6. A Verfed Sine is the Segment of the Diameter intercepted between the Arc and its Sine: Thus FA is the Verfed Sine of the Arc B A, and FD of the Arc BD. 7. Whatever Number of Degrees an Arc wants of a SemiCircle is called its Supplement. 8. That Part of the Radius which is betwixt the Center and Sine is equal to the Co-fine; thus CF is IB. = 9. If an Arc be greater or lefs than a Quadrant, the Sum or Difference of the Radius and Co-fine is equal to the Verfed fine. In a Triangle are fix Parts, viz. three Sides and three Angles: Any three of which being given, except the three Angles of a plane Triangle, the other three may be found either mechanically, by the help of a Scale of equal Parts and Line of Chords, or by an Arithmetick Calculation, if, fuppofing the Radius divided into any Number of equal Parts, we know how many of thofe equal Parts are in the Sine Tangent, or Secant of any Arc propos'd: The Art of inferring which is called Trigonometry, and is either Plane or Spherical. A Method of computing the natural Sine, Tangent or Secant of any Arc immediately, from the Length of the Arc being given. I The Length of any Arc is readily obtain'd from the Ratio of the Diameter of the Circle to its Circumference, exhibited by Van Ceulen, fince prolong'd and confirm'd by others, which is As 1 To 3 141592653589793238,'&c. [See the first Schol. to Sol. 2. Prob. 3. Chap. 3. Part III.] This Number, the Radius being 1, is the juft Length of the Semi-circle or Arc of 180°; whence any lefs Arc is eafily got by Divifion. Thus the Number of Minutes in 180° is 10800; by which 3.14159 &c. being divided, gives 00029088820866572159 + for the Length of the Arc of 1 Minute, which being multiplied by the Number of Minutes contained in any other Arc, ferves readily to give its Length. Hence, by Sir If. Newton's Series, publifh'd by Dr. Halley, in Phil. Trans. N° 219, the Sine, Cofine, Tangent, &c. of any Arc are had. I Thus if the Length of any Arc be puta, and Radius 1, then is the Natural Sinea-6) aaa-20)a a B-42) a a C-72) aa D-110) aa E, &c. putting B, C, D, E, &c. for the fecond, third, fourth, fifth, &c. Terms. [See Schol. 1. to Sol. 1. Prob. 3. Chap. 3. Part III.] Co-fine 1-2) a α-12) a a B-30)a a C-56) aa D-90) aa, &c. putting B, C, D, E, &c. for the fecond, third, fourth, fifth, &c. Terms. [See Schol. 2, to the last mention'd Prob.] *Y 2 Tangent Tangenta+3) a3 + 15 ) 2 as + 315) 17 a2 + 2835) 62 +155925) 1382 a11 + &c. Co-tangenta) 1 — 3) a — 45) a3945) 2 as 4725) a 93555) 2 a°. 638512875) 1382 a &c. Secant12) a2 + 24) sa + + 720) 61 a6 + &c. &c. - 2 Examples. 1. Let it be required to find the Sine and Co-fine of 5 Mi nutes. .co029088820866572 Arc of 00°. 01'. 5 .00145444104332860 Arc of 00°. 05' = a. The Sine of co°.05' = .0014544405 3054 adly, For the Co-fine of 5 Minutes, or the Sine of 89°. 55' The Co-fine of 5 Min. = .99999894250081 2. If the Sine of 29°. 55' be required. The Number of Minutes contained therein is 1795, by which ⚫000290888 &c. being multiplied, make the Length of the Arc = • 52214433455497 =a. •52246776602850 – •02372787732584= •49873988870266 the Sine of 29° 55'. : Since thefe Series converge the fwifteft near the Beginning and End of the Quadrant ; for raifing a Table no more than the firft or last 30 Degrees need be calculated. The rest are to be obtained from them by fuch Methods as fhall be fhewn farther on. 1. BF Elem.) Therefore:CBq¬BFq:= =CF, by Tranfpofition and Evolution; that is C √: Rq-Sq: =Σ. 2. BF the Sine of an Arc being given; to find AN the Sine of half the Arc. CF the Co-fine is known (by the firft) and confequently FA: Then:BFq+FAq: BA (by 47 1. Eucl. El) and BAAN (by the third Definition); i. e. √ : Sq +vg = S3 Arc. 3. To find the Sines of double, treble, quadruple, quintuple, &c. of any Arc, whofe Sine is given fucceffively. If from A, the End of the TV Diameter AB of a Circle 2, 222, and = any Co-fine of the Arc AD, as the Sine of n times the Arc AD is to the Sum of the Sines of n I times the Arc AD, and n+1 times the Arc AD. Draw the Chords AN, AM and AL, which AL produce to T, drawing MT=MA. Drawlikewife the Chords LM, MN, as alfo BE, and the Radius CE; and lastly CF L BE. = = By 20. 