draw the straight line MN. Through any point J in MN, draw KI perpen- dicular to MN. Make JI equal to GF, and JK equal to GH. Draw IL parallel to MN. Make IL equal to i l, and JM equal to jm, and join ML and MK ; then the two bevels for cutting the purlin are NMK and NML. 29. A method of finding the backing of the hip without the principles of the trehedral is as follows: -See Fig. 3, Plate XXVIII. Let it be required to find the bevel of the hip whose seat is Ct. Make the angle tCT equal to the angle which the back of the hip makes with its seat, or because the angle CtB is a right angle: in tВ make ⓇT equal to the height of the roof, and join CT ; then will tCT be the angle which the hip makes with its seat. Draw a line parallel to Ct, at a distance equal to half the thickness of the hip, meeting CD in R, and draw RS perpendicular to tС, meeting tС in S. Draw SV parallel to CT, and QU at a distance from CT equal to the breadth of the hip, supposed to be measured in a vertical plane. Draw QO perpen- dicular to CT, meeting it in 0, and SV in P. Having made this preparation, we shall now show how to apply the lines to the hip itself. Let No. 2 represent a part of the hip-rafter at the lower end, made to the breadth and thickness of the hip-rafter. 98 [Part II PRACTICAL ARCHITECTURE. Let BHFE be a rectangular plane representing one face of the hip which is to stand in a plane perpendicular to the horizon, and let BHKC be another rectangular plane at a right angle with the plane BHFE, representing the top of the hip before it is backed, and let the plane FHKP be the end of the piece. In HF, No. 2, make HL equal to OP, and in KP make KM also equal to OP. Bisect HK in I, and join MI and LI; then LIM is the backing of the hip. Draw IA and LD parallel to HB cutting away the triangular prism AILD on the one side, and the similar one AIM on the other, we shall form the back of the rafter as required. 30. TRUSSES EXECUTED. [Plate XXX.] Fig. 1, truss of the roof of the chapel of Greenwich Hospital. Fig. 2, truss of the roof of St. Paul's church, Covent Garden. Fig. 3, truss of the roof of the late Theatre, Drury Lane. [Plate XXXI.] Fig. 1, truss of the roof of Covent Garden Theatre. Nos. 1, 2, 3, 4, exhibit the double posts, and the method of fixing the irons. Fig. 2, truss of the roof of the present Theatre, Drury Lane. No. 2, sections, at the wall, of wall-plates, &c. No. 3, the method of strapping collar-beam and queen-posts at each end. [Plate XXXII.] Fig. 1, truss of the roof of Islington Church. This roof has failed, and tie-beams have at last been introduced. Fig. 2, truss of the roof of St. Martin's Church in the Fields. [Plate XXXIII.] Truss of the roof of the new Church of St. Pancras. Fig. 1, entire truss. Fig. 2, half truss, showing the parts more distinctly. [Plate XXXIV.] Fig. 1, truss of the roof of the Bourbonic Theatre, at Parma. Fig. 2, truss of the roof of the Fenice Theatre, at Venice. Fig. 3, truss of the roof of the Basilica of St. Paul's, without the walls of Rome. No. 1, section of the purlin. Most of the principals are either in two or three pieces: these have two or three scarfs. When one, it is in the centre, one pair of principals having their tie in one piece that is 78 feet long, without the least support from the trusses of the roof. CIRCULAR ROOFS OR DOMES [Plates XXXV. and XXXVI.] 31. Plate XXXV. shows several modes of construction. (AR PENTRY. TRESSES EXECLTED. ᎢᎦᏓᏆᎵ1! w mappe 11- TDAU L A ANI . wi 12 NN. .... . ...... IT . www www. ES . www Fig.' . . om man .. - // Alt NA Ta www. www. namas com IR ARGER ! he NS NE SYSTEM ASA S *O ... SER SANASSA MAALASSA ASSESSE = RSS HIP 27. Chapel of Greenrich Hospital, . VI 111 . www . Www. . www . www . www. * SU ht www . N www. UK INT- UNIA N www . . W Fig. 2. .. . IND win ww W ww www 1:10 C - AM S - - ARAN A- . S ANA SA SA VITI UN 277 50.3 W Church of St. Paul, Covent Garden . .. " VIII TIL 2 MIVII NINI www 1 WY anteriormente W IWWII www pr . e ww INI minnet TIIVIIN www. n . . TI w a . w M www ww omentant wwwwwww . www www more ART/ . BAS MAN VIII VIII VIII SA . ES mendeare NA Fig. 3. VI IN . . RAM wwwwwwwwwwwwwww I EPI RYHMR . more than Maramures lanie e I 1 I w WMNS SARAS NII UU YYNTI 80. 3 Roof of the late Theatre. Drwy Lane. NA! WANITA VIIIARA VYRAVILI WWW LLE WWWWWWWWWW HI he Almak!!! Dininn í Minnistin. A Fullarton& C. I ondan/Edinburg CARPENTRY. TRUSSES EXECUTED. PZATE,XXX7 14. N° 3. 2s N. 2. ... S F S - ... .. . . . .. Fig. 1. : .. . ile Fig. I. RE 'T . 11, . AMBI Hi = . . IS - - - - - - . - - Fin A Roof of Covent Garden Theatre WIMMINEN: WIN! Fig. 2. N. 3. A Fig. 2. N° 2. lil SA : - TS -.. - - m A - - - Fig. 2. 9°1 S2 Sil NOL - SP S WATER . - . WHI E -::: ! . .... - - . -- . Root' of Dreary Lane Theutre. - brawn by PNicholson A Fulasini kindad Eunburgh CARPENTRY. TRUSSES EXECT TE]). PLATE. XXXII. N . www. ---- .. ww . . www ... -... WWW . . S . he . centre www. . . . www Fig. 2. III . . w www . ... mer WIITTI who e III . Women T .. .. . . . . . . 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W11? . www ANNA 1 NENS IN SER i- RSS LUI Ili WIE . - - - - ---- - - - - .. . - - - - - - - - - - - - - - - - - - - **- - 60 PE- RA SALE Roof of the Church of S? Martin in the fields. TY P.Nindulsin. B. Zirrull. A Fruilauks & [.. Editori TUNA CA CARPENTRY. TRUSSES' EXECUTED. PL27 XVIZIT Fig. 1. TATTUMTIITINIJUMTIMITETTIINTITUTITUTUNUTTIINNI HITTTTTTTTTTTTTHETITIETEIKTIMATUTIMITETIMIT IMMTIMITEIMUMETTITIITTITUITITANITITTEINTITOIMINTTNINEITTIITTIMET 2 TIITTIMIITTIIIMIITMIIIIIIIIIIIIIIIIIIIIIIITEITETITMEITTHUMTIMINTITIINITTETEMENTTIEN 2 BEN lum 74 12. D ..…..….........60 ...... Roof of St Pancras New Church.. . II POS. Fig. 2. Imitationen autmlOGIMINIMI LETIH 114 YA TINNITUMBO TODINU TWITTER 11111 hili. 11: 1 mm L C w ww. . . . . SS IMA. www. WWW !1! 1S 1,715, E - - - NA C AN 27 . Oce PER om NA - w 4 V ch IUNIL DIE . VILAS EN - IS : . III I/ CREATIVE II - e- Half of one of the Truses of Roof to a larger Sale *** . . Ce //:// www. - - - - - - UI E . 2. TOP . I 11 minuuniin . Fig.3. Iillll VIZ IRA K - SOSIAL ETTI UTIH N ISUZU 1. Morton, E.Turrell. A Fullarun (Londoli Edinburgh CARPENTRY, PLATE XXXIV. TRUSSES EXECUTED pw Memy . . . . .. . . .. . ... .. Fig. 1. . I www . www .com ww www.www.w www wwwwwwwwwwwwwwwwwwww ... 1. mm S . When wwwwwwww www LG www. www. wwwwwwwwww w ww MR ETTO El . IN IITTIS 11 . SINGLE BIRD + 4 . . LUI QUE - '. - W 11tW VAN 10111/nitt 10.110111 ML101 1111111111er . W THION KITA W hinja 20 the_ o 2 0_ 20_ 130 Bourbones Theotre at Parma. . WWW . w . ww Fig. 2. www. ww . . . www . - CUTE .. www . U w www ntuk ww ww. . . - - - Web Lom- .- - - -- SS * K . 1 + T- HEN SIL ein an RE 4 .- . . NG . - LIITIKA 12 T 272 SI TH IM INS V ܠܠܠܠܠܠܠܓ ! 30 _ to _ INITI ZIU Theatre at Venice. www. www . www www . . www. www ... . . . II 1 TEC . . 7 TI Www VINN h EU 1 . I . II th II nh . www. www. . erret HINN . . www m .. w ewed www. . w www . Fiq..3. . ww . h www * . . .. I . - - - - - - - - - . - , - - - - - AU- - we w HI AN . www .. . w Www w . . .. . . wwwwwwwww www www. me . wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww . www . ww. www . w . . www www. www. . . . w + = . - - = - = AN SER ANAL AVE - 11 - S NA SEL AY . - - ER - SRL . . IIII 1. ti ) 18/12 ? Basilica of St Paul. Rome : Presented hi Thomas Inverton Donaldson tisa.. Architect. Engras'ed by E. Turrell A Fullarton & Co Londan Edinburgh CARPENTRY, FORUS OF COVSTRICTION. Ꮮ1ᎢᎬ ᏦᏒᏗ Fig.1 Fiq.? /Illes who Lei there www . - SOSYS - - - .- S3 - - W WINNIN . . W - HWILI .! . - -- W SEE W ", - NAR 3 - AS w - .THENTNING - - - Hit! - IM .. Stall !! SX . . . WhIIKIMEWWANY will - SEBE S . illi .IN 2 Winter Antonio . Hvo Williwa b A W . EESSA HI . : T! Huillit Institution almartin MINUDDIN SEE - អាយុយទៅយកបរបរអរគឺ WHOITUMIA - Willkomlin W BILINICAL minilanman DUUNNHILDRAWI WAHUI iH! II/ Wiki WEDDIK -- WHILTUNEN - - WAHILIURETIMAMURI SUNNURU - ** - - . - - - - WW.311 , - F BA - - - S SER S RS G - - - - RE - . THITTAUTUMIN ntuk it -" ; NA S - - +- - B - - - _ HAR will SS * SAAT . - ..! HOLENDE - -- E U / VIP 11 L ' !! ... I.' ! am INI G 1 UND T ! TITLE I '. TI UJIT111 ... MINUL Un S .. . - - - - - -- -- - - - -- -..- - VO? . jos oca * - Fig.3 Fiq.& - ! . .. ANIU W WAN T .. A ' :'? SAMA. - - - 171 U INN llengua # 2 a H, Wismi 1 - - 2 : A S UA S .- UB, . + : WI - - iliminating WWW -1 - - - - = . W SIS Y S . AINS Wmu **** *!11!! Z = i . B . * u . WINRIHIHIWww . . Sys n . . - millal . . - - .. iliitatilinde VIIN SS wa N illit - Pri - MA . R - - A . . - . muslim dan Nyi VY ! A De Rii 1 * ili " . - T1411 .- TIITIN N - SS: -- . HRT - KINH millisinilo!!! - - - HUWUND . YV 3 ili . KY IUTARTIRILIE . 101!!91 ....! Walusiaci... vid iti, - UT IRU. WWLWWwills L N" ?. V. 2. P inhol.n. 1. Lors. A Fallsrton. London Edinbur-b CARPENTRY. TRUSSES EXECUTED. . PLATE XXXVI. SEE NA I ILLE- ZA . . TT/ SC hy E . bilmituntil 1991 militiattini TOKMUUTTIMIT WIE SAGA we - 11 / RE 11 JUS IN . WILL ........................... ... . CITI ISUZ | 111 VIII11 UN 11 22 VA Dome of the Cathedral of S. Paul, London, Introduurd hP..Vicholsn Erupnuved by E.Torrell. A Fallarton & CºLondon & Edinburgh NCH CARPENTRY. PLATE XXXVU ROOF. . NO 7. N. 2. ANO3. Fig. 1. no 4. --- om V!9. .. Figi - - ...... FHF ........ . -------- No4. E Ehf No3. NO 2 --- . :888::::: we N8. E 11 --- CH EN --- th -- 9 TI" - -.-- -.-. N.10. Engraved by Turrell. Designed & Introduced by. Collinge Esq?'Engineer. A Fullarton. COI ondan Edinburgh mic, CARPENTRY, . TRUSS PARTITION. PLATE.XXXVIII. Fig. 2. U11 BE MIN119 - IMENO - - RS ANA 4 DIA - - 272 ww.no NIMIT 150 CE - w 1 UNI S IN W 1 NA RE AL- Tot ME 2 TA IL th 22 KA222BEE • -- I AV11 2 1 MIVAL . 22 23:22 YA EEEEEEEE NA TA 22: . RE - - - MORE - VAATA MA UN es Fig.2. E w NE ull - - - www ti 1 ul . Nu PIW HI 1.li.i wi EES . H 111!1. TEDY 2222 Biz 200V HRH,. NE ALLAW UT IVIC ZUAR -94.2 - VIII SA L. PL YTI ANA 14:VELLA ? he MI . 37 BEZ - E- SUN ANG LE SER MU EAA.21 23. X Intrertunt l . Foster. 4 Fuilar von&Co London&Edinburgh ch CARPENTRY, TRUSS PARTITION. PLATEXRXY. IMIT NEW TU NEW SE ET . . . SA IIIIIIIIII. 111111 1 HA ma - - HI - UM RIS IIIIIIIII TIESA > * ANA WA V ARA SA V 11 - IS I - . -- - IR AN SRL EUR N - . H. AN PARA IM I SS IT SI LIIT V . - he MINI 1 , - Il 21 J- WS D W 11 w . www . . www www w . . _ _ www. . .. RE SER IL VW - . . .. . . Want HA .. ENEN ER . www R V the TEN Il W w ..... www .. . . .. . . . * . . . .. . www. AA . - . 1 ... . ww . www. w . 1 www . - AN WR _ www SA ITIES WE AT I . NA WINNI VILA IMA IONIS * ... . . WW RA www . www ... www wwwwwww . www. w 111 1111 A. LV - www .. 9 - NA w VIEWS - - - - . . . 4 WE Il LE NA . - - . . Introduced by:P. Nicholson. En pript In E wll. A Fazlar onCloucou aindu : ان تم Sect. III.) 9.9 CARPENTRY. Fig 1 is the most simple construction of a circular ribbed roof. Fig. 2 is a circular ribbed roof, with a purlin for a building of greater extent than Fig. 1. Fig. 3 is a different construction without purlins, where, in every three consecutive rafters which pass through the axis, the bays on each side of the middle one are filled with rafters which stand in equidistant planes parallel to that in the middle. The edges of the parallel rafters on each side are circles of a less radius than that middle one. The principles of constructing spherical roofs depend entirely on the sections of a sphere. Fig. 4 is another method where the bays are filled in with strutting-pieces which have the same curvature as the ribs themselves, and of which the sides are in planes passing through the centre of the sphere. This method of construction is easily executed, and well calculated to resist the effort of falling, or the violence of stormy weather. Plate XXXVI. is a section of the roof of the dome of St. Paul's Cathedral. 32. CONIC ROOF. [Plate XXXVII.] Plate XXXVII. is a plan and elevation of a conic roof, by Mr. Collinge. No. 1 is a cast-iron top cross : a plan and view drawn to a large scale, is shown at C and D. The dotted lines are the principal rafters E, which terminate at the points a, a, and are there bolted. No. 2, cast-iron sockets that receive the ends of the purlins a a at every angle, (the principal rafter lying between them,) and are bolted together as shown at F. No. 3, cast-iron sockets that receive the lower purlins, as No. 2. No. 4, cast-iron angles that receive the ends of the plates, on the under side of which is a socket for the upper part of the post a to slip into, as at G: 66, plates intersecting at c, where the principal rafter E pitches. The whole is tightly secured by bolts. No.5, a cast-iron step which rests on the surface with a socket, as H, to receive the bottom of the post. 33. In the 'Edinburgh Encyclopædia,' under the article CARPENTRY, p. 528, Mr. Nicholson describes the properties of a circular roof with regard to strength, in the following words :—"A circular roof may be executed with timbers disposed in vertical planes, whether the ribs or rafters are convex, concave, or straight, without any tie between the rafters or ribs, even though the wall-plate were ever so thin, provided that it be only sufficient to sustain the weight of the roof, by joining the wall-plate, so as to form a chain, a ring, or endless plate, and by strutting the rafters in one or more horizontal courses, without any danger of lateral pressure, or of the timbers themselves being bent by the weight of the covering. But the same cannot be done with the roof of a rectangular building, for single parallel rafters would not only obtain a concave curvature, but would thrust the walls outwards. Hence the means of executing circular roofs with safety are simple ; but those for straight-sided buildings are complex, and require much more skill in contriving, according to the utility of the space between the rafters, which may be found necessary in forming more lofty or more elegant apartments, as in concave or coved ceilings.” This description of circular roofs has been completely verified in the subsequent execution of the conic roof by Mr. Collinge. 34. TRUSSED PARTITIONS. [Plates XXXVIII. and XXXIX.] Plate XXXVIII., Fig. 1, is a trussed partition with two small doors. 100 [PART II. PRACTICAL ARCHITECTURE. Fig. 2 is a trussed partition with folding doors. Plate XXXIX. is a trussed partition, with an aperture for folding doors, and another for a common door. CONSTRUCTION OF NICHES. [Plates XL., XLI., XLII., XLIII., and XLIV.] UN 35. A Niche is a recess in a wall for the purpose of placing some ornament in it. 36. The backs of niches are generally formed of cylindric and cylindroidic surfaces; and the heads of spherical, conoidal, or ellipsoidal surfaces. 37. When a conoidal or spherical surface is employed in the formation of the head of a niche, with the axis of the conoid vertical, a cylindric surface, having for its axis the continuation of the axis of the conoid, and for radius that of the base of the conoid, is employed. 38. The principles of spherical niches are the most simple possible. As every section of a sphere is a circle, the form of the edges of the ribs for sustaining the lath and plaster of a spherical niche must be circular. With regard to the position of the ribs of a spherical niche, when the sections of a sphere all inter- sect each other in a line passing through the centre, the radial sections are all equal; therefore if the head of a niche be spherical, one of the easiest methods of constructing the ribs is to make them all intersect each other in a horizontal line passing through the centre of the sphere.—See Plate XL., Fig. 1, where the ribs of the niches all meet in a horizontal line passing through the centre of the sphere. Here all the ribs have the same chords as the half plan. In Fig. 2, all the ribs meet in a vertical line which divides the front rib into two equal parts. To find the radius of curvature of any rib, complete the circle of which the inside of the plan is an arc; produce the middle line of the plan of any rib, as of di, to meet the opposite side of the circumference in g; on the whole line d g, or its equal fk, as a diameter, describe a semicircle, and from the point i, where the ribs intersect, draw a perpendicular į k to meet the semicircular arc in k; and the portion fk of the arc intercepted between the perpendicular and the back of the niche will be the curve of that rib. In Fig. 3 all the ribs are in vertical planes, and perpendicular to the front. Draw a line Dg through the centre D of the plan parallel to the seat of the front rib. Continue the line of the seat of any rib, suppose that of f h to g. Then from g, with the radius gf, describe the arc f k meeting the inside of the front rib in k. Draw De perpendicular to f g, meeting f g in e. From g, with the radius g e, describe an arc meeting the seat of the inside of the front rib in i; then the arc f k is the curve of the one side of the rib represented by the seat h f, and e i is the curve on the other side of the rib, and the distance between these two curves f k and e i exhibits the quantity of bevel which the edge of the rib requires. In Fig. 4 the ribs are all parallel to the front rib; consequently the diameter of any rib is the in- tercepted portion of that rib by the inner line of the plan. 39. Plate XLI. exhibits the plan and elevation of a spherical niche. Here the ribs are all in vertical planes passing through the centre of the sphere. Let lg be the middle line of the seat of a rib, meeting the centre of the sphere in g on each side of the line lg; draw a line at a distance equal to half the thickness of a rib, and let these lines meet the seat of the inside of the front rib in the two points o, P. Draw 00, PP perpendicular to lg, meet- ing the inner edge of the arc of the plan in two points o, p; then lop is the length of this rib, and S CARPENTRY. CONSTRUCTION OF NICHES. PLATE XL Fig. 1. - Fig. 2. yo 2. NO 12. B 12: . Frig: 3. Fig. 4. NO 12. NO: Na 1. Invented & Druwn by Nicholson.. À Fullarıon & CL näon&Edmborgh. . ion Wiv - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - I OAI 11 It - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - HAWA --- - - - - - - - - - - - - -- - - - -- - - - - i ? M - - - - - - - - - - - - - - - - - - - - - - - - -------- -------- - - - - - 1 - - -- - - - - - - --- - - - - - - -- -- - - - - - - - - - - - - - - * - - - - - - - - - - - - - - - - + - - - - - - - -- - - - - - - - - 10 - - - - - - - - - - - - - - - - - - - - - - Imania - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - . - FRONT - - - OF -- - - - - - -- - -- - - --- - - - - - . Apillartoni&C London & Ediaburgh --- ------------------- - - - - - - - ... - - -- -- - - - - - - - - - - - . . . - - - - - - - - - - - -- CONSTRUCTION OF NICHES, - - -- - - - -- -- N. 2. - CARPENTRY. - & Down by P. Nicholsun. - - - NICHE -- RIB. . --- --- --- --- --. . - - - -- -- - - - - -- - - - - It' - - 1 HANHIMOINE - - : . -- ----- - ------ - - - ---- -. - .--. -- - - - - --- - .- SS . -- AVY 1 -- * - -... ܫ ܫ ܝ ܝ ------------ ܝ - - - - - - - - - - -- - - - -- - - - -- - N: 3. . PLATE XLI "1: ' . L L.T. TV. -...-- --- PNicholsoni.. Mr 4 . 2 . - - -....-.-.-.- . --. . ... ...--. .... . . . . . . -- <3 - . . . . . . - . .. .. .. .. * . .. - - - - - - Abularwn&Co. Landon&Edinburgh PLAN OF NICHE. 1. ELEVATION. CONSTRUCTION OF NICHES. CARPENTRY. ..-.-.-.-.- .. . ! --- - - - - ---- PLATE XLII. ..." Oo H. Lowry. CARPENTRY. CONSTRUCTION OF NICHES. PLATE XLIII RE 2 . *** NE ce f R . NO 2 PLAN RE 1.1. www . mi MESTOSTA - www. A ra 2 21 we UN - - - - - - - - - - - . - - - - - - - II - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = GRASS NA NE WHY M LR - - - - - - - - - - - - - - - - -- - VY - - CC - ZA - N ZOO - 22 - si w I NO 3 ST CE RS XABI IR CE w ZONA . V AS A WON S B JA M NE . ZA CAL SA VON NO 2. . Rx S R F W F VAN wwwvw WUMINIUM ES SIE. ... SCH 1 ti MMM- SRL 4 q: COCCOLES - HO - S - - - - Iti. WWII!! 2 Feet Inches Improved & Drawn by T. Profser. AFnllarton & Cº London& Edinburgh CARPENTRY, · CONSTRUCTION OF NICHES PLATE XLIV. Fig. 1. Fig. 2. NU NO | 2. .. . . . . . NO 11 . . . . . . .. . . . . . No 11. . . . . . . . Fig. 3. Fig.4 - ********** .......... Nola N 2. - - No 1. W Introduced by M. Nicholson. Engraved by E.Iurrel. A Fullarion. C'London, Edinburgh SECT. III.] 101 CARPENTRY. the distance between the lines o d', p p shows the quantity of bevel to be taken off the top to fit against the front rib. Nos. 1, 2, 3, show the ribs by themselves as found by their seats. 40. In Plate XLII. the plan and elevation are each the segment of a circle, and consequently the two chords must be equal. In order to describe any rib: Through the centre H of the plan draw a straight line, and produce the arc of the inside of the plan to meet this line, and the intercepted distance will be a chord. Draw the line HZ perpendicular. Make HZ equal to the difference between the radius and versed sine of the front rib. Join the extremity W of the chord and Z ; then, with the radius ZW, from Z describe an arc which is the curvature of all the ribs ; the length of any rib may be found by applying the length of the seat of that rib from the extremity X of the chord upon the half chord WH, and, drawing a perpendicular to meet the arc, the portion of the arc between W and the point where the perpendicular meets it, is the length required. S 19 spherical niche in a circular wall on true principles, and in a more comprehensive and simple manner than any thing of its kind ever before published. No. 1 represents the plan, and dge the back or base rib; d be is the plan of the front rib, which should be got out in two or more parts, and jointed perpendicularly over the letter b. Let the distance between the lines b e and GO, Fig. 2, represent the thickness of the plank required to cut out the front rib; let dhMN represent the mould for marking the front side of the plank, which may be obtained as follows: Draw db, and continue it to I; draw bM perpendicular to dI ; let F be the centre, and draw FL parallel to 6M, crossing the line dI at E ; then E will be a centre for describing the mould d hMN. The mould for the back side of the rib may be obtained by continuing the line OG to K; then G is the centre required for describing the mould OPQR. Take the mould d hMN, and mark each side and end on the face of the plank; then cut it off truly to the line d h, and at right angles to the face of the plank: from the line be take the bevel at o, and apply it to the end of the plank at h, which will give one point at the back edge of the plank for applying the mould. From the line d b take the bevel of the joint at b: cut the end of the rib to the line MN, and to the bevel found at b. From 6, No. 1, draw a line at right angles to the line FG until it cuts the inside of the plan, which line and the curve of the plan will give the bevel of the under side of the rib at M, which will give another point at the back side of the plank to apply the mould OPQR; then work to the lines so obtained, which will give the upper and under side of the rib. Divide the curve line dN, No. 2, into any number of equal parts, and from those points draw lines parallel to EL across the face of the mould, and continue them to the line db; transfer the lines crossing the mould to the face of the rib itself, and take the corresponding distances from between the straight line db and the curve line db, and transfer them to the upper and under sides of the rib; trace curves through the points so found, which will stand perpendicular over the plan. Or, Take the stretch out of the curve line dN, and lay it out on a line as at No. 3 ; take the cor- responding distances from the line at No. 2, and trace a curve through them; then No. 3 will be a mould that will bend over the top of the rib, and give the curve required. Another may be obtained for the under side of the rib by proceeding in the same manner. 42. Plate XLIV., Fig. 1, a niche upon a circular plan and elliptic elevation. In this the ribs are set in vertical planes, and, though elliptic, are all of the same curvature. In Fig. 2 the plan is an ellipsis, and the head a semicircle. The ribs are here fixed in planes per- 102 [PART II. PRACTICAL ARCHITECTURE : pendicular to the plane of the wall. The position of the ribs in this diagram is exactly the reverse of that in No. 1. The ribs of this niche are all of the same curvature and the same length, being each a quadrant of an ellipsis. In Fig. 3 both plan and elevation are elliptic. The ribs are vertical and perpendicular to the face of the wall. The method of finding the ribs is shown in the Problems of STEREOTOMY. In Fig. 4 the ribs are disposed in vertical planes parallel to the front rib. The plan is a semi- ellipsis, and the elevation a semicircle. PENDENTIVE CEILINGS. [Plates XLV., XLVI., and XLVII] : 1, be the ature of scentive 43. Let abcd, Plate XLV., Fig. 1, be the plan of a room or staircase to be bracketed, so as to form the surface of a pendentive ceiling. Let aSVe be the section across the diagonal; it is required to find the curvature of the springing ribs. Draw iL perpendicular to a b meeting a b. Take the distance from i to the line ab, and set it from i on the line i a, and from this point draw a perpendicular to meet the curve as of the diagonal rib. Make the versed sine of the segment alb equal to the perpendicular, and describe the segment alb, which is the springing line required. If from the centre i an arc be described with a radius equal to the length of the seat of a rib to meet the seat of the diagonal rib i a, and if from the point of meeting a perpendicular be drawn to meet the curve as, the portion of the arc of the diagonal rib between a and the perpendicular will be the length of the rib corresponding to the seat which was taken. In Figs. 2 and 3 the diagonal section is a semicircle. In Fig. 4 the plan is an octagon, and therefore the springing lines on the sides are semicircles. 44. In Plate XLVI. pendentive ceilings, the same principle is observed as in Plate XLV. and the bracket from which all the others are taken is the quadrant of a circle. The portions of the brackets to be used are exhibited at Nos. 1, 2, 3, 4, 5, 6, found according to the length of their seats. In Fig. 2 the bracketing is upon an elliptic plan, and the general bracket, from which all the others are portions, is the quadrant of a circle as before. The portions of the brackets to be used over each seat are shown at Nos. 1, 2, 3, 4, 5, 6, 7. 45. In Plate XLVII. the springing lines are in the sides of a square pyramid, of which its vertex is above the base a b cd. The edges of the pyramid are tangents to the springing lines at the lower extremities, which are represented by the seats a, b, c, d. We shall now suppose that any one of the four angles contained by any face and the base of the pyramid is given, the angles which the edges of the pyramid make with the edges of the base will be readily found by the principles of the Trehedral. One of these angles is d al, and the angle which the seat ac makes with the adjacent angles is g aH. In order to find the springing curve upon the line a d, we have only to describe a circle aEd to touch the straight line aH' at the point a. Draw gE parallel to a b, meeting the arc of the circle in E; EI' parallel to da, meeting aH in I'; I'i parallel to ab, meeting the diagonal a c in é; and i e parallel to ad, meeting gE in e. We know that, since the springing curve a Ed is the arc of a circle, it will be projected into the arc of an ellipsis, of which one of the axes will be in the line Ee, and the vertex at e. If we now bisect O point where it meets will be the centre. This operation is shown upon the side c d by the line k o, drawn from k, the meeting of the tangents d k, k e". Through O, draw LM parallel to d C for the line CARPENTRY. PENDENTIVE CEILINGS. 72Y PLATE XLI' Fig... Fig. 2. = Wild W CE m . . AZIRI SM - - A - . AR ** . ** N 5. ****** ***** A - F . . SS - - CA -- - ---- A S S , - - - - - ---- - - - - - - --- - - - - - - - - - M- SA 1 2 P . -- -- - - S - A .-... --**** - SN . VA UA 4- SA . .........4 Ft Willett till C SIA EZ Ay CE COOL w ** -- -- - - - - - - - - CSS Fup.3. - - - A RAI LE DICIAL --- - GE - + - co A AL.. . S ** -* W RE SNA . - - - - " . - . - . . - . - . .. -- - - " . - . . - - M CS N .. . . N S LUES - - - Stum... - N Inventud ! b...Vicholson - - r. - 11:41 Tekni. C A R P ENTRY. PENDENTIVE CEILINGS. PLATE XLVI. Fig 1. Fig 1. NO4. 101. 1 Tigl. Fig 1. Fig 1. NO 5. Fig 1. NO 3. Fig 1. V.6. | 2 Fig 2. . . --- . .. O . ***......... NO 7. . . . Fig.2. Fig 2. Fig 2. N. 6. Jl. NO 1. i 1990 Fig 2. Fig 2. N.5. NO2.. wwcoram za - Fig 2. LE Fig 2. NO 4. N. 3. 4 G.H. Swanston P. Vicholson. A Faillarton & CºLondon & Edinburgh CARPENTRY. PENDENTIVE CEILINGS. PLATE ICVI. : Hi -- - . . - . - . -. -. . AG. - - . ---- .............. ...... .......................... ------------.. ::.-'M .---M .... - .. - Invented by P. Vicholsyrit. Engraved by W. Lowa A Frillamos& Leibungành:Irge Sect. III.) 103 CARPENTRY. · of the axis major ; then by Prob. iv. CONIC SECTIONS, describe a semi-ellipsis Ld &*C, of which one of the axes is O em, and d or c a point in the curve. TIMBER BRIDGES. 46. A wooden-bridge is composed of three distinct parts : 1st. The points of support of the super- structure : 2d. The frame-work which sustains the road-way: 3d. The road-way. The points of support for the superstructure, which are the abutments and piers, are formed either of wood or stone; the choice of the two systems depending, principally, on the character of the stream, and the nature of the bed. Stone-piers and abutments are generally preferable to those of wood, being of a less perishable nature, and offering more resistance to floating bodies and the action of freshets. The general arrangement of the stone-piers and abutments of wooden bridges differs in nothing, except in a few details required by the character of the superstructure, from those of stone-bridges. The starlings are carried up above the level of the highest water; and the portion of the supports above them, on which the road-way rests, is a simple pillar with plane faces. The lowest points of the frame-work, abutting against the pillars, should be above the level of the highest water to preserve the wood-work from decay, arising from exposure to alternate dryness and moisture. Wooden abutments may be formed by constructing what is termed a crib-work, which consists of large pieces of square timber laid horizontally over each other, to form the upright or sloping faces of the abutment: these pieces being halved into each other at the angles, and otherwise firmly connected together by iron-bolts. The space enclosed by the crib-work-which is usually built up in the manner just described only on three sides—is filled with earth carefully rammed, or with dry stone, as circum- stances may seem to require. A wooden abutment of a more economical construction may be made, by partly imbedding large pieces of timber placed in a vertical or an inclined position, at intervals of about four feet from each other, and forming the facing to sustain the earth behind the abutment of thick plank. Wooden piers may also be made according to either of the methods here laid down, and be filled with loose stone to give them sufficient stability to resist the forces to which they may be exposed ; but this method is very clumsy, and inferior, in every point of view, to stone-piers, or to the methods which are about to be explained. 47. The simplest arrangement of a wooden-pier consists in partly imbedding large pieces of timber which are placed in a vertical position from two to four feet apart. These upright pieces are connected at top by a horizontal beam, termed a cap, which is either mortised to receive a tenon made in each upright, or otherwise fastened to the uprights by bolts or pins. Other pieces, which are notched and bolted in pairs on the uprights, are placed in an inclined or diagonal position, to brace the whole system firmly. These several uprights of the pier are placed in the direction of the thread of the current; and, if thought necessary, two horizontal beams, arranged like the diagonal pieces, may be added to the system just below the lowest water-level. In a pier of this kind, the place of the star- lings is supplied by two inclined beams on the same line with the uprights, which are termed fender- beams, A great objection to the system just described arises from the difficulty of replacing the uprights when in a state of decay. To remedy this defect, it has been proposed to drive large piles in the posi- tions to be occupied by the uprights, and to connect these piles below the low water-level by four hori- zontal beams firmly fastened to the heads of the piles, which are sawed off at a proper height to re- ceive the horizontal beams. The two top-beams are arranged with large square mortises to receive 104 [PART II. . PRACTICAL ARCHITECTURE. the ends of the uprights which rest on those of the piles; the rest of the system may be arranged as in the former case. By the arrangement here explained two points are gained ; the uprights when decayed can be readily replaced, and they rest on a solid substructure not subject to decay; and shorter timber can be used for the piers than when the uprights are driven into the bed of the stream. 48. In deep water, and especially in a rapid current, it is thought that a single row of piles would prove insufficient to give stability to the uprights; and it has therefore been proposed to give a suf- ficient spread to the substructure to admit of bracing the uprights in two directions; that is, to add, besides the diagonal braces already described, struts on each of the other sides. To effect this, three piles should be driven for each upright; one just under its position, and the other two on each side of it, on a line perpendicular to that of the pier. The distance between the three piles will depend on the inclination and length that it may be deemed necessary to give the struts. The heads of the three piles of each upright are sawed off, and connected by two horizontal clamping-pieces below the lowest water-level ; and a square mortise is left in these two pieces over the middle pile to receive the uprights, which are fastened together at the bottom by two clamping-pieces resting on those of the heads of the piles, and are rendered more stable by the two struts. In localities where piles cannot be driven, the uprights of the piers may be secured to the bottom by means of a grillage, arranged in a suitable manner to receive the ends of the uprights. The bed on which the grillage is to rest having been suitably prepared, the grillage is floated to its position, and sunk either before or after the uprights are arranged to it, as may be found most convenient; the grillage being retained in its place by an enrockment. As a further security for the stability of the piers, the uprights may be covered by a sheeting of boards, and the spaces between the sheeting filled in with gravel. As wooden-piers are not of a suitable form to resist heavy shocks, ice-breakers should be placed in the stream opposite to each pier, and at some distance from it. In streams with a gentle current, a simple inclined beam covered with thick sheet-iron, and supported by uprights and diagonal pieces, will be all that is necessary for an ice-breaker. But in rapid currents a crib-work, having the forma of a triangular pyramid, the up-stream edge of which is covered with sheet-iron, will be required to offer sufficient resistance to shocks. The crib-work may be filled in, if it be deemed advisable, with loose blocks of stone. 49. Frames.]-In no branch of constructions has more diversity of arrangement, or greater bold- ness of design been shown, than in the frames of wooden bridges. Wherever ingenious practical car- penters have been found, structures of this kind have been raised, which, for judicious arrangement throughout, have called forth the admiration of the most scientific. Whatever, however, of apparent diversity may seem to exist in the great number of wooden bridges, they can all be reduced to two general classes ; each of which admits of two subdivisions. In the first class may be placed all the combinations which are composed of straight timber; and in the second, those which form wooden arches. The subdivisions of each arise from the position of the road-way, which may rest either on the frame, or be suspended from it. 50. First Class. —The simplest arrangement of the first subdivision of this class consists in a system of longitudinal beams termed sleepers or string-pieces, laid parallel to the axis of the road-way; they are slightly notched on the caps of the piers, and are fastened to them with bolts; and the sleepers receive the cross-joists of the road-way on which the boards and other road-covering are laid. The distance between the sleepers will depend on their strength, and on that of the cross-joists, which will seldom admit of being more than six feet. This system can seldom be used with safety for a carriage-road, where the width of the bays is beyond twelve feet. In bays of twelve to twenty feet wide, short pieces, termed corbels, must be placed on the caps of + I 4年生 ​, 學生​。 . WWWWWW lr 4 , 1- ff19 . - - 生生 ​- …. 上 ​- - PLATEXLVIII. - 1. it…. 上 ​F11111 -.-l ifi “ 一直 ​賽車手​, ---書畫等書書 ​十一 ​一 ​lol X F1-FFICE - - 事​, "中 ​老 ​m - --- 十一 ​Thirter--.- - - rt - - - - 十一 ​11 , -- - RSSSSSS - 一 ​./ 1 区县团 ​-r.… 上小事 ​\ 年十一 ​| TS- tist; r FILT 山十​.. 一 ​# TIMBER BRIDGES # . CARPENTRY 是在 ​: 一人​, A 北 ​A Fallarton&C London& Edinburgh 要 ​其 ​* 有一 ​一一一 ​一一一一一一 ​*** ****** - - 在中华大学 ​, 一生一世 ​年11 14 。 - . ., : +: -- -- ..TW - -- -省重生者​。 WAN 上一 ​- 其中 ​III - 主 ​学校 ​- - . - - 一 ​。 11 - TH * -- FAA_A 11 - -- TITUALITI LILLINEY1ff事中生 ​- --- 4 美 ​, - 一 ​ffffff 在一 ​* 一一一 ​上一 ​- + 主 ​- - 十一 ​- - KITTAILA 。 -- - DMININNINI 一一一 ​第二 ​- - - - - 长三 ​文 ​: Sect. III ] 105 CARPENTRY. the piers. The sleepers rest on the corbels, which serve the purpose of diminishing the bearing, which may be considered in this case as the distance between the ends of the corbels. When the bay exceeds twenty feet wide, the corbels will be subject to spring, or bend downwards, if made sufficiently long to give an effectual support to the sleepers; they must therefore, in this case, be strengthened by struts (Plate XLVIII., fig. 1.) mortised into them near their ends, and abutting against the uprights of the piers. This method may be used for bays thirty feet wide. Beyond thirty feet and within forty, it will be better to replace the corbels by a beam placed under being sustained by two struts which abut against it. The struts in this system may be so long as to be liable to sag, or bend downwards; to prevent which, inclined clamping-pieces, termed stirrup-pieces, are fastened to them and to the string-pieces. For bays above forty feet, it will be necessary to use both corbels and straining-beams, (fig. 3.) with a suitable combination of struts and stirrup-pieces. The struts in this case may be parallel to each other or not, as may be found most suitable; the angle, however, between them and the straining-beam should not be greater than 150°, to give sufficient stability to the system. 51. The simplest arrangement of the frames in the second subdivision consists (fig. 4.) in a hori. zontal sleeper termed a tie-beam, the extremities of which rest on the points of support of two inclined pieces which are mortised into the principal tie-beam near the points of support; and abut end to end at the other extremities, or otherwise against an upright, termed a king-post, placed at the middle of the principal tie-beam, to which it is fastened by a stirrup of iron, or by a tenon and mortise. The cross-joists are laid on the tie-beams, and with it are suspended from the inclined pieces, by the inter- notched on the tie-beam. A combination of this kind cannot be well-arranged for bays beyond forty feet in width. 52. For bays between forty and one hundred feet, a similar system (fig. 5.) may be arranged by placing a straining beam between the upper ends of the inclined pieces, and suspending the tie-beam and road-way from these points by two stirrup-pieces, termed queen-posts; but it is necessary in widths of more than forty feet to place diagonal braces in the space between the queen-posts, stirrup-pieces, and tie-beam. When the bay is beyond one hundred feet, a longitudinal beam (fig. 6.) is placed at top parallel to the tie-beam; and at a suitable distance from it, several inclined pieces abutting against several straining-beams placed under this longitudinal beam ; and stirrup-pieces are placed at all the points of junction between the inclined and straining-pieces. This combination may be extended to very wide bays. It is the same as that used for the celebrated bridge of Schaffhausen, one of the bays of which was 193 feet span. 53. Another combination of straight pieces for wide bays, consists of a large built-beam, which is formed of two parallel built-beams, firmly secured in their places by uprights and diagonal braces; or simply of a series of diagonal pieces placed close together, and firmly secured to each other, forming a kind of lattice-work. Each of these systems is now in extensive use; and the road-way in both systems may be placed either at top, at bottom, or at any intermediate point between these two. In frames belonging to this subdivision, not more than four frames, or ribs, can be set up for the road-way; two exterior, and one or two interior, dividing the total width into two equal parts, each of which may be only wide enough for the passage of one carriage and for foot-passengers. 54. Second Class.]—The simplest arrangement of the frames of this class consists (fig. 7.) in slightly bending a large beam, and confining it in this state between two fixed points of support, so that it may be made to sustain the middle point of the sleepers for the cross-joists. This system may be 106 [PART II. PRACTICAL ARCHITECTURE. applied to bays from twenty to forty feet wide, where timber of sufficient dimensions can be pro- cured. The most usual form of wooden-arches consists (fig. 8.) in making the arch of several concentric courses of timber bent to a suitable curvature ; the different courses being firmly united by keys of hard wood, and stirrups, or hoops of iron. The sleepers rest on the crown of the arch; and are sup- ported, between this point and their extremities, by vertical or inclined stirrup-pieces, which transmit their weight and that of the road-way to the arch. The position of the stirrup-pieces is not a matter of indifference ; the upright, or vertical position, being superior to the inclined, in which the pieces are perpendicular to the arch; for, in this last case, the inclined pieces not only transmit the vertical weight of the road-way to the arch, but also a hori- zontal compression, caused by the tension on the sleepers, which, acting on the upper ends of the stirrup-pieces, causes an equal action on the arch at their lower extremities; and besides this, the crown of the arch is subjected to a greater portion of the weight of the bridge when the pieces are in- clined than when they are in a vertical position; and as this part is the weakest, it will require addi- tional strength to resist this additional pressure. The arch may be relieved of a portion of the weight of the road-way by inclined pieces or struts, which abut against the points of support, and sustain the weight on the portions of the sleepers near- est those points. This arrangement, which is very judicious, can he made when the road-way rests on the arch. For very large spans the arch may be formed of two concentric built-beams, united by cross stirrup- pieces and diagonal braces. 55. The second subdivision of this class embraces those cases where the sleepers and road-way are suspended from the arch by upright stirrup-pieces of wood or wrought-iron. Here each arch may abut against two fixed points of support (fig. 9); in which case the sleepers may be either on a level with the points of support, or be suspended at some point above them ; or the arches may be let into the sleepers near their extremities. The sleepers in this last arrangement (fig. 10) acting as ties, must either be of one entire length, or be formed of a strong built-beam, arranged with indents, or with wooden keys and iron fastenings. In this subdivision, the road-way must be divided into two paths by one or two arches which rise in the centre of it. 56. Wooden arches may be made to span very wide bays; but in general it would be well to restrict the span to 300 feet; and for streams more than this width, to divide the space into two or more bays, as circumstances may point out; but the arrangement of the frame-work of such considerable struc- tures requires great judgment, skill, and care, on the part of the engineer. Simplicity should be regarded as an essential condition, so that any part may be easily taken out, and be replaced, without deranging the rest. The points where the frames rest against the support should be above the highest water-level, to preserve the essential parts from decay; and it would be a judicious arrangement to leave an open joint between all the courses of built-beams, or other heavy essential parts, which rest on each other, and are connected by bolts or hoops, to allow a free circula- tion of air around the pieces, in order to prevent the accumulation of moisture between them. The different ribs must be firmly connected by horizontal ties formed as clamping-pieces, which are bolted in pairs to the frames; and by diagonal braces, to prevent lateral motion, caused by the action of the wind, or the warping of the frames. 57. Road-way. The road-way is variously formed, according to the greater or less care which it may be deemed necessary to bestow on the structure. The best arrangement consists in laying a system of cross-joists on the sleepers, to receive a flooring of thick boards on which the road-covering rests, CARPENTRY. TIMBER BRIDGES 产 ​2 现代 ​Designed & Superintended hu P Nicholson Engrored by G. . Swanston Iden. Scale of feet 43 feet 月 ​A FallartondecLondon & Edinburph PTAIEJ, N Y.. .. Miil. ;:. 27 : Y A Fullarton& C: London&Edinburgh , . . . . : ER: S I ** * ITD IT . LHHH ! !!!!NINI ii IlW 11 . EUR .. a HUUUU'IL SA HITIMU min Tu.SE itt: L T UWONI . I . . - -.. he - HE . - . IW - - - . - - A . . N TOON LIN INI wth . - n - - Cross Section at the Centre of the Bridone, lirsis section through the Towers. 11 - - - - - . - - - - - - - -- - .- . -. - - . S1 JET 7 WWW .:WIZ Y HIIULUI . II A1 UWOY VIIMIIIII VIR 10 TI V ca Il WY VVN WA WWWU NA Vla 22 . * RADIO UU En Ull WIRKUNUDIMIRA 1,1 WWNWIN ! TS INTI 81YCIA .ZA ini 2 INSTI WHO l' UMH17 WWW . VW T1 VIZ E si : :: . . . - 1 STUDYM LATEX . ' 12 S WIR ITI ZEPHALLOWASSA it! . RADIO "IMIN PS IllIIVIT MilfTINUIHII WWW . L / / ALE IN . HRHRIN IN Wiki Vio WNIN .. HIS HUTHINDAN INI A 2 W / KLEILLA . Me milli mum W till !! 1 TOHU !!! 1 : IH :11! :. :: ' 17 litir Bilmltti . NE 4 .' TI . Att , INI? - - PE . 11! . NNNN WWWWWWWW MW IN E lli // 21 VI kit Ke Ye DHE . Oli 227 WIMMINA DWUNASTIRIM T. . 10 Nill llll POJ O whe This plu 3 SER Horizontul Sition of the Spandrells. . LAR . SINU ME 111111 111 Biw SI - - FM.butunnistumim ES ITUT. L I.11!1!11MMATT LUHI HINT Longitudinal sertion. 2 IIIIIIIII VIDEO . VILIN KI C III S7 // M E MNIEKU WWW. 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The foot-paths (fig. 11) receive a necessary elevation above the road-surface, and consist simply of a common flooring of boards laid on joists. The parapet of the bridge may consist of a simple hand-railing supported on uprights and braced by inclined pieces ; or something of a more ornamental character may be arranged either of wood or iron, according to the locality. 58. The principal objection to wooden bridges is the perishable nature of the material; but as this objection applies only to structures which are alternately exposed to moisture and dryness, a remedy may be found in covering the bridge with a roof and sides of shingles, or with the common weather-boarding of frame-houses. In bridges where the road-way rests on the frames, some diffi- culty might arise in arranging the roof, but a substitute could be found for it, by covering the plank- ing of the road-way with a metallic covering, to protect the frames at top, and by covering the sides in the ordinary way. Besides these conservative means, the parts of the structure most exposed should be covered with paint, pitch, or any other coating which may be found most efficacious. 59. As an example of a work of this kind, we give the bridge designed by Mr. Nicholson, and erected over the River Clyde, at Glasgow, for foot-passengers. In Plate XLIX. Fig. 1 is a general elevation of the bridge. Fig 2 is a longitudinal section. Fig. 3 is the joisting with the fender piles. Fig. 4, a transverse section. Fig. 5, the method of joining the beams over the posts. Fig. 6, the method of joining the trusses of the railing, by which the bridge is supported. 60. With the exception of drawings made by Palladio and others, from the descriptions given in Cæsar's Commentaries, of his bridge over the Rhine, we have no satisfactory account of any ancient wooden-bridge. Of those of more modern times, there is one described by Palladio, said to be situ- ated upon the Cismone, at the foot of the Alps, between Trente and Bassane in Italy. It is of very simple construction; the whole being suspended by the framing, which forms the sides; the opening between the abutments is 109 feet. Palladio also gives sundry designs for wooden bridges formed in different ways, some of which are supported by the sides only; and one is in the form of an arch. In Plate L. we have given four examples of wooden bridges from Palladio. We are informed that there was formerly a stone-bridge at Schauffhausen, that the Rhine injured the piers, and that in the year 1754 three arches fell; that the depth of water immediately on the upper side of the old piers being, during summer, from 18 to 20 feet, and from 28 to 30 feet below them, the idea of rebuilding a stone-bridge was abandoned, and that the old piers, excepting one near the middle, were taken away; that Ulric Grubenmann, a common carpenter of Tueffen, produced a model for a wooden-bridge, supported only by the abutments on the banks of the river; that after some hesitation on the part of the committee of Schauffhausen, his proposal was adopted, and that he completed this truly extraordinary work in the year 1758. The total length of the bridge was 364 feet, and its breadth 18 feet. It was eight feet out of a straight line, and the angle pointed down the river; the distance from the abutment next the town to the angle was 171 feet, and from the angle to the opposite shore 193 feet. When Grubenmann was applied to to erect this bridge, he was desirous of making it a single span of 365 feet from one side of the river to the other, but the magistrates deemed so large an arch in timber to be unsafe, and compelled him to take a bearing upon the central rock, which he unwillingly complied with. The bridge being finished, and publicly opened by the passage of loaded carriages, was declared to give complete satisfaction, and that being acknowledged, it is said that Grubenmann, in presence of his employers, took a thin board and passed it between the apparent abutment and the top of the rock, to show them that although it appeared to rest upon it, it did not in fact touch it, and thus that he had accomplished his desire of making it 108 [Part II. PRACTICAL ARCHITECTURE. a single span. However this may be, it afterwards sunk by the compression of the timber, and took a solid bearing upon the rock, which no doubt afforded very material assistance in its support. This bridge was much admired as a most excellent piece of carpentry, but owing to the oak beams that came in contact with the stone foundations being placed too low, and not being exposed to air, they rotted, and the bridge began to settle. Grubenmann being dead, it was repaired in 1783 by Georges Spengler, another ingenious carpenter of Schauffhausen, who raised the whole bridge by means of screw jacks, and replaced the decayed timber. This was the only repair done to it during the forty. two years it existed, and it would probably have been in good condition at the present day, had it not been burnt by the French army in 1799. This bridge was not composed of arches, but was on the principle shown in Plates LI. and LII. ; viz. a number of diagonal struts or braces with straining beams between them; but the road-way did not pass over the framing, but was suspended between separate parallel trusses by means of perpendicular ties called stirrups, united at their lower ends to straight beams appearing like tie beams, but having neither a strain of extension or com- pression upon them, since their ends may hang free of the abutment walls, their only office being to support transverse joists upon which the planks of the road are fixed. Fig. 1, Plate LI., exhibits an elevation of one side, including the roof, which was covered with shingles. Fig. 2 is a cross section at AAA, showing the uprights which are placed on the pier, the framing under the level of the road-way, the points from whence the braces proceed, the mortices for the beams which support the road-way, and the interior construction of the roof at these uprights. Fig. 3 is also a cross section at B, showing in what manner the aforesaid road-way beams and the braces pass through the other uprights, how the uprights are connected immediately below the roof, and also how the two pieces of which they are composed are bolted together. Fig. 4 shows the form of the roof at that place. Fig. 5 shows the manner in which the road-way beams, and those along the top of the uprights, are united. And Fig. 6 explains the nature of the points at C and D, by which the several pieces which compose the beam are connected together lengthwise. In Plate LII. Fig 1, is a longitudinal section including the lower part of the roof, and in which the situations of all the uprights, beams, braces, and iron ties, are distinctly shown. Fig. 2 is a plan of the floor, with every part of its framing; and Fig. 3 is a similar plan of the roof. In these Figures every part of the construction is so particularly delineated, as to render its office evident by inspec- tion. The braces proceeding from each abutment, are continued to the beam which passes along the top of the uprights, and the lowest of these general braces are actually united under that beam, thereby forming a continued arch between the abutments, the chord line of which is 364 feet, and the versed sine about 30 feet. These braces are kept in a straight direction by the uprights; which are placed 17 feet 5 inches apart. If this bridge had been formed in a straight line between the abutments, we can see no reason why this form of construction should not have supported a road-way of about 18 feet in breadth, as well as a slight roof; because, in that case, all the weight arising from the braces which proceed from the middle pier would have been saved, and the roof might have been made much simpler and lighter; but the general direction being 8 feet out of a straight line, and being loaded with an unnecessarily heavy roof, it was certainly advisable to make use of the braces from the middle pier, and thereby composing two distinct arches. Although the principles, and even the form of constructing this bridge, might have been drawn from Gautier's publication, or even Palladio's designs for wooden bridges, yet from the account of Ulric Grubenmann, being an illiterate man, there is reason to think it was from his own inventive genius that the whole design originated. There is not only a great boldness in the principal mem- being overdone in aiming at excellence and security, it is evident this was a first attempt, and that . - TUNNY A Fullart0n&London Edurbüro ligi á - . - - - ... . İ SE 1 TA i TL . 7: 2 H .IT . . S . 10 . ILITA' 1 Fig:: 0 IL WS PAO W ! - ! . VY SESE - DO www - - ZA - 22 / . ii . 1 S II- . . - - NG - NO - MUME - . ER- S - - AL - - LES in Switzerland. NES RET - Fig: 2 - - S - - al Schaffhausen The Wooden Bridge - - - AR - .. 167!!! SS . . . .. . Will 2 Fig. 1 . It has SUE FATRASTS. TT***. tv when neuem all . .. - . - .. * *. - - - - - .. . - ... . - - - . . . . - A UMBI III - . * - - - - . - - - .. * . - - - - - - - ... . . - -- mutta - - - . Se. - - - SASS.OR - - - GADE et ter SEA !!!" . - - - - -- - - . - - - . - - - - - - - . - I . - - - - - - - 1 - 1. EMANUEuro 19 10. La - - - . . - IM - - - -- . : -. bi in EILA - - - .. . .. . . O . UN T . 2 . RESULT : - ** - . . 2 S . 11 . Fig: 1 P'LITE 1.1. TLMBER BRIDGES CARPENTRY CARPENTRY TIMBER BRIDGES PLATE LII TIMBER BRIDGE over the RIVER DON at DYCE in ABERDEENSHIRE. Elevation. ee C AntiguNunnukilta Ilinni Intl Delton WTUDIRITTI Tini .. li.DILICI 2 - hit.tutti Uituuttuminent . AALALAR! Tihend 111 AREN A tau TIT muutVRUT MUTATIT LAGU TANT: . www: InviMTIBWTIILUNMUN ! Autour HI . __ S HUMIDID uslim W . W 22 + V Nig! - til. - 13.4 LE+ - - - - - ZEL ; : W MORBI PP! Spun log feet 1!) . 11 fill 11: MITI 21. .. . . . . Ex . - - - W - SA - - + - - 1 : WIR! C- Sin . UN Plan of the Roadway Plan of Soffit 11. SS N RE AR . 1 . HT! GS God . , X 141 : . . huta li . . . ETT *: ES SA - 2 VW RA . 7.1111 S - WOODEN BRIDGES from PALLADIO. SI WEB DINISIA www IS BE WI VE - ES - - www www wer 17 IN m www IN ART WIN ullin SIA IR Ulllll = es ** - - - - - - - - 9 . 7. Mottl 11.1.nis A Fullarton & Co London&Edinburgh SECT. III.] 109 CARPENTRY. there was an anxiety to avoid the possibility of failure, in what he conceived, and what, as far as regards him, was really a totally new project. We are informed that John Grubenmann constructed a bridge upon the same principles, of 240 feet span, over the Rhine, near Richenau; also that the two brothers erected one 200 feet span over the river Limmat, near Baden. And that the last work of Ulric was a bridge of 230 feet span at Wittingen. In this last, the form of construction was varied: instead of placing the braces diverging from each other, seven beams were built close upon each other, forming a catenarian arch between the abutments, of which the rise was 25 feet. These beams were of oak, in lengths of 12 or 14 feet, breaking joint in the manner of masonry. They were not fastened by pins, bolts, or scarfings; but were kept together by iron straps, placed five feet distant from each other, and fastened by bolts and keys. The road-way intersects them about the middle of their rise. Over the river Portsmouth, in North America, a Mr. Bludget constructed a wooden bridge 250 feet span, nearly in the same form as the last mentioned of Grubenmann; that is to say, each truss or arch consists of three rows of beams placed parallel with, but at some distance from, each other, and each beain consists of two halves, connected by dovetailed keys passing through them horizon- tally; and similar keys are also passed vertically through all the three beams. This has a more elegant appearance, than where the beams are laid close together; but we doubt if the frame is equally firm. Though of inferior magnitude, several upon principles equally simple and effective, have been erected upon rivers in Scotland. The largest is over the river Don, about seven miles from the city of Aberdeen, upon the road which leads from that place to Banff; the extent between the abutments is 109 feet 3 inches, and the breadth 18 feet. The frames which support the road- way are composed of short pieces of timber, but instead of being elevated above the level of the road-way in order that it may be suspended from them, they here support it after the manner of stone voussoirs. This bridge was erected in 1803. Small timber bridges being, in all countries abounding in wood, so obvious a means for crossing streams, it is impossible to trace their origin and progress; and those consisting of rows of piles driven into the bed of a river, and supported by common trussings and bracings, being found in most coun- tries, and being familiar to every body, it is only necessary, in what regards them, to refer to the Plates, and what has been said above. 61. Many other timber bridges might be described, but our limits prevent a description of the details such as the manner of joining the timbers, introducing and fixing the bolts and iron-work, and many other particulars which alone might fill a volume. These matters may be safely left to the judgment of the Engineer, when he has obtained clear ideas of the manner in which pressures will act and the best means of opposing them. But to strengthen his confidence in his own opinions and plans, it would be well that he should carefully inspect, measure, and take drawings from some of the best bridges built. 62. The greatest load to which a bridge is subject, is when it is crowded by human beings, and such a load amounts to about 120 lbs. to every square foot, independent of the weight of the bridge materials themselves; so that the actual load to be sustained should not be considered less than about 300 lbs. for every square foot of road-way; and such strength ought accordingly to be provided in every bridge that occurs in great public thoroughfares. Timber bridge building for such situations has long since been given up in England; but still they must always be used in certain positions, and for certain uses ; and the most frequent occasion that the Engineer will have for them is in what are called occupation bridges and shifting bridges in canal work. 63. In the formation of a navigable canal, it very frequently happens that the cutting may run through a portioir of a man's estate or farm, thus cutting off all communication between one part 110 [PART II. PRACTICAL ARCHITECTURE. and another by the intervening water, and in such cases the land-owner has a right to insist on the canal company putting him up such a bridge as shall enable him to have free access to, or occupation of his land. Such bridges not being on public roads, nor requiring any extraordinary degree of strength or elegance, are usually made of wood; and of course are maintained and kept in repair by the canal owners, unless a specific agreement is made to the contrary. When the canal is not in deep cutting, and its water comes nearly to the same level as the surface of the adjacent land, occu- pation bridges require considerable elevation in order that boats with high loads may pass under them; but to render them available, it becomes necessary to construct long hills or inclined planes at the abutments in order to obtain easy accessible roads to them. Such hills are expensive in their con- struction, dangerous in their use, and often unsightly and inconvenient; therefore shifting bridges are frequently used in place of them. These bridges are not raised up above the ordinary level of the land and roads, but they shift or move away in order to let highly loaded boats, or vessels with masts pass through them. 64. Shifting bridges are of two kinds, the draw-bridge and the swing-bridge. The draw-bridge is merely a wooden platform of sufficient width to allow the passage of horses, wheel-carriages, and passengers, and of sufficient length to reach from one side of the canal or water-course to the other, or rather to reach from a jetty or projecting abutment built on one side of the water to a similar projection on the opposite side, because when these bridges are used it is customary to contract the water passage to the smallest extent that will permit the necessary vessels to pass, in order to make the bridge platform as short and light as possible ; for a platform strong enough to bear heavily laden waggons must necessarily be heavy. This platform is so fastened by pivots or hinges at one of its ends to the jetty or projection, that it can be raised from its horizontal into a vertical position while vessels are passing, and this is done in two ways. One of these is to attach a chain to each of the corners of the unhinged side, to lead these chains over two iron pulleys fixed on the hinged side at a height exceeding the length of the platform, and to let them terminate at a cylinder or windlass, having a cogged wheel and pinion to gain power, fixed on the hinged side, for raising and lowering the platform by the turning of a handle. The other method is to fix a compound framed lever of wood or cast-iron, over the platform, such beam having strong iron pivots, which revolve on the tops of two posts fixed firmly in the ground at the two sides of the bridge. Two chains attached at their tops to the corners of the framed part of the lever, and at their bottoms to the sides of the platform, rather beyond the middle of its length; and two cast-iron weights are fixed on to the two arms of the lever for the purpose of nearly balancing the weight of the platform, and making it more easy to move. The platform should, however, have a preponderance, in order that it may lie steadily, when down, but the counterpoising weights assist materially in the ease and expedition of using the bridge, and are as important in lowering as in raising it; for if the platform is permitted to descend without counterpoise, it falls with a force that soon deranges itself, as well as the sill that is to receive it, and will stand in need of constant repair. 65. When the canal or water-course is very wide, two flaps or platforms, with separate apparatus for moving them, are hinged, one on each shore, and in this case they meet and abut against each other in the centre, and they must be curved and have good abutments on their land sides, to prevent them from spreading. Such drawbridges are very frequently introduced into the central or widest opening of stone, or other permanent river-bridges, to permit tall masted vessels to pass through them. 66. The swing-bridge is much more expensive in its construction than the drawbridge, but is very much used in canals, particularly for roads, when they are not sufficiently frequented to warrant the coustruction of a permanent brick or stone bridge. Their construction is such that the bridge never Sect. III.] 111 CARPENTRY. changes its horizontal position, but turns a quarter round upon a pivot, so that it may be turned over the water, or be pushed on one side so as to come over the dry shore. 67. The principles of carpentry already laid down apply to the construction of floors, partitions, frame buildings and roofs of every description ; but we have yet to notice another important appli- cation of them, which is the construction of centring for the support of arches while they are build- Whoever contemplates the nature of an arch formed of bricks, stones, or other separate pieces of material, must be convinced that they could not be placed in the positions they are intended to maintain, without some artificial support for them to rest upon, until the arch is completed and made capable of supporting itself. And when that is done, this artificial support has to be removed in order to throw open the space, that has been arched over, without any impediment. This applies equally to all arches or vaults from the smallest to the largest, and the artificial support that is thus made use of, is, as a whole, called the centre, or more properly the centring of the arch. 68. In the construction of centring for small arches, little or no skill is necessary; and in all centring the two chief points to be attended to are, that its upper or bearing surface shall be very correctly formed to the figure assigned to it, whether it be a portion of a circle, ellipse, or any other curve; and that it shall be sufficiently strong to bear the weight of the materials the arch is to be composed of, together with the workmen, tools, and other things that may be placed upon it, without sinking or changing its form. The first is necessary, because as the bricks or stones are in succes- sion placed immediately upon the upper surface of the centring, so of course the work, when finished, will exactly coincide with the form of such surface, and if any irregularities or inequalities exist in the centring, they will be transferred to the work, making it unseemly to the eye, and perhaps endangering its stability. The second qualification, strength and stability, is necessary to the per- fection of the first condition, because if a centre is made in the most true and perfect manner before it is loaded, and from weakness or bad fixing it changes its form or position from the gradually increasing load that is brought upon it, it will have just as bad an effect, or indeed worse, than if it had been badly formed in the first instance. Because bricks, mortar, and stone are inflexible mate- rials, which admit of no change of form after they are set, without breaking. The first portion of work that is placed upon the centring will of course adapt itself, or will be adapted, to the exact form of the centre. But if by continuing the work and increasing the load, the centre is made to alter its figure, it may in the one case produce a thrusting or expanding force against the new work, and thereby cause its joints to crack, or it may sink or recede from that work and leave it without support, which will cause its settlement, thereby destroying its symmetry and beauty, as well as its stability. 69. The principles of the construction of roofs already explained apply with certain modifications to those of centres; but the centring for arches is subject to many more difficulties than roofs, and require the exertion of skill and judgment to guard against them. The roof is a fixed construction, which when once put in its place is never afterwards to be disturbed, therefore any necessary pre- cautions may be resorted to for making it stable and secure. It has only to be covered with such slight materials as will resist the action of rain, snow, or heat; and consequently they are never very heavy, and whatever their weight may be, it is distributed with equality over the whole surface. No change takes place in the load of a roof, except that which arises from snow lying upon it, or perhaps, occasionally, persons standing upon it. With arch centring the case is quite different. The centre is not a fixed or stable erection, but is one that has to be moved and taken away, as soon as it has performed its office, and that without injury or disfigurement of the adjacent work; conse- 112 [Part II PRACTICAL ARCHITECTURE. quently it cannot be let into it, or form any part of it. The work or covering that is placed upon it, so far from being light, is massive and heavy in the extreme ; for, in the arches of large bridges, it is no uncommon case to have hundreds of tons of stone-work thrown upon the centre and depend- ing wholly upon it for support. And the load cannot be equally distributed over the whole surface, because it has to be built up gradually from its lowest to its highest point, and is, therefore, a con- stantly increasing series, incapable of aiding or supporting itself until the key-stones are introduced ; and even then, although the arch máy be capable of standing by itself, the centre is not relieved from its weight, until it is lowered, or permitted to recede from the superincumbent work. For these reasons the construction of a good and effective centring, and the manner of fixing it so that it shall be perfectly stable and firm while in use, and yet be easily lowered a small quantity without changing its figure or its original firmness and stability, and finally, that it may be entirely taken down and moved without injury or danger to the work built around it, is considered as one of the most difficult tasks the Engineer has to perform, and as a masterpiece of workmanship in the artists employed in its execution. In fact, the beauty, stability, and duration of an arch constructed of proper materials and with good workmanship, is dependent entirely on the perfection of the centring upon which it has been built. 70. Every centre consists of two principal parts or elements. One is called the rib, and this answers the important part of determining the form or curve of the arch, and of giving strength and support to the whole fabric; and the other is the covering, or lagging (as it is technically called), and consists of parallel boards, planks, or timbers, extending from one rib to another, or over several ribs according to the extent of the arch, and which is for the purpose of connecting the ribs together, and forming an extended smooth surface, upon which the bricks, stones, or other materials of the arch are to be built and put together. 71. In small centres these parts or elements are usually nailed together and combined, so that the ribs and lagging constitute but one piece, and the whole is moved and set up together. But in large centrings, the weight and magnitude of the materials renders it necessary that they should be separate. Nay, even more, for a single ríb of the centring of a large arch is so large and pon. derous that it can seldom be moved in its entire state, but requires to be taken asunder, and carried in detached pieces to the place where it has to be used, and it must there be put together or rebuilt, taking care to place it so exactly in its proper position that it will require no alteration after its erection. Of course as many ribs as are needful will require the same treatment, and after they are all set up parallel to each other and adjusted so that their upper surfaces may bone perfectly in the direction of horizontal lines strained across them, and they are found perfectly out of winding, they are to be secured in their places by sloping or diagonal braces from one rib to the other, and then the lagging or covering may be placed and fixed upon them. 72. No centre can be formed even for an arch of a single brick, or nine inches in thickness, with- out two ribs, viz. one at each end of the lagging; and the additional number of intermediate ribs must depend conjointly on their own strength, and the weight of materials they have to support. This problem can only be solved by the practical skill of the Engineer, aided by the rules that have already been given for determining the strength of materials, observing, in all cases, that it is better to err on the side of too much strength, than to trust to weak or nearly calculated structures. A large rib is always an expensive construction, but its first cost is nothing in comparison to the expense that will be incurred by its failure when in use, because if it sinks or changes its figure during the construction of the arch, there is no way of repairing the mischief that may arise, except that of pulling down all the previous work, strengthening the centre where it fails, and beginning the whole operation again. In large bridges it is customary to place the ribs parallel to each other, and from Sect. III.] 113 CARPENTRY. 1 three to six feet asunder, according to circumstances; and from what has already been observed upon them, it will be seen that their use and action is similar to the trussed principals of a roof; and as all the weight of the superstructure, including the lagging, is to be supported by the ribs, it will be equally evident, that the support of the whole centring, while it is in use, must be applied under the feet or abutment of the ribs and nowhere else. 73. In all regular arches, such as portions of cylinders, or any curves which have the same dimen- sion throughout, all the ribs made use of must be precisely of the same size; and as the load of work to be placed upon them will be very nearly equal in all parts, when the arch is finished, so the ribs should be of equal strength. It follows from this, that any mode of construction that is adopted for one rib will equally apply to all the others made use of; and hence in describing centres, we shall adopt the usual plan of giving a description of one rib only, and saying how often that rib is to be repeated, keeping in mind that however long the arch may be, the lagging runs the whole length of it upon the curved surface of the ribs, and in most cases at right angles to them ; conse- quently no particulars, except dimensions, will be necessary in describing the lagging. 74. In some cases it may be necessary to construct what are called flewing arches, or arches that are wider at one end than the other; or whose figure resembles a part of the superficies of a trun- cated cone. In this case the ribs must evidently not be of the same size, but must be formed with different radii of curvature. One rib must be made for the large end of the arch, and another for the small one, and these being set up parallel to each other, or in such other position as the two ends of the arch are meant to have in respect to each other when finished, lines must be fixed and stretched upon the curved surfaces of the two ribs at proportional distances from each other, and these lines will give the magnitudes of any number of intermediate ribs which it may be thought necessary to construct and place at any assigned distances between the two exterior ribs. The lag. ging must of course run in right-lined directions over the outsides of all the ribs, and will complete the conical or other diminishing surface which has to be given to the finished arch. 75. It may here be observed, that with the exception of building bridges of one or of two arches, there is no necessity for providing a quantity of centring equal in superficies to the quantity of arch to be constructed, because centres may be shifted and carried from one place to another, as the work proceeds, which saves a great expense. Thus in constructing a culvert or cylindrical drain for carrying water, or making a long vault or passage under ground between two parallel walls that have to be arched over, a centring of from six to nine feet long, or even less if the opening be large and the centring heavy, will be all that is necessary. This centring is first to be fixed in its proper place at one end of the work, and the arch is then worked over its whole extent. That done, the centre is struck (which is the technical expression for releasing and taking down centring,) and it is moved forward very nearly its own length, taking care to leave an inch or two of one of its ends underneath, but in contact with the underside of the portion of arch that has been built. In this new position it is to be made straight and level and again fixed; when a second quantity of arch work equal to its length may be built upon it, when it is again struck, advanced, adjusted, and fixed, and is ready for a third length of work, and by this process the arched vault may be continued any required distance with only one short centring. The only rule that can be given for its dimensions in such cases is, that it must not be so long or so heavy as to require any extraordinary exertion to move or refix it, because, in these cases, there is generally not much room for action, and a danger of disturbing and breaking part of the former work, unless the shifting and refixing can be performed with great facility. 76. In building a brick or stone bridge, of a single arch, over a river, it is obvious that we must fix the entire centring at once, before the arch is commenced, and likewise in building a similar 114 [PART II. PRACTICAL ARCHITECTURE. bridge of two arches, where the meeting or springing of both arches rests upon a pier in the middlo of the river, and the other feet of the arches abut on the banks, two centrings, one for each arch, will be necessary, and they must both be fixed in their places before the arch is commenced. The reason of this is, that if only one centre should be used, resting upon the bank or shore at one end, and upon the pier, in the centre of the river, on the other, and an arch should be constructed from one to the other, it would be impossible to strike and move the centring when the arch is finished, without endangering its downfall, unless the central pier is exceedingly strong. For so soon as the centring is removed the lateral spread of the arch will come into play, and as it will be incapable of acting upon the solid stone abutment, all its lateral expansive force will be exerted against the pier in the direction in which it is least capable of resisting pressure, and it will be overset. But by having the two centrings fixed at once, both arches will proceed at the same time an equality of load is induced on the two opposite sides of the pier, and when the centres are removed two thrusting forces, which are equal and diametrically opposite to each other, come into action at once and neutralize each other, and the pier, therefore, remains undisturbed. 77. If a bridge of many arches has to be constructed, for the reasons above assigned, two centres only will be absolutely necessary, though the use of a third will be advantageous. In this case, the two or three centres of the arches of one end or abutment of the bridge are first fixed, and the land, or abutment arch is first commenced from both its feet or abutments, and in doing this a part of the second arch must also be commenced upon the first pier so as to weigh it down, and at the same time to throw an equal weight on both sides of it, by which its stability is much increased; and this effect will be further augmented by also working up a small part of the second arch upon the second pier, so as to balance the second centring and prevent its sliding away from the lateral pressure of the load placed upon it. By judicious and skilful management the loading may be so arranged that the first or land arch may be completed, when its centring may be struck and carried to the position of the third or fourth arch as the case may be, and then the second arch from the land may be com- pleted, and in this way a bridge may be carried over a river until it arrives within two arches of the opposite side, when, if three centrings have been formed, one or two, as the case may be, can be fixed at once and worked up from the opposite shore to complete the work. 78. This mode of proceeding does not often apply, because it is generally the case that the central arch of a bridge is larger than any of the others, and that they diminish in size as they approach the shores; and whenever this is the case the centrings cannot be interchanged, except by com- mencing operations on one side of the river and then moving to the other, and so on alternately, until the meeting is made at the central arch. It was, however, adopted by Mr. Rennie, the English Engineer, in the construction of Waterloo Bridge across the Thames, in London, which consists of nine semi-elliptic stone arches all precisely alike, each rising 35 feet and having a span or opening of 120 feet. The object of this similarity of dimensions in all the arches, is to obtain a perfectly level road or causeway over the bridge, instead of ascending and descending two hills or inclined planes, as is the case in most bridges. 79. It sometimes happens that arches have to be constructed, for the sake of their strength alone, in places where they will never be seen; consequently their symmetry and beauty is not important, and in some cases the introduction and taking away of centring might be difficult, or even impossible. Thus, for example, if a very large window or opening has to be left in a high and heavy wall, but which we require to have a straight or right lined, instead of an arched appearance, there is no way of doing this except by throwing a timber or stone lintel or breast-summer across the opening, and building upon it. That lintel may be consumed by fire, may break, or will decay in time, and may thus endanger the wall. To guard against this we have only to build a quantity of work upon such Sect. III.) 115 CARPENTRY. S lintel with a curved or semicircular termination above, but without any regard to the bond of the rest of the work, such curve springing from the perpendicular sides of the opening, or what is still better, from the extreme ends of the lintels. This work is to be used as a centring for turning an arch upon, and this is next to be done, when the wall may proceed upwards upon the arch, just as it would have done upon the lintel; and it will be evident that where such a construction is resorted to, the whole lintel may decay, or may be taken away, as well as the quantity of work that was used as centring, because now the arch will sustain the load instead of the lintel, and a new lintel can at any time be introduced. 80. It frequently happens that arches are necessary for the support of the underground foundations of walls and other erections, and which might not be able to bear the great expense of regularly framed centring. Thus in digging the foundation for a long wall, the soil in general may be hard, solid, and trustworthy, but we occasionally meet with soft places, arising from springs, quicksands, or the ground having been before dug up, or recently made level, which would render it unsafe to carry the wall over them, and in this case it may be necessary to turn an arch from one hard place to another, thus passing over the soft and insecure places. This may generally be done by an earth- centre, i. e. by having a concave curved mould or pattern, formed out of thin plank to the shape we wish to give the arch, and then digging up and forming the ground so as to fit the curve of the mould. The earth so shaped must be well and solidly rammed down until the desired curve is obtained, and then a brick or stone arch may be built upon it just as well as upon a regular centring made by the carpenter, and upon the top of this the wall may proceed upwards with perfect security; for if the soft soil sinks away from under the arch, it will only be the same thing as taking the timber centre away from an arch constructed by the regular process. 81. This mode of centring is constantly resorted to for covering over furnaces, bakers' ovens, and cinder or coke ovens, the tops of which are an irregular kind of dome, which would be difficult to construct in carpentry. But the side walls being carried up perpendicularly, the body of the oven is filled with damp sand, which is raised up in a convex form to the exact shape the dome is intended to have, and the bricks are then laid as over any other centring When sufficiently dry and set, the sand is dug away and moved through the door or mouth of the oven.* * The paragraphs of this Section, from 67 to 8l inclusive, are taken from Mr. Millington's excellent . Elements cf Civil Engineering.' Philadelphia, 1843. SECTION IV. JOINERY.* Definitions, 1–4. Scribing and Mitreing, 5–9.---Formation of Curved bodies, 10.- Formation of Bodies in Parts by joining them with Glue, 11, 12.- Bending Machine, 13.— Hinged Joints, 14–21.- Door-joints, 22, 23. -Folding doors, 24.- Window Frames, 25—31.- Sashes, 32.-_Skylights, 33, 34.- Elliptic Arcbivolt, 35. - Mouldings, 36–58. Stairs, 59-68.-Hand-Railing, 69–83. 1. JOINERY is the art of employing wood in the finishing of rooms, and other parts of buildings. It is a difficult art to acquire, yet the convenience and comfort of dwellings depend more on it than on any other; hence the respect which is universally bestowed on a good joiner. 2. The smoothing of wood, by cutting the superfluous parts away in thin equal slices, is called planing ; and the tools used for this purpose are called planes, whether they are employed in reducing the surface to a plane, or to a convex, a concave, or an undulated form. 3. The wood is called stuff, and is previously formed into rectangular prisms by the saw. These prisms are denominated deals, boards, battens, planks, &c. according to their dimensions in breadth and thickness. So that in this article, whenever a piece of wood is spoken of, it is understood to be bounded by six planes, and to have all its angles right angles. 4. The arrises are the lines of concourse formed by every two planes, and are therefore eight in number. OF THE CONNECTION OF ANGLES. [Plate LIII.] 5. When two bodies are so fitted together that their surfaces intersect or meet each other, they are in general said to mitre or scribe. 6. Two bodies are said to mitre together in a plane passing through the common intersection of their surfaces. 7. One body is said to scribe upon another, when the two surfaces intersect each other, and when so much of the one body is cut off to make way for the other body entire. 8. In finishing, whether the bodies are mitred or scribed, the external appearance is the same. 9. The most approved modes of joining angles, are shown in Plate LIII. Figs. 1 and 2, exhibit methods of joining boards, framing, or dado at an internal angle. Figs. 3, 4, 5, 6, 7, 8, 9 and 10, exhibit the joining of boards, &c. at an external angle. In Figs. 1, 2, 3, the position of the grain of the wood is most frequently parallel to the direction of the edges of the section; but when employed in the construction of troughs, it is parallel to the mitre line In Figs. 4 and 5, the external angle, being that which is exposed to sight, is rounded or beaded. * This Article has been greatly enlarged and enriched by incorporating with it the principal part of Mr. Nicholson's contribution on Joinery to the Edinburgh Encyclopædia.. 4 95 CA -- - - - - - - - GIN = בא = ---ההר - PLATE LIN. - - - - - - ת ". - , - - -- . - - . - - .. ק אי - - - - ו י - - - - י - -. * - - - - - , - -* ** * - - - 2 י .. - י --- - . . ו ו ולא. ב.. ו - - א י 1 - - - אוווווו - - -- 1 ו א - - -- - י Introduced by ... liortiort.. . . 14 * - - 111 -- - - - i - - . // - ה י-"- - - * - - - - לה - / . - - . ----- . -------- -- - -- די 4 י - - - ": י י י - י י י י י - י- . - י י - -אייל - - י - - - - - - - - ש - -- -- / ... - - - - - י - - - - - - - - - - - / - - * י י * -- - - -- - - -- - --- --- - / - - - - - - - - - - - - - - - --- - - - - - - - - - - - -- - - - ם - י - - - - - - - - - - // - - -- - - - - -- - - - - - - - - ... . י- י- י --- - - - - - - -- - - - - - - ו - - ש-י - י - - - - - - - - -- - - - - Fig. 4. "zy. 10. - - - -- - - - - -- - - - - - - -- --- - --.- . - - - ה בלד - - - - - י- י י - - - - - - - - - - - - - - - -- ד ר ך י"" --- י - - - - - - - - - - - Fig. 8 ווו - - - - - - - - בז . הם נבט !!וווווווווווו;{ - - - - - -- -- Wel. - - ווינ - - - - - - - -- - - - - -- Fig.3. "- י -- י - * *. י - - . * - -- -. Fg.7. . י: .. . .. * * * * -- די - - - - - - - י : דן - - - - * * - - - - - - --- - - - - י י - י -- יי - - - - JOINERY. MITERING LLLL LLLL Introduced by P. Nicholson A Fullarion& COL andun Edinburgh - -- - - - --- - - - - -- - - זי - - - -- - - - - - -- - - -- - - - - - - - } ים ן Fig. 6. ------ - - - - . fg .2. נץ N° 2. ."- - - נון - - - - - - - -- - - - - - // - - ז -- * - א -- . " אמא . PIP - 11. " . . . איזי - קיז - • - ריש - : או : * . * - י . - * בין -- 14 לה - .. -- - . א P * , - י . - * כה --- - - --- -- Introduued by 4. Morton. - - -- - - - - . = - - - - - - -- י - - ל -- - - - - - - - -- - וי . -יי --י דרך : ג. מזל ל זו A - - - - --- - -- - ראשי -- - -- -- הדייר - - - - - - -- - - וזון - -- - - - - - - - 1*9.1. -- - - - - - - " ת - -יי - - . אביזרים זו בל - -- - * - - - NOZ - דרדר. .. - - - --- יל - Fig. 5. -וו הן - - ין - A - -- - - - - - -- * י - " י - - -- ל - י-: ** - -- - - - - -- - - - JOINERY. PLATE LIV INCURVATION OF BODIES BY GROOVING ROSA . SASA NI . ww Drawn by 1: Nicholson Eng? by.G.H. Swanston A Fullarton & Co.London.kEdinburgh 3.0 s CH. JOIVERY PLIJE ZV Fig L N I Fig2.N?II. Fig 2. Fig.3. Fig4. NOI. Fig4. NOI. Fig. N.I. Fig.5 NI. Figa.xm FigóNON I Illi - un . Fig.5 NOV. Fig.5.NOV. Fig.7NOI. Fig.6.N.1. 