metrical tables, together with the investigation of fome theorems, useful for extending trigonometry to the solution of the more difficult problems. SECTION I. A LEMMA I.. N angle at the centre of a circle is to four right angles as the arch on which it ftands is to the whole circumference. Let ABC be an angle at the centre of the circle ACF, ftanding 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 perpendicular to AE. Then, because ABC, 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 arch AD, (36.6.); and therefore alfo, 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 ftand on arches which have the fame ratio to their circumferences. For, if the angle ABC, at the centre of the circles ACE, GHK, ftand 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 ftraight lines interfect one another in the centre of 2 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 60 equal parts, each of thefe is called a minute; and, if a minute be divided into 60 equal parts, each of them is called a fecond, and fo on. And as many degrees, minutes, feconds, &c. as are in any arch, fo many degrees, minutes, feconds, &c. are said to be in the angle measured by that arch. 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 degree in the arch, which is the measure of that angle, is to 360. COR. 2. Hence alfo, 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 fame ratio to the number 360, that the angle which they measure has to four right angles (Cor. Lem. 1.) The degrees, minutes, feconds, &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. III. Two angles, which are together equal to two right angles, or two arches which are together equal to a femicircle, are called the supplements of one another. IV. H L K A ftraight line CD drawn through C, one of the extremities, of the arch AC, perpendicular to the diameter paffing through the other extremity A, is called the Sine of the arch AC, or of the angle ABC, of which AC is the measure. F B D 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 fegment DA of the diameter paffing 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 ftraight line AE touching the circle at A, one extremity of the arch AC, and meeting the diameter BC, which paffes 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 ftraight 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 fecant of any angle ABC, are likewise the fine, tangent, and fecant of its fupplement CBF. It is manifeft 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 alfo manifeft, that AE is the tangent, and BE the fecant, of the angle ABI, or CBF, from Def. 6. 7. E COR. to Def. 4. 5. 6. 7. The fine, verfed fine, tangent, and fecant of an arch, which is the measure of any given angle ABC, is to the fine, verfed fine, tangent and fecant, of any other archwhich is the measure of the fame angle, as radius of the first arch is to the radius of the second. the B OM D 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 fecant of the arch AC, according to Def. 4. 5. 6.7.; NO the fine, OM the verfed fine, MP the tangent, and BP the fecant of the arch MN, according to the U 4 fame fame definitions. Since CD, NO, AE, MP are parallel, CD: NO:: rad. CB: rad. NB, and AE: MP:: rad. AB: rad. BM, alfo 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 manifeft. And therefore, if tables be conftructed, exhibiting in numbers the fines, tangents, fecants, and verfed 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 fuppofed 1, or fome number in the feries 10, 100, 1000, &c. The ufe 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 complement of that angle, or of that arch. Thus, if BH be perpen dicular to AB, the angle CBH is the complement F bove a right angle; and the complement of the arch FC is HC. IX. The fine, tangent, or fecant of the complement of any angle is called the cofine, cotangent, or cofecant 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 |