Practical Carpentry WITH Steel Square Supplement ILLUSTRATED EY NEARLY 300 ENGRAVINGS. Practical Carpentry WITH STEEL SQUARE SUPPLEMENT BEING A GUIDE TO THE Correct Working and Laying Out of All Kinds of Carpenters’ and Joiners’ Work. With the Solutions of the Various Problems in Hip-Roofs, Gothic Work, Centering, Splayed Work, Joints and Jointing, Hinging, Dovetailing, Mitering, Timber Splicing, Hopper Work, Skylights, Raking Mouldings, Circular Work, Etc., Etc. TO WHICH IS PREFIXED A THOROUGH TREATISE ON “CARPENTERS' GEOMETRY." BY FRED T. HODGSON i.UTHOR of “The Steel Square and Its Uses, “The Builder’s Guide and Estimator’s Price Book,” “The Slide Rule and How to Use It,” Etc., Etc. MONTGOMERY WARD & COMPANY CHICAGO TABLE OF CONTENTS. PART I. PAG£ Geometry.— Straight Lines.— Curved Lines.— Solids.— Compound Lines. —Parallel Lines.— Oblique or Converging Lines.— Plane Figures.— Angles.— Right Angles.— Acute Angles.— Obtuse Angles.— Right- angled Triangles.— Quadrilateral Figu res.— Paral lelograms.— Rect- angles.— Squares.— Rhomboids.— Trapeziums. —Trapezoids.— Diag- onals. —Polygons.— Pentagons. — Hexagons. — Heptagons. — Octa- gons.— Circles.— Chords.— Tangents.— Sectors.— Quadrants.— Arcs.— Concentric and Eccentric Circles.— Altitudes.— Problems I. to XXIX. —Drawing of Angles.— Construction of Geometrical Figures.— Bisec- tion of Lines.— Trisection of Lines and Angles.— Division of Lines into any Number of Parts.— Construction of Triangles. Squares and Parallelograms.— Construction of Proportionate Squares.— Con- struction of Polygons.— Areas of Polygons.— Areas of Concentric Rings and Circles.— Segments of Circles.— The use of Ordinates for Obtaining Arcs of Circles.— Drawing an Ellipse with a Trammel. —Drawing an Ellipse by means of a String— Same by Ordinates.— Raking Ellipses.— Ovals.— Sixty-two Illustrations, ... $-34 PART n Arches, Centres.— Window and Door Heads.— Semi-circular Arch.— Segmental Arches.— Stilted Arches.— Horseshoe Arch.— Lancet Arch. —Equilateral Arch.— Gothic Tracery.— Wheel-Windows.— Equila- teral Tracery.— Square Tracery.— Finished Leaf Tracery.— Twenty- two Illustrations. • 35-42 VI TABLE OF CONTENTS. PART III. PAGE Roofs.— Saddle Roof.— Lean-to or Shed Roof.— Simple *Iip-Roof.— Pyramidal Roof.— Theoretical Roof.— Roof with Straining Beam- Gothic Roof.— Hammer-Beam Roofs— Curved Principal Roofs.— Roofs with Suspending Rods.— Deck Roofs.— King-post and Prin- cipal Roof.— Queen-post and Principal Roof.— Roofs with Laminated Arches.— Strapped Roof Frames.— Tie-beam Roofs.— Roofs for Long Spans.— Theatre Roof.— Church Roof.— Mansard Roof.— Slopes of Roofs.— Rules for Determining the Sizes of Timbers for Roofs.— Acute and Obtuse Angled Hip-Roofs.— Development of Hip-Roofs.— Obtaining Lengths and Bevels of Rafters.— Backing Hip-Rafters.— Lengths, Bevels and Cuts of Purlins.— Circular, Conical and Seg- mental Roofs.— Rafters with Variable Curves.— Veranda Rafters.— Development of all kinds of Rafters.— Curved Mansard Rafters.— Framed Mansard Roofs.— Lines and Rules for obtaining various kinds of Information.— Thirty-four Illustrations, - - - 43— PART IV. Covering of Roofs.— Shingling Common Roofs.— Shingling Hip- Roofs.— Method of Shingling on Hip Corner.— Covering Circular Roofs.— Covering Ellipsoidal Roofs.— Valley Roofs.— Four Illustra- tions. ---------- 65—68 PART V. The Mitering and Adjusting of Mouldings.— Mitering of Spring Mouldings.— Preparing the Mitre-box for Cutting Spring Mould- ings.— Rules for Cutting Mouldings, with Diagrams.— Mitre-boxes of various forms.— Lines for Spring Mouldings of various kinds.— Seven Illustrations. - 60—73 PART VI. Sashes and Skylights.— Raised Skylights.— Skylights with Hips.— Octagon Skylights with Segmental Ribs.— Angle-bars, with Rules and Diagrams, showing how to obtain the Angles, Forms, etc.— Sash- Bars, Hints on their Construction.— Twelve Illustrations. - - 74 78 TABLE OF CONTENTS. V/ PAET VII. PAGE Mouldings.— Angle Brackets.— Corner Coves.— Enlarging and Reduc- ing Mouldings.— Irregular Mouldings.— Raking Mouldings, with Rules for Obtaining. — Mouldings for Plinths and Capitals of Gothic Columns. — Mouldings around Square Standards. — Mitering Cir- cular Mouldings with each other — Mitering Circular Mouldings with Straight ones.— Mitering Mouldings at a Tangent.— Mitering Spring Circular Mouldings.— Description of Spring Mouldings.— Lines for Circular Spring Mouldings.— Seventeen Illustrations, - 79—67 PART VIII. Joinery.— Dovetailing.— Common Dovetailing.— Lapped Dovetailing.— Blind Dovetailing.— Square Dovetailing.— Splay Dovetailing.— Regular and Irregular Dovetailing.— Lines and Cuts for Hoppers and Splayed Work.— Angles and Mitres for Splayed Work.— Nineteen Illustrations. - -- -- -- -- 88—94 PART IX. Miscellaneous Problems.— Bent Work for Splayed Jambs.— Develop ment of Cylinders.— Rules and Diagrams for Taking Dimensions.— Angular and Curved Measurements.— Eight Illustrations. - - 96—99 PART X. Joints and Straps.— Mortise and Tenon Joints.— Toggle Joints.— Hoot Joints.— Tongue Joint.— Lap Splice.— Scarfing.— Wedge Joints.- King-bolts.— Straps, Iron Ties, Sockets, Bearing-plates, Rings, Swivels and other Iron Fastenings.— Straining Timbers, Struts and King- pieces.— Three Plates, Sixty-five Illustrations, - - 09- JO) PART XI. Hinging and Swing Joints.— D oor Hinging.— Centre-pin Hinging.— Blind Hinging.— Folding Hinging.— Knuckle Hinging.— Pew Hing- ing.— Window Hinging.— Half-turn Hinge.— Full-turn Hinge.— Back Flap Hinging.— Rule-joint. Hinging.— Rebate Hinging.— Three Plates. Fifty-one Illustrations. - 102— 1C2 TABLE OF CONTENTS. vm PART XII. PAGE Useful Rules and Tables.— Hints on the Construction of Centres.— Rules for Estimating.— Form of Estimate.— Items for Estimating,— Remarks on Fences.— Nails: sizes, weights, lengths and numbers.— Cornices, Proportions and Projections for Different Styles of Archi- tecture; and Tall and Low Buildings, Verandas, Bay Windows and Porches.— Proportion of Base-boards, Dados, Wainscots and Sur- bases.— Woods, Hard and Soft, their Preparation, and how to Finish.— Strength and Resistance of Timber of various kinds.— Rules, showing Weight and other qualities of Wood and Timber.— Stairs, Width of Treads and Risers ; their Cost; how to Estimate on them, etc.— Inclinations oj; Roofs.— Contents of Boxes. Bins and Barrels.— Arithmetical Signs.— Mensuration of Superficies.— Areas of Squares, Triangles, Circles, Regular and Irregular Polygons.— Properties of Circles.— Solid Bodies.— Gunter’s Chain.— Drawing and Drawing Instruments.— Coloring Drawings.— Coloring for Various Building Materials.— Drawing Papers.— Sizes of Drawing Papers.— Table of Board Measure.— Nautical Table.— Measure of Time.— Authorized Metric System.— Measures of Length.— Mea- sures of Surfaces.— Measures of Capacity.— Weights.— American Weights and Measures.— Square Measure.— Cubic Measure.— Cir- cular Measure.— Decimal Approximations.— Form of Building Contract, - J 04 — Hi PRACTICAL CARPENTRY PAST I.— GEOMETRY. i^jlEFORE a knowledge of geometry can be acquired, it wiH K55 be necessary to become acquainted with some of the terms and definitions used in the science of geometry, and to this end the following terms and explanations are given, though it must be understood that these are only a few of the terms used in the science, but they are sufficient for our purposes : 1. A. Point has position but not magnitude. Practically, it is represented by the smallest visible mark or dot, but geometrically understood, it occupies no space. The extremities or ends of lines are points ; and when two or more lines cross one another, the places that mark their intersections are also points. 2. A Line has length, without breadth or thickness, and, conse- quently, a true geometrical line cannot be exhibited; for however finely a line may be drawn, it will always occupy a certain extent of space. 3. A Superficies or Surface has length and breadth, but no thick- ness. For instance, a shadow gives a very good representation of a superficies : its length and breadth can be measured ; but it has no depth or substance. The quantity of space contained in any plane surface is called its area. 4. A Plane Superficies is a flat surface, which will coincide with a straight line in every direction IO PRACTICAL CARPENTRY. 5. A Curved Superficies is an uneven surface, or such as will not coincide with a straight line in all directions. By the term surface is generally understood the outside, of any body or object; as, for instance, the exterior of a brick or stone, the boundaries of which are represented by lines, either straight or curved, according to the form of the object. We must always bear in mind, however, that the lines thus bounding the figure occupy no part of the surface ; hence the lines or points traced or marked on any body or surface, are merely symbols of the true geometrical lines or points. 6. A Solid is anything which has length, breadth and thickness; consequently, the term may be applied to any visible object con- taining substance ; but, practically, it is understood to signify the solid contents or measurement contained within the different sur- faces of which any body is formed. 7. Lmes may be drawn in any direction, and are termed straight, curved, mixed, concave, or convex lines, according as they corres- pond to the following definitions. 8. A Straight Line is one every part of which A “ — D lies in the same direction between its extremities, Flg ' *' and is, of course, the shortest distance between two points, as from a to b, Fig. 1. 