150 A HINT TO MECHANICS INSTITUTIONS. Ecample. To be able to allow 20 per much would he gain if he sold them at cent. it is required to put on 25 percent. the gross price? for 25 +100=125 - 20 per cent. (or ) 120+ 21 per cent. (or ) = 123 = 100; and goods bought at 20 per cent. 1711. &s. 62d.=491. 8s. 6 d., being discount off, clears 25 per cent. profit; a gain of above 10 per cent. more for 100-20=80, nett cost, and 80+ 1 than the discount he was allowed. (or 25 per cent.) =100. Suppose, again, that the person he sells them to should (from certain cir-" To obtain the results in the preceding cumstances) be inclined to dispose of them at a loss of 30 per cent. what Table, divide the given sum by the should they produce, by public sale, decimal opposite to the rate per cent., to clear broker's commission, auction or multiply by denominutor, and divide duty, &c. amounting to 124 per cent. ? by the numerator, of the vulgar frac- 1711. 8s. 6 d. – 30 per cent. (or ) = tion. 120:,14=1421. 17s. lda 1421. 178. 12d. – 124 per cent. (or ) Rate per Cent. Dec. V. F. =120. And, again,-Suppose (a common 1,975 1.95 case) a manufacturer produces a new . article, which he sells at a profit of 40 ,925. 1,9 . per cent. but, competitors starting, he 1,875 . is compelled to lower his price; and, from time to time, as opposition increases, he comes down until he offers the article at 30 per cent. under his original charge, he will (it is very probable) tell you that he is still gaining 10 per cent., but, in fact, he is losing -he is selling at 981. what cost him 1001. * సొంతం A HINT Το MECHANICS' INSTITUTIONS. At the end of one of the gratuitous lectures which Dr. Gregory has lately Example.For 20 per cent. 100:8 been delivering to the Mechanics' In25; or, 100 x 5=100=4=25. stitution at Crayford, he suggested to them a plan by which they might As almost every person understands become their own lecturers, which the rates by aliquot parts, it would be was this :-Select some instructive needless to add a Table thereof. popular work, as Ferguson's Select Lectures, Millington's Lectures, the second volume of Gregory's MechaILLUSTRATIONS. nics, Bonnycastle's or Brinkley's A manufacturer's prime cost of goods Astronomy, Tilloch's Mechanics' being 1001. what should he charge for Oracle, the London and Glasgow them to clear 20 per cent. profit, and Mechanics' Magazines : let these, or (according to the custom in his branch appropriate portions of them, seof trade) allow a discount of 30 per lected by the Committee, be read in cent. ? order, and made the subject of con100 +- 20 per cent. (or ) = 120 •,7= versation at alternate meetings. To 1711. 8s. 63d. render the subjects equally plain to 1711. 8s. 630.-30 per cent. (or readers and auditors, let' three or =120. four boards be provided, each preSuppose expenses of carriage, &c. senting a face of about four feet by increase the price of the said goods to three, and let them be painted black. the wholesale dealer 21 per cent. bow Let some member or members of ON THE QUADRATURE OF THE CIRCLE. 151 the Committee draw such figures reads, and thus remove the great and diagrams of reference, with chalk, obstruction to the communication on a large scale, upon one or more of knowledge, where diagrams are of these boards, as may be required concerned, to a large auditory by a for the evening's reading. These single reader. Moulds of circles, boards, placed in suitable positions, ellipses, squares, parallelograms, and with a good light thrown upon &c. along the contours of which them, will render the respective chalk may be drawn, will much subjects as intelligible to the au- shorten the labour of sketching the ditors as to the individual who diagrams, SIR,_Although the celebrated problem of finding the quadrature, or the ratio, which the periphery bears to the diameter of the circle, has passed through the hands of many eminent mathematicians since its solution was first attempted by Apollonius Pergæus, and afterwards by Archimedes of Syracuse, the greatest geometricians of antiquity, it is still susceptible of various novel solutions, quite different from those previously given to it by any of the famous authors who have treated on the subject; and as I conceive the grand object of the Mechanics' Magazine is to give publicity to