thus:-Let it be required to reduce the lines of a figure to one fifth, for example; take with the ordinary compasses the length 1, and open the angle so that the distance of separation of the two arms at the divisions marked 5, is exactly equal to the interval between the points of the compasses. The instrument is now adjusted so that if any length be taken upon one of the

be taken as chords, corresponding radii.

40

sides of the angle, starting from

the apex, one fifth of the length I will be the distance from the end of this line to the corresponding point. Suppose for example, 46 is the length of the line, the interval between the two divisions marked 4'6 is the fifth part of the line required. This instrument can also be arranged so as to make a drawing five times as great every way as the model, only the lines of the model must and those of the copy as the

188. A second method, in some respects more convenient, is obtained from the principle demonstrated in § 174. The principle is applied by means of an instrument which we proceed to describe.

The proportional compass.

This instrument has some resemblance to the calibers. It has been shown that in the latter instrument, the distance betweeen the points at one extremity is always equal to that between those at the other. In the fourpointed compass these distances are not necessarily equal, but their ratio is constant. The instrument in its simplest form consists of two straight branches of equal length, which cut one another in the point O

(fig. 160), so that when the instrument is closed their extremities coincide; in other words

OA=0B, Oα= =0b Suppose OA be the double of Oa, AB will be the double of a b (§ 174), and this for every degree of opening of the angle A OB. If the distance A B (fig. 160) be taken between the long arms, we shall have between the small arms a distince, a b, equal to one half of A B. If, without changing the length of the arms, we can so arrange the instrument that a b may have any ratio whatever to A B, for example, 1 : 3, these compasses will serve for all reductions. The following ar- B rangement is adopted for this pur

a

Fig. 160.

b

A

pose. The instrument consists of two pieces of brass with a groove in each traversed by an axis round which they can turn. A screw-stud enables us to fix the instrument when properly adjusted. When the compasses are closed, a notch made in one of the arms exactly fits a catch in the other, and the two arms coincide when the closing is complete. By the help of a button, we can displace the axis and move it in the double groove. A scale is engraved upon the upper arm with the numbers,,, to b. Suppose we wish to arrange the instrument so as to reduce lines to one-fourth, we close it and bring the axis to the division marked, we then fasten the stud, and the instrument is arranged. It is evident that it can be regulated so as to make a drawing four times the size of the given drawing..

If the ratio of the size of the two drawings is not given in numbers, or if the numbers given be not engraved upon the instrument, the compass is arranged

I

by drawing a straight line a b which will represent a line AB of the model, and by bringing the button after several trials to such a position that when the line A B is taken between the two large points, the distance between the small points may equal a b.

189. A third method of reduction is furnished by the principle in § 180. Having drawn any two parallel straight lines, and measured off upon one of them a length A B, equal to a line of the model, and upon the other a length a b, equal to the length we wish to give A B in the copy, draw the lines A a, Bb, and produce them to the point O. If then A C be taken equal to a line in the model, by joining C O we shall obtain ac, the corresponding line of the copy. The application is facilitated by previously drawing from the point O a great number of lines in close proximity.

a

The pantograph.

190. The fourth method we shall describe is that afforded by the pantograph. By means of this instrument a reduced or increased copy of a design is obtained by simply following the outline with a point. It is composed of four rulers O A, AM, Bm, am (fig. 161) jointed at the points AB, a m. The figure BA am is always a parallelogram, and the three points, OM m, are always in the same straight line.

B

M

Q

Fig. 161.

It is evident since a m is parallel to A M, that OA: AM=0 a: a m, and since these lines are al

ways of the same length, this proportion holds good for every position of the instrument. Om M is always a straight line, and to A O, therefore always Om: OM

Again, since Bm is parallel Oa: O A.

But the ratio Oa: OA is invariable, for the terms are of constant length, and therefore the ratio Om: OM is invariable.

On this property the use of the instrument depends. Let the point O be fixed; at M attach a tracing-point to follow the outline of the figure P M Q, and at ma pencil; while the point M follows the line P M Q, the pencil at m will trace a line pmq similar to it. By varying the distances A a and A B, always preserving however the above conditions, we may make the ratio Om: OM any value we please less than unity.

There are several parts necessary to prepare the instrument for use. The centre O (fig. 162) is formed by the extremity of a clasp which will glide along the ruler O A, and may be fixed at any point by means of a screw. The support S of the clasp is fixed to the board by three pins. The four rulers are of constant length, but the tracing-point is borne by a clasp which may be fixed at any point of the ruler am by a screw. By this arrangement we can vary at will the ratios

B

Fig. 162.

of the distances a m and A M, and at the same time satisfy the condition that the three points, O, m, and M, shall be in the same straight line. The instrument is

supported on three small wheels or castors, , r, r. The lengths of the rulers usually vary from eighteen inches to a yard, according to the size of the designs to be reduced.

To measure the distance of an inaccessible object. 191. By means of the foregoing principles of proportion we may measure the heights and distances of inaccessible objects.

1st Method. Let AI be the distance required, I being the inaccessible object (fig. 163). Take a point B in the line AI, and any other point C, and fix a staff at each. Measure AC and BC and take these distances on the lines produced, thus obtaining the points, A' and B'. Find now a point I' in the line A'B' and also in the line CI. Then it is easily demonstrated that the triangle A'CI=ACI and A'I AI.