him the value of previously trying his pen on a piece of

waste paper.

He should also practise the imitation of printed letters. Plain Roman capital and small letters, and Italic capital and small letters are those generally used in military drawing. Any well printed book will furnish him with the most useful and best examples.

Besides different kinds of compasses, ruling pens, &c., good cases of instruments are usually provided with a protractor and a sector. Military boxes contain also a

set of Marquois scales.

PROTRACTOR.

A protractor sometimes consists of a horn or brass semicircle, divided into 180 equal parts, named degrees, and is used to determine the value of angles.

The best protractors, however, are made either of boxwood or ivory, and are in the shape of parallelograms, 6 inches long by 1.5 broad (vide Fig. 1, Plate I.); they contain, besides their graduated edge,

Two scales of chords of different sizes, the smallest marked C and the largest Cho, and whose purposes are the same as those of the graduated edge of the protractor, viz., to measure angles.

3

A scale of inches, I, I, I, 1, 1, 4, 4, of an inch duodecimally divided.

A series of plane scales divided in the ratios of 20, 25, 30, 35, 40, 45, 50, 55 and 60 units to one inch.

Diagonal scales divided in the ratios of 200 and 400 units to one inch.

A plotting scale, either divided at the rate of 40 units to one inch, or, in military protractors, at the rate of one mile to 4 inches, or 440 yards to one inch.

N.B.-The Student is expected to follow, with the help of his own instruments, the directions here given.

To construct an angle by the protractor. Draw a base line, BC, and determine on it the point at which the angle is to be constructed; place the centre A of the protractor on the given point, so that its lower edge coincides with the given base; determine the required angle with a finely pointed pencil, and through that obtained point draw a line from the original one.

PRACTICE.

1. On a base AB, two inches long, construct a triangle, so that the angle ABC equals 65°, and the angle BAC equals 47°. Vide Fig. 2. Place the protractor at the left extremity A of the given line, and determine the angle 47° inclined towards the right; remove the protractor to the right extremity B, and determine likewise the angle B, 65°, inclined towards the left.

N.B.-Remember that lines at angles less than 90° incline towards each other.

2. Construct a triangle ABC whose base AB = 1.7 inches, whose angle ABC 42°, and whose angle BAC = 57°.

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3. On a base AB, 1.5 inches long, construct a triangle, and let the angle BAC 105°, and the angle ABC = 37°. If with any radius we describe a circle, we shall find that the radius used as a chord will be contained exactly six times within the circumference; as a circumference is divided into 360°, it follows that the radius equals the chord of of the area, or 60°. The construction of the scales of chords, and of most sectoral lines, depend on that fact.

To construct a scale of chords.

Draw two lines, AB, BC, of any length, and at right angles to each other; from B as a centre describe with any radius the arc AC: respectively from A and C as centres with radius AB cut AC in D and E.

The triangle ABD will equal of 90°, or 30°.
The triangle ABE will equal, or 60°.

Trisect (by trials) each of these three sectors, and mark them 1, 2, 3, 4, 5, 6, 7, 8 and 9. The quadrand ADEC will be divided into 9 equal parts, each of which will be worth 10 degrees.

Divide therefore each of these nine parts into 10 subdivisions, to show single degrees (for the present purpose this may be omitted).

Draw the chord AC, and from A as a centre transfer the divided distance A1, A2, and A8 on the circumference (Fig. 3), to the chord AC, which number 10, 20, 30-80, &c.

Each of these divisions will equal 10 degrees, and the line AC will become a scale of chords of 90°.

To determine any angle, say 40°, by the scale of chords (see Fig. 4).

Draw any line long enough to contain the length A-60 taken on the scale of chords (Fig. 3), and with that length as a radius describe the arc BC; take in the compass the length A 40° (the value of the angle required) and adapt it from B to D (Fig. 4); the distance BD is the chord of the angle of 40o.

To obtain any other angle, say 70°. (Fig. 4.)

On the same arc, or on one similarly obtained, with the same radius, adapt the length A-70 obtained on the same scale of chords, from B to G.

Angles as great as 90° can be obtained in a similar

manner.

When they exceed 90° we must repeat the operation until the value of the angle is obtained: thus, let an angle of 100° be required. Determine the angle of 50° BE and repeat it on EF: BF will be the chord of an angle of 100°. (Fig. 4.)

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