Is one of the useful arts which contributes most materially to the comfort and convenience of man. As the arts of joinery and carpentry are often followed by the same individual, it appears, at first view, natural to conclude, that the same principles are common to both these arts. But a closer examination of their objects leads us to a different conclusion.

The art of Carpentry is directed almost wholly to the support of weight or pressure; and, therefore, its principles must be found in the mechanical sciences. In a building, it includes all the rough timber-work necessary for support, division, or connection; and its proper object is to give firmness and stability. See the Article CARPENTRY.

The art of Joinery has for its object the addition in a building of all the fixed wood-work necessary for convenience or ornament. It is the Intestinum opus of Vitruvius, and the Menuiserie des bâtiments of the French.

The joiner's works are many of them of a complicated nature, and require to be executed in an expensive material; therefore joinery requires much skill in that part of geometrical science which treats of the projection and description of lines, surfaces, and solids, as well as an intimate knowledge of the structure and nature of wood.

It may also be remarked, that the rough labour of the carpenter renders him in some degree unfit to produce that kind of accurate and neat workmanship which is expected from a modern joiner.

In early times, very little that resembles modern joinery was known; every part was rude, and joined in the most artless manner. The first dawns of the art appear in the thrones, stalls, pulpits, and screens of our cathedrals and churches; but, even in these, it is of the most simple kind, and is indebted to the carver for everything that is worthy of regard. Whether in these monuments, the carver and the joiner had been one and the same person we cannot now determine, though we imagine, from the mode of joining in some of them, that this was the case.

During several centuries joinery seems to have been gradually improving, but nothing appears to have been written on the art before 1677, when Mr Joseph Moxon, a Fellow of the Royal Society, published a work, entitled Mechanick Exercises, or the Doctrine of Handyworks. In this work the tools, and common operations in joinery, are described, with a collection of the terms then in use. It must have been a valuable work at that time, but to a master in the art it would convey little if any thing that was new. Sash-windows were introduced into England some time before the date of Moxon's work, but he has not noticed them. According to the observations of Dr Thomson this important improvement has not yet found its way into Sweden.

About the beginning of the last century several works of a most interesting kind made their appearance. Forms began to be introduced in architecture, which could not be executed at a moderate expense without the aid of new principles, and these principles were discovered and published by practical joiners. As might naturally be expected, these authors had but confused notions, with a scanty portion of geometrical knowledge; and, accordingly, their descriptions are often obscure, and sometimes erroneous.

The hand-rails of stairs offered many difficulties, and an imperfect attempt to remove them was first made by Halfpenny, in his Art of Sound Building, published in 1725. Joinery Price, the author of the British Carpenter, published in 1733, was more successful, and his remarks show a considerable degree of knowledge of the true nature and object of his researches.

The publication of Price's work must have produced a considerable sensation among joiners, for it was soon followed by many other works of different degrees of merit. Of these the works of Langley and Pain were the most popular.

The establishment of the principles of joinery, on the sound basis of geometrical science, was reserved for Nicholson. In his Carpenters' Guide, and Carpenter and Joiners' Assistant, published in 1792, he has made some most valuable corrections and additions to the labours of his predecessors.

Corresponding improvements were also made in the practice of joinery, for which we are much indebted to the late Mr James Wyatt. This celebrated architect kept together some of the best workmen in London, who were looked up to with a degree of emulation by young men, which had a beneficial effect on the progress of joinery. But the art is still far short of perfection. We conceive that many of those operations, on which the soundness of work chiefly depends, might be done with greater exactness, and less labour, by means of tools contrived for these purposes. The truth and certainty which have been introduced in block-making, is sufficient to encourage someone to extend the same manner of mortising in joinery. See BLOCK-MACHINERY.

The principles of joinery were cultivated in France by Progress of a very different class of writers. In the extensive work of Joinery in Fresier, entitled Coupe des Pierres et des Bois, 3 vols. 4to, France, 1739, all the leading principles are given and explained with tedious minuteness, offering a striking contrast to the brevity of our English writers. The first elementary work on that part of geometrical science, which contains the principles of joinery, appeared in France in 1705, from the pen of the celebrated Gaspard Monge, who gave it the name of Géométrie Descriptive. Much of what has been given as new in English works, had been long known on the Continent; but there does not appear to have been much, if any, assistance derived from these foreign works by any writer prior to Nicholson.

