" ' - '' > y ,, / / / //// UC-NRLF C 2 THE LIBRARY OF f : THE UNIVERSITY OF CALIFORNIA DAVIS *,'*'-. . > a o If li s^ ^v f * r\ ~" : r& "T^ L> *U J& & . '|'ITK Mor.sK OK Ki'i.'ro.v Ai,i';.\.\>'Di':u ESQ! ^ ( / r / . K ^ fa T PaStmjittrr 'fain ARCH iCTON ICK. practical WORKMAN'S COMPANION CARPENTRY, MASONRY, Ac. Ac. [EORY AND PRACTICE THE NEW PRACTICAL BUILDER, AMD '5 Companion: CONTAINING A FULL DISPLAY AND ELUCIDATION < >f the most recent and skilful Methods, pursui'd by ARCHITECTS AND ARTIFICERS, CARPENTRY, JOINERY, BRICKLAYING, PAINTING, GLAZING, PLASTERING, &c. &c. IN THE VARIOUS DEPARTMENTS OF MASONRY, SLATING, PLUMBING, INCLUDING, ALSO, NEW TREATISES ON JiEOMETRY, THEORETICAL AND PRACTICAL, TRIGONOMETRY, CONIC SECTIONS, PERSPECTIVE, SHADOWS, AND ELEVATIONS; A SUMMARY OF THE ART OF BUILDING; COPIOUS ACCOUNTS OF BUILDING MATERIALS, STRENGTH OF TIMBER, CEMENTS, &c. ; A DESCRIPTION OF THE TOOLS USED BY THE DIFFERENT WORKMEN ; AN EXTENSIVE GLOSSARY OF THE TECHNICAL TERMS PECULIAR TO EACH DEPARTMENT ; AMD THE THEORY AND PRACTICE or THE FIVE ORDERS, AS EMPLOYED IN DECORATIVE ARCHITECTURE t WORK IS ILLUSTRATED AND EMBELLISHED WITH NUMEROUS PLATES, FROM ORIGINAL DRAWINGS AH1> DESIGNS, MADE EXPRESSLY FOR THIS WORK, BY MICHAEL AN6ELO NICHOLSON, R. ELS AM, W. INWOOD, AND OTHER EMINENT ARCHITECTURAL ARTISTS. PRINTED FOR THOMAS KELLY, 17, PATERNOSTER ROW. 1823. LIBRARY UNIVERSITY OF CALIFORNIA; P ' VTS Cl JL PRINTED Y wrrr> AND RIPER, J.llTr.F HUM MX, I.ONDOK. PREFACE. 1 HE Work here respectfully submitted to the Public will be found to comprehend the PRESENT PRACTICE OF THE ART OF BUILDING, reduced to PURELY SCIENTIFIC and GEOMETRICAL PRINCIPLES, and yet explained in a manner so simple, as to be easily intelligible to any attentive Reader. To facilitate this important object, the Authors have commenced with a short Treatise on GEOMETRY, theoretical and practical, peculiarly adapt- ed to the general object of the Work, and containing such theorems and problems only as are absolutely necessary to be understood by every per- son connected with the leading departments of the art. And here the Authors cannot refrain from adding a few words, with a view to impress upon the minds of WORKMEN engaged in the construction of buildings, whether CARPENTER, JOINER, MASON, or BRICKLAYER, the PARAMOUNT and UNSPEAKABLE IMPORTANCE of obtaining some knowledge of the PRINCIPLES of GEOMETRY; since, of all the numerous classes, con- cerned in mechanical arts, THEY require the most intimate acquaintance with this science. The execution of the design of the architect is generally left to the skill of the Workman ; who is, of course, presumed to be fully competent to the performance of the task which he undertakes. Now, if he be not practically acquainted with the geometrical construction of the object to be executed, he is not only unfit for the undertaking, but, at every step 1. B 6 PREFACE. that he takes, he will manifest his ignorance and inability, and eventually overwhelm himself with confusion and disgrace. While persons of this description draw down upon themselves such merited degradation, those who, by assiduous application, have made themselves masters of the prin- ciples of geometry, and have obtained a clear and comprehensive view of the practical application of these principles, will not fail to enjoy that in- tellectual satisfaction which results from a successful termination of efforts, conducted with scientific skill, and crowned with general approbation; and, at the same time, open for themselves a legitimate path to that repu- tation which directly and naturally leads to opulence and independence. The articles on CARPENTRY and JOINERY are treated at great length, as their superior importance demand. Indeed, it has been the chief study of the Authors to give to these two branches the utmost degree of scien- tific connexion and development of which they are susceptible. In MASONRY, the artist will find an ample detail of the methods of cutting stone, illustrated by several plates, answering to most purposes which present themselves. And since the principles laid down in this Work are every where of a general tendency, the judgement of the work- man will enable him to apply them wherever difficulties may occur. The art of BRICKLAYING is but little connected with the study of geometrical lines; since the texture of bricks is such as will not admit of their being moulded to the different shapes which the ingenuity of the architect might devise. However, in order to render the present Work complete, and to obviate, as far as possible, every difficulty, several plates are introduced, illustrative of the various forms of arches, niches, ovens, &c. Few things are more important than a clear idea of the mutual con- nexion of the various parts of a building. The authors have, therefore, introduced a section, in which they have endeavoured to show, from first principles only, the dependance which each part has upon some other. PREFACE. 7 The various trades connected with building, as PLASTERING, PAINT- ING, GLAZING, PLUMBING, &c. will be found to be treated of in as com- plete a manner as was practicable ; these branches not admitting of any very scientific development. . A comprehensive Treatise on the FIVE ORDERS is subjoined to the trades accessary to building: these Orders, with their appropriate embel- lishments, form the basis and superstructure of architectural decoration. The parts of the Orders are drawn on a scale which speaks to the eye, and renders all farther detail unnecessary. The parts are given in modules and minutes; this being the best mode of exhibiting their proportions, so as to be most readily and clearly comprehended by the workman and student. In order to increase the utility of the Work, to the BUILDER and CONTRACTOR, a select series of designs, in the modern style, accommo- dated to the various ranks of society, have been introduced ; and, for the use of the ARCHITECTURAL STUDENT, that no accomplishment which might facilitate the operations of the draughtsman, or furnish the designer with more correct ideas, or more extensive views, may be wanting, the RULES of PROJECTION, and the PRINCIPLES of PERSPECTIVE, are pre- sented; and in the most familiar and simple manner in which the subject could be conceived. The Work concludes with a copious GLOSSARY of the most useful terms employed by architects and builders. On the whole, the following Treatise will be found to contain a much greater variety of subjects than any similar work; and, in the method of treating the various articles, the studious reader will discover many things entirely new. Thus, for example, in the designs for roofs, several modes are brought forward, for the first time, interesting, both with respect to the disposition and joining of the timbers ; and the examples which are given 8 PREFACE. . will be found of the greatest utility to the practical builder, in regulating his ideas with respect to any design under consideration, however much it may differ from any of the forms exhibited here. The schemes or diagrams are proportioned in their size to their probable utility; and the strictest regard has been paid to giving to all the parts of each figure their respective and just proportions. Finally, from the important information collected, the natural ar- rangement adopted, and the numerous and valuable illustrations exhi- bited in the course of this Work, the Authors flatter themselves that they will be found to have rendered an important service to a numerous and highly meritorious class of their fellow-subjects; whilst even the most inattentive observer cannot but acknowledge that the Publisher has spared no expense to render the Work deserving of extensive patronage and general approbation The grand principle of the under- taking is obvious : it is equally calculated to instruct the untaught, and to assist the intelligent; to promote a generous emulation, and at once to incite and satisfy inquiry into the elements and practice of those branches of science, than which no others are more conducive to the comfort and happiness of mankind. THE NEW PRACTICAL BUILDER, &c. CHAPTER I. THE ELEMENTS OF GEOMETRY. GEOMETRY is a science which considers the properties of lines and angles, as formed according to some certain law ; as, also, the construction of all manner of figures, according to given data. It is divided into two branches ; one of which considers the relations, po- sitions, and properties, of lines, so as to render a proposition clear to the understanding without the aid of compasses or other instruments; being de- monstrated, by a continued chain of reasoning, from certain principles pre- viously established and laid down as axioms; so that the conclusions from one truth become part of the data for the proof of a succeeding proposition. This, which is called the Theory of Geometry, is fully explained by EUCLID, in his celebrated " ELEMENTS," which have served as the basis of all succeed- ing treatises on the subject: and so much of those Elements as may be re- quired in the practice of Architecture will be found included in the present work. The other branch of geometry is entirely practical, and may be acquired without the theory, according to the directions hereafter given ; although with a knowledge of the reasons of the rules it will be more satisfactory. It is this practical branch that enables the architect to regulate his designs, and the artizan to construct his lines, so as to enable him to execute the work. Without the aid of this branch of knowledge, the workman will be unfit for any undertaking whatever; and, so long as he is ignorant of the methods of 1. c 10 THE NEW PRACTICAL BUILDER, &C. geometrical construction, he must remain under the control and direction of a superior in his own class. The definitions and problems, which follow, are calculated to instruct the uninformed mechanic, and will qualify him for proceeding to the remaining parts of this treatise, wherein it will be found that the application of this branch of science is absolutely necessary. The uses of Geometry are not confined to Carpentry and Architecture: Astronomy, Navigation, Perspective, and numerous other branches, are entirely dependent upon it. " It conducts the soldier in the field, and the seaman on the ocean; it gives strength to the fortress, and elegance to the palace." In short, there is no mechanical profession that does not derive considerable advantage from it. One workman is superior to another, in proportion to his knowledge of the subject we are now commenting upon, and which we are about to explain. The Terms are here as clearly defined as the nature of the subject will ad- mit, and the Problems are put in a regular succession; so that nothing is introduced, in any problem, as taken for granted, but what has been explained in some problem previously given . This selection, though not very numerous, is sufficient to enable the student to proceed with the remaining parts of the work, to which it is specially adapted: and every attention has been paid to divest the diagrams of superfluous lines, without rendering them less intel- ligible. DEFINITIONS OF TERMS IN PLANE GEOMETRY. 1. A POINT. Abstractedly considered, a point is said to have position without magnitude; and it is, therefore, represented to the eye by the smallest visible mark ; that is, simply, by a dot. 2. A LINE is considered as length only, without dimension of breadth or thickness : it is therefore represented by the motion of a visible point. 3. A RIGHT LINE is the shortest that can be drawn between two given points ; it being what is commonly called a Straight Line. ELEMENTS OF GEOMETRY. 11 A Curve or Curved Line is any other than a right line, from which it dif- fers by inflexion, either regular or irregular. 4. A SUPERFICIES, or SURFACE, is considered as an extension of length and breadth, without depth. 5. A PLANE SUPERFICIES is a flat surface, which will coincide, in every point, with a right line. 6. A PLANE FIGURE, SCHEME, or DIAGRAM, is the lineal representation of any object on a plane surface. If the lines forming the figure be straight, the figure is said to be rectilineal; being composed of right lines. 7. An ANGLE is the space or corner between two lines meeting in a point. It is expressed by three letters, as ABC ; and the middle letter, as B, always denotes the vertex, or angular point. The figure in the margin represents a plane rectilineal angle, it being formed by two right lines, meeting in a point (B). 8. CONVERGING LINES are right lines so inclined to each .^ other as to meet, if continued, or produced, to a certain point. Thus, AB and CD, converge to each other ; because, if produced, or continued, they meet in the angle O. 9. RIGHT and OBLIQUE ANGLES. If one right line stands upon another, so as to make the angles on each side equal, each angle is called a RIGHT ANGLE, and the line which stands upon the base, or lower line, is called a perpendicular. Thus, in fig. 1, annexed, if the line CD stand upon AB, and make the angles on both sides of CD equal, each of these angles is a right angle. In fig. 2, the line A.- ** *' ' ' CD does not make the angles on each side of it equal to each other ; and therefore CD is said to stand at oblique angles to AB ; while in fig. 1, CD is at right angles to AB. 10. AN ACUTE ANGLE is less than a right angle. 11. An OBTUSE ANGLE is greater than a right angle. In fig. 2, above, the line CD makes the angles on each side of it unequal : and, therefore, one must be greater than the other : the greater is, of course an obtuse angle, and the lesser an acute angle : for it is clear, by inspection, 12 THE NEW PRACTICAL BUILDER, &C. that, whatever be the position of the line CD, relative to AB, what the one angle has in excess, more than a right angle, the other must want, in order to be equal to the same. EXAMPLES : An Acute Angle. A Right Angle. An Obtuse Angle. 12. A PLANE TRIANGLE is a space inclosed by three right lines, as ABC. 13. A RIGHT-ANGLED TRIANGLE is that which has one right angle ; as ABC. 14. An ACUTE-ANGLED TRIANGLE is a triangle which has all its angles acute ; as figures A and B. 15. An OBTUSE-ANGLED TRIANGLE is a triangle having one obtuse angle ; - as figure C. 16. An EQUILATERAL TRIANGLE is a triangle having all its sides equal ; as figure A. 17. An ISOSCELES TRIANGLE is a triangle having two equal sides ; as figure B. 18. A SCALENE TRIANGLE is a triangle having no two of its sides equal ; as figure C. /_ 19. PARALLEL LINES are lines equally distant from each other A B in all parts, and which, though produced or continued ever so c far, could never meet. Such are the lines AB and CD. 20. A PARALLELOGRAM is a figure of which the opposite sides are parallels. Thus, the figures \~ ~\ V~ \ \~ D, E, F, G, are pa- E \ F \ \ * rallelograms. ' 21. When the parallelogram has a right angle, it is called a RECTANGLE. Thus the figures D and E are rectangles. 22. If the sides of the rectangle be equal, it is called a SQUARE. Hence, figure D (above) is a square. 23. If the two adjacent sides be unequal, the rectangle is termed an OBLONG ; as the figure E. ELEMENTS OF GEOMETRY. 13 24. If only two opposite angles of a parallelogram be equal, the called a RHOMBUS ; as, figures F and G. 25. If two adjacent sides of a rhombus be equal, the figure is RHOM-BO-ID ; as fig. G. 26. Every figure, inclosed by four right lines, is called a QUA- DRANGLE, or QUADRILATERAL. Thus, figures D, E, F, G, H, and I, are quadrangles, or quadrilaterals. 27. When all the sides of a quadrilateral are unequal, it is called a TRAPEZIUM. 28. If two sides of a trapezium be parallel, it is called a TRA- PE-ZO-ID ; as figure I. 29. Figures having equal sides and equal angles, or equi- lateral and equiangular figures, formed by more than four right lines, are called REGULAR POLYGONS. 30. A regular polygon of five sides is called a PENTAGON ; as figure K. 31. A regular polygon of six sides is called a HEXAGON ; as figure L. 32. A regular polygon of seven sides is called a HEPTAGON ; as figure M. 33. A regular polygon of eight sides is called an OCTAGON ; as fig. N. Of nine sides, an eneagon or nonagon ; of ten sides, a decagon ; of eleven sides, a undecagon ; of twelve sides, a duodecagon; of fifteen sides, a quindecagon: but polygons having more than twelve sides are commonly expressed as such, with the number of sides. 34. A CIRCLE is a plane figure formedj by one uniform curved line only, which is called its circumference. (Fig. O.) 35. The CENTRE of a circle is the point in the middle of it, as c in fig. O ; and the line cd, drawn from the centre to the circumference, is the radius of the circle ; all lines, radii, thus drawn, are equal. 2. D 14 THE NEW PRACTICAL BUILDER, &C. 36. The DIAMETER of a circle is a right line, drawn through the centre, and terminated on both sides by the circumference ; as ab, figure P. 37. A CHORD of a circle is a right line, drawn from one point of a circle to another, and dividing it into unequal or equal parts, or segments. In the latter case, the chord is also the diameter. Thus, cd,fig. Q, is a chord, as well as ab, fig. P. 38. A SEMI-CIRCLE is one-half of a circle, as divided into two f equal parts by the diameter. \ 39. A SEGMENT of a circle is that portion which is cut off by a chord. Thus, in figures Q and R, cde antifgh are .. - x segments ; and fig. S, though a semi-circle, is still a segment, / \ terminated by the diameter. 40. A SECTOR is the portion of a circle formed by two radii and the intercepted part of the circumference; as abc, figure T. 41. A QUADRANT is the fourth part or quarter of a circle ; or, in other words, a sector contained by two radii, forming a right angle at the centre, and the intercepted part of the cir- cumference ; as acb, fig. U. 42. An ARC or ARCH is any portion of the circumference of a circle. 43. The ALTITUDE of a figure is a right line drawn from the top or vertical angle perpendicular to the base or op- posite side, or to the base produced or continued. Thus CD, in the annexed figure, is the altitude of the triangle ABC. 44. NOTATION. It has been already shown (Art. 7) that an angle is generally expressed by three letters, as ABC, of which the middle one denotes the vertex, or angular point. The signs used in Algebraic Notation will be explained hereafter, under the proper head ; but as several therein explained s C T ELEMENTS OP GEOMETRY. 15 may previously occur, it is proper here to notice that, + (plus) signifies more, or one quantity or thing added to another : The sign (minus) signifies less, or one quantity subtracted from another : = means is, or are, equal to : x de- notes multiplication, as A x B, that is, A multiplied into B, or as 4 x 5 = 20 : -~ is the sign of division, as 36 -=-3 or f = 12. : : : : denotes equality of pro- portion ; thus, a : b : : c : d, signifies that a bears the same proportion to b as c to d; or, as 6 to 9 :: 2 : 3. /. signifies an angle: ^.s, angles: A a tri- angle : -L a perpendicular. 45. GENERAL TERMS USED IN GEOMETRY. An AXIOM is a truth so evident as to require no demonstration. A THEOREM is a truth or proposi- tion to be demonstrated by a process of reasoning. A PROBLEM is something proposed to be done. A LEMMA is a truth premised, in order to facilitate a following demonstration or solution : this term is rejected by some Geome- tricians as unnecessary! A PROPOSITION is a common term for a Theorem or Problem. A COROLLARY is a truth or deduction which arises from a pre- ceding demonstration. A SCHOLIUM is a remark explanatory of some de- duction from a preceding proposition. An HYPOTHESIS is something sup- posed or premised in a proposition, and from which some certain consequence is deduced. 46. AXIOMS. The AXIOMS are as follow : 1. Things that are equal to the same thing, or to equal things, are equal to one another. 2. If equal things be added to equals, the sums or wholes will be equal. 3. If equal things be taken from equals, the remainders will be equal. 4. If equal things be added to unequals, the sums or wholes will be unequal. 5* The halves of equal things are equal, and the doubles of equal things are equal. 6. Mag- nitudes that mutually agree, or fill equal spaces, are equal to one another. 7. The whole is greater than a part, and equal to all its parts. 8. Only one right line can be drawn from one point to another. 9. Two right lines can- not be drawn through the same point parallel to another right line, without coinciding with each other. 10. All right angles are equal to each other.. 11. Equal circles have equal semi-diameters.- 16 THE NEW PRACTICAL BUILDER, &C. 47. POSTULATES, or DEMANDS. A POSTULATE signifies something which may be assumed as granted. Hence it may be granted, 1st, That a right line may be drawn from any one point to any other point : 2dly, That a line may be produced, that is continued or lengthened at pleasure : 3dly, That a circle may be described from any centre, at any distance therefrom, or with any radius. THEOREMS. THEOREM 1. 48. If one right line, as CD, touch or cut another right line, AB, at any point between its two ends, it makes two adjacent angles, BCD, ACD, which, together, are equal to two right angles. DEMONSTRATION. If CD were perpendicular to AB, each of the angles, DCB and DCA, would be a right angle ; but, as BCD is the excess of DCA above a right angle, and DCB is less than a right angle by the same quantity, the angles DCB and DCA must be equal to two right angles. 49. COROLLARY. If one of the angles be a right angle, the other also must be a right angle. Again, if ever so many right lines stand thus on one point C, on the same side of a right line AB, the sum of all the angles are equal to two right angles ; and all the angles formed about the same point, by any number of lines, are altogether equal to four right angles. THEOREM 2. 50. If the sum of two adjacent angles ACD, DBC, be equal to two right angles, the exterior or outer sides, ACB, will form one right line. For, if CB be not the continuation of AC, assume CE as its continuation. In this case the sum of the angles ACD, DCE, (by theorem 1,) must be equal to two right angles ; but, by hypothesis, the sum of ACD, DCB, is equal to two right angles ; therefore the angles ACD, DCE, are equal to the two angles ELEMENTS OF GEOMETRY. 17 ACD, DCB ( Ax. 1, page 15) ; and, taking from each of these equal angles the angle ACD, there will remain the angle DCE, equal to DCB, or a part equal to the whole, which is impossible. THEOREM 3. 51. When two straight lines, AB,DE, cut each other, the opposite angles are equal. For, since DE is a straight line, the sum of the two angles ACD, ACE, is equal to two right angles (theorem 1) ; and, because AB is a straight line, the sum of the angles ACE, ECB, is equal to two right angles (theorem 1); therefore the sum of the angles ACD, ACE, is equal to the sum of the angles ACE, ECB ; and, taking away from each the common angle ACE, there will remain the angle ACD, equal to the vertical opposite angle ECB. THEOREM 4. 52. Two straight lines, which have two common points, coincide entirely throughout their whole extent. Let A and B be the two common points ; in the first place, the two lines can make but one from A tp B, (Ax. 10, p. 15). If it were possible that they could separate, let C be the point of separation, and let us suppose that one of them takes the ^ B direction CD, and the other CE. At the point C suppose CF to be drawn, perpendicular to AC ; then, be- cause ACD is, by hypothesis, a straight line, the angle FCD is a right angle ; (Def. art. 9 ;) in like manner, because ACE is supposed to be a straight line, the angle FCE is a right angle ; therefore the angles FCD, FCE, are equal ; but this is impossible (Ax. 9, page 15) ; therefore the two straight lines, which have two common points, A and B, cannot separate, but must form one continued line. 2. E 18 THE NEW PRACTICAL BUILDER, &C. \ THEOREM 5. 53. Two triangles are equal when, in the one, an angle and the two sides which contain it are equal, in the other, to an angle and the two sides which contain it. Let the angle A be equal to the angle D, the side AB equal to DE, and the side AC equal to DF ; then the triangles ABC, DEF, shall be equal. Suppose the triangle ABC to be placed upon the tri- angle DEF, so that AB may be upon DE ; then, because the angles A and D are equal, AC will fall upon DF ; and, because AB is equal to DE, and AC equal to DF, the point B will coincide with E, and C with F : therefore, the base BC will coincide with the base EF (theorem 4) ; and, since the sides of the triangles coincide, the other two angles must also coincide ; that is, they must be equal to each other. THEOREM 6. -54. Two triangles are equal when a side and two adjacent angles of the one are respectively equal to a side and two adjacent angles of the other. Let the side BC be equal to the side EF, the angle B A equal to the angle E, and the angle C equal to the aiigle F, the triangles shall be equal. For, suppose the triangle ABC to be placed upon the triangle DEF, so that their bases, BC and EF, may coincide ; then, because the angles B and E are equal, the straight line BA will fall upon ED; and, because the angles C and F are equal, the straight line CA will fall upon FD : therefore, the three sides of one triangle will coincide with the three sides of the other ; and, consequently, the triangles themselves will be equal : and, since therefore the sides of the triangles coincide, the corresponding angles will be equal. ELEMENTS OF GEOMETRY. 19 THEOREM 7. . 55. Any two sides of a triangle are together equal to more than a third side. For, in the triangle, ABC, the straight line BC is the shortest line that can be drawn from B to C ; therefore the sum of the two sides, BA, AC, is greater than BC. AT THEOREM 8. 56. If from any point, as O, within a triangle, ABC, there be drawn two straight lines, OB, OC, one to each extremity of any side, as BC, their sum is less than the sum of the other two sides of the triangle. Produce BO till it meet AC in D ; the line OC is less than the sum of the two lines OD, DC (theorem 7) ; and, adding to these unequals the line BO, the sum of the two lines,, BO, OC, is less than the sum of the three lines BO, OD, DC (ax. 4, p. 15) ; that is, the sum of the two lines BO, OC, is less than the sum of the two lines BD, DC. In like manner, BD is less than the sum of the two lines BA, AD ; and, adding DC to these unequals, the sum of the two straight lines, BD, DC, is less than the sum of the three straight lines BA, AD, DC ; that is, the two straight lines BD, DC, are less than the two straight lines BA, AC ; but the two straight lines BO, OC, have been shown to be less than the two straight lines BD, DC ; and, therefore, much less is the sum of the two straight lines BO, OC, than that of the two sides BA, AC, of the triangle ABC. THEOREM 9. 57. If any two sides AB, AC, of a triangle, ABC, are equal to two sides DE, DF, of another triangle DEF, each to each, and if the angle BAC, contained by the sides, AB, AC, be greater than the angle EDF, contained by the sides ED, DF, the base BC of the triangle which has the greater angle shall be greater than the base EF of the other triangle. 20 THE NEW PRACTICAL BUILDER, &C. Make the angle CAG equal to D, take AG equal to DE or AB, and join CG ; and because the two triangles CAG, DEF, have an angle of the one equal to an angle of the other, and the sides which contain these angles are equal, CG shall be equal to EF (tJieorem 5). Now there may be three cases, according as the point G falls without the triangle ABC, or on the side BC, or within the triangle. CASE 1. Because GC is less than the sum of the two straight lines GI, 1C ; and AB less than the sum of the two straight lines AI, IB : therefore, the sum of the two straight lines GC, AB, is less than the sum of the four straight lines GI, 1C, AI, IB ; that is, the sum of the two straight lines GC, AB, is less than the sum of the two straight lines AG, BC ; but AG is equal to AB, therefore GC is less than BC ; but GC = EF, therefore EF is less than BC. CASE 2. If the point G fall on BC, it is evident that GC, or its equal EF, is less than BC. CASE 3. Lastly, if the point G fall within the triangle ABC, by theorem 8, we have the sum of the two straight lines AG, GC, less than the sum of the two straight lines B THEOREM 21. 72. If a straight line, EF,meet two parallel straight lines, AC, BD, the sum of the inward angles CEF, EFD, on the same side, will be equal to two right angles. For, if not, suppose EG to be drawn through E, so that .A the sum of the angles GEF and EFD may be two right angles ; then EG will be parallel to BD (theorem 18) ; and g r j> thus, through the same point E, two straight lines, EG, EC, are drawn, each parallel to BD; which is impossible (ax. 11, p. 15); therefore no straight 3. G 26 THE NEW PRACTICAL BUILDER, &C. line that does not coincide with AC, is parallel to BD ; wherefore the straight line AC is parallel to BD. 73. COROLLARY. If a straight line is perpendicular to one of two parallel straight lines, it is also perpendicular to the other. THEOREM 22. 74. If a straight line, HK, meet two parallel straight lines, AC, BD, the alternate angles, AHK, HKD, shall be equal. For, the sum of the angles CHK, HKD, is equal to two ^- -f e. right angles ; and the sum of the angles BKH, AHK, is also / equal to two right angles ; therefore the angle HKD must B K ~~35 be equal to AHK. THEOREM 23. 75. If a straight line FG, cut two parallel straight lines, AC, BD, the ex- terior angle, FHC, is equal to the interior and opposite angle HKD. For, since the angle FHC is equal to the angle AHK A a / F c (theorem 3), and the angle AHK equal to the angle HKD ; / therefore the angle FHC is equal to the angle HKD. w~rs. 5" & THEOREM 24. 76. If a straight line, EF, meet two other straight lines, EG, FD, and make the two interior angles, EFD, PEG, on the same side, less than two right angles, the lines EG, FD, meet, if produced, on the side of EF, on which the angles are less than two right angles. For, if they do not meet on that side, they are either parallel, or else they meet on the other side. Now they cannot be parallel, for then the two in- terior angles would be equal to two right angles, instead of being less. Again, to show that they cannot meet on the other side, suppose EA to be parallel to DFB ; then, because the sum of the angles EFD, FEG, is, by hypothesis less than two \ right angles, that is, less than the sum of the two angles, ELEMENTS OF GEOMETRY. 27 FEK, FEG (theorem 1), and EFD is equal to FEA (theorem 20); therefore the sum of the two angles FEA, FEG, is less than the sum of the two angles FEK, FEG ; and, taking FEG from both, FEA is less than FEK : hence, FB and EK must be on opposite sides of EA ; and, therefore, can never meet. The truth of this proposition is assumed as an axiom in the Elements of Euclid, and made the foundation of parallel lines. THEOREM 25. 77. Two straight lines, AB, CD, parallel to a third, EF, are parallel to one another. Draw the straight line PQR, perpendicular to EF. Be- ERF cause AB is parallel to EF, the line PR shall be perpen- ^_ dicular to AB ; and, because CD is parallel to EF, the line PR is also perpendicular to CD : therefore AB and A_ & E CD are perpendicular to the same straight line PQ ; hence they are parallel (theorem 17). THEOREM 26. 78. Two parallel straight lines are every where equally distant. Let AB, CD, be two parallel straight lines. From any points, E and F, in one of them, suppose perpendiculars EG, FH, to be drawn; these, when produced, will meet the others at right angles, in H and G. Join FG; then, because FH and EG are both perpendicular to AB, they are parallel (theorem 17) ; there- fore, the alternate angles, HFG, FGE, which they make with FG are equal (theorem 22) : and, because AB is parallel to CD, the alternate angles, GFE, FGH, are also equal; therefore the two triangles GEF, FHG, have two angles of the one equal to two angles of the other, each to each ; and the side FG, adjacent to the equal angles, common ; the triangles are therefore equal (theorem 6) ; and FH is equal to EG ; that is, any two points, F, E, on the one of the lines, are equidistant from the other line. 28 THE NEW PRACTICAL BUILDER, &C. THEOREM 27. 79. In any triangle, if one of the sides be produced, the exterior angle is equal to both the interior and opposite angles ; and the three interior angles are equal to two right angles. Let ABC be a triangle ; produce any one of its sides, AC towards D ; and, from the point A, let AE be drawn, parallel to BC ; and, because of the parallels CB and AE, and the c~ ~~A 5" angle EAD = C, and the angle EAB = B (theorems 22, 23) ; therefore the sum of the two angles, EAD, EAB, is equal to the sum of the two angles C and B ; that is, since the angle BAD is equal to the sum of the two angles BAE, EAD, the angle BAD is equal to the sum of the angles B, C. Hence the outward angle is equal to the s,um of the inward opposite angles. Again, because the angle BAD is equal to the sum of the angles B and C, add to each the angle BAC, and the sum of the two angles BAC, BAD, will be equal to the sum of the three angles BAC, B, C, or the three angles of the triangle ; but the sum of the two angles BAC, BAD, is equal to two right angles (theorem 1) ; therefore the sum of the three angles of a triangle is equal to two right angles. 80. COROLLARY 1. If two angles of one triangle be equal to two angles of another triangle, each to each, the third angle of the one shall be equal to the third angle of the other, and the triangles shall be equi-angular. 81. COROLLARY 2. A triangle can have only one right angle. 82. COROLLARY 3. In any right-angled triangle the sum of the two acute angles is equal to a right angle. 83. COROLLARY 4. In an equilateral triangle, each of the angles is one- third of two right angles. ELEMENTS OF GEOMETRY. 29 THEOREM 28. 84. The opposite sides of a parallelogram are equal, as well as the opposite angles. Draw the diagonal BD. The triangles ADB, DEC, c have the common side DB ; also, because of the parallels, / / AB, CD, the angle ABD is equal to CDB (theorem 22) ; A B and, because of the parallels AD, BC, the angle ADB is equal to DBC ; there- fore the triangles (theorem 6) and the sides AB, DC, which are opposite the equal angles, are equal. In like manner AD and BC are equal ; therefore the opposite sides of the parallelogram are equal. Again, from the equality of the triangles, it follows, that the angle A is equal to the angle C ; and it has been shown that the angles ADB, BDC, are respectively equal to the angles CBD, DBA ; therefore the whole angle ADC is equal to the whole angle ABC, and thus the opposite angles are equal. 85. COROLLARY. Two parallels, AB, CD, comprehended between two other parallels, AD, BC, are equal. THEOREM 29. 86. If the opposite sides of a quadrilateral be equal, the figure is a parallelo- gram. For, drawing the diagonal BD, (as above,) the triangles ABD, BDC, have the three sides equal, each to each ; therefore the angle ADB, opposite to the side AB, is equal to the angle CBD, opposite to the side CD (theorem 10); hence the side AD is parallel to BC (theorem 19). For the like reason AB is parallel to CD : therefore the quadrilateral, ABCD, is a parallelogram. THEOREM 30. 87. A straight line, BD, drawn perpendicular to the extremity of a radius, CA, is a tangent to the circumference. For every oblique line, CE, is longer than the perpendicu- -Q -A. JB P lar CA (theorem 15) ; therefore the point E must be without the circle ; and since this is true of every point in the line BD, except the point A, the line BD is a tangent (def.9, p.ll). 4. H 30 THE NEW PRACTICAL BUILDER, &C. THEOREM 31. 88. Only one tangent can be drawn from a point, A, in the circumference of a circle. A J> /F Let BD be a tangent at A, in the circumference, de- scribed with the radius CA ; and let AG be another tan- gent, if possible; then, as CA would not be perpendicular to AG, another line, CF, would be perpendicular to AG, and so CF would be less than CA (theorem 14) ; therefore F would fall within the circle, and AF, if produced, would cut the circumference. THEOREM 32. 89. If two circumferences cut each other, the straight line which passes through their centres shall be perpendicular to the chord which joins the points of intersection, and shall divide it into two equal parts. For the line AB, which joins the points of intersection, being a common chord to the two circles ; if, through the middle of this chord, a perpendicular be drawn, it will pass through the points C, D, the centres of the two circles. But only one line can be drawn through two given points ; therefore the straight line which passes through the centres is a perpendicular to the middle of the common chord. THEOREM 33. 90. In the same circle, or in equal circles, equal angles ACB, DCE, at the centre, intercept equal arcs, AB, DE, on the circumference ; and conversely, if the arcs AB, DE, be equal, the angles ACB and DCE are also equal. If the angle ACB be equal to DCE, these two angles may be placed on each other; and, as their sides are equal, the point A will fall on D, ^ \\J i>\ and the point B on E ; but then the arc AB must also fall on DE ; for, if the two arcs did not coincide, there would be, in one or the other, points unequally distant from the centre ; therefore the arc AB is equal to DE. ELEMENTS OF GEOMETRY. 31 Next, if the arc AB be equal to DE, the angle ACB shall be equal to DCE ; for, if they are not equal, let ACB be the greater, and take AC I equal to DCE ; then, by what has been demonstrated, AI is equal to DE ; but, by hy- pothesis, the arc AB is equal to DE ; therefore the arc AI is equal to AB, which is impossible : therefore the angle ACB is equal to DCE. THEOREM 34. 91. An angle, ACB, at the centre of a circle, is double of the angle at the cirumference, upon the same arc, AB. Draw DC, (fig. 1,) and produce it to E. First, let the angle at the centre be within the angle at the circumference, then the angle ACE is equal to the sum of the angles CAD, CDA (theorem 27) ; but, because CA is equal to CD, the angle CAD is equal to CDA (theorem 11) ; therefore the angle ACE is equal to twice the angle CDA. By the same reason the angle BCE is equal to twice the angle CDB ; therefore the whole angle ACB is double the whole angle ADB. Next, let the angle at the centre (fig. 2,) be without the angle at the cir- cumference. It may be demonstrated, as in the first case, that the angle ECB is equal to twice the angle EDB, and that the angle EGA, a part of the first, is equal to twice EDA, a part of the second; therefore, the remainder, ACB, is double the remainder ADB. THEOREM 35. 92. The angles, ADB, AEB, in the same segment, AEB, of a circle, are equal to one another. Let C (fig. 1) be the centre of the circle ; Fig. i. Fig. 2. and, first, let the segment AEB be greater than a semi-circle. Draw CA, CB, to the ends of the base of the segment; then each of the angles, ADB, AEB, will be half of the angle ACB (theorem 34) ; therefore the angles ADB and AEB are equal. 32 THE NEW PRACTICAL BUILDER, &C. Next, let the segment AEB (Jig. 2,) be less than a semi-circle ; draw the diameter DCF, and join EF; and, because the segment ADEF is greater than a semi-circle, by the first case, the angle ADF is equal to AEF. In like manner, because the segment BEDF is greater than a semi-circle, the angle BDF is equal to the angle BEF ; therefore the whole angle ADB is equal to the whole angle AEB. THEOREM 36. 93. The sum of the opposite angles of any quadrilateral, ABCD, inscribed in a circle, is equal to two right angles. Draw the diagonals AC, BD. In the segment ABCD, the angle ABD is equal to ACD ; and, in the segment CBAD, the angle CBD is equal to CAD (theorem 35) ; therefore the whole angle ABC is equal to the sum of the two angles ACD, CAD ; and, adding ADC, the sum of the two angles, ABC, ADC, is equal to the sum of the three angles. Now these three angles are the angles of the triangle ADC, and therefore equal to two right angles (theorem 27) : therefore the sum of the two angles ABC, ADC, is equal to two right angles. In the same manner it may be demonstrated that the sum of the two angles BAD, BCD, is equal to two right angles. THEOREM 37. 94. An angle ABD, in a semi-circle, is a right angle ; an angle BAD, in a segment greater than a semi-circle, is less than a right angle ; and an angle, BED, in a segment less than a semi-circle, is greater than a right angle. Produce AB to F, draw BC to the centre, and, because CA is equal to CB, the angle CBA is equal to CAB (theorem 1 1). In like manner, because CD is equal to CB, the angle CBD is equal to CDB ; therefore the sum of the two angles CBA, CBD, is equal to the sum of the two angles CAB, CDB ; that is, the angle ABD is equal to the sum of the two angles CAB, CDB : but this last sum is equal to the angle DBF (theorem 27) ; therefore the angle ABD is equal to the angle DBF : but, when the angles are equal on each side of a ELEMENTS OF GEOMETRY, &C. 33 straight line which meets another, each of these angles is a right angle ; therefore each of the angles ABD and DBF is a right angle ; and, conse- quently, the angle ABD in a semi-circle is a right angle. Again, because in the triangle ABD, the angle ABD is a right angle ; there- fore BAD, which is manifestly in a segment less than a semi-circle, is less than a right angle : and, lastly, because ABED is a quadrilateral in a circle, the sum of the two angles A, E, is equal to two right angles ; but the angle A is less than a right angle ; therefore E, which is in a segment less than a semi-circle, is greater than a right angle. THEOREM 38. 95. The angle BAE, contained by a tangent AE to a circle, and a chord AB, drawn from the point of contact, is equal to the angle AGB in the alternate segment. Let the diameter ACF be drawn, and GF be joined ; and, because the angles FGA, FAE, are right-angles (theorems 37, 30), and of these FGB, a part of the one, is equal to FAB, a part of the other, (theorem 35,) the remainders BAE, BAG, ^- are equal. ALGEBRA. 96. ALGEBRA is a method of demonstrating propositions, and resolving questions, by means of the letters of the alphabet used as symbols. LETTERS are employed to denote angles, lines, surfaces, or solids ; each being considered as a multitude of units of the kind to which it belongs ; and, consequently, as number, abstracted from figure. Thus, a letter may denote the number 3, or a line of five equal parts of any measure ; as inches, yards, miles, &c. and so on. Letters which are thus generally employed are called quantities ; they being the representation of quantities. 4. I 34 THE NEW PRACTICAL BUILDER, &C. It frequently happens that a quantity consists of several quantities of the same kind, as of two or more distances to make one distance; these must, therefore, be joined by addition, or by addition and subtraction. In order to indicate this junction, two distinct signs will be necessary. The sign + (plus) implies that the quantity which follows it is to be added to that which goes before, and that all the quantities, when more than one, are to be added together into one sum. Thus,a + i shows that b is to be added to a, or that a and b are to be added together. Again, a+b+c+d implies that b is to be added to a ; c, to the sum of a and b; d, to the sum of a, b, c. The sign (minus) placed between two quantities, denotes that the quan- tity which follows it is to be subtracted from that which precedes it. Thus, m n denotes that the quantity represented by n is to be subtracted from that represented by m. Suppose, for instance, that m is 7, and /* 3 ; then 7-3 will be 4. Therefore, m n denotes the remainder arising by sub- tracting n from m. Hence SUBTRACTION is an opposite operation to ADDITION ; and, therefore, if any quantity be both added to and subtracted from the same quantity, the quantity thus added and subtracted may be taken away entirely, by which the expression will be in its most simple form: thus, m+aa is equivalent to m. When two quantities are equal to each other, this equality is implied by the interposition of the double bar = between each of the quantities. Thus, m + a a-m\ as, also, 4 + 3 + 6 2 = 11. Equal quantities, thus connected, are called Equations. TERMS are all those parts of an expression that are separated by the signs of Addition and Subtraction. Thus, a + b + c d, is a quantity consisting of four terms. Quantities which contain two terms are called binomials: thus, a +b, or, a b, are binomials. The MULTIPLICATION of two or more factors is indicated by connecting the letters representing the factors ; as, ab denotes a product of two factors ; abx a ELEMENTS OF GEOMETRY, &C. 35 product of the three factors, a, b, and x. Thus, let a = 2, b = 3, and x=5 ; then abx = 30 : Again, mmmm signify a product of four factors, of which all are equal : suppose m = 2, then mmmm = 16. When any number of factors are equal to each other, instead of repeating them to that number in the representation of the product, the product will be indicated with less trouble by writing only one of the equal factors and a digit ; the latter containing as many units as the factors are in number, over the right-hand side of the factor so written. Thus, instead of aa, bbb, xx, xxx, the same idea will be more conveniently expressed thus, z , b 3 , x 1 , x 3 . The continued product of equal quantities is called a Power ; the quantity itself is called the Base of that power ; and the digit, which indicates the number of factors, is called the Index or Exponent of that power. When factors of a product consist of compound terms, each compound factor is enclosed within brackets, or parentheses, and the factors thus included are joined to each other by bringing the bracket on the left hand of the one nearly close to that on the right-hand of the other. Thus, the product of a + b + c, x + a + c + d, and a + x, is represented by (a + b + c) (x + a + c + c?) (a +x). Let a = 1, b = 2, c = 3, d= 4, and x =5, then will( + * + c) (x + a + c+d) ( + *) = (l + 2+3) (5 + 1 + 3+4) (1 + 5) = 468. Any Power of a compound quantity is represented in a similar manner to that of representing a simple quantity, by inclosing the compound within brackets, and writing the number which indicates the power over the right- hand bracket, and on the right-hand side of that bracket : thus, (a + b) 3 de- notes the cube of a + b, and (x+y + x)* denotes the fourth power of x + y+x. DIVISION is represented by placing the dividend above the divisor, with a short line between them, as ~ ; which expression shows how often the quan- tity a contains the quantity b ; or how often the dividend contains the divisor. Let a be 12, and b be 3, then will be 4. FRACTIONS are represented in the same manner as Division, by placing the numerator above and the denominator below a short line. Thus, indicates a fraction, whose numerator is m, and denominator n. Let m be equal to 2, and n equal to 3 ; then * is equivalent to f or two-thirds : or, if we suppose 36 THE NEW PRACTICAL BUILDER, &C. m to be equal to 17, and n equal to 5, then would be 17-fifths of unity, or by dividing the numerator m, which is equivalent to 17, by the denominator n, which is equivalent to 5 : the quotient will be 3 and f . All expressions of quantity are said to be Simple when the operations are indicated by one or more letters, either in Multiplication or Division, without the intervention of the signs + or , as in the following : a, db, ?, ; which are all Simple expressions. Known quantities are generally represented by the initial letters, a, b, c, &c. of the alphabet, or by numbers ; and the unknown quantities by the final letters, v, w, x, y, 8. A Co-efficient is the number prefixed to any quantity. Thus, in the expres- sion 5x, the number 5 is the co-efficient of x ; or, if x represent a quantity to be discovered by an operation, and a a quantity already known, then, in the expression ax, the quantity a is called the co-efficient of or. Having explained the forms which indicate the operations of Simple Quan- tities, we shall now explain the rules for those performed upon Compounds. ADDITION OF ALGEBRA. 97. To add any number of simple affirmative quantities, which are of the same kind, together, or any number of quantities that have a common factor : Prefix the sum of the co-efficients to the quantity, and the product will represent the sum ; observing that, when no co-efficient is written, the co- efficient is understood to be unity : and, when the co-efficients are expressed by letters, these letters are to be joined with the sign + within brackets, and the common quantity adjoined or subjoined. In the following examples let the sum be put equal to S. Example 1. Add a, a, a, a, a, together; then 5a = S. Example- 2. Add ax, 2 ax, 3 ax, together ; then 6 ax = S. Ex.3. Add ax, bx, ex, dx, ex,fx, together; then (a+b+c+d+e+f)x=S. 98. To add any number of simple affirmative quantities of different kinds together : ELEMENTS OF GEOMETRY, &C. 37 Connect the whole to be added by the sign + ; and, if two or more of these quantities are to be found, of the same kind, they must be united into one simple quantity, or term, as above. Example 1 . Add a, b, c, d, together ; then a + b + c + d = S. Example 2. Add a, b, i, c, b, dx> ey, together ; a + 3b + c + dx + ey = S. 99. To add quantities together which have different signs : Join all the quantities into one expression for the sum ; observing to prefix the same sign to each quantity that it had before the whole were united. Example. Add a, bx, cd, 2bx, together ; then, a + c d 3 bx = S. 100. To add Compound Quantities together : Connect all the quantities, in the several parts, to be added into one expres- sion, giving each quantity the same sign that it had before, in each separate part, and observing to unite such terms as may be found of the same kind. Example. Add 5 bx + j-> 5 ab j, 2 bx - 3 ah + 3 ex, together. The answer will be Tbx + 2ab + - + 3ex = S. 9!tt urft 11 SUBTRACTION OF ALGEBRA. . 101. To subtract one simple quantity from another: Join the quantity to be subtracted to that from which the subtraction is to be made, with a different sign to the original one. Let the difference be put equal to D. Example 1. Subtract n from m ; then m - n - D. Example 2. Subtract Sab from Tab ; then Tab-Sab = 4ab = D. 102, To subtract a compound quantity either from a simple or compound quantity. Subjoin the terms of the quantity to be taken away, with their signs changed, to that from which they are to be taken ; observing that, when two terms are of the same kind, they must be united. Ex. Froma;y+4i-3c, subtract bx- 5 i+4c; then xy+9b- 7c-bx = V. It is evident that changing the signs of the terms of the quantity to be taken away cannot affect its aggregate or value, considered independently of 5. K 38 THE NEW PRACTICAL BUILDER, &C. its signs. For, if they are of different kinds, the difference must be the same after the change as before it took place. Thus, let 5 - 2 + 7 - 3, be a quantity to be subtracted. Then 5-2 + 7-3 = 7 and -5 + 2-7 + 3= -7 Hence we see the reason for changing the signs of the quantity to be sub- tracted. MULTIPLICATION OF ALGEBRA. 103. To find the algebraic product of two compound factors, or of one simple and the other compound. If one of the factors be a simple quantity, let that factor be made the mul- tiplier; then join the multiplier to every term of the multiplicand, and prefix the sign + to each product when its factors have like signs ; but prefix the sign when its factors have unlike signs ; then the sum of all the products is the total product. If the multiplier consist of more than one term, proceed with every term of the multiplier in the same manner as if it had but one term ; then the sum of all the simple products is the whole product ; observing that, all such simple products, as have a common factor, may be united together. Example 1. Multiply a + b by a + b. Operation ... a + b a + b a 2 + ab 104. So that the square of any binomial consists of the square of each part and twice their product. Hence, if we see such a quantity as #* + 2ax + a 2 , we shall immediately know that it is the square of the binomial x + a. ELEMENTS OP GEOMETRY, &C. 39 Example 2. Multiply ab by a ft, or find the square of a-rft. Operation. > . a b ab d'ab -ab+b* 105. Hence the square of any binomial, which has one of its parts negative, is the same which ever of the parts be negative ; for the square of each of the parts is always affirmative, and twice the product of the two parts is negative ; so that the square of a b is the very same as the square of b a. Example 3. Multiply a+b by a - b. Operation. . . a+b a-b a*+ab -ab-b* ' *-b*=(a+b)(a-b) 106. Hence we shall always know, by bare inspection only, that the differ- ence of two squares is the product of the sum and difference of the roots : hence rr 2 tf (x+y) (xy), and, reciprocally, that the product of the sum and difference of any two quantities is equal to the difference of their squares. Thus (a+%) (a -%)=(?- x\ ALGEBRAIC DIVISION AND FRACTIONS. 107. DIVISION is the converse of Multiplication ; therefore, if the signs be alike in the divisor and dividend, the quotient will be affirmative ; but if unlike, the quotient will be negative. The general rule is to place the dividend in the form of a numerator, and the divisor in that of a denominator ; expunge like quantities from both, and divide the co-efficients by the greatest common measure. 40 THE NEW PRACTICAL BUILDER, &C. * Example 1. Divide Sab by b ......... 1 = 3a Example 2. Divide -abc-by -be . . ~|| = | Example 3. Divide 4c by 16 ia . . . . ,-g^ = ^ 108. Powers of the same root are divided by subtracting their exponents. Example I.- Divide b 3 by b\ . . ..... = b. Example 2. Divide a 4 6V by 2 6 2 . . . . a ~ 109. A FRACTION is multiplied by any quantity by joining it to the nume- rator ; thus, the product of r and is represented by ^ which is the product of rm divided by n, or the number of times that the product rm contains n. Since the operation of division is opposite to that of multiplication, if a fraction be multiplied by a quantity equal to its denominator, both the de- nominator and the multiplier may be taken away from the result ; thus, if \ be multiplied by b, the result is ^, which is equivalent to the quaptity a only. 110. The Terms of a fraction are its numerator and denominator. If the terms of a fraction be equally multiplied, that is, multiplied by the same quantity, the value of that fraction will be the same as before ; thus I = -JJ : and, if the terms of a fraction be equally divided by the same quantity, the result will be equal to the original quantity. ALGEBRAIC EQUATIONS. 111. The resolution of an equation is the mode of finding the value of the unknown quantity, in terms of those which are given. The principles of this resolution depend on the following AXIOMS, which are similar to those already given at the beginning of the Geometry, in page 15. 1. If to each side of an equation the same quantity be added, the sums will be an equation still. 2. If from each side of an equation the same quantity be subtracted, the remainders will still be an equation "ELEMENTS OF GEOMETRY, &C. 41 3. If each side of an equation be multiplied by the same quantity, the pro- ducts will be an equation. 4. If each side of an equation be divided by the same quantity, the quotients will still be an equation. OF PROPORTION. 112. DEFINITION. Four quantities are proportionals when the first con- tains some part of the second, as often as the third contains the like part of the fourth. THEOREM 39. 113. If four quantities, a, b, c, d, are proportionals, the product of the two . ., /. . i , extremes will be equal to the product of the two means. Let the first, a, contain the wth part of the second b, m times ; then, by the de- finition, the third, c, will contain the wth part of d also m times ; now the wth part of b is - n , and the wth part of d is \ ; therefore | = m ; and J = m ; wherefore? = ' ';;j"JJj.- Iffioii'i'," Multiply each side of this equation by bd, and ^ = ^ ; therefore, dividing the terms of the fraction on the first side by b, and the terms of the fraction on the second side by d, which are common, ad -be. THEOREM 40. 114. If any equation consist of the product of two quantities on each side, and if the four factors be placed in a row, so that the two factors on either. k side may occupy the middle place of the four, and the other two each one of the extreme places; the four factors, thus taken in order, will be pro- portionals. Let ad=bc: Now here are four different ways of taking out the quantities ; but in which ever of these ways they are taken, we shall always have ad= be, by multiplying the two extreme terms together, and the two middle terms together. Let, therefore, a, b, c, d, be one of the four ; then, since ad= be, divide both sides by bd and 33 = ^ ; that is, j = - d ; now let h be the common measure of a 5 L 42 THE NEW PRACTICAL BUILDER, &C. and b, supposing h contained in a, m times, and in b, n times ; then | = ^ = * ; therefore f = m ; but is the rath part of b ; therefore the wth part of b is con- tained in a, m times. Again, let c = mi, and we shall have^ = ^; wherefore, by this equation, d=ni; therefore =^' = ^; and, consequently, |=z: wherefore, also, c con- n tains the nth part of d, m times. 115. COROLLARY. Hence,if two fractions are equal, the numerator of the one will be to its denominator, as the numerator of the other is to its denominator. To indicate that four quantities, a, b, c, d, are proportionals, the sign : is placed between the first two and between the last two ; and the sign : : is placed between the two terms that stand in the middle ; thus, a : b : : c : d, is read, as a is to b, so is c to d. 116. Four proportionals are termed a proportion. Of four proportional quantities, the last term is called the fourth propor- tional to the other three. The first and third terms of a proportion are called the antecedents, and the second and fourth terms the consequents. SCHOLIUM. 117. Since, from ad=bc, we may choose four different ways of taking out the first term, in order to make a proportion ; and since there are two ways of making choice of the second term after the first, there are eight ways in which the positions of the terms will be different these are a b c d a c b d b a d c b d a c c a d b c d a b d b c a d c b a . , -'' ELEMENTS OF GEOMETRY, &C. 45 118. So that, in every two sets of proportionals, "where the same quantity stands first, the terms of the one set are placed alternately to those of the other ; and we may also observe that, four of these sets of proportionals have their terms inverted with regard to the other four, and are therefore the same when read in contrary order. - THEOREM 41. 119. If the corresponding terms of any number of proportions are multi- plied together, the products, taken in the same order, will be proportionals. Thus, as a : b : : c : d e : f : : g : h i : k :: I : m then, as aei : Ifk : : cgl : dhm For ad be eh-fg im Ttl therefore, adehim = 6cfgkl. Consequently, aei : bfk : : cgl : dhm ; therefore the proposition is manifest. 120. COROLLARY. Hence, if the proportions are the same as the first, we shall have a* : i 3 : : c 3 : d 3 . 121. The proportion which is formed by the multiplication of the corres- ponding terms of two or more proportions is said to be compounded of these proportions. When two of the proportions to be compounded are the same, the propor- tion compounded of the two is said to be the duplicate of either: thus, a 1 : i 2 : : c' : rf a is in duplicate proportion to that of a : b : : c : d. When three proportionals, which are the same, are compounded, the com- pound is said to be triplicate to either of the simple ones : thus, a 3 : b 3 : : c 3 : d*, is the triplicate proportion of a : b : : c : d. i .1. j- j And so on to the succeeding orders. M THE NEW PRACTICAL BUILDER, &C. THEOREM 42. 122. If four quantities, a, b, c, d, be proportionals, the first will be to the sum of the first and second as the third is to the sum of the third and fourth. For, since a, b, c, d, are proportionals, ad=bc. To each side of this equa- tion add the product ac, and we have ad+ac=bc+ac ; that is, a(c+d)=c(a+b'). Therefore, a : a+b : : c : c+d. 123. COROLLARY. Since out of the two equal products ad, be, four com- binations in twos may be chosen, viz. ab, ac, bd, cd, we may therefore have the four following equations all different, by adding each combination to each side of the equation ad = be, viz. No. 1 .... a (c+d) = c(a+b) 2... .a(b+d) = b(a+c) 3 ---- d(a+b) = b(c+d) =c(b+d) Since each of these equations will give eight sets Of proportionals ; therefore the whole four will give thirty-two. THEOREM 43. 124. If four quantities, a, b, c, d, be proportionals, as the first is to the difference of the first and second, so is the third to the difference of the third and fourth. For, since a, b, c, d, are proportionals, ad be. Subtract each side of this equation from the product ac, and we have acad acbc; that is, a(c c?)=e(a-5), whence a : ab:: c : c-d. 125. COROLLARY, Since out of the two equal products ad, be, we may choose the four combinations in twos, viz. ab, ac,bd, cd; and may therefore have the four following equations, by subtracting each side of. the original equation, ad=bc t from each combination. ELEMENTS OF GEOMETRY, &C. 45 No. 1 ---- a(c d} c(a V) 2. . . .a(bd}b(a -c) 3. . . .d(ba)=.b(dc} 4 ---- d(ca} c(db) 126. Again, by subtracting each of the products, ab, ac, bd, cd, from each side of the original equation, ad..bc, we have No. 1 ---- a(d c)=c(b a) 2 ---- a(db] = b(c a) 3 ---- d(a-b) = b(cd) 4 ---- d(a c}=c(b d) Now, since every one of these eight different equations will give eight sets of proportionals, in each set of which the situations of the terms will be varied; therefore the whole will give sixty-four sets of proportionals, in which the terms will have different situations in every two sets. THEOREM 44. 127. If four quantities, a, b, c, d, be proportionals, it will be, as the sum of the first and second is to their difference, so is the sum of the third and fourth to their difference. For, dividing the equation, No. 3, pr. 123, by the equation, No. 3, pr. 126, we shall have a+b _c+d ab~ cd And, by dividing the equation, No. 3, in 123, by the equation, No. 3, in pr. 125, we have, a+b_c+d ba~ dc Wherefore a + b : a^b :: c + d : c 128. Three quantities are said to be proportionals when the first is to the middle quantity as the middle quantity is to the third ; thus, let o, b, c, be three proportionals ; then a : b : : b : c. * The character ^ between two quantities signifies the difference, as 5~9 z= 4. 6. M 46 THE NEW PRACTICAL BUILDER, &C. THEOREM 45. 129. If three quantities be proportionals, the product of the two extremes is equal to the square of the mean. Let a, b, c, be the three proportionals ; then, ac = b*, for, since - = ^, mul- tiply both sides of this equation by ab, and ^=^f, that is, i a =ctc. bnj THEOREM 46. 130. If the first and second terms of two proportions are alike, the third and fourth terms of both, placed in the order of their antecedents and con- sequents, will be proportionals. Let a : b :: c : d, and a : b : : e :f-, then will c : d :: e :f. From the first proportion we have ad=bc t and from the second we have be = qf; therefore, multiplying the corresponding sides of these equations, we have adbe = bcaf; and, throwing out the common factors on each side of this equation, there will remain de~cf\ wherefore, c : d :: e :f. THEOREM 47. 131. In any number of proportionals, of which all the ratios are equal, it will be, as the antecedent of any ratio is to its consequent, so is the sum of all the antecedents of the other ratios to the sum of all the consequents. For, let = 3, 3 = 7, j = f, then will 2 = |=J^=f. Therefore, ad be af=be ah bg M e Whence a(d+f+h) = b(c+e+g.) Wherefore, a : b :: c + e+g : d+f+h, as was to be shown. ; : ELEMENTS OF GEOMETRY, &C. 47 "~ f>; WM -*J. GEOMETRY, CONTINUED. ,*** *jod^ HAVING now explained so much of the principles of Algebra and Propor- tion as may be requisite to elucidate the subsequent propositions, we again proceed with the ELEMENTS OF GEOMETRY. The geometric definitions, &c. have been given, generally, in pages 10 to 14 ; but to those already explained are to be added several which follow, as it now becomes necessary that they, also, should be known. 132. EQUIVALENT FIGURES are such as have equal surfaces, without regard to their form. 133. IDENTICAL FIGURES are such as would entirely coincide, if the one be applied to the other. 134. In EQUIANGULAR FIGURES, the sides which contain the equal angles, and which adjoin equal angles, are homologous. 135. Two figures are similar, when the angles of the one are equal to the angles of the other, each to each, and the homologous sides are proportionals. 136. In two CIRCLES, similar sectors, similar arcs, or similar segments, are those which have equal angles at the centre. Thus, if the sector ABC be similar to the sector DBF, then the angle ABC will be * equal to the angle DBF ; or, if the arc AC be similar to the arc DF, then the angle at B will be equal to the angle at E. Also, if the segment GMH be similar to the segment KNL, the angle I will be equal to the angle R. 137. The AREA of a figure is the quantity of surface, containing a certain number of units of any given scale ; as of inches, feet, yards, &c. 48 THE NEW PRACTICAL BUILDER, &C. THEOREM 48. 138. Parallelograms which have equal bases and equal altitudes are equal. Take the two parallelograms, ABCD, and ABEF, upon the same base, AB, and between the same parallels, AB and DE ; these parallelograms are equal. For, in the parallelogram ABCD (fig. 1), the opposite sides CD and AB are equal ; and in the parallelogram ABEF, the opposite sides EF and AB are equal ; therefore EF is equal to CD (84)*. Again, in the parallelogram ABCD (fig. 2), the side CD is equal to AB ; and, in the parallelogram ABEF, the side FE is equal to AB ; therefore EF is equal to CD. Now, inj^g-. 1, since CD is equal to EF, add CF to both; then will DF be equal to CE. In fig. 2, take away the common part CF, and there will remain DF equal to CE. Therefore, in each of these figures, the three straight lines AD, DF, FA, are respectively equal to the three straight lines BC, CE, EB ; and consequently, the triangle ADF is equal to the triangle BCE: there- fore, from the quadrilateral ABED take away the triangle BCE, there will remain the parallelogram ABCD ; and, from the same quadrilateral, take away the equal triangle ADF, and there will remain the parallelogram ABEF: therefore the parallelogram ABCD is equal to the parallelogram ABEF. 139. COROLLARY. Every parallelogram is equal to a rectangle, of the same base and altitude. THEOREM 49. 140. Any triangle is equal to half a parallelogram of the same base and altitude. For the triangle ABC is equal to the triangle ACD, and the parallelogram ABCD is equab to the sum of both triangles ; and, consequently, double to one of them. * The figures thus inserted in a parenthesis refer to a preceding or a following paragraph ; as, in this instance, to 84, on page 29. ELEMENTS OF GEOMETRY, &C. 49 1. COROLLARY 1. Hence every triangle is half a rectangle, having the same base and altitude. 142. COROLLARY 2. Triangles which have equal bases and equal altitudes are equal. THEOREM 50. 143. Rectangles, of the same altitude, are to one another as their bases. Let ABCD, AEFD, be two rectangles, which have a com- mon altitude AD ; they are to one another as their bases AB, AE. For, suppose that the base AB contains seven equal parts, and that the base AE contains four similar parts; then, if AB be divided into seven equal parts, AE will contain four of them. At each point of division draw a perpendicular to the base, and these will divide the figure ABCD into seven equal rectangles (138) ; and, as AB contains seven such parts as AE contains four, the rectangle ABCD will also contain seven such parts as the rectangle AEFD contains four ; therefore the bases AB, AE, have the same ratio that the rectangles ABCD, AEFG, have. THEOREM 51. 144. Rectangles are to one another as the products of the numbers which express their bases and altitudes. Let ABCD, AEGF, be two rectangles, and let some line taken, as a unit, be contained m times in AB, the base of the one, and n times in AD, its altitude ; also p times in AE, the base of the other, and q times R in AF, its altitude ; the rectangle ABCD shall be to the rectangle AEGF, as the product mn is to the product^. Let the rectangles be so placed that their bases AB, AE, may be in a straight line ; then their altitudes AD, AF, shall also form a straight line (48). Complete the rectangle EADH ; and, because this rectangle has the same alti- 6. N 50 THE NEW PRACTICAL BUILDER, &C. B tude as the rectangle ABCD, when EA and AB are taken as their bases ; and the same altitude as the rectangle AEGF, when AD, AF, are taken as their bases ; we have the rectangle ABCD : ADHE :: AB : AE : : m : p (143) But m : p :: mn : pn therefore, ABCD : ADHE :: mn : pn. In like manner, ADHE : AEGF : : pn :pq But, placing the terms of these two sets of proportionals alternately, we have ADHE: pn :: ABCD : mn and ADHE: pn ::AEGV:pq therefore, by equality, ABCD : mn : : AEGF : pq therefore, alternately, ABCD : AEGF :: mn : pq. 145. OBSERVATION. If ABCD, one of the rectangles, be a square, having the measuring unit for its side; this square may be taken as the measuring unit of its surfaces ; because the linear unit, AB, is contained p times in EF, and q times in EH, by the proposition. 1x1 : pq :: ABCD : EFGH. Hence the rectangle EFGH will contain the superficial unit ABCD, as often as the numeral product pq contains unity. Consequently, the product pq will express the area of the rectangle, or will indicate how often it contains the unit of its surfaces. Thus, if EF contains the linear unit AB four times, and EH contains it three times, the area EFGH will be 3x4 = 12: that is, equal to twelve times a square whose side AB is = 1. In consequence of the surface of the rectangle EFGH being expressed by the product of its sides, the rectangle, or its area, may be denoted by the symbol EF x FG, in conformity to the manner of expressing a product in arithmetic. However, instead of expressing the area of a square, made on a line AB, thus, AB x AB ; it is thus expressed AB 2 . ELEMENTS OF GEOMETRY, &C. 51 146. NOTE. A rectangle is said to be contained by two of its sides, about any one of its angles. THEOREM 52. 147. The area of a parallelogram is equal to the product of its base and altitude. For the parallelogram ABCD is equal to the rectangle ABEF, which has the same base AB, and the same alti- tude (138), and this last is measured by AB x BE, or by AB x AF ; that is the product of the base of the parallelo- gram and its altitude. 148. COROLLARY. Parallelograms of the same base are to one another as their altitudes ; and parallelograms of the same altitudes are to one another as their bases. For, in the former case, put B for their common base, and A, a, for their alti- tudes ; then we have BxA:Bxa::A:a. And, in the latter case, put A for their common altitude, and B, b, for their bases ; then BxA:ixA::B:5. THEOREM 53. 149. The area of a triangle is equal to the product of the base by half its altitude. For the triangle ABC is half the parallelogram ABCE, which has the same base, BC, and the same altitude AD (140); but the area of the parallelogram is BC x AD (147), therefore, the area of .the triangle is | BC x AD, or BCx|AD. 150. COROLLARY. Two triangles of the same base are to one another as their altitudes ; and two triangles, of the same altitude, are to one another as their bases. THEOREM 54. 151. The area of every trapezoid, ABCD, is equal to the product of half the sum of its parallel sides, AB, DC, by its altitude, EF. * 52 THE NEW PRACTICAL BUILDER, &C. Through I, the middle of the side BC, draw KL, parallel to the opposite side AD, and produce DC until it meet KL / in K. In the triangles IBL, ICK, the side IB is equal to J /\ 1C, and the angle B equal to C (74), the angle BIL equal to -^ ^ CIK ; therefore the triangles are equal (54), and the side CK equal to BL. Now the parallelogram ALKD is the sum of the polygon ALICD ; and the triangle CIK, and the trapezoid ABCD, is the sum of the same polygon and the triangle BIL ; therefore, the trapezoid ABCD is equal to the parallelogram A-LKD, and has, for its measure, AL x EF. Now AL is equal to DK, and BL equal to CK; and CD = DK CK ; but DK is equal to AL, and CK equal to BL ; Therefore, CD = AL-BL. But AB = AL+BL. Therefore, AB + CD = 2AL Consequently \ (AB + CD) = AL. It follows that . . AL x EF = | (AB+CD) x EF. THEOREM 55. 152. A straight line, DE, drawn parallel to the side of a triangle ABC, di- vides the other sides AB, AC, proportionally, or so that AD : DB : : AE : EC. Join BE and DC : the two triangles BDE, CDE, have the same base, DE, and they have also the same altitude, because BC is parallel to DE ; and, consequently, the triangles DBE and DCE are equal. Since the triangles BED and AED have the same altitude, they are to one another as their bases ; and since the tri- angles CED and AED have the same altitude, they are to one another as their bases. Therefore, the triangle BDE : ADE : : BD : DA. But since the triangle BDE is equal to the triangle CED, therefore the triangle CED : ADE :: BD : DA. But the triangle CED : ADE : : CE : EA. It follows that, BD : DA :: CE : EA. ELEMENTS OF GEOMETRY, &C. 53 THEOREM 56. 153. If the two sides, AB, AC, of a triangle be cut proportionally by the line DE, so that AD : DB : : AE : EC, the line DE shall be parallel to the remaining side of the triangle. For, if DE be not parallel to BC, some other line, DO, will be parallel to BC ; then, by the preceding proposition, AD:DB:: AO : OC. And, by hypothesis, AD : DB : : AE : EC. Therefore, AO : OC :: AE : EC. And, by addition, AC : OC :: AC : EC. And hence OC must be equal to EC ; which is impossible, unless the point O coincide with E ; therefore no line besides DE can be parallel to BC. THEOREM 57. 154. A line, BD, which bisects any angle, ABC, of a triangle, will divide the opposite side AC into two segments, AD, DC, which shall have the same ratio as the other two sides, AB, BC, of the .triangle. From C, one extremity of the base, draw CE, parallel to BD, meeting AB produced in E. Then the angle ABD is equal to the angle EEC (75), and the angle CBD equal to BCE (74) ; but, by hypothesis, the angle ABD is equal to CBD ; therefore the angle EEC is equal to BCE : hence the side BC is equal to BE (63). Now, because ACE is a triangle, and BD is drawn parallel to one of its sides, AD : DC :: AB : BE (152) ; but, since BE is equal to BC ; therefore AD : DC : : AB : BC. THEOREM 58. 155. Two equiangular triangles have their sides proportional, and are similar to each other. 7. o 54 THE NEW PRACTICAL, BUILDER, &C. Let ABC, DCE, be two triangles, which have their angles equal, each to each ; viz. BAG equal to CDE, ABC equal to DCE, and ACB equal to DEC; the homologous sides, or the sides adjacent to the equal angles, shall be proportionals ; that is, BC : CE :: BA : CD, and BA : CD :: AC : DE. Place the homologous sides BC, CE, in a straight line ; and, because the angles B and E are together less than two right angles, the lines BA and ED shall meet if produced (76) : let them meet in F. Then, since BCE is a straight line, and the angle BCA equal to E, AC is parallel to EF. In like manner, because the angle DCE is equal to B, the straight line CD is parallel to BF; therefore ACDF is a parallelogram. 156. In the triangle BFE, the straight line AC is parallel to FE ; wherefore BC : CE : : BA : AF (152). Again, in the same triangle, BFE, CD is parallel to BF; therefore BC : CE :: FD : DE ; but, by substituting CD for its equal AF, in the first set of proportionals, and AC for its equal FD in the second set, we have BC : CE :: BA : CD by the first, and BC : CE :: AC : DE by the second, therefore, by equality, BA : CD : : AC : DE ; therefore the homologous sides are proportionals ; and, because the triangles are equiangular, they are similar. SCHOLIUM. It may be remarked that the homologous sides are opposite to the equal angles. THEOREM 59. 157. Two triangles, which have their homologous sides proportionals, are equiangular and similar. Suppose that BC : EF :: AB : DE and that AB : DE :: AC : DF the triangles ABC, DEF, have their angles equal : viz. 3 A equal to D, B equal to E, and C equal to F. At the point E make the angle FEG equal to B, and at the ELEMENTS OP GEOMETRY, &C. 55 point F make the angle EFG equal to C, then G shall be equal to A (80), and the triangles GEF, ABC, shall be equiangular ; therefore, by the preceding prop. BC : EF : : AB : EG and, by hypothesis, . . BC : EF :: AB : DE; therefore EG is equal to DE. In like manner BC : EF :: AC : FG and, by hypothesis. ... BC : EF :: AC : DF; therefore FG is equal to DF. Thus, it appears that, the triangles DEF, GEF, have their three sides equal, each to each, therefore they are equal (58). But, by construction, the triangle GEF is equiangular to the triangle ABC; therefore, also, the triangles DEF, ABC, are equiangular and similar. THEOREM 60. 158. Two triangles which have an angle of the one equal to an angle of the other, and the sides about them proportionals, are similar. Let the angle A equal D, and suppose that AB : DE :: AC : DF, the triangle ABC is similar to DEF. Take AG equal to DE, and draw GH parallel to BC, the angle AGH shall be equal to ABC (75), L and the triangle AGH equiangular to the triangle ABC; therefore AB : AG : : AC : AH ; but AG is equal to DE; therefore AB : DE :: AC : AH, but, by hypothesis, AB : DE :: AC : DF; therefore AH is equal to DF. The two triangles AGH, DEF, have therefore an angle of the one equal to an angle of the other, and the sides containing these angles equal ; therefore they are equal (53) ; but the triangle AGH is similar to ABC. THEOREM 61. 159. A perpendicular, AD, drawn from the right angle, A, of a right-angled triangle, upon the hypothenuse, or longest side, BC, will divide that triangle into two others, which will be similar to each other, and to the whole. 56 THE NEW PRACTICAL BUILDER, &C. The triangles BAD and BAG have the common angle B ; and, besides, the right angle BDA is equal to the right angle BAG ; therefore, the third angle BAD of the one is equal to the third angle C of the other (80) ; therefore the two triangles are equiangular and similar. In like manner it may be demon- strated that the triangle DAG is equiangular and similar to the triangle BAG ; therefore the three triangles are equiangular and similar to one another. j> THEOREM 62. 160. The square described upon the hypothenuse, or longest side, is equal to the squares described upon the other two sides. From the right angle C draw CD, perpendicular to the hypothenuse AB ; then the triangle ABC is divided into two triangles, ADC, CDB, which are similar to one an- other, and to the whole triangle ABC (159) ; therefore, by the similar triangles, ABC, CBD, .. AB : BC :: BC : BD; again, by the similar triangles, BAG, CAD, .... AB : AC : : AC : AD ; therefore, reducing the first to an equation, AB x BD = BC 2 and, reducing the second analogy to an equation, AB x AD = AC 2 then, adding these two equations AB x (AD + BD)= AC 2 + BC 2 but, since AB is equal to the sum of the two lines AD, DB, therefore AB 2 = AC 2 +BC 2 . THEOREM 63. 161. Two triangles, which have an angle of the one equal to an angle of the other, are to each other as the rectangle of the sides about the equal angles. Suppose the two triangles joined, so as to have a common angle, and let the two triangles be ABC, ADE. Draw the straight line BE. ELEMENTS OF GEOMETRY, &C. 57 B Now the triangle ABE : tria.ADE :: AB : AD ; Therefore the triangle ABE : tria.ADE :: AB x AE : AD x AE. Or, alternately, the triangle ABE : AB x AE :: the triangle ADE : AD x AE. In like manner, the triangle ABE : AB x AE :: the triangle ABC : AB x AC. Therefore, by equality, the tria. ABC : tria.ADE :: AB x AC : AD x AE. THEOREM 64. 162. Similar triangles are to one another as the squares of their homologous sides. Let the angle A be equal to the angle D, and A the angle B equal to E. Then AB : DE :: AC : DF (155) and AB : DE :: AB : DE. therefore, by multiplying the corresponding terms, we have AB 2 : DE 2 : : AC x AB : DF x DE. But the triangle BAC : triangle EOF : : AC x AB : DF x DE (162). Therefore the triangle ABC : triangle DBF :: AB 2 : DE 2 . Or thus, let A signify A triangle; then (162) A ABC : ADEF :: AB x AC : DE xDF. A ABC: ADEF :: AB xBC : DE xEF. ACxBC:DFxEF:: AABC : ADEF. Therefore, by multiplication, A ABC : ADEF :: AB 2 : DE 2 . THEOREM 65. 163. Similar polygons are composed of the same number of triangles, which are similar, and similarly situated. In the polygon ABCDE, draw from any angle, A, the diagonals AC, AD; and, in the other polygon, FGHIK, draw, in like manner, from the angle F, which is homologous to A, the diagonals FH, FI. Since the polygons are similar, the angle B is equal to its homologous angle G; and, besides, AB : BC :: FG : GH; therefore, the triangles ABC and FGH 7. p 58 THE NEW PRACTICAL BUILDER, &C. are similar (158), and the angle BCA is equal to GHF; these equal angles being taken from the equal angles BCD, GHI, the remainders ACD, FHI, are equal: but, since the triangles ABC and FGH are similar, we have AC : FH : : BC : GH ; and, because of the similitude of the polygons, we have BC : GH :: CD : HI ; therefore, AC : FH : : CD : HI. Now it has been shown that the angle ACD is equal to FHI : therefore the triangles ACD, FHI, are similar (158). In like manner, it may be demonstrated that, the remaining triangles of the two polygons are similar ; therefore the polygons are composed of the same number of similar triangles, similarly situated. THEOREM 66. 164. The perimeters of similar polygons are to one another as their homo- logous sides. AB :FG :: BC : GH. BC:GH::CD:HI. Therefore, AB : FG :: AB + BC + CD : FG + GH + HI (131) ; wherefore AB is to FG as the g perimeter of the polygon ABCD is to the perimeter of the polygon FGHIK. THEOREM 67. 165. The areas of similar polygons are as the squares of their homologous sides. Let the polygons be ABCDE and FGHIK ; from any angle, A, draw the diagonals AC, AD ; and, from the homologous angle F, draw the diagonals FH,FI ; then the triangles ABC, ACD, ADE, are respectively equal and similar to the triangles FGH, FHI, FIK. Therefore the triangle ABC : triangle FGH :: AC 2 : FH 2 And .... the triangle ACD : triangle FHI :: AC 2 : FH S . Therefore the triangle ABC : triangle FGH :: ACD : FHI. In the same manner it may be demonstrated that the triangle ACD : triangle FHI : : ADE : FIK, and so on, if the polygons consist of more triangles. ELEMENTS OF GEOMETRY, &C. 59 Hence (131) the triangle ABC is to the triangle FGH as the sum of the triangles ABC, ACD, ADE, to the sum of the triangles FGH, FHI, FIK ; but the sum of the triangles ABC, ACD, ADE, compose the whole polygon ABCDE, and the sum of the triangles FGH, FHI, FIK, compose the polygon FGHIK ; wherefore the triangle ABC is to the triangle FGH as the polygon ABCDE is to the polygon FGHIK ; but the triangle ABC is to the triangle FGH as AB 2 is to FG 2 ; therefore the similar polygons are as the squares of their homologous sides. 166. COROLLARY. If three similar figures have their homologous sides equal to the three sides of a right-angled triangle, the figure made on the side opposite to the right angle shall be equal to the other two. THEOREM 68. 167. In any triangle, ABC, the square of AB, opposite to one of the acute angles, is equal to the difference between the sum of the squares of the other two sides, and twice the rectangle BD x DC, made by the perpendicular AD, to the side BC. A A There are two cases, according as the perpen- dicular falls within or without the triangle. In the first case, BD = BC CD ; and, in the second case, BD = CD-BC. r> 15 5" In either case BD 2 = BC 2 + CD 2 -2BCxCD. But (160) AB 2 = AD 2 +BD 2 and (160) AD 2 +CD 2 = AC 2 . Therefore, by addition AB 2 = AC 2 + BC - 2B x CD. THEOREM 69. 168. In any obtuse-angled triangle, the square of the side opposite to the obtuse angle is equal to the sum of the squares of the other two sides, and twice the rectangle, BCxCD, made by the perpendicular, AD, upon the side BC. 60 THE NEW PRACTICAL BUILDER, &C. For BD = BC + CD; Therefore, BD 2 = BC 2 + CD 2 +2BCxCD; But, (160) ...... AB 2 =AD 2 + BD ! and (160) . . AD 2 + CD 2 = AC 2 Therefore, by adding these three equations together, AB 2 = AC 2 + BC 2 + 2BC x CD. THEOREM 70. 169. If any two chords, in a circle, cut each other, the rectangle of the segments of the one is equal to the rectangle of the segments of the other. Let AB and CD cut each other in O ; then OA x OB = OD x OC. For, join AC and BD : then, in the triangles AOC, BOD, the vertical angles at O are equal; also the angle A = D and C = B (92), consequently the triangles AOC and DOB are simi- lar, and their homologous sides proportional. Whence AO:OC::DO:OB Wherefore AO x OB = OD x CO. THEOREM 71. 170. If any two chords, in a circle, be produced to meet each other, the rectangle of the two distances, from the point of intersection to each ex- tremity of the one chord, is equal to the rectangle of the two distances from the point of intersection to each extremity of the other chord. Let AB and CD be two chords, and let them be produced to meet in O ; OA x OB = OD x OC. For, join AC and BD ; then, in the triangles AOC and DOB the angle at O is common, and the angle A = D (92); therefore the third angle, ACB, of the one triangle is equal to the third angle, DBO, of the other ; consequently the triangles AOC and DOB are similar, and their homologous sides proportional. Whence AO : OC :: DO : OB Wherefore, AO x OB = OD x OC. ELEMENTS OF GEOMETRY, &.C. 61 THEOREM 72. 171. In the same circle, or in equal circles, any angles, ACB, DEF, at the centres, are to each other as the arcs AB, DF, of the circles intercepted between the lines which contain the angles. Let us suppose that the arc AB contains three of such parts as DF contains four. Let Ap,pq,qB, be the equal parts in AB, and T>r, rs, &c. the equal parts in DF : draw the lines Cp, Cq,Er,Es, &c. ; the angles ACp,pCq, qCB, DEr, &c. all are equal; therefore, as the arc AB contains th part of the arc DF three times, the angle ACB will evidently contain 1 of the angle DEF also three times ; and, in general, whatever number of times the arc AB contains some part of the arc DF, the same number of times will the angle ACB contain a like part of the angle DEF. THEOREM 73. 172. In two different circles, similar arcs are as the radii of the circles. Let the circles AFG, afg, be each described from the centre C. Draw the radii CA, CF, then the arcs Af and of are similar. Draw CB, indefinitely, near CA, and the sectors C ab, CAB, will approach very nearly to isosceles triangles, which are similar to each other; therefore^ Ca : CA :: ab : AB; let BF be divided into small arcs, each equal to AB, and draw the radii from each point of division ; then bf will contain as many arcs, each equal to ab, as the arc BF contains arcs equal to AB ; therefore of is the same multiple of a b that AF is of AB ; whence Ca : CA : : af : AF. THEOREM 74. 173. The circumference of circles are to one another as their diame- ters. 8. Q 62 THE NEW PRACTICAL BUILDER, &C. For, let the circumferences ABCD, abed, be divided into quadrants by the radii OA, OB, OcC, OdD, then the quadrants AB, ab, will be similar arcs ; therefore, OA : Oa :: AB : ab wherefore, OA : Oa :: 4AB : PRACTICAL GEOMETRY. Fig. 1. PROBLEM 1. 174. To make an angle at a given point E, fig. 1, in a straight line, DE, equal to a given angle ABC, fig. 2. From the centre B, with any radius, describe an arc gh, cutting BA at g, and BC at h ; from the point E, with the same radius, describe an arc, ik, cutting ED at i; make ik equal to gh, and through the point k draw EF : then the angle DEF will be equal to the given angle ABC. PROBLEM 2. 175. To bisect a given angle ABC. From BA and BC cut off Be and By equal to each other ; from the points e andy as centres, with any radius greater than the distance ef, describe arcs, cutting each other at G, and join BG, which will bisect the angle ABC, as required. PROBLEM 3. 176. Through a given point, g 1 , to draw a straight line parallel to a given straight line, AB. From g, draw ge, to cut AB at any angle in the point e : in AB take any other pointy; make the angle A / / B By A equal tofeg, and make fh equal to eg, and through the points g and h draw the line CD ; then CD will pass through g parallel to AB, as required. -*- ELEMENTS OF GEOMETRY, &C. 63 PROBLEM 4. 177. At a given distance, parallel to a given straight line, AB, to draw a straight line, CD. *~ /^\ *~~\ 7i\. S In the given straight line AB, take any two / points, g and e ; and, from the centres g and e, A ^ with the given distance, describe arcs at h and * ; draw the line CD, to touch the arcs h and i ; then CD will be parallel to AB, at the distance required. PROBLEM 5. 178. To bisect a given straight line, AB, by a perpendi- cular. From the points A and B, with any distance greater A _ than the half of AB, describe arcs cutting each other in C and D ; join CD, and this line will bisect AB perpen- dicularly. PROBLEM 6. 179. From a given point, C, in a given straight line, AB, to erect a per- pendicular. In the straight line, AB, take any two points e and/, equally distant from C : from the points e and /, with any equal radius, greater than the half of ef, describe arcs, cutting each other at D, and draw CD, which will be perpendicular to AB. PROBLEM 7. 180. From a given point, B, (Geometry, plate II, fig. 1,) at the extremity of a given straight line, AB, to draw a perpendicular. Take any point, E, above the line AB, and, with the radius BE, describe the arc dEC, cutting AB in d: draw the straight line dEC, and join BC, which will be the perpendicular required. PROBLEM 8. 181. From a given point C, (fig. 2, pi. II,) to let fall a perpendicular to a given straight line, AB. 64 THE NEW PRACTICAL BUILDER, &C. From the point C, with any radius greater than the distance of AB, describe an arc cutting AB at e and,/; from the points e and,/, as centres, with any equal radius greater than the half of A, describe arcs cutting each other in D, and draw CD, which will be the perpendicular required. PROBLEM 9. 182. To describe the segment of a circle, which shall have a given length or chord, AB, {fig. 3 pi. II,) and a given breadth, or versed sine, CD.* By problem 2, bisect the straight line AB, by a perpendicular CE ; from the point D, where the perpendicular cuts the chord AB, make DC equal to the breadth, or versed sine : join AC ; and, by problem 1, make the angle CAE equal to the angle ACE : from E, as a centre, with the radius EA or EC, de- scribe the arc ACB, which will be the segment required. PROBLEM 10. 183. Through three given points, A, B, C, (fig. 4, pi. II,) to describe the circumference of a circle. Join AB, BC ; and, by problem 2, bisect each of the lines AB and BC by a perpendicular, and let the perpendiculars meet each other in I : from the centre I, with the distance I A, IB, or 1C, describe the circle ABC, which is that required. PROBLEM 11. 184. Upon a given straight line, AB, (fig. 5, pi. II,) to describe an equi- lateral triangle. From the centres A and B, with the radius AB, describe arcs cutting each other at C. Join AC and BC; then ABC will be the equilateral triangle required. PROBLEM 12. 185. Upon a given straight line, AB, (Jig. 6, pi. II,) to describe a square. From the point B, by problem 5, draw BC perpendicular to AB ; make BC equal to AB : from the points A and C, as centres, with a radius equal to AB * The meaning of sine, versed sine, &c. is given in Trigonometry, hereafter. ELEMENTS OF GEOMETRY, &C. 65 or B, describe arcs cutting each other in D, and join AD, and DC ; then A BCD is the square required. PROBLEM 13. 186. Upon a given straight line, AB, (figures 7 and 8, pi. II,) to describe a regular polygon of any number of sides. Produce the side AB to P, and on AP, from the centre B, describe a semi- circle ACP ; divide the semi-circumference ACP into as many equal parts as the number of sides intended ; through the second division, from P, draw the line BC ; bisect AB and BC by perpendiculars cutting each other in S ; from * S, with the radius AS, BS, or CS, describe a circle ABCDE, then carry the side AB or BC round the remaining part of the arc, which will be found to contain the remaining sides of the number required. Figure 7 is an example of a pentagon. Figure S is an example of a hex- agon : but, in this figure, we need not proceed by the general method ; we have only to make a radius of the given side AB ; and take the points A and B as centres ; and form the arcs AG and BG, and strike a circle with the radius GA or GB, which will contain the side AB six times. PROBLEM 14. 187. \n a given square, ABCD, (fig. 9, pi. II,) to inscribe a regular octagon, so that four alternate sides of the octagon may coincide with four sides of the square. Draw the diagonals AC and BD, cutting each other in S ; on the sides of the square make AL, AF, BE, BH ; CG, CK ; and DI, DM, each equal to half the diagonal; join ME, FG, HI, KL; then will FGHIKLMEF be the octagon required. PROBLEM 15. 188. In a given triangle ABC, (fig. 10, pi. II,) to inscribe a circle. Bisect any two angles, A and B, by the straight lines AE and BE, and the point E, the intersection of these two lines will be the centre of the inscribed circle : draw ED perpendicular to AB, cutting AB in D ; from E, with the radius ED, describe the circle DFG, which will be inscribed in the triangle ABC, as required. 8 R 66 THE NEW PRACTICAL BUILDER, &C. PROBLEM 16. 189. A circle, DBF, (fig. 11, pi. II,) and a line AB, touching it, being given, to find the point of contact. From the centre C draw the perpendicular CD, cutting AB in D, which is the point of contact required. PROBLEM 17. 190. Two straight lines, AB, BC, (fig. 12, pi. II,) forming any angle, being given, to describe a circle to touch each of these lines at a given point, A, in one of them. MakeBC equal to BA, and draw AD perpendicular to AB,and CD perpen- dicular to BC ; from the point of intersection D, with the radius DA or DC, describe the circle ACE, which is that required. PROBLEM 18. 191. In a given circle, ABCD, (fig. 13,j)l. II,) to inscribe a square. Draw the diameters AC and BD at right angles, and join AB, BC, CD, DA ; then ABCD will be the square required. PROBLEM 19. 192. To describe a segment, ABC, of a circle, by means of an angle. Let AC (fig. 14, pi. II,) be the length or chord, and DB the versed sine. Join BA and BC ; produce BA to E, and BC to F, making BE and BF of any length, not less than the chord AC. Prepare two straight edges, BE and BF, and fasten them together at the angle B, so that their outer edges may form the angle ABC ; and, to keep them to the extent, fix another slip, GH, to each straight edge at G and H. Bring the angular point B to A, then move the angle thus formed by the straight edges, so that the edge BE may always move upon the point A, and the edge BF upon the point C ; then if, during the time of moving, a pencil be held to the angular point B, and the point to trace over the plane, the segment of a circle will be described. Or thus 193. Let AC (fig. 15, pi. II,) be the length or chord, and BD the versed sine. Join AB, and draw BE parallel to AC, making BE of any length, not ELEMENTS OF GEOMETRY, &C. 67 less than AB. Form a triangular piece of wood, ABE; bring the angular point B, of the triangle, to the point A ; and move the triangle, so that the side BA may slide upon A, and the side BE upon B : then if, during the motion, a pencil be held at the angular point B, with its point tracing over the plane, the arc AB will be described by the point of the pencil. The arc AB being described, the arc BC will be described in a similar manner ; and, consequently, the whole segment of the circle, as required to be done. PROBLEM 20. To describe an ellipse to any length and breadth. 194. Method the first (fig. 16, pi. II). Draw the line AC, and make AC equal to the length required ; bisect AC by a perpendicular BD, and make EB and ED each equal to half the breadth. To find any point, g, in the curve ; with the difference of ED and EA, as a radius, from any pointy, in EB, describe an arc, cutting EC in h. Draw,/ h, and pr duce it to g, and make hg equal to EB or ED ; then g will be a point in th urve, as required. Or thus Divide AE and AF (fig. 17, pi. II,) each into the same number of )arts, as here into five. Through the points of section 1, 2, 3, &c. ic lines B^, Bi, Ek, &c. ; and through the points of section, 1, 2, 3, &c., draw the lines ID, 2D, 3D, &c., cutting the former lines drawn from he points, h, i, h, &c. ; then through the points A, h, i, k, &c. draw a curve ' and we shall have the fourth part, or quarter, of the whole curve. In ',ne same manner the other quarter DC may be found. And, by taking the point D, instead of B, and by describing the rectangle upon AC, so that the opposite side may pass through B, and dividing and drawing lines in the same manner, we shall have the whole curve. PROBLEM 21. 196. Having given three straight lines, any two of which, taken together, being greater than the remaining one, to construct a triangle. Let A, B, and C, (fig. 1, pi. Ill,) be the three lines given. Make DE equal to A ; and from D, as a centre, with a distance equal to C, describe an arc ; 68 THE NEW PRACTICAL BUILDER, &C. and from E, as a centre, with a distance equal to B, describe another arc, meeting the former in F ; join DF and EF ; and DEF will be the triangle required. PROBLEM 22. 197. Given a trapezium, to construct another every way equal to it. Let GHIK (fig. 2, pi HI,) be the given trapezium. Draw GI, the diagonal, and make LN equal thereto ; on LN describe the triangle LNM, equal to the triangle GHI, by the last problem. Describe, also, the triangle LON, equal to the triangle GKI, and the thing is done. PROBLEM 23. 198. Given a triangle, to make a rectangle equal to it. Let ABC (fig. 3, pi. Ill,) be a triangle. From C draw CD perpendicular to AB, which bisect in g, by the straight line EF, drawn parallel to AB ; from the points A and B draw the perpendiculars AE and BF ; and the rectangle ABFE is equal to the triangle ABC. PROBLEM 24. 199. To make a square equal to a given rectangle. Let ABCD (fig. 4, pi. Ill,) be a rectangle ; produce the side AB till Eh be equal to BC ; on A.h describe a semi-circle, and produce BC to meet the arc G ; on BG describe the square BGFE, and the thing is done. PROBLEM 25. 200. To find the side of a square equal to two squares whose sides are given. Let A and B (fig. 5, pi. Ill,) be the given sides. Make CD equal to A, and at C, draw CE perpendicular to CD, and make it equal to B ; join ED, and ED will be the side of a square equal to both the given squares. PROBLEM 26. 201. Given the sides of three squares, to find the side of another equal to the three. Let A, B, and C, (fig. 6, pi. Ill,) be the given sides. Find DF, as in the last problem, equal to both the squares, whose sides are A and B ; produce ED till EG equal DF, and make EH equal to C ; join GH, and the thing is done. ! ELEMENTS OF GEOMETRY, &C. 69 . PROBLEM 26. 200. To make a square equal to two given squares (fig. 8, pL III.) Draw the straight line AB, and BC perpendicular to AB. Make AB equal to the side of one of the given squares, and BC equal to the side of the other, and join AC : then the square described upon AC will be equal to the sum of two other squares described upon AB, BC : Thus, on AB describe the square ABGF ; on BC describe the square BHIC ; and on AC describe the square ACDE ; then will the square ACDE, described upon the hypothenuse, be equal to the two squares ABFG, CBHI, described upon the legs. \ PROBLEM 27. 201. To describe a square equal to any given number of squares {figures ' \ pi. III). L it be required to make a square equal to the three given squares, A, B, C. Make AB, fig. 10, equal to the side of the square A. Prolong BA to F, and draw BG perpendicular to FB, and on BG make BC equal to the side of the square B, and join AC ; then a square described on AC is equal to the two squares A and B. Make BG equal to AC, and FB equal to the side of the square C, and join FG, and on FG describe the square FGHI ; then the square FGHI is equal to the squares of FB and BG ; but the square of FB is equal to the square of C ; and, since AC is equal to the squares of A and B, the square of BG will be equal to the squares of A and B ; therefore the square FGHI is equal to the three given squares, A, B, C. PROBLEM 28. To divide a straight line in the same proportion as another is divided. 202. Method the first. Let AB, (fig. 11, pi. Ill,) be a given straight line, divided into the parts AF, FD, DB ; it is required to divide another straight line in the same proportion. Draw AC, making any angle with AB ; and make AC equal to the length of the line proposed to be divided. Join BC, and draw the straight lines DE, FG, parallel to BC, cutting AC in G and E ; then will the line AC be di- 9. s 70 THE NEW PRACTICAL BUILDER, &C. vided in the same proportion as the straight line AB ; as was required to be done. 203. Method the second. Let BC, (fig. 1, pi. IV,) be the given line. On BC describe the equilateral triangle BCA. Make, on the sides of this tri- angle, AD, A E, each equal to the length of the line to be divided, and join DE ; then DE will be equal to AD or AE : let the given line BC be divided in the points f, g, h, i. Draw A.f, Kg, A.h, A.i, cutting DE in the points f, g, h, i. Then the line DE will be divided by these points in the same pro- portion as the given line BC is divided by the points f, g, h, i. PROBLEM 29. 204. To describe an octagon whose opposite sides are at a given distance from each other {fig. 2, pi. IV). Describe a square, ABCD, of which each side is equal to the distance of the opposite sides of the octagon. Draw the diagonals AC and BD, cutting each other in P. From each of the points ABCD, with a radius equal to AP, BP, CP, or DP, one half of the diagonal, describe the arcs LPF, EPH, GPK, IPM, cutting AB, BC, CD, DA, in the points E, F, G, H, I, K, L, M, and join FG, HI, KL,ME ; then will EFGHIKLM be the octagon required. PROBLEM 30. % To describe an ellipse to any length and breadth. 205. Method tlie first (fig. 16, pi. II). Draw the line AC, and make AC equal to the length required ; bisect AC by a perpendicular BD, and make EB and ED each equal to half the breadth. To find any point, g, in the curve ; with the difference of ED and EA, as a radius, from any pointy, in EB, describe an arc, cutting EC in h. Draw fh, and produce it to g, and make hg equal to EB or ED ; then g will be a point in the curve as required. In the same manner we may find as many points in the curve as we please. 206. MetJiod the second (fig. 17, pi. II). Divide AE and AF each into the same number of equal parts, as here into five. Through the points of section ELEMENTS OF GEOMETRY, &C. 71 1, 2, 3, &c. draw the lines BA, Bi, B&, &c. ; and through the points of section, 1, 2, 3, &c. in AF, draw the lines ID, 2D, 3D, &c., cutting the former lines drawn from B, in the points h, i, k, &c. ; then through the points, A, h, i, k, &c. draw a curve, and we shall have the fourth part, or quarter, of the whole curve. In the same manner the other quarter DC may be found. And, by taking the point D, instead of B, and by describing the rectangle upon AC, so that the opposite side may pass through B, and dividing and drawing lines in the same manner, we shall have the whole curve. 207. Method the third, with a string (fig. 5, pi, IV). Having placed the axes at right angles, and bisecting each other, from one extremity, C, of the conjugate or shorter axis, with A E or BE, half of the transverse, or longer axis, describe arcs, cutting AB at F, f\ then F, f, are the two focal points. Round the points F, f, stretch a string, and fasten the ends of it ; carry the point forward, as to H, keeping the string always tense, and the point H, &c. will describe the curve as required. PROBLEM 31. 208. To represent an ellipse by means of the arcs of circles (fig. 6, pi. IV). Let AB be the length, and CD the breadth, as before. Draw BE perpen- dicular to CD. Make BF equal to EC. Bisect BF in /, and /C and ED, cutting each other ing. Bisect gC, by a perpendicular, cutting CD pro- duced in i. Make Ek equal to Ei, and join Fz, cutting AB in /. Make Em equal to El. Through m draw kp, and iq ; and, through /, draw kr, ih. From i, with the radius i C, describe an arc, qh ; and, from k, with the same radius, describe an arc, pr. From m, with the radius mp, describe an arc,pA.q; and, from I, with the same radius, describe an arc rEh: then will the com- pound figure, Aq, Ch, Br, Dp, represent the ellipse required. This method is used only for describing an arc on paper : it may also be used for drawing the lines, perpendicular to a real elliptic arch, for the joints of the stones. The first method of describing the curve may be used occa- sionally, from want of a proper instrument ; but, of all the methods, none is so accurate, in practice, as the trammel, which is next described. 72 THE NEW PRACTICAL BUILDER, &C. PROBLEM 32. A 209. To describe an ellipse with a trammel (fig. /4, pi. IV). Draw the straight line AB, and make AB equal to the length of the given ellipse. Bisect AB in E, by the perpendicular CD. Make ED and EC each equal to half the lesser axis. Let the trammel-rod \>e fgh. Suppose the nut f to be fixed, and the nuts g and h to be moveable. Move the nut g so that the distance fg may be equal to EC or ED ; and move the nut h, so that the distance fh may be equal to EA or EB ; then set g and h in the grooves of the trammel, and, by moving the end of the rod f, so that the pin g may slide along the line AB, and the pin h along the line CD ; the point or pencil will trace out the curve of the ellipse as required. PROBLEM 33. 210. A rectangle being given, to describe an ellipse, so that the two axes may have the same proportion as the sides of the rectangle (fig. 3, pi. IV). Let ABCD be the given rectangle ; it is required to describe an ellipse, that shall pass through the points A, B, C, D, and of which the axes shall have a ratio equal to the sides AB, BC, or to CD, DA, of the rectangle. Draw the diagonals AC, BD, cutting each other in I ; and, through the point I, draw EG, parallel to AB or DC, cutting one of the sides BC in P ; and HF, parallel to AD or BC, cutting one of the sides AB in K. From I, with the radius IK, describe an arc, KL, cutting IG in L. Bisect the arc KL in M, and draw mN, parallel to EG, cutting the diagonal DB in N. Join NP and NK, and draw BF, cutting HF in F, and draw BG, cutting EG in G. Make IE equal to IG, IH equal to IF ; then EG and FH are the two axes of the ellipse, which will be described as in the following problem. PROBLEM 34. 211. To describe an hyperbola, or a figure that may have any curvature at the summit that we please (Jig. 18, pi. II). Let AC be the base, or what is called a double ordinate: make ED equal to the height, in the middle of AC ; then, upon AC, as a side, describe a rectangle, ELEMENTS OP GEOMETRY, &C. 73 AFGC, so that the opposite side may pass through D ; produce ED to the point B ; take the point B, farther or nearer from D, according as the curva- ture at D is required to be flatter or quicker ; and observe that, the quicker the curve is at D the flatter it will be towards A and C. The point B being thus fixed, divide AE into any number of parts, as here into four ; also, divide AF into the same number of parts, viz. four ; through 1, 2, 3, &c., the points of section in AE, draw lines to B ; and through the points of section, 1, 2, 3, &c. in AF, draw lines to D, cutting the former lines drawn to B, in the points h, i, k, &c. ; and, through the points A, h, I, k, &c., draw a curve, which will be the half of an hyperbola, or an hyperbolic curve. PROBLEM 35. 212. To describe a parabola upon a given ordinate, AE, and a given abscissa, ED* (fig. 19, pi II). ,, .. ; : a , <- : :, ^ - .Make EC equal to EA, and complete the rectangle AFGC ; so that the op- posite side may pass through D. Proceed as in the two former problems, 33 and 34 ; excepting that, instead of drawing the lines to B, through the points 1, 2, 3, &c., in AE, to draw them parallel to ED. PROBLEM 36. 213. To describe the figure of the Sinest (fig. 20, pi II). Describe the quadrant FHG, equal to the height of the figure, and divide the arc HG into any number of equal parts ; the more of these the more per- fect the operation will be ; and extend the chords to double the number of parts upon the line AC, which is a continuation of FH, and mark the points of division. Draw the lines Ik, 21, 3m, &c. perpendicular to AC ; and, from the points 1, 2, 3, &c. of division in the quadrant, draw lines Ik, 21, 3m, &c. parallel to AC, and through the points A, k, I, m, &c., draw a curve, which will be the figure of the sines, as required. * The PARABOLA is a figure arising from the section of a cone, when cut by a plane parallel to one of its sides. This, with other terms in Conies, is fully described, under CONIC SECTIONS, hereafter. t The SINE, or right sine, of an arch, is a right line drawn from one end of that arch, perpendicular to a radius, drawn to the other end of the same. See TRIGONOMETRY, hereafter. 9. T 74 THE NEW PRACTICAL BUILDER, &C. GEOMETRY OF SOLIDS. DEFINITIONS OF SOLIDS. 214. A RIGHT CYLINDER is that which is formed by the revolution of a rectangle about one of its sides : the line round which the rectangle revolves is called the axis (plural axes) ; and the circles generated by the two opposite sides of the rectangle, perpendicular to the axes, are termed the ends or bases. The surface of the cylinder, generated by the line parallel to the axis, is termed the curved surface, which is either straight or convex, according as a straight edge is applied, parallel to the axis, or in any other direction. 215. A RIGHT CONE is that which is formed by supposing a right-angled triangle to revolve about one of its legs or perpendicular sides ; the fixed leg, or line, is called the axis ; the surface generated by the other leg is called the base ; and the surface formed by the hypothenuse, or side opposite the right angle, is denominated the curved surface, which is either straight or convex, according as a straight edge is applied upon the surface from the vertex, or in any other direction. 216. A SPHERE or GLOBE is that which is formed by supposing a semi-circle to revolve upon its diameter ; the diameter upon which the semi-circle re- volves is called the axis, and the surface formed by the arc of the semi-circle is called the curved surface, which is convex, in whatever way it may be tried by a straight edge. 217. An ELLIPSOID is formed or generated by supposing a semi-ellipse to revolve upon one of its axes ; the axis thus fixed is called the axis of the ellip- soid, and the surface generated by the curve is termed the curved surface. ELEMENTS OF GEOMETRY, &C. 75 PROBLEM 37. 218. To describe a conic section, from the cone, through a line given in position, in the section passing through the axis. Let ABC, (figures 1, 2, 3, pi. VI,) be the section of a right cone, and let DE be the line of section. Through the apex or top of the cone, C, draw CF, parallel to the base AB of the section, and produce ED to meet AB in D, as in figures 2 and 3, or AB produced in G, as in fig. 1, as also to meet CF in F. On AB describe a semi-circle, which will be equal to half the base of the cone. In the semi-circle take any number of points, a, b, c, &c. Draw Dd, in figures 2 and 3, and Gd in fig. 1, perpendicular to AB and Gd', in fig. 1, per- pendicular to GF ; as, also, DT, figures 2 and 3, perpendicular to DF. From the points a, b, c, &c. draw lines, ae, If, eg, &c., cutting Gd (figure 1) and Del (figures 2 and 3) in the points e,f, g, &c. In figure 1, make Ge', Gf, Gg', &c. equal to Ge, Gf, Gg, &c. ; and in Dd', (figures 2 and 3,) make De', Df, Dg', &c. equal to De, Df, Dg, &c. Through the points e', f, g', &c. draw lines to F. Through the points a, b, c, &c. draw lines perpendicular to AB. From the points of section, in AB, draw lines to the vertex C of the cone, cutting the sectional line, DE, in I, m, , &c. Through the points of sec- tion I, m, n, &c., draw Ih, mi, nk, &c. perpendicular to DE. Through the points D, h, i, Tt, &c. in fig. 1, or d' ', h, i, k, &c. figures 2 and 3, draw a curve, which will be the conic section required. OBSERVATIONS. 219. In the first of these figures, the line of section cuts both sides of the sec- tion of the cone ; in this case, the curve Dhik and eE is an Ellipse. In. fig. 2, the line of section DE is parallel to the side AC of the section of the cone ; in this case, the curve d'hi, &c. E, is a Parabola. In fig. 3, the line of section, DE, is not parallel to any side of the cone ; it must, therefore, when produced with the sides of the section through the axis, meet each of these two sides in different points : in this case, the section d', h, i, &c., E, is either an Ellipse or 76 THE NEW PRACTICAL BUILDER, &C. Hyperbola ; but the case is determined to be an hyperbola by the line of sec- tion meeting the opposite side BC at AC, where it cuts above the vertex at the point B'. Here we may observe that the line of section, DE, is the same as that which has before been called the abscissa, the part EB produced, contained between the two sides of the section, is called the axis major ; and the line Dd, perpen- dicular to DE, an ordinate. Hence the same section may be found by the method already shown in the problem ; viz. by drawing any straight line, deb, fig. 4- : make de equal to DE, fig. 3, and eb equal to EB, fig. 3. Through d draw the straight line DD at right angles to db' : make dT> equal to Dd', fig. 3 ; then, with the axis major b'e, the abscissa ed, and the ordinate ?D, on each side of the abscissa describe the curve of the hyperbola, which will be of the same species as that shown in fig. 3. PROBLEM 38. 220. To describe a cylindric section, through a line given in position, upon the section passing through its axis (fig. 4, pi. VI). This is no more than a particular case of the last problem. For a cylinder may be considered as a cone, having its apex at an infinite distance from its base ; or, practically, at a vast distance from its base. In this case all the lines, for a short distance, would differ insensibly from parallel lines ; and this is the construction shown at fig. 5, which is therefore evident. But as the section of a cylinder so frequently occurs, I shall here give a more practical description of it. Thus Let ABHI be a section of a right cylinder, passing through its axis, AB being the side which passes through the base, and let DE be the line of sec- tion. On AB describe a semi-circle ; and, in the arc, take any number of points, , b, c, &c. from which draw lines perpendicular to the diameter, AB, cutting it in Q, R, S, &c. : perpendicular to AB, or parallel to AI or BH, draw the lines Qq, Rr, Ss, &c. cutting the line of section, DE, in the points, q, r, s, &c. : from the points of section, q, r, s, &c. draw the lines qi, rk, si, &c. per- GEOMETRY OF SOLIDS. 77 pendicular to the line of section, DE. Make the ordinates qi, rk, si, &c. each respectively equal to the ordinates Qa, Rb, Sc, &c. ; and through the points D, i, k, I, &c. to E, draw a curve, which will evidently be the section of the cylinder, as required. The same may be done in this manner, viz. Bisect the line of section DE in the point /. Draw tm perpendicular to DE. Make tm equal to the radius of the circle which forms the end of the cylinder ; then, with the axis major, DE, and the semi-axis minor, tm, describe a semi-ellipse, which will be the section of the cylinder required. A DEFINITION. 221. A CUNEOID is a solid ending in a straight line, in which, if any point be taken, a perpendicular from that point may be made to coincide with the surface : the end of the cuneoid may be of any form whatever. The cuneoid, which occurs in architecture, has a semi-circular or a semi- elliptical end, parallel to the straight line to which the perpendicular is applied. PROBLEM 39. 222. To find the section of a cuneoid, with a semi-circular base, the given data being a section through the axis, perpendicular to the vertex, or sharp end, and the line of section upon that end. Let ABC, (fig. 6, pi. VIJ be the section through the axis, perpendicular to the sharp edge, and let DE be the line of section. This construction is similar to that of finding the section of a cylinder, ex- cepting that, instead of drawing parallel lines from the base AB, they are, in this figure, drawn from the points of section in AB to the point C, which is the vertex of the cuneoid : the ordinates Qa, Ri, Sc, &c., being transferred respectively to qi, rk, si, &c. ; and the curve D, i, k, I, &c. to E, being drawn through the points, D, i, k, I, &c., by hand. PROBLEM 40. 223. Given the position of the seats of three points, in the circumference of the base of a cylinder, and the lengths of the perpendiculars intercepted 10. u 78 THE NEW PRACTICAL BUILDER, &C. between the points and their seats, to find the section of the cylinder passing through these three points. Through the three points, A, B, C, (fig. 7, pi. VIJ describe the circum- ference of a circle. Join the two remote points, A and B, and draw AD, CF, and BE, perpendicular to AB. Make AD equal to the height upon A, BE equal to the height upon B, and CF equal to the height upon C. Produce BA and ED to meet each other in H : draw CG parallel to BH, and FG parallel to EH. Join GH. In GH take any point, G, and draw GK perpen- dicular to CG, cutting BH in K : from the point K draw KI, perpendicular to EH, cutting EH in L. From H, with the radius HG, describe an arc, cutting KI at I. Join HI. In the circumference, ACB, take any number of points a, b, c, &c., at pleasure, and draw ae, If, eg, &c. parallel to GH, cutting AB at e,f,g,&.c. Through the points e,f,g, &c., draw lines ei,fJt,gl, &c. parallel to GK, or AD, or BE, cutting DE at i,k, I, &c. ; from the points of section, *', Jc, I, &c., draw the lines in, Tto, Ip, &c., parallel to HI. Transfer the ordinates ea,fb, gc, &c. to in, Tto, Ip, &c. ; then, through the points D, n, o,p, &c. draw the curve Dnop, &c. to E, and it will be the section cut by the plane, as required, PROBLEM 41. 224. Given the great circle of a sphere, and the line of position of a section at right angles to that great circle, to find the form of the section. Let ABC, (Jig. 8, pi. VI,) be the great circle, and AB the line of section. On AB, as a diameter, describe a semi-circle, which will be the section re- quired : since all the sections of a sphere, or globe, are circles. PROBLEM 42. 225. Given the section of an ellipsoid, passing through the fixed axis, and the line of position of another section, at right angles to the first section, to find the form of the section through that line. Let ABCD, (Jig. 9, pi. VI,) be the section through the fixed axis, and EF the line of position. Through the centre of the ellipsoid draw AC parallel to EF. Bisect EF in H, and draw HG perpendicular to EF. Find HG a fourth PLANE TRIGONOMETRY. 79 proportional to AC, DB, HE. Then, with the axis major, EF, and the semi-axis minor, HG, describe a semi-ellipse, and it will be the section of the ellipsoid required. If AC be the axis major, BD will be the axis minor. In this case, join DC, and draw EG parallel to DC ; then HG will be the height found geometrically. PROBLEM 43. 226. To find the section of a cylindric ring, perpendicular to the plane pass- ing through the axis of the ring, the line of section being given. Let ABED, (fig. 10, pi. VI,) be the section of the ring, passing through its axis, and let AB be a straight line, passing or tending to the centre of the two concentric circles, AD and BE ; also, let DE be the line of section. On AB describe a semi-circle, and take , b, c, &c., any number of points in its cir- cumference ; draw the ordinates ae, bf, eg, &c. Through the points, e,f, g, &c., in the diameter AB, draw the concentric circles, ei,fk, gl, &c., cutting the sectional line DE in the points i, It, I, &c. Through the points i, k, I, &c. draw in, ho, Ip, &c., perpendicular to DE ; transfer the ordinates ea,/b, gc, &c. of the semi-circle, to in, Jco, Ip, &c. : and, through the points D, n, o,p, &c. draw the curve, DnopqlL, which is the section required. PLANE TRIGONOMETRY. DEFINITIONS OF TERMS IN TRIGONOMETRY.* 227. THE COMPLEMENT OF AN ARC is the difference between < that arc and a quadrant or quarter of a circle. Thus, the arc BC, which is the difference between AC and AB, is the complement of AB ; and AB is, in like manner, the com- plement of BC. * Trigonometry is that branch of Geometry which treats exclusively on the properties, relations, and measurement, of triangles. 80 THE NEW PRACTICAL BUILDER, &C. 228. The SUPPLEMENT OF AN ARC is the remainder be- tween that arc and a semi-circle. Thus, the arc given being AB, its supplement is BC. 229. The SINE OF AN ARC is a straight line, drawn from one extremity of the arc, upon and perpendicular to a radius or diameter. Thus, BM is the sine of the arc AB ; and here it is evi- dent that an arc and its supplement have the same sine. 230. The CO-SINE OF AN ARC is the sine of the complement of that arc. Hence, BO or IM is the co-sine of the arc AB ; and, therefore, the sine of the complement BC. 231. The TANGENT OF AN ARC is a straight line drawn from one extremity of the arc, where it touches it, to meet the pro- longation of the radius through the other extremity. The line AK, touching the arc at A, and extended to meet the radius IB produced, is the tangent of the arc AB. 232. The CO-TANGENT OF AN ARC is the tangent of the complement of that arc. Thus, CL is the co-tangent of the arc AB, or the tan- gent of the arc BC. In the annexed diagram let AB, AC, AD, AE, AF, AG, AH, to A, be the several portions of the circumference, by supposing the point B to re- volve round the circumference from A to B, C, D, E, F, G, H, the sine of any arc, in the first qua- drant, increases from A to C, where it is the greatest possible, and then decreases to E, where it becomes zero ; the sine will, therefore, be positive for the first semi-circum- ference, and in the other half it will be negative. PLANE TRIGONOMETRY. 81 The co-sine will be positive in the first quarter, negative in the second and third, and again positive in the fourth. The tangent will be positive in the first quarter, negative in the second, positive in the third, and negative in the fourth. TRIGONOMETRY. THEOREM 1. 233. If a perpendicular be drawn from an angle of a triangle, to the oppo- site side, which is the base ; then, as the base is to the sum of the two sides, so is the diiference of the sides to the difference of the segments of the base. For, (theorem 62, page 56,) AC 2 - CD 2 = AD 2 ; and, again, (theorem 62 J , BC 2 -CD 2 = BD 2 . Subtract the second equation from the first, and the ^ r> B result is = AC 2 - BC 2 = AD 2 - BD : but, since the difference of the squares of any two quantities is equal to a rectangle contained by their sum and difference, therefore (AC + BC) (AC - BC) = (AD + BD) (AD - BD) Whence, ( theorem 4,0, page 41, ) AD + BD : AC + BC : : AC-BC : AD-BD. ' TRIGONOMETRY. THEOREM 2. 234. The sum of the two sides of a triangle is to their difference as the tangent of half the sum of the angles at the base is to the tangent of half their difference. Let ABC be a triangle ; then, of the two sides, CA and CB, let CB be the greater. Produce CA to E, and make CE = CB, and join BE. Produce BC to F ; and, through A, draw FD perpendicular to EB, meeting it in D ; then FBD will be half the sum of the angles at the base, and ABD half their difference. Likewise, DF is the tangent of the angle FBD, and AD the tangent of the angle ABD : moreover BF is the sum of the two sides BC, CA, and AE is their difference. Then, by similar triangles, BFD, EAD, BF x AD = FD x E A. Where- fore BF : AE : : FD : AD ; which is the proposition to be demonstrated. 10. x THE NEW PRACTICAL BUILDER, &C. CONIC SECTIONS. DEFINITIONS OF CONIC SECTIONS. 235. A CONE is a solid body, terminating in a point, called its vertex, and having a circle for its base, connected to the vertex by a curved surface, which every where coincides with a straight line passing through its vertex, and through any point in the circumference of the base. If a cone be cut by an imaginary plane, the figure of the section so formed acquires its name according to the inclination or direction of the cutting plane. 236. A plane passing through the vertex of a cone, and meeting the plane of the base, is called a directing plane; and the line of common section is called a directing line. 237. If a cone be cut by a plane parallel to the directing plane, the section is denominated a conic section. 238. If the directing line fall without the base of the cone, the section is called an ellipse. 239. If the directing line touch the circumference of the base, the section is called a, parabola. 240. If the directing line fall within the base, the section is / called an hyperbola. 241. Equal opposite cones are those which have their axes in the same straight line ; and, if cut by a plane through their common line of axis, the sides of the section will be two straight lines cutting each other. Hence the two equal and opposite cones join each other at their vertices, and have their vertical angles equal. CONIC SECTIONS. 242. If the plane which produces the section of an hyber- bola be extended so as to cut the opposite cone, the two sections are denominated opposite hyperbolas. 243. If the plane of a conic section be cut perpendicu- larly by another plane, which passes through the axis of the cone, the line of common section, in the plane of the figure, is called the primary line. 244. A point where the primary line cuts a conic section is called a vertex of that conic section. Hence the ellipse has two vertices ; opposite hyperbolas have each one, and the parabola has one. OF THE ELLIPSE. 245. That portion of the primary line terminated at each extremity by the vertices of the curve, is called the axis major, or transverse axis. 246. A straight line, drawn perpendicularly to the axis major, from any point in it, to meet the curve, is called an ordinntc. 