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In the pages that succeed, the following instruments are essential :

1. A ruler or straight-edge,

on which are marked inches divided into sixteenths, and on which also is a scale giving millimetres.

This is used for drawing straight lines; for making them of any required length; and for measuring straight lines that are drawn.

2. A pair of compasses,

one leg of which is furnished with a pencil.

This is used for describing circles; also, with the help of the ruler, for laying off required distances; and for measuring distances that are laid off.

3. A protractor.

This is used for constructing angles of any given number of degrees; and for measuring the number of degrees in any given angle. It may also be used for determining whether one angle is greater than, equal to, or less than another.

For the more rapid and more accurate construction of figures, the following instruments are also desirable:

4. A pair of dividers,

both the legs of which terminate in fine points. These more accurately than the compasses will enable the pupil to measure and to transfer distances.

5. A set-square.

The right angle has very frequently to be constructed, and its construction can be more rapidly effected with the set-square than with the protractor.

6. A bevel.

This enables us very rapidly to determine the equality or inequality of angles, and to construct an angle equal to another.

7. Parallel rulers.

While for drawing lines parallel to each other nothing more is essential than a ruler along which the setsquare is made to slide, or a ruler and an instrument for measuring angles, or a ruler and compasses, these methods become tedious from the frequency with which the construction has to be made. Parallel rulers make the con

struction rapidly and accurately.

Care should be taken to use a pencil with a hard fine point, so that lines drawn may be narrow and well defined.

Smooth paper will be found better than rough.

Points and the ends of lines should be marked by indentations made with a needle or with the sharp points of the dividers.

A piece of smooth, perfectly flat board, about a foot square, will be found useful as a drawing board.

In all cases the pupil should construct for himself the necessary figures, and not content himself with those in the book, which are merely intended as suggestions. It will be usually found desirable to make figures on a larger scale than those in the text.

The chapters on similar triangles may be taken up, if thought desirable, as soon as the pupil has obtained an acquaintance with parallel lines, and knows that the opposite sides and angles of parallelograms are equal. Prominence may then be given to Exercise 17, Chapter xxi., which suggests a demonstration of the 47th, Book I., Euclid.


Geometrical Elements.

A straight line:

It is evidently the shortest distance between its ends.

A broken line :

A curved line:

An angle:

The size of the angle does not depend on the lengths of the bounding lines AB and AC, but on the amount of divergence of these lines from one another. Thus the angle P is greater than the angle Q, and the angle R is less than the angle Q.

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It is usual to indicate an angle by using one letter, as the angle P, or by using three letters, as the angle BAC. In the latter case the letter at the angle itself is in the middle, and the other two letters lie on the arms of the angle.

If ABC be a straight line, and the angles DBA, DBC be equal, then each of them is called a right angle, and the lines DB and ABC are said to be perpendicular to each other.

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Evidently at the point B there are four right angles.

An angle which is less than a right angle, as BAC, is called an acute angle.

An angle which is greater than a right angle, as EDF, is called an obtuse angle.

A circle is the usual figure described on a flat surface by means of the compasses.

Note the parts called centre, radius, and circumference.

All radii of the same circle are equal, since the ends of the compass legs remain the same distance apart while the circle is being described.





A line through the centre and terminated both ways by the circumference is called a diameter, as CD.

The part of the circle on each side of a diameter is called a semicircle.

A part of the circumference, as AB, is called an arc of the circle. The straight line joining A and B is called a chord.

Any line drawn from a point without the circle and cutting it, is called a secant.

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