With the centre A and radius equal to the chord of 60° describe the arc BC. Then, taking the chord of 47° from the scale, lay it off from B to C. Join AC, and BAC will be the required angle.

If an angle of more than 90° is required: first lay off 90°, and from the extremity of that arc lay off the remainder.

121. By the Table of Chords. The table of chords (page 97 of the tables) affords a much more accurate means of laying off angles.

Take for a radius the distance 10 from any scale of equal parts, to be described hereafter, and describe the arc BC, (Fig. 38.) Then, finding the chord of the required angle by the table, multiply it by 10, and, taking the product from the same scale, lay it off from B to C as before. Join AC, and the thing is done.

If the angle is much over 60° it is best to lay off the 60° first. This is done by using the radius as a chord. The remainder can then be laid off from the extremity of the arc of 60° thus determined.

122. Distances. Every line on a draft should be drawn of such a length as correctly to represent the distance of the points connected, in due relation to the other parts of the drawing. In perspective drawing, the parts are delineated so as to present to the eye the same relations that those of the natural object do when viewed from a particular point. To produce this effect the figure must be distorted. Right angles are represented as right, obtuse, or acute, according to the position of the lines; and the lengths of lines are proportionally increased or diminished according to their position. In drafting, on the contrary, every part must be represented as it is. The angles should be of the same magnitude as they are in reality, and the lines should bear to each other the exact ratio that those which they are intended to represent do. The plat should, in fact, be a miniature representation of the figure.

123. Drawing to a Scale. In order that the due pro

portion should exist in the parts of the figure, every line should be made some definite part of the length of that which it is intended to represent. This is called drawing to a scale. The scale to be used depends on the size of the map or draft that is required, and the purposes for which it is to be used. Carpenters often use the scale of an inch to a foot the lines will then be the twelfth part of their real length. In plats of surveys, or maps of larger tracts of country, a greater diminution is necessary. The scale should, however, in all cases, be adapted to the purpose intended and to the number of objects to be represented. Where the purpose is merely to give a correct representation of the plat, without filling up the details, the main object will be to make the map of a convenient size; but where many details are to be represented the scale should be proportionally larger.

Thus, for example, in delineating a harbor where there are few obstructions to navigation, a map on a small scale may be drawn; but where the rocks and shoals are numerous, the scale should be so large that every part may be perfectly distinct.

The scales on which the drawing is made should always be mentioned on the map. They may be expressed by naming the lengths which are used as equivalents, thus,— "Scale, 10 feet to an inch, 1 mile to an inch, 3 chains to a foot;" or better fractionally, thus,-1: 100, 1:250, 1: 10,000, &c.

124. Surveys of Farms. Where the farm is small, 1 chain* to an inch, (1:792,) or 2 chains to the inch, (1:1584,) may be used; but if the tract be large, as this would make a plat of a very inconvenient size, a smaller scale must be adopted. When, however, any calculations are to be based on measurements taken from the plat, a smaller scale than 3 chains to the inch (1: 2376) should not be employed.

* The surveyor's chain-commonly called Gunter's Chain-is 4 poles, or 66 feet, in length, and is divided into one hundred links, each of which is therefore .66 feet, or 7.92 inches in length.

125. Scales. Scales are generally made of ivory or boxwood, having a feather-edge, on which the divisions are marked. The distances can then be laid off by placing the ruler on the line, and pricking the paper or marking it with a fine pointed pencil; or the length of a line may be read off without any difficulty. Boxwood scales, if the wood is clear from knots, are to be preferred to ivory. They are less liable to warp, and suffer less expansion and contraction from changes in the hygrometric condition of the atmosphere.

Paper scales are often employed. These may be procured with divisions to suit almost any purpose, or the surveyor may make them himself. Take a piece of drawingpaper, and cut a slip about an inch in width; draw a line along its middle, and divide it as desired, either into inches or tenths of a foot. The end division should be subdivided into ten parts, and perpendiculars drawn through all the divisions, as represented in the figure, (Fig. 39.) Each of these parts may then represent a chain, ten chains, &c.

Fig. 39.

Paper scales, being subject to nearly the same expansion and contraction as the paper on which the map is drawn, are, on this account, preferable to those made of wood or ivory. They cannot, however, be divided with the same accuracy.

126. The plane diagonal scale (Fig. 40) consists of eleven

lines drawn parallel and equidistant. These are crossed at right angles by lines 1, 2, 3, drawn usually at intervals of half an inch. The first division, on the upper and lower lines, is subdivided into ten equal parts: diagonal lines are then drawn, as in the figure, from each division of the top to the next on the bottom,-the first, from A to the first division on the bottom line; the second, from the first on the top to the second on the bottom; and so on.

It is evident that, whatever distance the primary division from A to 1, or 1 to 2, &c. represents, the parts of the line AB will represent tenth parts of that distance. If then it were required to take off the distance of 47 feet on a scale of half an inch to 10 feet, the compasses should be extended from E to F.

The diagonal lines serve to subdivide each of the smaller divisions into tenths, thus:-The first diagonal, extending from A to the first division on the bottom line and crossing ten equal spaces, will have advanced of one of those. divisions at the first intermediate line, at the second, at the third, and so on. All the other diagonals will advance in the same manner.

If then the distance were taken from the line AC along the horizontal line marked 6 to the fourth diagonal, the distance would be .46, the division AB being a unit, or 4.6 if AB were 10. To take off, then, 39.8 feet on a scale of half an inch to 10 feet, the compasses should be extended to the points marked by the arrow heads G and H: similarly, 46.7, on the same scale, would extend from one of the arrow heads on the seventh line to the other.

In using the diagonal scale the primary divisions should always be made to represent 1, 10, 100, or 1000. When any other scale is required,―say 1: 300,-it is better to divide or multiply all the distances and then take off the results. Thus, if 83.7 were required to be taken off on a scale of

inch to 30 feet, first divide 83.7 by 3, giving 27.9, and then take off the quotient on a scale of inch to 10 feet. The other lines must all be reduced in the same proportion. The above method requires less calculation, and involves

less liability to error, than that of determining the value of each division on the reduced scale.

127. Proportional Scale. On most of the rulers furnished with cases of instruments there is another set of scales, divided as below, (Fig. 41.)

The figures on the left express the number of divisions to the inch. To lay off 97 feet on a scale of 40 feet to the inch, the compasses would be extended between the arrowheads on the line 40. Scales of this kind are very convenient in altering the size of a drawing. Suppose, for example, it is desired to reduce a drawing in the ratio of 5 to 3: the lengths of the lines should be determined on the scale marked 30, and the same number of. divisions on the scale 50 will give a line of the desired length.

128. Vernier Scale. Make a scale (Fig. 42) with inches divided into tenths, and mark the end of the first inch 0, of the second 100, and so on. From the zero point, backwards, lay off a space equal to eleven tenths of an inch, and divide it into ten equal parts, numbering the parts backwards, as represented in the figure. This smaller scale

is a vernier.

Now, since the ten divisions of the vernier

are equal to eleven of the scale, each of the vernier divisions