PROPOSITION X. THEOREM. Let the Arcs AB, BC, CD, DE, EF, &c. be equal ; and let the Subtenses of the Arcs AB, AC, AD, A E, &c. be drawn; then will A B :AC::AC:A B+AD:: AD:AC+AE ::AE:A D+AF:: AF: AE+AG. 1 L ETAD be produced to H, A E to I, A F to K, and AG to L, so that the Triangles ACH, ADI, AEK, AFL, be Isosceles ones : Then, because the Angle BAD is bisected, we shall have DH=A B (by the.last Prop.); fo likewise thall EI= A C, FK=AD, allo G L=A E. But the Itosceles Triangles A B C, ACH, ADI, A EK, A FL, because of the equal Angles at the Bases are equiangular : Wherefore it shall be, as A B: AC:: AC:(AHA) A B+AD::AD:(AI) ACHAE:: AE: (A K=) A D+AF: AF: (AL=) A E+AG. W. W. D. 1 2 1 :: Goroll, 1. Becaufe A B is to A C, as Radius is to douz ble the Cosine of the Arc A B, it shall also be (by Coroll. Prop. 4.) as Radius is to double the Cofine of the Arc A B, fo is LAB: LAC:: LAC: A B+ ADAD: ŽAC+ AĚ AE: AD+AF, C. Now let each of the Arcs AB, BC, CD, &c. be 2'; then will LAB be the Sire of one Minute, ; AC the Sine of 2 Minutes, A D the Sine of 3 Minutes, A E the Sine of 4 Minutes, &c. Whence, if the Sines of one and two Minutes be given, we may easily find all the other Sines in the following manner. Let the Cofine of the Arc of one Minute, that is, the Sine of the Arc of 89 Deg. 59', be called Q; and make the following Analogies; R.:2Q:: Sin. 2' :SI'+S. 3: Wherefore the Sine of 3 Minutes will be given. Alfo, R.:2Q:: S.3: S. 2+3.4'. Wherefore the S. 4' is given. And R.: 2 Q::S. 41:S. 3i+5.5'i and so the Sine of 3' will be had. Likewise, R.:2Q::S. 5': S. 4'+S. 6'; and so we Thall have the Sine of 6i. And in like manner, the Sines of every Minute of the Quadrant will be given. And because the Radius, or the first Term of the Analogy, is Unity, the Operations will be with great East and Expedition calculated by Multiplication, and contracted by Addition. When the Sines are found to 60 Degrees, all the other Sines may be had by Addition only (by Cor. 1. Prop.6.) The Sines being given, the Tangents and Secants may be found from the following Analogies (in the Figure for the Definitions); because the Triangles CED, CBG, CHI, are equiangular, we have CE:ED::CB:BG; that is, Col.: S.::R.:T. GB:BC::CH:HI; that is T.:R.:: R.; Cot. CE:CD::CB:CG, that is, Cof: R.::R. : Secant, DE:CD::CH:CI; that is, S.:R.::R. ; Cofec. Egc A6 SCHOLI U M. That great Geometrician, and incomparable Philosopher, Sir Haac Newton, was the firf that laid down a Series converging in infinitum ; from which, having the Arts given, their Sines may be found. Thus, if an Arc be called A, and the Radius be an Unit, the Sine thereof will be found ta be. A3 A5 A7 A9 A-+ + 1.2.3.4.5.6.7 1.2.3.4.5.6.7.8.9 AS - M&C. 1.2.3.4 1.2.3.4.5.6 1.2.3.4.5.6.7.8 These Series in the Beginning of the Quadrant, when the Arc A is but smalia Joon converge. For in the Series for the Sine, if A aves not exceed 10 Minutes, the two first Terms thereof, viz. A- A’, give the Sine 2015 Places of Figures. If the Arc A be not greater than in one Degree, ihe three firf Terms will exhibit the Sine to 15 Places of Figures; and fo the jaid Series are very useful for finding the fir; and last Sines of the Quadrant. But the greater the Arc A is, the more are the Terms of the Series required to have the Sine, in Num + 1.2 Numbers, true to a given Place of Figures. And then, when the Arc is nearly equal to the Radius, the Series converges very slow, and therefore, to remedy this, I bave devised other Series, similar to the Newtonian ones, wherein I suppose, the Arc, whose Sine is fought, is the Sum or Difference of two Ares, viz. Atz, or A-2: And let the Sine of the Arc A be called a, and the Cofine b. Then the Sine of the Arc Atz will be expressed thus : + az3 azs I bz az? bz3 bzs I. at + -EC. 1.2 1.2.3 1.2.3.4 1.2.3.4.5 And the Cofine is az bz? bz4 bz 2. b -+ 1:2 1.2.3 1.2.3.4 1.2.3-4.5 1.2.3.4.5.6 In like manner the Sine of the Arc A-zis bz a22 bz3 az4 bzs 3. a ta + 1.2 1.2.3 1.2.3.4 1.2.3.4.5 1.2.3.4.5.6 And the Celine is bza bz4 azs 4. bt + 36. azo az az3 Sines are bz az? has azt a26 + bzs 5. I 1.2 1.2.3 1.2.3.4 1.2.3.4.5 1.2.3.4.5.6 bz az? bz3 az4 bzs a26 6.