Fig. 33.
First, look at D of this figure. You have a line, a b, standing upon another C D, and perpendicular to it—that is, it leans neither to the right nor to the left. It makes two angles at b, one on each side of a b, and these are angles of 90°, or right angles, as I explained. Now, if one line like a b stands on another, these two angles are together equal to 180°, or twice 90°, whether this line is or is not upright or perpendicular to the other. Look at fig. C. Here you have the line x x, and standing on it several others; one, a b, is upright or perpendicular, making with it two angles of 90° each, or 180° together. Now, take f b, and suppose this to make 45° on the right-hand side, you see it makes therefore a proportionately larger angle on the other. It makes, in fact, an angle of 135°. But 135° added to 45° equals 180°, which is the same as before, and whichever line you take, the angles together made by it at b will equal 180° of the circle—that is, they will equal two right-angles.
Now, if I take the fig. D again, and carry on the line a b right through c d, where it is dotted, two angles will be made on the other side of c d, which will each be right angles of 90° as before, so that all the four angles thus made are equal. It follows from this, that whenever any two lines cut each other—E Q and R S for instance—the angles at T equal four right angles, no matter whether the lines are or are not perpendicular to each other: and what is more (and what I specially want you to note), the opposite angles are equal—i.e., the two small ones, or the two large ones.
The action of a mitre-block or mitre-box depends upon the principles here laid down, so you see that although few carpenters understand much about mathematics, and simply work as they were taught, without knowing or caring why, those who planned the method of work, and invented mitre-boards and such like devices to shorten work and lessen labour, must have understood a great deal about such things. And so it is generally, as you will find with inventions: things look easy enough, and natural enough, when we see them every day; but it has taken a great deal of thought and sound knowledge to invent them in the first place, and a great deal of practical experience to construct them so neatly. Even a common pin goes through such a number of processes as would surprise you, if you have never been able to see them made.
Look carefully at A. It represents a block of wood, about 1½ or 2 inches thick, and 3 or 4 wide, firmly screwed on the top of a board 1 inch thick. The length is about 18 inches. Two saw-cuts are made with a tenon-saw, right through the block to the board, at angles of 45° with the line a b. These are guides for the saw to work in. The wood to be cut is laid against the edge of the block, and rests on the board, and the saw is then applied in one of the grooves while the wood is being cut by it. Let H be such a piece. If the saw is put in the left-hand slit, it will cut it like y; if in the other, it will cut it the other way, like x; and thus, if a piece is taken off at each end, it will be as you see, ready to become one side of a frame. Now, examine K, which shows all the lines or edges of the mitring-board, as seen from above, with the strip a b sawn across in the line c a. The lines a b and c a cross each other, making the opposite angles equal; and as one angle is 45° the other must be 45° also, so that the right-hand side of the strip is correctly cut. But so also is the other end, and if we turn it over, it will exactly fit, and the two will form two sides of a square. I could prove to you that the second strip contains angles exactly similar to the first, but you ought to be able now to detect the reason for yourself, and I do not want to teach you more mathematics at present, as I am afraid you are tired of these, and will want to go on with the real work of fitting and making. I have, however, said enough, I think, to make you comprehend why the two saw-cuts must be at an angle of 45° with the edge of the top board.
Perhaps you wish to make your own board, however, and would like to know an easy way to get the saw-cuts at the right angle? I shall therefore show you how to do this, but you must be very exact in your workmanship. A B, Fig. 34, is the piece of thick board as seen from above, and close to it is a perspective view of the same which shows the thickness. Set off a distance, A E, equal to A C, and join C E. The dotted line shows you that C E is the diagonal of a square, and the angles at C and E are consequently each 45°; but we do not want this line to end at C, it is too exactly at the corner for convenience. Measure, therefore, a distance, E b and C a, equal, and join a b, which will be the place for the saw-cut; and the other can be marked out in exactly the same way. a x, in the perspective view, must be carefully marked by the help of the square. Take care to mark the line on the bottom board, where the edge of this upper thick piece will fall, and screw the two firmly together. If the edge and face of the thick piece are not truly square to each other, the mitres cut thus will not be correct; but, if all is well made, they may be glued at once together, no paring of the chisel being necessary or desirable.
The mitre-box, Fig. 33, B, is on precisely the same principle, but is chiefly used to cut narrow strips not over 2 inches wide; it should be neatly made of mahogany, half an inch thick. There is also generally made a saw-cut straight across, at right angles to the length of the box or board, which is convenient in sawing across such strips of wood, as it saves the necessity of marking lines against the edge of the square: of course, it is specially used where a large number of strips have to be cut square across. In all these you observe one saw-cut leaning to the right, the other to the left. This is necessary when picture-frames or moulded pieces have to be cut to 45°, because you cannot, of course, turn such pieces over and use either side, which you can do when the piece has no such mouldings.