247. The middle of the axis major is called the centre of the figure. In the figure here annexed, Aa is the axis major, PM an ordinate to it, and the point C, in the middle of Aa, is the centre of the ellipse, AMa. THE ELLIPSE. THEOREM 1. 248. The squares of the ordinates of the axis are to each other as the rectangles of the segments of the axis, from each ordinate to each of the two ver- tices of the curve. Let VDF be a plane passing through the axis of the cone, perpendicular to the cutting plane of the section AlMami, and let Aa be their common sec- tion, meeting the conic surface in the points A, a ; (then Aa will be the axis major,) and let QlRz, NMOz, be sections of the cone, parallel to the base. Then, because the base DGFE is perpendicular 84 THE NEW PRACTICAL BUILDER, &C. to the plane VDF, the three sections AlMami, QlRi, NMOwz, are all per- pendicular to the plane VDF ; therefore their common sections, li, Mm, are perpendicular to the plane VDF, and to the lines A, QR, NO ; but, because the plane VDF passes through the axis of the cone, it will divide all the circles parallel to the base into two equal parts ; therefore QR and NO are the diameters of the circles QlRi, NMOwz; and, because the chords li, MOT, are perpendiculars to the diameters, QR, NO, they will be bisected; let H be the point of bisection in li, and P the point of bisection in Mm. and HR=>. Then AP = CA + CP = a+x, P = Ca-CP =a-x, and AH=CA CH=- *, H = Ca + CH = +s:. ( APN, AHQ ..v (a + x) =t(a-x) Now, by similar triangles, < ( . . . . QIR y* = vw and, by the circle, < ( NMO tu=y\ Wherefore, eliminating t, u, v, w, by multiplying the given equations, the result will be y 4 (a + x} (-a-) = y 2 (a-ss) (a + x), or, by actual multiplication, 249. COROLLARY 1. Hence, every chord perpendicular to the axis major is bisected by the axis major. 250. COROLLARY 2. Hence the tangents at the extremity of the axis major are perpendicular to the axis major, DEFINITIONS RELATING TO THE ELLIPSE, CONTINUED. 251. A straight line drawn through the centre, perpendicularly to the axis major, and terminated by the curve, is called the axis minor, or conjugate axis. 252. A third proportional to the axis major and minor, is called thepzra- meter, or the latus rectum of the axes. Thus a and b being the semi-transverse and semi-conjugate axes, 2 : 2b :: 2b :p, the parameter ; therefore, ap = 2b*, or if/= \p, we shall have /"= V, therefore f-- J ~~ b" CONIC SECTIONS. 85 253. That point in the axis, cut by an ordinate, which is equal to half the parameter, is called the focus. In the figure here annexed, JBb, drawn through C, is the semi-axis minor ; and, if LI be a third proportional to Aa, B5, then LI is the parameter, and the point F, where it cuts Aa, is the focus. 254. Any line drawn through the centre, and terminated at each extremity by the curve, is called a diameter. 255. A diameter, which is parallel to a tangent at one extremity of another diameter, is called a conjugate diameter to that other diameter. 256. A straight line, parallel to a tangent, at the extremity of any diameter, terminated at one extremity by that diameter and the eurve at the other, is called an ordinate to that diameter. 257. The portion of a diameter between the centre and an ordinate, is called the abscissa of that ordinate, or of that diameter. In the figure here annexed, the straight line, Aa, drawn through the centre, C, is a diameter ; and, if ST be a tangent at A, and the diameter BA be drawn *r ' parallel to ST, the diameter is called the conjugate diameter of Aa ; and PM, parallel to ST or Bb, is an ordinate to the diameter Aa ; and the distance CP, on the diameter Aa, is called the abscissa. ELLIPSE. THEOREM 2. 258. The square of the axis major is to that of the axis minor as the rectangle contained by the two parts of the axis major, from the ordinate to each vertex, to the square of the ordinate. From the preceding theorem, 7 2 (o a ar z ) =y 2 ( 2 *). Now, let 6 represent the semi-axis minor, and let the ordinate y become b, then will its abscissa, %, become zero, and, consequently, y* (a* x*) =: y* (a 2 a 2 ) will be- come b* (a 1 a?} = ay ; whence a 2 : b* :: a* a? : y*. 11. Y 86 THE NEW PRACTICAL BUILDER, &C. 259. COROLLARY 1. Hence every ellipse has two focii at an equal distance from the centre; because y* = ^ i (a > a?). 260. COROLLARY 2. Hence the tangent at either vertex of the curve is parallel to the ordinates; and, consequently, perpendicular to the axis major. ELLIPSE. THEOREM 3. 261. The square of the axis minor is to that of the axis major as the rectangle contained by the two parts of the axis minor, from the ordinate to the extremity of the axis minor, to the square of the ordinate. For, by theorem 2, ........ a 2 y 2 = A 2 (a 2 -a; 2 ) and, by transposition, ..... & 8 a* therefore, 6* : a* : : & a - y* : a*. 262. COROLLARY 1. Hence the tangent at each extremity of the axis minor is parallel to the ordinates ; and, consequently, parallel to the axis major. 263. COROLLARY 2. Hence the axis major and minor are reciprocally con- jugate diameters. 264. COROLLARY 3. 'If a circle be described on either axis of an ellipse, an ordinate of the circle will be to the corresponding ordinate of the ellipse as the axis of this ordinate is to the other axis. Let PM cut the inscribed circle in N, and let PM be produced to cut the cir- cumscribing circle in N; in both cases let CP = x, PM = y, PN = y, CA = a, and By theorem 2, (258) ........... ....... V (a'-a; 2 ) = 0^, and, by the right-angled triangle, CPN, ........ y t =o i -x i . Therefore, multiplying these two equations, we have b*y l = a*ip > and ex- tracting the root by=ay; wherefore a : b :: y : y. CONIC SECTIONS. 87 265. COROLLARY 4. Hence any two corresponding ordinates of the circle and ellipse are in the same constant ratio of the two axes. ELLIPSE. THEOREM 4. 266. The square of the distance of the focus from the centre is equal to the difference of the squares of the semi-axes. CF 2 =AC a -BC a Let the ordinate FG, which passes through the focus, be denoted byy"; and CF, the distance of the focus from A \ the centre, by e. Then, by the equation of co-ordinates .... 2 y a = A 2 (a 2 a; 2 ) and, since (252) AC : BC :: BC : FG, a 2 / 2 = i 4 . Now, in the first of these two equations, when the abscissa x becomes t, the ordinate y will become f, and, consequently, a 2 y 2 = i 2 ( 2 e 2 ). Whence A 4 = 4 2 ( 2 E 2 ) or J 2 =a 2 -E 2 ; and, by transposition, <= 2 = a 2 & 2 . 267. COROLLARY 1. Hence, because qf=W, therefore o/ r =a 2 2 . 268. COROLLARY 2. Hence i 2 =a a e a = o a -c 2 a 2 = a a (l e 2 X 269. COROLLARY 3. Because a* if = 2 (a 2 - # 2 ) and that i 2 = o 2 (1 - e 2 .) There- fore, by substitution, there will arise y* = (1 -c 2 ) (a 2 -ar*) = 2 -a; 2 -c 2 a 2 270. COROLLARY 4. The semi-conjugate axis CB is a mean proportional between AF, FB, or between Af, fB, the distances of either focus from the two vertices, for J 2 = a 2 - 2 = (a + (a ) ELLIPSE. THEOREM 5. 271. The sum of two lines drawn from the focii, to meet any point in the curve, is equal to the transverse axis. Let FM = R,/M = r, FC =/C = e = ca. Then will FP = CF + CP = ca +, Fp, Kp, and let MT cut K/, in . Then, in the triangles /Ma, KM*, /M being = KM, MX, common, and the included angles f Mas, KM*, equal ; the base /* is = K*, and the angle /*M = K x M. Again, in the triangles fp, Kp, f being = K*, pz common, and the angles fop, Kp, equal, the ba.se fp is = Kp. Therefore Fp +pK = p +pf= FM + M/=FM + MK = FK ; wherefore the sum of the two sides, Fp, pK, of the triangle FjK, is equal to the third FK; which is impossible; therefore as MT cannot cut the ellipse, it must be a tangent. 282. COROLLARY. Hence, because the angle KM/ is bisected, the angle FM/ will also be bisected by the normal MN. For since the angle TM/ is = TMK, and TMK equal to the opposite angle FML ; there- fore TM/ is = FML : and, because the angles TMN and LMN are each a right angle ; therefore, taking away the angles TM/, LMF, which are equal, from the right angles TMN, LMN, there will remain the angle NM/ equal to the angle NMF. ELLIPSE. THEOREM 9. 283. If there be any tangent meeting the four perpendiculars to the transverse axis, the one at the extremity of the transverse, next to the point of intersection, will be to that which terminates in the point of contact, as that which passes through the centre is to the remaining one at the other extre- mity of the tranverse. CONIC SECTIONS. 91 AL:PM::CN:aO. Let AL = It, PM = /, CN = m, and aO = n. Then, if we can show that the distances TA, TP, TC, T, are four proportionals, AL, PM, CN, O, T . being the homologous sides of similar triangles, will likewise be proportionals. By theorem 7, (278,) sx+x ll = a t ; therefore, by transposition, sx+x* a a =o; to each side of this equation add s 2 + sx, and we have ** + 2sx + x*d t = s? + sx; that is, because s* + 2sx + a? is a complete square, and that # 2 + sx = s (s + x) (s+xY a l =s(s+x'), that is, (s + x + a~) (s+x a} = s(s+x). Whence s + xa : s :: s + x : s + x + a. ELLIPSE. THEOREM 10. 284. Every parallelogram, UVWX, circumscribing an ellipse, having its sides parallel to two conjugate diameters, is equal to the rectangle of the two axes. fr If tangents be drawn at the extremities of two conjugate diameters, they form an ellipse, of which CMUI is a fourth part. f^ *\( Draw CK, perpendicular to MT. Let By theorem 7, (278,) ................ and by theorem 7, cor. 1, (279) and, by similar triangles, f CL C (ICH,MTP . vt=nu = i 2 ( 3 -* 2 ) and by theorem 2, (258,) and by similar triangles, TCK, CIH .... j?n*-i?y* and by theorem 7, (278,) .............. 2 a; s = a 4 . For, multiplying the first four equations, a 3 -^ = 2 ; or, by transposition, a* a =ar ! . Multiply this and the three remaining equations and/>w = ai. 285. COROLLARY. Hence a 2 = x* + x\ and b z = y* + y\ 92 THE NEW PRACTICAL BUILDER, &C. ELLIPSE. THEOREM 11. 286. The sum of the squares of any pair of conjugate diameters is equal to sum of the squares of the two axes. Let CM m, CI=. See the preceding figure. By theorem 10, cor. (285) and by Geometry, (160) Therefore, by adding these together, we have a 2 + i 2 =m t + w 2 . (. . . a z =af I. . . 2 =y 2 < N.B. All these theorems concerning the Ellipse, and their demonstrations, are in the very same words as the corresponding number of those for the Hyper- bola, next following ; having sometimes only the word sum changed for the word difference. v ELLIPSE. THEOREM 12. 287. If a chord be bisected, the tangent at the extremity of the diameter, passing through the point of bisection, will be parallel to that chord. Through M, m, draw LN, In, to meet the circumference of the circle described on the axis DE, in the points, N, n. Produce Mm and DE, to meet each other in S, and join Sw, SN, and let the chord Mm be bisected in P. Then, because SI : Im :: SL : LM and Im : In : : LM : LN therefore eliminating LM, Im, SI : In : : SL : LN. Therefore Sn and SN are in a straight line. Through P draw qr, meeting DE in q, and Nw in r ; then Nra is bisected in r. Join Cr, and produce it to meet the circumference in R, and draw the tangent RT, meeting DE produced in T ; then RT is parallel to NS. Draw RG, perpendicular to DE, meeting DE in G, and CA in A. CONIC SECTIONS. 93 Then, because LN : LM : : qr : qP qr : qP :: GR : GA therefore ____ LN : LM : : GR : GA. And, therefore, the point A is in the curve of the ellipse. Then drawing AT, AT is a tangent by the preceding proposition. Now, because CR : Cr : : CT : CS and ........ CR : Cr :: CA : CP therefore ---- CT Therefore AT is parallel to Mm. CS : : CA : CP. ELLIPSE. THEOREM 13. 288. The rectangle of the squares of any semi-diameter, and of an ordinate to it, is equal to the rectangle of the square of the semi-conjugate and the dif- ference of the squares of the semi-diameter of the abscissa, and of the abscissa itself. CA 2 x PM 2 = BC 2 x (CA 2 - CP 2 ). Draw 7-K parallel to PM, cutting MN in K, and draw CG parallel to Nw, cutting the circumference in G, and CB parallel to PM, cutting the ellipse in B, and join BG. LetCR = r, Cr = u, rN = m = v, CA = , CB = i, CP By the equation of the circle ............ * (BCG,KrN. . rV ) y UCR, PCr ...... r*' Multiply the first and second equations together, and r 2 y 2 = 6 2 r 2 #V: or, by transposition, &V =r 2 (5 2 y 2 ). Multiply this and the third equation, and iV = a 2 (& 2 -y 2 ); or, by transposition, y = i 2 (a 2 -^). ELLIPSE. THEOREM 14. 289. The semi-ordinate, together with its prolongation, to meet a tangent at the extremity of a lotus rectum, is equal to the radius vector, through the same focus with that of the latus rectum. 12. 2 A By similar triangles, 94 THE NEW PRACTICAL BUILDER, &C. Let CT = Then will and ---- FM = PQ. = w, and FT = q. CT-CF=a-e = By notation ( M "~ e I w and by theorem 7, cor. 2, ue=a z and by theorem 4, cor. 1, of a? e 2 and by similar triangles, TPQ, TFG,. ... qv =fw. Multiply the first and second of these equations by e, and the results are m e* = eq, and ws = ue + ex. In each of these two equations, for ut substitute its equal a 2 from the second, and we have a 2 f 2 = eq and we = a? + ex. Multiply these two equations, and the remaining two of the given equations, and the final result is av = a? + ex, therefore v = a+^x. But, by theorem 5, we have R=a+~x, therefore R = v. OF THE HYPERBOLA. DEFINITIONS RELATIVE TO THE HYPERBOLA. 290. That portion of the primary line between the vertices of the two opposite curves, is called the transverse axis. 291. A straight line, drawn perpendicularly to the transverse axis, between the transverse axis and the curve, is called an ordinate. 292. The middle of the transverse axis is called the centre of the figure. 293. In the annexed diagram, Aa is the transverse axis, C, the middle of Aa, the centre ; PM the ordinate, and CP the abscissa. HYPERBOLA. THEOREM 1, 294. The squares of the ordinates of the axis are to each other as the rectangles of the two segments of the axis, from each ordinate to each of the two vertices of the opposite curves. CONIC SECTIONS. 95 Let VRQ, passing through the common line of axis of two opposite cones, be a plane perpendi- cular to the cutting plane of the opposite sec- tions, and let AMIPM' be one of the sections, HA the common line of section of the two planes, and let HA cut the two conic surfaces in A, X ; then AX will be the primary axis ; and let OMNM' be a section of the cone parallel to its base, RIQP. Because the plane of the base, RIQP, is per- pendicular to the plane VRQ, the plane OMNM' will also be perpendicular to the plane VRQ ; and, since each of the two planes AMIPM', RIQF, and AMIPM', OMNM', are perpendicular to the plane VRQ, their common sections IP, MM', are perpendicular to the plane VRQ ; therefore IP, MM', are perpendicular to the lines RQ, ON, AH, in the plane VRQ ; and, because the plane VRQ passes through the axis of the cone, it will divide all the circles parallel to the base into two equal parts ; therefore RQ, ON, will be diameters of the two circles : and, since the chords IP, MM', are at right angles to tbe diameters RQ, ON, the chords IP, MM', will be bisected in H and P. Therefore HI = HP, and PM = PM'. Let CA = HQ=v, and Then AP = CP - CA= *- , = CH-CA=*-, Now, by similar triangles, , (XPO, (,... QIRP and, by the circle, ...... < APN, AHQ . . (.r -d)v = NMOM' .... tu = 96 THE NEW PRACTICAL BUILDER, &C. Therefore, eliminating t, u, v, w, by multiplying the given equations, there will result y 2 (.v a) (,v + a~)=y*(z a) (z + a), or, by actual multiplication, y 2 (.f s -a s ) = y 2 (* 2 -a a ); therefore y 2 : tf :: (s-a) (z + a) : Or- a) (.r+a). 295. COROLLARY 1. Hence -rz= '-r, is a constant quantity. za* x- a 296. COROLLARY 2. Hence if z be made constant, y will be constant also. Therefore, when s 2 -a 2 becomes 2 , let y = i, and, consequently $> jcb* 297. COROLLARY 3. Hence every chord, perpendicular to the transverse axis, is bisected by the same axis. 298. COROLLARY 4. Hence the tangent at the extremity of the transverse axis is bisected by the transverse axis. 299. COROLLARY 5. Hence ay^-W (p - a 2 ) = 6V - a*tf. HYPERBOLA. DEFINITIONS, CONTINUED. 300. The constant value b of y, when z* - a 2 becomes a 2 , is called the semi- conjugate axis, or twice b the conjugate axis. 301. A third proportional to the transverse and conjugate axes, is called the parameter, or latus rectum, Thus a and b being the semi-transverse and semi-conjugate axes, 2 : 2b :: 26 :p, the parameter ; therefore, ap = 26 2 , or if/= \p, we shall have of- W, therefore f-* /-a' 302. That point in the axis, cut by an ordinate, which is equal to half the parameter, is called the focus. 303. If there be two opposite hyperbolas, and two others having the same centre, and their common line of axis at right angles to that of the two for- mer, the transverse equal to the conjugate of the two former, and the con- jugate equal to the transverse of the two former ; these four hyperbolas are called conjugate hyperbolas. 304. Any straight line, drawn through the centre, and terminated by opposite curves, is called a diameter. A diameter, which is parallel to a tangent at the extremity of another diameter, is called a conjugate diameter to that other diameter. CONIC SECTIONS. 97 305. A straight line, parallel to a tangent, meet- ing the diameter in one extremity, and the curve in the other, is called an ordinate to that diameter. 306. The distance between the centre and an or- dinate is called the abscissa, In the diagram here annexed, the straight line Aa, drawn through the centre, C, is a diameter ; and, if AT is a tangent at A, then Bb, parallel to AT, pass- ing through the centre C, is the conjugate diameter ; PM parallel to AT, is the ordinate to Aa, and CP is the abscissa. HYPERBOLA. THEOREM 2. 307. The square of the transverse axis is to the square of the conjugate axis as the rectangle of the two distances from the ordinate to the vertex of each curve. CA* : CB 8 : PA x Pa : PM. 5 a?- a' (r a)(i + a) Therefore a 2 : 6* :: OP- a) (.r + a) : y 2 . 308. COROLLARY 1. Hence every pair of oppo- site hyperbolas has two focii at an equal distance from the centre; because y z =^ (a 2 or 2 : and, since the origin of the abscissa commences at the centre, therefore the ordinate y must be the same at the same distance on each side of the centre. 309. COROLLARY 2. Hence the tangent at the vertex of either curve is parallel to the ordinates, and, consequently, perpendicular to the transverse axis. 12, 2 B 98 THE NEW PRACTICAL BUILDER, &C. HYPERBOLA. THEOREM 3. 310. The square of the conjugate axis is to the square of the transverse axis, as the sum of the squares of the semi-conjugate axis ; and that of the ordinate is to the square of the abscissa as CB 2 : CA 2 :: CB 2 + PM 2 : CP 9 . For, (theorem 2,) 2 y 2 = i 2 # 2 o 2 6 2 ; and, therefore, by transposition, 2 (y* + i 2 ) = b*x?, consequently, i a : a 2 : : y*+ i 2 : a?. (See figure, theorem 2.) HYPERBOLA. THEOREM 4. 311. The square of the distance of the focus from the centre is equal to the sum of the squares of the two axes. Let the ordinate FG, which passes through the focus, be denoted by^ and CF, the distance of the focus from the centre, by f. Then, by the equation of the co-ordinates, . . And, since (301) ........................ (?/*= b\ Now, in the first of these equations, when the abscissa x becomes equal to e, the ordinate y will become,/ 1 ; and, consequently, a 2 / 2 = b* (e 2 a 2 ) ; whence b 4 s=i s (e a -a?; or i* = -, and, by transposition, 2 = a* + i 2 . 312. COROLLARY 1. The two semi-axes, and the distance of a focus from the centre, are the sides of a right-angled triangle, ACB, and the hypothenuse, AB, is equal to the distance of the focus from the centre. 313. COROLLARY 2. The conjugate axis, CB, is a mean proportional between AF and Fa, or fa, f A, the distances between either focus and the two vertices ; for i 2 = f 2 a 2 = (t a) (e + a) . 314. COROLLARY 3. Hence, if = ca, then will y* c?x i + c 2 x? c 2 a 2 ; for, (theorem 2,) a 2 ^ 2 = & 2 (a- 2 a 2 ). But, by this theorem, we have 6 2 = f 2 a 2 ; then, multiplying these two equations, we have a z y 2 =(f a # 2 )(# 2 a 2 ); or, substi- tuting ca for f, we shall have y 2 = (c 2 -!) (ar 2 -a 2 ;=a 2 -* 2 + CONIC SECTIONS. 99 HYPERBOLA. THEOREM 5. 315. The difference of two lines, drawn from the focii to meet any point in the curve, is equal to the transverse axis. Let FM = Then will FP = and By Geometry, (prop. 62J. And, by theorem 4, cor. 3, (314) ......... . . f^tf-a? + c 2 * 2 -cV, wherefore, eliminating y, by adding these equations together, we have the equation .................................... R*=e t x t +2acx+a*; then, extracting the roots of each side of this equation, we have R = cx + a = ~x+a. In like manner will be found r = cx a=^x a; therefore R r = 2. HYPERBOLA. THEOREM 6. 316. The line bisecting the angle, at any point in the curve formed by the two lines drawn from that point to each focus, is a tangent. The tangent MT at M will bisect the angle FM/. In MF take MK = M/, and in MT take any point L ; join/L, and let it meet the curve in Q ; join also KL, FL, FQ. Then, by hypothesis, the angle KML=/ML, KM=/M, and LM is common, the base LK is = L c /V but the difference of any two sides of a triangle is less than the third ; * therefore FL - LK, or FL \^f, is less than FK, or F/, or FM -/M, or FQ -/Q. Hence/L is greater than/Q ; for, since FL-/L is less than FQ-/Q, if/L were less than/Q, FL + LQ * It is shown by every writer of Elementary Geometry, that the sum of every two sides of a triangle are greater than the third. Let a, b, c, be the three sides of a triangle; then, a + 6r c, o+er 6, cr a: therefore, by transposition, at c b, ar 6 c, brc a, b< a c, c = 6 a, cr-a b. N.B. r signifies greater than, 100 THE NEW PRACTICAL BUILDER, &C. would be less than FQ ; which is impossible. Therefore every point, L, in MT, except M, is without the curve of the hyperbola ; and MT touches it at M. HYPERBOLA. THEOREM 7. 317. In the line of the axis major, the rectangle contained by the distance between the centre and the intersection of the tangent, and the distance be- tween the centre and the ordinate, is equal to the square of the semi-axis major. Let CT = w. Then will FT = CF + CT = e+M. and ..../T=C/-CT = e-M. By theorem 5, (315,) ~ x+a and by Geometry, (theorem 57,) .... r (e + u) = R ( u). Multiplying these three equations together, we have ux a*. HYPERBOLA. THEOREM 8. 318. The semi-transverse axis a mean proportional between the two distances in the line of the transverse axis ; the one from the centre to the ordinate, and the other from the centre to the intersection of the tangent. For, by the preceding proposition, ux = a 2 , therefore x : a : : a : u. (See figure, theorem 7.) Or, if the subtangent PT = , then will CT= CP- PT = x-s. Whence (xs) x t? or a? sx = o t ; expressed as in the same proposition of the ellipse. HYPERBOLA. THEOREM 9. 319. If there be any tangent, and four perpendiculars to the line of axis, contained between the tangent and the line of axis, the rectangle under two, which passes through the vertices, will be equal to the rectangle of the third, which passes through the centre, and the fourth which is the ordinate. CONIC SECTIONS. 101 GxAI = CHxPM or G : PM : : CH : AI. PT being =*. See the preceding theorem. Then ..... aP = CP-CA = x-a AT = CA + CP-PT =a+x-s T = PT-P= s - x + a. Now, by theorem 8, d?x(x s~) = x* sx. Therefore, by transposition, cf + sx # 2 =O ; add sx x* to each side of this equation, and a 2 - x 2 + 2sx&* = sx s*, or a 2 (x 1 - 2sx +* 2 ) = s(x s); and, since the difference of two squares is equal to the rectangle of the sum and difference of their roots, (a+xs) And, by similar triangles, Whence ............................ AT x T = PT x CT (TCH, TAI . . CT x AI = CH x AT TpM ^ ^ Tp= ^ x pM 6 Therefore, eliminating AT, T, PT, CT, by multiplying these equations and GxAI = CHxPM. HYPERBOLA. PROBLEM. 320. Given the transverse axis of an hyperbola and an ordinate, to find the conjugate axis and assymtotes, (which are two straight lines, such as, if pro- duced indefinitely with the curve, will never meet each other,) and thence to describe the curve itself. Let Aa, (Jig. 1, pi. V,) be the transverse axis, and let PM be an ordinate. Make PD equal to AP. Then on aD describe the semi-circle aND. Produce PM to N. Draw AR perpendicular to CD, and make AR equal to CA. Join NR, and produce NR and DA, if necessary, to meet each other in S ; and draw MS, cutting AR in Q. Produce QA to T, and make AT equal to AQ. Then QT will be the conjugate axis, or AQ, AT, will each be the semi-con- jugate axis. Through the points C, T, draw JH ; and through the points C, Q, draw IK : then JH and IK are the assymtotes, by which the curve may be described. 13. 2c 102 THE NEW PRACTICAL BUILDER, &C. Because CP = #, and CA. = a, AP = PD = x a, and aP equal to x+a; therefore PN is a mean proportional between x + a and x - a, or between aP and PD ; but, in the hyperbola, the transverse axis is to the conjugate as the mean proportion between x+a, and xato the ordinate y or PM ; but AR is divided in Q, in the same ratio as PN is in N. Therefore PN : PM : : AR : AQ. That is, as the mean proportional between x + a, and x a is to the ordinate PM, so is the semi-transverse axis to the semi-conjugate axis. OF THE PARABOLA. DEFINITIONS RELATIVE TO THE PARABOLA. 321. That portion of the primary line which is within the curve, and which is terminated at one extremity by the vertex, is called the axis. 322. A straight line, drawn perpendicularly to the axis, between it and the curve, is called an ordinate. PARABOLA. THEOREM 1. 323. The squares of the ordinates of the axis are to each other as their distances from the vertex. Let VRQ be a plane, passing through the axis of the cone, perpendicular to the cutting plane of the section AMII'M' ; and let AH be their common section; then AH will be the axis. Let QIRI' be a section of the cone parallel to the base : Then, because the base RIQI' is perpendicular to the plane VRQ, the two sections AMI I'M', OMNM', are perpendicular to the plane VRQ; therefore their common sections, MM', II', are also perpendicular to VRQ and to the lines AH, RQ, ON ; but, because the plane VRQ passes through the axis of the cone, it will divide every circle pa- CONIC SECTIONS. 103 raHel to the base into two equal parts ; therefore ON is a diameter of the circle OMNM'; and, because the chords MM', II', are perpendicular to the diameters RQ, ON, they will be bisected ; let H be the point of bisection in IT, and P the point of bisection in MM'. 324. Let KP-x, PM=y, AH=*, HI=y, HQ=r, HR=OP=w, and PN=#. By similar triangles, . . APN, AHQ ...... vz t% and, by the circle, .......... QIR ...... y* = vw and, by the circle, .... ...... NMO ...... tw = y*. By multiplying these equations, the result will be xy t = xy i . Therefore x : % : : y* : 7*. 325. COROLLARY 1. Hence j = r- 326. COROLLARY 2. Hence, because j= p whatever may be the values of x, , y, 7, therefore ^ or j is a constant quantity : hence, putting ^ = \p ; there- fore y 2 = \px. PARABOLA. DEFINITIONS, CONTINUED. 327. The constant quantity p is called the parameter or latus rectum. 328. COROLLARY. Hence the parameter is a third proportional to the distance of the ordinate from the vertex, and the ordinate itself; for 2x : 2y :: 2y : p, orpx^Zif, that is, y* = %px. THE THREE CURVES OF THE CONIC SECTIONS. PROBLEM. 329. The vertical section of a right cone being given, and the position of the axis of a conic section, to describe that section. Let AVB, (fig. 2, pi. V,) be the section of a cone through its axis ; let ig be the line of the axis, and let it cut the section AVB at h, and the opposite side BV, produced, at g. On gh describe the semi-circle hqsg. Draw V/> parallel to AB, cutting the axis in p. Bisect hg in r, and drawer*, per- pendicular to hg. Make pw equal to pV ; then, with the transverse axis, kg, and the ordinate, pw, describe the ellipse hwtg, cutting rs at t; then rt is the semi-conjugate axis. In fig. 3, pi. V, draw the line aA, for the transverse axis, equal to gh,fig. 5; and bisect Aa in C, the centre. Through A draw DE, perpendicular to Aa; 104 THE NEW PRACTICAL BUILDER, &C. make AD and AE each equal to rt, fig. 2. Through C and D draw JH, and through C and E draw IK ; then JH and IK are the assymtotes. Draw any line, az, cutting the assymtote IK at h, and the assymtote JH at g. Make hi equal to ag, and i will be a point in the curve. In the same manner we may find as many more points as we please. Let the axis be cf, fig. 2, cutting the sides of the section AV, BV, at c and,/. Draw cd, ef, parallel to AB, cutting AV at e and BV at d. In fig. 4, draw AB equal to cf, fig. 2. Bisect AB in C. Make CF, C/, each equal to the half of df, or the half of ce ; then, with the transverse axis AB, and focii F,y, describe the ellipse ADBE. Again, in fig. 2, let the axis be mn, and let mn be parallel to the side AV of the vertical section, cutting the base AB at m, and the side BV at . On AB describe the semi-circle AOB, and draw mO perpendicular to AB. In the straight line AA', fig. 5, take any point, D, and make DA, DA' each equal to mO,fig. 2. Draw DC, perpendicular to AA', and make DC equal to inn, fig. 2. Then, with the abscissa AB, and ordinates DA, DA', describe the curve ACA', which will be the parabola. 105 CHAPTER II. CARPENTRY. OARPENTRY is the art of applying timber in the construction of buildings. The CUTTING OF THE TIMBERS, and adapting them to their various situa- tions, so that one of the sides of every timber may be arranged according to some given surface, as indicated in the designs of the architect, requires pro- found skill in geometrical construction. For this purpose it is necessary, not only to be expert in the common problems, generally given in a course of practical geometry, but to have a thorough knowledge of the sections of solids and their coverings. Of these subjects, the first has already been explained in the series of Problems given in the geometrical part of this Work, and we are now about to treat on the other ; that is, the METHOD of COVERING them. As no line can be formed on the edge of a single piece of timber, so as to arrange with a given surface, nor in the intersection of two surfaces, (by work- men called a groin,} without a complete understanding of both, the reader is required not to pass them until the operations are perfectly familiar to his mind. For the more effectually rivetting the principles upon the mind of the student, it is requested that he should model them as he proceeds, and apply the sections and coverings found on the paper to the real sections and surfaces, by bending them around the solid. The SURFACES, which timbers are required to form, are those of cylinders, cylindroids, cones, cuneoids, spheres, ellipsoids, &c., either entire, or as ter- minated by cylinders, cylindroids, cones, and cuneoids. 13. 2 D 106 THE NEW PRACTICAL BUILDER, &C. The FORMATION of ARCHES, GROINS, NICHES, ANGLE-BRACKETS, LUNETTES, ROOFS, &c., depend entirely upon their Sections, or upon their Covering, or upon both. This branch of carpentry, from its being subjected to geometrical rules, and described in schemes or diagrams upon a floor, sufficiently large for all the parts of the operation, has been called DESCRIPTIVE CAPENTRY. In order to prepare the reader's mind for this subject, it will be necessary to point out the figures of the sections, as taken in certain positions. ALL THE SECTIONS OF A CYLINDER, parallel to its base, are circles. All the sections of a cylinder, parallel to its axis, are parallelograms. And, if the axis of the cylinder be perpendicular to its base, all these parallelograms will be rectangles. If a cylinder be entirely cut through the curved surface, and if the section is not a circle, it is an ellipse. ALL THE SECTIONS OF A CONE, parallel to its base, are circles: all the sections of a cone, passing through its vertex, are triangles : all the sections of a cone, which pass entirely through the curved surface, and which are not circles, are ellipses : all the sections of a cone, which are parallel to one of its sides, are denominated parabolas ; and all the sections of a cone, which are parallel to any line within the solid, passing through the vertex are deno- minated hyperbolas. ALL THE SECTIONS OF A SPHERE or GLOBE, made plane, are circles. The solid formed by a SEMI-ELLIPSE, revolving upon one of its axes, is termed an ellipsoid. ALL THE SECTIONS OF AN ELLIPSOID are similar figures : those sections, perpendicular to the fixed axis, are circles; and those parallel thereto are similar to the generating figure. CARPENTRY. 107 OF THE COVERINGS OF SOLIDS. PROBLEM 1. To find the covering of the frustum of a right cone. Let ABCD (fig. 1, pi. VII,) be the generating section of the frustum. On BC describe the semi-circle EEC, and produce the sides BA and CD, of the generating section ABCD, to meet each other in F. From the centre F, with the radius FA, describe the arc AH ; and, from the same centre, F, with the radius FB, describe the arc BG ; divide the arc, BEC, of the semi-circle, into any number of equal parts ; the more, the greater truth will result from the operation; repeat the chord of one of these equal arcs upon the arc BG, as often as the arc BEC contains equal parts ; then, through G, the extremity of the last part, draw GF, cutting the arc AH at H ; then will ABGH be the covering required. PROBLEM 2. To find the covering of the frustum of a right cone, when cut by two con- centric cylindric surfaces, perpendicular to the generating section. Let ABCD (Jig. 2, pi. VII,) be the given section, and AD, BC, the line on which the cylindric surface stands. Find the arc BG, as before, in problem 1, and mark the points, 1, 2, 3, &c., of division, both in the arc BG, and in the semi-circumference ; from the points 1, 2, 3, &c. draw lines to F ; also from the points 1, 2, 3, &c. in the semi-circumference, draw lines perpendicular to BC ; so that each line thus drawn may meet or cut it. From the points of division in BC, draw more lines to F, cutting the arc BC in a, b, c, &c. From the points a, b, c, &c. draw lines parallel to BC, to cut the side BA : from the centre F, through each point of section in BA, describe an arc, cutting the lines drawn from each of the points 1, 2, 3, &c., in BG, at a, b, c> &c. ; then will B cutting mq at n ; draw g3 parallel to mq, cutting the semi-circular arc ABC at 3. Between A and 3 take any number of intermediate points, 1, 2, &c., and, through the points 1, 2, &c., thus assumed, draw \e, If, &c., cutting the seat ag, of the first groin, in the points e,f, &c., and AC in b, c, d, &c. Perpendicular to ag draw eh t fi, &c., and make eh,fi, gk, each equal to bl, c2, d3, &c. ; then, through the points g, h, i, k, draw a curve, which will form the groin belonging to the seat ag. From the points e,f, &c., draw lines et,fs, &c., cutting qm in the points p, o, &c. ; and, through the points q, t, s, r,draw the curve qtsr, which will be one of the ribs of the lunette. GROINS AND ARCHES. PROBLEMS, (fig. 5, d, Be, cutting Ae, in a, b, c, d, e. Again, through the same points 1, 2, 3, 4, B, in the qua- drantal arc AB, draw straight lines lq, 2r, 3s, 4t, BD, perpendicular to AC. From the point A, as a centre, with the several distances A, Ab, Ac, Ac?, Ae, describe the arcs ek, di, ch, bg, of, cutting Ak \nf, g, h, i, Jc. Parallel to the diameter of the greater semi-circle, or parallel to Ae, (Jig. 1, No. 1,) draw fl, gm, hn, to, kp, cutting the greater semi-circular arc in the points /, m, n, o, p. Through the points I, m, n, o,p, draw lq, mr, ns, ot,pD, parallel to AC, cutting the perpendiculars lq, 2r, 3s, M, BD, in the points q, r, s, t, D. Through the points A, q, r, s, t, D, trace a curve by hand, or put in nails at the points A, q, r, s, t, D, and bend a thin slip of wood so as to come in 116 THE NEW PRACTICAL BUILDER, &C. contact with all the nails ; then, by the edge of this slip, which touches the nails, draw a line with a pencil, or find points ; and the curve thus drawn will be half the seat of the rib. The other half, being exactly the reverse, may be found by placing the distances of the ordinates at the same distance from the centre, upon the diameter AC, and setting up the perpendiculars by making them respectively equal to the others. It will perhaps be eligible to make the whole curve ADC at once. The mould for cutting the ends of the boards, which are to cover the centres of the lesser openings, will be found as follows : On any straight line, C5, as on the diameter AC produced, set off the equal partsAl; 2,3; 3,4; 4B; of the quadrant AT?, on the straight line C5, from C to 1, from 1 to 2, from 2 to 3, from 3 to 4, from 4 to 5, and draw the straight lines In, 2v, 3w, 4#, 5y, perpendicular to C5. Make lu, 2v, 3w, 4x, 5y, each respec- tively equal to each of the ordinates comprehended between the base AC, and the seat AD ; then, through all the points C, u, v, w, x, y, draw a curve CwflWtfy, as before; then the shadowed part, of which the curve line Cuvwxy is the edge, is the mould for one side, which may also be made use of for the other. To apply this mould, all the boards should be laid together, edge to edge, on a flat or plane surface, to the breadth C5. Draw a straight line C5, per- pendicular to the edge of the first board, at the distance of 5y from the end. At the distance C5 draw a perpendicular 5y, and set off the distance 5y. Then apply the proper edge of the mould from C to y, as exhibited in the plate, and draw a curve across the boards, and cut their ends off by the line thus drawn ; then the ends, thus formed of the remaining parts, will fit upon the boarding of the greater vault, after being properly bevelled, so as to fit upon the surface of the said boarding. No. 4, of fig. 1, exhibits the curve, in order to draw or discover the line on the boarding of the greater vault, in order to place the boarding of the lesser vault. Nos. 2 and 3, fig, 1, show the method of forming the inner edges of the ribs, so as to range with the small opening. The under edge of the rib must be CARPENTRY. 117 formed so as to correspond to the curve which is its seat ; and the little dis- tances, between the straight line and the curve, must be set off on the short lines, shown at Nos. 1, 2, and 3 ; then a curve may be drawn through the points of extension, and the superfluous wood taken away ; then, the rib being put in its real place, the angle will exactly fall over its seat. The diagram, figure 1, and its different numbers, answer both the purposes of a centering and of ribbing for plaster-ceilings. Figure 2, pi. IX, exhibits the method of forming the Cradelling, or ribs, for plaster-ceilings of cylindro-cylindric arches. Here principal ribs only are used across the piers. The ribs of double curvature, which form the groins, though here exhibited, in order to fix the ribs, may be done without, by men of experience : but young workmen require every assistance, in order to acquire a comprehensive idea of the subject; it is, therefore, proper to show how the groined ribs are to be found. The other ribs, for lathing upon, are made of straight pieces of quartering, fixed equidistantly. Figure 3, pi. IX, is a plan in which common groins and cylindro-cylindric arches both occur. See the gate-way leading from the Strand, in London, into the court of Somerset-house. GROINS AND ARCHES. PROBLEM 7. (fig. 4, pi. IX.) To find the seats of the intersections of groins formed by the intersection of an annular and a radial vault, both being at the same height, the section of the annular vault being a semi-circle, and that of the radiating vault a semi- circle of the same dimensions, the plan being given. Perpendicular to the middle line, or axis, of the radial vault, draw a straight line from any point of that middle line ; from the point thus drawn, set the radius of the circle of the annular vault ; from the point of extension draw a line, parallel to the axis of the radiating vault, to meet the side of the plan. From the point of meeting draw a straight line, perpendicular to the axis, to meet the other side of the plan of that radiating vault : on the perpendicular thus drawn, between the two sides, as a diameter, describe a semi-circle : divide each quadrantal arc of this semi-circle, and each quadrantal arc of the 15. 2 G 118 THE NEW PRACTICAL BUILDER, &C. semi-circle which is the section of the annular vault, into the same number of equal parts. Draw lines through the points of division in each arc, perpen- dicular to the base or diameter, to meet the said diameter. Through the points of section in the diameter of the annular vault, and from the point of concourse of the two sides of the radiating vault, describe arcs. From the same point of concourse, and through the points of section of the diameter of the semi-circle, which is the section of the radiating vault, draw lines from the point of concourse of the two sides of the radiating vault. Then, through the intersection of these lines, and the arcs drawn from the points of section in the diameter of the semi-circle, which is the section of the annular vault, trace a curve, which will be the seat of the groin. The method of fixing the timber is exhibited at the other end of the figure. The ribs of both the an- nular vault and the radiating vault are all fixed in right sections of these vaults, as must appear evident from what has been shown. NAKED FLOORING. FLOORS are those partitions in houses that divide one story from another. FLOORS are executed in various ways : some are supported by single pieces of timber, upon which boards for walking upon are nailed. Floors of this simple construction are called single-joisted floors, or single floors; the pieces of timber, which support the boards, being called joists. It is, however, cus- tomary to call every piece of timber, under the boarding of a floor, used either for supporting the boards or ceiling, by the name of joists, excepting large beams of timber into which the smaller timbers are framed. When the supporting timbers of a floor are formed by one row laid upon another, the upper row are called bridging joists, and the lower row are called binding joists. Sometimes a row of timbers is fixed into the binding joists, either by mortises and tenons, or by placing them underneath, and nailing them up to the binding-joists : these timbers are called ceiling-joists, and are used for the purpose of lathing upon, in order to sustain the plaster-ceiling. ' CARPEHTRY. 119 In forming the naked flooring, over rooms of very large dimensions, it is found necessary to introduce large strong timbers, in order to shorten the bear- ing of the binding-joists ; such strong timbers are called girders, and are made with mortises, in order to receive the tenons at the ends of the binding-joists, which, by this mean, are greatly stiffened, being much shorter. The bridging-joists are frequently notched down on the binding-joists, in order to render the whole work more steady. Figure 1, pi. X, is the plan of a naked floor ; b, b, b, &c. are the binding- t joists ; a, a, a, &c. are the bridging-joists ; d, a timber close upon the stair-case. This piece of timber is called a trimmer : its use is to receive and secure the ends of the joists, e, e, e, &c. upon the landing. C, C, C, &c. are wall-plates, upon which the ends of the binding-joists rest. In the construction of floors, great care must be taken that no timber come near to a chimney ; therefore, the ends of the timbers, as here shown, have no connection with the fire-place, nor with the flues. The flues, in this plan, are indicated by their being shadowed darker than the other parts of the plan. Figure 2, pi. X, is the plan of naked flooring with a girder. Figure 3, shows the manner of framing joists into a girder, with the form of the mortise and tenon. No. 1, is the part of a joist framed into the girder ; and No. 2, is a joist out of the mortise. Figure 4, shows the connection of binding-joists, bridging-joists, and ceiling-joists ; as, also, the manner of fixing the binding-joists upon the wall- plates, which manner is called cocking, or cogging. The long dark parts represent the mortises, into which one end of the ceiling-joists are fixed. These long mortises are called pulley-mortises, or chase-mortises. The ceiling-joists are introduced into common mortises at one end, and the other end of them are let into these long mortises obliquely, and slide along until they are perpendicular. Figure 5, shows how the bridging-joist is let down upon the binding-joist, and how the ceiling-joists are fitted into the binding-joists. 120 THE NEW PRACTICAL BUILDER, &C. TUMBLING IN A JOIST, is to frame a joist between two timbers, of which the sides, which ought to be vertical or square to the upper edges, are oblique to these edges. Figure 6, shows the method of fitting in a joist between the sloping sides of two others. The first thing done is, to turn the upper edge of the joist upon the top of the two pieces into which it is to be fitted, and brought over its proper place. The next thing is to turn the joist on its under edge, so as to lie over its place; then apply a rule, or straight edge, upon the side of the one piece where the shoulder of the joist is intended to come ; then slide the joist until the line drawn come to the straight edge of the rule so applied ; then draw a line by the edge of the rule. Do the same at the other end, and the two lines thus drawn will mark the shoulder of the tenon at each end. LENGTHENING TIMBERS. TIMBERS may be lengthened in various ways, either by making the piece of timber in two or more thicknesses ; or by securing one piece to another, with a piece on each side, in order to cover the joint; and by spiking or bolting each piece on both sides of the joint. Sometimes the pieces that are applied on the sides are made of wood ; in this case, it is called fishing the beam : such modes are used in ships, when their masts, beams, or yards, are broken, in order to mend them. Other modes of continuing the length of timbers or beams is, by splicing them with a long bevel-joint, ending in a sharp edge at the end of each piece. Sometimes the sharp edge of the end of each piece is cut off, so as to form an obtuse angle at the top. Sometimes the splice is so formed as that the two surfaces which come into contact are reciprocally indented into each other, which will add greatly to their security, when firmly bolted together. Every kind of scarf should have a strong iron-strap upon each opposite side, extending in length considerably beyond each joint. CARPENTRY. 121 Figure 1, pi. X, shows the manner of building a beam in three thicknesses ; which, being strapped with iron across every joint, and bolted, will be ex- ceedingly strong and firm. Figure 8 exhibits the method of joining timbers by two tables and a key. Figure 9, the method of lengthening timber by a plain scarf, being cut only with an obtuse angle at the ends. Figure 10, the same kind of scarf, with two tables and a key. ^ Timbers that are scarfed and strapped ought to be so applied, that the sides which are strapped should be the horizontal sides ; for, if otherwise applied, they will be liable to split at the bolting. But, if the surfaces of the joints are to be placed in a vertical position, there ought to be two straps upon the top and two upon the bottom ; each strap being brought close to the vertical face. By this method it will be much stronger than when set in the other position, or with the joint of the scarf horizontal. THE ROOFING. THE ROOF is that part of a building raised upon the walls, and extending over all the parts of the interior, in order to protect its contents from depreda- tion, and from the severities and changes of the weather. The ROOF, in CARPENTRY, consists of the timber-work which is found ne- cessary for the support of the external covering. The most simple form of roofs is that consisting of a level plane ; but this description of roofs is adapted only to short bearings, and is not at all calcu- lated to resist or prevent the torrents of rain or moisture from penetrating into the interior. The next simple form is that which consists of an inclined plane; and, though well calculated to resist the injuries of the weather, and to afford greater strength than a level disposition of the timbers would supply, it is far from admitting of the utmost strength that a given quantity of timber is 15. 2 H 122 THE NEW PRACTICAL BUILDER, &C. capable of affording : it occasions an inequality, and a want of uniformity and correspondence in the proportions of the fabric, and an unnecessary and unpleasant height of walling. The best figure for a roof is that which con- sists of two equal sides equally inclined to the horizon, terminating in the summit, over the middle of the edifice, in a horizontal line, or the ridge of the roof, as it is called ; so that the section made by a plane, perpendicular to the ridge, is every where an isosceles triangle, the vertical angle of which is the top of the roof. This form is very advantageous, as it regards saving of timber ; for it may be executed with the same scantlings, to span double the distance that the simple sloping roof admits ; or, in buildings of the same dimensions, the scantlings of the timbers will be very much diminished. The antient Egyptians, Babylonians, and other eastern nations of the re- motest antiquity, constructed their roofs flat, as do likewise the present in- habitants of these countries. The antient Greeks, though favoured with a mild climate, yet sometimes liable to rain, soon found the inconvenience of a platform covering for their houses, and accordingly raised the roof in the middle, declining towards each side of the building, by a gentle inclination to the horizon, forming an angle of from 13 to 15 degrees, or the perpendi- cular height of from one-eighth to one-ninth of the span. In Italy, where the climate is still more liable to rain, the antient Romans constructed their roofs from one-fifth to two-ninth parts of the span. In Germany, where the severities of the climate are still more intense than in Italy, the antient inhabitants, as we are informed by Vitruvius, made their roofs of a very high pitch. When the pointed style of architecture \vas in- troduced into Europe, high pitched roofs were thought consonant with its principles ; and they therefore formed, externally, one of the most striking characteristics of the Gothic style. In their usual proportions, the rafters were equal to the breadth or span of the roof, or the rafters were the sides of an equilateral triangle, of which the spanning line Avas the base. During the middle ages this form prevailed, not only in public but in private buildings, from the most stately and sumptuous mansion down to CAEPENTRY. 123 the humble cottage of the common labourer; and this equilateral tri- angular roof continued in request until, finally, the pointed style came into disuse. When the celebrated INIGO JONES introduced Roman architecture, the rafters were made three-quarters of the breadth of the building ; and this pro- portion, which was called true pitch, still prevails in some parts of the country where plain tiles are used ; subsequently, also, the square seems to have been considered as true pitch : but, in large mansions, constructed in the Italian style, roofs of & pediment pitch, covered with lead, were introduced. In the present day, where good slates are to be obtained in abundance roofs may be covered with them, from the pyramidal equilateral Gothic down to the gently inclined Greek pediment. With regard to the present practice, the proportion of the roof depends on the style of the architecture of the edifice ; the usual height varying from one-third to one-fourth part of the span. There are, doubtless, some advantages in high-pitched roofs, as they dis- charge the rain with greater rapidity ; the snow does not lodge so long on their surface ; also, they may be covered with smaller slates, and even with less care, and are not so liable to be stripped by high winds as the low roofs are : but the low roofs bear less weighty on the walls, and are considerably cheaper, since they require shorter timbers, and of course, smaller scantlings. The roof is one of the principal ties to a building, when executed with judgment, as it connects the exterior walls, and binds them together as one mass; and, besides the protection it affords the inhabitant within, it pre- serves the whole work from a state of decay, which would soon inevitably ensue, from the violence of rain or frost, which would operate in a way of rotting the timbers, of destroying the connexion in the walls, and would cause them ultimately to fall. The several timbers of a roof are, principal rafters, tie-beams, king-posts, queen-posts, struts, collar-beams, straining-sills, pole-plates, purlins, ridge- piece, common-rafters, and camber-beams. The uses of these will appear from the following description of them. 124 THE NEW PRACTICAL BUILDER, &C. The usual EXTERNAL FORM of a ROOF has two surfaces, which generally rise from opposite walls, with the same inclination ; and, as the walls are most commonly built parallel to each other, the section of the roof made by a plane perpendicular to the horizon, and to one of the walls, is an isosceles triangle ; the base being the extension from the one wall-head to the other. This extension is called the span of the roof. To FRAME TIMBERS, so that their external surfaces shall keep this position, is the business of trussing ; and the ingenuity of the carpenter is displayed in making the strongest roof with a given quantity of timbers. All beams, or pieces of timber, from their weight, when supported at the two ends only, acquire a concavity on the upper side ; and this concavity is the greater as the distance between the props is the greater. It is, therefore, a grand object to prevent this bending as much as possible. The curvature, will take place whether the position of beams be horizontal or inclined ; but the same beam will have less curvature, as the angle, to which it is inclined to the horizon, is greater. For, it is evident that, Avhen a beam is laid level, and supported at its extremities, its curvature will be greater than when in- clined at any angle, however small ; and, again, if it stand perpendicular to the horizon, its curvature will be nothing ; that is to say, its curvature will be nothing when the angle of inclination is the greatest. The curvature which timber obtains by bending is called sagging. To prevent timber from sagging, as much as is possible, it must be supported at a certain number of intermediate points or places, besides the two extreme- points or places. Now these supports must themselves be supported from some base or other ; but, if the resting points or places is upon the surface or surfaces of other timbers, the greatest care must be taken that they do not fall between the extremities of the supporting timbers intended to sup- port the other. That is to say, the lower end of every piece of timber, used as a prop, must rest upon some fixed point; or, otherwise, the propping piece of timber must be so disposed that the pressing forces at each end must be equal to each other. These are the general principles upon which the strength of roofs depend. CARPENTRY. 125 A FIXED POINT or PLACE, in Carpentry, is the point or place where two timbers are joined together ; and no roof or piece of framing is good, where the end of one piece, used as a prop to another, presses upon a third piece of timber, supporting that prop, between its extremities. The pressing end of every prop ought to rest upon some fixed point. PRINCIPAL RAFTERS are the two pieces of timber, in a framed roof, that form the two equal sides under the covering. It is evident that the greater the opening is, the more supports each prin- cipal rafter will require. A piece of timber may be supported either from some fixed point above it, or some fixed point below it : if the connecting support be above the piece to be supported, then it is evident that it will act as a string ; but, if below the piece to be supported, it is evident that the prop must be an inflexible bar or piece of wood. Now, therefore, as the principal rafters cannot be supported from above, they must be supported from below ; the props must, therefore, be made of some stiff material, as iron or timber, of sufficient thickness ; and, because the pressure occasioned by the weight of the covering is uniformly distributed over the principal rafters, these principal rafters must be supported at a certain number of equidistant points, which will depend upon the distance of the walls. A TIE-BEAM is a piece of timber connecting the feet of the principal rafters, in order to prevent them from spreading by the weight of the cover- ing. The tie-beam is therefore used as a string, and is in a state of tension. In order to prevent the sagging of the tie-beam, in very wide houses, it must be supported in one or more places in its length ; and, if it cannot be supported from the ground, it must be supported from some fixed point or points in the roof itself. The only fixed point is where the two principals meet each other ; but this one point will furnish as many supports to the tie- beam as we please ; then every one of these points in the tie-beam SKQ fixed points : from each of these fixed points, in the tie-beam, the principal rafters may be supported ; or in as many points, or from any convenient fixed point, 16. 2 i 126 THE NEW PRACTICAL BUILDER, &C. in the tie-beam, the principal rafter above may be supported, in as many points in its length as we please. No direct rule can be given for the disposition and position of supporting timbers : the best way to judge of this is, such a disposition as will make the connecting timbers as short as possible, and the angles as direct as possible. Oblique or acute angles occasion very great strains at the joints, and should therefore be avoided. One grand principle to be obtained, in every frame or roof, is, to resolve the whole frame into the least number of triangles, which must be considered as the elements of framing. Quadrilateral figures must be avoided, if possible; and this may be done by introducing a diagonal, which will resolve it into two triangles ; for, without this, a four-sided figure will be moveable round its angles. Sometimes it may be necessary to resolve a quadrangular piece of framing into four triangles, by means of two diagonal pieces, particularly when this figure occurs in the middle of a roof. The principles of framing being once understood, a little practice in de- signing will soon enable the artizan to judge of the proper disposition of tim- bers, so as to produce a good design, if not the best possible. A KING-POST, or PRINCIPAL POST, is a vertical piece of timber, extend- ing from the meeting of the two principal rafters to the tie-beam, for the pur- pose of supporting the tie-beam in the middle. The KING-POST is, therefore, in a state of tension ; and, consequently, a string, which will not lengthen, may be used as a tie-beam ; and this string may be a slender bar of wrought iron, or any tenacious material, that will not be liable to greater extension when stretched or drawn in length. The principal rafters are frequently supported from one or more points in the king-post : but it is evident that they must be supported exactly in the same manner when the supporting points in the king-post are between its two extremities; so that the principal rafters may produce equal and opposite pressures, on each side of the king-post. Each of the principal rafters may be supported in as many points as we please, either from one point in the king-post, or from as many points as the number of points to be supported ; or, as has been said before, either from one CARPENTRY. 127 fixed point, or from as many fixed points, in the tie-beam, as the number of points to be supported ; and, in short, the principal rafters may be supported from any fixed point whatever, from the king-post, or tie-beam, or from both. This very circumstance points out a vast variety of designs for roofs. QUEEN-POSTS are two pieces of timber, equidistant from the middle of the truss ; the one suspended from the head of one of the principal rafters, and the other from the head of the other, with a level piece of timber between them. The QUEEN-POSTS, therefore, divide the internal space of the frame into three compartments, of which the two extreme ones are right-angled triangles, and the middle pne a rectangle. The use of the queen-posts is similar to that of the king-posts ; vix. for fur- nishing a general support for the principals, at different points between the ends, by connecting timbers, and supporting the tie-beam between its ex- tremities. STRUTS are those props which support the principal rafters, in one or more points, so as to divide them into equidistant parts. STRUTS are generally disposed in pairs, equally inclined to the vertical line, which divides the truss into two equal and similar parts ; and which, there- fore, divides the two beams into two equal lengths. Struts are necessary in roofs where the span is great ; and the greater the span or distance of the walls, the greater the number of struts will be required ; for, in this case, more points in the principals will have to be supported. A COLLAR-BEAM is the piece of timber framed between two queen- posts, A collar-beam, roof is necessary where the roof is to have a platform, or flat for walking upon, or wherever rooms are required in the roof. A STRAINING-SILL is a horizontal piece of timber, disposed between the feet of the queen-posts, to counteract the efforts of the struts, in pushing the principals nearer to each other Having thus noticed the several parts of a truss, it may be proper to observe that all king-posts, queen-posts, and tie-beams, are ties; and, therefore, a 128 THE NEW PRACTICAL BUILDER, &C. string incapable of farther extension than is sufficient to bring it to a straight line. A chain, or a slender bar of iron, will answer the same purpose as well as a piece of timber, or other such inflexible material. It is also to be re- marked that all collar-beams, principal rafters, and struts, are straining pieces ; which are, therefore, necessarily constructed of an inflexible material, such as wood or a stiff piece of iron. It may be further observed that, in complex frames, such as centering to large arches or bridges, in the act of building, the same timbers, in different stages of the work, sometimes perform the office of ties, and sometimes that of straining pieces ; and, in the transition of office, must be sometimes in a neutral state. The material employed in such situation must, necessarily, be inflexible. This is to be recommended not only here, but in every doubtful case, or where it is uncertain whether the part of the truss requires to be a tie or a straining piece. A POLE-PLATE is a beam over each opposite wall, supported upon the ends of the tie-beam, or upon the feet of the principal rafters. PURLINS are horizontal pieces of timber, supported by the principal rafters. A RIDGE-PIECE is a beam at the apex of a roof, supported by the king- post, or by the heads of the principals. COMMON RAFTERS are inclined pieces of timber, parallel to the prin- cipal rafters, supported by the pole-plates ; the purlins and the ridge-piece, for supporting the covering, the material of which is sometimes large slates, extended from rafter to rafter, and sometimes smaller slates, nailed upon boarding, battening, or hung to lath, and these nailed upon the common rafters. JOGGLES are the joints at the meeting of struts, king-posts, queen-posts, and principal rafters ; and, indeed, all the joints of a roof may be termed joggles. The best form of the joggles is that which is at right angles to the lengths of the struts or rafters, or at right angles to the tenoned piece : but this position cannot, at all times, be obtained, from the want of sufficient sub- stance of timber : in this case, the joint is made either oblique, or the upper part in a line with the side of the piece which has the mortise, and the lower CARPENTRY. 129 part perpendicular to the sides of the tenoned piece : or the joint is sometimes made partly parallel, and partly perpendicular, to the mortised piece. When the joint is oblique, the force of the tenoned piece, in the direction of its length, causes the end to slide upon the abutment, towards the side which contains the obtuse angle : but this is, in some degree, counteracted by the resistance of the tenon on the lower end of the mortise. However, with regard to the stress of the timbers in a frame, the abutting joint is of little importance. M. Perronet, the celebrated French engineer, formed the abut- ments, and, consequently, the joggles, in the arches of circles, making the centre at the other extremity. CAMBER-BEAMS are those timbers which are supported upon purlins over the collar-beams, and support the boarding for a leaden platform ; for this purpose they have an equal declivity from the middle, in order to cause the water from rain or snow to descend equally on each side of the roof. COCKING, OR COGGING, is the form of the joints, which the tie-beams and wall-plates make with each other; and here, as in every other case of jointing, the parts must be reciprocally reversed to each other, so that the protuberant part or parts of the one are the raised or indented parts of the other. The parts which come in contact are plane surfaces ; those which form the bottom or bottoms of the recess or recesses are parallel to, and those which form the sides perpendicular to, the surfaces, from which the jambs are made. The best method is, by cutting a groove, across the fibres, in the beam to be let down, to correspond to a rising in the plate formed by recessing the plate, on each side of the rising. Another method is by an external and internal dovetail : but this method is almost antiquated. PARTICULAR OBSERVATIONS ON ROOFS. GABLE-ENDED ROOFS, unless properly supported by ties, are liable to thrust out the walls, and particularly when the walls are very thin, or the distance between them very great. A HIPED ROOF, over a rectangular plan, when the jack-rafters are well secured to the principals, produces less laternal pressure 16. 2 K 130 THE NEW PRACTICAL BUILDER, &C. on the walls than a gable-ended one upon the same plan. A hiped roof, upon a square plan, produces less lateral pressure upon the walls than an oblong of the same area. With regard to roofs executed upon regular poly- gonal plans of the same area, that which has the greater number of sides produces less lateral pressure on the walls than that which has fewer sides ; and, when the number of sides are very many, the polygon may be con- sidered as a circle; and, consequently a circular roof will give less lateral pressure upon the walls than a polygonal one of any number of sides, how- ever great. In the execution of all these, however, it will.be necessary that a wall-plate should be formed to the plan, and well connected, not only between each of the angular points, but also at the angular points themselves; for, when the building is carried up on a polygonal plan, every two adjacent hips will act in the same manner as the two principals in a framed roof; and, therefore, every side of the wall-plate will be a tie ; and, consequently, if not properly joined, the walls will be liable to be rent. COVERING OF CIRCULAR ROOFS. CIRCULAR ROOFS may be covered upon two different principles ; one is, by supposing the axal section to be divided into a number of small equal parts, and the roof cut by planes through the points of division, parallel to the base; and, by considering the frustums of the solid as so many frustums of a cone, the covering of each respective part will be found. The other principle is by dividing the circumference of the base into a number of small equal parts, and supposing axal sections to be made through the points of division ; then, by considering the surface of each axal portion as the surface of a cylinder, the covering will be found. The distance between the points of division, in the former case, must be less than the breadth of the boards which are to form the envelopes of the covering, in order to make the convex edge of the board : this distance must, therefore, be less, as the length of the boards is greater. In the latter case, the distances between the points of division may be exactly CARPENTRY. 131 equal to the breadth of the boards. It is true that the surface of each part is spherical or convex, and therefore can neither be considered as the frustum of a cone, nor that of a cylinder : but, if the distance between the divisions be small, the surfaces will be almost straight in all the axal sections : so that there will be no practical difference, even though the widest boards be used in moderate-sized works. The boards which thus form the envelopes must be thin, in order that they may comply with the surface of the circular roof to be covered. It is here proper to notice that, when boards are bent, so as to form a surface either concave or convex, they are much stronger than if the surface were a plane, even though the ribs were at the same distance in both: but, in order to make the boards bend regularly and truly, the ribs ought to be disposed at a nearer distance, even at the widest place, which is at the bottom, than the rafters of a common roof. When the ribs are disposed in axal planes, they will come in contact with each other at the top, unless they terminate upon a circular kirb, of a diameter sufficient to prevent their doing so ; but as the intervals at the top are always much less than at the bottom, the ribs are sometimes discontinued, in order to reduce the intervals nearer to an equality of breadth throughout the length of each : the execution in this way will save the timber- work, and, consequently, lessen the expenses. Sometimes the ribbing of circular roofs consists of only several principal axal ribs, and the intervals filled in with jack-ribs ; which, if the surface to be covered be spherical, are portions of less circles of the sphere, and are disposed in pa- rallel vertical planes. PROBLEM 1. To find the bevels for cutting the various timbers in a hiped roof, and the backing of the hips (pi. XI, figures 1, 2, 3). Let ABCD, figures 1, 2, and 3, be the outlines of the wall-plates, AF, DF, and BE, CE, the seats of the hips, and EF the seat of the ridge-piece. To find the length of any rafter, draw a line from the one extremity of the seat of that rafter, perpendicular to that seat, and make the height of the perpendicular equal to the height of the roof; join the point of extension 132 THE NEW PRACTICAL BUILDER, &C. and the other extremity of the seat, and the line thus joined is the length of the rafter as required. Example 1 ; figure 1. Find the length of the common rafters standing upon IK ; divide IK into two equal parts in the point F: draw FL perpendi- cular to IK ; make FL equal to the height of the roof, and join IL or KL ; then IL or KL is the length of each rafter. Example 2 ; fig. 2. Find the length of the hip-rafter standing upon AF. Draw FS, perpendicular to AF : make FS equal to the height of the roof; join AS, and AS is the length of the hip. PROBLEM 2. To find the bevels of a purlin upon a hip-rafter, giving the seat of a com- mon rafter, and the seat of the hip-rafter, and the angle which the common rafter makes with its seat. Place the section-purlin in its real position with respect to the common rafter. Produce that side of the section of the purlin, of which the bevel is required, upon the hip toward the seat ; from one extremity of the line thus produced, and, with the length of the said line as a radius, describe a circle. Draw three lines, parallel to the wall-plate, to meet the hiped line : viz. one from the centre of the circle, one from the point where the line meets the circle, and the third a tangent to the circle. From the point in the seat of the hip-rafter, where the middle line meets the said seat, draw a line perpen- dicular to that middle line, to meet the tangent ; join the point where this perpendicular meets the tangent to the point where the line drawn from the centre meets the seat of the hip-rafter, and the angle formed by the line thus joining, and the line drawn from the centre of the circle, will be the bevel of the purlin. Example; pi. XI, fig. 1. AF being the seat of a hip-rafter, IF that of a common rafter, and FIL the angle which the common rafter makes with its seat, and abed the section of the purlin. Now suppose it were required to find the bevel of that side of the purlin represented by ad. CARPENTRY. 133 Produce ad to any point,,/; and from a, with the radius of, describe a circle, fgh. Parallel to the adjacent wall-plate, AB, draw three lines to cut the seat, AF, of the hip ; viz., from the centre, a, draw at ; and from the pointy, where of meets the circle, draw/A, the former cutting the seat in i, and the latter in k. Draw kl perpendicular tofk, and draw the tangent el, cutting kl in /; and join il; then lia is the angle required. In the same manner, by producing ab, we shall find the angle formed upon the end of the side, of which its section is ab. In order that the different inclined planes, which form the sides of a roof, should have an equal inclination to the horizon, the seats of the hip-rafter s ought to bisect the angles of the wall-plates. When a roof is wider at one end than at the other, in order, as in Jig. 3, to prevent its winding, let IK and OP be the seats of the two common rafters, passing through each extremity of the ridge-piece, and let the rafters IL and KL be found as before ; divide OP into two equal parts, in E ; draw ER per- pendicular to OP. Make the angle EPR equal to the angle FKL ; then ER will be the height of the roof upon the seat OP. If this should be objected to, because it makes the ridge higher at one end than at the other, let E, fig. 4, be the end of the seat of the ridge next to the narrow end of the roof. Bisect all the four angles of the roof by the straight lines AF, BE, CE, DF; and, through E, draw EG, parallel to AB, cutting AF in G; and draw EH, parallel to CD, cutting DF in H ; and join GH : then GH will be parallel to AD. This is true, because, since all the angles are bisected, if we imagine perpendiculars drawn from E to the three sides, the three straight lines thus drawn will be equal : and because EG is paraUel to AB, the perpendiculars drawn from the points E and G, to the straight line AB, are equal ; from the the same reason, because EH is parallel to CD, the perpendiculars drawn from the points E and H, to the straight line CD, are equal ; therefore the perpendicular drawn from the point G, to the straight line AB, is equal to the perpendicular drawn from H to the straight line CD. And, since the angles BAD and CDA are bisected by the straight lines AG and DH, the two 17. 2 L 134 THE NEW PRACTICAL BUILDER, &C. perpendiculars, drawn from G, to the sides AB and AD, are equal ; as also the two perpendiculars from the point H to the sides DA and DC : but the perpendicular drawn from G, to the side AB, is equal to the perpendicular drawn from H to the side CD ; therefore the perpendiculars, drawn from the points G and H, to the straight line AD, are equal to each other ; but when the perpendiculars drawn between two straight lines are equal, these two straight lines are parallel : therefore the straight line GH is parallel to AD. Whence, if all the angles of a roof be bisected, and if any point be taken in any one of the bisecting lines, and if a line be drawn through the point thus assumed, parallel to one of the adjacent sides, to meet the next bisecting line, and so on from one to another, till only one line remains to be drawn ; then, if the point assumed be joined to the point where the parallel meets the last bisecting line, the line thus joining will be parallel. OF NICHES. NICHES are ornamental recesses formed in walls, in order to enshrine some ornament, as a statue, or elegant vase. They are often constructed in thick walls, in order to save materials in masonry, or brick-work. Niches are sometimes constructed of ribs of timber, and lathed and coated over with plaster, which forms the surface to be exhibited. 4 The plans or bases of niches are always some symmetrical figure; as a rect- angle, a segment of a circle, or one of an ellipse. All the sections of niches, parallel to the base, are similar figures ; and all the sections parallel to the base, to a certain distance, are equal. Niches sometimes terminate upwards in a plane surface, and sometimes in an ellip- soidal surface ; but most frequently in the portion of a spherical surface : so that, as the faces of walls are generally perpendicular to the horizon, the aperture in the face is either a rectangle, or a rectangle terminating in the CARPENTRY. 135 segment of a circle, or in the segment of an ellipse. Two of the sides of the rectangle are perpendicular to the horizon. Niches are always constructed in a symmetrical form; vix. if a vertical plane be supposed to pass through the middle point of the breadth, perpendicular to the surface of the wall, it will divide the niche into two equal and similar parts ; or, if any two points be taken in the breadth, equi-distant from the sides of the niche, and if two vertical planes be supposed to pass through these points, perpendicular to the surface of the wall, the sections of the niche will be equal and similar. Niches are placed either equi-distantly, in a straight wall, or round a cy- lindrical wall, dividing the circumference into equal parts : sometimes they are placed in an elliptic wall. In the latter case, however, they ought not to divide the circumference into equal parts, but to be at an equal distance from each extremity of the major axis. Niches are frequently constructed in poly- gonal rooms ; a niche being placed in the middle of each side of the prismatic cavity. The opposite sides of such rooms are always equal and similar rect- angles. The plans are either hexagonal or octagonal ; but, most frequently, of the latter form. THE PRINCIPLES of FORMING the RIBS, for the head* of spherical niches, are drawn from the following considerations : All the sections of a sphere, made by a plane, are circles ; therefore the edges of the ribs to be lathed ought to be portions of circles. The ribs of niches may be placed either in vertical planes, or in horizontal planes ; and, indeed, in any manner, so as to form the spherical surface as required: it will be most convenient, however, to dispose the ribs either in vertical planes, or in planes parallel to the horizon, as the case may require. One of the most easy considerations for the ribs of a niche, when placed in vertical planes, is to suppose them to pass through a common line of intersec- tion ; and, if this line passes through the axis of the sphere, the ribs will be all equal portions of the circumference of a great circle of the sphere : and will, in consequence, be very easily executed. In this case, the square edges of 136 THE NEW PRACTICAL BUILDER, &C. the ribs will range, or form the surface of the niche. This position of the ribs is therefore very convenient for forming them, as it not only requires less time to execute them, but much less wood will be required. There is another position of vertical ribs, which is frequently convenient ; that is, by placing the ribs in cqui-distant planes, perpendicular to the surface of the wall ; and, consequently, when the surface of the wall is a plane, the planes of the ribs will be all parallel. Figure 1, in pi. XII, exhibits the plan and elevation of a niche ; the ribs are disposed in vertical planes, which intersect in the centre of the sphere. The plan, No. 1, is the segment of a circle ; and, in consequence of this, the back ribs are of different lengths, and will therefore meet the front rib in different places, as shown in the elevation, No. 2. For, if the plan had been a semi-circle, all the back ribs would have necessarily met the front rib in the middle of its circumference. Numbers 3, 4, 5, 6, (fig. 1,) exhibit the ribs as cut to their proper lengths, according to the plan, No. 1. Thus, let it be required to find the rib standing upon the plan BCED, of which the sides BD and CE are equi-distant from the line that passes through the centre A. In No. 6 draw the straight line ad, in which make ac, ab, ad, equal to AC, AB, AD, No. 1 : in No. 6, from the point a, as a centre, describe an arc of a circle ; from the points b, c, draw two straight lines, perpendicular to ad, cutting the arc ; then the portion of the arc, intercepted between the point d and the perpendicular drawn from the point b, is the arris line next to the front, and the part intercepted between the point d and the perpen- dicular is the arc forming the arris line next to the back; so that the extremities of the perpendiculars drawn from b and c, give the extremities of the joint against or upon the front rib. As to the form of the back edges of the ribs, they may be curved or formed in straight portions. In this manner all the other ribs will be formed ; as is evident from the preceding explanation. Figure 2, pi. XII, exhibits the plan and elevation of a niche, with the method of describing the ribs when they are disposed in parallel planes. No. 1 is the plan, No. 2 the elevation, and Nos. 3 and 4 the method of CARPENTRY. 137 drawing the ribs. The lengths of the bases of the ribs, in Nos. 3 and 4, are taken from the plan No. 1 ; as AK, AI, AH, AF, AE, AC, AB, are respectively equal to ED, ac, ab, of, ae, ai, ah, in the base No. 1. The two distances which approach near to each other show the quantity of bevelling. With these distances, from the centre A, No. 3, describe as many semi-circles as there are points ; then the double lines will represent the quantity of bevelling, or the distance from the square edge. No. 4 shows one of the ribs alone by itself. To DRAW THE RIBS OF A SPHERICAL NICHE, in a circular wall. Plate XIII, Nos. 1,2,3, 4, 5. Let No. 1 be the plan of the niche, and that of the wall Abed, &c. the base line of the circular wall ; ABCD the base line of the spherical niche ; and A, B, C, D, &c. the bases of the ribs, of which the sides are all supposed to stand in a vertical plane ; and a plane, passing through the middle of the thickness of each rib, parallel to the sides of that rib, is supposed to pass through the centre of the sphere ; and, therefore, the bases of these planes will pass through the point E, which is the projection of the centre of the sphere, on the horizontal plane, where the cylindrical and spherical surfaces meet each other ; and this we may suppose to be the plane of the paper. Now, since all sections of the sphere are circles, all the edges of the ribs of the niche will be circular ; but, because all the circles pass through the centre of the sphere, the edges of the ribs of the niche must be all segments of great circles of the sphere; and, therefore, they must all be described with one radius, which is equal to that of the arc A, B, C, D, &c., and, conse- quently, with the radius EA, EB, EC, ED, &c., as at No. 3, No. 4, No. 5, &c.; therefore, from F, G, H, as centres, describe the arcs DN, CM, BK; and draw FD, GC, HB. Produce FD to S, GC to Q, and HB to O. In the radius FD, No. 3, make J?d equal to Ed, No. 1, and draw dT perpendicular to FD, cutting the arc DN at N : then DN will be the under edge of the rib which stands upon dD, its plan. In No. 4, upon the radius CG, make Ch, Cc, equal to the plan of each side of the rib which stands upon C, No. 1 ; and, in No. 4, draw cR, h\, cutting the arc CM at L and M. 17. 2 M 138 THE NEW PRACTICAL BUILDER, &C. In like manner, in No. 5, make Be, B4, each equal to the side of the rib B, in the plan, No. 1 ; and, in No. 4, draw bP, eU, perpendicular to BH, cutting the arc BK in I and K. Then the backs of these ribs may either be the arcs ST, QR, OP, or may have any outline whatever ; but, for the convenience of what will be presently shown, in the fixing of the ribs, it will be proper to make them all circular arcs of one radius, which will make them suffi- ciently strong, Then IPKU is the representation of the top of the rib, which top coincides with the face of the wall, and, consequently, the distance between the lines LV, MR, is the quantity which this rib, now under description, must be bevelled. In like manner, IKPU is the represen- tation of the upper end of the rib which stands upon its plan B, and where it falls in the surface of the wall. The back of the ribs being made circular, and of one radius, they will all coincide with another spherical surface : if, therefore, the back ribs are fixed at their bases, the inner edges will be brought to the spherical surface, by fixing a rib, whose inner concavity has the same radius as the backs of the ribs, upon the backs of these ribs ; so that the plane, passing through the middle of its thickness, parallel to its sides, may pass through the centre of the sphere. In other respects, the plane of this fixing rib may have any other position whatever, besides what has now been described. BRACKETING FOR COVES AND CORNICES. COVE-BRACKETING is the finish of the top of the faces of a room, adjacent to the cornice, and consists, generally, of the concave surface of a cylinder ; though it may be, occasionally, that of a cylindroid ; and the latter surface will, in many cases, have the appearance of greater ease than the surface of a cylinder. All the vertical sections of coved ceilings, perpendicular to the wall, are equal and similar figures, alike situated to the surface of the wall, and equi- distant from the floor. CARPENTRY. 139 The CORNICE of a room has the same properties ; that is, its vertical sec- tions, perpendicular to the surface of the wall, are equal and similar figures ; and their corresponding parts are equi-distant from the wall, and also from the floor. As the coves and cornices of rooms are generally executed in plaster, when they are large, in order to save the materials, the plaster is supported upon lath, which is fastened to wooden brackets, and these again to bond-timbers, or plugs in the wall : for this purpose the brackets are equi-distantly placed, and are about three-quarters of an inch within the line of the cornice ; and, in order to support the lath at the mitres, brackets are also fixed in the angles. In fig. 1, pi. XIV, ABCD is the plan of the faces of the walls of a room. The plan of the bracketing is here disposed internally, and the angle brackets are placed at B and C. In fig. 2, ABCD is the plan of part of one side and the chimney-breast ; and here, on account of the projection, we have one internal angle and one external angle. We may here observe, that the angle bracket of the external angle is parallel to that of the internal angle. Figure 3 exhibits a bracket upon a re-entering obtuse angle. In fig. 4, ABCDEF is part of the section of a room ; CD is the ceiling line ; CB and DE are the sections of the coves ; BA and EF are portions of the wall-lines. Figure 5 shows the construction of a cove-bracket at a right angle. Let AC be the projection of the cove, and let Aa be part of the wall-line : make A equal to AC, and join aC ; on the base AC describe the bracket AB, which is here the quadrant of a circle. In the arc AB take any number of points, d, e, f, Sac., and from these points draw lines parallel to A ; that is, perpen- dicular to AC, cutting both AC and aC in as many points ; from the points of section in aC draw lines perpendicular to C, and make the lengths of the perpendiculars respectively equal to those contained between the base AC and the curve AB; and, through the points of extension, draw a curve; and the curve, thus drawn, will form the cove in the angle, as required to be done. 140 THE NEW PRACTICAL BUILDER, &C. . Figure 6, exhibits the construction of a bracket for an external obtuse angle, AaK being the wall-line. Figure 7, exhibits the construction of a bracket for an external acute angle. Figure 8, exhibits the section of a large cornice, where the lines within the mouldings form the bracket required. Figure 9, shows the construction of the angle-bracket for a cornice in a right angle. To form the bracket in the obtuse or acute angle, take any pointy* (figures 6 and 7,) in the given cove, and draw FA parallel to A, cutting the base AC of the given bracket in g, and the base ac of the angle-bracket in h : draw hi perpendicular to ac, and make hi equal to gf; then will i be a point in the curve. In the same manner we may obtain as many points as we please. This description also applies to the construction of an angle-bracket of a cornice ; the only thing to observe with regard to this is, to make all the con- structive lines pass through the angular points in the edge of the common bracket. In the construction of angle-brackets, it will be the best method to get them out in two halves, and so range each half to its corresponding side of the room ; and, when they are ranged, we have only to nail them together. METHODS OF BOARDING CIRCULAR ROOFS. WITH regard to the boarding of roofs for slates, there are two principles ; first, it is evident that, if a round solid be cut by two planes, each parallel to the base, the portion of the surface of the solid, between these planes, will nearly coincide with a conic surface, contained between sections perpendicular to the axis of the cone, of the same diameter each as those made by cutting the round solid ; therefore the whole of the round solid may be looked upon as so many conic frustums, lying upon one another ; therefore, to cover all the conic frustums is to cover the round solid. CARPENTRY. The other method of covering a round solid is to suppose the base divided into equal parts, and the solid to be cut by planes passing through the points of division, and through the fixed axis ; then the surface of the body will be divided into as many equal and similar parts ; so that, if any one of these por- tions of the solid be covered, the cover will, of course, fit any other portion thus divided ; and, as all the horizontal sections of each portion of the solid is the sector of a circle, the chords of all the sectors will be parallel to each other ; therefore the curved surface will be nearly prismatic. This, therefore, affords another method of forming the boarding. The first of these methods is called the horizontal method, and the second the vertical method, of covering a dome. In fig. 1, pi. XV, let ABC be a vertical section of a circular dome, through its axis ; and let it be required to cover this dome horizontally ; bisect the base, AC, in the point H, and draw HI perpendicular to AC, cutting the semi-circumference in B. Divide the arc BC into such a number of equal parts that each part may be less than the breadth of a board ; that is to say, allowing the boards to be of a certain length, each part may be of the proper width, allowing for waste : Then if, between the points of division, we sup- pose the small arcs to be straight lines, as they will differ very little from them, and if horizontal lines be drawn through the points of division, to meet the opposite side of the circumference, the trapezoids will be the sections of so many frustums of a cone, and the straight line HI will be the common axis for every one of these frustums. Now, therefore, to describe any board, which shall correspond to the sur- face of which one of the parts, ab, is the section, produce ab to meet HI in c; then, with the radii cb, ca, describe two arcs ; then radiating the end to the centre, the lines thus drawn will form the board required. In the same manner any other board may be found ; as is evident from the principle described. To FIND THE FORMS of the BOARDS for covering an ANNULAR VAULT (pL XV, fig. 2). Let AD be the outer diameter of the annulus, CG the inner, E the centre, and AC the thickness of the ring. 18. 2 N 142 THE NEW PRACTICAL BUILDER, &C. On AC describe the semi-circle ABC : then, if ABC be supposed to be set or turned perpendicular to the plane of the paper, it will represent half the section of the ring. From E, with the radius EA, describe the semi-circle AFD ; and, from the same centre, E, with the radius EC, describe the semi- circle CHG; then AFD is the outer circumference, and CHG the inner cir- cumference ; and, consequently, AFDGHCA, is the section of the ring, per- pendicular to the fixed axis, and the section ABC of the solid itself is perpen- dicular to the section AFDGHC. To find the form of any board ; divide the circumference ABC of the semi-circle into such a number of equal parts as the boards or planks out of which they are to be cut will admit. Let ab be the distance between two adjacent points ; through the centre E draw HI, perpendicular to AD ; and through the points a and I, draw the straight line ac, meeting HI in the point c: from c, with the radius ca, de- scribe an arc ; and from the same centre, c, with the radius cb, describe another arc, and enclose the space by a radiating line at each end ; and the figure bounded by the two arcs, and the radiating lines, will be the form of the board required. In the same manner the form of every remaining board may be found. It is obvious that, as common boards are not more than nine or ten inches in breadth, the boards formed for the covering cannot be very long; or other- wise they must be very narrow, which will produce much waste. To COVER an ELLIPSOIDAL DOME, the major axis o/ the generating ellipse being the fixed axis (pi. XV, fig. 3). Let ABC be the section through the fixed axis, or generating ellipse, which will also be the vertical section of such a solid. Produce the fixed axis AC to I, and divide the curve ABC into such a number of equal parts that each may be equal to the proper width of a board. Then, as before, draw a straight line through two adjacent points and b, to meet the line AI in c ; then, with the radii ca and cb, describe arcs, and terminate the board at its proper length. No. 2, '(fig. 3,) is an horizontal section of the dome, exhibiting the plan of the boarding. CARPENTRY. 143 Figure 4 is a section of a circular roof. The principle of covering it with boards bent horizontally, is exactly the same as in the preceding examples. It is now necessary only to explain ONE GENERAL PRINCIPLE, which extends to the whole of these round solids. The planes which contain the conic frustums are all perpendicular to the fixed axis, which is represented by HC, in all the figures. Produce ab, to meet the fixed axis HI in c ; then, with the radius ca, describe an arc ; and, with the radius cb, describe another arc, which two arcs will form the edges of the boards ; the ends are formed by radiating lines. Now, which ever figure we inspect, we shall find this rule to apply. As the boards approach nearer to the revolving axis, they may be made either wider or longer ; but, as the boards approach nearer to the fixed axis, the waste of stuff will be greater, and, consequently, the boards must be shorter. r |v When the boards come very near to the bottom of the dome, the centres for describing the edges of the boards will be too remote for the length of a rod to be used as a radius. In this case we must have recourse to the following method. Let ABC, (fig. 1, pi. XVI,) be the section of the dome, as before, and let e be the point in the middle of the breadth of a board : draw ed pa- rallel to AC, the base of the section, cutting the axis of the dome in g, and join Ae, cutting the axis inf. Then, by problem 10, page 64, describe the segment of a circle, through the three points d,f, e, and this will give the curve of the edge of the board, as required. Figure 1, No. 2, exhibits the manner of using the instrument. Thus, sup- pose we make DE equal to de, No. 1 : Bisect DE in G, and draw GF, perpen- dicular to DE, and make GF equal to gf, No. 1. Draw FH parallel to DE, and make FH equal to FE, and join EH ; then cut a piece of board into the form of the triangle HFE ; then let HFE be that triangle ; then move the vertex F from F to E, keeping the leg FE upon the point E ; and the leg F, and the angular point F of the piece, so cut, will describe the curve, or per- haps as much of it as may be wanted. It must be here observed that the line described is the middle of the board ; but, if the breadth of the board is properly set off at each end, on each side of 144 THE NEW PRACTICAL BUILDER, &C. the middle, we shall be able to describe the arc with the same triangle ; or, if the concave edge of the board is hollowed out, the convex edge will be found by gauging the board off to its breadth. As all the conic sections approach nearer and nearer to circles, as they are taken nearer to the vertex ; a parabola, whose abscissa is small, compared to its double ordinate, will have its curvature nearly uniform, and will, conse- quently, coincide very nearly with the segment of a circle ; and, as this curve is easily described, we shall here employ it instead of a circular arc, as in Nos. 3 and 4. Draw the chord DE, as before, and bisect it in G. Draw GF perpendicular to DE, and make GF equal to gf, in No. 1 : so far the construction of the diagrams Nos. 3 and 4, are the same ; but, in what follows, they are different: we shall, therefore, take each of them separately, and first No. 3. Divide each half, DG, GE, into the same number of equal parts; and, through the points of division, draw lines perpendicular to DE ; also, from the points D and E, at the extremities, draw perpendiculars ; and make each of these perpendiculars equal to GF ; then divide each into as many equal parts as DG or GE is divided into, and, through the points of division, draw lines to F, intersecting the perpendiculars ; and, through the points of inter- section, draw a curve, on each side of the middle point F, and this will be the form of the edge of the board, nearly. In No. 4, make FH equal to gf, No. 1, and join DH and HE. Then divide DH and HE, each into the same number of equal parts ; then, through the corresponding points of division, draw straight lines, and the intersection of all the lines will form the curve sufficiently near for the purpose. The lines thus drawn being tangents to the parabolic curve. The arc of a circle may, however, be accurately drawn through points, by the following method : Let DE, (fig. 1, No. 5,) be the chord of the segment, and GF the versed sine. Through F draw HF, parallel to DE ; join DF, and draw DH perpen- dicular to DF. Divide DG and HF each into the same number of equal parts, as five, in this example ; draw DI perpendicular to DG, meeting HF in CARPENTRY. 145 in I ; and divide DI into the same number of parts as DG : viz. five. Join the points of division in DG to those in HF, and also through the points of division in DI draw straight lines to the point F, cutting the former straight lines, drawn through the points of division in the lines DG and HF : then trace a curve from the point D, and through the points of intersection to F, and we shall have one half of the circular arc. The other half is found in the same manner, as is obvious from inspection of the figure. The PRECEDING METHOD of covering round solids requires all the boards to be of different curvatures, and continually quicker as they approach nearer to the crown ; but, by the following method of covering a dome, with the joints in vertical planes, when the form of one of the moulds is obtained, this form will serve for moulding the whole solid. The waste of stuff in this case is not less than in the other. The method which we are about to explain is not only useful in the forma- tion of the boards of a DOME, but in the covering of NICHES. In figures 2, 3, 4, ( pi. XVI,) No. 1 is the plan, and No. 2 the elevation ; the contour of the latter being a vertical section passing through the axis. Figure 2 represents a dome, whose contour is a semi-circle ; figure 3 repre- sents a segmental dome ; figure 4 represents a round body, of which the ver- tical section is an ogee, or curve of contrary flexure. Through the centre of the plan, G, draw the diameter, AC ; and the dia- meter BD, at right angles to AC ; and produce BD to E. Let BD, figures 2 and 3, be the base of a semi-section of the dome : on BD apply the semi- section CFD ; and as the dome, represented by figure 2, is semi-circular, the point F will coincide with the point A in the circumference of the plan. In figures 2 and 3 divide the curve FD, of the rib, into any number of equal parts, and extend the curve DF upon the straight line DE, from D to E ; that is, make the straight line DE equal in length to the curve DF. Through the points of division, in the curve DA, draw lines perpendicular to DG, cutting it at the points a,b,c: then, extending the parts of the arc between the points of division upon the line DE, from D to 1, from 1 to 2, from 2 to 3, &c. : make 18. 2 o 146 THE NEW PRACTICAL BUILDER, &C. Dd equal to half the breadth of a board, and join dG ; produce the lines la, 2b, 3c, &c., drawn through the curve DF, to meet the line dG, in the points d, e,f, &c. Through the points 1, 2, 3, &c., in DE, draw perpendiculars Ig, 2h, 3i, &c. : make Ig, 2h, 3i, &c., respectively equal to ad, be, cf, &c. ; and, through the points d, g, h, i, &c., E, draw a curve, which will form one edge of the board. The other edge, being similar, we have only to describe a curve equal and similar, so as to have all its ordinates respectively equal from the same straight line DE. In Jig. 4, the form of the mould for the boards is found in a similar manner, except that the curve DE is one side of the elevation, No. 2 : Lines are drawn from the points of division in DE perpendicular to the diameter AC, which is parallel to the base of No. 2 ; and the points of division are transferred from the radius GC, to the radius GD, which is the base of the section. The re- maining part of the process is the same as in figures 1 and 2. In figure 2, the curved edge of the board is a symmetrical figure of the sines ; the curves of the mould, fig. 3, is a smaller portion of the figure of the same curve : and, in jig. 4, the mould is a curve of contrary flexure ; and if the curve DE be composed of two arcs of circles, the curve of the edges of the mould for the boards will still be compounded of the figure of the sines set on contrary sides ; and, if the curve DE be compounded of two elliptic seg- ments, the edges of the mould for the formation of the boards will still be of the same species of curve : vis. the figure of the sines. This figure occurs very frequently in the geometry of building. COVERINGS OF POLYGONAL ROOFS. The plans of these roofs are here supposed to be regular polygons, and all the sections parallel to the base, similar to the base, and, consequently, similar to one another. They are made of prismatic solids, meeting each other in planes perpendicular to the plane of the base ; and these mitre-planes meet each other in one common axis, which passes through the centre of each polygon. CARPENTRY. 147 In pi. XVII, fig. J, the plan is denoted by the letters ABCDEFA. Then the centre of the polygon being the point I, draw the lines AI, BI, CI, &c. Bisect any of the sides, as AB, in the point L, and draw LI ; then LI is per- pendicular to AB. Produce the line IL to M, and let ILN be the section applied upon IL. In the curve LN take any number of points 1, 2, 3, at equal distances, and transfer these distances to the line LM, so that LM may be equal to the arc LN. Through the points 1, 2, 3, &c. in LM, draw lines Ig, 2A, 3i, &c. parallel to AB ; and through the points 1, 2, 3, &c., in the arc LN, draw lines Id, 2e, 3f, &c., also parallel to AB, cutting LI at the points a, b, c, &c., and BI at the points d, e,f, &c. : Make Ig, equal to ad, 2h equal to be, 3i equal to cf, &c. Through the points g, h, i, &c., draw a curve, which will be the edge of the joint over the mitre. .To find the angle-rib, through the points d, e,f, &c., draw dk, el,fm, &c. perpendicular to BI. Make dk, el,fm, &c., respectively equal to al, 62, c3, &c. Through the points It, I, m, &c., draw a curve, which will be the edge of the angle-rib, as required. Figure 2 shows the manner of describing a polygon, to any given number of sides. Thus suppose, upon the side AB, it were required to describe a heptagon. Produce BA to K, and, with the radius AB, describe a semi-circle, BGK, of which the diameter is BK ; divide the arc BK into seven equal parts, and through the second division, G, draw AG ; then BA and AG are two adjacent sides of the heptagon. Bisect each of the sides AG and AB by a perpendicular, meeting each other at I. Then I is the centre of a circle that will contain either of the sides AB or AG seven times. The equal chords, being inscribed in the remaining part of the circle, will complete the polygon as required. In this manner we may describe a polygon of any given number of sides whatever ; by producing the given side, and describing a semi-circle on that side, and the part produced, and dividing the arc into as many equal parts as the polygon is to contain sides ; then, drawing a line from the centre, through the second point of division, will form two adjacent sides of that po- lygon. The remaining part of the process is to be completed as before. 148 THE NEW PRACTICAL BUILDER, &C. Figure 3, (pi XVII,) shows the manner of finding the covering of a roof, when the plan is a regular pentagon. Figure 4, exhibits the method of framing the ribs for such sorts of roofs. Figure 5, shows the manner of describing the covering and ribs of a domical roof. Figure 6, shows the manner of describing the covering and ribs of a roof whose vertical section is a figure of contrary curvature. Figure 7, shows the method of describing a regular octagon from a given square. Thus, draw the diagonals ; then, with half the diagonal, as a radius, and from each of the four angular points of the square, describe a quadrant or arc ; join the two adjacent points of intersection, in each two adjacent sides of the square, and you have the octagon required. Figure 8, exhibits the manner of forming one of the ribs for the ogee roof, or that of contrary curvature. The method of finding the coverings and ribs of figures 3, 5, and 6, is the very same as that described in figure 1. Such forms of roofs most frequently occur in temples or garden-buildings. POLYGONAL ROOFS and CIRCULAR DOMES are of the same nature ; as a dome cannot be covered upon any other principle than that of a polygon having a finite number of sides. t . PENDENTIVE BRACKETING. t ,A PENDENTIVE BRACKETING occurs when certain portions of a con- cave surface are carried from the sides of a rectangular or polygonal room to a level ceiling or cornice. The parts thus introduced, between the walls and the ceiling, are called PENDENTIVES. Pendentives are either spherical, spheroidal, or conical. The figure in the walls, from which they spring, depends entirely upon the following principles: ; CARPENTRY. 149 It is well known that, if a sphere be cut by a plane, the section will be a circle ; and, if a hemisphere be cut by a plane perpendicular to its base, the section will be a semi-circle. If a right cone be cut by a plane, perpendicular to its base, the section will be an hyperbola ; and, generally, if any conoid, formed by the revolution of a conic section about its axis, be cut by a plane perpendicular to its base, the section will always be similar to the section of the solid passing through the axis ; and every two sections of a conoid, cut by a plane perpendicular to the base, at an equal distance from the axis, are equal and similar figures. Therefore if, on the base of a hemisphere, we inscribe a square within the containing circle, and cut the solid by planes perpendicular to the base, through each of the four sides of the square, the four sections will represent the four portions of each wall, and the arcs will represent the spring- ing lines for the spherical surfaces. Qn pi. XVIII, Jig. 1, No. 1 is the plan of a room, with the ribs which form the pendentive ceiling ; the semi-circles on the sides are supposed to turn up perpendicular to the plan bnmo, which will form the terminations of the four walls ; No. 2 is the elevation. Numbers 3, 4, 5, 6, and 7, exhibit the ribs for one eighth part of -the whole ; and, as these ribs are all in planes passing through the axis, they are all great circles of a sphere, of which the diagonals of the square is a diameter ; therefore, though the ribs are shorter in the middle of each side, and increased towards the angles, they are all described with the same radius, which is half the diagonal of that square. The whole of the scheme may be formed in paste-board. Thus, in figure 2, let ABCD be the plan ; on each of the sides, AB, BC, CD, DA, describe a semi-circle ; then let each semi-circle be turned round its respective diameter until its plane becomes perpendicular to the plane ABCD ; then the sides, thus turned up, will represent the sections of the sphere, and ABCD the base of the solid ; then the surface extending between the semi-circular arcs is entirely spherical. In figure 3, the pendentives are supposed to be placed on a conic surface, and the sides of the square not perpendicular, but equally inclined on every side, approaching nearer together as they ascend. 19. 2p 150 THE NEW PRACTICAL BUILDER, &C. Thus, let ABCD be the plan, and the circumscribing circle the base of the cone, and EOF a section of the cone through its axis. Then, if the inclination of each of these four planes be the angle EHI, making HI parallel to FG, then the conic section is a parabola, and may be drawn as shown at fig. 3, No. 2, and as described in the Practical Geometry of this Work. Figure 4, {pi. XVIII,) shows the method of describing the springing lines, when the sides are perpendicular to the plane ABCD. From the centre of the square, and through the angular points, describe the circle ABCD, and draw the diameter EF, parallel to any one of the sides, cutting AD and BC in c and H. In Ee take any number of points, a, b, &c., and draw ad, be, cf, per- pendicular to EF, cutting the side EG, of the section of the cone, in the points d, e,f, &c. From the centre of the plan describe the arcs ci, bh, ag, cutting the side DC in g and h, and the arc ci touching it in a. Perpendicular to DC, draw im, Td, gh, and make im. Id, gh, respectively equal to cf, be, ad. Then, upon the given base, DC, describe the symmetrical figure, Dmc, which will form the springing line, in order to set the ribs upon the wall. As this figure is an hyperbola, it may be described, independently of tracing it from the plan, thus: In fig. 4, No. 1, draw HK perpendicular to EF, cutting the side GF of the cone in I, and meeting the other side EG, produced in K, and IK will be the axis, IH the abscissa, and HC or HB the ordinate : then describe an hyperbola, fig. 4, No. 2, which has its axis, ab- scissa, and ordinate, respectively equal to IK, IH, HB, or HC. Figure \, (pi. XIX,) is the elevation of conical pendentives. In order to form the conic surface, the figure of an hyperbola must be described upon each side of the room. The figure in the plate exhibits two sides of the room. In this diagram aglib represents the springing line on one side of the room, and bhc that on the other side; the former agreeable to the straight line AB on the plan, No. 1, and the latter agreeable to the straight line BC on that plan. Figure 2 is a section and angular elevation of spherical pendentives ; the plan being exhibited by No. 1. CARPENTRY. 151 Figure 3 shows the method of drawing the springing lines on the walls ; the plan and the rib over the diagonal of the plan being given agreeably to the elevation, fig. 2. Here the plan is the square ABCD, and the rib over the diagonal of the square is DEFB. From the centre V, with a radius equal to half the side of the square, describe the arc gPG, which will touch the two sides DC, CB, of the square, at P and G, which are each in the middle of these sides ; and let the arc, thus described, cut the line BD atg-. Draw gx, perpendicular to DB, cutting the curve DE at h. lalleifiq ,'i3. laiaracifi orfJ wBtb Let QG, RI, SL, TN, UC, be the seats of the ribs for one-eighth part of the whole ; and since these are similar to those in every other eighth part, their formation will be sufficient for the whole of the ribs ; since there will be four ribs, for every one of those in the eighth part, exactly alike, so that each rib becomes a mould for three more. The plans QG, RI, SL, TN, UC, divide any arc described from the centre, V, into four equal parts, and termi- nate upon the side BC of the square, in the points G, I, L, N. Draw GH, IK, LM, NO, perpendicular to BC ; also draw iy, 1%, nfy, perpendicular to DB, cut- ting the under edge D E of the rib over the diagonal in the points k, m, o. Make GH, IK, LM, NO, each respectively equal to gh, ik, Im, no ; then the curve HKMOC being drawn, will be half the springing line over BC ; the other half, being made similar, will be the whole of the springing line. This springing line will serve as a mould for drawing the springing lines upon each of the four walls. As all the ribs are portions of a circle of the same radius, that is, they will have the same curvature as the edge DE of the rib which stands upon the diagonal ; the portion of each rib will be Dh, D&, Dm, Do, cut by the lines hx, ky, mz, 08?. Qofigure 4 shows the springing lines for each wall, agreeably to the plan and elevation,^. 1. The method is exactly the same as that described for fig. 3 ; and thus any farther description will not be necessary. i ; a^&asfc on* iioitiwB c &i S ^wtgi'i sJidulxti gaiod mefu 152 THE NEW PRACTICAL BUILDER, &C. afl ?4tsd, per- pendicular to BD, cutting BD at the points a, b, c, d. Then, upon AC, as an axis-major, and upon E#, Ei, EC, Ed, as so many axes-minor, describe the semi-ellipses, AaC, A.bC, AcC, AdC, which will represent the joints of the boards upon one side of the dome. Now, since all the sections of this dome, through the line AC, are identical figures, the vertical section, upon the line AC, will be identical to the half plan ABC, or ADC. Divide, therefore, BA into any number of equal parts, by the points of division e,f, g, h, i, It, I; the more, the truer the operation. Draw the straight lines em,fn, go, hp, iq, kr, Is, perpendicular to AC, cutting AC at the points m, n,o,p, q, r,s, and the semi-ellipse AdC, in the points t, u, v, w, x, y, z. On the straight line, GH, No. 2, set off the equal parts, Em, mn, no, &c., from each side of the centre E, each equal to one of the equal parts Be, ef,fg, &c., in the semi-elliptic curve, ABC, in the plan No. 1. Through the points m, n, o,p, &c., No. 2, draw tt, uu, vv, Sac., perpendicular to GH. Make mt, mt, each equal to mt in the plan No. 1 ; and nu, nu, No. 2, each equal to nu in the plan No. 1 ; then, through all the points t, u, v, &c., draw a curve on each side of the line AC, to reach from A to C, and each curve will be the edge of a board. No. 3 shows' the longitudinal elevation; viz. on the line AC of the plan. No. 4 exhibits the transverse elevation, the contour being identical to that of the section on the line AB. CARPENTRY. 155 di &aob kfiioeqilfo MB 10^ ebtBod 9ili Id nhol orfl baft oT obo^a Isorhrev arfo ba ^~ DESIGNS FOR PARTITIONS. H ,(IOSA Ltiw 1 moil sita9\ , i 9f fr ^*? .^ nc PARTITIONS, in Carpentry, are the ribs of timber used for sustaining the lath and plaster. It is evident that all single pieces of timber, when supported only at each extremity, will descend more and more towards the middle, and will obtain a curvature : but, if supported from any fixed points, will prevent that deflexion rr r m-3iX from the straight line. Figure 1, pi. XXIII, is a design for a TRUSS PARTITION, with a door in the middle. In order to keep the timbers from descending, two braces are in- troduced, one on each side of the door-way, and the weight is discharged at each extremity of the sill. The two struts, which support the middle of these braces, are supported at the lower extremities on the bottom of the door-posts. Now the door-posts cannot descend, without pressing down the braces, and the braces cannot descend without forcing down the extreme post; but, as each end of the foot-beam, or sill, is supported, the extreme posts cannot descend ; therefore the two braces cannot descend, and the posts on each side of the door-way cannot descend ; consequently, the timbers will keep straight. But the weight of the quarters will still have a tendency to bind the braces : in order to prevent this effectually, the parabolic arch is here introduced. Figure 2, pi. XXIII, is a design for a partition with two door-ways, one of them being a folding door. Here the braces on each side of the large opening not being "each supported at each extremity of the sill, and as the space is not interrupted by openings, a complete truss is introduced above the two apertures, particularly as there is sufficient height for the action of the braces. 156 THE NEW PRACTICAL BUILDER, &C. DESIGNS FOR ROOFS. Figure 1, pi. XXIV, is a design for a roof, of a very narrow span, having only one collar-beam, without a tie at the bottom. In this example the collar- beam acts as a tie, and is, therefore, in a state of tension : this ought not to be employed over a space exceeding fifteen or twenty feet wide. Figure 2 is a design for a roof with a tie-beam at the foot, and a collar- beam. Here the case is different ; since the tie-beam is in a state of tension ; the collar-beam is merely employed in keeping out the rafters, and is, there- fore, in a state of compression. This truss may be employed where the span is from twenty to twenty-eight feet in width. If the two sides of this truss are equally loaded, it will remain stationary ; but, if unequally, the equi- librium will be destroyed. This truss, without additional timbers, does not afford any support to the tie-beam. Figure 3 is a truss free from these inconveniences ; the tie-beam being supported by two struts or braces. Figure 4, No. 1, mpL XXIV, represents the side of a truss, with the ends of a longitudinal frame for supporting the tops of the rafters, which are here exhibited. No. 2 exhibits the frame as seen in the length of the roof, and is divided into several compartments, by means of a middle and two side posts. The ends of this longitudinal truss are fixed in the gables or cross-walls. Figure 5, No. 1, is a design for a truss, with a king-post and two queen- posts. Here the manner in which the tie-beam is supported upon the wall- plates is shown. The sections of the pole-plates and purlins are also exhi- bited. No. 2, of this figure, shews the manner of notching down the small rafter upon the purlins, and the manner of notching the purlins upon the principals. No. 3 exhibits the manner in which the purlins are notched, to receive the small rafters, and in which the principals are notched, in order to receive the purlins. No. 4 shows the form of the tenon at the end of the wave HHT CARPENTRY 157 principal rafters, and the form of the rafter-feet ; and No. 5, the manner in which the beam is mortised, in order to receive the feet of the principals. This roof is well adapted for the covering of an area of sixty feet span. Here the braces are all directed to the purlins which support the small rafters, and the small rafters the covering ; so that the whole is firmly sup- ported without danger. Figure 1, pi XXV, exhibits the design of a truss adapted to a sixty feet span. But, in this design, there is an opening in the middle, which is useful when rooms in the roof are required. Figure 2, pi XXV, is another design for a truss, with four purlins. This design may be employed in the covering of a span which extends between sixty and seventy feet. Figure 3 is a design for a truss, which may be employed in a span extend- ing from sixty to seventy feet, or even more, depending on the strength of the timbers. Instead of the posts, oblique ties are used. Figure 4 is the design for a truss, where the purlins are supported by uprights, which rest upon arches : the tie-beam may be also suspended from the arches. Figure 1, pi XXVI, is another design for a truss. Here the posts are all made double and bolted ; the braces, ab, db, on each side, are supposed to be very stiff pieces of iron, shouldered on the inside, and nutted under the tie- beam, and under the principal rafters. This makes a very strong and good roof. -bra owi baa oiLJcuox fl 1 anaamnjd .ittnotiiisqmda Iai9 . In order to make room for lofty apartments, when the building is of a limited height, and when a coved ceiling is required, the tie-beam must be placed above the feet of the principal rafter. Figures 2, 3, and 4, are designs for this purposetftfmq bu In fig. 2 the bracing timbers are halved together, and the posts are made double and bolted together. In fig. 3 the two braces, which perform the office of a king-post, or two queen-posts, are made double and bolted together. In fig. 4 the braces which join the rafter-feet and the king-post are made double, and bolted together : the queen-posts are mortised into the tie-beam and into the principal rafters, and the collar-beam into the queen-posts. 20. 2R 158 THE NEW PRACTICAL BUILDER, &C. Figure 1, pi. XXVII, is the design for the truss of a kirb-roof, with a door in the middle. Figure 2 is the design for the truss of a kirb-roof, with two doors, one at each extremity. In this example the king-post is made double, and the two braces nearest the bottom. This mode of fixing posts and braces is much firmer than bare mortising and tenoning. Figure 3 is a design for a flat roof, to be covered with lead. Figure 1, pi. XXVIII, is the design for a roof in several stages, adapted to warehouses. The explanation of the several parts are as follow : Figure 2 is the longitudinal truss for the upright frames, which rest upon the main tie-beams, and which support the oblique parts that connect the upper stage. Figure 3 exhibits the longitudinal truss in the side of the roof, and which may be made into windows. Figures 4 and 5 show the abutting joints, made of cast-iron, for receiving the lower and upper ends of the braces. Figure 6 shows the cast-iron abutments used in the truss, fig. 2. Plate XXIX exhibits the various methods for TRUSSING GIRDERS. Figure 1, No. 1, exhibits the side of a trussed girder with two braces, or trusses, as they are called. Here it is proper to observe that, though the sides should be firmly bolted together, the two braces should be entirely clear of the bolts, except at their lower extremities. No. 2 is the plan of the truss. Figure 2 is the design for a truss-girder in three parts, with two queen-bolts. No. 1 is the side, or elevation, of the truss ; and No. 2, the plan of the same. Figure 3 is a section of either of the above-mentioned truss-girders, at the abutments. Figure 4, a section of the king-bolt, nut, and washer, across the girder. Figure 5 is a section of the king-bolt, fig. 1, No. 1, taken longitudinally, and exhibiting the nut and part of the braces. Figure 6 is another design for a truss-girder, divided into four parts. But here it must be observed that, as the depth of the timber is not great, the effort of the braces to prevent the falling of the girder must be greatly insuf- ficient ; and, therefore, if height will allow, the proportion shown in fig. 7 will fully answer the purpose. 'Il3f* r }tf *Y* V/ ? V TUT JOINERY. i 3|i m .1 8iii e i; vw . CHAPTER III. icti JOINERY.* JOINERY is the art of uniting and framing wood, for the internal and external finishing of buildings. In Joinery, therefore, it is requisite that all the parts shall be much more nicely adjusted to each other than in carpentry, and all the surfaces which are to be exhibited to the eye shall be made per- fectly smooth. The wood employed in Joinery is denominated STUFF ; and of this there are BOARDS, PLANKS, and BATTENS; thus distinguished according to their breadths : BATTENS are from two to seven inches wide ; BOARDS, from seven to nine inches ; and PLANKS, from nine inches to any indefinite breadth. The OPERATIONS of JOINERY consist in making surfaces of various forms ; also Of GROOVING, REBATING, MOULDING, MORTISING, and TENONING. SURFACES, in Joinery, may be either plane or curved ; but they are most frequently plane. Every kind of surface is first formed in the rough, and then finished by means of tools, which will be described hereafter. GROOVING and REBATING consist in taking or abstracting a part which is every where of a rectangular section. A REBATE is formed close to the edge of the stuff; and a GROOVE, at some distance from the edge. A MORTISE is a cavity formed within the surface, for the purpose of re- ceiving the end of a piece of timber, to be joined at a given angle. The end, which must be very nicely fitted into the mortise, in order to make the two * CENTERINGS, STRENGTH OF TIMBER, &c. It may here be proper to notice that a Chapter on CENTERINGS, particularly the CENTERINGS of BRIDGES, with the application of timber to other purposes than those already explained, will follow the present Chapter on Joinery : in the same Chapter we shall, also, treat on the COMPARATIVE STRENGTH of TIMBER, &c. 160 THE NEW PRACTICAL BUILDER, &C. pieces as strong as possible, is called a TENON. As the sides of the mortise are generally perpendicular to the sides of the stuff, and at some distance from the sides of the piece in which the mortise is, a tenon is generally stopped by projecting sides, which are closely fitted upon the side of the piece of wood in which the mortise is made; and the parallel faces of both are made flush, and so closely united, as to appear almost like one single piece. The surface of the piece which has the tenon, and which comes in contact with the surface of the piece in which the mortise is made, is called the shoulder of the tenon. FRAMES are joined together, so as generally to form externally a rectangle, and i internally one, two, or more, rectangular openings : these openings are closed with thin boards, fitted into grooves round the edges, called PANELS. In ornamental work, the edges of the frame next to the panels, the two extreme vertical pieces of the frame, are denominated the STILES ; all the cross-pieces are denominated RAILS ; and vertical pieces, that separate the panels, MOUNTINGS. PLANKS are joined together by planing the edges straight and square, and rubbing them together with hot glue until the glue has been almost forced out of the joint ; then the ends and the proper faces being brought to their places, the rubbing is stopped, and, when the glue is quite dry, the two boards thus fixed will be almost as strong as one entire board. MOULDINGS have several names, according to their forms, connexion, situa- tion, or size. When the edge of a thin slip of wood is semi-circular, it is said to be rounded. Figure 1, plate XXX, represents the section of a piece rounded on the edge. When a semi-cylinder is formed on the edge of a piece of wood, within both surfaces, so that the diameter may be parallel to one side, this semi-cylinder is called a BEAD ; and the recess, between the surface of th cylinder and the solid wood upon the side, which is parallel to its diameter, is denominated a QUIRK ; and the whole part thus formed is called a BEAD and QUIRK. aS JOINERY. 