Number that Van Ceulen had done, which not only confirms Mr. Sharp's Quadrature, but shews us, that if the Diameter be 100poo, &c. the Circumference will be 3,14159. 2653589793.23846.26433.83279.50288. 41971. 69399. 37510. 58209. 74944. 59230. 78164. 05286. 20899. 86280. 34825. 34211. 70679 x of the fame Parts. Which is a Degree of Exactnefs far furpaffing all Imagination, being by estimation more than fufficient to calculate the Number of Grains of Sand that may be comprehended within the Sphere of the fixed Stars. The late Mr. Cun's Series for determining the Periphery of an Ellipfis (who was my Predeceffor in the Mathematical School erected by Frederic Slare, M. D. and established by a Decree of the High Court of Chancery for qualifying Boys for the Sea-Service) being new and curious, this Opportunity is taken of making it Publick. Let A be equal to a Quadrant of the Circle circumfcribing the Ellipfis whofe Periphery is requir'd. Then Ax eto 1. 3.5.7.97 to 2. 4. 6.8. 10. 256 2.4.6.16 2.4.6. 8. 128 I. 3. 5. 7. 9. 11. 21 é' 2, &c. 2. 4. 6. 8. 10. 12. 1024 is the Periphery of a Quadrant of the Ellipfis where ee= tt-cc tt t being the Semi-tranfverfe Diameter, and the Semi-conjugate, When this Series came to hand, it was imperfect, inasmuch as there were only the first five Terms without the Law of Continuation: But being defirous of rendring it compleat, after fome Confideration I found the Law to be as follows: It is plain by Infpection that the Numerators and Denominators of each Term, are compos'd of Numbers that run in arithmetical Progreffion, except the laft in each Term, viz. 1, 8, 16, TAR &c, and thofe being found by the continual Multiplication of thefe Fractions, ×××× 3 × 2 × 12, &c. the Law of continuing the whole Series as above, is evident, Whence, by a well known Method of fubftituting Capital Letters for each Term refpectively, the I following Series is deduc'd, viz. Ax 1-1еe. A a 4 1.3 4.4 B &c. where the Law of Continuation is evident also, fince each Capital Letter is equal to its precedent Term, 1.3 viz. Bee, C= e B, &c. and without doubt 4.4 in Practice is preferable to the former Series: But the Investigation of that, on which this laft depends, is omitted; purely on account of its being foreign to the prefent Subject. But to return; if the Series expreffing the Length of the Arch, viz. 5+ 5+ 40 s, &c. be revers'd, we fhall have the Value of s in the Terms of a, and confequently a direct Method for finding the Sine of any Arch from its Length given. Thus. If ass +45 +125" &c. Or sa For put s Then And s I 140 a', &c. 2. 3. 4. 5. 6. 79 c. Aa+ Ba+Ca3, &c. A3 a + A, B a3, &c. = a. 4% As, a3, &c. 3 And confequently Aaa, and A1, alfo B+} Ao, and B, A', alfo C+ A B+ Aso, and C-AB-A-x --4-26 = 12 -- Wherefore A = 1, B = - %, C = 0, &c. and confequently, s=a 1209 &c. From which three Terms the Law of a3 Continuation is easily discover'd, making s=a 2.3 as + - a7 + 2.3.4.5 2.3.4.5.6.7. 2. 3. 4. 5. 6. 7. 8. 9 Whence fubftituting A for a, and we shall have A A3 &c. for the Newtonian Series, according to our Author's Form; for finding the Sine of any Arch, its Length being given.Q. E. I. Again, becaufe the Square of the Radius made less by the Square of the Sine, is equal to the Square of the Cofine, by the second Propofition of our Author's Elements of plain Trigonometry; it follows, that if from the Square of the Radius 1; be taken the Square of the Sine--a+a, &c. the Square Root of the as Remainder, will be the Cofine-1-a2+ a3———a3 &c. thus. I Saza + — a3, &c. 120 Ssa2 as+a, &c. which being taken from the Square of the Radius 1, leaves 1-aa+as ~a, &c. the Square Root of which will be the Cofine. x − a a +¦ a3 — § d3, &c. ( 1 =) · a2 A2, A4 Wherefore putting A for a, we shall have for the Cofine A6 A2 A4 + 2 720' 24 &c. or I + I. 2 1.2.3.4. A6 A8 + 1.2.3.4.5.6. 1.2.3.4.5.6.7.8. Author's Form. Q.E. I. &r. according to the But because thofe Series, as our Author obferves, converge very flow, especially when the Arch is nearly equal to the Radius, he therefore devis'd (Page 287) other Series, whose Investigation may be as follows. Let Let the Arc, whofe Sine is fought, be the Sum or Difference of two Arcs, viz. Az, or A—z: And let the Sine of the Arc A be called a, and the Cofine b. Now if the Arc DFDE, Prop. 5th of the Elements of Trigonometry be call'd z, then its Sine FO, + will by the Newtonian Series be=% 1..2. 3 &c. and because 1. 2 1.2.3.4 1.2.3.4.5.6 CD: DK::CO: OP. Therefore OP=a Again, because the Triangles CDK, FOM are fimilar, it will be as CD: CK :: FO: FM; whence But OP+FM b% FM= + &c. 1.2.3.4.5 1.2.3.4.5.6.7 IF, the Sine of the Arc BF, viz. a+ I viz. A+z; confequently the Sum of thofe Series, + ++ 1.2. 1.2.3 1.2.3.4 1.2.3.4.5 is the Sine of the Arc Az. And because FM= And again, becaufe CD: CK:: CO: CP. &c. And by reafon of the fimilar Tri angles CDK, FMO, it will be as CD: DK:: FO: MO. Whence MOaz - + But CP-MO=CI, the Cofine of the Arc A-+2. Wherefore the Cofine of the Arch Az is b And because MO=PL, therefore CP+MO CL, and confequently the Cofine of the Arc A-z bz+ azs + + Q. E. I. &c. Now the Arc A is an arithmetical Mean between the Arcs A-z and A-tz, and the Difference of their &c. or za xx2 Z+ + 1.2.3.4.5.6 the Differences, or fecond Difference is + 1.2 1.2.3.4 1.2.3.4.5.6 &c. Which Series is equal to double the Sine of the mean Arc, drawn into the verfed Sine of the Arc z, and converges very foon; fo that if z be the Arch of the first Minute of the Quadrant; our Author fays the firft Term of the Series gives the fecond Difference to 15 Places of Figures, and the second Term to 25 Places. Whence, the following Rule is deriv'd for finding the Sine of the Arc Az or A-z. RULE. From double the Sine of the mean or middle Arc, fubtract the fecond Difference found by the Theorem ; and from the Remainder, fubtract the Sine of the given Extreme, whether it be the greater or least, and the Remainder will be the Sine of the other Extream. EXAMPLE. Let it be required to find the Sine of 30° 01' the Sines of 30 00, and 29° 59' being both given. accord |