9. A Curved Liiie is such that it does not lie in a straight direc- tion between its extremities, but is continually changing by inflec- tion. It may be either regular or irregular. 10. A Mixed or Compound Line is composed of straight and curved lines, connected in any form. 11. A Concave or Cojivex Lme is such that it cannot be cut by a straight line in more than two points ; the concave or hollow side is turned towards the straight line, while the convex or swell- ing side looks away from it. For instance, the inside of a basin is concave — the outside of a ball is convex. 12. Parallel Straight Lines have no inclination, but are every- where at an equal distance from each other; consequently they can never meet, though produced or continued to infinity in either or both directions. Parallel lines may be either straight or curved, PRACTICAL CARPENTRY. II provided they are equally distant from each other throughout their extension. 13. Oblique or Converging Lines are straight lines, which, if con- tinued, being in the same plane, change their distance so as to meet or intersect each other. 14. A Plane Figure , Scheme , or Diagram , is the lineal representa- tion of any object on a plane surface. If it is bounded by straight lines, it is called a rectilineal figure; and if by curved lines, a curvilineal figure. 15. An Angle is formed by the inclination of two lines meeting in a point: the lines thus forming the angle are called the sides; and the point where the lines meet is called the vertex or angular point . When an angle is expressed by three letters, as a b c, Fig. 2, the middle letter b should always denote the angular point: where Fig. 2- there is only one angle, it may be expressed more concisely by a letter placed at the angular point only, as the angle at a, Fig. 3. 16. The quantity of an angle is estimated by the arc of any circle contained between the two sides or lines forming the angle; the junction of the two lines, or vertex of the angle, being the centre from which the arc is described. As the circumferences of all circles are proportional to their diameters, the arcs of similar sectors also bear the same proportion to their respective circum- ferences; and, consequently, are proportional to their diameters, and, of course, also to their radii or semi-diameters> Hence, the 12 PRACTICAL CARPENTRY. proportion which the arc of any circle bears to the circumference of that circle, determines the magnitude of the angle. From this it is evident that the quantity or magnitude of angles does not de- pend upon the length of the sides or radii forming them, but wholly upon the number of degrees contained in the arc cut from the circumference of the circle by the opening of these lines. The circumference of every circle is divided by mathematicians into 360 equal parts, called degrees ; each degree being again subdi- vided into 60 equal parts, called minutes, and each minute into 60 parts, called seconds. Hence, it follows that the arc of a quarter circle or quadrant includes 90 degrees ; that is, one-fourth part of 360 degrees. By dividing a quarter circle, that is, the portion of the circumference of any circle contained between two radii form- ing a right angle, into 90 equal parts, or, as is shown in Fig. 4, into nine equal parts of 10 degrees each, then drawing straight lines from the centre through each point of division in the arc; the right angle will be divided into nine equal angles, each containing 10 degrees. Thus, suppose b c the horizontal line, and a b the per- pendicular ascending from it, any line drawn from b — the centre from which the arc is described — to any point in its circumference, determines the degree of inclination or angle formed between it and the horizontal line b c. Thus, a line from the centre b to the tenth degree, separates an angle of 10 degrees, and so on. In this manner the various slopes or inclinations of angles are defined. 17. A Right Angle is produced by one straight line standing upon another, so as to make the adjacent angles equal. This is what workmen call “ square,” and is the most useful figure they employ. 18. An Acute Angle is less than a right angle, or less than 90 degrees. 19. An Obtuse Angle is greater than a right angle or square, or more than 90 degrees. The number of degrees by which an angle is less than 90 de- grees is called the complement of the angle. Also, the difference between an obtuse angle and a semicircle, or 180 degrees, is called the supplement of that angle. PRACTICAL CARPENTRY. l 3 20. Plane Figures are bounded by straight lines, and are named according to the number of sides which they contain. Thus, the space included within three straight lines, forming three angles, is called a trilateral figure or triangle. 21. A Right-Angled* Triangle has one right angle: the sides forming the right angle are called the base and perpendicular; and the side opposite the right angle is named the hypothenuse. An equilateral triangle has all its sides of equal length. An isosceles triangle has only two sides equal; a scalene triangle has all its sides unequal. An acute-angled triangle has all its angles acute, and an obtused-angled triangle has one of its angles only obtuse. The triangle is one of the most useful geometrical figures for the mechanic in taking dimensions; for since all figures that arc bounded by straight lines are capable of being divided into tri- angles, and as the form of a triangle cannot be altered without changing the length of some one of its sides, it follows that the true form of any figure can be preserved if the length of the sides of the different triangles into which it is divided is known ; and the area of any triangle can easily be ascertained by the same rule, as will be shown further on. Quadrilateral Figures are literally four- sided figures. They are also called quadrangles, because they have four angles. 22. A Parallelogram is a figure whose opposite sides are parallel, as a b c d, Fig. 5. 23. A Rectangle is a parallelogram having four right angles, as a b c d, in Fig. 5. 24. A Square is an equilateral rectangle, having all its sides equal, like Fig. 5. 25. An Oblong is a rectangle whose adjacent sides are unequal, as the parallelogram shown at Fig. 10. 26. A Rhombus is an oblique-angled figure, or parallelogram having four equal sides, whose opposite angles only are equal, as C, Fig. 6. 27. A Rhomboid is an oblique-angled parallelogram, of which ths, adjoining sides are unequal, as d, Fig. 7. H PRACTICAL CARPENTRY. 28. A Trapezium is an irregular quadrilateral figure, having no two sides parallel, as e, Fig. 8. 29. A Trapezoid is a quadrilateral figure, which has two of its opposite sides parallel, and the remaining two neither parallel nor equal to one another, as f, Fig. 9. Fig. 8. Fig. 9. 30. A Diagonal is a straight line drawn between two opposite angular points of a quadrilateral figure, or between any two angular points of a polygon. Should the figure be a parallelogram, the diagonal will divide it into two equal triangles, the opposite sides and angles of which will be equal to one another. Let abcd, Fig. 10, be a parallelogram; join a c, then a c is a diagonal, and the triangles adc,abc, into which it divides the parallelogram, are equal. 31. A plane figure, bounded by more than four straight lines, is called a Polygon . A regular polygon has all its sides equal, and consequently its angles are also equal, as k, l, m, and n, Figs. 12-15. An irregular polygon has its sides and angles unequal, as H, Fig. 11. Polygons are named according to the number of their sides or angles, as follows 32. A Pentago?i is a polygon of five sides, as h or k, Figs. 11, 12. 33. A Hexagon is a polygon of six sides, as l, Fig. 13. 34. A., Heptagon has seven sides, as m. Fig. 14. PRACTICAL CARPENTRY. 1 s 35. An Octagon has eight sides, as n, Fig. 15. An Enneagoti has nine, a Decagon ten, an Undecagon eleven, and a Dodecagon twelve sides. Figures having more than twelve sides are generally designated Polygons , or many-angled figures. 36. A Circle is a plane figure bounded by one uniformly curved line, bed (Fig. 16), called the circumference, every part of which is equally distant from a point within it, called the centre, as a . 37. The Radius of a circle is a straight line drawn from the centre to the circumference ; hence, all the radii (plural for radius) of the same circle are equal, as b a y c a, e a,f a, in Fig. 16. 38. The Diameter of a circle is a straight line drawn through the centre, and terminated on each side by the circumference; conse- quently the diameter is exactly twice the length of the radius ; and hence the radius is sometimes called the semi-diameter. (See bae t Fig. 16.) 49. The Chord or Subtens of an arc is any straight line drawn from one point in the circumference of a circle to another, joining the extremities of the arc, and dividing the circle either into two equal, or two unequal parts. If into equal parts, the chord is also the diameter, and the space included between the arc and the di- ameter, on either side of it, is called a semicircle, as baem Fig. 16. If the parts cut off by the chord are unequal, each of them is called a segme?it of the circle. The same chord is therefore common to two arcs and two segments ; but, unless when stated otherwise, it is always understood that the lesser arc or segment is spoken of, as in Fig. 16, the chord c d is the chord of the arc c e d. If a straight line be drawn from the centre of a circle to meet the chord of an arc perpendicularly, as a / in Fig. 16, it will divide the chord into two equal parts, and if the straight line be produced ? 