161 Figure 2, plate XXX, is the section of a piece of wood, where a bead and quirk is run on the edge. BEAD and DOUBLE QUIRK is when a three-quarter cylinder is run on the edge, so that the surface of the cylinder may touch each adjoining face. Figure 3 exhibits the section of a bead and double quirk. A TORUS-MOULDING consists of a semi-cylinder, and two rectangular surfaces, one perpendicular to the diameter, and the other in the diameter produced. Figure 4 is a torus moulding : the small rectangular surface in the plane of the diameter is denominated a FILLET. Figure 5 exhibits the section of a DOUBLE TORUS. A FLUTE is the concave surface of the section of a cylinder or cylindroid, depressed within the surface of a piece of wood. Figure 6 exhibits the section of a piece of wood with three flutes. When a piece of wood is formed into two or more semi-cylinders, touching each other, the semi-cylinders are called REEDS, and the piece of wood is said to be reeded. Figure 7 exhibits the section of a piece of wood with four reeds wrought upon it. Figure 8 is the section of a moulding denominated a QUARTER ROUND. It consists of the fourth part of the convex surface of a cylinder. Figure 9 is the section of a moulding denominated a CAVETTO, consisting of the fourth part of the concave surface of a cylinder. Figure 10 is the section of a CYMA-RECTA, consisting of a quarter round and cavetto, joined together by one common tangent plane ; each part being the quarter of the surface of a cylinder, the one concave, and the other convex. To draw this curve, join the extremities a and c of the moulding : bisect ac in the point b, and draw de parallel to the longitudinal direction of the moulding ; make ad and ce perpendicular to de ; from d, with the radius da, describe the arc ab ; from e, with the radius ce, describe the arc be ; and abc is the moulding required. 20. 2s 162 THE NEW PRACTICAL BUILDER, &C. Figure 11 is also the section of a cyma-recta, of which the concave and convex parts are equal portions of a cylinder, but each portion less than the quartertoa iaaes To draw this curve, join the extremities a and b, and bisect ab in c : from a, with the radius ac, describe an arc ce ; and from b, with the radius be, describe an arc, cd',:from c, with the radius ca, describe an arc, ae, as also the arc bd. With the same radius, from the centre e, describe the arc ac ; and with the same radius, from the centre d, describe the arc cb ; then acb is the the curve, which is the section of the surface of the moulding. Figure 12 is the section of a moulding of the OGEE kind, called a CYMA- REVERSA : this moulding is of the same form as the cyma-recta, except that, in the cyma-recta, the concave portion of the moulding is the most remote from the eye ; whereas, in the cyma-reversa, the convex part is the most remote from the eye. Figure 13 is the section of a moulding called a SCAPE, which is composed of the quarter of the circumference of a cylinder, and a plane surface, which is a tangent to the cylindric surface, in the line of their meeting. Figure 14, part of the section of an ovolo with three fillets, which, when circular, or encompassing a column, are called ANNULETS. Figure 15 is the section of a moulding denominated a QUIRKED OVOLO. The method of drawing it is thus : Suppose it were required to touch the line de at the point d : draw dg perpendicular to de ; describe the circle bci ; 'make df equal to the radius of the circle bci, and join af. Bisect af by a perpendicular, gh, meeting qfinh', then, with the radius dg, describe the arc db: dbc will then be the ovolo required. Figure 16 is the section of a concave moulding called a SCOTIA. To form this moulding, describe the circle dabf, and draw cd perpendicular to the fillet. Make eg equal to the radius of the circle to be described, and let e be the centre of that circle : join ge, and bisect ge by the perpendicular df: from d, with the radius dc, describe the arc cb, and cba will be the scotia required. Figure 17 exhibits the METHOD OF DRAWING AN OVAL, to any length and breadth required. Draw the greater axis AC, and let r be the centre; .ojfc ;HHcaiiy< JoiNERXtni vravi am 163 through r draw BD, perpendicular to AC. Make rA and rC each equal to the semi-greater axis, and rB and rD each equal to the semi-lesser axis, Take half the difference of the two semi-axes from the lesser semi-axis, and, with the remaining part, as a radius, from the centre e, describe the arc gAf. Make En equal to Ae, and join en. Bisect en by the perpendicular om, meet- ing BD at m : join me, and produce me tof. Make rl equal to rm, and join le, and produce le to g. Make rh equal to re ; join /A, and produce Ihtok: also, join mh, and produce mh to z. With the centre m, and the radius ntf, describe the arc/U i ; and, from the centre /, with the radius Ig, describe the arc gDk ; lastly, from the centre h, and with the radius hk, describe the arc kd, which will complete the oval required. Figure 18 represents the section of a piece of wood which is said to be rebated. Figure 19, the section of a piece of wood said to be grooved. Figure 20, the sections of two pieces of stuff, grooved and tongued together : No. 1 shows the tongue, and No. 2 the groove, so adapted to each other that they may be joined closely together. This method is used where it is re- quired to join many boards together, so as to make one board, in order to prevent wind or air from coming through the joints between every two boards. nraiiloa gniaasqaiGona 10 jfili/cuio Figure 21 represents the section of a piece of stuff said to be rebated and beaded. aiw*tf> lo ^orftem 9nT ffBfb : ^> imctq gdJ JB & ontf ON THE VARIOUS FORMS OF FRAMING BOARDS WITH THEIR EDGES JOINED, SO AS TO FORM A RIGHT ANGLE, ONE BOARD WITH ANOTHER. PI. XXXI. Figure 1 shows the method of mitreing the ends of boards for dado,* or the like, at an external angle. * Ji Arf fwr\-jcif\ Y*i ii-LE> I M ~\ : T O fff ^ Figures 2 and 3, the method of joining troughs together. Figure 3 may also be applied to joining dado together, at an internal angle. Figure 4, the method of joining any kind of linings together, at an external angle ; a bead being stuck on the edge, in order to conceal the joint. : "taa- dJua->[ '(flu oi ,JAVO KA ottiwAaa HO conraM sdl eiididxa vl ^w, * By dado is meant the plane surface between the base and surbase of a room, or between the base and cornice of the pedestal of a column. 164 THE NEW PRACTICAL BUILDER, &C. Figure 5, a plane mitre, used for various purposes ; but, on account of its weakness, the method in fig. 1 is to be preferred. Figure 6 is another method of joining angles by a plane rebate. Figure 7 exhibits the method of dove-tailing : No. 1 represents the pins or male part, and the other the female dovetails. Figure 8 shows the method of making secret dovetails. At No. 1 the ends of the male dovetails are shown ; fig. 8 itself is the outside. No. 2 shows the section of both parts. OF DOORS. (PI. XXXII.) Figure 1 represents a SIX-PANEL DOOR, having ovolo and fillet on the stiles, with plane panels. Figure 2 represents FOLDING DOORS, which meet together upon a lap-joint, exhibiting a bead on both sides of the door. Figure 3 exhibits BEAD and BUTT. Figure 4, BEAD and FLUSH. The difference between bead and butt and bead and flush is this : In bead and butt, the bead is run on the edges of the panel ; but in bead and flush, the bead is run round all the four edges of the frame. Figure 5, section of part of the stile and panel of a square frame. Figure 6, section of part of a stile and panel moulded with quirk-ovolo and fillet ; the panel being flat on both sides. Figure 7, section of part of a stile and panel, having quirked ovolo and bead on the framing, with square panel. Figure 8, section of a part of the stile and panel of a door, with quirked semi-reversa on the framing. JOINERY. 165 + Mn*9*+9 Jo4}timiimm mtimi wk fcn vtftttaMfct ft ,3 w OF WINDOWS. (PL XXXIII.) .ciic&YQb dtafDSrl M& IMtM Skitl JMIft tJTt&fit Figure 1, No. 1, the elevation of a window. '-i *$nMi"%Et ' ' . ' v No. 2, section of the meeting bars, with a small portion of the stiles fixed to each. No. 3 shows the method of joining the intersecting bars, with the method of doweling them together. No. 4, the elevation of the intersection ; showing part of each branch or bar. No. 5, the section of one of the bars, being astragal and hollow. No. 6, a plane diminished bar for shop-fronts. No. 7, section of a bar with a Gothic point, instead of an astragal. No. 8, section of a Gothic bar. No. 9, section of a clustered bar, with two reeds on each side of the centre. No. 10, section of the stile of the sash-frame, with the astragal and hollow mouldings. < ' 9MfiftJw^ln f? **'' BASES AND SURBASES FOR ROOMS. (PL XXXIV.) Nos. 1, 2, 3, and 4, are all sections of SURBASE-MOULDINGS, differently designed. Nos. 5, 6, 7, and 8, are four different designs for BASE-MOULDINGS. ARCHITRAVES AND PILASTERS. (PL XXXV.) Nos. 1, 2, 3, 4, 5, 6, 7, 8, are sections for different designs of ARCHITRAVES. Nos. 9, 10, 11, 12, 13, 14, and 15, sections for different designs of PILASTERS. 21. 2i 166 THE NEW PRACTICAL BUILDER, &C. DIMINUTION OF COLUMNS. (PL XXXVI.) To diminish a column is to give such a form to the surface, that the sec- tions through the axis will all form convex curves on their edges. To DIMINISH THE SHAFT OF A COLUMN, as vhfig- 1. Draw the line repre- senting the axis, on which set off the height of the column ; then, from any point in the continuation of this line, at the bottom, describe a semi-circle, of a radius equal to that of the bottom of the column, and let the diameter of this semi-circle be perpendicular to the line of the axis. Through each extremity of the axis, draw the diameter of the shaft, at the top and bottom of the column. Through one extremity of the diameter of the top of the column draw a line parallel to the axis, to meet the circumference of the semi-circle. Divide the portion of the arc, between the point of section and the diameter, into any convenient number of equal parts; the more, the truer the result will be. Divide the height, or axis, of the column into as many equal parts as those contained in the arc of the semi-circle ; and through the points of division, both in the semi-circle and in the axis, draw lines per- pendicular to that axis. Through each point, beginning at the bottom of the arc, draw a line, parallel to the axis of the column, to meet its respective diameter ; then, through all the points of section, draw a curve, which will form the contour of the shaft, or a section of the column through its axis. In figure 2, instead of dividing the arc into equal parts, divide the dis- tance intercepted on the axis and on the radius of the semi-circle into equal parts, and proceed as in^. 1. To DRAW THE FLUTES OF A COLUMN, as in fig. 3. Draw a semi-circle and the axis of a column, as before. Divide the arc of the semi-circle into as many equal parts as it is to contain flutes; and, because the number of flutes are twenty, and the middle of a flute in the middle of the elevation of JOINERY. 167 the shaft, the semi-circle is divided into ten equal parts, and half a part, at the extremity of the diameter. Then, having found the diameters at the different heights of the shaft, as in Jig 1, on the lower diameter describe an equilateral triangle. Draw lines through each point of division in the semi- circle, parallel to the axis, to cut the base of the equilateral triangle. From the points of section in the base, draw lines to the vertex. Apply each diameter from the vertex, on the side of the equilateral triangle, and mark the extremity of that side. Through each point of section, on the side, draw lines parallel to the base ; and these lines will give the divisions of the flutes, which, being placed on the lines passing through the axis of the shaft, will give the points through which the flutes are to be drawn. In the corresponding points stick in pins, or nails, and bend a pliable slip of wood round the nails, which will form the curve-line of one side of a flute ; do the same for all the remaining sets of points, and we shall obtain the representation of the flutes required. The representation of the shaft thus fluted is shown mfig. 4. Figure 5 exhibits the diminishing rule, which is to be applied to the side of the shaft, in order to form the curve of the surface of the shaft, as required. ON THE METHOD OF SETTING-OUT THE FLUTES AND FILLETS OF PILASTERS AND COLUMNS. (PL XXXVII.) To SET OUT the FLUTES and FILLETS of a PILASTER; figure 1. Let us now suppose that the breadth of a flute is to that of a fillet as four to one ; and, that the breadth of the fillet, at the corner, is double to one of the in- termediate fillets, or half the breadth of one of the flutes. In the line BC, set off any convenient distance; set another interval, double of that distance; then the next part equal to half: repeat the in- tervals, so that an interval of one may be between every two intervals of 168 THE NEW PRACTICAL BUILDER, &C. four ; and that the intervals of four may be seven in number, and the in- tervals of one six in number ; and that there may be an interval of two upon each extreme. Then, with the distance BC, containing all the intervals, describe an equilateral triangle ABC. Draw lines to A from every point of division in BC. Now, it being understood that the distance BC is not less than the breadth of the pilaster to be divided ; therefore, on each of the sides of the equilateral triangle, make Ad and Ae equal to the breadth of the pilaster, and draw the line de ; then will de be equal to the breadth of the pilaster, and will be divided into the number of flutes and fillets required. Nothing farther is necessary than to lay the parts of the line thus divided upon the edge of a rod ; and thus the divisions may be transferred to each end of the pilaster. Figure 2 exhibits ANOTHER METHOD of DIVIDING the FLUTES and FILLETS of a PILASTER. The parts being placed on the line AB, in the manner before described, draw BC at any angle, so that the distance between the point A and the line BC may be less than the breadth of the pilaster. Through all the points of section in AB, draw lines parallel to BC : then take the breadth of the pilaster, with a rod or compasses ; and with that distance, as a radius, describe an arc from A, cutting the line BC in D, and draw AD : then apply AD as before. If the sides of the pilaster are convex, the several breadths must be taken equi-distantly throughout its length, and the divisions applied to the lines in the breadth of the pilaster at each part. The flutes and fillets may be tranferred to different circles, taken equi- distantly on the surface of a column, in the same manner ; but, instead of applying the parts upon a rod, they must be applied upon as many slips of parchment as the number of circumferences taken in the height of the column. But, in order to regulate the flutes, so that their edges may be all in planes passing through the axis of the column, draw a line on the surface of the column, so as to form the edge of a flute or fillet, or the middle of a flute or fillet ; then one of the ends of the slip may be applied to the line JOINERY. 169 where each circle intersects it, and the slip itself stretched round the circum- ference till the other end meets the line again ; then mark the divisions of the flutes and fillets on the circumference of the column. Through every row of equi-distant points, on the circumference of the column, from the line previously drawn, draw a line, which will he the line of demarcation of a flute and fillet. Figure 3 shows the METHOD of DRAWING the FLUTES and FILLETS of a DORIC COLUMN. One particular to be observed in this, is, that a plane, pass- ing through the axis of the column, must pass through the middle of a flute, when that plane is perpendicular to the front, and when it is parallel to the front. The whole circumference of the Doric column is generally divided into twenty equal parts, by the arrises of the flutes, which terminate upon each other, without the intervention of fillets. Figure 4 shows the METHOD of DESCRIBING the FLUTES and FILLETS on the shaft of the IONIC, CORINTHIAN, and COMPOSITE COLUMNS : and here we must observe that, as in the Doric, a plane passing through the axis, perpen- dicular to the front, in which the column stands, generally passes through the middle of a flute ; and, that a plane, passing through the axis, parallel to the front, passes also through the middle of a flute. The number of flutes and fillets in the Ionic, Corinthian, and Composite orders, are generally twenty-four each ; the fillet being about one-fifth part of the breadth of a flute. Figures 5 and 6 show the METHOD of GLUING UP the SHAFT of a COLUMN in staves. The number may be more as the diameter of the column is greater. In the example before us, the number of staves is eight ; therefore we must describe circles, one to the diameter of the top, and another to the diameter of the bottom, of the column, and circumscribe an octagon round each circle ; then draw another octagon, of which the sides are parallel to those of the octagon already drawn ; so that the distance of the parallel sides of the octagons may be equal to the proper thickness of stuff required to make the 21. 2 u 170 THE NEW PRACTICAL, BUILDER, &C. shaft of the column. From the angles of the octagon draw lines to the centre, which will give the directions of the joints ; but, though the angle shown by the bevel in fig. 5, would not differ sensibly from the truth, the proper method to find it is the same as finding the backing of a hip-rafter ; and, if the outside of the column is curved, it will be eligible to apply the bevel from the inside ; because, if applied to the convex side of the column, every different place would require a different angle. METHODS OF TAKING DIMENSIONS AND HINGEING. THE TAKING OF DIMENSIONS depends upon the method of describing a triangle, from the three sides being given: therefore, let ABC, (Jig. 1, pi. XXXVIII,) be the three given sides of a triangle : draw the straight line A in any convenient place, as DE : from D, with the straight line B, as a radius, describe an arc at F ; and, from E, with the straight line C, as a radius, describe another arc, cutting the former in F ; join DF and FE : then will DEF be the triangle required. In order to take the dimensions of a [place which is to receive a piece of framing, make an eye-draught, as in fig. 2, No. 1 ; and upon each line mark the dimensions of the sides ; then take the lengths of these sides from the scale, Jig. 3, and find the angular points of the triangle, No. 2 : having cut each piece of stuff to its proper length, scribe each edge down to its place ; then lay together the ends which are to meet, one piece being on the top of the other, and draw the shoulders of the joints. When the place which is to receive the framing consists of more than three sides, sketch the figure, as before ; draw lines from one corner to every other corner, and mark the dimensions upon these lines, and the dimensions of each side of the figure upon its respective line. This is fully exemplified in figures 4 and 5, taken from the same scale. JOINERY. 171 Figure 6 exhibits the METHOD of HINGEING ONE FLAP TO ANOTHER, the joint being what may be termed a lap-joint. The centre of the hinge is placed opposite to the joint. When one flap revolves upon another, it is sometimes required to throw the edge of the flap, when folded close upon the back of the other, at a given distance : to do this, the centre of the hinge must be placed at half that distance from the joint : this is exemplified in fig. 7. Figure 8 SHOWS the METHOD of HINGEING a RULE-JOINT, the axis of the hinge being in the axis of the cylindric surface that forms the rule-joint. Figure 9 shows one of the flaps folded upon the other. This method is sometimes used in window-shutters, though mostly in furniture. HINGEING DOORS AND SHUTTERS. (PI. XXXIX.) Figure 1, Nos. 1 and 2, represents the form of the joint of two stiles, in order to fit each other. No. 3 shows the same when hinged together. Figure 2, Nos. 1 and 2, exhibits a plane-joint, beaded alike on both sides : No. 3 shows the same when hinged together. Figure 3, Nos. 1 and 2, exhibits the same thing with a double-lapped joint. No. 3 shows the two parts put together. Figure 4, Nos. 1 and 2, shows the same thing, with a single-lapped joint. Figure 5 exhibits the manner of hingeing the shutter to the sash-frame. Figure 6 exhibits the manner of hanging the door upon centres. Figure 7 shows the method of hingeing shutters, so as to conceal the hinges. 172 THE NEW PRACTICAL BUILDER, &C. MOULDINGS ON THE SPRING. (PL XL.) Figure 1 is the elevation of a cylindric body. At the upper end, G and I are the profiles of a cornice, or section of a cornice, to be put round the cylindric body. If the moulding is formed from a solid piece, it must be formed in short lengths ; for, if the pieces are very long, the grain will run across them, and will render the pieces weak, and make the moulding very ugly. In order to prevent the crossing, as much as possible, the best way is to reduce it, by cutting off the right angle by a plane, as nearly parallel to the face as possible ; and the moulding may be bent in the same manner as in covering the frustum of a right cone. Thus may the moulding be got out of a thin board. Bisect the breadth DF, in the point E, and draw EH parallel to FG or DI. Produce the back of the cornice, Td, to meet the line of axis in m ; then, from the point m, as a centre, with the distances from m to each edge of the fillets and moulding, describe arcs, and these will represent the lines for working the mouldings on the board. Figure 2 shows the method of describing the mouldings when put round the segment of a cylinder. Here the whole semi-circle must be completed, and the moulding placed as before, and described after the same manner. Figure 3 exhibits the method of describing the moulding for the interior surface of a cylinder. Figure 4, the section of a base moulding, to be bent on the spring. Figure 5, a cornice, where the mouldings are almost in a straight line. This is well adapted for the surface of a cylindric body. Figure 6 is the profile of a cornice, where the cyma-recta only is intended to be sprung. Here this part must be bracketed behind, in order to keep it firmly in its place. The corona and the bed-moulding are made of a solid piece. JOINERY. 173 S RAKING MOULDINGS. (PL XLT.) WHEN a building has a pediment, with mouldings or a horizontal cornice, crowning the walls, and a pediment, with a similar cornice, upon the rake, the upper mouldings are mitred together, so that the mitre-plane may be perpendicular to the horizon : this renders the sections of the upper member of the horizontal cornice, and that of the pediment, dissimilar in their right section : the question then is, having the section of the one, how to find the section of the other. But, since the horizontal cornices are generally wrought first, the section of the horizontal moulding at the top is given, in order to find that of the pediment. Therefore, in Jig. 1, there is given the horizontal section, abcdefg, to find the section of the inclined moulding. Let the points a, b, c, d, e,f, g, be any number of points taken at pleasure ; draw lines through these points, parallel to the rake ; and, also, draw lines through the same points, perpen- dicular to the horizontal cornice, so that all shall cut any horizontal line in the points h, i, j, k, I, m, n. Transfer the distances between the points h, i,j, k, l,m,n, any where upon the raking line, to h', i',f, k', I', m',n'; and, from these points, draw lines perpendicular to the rake, cutting the inclined lines at the points, a',b',c',d', e',f',g' \ then, through the points ', V, c', d', e', f, g', draw a curve, which will be the section of the inclined moulding. Again, suppose it were required to return the moulding upon the rake to a level moulding at the top: Upon any horizontal line, transfer the distances between the points h, i,j, k, I, m, n, to h", i",j", k", I", m", n"\ and, from these points, draw lines perpendicular to the level cornice, cutting the raking-lines before drawn at the points a", b", c", d", e" ,f" , g" ; then, through the points a", b", c", d", e",f", g", draw a curve, which will form the return-moulding at the top. 22. 2x 174 THE NEW PRACTICAL BUILDER, &C. Figure 2. The lower section is the horizontal part, and the upper section is that of the upper return-moulding, found in a similar manner to Jig. 1, excepting that, as the mouldings themselves are circular, the lines drawn through the points b, c, d, must also be circular ; and, instead of laying the parts between the points, b, c, d, &c., upon the raking-line, they must be laid upon a straight line, which is a tangent to the circle. Figure 3 shows the method of finding the return-mouldings for a raking ovolo ; the lower section being the given moulding, and the upper one that of the return horizontal moulding. Figure 4 exhibits the method of finding the return of a raiting cavetto. PLATE XLII, fig. 1, shows the steps of a stair, where the base-moulding continues along the rake, and returns both at the bottom and top of the stair. Figure 2 exhibits the moulding upon the nosing ; jig. 3, the raking- mouldings, found as in fig. 1, pi. XLI. Figure 4 shows the same thing, when the mouldings are to be placed around an internal space. Figure 5 represents raking-mouldings for angle-bars of shop-fronts: abed, &c., is the given moulding. Take any number of points, a, b, c. d, &c., in the curve, and draw lines, aa' ', bb', cc' , dd' , &c., parallel to the face of the window ; draw a line perpendicular to the mitre line ; then, through the points a, b, c, d, &c., draw lines perpendicular to the line of the face of the window, cutting it at the points e, f, g, h, i. Transfer the distances between the points e,f, g, h, i, upon the line which is drawn perpendicular to the mitre- line, at e',/', g', h', i' ; then draw lines parallel to the mitre-line, cutting the lines drawn parallel to the front at the points a', V , c', d', &c., and, through the points, a', b', c', d', &c. draw a curve, and it will form one side of the angle-bar:, then, making the other similar, the whole angle-bar will be formed. Figure 6 shows another design of a bar, where the window returns at an obtuse angle. The method of forming the angle-bar is the same as in fig. 5. JOINERY. 175 V METHOD OF ENLARGING AND DIMINISHING MOULDINGS. (PL XLIII.) THIS depends entirely on the proportion which any two lines, of different lengths, have to one another, when divided in the same ratio. Euclid proves, and indeed it is self-evident, that, if a line be drawn parallel to one side of a triangle, and if lines be drawn from the opposite angle, through any number of points taken in one of the parallels, to cut the other, these two parallels will be divided in the same ratio. This is one of the principles of proportioning cornices. Another method of proportion, emanating from a similar principle, proved by the same geometrician, and which is equally evident, is, when any number of straight lines are drawn parallel to one side of a triangle, so as to cut each of the other two sides in as many points, each of the two sides thus divided have their corresponding segments in the same proportion. Hence we have only to construct a triangle, which shall have two of its sides given ; for, if the divisions in one of these lines be given, we may divide the other in the same ratio, by drawing lines parallel to the third side of the triangle : or, according to the first principle, if a straight line be drawn parallel to one side of a triangle, this straight line will divide the triangle into two similar triangles ; therefore, if the triangle to be divided be equilateral, the smaller triangle, when divided, will also be equilateral. Therefore, if the divided line be greater than the undivided line, we have only to construct an equilateral triangle, and set the length of the undivided line from one of the angular points upon one of the sides, and draw a straight line through the point of extension, parallel to the side opposite to that angular point ; then, placing the parts of the divided line on the greatest of the two parallel lines, we have only to draw lines, through the points of division, to the opposite angle, and the lesser parallel line will be divided in. the same proportion. 176 THE NEW PRACTICAL BUILDER, &C. Let AB, (fig. I, pi. XLIII,) be the height of a cornice, divided by the height of the members into as many segments. Upon AB describe the equilateral tri- angle ABE ; from the points of division in AB draw lines to E. On the side EB or EA, of the triangle, make EH or EG equal to the height of the intended cornice, and draw GH parallel to AB ; then GH will contain the heights of the members of the new cornice. The projections are found thus: AC, being the lower line of the cornice, produce AC to D ; and, from all the points of projecture, draw lines perpendicular to CD, cutting CD in as many points as the lines thus drawn. The point D, being the extreme projecture, produce the line downwards from D to F, and make DF equal to the perpendicular of the equilateral triangle AEB. Draw lines from the divisions in CD to the point F. Make FK equal to the perpendicular El, as terminated by the line HG, and draw KL parallel to CD, cutting all the lines drawn from the points of division in CD ; then KL will contain the projectures of the new cornice, of which the height is GH : and thus, having the heights and projectures of the members of the new cornice, it may be drawn by the usual rules. The mouldings of the architrave are proportioned in the same manner. Thus, describing an equilateral triangle MNO ; on the height, MN, of the architrave, produce the lines of division in the height to meet the line MN ; from the points of division in MN draw lines to O, the apex. Make OQ equal to the height of the new or intended architrave, and draw PQ parallel to MN, and PQ will be divided in the same proportion as MN. To find the projectures, draw RS perpendicular to MN, and describe the equilateral triangle RST. From the points of projecture, in the lines dividing the height, draw lines parallel to MN, cutting RS, the side of the equilateral triangle. Draw lines from the divisions in RS to the opposite angle T. Any where in the line MN make YZ equal to the side RS of the equilateral triangle, and draw YO and ZO, cutting PQ in a and b. On the side TS, of the small equilateral triangle, make Tc equal to ab, and draw cd parallel to RS ; then cd will be divided in the same proportion as RS. To ENLARGE A CORNICE, according to any given height : Figure 2. From any point, V, with a radius equal to the intended height of the cornice, JOINERY. 177 describe an arc cutting the opposite edge in U ; and the line VU, being drawn, will contain the heights of the members of the new cornice. To find the Projections. Draw any line, WX, perpendicular to UV, be- tween the extreme vertical lines of projecture ; and, from all the extreme points of projecture, draw verticals to cut the line WX, which will be divided into the proper projectures. CONSTRUCTION OF CIRCULAR SASHES IN CIRCULAR WALLS. (PL XL1V.) To FORM THE COT-BAR OF THE SASH-FRAME. Figure 1 is an elevation of the window. Divide the half arc of the cot-bar, PR, into any number of equal parts, as six ; and, from the points of division, draw perpendiculars to the horizontal line ag ; transfer the parts of the horizontal line ab,bc,cd, &c. to^g-. 2, from to 1 ; 1, 2; 2, 3; 3, 4; &c., to 6; and reverse the order from the central point 6, and draw perpendiculars upwards from these points : make the heights of the perpendiculars, Jig. 2, to correspond to those taken from the plan, fig. 1. Through all the points draw curves, which will be the form of the mould for the veneers to be glued in thickness. To form the Head of the Sash-frame. Divide the elevation round the outer edge into any number of equal parts, and draw perpendiculars to the chord of the half plan. From the points where these perpendiculars intersect the chord of the half plan, draw ordinates, perpendicular to the chord of the half plan. Make the ordinates FI, GK, HL, &c. equal to Bl, C2, D3, &c. ; and the ordinates of the inner curve equal to CM, DN, &c. ; then, through the points E, I, K, L, &c., draw a curve ; and within, through the other points, draw another curve ; and this will form the face-mould for the sash-head. Figure 3 is the development of the soffit of the under side of the sash-head. Figure 4, that which is applied to the convex side of the same : so that, by 22. 2 Y 178 THE NEW PRACTICAL BUILDER, &C. cutting away the superfluous wood on the outside of the space contained by the two lines, the sash-head will fit the surface of a cylinder made to the radius of the plan of the window. To find the Radial Bars. Let ny, fig. 1, be the place of a radial bar. In ny take any number of points, n, o, p, q, Sac., and draw the perpendiculars nv, ow, px, qy, &c., to ny. Draw, also, ni, ok, pi, qm, &c., perpendicular to the base-line of the elevation, cutting the base-line at the points a, b, c, d, &c. ; and cutting the convex side of the sash at the points e,f, g, h, and the concave side at i, k, /, m, &c. Make the distances nv, ow, px, qy, &c., re- spectively equal to ai, bk, cl, dm, &c.; and, through the points v, w, x, y, &c., draw a curve, which will form the convex edge of the radial bar. In the lines nv, ow, px, qy, &c., make the distances nr, os.pt, qu, &c., each respectively equal to ae, bf, eg, dh, &c.; and, through the points r, s, t, u, &c., draw another curve, which will give the inner edge of the radial bar. OF SKYLIGHTS. (PI. XLV.) SKYLIGHTS are windows in the roofs of houses, and are necessary when light cannot be had in the sides of the room or apartment required to be lighted. figure 1, No. 1, is a portion of the Plan of a Square Skylight. No. 2, the Elevation. To find the Seats of the Hips. Bisect the angle ABC, No. 1, by the straight line BE ; and bisect the angle BCD by the straight line CE : then BE and CE are the seats of the hips. To find the Length and Angles of the Hips. Draw EF perpendicular to CE, and make EF equal to the height of the skylight, and join CF; then CF is the length of the hip, ECF the angle which it makes with its seat or base, EFC the angle which it makes with the perpendicular. JOINERY. 179 To find the Backing of the Hips. From any point G, in EC, draw GL per- pendicular to CF, cutting FC in L; and, through G, draw HI, perpendicular to CE, cutting BC in H, and CD in I. In CE make GK equal to GL, and join HK in IK : then will HKI be the angle required. Figure 2 is the plan and elevation of an octagonal skylight. The method of finding the seats of the hips, their lengths, angles, and backing, is the very same as in the preceding example; and, therefore, any farther explanation will be unnecessary. Figure 3, an octagonal skylight, cut by the inclined side of a roof. The method of drawing the curb for such a roof requires some explanation, in order to be understood. No. 1 is tlaeplan, No. 2, the elevation, showing the part above the roof, which is exactly the same as that of a common skylight. In No. 2, through the summit S, draw ST, parallel to the base, cutting the sloping line of the roof in T. In No. 3 draw PQ equal to the breadth of the octagon, No. 1. Bisect PQ in U, by a perpendicular, UL. Make UL equal to RT, and join PL and QL. Through L draw KM, parallel to PQ, and make LK and LM each equal to UP or UQ. Draw PM and KQ, cutting each other in I, PL in V, and QL in W. Join VW. Make UC and UD each equal to half the side of the octagon. Draw CK, cutting PL in B, and DM, cutting QL in E. Through I draw NO, parallel to PQ, meeting CK in N, and DM in O. Draw NM, cutting BL in A, and VW in H : also, draw OK, cutting QL in F, and VW in G ; then ABCDEFGHA is the outline or exterior edge of the curb. The interior edge, abcdefgha, is drawn by means of the same points, K, L, M, and the edges, cd and kg, parallel to CD, so as to meet each other upon the diagonals al,bl,cl, dl, &c. This is the same as the perspective representation of the plan, by supposing the eye of the spectator fixed at S, No. 2, RT the section of the picture, and RX a section of the original plane. In this case, K, L, M, No. 3, are called the vanishing points ; L is the centre of the vanishing line ; PQ is what is called the intersecting line; PK, or QM, is the height of the picture ; and ST, No. 2, is the distance of the eye from the picture. 180 THE NEW PRACTICAL BUILDER, &C. The points K and M, No. 3, would have been ascertained in perspective, thus: Produce UL to Y, and make LY equal to ST. Now, supposing the plan, No. 1, on the other side of PQ, No. 3, with two sides of the octagon parallel to PQ, one of them may coincide with the part CD of PQ. Then, through Y draw a line YK and YM, parallel to the diagonals of the square which forms the sides of the octagon; then the points K and M are the vanishing points. DESIGNS FOR SHUTTING-WINDOWS. (PI. XLVI.) THE parts A, B, C, D, E, F, G, H, fig. I, form a SECTION of the SASH-FRAME. A, Pulley-stile of the sash-frame. B, Inside Lining. C, Outside Lining. D, Back Lining. E and F, Weights to- balance the sash-frame. Here the pulley-stile, A, is tongued into the inside lining, B, and into the outside lining, C. The back lining, D, laps over the edge of the outside lining, C, and is tongued into the inside lining, B. The parts K, I, B, form a recess for receiving the shutters, which recess is tailed the BOXING. t&iter I is called the back lining of the boxing, and is tongued into the inside ining of the sash-frame. K is a ground, of which the outside is flush with the plaster. The inside lining of the boxing is also tongued into the ground K. H, Section of the inside bead of the sash-frame. G, Parting-bead, and serves to separate the upper from the lower sash, in order that they may work freely and independently of each other. R, archi- trave, or pilaster. Figure 2, A VERTICAL SECTION of the WOOD-WORK of the SASH-FRAME, and the parts attached to it. JOINERY. 181 H, the Inside Bead, forming one side of a groove for the lower sash to run in. N, The Bottom Rail of the lower sash. O, Sill of the sash-frame, being of a different section to the uprights. P, Section of the back lining of the window. Figure 3, Section of the sash-frame and shutters, where the wall is not sufficiently thick to admit of boxing-room for the shutters. Here ABECD is a casing of wood, in order to receive the shutters, which is made in two folds, F and G, the one, F, being parallel to the wall, and connected to the other, G, by a rule-joint, and G is connected to the sash-frame by butt-hinges. Figure 1, (pi. XLVII,) is a HORIZONTAL, SECTION of a SASH-FRAME and SHUTTERS. Here, instead of the rule-joint, shown in the preceding plate, the shutters are made to draw or slide out of the wall, as shown, and the end is covered by a flap, as if it were a front shutter ; and this flap is hung to the sash-frame in the usual manner, and may be so framed together. Figure 2 is a vertical section through the head of the sash-frame. D, Head of the sash-frame. E, Inside Bead. F, Parting Bead of the sash-frame. C, The Soffit, tongued into the head of the sash-frame. B, Ground, flush on the front side with the plaster. A, Architrave Moulding. Figure 3, VERTICAL, SECTION through the window. K, Sill of the sash-frame. E, Inside Bead or Stop of the sash-frame, forming one side of a groove. I, Window Sill. H, A small flap hinged to the back, M, of the recess under the window. Figure 4, a HORIZONTAL SECTION through the sash-frame and shutters. O, Inside Lining of the sash-frame. P, Pulley-stile. E, Inside Bead, forming a stop for the lower sash. 23. 2z 182 THE NEW PRACTICAL BUILDER, &C. F, Parting Bead. N, A Door hung to the sash-frame, in order to conceal the end. T, Part of the section of the real shutter. U, Architrave Moulding. Q and R, Weights for balancing the sashes. &l ad ii ..I i ON THE FORMATION OF THE SHUTTING-JOINTS OF Figure \, pi. XLIX, represents a common door, supposed to be hinged at . The face of the door is in the line ac, and the breadth is terminated at a and c. On ac, as a diameter, describe a semi-circle, abc, cutting the other face of the door at b. Figure 2, on the same plate, represents the section of folding doors with jambs, upon a straight plan. Here we must suppose that one of the doors is shut, while the other opens. Let the half which is shut be edcbhgf, and let aedcb be the other half, which opens. Draw the line dc, parallel to the face of the door, bisecting the thickness, so that the middle of cd may be in the middle of the breadth of the door. Draw the line ad, and draw de perpendicular to ad; also draw the line ac ; and on ac, as a diameter, describe the semi-circle abc, cutting the other line of the door at b, and join cb; then will edcb be the form of the joint. The principle of opening is evident ; since no length can be applied in a place shorter than itself. The most remote point of the thickness in the moving part must pass, in the act of opening, every other part of the half that is stationary. The principle, therefore, amounts to this : that, since in the opening every point in the edge of the moving door describes the cir- cumference of a circle, every line drawn from the point a to the line be, ought not to be less than the line ab; and, because the angle abc is a right- angle, every line drawn from a, to meet the line be, will be the hypothenuse JOINERY. 183 of a right-angled triangle, of which one of the legs is the line ab ; therefore, ab is shorter than any line that can be drawn from the point a to the line be ; consequently, the point b, in the act of opening, would fall within the ex- tremities of every line drawn from a to the line be. It may, also, be shown that, if any point be taken in be, and a line be drawn from that point to the point a, the line thus drawn will be less than any other line drawn from to any other point of the line be, between the former point and the point c. In the same manner, because ade is a right-angle, the line ad is less than any other line that can be drawn from a to any point of the line de, between d and e ; and every other line drawn from a, to any point between d and e, will be less than any other line drawn from a to any other point in de, between that point and the point e. Having thus shown the reason of the method, the principle of figures 3, 4, 5, 6, will be evident on inspection ; or, at least, by comparing it with the former part of this description. Figure 7 is a section of the jambs of a pair of folding doors, with part of the section of the door : fig. 8, the section of the meeting-stiles of the doors. rill 9d 5otoao u( - OF JIBB-DOORS. ( See Plate L. ) 99 ^ A JIBB-DOOR is one that has no corresponding door, and which is flush with the surface of the wall, being generally papered over, the same as the room ; the design being to conceal the door as much as possible, in order to preserve the symmetry of the side of the room which it is in. Let fig. 1 represent the side of a room, in which KLMN is a jibb-door, I the base of the room, and H the surbase, extending across the door. Now, in order to make the jibb-door open freely, the mouldings must so be cut that no point of the moving part may come in contact with the jamb of the fixed part. This may be done by forming the end of the moving part, 184 THE NEW PRACTICAL BUILDER, &C. and the end of the jamb or stationary part, in such a manner that all the horizontal sections may be circles described from the centre of the hinge. In short, by making the end of the base and surbase, and the edge of the jamb, the surface of a cylinder, of which the axis line of the hinges is the axis of the cylinder. This is shown by fig. 4, where A is part of the jamb ; B represents a section of the door, upon which the iron containing the centre is fixed. C is the centre. The parallel lines in front represent the projec- tions of the mouldings. Draw Ccl perpendicular to the front line, and make de equal to Cd : from C, with the radius Ce, describe the circular line ef, and where the points of ec cut the parallel lines will be the extremities of the radii of the other circles. Figure 2 exhibits the section of the surbase, marked H, in Jig. 1 ; and B, Jig. 3, is the elevation of the base, shown at I, Jig. 1. OF STAIRS AND STAIR-CASING. DEFINITIONS OF THE PARTS OF STAIRS. 1. BY STAIRS we mean an assemblage of steps, so formed and united, that, by walking on them, we ascend or descend from one height to another. 2. The surfaces on which we set our feet are called TREADS ; and these, for the convenience of walking, are set at equal distances, and parallel to each other. 3. In order to give a solid appearance to the whole, every adjacent pair of treads are connected by a third and vertical piece, called a RISER. Each riser and tread, when fixed together, is called a STEP. 4. The wall which supports the ends of the steps is called the STAIR-CASE. 5. When the ends of the steps terminate upon a vertical prism or pillar, the prism or pillar is called a NEWEL. JOINERY. 185 6. If the ends of the steps be cut through in the surface of the newel, and the pillar or prism be supposed to be removed, the space left open by the removal of the solid is called the WELI--HOLE. 7. Stairs that have a well-hole, or hollow in the centre, are called GEO- METRICAL STAIRS. 8. The meeting of the sides which form the external angle of the steps is called the LINE OF NOSING ; but sometimes the line of nosing is covered with a moulding, and then this moulding is called the nosing. 9. When the steps are of equal breadth ; that is, when the distance from the line of nosing to the riser is every where equal, the steps are denominated FLYERS. 10. When the treads of the steps diminish in breadth toward the well-hole, the steps are called WINDERS. As the ends of the steps generally terminate upon a surface which is per- pendicular both to the risers and treads, the surface on which they thus ter- minate is generally that of a cylinder. 11. A number of contiguous flyers are called a FLIGHT. 12. When the tread of a step is so broad as to be equal to two or more of the other steps, and situated between floors, it is called a RESTING-PLACE. 13. If the tread of the resting-place form a right-angle, that is, if the two risers be perpendicular to each other, the resting-place is called a QUARTER- SPACE Or QUARTER-PACE. 14. When the breadth of the tread of a step is contained between the same vertical plane, or makes two right angles round the axis of the well-hole, the tread is called a HALF-SPACE or HALF-PACE. 15. Half-spaces and quarter-spaces are generally made on floors ; and, in this case, are called landing-places. PROPORTIONS, &c. The breadth of the steps of common stairs is from nine to twelve inches. In the best stair-cases of noblemen's houses and public edifices, the breadth ought never to be less than twelve, nor more than fifteen, inches. 23. 3 A 186 THE NEW PRACTICAL BUILDER, &C. A step of greater breadth requires less height than that of a less breadth. The general rule may therefore be as follows : Multiply the breadth and height of a given step together, and divide the product by the breadth of the required step, and the quotient will be the answer: or, by reciprocal proportion, as the given breadth belonging to the height required is to the breadth of the given step, so is the height of the given step to the height of the required step. For example, taking as a standard a step of 12 inches in breadth, and 5| inches in height, we may easily find the height of another of a given breadth, which we shall suppose to be 10 inches. The operation is 10 : 6} :: 12 : l ^ = 6 inches. Again, suppose we would find what ought to be the height of a step of which the breadth is nine inches ; the operation will be thus : 9 : 12 :: 5| _| 60 6 9)66 7|- inches, which agrees with what would be allowed in common practice. Before we lay out the stairs in a building, we must consider the height of the story, and determine upon the height of the steps ; which, being done, we must throw the height of the story into inches, and divide the number of inches in the height of the story by the height of the step. Thus, for ex- ample, suppose the height of the story to be ten feet four inches, and the height of the step to be seven inches, how many steps will be required in order to ascend to the given height ? Here (IQft. 4, in.) x 12 s= 124 inches. Now 124 -=-7 = 17^, which is the number of steps required ; but, as we must not have a fractional part, the stairs must consist of either seventeen or eighteen steps : if the plan admits of sufficient room, eighteen steps would be preferable to seventeen. But, if there are no winders in the stairs, an even number of steps will be more con- JOINERY. 187 venient than an odd number: therefore, if we cannot have eighteen, we must have sixteen : now, 124-r 16 = 7|- inches ; which may answer very well : but, if we are still confined for room on the plan, we must throw the semi- circumference round the newel into winders. The breadth of stair-cases may be from five to twenty feet, according to the destination of the building ; but if the steps be less than two feet four inches in length, they become inconvenient for the passing of furniture, as is generally the case in small houses. When the height of the story is very considerable, resting-places become necessary. In very high stories, that admit of sufficient head-room, and where the plan or area for the stairs is confined, the stairs may make two revolutions in the height of the story ; that is, the ascendant or descendant may go twice round the newel or well-hole; and this becomes necessary, otherwise the steps would be enormously high, or extravagant floor-room must be allowed for the stairs. As grand and principal stair-cases require broad and low steps, they there- fore require to be numerous, and admit of only one revolution in the height of the story; the plan being always proportioned to the height of the building. It may not be amiss to give an example here for a principal building, in order to show the number of steps both in the grand and in the common stair-case. For this purpose, suppose the story of a house to be sixteen feet high from floor to floor, the height of the steps of the servants' stair-case to be seven inches, and that of the grand stair-case to be six inches. Now the height of the story reduced to inches is 192, and first dividing by 7, thus 7) 192 27f then, dividing by 6, thus 6)192 ~32 188 THE NEW PRACTICAL BUILDER, &C. So that the servants' stair requires very nearly twenty-eight steps, and the grand stair-case thirty-two : but the space or area required to execute the common would not be much less than that required to execute the grand stair-case ; the common stairs must therefore have two revolutions in the height. This being allowed, will reduce the area to half of what it otherwise would have required. We must, however, observe that, when the height of the story is less than fourteen feet, the stairs will not admit of two revolutions. In planning a large edifice, particular attention must be paid to the situa- tion of the stairs, so as to give the most convenient and easy access to the several rooms. With regard to the lighting of a grand stair-case, a lantern-light is the most appropriate. By introducing this, more elegance is acquired, and the light admitted is more powerful ; but, indeed, where one side of the stair-case is not a portion of the exterior wall, a lantern or skylight is the only way in which the light can be admitted. In stairs constructed of stone, the steps are made of single blocks ; quarter- spaces and half-spaces are, however, made in two or more pieces, and joggled together : but, when the material is wood, the risers and treads must be made of boards, which are fastened together with glue, brackets, and screws ; and these, though done with the utmost care, can never be made so firm as not to yield to the passenger. To prevent stairs from becoming ricketty, in length of time, the steps must have an additional support under them ; and that the appearance may be both light and pleasant, the whole must be confined to as small a space as possible. This additional wood-work, which is necessary to the firmness and durability of the construction, is called the CARRIAGE OF THE STAIRS. The carriage of a stair consists of several pieces joined together. A flight of steps is generally supported by two pieces of timber, placed under the steps, and parallel to the wall, being fastened at one or both ends to pieces perpendicular thereto. JOINERY. 189 The pieces of timber which are thus placed under the steps are called ROUGH STRINGS. DOG-LEGGED STAIRS are those which have no well-hole, and consist of two flights, with or without winders. The hand-rail, on both sides, is framed into vertical posts, in the same vertical plane, as well as a board which supports the ends of the steps. The boards are called STRING BOARDS, and the posts are called NEWELS. The newels not only connect the strings, but they afford the principal support to the rail ; and thus it may be affirmed that the newels, posts, and hand-rail, are all in one plane. OPEN-NEWELLED STAIRS are those which have a rectangular well-hole, and are divided into two or three flights. (See Jig, 2, pi. LI.) BRACKETED STAIRS are those where the string-board is notched, so as to permit the risers and treads to lie upon the notches, and pass over beyond the thickness of the string-boards : the ends of the steps are concealed by means of ornamental pieces called BRACKETS. Geometrical stairs are generally bracketed ; but, in dog-legged and open- newelled stairs, only those of the best kind are bracketed. A PiTCHixc-PiECE is a piece of timber wedged into the wall, in a direction perpendicular to the surface of that wall, for supporting the rough strings at the top of the lower flight, when there is no trimmer, or where the trimmer is too distant to be used for the support of the rough strings. BEARERS are pieces of timber fixed into, and perpendicular to, the surface of the wall, for supporting winders when they are introduced ; the other end of the bearers is fastened to the string-board. A NOTCH-BOARD is a board into which the ends of the steps are let : it is fastened to the wall, or one of the walls, of the stair-case. CURTAIL-STEP is the lowermost step of stairs, and has one of its ends, next to the well-hole, formed into an ornament representing a spiral line. These are the principal parts which belong to a stair or stairs ; other parts connected with it belong to the HAND-RAIL, and will be all defined in the fol- lowing article. 24. 3 B 190 THE NEW PRACTICAL BUILDER, &C. CONSTRUCTION OF DOG-LEGGED STAIRS. HAVING taken the dimensions of the stair, and the height of the story, lay down a plan and section upon the floor, to the full size, representing all the newels, string-boards, and steps ; by this method the lengths and distances of the parts will be ascertained, as also the angles which they make with each other. The quantity of head-room, the situation of apertures and passages, will also be ascertained, and will determine whether quarter-spaces, half- spaces, or winders, are to be introduced. But, in order to give the most variety in the construction, we shall suppose the stairs to have two quarters of winders ; the whole being represented as framed together, the string-board will show the situation of the pitching- pieces, which must be put up next in order, by wedging the one end firmly into the wall, and fixing the other end into the string-board ; which, being done, put up the rough-strings, and put up the carriage part of the flyers. In dog-legged stair-cases the steps are seldom glued up, except in cases where the nosings return ; we shall therefore suppose them in separate pieces, and proceed to put up the steps. Place the first riser in its intended situation, fixing it down close to the floor, the top at its proper level and height, and the face in its true position. Nail it down with flat-headed nails, driving them obliquely through the bottom part of the riser into the floor, and then nailing the end to the string- board. Place the first tread over the riser, observing to give the nosing its proper projecture over the face of the riser ; and, to make it lie more solid upon the string, notch out the wood at the farther bottom angle of the riser, where it is to come in contact with the rough-strings, so as to fit it closely down to a level on the upper side, while the under edge beds firmly on the rough-strings at the back edge, and to the riser at the front edge : nail down the tread to the rough-strings, by driving the nails from the place on which the next riser stands, through that edge of the riser, into the rough-strings, and then nailing the end to the string-board. JOINERY. 191 Begin again with the second riser; which, being brought to its breadth, and fitted close to the top of the tread, so that the back edge of the tread below it may entirely lap over the back of the riser, while the front side is in its real position. Then nail the tread to the riser from the under side, taking care that the nails do not go through the face, which would spoil the beauty of the work. Proceed with riser and tread, alternately, until the whole of the flyers are set and fixed. Having finished the first flight of steps, fix the top of the first bearer for the winding-tread on a level with the last parallel riser, so that the farther edge of this bearer may stand about an inch forward from the back of the next succeeding riser, for the purpose of nailing the treads to the risers, up- wards, as was done with the treads and risers of the flyers. The end of this bearer being fitted against the back of the riser, and having nailed or screwed it fast thereto, fix then a cross-bearer, by letting it half its thickness into the adjacent sides of the top of the riser, and into the top of the long bearer, so as not to cut through the horizontal breadth, nor through the thickness of the riser, which will weaken the long bearer, and injure the appearance of the work: then fix the riser to the newel. Try the first winding step-board to its place ; then, having fitted it to its bearings, and to the newel, with a re-entrant angle, or bird's mouth, fix it fast. Proceed with all the succeeding risers and step-boards until the winders are complete. Having finished the winders, proceed with the retrogressive or upper flight, exactly in the same manner as has been done with the lower flight. The workman must then proceed to strengthen the work in the following manner : fix rough brackets into the internal angles of the risers and step- boards, so that their edges may join upon the sides of the rough-strings, to which they are fixed by nails, and thus the work is completed. N.B. In the best dog-legged stairs the nosings are returned, and sometimes the risers mitred to the brackets, and sometimes mitred with quaker-strings. In this case, a hollow must be mitred round the internal angle of the under 192 THE NEW PRACTICAL BUILDER, &C. side of the tread and face of the riser. Sometimes the string is framed into a newel, and notched to receive the ends of the steps ; and, at the other end, a corresponding notch-board, and the whole of the flyers are put up in the same manner as a step-ladder. By paying proper attention to what has here been said, a workman of good understanding will be able to execute such stairs, and put them up in the most sufficient manner, although he might never have seen one made or put up before. BRACKETED STAIRS. Here the same method of laying down the plan and section must be ob- served as in dog-legged stairs. The balusters must be neatly dovetailed into the ends of the steps, two dovetails being put in each, in such a manner that one of the balusters may have one of its faces in the same plane with the riser, and the other face in the same plane with the face of the bracket. GEOMETRICAL STAIRS. The steps of Geometrical Stairs ought to be neatly finished, so that they may .present a handsome appearance. The risers and step-boards ought not to be less than one inch and a quarter thick. The risers and step-boards ought to be well glued and secured together, with blockings glued in the internal angles. When the steps are set, the risers and step-boards must be fixed together by screws, passing from the under side of each horizontal part into the riser. The brackets must be mitred to the risers ; and the nosings, with a cavetto underneath, should be returned upon the brackets, and stopped upon the string-board. The under side may be finished with lath and plaster. In many old buildings, where the principal stairs were constructed of wood, it was customary to panel the soffit; but this is now very seldom done, except in pulpit-work, as the difficulty and time required to execute the work occasions very great expenses to the proprietor. JOINERY. 193 Geometrical Stairs are sometimes finished without brackets, the risers being mitred to the string-boards, instead of brackets ; and this mode is mostly practised in ordinary works. As the parts of a geometrical stair form the subject of the following article upon hand-railing, the other connecting parts will be specified in their proper place. Geometrical Stairs are mostly on a semi-circular plan, with a flight below, and another above; sometimes the plan is formed by four lines, at right angles with each other, and connected by quadrants of circles. Figure 1, pi. LI, represents DOG-LEGGED STAIRS with flyers only ; that is, which consist of steps of equal breadth. No. 1 is the Plan, showing the length and tread, or breadth, of the steps ; or that which shows their exact area. No. 2, the Elevation, shows the sections or ends of the steps. Figure 2 represents DOG-LEGGED STAIRS with winders, connecting the two straight flights. No. 1 is the Plan, showing the areas of the heads of the steps, as before. No. 2, the Elevation, shows the ends of the steps in the two flights, and the risers in both quarters of the winders. Figure 1, pi. LII, is a GEOMETRICAL STAIR-CASE, without winders. No. 1 is the Plan, and No. 2, the Elevation of the same. Figure 2, a GEOMETRICAL STAIR-CASE, with winders in both quarters. No. 1 is the Plan, showing the areas of the steps ; and No. 2 is- the Elevation, showing both the height and breadth of the steps, as, also, the proper turnings of the rail. Figure 1, pi. LIII, represents Geometrical Stairs, with winders in one quarter. No. 1 is the plan ; and No. 2, the Elevation, shows the turnings of the rail, agreeably to the plan. Figure 2 is a Plan and Elevation of a Geometrical Stair-case, with winders adjoining each flight, and a space between the winders. No. 1 is the Plan ; and No. 2, the Elevation of the same. Figure 1, pi. LIV, is a representation of an Elliptic Stair-case. No. 1, the Plan ; No. 2, the Elevation. No. 3 is the Elevation of the Twist and Scroll, which terminates the rail. 24. 3c 194 THE NEW PRACTICAL BUILDER, &C. No. 4<, Plan of the Scroll, agreeing an size with No. 3. In No. 1 the steps of the stair are equally divided round the wall-line, and equally divided round the inside of the rail. This mode of division lessens the acuteness which would otherwise be occasioned at the angles when the lines are drawn to the centre of the plan. The effect of drawing the lines which represent the risers of the steps is exhibited VOL fig. 2, where the half of the outside ellipse is divided equally, and the lines which represent the risers are drawn to the centre of the plan. When the steps are equally divided at the well-hole, the development of the external as well as the internal points of the steps will form a straight line. But, if they are divided as in fig. 2, as the breadth of the steps in- creases towards the extremities of the major-axis, the points of the external and internal angles of the development will be in a curve of contrary flexure. There are several ways in which we can conceive the steps to be divided : one of these is to suppose them perpendicular to one of the curves ; for they cannot be perpendicular to both: but this method renders the steps very unpleasant to the eye, as it makes them so very broad next to the wall, and so very narrow at the rail, that it has a very unpleasant effect in comparing the whole together. Another method is to proportion their breadths to the curvature of the wall and rail-lines: but this must be done only with regard to the inner curve, as the steps ought to be of an equal breadth at the middle. Figure 1, pi. LV, is the representation of GEOMETRICAL STAIRS, consisting of a series of flyers between two quarters of winders. No. 1 is the Plan ; No. 2, the Elevation. No. 3 is the development of the steps next to the well-hole. This is found by extending the base-line upon a straight line ; as AB, No. 1, upon AB, No. 3; the arc BC, No. 1, upon BC, No. 3; the straight line CD, No. 1, upon CD, No. 3; DE, No. 1, upon DE, No. 3 ; and EF, No. 1, upon EF, No. 3: so that ABCDEF, No. 3, will be the line ABCDEF, No. 1, extended in a straight line. Through all the points of division draw lines perpendicular to AF. JOINERY. 195 Set the heights of the steps upon the perpendicular line FG, and through the points of division draw lines parallel to AF ; then the horizontal parts meet- ing their corresponding heights will form the steps. Figure 1, No. 4, is a development of the steps and rail next to the well- hole, found exactly in the same manner as No. 3. These developments are of the greatest utility to the workman, as they enable him to judge of the proper form of his rail, so as to make it in the most agreeable and easy manner. Wherever there is an angle, that angle must be reduced, by taking it off in the form of a curve, which is more pleasant to the eye than a sudden change in the direction of the line. The taking away of an angle, either of the rail or string-board, is called by workmen the easings of the rail, or of the string. Figure 1, No. 5, is the Plan of the Scroll to about one-sixth part of the real size. Figure 1, No. 6, the side Elevation of the Scroll. Figure 1, No. 7, the wreath or twisted part, at the turn of the rail above, between the winders and the upper flight of steps. THE METHOD OF DESCRIBING THE SECTION OF THE HAND-RAIL AND MITRE- CAP FOR DOG-LEGGED STAIRS ; (fig. 1, pi. LVI.) Draw the straight line ab, and make it equal in length to the breadth of the rail. Bisect ab by the perpendicular i G, cutting ab at w ; make mi equal to two-sevenths of the depth of the rail, and wd equal to five-sevenths of the said depth. Draw dn perpendicular to di. Produce di to c, and make we equal to wd. From d, with the radius di, describe the arc ign. Join nb, and produce nb to g. Draw gd, cutting ab in r. Make ws equal to wr. Join ds, and produce ds tof. Join cr, cs, and produce cr to h and /, and c* to e and k. From d, with the radius di, describe the stcfg. From r, with the radius rg, describe the arc gbh-, and from s, with the radius sf, describe the arcfae. Make hi equal to hr, ek equal to es, and join kl. From k, with the radius ke, describe the arc et; and from /, with the radius 196 THE NEW PRACTICAL BUILDER, &C. > ftoqu t d.t9iv-~ Hi, describe the arc hi, and draw tu and tv, parallel to id; which completes the geometrical part of the section. From any centre, as G, in the line ED, and id produced, with a radius equal to that of the mitre-cap, describe the circle ABEFD. Draw a A., iB, parallel to iC. Join AC and BC, for the angle of the mitre ; C being taken at pleasure. Draw DE the diameter of the mitre-cap. In the curve of the section of the rail, take any number of points, m, m. Draw the lines p x parallel to *G, cutting AC in the points C, x. From G, with the radii Ga;, Gx, &c., describe arcs #P, xP, &c. Draw the ordinates PM, PM, &c., per- pendicular to DE. Then, when as many of the points are found as is neces- sary, draw a curve through them, and this curve will be the contour of the section of the mitre-cap. Figure 2 exhibits the method of drawing the Swan-neck of a dog-legged stair. No. 1, the Swan-neck and part of the capital ; No. 2, the base ; No. 3, the base, more at large. Figure '3, the method of tracing the brackets round the winding-steps from the flyers. Supposing CD equal to their length, and AB equal to those of the flyers. Having divided CD in the same proportion as AB, make the I ordinate pm equal to PM. HAND-RAILING. THE ART OF FORMING HAND-RAILS round circular and elliptic well-holes, without the use of the cylinder, is entirely new. Mr. Price, the author of ' The British Carpenter? is the first who appears to have had any idea of forming a wreath-rail. Subsequent writers have contributed little or nothing towards the advancement of this most useful branch of the Joiner's profes- sion, and have contented themselves with the methods laid down by Price, which were very uncertain in their application ; and, consequently, led to very erroneous results in the practice. JOINERY. 197 The first successful method of squaring the wreath, upon geometrical principles, was invented and published by Mr. Peter Nicholson, in 1792, in a work called ' The Carpenter's New Guide,' a book well known to architects and workmen. No previous author seems to have had any idea of describ- ing the section of a cylinder through any three points in space, making a mould to the form of the section, and applying it to both sides of the plank, by the principles of solid angles ; so that, by cutting away the superfluous wood, the piece thus formed might have been made to range over its plan. Since the first invention of the method, the author's experience and re- searches have produced many essential requisites, which were not thought of at first ; so that this branch, as here presented, is now much improved. The principle of projecting the rail furnishes the workman with a method by which he can ascertain, with great precision, the thickness of the plank out of which the rail must be cut. To do this in the most convenient way, the diagram must be made to some aliquot portion of the full size, which will supersede the necessity of laying it down on a floor. It must, however, be observed that the thickness of stuff found by this method is what will com- pletely square the wreath or piece. But, as the rail is reduced from the square to an oval or elliptic section, much thinner stuff may be made to answer the purpose ; so that, generally, for rails of the common size of 2| or 2-j inches thick, instead of requiring a three-inch plank, one of two and a half may be made to answer the purpose. 1. The HAND-RAIL of a stair is that which is put up in order to prevent accidents, by falling into or through the well-hole. 2. The PRISMATIC MOULD, formerly used for forming the wreaths, and fitting the rail together, is now of no other use than merely to help the cdn- ception of the learner. In this case, we shall still be obliged to use the idea of such a prism, which was called by workmen a CYLINDER, whatever might be the form of the base or right section. But, as the word cylinder is used to define a geometrical solid, and has had the sanction of the learned for upwards of two thousand years, we must not use the word cylinder in two different senses without some word of qualification ; as, otherwise, it will be 25. 3o 198 THE NEW PRACTICAL BUILDER, &C. impossible to know which of the two bodies is meant. We shall therefore call the cylinder used by workmen a cylindrome or working-cylinder. 3. Supposing the working-cylinder to be covered with a thin pliable sub- stance, as paper, and to be inserted in the well-hole, as if it were a newel, and the planes of the risers and treads to be continued, so as to intersect the covering; the indented line, formed by the intersections of the risers and treads, in the development of the covering, supposing it extended on a plane, is called the envelope of the well-hole. 4. The straight line formed on the envelope with the base of the cylinder is called the base of the envelope. 5. The straight line passing through the points of the external angles, on the development of the steps, is called the line of nosings. THE THEORY OF HAND-RAILING. Suppose any line to be drawn on the surface of the working-cylinder, and the working-cylinder to be cut entirely through, from this line to the opposite surface, so that a straight line, passing through any point of the line, drawn perpendicular to the surface, will coincide with the section made by cutting the solid, through the line thus drawn ; every such line will be parallel to the base of the working-cylinder. Suppose that the upper portion of the working-cylinder, separated from the lower, to be removed, and the lower to be inserted in the well-hole; then, if the surface of separation coincide with the nosings of the steps, while the base rests on the floor ; and, if we again suppose the whole to be elevated to' a certain height without turning, so that the base may be parallel to the floor, the surface of separation will form the top of the hand-rail in the square ; and the two vertical sides of the hand-rail will be a portion of the vertical surfaces of the working cylinder. Again, suppose that, while the portion of the working cylinder, thus formed, remains in the situation now described, another portion next to the top is again separated from the lower portion, but not removed, in such a JOINERY. 199 manner that the uppermost part may be every where of a certain thickness between the surfaces of separation ; the upper part, thus separated, would exactly form the hand-rail in the square ; and this the solid which we would wish to form, first in parts, and then to put the parts together, so as to con- stitute the whole solid Helix, as if it had been cut out of the solid of the .'_ rvji/TTj working-cylinder. ,, ro The form of the solid helix, now defined, is called by workmen a square rail ; the method of preparing the rail, in parts, of this form, is called the squaring of the rail. The square rail is therefore contained between two opposite surfaces, which are portions of the surfaces of the working-cylinder, and two other winding-surfaces, contained between each pair of curves of the helix. The PLAN OF A HAND-RAIL is the space or area which the base of the work- ing-cylinder would occupy on the floor. This area is therefore bounded by two equi-distant lines, on which each of the working-cylindric surfaces stand erect, and the breadth of the space between these two equi-distant lines is called the breadth of the rail. The STORY-ROD is a rod of wood, equal in length to the height of the stairs, or the distance between the surface of one floor and that of another. It is divided into as many equal parts as the number of steps in the height of the story : its use is to try the steps as they are carried up. For the conveniency of forming a square rail out of the least quantity of stuff, and in the shortest time, the rail is made in various lengths ; so that, when joined together, the whole may form the solid intended. If the rail thus joined be set in its true position, and if the joints be in planes perpendicular to the horizon, and to the surface of the working- cylinder, the joints are called splice-joints. But if each joint be in a plane perpendicular to one of the arrises, the joint is called a butt-joint. It is evident that any portion of a hand-rail may be made of plank-wood of sufficient thickness and length : because such a portion of the rail may be considered as a portion of the working-cylinder, contained between two parallel planes, which may represent the faces of the plank ; and the two 200 THE NEW PRACTICAL BUILDER, &C. sections of the working-cylinder, made on these planes, will represent a mould to draw the same upon the surfaces of the plank : then, if the sur- rounding wood be cut away in the same manner, the vertical sides of the rail will he formed agreeably to the definitions which we have given. A mould made to form a section of the working-cylinder is called a face- mould, A mould made to cover one of the vertical surfaces of a square rail is called a falling mould. A falling mould made to cover the convex surface of the square rail is called the convex falling mould; and that which is made to fit the concave surface is called the concave falling mould. The form of the falling moulds can be ascertained by geometrical rules; and, consequently, if the proper portion of the falling mould be rightly applied to the rail-piece, when cut out of the plank, and if lines be drawn by the two edges of the falling mould, and the superfluous wood cut away, so as to be every where perpendicular to the surface of the working-cylinder, that portion of the rail will be formed. The MITRE-CAP of a rail is a block of wood, turned to some agreeable figure, of a greater diameter than the breadth of the hand-rail. It is used in dog-legged stair-cases, for the purpose of giving a neat appearance to the termination of the rail, at a very little expense. The SCROLL is the termination of the hand-rail of a geometrical stair, in the form of a spiral, and is placed above the curtail-step, which is made to correspond with the scroll. BALUSTERS are vertical pieces fixed on the steps for supporting the hand- rail. In flights, the balusters are placed equi-distant, so that every step may have two balusters ; and that one side of each baluster may be in the plane of each riser, and the whole thickness of each baluster so placed that it may stand within the solid of the riser. In order to keep the hand-rail steady, and to provide against accidental violence, iron balusters must be inserted into the range, at equal distances, and strongly fixed to the steps. In common stairs, where wooden balusters only are used, the balusters are placed against the nosings, and let into JOINERY. 201 the step-board below ; and, being fastened to the nosings with nails, will be very secure. The RAMP, in a dog-legged and open-newel stair-case, is the upper end of a hand-rail adjoining to the newel, formed, on the upper side, into a concavity, but straight with regard to the plan. A SWAN-NECK, in dog-legged and open-newelled stair-cases, is a portion of the rail, consisting of two parts, the lower being concave and the upper convex on the top, and terminating in a framed newel, so as to be parallel to the horizon. A KNEE, in a dog-legged and open-newelled stair-case, is the lower end of a hand-rail, next to the newel, formed either into a concavity on the upper side, or made to terminate upon the newel with a short level, mitred into the raking or sloping part of the rail, which follows the curvature of the steps. The same part of a rail may therefore be both ramped and knee'd; that is, ramped at the upper end, and knee'd at the lower ; or it may be swan-necked at the upper end, and knee'd at the lower. When there is any sudden rise in the balusters, the top of the rail ought to be kept to the same height throughout, as nearly as possible ; but, should the height of the steps lead the top of the rail to irregularity, in the curvature of the rail, the line of fall must be rendered agreeable, by taking away the angles, and reducing the whole to a uniform curve. These curves are called the easings of the rail By this mean, the impediments which the hand might meet, in passing along the back of the rail, will be removed, instead of being suddenly interrupted at every junction. When the risers of three or more steps terminate in the same vertical line, in order to connect the lower and upper ends of the rail in the most agreeable manner, the intermediate part will require to be ramped, as is done in dog-legged stair-cases. But where not more than two risers terminate in the vertical line, the rail is frequently continued, so as to form an elbow in the intermediate part, in such a manner, that the top of the three parts thus connected may be all in one 25. 3 E 202 THE NEW PRACTICAL BUILDER, &C. plane ; which will be as pleasant to the eye as it is convenient ; since the parts, thus joined, may be cut out of the thinnest wood possible. To fix the rail in such a position as to differ the least from being parallel to the line of nosings, or the string-board, the top over the upper part may be depressed half the height of a step, while that over the lower part may be as much elevated ; so that, though the rail may not be entirely parallel with the line of the nosings, the lower part rising higher as it advances towards the intermediate part, or turn, and the upper part approaching nearer to the line of nosings, as it is more remote from that part, the whole appearance will be more agreeable to the eye. Where winders are necessary, and the well-hole very small, the top of the rail must be kept higher over the winders than over the straight part ; as, otherwise, the person who ascends or descends will be in danger of tumbling over into the well of the stairs. The rise of the rail, over the circular plan, cannot be regulated by geo- metrical principles, but must be left to the discretion of the workman. The rail ought not to be raised to any considerable degree over the winders, unless in very extreme cases, as it occasions not only a deformity in the fall of the rail, but a great inconvenience to the workman, by obliging him to prepare the semi-circular part of the rail to different face-moulds ; whereas the same moulds might be alike applied to both parts of the rail. Where this practice is necessary, the easing of the rail, at the upper end, will be over the circular part of the plan ; while that at the lower end will be entirely on the straight part below the winders ; and thus, as it occasions the upper part to have less slope than the lower, so it occasions also the face-mould of the upper part to be considerably shorter than that of the lower part. To FIND THE SECTION OF A CYLINDER, when it is cut by a plane per- pendicular to a given plane, parallel to the axis of the cylinder. (See Fig. \, pi LVII.) Let ABC, (fig. 1,) be the base, and ACRP the plane parallel to, or passing along, its axis, and PR the line of section. JOINERY. 203 In ABC take any number of points, e,f, g, and draw the lines he, if, jg, &c., parallel to AP or CR, cutting AC in a, b, c, d, &c., and PR in h, i,j, k, &c. : perpendicular to PR draw hi, im,jn, Jco, &c. Make hi, im,jn, 8rc., respec- tively equal to ae, bf, eg, &c. Through all the points /, m, n, o, &c., draw a curve, which will be the section required. If the cylinder be hollow, the inner curve will be described in the same manner. When the plane of section makes an acute angle, as in fig. 2. Let STU, (fig. 2, No. 2,) be the angle which the cylinder is to be cut at. Draw DU parallel to ST, at that distance from ST which is equal to the radius of the semi-circular base. Draw Er parallel to AC. From the centre, Z, of the semi-circle, draw ZW, parallel to AP or CR, cutting PR in W. Draw WX parallel to AC, and draw Xr parallel to AP or CR. Join Zr. Draw XY perpendicular to PR. From W, with the radius Zr, describe an arc, cutting XY at Y, and join WY. In the arc ABC take any number of points, e,f, g, &c., and draw the ordi- nates ea,fb, gc, &c., cutting AC in a, b, c, &c. From the points a, b, c, &c., draw ah, bi, cj, &c., parallel to AP or CR, cutting PR in h, i,j, &c. Draw hi, im,jn, &c., parallel to WY, and make hi, im,jn, &c., respectively equal to ae,fb, eg : then, through all the points /, m, n, &c., draw a curve, and it will be the section required. If the cylinder be hollow, the inside curve will be traced in the same manner, and from the same parallels Zr and WY, and by the very same ordinates ; only observing the points where the inner semi-circle cuts these ordinates. Upon the principle here shown, the sections of figures 3 and 4 are to be found : both these figures are segments of cylinders j fig. 3 being cut at an obtuse angle, and fig. 4, at an acute angle. To FIND THE SECTION OF A CYLINDER, so as to pass through three given points in its surface, (figures 1 and 2, pi. LXIV.) Let the seats of the three given points be ABC, in the base ABC of the cylinder, and let ADEC be the plane standing upon AC perpendicularly. 204 THE NEW PRACTICAL BUILDER, &C. Join the points A and C, by the straight line AC, and produce AC to F, and draw BH parallel to AF. Draw AD, BG, and CE, perpendicular to AF or BH. Make AD equal to the height of the point, over its seat A, BG equal to the height of the point over its seat B, and CE equal to the height of the point over its seat C. Join DE and produce DE to F. Draw GH parallel to DF ; and, through the points H and F, draw FI. In AF take any point, J, and draw JI perpendicular to AF ; and draw JK perpendicular to DF. From J, as a centre, with the radius 31, describe an arc cutting JK at 1. From J, with the radius J /, describe the arc Im, cut- ting AF in m, and join ml. In the base ABC take any number of points, a, b, c, d, &c., and draw the lines ae, bf, eg, dh, &c., parallel to FI, cutting AF in the points e,f,g, k, &c. Draw ei,fj, gk, hi, &c., parallel to AD, BG, or CE, cutting DE in i,j,k, I, &c. Through the points i,j, k, I, draw ip,jq, Ttr, Is, &c., and make ip,jq, kr,ls, &c., each respectively equal to ea,fb, gc, kd, &c. ; then, through the points E, p, q, r, s, &c., draw a curve, which will be the section of the cylinder as required. The angle 3ml is that which the plane of section now found, makes with the vertical plane ADEC. The whole of the art of hand-railing depends on finding the section of a cylinder to pass through three given points on its surface; the reader is therefore requested to understand this thoroughly, before he actually com- mences his study upon hand-rails ; for, if the principles are not comprehended, he will always be in difficulties, and liable to spoil his work. The hand-rail of a stair is made in various lengths, and each portion is got out upon the principle of its being the section of a cylinder, and is cut out of a plank not exceeding two and a half inches in thickness. It would not be practicable to get a rail out in pieces for more than a quarter of a circle : a portion of the rail got out in one length, answering to the serni- circumference of the plan, would require a very great thickness of stuff'; how much more then would a whole circumference require ? for the thick- ness of stuff increases in a much more rapid ratio than the circumference. JOINERY. 205 Figure 3 (pi. LVIII) is the method of drawing a curve, which shall touch two straight lines, AB, BC, in two given points A and C. Divide AB, BC, each into any number of equal parts ; beginning at A and B, draw the straight lines 11, 22, 33, 44, &c., through the corresponding points of division, and the lines thus drawn will be tangents to the curve. The reducing of a piece of wood from an angle to a round is called by workmen the easing of the angle. Figure 4 is an application of this to the hand-rail of a stair. Figure 5 shows how the same may be done when AB and BC are unequal. Figure 6 is another method of easing the rail at an angle. Let ABC be the angle, and let it be required to round it in a very small degree. Make Ed equal to Be. From d draw d/ perpendicular to AB, and ef per- pendicular to BC. From,/, with the radius/tf or/V, describe the arcs de and gh, which will form what workmen call a knee in the rail. THE METHOD OF DRAWING SCROLLS FOR HAND-RAILS, ANSWERING TO EVERY DESCRIPTION OF STAIRS. ( fl. LIX.) First, Let it be required to describe a spiral to any number of revolutions, between two given points, in a given radius. Let E (fig. 1) be the centre, EB the given radius, and let the two given points be A and B, between which it is required to draw any number of revolutions. Divide AB into two equal parts, in the point C, and divide AC or CB into equal parts, consisting of one part more than the number of revolutions ; then, in the line EB, make EF and El each equal to the half of one of these parts; then, upon FI, construct the square FGHI. Draw GE and HE. Divide GE and HE into as many equal parts as the spiral is to have revo- lutions ; through these points construct as many squares, of which one side will be in the line FI, and the other sides terminate in the lines GE and HE ; then the angular points of the outer square will be the centres for the first revolution, the angular points of the next less square the centres for the second 26. 3 P 206 THE NEW PRACTICAL BUILDER, &C. revolution ; the angular points of the next less square will be the centres for the third revolution, and so on ; one quadrant of a circle, which is one quarter of a revolution, being described at a time. To apply this to the present example, which is to have two revolutions, divide, therefore, AB into two equal parts, as before, in the point C, and divide AC or CB into three equal parts ; that is, into one part more than the number of revolutions. Make EF and El each equal to half a part, and on FI describe the square FGHI. Join GE and HE, and divide GE or HE into two equal parts, being as many as the number of revolutions, and complete the inner square klmn. Produce FG to Q. From the centre, F, with the distance FB, describe the quadrant BQ. Produce GH to R; then, from the centre G, with the radius GQ, describe the arc QR. Produce HI to S ; then, from the centre H, and, with the radius HS, describe the arc RS. From the centre I, with the radius IS, describe the quadrant ST ; which finishes the first revolution of the spiral. Again, produce kl to U, Im to V, mn to W ; then, from k, with the radius /5-T describe the arc TU ; from /, with the radius U, describe the arc UV; from m, with the radius mV, describe the arc VW ; from n, with the radius nW, describe the arc WA, which will complete the second revolution, and terminate in the point A, as required. To draw the second spiral, and thereby to draw the scroll complete. In the straight line EB set Bp towards E, equal to the breadth of the rail ; then from F, with the radius Fp, describe the avcpq ; from G, with the radius Gq, describe the arc qr ; from H, with the radius Hr, describe the arc rs ; from I, with the radius I*, describe the arc A*, which will complete the scroll. The whole breadth of this scroll, now drawn, is about twelve inches and five-eighths of an inch, and is intended for a very large step. The breadth of the scroll, Jig. 2, is ten inches and a half; the breadth of the next scroll, not shown here, is eight inches and three-eighths of an inch ; which will be a proper breadth for an ordinary-sized step. The breadth of fig. 3 is six JOINERY. 207 inches and seven-eighths of an inch ; which is applicable to hand-rails where there is very little room. All these scrolls are only portions of the great scroll, fig. 1, and show that one mould may be made so as to answer any number of revolutions ; these scrolls are all drawn from the same centres as that of fig. 1. Fig. 1 consists of eight quadrants ; fig. 2, of seven quadrants : the other, consisting of seven quadrants, is not shown, from want of room. Fig. 3 consists of five qua- drants ; and fig. 4, of four quadrants, and may be drawn, independent of fig. 1, by the very same rule, which must be adapted to one revolution : di- vide AB into two equal parts, in the point C, as before. Divide AC into two equal parts, that is, into one part more than the number of revolutions ; because it is to consist of only one revolution. Then, making E the middle of the side of the square, describe the square 1234. From the centre 1, with the radius IB, describe the arc BQ ; from the centre 2, with the radius 2Q, describe the arc QR ; from the centre 3, with the radius 3R, describe the arc RS; and, from the centre 4, with the radius 4S, describe the arc SA, which will complete the scroll, as required. TO FIND THE MOULDS FOR EXECUTING A HAND-RAIL. Let fig. 1, pi. LX, be the plan of the rail ; AB and DE being the straight parts ; BCD the circular part, which is divided into equal parts by the point C ; and, consequently, BC and CD are each quadrants. TO CONSTRUCT THE FALLING MOULD. In fig- 2 draw the straight line AB; and through A draw CE, perpen- dicular to AB. . Make AC and AE each equal to the stretch out of the quadrant BC or CD, (fig. 1) together with the breadth Eg or Dh of a flyer. Draw CD (fig. 2,) parallel to AB. Make CD equal to the height of twelve steps ; that is, equal to the ten winders, together with the breadth of the two flyers, one on each side of the winders. Make Eh equal to the breadth of one of the flyers, and draw hi perpendicular to CE, and make hi equal to the 208 THE NEW PRACTICAL BUILDER, &C. height of a step. Join *E. In like manner, in the straight line CD, make Dj equal to the height of a step. Draw.;/* parallel to CE, and makejfc equal to the breadth of one of the flyers. Join D&, and ki. As it is customary with some workmen to raise the rail higher upon the winders than upon the straight part ; and, as this is altogether arbitrary, draw Im parallel to ki, at such a height as the workmen may think proper. Make ln,no,mp, each equal to mE; then ease or reduce each of the angles Ino andjtmE to curves nwo,pqE; then draw a line rstuv parallel to DnwopqE, .comprehending a distance equal to the depth of the rail, and this completes the falling mould of the rail. . J._4 ,' A,iti\u a\tt,*f. fi *i . ' TO FIND THE FACE-MOULD OF THE RAIL. oa :;: esj.hfli' . It is customary to execute a small portion of the straight rail along with the wreath, or twist, for each quadrant of the rail ; and, as this distance is arbitrary, set off Be and Df,fig. 1, each equal to three inches. Transfer this distance to hx and by, fig* 2. \nfig. 2, draw xu perpendicular to CE, cutting the upper edge of the falling mould at M; also draw yssB* perpendicular to CE, cutting the under edge of the falling mould at n, and the upper edge of the same at ^ Through any point B* draw B*B, parallel to CE, and divide A.X B*B each into two equal parts at the points B, A. Draw BC perpendicular to CE, cutting the top of the rail at C, and draw Ao perpendicular to B*B, cutting the under edge of the rail at o. In Jig- 3 lay down the plan for one quarter of the rail, fig. 1, taking in the straight part. ABC, fig. 3, being the quadrant, and CD the straight part of the rail. . It is found that, if the part of the rail over the plan ABCDFE, /#. 3, were actually executed, that a plane would touch the three points of the rail in the perpendiculars erected upon the plan EBD. Join EF, (fig. 3,) and produce EF to G. Draw EH,BI, and DK, perpen- dicular to EG. Make EH equal in height to At, Jig. 2; BI equal to JOINERY. 209 jig. 2; and DK equal to xu, fig. 2. Join ED, and produce ED to L. Draw BM parallel to EL. Join HK, and produce HK to L ; and draw IM parallel to HL. Join ML, and produce ML to meet EG in G. In GM, take any point, q, and draw pq perpendicular to EG, cutting EG inp; and draw jar perpendicular to HG. From G, with the radius Gq, describe an arc cutting pr in r ; join Gr. In the curve CA take any number of points a, b, c, &c., and draw ad, be, cf, &c., parallel to GM, cutting EG in the points, d, e,f, &c. Draw dg, eh,fi, &c., perpendicular to EG, cutting HG in the points g, h, i, &c. Draw gk, hi, im, &c., parallel to Gr. Make gk,hl, im, &c. each respectively equal to da, eb,fc, &c. ; and, through all the points Jc, I, m, &c., draw a curve, which will give the outer edge of the falling mould. The inner edge, q, r, s, &c., is found in the very same manner : viz. by transferring the ordinates dn, eo,fp, &c. to gq, hr, is, &c., and drawing the curve qrs, which is the inner edge of the falling mould. Though the same principle serves to find the straight part of the rail, as well as the circular part, the straight part of the face-mould will be more ac- curately ascertained thus : Let FD (fig. 3) be the end of the straight part on the plan, and C n the line which divides the straight and circular parts of the rail. Through the points CD draw the ordinates nd, Cv, and D*, parallel to GM, cutting EG in the points d, v, x. Draw dg, vA, and %B, parallel to EH, cutting HG in the points g,A, B. Draw gq, An, and Bt, parallel to Gr. Make gq equal to dn, Au equal to vC, and Bt equal to zD. Draw FC parallel to EH, cutting HG in C. Join Cq and Ct. Draw qu parallel to Ct, and tu parallel to Cq ; then Cqut will be that portion of the face-mould answering to the straight part wCDF on the plan. The joint-line of the face-mould will be very accurately found thus : Produce (fig. 3) GE to w. Draw AM; parallel to MG, and wx parallel to EH, cutting GH produced in x. Draw xy parallel to Gr, and make any equal to wA.; and by this method the whole of the face-mould is found in the most accurate manner. 26. 3c 210 THE NEW PRACTICAL BUILDER, &C. To show the manner in which fig. 4 is determined, it is proper to observe here that the operation is inverted for the conveniency of finding the mould from the under side of the falling mould, instead of from the upper side. In fig. 4, draw the plan ABDFE, arid let the resting points be E, B, D. Join EF, and produce EF to G. Draw DK, BI, and EH, perpendicular to EF. Make DK equal to B*n,fig. 2; BI equal to A o,fig. 2, and EH equal to ED, fig. 2. Join ED, and produce ED to L ; join HK, and produce HK to L. Draw BM parallel to EL, and IM parallel to HL. Join ML, and produce ML to G, and join GH. In GE take any point, p, and draw pq perpendicular to GE, cutting GM in q. Draw pr perpendicular to HG. From L, as a centre, with the radius Lq, describe an arc, cutting pr at v, and join Gv : then com- plete the falling mould as before. 'irniiiMO iftctarf&to srfHuitdftOMiitat TO FIND THE MOULDS FOR EXECUTING A HAND-RAIL, WITH FOUR WINDERS IN ONE QUARTER, THE OTHER BEING FLAT, ROUND A SEMI-CYLINDRIC WELL- HOLE, HAVING FLYERS ABOVE AND, BELOW. To find the Falling Mould, (fig. 1, pi. LXI.) Draw the straight line JI, in which take any point, d, and draw dC perpendicular to JI. In dC make dt equal to the greater radius of the rail, and, through t, draw ab pa- rallel to JI. From the centre t, with the radius td, describe the semi-circle adb. In tC make tV equal to tb, together with three-fourths of tb. Join U and Vb. Produce U to meet JI in T, and Ub to meet JI in H. In JI make HI equal to the breadth of one of the flyers. Draw HG perpendicular to HI, and make HG equal to the height of a step, and join GI. Make TJ equal to the breadth of two of the flyers. Draw JN perpendicular to JI. In JN make JK equal to the height of a step, K4 equal to the height of four steps, KL equal to the height of five steps, KM equal to the height of six steps, and KN equal to the height of seven steps. Draw 4/ parallel to JI, meeting dC in/. Draw Lc parallel to JI, and 1e perpendicular to JI. Draw Mh parallel to L- Teak 745 818 9657802 2462 2488 15550 Poon ................. 579 596 6759200 2221 2266 14787 934 435 5806200 1672 1736 10853 Canadian Oak ......... 872 588 8595864 1766 1803 11428 756 724 4765750 1457 1477 7386 Adriatic Oak .......... 993 610 3885700 1383 1409 8808 Ash 700 395 6580750 2026 2124 17337 Beech ................ 69G 615 5417266 1556 1586 J 9912 Elm 553 509 2799347 1013 1042 5767 Pitch-Pine ..... 660 588 4900466 1632 1666 10415 Red-Pine 657 605 7359700 1341 1368 10000 553 757 5967400 1102 1116 9947 753 588 5314570 1108 1131 10707 Ditto second specimen 738 3962800 1051 1081 Mar Forest Fir- ........ 696 588 2581400 1144 1168 9539 Ditto, second specimen .. 693 531 403 411 3478328 2465433 1262 853 1310 890 10691 522 518 3591133 832 850 Ditto, third specimen Ditto, fourth specimen. . . 556 560 577 518 518 648 4210830 4210830 5832000 1127 1149 1474 1149 1172 1492 7655 7352 12180 STRENGTH OF TIMBER. 271 PROBLEMS. PROBLEM 1. To find the strength of direct cohesion, of a piece of timber, of any given dimensions. RULE. Multiply the area of the transverse section, in inches, by the value of C in the preceding table of data, and the product will be the strength required. Note. If the specific gravity be not the same as the mean tabular specific gravity, say, as the latter is to the former, so is the above product to the correct result. EXAMPLES. Example 1. What weight will it require to tear asunder a piece of teak three inches square, the specific gravity being 745 ? Here 15550 is the tabular value of C. 9 the area of the section. 139.950 Ibs. the weight sought. Example 2. What weight will break a cylinder of ash two inches in dig- meter, the specific gravity being 700 ? .7854 760 : 700 :: 54466 : 50166 Ibs. 4 700 3,1416 area of section. 760) 38126200 (50166 17337 tabular value of C. 3800 219912 1262 94248 760 94248 219912 31416 1GOO 54465,9192 the weight in Ibs. very nearly. 40 272 THE NEW PRACTICAL BUILDER, &C. PROBLEM 2. To compute the deflection of beams fixed at one end, and loaded at the other, with any given weight. RULE. Divide the continued product of the cube of the length in inches, the given weight in Ibs., and the number 32 by the continued product of the tabular value of E, the breadth and cube of the depth in inches, and the quotient will be the deflection required. Note 1. The same rule will apply if the beam is uniformly loaded, by employing the constant number 12 instead of 32. Note 2. If the specific gravity of the given pieces do not agree with the tabular specific gravity, we must operate as taught in the note to problem 1. EXAMPLES. Example 1. An ash batten, three inches square, is fixed in a wall, and projects from it four feet. If a weight of 200 Ibs. be hung on its extremity, how much will it be deflected ? 6580750 tab. value of E. Again, 4 feet = 48 inches. 81.=3 3 x3. 48x48x48=110592. 6580750 Therefore 110592 5-2646000 200 533040750 first product. 22118400 32 41236800 66355200 533040750) 707788800 (1.3, or 1| inches, nearly. 53304-0750 1747480500 1599122250 STRENGTH OF TIMBER. 273 Example 2. What would the same beam be deflected if supported by a prop at half its length ? Note. We know that the deflections of beams of the same scantling are as the cubes of the lengths ; therefore, since the prop reduces the beam to half its length, we have 4 3 : 2" : : 1| : of an inch, the answer. That is, 4 3 =64 2 3 = 8 Therefore 8 x li-j-64 = -J. Example 3. A batten of New-England fir, six feet long, four inches deep, and two and a half inches in breadth, is fixed at one end, and loaded uni- formly throughout its length with 200 Ibs. ; how much will its extremity be deflected ? Tabular value of E = 5967400 bd*= 160 358044000 5967400 954784000 Again, 6 feet = 72 inches. 72 3 = 373248 200 74649600 12 954784000) 8957952000 (.9 inch, or 1 inch, nearly. 8593056000 364896000 PROBLEM 3. To compute the deflection of beams supported at each end, and loaded in the middle, with any given weight. RULE. Divide the product of the cube of the length in inches, and the weight in Ibs., by the continued product of the three numbers, viz., the 34. 3 z 274 THE NEW PRACTICAL BUILDER, &C. tabular value of E, the breadth and cube of the depth, both in inches, and the quotient is the deflection sought. EXAMPLES. Example 1. A square beam of English oak, whose side is six inches, is supported on two walls twenty feet distant, and is loaded on the middle with 1000 Ibs. ; how much will it be deflected ? 5806200 tabular value of E. Again, 20 feet = 240 inches. 1296 = bd\ 240 34837200 240 52255800 57600 69674400 240 2304000 115200 13824000 1000 7524835200) 13824000000 (1.8 7524835200 62991648000 60198681600 2792966400 Example 2. A beam of red pine, eight inches in breadth, and one foot deep, is supported on two walls, distant thirty-three feet four inches ; how much will it be deflected with 2000 Ibs. suspended at its centre ? 7359700 tabular value of E. 13824 = 12 3 x8. 29438800 14719400 58877600 Again, 33ft. 4 in. = 400 inches, 22079100 and 400' = 64000000 7359700 2000 101740492800 128000000000 STRENGTH OF TIMBER. 275 Let, now, a number of these cyphers be rejected, and an equal number of places from the divisor, and we shall have 10174) 12800 (1.25 inches, nearly. 10174 26260 20348 59120 50870 PROBLEM 4. To compute the deflection of beams supported at each end, and loaded uniformly throughout their length with a given weight. RULE. Compute the deflection as before, and five-eighths of the result will be the answer. EXAMPLES. Example 1. A uniform bar of Adriatic oak, two inches square, is rested upon two pillars, distant twenty-four feet ; how much will it be deflected by its own weight, its specific gravity being 960, or 60 Ibs. to a cubic foot ? Tabular value of E is 3885700 16 = 2x2 23314200 3885700 62171200 The above data give 40 Ibs. for the weight of the bar, and 24 feet =288 inches. 288 3 = 23887872 40 621712) 9555148,80 (15.36 621712 5 3338028 8)76.80 3108560 9.6 inches, nearly. 2294688 1865136 4295520 3730272 276 THE NEW PRACTICAL BUILDER, &C. Example 2. A beam of Riga fir, twelve inches square, is to support the brick-work over a gateway, twelve feet wide ; the computed weight of the brick-work is 30,000 Ibs. ; what deflection may be expected ? 4638685 mean tabular value. 20736=12x12'. Again, 12 = 144. 27832110 144 3 = 2985984 13916055 30000 32470795 9618) 8957,9520000 (.93 92773700 86562 5 96187772160 ~^To 8)^ 28854 .58 in. nearly. 1321 Note. When the beam is fixed at each end, the deflection will be two- thirds of that found by rule. PROBLEM 5. To compute the ultimate deflection of beams before rupture. RULE. Multiply the value of U, found in the table of data, by the depth of the beam in inches, and divide the square of the length, also in inches, by that product, for the ultimate deflectioji sought. EXAMPLES. Example 1. A square inch rod of ash, six feet long, is broken by a weight applied to its centre ; how much is it deflected before rupture ? 395 tabular value of U. 6 feet = 72. 72 s = 51 84. 395) 5184 (13.1 inches, nearly. 395 1234 1185 490 395 95 STRENGTH OP TIMBER. 277 Example 2. What will be the ultimate deflection of a similar rod, twelve feet long? Here 12 feet =144 inches. I44 2 =72 2 x4 = 20736. 395) 20736 (52.4 1975 = 4.36 feet, nearly. Ans. ^ \-f* 986 790 1960 Example 3. A half-inch plank of larch, similar to our third specimen, being ten feet in length; how many inches will the centre descend before it fractures ? 518 tabular value of U. .5 depth. 259.0 10 feet =120. 120* =14400. 259) 14400 (55.5 inches, nearly. 1295 . 4 1450 1295 1550 1295 255 Note 1. The same rule applies to beams or rods fixed at both ends. Note 2. According to this rule, our deflection may exceed half the length of the piece. When this takes place, it shows that the ends may be brought together, with a fracture taking place in the beam. Note 3. When a piece is fixed at one end only, the deflection will be eight times what is given by the rule. 35. 4 A 278 THE NEW PRACTICAL BUILDER, &C. PROBLEM 6. To find the ultimate transverse strength of any rectangular beam of timber, fixed at one end, and loaded at the other. RULE. Compute the ultimate deflection by Note 3, last Problem ; divide this deflection by the length, which will give the sine of the angle of deflec- tion ; whence, by a table, find its secant. Multiply this secant by the breadth and square of the depth in inches, and the product again by the value of S x , in the table of data. Divide this last product by the length in inches, and the quotient will be the answer in Ibs. EXAMPLES. Example 1. What weight will it require to break a piece of fir from the forest of Mar, fixed by one end in a wall, and loaded at the other; the breadth being two inches, depth three inches, and length four feet ? The third Note to the last Problem gives the ultimate deflection, fifteen inches* Therefore, ^=.3125 = sine 18 13'. The secant of which is .... 1.0527 18 = 2 x 3 2 84216 10527 189486 Value of S' in tab, = 1310 1894860 5G8458 189486 48) 24822,6660 (517 Ibs. Ans. 240 82 48 342 336 STRENGTH OP TIMBER. 279 Example 2. A square oaken plank of twelve inches the side, projecting eight feet four inches from a solid wall, in which it is fixed ; what weight will be required to break it ? The deflection before fracture is 15.325 inches, therefore 15.325 100 = .15325 = sine 8 49', the secant of which is 1.0119; = 30345 Ibs. nearly. Note. When the beam is loaded uniformly throughout its length, the last result must be doubled. PROBLEM 7. To compute the ultimate transverse strength of any rectangular beam, when supported at both ends, and loaded in the middle. RULE. Multiply the tabular value of S' by four times the breadth and square of the depth in inches, and divide that product by the length, also in inches, for the weight. EXAMPLES. ' ', What weight will be necessary to break a piece of larch, similar to our third specimen, the length being eight feet four inches, the breadth eight, and depth ten inches, being supported at each end, and loaded in the middle ? Here, tabular value of S' is 1262 3200 = 4x8x10* 252400 3786 8/3?. 4 in. as 100) 40384,00 40384 Ibs. Ans. Note 1. If the beam be loaded uniformly throughout its length, the result must be doubled. Note 2. If the beam be fixed at each end, and loaded in the middle, then the result obtained by the rule must be increased by its half. Note 3. If the beam be fixed at both ends, and loaded uniformly through- out its length, the same result must be multiplied by three. 280 THE NEW PRACTICAL BUILDER, &C. III. ON SCARFING AND LENGTHENING BEAMS. WHEN timber cannot be procured of sufficient length to answer a required purpose, it becomes necessary to join two or more pieces together, in order to obtain the extent required, and the mode of uniting the pieces is called Scarfing. SCARFING is, therefore, the art of connecting two pieces of timber together, in such manner as to appear like one piece, and possess suf- ficient strength to answer the purpose which renders this connection necessary. In scarfing timber it is not requisite to pay particular attention to the form of the joint, as that can be altered at pleasure, to meet the views of the mechanic. In each piece of timber to be joined, the parts of the joints that come in contact are called scarfs. Scarfs are formed either by a slanting joint, or by notching the two parts together ; and, sometimes, by a third short piece, which has a mutual connec- tion with the two. The projecting parts on each piece are called tables. When the tables are brought in contact, they are firmly secured in that position by bolts passing through the joints. Some of the most useful methods of scarfing beams will be understood by a reference to the plate LXXI. Figure I, on this plate, shows the method of lengthening beams, without shortening the pieces, by applying an intermediate piece, and connecting the three by means of steps. CENTERINGS FOR BRIDGES. 281 Figure 2 represents a method of joining beams by steps. This method, as well as all that follow, shortens the beam by the entire length of the scarf. Figure 3 is a method of building beams in halves, and locking the two ends that meet each other, by means of a tabulated clamp, indented into the end of each piece. Figure 4 exhibits a method of extending beams by means of indentations, the two parts being locked together by a key or wedge, driven into a mortise left in the middle between the abuting parts of each piece. No. 2 is a per- spective view of the scarf and tables, on one of the pieces. Figure 5 exhibits the junction of two pieces of timber by means of an oblique scarf, which is shown distinctly, at No. 2, by a perspective de- lineation. The whole of these scarfings should be firmly bolted together, with bolts passing alternately from one side to the other ; and, when tabulated, the bolts should be put in the middle of the tables. - IV. CENTRINGS, OR CENTERINGS, FOR BRIDGES. ; IN carpentry, a centre means a combination of timber-beams, so disposed as to form a frame, the convex side of which, when boarded over, corresponds to the intended concavity of an arch. Having carried the piers or abutments to the height designed for the arch to spring, the next object is to set up the centre, the proper construction and erection of which may well be considered as the most masterly operation in the building of arches. In constructing the centre for an arch, the principal object to be kept in view is, to fix the beams in such a manner as to support (without changing 35. 4 B , 282 THE NEW PRACTICAL BUILDER, &C. shape) the weight of the stones and other materials that are to come upon them, through the whole progress of the work, from the springing of the arch to the fixing of the key-stone. This object has not always been suffi- ciently attended to by the architects of foreign countries; for, in many instances, it has been known that the centres of bridges, from the injudicious principles of their construction, have changed their shape considerably ; and, in consequence, the arches built upon them have varied both in form and strength from the intention of the engineer. In Britain, however, no great inconvenience has ever been known to arise from this circumstance; our engineers having constructed their centres on principles calculated to sup- port every weight, and resist every strain to which they might be exposed, and hence have arisen the most perfect models of masonic art that ever marked the progress of human industry. DESCRIPTION OF PLATE LXXII. . Figure 1 shows the centre of Westminster Bridge, \vhich is partly sup- ported by pieces strutting from the footings, and partly by piles driven into the bed of the river. Figure 2, the centre of Blackfriars' Bridge, supported entirely by pieces strutting from the footings and pier. fyf Figure 3, a longitudinal section of an arch of Waterloo Bridge, showing the piles on which the piers are raised, the masses of bricks composing the spandrels, and the centre supporting the arch. The dotted line shows the direction of a curve in which the weight is so distributed, that the different pressures to which the edifice is exposed have no tendency to change the form of the arch; and this curve, from the property just mentioned, has received the appellation of the curve of equilibrium. TIMBER BRIDGES. 283 ' . ' V. CONSTRUCTION OF WOODEN BRIDGES.-(P/. LXXIII.) Figure 1 is the elevation of a wooden bridge supported on the principle of an arch, and may be used where the ground rises on one side more than the other. In order that this bridge may be sufficiently strong, and the road or path-way easily surmounted by passengers and carriages ; the curvature of the lower or supporting arch is much greater than that above, forming -the road or path-way. Figure 2 is a design for a wooden bridge, supported by brackets, pro- jecting more and more as they rise. This design, as well as the following one, is adapted to a straight road or foot-way. Figure 3, design of a bridge, in which the intrados is the arc of a circle ; it is supported by wooden beams over the posts, acting as brackets. To prevent the ends of the supporting brackets from coming to a sharp edge, small keys are let in from the underside. In order that this bridge may be sufficiently strong, when the space between the posts is very great, a truss is placed in the middle, which increases the strength, so that the span may be extended to two, or even three, times the length that it could be without it. THE TIMBER FOOT-BRIDGE, over the Clyde, at Glasgow, is represented in the plate bearing this title, which may be numbered LXXIV. It was designed and superintended by Mr. Nicholson, in the year 1808. Figure 1 exhibits the elevation of the bridge. The curve of the road-way is the arc of a parabola. The land abutments are strong masses of masonry, to which the timbers of the floor, or foot-way, are well secured, by cramps of iron bolted to the stone-work. 284 THE NEW PRACTICAL BUILDER, &C. This bridge was constructed with the view of admitting a certain class of vessels to pass under it. Therefore, to keep the opening between the posts clear, the foot-way is suspended by trusses, which also support the rail-way. The breadth of the footway is about ten feet. Figure 2 is a plan of the beams. Figure 3, elevation of the middle opening. Figure 4, elevation of the opening adjoining one of the land abutments. Figure 5, a transverse section, showing also the elevation of the posts and braces. Figure 6 exhibits the scarfing of the beams, the manner of bolting them together, and their junction with the post by which they are supported. Figure 7 exhibits the manner of joining the braces and posts in the rail-way. This bridge has resisted the most tremendous ice-floods. Floods that have risen to such a height, as only to leave a small part in the middle of the road-way dry. Both Mr. Rennie and Mr. Telford, the most eminent engi- neers in the country, have given their approbation of its construction, with regard to the intention, its simplicity, and strength. MASONRY. 285 CHAPTER VI. MASONRY. MASONRY, practically considered, is the art of shaping and uniting stones for the various purposes of building. It, therefore, includes the hewing of stones into the various forms required, and the union of them, either by joints, level, perpendicular, or otherwise ; or by the aid of cement, or of iron, lead, &c. The operations of masonry require much practical dexterity, with some skill in geometry and mechanics. In treating on this subject, it will be necessary to divide it into several branches, and first we shall notice that local necessity was its parent, and that the fluctuations of the art have always marked the rise and fall of empires. In Egypt, Greece, and Italy, the greater works of masonry included some which were almost incredible in their extent; and their materials were equally so, if considered in detail. These countries seem to have been favoured in every way, in order to be eternized ; for they abounded in por- phyries and marbles, which promoted magnificence in their structures, without peculiar contrivance in arrangement. Modern masonry consists rather in the piling stones to a great height, than in covering an extent of plan; but this requires equal skill, if it be not productive of equal magnificence. 36. 4 c 286 THE NEW PRACTICAL BUILDER, &C. I. OF THE MATERIALS EMPLOYED IN MASONRY. MASONRY, at the present time, and particularly in Great Britain, is confined more to working in free-stone than in marble ; in the former of which our islands greatly abound. The nature or quality of this stone yields with facility to the artificial reduction of it to any required shape. Near Bath, and in several western counties of England, it is sawed into scantlings, by means of a toothed saw ; it is afterwards cut with a hand-saw, hewn by an axe, and dragged and smoothed according to the work it is intended for. PORTLAND FREE-STONE, the most general kind made use of by the London masons, is brought from the Isle of Portland, on the coast of Dorset, in blocks of all dimensions. Its hardness gives it many requisites for producing exquisite masonry. Most of our public buildings are composed entirely of this stone ; and it is frequently made use of in dwelling-houses for kirbs, strings, fascias, columns, cornices, floors of halls, vestibules, stair-cases, &c. The Portland free-stone is decidedly the most handsome free-stone known, being capable of bearing an arris in moulding equal to marble ; but the great expense of its freight and duty, has lately made the Gloucestershire stone its competitor; and it will, perhaps, in a few years, be entirely superseded thereby, except for internal work, where its superior neatness will always gain it the preference. The GRANITES of Cornwall, and the Dundee and Aberdeen stone, are much employed for works requiring great strength and solidity ; as for docks, bridges, &c. These stones are so remarkable for their excessive hardness, that it has been found necessary to bring workmen with the stone to London, as few native masons would undertake to work it. The superior strength and durability of this stone over free-stone, will give masons an opportunity, by judiciously blending and arranging the several qualities of stones, to their various purposes, of producing works far more durable than their former materials gave them the power of doing. MASONRY. 287 SAND-STONE is also common to most counties of England, as well as Scot- land. The quarries of sand-stone about the cities of Glasgow and Edinburgh are, however, not equalled in the kingdom ; this is evinced by the elegance of the buildings in these places, the superiority of which is to be found in mate- rial only, as the taste with which some towns built of brick have been laid out and executed, is not inferior to that displayed by the architects of the north. In the parish of Sproustone, near Kelso, is a sand-stone quarry, belonging to the Dowager Duchess of Roxburgh, of immense value. The stone is of a beautiful silver grey, but does not, when exposed to the weather, retain its colour, soon becoming of a smokey hue. It is remarkably fine, indeed, so fine, that it is unfit for the purpose of whetting an edged instrument. Most part of Kelso, as well as the whole of the bridges in its vicinity, is built of it. Although not a hard stone, it is very durable, and the seam is so deep and long that stones of any dimensions whatever can be procured from it. This stone has not been much used for a number of years, as some misunderstand- ing has subsisted between the noble doweress and the present proprietors, respecting the proper right to the use of the mine. It is hoped, however, that the contest will soon cease, and the adjacent country again have the use of a material which conduces so greatly to its neatness and elegance. There are many excellent beds of sand-stone in the course of the Tweed, but that which we have just mentioned is perhaps the best. A little farther up, on the lands of Dryburgh, the property of the Earl of Buchan, there is plenty of sand-stone, but it is of a deep red colour, and much indented by cross scars and seams ; it is not, therefore, so generally useful as some other at no great distance from it. In the body of this stone there are many small pieces of a softer texture, and a deeper red, that do much injury to it as a durable material. The celebrated statue of Sir William Wallace, the defender of Scotish liberty, is cut out of a mass of this stone; but the sculptor, Mr. John Smith, of Darnick, has shown his skill in selecting the piece which he has used, as it appears too hard to yield at all to the ravages of time and of weather. 288 THE NEW PRACTICAL BUILDER, &C. On the banks of the Tiviot, in the parish of Roxburgh, there .is a bed of sand-stone, of excellent quality, fit for almost any purpose to which it may be applied. This has been used to a great extent, by the proprietor, Sir George Douglas, of Springwood Park, but has not been brought into general use. It is of a white yellow colour, and rises from the bed in masses sufficiently large for every purpose of masonry. In the old buildings, in this neighbourhood, there is a species of sand-stone of a beautiful yellow and uncommonly soft ; but where the builders found it cannot now be ascertained, as no vestige of a seam remains wherein the stone is of such a deep and beautiful colour. Should such be discovered, it will be of immense value to the proprietor, as its beauty would bring plenty of pur- chasers : for, although the fragments that now remain in the old buildings are soft and easily calcined, yet we are not to suppose that it is naturally so ; and must rather conclude that it is of a nature not adapted to resist the weather, and that, in process of time, by being exposed to wetness and drought, it becomes of that softness which we have described. A little farther up, on the precipitous banks of the Tiviot, there are extensive beds of this stone ; but, from their situation, they cannot be wrought with success. In Dumfriesshire are great quarries of sand-stone, both of a red and white colour, exceedingly good ; but the towns of Dumfries and Maxwelton are mostly composed of the red kind. The masses here are very small, and pre- vent the mason's exhibiting their skill, to any great extent, in the cutting of figures, and other ornaments of architecture. If we mistake not, there are numerous and excellent beds of this stone about the town of Kirkcudbright, and the surrounding country : but that which is found abundantly in Ayrshire, on account of its white colour and durable nature, is to be preferred to any that we have yet mentioned, Sprou- stone only excepted. The mines in the other parts of Scotland, being nearly of the same nature with those already described, and this remark applying also to the mines of this sort of stone in England, we need not take any farther notice of it; since ' MASONRY. 28J> its utility, as a building material, is too well known by almost every person to require further detail. The next sort of stone we shall mention is the COMMON WHIN-STONE, which is so well known to almost every individual as to need no description ; but, since we are passing our opinion on stone, we shall say a few words of this also. It is common to almost every country where stone is to be found, but in no place is it so plentiful as in the Highlands of Scotland ; and the best for the purpose of the architect is that which is to be found in the beds of rivers, and places adjacent thereto. When broken, it is generally of a blue colour, of close grit, and remarkably hard ; but there are species of it subject to rot, and to be otherwise injured by exposure to the weather. Almost all rocks are composed of this stone, from which it rises in great masses, too clumsy and too hard for the polishing influence of the mason's steel. This stone is excellent in buildings, and many of the best houses in Scotland are composed of no other material : it gathers a sort of coat when it has been long exposed to alternate wetness and drought ; this coat is of a grey colour, and tends to bring on a decay of the stone, the durability of which may be considerably increased by frequently cleaning, which is prac- ticable in gentlemen's houses, and the like. In Scotland, whole towns are built of whin-stone, and this even when a good access to the best of free-stone can be had. This is shown in villages around Kelso, and at Sproustone, in the county of Roxburgh, where, as we have already shown, the best sand-stone in the kingdom is to be found. II. THE CONSTRUCTION OF BRIDGES. MATHEMATICAL SCIENCE, in all its dignity, is exercised in the theory and construction of bridges. To the enlightened mechanical mind, this, there- fore, has always been a favourite subject. To him especially who delights in ranging the field of geometry and the mazes of analysis, and who feels, as he 36. 