To facilitate the construction of geometrical figures, we add a short description of a few useful instruments which do not belong to the common pocket-case.

Let there be a flat ruler, AB, from one to two feet in length, for which the common Gunter's scale may be substituted; and, secondly, a triangular piece of wood, a, b, c, flat, and about the same thickness as the ruler: the sides, ab and bc of which are equal to one another, and form a right angle at b. For the convenience of sliding, there is usually a hole in the middle of the triangle, as may be seen in the figure.

By means of these simple instruments many very useful geometrical problems may be performed. Thus, to draw a line through a given point parallel to a given line. Lay the triangle on the paper so that one of its sides will coincide with the given line to which the parallel is to be drawn; then, keeping the triangle steady, lay the ruler on the paper, with its edge applied to either of the other sides of the triangle; then, keeping the ruler firm, move the triangle along its edge, up or down, to the given point; the side of the triangle which was placed on the given line will always keep parallel to itself, and hence a parallel may be drawn through the given point.

To erect a perpendicular on a given line, and from any given point in that line, we have only to apply the ruler to the given line, and place the triangle so, that its right angle shall touch the given point in the line, and one of the sides about the right angle, placed to the edge of the ruler-the oilier side will give the perpendicular required.

If the given point be either above or below the line, the process is equally easy. Place one of the sides of the triangle about the right angle on the given line, and the ruler on the side opposite the right angle, then slide the triangle on the edge of the ruler till the given point from which the perpendicular is to be drawn is on the other side, then this side will give the perpendicular.

Other problems may be performed with these instruments, the method of doing which it will be easy for the reader to contrive for himself.

When arcs of circles of great diameter are to be drawn, the use of a compass may be substituted by a very simple contrivance. Draw the chord of the arc to be described, and place a pin at each extremity, A and B, then place two rulers jointed at C, and forming an angle, ACB equal to the supplement of half the given number of degrees; that is to say, the number of degrees which the arc whose chord given is to contain, is to be halved, and this half being subtracted from 180 degrees, will give the degrees which form the angle at which the rulers are placed, that is, the angle ACB. This being done, the edges of the rulers are moved along against the pins, and a pencil at C will describe the arc required.

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Large circles may be described by a contrivance equally simple. On an axle, a foot or a foot and a hall long, there are placed two wheels, M and F, of which one is fixed to the axle, namely F, and the other is capable of being shifted to different parts of the axle, and, by means of a thumb-screw, made capable of being fixed at any point on the axle. These wheels are of different diameters, say of 3 and 6 inches, the fixed wheel F being the largest. This instrument being moved on the paper, the circles M and F will roll, and describe circles of different radii: the axle will always point to the centre of these circles, and there will be this proportion:

As the diameter of the large wheel is to the difference of the diameters of the two wheels, so is the radius of the circle to be described by the large wheel to the distance of the two wheels on the axle.

If the diameters of the wheels are as above staled, and it is required to describe a circle of 3 feet radius, then from the above proportion we have 6:6 - 3:: 3 feet or 36 inches: 18 inches = the distance of the two wheels, to describe a circle 6 feet in diameter.

It may be observed, that it will be best to make the difference of the wheels greater if large circles are to be described, as then a shorter instrument will serve the purpose.

We will conclude these instructions, by making a few remarks on the Diagonal Scale and Sector, the great use of the latter of which, especially, is seldom explained to the young mechanic.

The diagonal scale to be found on the plain scale in common pocket-cases of instruments, is a contrivance for measuring very small divisions of lines; as, for instance, hundredth parts of an inch.

Suppose the accompanying cut to represent an enlarged view of two divisions of the diagonal scale, and the bottom and top lines to be divided into two parts, each representing the tenth part of an inch. Now, the perpendicular lines BC, AD, arc each divided into ten equal parts, which are joined by the crossing lines, 1,2, 3, 4, etc, and the diagonals BF, DE, 7/8 drawn as in the figure. Now, as the division IV is the tenth part of an inch, and as the line FB. continually approaches nearer and nearer to BC, till it meets it in B, it will follow, that the part of the line 1 cut off by this diagonal will be a tenth part of FC, because Bl only one-tenth part of BC; so, likewise, 2 will represent two-tenth parts, 3 three-tenth parts, and so on to 9, which is nine-tenth parts, and 10, ten-tenth parts, or the whole tenth of an inch; so that, by means of this diagonal, we arrive;at divisions equal to tenth parts of tenth parts of an inch, or hundredths of an inch. With this consideration, an examination of the scale itself will easily show the whole matter. It may be observed, that if half an inch and the quarter of an inch be divided, in the same man-ner, into tenths and tenths of tenths, we may get thus two-hundredth and four-hundredth parts of an inch.

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