41 152. Figure in general is a space enclosed by lines upon all sides. 153. The least number of right lines which can include a space are three. 、154. A figure of three sides is called a triangle. 155. A figure of four sides is called a quadrilateral. 146. Figures of more than four sides are in general called polygons. 157. If three points be placed in any position, not in a straight line, and these points be connected by straight lines, a plane triangle is formed. 158. A CIRCLE is a plane round figure, bounded by a uniform curve line. 159. Geometers also consider a circle as a polygon, whose sides are infinite. 160. From definitions 153, 154 and 159 it follows that the triangle, and the circle are extreme figures; the one having the least number of sides possible, and the other the greatest possible number. By which it would seem that their properties and their uses would be very remote; which however is not the case; for the. properties of the one can hardly be shewn but by those of the other.* 161. The CIRCUMFERENCE of a circle is the uniform curve line which encloses the space called the circle. The circumference is also called the circle. 162. The CENTRE of a circle is a point within the circle, equally distant from all parts of the circumference. 163. The circumference may be divided into any number of equal parts. For the purposes of astronomy, navigation and surveying, geometers divide the circle into 360 equal parts called degrees each degree is subdivided into 60 minutes-each minute into 60 seconds -each second into 60 thirds, and so on. 164. The RADIUS of a circle is a right line drawn from the centre to the circumference; and is half the diameter. 165. The DIAMETER of a circle is a right line drawn from one point in the circumference to the opposite * Deparcieux. 6. point through the centre, and is equal to twice the radius. 166. An ARC is any portion of the circumference, and is the measure of the angle comprised by two radii, drawn from the centre, one to each extremity of the arc. 167. The COMPLEMENT of an arc, or angle is what the arc or angle wants of a quadrant, or quarter circle, and it is known by subtracting the arc or angle from 90 deg. 168. The SUPPLEMENT of an arc or angle is what the arc or angle wants of a semicircle, and is known by subtracting the arc or angle from 180 degrees. 169. The CHORD of an arc is a right line drawn from one extremity of the arc to the other. 170. The SINE of an arc or angle is a right line drawn from one extremity of the arc, perpendicularly upon the radius, or diameter which is drawn to the other extremity of the same arc. 171. The CO-SINE of an arc or angle is the sine of the complement of that arc or angle-thus the co-sine of 29 degrees is the sine of 61 degrees. The co-sine is equal to that part of the radius contained between the angular point and the foot of the sine. 172. The VERSED SINE of an arc or angle is that part of the radius which is contained between the foot of the sine and the extremtiy of the arc. Therefore the cosine and versed sine are equal to radius. 173. The TANGENT of an arc or angle is a right line drawn perpendicularly from the extremity of the radius, which meets one extremity of the arc. The tangent is terminated by the secant of the same arc or angle. 174. The SECANT of an arc or angle is a right line drawn from the centre through one extremity of the arc, till it meets the tangent of the same arc or angle drawn from the other extremity of the same arc or angle. By the two last definitions it is seen that the tangent and the secant terminate each other. 175. The CO-TANGENT of an arc or angle is the tangent of the complement of that arc or angle. 176. The CO-SECANT of an arc or angle is the secant of the complement of that arc or angle. 177. A SEMICIRCLE is half a circle or 180 deg. 178. A QUADRANT is a quarter circle or 90 deg. 179. A SEXTANT is the sixth part of a circle or 60 deg. 180. An OCTANT is the eighth part of a circle or 45 deg. 181. A SEGMENT is any part of the circle bounded by an arc and its chord. 182. A SECTOR is an arc bounded by two radii. 183. An ANGLE is the space comprised by the meeting of two lines which are not in the same direction.See No. 149. 184. All angles are right or oblique. 185. A RIGHT ANGLE is formed by two lines perpendicular to each other, and contains 90 deg. 186. An OBLIQUE ANGLE is greater, or less than a right angle or 90 deg.-if greater it is called obtuse-if less it is called acute. 187. As great use is made of the chords, sines, tangents and secants, it may aid the learner to refer to the diagram to ascertain the positions of these important properties of the circle. 