JOINERY

Joinery. 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.

Carpentry. 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.

Joinery. 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.

Progress of Joinery in England. 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 every thing 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.¹

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 Half-

penny, 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 some one to extend the same manner of mortising in joinery. See BLOCK-MACHINERY.

The principles of joinery were cultivated in France by a very different class of writers. In the extensive work of Friez, entitled Coupé des Pierres et des Bois, 3 vols. 4to, 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.

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

Geometrical knowledge necessary. 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.

Practice of Joinery. 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.²

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. 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'ameublement, 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 lines

Fig. 1.

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

¹ The method of making and using the French polish is minutely described in Dr Thomson's Annals of Philosophy, vol. xi. p. 119 and 371.

² Descriptions of the tools, with instructions for using them, may be found in Moxon's work before quoted, and in Nicholson's Mechanical Exercises, Taylor, London, 1812.

Joinery. 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 $a\text{'}d\text{'}$ and $b\text{'}b\text{'}$ perpendicular to $EC$, and setting off the points $a\text{'}$ and $b\text{'}$ at the same height above $EC$ respectively, as $a$ and $b$ are above $CD$, then the line $a\text{'}b\text{'}$ is the projection required.

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 $eb$ be parallel to $CD$, meeting $IC$ in $e$, then a line drawn parallel to $EC$, from the point $e$, will give the height of the point $b\text{'}$, 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 $a\text{'}e$ parallel to $CD$, and with the radius $ab\text{'}$ describe the arc $be$ cutting $a\text{'}e$ in $e$; draw $de$ perpendicular to $CD$, cutting the line $eb$ in $d$; join $ad$, and it is the length of the given line.

The real lengths of lines frequently are not given, therefore 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 hypotenuse of a right-angled triangle, of which one side is the projection upon the plane, and the other side is the difference between the perpendicular distances of the extremes of the line from the plane.

7. In fig. 2, $a\text{'}b\text{'}cd$ represents the horizontal projection, or plan, of a rectangular surface, and the elevation $ab$ shows its inclination; and its projection against another vertical plane, making any angle $ECD$ with the former, or plane of elevation, is shown by $a\text{'}d\text{'}b\text{'}c\text{'}d\text{'}$. $GC$ 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

Fig. 2.

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 desirable 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 tolerable 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 projection 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 shew 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 perpendiculars, 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 corresponding 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 complete 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 assistance of the lath, the curve may be drawn with more regularity.¹

If the curve $FD$ were very irregular, or a mixed line of straight parts and curved ones, the same method would determine the section; all the caution required is, that a sufficient 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 cylinder is an ellipse, and the sections of a cone are certain figures 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; therefore 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, excepting 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 $a1,\; b2$, &c. in the plan, be drawn perpendicular to $AB$, and continued till they cut the inclined line $DE$. Also draw the ordinates $a\text{'}1\text{'},\; b\text{'}2\text{'}$, &c. perpendicular to the line $DE$, and make the distances $a\text{'}1\text{'},\; b\text{'}2\text{'}$, &c. respectively equal to the corresponding distances $a1,\; b2$, &c. upon the plan. Through the points $E,\; 1\text{'},\; 2\text{'},$ &c. 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 trans-

Fig. 4.

¹ A simple and convenient instrument for this purpose is described in the Transactions of the Society of Arts, for 1817, vol. xxxv. p. 100.

Joinery. mel, or any other machine that produces the curve by a continued motion. (See the article ELLIPTOGRAPH.) DE is the transverse, and F' 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 e' 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 $\backslash frac\{360\; \backslash times\; 12\}\{40\}\; =\; 108$ 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 cb parallel to AB, and from the centre C describe the arc ed. Then half the development AEcd will be the mould for the soffit; or the form of a venter 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

Fig. 7.

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, to develop 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

Fig. 8.

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 $Bba\text{'}A\text{'}$ will be the covering of a soffit for a conical arch cutting obliquely into a straight wall.

15. 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 $Fc$ be the height of the arch over AB. Produce AC and BD to meet at G; set off $cm$ equal to $cF$, and $3n$ equal to $3E$, then draw a line through the points $mn$, 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 $Ga$, $Gb$, cutting the line AB in the points $ab$, and from each point erect a perpendicular to AB. Set off, on $3n$, the heights of the points in the curve ED, and divide the line $mc$ in the same proportion as $n3$, 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 $mn$ in $o$, and draw lines through the corresponding points of division in the lines $mc$, $n3$, so that $Go$ may be divided in the same proportion as $n3$. Draw $Gp$ perpendicular to GD, and equal to $Go$, and set off upon it the same distances as are upon $Go$. Then, with a radius $G1$, and the first division on $Gp$, 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 $G2$, and the second division on $Gp$ 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 $Gp$, and making $sw$ equal to $al$; $tx$ equal to $b2$, &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 $Bwxy$.

16. 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 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.

17. 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, $ABd$ 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.

18. The angle made by two planes which cut one another, is the angle contained by two straight lines drawn angle from any, the same, point in the line of their common section, at right angles to that line; the one in the one plane, and the other in the other.¹ 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.