6 PRACTICAL CARPENTRY. to meet the arc, it will also divide it into two equal parts, as cfjd. Each half of the chord is called the sine of the half-arc to which it is opposite ; and the line drawn from the centre to meet the chord perpendicularly, is called the co-sine of the half-arc. Con- sequently, the radius, the sine, and co-sine of an arc form a right angle. 40. Any line which cuts the circumference in two points, or a chord lengthened out so as to extend beyond the boundaries of the circle, such as g h in Fig. 17, is sometimes called a Secant . But, in trigonometry, the secant is a line drawn from the centre through one extremity of the arc, so as to meet the tangent which is drawn from the other extremity at right angles to the radius. Thus, Feb is the secant of the arc c e , or the angle cf e,\n Fig. 17 41. A Tangent is any straight line which touches the circumfer- ence of a circle in one point, which is called the point of contact, as in the tangent line e b , Fig. 17. 42. A Sector is the space included between any two radii, and that portion of the circumference comprised between them: ce f is a sector of the circle a f c e , Fig. 17. 43. A Quadrant , or quarter of a circle, is a sector bounded by two radii, forming a right angle at the centre, and having one- fourth part of the circumference for its arc, as f fd , Fig. 1 7. 44. An Arc , or Arch , is any portion of the circumference of a circle, as c d e , Fig. 17. It may not be improper to remark here that the terms circle and circumference are frequently misapplied. Thus we say, describe a circle from a given point, etc., instead of saying describe the cir- cumference of a circle — the circumference being the curved line thus described, everywhere equally distant from a point within it, called the centre : whereas the circle is properly the superficial space included within that circumference. 45. Concentric Circles are circles within circles, described from the same centre; consequently, their circumferences are parallel to one another, as Fig. 18. 46. Eccentric Circles are those which are not described from the PRACTICAL CARPENTRY* *7 same centre; any point which is not the centre is also eccentric in reference to the circumference of that circle. Eccentric circles may also be tangent circles ; that is, such as come in contact in one point only, as Fig. 19. 47. Altitude . The height of a triangle or other figure is called its altitude . To measure the altitude, let fall a straight line from the vertex, or highest point in the figure, perpendicular to the base or opposite side; or to the base continued, as at b d, Fig. 20 , should the form of the figure require its extension. Thus c d is the altitude of the triangle abc. We have now described all the figures we shall require for the purpose of thoroughly understanding all that will follow in this book ; but we would like to say right here that the student who has time should not stop at this point in the study of geometry, for the time spent. in obtaining a thorough knowledge of this useful science will bring in better returns in enjoyment and money, than if expended for any other purpose. We will now proceed to explain hew the figures we have de- scribed can be constructed. There are several ways of constructing nearly every figure we produce, but we have chosen those methods that seemed to us the best, and to save space have given as few examples as possible consistent with efficiency. Problem I. — Through a given point c (Fig. 18 a), to draw a straight line parallel to a given straight line a b. In a b (Fig. 18 a) take any point d , and from d as a centre with the radius d c, describe an arc c e , cutting a b in e, and from c as a i8 PRACTICAL CARPENTRY. centre, with the same radius, describe the arc */d, make dv> equal t c f and h, drav( d k, e /, /w, g n , h 0 , parallel to a c ; and the parts c k, k /, l m, etc., will be to each other, or to the whole line b c, as the lines a d, d e, e f, etc., are to each other, or to the given line or scale a b. By this method, as will be evident from 22 PRACTICAL CARPENTRY. the figure, similar divisions can be obtained in lines of any given length. Problem VIII, — To describe ah equilateral triangle lipon a given straight line. Let a b (Fig. 27) be the given straight line. From the points a and b, with a radius equal to a b, describe arcs intersecting each other in the point c. Join c a and c b, and abc will be the equilateral triangle required. Problem IX. — To construct a triangle whose sides shall be equal to three given lines , F, E, D. Draw a B (Fig. 28) equal to the given line f. From a as a centre, with a radius equal to the line e, describe an arc; then c A E — F — from B as a centre, with a rad ms equal to the line d, describe an- other arc intersecting the former in c; join c a and c b, and abc will Be the triangle required. Problem X. — To describe a rectangle or parallelogram having one of its sides equal to a given li?ie, and its area equal to that of a given rectangle . Let a b (Fig. 29) be the given line, and c d e f the given rectangle. Produce c e to g, making e g equal to ab; from g draw g k parallel to f f, and meeting d f produced in 11. Draw the diagonal g f, extending it to meet c d produced in l ; also draw l k parallel to d h, and produce e f till it meet l k in m ; then f m k h is the rectangle required. PRACTICAL carpentry. 2 3 Equal and similar rhomboids or parallelograms of any dimen- sions may be drawn after the same manner, seeing the comple- ments of the parallelograms which are described on or about the diagonal of any parallelogram, are always equal to each other; while the parallelograms themselves are always similar to each other, and to the original parallelogram about the diagonal of which they are constructed. Thus, in the parallelogram c c k l the complements cefd and f m k h are always equal, while the parallelograms efhg and d f m l about the diagonal g l, are always similar to each other, and to the whole parallelogram CGKL Problem XI. — To describe a square equal to two given squares. Let a and b (Fig. 30) be the given squares. Place them so that a side of each may form the right angle d c e ; join d e, and upon this hypothenuse describe the square degf, and it will be equal to the sum of the squares a and b, which are constructed upon the legs of the right-angled triangle dce. In the same manner, any other rectilineal figure, or even circle, may be found equal to the sum of other two simi- lar figures or circles. Suppose the lines c d and c e to be the diameters of two circles, then d e will be the diameter of a third, equal in area to the other two circles. Or suppose c d and c e to be the like sides of any two similar figures, then d e will be the corresponding side of another similar figure equal to both the former. Problem XII. — To describe a square equal to any number of given squares. Let it be required to construct a square equal to the three given squares a, b, and c (Fig. 31). Take the line d e, equal to the side of the square c. From the extremity d erect d f perpendicular to d e, and equal to the side of the square b ; join e f; then a square described upon this line will be equal to the sum of the two given squares c and b. Again, upon the straight line e f erect the per- 24 PRACTICAL CARPENTRY. pendicular f g, equal to the side of the third given square a ; and join g e, which will be the side of the square g e h k, equal in area to a, b, and c. Proceed in the same way for any number of given squares. Problem XIII. — Upon a given straight line to describe a regular polygon . To produce a regular pentagon draw A b to c (Fig. 32), so that b c may be equal to a b ; from b as a centre, with the radius b a or b c describe the semicircle A d c; divide the semi-circumference adc into as many equal parts as there are parts in the required polygon, which, in the present case, will be five ; through the second division from c draw the straight line b d, which will form another side of the figure. Bisect a b at e > and bd at f and draw e g and / G per- pendicular to a b and b d ; then g, the point of intersection, is the centre of a circle, of which a b and d are points in the circumference. From g, with a radius equal to its distance from any of these points, describe the circumference a b d h k ; then producing the dotted lines from the centre b, through the remaining divisions in the semicircle a d c, so as to meet the circumference of which g is the centre, in h and k, these points will' divide the circle a b d h k into the number of parts required, each part being equal to the given side of the pentagon. From the preceding example it is evident that polygons of any number of sides may be constructed upon the same principles, be- cause the circumferences of all circles, when divided into the same number of equal parts, produce equal angles ; and, consequently, by dividing the semi-circumference of any given circle into the H D G PRACTICAL CARPENTRY. *S number of parts required, two of these parts will form an angle which will be subtended by its corresponding part of the whole circumference. And as all regular polygons can be inscribed in a circle, it must necessarily follow, that if a circle be described through three given angles of that polygon, it will contain the number of sides or angles required. The above is a general rule, by which all regular polygons may be described upon a given straight line; but there are other methods by which many of them may be more expeditiously con- structed, as shown in the following examples : — Problem XIV. — Upon a given straight line to describe a regular pentagon . Let a b (Fig. 33) be the given straight line; from its extremity B erect Be perpendicular to a b, and equal to its half. Join a c , and produce it till c d be equal to b c , or half the given line a b. From a and b as centres, with a radius equal to b d, describe arcs inter- secting each other in e, which will be the centre of the circumscribing circle abf g h. The side a b applied successively to this circumference, will give the angu- lar points of the pentagon; and these being connected by straight lines will complete the figure. Problem XV. — Upon a given straight line to describe a regular hexagon . — Let a b (Fig. 34) be the given straight line. From the extremi- ties a and b as centres, with the radius a b describe arcs cutting each other in g. Again from g, the point of intersection, with the same radius, describe the circle abc, which will contain the given side a b six times when applied to its circumference, and will he the hexagon required. Problem XVI. — To describe a regular octagon upon a given straight line. Let a b (Fig. 35) be the given line. From the extremities a and b erect the perpendiculars a e and b f ; extend the given PRACTICAL CARPENVRY. 