DEF. 24. A diagonal is the straight line joining the vertices of any angles of a polygon which have not a common arm.

DEF. 25. The perimeter of a rectilineal figure is the sum of its sides. DEF. 26. A quadrilateral is a polygon of four sides, a pentagon one of five sides, a hexagon one of six sides, and so on.

DEF. 27. A triangle is a figure contained by three straight lines. DEF. 28. Any side of a triangle may be called the base, and the opposite angular point is then called the vertex.

DEF. 29. An isosceles triangle is that which has two sides equal; the angle contained by those sides is called the vertical angle, the third side the base.

DEF. 30. A triangle which has one of its angles a right-angle is called a right-angled triangle. A triangle which has one of its angles an obtuse angle is called an obtuse-angled triangle. A triangle which has all its angles acute is called an acute-angled triangle.

DEF. 31. The side of a right-angled triangle which is opposite to the right-angle is called the hypotenuse.

DEF. 32. The perpendicular to a given straight line from a given point outside it is called the distance of the point from the straight line.

DEF. 33. Parallel straight lines are such as are in the same plane and being produced to any length both ways do not meet. DEF. 34. When a straight line intersects two other straight lines it makes with them eight angles, which have received special names in relation to the lines or to one another.

2/1 3/4

6/5

7/8

H

Thus in the figure 1, 2, 7, 8 are called exterior angles, and 3, 4, 5, 6 interior angles; again 4 and 6, 3 and 5, are called alternate angles; lastly, 1 and 5, 2 and 6, 3 and 7, 4 and 8, are called corresponding angles. DEF. 35. A parallelogram is a quadrilateral whose opposite sides are parallel.

DEF. 36. A trapezium (or trapezoid) is a quadrilateral that has only one pair of opposite sides parallel.

DEF. 37. A parallelogram, one of whose angles is a right angle, is called a rectangle.

DEF. 38. A rhombus is a parallelogram that has all its sides

equal.

DEE. 39. A square is a rectangle that has all its sides equal. DEF. 40. The orthogonal projection of one straight line on another straight line is the portion of the latter intercepted between perpendiculars let fall on it from the extremities of the former.

DEF. 41. 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. This point is called the centre of the circle.

DEF. 42. A radius of a circle is a straight line drawn from the centre to the circumference.

DEF. 43. A diameter of a circle is a straight line drawn through the centre and terminated both ways by the circumference. DEF. 44. If any and every point on a line, part of a line, or group

of lines (straight or curved), satisfies an assigned condition, and no other point does so, then that line, part of a line, or group of lines, is called the locus of the point satisfying that condition.

AXIOMS AND POSTULATES.

99

GEOMETRICAL AXIOMS.

AXIOM 1. Magnitudes that can be made to coincide are equal. AXIOM 2. Through two points there can be made to pass one, and only one, straight line; and this may be indefinitely prolonged either way.

Hence,

1

a. Any straight line may be made to fall on any other straight line with any given point on the one on any given point on the other;

B. Two straight lines which meet in one point cannot meet again, unless they coincide.

AXIOM 3. Through the same point there cannot be more than one straight line parallel to a given straight line.

POSTULATES OF CONSTRUCTION.

Let it be granted that

1. A straight line may be drawn from any one point to any other point.

2. A terminated straight line may be produced to any length in a straight line.

3. A circle may be drawn with any centre, with a radius equal to

any finite straight line.

BOOK II.

EQUALITY OF AREAS.

SECTION I.

THEOREMS.

DEF I. The altitude of a parallelogram with reference to a given side as base is the perpendicular distance between the base and the opposite side.

DEF 2. The altitude of a triangle with reference to a given side as base is the perpendicular distance between the base and the opposite vertex.

It follows from the General Axioms (d) and (e), as an extension of the Geometrical Axiom 1, that magnitudes which are either the sums or the differences of identically equal magnitudes are equal, although they may not be identically equal.

THEOR. I. Parallelograms on the same base and between the same parallels are equal.

Let ABCD, EBCF be two parallelograms on the same base BC, and between the same parallels AF, BC: