22 PYTHAGOREAN THEORY OF THE NUMBER NINE. and n p= m т PYTHAGOREAN THEORY OF THE NUM BER NINE. paddle z e. Consequently, the last evident from the regression of numtwo paddles tend to propel the vessel bers. For the natural progression of in an oblique direction to its course, them is as far as to 9, but after it their each force cutting its respective pad- retrogression takes place. For 10 dle at right angles. In order to find becomes as it were again the monad. the composition or revolution of the Thus, if from each of the numbers forces, (paddles) in their respective 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, positions, (as in the fig.) take any the number 9 is subtracted, the numpoint m, when the tendency of the bers that remain will be 1, 2, 3, 4, 5, three forces, y m, u m, and n 6, 7, 8, 9, 10. And vice versá, the meet; draw m n=c, progression will receive an increase the sine f h, complete the parallelo- by the addition of 9. For if to each gram mp s n, and ms will be the of the numbers, 1, 2, 3, 4, 5, &c. direction and measure of the com 9 is added, the numbers produced bined forces of the paddles f, and will be 10, 11, 12, 13, 14, &c. Like. % c. Secondly, draw k=et, wise, by subtracting from 20, twice the sine of the angle at e, and com. 9, from 30, thrice 9, from 40, four plete the parallelogram mkrs, and times 9, from 50, five times 9, &c. m po will be the direction and mea. the numbers 2, 3, 4, 5, 6, &c. will be sure of the combined forces of the produced. By taking likewise from three paddles, % e, f, and 2 c. 100 eleven times 9, we again return and so on for any number of paddles. to the monad. And after the same I am, Sir, yours, &c. manner, we may proceed to infinity. WM. TONKIN, Mine Agent, Hence it is not possible there should Fowey, Cornwall. be any elementary number beyond the July 11, 1829. ennead. Hence, too, the Pythagoreans, called it ocean, and the horizon, because all numbers are comprehended by, and revolve within it. On this account, likewise, it was called by Sir,-In'the communication which them Halios,* concord, and Pers.rart I sent to your valuable Magazine, because it congregates all numbers, dated the 8th of this month, I ob and collects them into one, and does served, that in circulating decimals, not permit the conspiration of the the sum of the digits of the circulator numbers beyond it to be dissipated. I will be found to be 9. And it appears Though your correspondent, G. S. to me, that the cause of this singular has very ably shewn, that the sum of property may be satisfactorily obtained the digits of the circulator in every from what the Pythagoreans have said circulating decimal is not always 9; respecting this number. For the cele yet he seems not to have been aware, brated Pythagorean Nicomachus, in the that when this is the case, the sum of extracts from one of his lost writings the digits of the denominator of the preserved by Photius and entitled fraction from the expansion of which Θεολογουμενα αριθμητικης*, informs us the circulating decimal is produced, that the above mentioned philosophers, will be 9. And, if this be true, by celebrated the ennead-as flowing round the other numbers within the * i. e. maga to aliget from congredecad like the ocean, and that it was gating and collecting. also called by them the horizon, An + An epithet of Diana or the Moon, anonymous Greek author who wrote a who, as Diodorus Siculus informs us, treatise which bears the same title was much worshipped by the Persians, with that of Nicomachus, and was and was called in his time Persra. But reprinted at Leipsic in 1817, explains the Moon in the Orphic hymn to her, is these appellations as follows: called Tedeo pogos, i. e. bringing to a per“That there can be no number fect end, which accords with the nature of the ennead. See the second edition beyond the ennead, but that it circu. of my translation of the Orphic hymns, lates all numbers within itself, is * i. e. A Developement of the Theory # See more on this subject in my of Arithmetic. Theoretic Arithmetie, p. 206, &c. p. 27. CIRCULAR LOGARITHMIC SCALE. 23 making the circulater to be a divisor form can hardly be said to have been instead of a quotient, a circulating reduced to actual practice. For this decimal will be produced, and the step, we are, I believe, indebted to sum of the digits of the circulator will Mr. Joseph Lamb, late of the Royal be 9. Thus, in the instances which he Artillery, under whose directions they has adduced, o'z when expanded into are now made. The following is the a decimal will, as he says, be title of his pamphlet, “Description *015873,015873, &c.; and the sum of a Concentric Circular Proportioner, of the digits of the denominator 63 is an Instrument for abridging and fa9. Likewise 73675 when expanded, cilitating calculations : particularly gives the circulating decimal, adapted to the use of officers of the 000063,000063,000063, &c. The Army, Civil Engineers, and all persons same thing will likewise be found to engaged in Chymical and Mechanical take place, in os, o's, dis, o's, &c. Inquiry, London, 1827,” 8vo. 20 when expanded into a decimal series. pages. The instrument which I possess, is Thus too z's when expanded, will be of brass, consisting of a circle move185,0185,0185, &c. and the sum of the able within a narrow ring, and gradudigits, of the denominator 54, is 9. ated on both surfaces. The effective Also its will produce when expanded, diameter is 2.5 inches, equivalent to a the circulating series, .0074,0074,0074, single straight line of 7.85 inches, or to &c.; and, the sum of the digits of a double one of 1:57. On one side. the denominator 135 is 9. both scales are single; on the other, Again to when expanded, will one is single and the other double. give the circulating decimal series, The instrument is therefore adapted 0065359477124183,0065359, &c.and to the same computations as the lines the sum of the digits of the denomina A, B, C, and D, of Bevan's or Routtor is 9. ledge's rules, except those which reIf this should be found to be uni quire the slide to be reversed. versally true, there will be as many At the end of the pamphlet above circulating decimals, having 9 for the mentioned, it is stated that the instrusum of the digits of their circulators, ment may be had in silver, brass, or as there will be of those that have ivory, and with a wheelwork adjustnot. Your correspondent, G. S. will ment, if required. It may also be much oblige me, by any additional had with all the divisions on information on this subject; for I surface. write not for victory, but for truth. The graduation of my instrument is Your's, respectfully, very accurate and the subdivisions (except in one instance) are regulated Manor Place, Walworth. upon the principles laid down in Mec. power of reversing the lines is CIRCULAR LOGARITHMIC SCALE. a very important one in sliding rules. In the volume of Nicholson's Journal, Dr. Roget's Logometric Scale. before referred to, is a paper, illusSir,—Your correspondent Mr. Pea trative of the advantages attendant on cock, (vol. 10, p. 246), speaks of the that method; which are also dwelt circular form of the logarithmic scale, upon in a communication by Mr. as a new idea. In this he is mistaken, Silvanus Bevan, in vol. 49, of the it having been described more than a Philosophical Magazine and Journal. century ago, in “ Bion's Construction Both these papers are recommended and Use of Mathematical Instruments, to the perusal of persons who wish to by Stone." A particular account of be well-grounded in the theory and this form, with a description of some practice of the sliding rule. instruments actually made, will be In the Philosophical Transactions found in the first volume of " Nichol. for 1815, part 1, is a “Description son's Journal of Science," quarto series, of a new instrument for performing 1797. mechanically the involution and evoUntil lately, however, the circular lution of numbers,” by Dr. Roget. one PRESS. The instrument is similar in form to in the ensuing session, I do not know the common sliding rule, and is adapt of a more profitable subject to interest ed to perform involution and evolu the public attention during this vation, both with integral and fractional cation, than the reciprocity of accountindices, with the same facility as mul ability, which you have proposed; tiplication and division by the com and which must precede any attempt mon one. The substance of the paper, at reciprocity of trade with the United together with the engraving, is copied States, France, and Russia, even with into Rees' Cyclopædia, Art. Sliding our own colonies. Rules. I am, Sir, As in the case of the circular in Your strument, this ingenious contrivance of very obliged servant, HINDOSTANEE. Dr. Roget has hitherto been lost to the practitioner. Recently, however, Maida Hill, Aug. 25th. a twelve-inch pattern has been laid down at the expence of Messrs. W. and T. Gilbert, opticians, Leadenhallstreet, of whom copies may be ob IMPROVEMENT ON THE HYDROSTATIC tained. Last year there was published “The Sir, I have often heard it menSlide Ruler's Guide, being a prac tioned that the Hydrostatic Press has tical Treatise on the use of the Artj. not been found to answer by bookficer's Common Slide Rule," by binders so well as simple beating. George Oakley Lucas, 12mo. pp. 40. Though no bookbinder myself, the This little work affords an instance of reason which suggested itself to my what is but too common, viz. a per mind is this, (the fault complained of son's undertaking to assist the public, being that it does not expel the air) before he has made himself acquaint that in beating, the instrument used ed with the extent to which others does not fall with such uniform pres. have preceded him in the same de sure on the edges and outside of a partment. book, as the press does, and so leaves I am, &c. more room for the air to escape. I J. W. WOOLIGAR. propose, therefore, that the press be made in a very trifling degree convex, Lewes, Aug. 24, 1829. whereby the air would be gradually pressed from the centre to the open ings between the leaves and escape. ACCOUNTABILITY EAST This to me appears so obvious a reINDIA COMPANY'S CHARTER. medy, that I am induced to