26 line both ways to k and /, forming external right angles with the lines a e and b f. Bisect these external right angles, making each of the bisecting lines a h and b c equal to the given line a b. Draw h G and c d parallel to a e or b f, and each equal in length to a b. From g draw g e parallel to b c, and intersecting a e in e, and from d draw d f parallel to a h, intersecting b f in f. Join e f, and abcdfeghIs the octagon required. Or from d and g as centres, with the given line a b as radius, describe arcs cutting the perpendiculars a e and b f in e and f, and join g e, e f, f d, to complete the octagon. Otherwise , thus . — Let a b (Fig. 36) be the given straight line on which the octagon is to be described. Bisect it in a , and draw the perpendicular 'a b equal to a a or b a. Join a b , and produce a b to c, making b c equal to a b ; join also a e and b c , extending them so as to make c e and c f each equal to a c or b c. Through c draw c c g at right angles to a e. Again, through the same point c . draw d h at right angles to b f, making each of the lines fC, f D. c g, and c h equal to a c or c b, and consequently equal to one another. Lastly, join bc,cd,de,ef,fg,gh,ha; abcdefgh will be a regular octagon described upon a b, as required. Problem XVII . — In a given square to inscribe a given octagon . Let a b c d (Fig. 37) be the given square. Draw the diagonals a c and b d, intersecting each other in e ; then from the angular points abc and d as centres, with a radius equal to half the diagonal, viz., a e or c Fig. 45. right angles to a c, or divide a d and e b into the same number of equal parts, and number the divisions from a and e respectively, and join the corresponding numbers by the lines 1 1, 2 2, 3 3. Divide also a g into the same number of equal parts as a d or e b, numbering the divisions from a upwards, 1, 2, 3, etc.; and from the points , 2 and 3, draw lines to b ; and the points of intersection ot these, with the other lines at h 9 h, /, will be points in the curve re- quired. Same with b c. Another Method, — Let a c (Fig. 46) be the chord and d b the versed sine. Join a b, b c, and through b draw e f parallel to a c- PRACTICAL CARPENTRY. 3 1 From the centre b, with the radius b a or b c, describe the arcs a e, cf and divide them into any number of equal parts, as i* 2 , 3: from the divisions 1, 2, 3, draw radii to the centre B, and divide each radius into the same number of equal parts as the arcs a b Fig. 46. and c F ; and the points g , h , /, n , 0 , thus obtained, are points in the required curve. These methods, though not absolutely correct, are sufficiently acqurate when the segment is less than the quadrant of a circle. Problem XXV. — To draw an ellipse with the trammel . The trammel is an instrument consisting of two principal parts, the fixed part in the form of a cross efgh (Fig. 47), and the moveable piece or tracer klm. The fixed piece, is made of two rectangular bars or pieces of wood, of equal thickness, joined to- gether so as to be ii. the same plane. On one side of the frame so formed, a groove is made, forming a right-angled cross. In the groove two studs, k tnd /, are fitted to slide freely, and c trry attached to them the tracer kin . The tracer is generally made to slide ' through a socket fixed to each stud, and pro- vided with a screw or wedge, by which the distance a\ >art of the studs may be regulated. The tracer has another slider also adjustable, which carries a penc l or point. The instrument is used as fol- lows: — Let a c be tl e major, and h b the minor axis of an ellipse: lay the cross of the ti imrnel on these lines, so that the centre lines of it may coincide with them; then adjust the sliders of the tracer, so that the distance betveen k and m may be equal to half the major axis, and the distance between / and m equal to half the minor 3 2 PRACTICAL CARPENTRV axis ; then by moving the bar round, the pencil in the slider will describe the ellipse. Problem XXVI. — An ellipse may also be described by means of a string. Let a b (Fig. 48) be the major axis, and d c the minor axis of the ellipse, and f g its two foci. Take a string egf and pass it over the pins, and tie the ends together, so that when doubled it may be equal to the distance from the focus f to the end of the axis, b; then putting a pencil in the bight or doubling of the string at h and carrying it round, the curve may be traced. This is based on the well known property of the ellipse, that the sum of any two lines drawn from the foci to any points in the circumference is the same. Problem XXVII. — The axes of an ellipse being given , to draw the curve by intersections . Let a c (Fig. 49) be the major axis, and d b half the minor axis. On the major axis construct the parallelogram aefc, and make its height equal to d b. Divide a e and e b each into the same number of equal parts, and number the divisions from a and e respectively; then join ai, 12, 23, etc., and their intersec- tions will give points through which the curve may be drawn. The points for a “ raking ” or rampant ellipse may also be found by the intersection of lines as shown at Fig. 50. Let a c be the major and e b the minor axis : draw a g and c h each parallel to b e, and equal to the semi-axis minor. Divide a d, the semi- axis major, and the lines a g and c h each into the same number of equal parts, in r, 2, 3 and 4; then from e, through the divisions 1, 2, 3 and 4, on the semi-axis major a d, draw the lines e h, e k, e /, and e m; and from b, through the divisions 1, 2, 3 and 4 on the line a g, draw the lines 1, 2, 3 and 4 b; and the intersection of ** Fig. 48. PRACTICAL CARPENTRY. 33 these with the lines e i, 2, 3 and 4 in the points h kl m, will be points in the curve. Problem XXVIII. — To describe with a compass a figure resem- bling the ellipse . Let a b (Fig. 51) be the given axis, which divide into three equal parts at the points fg. From these points as centres, with the radius /a, describe circles which intersect each other, and from the points of intersection through /and g, draw the diameters cgE y c/d. From c as a centre, with the radius c d, describe the arc d e, which Fig. 51. Fig. 52. completes the semi-ellipse. The other half of the ellipse may be completed in the same manner, as shown by the dotted lines. Problem XXIX. — Another method of describing a figure ap- proaching the ellipse with a compass . The proportions of the ellipse may be varied by altering the ratio of the divisions of the diameter, as thus Divide the major ax^ of the ellipse a b (Fig. 52), into four equal parts, in the points f g h. O nfh construct an equilateral triangle /c h , and produce 34 PRACTICAL CARPENTRY. the sides of the triangle cf c fi indefinitely, as to d and e. Ther from the centres f and h y with the radius a f describe the circle a d g y b e g\ and from the centre c, with the radius c d, describe the arc d e to complete the semi-ellipse. The other half may be completed in the same manner. By this method of construction the minor axis is to the major axis as 14 to 22. **Ai;TlCAL CARPENTRY. 