metrical tables, together with the investigation of some theorems, useful for extending trigonometry to the solution of the more difficult problems. SECTION 1. L E M MA I. . N angle at the centre of a circle is to four right angles as the arch on which it stands is to the whole circumference. A 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 D H E JA We arch AD, (36. 6.); and KI B therefore also, the angle F fore, G 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 same 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 same ratio. Therefore, &c. DEFINITIONS. I. IF two straight lines intersect one another in the centre of 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. II. 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 6o 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. degrees, minutes, feconds, &c. as are in any arch, so many degrees, minutes, feconds, &c. are said to be in the angle measured by that arch. And as many Cor. I. 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 de. gree in the arch, which is the measure of that angle, is to 36c. Cor. 2. Hence also, the arches which measure the same angle, whatever be the radii with which they are described, contain the same 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.) The degrees, minutes, seconds, &c. contained in any arch or angle, are usually written as in this example, 49o. 36'. 24". 42""; that is, 49 degrees, 36 minutes, 24 seconds, and 42 thirds. IIL 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, perpen H K dicular to the diame. ter passing through the E other extremity A, is L called the Sine of the arch AC, or of the angle ABC, of which AC is F D the measure. Cor. 1. The fine of a qua drant, or of a right angle, is equal to the radius. Cor. 2. The fine of an arch is half the chord of twice that arch : this is evident by producing the line 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 hne 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 pai. ses 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 rac dius. VII. The straight line BE, between the centre and the extremity of the tangent A E, 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 manifeft, 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 E the measure of any given angle ABC, is to the fine, Р. versed fine, tangent and fecant, of any other arch which is the measure of the same angle, as the radius of the first arch is to B the radius of the second. OM D р N Let AC, MN be measures of the angle ABC, according to Def. i.; CD the fine, DA the versed fine, AE the tangent, and BE the fecant of the arch AC, according to Def. 4. 5. 6.7.; NO the fine, OM the versed fine, MP the tangent, and BP the fecant of the arch MN, according to the U 4 same fame definitions. Since CD, NO, AE, MP are parallel, CD: NO :: rad. CB : rad. NB, and AE: MP :: rad. AB: rad. BM, also BE: BP : :ABBM; likewise because BC: BD :: BN : BO, that is, BA : BD : :BM : BO, by conversion and alternation, AD: MO:: AB: MB. Hence the corollary is manifeft. 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 same angles to any radius whatsoever. In such tables, which are called Trigonometrical Tables, the radius is either supposed i, or some number in the series Io, 100, 1000, &c. The use and conftruction of these tables, are about to be explained. VIII. The difference between any angle and a right angle, or be tween any arch and a quadrant, is called the H K complement of that angle, or of that arch. L A of the angle ABC, and B В D the arch HC the complement of AC; also the complement of the obtuse angle FBC is the angle HBC, its excess above a right angle ; and the complement of the arch FC is HC. IX. The fine, tangent, or secant 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. CORO |