offer it to Sir,--An accountant cannot be a the notice of bookbinders through the partizan, therefore the decimal account medium of your work. has been submitted with great pro I am, Sir, priety to the readers of the “Mechanics' Magazine,” (No. 316.) Yours respectfully, It is asserted, and most truly, that ALPRED HOLDEN. Tca is cheaper in the United States, Liverpool, July 6, 1829. and in France, than in England. France exchanges 300 francs of her industry, for 120 rupees sicca, of East Indian. Were France to employ the QUICK COINING. English accountability, the French par There are eight presses at the Mint, of exchange could be no more than which, on cases of emergency, can all 250 francs. be put in action, and each press coins The English exchange is 24 pence 40 sovereigns in a minute, making 320 sovereigos by the whole eight presses in farthing ; with the decimal reckoning, a minute, or equal to 19,200 in an it should be, 2s. 5d. I believe forty hour. Allowance must, however, be rupees sicca weigh exactly one pound made for the breaking of dies, &c. As avoirdupois. As it is agreed, to inves many as 150,000 sovereigns per day have tigate the East India Company's Charter been coined. INGLISH 26 DR. LARDNER'S LECTURES ON MECHANICS. C. A toothed wheel revolving within 2. They are so minute as not to be the brass frame B. distinguishable. D. A rack working into the teeth of 3. They are indestructible by human C, and of sufficient length to turn C or natural agency. once round ; it has a pin 5, working in To prove the first it need only be a slit in B, and has a slit in which a shown that a body is compressible, pin 6 on B works, and thus its parallel for it cannot be said that the particles motion is preserved. themselves are compressed. Thus, F. A ruler, which may be retained when a piece of metal is contracted in any_position by the tightening hy cold, the weight remains the same screws F F, its upper corner being although the size is different. It is the brought to coincide with the line H, of pores, or spaces between the particles the scale of divisions, and its side with that are contracted. the point K. The distance from H to The second observation can easily K, must be exactly equal to the length be imagined, by reference to the deof the rack, or to the circumference of scription of divisibility before given. the wheel C. The third property is indestruct. L. A bar fixed on to a square part ibility.-It has been very justly and of the axis of C, below the frame, and truly observed, that “ It is as im. carrying a sliding tracer, and being possible to annihilate one of the pardivided into any number of equal parts. cles, of which a body is composed, as The scale H, must be divided into a to form one, and there is not one similar number of parts, but the length single particle less in the world now of a division on H, must be to that of than there were at its creation.” 'The a division on K, as 628 to 1, or as the whole power of man consists in makcircumference of a circle to a radius. ing three changes in matter, viz. by To use the instrument, set the tracer combustion, separation, or evaporation. on L, to the diameter of the generating Thus fuel is changed by combustion, circle, and the bar E, to a division on some of the particles escape up the H, corresponding with the division on chimney in the form of smoke, (and are L, at which the tracer stands, then dispersed over the earth to conduce to bring the angular point of the rack to the use or pleasure of man) others the point K, when the tracer will be remain in the chimney in that of soot, immediately below the axis of C, and while the rest fall in ashes below, or slide the point of the rack along the are dispersed about the room edge of the bar E, and the tracer will dust. describe a cycloid by the time that the All substances or materials are di. point of the rack arrives at the line H. vided into three kinds according to I am, Sir, the different strengths of cohesion. Your obedient servant, 1. When the cohesion of a body is J. MURDOCH, so much stronger than its repulsion as Mech. Draftsman. to keep it in a firm position, it is called 4, Vittoria-place, Mile-End-road, a solid. August 19, 1829. 2. When the cohesion is equal to the repulsion, or the difference between them on either side is very small, it is a liquid. 3. But if the cohesion be much less ON MECHANICS, DELIVERED BY DR. LARDNER, AT THE LONDON UNI than the repulsion, the particles will actually repel each other, in which case they are called gases. (Continued from last No. p. 12.) Every thing in nature may be classed It is supposed that all matter is under one of these three heads, with composed of elementary monicules, only four exceptions, viz. heat, light, which are quite hard and infinitely electricity, and magnetism ; the study minute, and of which there are three of which is too abstract to be treated things to be observed. of in the present course of lectures. 1. No particles are in actual contact A solid is the only class required with each other. to be understood in mechanics; the as NOTES OF COURSE OF LECTURES VERSITY |