3 . PART II— ARCHES, CENTRES, WINDOW AND DOOR HEADS. N order that the reader may be able to “lay out” and construct centres for arches, window and door heads, it is necessary he should have a clear conception of what an arch really is. For if a positive conclusion has not been ar- rived at, and if the “ arch principle ” is not fairly understood, he cannot be expected to design an arch, or to construct it with accuracy or intelligence, even if designed by another. Let us then state once for all, that every curved covering to an aperture is not necessarily an arch. Thus, the stone which rests on the piers shown in Fig. 53 is not an arch, being merely a stone hewn out in anarch-like shape; but at its top, the very point (A) at which strength is required, it is the weakest, and would fracture the moment any great weight were placed upon it. It is not the province of this work to enter into a scientific disquisition on the arch, but some of its Fig. 5a properties must be known to the mechanic before he will be able to construct* centres understandingly ; and the general principles here laid down will help the workman materially to form correct ideas concerning the work in hand. In all cases, however, we advise the student to arm himself with a thorough knowledge of 'be arch and the principles involved. Elementary works on the subject can be easily obtained, and all who would really study principles, and appreciate the exquisite refinement of the examples herein given, are strongly urged to read them. The semi-circular arch shown at Fig. 54 is self-explanatory so far as the divisions are concerned. The under surface is called the in- trados, and the outer the ^trados. The supports are called the 36 PRACTICAL CARPENTRY. piers or abujtments, though the latter term is one of more extensive application, referring more generally to the supports which bridges obtain from the shore on each side than to other arches. The term “ piers ” is, as a rule, supposed to imply supports which receive vertical pressure, whilst abutments are such as resist outward thrust. The upper parts of the supports on which an arch rests are called the imposts. The span of an arch is the complete width between the points where the intrados meets the imposts on either side; and a line connecting these points is called the “ springing ” or spanning line. The separate wedge-like stones composing an arch are called Youssoirs, the central or uppermost one of which is called the Key- stone ; whilst those next to the imposts are termed “ springers.” The highest point in the intrados is called the vertex or crown , and the height of this point above the springing line is termed the “rise ” of the arch. It will be evident that in a semi-circular arch, such as Fig. 54, this would be the radius with which the semi-circle is struck. The spaces between the vertex and the springing line are called the flanks or haunches. The following are the varieties of arches used : — The Semi-circular , as shown in Fig. 54. The Segment (Fig. 55), in which a portion only of the circle is used ; the centre c is therefore not in the springing line Sp, Sp. There are several other kind of arches besides the ones here de- scribed, but they are seldom made use of by the carpenter; there are the parabolic, hyperbolic, catenarian, and cycloidal. We have given the methods of describing the ellipse, which, next to the circle, is the most used in building. $ Pig. 55. Pig. 54. PRACTICAL CARPENTRY. 37 The Semi-circular Arch was that principally used by the Romans, who employed it largely in their aqueducts and triumphal arches. The others are, however, mentioned by some writers as having been occasionally employed by the ancients. During the middle ages other forms were gradually introduced. The Stilted Arch is an adaptation of the semi-circular arch, in which the springing line is raised above the top of the column, on a pedestal not much larger in diameter than the width of the vous- soirs of the arch. The Horse-shoe Arch . — This is almost restricted to the Arabian or Moorish style of architecture. In this form of arch the curve is carried below the line of centre or centres ; for in some cases the arch is struck from one centre, and in others from two, as in Fig. 56. Now it must not be supposed that the real bearing of the arch is at the impost a a ; for if this were really so, it must be seen that any weight or pressure on the crown of the arch would cause.it to break at b, but the fact is simply that the real bearings of the arch are at b b, and the prolongation of the arch beyond these points is merely a matter of form and has no structural significancy. The Horse-shoe arch belongs especially to the Mohammedan architec- ture, from its having originated with that faith, and from its having been used exclusively by its followers. Next in point of time, but by far the most graceful in form, is the pomted arch, which is essentially the mediaeval (or middle age, style, and is capable of almost endless variety. The origin of this form of arch has been the subject of much antiquarian discussion ; but it is certain that although the pointed arch was first generally used in the architecture of the middle ages, recent discoveries have shown that it was used many centimes previously in Assyria. 3 » PRACTICAL CARPENTRY, The greater or less acuteness of the pointed arch depends on the position of the centres from which the flanks are struck. The Lancet Arch . — This arch, Fig. 57, is constructed by placing the centres c c outside the span, but still on the same line with the imposts. This form of arch was first used in the Gothic, and as a rule indicates the style called “ Early English.” Equilateral Arch. — Fig. 58 shows the Equilateral arch, the ra- dius with which tiie arcs are struck being equal to the span of the arch, and the centres being the imposts; and thus, the crown and the imposts being united, an equilateral triangle is formed. This form was principally used in the “ Decorated” period of Gothic architecture from about 1307 until about 1390, at which time the Ggee arch (Fig. 59) was also occasionally used. At a later date, during the existence of the “ Perpendicular ” style of Gothic architecture, viz., from the close of the 14th cen- tury to about 1630, we find various forms of arch introduced, such as the Segmental (Fig. 60), formed of segments of two circles, the centres of which are placed below the springing; and still later on we find the Tudor \ or four-centred arch (Fig. 61), in which two of the centres are on the springing and two below it. The arches at the later period of this style became flatter and flatter, and this forms one of the features of Debased Gothic, when the beautiful and graceful forms of that style gradually decayed, and for a time were lost. Happily, in the present century there has been a grad- ual and spirited revival of the Gothic style, and works are now be- ing produced which bid fair to rival in beauty of form and in prin- ciples of construction the marvellous buildings of the middle ages. From the examples given, the workman will be able to lay out any of the usual arches required in building. Fig. 59. Fig. 60. Fig. 61. PRACTICAL CARPENTRY. 39 There are combinations, however, of these curves which the car- penter may be called upon to construct, such as the ones given herewith. Fig. 62 is the elementary study upon which the subsequent fig- ure is based. Having drawn the circle, describe on the diameter two opposite semicircles, meeting at the centre, a. Divine one of these into six equal parts, and set off one of these sixths from i to //. Draw a n , and divide it into four equal parts. From the middle point of a n draw a line passing through the centre of the semi- Fig. 62. Fig. 63. Fig. 64. circle, and cutting it in From c set off on this line the lengtn or one of the fourths of a //. This point and the two in a n will be the centres for the interior curves. Fig. 63 is the further working out of this elementary figure. It is desirable that a larger circle should be drawn. Then, when the figure has been carried up to the stage shown in the last, all the rest of the curves will be drawn from the same centres. Fig. 64 is the elementary form of the tracery shown in Fig. 66. We show the method of obtaining these curves in Fig. 67 : At any point, as at a, draw a tangent, and ag at right angles to it. From a, with radius o a, cut the circle in b and c, and the tangent in the point f, using b as a centre. Bisect the angle b at f, and produce the bisecting line until it cuts a g in h. From o, with radius o h, cut the lines d c and e b in 1 and j. From h, i and j. 40 PRACTICAL CARPENTRY. with radius H a, draw the three required circles, each of which should touch the other two and the outer circle. Returning now to Fig. 64, having inscribed three equal circles in a circle, j ’•* their centres, thus forming an equilateral triangle. From the centre of the surrounding circle draw radii passing through the angles of the triangle and cutting the circle in points; as d and two others. Draw e d and bisect it by eg; then thb centres for the curves which are in the semicircle will be on the three lines d c f € g and c e.. These curves, in Gothic architecture, are called “ foliations, 1 ” or " featherings, 1 ” and the points a which they meet are called “cusps." The completion of this study is shown at Fig. 66. Fig. 65 shows the elementary construction f Fig. 63 . Draw two diameters (Fig. 65) at right angles to each other, and join their ex- tremities, thus inscribing a square in the circle. Bisect the quad- rants by two diameters, cutting the sides of the square in the points, as g; join these points, and a second square will be inscribed within the first. The middle points of the sides of this inner square, as bed, are the centres of the arcs which start from the extremities of the diam- eters. From b , with radius b d , describe an arc, and from g, with radius g c y describe another cutting the former one in e. Then e is the centre for the arc i g y which will meet the arc struck from b y in i. Of course, this process is to be carried on in each of the four lobes. PRACTICAL CARPENTRY. 41 Fig. 68 is the completed figure. The method of drawing the foliation will have been suggested by Fig. 63, and is further shown in the present illustration. Fig. 69 shows the skeleton lines of Fig. 70. Divide the diameter into four equal parts, and on the middle two, as a common base, construct the two equilateral triangles oi n and o i m. Draw lines through the middle points of the sides of the triangles, which, intersecting, will complete a six-pointed star in the circle, the angles of which will be the centres for the main lines of the tracery. Fig. 70 is the completed figure. The small figures, 71 and 72, will be understood without further instruction than is afforded by the examples. Fig. 73 shows the construction of the tracery in a square panel. 