The latest French work which treats of joinery is Rondelet's L'Art de Bâtir. It is also the best foreign work on the subject that we have seen; but it is not at all adapted to the state of joinery in England. In practice, the French joiners are very much inferior to our own. Their work is rough, slovenly, and often clumsy, and at the best is confined to external effect. The neatness, soundness, and accuracy, which is common to every part of the works of an English joiner, is scarcely to be found in any part of the works of a French one. The little correspondence, in point of excellence, between their theory and practice, leads us to think that their theoretical knowledge is confined to architects, engineers, &c. instead of being diffused among workmen, as it is in this country.

In cabinet-work the French workmen are certainly superior, at least as far as regards external appearance; but when use, as well as ornament, is to be considered, our own countrymen must as certainly carry away the palm.

1 Travels in Sweden, p. 8. The appearance of French furniture is much indebted to a superior method of polishing, which is now generally known in this country. For many purposes, however, copal varnish (such as coachmakers use) is preferable; it is more durable, and bears an excellent polish.

Geometry is useful in all, and absolutely necessary in some, parts of a joiner's business; but it is absurd to encounter difficulties in execution, and to sacrifice good taste, convenience, economy, and comfort, merely for the purpose of displaying a little skill in that science. It is, however, a common fault among such architects as are better acquainted with geometrical rules than with the production of visible beauties, to form designs for no other purpose than to create difficulties in the execution.

But, when geometrical science is properly directed, it gives the mind so clear a conception of the thing to be executed, that the most intricate piece of work may be conducted with all the accuracy it requires.

The practice of joinery is best learned by observing the methods of good workmen, and endeavouring to imitate them. But the sooner a workman begins to think for himself the better; he ought always to endeavour to improve on the processes of others; either so as to produce the same effect with less labour, or to produce better work.

We intend, in this article, to give a plain and simple exposition of the most valuable principles of the art of joinery, which will, we hope, place many parts of the practice under a new point of view, and ultimately tend to improve them.

Cabinet-making, or that part of the art of working in wood which is applied to furniture, has little affinity with joinery, though the same materials and tools be employed in both. Correctness and strict uniformity are not so essential in moveables as in the fixed parts of buildings; they are also more under the dominion of fashion, and therefore are not so confined by rules as the parts of buildings.

Cabinet-making offers considerable scope for taste in beautiful forms, and also in the choice and arrangement of coloured woods. It requires considerable knowledge in perspective, and also that the artist should be able to sketch with freedom and precision.

If the cabinet-maker intend to follow the higher departments of his art, it will be necessary to study the different kinds of architecture, in order to make himself acquainted with their peculiarities, so as to impress his works with the same character as the rooms they are to furnish.

In as far as regards materials, and the principles of joining work, the cabinet-maker will find some useful information in the second and third sections of this article. In ornamental composition, he may derive much benefit from Tatham's Etchings of Ancient Ornamental Architecture, London, 1799; Percier and Fontaine's Recueil des Décorations Intérieures comprenant tout ce qui a rapport à l'Aménagement, Paris, 1812; and, for general information, the Cabinet Dictionary, and the Cabinet-Maker and Upholsterer's Drawing-Book of Sheraton, may be consulted.

Sect. I.—On making Working Drawings.

1. In this section we propose to lay before the reader the most important part of the principles of describing, on a plane surface, the lines necessary for determining bevels, forming moulds, or any other purpose required in the practice of joinery. The limits within which such an article as joinery must be confined, in a work like this, will not permit us to enter much into detail on the various points to be illustrated in this section; but we hope, by judicious selection, to place under one point of view the principles that are most useful to the joiner.

Projection of Bodies.

2. A clear idea of the nature of projection is so essential in making working drawings, that, in our endeavours to illustrate it, we cannot proceed upon principles too simple. In the first stage of such an inquiry, experiment furnishes at once the most clear and satisfactory evidence, particularly to those who are not familiar with mathematical subjects.

