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points, or points of intersection : around which he lied silk threads,which finished the formation of the limits of his figures.” See tig. 4.It may be added bv way of improvement, that for the division ofone series of numbers from another, a thin piece of timber in theform of a rider with which lines are drawn, having a pin at eachend for the holes in the squares, might be interposed between thetwo series to be distinguished. By the notation above exhibitedevery modification of number may be expressed, and consequentlyevery' arithmetical operation successfully performed ; but there isanother form of palpable arithmetic, equally comprehensive, andmuch more simple, than that of Sanderson, originally invented,and used in calculation, bv Dr. Henry Moves; a gentleman,whose uncommon attainments we have already endeavoured to il-lustrate. In a letter addressed to the Editor of the EncyclopadiaBritannica, the Dr. gives the following brief account of a palpablenotation, which he had used for 29 years, to assist his memoryin numerical computations. “ When 1 began to study the princi-ples of arithmetic, which I did at an early period of life, being un-acquainted with the writings of Sanderson, in which a palpable no-tation is described, I embraced the obvious, though imperfect ex-pedient of cutting into the form of numerical characters thin piecesof wood or melal. Bv 'arranging these on the surface of a board,

1 could readily represent any given number, not only to the touch,but also to the eye; ami bv covering the board with a lamina ofwax, my symbols were prevented from changing their places, theyadhering to the board from the slightest pressure. By this contri-vance, I could solve, though slowly, any problem in the scienceof numbers: but it soon occurred to me, that my notation, con-sisting of 10 species of symbols or characters, was much more com-plicated than was absolutely necessary, and that any given numbermight be distinctly expressed by three species of pegs alone. Toillustrate my meaning, let A, B, C, D, (tig. 5,) represent a squarepiece of mahogany a foot broad and an inch in thickness ; let thesides A B, B C, C D, D A, be each divided into 24 equal parts;let every two opposite divisions be joined by a groove cut in theboard sufficiently deep to be felt with the finger, and let the boardbe perforated at each intersection, with an instrument a tenth of aninch in diameter. The surface of the hoard being thus dividedinto 576 little squares, with a small perforation at each of their an-gles, let three sets of pegs or pins, resembling those represented inthe plate at the figures 6, 7, 8, be so fitted to the hole, in the board,that when stuck into them they mav keep their positions like thoseof a fiddle, and require some force to turn them round. The headof each peg belonging to the first set is a right-angled triangle,about one-tenth of an inch in thickness; the head of each peg be-longing to the second set differs only from the former, in having asmall notch in its sloping side, or hvpothennse; and (he head ofeach peg belonging to the third set is a square, ofwhicli the breadthshould be equal to the base of the triangle of the other two. Thesepegs should be kept in a case consisting of three boxes or cells,each cell being allotted to a set, and the case must be placed closeby the board previous to the commencement of every operation.Each set should consist of 60 or 70 pegs (at least when employedin long calculations;) arid when the work is finished, they shouldbe collected from the board, and carefully restored to their respec-tive boxes. Things being thus prepared, let a peg of the first setbe fixed into the board, and it will acquire four different valuesaccording to its position respecting the calculator. When its slop-ing side is turned towards the left, it denotes 1, or the first digit;when turned upwards, or from the calculator, it denotes 2, or the2nd digit; when turned to the right it represents 3 ; and whenturned downwards, or towards the calculator, it denotes 4, or the 4thdigit. Five is denoted by a peg of the second set, having its slopingside or hypothemise ttirneiUo the left; 6, by the same turned up-wards; 7, by the same turned to the right; and 8, by the sameturned directly down, or towards the body of the calculator. Nineis expressed by a peg of the 3d set when its edges are directed toright and left;"and the same peg expresses the cypher when itsedges are directed up and down. By three different pegs the re-lative values of the ten digits may therefore be distinctly expressedwith facility ; and by a sufficient number of each set, the steps andresult of the longest calculation may be clearly represented to thesense of feeling. It seems unnecessary to illustrate this by an ex-ample; suffice it to express in our characters the year of theChristian :era 1788: Take a peg of the first set, and fix it inlilt: board with its sloping side turned towards the left equal to one;