¹ 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 AfC 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.¹

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 show 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 FBG is the angle formed by two, the inclined faces of the pyramid.

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.

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, General method.

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 dC perpendicular to bC. 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 A6' perpendicular to bd. Set off the points 1', 2', 3', 4', &c. at the same distances respectively from the line A6', 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.

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 st 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.

means of arcs of circles, transfer the distances on $rs$ to the line $rs$, and from the points thus found, draw lines parallel to $st$; the intersection of these, with the inclined lines, will determine the form of the moulding, as is indicated by the letters.

When a pediment has a cornice with modillions, the caps of the modillions require to be traced by the same method.

23. It sometimes happens, that an inclined base-moulding has to mitre with a level one at an angle; and as the same thing occurs still more frequently with other moulding, such as cornices under the steps of stairs, &c. we shall give another example, which will serve still farther to illustrate the method of proceeding in such cases.

In fig. 15, a raking base-moulding is shown, where the

inclined moulding B is traced to mitre with the horizontal moulding C; and the horizontal moulding A is traced to mitre with the inclined one B. The preceding examples being understood, the lines and letters in the figure will be sufficient to show the mouldings are traced.

24. Mouldings being almost the only part of modern joiners' work, which can, in strictness, be called ornamental, and consequently that in which the taste of the workman is most apparent, we shall offer a remark or two that may have their use. The form of a moulding should be distinct and varied, forming a bold outline of a succession of curved and flat surfaces, disposed so as to form distinct masses of light and shade. If the mouldings be of considerable length, a greater distinction of parts is necessary than in short ones.

Mouldings for the internal part of a building should not, however, have much projection; the proper degree of shade may always be given, with better effect, by deep sinkings judiciously disposed. The light in a room is not sufficiently strong to relieve mouldings, without resorting to this method; and hence it is that quirked mouldings are so much esteemed.

SECT. II.—ON THE CONSTRUCTION OF JOINERS' WORK.

25. The goodness of joiners' work depends chiefly upon the care that has been bestowed in joining the materials. In carpentry, framing owes its strength to the form and position of its parts; but in joinery, the strength of a frame depends upon the strength of the joinings. The importance, therefore, of fitting the joints together as accurately as possible, is obvious. It is very desirable, that a joiner should be a quick workman; but it is still more so that he should be a good one; that he should join his materials with firmness and accuracy; that he should make surfaces even and smooth, mouldings true and regular, and the parts intended to move so that they may be used with ease and freedom.

Where dispatch is considered as the chief excellence of a workman, it is not probable that he will strive to improve

himself in his art, further than to produce the greatest quantity of barely tolerable work with the least quantity of labour. In some articles of short duration, dispatch in the manufacture may be of greater importance; but in works that ought to remain firm for years, it certainly is bad economy to spare a few shillings' worth of labour at the risk of being annoyed with a piece of bad work as long as it will hold together.

We have seen, with no small degree of pleasure, the effect of encouraging good workmanship in the construction of machinery, and would recommend that a like encouragement should be given to superior workmen in other arts.

Joining Angles.

26. When the length of a joint at an angle is not considerable, it is sufficient to cut the joint, so that when the parts are joined, the plane of the joint shall bisect the angle. This kind of joint is shown for two different angles, by fig. 16.

When an angle of considerable length is to be joined, and the kind of work does not require a joining should be concealed, fig. 17 is often employed; the small bead renders the appearance of the joint less objectionable, because any irregularities, from shrinkage, are not seen in the shade of the quirk of the bead.

A bead upon an angle, where the nature of the thing does not determine it to be an arris, is attended with many advantages; it is less liable to be injured, and admits of a secure joint, without the appearance of one. Fig. 18 shows

a joint of this description, which should always be used in passages.

Fig. 19 represents a very good joint for an exterior angle, whether it be a long or short one. Such a joint may be nailed both ways. But the joint represented by fig. 20 is superior to it; the parts being drawn together by the form of the joint itself, they can be fitted with more accuracy, and joined with certainty. The angles of pilasters are often joined, as fig. 20.

Interior angles are commonly joined, as shown by fig. 21. If the upper or lower edge be visible, the joint is mitred, as in fig. 16, at the edge only, the other part of the joint being grooved, as in fig. 21. In this manner are put together the skirting and dado at the interior angles of rooms, the backs, and back-linings of windows, the jambs of door-ways, and various other parts of joiners' work.

Framing.

27. Frames in joinery are usually connected by mortise and tenon joints, with gooves to receive pannels. Doors, window-shutters, &c. are framed in this manner. The object in framing is, to reduce the wood into narrow pieces, so that the work may not be sensibly affected by its shrink-

Joinery. age; and, at the same time, it enables us to vary the surface without much labour.

From this view of the subject, the joiner will readily perceive, that neither the parts of the frame nor the panels should be wide. And as the frame should be composed of narrow pieces, it follows, that the panels should not be very long, otherwise the frame will want strength. The panels of framing should not be more than 15 inches wide, and 4 feet long, and panels so large as this should be avoided as much as possible.¹ The width of the framing is commonly about one-third of the width of the panel.

