IV. QUESTION 569 anfwered by Mafters J. Paty and J. Ofborce, Youths of about 13 Years of Age, at the Mathematical Academy, Bristol. LE ETx the Annuity, y the Amount of 1. in one Year, a = 500, m = 1100; then, by the Doctrine of Compound Intereft and Annuities (See Note, P. 350 of Donn's * Arithmetic). ay - a my -m -I = (x), and confequently (putting 10 -I a 2n) y 10 -2nys = c; whence (completing the a Rate of Intereft = 3 £. 145. 3 d. 4 per Cent. per Annum, and the Annuity required 92. 16 s. 10d.. And thus it is, alfo, answered by Mefirs. 7. Addifon, T. Barker, J. Bennett, T. Bofworth, N. Brotunell, . Buddie, J. Dalby, Nujo Dargnas, W. Dent, W. Dixon, 7. Dymond, T. Hague, E. Jones, G. Lodge, J. Nordon, J. Probert, W. Rawle, T. Robinson, J. Roper, A Rowe, W. Servell, P. Sharp,. C. Smith, W. Spicer, W. Stoker, T. Todd, S. Vince and J. Toung.. very nearly. Pamphagus obferves that "fince the Annuity for born 5 Years amounts "to 500. and when forborn 10 Years to 1100. it is manifeft that "the Intereft of 500 £. for 5 Years must be = 100. Therefore the "Rate per Cent. will be eafily found 3. 145. 3 d. 4.19; and "thence, by a well known Theorem, the Annuity 92 £. 16 s. 10d. I .76."—And, from this Confideration of the Matter, Meffrs. Jer. Ainfworth and W. Wales have, likewife, answered it. CR N G M C AK B V. QUESTION 570 answered ly Mr. Wm Cole. ONSTRUC. A B Right-line joining the Centers of the given Circles interfecting each other in I) conceive a Semi-cir. to be defcribed, and apply therein, from either A or B, the Chord A C=9.5 (half the given Line), and parallel thereto let F G be drawn through I, terminating in their Peripheries, and it will be the Line required. F AC DEMON. Through C draw BD, and parallel to it, A E, meeting FG at right Angles (by 31.3 and 29.1 Eu.) and bifecting the Chords IG, IF (by 3.3. Eu.), and then FE+DG will be ED (9.5, per Conftruc.), and confequently F G 2 AC 19. CALCULA. Join the Points A, I and B, 1: Then, in the A AIB the Sides being given, the LABI is found 36° 52′ 12′′; which taken from the ABC (found 71° 48' 18", from the right angled A ACB) leaves 34° 56′ 6′′ L. IBD: Whence, from the right-angled AIDB, is found I D 4.5811; and therefore the Segment in the greater Cir. is 9.1622 &c. and that in the *NOTE, This ingenious Gent. intends, shortly, to favour the Public with a new, complete Courfe of the Mathematics in 8vo. a Thing very much wanted; and whereof the 1st and 2d Vols, are already published. 1 Jeffer 9.8377 &c. Inches.If the Chord A C be infcribed the other Way from B, the Conftruc. and Method of Calcula, will be the fame, and the Segment in the greater Circ, will, in that Cafe, be found 15,1684 &c. and that in the leffer = 3,8318&c. Mellrs. J Ainsworth, R. Butler (the Propofer), T. Bofworth, J. Buddle, J. Dalby, Nujo Dargnas, J. Dymond, R. Gibbons, C. Hutton, E. Jones, Mifs Ann Nicholls, Pamphagus, J. Paty, R. Pulman, A. Rowe, W. Sewell and W, Wales conftruct it in this Manner likewise, very nearly, Algebraic Solution to the fame by Mr. John Addifon. FI; then will IG a x, and (per 3. e. 3) E I x, and ID = ax. Then, the Sides of the ▲ AIB. being in the Ratio of 3, 4 and 5 (per Queft.), the AIB, oppofite the longeft of them, will, therefore, be a right Angle, and, confequently, the right-angled As AEI, BDI fimilar; whence b.. (AI): * (EI) : { c (BI) : ➡BD, and (pr. 47. e. 1) 2 , or (in Numbers), x cx 13.68x= 37.8; folved, x (FI) 9.8375,. or 3.8423; and confequently I G, 9.1625, or 15.1577, &c. In this Manner, nearly, it is, alfo, anfwered by Meffrs. T. Barker, W Dixon, J. Probert, W. Rawle, T. Robinfan, J. Roper, S. Vince and 7: Young. VI. QUESTION 571 anfwered by Mr. J, Chipchase. TONSTRUC. From P, the Port failed be drawn to reprefent the magnetical or N P defcribe a Semi-cir. and infcribe therein the Chord PC 25 Miles (the given Difference of Lat.), and PC will be the Direction of the true Meridian, AC the Departure and the NPC the Variation required; the Demonftration of which is evident from the Nat. of turning to Windward, &e. CALCULA, In the AAPT are given PT, TA together with their included Angle, whence AP is found 41.6694, and the L APT 1 90° 29', nearly; from which take the APC (found 53° 8', from the right-angled A ACP) and the Remainder 37° 21 (CPT) taken from the NPT ( 50° 37° 1, per. Confruc.) leaves the NPC 13° 15'4 the Variation required, which is Wefterly. In this Manner, nearly, it is alfo conftructed by Meffrs. 