188. On the diagram of navigation at the division of 100 as a radius, is a fine dotted curve line, which represents the arc or angle of 90 deg. If a line be drawn within this arc from one end of it to the other, that line will be a chord to the arc of 90 deg. and a line drawn from one end of this arc to any point of it will be a chord to that section of it which it subtends.-On the side marked meridian, and between the divis ions 87 and 88 is a fine dotted straight line, which extends to the dotted curve line, and intersects it at the angle of 29 deg. to which set the index; the straight dotted line represents the sine of 29 deg. which being parallel to the side marked equator, its value is there found to be 48, which is the length or ratio of the sine of 29 deg. to the radius of 100. The other dotted straight line which leads from the same point of intersection at the angle 29 deg. and to the side marked equator represents the cosine of 29 deg. which being parallel to the side marked meridian its value is there given 87, which is the length, or ratio of the co-sine of 29 deg. to the radius of 100. If the length of the radius be increased, or diminished, the length of the sine and the co-sine will in like ratio be increased or diminished; for if the radius be 200 on the index, the sine of 29 deg. will be 97, and the co-sine will be 175. In this example the side marked meridian is used as base. Now take the side marked equator as base, and count the degrees from right to left; the index will be on 61 deg.-and it is seen that the co-sine of 29 deg. becomes the sine of 61 deg. and the sine of 29 deg. becomes the co-sine of 61 deg. 189. The line drawn perpendicular from the radius 100, at one end of the arc of 29 deg. is its tangent, which is terminated by the secant drawn through the other end of the same arc, which secant is represented by the graduated edge of the index, and its value is found on the index, if set on 29 deg. to be 114 nearly; and the tangent being parallel to the side marked equator, is there found to be 55 nearly. 190. As the angle increases, the sine, tangent, and secant will also increase; but the co-sine, co-tangent and co-secant will decrease. Example.-Set the graduated edge of the index to the side marked equator on 0. Let the radius be 100-of 0 the sine and the tangent will be negative, the co-sine and secant will be cqual to the radius, and the co-tangent and the cosecant will be infinite. Now set the index on 8 deg. the sine will be 14 nearly, the tangent 14 nearly, and the secant 101; but the co-sine will be only 99. Then set the index on the angle 18 deg. and with the same radius of 100, the sine will be 31, the tangent 32, and the secant 105; but the co-sine will be only 944Now set the index on 45 deg. and to the same radius of 100, the sine and co-sine are each 70.7, the tangent and co-tangent are each 100, and the secant and cosecant are each 141. 191. In all circles, great or small, the sine of 90 deg. (called the sine total,) the tangent of 45 deg. the chord of 60 deg. and half the secant of 60 deg. are each equal to the radius which describes the circle. 192. The chord, sine, tangent, and secant of an arc in one circle is to the chord, sine, tangent, and secant of the same arc in another circle, as the radius of circle one is to the radius of the other. 193. In any plane triangle, the sides are proportional to the sines of their opposite angles. 194. In any right-angled triangle, the square of the longest side is equal to the sum of the squares of the other two sides. 195. In all plane triangles the sum of the three an gles is equal to two right angles, or 180 deg. 196. The longest side of any triangle is opposite the greatest angle. 197. An angle in a semicircle is a right angle. 198. An angle in a segment less than a semicircle is greater than a right angle. 199. An angle in a segment greater than a semicircle is less than a right angle. 200. An angle at the circumference of a circle is half the angle at the centre, standing on the same arc; and it is measured by half the arc it subtends. TRIGONOMETRY. 201. Trigonometry is the application of the properties of the circle to ascertain the unknown parts of triangles, some of the parts being given; and it is spheric, or plane. 202. In spheric trigonometry the sides of triangles are formed by the intersection of the arcs of three great circles. 203. In plane trigonometry the sides of triangles are formed by the meeting of three straight lines; and it is divided into right angled, and oblique angled. |