take a new peg of the second set and fix it in the next hole in thesame groove; proceeding as usual from left to right, with itssloping side turned to the right equal to 7 ; next take a peg of thesame sit and fix it in the next hole, with its sloping side turneddownwards, equal to 8; lastly, take another peg of the same set andplace it in the next hole in the same position, equal to 8 ; ai.d thewhole will express the number required. When it is necessary toexpress a vulgar fraction, l place the numerator in the grooveimmediately above, and the denominator in that immediately be-low the groove in which the integers stand ; and in decimal arith-metic an empty hole in the integer groove represents the commaor decimal point. By similar breaks 1 also denote pounds, shil-lings, pence, &c. and by the same expedient I separate in divisionthe divisor and quotient from the dividend. This notation, whichsupplies me completely with co-efficients and indices in algebraand lluxioivq seems much superior to any of the kind hitherto madepublic in the vest of Europe . That invented and described byMr. Grenville, having no less than ten sets of pegs, is by muchtoo complicated for general practice ; and that which we owe totlic celebrated Saunderson is apt to puzzle and embarrass the cal-culator, as the pegs representing the numerical digits can seldomor never be in the same straight line.” Mr. Cheese ’s Cushionappears to have more powers than any other instrument hithertoinvented for the purpose of teaching music to the blind ; andtherefore v.e here insert his description of it. It may possibly,however, be best for every blind adept in the musical art, alterbeing sufficiently instructed in its theoretical and practical princi-ples, to invent for himself a table, by which may be expressed allthe various phenomena of music; in which, by varying the formsand positions of his pegs, lie may habitually associate them withsounds, durations, rests, intervals, chords, cadences, da capos, re-peats, and all the various graces which give animation and expres-sion to musical sounds: for thus, being the immediate creaturesof his own imagination, they will easily become familiar to hismemory, and be more strongly and readily associated with thephenomena which 6iey are intended to express, than the inven-tions of any other. Mr. Cheese ’s machine, (see Plate XXVIII.fig. 9.) is intended, in “ teaching music to people deprived ofsight, to enable them to preserve their compositions, in the act ofcomposing,without the assistance of a copyist.”—“ That part of themachine, which represents the book, or paper, is a small cushionstuffed, on a little frame ; along which, is sewed a number ofpack-thread strings at equal distances from eacli other: theserepresent the lines in a music book : the live which compose thestave are made of large twine ; and those which represent the legeror occasional lines, drawn through the heads of the notes, wherethe music exceeds the compass of the established slave, are madeof small twine, and are on this machine of the same length as theothers. If the practitioner only wishes to write harpsichord mu-sic, the cushion may be what length he pleases, and about five orsix inches wide: the strings must be sewed in the follow ingorder ; beginning with the first or lowest, near the edge of thecushion ; 4 small ones, which correspond with the notes in the baseof the instrument ff, rr, cc, ee : next five large ones for thestave which correspond with the lines in the book, or notes in tiieinstrument, g, b, d,f, r; one small one, which represents the oc-casional line between the base and treble, or middle c; five largeones for the treble stave, which make the notes e, g, b, d, f;three small ones, which represent the leger lines when the music;goes in alt. These provide for the note a in alt, c in alt, and e inalt; in the space above which, next the edge of the cushion, the/in alt is wrote, when it is wanting, which completes the compassof the instrument. Those who only sing or play on single instru-ments, such as violins, &c. should have their cushions not abovehalf the w idth of those before-mentioned, upon which there shouldbe but one stave, and that in the following order: two small linesat bottom, five large ones in the middle, and three small ones attop. Neither of the outside lines of these small cushions shouldbe sewed close to the edge, as there are notes supposed aboveand below. At either end of these small cushions, there shouldbe a small wire staple, in order that anv number of them maybe combined together at pleasure, by miming a rod through thestaples: this will enable the practitioner to write what musicianscall Score, in any number of parts lie pleases ; and by thismeans a thorough knowledge of the great works of Handel , : ndall other classical authors, may be acquired, as well without sight

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