4 D 290 THE NEW PRACTICAL BUILDER, &C. contemplates them, that enthusiastic emotion which poets and philosophers alone can know. There are few operations of art to which the man of science can apply his speculative principles so successfully as to the building of bridges ; theory, however, has not always kept pace with practice, but now we are flushed with hopes, arising from recent examples, that science will hereafter have a potent influence in directing the operations of the mechanic, and that the carpenter and builder will lean, with full confidence, on the deductions of the mathematician. The construction of bridges is a subject in itself highly interesting ; the hazard attending their erection, and the ingenuity displayed, have such power over a contemplative mind, as to involve it in extreme astonishment. What, indeed, can be more striking than to see a huge mass of ponderous stones, suspended, not only over small brooks and rivulets, but over the mightiest rivers in the world ? Yea, so far has the intrepidity of the engineer carried him, as to attempt throwing bridges even over arms of the sea ; and here, also, has success been complete. Another source of astonishment is the neatness and elegance of the structure itself. Holding at nought the danger attending their labour, the workmen must show their skill in preparing, plac- ing, and embellishing, their materials, as if ornament, more than utility, were the object of their wishes: but, generally, it will be found that, while orna- ment is obtained, utility is not overlooked ; for the stones are so artfully constructed and laid together, that, in combination, they give mutual sup- port, and the heaviest load can be carried over with perfect safety. The construction of a perfect bridge is, however, a very complex operation; it could not be accomplished by a rude and unintelligent people ; and we find, in the history of bridges, that the erection of arches did not always correspond with the progress of other arts, even where an advantageous intercourse subsisted. The SIMPLEST BRIDGE is obviously that which is composed of a single tree, thrown across a small stream, whose width is not too great for the length of the tree ; but, when this is the case, a higher effort of inventive power is MASONRY. 291 necessary, and this is also soon supplied by stretching another tree from the opposite side, and fastening them together in the middle, by some means or other, such as twisting the branches. This sort of bridge must frequently occur by chance. Mr. Park found such in the interior of Africa. The next step is not much more complex; for the process of twining ropes of rushes or leathern thongs is very simple ; and, when the stream is too wide for one length, we have only to stretch as many as may be neces- sary, from one tree to another, on the opposite banks of the stream, and cover them so as to answer the purpose required. Bridges of this kind have been constructed in South-America. The next mode of forming bridges is to construct piers of stone, at such a distance from each other, as to admit a beam of timber, or a long single stone, to stretch over the width of the stream. This, if the water be shallow, is a very simple operation ; for the piers may be built of rough stone without mortar, and such a process would soon present itself to a rude people. : , But, if the stream be at all times rapid and deep, and the piers built of hewn stone laid with mortar, we may infer that the people who formed such a structure are a people acquainted with the useful arts ; for it is clear that the stones must previously have been quarried and hewn, and before a proper foundation for the pier could be had, the union and experience of various arts were required : hence, then, we conclude that the society in which a work of this sort, of any considerable magnitude, is accomplished, is far advanced in civilization, and has the command of much well-regulated labour. With respect to the mode, now commonly adopted, of constructing arches between piers of stone, the history of the Chinese leads us to understand, that they erected bridges in this manner many centuries before arches were known in Europe, or to the inhabitants of any part of the western world : but, when we consider the many specious claims that this empire has held forth for the high antiquity of its improvements and inventions, we may perhaps feel but little disposed to credit the assertion. Certain, however, we are, that arches are numerous as connected with their inland navigation. *so aAs ar girib ;99iJ ;< 292 THE NEW PRACTICAL BUILDER, &C. In Egypt and in India, the gardens which have produced many useful inventions, both in science and in art, the construction of the arch was totally unknown ; for the temples of the one, and the tombs of the other, were pro- duced by cutting matter away in the manner of sculpture : and further, in the antient works of Persia and Phoenicia no trace of an arch can be found. The Greeks created a school of architecture and sculpture, and carried their knowledge very far in 'these departments, yet even they have but an obscure claim to the knowledge of the arch. It is, at least, certain that they never used it externally in their temples, much less in the construction of bridges. It is to the Romans, then, that we are indebted for this useful application of a great principle in architecture ; but there is no certainty as to the time when it was first practised. It is asserted, by some, that the Romans derived their knowledge of the arch from the Tuscans ; but, if this were admitted, the first knowledge of the art is at least very intimately connected with Greece, for we believe it is not disputed that the Tuscans were a colony of Dorians. But, however doubtful may be the claim to the invention of the arch, we know, from history, that the Romans were the first to apply it to useful pur- poses ; such as forming aqueducts for conveying water to cities, erecting bridges over rivers, vaulting temples, and the like, and in erecting monuments to record the exploits of their heroes. Having thus touched on the rude bridges of uncivilized nations, and shown that the Romans were the first to bring the arch into general use, we shall pass over the description of the massy structures of this kind that have been reared in other countries, and confine ourselves to those only which are con- spicuous in our own kingdom, beginning with the BRIDGE of LONDON. This bridge was originally begun in the year 1176, by a priest, called Peter, curate of St. Mary Colechurch, a celebrated architect of those times, and occupied thirty-three years in building ; but this period will not appear surprising, when it is considered that it was built over a river in which the tide rises, twice every day, from 13 to 18 feet. The bridge at first consisted MASONRY. 293 of twenty arches, but, in 1758, the middle pier was taken down, and the two adjacent arches were converted into one, the span of which is seventy-two feet; its breadth is forty-five feet, and for many ages there were houses along each side of it, but these were removed, when the middle pier was taken down, in 1758. The remaining arches are very narrow, and the piers incon- veniently large, being from fifteen to twenty-five feet in thickness. The passage over the bridge is very commodious, but in other respects there is nothing remarkable about it. The foundation-stone of WESTMINSTER BRIDGE was laid by the Earl of Pembroke, (a nobleman distinguished by his taste in architecture,) on the 24th January, 1739. It consists of thirteen large and two small semi-circular arches, of which the middle one is seventy-six feet span, and the parapets forty-four feet. The engineer was M. Labelye, a native of France; but a Mr. James King, it is believed, had some hand in the design of it. About ten years after this magnificent edifice was completed, another was begun, at a mile lower down the river, known by the name of BLACKFRIARS' ; the design was by Mr. Robert Mylne : it consists of nine arches, of an elliptical form, of which the middle one is one hundred feet in span, and the breadth across the bridge is forty-three feet six inches. The arches being elliptical, and of wider span than those of Westminster, the bridge, of course, has a lighter I appearance. The general stile of it bespeaks a mind emboldened by the suc- cess of his predecessors, to advance a cautious step in the practice of bridge- building. It is a work of very great merit, and will stand a comparison with any other constructed in the same age. It was finished in lOf years. But the glory of England, in bridge-building, may be seen in the Strand or Waterloo Bridge, recently erected by Mr. Rennie, between Westminster and Blackfriars' Bridges ; and in the Southwark Bridge, between London and Blackfriars' Bridges. Many other excellent bridges have also been constructed in Britain, both of wood, iron, and stone, but to enter into a full detail of them here, would swell this article far beyond our prescribed bounds ; we must, therefore, drop this interesting subject, and proceed to give a slight sketch of the theory, as connected with the principles of Mechanical Philosophy. 37. 4 E 294 THE NEW PRACTICAL BUILDER, &C. In the THEORY of BRIDGES we have first to examine the nature of equili- bration ; as, by a proper knowledge of, and attention to, this important sub- ject, the beauty and stability of bridges are to be secured. The celebrated Dr. Hook proposed, as a proper form for an arch, the curve into which a rope or chain would arrange itself, if suspended at the two extremities by pins or nails fixed in a wall ; this curve is commonly called the catenaria, the properties of which have been investigated by different mathe- maticians. We do not, however, mean to give a full development of its principles in this place, but confine ourselves to what is necessary for our purpose, in order to understand the equilibrium of an arch. Let a chain, or string of beads, equal in size and density, be suspended at its extremities by two pins or nails. It will form itself into a curve line. Suppose this curve to be turned steadily round till it obtain a position in the same plane that the beads formed themselves into in the curve ; then all the beads in the arch will, by gravity and an equal pressure, retain the same position; and, consequently, the arch formed will be the catenaria. This arch, however, would support no weight, a mere breath being sufficient to destroy the equipoise. But, should we suppose the beads, in place of being small globes or spheres, to become pieces of a cubical form, equal in height to the diameter of the globes, and retaining the same position, the stability of the arch would be more considerable. The arch is now formed of a mass of truncated wedges, arranged in the catenarian curve, which passes through their centres ; and from this we infer that, when arches of other forms are supported, it is because some catenaria exists some where in their thickness. If the stones are all of the same weight, and touch only in a single point, this curve is, indeed, the only one proper for arches that have to support themselves without being disturbed by different weights or pressures ; but, for arches that, besides the loads continually passing over them, have their haunches filled up, it is not so well adapted till farther modifications take place. Suppose it were required to determine the form of an arch, of a given span and height, proper to carry a road-way of a given form. MASONRY. 295 Let the proposed span be marked horizontally, on a vertical plane of any substance that may answer the purpose ; bisect the span by a perpendicular directed downwards, and equal to the given height ; from the extremities of the span suspend a rope or chain, so that its middle point may be a little below the point marking the intended height of the arch ; divide the span into any number of equal parts, and at the points of division raise perpendi- culars cutting the suspended chain in particular points ; from these points suspend pieces of chain, so adjusted that their ends may meet the line of road-way : and it may be observed that, as those which hang near the haunch bring it down, the crown will rise to its proper position. If the sum of the small chains has, to the large one suspended from the extremities of the span, the same ratio that the material to be filled into the haunches has to the whole weight of the arch stones, this will be the exact form of the arch required to support the given road-way. Having given this mechanical method of determining the form of the arch, under the conditions proposed, we shall next show the requisites of the piers to support the arches. And here we must consider, that the piers and abut- ments of bridges should be so constructed that each arch may stand alone, independently of any other support but its own weight resisted by the piers. Many writers on the theory of bridges have found it necessary to determine the centre of gravity of the semi-arch, by means of the higher calculus ; but we shall here content ourselves with principles of an easier nature, as the deductions will be the more easily applied to practice. The ultimate pressure of arches may, in all cases, be reduced to two others ; viz., the horizontal thrust, and the weight of the semi-arch : but in the arch of equilibration, this pressure is directed perpendicularly to the joints of section, and, when we make an allowance, in other cases, for the effects of friction, the horizontal thrust is not to be feared. Therefore, we have chiefly to attend to the pier itself, taking care to have it of sufficient weight, that it may not overset by turning on its lower point, as a fulcrum. Take a point in the horizontal joint, to represent the centre of pressure, and, at this point, raise a line perpendicularly, to represent the weight of the semi- 296 THE NEW PRACTICAL BUILDER, &C. arch ; at the extremity of this draw another line, perpendicularly, to represent the horizontal thrust, and let the distant extremity of this line be connected with the assumed point ; the connecting line will show the ultimate pressure, and if, when produced, this line fall within the base of the pier, it is evident that it cannot overturn it. But, even though the production of this line fall without the base of the pier, the tendency that the pier has to be overturned may be destroyed by its weight ; for this weight may be supposed concen- trated in the centre of gravity of the pier, and therefore to act in a vertical line bisecting it. We shall now give analytical expressions for these forces, in order to which, let t horizontal thrust ; w =? weight of the half-arch ; p = weight of the pier ; h = height of the pier to the springing of the arch ; b sa the breadth at the base. Then, ht = %bp+%bw ; from which equation any one of the quantities can be expressed in terms of the rest. Thus, suppose it were required to find the breadth of the base of the pier, that is, to express it in terms of the other quantities which enter into its construction. Recurring to our equation, we have b (| p+^w} ht; then, dividing by (ij + | *>), it becomes *= Again, to find the height of the pier, it is h = * 3 . To find the weight of the pier, it jap = ^ To find the weight of the half arch, we have w = 4ft< ~ 6 26p . To find the horizontal thrust, we have t = ft ( 2 P+ 3tt . We shall show the numerical application of these formulae, by assuming numbers to represent the different quantities, without regarding whether they be or be not such as would actually occur in the construction of bridges. Let t- 250 w - 1000 p= 166 h= 15 MASONRY. 297 Now, suppose we want to discover the value of b, the base of the pier; the 4*< *U 4- V J.- 4X15X850 formula is b = ^+^; that is, b - (2 x 166) + (3 x 1000 r The actual operation is as follows : 166 1000 250 234 332 3000 1000 332 15 3332 ) 15000 ( 4.5 = b. 13328 16720 16660 60 b(Zp + 3vi) = To find h, the height of the pier, the formula is h = pvi = 4 X 250 166 1000 2 3 332 3000 332 1000 ) 14994 ( 14.994, or nearly 15. 1000 4994 4000 994 37. 298 THE NEW PRACTICAL BUILDER, &C. To find the weight of the pier, the formula isjo = ~~ = 4X ' 5X25 -^ X4 * X100 . 250 1000 4 4 1000 4000 15 500 4| 15000 4500 2 13500 3 9 ) 1500 13500 166 -p. To find the weight of the half arch, the formula is w = 4M ~* b ? = 4x15X850 8X4JX 166 250 166 4 4| 4| 1000 664 3 15 84 13| 15000 748 2 1496 2 27 ) 13504 ( 500 1496 135 2 04 1000 = w. To find the horizontal thrust, the formula is t = *g**> - * ( x 166 + x IQQQ; 4* 4x10 166 1000 832 + 3000 = 3332 15 13328 4 1666 60 ) 14994 ( 24.99 120 299 240 594 540 540 540 MASONRY. 299 There are other circumstances which enter into the construction of bridges, that would, if considered, give a change to the general equation which we have used ; but as these have been so fully elucidated in other works, written on bridges, it is needless to dwell longer on this subject here : we proceed, there- fore, to give a brief view of the fall of water under the arches of a bridge, it being necessary to pay some attention to this, also, in rearing the edifice. It is known that, if the same quantity of water, that flows in an open extended channel, have to pass through a space any way contracted, it rises above the general level ; and, consequently, has the velocity of its current increased. The piers of a bridge form obstacles in the way of running waters, and in proportion as they contract the channel, the water will rise above the usual level, and the rapidity of the current be increased ; but the investigation of the law of increase, agreeing to a given contraction, would occupy too much room for our present purpose, and would, also, be too refined a speculation for this work ; we shall, therefore, present a table which may be of considerable use to men of practice, as the desiderata is brought at once under his view. THE RISE of WATER produced by OBSTRUCTIONS to the CURRENT, as by Square-ended Piers, or abrupt Projections. VELOCITY. DESCRIPTION OF RIVtRS. OBSTRUCTIONS. Per Second . Fl. In. Per Hour. Miles The Current usu- ally termed. The Bottom which juft bears such Velocities. II -h || 1 || i II i Head of Water, and Velocity produced at the Obstruction, in Feet. Head. Vel. Head Vel. Head. Vel. Head Vel. Head. Vel. ^or3 5 Dull, 5 Gaze and > < Mud, $ .0008 .34 .0009 .35 .0010 .36 .0012 .37 .0017 .42 iorff i Gliding, Soft Clay, .0034 .68 .0036 .69 .0041 .73 .0049 .75 .0069 .83 1 ffor} Smooth, Sand, .0134 1.30 .0145 1.39 .0162 1.46 .011)7 1.5 .0276 1.66 2 IT\ j Uniform ) ( Tenors, ) Gravel, .0536 2.73 .0580 2.68 .0650 2.93 .0788 3. .1104 3.33 3 2A (Ordinary^ r Pebbles, -j 1 Shivers, f .1207 4.09 .130& 4.06 .1462 4.39 .1773 4.5 .2484 5. 4 2 T V ( Freshes, f ) and ( ' Shingle, 3 .2146 6.45 .2320 5.45 .2600 5.86 .3152 6. .4416 6.66 /-Extraor--v (Boulders} 5 3*\ \ dinary / land 8oft> .3353 6.82 .3625 6.94 .4062 7.32 .4925 7.5 .6900 8.33 <^ Floods S. ^Schistus,) G 4 T 'r / and i L Rapids, J $ Stratified J < Rocks, > .4828 8.18 .5320 8.33 .5840 8.78 .7092 9. .9936 10. r Torrents \ ( Indurat- ") 10 ^r } and i ed 1.341 13.64 1.45 13.9 1.625 14.6 1.97 15. 2.76 16.3 CCataracts) ( Rocks, J i {OS 300 THE NEW PRACTICAL BUILDER, &C. The Rise of Water produced by Obstructions to the Current, $c. continued. VELOCITY. DESCRIPTION OF RIVERS. OBSTRUCTIONS Per Second Pa- Hour. The Current usu- ally termed. The Bottom which just beam such Velocities. 1 11 t II i il i II * Head of Water, and Velocity produced at the Obstruction, in Feet. Ft. In Mile.. Heart. Vel. Head. Vel. Head. Vel. Head Vel. Head. Vl. 4 or 3 B Dull, J Oaze and 7 I Mud, S .0024 .468 .005 .87 .010 .83 .023 1.75 .096 2.5 4or6 i Gliding, Soil Clay, .0098 .937 .020 1.75 .039 1.66 .094 2.5 .386 5. 1 Hur$ Smooth, Saud, .0393 1.875 .082 2.5 .158 3.33 .375 5. 1.546 10. 2 IrV 5 Uniform }. ( Tenors, ) Gravel, .1572 3.75 .328 5. .632 6.66 1.500 10. 0.184 0* 3 2^ ( Ordinary ( r Pebbles,-) ^ Shivers, ( .3537 5.62 .738 7.5 1.422 10. 3.375 13.914 4 2A ( Freshes, ( 1 and f (Shingle, J .6238 7.5 1,312 10. 2.528 6.00 25. /-Extraor--\ r Boulders) 5 3* \ dinary / 3.930 . . 8.20 . . 15.8 37.5 (Cataracts) t in all probability, it will be found to answer many useful purposes ; but, in order that the table might be as perfect as possible, it should be divided into two parts : one referring to the velocity, and the other to the difference of level of the river's surface, for a space equal to the breadth of the bridge. The depth should also be particularly attended to, in discovering the ac- quired velocity; but we will not enter into these delicate circumstances, as the data cannot always be obtained with similar exactness. We shall now give an example to show the use of the table. Suppose it were required to build a bridge over a river, 100 feet wide, and velocity three feet per second ; but the abutments and piers together reduce the water-way to 75 feet, that is, diminishing the original width by one-fourth. Look in the table, opposite to 3 feet, and under the obstruction ^, and .2484 will be the head found ; this is about three inches, and therefore is not likely to encroach on the crown. MASONKY. 301 THE RISE of WATER produced by OBSTRUCTIONS to tJie CURRENT, when formed to diminish Contraction, as Piers, with Pointed Sterlings, 8fc. VELOCITY. UliSCRIl'TlON OF RIVERS. OBSTRUCTIONS. Per Second. Ft. In Per Id. i,i. Miles. The Curreut usu- ally termed. Tbe Bottom which just bears such Velocities. A II A II * || * 11 t Head of Water, and Velocity produced at the Obstruction, in Feet. Head. Vel. Head. Vel. Head. Vel. Head. Vel. Head. Vel. Jor3 if Dull, S Oazeand J * Mud, > .0003 .28 .0004 .29 .0004 .30 .0006 .32 .001 .35 forC f Gliding, Soft Clay, .0011 .56 .0014 .58 .0017 .60 .0023 .63 .004 .70 1 **or$ Smooth, Sand, .0045 1.13 .0056 1.16 .0069 1.20 0091 1.26 .015 1.40 2 M ( Uniform ? \ Tenors, ) Gravel, .0182 2.27 .0225 2.33 .0276 2.40 .0364 2.6-2 .060 2.80 3 2A ( Ordinary ( r Pebbles,) ) Shivers, ( .0409 3.40 .0507 3.39 .0621 3.60 .0819 3.78 .135 4.20 4 2 T " T T Freshes, f ) and f C Shingle, 3 .0728 4.54 .0902 4.66 .1104 4.80 .1466 5.04 .240 5.60 ,-Extraor--\ f Boulders} 5 3*r \ dinary / jand Soft .1137 5.68 .1410 5.83 .1725 6.00 .2275 6.30 .375 7.00 ^ Floods v CSchistus,' 6 4 T 'r / and \ L Rapids, J (Stratified? c Rocks, S .1638 6.81 .2030 6.99 .2484 7.20 .3276 7.56 .540 8.40 r Torrents ^ 10 GT'T and * .4550 11.36 .5640 11.66 .6900 12.00 .9100 12.60 1.500 14. v Cataracts) VELOCITY. DESCRIPTION OF RIVERS. OBSTRUCTIONS. Per Per The Current usu- Tbe Bottom which i 11 i II i II i II * ally termed. Velocities. Head of Water, and Velocity produced at the Obstruction, in Feet. Ft. In Miles. Head. Vel. Head. Vel. Head. Vel. Head. Vel. Head Vel. ior3 J Dull, 5 Oaze and ) I Mud, J .0014 .394 .0033 .52 .0067 .7 .0162 1.05 .068 2.1 jor6 i T Gliding, Soft Clay, .0058 .787 .0133 1.05 .0267 1.4 .0647 2.1 .274 4.2 1 Smooth, Sand, .0231 1.575 .0532 2.1 .1069 2.8 .259 4.2 1.086 8.4 2 2* ( Uniform ) ( Tenors, 5 Gravel, .0924 2.75 .2128 4.2 .4276 5.6 1.038 8.4 4.344 16.8 3 2^ ( Ordinary ( ) Shivers, ( .2079 4.325 .4788 6.3 .9621 8.4 2.331 12.6 9.774 25.2 4 2A ( Freshes, f j and f (-Shingle, 3 .3696 5.5 .8412 8.4 1.7104 11.2 4.144 16.8 17.376 33.6 /-Extraor--\ r Boulders j 5 3& \ dinary / jand Soft[ .5775 7.875 1.3200 10.5 2.672 14.0 6.475 21.0 27.15 42.0 < Floods V CSchistus, J 6 ed f 2.3100 15.75 5.32 21. 10.69 28.0 25.9 42.0 108.6 84.0 ' Cataracts) ( Rocks, ) 38. 4 G 302 THE NEW PRACTICAL BUILDER, &C. The effects of accidents to which rivers are liable may be discovered by this table. By inspection, it will be seen that, when the velocity is 10 feet per second, or above it, the inundation is such as to carry away every thing before it. The numbers in the latter part of the table are those which the theory assigns ; but, in fact, they are impracticable. The next subject to which our attention shall be directed is the Water- breakers, or, what the workmen call Sterlings, being the extremities of the piers which meet and divide the water in its course. The opinions of engineers respecting the proper form of this part of a bridge are various, some contending for one form, and some for another; but, it is obvious that, the form which divides the water with least resist- ance, and least contracts the water-way, must be the best ; if, at the same time that these advantages are attained, it retains sufficient strength to sup- port the structure. The right-angled isosceles triangle has been employed by some bridge- builders as a form the most suitable ; this opinion arises from the consi- deration of the right-angle being the strongest of all others ; that is, if two bodies meet each other at right-angles, they will overcome a greater resist- ance than they would do at any other inclination. Others, again, contend for a form of two arcs, containing each 60, de- scribed from the two angles of the pier, and meeting each other in the line dividing the water ; sometimes a semi-ellipse, described on a conjugate equal to the width of the pier, is used; and, not unfrequently, others of the conic sections, or some whimsical figure that pleases the fancy of the architect. The reasons given by the different individuals who have adopted these forms, may be stated as follows : Those who adopt the right-angle contend that it divides the stream best, and that a more acute angle would be too weak. Those who use the semi-circle and semi-ellipse, consider that they are the best calculated to resist the shock of a loaded barge, or other float- ing body, that may come upon them, and that the Gothic arches combine the advantages of both : But we have already observed, that that form must be the best which is best adapted to the figure of the contracted stream, retaining sufficient strength, and allowing the water to pass most freely. MASONRY. ( W3tt 3HT 303 > -.aftf** siAt 9vocffl to tbftdose i9q III.-OF FOUNDATION^'" A SUBSTANTIAL foundation is of the first importance in masonry, as with- out it no work can be durable ; and yet its construction is usually intrusted to the carpenter and bricklayer: the former for piling inferior, soft, and marshy, grounds; and the latter for raising the wall with little or no masonry. PLANKING consists in bedding strong boards of oak or fir, the whole length and breadth of the foundation : the former should never be less than three inches, and the latter five, in thickness ; and they should be scorched all over, previously to being laid down. Many dilapidations having been occasioned by the decay of planks, the use of them has, by some archi- tects, been lately abandoned. PILING is had recourse to, where the magnitude of the superstructure requires that the solid earth should be pierced. The piles, which are forced into the earth, are made of fir, oak, &c., usually about nine or ten inches square. Their length is ascertained by boring the ground ; the ends of the piles are cased or shoed with pointed iron, and the tops surrounded with the same metal. The machine for forcing these piles consists of a frame of wood, braced with strong pieces of timber, and secured by ledgers and feet ; with a cast-iron wheel at the top, about eighteen inches in diameter, and fluted on the outside for a rope, or chain, to move in. This rope, or chain, is attached to the axis of a heavy iron cylindrical beater, which, for general use, weighs about five or seven hundred weight. This cylinder slides some- times in grooves, in the upright frame, and often on the frame of the upright. A ladder is attached for adjusting the chain, and oiling the machine. It is worked by twenty or more men, each taking hold of a rope for that purpose, and thus raising the beater up and down in the frame. When many piles are to be driven, the great labour will frequently require double sets of men to vboii Jeoor esjsq o* 19)BW srfi nte inaioifiug 304 THE NEW PRACTICAL BUILDER, &C. work the beater alternately. The piles are driven as close together as can be ; the tops sawn off, and the intervals filled up, by the Romans with coals, and by us with chalk and rubble ; and the tops planked in the usual manner. In the construction of the London Docks, the piles were all grooved on their opposite sides; and, when driven close to each other, a tongue was forced between to bind the whole together, so as to produce a close chain of wooden piling, from one end of the foundation to the other. Some architects have not deemed either planking or piling eligible for foundations, within infirm or swampy grounds; and have, therefore, had recourse to a cradle of oak, or fir, in quartering, strongly frahied and braced together in bays, and in lengths of from five to ten feet, and of widths proportionate to the superstructure : these frames are again covered over by cross-pieces, or joists, and the whole bedded firmly on the ground, and filled up flush with chalk or rubble. For receiving the foundation of brick or stone walls, this has been found safer than planking ; because, if the quar- ters of the cradle should decay, the rubble would still remain united, and consequently the sinking of the building would be prevented. THE FOUNDATIONS of BRIDGES are generally laid with dry piers, by the water's being, for a time, turned into a new course, or by erecting a coffer- dam. A coffer-dam consists of a double chain of piles, driven into the ground, at a sufficient distance from the intended pier, to admit the works' being conveniently proceeded in ; when the piles are all firmly fixed in the earth, strong horizontal beams are framed and bolted to them with braces to stiffen the intermediate parts ; they are then finally planked inside and out, so as to form a complete case. The void between each casing is then filled with fat mould, or strong earth, so that very little water can percolate, and this is removed by pumping. A more ingenious method has likewise been practised. It consists of forming a strong grating of timber, covered with planks, which at once forms a floating-raft, and the floor upon which the stone pier is to be erected : the pier built on the raft is composed of stones, amply secured, and rendered, by cement, water-tight ; and the whole is so arranged as to float upon the water till it has advanced in height ; so that, a i wan SHT nnr MASONRY. 305 ... , ., , ,, , , ixiWi . >ixg 9/i ,. vBatsiliiotJSadariijfiovf if sunk, it should be above low-water mark, or higher, as found expedient. U PiJty^tvifLi. ftfvt' fifffi ifto flY? J5<4 ./ilQOJ QliJI This levity is obtained either by attaching the raft by ropes to vessels, or by the pier's being worked with vacuities sufficient to render it specifically lighter than an equal bulk of water. The pier is sunk either by letting the water into the vacuities, or by loosening the ropes ; but the bed of the river should be previously prepared for its reception, by machines of the descrip- tion of ballast-heavers. Should the bottom of the masonry-ground prove not to be level, it must be raised by pumping the water out, or by means of the machines in the vessels, and the ground then satisfactorily levelled. In the erection of Westminster Bridge, M. Labelye erected the piers in caissons, or water-tight boxes ; the bulk of the box, though loaded with the pier, producing a mass specifically lighter than an equal bulk of water : after each pier had been erected, the sides of the box served again for boxes of other piers ; the pier was sunk, and raised as above. Similar caissons vi j j.- T>I i f > -o -j -d dzd- were likewise used in erecting Blacktriars Bridge. Till of late years the foundation of bridges was erected in the following manner : The piles were driven into the bottom of the river, in the site of the intended pier, and then cut off a little below low-water mark ; the inter- stices being filled with stone and strong cement ; on these piles a grating of timber was laid, boarded with thick boarding, and thus was formed the floor for the intended pier. The work was then continued, at low water. This is a very simple method, requiring no machine beyond a pile-driving Ol a3bi i.'j';.',' uxoiJ d -^ r - engine. The foundations of the piers of London Bridge, as appeared from that which was removed, when the two small arches were converted into one, was composed of a quadruple row of piles, driven in close together on the exterior site of the pier, and forming a case to receive the stone and cement. So soon as the exterior piles were taken away, the force of the water cleared away the remainder, so that it could not be ascertained whether there were piles in the heart of the pier. To protect the piers of this bridge, sterlings were constructed round them. A sterling consists of an enclosure of piles driven close together into the bed of the river, and secured by horizontal 38. 4 H 306 THE NEW PRACTICAL BUILDER, &C. pieces of timber, bolted by iron to the tops of the piles ; and the void within, to the piling of the pier, filled with chalk, gravel, stone, &c., so as to form a complete defence to the internal piling, upon which the stone piers are erected. IV. OF WALLS. (PLATE LXXV.) THE antients used several sorts of Walls, in which more or less masonry was always introduced. They had their recticular, or reticulated, walls, and also the incertain : of these, the recticulative kind (fig. 2J was the most handsome ; but the joints are so ordered, that, in all parts, the courses have an inferior position ; whereas, in the incertain, (fig. 1.) the materials rest firmly one upon another; and are interwoven together, so that they are much stronger than the reticulated, though not so handsome. In this kind of wall the courses were always level; but the upright joints were not ranged regularly or perpendicularly to each other in the alternate courses, nor in any other respect correspondently ; but, uncertainly, according to the size of the brick or stone employed. Thus our bricks are arranged in ordi- nary walls, in which all that is regarded is, that the upright joints, in two adjoining courses, do not coincide. Walls, of both sorts, are formed of very small pieces, that they may have a sufficient quantity of, or be satu- rated with, mortar, which adds greatly to their solidity. To saturate, or fill up, a wall with mortar, is a practice which ought to be had recourse to in every case, where small stones, or bricks, admit of it. It consists in mixing fresh lime with water, and pouring it, while hot, among the masonry in the body of the wall. The walls called by the Greeks Isodomum, (fig. 4>.) are those in which all the courses are of an equal thickness ; and Pseudo-isodomum, or false, (fig. 3.) when they are unequal. Both these walls are firm, in proportion to the com- pactness of the mass, and the solid nature of the stones, so that they do not level absorb the moistness of the mortar ; and, being situated in regular and MASONRY. 307 courses, the mortar is prevented from falling, and thus the whole thickness of the wall is united. In the wall called emplection, (fig. 6.) the faces of the stones are smooth ; the other sides being left as they come from the quarry, and are secured with alternate joints and mortar. This kind of building admits of great expedition, as the artificer can easily raise a case, or shell, for the two faces of the work, and fill the intermediate space with rubble-work and mortar. Walls of this kind, consequently, consists of three coats ; two being the faces, and one the rubble cone, which is in the middle : but the great works of the Greeks were not thus built, for, in them, the whole intermediate space between the two faces was constructed in the same manner as the faces themselves (Jig. 5.) ; and they, besides, occasionally introduced diatonos, or single pieces, extending from one face to the other, to strengthen and bind the wall. These different methods of uniting the several parts of the masonry of a wall, should be well considered by all persons who are intrusted with works requiring great strength and durability. The existing examples of Roman emplection, with partial cores of rubble work, or brick, sufficiently prove its durability; but that of the Greek was worked throughout the whole thickness of the wall, in the same manner as the facings or fronts, as their temples, now existing, testify. The walls of modern buildings are sometimes built for ornament, but generally combined with solidity. In London, they are regulated by a specific Act of Parliament; but, to prevent dilapidation, it is necessary to strengthen the walls beyond what the law requires, as this law was framed only as a protection from fire. The thicknesses of walls should be regulated according to the nature of the materials, and the magnitude of the edifice; Walls of stone may be made one-fifth thinner than those of brick ; and brick- walls, in the basement and ground-stories of buildings of the first-rate, should be reticulated with stones, to prevent their splitting ; a circumstance which has been too much disregarded by our present builders. Having considered those parts of masonry which are essential to all build- ing in general, we shall now proceed to the ornamental or decorative parts of architecture in which masonry is concerned. 308 THE NEW PRACTICAL BUILDER, &C. V. ORNAMENTAL MASONRY. COLUMNS. These comprise, generally, a conoidal shaft, with a small diminution towards their upper diameter, amounting, generally, to about one-sixth less than the lower diameter. The proportion of columns, from the Egyptians, varied but little ; the columns of this people, in their larger temples, amounting only to about four and a half diameters in height. Those of Greece, as in the Parthenon, at Athens, are little more than five. In the best Roman examples, the proportion was increased to upwards of seven diameters. The columns of all the Grecian remains are fluted, though in different manners. The Doric shafts have their flutes in very flat segments, finished to an arris : sometimes flutings of the semi-ellipsis shape, with fillets, were adopted. The genius of an architect is generally displayed in the application of columns. The Greeks surrounded their public walks with them; their porticos carried this kind of splendour to its highest pitch ; as in them may be found the whole syntax of architecture and masonry. To construct a temple, in the Greek manner, required the greatest taste and judgment, combined with a perfect knowledge of architecture. The Parthenon, at Athens, exhibits, or rather did exhibit, the most elaborate display of ma- sonry in the world. The comparatively perfect state in which the monuments of Greece re- main, is a proof of the great judgment with which they were constructed. The famous Temple of Minerva would have been entire to this day, if it had not been destroyed by a bomb. The Propylea, which was used as a magazine for powder, was struck by lightning and blown up. The Temple of Theseus, having escaped accidental destruction, is almost as entire as when first erected. The little choragic monument of Thrasybulus, as well as that of Lysicrates, are also entire. These instances should impress on mo- dern architects the utility of employing large blocks, and of uniting them MASONRY. 309 with the greatest accuracy ; without which masonry is not superior to brick- laying. The core of the rubble-work of the Grecian walls is impenetrable to a tool ; which is an additional proof of the care which was taken in cementing their masonry. The joining of columns in free-stone, has been found more difficult than in marble ; and the practice used by the French masons, to avoid the failure of the two arrisses of the joint, might be borrowed with success for con- structing columns of some of our softer kinds of free-stone. It consists in taking away the edge of the joints, by which means a groove is formed at every one throughout the whole column. This method is employed only in plain shafts. It appears to have been occasionally used by the antients, though for a different purpose ; vix. to admit the shaft to be adorned with flowers, and other insignia, on the occasion of their shows and games. In the French capital, they affix rows of lamps on their columns, making use of these grooves to adjust them regularly, which produces a very good effect. The shafts of columns, in large works, intended to be adorned by flutes, are erected plain, and the flutings chiselled out afterwards. The antients commonly formed the two extreme ends of the fluting previously, as may be seen in the remaining columns of the Temple of Apollo, in the Island of Delos ; a practice admitting great accuracy and neatness. The finishing the detail of both sculpture and masonry on the building itself, was an universal practice among the antients: they raised their columns first in rough blocks, on them they placed the architraves and friezes, and sur- mounted the whole by the cornice ; finishing down only such parts as could not be got at in the building ; hence, perhaps, in some measure, arose that striking proportion of parts, together with the beautiful curvature and finish given to all the profiles in Grecian buildings. PILASTERS, in modern design, are frequently very capriciously applied. They are vertical shafts of square-edged stone, having but a small projec- tion, with capitals and bases like columns ; they are often placed by us on the face of the wall, and with a cornice over them. In Greek architecture, 39. 4 i 310 THE NEW PRACTICAL BUILDER, &C. they are to be met with commonly on the ends of the walls, behind the columns, in which application their face was made double the width of their sides ; their capitals differing materially from those of the accompanying columns, and somewhat larger at bottom than at top, but without any entasis or swell. PARAPETS. Parapets are very ornamental to the upper part of an edifice. They were used by the Greeks and Romans, and are composed of three parts ; viz. the plinth, which is the blocking course to the cornice ; the shaft or die, which is the part immediately above the plinth ; and a cornice, which is on its top, and projects in its moulding, sufficiently to carry off the rain-water from the shaft and plinth. In buildings of the Corinthian style, the shaft of the parapet is perforated in the parts immediately over the aper- tures in the elevation, and balustrade-enclosures are inserted in the perfora- tions. The architects have devised the parapet with reference to the roof of the building which it is intended to obscure. ASHLARING is a term used by masons to designate the plain stone work of the front of a building, in which all that is regarded is getting the stone to a smooth face, called its plain work. The courses should not be too high, and the joints should be crossed regularly, which will improve its appearance, and add to its solidity. GILLS. These belong to the apertures of the doors and windows, at the bottom of which they are fixed ; their thickness varies, but is commonly about one inch and a half; they are also fluted on their under edges, and sunk on their upper sides, projecting about two inches, in general, beyond the ashlaring. CORNICE. This forms the crown to the ashlaring, at the summit of a building ; it is frequently the part which is marked particularly by the archi- tect, to designate the particular order of his work ; hence Doric, Ionic, and Corinthian, cornices are employed, when, perhaps, no column of either is used in the work ; so that the cornice alone designates the particular style of the building. In working the cornice, the top or upper side should be splayed away towards its front edge, that it may more readily carry off the MASONRY. 311 water. At the joint of each of the stones of the cornice, throughout the whole length of the huilding, that part of each stone which comes nearest at the joints, should be left projecting upwards a small way ; a process by workmen called saddling the joints ; this is done to keep the rain-water from entering them, and washing out the cement. These joints should be chased or indented, and such chases filled with lead, and even when dowels of iron are employed, they should be fixed by melted lead also. RUSTICATING, in architecture and masonry, consists in forming hori- zontal sinkings, or grooves, in the stone ashlaring of an elevation, intersected by vertical or cross ones ; perhaps invented to break the plainness of the wall, and denote more obviously the bond of the stones. It is often formed by splaying away the edge of the stone only ; in this style, the groove forms the elbow of a geometrical square. Many architects omit the vertical grooves in rustics, so that their walls present an uniform series of horizontal sinkings. There are many examples, both antient and modern, of each kind. ARCHITRAVES adorn the apertures of a building, projecting somewhat from the face of the ashlaring ; they have their faces sunk with mouldings, and also their outside edges. When they traverse the curve of an arch, they are called archivolts. They give beauty to the exterior of a building, and the best examples are among the Greek and Roman buildings. BLOCKING COURSE. This is a course of stone, traversing the top of the cornice to which it is fixed ; it is commonly, in its height, equal to the projection of the cornice. It is of great utility in giving support to the latter by its weight, and to which it adds grace. At the same time, it ad- mits of gutters behind it to convey the superfluous water from the covering of the building. The joints should always cross those of the cornice, and should be plugged with lead, or cramped on their upper edges with iron, The Romans often dove-tailed such courses of stone. FASCIA is a plain course of stone, generally about one foot in height, projecting about an inch before the face of the ashlaring, or in a line with the plinth of the building : it is fluted or throated on its upper edge, to prevent the water from running over the ashlaring ; its upper edge is sloped 312 THE NEW PRACTICAL BUILDER, &C. downwards for the same purpose. It is commonly inserted above the win- dows of the ground-stories ; vix. between them and those of the principal story. A PLINTH, in masonry, is the first stone inserted above the ground : it is in one or more pieces, according to its situation, projecting beyond the walls above it about an inch, with its projecting edge sloped downwards, or moulded, to carry off the water that may fall on it. IMPOSTS. These are insertions of stone, with their front faces generally moulded; when left plain, they are prepared in a similar manner to the facias. They form the spring-stones to the arches in the apertures of a building, and are of the greatest utility. VI. THE CONSTRUCTION OF DOMES. THE CONSTRUCTION of a DOME is less difficult than that of an arch, be- cause the tendency, which each part has to fall, is not only counteracted by the pressure of the parts above and below, but also by the resistance of those on each side. Thus a dome may be erected, not only without the centring which an arch requires, but it may likewise be left open at the top, without any key-stone. The masonry of domes differs from that of arching, in the figure of each voussoir, which must fit the void in a sphere instead of its sections. If a dome rises nearly vertical, with its form spherical, and of equal thickness, it should be confined with a chain, or hoop, as soon as the rise reaches to about J4ths of its whole diameter, in order that the lower parts may not be forced out : but, if the masonry is diminished in thickness as it rises, this precaution will not be necessary. The dome of the antient Pantheon, at Rome, built by order of Augustus, is nearly circular, but its lower parts are made sufficiently thick to resist the pressure of the upper parts ; moreover, the spreading is prevented by several projections, which answer the purpose of abutments and buttresses. MASONRY. 313 At the restoration of the arts in Italy, many attempts were made to imitate the dome of the Pantheon ; but the skill of the architect was, in general, insufficient for ensuring success. Anthemius, a Grecian architect, was selected by the Emperor Justinian, for erecting a dome to the church of St. Sophia, at Constantinople, which church was to be in form of a cross, and vaulted with stone ; this dome he attempted to raise on the heads of four piers, about 115 feet high, and at the same extent from each other, with four buttresses to aid the piers. The buttresses were solid masses of stone, extending, at least, ninety feet from the piers to the north and south, so as to form the side walls of the cross. These effec- tually secured the piers from the thrust of the two great arches of the nave which supported the dome. But, when the dome was finished, and had stood a few months, as there was no provision against the thrust of the great north and south arches, the two eastern piers, with their buttresses, were soon pushed from their perpendicular; and, consequently, the dome and half-dome fell in. On the death of Anthemius, Isidorus, another Grecian, succeeded to the work, and, having filled up some hollows to strengthen the piers on the east side, again began to raise the dome ; but, while one part was building, another part fell in. It was now found that the pillars and walls of the eastern semi-circular end were too much shattered to give any resistance to the push which was directed against them ; and therefore seve- ral clumsy buttresses were erected on the eastern wall of the square which surrounded the Greek cross. These were roofed in, so as to form a kind of cloister, and lean against the piers of the dome, and thus oppose the thrust of the great north and south arches. The dome was now turned for the third time ; and though it was extremely flat, and, except the ribs, roofed with pumice-stone, it was soon found necessary to fill the whole, from top to bottom, with arcades, in three stories, to prevent the dome falling a third time. Thus a dome, which was intended to be a beautiful specimen of architecture, was rendered a mis-shapen mass of deformity and ignorance. This example should warn the architect not rashly to undertake what he has not sufficient science, in a proper manner, to perform. 39. 4 K 314 THE NEW PRACTICAL BUILDER, &C. Since that time, several domes have been erected in various parts of Europe, that display every requisite of beauty and strength, peculiar to this species of building. St. Peter's, at Rome, is a superb specimen ; and St. Paul's, at London, though it is more remarkable for its carpentry than its masonry, developes equal talent. The dome of the Hospital of Invalids, at Paris, is a beautiful work ; and that of the church of St. Genevieve, in- cluding the peristyle on which it rests, is, perhaps, the most beautiful speci- men in existence, as to form and composition. The peristyle is formed by fifty-two columns of the Corinthian order, each about fifty feet high, com- pletely insulated, and standing on a circular stylobat. Above the cornice of the peristyle, the dome arises in a beautiful curved line to the top, on which is formed a pedestal and gallery. But, when the dome was raised, the columns composing the interior began to sink with the weight ; and some of the shafts of the columns decorating the interior, which consisted of four naves, with the lantern and dome over them, began to fracture at the join- ings ; this defect was removed by walling up the inter-columniation at the four quarters of the screen, which thus preserved from dilapidation one of the finest monuments of taste and genius. The last specimen which we shall mention is the rotunda of the Bank of England, erected on the principles of the Pantheon, at Rome. The dome takes its spring from a wall of great thickness, and is furnished with several projections externally, which answer the purposes of buttresses. It is open at the summit, which lights the whole of the interior of the build- ing. This is an unique specimen, and bears very honourable testimony to the talents of its constructor, Sir Robert Taylor. VII. VALUATION OF MASONS' WORK. To all the distinct parts of masons' work, a certain value is assigned for the labour and expense of erection and execution. Masons' work is generally measured with two rods, about five feet long, each divided into feet, halves and quarters of feet, and sometimes inches ; but MASONRY. 315 the common rule is generally applied to measure the smallelr fractions. Stone-work, exceeding two inches in thickness, is valued by cubic feet ; if less than two inches, it is reckoned as slabs, and valued by superficial feet. All kinds of ornamental work, as groovings, flutings, joints, rebats, throat- ings, copings, &c., are valued by the running foot. The dimensions are put down in a book, ruled in three divisions on its left-hand side, the middle being about one-third of those on its sides, and is used for the inches and parts : the left-hand column is for the feet and the number of times the dimensions are to be repeated ; and the right-hand column for the quantities, when cubed and squared ; for it often happens that there are several pieces of the same size, and these are marked down, as well as the the nature of the stone, and the species of labour required for working it. The dimension-book will thus stand in the following manner : 3 6 (J 3 3 : *i Portland Landing 9. l 6 7 6 84 : 4 Plain Work Do. 3 9 3 7 6 22 : 6 Groove Do. Thus every portion of material and labour is accurately ascertained. After this has been done, a loose sheet of paper is ruled into as many columns as there are species of work, which is written over the head of each : as, beginning with cube of Portland-stone, is placed in the column under that head ; and the same for plain-work, sunk-work, moulded-work, and each species of running-work, separately. They are cast up at the bottom of each column, and from them made out into bills, beginning with cubes, then superficies ; and, lastly, ornamental-work. For measuring, cubing, squaring, valuing, and finishing the account, surveyors are allowed two and a half per cent, on the gross amount. Plain-work consists merely in the cleaning up of its surface, and all is measured which is seen. I 316 THE NEW PRACTICAL BUILDER, &C. Sunk-work is that which has been partly chiselled away, as the tops of window-cills, &c. Moulded-work is that which is formed into various forms on the edges, as cornices, architraves, &c. The dimensions of moulded-work are ascertained by girting the whole round with a piece of tape, over and into all its several parts ; the length of the tape will give the width of the moulded- work, and then taking its length, and squaring them together, the superficial quantity of moulded-work will be given. A distinct valuation is attached to each kind of labour ; and, as this varies in different places, it would be of little use to insert any here. In London, however, the prices are uni- form for each separate kind. VIII. DESCRIPTION OF THE SECTIONS OF ARCHES. (PLATE LXXVI.) To DESCRIBE A PARABOLIC ARCH, the span and height of the arch being given METHOD 1. Figures 1 and 2. Let AB be the span : Bisect AB in the point C, and draw CD perpendicular to AB. Make CD equal to the height of the arch. Produce CD to E ; making DE equal DC, join EA, EB. In AE set off the distances Al, A2, A3, &c., so that the parts may be all equal ; and, in EB, set off the parts El, E2, E3, &c., so that the differences may be all equal. Join the corresponding points 1,1 ; 2,2 ; 3,3 ; &c. and the inter- sections of the several lines will form the parabola required. Fig. 1 is adapted to a segment, where the rise of the arch is considerable. Fig. 2 is adapted to the head of an aperture, where the radius of curvature of the arch is very great, or where the deflection of the curve from a straight line is but small. METHOD 2. Figure 3. AB and CD being as before, draw DE parallel to AB, and AE parallel to CD. Divide AC into any number of equal parts, and AE into the same number. From the points 1, 2, 3, &c. in AE, draw MASONRY. 317 lines to D, intersecting perpendiculars to A, drawn from the points 1, 2, 3, &c., in AC. Through the points of intersection draw the curve AD. In the same manner draw the curve BD. To draw a straight line perpendicular to the curve from a given point li, (fig- 3J as a joint. In the curve take any other point, as B, at pleasure, and join B/z. Bisect BA in e, and draw eg perpendicular to AB, intersecting the curve in the intermediate pointy. Make fg equal tQ/e; join eg, and draw hi perpen- dicular to eg ; then hi will be the joint required. A PRACTICAL METHOD of describing the CURVE of an ARCH, and of drawing the joints. (Figure 4i.) Let AB be the axis major. Bisect AB in C, and draw CD perpendicular to AB, and make CD equal to the semi-axis minor. Draw DE and AE, res- pectively parallel to AB and CD. Divide AC and AE each into three equal parts. Produce DC tog, and make Cg equal to CD. Draw lines from the points of division in AE to the point D, to intersect other lines in ef, drawn from g through the points of division in AC ; then, e andy will be points in the curve. Bisect /'D by a perpendicular, meeting Dg produced in h, and join hf, intersecting AC in L Bisect ef by a perpendicular, meeting gh in '. Draw * q parallel to AB. From i, with the radius if, describe the arcfq. Join q A, and produce q A to meet the arc fq : join the point of meeting, and the point i intersecting AB in /. From h, with the radius h D, describe an arc D/; from i, with the radius if, describe an arc fe ; and from /, with the radius Ic, describe the arc cA. By placing the centres in the same position, the other half, DB, of the semi-elliptic arc, ADB, will be described. To DRAW a TANGENT to a semi-elliptic arch, the axis-major being hori- zontal. (Figure 5.) Find the focii u and v. Let s be a point in the curve : join su and sv. 40. 4 L 318 THE NEW PRACTICAL BUILDER, &C. Draw st, bisecting the angle usv, and st will be the joint required. In the same manner any other joint, qr, will be found. Or, by finding the position of the centres, and that of the lines for describ- ing the curve, as in fig. 4, the joints may be drawn, as qr from k, st from '. Figure 6 is a semi-circular arch, with the joints marked out. Figure 7 a semi-elliptic arch, with the joints also marked out, and the centres for drawing them as before. IX. STONE-CUTTING, &c. A SEMI-CIRCULAR RIGHT ARCH. (Plate LXXVII.J LET ABCD, (figure 4>,) be the plan of the arch. Divide the opening into two equal parts by the perpendicular EF ; from E, with the radius of the intrados, describe the semi-circle AFB ; and, from the same point E, with the radius of the extrados, describe the semi-circle GHI. Divide the arch GHI, of the extrados, into five equal parts, and draw the radiating lines ko, Ip, mq,nr, for the joints, which will form the heads of the arch-stones. The horizontals and perpendiculars are not drawn as in the last example, because the stones may be formed without making a mould for each stone, by having the head of the arch-stone and thickness of the wall only, in the following manner : Choose a stone of sufficient length to answer the thickness of the wall* and of such breadth and depth as to answer the other dimensions. Reduce the side intended for the intrados to a plane surface, on which draw the two parallel lines ub,cd, (fig. 1,) distant from each other the space between the joints of the intrados; then square one end, as, ac to abed, and parallel to ac draw bd, at a distance equal to the thickness of the wall. Square the other end of the stone, and on the head apply the mould pqml, (fig. k,) so that its extremities pq may coincide with ca, (fig. I,) when applied to one head, and with bd when applied to the other ; then hollow out the intrados, and cut the joints or beds according to the traces, as exhibited at figure 2. MASONRY. . 319 Figure 3 exhibits a stone entirely finished, and all the others are formed after the same manner ; but, instead of forming the heads on the stones them- selves, a bevel, such as shewn in^. 4, koA', may be used with advantage. The higher part of fig. 4 represents an arch-stone, accompanied with the moulds of each side, which will explain the application more particularly. The middle part of figure 4 shows fhe arch complete, with all the stones supporting one another. THE ELLIPTICAL ARCH, WIT*H SPLAYED JAMBS. (Plate LXXVIILJ To find the angles of the joints formed by the front and intrados of an Elliptical Arch, erected on splayed jambs. Figure 1, on the plate, is the plan of the Imposts. Figure 2, the Elevation. The impost A'B'C'D'E' is the first bed;fghik, the second; Imnop, the third; qrstu, the fourth ; vwxyx, the fifth : The other beds are the same in reverse order. The breadth of all these beds is the same as that of the arch itself. The lengths kK, nP, *U, xZ, of the front lines of the moulds of the beds are respectively equal to the lines HF, NL, SQ, XV, on the face of the arch. And also, hg, nm, sr, xw, on the fronts of the moulds equal to the corresponding distances HG, NM, SR, XW, on the face of the arch. The distances kf,pl, uq, zv, are each equal to the perpendicular part AE of the impost. TO FIND THE JOINTS OF AN ARCH IN MASONRY. Let ABC (Plate LXXIXJ be the intrados, and DEF be the extrados, of the arch. Draw DK, AL, CM, FN, perpendicular to the base DF, of the arch. Make the angle DFG equal to the angle which the wall makes with the jambs of the arch, and draw KN, at a distance from GF, equal to the thickness of 320 THE NEW PRACTICAL BUILDER, &C. the wall ; then the plaii of the wall is represented by GFNK ; the abutment on one side, or springing base, is represented by GHLK, and that on the other side by FIMN. Let J be the centre of the given arch. Divide ABC, the intrados of the arch, into as many equal parts as the arch-stones are in number. Through the points 1, 2, 3, &c., draw lines 1 q, 2r, 3*, &c., cutting the one side KN of the wall at q,r,s, &c., and the other side at n,o,p, &c. In figure 2, draw the straight line AB, and make AB equal to the stretch out of the arc ABC, and let Ag, gh, hi, be equal to the distances on the arc ABC. Draw the perpendiculars ACE, gkn, hlo, &c. Make AC equal to AH, figure 1 ; gk equal to Jen, figure 1; hi equal to I o, figure 1 ; im equal to mp, figure 1; and so on : then through the points C,k,l.m, &c., to D, draw the curve line CD. Likewise make AE, gn, ho, ip, &c. respectively equal to AL, kq, ms, &c. (fig. 1.) ; then, through all the points E, n,o,p, &c. to F, draw the curve EF, which will complete the whole developement of the in- trados of the arch. The parts CAwE, klon, Impo, &c., are the heads, or exact forms, of the ends of the stones of the arch, and therefore the moulds of the ends of the arch-stones must be made to correspond to these figures. To find the bevels of the joints of the stones. Let q,s,u, (fig- 1-J be the points of division on the intradosal line, next to the crown of the arch. From the centre J, and through the points q,s, u, &c., draw the lines Jr, J t, J v, &c., then qr, s t, uv, c. will represent the joints of the arch. From the points q,s,u, &c., draw qa, sd, ug, &c., parallel to AH, cutting GF in the points a,d,g, &c. ; and, from the points r,t,v, &c., draw re, tf, vi, &c., pa- rallel to AH, cutting GF in the points c,f, i, &c. Draw be, ef, hi, &c. parallel to DF. Then, to find the bed of the stone answering to the joint qr, draw the straight line abc, No. 1, and make ab equal to ba, figure 1. Draw ae per- pendicular to ac, and make ae equal to qr, figure 1 ; and make ac, No. 1, equal to GK or HI, &c. In No. 1, draw cd parallel to be, and ed parallel to ac ; then will the figure be formed of the bed of the stone. MASONRY. 321 In the same manner we may form No. 2, to the joint st ; and No. 3, to the joint uv; and so on. Nos. 1, 2, 3, 4, 5, &c. are the forms of the moulds for the beds of the right- hand arch-stones ; and Nos. 1, 2, 3, &c. reversed on the right-hand, are the moulds for the beds of the arch-stones on the left-hand : viz. No. 1, and No. 1, are the two bed-moulds next to, and on each side of, the crown ; No. 2, and No. 2, are the second equi-distant bed-moulds from the crown. In working the stones, the beds ought to be numbered the same as the moulds, in order that the arch-stones may be readily applied together. The stone-cutter may first work one of the beds, and cut the form of the mould upon the bed of the stone. OBLIQUE CIRCULAR ARCH. (Plate LXXX.^) Let ABNO (figure 1.) be the face of an oblique arch, of which rsmn is the plan ; AMrm, OQsn, the imposts ; rm and sn being the jambs. Suppose, then, that the obliquity of the arch were given, with the number of stones requisite for its construction, the figure of the stones may be ob- tained by the following construction : Find the centre C, of the span rs, which join with the points of division in the arch, by the straight lines CB, CN, &c. At the point C, in rs, make the angle rCD equal to the given obliquity ; in CD take any point, as P, and from P, draw PE, meeting rC perpendicularly in E : Upon EC describe the semi-circle EaicC, cutting the joints produced in the points a,b,c: with the distances E a, E b, EC, describe arcs meeting EC in the points a!,V,c' ; join Pa', Pb', and PC', then will Pa'r, Pb'r, and Pc'r, be the an- gles of the faces of the stones to which they are referred. Again, to find the angle of the bed: upon PC describe the semi-circle Pa"5"c"C; and, from C, with the distances Ca,Cb,Cc, cut the semi- circle in a"V c" ; join Pa", Pi", and PC", then PC a", PC b", and PCc", will be the angles of the bed. 40. 4 M 322 THE NEW PRACTICAL BUILDER, &C. CALCULATION. In the triangle PCE we have given the angle at C equal to the obliquity, and the side, CP, any magnitude at pleasure ; hence the sides PE and EC can be found: then, in the triangle EC a, we have given the angle at C, and the side EC to find E a equal to E a 1 ; lastly, in the tri- angle EPa' we have given the sides PE, Ea', to find the angle Pa'E, which is the angle made by the face of the stone and its bed. The general formula is cot. req. ang. = cot. obliquity x sin. - 80 n ^ w , where n is the number of stones in the arch, and m any multiplier in the natural series, 1, 2, 3, &c. The angle formed by two contiguous boundaries of the bed is found ex- actly as the last. The formula is cos. req. ang. = cos. obliquity x cos. - ^~. As a particular example, suppose the obliquity to be 73, and the number of stones 11, and the respective angles will be exhibited in the following Table : Divisions of the Arch. Face Angles. Bed Angles. 1 16,, 21',, 49-rV" 85,, 4',, 37" 73,, 42',, 29" 2 32 43 38-^r 80 36 52 75 45 41 3 49 5 27-A- 76 59 23 78 57 42 4 65 27 16 T V 74 27 31 83 1 25 5 81 49 5-r> T 73 9 46 87 36 55 The angles for the remaining divisions being the supplements to those in the table, it is unnecessary to give them here. To apply the formulae to a numerical Example, we will take the first divi- sion in the table CCot. 73 } Log. 1.485339 Formula first: -.) Then work off the four adjacent sides to a right angle with it, then guage the stone to its depth, and work off the lower horizontal surface ; apply the mould of the head to the ends of the stone, as in working the first arch-stone, and draw the receding lines, then work the joints and the intrados as before, and the stone will be finished. The key-stone, exhibited in figure 5, is wrought in the same manner, and the whole arch, as completed, is represented by figure 6. 324 THE NEW PRACTICAL BUILDER, &C. A SEMI-CIRCULAR ARCHED PASSAGE, BETWEEN TWO SEMI-CIRCULAR ARCHED VAULTS. (Plate LXXXII.; To form the curve of intersection, and cut the stones for this arch. Let AB be the thickness of the wall in which the passage is to be made, BC and A.t the two semi-circular arches ; EL the opening of the passage; by which the arch GHc is described, which divide into any number of parts, at plea- sure. Through the points of division draw the lines H*, Kp, tpN, which, with the semi-circular arches, will form mixt angles, that serve to give the heads of the stones the proper projection, to intersect the semi-circular vaults. To mark in the plan, the meeting of the passage with the said semi-cir- cular vaults, let fall the perpendiculars CO, SI, QR, &c., which, by their intersecting or meeting the lines RT, IV, OF, &c., will give the points R,I,O, &c., through which trace the curves FOR and EFL. To trace one of the first stones ; square the bed and one side of a stone, take the thickness of the wall AB, which lay along the arris of the stone, and, at each of the said lines draw two others on the bed, square to the arris ; with a bevel take the mixt angle ZAN, by which dress the two heads, square in themselves ; on the upper bed trace the versed sine G b, and on the side the sine bw, then work out the sheeting with the curve Gw, and cut the joints square to the sheeting curves ; that is, the joint of the semi-circular vault by the mixt angle ANr, and that of the passage by the bevel MG. The other stones are cut exactly in the same manner, excepting that the bevel QC is used to cut the sweep of the second stones, and the bevel sp v for the key. The rest is so plain as to require no explanation. MASONUY. 325 AN ARCHWAY REVEALED and SPLAYED, and the SPLAY ARCHED with a SEG- MENT, in order to give room for Gates to open when they are made to the height of the front Arch. (Plate LXXXIII.; See the elevation, (figure I,) where A is the impost, BB the reveal, C the splayed retreat. Let ABDC, (figure 2,) be the plan, Ke the depth of the impost, efg the reveal, gc the splay. Describe the arch of the gate-head A'EB', and that of the reveal a'e'b'; and, at the extremities C and D of the splay, draw the perpendiculars CF' and DG', in which find the points F and G in the following manner : Describe the arch of the splay I'K', (fig. 3,) make I'L' equal to gC, (fig. 2.) ; perpendicuhr to I'L' draw L/K/ ; make M'F' and N'G' each equal to L/K/, and through the points F'G' trace the arc F'G', as flat as may be necessary for the gates to swing open. The most compounded joint in this gate is OP, formed by the arc of the reveal and that of the splay. To draw the joint-mould for this : from the point h, (fig. 2,) draw AQ' perpendicular to AB, meeting O'P' in Q'. Draw I'S' perpendicular to IL, and Q'S' and P'T parallel to IL' : join TS' inter- secting the arc in U'; draw U'V parallel to P'T, meeting the joint-line O'P' in V, and V is the point in which the stone will form an angle. Draw the line of the impost ab, and the reveal bed (Jig. 2); draw U'W perpen- dicular to I'L', make hi, on the splay of the jamb, equal to IW, and draw ik parallel to AB. Make kl equal to O'V, mn equal to O'P', join dl, In, and abcdln will be the form of the joint ; and all the joints which are cut in this forked angle are found in the same manner. For the mould of the second joint. Make mp equal to XY, and join dp. To cut one of the first stones. With the head-mould, B'O'P'I', prepare an arch-stone, as No. 1, whose length is equal to am on the plan ; apply the 41. 4 N 326 THE NEW PRACTICAL BUILDER, &C. mould of the plan, IkefgCK, on the under-bed, and, on the upper bed, the joint-mould, xabcdlnx. On the soffit of No. 1, draw ab, to mark the thickness of the impost, and, on the rear or tail of the stone, draw c d, representing NT' on the elevation. Then, to hollow out the concave sur- face of the reveal, with a curved bevel, V V, (fig. 3 J draw the curves e f, gh, No. 1. By the lines, be, cd, dk, dress that side which will be terminated by kh, making use of the curved templet cut by b'l', (fig. 3,} which apply, from time to time, till the forked-joint is formed, and the whole of the super- fluous stone being cut away, it will appear in the form of No. 2. The second stone, No. 3, is traced in the same manner. X. RAKING MOULDINGS. (Plate LXXXIVJ RAKING MOULDINGS more frequently occur in Masonry than in Joinery : hence we give the following design in consistency with the plan of our work. Figure 1 is a complete design of a cornice, having part of the ogee level, and part inclined, as happens in the level cornice of a building, with a pedi- ment in the front : a, b, c, d, e, is the moulding at the angle of the break, or projecture, of the pediment, which moulding is given in order to find the right section of the inclined ogee in the pediment. Let of and e e be the two parallel lines which terminate the breadth of the raking or inclined moulding; and let ak and el be the parallel lines terminating the breadth of the moulding which is level. At a convenient place draw p t parallel to the edge a k of the level ogee : In the given moulding, a, b, c, d, e, take any number of points, b, c, d, and draw bg, ch, di, parallel to af, or ee; also draw ap, bq, cr, ds, and et, perpen- dicular to alt, or el, meeting pt in the points q,r,s: also, at any convenient distance from af, draw p' t' parallel thereto, and transfer the distances p q, qr,rs,st, to p'q', qV, rV, and s't', and draw p'A, q'B, r'C, s'D, t'E, MASONRY. 327 perpendiculars to t'p', or af, meeting the lines af, bg, c/i, di, ee, in the points A, B, C, D, and E, and a curve, being drawn through these points, will be the right section of the raking-moulding. To find the section through mitre t"e of the two inclined sides, draw t"p" perpendicular to t" e, and transfer the distances ts,sr,rq,qp, to t" s" , s" r" , r"q",q"p". Draw s"d, r"c, q"b, p"a, perpendicular to t"p", also draw fa, qb, he, id, perpendicular to t" e. Then the curve through the points a b c d e will be the common section of the two raking-mouldings, as re- quired. Figure 2 is a reverse ogee, which is traced on the same principle as the ogee, figure 1. 328 THE NEW PRACTICAL BUILDER, &C. CHAPTER VII. AN EXPLANATION OF TERMS, AND DESCRIPTION OF TOOLS, USED IN MASONRY; INCLUDING THE COMPOSITION OF CEMENTS, OR MORTAR, &c. ABUTMENT. A term used both in carpentry and masonry. An explana- tion of it, as used in carpentry, may be found on referring to page 218. In masonry, the abutments of a bridge mean the walls adjoining to the land, which support the ends of the extreme arches or roadway. APERTURE. An opening through a wall, &c. which has, generally, three straight sides ; of these, two are perpendicular to the horizon, and the other parallel to it, connecting the lower ends of the vertical stones or jambs. The lower side is called the till, and the upper side the head. The last is either an arch or a single stone. If it be an arch, the aperture is called an arcade. Apertures may be circular or cylindrical ; but these are not very frequent. ARCH. Part of a building suspended over a hollow, and concave towards the area of the same. ARCHIVOLT of the Arch of a Bridge. The curved line formed by the upper sides of the arch-stones in the face of the work ; by the archivolt is also understood the whole set of arch-stones that appear in the face of the work. ASHLAR. A term applied to common or free-stones, as they come out of the quarry. By ashlar is also meant the facing of squared stones on the front TERMS, &C. IN MASONRY. 329 of a building. If the work be so smoothed as to take out the marks of the tools by which the stones were first cut, it is called plane-ashlar : if figured, it may be tooled ashlar, or random tooled, or chiselled, or toasted, or pointed. If the stones project from the joints, it is said to be rusticated. BANKER. The stone bench on which work is cut and squared. BANQUET. The raised footway adjoining to the parapet on the sides of a bridge. BATTARDEAU. See Coffer-dam. BATTER. The leaning part of the upper part of the face of a wall, which so inclines as to make the plumb-line fall within the base. BEDS of a Stone. The parallel surfaces which intersect the face of the work in lines parallel to the horizon. In arching, the beds are, by some, called summerings ; by others, with more propriety, radiations or radiated joints. BOND. That regular connection, in lapping the stones upon one another, when carrying up the work, which forms an inseparable mass of building. BUTMENT. See Abutment. CAISSON. A chest of strong timber in which the piers of a bridge are built, by sinking it, as the work advances, till it comes in contact with the bed of the river, and then the sides are disengaged, being constructed for that purpose. CEMENT and MORTAR, Composition of. It is almost superfluous to say, that cement, or mortar, is a preparation of lime and sand, mixed with water, which serves to unite the stones, in the building of walls, &c. On the proper or improper manner in which the cement or mortar is pre- pared and used, depends the durabili ty and security of every building : we shall, therefore, here introduce many particulars on this head, discovered by Dr. Higgins, but which, not being generally known, have never been re- duced into general practice. For the preparation of every kind of mortar, or cement, the subsequent remarks should always be known. Of Sand, the following kinds are to be preferred ; first, drift-sand, or quarry-sand, which consists chiefly of hard quartose flat-faced grains, with sharp angles ; secondly, that which is the 41. 4 o 330 THE NEW PRACTICAL BUILDER, &C. freest, or may be most easily freed by washing, from clay, salts, and calca- reous, gypseous, or other grains less hard and durable than quartz ; thirdly, that which contains the smallest quantity of pyrites or heavy metallic-matter, inseparable by washing ; and, fourthly, that which suffers the smallest dimi- nution of its bulk in washing. Where a coarse and fine sand of this kind, and corresponding in the size of their grains with the coarse and fine sands hereafter described, cannot be easily procured, let such sand of the fore- going quality be chosen as may be sorted and cleansed in the following manner : Let the sand be sifted in streaming clear water, through a sieve which shall give passage to all such grains as do not exceed one-sixteenth of an inch in diameter ; and let the stream of water, and the sifting, be regulated so that all the sand which is much finer than the Lynn-sand, commonly used in the London glass-houses, together with clay, and every other matter spe- cifically lighter than sand, may be washed away with the stream; whilst the purer and coarser sand, which passes through the sieve, subsides in a con- venient receptacle, and the coarse rubbish and rubble remain on the sieve to be rejected. Let the sand, which thus subsides in the receptacle, be washed in clean streaming water through a finer sieve, so as to be further cleansed, and sorted into two parcels ; a coarser, which will remain in the sieve, which is to give passage to such grains of sand only as are less than one-thirtieth of an inch in diameter, and which is to be saved apart under the name of coarse sand; and a finer, which will pass through the sieve and subside in the water, and which is to be saved apart under the name of fine sand. Let the coarse and the fine sand be dried separately, either in the sun, or on a clean iron-plate, set on a convenient surface, in the manner of a sand-heat. Let stone-lime be chosen, which heats the most in slaking, and slakes the quickest when duly watered ; that which is the freshest made and closest kept ; that which dissolves in distilled vinegar with the least effervescence, and leaves the smallest residue insoluble, and in the residue the smallest quantity of clay, gypsum, or martial matter. Let the lime, chosen accord- TERMS, &C. IN MASONRY. 331 ing to these rules, be put in a brass-wired sieve, to the quantity of fourteen pounds. Let the sieve be finer than either of the foregoing ; the finer the better it will be : let the lime be slaked, by plunging it into a butt filled with soft-water, and raising it out quickly, and suffering it to heat and fume ; and, by repeating this plunging and raising alternately, and agitating the lime until it be made to pass through the sieve into the water ; and let the part of the lime which does not easily pass through the sieve be rejected : and let fresh portions of the lime be thus used, until as many ounces of lime have passed through the sieve as there are quarts of water in the butt. Let the water, thus impregnated, stand in the butt closely covered until it becomes clear, and through wooden cocks, placed at different heights in the butt, let the clear liquor be drawn off, as fast and as low as the lime subsides, for use. This clear liquor is called lime-water. The freer the water is from saline matter, the better will be the cementing liquor made with it. Let fifty-six pounds of the aforesaid chosen lime be slaked, by gradually sprinkling the lime-water on it, and especially on the unslaked pieces, in a close clean place. Let the slaked part be immediately sifted through the last mentioned fine brass-wired sieve : let the lime which passes be used in- stantly, or kept in air-tight vessels ; and let the part of the lime which does not pass through the sieve be rejected. This finer and richer part of the lime, which passes through the sieve, may be called purified lime. Let bone-ash be prepared in the usual manner, by grinding the whitest burnt bones; but let it be sifted, so as to be much finer than the bone-ash commonly sold for making cupels. The best materials for making the cement being thus prepared, take fifty- six pounds of the coarse sand, and forty-two pounds of the fine sand ; mix them on a large plank of hard wood placed horizontally; then spread the sand so that it may stand to the height of six inches, with a flat surface on the plank, wet it with the lime-water, and let any superfluous quantity of the liquor, which the sand in the condition described cannot retain, flow away off the plank. To the wettest sand add fourteen pounds of the purified lime, in several successive portions ; mixing and beating them up together, 332 THE NEW PRACTICAL BUILDER, &C. in the mean time, with the instruments generally used in making fine mortar : then add fourteen pounds of the bone-ash, in successive portions, mixing and beating all together. The quicker and the more perfectly these materials are mixed and beaten together, and the sooner the cement thus formed is used, the better it will be. This may be called coarse-grained water-cement, which is to be applied in building, pointing, plastering, stuccoing, or other work, as mortar and stucco generally are; with this difference chiefly, that, as this cement is shorter than mortar, or common stucco, and dries sooner, it ought to be worked expeditiously in all cases ; and, in stuccoing, it ought to be laid on by sliding the trowel upwards on it. The materials used along with this cement in building, or the ground on which it is to be laid in stuccoing, ought to be well wetted with the lime-water in the instant of laying on the cement. The lime-water is also to be used when it is necessary to moisten the cement, or when a liquid is required to facilitate the floating of the cement. When such cement is required to be of a still finer texture, take ninety- eight pounds of the fine sand, wet it with the lime-water, and mix it with the purified lime and the bone-ash, in the quantities and in the manner above described ; with this difference only, that fifteen pounds of lime, or there- abouts, are to be used instead of fourteen pounds, if the greater part of the sand be as fine as Lynn sand. This may be called fine-grained water-cement. It is used in giving the last coating, or the finish, to any work intended to imitate the finer-grained stones or stucco. But it may be applied to all the uses of the coarse-grained water-cement, and in the same manner. When, for any of the foregoing purposes of pointing, building, &c., a cement is required much cheaper and coarser-grained than either of the foregoing, then much coarser clean sand than the foregoing coarse sand, or well-washed fine rubble, is to be provided. Of this coarse sand, or rubble, take fifty-six pounds, of the foregoing coarse sand twenty-eight pounds, and of the fine sand fourteen pounds ; and, after mixing these, and wetting them with the cementing-liquor, in the foregoing manner, add fourteen pounds, or TERMS, &C. IN MASONRY. 333 somewhat less, of the purified lime, and then fourteen pounds, or somewhat less, of the bone-ash, mixing them together in the manner already described. When the cement is required to be white, white sand, white lime, and the whitest bone-ash, are to be chosen. Gray sand, and gray bone-ash formed of half-burnt bones, are to be chosen to make cement gray ; and any other colour of the cement is obtained, either by choosing coloured sand, or by the admixture of the necessary quantity of coloured talc in nowder, or of coloured, vitreous, or metallic, powders, or other durable colouring ingre- dients, commonly used in paint. This water-cement, whether the coarse or fine-grained, is applicable in forming artificial stone, by making alternate layers of the cement and of flint, hard stone, or bricks, in moulds of the figure of the intended stone, and by exposing the masses so formed to the open air, to harden. When such cement is required for water fences, two-thirds of the pre- scribed quantity of bone-ashes are to be omitted ; and, in the place thereof, an equal measure of powdered terras is to be used ; and, if the sand em- ployed be not of the coarsest sort, more terras must be added, so that the terras shall be one-sixth part of the weight of the sand. When such a cement is required of the finest grain, or in a fluid form, so that it may be applied with a brush, flint-powder, or the powder of any quartose or hard earthy substance, may be used in the place of sand ; but in a quantity smaller, in proportion as the flint or other powder is finer; so that the flint-powder, or other such powder, shall not be more than six times the weight of the lime, nor less than four times its weight. The greater the quantity of lime within these limits, the more will the cement be liable to crack by quick drying, and, vice versd. Where the above described sand cannot be conveniently procured, or where the sand cannot be conveniently washed and sorted, that sand which most resembles the mixture of coarse and fine sand above prescribed, may be used as directed, provided due attention be paid to the quantity of the lime, which is to be greater as the quality is finer, and, vice versd. 42. 4 P 334 THE NEW PRACTICAL BUILDER, &C. Where sand cannot be easily procured, any durable stony body, or baked earth, grossly powdered, and sorted nearly to the sizes above prescribed for sand, may be used in the place of sand, measure for measure, but not. weight for weight, unless such gross powder be specifically as heavy as sand. Sand may be cleansed from every softer, lighter, and less durable, matter, and from that part of the sand which is too fine, by various methods pre- ferable in certain circumstances, to that which has been already described. Water may be found naturally free from fixable gas, selenite, or clay ; such water may, without any great inconvenience, be used in the place of the lime-water; and water approaching this state will not require so much lime as above prescribed to make the lime-water; and a lime-water sufficiently useful may be made by various methods of mixing lime and water in the described proportions, or nearly so. When stone-lime cannot be procured, chalk-lime, or shell-lime, which best resembles stone-lime, in the foregoing characters of lime, may be used in the manner described, excepting that fourteen pounds and a half of chalk- lime will be required in the place of fourteen pounds of stone-lime. The proportion of lime, as prescribed above, may be increased without inconve- nience, when the cement of stucco is to be applied where it is not liable to dry quickly ; and, in the contrary case, this proportion may be diminished. The defect of lime, in quantity or quality, may be very advantageously sup- plied, by causing a considerable quantity of lime-water to soak into the work, in successive portions, and at distant intervals of time ; so that the calcareous matter of the lime-water, and the matter attracted from the open air, may fill and strengthen the work. The powder of almost every well-dried or burnt animal substance may be used instead of bone-ash ; and several earthy powders, especially the mica- peous and the metallic ; and the elixated ashes of divers vegetables, whose earth will not burn to lime, as well as the ashes of mineral fuel, which are of the calcareous kind, but will not burn to lime, will answer the ends of bone-ash in some degree. TERMS, &C. IN MASONRY. 335 The quantity of bone-ash described may be lessened without injuring the cement ; in those circumstances especially which admit the quantity of lime to be lessened, and in those wherein the cement is not liable to dry quickly. The art of remedying the defects of lime may be advantageously practised to supply the deficiency of bone-ash, especially in building, and in making artificial stone with this cement. As the preceding method of making mortar differs, in many particulars, from the common process, it may be useful to enquire into the causes on which this difference is founded. When the sand contains much clay, the workmen find tjiat the best mortar they can make must contain about one-half lime ; and hence they lay it down as certain, that the best mortar is made by the composition of half sand and half lime. But with sand requiring so great a proportion of lime as this, it will be impossible to make good cement ; for it is universally allowed that the hard- ness of mortar depends on the crystallization of the lime round the other materials which are mixed with it ; and thus uniting the whole mass into one solid substance. But, if a portion of the materials used be clay, or any other friable substance, it must be evident that, as these friable substances are not changed in one single particular, by the process of being mixed up with lime and water, the mortar, of which they form a proportion, will conse- quently be, more or less, of a friable nature, in proportion to the quantity of friable substances used in the composition of the mortar. On the other hand, if mortar be composed of lime and good sand only, as the sand is a stony substance, and not in the least friable, and as the lime, by perfect crys- tallization, becomes likewise of a stony nature, it must follow, that a mass of mortar, composed of these two stony substances, will itself be a hard, solid, unfriable, substance. This may account for one of the essential varia- tions in the preceding method from that in common use, and point out the necessity of never using, in the place of sand, which is a durable stony body, the scrapings of roads, old mortar, and other rubbish, from antient 336 THE NEW PRACTICAL BUILDER, &C. buildings, which are frequently made use of, as all of them consist, more or less, of muddy, soft, and minutely divided particles. Another essential point is the nature and quality of the lime. Now, experience proves that, when lime has been long kept in heaps, or untight casks, it is reduced to the state of chalk, and becomes every day less capable of being made into good mortar ; because, as the goodness or durability of the mortar depends on the crystallization of the lime, and, as experiments have proved, that lime, when reduced to this chalk-like state, is always inca- pable of perfect crystallization, it must follow that, as lime in this state never becomes crystallized, the mortar of which it forms the most indispensable part, will necessarily be very imperfect ; that is to say, it will never become a solid stony substance ; a circumstance absolutely required in the formation of good durable mortar. These are the two principal ingredients in the formation of mortar ; but, as water is also necessary, it may be useful to point out that which is the fittest for this purpose ; the best is rain-water, river-water the second, land-water next, and spring-water last. The ruins of the antient Roman buildings are found to cohere so strongly, as to have caused an opinion that their constructors were acquainted with some kind of mortar, which, in comparison with ours, might justly be called cement ; and that, to our want of knowledge of the materials they used, is owing the great inferiority of modern buildings in their durability. But a proper attention to the above particulars would soon show that the dura- bility of the antient edifices depended on the manner of preparing their mortar more than on the nature of the materials used. The following obser- vations will, we think, prove this beyond a possibility of doubt : Lime, which has been slaked and mixed with sand, becomes hard and consistent when dry, by a process similar to that which produces natural stalactites in caverns. These are always formed by water dropping from the roof. By some unknown and inexplicable process of nature, this water has had dissolved in it a small portion of calcareous matter, in a caustic state. So long as the water continues covered from the air, it keeps the earth dissolved TERMS, &C. IN MASONRY. jJOlrfw 337 in it; it being the natural property of calcareous earths, when deprived of their fixed air, to dissolve in water. But, when the small drop of water comes to be exposed to the air, the calcareous matter contained in it begins to attract the fixable part of the atmosphere. In proportion as it does so,' it also begins to separate from the water, and to re-assume its native form of lime-stone or marble. When the calcareous matter is perfectly crystallized in this manner, it is to all intents and purposes lime-stone or marble of the same consistence as before. If lime, in a caustic state, is mixed with water, part of the lime will be dissolved, and will also begin to crystallize. The water which parted with the crystallized lime will then begin to act upon the remainder, which it could not dissolve before ; and thus the process will con- tinue, either till the lime be all reduced to an effete, or crystalline state, or something hinders the action of the water upon it. It is this crystallization which is observed by the workmen when a heap of lime is mixed with water, and left for some time to macerate. A hard crust is formed upon the surface, which is ignorantly called frostling, though it takes place in summer as well as in winter. If, therefore, the hardness of the lime, or its becoming a cement, depends entirely on the formation of its crystals, it is evident that the perfection of the cement must depend on the perfection of the crystals, and the hardness of the matters which are entangled among them. The additional substances used in making of mortar, such as sand, brick-dust, or the like, serve only for a purpose similar to what is answered by sticks put into a vessel full of any saline solution ; namely, to afford the crystals an opportunity of fastening themselves upon it. If, therefore, the matter inter- posed between the crystals of the lime is of a friable brittle nature, such as brick-dust or chalk, the mortar will be of a weak and imperfect kind ; but, when the particles are hard, angular, and very difficult to be broken, such as those of river or pit-sand, the mortar turns out exceedingly good and strong. That the crystallization may be the more perfect, a large quantity of water should be used, the ingredients be perfectly mixed together, and the drying be as slow as possible. An attention to these particulars would make the buildings of the moderns equally durable with those of the antients. In the 42. 4 Q 338 THE NEW PRACTICAL BUILDER, &C. old Roman works, the great thickness of the walls necessarily required a vast length of time to dry. The middle of them was composed of pebbles thrown in at random, and which, evidently, had thin mortar poured in among them. Thus a great quantity of the lime would be dissolved, and the crys- tallization performed in the most perfect manner. The indefatigable pains and perseverance, for which the Romans were so remarkable in all their undertakings, leave no room to doubt that they would take care to have the ingredients mixed together as well as possible. The consequence of all this is, that the buildings formed in this manner are all as firm as if cut out of a solid rock ; the mortar being equally hard, if not more so, than the stones themselves. CENTRES. The frame of timber-work for supporting arches during their erection. COFFER-DAM, or BATTARDEAU. A case of piling, without a bottom, con- structed for inclosing and building the piers of a bridge. A coffer-dam may be either single or double, the space between being filled with clay or chalk, closely rammed. DRAG. A thin plate of steel indented on the edge, like the teeth of a saw, and used in working soft stone, which has no grit, for finishing the surface. DRIFT. The horizontal force of an arch, by which it tends to overset the piers. EXTRADOS of an Arch. The exterior or convex curve, or the top of the arch-stones. This term is opposed to the Intrados, or concave side. EXTRADOS of a Bridge. The curve of the road-way. FENCE-WALL. A wall used to prevent the encroachment of men or animals. FOOTINGS. Projecting courses of stone, without the naked superincum- bent part, and which are laid in order to rest the wall firmly on its base. HAMMER. See Tools. HEADERS. Stones disposed with their length horizontally, in the thick- ness of the wall. JETTEE. The border made around the stilts under a pier. TERMS, &.C. IN MASONRY. 339 IMPOST or SPRINGING. The upper part or parts of a wall employed for springing an arch. INTRADOS. See Extrados. JOGGLED JOINTS. The method of indenting the stones, so as to prevent the one from being pushed away from the other by lateral force. KEY-STONES. A term frequently used for bond-stones. KEY-STONE. The middle voussoir of an arch, over the centre. KEY-STONE of an Arch. The stone at the summit of the arch, put in last for wedging and closing the arch. LEVEL. Horizontal, or parallel to the horizon. MALLET. See Tools. MORTAR. See Cement. NAKED, of a Wall. The vertical or battering surface, whence all projec- tures arise. OFF-SET. The upper surface of a lower part of a wall, left by reducing the thickness of the superincumbent part upon one side or the other, or both. POINT. See Tools. PARAPETS. The breast-walls erected on the sides of the extrados of the bridge, for preventing passengers from falling over. PAVING. A floor, or surface of stone, for walking upon. PIERS in Houses. The walls between apertures, or between an aperture and the corner. PIERS of a Bridge. The insulated parts between the apertures or arches, for supporting the arches and road-way. PILES. Timbers driven into the bed of a river, or the foundation of a building for supporting a structure. PITCH of an Arch. The height from the springing to the summit of the arch. PUSH of an Arch. The same as Drift; which see. QUARRY. The place whence stones are raised. RANDOM COURSES, in Paving. Unequal courses, without any regard to equi-distant joints. 340 THE NEW PRACTICAL BUILDER, &C. SAW. See Tools. SHOOT of an Arch. The same as Drift; which see. SPAN. The span of an arch is its greatest horizontal width. STERLINGS. A case made about a pier of stilts in order to secure it. See the following article. STILTS. A set of piles driven into the hed of a river, at small distances, with a surrounding case of piling driven closely together, and the interstices filled with stones, in order to form a foundation for building the pier upon. STRAIGHT-EDGE. See Tools. STRETCHERS. Those stones, which have their length disposed horizontally in the length of the wall. THROUGH STONES. A term employed, in some countries, for bond-stones. THRUST. The same as Drift; which see. TOOLS used by Masons. The masons' Level, Plumb-Rule, Square, Bevel, Trowel, Hod, and Compasses, are similar in every respect to those tools which bear the same name among bricklayers ; and which are described here- after. Those tools, which differ from such as are used by the bricklayer, are as follow : The Saw used by masons is without teeth, and stretched in a frame nearly resembling the joiner's saw-frame. It is made from four to six feet, or more, in length, according to the size of the slabs, which are intended to be cut by it. To facilitate the process of cutting slabs into slips and scantlings, a portion of sharp silicious sand is placed upon an inclined plane, with a small barrel of water at the top, furnished with a spiggot, which is left suffi- ciently loose to allow the water to exude drop by drop ; and thus, by running over the sand, carries with it a portion of sand into the kerf of the stone. The workman sits at one side of the stone, and draws the saw to and fro, horizontally, taking a range of about twelve inches each time before he returns. By this means, calcareous stones of the hardest kinds may be cut into slabs of any thickness, with scarcely any loss of substance. But, as this method of sawing stone is slow and expensive, mills have been erected in various parts of Great Britain, by which the same process is performed at a TERMS, &C. USED IN MASONRY. 341 much cheaper rate, and in some of these mills every species of moulding upon stone is produced. Masons make use of many chisels, of different sizes, but all resembling, or nearly resembling, each other in form. They are usually made of iron and steel welded together; but, when made entirely of steel, which is more elastic than iron, they will naturally produce a greater effect with any given impulse. The form of masons' chisels is that of a wedge, the cutting-edge being the vertical angle. They are made about eight or nine inches long. When the cutting-edge is broader than the portion held in the hand, the lower part is expanded in the form of a dove-tail. When the cutting-edge is smaller than the handle, the lower end is sloped down in the form of a pyramid. In finishing off stone, smooth and neat, great care should be taken that the arris is not splintered, which would certainly occur, if the edge of the chisel were directed outwards in making the blow : but, if it be directed inwards, so as to overhang a little, and form an angle of about forty- five degrees, there is little danger of splintering the arris in chipping. Of the two kinds of chisels, which are the most frequently made use of, the tool is the largest ; that is to say, in the breadth of its cutting-edge ; it is used for working the surface of stone into narrow furrows, or channels, at regular distances ; this operation is called tooling, and the surface is said to be tooled. The Point is the smallest kind of chisel used by masons, being never more than a quarter of an inch broad on its cutting-edge. It is used for reducing the irregularity of the surface of any rough stone. The Straight-Edge is similar to the instrument among carpenters of the same name ; it being a thin board, planed true, to point out cross-windings and other inequalities of surface, and thus direct the workmen in the use of the chisel. The Mallet used by the mason differs from that of any other artisan. It is similar to a bell in contour, excepting a portion of the broadest part, which is rather cylindrical. The handle is rather short, being only just long enough to be firmly grasped in the hand. It is employed for giving percus- 43. 4 R 342 THE NEW PRACTICAL BUILDER, &C. sive force to chisels, by striking them with any part of the cylindrical surface of the mallet. The Hammer used by masons is generally furnished with a point or an edge like a chisel. Both kinds are used for dividing stones, and likewise for producing those narrow marks or furrows left upon hewn-stone work which is not ground on the face. VAULT. A mass of stones so combined as to support each other over a hollow. UNDER BED of a Stone. The lower surface, generally placed horizontally. UPPER BED of a Stone. The upper surface, generally placed horizontally. VOUSSOIRS. The arch-stones in the face or faces of an arch; the middle one is called the key-stone. WALL. An erection of stone, generally perpendicular to the horizon ; but sometimes battering, in order to give stability. WALLS, Emplection. Those which are built in regular courses, with the stones smoothed in the face of the work. They are of two kinds, Roman and Grecian, as already noticed. The difference is, that the core of the Roman emplection is rubble ; whereas in the Grecian emplection, it is built in the same manner as the face, and every alternate stone goes through the entire thickness of the wall. See pages 306, 307. Walls, Isodomum ; those wherein the courses are of equal thickness, com- pact, and regularly built ; but the stones are not smoothed on the face. Walls, Pseudo-Isodomum ; those which have unequal courses. See page 306. BRICKLAYING. 343 ' CHAPTER VIII. BRICKLAYING : INCLUDING AN ABSTRACT OF THE BUILDING ACT, 14th GeO. III. C. 78. BRICKLAYING is the art of Building with Bricks, or of uniting them, by cement or mortar, into various forms for particular purposes. The BRICKS of the antients were of various forms and sizes, and their tri- angular bricks were peculiarly adapted to certain figures, but modern bricks, of English make, are commonly of one form, 9 inches long, by 4| broad, and 2| deep. Bricks are made of a species of clay or loamy earth, either pure or with various mixtures ; they are shaped in a mould, and, after some drying in the sun or air, are burnt to a hardness. The more pure the earth of which it is formed, the harder and firmer the brick will be. The bricks generally known to our modern builders are of several sorts : that is to say, Marls, of two qualities, Gray-Stocks, and Place-Bricks, besides two or three foreign kinds, occasionally imported. Bricks vary in quality, according to the quality of the material of which they are composed, the manner in which the clay is tempered, and the diffusion of the heat while burning. The finest kind of Marls, called Firsts, are those usually selected for arches over doors and windows : those less fine, called Seconds, are com- monly used for the fronts of buildings. The Gray-Stocks are of the next quality, and are generally of a good earth, well wrought, with little mixture, sound, and durable. Place-Bricks are too frequently poor and brittle, badly 344 THE NEW PRACTICAL BUILDER, &C. burnt, and of very irregular colour. Burrs or Clinkers are such as are so touch overburnt as to vitrify, and run two or three together. Red Stocks and the Red Bricks, called also, from their use, Cutting Bricks, owe their colour to the nature of the clay of which they are made ; this is always used tolerably pure, and the bricks of the better kind are called by some Clay Bricks, because they are supposed to be made of nothing else. The Gray Stocks, being made of a good earth, well wrought, are com- monly used in front in building : the Place Bricks being made of clay, with a mixture of dirt and other coarse materials, and more carelessly put out of hand, are therefore weaker and more brittle, and are introduced where they cannot be seen, and where little stress is laid upon them : the Red Bricks, of both kinds, are made of a particular earth, well wrought, and little injured by mixtures ; and they are used in fine work, in ornaments over windows, and in paving. These are frequently cut or ground down to a perfect even- ness, and sometimes set in putty instead of mortar ; and thus set they make a very beautiful appearance. These are the kinds of bricks commonly used by us in building, and their difference is owing to variety in the materials. The Place Bricks and Gray Stocks are made in the neighbourhood of London, wherever there is a brick- work ; the two kinds of red brick, depending upon a particular kind of earth, can be made only where that is to be had ; they are furnished from several places within fifteen or twenty miles of London. We have already observed, that there are two or three other kinds of brick to be named, , which are imported from other countries ; and there is also one of the red or cutting brick sort, that is of our own manufacture, and for its excellence deserves to be particularly mentioned ; this is the Hedgerly Brick : it is made at a village of that name, of the famous earth called Hedgerly loam, well-known to the glass-makers and chymists. The loam is of a yellow-reddish colour, and very harsh to the touch, containing a great quantity of sand ; its particular excellence is, that it will bear the greatest violence of fire without injury : the chymists coat and lute their furnaces with this, and the ovens at glass-houses are also repaired or lined with it, :T BRICKLAYING. 345 where it stands all the fury of their heat without damage. It is brought into London for this purpose, under the name of Windsor loam, the village being near Windsor, and is sold at a high price. The bricks made of this are of the finest red that can be imagined. They are called Fire-Bricks, because of their enduring the fire ; and are used about furnaces and ovens in the same way as the earth. The foreign bricks above mentioned are the Dutch and Flemish Bricks and Clinkers : these are all nearly of a kind, and are often confounded toge- ther ; they are very hard, and of a dirty brimstone colour ; some of them not much unlike our Gray Stocks, others yellower. The Dutch are generally the best baked, and Flemish the yellowest. As to the Clinkers, they are the most baked of all, and are generally warped by the heat. These bricks are used for peculiar purposes ; the Dutch and Flemish for paving yards, stables, and the like ; and the clinkers for ovens. The fine red cutting Engfish Bricks are twice, or more than twice, the price of the best Gray Stocks ; the Red Stocks half as dear again as the gray ; and the Place Bricks, as they are much worse, so they are much cheaper, than any of the others. The Gray Stocks and Place Bricks are employed in the better and worse kinds of plain work ; the red stocks, as well as the gray, are used sometimes in this business, and sometimes for arches, and other more ornamental pieces : the fine red cutting bricks are used for ruled and gauged work, and sometimes for paving ; but the red stones are more frequently employed when a red kind is required for this purpose. The Red Cutting Brick, or fine red, is the finest of all bricks. In some places they are not at all acquainted with this; in others, they confound it with the red stock, and use that for it ; though, where the fine red brick is to be had pure and perfectly made, the difference is five to three in the sale price between that and the red stock. The Red and Gray Stock are frequently put in gauged arches, and one as well as the other set in putty instead of mortar : this is an expensive work, 43. 4s 346 THE NEW PRACTICAL, BUILDER, &C. but it answers in beauty for the regularity of the disposition and fineness of the joints, and has a very pleasing effect. The fine Red Brick is used in arches ruled and set in putty in the same manner ; and, as it is much more beautiful, is somewhat more costly. This kind is also the most beautiful of all in cornices, ruled in the same manner, and set in putty. The Gray Stocks of an inferior kind are also used in brick walls. The Place Bricks are used in paving dry, or laid in mortar, and they are put down flat or edgewise. If they are laid flat, thirty-two of them pave a square yard ; but, if they are placed edgewise, it takes twice that number : in the front work of walls the Place-Bricks should never be admitted, even in the meanest building. That consideration, therefore, only takes place in the other kinds : and the fine Cutting Bricks come so very dear this way, that few people will be brought to think of them ; so that it lies, in a great mea- sure, between the Gray Stocks and Red Stocks. Of these the gray are most used ; and this not only because they are cheaper, but, in most cases where judgement is preferred to fancy, they will have the preference. We see many very beautiful pieces of workmanship in Red Brick ; but this should not tempt the judicious architect to admit them into the front walls of buildings. In the first place, the colour itself is fiery and disagreeable to the eye ; and, in summer, it has an appearance of heat that is very dis- agreeable; for this reason it is most improper in the country, though the oftenest used there, from the difficulty of getting gray. But a farther con- sideration is, that, in the fronts of most important buildings, there is more or less stone-work ; now, as there should be as much conformity as can be attained between the general nakedness of the wall and those several orna- ments which project from it ; the nearer they are of a colour, the better they always range together ; and if we cast our eyes upon two houses, the one of red, and the other of gray brick, where there is a little stone-work, we shall not be a moment in doubt which to prefer. There is something harsh in the transition from the red brick to stone, and it seems altogether unnatural; BRICKLAYING. 347 in the other, the Gray Stocks come so near the colour of stone, that the change is less violent, and they sort better together. Hence, also, the Gray Stocks are to be considered as best coloured when they have least of the yellow cast ; for the nearer they come to the colour of stone, when they are to be used together with it, it is certainly the better. Where there is no stone-work, there generally is wood ; and this, being painted white, as is commonly the practice, has yet a greater effect with red brick than the stone-work : the transition is more sudden in this than in the other ; but, on the other hand, in the mixture of gray bricks and white paint, the colour of the brick being soft, there is no violent change. The Gray Stocks are now made, of prime quality, in the neighbourhood of London. The late Duke of Norfolk had the bricks brought from his estate, in that county, for building the front of his house, in St. James's- square ; but the event shews that his Grace might have been better supplied near at hand, as to colour, with equal hardness. The greatest advantage that a Gray Stock, which is the standard brick, can have, is in its sound body and pale colour ; the nearer it comes to stone the better; so that the principal thing the brick-maker ought to have in view, for the improvement of his profession, is the seeking for earth that will burn pale, and that will have a good body, and to see it has sufficient working. The judicious builder will always examine his bricks in this light, and be ready to pay that price which is merited by the goodness of the com- modity. The utility and common practice of building all our edifices of brick, both in London and the country, arises from motives too obvious to need a defi- nition; since it is generally considered to be much the cheapest, as well as the most eligible substance that can be invented for the purpose, both in point of beauty and duration, and inferior to nothing but wrought stone. BRICKS ARE LAID in a varied, but regular, form of connection, or Bond, as exhibited in Plate LXXXV. The mode of laying them for a 9-inch walling, shown in figure 1, being denominated English Bond; and Jigure 2, Flemish Bond. Figure 3 is English Bond, in a brick and a half, or 14-inch walling ; 348 THE NEW PRACTICAL BUILDER, &C. and figure 4, Flemish Bond, in the same. Figure 5 represents another method of disposing Flemish Bond in a 14-inch wall. Figure 6, English Bond, in an 18-inch, or two brick thick, wall ; and figure 7, English Bond, in a two and half brick thick wall. Figures 8, 9, 10, 11, represent square courses, in pairs, of Flemish Bond. In each pair, if one be the lower course, the other will be the upper course. The Bricks, having their lengths in the thickness of the wall, are termed Headers, and those which have their lengths in the length of the wall are Stretchers. By a Course, in walling, is meant the bricks contained between two planes parallel to the horizon, and terminated by the faces of the wall. The thickness is that of one brick with mortar. The mass formed by bricks laid in concentric order, for arches or vaults, is also denominated a Course. The disposition of bricks in a wall, of which every alternate course con- sists of headers, and of which every course between every two nearest courses of headers consist of stretchers, constitutes English Bond. The disposition of bricks, in a wall, (except at the quoins,) of which every alternate brick in the same course is a header, and of which every brick between every two nearest headers is a stretcher, constitutes Flemish Bond. It is, therefore, to be understood that English Bond is a continuation of one kind throughout, in the same course or horizontal layer, and consists of alternate layers of headers and stretchers, as shown in the plate ; the headers serving to bind the wall together, in a longitudinal direction, or lengthwise, and the stretchers to prevent the wall splitting crosswise, or in a transverse direction. Of these evils the first is of the worst kind, and .therefore the most to he feared. A respectable writer on this subject has said, that the old English mode of brick-work affords the best security against such accidents ; as work of this kind, wheresoever it is so much undermined as to cause a fracture, is not subject to such accidents, but separates, if at all, by breaking through the solid brick, just as if the wall were composed of one piece. The antient brick-work of the Romans was of this kind of bond, but the existing specimens of it are very thick, and have three, or sometimes more, BRICKLAYING. 349 courses of brick, laid at certain intervals of the height, stretchers on stretchers, and headers on headers, opposite the return wall, and sometimes at certain distances in the length, forming piers, that bind the wall together in a transverse direction ; the intervals between these piers were filled up, and formed panels of rubble or reticulated work ;* consequently great substance, with strength, were economically obtained. It will, also, be understood that Flemish Bond consists in placing, in the same course, alternate headers and stretchers, a disposition considered as decidedly inferior in every thing but appearance, and even in this the dif- ference is trifling ; yet, to obtain it, strength is sacrificed, and bricks of two qualities are fabricated for the purpose ; a firm brick often rubbed and laid in what the workmen term a putty-joint for the exterior, and an inferior brick for the interior, substance of the wall ; but, as these did not corres- pond in thickness, the exterior and interior surface of the wall would not be otherwise connected together than by an outside heading brick, here and there continued of its whole length ; but, as the work does not admit of this at all times, from the want of agreement in the exterior and interior courses, these headers can be introduced only where such a correspondence takes place, which, sometimes, may not occur for a considerable space. Walls of this kind consist of two faces of four-inch work, with very little to connect them together, and what is still worse the interior face often con- sists of bad brick, little better than rubbish. The practice of Flemish Bond has, notwithstanding, continued from the time of William and Mary, when it was introduced, with many other Dutch fashions, and our workmen are so infatuated with it, that there is now scarcely an instance of the old English Bond to be seen. The frequent splitting of walls into two thicknesses has been attributed to the Flemish Bond alone, and various methods have been adopted for its prevention. Some have laid laths or slips of hoop-iron, occasionally, in the horizontal joints between the two courses ; others have laid diagonal courses of bricks at certain heights from each other ; but the effect of the last me- t * See Masonry, pages 306 and 307. 44. 4 T 350 THE NEW PRACTICAL BUILDER, &C. thod is questionable, as, in the diagonal course, by their not being con- tinued to the outside, the bricks are much broken where the strength is required. Other methods of uniting complete Bond with Flemish facings have been described, but they have been found equally unsuccessful. In figures 2 and 4, (Plate LXXXVJ the interior bricks are represented as disposed with intention to unite these two particulars ; the Flemish facings being on one side of the wall only ; but this, at least, falls short of the strength obtained by English Bond. Another evil attending this disposition of the bricks is, the difficulty of its execution, as the adjustment of the bricks in one course .must depend on the course beneath, which must be seen or recollected by the workman ; the first is difficult from the joints of the under-course being covered with mortar, to bed the bricks of the succeeding course ; and, for the workman to carry in his mind the arrangement of the preceding course can hardly be expected from him ; yet, unless it be attended to, the joints will be frequently brought to correspond, dividing the wall into several thick- nesses, and rendering it subject to splitting, or separation. But, in the English Bond, the outside of the last course points out how the next is to be laid, so that the workman cannot mistake. The outer appearance is all that can be urged in favour of Flemish Bond, and many are of opinion that, were the English mode executed with the same attention and neatness that is bestowed on the Flemish, it would be con- sidered as equally handsome ; and its adoption, in preference, has been strenuously recommended. In forming English Bond, the following rules are to be observed : 1st, Each course is to be formed of headers and stretchers alternately. 2d, Every brick in the same course must be laid in the same direction : but, in no instance, is a brick to be placed with its whole length along the side of another ; but to be so. situated that the end of one may reach to the middle of the others which lie contiguous to it, excepting the outside of the stretching-course, where three-quarter bricks necessarily occur at the ends, to prevent a continued upright joint in the face-work. BRICKLAYING. 351 3d, A wall, which crosses at a right-angle with another, will have all the bricks of the same level course in the same parallel direction, which com- pletely bonds the angles, as shown lay figures 1, 3, and 6. The GREAT PRINCIPLE in the PRACTICE of BRICK-WORK lies in the proclivity or certain motion of absolute gravity, caused by a quantity or multiplicity of substance being added or fixed in resistible matter, and which, therefore, naturally tends downwards, according to the weight and power impressed. In bricklaying, this proclivity, chiefly by the yielding mixture of the matter of which mortar is composed, and cannot be exactly calculated, because the weight of a brick, or any other substance, laid in mortar, will naturally decline according to its substance or quality ; particular care should, therefore, be taken, that the material be of one regular and equal quality all through the building ; and, likewise, that the same force should be used to one brick as another ; that is to say, the stroke of the trowel : a thing or point in prac- tice of much more consequence than is generally imagined ; for, if a brick be actuated by a blow, this will be a much greater pressure upon it than the weight of twenty bricks. It is, also, especially to be remarked, that the many bad effects arising from mortar not being of a proper quality should make masters very cautious in the preparation of it, as well as the certain quantity of materials of which it is composed, so that the whole structure may be of equal density, as nearly as can be effected. Here we may notice a particular which often causes a bulging in large flank walls, especially when they are not properly set off on both sides; that is, the irregular method of laying bricks too high on the front edge : this, and building the walls too high on one side, without continuing the other, often causes the defects. Notwithstanding, of the two evils, this is the least; and bricks should incline rather to the middle of the wall, that one-half of the wall may act as a shore to the other. But even this method, carried too far, will be more injurious than beneficial, because the full width of the wall, in this case, does not take its absolute weight, and the gravity is removed from its first line of direction, which, in all walls, should be perpendicular and united ; and it is farther to be considered that, as the walls will have a superincumbent 352 THE NEW PRACTICAL BUILDER, &C. weight to bear, adequate to their full strength, a disjunctive digression is made from the right line of direction; the conjunctive strength becomes divided ; and, instead of a whole or united support from the wall, its strength is separated in the middle, and takes two lateral bearings of gravity ; each insufficient for the purpose ; therefore, like a man overloaded either upon his head or shoulders, naturally bends and stoops to the force impressed; in which mutable state the grievances above noticed usually occur. Another great defect is frequently seen in the fronts of houses, in some of the principal ornaments of Brick-work, as, arches over windows, &c., and which is too often caused by a want of experience in rubbing the bricks ; which is the most difficult part of the branch, and ought to be very well con- sidered : the faults alluded to, are the bulging or convexity in which the faces of arches are often found, after the houses are finished, and sometimes loose in the key or centre bond. The first of these defects, which appears to be caused by too much weight, is, in reality, no more than a fault in the practice of rubbing the bricks too much off on the insides ; for it should be a standing maxim (if you expect them to appear straight under their proper weight) to make them the exact guage on the inside, that they bear upon the front edges ; by which means their geometrical bearings are united, and all tend to one centre of gravity. The latter observation, of camber arches not being skewed enough, is an egregious fault ; because it takes greatly from the beauty of the arch, as well as its significancy. The proper method of skewing all camber arches should be one-third of their height. For instance, if an arch is nine inches high, it should skew three inches ; one of twelve inches, four ; one of fifteen inches, five ; and so of all the numbers between those. Observe, in dividing the arch, that the quantity consists of an odd number : by so doing, you will have proper bond ; and the key-bond in the middle of the arches ; in which state it must always be, both for strength and beauty. Likewise observe, that arches are all drawn from one centre ; the real point of camber arches is got from the above proportion. First, divide the height of the arch in three parts ; one is the dimensions for the skewing ; a line drawn from that through BRICKLAYING. 353 the point at the bottom to the perpendicular of the middle arch, gives the centre : to which all the rest must be drawn. Of FOUNDATIONS. Rules to be observed in laying Foundations. If a projected building is to have cellars, or under-ground kitchens, there will commonly be found a sufficient bottom, without any extra process, for a good solid foundation. When this is not the case, the remedies are to dig deeper, or to drive in large stones with the rammer, or by laying in thick pieces of oak, crossing the direction of the wall, and planks of the same timber, wider than the intended wall, and running in the same direction with it. The last are to be spiked firmly to the cross-pieces, to prevent their sliding, the ground having been previously well rammed under them. The mode of ascertaining if the ground be solid is by the rammer ; if, by striking the ground with this tool, it shake, it must be pierced with a borer, such as is used by well-diggers ; and, having found how deep the firm ground is below the surface, you must proceed to remove the loose or soft part, taking care to leave it in the form of steps, if it be tapering, that the stones may have a solid bearing, and not be subject to slide, which would be likely to happen if the ground were dug in the form of an inclined plane. If the ground prove variable, and be hard and soft at different places, the best way is to turn arches from one hard spot to another. Inverted arches have been used for this purpose with great success, by bringing up the piers, which carry the principal weight of the building, to the intended height and thickness, and then turning reversed arches from one pier to another, as shown in figure 18, (Plate LXXXV.^) In this case, it is clear that the piers cannot sink without carrying the arches, and consequently the ground on which they lie, with them. This practice is excellent in such cases, and should, therefore, be general, wherever required. Where the hard ground is to be found under apertures only, build your piers on these places, and turn arches from one to the other. In the con- struction of the arches, some attention must be paid to the breadth of the insisting pier, whether it will cover the arch or not ; for, suppose the middle 44. 4u 354 THE NEW PRACTICAL, BUILDER, &C. of the piers to rest over the middle of the summit of the arches, then the narrower the piers, the more curvature the supporting arch ought to have at the apex. When arches of suspension are used, the intrados ought to be clear, so that the arch may have the full effect : but, as already noticed, it will also be requisite here that the ground on which the piers are erected be uniformly hard ; for it is better that it should be uniform, though not so hard as might be wished, than to have it unequally so : because, in the first case, the piers would descend uniformly, and the building remain uninjured ; but, in the second, a vertical fracture would take place, and endanger the whole structure. WALLS, &c. The foundation being properly prepared, the choice of materials is to be considered. In places much exposed to the weather, the hardest and best bricks must be used, and the softer reserved for in-door work, or for situations less exposed. In slaking lime, use as much water only as will reduce it to a powder, and only about a bushel of lime at a time, covering it over with sand, in order to prevent the gas, or virtue of the lime, from escaping. This is a better mode than slaking the whole at one time, there being less surface exposed to the air. Before the mortar is used, it should be beaten three or four times over, so as to incorporate the lime and sand, and to reduce all knobs or knots of lime that may have passed the seive. This very much improves the smooth- ness of the lime, and, by driving air into its pores, will make the mortar stronger : as little water is to be used in this process as possible. Whenever mortar is suffered to stand any time before used, it should be beaten again, so as to give it tenacity, and prevent labour to the bricklayer. In dry hot summer-weather use your mortar soft ; in winter, rather stiff. If laying bricks in dry weather, and the work is required to be firm, wet your bricks by dipping them in water, or by causing water to be thrown over them before they are used, and your mortar should be prepared in the best way. Few workmen are sufficiently aware of the advantage of wetting bricks before they are used ; but experience has shown that works in which this practice has been followed have been much stronger than others wherein BRICKLAYING. 355 it has been neglected. It is particularly serviceable where work is carried up thin, and in putting in grates, furnaces, &c. In the winter season, so soon as frosty and stormy weather set in, cover your wall with straw or boards; the first is best, if well secured; as it pro- tects the top of the wall, in some measure, from frost, which is very preju- dicial, particularly when it succeeds much rain ; for the rain penetrates to the heart of the wall, and the frost, by converting the water into ice, expands it, and causes the mortar to assume a short and crumbly nature, and alto- gether destroys its tenacity. In working up a wall, it is proper not to work more than four or five feet at a time ; for, as all walls shrink immediately after building, the part which is first brought up will remain stationary; and, when the adjoining part is raised to the same height, a shrinking or settling will take place, and separate the former from the latter, causing a crack which will become more and more evident, as the work proceeds. In carrying up any particular part, each side should be sloped off, to receive the bond of the adjoining work on the right and left. Nothing but absolute necessity can justify carrying the work higher, in any particular part, than one scaffold ; for, wherever it is so done, the workman should be answerable for all the evil that may arise from it. The distinctions of Bond have already been shown, and we shall now detail them more particularly ; again referring to Plate LXXXV, in which the arrangement of bricks, in depths of different thicknesses, so to form English Bond, is shown in figures 1, 3, 6, and 7. The bond of a wall of nine inches is represented by fig. 1. In order to prevent two upright or vertical joints from running over each other, at the end of the first stretcher from the corner, place the return corner-stretcher, which is a header, in the face that the stretcher is in below, and occupies half its length ; a quarter-brick is placed on its side, forming together 6f inches, and leave a lap of 2 inches for the next header, which lies with its middle upon the middle of the header below, and forms a continuation of the bond. The three-quarter brick, or brick-bat, is called a, closer. 35G THE NEW PRACTICAL BUILDER, &C. Another way of effecting this is, by laying a three-quarter bat at the corner of the stretching course ; for, when the corner-header comes to be laid over it, a lap of 2 inches will be left at the end of the stretchers below for the next header ; which, when laid, its middle will come over the joint below the stretcher, and in this manner form the bond. In a fourteen-inch or brick-and-half wall, (fig. 3 J the stretching course upon one side, is so laid that the middle of the breadth of the bricks, upon the opposite side, falls alternately upon the middle of the stretchers and upon the joints between the stretchers. In a two-brick wall, (fig 6J every alternate header, in the heading course, is only half a brick thick on both sides, which breaks the joints in the core of the wall. In a two-brick and a half wall, (fig. 1,) the bricks are laid as shown in figure 6. Flemish Bond, for a nine-inch wall, is represented in figure 2, wherein two stretchers lie between two headers, the length of the headers and the breadth of the stretchers extending the whole thickness of the wall. In brich-and-half Flemish bond, (fig. &,) one side being laid as in figure 2, and the opposite side, with a half-header, opposite to the middle of the stretcher, and the middle of the stretcher opposite the middle of the end of the header. Figure 5 exhibits another arrangement of Flemish Bond, wherein the bricks are disposed alike on both sides of the wall, the tail of the headers being placed contiguous to each other, so as to form square spaces in the core of the wall for half-bricks. The Face of an upright-wall, English Bond, is represented by figure 19, and that of Flemish Bond, by figure 20. BRICK-NOGGING is a mode of constructing a wall with a row of posts or quarters, disposed at three feet apart, with brick-work filling up the intervals. In this mode the wall is, generally, either of the thickness or breadth of a brick, and the wood-work flush on both sides with the faces of the bricks. Thin pieces of timber, laid horizontally from post to post, are so disposed BRICKLAYING. 357 as to form the brick-work, between every two posts or quarters, into several compartments in the height of the story ; each piece being inserted between two courses of brick, with its edges flush with the faces of the wall. Figure 13 is that of the head of an aperture, with a part of each jamb, the arch being straight. The manner of drawing the joints are as follow : (see figure 17.) Describe an equilateral triangle upon the width, AB, of the aperture ; and, from the vertex, C, or opposite angular point, describe a small circle, with a radius equal to the thickness of a brick. Draw the upper line, DE, forming the extrados, at a distance equal to the height of four bricks from the in- trados. Draw the skew-backs, AB, DE, one upon each side ; and draw a line parallel to one of the skew-backs touching the small circle, and cutting the extrados line in F, and draw the line CF. Find the point G, in the same manner, and draw GC, and so on to the middle, when the operation will be complete. "By figure 14, is represented the head of an aperture with a part of each jamb, the head being an arch formed by two concentric arcs, less than a semi-circle. This kind of arch is called a Scheme or Segment Arch. Figure 15 is a semi-circular arch. Figure 16 is a semi-elliptic arch, struck from the two centres, A and B, and having the longer axis in a horizontal position. GROINS. In figure 12 is represented the mode of constructing Groins rising from octagonal piers, and which are very convenient in cellars, where the removal of great weights are required. This is Mr. Tapper's improve- ment on the common four-sided groin. The improvement consists in raising the angle of the groin from an octagonal pier, instead of a square one, which gives more strength, and, from the corners being removed, renders it more commodious for turning any kind of goods around it : and it is farther to be observed that, in this construction, the angles of the groin are strengthened by carrying the band round the diagonals of equal breadth, and affording better bond to the bricks. CONSTRUCTION OF BRICK GROINS. The construction of groins and arches has already been shown in this work, under the head of Carpentry, pages 45. 4 x 358 THE NEW PRACTICAL, BUILDEU, &C. 109 to 118. We shall, therefore, only add that the difficulty attending the execution of a brick groin lies in the peculiar mode of appropriating proper bond at the intersecting of the two circles, as they gradually rise to the crown to an exact point. In the meeting or intersecting of these angles, the inner-rib should be perfectly straight, and perpendicular to a diagonal line drawn upon the plan. But no definition, either by lines or description, can be given equal to what may be gained by a little practice. After the centres are set, let the bricklayer apply two or three bricks to an angle, by which he will effectually see how to cut them, as well as the requisites of bond. The workman must observe, that the manner of turning groins with respect to the sides, is the same as in other arches and centres, except in the angles, which must be traced by applying the bricks. If the arch is to be rubbed and guaged, you must divide each arch into an exact number of parts, and extend the lines till they meet in the groin ; by which means you will easily find the curve for the angle, from which you must make your templets; observe, in fixing the centres, that the carpenters raise them something higher at the crown, to allow for settling, which frequently happens, sometimes by the pressure upon the butments, otherwise from the length of the crown. Be cautious, in building of vaults, that the piers or butments are of suffi- cient strength : all butments to vaults, whether groined, or only arched, should be one-sixth part of the width of the span ; and moreover, if there is any great weight to be sustained, bridgings of timber should be framed, to discharge the weight from the crown of the arch ; after a vault or groin is finished, it is highly necessary to pour on a mixture of terras, or lime and water, on the crown ; and give it some little time to dry, before you strike the centres, in order to cement the whole together. CORNICES. Ornamental brick cornices are represented in figures 21 and 22. The first shows the rudiments of the Doric entablature, and the second is a Dentil Cornice. Many pleasing dispositions of bricks may thus be made, frequently without cutting, or by chamfering only. BRICKLAYING. 359
