1. Magnitudes which are equal to the same magnitude, or equal magnitudes, are equal to each other.

2. If equals are added to equals, the sums are equal.

3. If equals are taken from equals, the remainders are equal. 4. If equals are added to unequals, the sums are unequal in the same order; if unequals are added to unequals in the same order, the sums are unequal in that order.

5. If equals are taken from unequals, the remainders are unequal in the same order; if unequals are taken from equals, the remainders are unequal in the reverse order.

6. The doubles of the same magnitude, or of equal magnitudes, are equal; and the doubles of unequals are unequal.

7. The halves of the same magnitude, or of equal magnitudes, are equal; and the halves of unequals are unequal. 8. The whole is greater than any of its parts. 9. The whole is equal to the sum of all its parts.

Q.E.D. stands for quod erat demonstrandum, which was to be proved. Q.E.F. stands for quod erat faciendum, which was to be done.

The signs +, ? X,,, have the same meaning as in Algebra.

PLANE GEOMETRY.

BOOK I.

RECTILINEAR FIGURES.

DEFINITIONS.

36. A straight line is a line such that any part of it, however placed on any other part, will lie wholly in that part if its extremities lie in that part, as AB.

37. A curved line is a line no part of which is straight, as CD.

A

38. A broken line is made up of dif- Eferent straight lines, as EF.

NOTE. A straight line is often called simply a line.

FIG. 4.

B

D

39. A plane surface, or a plane, is a surface in which, if any two points are taken, the straight line joining these points lies wholly in the surface.

40. A curved surface is a surface no part of which is plane.

41. A plane figure is a figure all points of which are in the same plane.

42. Plane figures which are bounded by straight lines are called rectilinear figures; by curved lines, curvilinear figures.

43. Figures that have the same shape are called similar. Figures that have the same size but not the same shape are called equivalent. Figures that have the same shape and the same size are called equal or congruent,

THE STRAIGHT LINE.

44. Postulate. A straight line can be drawn from one point to another.

45. Postulate. A straight line can be produced indefinitely.

46. Axiom.* Only one straight line can be drawn from one point to another. Hence, two points determine a straight line.

47. COR. 1. Two straight lines which have two points in common coincide and form but one line.

48. COR. 2. Two straight lines can intersect in only one point.

For if they had two points common, they would coincide and not intersect.

Hence, two intersecting lines determine a point.

49. Axiom. A straight line is the shortest line that can be drawn from one point to another.

50. DEF. The distance between two points is the length of the straight line that joins them.

51. A straight line determined by two points may be considered as prolonged indefinitely.

52. If only the part of the line between two fixed points is : considered, this part is called a segment of the line.

53. For brevity, we say "the line AB," to designate a segment of a line limited by the points A and B.

54. If a line is considered as extending from a fixed point, this point is called the origin of the line.

* The general axioms on page 6 apply to all magnitudes, Special geometrical axioms will be given when required.

55. If any point, C, is taken in a given straight line, AB, the two parts CA and CB are said to have opposite directions from the point C (Fig. 5).

A←

B

FIG. 5.

Every straight line, as AB, may be considered as extending in either of two opposite directions, namely, from A towards B, which is expressed by AB, and read segment AB; and from B towards A, which is expressed by BA, and read segment BA.

56. If the magnitude of a given line is changed, it becomes longer or shorter.

==

Thus (Fig. 5), by prolonging AC to B we add CB to AC, and AB = AC + CB. By diminishing AB to C, we subtract CB from AB, and ACAB CB.

If a given line increases so that it is prolonged by its own magnitude several times in succession, the line is multiplied, and the resulting line is called a multiple of the given line.

Lines of given length may be added and subtracted; they may also be multiplied by a number.

THE PLANE ANGLE.

57. The opening between two straight lines drawn from the

same point is called a plane angle. The two lines, ED and EF, are called the sides, and E,

F

the point of meeting, is called the vertex of the angle.

FIG. 7.

The size of an angle depends upon the extent of opening of its sides, and not upon the length of its sides,

58. If there is but one angle at a given vertex, the angle is designated by a capital letter placed at the vertex, and is read by simply naming the letter.

If two or more angles have the same vertex, each angle is designated by three letters, and is read by naming the three letters, the one at the vertex between the others. Thus, DAC (Fig. 8) is the angle formed by the sides AD and AC.

An angle is often designated by placing a small italic letter between the sides and near the vertex, as in Fig. 9.

E

CD B

A F

FIG. 8.

cb

α

A

B

FIG. 9.

59. Postulate of Superposition. Any figure may be moved from one place to another without altering its size or shape.

60. The test of equality of two geometrical magnitudes is that they may be made to coincide throughout their whole extent. Thus,

Two straight lines are equal, if they can be placed one upon the other so that the points at their extremities coincide.

Two angles are equal, if they can be placed one upon the other so that their vertices coincide and their sides coincide, each with each.

61. A line or plane that divides a geometric magnitude into two equal parts is called the bisector of the magnitude.

If the angles BAD and CAD (Fig. 8) are equal, AD bisects the angle BAC.

62. Two angles are called adjacent angles when they have the same vertex and a common side between them; as the angles BOD and AOD (Fig. 10).