If some small pieces of wood, or pieces of wire, were joined together, so as to represent the form of a solid body, a cube for example, and if this figure were held between the sun and the surface of a plane board, then the shadow of the figure upon the board would be its projection upon that plane. From this simple experiment, it will appear, that the projection of any line placed in the direction of the sun's rays will be a point; the projection of any line parallel to the plane will be of the same length as the line itself; and the projection of any line inclined to the plane will be always shorter than that line.

3. We have supposed the board to be placed at any angle with the direction of the rays of the sun; but, for our present purpose, it is sufficient to consider them to fall perpendicularly upon it; hence it is obvious, that to project a straight line upon a plane, a perpendicular to the plane should be let fall from each end of the line, and the line joining the points where the perpendiculars meet the plane will be the projection required.

When a projection is made upon a horizontal plane, it is usually called a plan of the body. When the projection is upon a vertical plane, it may be an elevation or a section of the body; it is a section when a portion is supposed to be cut off; and the plane of projection is usually parallel to the plane of the section.

4. Bodies may be divided into three classes, according to the kinds of surfaces by which they are bounded. The first class, comprehending those which are bounded by plane surfaces, such as cubes, prisms, pyramids, and the like. The second class contains those which are bounded in part by plane surfaces, and the rest by curved surfaces, as cylinders, cones, &c. The third, including those which are bounded by curved surfaces only, as spheres, spheroids, &c.

The projections of the first class of bodies will consist of straight lines; those of the second class, of curved as well as straight lines; and those of the third class, of curved lines only.

4. Let ABCD, and CDEF, Fig. 1, be two plane surfaces, connected by a joint at CD, so that while the plane of CDEF remains horizontal, the plane ABCD may be placed perpendicular to it, and thus represent a vertical plane. Then, if a line be so placed in space that ab is its projection on the vertical plane, and a'b' its projection on the horizontal plane, its projection on any other vertical plane, HGEC, may be determined. This is easily effected, for we have seen, that if a perpendicular be drawn... to the plane from each end of the given line, they will give the positions of the ends of the line in the projection (Art. 3). Now, the same thing will be done, by drawing \(a'a''\) and \(b'b''\) perpendicular to EC, and setting off the points \(a''\) and \(b''\) at the same height above EC respectively, as \(a\) and \(b\) are above CD, then the line \(a''b''\) is the projection re- quired.

The heights may be transferred from one vertical plane to another when they are both supposed to be laid flat, by drawing the line IC, so as to bisect the angle ECD, and if \(c'd'\) be parallel to CD, meeting IC in \(c'\), then a line drawn parallel to EC, from the point \(c'\), will give the height of the point \(b''\), and so may be found the height of any other point.

6. In the particular case we have drawn, none of the projections represents the real length of the given line. To obtain this length, draw \(a'e'\) parallel to CD, and with the radius \(a'b'\) describe the arc \(b'e'\) cutting \(a'e'\) in \(e'\); draw \(d'e'\) perpendicular to CD, cutting the line \(c'b'\) in \(d'\); join \(a'd'\), and it is the length of the given line.

The real lengths of lines frequently are not given, there- fore another general method of finding them will be found useful, and which may be stated as follows: the length of an inclined line projected upon a plane is equal to the hy- potenuse of a right-angled triangle, of which one side is the projection upon the plane, and the other side is the dif- ference between the perpendicular distances of the extremes of the line from the plane.

7. In fig. 2, \(a'b'cd'\) represents the horizontal projection, or plan, of a rectangular surface, and the elevation \(ab\) shows its inclination; and its pro- jection against another vertical plane, making any angle ECD with the former, or plane of eleva- tion, is shown by \(a''b''c'd''. GC\) being perpendicular to EC, and AC perpendicular to CD, the heights may be transferred by means of arcs of circles described from C as a centre. This is a better method than that by bisecting the angle given in fig. 1; but neither of them so good, in practice, as setting of the heights with the compasses, or with a lath. In our figures it is desira- ble to show the connection of corresponding parts as much as possible; therefore, the reader will bear in mind that many of the operations we describe may be done with fewer lines when the operator is fully master of his subject.

8. It may be further noticed in this place, that when a point is to be determined in one line by the intersection of another, the lines should cross each other as nearly at right angles as possible; for, when the intersecting lines cross very obliquely, a point cannot be determined with any to- lerable degree of accuracy.

