Mechanics' Magazine, MUSEUM, REGISTER, JOURNAL, AND GAZETTE. ARTIST'S STUDY. Sir, In the practice of portrait painting, it is essential to have a good light thrown on the face, for, such shadows as are produced on the countenance by the light admitted into the artist's room, the same he must necessarily transfer to his canvas. We see therefore a material difference between the portraits by Rembrandt and those by Rubeus; the former painted by the light of the small window of a mill, while the latter had light from nearly the whole side of the apartment. The plan most in vogue at the present day is to darken all the window, except the top row of panes, but this, at certain seasons, is attended with the dis VOL. XII. advantage of darkening the room so much, as often to deprive the painter of a sufficiently clear light on his canvas. I think it would not be impossible to arrange an artist's study, so that the person, sitting for a likeness, should be in strong light and shade, and yet the artist have a broad light for the advantage of seeing his work The above engraving is intended to illustrate my plan. It represents an artist's study, having two windows, between these there is a partition A, of wood, or a thick curtain, dividing the room into separate apartments. In the partition, near the window, and about 4 feet from the floor, there is to be an opening C, of 3 or 4 feet square, с 18 GEOMETRIGAL RECTIFICATION OF THE CIRCLE. through which the artist, standing in Finsbury, August, 1829. fication of the circle, published by Mr. Sankey. As his method is as simple in practice as it is beautiful and satisfactory in theory, and leads to some curious deductions which are within the reach of students only moderately versed in geometry and trigonometry, I now transcribe it for the use of that class of your readers to whom such researches are always instructive. "To find a line which shall ultimately be equal to any given arc of a circle. "Let AB be the given arc of a circle AGB; draw the chord AB, also the indefinite right line AZ, touching the circle in A, one of the extreme points of the given arc AB; at B, the other extremity of the given arc, draw the right line BD perpendicular to the chord AB, aud meeting, in the point D, a line AD, which bisects the angle BAZ under the tangent AZ, and the chord AB; at this point of intersection D draw DE perpendicular to the bisecting line DA, and meeting in E, a line AE bisecting the angle DAZ under the tangent AZ, and the bisecting line AD; and so on continually at the points of intersection of the perpendiculars and bisecting lines, raise perpendiculars to the bisecting lines which shall again meet new lines bisecting the angles under the tangent and former bisecting lines; then the angle under the tangent and bisecting line will ultimately become less than any given, and consequently the last bisecting line will coincide with the tangent. In this case, I say, AY 'the' ultimate intercept of the tangent between A, the point of contact, and Y its intersection with XY, the last perpendicular raised to the bisecting lines, will be equal to the given arc AB. For, at L, M, N, &c., the points of intersection of the bisecting lines with the given arc, draw the lines LR, MS, NT, &c., perpendicular to the bisecting lines AD, AE, AF, &c., and meeting the bisecting lines AE, AF, AH, &c., in the points R, S, T, &c.; draw also the lines LB, ML, NM, &c. Now, on account of the equal angles LBA and LAB, and the right angle DBA, it is evident, that the line DA is bisected in L, its point of intersection with the arc AB; therefore the entire bisecting line DA: LA, the part which GEOMETRICAL RECTIFICATION OF THE CIRCLE. is intercepted by the circle AGB :: the given arc AB: its intercepted segment LA. And in like manner, and on account of the parallels LR, DE, MS, EF, NT, FH, &c., it can be shown in general, that any bisecting line that part of it which is intercepted by the circle: the given arc : its corresponding intercepted segment; but the intercept of the bisecting line will ultimately coincide with and become equal to the intercepted segment of the arc; therefore, the bisecting line itself will ultimately become equal to the given arc: this, of course, will take place, when the angle under the bisecting line and tangent vanishes, consequently AY, the ultimate intercept of the tangent, between XY the vanishing perpendicular, and the point of contact A, will be equal to the given arc AB. Your readers will perceive, that in the practice of this method, nothing is required in approximating to the length of any arc of any circle, but a succession of the simple operations of bisecting angles and raising perpendi culars; and that even a few such operations will, in any specific case, give a valuable approximation. It will also be seen, that the ultimate result contemplated in the theory would actually rectify the poposed arc, that is to say, would produce and present a right line that should be equal to it. This in theory is a beauty that Archimedes and Newton would have admired. Mr. Sankey subjoins to this construction some series, and dependent computations for the length of any arc, as well as some connected inquiries on the subject of the quadratrix, with 19 which mathematicians will be gratified. But it does not seem to have occurred to him that his method involves a geometrical demonstration of a long and greatly admired series of Euler, in which he exhibits an arc in a continued product of its sine and secants of submultiple arcs. As the research is not difficult, and will lead to a practical method of computing the length of an arc, perhaps you will allow its insertion in the "Mechanics' Magazine.” Recurring to the preceding diagram, it is evident, that CB is the sine of the proposed arc AB, to the radius OA or OB. Also, since the angle included between a tangent and a chord, is measured by half the arc of that chord (Geom. Hutton's Course, th. 48.) therefore the angle bAB=ABC (by the parallels Ab, CB,) angle AOB; and therefore AB is the secant of AOB to the radius BC. So again, AD, which is the secant of bAB to the radius AB is the secant of AOB, to the same radius. And, in like manner AE secant & AOB to radius AD; AF secant AOB to radius AE; and so on. Consequently, if a be any arc or angle, we have arc a sin. a. sec. a. sec. a sec. §a. sec. a., &c. which is Euler's series of dependent factors. Now, to evince the utility of this in practice, suppose we compute, by means of the trigonometric tables of logarithmic sines and secants, which will effect the successive multiplications by so many additions, the arc of 80 degrees to the radius 1. In doing this, I will, after the first four terms, set the numeral value of each new product, that the steps of the approximation may be readily traced. These two last values of the arc of 80° are true in the sixth place of decimals. And if we multiply the last of them by 9, and divide the product by 4, the result, 3.14159254, differs not before the seventh place of decimals, from the true value of 3.1415926535, &c. of the circumference of a circle whose diameter is 1. I know of no other series by which so accurate an approximation to a cirIcular arc and circumference can be made with so little labour. And I cannot but hope that, taken together, the geometrical process announced by Mr. Sankey, and my computation, referred to the same principles, will go far toward satisfying your correspondent, M. W. (No. 304.) August 20, 1829. O. C. F. GEOGRAPHICAL CARPETS. Sir, The endless improvements which are daily taking place in every branch of knowledge, and the rapid diffusion of them in every quarter of the kingdom, have been not a little promoted by the establishment of your truly valuable Magazine. Many useful hints and much information will be thus preserved for the consideration and advantage of posterity, which might otherwise have been buried in oblivion; and the easy way in which such hints may be communicated through the medium of your Magazine, greatly enhances its value, and has doubtlessly encreased 10th approx. their number. If you consider the following worthy a place in its pages, you may by inserting them, enable those whom it more immediately concerns, to judge of their utility and importance. The use of carpets is now become so universal, that every family which can afford one, makes it an indispensable article of comfort; and nothing can excel many of those of our own manufacture, in beauty of design or execution. But I do not recollect having seen any calculated to communicate any kind of useful knowledge, although they are susceptible of it, to a very considerable extent. I think a carpet is so admirably adapted to Geographical instruction, that it may be almost said to be a natural article for the purpose. A map is a picture of the surface of the earth, and on the ground is the place to view it. One on so large a scale as a carpet would admit, is calculated to give a more correct idea of the relative position of places than could be effected by the largest map now extant. A family in the daily occupation of a room furnished with such à carpet, would acquire unavoidably a more permanent knowledge of a given portion of the earth than could be obtained by any other means; and when the local position of the room would admit, it might be placed agreeably to the bearings of the compass, and it would thereby give a correct idea of the real direction of the places on the map. PADDLE-WHEELS. A moderately sized carpet would admit very distinctly of a fair representation of all the counties of England, or all the kingdoms of Europe. Perhaps a minute delineation of places could not be admitted, but an outline of the different counties and their names, in the case of England would be sufficient to give a general and correct idea of the kingdom; aud of the relative or contigious position of each county. General marks, as the spire of a Church, to denote the situation of the country town; darts to denote the direction of a river, and many other general features of a map, might be introduced without adding much to the complication of the design; and each county or kingdom might be rendered more distinct by giving it a different colour as in other maps now in common use. A portion of the coasts of France or Ireland 21 would likewise be of great advantage. Such a representation of any part of the world being continually in view, would give children a more correct and permanent knowledge of the geography of such places, than any references to maps or globes in common use; and would be more frequently glanced at, as there would be little or no trouble attending it. Should these hints catch the notice of Carpet manufacturers and be thought worthy of their consideration, their own ingenuity will suggest improvements. I think the demand for such an article would become extensive, and the mutual benefit of trade and society would be the result. I am, Sir, Respectfully yours, &c. A. TAYDHILL. Birmingham, April 13, 1829. 109 mit 293cl loa Sir,-I beg to offer a few remarks respecting the paddle-wheel question, vol. xi. p. 313. Let a b c d represent the paddle-wheel, and db cd, the arc of 180°. Hence d b, will be the water line. Let us also suppose the quadrant c z b, to contain 4 paddles, b, e, f, c, c being the maximum paddle, and b the minimum. It is evident that the increase of propelling power of each paddle, from the minimum point b, to the maximum point c, will be in ratio to the increment of the cosine gf, or to the decrement of the sine fh. Hence the propelling power will never cease, before the cosine gf, is absorbed in the radius at d. And the maximum force will take place when the sine f h vanishes at the point c; c n will be the tendency of the propelling power of the paddle z c; and mr that of the paddle & f and m y, that of the |