It is of the utmost importance, in framing, that the tenons and mortises should be truly made. After a mortise has been made with the mortise chisel, it should be rendered perfectly even with a float; an instrument which differs from a single cut, or float file, only by having larger teeth. An inexperienced workman often makes his work fit too tight in one place, and too easy in another, hence the mortise is split by driving the parts together, and the work is never firm; whereas if the tenon fill the mortise equally, without using any considerable force in driving the work together, is found to be firm and sound. The thickness

of tenons should be about one-fourth of that of the framing, and the width of a tenon should never exceed about five times its thickness, otherwise, in wedging, the tenon will become bent, and bulge out of the sides of the mortise. If the rail be wide, two mortises should be made, with a space of solid wood between; fig. 22 shews the tenons for a wide rail.

In thick framing, the strength and firmness of the joint is much increased by putting a cross or feather tongue in on each side of the tenon; these tongues are about an inch in length, and are easily put in with a plough proper for such purposes. The projected figure of the end of a rail, fig. 22, shews these tongues put in, in the style there are grooves ploughed to receive them.

Sometimes, in thick framing, a double tenon in the thickness is made; but we give the preference to a single one, when tongues are put in the shoulders, as we have described; because a strong tenon is better than two weak ones, and there is less difficulty in fitting one than two.

The panels of framing should be made to fill the grooves, so as not to rattle, and yet to allow the panels to shrink without splitting.

28. When a frame consists of curved pieces, they are

often joined by means of pieces of hard wood called keys. Fig 23 is the head of a Gothic window frame, joined with a key, with a plan of the joint below it. A cross tongue is put in on each side of the key, and the joint is tightened by means of the wedges aa.

It is, however, a better method to join such pieces by means of a screw bolt instead of a key, the cross tongues being used whichever method is adopted.

Joining with Glue.

Joining with glue. 29. It is seldom possible to procure boards sufficiently wide for panels without a joint, on account of heart shakes,

which open in drying. In cutting out panels, for good Joinery work, shaken wood should be carefully avoided. That part near the pith is generally the most defective.

If the panels be thick enough to admit of a cross or feather tongue in the joint, one should always be inserted, for then, if the joint should fail, the surfaces will be kept even, and it will prevent light passing through.

Sometimes plane surfaces of considerable width and length are introduced in joiners' work, as in dado, window backs, &c.; such surfaces are commonly formed of inch, or inch and quarter, boards joined with glue, and a cross or feather tongue ploughed into each joint. When the boards are glued together, and have become dry, tapering pieces of wood, called keys, are grooved in, across the back, with a dovetail groove. These keys preserve the surface straight, and also allow it to shrink and expand with the changes of the weather.

30. It would be an endless task to describe all the methods that have been employed to glue up bodies of such varied forms as occur in joinery; for every joiner forms work. methods of his own, and merely from his being most familiar with his own process, he will perform his work, according to it, in a better manner than by another, which, to an unprejudiced mind, has manifestly the advantage over it. The end and aim of the joiner, in all these operations, is to avoid the peculiar imperfections and disadvantages of his materials, and to do this with least expense of labour or material. The straightness of the fibres of wood renders it unfit for curved surfaces, at least when the curvature is considerable. Hence short pieces are glued together as nearly in the form desired as can be, and the apparent surface is covered with a thin veneer; or the work is glued up in pieces that are thin enough to bend to the required form. Sometimes a thin piece of wood is bent to the required form upon a cylinder or saddle, and blocks are jointed and glued upon the back; when the whole is completely dry it will preserve the form that had been given to it by the cylinder.

The proper thickness for the pieces to be bent may be easily determined by an easy experiment on a piece of the same kind of wood. Thus, select a piece of wood, of the same kind as that to be used, and bend it as much as it will bear without injury; then ascertain the radius of curvature, and also the thickness of the piece, at the most curved part of it. From these data the proper thickness for any other curve will be determined by the following proportion:

As the radius of curvature, found by experiment, is to the thickness of the piece tried; so is the radius of any other curve to the thickness of the piece that may be bent into it.¹

For example, we have found that a piece of straight grained white deal, of an inch in thickness, may be bent, without injury, into a curve of which the radius is 120 inches, therefore, $120\; :\; 1\; ::\; \backslash text\{radius\}\; :\; \backslash text\{thickness\}\; =\; \backslash frac\{\backslash text\{radius\}\}\{120\}$. That

is, a piece of deal of the same quality may be bent into any curve, of which the radius is not less than 120 times its thickness.

A piece of work glued up in thicknesses should be very well done; but it too often happens that the joints are visible, irregular, and in some places open; therefore other methods have been tried.

31. If a piece of wood be boiled in water for a certain time, then taken out and immediately bent into any particular form, and it be retained in that form till it be dry, a

¹ Panels of external doors and shutters may be rendered more secure by boring them, and inserting iron wires. See Trans. of the Society of Arts, vol. xxv. p. 106.

