a triangle to the base bisect either the base or the vertical angle, the triangle will be isosceles.

For if it bisect the base, let the part of the triangle to the left of c C be folded over that part to the right, since the angles at C are equal, the part of the base to the left of C will fall upon the part to the right; and since these parts are equal, the vertex of the angle b will fall upon the vertex of the angle a, and the side A will coincide with the side B, and will therefore be equal to it.

If the perpendicular to the base bisect the angle c; then doubling over the part to the left of c C upon the part to the right, the side A will fall upon the side B, because the angle at C is bisected by the perpendicular, and the part of the base to the left of C will fall upon the part of the base to the right of C, because the angles at C are equal; therefore the vertex of the angle b will fall upon the vertex of the angle a, and the side A will fall upon the side B, and will be equal to it.

(67.) These properties furnish the means of solving the problem to bisect an angle.

If c be the angle to be bisected, take equal parts A and B upon its sides, and draw a base C, so as to form an isosceles triangle; from the vertex of the angle c draw a line at right angles to this base, which may be done by a square; this line will, by what has already been proved, bisect the given angle.

(68.) The same principles furnish a solution of the problem to bisect a given straight line.

If the base C (fig. 33.) be the proposed straight line which is to be bisected, draw at its extremities any two equal acute angles, which may be done by the pattern of one acute angle, the sides of these acute angles will form an isosceles triangle (63); and if the perpendicular be drawn from the vertex c to the base of this isosceles triangle, that perpendicular will bisect the base (65.).

(69.) If the vertical angle, c, of an isosceles triangle were right, the base angles, a, b, would be each 45°, since

(70.) The angles at the base of an isosceles triangle must always be acute, since they are equal, and since more than one right or obtuse angle cannot exist in the same triangle (58.).

(71.) A triangle having three equal sides is called an equilateral triangle (fig. 34.).

fig. 34.

An equilateral triangle may be regarded as an isosceles triangle, any one of the three sides being taken as base; and as it has been proved that the angles at the base of an isosceles triangle are equal (63.), it follows that the three angles of an equilateral triangle are equal.

(72.) Also, if the three angles of any triangle are equal, the three sides must be equal, because it will be an isosceles triangle, according to what has already been proved, in whatever position it may be placed (63.).

Thus an equilateral triangle is equiangular, and an equiangular triangle is equilateral.

Since the three angles are together equal to 180°, each angle of an equilateral triangle must be 60°, or two thirds of a right angle.

The equilateral triangle presents the first example in geometry of a symmetrical figure.

Since a perpendicular from the vertex of an isosceles triangle upon the base divides it symmetrically (64.), an equilateral triangle will be divided symmetrically by a perpendicular from the vertex of any angle on the opposite side.

The isosceles triangle is extensively used in architecture and in carpentry. It is the form usually given to the roofs of buildings, and to the pediment which surmounts and adorns porticos, doors, and windows. In the Greek architecture, the character of the isosceles is obtuse; in the Gothic, acute.

CHAP. V.

OF CIRCLES.

D fig. 35.

B

(73.) IF a straight line have one of its extremities placed at a fixed point, C (fig. 35.), and be made to revolve round that point as a pivot, the other extremity will trace a line, every point of which will be equally distant from the point A C. Such a line is called a circle, the point C is called its centre, and the line CB its radius; the space inclosed within the curve is called the area of the circle, and the curved line itself is called the circumference of the circle.

E

(74.) A straight line extending across the circle, through its centre, and terminated in its circumference, is called a diameter.

A diameter consists evidently of two radii placed in the same straight line, and it is therefore equal to twice the radius of the circle; all diameters are therefore equal to each other.

The art of turning consists in the production of this figure by mechanical means. The substance on which the circular form is required to be conferred is placed in a machine called a lathe, which gives it a motion of rotation round a certain point as a centre; the edge or point of a cutting tool is placed at a distance from this centre, equal to the radius of the circle which it is desired to form; as the substance revolves, the edge or point removes every part of it which is more distant from the centre than the proposed radius, and consequently the

(75.) The circle is a perfectly symmetrical figure; for if it be made to revolve round its own centre no change whatever will take place consequent on the change of position of the parts: every part of its circumference being at the same distance from the centre, each point as it revolves takes the place of the preceding point, and no new portion of space is either vacated or occupied during this motion. The circle is unique in this property, which is possessed by no other figure what

ever.

It is in virtue of this property that the axles of wheels, shafts, and other solids which are required to revolve within a hollow mould or casing of their own form, must be circular. If they were of any other form, when placed in the mould or casing they would be incapable of revolving without carrying the mould or casing round with them.

Wheels, which are intended to maintain a carriage supported by them always at the same height above the road on which they roll, must necessarily be circles, with the axle of the wheel in their centre. The distance of the centre of the axle from the road will be equal to the distance of the centre of the wheel from its edge. the circle, this distance is always the same, and it is the only figure which has a point within it possessing this property.

In

(76.) The instruments by which circles are most commonly described are called compasses, and consist of two straight and equal legs connected together at one end by a joint, on which they are capable of moving, and terminating at the other ends in points, one of which carries a pen or pencil; the point of one leg is placed at the centre of the circle which it is intended to describe, while the other leg, carrying the pen or pencil, is made to revolve round, pressing the pen or pencil on the paper intended to receive the trace of the circumference.

When it is required to describe a circle with a radius too great for the space of the compasses, it may be done by attaching a piece of string with a pin to the

proposed centre, and looping into the string a pen or pencil at the proper distance for the required circle.

(77.) An instrument called a beam compass is also intended for describing circles of greater radius than those to which ordinary compasses can be conveniently applied. The beam compass consists of a straight bar A B (fig. 36.) usually divided into inches and parts of fig. 36.

A

c

S

B

P

an inch. At the commencement of the divisions there is a steel point C fixed projecting from the lower face of the bar. This point is intended to mark the centre of the circle to be described. A brass slider S is placed upon the bar, furnished with a clamping screw to fix its position at any required distance from the point C, which slider carries a point or pencil P, projecting downwards from the lower side of the slider.

In the application of the instrument to describe circles, the slider is moved along the bar until the distance of the describing point P, from the central point C, shall be equal to the radius of the required circle. The sliding piece is then fixed in its position by the clamping screw, and the central point C being placed at the centre of the proposed circle, the bar is moved round, the describing point P being pressed upon the paper so as to leave the trace of the circumference of the required circle.

(78.) If two circles have equal radii they will be equal in every other respect; for if the centre of the one be imagined to be placed on the centre of the other, the circumference of the one must coincide in every point with the circumference of the other, since every part of the circumference of each will be at the same distance from their common centre.

(79.) If two circles with different radii, be drawn round the same centre, every part of the circumference of one will be at the same distance from the circum

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