TRANSLATION AND IMITATION OF A LATIN POEM. But ah ! too soon will autumn quit the field, The birds sit moping on the naked trees, But, soon will spring in beauteous form appear, : J. BROWN, . Surfleet, May 11, 1811, ADDRESS TO MY FIRST GRAY HAIR. BY MR. THELWALL. “ And thou hast chang’d thy hue, companion staid, Of forty varying years ; thy darkest brown Shifting to silvery whiteness. Be it so: It is not the first time that I have met An old acquaintance with an altered face, And 'twill again betide me: or the wheel Of ever giddy Fortune must forego Her old propension, and no more invert My oft deluded hopes. But, of thy kind, Not Fortune's steadiest favour, nor her hate, Can stay the destin'd course. Mute monitor! Thou art, indeed, but as the harbinger Of many a change approaching; that shall soon, To all thy numerous tribe impart thy hue : Dappling at first with many a wintry spot, Till all is equal snow. Well! my firm mind, That many a less expected change hath borne, Can bear that also. Hovering Winter, hail ! Oh! give me yet, The sheltering fence of competence, to guard S mathematical Department. MATHEMATICAL QUESTIONS IN NO. IV. ANSWERED. X-10 Put x = number of gallons at first, then what the officer seized; and £9.185. — £8. 25. = o£ 1. 16s. is their value; therefore, x : £ 9. 188. ::: x-10 : £1.16s. Now multiply means and extremes, by reduction x = 22, and 9s. is the price per gallon. True answers were also sent by Messrs. Anglicus, J. Baines, B. Brooke, W. Bruster, W. Dunn, J. Gawthrop, A. Hirst, W. Harrison, J. Hine, Lucinda, R. Maffett A. Nesbit, Philpot, Rylando, J. Tomlinson, and J. Win. ward. 2x :: 1: 2. Qu. (32) answered by Mr. W. Bruster, Donningtot. By the nature of compound interest as 100:16, + +2*= the amount of £ 1. for one gear. Consequently = x (109421) + = the amourt of r pds. at 2x per cent. for 3r years, and « (100*-*)3r = 5x by the, question; hence ( and extracting the cube root (100.426) " '=1.709976, from whence, by trial and error, x cimes out = /100 + 2x 3r = 5, 100 5.312728 = the principal sought. The same by Mr. J. Baines, jun. Hort,, ry Bridge. By the question, (1 +.02x)3* x x =iir, or (1 +.02x)3• = 5, whence 3.x x log. (1 + .022", = .69897, and by approximation x = 5.3127. Other solutions were sent by Med rs. Anglicus, Dunn, Foril, Gawthrop, Hine, Hirst, Ma ett, Nesbit, Putsey, Rylundo, Tomlinson, and Winward. - 3. Qu. (33) answered by Anglicus, and Mr. Bruster. Put the versed sine A E = 50 = 1 a, and DE = r; then DA = (AE+DE)=v(a' + x^). Also, A È = a:AD:: AD: AB = a + = the diameter. Now, by mensuration and the question, \ 8 (a + z).– 2x a = 100, and by reduction 60x' - 1920x = 70400" from whence *= 53.806525, and A B = a = 107.90284265 is the diameter required. Otherwist : Mr. Ford, ar Mr. J. Gawthrop. Put the chord suweng the given arc = 2x, 0 = 30, and a = 160; then by known rules wh8V (r?_0)—20 = u, whence x = V(600 + 100) + = 53.806; whence to= 108 nearly. In the same manner the question was answered by Mr. Baines, Mr. Dunn, Mr. Hine, Mr. Hirst, Mr. Maffett, Mr. Nesbit, Rylando, Mr. Tomlinson, and Mr. J. Winward. 4. Qu. (34) answered by Mr. Putsey, and Rylando. Let D E F be a section of the given sphere, and ACB a section of the circumscribing cone. Put OD=r, and CD= x; then CE=VCD X CF) = V(x? — 2rr), and by similar tri. angles CE: EO :. ČD: DBA Armor Teore, i? — 2rc X 5 by the ques. tion is a minimum, which in fluxions, and reduced, we get x = 4r; hence A B = 472, and the solidity of the cone = 32 X 8 X .2618 = 67.0208, as required. Cor. The sphere is to its least circumscribing cone as 1 to 2. v (r? — 2rx); therefore, The same by. Messrs. Eruster., Dunn, Hirst, and Nesbit. Let DEF be a section of the sphere, and A CB that of the cone, which will be the least when Cn = 2n D, or Dn= 2n F; therefore, 2FD=8=CD, and CE =V(CO’ — EO2) =(36 — 4) = 5.6568542, which is also = AB, the base of the cone; hence its solidity is easily found = 67.0208 ; whence it appears that the solidity of the cone is = twice that of the sphere. . Answers to this question were sent also by Anglicus, Mr. Baines, Mr. Ford, Mr. Gawthrop, Mr. Hine, Mr. Maffett, Mr. Tomlinson, and Mr. Winward. |