² The reader will find some interesting propositions relating to fixture in the Article CARPENTRY, p. 624, vol. ii.

permanent change takes place in the mechanical relations of its parts; so that though, when relieved, it will spring back a little, yet it will not return to its natural form.

The same effect may be produced by steaming wood; but though both these methods have been long practised to a considerable extent in the art of ship-building, we are not aware that any general principles have been discovered, either by experiment or otherwise, that will enable us to apply it to an art like joinery, where so much precision is required. We are not aware that it has been tried; but, before it can be rendered extensively useful, the relation between the curvature to which it is bent, and that which it assumes, when relieved, should be determined, and also the degree of curvature which may be given to a piece of a given thickness.

The time that a piece of wood should be boiled, or steamed, in order that it may be in the best state for bending, should be made the subject of experiments; and this being determined, the relation between the time and the bulk of the piece should be ascertained.

For the joiner's purposes, we imagine, that the process might be greatly improved, by saturating the convex side of each piece with a strong solution of glue, immediately after bending it. By filling, in this manner, the extended pores, and allowing the glue to harden thoroughly before relieving the pieces, they would retain their shape better.

32. Large pieces of timber should never be used in joinery, because they cannot be procured sufficiently dry to prevent them splitting with the heat of a warm room. Therefore, the external part of columns, pilasters, and works of a like kind, should be formed of thin pieces of dry wood; and, if support be required, a post, or an iron pillar, may be placed within the exterior column. Thus, to form columns of wood, so that they shall not be liable to split, narrow pieces of wood are used, not exceeding five inches in width. These are jointed like the staves of a cask, and glued together, with short blocks glued along at each joint.

Fig. 24 is a plan of the lower end of a column glued up in staves; the bevel at A is used for forming the staves, that at B is used for adjusting them when they are glued together. A similar plan must be made for the upper end of the column, which will give the width of the upper end of the staves. The bevels taken from the plan, as at A and B, are not the true bevels; but they are

those generally used, and are very nearly true, when the columns are not much diminished. To find the true bevels, the principle we have given in art. 19 should be applied. The same method may be adopted for forming large pillars for tables, &c.

If a column have flutes, with fillets, the joints should be in the fillets, in order to make the column as strong as possible; also, if a column be intended to have a swell in the middle, proper thickness of wood should be allowed for it.

When columns or pillars are small, they may be made of dry wood; and to secure them against splitting, a hole may be bored down the axis of each column.

Fixing Joiners' Work.

33. We have hitherto confined our remarks to that part of joinery which is performed at the bench; but by far the most important part remains to be considered. For, however well a piece of work may have been prepared, if it be not properly fixed, it cannot fulfil its intended purpose. As in the preceding part, we shall state the general principles that ought to be made the basis of practice, and illustrate those principles by particular examples.

If the part to be fixed consist of boards jointed together,

but not framed, it should be fixed so that it may shrink, or swell without splitting. The nature of the work will generally determine how this may be effected. Let us suppose that a plain back of a window is to be fixed. Fig. 25 is a

section showing B the back of the window, A the window-sill, D the floor, and C the skirting. The back is supposed to be prepared, as we have stated in art. 29, and that it is kept straight by a dovetailed key a. Now, let the back be firmly nailed to the window-sill A, and let a narrow piece d, with a groove, and cross tongue, in its upper edge, be fixed to bond timbers or plugs in the wall; the tongue being inserted also into a corresponding groove in the lower edge of the back of B.

It is obvious, that the tongue being loose, the back B may contract or expand, as a panel in a frame. The dado of a room should be fixed in the same manner. In the principal rooms of a house, the skirting C is usually grooved skirting into the floor D, and fixed only to the narrow piece d, which is called a ground. By fixing, in this manner, the skirting covers, the joint, which would otherwise soon be open by the shrinking of the back, and from the skirting being grooved into the floor, but not fastened to it, there cannot be an open joint between the skirting and floor. When it is considered, that an open joint, in such a situation, must become a receptacle for dust, and a harbour for insects, the importance of adopting this method of fixing skirting will be apparent.

In fixing any board above five or six inches wide, similar precautions are necessary; otherwise it is certain to split when the house becomes inhabited. We may, in general, either fix one edge, and groove the other, so as to leave it at liberty, or fix it in the middle, and leave both edges at liberty.

Sometimes a wide board, or a piece consisting of several boards, may be fixed by means of buttons, screwed to the landing of back, which turn into grooves in the framing, bearers, or stairs, joists, to which it is to be fixed. If any shrinking takes place the buttons slide in the grooves. In this manner the landing of stairs are fixed, and it is much the best mode of fixing the top of a table to its frame.

34. The extension of the principle of ploughing and tonguing work together is one of the most important of the improvements that have been introduced by modern joiners. It is an easy, simple, and effectual method of combination, and one that provides against the greatest defect of timber work, its shrinkage. By means of this method, the bold mouldings of Gothic architecture can be executed with a comparatively small quantity of material; and even in the mouldings of modern architecture it saves much labour. For example, the moulded part of an architrave may be joined with the plain part, as shewn by fig. 26. If this method be compared with the old method of glueing one piece upon another, its advantage will be more evident.