71 • Fig. 72. Pig. 73. From each of the angles of the square (the inner one in this figure), with a radius equal to the length of the side of the square, 42 PRACTICAL CARPENTRY. describe arcs ; these intersecting will give a four-sided curvilinear figure in the centre. Draw diagonals in the square. From the point where the diagonal intersects the chrve b (the middle line of the three here shown) set off on the diagonal the length c b, viz., b m. From q , with radius in q y de- scribe an arc cutting the ori- ginal arc in o. Make in r equal to m • With the aid of this table, and taking into account the pressure of the wind and the weight of snow, the strength of the different parts may be calculated from the following em- pirical rules, which were deduced by Mr. Tred- gold from experience. They are easy of applica- tion, and useful for simple cases. Mr. Tredgold assumes 66 J lbs. as the weight on each sq. ft. It is customary to make the rafters, tie-beams, posts and strn; ail of the same thickness. IN A KlNC-POSr ROOF OF PINE TIMBER. To find the dimenzic.i r oj principal rafters . Rule . — Multiply the square of the length in feet by the span in feet, and divide the prciuct by the cube of the thickness in incVs \ then multiply the quotient by 0*96 to obtain the depth in inches Mr. Tredgold gives also the following rule for the rafters, as i&ccc general and reliable : — Multiply the square of the span in feet by the distance between the principals in feet, and divide the product by 60 times the rise to feet: the quotient will be the area of the section of the rafter in ins. If the rise is one-fourth of the span, multiply the span by the dis- tance between the principals, and divide by 15 for the area of section. ti of ill us . F'V.CTVJa^ carpentry. 52 When the distance between the principals is 10 feet, the arm of sect’on is two-thirds of the span. To find the dimensions of the tie-beam , when it has to support c ceiling only . ; ./?/// h o, equal to i k, the rise of the roof, and join ao, bo, c be turned round thtir seats, AOjBG, until their perpendiculars are perpendicular to the plane of the plan, the points, o o, and the lines, go> go> will coincide, and the rafters, a o, bo> be in their true positions. If the roof is irregular, and it is required to keep the ridge level, we proceed as shown in Fig. 96. Bisect the angles of two ends by the lines a by b by cg,dg, in the same manner as in Fig. 95 ; and through g draw the lines G E, g f, parallel to the sides, cb,da, respectively cutting a by b b y in e and f; join e f ; then the triangle, e g f, is a flat, and the remaining triangle and trapeziums are the inclined sides. Join g b y and draw H 1 perpendicular to it ; at the points m and N, where h i cuts the 56 PRACTICAL CARPENTRY. } N $ /*. NT VS...- yC'"."'X 7 — lines G e, g f, draw m k, n l perpendicular to h i, and make them equal to the rise ; then draw h k, i l for the lengths of the common rafters. At£,set up e m perpen- dicular tO B Ej make it equal to m k or n l, and join b m for the lengtn of the hip- rafter, and pro ceed in the same manner to ob- tain A vi, c m, 97^ D vi. To find the backing of a hip-rafter, when the plan is right-angled, we proceed as shown in Fig. 97. Let b b, b c be the common rafters, a d the width of the roof, and a b equal to one half the width. Bisect B c in a , and join a a, d a. From a set off a c, a d equal to the height of the roof a b , and join a d. d c ; then a d , 0 c , are the hip- rafters To find the backing* from any poufi h in a d. d raw PRACTICAL CARPENTRY. 57 w.d through g draw Piff. » the perpendicular h g, cutting &Gin g; pendicuiar to a a the line e /, cut- ting a b, a d in e and /. Make g k equal to g h, and join ke, kf\ the angle e k / is the angle of the backing of the hip-rafter c. Fig. 98 shows the method of ob- taining the back- ing of the hip where the plan is not right angled. Bisect ad in a , and from a describe the semicircle a^d; draw a b parallel to the sides a b, dc, and join a b, d b, for the seat > of the hip-rafters. From b set off on b a, b d the lengths b d, b c, equal to the height of the roof b c, and join A e f d d ’ for the lengths of the hip-rafters. To find the backing of the rafters— In a c, take any point and draw k A perpendicular to A Through h draw fhg\&s- 5 » PRACTICAL CARPENTRY pendieular to a b y meeting a b, a. p in / and^-. Make k / equal to h k y and join f l y g /; then / /, i is the backing of the hip. Fig. 99 shows how to find the shoulder of purlins: First, where the purlin has one cf its faces in the plane of theioof, as at E. From c as a centre, with any radius, describe the arc dg\ and from the opposite extremities of the diameter, draw dh y ^^per- pendicular to B c. From e and f where the upper adjacent sides of the purlin produced cut the curve, draw dc.> will then by construction be equal to the triangle V t m, and will give tl e seat and the length and pitch of the com- men rafter of the smaller roof b. Divide the lines of the seats in b>Ji figures, d c> k m, into the same number of equal parts; and through the points of division in e, from g as a centre, describe the curves c a, 2 g, if and through those in b , draw the lines 3 /, 4 g 9 M a, parallel to the sides of the roof, and intersecting, the curves in f g a. ' Through these points trace the curves c fga 9 a fg a } which give the lines of intersection of the two roofs. Then to find the valley rafters, join c a y a a\ and on a erect the lines a b 9 a h per- pendicular to c a and a a , and make them respectively equal to M L; then c b y a b is the length of the valley rafter, very nearly. Fig. 102 shows how a, curved hip-rafter may be obtained. The softer shown in this instance is ogee in shape, but it makes no dif- ference what chape the common rafter may be, the proper shape 6o PRACTICAL CARPENTRY. and length of hip may be obtained by this method. It will be no- ticed that one side of the example shown is wider than the other; this is to show that the rule will work correctly where the sides are unequal in width, as well as where they are equal. Let a bcfec represent the plan of the roof, fcg the profile of the wide side of the rafter. First, divide this rafter, g c, into any number of parts — - in this case six. Transfer these points to the mitre line e h, or what is the same, the line in the plan representing the hip-rafter. From the points thus established in e b, erect perpendiculars indefinitely. With the dividers take the distance from the points in the line f c, measuring to the points in the profile g c, and set tne same off on corresponding lines,, measuring from e b, thus establishing the points i, 2, 3, etc.; then a line traced through these points will be the required hip-rafter. For the common rafter on the narrow .side, continue the lines from e b parallel with the lines of the plan D e and a d. Draw a d at right angles to these lines. With, the dividers as before, meas- uring from f c to the points in g c, set off corresponding dis- tances from a d, thus establishing the points shown between a and h. A line traced through the points thus obtained will be the line cf the rafter on the narrow side. This is supposed to be the re- turn roof of a veranda, but is only shown as an example, for it is not customary to build verandas nowadays with an ogee roof, but with a rafter having a depression or cove in it. For accuracy it would be as well to make nearly twice the number of. divisions shown from i to 6, as are there represented. Fig. 103 shows a section of a Mansard roof with concave sides, and the manner of framing the same when it is to be erected on a brick or stone building, p c is the wall; c the wall-plate; ab the floor-joist; hi is the side rafter; aie the ceiling-joist; a o the top rafter; b b d the bracket to nail cornice to ; b the gutter, and ri the studding, which will be required if it is desirable to finish the roof- story for sleeping-rooms. The wall-plate is made of two thicknesses of two-inch plank nailed together, and lap jointed at the ends. The joists should be notched out to receive the longitudinal piece k t and the ends PRACTICAL CARPENTRY. 61 of each should be sawed off square at or near the dotted line k. They should then be put into place, nailed to the wall-plate, and the piece h should be firmly nailed to each. The lower end of the side rafters are cut out at the toe to rest on the piece h . The upper ends are also cut to receive the piece t, to which they should be firmly nailed. If it is required to lath and plaster on the ceiling-joists, they should be notched to rest on the piece i ; but if the room is to re- main rough, it will be as well to nail beveled pieces on each as shown by the dotted line at j. The end of each ceiling-joist should be sawed in shape to re- ceive the mouVngtf. with which it is usual to finish the upper part of the roof. The top rafters may rest either on a longitudinal piece laid on the ceiling-joists, or on the piece i — the latter being the better method. The curved portions of the side rafters are made separate from the straight part, and are most generally formed of two thicknesses of inch stuff, first sawed the right shape and nailed together, and then spiked to the straight part of the rafter. When so much of the roof has been put up, it will be well to mark on the ends of the floor-joists the proper depth for the gutter. This will be best done by holding a straight-edge on the ends of the joists, with incline sufficient to allow water to run off, and marking on each joiss the depth it will require to be cut down. The vertical part of the gutter is cut down in a line with the lower ends of the side rafters. The cornice brackets, which are cut of a shape suitable for isceiving the different parts of the cornice, are made of inch stuff, And are nailed to the floor-joists as shown by the dotted Imes 6<* PRACTICAL CARPENTRY. artd nail-marks at d k. The best method to pursue in p^ttijg diem up is to first nail one on to the joist at either extremity of the roof, then stretch a line tight between the same points on each, and nail up the intervening brackets, with the same points touching the line. If the line is tightly stretched, and proper care is taken in nailing up the brackets, the cornice will be perfectly straight. In Fig. 104 we have a section of a similar roof with straight sides. The different parts are lighter than those of Fig. 103, and the construction is adapted for a balloon frame building. The letters in Fig-. 