9. A curved line can seldom be projected by any other means than by finding a number of points through which the projected line must be drawn, or finding a series of tangents to the section. In giving an example of the pro- jection of a body bounded by a curved surface, we shall select a case of frequent occurrence in practice, referring to the Geometrie Descriptive of Monge, for more general methods.

Let ABC be part of the plan of the base of a solid, fig. 3, and FED its end elevation; the upper side of the solid being bounded by the curved surface FD. This solid is supposed to be cut at AB by a plane perpendicular to the base, and our intention is to show the form of the section. Draw EH parallel to BA, and GIHE will represent the plane upon which the section is to be projected. Set off

any convenient number of points, 1, 2, 3, 4, &c. in the given curve FD, from each of these points draw a line perpendicular to ED, to meet BA; and from the points in BA, thus determined, erect per- pendiculars, which will cut HE at right angles. Make GH equal to FE, and set off the points 1, 2, 3, &c. in GHE at the same distances respectively from HE as the cor- responding points 1, 2, 3, &c. in EFD are from the line ED. A curve being drawn through the points E, 1, 2, 3, 4, 5, G will com- plete the section. In large works, the joiner will often find it useful to put nails in the points, and to bend a regular lath against the nails; with the as- sistance of the lath, the curve may be drawn with more regularity.

If the curve FD were very irregular, or a mixed line of straight parts and curved ones, the same method would de- termine the section; all the caution required is, that a suf- ficient number of points should be fixed upon in the given curve; and upon the proper selection of these points much of the accuracy of the section will depend.

The angle ribs of groined ceilings, the angle ribs for coved ceilings, or brackets for large cornices, and the angle cantilevers for balconies or other works of a similar kind, are found by this method. If FD be the cross rib of a groin, then GE will be the form of the corresponding angle rib. Also, if the angle of a room be represented by LAC, and FD be the cove for the ceiling, then GE will be the proper angle rib for such a cove.

In some cases, the section may be determined by means of the properties of the given curve, when the nature of that curve is known. Thus the oblique section of a cylin- der is an ellipse, and the sections of a cone are certain fi- gures depending on the direction of the plane of section (see the article Conic Sections); but if an architect were confined to the use of geometrical curves, there would be small scope, indeed, for a display of taste in his art; there- fore the joiner must generally have recourse to the simple method we have described.

10. The section of a body may often be drawn by a more simple and direct process; and yet where the principle is still the same. Thus the section of the half cylinder ACB, in fig. 4, being compared with the process in fig. 3,

will be found to be the same in every respect, ex- cepting in the position of the parts of the figure. In fig. 4, ACB is the end or plan of the cylinder, and DE the inclination of the plane by which it is cut. Let the ordinates \(a_1, b_2, \ldots\) in the plan, be drawn per- pendicular to AB, and con- tinued till they cut the in- clined line DE. Also draw the ordinates \(a'_1, b'_2, \ldots\) perpendicular to the line DE, and make the distan- ces \(a'_1, b'_2, \ldots\) respectively equal to the corresponding distances \(a_1, b_2, \ldots\) upon the plan. Through the points E, \(1', 2', \ldots\) draw the curve DFE.

As the curve DFE is an ellipse, when ABC is a circle, in that case it will be better to draw an ellipse with a tram-

---

1 A simple and convenient instrument for this purpose is described in the Transactions of the Society of Arts, for 1817, vol. xxxv. p. 109. mel, or any other machine that produces the curve by a continued motion. (See the article Elliptograph.) DE is the transverse, and Fe' the semi-conjugate axis of the ellipse.

The most important application of the case, in fig. 4, is to the hand-railing of a staircase, with a curvilinear well-hole, or opening down the middle. For, if A e, or a B, show the breadth of the rail, A e C a B would be its plan; and D' F' a' E' the form of a mould, commonly called a face mould, for cutting out the rail by, when DE is the inclination of the plank. We cannot, however, proceed directly to the subject of stair-rails, without considering the development of the surfaces of bodies.

Development of Surfaces.

11. To develop the surface of a solid, is to draw, on some plane surface, a form that would cover it. If this form were drawn upon paper, and the paper were cut to it, the paper, so cut, ought to cover exactly the surface of the solid. Now, in joinery, it is often required that a mould should apply to a curved surface; and, therefore, the development of that surface upon a flexible material gives the form of the mould.

