8. When one line stands upon another, so as not to lean more to one side than to another, both the angles which it makes with the other are called right angles, as the angles ABC and ABD, Fig. 3, and all right-angles are equal to each other, being all equal to 90°; and the line AB is said to be perpendicular to CD.

Beginners are very apt to confound the terms perpendicular, and plumb or vertical line. A line is vertical when it is at right-angles to the plane of the horizon, or level surface of the earth, or to the surface of water, which is always level. The sides of a house are vertical. But a line may be perpendicular to another, whether it stands upright or inclines to the ground, or even if it lies flat upon it, provided only that it makes the two angles formed by meeting with the other line equal to each other; as for instance, if the angles ABC and ABD be equal, the line AB is perpendicular to CD, whatever may be its position in other respects.

9. When one line, BE (Fig. 3,) stands upon another, CD, so as to incline, the angle EBC, which is greater than a right-angle, is called an obtuse angle; and that which is less than a right-angle, is called an acute angle, as the angle EBD.

10. Two angles which have one leg in common, as the angles ABC, and ABE, are called contiguous angles, or adjoining angles; those which are produced by the crossing of two lines, as the angles EBD and CBF, formed by CD and EF, crossing each other, are called opposite or vertical angles.

11. A figure is a bounded space, and is either a surface or a solid. 12. A superficies, or surface, has length and breadth only. The extremities of a superficies are lines.

13. A plane, or plane surface, is that which is every where perfectly flat and even, or which will touch every part of a straight line, in whatever direction it may be laid upon it. The top of a marble slab, for instance, is an example of this, which a strait edge will touch in every point, so that you cannot see light any where between. 14. A curved surface is that which will not coincide with a straight line in any part. Curved surfaces may be either convex or concave. 15. A convex surface is when the surface rises up in the middle, as, for instance, a part of the outside of a globe.

16. A concave surface is when it sinks in the middle, or is hollow, and is the contrary to convex.

A surface may be bounded either by straight lines, curved lines, or

both these.

17. Every surface, bounded by straight lines only, is called a polygon. If the sides are all equal, it is called a regular polygon. If they are unequal, it is called an irregular polygon. Every polygon, whether equal or unequal, has the same number of sides as angles, and they are denominated sometimes according to the number of sides, and sometimes from the number of angles they contain. Thus a figure of three sides is called a triangle, and a figure of four sides a quadrangle.

A pentagon is a polygon of five sides.,

A hexagon has six sides.
A heptagon seven sides.
An octagon eight sides.
A nonagon nine sides.
A decagon ten sides.

An undecagon eleven sides.

A duodecagon twelve sides.

When they have a greater number of sides, it is usual to call them polygons of 13 sides, of 14 sides, and so on.

Triangles are of different kinds, according to the lengths of their sides. 18. An equilateral triangle has all its sides equal, as ABC, Fig. 4. 19. An isosceles triangle has two equal sides, as DEF, Fig. 5. 20. A scalene triangle has all its sides unequal, as GHI, Fig. 6. Triangles are also denominated according to the angles they contain. 21. A right-angled triangle is one that has in it a right angle, as ABC, Fig. 7.

22. A triangle cannot have more than one right-angle. The side opposite to the right-angle B, as AC, is called the hypothenuse, and is always the longest side.

23. An obtuse-angled triangle has one obtuse-angle, as Fig. 8. 24. An acute-angled triangle has all its angles acute, as Fig. 4. 25. An isosceles, or a scalene triangle, may be either rightangled, obtuse, or acute,

26. Any side of a triangle is said to subtend the angle opposite to it: thus AB (Fig. 7), subtends the angle ACB.

27. If the side of a triangle be drawn out beyond the figure, as AD (Fig. 8), the angle A, or CAB, is called an internal angle, and the angle CAD, or that without the figure, an external angle. 28. A quadrangle is also called a quadrilateral figure. They are of various denominations, as their sides are equal or unequal, or as all their angles are right-angles or not.

29. Every four-sided figure whose opposite sides are parallel, is called a parallelogram. Provided that the sides opposite to each other be parallel, it is immaterial whether the angles are right or not. Fig. 9, 10, 11, and 12, are all parallelograms.

30. When the angles of a parallelogram are all right-angles, it is called a rectangular parallelogram or a rectangle, as Fig. 11 and 12. 31. A rectangle may have all its sides equal, or only the opposite sides equal. When all its sides are equal, it is called a square, as Fig. 12.

32. When the opposite sides are parallel, and all the sides equal to each other, but the angles not right-angles, the parallelogram is called a rhombus, as Fig. 10.

33. A parallelogram having all its angles oblique, and only its opposite equal, is called a rhomboid, as Fig. 9.

34. When a quadrilateral or four-sided figure has none of its sides parallel, it is called a trapezium, as Fig. 13; consequently every quadrangle, or quadrilateral which is not a parallelogram, is a trapezium.

35. A trapezoid has only one pair of its sides parallel, as Fig. 14. 36. A diagonal is a right line drawn between any two angles that are opposite in a polygon, as IK, Fig. 15. In parallelograms the diagonal is sometimes called the diameter, because it passes through the centre of the figure.

37. Complements of a parallelogram. If any point, as E (Fig. 15), be taken in the diagonal of a parallelogram, and through that point two lines are drawn parallel to the sides, as AB, CD, it will be divided into four parallelograms, DD, L, F, GG. The two divisions, L, F, through which the diameter does not pass, are called the complements.

38. Base of a figure is the side on which it is supposed to stand erect, as AB, and CD, Fig. 16.

39. Altitude of a figure is its perpendicular height from the base to the highest part, as EF, Fig. 16.

40. Area of a plane figure, or other surface, means the quantity of space contained within its boundaries, expressed in square feet, yards, or any other superficial measure.

41. Similar figures are such as have the same angles, and whose sides are in the same proportion, as Fig. 17.

42. Equal figures are such as have the same area or contents.

43. A circle is a plane figure, bounded by a curve line returning into itself, called its circumference, ABCD (Fig. 18), every where equally distant from a point E within the circle, which is called the

centre.

44. The radius of a circle is a straight line drawn from the centre to the circumference, as EF (Fig. 18). The radius is the opening of the compass when a circle is described; and consequently all the radii of a circle must be equal to each other.

45. A diameter of a circle is a straight line drawn from one side of the circumference to the other through the centre, as CB (Fig. 18). Every diameter divides the circle into two equal parts.

46. A segment of a circle is a part of a circle cut off by a straight line drawn across it. This straight line is called the chord. A segment may be either equal to, greater, or less than a semi-circle, which is a segment formed by the diameter of the circle, as CEB, and is equal to half the circle.

47. A tangent is a straight line, drawn so as just to touch a circle without cutting it, as GH (Fig. 18). The point A, where it touches the circle, is called the point of contact. And a tangent cannot touch a circle in more points than one.

48. A sector of a circle is a space comprehended between two radii and an arc, as BIK (Fig. 19).

49. The circumference of every circle, whether great or small, is supposed to be divided into 360 equal parts, called degrees; and every degree into 60 parts, called minutes; and every minute into 60 seconds. To measure the inclination of lines to each other, or angles, a circle is described round the angular point, as a centre, as IK, Fig. 19; and according to the number of degrees, minutes, and