104 denote the same parts as the same letters in Fig. "03* and ?te explanation of Fig. 103 will answer for Fig. 104 so far as the same letters are con* cerned. p c is the balloon frame studding; c y a longi- tudinal piece for the floor- joists to rest upon. The studs are cut out at the top to receive the piece c , which must be firmly nailed to each. The floor-joists are notched to rest on the piece c t and will thus prevent the flams from spreading. Smce there is no curve on the rafter, the face of it comes flush with the inside of the gutter. Hence the side rafters ? re cut out at the heel to rest on the piece //, instead of the toe, as in Fig. 103. The piece h is beveled in order that the thrust on the side rafters shall not throw the lower ends out. The inside of the gutter is also made inclining so as to give as much substance as possible between the gutter and the piece h. The remaining parts are the sane as those Fig. 103, and the same description of those parts will answer foi both cuts. PRACTICAL CARPENTRY. 63 Fig. 505 shows how to find the angle-rafter and angle-cornice bracket, when the section as above described has been drawn. Let abc represent the given section on the draughting-board or floor, in which the same letters denote similar parts in Figs. 103 and 104. Draw the line A o at an angle of 45 0 with a f. Then from any points c, /, o y etc., of the section as shown, draw lines perpendicular to a f, and intersecting a o . In order to transfer the distances a e, a etc., on a 0 to a h, it is most convenient; in our small illustration, to describe arcs with a as a centre; but in practice, since the distance a o will be several feet, it will be best io lay a straight-edge along the line a o y and mark the points a, e, P , etc., on it ; then change the position of the straight-edge, and lay it along a h — the point before on a being made to coincide \vith it again, and transfer the marks to the floor or board on the line a h at e',/", etc. When this has been done, draw lines from 64 PRACTICAL CARPENTRY. these marks and perpendicular to ah. Now draw lines from the points Cypy o, etc,, on the section a b c, but parallel to f h, and in- tersecting the lines which are perpendicular to a h. Note the in- tersection of any two of these lines which were produced from the same point of the section, and this intersection will be the similar point of the angle-rafter. Perhaps the subject will be better under- stood if we follow the details of finding a single point of the angle- r;after ; such, for instance, as that corresponding to the point p oi the given section. From p draw pp' perpendicular to a f, and in- tersecting a o at p '. Make the distance a p f on ah equal to a p' on a Oy either by describing an arc with a as a centre and a p' as radius, or by transferring the point p to p" on a straight- edge, as before stated. From p" draw p" p" perpendicular to a h. Then from p on the section draw a line pp” parallel to f h, until it intersects the line f p" in the point This point p” will be the point of the angle-rafter corresponding to the point p of the section. After finding all the points in a similar manner, they must be joined by the requisite curved line, and a pattern-rafter cut to fit It will be apparent from inspection that the angle-bracket is found in the same manner. PRACTICAL CARPENTRY. 6 S PART IT- COVERING OF ROOFS. N slating or shingling a roof, great care should be taken at the hips, ridges and valleys. Where the roof is shingled, two or three courses should be left off at the ridge until the two sides. are brought up, then the courses left oft should be laid on together, and in such a manner as to have them “ lap ” over each other alternately. This can be easily done if the workman uses a little judgment in the matter; and a roof shingled in this manner will be perfectly rain-tight, without the ridge-boards or cresting. In valleys, the tin laid in should be sufficiently wide to run up the adjacent sides far enough to prevent “ back-flow ” from running over it. Ample space should also be left in the gutter to permit the water to flow off freely. There is a general tendency to make these waterways too narrow, which is frequently the cause of the water backing up under the shingles, causing leakage and premature decay of roof. There are several methods of shingling over a hip-ridge ; we prefer, however, the old and well-tried method of shingling with the edges of the shingles so cut that the grain of the wood runs parallel with the line of hip, as shown in Fig. xo6. Here it will be seen PRACTICAL CARPENTRY. 6G that the shingles next to those on the hip have the grain running up and down at right angles with the eave. On Fig. 107 we show a front view of the same hip, which will give a better idea of what is meant by having the grain parallel with the line of hip. abed show the cut or hip shingles, and n n n n the common shingles. The proper way to put in these shingles is to let the ends run over alternately and then dress them to the bevel of the opposite side of the roof ; this is shown better at o b d> where the edges of the shingles are shown that are laid on the other side of the roof. The edges of a and c show on the other side of the roof, and are not seen in the drawing. To cover a circular dome with horizontal boarding, proceed as follows : Let abc (Fig. 108) be a vertical section through the axis of a circular dome, and let it be required to cover this dome horizon- tally. Bisect the base in the point d, and draw p b e perpendicular to a c, cutting the circumference in b. Now divide the arc b c PRACTICAL CARPENTRY, 6 * too equal parts, so that each part will be rather less than the width of a board ; and join the points of division by straight lines, which will form an inscribed polygon of so many sides ; and through these points draw lines parallel to the base a c, meeting the opposite sides of the circumference. The trapezoids formed by the sides of the polygon and the horizontal lines, may then be re- garded as the sections of so many frustrums of cones; whence results the -following mode of procedure; produce, until they meet the line d e, the lines nf 9 fg 9 etc., forming the sides of the polygon. Then to describe a board which cor- responds to the surface of one of the zones, as f g 9 of which the trape- zoid is a section — from the point h 9 where the line fg produced meets d e, with the radii h f 9 h g 9 describe two arcs, and cut off the end of the board k on the line of a radius k k. The other boards are de- scribed in the same manner. To describe the Fig. 109. (Servering of an ellipsoidal dome with boards of equal width . Let a bcd (No. i, Fig. 109) be the plan of the dome, abc {No. 2) the section on its major axis, and l m n (No. 3) the section 68 PRACTICAL CARPENTRY. on its minor axis. Draw the circumscribing parallelogram of the ellipse f g h k (No. i), and its diagonals f h g k. In No. 2 divide the circumference into equal parts 1234, representing the number of covering* boards, and through the points of division 1 8, 2 7, etc., draw lines parallel to a c Through the points of division draw 1 /, 2 /, 3 x, etc., perpendicular to a c, cutting the diagonals of the circumscribing parallelogram of the ellipse (No. 1), and meeting its major axis in p t x y etc. Complete the parallelograms, and their inscribed ellipses corresponding to the lines of the cover- ing, as in the figure. Produce the sides of the parallelograms to intersect the circumference of the section on the transverse axis of the ellipse in 1 2 3 4, and lines drawn through these, parallel to l n, will give the representation of the covering boards in that sec- tion. To find the development of the covering, produce the axis d B, in No. 2, indefinitely. Join by a straight line the divisions 1 2 in the circumference, and produce the line to meet the axis pro- duced; and 12 e kg will be the axis major of the concentric ellipses of the covering i f g, 2 h k . Join also the corresponding divisions in the circumference of the section on the minor axis, and produce the line to meet the axis produced ; and the length of this line will be the axis minor of the ellipses of the covering boards. Before leaving the subject of roofs, it may be a? well to remark that the framing of valley roofs is so very much like that of hip- roofs, that it was not necessary to make special engravings for the purposes of showing how a valley-roof is constructed or “ laid out.” The cuts, bevels, lengths and positions of rafters and jacks may be easily found if the same principles that goierr Vip-roofs cxefcl- lowed, as a valley rafter is simply a hip reversed. PRACTICAL CARPENTRY. 