To take a Distance with the Hair Compasses.-Open them as nearly as you can to the required distance, set the fixed leg on the point from which the distance is to be taken, and make the extremity of the other leg coincide accurately with the end of the required distance, by turning the screw.

COMPASSES WITH MOVEABLE POINTS.

If an arc or circle is to be described faintly, merely as a guide for the terminating points of other lines, the steel points are generally sufficient for

the purpose, and are susceptible of adjustment with greater accuracy than a pencil point; but, in order to draw arcs or circles with ink or black lead, compasses with a moveable point are used. In the best description of these compasses the end of the shank is formed into a strong spring, which holds firmly the moveable point, or a pencil or ink point, as may be required. A lengthening bar may also be attached between the shank and the moveable point, so as to strike larger circles, and measure greater distances. The moveable point to be attached to the lengthening bar, as also the pen point and pencil point, are furnished with a joint, that they may be set nearly perpendicular to

the paper.

A, the compasses, with a moveable point at B.

c and D, the joints to set each point perpendicular to the

paper.

E, the pencil point.

F, the pen point. (This is represented with a dotting wheel, the pen point and the dotting point being similar in shape to each other.)

G, the lengthening bar

To describe small arcs or circles a small pair of compasses, called bow compasses. with a permanent ink or pencil point, are used. They are formed with a round head, which rolls with ease between the fingers. The adjoining figures represent two varieties of bow pen, fig. 1 being well adapted to describe arcs of not more than one inch radius, and fig. 2 to describe arcs of small radii with exactness by means of the adjusting screw

C.

For copying and reducing drawings, compasses of a peculiar construction are used; the simplest form of which is that called wholes and halves, because the longer legs being twice the length

of the shorter, when the former are opened to any given line, the shorter ones will be opened to the half of that line. By their means, then, all the lines of a drawing may be reduced to one-half, or enlarged to double their length. These compasses are also useful for dividing lines by continual bisections.

PROPORTIONAL COMPASSES.

By means of this ingenious instrument drawings may be reduced or enlarged, so that all the lines of the copy, or the areas or solids represented by its several parts, shall bear any required proportion to the lines, areas, or solids of the original drawing. They will also serve to inscribe regular polygons in circles, and to take the square roots and cube roots of numbers. In the annexed figure the scale of lines is placed on the leg A E, on the left-hand side of the groove, and the scale of circles, on the same leg, on the right-hand side of the groove. The scales of plans and solids are on the other face of the instrument.

To set the instrument it must first be accurately closed, so

that the two legs appear but as one; the nut c being then unscrewed, the slider may be moved, until the line across it coincides with any required division upon any one of the scales. Now tighten the screw, and the compasses are set.

To reduce or enlarge the Lines of a Drawing.The line across the slider being set to one of the divisions, 2, 3, 4, &c., on the scale of lines, the points A, B will open to double, triple, four times, &c., the distances of the points D, E (Euc. vi. prop. 4). If, then, the points A and B be opened to the lengths of the lines upon a drawing, the points D and E will prick off a copy with the lines reduced in the proportions of to 1, 3 to 1,4 to 1, &c.; but, if the points D and E be opened to the lengths of the lines upon a drawing, the points A and B will prick off a copy with the lines enlarged in the proportions of 2 to 1, 3 to 1, 4 to 1, &c.

To inscribe in a Circle a regular Polygon of any number of Sides from 6 to 20.-The line across the slider being set to any number on the scale of circles, and the points A and B being opened to the length of any radius, the points and E will prick off a polygon of that number of sides, in

B

the circle described with this radius; thus, if the line across the slider be set to the division marked 12 on the scale of circles, and a circle be described with the radius A B, D E will be the chord of ath part of the circumference, and will prick off a regular polygon of 12 sides in it.

To reduce or enlarge the Area of a Drawing.—The numbers upon the scale of plans are the squares of the ratios of the lengths of the opposite ends of the compasses, when the line across the slider is set to those numbers; and, the distances between the points being in the same ratio as the lengths of the corresponding ends (Euc. vi. prop. 4), the areas of the drawings, and of the several parts of the drawings, pricked off by these points, will have to one another the ratio of 1 to the number upon the scale of plans to which the instrument is set (Euc. vi. props. 19, 20; and xii. prop. 2). Thus, if the line across the slider be set to 4 on the scale of plans, the distance between the points A and B will be twice as great as the distance between D and E; and, if A and B be opened out to the lengths of the several lines of a drawing, D and E will prick off a copy occupying 4th the area; if the line across the

slides be set to 5 on the same scale, the distances between the points will be in the ratio of 1 to 5, and the area of the copy pricked off by the points D and E will be 4th of the area of the drawing, of which the lines are taken off by A and B conversely, if the lines of the drawing be taken off by the points D and E, the points A and B will prick off a copy, of which the area will be 4 times or 5 times as great, according as the line across the slider is set to the division marked 4 or 5 on the scale.

To take the Square Root of a Number.-The line across the slider being set to the number upon the scale of plans, open the points A and B to take the number from any scale of equal parts (see page 9), then the points D and E applied to the same scale of equal parts will take the square root of the number. Thus, to take the square root of 3, set the line across the slider to 3, open out the compasses, till ▲ and B take off 3 from any scale of equal parts, and the points D and E will take off 1.73, which is the square root of 3, from the same scale of equal parts. A mean proportional between two numbers, being the square root of their product, may be found by multiplying the numbers together, and then taking the square root of the product in the manner explained above.

The numbers of the scale of solids are the cubes of the ratios of the lengths of the opposite ends of the compasses, when the line across the slider is set to those numbers; so that, when this line is set to the division marked 2 upon the scale of solids, the distance between the points A and B will give the side of a solid of double the content of that, of which a like side is given by the distance of the points D and E when the line is set to 3, the respective distances of the points will give the like sides of solids, the contents of which will be in the proportion of 3 to 1; and so on.

The Cube Root of a given number may be found by setting the line across the slider to the number upon the scale of solids, and, opening the points A, B, to take off the number upon any scale of equal parts, the points D, E, will then take off the required cube root from the same scale.

THE TRIANGULAR COMPASSES.

One of the best forms of these instruments is represented in the annexed figure. abc is a solid tripod, having at the extremity of the three arms three limbs, d, e, and f, moving freely upon centers by which they may be placed in any po

sition with respect to the tripod and each other. These limbs carry points at right angles to the plane of the instrument, which may be brought to coincide, in the first instance, with any three points on the original, and then transferred to the copy. After this first step two of these points must be set upon two points of

the drawing already copied, and the third made to coincide with a new point of the drawing, that is, one not yet copied : then, by placing the two first points on the corresponding points in the copy, the third point of the compasses will transfer the new point to the copy.

Another form of triangular compasses is represented in the annexed figure.

THE DRAWING-PEN.

This instrument is used for drawing straight lines. It

consists of two blades with steel points fixed to a handle; and they are so bent, that a sufficient cavity is left between them for the ink, when the ends of the steel points meet close together, or nearly so.. The blades are set with the points more or less open by means of a millheaded screw, so as to draw lines of any required fineness or thickness. One of the blades is framed with a joint, so that by taking out the screw the blades may be completely opened, and the points effectively cleaned after use. The ink is to be put between the blades by a common pen, and in using the pen it should be slightly inclined in the direction of the line to be drawn, and care should be taken that both points touch the paper; and the observations