7. Addifon, T. Barker, T. Bosworth, J. Buddle, J. Dalhy, W. Dixon, R¿ Gibbons, C. Hutton, E. Jones, Mifs Ann Nicholls, Pamphagus, J. Probert, W. Rawle, T. Robinfon, J. Roper, A. Rowe, Mrs. E. Suggett, H. Watkins jun. J. Young and W. Wales. Meffrs. W. Spicer and W. Stoker have given neat algebraical Solutions to it. VII. QUESTION 572 answered by Mr. W. Wales. Co-f. of roo(the given Diff. of Lat, of the two Places), Pusof. of 560 45 2 (their given neareft Dift.) and & Co-f. of their Difference of Longitude required (Rad. 1); then, the Sine or Co-f. of the Sum of two Arcs being every-where equal to the Sine or Co-f, of the Sum of their Complements, the Co-f of the Sum of the Co-latitudes of the faid Places will bex (per Queff.); and 2c24 C* (per Simp. Trig. Prop. 27. P. 74) — 1 — ≈ (the VerfedSipe of their Diff. of Long.): Whence, by Reduction and putting & But +25, there results x = ¿2+2a-1+ &= .34199, VIII. QUESTION 573 an/wered by Pamphagus. W The fame answered by Mr. N. Brownell. THE given Equat, being 0.8 Log. of =y, put n=4, ≈➡ Hyp. · and m = 0.43429448, and then 2 zx” (— y) is a Man. 0.8 Vis and 0.5790593 Tabular Log, of, or the Log. of x Log of o.—.5790593 X3 7.9720921, and the Natural Number, answering thereto, is 0.009377608; which, multiplied by 100000, gives 937 £. 15 s. 2 d. 1 the Lady's Fortune, Meffrs. 7. Ainsworth, T. Barker, J. Bennett, J. Buddle, J. Dally, W. Rawle, T. Robinson, J. Roper, W. Sewell, P. Sharp, E. Smith, W. Spicer (the Propofer), and S. Vince have alfo anfwered it in this Manner nearly, and bring out the fame Conclufion. IX. QUESTION 574 answered by Mr. J. Leader. SUBSTITUTE x for y in the gi C UBSTITUTE vx for y in the given Equation, and it becomes a 3 -x11+do10 x10; and thence x = d v 10 + c v 6 — a v 3 + b v 2; y ( = v x ) = dv11 + c v 7 — eut+b3, and confequently, yx (the Flux. of the Area) = 9 7 a207 -20000426205 5 fis the Area required (f being the Sum of the Coefficients of the Powers v).— Then, from the given Equation in Fluxions (by Tranfpofition, Divifion and multiplying by y) is had 2 b x 8 y 2 — 3 α x 7 y 3 + 6 c x 4 y 6 + sody II X · 8 b x 7 y 2 + 7 u xбy 3 — 4 cx 3 y quired. 10 6 уй y the Subtangent re Anfwers to this Quest. have, alfo, been received from Meffrs. J. Ainsworth, T. Allen, T. Barker, Pamphagus, T. Robinson, W. Sewell, P. Sharp, and S. Vince (the Propofer). X. QUESTION 575 answered by Pamphagus. the Cent. of Grav. of the faid Solid from its Vertex, which (per 25 Queft.) is == x; and confequently n➡8.- -Having thus ob 34 tained the numerical Value of n, we fhall, in the next Place, have =c) j√1+cy3 = the Flux. of the Curve's Length; and the correct Fluent * thereof See Simpfon's Flux. Art, 84. Curve required. Meffrs. 7. Ainsworth, T. Allen, T. Barker (the Propofer), J. Bennett, T. Bofworth, N. Brownell, T. Todd, W. Rawle, W. Servell, W. Spicer and S. Vince answer it in the fame Manner, very nearly. XI. QUESTION 576 answered by Mr. T. Todd. c3.1416 (the Circumf. of the Circ. whofe Diam. is r), 11718 (the given Solidity of the required Cone), x its Alt. y the Rad. of its Bafe and p 39.2 Inches (the Length of a Pendulum vibrating Seconds); then (per P. 225 of Simp. Flux.) 4x2+ y2 the Dift. of the Cent. of Ofcillation from its Vertex (ar 5 x Point of Sufpenfion), and (by the Doctrine of Pendulums, their Lengths being inversely as the Squares of the Numb. of Vibrations made by 60" 12 x 5px them in the fame Time) Vibrations performed by (the Square of the Numb. of the Cone in 1) = x2 (per Quest.); from which, and the Equation cxy2=s, is found x ( 4500 p· 35 4 3 S 56.039 &c. and y ( 5.426 &c. Inches. CX 39.2 Inches. 1728, p = 3.1416, Mr. Ja. Young puts a Rad. of the Bafe of the required Cone, in Inches: then pxys, and (per Pa. 239 of Emer. Flux. 1A. the Height and y Dift. of the Cent: of Ofcillation from its Vertex, or the Length of a Pendulum Ifochronal to the Cone: Whence, and the preceding Equation, x comes out 56.0394 &c. and y = 5.4264 &c. Inches. In the fame Manner as above, nearly, the Answer is, alfo, given by Mers. 7. Addison, J. Ainsworth, T. Allen, T. Barker, T. Bafworth, N. Brownell, J. Buddle, W. Dawes, W. Dent, W. Dixon, E. Jones, Pampbagus, W. Rawle, T. Robinson, W. Sewell (the Propofer), C. Smith, E. Smith, S. Vince, and W, Wales. |