1 di!. Lili, . Fig.g. VIL. LUI attina Fig 8.NO. PRIE Fig.&NONI. . . . AC . . . . . A . . ! . , A . . . . - . - . .. nitrosoma 211UNhi. programa .. . Fig. 8.NOII. Fi).6. NOU arts 36 1 - - . 1 ... - Fig.10.NO). -- ..!. . Fig.Il.NO. Fig.12. NOII. Fiy.73NOI. ST IH -- MTWT Fig.10.N.11. - Isto: AR - - . - . vi Will Fig. INI. 13 22.NL 73. NOT Fig. 15. - Fig.14.NOT. SS . Fig.16. 111 . WIRINDHUNIDA .. Fig.17. INDIR MIRUNNUNTURI Fig.27. NIV. WANIRAIMADAARNIMATIN PS www . . ARENT ** * LE UUNINMI 11 Wir S . S - - HI! - = - - w - Fig.14. NOI, TE - A Fig.17. NOU ** - - * - . - - -- - PIEcholson S. Mollants Alarn101.2018 Eauburgh CH Sect. IV.] 117 JOINERY. Figs. 6, 7, 8, and 10, are what are properly denominated mitres. Fig. 6 is the most common form of a mitre. Fig. 7 a lapped mitre, which is much stronger than Fig. 6. Fig. 8 a lapped and tongued mitre used in the construction of the pews of St. Pancras church. Fig. 9 represents common dove-tailing. Fig. 10, secret or mitre dove-tailing. OF THE FORMATION OF CURVED BODIES. [Plate LIV.] 10. When a curved surface, or a curved bar, is to be formed, the effect may be produced in the one case by grooving, and in the other by bending. The incurvation of bodies by grooving is shown in Plate LIV., where Fig. 1 exhibits the method of forming a concave cylindric surface. ABC is a transverse section; and DEFG the elevation of the back. Fig. 2, the method of forming a concave conic surface for a half cone; ABC being the transverse section, and DGHI the elevation of the back. Fig. 3 exhibits the method of forming a concave conic surface for a segment less than a semi- cone ; ABC being the portion required, and DGHI the elevation. The remaining figure shows how a concave spherical surface is formed in gores; AD being the section, FGH the gore, and ABC the templet to bend it upon. . [Plate LV.) 11. Fig. 1, No. 1, is a section of two boards glued up edge to edge. No. 2, face of the same. Fig. 2, a section of two boards glued edge to edge, with a tongue inserted in a groove in each piece. By these means, a board may be made to any breadth, though the pieces which compose it be over so narrow. Fig. 3, two boards fixed at right angles, the edge of the one being glued upon the side of the other. They are strengthened by a block, which is fitted and glued to the interior sides. Fig. 4, No. 1, a section of two boards at an oblique angle, mitred and glued together, with a block in the angle. No. 2 shows the inner sides of the boards thus fixed. By this method columns are glued up. Fig. 5, No. 1, section of an architrave. As the moulding is generally, if not always, glued to the plate or board, the dotted line circumscribing the moulded part shows the section of the piece to be glued. No. 2, face of the architrave. No. 3, a section of the architrave before it is moulded. No. 4, a front of the same. No. 5, a section of the same to a reduced size, with the button and nail, showing the manner in which the two parts are glued together. No. 6 shows the back of the archi- trave with the buttons. The black dots show the heads of the nails. The buttons are used, in order to bring the two surfaces which are glued together in contact, after the pieces have been set and held together, and are afterwards knocked off when the glue becomes dry, and then the moulding is stuck, as shown by the section, No. 1, and elevation, No. 2. Fig. 6 shows the method of glueing up a solid niche in wood. No. 1 is the elevation. Here the work is constructed in the same manner as if it were stone or brick, except that the joints are all ! . 11 + 118 [Part II. PRACTICAL ARCHITECTURE. S parallel to the plane of the base ; for it is difficult to make a joint with curved surfaces, as would necessarily be the case if they all tended to the centre of the sphere. No. 2 and No. 3 show the two bottom courses, where the vertical joints are made to break, and not to fall in the same planes. This is distinctly seen in the elevation, No. 1. Fig. 7 shows the manner of glueing veneers together, so as to form a cylindrical surface. This is done by nailing brackets to a board, with their faces upwards, and their ends perpendicular, leaving a cavity sufficient for the veneers and wedges between the ends. In No. 1, the thin part in the form of an arc shows the veneers in the state of being glued, and the wedges are shown upon the convex side. No. 2 is a section of the board and bracket. The veneers ought to be heated before a large fire, and the glue laid on the surfaces that are to come in contact as hot as possible, to prevent the glue from setting, observing to glue only a small portion at a time, and then wedge it up. When the glue is dry, the wedges must be slackened, and the veneers, which will then form one solid, taken out. Fig. 8 shows a very strong method of forming a concave surface, by laying the veneer upon a cylinder, and backing it with blocks in the form of bricks, which are glued to the convex side of the veneers, and to each other. The fibres of the blocks must be as nearly parallel as possible to the fibres of the veneers. No. 1 shows a section of the cylinder, veneer, and blocks. No. 2 shows the convex side of the blocks. Fig. 9 shows another method of glueing veneers together with cross pieces screwed to a cylinder, the veneers being placed between the cross pieces and the cylinder. Fig. 10 shows the method of glueing up columns in eight staves or pieces, the whole being glued together in the manner of Figure 4. We must here observe, that the workman must be careful to keep the joints out of the flutes ; for being in the fillets, there will be more substance to prevent them from giving way. No. 1 is a section of the column at the top; and No. 2 a section at the bottom. After being supposed to be glued together, the octagons and mitres must be laid down correctly, in order to form the joints truly. Here are two bevels shown, one for trying up the mitres, and the other for trying the work when put together. Fig. 11 shows the method of glueing up the base of a column, according to the following descrip- tion. Let a course, consisting of pieces of equal lengths, be closely jointed together upon a plane surface or board, so as to be something more than the diameter of the most projecting moulding in the base, then glue the joints firmly together, and plane the upper surface smooth. Upon this course lay a second, with the same number of pieces as the first, closely jointed at the ends as before, and also to the upper surface of the lower course ; glue down one of the pieces, so that the middle of its length may fall upon the joint of the two under pieces; then the others being glued on suc- cessively till the space is closed, a third may be repeated in the same manner. The horizontal joints of these courses must be so regulated, as to fall at the junction of two mouldings, forming a re-entering angle. When the glue is thoroughly hardened, the base may be sent to be turned. A base, glued up in this manner, will stand much better than one which has the fibres of the wood perpendicular. No. 1 is the plan of the base. The whole lines directed to the centre, show the joints of the upper course; and the dotted line tending to the same point, show the joints of the course below. Fig. 12 shows the method of glueing up the modern Ionic capital. No. 1 is the plan exhibiting the manner of placing the blocks. No. 2 is the elevation of the same. The plan is here inverted. Fig. 13 shows the manner of glueing up the Corinthian capital for curving of the leaves. No. 1 is the plan inverted. No. 2 the elevation. The abacus is glued up in the same manner as the Ionic capital, Fig. 12. al 11 JOINERY. BENDING MACHINE. PLATE.LV. I . int Om - - - Y- - - - " n n n w d B A mor.com corte de 1.Osbom. Elurrell. Fullorton & C. London & Edinburen OF. Sect. IV.] 119 JOINERY. Fig. 14 is the method of forming a cylindrical surface, without veneers, by equidistant parallel grooves, and by inserting slips of wood in the grooves. No. 1 exhibits the elevation, and No. 2 the plan. Fig. 15 shows the method of forming a conic body. The theory of this is no more than covering the frustrum of a cone; the covering is formed by two concentric arcs, and terminated at the ends by the radii ; the radius of the one arc is the whole slant side of the cone, that of the other is the slant side of the part cut off. Here the grooves are all directed to the centre, and filled in with slips of wood glued as before, the semicircle ABC below, is the plan: the arc HI must be equal to the semicircumference ABC. Fig. 16 is the same for a smaller segment. Fig. 17 shows the method of glueing up a sphere or globe, by the same method. No. 1, the face of the piece; No. 2, the edge, showing the depth of the grooves; No. 3 shows the mould for forming the pieces to the true curvature; No. 3 exhibits the faces of two pieces put together. 12. The principle of a circular headed sash frame in a circular wall, depends upon the section of a cylinder, and the development of the surface as cut by another cylinder. In the formation of the radial bars, two of the sides are parallel planes, and the edges are portions of cylindrical surfaces, contained between the exterior and interior faces of the wall. To form the cylindrical surfaces of the concave and convex sides of the radial bars, it will be necessary to be informed, that the curves which direct the shape of the edges are portions of two different ellipses, formed by cutting two different cylindrical surfaces contained between the two sides of the cylindrical wall, and concentric therewith by two parallel planes, inclined at the same angle as the planes of the bar, and having their distance from one another equal to the thickness of the said bar. And, consequently the ellipse, which directs the form of the concave edge of the bar, will have its lesser axis equal to the diameter of the interior cylindrical surface, and that which forms the convex edge equal to the diameter of the exterior cylindrical surface." The circular bar, or, as it is improperly called, cod bar, depends on the development of the part of the cylindrical surface, formed by cutting a vertical cylinder by a number of horizontal concentric cylindrical surfaces, which gives the form of the veneers, or thin slices of wood to be bent in thick- nesses. The head of the sash depends on the cutting of a hollow cylinder, so that the side contained between the two cylindrical surfaces, that stand upon the exterior and interior sides of the plane of the sash, may be everywhere perpendicular to these surfaces, and to follow the true shape of the elevation of the window, and thus the angles will be easily moulded. But in order that there may be no variation of the mouldings in a circular sash frame, it is necessary that both the radial and circular bars, as well as the head, should be moulded upon the same principle as a hand-rail, viz. by means of face and falling moulds; the face mould for the radial bars will be as before observed, and the falling mould will be a parallel slip of wood, straight in the edges, in breadth equal to the thick- ness of the bar. The falling moulds of the other parts must be made according to the development of the cylindrical surfaces. BENDING MACHINE. [Plate LVI.] 13. A machine of a more or less complex kind is required to bend wood to the proposed form, after it has been boiled, or steamed, or glued up in thickness. The Plate LVI. represents a machine for bending sash bars, styles, beads, &c. 120 [PART II. PRACTICAL ARCHITECTURE. The plan is represented on the left side of the plate. AA represents the bed of the machine, which may be a plank suitable to the articles to be bent; f, f, f, f, represents dearers screwed to the bed, and likewise screwed down to a work bench, as shown at the section on the right hand; m m represents the heads of the screws; B shows a templet (commonly called a cylinder by workmen), the centre of which is at d, d, and is supposed to be employed bending a sash-style and bead at the same time, as shown in the section. Suppose the style intended to be bent to be worked to its proper rabbet and mouldings, and the templet rabbeted to receive it and the bead also; then suppose the style to be fastened to the straight part of the templet by means of small cramps, as represented at kk, nnnn represents a piece of iron hoop which is pressed close to the templet by means of the wheel i i and the screw gg; the cylinder is supposed to be in the act of being moved round by means of the lever CC, and when brought far enough round, may be confined by cramps as on the other side. FORMS OF HINGED JOINTS, AND THEIR HINGES. [Plates LVII., LVIII., and LIX.] 14. The forms of joints for folding and hinging is essential to the beauty of the work. Such joints ought to be so made, as to preserve the uniformity of the door or shutter on both sides ; and to exclude as much air as possible from rushing through between the edges of the two bodies to be hinged, and thereby rendering the apartments cold in winter. 15. In the joints of doors which are to be hinged together, both angles of one of the bodies are usually beaded, in order to conceal the open space which would be seen from every point of view; and to preserve the regularity of the work, the hinges employed to couple them together are made exactly to the size of the bead, on the side on which the knuckle is to be placed; so that, when they are hung, the knuckles of the hinges and the wooden bead forms one continued staff or cylinder. 16. The motions of hinged joints are easily traced, and though there be an immense variety of cases, the same general principle determines every one ; hence, it will not be necessary to collect more than a few useful forms for illustration. Accuracy of workmanship is indispensable in the formation of a good joint; to move with ease and freedom, the axis of each hinge should be straight, and all the axes of the hinges of the joint in a straight line; these are apparently simple conditions, but it requires considerable care to fulfil them. 17. Plate LVII. Fig. 1, No. 1, exhibits two parts hinged together; and Nos. 2 and 3, exhibit the two parts before being hinged, applicable to doors and their hanging styles. Fig. 2, 3 and 4 show varieties, the object of which is to prevent the joint being seen through. 18. Plate LVIII. shows the method of placing hinges on shutters, back-flaps, &c. In Fig. 1, the axis of the knuckle of the hinge is exactly opposite to the joint. In Fig. 2, when it is required to throw the back-flap at a given distance AB, we must place the centre of the hinge at half that dis- tance. No. 2 shows the same joint opened to a right angle. Fig. 3 is a rule joint; No. 1, the two parts hinged together in a line ; No. 2, the one opened to a right angle upon the other. Fig. 4 shows another method in order to throw the back-flap the contrary way to Fig. 2. Let e b and f d be the parallel lines of the joint; and let b d meet f d and e b perpendicularly, so as to make e b and f d equal. Bisect 6 d in c, and draw o c, and describe the semicircle cgo, meeting be in g. Through c draw gh, meeting f d in h; then o g h is a right angle: and as Og is perpendicular to 9 h, og is the shortest of all the lines that can be drawn from o to the straight line g h.-No. 2 is the same joint folded backwards. S JOINERY. FORMS OF HINGED JOINTS & OF THEIR HINGES. PLATE LPI.. - V- KUR WA . EL M - ITT . - ++ S _ _ - S + WITH L OLA - -. _ - -- _ EN SER RSS HR : - THE THE M UIT IW :: F - LG os . IH UL UNI . - WWW - BUTI YLE LEHEN IIIIIII Hiili UHR NW till - WH WINT WIDTH WIRELLI A LA LULUI AN -- - TO BARA - - Wiliit KUI TE ET * - - - - = TI!!!!! - - - - - - E 222 WWW: H - . W HARUHI till - - -- !! BRE AWIT it THURDI - - - -1 hn Hill! ADITHIRIMITIH - - TE TAS- TY DI IUNI IN WO - - - - -. --- WURM - - - MEGA Bill H . - NE V - - - AN NRK al HR AL YHL ER . UR VW - - - AU WEEK + . . - R IN CALL IT UNG . . UN! M NEW MY IN EN _ _ _ _ DAHL HEN SL - . S _ - V E- + - + Eng. by (5.11. Swanston Edin. Drawn bu P. Nicholson, 1. Full bundan da Edinburyrh. JOINERY. FORMS OF HINGED JOINTS & OF THEIR HINGES. PLATE EVIN. Fig 1. Fig.2.11. - LUN -- LE IL - - - - - - - - ..... - -- Fig 2. NO2 Fig. 3.1.1. -. -. ... -... - m Fig. 4. NOI. Bi.. ., Fig.3. N° 2 Fig. 4. N°2. - - - - Engraved biE. Tumell. Introduced by P.1hholson. # كه ت A Fallarton&Cº London & Edinburgh in to VCH 19 JOINERY, 11111111 11111111111 T FORMS OF HLVGED JOINTS & OF THEIR HINGES.. PLATE LIX Fig. 1. Fig. 2. Fig.3. DS M OUNT பாயாம பயப்படாயாயாயாயாயாயம் Fig.4. Fig.5. Pig.6 - Fig.:. N.07. Fig. 7. . PA - -- - Fig. 7.1 Fig. 8. N.° 2. 71 VA - -NM - - - Introduced by P. Nicholson. Engraved E.Turrell. A Fallateu 10 Lorum • Edinburgh. JOINERY. HANGING OF DOORS. PLATE LX Fig. I 2 Fig. 4. Introduced by PNicholson Engraved b R Rofre A Flarts & London & Edinburgh M Sect. IV.] JOINERY 121 S 19. Plate LIX. shows other forms of hinged shutters and their joints. Fig. 1, joint with the proper hinge for the doors of pews, so as to clear a cornice or other moulding. Fig. 2 is the same opened. Fig. 3, joint and hinge connecting the sash-frame and shutters. Fig. 4 and 5, concealed joints. Fig. 6, centre for a door. Fig. 7, another centre for a door. In No. 1, draw AB parallel to the jamb, meeting the other side in B. Make BD equal to BA, and join AD and AC, Bisect AC by a perpendicular EF, meeting AD in F; then F is the centre of the hinge. No. 2 is the joint in the act of opening. Fig. 8 is another form in order to accomplish the same object. 20. Shutters are always within the apartments, wherever beauty is aimed at; those on the outside destroying the appearance of the front. They are divided into several vertical slips folding behind each other, for the conveniency of concealing them within the thickness of the wall. Each slip or fold is framed and composed of several panels, either raised, or flat, surrounded with small mouldings contained within the thickness of the framing. 21. The case in which the shutters are enclosed is called the boxing. The parts of the sash-frame in connection with the shutters, are the inside-lining which forms one side of the boxing, and to which the front shutter is hung. The vertical piece of wood which adjoins the edges of the sashes and the inside lining, is called the pulley style. The vertical piece of wood which joins the pulley style on the outside, parallel to the inside lining, is called the outside lining. That side of the boxing which is parallel to the face of the shutter, is called the back of the boxing. The remaining third side of the boxing is either formed by the architrave which surrounds the aperture within the room, or, in very good houses, by a groined flush on one side with the plaster of the wall. The parts of the sash-frame which are parallel to the horizon, are the sill and top, which names bespeak the situation in which they are placed. Inside beads are those slips of wood, rounded on the edges, which form one side of the race or groove for the sashes to run in. Parting beads are those slips of wood which separate the upper and lower sashes. JOINTS OF DOORS. [Plate LX.] 22. In hanging doors all the points of the moving edge of the door should pass the surface of the edge of the jamb, or the surface of the edge of the fixed door, in the act of opening or shutting. The method of making doors open exactly, so as to cut away the least quantity of wood, or to keep the narrow planes of the edges as nearly perpendicular to the face of the work as possible, depends upon the following principle. Supposing a correct section to be drawn ; then if the aperture be shut with a door to open in one breadth, draw a straight line from the centre of the hinge to the opposite angle of the plane ; per- pendicular to which, draw another straight line, and this perpendicular will give the splay of the jamb which comes in contact with the edge of the door which is to be fastened or locked therein. If the aperture is closed with two doors, the principle is still the same, as it is only necessary to con- sider one of them to open at a time, while the edge of the other, which is bolted to the floor and soffit, is considered as a jamb; then proceeding with the other half, which is thus left to turn on its hinges, as if it were a whole, in the same manner that we have now described. 23. Let A, Fig. 1, Plate LX., be the centre of the hinge, and B the remote point on the other 122 [Part II PRACTICAL ARCHITECTURE. side and moving edge of the door. Join AB, and draw BC perpendicular to AB; then ACB will be the bevel required. If the surface of the edge of the door be formed into a rabbet, so that the ledge may stop the door from passing to the contrary side to that on which it revolves, each of the planes which is connected by the middle one must be formed in the same manner as before. Thus, in Figs. 2, 4, 5, 6, the parts BC, DE, are each formed separately, as No. 1. FOLDING DOORS. [Plate LXI.] 24. The method of forming the joints of folding doors, so that they may open and fold back against the wall, is shown in Plate LXI. Fig. 1 shows the jamb lining, grounds, and architraves, with hanging style, and part of the doors hinged to the hanging style. Fig. 2 shows one half of the jamb, with the part of the door opened and turned parallel to the wall. Fig. 3 shows the two meeting-styles. WINDOW FRAMES AND SHUTTERS. [Plates LXII.-LXVI.] 25. In Plate LXII. Fig. 1, is a horizontal section of a sash-frame and shutters through one side. A, the inside lining of the sash-frame. B, pulley-piece or pulley-style. C, outside lining. D, back lining. E, inside bead. F, parting bead of the sashes. G, G, weights to balance the sash-frame. H, parting strip. I, back lining of boxing. J, ground. K, front shutter hung to the inside lining A of the sash-frame by the hinge u. L, M*, back-flaps hinged together at w, and to the front shutter at v. N, architrave-pilaster. No. 2 is a vertical section through the bottom-rail of the sash, through the sill of the sash-frame, - and through a part of the back of the window. O, bottom rail of the sash. P, sill of the sash-frame. Q, back of recess of window. R, coping bead. Fig. 2, sections of a more common sash-frame. No. 1, horizontal section through one side of the sash-frame. No. 2, part of the elevation of the head and pulley style of the sash-frame. * By a mistake in the drawing, the sides of the flap M are reversed.' 。 . . ・いちば ​m 十至十 ​” “' C Mange Dr'} R unwan, 一一一 ​* * * * v/ v* * - 一 ​* *. 1 ** 作 ​* i n . . . .. .. *s*1 & ft 32 H *:" 4" , "l JOINERY. FOLDING DOORS. PLATE LXI. Fig.1. TAT Il C. . ZA . . H. 2 21 Ty . ON Pa **** - 1. PE II 1. . WWWMWWWW /.. .. 1 QI IVKA NIN 117 WAV 1: . H UK:1 ba 11 w Ae 1. 11 01: VASTEN si 1 293 SEUS . RE 1:21 0 ** . E. DET 1 ES . w **** WH .... .. *** . UAIA L . ........ AWIE A . JI . 1277.. . . S ET !. LI Site..... www ! . ht A . , 12: .. 11 4 . . . . . . . IZ 2 . Fig.3. .1 V V .. 12 . 4. NO . i. . L LA JOS .. * 1 ...... . . .se . th . * . . . . . 1 U . D , .... * ** AL! . WA! . R . 1. ** I . www . S 3 12 SA SEL . WW ... 4 .. 413 www 2 V . -. . . WWW . . . we . .. www. vis . . .. .. ******* . AS 4 . . . . " . 14... wwwwwwwwwww .. stron **** . D ... . . ** . www . SI . www WWWVK . . . VINNNNN VVVV . VVVV .. LIS . . . Www - w . ... N . . . .. . -- * . . 1.1 ** al . - . .! . . . ... ... . 1 HAURI . *** *.. .. * . . .. 11 wwwwwwwwww 2. 4. OW . W 0 .. WW CY VO . . www Fig.2. .. 11- w ... - on . NAR EN ... .. . . tr. . . VUUHLM w L mm 1 . 1 . EL emes . www www . . wwwm *** . we . . S A w Am 11 1 . I We " W . WW w . 11 . . + 2 WY . SO . 11 V * www * ! - . . * w 1:51: www "L . * . . . | . . . |ị . : . . . . . . . MANCA . AR ni . . . .. . :- 4. Se . > 24 LV . Oy * . . S PAühlson, E Turrell A Faillarton &C London& Edinburghi JOINERY. 4 CONSTRUCTION OF SHUTTERS. PLATE LX71. Fig. 1. N? 1. Frig. 1. 1. 2. LE R = AN M 11 I I NEN C EP - -- - NA . re DAN AE NA TA MAN Fu Su -- - - - - - - ----- ----- Fig. 2. NOL. CARE SEE ARA ELSH EN h LE V - - ERRE ES . Fig. 2. NO2. N MS AV NAS N - 11 EN w - SL M U DOVER NE - Drewn IP. Nicholson. A Fullartus C. London& Edinburgh JOINERY. CONSTRUCTTON OF SHUTTERS. PLAT'LL XIR Fig 1 Fra 2 PNicholson E.Turrell A Fullarton&C. Londom & dan burgh JOINERY. CONSTRUCTION OF SHUTTERS. PLATE LXI!! NII. PI 12 SUL TUIN W WW WWW WW E VICT TIINA WWW WIN I VEITTAIN V W CIIIIII ..1:"2 27 UNI 11 INUTI 1112 W WWW View .. ... .. ... .:-*. --- - Drown h P Nicholson. Engraved by C:Armstrong. AF:i.!"..::19:"Ludvidinburgh COM via JOINERY CONSTRUCTION OF SHUTTERS PLATE LXV Fig2 Fial NOL. NO2, Drzwa by P Nicholson A Fullarlon & C'Landon& Eamburgh INIL 30 { 0 1 Ꭼ Ꭱ . CONSTRICTION OF SAL'TTERS.. PLATE LITI. .. - - . . ... WOH LIN WWW CILITATEAURELATION 1/2 KR VW TTTA WIUM WWW WIL TUDOWARUNNAH . SA UI Will WHI With Wh IBIT VIIM WWW WITHIN Miccicctilitlil Hill M VW RU Iraroduced hy J. Osborne Final by G. I Stranston Edin? A. Fuilarton Š Co.. London & Edinburgh. JOINERY. FITTING & CUTTING WINDOW SHTTERS. PLATE LXVII. Pig. 2 Fig. I. - . Inir.duril hoplihulson. A Fridarton & C London& Edinburgh UN P OF PLATE TXVIII www / / .................... ::::::::::::::::::::::::::::::::::::::::::::::::::::::. . . . . . . . . . . . bo u ud ... .................... JOINERY, FITTING & CUTTING WINDOW SHUTTERS. .. . Introduced by:P Nicholson . A Fullarton&Co London & Edinburgh a . .....: il ....... mk e BUD I "... .... ............... ...... . WWW се. 3 Of Syst. IV. 123 JOINERY. LI S, part of the head. T, part of the pulley style. 26. When the wall is not of sufficient thickness to admit of the shutters on each side being con- tained in a boxing which does not project from the inside of the wall into the room, it is usual to make a sliding shutter within, and parallel to the surface of the wall; and the sliding shutter on each side is covered by a piece of framing, of which the outer face is flush with the surface of the plaster. See Plate LXIII. Fig. 2, horizontal section through the sash-frame and shutters. Fig. 3, vertical section through the sash-frame. 27. In Plate LXIV. the parts of the preceding construction are shown on a large scale. The corresponding parts have the same letters of reference as in Plate LXII. In No. 1, horizontal section through sash-frame; L, M, casing of wall for the shutter. K, a door hung to the sash-frame. N, part of the sliding shutter. O, part of the framed board covering the shutter. Q, architrave moulding. No. 2, a vertical section through the head of the sash-frame and soffit of the window. R, soffit. S, P, part of ground. 28. The design in Plate LXV. of the sections of shutters, is to exhibit the same appearance, whether the shutters are folded together and enclosed in the boxing, or unfolded and extended over the aperture, so as to exclude the light. This is obtained by means of a door hung to the architrave, which is opened when the aperture of the window is required to be closed; which being done, the door is again shut. Fig. 1, design for a straight door. Fig. 2, design for a door circular on the plan. 29. In some instances a space equal to the thickness of the sash-frame may be added to the recess, or boxing for the shutters, by leaving a sufficient reveal in the brick-work (see Plate LXVI.]; so that the shutters may fall in the space between the back of the sash-frame and the jamb. OBLIQUE WINDOW SHUTTERS. [Plates LXVII. and LXVIII.] 30. Window-frames are sometimes deranged by settlements and other causes, and a slight degree requires attention in fitting the shutters, Plate LXVII. Fig. 1 exhibits the inside elevation of a window, where the sash and sash-frame are out of the square. Fig. 2, a vertical section through the window. 31. Young men, who are not aware of the difficulty which the obliquity of the sash-frame occa- sions, are liable to spoil their work by cutting the shutters square to the joints, which run upwards ; by this means, the transverse joints will not be parallel to the horizontal bars of the window, as they ought to be. The proper method is shown on the next plate. Plate LXVIII. exhibits the method of cutting the shutters, which is as follows:- Having first fitted the shutters which are hung to the sash-frame in the boxings to the full length, revolve them on their hinges on the face of the window; fit in the intermediate parts or back-ilaps for each half of the aperture, with the proper rabbets on the edge of each flap. Draw a straight line 124 [Part II. PRACTICAL ARCHITECTURE. parallel to the top and bottom ends of the shutters in the middle of the breadth of the meeting bars of the two sashes; the hinges must then be placed in a straight line perpendicular to this line or to the ends, and as near to each joint as may be found convenient. CONSTRUCTION OF SASHES. [Plate LXIX.] 32. In sashes, the horizontal bars are strengthened by dowels. The elevations and sections of the various parts of a sash, are shown in Plate LXIX. Fig. 1. No. 1 is a section of part of the sash-frame. No. 2, the section of one of the styles of the sash, the moulding being denominated astragal and hollow. No. 3, a section of the sash-bar. No. 4 and 5, sections of meeting rails of the sash. No. 6, an elevation of the crossing bars according to the section No. 3. No. 7, a longitudinal section of the horizontal bar, and of the dowel, and the transverse section of the vertical bar, showing the best method of joining the two together. This method is called franking. Fig. 2 is a section of a sash with an astragal only, exhibiting the method of preventing the sashes from shaking SKYLIGHTS. [Plate LXX.] 33. The various forms of skylights are represented in Plate LXX. Fig. 1 is a square skylight. Fig. 2, an octagonal ditto. Each of these figures shows the backing of the hips in the same manner as the hip of a roof. Figs. 3 and 4, are skylights upon elliptic plans. 34. In order to show how to space out the ribs for each quarter. Join AB, and bisect AB by a perpendicular, which make equal to the half of AB; then C being the extremity, join AC and BC. From C, with a radius equal in length to the perpendicular, describe an arc, meeting AC in 0, and BC in 5. Divide the arch 075 into five equal parts, the number intended to be in each quarter, and through the points of division 1, 2, 3, 4, draw the lines meeting AB in the points d, e, f, g. Through the points d, e, f, g, draw lines from the centre of the figure to meet the elliptic curve, and the points of intersection are the places of the feet of the bars as required. Fig. 4 is seldom executed, on account of the difficulty of bending the glass. The plan in No. 2 is equally divided. AN ELLIPTIC OR CIRCULAR ARCHITRAVE ON A CIRCULAR PLAN. [Plate LXXI.] 34. Fig. 1 shows the plan and elevation. Divide the inner curve of the elevation into any number of equal parts at the points 1, 2, 3, &c and draw the line s a, 16, 2c, 3d, &c. perpendicular to the chord of the opening, meeting the wall line in the points a, b, c, d, &c. In the line MN, Fig. 2, set off the distances ab, bc, cd, &c. equal JOINERY, 1 PLATE LXIX, SASHES. · N. 7. 11 11: SININ _ tertintitu Fig. 1. / 7 . . A NAR- TV LES TE - * Nr IIIIIIIIIIIIIIIIIIIIIII.11PIL 1 III . n 014 ATTITUITIITTO WV 6 - - - - INXHINI UINTILIITTITUUT TIITIUMIC MULTIMEDIM Umumi TITISAN MINIT IL = = IILIL. COM inIIIIlIlIIII! '' T T10111 111111111111111111111* 1 IIII|1111111111 1'111111111111111 II. - - - 1111111 - WE V WY SA olt tuntil FTLIT 111111111111111111111110111111'111!11111. 11 ! Uli Fig.2. No 5. NO4 w S WA llllll Til!liitliile 11 - - . 1 S IS Vos - DOS - -- - - - - - NO - -- SUR .. . . . i UR w E S. O ARAFAvant . ARA NOTRESS V.1. R Rotte P.Nicholson AFullatan & CO Lundu & Edititugh. i Un c JOINERY SKYLIGHTS PLATE LXX Fig 1. Fig 2. . :11. N. 2. w SI DES BE V.2. το και ' S - - - Anar- - . - - - SR SHES Fhg 4. Fie 3. he ANNELSE DININI1110 2 . N°11. . - - - E WRITORIENTATORTTI UMANIPU Ilib. KIKUUHUHU WIKILETTE INW umumu . INT NO :.. NO2. 122 A T IMUN Ku Introduced by Pulicholsun. Engrared by Gladirin. Alularion&0. London& Edinburgh LIV. JOINERY, ELLIPTIC OR CIRCULAR ARCHITRAVE ON A CIRCULAR OR ELLIPTIC PLAN. PLATE.IXIT Fig. 1. II WY SVN WIN 21 T/ BW VA MY I HMMMIIHIIHIMMWWMWITHIMINIMIIMMUNITI WILL HUWINNINUM TO . . XIIIIIIIIIIII NUWINIMUI VW M WODNIU UND WIT IN : e Odd WIMMINIIIIIIIIti DIDUNT UT JUHUMIWUM 11 111 She IRUTTU 1111 NMWWUWUN Fig. 2. é a E. Zurrer PNicholson. A Fullartan & COLondon&Edinburgh JOINERY PLATES LXXI. XIIII. MOULDINGS Fig.3. : Fig.4. Fig.1. I Fig.2. CREATER RADAR Fig.8. Fig.10. Fig.g. HAL 1 Fig.13. Fig.11. Fig.12. Fig.15. Fig. 14. TI Fig.18. Fig.19. P.16. Fig.17. Fig.5. Fig.6. Fig. 7. S M. - A u - - - . Y . +- - - ST AND inariin Londonda tinburgh Bich Sect. IV.) 125 JOINERY, to the developments of a b, bc, cd, Fig. 1. To the line MN draw the perpendiculars 61, c2, d3, &c. and make the heights of these perpendiculars equal to those in the elevation, Fig. 1; through all the points a, 1, 2, 3, &c. draw a curve. Make aM equal to the breadth of the archivolt, and draw the outer curve MHPIN parallel to the inner curve. Here the veneers are made in three lengths. OF MOULDINGS. [Plate LXXII.] we tenen d 35. Wood is generally much thinner than the dimension of its breadth, reckoning the breadth and thickness on the sides of the rectangular section made by cutting it perpendicular to the fibres, the length being understood to be parallel to the fibres. The faces are the two broad planes that run in the direction of the fibres; and the edges are the two narrow planes which also run in the direction of the fibres. The ends are the two planes perpendicular to the fibres. 36. When the wood has been reduced to the rectangular shape by the square and plane, so that the sides may be planes, and the angles right angles, the next operation is to take away the right angles, and reduce the wood to mouldings, which is called sticking, and the moulding is said to be stuck. 37. When the edge of a piece of wood is reduced to a cylindrical form, it is said to be rounded, which is the simplest species of moulded work. 38. When a part of the arris is reduced to a semi-cylinder, so that the surface of the cylindrical part may be flush, both with the face and edge of the wood, and that a groove or sinking may be made in the face only, the cylindrical part is called a bead, and the sinking a quirk, so that the moulding is called a quirked bead. 39. When a quirk is also formed in the narrow plane, or edge, so as to make the rounded part at the angle three-fourths of a cylinder, the moulding obtains the name of bead and double quirk. 40. When there are two semi-cylindrical mouldings, rising both from a plane parallel to the face; and when one comes close to the edge of the piece, and the other has a quirk on the farther side, and its surface flush with the face of the wood, the combinations of these mouldings are termed a double bead, or double bead and quirk. In this combination, the bead which is next to the edge of the stuff is much less than the other. 41. Mouldings are generally separated from one another, and frequently terminated by two narrow planes, at right angles to each other, called fillets, which show two sides of a rectangular prism. 42. Mouldings, as well as fillets, are called members. 43. When a semi-cylindrical moulding, which rises from a plane parallel to the face, is terminated on the edge by a fillet, the two members thus combined are called a torus. 44. If there be two semi-cylindrical mouldings springing from a plane parallel to the face, termi- nated on the edge by a fillet, this combination of members is called a double torus. 45. A repetition of equal semi-cylindrical mouldings, springing from a plane or cylindrical surface, is called reeds. 46. The cima recta, and cima reversa, are called in joinery ogee. The former is called ogee, and the latter ogee reverse. 47. Ovolo presents a convex conic section. 48. A quarter round is the fourth part of a cylindrical surface, but has no quirk on either side. 126 (Part II. PRACTICAL ARCHITECTURE. MOULDINGS FOR FRAMING. [Plate LXXIII.] 49. In framed work, as doors, shutters, wainscotting, &c., the edges of the framing is generally reduced at the angles to mouldings. The mouldings for this purpose are the ovolo, or the ogee, with or without a bead next to the panel; but when the ovolo is employed, a bead or a fillet becomes necessary. The ogee is either common or quirked, with a bead at the bottom. 50. When the margins of the framing terminate on the edges next to the panel, with one or more mouldings, which both advance before, and retire from the face of the framing to the panelling. The mouldings thus introduced are called bolection mouldings. 51. The panelling of framed work is generally sunk within the face of the framing; sometimes, however, for outside work, it is made flush. In the best flush work, the panels are surrounded with a bead, formed on the edge of the framing, and the work is called bead and flush. In the more common kind of flush framing, the bead is run on the two edges of the panel in the direction of the fibres, and is called bead and butt. Fig. 1. Plate LXXIII. Fillets. Fig. 2. Edge rounded. This simple moulding is also sometimes called a bead; but not unless it is fixed to one side of a rectangular piece of wood, and the rounded part made flush with the other side. Fig. 3. Flush bead, or bead and quirk. Fig. 4. Bead and double quirk. Fig. 5. Double bead. Fig. 6. Torus. The torus in joinery differs from the bead, in having a fillet. Fig. 7. Double torus. Fig. 8. Reeded moulding on the edge. Fig. 9. Reeded moulding on the face, which may apply to bands, architraves, and pilasters Fig. 10. Reeded mouldings round a cylinder or staff. These will apply to columns, or other cir- cular bodies. Fig. 11. Semicircular flutes, which may apply to bands, pilasters, and columns. Fig. 12. Shallow flutes, which may also be applied to columns, pilasters, and flat bands. Fig. 13. Style of a door or shutter, with part of the panel, showing the mouldings which are here termed quirk, ogee, and bead. Fig. 14. Style and part of the panel of a door or shutter, showing the mouldings which are in this example termed quirked ovolo and bead. Fig. 15. Section of a door style, with part of the panel, showing the mouldings which are here termed bolection mouldings. Figs. 16, 17, 18, and 19, are various forms of sections for sash bars. RAKING MOULDINGS. [Plate LXXIV.] 52. Given the inclination of a moulding to the horizon, and a section of that moulding, to find the section of the return moulding. Let BE, Fig. 1, Plate LXXIV., be the line of inclination, and let CDM be a section of the moulding Draw a line through A parallel to EB. Through M draw MD perpendicular to the JOINERY RAKING MOULDINGS. PLATE LXXZV. . . Pig. I. Fig. 2. - - - - - - - - - Prig. 3. - - Y - Fig. 4. - - - - - - - - - - - - - - - - - - - - - - - - - - - * - - - - - - - - - - - - - - - + - - - - - - - - - - - - - - - - - - HL. ***------------- ---------------- - - - - - - - - - - - - - - - - - - - - - - ------ N92. Invented by P.N. N. 1. 193. - - - - - - - - Drunn br P Vicholson. Arnarton. Lonion), Earburgt SECT. IV.] 127 JOINERY. S line passing through A, meeting it in P. Draw Mm parallel to EB, and draw any line a f parallel to the horizon above the moulding. In a f make a p equal to AP, and draw p m perpendicular to a f, and m will be a point in the curve. This may be applicable to the skirting of a stair, where the passages return both above and below. In order to avoid the trouble of making mouldings to various sections, torous skirting is most frequently executed, as is shown in Figs. 2 and 3. Fig. 4 shows the method of finding the angle bars of shop fronts. No. 1 is the given bar, Nos. 2 and 3 are angle bars found from No.1. Thus rm, No. 3, is equal to RM, No. 1. In No. 2, make a p equal to AP, No. 1, and draw p m parallel to aE. 53. Raking mouldings depend upon the principle of a solid angle consisting of three plane angles, or what may be called a trihedral; and this may be considered either as a hollow or as a solid, according as it may be used externally or internally. The mouldings are supposed to be placed in two of the lines of concourse, and to meet each other in a plane passing through the other line of concourse. 54. The raking mouldings of a pediment are placed upon a solid trihedral, the horizontal moulding being disposed upon the obtuse angle, and the raking mouldings upon the top of the tympanum. In this case, the mitre of the mouldings is in the same plane with the line of concourse of the two sides of the building. 55. The three planes which terminate in a point in the inside of a rectangular room, may be considered as a hollow trihedral. Now, if these three planes which constitute the trihedral be at right angles, no difficulty can occur in constructing the mouldings, as each cornice may have the same section, and as the direction of both cornices are perpendicular to the line of concourse of the two vertical sides of the room; but where the one cornice is perpendicular, and the other oblique, the case becomes the same as the preceding. 56. The same principle is also applied to the bars of a bow window, of which the sides form a polygonal prism. In this, the trihedral is considered as formed by the face of one of the vertical planes ; a vertical plane bisecting the two adjoining faces and a horizontal plane. Let us suppose the inclination of the two planes through which the plane of the mitre passes, and the other two angles of the trihedral to be given. The projection of each mitre, and the figure of the mitre, or the section of one of the mouldings and the mitre line, must also be given, and we shall have suffi- cient data in order to ascertain the section of the other moulding. 57. This construction becomes very easy, where the inclination of the two planes is a right angle, and when the angle contained by the edges of the one plane is a right angle, and that contained by the edges of the other an obtuse angle, as is the case with a pediment. The plane of the two adjoining walls is generally a right angle, and the angle contained by two of the edges of one of the planes is an obtuse angle, and that contained by the two edges of the other a right angle. This case affords a very easy construction; it being only necessary to lay down the side of the building on which the pediment or inclined cornice is to be made, with a projection of the mouldings at the lower end, without any plan whatever, provided that the mouldings have the same projecture on both sides. 58. The same is also the case with regard to the two sides of a bow window, where the sides are vertical planes at any angles with each other. 128 [Part II. PRACTICAL ARCHITECTURE. OF STAIRS. [Plate LXXV.] 59. Stairs and hand-rails are most important branches in joinery; but before we enter upon their construction, it will be useful to point out some of the leading principles, without regarding the materials of which stairs are constructed. 60. The breadth of steps in general use is from 9 to 12 inches, or about 10 inches at the medium. In the best staircases, the breadth ought never to be less than 12 inches, nor more than 18. It is a general maxim, that a step of greater breadth requires less height than one of less breadth: thus a step of 12 inches in breadth will require a rise of 51 inches; which may be taken as a standard by which to regulate those of other dimensions ; so that multiplying 12 inches by 51, we should have 66; then supposing a step to be 10 inches in breadth, the height should be 18 = 65 inches, which is nearly, if not exactly, what common practice would allow. The proportion of steps being thus regulated, the next consideration is the number requisite between two floors or stories; to ascertain this, we have only to suppose the breadth of the steps to be given, say 10 inches each, as depending on the space allowed for the staircase, and this, according to the rule laid down, will require a rise of 7 inches nearly. Suppose then the distance from floor to floor to be 13 feet 4 inches = 160 inches; then 160 = 229, which would be the number required. But as the steps must be equal in height, we should rather take twenty-three rises, provided the staircase room would admit of it. 61. Stairs have several varieties of structure, which depend principally on the situation and desti- nation of the building. 62. Dog-legged stairs, are those that have no opening or well-hole; the rail and balusters of both the progressive and returning flights fall in the same vertical planes. 63. Geometrical stairs, are those which have an opening down the middle, and of which every step derives its support from that immediately below, and from the wall of the staircase. 64. The steps of a stair consist of two parts, one being parallel, and the other perpendicular to the horizon. The part which is parallel is called the tread of the step, and the other part which is perpendicular, is called the riser. 65. The rough timber work which is used in the support of a stair, is called the carriage. 66. The string board, is a board fitted against'the ends of the steps next to the well-hole, so as to make a complete finish ; and the string which terminates the ends of the winders, is a veneer made in the form of a spiral back, with thick wood, so as to make it sufficiently strong. 67. The most certain method of carrying up a stair, whether of stone or wood, is to provide a rod of sufficient length to reach from one floor to the other, divided into as many equal parts as the risers are in number, and thereby to try every step as the work advances. 68. In Plate LXXV. Fig. 1 is a plan and elevation of a dog-leg stair without winders. Fig. 2 is a plan and elevation of a dog-leg stair, with winders in one half of the turning. OF HAND-RAILING. [Plate LXXVI.) 69. A hand-rail is the upper part of the fence in a geometrical stair. In order that the hand may glide easily along the rail without straining the body, it is evident that the rail ought to follow the general line of the steps, and to be quite smooth and free from inequalities. JOIN E RY. STAIRS. PLATE LXXV .- . -- - - Fig. 1. Fig. 2. ELEVATION. ELEVATION. - - - - ----- -- -- - -- - --- - ---- - - -- --- . . . . . - - |-- L - - Fig. 1. Fig. 2. PLAN. PLAN. F ! Drawn by P. Nicholson: A Fularton&C°London& Edinburgh JOINERY. HAND RAILING. PLATE TXXVI. Fig. 2 Fig 3. fint. W S : II . N1 11 -- 2 UNTV CC A d b hi Ut i. S HAN HOW 2 . - . . - 1 . WUNG YAN - - . - w - - G NA I A IIIIIIIIIIIM ... SCC IS WS - 1011 -- 1+ AI - 91 Wwtdi. Iki 22% HUUMS - G6 - N FC IT Wulionu . are LA T 1 Fig.&. Fig.5. Fig.6. Fiq.7. Engraved h.G.Gladwin. Introduced by P. Nicholson. Pot I A Frillarimn&C London& Edinburgh chino Sect. IV.] 129 JOINERY. 70. The principle of hand-railing depends on the method of finding the section of a right cylinder, cylindroid, or prism, according to three given points in or out of the surface, that is, the section made by a plane through three given points in space. 71. The cylinder, cylindroid, or prism, is hollow, and equal in thickness to the breadth of the rail that is to the horizontal dimension of its section, and the ends or bases, the same as the plane or projection upon the floor. 72. The hand-rail of a stair may always be formed of a portion of this cylinder, cylindroid, or prism, the base of which is the plane of the stair; for the hand-rail itself must stand over the plane, it will therefore be contained between the vertical surface of the cylinder, cylindroid, or prism. And as the hand-rail is got out in portions, so that each portion may stand over a quadrant of the circle, or ellipse, which forms the plane, we may also suppose such a portion contained between two parallel planes, so that the portion of the hand-rail may be thus contained between the two cylin- 2 together to form the rail, are to be prepared in such a manner, that when set upon their place, all the sections which may be supposed to be made by a vertical plane passing through the axis of the cylinder, or cylindroid, may be rectangles, and this is called the squaring of the rail ; which is all that can be done by geometrical rules. 73. Now, as hand-rails are not made of such portions of hollow cylinders or cylindroids, but of plank wood, we have only to consider how such portions may be formed from a plank sufficiently thick. As the faces of the plank are planes, we may suppose the rail contained between two parallel planes, that is, between the two faces of the plank. Then such figures are to be drawn on the sides of the plank, that, when the superfluous parts are cut away, the surfaces that are formed between the opposite figures are portions of the external and internal cylindrical or cylindroidic surfaces. A mould made in the form of these figures, is called the face mould, which is only a section of the cylinder or cylindroid through three points in space. 74. The vertical, or cylindrical, or cylindroidic surfaces being formed, the upper and lower sur- faces must next be formed. This is done, by bending another mould round one of the cylindrical or cylindroidic surfaces, generally made to the convex side, and drawing lines on the surface round the edges of this mould. Then the superfluous wood is cut away from the top and bottom, so that if the piece were set in its place, and a straight edge applied upon the surfaces now formed, and directed to the axis of the well-hole parallel to the horizon, it would coincide with the surface. The mould thus applied upon the convex side to form the top and bottom of the piece, is called the fulling mould. 75. To find these moulds, the plan of the steps and rail must first be laid down; then the falling mould, which must be regulated by the heights of the steps; and lastly, the face mould is ascer- tained by the falling mould, which furuishes the three heights alluded to. 76. The illustrative Plate LXXVI. will render the nature of a hand-rail winding round the cylinder, of the size of the well-hole, easily understood. Fig. 1 exhibits the cylinder with the rail squared. Fig. 2 exhibits the method of describing the rail on the elevation, Fig. 1. Fig. 3 shows the position of a portion of the cylinder contained between two parallel planes, when seen in the direction of the cutting plane; and as the cylinder is supposed to be hollow, Fig. 4 shows the entire cylinder, and the half when seen obliquely. Figs. 5 and 6 show the concave and convex side of the same for one quarter, taking in a small portion of the straight rail. Figs. 7 and 8 exhibit Figs. 5 and 6 completely squared, or formed to the exact dimensions of the rail. II 1 130 PRACTICAL ARCHITECTURE. [PART II. Figs. 5 and 6 show the first state of the pieces as cut from the rough plank. Figs. 7 and 8, the pieces entirely squared. OF FINDING THE MOULDS FOR HAND-RAILS. [Plate LXXVII.-LXXXIII.] 77. To find the moulds of a land-rail with 8 winders in the turning or 4 in each half. Plate LXXVII. Let MMM, &c. be the outside of the plan, and M'M'M', &c. the inside of the same, ABCD a line passing through the middle of its breadth, the part AB being straight, and BCD one-fourth of the circumference of the circle ; the point C in the middle of the arc BD, A at one extremity of the line ABCD, and D at the other. Divide the quadrant BCD into any number of equal parts, which in this example are 4. Draw the straight line RK, and make RK equal to the development of the quadrant MMM, &c. on the convex side. Draw KH perpendicular to RK, and make KH equal to the height of a step. Draw HF parallel to RK, and make HF equal in length to the breadth of a flyer, and join FK. Draw Rt perpendicular to RK. In Rt make Ru* equal to the height of 4 winders, and join uK, curve of the angle at K. Through u, draw v w perpendicular to uK. Make u v and u w each equal to half the breadth of the falling mould FN, and draw the upper and lower edges of the falling mould, as seen in the figure. Join DC, and produce DC to E. Draw DE and CF perpendicular to DE. Make DE equal to one-fourth (or any part) of the height from R to the upper edge of the falling mould in the perpen- dicular Rt, and CF equal to one-fourth (or the same part) of the height from Q to the upper edge of the falling mould in the perpendicular QP, that DF was of the height in the perpendicular Rt. Join EF, and produce it to meet DC in E. Join the dotted line EA. Draw GP through the centre I, or at any convenient distance from the plan perpendicular to EA, and draw Mp parallel to EA, meeting GP in P. At any convenient distance, draw Hp parallel to GP. Make the perpen- dicular of the face mould equal to its corresponding height on the falling mould, and draw the straight line pl; then drawing ordinates PM, PM, &c. and p m, pm, &c. from the points P, P, P, &c. and p, p, p, &c. where lines parallel to HI meet PG and PI, and making p m equal to the corresponding PM on the plane both outside and inside, we shall have the face mould required. By making the middle of the falling mould at the heights, as the nosings of the steps, the same face mould will apply to the upper and lower parts of the rail over the circular part. The top line PPP, &c. is left on the falling mould to regulate its position, when bent upon the convex surface, as the line PPP, &c. will fall into the plane surface of the top of the plank. The line PPP, &c. is obtained by making the perpendicular fP, fP, fP, &c. equal to the corresponding perpendiculars f p, f p, $ p, &c. 78. Plate LXXVIII. exhibits the construction of the face mould and falling mould, in the case when the rail is to be raised higher upon the winders than upon the flyers. Here the curve line stu v w x will coincide with the top surface of the plank, and the lower edge of the falling mould above. The method of proceeding to find the falling and face moulds is so similar to the description of Plate LXXVII. that any further explanation is not required. 79. Plate LXXIX. shows the application of the trammel to the description of the face mould.--- See the article STEREOTOMY. * u, by mistake, is placed too far up. JOINERY, FLI.17 RAILING.. PLATE.LXXVII. .. SI m24 . - m 122 TETO . - - E . Engmred by E. luurell Turental b löchulsom.. A Fullarion & *. London& Käinburgh Vor Invented by P. Nicholson, - - - - .. III . . . . multllllllllll . www IUNTINI W Ittli wird V MNHUN .... 113 ..... W T111 SUNT: I UNUI AM ..... whic JUUNI IT MINU UNIT SIA III III/1 N WO? 11 HAI SUUNTOWN City Show WITHIURA WANINI . L NO // INWRITIN N w KM VIN NU www. eur H www SH WWWW he . UL .. . ... . ....... . ..... B ITINUNIUNIUM S w M li WW Wilt 1111 WINNI TINTINDD HUHU 111111 11 MULIINIHIN Hill . manten he www . W VIDEO UNTINLUND ID OUVINININNI . NET V Sie WIDUMMY IMUNINNUNNINUMINIU MIT www w * N W IUNI MINI MUHANNUR WINNIHALDINININ IHMONIIIIIIIII NIMITUWINNIN UMW 45 WWIIIIIIIIIIIIIIIIII WWWWWWWW do u IUCINIIIII NETINIAI U21 TIENT HUNNIKOVITIJUUNIT S 1 WINNI M INUNI! Ini WODNI HU - : . : - - --- - - -- - - - -- . .. - - --- - ------ .. . - ----- . HAND RAILING. A Fullarton&Co Landon& Edinburgh JOINERY, INT ! TIMIIIMIIMMMWWII OHIN NINO S IIIII AITI WINNI MINUUTINIO INITI M III w S4 TIT llllll IN INITIVNIMIN IIIIIIIIIII INITI m mID.NT fillimim ITUTU INUNDUNUITM NS 7 HUN UN 19 WITA UMNO 1 1 NIIIII IN * R: NOTIUNI UNIUNIUI 71 - - - - -- W LIINUMIIDIINUUNNIIII A1 L IIUIUUHUOWUH! V th VINNU . L 9 WWDWRNHUBUNNUNTUT . were th HKT111 WS WINNIN INTUIT www. 1 II NIINIT V ht . 1 . DINIWINUNNI! DIDUNUMMIT N . 21// M ! M MIN SIN .. ... . . . . . . . ' . 2 WW . inut M IT INIMUM WIIHIN WINNINDUTINIUWIN! MW BIL WWWWWWWU11 IMUNIKUTUMNI 11UUNNAN MUHTAUTUU NI HUUI1 III W UNIWWANININDINUMBUH IIIIIIIII11 W TANIN WINWIN WY W WIDHIWANI . E ** N 17/III 1 lilitis$1 BUTT / / . PA VI wewe IT111 JURIDIUNillkindlu MUUTUMIKULtu nh lllll I 11. O until + - - - - --- - -- -- - - - - - - - - - - - - - - - - . th NU NEWS WW I WIKIN ! Millallilllllllllllllllllllllll llll N - --- - - . . - - - lllllllllllllllllllllll W TV WWW PLATE LXXVII WWW - - - - .-... — ---- URL -------- -...- ... ---- - - -- - - - - -- ... - - -- Enureved by C. Armstrong. w Imamited by: P Nicholson. A Foliartside'London& Edinburgh HAN) ኢ1/11/'d; 2 de TEN AR . . . . . .. . . . . . - - 1 . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . Engraved by E Tuoreid. PLATE LXXIX. ilgio JOINERY. HAND RAILING. PLATE LXXX 's Most ch NO 2 N:3 DIE TPVMI 671 NA . Invented by:P Nicholson. A Fullarton&C°London& Edinburgh Chiba JOINERY-RR , WALL CA 21 STA 4 JU / . N° 2. A. Fullarton & Co London&Edinburgh Juventud Plicholson. - - - - -- - - - - - - + . . - - - M ATUL H-ND RAILING. - - - - - - - - - - - - - IXXXI LCFIA STAR . Ipil JOINERY1 /2 , HAND RAILING PIATE ΙΧΧΙΙ. . . . Fig. 3. SA S AN - THE SY - - - OS OLA - - . - TER < Tan - - *. . - - -- L. + ...... . . .. ........... . . .. . .... - .. ........ .. I D --- ... NA - Le .. RH HUN M 15 he 12 PS ! . S . . .. WA . . . M w : . SU Fig. 1. - - - - 1 I . - - ..... - - C . TR NR - - = = Fig. 1 A A Fullarton&c.London & Edinburgh JOINERY. Hullli , HAND RAILING. PLATE LXXXIII. NI ITUNT TA1 IN 7 WINTnto MIIINNNIII Hili lilililili Fig. 3. . 1 NNUHlli HET WHEN - . FA VE Fio. 1. ........... Fig. 2. Invented by P. Nicholson. A.Fullarton & Co London & Edinburgh. P of i JOINERY PLATE LXXXU NO IL Fig. 3 Fig. 7 Fig. 6. Fig. 5. Fiul Nº IT NON-- ΝΠ NI Voftit P Nzeholm A Flatten ondank Bambu Sect. IV.] 131 JOINERY. 80. Plate LXXX, exhibits the method of finding the elevation of the squared piece so as to get it out of the least quantity of stuff. The position of the plan is found in the same manner as in Plate LXXVII. and the heights are taken from those of the falling mould. 81. Plate LXXXI. exhibits the elevation of the squared piece of the rail where the ordinates are taken perpendicular to the chord joining the extremities of the concave side of the piece. The heights are taken from the corresponding heights on the falling mould in the same manner as in the preceding plate. The intention of this plate is to show that the rail will require much thicker stuff than if the position of the plan had been found as in Plate LXXVII. 82. Plate LXXXII. Fig. 1, exhibits the plan and elevation of a rail to go round a level half space, in order to show how to get it out of the least quantity of stuff. Fig. 2 is the common method of finding the elevation of the rail piece by joining the points at the extremities of the concave side for a base line, and tracing it from the falling mould, No. 3. This position requires stuff considerably thicker than in Fig. 1. 83. Plate LXXXIII. is a method of tracing the face mould so as to apply only to one face of the plank, and cut the piece out square to the face, or in lines which are perpendicular to the face. Find the elevation of the rail piece, and let BCIG be a section of the rail in the elevation. Pro- duce CB to meet the base OP in P, and the concave side of the plan in b. From the centre 0. and through b, draw the straight line Og, meeting the convex side of the rail in g. Draw gi parallel to bP, and b i perpendicular to OP. Let O be the point in the line OF which comes in contact with the extremities of the lower edge of the elevation corresponding to the centre 0 in the plan, and let the line CQ meet OF in A. Draw GH, BD, CE perpendicular to OF, meeting OF in the points H, D, E. In Fig. 3, draw a straight line, in which take the distance o p equal to OA, Fig. 1. Draw pg perpendicular to op. In pg, take p a equal to Pb, Fig. 1, and a g equal to i g, Fig. 1. Draw a e and gf parallel to po. In a e, make a d equal to AD, Fig. 1, and a e equal to AE, Fig. 1. Join d g, and complete the parallelogram e dgf; then e is a point in the concave side of the face mould, and g a point in the convex side, so that, by finding all the sections in the same manner, we shall obtain as many points both in the concave and convex side of the falling mould as we please. 84. A description of Plate LXXXIII., No. 2, containing directions for drawing the scroll for terminating the hand-rail' and other minor details, will be found in Appendix I., p. 472. SECTION V. PLASTERING. Value of the Plasterer's Art, 1, 2.- Internal Plastering, Cements, 2, 3. Lime, 4.- Plaster of Paris, 5.--- Lathing, 6–12.- Cornices, 13.- Ornamented Cornices and Enrichments, 14.- External Plastering, 15, 16.- Stucco, 17._ Roman Cement, 18. Bayley's Composition, 19.—Mastic Cement, 20. Keene's Marble Cement, 21.--Explanations of Terms, 22. 1. PLASTERING is an art of primary importance in building, more particularly as respects the interior of edifices, in the finishing and decoration of which it has long formed a very prominent feature. Of late years, indeed, especially in and immediately around London, in consequence of the great improvements that have been made in the composition and execution of stucco work, combined with the high price of stone and labour, the exterior of many of our buildings are, equally with the inte- rior, indebted to the skill and dexterity of the Plasterer for their principal decorations; and such is the perfection to which this species of work has been brought, that the ornamental works of the Plasterer, not only often rival in beauty, but in some cases, when well-executed, equal in durability, those of the Mason and Sculptor. The ordinary work of the Plasterer consists in preparing and covering the walls and ceilings of buildings with a composition, of which the ground-work is lime and hair mortar, finished with one or more coats of finer materials of different kinds, according to the description of work intended to be produced; but under the general term plastering is likewise included the execution of the several kinds of ornamental stucco work, together with the modelling and casting in Plaster of Paris, the various enrichments employed in the decoration of buildings. Formerly, the most elaborate enrich- ments of ceilings, cornices, &c., were executed by hand, but from the facility with which cast work is executed, and its consequent comparative cheapness—especially where a considerable quantity is required—it has almost entirely superseded the former method. INTERNAL PLASTERING. 2. Internal plastering is of two kinds; the one being executed on laths, the other on walls. The latter description is termed rendering. 3. The cements or mortars made use of by plasterers are of several kinds. The first is called Coarse stuff. This is prepared in the same way as common mortar, with the addition of a little hair from the tan-yards, which is incorporated therewith by the use of a three- pronged rake. The hair used should be long and fresh, and equally distributed throughout the stuff, so that if a finger be drawn across it, the hairs shall appear at least the sixteenth part of an inch asunder. Fine stuff is pure lime, slaked with a small quantity of water, and afterwards saturated to excess, and put into tubs in a semi-fluid state, when it is allowed to settle and the water to evaporate. Sometimes a small quantity of hair is added, dar Sect. V.] 133 PLASTERING. Putty is lime dissolved in a small quantity of water ; fresh lime being added from time to time, and the mixture stirred with a stick until the lime is entirely slaked, and the whole becomes of the consistence of mud: it is next sifted or run through a hair sieve, in order to separate the grosser parts of the lime, and is then fit for use. Putty differs from fine stuff inasmuch as it is prepared in a different manner, and always used without hair. Stucco, for internal work, is composed of fine stuff and very fine washed sand, in the proportion of one part of the latter to three of the former. All walls intended to be painted are finished with this description of stucco. 4. LIME.—The general nature and quality of lime, which forms the most important ingredient in all the operations of the Plasterer, has been already treated of at considerable length under the head of PRINCIPLES OF PRACTICAL ARCHITECTURE, Sect. II. § 47–63 ; and it is therefore only necessary to observe in this place, that, for the purposes of the Plasterer, chalk-lime is commonly used, being most suitable for internal works. The lime generally used in London comes from Purfleet, in Kent; but for external stucco and other works, where strength and durability are required, that made at Dorking, in Surrey, is pre- ferred. In some later works in London, the blue lias lime, brought from Devonshire, has been used both for internal and external stucco; and where due attention is paid to the preparation, and sufficient labour is bestowed in the use of it, it certainly forms a most excellent cement, being extremely hard and durable. Much, however, depends on the requisite labour being given to it, and as that is considerable, the expense of the work is materially increased thereby. Great attention is required on the part of the Plasterer in the preparation of his materials, more especially as regards the proper slaking of the lime; otherwise, nearly as soon as the work is finished, it is liable to be disfigured by numerous blisters on the surface. On opening these, a small particle of unslaked lime will be discovered in each, which must be removed, and the defect made good: this causes much additional labour, and, if the wall has been painted, occasions a considerable additional expense. 5. PLASTER OF PARIS.—Another important article to the Plasterer, in the execution of his best works, is Plaster of Paris, of which mouldings and other decorations are made. When expedition is required, it is of the greatest service to mix with the mortar, as it causes the work to set as soon as it is laid on. This material is found in great quantities in the environs of Paris, but that which is principally used in London is prepared from Gypsum,-a sulphate of lime dug in Derbyshire. It is calcined in a kiln, and then beaten into a powder, which being sifted, and mixed with water, is used either for casting ornaments, or with the mortar in such proportions as may be necessary to produce the effect required. The operation of mixing Plaster of Paris with mortar is termed gauging; and the material so compounded is called Gauged stuff. Plaster of Paris being formed into a thin paste by the addition of water, quickly sets or becomes hard, and at the instant of its setting its bulk is increased ; this expansion of the plaster, in passing from a soft to a hard state, is one of its most valuable properties, rendering it an excellent material for filling cavities where other mixtures would shrink and leave vacuities, or entirely separate from the adjoining parts. This material is used for taking casts and impressions of figures, busts, medals, ornaments, &c., and likewise for making the moulds in which they are formed. A coarse kind of plaster, which is much used for forming the floors of granaries, cottages, and farm-houses, is made from a blueish stone, much resembling the stone of which Dutch tarras is made. The stone is burnt like lime, and becomes white thereby, but when mixed with water it 134 [PART II. PRACTICAL ARCHITECTURE. does not ferment like lime. When cold it is beat into a fine powder, and, in order to prepare it for use, a quantity is put into a tub and water added till the whole becomes liquid; in this state it is stirred with a stick, and should be used immediately, or it will become hard and useless, as it will not bear to be mixed a second time like lime. In constructing this description of floors, the joists are laid in the usual manner, and a strong sort of reed, which grows in Huntingdonshire, is nailed thereto, on which the plaster is spread with as much expedition as possible, to the thickness of about two inches. In those parts where reeds cannot be obtained laths are made use of, but they are more expensive. These floors are very useful, inasmuch as they afford great security both against fire and vermin. 6. LATHS.- Laths are of various sizes and qualities. In and about London they are usually cut out of Baltic or American timber, in lengths of three and four feet, and are known in the trade by the name of lath, lath and half, and double lath. The common laths are a bare quarter of an inch in thickness; the others are thicker, as their names import, and increase in value accordingly. The thin laths are most applicable for partitions, and the thicker ones for ceilings. In the use of them, care should be taken so to dispose them that the joints may be as much broken as possible, by which means the plastering with which they are covered is much strengthened. Laths are occasionally made of oak; in which case it is necessary to use wrought-iron nails to fix them, instead of cast-iron nails, which are more commonly used with fir laths. LATHING. 7. After the lathing is completed, the next process is the layiny, which is performed by spreading a single coat of lime and hair all over the ceiling or partition to be plastered, keeping it as smooth 8. Lathing, laying, and setting, is a superior description of work to the foregoing. The first coat being laid on, is crossed all over with the end of a lath, to give it a key to receive the next coat, this is called pricking-up." "When this coat is sufficiently dry, a thin and smooth coat of putty is spread oyer it with a smoothing trowel, which the workman uses in his right hand, having in the other a large flat brush of hogs' bristles, which he occasionally dips in water, drawing it backwards and forwards over the work, thus producing a smooth and even surface. 9. Lathing, floating, and setting, differs from the foregoing in having the first coat pricked-up to receive the set, which is here called floating. To perform this operation, the workmen are provided with a substantial straight edge from six to twelve feet in length, varying according to the dimen- sions of the room in which the work is to be executed, and which must be used by two workmen. The parts to be floated are first tried by a plumb-line, to ascertain whether they are perfectly fiat and level ; and wherever any deficiency appears, it must be filled up with lime and hair; this is termed filling out, and when the whole surface is made sufficiently level the screeds are begun to be formed. 10. A screed is a style of lime and hair of about seven inches in width, gauged quite true by means of a straight edge. The screeds are to be formed at about three or four feet from each other, in a vertical direction on all the sides of the room; and the intervals between them are to be filled up with lime and hair till they are flush with the face of the screeds. The straight edge is then worked horizontally over the screeds, by which means any superfluous stuff, which may project beyond them in the intervals, is removed, and a perfectly plain surface is produced. This operation is termed floating, and is occasionally applied to ceilings as well as to walls, the screeds being formed Sect V] 135 PLASTERING, in the direction of the width of the apartment. This process requires considerable care and skill in the execution. The setting to floated work is performed in the manner already described for laying, but, as it is employed for the best apartments, greater care is required, and about one-sixth part of Plaster of Paris is added in order to make it set more speedily, and give it a more close and compact appear- ance; it likewise renders it more firm, and better fitted to receive such colouring as may be required when finished. For floated stucco work the pricking-up coat cannot be too dry; but if the floating which is to receive the setting coat be too dry before the set is laid on, it will be liable to peel off or crack, so as to disfigure the work; particular attention should therefore be given to have the under coats in a due state of dryness when the external coat is laid on. 11. Trowelled stucco—which is a very neat kind of work, and commonly used in dining rooms, balls, galleries, staircases, &c., where the walls are intended to be painted-must be worked on a floated ground, and the floating should be perfectly dry before the stucco is applied. In this process, the plasterer is provided with a wooden tool called a float, consisting of a piece of thin deal, about nine inches long and three inches wide, planed smooth, with its lower edges a little rounded off, and having a handle fixed on its upper surface. The stucco being prepared as already described, is afterwards well-beaten and tempered with clean water. The ground intended to be stuccoed is first prepared with a large trowel, and made as smooth and level as possible, and the stucco having been spread upon it, to the extent of four or five feet square, the workman, with the float in one hand and a brush in the other, begins to rub it smooth with the float, having first sprinkled it with water from the brush; this he does in small portions at a time, and proceeds alternately sprinkling and rubbing the face of the stucco till the whole is completed. The water has the effect of hardening the face of the stucco, and when well-floated, the stucco feels to the touch as smooth as glass. This work is executed both on walls and on laths. 12. Render and set, or rendering floated and set, combines the foregoing processes, only no lathing is required. Rendering is to be understood of a wall, whether of brick or stone, being covered with a coat of lime and hair. By set, is denoted a thin coat of fine stuff or putty laid upon the rendering. These operations are similar to those described for setting ceilings and partitions; and the floated and set is laid on the rendering in the same manner as before described on lathed work. CORNICES. 13. Cornices are either plain or ornamented. In order to execute a cornice according to a given design, it is necessary to prepare a mould of the several members, which mould is usually made of sheet copper or brass, indented so as to repre- sent exactly the forms and projections of the said members, and fixed into a wooden handle. If the projection of the cornice exceeds eight inches, it is requisite to fix bracketting to receive the same. This consists of pieces of wood fastened to the wall, on which the cornice is to be formed, about a foot apart, to which laths are affixed; the whole is then covered with a coat of rough plaster, allow- ance being made for the thickness of the stuff necessary to form the cornice, for which about one inch and a quarter is generally sufficient. To run the cornice two workmen are necessarily employed, who must be provided with a tub of set or putty, and a quantity of Plaster of Paris. Previously to using the mould, they gauge a straight line or screed on the wall and ceiling, formed of putty and plaster, and extending so far on each as to answer to the bottom and top of the cornice 136 (PART II. PRACTICAL ARCHITECTURE. to be formed. On the screed thus formed on the wall, one or two slight deal straight edges are nailed, and a notch or chase being likewise cut in the mould forms a guide for it to run upon. When all is so far ready, the putty is to be mixed with about one-third of Plaster of Paris, and rendered of a semi-fluid consistence by the addition of clean water. One of the workmen then takes two or three trowels full of the prepared putty on his hawke, which he holds with one hand, whilst, with the other, he spreads the stuff on the parts where the cornice is to be worked: the other work- man occasionally applying the mould to see where more or less of the material is required. When a sufficient quantity has been put on to fill up all the parts of the mould, the mould is worked back- wards and forwards, being at the same time held firmly to the ceiling and wall, by which means the superfluous material is removed, and the contour of the cornice completed of the form required, Sometimes it is necessary to repeat this operation several times, in order to fill up such parts as are deficient in the former applications. A piece of cornice is thus soon formed of from ten to twelve feet in length, according as may be required, as it is desirable that the whole of the cornice between any two breaks or projections, should be completed at the same time, in order that the entire length may be correct and true. When the stuff gets too stiff-which, while working large cornices is frequently the case, from the Plaster of Paris causing the putty to set very quickly-it must be sprinkled with water from a brush, as often as may be necessary to keep it moist. When the cornice is of large dimensions, three or four moulds are necessary, which are applied in succession, in the manner already described, until the whole is formed. The mitres, both internal and external, are afterwards modelled and filled up by hand; and, in this operation, the effect produced by a skilful workman is easily discernible. ist. ORNAMENTED CORNICES AND OTHER ENRICHMENTS. 14. Ornamented cornices are formed, in the first instance, in the manner already described, excepting that additional indents or sinkings are left in parts of the plain mouldings to receive the several enrichments, which are now cast in Plaster of Paris, by means of moulds prepared for the purpose. A considerable saving is thus effected, as formerly all species of enrichments were worked by hand, by artisans known in the trade by the name of ornamental plasterers. All the ornaments which are cast in Plaster of Paris, are previously modelled in clay, from designs made for the purpose. In forming the clay model, the taste and judgment of the artist is put to the test. When the model is finished, and become somewhat firm, it is oiled all over and put into a wooden frame adapted to it; the several parts are then retouched and perfected, to prepare them to receive a covering of melted wax, which is poured warm into the frame, and over the clay model, where it is left to cool and fix itself. When cold it is turned upside down, and the wax comes easily away from the clay, and forms a mould for future castings. In this mould the enrichments are cast, which is generally done by a common Plasterer. The wax moulds are usually made so as to cast about a foot in length of the ornament at a time, greater lengths being very difficult to get out of the mould. The casts are made with the finest and purest Plaster of Paris, saturated with water; and the wax mould is oiled previously to pouring in the plaster. The intaglios or casts, when first taken out of the mould, are not very firm, and are placed either in the air or in a hot oven to dry them; SECT. V.] 137 PLASTERING. when hard enough to bear handling, they are scraped and cleaned up, preparatory to being fixed in the places for which they are intended. Enriched friezes and bas reliefs are formed in a similar manner to the ornaments just described, excepting that the wax mould is so formed as to allow of a ground of plaster being left behind the ornament, of about half an inch in thickness; this is cast to the ornament or figures, and strengthens and defends their proportions, as well as promotes their general effect, when fixed in the situation for which they are intended. Capitals to columns are produced by a similar process, but they require several moulds to complete them. In forming the Corinthian capital, it is necessary first to prepare the bell or vase, which must be so shaped as to promote a graceful effect in the foliage and volutes, for each of which, as well as the other details, separate and distinct cameos will be required. These are all cast separately, and afterwards attached to the bell or vase, by means of liquid Plaster of Paris. EXTERNAL PLASTERING. 15. External plastering is either rough-cast, or stucco. The first is a description of covering much cheaper than stucco, and therefore for the most part employed in cottages, farm-houses, and such buildings. 16. Rough-cast is executed in the following manner. The building intended to be rough-cast is first covered, or as it is technically called pricked-up, with a coat of lime and hair. When this is sufficiently dry, a second coat is laid on of the same materials as the first, but as smooth as it can be spread: as fast as the workman finishes this surface, he is followed by another with a pail full of rough-cast, with which he bespatters the new plastering, and the whole is left to dry together. The rough-cast is composed of fine gravel, and clean sand washed from all earthy particles, and mixed with pure lime and water till the whole is of a semi-fluid consistency: this is thrown on the wall with round deal handle. Whilst with this tool the workman throws on the rough-cast with his right- hand, he holds in his left a common white-washing brush dipped in the rough-cast, and with this he brushes and colours the mortar, and the rough-cast he has already spread, to give them when finished a regular uniform colour and appearance. 17. STUCCO.- Stucco is a description of plastering executed so as, when finished, to resemble stone. It is composed of various kinds of materials, and distinguished by different names, as Roman cement, Bayley's composition, and Hamelein's mastic. See PRINCIPLES OF PRACTICAL ARCHITEC- TURE. Sect. II. § 67—87. 18. ROMAN CEMENT.-— Roman cement was first introduced to the public in the year 1796, when Mr. Parker obtained a patent for the invention. Mr. Parker in his specification says, “ Nodules of clay, or argillaceous stone, generally contain water in their centre, surrounded by calcareous crystals, and having veins of calcareous matter; they are formed in clay, and are of a brown colour like the clay.” These nodules are directed to be broken into small pieces and burned in a kiln like lime, with a heat sufficient to vitrify them, and then to be reduced to a powder. Two measures of water to five of the powder make tarras ; lime and other matters may or may not be added, and the pro- portion of water may be varied. The process of working this cement is nearly similar to that of other stucco, but no roughing-in coat is required, as with this cement the work is done by the finishing process only of the other kinds. 138 [PART II. PRACTICAL ARCHITECTURE. O The staining of this stucco to represent masonry, is done by diluting sulphuric acid (oil of vitriol) with water, and putting into the liquid ochres or other colours, to vary the tint according to the taste of the individual using it. 19. BAYLEY'S COMPOSITION.—A stucco now in general use for external work, is that description known to Plasterers by the name of Bayley's compo. It consists of Thames sand washed clean, and ground Dorking lime, mixed dry in the proportion of three of the latter to one of the former. These ingredients, when well incorporated together, should be secured from the air in good tight casks, till the moment it is wanted to be used. The process of preparing and using this stucco is as follows: first, the wall intended to be covered with it, is to be prepared by raking the mortar out of all the joints and indenting the surface of the bricks; all dust and other superfluous matter must be completely removed, and the surface sprinkled with clean water; the wall is then ready to receive the first coat, called by the workmen the rough- ing-in, which is effected by diluting to excess a quantity of the stucco in pails of water, till it is little stiffer than common whitewash, and then covering the whole surface of the wall therewith, by means of a flat hogs-hair brush, after which it must be left till it becomes tolerably dry and hard: this will be apparent from its getting whiter and more transparent than when it is first laid on. A quantity of the stucco having been tempered and saturated with water, when deemed of a pro- per consistence, screeds, or styles, of about eight inches in width, must be formed at the extremities of the wall, from top to bottom; this should be done with accuracy, which must be proved by means of the plumb-rule and straight edge, and subsequently the space between the screeds so formed, must be filled in with similar ones, at distances of about five feet apart over the entire surface of the wall, excepting where apertures are required to be left, when such an arrangement of the distance of the screeds must be made as may be found necessary. The next operation is to fill in the spaces between the screeds with the stucco, one after the other, applying the straight edge from time to time in order to remove the superfluous stucco, and leave the whole surface true and even. Should any of the spaces between the screeds be hollow and uneven after the first operation, more stucco must be added, and the application of the straight edge repeated as often as may be found necessary, till the entire surface is made perfectly true. The finishing of this species of stucco consists in floating, or hardening the surface by rubbing it with a wooden float, and sprinkling it with water meanwhile, till the stucco becomes perfectly smooth and hard. Cornices, mouldings, and fascias, are formed in a similar manner to that usually adopted in com- mon plastering, which has been already described; but it will be found necessary in running mould- ings with this stucco, to add a small quantity of Plaster of Paris, in order to produce a greater degree of fixation whilst running or working the mould. 20. HAMELEIN'S MASTIC CEMENT.—This cement consists of earth and other substances that are insoluble in water, or nearly so, either in their natural state, or such as have been manufactured, as earthenware, china, &c. To these materials, when pulverized, are added the different oxides of lead,---as litharge, grey oxide, and minium,-all reduced to a fine powder ; to which again is added a quantity of pulverized glass or flint stones. The whole being intimately incorporated, and made of a proper consistence with some vegetable oil, as that of linseed, will form a durable stucco or plaster impervious to wet. The following is the proportion of the several ingredients. To any given weight of pit or river sand, or pulverized earthenware, or porcelain, add two-thirds of the weight of Portland, Bath, or any other stone of the same nature, pulverized. Then to every five hundred and sixty pounds of this mixture, add forty pounds of litharge, two pounds of pulverized glass, or flint stones, one pound Sect. V.) 139 PLASTERING. of minium, and two pounds of grey oxide of lead. The whole should be thoroughly mixed together, and sifted through a sieve, the fineness of which will depend on the different purposes for which the cement is intended; it should then be put into casks, and it will keep for any length of time. The following is the method of using it. To every thirty pounds weight of the cement, add one quart of oil,—either linseed, walnut, or any other vegetable oil, -and mix it together as any other mortar, pressing it gently together, either by treading on it, or with a trowel: it will then have the appearance of moistened sand, and as it soon hardens, care must be taken that no more be mixed at -one time than is requisite for the present use. Previously to applying the cement to the wall, especially if of brick, it must be brushed with oil. If the cement is intended to be applied to wood, lead, or any thing of a similar nature, less oil may be used in the mixing of it. 21. KEENE'S MARBLE CEMENT.-A new patent cement is at present attracting considerable atten- tion under the name of Keene's Patent Marble cement. It is understood to be composed of gypsum and alum; and is remarkable for its power or susceptibility of polish. There are several qualities of it. The coarser qualities form a paving scarcely distinguishable from stone in colour and hard- ness, but of one-third the price; the best white quality produces excellent imitations of Florentine and other mosaics. EXPLANATION OF TERMS. 22. Besides the several terms already explained in the foregoing general description of plastering, many others occur connected therewith, the principal of which are arranged alphabetically, with explanations, in the following pages, together with a description of the several tools and implements used by the Plasterer in the execution of his work. Angle float, a float made to any internal angle. Arris, the line on which two surfaces of a wall meet each other, forming an external angle. The working of these intersections requires extra labour and attention, and is paid for accordingly by the lineal foot. Alabaster, see Gypsum. Asphalt um, a kind of bituminous stone, found near the ancient Babylon, and lately in the province of Neufchâtel. When asphaltum is mixed with some other materials, a cement is formed impervious to air or water. It is sup- posed to be the celebrated cement used in the walls of Babylon, and in the temple at Jerusalem. B. Bastard stucco, is three-coat plaster; the first generally roughing-in or rendering, the second floating as in trowelled stucco, but the finishing coat contains a little hair besides the sand. It is not hand floated, and the trowelling is done with less labour than what is denominated trowelled stucco. Bay, a strip or rib of plaster between screeds, for regulating the floating rule. Beater, an implement used by the labourers for tempering or incorporating the lime, sand, and hair together. Bracketting for plastering is variously named according to the form of the ceiling which it sustains, as groin-bracketting, dome-bracketting, cove-bracketting, spandrell-bracketting, spherical-bracketting, bracketting for the heads of niches, &c. In all these, the brackets are so disposed that their edges will be parallel to the surface of the plastering when finished. The distance between the edge of the brackets and the surface of the plaster, is three-fourths or seven-eighths of an inch. 140 [Part II. PRACTICAL ARCHITECTURE. Ceiling, the upper siđe of an apartment opposite to the floor, and usually finished with plaster work. Ceilings are set in two ways; the best where the setting coat is composed of plaster and putty. Common ceilings are finished the same as walls set for paper. Coarse stuff, see page 132. Coat, a stratum or thickness of plaster work done at one time. D. Derby, a two-handed float. Die, is when plaster loses its strength. Dots, patches of plaster put on to regulate the floating rule. Dishing-out, any kind of coved work formed by wooden ribs for plastering upon; the term is of the same import as cradling. Dubbing-out, is the making out with tiles, or any other material, a deficiency in thickness. F Finishing, is the best coat of three-coat work when done for stucco. The term setting is commonly used when the third coat is made of fine stuff for paper. Float, an implement for forming the second coat of three-coat work to a given form of surface. Floats are of three kinds; the hand-float, the quick.float, and the Derby. Floating screed, a wooden rule for running cornices. G. Gypsum, a substance formed by the combination of sulphuric acid with calcareous earth. It is the material from which Plaster of Paris is formed, being calcined in kilns. It is of various kinds, known by the name of alabaster and talc. H. Hawke, a square board with a handle projecting from the under side, for holding the stuff which the plasterer is using, with which he is supplied from time to time by a labourer or a boy. Hacking, roughing the surface of a wall previously to plastering it, for the purpose of causing the plaster to adhere more firmly. Hundred of lime, a denomination of measure, denoting in some places thirty-five, and in others twenty-five heaped bushels or bags. Joint rules, are narrow trowels and rules for making good niches, &c. L. Laying on trowels, see Trowels. Limewash. This is the most common and cheap colouring, and is generally used for such purposes as do not require any great picety; but a very superior whitewash may be made with it alone if proper care is taken in its prepara- tion, as it is less liable to peel off than that which is made with whiting and size. Take pieces of the best chalk lime, the clearest that can be selected, break them small, and pour clean water on them, stirring the mixture for a short time; let it then remain for a few seconds, and afterwards pour it into another vessel, leaving the heavier particles behind; add more water, stirring it as before, and let it again settle; then pour off the water from the top, strain the mixture through a very fine sieve, and keep it covered till wanted; then add a sufficient quantity of water to make it of a proper consistence, and a very superior whitewash will be produced applicable for the most delicate purposes. In using it, it is better to apply two thin coats than one thick one, as it will in the latter case be liable to look smeary. SECT. V.) 141 PLASTERING. M. Mitreing angles, making good internal and external angles of mouldings. P. Pail, a vessel for holding water to moisten the plaster. Pargetting, a term used for plastering walls, and sometimes for the plaster itself. Picking out tools, are made of steel and polished; they are of various sizes, commonly about seven or eight inches long, and about half-an-inch wide, flattened at both ends, and ground away till they are somewhat rounding. They are used to finish mitres, the returns of cornices, and to fill up the joinings of ornaments, &c. These tools are required to be kept perfectly clean and free from rust. Pugging, a coarse kind of mortar laid upon wood fillets, between joists, in order to prevent sound reaching from one apartment to another. R. Rake, an iron instrument with two or three prongs fixed to a wooden handle, used for incorporating the hair with the mortar. Scratcher, an instrument for scratching the plaster. Screed, see Floating Screed. Stopping, making good holes in the plastering. T. Tarras, a strong mortar or cement, of great use in lining cisterns and tanks, and in other descriptions of aquatic works. Tessera, a composition sometimes used for covering flat roofs. Truversing the screeds for cornices, is putting on the gauged stuff on the ceiling screeds for regulating the running mould of the cornice. Trowels are of various sorts and sizes: such as the laying and smoothing trowel, consisting of a fat plate of hardened iron, very thin, about ten inches in length, and two inches and a half in width, ground to a semi-circular shape at one end, the other being left square. On the back of the plate, and nearest to the square end, is rivetted a piece of small rod iron, to the other end of which a round wooden handle is adapted. With this tool the several coats of plastering are put on. The trowels used by the Plasterer are made with a superior degree of neatness to the tools known by the same name and used by other artificers. Those of the largest size are about seven inches long on the plate, which is formed of polished steel, two inches and three quarters wide at the heel, and converging to an apex or point; the handle is usually of mahogany with a wide brass ferrule. With this trowel the Plasterer gauges all his fine stuff, and plaster for the purpose of forming mouldings, &c. The other trowels are fitted up in & similar manner, varying gradually in their size, the smaller ones being only two or three inches in length. SECTION VI. PLUMBERY. Definition, 1.