-+ -+ -830. I 1.2 1.2.3 1.2.3.4 1.2.3.4.5 1.2.3.4.5.6 Whence the Difference of the Differences, or the second Difference, will be 2az? 2a24 7. €50. 1.2.3.4.5.6 2azó 25 Places. iDbich Series is equal to double the Sine of the mean Arc; drawn into the Verfid Sine of the Arc z, and converges very foon. S. that if z be the firp Minute of the Quadrant, the first Term of the Series gives the second Difference to 15 Places of Figures, and the second Term to From hence, if the Sines of the Arcs, diftant one Mirute from each other, be given ; the Sines of all the Arcs, that are in the same Progrifian, may be found by an exceeding easy Operation. In the fir fi and second Series, if A=0; then shall a=0, and bits Cine will become Radius, or 1. And hence if the Tens herein ais, are taken away, and i be put instead of b, tbe Series will become the Newtonian. In the third ard fourib Series, if A be 90 Degrees, we shall have b=o, and ai. IV hence, again taking away all the Terms wherein b is, and putting i inftead of a, we shall have the Newtonian Series arise. Note, All the said Series easily flow from the Newtonian ones. By the fifth Proposition. PROPOSITION XT. . THEOREM. In a Right-angled Triangle, if the Hypothenuse be made the Rodius, then are the sides of the Sines of their opposite Angles ; and if either of the Legs be made the Radius, then the other Leg is the Tangent of its opp frie Angle, and the Hypothe nuje is the Secunt of that angle. IT Tis manifeft, that C B is the Sine of the Arc D, and A B the Cofine thereof; but the Arc C D is the Measure of the Angle A, and the Complement of the Measure of the Angle C: Moreover, if A B in the second Figure to this Propofition, be supposed Radius, then BC is the Tangent, and AC the Secant of the Arc BD, which is the Measure of the Angle A. So also, if BC be made the Radius, then is BA the Tangent, and AC the Secant, of the Aic BE, or Angle C. W.W.D. Therefore, as AC, being taken as some given Measure, is to BC taken in the fame Measure; fo Thall the Nomber 10,000,000 Parts, into which the Radiús is supposed to be divided, be to a Number expressing, in the fame fame Parts, the Length of the Sine of the Angle A; that is, it will be, as AC:BC::R:S, A. by the same Reason, as AC : BA :: R:S, C. also, as A B : BC ::R: T, A. and, as BC:BA::R: T, c. And so, if any three of these Proportionals be given, the fourth may be found by the Rule of Three. PROPOSITION XII. THE O R E M. Sides of plane Triangles are as the Sines of their opposite Angles, I F the sides of a Triangle, inscribed in a Circle, be bisected by perpendicular Radii ; then shall the half Sides be the Sines of the Angles at the Periphery; for the Angle B DC, at the Centre, is double of the Angle BAC at the Periphery (hy 20 El. lib. 3.); and so (the Half of B D C, viz.) B D E=B AC, and B E is the Sine of (B D E, or) B A C. For the same Reason, BF Thall be the Sine of the Angle B C A, and A G the Sine of the Angle A B C. In a Right angled Triangle we have B D=BC Radius (by 31 Eucl. 3); but Radius is the Sine of a Right Angle : Whence half B C is the Sine of the Angle A. In an obtule-angled Triangle, let B I, CI, be drawn; and then the Angle I thall be the Complement of the Angle A to two Right Angles (by 22 El. 3.); and so they shall both have the faine Sine. Angle B D E (whole sine is BE,=Angle l; therefore B E shall be the Sine of the Angle B AC. And so in every Triangle, the Halves of the Sides are the Sines of the opposite Angles: But it is manifest, that the sides are to one anocher, as their Hálves. And therefore, the sides of plane Triangles are as the Sines of their opposite Angles. W. W. D. But the |