Pl. Trig. LEMMA II. 'N unequal circles, the arches, upon which equal angles at the centre ftand, have the fame ratio to the circumferences of the circles. Let ABC, GBH be equal angles at the centre of the circles ACD, GHK, ftanding upon the arches AC, GH: and because ABC is an angle at the centre, it is to four right angles, a LEM. 1. as a the arch AC to the circumfe rence ACD. For the fame reason, b11. 5. ference ACD, as b the arch GH to NOTE. The arches AC, GH are C H faid to be fimilar, when they have the fame ratio to the circumferences ACD, GHK. F two magnitudes confift of parts each equal to the fame magnitude; they have to one another the fame ratio which the numbers denoting the parts of which they confift, have to one another. Let A, B confift of parts each equal to C: and let I be unity, and M the number of parts equal to C, in A; and N the number of parts equal to C, in B: A is to B, as M to N. Because there are as many parts equal to C, in A, as there are units I in the number M; A, M are A CB MIN a C. 5. equimultiples of C, I; therefore A is to C, as a M to I: and b Cor. 4.5. B, N are equimultiples of C, I; therefore A is to B, as M to N. LEM MA HAL ALF the fum of two unequal magnitudes, together with half their difference, is equal to the greater; and half the difference taken from half the fum is equal to the lefs. Let Let AB be the greater of two unequal magnitudes, and BC the lefs; therefore AC is their fum: make AD equal to BC; therefore BD is their difference: and if BD be bisected in E, BE or ED is half their difference: and because AD is equal to BC, and DE to EB; the whole AE is equal to EC; therefore AE DE B C or EC is the half of the fum AC. But the greater magnitude AB is equal to AE, EB together; that is, to half the fum added to half the difference; and the lefs BC is equal to the excess of EC half the fum, above EB half the difference. AN A DEFINITIONS. I. N arch of a circle is called the measure of the angle at the centre ftanding on that arch. Thus, the arch AC is the measure of the angle ABC. II. Pl. Trig. Y The circumference of every circle is fuppofed to confist of 360 equal parts, each of which is called a degree: and a degree is fuppofed to confift of 60 equal parts, called minutes; and a minute of 60 equal parts, called feconds; and fo on. Also, an angle at the centre is faid to be of as many degrees, minutes, &c. as there are in the arch, which is the measure of that angle. COR. I. A right angle is an angle of 90 degrees. For the measure of it is the fourth part of the circumference. COR. 2. Arches of different circles, which measure the fame angle, contain the same number of degrees, and parts of a deFor the number of degrees and parts of a degree in gree.. each of them is to 360, as each of the arches to its circumference; that is, as the angle which they measure to four a LEM. 3right angles ». III. The fine of an arch is a straight line drawn from one extremity COR. 1. The fine of AH, the fourth part of the circumference, is COR. 2. The fine of an arch is the half of the chord of twice IV. The verfed fine of an arch is the fegment of the diameter, to which the fine is perpendicular, between the fine and the arch. Thus, AD is the versed fine of the arch AC. b LEM. 1. € 3.3. V. The tangent of an arch is a ftraight line touching the circle at one extremity, and meeting the diameter that paffes through the other extremity. Thus, AE is the tangent of the arch AC. COR. The tangent of the eighth part of the circumference is VI. The fecant of an arch is the ftraight line drawn from the F VII. The fine, verfed fine, tangent, G M and fecant of an arch, are alfo called the fine, verfed fine, tangent, and fecant of the angle, of which that arch is the meafure. Thus, CD is the fine of the angle ABC; AD is its versed fine; AE is its tangent; and BE its fecant. COR. The radius is equal to the fine, or verfed fine of a right angle; or to the tangent of half a right angle. VIII. The difference between an arch and the half of the circumference, or between an angle and two 1ight angles, is called the fupplement of that arch, or of that angle. Thus, the arch CF is the supplement of the arch AC; and the angle CBF is the fupplement of the angle ABC. COR. The fine, tangent, and fecant of any angle ABC, are also the fine, tangent, and fecant of its fupplement: For, by Def. 4th, CD is the fine of the arch CF, that is, of the angle CBF, by Def. 7th: and if CB be produced to G, it is manifeft, from Def. 5th and 6th, that AE is the tangent, and BE the fecant of the arch AG, that is, of