For drawing geometrical figures, the most essential instruments are the compasses and the ruler.

1. The Compasses. These are of various kinds, e.g., we have (1) a pair of dividers, having two steel points. These are chiefly used for measuring distances. (2) A pair of bow-pencil compasses, having one of its legs furnished with a holder for a pencil. These are used for describing circles. (3) A pair of bow-pen compasses, having one of its legs furnished with a mathematical pen. These are used for describing circles in ink.

2. The Ruler.—These may be of any length, though for the most part they are either 6 inches or a foot. They should have a bevelled edge, and should be divided into inches. They are used for drawing straight lines.

In addition to the foregoing, the student will also require

3. The Drawing-Board.—[Its construction is so well known, that any description of it is unnecessary.]

4. The T Square. This instrument consists of two straight

rulers fixed at right angles to each other, as shown in the foregoing diagram. It is used for drawing perpendicular and parallel lines, the cross piece, stock, or hilt, AB, being made to slide along the edge of the drawing-board, all straight lines drawn along the edge of the blade, CD, will be parallel to one another.

Note 1. A shifting bevel piece, EP, with clamping screw, is sometimes attached to the hilt of the square, which enables us to draw parallel lines having any given inclination to the sides of the drawing-board or to the base line of the drawing. Note 2. In the figure, GH represents a drawing-board.

5. The Set-Square.-- This is a triangular piece of wood ABC, having the edge BC at right angles to AB. It forms a cheap and convenient instrument for drawing perpendicular or parallel lines

on paper. Set-squares are of two kinds, e.g., we have (1) the set. square of 45° ; having a right-angle at B, and each of the angles at Á and C 45° ; (2) the set-square of 60° having a right-angle at E, an angle of 60° at D, and hence an angle of 30° at F. (Euc. 1. 32.)

6. The Parallel Ruler. - This is a very simple instrument for drawing parallel lines. It consists of two rulers fixed parallel to each other by means of two equal brass links which are fastened to

the rulers at equal distances by pivots. The edge, CD, of one ruler being placed along a straight line, a pencil mark drawn along the edge, AB, of the other ruler will trace a parallel straight line.

7. The Rolling Parallel Ruler. This instrument consists of a

ruler, AB, generally divided into inches and tenths ; near the

extremities are placed two rollers turning on an axis parallel to the edge AB, so that the ruler is capable of moving at right angles to the direction of the edge AB. By the aid of this instrument, any number of lines may be drawn parallel to a given line, and at any given distance from each other.

8. The Protractor.—This instrument is used for measuring angles, and for laying down angles on paper of any proposed magnitude. It consists of a brass semicircle, ACB, the circumference of which is divided into degrees. To lay down any proposed angle, say

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60 degrees, draw a line along the edge, AB; place a mark coincident with the centre, C, and another mark coincident with 60°, as figured on the brass circle, then a line drawn on the paper between these two points will give two straight lines inclined to each other at an angle of 60°.

9. Proportional Compasses.—This instrument is used for reducing or enlarging a figure in any required proportion. It consists

of two brass legs, AB and CD, terminating at both ends with fine steel points, E, F, G, H. The legs turn on the pivot 0, which may be adjusted so as to divide the length of the legs from point to point in any proportion. Now, whatever the opening of the compasses inay be, the distances GE, and HF, between the points will be in the same proportion to each other as the lengths OG and OH, into which the legs are divided by the pivot. For example, if OG be double OH, then GE will be double HF, and the legs so divided would enable us to enlarge a figure to double its size, or to reduce it to half its size.

10. Diagonal Scale. 11. Scale of Chords.

Note. For a description of the last two instruments, see pages 182 and 183 respectively.

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