FOR THE PORT FOLIO. TO MALVINA, IN RETURN FOR A WREATH OF WILD FLOWERS. THE wilding wreath though faded, love, A bright remembrance still doth prove, Oh dear to me their native fields, How often on their banks we've stray'd, Or sought the shady grove, When on the lake bright sun-beams play'd, And fragrant garlands wove. SCIENCE. FOR THE PORT FOLIO. MR. EDITOR, A FEW weeks ago, I forwarded you solutions, to the two ques tions inserted in the fourth number of the fifth volume of The Port Folio. I, soon after, concluded, that, to so beautiful a question as the second, a nearer approximation ought to be made, before it was given to the public; annexed you have the result of my labours. Solution. Let A D B X be a semicircle, of which, A X is the diameter. Draw the chord BX = 10 = a. Join A B, and bisecting A B in E, draw CED, from the centre C, through E, and terminating in the circumference, at D. A D E Now, since CD is at right angles to A B, by the 3d Prop. 3d b. Eu., and X B, by the 31st of the same book; the triangles AEC and A B X are equiangular, and consequently, similar; by the 4th Prop. 6th b. Eu. therefore A C: AX:: EC: B X; and since AC= A X; E C = ; B X = 5 = a Now, put A X = 2 x; then A C, or radius = x; and by 47th of the 1st b. Eu. A B2 = A X2 - B X2; consequently, A B = 4x2-a2; A E A B, and consequently A B, and consequently = √4x2 —a2; = √ = these expressions, A D will easily be found Again; by mensuration, halfthe arc ADB= This equation being twice squared, for the purpose of ridding it of the radicals; being cleared of all fractions and reduced to order for solution, will be found to be 961 x3-9920 x 32250x280000 x3 + 2340625x-7200000 x3-41625000x2 506250000. Now, by inspection, I discover, that the value of x is something more than 8.5; I therefore, put r = maker +z = x. Then, by a general theorem, for the resolution Q—ar” —- brn−1 —- c r n−2 —- &c. of these high orders, we have z = - 8.5, and 71-1 nur +11-1x br. n-2 + &c. - in which n = 8, a = 961, b = — 9920, c = 32250, &c. on to g, and Q =- 506250000. In this last equation, the values of the various powers of r being obtained and multiplied by their respective coefficients, we shall have this final equation 8.589126 + &c. = AC =rahypothenuse of the triangle The hypothenuse of the tri was r+z, we now haver dius; and consequently, A X, the ABX = 2 x = 17.178252 + &c. angle being now ascertained, it will be easy to obtain A B, the perpendicular; which, upon calculation, will be found = 13.967546; and consequently, the area of the triangle A B X = AB X B X = 69.83773+ &c. Q. E. I. Err. Ex. By substituting this value of x, for r, in the above equation, we could approximate still nearer to the real value of x; but as this is merely a question of pleasure, and absolute exactitude is not necessary, I hope the proposer will be satisfied with this approximation. Should he not; I will substitute this value of x, repeat the operation and give you, Mr. Conductor, the result, in some future number of your useful, instructive, and entertaining miscellany. THOMAS P. IRVING. Newbern, North Carolina. MR. EDITOR, Chapel Hill, August 21st, 1811. I have waited some time for the solution of the second question of Philo Mathematicus, proposed in your number for April last. Since there exists a doubt as to the propriety of the statement of the question, which I observed in one of your numbers, a friend has given the following solution, herein inclosed, which you are at liberty to insert, if thought worthy. A B C; and each is double of the triangle A D C. G The seg. ment A G C is equal to the triangle A D C or AD B. The tri VARIETY. FOR THE PORT FOLIO. A candid observer must be astonished at the ingenuity of the French nation, and the multitude of the literary productions in every department, when the smallness of the pecuniary emolument is considered. For though the honest desire of lasting fame, That last infirmity of noble minds! and the wish of being useful to mankind, be the first and most powerful incentives in the breast of an eminent writer, yet the means of procuring superior ease and convenience, and of scattering a few flowers on his brief passage through life, may also be regarded as a not illaudable, though secondary motive. No French writer has received such pecuniary rewards as our Robertsons and Gibbons; not to mention the prices paid by the booksellers in London for works of a less marked distinction. Fourcroy is said to have received thirty thousand francs, or about thirteen hundred pounds, for a work on chemistry, in twelve or thirteen volumes; and a bookseller boasted of having offered Segur the elder, twenty thousand francs for an abstract of the history of France, in two octavo volumes. The lowness of literary emoluments at Paris may be partly ascribed to a very simple circumstance, that since the death of the celebrated Pancoacke, the booksellers rarely print editions in quarto, so that the computations are made on a form of inferior size and price, and are usually scattered over two or three editions, instead of a large sum paid for the entire right of the copy. Dutens, the traveller, declares that the Bostonians a long time ago, entertained the design of withdrawing themselves from the authority of England: for M. de Bougainville told him, that when he was in Canada, he translated a letter from them to Mons. de Montcalm, in which they proposed to give themselves up to France. COWLEY, who speaks experimentally, having been exposed to many vicissitudes, both as a man and as an author, thus insists |