- Lead, 2.—-Lead-mines, 3. - Building lead, 4.- Smelting operations, 5. — Casting sheet lead, 6. -Casting Cisterns, 7.- Laying sheet lead, 8–15.— Solder, 16, 17.— Pipes, 18—21.- Pumps, 22. Tools, 23. — Table of Weights, 24.-_Sheet copper, 25.-- Dotting, 26.--Tin plates, 27. S 1. PLUMBERY, or plumbing, is the art of preparing and working lead, and applying it to various uses in buildings. It likewise embraces a certain portion of the science of hydraulics : for the construct- ing of pumps and other engines for the purpose of raising water, the forming of cisterns and re- servoirs, and the providing and arranging the requisite pipes, cocks, valves, and other apparatus connected therewith, is usually intrusted to the Plumber. 2. Lead, which from its durability, malleability, and other valuable properties, is of the greatest use in building, is of a blueish-white colour, and, when recently melted, very bright, but it soon becomes fàrnished if exposed to the air. Its specific gravity is 11.3523; it melts when heated to the temperature of 612° of Fahrenheit; when a very strong heat is applied, the metal boils and evaporates; if it be cooled slowly it crystallizes. It may be reduced by the hammer into very thin plates, and may likewise be drawn out into wire; but its tenacity, as compared with other metals, is not great. A leaden wire of one-twelfth of an inch diameter, is not capable of sustaining a greater weight than eighteen pounds avoirdupois without breaking; whilst a copper wire of similar diameter will sustain upwards of three hundred pounds; and one of iron, nearly six hundred pounds. 3. Lead, as obtained from the mines, is almost always combined with sulphur, and is therefore termed a sulphuret : it is found in most of the countries in Europe, more especially in France, Spain, and Germany. There are very extensive mines in Durham, Yorkshire, and Derbyshire, and likewise in some parts of Wales, Ireland, and Scotland. 4. Lead, employed for building purposes, is of three kinds : First, ingot, or pig lead, as it comes from the smelting-house. Secondly, cast lead, being, as the term imports, formed into sheets by fusion. Thirdly, milled lead, so called from its being formed into sheets by being passed between cylindrical rollers, most commonly, though not invariably, at works in the immediate vicinity of the mines where it is found. 5. The operations of roasting, or, as it is termed, smelting the ore to obtain the pure metal, consist, 1st, in picking up the mineral, and separating the unctuous, rich, or pure ore, from the stony matrix and other impurities ;—2dly, in pounding the picked ore under stampers ;-—3dly, in washing the pulverized ore to carry off the matrix by the water ;-4thly, in roasting the mineral in a reverbera- tory furnace, taking care to stir it to facilitate the evaporation of the sulphur. When the surface begins to assume the consistence of paste, it is covered with charcoal, the mixture is then shaken, and the fire increased, and the lead flows down on all sides to the bottom of the basin of the furnace, SECT. VI.] 143 PLUMBERY. whence it is drawn off into moulds prepared to receive it. These are called pigs, and are sold by the merchants under that name, varying in price according to the quality of the lead. * 6. Cast lead is formed into sheets by the plumber, either from pigs, or from old lead which he may have taken in exchange, † and usually in the following manner : A cast-iron pot, fixed in brickwork, is placed at one end of the shop, near to the casting table, into which pot the lead to be melted is thrown. The casting table is in its form a parallelogram, varying in size, according to circumstances, from four to six feet in width, and from twelve to twenty in length. It is raised from the ground so as to be about six or seven inches below the top of the pot which contains the metal, and stands on a strong wooden frame so as to be quite steady and firm. The top of the table is formed of deal boards made perfectly true and smooth, and has a thick wooden rim, projecting upwards four or five inches at the head and sides, called the shafts. At the end of the table nearest to the pot in which the lead is melted—and which is raised about an inch higher than the other end—is placed, what is termed, the head-pan, formed either of two planks of wood fastened together at right angles in their length, with two triangular pieces fitted in between them at their ends, or of strong sheet-iron wrought into a similar shape. This head-pan stands with its bottom-which is a sharp edge—on a bench at the head of the table, leaning with one side against it; on the opposite side is a handle to lift it up by, in order to pour out the liquid metal. It is in length equal to the width of the mould; and on the side next the mould are two iron hooks to hold it to the table, and to prevent it from slipping whilst the metal is pouring out. The head-pan is made of sufficient capacity to contain as much melted lead as is required to form one sheet. The whole surface of the mould is then covered with a stratum of fine sand of about two inches in thickness, slightly moistened and pressed firm, and rendered perfectly smooth by means of a tool called the strike, and by the use of a plane of thick polished brass adapted for the purpose. Previously to the casting being commenced, the strike--which consists of a board about five inches in width, and rather longer than the inside of the mould, so that its ends, which are notched about two inches deep, may ride upon the shafts—is prepared for use, by tacking two pieces of old hat on the notches, or by covering them with leather cases, so as to raise the under edge of the strike about one-eighth of an inch or more above the sand, according to the thickness that the sheet of lead is intended to be made. The edge of the strike is then smeared with tallow, and laid across the mould with its ends resting on the shafts. When the metal in the pot is adequately heated, the head-pan is filled therefrom by means of ladles, and the scum having been removed, the contents of the pan are poured into the mould, and forced to the further end of it by means of the strike; the superfluous lead being received by a trough fixed at the end of the mould for that purpose, whence it is conveyed into the pot again to be re-melted. The sheet is then rolled up and weighed, and the process repeated to produce other sheets. * The merchants' price for the best WB. refined pig lead, at the present time, (October 1843,) is £15 15s. per ton; in October 1830, it was £16. Milled lead is usually charged about 10s. to 15s. per ton more than pig lead of the same quality: formerly the differ- ence was much greater, but the expense of manufacturing has been reduced by the introduction of machinery. The expense to the plumber of converting pig lead into sheets, including waste, is about 2s. per cwt. † When plumbers take old lead from a building in exchange for new, it is customary with them to allow for it, from 4s. to 5s. per cwt. less than the price agreed upon for the new cast lead. A further allowance of four pounds per cwt. is likewise made on the gross weight of the old lead for dirt. The quality of lead being considerably deteriorated by long exposure to the vicissitudes of the weather, and other casualties, whilst in use on a building,—it is desirable that when old lead is recast, a considerable proportion of new pig lead should be used with it, as the sheets otherwise become hard and brittle. In contracts, it is usual to stipulate that at least one-half shall be new pig lead of the best quality. 144 [PART IL PRACTICAL ARCHITECTURE. By these means lead is cast of different thicknesses, varying from five to fourteen pounds to the square foot. That used for covering flats and gutters seldom exceeds eight pounds to the foot; the thicker kinds being applied to lining cisterns, reservoirs, and similar purposes. 7. When it is intended to cast a cistern, the size of the four sides is measured out, and the dimen- sions of the front having been taken, long slips of wood, on which the mouldings required to be in- troduced have been formed, are pressed upon the sand, and leave their impressions; various orna- ments also are stamped upon the internal area by means of lead moulds ; such of the sand as may have been disturbed during this process is then made smooth, and the casting is performed in the same manner as for plain sheets. When the casting is completed, the lead is bent into the shape required, so that the ends come at the back of the cistern, where they are soldered together. The bottom, which is usually made of thicker lead than the sides, is afterwards soldered to them, 8. In laying sheet lead care should be taken that the surface on which it is laid be made as even as possible, otherwise the lead soon becomes cracked and unsightly: this is usually done by means of boarding, but plastering is sometimes used. When boarding is employed it should be well-seasoned, and of sufficient thickness to prevent its warping and twisting upwards. Good yellow deal, an inch and a quarter in thickness, is the best for the purpose, but inch deal is often used. The sheets of lead are seldom more than six feet in width; consequently, when a large surface is to be covered, it becomes necessary to have joints, and these the plumber must so construct as to prevent their leaking: the best way to effect this object is by forming what are called rolls. 9. A roll consists of a piece of wood of about two inches square, but rounded on its upper side, which is fastened under the joint of the lead, between the edges of the two sheets, which meet to- gether, so as to overlap each other two or three inches: one of these laps is dressed over the roll on the inside, and the other over both roll and lead on the outside, by which means the water is pre- vented from penetrating the joint. No other fastening is requisite than the adherence of the lead to the roll, the edges being closely hammered together, and dressed down closely to the boarding; indeed, all fastening to sheet lead exposed to the action of heat and cold ought to be avoided, as it expands and contracts considerably by the vicissitudes of temperature, and if it is fastened so as to prevent such changes from affecting it uniformly, it will soon become cracked and dilapidated. 10. When rolls are not used—which sometimes happens from their projection being inconvenient -seams are employed, which are formed by simply turning up the two edges of the lead which approach each other, one being lapped over the other, and then dressing them down close to the boarding of the flat throughout their whole length; but this plan is by no means equal to the roll either for neatness or durability. 11. Soldering the joints is sometimes had recourse to for such kind of work, but it is a very bad practice, as lead so fastened will very easily crack. 12. Lead flats should always be laid with a sufficient current to keep them dry, which is done by giving a fall, from back to front, of about one quarter of an inch to each foot of width of the flat; so that, if the sheets are twenty feet in length, they will be so laid as to be five inches higher at the one end than the other: this is termed giving a current, and is generally determined by the carpen- ter and plumber, previously to the commencement of the laying of the lead, and whilst the former is laying the boarding to receive it. 13. Flashings are pieces of lead-most commonly milled lead—about eight or nine iuches wide, varying somewhat according to circumstances, fixed all round the extreme edges of a flat or gutter after the lead has been laid, the edges of which should always be turned up a few inches. Where flashings are fixed to brickwork, one edge thereof is inserted into the joint between the courses of Sect. VI.) 143 PLUMBERY bricks, and then dressed over the edge of the lead of the flat or gutter that has been turned up. When fixed to stone, a channel or groove is cut in the stone, into which the edge of the flashing is inserted and burned in; it is then dressed over the other lead as already described. 14. Drips, in flats and gutters, are formed by raising one part above the other; the lead of the lower part being turned up against the edge of the part so raised, and the edge of the lead of the higher part being dressed down over it. Recourse is had to drips when the length of the gutter or flat exceeds that of the sheet of lead, in order to avoid the necessity of soldering the joint, which should always be avoided. 15. Milled lead is mostly used for covering the hips and ridges of roofs, and for making some description of pipes; but it is not adapted for gutters, or any part of the building much exposed to wear and tear, or to the action of the sun. It is usually very thin, seldom exceeding one-sixteenth of an inch in thickness, or four pounds to the square foot. Milled lead is preferable to cast lead, because it can be made thinner and more uniform in thickness, and is not subject to the small air- holes or bubbles that cannot be avoided in casting sheet lead. 16. Solder is used by the plumber for the purpose of securing the joints of lead work, in cases where the lap or roll-joint cannot be applied; for joining pipes, fastening cocks, and a variety of other purposes. It is a general rule with respect to solder, that it should be easier of fusion than the metal intended to be soldered by it; it should likewise be as nearly as possible of the same colour. Solder is composed of lead and tin fused together, after which it is run into moulds. 17. In the operation of soldering, the surfaces of the metal intended to be joined are scraped and rendered perfectly clean ; they are then brought close up to each other, and so held, whilst a little resin or borax is applied in order to defend the metal from oxidation whilst soldering. The heated solder is then brought in a ladle and poured on the joint to be soldered, and then smoothed and finished by being rubbed with a grozing-iron. When complete, it is filed or scraped off, and made even with the joint and contiguous surface of the lead. 18. The pipes used by plumbers are of various descriptions and sizes, and are usually designated by their respective diameters, as half inch, three-quarter inch, inch, and one and a half inch, up to which size they are charged by the foot lineal; above that size they are charged by weight. 19. Pipes of this description were formerly cast on a core of their respective diameters; but, for some years past, they have been made by a machine worked by steam, which furnishes a superior and cheaper article, of almost any length. 20. The rain water pipes attached to the outsides of buildings, for the purpose of conveying the rain water from the roofs, are called socket pipes. These are commonly made of milled lead, and are of various diameters, from three to five inches, and formed in lengths of from eight to ten feet each. The sheet lead for making them is dressed on a rounded core of wood, and the vertical joint at the back is fastened and secured by solder. The horizontal joints are formed by an astragal moulding in a separate piece of lead, about two or three inches wide, which laps completely over it, both above and below, and is called the lap joint or collar of the socket pipe. Two broad pieces of lead are attached to the backs of the lap joints, called the tacks ; these are spread out right and left of the pipe upon the wall, to which they are hammered quite close, and afford the means by which the pipe is fixed to the building with iron wall-hooks. 21. The cistern head, which is fixed at the top of a rain water pipe, is sometimes made of shcet lead, but is more commonly cast in a mould. It is fastened to the wall by tacks, in a similar man- ner to the water-pipes, and should always be covered by a grating, formed either of lead or iron, to prevent the pipe being stopped up. 22. The plumber's employment in pump-making is generally confined to two or three kinds re. quired for domestic purposes, of which the suction and lifting pumps are the principal. These, as 146 [Part II. PRACTICAL ARCHITECTURE. . well as the apparatus for water closets, are most commonly manufactured by a particular set of work- men, and sold to the plumber, who furnishes the requisite lead pipes, and fixes them in their places. 23. The plumber has not occasion for a great variety of tools, on account of the ductility of the metal in which he works: such as are required are usually supplied by the master-tradesman. They consist of an iron hammer, made rather heavy, having a short but thick handle, two or three different sized wooden mallets, and a dressing or fatting tool. The latter instrument is made of beech, commonly about eighteen inches long, and two inches and a half square, planed quite smooth on one side, and rounded into an arch on the other; one of its ends is tapered and rounded, to make it convenient to be held in the hand of the workman. With this tool the plumber stretches out and flattens all the sheet lead, as well as dresses it into the shape it may be wanted, in the various pur- poses to which such lead is applied; using first the flat side of this tool, and then the round side, as may be required. The plumber likewise uses a jack and trying plane, similar to those tools used by carpenters, for the purpose of planing the straight edges of the sheet lead, when it is required to pre- sent a very regular and correct line, which is sometimes necessary. A chalk line is necessary also in order to line out the lead to the different widths it may be required. The cutting tools consist of chissels and gouges of different sizes, and knives ; these latter are used for the purpose of accurately cutting the sheet lead into strips and pieces, of such different sizes and figures, as may have been drawn on the lead by the chalk line. Files and rasps of different sizes are used in manufacturing cisterns, pipes, pumps, &c.; and a stock and centre bits are occasionally required for making perforations through lead or wood, for the purpose of inserting pipes, &c. For soldering, a variety of different sized grozing-irons are used: they are commonly about twelve inches long, and tapered at both ends, the handle end being turned quite round, to admit of its being held firmly in the hand whilst in use. The opposite end is sometimes made spherical, and some- times spindle-shaped; the size being regulated by the soldering to be done with them. These irons are heated to redness when used. Iron ladles are used for the purpose of melting the solder, and are of various sizes. A measuring rule is likewise indispensable to the plumber, which is usually made in a peculiar manner; being, when open, two feet in length, but folding into three parts of eight inches in length each. Two of the legs are of box wood, and divided duodecimally; the third piece is of steel; this is attached to one of the box legs by a pivot on which it turns, and the same leg being grooved out on its side, it receives the steel leg when not in use in the groove. Scales and weights are of course necessary implements to the plumber, as most of the articles he supplies are charged by weight to the employer. 24. The following table will be found useful in determining the weight of lead, when the thickness of the sheet is ascertained. A cubic foot of cast lead weighs 7091 lbs. S Thickness. Inch. Thickness. Inch. aftale cole lopen unter Pounds to the square foot 59.125 44.343 36.955 29.562 22.173 14.782 11.825 11.086 9.854 8.446 7.391 6.569 Pounds to the square foot. 5.912 5.375 4.927 4.548 4.223 3.942 3,695 3.478 3.284 3.112 2.956 conta del op het osten waren egin Sect. VI.] 147 PLUMBERY. 25. Sheet copper, being a more tough and costly metal, is rolled much thinner, and is designated by the number of ounces in a square foot. Six ounce copper is very thin; but from eight to ten ounces forms a good covering. 26. Sheet metals, particularly in forming gutters, are frequently nailed to the small boarding that is placed under them for their support, and, in order to prevent the nail hole suffering water to pass, the nail and hole are covered by a small patch of solder ; this is called dotting. The use of nails, in this manner, requires some judgment and foresight, as to the effect of expansion and contraction, because these very frequently draw the nailş. 27. Tin plates soldered together are very extensively used in the United States for gutters, rain- pipes, valleys of roofs, and even for covering buildings; but their cheapness is their only recommen- dation; they are soon rendered useless by oxidation, unless protected by frequent painting. SECTION VII. HOUSE PAINTING. Definition, 1.--Nature and object of painting, 2. —Different kinds of House painting, 3. White lead. 4. Red lead, 5.— Litharge, 6. —Linseed oil, 7.- Drying oils, 8.— Turpentine, 9.--List of colours, 10.- Painter's tools, 11-16.- General directions in preparing colours, 17, 18.- General directions in laying on paint, 19, 20.-_ Graining, 21-26. Ornamental painting, 27-29.- Inscription painting, 30. 1. Painting, as applied to buildings, is the art of covering the several kinds of wood, plaster, ironwork, and other materials employed therein, with mineral colours, rendered fluid by being satu- rated with oils, oil of turpentine, or sometimes water; for the purpose both of decoration, and in order to preserve the different materials from the action of air and damp. 2. A pigment, properly prepared, is spread over the several objects with brushes; and by the re- petition of different coats successively, as the previous one becomes dry, a coating or glazing of paint is formed, which operates as a protection to the material itself, renders it easier of being cleaned, and at the same time gives a variety and neatness to the general appearance of the building. 3. This species of artificers' work may be divided into four branches : viz., Common Painting, Graining, Ornamental Painting, and Inscription Painting, 4. The whole of the prismatic colours are occasionally called into use by the painter, and are varied into an almost endless number of tints ; but the ground-work of all house painting is formed by a paint prepared from lead, and known in the arts by the name of ceruse, or white lead. Ceruse is a carbonate of lead, and is prepared in the following manner :- Thin plates of lead, rolled up spirally so as to leave a space of about an inch between each coil, are placed vertically in earthen pots, at the bottom of which is a quantity of strong vinegar: the pots are then covered, and exposed to a gentle heat, in a sand bath, or bedded in hot horse dung. The vapour of the vinegar, assisted by the tendency of the lead to combine with the oxygen that is present, corrodes the lead, and con- verts the external portion of it into a white substance, which comes off in flakes when the lead is uncoiled. The plates are thus repeatedly treated till they are completely reduced to an oxide. The ceruse is afterwards bleached, ground, and saturated with linseed oil. It is then put into tubs resembling butter-firkins, each containing about three hundred pounds weight; and in such tubs it is dispensed at the colour-shops. It is frequently adulterated either with whiting or Paris white ; a painter, therefore, who is desirous that his work should retain its colour and useful properties, is careful to procure his lead from quarters where he can most depend on having it pure ; and as lead improves by age, a tradesman with an extensive business usually makes a practice of keeping by him a considerable stock of this article, so as not to be under the necessity of using it till it has been in his possession two or three years. In a paper lately read before the Academy of Sciences in Paris, it is stated that the flowers of antimony yield a finer white, mix more freely with other colours, are more durable, and cost only one-third of the price of white lead; while, neither in the process of manufacture, nor in use, does this substance affect the health of the workmen, Sect. VII.] 149 HOUSE PAINTING. 5. The red oxide of lead is likewise extensively used in painting, and is called red lead. With a small admixture of white lead, it answers very well for the first coats of iron, or other external work. 6. Litharge is employed by painters to facilitate the drying of their colours, and is composed of the ashes of lead, or a kind of dusky powder that first appears in the process of oxidation. In this state it is denominated by chemists a sub-carbonate of lead; by painters it is technically known as dryers. It is afterwards saturated with linseed oil to render it more drying. 7. Linseed oil is obtained by pressure from the seed of flax : it is afterwards filtered to clear it nearer the oil becomes colourless, the better is the quality; and this is greatly promoted by keeping. Linseed oil will, when it has been kept a year or two, deposit all its colouring particles, and become nearly as transparent as water. 8. Drying oils are made by mixing red lead, litharge, and umber, with linseed oil, in the propor- tion of half-an-ounce of each to one pound of oil. The composition should be boiled over a slow fire, care being taken to skim it from time to time. The matter skimmed-off, together with the dregs, is a good dryer for dark colours : it is of a lead colour, and is often used in outside-work. As soon as this scum begins to rarify and become red, the fire is lowered, and the oil being left at rest, settles and clarifies. The method of ascertaining whether the mixture is sufficiently heated is to put a quill into it, which will in that case be immediately consumed; and this is moreover a sure test of the goodness of its quality. Linseed oil so prepared is vended at the shops under the name of boiled oil, and the best description of painting is executed therewith.* 9. Oil of turpentine-or, as it is termed by the painters, turpsmis likewise in general use for house painting, both for hardening the colour, and for what is technically termed flatting. All the larch and fir trees furnish a resin known by the general name of turpentine ; of which the different kinds are distinguished in commerce according to their several degrees of goodness. The larch tree furnishes what is called Venice turpentine, which is obtained by being made to flow from the trunk of the tree through holes, made with an augur, in which small pipes are fixed that conduct the juice into buckets placed to receive it. This turpentine has a yellowish and limpid colour, a strong aromatic smell, and bitter taste. It is afterwards subjected to the process of distillation, by which it liberates an oil more or less volatile according to the degree of heat employed: this white, limpid, odoriferous oil is called Essence of turpentine. The residuum from this distillation forms the boiled turpentine of commerce, which is sold in shops by the gallon, in the same way as oil. Turpentine, as well as oil, improves greatly by age ; hence painters in a large way of business always keep a considerable stock, which enables them to depend on the work retaining its colour,-a quality of the utmost importance in our usual practice of house painting. 10. The several colours are prepared for use by being ground on slabs of porphyry, marble, or other hard and smooth substances, till the particles are reduced to an extremely fine state, being saturated with oil or water according as the colour ground is to be used with the one or the other. Having mixed together his white lead and turps, the workman adds dryers, and one or more colours, according to the tint he wishes to produce. In painting in distemper, as it is called, whiting is used for white lead—the colours are ground in water. The colours, or stainers, used by house painters may be thus classed : * Attempts have been made to render fish-oil applicable for the purposes of painting, but without much success, as it not only always retains a disagreeable smell, but likewise dries clammy. 150 [PART II. PRACTICAL ARCHITECTURE. Green 1. Vermilion 2. Native cinnabar 3. Red lead 4. Scarlet ochre 5. Common Indian red 6. Spanish brown 17. Terra di Sienna, burnt Red (31. Verdigris 32. Crystals of verdigris 33. Prussian green 34. Terra verte 35. Sap green Colour, red tending to orange Orange 36. Orange lake Red Colour, crim- son tending to purple Purple ( 37. True Indian red 38. Archil 39. Logwood wash [ 8. Carmine 19. Lake < 10. Rose pink 1-11. Red ochre (12. Venetian red 13. Ultramarine 14. Ditto ashes | 15. Prussian blue 16. Verditer 17. Indigo (18. Smalt Blue Brown 40. Brown pink | 41. Bistre 42. Brown ochre 43. Umber 44. Cologne earth (45. Asphaltum White 19. Kings yellow 20. Naples yellow 21. Yellow ocbre 22. Dutch pink 23. English pink 24. Light pink 25. Gamboge 26. Masticot 27. Common orpiment 28. Gallston 29. Chromate of iron and lead (30. Terra di Sienna r 46. Flake white 47. White lead 48. Troy white ( 49. Flowers of bismuth Yellow Black ( 50. Lamp black < 51. Ivory black 52. Blue black The foregoing list embraces nearly the whole of the colours used by the house painter; and these heing mixed in due proportions, will produce any tint that may be required. 1. Vermilion is a bright scarlet pigment, formed of common sulphur and quicksilver by a chemical process. The best comes from China. This colour is very expensive, and many imitations of it are made, particularly by the Dutch 2. Cinnabar is a similar pigment, differing from vermilion only by a more crimson colouring. 3 Red lead, or minium, is lead calcined till it acquires a proper degree of colour, by being exposed with a large surface to a fire with a current of air passed over it. 4. Scarlet ochre is an earth having a base of green vitriol, and is separated from the acid of the vitriol by calcination 5. Common Indian red is of a hue verging towards scarlet. It is imported from the East Indies. 6. Spanish brown is a native earth, found in the state in which it is used. 7. Terra di Sienna is a native ochre, and is brought from Italy in the state in which it is found. It is originally yellow, and is often used in that state. It changes by calcination to an orange red, though not of a very bright tint; and is then frequently used for graining mahogany. 8. Carmine is a bright crimson colour, and is formed of the tinging substance of cochineal with uitric acid. It is not well-calculated to mix up with oil, as its colour changes rapidly by exposure to the light and air. 9. Lake is a white earthy body, formed from cuttle-fish bone, or the basis of alum, or chalk tinged with some vegetable dye, such as is obtained from cochineal or Brazil wood, taken up by an alkali, and precipitated on the earth by the addition of an acid. 10. Rose pink is a lake like the former, excepting that the earth or basis of the pigment is principally chalk, and the tinging substance is extracted from Brazil or Campeachy wood. Sect. VII.) 151 HOUSE PAINTING. 11. Red ochre is a native earth; but that which is in common use is coloured red by calcination, being yellow when dug out of the earth. The yellow ochre is brought chiefly from Oxfordshire, where it is found in great abun. dance. 12. Venetian red is a native ochre, rather inclining to scarlet. 13. Ultramarine is, when perfect, of a brilliant blue colour, of an extremely beautiful and transparent effect in oil, retaining this property with whatever vehicle or pigment it may be mixed. As formerly prepared from lapis lazuli, it was excessively dear when good, and was frequently sold in an adulterated state; it is now, by an ingenious chemical process, manufactured at a comparatively trifling expense, and its use is consequently becoming much more general. 14. Ultramarine ashes are the residuum or remains of the finer ultramarine. 15. Prussian blue, a brilliant pigment, is the fixed sulphur of animal or vegetable coal, chemically combined with the earth of alum. 16. Verditer is the mixture of chalk with precipitated copper which is formed by adding a due proportion of chalk to the solution of copper made by refiners, in precipitating silver from nitric acid, in the operation called parting. 17. Indigo is a tinging matter extracted from certain plants. The indigoes of commerce are chiefly imported from the East Indies. 18. Smalt is glass, coloured with zaffre, and afterwards ground to a powder. 19 Kings yellow is a pure orpiment, or arsenic, coloured with sulphur. 20. Naples yellow is a native ochre, of a colour rather inclining to orange. It stands remarkably well. 21. Yellow ochre is a mineral earth, which is found in many places, but in very different degrees of purity. See No. 11. 22. Dutch pink is a pigment formed of chalk coloured with the tinging particles of French berries. This colour is apt to fly. 23. and 24. English, and Light pink are merely lighter and coarser kinds of Dutch pink. 25. Gamboge is a gum brought from the East Indies. It is dissolved in water to a milky consistence, and is then of a bright yellow colour. 26. Masticot, as a pigment, is fake white, or white lead gently calcined, by which it is changed to a yellow, wbich varies in tint according to the degree of the calcination. 27. Orpiment is a fossil body of a yellow colour, composed of arsenic and sulphur, with a mixture of lead, and some- times other metals. 28. Gallstone is a concretion of earthy matter formed in the gall bladders of beasts. 29. The Chromates of lead and of iron are excellent yellow colours, and much in use. 30. Terra di Siennn has been already noticed under No. 7. 31. Verdigris is an oxide of copper formed by a vegetable acid. It is much used as a green paint. 32. Crystals of verdiyris is the salt produced by the solution of copper, or common verdigris, in vinegar. 33. Prussian green is in composition similar to the blue of that name. See No. 15. 34. Terra verte is a native earth. It is of a blueish green colour resembling the tint called sea-green. It is a colour that stands well. 35. Sap green is the concreted juice of the buckthorn berry. 36. Orange lake is the tinging part of annatto, precipitated together with the earth of alum. 37. True Indian red is a native ochrous earth of a purple colour. It is imported from the East Indies, and is an excellent and extremely useful colour. 38. Archil, or Orchil, is a purple tincture made from the Lichen Roccella. 39. Logwood is a well-known wood brought from South America, and affords a strong purple tincture. 40. Brown pink is the tinging part of some vegetable of an orange colour, precipitated on the earth of alum. 41. Bistre is a brown transparent colour, made from soot boiled and diluted. 42. Brown ochre is an earth of a warm brown or foul orange colour. 43. Umber is an ore of iron of a blackish brown colour, so called from Ombria in Italy. 44. Cologne earth is a fossil substance of a dark, blackish brown colour, a little inclining towards purple. 45. Asphaltum is sometimes used by painters in lieu of brown pink. 46. White flake is a ceruse prepared with the acid of grape. 47. The process of preparing White lead has already been described in this section, Art. 4. 48. Troy white is chalk neutralized by the addition of water in which alum has been dissolved. 152 [Part II. PRACTICAL ARCHITECTURE. 49. Flowers of Bismuth is an oxide of that metal, little used. 50. Lamp black is the soot from oil collected as it is formed by burning. 51. Ivory black is the charcoal of ivory, or bone, formed by subjecting it to a great heat and excluding all access of air. 52. Blue black is the coal of any kind of wood-maple is the best-burnt in a close fire, where the air can bave no access. i 11. The tools used by painters are few in number, and are usually supplied by the masters to the journeymen. They consist chiefly of brushes made of hogs' bristles, and sash pencils, or sash tools, of different sizes, and finer hair. 12. The tool termed the pound brush is composed of hogs' bristles. It is used as a duster until the ends of the hair are worn away and the brush becomes soft. It is then used in the colour, being better adapted to spread it evenly after such wear. 13. Ground brushes have of late been used, which are prepared for the purpose, and are much more convenient. The brushes vary in size, as may be necessary for the purpose of painting the several mouldings, or bars of sashes, and generally for picking out lines intended to be left of a different tint from the general body of the work. 14. The pencils are made of badger's hair, or any fine hairs, fixed in quills of various sizes. 15. The pallet is made of either apple or pear tree, very thin, but somewhat thicker in the centre than at the extremities. It is either of a square or oval shape, and has a hole near the edge large enough to admit the thumb by which it is held. When it is new, it is well saturated with oil of walnuts, and afterwards polished. 16. The pallet-knife is a thin flexible plate of steel, rounded at the end, the other extremity being fixed into a wooden handle. 17. It is of great importance in painting that the various vessels and pots should be kept clean. It is desirable also that those containing the colours should be varnished, as this prevents their dry- ing too quickly. Due attention must likewise be given to the grinding and diluting of the colours, that they may be neither too thick nor too thin. In grinding colours, no more liquid should be used than is necessary to make the substance yield easily to the mullet; and no more colour should be mixed than is necessary for the work to be performed ; for colours newly mixed are more vivid and brilliant. When colours are ground with oil of turpentine, and diluted with varnish, they should be applied immediately, as they dry more speedily than those prepared in oil. The colours thus prepared possess great brilliancy, but they require more skill to manage them. 18. House painting is, for the most part, executed in oil colour ; or--when no gloss is required- the colour is mixed with turpentine, in which case it is denominated flatting. In mixing up the colours for oil painting, white lead forms the base ; which is changed and modified by the addition of coloured substances till it is brought to the tint required. Those colours that are prepared from vegetable bodies produce, when first used, a much more brilliant effect than those made from inineral substances; but they will not stand the combined effects of air and light in the same degree; under exposure they soon fado and turu black, whilst the mineral colours will remain for a long time unchanged. 19. Good painting is known by the solidity and fulness of its appearance, without slowing any marks of the brush. In no branch of artificers' work is there a greater variety in the quality, or greater deception practised by unprincipled members of the trade than in this. The several materials admitting of easy adulteration are used of various degrees of purity, which circumstance will alone account for the very low rate at which painters' work is sometimes undertaken ; and though, to the eye of an inexperienced person, work performed with inferior colours may at first present but HOUSE PAINTING. 153 . little difference from the best work, yet such paint is neither useful in preserving the work to which it is applied, nor does it even retain its colour for any length of time. 20. If the work has ever been previously painted, it should in the first place be entirely divested of all grease : this may be effected by scouring it with pearlash and sand, which, though rather an expensive process, will be amply repaid by the appearance and increased durability of the paint. The pearlash being thoroughly cleaned off, the work is then to be primed and stopped with putty, which consists in filling up any nail holes or other defects in the surface of the work with putty, the finer work being stopped with white lead putty; any inequalities should be rubbed down with pumice-stone, and the several mouldings thoroughly cleaned out. If any portion of the work is new, it must be what is termed brought forward, which consists in giving it such a number of coats as to make it correspond with the old work. If the work has not been painted before, the first operation is what is termed knotting, which consists in covering over all the knots in the wood-work with a preparation of white lead, red lead, and a little copperas mixed with oil, Care should be taken to perform this operation effectually, otherwise the knots will soon appear through the paint, and produce a most unsightly effect. After the knotting is completed, a thin coat of oil colour, called the priming, is to be laid over the whole. The coats of paint are then to be repeated as often as may be considered necessary, which in new work is usually four times for wood-work. Stucco requires an additional coat. In old work, the number of coats must of course depend on the state it is in at the time. Care should be taken that each coat is laid on of a proper consistence : if too thick, it will crack. Indeed the first coat should always be laid on thin, in order that the oil may sink into the wood or stucco. In common work it is not unusual to recommend what is called clearcole and finish, which consists in giving the work a coat of white lead and size, to form the ground, instead of a coat of oil paint. This is a method, how- ever, which has nothing but its cheapness to recommend it: it may, perhaps, save the expense of a coat of oil at the time, but there is no real economy in the end, as the work is never so durable or useful as a preservative. When work is flatted, great care should be taken to keep the colour in a wet state whilst in progress, otherwise it will dry streaky. In painting in distemper, it is a good practice to use oil colours first, then flatten, and finish in distemper. 21. Graining is understood amongst painters to be the imitating of the several different species of scarce woods, such as mahogany, wainscot, rose-wood, sattin-wood, maple-wood, &c., and likewise of the different kinds of marble. Of late, from the increased demand for this species of decoration, great attention has been given to the improvement of this branch of the art; and some artists are so expert in executing these kinds of imitations, that it is not easy to distinguish their performances from the real substances, except by the touch. We need scarcely add, that success in this depart- ment of house painting is peculiarly dependent upon the individual artist's taste. Mr. Charles Moxson's work, entitled “The Grainer's Guide,' will be consulted on this branch of decorative painting with the greatest advantage. 22. The graining of wainscot is performed in the following manner: A ground is first laid of a good warm colour made of ochre, raw Terra di Sienna, and umber. The painter then prepares his pallet-board with small quantities of fine Oxford ochre, raw Terra di Sienna, burnt Terra di Sienna, and umber ; having some boiled oil and oil of turpentine mixed together, wherewith to saturate the colours, which are made very thin, in order that when applied they may produce a transparent effect. He is likewise provided with several kinds of camel hair pencils, and with two or three flat hogs' hair brushes. When the colour is properly mixed, he applies it over a pannel, or any other small piece of the work, to judge of the effect of the tint; he then proceeds first with the pannels, and afterwards with the rails and styles of the framing, doors, or shutters, as the case may be. The 154 [Part II. PRACTICAL ARCHITECTURE. flat hogs' hair brush being dipped in the mixture of oil and turpentine, and then drawn down the newly laid colour, occasions the shades and grainings in it, which arise from the brush supplying an excess of saturation to the colour it touches. The mottled appearance is produced by a camel hair pencil dipped in turpentine and then dusted off. When the whole is finished, it is left to dry; after which it is varnished with a coat or two of fine light copal varnish. 23. The graining of sattin-wood is performed in a similar manner, only the ground and graining colour is prepared of different colours. The ground for sattin-wood is usually composed of ceruse and Naples yellow, diluted with oil of turpentine ; the graining colour of raw Terra di Sienna, burnt Terra di Sienna, and a little Vandyke brown. 24. For mahogany, the ground is made with raw Terra di Sienna, burnt Terra di Sienna, burnt ochre, and yellow ochre, or with red lead and Oxford ochre: the graining colour with burnt Terra di Sienna, raw Terra di Sienna, and rose pink, mixed with beer, which adds to the transparency of the colour. Mahogany should be varnished at least twice with good copal varnish. 25. For marbling, the white ground should be prepared with white lead mixed with linseed oil and turpentine, in the proportion of two parts of the former to one of the latter. The veining is done with a little blue black, and a small quantity of Prussian blue. The other kinds of marble are pre- pared in a similar manner, with such colours as are best suited to represent the particular species of marble required. The white and dove marbles should not be varnished, as it has a tendency to increase the effect of the change of the colours. Sienna marble, black, or porphyry, should be var- nished with copal varnish of the best quality. All marbles are better imitated on plaster, stone, or slate, than on wood. 26. Ornamental painting embraces the executing of the various species of architectural decoration, such as friezes, pannels, foliage, &c., in chiaroscura, or light and shade, on walls and ceilings. In performing this description of work, it is necessary, in the first place, that a ground be prepared of the form and dimensions the proposed decoration is to occupy; and this is to be painted of the tint it is intended that it should remain. The ornament or enrichment to be executed thereon should be marked out with a black-lead pencil, and then painted and shaded to give it its due effect. Such decorations are sometimes painted on slips of paper or Irish linen, and pasted up afterwards ; but this is a plan by no means to be recommended. 27. Some artists, to facilitate their work, and when the ornament is required in large quantities, do it by the method termed stencilling, which consists in drawing out a certain length thereof very accurately on thick paper, then pricking holes with a large-sized needle at regular distances all round the pattern, which is afterward laid flat against the wall to be ornamented. A small linen bag containing powdered chalk being then gently struck against the outer surface of the paper so perforated, the powder enters the apertures, and fixes itself against the wall, exhibiting the exact outline of the ornament, which the painter immediately fixes by painting it on the wall. By this means a considerable saving of time is effected. 28. Sometimes painting of this description is enriched by gold, which is applied after the ornament has been painted in, as it is termed, by the process known by the name of gilding in oil. 29. Within these few years, a very remarkable improvement in decorative house painting has begun to appear; still, we rarely see, in the painting of dwelling houses, any combination of colours tliat does not in some respects offend against the laws of colour, though these are clearly pointed out and established by the science of optics. The unpleasant effects thus frequently produced may account for the prevalent use of neutral colours as they are called-in ornamental painting : for these are less liable to offend the eye by their unskilful juxtaposition. Young men intending SECT. VII] 155 HOUSE PAINTING. to prosecute this branch of art will receive incalculable benefit from the study of Mr. Hay s different treatises, particularly that on the laws of harmonious colouring. 30. Inscription writing is executed by persons known in the trade as letter-writers. The process consists in sketching out, with pencil, the words or figures required to be written, and afterwards applying colour over the same with camels' hair pencils, adding shadows thereto according as it may the process is similar as to the sketching out; but when the letters are painted, they are covered with leaf gold, which is left to fix itself by the drying of the paint on which it has been laid, after which a sponge and water is used to clear away the superfluous gold. The whole of the inscription part Third. ON THE GRECIAN ORDERS OF ARCHITECTURE, WITH GLOSSARY OF TERMS. *A/XXXTᏄᏓᎨᎢd EGYPTIAN CIPIT. ILS. 1ᏍᏗᏗᏎᏗe&a«. THE ORDERS OF ARCHITECTURE PLATE LXXXV GREEK AND ROMAN EXAMPLES. OWO ----* tapital...-----Entallas Shaft. ------ CORINTHIAN Jord ZESZESESZS napusten Shaft DORIC BONIC from the Parthenon a thens. From the the Theatre of the imple or the lor Siod. Marcellus. Minery: Polias. Fortuna Virilis wat Home at th . koma Note. As die dimensions of the SB3ral Ewoples of the orders given in this plante y considerat ly the Seales w wcho They are drawne vaid lause d that the determine the portions De Searal para may be more obvious the Fotoooo this Pantheon * Fone A Fullarton&C London & Edinburgh PART III. ON THE ORDERS OF ARCHITECTURE. SECTION I. DEFINITIONS. [Plates LXXXIV.-LXXXVI.] Capital, 5.-- Shaft, What constitutes an Order, 1.--Egyptian and Greek Orders, 2.--Columns, 3.-Base, 4.- 6. Entablature, 7.- General remarks on the Orders, 8-11. 1. The moderns have applied the term order to those architectural forms with which the Greeks composed the facades of their temples. The principal members of an order are, Ist, A platform ; 2d, Perpendicular supports; and, 3d, A lintelling or covering connecting the tops of these supports, and crowning the edifice. The proportioning these essential parts to the edifice and to each other, and adapting characteristical decorations, constitutes an order, canon, or rule. 2. The Egyptians employed all the principal members of an order, and also in some instances adapted very fine decorations, but they never varied the general character,--it continued uniformly to express dignified gravity. The Greeks, in the spirit of freedom, and with their peculiar facility of invention, varied the expressions of their architecture as well as sculpture, and produced three species of composition, which are denominated orders. See Plate LXXXV. 3. The principal member of an order, is the perpendicular support or column : the accompani- ments being subservient to this leading feature. The bottom of the column is placed either on a general artificial platform, or upon a particular plinth, or both. 4. The lower part of the column, which rests upon the square plinth, is sometimes encompassed with mouldings, which, in allusion to their position, are, in conjunction with the plinth, termed a base. 5. The top part of the column is also covered with a square plinth, with its sides straight or curved, and is generally accompanied by circular mouldings, or sculptured decorations, upon the top part of the column which is immediately underneath it: this, taken together, is called the capital. In Plate LXXXIV. we have given examples of Egyptian capitals. 6. The body of the column, which reaches between the base and capital, is termed the shaft: it ings, either meeting in an edge, or leaving a small plane space between them. 7. The lintelling, or covering, which lies upon and connects the columns, is termed the entabla- ture; and is subdivided into three parts, named architrave, frieze, and cornice. The architrave con- sists of a mere lintel laid along the tops of the columns; the frieze represents the ends of the cross beams resting upon the former, and having the spaces between filled up, having also a moulding fixed to conceal the horizontal joint, and divide it from the architrave; and the upper member, or cornice, represents the projecting eaves of a Greek roof, showing the ends of the rafters. 160 | PART III. ORDERS OF ARCHITECTURE. which exhibit Egyptian character, and proportions adapted to the climate and materials of Greece; the column of the Doric being as gross, in proportion to its height, as the pillars of the Thebaid; but Greece being subject to rains, it was found necessary to elevate the whole edifice on an artificial different from Egyptian. 9. The three Greek orders, named DORIC, IONIC, and CORINTHIAN, have the same principal members, and subdivisions; but the dimensions, mouldings, and decorations, vary very considerably, as will be seen by the specimens of each, which will, in the course of this investigation, be produced. 10. It is only in Greece, or in the territories of Greek colonies, that pure specimens of these orders have been found. Their mouldings exhibit every specimen of conic section, as elliptical, parabolical, hyperbolical; some are merely champhered: the circle was seldom employed excepting These 11. There are other two orders, known to architects as the Tuscan and COMPOSITE orders. are supposed to be of Italian or Roman origin. See Plate LXXXVI. PLATE LXXXVI from roof" Tiais a Rome: Shart COMPOSITE OR ROMAN ORDER THE ORDERS OF ARCHITECTURE. THE REGULAR MOULDINGS. GREEK ROMAN De MOHDOT OLO Scotiat of Case Torus on Tore TUSCAN ORDER. A Fallarton C'London& Edinburgh GRECIAN MOULDINGS. PLATE ZXXXVIL Fig.1. Fig.2.NP1. ___Pig. 2.N? 2. Fig.3. Fig.4. Fig.9. Fig. 5. Fig.6. Fig. 8. | Fia.7. Fig.10. Fig.m. Fig.12. . । Fia..1.3. Fid.14. - - --- - - Fig.15. -- . 4 . .... . .. --. - ..- . ... ... ... Fig.16. . .. - Fig.17. - - -- - - - A FillartarikCLondork Edinburgh SECTION II. MOULDINGS. Definitions, 1.-—-Names and shapes of Mouldings, 2-12_ Rules for describing Mouldings, 13-28. [Plates LXXXVI. and LXXXVII.] 1. MOULDINGS are prismatic or annular solids, formed by plain and curved surfaces, and employod as ornamental parts in most architectural operations. All parallel sections of straight mouldings, and all sections passing through the axis of annular mouldings, are equal similar figures. The forms of all mouldings are referred to a section at right angles to their longitudinal direction, when pris- matic, or passing through the axis, if annular; and this is simply denominated the section, on account of its frequent use, as oblique sections only occur in mitres. The names of mouldings depend upon their form and situation. We have already given a few brief descriptions of the principal mouldings PACE: 1254 used in Joinery work [Part II. sect. iv. Art. 37–51]; but a more minute description of those used in architectural work is here desirable. 2. If the section is a semicircle which projects from a vertical diameter, the moulding is called an astragal, bead, or torus. If a torus and bead be both employed in the same order of architecture, they are only distinguished by the bead being the smallest. The tori are generally employed in bases, but the bead both in bases and.capitals. See on this, and the ten succeeding articles, Plate LXXXVI. 3. If the moulding be convex, and its section be the quarter of a circle or less, and if the one extremity project beyond the other equal to its height, and the projecting side be more remote from the eye than the other, it is termed a quarter round. This, in Roman architecture, is always employed above the level of the eye. 4. If the section of a moulding be concave, but in all other respects the same as the last, it is denominated a cavetto. The cavetto is never employed in bases or capitals, but frequently in entablatures. 5. If the section of a moulding is partly concave and partly straight, and if the straight part be vertical and a tangent to the concave part, and if the concavity be equal to or less than the quadrant of a circle, the moulding is denominated an apophyge, scape, spring, or conge. It is used in the Ionic and Corinthian orders for joining the bottom of the shaft to the base, as well as to connect the top of the fillet to the shaft under the astragal. 6. If the section be one part concave and the other convex, and so joined that they may have the same tangent, the moulding is named a cymatium ; but Vitruvius calls all crowning or upper mem- bers cymatiums, whether they resemble the one now described or not. 7. If the upper projecting part of the cymatium be a concave, it is called a cima-recta. This moulding is generally used in the crowning members of cornices, but seldom found in other situations. 8. If the upper projecting part of the cymatium be convex, it is called a cima-reversa, and is the smallest in any composition of mouldings, its office being to separate the larger members. It is seldom used as a crowning member of cornices, but is frequently employed, with a small fillet over it, as the upper member of architraves, capitals, and imposts. 162 [PART III. ORDERS OF ARCHITECTURE. 19 uu ES D 9. If the convex part of a moulding recede and meet a horizontal surface, the recess formed by the convexity and the horizontal surface is termed a quirk. 10. If the section of the moulding be a convex conic section, and if the intermediate part of the curve project only a small distance from the greatest projecting extremity, and if the tangent to the curve at the receding extremity meet a horizontal line produced forward without the curve at the upper extremity, the moulding is called an ovolo. It is generally employed above the eye, as a crowning member in the Grecian Doric. Ovolos may be used in the same composition of different sizes; it is sometimes cut into egg and tongue, or egg and dart, when it is termed echinus. It is employed instead of a torus in the base of Lysicrates at Athens. The contour of ovolos are gen- erally elliptical or hyperbolical curves. These curves can be regulated to any degree of quickness or flatness; the parabola can also be drawn under these conditions, but its curvature does not afford the variety of change of the other two species. 11. If the section be a concave semi-ellipse, having its conjugate diameter such that the one may unite the extremities of its projections, and the other diameter may be parallel to the horizon, the moulding is termed a scotia. They are always employed below the level of the eye; their situation is between two tori. The one extremity has generally a greater projection than the other, and the greater projection is nearest to the level of the eye. 12. If the section of the moulding be the two sides of right angles, the one vertical, and the other of course horizontal, it is termed a fillet, band, or corona. A fillet is the smallest rectangular member in any composition of mouldings. Its altitude is generally equal to its projection; its purpose is to separate two principal members, and it is used in all situations under such circumstances. The corona is the principal member of a cornice. The beam or fascia is a principal member in an archi- trave as to height, but its projection is not more than that of a fillet. 13. In the following descriptions, the projections and heights are always supposed to be given in position to the extremities of the curve. To describe the torus, Plate LXXXVII. Fig. 1. Let a b be the vertical diameter whence the torus projects; bisect ab in c; from c, with the radius ca or cb, describe the semicircle 6 da, which will be the profile of the torus. 14. To describe the ovolo, the height and projection being given Fig. 2. First, let the height and the projection be equal to each other. Draw a b equal to the height, and b c at a right angle with, and equal to, a b, for the projection; then with the radius ba or b c describe the arc a c, which is the contour of the ovolo. But if the projection is not equal to the height, but less, as in Fig. 2, draw ab and b c forming a right angle as before, a b being made equal to the height, and b c equal to the projection ; from the point of recess a, with the height a b, describe an arc bd; and from the point c of projection, with the same radius, describe another arc cutting the former at d; lastlv, from d, with the radius d a or d c, describe the arc a c, which is the contour required. 15. The methods of describing the cavetto, Figs. 3 and 4, are the same as that for describing the ovolo, the one being the same as the other reversed. 16. To describe the cima-recta, Fig. 5. Join the point of recess a to the point of projection 6 by the line a b; bisect a b in c, with the distance bc from the points c b; describe the intersection e, and from the points a c, with the same distance, describe the intersection d; from d, with the dis- tance da or d c, describe the arc ac; and from e, with the distance eb or ec, describe the arc bc; and a cb will be the contour of the cima-recta required. If the curve is required to be made quicker, we have only to use a less radius than that of ac or cb, in order to describe the two portions of its contour. 17. The same description applies to the cima-reversa, Fig. 6, by the same letters of reference. SECT. II.] 163 MOULDINGS. 18. fo describe the apophyge, Fig. 7, the projection being given. Let a b be the projection, and ace a line which it is required to touch. Make a c equal a b, and with the distance ac or a b from the points b and c describe the intersection d; from the point d, with the radius db or d c, describe the arc b c, which is the contour of the apophyge. 19. To describe the apophyge so as to touch a right line given in position at the point of projec- tion, Fig. 8. Let b c be the right line; and a b the projection of the moulding; draw a cd f at a right angle with ab; make cd equal to cb; draw b e perpendicular to b c, and d e perpendicular to cd; from the point e describe the arc 6 d, which is the contour of the moulding. 20. To describe the scotia, Fig. 9, the extremities a and b of the curve being given. From the projecting point b erect bde, and from the receding point let fall a gc perpendicular to bc, the horizontal of the moulding; add the half of a c and two-thirds of b c into one length, which set from 6 to d; from the centre d, with the distance db, describe the semicircle b fe; draw the straight line eaf, and dg f; from the point g, with the distance ga or gf, describe the arc af: then will a f h be the contour of the scotia required. 21. To describe an ovolo, the tangent a c at the receding extremity a, and its projection at 6 being given, Figs. 10 and 11. Draw the vertical line cbd; draw be parallel to ca, and a e parallel to cb; produce a e to f, making e f equal to ea; divide e b and b c each into the same number of equal parts; from f, and through the points of division in b, draw right lines; also from a, and througla each of the divisions in b c, draw another system of lines, and the corresponding intersections of each pair of lines will be as many points in the curve as there are pairs; then a curve being drawn through the points, will be the greater part of the contour. The remaining part bg may be found in the same manner, by drawing lines from a through the points in be instead of f, and drawing lines from bd to f, instead of b c to a. The curve drawn in this manner is a portion of an ellipsis, something greater than the quarter of the whole. The recess of the moulding at its projecting point is denominated the quirk. Fig. 10 is adapted for entablatures; and Fig. 11 having a large projection, to capitals of Doric columns, such as may be seen in the temple of Corinth, and in the Doric portico at Athens. This method, though easy, gives the extremity of the conjugate axis, between the receding extremity a, and the point of projection b; but the following method gives the extremity of the shorter axis, where the tangent commences, at the receding extremity of the con- tour of the ovolo. 22. To describe the ovolo, supposing the extremity of the conjugate axis to be at the point of contact a, Fig. 12. Join ab, which bisect in e, and draw cef: make a flk perpendicular to the tangent a c, then the point f will be the centre of the ellipsis: draw fhi parallel to ac; take the distance f a, and from the point b cross the line fi at h; produce b h to l, and make fi equal to bl; then with the semi-transverse fi, and the semi-conjugate f a, describe an ellipsis, and the portion of the curve contained between the extremities a and g, will be the contour of the moulding required. This method is recommended as producing the most graceful form of an ovolo, as the lower extremity of the curve begins at the point of contact. From the large projection here given, the moulding is adapted to Doric columns. 23. To describe the hyperbolical ovolo, as used in Doric capitals, the same things being given as before, Fig. 13. Erect ad e f g perpendicular to the horizon, and draw cd and be at right angles to a defg: make eg equal to a e, and e f equal to ad; join bf, divide bf and b c into the same number of equal parts, and draw lines from g through the divisions of bf, also lines from a through the divisions of bc: each corresponding pair meeting as before will give the points in the curve of the hyperbolical moulding. This is the general form of the ovolos in the capitals of the Greciau Doric. 164 [PART III. ORDERS OF ARCHITECTURE. 24. To describe a scotia, Fig. 14. Join the extremities a and b of the moulding ; bisect ab in c; draw eod parallel to the horizon; make cd equal to the recess of the curve, and ce equal to cd; then with the conjugate diameters a b and ed describe the curve adb, which will be the contour of the moulding required. 25. Fig. 15 represents the form of the annulets as applied in Fig. 13, where the receding parts are in the tangent at the bottom of the curve of the ovolo. 26. Fig. 16 represents another kind of annulet, which has a vertical position. This form is only to be found in the Doric portico at Athens. 27. Fig. 17 represents a curious Grecian moulding, to be found under coronas. 28. Sir William Chambers observes that all these different mouldings have a definite use and propriety in correct architectural practice; and that, on examining the best remains of ancient architecture, we find that, in all their profiles, the cima and cavetto are never employed where strength is required, but only as finishings ; while the ovolo and talon are always employed as sup- porters to the essential members of the composition ; and the chief use of the torus and astragal is to fortify the tops and bottoms of columns, and sometimes of pedestals. The same authority, there- fore, considers Palladio to have erred when he employed the cavetto under the corona, and used the cima so frequently as a supporting member; and Vignola to have been equally in fault when he finished his Tuscan cornice with an ovolo, which gives a mutilated air to the whole profile, " as it resembles exactly that half of the Ionic cornice which is under the corona.” - - - - ' - - # - - - - - +++ - -- - - . - - ++ - - - - - - - - - - - - - - - - - - --- - - - - -- - - -- ..- - . . - - - - - - --- - - - - - - - - - - - - - - - - - - - PLATE LXXXVII. .... Fig.3.NC 3. Fig.3.1?1. Fig 3. N?? Fig. .5. ka, 4. NP . Fig. 141. 2. Fig. 41N!3. Fig.4.NO 1. UUUUUUUUUU - - - - - - - -- - - - - - - - - - - - .- - - - - - - - - - - 'I - - . *.. - - - . - - V . . - - - - - + - .- * - ...... . - -- - - - - -- - - . - . . ..------- - - - - --- ---- - ------- - --- - - - - - -- - - - - . . . - - . - - . -- - - - -- - - - I - - - - - - - - - - - - - - - - --- - - - - . M 4 A - - .- E + - + - - - - - + + V - - - . R SE - - - - - GRECIAX ARCHITECTURE. A Frlarian&C London& Edinburghi 2121 Frig. 2. ELE E Fig. 7. R72345C Line RE of I SECTION III. OF DESIGNING COLUMNS. To diminish the shaft of a column, 1.- To give less swell to a column, 2. To describe the flutes of a column without fillets, 3.---To describe flutes with fillets on the shaft of a column, 4. [Plate LXXXVIII.] 1. To diminish the shaft of a column, Plate LXXXVIII., Fig. 1. Let AB be the altitude of the column, and BC the diminution at the upper end of the shaft; divide AB into any number of equal parts, and divide the projection BC into the same number ; draw the lines 16, 2d, 3c, &c., at right angles to the altitude ; and draw other lines from the points 1, 2, 3, &c., in BC towards A, to inter- sect with the former parallel lines at the respective points b, c, d ; then A b c d e f C, will be the curve line of the section of the column. 2. But suppose it were required to give less swell to the column as in Fig. 2. Divide AB as before, and DC into two equal parts at D; divide DC into as many equal parts as AB; then proceed as in Fig. 1. Or thus : suppose EF to be the axis of the column, EG, EA the semi-diameters at the bottom, and FN, FC the semi-diameters at the top ; on AG, as a chord, describe the segment AOPG of a circle, proportionably less than a semicircle, as the swell is intended to be less ; draw NP parallel to FE ; divide the arc GP into any number of equal parts, and divide the altitude EF into the same number of equal parts: through the points of division draw the lines a h, bi, ck, dl, em, parallel to AG ; also draw G 1h, 2 i, 3 k, 41, 5 m, PN, perpendicular to A; then through the points G, h, i, k, l, m, N, draw a curve, which will be the contour required. 3. To describe the flutes of a column without fillets, Fig. 3. Let AB, No. 1, be the diameter, which bisect in G; draw AD and BC perpendicular to AB, and describe the semicircle AEFB; draw DC to touch the circle, and DEG and CFG to the centre G; divide the arc EF into five equal parts, and run the same part on the arcs EA and FB, so that the whole will be divided into nine equal parts, and two half parts at each extremity; then the points of division will mark the arris of the flutes. Their concavity will be found by an arc described from the summit of an equilateral triangle. The fluting at the upper end of the shaft, shown in the concentric circle, is described in the same manner. No. 2 represents the bottom elevation, and No. 3 the top elevation of the shaft, as drawn from the section. 4. To describe futes with fillets on the shaft of a column, Fig. 4. Supposing every thing is done as in Fig. 3, before the division of the circle. Divide EF into six equal parts, and run the chord upon the arcs EA and FB, each of which will contain it three times, so that the whole semi-circumference will be divided into twelve equal parts, the points of division marking the centres of the flutes : divide the chord of one of these small arcs into five equal parts; then with three of these parts as a Dianne ram radius, from each of the aforesaid centres describe a semicircle, which will be that representing the section of the flute. Those of the interior circle representing the top of the shaft, are found by drawing the lines to the centre, as appears sufficiently by the figure. No. 2, the elevation of the fluting at the bottom of the shaft, as in the temple of Vesta at Rome. No. 3, the elevation of the fluting, as in the temple of Bacchus at Teos. Nos. 4 and 5, the common way in which the flutings of columns are terminated at the bottom and top of the shaft. SECTION IV. OF THE DORIC ORDER. Shaft and capital, 1.- Architrave, 2.- Frieze, 3.- Cornice, 4.- Tables of comparative proportions, 5. Observations thereon, 6–10. - Table of columnar proportions, 11-14.— Proportions established, 15, 16. Roman Doric, 17. [Plates LXXXV., LXXXIX., and XC.] 1. This is the most ancient of the three orders, and, while employed by the Greeks, was without a base. The surface of its shaft is usually found worked into twenty very flat flutes, meeting each other at an edge, which is sometimes a little rounded. The upper member of the capital is a square abacus or thin plinth, under which is a large and elegantly formed ogolo, with a great projection ; immediately under the ovolo, there are three fillets or annulets which project from the continued line of the under part of the ovolo, and have equally recessed spaces betwixt them; the flutings of the column are terminated by the under side of the last of these fillets, and either partly or entirely in a plain surface at right angles with the axis of the column. 2. The architrave is composed of one vertical face, with a band or fillet at its upper edge; to the under side of this band are suspended a small fillet and conical drops or guttæ, which, for their position, are dependent upon the ordnance of the frieze. 3. The frieze consists of rectangular projections and recesses placed alternately. The height of each projection or tablet is rather more than its breadth. The recesses are either perfectly or nearly square. The tablets are each cut vertically into two angular channels, with two half ones on the extreme edges; each channel is formed by two planes meeting at its bottom at a right angle, and each forming an angle of 135 degrees with the face of the tablet. The upper ends of the channels are terminated in various forms; the tablets are, from their channellings, named triglyphs. In a direction immediately under each triglyph, and equal to its breadth, a small fillet is attached to the lower side of the architrave crowning band, and from it depend six guttoe or drops, which are gen. erally the frusta of cones with their bases downwards, though they are sometimes of a cylindrical shape. The square spaces in the frieze between the triglyphs, are named metopes, and are frequently decorated with sculptures. 4. The cornice is strongly marked by a corona of great projection, forming a very distinct separa- tion between its upper and lower parts ; and by having, below the corona, and immediately over the triglyphs, blocks named mutules, which also project considerably, and have the plane of their soffits with an inclination from their roofs towards the horizon, and these have likewise guttæ or drops depending from their soffits. 5. That the proportions which the different members of the Doric order bore to each other were practised by the Greeks with considerable latitude, will be evident from the following Tables, which exhibit the dimensions of the principal parts of this order in all the ancient Greek edifices which have been examined with accuracy. The dimensions here put down are in feet, inches, and decimal parts. DORIC ORDER FROM THE PARTHENON ATHENS, PLATE I.XXXLX. ------ - - ---- -------- . . . 111 Birs'.:,: h---7.,-.. 4834 ---- Ae 46% ---- - * C a . . .. . . . .. .. .. ... . . .. . ... .. . .. .. . .. R . . . ON -- 017 -in.--26" ---.. - -----1800 con... + - ----- ... ---------. Wod. 745 p! -------..Ilod----------- -------- I Mod.270..-- 0 0 0 Doel Loo 0 0 0 oll 0 0 0 осссссеер lo 01 17777777----- -- - -- ------- M2 26.1348 29* 1333 2014 29.64 ---------1. Mod. izap...----------- 313 I 0 25 V Dule 2 01 .---- .-- Mulp ----..! 1/..12 91; --- VAAMAAVAALIJALAUTTUA ----- --- - PATE и ҳҳxy ER .-------- - -2016- ka----- 25p?S-------- the then-Ting-of hit? Mod.x240S-------- 34 ----- ------- 2015 -------- - 25 -R ----•zzj-------- XESSE ES * -- - -----Zlod ----------- -- - ------------ --- sa------...33.- ------ 6.2.8L- --24.p?.-- -10.Hort2017 294 * --- --- ------ ------- Lotule.------ --- 2.2 ----- ----- -- TA ---*- ----- - ------- 22.--- ----- 174p --------- TV3 -*- . --- 131 -- ----10 ----.22. ----- ----1045 pm -- ----------------llod. 20 Inpatiti pofitez EMT W. A.Beever Sculp? A Fullaricn&C London & Edinburgh 02 SO ROMAN ARCHITECTURE. DORIC ORDER, FROM SIR WW CHAMBERS. 95 Nin Sarietathe corona de Mutlules 45 Men 30 Min 304 32 Muz 27 Dra uw gemaa & 35.2 2012 2323 UCS 133 Introduced by John Day NI A Fruiartenke London Edinburgh SECT. IV.] 167 OF THE DORIC ORDER. No. I. -DIMENSIONS OF COLUMNS. OF THE SHAFT. CAPITAL. NAMES OF EDITICES. Тор. 19 Heights. 15' 9.55" 17 4.25 18 8.6 4.0 19 9.150 20 4.25 210.0 21 2.0 0.75 23 8.575 26 2.5 28 8.0 . . . 28 9.85 28 11.42 31 0.75 32 8.13 33 11.2 34 2.8 48 7.0 Temple of Jupiter Panellenius at Ægina, . . Temple of Ægina, . . . . . Temple of Theseus, . . . . . . Portico of Philip king of Macedon, Temple of Minerva at Sunium, . . . . Hexastyle temple at Pæstum, . . . Epneastyle temple at do., . . . . Temple of Juno Lucina at Agrigentum, . Temple of Concord at do., . . Temple of Corinth, . . . Doric portico at Athens, . . . . . Temple of Minerva at Syracuse, peristyle, Lesser hexastyle temple at Silenus, . . . Propylea at Athens, . . . . . Hypethral temple at Pæstum, . . . . Temple of Minerva at Syracuse, Pronaos, . Greater hexastyle temple at Silenus, . . . Temple of Jupiter Nemeus, . . . Parthenon at Athens, . . . . . Octostyle hypethral temple at Silenus, . . Temple of Jupiter Olympus at Agrigentum, . Diameters. Bottom. 3 2.6" 2 4.65" 3 26 2 4.65 3 3.65 2 6.63 2 11.5 2 5.3 3 4.2 2 6.65 4 2.93 3 0.52 4 9.75 3 2.0 4 6.1 3 4.875 7.7 3 6.75 5 10.0 4 4.1 4 405 3 4.6 6.04 0.5 6 6.9 4 1.9 . . . 3 11.0 0.03 4 98 6 9.3 5 2.0 7 5. 9 5 0. 2 4 3.0 6 1.8 4 9.75 10 7.5 6 3.6 9 11.5 1' 6.7" l 6.7 1 7.9 0.6 1 2.966 2 4.75 2 483 . . . 2 3.125 2 4.375 3 5.325 . . . 2 665 3 10.17 3 35 8.08 3 CAN COCO 9.9 9.5 25 No. II.-HEIGHTS OF ENTABLATURE. Cornice. . . . . 1 2 2 . . 1 3.9 0.45 2.8 0.6 9.0 * . . . . 1 I * . . NAMES OF EDIFICES. Temple of Jupiter Panellenius at Ægina, . Temple of Ægina, . . . . . Temple of Theseus, . . . . . Portico of Philip king of Macedon, . Temple of Sunjum, . . . . . . Hexastyle temple at Pæstum, . . . Enneastyle temple at do., . Temple of Juno Lucina at Agrigentum, . Temple of Concord at do., . . . Temple of Corinth, . . . . . Doric portico at Athens, . . . . . Propylea at Athens, . . . . . Hypethral temple of Pæstum, . . . . Parthenon at Athens, . . . Temple of Juniter Olympus at Agrigentum, . CO CO CO . . Architrave. 2' 9.05 29 .05 2 9 8. 1 7 10. 2 8.45 3 2.625 3 9.75 4 1.75 3 7.375 4 8.75 2 6 10. 3 9.0 4 11.25 9 5. 1 10 4.18 . . . . . . . . . . . . . . Frieze. 2' 9.1" 2 9. 1 8.55 4. 9 2 8.45 3 2.0 3 4.0 3 4.25 3 6.95 * * 0.7 3 9.85 4 8.8 5.05 10 2.5 . . 1 11.25 * 6 . . . . 3 6.45 . . AN A CO CO . . l . 2 3 * . . 4 6.0 3.8 . . . 6. In the temples of Jupiter Panellenius at Ægina, Theseus at Athens, Minerva at Sunium, Jupiter Nemeus between Argos and Corinth, the cornice has lost the ovolo or crowning member, the dimensions in the table are therefore taken without it; the temple of Minerva Parthenon at Athens is the only ancient Greek edifice in which this member exists, and in this instance the lower extremity is recessed within the fillet which is immediately under it. Those parts marked with a star in the table are totally gone. The crowning member of the cornice of the portico of Philip king of Macedon 168 (PART III. · ORDERS OF ARCHITECTURE. 1 is a cima recta, having its lower extremity also recessed within the fillet which is below it. The cornice of the hexastyle temple of Pæstum is crowned with a cavetto. Being a singular circum- stance in the Greek Doric, it leads to a suspicion that it may have been added in some subsequent repair. 7. From consulting these Tables, it will be seen, that in the best examples, the height of the architrave and frieze are nearly equal ; in the temple of Sunium they are precisely so. The only exception of consequence is in the temple of Juno Lucina at Agrigentum, where the architrave exceeds the frieze 9} inches : in no other instance is the difference at all material, and any that does exist has most probably arisen from the impracticability of procuring so many marbles of the same dimensions. In comparing the tabular dimensions, it is to be observed, that the band or capital of the triglyph is included in the height of that member. This, in Stuart's • Athens' and the 'Ionian Antiquities,' projects only in front, but never returns upon the flanks but at the external angle of the edifice. With the exception of the Doric portico and the temple of Jupiter Nemeus, where the heads of the glyphs are terminated by planes parallel with the horizon, the middle part of the heads, in all the purest examples, is a horizontal plane, and the surface of the sides cylindrical, so as to form a tangent with the intermediate plane, and likewise the arc on the face a tangent with the intersection of each vertical plane of the glyph. Each semi-glyph is terminated by two semi- cylindrical surfaces, the axis of each cylinder being perpendicular to each return of the glyph, and thereby forming a semi-cylindric groin, and a pendent with each angle above the semi-glyph. The general height of the epistyle or architrave is equal to the superior diameter of the column, though in some cases a little more or less ; the height of the zophorus or frieze is equal to that of the epistyle; the mean breadth of the triglyph tablet is equal to half the inferior diameter of the column; the mean height of the cornice is half the diameter; so that the architrave, frieze, and cornice, are respectively to each other as the numbers 3, 3, and 2. If the whole entablature is divided into 8, the breadth of the triglyph is two of these parts. 8. In all examples of this order, except the temple of Apollo at Delos, the hexastyle temple at Pæstum, the portico of Philip king of Macedon, and the Doric portico at Athens, the face of the triglyph tablet, and that of the epistyle or architrave, are in one vertical plane ; so that the fillet named regula, and the gutta under the cup of the epistyle, being regulated by the breadth of the triglyph, will, at each external angle, only touch at their internal points, and leave a void space at the external angle of the epistyle cups. This is exemplified in the temples of Minerva, Theseus, and the propylea at Athens; the temple of Minerva at Sunium ; Jupiter Nemeus, Jupiter Panellenius, Minerva at Syracuse ; Concord at Agrigentum ; the hypethral temple at Pæstum, and also of Silenus and Jupiter at the same place. And where the face of the epistyle, and that of the triglyph tablet, are in one vertical plane, the guttæ will be six in number under each regula, at every external angle or return, that is, making twelve on the two sides. In the Doric portico at Athens, the temple of Apollo at Delos, and the hexastyle temple at Pæstum, the face of the metopes and that of the epistyle are in one vertical plane. The triglyphs, regula, and guttæ, project from the plane of the epistyle, and at the returns meet at the external angles; and though the guttæ appear six on each face, yet the guttæ at the angle being common to both faces, the whole make only eleven. 9. In the cornice, the corona forms the most prominent feature in the temple of Concord at Agrigentum, and of Jupiter at Silenus. Instead of having the crowning ovolo, the cornice termi- nates with a face receding within the corona. This is so contrary to usual practice and propriety, that we are led to suspect, that a defect in the stones, which formed the upper division of the cornice, may have been supplied by having an ovolo fixed in this recess. In every specimen of pure Doric, the cornice has mutules. In examples to be found in Sicily, the drops from the soffit of the mutules, Sect. IV.7 169 OF THE DORIC ORDER. are cylinders of greater height than diameter ; but in all the best examples, they are not more than half their diameter in height, and in some instances considerably less. In the temple of Theseus, the drops are frustums of cones ; but, in the same specimens, those under the regula upon the epi- style have both a concave and convex flexure. 10. The hexastyle temple at Pæstum, instead of mutules, has the soffit formed into coffers; but, in this specimen, it is only the capital and triglyphs which bear any affinity to the Greek Doric; and even on the capital, in place of annulets, there is a row of delicate leaves crushed together between two astragals; and the triglyph being placed at the returning angle, they are over the centre of the columns ; so that this specimen partakes more of the degenerate Roman than the pure Greek. 11. Having investigated what relates to the primary and secondary divisions of this order, and also noted the positions and relative proportions of the leading features, we shall add a farther state- ment respecting its columns, founded upon Table No. I. In this additional Table, the diameter at the base is considered the same in all, viz, unity. The whole numbers represent diameters. The figures to the right hand of the point are decimal parts of the diameter. The examples are arranged increasing in altitude. No. III. Diameters. 4.065 Top of Shaft. .73 .687 .661 769 4.134 4.329 4.361 4.410 4.572 4.605 .762 .592 .755 4.753 NAMES OF EDIFICES. Temple of Corinth, Hypethral temple at Pæstum, . Enneastyle do. do., . Greater hexastyle do. at Selinus, . Temple of Minerva at Syracuse, Octostyle hypethral temple at Selinus, Temple of Juno Lucina at Agrigentum, Temple of Concord at Agrigentum, Hexastyle temple at Pæstum, . . Temple of Jupiter Panellenius, . Temple of Minerva at Athens, . Temple of Theseus, do., . Temple of Minerva at Sunium, . Temple of Apollo in the Island of Delos, Doric portico at Athens, . . Temple of Jupiter Nemeus, Portico of Philip king of Macedon, . .767 .717 4.795 .742 .782 5.397 5.566 5.669 5.899 5.931 6.042 6.515 6.535 . .772 .762 .754 .780 .816 . . . . . , .825 12. From this Table it is evident, that the ancients did not scrupulously adhere to any precise proportions in their columns for different edifices; but not knowing the dates of the construction of the several specimens, we are unable to determine whether these differences existed at the same time, or succeeded each other in consequence of a change of taste. It may also be observed, that, of seventeen examples, the upper diameters of six are less than three-fourths, and eleven greater. The diminution of the superior diameter in the temple of Theseus is .772, which is something less than a mean between three-fourths and four-fifths: the half sum of these fractions being .775. This example of the temple of Theseus is one of the best of the Greek Doric, and may be taken as a rule ; or in practice, to make the superior diameter three-fourths of the inferior, is still more simple, and sufficiently correct. 13. In every Greek Doric, the vertical face of the epistyle or architrave projects beyond the supe- rior diameter, but is within the inferior one. 170 [Part III. ORDERS OF ARCHITECTURE. 14. In the temple of Theseus, the height of the abacus is nearly one-fifth of the diameter, and the ovolo and annulets together are very nearly equal to the abacus. The height of the annulets is very nearly one-fifth of the ovolo. The horizontal dimension of the abacus extends, on each side, very nearly six times its height. The neck of the capital is nearly half the height of the annulets. In the temple of Corinth, and the Doric portico at Athens, the ovolo or echinus is of an elliptical shape; but in every other instance of Greek capitals it is hyperbolical, excepting the single instance of the portico of Philip king of Macedon. į 15. From the preceding dimensions and observations, we establish the following proportions for the construction of the Doric order: Considering the diameter that of a circle, at the lower end of a shaft the column is six diameters in height. The thickness of the upper end of the shaft is three- fourths of the lower, or it diminishes one-fourth of the diameter. The height of the capital is half a diameter. That of the ovolo, with the annulets, and that of the abacus, are each one-quarter of the upper diameter. The annulets are one-fifth of one of the parts. The horizontal dimension of each face of the abacus is six times its height. The entablature is divided into four equal parts ; the upper one is the height of the cornice; the remaining are divided equally between the architrave and frieze. The inner edge of the angular triglyph is placed in a vertical line with the axis of the column. The height of the triglyph is divided into five equal parts ; three of these parts give the distance of its returning face, and determine also that of the epistyle, and consequently include the breadth of the triglyph. The height of the capital of the triglyph is one-seventh of its whole height, and the capital of the metope one-ninth. The breadth of the triglyph is divided into nine equal parts, giving two to each glyph, one to each semi-glyph, and one to each of the three inter-glyphs , The metopes are square. The height of the cornice is divided into five equal parts; the lower is , given to the fillet, the mutule, and drops ; the next two to the corona ; and the remaining two parts are subdivided and disposed amongst the several members. The projection of the cornice is equal to its height; it is divided into four equal parts, giving three to the projection of the corona. The number of annulets in the capital vary from three to five ; and the number of horizontal grooves, which separate the shaft from the capital, vary from one to three.