35. The architraves, skirtings, and surbase mouldings, are fixed to pieces of wood called grounds; and as the straightness and accuracy of these mouldings must depend upon the care that has been taken to fix the grounds truly; it will appear, that fixing grounds, which is a part often left to inferior workmen, in reality requires much skill and attention; besides, they are almost always the guide for the plasterer. Where the plasterer's work joins the grounds, they should have a small groove ploughed in the edge to form a key for the plaster.

36. In our remarks on construction, we must not omit to say a few words on laying floors, because it will give us

Joinery. an opportunity of pointing out a defect which might be easily remedied. The advice of Evelyn, to tack the boards down only the first year, and nail them down for good the next, is certainly the best, when it is convenient to adopt it; but, as this is very seldom the case, we must expect the joints to open more or less. Now these joints always admit a considerable current of cold air, and also, in an upper room, unless there be a counter floor, the ceiling below may be spoiled by spilling a little water, or even by washing the floor. To avoid this, we would recommend a tongue to be ploughed into each joint, according to the old practice. When the boards are narrow, they might be laid without any appearance of nails, in the same way as a dowelled floor is laid, the tongue serving the same purpose as the dowels. In this case we would use cross or feather tongues for the joints.

Folding floors censured. There is a method sometimes used in laying floors, which workmen call folding; according to this method, two boards are laid, and nailed at such a distance apart, that the space is a little less than the aggregate width of the boards intended for it; these boards are then put to their places, and, on account of the narrowness of the space left for them, they rise like an arch between its abutments. The workmen force them down by jumping upon them. Accordingly, the boards are never soundly fixed to the joists, nor can the floor be laid with any kind of evenness or accuracy. We merely notice this method here, in order that it may be avoided.

Heading joints. As boards can seldom be got long enough to do without

joints, it is usual, except in very inferior work, to join the ends with a tongued joint, as shewn in fig. 27, where B is the joist. The etched board is first laid, and nailed to the joist.

In oak floors, the ends are forked together sometimes, as shewn at A, fig. 28, in order to render the joints less conspicuous.

The joints should be kept as distant from one another as possible.

Hinging.

Hinging. 37. It requires a considerable degree of care to hang a door, a shutter, or any other piece of work in the best manner. In the hinge, the pin should be perfectly straight, and truly cylindrical, and the parts accurately fitted together.

The hinges should be placed so that their axes may be in the same straight line, as any defect in this respect will produce a considerable strain upon the hinges every time the hanging part is moved, which prevents it from moving freely, and is injurious to the hinges.

In hanging doors, centres are often used instead of hinges; but, on account of the small quantity of friction in centres, a door moves too easily, or so that a slight draft of air accelerates it so much in falling to, that it shakes the building, and is disagreeable. We have seen this in some degree remedied by placing a small spring to receive the shock of the door.

The greatest difficulty, in hanging doors, is to make them to clear a carpet, and be close at the bottom when shut. To do this, that part of the floor which is under the door, when shut, may be made to rise above a quarter of an inch above the general level of the floor; which, with placing the hinges so as to cause the door to rise as it opens, will be sufficient, unless the carpet should be a very thick one. Several mechanical contrivances have been used for either raising the door, or adding a part to spring close to the floor as the door shuts. The latter is much the better

method. The reader who may be desirous of examining this method, may consult the Transactions of the Society of Arts, (vol. xxvi. p. 196.)

38. Various kinds of hinges are in use. Sometimes they are concealed, as in the kind of joints called rule joints; others project, and are intended to let a door fold back over projecting mouldings, as in pulpit doors. When hinges project, the weight of the door acts with an increased leverage upon them, and they soon get out of order, unless they be strong and well fixed.

The door of a room should be hung so that, in opening the door, the interior of the room cannot be seen through the joint. This may be done by making the joint according to fig. 29. The bead should be continued round the door, and a common but-hinge answers for it.

The proper bevel for the edge of a door or sash may be found by drawing a line from the centre of motion C, fig. 30, to e, the interior angle of the rebate, draw ed perpendicular to Ce, which gives the bevel required. In practice, the bevel is usually made less, leaving an open space in the joint when the door is shut; this is done on account of the interior angle of the rebate often being filled with paint.

Stairs.

39. The construction of stairs is generally considered the highest department of the art of joinery, therefore we treat of it under a distinct head.

The principal object to be attended to in stairs is, that they afford a safe and easy communication between floors of different levels. The strength of a stair ought to be apparent as well as real, in order that those who ascend it may feel conscious of safety. In order to make the communication safe, it should be guarded by a railing of proper height and strength; in order that it may be easy, the rise and width, or tread, of the steps should be regular and justly proportioned to each other, with convenient landings; there should be no winding steps, and the top of the rail should be of a convenient height for the hand.

