former one in A. Join BA, AC. ABC is the triangle required.

B

When and how would this construction fail?

To construct an isosceles triangle, or an equilateral triangle.

7. (2) To construct a triangle, having given two sides and an angle opposite to one of them.

A

B

Draw a line BC of unlimited length, and at B make an angle CBA equal to the given angle; take BA equal to the side not opposite to the given angle. With centre A and radius

equal to the side opposite the given angle describe an arc. If this meet BC in C, ABC is a triangle satisfying the given conditions.

The problem however requires further discussion.

For, first. If the side given as opposite to B be less than the perpendicular from A to BC, the arc will not meet BC, and the triangle is impossible.

Secondly. If the given side be equal to the perpendicular from A to BC, the arc will touch BC in one point, and the triangle will have a right angle.

Thirdly. If the given side be greater than the perpendicular from A to BC, the arc will meet BC in two points, as C and D.

We must now further distinguish two cases.

First. If the given side be less than AB, then C and D will both lie on the same side of B, and two triangles ABC, ABD may be constructed, both of which indifferently satisfy the given conditions.

Secondly. If the given side be greater than AB, it must therefore be more remote from the perpendicular than AB. Hence C and D must lie on different sides of B. As before, two triangles may be constructed, but one alone satisfies the given conditions, for the other does not contain the angle CBA, but its supplement.

We have supposed CBA acute; discuss the construction when CBA is obtuse.

7. (3) To construct a triangle, having given two sides and the included angle.

4. To construct a triangle, having given two angles and a side.

8. To construct a parallelogram, knowing, 1. Two adjacent sides and a diagonal.

3. The two diagonals, and the angle between them.

4. The perimeter, a side, and any angle.

9. To trisect a right angle.

10. To construct a triangle, having given,

1. The base, an angle at the base, and the sum difference of the sides.

2. The base, the difference of the sides, and the difference of the angles at the base.

3. The base, the angle at the vertex, and the sum or difference of the sides. [7. 2].

4. Two sides and the line drawn from the vertex to the middle point of the base.

5. One side and two of the lines from the angles to the middle points of the opposite sides.

6. The three lines from the angles to the middle points of the opposite sides.

It may be assumed that these lines all pass through one point which cuts off from each a third part of its length,

11. To construct a square on a given line.

12. To construct a regular hexagon on a given line.

13. To construct a regular octagon on a given line.

BOOK II.

THE CIRCLE.

THE radius of a circle is a straight line drawn from the centre to the circumference.

Circles which have equal radii are equal.

Any straight line drawn from one point of the circumference to another is called a chord; as AB.

The part of the circumference cut off by the chord is called an arc; it is denoted either by two letters AB with a curved mark over them, or by three letters A CB, the first and last of which are at the extremities of the chord.

The figure ACB contained by the arc and chord is called a segment.

It will be seen that every chord divides the circumference into two arcs ACB, ADB; and the circle into two segments denoted by the same letters.

THEOREM I.

The diameter EOC, drawn perpendicular to a chord AB, bisects that chord.

Let O be the centre. Join OA, OB.

Then OA, OB being equal are equally distant from the perpendicular.

Therefore AD = DB.

D

Conversely, if EOC bisect AB it shall cut it at right angles.

COR. Similarly it may be shewn that EOC bisects all chords parallel to AB.

Thus corresponding to every point, as A, on the circumference on one side of any diameter, there is another point, as B, at an equal distance on the other side.

Hence, the circumference is symmetrical with regard to any diameter, and if one semi-circumference be turned about the diameter till it fall on the other, the two will coincide throughout, each point coinciding with its corresponding point.

THEOREM II.

The shortest line that can be drawn from a point A to meet the circumference of a given circle lies on the line AO, which joins A with the centre of the circle; so also does the longest.