9. When a line is not stated to be crooked, or curved, it is under

stood to be straight.

10. Parallel Lines are those which being in the same plane, will not meet, though produced to any distance.

11. Oblique Lines are those which will meet if produced.

12. A Line is a Tangent to a circle, or other curve, when the line touches the curve without cutting it, when both are produced.

13. An Angle is the inclination of two lines which diverge from the same point.

14. A Perpendicular is a line meeting another line

so that the angles on each side of it are equal.

15. A Right Angle is the angle formed by a perpendicular with the line which it meets.

16. An Oblique Angle is one which is formed by the meeting of two oblique lines.

17. An Acute Angle is less, and an

Obtuse Angle is greater than a right

angle.

18. A Plane Surface, or a Plane, is that with which a right line may every way coincide.

19. A Curve Surface is that with which a right line will not every way coincide.

20. Plane Figures are bounded either by right lines or curves, and they have as many sides as they have angles.

21. A Triangle is a figure of three sides.

22. An Equilateral Triangle is one whose sides are all equal to each other.

23. An Isosceles Triangle has only two of its sides equal.

24. A Scalene Triangle is one whose sides are all unequal.

25. A Right Angled Triangle has one of its angles a right angle.

26. An Oblique Angled Triangle is one whose angles are all oblique.

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27. An Obtuse Angled Triangle has one obtuse angle.

28. An Acute Angled Triangle has all its angles

acute.

29. A figure of four sides is called a Quadrangle, or a Quadrilateral.

30. A Parallelogram is a quadrilateral having each pair of opposite sides parallel.

31. A Rectangle is a parallelogram having one of its angles a right angle; and it is said to be contained by any two of its adjoining sides.

32. A Square is a rectangle having all its sides equal.

33. A Rhomboid is a parallelogram, one of whose angles is oblique.

34. A Rhombus is a rhomboid whose sides are all equal.

35. A Trapezium is a quadrilateral which has not its opposite sides parallel.

36. A Trapezoid is a quadrilateral having only one pair of opposite sides parallel.

37. A Diagonal is a line joining the opposite angular points of a quadrilateral.

38. Plane figures of more than four sides receive the general denomination of Polygons.

39. A Polygon of five sides is called a Pentagon; one of six sides, a Hexagon; one of seven sides a Heptagon; one of eight sides an Octagon, &c.

40. A Regular Polygon is one which has all its sides equal, and also all its angles equal.

41. An Irregular Polygon is one which has not all its sides and all its angles equal.

42. A Circle is a plane figure bounded by a curve line, every part of which is equally distant from a certain

point within the figure, called the centre.

43. The curve line which bounds the circle is called the Circumference. Note, the Circumference is sometimes called the Circle.

44. The Radius of a circle is a line drawn from the centre to the circumference.

45. The Diameter of a circle is a line drawn through the centre and terminating on both sides at the circumference.

46. An Arc of a circle is any part of the circumference.

47. A Chord is the straight line joining the extremities of an arc.

48. A Segment is a part of a circle bounded by an arc

and its chord.

49. A Semicircle is a segment cut off by a diameter.

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50. A Sector is a part of a circle bounded by two radii and their intercepted arc.

51. A Quadrant is a sector whose bounding radii are perpendicular to each other.

52. In a right angled triangle, the side opposite the right angle is called the Hypothenuse, and the side including the right angle are called the Base and Perpendicular, and sometimes the legs of the triangle.

53. When an angle is denoted by three letters, one of them is placed at the angular point, and the other two on the lines which include the angle; and that which stands at the angular point is read in the middle, as A B C, or CBA.

54. An angle in a segment of a circle is contained by two lines drawn from any point in the arc of the segment to the extremities of the same arc.

B

D

55. An angle on a segment, or on an arc, is contained by two lines drawn from the extremities of the arc to any point in the opposite part of the circumference.

56. An angle at the circumference of a circle is one whose angular point is in the circumference; and an angle at the centre, is one whose angular point is at the centre.

57. A rectilineal figure is inscribed in a circle, or the circle circumscribes the rectilineal figure, when all the angular points of the figure are in the circumference of the circle.

58. A rectilineal figure circumscribes a circle, or the circle is inscribed in the rectilineal figure, when each side of the figure touches the circumference of the circle without cutting it.

59. A rectilineal figure is inscribed in another, or the latter circumscribes the former, when all the angular points of the former are placed on the sides of the latter.

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60. Identical Figures are those which are mutually equal in all their parts.

61. Similar Figures are those which have all the angles of the one equal to all the angles of the other, and the corresponding sides about the angles of each figure, proportional.

62. The Perimeter of a figure is the sum of all its sides. 63. A Problem is an operation proposed to be performed. 64. A Theorem is a truth which it is proposed to prove.

65. A Proposition is a general term signifying either a problem or a theorem.

66. A Lemma is a preparatory proposition to render what follows more easy.

67. A Corollary is an obvious consequence resulting from a preceding proposition.

68. A Scholium is an observation, or remark upon something preceding it.

69. An Axiom is a self evident truth.

70. A Postulate is a request to admit the possibility of performing a certain operation.

AXIOMS.

1. Things which are equal to the same thing, are equal to each other.

2. When equals are added to equals, the wholes are equal. 3. When equals are taken from equals, the remainders are equal. 4. When equals are added to unequals, the wholes are unequal. 5. When equals are taken from unequals, the remainders are unequal.

6. Things which are like multiples of the same, or of equal things, are equal to each other.

7. Things which are like parts of the same, or of equal things, are equal to each other.

8. The whole of any thing is equal to the sum of all its parts.

9. The whole of any thing is greater than a part of it.

10. Magnitudes which coincide with one another, or fill the same space, are identical or mutually equal in all respects.

11. All right angles are equal to one another.

12. If two straight lines are parallel to each other, a line which meets one of them will, if produced, meet the other.

13. If two straight lines intersect each other, they cannot both be parallel to the same straight line.

POSTULATES.

1. Let it be granted that a straight line may be drawn from one point to any other point.

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2. That a straight line may be produced to any length in the same direction.

3. That a circle may be described round any point as a centre, and with any radius.

4. That from any point in a given straight line a perpendicular to that line may be drawn.

5. That a right line, or rectilineal angle may be made equal to any given right line, or rectilineal angle; that any right line or rectilineal angle may be bisected; and that through a given point a line may be drawn parallel to any other line.

THEOREMS.

THEOREM I.

In any two triangles A B C, D E F, if two sides CA, CB, in the one, be respectively equal to two sides F D, F E in the other, and the angle C included by the sides C A, C B, be equal to the angle F, includea by the sides FD, FE; then the two triangles will be identical, or equal in all respects; and have the angles equal which are opposite to the equal sides.

For conceive the point C to be laid on the point C F, and the line C A on the line F D, then, because these lines are equal, the point A will coincide with the point D. And as C A coincides with F D, and

F

BD

E

the angle C is equal to the angle F, the line C B will A fall on the line FE; and because C B and FE are equal, the point