The first person that attempted to fix the relation between the height and width of a step, upon correct principles, was, we believe, Blondel, in his Cours d'Architecture. If a person walking upon a level plane move over a space, P, at each step, and the height which the same person could ascend vertically, with equal ease, were H; then, if h be the height of a step, and p its width; the relation between p and h must be such, that when p = P, h = o; and when H = h, p = o. These conditions are satisfied by an equation of the form $h\; =\; H\; \backslash left(\; 1\; -\; \backslash frac\{p\}\{P\}\; \backslash right)$. Blondel assumes 24 inches for the value of P, and 12 inches for that of H; substituting these values in our equation, it becomes $h\; =\; \backslash frac\{1\}\{2\}\; (24\; -\; p)$, which is precisely Blondel's rule. We do not think these the true values of P and H; indeed, it would be difficult to ascertain them; but they are so near, and agree so well with our observations on stairs of easy ascent, that they may be taken for the elements of a practical rule. Hence, according as h or p is given, we have $h\; =\; \backslash frac\{1\}\{2\}\; (24\; -\; p)$, or $p\; =\; 24\; -\; 2h$.

Thus, if the height of a step be six inches, then $24\; -\; 12\; =\; 12$, the width or tread for a step that rises six inches.

40. The forms of staircases are various. In towns, where

ery. space cannot be allowed for convenient forms, they are often made triangular, circular, or elliptical, with winding steps, or of a mixed form, with straight sides and circular ends. In large mansions, and in other situations, where convenience and beauty are the chief objects of attention, winding steps are never introduced when it is possible to avoid them. Good stairs, therefore, require less geometrical skill than those of an inferior character.

The best architectural effect is produced by rectangular staircases, with ornamented railing and newels. In Gothic structures scarcely any other kind can be adopted, with propriety, for a principal staircase. Modern architecture admits of greater latitude in this respect; the end of the staircase being sometimes circular, and the hand-rail continued, beginning either from a scroll or a newel.

41. When a rectangular staircase has a continued rail, it is necessary that it should be curved so as to change gradually from a level to an inclined direction. This

curvature is called the ramp of the rail. The plan of a staircase of this kind is represented by ABCD, fig. 31, and fig. 32 shows a section of it, supposing it to be cut through at ab, on the plan.

The hand-rail is supposed to begin with a newel at the bottom, and the form of the cap of the newel ought to be determined, so that it will mitre with the hand-rail. Let H, fig. 33, be the section of the hand-rail, and ab the radius of the newel; then the form of the cap may be traced at C by the method we have already described. (Art. 9 and 10.)

The sections of hand-rails are of various shapes; some of the most common ones are too small; a hand-rail should never be less than would require a square, of which the side is 2½ inches, to circumscribe it.

For the level landings of a staircase the height

of the top of the hand-rail should be about 40 inches, and in any part of the inclined rail the height of its upper side above the middle of the width of the step should be 40 inches less the rise of one step, when measured in a vertical direction.

To describe the ramps, let rs be a vertical line drawn through the middle of the width of the step, fig. 32; set ru equal to rs, and draw ut at right angles with the back of the rail, cutting the horizontal line st in t. From the point t, as a centre, describe the curve of the rail. When there is a contrary flexure, as in the case before us, the method of describing the lesser curve is the same.

42. The hand-rail of a stair often begins from a scroll; and that kind of spiral which is called the logarithmic spiral, has been proposed as the best for the purpose. It is shown by writers on curve lines, that any radial lines drawn from the centre will be cut by the logarithmic spiral in one

and the same angle. By means of this property of the curve, it may be described as follows:

Let C be the centre, fig. 34, and draw AB perpendicular to DE, crossing it in C. Bisect the angles by the lines ab, cd. Draw eBb to cut CB at the angle proposed for the curve, and to meet Cb in b; draw eb perpendicular to be, cutting Cc in c; draw ca perpendicular to eb cutting Ca in a; and proceed round with as many revolutions as may be required in the same manner. Then B, E, A, D, F, G, &c. are points in the curve, and the lines

eb, cb, oa, ad, &c. are tangents to the curves at these points. Therefore, the curve may be either drawn by hand, or by means of circular arcs. Also, any number of interior or exterior spirals may be drawn by drawing lines parallel to the tangents, as xy, yz, &c.

If eb were to cross BC at a right angle, the curve would be a circle.

43. The scrolls and volutes used in architecture are always made to terminate in a circle at the centre; consequently none of the curves described by mathematicians are adapted for these purposes. But the construction we have employed for the logarithmic spiral readily leads to a species of spiral that appears well suited for scrolls or volutes. In the logarithmic spiral the angle of the curve is constant; but imagine the angle to change regularly, and to become a right angle at the point where the circle called the eye begins. This would afford us a regular and pleasing curve, unfolding itself from a circle in the centre. This curve might be called the Architectural Spiral.

Let C be the centre, fig. 35, and round this centre de-

scribe a circle for the eye of the scroll, or volute. Divide this circle into eight equal parts, and draw lines from the centre through the points of division.