-In Plate LXXXIX. are shown the proportions of the Doric order as used in the Parthenon at Athens: viz. Section through the 1 entablature ; Plan of the soffit of the corona ; Columns showing the proportion in modules and minutes ; Capital of Antæ; The same on a large scale ; and Section through the annulets of the capital of the column. 16. In the application of the Doric order to temples, the shafts of the columns are generally placed upon three steps, which are not proportioned like these in a common stair, but to the magnitude of the edifice. 17. The best example of Roman practice is that taken from the theatre of Marcellus, by Sir | W. Chambers, of which we have given the sections in Plate XC. W 7 FROM THE TEMPLE ON THE RIVER ILYSSYS NEAR ATHENS. PLIZE XO 897 Port Tom Centre of Column Pia. 1. 10.3 Centre of Column from Fig. 5. ---4 Mod. 104 I------- 1 Module 10 POR from Centre of Column - 1 Mochile - 1 Marche -- 1 Module ------ W A Heever Sculp 11. Purser.del! scale of Feet A Foliarto C Londen. Edinburgh || A PLATE XCII, IONIC ORDER. FROM GATE WAY AT ELETSIS. 19 20 2 5625. காரன் 1.85 - - -- - -- - - - - - - - - - - - - - - - - - 5.18 | 2.77. அ. 13.27 40.19 50.55 17 36.87 0.02 ----------------- -------.. 2870 3382 274 2.91 -- == 137.64 30.92 43 - = ----- - os 27.92 3740 - - -- - -- 949 - -- -- - - - - - - - - - - -- -- - 885 26.63 37.28 12 230 104 29,02 11016 23.09 -- - -- - . SK .. I - - - 43.49 - 230 1.2 - 6.28 - - - - - - - 25.66 17. 32 Enurarul lor Colruutrolig 77071 A Fuillarton & C'London& Edinburgh IONIC ORDER. PLATE XCII. PL 1V.PROFILE, AND SECTIONS OF CAPITAL Fig. 1. Fig. 2. Fig. 4. 73.22 1367 8.72 37.33 7.77 తెలు తెరవ 9.77 36,2 9 36.2 6042 Faureret hor tomtreno A Futiarton tec'londoni Edinburgh UN IONIC ORDER. : PLATE XCIV. VOLUTE AND BASE OF COLUMN.. 7.28 2X 2.32 29,02 -2008 1,03 99 1,67 have 23.09 70.76 ........ ...... .....'. 39981... ............. 246 --7471 2.03 ........ Z/514 K...............-5.75 ............ ...... 64......... 6.67 HOLZ ...... 2 ---,-..9.4 ........ ....267 ........... 3.24 ..... K-321..... .......256..................5 ..... ... 3;05...236.pmok 5.38 1,92 ................ .........84......... i...3.04... 38.... .. y . ........ ...........12, UZ .......... ..........778.6..... ..... ....... ..................... 60 | 1411 793 010) 733,08 12:02 36,37 15,73 577-92. ..... 34.72 136,37 29,84 1 5.00 34.64 2.4. 4203 28,26 A Fullarton & Co Lendan (Edinburgh GRECIAN ARCHITECTURE. IONIC ORDER. VOLUTE. PLATE XCV. Radi for three Revolutions Fast Second ! Third 20 16.744 15.321 14.019 12.827 1.737 10.739 9.826 8.991 8.227 7.527 6 888 6.302 5.766 5.276 4.828 4.417 4·042 3.698 3:384 3.096 2.833 2.592 2.372 22.7 78.2 ----- 116 134 6.9 12.8 753 1 1 2 3 4 5 6 7 8 9 10 a 22 23 24 25 26 27 28 29 30 . L PSUNK P. Yichelsen W. Lorry A Fullartan. C'London & Edinburgh GRECIAN ARCHITECTURE. IONIC ORDER. VOLITE. . PLATE XCVI. _ . - - - -... - - - - --- ---- 18.3 10. 2.07 21. HE- --- - 1 N ... . .. .... 12.8 24 2 2 3 of 5 6 7 7 8 9 10 Iraantal *:Dir71171 hit.Milution Enumered by W. Luwry. 9 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 0 1 2 3 4 5 6 i 8 o 20 21 2 3 lat di 16 17 18 19 20 0 1 2 3 b C y 20 11 12 13 14 15 16 17 18 19 20 OF CH AFülhartonk CʻLondon: Eainburgh GRECIAN ARCHITECTURE. IONIC ORDER VOLUTE. PLATE XCVII . ---------- --- No 2. 70 ? WUM Z . INI . AW SIIIIIIIIIIIIIIIIIIII PuNicholson. 1. Lowry. A Fullarton & C.I ondan Edinburgh SECTION V. OF THE IONIC ORDER. General remarks, 1. Origin of capital, 2.- Temple of the Muses, 3. --Temple of Minerva Polias, 4.- Cornice: 5.- Frieze, 6.—--Asiatic Ionic cornice and frieze, 7. Column, 8.— Base, 9.—-Volute, 10–13.- Flutes, 14. - Description of Plates, 15.-General proportions, 16. [Plates XC.—XCVII.] 1. ALTHOUGH, in the Ionic, the primary division, into columns and entablature, and the secundary, of shaft, capital, architrave, frieze, and cornice, remained the same, yet the omission of some of the principal subdivisions of those parts, and the introduction of others entirely new, as well u the greater degree of delicacy in all, distinguishes this sufficiently from the Doric order, and estabxishes its claim as a distinct order or canon of Greek architecture. 2. The Ionian Greeks having become wealthy, and being, no doubt, influenced by the manners of their Asiatic neighbours, refined upon the simplicity of Attic architecture, and invented a capital totally different from that employed in the mother country. Its origin is, however, problematical. Vitruvius reports it to have been made in representation of the curls in the head-dress of females ; but other hints are quite as probable, as the spiral shape of the horns of rams, used in their sacrifices; or that assumed by the barks of some trees, when dried in the sun; or that of certain delicate vegetables, such as the slender fern, before it is quite unfolded ; or the beautiful spiral forms of various sea-shells, any of which are sufficient to guide the fancy of an ingenious artist in composing the volutes of the capital of Ionia. In the architrave and frieze, all appearances of triglyphs and guttæ are omitted; and in the cornice, instead of the bold mutules of the Doric, the ends of smaller pieces of wood, to which the covering tiles were fixed, are represented by what are termed dentils or teeth. This order also differed from the Doric, by having a base at the lower extremity of the shaft; the propriety of this might have arisen from the diameter of the shaft being much less than that of the Doric, in proportion to the height of the order, or the weight it had to sustain. 3. The rest of the Ionic order is not so precisely defined, or so uniformly adhered to, as similar parts in the Doric. The Temple of the Muses, on the Ilyssus, is composed of few, but very distinct and bold parts. The height of the volutes is three-fifths, and that of the whole capital two-thirds of the diameter of the shaft. If the entablature is divided into five parts, two are occupied by the architrave, which consists of a single fascia, crowned with a cymatium ; the remaining three are divided into five other parts, three of which are occupied by the plain part of the frieze, and the remaining two by the cornice. The cymatium of the frieze, which consists of a cima-reversa stand- ing upon a bead, is worked out of the cornice. The cornice, as viewed in front, is composed of a corona, cymatium, and cima. 4. In the temple of Erectheus and Minerva Polias at Athens, the architrave has three fasciæ and a cymatium, and the cymatium of the frieze is worked chiefly out of the cornice. The height of the entablature, from the bottom of the lower fascia of the architrave to the top of the cymatium upon the corona, if divided into nineteen parts, the architrave and frieze will each have eight, and the corona, larima, and cymatium will occupy the remaining three parts. In these specimens, the volutes have a singular degree of symmetry and beauty. 172 [PART III. ORDERS OF ARCHITECTURE. 5. In all the Greek Ionics, the height of the cornice, measured from the lower edge of the corona upwards, appears to have a constant ratio to the total height of the entablature, viz., nearly as 2 to 9, which seems the true one to accord with the character of the order. The great recess of the mouldings, under the corona, gives it a striking prominence, and prevents the cornice from appearing too heavy, though both the dentil band and cymatium of the frieze are introduced under it. 6. On account of the frieze being wanting in most of the Asiatic remains, although the architrave and cornice have been accurately measured, the height of the entablature cannot be ascertained. The only instance in which a frieze has been discovered is in the theatre of Laodicea ; and there it is rather less than one-fifth of the entablature. In the temple of Bacchus at Teos, and Minerva Polias at Priene, the architraves are divided into three fasciæ below the cymatium. Their proportions are very different from those at Athens, though also elegant in character and effect. 7. In all the Asiatic Ionics, the crowning mouldings of the cornices are cima-recta, less in pro- jection than height. The dentil bands are never omitted, and their height is a mean between that of the cima-recta and of the larima, corona, or drip, being uniformly greater than that of the corona, and less than that of the cima-recta. The cyniatium of the denticulated band is recessed upwards, being almost entirely wrought out of the soffit of the corona, which nearly conceals its height. The height of the cornice, from the top of the cima to the lower edge of the dentils, is equal, or nearly so, to that of the architrave. The height of the frieze, without its cymatium, may be about one- fourth of the whole entablature. 8. We have observed that the height of the Ionic column was originally eight diameters; the moderns have increased it to nine. The shaft is generally cut into 24 flutes, with as many fillets. The altitude of the entablature may, in general, be two diameters; but it may be increased, and should not be less than one-fourth of the height of the column in works of magnificence. 9. The base of the Athenian Ionics consists of two tori, having a scotia or trochilus between them, separated from the tori, above and below, by two fillets ; the fillet above the inferior torus projects, in general, as far as the extremity of the superior torus, and the fillet beneath the upper torus pro- jects beyond both. The scotia is very flat, its section forming an elliptic curve, which joins the fillet on either side. The tori and scotia are nearly of equal altitudes. In the temple on the Ilyssus there is a bead and fillet on the upper torus, joining the fillet to the scape of the column; in the same temple, as well as in that of Erectheus, the upper torus of the base is fluted; but the lower part, which joins the upper surface of the fillet above the scotia, is left entire. In the latter temple the lower torus of the base of the Antæ is receded; and in that of Minerva Polias it is fluted, each flute being separated from those on either side by two small cylindric mouldings of a quadrantal section, joining at their convexities. The upper scotia of the temple of Minerva Polias is also enriched with a guilloche. Vitruvius has, very properly, termed this the Attic base, it having been employed by the Athenians in all their Ionics; and, although the Ionians had another, more par- ticularly appropriated to this order, they sometimes employed the Attic, as in the temple of Bacchus at Teos. In the temple of Minerva Polias at Priene, and that of Apollo Didymæus near Miletus, the bases consist of a large torus, three pair of astragals, and two scotiæ, inverted towards each other; the upper pair of astragals is below the torus, and the scotia intervenes between each pair. In the former of these temples, an additional astragal separates the torus from the shaft. The torus is elliptical with its under part fluted; there is likewise a flute in the upper part of the base. In the temple of Apollo Didymæus, the upper torus is semicircular, and each bead of every pair is separated by a narrow fillet. This base differs but little from that assigned to this order by Vitruvius, only in the former the scotiæ are inverted, and present a greater variety of profile. 10. The volute, which is the great distinguishing feature of this order, in the Athenian Ionics, S SECT. V.] 173 OF THE IONIC ORDER. and the temple of Minerva Polias at Priene, the lower edge of the channel which runs between them is formed into a curve, bending downwards in the middle, and revolving about the spirals on either side. In the temple of Erectheus, and Minerva Polias at Athens, each volute has two channels formed by two distinct spiral borders; the borders forming the exterior volute, and the under side of the lower channel, have between them a deep recess or spiral groove, which diminishes gradually in breadth, till it loses itself in the centre of the eye. In the temple of Bacchus at Teos, the great theatre of Laodicea, and in all the Roman Ionics, the channel whereby the two volutes are connected has no border on the lower edge, but terminates with a horizontal line, falling in a tangent to the commencement of the second revolution of each volute. 11. In the temple of Erectheus, the column terminates with a fillet and astragal a little below the eye of the volute; and in the temple of Minerva Polias, it is terminated with a single fillet. In both instances, the colerino or neck is decorated with honeysuckles. The upper annular moulding of the column is of a semicir«ular section with a guilloche. 12. The capitals of both Greek and Roman Ionics have the eschinus, astragal, and fillet ; the eschinus is always cut into eggs, surrounded with borders of angular sections with tongues between them; the astragal consists of a row of beads, having two small ones inserted between every two large ones. In all the Roman buildings, except the Coliseum, these mouldings are cut in the same manner. 13. When Ionic columns stand in the flanks as well as the fronts of a building, two volutes at the corner of each angular column are contrived to present the same form in the flank as in the front, as in the temple of Bacchus at Teos, of Minerva at Priene, Erectheus, and that of the Muses at Athens, and likewise of Fortuna Virilis at Rome: the angular capitals have, in all these instances, one volute on each side, projected in a curve towards the angle. Amongst the ancient Romans, as at the temple of Concord at Rome, the capitals of all the columns are made to face the four sides of the abacus; and it was from this specimen that Scammozzi, encouraged by the example of Michael Angelo, composed the capital upon this principle, which bears his name. 14. The following examples will show the number of flutes and their form in the temples of the Ilyssus. Erectheus and Minerva Polias at Athens, Bacchus at Teos, Minerva Polias at Priene, and Apollo at Didymæus, near Miletus ;—each column has 24 flutes, and as many fillets. The columns of the aqueduct of Adrian, the Ionic colonnade near the Lantern of Demosthenes, and the great theatre at Laodicea, also at the temple of Concord, the second order of the Coliseum, and the theatre of Marcellus at Rome, have their shafts plain; but the columns of the temple of Fortune, at the same place, have 24 flutes. The section of the flutes of the columns, in the temple on the Ilyssus at Athens, is elliptical; the flutes descend and follow the curve of the scape of the column in thie following specimens, viz. the temple of Minerva Polias, and of Erectheus at Athens, the temple of Bacchus at Teos, and Minerva Polias at Priene. 15. Plate XCI. exhibits the Ionic order from the Temple on the Ilyssus at Atheus. Fig. 1, the order complete, with the dimensions of the different parts. Fig. 2, plan of one quarter of the capital. Fig. 3, the same showing the angular volute. Fig. 4, section through the centre of the front face of the capital. Fig. 5, section through centre of flank of column. Fig. 6, elevation of half of the flank of the capital. Fig. 7, elevation of flank of antæ. Fig. 8, painted ornaments of the upper fascia of the architrave of the Pronaos. Fig. 9, plan of one of the flutes of column. 174 [PART IIT. ORDERS OF ARCHITECTURE. Plate XCII. exhibits the Ionic order from the gateway at Eleusis: Entablature and capital of column. Plate XCIII. the same. Fig. 1, section through architrave, showing the flank of the capital. Fig. 2, section through the flank of the capital. Fig. 3, plan of the capital, showing likewise a plan of the flutes. Fig. 4, section through the centre of the part of the capital. Plate XCIV. Ionic order from the gateway at Eleusis. Fig. 1, half of the capital, showing one of the volutes to a larger scale. Fig. 2, the base of the column. Plate XCV. represents the method of striking the spiral of an Ionic volute. The spiral shown in this plate is denominated the logarithmic spiral. The way to describe this spiral—the different radii having been previously calculated and given, as in the table of figures by the side thereof—is as follows: Draw a straight line, and take any point therein for the centre of the eye, or cathetus: through this point draw another straight line at right angles thereto, and these two straight lines cutting each other will of course form four right angles; bisect any two adjacent right angles, and let the bisect. ing lines be produced on the other side of the centre, and the whole will be divided into eight equal angles by as many lines, upon which the radii are to be placed. From the centre of the eye set up 20, which will give the extremity of the first radius, upon the second line ; towards the right, from the same point set 18.3, and following round in succession 16.744, 15.321, &c., for the three revolu- tions of the spiral. In the first quadrant, take the length of the middle radius, viz. 18.3, set one foot of the compasses in 20, and describe an arc near to the centre, and then, with the same radius, set the foot of the compasses in 16.7 and describe an arc, cutting the former arc; then from the point of intersection, as a centre, describe an arc through all the three points 20, 18.3, 16.7: Pro- ceed in the same manner with each of the quadrants in rotation, by taking the middle radius, and from the extremity of each outer radius describe two arcs intersecting each other; and from the points of intersection as a centre with that radius, describe an arc through the two extremities till you arrive at 2.4: then with the radius 2.4 describe a circle for the eye, and the whole spiral will be completed. The dimensions in the table are calculated to three places of decimals, but, in order to avoid con fusion, only one decimal place is figured on the plate. Plate XCVI. shows a volute drawn according to the principles shown in the preceding examples, and consists of three spirals: the figures affixed round the outer spiral are supposed to be minutes, and consequently no other scale is required for that spiral than that of the order itself, but to draw the two interior spirals two new scales must be found, as shown at the bottom of the plate, unless the use of proportional compasses is adopted, which is perhaps the readiest way for those who are in the habit of using them. In Plate XCVII. Fig. 1 shows a volute in imitation of that of the temple of Erectheus at Athens, drawn according to the principles shown in the preceding examples. It consists of eleven spirals, which may likewise all be drawn from scales, as shown in the plate, or by proportional compasses as before noticed. Fig. 2 shows the section of the volute. Fig. 3 shows a method of forming a proportional scale for the interior spirals. 16. The general proportions of the Ionic order for practice, is as follows. Divide the whole height into twenty-one equal parts ; give four to the height of the entablature. Divide the height SECT. V.) 175 OF THE IONIC ORDER. of the entablature into three equal parts; make the cornice, frieze, and architrave, each one part: divide the height of the architrave into four equal parts ; give one to the mouldings of the upper part or capital: divide the capital of the architrave into nine equal parts; give one to the upper fillet, three to the cavetto, four to the ovolo, and one to the bead : divide the height of the frieze into six equal parts; and give the upper one to its capital: divide the height of the cornice into three equal parts; divide the upper part into six parts, give one to the upper fillet, four to the cima- recta, and one to the lower fillet, and turn one downwards for the ovolo under; divide the lower third of the cornice into six equal parts, and dispose of the parts as appears by the scale. The height of the base, including the plinth, is half the diameter ; the parts are proportioned in height as appears by the scale. The whole height of the capital is three-fourths of the upper diameter ; the height of the volute 7-12ths of the lower diameter. Dividing the height of the volute into three equal parts; the top of the lower one reaches to the bottom of the ovolo, the second division upon the top of the festoon: the smaller members will be found by subdivision. The juttings of the members are as follows: the cornice projects equal to its height; the projections of the intermediate members will appear sufficiently clear by the horizontal scales affixed to the Plate. The general projection of the base is one-sixth of the lower diameter of the column. SECTION VI. OF THE CORINTHIAN ORDER. By whom introduced, 1.— Has an Attic base, 2.— Shaft compared with the Ionic, 3.- Capital, 4.- General remarks, 5-8.- Notices of different examples of this Order, 9-25.— Notices of modern practice in this Order, 26.- General observations, 27._ Table of Proportions, 28.-- Projection of the capital, 29. [Plates XCVIII.-CV.] 2 1. UNLESS we admit the account given by Vitruvius respecting the invention of the capital by Callimachus, who is said to have been an Athenian sculptor, and a contemporary of Phidias about 540 B. C., there is no certain evidence with regard to the time when this order was established. Pausanias (book viii.) says, that in the fourth century before the Christian era, it was introduced by Scopas of Paros, in the upper range of columns in the ancient temple of Minerva at Tegæa ; but it has been alleged, that this temple was only begun by Scopas, and being left unfinished, had this upper range added, upon the lower ancient Doric, under Roman influence. There must cer- tainly have been some particular reason why this order was called Corinthian; but Doric remains only have been discovered on the site of that city by modern visiters. 2. In all the examples in Stuart's • Athens,' this order has an Attic base ; the upper fillet of the trochilus or scotia projects as far as the upper torus. In the monument of Lysicrates, the upper fillet of the base projects farther than the upper torus, which is an inverted ovolo. 3. Vitruvius observes, that the shaft has the same proportions as the Ionic, except the difference which arose from the greater height of the capital, it being a whole diameter, whereas the Ionic is only two-thirds of it. But this column, including the base and capital, has, by the moderns, been increased to ten diameters in height. If the entablature is enriched, the shaft should be fluted. The number of flutes and fillets is generally twenty-four; and the lower one-third of the height sometimes has cables, or reeds, or spirally twisted ribbands, inserted on them. 4. The great distinguishing feature of this order is its capital, which has for two thousand years been acknowledged the greatest ornament of this school of architecture. The height is one diameter of the column; to which the moderns have added one-sixth more. The body, or nucleus, is in the shape of a bell, basket, or vase, crowned with a quadrilateral abacus, with concave sides, each dia- gonal of which is equal to two diameters of the column. The lower part of the capital consists of two rows of leaves, eight in each row; one of the upper leaves fronting each side of the abacus. The height of each row is one-seventh, and that of the abacus one-eighth of the whole height of the capital. The space which remains between the upper leaves and the abacus is occupied by little stalks, or slender caulicoli, which spring from between every two leaves in the upper row, and volutes. The sides of the abacus are moulded, as in the Stoa, or portico, and arch of Adrian at Athens, and also the ruin at Salonica : the curves of the sides are continued until they meet in a sharp horn or point. In the Attic capital, the small divisions of the leaves were pointed in imitation CORINTHIAN ORDER, FROM THE MONVENT OF LYST CRATES AT ATHENS. PLATE XCVIII. . WS B 4 1278 ------- 2016 --- - ------ 1.1545 ---------- 1 Mod?g--- Fig 4. Vt I-------- ---- Spon to -------2. ----- --- ----- --.L. 10 - - 2 - ------1.1404.----- - V 28" - --..1..77'3 ----- -------- 1.83 ------------ Fig.1 from Centre of lolumn 2473 --- 23 ---- 2074 - - - - - - - - - - - met ----1.14) --------- ----1..22'g --------- SA -----. 1. 20- ska--- ---1.. 23% -----2.16 ---- --- . -----8261*POH I------- --...45** ----2.16'y ---- wae K - -- - ------ 1 ------SE1---- MS 4-os --- -- -- -------- 2 3 ------- ------- lodules - - - - - - - - - - - ----- ---- 1.257 ------ L- 1.15% J.--. 1..733 ------1.544 1.8 ²/3 ------ 2 ----I. 72 centre of column. -- -- - - - - - - < / CA from T- -------- 2 Mod, 21.-mama. -----.1.7318 Fig. 2. 1 ch A H - ------ VE 97 ------- Wie from Cintir or column. (1.3718 1.778 ----- 11,813 ---- 1.63/5 --- nord POST -- -- (Z. 24% EE-- - ----------292--- ------ -------------.1.13'*----- ----- - ----Z7A. --234--- 3571 - - ***1897---- -- ---- W.Kurser. De? WA Beerer Sculp! espin scale or feet A Fullarvon&Co Landon&Edinburgh CORINTHIAN ORDER. PLATE XC CAPITAL & BASE FROM THE PORTICO OF THE PANTHEON AT ROME. Measure & Dran at Pomezov Thomas leverta Dmaldian Buy Engrami by H. Lanty A Barton & Cº Indo Edimburg CORINTHIAN ORDER, PLATE C ENTABLATURE FROM THE PORTICO OF THE PANTHEON AT ROME. 68% 671 0000000000000000000000000ODOODDON 347 22 333 293 Measured & Drawn by Thomas Leverton Donaldson Esq?" Archt Enged by G. H Swanston Edan! A Fullarton & Cº London & Edinburgh CORINTHIAN ORDER, PLATE CI CAPITAL & BASE FROM THE INTERIOR OF THE PANTHEON AT ROME. 68 45% 144 los 413 tot 30 20 73 126% 75 130 32 30 353 25 237 352 194 183 133 (323 364 7355 (343 7383 Measured and Drawn at Rome, by Thomas Leverton Donaldson Esq Architect. Ened by G.H.Swanston Edin! A Fullarton & Co London & Edinburgh Mi Ꮯ ᎤᎡᏆᎢᎻᎥ ᎪᎢ Ꭴ Ꭱ Ꭰ Ꭼ Ꭱ . 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HO Peter Michodron - - - L - . -- - - - Fig.1.1:1. - - - - - - -- - - - - - - - - - - - - - - - ---- - - - - - -- - - - - .- - - . - - - - - ----- - - --- - - Fig. 1.N? 2. - - - --- - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - S . . . -- - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - more - - .. - --- - - . - - - - - -- - . - - - - - - - - -- - - - - - - - - - - -- - - - - ---- ----- - - - - - --- -- - - - - - -- - - --- -- - - -- - - - - - - - - - - .. -- -- - - -- - - - - - - - - - - - - - -- - --- - -- - - - -- - - - - - *** - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - -- - -- - - - - -- - - - - - - - -- - - - - . . . - - - . A FrillartonkCʻLondon. Edinburgh CAPITAL.. CORINTHIAN ORDER. } Fig. 2.Nº2. . .. . - -- . . www . . - - - . Fun.4.1! 2. Fig. 4. NOI. TI ESCO - - - - - Fiy. i3. rasas Fig.2.1.1. V . www PLATE 104 mm J. Moffit, ( ORIVTHIAY ORDER. PLATE 105. MODILLION Fig. 1. . SWN UR WWW 42 ZA . . VW W W Hill' WWW U MY W X 1.HU WHITE PI 11 ITA ti 11/12 WH Willid . . WE W - - - .-.--. .. . .. -- WWMWM - - - . . . . . 1111111 Allt - - .--- Fig. 2. .- -- - - - . 1111 1 Engrared by II. Louri. A Fullarion C London& Edinburgh Sect. VI.] 177 OF THE CORINTHIAN ORDER. 5. The best specimens of this order are the monument of Lysicrates, the Stoa, and arch of Adrian at Athens; and the pantheon of Agrippa, and the three columns of the Campo Vaccino at Rome. In the monument of Lysicrates, [See Plate XCVIII.,] the lower part of the capital consists of two rows of leaves; the lower row is plain, and the upper one ruffled; and the latter is nearly twice the height of the former. The number of leaves in the upper row is eight; in the lower, sixteen. In the upper row, the sides of the middle leaf, upon each front of the capital, are covered by the flank-leaves; and the whole of the leaves appear as if fastened to the body, or bell part, by a rose-headed pin on each side of the flank-leaves. Each of the helices, or stalks, proceeds from the sides of the middle leaf at the top of each row, as if they sprung from one common vertical stalk; then rising upwards, they take a direction towards the left, in a line of contrary curvature, and terminating in foliage. The volutes in the middle of the capital are quadruple, each one of each pair of one side, meeting each one of the other pair upon that side. And as each pair of volutes spring from the same trunk, and begin at the same horizontal position of the curve, each one of each pair is turned outward, and varies in size, the lesser being above, and the greater below. The four volutes form a curvilinear quadrilateral figure, by having their convex sides presented to each other. The upper or lesser pair support the honeysuckle which covers the middle part of the abacus. The corners of the abacus are cut off; and the hollow or lower member forms an inverted scotia, which is nearly four times the height of the crowning ovolo. 6. It may be observed generally, in the Greek Corinthian, that the volutes terminate in a point in the natural spiral, without either coiling round a circular eye, or bending backwards in a serpentine form, as in most of the Roman specimens. 7. The Corinthian order seems never to have been much employed in Greece before the time of the Roman conquest; but this powerful people employed it almost exclusively in every part of their extensive empire ; and it is accordingly in edifices constructed under their influence that the most perfect specimens are found. 8. It has been remarked by Vitruvius, that Corinthian columns were sometimes surmounted by a Doric entablature, which practice, besides that it is in itself very extraordinary, is not supported by any antique example now to be found. His observation respecting the Ionic entablature over the same kind of columns is verified in a number of instances. 9. The arch of Adrian at Athens, has a cornice with dentils, a plain frieze, an architrave with two plain fasciæ, and an Attic base. 10. A temple at Jackly, near Mylassa, has a cornice with dentils, a swelled frieze, an architrave with three plain fasciæ, and an Attic base. 11. At Salonica, (the ancient Thessalonica,) a building called the Incantada, has a cornice with dentils, a swelled frieze ornamented with flutings, an architrave with three plain fasciæ, and an Attic base. 12. The temple of Vesta, or Tivoli, has a plain cornice, with a dentil band uncut, an ornamented frieze, an architrave with two plain fasciæ, and an Attic base. 13. At Rome, the temple of Antoninus and Faustina has a plain cornice, with the dentil band uncut, an ornamented frieze, an architrave with two fasciæ divided by an astragal, and an Attic base. 14. The portico of Septimius Severus, in the same city, has a plain cornice, with a small uncut dentil band, a plain frieze, and an architrave with three fasciæ divided by mouldings. 15. In all these instances, the entablature and base are similar to those generally observed in the Ionic order, from which these Corinthian examples differ only in the form of their capitals. But in those which we are now about to cite, it will appear, that the Romans attempted to give the Corinthian order a more distinct character, by appropriating to it a peculiar entablature and 178 [Part IIL ORDERS OF ARCHITECTURE. 11 base, and thus making a complete order of what might be previously regarded as a composition, in which light Vitruvius seems to have considered it. 16. The portico of the Pantheon has a cornice with modillions, and an uncut dentil band, a plain frieze, an architrave with two fasciæ divided by mouldings, and a Corinthian base. See Plates XCIX. to CII. 17. The Temple of Peace, at Rome, has a cornice with modillions and dentils, a plain frieze, and an architrave with three fasciæ divided by mouldings. 18. In the Campo Vaccino, the three columns supposed by some to have belonged to a temple of Jupiter Stator, and by others to a temple dedicated to Julius Cæsar, have a cornice with modillions and dentils, a flat frieze, an architrave with three fasciæ divided by mouldings, and a Corinthian base. See Plate CIII. 19. The temple of Jupiter Tonans, at Rome, has a cornice with modillions and dentils, a flat frieze, and an architrave with three fasciæ divided by mouldings. 20. The arch of Constantine has a cornice with modillions and dentils, a plain frieze, an archi- trave with three plain fasciæ, and an Attic base. 21. At Ephesus, the temple supposed by Chandler to have been erected by permission of Augus- tus, in honour of his uncle Julius, has a cornice with modillions and dentils, a swelled and orna- mented frieze, an architrave with three fasciæ divided by mouldings, and an Attic base. 22. The exquisite and unique Maison Quarré, at Nismes, has a cornice with modillions and dentils, a flat frieze, an architrave with three fasciæ divided by mouldings, and an Attic base. 23. To these we may add the following, in which the alteration seems but partially to have taken place; there being neither dentils nor dentil bands in the cornices, and the mutules, from their situation, appearing rather like a variation from the proper Ionic dentil, than a new member. 24. A portico at Athens, supposed by Mr. Stuart to be the ancient Pæcilè: a cornice with mutules of two square faces, an architrave with two plain fasciæ, and an Attic base. 25. The frontispiece of Nero, at Rome: a cornice with mutules of two square faces, an ornamented frieze, and an architrave with two fasciæ, divided by an ogee. 26. Of the modern architects who have treated of this order, Palladio makes the column 9] diameters high, one-fifth of which he gives to the entablature, consisting of a cornice with modillions and dentils, a flat frieze, and an architrave with three fasciæ divided by astragals; the base is Attic. The design of Scamozzi bears a general resemblance to that of Palladio, but his column has ten diameters in its altitude ; his entablature is one-fifth of this height; the cornice has modil- lions, the architrave consists of three fasciæ divided by astragals, and the base is Attic. Serlio, following Vitruvius, has given this order an Ionic entablature, with dentils, and the same proportion of the capital; his column is nine diameters high, and has a Corinthian base. Vignola's Corinthian is a grand and beautiful composition, chiefly imitative of the three columns. He makes the column ten diameters and a half in height; the entablature is a fourth of that altitude ; the cornice has modillions and dentils; the frieze is plain ; the architrave is of three fasciæ divided by mouldings, - and the base is Attic. 27. Sir William Chambers has observed, that “the Corinthian order is proper for all buildings where elegance, gaiety, and magnificence are required. The ancients employed it in temples dedi- cated to Venus, Flora, Proserpine, and the nymphs of fountains ; because the flowers, foliage, and volutes, with which it is adorned, seemed well-adapted to the delicacy and elegance of such deities.” This theory, however plausible, is unsupported, or rather contradicted, by facts. The Romans—who appear, as already hinted, to have adopted the Corinthian order in preference to the others-employed it indiscriminately, and erected Corinthian temples to Jupiter, Mars, and Neptune, to whom the SECT. VI.] 179 OF THE CORINTHIAN ORDER. Greeks dedicated temples of the Doric order. The temples of Minerva at Athens and at Sunium are Doric ; that of Minerva Polias at Priene is Ionic. The temple of Jupiter Olympus at Elis was Doric; that at Athens, built by Adrian, is Corinthian. The numerous temples of the Grecian colonists in Sicily and Italy, are uniformly Doric, marked by the most severe and massive simplicity. The cities of Ionia present the best examples of a chaste and elegant Ionic: and the magnificent structures of Balbee and Palmyra are wholly of the Corinthian order, and in the most florid style of ornament. Whence we may conclude, that the choice of the orders of architecture was rather governed by national taste, than by any ideas of identity between the character of the style, and that of the object to which the building was to be devoted. 28. In the following Table will be found the proportions of some of the principal examples of the Corinthian order. It is to be recollected, that the several members are measured by the lower diameter of the shafts, which is divided into 60 parts. EXAMPLES. Cornice. Frieze. 39} 54 . 30 372 40% . 684 431 43] 6921 . 692 321 65% 5872 . Portico of the Pantheon, . . . Temple of Vesta at Rome, . . Temple of Vesta at Tivoli, . . Temple of Antoninus and Faustina, Three columns in the Campo Vaccino, The Basilica of Antoninus at Rome, The arch of Constantine, . . . Temple at Ephesus, . . . Temple at Jackly, near Mylassa, Pæcilè at Athens, . . .. Arch of Adrian at Athens, . . The Incantada at Salonica, .. Palladio, . . . . . . Scamozzi, . . . . . Serlio, . . . . . . Vignola, . . . . . Height of Column. Height of Capital. Architrave. , 9 344 67% 428 10 58 775 . 9 214 57 9 361 . 1066 664 434 . 10 15 63. 433 647 . 952 72 41 663 . 9 30 10 . 9 10 40 389 384 46 411 47] . 32 30 37 . 45 60 29. In Fig. 1. of Plate CIV. No. 1 shows the plan of the Corinthian capital; and No. 2, the elevation. No. 1 of Fig. 2, the semi-plan, is divided into eight equal parts, which, being carried up perpendicularly to the elevation, gives the centres of the leaves of which the projections are formed by those upon the plan. The length of the diagonal of the abacus is two diameters; the centre of each side is determined by the vertex of an equilateral triangle. The elevation shows the general outlines of the leaves before the foliage is cut. No. 2. Fig. 2 shows the general form and manner of raffling the leaves. Fig. 3 shows the front of the leaf according to the three columns in the Campo Vaccino. Fig. 4 is a modillion; No. 1 being the side, and No. 2 the profile. In Plate CV. Fig. 1 shows the side elevation of the modillion in the temple of Jupiter Stator, with a section of the coffer and flower therein ; and Fig. 2 is a plan of the modillions, showing the soffit. SECTION VII. ROMAN ORDERS. General observations, 1.— The Tuscan order, 2.- Composite, 3. [Plates CVI.-CVIII.] 1. BESIDES the three Greek orders now described, two others were introduced in ancient Italy, viz. the Tuscan and Composite. In their general character they are governed by the canons of the Greek school, and have indeed very little claim to be separately classed : our notice of them will therefore be proportionally confined. 2. The title of the first leads us to assign its origin to Tuscany; and this conjecture is strengthened by that people being admitted as the offspring of Dorians. No ancient remains of this order having been discovered with entablatures, it is only from the accounts given by Vitruvius that the form and ratio of its members can be determined. He allows seven diameters for the height of the columns, and diminishes the upper part one-fourth half of the diameter; the base is half a diameter in height, one of which is given to a circular plinth, and the other to a torus ; the capital is also half a dia- meter in height, and one in breadth upon the abacus; the height is divided into three parts, one of which is given to the abacus, one to the echinus, and the third to the hypotrachelion and apophygis; the architrave has two faces, with an aperture between them of about 1} inch for the admission of air to preserve the beams; the lower face is vertical upon the edge of the top of the column; the frieze is plain and flat; the mutules of the cornice project over the beams and walls, equal to one- fourth of the height of the column. The columns of Trajan and Antonine are specimens of the Tuscan, though being eight diameters high, they exceed, by one diameter, what is generally assigned to this order. St. Paul's church, Covent Garden, in London, is the best modern example. In Plate CVI. we have given an example from Sir William Chambers. 3. In the Composite order the upper part of the capital is that sort of Ionic which presents a similar face on each of the four sides. The lower part consists of two rows of acanthus leaves, as in the Attic Corinthian. The column of the Roman edifices, with composite capitals, have, in general, Corinthian entablatures; the arches of Septimius Severus, and the Goldsmiths at Rome, have Ionic entablatures. See Plate CVII. Modern architects have generally adopted the entablature of the frontispiece of Nero, Plate CVIII., or introduced adventitious members of other orders, as the denticulated band of the Ionic with its cymatium, between the modillions and cymatium of the frieze of the Corinthian. The modillions employed in the Composite order differ from those of the Corinthian in being more massive, composed of two faces, and having a cymatium like an architrave. Indeed the capital being much bolder than the Corinthian, all the other members should be made of suitable magnitude. The Romans employed this order chiefly in triumphal arches. The moderns have introduced it in various sorts of works of the greatest magnificence. ᎢᏓs CAN 0ᎡᎠᎬᎡ , PLATE, 106. ENTABLATURE BASE AVD CAPITAL. T 35 I 1667 BS 1372 -38 36 3. | 22 % 225 132 Z/ 31? _ 125 20 | 22 25 17* | 22% w 30 30 1*2* 30 WA2 tingraral by Catrmstrong. A Fullar tou& Co London (Edinburgh mic COMPOSITE ORDER PLATE CTU CAPITAL & BASE FROM THE ARCH OF TITUS AT ROME. 263 1373 2.5 23 222 20 Engraved by H. Lowry. A Fazlarton c'Londonk Edinburgh NI COMPOSITE ORDER. PLATE 108 ENTABLATURE FROY TH FRONTLSPIECE OF NERO AT ROME. 2 KUWAVAAMONUMUNNMOWOWOWOWOWOWOWOWMOM . F ZE . th - - -- - - - - - - - - - - - - - - -- - -- - - - - - - - - - . - - - - . re L th . f 17 A1 PLC . A A INI . 2 ) w --------..-..---- 40% 38% UAULUWUMONOLUNURULUMULUULURULUAUNUMUOAUNUIUIUIULUIQUION 1369 3675 7 675 ---- ----- MOQUIUOQQUOODUMOULOOQOOQUOMODONMUODOITAA 479 V ------ 13 nic 117 D45% 4474 SURAUAY ZAN 2 YALVANI AVY *** 1378 376 Des. Yunu. Yenke W e pers Y OLY-50 461 46k Y-01-10tY2 e lse og te istutathie (1 18282 WXWARINOV 23% Liz V . SA PS . . will . - . TES - - VI. - W 2 . 0 . MUST 1 * . WA . ' VAN RE 1 V . de .. ) Vet . OR 22 CA YOR . he St. . uur . DU CO 1 . . w NE we SS 13: . AIRY Il TA KW AL S . . Y 0. 1 . RA VC AVOM PI W _ . .. A E IN 11: 1 T . . > ! 1 31 DINOMOMONOMOVOJNOJOKONUMUNUMONONO37% 365 DNI SISISISISISISISI 3300 33% 2834 11371 1.26% - . Fingrared by Ir. Louvr*. A FLISDÉC10udont ninjurgh SECTION VIII. GENERAL REMARKS. A sixth order, 1.- General remarks on columns, 2, 3; as engaged or insulated, 4.- Intercolumniations, 5-8. Laws of superposition of orders, 9-15. Of the pedestal, 16–18._Of Pilasters, 19, 20. 1. VARIOUS unsuccessful attempts have been made in different countries to invent a new, or sixth order; but such is the limited extent of the human imagination, that the Doric, Ionic, and Corin- thian, have ever floated uppermost, and all that has been produced amounts to nothing more than different arrangements and combinations of their several parts, with some trifling variation, scarcely deserving notice, the whole generally tending rather to diminish than increase the beauties of the ancient orders. Indeed those three orders possess the entire means whereby the art renders sensible the various grades of character or expression ; and though that character and expression may be considerably varied, and something novel produced, it will be found upon close examination to result only from a combination of the means contained in the three primitive orders, and not from invention.* 2. The shaft of the column of each of the orders is diminished towards the top, but the diminution varies in different examples, and is greater in the Doric than in the other orders, being in the former one-fourth or one-fifth of the lower diameter, whilst in the Ionic and Corinthian orders it seldom exceeds one-sixth; it is however in all cases regulated, in some degree, by the height of the column. In some examples the shaft is regularly diminished from bottom to top, being a frustum of a cone : in others, the surface of the shaft is curved, and it then assumes a conoidal shape: this curvature or swelling of the shaft is termed the entasis, and is produced by two different methods : in the one case the curved line is throughout the whole height, and falls within the circumference of the inferior diameter of the column ; in the other, the shaft is somewhat increased at about one-third or one- fourth of the distance between the base and necking of the column; the former practice is most consonant with the best examples of antiquity. 3. The shaft of the column is oftentimes fluted, but the flutings vary both in form and number in different examples. In the Doric order, the section of the flute is a segment of a circle, the flutes meeting each other in an arris, and with few exceptions are twenty in number: those of the Ionic and Corinthian orders have a semicircular, or semi-elliptical section, and are usually twenty-two in number, with a fillet between each flute, which is diminished in the same proportion as the flute itself. The termination of the flutes are most commonly of a spherical or spheroidal form; but there are some exceptions, amongst which may be instanced those of the order of the Sybil's Temple at Tivoli, and that of the Choragic monument of Lysicrates at Athens. Sometimes the lower parts of * Thus Peter de la Roche, in an 'Essay on the Orders of Architecture,' published in 1768, proposed the introduction of "a new Great order,” called the Britannic order, the capital of which was to be ornamented with mouldings of eight ostrich feathers, so that three composing a Prince of Wales's plume, should appear in whatever direction it was viewed. The height of the shaft was to be 26. modules; of the column, 31 modules; of the pedestal, 10; and of the entablature, 6. The capital, whether on columns or pilasters, was to bave an angular member. The column was to have numerous shallow flutings; and the bottom of the pannel to be ornamented with the Englisk rose. 182 [Part III. ORDERS OF ARCHITECTURE. moulding, having , insulated. Whos then said to be cabled S DIAM.CT: C the flutes, to the he ylit of about one-third of the shaft, are filled up with a moulding, having a convex section representing a rope or staff; the shaft is then said to be cabled. 4. Columns are either engaged or insulated. When insulated they are placed either very near the walls, or at a considerable distance from them. When the columns are engaged, or very near to the walls of a building, the intercolumniations are not limited, but depend on the windows, arches, niches, or other objects that are placed between them ; but columns that are entirely detached, and alone perform the office of supporting the entablature, as in peristyles, porches, galleries, &c., must be so placed as to possess both real and apparent solidity. 5. The ancients had several different manners of spacing their columns, which are described by Vitruvius in his third and fourth books, and were thus named ; pycnostyle, of which the interval was equal to one and a half diameter of the column; the systyle interval of two diameters; the custyle of two and a quarter diameters; the diastyle of three diameters; and the areostyle of four diameters. These intercolumniations were applied particularly to the Ionic and Corinthian orders ; the Tuscan and Doric being regulated in other ways. 6. In the Doric order the intercolumniations were regulated by the triglyphs, one of which was always placed over the centre of each column, having one, two, or more triglyphs between the columns, as circumstances might render necessary: the Tuscan intervals are described as being very wide, some exceeding seven diameters ; the architrave being of wood, made this arrangement practicable. SADARM, etc. 7. The eustyle intercolumniation being a medium between the narrow and wide intervals, and being at the same time spacious and solid, has been more used, both by the ancients and moderns, than any of the others. 8. There is likewise another species of intercolumniation occasionally used where the columns are 772 Dipinong grouped or coupled, in which case two columns are placed within half a diameter of each other, and the interval between them and the next pair is equal to three and a half diameters. This disposition of columns has been employed by Sir C. Wren in the western front of St. Paul's Cathedral; by Perrault in the facade of the Louvre ; and is likewise very commonly used in the porches and por- ticoes of modern buildings. 9. When two or more orders are placed upon each other in a building, the laws of solidity require that the strongest should be placed lowermost, wherefore the Doric should support the Ionic, the Ionic the Composite, or the Corinthian, and the Composite the Corinthian. 10. This rule is not however always strictly attended to; most authors having placed the Com- posite above the Corinthian, and we find it so disposed in many modern buildings. 11. There are likewise instances where the same order as is used below is repeated above ; there are others where an intermediate order is omitted, and the Ionic placed on the Tuscan, or the Corinthian on the Doric, but none of these practices are correct. The first of them is an evident trespass against the principles of solidity, and should never be imitated; the second occasions a tiresome monotony : and the last cannot be effected without several disagreeable irregularities. 12. In placing columns above each other, it is always to be observed, that the axes of all the columns should correspond, and be in the same perpendicular line, more especially where the columns are detached, as the structure will then be solid, which it cannot be when the superior column is placed considerably within the inferior one, as in that case a great part of it can have no other sup- port than the entablature of the order below it. 13. Many and some considerable difficulties occur in placing columns upon columns, and indeed the practice should, as much as possible, be avoided. The most simple rule, and that which is on SECT. VIII.) 183 GENERAL REMARKS. the whole attended with the fewest difficulties, is that given by Scamozzi, founded on a passage in the first book of Vitruvius, importing that the lower diameter of the superior column should con- stantly be equal to the upper diameter of the inferior one, as if all the columns were formed of ono long tapering tree, cut into several pieces. Many architects-amongst which number are Palladio and Scamozzi-place the second order of columns upon a pedestal. 14. In compositions consisting of two stories of arcades this cannot be avoided ; but in colonnades it may and ought, for the addition of the pedestal renders the upper ordonnance too predominant, and the projection of the base of the pedestal is both disagreeable to the eye, and much too heavy a load for the inferior entablature. In the Barberine palace at Vicenza, Palladio has placed the columns of the second story on a plinth only, and this disposition is the best, the height of the plinth being regulated by the point of view, and made sufficient to expose to sight the whole of the base of the column. 15. Other difficulties likewise occur in the application of orders above orders, arising out of the entablature of the inferior one ; many architects holding it to be improper to employ in such situa- tions those parts thereof as have a reference to the roof of the building, such as the upper member of the cornice, the dentils, mutules, &c. 16. Some writers on architecture have considered the pedestal as a necessary part of an order, but seeing that in the particular description given by Vitruvius of the Doric, Corinthian, and Tuscan orders, no notice is taken of the pedestal; and that, in speaking of the Ionic order, he only mentions it as a necessary part in the construction of a temple, without intimating that it belongs to the order; it may perhaps be proper to consider the pedestal as a separate body, having no more con- nection with an order than an attic or basement, or any other part with which it may occasionally be accompanied 17. The pedestal, like the column, or entablature, is divided into three parts, the base, the die, and the cornice. The die is almost always of the same figure, being either a cube, or parallelopiped ; but the cornice and base are varied, and adorned with more or fewer mouldings, plain, or carved, according to the simplicity or richness of the order with which it is employed. There are no fixed proportions for pedestals, but their heights are regulated according to the nature of the design in which they are employed; the width is, however, always the same as the lower torus of the base of the column. 18. Pedestals are sometimes useful in order to lessen the height and diameter of the column, and consequently the proportions of the other part of the composition ; but they should never be used without absolute necessity, particularly when the columns are detached from the building, as they always give the order a ricketty appearance : indeed a column raised on a pedestal has been justly compared to a man mounted on stilts. Occasionally, in the interior of churches, courts of justice, and other buildings, where a large number of persons assemble, and it is desirable that the whole order should be viewed at once, they may be used with good effect. 19. Pilasters differ from columns in the plan only, which is square, whilst that of the column is circular. Their bases, capitals, and entablatures, have the same parts, with the same heights and projections as those of columns, and have sometimes the same diminution and flutings. They are most commonly attached to walls, having a projection of one-sixth or one-fourth of their diameters, but they are sometimes insulated. Pilasters frequently occur in the remains of Roman works; the Greeks, on the contrary, more generally used antæ, having one face, the width of the column they accompanied; whilst on the flank they were extremely narrow, and the capitals and bases were totally dissimilar to those of the columns. 20. Some writers on architecture-amongst whom may be more especially mentioned the Abbé 184 [PAвт ІІ. ORDERS OF ARCHITECTURE. Laugier-have inveighed most strongly against the use of pilasters on any occasion; contending that they are nothing better than bad representations of columns; that their angles indicate the formal stiffness of art, and are a striking deviation from the simplicity of nature; that they are not suscep- tible of diminution, one of the most pleasing features of the column; and that their use is altogether unnecessary, inasmuch as columns will at all times answer the purpose. Notwithstanding all these objections, pilasters have been, and doubtless will continue to be, much used; they are to be found in the remains of Grecian and Roman works, both associated with columns and otherwise ; were much used by the great restorers of architecture in the fifteenth century, likewise by modern architects of all nations; and their application may certainly be vindicated on every principle of reason and good taste, especially where economy is requisite ; for although it must be acknowledged that their effect is inferior to that of columns, yet they are often of the greatest use, especially at the angles of buildings; even an entire front, decorated with pilasters, if designed with taste and judgment, may be made to produce a most striking and beautiful effect. TO SECTION IX. A GLOSSARY OF TERMS PECULIAR TO THE ORDERS OF ARCHITECTURE, OR HAVING IMMEDIATE RELATION THERETO. A. Abacus, the upper member of the capital of a column. In the Grecian Doric order it is a massive square plain tablet which serves as a crowning piece. In the Ionic order, the edges of the abacus are moulded, either with an ovolo, or ogee moulding, and in some examples enriched. In the Corinthian order, each face of the abacus is of a concave form and moulded, the angular points being cut off, except in some few instances; in some examples, the faces are enriched with carving, in others they are plain. The word abaciscus is sometimes used as synonymous with abacus. Acroterium, the extremity or vertex of any thing. It usually signifies a pedestal or base, placed on the angle or on the apex of a pediment, for the purpose of supporting a vase or statue. Adit, or aditus, the approach or entrance to a building. Anchor, an ornament in the form of an anchor, or arrow-head, often employed in the echinus or ovolo, between the borders which surround the eggs. It is sometimes called a tongue. Mesa D e 7. " Angular Capital, a term generally applied to the modern Lonic Capital, invented by Scamozzi, having the four faces alike. It is likewise used to designate the capitals of a Grecian edifice, which have two fronts alike, placed at the angles of porticoes, in order that the capitals of the angular columns may correspond with those of the columns in the flanks as well as the front of the building. Angular Modillions, the modillions which are placed at the return of a cornice, in the diagonal vertical plane passing through the angle or mitre of the cornice. They frequently occur in the remains of the buildings at Balbec and Palmyra, and at Spalatro. Annular Mouldings are such as have vertical sides, and horizontal circular sections. Annulets, the annular fillets between the hypotrachelion and echinus of a Doric capital. In the Roman Doric order, they are usually three in number, and of equal size, with rectangular sections. In the Grecian examples, they vary in number from three to five; the sinking between each two follows the line of the echinus, and the outer sides of the fillets form a curve parallel to that of the sinking; the upper side of each is perpendicular to the curve, and the lower side is concave towards the space between each two; the concavity begins in a direction perpendicular to the curve of the moulding; the futings of the shaft terminate under the lowest annulet. Ante, the terminations of the projecting flank-walls of a Grecian temple which receive or support the architrave. The antæ have slight projections on each side of the wall. The face next the column being usually the mean width between the superior and inferior diameter of the column, the breadth of the face of the antæ on the flanks is always much narrower than the other faces. The antæ have capitals and bases totally dissimilar to those of the column, formed of horizontal mouldings, more or less enriched according to the order they accompany. It is a rule in the use of antæ, that no moulding used on them shall exceed their projection. Apophyge, a concave quadrantal moulding, joining two vertical members of different projections, and forming an exte- rior angle with that which has the greatest projection, and a tangent with the other. A concave quadrantal moulding is used in the Ionic and Corinthian orders to join the bottom of the shaft to the base, likewise to connect the top of the shaft with the fillet under the astragal. It is sometimes called the scape or spring of the column. Arcostyle, a species of intercolumniation in which the distance between the columns is four diameters. It is chiefly used in the Tuscan order. Architrave, the division of the entablature which rests upon the column, and which represents the lintelling beam placed over the columns, and the outer columns for supporting the cross beams. In the Grecian remains, the architraves are always much higher than in the Roman examples: the soffites of the architraves in Grecian exam. ples always exceed the upper diameter of the column, but in the Roman they are equal thereto. Architrave Cornice. See Cornice. 2 A 186 [PART IIL ORDERS OF ARCHITECTURE. Astragal, a moulding having usually a semicircular section projecting from a vertical diameter. It is frequently applied not only at the upper end of the shafts of columns, but likewise in their bases and entablatures; and is sometimes plain and sometimes ornamented, being cut into beads formed alternately of oblate and prolate spheroids; in other instances, into figures consisting of double cones with cylindrical parts between them. Attic Base, is that base of a column which consists of an upper and lower torus, a scotia, and fillets between them. It is much used in modern, as it likewise was in ancient works. Attic Order, a term used by some writers on architecture to denote the style of building frequently employed in the decoration of an attic, in which antæ or small pilasters are used. •B. Baguette, a small astragal moulding, sometimes carved and enriched with beads, ribands, laurel, &c. When the baguette is enriched it is called a chaplet, and when ornamented a bead. Baluster, the lateral side of an Ionic capital, contained between the front and rear volute; also the small column used in a balustrade. Band, a narrow flat surface, having its face in a vertical plane, as the band of the Doric architrave, and the dentil band, which is the square out of which the dentils are cut. The word fascia or plat-band is usually applied to broad members, and band to narrow ones wider than fillets. Band is likewise the cincture round the shaft of a rusticated column. Base, that part of a column between the shaft and the pedestal. Cable, a moulding of a convex circular section, representing a rope or staff, laid in the fute of a column. It is always shorter than the fute, and placed at the lower end of it; the cabling is usually about one-third of the height of the shaft. This species of ornament is of rare occurrence in classical works. Canal of the Ionic volute, the spiral channel, or sinking on the face, which begins at the eye in a point, and expands in width until the whole number of revolutions are completed. The term canal is also used sometimes for flute. Canal of the Earnier, the channel recessed upwards on the soffite, for preventing the rain water from reaching the bed or lower part of the cornice. Cant Moulding, a bevelled surface, or one that is neither perpendicular to the horizon, nor to the vertical surface of the body of the building. These mouldings are of very remote antiquity; a cant moulding is applied instead of the echinus to the capitals of the columns of the portico of King Philip, and in many other examples of Grecian and Roman antiquity. Canted Columns, a column wherein the horizontal sections are polygons, consisting of straight sides instead of concave sides or Autes. Examples may be seen in the columns of the portico of King Philip, and those of the temple at Cora; but they are rarely met with. Capital of a Column, the assemblage of mouldings or ornaments above the shaft of a column on which the entablature rests; in other words, the head or uppermost member of the column. The capital is one of the principal features by which the order is distinguished. Capital, Angular. See Angular Capital. Capital of a Triglyph, the projecting band which surmounts the plain vertical area or face. In the Grecian examples the capital of the triglyph projects but a very little, and is not returned on the flanks, except at the angular tri- glyphs, and this only upon each face of the building; but in the Roman Doric examples, the capitals of the triglyphs project more than in the Grecian, and have the same projections on the flanks as on the face. Caryatic Order, an order of architecture, the entablature of which is supported by female figures instead of columns; the figures themselves are called Caryatida, Caryates, or Caryans. The origin of this order, as related by Vitruvius, was as follows: “ Carya, a city of Peloponnesus, joined the Persians in their war against the Greeks; these, in return for the treachery, after baving freed themselves by a most glorious victory from the intended Persian yoke, unanimously resolved to levy war against the Caryans. Carya was in consequence taken and destroyed, its male population extinguished and its matrons carried into slavery. That these circumstances might be better remem- bered, and the nature of the triumph perpetuated, the victors represented them draped, and apparently suffering under the burthen with which they were loaded to expiate the crime of their native city. Thus in their edifices did the ancient architects, by the use of their statues, hand down to posterity a memorial of the crime of the Caryans." There is a beautiful example of this order in the temple of Pandrosus at Athens. Sect. IX.] 187 GLOSSARY OF TERMS. Caulicoli, or caulicola, in the Corinthian capital, the staiks between the upper row of leaves, ramifying upwards, each into two foliated branches, and seeming to support the volutes under the abacus, each branch supporting one of the sixteen volutes, or helices, two of which are placed at each angle, and two in the middle of each face of the abacus. Cavetto, a concave moulding or cove, the curvature of the section of which does not usually exceed the quarter of a circle. It is principally used in cornices. Chapiter. See Capital. Chaplet, a small carved ornamented fillet. Coffer, a recessed pannel, of a square or polygonal figure, anciently used in level soffites, and in the intradoses of cylindrical vaults. Coffers are often employed in the soffites of the cornice of the Corinthian and Composite orders, between the modillions. The term is often applied to any sunk pannel in a ceiling. Column, in a general sense, is a vertical support of a body, diminishing upwards in its horizontal dimensions. Besides the several species of columns already described under the respective orders, there are several others known to architects. Carolytic columns have foliated shafts decorated with leaves and branches winding spirally round them, or disposed in the forms of crowns and festoons. They were used by the ancients to support statues. Twisted columns, making several circumvolutions in the height of the shaft, after the manner of a screw, and some- times having several threads or screws following one another in the same circumference, are sometimes used. These are likewise called spiral columns. Niched columns, are such as are placed in a niche, with the axis of the column in the plane of the wall. The ancients had likewise their agricultural, astronomical, boundary, chronological, funereal, gnomonic, historical, indicative, itinerary, lactary, legal, manubiary, menian, military, milliary, phosphorical, rostral, statuary, symbolical, or zoophoric columns. Conge. See Apophyge. Cornice of an order, the secondary member of the order itself, or a primary member of the entablature. The entablature being divided into three principal parts, the upper one is the cornice. Cornices are of several kinds, viz. :- An architrave cornice rests upon the architrave, the frieze being omitted. An example of this species of cornice may be seen in the temple of Pandrosus at Athens, over the Caryatic order; cornices of this description are adapted to situations where a regular entablature would be out of proportion to the body which it crowns. A mutule cornice is appropriate to the Doric order, the mutules having inclined soffites. A dentil cornice has a denticulated band and is usually used with the Ionic order, sometimes with the Corinthian. A modillion cornice, is one in which modillions are employed to support the corona instead of mutules: it is generally used with the Corinthian order. A block cornice, is that where plain rectangular prisms with level soffites are placed to support the corona. Corona, one of the principal members of a cornice, having a broad vertical face and a bold projection. The solid out of which it is formed is generally recessed upwards from its soffite: it is often called the drip from the circumstance of its discharging the rain water in drops from its edge, and by this means sheltering the subordinate parts below: it is likewise sometimes termed the larmier. Coupled Columns, such as are disposed in pairs, so that the capitals and bases almost touch, there being but a space equal to half a diameter between the shafts. Cyma-recta, a moulding or member of a cornice, the profile of which is waved, being concave at top and convex at bottom. Cyma-reversa, a moulding formed of two members, being convex at top and concave at the bottom, consequently the reverse of a cyma-recta. It is commonly called the ogee moulding. Cymatium, the uppermost moulding of a cornice, the section of which is usually a curve of contrary flexure. Dado, that part of the pedestal of a column which is between the base and cornice. Decastyle, a species of temple having ten columns in front. Dentils, similar and equal solids disposed in a row at equal intervals in a cornice, each presenting four sides of a rectangular prism; the side parallel to the vertical face, and the one parallel to the soffite, being attached to the vertical and horizontal planes of an internal right angle. The whole series of dentils in the same range is called the denticulated band. The proportions for dentils, according to Vitruvius, are that the dentil band is to be equal 188 [Part III. ORDERS OF ARCHITECTURE. in height to the middle fascia of the architrave, and the projection to be the same as the height; the width of the dentil to be one-half its height, and the intervals between them to be two-thirds of the width of the dentil. Dentils are most commonly used in the Ionic and Corinthian orders; the Doric order of the theatre of Marcellus at Rome likewise has dentils in the cornice. There are some ancient examples where the dentil band is introduced into the cornice, but the dentils are not cut. Diameter of a Column, the thickness of the lowest part of the shaft of a column. The proportions of the several parts of an order are regulated by a measure formed by the lower diameter of the column called a module, and usually Diastyle, that method of spacing columns in which the intercolumniation is equal to three diameters of the column. Die of a Pedestal, that part contained between the base and cornice. Diglyph, a tablet, or projecting face, with two indentations or channels placed in the frieze of a cornice. Diminution of Columns, the continued contraction of the diameter from the base to the top of the shaft. Sometimes columns diminish gradually from the bottom to the top; in other examples, the diminution commences from one- third of the distance from the bottom. Ditriglyph, an arrangement presenting two triglyphs over the intercolumn, that is, if in two adjoining columns a tri- glyph be placed with its middle over each, the ditriglyph will contain three metopes or spaces, two half triglyphs, and two whole triglyphs. Dodecastyle, a species of temple having twelve columns in front. Drip. See Corona. Drops, small pendant cylinders, or the frustums of cones attached to a vertical surface; the axis of the cylinders or cones having also a vertical position, and their upper ends attached to a horizontal surface. Drops are used in the cornice of the Doric order under the mutules, and in the architrave under the triglyphs: each mutule bas three rows from front to rear, with six drops in each row, disposed at equal distances in lines parallel to the front; the drops on the architrave are likewise six in number under each triglyph, disposed at equal distances from each other. In the Grecian examples the drops are almost always of a cylindrical form: in the Roman examples they are frustums of cones, as they are likewise in some of the temples at Pæstum. Echarpe, the small moulding which marks the middle and forms the band of the balustre in the volute of an Ionic capital. Echinus, a convex moulding much employed in cornices, and in the Doric and Ionic capitals, and usually having orna- ments carved therein representing eggs. In the Roman examples it is most frequently a quarter of a circle, and is called ovolo. Ecphora, the projection of any moulding before the face of the one below it. Eggs, ornaments in the form of oblong spberoids, having their greater axis inclined, projecting at the top and receding at the bottom, but each axis in a plane perpendicular to the surface of the ovolo. The eggs are generally truncated, and have their upper part cut off by a plane parallel to the horizon. Each truncated spberoid is surrounded by a border of an elliptic figure, in close contact, showing somewhat more than a semi-ellipsis, the shorter axis being horizontal. In the space between each egg is a tongue, or as it is sometimes termed anchor, the front edge of which comes in contact with the surface of the original moulding. Entablature, the whole of that part of an order which is supported by the column. It consists of three parts, the architrave or beam which rests upon the capitals of the column, the frieze formed by a row of cross beams which are supposed to support the ceiling, and the cornice the various members of which represent the timbers of the roof. Entasis, the enlargement or swelling of the shaft of a column, found in almost all Grecian examples. Epistyle, or Epistylium, the architrave. Eustyle, the disposition of columns in which the intervals are exactly two diameters and a quarter. This intercolum- niation has been much used both by the ancients and moderns Eve, of a volute, the circle at the centre, from the circumference of which the spiral commences : called likewise the cathetus. Eyebrow, a name sometimes given to the fillet. F. Fascia, a vertical member in the combination of mouldings, having a very small projection, but considerable breadth, such as the bands of an architrave. Sect. IX.] 189 GLOSSARY OF TERMS. Fillet, a small member consisting of two planes at right angles, used to separate two larger mouldings, or to form a cap or crowning to a moulding. The intervals or small bands between the flutes of a column or other body are termed fillets Flutes, or Flutings, prismatic cavities depressed within the surface of a column, or any piece of architecture, at regular distances, and generally having a circular or elliptical section, either meeting each other in an arris, or meeting the surface in an arris, and leaving a portion of the surface between either two cavities of an equal breadth, or diminishing in regular progression. Frieze, or Frize, the middle principal member of the entablature which separates the cornice from the architrave. Friezes are of various forms, usually flat, but sometimes convex, and in a few instances formed of two curves of a contrary flexure. Friezes are often ornamented with foliage, sculpture, and symbolical devices of various descrip- tions, according to the nature and destination of the structure. Fusarole, a semicircular member cut into beads, generally placed under the echinus of the lonic and Composite capitals. Fust, the shaft of a column: the part comprehended between the base and capital. G. Gorge, a concave moulding less recessed than a scotia. This word is sometimes used for the cyma-recta: likewise the space between the astragal, at the top of the shaft of a Doric column, and the annulets. Grouped Columns, or Pilasters. Columns are said to be grouped when two or more are connected together, or placed upon the same pedestals. Gula, or Guele, the same as cyma-reversa or ogee. Gutte. See Drops. Helices, little scrolls in the Corinthian capital, called also urilla. Hem, the protuberant part of the Ionic capital formed by the spiral fillets. Hunitriglyph, the half triglyph. Hypæthral, a building or temple without a roof. Hypotrachelion, that part of a Roman Doric capital comprebended between the astragal at the top of the shaft, and the fillet or annulets under the ovolo. Impost, the capital of a pier, or pilaster, which receives an arch Inserted Column, a column let into the wall. An example of this disposition of columns may be seen in the Vatican, designed by M. Angelo. Insulated Column, a column that stands quite clear of a wall. Intercolumniation, the distance between columns, measured by their lower diameters. Interdertil, the space between the dentils. It is sometimes two-thirds, sometimes nearly three-fourths of the breadth of the dentil. Intermodillion, the space between two modillions. Isagon, a figure with equal angles. Larmier. See Corona. Lozenge, a quadrilateral figure of four equal sides, with oblique angles. M. Meros, the middle part of a triglyph. Metoche, a term used by Vitruvius to signify the space or interval between the dentils of the Ionic, or triglyphs of the Doric order. Perhaps the word should be metatome. Metope, the interval between the triglyphs in the Doric frieze. It may be either plain or decorated. In some instances it appears to have been left quite open. 190 [PART III ORDERS OF ARCHITECTURE.' Minute, an equal portion of a module, sometimes one-sixtieth part, but usually only one-thirtieth part. Modillion, unutules carved into consoles, placed under the soffite or bottom of the corona in the Corinthian and Com- posite orders at equal distances, giving support to the larmier and corona. There are many different examples of modillions amongst the remains of antiquity Module, a certain measure taken at pleasure for regulating the proportions of columns, and other parts of a building. Architects usually choose the diameter or semi-diameter of the bottom of the column for their module, and this they subdivide into parts called minutes. Vignola divides his module, which is a semi-diameter, into twelve parts for the Tuscan and Doric orders, and into eighteen for the other orders. The module of Palladio, Scamozzi, Freart de Chambray, Desgodetz, and Le Clerc, which is likewise a semi-diameter, is divided into thirty parts or minutes in all the orders. Some divide the whole height of the column into twenty parts for the Doric, twenty- two and a half for the Ionic, and twenty-five for the Roman, and one of those parts they make a module. Mono-triglyph, having only one triglyph between two columns. Mouldings. See SECTION II. Mutules, the projecting blocks in a Doric cornice which appear to support the corona and superior members of the cornice, the soffite being an inclined plane and representing the ends of the rafters in the original modern structures. N. Neck of a Capital, the space between the channelures and the annulets of the Grecian Doric capital. In the Roman Doric it is the space between the astragal and the annulet: some examples of the Ionic order likewise have neck- ings, though most of the antique examples are without them. Niche, a cavity or hollow in the thickness of a wall. 0. Obelisk, a pillar of a rectangular form, gradually diminishing towards the summit, and terminating in a pyramidior, or small flat pyramid. Ogee, a compound moulding, the upper part being convex the lower part concave. Ovolo, a moulding the profile of which is a quarter of a circle ; called likewise the quarter-round P. Patera, an ornament frequently introduced in friezes, fascias, and imposts, over which are hung festoons of husks and flowers. Pateras are sometimes much enriched with foliage, and have a mask or head in the centre. Paternosters, ornaments in the form of beads, either round or oval, used on baguettes, astragals, &c. Peristyle, a continued range of columns of a straight or circular form. Persian order, an order in which the representations of men are introduced supporting the entablature, in the same manner as the figures of women are in the Caryatic order. This order was first used by the Athenians on the occasion of a victory that Pausanias obtained over the Persians, representations of whom were introduced with their hands bound and other characteristic marks of slavery supporting a heavy entablature. Pilaster, a square column sometimes insulated, but more frequently let into a wall, and only projecting one-fourth or one-fifth part of its thickness. The pilaster borrows its name from the order it accompanies, or whose character and proportions it resembles. Pillor, a kind of irregular column round and insulate, but deviating from the just proportions of a column. Plafond, the bottom of the projecture of the larmier of the cornice, called also the soffite. Platband, any square flat moulding the height of which much exceeds its projection; such are the faces of an archi. trave and the platbands of the modillions of a cornice. Plinth, a square piece under the base of a column. Profile, the contour or outline of a figure, such as a collection of mouldings in a cornice or architrave, &c. Pucnostyle, the smallest intercoluniniation mentioned by Vitruvius, being an arrangement of columns wherein the space between them does not exceed one diameter and a balf. Quarter-round, a moulding approaching to a quadrant, such as the ovolo or echinus. SECT. IX] 191 GLOSSARY OF TERMS, R. Rustic order, a species of building decorated with square blocks, the faces of which are hatched or roughly picked with a hammer. Scotia, a recessed moulding of an elliptical or circular section, placed between the upper and lower torus in the bases of columns. It is called likewise trochilus. Scroll. See Volute. Shaft of a Column, that part of a column which may be denominated the frustum of a conoid, between the base and the capital Sima. See Cyma. Sima-recta. See Cyma-recta. Sima.reversa. See Cyma-reversa. Soffite, the underside of mouldings which are projected upon a horizontal plane, such as the corona. Systyle, the intercolumniation in which the columns are placed at two diameters apart. TUA T. Tænia, a small square fillet at the top of the architrave in the Doric capital. Taloir, a term sometimes used for the abacus. Talon, the French name for the ogee. Terminus, a trunk or pedestal adorned at the top with a figure of the head of a man, woman, or satyr, whose body seems to be enclosed in the trunk as in a sheath which usually tapers downwards. Torus, à moulding having usually a semicircular or semi-elliptical section, used in the bases of columns. Triglyphs, the tablets in the Doric frieze champhered on the two vertical edges, and having two channels in the middle, called glyphs. Trochilus. See Scotia. V. Volute, one of the principal ornaments in the Ionic capital, composed of two or more spirals of the same species, having one common eye and centre, variously channelled and hollowed out in the form of mouldings. Zocle, or Socle, a low square member serving to support a column instead of a pedestal. Zophorus, the frieze, so called by the ancients, because that part of the order was usually decorated with the repre- sentation of animals. Vart Fourth. ELEMENTS AND PRACTICE OF GEOMETRY. 2 B *** It has been found advisable to alter the arrangement of the different Parts of the present edition of 'The Builder's and Workman's Director,' as indicated in the Prospectus ; and to make the Historical Notices of Domestic Architecture the Seventh Part of the Work. S PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. SECTION I. ELEMENTS OF GEOMETRY. GEOMETRY is the science of quantity, form, and dimension, and has for its object the development of those principles by which solids, surfaces, lines, and angles may be measured and compared. Practical geometry teaches the method of constructing figures, and describing angles and lines, according to certain conditions, which are mostly assumed with reference to practical application. Axioms are propositions of which the truth at once appears. Demonstration is an argument founded upon axioms, or other facts already known, in order to ascertain the truth of some assertion, or the correctness of some operation proposed to be performed. When any operation is proposed to be performed, the proposition is called a problem ; but when merely the truth of some assertion is proposed to be demonstrated, the proposition is called a theorem. Postulates are petitions respecting the possibility of certain operations that may evidently be per- formed. A reductio ad absurdum is a proposition of which the truth can only be elicited by indirect means, with the assistance of axioms only; and this method consists in showing that every other supposition besides that proposed is absurd. A lemma is the demonstration of some premise, in order to render the thing proposed more easy to be comprehended. A scholium is a remark or observation made upon something going before. A corollary is a consequent truth gained by a definition, or some preceding truth in a demon. stration. An hypothesis is a supposition or assumption made, either in the enunciation of a proposition, or in the course of a demonstration. . SIGNIFICATION OF SIGNS. The sign = denotes that the quantities between which it is placed are equal; and is therefore read “ equal to,” or “ is equal to." > The - denotes that the quantity which precedes it is greater than that which follows it; and is therefore read “greater than," or " is greater than.' < The denotes that the quantity which precedes it is less than that which follows it; and is read "less than," or " is less than." The sign + denotes that the quantity which follows it is to be added to that which goes before it. T 196 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. The sign - denotes that the quantity which follows it is to be subtracted from that which is placed before it. The sign i signifies an angle; and is therefore read “angle.” The sign a signifies a triangle; and is therefore read " triangle." AXIOMS. 1. Things which are equal to the same thing are equal to one another. 2. The whole is equal to the sum of all its parts. 3. If to equal things equal things be added, the wholes will be equal. 4. If from equal things equal things be taken away, the remainders will be equal. 5. If to unequal things equal things be added, the sums will be unequal. 6. If from unequal things equal things be taken away, the remainders will be unequal. TITOMY POSTULATES. 1. A straight line may be drawn from one given point to another. 2. A straight line may be prolonged or continued at pleasure. 3. From any centre and with any given radius a circle may be described. 4. A straight line may be drawn perpendicular to another from any assigned point in that other. 5. A straight line may be made equal to another. 6. A straight line may be made equal to any part of another, or any part of a straight line may be cut off. 7. A rectilineal angle may be made equal to any part of another, or any part of a given angle may be found. DEFINITIONS. 1. A solid is that which fills a certain portion of space, so as to prevent every other solid from occupying that space. 2. A surface, or superficies, is the boundary of a solid; or the surface of a solid is that part of it that may be seen and felt. 3. A point is that which indicates a certain position on a surface, or in a solid, but is no portion of such surface or solid. * 4. If a pencil or other pointed instrument be drawn upon a surface, the trace it leaves behind upon that surface is called a line. Fig. 1. Coroll. Hence a line has continuity, but if divided into two parts, each extremity thus cut is a point. • The smallest mark that can be made so as to be seen is in practice called a point, and is usually formed with a pen or pencil, or in a drawing with one extremity of a compass. Sect. I.] 197 ELEMENTS OF GEOMETRY. 5. Straight lines are such as being applied to each other in every direction, touch, or coincide throughout their whole length. Coroll. Hence two straight lines cannot enclose a space.* 6. If a straight line touch a surface throughout the whole length of the line in every direction in which it can be applied, such surface is called a plane surface, or simply a plane. 7. The distance between two points is the straight line reaching from one of the points to the other. 8. One line AB is said to meet another CD when the extremity B of the one falls on the other line CD. Fig. 2. 9. One line is said to cut another, when either of these two lines divides the other into two parts. 10. An intersection is the cross formed by two lines cutting each other. Fig. 3, No. 1. Fig. 3, No. 2. Fig. 3, No. 3. Once for all, it is only necessary to state, that, when lines are mentioned without any other word to restrain their signification than the simple word line, a straight lino is implied; and thus all lines in the definitions or construction of diagrams are understood to be straight lines. When lines of any other kind are meant, a word of distinction will always be introduced. 11. Two straight lines are said to be parallel when there is always the same distance between them, and if produced ever so far at either end would never meet. Fig. 4. 12. When any two lines of any number are parallel, the whole number are called parallels, and any one of them is called a parallel. Fig. 5. * Thus suppose two very thin boards are set together in contact face to face, and their edges shot; then let the edge of the one board be brought into contact with the edge of the other board; if no light appears between the two edges, each edge is straight, and is therefore called a straight edge. Another method of proving a straight edge upon the same principle, is as follows: shoot the edge of a board as before, lay its face upon the planed face of another board, and draw a line along the edge thus shot upon the planed face of the board under it. Mark two points in the line so drawn, and upon the edge of the board over the line make marks at the same two points. Lay the upper side of the upper board upon the planed face of the under board, and upon the other side of the line, and bring the two marks or points in the planed edge to the two points in the line; then, if the edge coincide with the line, the line is straight; but if it does not, the operation of shooting the edge must be repeated. The term straight edge is, however, applied by workmen to any board which has one straight edge, and which is used for the purpose of drawing straight lines; since the edge cannot exist without the board, which must be used as a handle in order to apply its straight edge in drawing. 198 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. 13. If any given number of straight lines, when produced, all meet in one point, they are called converging lines, or centrolineals. Fig. 6. - -- 14. An angle is the space contained between two straight lines which meet each other, but not in the same straight line; and the point where the two straight lines meet is called the point of the angle, or point of meeting, or point of concourse, or simply concourse. Fig. 7, No. 1. Fig. 7, No. 2. 15. Each of the two straight lines which form an angle is called a leg, and sometimes a side. 16. One angle A is greater than another B, if when A be applied to B, so that the point of the angle A may be upon that of B, and one of the legs of A upon one of the legs of B, the remaining leg of B falls without the space contained by the legs of A. Fig. 8, No. 1. Fig. 8, No. 2. B 17. One angle A is less than another angle B, if when A is applied to B, so that the angular point of A may be upon that of B, and one leg of A upon one leg of B, the remaining leg of A falls within the space contained by the legs of B.* Fig. 9, No. 1. Fig. 9, No. 2. B B ....... 