metrical tables, together with the investigation of fome theorems, useful for extending trigonometry to the solution of the more difficult problems. N angle at the centre of a circle is to four right A angles as the arch on which it stands is to the whole circumference. Let ABC be an angle at the centre of the circle ACF, standing on the circumference AC: the angle ABC is to four right angles as the arch AC to the whole circumference ACF. Produce AB till it meet the circle in E, and draw DBF ABD are two angles at the centre of the circle ACF, the angle ABC is to the angle ABD as the arch AC to the Varch AD, (36.6.); and therefore also, the angle ABC is to four times the angle ABD as the arch AC to four times the arch AD (4.5.). But ABD is a right angle, and therefore, four times the arch AD is equal to the whole circumference ACF; there, fore, fore, the angle ABC is to four right angles as the arch AC to the whole circumference ACF. Cor. Equal angles at the centres of different circles stand on arches which have the fame ratio to their circumferences. For, if the angle ABC, at the centre of the circles ACE, GHK, stand on the arches AC, GH, AC is to the whole circumference of the circle ACE, as the angle ABC to four right angles; and the arch HG is to the whole circumference of the circle GHK in the fame ratio. Therefore, &c. DEFINITIONS. I. IF two straight lines interfect one another in the centre of a circle, the arch of the circumference intercepted between them is called the measure of the angle which they contain. Thus, the arch AC is the measure of the angle ABC. If the circumference of a circle be divided into 360 equal parts, each of these parts is called a degree; and, if a degree be divided into 60 equal parts, each of these is called a minute; and, if a minute be divided into 60 equal parts, each of them is called a second, and so on. And as many degrees, minutes, seconds, &c. as are in any arch, fo many degrees, minutes, seconds, &c. are said to be in the angle measured by that arch. Cor. 1. Any arch is to the whole circumference of which it is a part, as the number of degrees, and parts of a degree contained in it, is to the number 360. And any angle is to U 3 four four right angles as the number of degrees and parts of a degree in the arch, which is the measure of that angle, is to 36e. COR. 2. Hence also, the arches which measure the same angle, whatever be the radii with which they are described, contain the fame number of degrees, and parts of a degree. For the number of degrees and parts of a degree contained in each of these arches has the same ratio to the number 360, that the angle which they measure has to four right angles (Cor. Lem. 1.) A The degrees, minutes, seconds, &c. contained in any arch or angle, are usually written as in this example, 49°. 36'. 24". 42'""; that is, 49 degrees, 36 minutes, 24 feconds, and 42 thirds. Two angles, which are together equal to two right angles, or two arches which are together equal to a semicircle, are called the supplements of one another. IV. A straight line CD drawn through C, one of the extremities, of the arch AC, perpendicular to the diame COR. 2. The fine of an arch is half the chord of twice that arch: this is evident by producing the fine of any arch till it cut the circumference. V. The segment DA of the diameter passing through A, one extremity of the arch AC, between the fine CD and the point A, is called the Verfed fine of the arch AC, or of the angle ABC. VI. VI. A straight line AE touching the circle at A, one extremity of the arch AC, and meeting the diameter BC, which pafses through C the other extremity, is called the Tangent of the arch AC, or of the angle ABC. Cor. The tangent of half a right angle is equal to the ra dius. VII. The straight line BE, between the centre and the extremity of the tangent AE, is called the Secant of the arch AC, or of the angle ABC. Cor. to Def. 4. 6. 7. the fine, tangent, and secant of any angle ABC, are likewise the fine, tangent, and secant of its supplement CBF. It is manifest from Def. 4. that CD is the fine of the angle CBF. Let CB be produced till it meet the circle again in I; and it is also manifest, that AE is the tangent, and BE the secant, of the angle ABI, or CBF, from Def. 6. 7. Cor. to Def. 4. 5. 6. 7. The fine, versed fine, tangent, and secant of an arch, which is the measure of any given angle ABC, is to the fine, versed fine, tangent and N fecant, of any other arch which is the measure of the same angle, as the radius of the first arch is to the radius of the second. OMD Let AC, MN be measures of the angle ABC, according to Def. 1.; CD the fine, DA the versed fine, AE the tangent, and BE the secant of the arch AC, according to Def. 4. 5. 6.7.; NO the fine, OM the versed fine, MP the tangent, and BP the secant of the arch MN, according to the U 4 fame same definitions. Since CD, NO, AE, MP are parallel, CD: NO:: rad. CB: rad. NB, and AE: MP:: rad. AB: rad. BM, also BE: BP :: AB: BM; likewise because BC: BD :: BN : BO, that is, BA: BD :: BM: BO, by converfion and alternation, AD: MO:: AB: MB. Hence the corollary is manifest. And therefore, if tables be constructed, exhibiting in numbers the fines, tangents, secants, and versed fines of certain angles to a given radius, they will exhibit the ratios of the fines, tangents, &c. of the fame angles to any radius whatsoever. In fuch tables, which are called Trigonometrical Tables, the radius is either supposed 1, or some number in the feries 10, 100, 1000, &c. The use and construction of these tables, are about to be explained. VIII. The difference between any angle and a right angle, or be C F tween any arch and a quadrant, is called the complement of that angle, or of that arch. Thus, if BH be perpen angle HBC, its excess a bove a right angle; and the complement of the arch FC is HC. The fine, tangent, or fecant of the complement of any angle is called the cofine, cotangent, or cosecant of that angle. Thus, let CL or DB, which is equal to CL, be the fine of the angle CBH; HK the tangent, and BK the fecant of the fame angle; CL or BD is the cofine, HK the cotangent, and BK the cofecant, of the angle ABC. : COR |