With any radius aC, and C as a centre, describe the arc ac, and upon this arc set off any number of equal divisions. The extent of a division must be regulated by the quantity the curve may unfold at each revolution, and the number depends on the number of revolutions.

Then, beginning at A, draw Ab perpendicular to Ca; db parallel to C'; de perpendicular to C2; ef parallel to C3; and so on for any number of revolutions. The points A, B, D, E, F, G, and H, in the curve, and the tangents to these points, are found; therefore the curve may be described by hand, or by means of circular arcs.

The tangents to any interior or exterior spiral will be parallel to the ones first found, and, therefore, any number may be drawn with the greatest facility.

Joinery. Neither the logarithmic nor the architectural spiral can be drawn truly by circular arcs; but we shall here point out the principle by which such spirals may be drawn. When a spiral is drawn by means of circular arcs only, the centres of the adjoining arcs must always be upon the same straight line; and the regularity of the curve will depend on the number of arcs employed to describe one revolution. Let the proposed distance between the revolutions be divided into as many equal parts as there are to be circular arcs in one revolution; and, on the eye as a centre, construct a regular polygon of the same number of sides as the number of divisions, and on each side equal to one division. Then the angles of the polygon will be the centres for describing the spiral, as shewn by the figures below, where the triangle, square, and hexagon, are given as examples:

If a spiral be drawn to begin from a circle at the centre, let the arcs be described from the angles of a rectangular fret, as in fig. 39, the sides of which may increase in any regular proportion. Or, a figure may be drawn in the same manner as the tangents of the spiral, fig. 35, and the arcs described in the angle, as in fig. 40. By either of these methods a pleasing curve may be obtained.

44. Fig. 41 represents the plan of a staircase, beginning

with a scroll, and having steps winding round the circular part of the well-hole.

In the first place, let the end of the steps be developed according to the method we have given in Art. 13. Fig. 43 shews this development. Now, the hand-rail ought to follow the inclination of a line drawn to touch the nosings of the steps, except where there is an abrupt transition from the rake of the winding to that of the other steps; at such places it must be curved; the curve may be drawn by the help of intersecting lines, as in fig. 44, if the workman cannot trust to his eye.

The part which is shaded in fig. 43, represents the hand-rail and ends of the steps, when spread out, and the hand-rail is only drawn close to the steps for convenience, as it would require too much space to raise it to its proper position. This development of the rail is called the falling mould.

The wood used for hand-rails being of an expensive kind, it becomes of some importance to consider how the plank may be cut so as to require the least quantity of material for the curved part of the rail. Now, if we were to suppose the rail executed, and a plain board laid upon the upper side of it, the board would touch the rail at three points; and a plank laid in the same position as the board would be that out of which the rail could be cut with the least waste of material.

Let it be required to find the moulds for the part ab of the rail, fig. 41, and to avoid confusing the lines in our the small figure, the part ab has been drawn to a larger scale in fig. 42. The plain board, mentioned above, would touch

the rail at the points marked C and B in the plan; draw the line CB, and draw a line parallel to CB, so as to touch the curve at the point E. Then E is the other point on the plan; and a', e', and b', are the heights of these points in the development, fig. 43.

Erect perpendiculars to CB, from the points C, E, and B, fig. 42, and set off Ca, on fig. 42, equal to a'e', fig. 43; Ee equal to de', and Bb equal to fb'. Through the points C and E, draw the dotted line Ch; through ae draw a line to meet CE in h; and through the points ab, draw a line to meet CB in g; then join hg, and make Ci perpendicular to hg.

Now, if Cd be equal to Ca, and perpendicular to Ci; and di be joined, it will be the angle which the plank makes with the horizontal plane, or plan. Therefore, draw FD parallel to Ci, and find the section by the process described in Art. 10. This section is the same thing as would be obtained by projecting vertical lines from each point in the hand-rail against the surface of a board, laid to touch it in three points. The inexperienced workman will be much assisted in applying the moulds if he acquires a clear notion of the position when executed.

To find the thickness of the plank, take the height to the under side of the rail er in the development, fig. 43, thickness, and set it off from s, in the line Ci, to r, in fig. 42; from the

the point $r$ draw a line parallel to $di$, and the distance between those parallel lines will be the thickness of the plank.

The mould, fig. 42, which is traced from the plan, is called the face mould. It is applied to the upper surface of the plank, which being marked, a bevel should be set to the angle $idC$, and this bevel being applied to the edge will give the points to which the mould must be placed to mark out the under side. It is then to be sawn out, and wrought true to the mould. In applying the bevel, care should be taken to let its stock be parallel to the line $di$, if the plank should not be sufficiently wide for $di$ to be its axis.

After the rail is truly wrought to the face mould, the falling mould, fig. 43, being applied to its convex side, will give the edge of the upper surface, and the surface itself will be formed by squaring from the convex side, holding the stock of the square always so that it would be vertical if the rail were in its proper situation. The lower surface is to be parallel to the upper one.