  • ,) and elevation, (fig. 10 J and which is so shaped in order to make a proper passage for the smoke to the flue, C. This door, or blower, is not hung, but is put up and taken away by hand. When the oven is sufficiently hot, a man, placed at the mouth of the oven, with the iron bar, E, (fig. 5,) slides the cast-iron stopper, D, (fig. 1 and 6,) to the angle, F, where it stops, as shown by the dotted lines ; then going to the mouth of the furnace, he hooks the crooked part, G, of the same iron bar, (fig. 5 ,) into the circular hole of the stopper, H, (fig. 7,) and pulls the fillet, IIIII, (Jig. 8,) into the frame of the furnace, whereto it fits. This stopper is made to slide, but not in a groove, as the cinders might sometimes prevent its being shut. Figure 8 represents an iron frame, to be fixed round the mouth of the furnace in the inside. The opening of the mouth to be one foot two inches wide, and one foot high, which receives the fillets of the stopper, III. The door, K, (fig. 4,) is fastened to an iron chain, and is raised or let down, at pleasure, by turning the lever, L, (figures 4, 10.) This door is used to diminish, as much as possible, the mouth of the oven, while the bread is putting in, and to prevent the heat from escaping. To the handle of the lever is hung an iron pin, with a chain, and over it is a semi-circular iron plate, fastened to the wall, with five holes drilled in it to receive the pin, which will regulate the height of the door, K, at pleasure. 45. 4 Y 362 THE NEW PRACTICAL BUILDER, &C. When bread is baking, the curved door, B, not being then wanted, is taken away ; and the two doors of the oven, with the two doors of the furnace, are shut up. At the top of the furnace, M, (fig. 4-,) is a small flue, about three inches square, which communicates with the flue of the oven. The use of this small flue is to leave a passage for the sulphur that may remain in the ashes, and might injure the bread while baking. The communication of this small flue of the oven is opened or shut by mean of an iron slider, N, (fig. 10.) Over the furnace is a niche, (Jig. 3,) with a boiler of hot water. It has been noticed that, in ovens on this construction, whatever be the size of the fire-place, it is always proper to set the bars eight or ten inches in from the door, which will keep a supply of coals warming before they are pushed forward into the fire. The importance of this is known to those who have attended to the effect of every fresh supply of coals to the boilers of steam-engines, as it instantly stops the boiling, unless this precaution is attended to. It also prevents, in a great measure, the cold air getting in between the door and frame of the fire-place, which frequently happens, from the difficulty of fitting iron doors to iron frames. Ovens, on the improved construction, will hold, according to their size, as follows : Eight feet wide, and seven feet deep, eight bushels of bread. Nine feet wide, and seven and a half feet deep, ten bushels. Ten feet wideband eight and a half feet deep, twelve bushels. If required to hold less than eight bushels, or more than twelve, the proportions, of course, must vary accordingly. The oven represented in the plate is eight feet wide and seven deep ; and, therefore, as stated above, it is adapted for eight bushels of bread. The fire- hole, or furnace, (Jig. I,) enters the oven in a direction, diagonal with the farthest corner ; the sides of the oven are carried nearly straight, and turned as sharp as possible at the haunch and shoulder, this form being supposed better calculated to retain the heat than any other : the flue is immediately over the entrance, as shown in figures 1 and 4. Welsh lumps, or fire-bricks, BRICKLAYING. 363 are used for the furnace. In a work of this nature, it is usual to introduce a considerable quantity of old iron hoops, more especially around and over the oven, in order to keep the work together. This precaution is adviseable on all occasions where great heat is required. In building the oven, the crown is turned with the bricks on end, as shown in the section, (fig. 4) ; and, instead of centering, the custom is to fill the whole space with sand, clay, or rubbish, which is well trodden down, and fashioned to the shape which it is intended the crown shall be of. When the upper work is finished, the sand is dug out or removed by the mouth of the oven. Other particulars, interesting to the bricklayer, will be found in the Explanation of Terms, used by Bricklayers and Plasterers, hereafter. SUBSTANCE OF THE BUILDING ACT, &c. BY the Building-Act, of the 14th of Geo. III., every master-builder, or owner, in the cities of London and Westminster, the Weekly Bills of Mor- tality, the Parishes of Saint Mary-la-Bonne, Paddington, St. Pancras, and St. Luke's, Chelsea, prior to beginning any building, within the first and seventh rates, must give twenty-four hours notice of his intention to the district-surveyor, descriptive of the edifice to be erected or altered. And the proprietors of houses and grounds must also give three months' notice to pull down old party-walls, party-arches, party fence-walls, or quarter parti- tions, when decayed, or of insufficient thickness, and to be left with the owner or occupier of such house ; and, if empty, such notice to be stuck up on the front door, or front of such house. The same Act also recites, that every front side or end wall, not being a party-wall, shall be deemed an external wall. The thicknesses of party and external walls are described only of the first, second, third, and fourth, rates of building, the thicknesses of which, together with the prescribed thicknesses of the walls to the backs of the chimnies, in the several party and external walls, are described in Plates LXXXVII, 3G4> THE NEW PRACTICAL BUILDER, &C. LXXXVIII, and LXXXIX, which will fully elucidate the subject, as regards the construction of party and external walls, with the chimnies therein. The footings of external walls must have equal projections, except where the adjoining buildings will not admit of it, in which case the builder must be guided by circumstances, and conform, as nearly as possible, to the directions of the Act, under the guidance of the district surveyor. Walls, and other external inclosures to buildings, of the first, second, third, fourth, and fifth, rates, must be of brick, stone, artificial stone, lead, copper, tin, slate, tile, or iron, or a combination of those articles; but for foundations, wood-planks may be used. If it should be required to make recesses in external walls, they must be arched, in order that the arches and recesses may each be equal in thickness to one brick : of course recesses cannot be formed in walls which are only one brick thick. External walls of buildings, of the first, second, third, and fourth, rates, cannot be converted into party-walls, unless they are of the heights and thicknesses above the footings required for party-walls. Party-walls appertain to such persons who are the owners of first, second, third, or fourth, rate buildings, and who do not intend to have distinct and separate walls on such sides as are contiguous to other buildings. Party- walls must be placed half and half on the ground of each of the owners, without any notice being given, provided it adjoins vacant ground. Party-walls, or additions thereto, must be carried up at least eighteen inches above the roofs of the adjoining premises, measuring at right angles with the back rafters, and twelve inches above the gutters of the highest buildings against which they may abut, unless the height of the party-walls shall exceed those of the parapets, or blocking courses; in which events, they may be left less than twelve inches above the gutters, at distances of two feet six inches from the fronts of the parapets and blocking courses. Where it occurs that dormer-windows, or the heads of trap-doors, &c. are fixed upon flats, or roofs, within four feet of party-walls, such party-walls must be carried two feet higher than such dormer or erections previously constructed. BRICKLAYING. 365 Recesses must not be made in party-walls, that will reduce any parts thereof to thicknesses less than the Act requires for the highest rates of buildings, to which such walls shall appertain, except for chimney-flues, girders, &c., or for embracing the ends of piers or walls. Openings must not be made in party-walls, unless to communicate between two or more stacks of warehouses, or between one approved stable- building and another, in which cases, the communications must have iron- doors, not less than a quarter of an inch thick in the panels, fixed in stone frames and sills. Openings, also, may be made for public passage-ways on the ground-floor for foot passengers, or sufficiently wide for the egress and regress of carriages of every description ; as likewise for cattle, provided such passages or carriage-ways are arched over with brick or stone, or brick and stone united, of the thickness of one brick and a half at the least, for first-rate and second-rate buildings, and not less than one brick for the third and fourth rates ; and where cellars, or vaults, are required under such passages, or arch- ways, they must be arched over throughout in the same manner, and accord- ing to the same proportions before-mentioned. Party-walls, party-arches, or chimney-shafts, old or new, may be cut into, except in the following instances. Where the fronts of buildings are in lines with each other, incisions may be cut, either in the rear or principal fronts, for the purposes of inserting the ends of any new external walls of any intended adjoining building, but such incisions must not be made more than nine inches deep from the outward faces of the external walls, nor beyond the centres of the party-walls. And for the more convenient insertion of bressumers and story-posts, to be fixed on the ground-floors, either in the back or principal front walls, recesses may be cut from the foundations of the new walls to the heights of the bressumers, fourteen inches deep from the outward faces of such walls, and four inches wide in the cellar stories, and two inches wide in the ground stories. Party-walls may also be cut into for the purposes of tailing-in stone steps or stone landings, or for bearers to wood-stairs, or for laying in stone corbels, for supporting chimney-jambs, girders, purlines, binding or trimming joists, or any other principal timbers. 46. 4 z 366 THE NEW PRACTICAL BUILDER, &C. Perpendicular recesses may also be cut in party-walls, which are not less than one brick and a half thick, for the purposes of inserting walls or piers ; but the incisions must not be more than fifteen inches deep ; and where two or more incisions are made in the same description of walls, they must not approximate nearer to each other than ten feet. Incisions may be made in party-walls for any of the preceding purposes, that is, provided they are not likely to injure, remove, or endanger the timbers, chimnies, flues, or internal finishings, of the adjoining buildings. The incisions, indents, cuts, or recesses, before-mentioned, must be made good as soon as possible ; and during the time such operations are carrying into effect, the buildings adverted to must be carefully pinned up, where required, with bricks, stone, slate, tile, or iron-bedded, in mortar. The portions of houses, or other buildings, which are contrived to be built over public passages, or gateways, being divided, and again sub-divided, into rooms and stories, the property of different owners, when built or re-built, must have party-walls and party-arches, at least a brick and a half thick, for first and second-rate buildings, and one brick thick for second and third-rate buildings, between the several rooms or stories belonging to the different proprietors ; but the buildings to the Inns of Court are exempt from the latter regulation, it being deemed sufficient to build party-walls where any rooms or chambers communicate with the stair-cases, the walls in which instances are subject to the same regulations as other party-walls. Where the lower-rate buildings adjoin those of higher rates, the additions intended to be made thereto must be built according to the rules prescribed for the higher rates. Party-walls, which are built against other buildings, must be of the same thicknesses with the walls of the highest adjoining buildings in the stories next below the roofs ; but such party-walls must not be erected, unless they can be done with the greatest safety to the adjoining walls or buildings. Dwelling-houses, or other buildings, four stories high from the founda- tions, exclusive of the rooms in the roofs, must have party-walls, according to the third-rate of buildings, although such houses, or other buildings, estimated by the numbers of squares on the ground-floor, are of the fourth- BRICKLAYING. 367 rate. And every dwelling-house, or other building, exceeding four stories in height from the foundations, exclusive of rooms in the roofs, must have party-walls according to the first-rate, although such houses, or other buildings, are not of the first-rate, that is, according to the number of squares on the ground-floor. Chimnies must not be erected on timbers, unless it be indispensably neces- sary to pile and plank the foundations. Chimnies may be built in party-walls back to back ; but they must not be less in thicknesses, from the centres of such walls, than are described in the plate of plans elucidating the intention of the Act of Parliament, as regards the construction of chimnies in party and external walls. The breasts of chimnies in party-walls must not, in any instance, be less than one brick thick in the cellar, and half a brick in every other story, and the divisions or widths, between the flues, must never be less than half a brick thick. The flues in party-walls may be built against each other, but must not approach nearer than two inches to the centres of such walls. The breasts of chimnies and backs, together with the flues, must, in all cases, be rendered or pargetted, care being taken that timbers are not any where introduced for supporting the breasts, which must be carried by strong brick or stone arches, assisted by wrought-iron bars, as many as are requisite to each opening. The flues in party-walls against vacant ground, must be lime-whited, or marked in some durable manner, and immediately after any other houses are erected against them, whatever is deemed requisite to be done to the flues must, as soon as possible, be completed. Brick-flues, or funnels, must not be made on the outside of any building, of the first, second, third, or fourth, rates, next to any street, square, road, or passage, so as to project beyond the general line of the buildings ; nor must any funnel, or pipe of iron, copper, or tin, for the conveyance of smoke or steam, be fixed near any public street, or court, &c., to buildings of the first, second, third, or fourth, rate description; nor must any pipes, for the conduct- ing of smoke, be fixed within any of the before-mentioned buildings, nearer than fourteen inches to any timber, or other materials subject to take fire. 368 THE NEW PRACTICAL BUILDER, &C. Every dwelling-house, which exceeds! NINE squares of building on the ground-floor, including internal and external walls, are deemed the first-rate or class of buildings, and subject to be built in manner before-described; and, likewise, according to the thicknesses of the walls, drawn and figured in the sections and plans hereunto referred. Dwelling-houses, also, which exceed FIVE squares of building, and are not more than NINE, are deemed second-rate houses. Dwelling-houses, also, which exceed THREE squares and ONE-HALF, and are not more than FIVE squares, are deemed third-rate houses. Dwelling-houses, also, which shall not exceed THREE and ONE-HALF squares, are deemed fourth-rates. Fifth, sixth, and seventh, rate buildings, may be built of any dimensions, provided the conditions, stipulated in the Act of Parliament, are adhered to. DISTRICT-SURVEYORS' FEES. . s. d. For every first-rate building. 3 10 For every alteration or addition 1 15 For every second-rate 3 3 o For every alteration or addition 1 10 For every third-rate 2 10 For every alteration or addition 1 5 For every fourth-rate 2 2 For every alteration or addition . * 1 1 For every fifth-rate 1 10 For every alteration or addition 15 For every sixth-rate 1 1 o For every alteration or addition 10 6 For every seventh-rate 10 6 For every alteration or addition 5 The District-Surveyors are elected by the Magistrates for the Counties of Middlesex and Surrey. PLASTERING. 369 CHAPTER IX. PLASTERING. IN modern practice, PLASTERING, by its recent improvements, occurs in every department of architecture, both internally and externally. It is more particularly applied to the sides of the walls, and the ceilings of the interior parts of buildings, and, also, for stuccoing the external parts of many edifices. In treating on this subject, we shall divide Plastering under its several heads : as plastering on laths, in its several ways ; rendering on brick and stone; and, finally, the finishing to all the several kinds of work of this description ; as well as modelling, and casting the several mouldings, both ornamental and plain; stuccoing, and other outside compositions, which are applied upon the exterior of buildings ; and, the making and polishing the scagliola, now so much used for columns, and their antaB, or pilasters, &c. LIME forms an essential ingredient in all the operations of this trade. This useful article is vended at the wharfs about London in bags, and varies in its price from thirteen shillings to fifteen shillings per hundred pecks. Most of the lime made use of in London is prepared from chalk, and the greater portion comes from Purfleet, in Kent ; but, for stuccoing, and other work, in which strength and durability is required, the lime made at Dorking, in Surrey, is preferred. The composition, known as PLASTER OF PARIS, is one on which the Plasterer very much depends for giving the precise form and finish to all the better parts of his work; with it he makes all his ornaments and cornices, besides mixing it in his lime to fill up the finishing coat to the walls and ceilings of rooms. 46. 5 A 370 THE NEW PRACTICAL BXIILDER, &C. The stone, from which the plaster is obtained, is known to professional men by several names, as sulphate of lime, selenite, gypsum, &c. ; but its common name seems to have been derived from the immense quantities which have been taken from the hill named Mont-Martre, in the environs of Paris. The stone from this place is, in its appearance, similar to common free-stone, excepting its being replete with small specular crystals. The French break it into fragments of about the size of an egg, and then burn it in kilns, with billets of wood, till the crystals lose their brilliancy ; it is then ground with stones, to different degrees of fineness, according to its intended uses. This kind of specular gypsum is said to be employed in Russia, where it abounds, as a substitute for glass in windows. According to the chymists, the specific gravity of gypsum, or Plaster of Paris, is from 1.872 to 2.311, requiring 500 parts of cold and 450 of heat, to dissolve it ; when calcined, it decrepitates, becomes very friable and white, and heats a little with water. In the process of burning, or calcination, it loses its water of crystallization, which, according to Fourcroy, is 22 per cent. The plaster commonly made use of in London is prepared from a sulphate of lime, produced in Derbyshire, and called alabaster. Eight hundred tons are said to be annually raised there. It is brought to London in a crude state, and afterwards calcined, and ground in a mill for use, and vended in brown- paper bags, each containing about half a peck ; the coarser sort is about fourteen-pence per bag, and the finest from eighteen to twenty-pence. The figure-makers use it for their casts of anatomical and other figures ; and it is of the greatest importance not only to the plasterer, but to the sculptor, mason, &c. The working-tools of the plasterer consist of a spade, of the common sort, with a two or three-pronged rake, which he uses for the purpose of mixing his mortar and hair together. His trowels are of two sorts, one kind being of three or four sizes. The first sort is called the laying and smoothing-tool ; its figure consists in a flat piece of hardened iron, very thin, of about ten inches in length, and two inches and a half in width, ground to a semi-circular shape at one end, while the other is left square ; on the back of the plate, and PLASTERING. 371 nearest to the square end, is riveted a piece of small rod-iron, with two legs, one of which is fixed to the plate, and the other, adapted for being fastened in a round wooden handle. With this tool all the first coats of plastering are put on ; and it is also used in setting the finishing coat. The trowels of the plasterer are made more neatly than the tools of the same name used by other artificers. The largest size is about seven inches long on the plate, and is of polished steel, two inches and three-quarters at the heel, diverging to an apex or point. To the wide end is adapted a handle, commonly of mahogany, with a deep brass ferrule. With this trowel the plasterer works all his fine-stuff", and forms cornices, mouldings, &c. The other trowels are made and fitted-up in a similar manner, varying gradually in their sizes from two to three inches in length. The plasterer likewise employs several small tools, called stopping and picking-out tools ; these are made of steel, well polished, and are of different sizes, commonly about seven or eight inches long, and half an inch wide, flattened at both ends, and ground away till they are somewhat rounding. With these he models and finishes all the mitres and returns to the cornices, and fills up and perfects the ornaments at their joinings. The workman in this art should keep all his tools very clean ; they should be daily polished, and never put away without being wiped and freed from plaster. In the practice of plastering many rules and models of wood are required. The rules, or straight-edges as they are called, enable the plasterer to get his work to an upright line ; and the models guide him in running plain mouldings, cornices, &c. The CEMENTS made use of, for the interior work, are of two or three sorts. The first is called lime and hair, or coarse stuff; this is prepared in a similar way to common mortar, with the addition of hair, from the tan-yards, mixed in it. The mortar used for lime and hair is previously mixed with the sand, and the hair added afterward. The latter is incorporated by the labourers with a three-pronged rake. 372 THE NEW PRACTICAL BUILDER, &C. FINE-STUFF, is pure lime, slaked with a small portion of water, and after- wards well saturated, and put into tubs in a semi-fluid state, where it is allowed to settle, and the water to evaporate. A small proportion of hair is sometimes added to the fine-stuff. STUCCO, for inside walls, called trowelled or bastard stucco, is composed of the fine-stuff above described, and very fine washed sand, in the proportion of one of the latter to three of the former. All walls, intended to be painted, are finished with this stucco. MORTAR, called gauge-stuff, consists of about three-fifths of fine-stuff and one of Plaster of Paris, mixed together with water, in small quantities at a time : this renders it more susceptible of fixing or setting. This cement is used for forming all the cornices and mouldings, which are made with wooden moulds. When great expedition is required, the plasterers guage all their mortar with Plaster of Paris. This enables them to hasten the work, as the mortar will then set as soon as laid on. PLASTERERS have technical words and phrases, by which they designate the quality of their work, and estimate its value. By LATHING is meant the nailing up laths, or slips of wood, on the ceiling and partitions. The laths are made of fir or oak, and called three-foots and four- foots, being of these several lengths : they are purchased by the bundle or load. There are three sorts of laths ; viz. single laths, lath and half, and double laths. Single laths are the cheapest and thinnest ; lath and half denotes one third thicker than the single lath; and double laths twice their thickness. The laths generally used in London are made of fir, imported from Norway, the Baltic, and America, in pieces called staves. Most of the London timber- merchants are dealers in laths; and there are many persons who confine themselves exclusively to this branch of trade. The fir-laths are generally fastened by cast-iron nails, whereas the oaken ones require wrought-iron nails, as no nail of the former kind would be found equal to the perforation of the oak, which would shiver it in pieces by the act of driving. PLASTERING. 373 In lathing ceilings, it is adviseable that the plasterer should make use of laths of both the usual lengths, and so manage the nailing of them, that the joints should be as much broken as possible. This will tend to strengthen the plastering laid thereon, by giving it a stronger key or tie. The strongest laths are adapted for ceilings, and the slightest or single laths for the parti- tions of buildings. LAYING consists in spreading a single coat of lime and hair all over a ceiling or partition ; taking care that it is very even in every part, and quite smooth throughout : this is the cheapest manner of plastering. PRICKING-UP is similar to laying, but is used as a preliminary to a more perfect kind of work. After the plastering has been put up in this manner, it is scratched all over with the end of a lath, in order to give a key or tie to the finishing coats, which are to follow. LATHING, LAYING, and SET, are applied to work that is to be lathed as already described, and covered with one coat of lime and hair ; and, when sufficiently dry, finished by being covered over with a thin and smooth coat of lime only, called by the plasterer putty, or set. This coat is spread with the smoothing trowel, and the surface finished with a large flat hog's-hair brush. The trowel is held in the right hand, and the brush in the left. As the plasterer lays on the set, he draws the brush backwards and forwards over it, till the surface is smooth. LATHING, FLOATING, and SET, consists of lathing and covering with a coat of plaster, which is pricked up for the floated work, and is thus performed : The plasterer provides himself with a strong rule, or straight-edge, often from ten to twelve feet in length ; two workmen are necessarily employed therein. It is began by plumbing with a plumb-rule, and trying if the parts to be floated are upright and straight, to ascertain where filling up is wanting. This they perform by putting on a trowel or two of lime and hair only : when they have ascertained these preliminaries, the screeds are prepared. A SCREED, in plastering, is a stile formed of lime and hair, about seven or eight inches wide, guaged exactly true. In floated-work these screeds are made, at every three or four feet distance, vertically round a room, and 47. 5 B 374* THE NEW PRACTICAL BUILDER, &C. are prepared perfectly straight by applying the straight-edge to them to make them so ; and, when all the screeds are formed, the parts between them are filled up flush with lime and hair, or stuff, and made even with the face of the screeds. The straight-edge is then worked horizontally upon the screeds, to take off all superfluous stuff. The floating is thus finished by adding stuff continually, and applying the rule upon the screeds till it becomes, in every part, quite even with them. Ceilings are floated in the same manner, by having screeds formed across them, and filling up the intermediate spaces with stuff, and applying the rule as for the walls. ^aHo^r Plastering is good or bad, in proportion to the care taken in this part of the work ; hence the most careful workmen are generally employed therein. The SET to the floated work is performed in a similar way to that already described for the laid plastering ; but floated plastering, for the best rooms, is performed with more care than is required in an inferior style of work. The setting, for the floated work, is frequently prepared by adding to it about one-sixth of Plaster of Paris, that it may fix more quickly, and have a closer and more compact appearance. This, also, renders it more firm, and better adapted for being whitened, or coloured, when dry. The drier the prick- ing-up coat of plastering is, the better for the floated stucco-work ; but if the floating is too dry before the last coat is put on, there is a probability of its peeling off, or cracking, and thus giving the ceiling an unsightly appearance. These cracks, and other disagreeable appearances iu ceilings, may likewise arise from the weakness of the laths, or from too much plastering, or from strong laths and too little plastering. Good floated work, executed by a judicious hand, is very unlikely to crack, and particularly if the lathing be properly attended to. RENDERING and SET, or RENDERING, FLOATED, and SET, includes a portion of the process employed in both the previous modes, with the exception that no lathing is required in this branch of the work. By rendering is meant that one coat of lime and hair is to be plastered on a wall of brick or stone; and the set implies that it is again to be covered and finished with fine stuff, or PLASTERING. 375 putty. The method of performing this is similar to that already described for the setting of ceilings and partitions. The floated and set is performed on the rendering in the same manner as it is on the partitions and ceilings of the best kind of plastering, which has been described. TROWELLED-STUCCO is a very neat kind of work, much used in dining- rooms, vestibules, stair-cases, &c., especially when the walls are to be finished by painting. This kind of stucco requires to be worked upon a floated ground, and the floating should be as dry as possible before the stuccoing is began. When the stucco is made, as before described, it is beaten and tem- pered with clean water, and is then fit for use. In order to use it, the plas- terer is provided with a small float, which is merely a piece of half-inch deal, about nine inches long and three inches wide, planed smooth, and a little rounded away on its lower edge ; a handle is fitted to the upper side, to enable the workman to move it with ease. The stucco is spread upon the ground, which has been prepared to receive it, with the largest trowel, and made as even as possible. When a piece, four or five feet square, has been so spread, the plasterer, with a brush, which he holds in his left hand, sprinkles a small part of the stucco with water, and then applies the float, alternately sprinkling and rubbing the face of the stucco, till he reduces the whole to a perfect smooth and even surface. The water has the effect of hardening the face of the stucco ; so that, when well floated, it feels to the touch as smooth as glass. CORNICES are plain or ornamented, and sometimes include a portion of both ; in the ornamented, superior taste has latterly prevailed, on princi- ples derived from the study of the antique. The preliminaries, in the forma- tion of cornices in plastering, consist in the examination of the drawings or designs, and measuring the projections of the members : should the latter be found to exceed seven or eight inches, bracketing will be necessary. BRACKETING consists of pieces of wood fixed up at about eleven or twelve inches from each other, all round the place intended to have a cornice ; on these brackets laths are fastened, and the whole is covered with one coat of plastering, making allowance in the brackets for the stuff necessary to form the cornice; 376 THE NEW PRACTICAL BUILDER, &C. for this about one inch and a quarter is generally found sufficient. When the cornice has been so far forwarded, a mould must be made of the profile or section of the cornice, exactly representing all its members ; this is generally prepared, by the carpenters, of beach-wood, about a quarter of an inch in thickness ; all the quirks, or small sinkings, being formed in brass. When the mould is ready, the process of running the cornice begins : two workmen are required to perform this operation ; and they are provided with a tub of set, or putty, and a quantity of Plaster of Paris; but before they begin with the mould, they guage a straight line, or screed, on the wall and ceiling, made of putty and plaster, extending so far on each as to answer to the bottom and top of the cornice, for it to fit into. This is the guide for moving the mould upon. The putty is then mixed with about one-third of Plaster of Paris, and incorporated in a semi fluid state, by being diluted with clean water. One of the workmen then takes two or three trowels full of the prepared putty upon his hawk, which he holds in his left hand, having in his right hand a trowel, with which he plasters the putty over the parts where the cornice is to be formed ; the other workman applying the mould to ascer- tain where more or less of it is required ; and, when a sufficient quantity has been put on to fill up all the parts of the mould, the other workman moves the mould backwards and forwards, holding it up firmly to the ceiling and wall ; thus removing the superfluous stuff, and leaving in plaster the exact contour of the cornice required. This is not effected at once, but the other workman keeps supplying fresh putty to the parts which want it. If the stuff dries too fast, one of the workmen sprinkles it with water from a brush. When the cornices are of very large proportions, three or four moulds are requisite, and they are applied in the same manner until the whole of their parts are formed. The mitres, internal and external, and also small returns or breaks, are afterwards modelled and filled up by hand. ORNAMENTAL CORNICES are formed previously, and in a similar way to those described, excepting that the plasterer leaves indents or sinkings in the mould for the casts to be fixed in. The plasterers of the present day cast all their ornaments in Plaster of Paris ; whereas they were formerly the work PLASTERING. 377 of manual labour, performed by ingenious men, then known in the trade as ornamental-plasterers. The casting of ornaments in moulds has almost super- seded this branch of the art ; and the few individuals now living, by whom it was formerly professed, are chiefly employed in modelling, and framing of moulds. All the ornaments which are cast in Plaster of Paris, are previously mo- delled in clay. The clay-model exhibits the power and taste of the designer, as well as that of the sculptor. When it is finished, and becomes rather firm, it is oiled all over, and put into a wooden frame. All its parts are then re- touched and perfected, for receiving a covering of melted wax, which is poured warm into the frame and over the clay-mould. When cool, it is turned upside down, and the wax comes easily away from the clay, and is an exact reversed copy. In such moulds are cast all the enriched mouldings, now prepared by common plasterers. The waxen models are made so as to cast about one foot in length of the ornaments at a time; this quantity being easily removed out of the moulds, without the danger of breaking. The casts are all made with the finest and purest Plaster of Paris, saturated with water. The casts, when first taken out of the moulds, are not very firm, and are suffered to dry a little, either in the air, or in an oven adapted for the purpose ; and when hard enough to bear handling, they are scraped and cleaned up for the workmen to fix in the places intended. The FRIEZES and BASSO-RELIEVOS are performed in a manner exactly similar, except that the waxen moulds are so made as to allow of grounds of plaster being left behind the ornaments, half an inch, or more, in thicknesses; and these are cast to the ornaments or figures, which strengthen and secure the proportions. CAPITALS to columns are prepared in a similar way, but require several moulds to complete them. The Corinthian capital will require a shaft or bell to be first made, exactly shaped, so as to produce graceful effects in the foliage and contour of the volutes ; all of which, as well as the other details, require separate and distinct reversed moulds when intended for capitals made to order. 47. 5c 378 THE NEW PRACTICAL BUILDER, &C. The plasterers, as before-mentioned, in forming cornices, in which orna- ments are to be used, take care to have projections in the running moulds ; which have the effect of grooves or indents in the cornices ; and into these grooves are put the ornaments after they are cast, which are fixed in their places by having small quantities of liquid Plaster of Paris spread at their backs. Friezes are prepared for cornices, &c., in a similar way, by leaving projections in the running moulds, at those parts of the cornices where they are intended to be inserted, and they are also fixed in their places with liquid plaster. Detached ornaments, when designed for ceilings, or any other parts, to which running moulds have not been employed, are cast in pieces exactly corresponding with the designs, and are fixed upon the ceilings, or other places, with white-lead. Plasterers require numerous models in wood, and very few or any of their best works can be completed without them. But, with moulds, good plas- terers are capable of making the most exquisite mouldings, possessing sharpness and breadths unequalled by any other modes now in practice. This, however is, in some measure, dependent on the truth of the moulds. Good plastering is known by its exquisite appearance> as to its regularity, correctness, solid effect, and without any cracks or indications of them. ROMAN CEMENT, or OUTSIDE STUCCO. The qualities of this valuable cement is now generally known in every part of the United Kingdom. It was first introduced to public notice by the late JAMES WYATT, Esq. eminent for having planned and executed some of the most magnificent and useful structures in these countries. It was originally known as Parker's Patent Cement, and was sold by Messrs. Charles Wyatt and Co., Bankside, London, s.t Jive shillings and sixpence per bushel : it is now vended by different manu- facturers in the Metropolis at three shillings per bushel, and even less, when the casks are returned. Equal quantities of this cement and sharp clean grit- sand, mixed together, will form very hard and durable coverings for the out- sides of public and private edifices. If the sand is wet or damp, the compo- sition should be used immediately. When the works are finished, they should be frescoed, or coloured, with washes, composed in proportions of five ounces PLASTERING. 379 of copperas to every gallon of water, and as much fresh lime and cement as will produce the colours required. Where these sorts of works are executed with judgment, and finished with taste, so as to produce picturesque effects, they are drawn and jointed to imitate well-bonded masonry, and the divisions promiscuously touched with rich tints of umber, and occasionally with vitriol; and, upon these colours mellowing, they will produce the most pleasing and harmonious effects ; especially if dashed with judgment, and with the skill of a painter who has profited by watching the playful tints of nature, produced by the effects of time in the mouldering remains of our antient buildings. The following are the average prices for Roman Cement, as charged in the Metropolis, at per yard superficial. s. d. Roman Cement, on the outsides of buildings, including all materials, and colouring in the manner described 4 9 Ditto, without colouring 4 Ditto, on strong laths 5 3 Plain fascias, pilasters, and belting-courses, or strings, per foot super 6 Plain cornices, ditto. , 2 2 Sinkings, per foot run 5 Arris', ditto 2 Nine-inch reveals to windows, ditto 8 Four inches and a half reveals, ditto 5 Where the works are circular, one-fourth, one-fifth, or one-sixth, should be added to the above prices, in proportions to the quickness of the cur- vatures. The expenses of dubbing-out, to make the works fair, must be added to the before-mentioned prices ; also the value of the cement consumed in per- forming the same ; together with all the time in erecting and taking down the scaffolding ; as, also, the expenses of iron spikes, for twine, and other requisite materials* for creating extra projections to cornices, pilasters, &c- 380 THE NEW PRACTICAL, BUILDER, &C. One bushel of cement, used with discretion, care, and judgment, will perform from three to four yards superficial ; that is, mixed with an equal portion of clean sharp grit-sand ; and, in procuring the latter article, great pains should be taken to select such qualities as are of a lively and binding description, and free from all slime or mud. As soon as the sand and cement is mixed with clean water, the composition should be used as quick as possible, and not a moment lost in floating the walls, which will require incessant labour, until the cement is set, which is almost instantaneous. ROUGH-CASTING is an outside finishing cheaper than stucco. It con- sists in giving the wall to be rough-casted a pricking-up coat of lime and hair; and when this is tolerably dry, a second coat of the same material, which is laid on the first, as smooth and even as can be. As fast as this coat ig finished, a second workman follows the other with a pail of rough-cast, which he throws on the new plastering. The materials for rough-casting are composed of fine gravel, with the earth washed cleanly out of it, and after- wards mixed with pure lime and water, till the entire together is of the con- sistence of a semi-fluid ; it is then spread, or rather splashed, upon the wall by a float made of wood. This float is five or six inches long, and as many wide, made of half-inch deal, to which is fitted a rounded deal handle. The plasterer holds this in his right hand, and in his left a common white- wash brush ; with the former he lays on the rough-cast, and with the latter, which he dips in the rough-cast, he brushes and colours the mortar and rough-cast that he has spread, to make them, when finished and dry, appear of the same colour throughout. SCAGLIOLA. The practice of forming columns with SCAGLIOLA is a distinct branch of plastering. It originated in Italy, and was thence intro- duced into France, then into England. For its first introduction here our country is indebted to the late Henry Holland, Esq. who was for many years the favourite architect of his present Majesty, who caused artists to be invited from Paris to perform such works in Carlton-Palace ; some of whom, from finding a considerable demand for their works, remained with us, and taught the art to our British workmen. PLASTERING. 381 In order to execute columns and their antae, or pilasters, in Scagliola, the following remarks and directions are to be observed : when the architect has furnished the drawing, exhibiting the diameter of the shafts, a wooden cradle is made, about two and a half inches less in diameter than that of the pro- jected column. This cradle is lathed all round, as for common plastering, and afterwards covered by a pricking-up coat of lime and hair : when this is quite dry, the workers in Scagliola commence their peculiar labour. The Scagliola is capable of imitating the most scarce and precious marbles; the imitation taking as high a polish, and feeling to the touch as cold and solid as the most compact and dense marble. For the composition of it the purest gypsum must be broken in small pieces, and then calcined till the largest fragments have lost their brilliancy. The calcined powder is then passed through a very fine sieve, and mixed up with a solution of Flanders glue, isinglass, &c., with the colours required in the marble they are about to imitate. When the work is to be of various colours, each colour is prepared se- parately, and they are afterwards mingled and combined nearly in the same manner as a painter mixes, on his pallet, the primitive colours which are to compose his different shades. When the powdered gypsum, or plaster, is prepared, and mingled for the work, it is laid on the shaft of the column, &c., covering over the pricked-up coat, which has been previously laid on it, aud is floated, with moulds of wood, to the sizes required. During the floating, the artist uses the colours necessary for the marble intended to be imitated, which thus become mingled and incorporated in it. In order to give his work the polish or glossy lustre, he rubs it with a pumice-stone, and cleanses it with a wet sponge. He next proceeds to polish it with tripoli and char- coal, and fine soft linen ; and, after going over it with a piece of felt, dipped in a mixture of oil and tripoli, finishes the operation by the application of pure oil. This is considered as one of the finest imitations in the world ; the Scagliola being as strong and durable as real marble for all works not exposed to the 48. 5 D THE NEW PRACTICAL BUILDER, &C. effects of the atmosphere, retaining its lustre as long and equal to real marble, without being one-eighth of the expense of the cheapest marble imported. COMPOSITION. Besides the composition, before adverted to, for co- vering the outsides of buildings, plasterers use a finer species of composi- tion for inside ornamental works. The material alluded to is of a brownish colour, exceedingly compact, and, when completely dry, very strong. It is composed of powdered whitening, glue in solution, and linseed-oil ; the proportions of which are, to two pounds of whitening one pound of glue, and half a pound of oil. These are placed in a copper and heated, being stirred with a spatula till the whole becomes incorporated. It is then suffered to cool and settle; after which it is taken and laid upon a stone, covered with powdered whitening, and beaten till it becomes of a tough and firm consistence. It is then put by for use, and covered with wetted cloths to keep it fresh. The ornaments to be cast in this composition are modelled in clay, for common plastering, and afterwards carved in a block of box-wood. The carving must be done with great neatness and truth, as on it depends the exquisiteness of the ornament. The composition is cut with a knife into pieces, and then closely pressed by hand into every part of the mould ; it is then placed in a press, worked by an iron screw, by which the composition is forced into every part of the sculpture. After being taken out of the press, by giving it a tap upside down, it comes easily out of the mould. One foot in length is as much as is usually cast at a time ; and when this is first taken out of the mould, all the superfluous composition is removed by cutting it off with a knife; the waste pieces being thrown into the copper to assist in making a fresh supply of composition. This composition, when formed into ornaments, is fixed upon wooden or other ground, by a solution of heated glue, white-lead, &c. It is afterwards painted or gilded, to suit the taste and style of the work for which it is intended. PLASTERING. 383 Mfl9 .d .ncyiiaoqflioo ortJ 39bia..v ')D PLASTERER'S MEASURING AND VALUATION. The measuring and valuation of plasterer's work is conducted by surveyors. All common plastering is measured by the yard square, of nine feet ; this includes the partitions and ceilings of rooms, stuccoing, internally and exter- nally, &c. &c. Cornices are measured by the foot superficial, girting their members to ascertain their widths, which, multiplied by their lengths, will produce the superficial contents. Running measures consist of beads, quirks, arris's, and small mouldings. Ornamental cornices are frequently valued in this way ; that is, by the running foot. The labour in plasterer's work is frequently of more consideration than the materials ; hence it becomes requisite to note down the exact time which is consumed in effecting particular portions, so that an adequate and proper value may be put upon the work. wtnr- 384 THE NEW PRACTICAL BUILDER, &C. CHAPTER X. AN EXPLANATION OF TERMS, AND DESCRIPTION OF TOOLS, USED IN BRICKLAYING AND PLASTERING. ANGLE FLOAT. See Float. BASS. A short trough for holding mortar, when tiling the roof: it is hung to the lath. BASTARD STUCCO. See Stucco. BAY. A strip or rib of plaster between screeds, for regulating the floating rule. See Screeds. BED OF A BRICK. The horizontal surface as disposed in a wall. BEDDING STONE. A straight piece of marble used to try the rubbed side of a brick. BRICKLAYER'S TOOLS. The principal tools used by bricklayers are repre- sented in plate XC. The Brick-trowel, (fig. 1 J is used for taking up and spreading mortar, and likewise for cutting bricks to any required size : this last use renders it necessary that they should be made of the best steel, well tempered. The Hammer used by bricklayers (Jig. *!,) is adapted either for driving, or dividing bricks, and for cutting holes in brick-work, &c. To suit it for these different purposes, one end of the head is formed like the common hammer ; and the other end is furnished with a kind of axe, similar to that used by carpenters, only much narrower in proportion to its length. The handle is TERMS, &C. IN BRICKLAYING AND PLASTERING. 385 placed much nearer to the striking than to the cutting part of the head. The hammer employed for dividing and pulling down old brick walls, instead of the axe part, more nearly resembles an adze, but is not so broad in porpor- tion to its length. The Plumb Rule is about four feet in length, and it is furnished with a line and plummet, to direct the workman in carrying up walls vertically. It con- sists of a well-seasoned board, of moderate thickness and breadth, and about four feet long. Down the middle of one side of this board a straight line is drawn, so that a cord which is fastened to the top, equi-distant from the arris on each side of it, may hang straight therein. To this cord a weight is attached, in order to shew the inclination of the cord more correctly, and a hole is cut in the bottom to admit it. The workman, in order to carry up his wall perpendicularly, applies either of the sides of this rule to the wall, so that the plummet and its cord may face him ; and if the cord of the plum- met does not coincide with the line on the rule, before the mortar becomes dry, he sets the bricks farther in or out, according to circumstances. The Level, used by bricklayers, is similar to that of the carpenter, described in page 233. Its length varies from six to twelve feet. If to the middle of a long narrow edge of a board of the same thickness, but about double the length of the plumb-rule, one end of the plumb-rule were joined at right angles, as the lower edge or side of the piece thus added to the rule would become the surface placed on walls, to ascertain whether they are horizontal or not, the plumb-line would thus become a level. The correctness of a level is determined by placing it vertically on any flat surface, and lowering or raising the support till the cord of the plummet exactly coincides with the line on the perpendicular rule. When this is done, reverse the ends, and if the same effect takes place, the level is true ; but if it does not, the bottom of the board must be shot, till this coincidence takes place. The perpendi- cular and horizontal parts of the level are not only fastened together by a tenon and mortise, but two braces are likewise added, in a slanting direction, from the horizontal piece nearly to the top of the perpendicular one, in order to give it greater firmness. 48. SE 386 THE NEW PRACTICAL BUILDER, &C. The Large Square is employed for setting out the sides of a building at right-angles. The Small Square, (fig. 3,) is used for trying the bedding of bricks, and squaring the soffits across their breadth. The Bevel, (fig. 4J is employed in drawing the soffit line on the face of the bricks. The Rod is from five to ten feet in length, and is used for measuring lengths, heights, and breadths, with more facility than could be done by a pocket-rule. The feet are divided by notches, and one of those next to the extremity of the rod is subdivided into inches. The Measuring-Tape is a kind of strong tape, graduated, marked, and coiled up by a little winch into a cylindrical box ; and thus it unites the portableness of the pocket-rule, with the greater despatch of the rod. The Jointing-Rule (fig. 5,) is about eight or ten feet long, and about four inches broad. It is used for running the joints of brick-work. When it is designed for the use of two bricklayers, the latter length is employed. The Jointer (jig. 6,) is an iron tool, used along with the jointing-rule, to mark the joints of brick work horizontally and vertically. Its form nearly resembles an 0D, with less flexure in proportion. The Raker, (fig. 8,) is a piece of iron, pointed with steel, with two knees or angles, so as to resemble a Z, if the connecting stroke of the top and bottom were perpendicular instead of slanting ; so that the pointed parts are of equal lengths, and stand at contrary sides of the middle part. It is used to pick or rake loose and decayed mortar out of the joints of old walls, that are intended to be re-filled with new mortar, or fresh pointed. The Hod resembles the half of a rectangular box, divided so as to consist of two boards at right angles to each other, and a third board to shut up one end. From the middle of the angular ridge, formed by the meeting of the two sides, projects the handle, which is a pole of about four feet long. That part of the ridge which is between the back or end board, is generally covered with several thicknesses of leather ; or with leather stuffed with wool, in order that the angle of the ridge may not cut the shoulder of the labourer. TERMS, &C. IN BRICKLAYING AND PLASTERING. 387 This utensil is used for carrying bricks or mortar for supplying the brick- layer. When used for mortar, it is customary to strew the internal surface of the hod with clean dry sand, to prevent the mortar from adhering to the wood. The Line Pins, (Jig. 9,) consist of two iron pins, with a line of about sixty feet, fastened by one of its extremities to each. It is used by being stretched close to the wall, and removed at proper intervals as the work advances, in order to guide the workmen in laying the different courses of bricks exactly straight. The Rammer, used by bricklayers, (fig. 10J is similar to that of paviors, and it is used for ascertaining whether ground is sufficiently strong to bear a foundation. It is also used for giving unstable ground its proper degree of compression, in order to prevent fractures in the walls ; which would natu- rally ensue, if the foundation were laid when the ground is in a loose state. The Iron Crow is a long bar of iron used for breaking through old walls, or raising heavy bodies out of the ground. The Pick-axe, (fig. 11,) consists of a long bar of iron, of considerable thickness in the centre, which is furnished with a hole or mortise for receiving a handle. The two arms on the sides of the centre and handle generally diminish or taper considerably towards their extremities, in order that their points may more readily enter the earth in digging ; or be inserted between bricks or stones to separate them. The arms, likewise, incline towards the handle,, so as to form the segment of a circle. The pick-axe is frequently used in conjunction with the iron crow. The Compasses, used by the bricklayer, are similar to those used by the carpenter and joiner, and are employed for traversing arches. The Grinding- Stone is used for sharpening axes, hammers, &c., and scarcely, perhaps, requires to be mentioned. The Scribe is a pike, or large nail, ground to a sharp point, and used for marking the bricks in the part where they are to be cut. The Chopping-block is made of any chance piece of wood that can be obtained, of about six or eight inches square, when for two men to work 388 THE NEW PRACTICAL BUILDER, &.C. thereon; and lengthened in proportion for four or more. It is generally supported, about two feet three inches from the ground, upon two or more fourteen-inch brick piers. It is better to have several blocks when they can be obtained, in preference to allowing many hands to be employed at one ; because the vibrations communicated by one workman are liable to inconvenience another. The Chopping-block is used for reducing bricks to any required form by means of the axe. The Banker (fig. 13,) is a bench, from six to twelve feet in length, accord- ing to the number of men it is intended to accommodate ; from two to three feet in breadth, and about one inch thick. An old ledged door may easily be converted into a banker, by placing its back edge against a wall, and supporting the front with three, four, or five posts : it is used for rubbing and guaging bricks for arch-work. B, in the figure, represents the rubbing-stone. The Camber-slip (fig- 12,,) is a piece of board of any length or breadth, made convex on one or both edges, and generally something less than an inch in thickness : it is made use of as a rule. When only one side or edge is cambered, it rises about one inch in six feet, and is employed for drawing the soffit lines of straight arches : when the other edge is curved, it rises only about one half of the other, viz. about half an inch in six feet, and is used for drawing the upper side of the arch, so as to prevent its becoming hollow by the settling of the arch. But some persons prefer having the upper side of the arches straight ; and, in this case, the upper edge of the arch is not cam- bered. When the bricklayer has drawn his arch, he gives the camber-slip to the carpenter, who by it forms the centre to the curve of the soffit. The bricklayer, in order to prevent the necessity of having many camber-slips, should always be provided with one which is sufficiently large for the widest aperture likely to be arched. The Mould is used for forming the face and back of the brick to its proper taper; and, to this end, one edge of the mould is brought close to the bed of the brick previously squared. The mould has a notch for evet y course of the arch. TERMS, &C. IN BRICKLAYING AND PLASTERING. 389 The Templet is an instrument for taking the length of the stretcher and the width of the header, in building walls, &c. The Tin Saw is used for cutting, to about the eighth of an inch deep, the soffit lines made, first by the edge of the bevel on the face of the brick, and then by the edge of the square on the bed of the brick. This incision is made in order to form an entrance for the brick-axe, that it may reduce the bricks to the proper form for arches, without splintering or jagging their edges. This saw is likewise used for cutting the false joints of headers and stretchers. The Brick-axe is used for reducing or cutting off the soffits of bricks to the saw-cuttings, and the sides to the lines drawn by the scribe. Much of the labour required for rubbing the bricks may be removed by the axe being managed with dexterity. The Rubbing-Stone, B, (fig. 13,) is a rough grained stone, of about twenty inches in diameter, generally fixed at one end of the banker, on a bed of mortar. It is used for smoothing those bricks which have been brought to their proper form by the axe. The headers and stretchers in returns, which are not axed, are likewise dressed upon the rubbing-stone. If the grain of the stone is found not to be sufficiently sharp to reduce the bricks with the necessary expedition, a little sand will tend to remove this deficiency. The Bedding-Stone consists of a marble slab, from eighteen to twenty inches in length, and from eight to ten wide, and of any thickness. It is used to try whether the surface of a brick, which has been already rubbed, be straight, so that it may fit upon the leading skew back, or leading end of the arch. The Float-stone is used for taking out the axe-marks, and smoothing the surfaces of curved work, as the cylindrical backs and spherical heads of niches. But, for this purpose, the Float-stone must itself be curved in the reverse form, though of a radius equal to that intended for the brick, so that it may coincide, as nearly as possible, with the brick. CEILING. The upper side of an apartment, opposite to the floor, generally 49. 5F 390 THE NEW PRACTICAL BUILDER, &C. finished with plastered work. Ceilings are set in two different ways, the best is where the setting coat is composed of plaster and putty, commonly called guage. CLINKER. A portion of a brick, used where the distance will not permit a whole one in length. COARSE STUFF. See Lime and Hair. COAT. A stratum, or thickness, of plaster-work, done at one time. COMPO, or Compos, used in outside stucco-work, implies the materials with which Roman or any other similar cement is composed : that is, when the component parts are all incorporated. To ascertain their different qualities, the same must be decomposed, or analysed. COURSE. A horizontal stratum, layer, or row, of bricks, extending along the whole length of a wall. DERBY. A two-handed float. DIE. Plaster is said to die when it loses its strength. DOTS. Patches of plaster put on to regulate the floating-rule in making screeds and bays. FINE STUFF, for plastering, is made of lime slaked and sifted through a fine sieve, and mixed with a due quantity of hair, and sometimes a small quantity of fine sand. FINISHING, in plastering, 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 papering. FIRST COAT, of two-coat work, in plastering, is denominated laying when on lath, and rendering when on brick : in three-coat work upon lath it is denominated pricking-up; and upon brick, roughing-in. FLOAT. An implement used in plastering for forming the second coat of three-coat work to a given form of surface. An Angle-Float is a float made to any internal angle to the planes of both sides of a room. The Derby is a two-handed float. FLOATED LATH and PLASTER, set fair for paper, is three-coat work ; the first pricking-up, the second floating, and the third, or setting coat of fine- TERMS, &C. IN BRICKLAYING AND PLASTERING. 391 stuff, understood to be pricked-up, as there is no floated work without pricking-up. FLOATED, RENDERED, and SET. A common term explained in the previous definitions. FLOATED WORK, in plastering. That which is pricked-up, floated and set, or roughed-in. FLOATING, in plastering. The second coat of three-coat work. FLOATING-RULES, used by plasterers, have been already noticed. They are of every size and length. FLOATING-SCREED, in plastering, differs from cornice-screeds, in this, that the former is a strip of plaster, and the latter wooden rules for running the cornice. See Screed. GAUGE, in plastering. A mixture of fine-stuff and plaster, or of coarse- stuff and plaster, used in finishing the best ceilings, and for mouldings, and sometimes for setting walls. HAWKE. A board, with a handle projecting perpendicularly from the underside, for holding plaster. LATHS. Small slips of wood nailed to rafters, for hanging the tiles ot slates upon. For laths used in plastering, see page 372. Double Fir Laths are laths three-eighths of an inch thick, single laths being a bare quarter, or less than a quarter of an inch. LATH, FLOATED and SET FAIR. These words bear the same meaning as lath pricked-up, and floated and set, to which the reader is referred. LATH, LAYED and SET, in plastering, is two-coat work ; only the first coat, called laying, is put on without scratching, unless it be with the points of a broom. This is generally coloured on walls, and whited on ceilings. LATH, PLASTERED, SET, and COLOURED, is the same with lath, layed, set, and coloured, to which the reader is also referred. LATH, PRICKED-UP, FLOATED and SET, for PAPER, is three-coat work ; the first is pricking-up, the second floating, and the third the finishing of fine-stuff'. LAYING, in plastering. The first coat on lath of two- coat plaster, or set-work. It is not scratched with the scratcher ; but its surface is roughed 392 THE NEW PRACTICAL BUILDER, &C. by sweeping it with a broom; it differs only from rendering on its application. Rendering is applied to the first-coat work upon brick; whereas laying is the first of two-coat work upon lath. LAYING ON TROWELS. The trowels used for laying on plaster. LIME and HAIR, in plastering. A mixture of lime and hair used in first- coating and in floating : it is otherwise denominated coarse-stuff. In floating, more hair is used than in first-coating. MITERING ANGLES. Making good the internal and external angles of mouldings. MORTAR. A preparation of lime and sand mixed with water, and used as a cement, as already explained in pages 329, 371, &c. MOULDINGS, in plastering. When not very large, mouldings are first run with coarse guage to the mould, then with fine-stuff, then with putty and plaster, and, lastly, run off, or finished, with raw putty. When mouldings are large, coarse-stuff is first put on, then it is filled with tile-heads or brick- bats, and run off successively with coarse gauge, fine-stuff gauge, putty guage, and finished with raw putty. In running cornices, there must always be screeds upon the ceiling, whether the ceiling is floated or not. PARGETING. A term derived from, parget, or the plaster-stone, and used for the plastering of walls ; sometimes for the plaster itself. PLASTER. The material with which ornaments are cast, and with which the fine-stuff of guage for mouldings and other parts are mixed. PRICKING-UP, in plastering. The first coating of three-coat work upon laths. The material used is coarse stuff, sometimes, in London, mixed up with drift road sand, scrapings, or Thames sand, and its surface is always scratched with an instrument used for that purpose. PUGGING. The materials composed of bricks and mortar, &c. introduced between the joists of floors, in order to prevent the communication of sound, or to deaden it in the interval from one story to another. PUTTY. A very fine cement, made of lime only. RBNDERED and FLOATED, in plastering, is three-coat work, more commonly called floated, rendered, and set. TERMS, &C. IN BRICKLAYING AND PLASTERING. 393 RENDERED, FLOATED, and SET, for paper, should be termed roughed-in. Floated and set, for paper, is three-coat work ; the first, lime and hair upon brick-work ; the second, the same stuff, with a little more hair, floated with **^* r * a long rule ; the last, fine stuff, mixed with white hair. RENDERED and SET. The same as set-work. See Set^work. ROUGH-CAST. The overlaying of walls with mortar, without smoothing it with any tool whatever. This has been already noticed. See page 380. ROUGHING-IN, in plastering. The first coat of three-coat work. ROUGH-RENDERING, in plastering, means one coat rough. ROUGH-STUCCO. See Stucco. SAIL-OVER. A term denoting the over-hanging of one or more courses of bricks beyond the naked of the wall. SECOND COAT, in plastering. Either the finishing coat, as in layed and set, or in rendered and set ; or it is the floating, when the plaster is roughed-in, floated, and set, for paper. SCREEDS. Wooden rules for running mouldings, and also the extreme guides upon the margins of walls and ceilings for floating. SET-FAIR. A term in plastering, used after roughing-in and floated, or pricked-up and floated work. SETTINP. The quality that any kind of stuff has to harden in a short time. SETTING-COAT, on ceilings or walls, in the best work, is guage, or a mixture of putty and plaster ; but in common work it consists of fine-stuff. SET-WORK, in plastering. Two-coat woi-k upon lath. SKEW-BACK. The sloping abutment for the arched head of a window. STOPPING. Making good apertures and cracks in the plaster. STUCCO, or FINISHING. The third coat of three-coat plaster. Rough Stucco is that which is finished with stucco, floated, and brushed. BASTARD STUCCO. 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 trowel- ling is done with less labour than what is denominated trowelled-stucco. THIRD COAT, in plastering. The stucco for paint, or setting for paper. 49. 5 G 394- THE NEW PRACTICAL BUILDER, &C. THREE-COAT WORK, in plastering, is that which consists of pricking-up, or roughing-in, floating, and a finishing coat. TOOLS, BRICKLAYERS'. See Bricklayers' Tools, pages 384, &c. TOOLS, PLASTERERS. See pages 370, 371, &c. TOOTHINGS. Bricks projecting at the end of a part of a wall, to bond a part of the said wall, not yet carried up. TRAVERSING THE SCREEDS FOR CORNICES, is putting on gauge-stuff on the ceiling screeds, for regulating the running mould of the cornice above. TROWELLED-STUCCO, for paint. The same as roughed-in for brick-work. TWO-COAT WORK, in plastering, is either layed and set, or rendered and set. See those articles. WATER TABLE. The bricks projecting below the naked of a wall to rest the upper part firmly upon. WORK. The coating of plaster layed and set, and applied to brick-work where there are two coats only. SLATING. 395 CHAPTER XL SLATING, WITH AN EXPLANATION OF TERMS, &C. SLATING, till lately, was employed only for covering the roofs of build- ings ; but it is now used for forming balconies, chimney-pieces, castings to walls, skirtings, stair-cases, &c. The slates most in use about London are the Welsh and Westmoreland, but the latter are now almost superseded by the former. About eighty years ago French slates were much used : they are smaller than Welsh slates, and, being very thin, are consequently extremely light ; and their thinness renders them liable to be easily penetrated by wet, and are, of course, considered unfit for our climate. Slates of this kind, after they have been long exposed to the atmosphere in these countries, are often shivered in pieces; and they may sometimes be found completely decomposed, or reduced to a pow- dered state. This imperfection, since it has been known, has prevented their being used in 'this country ; nor is it probable they will be again intrpduced. The WELSH SLATES are generally classed in the following order. ft. in. ft. in. DOUBLES, average size 1 1 by 6 LADIES 1 3 8 COUNTESSES , 1 8 10 DUCHESSES .' 2 1 WELSH RAGS SO 20 QUEENS ,. 30 20 IMPERIALS 2 6 2 PATENT SLATE. . .>. . 26 20 396 THE NEW PRACTICAL BUILDER, &C. The DOUBLES are so called from their small size. These are made from the fragments of the larger qualities, as they are sorted, respectively. The LADIES are made up from fragments, as above, in pieces that will square up to the size of such a description of slate. COUNTESSES are in size the next gradation above ladies; and DUCHESSES still larger. SLATE is extracted from the quarries in the same manner as other stony substances; that is, by making perforations between its beds, into which gunpowder is placed and fused. This opens and divides the beds of the slate, which the quarry-men remove in large blocks. These blocks are afterwards split by wedges of iron, driven between their layers, which separate the blocks into such scantlings as may be required. When slate is to be exported in this state, its edges are sawn to the sizes ordered ; for it is not necessary to use the saw to the horizontal stratum of the slate, as that can be divided nearly as correct by the ready method above-mentioned. The works in Wales, for sawing slate, are furnished with excellent machinery, which is set in motion either by steam or by water, and keeps in action a vast number of saws, each sawing the scantlings of slate into pieces adapted to their respective purposes. The IMPERIAL SLATING, for roofs, is particularly neat, and is known by having its lower edge sawn ; whereas all the other slates, used for covering, are only chipped square on their edges. PATENT SLATING is so called, among the slaters, from the peculiar mode of laying it on roofs, but we are not aware of any patent for it being ever obtained. It was first brought into notice by the late James Wyatt, Esq. the architect of our late revered king, George the Third. Patent slating may be laid on rafters of much less elevation than any other kind of slating ; it is considerably lighter ; because the breadths of the laps are much less than the slates adopted for the common sort of slating. Patent slating was originally composed of slates called the Welsh Rags, but at the present time it is composed of the Imperials, which are lighter, and much neater in appearance. On the WESTMORELAND SLATE some experiments were made by Dr. Wat- son, the late Bishop of Llandaff, whence it appears that there is very little SLATING, 397 difference in the natural composition of this kind of slate and that of Wales. The Bishop's comparison of their absolute weight, as compared with the weight of other materials made use of as a covering to buildings, may be of great utility, inasmuch as it may tend towards forming a datum for adding to, or diminishing from, the quantity of timber employed in roofs of different spans and elevations. " That sort of slate," says his lordship, " other circum- stances being the same, is esteemed the best, which imbibes the least water ; for the imbibed water not only increases the weight of the covering, but, in frosty weather, being converted into ice, it swells and shivers the slate. This effect of the frost is very sensible in tiled houses, but it is scarcely felt in slated ones ; for good slate imbibes but little water ; and, when tiles are well glazed they are rendered, in some measure, with respect to this point, similar to slate." He adds, " I took a piece of Westmoreland slate and a piece of common tile, and weighed each of them carefully ; the surface of each was about thirty square inches ; both the pieces were immersed in water for ten minutes, and then taken out and weighed as soon as they had ceased to drip, and it was found that the tile had imbibed about one-seventh part of its weight of water, and the slate had not imbibed a two-hundredth part of its weight. Indeed the wetting of the slate was merely superficial, while the tile, in some measure, became saturated with the water. I then placed both the wet pieces before the fire ; in a quarter of an hour's time the slate was become quite dry, and of the same weight it had before it was put into the water ; but the tile had lost only about twelve grains of the water it had imbibed, which was, as near as could be expected, the same quantity that had been spread on its surface ; for it was this quantity only which had been imbibed by the slate, the surface of which was equal to that of the tile. The tile was left to dry in a room heated to 60 of Fahrenheit, and it did not lose all the water imbibed in less than six days." He adds, further, " that the finest sort of Westmoreland slate is sold at Kendal at 3*. Qd. jper load, which will amount to 11. 15s. per ton, the load weighing two hun- dred weight. The coarser sort may be had at 2*. 4d. a load, or ll. 3s. 4>d. per ton. Thirteen loads of the finest sort will cover forty-two square yards of roofing, and eighteen loads of the coarsest will cover the same quantity ; 50. 5 H 398 THE NEW PRACTICAL BUILDER, &C. so that there is half a ton less weight put upon forty-two square yards of roofs, when the finest sort of slate is used, than if it was covered with the coarsest kind, and the difference of expense only three shillings and sixpence." It must be remarked, that this slate owes its lightness, not so much to any diversity in the component parts of the stone from which it is split, as to the thinness to which the workmen reduce it, and it is not so well calculated to resist violent winds as that which is heavier. COMPARISON IN WEIGHT OF THE SUNDRY COVERINGS EMPLOYED ON ROOFS. A Common Plain Tile weighs thirty-seven ounces, and they are used, at a medium, seven hundred to cover a single square of roof of one hundred superficial feet. A Pan-tile weighs seventy-six ounces, or four pounds and three-quarters, and one hundred and eighty are required to lay on a single square of roof. Both the plain and pan tiles are commonly bedded in mortar ; indeed the former cannot be well laid on a roof without it. The mortar used may be about one-fourth of the weight of the tiles. COMMON LEAD or COPPER, for covering roofs, generally requires seven pounds of the former, and seven ounces of the latter, for each superficial foot. A square of one hundred feet covered with the above materials stand rela- tively thus : cwt. qrs. Ibs. For Copper, per square 3 16 Lead.. 6 1 Fine Slate 6 21 Coarser ditto 8 1 8 Plain Tiles 18 Pan-Tiles 9 2 Hence a careful builder may select such a covering as his building may be best adapted to support. SLATING. 399 OF THE MANNER OF LAYING SLATES. ALL kinds of slate have a lap of each joint, generally equal to one-third of the length of the slate. The largest slates are reserved for the eaves. The slater, first picking and examining the slates to discover the strongest and squarest end, holds the slate over the edge of a small block of wood, cuts one of its edges straight, and having gauged the other thereby, cuts off that also; and, making two small holes on its opposite side, finishes the slate for the roof. All quarry slates require this preparation from the slater. The above holes are for the reception of the nails intended to fasten the slates to the roof. The copper and zinc nails are esteemed the best. When iron nails are used, they should be previously put into a tub of fluid white lead, till they are completely covered, and then left to dry. Iron nails, plated with tin, have lately been introduced, and are much cheaper than copper ones. A base or floor is laid for receiving the slates. Boarding is essential for Doubles and Ladies, and this should be laid with even joints and well nailed down to the rafters. Then the slater, having provided himself with slips of wood, two inches and a half wide and one inch thick on one side, and chamfered away to an arris on the other, nails down these tilting fillets, as they are called, beginning with the hips, then the sides, eaves, and ridge. He next lays the eaves, setting their lower edge to a line, and nails them down to the boarding. A bond is then formed to the under sides of the eave, by placing another row under, so as to cross all the joints ; this row is not nailed, but tightly pushed up and left dependent for support on the pressure of those above them and their own weight on the boarding. The slater then draws a straight line on the upper part of the row laid for a bond, and lays another course of slate crossing the joints of the other, and nails them down even with the line marked. This process is repeated till the whole roof is finished. All the larger kinds of slate may be made to lay upon battens, which saves ar> expense of about twenty shillings per square. A batten is a narrow piece of deal, about two inches and a half or three 400 THE NEW PRACTICAL BUILDER, &C. inches wide. For Countesses the bottoms may be about three-quarters of an inch thick, but for the larger kind of slate they should be an inch. The slater is the fittest person to lay them, as he may thus suit the length of his slates, and other particulars, which he best understands. This is the general method of laying all kinds of slate, except the patent slating. The PATENT SLATING, so called, consists of the largest slates of an uniform thickness ; Imperials are now generally taken. Neither battens nor board- ing is required for these slates ; but the common rafters should be left loose on the purlins, so that a rafter may be fixed under each of the meeting joints of the rows of slates. The work of covering is then performed, as above described, except that no bond is required, the slates being laid uniformly, and screwed down by two or three strong one and a half inch screws at each end into the rafters beneath. Filleting is now commenced, which consists in covering the meeting-joints with fillets of slates, bedded in putty, and screwing them down through the whole into the rafters. These fillets are then neatly pointed up with more putty, and then painted, to resemble the slates. The hips and ridges of roofs are sometimes filleted, but lead is preferable. The Patent Slating may thus be laid, perfectly water-tight, with a rise of two inches in one foot of a rafter. These are the general modes of laying slates ; and for peculiar neatness they are sometimes laid in a lozenge form, but as, in this form, only one nail can be used to each slate, thus laid they soon become dilapidated. The SLATERS' TOOLS are the Saixe, Ripper, Hammer, Shaving-Tool and various kinds of Chisels, Guagcs, and Files. The SAIXE is of steel, and not unlike a large knife, except having on its back a piece of iron, projecting about three inches, and drawn sharp to a point. It has a handle of beech, and is used for chipping and cutting slates. The RIPPER is formed of iron, about the same length as the Saixe, with a very thin blade, tapering towards the top, where a round head projects about half an inch, with two little notches at the intersection of one with the other. SLATING. 401 This tool is used for lifting up and removing the nails out of old slating, when it is to be repaired. The HAMMER differs but little from the common tool of the same name, except that the upper portion of the driving part is higher and bent towards the handle, so as to form a claw, by having a notch in its centre. It is used for extracting nails that do not drive satisfactorily. The SHAVING TOOL consists of a blade of iron, about eleven inches long and two wide, sharpened at one of its ends, like a chisel, and mortised into two round wooden handles. It is used for smoothing the surface or face of a slate, for skirtings, floors of balconies, &c> The CHISELS, GAUGES, and FILES, need no description. They are used for finishing the better parts of the slater's work, as mouldings, chimney- pieces, skirtings, casings, &c. The STRENGTH of SLATE to Portland Stone has been calculated to be as five to one, and consequently slate should be used where lightness is required combined with strength. VALUATION OF SLATERS' WORK. SLATERS' work is measured by Surveyors by the square of 100 feet super- ficial. Besides the nett dimensions of their work, slaters are allowed six inches for eaves, and four for hips, for common slating; and nine inches in addition 'for rags or imperials. Slating for roofing may be averaged thus per square. . s. d. Doubles 220 Countesses 2 12 6 Welsh Rags and Imperials 3 13 6 Westmoreland . . . . , 4 14 6 Slaters' work, for galleries, varies, according to the mouldings, from 4*. 6d. to 5s. 6d. per foot superficial ; skirtings and facings from Is. 6d. to 2*. per foot. Chimney-pieces, &c. are sold at so much per piece. 50. 5 1 *02 THE NEW PRACTICAL BUILDER, &C. EXPLANATION OF THE TERMS USED IN SLATING. BACK OP A SLATE. The upper side of it. BACKER. A narrow slate put on the back of a broad square-headed slate when the slates begin to get narrow. BED OF A SLATE. The lower side of it. BOND OR LAP OF A SLATB. The distance between the nail of the under slate, and the lower end of the upper slate. EAVE. The skirt or lower part of the slating hanging over the naked of the wall. HOLING. The piercing of slates for the admission of nails. MARGIN OF A COURSE. Those parts of the backs of the slates exposed to the weather. NAILS. Pointed iron, or copper, or zinc, of a pyramidal form, for fastening the slates to the laths or boarding. They are commonly of the description or shape of Clout-nails. PATENT SLATES. Those which are used without boarding, and screwed to the rafters, with slips of slates, bedded in putty, to cover the joints. SCANTLE. A gauge by which slates are regulated to their proper length. SORTING. Regulating slates to their proper length by means of the scantle. SQUARING. Cutting or paring the sides and bottom of the slates. TAIL. The bottom or lower end of the slate. TRIMMING. Cutting or paring the side and bottom edges of a slate, the head of the slate never being cut. PLUMBING. 403 CHAPTER XII. PLUMBERY, OR PLUMBING. THE ART of PLUMBERY comprehends the practice of casting and laying sheet-lead, with the making and forming of pumps, cisterns, reservoirs, water-closets, &c. The ductility of the metal in which he works enables the Plumber to effect his operations by means of tools, few in number and simple in construction. Of these the principal are as follow : A HAMMER, which is made of iron, and which differs from the one in common use only in being somewhat heavier. A JACK-PLANE, which is similar to that used by carpenters, and employed by the plumbers for smoothing the rough parts of the edges of sheet lead. The TRYING-PLANE is similar to the carpenter's instrument of the same name ; and is used for trying or finishing the edges of sheet lead, when they have been already smoothed by the jack-plane. A CHALK-LINE is a line which, being rubbed on chalk, is used for marking out the lead into the different widths which may be required. The DRESSING and FLATTING TOOL is made of beech, about eighteen inches long, and two and a half square ; planed smooth on one side, and rounded into an arch on the other. It is used for stretching out and flattening the sheet lead, or dressing it into any required shape. MALLETS are similar to those of the carpenter, and are of various sizes to suit different kinds of work. 404 THE NEW PRACTICAL BUILDER, &C. GLASING or HEATING IRONS are of various sizes, generally about twelve inches long and tapering at both ends ; the handle end being turned quite round, that it may be held firmly in the hand, and the opposite end spherical or of a spindle shape. They are used red-hot in soldering. LADLES are of three or four different sizes, and are used for melting the solder. CHISELS are of various sizes, according to the purposes to which they are applied. GAUGES also vary in size according to their uses. CUTTING KNIVES are of various sizes, and are used for dividing sheet lead at the mark left by the chalk-line. FILES are likewise of various sizes. The name points out their use. The MEASURING RULE is two feet in length, and is divided into three equal parts. Two of these legs are of box-wood, duodecimally divided, and the third of slow tempered steel. Its name points out its use. CENTRE-BITS are of various sizes, according to the perforation they are intended to make. COMPASSES are used for striking out circular portions of lead. These are the principal tools ; but the plumber must also be provided with weights and scales, as most of his work is charged by the weight. LEAD, the metal in which the plumber chiefly works, is distinguished for its durability, maleability, and other properties, which render it of the highest importance. It is of a bluish-white colour, and, when newly melted, very bright, but it soon becomes tarnished by exposure to the air. Its specific gravity is 11,3523. It may be reduced, by the hammer, to very thin plates, and may also be drawn out into wire ; but its tenacity is not great if com- pared with many of the other metals. A lead wire of one hundred and twen- tieth of an inch diameter is capable of supporting 18*4 pounds only without breaking. Lead melts when heated to the temperature of 612 of Fahrenheit, and when a stronger heat is applied it boils and evaporates. If it be cooled slowly, it crystallizes. When exposed to the air it soon loses its lustre, and PLUMBING. 405 acquires at first a dirty grey colour ; and, finally, by corrosion, its surface becomes almost white. When taken from the mines, lead is almost always combined with sulphur, and hence it is called a sulphuret. The operation of roasting the ore, or smelting, consists : 1st. In picking up the mineral, to separate the unctuous, rich, or pure, ore, from the stony matrix, and other impurities. 2d. In pounding the picked ore under the stampers. 3d. In washing the pulverized ore to carry off the matrix by the water. 4th. In roasting the mineral in a reverberatory furnace, taking care to stir it, in order to facilitate the evaporation of the sulphur. When the surface begins to have the consistence of paste, it is covered with charcoal; the mixture is shaken, the fire increased, and the lead then flows down on all sides to the bottom of the basin of the furnace, from which it is drawn off into moulds or patterns, prepared to receive it. The moulds are made so as to take a charge of metal equal to one hundred and fifty-four pounds ; these are called, in commerce, pigs, or pigs of lead. The plumbers use lead in sheets of two kinds ; the one called cast, and the other milled lead. The cast lead is used for the purpose of covering the flat roofs and terraces of buildings, forming gutters, lining reservoirs, &c. In architecture it is technically divided into 5, 5|, 6, 6|, 7, 1\, 8, 8^, Ibs. cast- lead ; by which is understood, that every foot superficial of such cast-lead is to contain one or other of these weights ; so that an architect, when directing a plumber to cover or line a place with cast sheet-lead, tells the workman that it is to be done with 6 or 7lb. lead ; meaning by it, that he expects each foot superficial of the metal to be equal in weight to six, seven, or other, number of pounds. CASTING. Every plumber, who conducts business to any extent, casts his sheet-lead at home ; this he does from the pigs, or from old metal, or both. To perform this he provides a copper, well fixed in masonry, and placed at one end of his casting-shop, near to the mould or casting-table. The casting-table is, generally, in its form, a parallelogram, well jointed, and bound with iron at the ends, and varying in its size from six feet in width to eighteen or more feet in length. It is raised from the ground so high as to be about six or seven inches below the top of the copper, which contains 51. 5 K 406 THE NEW PRACTICAL BUILDER, &C. the metal, and stands on strongly framed legs, so as to be very steady and firm. The top of the table is lined with deal boarding, laid firmly and very even, and it has a rim projecting upwards, four or five inches all round. At the end of the table, nearest to the copper, a box, called the Pan, is adapted, which box is equal in length to the width of the table ; at the bot- tom is a long horizontal slit, for the heated metal to issue from. This box moves upon rollers along the edges of the projecting rim of the table, and is set in motion by ropes and pulleys, fixed to beams over the table. So soon as the metal is found to be adequately heated, the bottom of the table is covered by a smooth stratum of dry and clean sand. These boxes are made to contain as much melted lead as will cast the whole of the sheet at the same time, and the slit in the bottom is adjusted so as to let through a suffi- cient quantity to cover it completely, of the thickness and weight per foot required. When the box has dispersed its contents upon the table, it is suffered to cool and congeal ; after which it is rolled up and removed away. Other sheets are made in succession till all the melted metal in the copper be cast up. The sheets thus formed, when rolled up, are weighed; as it is by the weight that the price is adjusted. The other kind of sheet-lead made use of by plumbers, and called MILLED LEAD, is not manufactured at home, it being purchased of the lead merchant, as it is commonly cast and prepared at the ore and roasting furnaces. It is very thin, its weight being not more than four pounds in the foot superficial. Milled Lead is used by architects for covering the hips and ridges of roofs, or any part exposed either to great wear and tear, or to the effects of the sun. It is laminated in sheets of about the same size as that of cast sheet-lead ; and, in the operation of making it, a laminating-roller is used, or a flatting- mill, which reduces it to the state in which it is used. SOLDER is used for uniting the joints of leaden work, and it should more readily acquire a state of fusion than the metal intended to be soldered thereby, and be of the same colour. The solder generally made use of by the plumber is called soft solder, and is made of tin and lead, in equal parts, fused together, and run into moulds not unlike a common gridiron. In this state it is purchased of the manufacturer by the plumber, at so much per pound. PLUMBING. 407 LAYING SHEET-LEAD. The ground for sheet-lead, whether it be of plaster or of boards, should be perfectly even, otherwise the work will be bad, unsightly, and liable to crack. The sheets not being more than six feet in width, make it necessary that they should sometimes be joined. This is performed -either by seams or rolls. The seams are formed by bending the two edges of the lead up and then over each other, and then dressing them down close. The rolls are formed by fastening a piece of wood, about two inches square, under the joints of the lead, and dressing one of the edges of the lead over the roll on the inside, and the other edge over both of them on the outside, and then fastening them down by hammering. Soldering is sometimes used for joining two sheets, but no good plumber would recom- mend it, as it always cracks after being exposed to one summer's sun. All sheet lead is laid with a current to keep it dry. This is done by the carpenter raising the boarding about one half or a quarter of an inch to every foot upon which the lead is laid. DRIPS on FLATS or GUTTERS are formed also as in the preceding manner, and by dressing the joints of the lead as described for rolls. This is used to avoid solder, and keep the work dry. FLASHINGS are pieces of milled lead, about eight or nine inches wide, and are fixed round the extreme edge of a flat or gutter, in which lead has been used. One edge is dressed over the lead of the flat or gutter, and the other fastened, either by passing it into the joints of brick-work, or by means of wall-hooks. The PIPES used by the plumber are of various sizes and descriptions. The small sizes are called by their calibre, or bore, \, f, 1, 1 J, If, and 2, inch pipes. Pipes below 1| inch calibre are charged by the foot run, and the larger size sometimes by the hundred weight. Socket Pipes are those which are used for conveying superfluous water from roofs, &c., and are called 3, 3|, 4, or 5-inch. They are generally made of milled lead, in lengths of eight or ten feet, dressed on a rounded core of wood, and fastened at the vertical joint with solder. The horizontal joints are formed by an astrigal moulding, in a separate piece of lead, about two or three inches wide, which laps completely over it, both above and below the joint, and hence it is called a lap-joint. Two broad pieces of lead, called 408 THE NEW PRACTICAL BUILDER, &C. X tacks, are attached to the back lap-joints and spread out, right and left, for fastening the pipes to the wall by means of wall-hooks of iron. The cistern- head, which is fixed at the head of rain-water pipes, is either made of sheet- lead or cast in a mould, and is fastened by tacks, as above. RESERVOIRS are generally formed of wood, or masonry, lined with sheet-lead united with solder. PUMPS, made by the plumber, may be divided into Sticking, Lifting, and Forcing, Pumps. The last is now but little used. The Sucking-Pump consists of two pipes, the barrel and suction-pipe, the latter being smaller in diameter. These are joined by flanches, filled with leather, and pierced with holes, to fasten them with screw-bolts. The lower end of the suction-pipe is spread out to facilitate the entry of the water, and it frequently has a grating to keep out filth or gravel. The working barrel is cylindrical, and as evenly bored as is practicable, to give the piston the least possible friction. The Piston is generally made of wood, in the form of a truncated cone, the small end being cut off at the sides, so as to form a kind of an arch, by which it is fastened to the iron-rod or spindle. The two ends of the conical part may be hooped with brass, and the larger end of the cone uniformly sur- rounded with leather to some distance below its base. This leather band should be sufficiently large to render the piston air-tight, without causing much friction. This is the pump which is generally used for raising water for the common purposes of life. The Lifting-Pump consists, as in the former, of a working-barrel, which is closed at both ends. The piston is solid, and its rod passes through a collar of leather in the plate, which closes the upper end of the working barrel. The barrel communicates laterally with the suction-pipe, and above with the rising main. This pump differs from the preceding only in having two valves, the lower one moveable and the upper one fixed. The Forcing Pump consists of a working-barrel, a suction-pipe, and serving-main, or raising-pipe. The last is usually in three parts, the first consisting of one piece, and making part of the working-barrel ; the second is joined to it by flanches forming an elbow with it; and the third is the be- PLUMBING. 409 ginning of the main, and is continued to where the water is delivered, where it is furnished with two moveable valves. The perfection of the barrel and piston of this pump is so great, as to require neither wadding nor leather. It is used for forcing water to any great height. There are two or three other kinds of pumps which require a short description, and they are as follows : Mixed Pumps are formed by combining the principles of the forcing and sucking-pumps together : when the lower valve of a forcing-pump is above the surface of the water it can raise it only by suction, the manufacture re- maining as before. The mechanism of a pump may be employed for convert- ing the weight of water descending in its barrel so as to work another pump. The Spiral Pump is formed by winding a pipe round a cylinder with an horizontal axis, and then connecting one end with a vertical tube, while the other is at liberty to turn round and receive water and air at each revolu- tion. Such a pump is said to be able to raise a hogshead of water in one minute 74 feet high, through a pipe 760 feet long. The Water Screw Pump consists either of a pipe wound spirally round a cylinder ; or of one or more spiral excavations, formed by means of spiral projections, from an internal cylinder covered by an external coating, so as to be water-tight. These pumps are used for removing water out of the foundations of bridges, and for supplying pieces of artificial water, as, by means of the Archimedian screw, in the Royal Gardens, at Kew. The materials of these pumps are manufactured to almost every required purpose, and thus sold to the plumber, who only put them together, so as to make them produce their desired effects. The different parts of water-closets are made in a similar way, and sold to the plumber, who places the basin, apparatus, traps, socket-pipe, cistern, and forcing or lifting pump together, so as to put them into action. Plumbers charge their sheet lead by the hundred weight, according to prices arranged at intervals by the Warden and Court of Assistants of the Plumbers' Company. Milled lead is two shillings per hundred weight more than the cast-lead. 51. 5L 410 THE NEW PRACTICAL BUILDER, &C. CHAPTER XIII. HOUSE-PAINTING. HOUSE-PAINTING is the art of colouring and covering with paint all the several kinds of wood, iron-work, &c. either as an external finish or for protection from the weather. This practice may be divided into four branches; that is to say, COMMON PAINTING, GRAINING, ORNAMENTAL PAINTING, and INSCRIPTION WRITING. WHITE LEAD, is the principal ingredient used in house-painting ; and this is a calx, obtained by rolling sheets of lead into coils, with their surfaces, about half an inch distant from each other, and then placing them vertically in earthen pots, with a portion of good vinegar at the bottom, in such a way that, when set in a moderate heat, the vapour of the vinegar corrodes the lead, so that the external portion will come off in white flakes when the lead -is beaten or uncoiled. These flakes, being bleached, ground, and saturated with linseed oil, form the white lead of the shops. This composition, if genuine, improves by keeping, and for the best whites it should be, at least, two or three years old. The Nottingham white lead is the most esteemed for what is called flatting, or dead-white. This article is so frequently adulr terated with common whiting, that it cannot be too carefully examined, before it be used. LITHARGE is composed of the ashes of lead, and is a kind of dusky powder that first appears in the oxydation of the lead. It is mixed with linseed oil, and used to give the paint a greater capability of drying quickly. HOUSE-PAINTING. 411 LINSEED OIL, which is used in every kind of house-paint, is obtained, by pressure, from the seed of flax, and then filtered to free it from feculse. This oil should be kept for two or three years, that it may precipitate its colouring particles ; as the more transparent it is, the better is the paint, of which it forms a component part. In Holland they whiten linseed oil by the following process, which gives it the effect of age. Having put the linseed oil into a well-glazed pot, they add to it fine sand and water, each of the same quantity as the oil, and having covered the vessel with glass, expose it to the sun, and stir it once every day. This process soon renders it very white ; after which it should rest for two days, and is then fit for use. DRYING OILS are usually prepared from litharge, white lead, plaster, and umber, in the following manner : to one pound of linseed oil add half an ounce of each of the above ingredients, and then boil the composition over a gentle fire, taking care to skim it from time to time. When the scum begins to rarify, and becomes red, the fire should be stopped, and the oil left to clarify and settle, when it is fit for use. The scum is called smudge, and is used for outside work. Oil, thus prepared, may be bought at colour shops under the name of toiled oil, and it is always used for the best house-painting. OIL OF TURPENTINE, or TURPS, is made from the resin of that name, which is obtained from all larch and fir-trees. Venice Turpentine is obtained from the larch, by fixing pipes in a hole made in the trunk, and thus con- ducting the liquid turpentine into buckets. In this state the turpentine has a yellowish limped colour, a strong aromatic smell, and a bitter taste. When this liquid or juice is distilled, by means of a bath, an oil is obtained, which is called Essence of Turpentine. The residue of the distillation is the boiled turpentine of commerce. This oil improves much by age, as the older it is the longer will the work performed with it retain its colour. It is much used in what is generally calleAflatting. All the prismatic colours are occasionally used by the . painter, and he varies these, according to circumstances, into almost every gradation of tint. They may be thus classed : 412 n O i 4' O s o s" crc w Ij m I J "Ultramarine, Ditto, ashes, Prussian-Blue, Verditer, Indigo, Smalt. King's-Yellbw, Naples ditto, Yellow-Ochre, Dutch-Pink, English ditto, Light ditto, Gamboge, Masticot, Common Orpiment, Gall-Stone, -Terra di Sienna. JE W W - tf O THE NEW PRACTICAL BUILDER, &C. 'Vermilion, Native Cinnabar, Red-Lead, Scarlet-Ochre, Common Indian-Red, Spanish Brown, Terra di Sienna, burnt, Carmine, Lake, Rose-Pink, Red-Ochre, .Venetian-Red. Verdigris, Crystals of ditto, Prussian-Green, Terra Verte, .Sap Green. v Orange Lake. True Indian-Red, Archil, .Log-Wash. fc o < PS PQ Brown-Pink, Bistre, Brown-Ochre, Umber, Cologne Earth, Asphaltum. f White-Flake, White-Lead, Calcined Hartshorn, Pearl White, Tray White, Eggshell White, -Flowers of Bismuth. ^ Lamp-Black, < Ivory-Black, CBlue-Black. PAINTING. 413 These colours are nearly all that are employed by the house-painter, and are those which experience has taught him to mix so as produce almost every teint that can be required. VERMILION is a bright scarlet pigment, formed of common sulphur and quicksilver, prepared for use by a chymical process. It has a delicate body, and will grind as fine as the oil itself, but is too expensive for common use. CINNABAR is a similar pigment, differing only from vermilion by having a more crimson teint. RED-LEAD, or MINIUM, is lead calcined till it acquires a proper degree of colour, by exposing it with a large surface to the fire. SCARLET-OCHRE is an earth, with a base of green vitriol, and is separated from the acid of the vitriol by calcination. COMMON INDIAN-RED, is of an hue verging to scarlet, and is imported from the East Indies. VENETIAN-RED is a native ochre, rather inclining to scarlet; this is the pigment which is selected for graining of doors, &c., in imitation of ma- hogany. SPANISH-BROWN is a native earth, found in the very state and colour in which it is used. TERRA DI SIENNA is a native ochre, and is brought from Italy, where it is generally found. It is yellow originally, and in this state is often made use of, and is accordingly placed among the yellow colours. It changes to an orange-red by calcination, though not of a very bright teint, for which property it is sought to produce a pigment of that colour. CARMINE is a bright crimson colour, and formed of the tinging substance of cochineal, with nitric-acid. It is not well calculated to mix up with oil, as its colour changes rapidly by exposure to the air and light. LAKE is a white earthy body, as cuttle fish-bone, or the basis of alum or chalk, tinged with some vegetable dye, such as is obtained from cochineal, or Brasil-wood, taken up by an alkali, and precipitated on the earth by the addition of an acid. ROSE-PINK is a lake, like the former, except that the earth or basis of the 52. 5 M 414 THE NEW PRACTICAL BUILDER, &C. pigment is principally chalk, and the tinging substance extracted from Brasil or Campeachy wood. 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, and the same with the yellow-ochre commonly used. This latter substance is chiefly brought from Oxfordshire, where it is found in great abundance. ULTRAMARINE is a preparation of calcined lapis-lazuli, which is, when perfect, of a brilliant blue colour ; it has an extremely beautiful and trans- parent effect in oil, and will retain this property with whatever vehicle, or pigment, it may be mixed. It is excessively dear, and is frequently sold at the colour shops in an adulterated state. ULTRAMARINE ASHES are the residuum, or remains, of the calcined lapis- lazuli. PRUSSIAN-BLUE is a brilliant pigment ; it is the fixed sulphur of animal or vegetable coal, chymically combined with the earth of alum. VERDITER is composed of a mixture of chalk with the precipitated copper, formed by adding the due proportion of chalk to the solution of copper, made by the refiners in precipitating silver from the nitric acid, in the opera- tion called parting, in which they have occasion to dissolve it in order to purify it. INDIGO is a tinging matter, extracted from certain plants, which are found in both the Indies, from whence the Indigos of commerce are mostly im- ported. SMALT is a species of glass, of a dark blue colour, being coloured with zaffre, or oxyde of cobalt, and afterwards ground to a powder. This beau- tiful colour carries no body in oil, and can be strewed only upon a ground of white lead. KING'S-YELLOW is a pure orpiment, or arsenic, coloured with sulphur. NAPLES-YELLOW is a warm yellow pigment, rather inclining to orange. YELLOW-OCHRE is a mineral earth, which is found in many places, but of different degrees of purity. See Red-Ochre, above. DUTCH-PINK is a pigment formed of chalk, coloured with the tinging par- PAINTING. 415 tides of French berries. It is not well adapted for work in oil, because it fades speedily. ENGLISH and LIGHT PINK are merely a lighter and coarser kind of Dutch pink. 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. MASTICOT, as a pigment, is flake-white, or white-lead gently calcined, by which it is changed to a yellow, which varies in teint according to the degree of calcination. ORPIMENT is a fossil body, of a yellow colour, composed of arsenic and sulphur, frequently with a mixture of lead, and sometimes of other metals. GALL-STONE is a concretion of earthy matter, formed in the gall-bladder of beasts : it is but little used. VERDIGRIS is an oxyde of copper, formed by a vegetable acid : it is used in most kinds of painting where green is required. CRYSTALS of VERDIGRIS is the salt produced by the solution of copper, or common verdigris, in vinegar. PRUSSIAN-GREEN is, in composition, similar to blue of the same name. TERRA-VERTE is a native earth ; it is of a bluish green colour, resemblihg the teint called sea-green. SAP-GREEN is the concreted juice of the buck-thorn berry. ORANGE-LAKE is the tinging part of anata, or annotto, precipitated together with the earth of alum. TRUE INDIAN-RED is a native ochrous earth of a purple colour, but so scarce as seldom to be met with at the shops. ARCHIL is a purple tincture, prepared from a kind of moss. LOGWOOD is brought from America, and affords a strong purple tincture. BROWN-PINK is the tinging part of some vegetable, of an orange colour, precipitated upon the earth of alum. BISTRE is a brown transparent colour of yellowish tint. BROWN-OCHRE is a warm brown or foul orange colour. COLOGNE EARTH is a fossil substance of a dark blackish brown colour, a little inclining towards purple. 416 THE NEW PRACTICAL BUILDER, &C. ASPHALTUM is sometimes employed by the painters to answer the end of brown-pink. WHITE-FLAKE is a ceruse prepared by the acid of the grape. TROY-WHITE is simply chalk, neutralized by the addition of water in which alum has been dissolved. LAMP-BLACK is properly the soot of oil collected as it is formed by burning ; but, generally, no other than a soot raised from the resinous and fat parts of fir-trees. IVORY-BLACK is composed of fragments of ivory or bone, burnt to a black coal, in a crucible or vessel, from which all access of air is excluded, and then ground very fine for use. BLUE-BLACK is the coal of some kind of wood burnt in a close heat, to which the air can have no access. Of the COMPOUND COLOURS, Lead colour is of indigo and white : Ash colour, of white-lead and lamp-black : Stone colour, white, with a little stone-ochre : Buff, yellow-ochre and white-lead : Light willow green, verdigris and white : Grass green, verdigris and yellow pink : Carnation, lake and white : Orange colour, yellow-ochre and red-lead : Light timber colour, spruce ochre, white, and a little umber : Brick colour, red-lead, with a little white and yellow- ochre. All the simple colours may be bought at the colour shops, either in a crude or prepared state. They are prepared by saturating them with linseed oil or water, as either is to be used with them, and then grinding them on a slab of porphyry, marble, or granite, till they are perfectly levigated. All colours are derived from either vegetable or mineral substances ; but though the former kinds have a more brilliant effect at first, they soon change when exposed to the atmosphere, which the latter never do. The painters' tools are few in number, being almost entirely brushes. The pound brush is made of hogs' hair, and is used as a duster, till the soft part is worn away. This previous wear adapts it better for spreading the colours for which it is afterwards used. The other brushes vary in size according to the work they are intended for. PAINTING. 417 COMMON HOUSE-PAINTING may be divided into the following kinds : CLEARCOLE AND FINISH, which is the cheapest kind of painting ; it is per- formed by first dusting and cleaning what is to be painted, and filling up cracks and defects with putty, called stopping. The whole is then painted over with a preparation of whiting and size to form the ground. Over this a coat of oil-colour, prepared with lead, called the finish, is laid. Where work is not very dirty, this may answer pretty well, but not for outside work. TWICE IN OIL means when the work is twice painted over. THRICE IN OIL, means when the work has been twice painted over with oil-colours, and once in colours prepared in turpentine. THREE TIMES AND FLAT means three coats of oil colour and one of turpen- tine. This is generally used for new work. BRINGING FORWARD is a term applied to priming and painting new wood added to old work, or old work which has been repaired, so that the whole shall appear alike when finished. GRAINING is the imitating, by means of painting, various kinds of rare woods ; as satin-wood, rose-wood, king-wood, mahogany, &c., and likewise various species of marble. For this kind of work the painter is furnished with several camel's-hair pencils, and with one or more flat hogs' hair brushes. An even ground is first laid of a composition formed of ceruse, and the colour required diluted with oil of turpentine. This is then left for a day or two to get fixed and dry. The painter then prepares his pallet-board with small quantities of the colour required ; and, being furnished with some boiled oil and oil of turpentine mixed together, tries the effects of the teint by spreading it over a panel, and if it suits, perseveres by doing a panel at a time. The shades and graining is then produced by dipping the flat hog's-hair brushes in the mixture of oil and turpentine, and drawing it down the newly-laid colours. The other particular appearance required is produced by means of the eemel's-hair pencil. When all is fixed and dry, the whole is covered with one or two coats of good oil varnish. This kind of painting is not much 52. 