69 PART T.— MITERING MOULDINGS, NE of the most troublesome things the carpenter meets with is the cutting of a spring moulding when the hori- zontal portion has to mitre with a gable orraking mould- ing. Undoubtedly the best way to make good work of these mouldirgs is to use a mitre-box. To do this make the down cuts B, b (Fig. no) the same pitch as the plumb cut oh the rake. The over cuts o, o, o, o should be obtained as follows : Suppose the roof to be a quarter pitch— though the rule works for any pitch wnen followed as here laid down — we set up one foot of the rafter, as at Fig. in, raising it up 6 inches, which gives it an inclination of quarter pitch ; then the diagonal will be nearly 13*4 inches. Now draw a right-angled triangle whose two sides forming the right angle, measure respectively 12 and 13J4 inches, as shown in Fig. 112. "The lines a and b show the top of the mitre-box with the lines 70 PRACTICAL CARPENTRY, marked on. The side marked 13^ inches is the side to mark from ; this must be borne in mind, and it must be remembered that this bevel must be used for both cuts, the 12 inch side not being used at all. Another excellent method for obtaining the section of a raking mould that will intersect a given horizontal moulding, is given below, also the manner of finding the cuts for a mitre-box for same. The principles on which the method is based being, first, that similar points on the rake and horizontal parts of a cornice are equally distant from vertical phnes represented by the walls ot a building; and, second, that such similar points are equally distant from the plane of the roof. Representing the wall faces of a build- ing bv the line d b (Fig. 113), and a section of the horizontal cor- nice by d b ab cde f— b a b c being the angle of the roof pitch— and following the idea given in Figure 105, draw lines aa',c f\ parallel to d b and intersecting the line k a\ which is drawn at right angles to d b through the point b ; then, with b as a centre, describe the arcs a' c' /', f etc., intersecting the same line k a' on the opposite side of d b ; after which extend lines from the points d b k , parallel to d b. This gives the point k at the same distance from d B-as the points a and a\ and the line lb at the same distance as c d. The rest of the same group of parallel lines are found to be similarly situated with respect to a b. From Descriptive Geometry we have the principle, that if we have given two intersecting lines contained in a plane, we know the position of that plane ; hence we may represent the plane of a roof by the line b a and b k (Figs. 113 and 114)1 anc ^ s * nce be most convenient to measure the distances required in a direction perpendicular to that plane, in following out the principle draw lines from the points c e f t etc., parallel to b a and intersecting the line b g, which is made perpendicular to b a. This gives us on b^ the perpendicular distance of the points c e f etc., from the line b < 7 . From the intersections of these lines with b g , and with b as a centre, describe arcs intersecting the line d b at i h r g\ etc.; from Fig. 112. PRACTICAL CARPENTRY. V these intersections with d b draw lines i' /, h' p } g' r , etc, parafiel to b k y until they in- tersect the first group of lines drawn perpendicu- lar to b ky and the intersection of each set of two lines drawn from the same pomt on the horizontal section will give the simi- lar point of the rake section. Takin g the point /, for ex- ample. we have, as before proved, / at the same distance from d e as c, and i being at the same distance from b a as a d i being equal to b i\ and b i = 1 t, i ( is equal to b : and consequently, / is the same distance from b k as c is from b ay which is in ac- cordance with princi- ple already shown. The intersection of each set of lines be- ing found and marked by a point, the con- tour of the moulding may be sketched in. Fig. H 4 . and the rake mould- ing, of which the scc- $nn is thus found, will intersect the given horizontal moulding, 4 proper care has been taken in executing the diagram. P^LllCAL CAIU&NTMY* i* t: * Fig. 115. Fig. 1 15 shows ho^ to find die mitre cut for the rake moulding, the cut for the horizonta ; cne being the same as for any ordinary 9tg in PRACTICAL CARPENTRY. 7 * moulding. Take an ordinary plain mitre-box, n j l, and draw the line a b, making the angle abj equal to the pitch angle of the roof. Draw b d perpendicular to a b, and make it equal to the width of the box k j ; make d e parallel to a b, and extend lines from b and p square across the box to k and c ; join b c and e k. abc will be the mitre cut for two of the rake angles; hek will be the cut for the other two angles, the angle hen being equal to the ^ngle abj. In mitering, both horizontal and rake moulding, that part of the moulding which ‘is vertical when in its place on the cornice, must be placed against the side of the box. Lines for the cuts m a mitre box, for joining spring mouldings may be obtained as follows: If we make b Fig. 216, the moulding showing the spring or lean of the member, and d e the mitre re- quired, then proceed as follows : With a as a centre, and the radius a g, describe the semicircle fhg c; then drop perpendiculars Srcm the line f c, at the points f, a, h, g and c, cutting the mitre line as shown on the line 1 d. Draw 1 e parallel to f c, then from 2 draw i s, which will be the bevel for the side of the box, and the bevel o r will be the line across the top of the box. The mitre line, as shown here, is for an octagon, but the system is applicable to any figure from a triangle or rectangle to & polygon with any number of sides. PRACTICAL carpentry; PART VI -SASHES AND SKYLIGHTS. N the skylight, Fig. 117, of which No. i is the plan, and No, 2 the elevation, it is required to find the length and backing of the hip. Let a b be the seat of the hip ; erect the perpendicular a c, and Pig. n ? . PRACTICAL CARPENTRY. 75 make it equal to the vertical height of the skvlighi, and draw b c, which is the line of the underside of the hip. The dotted hne^ k Fig. us. To find the backing, from any point in b c, as^: draw perpen- dicular to BC,a line g f meeting a b in f, and through f draw a i7iT> e/tm e> /it PRACTICAL CARPENTRY. 16 line at right angles to a b, meeting the sides of the skylight in d and e. Then from f as a centre,, and .with F£-as radius, cut the line a b in //, and join d //, e //. The angle d^e is the backing of the hip, and th* bevel d h e will give the^ angle of backing when applied to the perpendicular side of the hip bar. In Fig. 1 1 8, in which No. i is the plan, and No. 2 the elevation of a skylight with curved bars, to find the hip: let a b be the seat of the centre bar, and d e the seat of the hip. Through any divisions 1234c of the rib, over a b draw lines at right angles to a b and pro- duce them to meet e d in porsv. From these points draw lines per- pendicular to e D, and set up on them the corresponding heights from a b t m n 1 0 t, m 2 in \ p a, eitv 1P21ACT3CAL CARPENTRY. u Fig. £19 shows several ribs suitable for skylights. They aie de- signedly made complicated so as to exemplify the manner of getting the shapes of the mouldings. No. 1 shows the section of a rib; these ribs may be moulded as shown, or they may be chamfered from the glass line down to the point a. No. 2 shows a hip 01 angle rib; the backing, qds, is obtained as shown in Figs. 1 17 and r s 3 . No. 4 is Another hip made of larger section than No. 2. No& ? and 5 show sections of bars that may be used in connection with the ribs where required. No. 3 is drawn on an angle and let* tered for reference, so as to show the workman how such bars, mouldings or other work can be manipulated when the necessity for their use arises. Figs. 120, 12 1 and 122 shew how an angle, bar for ordinary Si*. 120. PRACTICAL CARPENTRY. 78 sashes may be obtained. Fig. 121 exhibits a section of the regular bar, which may be any shape. The lines abed are drawn from fixed points of the moulding. These lines are continued ; they cut the lines o, o in Figs. 120 and 122. Make the distances on the Hues abed , etc,, in Figs. 120 and 122, the same as in Fig. 121, »*om the line o. , The points of juncture of these lines with the Junes parallel with the central sectional lines o o, will be the points through which to describe the angle bar. Fig. 120 shows a bar set on an angle of 45 0 , or, as workmen term it. “ it is a mitre bar.” Fig. 122 is set on a more oblique angle. The rules given in the foregoing will apply to any angle A very ready way to find the shape of an angle bar is to take a piece of the straight bar and stand it on edge in the mitre box, and saw off a thin section of the bar to the same angle as the bar re- quired; then the outlines of this thin section will be very nearly the shape wanted. Some workmen adopt this method altogether of finding the section of their angle bars, but we do not recommend it £S it is faulty in more than one respect, and is unscientific. PRACTICAL CARPENTRY. 79 PART VIL— MOULDINGS. NGLE brackets for coves or any other mouldings may be laid off by proceeding as follows (Fig. 123) : First, when it is a mitre bracket in an interior angle, the angle being 45 0 , divide the curve c b into any number of equal parts 12345, h/id draw through the divisions the lines t d t 3/ 4