The sudden change of the width of the ends of the steps causes the soffit line to have a broken or irregular appearance; to avoid it, the steps are made begin to wind before the curved part begins. Different methods of proportioning the ends of the steps are given by Nicholson, Roubo, Rondelet, and Krafft. We cannot in this place enter into a detail of these methods, but for the reader's information a list of the principal writers on staircases is subjoined.

Price, in his British Carpenter, 4to, 1735; Langley, Builders' Complete Assistant, 8vo, 1738; Frezier, Coupe des Pierres et des Bois, 4to, 1739; Roubo, L'Art du Menuisier, folio, 1771; Skaife, Key to Civil Architecture, 8vo, 1774; Nicholson, Carpenters' New Guide, 4to, 1792; Carpenters' and Joiners' Assistant, 4to, 1792; Architectural Dictionary, 4to; Transactions Society of Arts, &c. for 1814; Treatise on the Construction of Staircases and Handrails, 4to, 1820; Rondelet, Traité de l'Art de Bâtir, tome iv. 4to, 1814; and Krafft, Traité sur l'Art de la Charpenter, part ii. folio, 1820.

SECT. III.—ON MATERIALS.

45. There is no art in which it is required that the structure and properties of wood should be so thoroughly understood as in joinery. The practical joiner, who has made the nature of timber his study, has always a most decided advantage over those who have neglected this most important part of the art.

In the article ANATOMY, VEGETABLE (vol. iii. p. 61 and 82), the structure of wood is described; in this place, therefore, we shall only show how the joiner may, in a great measure, avoid the warping caused by its irregular texture.

46. It is well known that wood contracts less in proportion, in diameter, than it does in circumference; hence a whole tree always splits in drying. Mr Knight has shown that, in consequence of this irregular contraction, a board may be cut from a tree that can scarcely be made, by any means, to retain the same form and position when subjected to various degrees of heat and moisture. From the ash and the beech he cut some thin boards, in different directions relatively to their transverse septa, so that the septa crossed the middle of some of the boards at right angles, and lay nearly parallel with the surfaces of others. Both kinds were placed in a warm room, under perfectly similar

circumstances. Those which had been formed by cutting across the transverse septa, as at A in fig. 44, soon changed their form very considerably, the one side becoming hollow, and the other round; and in drying, they contracted nearly 14 per cent. in width.

The other kind, in which the septa were nearly parallel to the surfaces of the boards, as at B in fig. 44, retained, in shrinking with very little variation, their primary form, and did not contract in drying more than three and a half per cent. in width.¹

As Mr Knight had not tried resinous woods, two specimens were cut from a piece of Memel timber; and, to render the result of our observation more clear, conceive fig. 45 to represent the section of a tree, the annual rings being shewn by circles. BD represents the manner in which one of our pieces was cut, and AC the other. The board AC contracted 3.75 per cent. in width, and became hollow on the side marked $b$. The board BD retained its original straightness, and contracted only 0.7 per cent. The difference in the quantity of contraction is still greater than in hard woods.

Fig. 45.

From these experiments, the advantages to be obtained merely by a proper attention in cutting out boards for panells, &c. will be obvious; and it will also be found that panells cut so that the septa are nearly parallel to their faces, will appear of a finer and more even grain, and require less labour to make their surfaces even and smooth.

The results of these experiments are not less interesting to cabinet-makers, particularly in the construction of billiard-tables, card-tables, and indeed every kind of table in use. For such purposes, the planks should be cut so as to cross the rings as nearly in the direction BD as possible. We have no doubt that it is the knowledge of this property of wood that renders the billiard-tables of some makers so far superior to those of others.

In wood that has the larger transverse septa, as the oak, for example, boards cut as BD will be figured, while those cut as AC will be plain.

47. There is another kind of contraction in wood whilst drying, which causes it to become curved in the direction of its length. In the long styles of framing we have often observed it; indeed, on this account, it is difficult to prevent the style of a door, hung with centres, from curving so as to rub against the jamb. A very satisfactory reason for this kind of curving has been given by Mr Knight,² which also points out the manner of cutting out wood, so as to be less subject to this defect, which it is most desirable to avoid. The interior layers of wood, being older, are more compact and solid than the exterior layers of the same tree; consequently, in drying, the latter contract more in length than the former. This irregularity of contraction causes the wood to curve in the direction of its length, and it may be avoided by cutting the wood so that the parts of each piece shall be as nearly of the same age as possible.

48. Besides the contraction which takes place in drying, wood undergoes a considerable change in bulk with the variations of the atmosphere. In straight-grained woods the change in length is nearly insensible;³ hence they are sometimes employed for pendulum rods; but the lateral dimensions vary so much, that a wide piece of wood will serve as a rude hygrometer.⁴ The extent of variation de-

¹ Philosophical Transactions, part ii. for 1817, or Philosophical Magazine, vol. i. p. 437.

² Mr Ramsden and General Roy made some experiments on the expansion in length. See Account of the Trig. Survey, vol. i. p. 46 and 49.

³ See Phil. Trans. Lowthorpe's Abridg. vol. ii. p. 37.

VOL. XII.