42. A Regular Polygon has all its sides and all its angles equal. If they are not both equal, the polygon is Irregular.

43. An Equilateral Triangle is also a Regular Figure of three sides, and the Square is one of four: the former being also called a Trigon, and the latter a Tetragon.

44. Any figure is equilateral, when all its sides are equal: and it is equiangular when all its angles are equal. When both these are equal, it is a regular figure.

45. A Circle is a plane figure bounded by a curve line, called the Circumference, which is every where equidistant from a certain point within, called its Centre.

The circumference itself is often called a circle, and also the Periphery.

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

47. The Diameter of a cirle is a line drawn through the centre, and terminating at the circumference on both sides.

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

49. A Chord is a right line joining the extremities of an arc.

50. A Segment is any part of a circle bounded by an arc and its chord.

51. A Semicircle is half the circle, or a segment cut off by a diameter.

The half circumference is sometimes called the Semicircle.

52. A Sector is any part of a circle which is bounded by an arc, and two radii drawn to its extremities.

53. A Quadrant, or Quarter of a circle is a sector having a quarter of the circumference for its arc, and its two radii are perpendicular to each other. A quarter of the circumference is sometimes called a Quadrant.

OQ DODD DOD

DEFINITIONS.

54. The Height or Altitude of a figure is a perpendicular let fall from an angle, or its vertex, to the opposite side, called the base.

55. In a right-angled triangle, the side opposite the right angle is called the Hypothenuse; and the other two sides are called the Legs, and sometimes the Base and Perpendicular.

56. When an angle is denoted by three letters, of which one stands at the angular point, and the other two on the two sides, that which stands at the angular point is read in the middle.

57. The circumference of every circle is supposed to be divided into 360 equal parts called degrees; and each degree into 60 Minutes, each Minute into 60 Seconds, and so on. Hence a semicircle contains 180 degrees, and a quadrant 90 degrees.

58. The Measure of an angle, is an arc of any circle contained between the two lines which form that angle, the angular point being the centre; and it is estimated by the number of degrees contained in that arc.

59. Lines, or chords, are said to be Equidistant from the centre of a circle, when perpendiculars drawn to them from the centre are equal.

69. And the right line on which the Greater Perpendicular falls, is said to be farther from the centre.

61. An Angle In a Segment is that which is contained by two lines, drawn from any point in the arc of the segment, to the two extremities of that arc.

62. An Angle On a segment, or an arc, is that which is contained by two lines, drawn from any point in the opposite or supplemental part of the circumference, to the extremities of the arc, and containing the arc between them.

63. An Angle at the circumference, is that whose angular point or summit is any where in the circumference. And an angle at the centre, is that whose angular point is at the

centre.

64. A right-lined figure is Inscribed in a circle, or the circle Circumscribes it, when all the angular points of the figure are in the circumference of the circle.

65. A right-lined figure Circumscribes a circle, or the circle is Inscribed in it, when all the sides of the figure touch the circumference of the circle.

66. One right-lined figure is Inscribed in another, or the latter circumscribes the former, when all the angular points of the former are placed in the sides of the latter.

67. A Secant is a line that cuts a circle, lying partly within, and partly without it.

66. Two triangles, or other right-lined figures, are said to be mutually equilateral, when all the sides of the one are equal to the corresponding sides of the other, each to each : and they are said to be mutually equiangular, when the angles of the one are respectively equal to those of the other.

69. Identical figures are such as are both mutually equi lateral and equiangular; or that have all the sides and all the angles of the one, respectively equal to all the sides and all the angles of the other, each to each; so that if the one figure were applied to, or laid upon the other, all the sides of the one would exactly fall upon and cover all the sides of the other; the two becoming as it were but one and the same figure.

70. Similar figures, are those that have all the angles of the one equal to all the angles of the other, each to each, and the sides about the equal angles proportional.

71. The Perimeter of a figure, is the sum of all its sides taken together.

72. A Proposition, is something which is either proposed to be done, or to be demonstrated, and is either a problem or a theorem.

73. A Problem, is something proposed to be done.

74. A Theorem, is something proposed to be demonstrated. 75. A Lemma, is something which is premised, or demontrated, in order to render what follows more easy.

76. A Corollary, is a consequent truth, gained immediately from some preceding truth, or demonstration.

77. A Scholium, is a remark or observation made upon something going before it.

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 double of the same thing, or equal things, are equal to each other.

7. Things which are halves of the same thing, are equal. 8. Every whole is equal to all its parts taken together. 9. Things which coincide, or fill the same space, are identical, or mutually equal in all their parts.

20. All right angles are equal to one another.

21. Angles that have equal measures, or arcs, are equal.

THEOREM I.

Ir two triangles have two sides and the included angle in the one, equal to two sides and the included angle in the other, the triangles will be identical, or equal in all respects.

In the two triangles ABC, DEF, if the side AC be equal to the side Dr, and the side BC equal to the side EF, and the angle c equal to the angle r; then will the two triangles be identical, or equal in all respects.

F

AA

BD E

For conceive the triangle ABC to be applied to, or placed on, the triangle DEF, in such a manner that the point c may VOL. I.

37

coincide with the point r, and the side ac with the side DF, which is equal to it.

A

Then, since the angle F is equal to the angle c (by hyp.), the side BC will fall on the side EF. Also, because AC is equal to DF, and BC equal to EF (by hyp.), the point a will coincide with the point v, and the point B with the point E; consequently the side AB will coincide with the side De. Therefore the two triangles are identical, and have all their other corresponding parts equal (ax. 9), namely, the side AB equal to the side DE, the angle A to the angle D, and the angle B to the angle E. Q. E. D.

THEOREM II.

WHEN two triangles have two angles and the included side in the one, equal to two angles and the included side in the other, the triangles are identical, or have their other sides and angle equal.

Let the two triangles ABC, DEF, have the angle equal to the anglo D, the angle B equal to the angle E, and the side AB equal to the side DE; then these two triangles will be identical.

A

For, conceive the triangle ABC to be placed on the triangle DEF, in such manner that the side AB may fall exactly on the equal side De. Then, since the angle a is equal to the angle D (by hyp.), the side AC must fall on the side DF; and, in like manner, because the angle R is equal to the angle, the side BC must fall on the side EF. Thus the three sides of the triangle ABC will be exactly placed on the three sides of the triangle DEF consequently the two triangles are identical (ax. 9), having the other two sides AC, BC, equal to the two DF, EF, and the remaining angle c equal to the remaining angle r. Q. E. D.

THEOREM III.

In an isosceles triangle, the angles at the base are equal.. Or, if a triangle have two sides equal, their opposite angles will also be equal.

If the triangle ABC have the side ac equal to the side BC: then will the angle в be equal to the angle A.

For, conceive the angle c to be bisected, or divided into two equal parts, by the line CD, making the angle ACD equal to the angle

BCD.