CHAPTER II.

THE PRIMARY CONCEPTS OF GEOMETRY.

I. Definitions of Geometric Magnitudes.

34. Geometry is the science which treats of the properties of space.

35. A part of space occupied by a physical body, or marked out in any other way, is called a Solid.

B

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C

36. The common boundary of two parts of a solid, or of a solid and the remainder of space, is a Surface.

37. The common boundary of two parts of a surface is a Line.

38. The common boundary of two parts of a line is a Point. 39. A Magnitude is any thing which can be added to itself so as to double.

40. A point has position without magnitude.

41. A line may be conceived of as traced or generated by a point on a moving body. The intersection of two lines is a point.

42. A line on a moving body may generate a surface. The intersection of two surfaces is a line.

43. A surface on a moving body may generate a solid. 44. We cannot picture any motion of a solid which will generate any thing else than a solid.

Thus, in our space experience, we have three steps down from a solid to a point which has no magnitude, or three steps up from a point to a solid; so our space is said to have three dimensions.

45. A Straight Line is a line which pierces space evenly, so that a piece of space from along one side of it

will fit any side of any other portion.

46. A Curve is a line no part of which is straight.

A line.

A curve.

47. Take notice: the word "line," unqualified, will henceforth mean "straight line."

48. A Sect is the part of a line between two definite points.

49. A Plane Surface, or a Plane, is a surface which divides space evenly, so that a piece of space from along one side of it will fit either side of any other portion.

50. A plane is generated by the motion of a line always passing through a fixed point and leaning on a fixed line.

51. A Figure is any definite combination of points, lines, curves, or surfaces.

52. A Plane Figure is in one plane.

The

53. If a sect turns about one of its end points, the other end point describes a curve which is called a Circle. fixed end point is called the Center of the Circle.

54. The Radius of a Circle is a sect drawn from the center to the circle.

55. A Diameter of a Circle is a sect drawn through the center, and terminated both ways by the circle.

56. An Arc is a part of a circle.

57. Parallel Lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet.

58. When two lines are drawn from the same point, they are said to contain, or to make with each other, an Angle.

The point is called the Vertex, and the lines are called the Arms, of the angle. A line drawn from the vertex, and turning about the vertex in the plane of the angle from the position of coincidence with one arm to that of coincidence with the other, is said to turn through the angle; and the angle is greater as the quantity of turning is greater.

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59. Since the line can turn from the one position to the other in either of two ways, two angles are formed by two lines drawn from a point.

Each of these angles is called the Explement of the other. If we say two lines going out from a point form an angle, we

B

A

are fixing the attention upon one of the two explemental angles which they really form; and usually we mean the smaller angle.

60. Two angles are called Equal if they can be placed so that their arms coincide, and that both are described simultaneously by the turning of the same line about their common

vertex.

61. If two angles have the vertex and an arm in common, and do not overlap, they are called Adjacent Angles; and the angle made by the other two arms on the side toward the common arm is called the Sum of the Adjacent Angles. Thus,

using the sign for the word "angle," the sign of equality (=), and the sign of addition (+, plus),

62. A Straight Angle has its arms in the same line, and on different sides of the vertex.

63. The sum of two adjacent angles which have their exterior arms in the same line on different sides of the vertex is a straight angle.

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64. When the sum of any two angles is a straight angle, each is said to be the Supplement of the other.

65. If two supplemental angles be added, their exterior