3 Euch El. LECA2 LÉBA; therefore SLEBA SArc DA; confequently the Co-fine of the Arc DA, or of EBA is BF, which is FE; wherefore BE is 2 Arc DA. Again, the Angle LAM MAN= EBA (by 27. 3 Eucl. El.) CEB LTM (by 5. 1. Eucl. El And the Sum of the Angles A NM and ALM is (by 22. 3 Eucl. El.) = 2 L=ALM+TLM; wherefore the Angle ANM TLM; alfo LM NM: Confequently the Triangles AN M and TLM are fimilar and equal to one = a rother another: Whence TL-AN. The Triangles BCE and AMT are fimilar: Wherefore CE BEAM. AT:: AM · AT; that is, (by 3d and 4th preceding Definitions, and by what is above faid) R (CE) • 2 2 Arc DA (BÉ):: Sine of 2 times the Arc AD(or Sine of Arc A M, which is AM). Sine of n I times the Arc AD (or Sine of Arc AL, which is AL) + Sine of 2+1 times the Arc AD (or Sine of Arc A N, which is A N) AT. W.W.D. By this one Rule, after the firft Sine, and confequently (by the firft) double its Co-fine are obtain'd, the whole Work may (if need) be accomplish'd. Thus ; SDA..$2DA. Note DA, or Arc DA. R2EDA:: S2DASDAS 3 DA. Then SDA+S3 DA-SDA-S3DA. S3DA S2DA+S4DA. &c. S2DA+S4DA-S2DA S4DA. = And it has thefe Advantages, the two firft Tems are inva riable, the first being the Radius = 1, Divifion therefore is wholly excluded; the second being fix'd, a fmall Table of its Products, to 10, turns Multiplication to Addition. 4. Having given the Sines of all the Arcs, from the Beginning of a Quadrant, to any Part thereof diftant from each other by an equal Interval; thence to find the Sines of all Arcs to the Double of that Part. K I D B E 麺 M H F Line DE in I. Let BF, the Sine of the Arc AB, be given; as alfo E G, the Sine of any Arc A E lefs than A B, and EI ID, the Sine of the Arc BE(Arc BD) the Difference of the Arcs A B and AE; and let DH, the Sine of the Arc AD, the Sum of the Arcs A B and BE, be required. The Radius CB being drawn will interfect D H in M, and will GA be to, and bifect the straight Draw IK and EL Is the Radius A C. Then the As DIK and DEL being fimilar, and DE= 2 DI, DL must therefore be 2 DK. The Ld As CBF (CMH, DMI) and DIK are fimilar; wherefore CB - R CF DIEI DK; therefore R 2 CF:: EI 2DKDL-DH-ÉG: Confequently EG + =DH. EIX 2 CF R Scholia. 1. If A B be 30 Degrees, then 2 CF(=2E 30° 2 S60°) will be Rx3, and EGEIX√3 = DH: That is to fay, the Sine of an Arc lefs than 30 Degrees, added to the Sine of its Defect x 3, is the Sine of an Arc fo much exceeding 30 Degrees, as the other wanted of 30 Degrees. Ex. gr. S 19°S 11° x √3 = S 41°. 2. If AB be 60 Degrees, then 2 CF will be = = R (for 2 Σ 60° 2 S 30° Chord of 60° is R, by 15. 4 Eucl. El) and EGEIDH: That is to fay, the Sine of an Arc lefs than 60°, added to the Sine of its Defect, gives the Sine of an Arc as much exceeding 60 Degrees. Ex. gr. S41°+S19°S 79°; S79-S41°-S19°; Or S 79°-S19° S41°; that is, if from the Sine of an Arc exceeding 60 Degrees, the Sine of the Excess be subtracted, there will remain the Sine of an Arc wanting fo much of 60 Degrees. Having found the Sines and Co-fines, the Tangents, Secants, &c. may be found by the following Proportions. The As CFB and CAG(See Fig. in Page 322) are fimilar; and the As CIB and CHK are alfo fimilar; there CI(FB) IB(CF):: CHHK. Plane Trigonometry is folv'd by the Help of four fundamental Propofitions, call'd Axioms. A |