5 N 4rl8 THE NEW PRACTICAL BUILDER, &C. dearer than good work in the common way, but it will last ten times as long, by being occasionally re-varnished, without losing any of its freshness. ORNAMENTAL PAINTING embraces the executing of friezes and the decorative parts of architecture, in chara-obscura, or light and shade, on walls or ceilings. It is performed by first laying a ground of the colour required, then sketching the ornament with a black-lead pencil, and after- wards painting and shading it, so as to give the required effect. INSCRIPTION WRITING is similar in process to ornamental painting ; the painter sketching out the letters in pencil and finishing the outline with colour. If the letters are to be gilt, they are covered with leaf gold, while the paint is wet. After the whole is dry, the superfluous gold is removed with a moist sponge, and the work covered with a coat of good oil varnish. VALUATION OF PAINTERS' WORK. PAINTERS' work, in general, is valued at the yard superficial of nine square feet. Sash-frames are valued at per piece, and sash squares at per dozen. Inscription writing is charged by the inch, viz. the height of one letter being taken, and multiplied by the whole number of letters, will give the quantity of inches. Shadowed letters are a halfpenny more than plain ones, and gilt letters treble the price. The charges for painting are regulated in London by intelligent surveyors ; but, as colours and oils can be purchased at any degree of purity, painting is often done at twenty per cent, less than the prices so regulated ; but the value of painters' work, in all cases, depends upon the quality of the materials and goodness of workmanship, and in reference also to the different species of works. GLAZING, 419 CHAPTER XIV. GLAZING. ON glass very little has been communicated in the works of the antients. It, indeed, appears, from the ruins of several Grecian buildings, that they had apertures or windows ; and it would seem, from the nature of their con- struction, they were adapted to receive a frame filled with some transparent substance. In some of the apertures discovered at Pompeii and Hercula- neum, squares of amber were found ; yet, though many of the Roman authors mention glass, it was so rare as to be employed only in the mansions of the opulent. BEDE is the first who mentions glass as applied to glazing windows. He likewise informs us that Abbot Benedict was the first who introduced the art of making glass into this kingdom, about the year 669 ; and, from the speci- mens that now remain, it is evident that not only the making of glass, but the art of staining it, made rapid strides towards perfection in a very short time after. Glazing, as now practised, embraces the cutting of glass, and fixing it into sashes of wood or casements of lead ; and likewise the ornamenting of win- dows with stained glass. Plain coloured glass may frequently be used with a very pleasing effect, and is very little more expensive than good common glass. Coloured glass is charged by the pound; Claude Lorraine, green, red, &c., at about six or seven shillings, and blues somewhat more. 420 THE NEW PRACTICAL BUILDER, &C. Glazing in lead-work is of the most antient description ; sashes being of mo- dern date. These sashes are formed with a groove or rebate, on the back of their cross and vertical bars, for the reception of the glass, which the glazier cuts to its proper size, and beds in the composition called putty. PUTTY is made of pounded whiting, beat up, with linseed oil, into a tough tenacious cement. When used for mahogany frames, a little red-ochre is mixed with it to suit the colour of the wood. The beauty of glazing depends principally on the colour of the glass. Glaziers now use chiefly what is called crown glass, which is divided into three kinds, called firsts, seconds, and thirds, according to their qualities, on which its value depends. The glass is bought by the crate, which consists of twelve tables of the best, fifteen of the seconds, and eighteen of the thirds. These tables are each about three feet in diameter. A crate of the best glass is valued at about four guineas ; of the seconds, about three ; and the thirds, two guineas. The crown glass manufactured at Newcastle and its neigh- bourhood is esteemed the best : the prices of glass are various. GREEN-GLASS is another sort, much used for common purposes, being not more than half the price of the crown-glass. From many old windows, it appears that this kind of glass was the most antient made use of. WAVED PLATE-GLASS is very thick and strong, presenting an uneven sur- face, as if indented all over with wires, so as to leave the intermediate spaces in the form of lozenges ; i{ was formerly much used in counting-houses, c. to prevent the inconvenience of being overlooked ; but though it has lately been superseded by ground-glass, it is still to be obtained in London. Ground or Rough Glass is used for the same purpose as the above, and is no other than the common crown glass, rendered opaque by having its polish taken off, and rubbing it with sand and water or emery. PLATE-GLASS is the most beautiful glass made use of, being nearly colour- less, and sufficiently thick to admit being polished in the highest degree. Tlie tables of this glass will admit of pieces being taken out of them much larger than can be obtained from any other kind of glass. The British Plate-Glass Company, in Albion-Place, London, is the most celebrated depot GLAZING. 421 for this species. There it is sold by the inch, in proportion to its size, the value increasing accordingly. Though the expense of this glass, by far, exceeds that of any other, yet is now so much preferred, as to be used in many shop-windows in the leading streets. GERMAN SHEET is another species of glass much esteemed, and would be superior to the preceding, had it not a disagreeable appearance, from being very wavy or uneven. BOHEMIAN PLATE-GLASS is similar to the above, only possessing a .red teint ; and though much used about thirty years ago, it is now quite rejected. GLAZIERS value their work by feet, inches, and parts, according to the size of the panes, or squares, employed. The charges are regulated by the Masters, Wardens, and Court-Assistants of the Company of Glaziers, and at present run thus : s, d. Best crown, not exceeding 3 feet, per square 3 10 Ditto, ditto 2 ft. 6 in., ditto 3 4 Ditto, ditto 2 ft. ditto 3 2 Ditto, under 2ft. ditto 3 Seconds of the same dimensions are about ten per cent, cheaper ; and Thirds, of similar dimensions, are 10 per cent, cheaper than the seconds. Green glass is the cheapest, never exceeding eighteen pence per foot. The price of all kinds of bent glass, for circular and other windows, varies according to the size, the trouble of obtaining it, and fitting it in. Cottage and some kinds of church windows are glazed in squares, or other figures, in leaden rebates, which are cast and drawn for the purpose, and soldered together at the interstices. This leaden work is of various sizes, according to the strength required, and is used instead of the cross bars of sashes. The grooves left in it for the glass have their cheeks sufficiently soft to be pressed down all round to admit the glass, and again raised up, when the glass has been put in, to keep it firm. Such windows are strengthened by vertical and cross bars of iron, with bands, which, having 53. 5o 422 THE NEW PRACTICAL BUILDER, &C. been soldered to the lead, are twisted round the iron. In cottage windows the bars, instead of being of iron, are often of wood. Glaziers formerly cut their glass out with an instrument called a groxing- iron; but this process was not only tedious but difficult, and has therefore been entirely superseded by the introduction of the diamond, which is as complete a tool for the purpose as can possibly be required. This instru- ment consists of a diamond spark, in its natural unpolished state, fixed in lead, and fastened to a handle of some hard wood by means of a brass ferrule. The handle is about the size of a moderate drawing pencil. The diamond is the principal working tool of the glazier. His other tools are a rule and several small straight edges. The former is divided into thirty-six parts, cl- inches, and is used for dividing the tables of glass into any required size. The straight edges are merely thin pieces of some hard wood, about two inches wide, and one quarter of an inch thick, and are used for the diamond to work against. Glaziers are likewise furnished with stopping knives, which resemble dinner knives, with the blade reduced to about three inches in length, and ground away on each side of its edges to an apex. They are used for bedding the glass in the rebates, and for spreading and smoothing the putty. A Hac/eing-out Tool is an old broken knife, ground sharp on its edge, and used for removing old putty out of the rebates, which are to be filled with new squares of glass. The glazier's hammer is similar to the smaller kinds used by other artificers. Glaziers are also furnished with a pair of compasses, which has one of its legs formed with a socket to receive the handle of the diamond, for drawing and cutting out any peculiar shapes of glass for fan-lights, &c. GOOD GLAZING requires that the glass be cut full into the rebates; for, when too small or too large, it is liable to be broken by the least pressure within, or even the wind from without : moreover, the putty should never project beyond the line of wood in the inside, and large squares should be further secured by small sprigs being driven into the rebates of the sash, and covered over with another coat of putty. GLAZING. 423 The business of a glazier includes the cleaning of windows, and this forms no inconsiderable portion of the trade in London ; some masters keeping one or two men constantly employed therein. The charge is regulated by the number of windows cleaned, and the number of squares in each frame. Windows, exceeding twelve squares, are charged at from 6d. to 8d. per dozen, the large squares of French sashes being raised about one-third more. The master-glazier takes upon himself the risk of windows being broken by his men, when employed in cleaning them. In many parts of the United Kingdom it is 'the custom to measure all the wood-work appertaining to the sashes, for the quantities of glass contained in the respective squares ; also, the lead-work. And such is the prejudice in favour of the practice in some places, that if any intelligent person was to attempt to reason them out of it, he would be considered a most inequitable valuator, and unworthy of being countenanced. Time and concurring cir- cumstances, it is presumed, may, at some period or other, equalize our customs, weights, and measures ; but until that period arrives, the system of valuation must be dependent upon local customs. The net quantities of glass should, in all cases, be measured, except in circular fan-lights and similar works, where the glass should be measured in the widest part ; and because the pieces cut off to make the glass fit the apertures can be considered only as waste glass, the price or allowance for which is not embraced in the value charged by the glazier for his glass so consumed. 424 THE NEW PRACTICAL BUILDER, &C. CHAPTER XV. OF BUILDING IN GENERAL. HAVING already treated on the component parts of Building, under the respective heads of CARPENTRY, MASONRY, BRICKLAYING, &c., we shall now attempt to give a GENERAL VIEW OF THE SUBJECT ; and, that the reader may be enabled to follow us, in fully comprehending our description, we shall explain, methodically, those technical terms, used by architects and builders, which require to be previously understood ; and then present the various parts of a building, in their natural order, with such remarks as may be practically useful. To those who, by practice and experience, are already acquainted with the subject, such an arrangement and explanation cannot but be agreeable ; and to those who do not possess such advantage, this mode of proceeding will be found materially useful, as it will imperceptibly and effectually conduct them to an acquaintance with the most prominent features of the art. The student is presumed, of course, to be acquainted with the geometrical parts of this work ; and, more particularly, all the definitions of superfices and solids. This information is indispensable, as the parts of a building are either the simple forms of some one geometrical solid, or compounded of portions of several : and, although the forms of objects are most easily understood by a reference to such as we have been accustomed to see, yet, as the same forms are not known to all classes of men under the same names, the language of geometry is the only language which can be employed to convey ideas of magnitude, form, and position. OP BUILDING IN GENERAL. 425 In speaking of Length and Breadth, as applied to any object fixed to the horizon, as a building, we mean such dimensions as are measured in an hori- zontal plane. Depth is the distance which some part recedes from a surface, without regard to the position of that surface. Height is estimated in a vertical line ; that is, in a line perpendicular to the horizon. ROOMS are the interior vacuities or habitable parts of a building. They must be closed on all sides for security, lodging, and comfort, and for all manner of purposes to which they may be applied. For the convenience of entering rooms, or of passing from one room to another, apertures should be made on one or more sides, which may be closed at pleasure, by means of moveable parts known by the name of doors. Rooms are disposed one above another, or by the side of each other, as may be found convenient. One side of every room must be an horizontal plane, for the convenience of walking upon, and for placing various articles of furniture in the most secure position ; this side is denominated the floor. When several rooms are placed side by side of each other, the passages or communications will be most conveniently effected when all the floors are on the same level. Again, for the convenience of the inhabitants, and to add to the strength of the building, not only the exterior, but likewise every side, of a room which appertains to the floor, should be carried to a certain height from it, in a line perpendicular to the horizon. Hence, it is clear the floor must be in a straight line, and the side thus carried up must be a plane surface. The solid parts contained between the vertical surfaces are the walls. The remaining sides, opposed to the floor, are denominated the ceilings. The latter parts may have such forms which the materials employed in the con- struction will admit of. The forms of many parts of buildings are alto- gether arbitrary, as to their use ; they are, therefore, determined by the ideas of economy or beauty; upon such principles as these the formation of ceilings are regulated. If economy be the chief object, the celling is a plane surface as well as the floor ; and hence, when several rooms are placed one over the other, with plane ceilings, the surfaces of the solid, contained between the floor and the ceiling of the room below, are parallel planes. 53. 5 P 426 THE NEW PRACTICAL BUILDER, &C. Keeping the same object in view, the surface of walls will not only be planes, but they will also be parallel to each other, and equally thick throughout. The forms of walls are determined by the line upon which they rise from the ground : thus, if the line be straight, the wall is called a straight wall; or, if it be circular, the wall is said to be circular, and so on. Straight walls are generally built ; circular walls occasionally ; but those of other forms are elliptic, &c., but they seldom occur in the common prac- tice of building, at least in exterior walls. A wall which divides one apartment from another, is called a partition or division wall. When two walls form an external or internal angle, the line of concourse made by the two sides is termed an external or internal quoin, or an external or internal angle. To combine strength and stability with economy in an edifice, the walls, as they rise, should diminish in thickness ; keeping in view that a straight line, ascending from any point of one of the surfaces, ought not to fall without that surface. If the line thus ascending fall entirely within the solid mass, the face of the wall is said to batter ; and the horizontal distance between the vertical line, ascending from any point in the line terminating the upper extremity of that surface, and the line terminating the lower extremity of the same, is called the battering of the wall. The exterior sur- faces of walls are generally executed in planes perpendicular to the horizon. When the face of a wall is not one entire or continued surface, or when it is formed by two or more continued surfaces, each rising from the horizontal base which forms the top part of the wall below ; the part thus connecting the two surfaces is called an off-set. When a house consists of several apartments one above another, each apartment, or as many as may be on one floor, is called a story; and when each story of a house has a number of apartments, the aggregate of the apart- ments on each story is called a suite of apartments, that is, when the rooms are connected by folding or other sorts of doors. When a building consists of two or more stories, it is usual to diminish the walls from the inside faces, by breaking each side into several surfaces, so OP BUILDING IN GENERAL. 427 that the off-sets may be immediately under the ceilings ; the sets-off thus made will not only prevent them from being seen, but will also afford a secure support to the floors. Hence, also, the floors, extending from one side of the building to the opposite side, will tie the walls firmly together, and prevent them either from spreading or approaching towards each other. The exterior walls of a building are not only connected by the floors, but likewise by the division or partition walls. The distance between the two nearest surfaces of parallel and opposite walls is called the span, or between external walls, the span of the building. The exterior parts of a house, by which its upper extremities terminate and extend over the area of each of its floors, in order to protect and secure the interior, is denominated the roof. The forms of roofs are various, and depend upon the nature of the climate, and prevailing custom of the country. Flat roofs were mostly used by the antients ; but the Greeks, from their country being at times exposed to heavy rains, soon perceived the inconvenience of flat or horizontal roofs. They accordingly constructed their roofs in two inclined rectangular planes, sloping from the middle towards the sides, and inclining equally to the horizon, so as to terminate each wall in an isosceles triangle with an horizontal base. The proportion of the height of this triangle to its base was in the ratio of one to eight, or one to nine ; or, speaking technically, the height was one-eighth, or one-ninth, part of the span. The Romans, who had still more occasion than the Greeks to provide for the speedy discharge of rain from their houses, did not alter the Grecian form of the roof, but varied its proportion, making the height of their roofs form one-fifth to two-ninths of the span. After the decline of the Roman Empire, high-pitched roofs were very generally introduced; the standard form being that of an equilateral triangle. No part of the practice of building has been more the object of caprice than the proportions of the roof. Even in the present day, we meet with almost every variety of proportion which can subsist between the height of the roof and its span. In ordinary dwell- ings the height varies from one-third to one-fourth part of the span ; in other 428 THE NEW PRACTICAL BUILDER, &C. cases, the proportion depends upon the taste of the architect and the style of the building. Roofs that are high-pitched discharge rain and snow more quickly than those which are lower ; they are less liable to be stripped of their covering by the wind, and the rain is not so easily blown through their joints ; but they are more expensive than low roofs, as they require longer timbers, and a greater quantity of covering. But, though low roofs possess the advantage in point of economy, they require large slates and greater care in the execution. Apertures built in walls for the admission of light are called ivindows. Those for egress and regress are termed door-ways. The framed-work for closing the aperture is called the door. The surfaces which surround the aperture, and which are contained within the two surfaces of the wall, are called the sides of the aperture. The lowest side of every aperture is termed the sill, and is always parallel to the horizon; the two sides which spring upwards from the sill are called the jambs, and these should be perpendicular to the horizon, and of an equal height. The remaining side, opposed to the sill, and which connects the tops of the jambs, is termed the soffit. The soffit is generally a plane surface, consisting of parallel planes ; and since each jamb rises to the same height, the soffit will, in this case, be parallel to the horizon. Occasionally, however, the section of the soffit, parallel to the sur- face of the wall, is some part of a circle, not exceeding the semi-circum- ference. The points from which the section rises from the tops of the jambs are called the springing-points. The solid contained in the soffit and the vertical surface of the wall is denominated an arch ; in this case, the soffit is termed the intrados of the arch. If the section of the soffit be less than a semi-circle, the arch is called a scheme-arch ; but if a semi-circle, it is termed a semi-circular arch. In some cases, the jambs do not ascend perpendicularly to the horizon, but stand at equal angles therewith, so as to approach nearer towards each other at the summit than at the bottom : and examples of this sort are not uufrequent in antient and modern works. The sections of apertures, parallel to the face of the wall, are sometimes entire circles, or entire ellipses, having one axis parallel to the horizon ; and, OF BUILDING IN GENERAL. 429 consequently, the other perpendicular thereto: in this case, the aperture cannot properly be said to have sills, jambs, or soffits. Occasionally, the sec- tion of the soffit, parallel to the surface of the wall, is constructed of two equal and similar concave curves, meeting each other in a receding angle, and in a vertical line which divides the aperture into equal and similar parts, so that the whole opening may form a symmetrical figure : arches of this description are termed Gothic arches. Semi-circular-headed windows have a pleasing effect in circular walls, par- ticularly when the exterior surface is an entire cylinder, and the diameter of considerable magnitude ; but, in a small building, they greatly weaken the construction of the walls, and produce a disagreeable effect by the over- hanging of the head. Since apertures for the admission of light are indispensable, it is essential, as well for the strength as for the beauty of the edifice, that the sills of all the windows in the same story should be in the same straight line ; and, also, that each jamb of a window in any story should be in a straight line with the jamb of a window in every other story. The solid parts of walls between apertures are termed piers, as also the external parts annexed to such apertures, which are identified as external piers, or quoins, commonly so called. When any uninterrupted part of the exterior walls of a building contains more than two equal and similar windows on one level, the beauty of their arrangement will require that the intermediate piers be equal in breadth ; for the same reason, the extreme piers in any apartment, which has one or more such windows, ought to be equal to each other. In all cases, where fanciful dispositions of the apertures are attempted, they ought to be disposed in symmetrical order. Windows, in modern rooms, in which they are introduced to light the apartments, are, by means of jambs standing obliquely to the faces of the walls, made widest within ; and these obliquities are called the splays or splayings of the jambs, or more simply the splays: and in these cases the windows, or jambs, are said to be splayed. 54. 5 Q 430 THE NEW PRACTICAL BUILDER, &C. Each jamb of a window will generally consist of four vertical planes within the two surfaces of the wall. The first plane from the exterior surface of the wall forms an external right angle with that surface; the first and second planes make with one another an external right angle ; the second and third, an internal right angle; and the third and fourth, an obtuse angle. The fourth plane is generally continued down to the floor, in order to form the side of the recess made from the inside of the wall below the sill of the window ; so that two of the three sides of this recess are formed by the con- tinuation of the two planes of the jambs next to the inner surface of the wall, and the third face is a surface parallel to that surface joining the continua- tion of the line of concourse at the meeting of the planes which form the obtuse angle in each jamb. The most usual form of a room is that of a rectangular prism ; and, conse- quently, it consists of four vertical sides, and of two sides parallel to the horizon. In large edifices, for the sake of beauty and variety, the forms of apart- ments, as they rise from the floor, are sometimes made circular or elliptic ; and sometimes, they are compounded of a rectangular portion in the middle, and of symmetrical circular segments at the ends, having the chord or dia- meter extending either the whole breadth of the end to which it is attached, or only to a part of such breadth, leaving an equal portion of it at each ex- tremity of the chord. In this case the faces of the walls consist of two oppo- site rectangles, and the ends of equal portions of a cylindric surface. When the floor of a room is rectangular, the ceiling is usually of the same form. In magnificent apartments, however, ceilings are often made to rise from each side to the middle in a concave shape, presenting the surface of a cylinder, having its axis parallel to the horizon. The chord of the cylindr ic section is extended to 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 an equal segment of a circle. Ceilings, formed simply of cylindric surfaces, are termed, by mechanics, waggon-heads, in imitation of the coverings to broad-wheeled carriages ; but OF BUILDING IN GENERAL. 431 the proper term for soffits to rooms, whether to halls, churches, chapels, or domestic apartments, is coved ceilings, which are of various forms. The cylindric or coved ceiling is not the only form which a rectangular apartment will admit of. It may also consist of a quadrantal portion of a cylindric surface, rising from each vertical face to a level oolong in the middle ; and this compound form, also, is termed a coved ceiling. The mul- tifarious dispositions of which are infinite. And without altering the figure of the floor, the ceiling may be formed of cross arches ; that is, such that each arch may coincide with the surface of a cylinder, having its axis parallel to the horizon, and to the sides of the apart- ment. When the summits of each arch rise to the same height, this kind of ceiling is called a groin, or is said to be groined ceiling. The rectangular floor will also allow of a concave ceiling in an ellipsoidal or spherical form, with one of its axis' in the line of intersection of the two diagonal planes, which may be conceived to pass through each of the four lines terminating the vertical sides, two by two. In this case, the planes which form the four vertical sides of the room, will not meet the ellipsoidal or spherical surface in a straight line, but in a curve of an elliptic form, or in a circular segment rising from each corner to the middle of each vertical adjacent face. As the circles or arcs, which form the lines of demarcation between the ceiling and the vertical faces of the apartment, descend from the middle of each upright face to the vertical line, formed at the angle, by those faces, the ceiling is termed a pendulous ceiling ; or it is denominated a pendcntme one, being suspended in the manner described. The floor of an apartment may have any geometrical figure that the fancy of the architect may suggest. A building occupying the site of an equilateral triangle, or of a regular hexagon, will divide without any loss of space into apartments, having also their floors in the forms of equilateral triangles or hexagons, and doors may be inserted in the middle of the side of one apart- ment without destroying the symmetry of the adjacent room ; yet none of those figures are so convenient as the square or rectangle : however, they are 432 THE NEW PRACTICAL BUILDER, &C. sometimes employed for the sake of variety, where motives of economy do not restrain the expense. When the form of an apartment is spoken of, we mean that of the floor; since all the horizontal sections below the ceiling are equal and similar figures. Forms of all manner of rooms, besides the square, or such as are rectan- gular, are denominated fanciful, or fancy rooms, which may be made very beautiful. The floors of apartments are, however, seldom regulated by the forms of the buildings. But, as the sites of edifices are generally rectangular, apart- ments constructed in the forms of polygons occasion not only great waste of space, but also of materials ; and such forms likewise require to be fitted with more than ordinary attention to elegance ; hence, when employed, no restraint should be laid as to expenses. It, however, rarely occurs that a suite or series of rooms have more than one polygonal apartment, unless it may be in cor- responding parts, in order to render the whole disposition symmetrical. Apartments having others immediately above them, and on every side, cannot be sufficiently lighted. Hence it would be improper to build a house of a rectangular form, and of two or more stories, with more than two apart- ments in breadth. If, therefore, it is desirable to build upon a large scale, the fronts of the edifice, on all sides, must be extended, or the middle of the area be left open without any building, or certain parts must be left entirely uncovered, or at least with sufficient exposed apertures, to be filled with glass for the admission of light. As it is desirable to render every apartment conveniently accessible, without passing through any other, a certain portion of every building should be taken for this purpose ; and the parts thus occupied, when not very wide, are termed passages. Passages of communication may be lighted in various ways. If there be only one series of rooms in the breadth, their situation must then be on the sides of the building, and consequently they may be lighted by apertures in the exterior walls, to any degree that may be desired : but if the whole, or any part, of the building has two rooms in breadth, and more than two in length, OP BUILDING IN GENERAL. 433 the situation of the passage must be uaturally in the middle, with one room in depth on each side of them. If the rooms are not very numerous, the passage may be lighted from one of the ends, but when the length is considerable, it ought to be lighted from both : but, when the edifice is very long, cross passages should extend to the sides ; and these will afford the most convenient and private entries to the rooms on each side of them. The apartment for the stairs must have apertures through all the floors, or be entirely open to the top, except the solid parts for walking upon, in order to give the occupants sufficient light. The flat parts on the same level with the floor are called landings, and these afford passages into the surrounding rooms. This mode of approaching them is adopted in all descriptions of edifices ; but, in small buildings, it is the only way of entering the rooms, particularly in the upper stories. When the apertures for the stairs are surrounded on all sides with rooms, the passages are usually made entirely round the stairs, with apertures in the walls of enclosure, in order to light the adjacent passages. In large buildings, the number of stair-cases should be proportioned to the magnitude of the edifices ; and the former should be regulated so as to require the fewest passages, and by these means due portions of the occu- pants may be kept desirably separate from each other ; but with such means of communication as to be accessible to certain individuals of the house, at the will of the proprietor. In the uppermost stories of buildings, passages may be introduced almost in any direction which may give the most convenient access to the apart- ments, as light, it is presumed, in most of such cases, may be easily supplied from apertures left in the ceilings or adjacent roofs. In some magnificent edifices, the rooms in the several heights are so con- structed that each story may be connected by one lofty apartment, that is, by means of large apertures in each floor ; and, in this manner, the rooms of the surrounding buildings may be accessible, that is, by means of the residual parts of the floors on the sides of the apertures adverted to. As it is necessary for the comfort of the inhabitants that rooms should be 54. 5 R 434 THE NEW PRACTICAL BUILDER, &C. kept of proper temperature at certain seasons of the year, fires must be introduced, for the purpose of communicating warmth to them. To effect this in the most convenient manner, certain recesses, commonly of a rectan- gular form, are made in the walls on the sides of the rooms ; and these serve as receptacles for fuel; and, to take away the smoke, tubes, or flues, are carried up from these recesses, in the thickness of the wall, to the summit of the building. The openings from the sides of the rooms are called ihejire- places ; and the tubes, or flues, for carrying off the smoke are denominated the funnels, or flues. The solid parts of the walls, between the funnel, or flues, and the rooms, are called the breasts of the chimnies. As the forms of the apertures for the fire-places are generally rectangular, the vertical sides are called jambs ; and the sides which stretch over the tops of the jambs, parallel to the horizon, are called the mantles. The portions of walls, con- taining the terminations of the chimnies, which are always carried above the roofs, are called the chimney-shafts. When walls contain a great number of flues, they are called stacks of chimnies: and, if the funnels, or flues, approach very near to each other, the solids which divide them are termed withs. To prevent chimnies from smoking, they ought to be so constructed that the current of air which naturally presses towards the fire-places, may not proceed entirely through the fires, but in such manner that portions of the air may pass immediately over the flame, so as to force the smoke up the chimnies ; for as the air is heated it becomes more volatile and desirous of ascension. With regard to the construction of the funnels, or flues, sharp angular turns must be avoided, as they retard the ascent of the smoke. The form of the chimney is not always the cause of apartments being annoyed by smoke ; it has been proved, beyond doubt, that much depends upon the situation of the building, as similar forms, in different situations, will often have different effects. Consequently, in building chimnies, no effectual precaution can be taken, nor can any remedy be applied until the fires are made. Without the agency of contrary currents of air, smoke may be occasioned from the mere form of the chimney; for if the flues are too narrow at the top, the smoke will not be discharged so quickly as it is generated ; and in time, therefore, it OF BUILDING IN GENERAL. 435 must fill the passage or flue, and ultimately the apartment itself. To provide, as much as possible, against smoke, in the building of chimnies, the apertures should be carried up in gentle windings or curvatures, particularly near the top, as all angles and sharp turnings are obstacles. For the same reason the interior surfaces should not be left rough, but should be smoothed over with plaster ; and all pieces of lime, shivers of stone, or broken brick, left by the carelessness of workmen, should be removed, as the wall is carried up. On this account, too, circular flues permit the smoke to pass more freely than those which have rectangular sections ; and, therefore, if expenses are not primary considerations, they should be so constructed that their surfaces may be those of hollow cylinders. If these precautions are found insufficient, the narrowing of the apertures above the fires should be tried; but as these remedies are sometimes inconvenient, they ought, perhaps, to be the last of the chimney-doctors' resources. The doctrine of smokey fire-places is as yet very imperfect ; many experi- ments have been tried, but still we have not obtained any general and satis- factory results, notwithstanding the celebrity of those who have made them. The writings of Dr. Franklin, Count Rumford, and Mr. Clavering, contain details of experiments on smokey chimnies, and the works of those authors may be consulted by such as wish to enter more fully into the subject. The superficial surfaces for walking upon we have hitherto called the floors; but this term, more properly speaking, is applicable to the thicknesses of timber or stone floors. So that, when apartments are placed over each other, the solid parts contained between the ceilings of the lower and upper apart- ments, and the surfaces for walking upon in those above, are called the floors. FLOORS constructed of stone are more particularly denominated pavements. The surfaces upon which we walk may, therefore, with propriety, be called the levels of the floors or pavements. WALLS underneath windows are generally recessed from the bottoms of the windows to the levels of the floors ; and these increase the areas of the floors. Each vertical side in the same plane with the vertical side or jamb of the window is termed the side or elbow of the recess ; and that surface of the recess whieh is parallel to each face of the wall, is denominated the back. 436 THE NEW PRACTICAL BUILDER, &C. PROPORTIONS OF THE APERTURES OF DOORS AND WINDOWS. DOOR-CASES were formerly constructed of trapezoidal forms, that is, wider below than above ; this probably arose from such doors possessing the inhe- rent power of shutting without the aid of the passengers. Instances of these forms are not common, but may be seen in the Bank of England, and several other modern structures, where the doors are hung upon such scientific principles as not to require any trouble in shutting. The dimensions of the apertures of doors should be regulated by the mag- nitude of the buildings. All the doors in the same stories are generally made to correspond to each other in heights. In private houses they may extend from three to four feet in widths, and from seven to eight feet in heights, or more, in proportions to the heights of the stories. When doors exceed three feet in width they should be closed with folding leaves, in order that the rooms which they separate may be occasionally united : this practice is pre- valent, and found very convenient for the reception of company. When the front of a building has only one door, it is generally placed in the middle, not only to preserve the symmetry of the front, but also to shorten the access to the various apartments. But in town houses this dis- position is not convenient, since most dwellings have only one apartment in the extent of the front, consequently the external doors are, in most cases, if not always, at one end. With regard to windows, the heights of the sills were formerly from three to three feet and a half high from the floors, which was convenient for leaning upon ; but of late years, in imitation of the French, it has become the fashion to make the sills of the windows equal with the levels of the floors, parti- cularly in the principal stories ; and in each of the others, the heights of the sills are now reduced to two, or two feet six inches, or thereabouts. OF BUILDING IN GENERAL. 437 In regulating the dimensions of windows, it may not be superfluous to remind the proprietor, or builder, that, although it may be very pleasant in summer to have the apertures for the windows either very large or very numerous, yet the extraordinary area occupied by them will render the apartments very cold in winter ; and to provide against these inconveniences, double glass frames are sometimes deemed indispensably necessary. In France, and in Italy, where the climate is warm, such windows are admiss- able ; but, in the streets of London, we are at a loss to present any apology for their introduction : they seem to be of little use, except to inhale the muddy exhalations of dirty streets. Verandas are equally absurd, which are also to be observed on the northern sides of our streets and squares, which the rays of the sun seldom or ever visit. PROPORTIONS OF APARTMENTS. THE proportions of apartments depend very much upon the use to which they are intended to be applied. The lengths of dwelling-rooms may extend from one to one and a half, or to twice their breadths ; and in galleries even four times. In general, however, it is to be observed, that the greater the capacities are, the more the lengths may exceed the breadths. Thus, in small houses, the dining and drawing rooms may be square ; but, in larger edifices, they may extend even to be a double square. With regard to the heights, three-fourths of the breadths are esteemed good proportions to the other dimensions. When the ceilings are coved or arched, the heights may be equal to the breadths, or to one and one-quarter the breadths. The breadths of the principal passages may be one-third of those of the principal rooms ; and the breadths of passages, or of passages belonging to a common house, may be one-fourth of the breadths of the principal rooms : the heights of passages should be the same as those of the rooms, but the lengths must be regulated by the buildings. 55. 5 s 438 THE NEW PRACTICAL BUILDER, &C. When the heights of rooms exceed the proportions adapted to the other dimensions, cylindrical vaultings, coves, domes, groins, &c., may be intro- duced, as may be best suited to the use made of the apartments. The heights of ceilings are necessarily regulated by those of the principal rooms of a story ; hence, apartments of an inferior description will often have their ceilings disproportionately high. To remedy this, floors may be intro- duced so as to divide the heights into two stories ; the lower one of a height proportioned to the length and breadth of the room, and an upper one suffi- ciently high to admit of persons walking erect. In this case, the upper story is termed a mezzanine or inter sole. Mezzanines, when they can be introduced, are exceedingly convenient for servants, lodging-rooms, powdering-rooms, wardrobes, &c. In buildings where beauty and magnificence are studied rather than economy, the halls and galleries may be raised to the height of two stories. Saloons are frequently raised the whole height of the building, and have gal- leries at the height of the stories around the interior circumference, for the purpose of communication with the various apartments. A Stair, or set of steps, is a structure by which persons may ascend to, or descend from, any story with ease and pleasure. The surfaces upon which our feet are set, in the act of passing up or down, are called treads; these should be equi-distant horizontal planes, placed at convenient distances. But when two treads are joined together by a third plane, rising perpen- dicularly from the plane of the treads, so as to have the breadths of the lower treads on the front, and the breadths of the upper treads on the back of these new planes, the third series of planes thus introduced are called risers. Every riser and tread forming an external right-angle is called a step. The line of concourse formed by the meeting of the riser and tread of a step, is called the nosing. Sometimes the tread of a step projects, in a small degree, beyond the riser below it, and is rounded so as to make it at once strong, and agreeable to the eye ; this termination is also termed the nosing of the step. OF BUILDING IN GENERAL. 439 Steps may be supported at each end by walls, and this is very frequently done in common houses. A series of steps, having all the treads equal, is called & flight of steps. It is easy to conceive that a stair, consisting of one flight, must be very inconvenient, on account of the great length of the area which would be required to contain it. It will be proper, therefore, to show in what manner the direction of the ascent may be changed, in order to prevent the length from exceeding the dimensions of the rooms adjacent. And this difficulty may be overcome by allowing a double area for the stair, with a floor between them, extending from the entrance where the stair begins to such a distance as will leave sufficient breadth for the passenger to pass from one stair-apartment to the other. Now it only remains to con- struct two equal flights, each half the height of the story ; the first being placed on a proper floor, and made easy of access ; and having a surface for walking upon, on a level with the top of the highest riser, and extending over the whole breadth of the two stair-apartments. The next flight may be raised upon this floor, and may have a similar floor at the top with its surface on a level with the floors of those apartments in the story to which a com- munication is required. The floor between the two flights is termed a half space or resting place ; and that upon a level with the floor of the story, at the head of the second flight, is denominated a landing, And here the landing naturally points out the situation of the door or doors. In this manner we may ascend to the uppermost stories of buildings, or to as many stories in them as are deemed requisite. The two apartments, or areas, each containing flights of steps, are collec- tively termed the stair-case. The wall which divides the two flights is called the newel of the stair- case. As walls surrounding the stair are commonly carried up in the form of rectangular prisms, by way of rendering the turning easy to the pas- senger, the angles of the newels are generally reduced by forming them into semi-cylinders. 440 THE NEW PRACTICAL BUILDER, &C. When the apartment, or arena, allowed for the stair-case, require the steps to be inconveniently high, instead of the half-space, steps are introduced, with their treads diminishing in breadths towards the newel. The diminished steps, thus introduced, are denominated winders ; and by their means the stair is made to consist of two equal flights, with a series of winders between them. When the thickness of the newel, or wall between the flights, is very con- siderable, another flight of steps is sometimes inserted between the planes or faces of the newel, extended into the passage of the stair, round which the person ascending has to turn, in order to proceed to the upper flights. By this introduction, the stair is made to consist of three flights, and two level parts, called quarter-spaces, between each of the two flights.. These quarter-spaces are also occasionally occupied with winders, by which means an easier ascent is gained, and the want of room compensated for as much as possible. From what has been said, it may be easily conceived that floors of stairs may have many different forms, and that the steps may always be properly adapted to those forms, with convenient quarter-spaces, half-spaces, and landings, where required. The apartment, or arena, for the stairs being a prismatic cavity, the form of the stair-case is denominated from the area contained within its walls, at the base or floor from which the adjacent walls rise. Therefore, if the floor underneath is rectangular, quadrangular, circular, or elliptical, the stair-case must also be rectangular, quadrangular, circular, ellip- tical, or as the case may happen. Now let us conceive the newel to be disengaged from the ends of the steps, so as to form a complete prism, and to be taken out from the level of the floor, and the steps to remain firm ; the stair thus constructed is a geometrical stair. The open space left by the newel is called the well-hole. The continuation of the ends of the steps next to the well-hole is denomi- nated the string. The surface of the stair, opposed to the floor or to the steps, as seen from below, by those ascending, is termed the soffit. OF BUILDING IN GENERAL. 441 The newels of stair-cases are sometimes constructed hollow, with apertures through the sides. Examples of this kind may be seen in St. Paul's Cathedral, where the stairs are cylindric, or stand upon circular bases ; the examination of which will afford great satisfaction to those who derive pleasure from scientific research. The grandest example of a geometrical stair-case, upon a circular base, may be seen in St. Paul's, which cannot fail to attract the attention of the mathematician, geometrician, and architect. A stair-case having a newel in the middle, is called a. pillared or newelled stair. A stair contained within a circular or elliptical wall is called a winding stair ; or one which is circular or elliptical, as the case may be. Geometrical stairs are much more elegant than pillared stairs; and those that have rectangular bases, when constructed without - winders, are deemed handsomer than geometrical stairs which rise from a circular or elliptic base; unless, indeed, the diameter be very large, as in the case adverted to, in the sublime structure of St. Paul's. In ordinary houses the breadths of the steps are generally from 10 to 12 inches, the heights from 6 to 7| inches, and the lengths from 2 feet 6 inches to 4 feet. In more stately mansions the steps may be from 4 to 6 inches high, from 12 to 15 inches broad, and 6 feet long and upwards, in propor- tions thereto. Any portion of the exterior side of a building which protrudes itself towards the spectator, is denominated a. projection or break; and a part that recedes from him, is termed a recess. When deep semi-circular recesses are made for ornament, they are termed niches, for the reception of busts or statues. To prevent the disagreeable sameness that would arise from the view of the plain and regular surfaces of walls and ceilings, certain ornaments are introduced, which compose or divide the surfaces. When these dividing parts are formed of curved surfaces and planes, so as to meet in edges, forming straight lines parallel to each other, and to the sur- 55. 5 T 442 THE NEW PRACTICAL BUILDER, &C. faces which they compart; and, at the same time, the curved parts are portions of cylinders or cylindroids ; the surfaces thus formed are termed mouldings. If these mouldings be conceived to be bent round a cylinder or cylindroid, or round a building in either of these forms, so that the edges of the mould- ings may be in planes perpendicular to the axis of the cylindric or cylin- droidic surface, they are called circular or elliptic mouldings. In ordinary cases of modern building, the general position of mouldings upon the walls of edifices, either within or without, are horizontal. When mouldings are applied to a wall, for the purpose of dividing its sur- face, the entire should be so constructed that the parts shall not hide each other, unless with the intention to produce a shade. Every separate part is termed a member. Hence, in the same collection of mouldings, those members which are most remote from the horizontal plane, passing through the eye, must be the most remote from the surface of the wall, in order to be seen to the best advantage. In speaking of the section of a moulding, we mean the surface which would arise from cutting it by a plane, perpendicular to one of its edges. All edges formed by the angular meeting of any two surfaces, are called arrises, when they terminate external angles. The plane surfaces, which enter into the combination of mouldings, are generally either parallel or perpendicular to the horizon. When this is not the case, the deviations from this position will be noticed. In any curved moulding, terminated by two edges, or arrises, that edge which is the most remote from the eye is also most remote from the surface of the wall ; except in mouldings of a cylindric form, which, in general, must be very small when above the eye, as otherwise they will conceal those which are above them. When cylindric mouldings are situated below the eye, semi- cylindric forms are occasionally introduced : these are made so large as to constitute the principal features of the combination, because mouldings in these depressed situations are never so far removed from the eye as to be out of the reach of near inspection. OP BUILDING IN GENERAL. 443 Hence, to produce the best effect, the situation of mouldings above the eye, the manner in which they are illumined, and even the forms them- selves, will differ from the situation, form, &c. of those mouldings which are below the horizontal plane passing through the eye. Small mouldings of a semi-circular section are called Beads; and these are sometimes introduced in a combination of mouldings above the eye, but much oftener below it. A large semi-circular moulding is termed a Torus ; its situation is naturally below the eye. When the form, or section, of a moulding is concave, it is called a Cavetto ; but below the eye may be either a Scotia or Cavetto, according as the cur- vature varies from both its extremes. A convex moulding above the eye is denominated an Ovolo. A moulding which has one part concave and another convex, or that which has its section composed of two curves of con- trary curvature, is called a Cymatium, a term which literally signifies a wave. When the concave part of a cymatium is more distant from the horizontal plane of the eye than the convex part, it is called a Cyma-recta ; but if the contrary be the case, it is styled a Cyma-reversa. It may be proper to recall to the student's recollection, that, in every moulding above the eye, the upper edge recedes farther from the face of the wall than the lower edge. Curved Mouldings are always separated from each other by a small member, called a Fillet. This member consists of two plane visible surfaces, one perpendicular, and the other parallel, to the horizon. If one of the intermediate members of a collection of mouldings above the eye consist of a lofty vertical plane, with a considerable projection under- neath, either a level or inclined plane, it is called a Corona, Larmer, or Drip. The use of ^this member is to throw the water from the lower edge of the vertical surface, without changing its course to that underneath, to protect the lower parts. Mouldings, which may be generated by planes carried round their axis' in those planes, are called rotative mouldings. 444 THE NEW PRACTICAL BUILDER, &C. Where buildings have a series of mouldings at or near their tops, such series are termed Cornices. Mouldings below the plane passing through the eye, or at the bottom of a building, is called the Base ; and here we may observe, that the part of the building under the base protrudes beyond the surface of the wall above the base. Any intermediate collection of mouldings is denominated a moulded string course. A series of mouldings, placed over the top of each vertical side of a room, is termed a Cornice ; but when the series is situated near the bottom of the room, it is styled a Base. A series of mouldings placed at the springing of an arch, or where each end begins to rise from the vertical surface, is called an Impost. A series of mouldings round a cylindrical arch, having their edges circular, and these edges described from a certain point in the axis of the cylinder, is called an Archivault. Or, if the arch be elliptic, with a series of mouldings around it, and if the mouldings have their edges equi-distant from each other, and from the wall, such a collection is also called an Archivault. The surface of a wall is technically denominated the Naked of the wall. A surface, parallel to the surface of a wall, and enclosed on all sides by mouldings, is called a Panel. By means of bases, strings, cornices, panels, &c. we may easily conceive that a wall is susceptible of being comparted into very elegant symmetrical forms. When the face of a wall is decorated with mouldings, bordering on the margin of an aperture, for a door or window, such disposition of moulding is called an Architrave. Mouldings, or solid parts, which protrude towards the spectator, are called by the general name of projections ; and whatever distance any part is from the wall, that distance is termed the projection of that part. A Tablet is a projection fixed in a wall, with one face parallel to the sur- face; and the sides of the tablet, which connects its face with the surface of OF BUILDING IN GENERAL. 445 the wall, are generally planes perpendicular to that surface. The forms of tablets, though occasionally circular, are generally rectangular. However, they are susceptible of every regular geometrical figure, or combination of figures, symmetrically disposed ; that is, so that the parts on the right and left correspond, which should be the case, not only with the subordinate' but also with the several component parts of architectural decorations. A series of mouldings is said to return, when a vertical section of them, parallel to the face of the wall, produces the same forms as a vertical section perpendicular to that face. When return mouldings do not reach the extent of the wall, or when they ex- tend to a very small distance only, they are said to profile, or terminate, upon f he wall; that is to say, the form or impression in which the surfaces of the mould- ings would intersect the surface of the wall, will be equal and similar to the section of the mouldings on the face or intermediate parts between the returns. In all insulated buildings, where cornices are employed, they should be carried round the entire edifices ; unless, in so doing, it is thought too ex- pensive. But strings, and other collections of horizontal mouldings, are frequently interrupted, even in the lengths of the fronts, by certain vertical projectures which are more prominent. Horizontal mouldings, placed over the architraves of apertures, and pro- jecting over the extreme parts of the architraves where they terminate on the walls, or over the superior terminating members of the fronts of those archi- traves, are called the cornices of the windows or doors over which they are introduced. And if the architraves be joined to such cornices, the entire are called architrave cornices, or imperfect entablatures. Any face of a projection, which is parallel to the surface of the wall, is termed the/row^ of that projection. When any projection exhibits three vertical faces ; the middle one, being parallel to the surface of the wall, and the other two perpendicular to it, the two faces which adjoin the wall are called flanks. Hence the three faces exhibited are the front and the two flanks. 56. 5 v 446 THE NEW PRACTICAL BUILDER, &C. Here it will be necessary to observe that, when the lengths, breadths, and heights of projections are mentioned, the lengths are measured in horizontal lines parallel to the naked faces of the walls from which they protrude; and the breadths are measured parallel to those faces; consequently the breadths, in the proper sense, is the same as the projecture. Hence we perceive that the greatest extension of a body is not always its length. When a projection, exhibiting a face and two flanks, is placed between the cornice and the architrave of an aperture, so that each flank may be in the same plane with each side of the architrave, which terminates against the wall, the projection thus described is called a Frieze. The cornice, frieze, and architrave, are collectively termed the entablature of the door, window, or other aperture, to which they belong. In this enumeration we include only the horizontal part of the architrave, and not the vertical mouldings belonging to it. In buildings, upon magnificent scales, projections, similar to the entabla- tures just described, are carried round the edifices ; and where the expenses are limited, along the front only, these projections are also termed Enta- 77 , Matures. But where entablatures of this latter description are executed, it is usual to introduce other equal and similar projections, exhibiting the fronts and flanks; these terminate immediately under the entablatures at the upper extremities, and upon the horizontal surfaces of projections at their lower extremities. The projections thus terminated are called pilasters. PILASTERS generally terminate at the top, with mouldings, which return on the sides adjacent to the wall in the same manner as the fronts, which are parallel to them ; the mouldings which thus terminate the upper extre- mities of the pilasters are called capitals : and very frequently these capitals consist both of mouldings and ornaments. Occasionally, return mouldings are introduced also at the lower ends of the pilasters ; and the mouldings thus situate are called the bases of the pilasters. OF BUILDING IN GENERAL. 447 The surfaces of pilasters are often ornamented with a series of equal and similar concave mouldings, terminating in vertical edges; mouldings thus introduced are called Flutes. Flutes terminate in different manners at their upper and lower extremities, or where they enter the faces of the pilasters, so as to preserve entire the parts of these faces, whether above or below. But the forms in which they terminate are generally the same as those of their horizontal section, sup- posing the pilasters to be cut through any intermediate parts of the heights. In the recess of an internal angle, formed by two planes of a cornice, or even in a string course, a series of equal and similar prismatic solids are sometimes attached to each of those planes, so that every solid may exhibit four entire faces. The solids thus attached are called, according to their forms, dentils, blocks, mutules, modillions, or eantalivers. When they are very small and near to one another, so that their heights may exceed their lengths, and also, that the horizontal distances between each and those immediately adjacent may be less than their lengths, they are called Dentils ; and the cornices are said to be denticulated. If the length of each solid be greater than its height or breadth, the solid is called a Mutule, and the cornice itself a mutule cornice. If the distance between each solid exceed twice its length, and if the pro- jecture be from twice to three times the length, each solid is called a Modil- lion; and the cornice thus ornamented amodillion cornice. When the projecture of each solid is more than thrice the length, each solid is called a Cctntaliver ; and the cornice itself a cantaliver cornice. PROPORTIONS OF MOULDINGS. WHEN a room is adorned with an entire entablature, its height may be about one-sixth part of the height of the room ; or, in some cases, one- seventh. If a cornice only is executed, its height may be about one- 448 THE NEW PRACTICAL BUILDER, &C. twentieth, or one-thirtieth part of the height of the room; the proportion depending entirely on the combination, number of mouldings introduced, and their purpose or design. Thus, in the houses of middling classes of people, the cornices may be made light ; but, in the houses of the nobility, where grandeur is an object, they should have more massive appearances ; and it may be laid down, as a general principle, that all interior decorations and proportions of mouldings must be lighter than those employed in the exterior parts of buildings. The reason of this is obvious ; for, in a room, where the eye is confined within a certain distance, and the ornamental parts large, they will naturally appear heavy ; but, on the exterior of edifices, the mould- ings are viewed with reference to the sizes of the entire buildings, which proportionately diminishes their magnitude; and the advantage of seeing the object at different distances, tends still further to lessen the effect of projectures. ON THE BEAUTY OF BUILDINGS. IN the arrangements and proportions of the parts of buildings, they ought to be such as to elevate the pleasures of imagination to the sublime, (which is superlatively explained by a great writer,) to bear the features characteristic of its destination. Grandeur cannot exist without magnitude, both in the entire and in its parts ; but a building may be great, and the parts massive, without its being grand. Besides magnitude, a considerable degree of ele- gance is required to produce grandeur. Sameness of surface, without di- versities of figure symmetrically disposed, will not raise any pleasure in the mind. But, when the parts of an edifice follow each other alternately, or repeat a series of symmetrical portions in succession, the combination is easily understood, and the imagination pleased with the uniformity and variety of the compositions. OF BUILDING IN GENERAL. 449 As a cylinder is a pleasing object, since no two parts of its surface, nor of its circular ends, are alike opposed to the eye ; so, in executing a circular building, the terminations ought to be in continued circles. Hence the enta- blature of such an edifice should not, on any account, be broken ; though the intermediate parts of the wall may be decorated with well-proportioned sym- metrical parts, as columns, pilasters, or ornamented windows; which, if judi- ciously introduced, cannot fail to produce the most pleasing effects. Though definite rules cannot be laid down for the formation and contours of buildings, yet the centre parts ought to assume commanding features, and the grand outlines of the entire should, generally speaking, approach to those of the pyramidal forms. Lodges, and small houses standing alone, show, with good effect, when their figures approach to the forms of cubes : but, in large mansions, rectan- gular prisms, with oblong bases, have more pleasing appearances than cubes. The proportions of large buildings may be termed good when the lengths do not exceed the breadths, in greater portions than when they are about four to three. SITUATIONS FOR COUNTRY RESIDENCES. THE most essential qualities of good situations are those that are most conducive to health. Where persons intending to build, enjoy the power of fixing upon situations, they should sedulously avoid the proximity of marshes, fens, boggy ground, or stagnant water : and, if rivers are very near, the sites of the houses should be on elevated ground, so as to be out of the reach of unwholesome fogs, which rise from the water at particular periods. In neighbourhoods where the inhabitants are healthful, cheerful, and remarkable for longevity, these may be regarded as possessing salubrious air. Easy accesses to public roads, with supplies for water and fuel, are indispensable requisites in the choice of situations. 56. 5 x 450 THE NEW PRACTICAL BUILDER, &C. f Again, it is much better to have a house sheltered by trees than by moun- tain scenery ; because the former yield a cooling and refreshing air, which, during the heat of the summer months, is not only pleasing but animating ; and, in winter, they serve, in some degree, to break off the keenness of the blasting winds and tempests; while mountains, according to their position, protect only from certain winds ; and if the situations are directly east or south, they will be found extremely unpleasant at particular seasons of the year. And with regard to the positions of apartments, such as studies, libraries, dining and drawing rooms, boudoirs, principal and inferior bed-chambers, dressing-rooms, &c., all these should be arranged according to existing cir- cumstances ; taking care, if possible, to place the studies or libraries to the north, and the dining and drawing rooms in such places as to avoid too much of the morning and evening sun : and with regard to the residue, such as kitchens, sculleries, &c. they should be situate so as to prevent the fumes of stews and offensive smells from approaching any of the best apartments, yet, at the same time, be so contiguous as not to create unnecessary trouble to servants, whose comforts should be studied, which, if neglected, the house is o sure to be comfortless. THE FIVE ORDERS. 451 CHAPTER XVI. THE THEORY AND PRACTICE OF THE FIVE ORDERS OFARCHITECTURE, WITH THE Opinions thereon of the most distinguished Professors in the Art of Building, AS PRESENTED, AT DIFFERENT PERIODS, TO THE SCIENTIFIC WORLD, BY SIR WILLIAM CHAMBERS, AND OTHER EMINENT ARCHITECTS, ANTIENT AND MODERN; FROM WHICH THE STUDENT, AS WELL AS THE AMATEUR, MAY COLLECT EVERY INFORMATION REQUISITE TO PROPORTION; AND DRAW AND EXECUTE EVERY PART OF THE SEVERAL ORDERS, WITH CORRECTNESS, SKILL, AND TASTE, IN THE PUREST STYLE. THE ORDERS of ARCHITECTURE constitute the basis upon which, chiefly, the decorative part of the science is built, and towards which the attention of the architect must ever be cherished, even where Orders are not introduced : for, in them originate most of the forms used in decoration ; they regulate most of the proportions ; and to their combination, multiplied, varied, and arranged, in a thousand different ways, architecture is indebted for its most splendid productions. These Orders are different modes of building, supposed to have been originally imitated from the primitive huts ; being composed of such parts as were essen- tial in their construction ; and, afterwards, adopted in the temples of antiquity, whicbj, though at first simple and rude, were, in the course of time, and by the ingenuity of succeeding architects, wrought up and improved to such a pitch of perfection, on different models, that each was, by way of eminence, denominated an ORDER., Of these there are Five : three, said to be of Grecian origin, are called Grecian Orders; being distinguished by the names of Doric, Ionic, and Corinthian; they 452 THE NEW PRACTICAL BUILDER, &C. exhibit three distinct characters of composition, supposed to have been suggested by the diversity of character in the human frame. The remaining two, being of Italian origin, are called Latin or Roman Orders ; they are distinguished by the names of Tuscan and Roman, and were probably invented with a view of extending the characteristic bounds, on one side still farther towards strength and simplicity, as on the other, towards elegance and profusion of enrichments. At what periods the Orders were invented, or by whom their improvement was advanced, we can now conjecture only from the structures and fragments of anti- quity, built in different ages, and still remaining to be seen in various parts of Europe, Africa, and Asia. Of their origin, little is known but from the relation of Vitruvius, the veracity of which has "been much questioned, and it is probably not altogether to be depended upon. Of the two Latin Orders, the Tuscan is said to have been invented by the inhabitants of Tuscany, before the Romans had intercourse with the Greeks, or were acquainted with their arts and sciences. Probably, however, these people, originally a colony of Greeks, only imitated, in the best manner they could, what they remembered of the state of building, as it existed in their own country : sim- plifying the Doric, either to expedite their work, or, perhaps, to adapt it to the abilities of their workmen. The second Latin Order, though of Roman production, is but of modern adoption ; the antients never having considered it as a distinct Order. It is a mixture of the Ionic and Corinthian, and is now distinguished by the names of Roman, or Composite. The ingenuity of man has, hitherto, not been able to produce a Sixth Order, though large premiums have been offered, and numerous attempts have been made, by means of first-rate talents to accomplish it. Such is the fettered state of human imagination, such the scanty store of its ideas, that Doric, Ionic, and Corinthian, have ever floated uppermost ; and all that has been produced amounts to nothing more than different arrangements and combinations of their parts, with some trifling deviations scarcely deserving notice ; the whole generally tending more to diminish than to increase the beauties of the antient orders. The suppression of parts of the antient orders, with a view to produce novelty, has, of late years, been much practised among us, but with very little success. And, though it is not wished to restrain sallies of imagination, nor to discourage THE FIVE ORDERS. 453 genius from attempting to invent; yet it is apprehended that attempts to alter the primary forms invented by the antients, and established by the concurring approbation of many ages, must ever be attended with injurious consequences, must always be difficult, and seldom or never successful. It is like coining words, which, whatever may be their value, are at first but ill received, and must have the sanction of time to secure them a current reception. An ORDER is composed of two principal members, the Column and the Enta- blature; each of which is divided into three principal parts. Those of the COLUMN are the Base, the Shaft, and the Capital; those of the ENTABLATURE, are the Architrave, the Frieze, and the Cornice : all these again are sub-divided into many smaller parts, the disposition, number, forms, and dimensions, of which characterize each order, and express the degree of strength or delicacy, richness or simplicity, peculiar to it. Columns of the Five Orders, with their respective names, are represented in Plates II. and III. The simplest and most solid of these orders is the TUSCAN ORDER. It is com- posed of few and large parts, devoid of ornaments, and is of a construction so massive, that it seems capable of supporting the heaviest burdens ; whence it is, by Sir Henry Wotton, compared to a sturdy labourer dressed in homely apparel.* The DORIC ORDER, next in strength to the Tuscan, and of a grave, robust, or masculine, aspect, is by Scamozzi called the Herculean. Being the most antient of all the orders, it retains more of the primitive hut style in its form than any of the rest, having trigliphs in the frieze, to represent the ends of joists, and mutules in its cornice, to represent rafters, with inclined soffits, to express their direction in the originals, from which they were imitated. The Doric columns are often seen in antient works, executed without bases, in imitation of trees ; and, in the primitive buildings, without any plinths to raise them above the ground. Freart de Cambrai, in speaking of this order, f observes, that delicate ornaments are repugnant to its characteristic solidity, and that it succeeds best in the simple regularity of its proportions : " Nosegays and garlands of flowers," says he, " grace not a Hercules, who always appears more becomingly with a rough club and lion's skin ; for there are beauties of various sorts, and often so * Vide Sir H. Wotton's Elements of Architecture. t Freart cle Cambrai was a learned architect of the seventeenth century, who died in 1676. He was employed by Louis XIII. to collect antiquities, and engage the ablest artists to reside in France. 57. 5 Y 454 THE NEW PRACTICAL BUILDER, &C. dissimilar in their natures, that those which may be highly proper on one occa- sion, may be quite the reverse, even ridiculously absurd, in others." The IONIC, being the second of the Grecian orders, holds a middle station between the other two, and stands in equipoise between the grave solidity of the Doric, and the elegant delicacy of the Corinthian. Among the antiques, however, we find it in different dresses ; sometimes plentifully adorned, and inclining most towards the Corinthian; sometimes more simple, and bordering on Doric plain- ness ; all according to the fancy of the architect, or nature of the structure where employed. It is throughout of a more slender constructure than either of the afore-described orders ; its appearance, though simple, is graceful and majestic ; its ornaments should be few, rather neat than luxuriant ; and, as there ought to be nothing exaggerated, or affectedly striking in any of its parts, it is not unaptly compared, by Sir Henry Wotton,* to a sedate matron, rather in decent than mag- nificent attire. " The CORINTHIAN ORDER," says Sir Henry Wotton, " is a column lasciviously or extravagantly decked, like a wanton courtezan or woman of fashion. Its pro- portions are elegant in the extreme ; every part of the order is divided into a great variety of members, and abundantly enriched with a diversity of ornaments." " The antients," says De Cambrai, " aiming at the representation of a femi- nine beauty, omitted nothing either calculated, to embellish, or capable of per- fecting, their work;" and he observes, "that, in the many examples left of the Order, such a profusion of different ornaments is introduced, that they seem to have exhausted imagination in the contrivance of decorations for this masterpiece of the art. Scamozzi calls it the Virginal, and it certainly has all the delicacy in its form, with all the gaiety, gaudiness, and affectation in its dress, peculiar to young women." f *ffOT 488 THE NEW PRACTICAL BUILDER, &C. column round: the latter order being, according to his precepts, especially- adapted to servile and commercial purposes, where convenience should always give way to beauty. But, whatever may be the opinion of the vulgar, it is pre- sumed that men of good taste will allow that, in most cases, a well-proportioned graceful base is very handsome; and not only so, but also of real utility, serving to keep the column firm in its place; and that, if columns without bases are entirely set aside, it will be a mark of wisdom in architects rather than an indication of their being swayed by prejudice, as some blind adorers of the antients would insinuate. The latter are the sentiments of Sir William Chambers, who had a rooted aversion to every thing which was Grecian : nevertheless, it must be granted that he was, take him " all in all," a man of considerable judgement, and reasoned well upon his art. In imitation of Palladio, and all the modern architects, except Vignola, he has made use of, the attic base in this order ; which base certainly is the most beautiful of any. Yet, for the sake of variety, when the Doric and Ionic orders are employed together, the base invented by Vignola should be adopted, as shewn in the Doric Order, Plate II. This base Bernini has employed in the colonnade of Saint Peter's, at Rome; and it has also been very successfully applied in many other buildings, Vitruvius gives to the height of the Doric capital one module ; and all the moderns, except Alberti, have followed his example. Nevertheless, as the capital is of the same kind with the Tuscan, they should be nearly of the same proportion, in reference to the heights of their respective columns ; and, under these circum- stances, the Doric capital should be more than one module ; which, indeed, it is, both at the Coliseum and the Theatre of Marcellus, at Rome ; being in the first of these buildings upwards of thirty-eight minutes, and in the latter thirty-three minutes, high. In the design, Orders, Plate V, the example adverted to after Sir William Chambers, the height of the entire capital is thirty-two minutes; and, in the form and dimensions of the several members, it seems that he deviated but little from the Theatre of Marcellus, at Rome. The frieze or neck of the capital is enriched with husks and roses, as in Palladio's design, and as it has been ex- ecuted by Sangallo at the Farnese Palace.* The projections of the husks and flowers should not exceed the upper cincture of the column. * Sangallo was one of the archkects employed in building St. Peter's, at Rome. THE FIVE ORDERS. 489 The architrave is only one module in height, and is composed of one fascia and a fillet, as at the Theatre of Marcellus. The drops in this, the Roman Doric, are conical, as they are in most of the Roman buildings ; and not pyramidal, as they are generally executed by our English artisans. They are presumed, says Chambers, to represent drops of water that have trickled from the trigliphs; and, consequently, they should be cones, or parts of cones, and not pyramids ; but the Grecian architects, who were better versed in the minutiae and details of architec- ture, thought very differently, and made these drops portions of cylinders, the plan being rather more than a semi-circle, and those in the soffits of the mu- tules perfectly round ; and, instead of being inserted in the solid of the mutules, they are described, in the Grecian Doric, as so many pendents; which, in execution and in effect, is infinitely superior to the cold Roman style of finishing the same parts. i-arf ... The Doric frieze and cornice by Sir William Chambers, as given in this work, are, each of them, one module and a half in height, the metope is square, and enriched with a bull's skull, decorated with garlands of beads, in imitation of those in the Temple of Jupiter Tonans. In some antient fragments, and in a large portion of our modern edifices, the metopes between the trigliphs are alternately ornamented with ox-skulls and with patteras ; but they may, with great propriety, be filled with any other ornaments of suitable forms, and frequently with such as are appropriate to the buildings they decorate. For example : in military structures, the head of Medusa, or the Furies, thunderbolts, and other symbols of horror, may be correctly introduced : also helmets, daggers, garlands of laurel or oak, and crowns of various sorts, such as those used among the Romans, and presented as rewards for various mili- tary achievements : but spears, swords, quivers, bows, cuirasses, shields, and the like, should be avoided ; because the actual dimensions of these instruments are too great to find admittance in such limited spaces as the compartments adverted to, and as diminutive representations always convey ideas of triviality, they should, consequently, be wholely avoided. In our churches, dedicated to the saints, and set apart for Christian worship, cherubs, chalices, and garlands of palm or olive, may be introduced ; likewise doves, and other symbols of moral virtues. In pri- vate houses, crests or marks of dignity conferred, may, on some occasions, be permitted ; but seldom, and indeed never, where they are of such stiff insipid 61. 6 H 490 THE NEW PRACTICAL BUILDER, &C. forms as stars and garters, modern crowns, coronets, mitres, and similar graceless objects, the tasteless effects of which may be seen at the Treasury, in St. James's Park, and on various other buildings in the metropolis. Among all the entablatures of the Fr lers, the Doric is the most difficult to distribute : that is, on account of the intervals between the centres of the trigliphs, which will not admit of being increased or decreased, without mate- rially injuring the symmetry and characteristic beauty of the composition : and hence it is that the composer must be fettered by intercolumniations, devisable by two modules and a half, or of 250 minutes from centre to centre, which entirely excludes coupled columns, and produces spaces which, in general, are either too wide or too narrow for the purpose ; and, to remove these difficulties, the trigliphs have been often omitted and the entablature made plain, as at the Coliseum in Rome, at the Custom-House in Dublin, and in many other magnificent buildings, not only in this country, but abroad. It is an easy expedient; but, at the same time, it de- prives the order of its principal and primitive characteristic, and leaves it very poor and so much impoverished, as to be very little, if at all, superior to the Tuscan order; the remedy therefore seems desperate, and ought never to be adopted but in extreme cases, as the very last resource. Chambers says that the antients employed the Doric order in temples dedicated to Minerva, to Mars, and to Hercules, whose grave and manly dispositions suited well with the character of the order : and Serlio says, it is proper for churches dedicated to Christ, to Saint Paul, Saint Peter, or any other saints, remarkable for their fortitude, in exposing their lives and suffering for the Christian faith : and Le Clerc recommends the adoption of it, in all sorts of military buildings, in the entrances to cities, arsenals, gates of fortified places, guard-rooms, and in all manner of similar edifices. It may also be employed in private houses ; and, in particular, in the dwelling-houses of generals or other martial men : it may like- wise be introduced in mausoleums erected to their memory, or in triumphal bridges and arches built to celebrate their victories. The height of the Roman Doric column herein referred to is sixteen modules; which in buildings where majesty and grandeur is required, is a suitable propor- tion ; but in an infinity of other instances it may be made more delicate. Vitru- vius makes the Doric column in porticos loftier by half a diameter than in temples, and many of the modern architects have followed his example. In private THE FIVE ORDERS. houses, therefore, it may be 16J, 16, or 16f , modules high : .and for interior decorations even 17 modules, and sometimes perhaps a little more, which increase in the height may be carried entirely to the shaft, as described in the Tuscan order, without altering, in the smallest degree, either the base or the capital. The entablature may also remain unaltered in all its parts, for what is good in the one case applies to the other. The ROMAN DORIC ORDER stands second in the list of the Five Orders ; but the Grecian Doric, from which the former emanates, stands first among the three Greek orders, and is the most antient of those so called in architecture, being evidently derived from the Egyptians, of which little doubt can be entertained by those who have examined that great national work at the British Museum, which was published under the auspices of Buonaparte, at the time he was identified as NAPOLEON LE GRANDE. The Temple of Minerva, at Athens, commonly called the Parthenon, is considered, by the most learned architects and philosophers, as the boldest specimen of Grecian architecture that ever was constructed ; the style of this structure is now generally known to be what is termed the Grecian Doric: i but, besides this magnificent temple, the beauties of which have been explored by Stuart and others, there are several temples and buildings of great interest, well worthy the consideration of the architectural student, connoisseur, and draughts- man ; particularly the Ionic Temples of Erectheus, Minerva Polias, the small Temple on the river Ilissus, the Temple of the Winds, the Choragic Monument of Lysicrates, commonly called the Lantern of Demosthenes, as well as the Choragic Monuments of Trysallus and others ; among some of the last-mentioned may be collected almost every thing which is great and good in Grecian archi- tecture. The numerous examples of Roman and Grecian ornaments, mouldings, bases, capitals, and cornices, given in this work, have been selected as specimens of the pure style ; and arc, therefore, recommended, with some degree of confidence, to the attentive consideration of our readers. The manner of reducing the GRECIAN DORIC ORDER to practice, is defined in the representation of the plates ; which, we hope, will facilitate the labours of those who are anxious to acquire so much practical information as will enable them to reduce the order to such proportions, as, under all circumstances, will be pleasing and agreeable. 492 THE NEW PRACTICAL, BUILDER, &C. REFERENCES TO THE SEVERAL PLATES EXPLANATORY OF THE DORIC ORDER. ORDERS, PLATE V. Fig. I. The entablature and capital of the Roman Doric Order, on a large scale, wherein the heights and projections of the respective members are correctly proportioned by a scale of modules and minutes, as ex- plained in the Theory of the Five Orders, and which method equally applies to the lower and upper diameters of all the Orders. Fig. 2. Elevation of half of the attic base, the most esteemed among the antient examples. Fig. 3. Plan of the soffit ; exhibiting the various ornaments appropriate to the mutules and spaces between. ' ORDERS, PLATE XI. Fig. 1. Grecian Doric Entablature, accompanied with an imitation of one of the capitals of the columns in the Temple of Theseus. Fig. 2. Plan of the soffit in the last-mentioned entablature, showing the mu- tules, with the bells, or circular drops, appertaining thereto. Fig. 3. Plan of the angular trigliphs and the forms of the residue. ORDERS, PLATE XII. Fig. 1. Grecian Doric Entablature, showing part of a pediment imitated from the Temple of Minerva, at Athens ; with one of the ca- pitals and bases of the columns appertaining to that magnificent temple. A view of the above entablature and capital is also given, under the article of Perspective, in this work, by M. A. Nicholsen. Fig. 2. Plan of the soffit, in the above entablature, showing the mutules, with the bells or circular drops. ORDERS, PLATE XIII. Fig. 1. Grecian Doric Entablature, showing the appli- cation of the antae at the angles of buildings. Fig. 2. The profile of the foregoing entablature and antae. ORDERS, PLATE XIV. Fig. 1. Grecian Entablature, with antique wreaths ; showing, also, the application of the antse at the angles of buildings, and as exe- cuted in the Choragic Monument of Trysallus, at the foot of the Acropolis, or Citadel of Athens. Fig. 2. Profiles of the entablature and antae, both of which are highly esteemed as Grecian examples, for their correct proportions, and decided effects when car- ried into execution. THE FIVE ORDERS. 493 PRACTICE OF THE IONIC ORDER. THIS order is identified as the third in the list of the five antient orders of Roman architecture, and is proportioned by Sir William Chambers, as described in Orders, Plate II. The Ionic is the second, also, in the list of the three Grecian orders, as described in Plate III. It is necessary that the distinction should be made and well known, as there is a difference in the character of the Roman and Grecian Ionic, although both are recognized under the same general name. The general proportions of the antient Ionic order, as adopted by the Grecian and Roman architects, are nearly alike; but the minutiae and detail are very different ; which will be presently discovered, by an attentive examination of the subsequent plates. Among the antients, says Chambers, who always refers to the Roman archi- tects, the form of the Ionic profile appears to have been more positively deter- mined than that of any other ; for, in all the antiques, the Temple of Concord excepted, it is exactly the same, and conformable to the description given by Vitruvius. In Plate VI, of Orders, is represented the design of the antique profile, col- lected by Sir William Chambers, from different antiquities at Rome. The height of the column is eighteen modules, and that of the entablature four and a half, or one quarter of the height of the column, as in the other orders ; which is a trifle less than in any of the antient examples. The base is attic, as it is in most of the Roman antiques, and the shaft of the column may be either plain or fluted, with twenty-four or twenty flutings only, as at the Temple of Fortune, the plan of which flutings should be a trifle more than semicircular, as in the Temple of Jupiter Tonans, and at the Forum of Nerva ; because, when so executed, then they are more distinctly marked. The fillets, or intervals between the flutes, should not be much broader than one-third of their widths, nor narrower than one quarter. 62. 6 i 494 THE NEW PRACTICAL BUILDER, &C. The ornaments of the capital should correspond with the flutes of the shaft, and there should be an ove, or a dart, according to the strict rules of the Romans, over the middle of each flute: but, in the Roman Ionic volute, described in Plate IX, we have made some deviations from the general rule, and have intro- duced the contour and proportions of a Roman Ionic volute, which is considered as an improvement upon Goldman's and Delorme's method of describing the principal characteristic of this order. It is, therefore, deemed unnecessary to enter into further details upon the various opinions of different authors, on a subject which will be best comprehended and felt by a comparative view of the several diagrams for describing volutes. The Roman volute is by some architects preferred ; which will be apparent to the scientific observer on a cursory view of the magnificent street, now leading from Carlton-Palace up to Portland Place, in London. In the Ionic facade, opposite the same palace, the volutes of the capitals are Grecian, and are proportioned in the manner of those in the small temple on the River Ilissus, and as described in this work. This capital is justly esteemed ; nor can it be sufficiently appreciated by those who entertain a true love for architecture. It is, therefore, surprising that the ingenious architects employed in the new street should, in any part of it, have adopted inferior specimens of the Roman Ionic capitals, as in the quadrangle, opposite the facade before-mentioned. In passing up the street, however, towards Portland Place, it is observable that, ^wherever the Ionic order has been subsequently introduced, improvements have taken place in the adoption and style of the Ionic capitals, except in the finale to the street, which presents to the eye of the inquisitive spectator a circular Ionic portico, terminated by a fluted conical spire of the same form ; the meta- phorical intention of which is not clearly understood, unless it is meant to convey, by a well-proportioned geometric figure, a new species of metaphysics, deducible, but which can be comprehended only by those who are deeply versed in mathematics. The effect of this spire is stated to be sublime; but what is not generally comprehended must be injudiciously applied; and, therefore, we la- ment that a magnificent street, so justly distinguished for its picturesque and architectural beauties, should be terminated by a conical finial, in no respect correspondent with the bold and intelligent metaphors usually applied by tho Genius, of Architecture. THE FIVE ORDERS. * 495 The three parts of the Ionic entablature, as represented in Plate VI, of Orders, bear the same proportion to each other in this as in the Tuscan order : the frieze is plain, as being the most suitable to the simplicity of the rest of the composition ; and the cornice is almost an exact copy from Vignola's design, in which there is a purity of form, a grandeur of style, and close conformity to the most approved specimens of the antients, not to be equalled in any of the profiles of his com- petitors. If it be requisite to reduce the Ionic entablature to two-ninths of the height of the column, which on most occasions is preferable to that of one quarter, espe- cially where the eye has been accustomed to contemplate diminutive objects, it may be easily accomplished by making the module of the entablature less, by one- ninth, than the semi-diameter of the column ; afterwards dividing it as usual, and strictly observing the same dimensions as are figured in the engraved plate VI. The distribution of the dentil-band will, in such case, answer very nearly in all the regular intercolumniations, and in the extreme angle there will be a dentil, as there is in the best examples of the antique. In the decorations of the interior of all apartments, where much delicacy is requisite, the height of the entablature may be reduced even to one-fifth of the column, by observing the same method, and making the module only four-fifths of the semi-diameter. The ANTIQUE IONIC CAPITAL, not only in the Grecian but Roman style, differs from all others ; inasmuch as the front and side forms are not similar. This par- ticularity occasions great difficulty, whenever breaks are introduced in the enta- blature, or where the decorations are continued in flank as well as in front : for, either all the capitals in the returns must have the baluster side outward, or the angular capital will have a different appearance from the rest, neither of which is admissible where good taste prevails. The architect of the Temple of Fortune, at Rome, as likewise the scientific artist who designed the small temple on the River Ilissus, have each fallen upon expedients which, in some degree, remedy the defect above-mentioned. In each of those buildings, ,as well as others, the corner capitals have their angular volutes in oblique positions, inclining equally to the front and side, and presenting volutes both ways : and, says Chambers, where persons are violently attached to the an- tique, or furiously bent on rejecting all modern inventions, however beautiful, this 496 THE* NEW PRACTICAL BUILDER, &C. is the only way to gratify them; but, when such is not the case, the angular capital invented by Scamozzi, and lately imitated in the circular portico of Langham Chapel, may be introduced ; for it must be allowed that the distorted figure of the antique capital, as represented in Plate XXI, of the Orders, with one straight volute and the other twisted, is very objectionable, and far from being pleasing to the eye ; yet we are of opinion that the Grecian antique volutes, as carved at the East-India House, in Leadenhall-Street, at the Saint Pancras new church, at the College of Surgeons, in Lincoln's-Inn Fields, and in various other public buildings, are worthy of imitation; and therefore we cannot better discharge our duty than by recommending the student first to draw all the specimens given in this work, and as he proceeds, if opportunities permit, to examine the buildings above-mentioned, or such as are of a similar description. As the Doric order, says Chambers, is particularly affected in churches and temples dedicated to male saints, so the Ionic is chiefly used in such as are conse- crated to females of the matronal state. It may, likewise, be employed in Courts of Justice, as well as the Roman or Grecian Doric ; it may also be introduced in libraries, colleges, seminaries, and other structures having relation to arts or letters, and also in private-houses, and in palaces to adorn the ladies' apartments : and, says Le Clerc, in all places dedicated to peace and tranquillity. The antients employed it in temples sacred to Luna, to Bacchus, to Diana, or other deities, whose dispositions hold a medium between the severe and the effeminate. The Grecian Ionic specimens of capitals, cornices, friezes, and architraves, are, generally speaking, better profiled than those of the Romans : the judicious com- poser should, therefore, contemplate the several parts appertaining to each style ; and, by alternately rejecting and adopting, he will, by degrees, improve his taste : but, as regards the bases of the Grecian Ionic order, usually employed in the antique, we cannot recommend them, although most slavishly adopted by many of our modern practitioners. The attic base of the Romans is the best, sim- plest, and most natural ; and, wherever applied, is sure to give satisfaction : it is, therefore, recommended to the serious consideration of the student. THE FIVE ORDERS. 497 REFERENCES TO THE PLATES APPERTAINING TO THE IONIC ORDER, WHICH INCLUDE THE ROMAN AND GRECIAN EXAMPLES. ORDERS, PLATE VI. Fig. 1 represents the entablature and capital of the Roman Ionic order, on a large scale, proportioned by modules and minutes. Fig. 2 represents the attic base to the same scale. Fig. 3. Plan of one quarter of the capital. Fig. 4. Section of the capital. Fig. 5. Half the elevation of the barrel. ORDERS, PLATE IX. The contour and proportion of a Roman Ionic Volute, with the appendages; from a description by Mr. R. Elsam, architect, on a large scale. ORDERS, PLATE XV. Fig. 1. The entablature and capital of the Grecian Ionic Order, in imitation of the Ionic portico to the small temple on the River Ilissus.* Fig. 2. Half the base. Fig. 3. Half the section of the capital. Fig. 4. Altitudinal scale of the base. ORDERS, PLATE XVI. Grecian Ionic capital at large, in imitation of the last example, by Mr. R. Elsam. ORDERS, PLATE XVII. Grecian Ionic capital at large, in imitation of the example in the Minerva Polias, at Preene, by Mr. R. Elsam. ORDERS, PLATE XVIII. Fig. 1. Plan of the Ionic capital from the Temple of Erectheus, at Athens. Fig. 2. Elevation of the last-mentioned capital. Fig. 3. Plan of the Ionic capital, from the Temple of Minerva Polias, at Athens. Fig. 4. Elevation of the last-mentioned capital. Fig. 5. Diagram, on a large scale, shewing the minutiae of finding the different centres for striking the two last described volutes ; which, by an attentive exami- nation, will teach the inquisitive student every thing which is requisite on the subject. * An interesting view of this capital is given hereafter, under the article of " PERSPECTIVE," from a drawing by Mr. Michael Angelo Nicholson. 62. 6 K 498 THE NEW PRACTICAL BUILDER, &C. ORDERS, PLATE XIX. Fig. 1. Flank elevation of half the Ionic capital, as executed in the Temple of Minerva Polias, at Athens. Fig. 2. Section of the same, shewing the barrel of the volute. Fig. 3. Transverse section of the same capital. Fig. 4. Transverse section of the Ionic capital, in the Temple of Erectheus, at Athens. Fig. 5. Flank elevation of the same capital. Fig. 6. Section of the same, showing the barrel of the volute. ORDERS, PLATE XX. Fig. 1. Grecian Ionic entablature, appertaining to the Temple of Minerva Polias, at Athens. Fig. 2. Base of the columns to the above order. Fig. 3. Half the base of the antae to the same example. Fig. 4. Half the capital of the antae to the same. ORDERS, PLATE XXI. Fig. 1. Plan of an angular Grecian Ionic capital, in imi- tation of those employed in the small temple on the River Ilissus. Fig. 2. Flank elevation of the capital. Fig. 3. Section of the capital. Fig. 4. Section, showing the barrel of the capital. ORDERS, PLATE XXII. Fig. 1. An antique Grecian Ionic base. Fig. 2. An antique Grecian Ionic capital. Fig. 3. An antique Grecian Ionic base. Fig. 4. An antique Grecian Ionic capital. oyioemto Jkifi . t (mv '/tcni or' I PRACTICE OF THE COMPOSITE ORDER. CORRECTLY speaking, the Grecians and Romans had but three recognized orders; the Composite and Tuscan not having been acknowledged by the an- tients : the moderns, however, have ranked the two latter with the three original orders, which are the Doric, Ionic, and Corinthian. It is the practice with most authors to give to the Composite the fifth or last place ; as being the last invented; and also from being a compound of all the rest, which, of course, ought to be pre- THE FIVE ORDERS. ceded by those which are the simplest. But Chambers has followed Scamozzi's arrangement; it being the most natural for the orders to succeed each other according to their degrees of strength, and in the progression that ought to be observed, when they are employed together. Palladio, Scamozzi, Vignola, De Cambrai, Serlio, and Delorme, together with other modern architects, have each followed the bias of their own inclinations in designing the Composite order : fac-similes, or counter-likenesses, are, therefore, seldom to be met with ; it is, however, manifest, from an examination of the best works, that the order now under consideration is compounded, chiefly, of the component parts of the Ionic and Corinthian, without possessing the native sim- plicity and elegance appertaining to the two latter classical orders. The Com- posite is, nevertheless, an order of considerable merit, which, on many occasions, will claim a decided preference, and cannot fail to be duly appreciated when judiciously introduced. The modern architects, says Chambers, have varied more in this than in any of the orders ; and De Cambrai observes, that, abandoned by their guide, Vitruvius, and left entirely at large, they have all taken different paths, each following the bent of his own particular fancy ; and among them, Serlio has been the least successful; having chosen for the model of his entablature that of the fourth order of the Coliseum : a composition too clumsy even for a Tuscan order. Delorme, however, has followed his example, and has mistaken the fourth order of the Coliseum, which is Corinthian, for the Composite; and Palladio, in his profile, has imitated the frontispiece of Nero, and corrected its defects with much judgement. His architrave is also copied from the same building : but he has omitted the beautiful frieze, and has submitted in its place a swelled one, similar to the Basilica of Antonius. His entablature is too low, being only one-fifth of the column ; and it is also singular that, although he has made the column more slender than in the Corinthian order, yet his entablature is far more heavy ; it being composed of fewer and larger parts. His capital and base are imitations from the arch of Titus, the latter of which is designed without a plinth, as it is executed in the Temple of Tivoli, and joined to the cornice of the pedestal, as may be observed by reference to Orders, Plate X, on which is also represented the capital adverted to, and which has been lately adopted in the hexastyle portico, by Mr. George Smith, in the front of Saint Paul's New School, opposite 500 THE NEW PRACTICAL BUILDER, &C. the east end of the Cathedral; it has been adopted, likewise, at the Bank^of England, by Mr. Soane. The capital here noticed is described as Corinthian, but with more correctness, perhaps, it might be termed a Composite capital; from its being a medium between the Corinthian and Composite referred to in Chambers' orders, which have been followed in this work, that is, as regards the Roman principles or style of architecture. Vignola's composition of this order has nothing remarkable about it. The ar- chitrave varies but in a very small degree from the frontispiece of Nero, and the cornice is nearly the same as the Ionic composed by him ; the chief difference consisting in the transposing of mouldings and the enlargements of dentils, each of which appear to be alterations much for the worse. And Scamozzi's entablature is like Palladio's; that is, only one-fifth of the height of the column ; and, being much divided, produces a very trifling appear- ance. The cornice, however, is well composed ; and is also, in a great degree, imitated from the third order of the Coliseum. The capital is similar to Palladio's, and the base is attic, enriched with astragals in accordance with the Basilica of Antonius. The engraved design here referred to, in imitation of Chambers' Composite order, is the invention of that learned architect, who states that he has attempted to avoid the faults, and to unite the perfections, of those before-mentioned ; but how far he has succeeded is left to those who have judgement to discriminate. Under any circumstances, however, the student may refer to Palladio, Scamozzi, or Vignola, which has been the practice heretofore. The height of the column is twenty modules ; the entablature five ; the base is attic, and its measures are precisely the same as in the Doric and Ionic orders ; but, as the module is less, all the parts are of course proportionately delicate. The shaft is enriched with flutings, which may be either to the number of twenty or of twenty-four, similar to the Ionic order ; for we cannot see any reason why, in the different orders, their number should be either augmented or diminished. The module being less, the flutes will likewise be less, and correspond exactly with the character of the rest of the composition. The general effect of the capital is very nearly of the same description as those of the moderns ; being enriched with leaves, in imitation of the acanthus, as most of the antique capitals of the Composite order are. But, as regards the method THE FIVE ORDERS. 501 of tracing the minutiae of these capitals, few instructions, it is presumed, will be deemed sufficient ; the design having been correctly drawn and figured by a very accurate draftsman. The curves of the abacus are described from the summits of equilateral tri- angles ; the projections of the volutes are determined by lines drawn from the extremities of the astragals to the extremities of the horns of the abacus ; and, the projections of the leaves are determined by other lines drawn parallel to those from the fillet under the astragal. The style and manner, says Chambers, of executing these and other enriched capitals in the metropolis, is, generally speaking, bad ; but it is evident that he did not mean, in the aggregate, to accuse our workmen of want of talent or capacity ; many of them, says he, cannot be excelled in neatness of execution ; and, perhaps, in point of workmanship, they do, for the most part, eclipse those of any other country : but it frequently happens, from the want of liberality in their employers, and, in some degree, perhaps, from want of thorough skill and facility in designing, that their performances are very insipid, tame, and without effect, and by no means expressive of the taste or intelligence intended to be con- veyed in the designs which have been ordered to be carried into execution. It is, also, to be regretted, that the greatest architects have too much neglected the detail ; having directed their attention chiefly to the general disposition of their compositions. This neglect, though authorized by some great men, ought not, by any means, to be imitated. It is not only the province, but the duty, of the architect strictly to attend to the minutest parts, as well as to the most consi- derable. Where the entire execution of a fabric is left to his direction, the faults that are committed will, of course, be carried to his account ; and, therefore, it behoves him to be choice and extremely select in employing the ablest workmen, and to furnish them with all manner of proper models and precise instructions ; in which he will at once announce the extent of his capacity, and distinguish himself from the common herd of those who, without the requisite qualifications, arrogate the title of architects. The most masterly disposition, says Chambers, incorrectly executed, can be considered only as a sketch in painting, or as an excellent piece of music, miserably murdered by village fiddlers, equally destitute of taste and powers of execution. 'W>hrfiwr!iwfciTM-' The Ionic, Composite, and Corinthian, capitals, to be seen in various parts of 63. 6 L 502 THE NEW PRACTICAL BUILDER, &C. the new buildings at Somerset-House, London, were copied by the author last mentioned, from models executed under his directions at Rome, and imitated, both in point of form and manner of workmanship, from the choicest antiques ; hence they will serve as guides to those who are anxious to obtain information at the fountain head ; and, by a careful examination of them, the student may obtain such information as cannot be conveyed by the most powerful language, or the most intelligent designs. The entablatures of the Composite order, in Plates II. and VII., bear the same proportion to each other as the Ionic and Tuscan orders. The architrave is nearly of the same form with those of Palladio and Vignola. The frieze may be enriched with foliages in imitation of the antique ; but the prominent parts should never project more than the uppermost moulding of the architrave under them. The cornice is imitated from Scamozzi, and differs from the Corinthian only in the modillions, which are square, and composed of two fascias. The soffit of the intervals between the dentils should be hollowed out upwards, behind the little fillet in front, which is the case in most of the antiques ; and the incisions produce dark shades, tending to mark the dentils more distinctly. From the same reasons similar methods should also be observed in the Ionic and Corinthian orders. The roses, in the soffit of the corona, should not project beyond the horizontal sur- face ; and care should be taken not to vary them so much as at St. Peter's, of the Vatican, and some modern buildings, because the unity suffers thereby : the mo- dillions or dentils might, with almost as much propriety, be varied. It will, therefore, be proper, in small compositions, to make them all alike, as they are in most of the antiques ; to the end that they may not arrest or occupy too much the attention of the spectator, as objects intended for distinct contemplation ; but, simply, as parts of the entire. In larger compositions, they may be of two kinds, as in the Long-room of the new Custom-House, in London, designed and carried into effect under the direction of Mr. David Laing, an architect not less eminent for having planned and executed some good and useful structures, than for his excellent treatise on building, contained in his work, entitled " Plans of Buildings, Public and Private." .,,<- 01 sWewiqq* aws- > In the elegant and spacious room just mentioned, the ornaments in the three domical, coved, or vaulted, soffits are of two sorts ; but similar in outline and dimensions, which occasions much variety, but without confusion ; for the flowers THE FIVE" ORDERS. 503 succeed each other so rapidly, and are, from their similitude, so immediately and well comprehended, that the third impression takes place before the first is in any degree removed ; so that the same effect is nearly produced by alternate successions of the same objects. The Romans introduced the Composite order more frequently in their tri- umphal arches than in any other buildings ; meaning, as Serlio supposes, to express their dominion over other nations, the inventors of the orders of which this is composed. It may, says Le Clerc, with singular propriety be used wherever elegance and magnificence are to be united ; but, more especially, in buildings designed to commemorate great and signal events, or to celebrate the combined achievements of conquerors and legislators ; and, from these reasons, that the capitals or other ornaments may be composed of emblems and of allegorical representations, in conformity to the customs of the antients ; as appears by numerous fragments of capitals and entire members of architecture scattered about in different parts of Rome, and elsewhere. The entablature of the Composite order, as introduced in this work, may be reduced to two-ninths of the column; which, to avoid fractions, we will call four modules and one half, by making the module only nine-tenths of the semi- diameter ; at the same time observing the same measures as are figured in the design, in which case there will be a dentil in the outward angle similar to the Ionic order. It may also, if required, be reduced to one-fifth, by making the module four-fifths of the semi-diameter : but, in cases where it is requisite to diminish so much, it will be preferable to adopt the Ionic cornice ; which, being composed of fewer parts, will retain an air of grandeur without affecting the entire mass. ojsteaqa o f 'J "to aoiiaaJiti ^ REFERENCES TO THE PLATE ON THE PRACTICE OF THE COMPOSITE ORDER. ORDERS, PLATE VII. Fig. 1. Entablature and capital of the Composite order, wherein the heights and projections of the several members are proportioned in the manner already described. Fig. 2. The attic base applicable to this order, and similar to the Ionic and Doric in proportions. Fig. 3. Shows the plan of the soffit in the entablature, with the manner of finishing the modillions. " iud ^tehflv Aw fa wwynw douiw ,aaoiwi 504 THE NEW PRACTICAL BUILDER, &C. . PRACTICE OF THE CORINTHIAN ORDER. THIS order, says Chambers, is suitable and proper for buildings where elegance, gaiety, and magnificence, are required. The antients employed it in temples dedicated to Venus, to Flora, Proserpine, and the nymphs of fountains ; and be- cause the flowers, foliage, and volutes, with which it is adorned, seemed well adapted to the delicacy and elegance of such deities. Being the most splendid of the Five Orders, it is also extremely proper for the decorations of palaces, public squares, or galleries and arcades surrounding them; for churches dedicated to the Virgin Mary, or to other virgin saints : and, on account of its rich, gay, and graceful, appearance, it may, with singular propriety, be used in theatres, in banqueting-rooms or ball-rooms, and in all places consecrated to festive mirth or convivial recreation. It is an uncontroverted opinion, that the fragments of the three columns in the Campo Vaccino (the supposed remains of Jupiter Stator) were, when entire, and still are, the most perfect compositions of the Corinthian order among the Roman antiques ; and it is also the opinion of the most learned, that the best example of the Corinthian order among the Grecians, is that which is employed in the Choragic monument of Lysicrates, commonly called the Lantern of Demosthenes, and which capital is introduced in this work : but, independent of that relic, va- rious specimens of Grecian Corinthian capitals, as well as columns and bases, may, at proper times, be seen among the Elgin Marbles, at the British Museum, in London, where the industrious, in search of information, will find ample gratifica- tion. The study of these choice remains, now the sole property of the nation, will, it is presumed, contribute more to rouse the imagination of architects, sculptors, and painters, than all the books, prints, and drawings, which have been published, printed, and made, for the last hundred years. Let the student, there- fore, after having obtained the usual elementary knowledge of architecture, advantageously apply his genius and understanding to the consideration of those rare and invaluable materials, on which are engraven the talent, learning, history, and genius, of the Greeks, in characters which cannot fail deeply to affect and influence his imagination. THE FIVE ORDERS. 505 The possession of the Elgin collection is a treasure to the lovers of the Fine Arts; nor can the fragments, so judiciously selected, be sufficiently appreciated: for, during the few years in which they have been exposed to public view, they have produced such interesting and visible changes in architecture, sculpture, and painting, as cannot fail to impress every susceptible mind with a feeling of gratitude, for the liberal sum voted in Parliament to purchase some of the most valuable relics of architecture and sculpture in the civilized world. Palladio, in his fourth book, where he gives the entire profile at large of the three Corinthian columns in the Campo Vaccino, acknowledges that he never had seen any work better executed, or more delicately finished; and, furthermore, that the parts were beautifully formed, well-proportioned, and skilfully combined.* Now, upon an impartial view, with reference to the Grecian and Roman Corin- thian capitals, no doubt in the world can be entertained that the Roman Corinthian capital is as much superior to the Grecian as the latter Ionic capital is to the Roman. The absurdity, therefore, of prejudiced minds, in favour of every thing which is Grecian, must be manifest : the Romans excelled in some instances, and the Grecians in others ; it is, consequently, the duty of the student to avoid being seduced by, or entirely devoted to, Grecian or to Roman principles : in each style there is much to admire and much to condemn. By judicious selections, the man of correct taste, possessing integrity of mind, may evince his judgement by con- structing, altering, and changing, without violating the doctrine of good taste. Vignola's composition of the Corinthian order is uncommonly beautiful, and, without doubt, superior to that of any other master ; he having artfully col- lected all the perfections of the best originals, out of which he formed an entire composition far preferable to any of them. The design for the Corinthian order given in this work, from Chambers, differs very little from that of Vignola. The column is twenty modules high, and the entablature five, which proportions are a medium between those of the Pantheon, at Rome, and the three columns of the Campo Vaccino. The base of the column may be either attic or Corinthian ; each of them are beautiful. Palladio and * To afford the amateurs of architecture a full and comprehensive idea of the beauties of the Antique Roman Corinthian capital, we have given an interesting engraved view of one of the most approved, from an intelligent drawing by Mr. M. A. Nicholson, under the article of " PERSPECTIVE," and to which plate the reader is referred, as well as to Orders, Plate VIII, which represents the Corinthian order in detail, as proportioned by Sir William Chambers. 63. 6 M 606 THE NEW PRACTICAL BUILDER, &C. Scamozzi have employed the attic, enriched with astragals; but so frequent a repetition of the same semi-circular forms in junction, produces but an indifferent effect ; as may be observed in the bases of the columns at the parish church of Saint Martin in the Fields, and in several buildings of the metropolis, in which the profiles and forms of Palladio, good, bad, and indifferent, have been indis- criminately employed. If the entablature is enriched, the shaft of the column should be fluted, provided it is not composed of variegated marble, for a diversity of colours renders even smooth surfaces confused, and ornaments of sculpture serve only to make the confusion greater. The flutings may be filled to one-third of the height with cableiugs, as in the inside of the Pantheon, at Rome, which strengthen the lower part of the column, and renders it less liable to damage. But, when the columns are not within reach, nor subject to be hurt by passengers, the cables are better omitted, as the general hue of the shaft will then be the same throughout, and appear to be of a piece ; but, when parts of the flutes are filled, and the other parts left vacant, it is not the case ; for the shaft then appears divided, and is liable to produce a great defect. The capital is enriched with olive leaves, as are most of the antiques, at Rome, of tliis order ; the acanthus being seldom employed but in the Composite order. With respect to the manner of tracing and working this capital, the design, with what has been said on the same subject in the Composite order, will serve as a sufficient explanation. The divisions of the entablature bear , the same relative proportions to each other as in the Tuscan, Ionic, and Composite, orders. The frieze is sometimes enriched with bas-reliefs, as described in Chambers's Civil Architecture ; but it is more frequently left quite plain. The parts and ornaments in the cornice are all regularly disposed, and perpendicularly over each other. The coffers of the soffit of the cornice are square, and the borders round them equal on all sides, as they are in the arch of Titus, and as Palladio has made them ; a precaution neglected by Vignola, notwithstanding his usual regularity. ait-The antients, as well as the moderns, have frequently employed the Ionic enta- blature in the Corinthian order, as appears by many of the buildings : and this is remarkable in the Corinthian portico, erected by Mr. James Gandon, to the en- trance of the late House of Lords, in Dublin, now the approcht: to the Directors' THE FIVE ORDERS. 507 Court-Room of the National Bank ; and, according to Vitruvius, even the Doric entablature has been employed upon Corinthian columns ; though, of the latter practice, there is not, as we are aware of, any example extant. The same author furthermore observes, that the Grecians, in their works, never employed the dentils under the modillions. It is, however, certain that the Romans were not so fastidious or over nice ; for, in their most esteemed works, such as the Temple of Jupiter Stator, the Forum of Nerva, the Temple of Jupiter Tonans, and several others, we observe the dentils under the modillions ; and these examples, it is presumed, will continue to authorize the same practice. The origin and history of these things are extremely remote, and known to but few, while the general effects of such compositions are known to all. If deviating, therefore, from what is little known and less felt, will contribute towards the perfection of that which all see and all approve, it cannot be justly censured. The liberty, however, of deviating from the origin or reason of things, was by the antients, and must by us, be exercised with extreme caution ; as it opens a wide door to whim and extravagance, and leaves a latitude to the composer, which often betrays arid hurries him into ridiculous absurdities. When the modillion cornice is employed on large concave surfaces, the sides of the modillions and coffers of the soffit should tend towards the centre of the curve, as in the Pantheon, at Rome ; but, when the concave is small, it will be far the better way to direct them towards the opposite point in the circumference, that the contraction may be the less perceptible, and the parts dependent thereon suffer less deviation from their natural form. The same rules must also be observed with regard to dentils, to the abacus, and bases of columns and pilasters, and likewise to the flanks of the pilaster itself. But, on convex surfaces, the sides of all these should be pai'allel to each other, for it would be very unnatural, and extremely disagreeable, to observe them narrowest where they spring out of the cornice, and diverging as they advance forwards, forming sharp angles and a sort of mutilated triangular plan, with enlarged solids, and diminished intervals, each calculated to destroy the usual proportions and beauty of the composition. The entablature to the Corinthian order may be reduced to two-ninths, or one- fifth of the height of the column, by the same rules as are given in the Ionic and Composite orders : but, where it is rendered necessary, or it is deemed expedient, to make the entablature so small as one-fifth, it will be best to substitute the lonie 508 THE NEW PRACTICAL BUILDER, &C. entablature, as Palladio has done in the peristyle of his Olympic Theatre, at Vi- cenza, and in several of his buildings; or else to retrench the dentils of the cor- nice, as in one of Serlio's, and in Scamozzi's profiles, the part of the cornice under the modillion band remaining then composed of only the ovolo and ogee, sepa- rated by a fillet, as in the several temples mentioned in Palladio's fourth book. REFERENCES TO THE SEVERAL PLATES ON THE PRACTICE OF THE CORINTHIAN ORDER. ORDERS, PLATE VIII. Fig. 1. Represents, on an enlarged scale, the entablature and capital of the Roman Corinthian order, proportioned by modules and minutes in manner before described by Sir William Chambers. Fig. 2. Represents the attic base and part of the column, proportioned in a similar manner to the preceding. Fig. 3. The antique Corinthian base. Fig. 4. An oblique profile of the capital, showing the projection of the leaves, by a line drawn from the astragal to the extreme of the volute. Fig. 5. Semi plan of the capital, showing the number and arrangement of the leaves, as likewise the segmental curves forming the outlines of the capital. Fig. 6. Plan of the soffit of the entablature, shewing the modillions, and the manner of finishing the soffit at the angles. ORDERS, PLATE X. Fig. 1. A Roman Corinthian capital, from actual measure- ment, copied from the Temple of Vesta, at Tivoli. Fig. 2. Section or profile of the annexed capital. Fig. 3. Plan of the several parts of the above capital and base. Fig. 4. Elevation of the base and part of the cornice of the pedestal on which the order stands. ORDERS, PLATE XXIII. Fig. 1. A Grecian antique Corinthian capital apper- taining to a column. Fig* 2. Profile of the before-mentioned capital. Fig. 3. Grecian Corinthian antique capital to an antae. Fig. 4. Profile also of the last mentioned capital appertaining to the same. ORDERS, PLATE XXIV. Fig. 1. Grecian antique base to a Corinthian antse. Fig. 2. Grecian Corinthian capital to an antae. Fig. 3. Grecian antique base to a Corinthian antse. THE FIVE ORDERS. 