Now, all angles are thus measured by the divisions of a circle; the line at 45, which meets the line from 360 at the centre, makes with it an angle of 45°, which is half a right angle. A line drawn at 30° would make an angle of 30 with the same line from 360, and so on right round; only when two lines come exactly opposite one another, as 360 and 180, or 270 and 90, these make no angles—they are but one straight line passing through the centre, and are called diameters of the circle, a word which means measure through, or across the circle. Now, the corners of a square frame, or of a drawer or box, are right angles of 90°. At R, I have drawn such a corner of a frame, and if I place one point of a pair of compasses at e, and draw a circle cutting through the lines of the sides of the frame, you see I should make it 90°, or a quadrant, like N. Moreover, if I draw the sides of the frame as if they crossed as at e R, I draw a small square, and the line e R is the diagonal of such square: e R is the mitred joint I have to cut. Look at T S and you will see this, as here the two sides of the frame are represented as cut ready to be joined together.
A square has another quality: all its sides are equal, and this is very important, and will help us in cutting out the work. x Y represents the strip of wood to be properly sloped off for a mitred joint. With a gauge such as that just above x, or your regular marking gauge, set off on the side Y a distance equal to x x (the width of the pieces); join x b by a line, and you will have the right slope. Why? Because when you measured with the gauge you marked the two equal sides of a square, and x b is the diagonal of it, which is exactly the same as you had at e R. By measuring in this way, therefore, you can, if your strips are already truly squared up, always mark out a mitred joint correctly. The two little angles at x and b are also, I should point out, equal—each half of a right angle or 45°, and the other strip or side of the frame will make up the other half right angle, or complete the exact square of 90°.
In all this I have clearly laid down the principles of mitred joints, and given you a lesson in mathematics. I shall now, therefore, go on to the work of practical construction (Fig. 32). You must be very careful to make the edge B square to the side A, as in all other work which I have explained to you; or, if this side is moulded like the front of a picture-frame, you must square the edge with the back. After having cut all the pieces, you have to glue them and fasten them together. Warm them, and use the glue boiling, as directed before, and quickly lay the pieces together. To do so effectually, you must place them flat on a board or on your bench, and having adjusted them, you can tie a strong cord round the whole, putting little bits of wood close to the corners, so that the string shall not mark your work, if such marks would be of consequence. Or you can wedge up strongly in another way. If you look at C you will see a square representing a frame with eight spots round it. These are nail heads, and mark the position of eight nails driven round but not touching the frame into the bench. Then, having prepared eight small wedges, drive them in between the frame and the nails.
You will find this as simple and easy a way of keeping the frame together as any, and all must remain till the glue is dry and hard—probably till the same hour on the following day? Then remove the wedges and take up your frame, which should be trim and strong. Nevertheless, you are now to add considerably to the strength of it in one or both of the following ways.
Fig. 32.
With a mitre-saw or tenon-saw cut one or two slits at each angle, as seen at D, Fig. 32, e and f. Cut little pieces of thin wood, and having glued them, drive them into these slits. If you saw them slanting, some tending upwards and some downwards, it will be better than cutting straight into the frame. Then, when all is dry, neatly trim off these pieces even with the frame. You may also, if the work is of a more heavy kind, as a large picture-frame, finish with keyed mitres, g. Cut a place with a chisel of the shape here shown, about one-eighth of an inch deep, half into one piece and half into the other. Then cut out a key of the same form of thin hard wood, to fit exactly, and glue it in. The shape of the key prevents the joint from coming apart, and makes it very strong and durable. A very large number of light boxes are made with mitred joints, as workboxes, water-colour boxes, compass-boxes, and such like; and you can examine these for yourself; but you will not often see the keys at the angles, because most of such boxes are veneered, or covered when finished with a thin layer of some ornamental wood.
I shall now proceed to show you how these joints can be cut at once without the trouble of gauging and measuring to find the proper angle. Therefore I shall let you into the secret of mitring boxes and mitring boards, which, if you had much to do of this kind, would shorten your work considerably.
Fig. 33, A, represents a mitring-board, B a mitring-box. We must go into a little mathematics again, and try to understand these, because, if you do so, you may devise others, occasionally more suitable for any special work you have in hand.