18. When the two angles formed by a straight line crossing one extremity of another straight line are equal, each of the equal angles is called a right angle.t Fig. 10. * The magnitude of an angle is not dependent on the length of its legs (as young beginners are apt to suppose), but upon the opening. † This and the preceding definition are the criterion to prove the truth of the instrument called a square, which Carpenters and Joiners use for making the faces of their stuff at right angles with each other. The workman applies the inner edge of the stock of the square to the straight edge of a board, so that the under side of the blade may be coincident with the surface of that board; and by one of the edges of the bladc of the square, with a sharp-pointed Sect. I.) 199 ELEMENTS OF GEOMETRY. 19. In two right angles formed by a straight line crossing one extremity of another, the line which divides the other is called a perpendicular to the divided line. Fig. 11. A-- B Thus the straight line CD is called the perpendicular; and the other straight line AB is some- times called the base. 20. In two right angles formed by a straight line crossing one extremity of another, the dividing line is said to stand at right angles to the divided line. So that when it is said that such a line is at right angles to such another line, it is equivalent to saying, that line is perpendicular to the other line. 21. An obtuse angle is that which is greater than a right angle. Fig. 12. - - - - 22. An acute angle is that which is less than a right angle. Fig. 13. 23. A figure, or plane figure, is that which is enclosed by one or more lines. 24. The lines which enclose a figure are called the sides of that figure. The side of a figure next to the bottom of the paper is generally called the base, by way of dis- tinction from the other sides; though occasionally any side may be considered as the base. In this treatise, every side of a figure is considered to be a straight line, unless otherwise specified. In every subsequent definition, a figure is supposed to be drawn upon a plane surface, according to the conditions specified in the definition; or, if the figure limit the extension of the surface, it is supposed to be such as will coincide with a plane. 25. Equilateral or equal-sided figures are such as are enclosed by equal straight lines. instrument, draws a line on that surface: he then turns the square, so that the other side of the blade may be coincident with the surface of the board, and the inner edge of the stock of the square coincident with the edge of the board, and the same edge of the blade before used, upon some part or point of the line thus drawn; then, if this edge of the blade of the square coincide entirely with the line, the said square is adjusted, and the inner edge of the stock and that edge of the blade used will form a right angle; but if the edge of the blade cross the line, the square is not true, and requires adjustment. When the two faces of a piece of stuff form a right angle, the one face is said to be square to the other; and when every two adjoining faces are made square, the stuff is said to be squared. 200 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. 26. Equiangular figures are those which have the angles contained by their sides equal to each other. 27. Opposite sides of a figure are such two sides as have the same number of intermediate sides connecting each of their extremities. 28. Opposite angles of any figure are any two angles that have an equal number of sides on each side of the straight line joining their angular points, or supposed to join them. 29. A side is said to be opposite to an angle, when the figure has an equal number of sides joining each extremity of that side and the angular point. 30. A figure bounded by three sides is called a triangle. 31. A triangle which has no two equal sides is called a scalene triangle. Fig. 14. 32. A triangle which has only two equal sides is called an isosceles triangle. Fig. 15. 33. The side of an isosceles triangle, which is unequal to either of the other two, is called the base. 34. A triangle which has all three of its sides equal is called an equilateral triangle. Fig. 13 1 35. A triangle which has a right angle is called a right-angled triangle. Fig. 17. 36. A triangle which has an obtuse angle is called an obtuse-angled triangle. Fig. 18. 37. A triangle which has all its angles acute is called an acute angled triangle. Fig. 19. Sect. I.] 201 ELEMENTS OF GEOMETRY. 38. A figure enclosed or bounded by four sides, is called a quadrilateral or quadrangle. Fig. 20, No. 1. Fig. 20, No. 2. Fig. 20, No. 3. Fig. 20, No. 4. 39. A quadrilateral, which has each pair of its opposite sides converging, is called a trapezium. Fig. 21. 40. A quadrilateral, which has only one pair of its opposite sides parallel, is called a trapezoid. Fig. 22. 41. A parallelogram is a quadrilateral which has each pair of its opposite sides parallel to each other. Fig. 23, No. 1. Fig. 23, No. 2. 42. A rectangle is a parallelogram which has one right angle.* Fig. 24, No. 1. Fig. 24, No. 2. U 43. A rectangle, which has the two sides containing the right angle unequal, is called an oblong. Fig. 25. 14. A rectangle, which has two of its adjoining sides containing the right angle equal, is called a square. Fig. 26. 45. The length of an oblong is the distance between the extremities of the greatest side of the two that contain the right angle. * It may, however, be demonstrated, that if a parallelogram have one right angle, it will also have each of its other three angles a right angle. 2 C 202 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. · 46. The breadth of an oblong is the distance between the extremities of the least side of the two that contain the right angle. 47. A polygon is a figure enclosed by more than four sides. Fig. 27, No. 1. Fig. 27, No. 2. Fig. 27, No. 3. 48. A regular polygon is a figure which is both equilateral and equiangular, or that which has all its sides equal, and all its angles equal. Fig. 28, No. 1. Fig. 28, No. 2. Fig. 28, No. 3. Fig. 28, No. 4. 49. A pentagon 18 a polygon of five sides; a hexagon, a polygon of six sides; a heptagon, a polygon of seven sides; an octagon, a polygon of eight sides; and so on.* : 50. If a point A be assumed in a plane, and if the end of a straight line of any given length be fixed in that point A, and if, when the other end B is carried entirely round, a pencil or point be held at that end B, so as to trace a line on the plane, the figure thus described by the moving point is called a circle. Fig. 29. 51. The fixed point A, is called the centre of the circle. 52. The line described by the moving point B, is called the circumference or periphery. 53. Any straight line AB, drawn from the centre of a circle to its circumference, is called a radius. Coroll. Hence all the radiit of the same circle are equal. 54. If, in a given circle, a straight line AB be drawn so as to be terminated at each end A and B by the circumference, the line thus drawn is called a chord of that circle. Fig. 30. * These names are composed from the Greek numerals eis, duo, treis, teseres, pente, her, hepta, octo, enned, which signify one, two, three, four, five, six, seven, eight, and nine. Again, deca, icosi, triaconta, teseraconta, penteconta, hex- aconta, hepdomeconta, ogdoeconta, enneneconta, and hecaton, signify ten, twenty, thirty, forty, fifty, sixty, seventu, eightu. minetu, and a hundred, &c. Therefore, compounding the word which expresses the number of tens with that which expresses the number of units, we shall have the sum of both, or the particular number we desire; thus icosi eis signifies twenty-one. The other word which forms the last part of the name is derived from another Greek word gonia, wbich signifies an angle. + The term radii is the plural of radius, and is Latin. It is used in order to avoid the disagreeable pronunciation of the word radiuses, formed according to English construction. SECT. I.] 203 ELEMENTS OF GEOMETRY. 55. A chord AB, which passes through the centre C of the circle, is called the diameter. Fig. 31. Coroll. Hence the diameter of a circle is equal to twice its radius. 56. The arc of a circle is any portion AB of the circumference. Fig. 32. . Thus, if the arc of a circle be half its circumference, it is called a sernicircular arc; or, if it be one quarter, it is called a quadrantal arc. 57. A straight line AB, joining the extremities of an arc, is called the chord of that arc. Fig. 33. NO 58. A straight line AB drawn from the middle of a chord, perpendicular to that chord, and meet- ing the arc, is called the versed sine of that arc. Fig. 34. 59. A segment of a circle is a portion of a circle bounded by a chord, and the arc of the circle intercepted by the chord. Fig. 35, No. l. Fig. 35, No. 2. 60. A semicircle is a segment terminated by the diameter and the semi-circumference cut off by the diameter. Fig. 36. 204 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. 61. A quadrant of a circle is that portion of a whole circle which is contained by two radii crossing each other at right angles, and the part of the circumference comprehended between the extremities of the radii. Fig. 37. 62. A sector of a circle is the figure ABC comprehended between two radii CA, CB, and the portion AB of the circumference cut off by these radii. Fig. 38, No. 1. Fig. 38, No. 2. COA Coroll. Hence a quadrant is the sector of a circle comprehended by two radii at right angles with each other, and the arc between these radii. 63. A straight line AB is said to touch a circle when the line is produced on each side of the point where the line meets the circle, without cutting the circle. Fig. 39. LB 64. The point where a straight line touches a circle is called the point of contact. 65. A straight line AB, which touches the circumference or any arc of a circle, is called a tangent. • Fig. 40, No. 1. Fig. 40, No. 2. B - 66. A straight line AC, drawn from any point A without a circle to cut the circumference and to meet it in another point, is called a secant. Fig. 41. 67. One circumference of a circle is said to touch another when both circumferences meet each other in one point only. Sect. I. ] 205 ELEMENTS OF GEOMETRY. 205 Fig. 42, No. 1. Fig. 42, No. 2. 68. A diagonal is a straight line drawn entirely within a figure, from one angle to another. 69. One angle B is to another E, as the magnitude of the arc ac, described between the legs of the one angle, is to that of df, described between the logs of the other, with the same radius. Fig. 43, No. I. Fig. 43, No. 2. BL EL Thus, if the arc df be twice the arc ac, then the angle E is twice the angle B; or, if the greater arc be divided into equal parts, then, whatever number the lesser arc contains of the parts of the greater arc, the lesser angle will contain the like parts of the greater: this will be evident by draw- ing lines from each point of concourse to every division of each arc; then each angle will contain as many smaller angles, as the arc between the legs of that angle contains of the smaller arcs between the points of division. Thus, if the arc df be divided into five equal parts, and if the arc ac contain three of these parts, the angle E will be divided into five equal angles, and the angle B three equal angles: the angle B will, therefore, be three-fifths of the angle E. 70. If, from a given point in any line, a given distance be set off upon that line by a compass, the other extremity thus found by the extent of the compass, is called the point of distance, or point of extension. 71. To rectify or develop the arc of a circle is to find a straight line equal in length to that arc, supposing the arc to be extended or stretched out into a straight line. 72. A straight line equal to the arc of a circle when stretched out into a straight line, is called the rectification or development or the length of that arc. 73. The distance between a point and a straight line, is the perpendicular drawn from that point to the straight line. 74. Alternate angles are those made on the contrary sides and at each end of a line meeting two others. 75. Figures which have their angles respectively equal to one another are said to be equiangular. 76. Similar polygons are those which may be divided into as many equiangular triangles as the figure has sides. 77. The altitude of a figure is a straight line drawn perpendicularly upon its base, or upon the pro- longation of its base. 1 In order to obtain the results with greatest despatch, the symbolical method has been adopted rather than that of writing the whole in words, as the demonstration is at once brought under the notice of the eye in the form of equations ; and by a single glance, every part may be instantly compared with another, and the consequences immediately drawn by taking away the common TI 206 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. terms, in the expressions which stand in opposition to each other, and rejecting or dividing out com- mon factors. By tliese means conciseness and perspicuity are gained at the same time, without either relaxing from the strictness or deviating from the elegance of the method pursued by Euclid; and, indeed, the various equations, though symbolically expressed, may be read in the very same words as if the whole had been written ; for it must be remembered, that though a symbol only occupies the space of a single letter, it conveys the meaning of one or more words. THEOREM I. If two triangles have two sides and the included angle of the one respectively equal to two sides and the in- cluded angle of the other, then shall the third side of the one be equal to that of the other, the one tri- angle equal to the other, and the remaining angles of the one equal to those of the other, each to each, which subtend the equial sides. Let the two sides BA, AC of the a ABC be respectively equal to the two sides ED, DF of the triangle DEF, and the L A = D; then shall BC = EF, and the A ABC = DEF, the B = E, and the BL the point A may coincide with the point D, and the straight line AB with DE, because the A is equal D, the straight line AC will coin- cide with DF; and because AB coincides with DE, and AC with DF, and since AB is = DE, and AC = DF, the point B will coincide with E, and C with F; therefore the base BC coincides with EF, the A ABC with DEF, the L B with E, and the LC with F. THEOREM II. If two angles of a triangle be equal to two angles of another triangle, and the side of the one equal to the side of the other between the equal angles, each to each, the two triangles shall be equal, and the corre- sponding sides and angles shall be equal. : Let the two triangles be ABC, DEF, having the L A = D, the B = E, and the side AB = DE ; the A ABC will be = the A DEF, the LC = F, the side AC to DF, and BC to EF. For if the point A be applied upon D, so that the straight line AB may coincide with DE, the point B will coincide with E, and, the an. gles at A and B being respectively equal to the angles at D and E, the straight line AC will coincide with DF, and BC with EF; therefore the point C will coincide with F; and because the three points A, B, C coincide with the three points D, E, F, the A ABC will coincide with the A DEF, the side AC with DF, BC with EF, and the L C with F. Coroll. Hence, if two triangles be equiangular, and a side of the one be equal to a corresponding side of the other, the two triangles will be equal. Sect. I.) 207 ELEMENTS OF GEOMETRY. THEOREM III. The angles at the base of an isosceles triangle are equal to each other. Let ABC be an isosceles a having the side AB = AC, the L ABC is = the 6 ACB. For suppose the . A to be bisected ; then the side AB being = AC, and AD common, the two sides BA, AD will be = the sides CA, AD; and the BAD being = CAD, the A BAD will be = CAD (Th. i.); and the 4 B will be = C. Blo THEOREM IV. . If two angles of a triangle be equal, the sides which subtend the equal angles shall also be equal. Let the A ABC have the < B = the < C, the side AC shall also be = the side AB. For if AB be not = AC, one of the two sides AB, AC is greater than the other; let AB be — AC ; upon A B set off BD = AC, and join DC. Then, in the As BDC, BAC, as BD is = AC, the two sides AC, CB are respec- tively equal to the two sides DB, BC, and the < ACB = the 2 DBC; therefore (Th. i.) the a ACB is = the A DBC, the greater to the less, which is impossible. THEOREM V. The two angles made by a line touching one of the extremities of another, are together equal to two right angles. . Let the line AB touch the extremity C of the line CD. Then if the A ACD be = the BCD, each of them will be a right angle ; but if not, let CE make right Ls with AB; then, the LACE + ECB = ACE + ECD + DCB = two right Ls; and since - ACD = ACE + ECD, therefore (Ax. 1) the 6 ACD + DCB = to right Ls. AL THEOREM VI. The opposite angles formed by two straight lines cutting each other are equal. Let the straight line AC cut the straight line BD in E; the L AEB is = DEC, and the 2 AED to BEC. For the L AED + AEB is = two right Ls, and the L AEB + BEC is = two right Ls; therefore . AED + AEB is = AEB + BEC: from each side of this equation take away the common - AEB, and there will remain the . AED = BEC. 208 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. THEOREM VII. If one side of a triangle be produced, the exterior angle shall be greater than either of the two interior opposite angles. Let the side AB of the A ABC be produced to D; then DBC is E either of the two LS BAC, BCA. For suppose BC to be bisected in E: join AE, and produce the line to F. Make EF = EA, and join BF. Then, because EC is = EB, and EF = EA, and the 4 BEF = AEC, the 4 ECA = EBF. But the 4 EBD is EBF; therefore also the - EBD is = ECA. In the same manner, if CB be prolonged to G, it may be demonstrated that the exterior . GBA is E BAC; but the 2 GBA is = DBC (Th. vi.); therefore the 2 DBC E BAC. THEOREM VIII. The greatest side of every triangle is opposite the greatest angle. Let AB be the greatest side of the A ABC; then C is the greatest L. In AB take AD = AC, and join DC; then (Th. iii.) the 6 ADC = ACD; but ADC is E B; therefore the LACD is B(Th. vii.); therefore the whole - ACB is E B. In the same manner it may be shown that ACB is also É A. I A4 THEOREM IX. The greatest angle of every triangle is opposite to the greatest side. Let BAC be the greatest L of the A ABC ; then shall BC be the great- est side. For if BC is not = BA, it must either be equal to it or less; but BC can- not be = BA, for then the L A would be = C (Th. iii.), but it is not; neither can BC be * BA, for then the A would be $,C (Th. viii.), which is contrary to the hypothe. sis: then, since the side BC is neither = BA nor > BA, it must necessarily be greater. THEOREM X. The sum of any two sides of a triangle is greater than the tnird side. Let ABC be a a; then BA + AC is + BC. For produce BA to D, making AD = AC, and join DC. Then, because AD is = AC by construction, the L ADC = ACD; but the BCD is E ACD; therefore also the - BCD is É ADC: and since the greatest side of every A is opposite to the greatest < (Th. viii.), BD is Sc É BC; but BD is the sum of the sides BA, AC; wherefore the sum of the two sides BA, AC, is BC. SECT. I. 209 ELEMENTS OF GEOMETRY.. THEOREM XI. The alternate angles made by a straight line meeting two parallel lines are equal. Let EF meet the parallels AB, CD; then the AEF = EFD, and the L BEF = EFC. For if the angles AEF and EFD are not equal, one of them must be greater than the other: let EFD be AEF, and make the . EFB = AEF. Then the exterior . AEF of the A EFB will be the interior - EFB (Th, vii.); but by construction it is also equal to it, which is impossible; therefore the .AEF cannot be $ EFD. In the same manner it may be proved that it cannot be = EFD; therefore as the L AEF can neither be less nor greater than the EFD, the L AEF must be = EFD. THEOREM XII. Two straight lines passing through the extremities of another, and making the alternate angles equal, are parallel. : The two straight lines AB, CD passing through the two extremities E, F of the line EF, and making the L AEF = EFD are parallel. A E B C-ILD AB. Then, because AB is parallel to FG, the L AEF = the alternate - EFG (Th. xi.); but by hypothesis the . AEF = EFD; therefore (Ax. 1) the . EFD = EFG the less the greater, which is impossible. THEOREM XIII. If a line pass through one of two parallels, and meet the extremity of the other, the external angle is equal to the internal angle at the point of concourse. Let the line ED pass through B, and meet CD which is parallel to AB, in D; then the EBA = EDC. For produce AB to F. Then (Th. xi.), because AF and CD are parallel, the alternate Ls CDB and DBF are equal; but (Th. vi.) the angle DBF = EBA or ABE being opposite angles, therefore the 4 ABE = CDB (Ax. 1). 2 D - 210 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. THEOREM XIV. wol If a straight line passing through one line, and meeting a second line, make either angle at the intersec- tion equal to the angle on the same side at the point of meeting, then the two straight lines thus intersected and met are parallel. Let the straight line EF cutting AB in G, and meeting CD in F, make the L AGE = CFG; then AB is parallel to CD. Now, since (Th. v.) AC . AGE + AGF = two right Ls { L CFG + DFG = two right LS, E AGE + AGF = CFG + DFG; but by hypo- thesis, the L AGE = CFG; therefore AGF = DFG; wherefore (Th. xii.) AB is parallel to CD THEOREM XV. Two straight lines meeting the extremities of another, and making the angles on the saine side of that other line equal to two right angles, are parallel. Let the two straight lines AB, CD meet the two extremities E, F A-LB of the line EF, making the L AFE + CEF = two right Ls; then · AB is parallel to CD. C- For by hypothesis, the L AFE + CEF = two right Ls, and by (Th. v.)- - the 6 AFE + BFE = two right Ls; therefore (Ax. 1) - the L AFE + CEF = AFE + BFE ; then, taking away the common AFE, there will remain the L CEF = BFE. But these are alternate Ls; therefore AB is parallel to CD (Th. xii.). THEOREM XVI. The angles on the same side of a straight line meeting two parallels are equal to two right angles. A E B Let the straight line EF meet the two parallels, AB, CD in the two points E, F; the LAEF + EFC = two right Ls. For (Th. v.) - the LCFE + EFD = two right LS: but (Th. xi.). --- the L AEF = EFD; therefore - ... LCFE + AEF = two right Li. THEOREM XVII. : Straight lines which are parallel to the same straight line, are parallel to one another. Let the two straight lines AB, EF be each parallel to CD; then A- AB' and EF are parallel to each other. Draw GH, meeting AB in G and EF in H, and cutting CD in I. Then, because AB is parallel to CD, the LAGI = GID (Th, xi.); and because EF is parallel to CD, the 2 GID=GHF; therefore the LAGI or AGH = GHF, and therefore (Th. xii.) AB is parallel to CD. Sect. I.) 2013 ELEMENTS OF GEOMETRY. THEOREM XVIII. If one side of a triangle be produced, the external angle is equal to the sum of the two internal angles which are not contiguous or adjacent to the external angle. Let ABC be a triangle, and let BC be produced to D; then the exterior L ACD= CAB + ABC. Let CE be drawn parallel to AB. Then, since AB is parallel to CE, and AC joins them, the 4 ECA = B CAB (Th. xi.); and since BD passes through the two extremities B, C of the parallel AB, EC, the 4 ECD= ABC (Th. xiii.); therefore, by adding these two equations together, the ECA + ECD= CAB + ABC ; but ECA + ECD= ACD; therefore the < ACD = CAB + ABC (Ax. 1). THEOREM XIX. The sum of the three angles of every triangle is equal to two right angles. Let ABC be any a. Produce one of the sides, as BC to D; then the 2 ACD= CAB + ABC (Th. xviii.). To each side of this equation add the remaining L ACB; then the 2 ACD + ACB = CAB + ABC + BCA: but the 2 ACD + ACB = two right Ls (Th. v.); therefore the 2 CAB + ABC + BCA = two right L s. · THEOREM XX. The sum of all the angles inade within any straight lined figure, by its sides, is equal to twice as many right angles, wanting four, as the number of the sides of the figure. Let the figure be ABCDE. Take P any point within the figure, and join PA, PB, PC, PD, PE; then for every side of the figure there is a A ; and since the sum of the three angles of every A is equal to two right angles, therefore the internal angles of the figure, together with four right 2s at the centre, are equal to twice as many right angles as the figure has sides; and therefore the internal angles formed by the sides of the figure are equal to twice as many right angles, wanting four, as the figure has sides. Coroll. Hence the sum of the internal angles of a quadrilateral figure is equal to four right angles ; for, deducting the four right angles at the centre from eight, which is twice the number of sides, there remain four for the internal Ls of the figure. · 212 ГРАВт IV. ELEMENTS AND PRACTICE OF GEOMETRY. THEOREM XXI. The sum of the external angles of any rectilineal figure is equal to four right angles. . For when the sides are prolonged, for every side of the figure the external A, together with the internal , make two right Ls; therefore the sum of the external and internal angles make twice as many right angles as the figure has sides; but the internal angles, together with the angles at the centre, also make twice as many right Ls as the figure has sides: from each of these equal sums take away the internal Ls, and there will remain the external Ls equal to the angles at the centre; that is, equal to four right 2 s. THEOREM XXII. The opposite sides and also the opposite angles of a parallelogram are equal, and the diagonal divides the figure into two identical triangles. Let ABCD be a parallelogram, the LA = C, and the L B = D, and likewise the side AB = DC, BC = AD. Join AC; and since AB is parallel to DC, the - ABC + BCD = two right Ls (Th. xvi.); and D4 because BC is parallel to AD, the - BCD + CDA = two right Ls (Th. xvi.); therefore - ABC + BCD = BCD + CDA (Ax. 5): take away the common – BCD, there will remain the ABC = CDA. In the same manner the 4 BAD may be proved to be equal to BCD. With respect to the sides, because AC joins the two parallels AB, DC, the alternate Ls BAC, ACD are equal (Th. xi.), and for the same reason the alternate Ls CAD, ACB are equal; there- fore in the two AS ACD, ACB, the side AC is common, and the L ACD= CAB, and the - CAD = ACB; and therefore AB = CD (Th. ii.), and AD = BC. THEOREM XXIII. The diameter of a circle passing through the point of contact of a tangent, is perpendicular to that tangent. Let EDF be a circle, C its centre, GH a tangent to the point D; CD, the semidiameter, is perpendicular to GH. For if CD is not perpendicular to GH, let CG be perpendicular to GH: then CGD will be a right-angled A ; and consequently the - CDG will be acute; therefore since, by hypothesis, the < CDG E CDG, the side CD Ġ H É CG. But CD = CF; wherefore CF = CG, a part greater than the whole, which is absurd; therefore CD is perpendicular to GH. SECT. I.] 213 ELEMENTS OF GEOMETRY. THEOREM XXIV. The angle in a semicircle is a right angle. iii.) Let DEF be a semicircle, the < DEF is a right L. For draw the radius EC; then the - ....LCDE = CED> and the .....CFE = CEF therefore + CDE + CFE = CED + CEF; but the į CED + CEF = DEF therefore + CDE + CFE = DEF (Ax. 1). Now, since the three angles of every A are = two right Ls, the 4 DEF is the half of two right angles ; that is, DEF is a right angle. THEOREM XXV. The angle at the centre of a circle is double the angle at the circumference. Let DEF be a circle, the ECF = twice EDF. For Case 1. Where one of the lines containing the < at the circumference passes through the centre, the ECF = CDF + CFD (Th. xviii.); but CDF = CFD (Th. iii.); therefore 6 ECF = 2.CDF or 2.EDF. Case 2. Where a line DG drawn from the point D of the < at the circum- ference, and through the centre C, lies between the lines containing both the AGB and since .... the Z DHE = m; therefore, by equality, • the 7 DHE = the 6 AGB the arc AB the arc DE: therefore also (Th. xxv.) the 2 ACB – the arc AB the Z DFE the arc DEⓇ Coroll. Hence angles are measured by the arcs of circles of the same radii described between their legs from the angular points. THEOREM LV. The ratio of the angles contained by any two equal arcs is equal to the reciprocal ratio of their radii. Let BAC, EDF be any two Ls, and let the arc BC described . from the centre A be equal to the arc EF described from the centre D, HI. EL the LD - AB the Z Ā=DE Blo For in AB take AH = DE, and from the centre A with the distance AH describe an arc cutting AC at I. Let AB = R, AH = DE = r, the arc BC = EF = A, and the arc HI = a. Then, by similar sectors, ABC, AHI, ..... 2 А and (Th. liv.) .............. = therefore, by equality, ........ = = AB. Besides the characters already employed, we shall, in the sequel of these elements, introduce the signs I for perpendicular to, and || for parallel to; also, the letters expressing the sides and angles ut 2 F 226 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. of similar triangles will be written exactly in the order of the lines of the figure, as indeed they have been in the preceding part of this work. By these means, we may take the product of the extremes and means, or their ratios, at once: thus in the triangles ABC, DFT wa mav have ABÉF = BC.DE, or B6 = Ef, or AC = DF, OF DE = PF , &c. according as the one or the other may be the most convenient. AB CURVE LINES. GENERAL DEFINITIONS. M 1. A curve is a line, of such a nature that if any two points be taken in it, a straight line passing through these two points will not coincide with any portion of that line. 2. If PM, (a line of variable length,) be considered to move Fig. 1. parallel to itself, that is, always remaining at the same angle with the line AC, or AC produced, and if we have an equation expressed in terms of the constant quantity AC, and the two variable quantities CP, PM, the variable point M will trace outÖ À A P C a line AM, called the locus of the equation. 3. The curye traced by such point is also called the locus of the point. The variable line CP is called the abscissa, and the other variable line PM the ordinate ; also the fixed point C is called the origin of the abscissa, and the other fixed point A the vertex of the curve. 4. The straight line CP, or the line which contains CP, is called the line of the abscissa. 5. A tangential line is a straight line wbich touches à curve, and which cannot cut it, though produced at either of its extremities. 6. The point where a tangential line touches a curve, is called the point of contact. 7. The portion of a tangential line, intercepted between the point of contact at the extremity of an ordinate and the line of the abscissa, is called the tangent of the curve. 8. That portion of the line of the abscissa intercepted between the ordinate and a tangent to the curve at the other extremity of that ordinate, is called the subtangent. 9. A line drawn from the extremity of an ordinate, perpendicular to a tangent at the same point, to meet the line of the abscissa, is called the normal. 10. The intercepted portion of the line of the abscissa between the ordinate and the normal is called the subnormal. NOTATION. M Iu the figure here annexed, AC is the abscissa, MT the tangent, PT the subtangent, MN the normal, and PN the subnormal; also, AT is the complement of the subtangent. .. In the Algebraic Notation, let CA = a, CP = x, PM = y, MT = t, PT = s. Ο Α NC SECT. I.) . ELEMENTS OF GEOMETRY. 227 CURVES OF THE FIRST ORDER. DEFINITIONS. 1. A Curve of the first Order is the locus of a point of which the sum or difference of its distances from two fixed points is equal to a given straight line. 2. Each of the two fixed points is called a focus. 3. Each of the two straight lines expressing the distances from a point in the curve to each focus, is called a radius vector. 4. A straight line passing through each focus, if terminated at both ends by the curve or opposite branches of it, is called the axis major or transverse axis. 5. The middle point of the straight line between the two foci is called the centre of the figure. 6. The curve described by the sum of the two radius vectors, when it is equal to a given line, is called an ellipse. Thus, if FM, fM be any two radius vectors, and if FI, fl be any other two radius vectors; then if FM + fM = FI + fi, the curye is an ellipse, the straight line Aa is the axis major, and the point C in the middle of Ff is the centre. 7. The curve described by the difference of the two radius vectors, when it is equal to a given line, is called an hyperbola. Thus if FM, fM be any two radius vectors, and FI, fI any other two; then if fM - FM = fI — FI, the curve AIM is an hyperbola, and the equal and similar curve a im is called the opposite hyperbola. Coroll. Hence, if one of the focal points be infinitely dis- tant, any radius vector drawn from that point will be parallel to the axis major or line passing through the focal points ; whence we shall have another species of curve between the ellipse and hyperbola, called a parabola. Let R be any point in the axis; draw RP 1 Rf, and if Qi and Pm be any two lines parallel to Rf, and if Qi if= Pm - mf, the curve aim is a parabola. Hence if Pm = mf, Qi must be = if, as also Ra = a f. M CURVES OF THE FIRST ORDER. OF THE ELLIPSE. From the definition given of the ellipse, it is evident that the curve may be united so as to form one continued line, which must be understood in the following definitions, DEFINITIONS (continued). 8. A straight line passing through the centre at right angles to the axis major, and terminated at each end by the curve, is called the axis minor. 228 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. 9. A straight line passing through either focus parallel to the axis minor, and terminated by the curve, is called the latus rectum or parameter. 10. The distance from the centre to either focus is called the excentricity. 11. A straight line which touches the curve at the point when a diameter or any other line meets the curve, is called a tangent to the extremity of such diameter or other line. 12. Any straight line drawn through the centre, and terminated at each end by the curve, is called a diameter. 13. A diameter which is parallel to a tangent at one extremity of another diameter, is called the conjugate diameter of that other diameter. NOTATION OF THE FIGURE. In the figure, Aa is the axis major or transverse axis, Bb the axis minor or conjugate axis, C the centre, CP the abscissa com- mencing at the centre, PM the ordinate, F, f the two foci, CF or Cf the excentricity, FM, fM the radius vectors, MT the tangent, PT the subtangent, MN the normal, and PN the sub- normal. In the Algebraic Notation, besides what has already been defined of curves in general, let CF = Cf = e. Ek PROPOSITION I. THEOREM.. The sum of the two radius vectors is equal to the axis major. For (Def. 6) SFM + fM = Ff + 2AF FM + fM = Ff + 2af therefore - - AF = af wherefore FM + fM = F + AF + af = Aa = 2.CA. PROPOSITION II. THEOREM. The distance of either focus from'one extremity of the axis minor is equal to the semi-axis major. For the a BCF is equal and similar to the a BCf; and con- sequently BF = Bf, and since (Prop. i.) FB + Bf = 2:AC therefore ... 2.FB = 2.AC wherefore . . FB = AC. Sect. I.? 229 ELEMENTS OF GEOMETRY. PROPOSITION III. THEOREM. The difference of the squares of the semi-axis major and of the excentricity is equal to the square of the semi-axis minor. Here FB = AC (Prop. i.); therefore FB = a (see Notation). Since BC2 = FB – FC(Ge. Th. xxxv. Cor.) 12 = a? — 6%. Coroll. Hence e = al — 62. PROPOSITION IV. THEOREM. Half the difference of the two radius vectors is equal to the product of the excentricity and the abscissa divided by the semi-axis major. * (MF – Mf=. From M with the radius Mf describe the circle ISFR, cutting FM at I, and Aa at S, and produce FM to R: Then - - SF = 2Cf + 2Pf=2CP = 2x, and since - Aa = FM + Mf = FM + MR = FR; therefore . Aa= FR = 2a. Now let FI = V, and since - - - - - - - Ff = 2e, then (Ge. Th. xlii.) FR:FI = Ff:FS - - - - 2av = 40x; ee = . therefore, dividing by 4a, ............ Coroll. FM =a + one, and fM=a- er PROPOSITION V. THEOREM. . The rectangle of the square of the semi-axis major and the square of an ordinate, is equal to the rectangle of the square of the semi-axis minor and of the difference of the squares of the semi-axis major and the abscissa. XXXV. That is, CA2.PMⓇ = BCP (CAP - CP). Here FP=e + x and (Pr. iv. Cor.) FM =a + €* and (Ge. Th. XXXV. Cor.) y2 = (a +65)– (e + x)2 and (Pr. ii. Cor.) - .... ? = al — 62: whence, eliminating e, and by reduction, we obtain ay2 = 72 (a- mu?) which is the equation of the co-ordinates of the axis. By actual involution of the two terms on the second side of the second conditional equation, and by reduction, we obtain a yü = a* + e* (22 - am) - a² m2; then, by substituting al — b2 for e, we have a ya = ax + (a? — 6%) (20* — aʻ) — a x>, which, by multiplying the factors of the second term, and by reduction, becomes a ya = (a— x?). yn 230 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION VI. THEOREM. The ratio of two straight lines, ordinates to either axis of an ellipse, is equal to the ratio of the same tro lines, continued if necessary, as ordinates in a semicircle described upon that axis as a diameter. That is, CB – CD N For upon Aa, as a diameter, describe the semicircle ANDa, and produce the ordinates CB, PM of the ellipse to meet the circle in D and M. Let PN = go . By the co-ordinates CA, CB, PM of the ellipse a? y2 = b2 (a? — x2) and by the co-ordinates CA, CD, PN of the circle g? = a? — m2; therefore, eliminating 2, ........... wherefore, dividing by y2 and extracting, ....... =; The elimination of- & is performed by dividing the first equation by the second, * PROPOSITION VII. PROBLEM. To draw a tangent through any point in the prolongation of the axis. Ta Let M be the given point. On the axis major Aa describe the circle ANOa. Through M draw PN I Aa. Join CN, and draw NT I CN, and join TM; then TM is a tangent to the curve at M. For if TM is not a tangent at M, it will cut the curve in some other point K. Through K draw IK || PM, and prolong IK to meet the arc of the circle in L, and the tangent TN in 0; then IO will be greater than IL. STPM, TIK - - ... - - - TP.IK = PMÓTI By similar triangles, ) TPN. TIO . . . . . . . PN.TI = TP:10 and by corresponding ordinates (Pr. vi.) . . . . . . IL·PM = PN IK whence, eliminating PN, PM, PT, IK, IT, we have ... IL = 10, which is impossible, since IO is greater than IL. The elimination is performed by simply multiplying the three conditional equations in the order in which they stand, rejecting the common factors. • It is evident by this proposition that an ellipse may be readily described by means of proportional compasses ; for if one end of the instrument be opened to the radius of the circumscribing circle, while the other end is made to extend to the semi-axis minor of the ellipse, the compasses may be applied successively to any number of ordinates, drawn either regularly, or at random, as most convenient. The sector may also be used, though not so expeditiously, for the same purpose. SECT. I.] 231 ELEMENTS OF GEOMETRY. PROPOSITION VIII. PROBLEM. The curve, the axis major, and any diameter of an ellipse, being given, to find a conjugate to that given diameter. . . The construction of the figure (Pr. vii.) remaining, draw CE INC, meeting the circle at E, and ER I Aa, cutting the ellipse in D, and meeting Aa at R, and draw the diameter Dd; then Mm and Dd are conjugate dia- meters. Because NT, CE are each I NC, the radius CE is Il NT; and since RE is || PN, the as TPN and CRE are similar; Therefore, by similar as TPN, CRE, . - . . - - TPRE=PN-CR . PM PN and since (Pr. vi.) = . . . . . . . . . PM.RE=PN.RD wherefore, eliminating PN, RE TP_ CR . . . . - - - - - PM= consequently the two As TPM and CRD are similar, and therefore the diameter Dd is parallel to MT; wherefore the diameter Dd is conjugate to Mm. (Def. 13.) PROPOSITION IX. THEOREM. The rectangle of the squares of any semidiameter, and of an ordinate to it, is equal to the rectangle of the square of the semiconjugate, and of the difference of the squares of the semidiameter of the abscissa and the abscissa itself. TE The construction of the figure (Pr. viii.) remaining, let KG be an ordinate to the diameter Mm at K, and prolong. GK till it meet Aa in I and the curve in g, and through the point G draw QG I Aa, and prolong QG to meet the cir- cumference of the circle in H. Draw HII NC, cutting NC in L, and Aa in I, and through K draw JL I Aa, meeting CN and IH in L. Lastly, draw LB || IG, meet- ing QH at B; then shall CM2.KG² = CD (CM? — CK?). . For let CM. = a, CD = 6, CK = X, LB = KG = y, CN = CE = r, CL = U, and LH = 0; :- : guide y = 62202 Then by similar As MCN KCL. - - - - - aku — gu2 and by the co-ordinates of the circle - - - - - - - gode = pod m2: . whence, by eliminating r, u, v, we obtain - - - - - 2 y = 12 (a? — q). In the third equation, for u? substitute its equal aon from the second, and we have v2 = (a? —- <%). In the first equation, for v2 substitute its equal ) (22 12) now found, and we have y2 = (22 – 52), or, multiplying by a? · · · · · · 22 y2 = 62 (a? — 22). - Then by similar i SHLB, ECD: Ell , 72 x2 por 232 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. Coroll. 1. Hence, also, 62 m2 = a* (82 - y2), where ő is the semidiameter of the abscissa, y the abscissa itself, x the ordinate, and a the semiconjugate diameter. For a ya = a b2 — 62 m2; therefore, by transposition, 72 m2 = a? — az ya. That is, 62 x2 = a (6% — y), and this is the same as substituting a for b, 6 for a, w for y, and y for x in the original equation ax y2 = 0(a? — 22). Since the relation is the same if taken upon either semidiameter, the equation, which is the locus of the figure, is properly called the equation of the co-ordinates. Coroll. 2. Hence either diameter is conjugate to the other, since the same curve is produced between every two semiconjugate diameters, whether the abscissa is taken on the one or the other. Coroll. 3. Hence, in any equation expressed in terms of a, b, x, y, if the constants a, b, are exchanged for each other, and the variables x, y, for each other, the relations will be the same. NOTATION. M B We have just seen, that any two conjugate diameters have the same property as the two axes. We are now to understand in the sequel, that, in all conjugate diameters, the abscissal diameter is marked Aa, and its semiconjugate by CB or Cb, the abscissa by CP, the ordinate by PM ; and if any other ordinate be necessary on the same diameter, the abscissa of that ordinate is indicated by CH, and the ordinate itself by HI. In the algebraic data and demonstrations, CA = Ca is denoted by a, CB by b, CP by x, PM by y, as has been adopted in respect of the two axes. Moreover, we shall denote the abscissa CH by z, and its correspond- ing ordinate HI by y, so that the equation of the co-ordinates in refer- ence to PM is a' y = 62 (a? — «%), and the equation of the co-ordinates in reference to HI is a gl = 62 (a? — z). A PROPOSITION X. THEOREM. The ratio of the two rectangles contained each by the two segments of a diameter cut by an ordinate, is equal to that of the squares of the two ordinates themselves. P That is AP.Pa AH: a2 conseo (see Fig. in Notation) For (Pr. ix.) - - - - - - - - - a2 y = 62 (a? — *2) Again (Pr. ix.) • • • = 62(a2- wherefore, eliminating b, . . . . . . . . = = .......() ( al - 02 For dividing the first by the second of the given equations, we ootain = ; and by resolving each term of the fraction on the second side of this equation into its factors, we obtain y = (a + x) (a --X) (a + x) (a − 2) Coroll. Hence yi (am — ) = (a? — a?) is the equation of the co-ordinates CA, PM, HI, or ya (a + x) (a — x) = 72 (a + x) (a — «). Sect. I.] 233 ELEMENTS OF GEOMETRY. PROPOSITION XI. THEOREM. Given any diameter and an ordinate, to describe the curve of the ellipse by continued motion. Draw AD I Bb. From M, with the radius AD, describe an arc cutting Bb in I; join MI, and prolong MI to G: draw PG I PM, and join CG, producing the line both ways as far as necessary. Then, in moving the straight line GIM so that the point I may always be in the line Bb, and the point G in the line CG, or its pro- longation, the point M will trace out the curve. For draw IH || PG, cutting PM in H, and let AD = MI = 0, HI = PF = U, and GP = v, and we may observe, that since MG = BC, MG as well as BC will be denoted by 6 : Then, because of the right 2d MPG, -.... y = 62 - SDAC, FPC, - - - - c? x2 = az u? By similar triangles - MGP. MIH, .. . 2222 =2192 wherefore, eliminating C, U, o, we have .... a? ya = 62 (a? — x?). 62 22 : For multiplying the second and third equations together, we have al v2 =b2 x2; then, substituting "come in the first equation for v2, we have a2 y2 = 62 (a? — 32). PROPOSITION XII. PROBLEM. Given any two conjugate diameters and the position of any other diameter, to find the extremities of that diameter. - - Let Fm be any line given in position to the two conju- gate diameters Aa, Bb; it is required to find the portion of Fm which is a diameter of the ellipse. Draw FA || Bb, and AG I FA. Make AG = BC; from G, with the radius GA, describe the arc HAK: draw FG, cutting the arc in H, and join GC, and draw HM || GC, M is a point in the curve. For let CF = m, CM=LH = n, GF=p, GH=q, and GI =v; and because AG = BC, AG as well as BC will = b. CACF, PCM .. na = ma By similar triangles .. CFG, LHG ..... ma qe = 72 m2 (FGA, HGI ..... pa m2 = 622 and, by the co-ordinates of the circle, - ..... 62 — 392 = y2; therefore, eliminating m, n, p, q, v, we have - - - -. aya — 72 (a2 — x). 62 22 For multiplying the three first equations together, we have al v2 = 62x2 ; then, substituting from this equa- tion for v2 in the fourth, we have a2 y2 = 62 (a? - x2). 2 G 234 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION XIII. PROBLEM. Any diameter and an ordinate being given, to find any point in the curve. M Draw FI || Aa, and AF || HI: divide HI, FI each into two such parts (in g and h) that the ratio of HI, Hg may be equal to the ratio of FI, Fh: join hA, and through g draw aM, meeting hA at M: the point M is in the curvᎾ. For draw hK || FA, cutting AH in K; and let HI = Kh=y, Hg=m, Fh= AK = n, FI =AH =g, Ha=h, AP=v, and Pa = w. | APM, AKh, v y =ny By similar triangles - .. 1 aPM, aHg, mw=hy and since, by construction, = ....ngmg therefore, eliminating mn, - - .. . - - - 22 v w = 42 g h. which result is agreeable to the equation of the co-ordinates (Prop. x. Cor.). This result is simply obtained by multiplying the three given equations. PROPOSITION XIV. THEOREM. If through any given point two straight lines be drawn to cut the curve each in two points, the ratio of the rectangles of the segments of each line, from the given point to each point in the curve, is equal to the ratio of the squares of the semidiameters parallel thereto. Alt Let Dm, Dn be two straight lines drawn from D, so that Dm may meet the curve in M, m and Dn in N, n; ,,, DM.Dm CB2 then will DNiDn = CK2 For let us first suppose that one of the lines Di passes through the centre, and that it meets the curve in I, i; also let CI = Ci=d, PD=v, and CD = w. Then, by similar triangles, SHCI, PCD, d2.22 = 22 22 CHI, CPD, v222 = 72 72 Yn M B V again (Pr. ix.) ......... a?o? = 62 (a? - 24) 62 d2 therefore, eliminating* a, x, y, z, we have 72 — 22 22 — W2 w2 72 d2 2 * Substitute found from the first equation, and found from the product of the first and second equations respectively, for z, 72 in the fourth equation, and we have d2 72 -22 62 d2 x2 * From this equation, and also from a2 w2 the third equation find the value of 62 , and we at last obtain 22 BC2 = 72 7222 which is equivalent to : mD-DM= CK2 mD:D BC2 1 DN who consequently nD.DN = CK2 Sect. I.J 235 ELEMENTS OF GEOMETRY. that is, - - - - - - - - - - -moodM = BC2 C12 D.DI CK2 C12 therefore also, if CK be parallel to Dn. nD.DN — iD DI ... BC? - CK2 mD:DM BC2 and consequently mD:DM =nDDN OF nD.DN = CK2 Coroll. 1. Hence the ratio of two tangents meeting each other is equal to that of the semidiameters parallel to such tangents. For suppose Dm to touch the curve at A, and Dn at P, the rectangle DM.Dm will become DA', and the rectangle DN•Dn will become DP2; therefore the equation DM.Dm - CB2 : hocome DA. DA CB2 ecome Dp2 = 12; wherefore DN.Dn = TRO will become on DA СВ no=ma, or DACK = DP.CB. PROPOSITION XV. THEOREM. The rectangle under that part of a diameter produced which lies between the centre and the intersection of a tangent at the extremity of an ordinate and the abscissa, is equal to the square of the semidiameter of that abscissa. MA SP That is, CT:CP= CAP. Produce the tangent TM to H, and draw AK, a H tangents at A, a, meeting TH at K as well as H: draw the ordinate PM, and parallel to Aa draw KL, MN, cutting PM in L, and a H in N: draw the semi- diameter CD parallel to KH, and the semidiameter CE parallel to AK or aH. Let CD=f, CE =g, a H = m, AK =n, HM = 0, MK =w, and CT=0; then will aP = MN = a + x, KL = AP=a— , Ta = u + a, and AT=u—a.. (MK.CE = AK.CD, gw=nf Now, since (Cor. Pr 2aH.CD = MH.CE, m f =vg (TAK, TaH, n (u + a) = and by similar AS -... (u — a) HMN, MKL, (a - 2) =w (a + x) therefore, eliminating m, n, V, W, we have .... U x= a. Multiply the four equations together, and we have (u + a)(a − x) = (ü -- a) (a + x); and by actual multiplication, and rejecting the common terms, we have u x = a*. 236 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION XVI. THEOREM. If any semidiameter is situate between two semiconjugate diameters, and if ordinates be drawn from each extremity of these semiconjugate diameters to the intermediate semidiameter, the rectangle of the intermediate semidiameter, and one of its ordinates is equal to the rectangle of the semiconjugate to the intermediate semidiameter and the abscissa of the other ordinate. Let AC be a semidiameter between the two semiconjugate diameters IC, MC, then will CA.PM=CBCH. For draw the semiconjugate diameter CB to CA, and pro- long CA to T, and draw MT a tangent at M. We have here PT =U — X. Then (Prop. xv.) - - - - - -- u x = a and by similar triangles, MPT, IHC, 22 (u - 2)2 = x2y?, and (Pr. x. Cor.) ------- you (a? – z)=(a? — x2) again (Pr. ix.) - .......... a? yu = (a? — x2) wherefore, eliminating u, x, y, we have .... ay=6%. In the second equation, for u substitute on from the first, and we have gim (a2 — 2:2) = x2 12 22; multiply this and the third equation, and we have (a? — 2) (a? — 22) = x2x2, which by actual multiplication and reduction becomes a? — xé = z; and this being multiplied by the fourth equa- tion, becomes a y=b z. Coroll. 1. Hence ar= b x. Coroll. 2. Hence z = a? — 22, 22 = a? — 3*, y2 = 62 — , and g? = 62 - y2. PROPOSITION XVII. THEOREM. The rectangle under the two contiguous parts of a tangent, limited on each side of the point of contact by the prolongation of two conjugate diameters, is equal to the square of the semidiameter parallel to that tangent. N Let CI, CM be two semiconjugate diameters, meeting the tangent in T and N, which passes through A, the extremity of any semi- diameter, as CA, then AT-AN = CB². For let AT=t, AN = U. : SCPM, CAN, .--. u x = a y By similar triangles 1 CHI, CAT . . . t and (Pr. xvi.) ........... ay=bz and (Pr. xvi. Cor. 1) ....... - - ay = therefore, eliminating a, x, x, y, y, - - - - - tu= 62 The result is obtained by multiplying the four conditional equations as they stand, and expunging the common quantities. Sect. I.) 237 ELEMENTS OF GEOMETRY. PROPOSITION XVIII. THEOREM. If the angle contained by one of the radius vectors and the prolongation of the other be bisected, the bisecting line will be a tangent to the curve. JI Let FM be produced to H, and let the LfMH be bisected; the line MI which bisects the LfMH is a tangent to the curve at M. For make MH = Mf, and from any point I in MI draw IH and If, and join FI and Kf, and let FI meet the curve in K. Then, because MH= Mf, and MI common, and the LHMI= If = FH, and consequently greater than FM + Mf. Hence the point I is without the curve; and since every point in MI except M is without the curve, MI is therefore a tangent to the curve at M. PROPOSITION XIX. THEOREM. A If two straight lines be tangents at the extremities of a chord ordinately applied to a diameter, they will meet each other in the prolongation of that diameter. Let the chord MM' cut the diameter Aa in P. Draw II' || MM', and join MI, M'I'. Prolong a A to T', and let MI, M'I' prolonged meet aT in T and T. Draw IK, I'K', each - T ule o parallel to Aa, cutting MM' in K, and K'; the tangents MT, M'T cannot meet the prolongation of the diameter aA in two separate points T, T. For let IK= I'K' = U, KM= KM' =v, PT = s, and PT = S. (MKI, MPT - - - - - - Sv=UY Then, by similar triangles > M'K'I', M'P'T . .. - So=uu wherefore, eliminating u, v, y, we have - ...... S=s; therefore PT = PT'; consequently the points T, T' coincide, whatever may be the distance of MM', II. PROPOSITION XX. THEOREM. The rectangle under the semi-axis major and half the parameter is equal to the square of the semi-axis minor. Because FM + fM = 2a, therefore, when FM I Aa, FM will be = y; therefore y + fM = 2a, and by transposition fM = 2a -y; and since Ff = 2e, (Ge. Th. xxxv. Cor.) - - - y2 = (2a - y)2 – (2e), and since (Pr. ii.) - - - - .62 = a -6%; then, eliminating e, - - ay= 62. Therefore y is here a constant quantity. If we now put y = 1p, we shall have jap = 62. 238 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION XXI. THEOREM. The rectangle under the semi-axis major and the square of an ordinate to it, is equal to the rectangle under half the parameter and the difference of the squares of the semi-axis major and of the abscissa. For (Pr. ix.) .. ....... a ya = 62 (a? - x), and (Pr. xx.) - - - - - - - --- 62 = fap; therefore, eliminating b, ...... a ya = bp (a? — x2). PROPOSITION XXII. THEOREM. Every parallelogram of which the sides are parallel to two conjugate diameters, and are tangents to the curve, is equal to the rectangle of the two axes. NB U T . Let CI and CM be two conjugate semidiameters, and let Aa be the axis major, and Bb the axis minor, and let the two adjacent sides UV and UX of the parallelo- gram UVWX touch the curve at M and I. Produce Aa to cut UX in t and VU produced in T. Draw the ordinates of the axes HI, PM; also draw CN perpen- dicular to UV, cutting UV in N. Let Ct = t, TC = r, CN = p, and CI = n. Since (Pr. xvi. Cor.) - - - - - - - - - -. 22 = a? - 22 and (Pr. ix.) - ... - - ... - 62 (a? — 2?) = aya. and (Pr. xv.) . .. ...... ... a? = tz; SCI, CTM, - -..-- ty = py and by similar AS TCN, CIH - - - - - - - ry = pn therefore, eliminating r, t, x, y, z, q ....... ab = pn. Multiply the two first equations, and we have bz = ay; multiply this and the three remaining equations, and we have ab = pn. PROPOSITION XXIII. THEOREM. The sum of the squares of any two conjugate diameters is equal to the sum of the squares of the two axes Let CM, CI be two conjugate semidiameters on the same side of the axis minor Bb; then CMP + Cl2 = CAP + CB2. For draw the ordinates MP and HI to the axis major Aa, and let ALERE MC = m, and CH = n. sa² = x² + 2 By: (Pr. xvi. Cor. 2) ........3 262 = y2 + y2 (CM? = CP2 + PM, m2 = 22 + y2 and since (Ge. Th. xxxv.) CI2 = CH2 + H12 m2 = 22 + y2 therefore, eliminating x, y, z, y,..- m2 + m2 = a + b2. For addiug the two first equations together, we obtain az + 62 = + 2 + y2 + gom; and add ing the two last together, we have ma + m2 = 22 + z 2 + y + 3%; therefore ma + m2 = a + b2. _ _ Sect. I.] 239 ELEMENTS OF GEOMETRY: PROPOSITION XXIV. THEOREM. The rectangle under the axis major, and that part of it between the intersection of the normal and the Let FN = U, fM = V, and FM = w; then will fN = 2e - U. a + et = v By (Prop. iv. Cor.) 2e - u and by (Ge. Th. liii.) | PF NTCFya • TI whence, eliminating v, w, u=e- The elimination is obtained by dividing the first equation by the second; and multiplying the result by the third. By reduction we find u = e-boat PROPOSITION XXV. THEOREM. The rectangle under the square of the semi-axis major and the subnormal is equal to the rectangle under the square of the semi-axis minor and the abscissa. Now, PF = x - e. Let PN=w; consequently w=u + x — e, and by (Pr. xxiv.) - - - - - - - - - - - ure: also by (Pr. iii. Cor.) 62 = a 2 – 62 ; therefore, eliminating e, u, v, - - - - - - - - - - - - - - axw = bx. The elimination is obtained by substituting a? – 52 from the third, fore in the second equation, and we obtain u = e X + 62x 2, or by transposition, 64x u + x -er , and consequently w= ; therefore aéw = 62x. PROPOSITION XXVI. THEOREM. The rectangle under the abscissa and the subtangent is equal to the difference of the squares of the semi-axis major and of the abscissa. For PT being =s (see Notation, p. 228), and PN =w, and since TMN is a right-angled a, and PM 1 TN (see Fig. Pr. xxv.), the as TPM and MPN are similar. Then, by similar as TPM, MPN, ...... sw=ya and (Pr. ix.) ............ aya = 62 (a? — 22) and by (Pr. xxv.) ............ bax = a-w; · wherefore, eliminating b, w, and y, ...... sx =q? — . The resulting equation is found by simply multiplying the three given equations as they stand, and rejecting the common factors. 11 240 (PART IV ELEMENTS AND PRACTICE OF GEOMETRY. C CURVES OF THE FIRST ORDER. OF THE HYPERBOLA. DEFINITIONS (continued). * 14. A line drawn through the centre perpendicularly to the transverse axis, and limited at each extremity by the circumference of a circle described from the vertex with a radius equal to the distance of the focus from the centre of the figure, is called the conjugate axis. 15. Any straight line drawn through the centre and terminated by the opposite curves, is called a diameter. 16. A diameter which is parallel to a tangent at the extremity of another diameter, is called the conjugate of that diameter. 17. The distance of the centre from either focus, is called the excentricity. PROPOSITION XXVII. THEOREM. The difference of the two radius vectors in the hyperbola is equal to the transverse axis. - - Ha Ati Let Aa be the transverse axis, M any point in the curve, F the nearer, and f the remoter focus. SFM - FM=fA — AF = fa taA — AF for (Der.) FM - FM= Fa- af = FA + Aa – af therefore fa + aA — AF =FA + Aa - af; consequently . . 2af = 2AF, and - - - - af = AF; therefore . fM – FM=fA — AF=fA — af =aA. PROPOSITION XXVIII. THEOREM. The square of the excentricity is equal to the sum of the squares of the two semi-axes. Let CB be the conjugate semi-axis ; then, since aB= Cf (Def. 8) the excentricity, and since aB= aC2 + CB (Ge. Th. XXXV.), e = a + b2. JA * See definitions 1 to 7, Elements of Geometry, Curve Lines.. SECT. I.] 241 ELEMENTS OF GEOMETRY. PROPOSITION XXIX. THEOREM. Half the sum of the two radius vectors is equal to the product of the excentricity and the abscissa divided by the semi-axis major. # (FM + fM) = CF X CP. CA From M, with the radius Mf, describe the circle IfSR, cutting FM at I, and Aa produced at S, and produce FM to R. Let FR=v. Now SF= 2CF + 2Pf = 2CP= 2x; and since FI = 2a (Pr. xxvii.); and since Ff=2e; and since FR.FI= Ff:FS (Ge. Th. xlii.) - 2av = 4ex and therefore, dividing by 4a - - ... ex Coroll. FM = (x + a, and fM= a = ex = d. PROPOSITION XXX. THEOREM. The rectangle of the square of the semi-axis major and the square of an ordinate is equal to the rectangle of the square of the semi-axis minor, and of the difference of the squares of the abscissa and the semi-' axis major. A Ćate Here FP=CF + CP=e+ x. Let FM - v, and e + x = w; therefore FP = w = et X. Then, since P1.2 = F12 — FP2 (Ge. Th. XXXV. Cor.) ya = 102 — w2 and since FM = ** + a (Pr. xxix. Cor.) v = C + a)? and since w=e + x -.... w = (e + x)2 also, since (Pr. xxviii.) ... .. ef = a + 62 ; whence, eliminating e, v, w ... a® y = 62 (202 — a?). substitute For, substituting the expansions of + a)2 and (e + x)2 from the second and third equations, for their equals 02, w in the first, we shall have yo = (22-42) — 22 + c2. In this equation substitute a2 + 8? frəm the fourth, for e?, and we have ya = 42+8 (22 – 22) — 42 + a2 = (72 a?); or multiplying by a?, r2 y2 =W (2 -- 43). Corol. Whence 62 = 63.72 – . 242 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION XXXI. THEOREM. The square of the semi-axis major is equal to the rectangle of the two parts of the line of the axis, the one between the centre and the intersection of the tangent, and the other between the centre and the ordinate. Let fM = U, FM = V, fT = W, FT = %; then will w + z = 2e. fM fT , And, since FM= FTI (Ge. Th. liii.) . . . . Uz and since (Pr. xxix.) - - - - - - - fa ď Í ath ľ and since (Pr. xxix. Cor.) ....... v = fra? also, since - - - - - - - - - - w +z = 2e; whence, eliminating u, v, w - - - - - - a² = x (x – e). For to each side of the first equation add zv, and it will become by ordering % (u + 0) = 0 (6 + ). Multiply this and the second, third, and fourth equations together, and we obtain xz = ex — a?; whence a2 = x (e - %). · Coroll. 1. Hence z e Coro ence PT- FP. 22 am a Coroll. 2. Hence PT =- FT + FP =(-2) + (x −e) = <---a?. Coroll. 3. Hence AT = TP – AP= ** ;? – (2 –a)= 4 (0,5). Coroll. 4. Hence CT =CA – AT=a_ (8= a) = ? ce AT Coro ce C'T a PROPOSITION XXXII. PROBLEM. Given the two axes in position and magnitude, to find the position of the asymptotes.* Draw A D perpendicular to AC, and BD parallel to AC; and through the points C and D draw the straight line CN; CN will be an asymptote to the curve. For let s = the subtangent PT (see Notation) AT = %, and AS = 0.. By similar triangles TPM, TAS, - - - - $2v2 = y2z2, and by co-ordinates (Pr. xxx.) - - - - - ały2 = 62 (x2 − a2); and since AT =z= _,_a[x — a) (Pri " (Pr. xxxi, Cor. 3) - - 2222 = a2 (x – a)2; AÞ and since PT =s=+2 (Pr. xxxi. Cor. 2), (22 -- a2)2 = 8212; wherefore, eliminating s, y, z, we obtain ..... v=bveza. * The asymptotes are straight lines drawn through the centre, continually approaching nearer and nearer to the curve the farther they are carried, but which, if infinitely extended, would never touch the curve. SECT. I. 243 . ELEMENTS OF GEOMETRY. If we now suppose x infinite, the quantity will not differ from unity; we shall therefore have v = 0, 1 and because CT = (Pr. xxxi. Cor. 4), therefore since x by hypothesis is infinitely great, will be infinitely small; wherefore, the tangent at an infinite distance will pass through the centre C, and therefore the asymptote will pass through the points C and D. The above elimination is obtained simply by multiplying the given equations, and dividing out the common factors. PROPOSITION XXXIII. THEOREM. If an ordinate be produced to meet each asymptote, the rectangle of the two segments of the line between the asymptotes, as separated by either branch of the curve, is equal to the square of the semi-axis minor. For produce the ordinate PM at both ends to meet the asymptotes in N and n, cutting the other branch of the curve in m. By similar As CAD, CPN - - - - - - - - PN= therefore . . . . . . NM= PN- T and , - - - - - - - NM= + + m2 therefore - - - - - - - - • NM X Mn y?; a2 and since (Pr. xxx. Cor.) ...... 6? = — ya ; therefore · - . . . . . . - NM X Mn = 62 = AD2 = C6%. PROPOSITION XXXIV. PROBLEM. To find the equation of the co-ordinates as it relates to the asymptotes. IN Draw AL and MQ parallel to the asymptote Cu, cutting CN in L and Q; also draw AI parallel to CN, cutting Cu in I; then, (NMQ. DAL - - NM X AL = MQ X AD by similar as ASUMK, ADL - . uM x DL = MK X AD and by (Pr. xxxiii.) - -..-.- ADP = uM X MN; whence, eliminating AD,UM,NM, AL X DL = MK X MQ. But, because AL and DL are equal, the result is AL = MK:MQ; that is, by making AL = a, MK = CQ= x the abscissa, and making QM=y the ordinate, we shall have a' = xy, which is the equation of the co-ordinates as it relates to the asymptotes. 244 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. . PROPOSITION XXXV. THEOREM. If any two parallel lines be drawn so as to cut two connected branches and their asymptotes, the products of the two segments of each line, contained between the asymptotes, and separated by either of the branches of the curve, are equal. For let the two parallels Rr, Ss, terminated by the asymp- totes in R, S, r, s, cut the hyperbola in the points M, N, m, n, and let the lines Ee, Ff be drawn through the points M, N, perpendicular to the axis, to cut the connecting branch of the curve in i, k, and the asymptotes in E, F, e, f, and we shall have, CRME, SNF - RM X NF = ME XSN by similar As Me, sNb - Mr X Nf = Me x sN SME X Me= 32 by (Pr. xxxiii.)'. - - - . . . } 62 = FN X Nf therefore - - - - - - - - - RM x Mr = SM.Ns and therefore, also, - - - - - - rm x mR=sn x ns. rm X PROPOSITION XXXVI. THEOREM. If a straight line be drawn between the asymptotes to cut two branches of the curse, the parts of the straight line without the.curve are equal. GRM X Mr = TA At By (Pr. xxxv.) - - TA X At = rm x mR SMm+ mr = Mr and since - - - - 3 Rm= RM + Mm therefore, by multiplication, RM (Mm + mr) = (rm (RM + Mm), and this, by actual multiplication, becomes RM.Mm + RM'mr = rm:RM + rm.Mm; and throwing out the common quantities, we have RM.Mm = rm:Mm; and, dividing by the common factor Mm, RM = rm, CURVES OF THE FIRST ORDER. . OF THE PARABOLA. DEFINITIONS.* 18. The line RP which is at the same distance from any point of the curve as the focus is from that point, is called the directrix. 19. A straight line drawn through the focus perpendicular to the directrix, is called the axis. 20. The point of the curve cut by the perpendicular drawn to the directrix, is called the vertex. * See definitions 1 to 7, Elements of Geometry, Curve Lines. Sect. I.] 245 ELEMENTS OF GEOMETRY.' 21. A straight line drawn through the focus, perpendicular to the axis, and terminated by the curve, is called the parameter, or latus rectum of the axis. 22. Every straight line parallel to the axis, is called a diameter. 23. The point where a diameter meets the curve, is called the vertex of that diameter. 24. A straight line parallel to a tangent, and limited by a diameter passing through the point of contact and by the curve, is called an ordinate to that diameter. 25. The part of the straight line of a diameter limited by its vertex and an ordinate, is called the abscissa. 26. A chord drawn from the focus parallel to a tangent at the extremity of any diameter, is called the parameter of that diameter. U 1 PROPOSITION XXXVII. THEOREM. The parameter of the axis is equal to four times the distance of the focus from the vertex. Let RQ be the directrix, and Gg the parameter; and let QN be perpendicular to RQ: then GQ = GF; therefore QG = Gg; and consequently, because QG =RF, RF = {Gg ; wherefore AF = AR=1Gg, or 4AF = Gg. RA PROPOSITION XXXVIII. THEOREM. The line bisecting the angle made by the radius vector and a line from that point of the curve where the radius vector meets it, perpendicular to the directrix, is a tangent to the curve. Blas Let ML bisect the angle FMQ. In LM, or LM produced, take any point N, and draw NS perpendicular to the directrix. Then, since MF = MQ and 4 FML=QML, FL= QL, and ML is perpendicular to FQ; hence NF=NQ; consequently NF is greater than NS, and N is not in the curve, which therefore lies wholly on one side of NL. PROPOSITION XXXIX. THEOREM. The subtangent of the axis is double of the abscissa. Because the parallels TP and QM are joined by TM, the alternate angles PTM and QMT are equal ; but because the LFMQ is bisected, the LQMT FMT; therefore the - FMT = PTM; and therefore FM= FT. But PR = AP + AR = AP + AF; and since FT = FM, and FM = PR therefore FT = PR, consequently PF = TR; and since AF = AR, there- fore PF + AF = TR + AR. · But PF + AF = AP, and TR + AR = AT; whence AP AT, and consequently AP + AT = twice AP. PLUM 246 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITIOŃ XL. THEOREM. The rectangle under the parameter and the abscissa of the axis is equal to the square of the ordinate. That is, Gg X AP= PM2. Let AP= x, PM = y, AF = f, Gg =p; then will FP= AP -- AF = x — f, FM = PR= x + f. Since (Ge. Th. xxxv.) FP2 + PM? = FM”, (x - 5)2 + y2 = (x + 5)2 wherefore, by multiplication and reduction, - -- y2 = 4fx ; and since (Pr. xxxvii.) ---------- 4f=p, therefore, eliminating f, - - - - - - - - - - y = px. This will be found simply by actual multiplication, rejecting the common terms xa, f?, and trans- posing 2fx. PROPOSITION XLI. THEOREM. The rectangle under the parameter and a perpendicular from any point in the curve to a chord ordinately applied to the axis, is equal to the rectangle under the two segments of that chord. That is, P X QR = QM X Qm; where p is the parameter. Let A0 = %, OR = y, Qm=u, QM =v, and QR = w. Then will Qm= Pin + OR = y + y = U, QM= PM — OR= y - y = 0, QR= AP – A0= x — %=w. Spx = y2 Since (Pr. xxix. Cor.) p.A0 = ORP, and p•AP = 2pz = go? therefore · - - - p (- 2) = (y - 3) (y + y), or pw = uv For, by subtracting the second equation from the first, we have px - = (y + y) (y-2). pz sms.y - g?, or p (x-7) PROPOSITION XLII. THEOREM. The ratio of the two parts of a chord, perpendicular to the axis, on each side of a parallel to the axis, is equal to the ratio of the two parts of that parallel on each side of the curve between the chord and a tangent at the oxtremity of that chord. RS That is, Mama That RO SECT. I.1 247 ELEMENTS OF GEOMETRY. Let Qn = U, Qm =%, QR = w, and QS = s; then will MQ = Mm -- Qm= 2y — U, and RS = QS – QR=s-w; and since AT = AP = x, PT = 2x; and since PM =y, Mm = 2y. By similar as MQS, MPT ....... 2xv = sy by (Pr. xl.) ............ y2 = px and by (Pr. xli.) - - - - - - - - - - pw = Uv; therefore, eliminating F, 0, ll, - ... - 2xy = $u. Now, subtracting uw from each side, . - w (2y — u)=u (s --- w), and dividing each side by uw, ........... 2y — = $ 0. U W U II PROPOSITION XLIII. THEOREM. The rectangle of the parameter and the portion of a line, parallel to the axis, which lies between the curve and a tangent, is equal to the square of that part of a chord, drawn through the point of contact per- pendicular to the axis, between that point and the prolongation of the line which is parallel to the axis. That is, p X IE= CK2. Since (Pr. xlii.) = ...... EI:KL = EKKC and (Pr. xl.) ...... - - - - ... p:EK = CK KL therefore, eliminating EK, KL, --.-.-- piEI = CK?. PROPOSITION XLIV. THEOREM. oc S. A1 The ratio of two lines parallel to the axis, intercepted between the curve and a tangent, is equal to the ratio of the squares of the parts of the tangent intercepted between the point of contact and the intersec- tion of each of the parallels. That is, EI - CT is, AT = CTE For (Pr. xliii.) - ........... . 3 S p EI = CK? 2 CD2 = p.AT and by similar as TCD, ICK, ...... - CT2.CK? = CD.CI? wherefore, eliminating p, CD, CK, - - - - - - CT2.EI = CIPAT, and dividing by CIPEI ........... Cara CT2 AT . K 1 PROPOSITION XLV. THEOREM. The ratio of the abscissas of any diameter is equal to that of the squares of the ordinates. That is, CL, CN being the abscissas, LE, NA their corresponding ordinates, NA CN LEZ CL Draw the tangent CT, which will be parallel to NA, and draw EI and AT parallel to NC. Let CL = IE =%, CN= AT=%, LE = CI=y, and NA= AT CT2 Since (Pr. xliv.) T r ta - • • • • • - - - - 248 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION XLVI. THEOREM. A line parallel to the axis, terminated by a double ordinate and a tangent at the extremity of that ordi nate, is divided by the curve in a ratio equal to that in which the line divides the double ordinate. That is, = IR Let MS = V, sr = Q, SR = %, si = m, Ms = n, MR = t; and Ml = U. Then IR = MR — MI = t-U, and rl = ls — rs = m - X.. Then, by similar as RSM, ISM, - - - - - - - - - zn = mv, and by (Pr. xliv.) - = - - - - - - - - - - - - 204 = zn", L1 and by similar As, IMS, RMS, - - - ...-.. nt = wv; therefore, eliminating n, v, z, - - - - - - - - - - - tx = mu, or, subtracting ux from both sides, ....... (t – u) x = (in — x) U. Coroll. Hence the subtangent is, in all cases, double of the abscissa ; for when t = 2u, the ex- pression x (t – u) = u(m — x) becomes 2x = m, sr:MS = sl·Ms. PROPOSITION XLVII. THEOREM. If two tangents meeting each other be divided each into two parts, so that the ratio of the parts of the one may be equal to the ratio of the parts of the other in a contrary order, a line drawn between the points of division will touch the curve, and the two parts of such line, divided by the point of contact with the curve, will have a ratio to each other equal to that of the parts of the tangents. Let CD be divided in G, and DE in H, so that GD - HE, H = athe line GH m_ GD HE For through the points G, I, D, H, draw the diameters GK, IL, DM, "KLMN HN, as also the lines CI, EI, which will be double ordinates to the diameters GK, HN; therefore the diameters DM, GK, HN will bisect the chords CE, CI, EI, joining the several points of contact. Now KM = CM — CK = {CE -- ECL = {LE = LN = NE, and MN = ME - NE = ICE — İLE = ICL = CK= KL. Let MN = CK = KL = U, and KM = LN = NE --V ; u _GI _CG _DH then = TH = GD = HE SECT. I.) 249 ELEMENTS OF GEOMETRY. VARIOUS ORDERS OF LINES. CURVES are distinguished as of two kinds, viz.--Algebraical and Transcendental. Algebraical Curves are those in which the relation of the abscissas to the ordinates can be defined by a common algebraical expression. Transcendental Curves are those in which the abscissa or ordinate depends on some transcendental quantity; as an arc, sine, cosine, logarithm, &c. Lines are divided into various orders, according to the degree of the equation expressing the con- ditions of the locus. The locus of a simple equation is called a line of the first order. The locus of a quadratic equation is called a line of the second order. The locus of a cubic equation is called a line of the third order ; and so on. LINES OF THE FIRST ORDER. Fig. 1. M . Fig. 2. Let C be a fixed point, PM an ordinate to the abscissa CP; then CP being = 4, and PM = y, the equation which regulates the length of the ordinate PM may be ay = c + bx. Now, in order to ascertain the distance of the point A (where the locus AM cuts the abscissa) from the fixed point C, y or PM must be a = zero or nothing, and consequently c + bx = 0, and therefore x =- ; which indicates that. when y is zero, a must be set towards the left hand at the distance of ; but when x = 0, ay = 0, TO and consequently y=0 Therefore, if in the line AP we fix the point C, and make the distance CA = , and draw CD at a given angle to AC, and make CD = then CD will be the ordinate at the point C. Since AP = AC + CP, AP = 8 + x=9+ 6*; therefore we shall have and º + ba respectively for the two distances AC, AP, and y for the corresponding ordinates CD, PM. ct bac Now XY = "bna vax > or ay = c + bx, the proposed equation. It therefore appears that the locus AM is a straight line, for the ratio of AC, AP is equal to that of CD, PM. b : LINES OF THE SECOND ORDER. Let the equation of the co-ordinates be y = A + Bx + Coxa. Now in order to ascertain in terms of the given quantities A, B, C, where the locus AM cuts the abscissa, we must, as before, 21 250 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. make y = 0, and consequently A + Bx + Caca = 0: the solution of this quadratic equation gives - BEN (B2 — 4AC) >, which indicates that y may be = zero in two different places; there- 20 fore the locus of this equation must either be one continued line or two separate lines, as it will meet the abscissa in two different points. Since y2 = A + Bx + Cx?, we shall have y= IN (A + Bx + Ca>); from which it is evident that the greater x is, the greater also must be the value of y, and therefore there can be no real value of y between the two values of x; whence the curve can never return to itself: the equation, therefore, belongs to two separate curves. - B+ N (B2 – 4AC In the equation x=- A%, when both values of x are affirmative, the vertices 20 of both curves are on the right hand of the origin C of the abscissa ; if both are negative, they are on the left; and if one be affirmative and the other negative, the affirmative value is on the right of C, and the negative on the left. The value of y in relation to any value of x between the two ver- tices is imaginary. When B2 — 4AC is negative, the axis of the curve changes the direction assumed for it, and takes the parallelism of the ordinates ; for in this case no value of y can ever become zero. In order to exemplify what has now been advanced, let the locus of the curve be the equation y = 3x2 + 8x + 4. By making y=0, we shall have, from the equation 3x2 + 8x + 4 = 0, the two values of x, — $ and — 2; which indicate that the vertex of the one curve is of unity to the left hand of the assumed point C, and that the vertex of the other curve is 2 in the same direction; therefore the distance between the vertices is 13. In order to calculate the ordinates, since the vertices of both curves are towards the left, we shall begin with the greatest negative value of x, and proceed progressively by adding unity, which will of course diminish the negative values, and increase the affirmative ones. Let x = -5; then y = I N (3x2 + 8x + 4) = N 39 = 6:24 x = - 4 - - y = 4:47 x = - 3 - - y = 2.64 = - 2 - - y=0 0 . = -1 - - y = imaginary = - - y = 0 = 0 - - y = 2 = 1 - - y = 3.87 x = 2 - - y = 5.65 x = 3 - - y = 7.41 &c. IIIIII colhe Let y2 = 3x2 + 6x + 4 be proposed as another example: but since, making y = 0, the equation 3x2 + 6x + 4 = 0 has no real root, therefore every value of y is possible. Here it might be proper to inquire what might be the value of x, when y is the least possible ; but as this would lead us to the differential calculus, we shall drop this inquiry, and begin at pleasure, making the first negative value of 2, 3, and ascending to its positive values, by adding unity to every last value. The several values of y in the equation y=w (3x2 + 6x + 4) corresponding to the assumed values of x, are as stated in the following table :- Sect. 1.) ELEMENTS OF GEOMETRY. 251 When II | It . o caso do nosso y = 7.0 5•28 3.605 2.0 1, y = 1:0 2, y = 2.0 3, y = 3.605 y = 5.28 5, y = 7.0 &c. The curves may easily be drawn according to the values of the quantities thus found. As a further illustration, let it be required to trace the locus of the equation ya =4 - x - 30%. Making y = 0, we shall find x = - 4, or + 2; therefore one extremity of the axis is four equal parts on the left hand of the origin of the abscissa, and the other extremity two equal parts on the right of it; and consequently the whole axis is equivalent to six equal parts. It will therefore be proper to find the values of y, beginning the first when x= — 4, and y=0); therefore, making « successively - 4, — 3, — 2, - 1, $ 0, 1, 2, in the equation y=N (4 — - $), and aggregating the quantities within the + 0, 1, 2, &c. radical sign, and extracting the root each time, we shall find the values of y agreeable to the assigned values of x in the following table :- When x = - 4, y = 0 y = 1:58 = – 2, y = 2.0 = – 1, y = 2:12 x = I 0, y = 2:0 x = + 1, y = 1:58 + 2, y = 0. The curve will return to itself in all cases when the sign of the term which contains the square of x is negative, whatever may be the signs of the other two terms. Equations which have the sign of the term containing the square of & negative, can only be in one of the four following forms, viz. :- Py = A + Bx - Cx2 or Py2 = A - Bx – Cu2 or Pyz = - A + Bx – Cx2 or Py = 1 + 1 + III It is evident in the last of these equations that & cannot have any affirmative values, for then both values of y will be negative, and consequently impossible. Let it be required to trace the locus of the equation Py2 = A + Bx. Making y=0, the quantity x can only have one value, from the equation A + Bx = 0; there. fore the curve cannot cut the abscissa in more than one point; hence it is evident there can neither be a continued curve, nor two opposite ones. 6 ) 252 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. By way of illustration, let y2 = 3 + 2x, or y= v (3 + 2x). Making y = 0, we shall have =- = - 1:5; from which it appears that the apex of the curve is 1.5 on the left of the origin ; therefore, making x=-1, x= I 0, x= 1, x=2, &c. in the equation y=IN (3 + 2x), aggregating the quantities within the radical sign, and extracting the root each time, we shall find the values of y agreeable to the assigned values of x in the fol- lowing table :- =- = = mió i opos 1, y=1= 1.00 N3 = 1.73 1, y =N5 = 2.23 W7 = 2.64 3, y =n9= 3:00 We shall observe, to conclude, that all lines of the second order are curves of the first order, so that every quadratic equation is the locus of an ellipse, hyperbola, or parabola. SECTION II. PRACTICAL GEOMETRY. CHAP. I.-GEOMETRICAL PROBLEMS AS REGARDS PLANE FIGURES. NOTATION. Points are merely assumed, that is, taken at pleasure, or are given in position previous to the com mencement of any operation,* and are either made with a pen or pencil; but, in nice operations, they are formed with the pointed extremity of an instrument. The only instruments allowed in practical geometry, are a pair of compasses and a ruler, which is a bar of wood or metal, with one straight edge. The compasses or dividers, as they are called, serves to measure the distance between any two points, or to transfer the distance between any two points to any other place, and also round a given point to describe a circle which may have any given radius. The ruler is employed in directing the motion of a point in a straight line. When any number of letters indicating points are written in succession, each of such distinguish. ing letters is separated from the following one by a comma. A line which stands alone is sufficiently distinguished by placing a letter as near to one end of it as possible ; but when a line is required to be distinguished from others crossing it in one or more points, it is necessary to place a letter at each of its extremities: in this case, the letters which in. dicate the line are brought together in the text, without the intervention of any mark of division. When any two letters are thus placed together in the text, without having the word line or straight line, or arc, immediately preceding these letters, a straight line is implied. Thus, if it is said, draw CD parallel to AB; the meaning is, to draw the straight line denoted by CD parallel to that denoted by AB: or to draw the straight line which has C at one of its ends, and D at the other, parallel to the straight line which has A at one of its ends and B at the other. * In practice, we must absolutely begin with a point: then, if we wish to represent a line on a given surface, we must assume or give two points, through which the line is to pass, and draw the line through them: and if we would represent a surface, we must draw lines round it; but though a drawing may be made on any surface, it is convenient to use a plane, or plane surface. Since a point, a straight line, and a surface, only exist in the imagination, we must therefore use the surface of a solid for drawing upon; the straight edge of a ruler instead of a straight line; and the end of a sharp-pointed instrument, pencil, or pen, instead of a point. · Now, if a point is made with a sharp-pointed instrument through the surface of a solid, it will displace a small portion of the solid, and form a cavity of equal capacity; and if formed by a pen or pencil, it will leave a small solid upon the surface; so that in directing practical operations, as a point must be visible, it is either a small void, or a small solid. For the same reason as a line must be seen, it must have both breadth and thickness, whether considered as a small solid or void. From what has now been said, it is evident that, though geometrical operations are accurate in a theoretical point of view, they are only very near approximations in practice; therefore, the smaller we can assume points, and the finer we can draw lines, the more accuracy we shall obtain in our operations. 254 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. An angle which stands alone is indicated by a single letter. When two or more angles meet together at the same point, in order to distinguish any particular angle from the rest, it is necessary to place a letter somewhere in each line, and another letter at the point where the angles meet each other: the angle thus denoted will be expressed in the text by three letters, without any mark of division between any two of them. When three letters are placed together, without any word to denote the thing they point out, they mean an angle. Thus, if it were said, make ABC equal to EFG, the meaning is, to make the angle denoted by ABC equal to the angle denoted by EFG; or to make the angle formed by the two straight lines AB, BC, equal to the angle formed by the two straight lines EF, FG. A figure is denoted by placing a letter at the meeting of every two sides, and these letters follow each other in the text, without being separated by any mark. A circle, or an arc of a circle, is denoted by placing three or more letters either upon or as near to the circumference as they can be written, and in the text by placing them in succession without any mark between them. All given points and lines, whether straight or circular, are expressed by Roman letters ; and those points and lines which are either assumed, or found by the operation, are denoted by italics ; given points, unconnected with lines, are denoted by capitals. The extremities of lines are also denoted by capitals, and the intermediate points of division in lines, by small letters. The constructive lines of a diagram are either dotted or drawn very thin, in order to distinguish them from given and required lines, which are much more strongly marked. When the word join immediately precedes two letters, the meaning is to draw a straight line be- tween the points where these two letters are placed ; thus, if it be said, join A B, we are to understand that a straight line is to be drawn between the point at A and the point at B; and this amounts to the same thing as if it had been said, draw A B. When the word bisect is used with regard to a straight line, or the arc of a circle, or an angle, it means that the straight line is to be divided into two equal parts, or the arc is to be divided into two equal arcs, or that the angle is to be divided into two equal angles. And when the word trisect is used, it means to divide the thing to which it is applied into three equal parts, whether it be a straight line, the arc of a circle, or an angle. PROBLEM 1. Between two given points, A and B, to draw a straight line. Lay the straight edge of a ruler upon the point A; then if the edge of the ruler thus applied to the point A coincide also with the point B, draw the point of a pencil or pen from A to B, keeping it always close to the straight edge; then the trace which it leaves upon the surface is the straight line required. But if the straight edge coincides with the point A, and not with the point B, move the ruler round upon A, until the straight edge arrive at the point B; then draw the straight line from A to B as before. Fig. 41 A Sect. II.) 255 PRACTICAL GEOMETRY. PROBLEM II. To produce or extend a straight line to any distance from one of its extremities. Apply the straight edge of a ruler to the extremity of the straight line which is to be produced, and take any other point in the said straight line, so far from the end thus applied, that the distance may be less than the extent of the straight edge; then slide the edge of the ruler upon the two fixed points, until one end of the straight edge arrive at the point taken in the given line; then draw a line close to that part of the straight edge which is not upon the line from the end to be continued, and the given line will be produced as required. The operation may be repeated so as to lengthen a straight line to any extent. PROBLEM III. Round a given point to describe the arc of a circle, or the whole circumference of that circle, with any given radius. Press the legs of the compasses, so that the distance between their extremities may be equal to the radius given ; then place one of the extremities of the legs upon the given point, carry the other leg round; and the trace which it leaves will be the arc or circumference, according as the line traced out by the moving point has two extremities, or none. In this last case it will enclose a space, and will therefore be a plane figure. PROBLEM IV. At a given point C to draw a perpendicular to a given straight line AB. Fig. 45, No. 1. From the given point C, with any radius, describe an arc, so as to cut the line AB in two points d and e. From the points d and e with the same radius, or any other equal radii, describe arcs cutting each other in f, and draw the straight line Cf; then Cf is perpendicular to AB. T Fig. 45, No. 2 1 This problem supposes the line AB to be so long that both the points d and e may fall upon it; but the point*C, through which the perpendicular passes, is not limited to any situation, and con- sequently the problem divides itself into two cases; one where the point C is out of the line AB, and the other where it is in the line AB; but the same description serves alike for each case. In No. 1, the point C is in the line AB; but in No. 2 it is out of it. 256 [PART IV . ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM V. From a given point B, at or near the end of a straight line AB, to erect a perpendicular. METHOD 1. Take any point e above the line AB, and with a radius or dis- Fig. 46. tance eB describe an arc cВd, cutting the straight line AB in the point c: draw a straight line through the point c and the centre e, so as to meet the arc in d; draw the straight line Bd, and Bd will be perpendicular to AB, as required to be done. This problem may be performed on the ground, in the follow- ing manner :- A Take a tape of any convenient length, and double it; fasten one end in the point B, and the other end in e; take hold of it by the middle where it is doubled, and stretch each half, and put a pin in the point e. Loosen the end B, and carry it round the arc to d, so that the point c, e, d, may be in a straight line, and the line cd stretched between c and d: this may be adjusted by the eye of the operator when it comes into the straight line with cand e; then draw the line dB, which will be perpendicular to AB, as required. ل محات METHOD 2. Let AB be a five foot rod, or a line consisting of five equal parts, Fig. 47. and let it be required to draw a perpendicular from the end D of the straight line CD. Make DC equal to four parts; upon D, with a radius of three parts, describe an arc at e; from C, with a radius of five parts, describe an- other arc at e; then through the intersection e and the point D draw the straight line ed, and eD will be perpendicular to CD, as required. AB. This method depends on the 47th proposition of the first Book of Euclid. It is commonly called the Pythagorean theorem, being discovered by Pythagoras, who, it is said, sacrificed a whole hecatomb of oxen on the occasion. Euclid shows that the square of the hypothenuse, or longest side of a right-angled triangle, is equal to the sum of the squares of the other two sides. Now the side Ce being 5, its square is 25, which is equal to the sum of the squares of 4 and 3, which are 16 and 9, making together 25; agreeably to that celebrated theorem. METHOD 3. When the point through which the perpendicular is to pass is above the line. Let E be the point through which the perpendicular is to pass, Fig. 48. and AB the line to which it is to be drawn. In AB take any two convenient points c and d; from c, with the radius cE, describe the arc Ef; and from d, with the radius dE, describe an arc cutting the former arc in f, and draw the straight line Ef, which will be perpendicular to AB, as re- quired. SECT. II.] 257 PRACTICAL GEOMETRY. PROBLEM VI. At a given point E in the straight line DE, to make an angle equal to a given angle ABC. Fig. 49, No. 1. On From the point B with any radius, describe an arc cutting AB at g, and BC in h: from the point E with the same radius describe an arc ik, cutting DE in i; make ik equal to gh, and through the points E, k, draw the straight line EF; then the angle DEF will be equal to the angle ABC, as proposed to be done. La C Fig. 49, No. 2. E PROBLEM VII. Through any given point C, to draw a straight line parallel to a given straight line AB, METHOD 1. Fig. 50. ناھ Fig. 50. In AB take any point f, the more distant from C the better. Join Cf: from f with any radius describe an arc cutting AB at g, and fC at h; from C, with the same radius, describe an arc į k cutting Cf at i; make i k equal to gh; through C and k draw the straight line DE, and DE will be parallel to AB as required. It may be here observed, that this problem depends upon the last. 3 METHOD 2. Through C draw Cf perpendicular to AB, cutting AB Fig. 51. in f; take any point g, the more remote from f the bet- ter, and draw g h also perpendicular to AB; make g h equal to fC; and through the points C and h, draw the straight line DE, which will be parallel to AB as re- quired. N.B. Instead of fC and gh being perpendicular to AB, they may be drawn at any equal angles ; but as the intention of this Treatise is to assist the workman or builder, who is generally provided with a square, parallel ruler, and a right-angled triangle, it is better to draw them perpendicular. 