angles are to be obtained by the lines of chords, and the
polygons are likewise to be constructed by the lines of
polygons on the same instrument.

PRINCIPAL UNITS OF LINEAL MEASURES REDUCED TO ENGLISH
INCHES, FEET, YARDS, MILES, ETC.

Although I have already explained the general method of constructing plain and diagonal scales, page 6, I will not apologise for referring again to that subject, practice having convinced me that it very often may advantageously bear repetition.

A scale is a conventional object intended to represent artificially certain dimensions bearing some stated proportions to some given real magnitudes.

In Geometrical drawing the scales most generally used are the plain and the diagonal scales.

When a given space on a map or plan represents some known distance which has to be represented by a scale not requiring very minute subdivisions, a plain scale will suffice.

Let us suppose, for instance, that the space of one inch upon a map should represent a length of 10, 15, or 20 feet, yards or miles, &c. and not require any more minute

subdivision, we should construct a plain scale, because it is easy to subdivide the space of one inch into, or in the ratio of 10, 15, or 20 equal parts.

Should we, however, require to subdivide the feet into inches, the yards into feet, or the miles into furlongs, we should be compelled to construct a diagonal scale, because we would find it impossible, or very difficult, to obtain the scale by any other means.

PRACTICE.

Construct a scale in the ratio of 12 feet to 1 inch. Let 70 feet be required.

Then, 12 feet is to 1 inch as 70 feet is to 5.833 inches. Draw a line 5.83 inches long, divide it into seven equal parts, or primary divisions, each of which will represent 10 feet. Subdivide the left hand primary division into 10 equal parts (or single feet.) (Euc. 2. VI.)

Draw a thick second line at of an inch below, and parallel to the first, and number, as an example, Fig. 1, Pl. XIV. Print before the scale, "scale of," and at the further extremity, "feet," or whatever may be the denomination of the units intended to be represented by the scale.

As in this case the space of one inch represents 12 feet of real dimensions, the proportion between the scale and the original distance will evidently be the same as between one inch and as many inches as are contained in 12 feet, therefore 12 feet reduced to inches = 12 x 12 = 144 inches, which are represented by one inch, and consequently give the representative fraction of the scale

REPRESENTATIVE FRACTION.

A representative fraction, therefore, determines the proportion that exists between the length of the scale and the original dimension which it represents. In other words, it

E

shows the proportion between an inch, a foot, a yard, a mile, or any other denomination as represented on paper, with a real inch, foot, yard, or mile, &c. in real length on the ground. The representative fraction is usually placed over the scale. The scale is now completed, and all plain scales are constructed after this method. The secondary subdivisions varying only with the number of units represented by one primary division. Thus, if one of the primary divisions represents one foot to show inches, the secondary subdivision will be subdivided into 12 equal parts. If it represent fathoms, to show feet it will be subdivided into 6 parts. Furlongs to show chains, into 10 parts; miles into furlongs, into 8 parts; yards into feet, into 3 parts, &c.

Should we require a scale of 120 feet to 1 inch. It is evident that we could not divide one inch accurately into 120 parts by the method just explained. We must, therefore, construct a diagonal scale. Let us assume 70 feet then as in the former scale 120: 1:: 700: X = 5.833. Divide a line 5.83* into 7 equal parts, each representing 100 feet, and subdivide the first primary division into 10 equal parts, or secondary divisions, each representing 10 feet. Draw below this scale line .10 other lines parallel and equidistant from each other (any space will do, but one-tenth of an inch apart is usually found a convenient distance, the size of the scale, however, being the best guide to follow.) From the first secondary subdivision to the left of the lowest line draw a diagonal line to the upper intersection of the highest horizontal line with the left hand extremity of the scale, and from each of the other subdivisions of the base draw lines parallel to this, Fig. 2, Pl. XIV., and number the scale as in example.

*In practice it is seldom necessary to go beyond two or three places of decimals. Two places when the third decimal is less than 5; three places when it is more.