The covering of a square pyramid may be found by erecting a perpendicular from the middle of one of the sides of its base, as from a in the side AB, fig. 5. Upon this perpendicular set off a C equal to the slant height of the pyramid; then, with the radius AC and centre C describe the arc A3, and set off the distance AB three times upon the arc. Join the points C3, C2, C1, CA, and CB, and draw the lines 32, 21, 1A, which determine the covering required.

It is obvious, that we could develop a pyramid of which the base might have any number of sides, by the same method; and that a near approximation to the development of a right cone might be effected by the same means, which, in fact, is the means usually employed. But the following method of spreading out the surface of a cone will be found more correct.

12. Let ABC, fig. 6, be the elevation of a cone, and ADB half the plan of its base. With the radius AC describe the arc AE, which will be the line bounding the development; and, to find the length of the arc, or rather the angle containing it, multiply 360 by the radius Aa of the base, and divide the product by the slant height AC of the cone; the quotient will be the number of degrees in the arc AE, when the surface ACE exactly covers the whole cone. Thus, let Aa be 12 feet, and AC 40 feet; then \( \frac{360 \times 12}{40} = 108 \) degrees, and making ACE an angle of 108 degrees, we have the sector ACE that would cover the cone.

This applies to the soffits of windows, where they are enlarged towards the inside, to admit light more freely than square recesses would do. If ab be the width of the soffit, draw eb parallel to AB, and from the centre C describe the arc ed. Then half the development AE ed will be the mould for the soffit; or the form of a veneer that would cover it.

13. The development of a cylinder is also of use in forming the mould for soffits, but is still more useful in the construction of stairs; and, as we are obliged to consider it as a prism with numerous sides, it is obvious that any other body of a like kind may be developed by the same means.

Let ABC, fig. 7, be the plan of half a cylinder, and A'E its height. Divide the curve ACB into any convenient number of equal parts, and let these parts be set off from C to A, and from C to B'. When the curve is a semicircle, divide the diameter AB into the proposed number of parts, and make aD equal to three-fourths of the radius. From D, through the points A and B, draw the lines DA', DB'; then A'B' is nearly equal to the curve ACB stretched out; and, by drawing a line from D through each of the divisions in AB, the line A'B' will be divided into the same number of equal parts.

In either case, erect a perpendicular from each point of division, and EA'B'F will be the development of the surface.

If we suppose A'B' to be divided into the number of steps that would be necessary to ascend from B to A, in a circular staircase, the development of the ends of these steps may be drawn as in the upper part of the figure. The projection G of the cylinder, with the lines of the development drawn upon it, and the ends of the steps, shews the waving line formed by the nosings of the steps, and consequently by the hand-rail of a circular staircase.

When a part of a cylinder is cut off by a plane, the line of section will be a curved line upon the development, as is shewn in the lower part of the development, fig. 7. The faint lines shew the manner of finding the edge of the covering, and is the same as finding a mould for a soffit formed by an arch cutting obliquely into a straight wall.

14. In an oblique cone, the lines drawn on its surface, from its base to the vertex, would be of different lengths; and as those lengths are not shewn by the plan or elevation, they may be had by means of the principle stated in art. 6.

Let ABC, fig. 8, be the given cone, and AEB a plan of

---

1 This has been shown by Dr C. Hutton, in his Mathematical Tracts, vol. i. p. 169. JOINERY.

half its base; to find the development, produce AB, and from the vertex C let fall the perpendicular CD. Divide the circumference of the base into any number of equal parts, and from each point of division describe an arc from D, as a centre, to cut the line AB at 1, 2, 3, &c. From C, as a centre, describe the arcs AA', 11, 22, &c., and with a radius equal to one of the divisions of the circumference of the base, and the centre B cross the arc 55, which determines the point 5 in the development, with the same radius, and the point 5, as a centre cross the arc 44, and so on for the remainder of the arcs. Join A'C, and draw a curve line through the points A', 1, 2, &c., which gives the covering for half an oblique cone.

If the cone be cut by a plane, ab, parallel to the base, the surface Bb'a'A' will be the covering of a soffit for a conical arch cutting obliquely into a straight wall.