258 (PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM VIII. To draw a straight line at a given distance parallel to a given straight line AB. Fig. 52. ter In AB, take any two points c and d, the more remote the better : from each of these points, with a radius equal to the distance required, describe arcs e and f, and draw the straight line GH to touch the arcs e and f; then GH is parallel to AB. Α ο à B PROBLEM IX. To divide a given line AB into two equal parts by a perpendicular. Fig. 53. From the point A, with any radius greater than the half of AB, describe an arc ; from the point B with the same radius, describe another arc cutting the former in the points e, f, and draw the straight line e f, which will be perpendicular to AB as required. N.B. This operation is called bisecting a given line AB by a per- pendicular. UOJ PROBLEM X. Upon a given straight line AC, as the diameter of a circle, to describe the circumference of that circle. Fig. 54. Ꭰ . Divide AC into two equal parts in the point e: from e as a centre, with the radius e A or eC, describe the circumference ABCD, and the thing required is done. PROBLEM XI. To bisect any arc AB of the circumference of a circle. Fig. 55. Join AB, or suppose it to be joined, and bisect AB by a perpendicular, cutting the given arc AB in d; then the arc AB is divided into two equal arcs dA and dB, as required to be done. Sect. II.] 259 PRACTICAL GEOMETRY. PROBLEM XII. At a given point A in the circumference of a circle, to apply a straight line DE less than the diameter, as a chord to that circle. From the point A, with the radius DE, describe an arc cutting the cir- Fig. 56. cumference in B ; join AB, and AB is a chord to the circle ABC, equal to the straight line DE, as required. -- PROBLEM XIII. Through a given point A in the circumference or arc BAC of a circle, to draw a tangent. METHOD 1. When the centre of the circle is given... Draw a line from the given point A to the centre E, and through A draw FG perpendicular to EA, then FG is the tangent required. 7 Fig. 57. F METHOD 2. Fig. 58 When the centre is inaccessible or not within reach. From the given point A take any two successive arcs Ad, de, and with the chord of the arc Ad, as radius, describe the arc h i ; draw the chord Ae, cutting the arc h i at h: make d i equal to dh, and through the points A and i draw the straight line FG, which is the tangent required. PROBLEM XIV. Through a given point P, without the circumference of a circle ABC, to draw a tangent to that circle. Through the centre E and the given point P, draw EP; and Fig. 59. on EP as a diameter describe the semicircle EDP, cutting the circumference of the given circle ABC in D: join DP, and DP is the tangent required. 260 ART 1 ELEMENTS AND PRACTICE OF GEOMETRY. IV. ... [Part PROBLEM XV. Given a straight line FG, touching the circumference of a given circle ABC, to find the point of contact. Fig. 60. Through the centre E draw EP perpendicular to FG, and cutting FG in P; then P is the point of contact required. W AXIOMS. w Axiom 1. If a circle touch a straight line at a given point, the centre is in another straight line, drawn from that point perpendicular to the given straight line. Axiom 2. If one circle touch another, the centres of the two circles and the point of contact will be in one straight line. Axiom 3. If a circle pass through two given points, the centre is in the perpendicular bisecting the straight line terminated by these points. Axiom 4. If the centre of a circle is in two straight lines, it will be in the point where the two straight lines meet each other. Axiom 5. If any two given points be joined, and the straight line which is extended to these points be bisected by a perpendicular, any point in the perpendicular will be equally distant from each of the two given points. Axiom 6. If a straight line be drawn parallel to one of the sides of a triangle, the straight line will cut the other two sides of the triangle in the same proportion. Axiom 7. If two straight lines cut any number of given parallels, the two straight lines will be divided in the same proportion. Axiom 8. If two parallel straight lines be cut by any number of converging lines, or lines terminat- ing in a point, the intercepted parts of the one parallel will be proportionals to the intercepted parts of the other. D PROBLEM XVI, Through three given points A, B, C, not in a straight line, to describe the circumference of a circle. Fig. 61. Join AB and BC, or suppose them to be joined, and bisect each of the lines AB and BC by a perpendicular, and let the two per- pendiculars meet in d; from the point d with the radius dA, dB, or dC, describe the circumference A B C of a circle, which, pass- ing through any one point, will necessarily pass through the other Iden two. SECT. II.] 261 PRACTICAL GEOMETRY. PROBLEM XVII. Two straight lines AB and CD being given in position, but not in a straight line, to describe the arc of a circle that shall touch them both, and one of them AB in a given point E. Fig. 62. B If AB and CD do not meet, produce each, so that they may both meet in į; make i g equal to iE; and draw Eh perpen- dicular to AB, and g h perpendicular to CD: from h with the distance hE or hg, describe the arc Efg, which was required to be done. PROBLEM XVIII. To describe the arc or circumference of a circle that shall touch a straight line AB in a given point G, and that shall pass through a given point H, not in the same straight line with AB. Fig. 63. Draw Gc perpendicular to AB, and join GH, or suppose it to be joined: bisect GH by a perpendicular c d: from the point c, where the two perpendiculars Gc and dc meet, and with the radius CG, describe the arc fGH, and it will pass through the point H as desired. E DEMONSTRATION. Because Gc is perpendicular to AB, the centre of the circle that will touch the straight line AB is in the perpendicular Gc, by Axiom 1 ; and because the perpendicular cd bisects the straight line GH, the centre of a circle passing through the points G and H is in the line cd, by Axiom 3 ; there- fore, since the centre of the circle is in the two straight lines Gc and dc, it is in the point c of their intersection. PROBLEM XIX. To describe any portion of the circumference of a circle that shall touch a straight line AB in a given point P, and that shall also touch a given circumference or arc FGH. Fig. 64. G 11 Draw Pc perpendicular to AB, and through e the centre of the arc FGH draw Fe parallel to Pc, cutting the circle FGH in F: join FP, and produce FP to meet the arc FGH in q. Join q e, meeting Pc at c, and from c as a centre, with the radius cP, describe an arc which will touch the given circumference or arc FGH at q; then Pq r is the arc, or circumference required. 262 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. DEMONSTRATION. Because Pc is drawn from the point P perpendicular to AB, the centre of the circle which will touch the line AB in P is the straight line Pc ; and because Fe is parallel to Pe, the triangles qeF and qcP are similar ; but because F and q are in the circumference of the circle FGH, Fe and qe are equal, being radii of the circle ; therefore, since qe is equal to eF, the portion qc of eq must be equal to cP; whence a circle described from c with the radius cP will pass through the point q, and touch the straight line AB at P. Now, since the centres e and c of the two circles, and the point q are in one straight line, the two circles will also touch each other at q. PROBLEM XX. To describe two arcs that shall meet each other in the line of their centres, and that shall touch two straight lines DP and ER, at a given point in each line ; and that the arc belonging to the lesser circle shall have a given radius. Fig. 65. D Let R be one of the points of contact, and P the other. From the point of contact P, draw Pa perpendicular to the tangent PD; and from the other point of contact R, draw R6 perpendicular to the other tangent ER. Make Pa and Rb each equal to the given radius, and join a b: bisect a b by a perpen- dicular f c, cutting Pa produced at c: join c b, and produce cb to q. From c, with the radius cP, describe the arc Pq, and from b, with the radius 6 q, describe the arc qR; then will the arcs Pq and qR meet each other at q in the same straight line with their centres c and b, and touch the line ER at R, and PD at P. DEMONSTRATION. Because Rb is perpendicular to ER, the centre of the circle which touches the line ER at R is in the line Rb; for the same reason, because Pa is perpendicular to PD, the centre of the circle which touches the straight line PD at P is in the line Pa, or Pa produced. Again, because ab is bisected by the perpendicular fc, the points a and b will be equally distant from any point in fc ; therefore ca and cb are equal to each other : and because bq is equal to bR, and bR is equal to aP, bq is equal to aP; whence cq is equal to cP; therefore a circle described from c with the radius cP will touch the circle qR at q in the same straight line with the centres 6 and c; likewise the arc qR will touch the straight line ER at R, and the arc Pq, the straight line PD at P. PROBLEM XXI. To divide a straight line AB into any number of equal parts. Through one end B of the straight line AB, draw BC, making any given angle with AB, and through the other end A draw AD parallel to BC ; set as many equal parts upon each of the par- allels, beginning at the point in the given line, as the given line AB is to contain equal parts, and join the one extremity of the given line to the remote extremity of the line which forms an angle with it, and join every two corresponding points in succession of the parallel lines; the lines thus joining will divide the given line AB into the number of parts required. Sect. II.] 263 PRACTICAL GEOMETRY. EXAMPLE Divide the line AB into five equal parts. Fig. 66. Through B draw BC, making any angle with AB, and through A draw AD parallel to BC; from either end A of the line AB set five equal parts upon the parallel AD that joins that end, and from the other end B set five equal parts upon the parallel BC connected with the given line AB at B, and let C be the point where the last part reaches to : join AC; 1, 1; 2, 2 ; 3, 3, &c., and AB will be divided into equal parts at the points a, b, c, &c. Ya b c d PROBLEM XXII. To divide the circumference of a circle into any number of equal parts. It is rather unfortunate for the practice of geometry, that this division can be effected by direct geometrical principles only in a few cases, * and that all these cases cannot be resolved by the same method. The number of primary cases is five, viz. :-a circle may be divided into two, three, four, five, and six equal parts, but we cannot continue the progression any further; it is however evident, that when one of these divisions is found, any multiple of that division, by any term of the geome- may find certain divisions without limitation, the intervals to be filled will be much more numerous than the points of division which can be ascertained by rule: but as these rules, though limited, are of the greatest use to the workman, we shall proceed to show the particular method for effecting each case. * The following method may be seen in many of our modern publications of practical geometry; it is general, and in all cases near the truth, but not exact in any case; indeed not sufficiently exact for a correct drawing. To divide the circumference of a circle into parts which shall be nearly equal to each other from a given point A. Fig. 67. Draw the diameter Ac and the diameter bd at right angles to Ac. Produce od to f, and make df equal to three quarters of the radius: divide the diameter Ac into as many equal parts (by Prob. XXI.), as the number of equal parts required in the circumference; and from the point f, and through the second point 2 of division, draw the straight line fg; then the part Ag of the circumference will be nearly equal to the part required. If the line be drawn through the first point, of course it will cut off a corresponding portion of the 264 (PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. RULE 1. For dividing the circumference of a circle into 2, 4, 8, 16, 32, &c. equal parts, from a given point A in the circumference. Fig. 68. Draw the diameter Ae, and the circumference will be divided into two equal parts, Ae and eA: bisect each of the arcs Ae, eA, or draw the diameter gc at right angles to Ae, and the circumference will be divided into four equal parts Ac, c e, eg, gA: bisect each of these arcs of the fourth part, and the circumference will be divided into eight equal parts Ab, b c, cd, de, &c., and so on, as long as we please to double the last number of equal parts by continual bisection. RULE 2. For dividing the circumference of a circle into 3, 6, 12, 24, &c. equal parts, from a given point A in the circumference. * From the point A, with the radius of the circle, describe an arc cutting Fig. 69. the circumference in b; from b, with the same radius, describe an arc cutting the circumference in c, and so on continually one point after an- other is found, and the circumference will be divided into six equal parts, at the points A, b, c, d, &c. The circle may, by this means, also be divided into three equal parts, by passing over every other division. By bisecting each arc of the sixth part, the whole circumference will be divided into twelve equal parts; and if each arc of the twelfth part be bisected again, the whole circumference will be divided into twenty-four equal parts; and so on, as often as we please to double the last number of equal parts. RULE 3. To divide the circumference of a circle into 5, 10, 20, 40, &c. equal parts, from a given point A in the circumference. Fig. 70. Draw the diameter Ak and the diameter f g at right angles to Ak; bisect the radius Og in p, and from p, with the radius pa, describe the arc Ah cutting f g in h: then the straight line Ah will divide the circumference into five equal parts Ab, b c, cd, de, ea; and by continually bisecting the arcs, we shall arrive at the number of equal parts as proposed by the rule. By the three rules given, we can divide the circumference of a circle into any number of equal parts between one and seven: and it may be generally observed, that, when the primary number of divisions is effected, the circumference may be divided into any 11 Sect. II.] PRACTICAL GEOMETRY. 265 multiple of the parts which is produced by any term of the series 2, 4, 8, 16, &c.; thus, when the primary number is found, that number may be doubled, quadrupled, octupled, &c. by continual bisection of the arcs. RULE 4. To divide the circumference of a circle into any number of equal parts, by trials. Set the points of the compasses or dividers as near to the chord of one of the parts as can be con- jectured or guessed at; then apply this distance as successive chords round the circumference of the circle, and, if the circumference is divided into more parts than are required, the points of the dividers must be made to subtend a greater distance; but if the circumference contain fewer parts, the points of the compasses must be made to subtend a less distance. Proceed in this way by con- tinually correcting the distance last found of the points of the dividers, until the number of equal parts are found. We have no occasion to apply this rule for any number of equal parts less than seven. The greater the number that is required, the more difficult it will be to approximate to the true dis. tance.* * This method of approximation is, however, in practice, the best and most accurate that can be resorted to for dividing lines and circles; for if a line be divided as is shown in the example to Prob. XXI. though the points may be accurately marked in the two parallels, yet there is great chance that, in drawing the lines across, the pencil is not placed exactly at the points, or may not be held precisely upright, or at the same angle during the whole length of the line, which are considerable sources of error. In the practice of perspective, and in drawing and contriving machinery, it often bappens that lines and circles are to be divided into numbers, that render the method of simple trial scarcely practicable. Thus, suppose it were required to divide a circle into 37 parts; now to repeat by trial 37 divisions round the circle is very tedious, and it is hardly possible to effect the division correctly, because a very small inaccuracy in the opening of the compasses, when repeated 37 times, amounts to a great error. The best method in these cases is to subtract such a small number from the number proposed, as will leave one easily di visible, and then, from the length of the whole line of circumference, to calculate what will be the length of the small part so cut off. Set off this part from a scale, or protractor, and then divide the remainder of the circle or line into the parts required by trial. Example. 37 – 1= 36, therefore, by subtracting 1, we produce a number very easily divisible, for 3 X 3 X4= 36. Now suppose that the circle is 7 inches in diameter, the circumference will be (nearly enough for our present purpose) 22 inches, and the 37th part of such circumference will be 34, or ·59 of an inch: therefore, set off ·59 of an inch as one part, and divide the remainder of the circumference into 4 parts, each of these again into 3, and these last also into 3 parts, and the circle will thus be divided into 37 parts. When you have a protractor or sector at hand, all that is necessary is to calculate how many degrees are taken up by the small part of the circle you cut off, and then set such number of degrees on the circumference; after which proceed with the division as before directed. Thus, as the entire circle is 360°, the 37th part of it must be 1900, or 9° 44', or 98 degrees; therefore, set that portion on the circumference, and divide the remainder into 36, by the method already shown. The same method may obviously be pursued with respect to straight lines. Thus, if it be required to divide a line 11 inches long into 37 parts, each part must be H of an inch. Therefore, set off H of an inch for the first part, and proceed to divide the remaining 36 by the most convenient divisors of that number. It is sometimes more advisable to add to a number than to subtract from it in dividing a line; but the method of proceeding in both cases is essentially the same. 2 L 266 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM XXIII. From a given point A, in a given straight line AB unlimited towards B, to find the length of any arc CdE of a circle. Fig. 71. Take any small extent in the compasses, and repeat it . upon the arc CdE as often as it can be contained, and let f be the remote extremity of the last of these parts ; which being done, repeat the same extent from A towards B upon the straight line AB, as often as it has been con- tained upon the arc Cf, and let the extremity of the last Ats Lig be at g. In the straight line AB make gВ equal to the remainder fE of the arc ; then AB is very nearly equal in length to the arc, being somewhat less. It is obvious, that the greater the number of chords or parts contained in the arc, the more exactly will its length be obtained: but though the result of this operation is evidently defective, it is easier, and more to be depended upon, than any other method. It will be satisfactory to show the degree of accuracy obtained by this process. The sum of the sides of a 48-sided polygon inscribed in a circle, of which the diameter is unity, is 3.1393;* and this number is obviously less than the true circumference. The circumference of * It is proved in a Lemma, page 172, Simpson's Geometry, Coroll. the new Edition, 1821, that, if the diameter of a circle is 2, then if the supplemental chord of any arc be added to the number 2, the square root of the sum will be the sup- plemental chord of half that arc. If ACD be a semicircle, and AC the chord, then CD is the supplemental Fig. 72. chord; and if AC be equal to the radius, it will be equal to 1, because AD is equal to 2; therefore in the right-angled triangle ACD we have the hypothenuse AD = 2, and one of the sides AC=l; but it is proved in Euclid 47, book 1, that AD2 = AC2 + CD2; therefore CD2 = AD2 – AC2, that is, CD2 = 22 - 12 = 4 – 1= 3; whence CD =N3= 1.7320508075, which is the supplemental chord of of the semi-circumference; then, by the preceding corollary, 1 Sof the semi-circumference. 2 + 1.7320508075 = 1.9318516525 for the supplemental chord of N 2 + 1.9318516525 = 1.9828897227 for the supplemental chord of the ✓ 2 + 1.982889722%= 1.9957178465 for the supplemental chord of the N 2 + 1.9957178465 = 1.9989291749 for the supplemental chord of 4 &c. &c. Now subtract 3.9828897227, which is the square of the supplemental chord of the 24th part of the semi-circumfer- ence, from 4, the square of the diameter, and there remains 0.0171102773, the square of the chord itself. The square root of this last number is 0.1308062, which is therefore the chord of the 48th part of the whole circle: whence 0.1308062 X 48 = 6.27869, which is the sumn of the sides of a polygon inscribed in a circle consisting of 48 sides, the diameter being 2: therefore, when the diameter is 1, the sum of all the chords will be 3.1393 nearly. SECT. II.] PRACTICAL GEOMETRY. 267 a circle of which the diameter is unity is very nearly 3.1416, which exceeds the sum of the sides of the 48-sided polygon by less than 13bo part of the whole. Now let us suppose the diameter of a circle to be one foot, and, as workmen's rules are generally divided into inches and eighth parts, multiply the decimal part of each of these numbers by 12, and point off the decimals, and the re- mainder will be inches: the decimals of the inches of each number being multiplied by 8, and the decimals of each new product pointed off, we shall have the number of eighth parts; thus, 3.1393 12 3.1416 12 1.6716 1.6992 8 5.3728 5.5936 Therefore the sum of the sides of a polygon of 48 sides, inscribed in a circle whose diameter is 1 foot, is 3 feet, 1 inch and ß, with a fraction of an eighth; and the circumference of the circle is 3 feet, 1 inch and }, with a fraction: the difference of these fractions, however, does not amount to To find very nearly the length of the arc of a circle geometrically, ABC being the given arc. METHOD 1. Fig. 73. Join AC, and bisect AC by a perpendicular BF; find the centre E, if not given, and complete the circle ABCD. Divide the radius DE into four equal parts, and set three of these parts from D to F: through B draw GH parallel to AC; from F, through the points A and C, draw FG and FH: then the straight line GH will be the development or length of the arc, very nearly. METHOD 2. Fig. 74. Produce the chord AC to E, and bisect the arc ABC in B; set twice the chord AB of the half arc from A to D: divide CD into three equal parts, and make DE equal to one of the parts; then AE will be nearly equal to the length of the arc ABC. C DE METHOD 3. Fig. 75. Divide the chord AB into four equal parts; set one part on the arc from B to D: join the remote extremity C of the third part from B to the extremity D of the arc BD: then CD will be nearly equal to half the length of the arc. The first and second of these geometrical methods are much nearer to the truth than the third and last method; but neither of them is so correct, and convenient for practice, as that recommended in the text. 268 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. ts of an inch in the whole circumference. Therefore, by only dividing the circumference of a circle into 48 equal parts, or each quadrant into 12, we come sufficiently near the truth for the purpose of the workman; and were we to divide an arc of a circle (not exceeding a quadrant) of the same diameter, into twenty-four parts, and extend them upon a straight line, we should not lose one- twentieth part of an eighth part of an inch. And thus we see how far the rule may be depended upon. The methods generally given for this purpose, in books of Practical Geometry, are not only erroneous, but the same rule gives the length of the arc either in excess or defect, according to the proportion that subsists between the chord and the versed sine of that arc. APPLICATIONS. EXERCISES TO PROBLEMS XIX. AND XX. , [Plate CIX.] Fig. 1. is termed a lancet arch. It is composed of two curves, meeting in a point at the top, and joining a straight line at each side, in such a manner that the two straight lines are parallel to each other, and tangents to each curve; and that every portion of the two curves, terminated by equal chords from their point of meeting, are equal and similar arcs. Suppose now that it were required to draw each curve by means of circular arcs; and suppose the two arcs next to the point of meeting to be formed to the fancy of the draughtsman, how must the other part be described ? The problem to be performed is this:-- To describe the circumference of a circle that shall touch a straight line AB in a given point P, and that shall also touch a given arc FGH. To execute this problem, the reader is required to read Problem XIX. page 237, with reference to the figure now explained. Fig. 2. is what is commonly denominated an oval. An oval, generally speaking, is an oblong figure formed by a curve line which returns to itself: from this definition of an oval it is evident that there may be an infinite variety of curves so called. The oval which we are about to describe, is a figure resembling an ellipse, compounded of four circular segments, in such a manner, that the opposite segments may be equal portions of equal circles, and that the radii of the two arcs, at the point where they meet, may be in the same straight line. In order to describe this kind of oval, Draw the line PK equal in length to the longest dimension of the figure, and bisect PK by a per- pendicular HL, cutting it in p: make pH and pe each equal to half the breadth of the figure ; then describe a circle or the arc of a circle FGH, in such a manner that the centre may be in the line HL, or in HL produced, and that the circumference may not cut the line PK, but may fall as much without as the draughtsman may think proper ; and draw AB perpendicular to PK, by Problem V. Then, by applying the rules of Problem XIX. successively to each end, we shall be enabled to com- plete the figure, for we have only To describe such a portion of the circumference of a circle that shall touch a straight line AB in a given point P, and that shall also touch a given circumference or arc FGH. One end being described, the other will be found in the same manner, and thus we shall have three quarters of the figure. PRACTICAL GEOMETRY. PLATE 109. EXERCISES TO PROBLEMS, 19,& 20. . -- ---- 12. Fig. I. TIU n H Fig.4. P.lari. . ----TH Fig. 3. ja Fig.5. P. art. P. art. Fig. 6. Fig. 7. Fig. 8. P Invented. & Drawn byt. Nicholson. Engraved by W. Lowry A Fallartani. E'London& Edinburgh Sect. 269 i II. 1 PRÀCTICAL GEOMETRY. Let these three arcs thus found be qHo, the first part described, and qPn, oKn, the two parts now drawn ; these three arcs are terminated by drawing lines through every two of their centres so : as to meet the arcs : the remaining segment mLn being similar to the opposite one, qHo, must be described with the same radius, so as to have its centre i in the straight line LH, and that its curve may pass through the point L. Having now shown some applications of Problem XIX., we shall point out some uses of Problem XX., which applies to all the remaining figures in the plate. · Fig. 3 is another lancet arch, supposed to have the same properties as that already defined; but here the angle GPI at the vertex is given ; it must, however, always be greater than the angle formed by the lines extending from each extremity of the chord RH to the summit P. : To prepare the figure for the application of the above Problems, draw Pc perpendicular to PG, by Problem V., and let R be the point which divides the arch from the straight line on one side, or what is denominated one of the springing points of the arch, and let RH meet the tangent RE in R: then, to describe the half of the arch PqR, we have only, by Problem XX., To describe two arcs which shall meet each other in the line of their centres, and that shall touch two straight lines DP, ER, at a given point in each line, and that the arc belonging to the lesser circle shall have a given radius. The other half is described in the same manner; or through c, the centre of the arc Pq, draw cg parallel to RH the base of the arch. From the summit P, draw Pk parallel to Rk or Hl, meeting gc in k. Make kg equal to kc, and make Hn equal to Rb, and draw gnm : then g is the centre for the arc Pm, and n the centre for the arc in H. Fig. 4 is another oval, having the same properties as that which stands above it. To prepare this figure : Draw the straight line RH, equal to the length of the oval. By Problem IX., bisect RH by the perpendicular PK: then set off half the breadth on each side of the centre, and the remote extremities P and K are the extremities of the breadth. In order that our oval may be a tolerably good representation of the mathematical curve called an ellipse, divide the difference between half the length and half the breadth into two equal parts, and set three from the centre upon the line RH on each side of it to b and i ; then, by Problem XX., Describe two arcs that shall meet each other in the line of their centres, and that shall touch a straight line DP in P, and ER in R, and that the arc belonging to the lesser circle shall have a given radius. Having finished one end, the other will be described in the same manner, and the remaining side by the same radius as the first side opposite. Fig. 5 is a semi-oval, drawn in the same manner as the figure above it. The curves of the three figures at the bottom of the plate are each composed of two circular arcs. Fig. 6 is an imitation of the section of a Grecian ovolo; Fig. 7 and 8 are in imitation of the sec- tion of a scotia. In both these first figures the lower edge of the fillet is a tangent to the upper part of the curve ; and in Fig. 8, the upper edge of the fillet is a tangent to the curve at its lower extremity. Suppose now that the arc Rq is described to fancy ; then, if it touch the lower edge RE of the fillet at R, and if Rb be perpendicular to RE, the centre of the circle which will touch the line RE at R will be in the line Rb; therefore, from a centre in the line Rb, with any radius which the operator may think proper, describe the arc. Draw Pc perpendicular to DP, In Pc cut off Pa equal to the radius of the lesser circle, and join a b. Bisect a b by a perpendicular fc. Draw cq through the point of intersection c and the centre b. From c with the radius cq describe the arc qP, and this will complete the scotias: but without describing any figure particularly, as the lines DP and ER are supposed to be given, and the points of contact R and P, therefore in all the figures of this plate, except the two uppermost, we have only to execute Problem XX., which is, 270 '[Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. To describe two arcs that shall meet each other in the line of their centres, and that shall touch two straight lines DP and ER at a given point in each line, and that the arc belonging to the lesser circle shall have a given radius. PROBLEM XXIV. To divide an angle into any number of equal angles. This problem, like that of the division of the circle, cannot be generally effected except by con- tinual bisections; in this case we may divide an angle geometrically into two, four, eight, sixteen, &c. equal parts, by first describing an arc, and, if we wish to divide the angle into two equal parts, we must bisect that arc; if it were required to divide the angle into four equal parts, we must bisect each half arc; if it were required to divide the angle into eight equal parts, we must bisect each quarter of the arc, and so on to the proposed number of parts required. EXAMPLES Fig. 76. Ex. 1. Bisect the given angle ABC. From the angular point B describe an arc cd cutting AB at c, and BC at d, and bisect the arc c d by the straight line Bf : then will the angle ABC be divided into two equal angles ABf and fBC. . Bed Be Ex. 2. Quadrisect or divide the angle ABC into four equal angles. Fig. 77. From B, the point of the angle, as a centre, describe the arc de, cutting AB at d, and CB at e; and bisect the arc de by the line Bf, cutting the arc d e in g: bisect the arcs dg and ge by the lines Bh and Bi, then will the angle ABC be divided into the four equal angles A Bh, hBf, fBi, and iBC. Ex. 3. Divide the angle ABC into eight equal parts. Fig. 78. From the angular point B describe the arc d e, cutting BA at d, and BC at e, and bisect the angle ABC by the straight line Bf cutting the arc d e in g : bisect the arcs dg and ge by the straight lines Bh and Bi, cutting the are de in the points k and l: bisect the arcs d k, k g, gl, and le, by the straight lines Bm, Bn, Bo, Bp, and the angle ABC will be divided into the eight equal angles A Bm, mBh, hBn, nBf, fBo, oBi, iBp, and pBC, as required to be done. SCHOLIUM. angles in the geometrical progression, 2, 4, 8, 16, 32, 64, &c. Sect. II.) 271 PRACTICAL GEOMETRY. Though an angle cannot be geometrically divided into any number of equal parts, the problem is not impossible, as it may be done by the resolution of algebraic equations of the third, fourth, fifth, &c. degree. Analytical principles are, however, too difficult to be acquired by the generality of readers, who must content themselves with the method of approximation, as in the case of dividing a circle into equal parts. The practice of the operator will enable him to ascertain, by a few trials, the portion of the arc that will divide the whole into the required number of parts; and, conse- quently, to divide the angle into the same number of equal parts, by drawing straight lines from the angular point through every point of division. The trisection of a right angle is of considerable use, and, as it can be geometrically constructed, we shall show how it is to be done in the following pro- position. PROBLEM XXV, To trisect or divide a right angle ABC into three equal parts. Fig. 79. C1 From the angular point B describe the arc de, cutting BA at d, and BC at e. From d, with the radius of the arc, describe another arc, cut- ting the arc d e at f, and from e, with the same radius, describe another arc cutting the arc de in g: join Bf and Bg, and the angle ABC is divided into the three equal angles A Bg, gBf, fBC. . PROBLEM XXVI. To inscribe a polygon of any given number of sides in a given circle, and from any point in that circle. Divide the circumference of the circle into as many equal parts as the polygon is to contain sides, beginning at the given point: join any point thus found to the next point of division, and proceed progressively, always joining the next point to the end of the chord last drawn, until one chord only remains to be drawn; then draw that chord from the extremity of the chord last drawn to that of the chord first drawn, and the polygon will be formed. Examples in a hexagon, octagon, and decagon. Fig. 80, No. 1. Fig. 80, No. 2. Fig. 80, No. 3. ООО 272 [Part IV ELEMENTS AND PRACTICE OF GEOMETRY. Examples in a pentagon, heptagon, and enneagon. Fig. 81, No. 1. Fig. 81, No. 2. Fig. 81, No. 3. * *** PROBLEM XXVII. Upon a given straight line AB to describe an equilateral triangle. Fig. 82. From the centre A, with the radius AB, describe an arc; from the centre B, with the same radius, describe another arc cutting the for- mer; from the point C, where the two arcs meet, draw the straight lines CA and CB, and ABC is the equilateral triangle required. This is the first proposition in Euclid's Elements. PROBLEM XXVIII. Upon a given straight line AB to describe a square. Fig. 33. Draw the straight line BC perpendicular to AB; make BC equal to AB: from A, with the radius AB or BC, describe an arc; and from C, with the same radius, describe another arc, cutting the former in the point D: join AD and DC; then ABCD is the square required. This is the forty-sixth proposition of the first book of Euclid's Elements. After having drawn the right angle A BC, Euclid directs that the straight line CD be drawn parallel to AB, and AD parallel to BC. PROBLEM XXIX. Upon a given straight line AB to describe a polygon of any given number of sides. Produce the straight line AB to K, and on AK, with the radius AB, describe the semicircle ACK. Divide the arc ACK into as many equal parts as the number of sides in the proposed poly- gon: join the second point of division C to the centre B; bisect each of the sides AB and BC by perpendiculars meeting each other in I; from the centre I, with the radius IA, IB, or IC, describe a circle, which being thus made to pass through any one of the three points. A, B, C, will necessarily SECT. II.] PRACTICAL GEOMETRY. 273 pass through all the others; therefore we may begin at C or A, and apply the chords each equal to AB or BC to the remaining part of the circumference.* The following are examples in a pentagon, hexagon, and heptagon. Fig. 84, No. 1. Fig. 84, No. 2. .. . Ot.. 10. . Fig. 84, No. 3. . * There is another method of executing this problem; and though the principle is not correct, it may not be amiss to show the process and construction, as the beauty of the scheme has so frequently attracted the notice of the student. Let AB be the given side. Fig. 85. Upon the centre A, with the radius AB, describe an arc BK, and from B, with the same radius, describe an arc AK: through K draw PQ perpendicular to AB: divide either of the arcs, as BK, into six equal parts. Then if it were required to describe a circle that should contain AB six times, there is nothing more required than to describe a circle from K with the radius KA or KB; in this case, the rule is exact. But, if a greater or less number of parts were required, we must describe an arc from K with a radius equal to the chord of as many parts of the arc BK as the number required exceeds six, or equal to the chord of as many parts of the arc BK, as six exceeds the number of sides required, and cross the line PQ above or below K accordingly; then from the point thus found in PQ as a centre, with a radius extending to either of the points A or B, describe a circle, and it will AP B contain the side AB as required. 2 M 274 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM XXX. To describe a triangle of which the three sides shall be each respectively equal to each of three given lines, which lines must be such that any two of them taken together are greater than the third. Fig. 86. Let A, B, C, be the three given straight lines. Draw the straight line DE, and make it equal in length to A: with the radius B, from the centre D, describe an arc, and with the radius C from the centre E describe another arc, cutting the former arc at F: join DF and FE, and DEF is the triangle required. AL BA Pow PROBLEM XXXI. To describe a polygon equal and similar to a given polygon upon a given straight line, equal and corresponding to one of the sides of the given polygon. Fig. 87, No. 1. Divide the given polygon into as many triangles, wanting two, as the given figure has sides, so that no space may remain but what is entirely resolved into triangles. Upon the given side describe a triangle equal and similar to the triangle upon the corresponding side of the given polygon; then describe the re- maining triangles to succeed each other in the same manner as in the given figure, and the figure thus drawn will be equal and similar to the one proposed. In the example here given, ABCDE is the given polygon, and FG the side of the required polygon, corresponding to AB of the given polygon. Here the given polygon is resolved into the triangles ABC, ACD, and ADE; then, in the polygon to be constructed, the triangle FGH is described by Problem XXX. equal and similar to the triangle ABC; the triangle FHI, equal and similar to the triangle ACD; the triangle FIK, equal and similar to the triangle ADE; and thus the whole figure FGHIK is described equal and similar to the given figure ABCDE.* Fig. 87, No. 2. * It is by the application of this problem that we are enabled to make a plan of any proposed place, or to take the dimensions for executing any piece of work. Suppose, for instance, that it were required to make the plan of a quad- rangular room: a rough drawing must first be made of the four sides; then each side of the room must be measured, and the dimension placed upon each corresponding side of the draught: but these four sides will not be sufficient, as the angles of the figure may vary; we must therefore take one of the diagonals also, and this will resolve the figure into two triangles, which form a compound figure comprehended by the rule of this problem, SECT. II.] 275 PRACTICAL GEOMETRY. PROBLEM XXXII. Y To describe a triangle similar to a given triangle upon a given straight line, corresponding to one of the sides of the given triangle. Make an angle at each extremity of the given straight line equal Fig. 88, No. 1. to the angle at each extremity of the corresponding line (Prob. VI.) of the given triangle, and on the same side of the given line with the corresponding angles of the given triangle: produce the leg of each angle which is not the given side till both meet, and the triangle thus formed will be similar to the given triangle. Thus, let ABC be the given triangle, and DE the given straight B. line. Fig. 88, No. 2. At the point D, make the angle EDF equal to the angle BAC, and at the point E, make an angle DEF equal to the angle ABC; then, if the two legs of the angles meet in F, the thing required is done ; but if not, produce these two legs till they meet in F, and the triangle DEF will be the triangle required. If the two first corresponding sides are parallel, every other pair D4 of corresponding sides will also be parallel; and consequently in this case the required triangle may be constructed by drawing parallel lines. -- - PROBLEM XXXIII. To construct a rectilineal figure similar to a given rectilineal figure upon a given side, corresponding to a side of the given figure. Describe a triangle upon the given side similar to that upon the corresponding side of the given figure : then describe the next triangle similar to that in the given figure, and so on, one triangle after another, upon its succeeding corresponding line, until all the triangles are described; then the figure thus formed will be similar to the figure proposed. · EXAMPLE. Fig 89, No 1. Fig. 89, No. 2. In the two figures here exhibited, ABCDEF is that which is given; a b c d e f is the figure required to be constructed; and a b is a side corresponding to the side AB. First, the tri- angle a b.c is described similar to the triangle ABC; the triangle a c d is described next similar to the triangle ACD, and so on. 276 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM XXXIV. Upon a given straight line AC to describe the segment of a circle that shall contain a given angle H. Fig. 90, No. 1. Bisect AC by a perpendicular EF, and draw AG, making the angle CAG equal to H: draw AE perpendicular to AG. From the centre E, and distance EA, describe the arc ABC: then, if any point B be taken in the arc, and the lines BA and BC be joined, the angle ABC will be equal to the given angle H. For the angle contained by a chord and a tangent at the extremity of that chord is equal to the angle in the alternate segment. (Euclid, book iii. prop. 32.) Fig. 90, No. 2 B PROBLEM XXXV. In a given circle DGFE to inscribe a triangle similar to a given triangle ABC. Fig. 91, No. 1. Fig. 92, No. 2 From any point D in the circumference draw any chord DE, and draw the chord DF, making the angle EDF equal to the angle ABC. Join EF; then, if the angle FED be equal to the angle CAB, the thing is done ; but if not, draw the chord EG, making the angle FEG equal to the angle CAB, and join GF; then EGF is the triangle required. SCHOLIUM. Though the problem of describing the segment of a circle is comprehended in that of making the circumference pass through three given points (Prob. XVI.); yet, as it is more convenient in prac- tice to take the chord and versed sine of the arc, than to find the position of three points, we shall here show how this is to be done. Sect. II.) 277 PRACTICAL GEOMETRY. PROBLEM XXXVI. The chord AC and versed sine of the segment of a circle being given in magnitude, to describe the arc. METHOD 1. Fig. 92. Bisect AC in E by a perpendicular BD; make EB equal to the versed sine, and join AB. Make the angle BAD equal to the angle ABD. From the centre D, with the distance of any one of the two lines DA, DB, describe the arc ABC, which will necessarily pass through the third point C, and therefore must pass through all the three points. METHOD 2. Fig. 93. · Bisect AC in E by a perpendicular BD, and join AB as before. Bisect AB by a perpendicular, meeting BD in D: then with the radius equal to any one of the three distances DA, DB, DC, de- scribe an arc ABC, which will necessarily pass through all the three points. PROBLEM XXXVII. ww To find the point of prolongation of one extremity C of the arc ABC of a circle, so as to meet a given straight line AE passing through the other extremity A, without making use of the centre. Fig. 94. d From A draw any chord AB; and draw the line Ad to cut the arc in d, so that the angle BAd may be equal to the angle BAE. From the centre B, with the distance Bd, describe an arc cutting AE at E, and E will be the point required. - SCHOLIUM. By these means, an arc, which is too small a portion of the circumference to answer the intended purpose, may be prolonged when the centre is inaccessible, or when it is so remote that it would be inconvenient to use it. This is a case that frequently occurs in practice. 278 [Part IV ELEMENTS AND PRACTICE OF GEOMETRY. CHAP. II.-OF RATIO AND PROPORTION. DEFINITIONS. 1. A ratio is the relation between two things, such as lines or magnitudes of any description or quantity, found by considering how often the one is contained in the other, or how many equal parts the one may be of the other. 2. The first quantity which contains the other is called the antecedent or leading quantity; and that which is contained is called the consequent or following quantity. 3. An analogy or proportion is the equality of two ratios, or the comparison of four quantities, of which the first must contain the second, or the like parts of it, as often as the third contains the fourth, or the like parts of it. Thus, four lines A, B, C, D, are proportional, when if A and C Fig. 95. be divided into the same number of equal parts, B contains as many parts of the scale A, as D contains of the scale C. B__ D 4. When four lines are proportional, the lines themselves are called proportionals. 5. Each of the four proportionals is called a term. 6. Of four proportionals, the first and last terms are called the extremes, and the two terms between are called the means. 7. When the two middle terms happen to be equal, the four terms having thus the two middle ones equal, are called three proportionals. 8. A fourth proportional to three given lines, is a line of such length that it may contain the like parts of the third which the second does of the first. 9. A third proportional to two given lines, is a line of such length that it may contain the like parts of the second which the second contains of the first. This is, in fact, a fourth proportional to three lines, when the second and third lines are equal. 10. When four lines are proportionals, the first is said to be to the second as the third is to the fourth. 11. When three lines are proportionals, the first is said to be to the second as the second is to the third. 12. A mean proportional is one of the equal means in three proportional lines; that is, it is a line of such length, that the first line is to the mean, as the mean is to the third line. A knowledge of the proportionality of lines is of the greatest use not only to practical mechanics, but to architects and to builders' clerks: for it is upon this principle that all scales are made, and objects enlarged or diminished, and that drawings and models are made to represent buildings PROBLEM XXXVIII. Given three straight lines to find a fourth proportionat. Observe first, that all lines applied to the legs of an angle are applied from the point of concourse, and the points set off on the legs are called points of extension ; this being understood, we may proceed. - - - - - - Sect. 11.] PRÁCTICAL GEOMETRY. 279 Draw two straight lines at any angle, calling one of the lines the first leg, and the other the second leg. Apply the first and second of the lines upon the first leg, and the third line upon the second leg: join the points of extension of the first and third lines; and through the point of exten- sion of the second line, draw a line parallel to the line which connects the two legs, of sufficient length to reach the other leg; then the distance between the point of intersection now made and the point of concourse of the angle is the fourth proportional required. EXAMPLES Ex. 1. Given the three straight lines A, B, C, to find a fourth proportional. ther . A B Fig. 96. C Draw any angle FDH; apply the straight line A from D to F, and the straight line B from D to E: apply C from D to G; join FG, and draw Eh parallel to FG, cutting DG in H; then DH is the fourth proportional required. Ex. 2. Given the three straight lines A, B, C, to find a fourth proportional. AT III Fig. 97. A B C Having, as before, made the angle EDH, apply the straight line A from D to F, and the straight line B from D to E, and the straight line C from D to G: join FG, the points of extension of the first and third terms, and through E draw EH parallel to FG, cutting DG produced in H: then DH is the fourth proportional. PROBLEM XXXIX. To find a third proportional to two given lines. Having drawn an angle as in the case of four proportionals, apply the first and second of the two given lines upon the first leg, and the second line also on the second leg: join the point of extension of the first line to that of the second on the second leg; and through the point of extension of the second line on the first leg, draw a line parallel to the line extending between the two legs so as to cut the second leg; and the distance of the intersection from the point of concourse of the angle is the third proportional required. 280 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. EXAMPLE Find a third proportional to the lines A and B. Fig. 98. A B Having, as before, drawn the angle DCG; apply the two given lines A and B respectively from C to D and E upon the first leg, and the line B upon the second leg from C to G: join DG, and through E draw EF parallel to DG, cutting CG in F, and CF is the third proportional. U SCHOLIUM 1. The method of finding a third proportional, as has been observed, can hardly be called a distinct problem from that of finding a fourth proportional; the former being only a particular case of the latter, when the two middle terms happen to be equal; but from the frequent occurrence of three proportionals in practice, it is here given under a distinct head, as it is done in most mathematical works. Proportion is not only useful in finding a third and fourth term, but also in dividing a line, so that the ratio of every two parts may be equal to the ratio of every two correspondent parts of a given line; or, in other words, the dividing of a line into as many parts as another line, and in the same proportion as the corresponding parts of that other line, which is shortly and generally termed “ dividing a line in the same proportion as another." PROBLEM XL. To find a mean proportional between two given straight lines. LI Place the two straight lines in one straight line, and mark the point of division. On the line compounded of the two lines, as a diameter, describe a semicircle; from the point of division draw a perpendicular to the straight line to meet the arc, and the perpendicular will be the mean pro- portional required. EXAMPLE Find a mean proportional between the two straight lines A and B. Fig. 99. Draw the straight line CE, on which set CD equal to A, and DE equal to B. On CE, as a diameter, describe the semicircle CFE, and from the point D, where the two lines A and B join, draw DF perpen- dicular to CE, meeting the arc of the semicircle in F; then DF is the mean proportional required. BL Sect. II.] 281 PRACTICAL GEOMETRY. 281 PROBLEM XLI. To divide a straight line into as many parts as another given line, so that every pair of corresponding parts, one of them being taken from each line, shall have an equal ratio to any other such pair. METHOD 1. Draw two straight lines forming an angle as formerly; place the given divided line upon the first leg, and mark the points of division, and place the undivided line on the second leg: join the ex- tremities of the two lines, and through all the points of division in the first leg draw lines parallel to the connecting line, to cut the undivided leg : the leg now divided will have all its parts in the same proportion as the given divided line. EXAMPLE TO METHOD 1. Divide the line B in the same proportional parts as the line A. Fig. 100. Draw any angle HCN, and place the given divided line A upon the leg CH with its divisions, and place the other undivided line B upon the leg CN; join HN, and through the points d, e, f, g, of division, draw the lines di, ek, fl, gm, parallel to HN: the line CN, equal to the un- divided line B, will be divided in the same proportion as the divided line A. METHOD 2. D Place the two given lines parallel to each other; join each pair of their extremities, and prolong the lines thus joining them, till they meet : from the point of intersection draw lines through all the points of division of the divided line, and prolong them, if necessary, to meet the undivided line, and the undivided line will thus be divided into parts which shall have the same proportion to each other as the parts of the divided line. " EXAMPLE Fig. 101. Draw DE equal to A, and FG parallel to DE equal to B. Mark the points i, k, corresponding to the divisions of A: join DF and EG, and produce the lines DF and EG till they meet in H. Draw the lines Hi, Hk, cutting FG in 1, m; then FG, which is equal to B, is divided in the same pro- portion as the line DE, equal to A.-See Note* in next page. 1 2 N 282 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. METHOD 3. When the same proportion is frequently required in practice, the following method, which is founded on the same principle, will perhaps be the most eligible. Fig. 102. to the greato. Describe an equilateral triangle LAE, of which the side is equal to the greatest of any lines that it may be required to divide; then sup- pose the side AE to be divided in the proportion required, and the dividing lines bL, CL, DL, drawn: let it be required to divide the line E to be divided in I one required to divi LA and LE, mark off LF and LK, each equal to the line M, and join FK ; then FK will be divided in the same proportion as AE.* , А с а METHOD 4. Fig. 103. n Draw the parallels DE, FG, respectively equal to the lines A and B, and mark the divisions of the line A on DE: join DG and EF, cutting each other in H; from the points f, k, l, draw lines through H to meet the line FG in o, n, m; then will the line GF, equal to B, be divided in the same proportion as the line DE equal to A. 041 Az METHOD 5. BY PARALLEL LINES. When the undivided line is greater than the divided line. Upon any convenient surface place the divided line, and through its extremities and its points of division draw parallel lines: from any point in one of the extreme lines, with the radius of the undi- vided line, describe an arc cutting the other extreme line ; then a line drawn from the centre to the point of intersection will be divided into the proportion required. Fig. 104. * Demonstration of the second and third methods. Suppose the side AC of the triangle AGC to be divided in any given proportion AB, bC; and the line bG drawn; and let DF parallel to AC, cutting AG in D, and CG in F, be cut in the point e by bG; then will Ab be to bС, as De, eF For, by the similar triangles, GbA, GeD, Gb : bA :: Ge : eD, and by the similar triangles, GhC, GeF, Gb : 6C :: Ge : eF; therefore, by equality of ratios, Ab : 5C :: De : eF. Sect. II.] 283 PRACTICAL GEOMETRY. EXAMPLE Divide the line B in the same proportion as the line A. Fig. 105. . A B RM Draw the line CG equal to the divided line, and mark its divisions at the points d, e, f: through the points C, d, e, f, G, draw the lines CH, d i, e k, fl, GM, at any convenient angle parallel to each other. From any point N in CH, with the radius of the undivided line B, describe an arc cutting GM at R, and join NR, which is equal to the given line B; then vill NR be divided at the points o, p, q, in the same ratio as the line A. o SCHOLIUM It will be observed, that, when the contrary ends of the lines are joined, the point of intersection falls between the parallels, and therefore the divisions on the parallels have contrary positions to each other ; but this cannot be called an inconvenience, since the parts of the divided line can be made to proceed from either end of the line DE, so as to make the progression begin as may be required in FG. This method will be most convenient when the two lines are nearly of the same length; because, in this case, the space which contains the figure will not be very great. PROBLEM XLII. To divide the space between two parallel lines into any number of spaces of equal breadth, by parallel lines. Repeat any convenient distance upon a separate line as many times as shall equal the number of parts into which it is intended to divide the space between the two parallel lines, but so that the length of all such distances taken together shall not be less than the space between the two parallel lines. Take the whole length of such distances as a radius, and from any point in one of the parallel lines describe an arc cutting the other, and join the centre and the point of intersection ; then, upon the line thus joining the parallels, set off the divisions of the line of equal parts, and through the points of division draw lines parallel to either of the two given lines, and the distance between the two given parallel lines will be every where divided into the number of equal parts required. EXAMPLE Fig. 106. H_ B To divide the space between the two parallel lines AB, CD, into six equal parts. Set off six equal parts upon the line EF, so that the whole of the parts EF, may not be less than the distance between AB and CD. From any point G, in CD, with the radius EF, describe an arc meeting or cutting AB at H, and join GH; transfer the equal parts of the line EF to GH, and through the points of division draw lines parallel to AB or CD, and these lines will divide the space as required. Errentenentementenbond F 284 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. INN 11VAYIN SCHOLIUM. Upon this principle, a Carpenter may divide the breadth of a board into any number of parallel slips, all equal in breadth, or in any given proportion to each other. Thus, let ABCD be a board, Fig. 107. and let it be required to di- vide it into five laths or bat- tens of equal breadth. Upon the edge of the board set off five equal parts of any convenient length from E to É 1 2 3 # F F: from any point E, with the radius EF, describe an arc cutting the opposite side of the board at g, and join Eg. Transfer the divisions of the line EF to the line EG ; and through the points of division 1, 2, 3, 4, draw the parallel lines h i, k l, mn, o p, and the board will be divided into five equal slips as required. It is plain that the distance EF must be less than the breadth of the board: if any other proportions are required, the process will be exactly similar. Instead of using the line EF, the number of parts may be set off on a small slip of wood, or upon the edge of a two-feet rule, and transferred at once to the line Eg. MN PROBLEM XLIII. To find a series of lines in continued proportion from two given lines. Having formed an angle, place one of the given lines from the point of the angle upon one of the legs, and the other line upon the other leg; and join the points of extension. From the angular point cut off a distance from the leg on which the longest line was placed, equal to the shorter line ; and through the point of division draw a line parallel to the line joining the two legs, cutting the leg on which the shorter of the two given lines was placed. Proceed in the same manner, always cutting off a distance from the angular point upon the leg on which the longest given line was placed, equal to the distance from the angular point to the point on the other leg cut by the parallel last drawn: then, from the point of intersection draw another parallel as before, and proceed in the same way until the process is carried as far as may be required. The longest leg of the angle will thus be divided into a series of continued proportionals. EXAMPLE. Find a series of lines in continued proportion from the two given lines M and N. Having drawn the angle ALF, make LA equal to M, and LF equal Fig. 108. to N, and join AF. Make Lb equal to LF, and draw b g parallel to M N AF, cutting LF in g. Make Lc equal to Lg, and draw ch parallel to bg, and so on, as far as may be necessary.* k g * When LA, Lb, Lc, Ld, &c. are in continued proportion, their differences Ab, bc, cd, de, &c. are also in continued proportion. The truth of this assertion will be shown in the following demonstration : SECT. 11.) 285 PRÀCTICAL GEOMETRY. PROBLEM XLIV. From a given straight line to cut off any part of it. METHOD I. Draw two straight lines at any given angle as before. On one of the legs repeat as many times any convenient length, from the angular point, as the part required to be cut off the given line is contained in the whole line, and on the other leg set off the length of the given line; join the two the line extending between the extremities of the two legs, to cut the other leg; then the distance between the angular point and the intersection will be equal to the part required to be cut off from the given line. EXAMPLE Let it be required to take one-fourth part of the line F. Fig. 109. On the leg AC of the angle CAE, set off four equal parts from A to C, and make AE on the other leg equal to the given line F. Join CE, and through b, the first point of division, draw bd par- allel to CE, cutting AE in d; then Ad is one-fourth part of the given line F. METHOD 2. Draw any line parallel to the given line; and on the parallel thus drawn, repeat any convenient distance as often as the portion to be cut off from the given line is contained in the whole line. Let LA = 0, Lb = 6, Lc=c, Ld=d, Lee, &c.; then, hy the definition of continued proportion, a:b::b:c 6:0::c:d cid::d:e &c. Then, by subtracting the alternate terms, lab: bc:: bc:cmd b:c::b-:0-d}; therefore b-C:6-d::c-did- c:d::c-d:de) . &c. But a-b=La - Lb= Ab; b-=Lb — Lc=bc; c-d=L- Ld=cd, &c.; therefore Ab, b c, cd, de, &c. are in continued proportion, as well as La, Lb, Lc, Ld, Le, &c. This also admits of a demonstration by numbers : for, let the whole line AL be 32 inches long, and (to avoid fractions) we will suppose Lb equal to 16 inches; then will Lc be 8 inches; Ld 4 inches ; Le 2 inches, &c. Then the proportion is thus expressed, 32: 16:: 16:8::8:4:: 4:2, &c. Now if we subtract each term in this series from that next above it, we obtain the differences, which are also in con- tinued proportion, namely, 16:8::8: 4:;: 4: 2, &C. 286 PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. Draw a straight line through each two corresponding ends of the given line and its parallel; and from the point where the two lines meet, draw another straight line through the first division of the parallel, to cut the given line; and the part of the given line thus cut off, will be the portion required. EXAMPLE. Let it be required to cut off one-fifth part of the line AB. Fig. 110. Draw CD parallel to AB; from C to D set off five equal parts: draw AC and BD meeting at E; from E through f, draw the line Ef g, cutting AB at g; then will Ag be a fifth part of the whole line AB as required. PROBLEM XLV. To find any fractional part whatever of a given line, or to form an accurate scale. From the extremity of the given line draw a straight line at any angle with it. From the point of concourse on the line thus drawn, set off as many equal parts as the number of parts which the given line is intended to contain, and join the unconnected extremities of the last part, and that of the given line: through the points of division draw lines parallel to the given line to cut the con- necting line. EXAMPLES Ex. 1. Find any number of sevenths of the line AB. Fig. 111. Draw BC, and mark off seven equal parts from B to C; join AC; through the points of division draw lines parallel to AB to meet the line AC; then d e will be one-seventh part of the line AB, fg two-sevenths, h i three-sevenths, &c. f/-e hi AL Ex. 2. Find any number of tenths of the line AB. Draw AC, and set ten equal parts on the line AC; join BC, and through the points of division in AC, draw lines parallel to AB, to cut the line BC; Fig. 112. С аде It is upon this principle that diagonal scales are constructed. hui Sect. II.) 287 PRACTICAL GEOMETRY. PROBLEM XLVI. Given one side of a triangle, and one of the angles adjacent to that side, to complete the triangle, so that the other two sides may have a given ratio. Fig. 113. Let AB be the given side of the triangle, BAC the given angle, and let the lines M and N be the ratio of the other two sides. In the indefinite leg AC of the angle; make Ae equal to M; from the point e, with a radius equal to N, describe an arc cutting AB at d: join e d, and through B draw CB parallel to e d; then ABC is the triangle required. * M PROBLEM XLVII. Upon a given straight line to describe a trapezoid, so that the two sides which join the given side may be equal to each other, and the intermediate side may be parallel to the given side and at a given distance from it, and in a given ratio with either of its adjacent sides. Fig. 114. D8_f Let AB be the given straight line, and let the ratio which the middle side to be described is to have to each of its adjacent sides be as MN to PQ. Bisect AB by the perpendicular e f cutting it in e, and make e f equal to the distance of the opposite side ; join Af, and through f draw DC parallel to AB. Bisect MN in k, and from the point f in the line CD make f g equal to Mk or kN, and from g with the distance PQ, describe an arc cutting Af in h. Draw AD parallel to hg; make fC equal to fD, and join BC; then ABCD is the trape- zoid required. Pa SCHOLIUM. The operation, after having made the perpendicular e f, and having drawn DC parallel to AB, and having joined Af, is the same as in Problem XLVI.; for here are given the base Af, the angle AfD, and the ratio of the sides Mk, PQ, to describe the triangle AfD. The use of this Problem is to describe the plan of a prismatic bow of a building, which may either have its three visible sides equal to each other in breadth, or in any given ratio to each other. , * DEMONSTRATION. Because (in the figure given in the text) Ae is equal to M, and e d is equal to N, therefore Ae is to e d as M is to N; but because BC is parallel to de, Ae is to ed as AC to CB; therefore AC is to CB as M is to N; and the triangle ABC is described as required. 288 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM XLVIII. A square being given, to inscribe a regular octagon which shall have four of its sides portions of the four sides of the square. METHOD. 1. Fig. 115. h PY Let ABCD be the given square ; bisect any two adjacent sides AB and AD by the perpendiculars fh and eg, cutting each other in the centre s: make se, sf, sg, s h, each equal to sА, sB, SC, or sD: join e f, cutting the two adjacent sides AD, AB, in L and M; join fg cutting AB in N, and BC in 0 : join g h, cutting BC in P, and CD in Q; and lastly, join he, cutting CD in I, and DA in K; then will IKLMNOPQ be the octagon required. S METHOD 2. Fig. 116. Draw the two diagonals AC, BD, cutting each other in s, as before ; then each of the diagonals will be bisected by the point s. With a dis- tance equal to As half of a diagonal, cut off from the four angular points A, B, C, D, the distances AK, AP, BI, BM, CL, CO, DN, DQ, from each of the sides of the square, and join KL, MN, OP, QI; then will KLMNOPQI be the octagon required. * Fig. 117. Join Ks; then AK being equal to As, the two angles AKs, AsK are equal to each other; and, since the three angles of every triangle are equal to two right angles, and the angle KAs is half a right angle, each of the two angles AsK, AKs must be three quarters of a right angle; consequently the angle KsB must be a quarter of a right angle. For the same reason the angle AsI must also be a quarter of a right angle; therefore the two angles Asi, KsB, together make half a right angle; and since the angle AsB is a right angle, the angle IsK must be half a right angle, or the whole number of angles round the centre s equal to eight half right angles. D A :Ι PRACTICAL GEOMETRY INACCESSIBLE LINES. PLATE 110. Fig.1. a ca Fig.2. h Fig. 3. -------- H - Fig.4. - --- - H B Fig. 5. Fig.6. Invented by PNicholson. A Fuliart au &C London& Edinburgh SECT. II.] 289 PRACTICAL GEOMETRY. PROBLEM XLIX, [Plate CX] Given any two lines AB and CD tending to an inaccessible point, to draw a line through another point E, so that all the three lines may tend to the same point. In the line AB, take Ag any convenient part of AB, and C Fig. 118. any convenient point in CD, and join AC, gC, also join AE and gE: Through any point B at or near the other extremity of AB draw BD parallel to gC; and through the point D draw Dh paral- lel to CA, BF parallel to gE, and hF parallel to AE, and join EF: then AB, CD, EF, would all meet in the same point if produced. The operation is evidently the same whether the point E lies between or without the two given lines. When there are two given lines and several points, either with- out or between the two given lines, the same process must be gone through for each point; but the same bases Ag and hB may be made to serve for every point, as is plain by inspecting Plate CX. of Inaccessible Lines. See also Fig. 1 and 2, Plate CX. Inaccessible Lines, where the diagrams are in a proportion which is likely to occur in practice. PROBLEM L. To draw a straight line through a given point that shall make equal angles on the same side of it with two other given straight lines. Through the given point draw a perpendicular upon each of the two given lines, and produce one of the lines on the other side of the point; bisect the angle contained by the line thus produced and make equal angles with those two lines. EXAMPLE Draw a straight line through the point E to make equal angles on the same side of it with the two given lines A B and CD. Through E draw EF perpendicular to CD, and EH perpendicular Figs. 119, Nos. 1 & 2. to AB; then draw IK bisecting the angle FEH, and the angles BIK and DKI will be equal to one another. See also Fig. 3 and 4, Plate CX. Inaccessible Lines, where the diagrams are nearer to the proportion in which they would occur in practice.* In No. 2, the given point E, and the two other points E, G, where the perpendiculars cut AB, coincide. * DEMONSTRATION. В р Because in No. 1, EG is perpendicular to AB, and EF perpendicular to CD, the two triangles EGI, EFK are right- angled; and because the two opposite equal angles GEA, FEH are bisected, the two angles IEG, KEF must be equal; therefore in the two triangles EGI, EFK, the remaining angle EIG is equal to the remaining angle EKF. 290 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM LI. Given the chord and versed sine of a segment, to draw a tangent at either extremity of the chord, without making use of the centre or any part of the arc. Fig. 120. B Let AC be the chord, and DB the versed sine ; join BC, and draw the line EC making the angle BCE equal to the angle BCD. * See also Fig. 5, Plate CX. of Inaccessible Lines, where the diagram is nearer to the proportion in which it would occur in practice. PROBLEM LII. Given the chord and versed sine of the segment of a circle, to draw a line tending to the centre from the extremity of the chord, without making use of the centre. Fig. 121. Let AC be the chord, BD the versed sine given in position and magnitude. Find the tangent CE as in the last problem, and draw CF perpendicular to CE ; then CF will tend to the centre as required. See also Fig. 6, Plate CX. of Inaccessible Lines. . S PROBLEM LIII. [Plate CXI.] To describe the segment of a circle when the extension of the radius reaches beyond the space where the operation is to be performed. Bisect the chord AC (Fig. 1, Plate CXI. Inaccessible Lines) by the perpendicular DB, making DB equal to the versed sine of the segment: join AB and BC. Prolong. BA to E, and BC to F. Fasten two slips of wood BE and BF together, so that their outer edges may contain the angle EBF; then bring the angular point B to A, and move the in- strument so, that, while the point B is proceeding from A to C, the edge BE may slide upon a Fig. 122. * The above depends upon this proposition of Euclid, book iii. prop. 32, that the angle made by a chord and its tangent is equal to the angle in the alternate segment. Now in Fig. 121, BC being a chord, BAC and BFC are angles in the alternate segment; and since DB is perpendicular to AC, the isosceles triangle ABC is divided into two equal and similar triangles; therefore the angle BCA is equal to the angle BAC, and consequently the angle BCA is equal to the angle BCE. PRACTICAL GEOMETRY, INACCESSIBLE LINES. PLATE. 111 Frig. I. 2 Fig. 2. www . w .... .. . . wwmh . - NA AL V Vw ERNES . . Commen . www . www . www Fig. 2: . .. Fig. 4. WE HIBITITUTO OX M Fig.5. Fig. 6. S U www. UN 2 k! P Invented by PNicholson. A Futiarton & C'London& Edinburgh SECT. II.) 241 PRACTICAL GEOMETRY. point or pin fixed at A, and the edge BF upon a pin fixed at C; then the pressure of a pencil at the angular point B upon the plane of description will describe the segment required. * Here it is plain that neither of the legs BE and BF must be less than the chord AC, or the arc cannot be described at one movement of the instrument. SCHOLIUM. Though this method in the simple form now explained is exceedingly convenient in a variety of cases, there are many, however, that occur where it cannot be applied for want of room to work the instrument, without moving the apparatus and board or table on which the arc is described, as is already evident from the problem now given, and will be still more so by inspecting Fig. 2, Plate CXI., where it appears that the instrument requires much more space than the radius itself; there- fore some modifications will be necessary in order to meet every case that may occur in practice. This will be explained in the following problems. PROBLEM LIV. Given two tangents to the arc of a circle and their points of contact, to find any number of equidistant points in that arc without making use of the centre, or having the arc previously described. Fig. 123. Let AB and CD be the two tangents, A and C their points of contact: join AC, and divide the angles CAB and ACD each into as many angles as the number of equidistant parts required in the arc; then the points m, n, o, where the lines intersect, are the points re- quired. † SCHOLIUM. If the number of parts is even, this problem can be geometrically executed by continual bisec- tions. The practice may be as follows:—Having drawn the chord AC; from the centre A, with the distance AC, describe the arc CB, and from the centre C, with the same distance, describe the arc AD: divide the arcs AC and BD, each into as many equal parts as the number of equidistant points to be found in the arc between the points of contact A and C, which are here four; then, numbering the parts at one end from A to D, and the other in the contrary order from B to C, every pair of lines drawn from the same numbers to the extremities A and C of the chord will be a. point in the are required. as will be shown in the following proposition. * The principle of the above method is founded on this theorem, that all angles in the same segment of a circle are equal to each other, and is demonstrated in proposition xxi. book iii. of Euclid. † This method depends on the equality of the arcs in the circumference, when the angles at the circumference are equal; and here, since the angles CAO, o An, nAm, mCA, as also the angles ACm, mCn, nCo, o AC are equal, the points m, n, o, must be in the circumference of the same circle in which the points A and C are. The theorem on which this principle depends is, " In equal circles, angles, whether at the centres or circumferences, have the same ratio that the arcs on which they stand have to one another;" which is demonstrated in proposition xxxiii. book iii. of Euclid. 292 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM LV. OC Given the two extremities of an arc, and a tangent at one of them, to find any number of equidistant points in the arc, without having any part of the circumference, or making use of the centre. Let A, C, be the two extremities of the arc, and AB the Fig. 124. tangent at the point A. From the centre A, with the distance AC, describe the arc BC; divide the arc BC into as many equal parts, at the points 1, 2, 3, as the number of equidistant points required to be found in the arc. From the centre C, with the dis- All tance CA, describe the arc Af: make Af equal to one of the equal parts of the arc BC; draw 1A, 2A, 3A, and fC which will intersect 1 A at m; then from m, with the distance mA, describe an arc cutting 2A at n; from n, with the same radius, describe an arc cutting 3A at o; then the distances Am, mn, no, oC, will be all equal. SCHOLIUM. As the mechanical construction of a geometrical problem is never executed with the rigorous exactness of the theory, and is very frequently insufficient to the end proposed, unless proved by trial; so in this instance it may happen, that the distance Am, when repeated the requisite number of times upon the several lines, may either extend beyond the point C, or fall short of it, as, from the obliquity of the intersection, the exact point m cannot be easily seen: in this case we must extend or contract the distance a very small portion, until the number required fall upon the point C. This may be easily adapted to the following proposition. PROBLEM LVI. Given the chord and versed sine of the arc of a circle, to find any number of points in each half arc of that segment. Let AC (Fig. 3, Plate CXI.) be the chord, and DB the versed sine. Through B draw PQ parallel to AC. Here are now given the two extremities A, B, and the tangent BP, to find the number of equidistant points required: this being done by Problem LV, we shall have the arc. In like manner for the other half, there are given the two extremities B and C, and the tangent BQ, to find the number of equidistant points required; which being executed by the same problem, we shall have the other half, and consequently the whole arc of which the chord is AC and the versed sine DB. The diagram exhibited in the plate is drawn in proportions likely to occur in practice. PROBLEM LVII. Given the chord and versed sine of the segment of a circle, to describe the curve, without having recourse to the centre. Find the intermediate points m, n, o, (Fig. 4, Plate CXI.,) for each half of the arc, by Problon LV. Now, taking any three adjacent points m, B, m, draw the two lines Bm, Bm, and produce SECT. II.) 293 PRACTICAL GEOMETRY. Bm to k, and the other line Bm to i, so that neither Bk nor Bi may be less than the distance be- tween m and m: form now the edge of a board to the angle kBi; bring the point B of the angle of the board to m; then in moving the board so that the edge Bk may slide upon a pin at m, and the edge Bi upon a pin at the other point m, the angular point B will trace the arc mBm by means of a pencil. An arc being described in this manner for every two adjacent distances, the whole segment will be completed. SCHOLIUM. The use of this problem will render the operation of describing the arc of the segment of a circle a most agreeable undertaking, when the centre is inaccessible. The same method ought still to be followed in very large arcs, supposing the centre to be accessible, but beyond the reach of a rod, as it would not be easy to move a chain or a line so as to keep it straight. PROBLEM LVIII. Given the chord of an arc, and any intermediate point whatever in that arc, to find the versed sine of the segment without having the centre given. Let AC (Plate CXI. Fig. 5.) be the chord of the arc, and M the intermediate point; bisect AC by a perpendicular DB, and join AM, cutting DB at e; join eC and MC; draw BC bisecting the angle eCM, and DB will be the versed sine required. * PROBLEM LIX. Given any number of equidistant points in the arc of a circle, to draw a line through any one of these points which shall be a tangent to the arc at that point, without making use of the centre of the circle. Let M, s, u, (Fig. 6, Plate CXI.,) be any three points, of which the middle one s is equidistant from each of the other two, u and M, and let it be required to draw a line through the point M that * DEMONSTRATION. In addition to the constructive lines join AB: we must here prove that the Fig. 125. angle ABC is equal to the angle AMC; to this purpose, since all the angles of * в м every triangle are equal to two right angles, and since all the angles in the same segment are equal to each other, the sum of the angles BAC, BCA at the base of the triangle ABC, ought to be equal to the angles MAC, MCA at the base of the triangle AMC; therefore each of the two angles BAC, BCA, ought to be equal to the less angle MAC of the triangle AMC, together with half the difference of the two angles MAC, MCA of the triangle AMC. Now the angle eCA is equal to the angle e AC, and the angle e AC is equal to the angle MAC; therefore the angle eCA is enual to the angle MAC, and the two angles eCA, MCA are equal to the two angles MAC, MCA; consequently eCM is the difference of the two angles MAC, MCA; and since BC bisects the angle eCM, the angle BCA is equal to the angle MAC, together with half the difference of the two angles MAC, MCA. 294 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. shall be a tangent to the curve. Join Mu, and from M with the radius Ms describe an arc rst cutting Mu at r. Make s t equal to s r and draw Mt; then Mt is the tangent required. * PROBLEM LX. To draw lines through any given points in or out of an arc of a circle that shall tend to the centre, without knowing where the centre is. Find the equidistant points (Fig. 6, Plate CXI.) A, v, e, f, B, u s, M, C in the arc, by Problem LV. at such a distance as to be within the reach of the instrument now about to be described; but before we can apply this instrument conveniently, it will be necessary to draw lines through the points o, e, f, B, &c. tending to the centre: for this purpose, bisect the distance between v and f by the perpendicular w h, and that from e to B by the perpendicular x k; wh will pass through the point e, and x k through the point f: the two perpendiculars thus drawn to their imaginary chords will tend to the centre. Set off any convenient distance e g on the line wh; make f i equal to e g, and through the points g and i draw a line : draw i p parallel to g h, and make the angle agn equal to the angle k i p; then place several straight edges together in such a manner that the edge of one of them may be upon the line i q, another upon the line g h, and a third upon the line gn; then fasten these to- gether at the angle in which they are placed as one machine : put a pin in the point g, and another in the point i, and move the instrument so that the edge g n.may slide upon the pin at g, and the edge g i upon the pin at i ; then, when the edge g h falls upon any intermediate point between e and f, stop the motion of the instrument, and draw a line along the edge g h; the line thus drawn will tend to the centre. † Fig. 126. * DEMONSTRATION. Let Msu be the arc of a circle, and let it be bisected in s, and join Ms, s u. Draw the chord Mu; from M with the distance Ms describe the arc r s t, cutting Mu in r. Make s t equal to sr, and join Mt; then, because the arc st is equal to the arc s r, the angle sMt is equal to the angle sMr; that is, equal to the angle sMu; and because sM is equal to s u, the angle sMu is equal to the angle suM. But sMt is the angle made by the chord Ms and the tangent Mt, and Mus is the angle in the alternate segment; therefore Mt is a tangent to the circle. Fig. 127. | DEMONSTRATION. The angle CBD is equal to the angle CAD, because they stand upon the same arc DC; and the angle CDB is equal to the angle CAB, because they stand upon the same arc BC; therefore, if D, C, B, be three fixed points, the angles CAD and CAB will always continue to be the same wherever the point A is situated in respect of A and B; therefore, if AE, AC, AF, be three inflexible rulers always at the same angles, if the ruler AE slide upon the point D, and AF upon the point B, the middle ruler AC will always pass through the point C., PRACTICAL GEOMETRY, IVACCESSIBLE LINES. PLATE 112 Fig. 1. tutk time .........com Fig. 3. Fig. 2. w ws w ore . ------- .......... naraon.eo... ... svorcov....... 4. Invented by P. Nicholson: Engraved by F. Tural. A Fullarton & C. Landon kEdinburgh (CH. SECT. II.] 295 PRACTICAL GEOMETRY. PROBLEM LXI. [Plate CXII.] Given any two straight lines inclined to each other, to describe an arc of a circle which shall pass through a given point in one of the lines, and shall have a given chord, and the point of intersection of the two lines as a centre. Let AB, CD (Plate CXII. Fig. 1 and 2) be the two lines tending to the centre, and let C be the point through which the arc is to pass. Draw AC, by Problem L. so as to make the angles DCA and BAC equal to each other.* Divide AC into any convenient number of equal parts, as 3, and make CE equal to a third of CA. Draw EG parallel to AB, and from G, with the radius GC, describe the arc ECHF. From the point C, with a radius equal to a third of the extension of the given chord, describe an arc cutting the arc CHF at F. Join CF, and produce CF to I, making CI equal to three times CF. Biseot CI in K, and draw KL perpendicular to CI; also bisect the arc CHF in H. Through the points C and H draw the straight line CL; then, by Problem LVI., describe the arc of a circle, of which CI is its chord, and KL the versed sine, and the thing required will be done. Fig. 1 has its centre at so very remote a distance, that the lines are not so well adapted for de- scription as Fig. 2; but such a form is here shown because it is very likely to occur in practice. CHAP. III. – ARITHMETICAL OPERATIONS RESPECTING THE CIRCLE. PROBLEM I. Given the chord and versed sine of the arc of a circle, to find the radius. Divide the square of half the chord by the versed sine; add the versed sine to the quotient, and divide by 2, which will give the radius. I * This operation may be very neatly performed, as in Fig. 3. Draw CE parallel to AB, and bisect the angle DCE by the straight line CA, and CA will make equal angles with AB and CD. † In Fig. 1 EC is made one-tenth of AC, but the same letters of reference are retained in both figures. $ DEMONSTRATION. Let Mm be the chord, and AP the versed sine. Through the three points M, A, m, Fig. 128. draw the circle MA ma, (by Problem XVI.,) and produce AP to a. In the semicircle AMа, MP is a mean proportional between AP and Pa, that is, AP is to MP, as MP is to Pa, and therefore, by the nature of proportionals, AP multiplied by Pa is equal to MP multiplied by itself, or the square of MP: but MP is half the given chord, and AP and Pa together make the diameter; therefore the square of the half chord is equal to the product of the versed sine multiplied by the remainder of the diameter. Consequently, if the square of the half chord be divided by the versed sine, and the versed sine be added. to the quotient, it is evident we have the diameter of the circle, half of which is the radius. MK .. 296 [Part IV ELEMENTS AND PRACTICE OF GEOMETRY. EXAMPLE. Required the radius of a circle of which the chord is 16 feet, and the versed sine 5 feet. 5)64 = the square of 8, the half chord 12:8 5 = the versed sine 2)17.8 8.9 the radius required or 8 feet 10% inches radius. PROBLEM II. The radius and half chord of the segment of a circle being given, to find the versed sine of the arc of that segment. * RULE. From the radius subtract the square root of the difference of the squares of the radius, and that of the half chord, and the remainder is the versed sine. EXAMPLES. Ex. 1.' Required the versed sine to the segment of a circle of which the chord is 12 feet, and the · radius of the circle 250 feet. 250 radius 250 12500 50 62500 square of the radius 250 36 square of 6, the half chord 62464(249.927 44) 224 176 489) 4864 4401 from radius 250 subtract 249.927 0.073 the versed sine in decimals of a foot 12 4989) 46300 44901 49982) 139900 99964 .876 in decimals of an inch 499847) 3993600 3498929 7.008, so that the versed sine required is about { of an inch. 494671 In Fig. 128, as MCP is a right-angled triangle, the square of MC is equal to the sum of the squares of MP, and PC (Problem V.); but MC is the radius, and MP the half chord, therefore PC is equal to the square root of the differ- ence of the squares of the radius and half chord, and if PC be taken from the radius there remains the versed sine: whence we obtain a rule for this problem. SECT. II.] 297 PRACTICAL GEOMETRY. Ex. 2. Supposing the radius of the plan of a circular bow in a building to be 8 feet 104 inches, and that bow to contain windows each four feet wide; how far will the arc of the sash-frame project from its chord? 