new improvements in the Arts and Sciences- I will, with the Editor's permission, present its readers with one of these new methods of solution, inteuding to publish the whole myself in a separate pamplet at soine future period. This solution assigns the length of a given circular arc in an elegant and novel series, which is afterwards summed or converted into another series, for the purpose of numerical computation. Subsequently a new and general theorem is investigated, and applied to finding various serieses of very rapid convergency; among these are deduced, from our theorem, the best of those given by the celebrated authors, Euler, Machin, Hutton, and Bonnycastle. From the same source is also derived a new and quickly converging series in terms of the arc of 450, and consequently its root is the eighth part of the whole circumference of the circle. The same fertile theorem enables us to divide the whole periphery into 8 N equal parts, and having just glanced at the method of doing this, I shall conclude the Essay, not, however (I hope), without exciting some interest among the geometrical readers of your valuable miscellany, should you, Mr. Editor, deem it worthy of insertion. I am, Sir, yours sincerely, Horton-lane, Bradford, AMICUS. Yorkshire, 152 ON THE QUADBATURE OF THE CIRCLE, PROPOSITION 1.-PROBLEM. INVESTIGATION AND SUMMATION OF A NEW SERIES, EXPRESSING THE LENGTH OF A CIRCULAR ARC. Let BL, the tangent of the circular arc BN, be divided into n equal parts at the points C, D, E, F, &c., the radius AB being unity ; then, assuming the whole tangent BL equal 2, each of the parts, BC, CD, DE, EF, &c. will be nt nt equal the tangent of the angle BAC. Now, by means of the well-known trigonometrical formula, tan. (a - b) tan, a-tan. we readily deduce 2 1 3 2 nt tan, L. CAD = nt nt tan. L DAE nt 2 3 1 + 708 72 +62 n? 12 3 nt nt tan. LEAF nt 12 &c. Where the law of the contimo to +12 na 2 6 2 n2 1+ nuation is obvious, the general expression for the nth tangent being ao tot (m +1)^ in which n represents any integer whatever, finite or infinite, If we suppose n to be indefinitely great, then the equal parts into which the given tangent BL is divided, and likewise the angles BAC, CAD, DAE, &c. become indefinitely small. But the tangents and the arcs of such indefinitely small angles being ultimately equal to one another, it thus appears that the length of the arc BN is equal to, or expressed by, the sum of the series 1 1 1 + 2 m2 12 na infinitum); or, , since each term of this series is, bona fide, & mere point, we 1 are permitted to reject the first term, as being indefinitely small, which will rather simplify the series for the purpose of summation. Thus the rectifia cation of an arc of the circle, on which the quadrature depends, is reduced to + in Ć n n2 t% + 30% + &c. in nt finding the sum of the infinite series nt. (R ) +27776 72 12 + n2 fi + 20+ m2 2 + 30 + m2 742 + &c.) (1). I have not, however, been fortunate enough to assign'the sum of this elegant series in finite terms, though its simplicity is such as to hold out the hope, that this “ consummation, most devoutly to be wished," may, perhaps, be accomplished by abler hands at some future period. (Po be continued in our next.) SIR, -Seeing in your Magazine an inquiry by A.B.C: (p. 54, vol. 111.), for a Machine for Kneading Dough, I beg leave to inform him, that in the public baking-houses of Genoa, bread has been long manufactured in this manner, with a great saving both of time and labour. I will, with your leave, attempt a description of the machine. I remain, Sir, the distance of 21 palms. D, the axis, Description, Fig. 1. A is a frame of wood which supports the axis of the machine; a wall, 14 palms high from the ground, may be made use of instead of this frame. B, a wall, 31 palms thick, through which the aforesaid axis passés. C, another wall similar to the former, and facing it, to piece (which cannot be seen in the figure), tion shall be continued a few minutes 14 palms Jong, and half a palm thick, more or less, according to circumstances. placed close to the wall, B. K is a strong The measures in the preceding descripcurved piece of oak, fixed transversely in tion are given in Genoese palms, each of the side beams, H, to receive the axis of which very nearly equals 9.85 of