As it often happens that there is not a sufficient space between the head of a door, or a window, and the cornice of the ceiling, to admit of the same bevel being preserved at the crown or top, as at the sides of the window; in such cases the soffit is made level at the crown, or with such an inclination only as will prevent the architrave cutting into the cornice of the room.

Let ABCD, fig. 9, be the plan of the space to be covered with a soffit, ED the arch of half the opening, which is


in its proper position when set perpendicularly over the line CD; and let F e be the height of the arch over AB. Produce AC and BD to meet at G; set off c m equal to c F, and 3 n equal to 3 E, then draw a line through the points m n, which will give the inclination of the soffit at the highest part of it. Divide the arch ED into any number of equal parts (in our example we have only divided it into three parts), and from each point of division let fall a perpendicular to CD, meeting the line CD in the points 1, 2. Through these points draw the lines G a, G b, cutting the line AB in the points a b, and from each point erect a perpendicular to AB. Set off, on 3 n, the heights of the points in the curve ED, and divide the line m e in the same proportion as n 3, which will give the corresponding heights for the arch FD, and through the points thus found the arch FD should be drawn.

Make Go perpendicular to GE, cutting a line passing through the points m n in o, and draw lines through the corresponding points of division in the lines m c, n 3, so that Go may be divided in the same proportion as n 3. Draw G p perpendicular to GD, and equal to Go, and set off upon it the same distances as are upon Go. Then, with a radius G J, and the first division on G p, as a centre, describe an arc at s, and with a radius equal to one of the divisions of the arc ED and D as a centre, cross the arc s, which gives one point. Also, with a radius G 2, and the second division on G p as a centre, make an arc at t, which, being crossed by an arc described with a radius, equal to one of the divisions of the arc ED, and s as a centre, determines another point in the edge of the covering. Proceed in the same manner till half the development of one edge be completed; the other edge will be obtained by drawing lines through the points s, t, u, from the corresponding points in G p, and making sw equal to a l; tx equal to b 2, &c.

As both sides are the same, the soffit mould for one side requires only to be reversed for the other side. If the soffit be level at the crown, the process may be rendered shorter; but, where it is possible to get space for a slight inclination, the appearance of the soffit is always materially improved.

If the plan of the wall be circular, find the development of the arc ED as before, and transfer the distances from CD of the points in the curved wall, to the corresponding lines in the development, in the same manner as was done to find the edge B w x y.

The development of a sphere, or globe, can be effected only by an approximate process, as it is impossible to apply a plane surface so as to touch more than one point sphere at a time; but various methods may be employed which are useful in forming spherical surfaces.

A sphere may be divided into numerous zones, the surface of each zone may be considered as that of the frustum of a cone, and developed in the same manner as has been described for a portion of a cone in art. 12. The upper part of fig. 10 shews half a sphere developed in this manner; and when it is divided into very narrow zones, the covering found by this process has some advantages, in practice, that are peculiar to it.

The surface of a sphere may also be developed by inscribing it in a cylinder, LMNO, fig. 10, and considering a small portion, or gore, ABD, to coincide with the surface of the cylinder. Then, if the portion ABD, considered as part of a cylinder, be developed by the process described in art. 13, one gore, AB d will be obtained; and by dividing the circumference of the sphere into any number of equal parts, and making AB equal to one of these parts, the same mould will serve for the whole of the sphere.

Another method of developing a sphere consists in supposing it to be a polyhedral, or many-sided figure; but this method has no advantage over the preceding ones, while it has the inaccuracies of both of them.

In lining and boarding domes, the position of the ribs to which the boards are to be fixed will determine the method of development that ought to be adopted; but the form of the veneers for a spherical surface may be determined by either method.

To determine the Angle formed by two Inclined Planes.

The angle made by two planes which cut one another, is the angle contained by two straight lines drawn from any, the same, point in the line of their common sec-planes inclination, at right angles to that line; the one in the one plane, and the other in the other. This angle is the same as that which the joiner takes with his bevel, the bevel being always applied so that its legs are square from the arris, or common section of the planes.