8 feet 104 inches 12 radius 106-75 reduced to inches and tenths 106.75 53375 74725 64050 10675 11395.5625 576 = square of half chord in inches 10819.5625(104:017 204) 0819 816 106-75 104.017 208.01) 35625 20801 2:733 208.027) 1472400 1456189 5.864, versed sine as required. 16211 So that the versed sine, or the projection of the sash, is very nearly 24 inches. Ex. 3. Required the versed sine to a segment of which the chord is 200 feet, and the radius of the arc 250 feet. 250 250 12500 500 62500 10000 52500(229.128 42) 125 250.0 229.128 84 449) 4100 4041 20.872, versed sine as required. 4581) 5900 4581 45822) 131900 91644 458248) 4025600 3665987 359616 2 P 298 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM III. Given the radius of a circle, and the versed sine of a segment of that circle, to fond the distance between the centre of the circle and the chord of the segment. Fig. 129. 1 Let DBEF be the circle, Oits centre, DE a chord, C the middle of the chord, and BC the versed sine ; then, if BC be produced to F, BF will be a diame- ter, and OC will be the distance between the centre 0, and the chord DE and OB the radius of the circle. Now OC = OB — BC. raó RULE. From the radius subtract the versed sine, and the remainder is the distance between the centre of the circle and the chord. EXAMPLE Suppose the radius of a circle to be 250, and the versed sine 20.872 ; required the distance of the chord from the centre. 250.0 20.872 229.128, the distance between the centre and the chord, as required. In the segment of a circle, any line parallel to the versed sine, contained between the arc and the chord is called an ordinate, and the distance of that ordinate from the versed sine is called an abscissa. PROBLEM IV. Given the radius of a circle, the distance from the centre to a chord of that circle, and the distance of an ordinate in the segment from the versed sine, to find the length of the ordinate. From the square root of the differences of the squares of the radius and of the abscissa subtract the distance from the centre of the circle to the middle of the chord, and the remainder is the Fig. 130. B M. * Let DBEF be the circle, DE the chord, MP the ordinate, and 0 the centre. Draw the radius OM, and the diameter BF bisecting the chord DE: draw OQ parallel to DE, and pro- duce MP to Q. In the right-angled triangle OMQ, the square of the radius OM is equal to the sum of the squares of OQ (which is equal to the abscissa), and QM, and therefore QM is equal to the square root of the difference of the squares of OQ and OM. But QM is equal to the given ordinate, together with the distance from the centre of the circle to the middle of the chord; therefore, if from the square root of the difference of the squares of the radius and the abscissa, we subtract the distance from the centre of the circle to the middle of the chord, the remainder must be equal to the ordinate. . SECT. II.) 299 PRÁCTICAL GEOMETRY. EXAMPLE. Given the radius of a circle 250 feet, the distance between the centre and a chord 229.128 feet, to find an ordinate of the segment at the distance of 10 feet from the versed sine. 250 250 12500 50 62500 = the square of the radius 100 = the square of the abscissa 62400(249.799 The square root of the differences of the squares of the radius and the abscissa being now found to be ... 249.799 subtract .....229128 44) 224 176 489 4800 4401 20:671 and the remainder . . . is the ordinate required. 4987) 39900 34909 49949) 499100 449541 499589) 4955900 4496301 459599 1 UTY Upon the principles of these calculations the largest circles may be drawn, by calculating a suffi- cient number of ordinates. EXAMPLE. Let it be required to construct a segment of a circle of which the chord is 200 feet, and the radius of the arc 250 feet. The method of doing this is to find a sufficient number of points in the curve not exceeding the length of an ordinary board; say 12 feet. We may now calculate the ordinates for one-half of the curve at 10 feet distance from each other, in order that a twelve feet board may reach between any .. two adjacent points: this board must be curved on one of its edges to the arc of the circle required; then the curved edge, with its concavity towards the centre, being successively applied to every two adjacent points, and the curve drawn between them at every application to the surface, the entire arc will be described. The versed sine of the arc on the edge of the board will be found by Problem ii., and this arc being within reach, it may be described by Problem lvii., Practical Geometry. Here follows the calculation of the ordinates, 10 feet distant from each other, the middle one being at the head of the column, and the others in succession towards one of the extremities of the chord. 300 [PART. IV. ELEMENTS AND PRACTICE OF GEOMETRY. Feet In the first place, the versed sine of the arc must be found, by Problem 20•872 versed sine 20.671 first ordinate II., the operation being performed in Ex. 3, p. 297 ; the result, 20-872 20.070 second ordinate feet, is the versed sine, which subtracted from the radius gives 229.128, 19.065 third ordinate the distance between the centre and the chord ; and having this distance 17.597 fourth ordinate 15.821 fifth ordinate between the centre of the circle and the middle of the chord, and the radius 13:511 sixth ordinate of the circle, and the length of the chord, any ordinate is found by Problem 10.871 seventh ordinate iy. The calculation of the first ordinate at 10 feet distant from the middle 7672 eighth ordinate 4:056 ninth ordinate of the chord is exhibited in Ex. p. 299. Here follows the segment drawn according to this calculation. Fig. 131. AB is 200 feet CD = 20.872 e f = 20 671 gh = 20.070 &c. &c. f n f Dhk me q. go m. k h The following Table exhibits the versed sines for the segments of circles which have chords of 10 feet each, from a radius of 10 feet to 110, increasing successively by unity; and from 110 to 132 feet, increasing successively by two feet at a time; and from 200 to 300 feet, increasing successively by 10 feet. Radius. Versed sine. Radius, Versed sine. Radius. Versed sine. Radius. Versed sine. ft. ft. 1091 inches. 1•37688 Il 13 A CON 14 15 16 ft. inches. 1 4.07700 1 2.42520 1 1.09548 1 0.00000 0 11.07972 0 10.29444 0 9.61584 09.02316 0 8.50080 08.03640 0 7.62103 0 7.24716 0 6.90864 0660072 17 114 inches. 3-50028 3.42024 3.34380 3.27060 3.20064 3:13356 3.06924 3.00756 2:94828 2.89140 2.83656 2.78376 2.73300 2.68401 2.63676 2.59104 2-54700 2:50440 2.46324 2-42340 2:38476 2 34744 2.31120 2.27604 2.24196 2.20896 2.17680 2.14560 2-11536 2.08596 2:05728 2:02944 2.00232 06.06132 inches. 1.97592 1.95012 1.92516 1.90068 1.87692 1.85364 1.83108 1.80888 1.78740 1•76628 1.74576 1.72632 1.70604 1.68684 1.66706 1.632 1.630 1.613 '1:59696 1.58376 1.56360 1.54752 1.53168. 1.51620 1.50096 1.48608 1.47156 1.45728 1:44324 1.42944 1.41600 1.40268 1.38972 In the following the difference of the radic is 2 feet. 110 1.36461 112 1.34004 1.31652 116 1.29372 118 1•27176 120 1.25064 122 1.23012 124 1.21020 126 1•19100 128 1.17240 130 1.154 132 1.136 0 5.60412 0 5040060 0 5 21148 05.03532 04-87068 0 471648 0 4.57200 0 4.43592 0 4.30788 04.18704 0 4.07280 0 3.96468 0 3.84220 0 3.76488 0 3.67236 03-58448 100 101 102 In the following the difference of the radii is 10 feet. 200. 0.75000 210 L 0-71448 220 0.69196 230 0.67836 240 0.65016 250. 0.60000 260 0·56784 270 0:55572 103 104 105 40 an 42 0.49890 As the chords are proportional to the radii, we may multiply both by any number we please, and obtain a correct result. Thus, were it required to find the versed sine to a segment of which the Sect. II. 301 PRACTICAL GEOMETRY. chord is 30 feet and the radius 36 feet, we have only to multiply the versed sine in the table for a radius of 12 feet, namely 1 foot 1.09548 inches, by 3, and we obtain 3 feet 3.28644 inches, the versed sine required. As the intervals between the tabular lengths of the radii increase by multiplication, there will still often occur cases in which this table cannot be used, and we must then resort to the mode of calculation already explained. PROBLEM V. To construct the plan of a circus or semicircular crescent, or quadrant to a given radius. Fig. 132. Let ABCD be a circle, 0 its centre, and let ABCD be a square inscribed in the circle, and EFGH a square circumscribing the circle. Now, the inscribed square ABCD is half the circumscribing square EFGH, and therefore, De Af, which is a fourth part of the inscribed square, is the half of OAEB, the square of the radius; therefore the square of the half chord Af is half the square of the radius. Whence the radius and the half chord being given, the versed sine f g will be found by Problem ii., p. 296, and the segment AgB may be constructed by ordinates, by Problem iv., p. 298. EXAMPLE Here, by availing ourselves of the construction of the figure and its properties, the difference of the squares of the radius and that of the half chord is half the square of the radius, or (300)2 - 45000 feet. The square root of 45000 is 212.132 feet; this subtracted from the radius 300 feet, gives 87.868 feet for the versed sine of each quadrant; and 212.132 is the distance of each chord from the centre of the circle ; and lastly, having the radius of the circle, and the distance of each chord from the centre, we may find as many ordinates as we please in each of the four segments, by Problem iv., page 298. CHAP. IV.-CURVE LINES. OF THE ELLIPSE. If in any plane two points be assumed, and if a slender needle or pin be set up at each point, so that a thread joined at its extremities may be put loosely round them; and if a pencil, or other tracing point, be applied so as to stretch the thread tight, and in such manner be carried entirely round the pins, a curve will be described which is called an Ellipse. * This curve is always of an * The Ellipse, Parabola, and Hyperbola, as they result from particular sections of the cone, are frequently termed “Conic Sections," and treated of with reference to such origin; but the student should be informed that these curves may all be traced by operations which bear no obvious analogy to the cone (as we have already shown in the case of the ellipse); and as the useful application of these curves, and the development of their most remarkable properties, may be effected without presupposing the existence of a cone, we have thought it most advisable to describe them as gen- erated by such means as lead most readily to a clear perception of their properties. 302 [PÁRT IV. ELEMENTS AND PRACTICE OF GEOMETRY. oblong figure ; the greatest diameter is called the axis major, or conjugate diameter, and the least diameter, the axis minor, or transverse diameter. The points at which the pins are fixed are called the foci of the ellipse, and the length of the ellipse compared with its breadth will depend on the shortness of the thread; for the longer the thread is, the more will the curve approach the figure of a circle, et vice versa. The elongation of an ellipse is called its eccentricity, and if one ellipse be of a longer or shorter figure than another, it is said to be more or less eccentric. PROBLEM I. Fig. 133. The two axes Aa and Bb being given to find the foci. From the extremity B of the axis minor, with the distance CA, or Ca, (equal to the semi-axis major,) describe an arc meeting the axis major Aa in F, f; the points F, f are the two foci. DEMONSTRATION. As, in the definition we have given of the ellipse, one portion of the thread is constantly stretched in a straight line between the pins or foci, and the remaining portion extends from one focus to the pencil, and from the pencil to the other focus, it is obvious that such remaining portion must always be equal to the axis major of the ellipse, and therefore the sum of two lines meeting in one extremity of the axis minor, and terminated in the two foci, is equal to the axis major, and the distance of the foci from the centre is the leg of a right-angled triangle, of which the other leg is the semi-axis minor, and the hypothenuse the semi-axis major; therefore the distance of either focus from the semi-axis minor is equal to the semi-axis major.. See Elements of Geometry, Curve Lines, Prop. ii. PROBLEM II. Given the axis major Aa and the two foci F, f, to find any point in the curve of an ellipse. Between Ff take any point g; from F with the radius Ag describe Fig. 134. an arc at M, and from f, with the radius a g, describe another arc, meeting the former at M, and the point. M is in the curve. DEMONSTRATION. The reason of the method for finding any point M in the curve, is because the sum of the two lines drawn from the foci to any point in the curve is equal to the axis major.–See Curve Lines, Elements of Geometry, Proposition i. SCHOLIUM. It is evident that not only the same radii might have been employed in finding the point M on the other side of the axis major, but that they might have been used in finding two other points at the same distance from the centre ; so that by the two distances Ag, g a, four points in the curve might have been found. SÉCT. II.] 303 PRACTICAL GEOMETRY. PROBLEM III. Given the two axes Aa, Bb, of an ellipse, to find any point in the curve. METHOD 1. Fig. 135. Make Ch equal to CA. In Ca take any point d, so that Cd may be less than Bh. From d, with the radius Bh, describe an arc meeting BC in e. Produce de to M, and make eM equal to CA, or dM equal to BC, and the point M will be in the curve. В с D Fig. 136. DEMONSTRATION. For upon Bb describe the semicircle BRb; through the point M draw PR parallel to CA, meeting Bb in P, and join RC. Then, because de is equal to the difference of the two semi-axes, and eM equal to CA the lesser axis, dM is equal to BC the greater; and therefore dM is equal to CR. Now, by similar triangles, PRC, PMe, - - - PR X Me = RC X PM; and since - - - - - - - - - - - - - RC = BC, and . - - - - - - - - - . . . . Me = AC; therefore, by eliminating RC, Me, there will result PR X AC = BC X PM ; wherefore PR : PM :: BC: AC; and (Prop. vi. Curve Lines, Elements of Geometry) the curve is an ellipse. Upon this principle an instrument may be made to describe the curve by continued motion; for if dM be conceived to be an inflexible line, and the points d, e, M as fixed in this line, supposing the point e to be compelled to move in the line Bb, and the point d in Aa, and the point M to be moved from B to A ; then the point M will describe the curve BAba of the ellipse. This instrument may be constructed by making grooves in the plane of description instead of the lines Aa, Bb, of the position of the axis; and instead of the inflexible line deM, using a rod with three projecting cylindrical pins, which must have their diameters equal to the breadth of the grooves, so that, when the pins are inserted in the grooves, their axes may be perpendicular to the plane of description, and their lower 'ends in a straight line, and at the same respective distances from each other that the points d, e, M are. Such an instrument is called a trammel, and is generally made of two rectangular bars, with a groove in each, so fixed together, that when laid on the plane of description, the grooves may be at right angles to each other, the bottom of each groove parallel to the plane of description, and their other two sides perpendicular to it, and that the upper surface from which the groove recedes may be a plane parallel to the plane of description. The rod may be contrived with two moveable pieces called nuts, with one of the cylindrical pins fixed in each nut, and a pencil fixed at the remote extremity, so that the ends of the pins may be at the same distance from the under edge of the rod : but the pencil must be of sufficient length to reach the plane of description. Carpenters construct their trammels of wood; but the transverse piece is generally made entirely upon one side of the 304 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. other, so as to describe a semi-ellipse at a time, as they are mostly employed in drawing elliptic arches, or portions of ellipses not exceeding half the entire curve. It is by a contrivance on this principle that elliptic turning is performed. PROBLEM IV. Given the two ares in position, the vertex A of the one, and another point M in the curve, to find the limits of the other axis. . Fig. 137, No. I. PM From M, with the radius AC of the semi-axis given, describe an arc meet- ing the other unlimited axis at e, and let d be the point where Me either meets the axis Aa between the points M, e, as in No. 1, or, being produced, meets it beyond e, as in No. 2. Make CB, Cb each equal to dM, and Bb is the other axis, whether major or minor. Beeb cyb Aa is the axis major when the point d falls between the centre and I. Fig. 137, No. 2. one extremity A; but if it fall beyond C, it is the axis minor. This is too evident to require a demonstration. myM Cor. 1. Hence, since d e is the difference of the semi-axes, any : B point in the curve may be found without finding the other axis. : For taking the point d at pleasure, so that the distance Cd may be less than the given difference, from the point d as a centre, with a radius equal to the same difference, describe an arc meeting Bb at e, and produce á e to M, and make eM equal to AC; then M will be a point in the curve. PROBLEM V. Upon a given straight line a a as an axis, to find any point in the curve, the length of the other axis or semi-axis being known. Fig. 138. 아 ​Draw the straight line AA equal in length to the other axis, which is only given in length and not in position. Upon a a, as a diameter, describe the semicircle a mba. Bisect AA in C and a a in c. Divide CA in P and ca in P, so that CA: CP::ca:CP. Draw CB and PM perpendicular to AA, also c b and Pm perpendicular to a a. Make CB equal to cb, and PM equal to Pm, and M is a point in the curve answering to the axis AA and the semi-axis CB. M B 빠 ​A PC DEMONSTRATION. Let CA = a, CB = ca=cb = 6, PM = Pm = y, CP = x, and cP = By the right-angled triangle cPm (Th. xxxv. Geom.) - - - 12 = 12 - y2 and by construction, a : x::b; - - - - - 62 m2 = 22 y wherefore, eliminating v from these two equations, there will result a? (62 — y2), which is an equa- tion to the ellipse. SECT. II.] PRACTICAL GEOMETRY. 305 It makes no difference to the demonstration whether it be the axis major or minor that is given in position. PROBLEM VI. To trace or draw the curve of an ellipse, or that of any other figure, through points given or found in the curve. METHOD 1. Join two adjacent points by an arc drawn by the eye; join one extremity of this arc to the next point, in the same manner; and the last extremity to the next point in succession by another arc, and so on till the two ends of the line meet, or form one continued line of the required length: the curve thus drawn will be the more accurate as the perception of the eye is quick in its judgment of the figure, and the hand that draws it steady. The greater the number of points, and the nearer the points are at equal distances, the easier will the curve be drawn. METHOD 2. Put in pins în all the points, and bend a flexible elastic slip of wood or metal round the convex side, or outside the pins; then, when the side of the slip comes in contact with every pin, draw a curve on the inner or concave side ; and if the slip is not of sufficient length, repeat the operation as often as may be found necessary, till the curve forms one continued line, or an arc of the length intended. * PROBLEM VII. Given two diameters to find any point in the curve. Fig. 139. From the vertex A of the semidiameter AC, draw AL perpendicular to the semiconjugate BC, cutting it in D. Make AL equal to BC, and join CL. From any point G in CL with a radius DL describe an arc meeting BC in I; join GI, and produce it to M. Make GM equal to LA; then will M be a point in the curve. DEMONSTRATION. For join PG. Let GP = v, and because AL = BC = MG by construction, and BC = b by notation, BC = AL = MG = b. Because of the right-angled triangle MPG, - - - - - - 22 = 62 - y2 and of the similar triangles CAL, CPG, - ... ... 62 m2 = až 22; wherefore, eliminating v, there will arise 72 m2 = ax (62 - y2), which is the equation of the co- ordinates of the ellipse. * A very useful instrument is formed by fitting an elastic ruler of wood or metal, so that it may be bent by screws to any requisite degree, and thus adapted to every degree of curvature. Shipbuilders are in the habit of using a great variety of curved pieces of wood cut to circular arcs of different radii, and other. curves, and the piece nearest to any particular curve is chosen from among them, 2 Q 306 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. SCHOLIUM Upon this principle the curve may be described by the continued motion of a point. For suppose GIM to be an inflexible line, or the straight edge of a ruler, and G, I, M to be fixed points in it: then, suppose the point M to be moved, so that the point I may be always in the line Bb, and the point G in the line LC, the point M will describe the curve of the ellipse, or as much of it as may be found necessary. A machine so constructed is called an oblique trammel. PROBLEM VIII Given any two diameters Aa, Bb, to find the two points where a line drawn through the centre in a given position meets the curve. Fig. 140. Through the vertex A of the given diameter Aa, draw FA parallel to the other given diameter Bb, and draw AG perpen- dicular to AF. Make AG equal to BC. From G, with the radius GA, describe the arc AH. Draw FG meeting the arc AH in H. Join GC, and draw HM parallel to GC, meeting Fm in M. Make Cm equal to CM, and M, m are the points required. DEMONSTRATION. If Mm is a diameter, the point M is in the curve. To prove this, draw M:P and HI parallel to BC; MP meeting CA in P, and HI meeting AG in I; draw LH parallel to CM, meeting CG in L, and join PI. Let CF = m, CM = LH = n, GF = P, GH = 9, CA = a, CP = x, HG = AG = CB = b, GI = 0, and IH = PM = y. (PCM, ACF - . . . - - - - ma x2 = a? na | LHG, CFG - - - - - - - - - p2 n2 = m 2 and by the right-angled triangle GIH - - - - - - - - - - 22 = 62 - y2 Whence, eliminating m, n, p, q, v, we have 62 ml = a? (62 - y²), which is the equation of the co-ordinates of an ellipse. EXPLANATION OF THE PLATES OF CURVE LINES. : [Plates CXIII. and CXIV.] Plate CXIII.-Ellipse, exhibiting examples of the various methods of describing the curve by Ex. 1.-In Fig. 1, the axes Aa, Bb, are given to describe the curve.. Find the foci F, f; by Prob. i. ; taking the points g, h, i, k, &c. at pleasure, between F, f, by PRACTICAL GEOMETRY CURVE LINES, PLATE 113. ELLIPSE. Fig 2. Fig. 2. Fig.3. 7 ve INITIA Alth it to . w NA тъ Fig. 4. Fig. 5. Fig. 6. nh ML AMUTI NO 2. ZITTI C P P P were M M ' NO 1. p A P P' P" Fig. 7. Fig. 8. 1.Nicholson G.Gladwin. Av. A Fullarton& CO I don&Edinburgh for PRACTICAL GEOMETRY CURVE LINES,.. ' PLATE 114 Fig.2. Fig. 1. Fig.3. 2 Fig. 5 Fig.6. Fig.4. Fig. 9. Fig. 8. D Fig. 7. OKUMJUMBUMBURU BOWRIDI Fio. 12. Fig.10. G m P. Nicholson. 30 < CH A Fallarton. C'London&Fainburgh SÉCT. II. 307 - PRACTICAL GEOMETRY. C Prob. ii. find the points in the curve, viz.-m., m from g, m', m' from h, m", m" from i, &c. and through all the points.m, m', m", &c. draw the curve, by Prob. vi. Ex. 2.-In Fig. 2, the axes are given to describe the curve without the foci. Here, on the edge of an ivory rule or thick slip of paper, mark the distance of the points m, 0, equal to the semi-axis major, and the distance of the points m, n equal to the semi-axis minor. Place the point n in the axis major Aa, and the remote point o in the line Bb of the axis minor; then make a mark at m, and m is a point in the curve. Having, in this manner, found a sufficient number of points, draw the curve by Problem vi. Ex. 3.-Fig. 3 exhibits the drawing or tracing the curve of an ellipse by means of the trammel described in Prob. iii. Having set the distance m n from the pencil to the first sliding point equal to CB tlie semi-axis minor, and mo equal to the semi-axis major; then the sliding points being in the grooves, move the end R, and the pencil at m will describe the curve. Ex. 4.-Fig. 4 and 5 exhibit the method of drawing the curve when the semi-axis CA and an ordinate PM or Pm is given. Having drawn the unlimited axis Bb in its right-angled position to AC, and having, on the edge of an ivory rule or slip of thick paper, made the distance mo equal to the given axis AC, place the point m on the edge of the rule upon the given point m in the curve, and move the end Q, so that the point o in the edge of the rule may fall on the unlimited axis Bb. Mark the point' n on the edge of the rule where it meets the line Aa of the given axis ; then any point in the curve will be found, as in Example 2, Fig. 2. Ex. 5.--Fig. 6 exhibits th method of describing a semi-ellipse upon a given axis Aa, according to the principle in Problem v. Draw any angle ACH, No. 3. Make CA equal to the semi-axis CA, No. 1, and CH equal to the other semi-axis which is only given in length. In CA take any number of points P, P', PM, &c. Join AH, and draw the lines Pp, Pp, P" p, &c. parallel to AH, meeting CH in the points p, p, p. In the semi-axis Ca, No. 1, make CP, CP, CP", &c. respectively equal to CP, CP', CP", &c. No. 3. At any convenient place No. 2, draw the straight line GH. From any point c in GH, with the radius cH, No. 3, describe the semicircle GBH. Make cp, cp, cp", &c. respectively equal to Cp, Cp, Cp, &c. No. 3. In No. 1, draw the straight lines CB, PM, P'M', &c. perpendicular to Aa; and in No. 2, draw CB, Pm, P'm', &c. perpendicular to GH, meeting the circular arc at B, m, m', &c. In No. 1, make CB, PM, PM, &c. respectively equal to CB, Pm, P'm', &c. in No. 2, and the points B, M, M', &c. will be in the curve, by Prob. v. The curve itself is drawn by Prob. vi. Fig. 7 exhibits the application of the oblique trammel for describing an ellipse, of which the two diameters Aa, Bb are given in position and magnitude. From the vertex B of the minor or less diameter, draw BD perpendicular to the greater Aa, cut- ting it in E. Make BD equal to AC, and join CD: then the moving points of the trammel rod being set to the distances BE and BD, the curve may be described as shown, the grooves of the trammel being in the lines Aa, CD.-See the principle, Prob. vii. Fig. 8 is an example of the application of Prob. viii. to finding the points where a line drawn through the centre in a given position meets the curve from the diameters Aa, Bb given in position and magnitude. Through B, draw FG parallel to Aa, and draw BE perpendicular to FG, and make BE equal to AC. From E, with the radius EB, describe the arc HBI. Produce mM to F, and join FE, meeting the arc HBI at H. Join CE, and draw MH parallel to CE; make Cm equal to CM, and the points M, m are the vertices of the diameter Mm; so that the position of three diameters being given, of which two are conjugate to each other, and of a given magnitude, the vertices of the third will be found. 1 308 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. In this manner we may find the extremities of as many unlimited diameters given in position as we please, and a curve being drawn through their vertices will form the ellipse. PROBLEM IX. An ellipse being given, to describe a concentric ellipse through a given point in one of the semi-axes. Join AB in Fig. 1, (Plate CXIV. Curve Lines,) the two extremities of the semi-axes CA, CB, and through the given point E in CA draw ED parallel to AB, meeting CB in D; then with the semi-axes CE, CD, by Prob. iii. find a sufficient number of points, and trace the curve through them, by Prob. vi. PROBLEM X. To describe an ellipse about a rectangle so that the axes of the ellipse may have the same ratio as the sides of the rectangle.. Let GHIK, Fig. 2, be the rectangle. Draw the straight lines Aa, Bb, bisecting each other at C, and the sides of the rectangle at E, e, D, and d. In Bb make DP equal to DK, and draw PK cutting Aa at N. Make CB, Cb each equal to NK, and CA, Ca each equal to PK; then describe an ellipse upon the axes Aa, Bb, which will circumscribe the rectangle GHIK, and the axes of the ellipse will have the same ratio as the sides of the rectangle. DEMONSTRATION. Draw PQ parallel to Aa, meeting the side KI of the rectangle in Q. Let CA = PK = a, CB = KN = 6, KQ = DP = CE = p, and eK = CD= q. By similar triangles QKP, eKN, - - - - - - - - - bp= aq therefore a :b::p:q, or CA : CB :: CE: CD. PROBLEM XI. The curve ADGaHE of an ellipse being given, to find the centre. Draw any two parallel lines DE, GH (Fig. 3) to meet the curve in E, D, H, G. Bisect the lines DE and GH, and through the points of bisection draw Aa, meeting the curve in the points A, a ; bisect Aa in C, which is the centre required. PROBLEM XII. Given the ellipse A Bab (Fig. 4), to find the axes in position and magnitude. From the centre C, found by Prob. xi, describe a circle meeting a curve in D, E, F, G. Join DG. Bisect DG, and through the point of bisection and the centre draw Aa, meeting the curve in A, a: through C draw Bb perpendicular to Aa, meeting the curve in Bb; then Aa, Bb are the two axes as required. PROBLEM XIII. An ellipse mAM (Fig. 5) and a tangent Tt being given, to find the point of contact. Draw Mm parallel to Tt, meeting the curve in M, m. Bisect Mm in P, and draw AC through P to the centre C, and A is the point of contact. SKCT. II.) 309 PRACTICAL GEOMETRY. PROBLEM XIV. An ellipse MAM (Fig. 6) being given, to draw a tangent through a given point A in the curve without having any diameter or foci. Find the centre C, and draw CA; from C with any radius describe a circle within the curve of the ellipse. Draw LM and Km parallel to AC, tangents to the circle, and meeting the ellipse in M, m: join Mm, and through A draw Tt parallel to Mm, and T is the tangent required. PROBLEM XV. Given an ellipse A Bab (Fig. 7) and the two axes Aa, Bb, to draw a perpendicular through a given point M in the curve. Find the foci F, f, by Prob. i. Through M draw FH and fG, and draw MN bisecting the angle HMG, and MN is perpendicular to the curve. Upon this principle a trammel may be made to draw the joints of the youssoirs of any elliptic arch, as exhibited in Fig. 8. PROBLEM XVI. An ellipse AMa (Fig. 9) and the foci F, f being given, to draw a tangent through a given point M in the curve. Draw FM, fM, and produce FM to H. Draw QT bisecting the angle fMH, and QT will be a tangent to the curve at M. PROBLEM XVII. An ellipse A Bab (Fig. 10) and the axes Aa, Bb being given, to draw a tangent through a given point M in the curve without knowing the position of the foci. Join AM, and bisect AM in d. Through d draw CQ, and draw AQ parallel to CB. Draw QR through M, and QR is the tangent required. Or, U Join Ma, and bisect Ma in e. Through e draw CR, and draw aR parallel to CB. Draw QR through M, and QR is the tangent required. PROBLEM XVIII. Given three straight lines passing through the centre in position, of which two are conjugate diameters, * to find a conjugate diaineter to the third straight line. Let Aa, Bb (Fig. 11) be the two conjugate diameters: it is required to find a conjugate diameter to the third straight line Fm. Through B draw FG parallel to Aa. Draw BE perpendicular to FG, and make BE equal to CA. Join EF, and draw EG perpendicular to EF. Draw Gn through the centre C. From E, with the radius EB, describe the arc. KBL, meeting EF at K, and EG at L. Join EC. Draw KM and LN parallel to EC, meeting Fm and Gn in M and N. Make Cm equal to CM, and Cu * See definition 8, Elements of Geometry, Curve Lines. 310 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY equal to CN; then, if an ellipse be described through the points AMBNa m bn, Mm and Nn will be two conjugates as well as Aa, Bb. * PROBLEM XIX. Given two conjugate diameters Aa, Bb (Fig. 12), to find the two axes. Through B draw FG parallel to Aa. Make BE equal to CA, and join EC. Bisect EC. by the perpendicular QR, meeting FG at s. From s, with the distance SE, describe the semicircle FEG. Join FE and GE; also join FC and GC, and produce FC to m, and GC to n. From E, with the radius EB, describe the arc KBL, meeting FE and GE at K and L. Draw KM and LN parallel to EC, meeting Fm at M, and Gn at N. Make Cm equal to CM, and Cn equal to CN; then Mm and Nn are the axes. DEMONSTRATION. Because Qs is perpendicular to EC, the distances sE, SF, sG, SC are all equal; therefore the angle FCG is a right angle as well as FEG: but, because FEG is a right angle, Mm and Nn are conjugate diameters, by the preceding Problem; but when two conjugate diameters are at right angles to each other, such two conjugate diameters are the axes. OF THE HYPERBOLA. The hyperbola is the curve of the section of a cone when cut by a plane inclined to the base at a greater angle than the lines forming the sides of the cone. Thus, if a cone stand on a table, and be cut by a plane perpendicular to the table, or inclined to it at any angle between the perpendicular and the angle of either side of the cone, the section produced will be an byperbola. From the position of the plane, it is evident that the hyperbola cannot return into itself, or form a complete curve, like the ellipse ; because, if the cone were infinitely extended downwards, and the dividing plane likewise produced, it could never reach the opposite side of the cone, or leave it, otherwise than by cutting the base, and joining the sides of the curve by a straight line. The hyperbola, therefore, is a curve which may be produced indefinitely, and is always considered as capable of such extension, unless limited by particular circumstances. If a second cone be placed with its point downwards on the point of the first cone, the dividing plane may be made to pass through both cones, thus making two equal and similar sections, which are termed opposite hyperbolas, and many important considerations are suggested by the position of these opposite sections. "As the sides of the hyperbola are capable of indefinite extension, so they become gradually less curved as they are extended, and the curve is of such a nature that straight lines may be drawn from the centre, between the vertices of the two opposite curves, which shall continually approach the sides of the hyperbolas, and yet, if infinitely produced, can never touch them. Such straight Jines are called asymptotes. * DEMONSTRATION. As FEG is a right-angled triangle, FB X BG=BE2: but BE = AC, and it is shown in Prop. xvii. Elements of Geometry, Curve Lines, that, if FB X BG = AC2, then two lines drawn from F and G through the centre will be conjugate diameters. It is also proved by Prop. xii. Elements of Geometry, Curve Lines, that the points M, and N, are the terminations of such conjugate diameters, and are therefore points in the curve. SECT. II.] 311 PRACTICAL GEOMETRY. PROBLEM XIX, Given the asymptotes QR, UV, and a point M in one of the opposite curves, to find any point in the other opposite curve. Fig. 141. Draw Mm meeting UV in k, and QR in l. Make Im equal to kM, and m will be a point in the curve. Hence, from the same point M we may find as many points in the curve contained within the angle QCU as we please. In this manner, Fig. 1, Plate I. Curve Lines, Hyperbola, is constructed; all the points being found from the same point a in the opposite curve. KM TA DEMONSTRATION. Fig. 142. A Let M and N be two points in the same branch of the curve. Draw Mg, Nh parallel to each other, meeting QR in the two points d, k, and UV in the two points 1, e, and the opposite curve in the two points h, g. Bisect k l in 0. Through the two points 0, C draw Bb, and through the two points M, N draw the straight lines Rr, Ss parallel to Bb, meeting QR in the two points R, S, and UV in the two points r, s, and the curve in the two points m, n. TL By similar triangles SRMD, SNk, - -- - ....... Nk x RM = Md X SN | rMe, sNI, - - - - - - - . . . . NI X rM = Me x sN and by Prop. xxxv. Elements of Geometry, Curve Lines, ..... SN X Ns = RMX Mr wherefore, eliminating RM, SN, Mr, Ns, ---- ...Nk X NI = Md X Me In the same manner, ........ - - - ..: hl x hk = ge x gd hence --- ................. Nk X NI = hl x hk and since kl is bisected in 0ſ .............. Nk X NI = ON2 - "Ok? -..... ........ Oh - 012 = hlx hik and since it has been shown that ............ hk x hl = Nk X NI therefore, eliminating Nk, Ni, hk, hl, and we shall have Oh — 012 = ON? - Ok2, and since Ok = Ol, therefore Oh= ON. 312 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROBLEM XX. Given the transverse axis Aa of an hyperbola, and any point M in the curve, to find the conjugate axis. Fig. 145, No. 1. Bisect aA in C. Through the centre C, draw FG perpendicular to Aa, and through the vertex A draw HI parallel to FG, and draw Me parallel to Aa, meeting FG in e. From e, with the distance eM, describe an arc, cutting HI in k. Join ek, meeting CA in d, or, if necessary, as in No. 2, produce ek to meet CA in d; then Cd is the magnitude or length of the semi-conjugate axis : there- fore, make CB and Cb each equal to CD, and Bb is the conjugate axis. In No. 1, the conjugate axis is less than the transverse axis, and in No. 2, it is greater than the transverse axis. · F_ B_c. b _G HÀ Fig. 143, No. 2 If the point k coincide with A, both the axes are equal, and the hyperbolas to which they belong are termed equilateral. The reason of this method of finding the conjugate axis will be understood from the demonstration given at the end of Problem xxiii., in this section. . e o ba M Va. PROBLEM XXI. Given the transverse axis Aa, and the conjugate axis Bb, of an hyperbola, to find the foci. In the line of the transverse axis, make CF, Cf, each equal to AB or Ab; Fig. 144. then F and f are the foci. PROBLEM XXII. Given the transverse axis of an hyperbola in position, and the length of both axes, to find the asymptotes. Fig. 1453 . Through A, the vertex, draw HI perpendicular to the transverse axis Aa, and make AH, AI, each equal to the semi-conjugate axis. Through the points C, I, draw PQ, and through the points C and H draw RS; the straight lines PQ, RS, are the asymptotes. a . Fig. Fig. 1. -.- - - - - - - - - - --- - - - - - - - - - - .. . - - . sk - - --- A Fullarton. C'Landon & Eduburgh Engraved by Kidrmotring Fig. 2. HYPERBOLA. CURVE LINES, PRACTICAL GEOMETRY - --- - ----- - . - - - - - - - - - - - . + - - - - -+ - -- -- - - - - . Fig. Fig. 3. 6. - - -- - - ... - - PLATE.125, SECT. II.) 313 PRACTICAL GEOMETRY. PROBLEM XXIII. Given the transverse axis Aa of an hyperbola in position and magnitude, and the magnitude of the conju- gate axis, to find any point in the curve. [Plate CXV.] Fig. 146, No. 1. m Bisect Aa in C. Through the centre C draw FG per- pendicular to Aa, and through A draw HI parallel to FG. In CA, make cd equal to the semi-conjugate axis. Through any point e in FG, draw ed meeting HI in k, as in No. 2, or produce ed, if necessary, to meet HI in k, as in No. 1. Through e, draw Mm parallel to Aa. Make eM, em, each equal to e k, and M, m are points in the opposite hyperbolas. АК. — I Fig. 146, No. 2. H-Ilkal In this manner, all the points in the opposite curves, in Fig 2, (Plate CXV., Curve Lines, Hyperbola,) and in the succeeding diagram here exhibited, are found; the oppo- site hyperbolas shown in the plate, being constructed ac- cording to No. 1, and those in the diagram here shown according to No. 2: in the figure referred to in the plate, the transverse axis is greater than the conjugate ; but in the following diagram the conjugate axis is greater than the transverse. When both the axes are equal, as in the equilateral hy- perbolas, the points d and k both coincide with the vertex A; in this case, the construction to find any point in each opposite curye is simply as follows :- Fig. 146, No. 3. n2 AIVAT Fig. 146, No. 4. m Bisect the transverse axis Aa in C. Through the cen- tre C, draw FG perpendicular to Aa. Through e, any point in CF, draw Mm, and make em, eM each equal to eA: M. m are points in the opposite curves. The complete construction of equilateral opposite hyper- bolas is exhibited in Fig. 3 (Plate CX V., Curve Lines, Hyperbola). M 2 R 314 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. Fig. 147. GENERAL DEMONSTRATION. The construction being made as in No. 1, draw AF parallel to k e. Now, let CA = Ca = a, CD= b, and AF = ek=eM= 4, Ce=y, and d e = 0. By similar triangles Cde, CAF, ....... 62 x2 = a? 12 and by the right-angled triangle dCe ----... 22 = 12 + 22 therefore, eliminating v, there will arise 62 22 = a (12 + y2), or, by trans- position, a’ y2 = 62 (22 — a?), which is the central equation of the hyper- bola.— See Elements of Geometry, Curve Lines, Prop. xxx. Α PROBLEM XXIV. Given a diameter Aa, the abscissa CQ, and ordinate QN, to find any point in the curve. Draw NQ and HA perpendicular to AQ, and draw NH parallel to Aa. Fig. 148. Divide NQ and NH each in the same ratio in r and s. Draw r a and sA meet- ing each other at M, and M is a point in the curve. In this manner the opposite curves, Fig. 4, Plate I., Curve Lines, Hyperbola, are described. 4 DEMONSTRATION. Let C be the centre of the diameter Aa. Draw s t and MP parallel to NQ, meeting AQ in t and P. Let CA = Ca = a, CP = x, PM = y, CQ= %, QN=Y, At= Hs=v, and Qr = w; then will aP=CP + Ca = x + a, AP=CP – CA = x — a, aQ = CQ + Ca = m + a, and AQ CQ– CA = 2 – 3. By similar triangles aPM, aQr, - - - - - - (x + a) w = (z + a) y, and - - - - - APM, Ats, - - - - - (x - a) v = vy, and by construction HN : Hs :: QN : Qr - - - - - 0 y = (z — a) w; therefore, eliminating v and w, we have ? — a?) q = (22 — a) y, and therefore the curve is an hyperbola. (D PROBLEM XXV. Fig. 149. To find a point in the curve by another method. Find the conjugate axis by Prob. xx., and the foci by Prob. xxi. ; then the trans- verse axis Aa, and the foci F, f being now given, any point in the curve will be found by the following method : In ff produced, take any point q. From F, with the distance Aq, describe an arc at M, and from f, with the distance a q, describe another arc meeting the for- mer arc at M; then M is a point in the curve. Sect. II.] 315 PRACTICAL GEOMETRY. This is evident from Def. 7, Elements of Geometry, Curve Lines. In Fig. 5, Plate I., Curve Lines, Hyperbola, the points are found by this problem. PROBLEM XXVI. Given either curve of an hyperbola and the foci, to draw a tangent to the curve from any point M. Fig. 6 (Plate CXV., Curve Lines, Hyperbola). Join FM, fM, and bisect the angle fMF: the bisecting line MT is the tangent required. PROBLEM XXVII. Given either curve of an hyperbola and the foci, to draw a straight line from any point M in the curve perpendicular to that curve. Fig. 6 (Plate CXV., Curve Lines, Hyperbola). Find the tangent MT by Problem xxvi., and draw MN perpendicular to MT; then MN is perpendicular to the curve at M, as required. OF THE PARABOLA. PROBLEM XXVIII. . In a parabola are given in position the axes and its vertex, and any other point in the curve, to find the directrix* and focus. Fig. 150. Through the vertex A, draw AD perpendicular to the line TP of the axis, and from the given point M in the curve draw MP parallel to DA. Make AD equal to the half of PM. Join PD, and draw DT perpendicular to DP. Through T draw QR parallel to AD. In AP, make AF equal to AT; then QR is the directrix, and F the focus. This will be evident by considering the Coroll, to Def. 7, Elements of Geometry, Curve Lines. PROBLEM XXIX. METHOD 1. In a parabola are given the vertex A, the position TD of the axis, and a point N in the curve, to find a double ordinate. Fig. 151. I....... R Draw Nn perpendicular to TD, meeting TD in D. Find the focus F, and the point T, through which the directrix passes. Through any point P in AD or AD produced, draw Mm parallel to Nn. From the focus F, with the distance TP, describe two arcs meeting Mm at M, m, and the points M, m are in the curve, and consequently Mm is a double ordinate. IL * See definition 18, Elements of Geometry, Curve Lines. 316 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. DEMONSTRATION. Draw the directrix QR, and join MF. Now TP is equal to QM; but by Coroll. to Def. 7, Ele- ments of Geometry, Curve Lines, the point M is in the curve, whence QM is equal to MF. · METHOD 2. Fig. 152. From F, with the radius FA, describe a circle AGL, and from F with any radius describe another circle MKm. In AK, make AH equal to GK, and draw Mm perpendicular to AK, the points M, m are in the curve, and Mm is a double ordinate. LC DEMONSTRATION. Produce FA to T, and make AT equal to AF. Then T will be the point in the axis through which the directrix passes. We have now only to prove that FM or FK is equal to TH, Now, by construction, - - - - - - - - - - - GAK = H, and by the parabola, - - - - - - - - - - - - FG = AT; therefore, by addition, FG + GK= HA + AT. Now FG + GK= FK, and HA + AT=HT; therefore FM or FK = HT, as was to be shown. PROBLEM XXX. LA . Given a tangent NR, a double ordinate NG from the point of contact, and the position of a diameter, to find any point in the curve of the parabola. METHOD 1. Draw GR and q 1 parallel to the diameter, meeting NR in 1. Make Rh equal to 1 q, and draw hN meeting q 1 in M; then M is a point in the curye. Fig. 153. METHOD 2. Fig. 154. Draw any line Nh between NR and NG, and draw GR parallel to the diameter, meeting Nh in h. From q draw q h parallel to NR, and draw qM parallel to GR, and M is a point in the curve: DEMONSTRATION OF MÉTHOD.I. . Since (Prop. X. Elements of Geometry, Curve Lines) Nq: NG :: MI : q1, Nq:ql = NG.MI and since by parallel lines, Ml:ql:: hR: GR, - - - - - - - - MI.GR - ql'hR and by similar triangles Nql, NGR - ---- --- - - - - qI·NG = Nq.GR therefore, eliminating the common quantities, will be found qi =hR; therefore M is in the curve SECT. II.] 31? PRACTICAL GEOMETRY. DEMONSTRATION OF METHOD 2. Because q 1 has been shown to be equal to hR, and since q land h R are parallel lines, the straight lines NR and q h will also be parallel. PROBLEM XXXI. In a parabola are given an ordinate DN, and the abscissa AD, to find any point in the curve. METHOD 1. Draw any line Mq parallel to AD, meeting DN in q, and find the point 1, so that DN : Dą :: DA : AL, and draw Nl meeting Mq at M, then M is a point in the curve. Fig. 155, No. 1. Fig. 155, No. 2. PM A LN fo DEMONSTRATION. Draw PM parallel to ND, and let AD= a, DN=b, AP=x, PM 3 Dq = y, and PI = 0. then in No. 1 - - AI = PI - AP=v — X, and in No. 2 = PI + AP=0 + 2, and in No. 1 and 2 - PD = AD - AP = a - X, . and in No. 1 - - Nq = ND -qD = b – y, and in No. 2 = ND + Dq = 1 + y. Now by construction Al : AD :: Dq.: DN; .::6(0 - x)=ay, and by similar triangles NqM, MPI, ... 0b - y)=y(a — «). By finding the value of v in each of these equations, and by putting these values equal to each other, there will result 62 x ay, which is the equation of the parabola. The demonstration is the same for the second figure, except that Al=0 + x, and Na=b + y. METHOD 2. Fig. 156. Draw AG parallel to DN, and NG parallel to DA. In DN, take any point q, and draw qM parallel to DA. Divide GN in p in the same ratio that DN is divided in q, and join PA meeting M at M: the point M is in the curve. DEMONSTRATION. Draw MP and pf parallel to ND, meeting AD in P and f. Let AD or GN = a, DN or fp=b, APX, Dq or PM=y, Af or Gp=p. By the similar triangles APM, Afp, - - . bx=py, and by construction GN : Gp :: DN : Dq, .. bp=ay; therefore, eliminating p, there will be found bʻx = aył, which is the equation of the parabola. 318 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRI. DESCRIPTION OF THE DIAGRAMS, CURVE LINES, PARABOLA, [Plate CXVI.] Fig. 1 shows the method of finding the focus, the axis and ordinate being given. The principle is, that the distance of the focus from the vertex of the curve is a third proportional to Ap, Pq, or to the abscissa and to half the ordinate. This is the same in principle as Prob. xxviii. Fig. 2 shows the method of describing a parabola by means of points found as in Prob. xxix. viz.: any point M is thus found by drawing a perpendicular through M to the axis, and describing an arc or two arcs from the focus, with the distance between the perpendicular and directrix, meet- ing the perpendicular in M, m. In this case the directrix QR and the lines QM need not be drawn; for the distance TP being applied as a radius from the focus, the points M, m will be found as before. Fig. 3 depends upon the same principle as shown in Prob. xxix. Method 2. Fig. 4 depends upon the principle shown in Prob. xxx. where it is evident that the points q, h divide the lines GN and GR in the same ratio. Hence we may find as many points in the curve as we please by dividing the lines GR and GN, each into the same number of equal parts. Figs. 5 and 6 show the application of Prob. xxxi. Fig. 5, as described in No. 2, and Fig. 6, as in No. 1. Indeed, the method here applied is evident from Prob. xxx. Fig. 7 is described upon the principle shown in Prob. xxx. Method 2. Figs. 8 and 9 are described upon the principle demonstrated in the Elements of Geometry, Curve Lines, Prop. ii. viz. : by dividing each of the lines CD, DE in the same proportion, and drawing straight lines through the corresponding points of section. Fig. 10 is described upon this principle, viz. that the ordinates are as the squares of the abscissas. Fig. 11 is described by the principle shown in Prob. xxxi. Method 2. PROBLEM XXXII. Given the curve of a parabola (Fig. 12) and a diameter AP, and the vertex A, to find a double ordinate. Produce PA to 0, and through O draw JK perpendicular to OP. Take J at any distance from 0, and make OK equal to OJ. Parallel to OP draw JM, meeting the curve in M, and Km meeting the curve in m, and join M, m; then Mm is the double ordinate required. This is so evident as not to require demonstration. PROBLEM XXXIII. Given the parabola (Fig. 13), a point M in the curve, the axis AD, and the focus F, to draw a tangent through the point M. Join FM, and draw MG parallel to DA. Bisect the angle FMG, and the bisecting line MT is a tangent to the curve. DEMONSTRATION. For the line bisecting the angle made by the radius vector, and a line from that point of the curve where the radius vector meets its perpendicular to the directrix, is a tangent to the curve, Elernents of Geometry, Curve Lines, Prop. ii. PRACTICAL GEOMETRY CURVE LINES PARABOLA. · PL.NET16. Fig. 2. QQ:T- Fig. 3. Fig. 1. LAAFP N一 ​一 ​- -- Fig.4.. g.5. 重鲁番 ​TF 事 ​事 ​事业 ​事争 ​重量 ​g.16. N好一 ​N4 平台学者​:并在生产中 ​Do ,有一 ​* * - - Fig. 9. A Fig.7. 1 / / Fig. 8. NR Rg. 10. Fig. 11 Fig.72. --- - -. . .. . … 一年一​: Fig. 23. Fig. 14. Fig.15. M AN NA N - fingrared by C.Ar ro strong. A Hallarton&C London & Edinburgh w Sy PRACTICAL GEOMETRY ! CURVE LINES. PLATE 117, ELLIPSE, HYPERBOLA,& PARABOLA. Fig. 1. Fiq. 2. 7 6 ý 4 3 2 1 31 m Parabola Fig. 3. ---- ------ --- -- -------- -mom Ellipse Ityperbola Fig. 4 Fiq. . Q M Parabola Fig. 6. Qú Pay Elipse Hyperbola Engraret hv 11.11rti. A Fullarton & C°London. Edinburgh ІСТ Sect319 1 . II.) PRACTICAL GEOMETRY. . PROBLEM XXXIV. Given the parabola (Fig. 14), a point M in the curve, and the axis AD, to draw a tangent to the curve through the point M. Draw MP, meeting the axis AD in P. Produce AP to T, making AT equal to AP, and draw TM, which is the tangent required. DEMONSTRATION. Because the subtangents of the axis is double the abscissa, Elements of Geometry, Curve Lines, Prop. iii. PROBLEM XXXV. Given the parabola (Fig. 15), a point M in the curve, and a diameter AP, to draw a tangent through a given point M in the curve. Find the ordinate p m, by Prob, ii., and draw the ordinate PM parallel to p m. Produce AP to T, making AT equal to AP, and draw MT, which is the tangent required. PROBLEM XXXVI. Given the vertex, the focus, and the latus rectum in position and magnitude, to find any point in the curve of an ellipse, hyperbola, or parabola. [Plate CXVII.] METHOD 1. In Plate CXVII. Curve Lines, Figs. 1, 2, 3, let A be the vertex, F the focus, and FH the half of the latus rectum. From F, with the radius FA, describe an arc AG, meeting FH in G. Make FP to HQ as AF is to GH. Draw Mm parallel to FH, and from F, with the radius FQ, describe the arc QM, and the point M will be in the curve. If with the same distance and from the same point another arc be described, meeting Mm at m, m will be another point in the curve. Or, divide AF into any number of equal parts, as two, and set off any number of these parts from F in the line FP; also divide GH into the same number of equal parts, and set off as many of these parts from H in the line HQ; then proceed with any two corresponding points 2, 2, as has been done with the points P and Q, and we shall have one point M' at the extremity of each ordinate, and one in the extremity of each equal and opposite ordinate. The curve is that of an ellipse when GH is less than AF, or AF greater than the half of FH, as in Fig. 1. The curve is that of a parabola when GH is equal to AF, or AF the half of FH, as in Fig. 2; and the curve is that of an hyperbola when GH is greater than AF, or when AF is less than the half of FH, as in Fig. 3. METHOD 2. Figs. 4, 5, and 6. Draw AL parallel to FH. From A, with the distance AF, describe an arc AL. Through the two points L, H, draw LQ, produced at pleasure. Draw Qm parallel to *FH, meeting AF produced in P. From F, with the distance PQ, describe an arc meeting Qm at M; then M is a point in the curve. 320 [PART IV. ELEMENTS AND PRACTICE OF GEOMETRY. If from the same point F, with the same distance PQ, another arc be described meeting Qm in m, m will be another point in the curve. In the ellipse, Fig. 4, if the two axes Aa, Bb be given in position and magnitude, the curve may be described by the same method. For, find the focus F by Prob. i. Practical Geometry, Curve Lines, and draw AL parallel to FH, and produce CB to N. Make CN equal CA, and AL equal to AF, and join LN; then proceed as we have just shown.* PROBLEM XXXVII. Given the abscissa, an ordinate, and the axis of the curve of an ellipse, hyperbola, or parabola, to find any point or points in that curve. [Plate CXVIII.] 0 Let Aa be the axis (Figs. 1, 2, 3, Ellipse, Hyperbola, and Parabola, Plate CXVIII.) PM the ordinate either from some point in the axis, or from some point beyond it. Draw AD parallel to PM, and MD parallel to PA. Divide MP in h, and MD in g, each in the same proportion. Draw a h and gA, in Figs. 1 and 2, and in Fig. 3, h m and gA, meeting each other at m, and m is a point in the curve. Figs. 4, 5, and 6 show the method of describing the respective curves by finding a sufficient number of points. In the parabola, Fig. 3, and 6, the axis Aa is of infinite length; therefore only the point A can be given ; and since the axis is of infinite length, the line hm must be drawn parallel to the abscissa. PROBLEM XXXVIII. Given the vertex, the ordinate, and a tangent at the extremity of that ordinate of an ellipse, parabola, or hyperbola, to find the centre, and to determine the species of the curve. [Plate CXIX.] Figs. 1, 2, 3, Plate CXIX. Ellipse, Hyperbola, and Parabola. Through the vertex A, draw AT parallel to the ordinate PM, meeting the tangent MT in T. Join AM, and bisect AM in d. Through the point of concourse T, and the point of bisection d, draw Td which in Figs. 1 and 2 may be continued till it meets AP in C; then C is the centre, but in Fig. 3, Td is parallel to AP, and the centre is therefore at an infinite distance. If the centre fall on the same side of the line AT on which the ordinate lies, the curve is an ellipse; and if the centre fall on the contrary side of the line AT, the curve is an hyperbola ; but if the line dt happen to be parallel to the abscissa AP, the curve is a parabola. Hence, since the diameter is found by the above method, we may describe the curve by finding a sufficient number of points, as in 4, 5, and 6, by Prob. xxxvii. ; for then a diameter and double ordinate will be given to find these points. * With regard to CN being made equal to CA, and AL to AF, see Emerson's Conic Sections, Prop. xxxviii. page 23, Ellipse. RACTICAL GEOMETRY CURVE LINES. ELLIPSE HYPERBOLA E PARABOLA PLATE 118. Ellipse D ----- Parabola Fig.l. Hyperbola TA ----... -- l'ig.3 Fig. 2. D ' . - . -- - ***........ . . os AY - Inonduued by P. Nicholson. A Frilarw i n dü& Eainburgh CH PRACTICAL GEOMETRY CURVE LINES ELLIPSE HYPERBOLA | PARABOLA. PLATE 09. Fiy.3. Fig. 1. Elipse aj Parabola Fig.?.. F RA== *- | Hyperbola 12 . . A . . . . . .. . . .. . Fiq.4 1 " AKO som om ...... Areteted by P. Nicholson Engnewed by Syms. A Fullarlor&Co London & Edinburgh UN OF I PRACTICAL GEOMETRY, CURTE LINES PLATE 120 Fų.1. Fig. 2. ТВ Fig. 4 w 6 Enurrared by natural Aldiz Levis Ernuda M PRACTICAL GEOMETRY,. PLITE CURTE LIAES. Fig.I 1. tahanap ng Fiq.12. t attoos for forefoort Fig. 3. Fig. 4.. --- - ... . .. camin..... Fig.5. 2 NS Fig. 4 . K Y3 . Inimrel ''Carn.sirmu. Eullartan. & Lanson Ecarburga SECT. II.] 321 PRACTICAL GEOMETRY. CURVE LINES OF THE HIGHER ORDERS. [Plate CXX.] THE CONCHOID OF NICHOMEDES is a curve of such a nature, that if in Fig. 1, Plate CXX., Curve Lines, AB be a straight line, C a point out of it, and CE a line perpendicular to AB, cutting AB in d; and if dE be a given distance, and if any number of straight lines CM be drawn, all the dis- tances gM between the straight line AB and the curve will be equal to each other. The fixed point C is called the pole, and it is obvious that the curve may be described by a tram- mel, as in Fig. 2. PROBLEM XXXIX, To draw a tangent to the conchoid through any point M in the curve, Fig. 3. Produce CM to Q, and make MQ equal to Cg. Draw QT perpendicular to CQ, meeting AB in T, and join TM; then TM is the tangent required.---See Newton's Fluxions, Ex. 2, p. 64, and also p. 72, 8vo. edition. Oy THE SINIC CURVE. The Sinic Curve, or figure of the sines, is a curve of such a nature, that the abscissa AP, Fig. 4, is equal to the arc a p of a circle, and the ordinate PM is equal to the sine p q of that arc. The figure may therefore be easily described by dividing the arc into equal parts, and repeating one of such parts in the line Aa as often as necessary, for any part or the whole of the semicircumference. Figs. 5 and 6 are described from an elliptic curve in the same manner. The reader may here observe the very near coincidence of these curves with those of the first order. When the axis major is to the axis minor in the ratio of 4 to 3, as in Fig. 5, the curve is nearly that of a parabola. When the generating ellipse stands with its axis major parallel to the ordinates, the figure will approach to an hyperbola. The axis Ba of the nearest hyperbola may be found by Prob. xxxviii. p. 320, supposing BD the abscissa, DA the ordinate, and M a point in the curve given. SPIRALS. DEFINITIONS. A Spiral is a curve making any given number of revolutions round a fixed point without meeting itself. The fixed point is called the centre of the spiral. A line drawn from the centre of the spiral to the curve, is called an ordinate. SPIRAL OF ARCHIMEDES. [Plate CXXI.] DEFINITION. If the arc passed over by the radii be always in a given ratio to the difference of the ordinates, the spiral is called the spiral of Archimedes. Therefore to draw the spiral of Archimedes, we need only to draw lines forming equal angles round the centre, and fix upon one of these lines as the greatest ordinate; which being determined, 2 5 322 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. divide it into as many equal parts as the number of revolutions intended, and subdivide each part into as many smaller equal parts as the number of angles: make the second or next ordinate one part less; the third two parts less; the fourth three parts less, &c. than the first, and draw the curve through these points.-See Fig. 1, Plate CXXI., Practical Geometry, Curve Lines. PROBLEM XL. To draw a tangent through any point M in the curve, Fig. 1, Plate CXXI., Practical Geometry, Curve Lines. From the centre C, with the distance CM, describe a circle. Draw CT perpendicular to CM, and make CT equal to the circumference of the circle: then TM being drawn, is the tangent. LOGARITHMIC SPIRAL. If the ordinates form equal angles at the centre, and all the succeeding ordinates and chords form equal angles at the curve, the spiral is called the logarithonic or proportional spiral, as in Fig. 2, Spirals. Hence the description of the curve is evident by similar triangles. HYPERBOLIC SPIRAL. DEFINITIONS. If from any point in a straight line, arcs of circles of equal length be described on the same side of it and terminate in it, the curve passing through all the other extremities is called the hyperbolic spiral.—See Fig. 3, Spirals. The straight line in which the arcs terminate, is called the axis.-See Fig. 3. A straight line drawn on the other side of the curve, opposite and parallel to the axis, is called the asymptote. PROBLEM XLI. To describe the hyperbolic spiral, the axis, the centre, and the ordinate next to the axis, being given, Figs. 4 and 5. Let BS (Fig. 4) be equal to the given ordinate. Draw BG at any angle with BS, and SR par- allel to BG. In the straight line BG, make Bl equal to any distance, and make B2, B3, B4, &c. equal to twice, three times, four times, &c. that distance, and make SR equal to B2. Draw 1R, 2R, 3R, &c. meeting BS in the points C, D, E, &c. Having drawn the ordinates Ca, Cb, Cc (Fig. 5) at equal angles, make the ordinate Ca equal to twice SB, the ordinate Cb equal to Sc, the ordinate Cd equal to Sd, &c.; then the curve passing through all the points a, b, c, d, &c. is the hyperbolic spiral. In the diagram, Fig. 4, the line BG might require to be extended more than the space would allow. In this case we might draw any lines parallel to BG, and repeat one of the equal parts in- tercepted by any two adjacent lines drawn from R to the divisions in BG; thus making i k, k 1, 1 m, &c. each equal to i h, or h g, or g f, &c.; and should this be inconvenient, take another parallel line nearer to S. SECT. II.] 323 PRACTICAL GEOMETRY. - -- - . - . - - - - - - - - - - - - - - - - - - - - - - - PROBLEM XLII. To draw a tangent to any point e in the curve, Fig. 4. Describe the arc m n Fig. 5 between the axis and the curve. Draw CQ perpendicular to the axis Cm, and make CQ equal to the arc m n. From C, with the radius CQ, describe the arc QT, and draw CT perpendicular to Ce; then draw Te, which is the tangent required. CHAP. V.-PLANE TRIGONOMETRY. PLANE TRIGONOMETRY is the art of computing the sides and angles of a plane triangle from certain data derived from the relation of the sides and angles to each other. The sides and angles of a triangle are called its parts; and as every triangle has three sides, and three angles, it is said to consist of six parts. In every triangle, when any three of the parts are given, provided one of them be a side, the other three can be found.* The two legs of every angle may be made radii of a circle drawn from their point of meeting, and by comparing the length of the arc intercepted between the two legs, with the length of the whole circumference of the circle, we obtain a measure of the angle. If the circle be divided into an aliquot number of parts, the whole circumference is commonly into four quadrants, or quarters of a circle; or an angle is said to cut off the eighth or twelfth of a circle: but this method would be ill suited to the purposes of trigonometry where angles of every possible dimensions come under consideration. It is therefore convenient to divide the circumference into a number of parts, and these again into smaller parts, by which means fractions are avoided in most cases. With this object in view the ancient mathematicians divided the circumference into 360 equal parts, and this practice is still followed by the moderns, except the French, who divided the circumference into 400 equal parts. † Therefore, admitting the circumference of a circle to be divided into 360 equal parts, the quadrant will contain 90 of these parts. * It is obvious that when the three angles are the only data, we cannot ascertain the length of the sides, because all similar triangles must be equiangular, whatever may be their dimensions. † The student should bear in mind that this division of the circle into degrees is only equivalent to making a scale on a straight line, in order to measure small parts of the line. The number 360 was very wisely chosen by the ancients, because it is more divisible than any other near it, and at the same time large enough to render a further division un- necessary, in many cases that must have occurred before instruments were brought to great perfection. The number 400, :adopted by the French, is perhaps the only instance in which their admirable system of measures has failed to effect a practical improvement; for where the circle is divided into 400 parts, neither the 3rd, 6th, 7th, 9th, parts of the circle consist of an even number of degrees; whereas by the division into 360, these parts, with the excep- tion of the 7th, are all exactly di visible. The same observation applies to the subsequent division by 60 into minutes, seconds, &c. as contrasted with the decimal subdivision of the degree used by the French, 324 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. Each of the 360 equal parts into which the circumference of every circle is supposed to be divided, is called a degree; and in order to obtain the mensuration of angles to the greatest nicety, every degree is subdivided into sixty equal parts, called minutes; each minute again into sixty equal parts, called seconds ; each second into 60 equal parts, called thirds; and so on till the divisions become imperceptibly small. NOTATION. 201 30 20 JA Degrees are indicated by placing a small cipher on the right a little above the unit figure. Minutes are denoted by a small accent or dash in the same situation ; seconds by two such accents ; and so on for the other smaller divisions. Thus, 35° 31' 23" is read 35 degrees 31 minutes 23 seconds. * For the convenience of measuring angles on a small scale, describe the quadrant ACB with any convenient radius, and divide the arc AB into nine equal parts. Draw the chord AB, and from B as a centre, with the distances A 10, A 20, A 30, &c. describe arcs, cutting the chord AB in the points 10, 20, 30, &c. and the distances A 10, A 20, A 30, &c. measured on the chord AB, will be the chord of 10°, 20°, 30°, &c. Such a scale will be found in most cases of mathematical instruments, as well as scales of equal parts. As the circle is divided into 360 degrees, the chord of 60 degrees is obviously the side of an inscribed hexagon, which is equal to the radius. It is therefore necessary in measuring an angle by this scale of chords, first to draw an arc of a circle with an open- ing equal to the chord of 60 degrees, and then apply the scale upon such arc, for the measurement of the angle. DA PROPOSITION İ. At a given point in a straight line to draw another straight line that shall make an angle containing any proposed number of degrees. From the given point, with a radius equal to the chord of 60°, describe an arc meeting the straight line; and from the point of intersection as a centre, with the chord of the proposed number of degrees, as a radius, describe another arc cutting the former arc; draw a line to join the point of intersection of the two arcs with the given point, and the angle thus formed will contain the proposed number of degrees. * This division of the circle, as far as practical utility is concerned, finds a limit sooner or later in the inevitable imperfection of our instruments; for though we may carry it as far as we please in calculations founded upon assumed data, and arrive at conclusions proportionably exact, yet in the practice of Trigonometry, our calculations always depend upon actual observations, or measurements of angles, made with a theodolite or sextant, a compass, or some less accurate instrument; and as the angles indicated by these instruments are only such as can be made visible on the circumference of a circle of a few inches radius, the student may naturally wonder how so small a division as a minute, or the 3600th of the circumference can at all be seen or measured; but the moderns have arrived at such extraordinary perfection in dividing instruments that an error amounting to no more than the 3000th part of an inch, in placing any one division, could not escape detection in some instruments intended for accurate measurement of angles. Sect. II.) 325 PRACTICAL GEOMETRY. EXAMPLES. Ex. 1.--At the point B in the straight line BA, describe an angle that shall contain 40°. From the point B with the chord of 60° describe the arc ef, meeting AB Fig. 1. in e; from e, with a radius equal to the chord of 40°, describe another arc meeting the former in f; join Bf; then ABf is the angle required. Ex. 2.- At the point B in the straight line BA, to describe an angle which shall contain 90°. From B, with a radius equal to the chord of 60°, describe the arc ef, meeting Fig. 2. AB in e, and from e, with a radius equal to the chord of 90°, describe another arc, meeting the former in f. Join Bf, and ABf will contain 90°. Note.--As the three angles of every plane triangle taken together are equal to 90° X 2, or 180°, which is the number of degrees in the semicircle, we know té LB that when one angle contains 90°, the other two must contain 90° between them. To make at a given point in a straight line an angle which shall contain a number of degrees greater than 90, we have only to subtract the given number of degrees from 180, and at the given point make with the remainder an angle in the opposite direction. Ex. 3.--At the point B in the straight line BA, make an angle which shall contain 130 degrees. Here 180 — 130 = 50. Produce AB to e, and from B, with a Fig. 3. radius equal to the chord of 60°, describe the arc ef, meeting AB produced in e. From e, with a radius equal to the chord of 50°, describe another arc, meeting the former in f. Join Bf, and A Bf will contain 130 degrees. Or thus- From B, with the chord of 60°, describe the arc ghf, meeting BA in g. From g, with the chord of 90°, set off gh, and from h, with the chord of what the given angle exceeds 90°, set off hf. Join Bf, and A Bf will contain the proposed number of degrees. PROPOSITION II. To find the number of degrees which a given angle contains. From the point of concourse, with a radius equal to the chord of 60°, describe an arc meeting both lines which contain the angle; then apply the chord of this arc cut off by the lines, to a scale of chords, observing to place one foot of the compasses in zero : the other will point out the number of degrees the angle contains. EXAMPLES Fig. 1. Ex. 1. Suppose (in Fig. 1) the angle A Bf were given, and it were required to find how many degrees it contains. From the point B, with a radius equal to the chord of 60°, describe the arc ef, meeting AB and Bf in the points e and f: the chord ef applied to the scale will show the angle to contain 40 degrees. 326 [Part IV ELEMENTS AND PRACTICE OF GEOMETRY. Fig. Ex. 2. Suppose (in Fig. 2) the angle ABf were given, and it were required to Fig. 2. find how many degrees it contains. From B, with a radius equal to the chord of 60°, describe the arc ef, meeting the straight lines AB and Bf in e and f: apply the chord ef to a scale as before directed, and the angle will be found to contain 90 degrees. Ā Ex. 3. Suppose the angle ABf (Fig. 3, Prop. i.) were given, and it were required to ascertain how many degrees it contains. Produce AB to e, and from B with the chord of 60° for a radius, describe the arc ef, meeting AB in e, and Bf in f: apply the chord ef to the scale as before, and the angle eBf will be found to contain 50°, which must be taken from 180, and the remainder, 130, is the number of degrees in the angle ABf. Or thus, From B, with the chord of 60°, describe the arc ghf, meeting AB and Bf in the points g and f: apply the chord of 90° to gh, and the chord of hf to the scale, which will give 40° for the angle hBf, and this added to 90°, gives 130° for the angle ABf as before. Having given these examples on the construction and measuring of angles, we will now proceed to the construction and computation of the different cases that may occur in the practice of Trigonom- etry; but as some terms are used in Trigonometry which have not been hitherto defined, and some lines are drawn in relation to the circle which are not spoken of in the treatise of Geometry, and by the known lengths of which trigonometrical calculations are for the most part effected, we shall first require the reader's attention to certain definitions, together with the propositions on which the art of Trigonometry is founded. It may here be proper to observe that there are two methods by which the cases of Trigonometry are usually solved, namely, construction and calculation. The former method is generally used in such cases as admit of the triangle being drawn of its full size, or to a very large scale, as in carpentry, or masonry, because the errors inseparable from actual drawing are not increased at all, or at most not to an inconvenient degree by multiplication. But where the data are obtained by observation of angles made with accurate instruments, as in surveying land, measuring distances, &c., it is only by calculation that we are enabled to obtain a correct result. _ PLANE TRIGONOMETRY. DEFINITIONS. 1. The complement of an arc is another arc which is the difference between the first are and a quadrant. · Let BC be an arc, and BCD a quadrant; then CD is the complement of that arc BC, and BC is the complement of the arc CD. 2. The supplement of an arc is another arc which is the difference between the first arc and a semicircle. Let BC be an arc, and BCD a semicircle; then CD is the supplement of the arc BC, and BC is the supplement of the arc CD. D A B 3. The sine of an arc is a straight line drawn from one extremity of the arc upon the diameter passing through the other extremity, and perpendicular to such diameter. Let BC be the arc of a circle described from the point A ; then CE, drawn perpendicularly upon the radius AB or the diameter DB, is the sine of the arc BC or CD. DE AB 1 Sect. II.] 327 PRACTICAL GEOMETRY. 4. The cosine of an arc is the sine of the complement of that arc. Let BC be an arc, and BCD a quadrant; then CH, or AE is the cosine of the arc BC. 5. The tangent of an arc is a straight line, one end of which touches the circle at one extremity of the arc, and the other end meets a line drawn from the centre of the circle through the other extremity of the arc. Let BC be an arc; then BF, included between the point of contact B and the line AF, is the tangent of the arc BC. 6. The cotangent of an arc is the tangent of the complement of that arc. Let BC be an arc, and BCD a quadrant; then DG, the tangent of the comple- mental arc CD, is the cotangent of the arc BC. 7. The secant of an arc is a line drawn from the centre through one extremity of an arc to meet a tangent from the other extremity. AF is the secant of the arc BC. 8. The cosecant of an arc is the secant of the complement of that arc. Let BC be an arc, and BCD a quadrant; then AG is the cosecant of the arc BC, and Ad is the cosecant of the arc DC. The sine, cosine, tangent, cotangent, secant, cosecant of an angle, is the sine, cosine, tangent, cotangent, secant, cosecant of an arc described with any radius from the angular point, and terminated by the legs of that angle. 9. The complement of an angle is an angle which is equal to the difference between that angle and a right angle. Let ABC be an angle, and ABD a right angle; then CBD is the complement of the angle ABC, and ABC is the complement of the angle CBD. 10. The supplement of an angle is an angle equal to the difference between that angle and two right angles. Let AD be a straight line, and BC another meeting it at the point B; then the angle CBD is the supplement of the angle ABC, and the angle ABC is the supplement of the angle CBD. ----- в А. CONTRACTIONS. с. Sin. for sine, cos. for cosine, tan. for tangent, cot. for cotangent, sec. for secant, cosec. for cosecant, and rad. for radius. The sine and cosine of an arc are the same as the sine Fig. 1. Fig. 2. and cosine of its supplement; for, if ACD be a semicircle, and AC an arc less than a semicircle ; then (by Def. 3) CE is the sine not only of the arc AC, but of the arc CD, DE B DETA and BE is the cosine ; but with regard to Fig. 1, the co- sine of the arc AC being greater than a quadrant is negative, while in Fig. 2 it is affirmative, being less than a quadrant. 328 (PART IV ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION III. The ratio of the sides of a triangle is equal to that of the sines of their opposite angles. In the triangle ABC, draw CD perpendicular to AB, cutting AB in D. On AC and BC, with any distance, cut off Ak, Bh, equal to each other, and draw ki and hg parallel to CD, cutting AB in i and g. 4 i ] Let BC = a, AC =b, CD = p, Ak = Bh = r.. Aik, ADC, Sm; -= Then, by sim. AS - - - - - - - Bgh, BDC, sin, B = sin. A therefore, eliminating r, p, - - - - - - The elimination is made by dividing the higher equation by the lower. Bilino 218 olo PROPOSITION IV. The cosine of any angle is equal to the sum of the squares of the sides containing that angle, minus the square of the remaining side multiplied into the reciprocal of twice the product of the containing sides, radius being considered unity. For draw CD perpendicular to AB, cutting AB in D, and let the sides be denoted by the small letters corresponding to the capitals at the oppo- site angles, and let AD = the distance of the perpendicular CD from the + A, be called d. Then (Geom. Th. xxxvii.) we have a = 12 + c2 - 2cd 0 1 and therefore - - - - - - - d - 2c but since g = cold, the (cos. A) b=d; therefore, mult. the two eqs. a_62 +62 - a? and dividing by b, we find ) 2bc PROPOSITION V. The ratio of the sum and difference of the sides of a triangle is equal to the ratio of the tangents of the half sum and half difference of the angles at the base. Let ABC be a triangle. From A, with the distance AB, describe the circle FBD cutting AC in D, or AC produced in D. Produce CA to F; join DB and FB: draw DE parallel to BC, cutting FB in E, or FB produced in E; then the LABD = ADB is the half sum of the Ls ABC, ACB, and the L DBC = BDE half their difference; and since in the 'As ABC, ABD, the - [ DAB + ABC + ACB = ADB + ABD + DAB; therefore (Ax. 5) ABC + ACB = ADB + ABD = 2ADB therefore - - (ABC + ACB)= ADB = ADE + BDE; SECT. II.] PRACTICAL GEOMETRY. 329 therefore . • }(ABC + ACB) = ACB + BDE ; whence (Ax. 5) 1(ABC — ACB) = BDE. Now, in the diagram, FC is the sum of the two sides AB, AC, and CD is their difference; also BF is the tangent of the 4 ADB or BDF, and BE is the tangent of the - BDE; FC FB tan. BDF therefore, by the property of parallels, CÚ = BE = pan. BDE RIGHT-ANGLED TRIANGLES. In a right-angled plane triangle, the sides containing the right angle are called the legs, and the side opposite to the right angle is called the hypothenuse. In a right-angled triangle only five of the parts are varied, in consequence of the right angle being a constant quantity. In order, then, to ascertain the number of ways the data may be varied, taken two and two at a time, let A, B be the two acute angles, a, b the legs opposite them respectively, and c the hypothenuse. Now the number of ways which two things can be selected out of five is known, by the theory of combinations, to be 10. Let the above representatives of the angles and sides be combined accordingly, and they will stand as below :- s Aa, Ba, BC, ac, ab Bb, Ab, Ac, bc, BAS whence we may observe that the number of ways in which a leg and an opposite angle can be given is two; a leg and adjacent angle two; the hypothenuse and adjacent angle two; the hypothenuse and a side two; the two sides one, and the two angles one. Let the case in which the two angles are given be set aside, as these data are not sufficient to limit the problem; and of the other combinations, we shall have the five following cases; viz. :- a leg and an opposite angle; a leg and an adjacent angle; the hypothenuse and an adjacent angle; the hypothenuse and a leg ; the two legs. But because, when an acute angle is given, the other is found by subtracting the first given from 909, the two cases in which a leg and an angle are given may be reduced to one, and consequently the five cases will be reduced to the following four; viz.: a leg and an angle; the hypothenuse and a leg; the two legs. Hence the resolution of right-angled triangles may depend on four different propositions. PROPOSITION VI. Given one of the legs and an opposite angle, to construct the triangle and find the remaining parts. Ex.-Given the leg BC and its opposite angle BAC. Required AB and AC. Subtract the given angle from 90°, and the remainder will be the angle BCA, which is an angle adjacent to the given side. 2 T 330 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. Construction.— Draw the line AB of any convenient length, and at the point B draw BC perpendicularly to AB; and from a scale of equal parts lay off BC equal to the given length. At the point C, make with CB an angle BCA equal to the complement of the given angle BAC. Join CA, and measure AB and AC on the same scale from which BC was taken, and the lengths of the sides of the triangle will agree with the following calculations. BC sin. ACB As sin. BAC : sin. ACB :: BC : AB = ? sin. BAC BC sin. ABC sin. BAC : sin. ABC (90°) :: BC: AC = sin. BAC PROPOSITION VII. The hypothenuse and the adjacent angle being given, to construct the triangle and find the remaining parts. Ex.—Given the hypothenuse AC, and the angle BAC, to find the sides A B and BC. Construction.— Draw AC (Fig. to Prop. vi.) of the given length, taken from a scale of equal parts, and at the point A make the angle BAC equal to the given angle ; and from the point C draw CB perpendicular to AB; then AB and BC, measured from the same scale with AC, will have their lengths determined. Calculation. Find the angle ACB as before ; then, sin. ABC (90°): sin, BAC :: AC: BC= AC sin. BAC sin. 90° sin. ABC (900): sin. ACB :: AC: AB – AC sin. ACB sin. 90° PROPOSITION VIII. The hypothenuse and one of the legs being given, to construct the triangle and find the other parts. Ex.--Given the hypothenuse AC and the base AB, to find the angles and the other side. Construction.—Draw AB (Fig. to Prop. vi.) of the given length, measured from a scale of equal parts, and at the point B in AB draw BC perpendicular to AB; and from the point A, with a radius equal to the given hypothenuse, describe an arc cutting BC in C. Join CA ; then the side BC, measured from the same scale as AB and AC, will have its length determined. Calculation.--AC: AB : : sin. ABC (90°) : sin. ACB =- W AD_AB sin. 90° =-A ; :: BAC = 90° – ACB. To find BC:- AC sin. BAC sin. ABC (90°): sin. BAC :: AC: BC = sin. 900 Note. If we suppose the sine of 90°, which is equal to the radius of a circle, to be 1, * each of the preceding formulæ, where the sine of 90° enters, will be greatly simplified. * In the tables by which trigonometrical calculations are worked the radius is expressed by unity, with ciphers after it, so as to conform with the number of decimal places in the other parts of the table. SECT. 12.1 331 PRACTICAL GEOMETRY. PROPOSITION IX. Given the two legs, to construct the triangle and find the remaining parts. Ex.-Given the base AB, and the perpendicular BC, of a right-angled plane triangle. Required the angles and the hypothenuse. Construction.-Draw AB (Fig. to Prop. vi.) of the given length, and at the point B draw BC perpendicular to AB, making BC also of the given length. Join AC; then AC, applied to the same scale whence AB and BC were taken, will have its length determined, and the angles can be measured as before directed. The solution of this example will be obtained by Proposition v. Thus, AB + BC: AB BC:: tan. * (1800 – ABC) : tan. (BAC » BCA = (AB_" BC) X tan. # (180° — ABC) AB + BC = half the difference of the angles at the base, and the half sum is known; therefore the angles themselves are known, and the sides can be found as before. From what has been done above, it is evident that the propositions already given are sufficient to solve every case of right-angled triangles, without embarrassing the judgment, and loading the memory with a multiplicity of useless varieties. It may, however, be said, that another case may occur which has not been alluded to, viz. when the three sides are given to find the angles; but when it is considered that the right angle in this case also is known, the other angles may be found by an inversion of Proposition iii. OBLIQUE ANGLED TRIANGLES. As the propositions given with respect to right-angled triangles apply.equally to the same cases of oblique angled triangles, we shall, in the remaining part of this treatise, confine ourselves to such practical examples of construction as will, by reference to what has been already advanced, render the method of calculation sufficiently obvious. In order to ascertain the number of cases that may occur in the construction of oblique angled triangles, let A, B, and C denote the three angles, and a, b, c their respective opposite sides: now, as we must always have three of these six things given to determine the rest, we have to inquire how many ways three things can be selected out of six. The theory of combinations gives twenty, which may be arranged as below:- No. 1. ABC No. 6. abc No. 2. ABa АВЬ BC ACC BCc No. 3. B Ca ACO ABC No. 4. Aab Bab Aac Cac Bbc Cbe No. 5. Cab Bac Abc A Ca But although, in the same triangle, the number of ways which three parts can be given in order to find the rest is 20, yet, as six of these selections, No. 2, have this in common, viz. two angles and an opposite side ; and three, No. 3, have two angles and a contained side ; the nine combinations of data may be reduced to one case, because, when two angles of a triangle are given, the third is also given, being the supplement of the other two; therefore, setting aside No. 1 where the three angles 332 [Part IV. . ELEMENTS AND PRACTICE OF GEOMETRY, are given (which data only determine the ratio of the sides), we may draw this conclusion, viz.- that the construction must fall under one of the following cases, in which may be given two angles and a side . - two sides and an opposite angle two sides and the contained angle three sides - - - - . . - - No. 2 and 3 ; No. 4; No. 5; No. 6. - - - Though the construction will admit of these four cases, yet, with regard to the arithmetical solu- tion, there are only three; for Nos. 2 and 3, where two angles and a side are given, and No. 4, where two sides and an opposite angle are given, may be reduced to one case; as will appear by the property developed in Proposition iii. PROPOSITION X. Given the three sides of an oblique angled triangle, to find the angles. The solution of this proposition by construction must be sufficiently obvious to the student; we may therefore proceed at once to the calculation. Let fall a perpendicular from the greatest angle to the opposite side, or base, (which will be the greatest side,) so as to divide the whole triangle into two right-angled triangles. The proportions will then be As the base, or sum of the segments, Is to the sum of the other two sides ; So is the difference of those sides, To the difference of the segments of the base. The segments of the base are then found by adding half the difference to half the sum for the greater, and subtracting it for the less segment. We have thus two right-angled triangles, in each of which two sides and the angle opposite to one of them are given. The remaining angles may therefore be found by the rule deduced from Propo- sition iii. PROPOSITION XI. Given two angles and a side, to describe the triangle and find the remaining parts. S Note. If all the given parts are not contiguous, take the supplement of the two given angles, which will be one of the angles adjacent to the side given ; consequently the side will be situated between two angles. In the oblique angled plane triangle ABC let there be given the side AB, 34 feet, the angles ABC, BCA, 350 and 62º respectively, to find BC and AC. Construction.—Draw the straight line AB, and from a scale of equal parts take off 34, which apply from A to B. At the point B in the straight line AB, make the angle ABC equal to 350; and at the point A make an angle equal to 83°, the supplement of the sum of both given angles. Apply BC and AC to the same scale whence AB was taken, and they will measure 38 and 22 feet respectively. SECT. II. 333 PRACTICAL GEOMETRY. PROPOSITION XII. Given two sides and the contained angle, to construct the triangle and measure the other parts. Draw a straight line of any convenient length, and at any point in the same make an angle equal to the given angle; and from a scale of equal parts, lay off from the vertex of the angle, on the legs containing it, the respective lengths of these legs; join their remote extremities, and the line thus joining them will be the third side of the triangle, which must be measured on the same scale with the given sides and the angles. Ex.--Given the two sides AB, BC (Fig. to Prop. xi.) 34 and 38 feet respectively; and the angle ABC 35º; to find the other angles and the third side. Construction.-- Make AB = 34 feet, and at the point B in AB make the angle ABC = 350 ; and from the scale from which AB was taken, make BC = 38 feet. Join AC, which being applied to the same scale, will be found to measure 22 feet. The angles BAC, ACB, being measured as before, are 830 and 62º respectively. PROPOSITION XIII. Given two sides and an angle opposite to one of them, to construct the triangle, and measure the remaining parts. Draw a straight line equal to one of the given sides, at one extremity of which make an angle equal to the given angle ; at the other extremity, with a distance equal to the other given side, de- scribe an arc, which will either touch the unknown side in one point, or cut it in two points ; hence this case will, in certain instances, admit of two solutions. Ex.-Given the two sides AB, AC, equal to 34 and 22 feet respectively, and the angle ABC opposite the side AC equal to 35°, to find the third side and the other angles. Construction.--Draw AB, 34 feet, as before directed, and at the point B in AB make an angle equal to 350 ; then from the other extremity A of AB, with a distance equal to 22 feet, taken from the same scale as AB, describe an arc which will either touch the unknown side in D, or cut it in the points C, C. Join AD if it touch in D, or CA, O'A if it cut in the points C and C. The line BD, BC, or BC', will in either case be the third side, that is, BC= 20, BD= 28.4, and BC = 37 respectively; and the angles being measured as before directed, will, according to the respective sides, be BCA 113°, BAC 32°; BDA 909, BAD 55º; BCA 66°, and BAC 79°. . It is obvious that the ambiguity in this proposition is more likely to mislead in proportion as the angle ADB approaches a right angle ; but when that angle is either decidedly obtuse, or acute, the peculiar circumstances of every case will generally limit the proposition, so as to leave no doubt as to which solution must be adopted. In practice the student should be careful to prevent the pos- sibility of ambiguity in similar cases, or the most correct measurement of angles may be rendered useless. 