509 Fig. 4. Grecian Corinthian capital to an antae. Fig. 5. Profile of the preceding capital. Fig. 6. The base to the Grecian Corinthian pilaster, or antae appertaining to the latter. 1f jj ORDERS, PLATE XXV. Fig. I. The entablature and capital of the Grecian Corinthian order, imitated from the Choragic Monument of Lysicrates, commonly called the Lantern of Demosthenes. Fig. 2. Is a representation of half the base proportioned, as in the preceding examples. -.t . Fig. 3. Plan of the soffit of the entablature, showing the dentils, c. 1 *! ' THE PRACTICE OF PILASTERS, AND THE GREEK ANT2E. COLUMNS differ from pilasters in their plans only ; the latter being square or rectangular, whereas the former are round. Pilasters, when accompanied by columns in the Roman style, have their bases, capitals, and entablatures, the same as the columns, and their component parts are all of similar heights and projections : and, when complete, they are identified by the names of Tuscan, Doric, Ionic, Composite, and Corinthian, Pilasters.* Of the two opposite compositions, says Chambers, the column is, without any doubt, the most perfect as well as the most beautiful. Nevertheless, it would be im- possible for composers in architecture to dispense with pilasters ; and upon most, and if not upon all, occasions they may be employed with fitness and great propriety. In numerous instances, on various accounts, they are even preferable to columns. Pilasters are stated to be of Roman invention ; and, doubtless, the composition of them is an improvement upon the Greeks, who, instead of them, employed what is called the antte ; and, says Mr. Joseph Gwilt, very justly, one of the most objectionable practices of the day is, the servile imitation of this Greek antae. It is, adds he, quite inconsistent with any regard to primitive types, from which the Grecian architecture is supposed to have originated. Added to this, their application in such very thin laminae against the walls, as could be pointed to in * The reader is referred to the following Plates of Orders, for various examples of pilasters and the Greek antae, Plates XIII, XIV, XX, XXIV. 64. 6 N 510 THE NEW PRACTICAL BDILDER, &C. some pseudo-Grecian buildings about the metropolis, produces a remarkably silly and unmeaning effect. The Greeks employed these antse in their temples, to receive the architraves where they entered upon the walls of the building; and in most, if not in all, of the examples of the antique, the front of the antae is equal in diameter to the upper one of the adjacent column ; the antae being also of the same width at the top as at the bottom, and not diminished as in the Roman examples of pilasters. And these antae, now so much the fashion in the metropolis, are executed, in imitation of the originals, equal in one direction to the diameter of the column in front; but, in flank, extravagantly thin in proportion to their heights, and neither their bases nor capitals bear any resemblance to those of the columns they accompany, and, it is supposed, in reference to the Greeks, that the Roman architects, being disgusted with the poor meagre aspect of these antae, and the want of accordance in their bases and capitals, substituted pilasters in lieu of them ; which, being propor- tioned and decorated in a similar manner with the columns, are, in the eyes of the most thinking and unprejudiced persons, more fitting, applicable, and seemly, as tending at once to preserve the unity and harmonious effects of all manner of architectural compositions wherein Columns and Pilasters accompany each other. The compilers of this work are perfectly aware of the strong prejudice in favour of Grecian examples of every sort ; nevertheless, they conceive it their duty to advise the juvenile student against the adoption of whatever appears to be not only inconsistent with, but repugnant to, good taste. Several authors, says Chambers, are of different opinions about pilasters and their application, and to the end of the world such differences will exist in the minds of scientific men upon points of taste. A French Jesuit, says the same intel- ligent writer, many years ago, published an Essay on Architecture, which, from its plausibility, force, and elegance of diction, went through several editions, and operated very powerfully on the superficial part of European connoisseurs. The Abbe Laugier, who, it is understood, is the author adverted to, inveighs in the strongest terms against pilasters, and against every other architectonic form, ex- cepting such as were imitated by the first builders in stone, from the primitive wooden huts : as if, in the entire catalogue of arts, architecture should be the only one confined to its pristine simplicity, and debarred from any deviation or improvement whatever. THE FIVE ORDERS. 511 To pilasters the learned father objects, because they are, in his opinion, nothing better than bad representations of columns. Their angles, says he, indicate the formal stiffness of art, and are a striking deviation from the simplicity of nature ; their projections, sharp and inconvenient, offend and confine the eye; and their surfaces, without roundness, give to the entire order a tame insipid effect ; they are not, as he thinks, susceptible of diminution, one of the most pleasing pro* perties of columns : and, in his opinion, they never can be necessary. To sum up the whole, " he hates them ; his aversion was first innate, but it has been subsequently confirmed by the study of architecture." Now, as regards the reverend father's inborn aversion, much need not be said ; and as to the several other objections, as they consist more of words than meaning, on that account they seem not to require argumentative refutation: conviction on the face of his own showing is too evident. To assert that pilasters are not susceptible of diminution, at once, discovers very little acquaintance either with books of architecture or with buildings. Innumerable are the instances, in the remains of antiquity, of their being diminished; and, in particular, when associated with columns. They are so in the Temple of Mars the avenger, in the Frontispiece of Nero, in the Portico of Septimus Severus, and in the Arch of Constantine, at Rome. Scamozzi always gave to his pilasters the same diminu- tion as to his columns; Palladio has diminished them in all his buildings at Venice ; and Inigo Jones has likewise done the same in many of his designs, and in particular at Whitehall. And if we trace back to the origin, and consider pilasters either as the repre* sentation of the ends of partition-walls, or trunks of trees reduced to the diameter of the round trunks, but left square for greater strength, the reason for dimi- nishing them will, in either case, be manifest. It is, also, a strange error to suppose, or to assert, that pilasters are never necessary ; but that columns will, at all times, answer the same purpose : for, at the angles of most architectural fronts to buildings they are indispensably neces- sary, both for solidity and beauty. For the angular support, having a greater weight to bear than any of the rest, they should be so much the stronger, so that its diameter must be encreased, or its plan altered, from the circle to the square. The last is certainly the most reasonable expedient ; but, chiefly so, as it obviates a very striking defect, occasioned by employing columns at the angles of build- 512 THE NEW PRACTICAL BUILDER, &C. ings ; which is, that the angle of the entablature is left, as it were, apparently suspended in the air without any apparent support; a sight very painful and disagreeable in many oblique points of view, and at the same time very unsolid. It is customary in most, if not on all, occasions, to porticos and other detached compositions, to employ columns at the angles ; and it is extremely judicious so to do : for, of defects such as those described the least should be preferred ; and although, says Chambers, the reverend Father Laugier, whose objections have been cited, could not see any reason for rejecting detached pilasters when engaged ones are permitted, yet a very substantial reason may be assigned, which is, that detached pilasters, in certain oblique views, appear thicker than they do in front, and nearly in the ratio of seven to five ; but, of course, when seen in front, they will appear well-proportioned ; yet, with respect to the columns they accompany, they never can appear so when viewed upon the angle ; as may be observed in the colonnades at Burlington-House, in Piccadilly, London ; and in the portico of Saint George's, Hanover Square ; as well as in the extraordinary applique to the front of the new theatre in Drury Lane. Engaged pilasters may be employed, and appositely, in churches, galleries, halls, and in other exterior decorations, to save room ; for, as they seldom project more than one quarter of their diameters, they will not occupy near so much room as attached three-quarter columns. Pilasters are also frequently introduced, with great propriety, in exterior decorations; and, very frequently, to avoid super- fluous expense. The effects of this may be observed in many of the splendid mansions of the nobility in the metropolis, and in many other towns; at other times pilasters accompany columns, being placed beyond them, to support the springing of the architraves, as in the Corinthian portico of Saint Martin's Church, London, in which instance the pilasters are continued all round the building, and in the intervals between the several loggia porticos, as well as from the latter, up to the grand portico in front, allowed to be the most perfect of the description in the British Metropolis. Blondel also says, that pilasters may likewise be em- ployed instead of columns, to form porticos; but, among the Roman antiques, examples of this sort are not to be found. The Choragic Monument of Trysallus, at the foot of the Acropolis, may, however, be considered as a Grecian example, a fac-simile of which, as regards proportions, may be seen at the entrance to the Rev. Mr. Belsham's chapel, in Essex Street, leading from the Strand. THE FIVE ORDERS. 513 The proportions of the monument adverted to are very much and justly admired; but, as regards the square detached pillar, or pilaster, in the centre, it cannot receive the approbation of scientific men when applied as a portico round an entrance, the aperture of which is in the centre of the portico applique. When pilasters are introduced as chief ornaments in compositions, they should always project at least one quarter of their diameters beyond the walls, as Sca- mozzi teaches, and as they have been executed by Inigo Jones, at Whitehall ; which projection produces that degree of boldness requisite in buildings of a certain standard, and in the Corinthian and Composite orders is also more regular: and, because the stems of the volutes, and the small leaves in flank of the capital are then cut exactly through their centres. But, if 'the cornices of the windows should be continued in the inter-pilasters, which is sometimes the case ; or, if there should be cornices to mark the separations between the principal and second stories, as at the Mansion House, in London ; or large imposts of arches, the projections must then, and in such cases, be increased; provided they are not sufficient to stop the most prominent parts of such decorations, it being extremely offensive to the judicious to see several of the uppermost mouldings of an impost or cornice cut away perpendicularly, in order to make room for the pilaster, while the cornice or impost on each side projects considerably beyond it ; as is the case in many public buildings of great notoriety. Mutilations should, on all occasions, be studiously avoided, as being destructive of perfection, and strong indications either of inattention or ignorance. When pilasters are placed behind columns and very near to them, they should not project above one-eighth of their diameter, or even less, unless there should be imposts, or continued cornices, in the inter-pilasters ; in which case, whatever has been before observed must be particularly attended to. Where flutings are required to the shafts of pilasters, the same proportions should be followed as in the same ornaments to columns. 64. G o 514 THE NEW PRACTICAL BUILDER, &C. 100 .KiotraBf 'f.sa ,te..! at baflilfiiJ; Bgntbiir ~ iar OOIV198 Ifi9t 6 , hhfOV/ Tfo styte ^usnifa 1 * .'nBicfm^nto fidt oi rf^iiduq ,9cniJ 30? ON THE I'tsiJ' ANTIENT ARCHITECTURE OF GREAT BRITAIN, RECOGNIZED AS ^ THE GOTHIC, SAXON, AND NORMAN MODES OF BUILDING. cj ', \j ni clA gnijDS8Txni PROBLEM 2. Given the seat of a line, its intersecting point, the angle it makes with its seat, and the centre and distance of the picture ; to find the vanishing point and the representation of that line. Let DE be the seat of the line, D its intersecting point, and S the centre of the picture. Draw DC, making with DE an angle, equal to that which the original line makes with its seat. Draw SV parallel to __ "T / " \ "'"" j to DE, and SO perpendicular to SV, making SO equal to the AX: 1 distance of the picture. Draw OV parallel to DC, meeting ^^ SV in V, then will V be the vanishing point, OV its distance from the point of sight, and DV the representation of the line proposed. n; t nofta DEMONSTRATION. Imagine the planes OVS and CDE to be turned round on the lines SV and DE as axes, till they become perpendicular to the picture ; then will O be the point of sight, and DC the original line parallel to OV. V is its vanishing point (Def. 14), and, consequently, DV is the indefinite representation. Q.E.D. COR. Conceive DC to be the original line laid on the picture, by turning the plane CDE round the line ED. The representation of any part of it, as AC, may be found, by drawing AO and CO as visual rays, intersecting DV in the points a c ; as the points a c depend only on the parallelism and porportionality of the lines OV and DC. For aV : D :: VO : DA, and cV : cD :: VO : DC. These analogies arising from the similarity of the triangles VO, a DA ; as also cVO and cDC. PROBLEM 3. Given the representation of a line and its vanishing point, to find the repre- sentation of a point, whose original divides the original line in a given ratio. PERSPECTIVE. 525 Let AB be the representation of the line whose division is required, and V its vanishing point. Draw, at pleasure, VO, and ba parallel to VO. Through any point, O, in the line VO, draw OA and OB, intersecting ba in a and b. Divide ab in c, in the given ratio, and draw Oc, intersecting AB in C ; then will C be the representation required: the original of BC being to the original of CA as be to ca. DEMONSTRATION. OV being parallel to ba, ba may be considered as the original line, and OV as its parallel; and, consequently, O as the point of sight, and O, bO, cO, as visual rays, generating the points ABC. PROBLEM 4. Given the representation and vanishing point of a line, together with a point in that representation, to find another point in the representation, which, with the one given, shall intercept a line, whose original shall have a given ratio to the original of the given representation. Let AB be the given representation, V its vanishing point, and C the point given, which, with another point to be found, inter- cepts the required portion of AV. Draw VO, at pleasure, and let ad be drawn parallel thereto. From any point, O in VO, draw OA, OB, OC, meeting ad in the points a, b, and c. Make cd to ab, as the part represented by AB is to the required intercept, and draw Oc?, cutting AV in D ; then will D be the required point. For the original of CD is to the original of AB as cd is to ab. DEMONSTRATION . If OV be conceived as the vanishing line of a plane, passing through the original of AV, ad, being parallel to it, may be considered as the representation of a line parallel to the picture (Cor. 2, Theor. V.) ; and, therefore, its parts, ab and cd, will have the same ratio as their originals. (Theor. IV.) But, because of the vanishing point, O, the originals of O, Ob, Oc, and Od, are parallel (by Cor. I, Theor. III.) ; wherefore, the original of CD is to the original of AB as edtoab. Q.E.D. 66. 6 R 526 THE NEW PRACTICAL BUILDER, &C. sdJ ad Iliw OOl bria 3HI OJ 9A *te ;>'<) 528 THE NEW PRACTICAL BUILDER, &C. a, of these indefinite representations will be the representation of the original point A ; and, in the same manner, the representation, b, of the original point, B, may be found, and the figure completed, as before. DEMONSTRATION. Suppose the plane of the picture EF to revolve on the line EF, and the plane HOG on the line HG, as axes, so as to be parallel to each other, and both elevated above the original plane, which is comprehended between the directing and inter- secting lines HG and EF. Then O will be the point of sight, F the intersecting point of the original line AB, and G its directing point ; but, whatever be the angles which the plane of the picture and the directing plane make with the original plane, they will always be parallel. Therefore, Fa will be the indefinite representation of AB, (Theor.V,) and E the indefinite representation EA; and, consequently, the point a, is the representation of the point A. PROBLEM 7. The same data being given, as in Problem 6, to find the representation of any figure in the original plane. Example 1. Let it be required to find the re- presentation of the pentagon KLMNP. Draw OG, OH, OI, and OV parallel to KL, LM, MN, and KP, meeting HG in the points G, H, I, H < and V, which are the vanishing points of the lines KL, LM, MN, and KP. Produce KL, LM, MN, to their intersecting points, Q, R, and T. Draw QG, RH, and TI, inter- secting each other in the points I, m, which are the representations of the points LM. Join OK, ON, intersecting the indefinite representations QG, TI, in the points k and . Draw &V, which is the indefi- nite representation of KP. Lastly, draw OP, intersecting k\ inp; and pis the representation of P. The representations of curve-lined figures, are obtained by finding the repre- sentations of a sufficient number of points, and joining them neatly by the hand. PERSPECTIVE. 529 A if' OOH ayodte Example 2, by a Vanishing Point. Let FV be the vanish- ing line, GE the intersecting line, O the point of sight, and the original figure, ABC, a circle. Then the representation of any point, A, may be found thus. Through A draw AD, at pleasure, meeting GE in D ; and draw OV parallel to AD, meeting FV in V. Join DV and G AO, intersecting each other in a ; then a is the representa- tion of the point A. In finding the representations of the other points in the ^~^ circle, much labour may be saved by drawing lines through all the points parallel to AD, for then one vanishing point will serve for ascertaining the representations of as many points in the original figure as are necessary. Or, by a Directing Point, thus: Let VF be the directing o line, O the point of sight, and DG the intersecting line, AEB /\\ the original plane. Draw AV, at pleasure, meeting DG in D, and VF in V. Join OV, and draw Da parallel to VO, and let OA intersect Da in a. Then a is the representation of A. And as many points as may be thought necessary may be found in the same manner ; but much labour will be saved by employing the same vanishing point for all the points in the circumference. In this case, the indefinite representations will be parallel to Da. PROBLEM 8. To find the representation of any figure in a plane parallel to the picture. The representation being similar to its original (by Cor. 2, Theor. IV,) we have only to find the representation of one line of the original figure, and on that line, as a side, construct a figure whose homologous sides shall have the same ratio as those of the original figure. PROBLEM 9. Given the intersecting and vanishing lines, the centre and distance of the vanishing line, with the representation of a figure. To find the original of that figure, whose representation is given : 66. 6 s 530 THE NEW PRACTICAL BUILDER, &C. Let it be proposed to find the original of the pen- tagon klmnp. Produce the sides till they meet the intersecting line, in the points Q, R, T, and the va- nishing line in the points G, H, I ; and produce kp to its vanishing point, V. Draw OG, OH, OI, OV, and QK, RM, IN, parallel to the first three respec- tively, meeting in L and M, which will be the originals of / and m. Draw Ok and On, which will meet QL and TM in the original points, K and N. Then draw KP parallel to OV ; and let Op be also drawn, meeting KP in P, which is the original of p. Lastly, draw NP, and we shall have KLMNP the original of klmnp. This problem being the reverse of Problem 7, the truth of the construction is manifest. PROBLEM 10. The same data being given, to find the original of one representation only. Produce I, II, to its vanishing point V ; and draw VO. In the vanishing line take V3 equal to VO. Join 31, 311, which produce to the intersecting points, 1, 2. Then will 1, 2 be the length required of the original I, II. PROBLEM 11. Given the vanishing line, its centre, and distance, with the representation of a line. To find the representation of another line, so that the originals of the two lines shall contain a given angle. Let Gl be the vanishing line, O the point of sight, and ab the representation given. Produce ab to its vanishing point, G. Join GO, and make the angle GO1 equal to the given angle, and draw lea, which will be the line required. DEMONSTRATION. This is evident from Def. 14, and from Theor. Ill ; for the radials of two original lines which form an angle, make an angle equal to that of the originals. Then bac is the representation of that angle. PERSPECTIVE. 531 PROBLEM 12. Given the vanishing line, its centre, and distance, with the representation of one side of a triangle, whose species is given. To find the representation of the whole triangle. Let ab be the given representation, which pro- duce to its vanishing point, at G. Join GO, and make the angle GOI equal to the angle, which the side of the original triangle, whose representation is given, makes with the other side, terminating in the same point. Draw OH to make an angle equal to the obtuse angle of the triangle. Join Hb, which produce to c; then will abc be the representation required. PROBLEM 13. Given the vanishing line, its centre, and distance, with the representation of one side of any figure. To find the representation of the whole figure. Resolve the figure into as many triangles as it has sides, by diagonals drawn from the nearest angle to all the other angles ; then find the representations of these triangles one after another. This may be done otherwise, by the application of the preceding problems. PROBLEM 14. Given the centre and distance of the picture, with the vanishing line of a plane, to find the vanishing point of lines perpendicular to that plane. A 3 Let AB be the given vanishing line, and C the centre of the pic- ture. Through C draw AD, perpendicular, and CO parallel, to AB. Make CO equal to the distance of the picture. Join AO ; and draw OD perpendicular to AO. Then D is the vanishing point of the lines required. DEMONSTRATION. Suppose the triangle AOD to be turned round AD, as an axis, until its plane becomes perpendicular to the plane of the picture. This done, the plane, passing 532 THE NEW PRACTICAL BUILDER, &C. through the point of sight O, and the vanishing line AB, will be parallel to the original plane; and the line OD will be perpendicular to this plane, passing through the point of sight, and, consequently, will be parallel to lines which are perpendicular to the original plane. Therefore D is the vanishing point of these lines. COR. The original plane is perpendicular to the picture, when the vanishing line AB passes through its centre. In this case, the vanishing point D will be infinitely distant, and the representations sought, perpendicular to AB. PROBLEM 15. Given the centre and distance of the picture, with the vanishing point of pa- rallel lines. To find the vanishing line of a plane, whose original is perpendicular to the original parallel lines. Let C be the centre of the picture, and D the vanishing point of parallel lines. Through C draw DA, and let CO be perpendicular to DA. Make CO equal to the distance of the picture. Join DO ; draw OA per- pendicular to DO ; and, through A, draw AB parallel to CO. Then AB will be the vanishing line required, A its centre, and OA its distance. This construction necessarily follows from the preceding problem. PROBLEM 16. Given the centre and distance of the picture, with the vanishing line of parallel planes. To draw through a given point another vanishing line of a 'plane which is perpendicular to that plane, whose vanishing line is given; and to find the centre and distance of the vanishing line thus drawn. Let AB b the given vanishing line, C the centre of the picture, A__ , B and E the given point. Find (by Prob. 14.) the vanishing point D, of lines perpendicular to the original plane, whose vanishing line is AB. Join DE, which is the vanishing line required. Draw CF per- pendicular to DE, meeting it in F, and F is the centre of the vanish- ing line DE (by Theor. I). PERSPECTIVE. 533 With CF, as a base, and the distance of the picture as a perpendicular, draw a right-angled triangle ; then the hypothenuse is the distance of the vanishing line required. DEMONSTRATION. Because the plane, whose vanishing line is required, is perpendicular to the other, whose vanishing line is given, its vanishing line must pass through the point D. Q.E.D. PROBLEM 17. Given the centre and distance of the picture, the vanishing line of the common intersection of two planes, inclined at a given angle, and the vanishing line of one of them, to find the vanishing line of the other. Let C be the centre of the picture, BG the given va- nishing line of one of the planes, B the vanishing point of their common intersection, and H the angle of their inclination. Find the vanishing line, GD, of planes perpendicular to the lines whose vanishing point is B, (by Prob. 15,) and let that vanishing line cut the vanishing line given in G. In GD find the point E of lines, making a given angle, H, with the lines whose vanishing point is G, (lay Prob. 11,) that is, in BCF, which is perpen- dicular to GFD ; make FP equal to the distance of the vanishing line, GD ; and draw PG and PE, making the angle EPG equal to H ; and draw BE, which is the vanishing line required. DEMONSTRATION. Draw CA perpendicular to BF, and equal to the distance of the picture, and join AF and AB ; then will AF be equal to FP. Imagine the triangle, BAF, to be turned on BF, until its plane be perpendicular to the picture. Imagine, also, the triangle, GPE, to be turned round GE, until FP coincide with FA; and the planes BPG, DPB, will be parallel planes of the originals, whose vanishing lines are BG, GD, DB. Q. E. D. 67. 6 T 534 THE NEW PRACTICAL BUILDER, &C. EXAMPLES. In the application of Perspective to figures in the original plane, to save repeti- tion in describing the diagrams, VL is the vanishing line, IN the intersecting line, C the centre of the picture, in the vanishing line VL. CP is drawn perpendicular to VL, and equal to the distance of the picture. The space below the intersecting line, IN, is supposed to be the original plane. The space above the vanishing line, VL, may either be considered as a continuation of the picture upwards, or the plane which passes through the eye parallel to the original plane. Ex. 1. To find the representation of a point A. (Fig. 2, Plate V.*) In CP make PD equal to the height of the eye. Draw A/ towards D, meeting IN in/. Through A draw any line, AI ; and draw Pa parallel thereto, meeting the vanishing line in a ; and join la. Draw a perpendicular to IN, from/, to meet la ; and the point of intersection will be the representation of the original point A. Here the point D is used as a directing point ; and where it appears in the subsequent diagrams, it is used for the same purpose. Ex. 2. To find the vanishing point of an original line, AB. (Fig. 2.) Draw Pa parallel to AB, meeting the vanishing line in a; then a is the vanishing point required. Ex. 3. To find the intersecting point of a given line, AB. (Fig. 2.) Produce BA to meet the intersecting line in I ; then 1 is the point required. Ex. 4. To find the indefinite representation of the line AB. (Fig. 2.) Find the vanishing point a, (as in Example 2,) and the intersecting point d, (as in Example 3.) Join ac?, and ad will be the representation required. Ex. 5. An original line, AB, (Fig. 2,) a point in that line, and the indefinite representation, ad, being given ; to find the representation of the point Draw A/ towards D, meeting the intersecting line in/, and draw/ perpen- dicular to IN, meeting the indefinite representation at? in a; then a is the repre- sentation of the point A, as required. Hence we may find the representation of a line, AB, limited at both ends. Ex. 6. To find the representation of any plane figure. * This plate has been numbered V, by mistake. It is, properly, Perspective, Plate I. Hence, VI. should be II., &c. PERSPECTIVE. 535 Find the indefinite representation of all the sides, and the space enclosed by these representations will be the representation of the figure. Figure 3 shows the representation of a triangle : Jig. 4 the representation of a square ; and jig. 5 the representation of a hexagon. Ex. 7. To find the representation of a circle. (Fig. 6.) Find the representation, abed, of a, square, circumscribing the circle; draw the diagonal of the trapezium, which represents the square ; find the representation of two lines, touching the circle in E and F: and the representations, eandf, of these points, with the intersection of the two diagonals of the trapezium, is in the representation of the circle. Draw a line to each vanishing point of the square, through the representation of the centre, to cut or meet each line of the repre- sentation of the sides. Through the four points of the trapezium, and the two points e andy, describe an ellipse, which will be the representation required. Ex. 8, fig. 7, shows the representation of a circle, by drawing parallel lines through the figure, and finding the indefinite representation of these lines, with the representations of the points where the parallels cut the original circle. An ellipse being described through the points thus found, will be the representation of the circle. Ex. 9. Fig. 1, pi. VI, exhibits the representation of a rectangular block, as it would appear if placed on the ground, and also as it would appear if placed above the level of the eye. The heights are set off on the perpendicular,^, from the intersecting point, e. Ex. 10. To find the representation of a square pyramid. Let ABCD, (Jig. 2,) be the plan. Produce DA to its intersecting point e, and draw g/ parallel to DA. Find the representations of the sides AD and AB by the preceding examples. From the centre of the plan, g, draw a line to the directing point D; and from the point where this line cuts the intersecting line, IN, raise an indefinite perpendicular ; also, draw hf perpendicular to IN, and equal to the height of the pyramid. From h, draw a line to the vanishing point V, cutting the indefinite perpendicular in g; then g is the vertex of the pyramid ; and join gd, ga, andgb, which will complete the representation. Ex. 11. To find the representation of a frustum of a square pyramid,^-. 3. Find the representation of the entire pyramid, as in the last example ; and let 536 THE NEW PRACTICAL BUILDER, &C. efgh be a plan of the top of the frustum. Draw ef to its intersecting point m, and make mn perpendicular to IN, and equal to the height of the frustum. From , draw a line to the vanishing point V, which will give ef, the edge of the frus- tum; and from /draw, to the other vanishing point, L, the linefg, which com- pletes the frustum. Ex. 12. To find the representation of a cone,^. 4. Find the representation of the base, as in the example (fig. 6, pi. V.) ; and the representation of the vertex, as (in Example 10.) Then join om and on, which complete the picture of the cone. Ex. 13. Fig. 5, plate VI, is a representation of a wall, with a semi-octagon tower projecting from it. Towers of this kind are very common in Gothic buildings. Ex. 14. 'Fig. 6 shows an example of a round tower joining a straight wall. These two examples may be considered as exercises of the application of the rules already given. Ex. 15. To find the representation of a flight of returning steps. (Fig. 1, plate VII.) Produce the lines that form the boundary, and the lines that divide the steps on the plan to their intersecting points, e,f, g, and find the vanishing points of these lines. From the points e,f,g, draw perpendiculars to the intersecting lines. On each of these perpendiculars set the respective heights of the steps, and draw lines from the extremities of each to the vanishing point. Find the termination of each face by the perpendicular lines, as seen in the figure, and also the vanishing point of the other side. Draw lines to this point from the corners of the faces, which lines will represent the heights. Find the terminations of these faces ; then the returning lines being drawn to each vanish- ing point, will complete the representation of the steps as required. Ex. 16. Fig. "2 is the representation of a flight of steps, with kirbs at the ends. The perspective heights are found, by finding a section of the steps on the plane of the picture, and the lengths are found by the directing point D. Ex. 17. Fig. 3 is the representation of a square tower. The first thing to be done is to find a section of the face, as shown by a dark line. Set the heights of the door and windows upon this line, and draw lines from all the points of the section to the vanishing points of that side of the plan. By this mean, all the projecting PERSPECTIVE. 537 parts, as cornice and plinth, will be found. Then, having found the perspective breadth of the front, the terminations of the cornice, and breadth of the windows, the front will be finished. The return end will be found in a similar manner to the last two examples. Ex. 18. To find the representation of a tower and spire. Our first object is to find the representation of the rectangular part, and on this draw the perpendicular lines that represent the angles of the octagon. Produce DA, (fig. 4,) on the plan, to L, its intersecting point. Draw LU perpendicular to IN. Set off the height of the tower and pediment on LU. Draw lines from these heights to the vanishing point V. As the vertex of the pyramid is not in the same plane with the front, draw YP through the centre of the plan, parallel to the front line AD, meeting IN in P. Draw PR perpendicular to IN. Set the height of the spire from the ground on PR, and draw RV. Draw a line from Y to D, meeting IN ; and, from the point of meeting, draw a perpendicular to the intersecting line, and the point where it meets RV is the vertex of the pyramid. The sides of the spire are found by drawing lines from points in the lines, ad, ad, which represent the angles on the plan. Ex. 19. Fig. 5 represents an arcade, or range of arches. The pillars are drawn, like as many square towers, of a certain height. The arches are found by circumscribing a rectangle on each, drawing the diagonals, and making the heights of the points, r, q, m, I, double the heights of the arches ; then drawing lines to each of these points, from the chord line of each arch, to cut the diagonals. The elliptic heads will pass through these points, and through the points where the perpendiculars meet the top side of the representation of the rectangle. The circular heads of the inner face are drawn in the same way. The cornice and plinth are drawn as in the square tower, figure 3. Ex. 20. Fig. 6 exhibits the representation of a square tower, consisting of different stages. Of this nothing can be said more than what has already been explained, and what will be understood by the lines in the drawing. Ex. 21. Plate VIII shows the representation of a denticulated cornice. The profiles of the mouldings, in the representation, are found by means of the section, in dark lines, and the terminations, or lengths, are found as in the preceding examples. In this example the plan is above the representation ; an arrangement 67. 6 u 538 THE NEW PRACTICAL BUILDER, &C. which shows the relations of the lines in a more evident manner, and which, in some cases, is more convenient than placing it below. Ex. 22. Plate IX exhibits the representation of a house, with a cantailiver cornice. The heights of the mouldings and pediment are found from the line of heights, which is the intersection of a plane passing through the ridge of the roof: the heights of the steps may also be found from the same. The vertical lines which terminate the fronts, the breadth of the windows, and the chimney shaft, may be found from the plan, which is placed above the perspective drawing for the con- venience of drawing the perpendiculars, from the intersecting line, AD, to the several terminations of the representation. Through the oblique lines of the pediment may be drawn the two ends of each inclined cornice. They will, however, be more elegantly found by means of Problem 14. Ex. 23. Plate, Perspective Grecian Doric. This plate is a representation of the Grecian Doric capital and entablature in perspective, with a geometrical outline of the mouldings. It illustrates the use of perspective, in showing the manner of finishing the parts of this chaste and simple order, in a case where it is difficult to render it intelligible by geometrical drawings alone. A good perspec- tive representation has nearly the same advantages as a model in such cases. Ex. 24. Plate, Ionic Capital in Perspective. This is another example of the use of perspective in elucidating the disposition of parts, which are often not clearly understood by those who have not an opportunity of examining good specimens of the orders. Ex. 25. Plate, Corinthian Order. It is evident from the principles of Per- spective, that there is only one point from whence a picture can be seen in its true form, and that point is the point of sight. (Def. 3.) If the point of sight be made too near to the plane of the picture, the eye cannot take in all its parts at once; and, consequently, the picture cannot produce an agreeable effect as a whole. The representation is also distorted and unnatural, from the parts di- minishing too rapidly. The distance of the point of sight, from the plane of the picture, which is best suited to the power of the eye, is between two and three times the breadth of the picture. The distance most commonly adopted by artists of taste, is about two and a half times the breadth. PROJECTION. 539 ' .-. PROJECTION. THE most useful kinds of architectural drawing depend upon the Theory of Projection, and, consequently, its principles ought to form a part of that stock of knowledge which is essential to a student in architecture. Some of the principles of projection are so easily comprehended, that they are acted upon without previous study ; others are so difficult, that few are com- petent to apply them. For example, the plan of a building is a projection of it on a horizontal plane ; and an elevation of a building is its projection on a vertical plane. In these simple operations no difficulties occur ; but there are many cases which arise in carpentry, joinery, and masonry, where a profound experience in projection is required. To a workman, skill in projection is a great acquisition ; it enables him to form a clear conception of intricate forms, and to foresee how different parts will join or connect with each other. It enables him to understand drawings and designs with readiness, and to work to them with certainty and accuracy. The doctrine of shadows depends on the principles of projection, and the advantage of knowing how to shadow properly is so evident that we need not say more on the subject. But there is another application of the art of projection which is less generally understood, that is, as a mode of representing such objects as are always carica- tured in attempting to draw them in perspective. To this class of objects belong all small models, machines, pieces of furniture, and the like ; for such objects, projection is the most simple and convenient mode of representation. Maps and plans, of various kinds, are drawn by the rules of Projection ; and the use of these rules is extensive in many arts and sciences. We shall, however, confine ourselves to its principles and application to architectural subjects. 540 THE NEW PRACTICAL BUILDER, &C. DEFINITIONS. 1. If a perpendicular be let fall from a point to a plane, the place where the perpendicular meets that plane is the projection of that point. Hence, as lines, surfaces, and solids, may be conceived to be composed of points, they may be projected upon a plane. 2. A plane of projection is that plane on which the projection is to be made. It is also called the plane of representation. 3. When a projection is made on a horizontal plane, it is called the plan of the object. 4. When the projection is made on a vertical plane, it is called the elevation of the object. 5. When the projection exhibits an object, as it would appear if cut by a vertical plane, the representation is called a section. 6. A primitive plane is that which contains a point, a line, or a plane surface, of a given object. PROPOSITION. The projection of a straight line is a straight line. If the given straight line be parallel to the plane of projection, it is projected into an equal straight line ; but, if the given line be inclined to the plane of projection, the given straight line will be to its projection, as the radius is to the co-sine of the angle of inclination. Solution. Let a plane, perpendicular to the plane of projection, pass through the given straight line. The intersection of that plane, with the plane of projec- tion, will be a straight line, (Euclid, 3 prop., XI book,} and ( Def. 1,) the intersec- tion contains the projection of the given line. From each extremity of the given line, AB, (fig. 1, pi. I,) let fall a perpendicular to the intersecting line, I L ; then, the part of that line, intercepted between the perpendiculars A, Bi, is the projection of the given line. If the given line be parallel to the plane of projection, then, because the perpendiculars are parallel to each other, and the given line parallel to the intersecting line, the projection must be equal to the given line. If the given line be inclined to the plane of pro- jection, and AB, (fig. 1,) be that line, IL the intersecting line, and a b the projection intercepted between the perpendiculars A a and BJ. Draw BC pa- PROJECTION. 54,1 rallel to IL, and the angle ABC will be equal to the angle of inclination of the given line to the plane of projection. Now AB being the radius, BC, or its equal, ba, is the co-sine of the angle of inclination ; therefore AB is to a b as the radius is to the co-sine of inclination. COR. 1. The projection of a straight line, perpendicular to the plane of pro- jection, is a point. COR. 2. If several straight lines, having the same inclination to the plane of pro- jection, be projected, each of the originals will have to its projection the same ratio. COR. 3. A plane angle, parallel to the plane of projection, is projected into an equal plane angle. COR. 4. A plane angle, inclined to the plane of projection,is projected into an angle of which the sine is reduced in the ratio of the radius to the co-sine of inclination. COR. 5. Lines which are parallel in the original are parallel in the projection. COR. 6. Any plane figure, parallel to the plane of projection, is projected into an equal and similar figure. COR. 7. The area of any plane figure is to the area of its projection, as the radius is to the co-sine of its inclination to the plane of projection. COR. 8. The projection of a circle inclined to the plane of projection is an ellipse, of which the transverse diameter is equal to the diameter of the circle; and the conjugate diameter is to the diameter of the circle, as the co-sine of inclination is to the radius. PROBLEMS. PROBLEM 1. To find the projection of a point, situate in a plane inclined to the plane of projection in a given angle. Let IL (fig. 2) be the intersecting line of the two planes, and A the given point. From A draw A a perpendicular to IL. Make wVa equal to the angle of inclina- tion of the two planes, and Vw equal to VA. From w draw a line parallel to IL, intersecting Aa in a ; then a is the projection of the point A. Illustration. Conceive DLIC to be the plane of projection, and LEFI the plane, containing the point A ; and that these planes turn on the line I L as an axis. Also, let the triangle Vaw turn on the line aV, as an axis, till it be perpen- dicular to the plane of projection DI, and turn the plane El on IL, as an axis, till AV coincides with wV; then aw is obviously a perpendicular from the plane of projection to the given point ; and therefore a is the projection of A. 68. 6 x 542 THE NEW PRACTICAL BUILDER, &C. Remark. It is often necessary in practice to make all the projections of an object on the same sheet of paper, or on the same area ; therefore, we conceive all the planes to be spread out, or laid flat in one and the same plane ; and, when we wish to consider them in their true positions, we imagine them to revolve on their intersecting lines as axes. It is for this reason that Monge, a celebrated French author on this subject, very properly recommends that the intersecting lines should be drawn in a distinct manner. PROBLEM 2. To find the projection of a line situate in a plane, inclined to the plane of projection in a given angle. Let IL {figures 3 and 4,) be the intersecting line of the two planes, and AB the given line. From the extremities of AB draw Act and Eb, perpendicular to IL. Make bVw equal to the angle formed by the planes, and find the projection of the point B, (as in Problem 1.) Then, if the line be inclined to the intersecting line, (as 'm fig. 3,) produce AB, till it meets the intersecting line IL, in L. Join Lb, and ab will be the projection of AB. If the given line be parallel to IL, (as in fig. 4,) from b, found as before, draw a line, parallel to IL, meeting A a in a, and ab is the projection of AB. PROBLEM 3. To find the projection of a plane curve, situate in a plane, which is inclined to the plane of projection in a given angle. Let IL (fig. 5) be the intersecting line of the two planes, and ABCD points in the curve, the number of which may be increased at pleasure. Draw a line, ACL, to touch the curve at C ; and find the projection of this line, and of the points G, H, where it is crossed by lines drawn from D and B to V. Draw Dd, Cc, and B b, perpendicular to IL ; and from V draw hV, gV, produced to meet the perpen- diculars in d and b ; then the points a, b, c, d, are the projections of the points A, B, C, D, in the curve ; and a curve drawn through abed will be the projection required. Another Method. Let IG, (fig. 2, plate II,) be a line drawn in the same plane with the curve, and let HI be the projection of that line, found by Prob. 2. From any point, A, in the curve, draw a line parallel to IL, cutting it in k. Make kn and A a perpendicular to IL, the former meeting HI in n. From n draw a line parallel to IL; and the point a, where it meets A a, is the projection of the point A. The projections of any other points, B, C, D, E, being found in the same manner, the curve drawn through them will be the projection of the given curve. PROJECTION. 543 PROBLEM 4. To find the projection of a plane angle situate in a plane, which is inclined to the plane of projection in a given angle. Let I L (fig. 6, plate I,) be the intersecting line, and ABC the given angle. Determine the projections of the lines AB,AC, by Prob. II.; and the angle abc will be the projection of ABC. The projection of a plane triangle is found in the same manner, being completed by joining the points be, as shown by dotted lines in the figure. The projection of a parallelogram is also effected in a similar manner. For let the angle ABC, (fig. 7,) be projected by the last problem ; and draw ad parallel to be, and dc parallel to ab ; then abed is the projection of the parallelogram ABCD. PROBLEM 5. To find the projection of a pentagon, situate in a plane which is inclined to the plane of projection in a given angle. Let the pentagon, ABCDE, (fig. 8,) be described by Problem 13, of Practical Geometry, and project the side AE by Problem II. Produce ED, AD, BD, and BC, to the intersecting line ; and from the points of intersection to a, draw ac and ad; also, from the points of intersection, draw db parallel to ea, ed parallel to ac, and be parallel to ad. Join ab ; and the figure, abcde, is the projection of the pentagon ABCDE. PROBLEM 6. : To find the projection of a circle. Let IL (fig. I, plate II,) be the intersecting line, and HI the projection of any line, GI. Perpendicular to I L draw lines, Aa, Dd, &c. from as many points in the circle as may be considered sufficient ; and from the same points draw lines parallel to I L, to meet the line GI. From the points where these parallels meet GI, draw lines perpendicular to I L to meet I H ; and parallels to I L, drawn from the points of intersection in IH, will meet the perpendiculars A a, DC?, &c. in the points a, d, b, &c. in the projection, corresponding to the points ADB, &c. in the circle. Since the projection of a circle is an ellipse, (Cor. 8,) it will be sufficient to find the projection of the diameters, AC and BE ; and on these diameters describe an ellipse by any of the methods given in our Practical Geometry. (See Prob. D, p. 71.) PROBLEM 7. To find the projection of a line perpendicular to a plane, of which the position is given. Let I L be the intersecting line, (fig. 9, plate I,) and A the point in the given plane, to which the line is perpendicular. Draw A b perpendicular to IL, and make aVw equal to the angle the given plane makes with the plane of projection; 544 THE NEW PRACTICAL BUILDER, &C. also make \w equal AV. Let BD, drawn perpendicular to Vw, be the given line ; and from B and D draw lines parallel to 1 L, cutting A b in the points b and d ; then bd is the projection of the line BD. PROBLEM 8. To determine the projection of a line, which is inclined in a given angle, to a plane, of which the position is given. Let AB (fig. 10,) he the given line, and BAG its inclination to the primitive plane. Make BC perpendicular to AC, and AC is its projection on the primitive plane. Let cVw be the angle, which the primitive plane makes with the plane of projection, and find the projection of the line AC, by Problem II. Make wx perpendicular to wV, and equal to BC ; and from x draw a line parallel to I L, meeting the perpendicular Cb in b. Join ab, and it is the projection of AB. PROBLEM 9. To determine the projection of a triangular pyramid, (fig. 11.) Let the pyramid stand upon a plane, which is inclined to the plane of projec- tion, in the angle oVw, and let xy be its perpendicular height, and ABC the plan of its base. Project the plan of the base, by Problem 4, and draw Drf perpen- dicular to IL. From x draw a line parallel to IL, cutting DC? in d; then supposes the primitive plane to coincide with the base of the prism ; but the procesa is the same in both cases. Let ABCD represent the end of the prism which is given, and find its projec- tion, abed, by Problem 4. Also, find the projection by, of the angle or arris of the prism, by Problem 7, XY being the height. From the point y, draw lines parallel to be and ba, and complete the parallelogram yxwv, in fig. 3. PROBLEM 11. To determine the projection of an oblique prism, (fig. 5.) Let EGH be the inclination of the arris of the prism to its base, and BF the projection of that arris on the primitive plane. Find the projection of the base, ABCD, by Problem 4, and the projection of the arris by Problem 8, and complete the representation, as in the last Problem. PROBLEM 12. To determine the projection of a right cylinder, (fig. 6.) Find the projection of the base by Problem 6, and the projection of its altitude by Problem 7, as indicated by the figure. SHADOWS. 545 SHADOWS. THE Theory of Shadows is founded on the supposition that light is propagated in straight lines. This supposition is not strictly true ; but it does not sensibly differ from the truth in any case where we have occasion to apply it in finding shadows. As shadows from artificial lights are seldom introduced in architectural draw- ings, we propose to confine our rules to those produced by the sun ; and the sun's rays may be considered parallel to one another in consequence of its immense distance, compared with the distances of any objects on the earth's surface. DEFINITIONS. 1. Those parts of a body which receive the direct rays of the sun are said to be in light. 2. Those parts of a body which do not receive the direct rays of the sun are said to be in shade. 3. That part of a surface which is deprived of light, by another body inter- cepting the sun's rays, is said to be in shadow. The doctrine of shadows has two objects, vi. to determine the boundary of light and shade, and to find the form of the shadow. In architectural drawings, the breadth of a shadow is usually made equal to the depth of the projection which produces it ; and an adherence to this simple rule has several advantages, besides its convenience in application ; for, when it is attended to, the real quan- tities of projection or recession are shown by a shadowed elevation, rendering it at once ornamental and useful. But though the shadows are equal to the projections, and the drawing is said to be shadowed at an angle of 45, the inclination of the sun's rays to the plane of the horizon is only 35 16' ;* and it is the projection of the direction of the sun's rays against the vertical plane which make an angle of 45 with the horizon. * For the radius is to the tangent of the sun's inclination as ,y/2 : 1. Hence, the radius being made unity, the tangent is 7= ; and its logarithm 0-849485, which corresponds, nearly, to the tangent of 35 16'. 68. 6 Y 546 THE NEW PRACTICAL BUILDER, &C. EXAMPLES. Ex. 1. To find the shadow of a small rod projecting at right angles, from a vertical plane. Let b (fig- 1, plate IV, Shadows,') be the point in the plane from which the rod projects, and a A its plan; ZX being the base line, or intersecting line of the plan and vertical plane. From A, draw A c in the direction of the sun's rays, (which is 45 in these Exam- ples,) and raise the indefinite perpendicular cd. From 6 draw bd, in the projected direction of the sun's rays, (which is also 45 in these Examples,) intersecting cd in d; then bd will be the shadow of the rod. Ex. 2. To find the shadow of a vertical plane, situate at right angles to the vertical plane on which it forms the shadow. Let AB, (fig. 2,) be the plan of the plane, and ab its elevation. Draw Be, Ad, in the direction of the sun's rays ; and from c and d raise vertical lines. From b draw a line in the projected direction of the rays, cutting the vertical lines in e and,/; and cefd will be the boundary of the shadow against the vertical plane. Ex. 3. To determine the shadow which a rectangular plane will form against a wall, when the plane is inclined to the wall. Let AB, (fig. 3,) be the plan of the plane, and aakb its elevation. Draw Ac, B d, parallel to the direction of the sun's rays, and raise vertical lines from c and d. From a and h draw lines parallel to the projected direction of the rays, cutting the ver- tical lines in e,f. Join ef, and cefd is the boundary of the shadow on the wall. Ex. 4. To find the shadow projected, by a square pillar, against a wall. From A, B, and C, on the plan of the pillar, draw lines, parallel to the direction of the sun's rays, to the base line ; and from the points thus found, in the base line, raise vertical lines. Draw lines parallel to the projected direction of the sun's rays from the points g, h, in the elevation, meeting the vertical lines in n, m, and s. Join nm and ms, which determine the boundary of the shadow. The shadow of a triangular prism, (fig. 5,) is found by the same process, as will be obvious from the lines on the figure. Ex. 5. To find the shadow, projected against a wall, by a square pillar, with a square abacus or cap, and also the shadow of the cap against the pillar, (fig. 6.) Draw lines, parallel to the direction of the sun's rays, to the base line, from the points A, D, and C, in the plan of the cap ; and from the points E, H, in the plan SHADOWS. 547 of the pillar, extending the lines which pass through the points H and E, till they meet the plan of the cap. From each of the points thus found in the base line, draw indefinite vertical lines. Draw a vertical line from the point L, and also from the one adjoining E, to meet the under edge of the cap. From these points in the under edge of the cap, and from the angles a, d, e, r, draw lines parallel to the projected direction of the sun's rays, which will meet the verticals in the points m,p, o, &c. Join these points, and the boundary of the shadow against the wall will be obtained. From the point where the projected direction of the rays, from the under ed ge of the cap, cuts the angle e of the pillar, draw a line parallel to the edge of the cap, then join the point where this parallel meets the angle,,/, with the point where the ray, from the part of the under edge of the cap, corresponding to L on the plan, meets the angle h, and the boundary of the shadow of the cap will be determined. Figure 7 shows the lines for projecting the shadow of a horizontal cross, resting upon a square pillar ; and fig. 8 exhibits the shadows of an entablature, supported by four square pillars. Ex.6. To determine the shadow which a cylindrical abacus, or cap, casts upon its column; and also the shadow projected against a wall by the column and cap, ./Eg*. 9. Let a line, perpendicular to the direction of the sun's rays, be drawn through the centre of the column on the plan ; then, if vertical lines be drawn from the points where this perpendicular cuts the outlines of the column and cap, such vertical lines will determine the boundaries of light and shade ; and, on the light side of these boundaries, the form of the shadow will be determined by the under edge of the cap ; and on the shade side by the upper edge of the cap. To find the shadow of any point, E, of the edge of the cap against the column, draw EG parallel to the direction of the sun's rays, and draw a vertical line from E, to meet the under edge of the cap ; and one from G, upon the surface of the column. (This line is omitted in the figure.) Then, from where the vertical from E meets the edge of the cap, draw a line parallel to the projected direction of the rays to meet the vertical from G, in the point g, and g will be in the boundary of the shadow. In the same manner, several points in the boundary may be deter- mined, and the line of shadow drawn through them. Again, to find any point in the boundary of the shadow against the wall, as for example, the point C, draw C d parallel to the direction of the light ; and from d 548 THE NEW PRACTICAL BUILDER, &C. draw an indefinite vertical line. Since C is on the shade side, the shadow will be cast by the upper edge of the cap ; therefore, draw a vertical line from C, meeting the upper edge in c\ and from c draw a line parallel to the projected direction of the rays of light, meeting the vertical from d in the point e ; then e is in the boun- dary of the shadow, and any other points may be found in the same manner. The shadow projected by a square cap upon an octagon pillar is shown by fig. 10 ; also the shadow formed on a wall by the pillar and its cap. Figure 11 exhibits the shadow cast by a square abacus on a column, and the shadow they form on a wall. In both these figures the shadows are determined, as in the preceding examples. Ex. 7. Fig. 1, plate III, represents the shadow projected against a wall by a balcony, with its cantalivers. ABCD is the plan, and ab the elevation of the balcony, and IL the base line. From the points A, K, E, and F, draw lines parallel to the direction of the rays of light to intersect the base line, and raise indefinite perpendiculars from each of the points of intersection. Also, from the points a, k, e,f, g, and h, in the eleva- tion, draw lines parallel to the projected direction of the rays of light, meeting the indefinite perpendiculars in the points m, n, o,p. Draw ns and op parallel to the base line ; and when the shadow q r of the other cantaliver is found, by the same method as the one which is described, the boundary of the shadow on the wall will be determined. The shadow against the end of the cantaliver is found by drawing FG parallel to the rays of light ; and the ray, from the corresponding point g, in the elevation, meets the arris,,/, of the cantaliver, in the boundary of the shadow, which is pa- rallel to the edge k h of the balcony. Ex. 8. Figure 2 shows the shadows projected by a pedestal upon a flight of steps, both in plan and elevation. A is the elevation, and B the plan of the flight of steps. From the points C an the plan, and c in the elevation, draw lines parallel to the direction of the rays of light, intersecting the lines of the steps in/, e, d, and E, D. The vertical dotted lines from these points determine the boundary of shadow on the plan, and on the elevation, as will be evident from the figure. Ex. 9. Figure 3 represents the shadow projected by a pediment against a wall. IL is the base line, with the plan of the pediment below it, and the elevation SHADOWS. 549 above it. It will be obvious that the shadows of the arrises of the pediment, which meet in the point h, are bounded by lines parallel to those arrises meeting in n ; but when the projection of the upper members of the cornice is such that the arris $ s is in shadow, the shadow of the under edge of the fillet g must be found, which is mr in this example. The other parts of this example being similar to the preceding ones, it is unnecessary to repeat the construction. Ex. 10. To find the shadow of a rectangular niche, with a semi-circular head. ABCD (fig. 5,) is a plan of the niche, and adits elevation. Find the shadow m of the point e, the centre of the arch ; and from m, as a centre, describe the arc op, with the same radius as the arch. The rest of the process is evident from the figure. Ex. 11. In a portion of a pilaster, to find the shadow of the flutes in plan and elevation. From the centre O, (fig '.4,) draw OG perpendicular to the direction of the rays of light, then OG is the line of light and shade, and the vertical Gg gives the point #, from whence the shadow in the elevation commences ; and the shadows of the points, a, d, e,f, may be found in this manner, taking the point e for example. Draw Ez and ek parallel to the directions of the rays; and from i draw the vertical ik, meeting ek in It ; then Ic is the shadow of the point e. The shadow of the arris ah, is a portion of an ellipse cm', for the section through AC, in the plane of the arris, is a circle, and its projection is an ellipse, of which ch is the conjugate, and AC the transverse diameter; and if the ellipse be described on its conjugate ch, the line hm, parallel to the direction of the light, will cut the ellipse in the point where the shadow quits it. If era be drawn perpendicular to the direction of the rays, n is the termination of the shadow ; and it may be shown that the shadow mn, of the arc hn, is also a portion of an ellipse; but a more general method Is somewhat easier in practice, besides being applicable to any species of curve. Let the shadow of any point, v, be required ; q being the corresponding point on the plan. Draw qt parallel to the rays of light; and let tsr be a section through tq ; which, in this example, is an arc of a circle, of which w is the centre. From r, draw rs, to make an angle of 35 16' with qt; then a perpendicular, from the point * to qt, gives the point x, through which the shadow passes on the plan ; and if a line be drawn from v, parallel to the direction of the rays of light, 69. 6 z 550 THE NEW PRACTICAL BUILDER, &C. the point, where a vertical line from x meets that line, will be in the boundary of the shadow in the elevation. Ex. 12. Figure 6 represents the shadow thrown against the back of a circular niche by its arris. ABC is the plan, and acb the elevation of the niche, and d the centre. Draw d6 perpendicular to the direction of the rays of light, then the point 6 is the termi- nation of the shadow. From each of the points, A, 1, 2, and 3, on the plan, draw lines to the back of the niche, in a direction parallel to the rays of light ; and perpendiculars, from the points where they meet the back, will meet the directions of the rays from the corresponding points, a, 1, 2, 3, in the elevation, in the boundary of the shadow. To find the shadow of any other point, as 5, letyo be drawn parallel to the direction of the light, andfr perpendicular tofo, passing through the centre D ; and 5m parallel iofr. From the pointy describe the arc om ; and make mn inclined to of, in an angle of 35 16'. Draw np parallel iofr, and the vertical, from the pointy, will meet the direction of the light, from the point 5 in the elevation, in the boundary of the shadow. Any other points in its boundary may be found in the same manner. The shadows on the sections of domes, groins, and the like, are easily found by the same methods. Figure 1 shows the shadow on the section of a circular room, with a level ceiling. Having shown the methods of finding the shadows produced by the direct rays of the sun, the management of shadows, cast by reflected light on the parts of bodies in shade, will easily be obtained. The light must be considered to proceed from the reflecting surface, and the depth of shadow should be in proportion to the quantity of light it reflects. The direction of the reflected light may be determined from the well-known optical principle, that whatever angle the sun's rays make with the reflecting plane, they will be reflected in an equal angle from the plane in the opposite direction. Hence the shadows, from reflected light, are usually the reverse of the shadows produced by the direct rays of the sun ; that is, if the one be cast downward the other will be cast upwards. We shall close this part of our subject with recommending the student to study from nature. His knowledge of the geometrical description of shadows will aid him in his researches, and nature will offer him new examples to exercise his skill in geometry. DECIMAL FRACTIONS. 551 APPENDIX. DECIMAL ARITHMETIC, &c. DECIMAL FRACTIONS. AT what time, or by whose ingenuity, Decimal Arithmetic was first introduced, is a subject quite unknown ; but the perfection which it has now attained is, doubtless, owing to modern times. In Decimal Fractions, the integer or unit, (whether it be a unit of time, of weight, or of measure,) is sup- posed to be divided into ten equal parts ; and each of those parts is again supposed to be sub-divided into ten equal parts, and so on to infinity, according to the powers of ten. The integer thus divided is to be considered as the numerator of a fraction, while 10, and its successive powers, compose the denominator. Thus -j^,, -s-J^, - t 7 ga , Telou) & c - to infinity. But, in dividing by one with any number of cyphers annexed, it is usual to cut off from the dividend as many places towards the right as there are cyphers in the divisor ; therefore, since the denominator of a decimal fraction is always one with some determinate number of cyphers annexed, it may be rejected in every case, and a point or period used in place of it : thus, -^ may be denoted by -7, and i-J-^ by '07. Hence it appears, that cyphers placed to the left of a decimal fraction, decrease its value exactly in the same proportion that cyphers placed to the .right of whole numbers increase their value ; that is, in a porportion rising by the successive powers of 10. The following Table will exhibit the relation between the integral and fractional scales. Whole Numbers. 1 Decimal Fractions. H H ffl H | | i d s 3 2? 5 S % 1 S 3 S 2 i 1=L s, s, 2> 2, 2, o 1 H p P H H- B r a K H * t> d" o 3, o H H E B 00 B- tr O &* 3" 1 3 i o B p- I 5432 1 1 2 3 4 5 6 From this it appears that Decimal Fractions are really more like whole numbers than Vulgar Fractions are ; and the various processes to be performed on them are precisely the same, the place of the point, or period, that marks the fraction being the only difference to be attended to. ADDITION OF DECIMALS. RULE. Place the numbers under each other, according to the value of their places. Find their sum, as in whole numbers, and point off as many places for decimals as are equal to the greatest number of decimal places in any of the given numbers. EXAMPLES. 1. Find the sum of 25-636+7-0625+32-125+-006325+-754-11-010325. 25-635 7-0625 32-125 006325 75 11-010325 76-509150 Where the cypher on the right-hand side of the decimal may be omitted, as it does not alter the value of the fraction. 552 THE NEW PRACTICAL BUILDER, &C. 2. Find the sum of 376'25+86-125+637-4725+6-5-|-358-865+41-02. Ans. 1506-2326. 3. Find the sum of 3-5+47-25+927-01+2-0073+1-5. Ans. 981-2673. 4. Find the sum of 276+54-321+-65+112+1-25+-0463. Ans. 444-2673. SUBTRACTION OF DECIMALS. RULE. Subtract, as in whole numbers, and mark off the decimals, as in Addition. EXAMPLES. 1. Find the difference between 2464-21 and 327'07643. 2464-21 327-07643 2137-13357 2. Find the difference between 127'62 and 13-725. Ans. 113-895. 3. Find the difference between 603-5725 and 32-0012. Am. 571-5713. 4. Find the difference between -65325 and -0735. Ans. -57975. MULTIPLICATION OF DECIMALS. RULE. Multiply as in whole numbers, and cut off as many decimal places from the product as are con- tained in both factors. If there be not so many places in the product as there are decimal places in both factors, the deficiency must be supplied by prefixing cyphers to the left-hand side of the product. EXAMPLES. 1. Required the product of '325 and 32-5. 32-5 325 T625 650 975 2. Multiply -0375 by 33-75. Ans. 1-265625. 3. Multiply -63478 by -8204. Ans. -520773512. 4. Multiply -385746 by -00464./&W. -00178986144. DIVISION OF DECIMALS. RULE. Divide as in whole numbers ; and from the right-hand side of the quotient point off as many places for decimals, as the decimal places in the dividend exceed those of the divisor. If there are not so many places in the quotient, the deficiency must be supplied by prefixing cyphers to the left of the quotient. If there be a remainder, cyphers may be annexed to the dividend, and the division continued. EXAMPLES. 1. Divide 395-275 by 3-75. 3-75)395-275(105-406 375 2500 2250 250 Here the number 250 would recur at every step : hence the quotient figure would always be the same, and this kind of decimal is said to repeat. Whence the appellation of repeaters. 2. Divide 234-70525 by 64'25. Ans. 3-653. 3. Divide 217-568 by 1000. Ans. -217568. 4. Divide -408408 by 52. Ans. -007854. DECIMAL FRACTIONS. 553 REDUCTION OF DECIMALS. CASE 1. To reduce a vulgar fraction to a decimal of equal value. RULE. Multiply the numerator by 10, or its power, and divide by the denominator. EXAMPLES. 1. Reduce f to a decimal fraction. 2) 1-0 5 2. Reduce to a decimal fraction. 4) 1-00 25 3. Reduce to a decimal fraction. 5)1-0 2 4. Reduce } to a decimal fraction. 8) 1-000 "" Note, What number of cyphers more than one we have to annex before the division succeeds, so many cyphers must be placet on the left side of the first significant figure in the quotient. EXAMPLES. 1. Reduce -j^ to a decimal fraction. 16) 1-00 (-0625 96 40 32 80 80 2.^-Reduce ^J to a decimal. 200) 1-000 (-005 1000 Sometimes, in dividing, the same remainder successively arises, consequently the same figure must be successively obtained in the quotient ; when this is the case, the decimal is called a repeater ; when the repeater is not preceded by some figures that do not repeat, the decimal is called a pure repeater ; but if one or more figures precede the common figure, it is a mixt repeater. EXAMPLES. 1. Reduce f to a decimal fraction. 3)10 333, &c. a pure repeater. 2. Reduce $ to a decimal fraction. 6)1 166, &c. a mixt repeater. It sometimes also happens, that a certain number of figures recur, in this case the decimal is called a circulating one. EXAMPLES. 1. Reduce 7- to a decimal fraction. 7) 1-000000 -1428571, &c. a circulating decimal. 2. Reduce 7 \ to a decimal fraction. 11) 100 (-0909, &c. a circulate. EXAMPLES. 1. What is the decimal value of j. ? Ans. '375. 2. What is the decimal value of s ' r ? Ans. -04. 3. Reduce -rfj to a decimal? Ans. -015625. 4. Find the decimal value of #& ? Ans. '071577, &c. CASE 2. To reduce numbers of one denomination to decimals of another denomination retaining the same value. RULE. Reduce the integer to the same name with the given number, and divide the lesser by the greater, annexing cyphers to the dividend for the decimal. 69. 7 A 554 THE NEW PRACTICAL BUILDER, &C. EXAMPLES. 1. Reduce 9 shillings to the decimal of a pound. 1 20 20 shillings. o == '45 of a pound. 2. Reduce 2 feet 6 inches to the decimal of a yard. 2 ft. 6 in. = 30 inches. 3x12 = 36 inches. |f = \ = -833, &c. 3. - Reduce 6 inches to the decimal of a foot. Ans. '5. 4. Reduce 9rf. to the fraction of a shilling ? Ans. '75. CASE 3. To value any given decimal in terms of the integer. RULE. Multiply the decimal by the number of parts in the next less denomination, and cut off as many places for the right hand as there are places in the given decimal for a remainder. Multiply this remainder by the number of parts in the next inferior denomination, and cut off the same number of places as before, and so on. EXAMPLES. 1. What is the value of -625 of a shilling ? 625 12 Hence, Td. is the equivalent of '625 of a shilling. 2. What is the value of '75 feet? Ans. 9 inches. 3. What is the value of -125 feet ? Ans. li inch. 4. What is the value of '037-5j 1Ans. 9d. 5. What is the value of -333 feet ? Ans. 4 inches. NOTE. In those cases where repeaters occur, the steps are precisely the same as in finite decimals, only observing to carry for each 9 when operating on the first repeating figure. DUODECIMAL ARITHMETIC. As dimensions are generally taken in feet and inches, which are divided and subdivided by 12, and its powers, a peculiar kind of Arithmetic, adapted to subdivision by 12, is used by Artificers in computing the contents of their work ; it is called Duodecimals, or Cross Multiplication. To Multiply Feet, Inches, $c. by Feet, Inches, $c. RULE. Under the multiplicand, write the corresponding denominations of the multiplier. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and write the result of each under its respective term, observing to carry an unit for every 12 from each denomination to that next superior. In the same manner, multiply all the multiplicand by the inches in the multiplier, and set the respective results one place removed to the right of those in the multiplicand. Do the same with the seconds, and other lower denominations, and the sum of all the partial products will be the answer. EXAMPLES. 1. Multiply 6 ft. 3 in. 9 sec. by 6 ft. 9 in. and 3 sec. ft. in. sec. 639 693 37 10 6~ 4899 1 6 11 3 42 8 10 8 3 2. A garden wall is 254 feet long, 12 feet 7 inches high, and 3 bricks thick: how many rods are in it ! Ans. 23 rods and 136 feet. 3. A room is to be ceiled, whose length is 74 feet 9 inches, and width 1 1 feet 6 inches : what will it come to at 3*. 10|rf. per yard 1Ans. ^18. 10s. Irf. DECIMAL FRACTIONS. 555 4. If a house measures, within the walls, 52 feet 8 inches in length, and 30 feet 6 inches in breadth, and the roof of true pitch, or the rafters three-fourths of the breadth of the house ; what will it cost roofing, at 105. Qd. per square? Ans. 12. 12*. INVOLUTION, OR THE RAISING OF POWERS. A power is the product that arises by multiplying a number by itself as many times (wanting one) as tkere are units in the exponent of the power proposed. EXAMPLES. 1. What is the fifth power of 7 T 7 7 49 = second power. 7 343 = third power. 7 2401 = fourth power. 7 16807 = fifth power. 2. What is the third power of 35 ? Ans. 42875. 3. What is the fifth power of -015 ? Ans. '000000000759375. 4. What is the fourth power of 3'7? Ans. 187-4161. The first nine powers of the nine digits being arranged in a Table, are frequently found to be of con- siderable use in facilitating the computation of powers. Table of the first Nine Powers of Numbers. 1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. 1 1 1 1 1 1 1 1 1 2 4 8 16 32 64 128 256 512 3 9 27 81 243 729 2187 6561 19683 4 16 64 256 1024 4096 16384 65536 262144 5 25 125 625 3125 15625 78125 390625 1953125 6 36 216 1296 7776 46656 279936 1679616 10077696 7 49 343 2401 16807 117649 823543 5764801 40353607 8 64 512 4090 32768 262144 2097152 16777216 134217728 9 81 729 6561 59049 531441 4782969 43046721 387420489 EVOLUTION, OR THE EXTRACTION OF ROOTS. The root of any number or power, is such a number, as being multiplied into itself a certain number of times, will produce the power proposed. Thus, 3 is the square root of 9, because 3 x 3=s9 ; and 8 is the cube root of 512, because 8 x 8 x 8=512. The exact root of every number cannot be determined ; but, by means of decimals, we may approximate to any degree of exactness required. The roots thus approximated are called Surd Roots ; and those which can be exactly found, are called Rational Roots. To Extract the Square Root. RULE. Divide the given number into periods of two figures each, pointing towards the left in integers, but towards the right in decimals. 556 THE NEW PRACTICAL BUILDER, &C. Find the greatest square that is contained in the first period on the left hand ; (setting down its root like a quotient figure in division ;) subtract that square from said period, and to the remainder bring down another period for a new resolvend. Double the root of the first square for a divisor. Find how often this divisor can be got in the dividend, omitting the first figure on the right, and set the result in the quotient, and also annex it to the divisor. Subtract the product of this quotient figure, and the divisor so augmented from the dividend, and to the remainder bring down the next period for a new dividend. Find a divisor, as before, by doubling the figures that are already in the root ; and from these find the next figure of the root, as in the last step ; and so on till all the periods be brought down. If there be still a remainder, the root may be approximated by annexing periods of cyphers for decimals. Note. The reason for dividing the number into periods of two figures each, is, because the square of any single digit never amounts to more than two places. Hence there must be as many figures in the root, as there are periods of two figures in the given number. EXAMPLES. 1. What is the square root of 1225 ? 3)1225(35 65) 325 325 2. What is the square root of 723 ? 2) 723 (26-888669 _ 46)323 276 528) 4700 4224 5368) 47600 42944 63768) 465600 430144 537766) 3545600 3226596 5377725) 31800400 26888625 53777309) 491177500 483995781 ..7181719 3. Required the square root of "0729. 2 ) -0729 (-27 4 47) 329 329 4. Extract the square root of -00032754. Ans. -01809. 5. Extract the square root of 368863. Ans. 607'340092, &c, To Extract the Square Root of a Vulgar Fraction. RULE. Extract the root of the numerator for the numerator of the root sought, and the root of the de- nominator for the denominator. Or reduce it to a decimal and proceed as before. EXAMPLES. 1. Required the square root of -fyjj-. The square root of 729 is 27, and the root of 1225 is 35, consequently, the square root of the proposed fraction is ff . DECIMAL FRACTIONS AND EVOLUTION. 557 2. Required the square root of -||. 29) 160 (-5517241379, &c. 145 150 145 5 7) -5517241379 ("74278 ' ' 49 144) 617 576 1482) 4124 2964 14847)116013 103929 148548) 1208479 1188384 3. What is the square root of f ? Ans. 4. What is the square root of JJ = *. 20095 To extract the Cube Root. RULE. Separate the given number into periods of three figures each, putting a point over every third figure from the place of units. Find the greatest cube in the first period, and set its root on the right hand of the given number, after the manner of a quotient figure in division. Subtract the cube thus found from the said period, and to the remainder annex the next period, and call this the resolvend. Under the resolvend put the triple root and its triple square, the latter being removed one place to the left, and call their sum the divisor. Seek how often the divisor may be had in the dividend, exclusive of the place of units, and set the result in the quotient. Under the divisor put the cube of the last quotient figure, the square of it multiplied by the triple root, and the triple of it by the square of the root, each removed one place to the left, and call their sum the subtrahend. Subtract the subtrahend from the resolvend, and to the remainder bring down the next period for a new resolvend, with which proceed as before, and so on till the whole be finished. NOTE. Should there be a remainder after all the periods are brought down, the operation may be conti- nued by annexing periods of cyphers, as in the square root. EXAMPLES. 1. What is the cube root of 1953125 ? 3 1953125 (125 root. 3 1 Divisor 33 953 resolvend. 8 12 12 X3= 36 ~Q~ 12* X 3=432 728 subtrahend. Divisor 4356 225125 resolvend. 125=5' 900 =5* X12X3 2160 =12* x 3x5 225125 subtrahend. 70. 7 B 558 THE NEW PRACTICAL BUILDER, &C. 2. What is the cube root of 140708-483 ? Ans. 52-74. 3. What is the cube root of '0001357 ? Ans. -05138. 4. What is the cube root of 13| ? Ans. 2-3908. 5. What is the cube root of 27054036008 I -Ans. 3002. We shall now give a few examples to exercise the reader in the application of the square and cube roots. Given the hypothenuse and one leg of a right-angled triangle to find the other leg. RULE. Multiply the hypothenuse and leg each by itself, and the square root of the difference will be the length of the other leg. (This has been demonstrated in Theorem 62, Geometry.) Or thus, multiply the sum of the hypothenuse and leg by the difference of the same, and the square root of the product will be the other leg. Example. The length of the rafters is 18 feet, and half the width of the house 12 feet, What is the per- pendicular rise of the roof? 18x18 = 324 12x12 = 144 I80~~( 13-038 feet. ! 2 -^niws-rb ' ^ 2603)10000 7809 26068) 219100 208544 10550 Or thus, 18+12 = 30 1812 = 6 180 the root of which is 13-038 feet. Given the two legs to find the hypothenuse. RULE. Multiply the two legs each by itself, and the square root of the sum will be the hypothenuse. Example. The perpendicular height of a roof is 13-038 feet, and the width of the house 24 feet ; required the rafter ? 1 J24= 12 ~2 12 x 12 144 13-038 x 13-038=180 1) 324 (18 feet, the rafter sought. 1 28) 224 224 1. The diameter of a globular stone is 12 inches, What must be the diameter of another that contains 6 times the matter ? Ans. 21-7, &c. inches. RULE. Cube the diameter and multiply by 6, and the cube root of the product is the answer. MATHEMATICAL INSTRUMENTS, SCALES, &C. 55& MATHEMATICAL INSTRUMENTS, SCALES, &c., INCLUDING THE CENTROLINEAD AND CYCLOGRAPH. THE MATHEMATICAL INSTRUMENTS, commonly included in a case, for the purposes of drawing, mostly consist of the compasses, with their appendages, the steel drawing-pen, the pencil-holder, with black-lead pencil, the protractor, or graduated semi-circle, the plane-scales, and parallel-ruler ; to which are, at times, added other scales and implements, adapted to particular purposes, as in land-surveying, &c. The use of the COMPASSES is too well known to require particular explanation. Of these there are, in the cases, two sorts ; one with fixed steel points, and the other with one point fixed and the other point moveable. When the moveable point is taken off, there may be put in its place a steel drawing-pen point or a pencil- holder. The steel-pen point is put on the compasses, when it is intended therewith to describe circles, or arcs, in ink, which are to remain. Occult arcs, or such as are to be rubbed out again, are most conveniently described with the pencil-holder. The other steel-pen is used for drawing right lines from any given points, in any direction. An elucidation of the use of the compasses has been fully given under the head of PRACTICAL GEOMETRY, pages 62 to 70, &c. The SCALE is so called from a Greek word, which signifies a wooden measure of length, and is a thin broad rule of wood, ivory, or brass, divided into different lines, of various names and use. The best and most useful scales, for architectural purposes, are represented on the plate entitled Geometry, Plate I., of the exact size in which they are usually made.* The graduations in the plate have been made with such care, that we believe it may be relied on, for practice, by such as have not the instruments at hand. In this plate, figure 1 represents the PROTRACTOR or SEMI-CIRCLE, projected in form of a parallelogram, either for laying off or measuring angles, and numbered both from right to left, and from left to right, to 180 degrees. Figure 2, exhibits the back of the same scale, and it contains six lines of equal parts, with a Decimal Diagonal Scale, for plotting, or planning. The first have sub-divisions, both for decimals and inches ; and the larger figures at the end of the lines show how many decimal parts are contained in one inch, as from SO to 60. The diagonal scale is sub-divided to hundreth parts of one half and one quarter of an inch : its principle and use will be obvious on inspection ; as it may be seen that the perpendiculars are divided into ten equal parts, and through the divisions parallel lines are drawn, of the whole length of the scale. Again, the length of the first division is divided both at top and bottom, into ten equal parts, and the points are connected by diagonal lines, so as to take off dimensions or numbers of two or more figures. Examples. If the largest divisions be taken as units, the exterior smaller divisions will be tenth parts, and the divisions in the height will be hundredth parts. If the larger divisions be taken as tens, the next smaller will be hundredths, and the smallest thousands, &c. Each set of divisions being tenth parts of the former ones. To take the distance representing one and four tenths from the scale, (say the half-inch,) set one foot of the compasses on the upper line, to the larger division 1, and open the other leg to 4 in the subdivisions on the right. The extent will be the distance required. To take a distance equal to 25, set, in like manner, one foot of the compasses on the larger division 2, and extend the other to the sub-division 5, which will be the distance. * As this plate was given in the second edition of the first number, those who have not received it may obtain it on application to the publisher. f60 THE NEW PRACTICAL BUILDER, &C. For 34C, the larger divisions, being in this case taken as hundredths, set one leg in 3, upon the line marked 6 at the end, and extend the other to the diagonal 4, which will be the extent required. And, conversely, may the length of any line be measured, relatively to another of a determinate length. Figure 3 represents the face of another plotting scale, which contains lines of chords, of different radii; and equal parts, for feet and inches. The definition of a chord is given in Geometry, page 13, and the lines of this name are divided for the purpose of laying off and measuring angles, on the established principle that the radius or semi-diameter of a circle is equal to the side of a hexagon inscribed in the same circle ; or, in other words, to the chord of CO degrees. Hence, by taking the extent of the chord of 60 in the compasses, applying one foot to an angular point, and sweeping an arc with the other from leg to leg, (produced, if required,) the exact measure of the angle may be found. Examples. 1. To construct an angle at the point A, of the line AB, of any number of degrees, suppose 26. From the line of chords take in your compasses the extent to 60 ; and setting one foot in A, describe the arc CB : then ^^ \ take 26, the number proposed, from the same line of chords, ^^ \ in your compasses, and set it off from B to C. Join AC, and the angle BAG will contain 26 degrees, as required. 2. To find the quantity of a given angle, as CAB. This is the converse of the preceding example : for, if you take in your compasses the line AB with the chord of 60, and describe the arc BC, it becomes evident that the distance BC will be the quantity of the angle BAG ; which, measured on the line of chords, will show its quantity, or the number of degrees that it contains. 3. To describe any regular polygon in a circle. First find the angle of the centre, which is done by dividing 360, the number of degrees in a circle, by the number of sides of which the proposed figure is to consist : then, from the centre, draw two radii in the angle given by the quotient, and the chord of the angle included between them will be the side of the polygon required. Thus may any regular polygon be described, the radius being always 60". Figure 4 represents the back of fig. 3, and contains another set of plotting scales, for half an inch, one quarter of an inch, three-eighths, and one-eighth, of an inch, to the foot, &c. and sub-divided diagonally for greater accuracy. The uses of these are too clear to require further explanation. DESCRIPTION AND USE OF THE CENTROLINEAD. THE CENTROLINEAD is an instrument used in Perspective, &c. and invented by Mr. Peter Nichol- son, for drawing lines to an inaccessible or vanishing point. For this invention Mr. Nicholson was rewarded by the Society of Arts, in May, 1814, with the sum of twenty guineas ; and, in 1815, for an improvement, ftie Centrolinead now in general use, with the silver medal. Figure 1, of the plate entitled CENTROLINEAD, represents the instrument now in use. It is constructed with two legs, about a foot long, with a joint in the angle, similar to a carpenter's rule. The centre of the knuckle is pierced with a small hole, (Jig. 2,} in which a pin is rivetted, in order to admit of a blade being fitted on it, for drawing lines to a point. The drawing edge of this blade is made to pass through the centre of the joint ; and the blade, screwed to the two legs, may be fixed at any required angle. The edge of the blade and the edges of the two legs, all tend to the same point. The blade is made to reverse, so as to draw lines to either side of a building : the same legs will answer to blades of various lengths, as occasion may require. In complex drawings, it will be convenient to use a distinct centrolinead for each point, to prevent the trouble of frequently adjusting the instrument Figure 2 exhibits the instrument on a larger scale, with part of the legs and the blade broken off. Figure 3. The joint fixed to part of the legs. The figures No. 1, No. 2, No. 3, No. 4, No. 5, are the detail of figure 3. .07 MATHEMATICAL, INSTRUMENTS, SCALES, &C. Figure 4. Brass fixed to the blade and legs, by means of three screws, so as to fasten them together, as represented in figure 1. Figure 4, No. 1, edge of figure 4, with part of the edge of the blade fixed to the brass, by a screw, and two pins to steady it. Figure 5 shows how a T-square may be made into a centrolinead, by fixing a piece in the form of a wedge to the stock of the square. Figure 6. A diagram, showing the figure where the points form for working the instrument on, when the legs of the instrument are at different angles, to draw to the same vanishing point. . MODE OF SETTING THE CENTROLINEAD, WITH SCREWS, BY MR. P. NICHOLSON. (Second Plate.) DRAW two lines tending to the vanishing-point, or to the station point, which must be found by a pro- blem : then put two pins in vertically, over each other, on each side of the line without the space, making the distances on each side of the line nearly the same. Press the two edges of the legs, which run to the centre of the joint, against the pins, and move them along till the edge of the blade, which passes through the centre of the joint, coincides with the line or crosses it ; then loosen the screw that fastens the blade to the leg of the centrolinead, and press the legs gently against the pins till the blade coincides with the line : then fasten that screw, and loosen the other ; next move the instrument down to the other line ; and, if the same edge of the blade coincides with the line, fasten the screw, and the instrument is set : but if not, proceed, as above, till the edge of the blade coincides with the two lines; it generally requires three settings ; the least number it can have is two, that is, one for each angle. The centrolinead may be clamped at once, in the manner following : Draw a line between the two given lines, so as to be terminated by them ; set each of the angles of the centrolinead to each angle formed by the cross line ; put a pin in each angular point, formed by the cross line, with each of the given lines ; then placing the central edges of the legs upon the pins, one on each, draw lines by the central edge of the blade, and those lines will tend as accurately to the point as if they were drawn to it by a longer ruler. ME THOD of finding two Lines fending to the Station Point, and also of getting the Vanishing Point on your Plan. Let ABCD (fig. 1 .) be the size of your drawing board ; IJ your intersecting line ; and HEFG the plan of the remotest projections of the cornice of your building ; making the line EH, produced to e, an angle of about 43 degrees with the intersecting line IJ. Then draw a line, NM, perpendicular to IJ ; at one extremity of your paper set any equal distance, ML or MK, on each side of M, and make the distance MN, equal to 3 times the base of the triangle KL, or any other number, according to the distance at which you wish to view the object : then draw EO parallel to LN ; and GP parallel to KN ; and these are the two lines tending to the station point W, to set the centrolinead to, which falls a considerable way without the board. To find the vanishing points, divide OP into two equal parts, at Q; divide QO into four equal parts, and draw IS perpendicular to IJ, to cut the line EW, at S '; draw ST parallel to IJ, and TQ will be one fourth of the real distance W ; then draw TU parallel to /ft?; and TV parallel to EH: then will QC/and QFbe each one fourth of the real vanishing points, which should be drawn from W, if it could be got within the board. 70. 7 c 562 THE NEW PRACTICAL BUILDER, &C. MANNER of transfering the points from your plan, or to your Picture, in order to get the two lines tending to the vanishing points ; and, also, Mr. M. A. Nicholson's method of setting the Centrolinead without screws, as shown in the Plate. Figure 2. Draw your intersecting line LM, and then your vanishing line AB ; divide your paper into two equal parts at C ; draw any line CD perpendicular to AB; set up any four equal parts, CE, a little higher than you intend your object to be; make CG equal to QV, and CH equal to Q 7 on your plan,y?g. 1. Draw GF and HI, perpendicular to AB ; make GF and HI each equal to three parts of CE ; join EF and El, and these are the two lines tending to the real vanishing points, which should be drawn from the point W, to intersect the line 7J on the plan, fig. 1. Having found the two lines EF and El, tending to the vanishing points, the next thing will be to set the centrolinead, which is done by fixing in a pin at/, a little below the intersecting line, then placing the blade of the instrument to the line FE, at the same time pressing the leg to the pin/: then draw a line on the side of the leg cd; then move the blade to the vanishing line AB, making ab, coincide with AB ; at the same time pressing the leg ed to the pin /, as above, and drawing another line along the side of the leg cd, to intersect the former line ag ; then g is the point for fixing in another pin, and the instrument is set, and will draw all lines on the edge of the blade a b, between FE and AB, to a point, by pressing the legs dc, and de, against the pins g and/. A Centrolinead constructed in this manner, without screws, will draw lines to various inclinations, except- ing when the lines are at a very sharp pitch or a very flat one ; then the angle of the legs should be made accordingly. I have found, in practice, that three instruments, constructed in this manner to different angles, are sufficient to draw any building, whether octagonal or rectilinear ; as the angle under which we view the building, in general, is the same, in order to make an agreeable picture.* f DESCRIPTION AND USE OF THE CYCLOGRAPH. THE CYCLOGRAPH is an instrument for drawing arcs of circles, and is very useful in drawing the mouldings of the top of Bridges, &c. Fig. 1, of the plate entitled CYCIOGRAPH, is a representation of the instrument, which is used by placing in a point, as at A and C, at the springing of the arch, and adjusting the point of the pin or pencil, which is inserted in a tube at B, to the top of the arch, by means of one of the largest screws to the right hand side of the joint. Figure 1. No. 1, is a side view of Jig. 1, representing the tube fixed in the joint. Figure 2, exhibits the legs without the brass-work. Figure 3. No. 1, the brass-work, with the tube in the joint. No. 2. is a side view of fg. 3. No.l. Figure 4. No. 2, exhibits the joint when the pieces are separated. No. 1, and No. 3, sections of Jig. 4, and No. 2. Figure 5. Brass.piece for holding and adjusting the leg by means of a slit in the brass, struck from the centre of the joint, and fastened to the leg by a screw. Figure G. Section of the joint and tube to a larger scale, with the drawing pin in it, and fastened by means of a screw at the side of the tube. * Those Gentlemen who wish for the instrument, made agreeably to the above setting, may be supplied by Mr. M. A. Nicholson, on application to the Publisher of this work. ARRANGEMENT AND DESCRIPTION OF THE PLATES. 563 ARRANGEMENT AND DESCRIPTION OF THE PLATES GIVEN IN THIS WORK. IN turning over our numerous ENGRAVINGS, the reader will perceive that they consist of SEVERAL DISTINCT SERIES ; of which the first is regularly numbered, from PLATE I. to XCII. Next follow the plates entitled ORDERS, Plate I. to XXV., and succeeding these are four plates of ORNAMENTS, &c. In succession are thirteen plates to illustrate PERSPECTIVE, PROJECTION, SHADOWS, and INSTRUMENTS : to which follow a series of PLANS and ELEVATIONS, in forty-seven plates ; including seven designs for SHOP- FRONTS, in three plates ; four designs for WINDOWS, in two plates ; and a beautiful design for a Grecian Chimney-Piece, on one plate. / The FIRST SERIES of plates [I. to XCII.] are regularly referred to in the Book, within the first 450 pages. The ORDERS, or Second Series, are treated on under the proper head, pages 451 to 517. PERSPECTIVE, PROJECTION, and SHADOWS, pages 518 to 550. The plate of SCALES, Geometry, PI. I., of the Second Edition, is described in pages 559, 560, and the plates of the CENTROLINEAD and CYCLOGRAPH in the succeeding pages. The PLATES of ORNAMENTS, above mentioned, accompanying the plates of Orders, seem to require no particular description : yet it may be as well to notice that in Plate I, of Grecian Ornaments, figure 1 represents the ornament between the bead and small projecting band of the top of the column of the Temple of Minerva Polias, at Athens. Figure 2 is the ornament on the Cymatium of the door of the same temple, stretched out on a flat surface. Figure 3 is the enriched moulding of the cap of the same templCj showing the finishing of the external angle. Figure 4, Egg and Tongue, with beads below, belonging to the same cap. The SERIES of PLANS and ELEVATIONS are as follow : PLATES I to V., First, Second, Third, and Fourth, Rate Houses ; which require no other description than that already given in the work. PLATE VI. is the Plan and Elevation of a SMALL COUNTRY VILLA. Figure 1, Elevation of the villa, with a pediment in front, and another in the back. The construction of the roof, from its great projection of cornice, and the beautiful shadows which is thrown from the cantalivers has always a pleasing effect. The extent of this villa in front may be 48 feet 6 inches ; in the centre is a porch, with four columns, of the Ionic order, coupled, on each side of the entrance, with a regular entablature and pediment. The height of the columns 10 feet 6 inches; the entablature, 2 feet 6 inches; and the pediment, 2 feet 3 inches. The entrance is 3 feet 10 inches, raised 2 feet 10 inches above the level of the ground ; ascended by six steps, one of which is in the door-way. The windows on each side of the porch are 5 feet from the ground, 7 feet 9 inches high, 3 feet 8 inches wide, with horizontal rustics on each side, from the top of the plinth to the floor of the bed-chamber, above which are two bands, one of which the architraves of the chamber-floor windows stop against. The windows of the chamber-floor are of the same height and width as the lower ones, with an architrave round them, on the top of which is a frieze and cornice. Figure 2 is the Ground Plan of the above ; A, the porch. Hall, 12 feet 4 inches by 7 feet 10 inches ; lobby, 7 feet 9 inches square; B, stair-case; parlour, 20 feet 5 inches by 16 feet 6 inches; dining-room, 22 feet 10 inches by 16 feet 6 inches; library, 16 feet 6 inches by 14 feet; kitchen, 16 feet 3 inches by 15 feet 6 inches. PLATE VII. PLANS AND ELEVATIONS FOR A VILLA, WITH WINGS. Figure 1. Principal Elevation : this building, if properly situated, would command a fine prospect from the circular bows in the wings. The whole extent of the building, including the wings, may be 85 feet 6 inches ; each of the wings 24 feet 9 inches, and the body of the building 36 feet. The entrance is 3 feet 9 inches, raised upon three steps, of * Directions to the Book-binder, for arranging and placing the plates, will be givcu hereafter. 504 THE NEW PRACTICAL BUILDER, &C. 6 inches rise, on the top of which are two Doric columns, with pilasters behind them. The columns are six diameters high ; the entablature 2 feet 4 inches, and the blocking-course 1 foot. The windows on each side of the porch, in the centre part of the building, are 4 feet 3 inches from the ground, 7 feet 6 inches high, and S feet 8 inches wide, with a Grecian architrave around them ; diminishing on each side of the centre of the window, and parallel to the sides of the column. The ears at the top of the window are vertical with the bottom of the architrave ; the window in each wing, parallel to the front, is of the same dimensions as above : and the window in each bow is Venetian, placed between four columns of the Ionic order. The proportion of die columns is eight and a half diameters in height, and each diminishes about one-sixth ; the entablature is one-fourth of the height of the column, on the top of which is a blocking- course, 10 inches high. Figure 2. Ground Plan of the Principal Story. A, the Vestibule, 13 feet 3 inches by 6 feet ; BB, Pas- sage, 33 feet by 4 feet ; Parlour, 13 feet 3 inches by 12 feet 9 inches ; Library, 13 feet 3 inches by 12 feet 9 inches ; Dining-room, 1 8 feet, exclusive of the part in the bow, by 1 6 feet 6 inches ; Drawing-room of the same dimensions ; kitchen, 18 feet 8 inches by 12 feet 9 inches ; D, Scullery, 6 feet 6 inches square : E, Pantry, 6 feet 2 inches by 3 feet 9 inches ; C, Closet, 6 feet 4 inches by 5 feet 9 inches ; e, Entrance to the Chambers ; /, Entrance to the Wine-cellar, &c. Figure 3. Plan of the Chamber-Floor. AA, Passage to the Bed-rooms, 25 feet by 4 feet ; B, Bed- room, 13 feet by 11 feet 10 inches ; H, Water-closet, 6 feet 6 inches by 4 feet 2 inches ; C, Bed-room, 13 feet 6 inches by 12 feet 10 inches; D, Dressing-room to ditto, 7 feet 5 inches by 4 feet; E, Front Bed-room, 13 feet 3 inches by 12 feet 9 inches; F, Dressing-room to ditto, 13 feet 3 inches by 6 feet; G, Bed-room. PLATE VIII. PLAN AND ELEVATION OF A COUNTRY VILLA. Figure 1. Elevation of a Castellated Gothic Villa, with buttresses, &c. This building would be suitable for a gentleman of moderate fortune. Its whole length is 78 feet ; and its height, from the surface of the ground to the top of the battlements, 34 feet 2 inches. The battlements are continued all round the building, and the height of them is 2 feet 6 inches. The buttresses are 29 feet 3 inches high, with two water tables, on the top of which is a cornice. The cornice is continued all round the building. The windows on the ground-floor are 4 feet 3 inches from the ground, their height 9 feet, and width 4 feet. The top of each window is crowned with a tablet, which reaches to* a little below the top of the window, on each side. The chamber-floor windows are 19 feet from the ground, their height is 8 feet, and width 4 feet, and they are crowned with a tablet, as below. The entrance is on the flank to the right, raised 1 foot 6 inches above the level of the ground, and ascended by three steps ; it is enclosed within a porch, of 12 feet in front and 8 feet deep ; the openings of the front and sides of the porch are 8 feet and 4 feet 10 inches. The height to the springing of the arches is 8 feet, and to the top of the arch 3 feet 10 inches, over which runs a band, and is of the same height as that in the octagonal front of the building. On the left flank is a green-house, which will have a very beautiful effect on entering, as seen at one extremity of the passage through a sash-door. Figure 2. Ground Plan of the Principal Story. A, Porch, 8 feet by 6 feet ; B, Passage, communicating to the different apartments, 74 feet long by 6 feet wide ; C, Stone Staircase to the Bed-chambers, steps 3 feet 6 inches long, treads 1 1 inches, risers rather more than C inches ; Breakfast-Room, 20 feet by 1 7 feet 6 inches ; Dining-Room, 20 feet by 1 7 feet 6 inches ; Drawing-Room, 30 feet by 30 feet ; Library, 20 feet by 17 feet 6 inches; E, Parlour, 20 feet by 17 feet 6 inches ; D, Waiting-Room, or Dressing-Room, 19 feet by 15 feet; F, Water-Closet, which is entered by a door under the staircase; Green-house, 42 feet 6 inches by 12 feet G inches. The Servants' apartments, &c. are on the basement, which is entered by the staircase, C. PLATE IX. PLAN and ELEVATION of a COUNTRY VILLA. Figure 1. Principal Elevation of a castellated Gothic Villa, with Buttresses, and Pinnacles on a straight Front. The extent of this building, from the extremity of one wing to that of the other, is GO feet : extent of each of the wings, 11 feet 10 inches. The body of the building, 3G feet 4 inches. The entrance, 3 feet 4 inches wide, with a Gothic head, receding ARRANGEMENT AND DESCRIPTION OF THE PLATES. 565 from the central part of the front 3 feet 6 inches, forming a Porch, and raised, above the level of the ground, I foot 6 inches ; ascended by 3 steps of 6 inches rise. The entrance to the Porch is 4 feet 10 inches wide, and it rises 6 feet to the springing of the arch : the arch is 4 feet high, and is ornamented with mouldings and crockets on each side. The windows of the ground floor, on each side of the Porch, are 6 feet 6 inches high by 4 feet 6 inches wide ; and those in the wings are 6 feet high, and 4 feet 6 inches wide : those in the bed chamber 4 feet 3 inches high and 4 feet wide. ; Figure 2. Ground Plan of the Principal Story. A flight of three steps to the Porch, L, 7 feet 7 inches by 2 feet 6 inches. I. Hall, 9 feet 6 inches by 7 feet 7 inches. N, Staircase, of wood, steps 3 feet 3 inches, their rise is 6^ inches, and the tread 10 j inches. PP, Passage to the different apartments, 57 feet 8 inches by 4 feet. B, Breakfast-Room, 14 feet 3 inches by 12 feet. A, Dining-Room, with folding-doors, 14 feet 3 inches by 12 feet. F, 12 feet 3 inches by 10 feet 7 inches. D, Parlour, 12 feet by 10 feet 4 inches. C, Library, 12 feet by 10 feet 4 inches. H, Servant's Waiting-Room, 7 feet 6 inches by 6 feet 3 inches. E, Kitchen, 12 feet 3 inches by 10 feet 7 inches. G, Wash-house, 10 feet by 7 feet 6 inches. O, Water- closet, 4 feet 2 inches by 3 feet. PLATE X. LODGES and ENTRANCE to a MANSION. These lodges are in the castellated style, and would be suitable for the octagonal mansion represented in plate XII. Figure 1. The Elevation of the Lodges. The central part, betwixt the piers, for carriages, may be I 1 feet 9 inches in the clear, above the plinth. The height of the gateway 7 feet ; the piers 3 feet wide and 9 feet 10 inches high : the faces and sides forming the internal angles (see plan, jig. 2,) are ornamented with small recesses, terminating at the top with Gothic heads, filled in with three cusps and two semi-cusps, denominated by the name of a Quadrefoil arch. The part above that is carried solidly over the internal angles, forming a regular octagonal figure, with two water-table mouldings, and small panels in the faces betwixt the mouldings, filled in with cusps. The side-gate for passengers is five feet wide above the plinth, betwixt the porch and pier. The eleva- tions of the fronts of the lodges is of an octagonal form, with hexagonal buttresses on the angles, the sides of which are filled in with small recesses, with Gothic heads. The length of each lodge in front, including the porch, is 20 feet : its height, from the ground to the top of the battlements, 1 3 feet. The battlemerits, including the corbels, are 2 feet high : the windows are 3 feet wide and 6 feet high, from the sill to the top of the arch : standing in recesses 2 inches deep, 4 feet wide, and 8 feet high, from the top of the plinth to the top of the arch. The height, from the top of the roof to the ground, is 15 feet. The breadth of the battlements is 9 inches and a half. The embrasures, or openings, 5 inches wide. Figure 2. Plan of the Lodges and Gateways. The whole length in front, from the extremity of one lodge to that of the other, is 67 feet 9 inches : the distances betwixt the porches is 27 feet 9 inches : the size of the j;orch, in the clear, is 5 feet 3 inches by 3 feet. The Living-Rooms, 14 feet by 11 feet 7 inches. The Bed-Rooms 1 1 feet 7 inches, by 7 feet 6 inches. PLATE XI. GROUND PLAN of a DESIGN for a MANSION in the CASTELLATED STYLE. The entrance is by a Groined porch, ascended by 4 steps ; length of tread 7 feet, breadth 11 inches, and risers 6 inches, to be made of granite. I, Hall, 30 feet by 27 feet 6 inches. L, Passage, 6 feet 6 inches wide ; lighted by a borrowed light from the lantern of the Picture Gallery. H, Principal Staircase, 23 feet long by 16 feet 6 inches ; the length of the treads 7 feet, arid breadth one foot ; their rise, rather more than 6 inches. K, Back Staircase, 16 feet 4 inches by 10 feet ; length of tread 4 feet 4 inches, breadth lOf inches ; risers 6f inches. E, Breakfast-Room, 46 feet 6 inches by 46 feet 6 inches. F, Music-Room, 30 feet 6 inches by 17 feet. D, Dining-Room, 46 feet 6 inches by 46 feet 6 inches. G, Parlour, 19 feet by 17 feet. B, Drawing- Room, 46 feet 6 inches by 46 feet 6 inches. C, Library, of the same dimensions. A, Picture-Gallery, 43 feet by 43 feet, lighted by a lantern from the top of the building. The Water- Closet, 16 feet 8 inches by 6 feet 6 inches, lighted by a borrowed light through ground-glass, from the well-hole of the staircase. The plans of the rooms above are adapted to contain, at least, seven Bed-Rooms, with Dressing-Rooms, and Closets. 71. 7 D 566 THE NEW PRACTICAL BUILDER, &C. PLATE XII. PRINCIPAL ELEVATION for a MANSION in the CASTELLATED STYLE. This building, from its construction, would have a majestic appearance if placed on a hill, or elevated ground, and entirely detached from other buildings, more especially if surrounded by wood, and near the side of some river or stream The whole length in front maybe assumed as 131 feet G inches, exclusive of the buttresses. The body of the building, parallel to the front, and at right angles, passing through the centre of it, will thus be 102 feet 6 inches. Its whole height, from the ground to the top of the battlements, 88 feet. The height to the top of the second row of battlements of the larger octagonal body, 79 feet 8 inches. The turrets, 15 feet 6 inches high, 5 feet 3 inches diameter, and projecting three quarters of a circle below the corbels, from the angles of the plain faces of the octagonal body. The windows in the faces, from the top downwards, are all of one width ; each of their heights is, respectively, 5 feet, 6 feet, 7 feet, and 8 feet, lu'gh ; with water- tables and mouldings over them. The window-frames, of oak, consist of a Munnion Transom and bars ; the rectangular projections are 30 feet each in front, and 19 feet deep, with turrets at the angles 13 feet C inches high, and 5 feet diameter. The openings of the porch, in the clear, each 7 feet wide, and 17 feet high ; ornamented on the chamfers with mouldings. On the top of the arch are crockets, rising in the form of pinnacles. The doors that light the Hall, which is seen through the openings of the porch, are each 5 feet 7 inches wide in the clear, and 16 feet high, moulded all round. The frames of the aperturet are of oak, and are made to open in two halves ; the bottom of which is panelled. The front of the octa- gonal wings extends 50 feet 3 inches ; each of the faces is 20 feet 9 inches nearly, with buttresses on the angles. Their height, from the ground to the top of the enriched battlement, is 27 feet. The windows are each 7 feet 4 inches wide in the clear ; their height, from the cill to the top of the arch, is 14 feet. The reveals are chamfered about half-way in. The space between the cill and the plinth is filled in with orna- mented panels. The frames of the windows are of oak, with Gothic heads, and are naade to open in one of the compartments. PLATE XIII. GROUND PLAN of the SEAT of HENRY MONTEITH, Esq. at Carstairs, near the River Clyde. The space H represents the porch, with buttresses at the corners, ascended by three steps, and groined in the inside. The dimensions are 1 6 feet by 1 5 feet 6 inches. E, Hall, with two small windows, on each side of the- doorway ; 19 feet 4 inches by 18 feet 4 inches. D, Library, of an octagonal form, 22 feet 9 inches by 22 feet 9 inches, to the inside of the bay-window. Length of the Passage, 10 feet 3 inches. I, Grand Staircase, 25 feet 3 inches by 23 feet 3- inches, lighted by a large window in the side wall, and oppo- site to the window in the bottom of the passage ; length of the steps, 4 feet ; rise 6 inches and a half; tread, 1 1 inches. C, Breakfast-Room, 32 feet 3 inches, including the bay, by 25 feet. B, Dining-Room, 39 feet by 30- feet, including the bay. A, Drawing-Room, 34 feet by 29 feet 6 inches, including the bay. F, Bed- Room, 22 feet 9 inches by 18 feet 5 inches. V, Powdering-Room, 11 feet 3 inches by 9 feet 9 inches. N, Pariow, 16 feet 6 inches by 11 feet 3 inches. L, Nursery, 23 feet 7 inches by 20 feet 5 inches. G, Kitchen, below the level of the rooms on the ground-floor, and communicating to them by a winding stair- case at one corner, 23 feet 9 inches by 20 feet 6 inches. O, Bed-Room, 16 feet 9 inches by 14 feet. P, Bed- Room, 16 feet 9 inches by 10 feet. R, Housekeepers-Room, 16 foet 5 inches by 15 feet. U, Anti-room, 1 1 feet 4 inches by 9 feet 4 inches, lighted at the top, wkh a closet at one end. The largest winding staircase leads up to the servant's Bed-Rooms, which are carried lower than die Bed.Rooms over the body of the building, and communicate with the Bed-Rooms on the body of building by a door, leading off from the second flight of steps of the grand staircase. The parts which are broken, off at the farthest extremity to the left, are the offices connected with the Kitchen. PLATE XIV. PRINCIPAL ELEVATION of the SEAT of HENRY MONTEITH, Esq. The whole extent of this, building, exclusive of the offices, not shown in the plate, is 175 feet. The principal part, including the bays, with the Oriel windows on each side of the porch, is 64 feet 10 inches. The entrance, within the porch, is 5 feet 9 inches wide in the clear, and 12 feet 10 inches high to the top of the soffit of the arch, raised 1 foot 7^ in. above the level of the ground, and ascended by three steps of 6{ in. rise, tread 1 foot. The width of the opening in the porch is 11 feet 6 inches, and its height 13 feet 6 inches, from the top of the steps to the top of the soffit of the arch. ARRANGEMENT AND DESCRIPTION OF THE PLATES. 567 The height of the porch, from the ground to the top of the battlements, is 20 feet 9 inches. The height of the central part, behind the porch, from the ground to the top of the battlements, is 54 feet 6 inches ; its breadth, (including the octagonal buttresses, which rise at the top into pinnacles,) is 24 feet. The height of the bay, with the Oriel window, between the octagonal buttresses, from the ground to the top of the battlements, is 37 feet; its breadth 12 feet 6 inches. The central Oriel window, without the mun- nions, is 5 feet wide, and 9 feet C inches high : each of the diagonal openings is 2 feet 6 inches wide, in the clear of the reveals. The window above the bay is 5 feet 8 inches wide, and 6 feet 2 inches high, divided into munnions with Gothic heads. The bays, with the Oriel windows on each side of the porch, are 35 feet 6 inches high, from the ground to the top of the battlements, and 1 6 feet wide. Each of the Oriel windows, in the central part, on the ground-floor, is 7 feet 4 inches wide, and 12 feet high in the clear ; and each of the diagonal openings 2 feet 4 inches in the clear. The Oriel windows over the ground-floor, in the central part, are each 7 feet 6 inches wide, and 9 feet high, and those on the diagonals 2 feet 4 inches wide. The part behind the bays is of a pediment-form, with square buttresses, and small pediments, rising into pinnacles. The extreme wing, to the left, is 29 feet wide, and 39 feet high, from the ground to the top of the battlements. The turrets are each 11 feet 9 inches high, and project three-quarters of a circle out, from the angles below. The window on the ground-floor is 7 feet wide, and 9 feet 3 inches high. The Oriel window above, in the central opening, is 4 feet 9 inches wide and 7 feet high : the side openings are each I foot 7 inches wide. The space which recedes from the wing to the left, and to the octagonal- tower, is 24 feet 5 inches wide, and 31 feet high to the top of the battlements. The windows on the ground-floor are each 3 feet 9 inches wide, and 8 feet 5 inches high : the windows above are 3 feet 9 inches wide, and 5 feet 10 inches high. The octagonal tower, adjacent to the above, is II feet wide and 45 feet high; each of the small windows in the faces, from the top downwards, is 7 feet, 8 feet, and 8 feet 7 inches high : their width, in the clear, 1 foot. The space between the Oriel windows and the octagonal tower is 29 feet wide, and 36 feet 6 inches high, from the ground to the top of the battlements ; the windows on the ground-floor are 4 feet 10 inches wide, and 10 feet high ; the windows above those on the ground-floor are 3 feet 9 inches wide, and 6 feet 6 inches high. The octagonal tower at the extremity, to the right, is II feet wide, and 49 feet high from the ground to the top of the battlements ; each of the windows, from the top downwards, is 6 feet 9 inches, 7 feet 9 inches, and 9 feet 9 inches, high ; and each of their widths 13 inches. The small space between the tower and the Oriel windows is 5 feet 9 inches : the small windows between are blank. All the munnions and Gothic heads which divide the windows into compartments are of oak. PLATE XV. PLAN and ELEVATION of a MANSION. Figure 1. This building, though perfectly straight in front, would have a pleasing effect from its colonnade, and the portico projecting out from it. The whole length of the building may be 89 feet 6 inches, and its height, from the ground to the top of the blocking- course, 49 feet The columns are of the Doric order, without flutes, raised upon a plinth 18 inches above the level of the ground, and stand 4 feet from the wall. The height of the columns is 14 feet ; being six diameters and a half high. The entablature, 2 feet 10 inches ; above which is a blocking-course, about 10 inches high, on which the balcony is fixed. The windows under the colonnade are 8 feet 9 inches high, and 4 feet 5 inches wide, and come down within 5 j inches of the level of the colonnade, forming a step out. The architraves around the windows are one-sixth of their opening. The sashes are of the common kind ; but are so constructed as to make them go higher up than the top of the window, in order that a person may get out without stooping. The entrance is 4 feet C inches wide, with a small window on each side, and a fan-light over the top to lighten the vestibule. The two columns which stand before pilasters, and immediately behind the fwo extreme columns, in front of the portico, are intended to rest a beam upon the top of them, in order to support a viranda, wjiich 9ome,s a little before the balcony, and is composed of piers of 1 foot 2 inches 568 THE NEW PRACTICAL BUILDER, &C. square, and 9 feet 6 inches high ; which may be either of stone or wood, with bases and ornamented caps. On the top of this is an entablature, 2 feet high, with a small blocking-course. The windows on each side of the viranda are 9 feet high, and 4 feet 3 inches broad, with an architrave one-sixth of the width round them, and a frieze and cornice above, and come down to 6 inches from the level of the balcony, forming a step out. The sashes are of the same form as those below. Those in the attic are of smaller dimensions ; their height 6 feet 3 inches by 4 feet. That in the centre is a Venetian window, with a flat segment-head. Above the attic is a cornice, 1 foot 3 inches high, and 2 feet projection ; its parts are a fillet, ovolo, and corona, with a channel underneath it to carry off the rain. There is also a semi-reversa next to the wall, which is concealed in the channel, the bottom of which is on a level with the bottom of the corona, and cannot be seen unless in perspective. There is also a bead and fascia below that. Figure 2. Ground Plan, or Principal Story. A, Porch. BB, Colonnade. C, Hall, 19 feet 6 inches by 16 feet 2 inches. G, Stone-Staircase, with flyers and winders ; the length of the steps is 4 feet 3 inches, their tread 11 inches, and their rise rather more than 6^ inches. N, Back-Staircase. The Lining- Room, 32 feet 6 inches by 19 feet 6 inches. Library and Breakfast-Room of the same dimensions. Housekeeper's Room, 15 feet square. Steward's Room, 15 feet by 13 feet. Slrong-Room, 7 feet 3 inches by 6 feet 3 inches. Dressing-Room, 15 feet by 12 feet 9 inches. Stone-Room, 15 feet by 9 feet 10 inches. PLATE XVI. GROUND PLAN of a CHURCH in the GRECIAN STYLE. The PORTICO (a), of four columns, projects out from the front of the wall, 8 feet C inches. The Vestibule (b) leads to the body of the chapel and side stair-cases; diameter, 19 feet 9 inches within the columns. Staircases of an elliptical form (cc), leading to the gallery, 22 feet 8 inches by 20 feet 9 inches; length of treads, 4 feet 10 inches ; breadth, in the middle, 1 1 inches ; risers rather more than 6 inches. Side Entrances, ff. The body of the church may be 89 feet long and 54 feet 3 inches broad, and it will contain sixteen hundred sittings, (including free seats,) exclusive of seats for children in front of the organ. hhhh, represent Pews, 3 feet wide; seats, 1 foot; book-desk, 5 A inches; oo, Larger Pern; pp, spaces between the free-seats. The Pulpit (n), of an hexagonal form; i, Stair ascending to it. Reading-desk (m), with clerk's seat in front ; stair (i) ascending to it. The Communion-place, of a circular form, with four three-quarter columns, and two antae : betwixt the columns are two niches, and a window in the centre. The Vestry-Room (s) is 22 feet 2 inches by 15 feet 4 inches. The Entrance to Circular Stair (y), leading to library above. The Anti-Room under Portico (u) 10 feet 5 inches by 8 feet. The Robing-Room (t), 22 feet 2 inches by 15 feet 4 inches. . The Anti-Room (c) 10 feet 5 inches by 8 feet. X, Entrance to die Catacombs. W, Back Portico, of four columns, projecting out from the wall 5 feet 6 inches. PLATE XVII. FRONT ELEVATION OF THE SAME CHURCH, IN THE GRECIAN STYLE. The extent of the front of this building to the extremities of the antae is 64 feet 9 inches; the breadth of the portico, at the top of the columns, is 37 feet. The columns are raised 1 foot 1\ inches above the level of the ground, and are designed from the Monument of Lysicrates ; (See Orders, pi. XXV.) their diameter is 3 feet, and height, including the base and capital, 29 feet. The height of the entablature is 7 feet 4 inches ; the architrave, 2 feet 9 inches; the frieze, 1 foot 10 inches; and the cornice, 2 feet 9 inches. The ornament which stands on the top of the cornice is 13 inches high, and is continued all round the building. The antae are of the same width as the upper diameter of the columns, and do not diminish. The capitals of the antas are a composition, as there are ho antae to be found in this style of Grecian architecture. The principal entrance is ascended by three steps, in front of the portico, of 6 in. rise; tread, 1 foot. Its width at the bottom is 6 feet 11 inches; it diminishes to the top, and its height is 14 feet. The side entrances are each 6 feet 7 inches at bottom, and diminish to the top ; their height, 12 feet 9 inches, with an architrave round them and a cornice at the top, supported at each extremity by a console. The niches on each side of the principal entrance are 4 feet 3 inches wide, and 9 feet 10 inches high, and diminish on each side parallel to the sides of the columns. The Attic, which stands over the cornice of the entablature, is 5 feet 9 inches high, with a dentil cornice and three fascias below. The height of the pediment is 7 feet 1 inches, from the top of the cornice on the attic. ARRANGEMENT AND DESCRIPTION OF THE PLATES. 