53. The Circumference of a circle is the boundary line of the figure. (Fig. 29.)
54. The Center of a circle is a point within the circumference equally distant from every point in its circumference, as A, Fig. 29.
Terms and Definitions. 5
55. The Radius of a circle is a line drawn from the center to. any point in the circumference, as A B, Fig. 29, that is, hall' the diameter. The plural of radius is radii.
56. The Diameter of a circle is any straight line drawn through the center to opposite points of the circumference, as C D. Fig. 29.)
57. A Semicircle is the half of a circle, and is bounded by half the circumference and a diameter. (Fig. 30.) '
Fig 30. - A Semicircle.
Fig. 31 - Seynunts.
58. A Segment of a circle is any part of its sur-face cut off by a straight line, as A E B and C F D, Fig. 31.
59. An Are of a circle is any part of the circumference, as A B E and C F D, Fig. 32.
Fig. .32 - Arcs and Chords.
Fig. 33.Secto .s.
60. A Chord is a straight line joining the extremities of an arc, as A E and C D, Fig. 32.
61. A Sector of a circle is the space included between two radii and the are which they intercept, as A C B and D C E, Fig. 33, and B A C. Fig. 34.
62. A Quadrant is a sector whose area is equal to one-fourth of the circle. (B A C. Fig. 34.) The two radii bounding a quadrant are at right angles.
63. A Tangent to a circle or other curve is a straight line which touches it at only one point, as E D and A C, Fig. 35. Every tangent to a circle is perpendicular to the radius, drawn to the point of tan-gency. Thus E D is perpendicular to F D and A C to F B.
Fig 34. - A Quadrant.
Fig. 35. - Tangents.
64. Concentric circles are those which are described about the same center. (Fig. 36.)
65. Eccentric circles are those which are described about different centers. (Fig. 37.)
66. Polygons are inscribed in, or circumscribed by, circles when the vertices of all their angles are in the circumference. (Fig. 38.)
Fig. S6. - Concentric Circles.
Fig 37 - Eccentric Circles.
67. A circle is inscribed in a straight-sided figure when it is tangent to all sides. (Fig. 39.) All regular polygons may be inscribed in circles, and circles may be inscribed in the polygons; hence the facility with which polygons may be constructed.
Fig. 38 - An Inscribed Triangle.
Fig 30.-An Inscribed Circle.
68. A Degree. - The circumference of a circle is considered as divided into 360 equal parts, called degrees
(marked°). Each degree is divided into 60 minutes (marked'); and each minute into 60 seconds (marked"). Thus if the circle be large or small the number of divisions is always the same, a degree being equal to 1/360th part of the whole circumference; the semicircle is equal to 180° and the quadrant to 90°. The radii drawn from the center of a circle to the extremities of a quadrant are always at right angles with each other; a right angle is therefore called an angle of 90° (A E B, Fig. 40). If a right angle be bisected by a straight line, it divides the arc of the quadrant also into two equal parts, each being equal to one-eighth of the whole circumference, or 45°, (A E F and FEB, Fig. 40); if the right angle were divided into three equal parts by straight lines, it would divide the arc into three equal parts, each containing 30° (AEG, G E H, H E B, Fig. 40). Thus the degrees of the circle are used to measure angles, therefore by an angle of any number of degrees, it is understood that if a circle with any length of radius be struck with one foot of the compasses in its vertex, the sides of the angle will intercept a portion of the circle equal to the number of degrees given. Thus the angle A E H, Fig. 40, is an angle of 60°. In the measurement of angles by the circumference of the circle, and in the various mathematical calculations based thereon, use is made of certain lines known as circular functions, always bearing a fixed relationship to the radius of the circle and to each other, which gives rise to a number of terms, some of which, at least, it is desirable for the pattern cutter to understand.
Fig. 40. - A Circle Divided into Degrees for Measuring Angles.
Fig. 41. - Complement
Fig. 42. - Supplement.
69. The Complement of an arc or of an angle is the difference between that arc or angle and a quadrant. In Fig. 41, ADB is the complement of B B C, and vice versa.
70. The Supplement of an arc or of an angle is the difference between that arc or angle and a semicircle.
Fig. 45. - Diagram Showing the Circular Functions of the Arc A H or Angle A C H.
In Fig. 42, B D C is the supplement of ADB. and vice versa.
71. The Sine of an arc is a straight line drawn from one extremity perpendicular to a radius drawn to the other extremity of the arc. (H B, Fig. 43.)
72. The Co-Sine of an are is the sine of the complement of that arc. H K, Fig. 43. is the sine of the arc A H.
73. The Tangent Of an Arc is a line which touches the arc at one extremity, and is terminated by a line passing from the center of the circle through the other extremity of the arc. In Fig. 43, A E is the tangent of A H or of the angle A C H.
74. The Co-Tangent of an arc is the tangent of the complement. Thus F G, Fig. 43, is the co-tan gent of the arc A H.
75. The Secant of an are is a straight line drawn from the center of a circle through one extremity of that are and prolonged to meet a tangent to the other extremity of the arc. (EC, Fig 43.)
76. The Co-Secant of an arc or angle is the secant of the complement of that are or angle, as F C, Fig. 43.
77. The Versed Sine of an are is that part of the radius intercepted between the sine and the circumference. (A 15. Fig. 43.)
78. An Ellipse is an oval-shaped curve (Fig. 44), from any point in which, if straight lines be drawn to two fixed points within the curve, their sum will be always the same. These two points are called foci (F and H). The line A B, passing through the foci, is called the major or transverse axis. The line E G, perpendicular to the middle of the major axis, and extending from one side of the figure to the other, is called the minor or conjugate axis. There are various other definitions of the ellipse; besides the one given here, dependent upon the means employed for drawing it, which will be fully explained at the proper place among the problems. (See definition 118.)
Fig. 44. - An Ellipse.
Fig. 46. - A Hyperbola.
Fig. 45. - A Parabola.
Fig. 47. - Evolute and Involute.
Fig. 48. - A Triangular Prism.
79. A Parabola (A B, Fig. 45) is a curve in which any point is equally distant from a certain fixed point and a straight line. The fixed point (F) is called the focus, and the straight line (CD) the directrix. In this figure any point, as N or M. is equally distant from K and the nearest point in C D, as H or K. (See definition 1 13. )
80. A Hyperbola (A B, Fig. 46) is a curve from any point in which, if two straight lines be drawn to two fixed points, their difference shall always be the same. Thus, the difference between E G and G L is H L. and the difference between E F and F L is B L. H L and B L are equal. The two. fixed points, E and L, are called foci. (See definition 113.)
81. An Evolute is a circle or other curve from which another curve, called the involute or evolutent, is described by the aid of a thread gradually unwound from it. (Fig. 47.)
82. An Involute is a curve traced by the end of a string wound upon another curve or unwound from it. (Fig. 47.) (See also Prob. 84, Chapter IV (Geometrical Problems).)