our the trundle; its length 14 palms, and its inches. The machinery inay be varied thickness lú. Z is a trundle of 5) palms in its construction according to circumin diameter, and ld in height, which is stances, and the energy of the first moved by the cog-wheel, G. M is a trun- mover much better applied thau by meni dle, proceeding from the trundle, L, and walking in a common wheel. continued through the cross, N, to the bottom of the tub, P; its centre is made of iron, partly square and partly round, [in November, 1811, a patent was and it turns in a socket of brass. The granted to Mr. Joseph Baker, navy-confirst part of this axis, between the trun tractor, for a method of kneading dough dle, L, and the cross, N, is of square iron, surrounded by two pieces held to by means of machinery, which is thus gether by iron hoops, which may be re described :-An upright shaft, turning moved at pleasure to examine the iron on a pivot, is fixed in the centre of a within; its length is three palms, its circular trough, so that the dough placed diameter about one palın. The second in such trough may be kneaded by a part of the axis, which is within the tube, is made like the first part; its length is stone or iron roller on its edge, which a palm and a half, its breadth 1 1-6th passes over it in a rotatory motion, being palms. The wooden sheath of this part fixed at a due distance by an horizontal of the axis is fixed to the bottom of the bar or axle to the shaft, which is to be tub by means of three screws with nuts. This axis is distant one-third of a palın tal bars also fixed thereto, and worked, turned by means of one or more horizonfrom the nearest triangular beater of the cross. N, the cross, formed of two bars like a capstan, by a proportionate numof wood unequally divided, so that the ber of bipeds or quadrupeds. These horifour arms of the cross are of different zontal bars have small shares fixed to lengths; one of the two pieces of wood them, so as to run in the trough, and of which the cross is made is six palms in length, the other five; their thickness acting like a plough, they cause the dough is 7-12ths of a palm, and their breadth to present fresh surfaces for each succesone palm. O, four pieces of wood, called sive revolution.-EDIT.] beaters, of a triangular shape, fixed vertically into the extremities of, and underneath the arms of the foreinentioned cross; they are one and three-quarters WARM BATHING. palms in length, and half a palm in thickness, and beat or knead the dough Sir_Should Warm Baths be estain the tuh at equal distances from the blished contiguous to any of the centre. Pis a stout wooden tub, about great gas-works, upon the principle a quarter of a palm thick, well hooped recommended by Mr. Bell, in your with iron; its diameter is six palms, its height is in the clear. 92d Number, I would beg leave to inquire of your medical readers how Fig. 2 is a box or trough of wood, four palms long, and three wide, in which far it may be made to imitate, in the leaven is formed (in about an hour) efficacy, the saline springs, or sea in a stove, and in which it is afterwards water, as a bath, by the addition of carried to the tub, P. common salt (muriate of soda); and Fig. 3 exhibits a view of the trundle, if so, what proportion of salt should cross, &c. with a section of the tub. be added to the water for that pur Fig. 4 is a bird's-eye view of the cross pose. I would also wish to inquire, and rub, with the upper ends of the tri- if it could be improved by the addiangular beater. This tub, P, will con- tion of any other substance ? It tain about 18 rubb: (about 19 bushels) of would also be desirable to be imHour, which is carried to it in barrels; forined, to what degree of temperathe leaven is then carried to it in the box or trough, and when the whole is tem- ture the bath should be heated for pered with a proper quantity of warm general use. water, the men work in the wheel till There appear some obstacles in the dongh is compietely and properly the way to the general adoption of kpeaded. In general, a quarter of an hour is sufficient to make very good warm baths upon Mr. Bell's princidough ; but an experienced baker, who ple, which may prevent their besuperintends, determines that the opera- coming “co-extensive with gas illum ܪ |