extremity of the centre line or star marked on the instrument -coincides with a; and,

For very accurate measurements of angles, however, use a 6" circular protractor. The limit of error with this instrument may be only of a degree. Such angles as 30°, 45°, 60°, 90°, 120°, etc., may be set out directly with the set set-squares alone; and, as 45° — 30° 15°, and 45° + 30° 75°, it follows that lines including angles of 15° and 75° may likewise be drawn by a suitable adjustment of set-squares.

=

The mark() means feet, and (") inches: thus 2′ 3′′ means 2 feet 3 inches.

"; 47′′

=

410"; 1.25′′

= 1205

25/

=

1"

=

5 "/ 10

==

=

(=) means equals, or equal to. Fractions of an inch are given either in decimals or proper fractions: thus '5" = 11′′ ; 2·75′′ = 23′′: also, 125′′ =1′′ ; ·375′′ = 3′′; ·625′′ · g" and 875" =}". Points, lines, and plane figures are referred to in the text by letters or by numbers.

125" 1000

7. A set of Scales is now required for the examination. The most useful are those which are decimally subdivided. If both faces of a 12′′ ivory or boxwood "scale" be used, at least eight different scales, consisting of ", 1", ",", ", 1", 1" and 3" to the foot can be got on. A scale of centimetres with millimetre subdivisions is also useful (see fig. 49, Chap. V.).

4

DEFINITIONS.—A point is that which has position, but not magnitude. It is indicated thus: A or + A. A straight line (or briefly a line) has length without breadth. It is the shortest distance between two points. The extremities of a line are points; hence, when the positions of the extremities are known, the position of the line is likewise known.

AH

Fig. 4.

A definite line, AB, is indicated thus: An angle is the inclination of one line to another line. An angle is acute when less than a right angle, and obtuse when greater than a right angle. Angle CAB or angle A 40°, is indicated thus: When a straight line standing on another straight line makes the adjacent angles equal to one another, each of these angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

=

40

A

B

Fig. 5.

Parallel lines are such that, if produced ever so far both ways, do not meet.

The complement of an angle is that angle which it requires to complete a right angle: thus, the complement of 50° is 40°. The supplement of an angle is that angle which it requires to complete two right angles: thus, the supplement of 150° is 30°.

A triangle is a figure enclosed by three lines or sides. The sum of the angles of a triangle 180° thus, if two angles are 30° and 70° respectively, the third is 80°. When the three sides are equal, the triangle is equilateral; when unequal, it is scalene; when only two sides are equal, it is isosceles. A right-angled triangle has one of its angles a right angle, and the side opposite the right angle is called the hypotenuse. The altitude of a triangle is the perpendicular height of a corner (called the apex or vertex) above the side opposite (called the base).

Quadrilaterals are figures enclosed by four lines. The following are quadrilaterals :

(a) A parallelogram, which has its opposite sides parallel and equal.

(b) A square, which has all its sides equal and its angles right angles.

(c) A rectangle, or oblong, which is a parallelogram with all its angles right angles, but its sides not all equal.

(d) A rhombus, which has all its sides equal, but its angles not right angles.

(e) A rhomboid, which has its opposite sides equal, but its angles not right angles.

(ƒ) A trapezium, which is an irregular quadrilateral.

(g) A trapezoid, which is an irregular quadrilateral, but two of its sides are parallel.

A regular polygon is a figure having a number of equal sides and equal angles. If the angles or sides are unequal it is then said to be an irregular polygon.

The following are special names given to polygons, regular or irregular, according to the number of their sides or angles :

A diagonal of a figure is a line joining two opposite angular points.

The perimeter of a figure is the sum of all its sides.

The periphery is the boundary line of a circle or curved figure.

Adjacent means adjoining.

A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another. And this point is called the centre of the circle.

In Practical Geometry, the circumference of a circle is also called the circle.

A diameter of a circle is a line drawn through the centre, and terminated both ways by the circumference.

A radius of a circle is a line drawn from the centre to the circumference, and is equal to half the diameter.

An arc of a circle is any part of the circumference.

A line joining the extremities of an arc is called the chord of that arc.

A semi-circle is half a circle.

A segment of a circle is the figure contained by an arc and its chord.

A sector of a circle is the figure contained by two radii and the arc between them; when it is a quarter of a circle it is called a quadrant.

A tangent to a circle is a line which touches the latter in a point, but does not cut it.

The radius joining the centre of a circle and the point of contact of a tangent is at right angles to the tangent.

Concentric circles have the same centre.

When a point moves so as always to satisfy a given condition, or conditions, the path it traces out is called its locus under those conditions.

The angle which at the centre of a circle stands on an arc equal in length to the radius of the circle is called a Radian (see fig. 41).

CHAPTER II.

POINTS, STRAIGHT LINES, AND ANGLES.

THE student must draw all the figures in the book full size where dimensions are given; in other cases, as in Prob. 1, they should be increased to at least twice the size. The figures are usually reduced to save space in the book.

PROBLEM 1.

Given a line, AB, to bisect it, or divide it into two equal parts.

First copy the line AB twice the length given (fig. 6). With A as centre, and radius rather greater than half AB, describe an arc. With B as centre and an equal A radius, describe a second arc, intersecting the first one in points C and D. Join CD very accurately, cutting AB in P, the required point.

Fig. 6.

CD produced both ways indefinitely may be regarded as the locus of a moving point which is required to be

constantly equidistant from the extremities A and B. See the definition of locus.

PROBLEM 2.

Given two points, A and B, and a line, PQ, to determine a point c, in PQ, equidistant from A and B.

Join AB, and bisect it with the

C

Q

Fig. 7:

line CD, by Prob. 1, to meet PQ in c, the required point.