the angle ABG or CBF, by Def. 7th. IX. The difference between an arch and the fourth part of the circumference, or between an angle and a right angle, is called the complement of that arch, or of that angle. Thus, if BH be at right angles to AB, the arch CH is the complement of the arch CA, and the angle CBH is the complement of the angle ABC, or of CBF. X. The fine, tangent, or fecant of the complement of any angle, is called the cofine, cotangent, or cofecant of that angle. Thus, if CL, HK be perpendiculars to BH; CL is the cofine of the Pl. Trig. angle ABC, and HK its cotangent, and BK its cofecant. COR. 1. The fegment of the diameter to which the fine is per pendicular, between the fine and the centre, is equal to the cofine. Thus, BD is equal to CL, the cofine of the angle d 34. 1. ABC. COR. 2. The cofine of any arch is to the fine, as the radius to the tangent of the fame arch. For the triangles BDC, BAE are equiangular, because the angle ABE is common, and e 32. I. BDC, BAE are right angles; therefore BD is to DC, as BA to AE f. COR. 3. The radius is a mean proportional between the cofine and the fecant of any angle. For BD is to BC or BA, as BA to BE. £ 4. 6. COR. 4. The radius is a mean proportional between the tangent and cotangent of any angle. For, the alternate angles ABE, BKH are equal; and BAE, BHK are right angles; therefore g 29. 1. the triangles BAE, BHK are equiangular; wherefore EA is to AB, as BH or BA to HK. COR. 5. The tangent and cotangent of any angle are reciprocally proportional to the tangent and cotangent of any other angle. For the tangent of the firft angle is to the radius, as the radius to its cotangent; and the radius is to the tangent of the other angle, as the cotangent of that other angle to the radius: Therefore, by perturbate equality, the tangent of h 22. 5. the firft angle is to the tangent of the other, as the cotangent of the other to the cotangent of the first. IN PROP. I. Na right angled triangle, if the hypothenufe, or fide opposite to the right angle, be made the radius of a circle; the other fides are the fines of the angles oppofite to them, or the colines of the angles adjacent to them. And if either of the fides about the right angle be made the radius, the other fide is the tangent of the angle oppofite to it, and the hypothenufe is the fecant of the fame angle. Let ABC be a triangle, having the right angle BAC: and from the centre B at the diftance BC, defcribe a circle meeting BA produced in D: and because AC is drawn from C the extremity of the arch CD, perpendicular to the radius BD, which paffes through the other extremity D; CA is the fine of the Pl. Trig. the arch CD, or of the angle CBD boppofite to CA; and BA the fegment of the diameter, to which CA is perpendicular, a 3. Def. between the fine CA and the centre c 1. Cor. 10. Def. 7. Det. B, is the cofine of the angle ABC c. In the fame manner, if from the centre C, at the distance CB, the arch BE be defcribed, it may be proved, that BA is the fine, and AC the cofine of the angle ACB. Again, from the centre B, at the diftance BA, defcribe the arch AF. And because BAC is a right angle, fCor.16.3. AC touches the circle at Af ; and it meets the diameter paffing through F, in C; therefore AC is d 5. Def. the tangent of the arch AF, or of the angle ABC: and BC, which is drawn from the centre to the extremity C of the tane 6. Def. gent, is the fecant of the fame angle ABC. b 12. I. F two unequal arches measure the fame angle, the fine, verfed fine, tangent, and fecant of one of them, have to the radius of that arch, the fame ratios, which the fine, verfed fine, tangent, and fecant of the other arch, have to the radius of it. Let AB, DE be two unequal arches, which measure the angle a ri. or ACB: and draw BF, AG, EH, DK at right angles to AC: Then, BF is the fine ↳, AF the verfed fine, AG the tangent and CG the fecant of the arch AB: and EH is the fine 5, Def. the versed fine, DK the tangent a, and CK the fecant of the e 6. Def. arch DE. And because CFB, 3. C 4. Def. 5. Def. f 32. 1. 8 4. 6. e CHE are right angles, and FCB common to the triangles CFB, CHE; they are equiangular f; and therefore FB is to BC, as g HE to EC; that is, the fine of the arch AB is to the radius BC, as the fine of the arch ED to the radius CE. Alfo CB or CA is to CF, as CE or CD to CH; h E. 5. and, by converfion h, CA is to C e B K HD AF, as CD to DH; that is, the radius CA is to the verfed fine AF, as the radius CD to the verfed fine DH. In the fame manner, it may be proved, that the tangent AG is to the radius AC, |