334 [Part IV ELEMENTS AND PRACTICE OF GEOMETRY. OF THE RIGHT-ANGLED TREHEDRAL. VIDT DEFINITIONS. A right-angled trehedral is a solid formed by three planes meeting each other in a point, two of which planes are perpendicular to each other. The angle at the vertex of the figure in each of the perpendicular planes is called a leg. The angle at the vertex of the figure in the plane opposite the right angle, is called the hypothenuse. The angles formed by the plane opposite the right angle, meeting each of the perpendicular planes, are called the angles of the solid. PROPOSITION XIV. Given one leg and the adjacent angle of the solid, to find the other leg.* Fig. 2. Construction. Method 1. Let BAQ be the given leg. Fig. 1. From any point in AQ draw Qr, meeting AB or AB produced, perpendicularly in the point r; make the angle Qrs equal to the adjacent angle of the solid ; draw Qs per- pendicular to Qr, meeting rs in the point s; draw also Qt perpendicular to AQ, and make Qt equal to Qs. Join At, and Qat will be the required leg. Or thus, Method 2.-From any point Q in AQ draw @r, meeting AB or AB produced, perpendicularly in the point r. Produce AQ to s, and make Qs equal to Qr; draw Qt perpendicular to AQ; make the angle Qst · equal to the adjacent angle of the solid. Join At, and QAt will be the required leg. Fig. 3. * This is the same as if one leg and the adjacent angle of a right-angled spherical triangle were given, to find the other leg Formula tan. req. leg = tan. adj. angle x sin. given leg. This is the formula of Napier; but we shall show how the same may be derived from the figure itself, without referring to the principles of spherical trigonometry. For this purpose, let AQ be considered as given the radius of a certain circle. aAQ sin. Qar Then, per Trig. sin. QrA : sin. QAr :: AQ : Qr=4 sin. QrX® ; rad. (1) : tan. Qrs :: - AQ tan. Qrs sin. Qar Du n. Qar. But Qt is by construction = Qs; : Q = AQ tan. Qrs sin. QAr: A0 : AQ tan. Qrs sin. QAT sin. QrA sm. QrA sin. QrA :: rad. (1): tan. QAt=tan. Qrs X sin. QAr; because AQ in the denomivator destroys AQ in the numcrator, and sin. QrA= 1, the angle QrA being a right angle. sin. Qrð : Qs = Sect. II.] 335 PRACTICAL GEOMETRY. EXAMPLES Ex. 1. Given the angle formed by the wall plate of a roof, and the seat of a hip rafter, together with the angle which the roof makes with the wall plate, to find the angle which the hip rafter makes with its seat. This is the same as if the angle BAQ of one leg, and the adjacent angle of the solid were given, to find the other leg. The construction of Fig. 4, is according to Method 1, and the construction of Fig. 5, according to Method 2. Fig. 4. Fig. 5. Ex. 2. Given the angle which the style of a horizontal dial makes with the substyle, and the time of the day, to find the angle in the plane of the dial which the substyle makes with the style. Suppose it were required to find the shadow of the style of a horizontal dial four hours after noon, the latitude of the place being given. Construction.—Let A be the centre of the dial, AQ the sub- Fig. 6. style, or 12 o'clock hour line, and QAB the angle which the style makes with the substyle. From any point Q in AQ draw Qr, meeting AB, or AB pro- duced, perpendicularly in the point r. Produce AQ to s, and make Qs equal to Qr; draw Qt perpendicular to AQ. Make the angle Qst equal to 60° (allowing 15° to an hour), and join At; then Q At is the angle sought. This is the same in every respect as the second method. In the diagram the whole of the hour lines are exhibited, being all drawn in the same manner as the line At; and this is easily effected by dividing the quadrants QT and QU each into six equal parts. The very same lines apply to the joints of an oblique arch in Masonry, where the lines shown in TQ U are for those of the right arch. 336 [Part IV. ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION XV. Given one leg and the adjacent angle of the solid, to find the hypothenuse. * Fig. 1. Fig. 2. In AQ take any point Q; from Q draw Qy, cutting AB or AB produced, perpendicularly in r. Make the angle Qrs equal to the adjacent angle of the solid; draw Qs perpendicular to Qr; make ry equal to rs, and join Ay; then BAy is the hypothenuse required. Fig. 3. The principles of this problem may be successfully applied to the development of roofs, the cutting of timbers, &c. As an Example,-Given the angle of two faces of a piece of timber, and two lines drawn obliquely from the same point of the ridge line (or arris as it is called by workmen), to find the angle contained by these lines. PROPOSITION XVI. Given the two legs, to find the adjacent angles of the solidot Construction.-In AB, take any point r, and draw ru perpendicular to AQ, meeting AQ in W From u draw uv perpendicular to At, meeting At in v. From u towards A on AQ, make uw equal to uv. Join rw, and uwr is one of the angles required: the other may be found in the same manner. . * This is the same thing as having one leg of a right-angled spherical triangle and the adjacent angle given, to find the hypothenuse. Formula cot. hyp. = cos. given L X cot. given leg. The formula here given is that derived from Napier's circular parts; but we shall show how the same may be obtained immediately from the figure, without referring to the principles of spherical trigonometry. Let AQ be given as before = the radius of a certain circle; then, sin. Qra : sin. QAr :: AQ : Qr = AQ sin. Qarino ; sin. QrA : cos. QAr :: AQ : Ar = 4 in AQ cos. QAr iro sin. QrA sin. QTAC: AQ sin. Qar , AQ cos. QAT AQ sin. Qar tan. QAr root RANO sin. QrA cos. Ors; but ry=rs, :: sin. Qrð ::: rad. (1): tan. BAY= sin. QrA cos. Qrs":19 cos. Qrs cot. QAr. † This is the same as if the legs of a right-angled spherical triangled were given to find the acute angles. Formula cot. req. angle = sin. adj. side x cot. opp. side. Solution from the figure. sin. Qrð ; cos. Qrs : rad. (1) :: AQ sin. Qar - coe Ore, or cot. BAy = Sect. II.) 337 PRACTICAL GEOMETRY- Or thus, In AB take any point r; draw ru perpendicular to AQ, meeting AQ in u; draw also uv perpendicular to At, meeting At in o, and uw perpendicular to uv; make uw equal to ur, and join vw; then will uvw be one of the angles required, and in the same manner the other may be found. By either of these methods we may find the backings of hip rafters. PROPOSITION XVII. Given the legs, to find the hypothenuse.* Construction. In AB, with any convenient radius, describe a semicircle Azr, meeting AB in r. Draw rw perpendicu- lar to AQ, meeting AQ in u; draw also uv perpendicular to At, meeting At in v. From A with the radius Av describe an arc, meeting the semicircle in z, and through z draw AW; then will BAW be the angle required. PROPOSITION XVIII. Given the hypothenuse and either adjacent angle of the solid, to find the adjacent leg.t Construction.—Let BAY be the hypothenuse, and through any point r in AB draw YQ perpendicular to AB. Make the angle Qrs equal to the adja- cent angle of the solid, and rs equal to rY. Draw są perpendicular to YQ, and join AQ; then BAQ is the leg required. Let rA equal the assumed radius; then will rA sin, r Au = ru, and rA cos, rau = A rA sin. uAv cos. r Au = uv = uw rA sin. rAu t an. r Au talle riu – rA sin. UAV cos. Tau sim. udo or cot. rwu = sin. u Av cot. r Au. * This is the same as if the two legs of a right-angled spherical triangle were given, to find the hypothenuse. Formula, cos. hyp. = rect. cosines of legs. Solution from the figure. Let TA be the assumed radius; then Au=rA cos. r Au, and Av=rA cos. r Au cos, udv. But Az= Av; ::TA. rA cos. rAu cos. u Av:: 1 cos. rAz; that is, cos. rAz = cos. r Au X cos. u Av. † This is the same as if the hypothenuse and an acute angle of a right-angled spherical triangle were given, to find the side adjacent to that angle. Formula, tan. req. side =tan. hyp. X cos. given angle. Solution from the figure. Let A Y be the assumed radius ; then Yr=r8= AY sin. YAr, and Ar=AY cos. YAT Q = AY sin. YAr cos. Qrs; ::. tan. Qar=tan. YAr cos. QAr. . 20 338 [PART IV ELEMENTS AND PRACTICE OF GEOMETRY. PROPOSITION XIX. Given the hypothenuse and either angle of the solid, to find the opposite leg.* Construction.—Find the leg rA by the last proposition ; then through any point r in Ar draw YQ perpendicular to Ar. Make the angles Qrs equal to the given angle of the solid. Draw Qs perpendicular to Qr, and Qt to A2, making Qt = Qs. Join At, and Q At will be the leg re- quired. SCHOLIUM. Let A and B represent the two acute angles of the solid, a and b their opposite legs, and c the hypothenuse. Now here are all together five parts ; and since the number of ways which two things can be selected out of five is ten, we can have the following combinations of data, AB, Aa, Ba, Ab, Bb, ab, Ac, Bc, ac, bc. It therefore appears, that there are ten cases of the right-angled trehedral; we shall, however, dismiss the subject by observing, that the six cases which we have given are suffi- cient to answer the various examples which appear in the body of the work. Having now resolved all the cases that can possibly occur in Plane Trigonometry, it only remains for us to point out some of the applications of that valuable science. Trigonometry is eminently useful in surveying land ; in measuring heights and distances, which are either inaccessible, or do not present facilities for a more obvious mode of measurement, and in a variety of arts more limited in their operations. These are perhaps the only applications which it is necessary to speak of in a work like the present; but the intelligent student will readily conceive that the principles of a science so usefully applied to our immediate necessities, may be employed with equal advantage in the more extended scale of astronomical observations, the art of navigation, and the trigonometrical survey of kingdoms. The method of making the necessary calculations of the sides and angles of triangles has been already shown; but practical examples of working the questions could not well be given without the aid of tables of logarithms, and to have introduced such tables of sufficient extent to render them really useful, would have much increased the bulk of the present volume, without rendering it the more acceptable to a great portion of our readers. in volumes by themselves, and which are usually prefaced by a full explanation of their use, rather than attempt to work out solutions from tables not in this book, operations but little satisfactory, of the most useful tables of logarithms are the following :- The tables by M. de la Lande, in one small pocket volume, are exceedingly well adapted for common calculations. Where greater accu- racy is required the student may consult those by Dr. Hutton, or Caillet, each in one large volume * This is the same as if there were given the hypothenuse and an angle of a right-angled spherical triangle, to find the side opposite to that angle. Formula, sin. req. side =sin. opp. ang. X sin. hyp.: Solution from the figure. Let YA be the assumed radius; then YA sin. YAr = Yr Ers; YA sin. YAr sin. Qrs = Qs = Qt; but YA = tĄ, ::: sin. Qat=sin. Yar sin. Qrs. Sect. II.) 339 PRACTICAL GEOMETRY. 8v0.; and if any one shall desire to make the most elaborate calculations, the tables by Bagay, in one volume 4to., (where the sines, tangents, &c. are given to every second of the quadrant,) may be referred to. The measurement of lines and angles present more difficulty in practice than is generally imagined by persons who have only studied the theory of Trigonometry, particularly when operations are performed upon a large scale ; and it requires much judgment to form such an arrangement of parts which constitute the data of calculations, as may render the inevitable inaccuracy of our measure- ment least detrimental to the correctness of the result. With this object in view, the occurrence of very small angles should be as much as possible avoided, because a very trifling error in estimating their quantity, will make a surprising difference in the contiguous sides of the triangle : besides which source of inaccuracy, as the side opposite a very acute angle is generally the measured side, or, as it is termed, the base line, the error of the whole work is increased in as great a proportion by any trifling error in the length of this line, and to measure a straight line of considerable extent correctly, is frequently a much more difficult problem than the apparent simplicity of the operation seems to imply. . We could adduce other instances where precautions are necessary to precision in the practice of Trigonometry, but it may be sufficient by a single example to warn the young practitioner that the data, or substructions on which he grounds his calculations, are for the most part of his own choosing, and that the accuracy of the results will therefore chiefly depend on his judgment, and the perfec- tion of the instruments which he employs ; for without correct data the most elaborate calculations serye but to multiply errors. 塑y 實力​, THE GEOMETRICAL PRINCIPLES OF ARCHITECTURE AND THE BUILDING ARTS. PART V. THE GEOMETRICAL PRINCIPLES OF ARCHITECTURE AND THE BUILDING ARTS. THERE are several principles which are common to all the building arts, as the principles of finding the sections of solids, called Stereotomy; of drawing the exact form of a solid on a plane surface, called Projection; of finding the surfaces that exactly cover a solid, called Development; and of the construction of drawings for solids of double curvature. SECTION 1. STEREO TOMY. STEREOTOMY is that branch of Geometry which treats of the sections of solids; and of their appli- cation to the various purposes of architecture. OF PLANES. DEFINITIONS. 1. A straight line is perpendicular or at right angles to a plane, when it makes right angles with every straight line meeting it in that plane. 2. A straight line is parallel to a plane, when both the straight line and the plane may be extended indefinitely without meeting. 3. Two planes are parallel to each other when both are produced indefinitely without meeting. 4. The common section of two planes is a straight line. For any two points in the common sec- tion are in both planes; but a plane is that in which any two points being taken, the straight line between them is wholly in that plane; therefore the straight line must be in both planes; and since all straight lines coincide through the same two points, the common section is a straight line. 5. The angle contained by two straight lines, drawn each from any point in the common section of two planes perpendicular to the line of section in each plane, is called the angle or inclination of these two planes. 6. If this angle be a right angle, the planes are perpendicular. TA THEOREMS. 1. A plane which passes through two straight lines can have only one position. 2. If a straight line be perpendicular to two straight lines at the point of their intersection, it is perpendicular to the plane in which these lines are. 3. If a straight line be perpendicular to a plane, every straight line parallel to that straight line is perpendicular to the plane. 344 [PART V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. LT 4. Two planes perpendicular to the same straight line are parallel to each other. 5. The intersections of two parallel planes with a third plane are parallel. 6. A straight line perpendicular to one of two parallel planes is also perpendicular to the other. 7. Parallel straight lines intercepted between two parallel planes are equal. 8. If two straight lines meeting one another be each parallel to each of two others that meet one another, though not in the same plane with the first two, the first two and the last two contain equal angles, and the plane passing through the first two is parallel to the plane passing through the last two. 9. If three straight lines, not situate in the same plane, are equal and parallel to each other, the triangles formed by joining the extremities of these lines are equal, and their planes parallel. 10. If two straight lines be cut by three parallel planes, the middle plane will divide the finito lengths of the lines termiñated by the other two planes in the same ratio. 11. If a straight line be perpendicular to a plane, every other plane which passes along that straight line shall be perpendicular to the first plane. 12. If two planes be perpendicular to each other, and a straight line be drawn in one of them perpendicular to their common section, the straight line is also perpendicular to the other plane. 13. If two planes be perpendicular to a third, their common section is perpendicular to the third. OF SOLID ANGLES. DEFINITION. A Solid ANGLE is the surface formed by the meeting of the angular points of more than two plane angles which are not in the same plane, in one point; or it is the angular space comprised between several planes meeting in one point. THEOREMS. 1. If a solid angle be formed by three plane angles, the sum of any two of them is greater than the third. 2. The sum of all the plane angles which form any solid angle is less than four right angles. 3. If two solid angles be each composed by three plane angles, of which the plane angles of the one solid angle are respectively equal to the angles of the other solid angle, any two planes of the one solid angle shall have the same inclination as the corresponding planes of the other. OF SOLIDS. DEFINITIONS. 1. A solid is that which has length, breadth, and thickness. 2. A prism is a solid contained by parallelograms terminating at the ends in two parallel plane figures. 3. The parallel plane figures are called the bases of the prism, and the parallelograms are called the sides, which taken together constitute the intermediate surface. 4. The altitude of a prism is the distance between its bases. 5. A prism is right when the lateral edges are each perpendicular to the plane of its base; then each of them is equal to the altitude of the prism. Sect. I.] 345 STEREOTOMY. 6. A prism is oblique when the lateral edges are not perpendicular to the bases. 7. A prism is triangular, quadrangular, pentagonal, hexagonal, &c. according as the base is a triangle, a quadrilateral, a pentagon, a hexagon, &c. 8. A prism which has a parallelogram for one of its bases, is called a parallelopiped. 9. A parallelopiped is rectangular when all its faces are rectangles. 10. When the rectangles are squares, the parallelopiped is called a cube. 11. A pyramid is a solid formed by more than two plane triangles, of which one of the angular points of each triangle terminates in the same point, and the other angular points terminate in a plane figure. 12. The triangles which unite in a point and terminate in the plane figure, are called the sides of the pyramid ; the point where they meet, is called the vertex; and the plane figure in which the sides terminate, is called the base. 13. The altitude of a pyramid is the perpendicular drawn from its vertex to the plane of its base. 14. A pyramid is triangular, quadrangular, &c. according as the base is a triangle, a quadrangle, &c. 15. Two solids are similar when they are contained by the same number of similar planes, simi- larly situated, having like inclinations to each other. CYLINDER. 1. A solid generated by the revolution of a rectangle about one of its sides, which remains fixed, is called a cylinder. 2. The fixed straight line of the rectangle is called the axis of the cylinder. 3. The two circles described by the opposite ends of the rectangle which are perpendicular to the axis, are called the ends or bases of the cylinder. 4. The surface generated by the side of the rectangle which is parallel to the axis, is called the convex surface. 5. If a cylinder be cut by a plane parallel to the axis, the least portion is called the archoid of a cylinder. 6. The portion of a cylinder contained between two planes passing along the axis, is called i sectroid of a cylinder. 7. If the two planes of a sectroid of a cylinder form a right angle with each other, the sectroid of the cylinder is called a quadrantal sectroid of a cylinder. CYLINDROID, OR CYLINDRICAL SOLID HAVING AN ELLIPTICAL BASE. 1. A solid, generated by the motion of an ellipsis parallel to itself, and so that a straight line per- pendicular to the plane of the ellipsis will coincide with the convex surface, is called a cylindroid. 2. The two equal and similar ellipsis are called the ends or bases of the cylindroid. 3. The intermediate surface is called the curved surface. 4. A straight line passing through the centres of the two bases, is called the axis of the cylindroid. 5. If a cylindroid be cut by a plane along the axis or parallel to it, and through the axis of one of the ends or parallel to it, the half or the least portion of the cylindroid is called the archoid of a cylindroid. 2 x 346 [Part V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. CYLINOID, OR SEGMENT OF A CYLINDER, OR CYLINDROID. 1. A solid which is either the segment of a cylinder or of a cylindroid, or compounded partly of a prism, and partly of a cylinder or cylindroid, such that all the rectangular sides may be parallel to the axis of the cylinder or cylindroid, and that two of the surfaces of the solid may be perpendicular to that axis, is called a cylinoid. 2. The axis of the cylinder or cylindroid is also the axis of the cylinoid. 3. Each of the two surfaces which are perpendicular to the axis of the cylinoid, is called an end of the solid. 4. If a cylinoid be cut by a plane obliquely to the axis, but perpendicular to one of the plane sides which joins the ends, each of the parts is called the ungula of a cylinoid. 5. The plane of the cylinoid which joins the two bases, and which is perpendicular to the oblique section to the axis, is called the determinating plane. 6. The end of the ungula which is perpendicular to the axis, is called the right end of the ungula. 7. The end of the ungula which is oblique to the axis, is called the oblique end of the ungula. PROBLEM I. Given the right end of the ungula of a cylinoid, and the determinating plane, or plane perpendicular to the oblique end, to find the oblique end. GENERAL RULE. Fig. 1. ( Let kKDd be the determinating plane of the oblique end, KD being the line of position, and let kd be perpendicular to kK or dD; then on k d describe k b md, the right end of the figure. Draw mP parallel to D or kK, meeting d k in p, and DK in P, and draw PM perpendicular to KD. Make PM equal to p m. In the same manner will be found the point B, making QB equal to qb: a sufficient number of points being found through all these points, draw a curve, and we shall have the oblique end KBMD. SCHOLIUM Upon this principle, the angle ribs of groins and the angle brackets of coves are to be described, the right end being the given rib or bracket, and the oblique end the angle rib or bracket to be found. Thus let the determinating plane be AacC, and the given rib or bracket be a mnc, the angle rib or angle bracket will be AMNC. Fig. 2. Fig. 3. SECT. I.) . STEREOTOMY. . 347 Fig. 4. _ Fig. 5. The given angle rib or angle bracket may either be the quadrant of a circle, as in Fig. 2, or the quadrant of an ellipsis upon either axis, as in Figs. 3 and 4. In Fig. 3 the right end of the ungula stands on the semi-axis major, and in Fig. 4 on the semi-axis minor. In each of these three cases, the oblique end will also stand on the axis major or minor, and the curve of the oblique end is an ellipsis, and may therefore be described by a trammel. (See Conic Sections, Prob. iii.) If the end be the segment of a circle less than a semicircle, Fig. 5. Let kKDd be the determinating plane of the oblique end, and let k bd be the right end, and kD the line of posi- tion, and let c be the centre of the segment k b d. Through c, draw bR parallel to kK or dD, meeting K.D in R, and k d in r; and through R, draw hB perpendicular to KD. Make RB equal to rb, and make BC equal to bc: then we have given a semi-axis BC, and a point D in the curve of an ellipsis; therefore the semi-ellipsis may be described by Prob. iv. and iii. Conic Sections. If a portion m bd of the outline of the end or base k dbmnl be an arc of a circle equal to one-fourth of the entire circum- ference, and if m n be a straight line, and joined to the arc mbd as a tangent at m, the curved part of the outline of the oblique end may be described by a trammel, and the corresponding part to m n will be found by drawing a tangent to the curve ; thus, let kKLI be the determinating plane, and KL the line of position. Fig. 6. 01 Fig. 7. M M Through the centre c of the arc m b d, draw the straight line bR parallel to IL or kK, meeting kl in r, and KL in R. Draw RB and KD perpendicular to KL, and make RB equal to rb, and KD equal to kd; also make BC equal to bc: then here are given a semi-axis CB, and a point D in the curve, to de- scribe the ellipsis or the portion MBD required; which being done, * join CD, and draw MN parallel to CD, and draw LN perpendicular to KL. * Or through C, draw Aa parallel to LK, and from D as a centre, with the distance BC, describe an arc meeting Aa in g. Draw Dg to meet BC prolonged in b. Make CA and Ca each equal to Dh; then the ellipsis or a portion of it may be described by Prob. iji. Conic Sections. 348 (PART V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. PROBLEM II. Given the oblique end, and the determinating plane, to find the right end or base. GENERAL METHOD. Let kKDd be the determinating plane, and KBMD be the Fig. 8. oblique section. Draw PM perpendicular to KD the line of posi- tion, meeting KD in P, and draw Pm parallel to kK or dd, meeting dk in p. Make p m equal PM, and the point m will be one of the points in the base. In the same manner we may find as many points as we please. A sufficient number of points being thus found, the curve may be traced. k K Or, if the oblique end be an ellipsis, or any given portion of an ellipsis, or of a circle, the base or right end will also be the same portion of an ellipsis or circle, and may therefore be described by a trammel. PROBLEM III. The base or right end of the ungula of a cylinoid being given, and the length of two straight lines per- pendicular to the base contained between two points in the outline of the base and the oblique end, to describe that oblique end so that it may pass through another given point in the outline of the base. con Let dbfme be the outline of the base, and let d, f, be the Fig. 9. two points on which the perpendiculars stand, and e the point through which the oblique end is to pass. Through the point f draw d k, and draw d h and fi each perpendicular to d k. Make dh equal to the height of the straight line upon the point d, and fi equal to the height of the straight line upon the point f. Through i draw h k, and through e draw kG; and through any point G in kg, draw Gt perpendicular to kg. Draw dT parallel to kG, meeting Gt in t. Make tt equal to dh, and join TG. Then if tdbfme G be considered the end of the ungula of a prism, and tTG the determinating plane, the oblique end TDB MEG may be found by the General Method, Prob. I. Or if d bm be an arc equal to the fourth part of the whole circle, and me perpendicular to the radius c m; then, in the oblique end, the curve DBM will be a portion of the semi-ellipsis ADBMa, of which the centre is C, and ME parallel to CD will be a tangent to the curve at the point M. The most usual application of this problem is to find the moulds for the hand-railing of stairs. are We CONE DEFINITIONS. 1. A cone is a solid generated by the revolution of a right-angled triangle about one of the sides 2. The fixed side of the triangle, about which the side perpendicular to it and the hypothenuse revolves, is called the axis of the cone. SECT. I.] 349 STEREOTOMY. 3. The circle described by the revolution of the side of the triangle, which is perpendicular to the axis, is called the base of the cone. 4. The other extremity of the axis which does not terminate in the base, is called the vertex of the cone. 5. The surface generated by the hypothenuse of the revolving triangle, is called the convex surface of the cone. 6. The portion of a cone contained between two planes perpendicular to the axis, is called the frustum of a cone. 7. The portion of a cone contained between the base and a plane cutting the axis obliquely, is called the ungula of a cone. 8. The portion of the ungula of a cone contained by a plane which is perpendicular to the base and the oblique plane to the axis, and which also passes through the base, is the semi-ungula of a cone. SECTIONS OF A CONE. PROBLEM IV. Given the triangular section DEF of a right cone through its axis, and the line of position of any other section of the cone perpendicular to the plane of the triangular section, to find that other section of the cone. CASE 1. Fig. 10. Let GH be the line of position meeting both sides DE, DF of the cone, or DE DF produced ; then this position of the cutting plane will cause the section to be an ellipsis; therefore proceed as follows: 0 Bisect GH in i, Fig. 10, and through i draw k l parallel to EF, the base of the triangular plane, meeting DE in k, and DF in l. Fig. 11. In Fig. 12, draw the straight lines Aa, Bb at right angles, meeting each other in C. Make CA, Ca, each equal to iG or iH, Fig. 10, and make CB, Cb, each equal to a mean proportional between i k and il, Fig. 10; then, on the two axes Aa, Bb, describe the ellipsis AB a b, which will be the section of the cone for this position of the line GH, Fig. 10. In order to refresh the reader's memory, the method of finding the mean proportional is exhibited at Fig. 11, in this and also in the fol- lowing case. See Prob. xl. Geometry. And for the method of describ- ing the ellipsis, see Conic Sections, Prob. iii. . kih Fig. 12. . 350 [PART V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. CASE 2. Fig. 13. Let the straight line GH, Fig. 13, meet the section CEF, which passes through the axis of the cone in G, and the section C e f of the opposite cone in the same plane in H. Bisect GH in i, and draw il perpendicular to the axis of the cone, meeting Ff in k, and Ee in l. Find a mean propor- tional between ik and i l, and let in, Fig. 14, be the mean proportional. Draw Aa Fig. 15. From C in Aa make CA, Ca, each equal iG or iH, Fig. 13. Now having the transverse axis Aa, and the semi-conjugate i n, the opposite hyperbolas may be described. Fig. 14. Fig. 15. SCHOLIUM. Fig. 16. If the line GH be parallel to the axis R$, Fig. 16, then the points i, k will coincide with C, and Ci will then be the semi- conjugate axis; and if GA and Ha be drawn perpendicular to GH, meeting RS in A, a, then Aa will be the transverse axis. And because AG and aH are each equal to Ci, and that Aa is bisected in C, and the two straight lines Ee, Ff, pass through the points G, C, H, the two straight lines Ee, Ff, are asymp- totes to an hyperbola, of which the transverse axis is Aa, and the semi-conjugate Ci; therefore the curve may be described by Prob. xix., Conic Sections. Sect. I.1 351 STEREOTOMY. CASE 3. Fig. 17. If the line GK be parallel to one side CE of the cone, and CE and CF equal to each other, the line GK meeting EF in K; then if AB, Fig. 19, be made equal to GK, and Dd be drawn perpendicular to AB, and BD, Bd be made each equal to kn, Fig. 18, a mean proportional between EK, KF, Fig. 17; and, lastly, if a parabola be described upon the abscissa AB, and upon the semi-ordinate BD or Bd; this curve will be the section of the cone through GK, Fig. 17. See Conic Sec- tions, Prob. xxxi. Fig. 18. n. Fig. 19. CONOID. DEFINITIONS. 1. If the portion of a conic section contained by the axis, an ordinate, and the part of the curvo between them, be revolved round the axis which remains fixed, the solid generated is called a conoid. 2. The axis of the generating figure is called the axis of the conoid. 3. The surface of the solid generated by the curve, is called the curved surface of the conoid. 4. The side of the solid generated by the ordinate, is called the base of the conoid. 5. If the conoid be cut by a plane parallel to its base, the part of the solid contained between the base and the cutting plane is called a zone of a conoid. 6. If the curve of the generating figure be an arc of an ellipsis, the solid is called an elliptic conoid. • 7. If the curve of the generating figure be an arc of an hyperbola, the solid is called an hyperbolic conoid. 8. If the curve of the generating figure be an arc of a parabola, the solid is called a parabolic conoid. 9. A portion of a conoid contained between two planes which meet the axis, and which are perpen- dicular to the base, is called the sectroid of a conoid. These Definitions also include some of those belonging to the sphere and the cone. GENERAL PROPERTY. Hence, if any section of a conoid be Every two parallel sections of a conoid are similar figures. given, any section parallel to that section may be found. 352 IPART V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. CUNEOID, 0Ꭱ WED GE FOᎡᎷ Ꭼ D soLI D. DEFINITIONS. 1. A cuneoid is a solid terminating in a straight line at one end, and in a plane parallel to that straight line at the other, in such a manner that another straight line perpendicular to the first, from any point in it, may coincide with the intermediate surface. 2. The straight line in which the solid terminates, is called the directing edge of the solid. 3. The plane figure is called the base or end of the solid. 4. The straight line passing through the centre of the base perpendicular to the directing edge, is called the axis. 5. The triangular section which passes along the axis, is called the principal triangle. 6. The rectangular section which passes along the axis, is called the principal rectangle. 7. If a cuneoid having the principal triangular section for a side be cut by a plane perpendicular to that side, the portion of the solid which contains the base, is called the ungula of that cuneoid. 8. The side of the ungula which is oblique to the axis and to the principal triangle, is called the oblique section. 9. The part of the principal triangle contained between the base and the oblique end, is called the deterninating side. PROBLEM V. Given the determinating side of the ungula of a cuneoid in the plane of the principal triangle and the base of the solid, to find the oblique end. Fig. 20. Let aAdd be the determinating side of the solid, a b md the base, and AD the base line of the oblique end. Draw p m perpendicular to a d, meeting a d in p. Prolong dd, aA to meet in E. Draw pE, meeting AD in P, and draw PM perpendicular to AD. Make PM equal to pm, and M will be a point in the curve ; in this manner we may find as many points as will be necessary to describe the curve ABMD. .. . . . . . . . > . .... ... PROBLEM VI. Given the oblique end ABMD of the ungula of a cuneoid and the determinating side dDAа in the plane of the principal triangle, to find the base of the solid. In AD take any point P, and draw PM perpendicular to AD. Through P draw Ep, meeting a d curve of the base. In the same manner, as many points may be found as will be necessary to draw the curve a b m d of the base. SECT. 1.7 353 STEREOTOMY. PROBLEM VII. Given the determinating side of a semi-cuneoid upon the plane of the principal rectangle and the base of the solid, to find the section of the solid parallel to the base through a given straight line in the plane of the principal rectangle. Let a E Fd be the plane of the principal rectangle, and a m' b md Fig. 21. the base of the solid, and let AD parallel to a d be the given line of section. Bisect a d in c, and through c draw Gb parallel to Fd or Ea, meeting EF in G. In cb take c p any convenient distance from C, and through p draw m m' parallel to a d. In ca or cd make cq equal to c p, and draw Gq meeting AD in Q. In CG, make CP equal to CQ, and through P draw M M' parallel to AD. Make PM equal to pm, PM' equal to p m', and the points M, M', will be in the curve. In the same manner, as many points may be obtained as will be found necessary to trace the curve AM'BMD. If the base a m'b m d be an ellipsis, the curve AM'BMD will be an ellipsis also ; and therefore, if CB the semi-conjugate axis be found, the curve AM'BMD may be described by a trammel. That the curve is an ellipsis may be shown thus:- CB:CP ::cb:cp; and CD and PM are each equal to cd, pm; therefore, by Prob. V., Conic Sections, the curve is an ellipsis. This may be applied to the description of the ribs for a hollow in the ceiling, as of that under the gallery of the church, in Waterloo Place. SPHERE. DEFINITIONS. 1. A solid generated by the revolution of a semicircle about its containing diameter which remains fixed, is called a sphere. 2. The diameter of the semicircle which remains fixed, is called the axis of the sphere. 3. The centre of the semicircle is called the centre of the sphere. 4. The surface generated by the semicircular arc, is called the curved surface of the sphere. 5. The portion of a sphere contained by a part of the curved surface and a plane, is called the segment of a sphere. 6. The portion of a sphere contained between two parallel planes, is called a frustum or zone of the sphere. 7. The plane of the segment of a sphere is called the base. 8. If the base pass through the centre of a sphere, the segment is called a hemisphere. 9. The portion of a sphere contained between two planes passing through the centre, is called the sectroid of a sphere. 10. The portion of a sphere contained between any two planes of which their intersection does not pass through the axis, is called an imperfect sectrum of a sphere. GENERAL PROPERTY. Every section of a sphere is a circle. 2 Y 354 [PART V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. PROBLEM VIII. Given the line of common section of the two circular segments of the sectrum of a sphere, the inclination of their planes, and the versed sine of each segment, to find the radius of the great circle perpendicular to the line of common section. Fig. 22. Let AB be the line of common section. Bisect AB by a perpen- dicular CD, meeting AB in e. Draw the line e f, making the angle Cef equal to the angle of incli- nation of the two planes. Make eC equal to the versed sine of the one segment, and e f equal to the versed sine of the other. Through the three points A, C, B, describe the circumference ACBD of a cir- cle; also find the centre g of a circle that will pass through the three points D, f, C; then will gC or gD be the radius of the sphere required. SOLIDS OF REVOLUTION. DEFINITIONS. 1. If a plane figure which has one straight edge be revolved about that straight edge, so that any point in the figure may just describe a circle, the solid formed is called a solid of revolution. 2. The fixed straight line is called the axis of the solid. 3. If a solid have a side which is a plane figure perpendicular to the axis, this perpendicular side is called the base of the solid. 4. The section of the solid passing through the axis is called the axal section ; and each part of the axal section on each side of the axis, is called a radial or radial section. 5. A portion of the solid comprehended between two planes which meet together in the axis, is OS 6. If the solid be cut by any plane inclined or parallel to the axis, and if this inclined plane be cut perpendicularly by another plane passing through the axis, the line of section in the oblique or paral- lel section is called the axis of that oblique section, or of that parallel section. 7. A straight line drawn in the plane of the oblique section or parallel section, perpendicular to the axis of this section, and terminated by the outline of the section, is called a double ordinate. 8. The axal section perpendicular to the oblique or parallel section, is called the determinating section. GENERAL PROPERTIES. All the sections of a solid of revolution parallel to the base are circles. An ordinate of any oblique section is a mean proportional between the two segments of the straight line which passes through the intersection of the double ordinate, perpendicularly to the axis of the solid, and which is terminated by the surface of that solid. SECT. 1] 355 STEREOTOMY. PROBLEM IX. To describe any oblique section of a solid of revolution, the position of the cutting plane being given in the determinating section. ATX Fig. 23. Let VWX be the determinating section, sn the axis or line of the oblique section, VZ the axis. Through h, any convenient point in sn, draw fg perpen- dicular to VZ, meeting the opposite sides of the determinating section in f and g. Find h i in Fig. 24, a mean proportional between fh and hg. In Fig. 25, make Aa equal to sn, AP equal to sh, and Pa equal to hn. Through P draw MM' per: pendicular to Aa. Make PM and PM' each equal to the mean proportional hi, and the points M, M' will be in the curve : and thus the curve may be drawn through a sufficient number of points found in the same manner as the points M, M. This method may be applied to determine the form of oblique braces for a dome on a circular base. Fig. 24 Fig. 25. PROBLEM X. To find the section of a solid of revolution by a plane parallel to the axis; when the generating figure and the position of the line of section on the base of the solid are given. Let AaEFg be the base of the solid, Aa the line of section, og mbd Fig. 26. the generating figure, and og the line in which it meets the base, od being the altitude of the figure. · Draw oB perpendicular to Aa, meeting Aa in C. From the centre o with the radius oC, describe an arc Cc, meeting og in c, and draw cb perpendicular to og. Make CB equal to cb, and B is the summit or vertex of the curve. .. In like manner from o with any convenient radius greater than oC, but less than o g, describe the arc pPP', meeting og in p, and AC in P, P. Draw p m perpendicu- lar to og, and PM, PM' perpendicular to CA. Make PM, P'M' each equal to pm, and the points M, M' will be in the curve. A sufficient number of points being found in this manner, the curve AMBM'a may be traced through them. nd Bis the summit Rik :.SCHOLIUM. In a sphere, a spheroid, a cylinder, a cone, and the three condids, being solids of revolution, the section may be described either by the rules peculiar to each of these solids, or by the general rules for solids of revolution: 336 [PART V, GEOMETRICAL PRINCIPLES OF ARCHITECTURE. S DEFINITIONS. 1. A solid, of which all its sections are similar ellipses, similarly situated, and perpendicular to a straight line passing through their centres, so that every ellipsis may have one of its axes in a section of the solid which is also an ellipsis, is called an ellipsoid. 2. The section of the solid in which all the axes of the ellipses are situated, is called the determi- nating ellipsis. 3. The centre of the determinating ellipsis is called the centre of the ellipsoid. 4. The line passing through the centre perpendicularly to the determinating ellipsis, and termi- nated by the surface of the solid, is called the axis of the solid. 5. A portion of the solid cut off by the determinating ellipsis, is called a hemi-ellipsoid. 6. In a hemi-ellipsoid, the determinating ellipsis is called the base of the solid. 7. The portion of an ellipsoid or of a hemi-ellipsoid contained between planes meeting each other in the axis of the solid, is called a sectroid of an ellipsoid, or sectroid of a hemi-ellipsoid. 8. The portion of the base of the solid which remains with the sectroid, is called the base of the sectroid. 9. The two surfaces which meet the axis, are called the radial sides. GENERAL PROPERTIES OF TIIE ELLIPSOID. 1. Every section of the solid is an ellipsis. 2. Every section of the solid perpendicular to the determinating ellipsis, has one of its axes in the plane of the determinating ellipsis, and has the other perpendicular to that plane. 3. Any two sections of the solid cut by parallel planes, are similar ellipses and similarly situated. Hence, if one of the parallel sections be given, the other may be found by having one of its axes given. 4. If the sectroid of a hemi-ellipsoid be cut by a plane parallel to the axis of the solid, and par- allel to the chord which joins the two ends of the curve in the base of the sectroid, the points in which the cutting plane meets the curves of the radial sides are equally distant from the base. This principle is applied in Plate 5, plan and elevation, Figs. 3 and 4, which is of the greatest use in the construction of niches, domes, &c. where the ellipsoidal, or the hemi-ellipsoidal, form is employed. 5. If through any given point in the curve of one of the radial sides, a straight line be drawn parallel to the chord of the base, the straight line thus drawn will meet the curve of the other radial side. 6. Hence, a cylindric surface, or cylindroidic surfaces, may be made to pass through the curves of both radial sides. Sect. I.] 357 STEREOTOMY. PROBLEM XI. Given the base of a hemi-ellipsoid, and the height of the axis above the base, to find a section of the solid . parallel to the axis to pass along any given chord in the base. Fig. 27. og Penn El Let AaEFD be the base of the solid, o the centre of the base, and Aa the line of section. Through o, draw o q perpendicular to DE, and make oq equal to the height of the axis; join Eq. Bisect Aa in C, and draw CB perpendicular to Aa. Make the angle CaB equal to the angle o Eq. Upon the axis Aa and semi-axis CB describe the semi-ellipsis A Ba, which is the section of the solid required. Cab og PROBLEM XII. Given the base of a hemi-ellipsoid and the height of the axis, to find the radial section upon any given line in the base of the solid. Fig. 28. Let ADE be the base of the solid, AC the base line of the radial section. Draw CB perpendicular to CA, and make CB equal to the height of the axis. Upon the semi-axes AC and CB describe the elliptic arc AB, which is the quadrant of the whole curve, and ABC is the radial section required. DOMOID. DEFINITIONS. 1. If a solid has one platie side, and if any portion of the solid, contained between two planes which meet each other in a line perpendicular to that side, be cut by any two planes parallel to that same side, and the two sections be similar figures, the solid is called a domoid. 2. The perpendicular is called the axis of the domoid. 3. The side of the solid to which the axis is perpendicular, is called the base of the domoid. 4. A portion of a domoid contained between two planes passing along the axis of the solid, is called the sectroid of a domoid. 5. The two sides of the sectroid of a domoid which meet the axis of the solid, are called the radiat sides of that sectroid. 6. The plane of the sectroid which is perpendicular to the axis, is called the base of the sectroid.. 7. The two straight edges of the base which meet in the axis, are called the radials of the base of the sectroid. : 8. The portion of the surface of a domoid which is not the base, is called the curved surface of that domoid. 9. If a domoid be cut by a plane parallel to the plane of the base, the portion of the solid.com- prehended between the two parallel planes is called a truncated domoid. 10. In each radial side of the sectroid of a domoid, the curve at the meeting of that radial sido and the curved surface of the solid is called the curve of that radial side. 358 | Part V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. 11. If the curve of the radial section of a domoid be convex without the figure, and the base be a circle greater than any of its parallel sections, the solid is called a cupoloid. PROBLEM XIII. Given the base of a domoid, a radial section, and the position of the line of section in the base, to find another radial section passing through any given line in that base. Fig. 29 Let AaDEF be the base of the domoid, a m b C the given section, aC its line of section; and let AC be the line in which the required radial section meets the base. Join Aa. Draw pP parallel to Aas meeting aC in p, and AC in P. Draw p m perpendicular to aC, and PM perpendicular to AC; make PM equal to pm, and M is a point in the curve. Hence, if the curve of the given section am b be the quadrantal arc of an ellipsis, the curve AMB to be found will also be the quad- rantal arc of an ellipsis or of a circle; and therefore the curve may be described by a trammel, the axis CB being equal to Cb. But if the curve has no peculiar property, we may find as many points as will be necessary to trace the curve in the same manner as the single point was found. M Fig. 30. UNGULUS. DEFINITIONS. 1. The portion of a cylindroid comprehended between two planes which meet each other in a point of the axis, and which have their line of concourse parallel to one of the axes of the ends, is called an ungulus. 2. If an ungulus be cut into two parts by a plane perpendicular to the line of concourse, each of the parts is called a segment of an ungulus. 3. The side of the segment of an ungulus which is perpendicular to the line of concourse, is called the base of the solid. 4. The two sides which are perpendicular to the base, are called the radial sides. 5. The line of concourse in which the radial sides meet, is called the axis of the ungulus or of the section of the ungulus. 6. The straight line in which the curved surface and the base meet each other, is called the line of subtense. 7. When the two radial sides of an ungulus or of the segment of an ungulus are equal, the solid is called an isosceles ungulus, or an isosceles segment of an ungulus. 8. When an ungulus is cut by a plane parallel to its base, the part of the solid contained between the two parallel planes is called the frustum of an ungulus. We have so far confined our definitions to geometrical solids; to these however there are several to add which are peculiar to architecture: we shall commence with the species introduced by the Roman architects, and follow in the order of their introduction, or nearly so, till we arrive at the most complex species used in florid Gothic architecture. VI. ARCHITECTURAL SOLIDS. PLATE 122, ARCHOID DONOID DOVOD Ma.? Fig. 3. DOVOID Fig. 4. PENDENTOIDS. Fiqu. Fig. 6. GROINOID PEMDEN TOLDY. Fig. 7. LUNOLD Fig. 11 LUNETTED DOMOI. GROLNOID Fig. 10 Fig. 12. Engraved by Armstrong A.Fullarton & C Landon & Edinburgh ARCHITECTURAL SOLIDS. PLITE 23 . Fig. 1. Fig. 2. Fig. 3. . . . : . ..................... *** SU . N Fig. 4. Fig.. Fig.6. we . .. www. ... III : .! .. . . TS .. ... . M . ' . . . . . .. .. .. .... ... . . . . . . . . . ..... . ...... ... .. . ... ... i . . . .... . . ?. Fig. 7. Fig. 8. l'ig.2. ..... . . .. .. . . . . - .. ... KO . . . .. 2 . .... - - - ...:'. + .. W . LI M K . . . 12 . - ... . ! . . . . . . . .... . ' ' . .. - ... Eruraveil by Carmstrong. A Fullarton&C Londona Edinbufli is of I SECT. I.) 359 STEREOTOMYTT . ARCHITECTURAL SOLIDS. DEFINITIONS. 1. The art of combining the various geometrical figures and solids in the formation of the exterior and interior parts of an edifice, is called the geometry of architecture. 2. Plane figures which can be so divided into two parts, that, when the one part is applied to the other, their boundaries may coincide, are called symmetrical figures. 3. The straight line which divides a symmetrical figure into two equal and similar figures, is called the line of symmetry. 4. When one side of a room or the front of an edifice is so constructed that it can be cut into two such parts by a vertical plane, and that a straight line passing from any point in that front, perpen- dicularly to that plane, will meet another point of that front on the other side of the plane at the same distance from it as the first point, that side or front is said to be symmetrical. 5. The plane which thus divides the front of a building into two equal and similar parts, is called the plane of symmetry. In the decoration of walls, and in the construction of the front of an edifice, the plane of symmetry is always vertical. 6. The surface of a room which is opposite to the floor is called the ceiling. 7. When the ceiling of a room is composed of one or more of the concave surfaces of one or more geometrical solids, the ceiling is called a vault or vaulted ceiling. 8. When the vaulted ceiling is that of the concave surface of an archoid having its axal plane vertical, the ceiling is called an arched ceiling. 9. When the ceiling is the concave surface of a domoid having a circular or elliptic base, and its axis in a vertical position, the ceiling is called a dome or cupola. 10. When the ceiling is of the form of a pendentrix, it is called a pendented ceiling. 11. The parts which are terminated by the sides and the base of the solid, are called pendentives. 12. When the ceiling is in the form of a groinoid, it is called a groined ceiling, or cross vault. 13. When a ceiling which meets each of the walls in the concave surface of the segment of an archoid, and the inner edges of the surfaces of the archoids meet the edges of a plane surface form- ing the middle of the ceiling, so that the bases of the archioids may be horizontal, and each axal plane parallel to each wall, the ceiling is called a coved ceiling, and the archoidal surfaces coves. M ILLUSTRATION OF ARCHITECTURAL SOLIDS. [Plates CXXII. and CXXIII.] Fig. 1, Plate CXXII., is called an arched vault when applied to vaulting, but it is called an archoid when considered as a geometrical solid. Any perforation in a building covered with an arched vault, is called an arch-way ; but when the ends are shut and form a room, that room is said to be vaulted. Figs. 2, 3, and 4, represent domes; such figures are called domoids when considered as geometrical solids, but when applied to building they represent the external appearance of domes. Domes upon elliptic plans are only used internally with regard to their concave surface. Polygonal domes upon octagonal plans are used both externally and internally, but those upon any other plan very rarely, LO 360 [PART V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. S All circular domes are comprehended under the general name of conoidal domes; of these the hemi- spheric dome, exhibited in Fig. 2, is frequently used, as also the lesser segments of spheres and frustums of spheres. Figs. 3 and 4 represent domes upon polygonal plans; of these octagonal domes are most frequently executed. Figs. 5, 6, 7, 8, in geometry called pendentoids, from the hanging portions of the curved surface that are terminated against the sides, or the sides and base of the solid. These solids have no technical name in architecture, at least none used by our architects. Alberti, one of the famous restorers of Roman architecture, calls such a solid as Fig. 5, 6, or 8, when employed in the internal construction of an edifice, a vellar cupola, from its resembling the sail of a ship when fully extended by the wind. One of the pendentives of each of the solids represented by Figs. 5, 6, 8, is marked abcde, and the pendentive in Fig. 7 is marked abcdef. The vaultings of the nave and aisles of St. Paul's Cathedral, London, are formed by a combination of concave pendentoids. Fig. 7 occurs mostly in arched passages piercing a dome. Figs. 9 and 10 are termed in building groined vaults, geometrically called groinoids. These figures are either applicable to rooms of four or eight sides, or to cross passages. Fig. 9 is also frequently employed in the vaultings of cellers as well as in magnificent apartments. Here the lines in which the curved surfaces of the arched vaults meet each other, are called groins. Those who reside in London may avail themselves of the opportunity of seeing groined vaults, such as Fig. 9, executed in stone in the passage under the building called the Horse Guards, leading from Charing Cross to St. James's Park. Fig. 11, geometrically called a lunoid, from the lesser arched windows, called lunettes, penetrating a large arched vault. This species of groined vaults is commonly called by workmen Welsh groins. Lunette is the French name of an arched window.penetrating an arched ceiling. Such groins may be seen executed in stone in the gateway at Somerset House, leading from the Strand to the court. Fig. 12, geometrically called a lunetted domoid, from the lunetted windows piercing a dome. Polygonal domes and groined vaults are of the same species, the one convex, the other concave; for if the surfaces of all the archoids, that have their axal planes of the same breadth that form an equal pitched groined vault, were to remain entire, the space that is completely enclosed by the surfaces of the archoids is a domoid or dome in architecture ;* and if the external curved surfaces remain, and the curved surfaces forming the dome be cut away, the interior cavity will form a groined ceiling or vault: so that the solid domoid might be exactly comprehended between the groins or angles of a groined vault; and thus, if the cubature or solidity of the dome be known, that of the vacuity of the groined ceiling would be ascertained. [Plate CXXIII.] Fig. 1 is a quadrilateral pointed groined vault, such as may be seen in many ancient churches and cathedrals. Fig. 2, an octagonal pointed groined vault, such as may be seen in chapter houses belonging to ancient British cathedrals. - * In the mathematical names of the architectural solids, the termination oid from the Greek, signifying likeness, has been added to the architectural names which have been given to these solids, or to certain parts of them when the entire solid was without a name. The termination atrix might be applied to the internal cavities formed by these solids, ag domatrix, groinatrix, &c. Sect. I.] 361 STEREOTOMY. Fig. 3, a ribbed groined vault upon a rectangular plan, of which there are several examples in the pointed or Gothic style of architecture. Fig. 4, a ribbed groined vault upon a rectangular plan, where the apex of each vault is a concave line in a vertical plane. Fig. 5 is a pendent rib-ridged vault upon a square plan. The construction of the pendents is as follows:- Suppose the lower point of a simple curve to coincide with the upper extremity of the axis of a vertical pillar, which axis produced is a tangent to the curve in that point; and if the curve be revolved about the tangent as an axis, it will describe a solid, the horizontal sections of which are Fig. 6, truncated ribbed vault upon a square plan, the upper part being cut off by a plane parallel to the base through the vertices of the pointed arcs: the sections form a quadrilateral with four curved sides, of which each is the quadrant of a circle, and concave towards the interior of the figure. This quadrilateral is generally filled up with curious devices of ornamented polyfoils. Fig. 7, a truncated pendent rib vault upon an oblong plan. The vaults represented in these elementary figures may be seen in the Chapel of King Henry VII. at Westminster, and in King's College Chapel, Cambridge. Fig. 8, an octagonal dome groined vault with pointed arches. Fig. 9, a circular dome groined vault with pointed arches. Here the sides of the archies are in vertical planes. 2 z 362 SECTION II. ORTHOPROJECTION. ORTHOPROJECTION, or orthographical projection, is the method of finding the representation of an object on a plane, by drawing straight lines from every point of that object perpendicular to that plane. PRINCIPLES. The position of a point with regard to a line is given, when its distances from two fixed points in that line are known. The position of a point in space with regard to a plane is given, when its distances from any three points in that plane are known. The position of a point in space with regard to a plane is also given, when that point is in another plane of which its inclination to, and the line in which it meets the first plane, are known, and when the position of the point in regard to the line of section is given. DEFINITIONS. 1. The point in which a perpendicular from a given point in space to a given plane meets that plane, is called the seat of that point in space. 2. The perpendicular is called the height of that point from its seat. 3. The plane to which the perpendiculars are drawn, is called the plane of projection. 4. The seat of a line is a line in the plane of projection in which a perpendicular drawn from any point in that line will terminate. 5. A plain containing lines or points to be projected, is called an original plane. PROBLEM I. Given the inclination of the plane of projection to the original plane, and the situation of any point in the original plane, to find the seat of that point. Fig. 31. . Let A be the given point in the plane RSTU, and Z the angle of inclination of the planes, and let SPQT be the plane of projection, and the line of common section of the planes be ST. Draw Aa perpendicular to ST, meeting ST in m. At the point m in the straight line a m, make the angle a mn equal to Z, and make ma equal to mA. Draw n a perpendicular to a A, and a will be the seat of the original point A. Coroll. If the angle amn be a right angle, the seat a of the original point A will coincide with the point m. a ------- Sect. II.) 363 ORTHOPROJECTION. DEMONSTRATION. Suppose the triangle a mn to be turned round upon m a, till its plane be perpendicular to the plane SPQT, and let the plane RSTU be turned round ST, until it is inclined in the given angle Z: then the line mA will be in a plane perpendicular to ST, the line of common section; and since the plane am n is by construction also perpendicular to ST, MA must fall upon mn; and because mn is equal to mA, the point A will fall upon the point n. Also, because an is perpendicular to a m, and the plane man is by construction perpendicular to the plane SPQT, the straight line an will be also perpendicular to the plane SPQT. As A, now supposed to coincide with n, will be the original point in space in respect to the plane SPQT; and an being perpendicular to the plane SPQT, the point a will be the seat of the original point A, by the definition. "Ida Wo . SCHOLIUM. Hence, because the angle made by two straight lines drawn from the same point, perpendicular to the line of common section of two planes, one line in each plane, is equal to the angle contained. by two other straight lines drawn from any other point, perpendicular to the line of common section of two planes, one line in each plane; it is not necessary to make one leg of the angle a mn in ther' straight line perpendicular to the original point A, for any other line perpendicular to S T will answer the same purpose. Thus.-Draw Aa perpendicular to ST, meeting ST; and Fig. 32 from any convenient point m in ST, draw mL parallel to Aa. Make the angle Lmn equal to the angle of the in- clination of the planes, and make mn equal to i a. Draw na parallel to ST; then a will be the seat of A, as required. It is evident that the same angle Lmn will serve for obtaining the seats of any number of points whatever. .. EXAMPLES. Fig. 33. Ex. 1.-—Find the seats of any number of points in a given curve in a plane at a given angle Q with the plane of pro- jection. Here the points a, d', a", are the seats of the points of the original points A, A', A". In this example the original figure is a semicircle, and the original plane intersects the plane of projection in the diameter DE. -- Fig. 34. Ex. 2.-If the diameter DE be in the line of section ST, and the angle Q a right angle, we have nothing more to do than to draw AQ, A'a', A"a", &c. perpendicular to ST or DE, and the points a, a', a", &c. are the seats of the original A, A', A", &c. 364 [Part V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. PROBLEM II. Given the magnitude of a line and its position in respect to the plane of projection, to find the seat of that line. CASE 1. Fig. 35. ... vento . .. . . When the line is inclined in respect to the plane of projection. Find by Prob. i. the seat 6 of the original point B, and produce BA to meet ST in i. Join bi, and draw Aa parallel to Bb; then will a 6 be the seat of the original line AB. . Or thus:- Find a and b, the seats of the original points, and join a b. .. .. CASE 2. Fig. 36. _ B When the original line is parallel to the line of section. Find a, the seat of the original point A, and draw a b parallel to AB, and draw Bb parallel to Aa; then a b is the seat of the original line AB. T PROBLEM III. Find the seat of a straight line on the curved surface of a cylinder upon a given axal section, the point in the circumference of the base in which the original line terminates being given. 11 Fig. 37. Нf В ПР Let DD'H'H be the axal section, which in this case is the plane of projection, AB being the axis itself. On DD' as a diameter describe the semicircle DED', and let E be the given point in the semicircumference DED. Find e, the seat of the point E, the semicircle being supposed to be raised upon DD perpendicular to the plane DD'H'H. Draw e f parallel to AB, meet- ing HH' in f; then e f is the seat of the line required. DUBAD SECT. II. 365 ORTHOPROJECTION. PROBLEM IV. To find the seat of a straight line on the curved surface of a cone upon a section passing along the axis, a point in the circumference of the base or end in which the original line terminates being given. Fig. 38. . Let DD'C be the given section, which is to be the plane of projection, and AC be the axis itself. On DD, as a diameter, describe the semicircle DED, and let E be the given point in the semicircumference DED'. Find e, the seat of the point E; the semicircle being supposed to be raised upon DD' perpendicular to the plane DD'C. Join eC; then eC is the seat of the line required. DT A PROBLEM V. Find the seat of a straight line on the curved surface of the frustum of a given cone upon a section passing along the axis, a point in the circumference of one of the ends in which the original line terminatės being given. Fig. 39. Let DD'H'H be the given section, and, consequently, the plane of projection, and AB the axis. Produce DH, D'H', to meet each other in C. Then, by the preceding example, find eC the seat of the straight line passing through the point E in the base, and through C the vertex of the cone; and let eC meet HH' in f, and ef will be the seat of the line required. PROBLEM VI. Given the seat of a line, and the distance of each extremity of its original from each seat, to find the length of the original line. Draw two straight lines to form a right angle. Make one of the legs of the right angle equal to the length of the seat of the line which is given, and the other leg equal to the difference between the distances of the seats of the ends of the line from its original point; then join the extremities of these two lines which do not meet, and the hypothenuse will be the length of the original line. EXAMPLE. Let ab be the seat of an original line, and let C be the Fig. 40. distance between a and its original point, and D the distance CA between 6 and its original point. Construct the right angle EFG, and make one of the legs FG equal to ab, and the other FE equal to the difference between C and D, and join EG; then EG is the length of the straight line, of which a b is the seat. Or thus:- Draw aH perpendicular to a b, and make al equal to the difference between C and D, and join bH; then will bH be the length of the line which is the original of a b. la ---- alles 366 [Part V GEOMETRICAL PRINCIPLES OF ARCHITECTURE. PROBLEM VII. Given the projection abc of a triangle, and the height of each original point of concourse of that triangle from the plane of projection, to find the original figure. Fig. 41. A Draw two straight lines from each angle, one perpendicular to each side. Upon two lines drawn from each angle, set off the height of that point; join the two extremities of each pair of parallels, and with the three lines thus joined, describe a triangle. Here the corresponding parts of the figures are referred to by similar letters. CL PROBLEM VIII. Given the distances of a point in space from three fixed points in the plane of projection, to find the seat of that point and the distance between the point and its seat. Fig. 42. Let the three given points be B, C, D, and let the distances of these points from the original points be equal to the three straight lines E, F, G respectively. Join BD, BC, CD. From B, with a radius equal to the line E, de- scribe two arcs; and from C, with a radius equal to the line F, describe an arc meeting one of the arcs described from B in A; and from D, with a radius equal to the line G, describe an arc meeting the other arc described from B in the point A'. Draw the straight line Aa perpen- dicular to BC, and draw the straight line Ala perpendicular to BD; then the point a will be the seat of the original point required. Let Aa meet BC in i. Draw aA" parallel to BC. From i, with the radius i A, describe an arc meeting aA" in A". Then aA is the distance of the original point from its seat a. This will be evident by turning up the three triangles till the three points A, A', A" coincide. E- Gu PLANS AND ELEVATIONS. DEFINITIONS. 1. When two planes are at a right angle with each other, and when an original object between them is first represented upon one of the planes and then on the other, the two representations may be called the seats of that object. SECT. II.] 367 ORTHOPROJECTION. 2. The seat of the object which is parallel to the horizon, is called the plan of that object. 3. The vertical seat is called the elevation of that object. 4. The plane on which the plan is made, may be called the primary plane." 5. The plane on which the elevation is made, may be called the plane of elevation. 6. The planes on which the two seats are made, are called the orthographic planes, or planes of projection. 7. The line of separation between the orthographic planes, is called the base line of elevation, or the base of the plane of elevation. PROBLEM IX. Given the plan of a point, and the distance of the original point from its plan, and base line of the plane of elevation, to find the elevation of the point. Fig. 43. Let a be the plan of the point, and PQ the base line of the elevation. Draw aa perpendicular to PQ, meeting PQ in i. Make i a' equal to the distance of the original point from its plan. Then a will be the elevation of the point of which a is the plan. PROBLEM X. Given the seats of three of the angular points of a parallelogram in the plan, and the distances of the points from their seats, to find the elevation of the parallelogram, and to complete the plan. Fig. 44. Let a, b, c, be the seats of the three angular points in the plane. Draw a a', 66', có, perpendicular to PQ, meeting PQ in the points į, k, l. Make é a equal to the height of the angular point above a, 16' equal to the height of the angular point above 6, and k d' equal to the height of the angular point above c; join a b,bc in the plan. Draw od parallel to ba, and ad parallel to bc, and a b c d will be the plan. Again, join 6', 6' 6. Draw a ď parallel to b'c', and d' Ã parallel to B'd', and á b'd' I will be the elevation. 368 [PART Y GEOMETRICAL PRINCIPLES OF ARCHITECTURE. PROBLEM XI. to complete the plan of that figure, the plane of the figure being parallel to the primary plane, and its Fig. 45. ad te Let a b be the seat of the given side in the primary plane. On a b describe the figure abcd equal and similar to the original figure, and abcd will be the plan required. Draw a d' perpendicular to PQ, meeting it in r. Making ra' equal to the height of the figure above the primary plane, and draw a' é parallel to PQ. Draw bb', c', dď, parallel to ad', meeting ac in 6', , d', and the straight line ác' will be the elevation. PROBLEM XII. Given the seat in the primary plane of one of the edges of a given rectangle, of which one edge is parallel and the face perpendicular to the primary plane, to find the elevation of the rectangle, the height of the edge above its seat being given. Let ab be the given seat. Draw a d' perpendicular to PQ, meeting Fig. 46. PQ in m. In m d make ma' equal to the height of the edge above its seat, and aď equal to the vertical dimension of the rectangle. Draw LB b c parallel to a ď'; also draw b', đó parallel to PQ; ab is the plan and a b c d the elevation required. If a b be perpendicular to PQ, then both the plan and the elevation will be projected into straight lines. PROBLEM XIII. To find the elevation of a circular arch intersecting the surface of a cylinder, the base of the arch being parallel, and the axes of the cylinder perpendicular to the primary plane, the plan being given. Fig. 47. Let pqrst be the intersection of the surface of the cylinder with the primary plane, Dp and It the seats of sides of the arch. Draw DI perpendicular to Dp; on DI describe thesemicircle DABCI. In the semicircular arc take any number of points A, B, C; then suppose the arc DABCI to be perpendicular to the primary plane. Find the seats a, b, c, (by Prob. i. p. 362,) of the points A, B, C. Draw aq, br, cs, &c. parallel to Dp. Find the elevation pt of the base of the arch. Make ed, fr, gs', &c. respectively equal to a A, 6B, C, &c. and describe the curve pqrst, which will be the elevation required. -Q Oy SECT. II. | 369 ORTHOPROJECTION. PROBLEM XIV. To find the elevation of the arc of a semicircle or other curve bent into a surface which is perpendicular to the primary plane, the diameter or chord being in a plane parallel to the primary plane ; given the intersection of the surface and the seats of the extremities of that diameter or chord, and the heights of the two seats. Fig. 48. P- Let RS be the intersection of the surface, and the two points d, i, be the seats of the extremities of the diameter. Draw d d and ii perpen- dicular to PQ, meeting PQ in k and I. Make kd and li' each equal to the height of the diameter above the primary plane, and draw d' i' parallel to PQ. Divide the curve di of the intersection into any number of equal parts, as here into four, and set the distances d a, ab, bc, ci in the straight line DI from D to m, from m to n, and from n to o. Draw ma, nB, C, perpendicular to DI, meeting the arc in A, B, C. From the points a, b, c, in RS, draw a á', 66', cc, &c. meeting d' ï in m, n, o. Make m' a', n' b', óc respectively equal to mA, nB, oC, the ordinates of the semicircle, and through the points d', a', b', c', i, draw the curve d'á' 6' ó i, which is the elevation required. R/ mo Fig. 49. . . . . The easiest method of finding the elevation of the section of a cone cut by a surface perpendicular to the primary plane, the axis of the cone being parallel to the primary plane, and the plan of the axal section being given, is to find the elevation of the entire cone; then find plans and elevations of the lines on the conic sur- face, as mc, m' c', and the perpendicular to PQ, drawn from a to meet m'c', will give the elevation a' of the point above a in the plan. . " Y 370 [PART V. GEOMETRICAL. PRINCIPLES OF ARCHITECTURE. Fig. 50. SCHOLIUM. In finding the elevation of the whole cone, two sets of perpendicular lines are introduced ; one set only would be necessary, provided the perpendicular heights of the seats in the line r s of a sufficient number of points in the conic surface were known. This object being obtained lessens the labour, and prevents an unnecessary confusion of lines in the construction of complex eleva- tions. To attain this object, construct a right angle VUW. Make UV equal to mc, UW equal to mM, and join VW. In VU, take Væ equal to ca, and draw x y perpendicular to UV. Make c' á equal to xy, then the point a' will be the elevation of the point in the conic surface of which a is the plan. In this manner, as many points as will be necessary for drawing the curve may be found. PROBLEM XV. Given the plan and exterior elevation of a circular arch in a circular wall, to describe the line of intersection formed by the interior surface of the wall and the surface of the arch. Fig. 51. . ... . ..... .... Let abcdefg be the plan of the soffit of the arch, abcd being the seat of the exterior surface of the wall, and e f g that of the interior surface, de and a g the jambs, and let ab' ó đ be the elevation of the intersection of the soffit of the arch, and the exterior convex surface of the wall, and let the straight line ád be the representation of the plane on which the arch rests. Draw e é perpendicular to PQ, meeting áď in é. From any point c in the seat of the exterior surface, draw of parallel to de in the plan. Draw co parallel to e é, meeting the elevation of the intersection of the arch and the exterior surface of the wall in c', and draw c'f? parallel to PQ, and ff parallel to e é. Through the points é, f, &c. draw the curve e' f', which will be the inner edge of the intrados of the arch, as required. .... . .. . QUANTU Uitb ME PROBLEM XVI. Given the plan of a line, and the distance of each extremity of that line from its seat, to find the elevation of the line. Draw a d', 6 b' perpendicular to PQ, meeting PQ in the points Fig. 52. i, k. Make i a equal to the height of the one extremity of the original line above its seat a, and make k b' equal to the height of the other extremity of the original line above its seat b; join a'5', and a b' is the elevation of the line required. Secr. II.] 371 ORTHOPROJECTION. PROBLEM XVII. Given the two seats a b, a' b' of a straight line, to find the length of the original line. . Fig. 53. Through a', one extremity of the elevation, draw m n parallel to PQ, meeting k b the perpendicular from the other in the point n; make n m equal in length to b a, the plan of the line; then the dis- tance of the points m, b' will be the length of the original line. mo PROBLEM XVIII. When the original line is in a plane perpendicular to the line of common section. Fig. 54. Through a', one extremity of the elevation of the line, draw a'A parallel to PQ. Make d'A equal to ab, and the distance b'A is the length of the original line as required. 10. PROBLEM XIX. Given the two seats bac, b' a'c' of two straight lines meeting in a point, to find the angle contained by the two original lines. Fig. 55. Find Ab, a c, BC, the lengths of the three original lines, by the preceding Problem, and with the lengths of these three lines describe the triangle BAC, and the angle BAC will be the angle contained by the two original lines. A separate figure is used for the construction, which might have been dispensed with, as the lengths of the three seats on the plan might have been applied upon the horizontal lines of the elevation ; but the confusion which this would have produced in the figure is avoided, in order to explain the method more clearly. UDO- 372 [Part V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. Fig. 56. If one of the seats a d' be parallel to the primary plane, Fig. 56, the originals Ab, b'C of the two seats a b, b c on the plan may be found in the elevation without any confusion. Then in the tri- angle BAC, the side AB is equal to Ab', BC equal to BC, and AC equal to a c on the plan. AL Fig. 57. mn LOOD Coroll. If the two seats of an object be given, the object itself may be determined; for the lengths of the lines may be found by Prob. xvii., p. 371, and every angle by the last Problem, p. 371. Thus the length of the hip é m', the angle e' m' n which the hip makes with the ridge, and in short the distance between any two fixed points whatever, may be found, as the original distance represented by é i on the elevation, and e i on the plan. In Masonry, Carpen- try, Joinery, &c., every original length and angle may be found from its plan and elevation by this sim- ple method. PLATES ILLUSTRATIVE OF ORTHOPROJECTION, [Plates CXXIV.-CXXXIII.] Plate CXXIV. shows the application of Prob. xiii. p. 368, to describing the elevation of semicircu- lar arches in a circular wall. The seat of the thickness of the arch is given at s t on the plan, and its seat on the elevation is found at s t, as in Prob. xv. p. 370. Plate CXXV. exhibits the manner of describing the elevation of arches with conical soffits. The elevations of the two arches on the left side of the plate are described by Prob. xiv. p. 369, and that on the right hand by Prob. xiv. p. 369. Plate CXXVI. exhibits the manner of finding the elevation of the intersection of a cylindric sur- face with the surface of a solid of revolution, the axes of both solids being parallel, and the bases of the solids being parallel to the primitive plane. One general principle of finding a point in the elevation is this :- Draw i e perpendicular to a d, meeting the curve of the base in e. In the curve line a cd, which represents the intersection of the cylindric surface with the base of the solid, take any point b, and 3o. ) CH Fingrared by:c Armstrong. PLATE 14. per we . b A 1 . .. - -- - - - - .- - . - - . - - . / 1214.-.. . ///// ZIZ TILSIZ.X/ we WITAIN .. LIITTIINA - 117 *** " /// / / / + + + - - - - - ... ... .. - .-.- . . . -. .. - - -- - - - - - - - - - . . Y ....-L. . . -- -- - - - --- . 111 // PE / he / -- / - . / . i . . ORTHOGRAPHICAL PROJECTION, PLAN AND ELEVATION. PLAN. RE -- -- - - - ELEVATION - A Fullarton&Co Landonk Edinburske - * . - MORE HIER ZZZZ // ZT ZA/ ////// //////// //////// - te eta .. .-. - - - - - - -- . . . - - - - - . . - - - - - --- - -- - . -.-. - - - - - - -- + - - - - / U1227 C - - - - ---- 017 12 VILLIMIZI 20171117/ AV1111HITUR // - - --- + - - . - - . - . . - . - - -. - . - . 11 SZ w 1/11/ 2 - -- - 1 " ' " " + . . . .. . .. .. . . . . . . . . . . .. . - ..:::::8::1:1::: . - hrvented & Drawn by P. Nicholson. ORTHOGRAPHICAL PROJECTION. PLAN AND ELEVATION. PLATE 195. 10.3. -- - . . --. .-. ---------------- ---- mammam-macam mo naman me ----...- me more N°4. . n. nee j. . - - m - p .......... .............. ELEVATION. NO2. - . - - - -- . - - - . . ---- . .. . - - - - - - . . ....... . -- . -- -------- - . - . . . - 7 . ras...... ---- PLAN LOM. V S - - - 2 MINI - AN - - - - - - La - - - IIIIIIIIIIIIIMIIMTINIO w ca m Dicholsos G. Amstrong A Fullarton&C Londo&Einburgh ORTHOGRAPHICAL PROJECTION. utili Nh1.no 9 PLAN AND ELEVATION. PLATE. 126. Fig. 1. Fig. 2. ELEVATION. ELEVATION. 1 N PLAN. PLAN. Fig. 3. ELEVATION. Fig. 4. ELEVATION. ** PLAN. PLAN. P Nicholson. Cidrinstra A Fullar ton& Co Landon&Edinburgh ORTHOPROJECTION. PLATE 127. OF TUE SIRFACES OF TWO SOLIDS. Fig.1. Fig. 2. Fia.,5. .. Fig.3. Fig. 4. . .. . .. ..- ..- - ---- --- ... ****** ".......... **• -... ..........." . .- Einmal hiermitrone. A Fullarton&C Indon& Edinburgh ico .... ... ... . M . . . . ............. ... . . . . . . . . . . .. ...... Fig. 3. Fr. ..........one . .. .. ......... ........ ... ........... . .. .. *. A Fallarton & C London & Edinburgh Tinarared bl.frnutmang. .......... Fig.5. OF THE SURFACES OF TWO SOLIDS. OR THO PROJECTION. ...... . : - .... . ... ... .. . ... . .... ... . .......... ...... ... . ..... ..... Frig. 4. 1979. . . ... . .. .... . . . .. . . - X Tore ONA PLATE 128... ORTHOPROJECTION. PLATE 129 • OF THE STRFACES OF TWO SOLIDS. Fig. 1. Fig. 2, - when Fig. 3. Fig. 4. . . . . more www wwwwwwwwwww www. Engrared. br 1! Lowri. A Fullarton&C London & Edinburgh ᎤᏗᎴ HᎤ ᎡᎤ Ꭻ Ꭼ [ T ] C . PL 177.730. OF THE SURFACES OF TIMO SOLIDS. Fig. 1. Fig.?. . .... . . . . . ... 1 ....... . . .. .. Fig. 3, Fig. t. . . lindrared hr C. Irmatrono. AFrillartan &C London& Edinburgh * أم لا CHE ORTHOPROJECTION. PLATE.131 OF THE SURFACES OF TWO SOLIÐS. Fig. 1. Fig.2. A - L .. Fig. 3. Fig. 4. - Engrared bi W. Lowry. A Pullatton & Co London &Edinburgh AM ORTHOPROJECTION, PLATE 132 1 . 91 OF THE INTERSECTION OF THE SURFACES OF TWO SOLIDS. Fig. 1. Fig. 2. ........... qy II -- . . . . . . . . - . Fig. 3. Fig. 4. 7 1 NH . ES ... . - .. 1 .. 11 ........ - Fig. 5. mna ... -- . ". . - -- DI Emoraret hy .4matrang A. Fallartori & C. London & Edinburgh . L +/+ 22 Fa. :3 ... .... ... . .... ....... .......... ....... . Fig.l. A Frihartan&C‘Londenk Fitinburge Engrared by Cetrmstrong. OF THE SURFACES OF TWO SOLIDS, Ꭴ ᎡᎢHOPᎡ OJE CTION . 2. ...... Fig. 4. Fig. 2. 2 . 710 PLATE 133. SECT. II.] ORTHOPROJECTION. 373 draw b b perpendicular to the base line a d of elevation, meeting ad in K. Join Ib, and produce Ib to meet the curve a ed (which is the intersection of the curved surface of the solid and its base) in the point f: join ef, and draw bg parallel to fe, meeting Ie in G. Draw GH perpendicular to le, meeting the curve dHe in H: IeHd being considered as a section of the solid upon le perpendicular to a d, the chord of the base, or to the plane of the base itself. Make Kb equal to GH; then b is a point in the elevation. These figures apply to the elevation of niches in circular walls. Figs. 1, 2, 3, have circular bases; Fig. 4 has an elliptical base, and the solid is generated round an axis parallel to the primary plane'; Fig. 2 is a cone with its axis perpendicular to the base. Though the method of finding the elevation of the two surfaces is general, yet, when the perimeter is circular, a more simple method may be adopted; thus:- In Fig. 1, take any point b as before. From the centre I, with the radius Ib, describe the arc bG, meeting Ie in G. Draw GH perpendicular to Ie, and, b b being drawn as before, make Kb equal to GH, and 6 is a point in the elevation. In Fig. 2, let IFP be a section of the cone. From I, with the radius Ib, describe the arc BG meeting HP in G. Draw GH perpendicular to IP, meeting PF in H. Make Kb equal to GH, and b is a point in the elevation. PROBLEM XX. To find the equation of the curve which is the seat of the intersection of two semi-cylinders or two circular archoids, on a plane coinciding with their axes. Let CO, C'O be the axes of the two solids. Fig. 58. Through any point C in CO, draw Aa perpendicular to CO, and, with a radius equal to the radius of the less circular archoid, describe the semicircle AIBa. Draw AF and aG parallel to CO. Again, through any point C' in C'O draw A'á' perpendic- ular to C'O, and, with a radius equal to the radius of the greater circular archoid, describe the semicircle A'I'B'a'. Draw A'R parallel to C', meeting AF in F, and aG in G. In the lesser archoid, imagine the semicircle AIBa to be turned perpendicular to the plane of its base, and a plane drawn parallel to the axis CO, and perpendicular to the base, meeting the end AIBa in the line DI, and AFGa in Dm. . In like manner, in the archoid which has the greatest radius, imagine the semicircle A'I'B'a to be turned perpendicularly to the plane of its base, and a plane drawn parallel to the axis c'o, and perpendicular to the plane of the base, meeting the end A'I'B'a' in D'I', and the plane A'RS' in D'm, so that I'D' may be equal to ID. Join C'I' and CI. Produce D'm to meet CO in P. Let C'I' = R, CI = r, D'I' = DI = %, OP = C'D' = x, and Pm = CD = y. SA'B'á .............. R? — = ? Then by the semicircle (ABa - - - - - - - - - ... - - 22 = pi ya . Therefore, eliminating , by adding these equations together, we have R? — ** = gold — y?, from which it appears that the curve Fm G is an equilateral hyperbola, because the co-efficients of a and y are each unity, and have like signs. 374 [PART V. GEOMETRICAL PRINCIPLES OF ARCHITECTURE. Let y = 0, and R? — Q = md, or x? = R2 — på; whence it appears that x is the base of a right- angled triangle, of which the hypothenuse is R, and the base r. To find the equation of the curve which is the seat of the intersection of a circular and an elliptical archoid, upon a plane coinciding with the axes of the solids, and forming their base. 7 Let CO, C'O be the axes of the two solids. Fig. 59. Through any point C in CO, draw Aa perpendicular to Co. From C, with a radius equal to the circular archoid, describe the semicircle AIBa. Draw AE and aH paral- lel to CO. Again, through any point C in co, draw A'd perpen- dicular to C'O. Make C'A', C'd' each equal to the semi-axis major. Draw C'B' perpendicular to A'd', and make C'B' equal to the semi-axis minor, which we shall suppose to be greater than CB; then with the axis major A'd, and semi-axis minor C'B', describe the semi-ellipsis A'I'B'a'. Draw a'S parallel to CO, meeting AE in E, and aH in H, and draw A'R also parallel to C'O, meeting AE in F, and aH in G. Draw Dm parallel to CO, and produce mD to I, and let D meet Aa in D. In C'B' make C't equal to DI. Draw tI' parallel to A'á, meeting the elliptical arc in l', and draw I'P parallel to C'O, meet- ing CO in P, and Dm in m; then m is a point in the curve. DEMONSTRATION. For suppose the semi-ellipsis A'I'B'a' and the semicircle AIBa to be each raised perpendicular to the plane EFGH; then, the plane passing through D'I' and D'm, and the plane passing through DI and Dm, would intersect each other in a line perpendicular to the plane EFGH from m; and the distance between m and the surface on the perpendicular would be equal to D'I' or DI. Let C'A' = C'a = a, C'B' = 6, CA = r, OP = C'D' = x, and Pm = CD = y. and by the semicircle AIBA, . . . - - - - - -- - -- - = gole - y%. Multiply the second equation by aʻ, and aạ x = a que — 42 y®; therefore a pode - az ya = a 12 - 62 m2, which is an equation to an hyperbola. Now, making a infinite, y will also be infinite, and the equa- making u k and ul each equal to b = C'B, and drawing lo and k q through O, the straight lines lo and k q will be the asymptotes of the curve. The transverse axis will be found by making y = o in the equation a’ me - a2 y = a 62 62 x*, and finding the value of ~, which is x = (62 — m2). SECT. II.] 375 ORTHOPROJECTION. PROBLEM XXII. If the surface of an elliptical archoid meet the surface of a hemisphere, and if the bases of the two solids be in the same plane, the figure which is the seat of the intersection of the two surfaces in the plane of their bases will be a conic section. DC Let EFGH be the base of the hemisphere, AaGF the base of the archoid, Fig. 60. and AIBa the semi-ellipsis which forms the end of the archoid, and FMG be the seat of the intersection of the two curved surfaces. Suppose a plane drawn perpendicular to the base AaGF of the archoid along the axis CS, meeting A Ba in the semi-axis CB, and let a plane be drawn through m, parallel to this plane, meeting the semi-ellipsis ABa in the line DI. Also through the centre $ of the hemisphere, suppose a third plane to be drawn perpendicular to the last two planes, and to the plane of the bases of the two solids, meeting the base EFGH in SE. Draw Pm perpendicular to CS, meeting CS in P, and draw mq perpendic- ular to SE, meeting SE in Q, and join Sm. Then mQ= PS will be equal to the distance of the original point whose seat is m in the inter- section of the curved surfaces of the two solids from the plane passing through SE. In like manner Pm or SQ will be equal to the distance of the same original point from the plane passing CS. Let go be the radius of the hemisphere, h=CA = Ca the horizontal semi-axis of the archoid, and p= CB the perpendicular axis. Also, let SP = Qin = x, Pm = y; and let z = DI be the distance of the seat m from its original in the intersection of the two surfaces. By the semi-ellipsis ABa, - - - - - - - - - - - 1.222 =p(ha — y²), and since (Sm)2 = x2 + y2 .......... .. = gol . - ** — yº. Multiply the second of these two equations by hạ, and the first two sides will become identical, and consequently the second sides equal; therefore, hy2 — 12x2 — h?y2 = p