569 The height of the pedestal, from the bottom of the pediment to the top of the columns round the belfry, is 7 feet 10 inches. The columns and entablature round the belfry are 20 feet 10 inches high, and are similar to those in the portico ; the wall, which is seen between the columns, is rusticated above the two plinths. The apertures in the belfry, for letting out the sound, are 4 feet 2 inches wide, and 11 feet 3 inches high. The part where the dials of the clock are placed is of an octagonal form ; its height, including the two circular steps from the top of the cornice, round the entablature of the belfry, to the top of the cornice, above the dials, is 9 feet 10 inches. There are four dials in it, at right angles to each other, and four small apertures in the diagonal faces, each 3 feet wide and 4 feet high, filled in with perforated luffer boarding in the form of scales. The part over the dials, above the two circular steps, is of an octagonal form, with eight columns sup- porting an entablature. Its height, including the two circular steps at the top of the entablature, is 15 feet 8 inches. The diameter of each column is 1 foot 5-J- in., and 1 1 feet 7 inches high ; the entablature, 2 feet 7 inches. The height of the small pediments above the entablature is 1 foot 9 inches, with a honey-suckle betwixt each. The height of the spire above the top of the pediments to the top of the cross is 44 feet 9f inches, and this portion is ornamented with scales to the height of 28 feet 10 inches. The whole height of the steeple, from the ground to the top of the cross, is 152 feet. PLATE XVIII. FLANK ELEVATION of the same CHURCH, in the GRECIAN STYLE. The whole extent of this front, including the projecting porticos on the bottom line of the entablature, is 166 feet 3 inches. That between the two extreme half anta?, on each side of the bows, is 146 feet 8 inches ; and the plain part between the bows, is 88 feet 2 inches. Each of the bows is 26 feet 3 inches. The height, from the top of the steps to the top of the cills of the lower windows, is 3 feet 8 inches. The lower windows are 5 feet 2 inches wide, diminishing a little at the top, and their height is four feet 10 inches. The height between the under side of the lintel of the lower windows and the top of the cill of the upper windows is 6 feet 7 inches. The height of the windows above is 9 feet 6 inches ; and the breadth, at the bottom, 5 feet, diminishing to the top about 3f inches. The height from the under side of the lintel of the upper windows, to the lower line of the entablature, is 4 feet 5 inches. The height from the ground to the top of the roof is 50 feet 7| inches. The frames of the windows to be of metal. All the ornaments on the exterior of this building may be of terra-cotta, or of stone, if built in a country where both labour and stone are cheap. ^, PLATE XIX. BACK ELEVATION of the same CHURCH, in the GRECIAN STYLE. This view represents the entrance to the Catacombs, the Vestry-Room, and the Robing-Room. The columns, the width of the portico, and the spaces on each side of the portico, with the widths and heights, are the same as in the front elevation. The windows on each side of the portico are similar to those in the flank elevation, and are of the same height and breadth. The width of the entrance to the catacombs, in the clear, is 5 feet 8 inches at the bottom, diminishing upwards parallel to the sides of the columns; and its height is 10 feet 3 inches. The entrances on each side of the central one, are each .5 feet at the bottom and 10 feet 3 inches high. The doors are of oak, and open in two halves. The height of the projecting part, under the portico, is 15 feet 4 inches, from the top of the steps, to the top of the ornament which extends right and left. Over the central door are three urns, standing upon a small pedestal. PLATE XX. LONGITUDINAL SECTION of the same CHURCH in the GRECIAN STYLE. The lower part of this section represents the vaults, which are entered from the projecting part beneath the portico, at the east end, by a flight of steps descending downward. Their height from the ground to the top of the soffit of the arch i^7 feet 9 inches, and the width 6 feet. The height of the ceiling underneath the gallery, from the level of the floor of the body of the church, is 10 feet 5 inches: the small columns, which stand upon pedestals, to the height of the pewing, and which support the gallery, are 6 feet 4 inches high, and their diameter about a ninth part of the height. The front of the gallery, over the columns, is 3 feet 5 inches high. 71. 7 E 570 THE NEW PRACTICAL BUILDER, &C. The height from the floor of the body of the church to the top of the ceiling, beneath the roof, is 36 feet 8 inches. The columns in the communion-place are 21 feet 4 inches high, and their diameter 2 feet 4 inches. The shafts of the columns are represented in Scagliola: the entablature over the columns is 4 feet 11 inches high. The height of the columns in the Vestibule is 20 feet 8 inches, and their diameter 2 feet 5 inches. The height of the entablature is 5 feet, and the dome 9 feet 4 inches, from the cornice to the elliptical opening which admits the light into the vestibule from the small windows above. The height from the level of the vestibule to the top of the first platform, which leads up to the belfry, is 43 feet 6 inches ; to the second platform on the floor of the belfry, 57 feet ; and, to the third platform, where the clock-work is fixed, 78 feet 9 inches. The view of the part above the clock-work exhibits the manner of framing the different off-sets to the top of the spire. PLATE XXI. TRANSVERSE SECTION of the same CHURCH, in the GRECIAN STYLE. This section exhibits a view on looking towards the west end. The lower part shows the arrangement of the vaults, with the passages communicating to them. The depth of the vaults, on each side of the central passage, is 10 feet ; and of those on the right and left of the side passages, 9 feet. The width of the side passages is 3 feet 3 inches, and of the middle one 5 feet 9 inches. The openings in the exterior walls, which are sectioned, are of a circular form, and are to admit the light into the vaults, by means of circular walls built in front of them, and covered with a grating. Over the vaults is shown a section of the interior of the pews, the free seats down the middle aisle, and also the middle and side entrances to the body of the church. Upon the gallery is shown the Organ, and the side entrances from the staircases. The breadth of the gallery, which is shown in this section, is 14 feet 10 inches from the wall, to the front of the gallery. The width of the passage is 3 feet 4 inches, and each of the pews 2 feet 8 inches. The height of the seats in the pewing, is 1 foot 6 inches. PLATE XXII. GROUND PLAN of a CHAPEL. A, represents the Porch, recessed within two columns, 26 feet 4 inches by 4 feet 6 inches. B, an elliptical Vestibuk, with pilasters and niches, lighted from the top, 27 feet 3 inches by 16 feet 10 inches. D and C, Side Staircases to gallery, 25 feet by 13 feet ; with a circular staircase in one corner, leading to the children's gallery and tower. The size of the interior of the body of the chapel is 83 feet by 58 feet. The principal passage, repre- senting the free seats, is 8 feet within the clear of the pew-doors. The side passages are each 3 feet to die front of the seats next the walls. The pews are 3 feet wide ; the seats 1 foot ; the book-desk 5 j inches ; and the doors 1 foot 7 inches. The Pulpit, (n) is of an hexagonal form, with stairs ascending up to it. o, Reading-Desk, with clerk's seat in front. The little black circles represent the columns which support the gallery. The Communion-Place (H) is of an elliptical form, and raised one foot high. E, Vestry-Room, 18 feet 6 inches by 13 feet, with a fire- place, and small closets in the angles. The Strong Closet, (e) 6 feet 2 inches by 5 feet 2 inches. Water-" Closet, (g) 6 feet by 3 feet 9 inches, the mean proportion. F, Robing-Room, with fire-place, and closets of the same dimensions as Vestry-Room. G, entrance to the vaults. PLATE XXIII. PRINCIPAL ELEVATION of the same CHAPEL. This building is in the style of the Grecian Doric. The extent of its front, to the extremities of the pilasters, is 66 feet. Its height, from the ground to the top of the cross, is 112 feet. The entrance, or door, is raised 2 feet 8 inches above the level of the ground, ascended by 6 steps of rather more than 6 inches rise, which are continued all round the building. The opening of the door is 7 feet 3 inches in the clear at bottom, and 6 feet 10 inches at the top ; diminishing about one-seventeenth part of the breadth. The door is of oak ; it is divided into eight panels, and opens in two halves, to die height of the bead betwixt the third and fourth panel ; and is hinged to a vertical bead, which runs up by the side of the architrave of the door. The architrave is about two-ninths of the breadth ARRANGEMENT AND DESCRIPTION OF THE PLATES. 571 of the door. That part of the architrave which extends across the top of the door, is a little less. Over the architrave is a Cornice and Pediment, with an ornament at each corner, supported at each extremity of the cornice by a console. Over the door is a panel, which may be filled in with bas-relief, or an inscription. The Columns are 31 feet 5 inches high ; their diameter at bottom 5 feet 2f inches, and at top 4 feet. The pilasters are of the same width as the top of the column. The mouldings of the caps are similar to those of the Monument of Thrasyllus. The lower windows betwixt the pilasters, are 5 feet 5 inches wide at bottom, and at top 5 feet 1 1 in. The windows above are of the same width, at bottom as those below, and at top 5 feet 3y in. The architraves are 1 foot If in. with a break at top, of about 2 inches. The bars of the windows are of metal. The cills of the windows arell| inches. The height of the Architrave, Frieze, and Cornice, is 8 feet. The breadth of the triglyphs is 2 feet 5 inches. The height of the Pediment, from the top of the cornice to the top of the Cymatium, is 11 feet 6 inches : on the top, and at each extremity of which are placed Acroterice. That part which projects beyond the bottom of the Cupola, is to admit light into the vestibule by means of six small windows in the faces of the pedestal of the Cupola, which is concealed within it. The windows in. the belfry are 4 feet 5 inches wide, and 1 1 feet 6 inches high, to the top of the arch. The aperture of the latter is filled in with horizontal luffer-boarding. The pilasters round the belfry are 16 feet 6 inches high and 1 foot 1 1 inches wide ; the moulding in the caps, are the same as those in the front ; the bars are similar to the attic base ; the height of the entablature is 4 feet 2 inches, with wreaths in the frieze, and ornaments above the cornice. The part above die belfry, which contains the clock-work, is of an octagonal form, with a cornice and continued ornament above, similar to that on the top of the cornice of die Monument of Lysicrates. (Orders, Plate XXV.) The faces of the octagonal part is filled in widi four dials, at right angles to each other, and four small windows, 3 feet 6 inches wide, and 3 feet high : the apertures of which are filled in with luffer-boarding, in the form of scales. Above the octagonal part is a circular dome raised upon a step, with a ball and cross over it. PLATE XXIV. FLANK ELEVATION of the same CHAPEL. The whole length of this elevation is 142 feet between the two outer pilasters. That part between the pilasters, wherein the windows are, is 79 feet. The heights of the doors, pilasters, entablature, and cupola, are the same as those in the front elevation. The lower windows are 5 feet 4 inches wide and 4 feet 7 inches high, and dimmish at the top one inch and a half: the windows above these are 5 feet 4 inches wide, and 9 feet 8 inches high, and diminish 3 inches and three quarters at top. The architraves are 1 foot 1 inch and a half, with a break at top of about 2 inches on each side : over the top of the architrave is a cornice and a pediment, with a honeysuckle at each extremity. PLATE XXIV. A. THE BACK ELEVATION of the same CHAPEL. The general dimensions of this eleva- tion, in the heights and breadths, are the same as in the principal elevation, and differ only in the recesses, with the piers and arches turned over them between the antae ; and the entrance to the catacombs, with the recessed window within the arch, for lighting the Communion-place. The entrance between the columns is 5 feet 8 inches wide, and 9 feet 4 inches high, with folding-doors, and circular holes in them for ventilation. The height, from the top of the steps, round the building, to the top of the cornice over the entrance, is 12 feet 5 inches. The piers of the outer arch, behind the columns, are in the same face with the wall below the cornice over the door : the opening of the outer arch is 12 feet 8 inches wide, and 17 feet 8 inches high, to the soffit of the arch, the inner arch is recessed about 5 inches from die outer one, and forms a margin of 10 inches all round. Over the entrance is a sarcophagus standing on a plinth. The height of the piers, with the imposts and bases, which support the archivolts between the antae, is 21 feet 2 inches, the opening between the piers is 10 feet 9 inches, and the height, from the top of the steps to the top of the intrados, is 26 feet 6 inches. Over die centre of each window is a wreath. PLATE XXV. LONGITUDINAL SECTION of the same CHAPEL. This section is taken through the middle of the chapel from west to east. The lower part represents the brick vaults, whieli are entered beneath the 572 ,--. THE KEW PRACTICAL BUILBEFt, &C. porch, from the east end, by a flight of steps descending downwards. The height of the vaults is 8 feet from the ground to the top of the soffit of the arch ; the breadth 6 feet 6 inches, and the depth, 1 1 feet 6 inches. The thickness of the arches is a brick and a half. Above the arches is represented the dwarf wall, down the middle passage, (see transverse section, pi. XXVI,) for resting the Yorkshire slabs on, which cover the vacancy where the flues are laid for heating the chapel. The part which supports the Reading-Desk [is of an octagonal form, standing upon a small plinth, with eight insulated columns and entablature, similar to the Tower of the Winds ; the height of the columns is 5 feet ; their diameter, an eighth part of their height ; and the entablature a sixth part. Over the middle entrance, against the side of the Tower, is shown the side of the organ, raised upon a platform. The height of the pilasters in the Communion-Place, is 22 feet 6 inches, their breadth one-ninth part of the height, with plain caps and bases. The height of the entablature is 5 feet 6 inches. The general heights ot the interior of the body of the chapel, are given in the description of the transverse section. The height of the Vestibule, from the ground to the top of the elliptical opening in the dome, which admits the light from the small windows beneath the first platform of the tower, is 30 feet 6 inches ; the height of the pilasters is 18 feet, and of the entablature 4 feet 10 inches. Over the top of the cornice is a dome, whose transverse section is a semi-circle, and longitudinal section an ellipsis, with sunk panels, radiating towards the apex. The height, from the level of the vestibule to the flooring boards of the first platform in die tower, is 46 feet; the height, from the floor of the first platform to the floor of the belfry, is 15 feet; from the floor of the belfry to the third platform, 19 feet; from the third platform to the fourth platform, 11 feet ; and, from the fourth platform to the top of the soffit of the dome, 7 feet 4 inches. The dome is tied together by means of iron bars, radiating from a vertical spindle in the middle of it, and the ends of which are let into the stones and run in with lead. PLATE XXVI. TRANSVERSE SECTION of the same CHAPEL. This section represents the interior of the chapel, on looking towards the west-end, and shows the organ with the children's seats in front of it ; also the doors on the gallery, entering from the staircases. The side pews, passages, and truss to the gallery. The lower part, under the gallery, exhibits the principal entrance from the vestibule, the side entrances from the staircases, with a section of the interior of the pews, raised upon sleepers 5 inches above the level of the passages. The height of the gallery, above the level of the passages, is 10 feet. The height of the columns, which support it, is 6 feet 3 inches, and their diameter rather more than 9-J inches : they are raised upon pedestals to the height of the top of the pewing, are of the Grecian Doric, and may be either plain or fluted. The height of the front of the gallery is 3 feet 7f inches ; and the height, from the ceiling beneath the gallery to the ceiling of the roof, is 27 feet 9 inches. PLATE XXVII. PLAN of an OCTAGONAL MAUSOLEUM. A, the Porch, with two columns and pilasters in front, raised two feet above the level of die ground, ascended by 4 steps of C inches rise. The inter- columniation of the columns is of the Diastyk, with pilasters facing die flank of each column. The Entrance (/) is 4 feet 8 inches wide. B, Chapel, 19 feet diameter, from the front of the base of one column to the front of the opposite ; with eight columns, of the Doric order, and pilasters behind them. The intercolumniation of the columns is of the Systyk. c, c, c, c, c, are cells for the coffins, to be 6 feet 8 inches deep, and 2 feet 3 inches wide, with dwarf walls, 9 inches thick, carried up on each side, and arches turned over them ; (See the Section.) e, e, e, e, represent porticos, widi four columns, each projecting out from the body of the building 3 feet 9 inches ; the intercolumniation of these columns are of the Pycnostyle. a, a, a, denote pedestals for figures to stand on. PLATE XXVIII. PRINCIPAL ELEVATION of the OCTAGONAL MAUSOLEUM. This structure is in the Greek style, and would be suitable for a family mausoleum, placed in a retired situation on a nobleman's estate, especially if surrounded by water. The diameter of the body of the building, parallel to the front, is 49 feet 4 inches ; its height 43 feet 7 inches. The columns are designed from those of the Toner of the Winds. Those forming die porticos stand upon a plinth, 2 feet high. The height of the columns is ARRANGEMENT AND DESCRIPTION OF THE PLATES. 573 19 feet 3 inches. Their proportion, 8 diameters, and they diminish about a sixth. The entablature is 3 feet 1 1 inches, or nearly one-fourth of the height of the column. The architrave is 1 foot 6 inches ; the frieze, 10 inches; and the cornice, 1 foot 7 inches. The height of the pediment, exclusive of the cymatium, is one-fourth of the breadth of the frieze. For the real projections of the cornice and architrave, see the section through EF, on the preceding plan. In the central part of the front of the building is a sarco- phagus, fluted on the face, with a wreath in the centre, and over its top two reverse scrolls, with a honey- suckle between them. On each side of the sarcophagus is an urn. These emblems, supposed to contain the ashes of the dead, were used by the antients in their mausoleums. Over the central part of the body of the building is a segment dome, on the top of which is an opening to admit light into the interior. PLATE XXIX. SECTION of the OCTAGONAL MAUSOLEUM. This section is taken through EF, on the plan, and it exhibits the interior of the Chapel, and the cells, with the coffins in them. Over the top of the cells is a tablet, on which an inscription may be made. The columns are of the Grecian Doric, and they are raised upon a step of about 6 inches high ; their height is 17 feet 7 inches ; they are six diameters high, and diminished at the top 13 minutes of the base. The Entablature is designed from die Monument of Thrasyllus, the height of which is one-third of the column. Over the top of the entablature are semi-circular recesses, with busts, emblematic of those contained in the cells below. Over the top of these is a cornice, from which springs a semi-circular dome, with panels radiating from the apex. In the centre of the dome is a sky-light of a conical form, around which is a channel, with a hole perforated in it, for carrying off the rain. The soffits of the porticos are panelled, with roses in the centre. On the left is a female statue, standing on a pedestal, with an urn in her left hand, and a reversed torch in the other. PLATE XXX. GROUND PLAN of another MAUSOLEUM. (P.) BB, represent Porches, or entrances, with four columns, each 35 feet 6 inches by 8 feet 2 inches, in the clear ; dd, groined passages to the chapel and cemetry, with antae on the sides and columns at each extremity, 39 feet 4 inches in the clear of the columns, by 10 feet within the face of the antae. Chapel, 25 feet diameter, with niches and ante. Circular ceme- teries, 20 feet diameter, each containing 8 cells marked c. Flank cemeteries, 57 feet 5 inches in the clear of the columns, by 12 feet 4 inches within the face of the anUe, containing together 20 cells, d, winding staircase to the upper plan, eeeeeeee, pedestals for figures to stand on. CC, Flank Porches, with square pillars in front, 29 feet 6 inches, in the clear, by 10 feet 6 inches. The middle or largest rectangular cemetery, which is entered by a porch, from the flank cemetery to the right, between two pedestals supporting figures, is 60 feet within the face of the three quarter columns, by 38 feet, groined with four segmental arches, springing from the top of the entablature, which is carried round the top of the four insulated columns. The central part is crowned with a dome of an elliptical form, with panels radiating towards the apex. In the centre of the dome is an opening to admit the light, and from which a lamp may be suspended. The spandrels below the dome are fluted. PLATE XXXI. UPPER PLAN of the MAUSOLEUM. (P.) The arrangement of this plan, above the steps, within the parapet, is peripteral, being nearly surrounded by columns. The intercolumniations of the front and back are of the diastyle ; and those on the flanks and interior are of the systyle : f, shows the top of the winding staircase; a a a a, walk all round the'building ; gg, a flight of steps, leading to the interior,- which is entered between the antae within the flank columns; eeeeee, pedestals for figures to stand on; cc, rooms which may be appropriated to contain the relics of the deceased, 46 feet 3 inches by 10 feet 4 inches, in the clear. 72. 7 F 574 THE NEW PRACTICAL BUILDER, &C. PLATE XXXII. FRONT ELEVATION of the MAUSOLEUM. (P.) The whole extent of this building, above the sub-plinth at the base, is 118 feet 4 inches. The central part, forming the entrance, is 50 feet 4 inches ; the front of which is in a vertical face, and the sides of the projecting part in a sloping direction, having the appearance of an Egyptian entrance. The columns are designed from the Monument of Lysicrates; their height is 26 feet, and their diameter 2 feet 10| inches, raised 20 inches above the level of the ground, forming three steps. The entablature is 8 feet high, to the top of the ornament over the cornice. The frieze and architrave, 4 feet high, and converted into one face, in order to give it a more massy and solemn appearance ; upon it are figures in bas-relief, of various devices. The entrance, within the columns, is 15 feet high, 7 feet wide at the bottom, and 6 feet 2 inches at the top. The spaces on each side of the entrance, at the bottom, above the sub-plinth, are of 22 feet 6 inches, filled in with horizontal vermiculated rustics. The piers at each extremity are 1 1 feet 6 inches, diminishing <>n the front the same as on the flanks. Within the blocking-course, and 3 feet 6 inches below the top of it, is a walk all round the building, from which you ascend by a flight of steps, of 7 feet 9 inches high, to the inner part of the Mausoleum above. The projecting part of this, with the antse at the extremities, is, in extent, 52 feet 4 inches. The height of the columns, 26 feet 2 inches, and their diameter 2 feet 10f inches. The width of the antae is 2 feet 5 inches. The windows between the columns, in the clear, are 5 feet at the bottom, 4 feet 6 inches at the top, and 9 feet 4 inches high, filled in with trellis-work. In the centre intercolumniation is a figure, representing Hope or Charity, standing on a pedestal. The entablature over the columns is 6 feet 6 inches high, and the pediment 8 feet 4 inches. The height of the attic, (which is formed on the flanks by Caryatides holding reversed torches and urns, standing on a plinth, and crowned with an entablature,) is 12 feet 6 inches high. Above the attic rises a base, which is nearly square ; and, on the top of these, three steps are raised, with pedestals at each corner ; and on the latter, figures or urns may be placed. The height of the base, steps, truncated pyramid, and ornament at the top, is 68 feet 8 inches. PLATE XXXIII. FLANK ELEVATION of the MAUSOLEUM. (P.) The whole length of this front, above the sub-plinth, at the base, is 128 feet 5 inches. The central part, extending to the rustics, is 43 feet 9 inches. The parts, forming the rustics on each side of the centre, are each 31 feet 9 inches, and the extreme piers 10 feet 7 inches. The widths of the square pillars, forming the centre, are 2 feet 9 inches. The part which stands on the steps above show the entrance into the interior, and between the three central intercolumniations the tomb is seen, which denotes where the head of the family lies. PLATE XXXIV. represents a design for a MONUMENT, in the Grecian style, erected, in the year 1806, in the parish church of St. Benet, Gracechurch-street, London, at the expense of the late James Townley, Esq. of Ramsgate, and of Paul's Chain, Doctors' Commons, to the memory of an affectionate father and mother, the former of whom was many years the Rector of St. Benet's. The design is by Mr. Elsam, the architect, and it was executed, under his direction, by Richard Westmacott, Esq., Sculptor and Royal Academician, for the sum of fifty pounds. XXXV. REFERENCES TO THE PLATE WHICH REPRESENTS THE ROOFS OF ST. PANCRAS CHAPEL, OF ST LUKE'S CHURCH, AND OF CAMDEN CHAPEL. Figure 1. The Roof of the new Gothic Chapel of St. Pancras, Somers 1 Town, near London. Scantlings of Timbers. A, Oblique Tie-beam, 11 inches by 7 inches; at bottom, 10 in. by 7 in., top cut asunder : B, Collar-Beam, 10 in. by 7 in. : C, Principal rafter, 10 in. by 7 in. ; 9 in. by 7 in. at top : D, Common rafter, 5 in. by 2| in. : E, Purlins, 6 in. by 7 in. : F, Pole-Plates, 5| in. by 8| in. : G, Struts, 6 in. by 7 in. : H, Braces, 8 in. by 7 in. : I, Oblique braces, 6 in. by 7 in. ; K, Ribs to ceiling, 7 in. by 5 in. , ARRANGEMENT AND DESCRIPTION OF THE PLATES. 575 Figure I, No. 1, Section through EF, to a larger scale. y Kon , A7!rJ 3 TXoal.IIXXX Figure I, No. 2, Section through GH, to a larger scale. . { ^ }K r ,. Figure I, No. 3, Section through AB, to a larger scale. s3ji) lK>bl9V fi ni ^ fafa -) o lncnl 9it{ ej : F^wre 1, Afa 4, Section through CD, to a larger scale. - ^^ ^^3 KS -) Q OTa|rf , ': .Figure 2. Roof of St. Luke's Church, Old Street, London. Scantlings of Timbers. A, Double King-post of Oak, 9 inches by 5 inches each : B, Principal rafter, 8 in. by 7 in. at top, 6 in. by 7 in. at bottom : (N. B. Very slight :) C, Assisting rafter, 6 in. by 7 in. ; D, Tie-beams, 14 in. by 12 in. : E, Hammer-beams, 12 in. by 12 in. ; F, Stretching-beams, 10 in. by 8 in. : G, Queen-Posts of oak, 10 in. by 7 in. : H, Struts, 5 in. by 5 in. : I, Common rafters, 5 in. by 3 in. : K, Purlins, Sin. by C in. ; L, Wall-plates 12 in. by 8 : M, Poll-plates, 6 in. by 6 in. ; Ridge-boards, 12 in. by 2| in. ; Hips, 12 in. by 2| in. ,-' r o \i Figure 3. Roof of Camden Chapel, Camden Town, near London. Figure 3, No. 1, Section through ef, to a larger scale. _. - T r* 1 1 1 1 1 Figure 3, No.. 2, section through erf, to a larger scale. Scantlings of Timber. A, Tie-beam, 14 inches by 9 inches : B, Queen-posts of oak, 8 in. by 7 in. : C, Small Posts of Oak, 7 in. by 7 in. : D, Struts, 7 in. by 7 in. : E, Small Struts, 6 in. by 6 in. : F, Prin- cipal Rafters, at top, 9 in. by 7 in. ; at bottom, 1 1 in. by 7 in. : G, Common Rafters, 6 in. by 2| in. ; Horizontal Rafters to Flat, increased in depth to produce a proper current, 8 in. by 5 in. : H, Wall-Plate, 9 in. by 6 in. : J, Truss to Collar-beam, 5 in. by 3 in. : K, Collar-beam, 10 in. by 7 in. : L, Spiked to the sides of Tie-beam, 8 in. by 2 in. : M, Binders, 8 in. by 5 in. : N, Ceiling-joist, 3$ in. by 2| in. : O, Abut- ment-piece, 5 in. by 5 in. XXXVI. REFERENCE TO THE PLATE OF TRUSSES, LATELY EXECUTED IN CAMDEN TOWN CHAPEL, NEAR LONDON. Figure I.- Truss to the Gallery at the nest end, bearing on the Toner nail. H, Strut, 8 inches by G inches : J, Trimmer, 9 in. by 6 in. : K, Small Strut, 5 in. by 4 in. Figure 2. Truss to side Galleries. A, Bressummer, 10 inches by 8 inches : B, Girder, 10 in. by 8 in. : C, Binders, 13 in. by 3f in.; D, Bridging-joist, 4f in. by 3 in. ; E, Truss, 6 in. by 4 in. : F, Carriage, 8 in. by 4 in. : G, King (Oak), 4 in. by 4 in. : H, Binders to Ceiling-joists, 7 in. by 5 in. : J, Ceiling-joists 3 in. by 2| in. : K, Plate, 8 in. by 5 in. : L, Wall-plate, 12 in. by 6 in. Figure 3. End Gallery and Children's Gallery. A, Bressummer, 10 inches by 8 inches : B, Girders, 10 in. by 8 in. : C, Binders, 13 in. by 3 in. : D, Bridging-joist, 4 in. by 3 in. : E, Truss, 6 in. by 4 in. ; F, Carriage, 8 in. by 5 in. : G, Bearer, 7 in. by 4 in. : H, Puncheons, 3f in. by 3 in. : J, Binders to Ceiling- joists, 7 in. by 5 in. : K, Ceiling-joists, 3| in. by 2| in. : L, Plates, 9 in. by 6 in. : M, Oblique Girder, 12 in. by 8 in. : N, Carriage, 8 in. by 6 in. : O, Binders, 10 in. by 3 in. : P, Bridging-joist, 4 in. by 3 in. ; Q, Binders to Ceiling-joists, 7 in. by 5 in. : R, Ceiling-joist, 3| in. by 2| in. : S, Plate, 12 by 8 in. Figure 4. Truss to the nest end Gallery. T, Trusses, 6 inches by 4J inches: V, Strainers, 5 in. by 4{ in. : X, Abutment-piece, at each end, 9 in. by 6 in. ; ditto, in the middle, 7| in. by 6 in. XXXVII, XXXVIII, XXXIX ; DESIGNS FOE SHOP-FHONTS, PLATES I, II, and III. ; Windows, IV. and V, ; and a Chimney-piece, VI. These designs for SHOP FRONTS, &c. in the present improved style, are too clear to require more than a simple notice. The adoption of either, in preference to the others, will, of course, depend on taste and local circumstances. n am 1 -70 KQiTiiHDaaa OKA n am 1 -70 KQiTiiHDaaa OKA TazuEs 576 THE NEW PRACTICAL BUILDER, &C. 1o itdrmifi grcras, aril -janboi i -, eiau. TO aiui^r h**'** PLATE VI. of this series, represents a design for a Grecian Chimney-piece, adapted for a Dining Room. The ornaments in the pilasters represent ripe fruit : the tymphan of the pediment an antique wreath, with ribbons, the extremities of which are terminated by Grecian heads, taken from models in the British Museum. The design is by Mr. Elsam, the architect, under whose direction it has been executed by Peter Turnerelli, Esq., an eminent sculptor, many years patronized by the Royal Family and the most distinguished personages in these countries. .; tfion sir XL. DESIGN (on tmo plates) for a COUNTY COURT-HOUSE and PRISON ; by Mr, Elsam, the Architect. The DESIGN to which the explanation applies, has been studied with reference to all the conveniences which are usually required in Buildings of this description. It embraces the Civil and Criminal Courts of Justice, together with Grand and Petty Jury-Rooms, Judges' Retiring-Rooms, Water-Closets, Witnesses -Rooms, Clerks' '-Offices, &c. ; adjacent to which Building are the proposed PRISONS, consisting of a Gaoler' s-House, Turnkeys' -Lodge, Prison Buildings for Debtors and Felons, with a Chapel, Infirmaries, airing grounds, pas- sages, and avenues of communications to the Court-House, and every requisite appendage for 300 prisoners. The entire of the Prisons, it is proposed, should be enclosed by an external and an internal boundary wall, to prevent the possibility of intrigues being carried on by persons from the outside, to effect escapes. Esti- mated expense, 70,000. The arrangements of the interior of the Court-House, it is presumed, will not be considered injudicious ; inasmuch as the two Courts of Justice are detached from each other, by a spacious and well lighted circular Hall in the centre, which will admit of the most spacious communications with every part of the building, not only on the ground-floor, but also in the upper story, wherein a Gallery is proposed to communicate with all the upper apartments, consisting of the Grand Jury's Dining-Room, SherifFs-Offices, and such other con- veniences as may be deemed requisite for the County Records ; including rooms for the Housekeeper and attendants. The PRISON consists of four wings, springing from a central building, which comprises several well-siaed apartments, with a circular staircase in the centre, and passages of communication to every part of the build- ing ; each of which apartments, with the Chapel at the top, have windows diagonally situate, and commanding full views of the several Court- Yards ; so that each Class of Prisoners may bq inspected, not only on the ground-floor, but from each of the upper stories ; and, as this part of the building is intended chiefly for the use of the resident Gaoler, or Inspector, it is presumed, that a Prison built upon this plan could not fail, with due attention, to be minutely inspected, and consequently very secure : more especially, when it is con- sidered, that the approaches to the central point are, by means of the external avenues and the divisional pas- sages, to be ascended by flights of steps ; and, as shown, the internal Court- Yards in the form of quad- rants, are intended to be elevated several feet above the Prisoners' airing grounds, in order to afford the Inspector, or Gaoler, the most convenient opportunities of viewing the interior of the Prisoners' Court- Yards, between the circular iron railings. The Turnkey's-Lodge is proposed to be situate in the centre of the principal front of the boundary-wall' and is intended to comprise suitable apartments for the Principal Turnkey and his family, with a hot and cold bath, fumigating-room, and store-room for County cloathing, besides a place for public executions on a lead flat at the top. The FOUR WINGS, or Prison Buildings, are each of them divided in the centre by a building similar to the central building before described; the former of which are intended for the under Turnkeys, and the superior description of Prisoners ; each of which buildings have, also, circular staircases in their centres, with passages of communication to the several airing grounds, dining-halls, and work-rooms, which should be flagged or groined over in brick-work, and suitably fitted-up for the purposes intended. The upper stories of the several prison buildings are each alike ; and, in the event of the design, or any portions thereof, being carried into effect, the spaces above stairs should be divided into the same number of compartments as the ARRANGEMENT AND DESCRIPTION OF THE PLATES. 577 .038 .ftaajraa JAOITO/HI ws* SHT OTS groined arches or piers represent, which will produce the same number of sleeping cells on each of the three upper stories ; and these, being flagged or vaulted over, and secured with barred iron window gratings, and strong door locks and bolts, may be considered perfectly secure. The uppermost stories of the centre buildings comprise the Infirmaries ; and the lead flats, over the right and left hand wings, the airing places, for the convalescent to walk upon. Water-Closets should be provided within the buildings, and the utmost care taken to ventilate and well drain. If the Prison is situate near a running stream, it wiH be the more desirable, both as regards the health of the Gaoler and his- inmates. This, and the mode of drainage, are subjects of the greatest importance ; as is also the acquire- ment of good and wholesome spring water, which should be pumped daily into a central reservoir fo the use of the Gaoler and his family, as well as for the numerous persons who may at times be confined within the walls of the Prison. ;^ References to the Ground Plan. A, Great Hall : B, Criminal Court : C, Civil, or Nisi Prius Court : D, Judges' Retiring Rooms : E, Petty Jury's Retiring Rooms : F, Staircases : G, Grand Jury Room : H, Clerks' Room : I, Witnesses' Room : K, Waiting Rooms : L, Clerk of the Peace's Office : M, Witnesses' Room : N, Clerks' Rooms : O, Portico : P, Yard of communication, with Privies : Q, Yard for females about to be tried : R, Yard for males about to be tried : S, airing grounds for eight classes of prisoners : T, Inspection, yards : V, Gaoler's house : U, Under-Turnkey's, and superior prisoners : W, Dining Halls and Work Rooms : X, Gaoler's passages of communication : Y, avenues between the boundary walls : Z, Turnkeys' Lodge, c. XLI. DESIGN (on two plates) for a SMALL COUNTY PRISON ; by Mr. Elsam, the Architect. The DESIGN which accompanies this explanation, was executed at Cavan, in Ireland, in the year 1810, and cost jl2,000. Since that period it has been adopted, with some deviations, in various parts of the United Kingdom ; and latterly, the Design has been laid before the Prison Committee of the House of Commons, and other public bodies, who have been pleased to testify their approbation of the principle of its arrange- ments : and these circumstances have induced the author to extend its publicity, through the medium of this work, in the hope that it may induce other professional men to elicit their ideas upon the subject. The PLAN comprises three wings, besides a central building, called the Gaoler's house, with a Turnkey's Lodge, situate in the principal front of the boundary-wall, which encloses the different airing grounds and the several buildings, as before described. The centre building, on the ground floor, comprises the Gaoler's apartments, from whence the principal Court-Yards may be inspected. The first floor comprises the Chapel, to be divided and sub-divided as occasion may require ; and the second floor, the Infirmary, to be divided in like manner. This building will also afford suitable accommodation for two or three female debtors. The Left Wing will accommodate 1 6 debtors, with sleeping rooms in the upper stores ; besides a spacious day room on the ground floor, which should be fitted up with every convenience the place will admit of, to ameliorate the condition of this class of unfortunate persons. The Right Wing will also accommodate 1C prisoners, with sleeping cells in the upper stories, to be divided, as described in the references, for persons, under their respective sentences ; as likewise those for misdemeanors. The Rear Wing will also accommodate 16 prisoners, accused of felony, with separate sleeping cells in the upper stories, besides day rooms on the ground floor ; for each of which classes, the plan provides separate airing grounds, with detached privies. The Staircase, in the centre, will communicate with every part of the building, through which the dif- ferent classes of prisoners are proposed to be removed, alternately, from their respective day rooms and sleeping apartments, every morning and evening : by which arrangement it will be difficult for any prisoner- to make an escape, and in this respect, the plan is different from any other design of the same description. 72. 7 G 578 THE NEW PRACTICAL BUILDER, &C. The Turnkey's Lodge, provides apartments for that officer, with sleeping cells for female felons, King's evidences ; a room for county cloathing, a hot and cold bath, and a place for public executions in the most con- spicuous part of the building : attached to which, are two airing grounds under the inspection of the Turnkey. By the preceding distributions, it will be seen that a prison, built upon tliis plan, will conveniently lodge and secure from 50 to 60 prisoners, in the ratios of persons usually confined in small County Prisons, where the number of debtors, and the proportionate quantity of other prisoners, do not require an extensive classification. References to the Ground Plan. A, Gaoler's Apartments : B, Debtors' Day Room : C, Misdemeanors Day Room : D, Day Room for Prisoners under sentence : E, Day Room for accused Felons. F, G, H, I, K, L, M, N, Airing Grounds : F, for Debtors : G, for Misdemeanors : H, for Prisoners under sentence : I, for accused Felons : K, for King's evidence : L, for women Felons : M, for female Debtors : N, for refractory Prisoners : O, Turnkey's Hall, or Lodge : P, Hot and Cold Baths ; Q, Fumigating Room, with an oven : R, Staircases : T, Privies. The sleeping apartments for the Debtors, and cells for the Felons, are proposed to be in the upper stories ; this is recommended, not only for the sake of security, but likewise to produce a free ventilation throughout, which cannot be better effected, than by the alternate removal of each class of prisoners from their several day rooms and sleeping apartments every morning and evening, in the manner before described, by means of the central staircase. ma Jcil* oatifilaib -to IK -, .at ..I...,, .74, ..,,...., .88 ... _ , . - ' - 'rinso 10 - ,xori 10 . ;ubts *d. GLOSSARY AND GENERAL INDEX. 579 ttliqS' >iJRo )*A. , bloo fcoc ioA t- ^jm. iol moot * ; ;bi? 5 gm&Imd <;3 nioft 37i.':>5* fcne LO'feb:k-rfmua aAt aW* ^n-wh 5 GLOSSARY OF TECHNICAL TERMS AND GENERAL INDEX, 1 WITH REFERENCES TO THE PAGES IN WHICH THE VARIOUS PARTICULARS ARE EXPLAINED ; AND INCLUDING AN EXPLANATION OF SUCH TERMS AS HAVE NOT ALREADY BEEN PARTI- CULARLY DEFINED. M '< jjifojfl ,,O xancw -I y AARON'S ROD ; an ornamental figure, representing a rod with a serpent twined about it, and, called by some, though improperly, the Caduceus of Mercury. Abacus ; the upper member of a capital of a column, serving as a kind of crown piece in the Grecian Doric, and a collection of members or mouldings in the other Orders. See Q. Orders, plate L, and page 467. Abreuvoir or Abrevoir, in Masonry ; the joint or junction of two stones ; or the space or interstice to be filled up with mortar or cement, Abreuvoir also signifies a Bathing House or Place. Abutment or Butment ; 218, 328. Acanthus ; a plant, the English Bear's Breech, the leaves of which are represented in the capital of the Corinthian Order, &c. Acanthine means ornamented with leaves of the acanthus. Accessories ; in architectoral composition, those parts or ornaments, either designed or accidental, which are not apparently essential to the use and cha- racter of a building. Accompaniments ; subordinate buildings or ornaments. Accouplement, in carpentry ; a tie or brace, or the entire work when framed. Acropolis ; from the Greek : the highest part of a city, the citadel or fortress. The Acropolis, or citadel, of Athens is frequently noticed in the pre- sent work. Acroterium ; (plural, Acroteria) the extremity or ver- tex of any thing ; a pedestal or base placed on the angle, or on the apex of a pediment, which may be for the support of a vase or statue. ./Edes ; among the antients, an inferior kind of temple. JEgis; in decoration, a shield or breast-plate, par- ticularly that of Minerva. ^gricanes; sculptures representing the heads and skulls of rams ; commonly used as a decoration of antient altars, friezes, &c. .Eneatores ; sculptures representing military musi- cians. Aerial Perspective ; the representation of objects, weakened and diminished in proportion to the dis- tance from the eye. jEtoma ; a pediment, or the tympanum of a pediment. Aile or Aisle ; a walk in a church, on the sides of the nave ', the wings of a choir. Air-trap ; an opening for the escape and admission of air. Alcove ; a recess or part of a chamber, separated by an estrade, or partition of columns, and other cor- responding ornaments. Aroeostyle ; the greatest interval or distance that can be made between columns. ^Eosystyle, see page 466. Algebra, defined 33. Algebraic quantities 33. notation 34. equations 34, 40. fractions 34, 40. addition 36. subtraction 37. multiplication 38. division 39. proportion 41. Alto-relievo or High-relief; that kind or portion of sculpture which projects so much from the surface to which it is attached, as to appear nearly insu- lated. It is therefore used in comparison with Mezzo-relievo, or Mean-relief, and in opposition to Basso-relievo or Low-relief. Amphitheatre ; a spacious edifice, of a circular or oval form, in which the combats and shows of an- tiquity were exhibited. Amphora (plural, Amphorae) ; a vase or earthen jar, with two handles ; among the antients, the usual receptacles of olives, grapes, oil, and wine. Hence, in decoration, Amphoral means shaped like an amphora or vase. Amulet ; in decoration, a figure or character to which miraculous powers are supposed to be attached, and which particularly distinguished the buildings of Egypt. Ancon ; in decoration, a curved drinking cup or horn. The arm of a chair. Ancones ; ornaments depending from the corona of Ionic door-ways, &c. The trusses, or consoles, or brackets, sometimes employed in the dressings of apertures, as an apparent support to the .cornice, upon the flanks of the architrave. 580 THE NEW PRACTICAL BUILDER, &C. Angels ; brackets or corbels, with the figures or heads of angels. Angle-Bar ; the upright bar of a window, constructed on a polygonal plan, standing at the meeting of any two planes of the sides. A mullion. Angle-Bracket. See Bracketing. Angle-Chimney ; a chimney in the angle, or in a side formed at an angle of the apartment. Angle-Float. See Float, 390. Angle-Rafter; in carpentry, otherwise Hip-rafter. See Hip. Angular Capital ; the modern Ionic or Scammozian capital, which is formed alike on all the four faces, so as to return at the angles of the building. Annular- Vault ; a vault rising from two circular walls ; the vault of a circular corridor. Annulet or Fillet ; a small square member in the Doric capital, under the quarter-round. See Or- ders, O, Plate I. It is also used to imply a narrow flat moulding, common to the various parts of the columns, particularly their bases, capitals, &c. Antae ; a species of pilasters common in the Grecian temples, but differing from pilasters, in general, both in their capitals and situation. See page 509, and Orders, plates XIII, XIV, XX, XXIV. Apartments or Rooms, usual form of, 430. , fancy or fanciful, 432. , proportions of, 437. , suite of, 436. Apertures of Doors and Windows, proportions of, 436. Apron, in plumbing, the same as Flashing. Arabesque or Moresque ; something done after the manners of the Arabians or Moors, and destitute of human and animal figures. Arcade ; an aperture in a wall, with an arched head : it also signifies a range of apertures with arched heads. Arcades are frequently constructed as por- j ticos, instead of Colonnades, being stronger, and less expensive. In the construction, care must be taken that the piers be sufficiently strong to resist the pressure of the arches, particularly those at the extremities. The arcades of the Romans were seen in triumphal arches, in theatres, amphi- theatres, and aqueducts; and frequently in temples. They are common in the piazzas and squares of modern cities, and may be employed, with great propriety, in the courts of palaces, &c. For an application of the Orders to Arcades, see Plate XCII, annexed to the plates of Orders, but erro- neously numbered. Arc-boutants, or Boutants ; arch. formed props, in Gothic churches, &c. for sustaining the vaults of the nave; their lower ends resting on the pilas- tered buttresses of the aisles, and their upper ends resisting the pressure of the middle vault, against the several springing points of the groins. They are, at times, called flying-buttresses, arched but- tresses, and arch-butments. Arch ;. a part of a building supported at its extremi- ties only, and concave towards the earth or hori- zon : but arches are either circular, elliptical, or straight : the last being so termed, but improperly, by workmen. The terms arch and vault properly differ only in this, that the arch expresses a nar- rower, and the vault a broader, piece of the same kind. Arches ; definition, 328, 428 : Scheme Arch, 428 ; Semi-circular Arch, 428: Arch in Carpentry, 109. In Masonry, description of the sections of, 316 : pa- rabolic, 316 ; curve of, how to describe, 317. In Stone- cutting, a semi-circular right arch, 318 ; ellip- tical arch with splayed jambs, 319 ; to find the joints of, 319 ; oblique circular arch, 321 ; oblique arch, 323 ; a semi-circular arched passage, be- tween two semi.circular arched vaults, 324; an archway, revealed and sphyed, &c. 325. Arches, straight ; heads of apertures which have a straight intrados in several pieces, with radiating joints, or bricks tapering downwards. Archway. See Arch, above. Architectonic ; something endowed with the power and skill of building, or calculated to assist the architect. Architecture, antient, of Great Britain, recognized as the Gothic, Saxon, and Norman, modes of build- ing, 514. Architrave ; a beam ; that part of an entablature which lies immediately upon the capital or head of the columns. See pages 311, 444, 445, 453, 465, 467. Archivault or Archivolt ; 444 : of the arch of a bridge, 328. Arena for Stairs, 440. Arris, 218: Arrises, 442. Ashlar, 328 : Ashlaring, 310. Ash Timber, 260. Astragal, 468, 469 : and M and fig. 9, Orders, pi. I. Axiom, Axioms, Geometric, 14, 15: In Perspec- tive, 420. B. BACK ; generally that side of an object which is op- posite to the face, or breast : but the back of a handrail, is the upper side of it ; that of a rafter, is the upper side of it in the sloping plane of one side of a roof. Back-shutters, or back-flaps ; additional breadths hinged to the front-shutters, for completely closing the aperture, when the window is to be shut. Back of a slate, 402. Bagnio ; the Italian name for a bath, or bathing- house : answering to the Greek Balaneia, and the Latin Balneum. Balcony (from the French Balcon) ; an open gallery, projecting from the front of a building, and com- monly constructed of iron or wood. When a por- tico or porch is surmounted with a balcony, it is commonly of stone, with iron or wood. See Plates of Elevations, I, III, IV, V, XV. Baluster ; a small kind of column or pillar, belonging to a Balustrade. Balusters, in Joinery, 200. Balustrade ; a range of Balusters, supporting a cor- nice, and used as a parapet or screen, for conceal- ing a roof or other object. GLOSSARY AND GENERAL INDEX. 581 Bande or Band ; a narrow flat surface, having its face in n vertical plane: hence Bandelet, a little band, any flat moulding or fillet. Banded Column ; a column encircled with Bands, or annular rustics. Bank of England, dome of, 314. Banker, in Masonry, 329. Banquet, 329. Barge Course ; that part of the tiling which projects over the gable of a building, and is made up be- low with mortar. Barker, 402. Base ; the lowest part of a figure or body. Base of a Column, 453, 466 : Orders, Plates I. XXII. Bases and Surbases, 165. Bass, in Bricklaying, 384. Basso-relievo. See Alto relievo, Bastard Stucco. See Stucco. Batterdeau or Coffer-dam, 329. Battens, 1.59. Batter, 329, 42C. Battering, 426. Baulk, 219. Bay- Window ; a window projecting from the front, in two or more planes, and not forming the segment of a circle. Bay, in plastering, 384. Bead, joinery, 219. Bead and butt, 164. flush, 164. quirk, 160. double quirk, 161. Beads, 443. Beaking Joint, in Carpentry ; a provincial term, de- noting that the heading joints of the boards of a floor Jail in the same straight line. Beam, tie-beam, &c. 219. Beam-filling; filling up the space, with stones or bricks, from the level of the under edges of the beams to that of their upper edges, c. Bearers, 219. of stairs, 189. Bearing, 219. Bearing- Wall or Partition ; in a building, is a wall resting upon the solid, and supporting some other part, as another wall, &c. Beauty, or the harmonious disposition of the parts of buildings, 448. Bed of a brick, 384. slate, 402. -< stone, 329. Bedding-stone, 384. uiir jno-fl gnnj Beech timber, 259. Belfry, antiendy the campanile ; the part of a steeple in which the bells are hung. Belvedere ; a turret, lookout, or observatory, command- ing a flue prospect, and generally very ornamental. Bench, carpenter's, 219. Bevel, carpenter's, 219. Binding-Joists; those beams in a floor which support transversely the bridgings above, and the ceiling joists below. .: jpri' 73. Binding-Rafters. The same as Purlins. See Pur- lins. Bird's mouth. 219. Birch timber, 262. Blade of a Tool, 219. Blocking Course, 311. Blockings in Joinery, 219. Blocks in Building, 447. Board, 219. Boarding circular roofs, 140. Boasting ; in stone-cutting, paring the stone irregu- larly with a broad chisel and mallet ; in carving, the rough cutting of the outline, before the incisions are made for the minuter parts. Bohemian Plate Glass, 421. Bond, English and Flemish, 347 to 356. in masonry 329. or lap of a slate, 402. Bond- timbers, 219. Boning ; in carpentry and masonry, the art of making a plane surface by the guidance of the eye. Joiners try up their work by boning with two straight- edges, which determine whether it be in or out of Winding ; that is to say, whether the surface be twisted or a plane. Bosse or Boss, in sculpture ; relief or prominence : hence Hostage, the projection of stones laid rough, to be afterwards carved into mouldingc, capitals, or other ornaments. Bossage is also that which is otherwise called Rustic mark ; consisting of stones which seem to advance beyond the naked of a building, from indentures or channels left in the joinings ; these are used chiefly in the corners of edifices, and thence called Rustic quoins. Bottom-Rail, 219. Boulder-Walls ; those constructed of flints or pebbles, laid in strong mortar. Bow-Window ; a window forming the segment of a circle. , e^i Boxings of a window, 220. "\fa <)t{riud'j : Brick flues, or funnels, 367. ->Ai ;9vn srfj groins, construction of, 857. ^agaiijod hatai Bricklaying, 343 to 368. Bricklayer's Tools, 384.. -.;, iqg fsiavoi-; itij Bricknogging, 356. Bricks, different qualities .of, 343. .-(, hne , a course of, 348. ,;(f s 'io Jisq i: headers, 348. >i -Sf&nnm ban ,vlo so stretchers, 348. ..tots 1ad noi Brick- work, great principle of, 351. 7H 582 THE NEW PRACTICAL BUILDER, &C. ^A&'Ji^'.L ULJWltlr.^ai.' \-ir.*\ KAnavfew^fv Bridges, centrings for, 28 J. construction of, 289. foundation of, 804. London Bridge, 292. Westminster Bridge, 293^""i ;l Blackfriar's Bridge, 293. Waterloo Bridgef293. Southwark Bridge, 293. Theory of 293 wooden, construction of, 283. Bridging floors, 220. gutters, 220. n j . 'i LljM)p'w>vjj oifli 'Rvnini viova Broad-stone ; the same as Free-slope. Buffet ; an ornamented cupboard, or cabinet, for plate, glasses, china, &c. Building Act, 14 Geo. III. substance of, 363. in general, 424. lUlUll-l ,nr <|UOv. . length, depth, and height, in, 425. .BuUdings, on the beauty of, 448. T, 1- 1 1, - 1 ' Burrs ; clinker bricks. T> At Butment, see Abutment. Butt-end of timber ; the largest end next to the root. Buttery; a store-room for provisions. Butting-Joint, 220. Buttress or Plaster Bricks ; those made with a notch at one end, half the length of the brick, and used for binding work built with great brick. Buttresses, flying, &c. see Arc-boutants. -r '. j s> c. CAPUCEUS, an emblem or attribute of Mercury ; a rod entwined by two-winged serpents. Caisson, in Masonry, 329. Camber ; an arch on the top of an aperture, or on the top of a beam : whence Camber-rvindorvs, &c. Camber-beams, 129, 221. Campana ; the body of the Corinthian capital. Campanae, or Campanula, or Guttse ; the drops of the Doric architrave. Campanile ; antient name for a belfry. Cantalivers, 221, 447. Cant-moulding ; a bevelled surface, neither perpen- dicular to the horizon, nor to the vertical surface to which it may be attached. Cap, in joinery ; the uppermost of an assemblage of parts ; as the capital of a column, the cornice of a door, &c. Capital of a column, 46G. Carcase of a building, 221. Carcase roofing^ that which supports the covering by a grated frame of timber-work. Carpentry, in general, 105 to 158. Carriage of a stair, 188, 221 ; to carry up, 221. Caryatidae or Caryatides ; so called from the Cary- atides, a people of Caria ; an order of columns or pilasters, under the figures of women dressed in long robes, after the manner of the Carian people, and serving to support an entablature. This order is styled the Caryatic. Case of a Door ; the frame in which the door is. hung. interment iixt ,'. tbrfl IMIB no io1 Casements; sashes or glass frames, opening on hinges,. and revolving upon one of the vertical edges. Castellated ; built in imitation of an antient castle. Casting or warping, 221. Casting ornaments, 377. Casting, in plumbery, 404. Catacomb ; a subterraneous place for the inte of the dead. Cavetto, 161, 443, 468, 469. Ceiling, in plastering, 389. , cylindric coved, 431. , groined, &c. 431. , pendulous, 431. , waggon-headed, 430. Cements and Mortar, composition of, 329. Cements; lime and hair, 371 ; fine stuff, 372 ; stucco, S72; mortar, so called, 372; gauge stuff, 372; Roman or outside stucco, 378 ; charges for, 379. Centrings of Bridges, &c. 281, 293. Centrolinead, use of the, 559. Chain-timber, in brick building ; a timber of large' dimensions placed in the midule of the height of a story, for imparting strength. Chamfering, or cyphering, in carpentry, 221. Chancel ; the communion place, or that part of a Christian church between the altar and balustrade which incloses it. Chapel ; plans and elevations for a chapel in the modarn style, 570 to 572. Plates XXII. to XXVI. Chantry ; a small chapel, on the side of a church, &c. Chapiter ; the same as Capital. Chaplet ; a small carved or ornamented fillet. Chesnut timber, 262. Chimneys, 367 ; stacks of, 434 ; widths of, 434 ; breasts of, 434 ; construction of, 434 ; to prevent smoking, 434. Chimney-piece in the Grecian style, design for, 575; and Elevations, second series, plate VI. Chimney shafts, 434. Church, in die Grecian style, plans and elevations of, 5GS to 570. Plates XVI to XXI. Circular roofs, boarding of, 140. , purlins in, 152. Circular sashes, 177. Gills or Sills, in masonry, 310. Clamp, in carpentry, 221. Clamping, in joinery ; securing boards with clamps. Clear-cole and finish, 221, 417. Clear story windows, 221. Clinker, in bricklaying, 390. Cloacae ; the Roman name for sewers, drains, and sinks, conveying filth from the city into the river. Coarse stuff, in plastering, 390. Cocking or Cogging, 129. Coffer-dam ; a hollow space, formed by a double range of piles, with clay rammed in between, for the purpose of constructing an entrance-lock to a canal, dock, or bason. See also Batterdeau. Cogging. See Cocking. Coin or Quoin ; a corner or angle made by the two surfaces of a stone or brick building, whether ex- ternal or internal. . GLOSSARY AND GENERAL INDEX. Collar; a ring or cincture. Collar-beam, 127. Colonnade ; a range of columns, whether attached or insulated, and supporting an entablature. Column, 453, 465. , step of, 466. Columns, diminution of, 166, 459. ' , in joinery, flutes and fillets of, 167. and their proportions, in masonry, 308. Colours for painting, &c., table of, 412; catalogue and description of, 413. Comparted; divided into smaller parts; or parti- tioned into smaller spaces. Compo or Compos, 390. Composite Order, 452, 454, 498 : References to the plate of, 503. Composition, in plastering, 382. Cofferdam or Batterdeau, 328. Conic Sections, 82 ; definitions in, 82 ; the ellipse, 83; hyperbola, 94; parabola, 102. Conservatory; a superior kind of Greenhouse, for valuable plants, &c., arranged in beds of earth, with ornamental borders. Console ; a bracket or projecting body, shaped like a curve of contrary flexure, scrolled at the ends, and serving to support a cornice, bust, vase, or other ornament. Consoles are also called, according to their form, ancones or trusses, mutules, and modillions. Continued ; uninterrupted ; unbroken ; as a con- tinued attic, pedestal, &c., not broken by pilasters or columns. Contour ; a French word for Outline. Coping ; the stones laid on the top of a wall, to strengthen and defend it from injury. Corbeils ; carved work, representing baskets filled with fruit or flowers, and used as a finish to some elegant part of a building. This word is some- times used to express the bell or vase of the Corinthian capital. Corbels ; a horizontal row of stones or timber, fixed in a wall or on the side of a vault, for sustaining the timbers of a floor or of a roof; the ends, projecting out six or eight inches, as occasion may require, in the manner of a shoulder-piece, and cut at the end according to fancy, in form of an ogee, &c., the upper side being flat. In the castellated style of architecture, the Corbels are a range of stones, projecting from a wall, for the purpose of support- ing a parapet or superior part of the wall, which projects beyond the inferior part. Cornice; a crowning; any moulded projection which crowns or finishes the part to which it is attached. The Cornice of an order is a secondary member of the order itself, or a primary member of the enta- blature. The latter is divided into three principal parts, and the upper one is the cornice. Cornices, 310, 444, 445, 453, 466. Cornices, in bricklaying, 358, 375, 376. Corinthian Order, 451, 454, 504; references to the plate of the, 508. Cornucopia ; the horn of plenty ; represented in sculpture under the figure of a large horn out of which issue fruits, flowers, grain, &c. Corona, Larmier, or Drip, 443, 469, 470, 474. Corridor ; a long gallery or passage around a build- ing, and leading to the several apartments. Counter-forts ; projections of masonry from a wall, at certain regular distances, for strengthening it or resisting a pressure. Counter-guage, in carpentry ; a method of measuring joints by transferring the breadth of a mortise to the place of the other timber where the tenon is to be made. Counter-lath, in tiling; a lath placed, by eye, be- tween every two guaged ones, so as to divide every interval into two equal parts. Jj oJ ., Country-house.' See Villa, Country Residences ; best situations for, 448, 449 : arrangements of, 449. County Court-House. See Prison, and p. 575. Coupled Columns ; those disposed in pairs, so as to form a narrow and wide interval alternately. Couples ; rafters framed together in pairs, with a tie,, which is generally fixed above the feet of the rafters. Course ; a continued level range of stones or bricks, in a wall, &c. Course of Bricks, 390. Coursing Joint ; the joint between two courses. Cove ; any kind of concave moulding ; also the con- cavity of a vault. Hence, a coved and flat ceiling is a ceiling of which the section is a portion of a circle, springing from the walls, and rising to a flat surface. Covent-Garden, the church of, 482. Cover, in slating ; the part of the slate that is hidden : the exposed part being called the margin. Cover- way, in roofing; the recess or internal angle left to receive the covering. Covered-way ; a passage arched over. Covering of solids, 107; of circular roofs, 130; of polygonal roofs, 146. Coverings for roofs, comparative state of the weight of, 398. Coves and Cornices, bracketing for, 138. Coving ; an exterior projecture, in an arched form, now disused. The covings of a fire-place are the in- clined vertical parts on the sides, so formed for contracting the space, &c. tt dau^ Crockets; in the pointed style of architecture, the small ornaments placed equi-distantly along the an- gles of pediments, pinnacles, &c. See Elevations, pl.IX. .,;> Crosettes, in decoration ; the trusses or consoles on ; the flanks of the architrave, under the .cornice. Cross-grained stuff, 221. Cross-springers ; in groins of the pointed style, the ribs that spring from one diagonal pier to the other. Crown ; the uppermost member of a cornice, includ- ing the corona, &c. Of an arch, its most elevated line or point. Crown and other glass, 420. Crown-post, 221. Crypt ; an antient name for the lowest part or apart- ment of a building. Cube-root, extraction of, 557. Cupola ; a dome, arched roof, or turret.. 584 THE NEW PRACTICAL BUILDER, &C. Curling-stuff, 221. Curtail-step in stairs, 189. Cusps ; the pendents of a pointed arch, &c., two of which form a trefoil, three a quadrefoil, four a cinquefoil, &c. See Elevations, pi. X. and descrip- tion of the same, page 5C5. Cyclograph, use of the, 562. Cylindro-spheric groin. See Groin. Oymatium, or cyma, or summit of a cornice, 443, 468, 469, 470. Cyma-recta, 443, 469; in joinery, 161. Cyma- reversa, 443, 472. D. DADO. See Note, page 163. Dead-shoar ; an upright piece of wood, built up in a wall which has been broken through, in order to make some alteration in the building. Deal-timber, 222. Decastyle, or Diastyle, 466. Decimal Arithmetic, 5.51. Demi, or Semi, or Hemi, signifies one half. Hence Semi-circle, Hemi-sphere, &c. Demi-relievo, in carving or sculpture, denotes that the figure rises one half from the plane. See Alto-relievo. Denticulated, 447. Dentils, 447, 469. Derby, in plastering, 390. Die of a pedestal ; the part comprehended between the base and cornice. Die, in plastering, 890. Diglyph ; a tablet with two engravings or channels. Dimensions and Hingeing, 170. Diminished Bar, in joinery ; the bar of a sash that is thinnest on the inner edge. Discharge, in carpentry, 222. Dish-out ; to form coves by means of ribs, or wooden vaults for plastering upon. Distemper ; in painting, the working up of colours with something besides mere water or oil, as size, or other glutinous or unctuous substances. Ditriglyph; having two triglyphs over an intercolumn. Dog-legged Stairs, 189, 190, 193, 195. Domes, construction of, 312. Dome of St. Sophia, at Constantinople, 313; of St. Peter's, at Rome, 314; of St. Paul's, London, 314; of the Hospital of the Invah'ds, at Paris, 314 ; of St. Genevieve, at Paris, 314 ; of the Bank of England, 314. Door-frame, 222. Door-cill or sill, 428. Doors, 428. Doors, jambs of, 428. , soffit of, 428. , proportions of, 436. ., . -, with ovolo and fillet, 164. , folding, 164. , shutting joints of, 182. , jib, 183. Doric order, 451, 453, 486 ; references to the plate of, 492. Dormer or Dormant Window, 222. Dots, in plastering, 390. Double-hung sashes ; in joinery, those of which the window contains two, and each moveable by means of weights and lines. Dove-tailing ; in joinery, a method of fastening one piece of wood to another, by projecting bits, cut in the form of dove-tails in one piece, and let into corresponding hollows in another. Drag, in carpentry, 222 ; in masonry, 338. Dragon-beam, 222. Dressings ; all mouldings projecting beyond the naked of walls or ceilings. Drift, in masonry, 338. Drip. See Corona. Drips, in flats or gutters, 407. Drops ; in ornamental architecture, small pendent cylinders, or frustums of cones attached to a sur- face vertically, with the upper ends touching a horizontal surface, as in the cornice of the Doric order. See Orders, plates I, II, V. Drum or Vase, of the Corinthian and Composite capitals ; the solid part to which the foliage and stalks, or ornaments, are attached. Drying oils, 411. Duodecimal Arithmetic, 554. Dwarf-wainscotting ; that wainscotting which does not reach to the usual height. Dwarf-walls ; those of less height than the story of a building. Dye ; the plain part of a pedestal, between the base and cornice. E. EAVES ; the margin or edge of a roof, overhanging the walls. , in slating, 402. Echinus or ovolo, Roman, 472. Edging, in carpentry, 222. Elbows of a Window ; the two flanks of panelled work, one under each shutter, and generally tongued or rebatted into the back. Embattled; a building with a parapet, having em- brasures, and therefore resembling a battery or castle. See Elevations, plates VIII, IX, X. Embossing ; forming work in relievo, whether cast, moulded, or cut with a chisel. See Alto and Demi- relievo. Emplection Walls, 342. English Bond, in bricklaying, 348. Entablature, 446, 453, 465. Entablatures, imperfect, 445. Enter, in carpentry, 222. Elm timber, 261. Epistylium, or architrave of the entablature, 453, 465, 467. Estrade; a French word for a public walk. In a room, a small elevation of the floor, frequently encompassed with a rail or alcove. Eustyle, 466. Evolution or Extraction of Roots, 555. External Walls, 363. Extrados of an arch, 338 : of a bridge, 338. GLOSSARY AKD GENERAL INDEX. 585 F. FACADE ; the flice or front of a building. Face-mould, in joinery, 222. Facings ; in joinery, those fixed parts of wood-work which cover the rough work of the interior sides of walls, &c. Falling-moulds ; in joinery, the two moulds which are to be applied to the vertical sides of the rail- piece, in order to form the back and under surface of the rail, and finish the squaring. Fang of a tool, 222. Fasciae, 311,474. Feather-edged boards, 222. Fees, district surveyor's, 368. Fence-wall, 338. Fillet, 161, 167, 443, 468, 9, 474. Filling-in pieces, 222. Fine-set, in carpentry, 223. Fine-stuff, in plastering, 390. Fire-places, 434. First coat, in plastering, 390. ' Fir-poles, 223 : Fir-timber, 261. Five Orders of Architecture, 45 1 . Flanks of a projection, 445. Flashings, in plumbery, 407. Flemish Bond, in bricklaying, 347. Flight of steps, 439. Float, in plastering, 390. Floated, lath and plaster, 390. Floated, rendered and set, 391. Floating, Floated Rules, and Floating Screed, 391. Flooring, naked, 118. Floors of Rooms, 455. Flues or Funnels, 434. Flutes, or Fluting, in joinery, &c. 161, 167. Flyers ; steps, of which the treads are all parallel, 185. Flying buttresses. See Arc Boutant. Footings, in masonry, 338. Foundations, 303. , planking and piling, 303. r; of bridges, 304. Fractions, decimal, 551. Frames, in joinery, 1 CO. Framing, in joinery, 163. Framing pf a house; all the timber-work, compre- hending the carcase flooring, partitioning, roofing, ceiling, beams, &c. Franking ; in sash-making, is the operation of cut- ting a small excavation on the side of a bar for the reception of the transverse bar, so that no more of the wood be cut away than may suffice to show a mitre when the two bars are joined together. Free-stone of Portland and Bath, 286. Free-stuff, in carpentry, 223. Fret ; a species of ornament, commonly composed of straight grooves or channelures at right angles to each other. The labyrinth fret has many turnings or angles, but in all cases the parts are parallel and perpendicular to each other. See Plate of Frets, attached to those of tlie Five Orders, 73. Frieze, or Prize, or Zophorus, 446, 475. Front of a projection, 445. Frosted ; a species of rustic work, representing ice formed by irregular drops of water. Funnels or Flues, 434. Furrings, in carpentry, 223. Frowey timber ; such as works freely to the plane, without tearing. .OTi' - GABLE ; the triangular part of the wall of a house or building, immediately under the roof. Galilee ; a porch constructed at or near the west end of the great abbey churches, where the monks and clergy assembled on proceeding to, and returning from, processions, &c. Gangway ; in building, the temporary rough stair, set up for ascending or descending, before the regular stair is built. Gathering of the wings, in a chimney : the sloping part above the fire-place, where the funnel con- tracts or tapers. Gauge or Gage ; in carpentry and joinery, a tool for drawing a line or lines on any side of a piece of stuff, parallel to one of the arrises of that side. See Tools, Gauge, or Gage, in plastering, 391. Geometry defined, 1. Geometric definitions, 10, 47. notation, 14. Geometry, practical, 62. of solids, 74 : definitions in, 74, 77. Geometrical stairs, 192, 193, 194, 441. Girder, in carpentry, 223. Girt. The same as Fillet, Glass and Glazing, in general, 419, , coloured, 419. -, crown, green, &c. 420. , plate and waved plate, 420. , ground or .rough, 420. , German sheet and Bohemian plate, 421. Glazier's Tools, 422. Glazing, method of, 422. , in leaden rebates, 42 1 . , valuation of, 421,423. Glue, 223. Gorge ; a concave moulding, much less recessed than a scotia. This word is sometimes used for the cyma-recta. Gothic, more properly British, architecture, 514. Graining, 417. Granite, 286. Greek Orders of Architecture ; the Doric, Ionic, and Corinthian. See these names respectively. _!, Griffin, or Griffon ; a fabulous animal, sacred to Apollo, and mostly represented with the head and wings of an eagle, and the body, legs, and tail, of a lion. It was a common ornament of antient temples. Grindstone, 225. 7i 586 THE NEW PRACTICAL BUILDER, &C. Groin ; the hollow formed by the intersection of two or more simple vaults, crossing each other at the same height. Groins of different forms are distin- guished by particular designations, as follow : Conic Groin is a groin formed by the intersection of one portion of a cone with another. Cono-conic Groin is one which is formed by the inter- section of one conic vault piercing another of greater altitude. Cylindric Groin ; that which is formed by the inter- section of one portion of a cylinder with another. Cylindroidic Groin ; that which is formed by the in- tersection of one portion of a cylinder with another. Cylindro-cylindric Groin is that which is formed by the intersection of two unequal cylindric vaults. Etfui-angular Groin is that in which the several axes of the simple vaults form equal angles, around the same point, in the same horizontal plane. Multangular Groin is that which is formed by three or more simple vaults piercing each other. Rectangular Groin is that which has the axis of the simple vault in two vertical planes, at right angles to each other. Spheric Groin is that which is formed by the inter- section of one portion of a sphere with another. Cylindro-spheric Groin ; that which is formed by the intersection of a cylindric vault with a spheric vault ; the spheric portion being of less height than die cylindric. Sphero-cylindric Groin is that which is formed by the intersection of a cylindric vault with a spheric vault, the spheric portion being greater in height than the cylindric. Groined ceiling ; a cradling constructed of ribs, lathed and plastered. Groins and Arches, in carpentry, 109. Groins of Bricks, construction of, 357. Grooving, in joinery, 159. Grotesque; the light, gay, and beautiful, style of or- nament, practised by the antient Romans in the decoration of their palaces, baths, villas, &c. It is supposed to have originated from the hieroglyphics of Egypt, where the human body may be seen fantastically attached to foliage, vases, and other figures. Ground glass, 420. Ground-plate or sill, 225. Grounds, in carpentry, 225. Guttae. See Drops. H. HALF-SPACE or resting place, 439. Hall ; a word commonly denoting a mansion or large public building, as well as the large room at the entrance. Hence Guildhall, Town-hall, &c. Hammer-beam ; a transverse beam at the foot of the rafter, in the usual place of a tie. Hanging of doors and shutters. See Hingeing. Handspike, 225. Hanging stile, 225. Hand-railing, 196. Hand-rails, moulds for, 207. Hawke, in plastering, 391. Headers, in bricklaying, 348. , in masonry, 338. Heart-bond ; in masonry, the lapping of one stone over two others, together making the breadth of a wall. Helix; little scrolls in the Corinthian capital, also called Urillce. Hem ; the projecting and spiral parts of the Ionic capital. Hexastyle, 467. Hingeing, 170. Hingeing doors and shutters, 171. Hip-roof, 225. Hoarding, or Hording, 225. Holing, in slating, 402. Hollow-wall ) a wall built in two thicknesses, leaving a cavity between, which may be either for saving materials, or for preserving an uniform temperature in apartments. House-painting, 410. Houses, rates of, 368. See Elevations, &rc. plates I. toy. Housing ; the space excavated out of one body for the insertion of some part of the extremity of an- other, in order to unite or fasten the same together. Hoveling; carrying up the sides of a chimney, so that when the wind rushes over the mouth, the smoke may escape below the current or against any one side of it. I. IMPOST or springing, 339, 444. Intaglios ; the carved work of an order or any part of an edifice, on which heads or other ornaments may be sculptured. Intercolumn; the open area or space between two columns. Intercolumniation, 466. Inter-dentils ; the space between dentils. Inter-fenestration ; the space between windows. Inter-joist ; the space between joists. Inter-pilaster ; the space between pilasters. Inter-quarter ; the space between two quarters. Intersole or Mezzanine, 438. Intertie, 225. Intrados, 428 : of a vault, the concavity or interior surface. See Extrados. Involution or Raising of Powers, 555. Ionic Order, 451 to 454 : practice of, 493 : references to the plates of, 497. Isodomum. See Wall. J. JACK-PLANE. See Tools in Carpentry. Jack-rafters, 225. Jack-ribs, 225. Jack -timber, 225. Jambs, the vertical sides of an aperture, as of doors, windows, &c. of fire-places, 434. of windows, 429, 430. Jamb-lining ; the lining of a jamb. GLOSSARY AND GENERAL INDEX. 587 Jamb-post ; a post fixed on the side of a door, &c. and to which the jamb-lining is attached. Jamb-stones ; in walls, those used in building the sides of an aperture, and of which every alternate stone should have the whole thickness of the wall. Jettee or Jetty, in Masonry, 338. Jibb-doovs, 183. Joggle ; the joint of two substances, as of wood, &c. so formed as to prevent their sliding past each other. See page 128. Joggled-joints, 339. Joggle-piece, in carpentry, 225. Joinery, in general, 159. Jointer. See Tools in Joinery. Joists, 226. Juffers, 226. K. KEEP ; in a castle, the middle or principal tower. Kerf, in carpentry, 226. Keyed-dado; dado, secured from warping by bars grooved into the back. Keys ; in naked flooring, are pieces of timber framed in, between every two joists, by mortise and tenon. If driven fast between each pair, with the ends butting against the grain of the joists, they are de- nominated strutting pieces. Keys ; in joinery, pieces of wood let, transversely, into the back of a board, especially when made of several breadths of timber, either by dove-tailing or grooving. Key-stone, 339. King-post, 126, 226. Knee, 226. Knot, 226. Knotting ; in painting, the process for preventing knots from appearing in the finish. LABEL ; an ornament placed over a window or other aperture, generally in a castellated building, and consisting of a horizontal portion over the head, with a part at each end returning downwards at a right-angle : the latter may be terminated by a bead, but it more frequently returns again at a right-angle outwards or horizontally. See Plates of Elevations, VIII, IX, X, XIV. Labyrinth ; an intricate building, so contrived by its meandring form, as to render it difficult for those who have entered, to find the way out again. Hence a Labyrinth-fret, a fret with many turnings, which was a favourite ornamen* of the antients. See Fret. Lacunarias, or Lacunars ; panels or coffers formed on the ceilings of apartments, and sometimes on the soffits of coronae in the Ionic, Corinthian, and Composite, orders. Lancet-arch ; the same as pointed arch. Landing of Stairs, 433, 439. Lantern ; a turret raised above the roof, with win* dows round the sides, constructed for lighting an apartment beneath. Larmier or Larmer. See Corona. Lath-bricks ; a sort of bricks, much longer than the* ordinary sort, and used for drying malt upon. Laths and Lathing, 372, S91. Lathing, laying, and set, 373, 391. , floating, and set, 373, 391. , plastered, set, and coloured, 391. , pricked up, floated, and set, for paper, 391. Laying, in plastering, 373, 391. Laying on trowels, 392. Laying slates, manner of, 399. Lead ; its properties, 404 ; pigs of, 404 ; milled lead, 406 ; sheet lead, laying of, 407 ; how charged, 408. Lead, white, 410. Ledgers ; in scaffolding for brick buildings, the horizontal pieces of timber parallel to the wall, and fastened to the standards by cords, for supporting the putlogs. On the last are laid the boards for working upon. Lemma, definition of, 15. Length, breadth, depth, and height, in building, 425. Lengthening timbers, 120, 280. Level, in masonry, 339. Lever-boards ; a set of boards, parallel to each other, so connected together that they may be turned to any angle, for the admission of more or less air or light ; or so as to lap upon each other and exclude both. Lime, 369. Lime and Hair, 392. Lime-wood, 262. Lining; the covering of the interior surface of a hollow body, and used in opposition to casing the exterior surface. Lining of a wall, 226. Lining-out ; drawing lines on a piece of timber, &c. so as to cut it into boards, planks, or other figures. Linseed oil, 411. Lintels, in carpentry, 226. Listing ; in carpentry and joinery, the act of cutting away the sap-wood from one or both edges of a board. Litharge, 410. Lobby ; a small hall or waiting-room, or the entrance into a principal apartment. Lodges and Entrance to a Mansion, 565, and Eleva- tions, plate X. London Bridge, 292. Lower-rail, 226. Luffer-boarding ; a series of boards placed in an aperture, very frequently in lanterns, so as 40 admit air into the interior, and to exclude rain. Lunette ; an aperture in a cylindric, cylindroidic, or spherical, ceiling ; the head of the aperture being also cylindric or cylindroidic. Luthern ; a kind of window, over the cornice, in the roof of a building, formed perpendicularly over the naked of the wall, for the purpose of illumi- nating the upper story. They are denominated according to their forms, as square, semi-circular, bull's eyes, &c. Lying panel, 226. 588 THE NEW PRACTICAL BUILDER, &C. M. MAHOGANY timber, 262. Mansion ; a large dwelling house or habitation : the chief house of a manor, &c. Mansions, designs for, 565, 6, 7: Elevations, &c. plates XI. to XV. Mantles of fir-places, 434. Mantle-tree ; the lower part of the breast of a chim- ney, now by law, in disuse ; an iron bar, or brick, or stone, being substituted. Marble ; a species of lime-stone, too well known to require ^description. It is found in almost every part of the world, more especially Italy, but there are many fine varieties in Great Britain and Ireland. Margins or Margents, 226. Margin of a course, in slating, 402. Masonry, in general, 285. , materials employed in, 286. , ornamental, 308. Mason's work, valuation of, 314. plain work, 315. sunk work, 316. . . moulded work, 316. Mathematical Instruments, for planning, &c. 559. Mausoleums, 572, 3, 4. Elevations, &c. plates XXVII. to XXXIII. Mechanical Powers ; such implements or machines as are used for raising greater weights, or overcoming greater resistances, than could be effected by the na- tural strength without them. The simple machines, called Mechanical powers, are six in number ; viz., the lever, the wheel and axle, the pulleys, the in- clined plane, the wedge, and the screw: and of these all the most compound engines consist. The general principle is, that the power or advantage gained by any of these machines, be it ever so simple, or ever so compound, is as great as .the space moved through by the working power is greater than the space through which the weight or resistance moves during the time of working. Thus, if that part of the machine to which the working power is applied moves through 10, 20, or 1000, times as much space as the weight moves through in the same time, a person who has just strength enough to work the machine will raise 10, 20, or 1000, times as much by it as he could do by his natural strength without it : but then the time lost will be always as great as the power gained: for it will require 10, 20, or 1000, times as much time for the power to move through that number of feet or inches as it would do to move through one foot, or one inch, &c. Medallion; a circular tablet, ornamented with em- bossed or carved figures, bustos, &c. Member ; any part of an edifice or of a moulding. Members of which an Order is composed, 453. Meros; the middle part of a trigliph. See Trigliph. Metope ; in the Doric frieze, the square piece or interval between the trigliphs, or between one tri- gliph and another. The metopes are sometimes left naked, but are more commonly adorned with sculpture. When there is less space than the com- mon metope, which is square, as at the corner of the frieze, it is called a semi or demi-metope. See Orders, plates III. and V. Mezzanine or Intersole, 438. Mezzo-relievo or Demi-relievo ; sculpture in half relief. See Alto-relievo. Middle Post ; in a roof, the same as King Post. Middle Rail, 226. Milled Lead, 406. Minnaret ; a Turkish steeple with a balcony. Minutes, in architecture, 458, 466. Mitre, 226. Mitreing angles, 392. Modillions, 447, 469. Modillion Bands, 474. Modules and Minutes, 458, 466. Monopteron, or Monoptral Temple ; an edifice con- sisting of a circular colonnade, supporting a dome, without any inclosing wall. Monotrigliph ; having only one trigliph between two adjoining columns : the general practice in the Grecian Doric. Monument, sepulchral, design for, 474. Elevations, &c. plate XXXIV. Moresque, or Moresk. See Arabesque. Mortar, 392. See Cement. Mortise, 159 ; Mortise and tenon, 227. Mosaic, or Mosaic Work ; an assemblage or combi- nation of small pieces of marble, glass, stones, &c., of various colours and forms, cemented on a ground so as to imitate paintings. Mosaic work of marble, which is, from its nature, very expensive, may be frequently found in the pavements of temples, palaces, &c. Moulded string course, 444. Mouldings, 441, 442, 475 ; base, 444. , section of, 442 ; proportions of, 447. --, return, 445, 446 ; profile, 445. -, of the Five Orders, 467, 476. , in joinery, 160. on the spring, 1 72 ; curved, 443. -, raking, 173 ; rotative, 443.* , enlarging and diminishing of, 175. , in plastering, 392. -, raking, in stone-cutting, 326. Moulds for hand-rails, 207. Mullion or Munnion, 228. Muntins or Montants, 228. Museum; originally, a palace at Alexandria, which occupied a considerable part of the city ; it was thus named from its being dedicated to the Muses, and appropriated to the cultivation of the sciences and of general knowledge. Mutule, 447 : Mutule cornice, 447. N. NAILS, 402. Naked of a Wall or Column ; the plain surface, in distinction from the ornaments. Thus the Naked of a Wall is the flat plain surface that receives the mouldings ; and the naked of a column or pilaster is its base surface. See 339, 444. GLOSSARY AND GENERAL INDEX. 589 Naked flooring, 118,128. Nave ; the body of a church, reaching from the choir or chancel to the principal door. Nebule ; a zigzag ornament, but without angles, fre- quently found in the remains of Saxon architec- ture. Neck of a capital ; the space between the channelures and the annulets of the Grecian Doric capital. In the Roman Doric, the space between the astragal and annulet. See Orders, plates I, II, and III. Nerves ; the mouldings of the groined ribs of Gothic vaults. Newel of a stair-case, 228, 439, 441. Niche ; from an Italian word, signifying a shell ; a hollow formed in a wall, for receiving a statue, &c. 441. An 4ngular niche is one formed in the corner of a building : A Ground Niche, one hav- ing its rise from the ground, without a base or dado. Niches ; in carpentry, 134 : in brickwork, 359. Nogs ; a provincial term, signifying what are other- wise called Wood-bricks. Nogging. See Brick-nagging. Nosing of a step, 438. Notch-board, in stairs, 189. O. OAK timber, 257. Obelisk ; a quadrangular pyramid, high and slender, raised as a monument or ornament, and commonly charged with inscriptions and ornaments. Octastyle, 467. Odeum ; among the antients, a place for the rehearsal of music and other particular purposes. Off-set, 426 : in masonry, 339. Ogee ; a moulding of two members, one concave, the other convex. It is otherwise called a cymatium : in joinery, 228. Oil, linseed, 411 : drying, 411: of turpentine, 411. Opened-newelled stairs, 189. Orders of Architecture, origin of, 452 : proportions of, 455, 457 : ornaments of, 467, 476 : practice of, with illustrations, 480. Oriel-window ; a projecting angular window, com- monly of a triagonal or pentagonal form, and di- vided by mullions and transoms into different bays and compartments. Ornamental Masonry, 308. Painting, 418. Ornaments, casting of, 377. Orthogonal ; the same as Rectangular. Orthography ; an elevation, showing all the parts of a building in true proportion. Ova ; an ornament in form of an egg. Oviculum is its diminutive. Ovens, construction of, 360, and plate LXXXVI. Ovolo, 443, 468, 9. Out of winding; perfectly smooth and even, or form- ing a true plane. Out to out ; to the extremities or utmost bounds; as in taking dimensions. 73. P. PAGODA, or Pagod; an Indian temple, common in Hiudoostan and the countries to the east. These structures, dedicated to idolatry, are mostly of stone, square, not very lofty, without windows, and crowned with a cupola. Painter's tools, 416. Painter's work, valuation of, 418. Painting in general, 410. Painting, the several kinds of, 417. Palace ; a name generally given to the dwellings of kings, princes, bishops, &c. Pale ; a pointed stake, and piece of board, used in making enclosures. Hence a Paling Fence is that sort of fence which is constructed with pales. See Post and Paling. Paling for trees ; a sort of fencing for separate trees, formed by three small posts, connected with cross bars. Palisade; pales or stakes set up for an enclosure. Pallier, or Paillier j a French term, signifying a land- ing place in a stair-case, which, being broader than the rest of the stairs, serves as a resting-place. Pallification, or Piling; the act of piling ground work, or strengthening it with piles. Panel, 228, 444. Panels, in joinery, 160. Pantheon; a temple of a circular form, originally pagan. Parapet ; a dwarf wall, generally raised to prevent accidents. Parapets, 310, 339. Parget ; the several kinds of gypsum or plaster stone, of which Plaster of Paris is composed. Pargeting, in plastering, 392. Partition or Division Wall, 426. Partitions, designs for, 155. Party-walls, 363. Passages, 432, 433. Paternosters ; a sort of ornament in form of beads, round or oval, on astragals, c. Pavements, 435. Pavilion ; a kind of turret or building, usually insu- lated and contained under a single roof; sometimes square, and sometimes in the form of a dome : thus called from the resemblance of its roof to a tent. Paving, 359. Pedestal ; a square body of stone or other material, raised to sustain a column, statue, &c. It is, there- fore, the base or lowest part of an order of co- lumns. A Square Pedestal is that of which the height and width are equal: a Double Pedestal, that which supports two columns, and therefore is greater in width than height: a Continued Pedestal is that which supports a row of columns, without any break. Pediment; an ornament, properly of a low triangular figure, crowning the front of a building, and serv- ing often also as a decoration over doors, windows, and niches. Though the original and natural form of the pediment be triangular, it is sometimes 7K 590 THE NEW PRACTICAL BUILDER, &C. formed as a segment of a circle, and sometimes broke to let in busts or figures. The pediment consists of its tympanum and cornice; the tym- panum is the panel, which may be either plain or ornamented. The cornice crowns this tympanum. Pendent Bridge ; a wooden bridge supported by posts and pillars, and suspended only by butments at the ends. . Pendentive ; the whole body of a vault, suspended out of the perpendicular of the walls, and bearing against the arc-boutants. Pendentive Cradling ; the timber-work, in arched or vaulted ceilings, for sustaining the lath and plaster. The term Dishing-out is sometimes used instead of Cradling. Pendentive Bracketting, 148. Pentastyle ; a work containing five rows of columns. Periptere ; a building encompassed with columns, which form a kind of aisle all round it. It is thus distinguished from a building which has columns only before it, by the Greeks culled a Prostyle, and from one that has none at the sides, called an AmpM-prostyle. The space, or aisle, in a perip- tere, between the columns and the wall, was called the Peridrome. Peristyle ; among the antients, the converse of Perip- tere, a continued row of columns within the build- ings : among the moderns, a range of columns, either within or without the same. Persians ; statues of men, serving instead of columns, to support entablatures. They differ from the Carya- tides, inasmuch as the latter represent women only. Perspective; definitions in, 518: axioms in, 520 : theo- rems in, 520 : problems in, 523: examples in, 534. Piazza; a portico or covered walk, supported by arches. Pier; a square pillar, without any regular base or capital. Piers, 429 : in houses, 339 : of a bridge, 339. Pigs of lead, 404. Pilasters, 309, 446 : a Demi-pilaster is one that sup- ports an arch : pilaster, in joinery, 165. , flutes and fillets of, 167, 447. , practice of, and Greek anise, 509. Piles, 339. Pile-planks ; planks of which the ends are sharpened, so as to enter into the bottom of a canal, &c. Pillar ; a column of an irregular make ; not formed according to rules, but of arbitrary proportions ; free or insulated in every part, and always deviat- ing from the measures of regular columns. This is the distinction of the pillar from the column. A square pillar is commonly called a pier. A but- ting pillar is a butment or body of masonry, erected to prop, or to sustain, the thrust of a vault, arch, &c. Pinnacle ; the top or roof of a building, terminating in a point. Pipes of lead, 407. Pitch of an arch, 339 : of a roof, 228. Pitching-piece, in stairs, 189. Plank and planks, 159, 228. Planks, joining of, 160. Planting ; laying the first courses of stone in a foun- dation, with all possible accuracy. Plaster, 392. of Paris, 389. Plasterer's measuring and valuation, 383. Plastering in general, 369. , tools used in, 370. Platband ; any flat square moulding, of which the height much exceeds its projecture. See Fasciae. The Platband of a door or window, is used for the lintel, where that is made square or not much arched. Platbands ofjlutings are the lists or fillets between the flutings of columns. Plates, floor or roof, 228. Plate-glass, 420. Platform ; a row of beams, supporting the timber- work of a roof, and lying at the top of the wall where the entablature ought to be raised : also a flat terrace on the top of a building. Plinth ; the square piece under the mouldings in the bases of columns. The plinth terminates the co- lumn with its base at the bottom, as the abacus does with its capital at the top : but the abacus, in the Tuscan order, being plain, square, and massy, has been called the plinth of that capital. The plinth of a statue, &c. is a base serving to support ft and its pedestal. See, farther, pages 312, 467, 474. Plugs ; pieces of timber, driven perpendicularly into a wall, and having the projecting part sawn away, so to be flush with the face. Plumbery, or Plumbing in general, 403. Plumber's tools, 403. Pointed-arch ; an arch so pointed at the top as to resemble the point of a lance. Pointed architecture ; that style vulgarly called Gothic, more properly English. Pole-plate, 128. Poplar-wood, 262. Porch ; the kind of vestibule at the entrance of tem- ples, halls, churches, &c. Portail ; the face of a church, on the side in which the great door is formed ; also the gate of a castle, palace, &c. Portal; a little gate, where there are two of a dif- ferent size : also a kind of arch, of joiner's work, before a door. Portico ; a covered walk, porch, or piazza, supported by columns. Porticos, the most esteemed, 467. Post and Paling ; a close wooden fence, constructed of posts set into the ground and pales nailed to rails between them. The part of the post in- tended to be inserted in the ground should be charred, or superficially burnt, in order to prevent decay. Post and Railing ; an open wooden fence, consisting of posts and rails only. Posticum ; a postern gate or back-door. Postcenium ; in an antient theatre, a back room or place for dressing in, &c. Posts, in carpentry, 229. Postulates, 15. GLOSSARY AND GENERAL INDEX. 591 Powderings ; a species of device for filling up vacant spaces in carved works, &c. Prick-posts, 229. Pricking-up, 373, 392. ' Priming ; in painting, the laying on of the first colour. Principal Brace ; a brace immediately under the chief rafters or parallel to them. See Brace. Principal Rafters. See Rafters. Priory ; a religious house or institution, at the head of which is a prior or prioress. Prison and County Court House, design and plan for, 575. Elevation, plate XL, &c. Prison, design for a small county prison, 577. Ele- vation, plate XLI, &e. Problem, definition of, 15. Problems, 62, 75, 77. Profile ; the figure or draught of a building, &c. ; also the general contour or outline. Projection or Break, 441. Projection, treatise on, 539 : definitions, &c. 540 : problems, 541. Projections, 466. Prefecture ; the outjetting, or prominence, which the mouldings and other ornaments have beyond the naked of the wall, &c. Pronaos ; an antient name for a porch to a temple or other spacious building. Proscenium ; in a theatre, the stage, or the front of it. Prostyle ; a range of columns in front of a temple. Prothyrum ; a porch or portal at the outer door. Protractor, 559. Pudlaies, 229. Pugging, in bricklaying, 392. Pug-piling ; dove-tailed or pile planking. Pumps, leaden, 408. Punchions, in carpentry, 229. Purfled ; ornamented in a manner resembling dra- pery, embroidery, or lace-work. Purlins, 128, 229. , in circular roofs, 152. Putlogs or Putlocks ; in scaffolding, the transverse pieces, at right angles to the wall. See Ledgers. Push of an arch. See Drift. Putty, plasterer's, 392 : glazier's, 420. Puzzolana ; a substance composed of volcanic ashes, named from Puzsuolo, in Italy, where it abounds, and celebrated as a principal ingredient in cements. When mixed with a small proportion of lime it quickly hardens, and this induration takes place even under water. See Cements. Pycnostyle, 466. Pyramid; a solid massive structure, which, from a square, triangular, or other, base, rises diminishing to a vertex or point. Q. QUADRA ; any square border or frame encompassing a basso relievo, panel, &c. Quadrangle; a figure having four sides and four angles : a square is, therefore, a regular quad- rangle, and a trapezium an irregular one. Quadrefoil Arch, 565, and Elevations, &c., plate X. Quarry, 339. Quarry also means a pane of glass, in a lozenge or diamond form. Quarter-round, 161. Quartering, 229. Quarters, 229. Quarter- spaces, in stairs, 440. Queen-posts, 127. Quink ; a piece of ground taken out of any regular ground-plot or floor. Thus, if the ground-plot were oblong or square, a corner-piece separated from it,- to make a court, yard, &c. is called a quink. Quirk, in joinery, &c., 160, 161, 472 : Quirked ovolo, 162. See Ovolo. Quirk-mouldings are the convex parts of Grecian mouldings, where they recede at the top, and form a re-entiant angle with the soffit which covers the moulding. Quoin, external or internal, 426, 429. The name is particularly applied to the stones at the corners of brick buildings. When these stand out beyond the brick-work, with edges chamfered, they are called Rustic Quoins. R. RABBITING. See Rebating. Rafters, 229. , principal, 125. , common, 128. Rails, in joinery, 229. Raiser ; a board set on edge under the foreside of a step or stair. Raising-pieces ; pieces that lie under the beams and over the posts or punchions. Raising-plates or Top-plates, 229. Raking moulding ; a moulding whose arrises are in- clined to the horizon in any given angle, 173, 174. Ramp ; in hand-railing, a concavity on the upper side, formed over risers, or over a half or quarter space, by a sudden rise of the steps above. Rampant arch; an arch, of which, the abutments spring from an inclined plane. Random courses, in paving, 339. Rank-set, 229. Rates of Houses, 368, and Elevations, plates I. to V. Rebating, in joinery, 159. Recess, 441 : of windows, 435. Reed; Reeded; 161. Reglet, or Riglet ; a flat narrow moulding, used chiefly in compartments and panels to separate the parts or members, and to form knots, frets, &c. Regrating ; in masonry, taking off the outer surface of an old hewn stone, so as to make it look new again. Rejointing ; in masonry, the rilling up of the joints of stones in old buildings, when worn hollow by time and weather. Relievo, Relief, or Embossment. See Alto-relievo, Sec. Rendered and set, in plastering, 374, 393. 592 THE NEW PRACTICAL BUILDER, &C. Rendered and floated, 392. , floated, and set for paper, 393. Reservoirs, leaden, 408. Resault; a French word signifying projecting or receding from a line or general range. Return, 229. Return-bead ; a bead which appears on the edge and face of a piece of stuff in the same manner ; form- ing a double quirk. Revels, pronounced Reveals; the vertical retreating surface of an aperture, as the two vertical sides between the front of the wall and the windows or door-frame. Rib ; a curved or arch-formed timber. Ribbing ; the whole of the timber-work for sustaining a vaulted or coved ceiling. Ridge, 229 : Ridge-piece, 128. Risers, of stairs, 229. Roman Cement, or Outside Stucco, 378. , charges for, 379. Roman or Composite Order, 452, 454. Rood-loft ; in antient cathedral and abbey churches, the gallery over the entrance into the choir. Roof, 229. , and forms of, 427. , span of, 124. . , trussing of, 124. Roofs, gable-ended, 129. , designs for, 156. Roofs of St. Pancras' Chapel, St. Luke's church, and Camden chapel, 574, and Elevations, &c. plate XXXV. Roofs, comparative weight of different coverings for, 398. Room or Apartment, usual form of a, 430. Rooms defined, and how to be disposed, 425. , fanciful, 432. Rose; an ornament in the form of a rose, found chiefly in cornices, friezes, &c. Rotondo or Rotunda ; a common name for any cir- cular building. Rough-casting, 380, 393. Roughing in, 393. Rough-rendering, 393. Rough-strings, 189. Rough. stucco, 393. Rubble-wall ; a wall built of unhewn stone, whether with or without mortar. Rudenture ; the figure of a rope, or of a staff, whether plain or carved, with which a third part of the fluting of columns is frequently filled up. It is sometimes called cabling : hence the columns are said to be cabled or rudented. Ruderated ; in paving, &c., laid with pebbles or little stones. Rustic building; one constructed in the simplest manner, and apparently more agreeably to the face of nature than the rules of art. Rustic work and rustic quoins are commonly used in the basement part of a building. Rusticating, 311. Rustic-work ; that exhibited on the face of stones, which, instead of being smooth, are hatched or picked (frosted or vermiculated), with a point of a tool, &c. S. SAGGING ; bending downwards in the middle, from a horizontal direction ; as a long plank laid horizon- tally, and supported at each end only: 1J4. Sagitta ; a name by some used for the key-piece of an arch. Sail-over, 893. Sally or Prefecture. See Bird's Mouth. Saloon ; a spacious, lofty, and elegant, hall or apart- ment, vaulted at top, and generally having two ranges of windows. A state-room common in the palaces of Italy. Sand-stone, 286. Sapheta ; a soffit. See Soffit. Sarcophagus ; a tomb of stone, in general highly decorated, and used by the antients to contain the dead bodies of distinguished personages. Sash ; a frame for holding the panes or squares of glass in windows : too well known to require de- scription. See Windows. Sashes, circular, in circular walls, 177. Scagliola, 380. Scaffolding. See Ledgers. Scantle, in slating, 40$. Scantling, 229. Scape-moulding, 162. Scarfing, 229 : and Lengthening beams, 280. Scheme-arch, 428. Scotia. 444, 468, 469. Screed, in plastering, 373, 393. Scribing ; adjusting the edge of a board, so that it shall fit and correspond with a given surface. In joinery, the act of fitting one piece of wood upon another, so that the fibres of one may be perpen- dicular to those of the other. For a different method, see Mitring. Scroll. See Volute. Scrolls for hand-rails, 205. Sealing; fixing a piece of wood or iron in a wall, with mortar, lead, or other binding, for staples, hinges, &c. Seasoning timber, 259, &c. Second coat, in plastering, 393. Section of a building ; a representation of it, as ver- tically divided into two parts, so as to exhibit the construction of the interior. Semi-circular arch, 428. Sepulchral Monument, design for, 474, and Eleva- tions, plate XXXIV. Sesspool, or Cesspool ; a deep hole or well, under the mouth of a drain, for the reception of sediment, &c., by which the drain might be choked. Set-fair, Setting, Setting-coat, and Set-work, in plas- tering, 393. Sewer ; a common drain or conduct for conveying foul water, &c. GLOSSARY AND GENERAL INDEX. 593 Shadows, 545 : definitions of, 545 : examples of, 546. Shaft of a column, 453. chimney ; the turret above the roof. Shaken stuff, 230. Sham-door ; in joinery, a panel of frame-work that appears like a door, hut does not open. Shanks ; the intersticial spaces between the channels of the trigliph, in the Doric frieze : sometimes called Legs. Sheet-lead, laying of, 407. Shingles, 230. Shoar ; an oblique prop, acting as a brace upon the side of a building. Shoar, dead. See Dead-shoar. Shoe ; the part at the bottom of a water trunk or pipe, for turning the course of the water. Shoot of an arch. See Drift. Shop-fronts, designs for, 575, and Elevations, second series, plates I, II, III. Shreadings, 230. Shutting-joints of doors, 182. Shutting- Windows, designs for, 180. Side-posts ; in roofing, a sort of truss-posts, placed in pairs, each post being fixed at the same distance as the rest from the middle of the truss. Sills or Cills, 310. Single-hung ; in window-sashes, when one only is moveable. Single-measure; in doors, means square on both sides, in opposition to double-measure, which signi- fies moulded on both sides. If moulded on one side, and square on the other, it is expressed by measure and half. Skew-back, 393. Skirtings, or Skirting Boards, 230. Skirts of a roof, 230. Skylights, 178. Slates, names, qualities, and sizes, 395. , manner of laying, 399. , patent, 402. Slating, in general, 395. Slater's tools, 400. work, valuation of, 401. Sleepers, 230. Slit-deal ; an inch deal cut into two leaves or boards. Socle, or Zocle; a square piece, broader than it is high, placed under the bases of pedestals, &c., to support vases, and other ornaments. As there is a Continued Pedestal, so is there, also, a Continued Socle. See Pedestal. Soffita, or Soffit ; any timber ceiling, formed of cross- beams of flying cornices, the square compartments or panels of which are enriched with sculpture or painting. Soffit also means the under side of an architrave, and that of the corona, or drip, &c. ; also, the horizontal undersides of the heads of apertures, as of doors and windows. Soffit, of stairs, 441. Solder, plumber's, 406. Sommering ; the continuation of the joints of arches towards a centre or meeting point. Sorting, in slating, 402. 74. Southwark Bridge, 293. Span of an arch, 340. a building, 427. Span-roof; a simple roof, consisting of two inclined sides. Spars, 230, Spherical and Spheroidal Bracketing ; brackets formed to support lath and plaster, so that the outer sur- face shall be spherical, or spheroidal. Sphinx; a favourite ornament in Egyptian archi- tecture, representing the monster, half woman and half beast, said to have been born of Typhon and Echidna. Splayed ; t one side making an oblique angle with the other, as Splays or Splaying Jambs, 429. Springing-course ; the horizontal course from which an arch begins to spring, or the rows of stones upon which the first arch- stones are laid. Springing-points, 428. Square, in geometry, 12 : but, among workmen, it commonly means that one side or surface is per- pendicular to another. In joinery, -the work is said to be framed square, when the framing has all the angles of its styles, rails, and mountings, square, without mouldings. Square of building, 368, is 100 superficial feet mea- sured on the surface of the ground. Square-root, extraction of, 555. Squaring hand-rails ; the method of cutting a plank to the form of a rail for a stair-case, so that all the vertical sections may be rectangles. Squaring, in slating, 402. Stacks of chimnies, 434. Stair, or Set of Steps, 438. Staircase, elliptic, 193. Stairs and Staircasing, 184, 439. , definitions of the parts of, 184. , proportions of, 185. , carriage of, 188. , treads and risers of, 438. , construction of, 439. , dog-legged, 189, 190, 193, 195. ' : v , open-newelled, 189. , bracketed, 189, 192. , pitching piece of, 189, , bearers of, 189. , Botch-board of, 189. , curtail step, 189. , geometrical, 192, 193, 194,441. pillared or newelled, 441. , winding, 441. Standards ; the upright poles used in scaffoldisg-. In joinery, the upright pieces of a plate-rack. Staves ; in joinery, the boards that tfe united late- rally, in order to form a hollow cylinder, cone, &c. In stables, the cylinders or rounds forming the hay-rack. ' Step, nosing of a, 438. Steps, flight of, 439 : breadths of, 441. Sterlings, 340. Stiles of a door, 230. Stilts or Sterlings, 340. 7 L 594 THE NEW PRACTICAL BUILDER, &C. Stone-cutting, &c. 318. Stopping, in plastering, 393. Story, 426. Story-posts ; upright timbers, chiefly in sheds and workshops, and so disposed, with a beam over them, as to support the superincumbent part of the exterior wall. Straining cill, 127. Stretchers, in masonry and bricklaying, 340, 356. Striae ; the fillets or rays separating the furrows or grooves of fluted columns. Striges ; the channels of a fluted column. String-board ; in stairing, a board placed next to the well-hole, and terminating the ends of tlje steps. Strings of stairs, 441. Struts, 127, 230. Stucco, 372, 378, 393. , bastard, 393. Stud-work. See Bricknogging. Stuff, in joinery, 169, 280. Suite of apartments, 426. Summer, 230. Summer-tree ; a beam full of mortises to receive the . ends of joists, and to which the girders are framed. Sunk shelves ; in pantries, &c., shelves having a groove to prevent the plates, set up on edge, from sliding off. Surbaces and Bases, 165. Swallow-tail ; a mode of uniting two pieces of timber so strongly that they cannot fall asunder. See Dovetail. Sycamore wood, 262. Systyle, an intercolumniation of two diameters, 466. T TABLE, projecting or raised ; a flat surface, sometimes ornamented, which projects from the surface of a wall. See Tablet. Table, raking ; one not perpendicular to the ho- rizon. Table rusticated. See Rusticated. Table of Glass ; the circular plate, before it is cut or divided. Twenty-four such make a case. Tabled'; cut into, or formed like, tables. Tablet, what, 444. Tsenia, or Tenia ; a small square fillet, at the top of the architrave, in the Doric capital. Tail of a slate, 402. Tail in ; to fasten any thing into a wall at one end, as steps, &c. In joinery, commonly called housing. Tail-trimmer ; a trimmer next to the wall, into wkich the ends of joists are fastened. Tailing ; the/fart of a projecting brick, &c. inserted in a wall. Talon, 468, 469. Talus ; the slope or inclination of a wall,' among work- men called Battering. If the wall inclines beyond the perpendicular of its base, it is called hanging. Tambour ; from a word signifying a drum, and mean- ing the naked of a Corinthian or Composite capital : also the wall of a circular temple, surrounded with columns. The same word signifies a place inclosed with folding doors, to break the current of air from without, at the entrances of churches, &c. Taper, 230. Tarras, or Terras; a strong mortar or plaster, used in aquatic works. Tassels ; the pieces of timber that lie under the mantle- tree ; common in the country. See Torsel. Teaze-tenon ; a tenon upon the top of a post for supporting two level pieces of timber at right angles to each other. Telamones ; a Roman term for the figures of men supporting a cornice, &-c. The same as the Atlan- tida? and Persians of the Greeks. See Persians. Templet, 389. Tenon. See Mortise. Terminus (plural Terminii); a trunk or pedestal, sculp- tured at top into the figure of the head of a man, woman, or satyr, whose body seems to be inclosed .in the trunk, as in a sheath. The latter is called the Vagina. Terrace ; an elevated area for walking upon, and sometimes meaning a balcony. Terrace-roofs ; roofs flat on the top. Tesselated pavement ; a curious pavement of Mosaic work, composed of small square stones, bricks, &e. called tesselae. Tessera ; a cube or dye : also a modern composition for covering flat roofs. Testudinal Ceilings ; those formed like the back of a tortoise. Tetrastyche ; a gallery with four rows of pillars. Tetrastyle, 467. Theorem, definition of, 14. Theorems, geometric, 16, 48. Thorough-lighted ; in rooms, those having windows on opposite sides. Three-coat work, in plastering, 393. Three times and flat, in painting, 417. Through-stones, 340. Thrust or Drift. See Drift. Tie, 231. Tie-beam, 125. Tiles ; the artificial stones used in covering buildings. Plane-tiles and Crown-tiles are of a rectangular form, with a flat surface, of which the dimensions are about 10^ inches long, 6 broad, and five-eighths thick : weight from 2 Ibs. to 2j Ibs. Ridge-tiles, or Roof-tiles, are those of a cylindric form, and used for covering the ridges of houses. Of these the dimensions are 12 inches long, 10 broad, and five-eighths thick: weight, about 4^ Ibs. Those covering the angle formed by two sloping sides are called hip-tiles. Gutter-tiles, formed according to the purpose for which they are intended, are of the same weight as the ridge-tiles. Pan-tiles are those having each surface, both con- cave and convex ; they are 1 hung on the lath, by means of a ledge formed en the upper end. The usual size is 14y inches long and 10 broad. Weight, from 5 to Sjlbs.- GLOSSARY AND GENERAL INDEX. 595 Tile-creasing; two rows of tiles fixed horizontally under the coping of a. wall, for discharging rain- water. Timber, qualities of, &c. 257 to 263. , seasoning of, 263. , strength of, 264. , specific gravity of, c. 270. , problems on the strength of, 271. Timbers, lengthening of, 120. Ton of timber ; about 40 square feet. Tiondino ; a round moulding resembling a ring. See Torus. Tongue ; a. projecting part, on the edge of a board, to be inserted in a groove ploughed in the edge of another. Tools used in bricklaying, 384. carpentry and joinery, 232. vmro-rj' glazing, 422. masonry, 340. painting, 416. plastering, 370. plumbing, 403. slating, 400. Toothings, in plastering, 394. Top-beams ; the collar-beam of a truss ; the same as formerly called mind-beam or strut-beam, and now collar. beam. Top-rail ; the upper rail of a piece of framing or wainscotting. Torsel ; a piece of wood laid into a wall for the end of a timber or beam to rest on. Torus, 469 : Double torus, 161. Torus moulding, 161, 468. Trabs ; an antient name for wall-plates or rising- plates, for supporting the rafters. Transept ; the cross-ailes of a church of a cruciform structure. Transom ; a cross-beam : the horizontal piece framed across a double-lighted window. Transom windows, 231. Traversing the screeds for cornices, 394. Tread of a step ; the horizontal part of it. Trellis-work; reticulated or net-like framing, made of thin bars of wood. Trigliph or Triglyph, 467. Trigonometry, plane, 79. , definitions in, 79. , theorems in, 81. Trimmed ; cut into shape, or fitted in between parts previously executed, as in partition-walls, &c. Trimmers, 231. Trimming joists, 231. , in slating, 402. Tripod ; a three-legged seat, from which the priests of antiquity delivered their oracles, and frequently' represented in architectural ornaments. Trophy ; an ornament representing the trunk of a tree, supporting military weapons, colours, &c. Trowelled stucco, in plastering, 375, 6. . , for paint, 394. Truncated ; cut short or divided parallel to the base. The frustum of a cone, pyramid, &c., is therefore truncated. Truncated roof, 231. Truss, 231 : Trussed roof, 231. Truss-partition ; one with a truss, generally consist- ing of a quadrangular frame, two braces, and two queen-posts, with a straining piece between the queen. posts, opposite the top of the braces. Truss-post, 231. Trussels, 231. Trussing-pieces ; such timbers in a roof as are in a state of compression. Tumbling-in or Trimming-in. See Trimmed. Turning-piece; a board, with a circular edge, for turning a thin brick arch upon. Tuscan Order, 452, 453, 480: references to the plate of, 485. Turps. See Oil, 411. Tusk, 231. Two-coat work, in plastering, 394. Tympanum, or Tympan. See Pediment. i --- Tympan also signifies the panel of a door and the dye of a ' pedestal. TJ. i fiu.'JU c ;93iJ-rjfn;i.j!- UNDER bed of a stone, 342. Upper bed of a stone, 342. Uphers, 231. y. j)2ClI jul(:.i >'U.*'. VALLEY ; the internal angle of two inclined sides of a roof. , Valley-rafter, 231. The Valley-board is a board fixed upon this rafter, for the leaden gutter to lie on. Valuation of glazier's work, 421, 423. , . . , . mason s work, 314. painter's work, 418. plasterer's work, 383. slater's work, 401. Vault ; an interior concavity extending over two pa- rallel opposite walls. The axis of a vault is the same as the axis of a geometrical solid, 342. Sae Arch and Groin. The Reins of a vault are the sides or walls which sustain the arch. Vellar cupola ; a cupola or dome, terminated by four or more walls. Venetian Door; a door lighted on each side. Venetian Window ; a window having three separate apertures. Ventiduct ; a passage or place for wind or fresh air. Vermiculated Rustics ; stones worked or tooled so as to appear as if ea.ten by worms. Villas or Country-Houses, 563, 564, and Ekvations, plates \l,Vll,\lU, IX. Volute ; the scroll or principal ornament of the Ionic capital, 494. W. WAINSCOTING ; in joinery, the lining of walls ; mostly panelled. Wall, 342. Wall, partition or division, 426. Wall-plates, 231. Walnut-timber, 262. Walls, construction of, 306, 354. 596 THE NEW PRACTICAL BUILDER, &C. Walls, Isodomum, 306, 342. , pseudo-isodomum, 306, 342. , emplection, 307, 342. , thickness of, 307. , exterior, 427. , external and party, 363. , beneath windows, 435. Waterloo Bridge, 293, and plates LXXII. (A.) and LXXII. (E). Water-table, 394. Weather-boarding; feather-edged boards, lapped and nailed upon each other, so as to prevent rain or drift from passing through. Weather-tiling ; the covering of a wall or upright with tiles. Web of an iron, 231. Weight of coverings for roofs, 398. Well-hole of stairs, 441. Westminster Bridge, 293, and plate LXXII. Whinstone, 289". White lead, 410. i')i1 ban an U > , T ' Winders, in stairs, 440. Windows, 165, 428 : recess of, 435. , jambs of, 430. , proportions of, 436. , designs for, 575, and Elevations, second series, plate IV. -, cleaning of, 423. Withs of Chimneys, 434. Wood-bricks; blocks of wood, shaped like bricks, and inserted in walls as holds for the joiner)*. Wooden Bridges, construction of, 283. Work, in plastering, 394. Wreathed columns ; such as are twisted in the form of a screw. Now obsolete. Z. ZOCLE. See Socle. Zophorus. See Frieze. Zystos ; among the antients, a portico or aile of un- usual length, commonly appropriated to gymnastic exercises. I l .enoittm'I tol sn^w'- = , . . hrrn biaoM ,iui: .Uid ;?.3I .&' Ifth . -BneH ;SOI ,s-uaJ3 Lai- ERRATA. Page 203, line 3 from the bottom, for LXIV. insert LVIII. - 213, line 3, in some copies, for LXIV. insert LVIII. 243, last line but one should read, " trying-plane, long-plane, jointer-plane, and smoothing'-plane, hut- not in," &c. 244, line 5, erase the words, ' on the fore-end of the stock, or.' 534, line 4, add, Fig. 1, pi. V. Windows, PI. V, should be erased from page 575, near the bottom, having been inserted by ipistakt. eas ......... - *& . ; TJMTOSAM ni baeu .aiooT 1o vi ,TJ10M ,Bl! srii :,.,.,. io aMW> - . ,5n;adA a* y. Priuted by Ji RlDEE, Little Britain, London. taaaii COKTENTS ' 1> wai* STHT . CONTENTS. PAGE CHAPTER I. The ELEMENTS of GEOMETRY, including ALGEBRA, PLANE-TRIGONOMETRY, and CONIC SECTIONS '- Geometry in general Definitions of Terms 10, 14, 47 Notation 14 Axioms. &c 15 Theorems 16, 48 ALGEBRA 33 Addition of, 36; Subtraction, 37; Multiplication, 38; Division and Fractions, 39; Equations, 40 ; Proportion . , . . . 41 Practical Geometry 62 Geometry of Solids , , 74 Plane Trigonometry 79 Conic Sections 82 CHAPTER II. CARPENTRY, in general 105 Coverings of Solids, 107; Groins and Arches, 109; Naked Flooring, 118; Lengthening Timbers, 120; Roofing, 121; Covering of Circular Roofs, 130; Niches, 134; Bracketing for Coves and Cornices, 138 ; Boarding Circular Roofs, 140; Coverings of Polygonal Roofs, 146 ; Pendentive Bracketing, 148 ; Purlins in Circular Roofs, 152 ; Designs for Partitions, 155 ; Designs for Roofs 156 CHAPTER III. JOINERY, in general, Mouldings, &c 159 Doors, 164; Windows, 165; Bases and Surbases, 165; Architraves and Pilasters, 165; Dimi- nution of Columns, 166; Method of setting out Flutes and Fillets, 167; of taking Dimensions and Hingeing, 170; Hingeing Doors and Shutters, 171; Mouldings on the Spring, 172; Raking Mouldings, 173; Method of enlarging and diminishing Mouldings, 175; Circular Sashes in Circular Walls, 177; Skylights, 178; Designs for Shutting- Windows, 180; For- mation of the Shutting-joints of Doors, 182 ; Jibb-doors, 183 ; Stairs and Stair-casing, 184 ; Dog-legged Stairs, 190; Bracketed Stairs, 192; Geometrical Stairs, d 92; Hand-railing.... 19<5 CHAPTER IV. An EXPLANATION of TERMS, and Description of TOOLS, used in CARPENTRY and JOINERY 218 Terms, &c. 218 ; Description of Tools. 232 CHAPTER V. REMARKS on, and INSTRUCTIONS for choosing, the different sorts of TIMBER; the relative Strength of Timber, and Modes of ascertaining it ; the Scarfing and Combination of Beams; the Formation of the Centring of Arches, and the Construction of Timber Bridges, &c. 257 CHAPTER VI. MASONRY, in general 285 Materials employed in Masonry, 286 ; Construction of Bridges, 289 ; Foundations, 303 ; Walls, 306 ; Ornamented Masonry, 308 ; Construction of Domes, 312 ; Valuation of Mason's Work, 314; Description of the Sections of Arches, 316; Stone-cutting, &c. 318; Raking Mouldings ., 326 CHAPTER VII. EXPLANATION of TERMS, and Description of TOOLS, used in MASONRY ; including the Composition of Cements, Mortar, &c 328 CHAPTER VIII. BRICKLAYING ; including an Abstract of the Building-Act 343 Foundations, &c. 353; Niches, 359 ; Construction of Ovens, 360 ; Substance of the Building Act 363 B (2) CONTENTS. PAGE CHAPTER IX. PLASTERING, Treatise on. . . ; , 369 Materials and Working Tools, 369, 370; Lathing, &c. 372; Cornices, 375; Casts, &c. 377; Capitals to Columns, 377; 'Roman Cement, 378; Rough-casting, 380; Scagliola, 380; Composition, 382 ; Plasterer's Measuring and Valuation 383 CHAPTER X. EXPLANATION of TERMS, and Description of Toots, used in BRICKLAYING and PLASTERING 384 CHAPTER XI. SLATING, with an Explanation of TERMS, &c ; 395 Comparison, in weight, of the sundry Coverings employed on Roofs, S98; the Manner of laying Slates, 399 ; Slater's Tools, 400 ; Valuation of Slater's Work, 401 ; Explanation of Terms. . 402 CHAPTER XII. PLUMBERY or PLUMBING 403 Tools, &c. 403 ; Casting, 405 ; Solder, 406 ; laying Sheet Lead, &c. 407; Pipes, 407; Reser- voirs and Pumps 408 CHAPTER XIII. HOUSE-PAINTING 410 Materials and Colours, 410; Common House-Painting and Graining, 417 ; Ornamental Paint- ing, &c. 418 ; Valuation of Painter's Work 418 CHAPTER XIV. GLAZING, in general 419 ' ClaSsy 42CT ; Glazier's Valuation, &c. 421 ; Tools, &c t 422 ,.'-.. CHAPTER XV. Of BUILDING in GENERAL 425 Explanation of Terms in Building, 424 ; Proportions of the Apertures of Doors and Windows, 436 ; Proportions' of Apartments, 437 ; Stairs, 438 ; Mouldings, &c. 442 ; Proportions of Mouldings, 447 ; on the Beauty of Buildings, 448 ; Situations for Country Residences 449 CHAPTER XVI. The THEORY and PRACTICE of the FIVE ORDERS of ARCHITECTURE 451 The Orders, in general, 451 ; Proportions of the Orders, 457; Definitions, &e. 465; Mouldings, Ornaments, &c. 467; References to the Plates of the Five Orders, 479; PRACTICE of the Tuscan Order, 480 ; of the Doric Order, 486 ; of the Ionic Order, 493 ; of the Composite Order, 498 ; of the Corinthian Order, 504 ; of Pilasters and the Greek Antae 509 On the antient Architecture of Great Britain 514 CHAPTER XVII. PERSPECTIVE, PROJECTION, and SHADOWS, Treatise on 518 Definitions, 518 ; Axioms, 520 ; Theorems, 520 ; Problems 523 Projection, 539 ; Shadows 545 APPENDIX. Decimal Fractions, 551 ; Involution, or Raising of Powers, 555 ; Evolution, or the Extraction of Roots, 555 ; Mathematical Instruments, Scales, &c., including the Centrolinead and Cyclograph 559 ARRANGEMENT and DESCRIPTION of the PLATES given in this Work 563 GLOSSARY of TECHNICAL TERMS and GENERAL INDEX 579 DIRECTIONS THIS BOOK IS DUE ON THE LAST STAMPED BELOW RENEWED BOOKS ARE SUBJECT TO IMMEDIATE RECALL LIBRARY, UNIVERSITY OF CALIFORNIA, DAVIS Book Slip-50J-8,'66(G5530s4)45S m N9 525489 The new practical builder, and workman's companion* PHYSICAL SCIENCES LIBRARY The new practical builder, and workman's companion.