This is the definition given by Professor Playfair, in his Elements of Geometry, and it is better suited to our purpose than Euclid's definition. If two lines, AB and CD, be drawn upon a piece of pasteboard, at right angles to one another, crossing at the point E, and the pasteboard be cut half through, according to the line AB, so that it may turn upon that line as a joint; then, to whatever angle, CED, fig. 11, the parts may be turned, the lines EC and ED will be always in the same plane. Also, a line FD, drawn from any point D, in the line ED, to any point F, in the line EC, will be always in the same plane. From these self-evident properties of planes, it is easy to determine the angle formed by any two planes, when two projections, or one projection and the development of the surfaces, are given.

19. Let ABC, fig. 12, be the plan of part of a pyramid, and BD the elevation of the arris, or line formed by the common section of the planes in respect to the line EB; EB being the projection of that arris upon the plan.

Draw AC perpendicular to EB, cutting it in any point E, and from E draw EF perpendicular to DB. With the radius EF, and centre E, cross EB in f, and join Af and fC, then the angle A fC is the angle formed by the planes of the pyramid.

The angle may be constructed when the plan and elevation of any two lines drawn in the planes, so as to intersect in the arris, are given; but as these projections are not often given in drawings of joiners' work, we have inserted the preceding, though it be a less general method.

The backing, or angle for the back of hip-rafters in carpentry, and of hipped sky-lights, is found in this manner; ABC being, in that case, supposed to be the plan of an angle of the roof or sky-light, and DB the inclination of the hip-rafter.

20. To shew how the angle formed by two planes may be found when the plan and development are given, let it be required to find the angle contained by the two faces of a square pyramid, fig. 5.

Draw FB perpendicular to AC, and with the radius BF, and centre B, describe the arc FG. Then, with the radius DB, and centre F, cross the former arc in G, join BG, and EBG is the angle formed by two, the inclined faces of the pyramid.

Raking Mouldings.

21. When an inclined or raking moulding is intended to join with a level moulding, at either an exterior or an interior angle, the form of the level moulding being given, it is necessary that the form of the inclined moulding should be determined, so that the corresponding parts of the surfaces of the two mouldings should meet in the same plane, this plane being the plane of the mitre. It may be otherwise expressed, by saying that the mouldings should mitre truly together.

If the angle be a right angle, the method of finding the form of the inclined moulding is very easy; and as it is not very difficult for any other angle, it may perhaps be best to give a general method, and to illustrate it by examples of common occurrence.

General Method of describing a Raking Moulding, when the Angle and the Rake, or inclination of the Moulding, is given.

Let ABC, fig. 13, be the plan of the angle of a body, which is to have a level moulding on the side AB; and this level moulding is to mitre with an inclined moulding on the side BC. Also, let CBD be the angle the inclined moulding makes with a level or horizontal line BC.

Produce AB, and draw Cb perpendicular to AB; also make DC perpendicular to BC, and d C perpendicular to b C. Set off Cd equal to CD, and join bd; then the inclined moulding must be drawn on lines parallel to bd.

Let 1, 2, 3, 4, &c. be any number of points in the given section of the level moulding; from each of these points draw a line parallel to bd, and draw A G' perpendicular to bd. Set off the points 1', 2', 3', 4', &c. at the same distances respectively from the line A G', as the corresponding points 1, 2, 3, 4, &c. are from the line AB, and through the points 1', 2', 3', &c. draw the moulding. The moulding thus found will mitre with the given one; also, supposing the inclined moulding to be given, the level one may be found in like manner.

If the angle ABC be less than a right angle, the whole process remains the same; but when it is a right angle, BD coincides with bd; and the method of describing the moulding becomes the same as that usually given; as it does not then require the preparatory steps which are necessary when the angle is any other than a right angle.

22. It is in pediments, chiefly, that the method of forming raking mouldings is of use. Fig. 14 represents part of a pediment; AB is that part of the level moulding which mitres with the inclined moulding; all that part of the cornice below B; being continued along the front, the lower members of the raking cornice stop upon it; and, therefore, do not require to be traced from the other.

In that part of the cornice marked AB, set off a sufficient number of points; and from each of these points draw a line parallel to the rake, or inclination of the pediment. Also, let a verticle line be drawn to each of the same points from the horizontal line rs. Make s't perpendicular to the inclination of the pediment, and with a slip of paper, or by

---

1 On this subject the reader may consult Monge's Géométrie Descriptive, Art. 19 et 20, par. 23 and 24, 4th edition, Paris, 1820.