22 PYTHAGOREAN THEORY OF THE NUMBER NINE. paddleze. Consequently, the last WM. TONKIN, Mine Agent, July 11, 1829. PYTHAGOREAN THEORY OF THE NUM- Sir,-In the communication which I sent to your valuable Magazine, dated the 8th of this month, I observed, that in circulating decimals, the sum of the digits of the circulator will be found to be 9. And it appears to me, that the cause of this singular property may be satisfactorily obtained from what the Pythagoreans have said respecting this number. For the celebrated Pythagorean Nicomachus, in the extracts from one of his lost writings preserved by Photius and entitled Θεολογούμενα αριθμητικης*, informs us that the above mentioned philosophers, celebrated the ennead-as flowing round the other numbers within the decad like the ocean, and that it was also called by them the horizon. An anonymous Greek author who wrote a treatise which bears the same title with that of Nicomachus, and was reprinted at Leipsic in 1817, explains these appellations as follows: "That there can be no number beyond the ennead, but that it circulates all numbers within itself, is i. e. A Developement of the Theory of Arithmetic. Like evident from the regression of num- On Though your correspondent, G. S. has very ably shewn, that the sum of the digits of the circulator in every circulating decimal is not always 9; yet he seems not to have been aware, that when this is the case, the sum of the digits of the denominator of the fraction from the expansion of which the circulating decimal is produced, will be 9. And, if this be true, by * i. e, παρα το αλιζειν from congregating and collecting. CIRCULAR LOGARITHMIC SCALE. making the circulater to be a divisor instead of a quotient, a circulating decimal will be produced, and the sum of the digits of the circulator will be 9. Thus, in the instances which he has adduced, when expanded into a decimal will, as 23 form can hardly be said to have been reduced to actual practice. For this step, we are, I believe, indebted to Mr. Joseph Lamb, late of the Royal Artillery, under whose directions they are now made. The following is the title of his pamphlet, he says, be Description of a Concentric Circular Proportioner, an Instrument for abridging and facilitating calculations: particularly adapted to the use of officers of the Army, Civil Engineers, and all persons engaged in Chymical and Mechanical Inquiry, London, 1827," 8vo. 20 pages. 015873,015873, &c.; and the sum of the digits of the denominator 63 is 9. Likewise when expanded, gives the circulating decimal, The 000063,000063,000063, &c. same thing will likewise be found to take place, in,,,, &c. when expanded into a decimal series. Thus too when expanded, will be 185,0185,0185, &c. and the sum of the digits, of the denominator 54, is 9. Also will produce when expanded, the circulating series, 0074,0074,0074, &c.; and, the sum of the digits of the denominator 135 is 9. Again when expanded, will give the circulating decimal series, 0065359477124183,0065359, &c. and the sum of the digits of the denominator is 9. If this should be found to be universally true, there will be as many circulating decimals, having 9 for the sum of the digits of their circulators, as there will be of those that have not. Your correspondent, G. S. will much oblige me, by any additional information on this subject; for I write not for victory, but for truth. Your's, respectfully, THOMAS TAYLOR. Manor Place, Walworth. CIRCULAR LOGARITHMIC SCALE. Dr. Roget's Logometric Scale. Sir,-Your correspondent Mr. Peacock, (vol. 10, p. 246), speaks of the circular form of the logarithmic scale, as a new idea. In this he is mistaken, it having been described more than a century ago, in "Bion's Construction and Use of Mathematical Instruments, by Stone." A particular account of this form, with a description of some instruments actually made, will be found in the first volume of "Nicholson's Journal of Science," quarto series, The instrument which I possess, is of brass, consisting of a circle moveable within a narrow ring, and graduated on both surfaces. The effective diameter is 2.5 inches, equivalent to a single straight line of 7.85 inches, or to a double one of 1:57. On one side. both scales are single; on the other, one is single and the other double. The instrument is therefore adapted to the same computations as the lines A, B, C, and D, of Bevan's or Routledge's rules, except those which require the slide to be reversed. At the end of the pamphlet above mentioned, it is stated that the instrument may be had in silver, brass, or ivory, and with a wheelwork adjustment, if required. It may also be had with all the divisions on one surface. The graduation of my instrument is very accurate and the subdivisions (except in one instance) are regulated upon the principles laid down in Mec. Mag. vol. 10, p. 173. The power of reversing the lines is a very important one in sliding rules. In the volume of Nicholson's Journal, before referred to, is a paper, illustrative of the advantages attendant on that method; which are also dwelt upon in a communication by Mr. Silvanus Bevan, in vol. 49, of the Philosophical Magazine and Journal. Both these papers are recommended to the perusal of persons who wish to be well-grounded in the theory and practice of the sliding rule. In the Philosophical Transactions for 1815, part 1, is a Description of a new instrument for performing mechanically the involution and evolution of numbers," by Dr. Roget. c4 The instrument is similar in form to the common sliding rule, and is adapted to perform involution and evolution, both with integral and fractional indices, with the same facility as multiplication and division by the common one. The substance of the paper, together with the engraving, is copied into Rees' Cyclopædia, Art. Sliding Rules. As in the case of the circular instrument, this ingenious contrivance of Dr. Roget has hitherto been lost to the practitioner. Recently, however, a twelve-inch pattern has been laid down at the expence of Messrs. W. and T. Gilbert, opticians, Leadenhallstreet, of whom copies may be obtained. Last year there was published "The Slide Ruler's Guide, being a practical Treatise on the use of the Artificer's Common Slide Rule," by George Oakley Lucas, 12mo. pp. 40. This little work affords an instance of what is but too common, viz. a person's undertaking to assist the public, before he has made himself acquainted with the extent to which others have preceded him in the same department. I am, &c. J. W. WOOLLGAR. Lewes, Aug. 24, 1829. ENGLISH EAST ACCOUNTABILITY INDIA COMPANY'S CHARTER. Sir,-An accountant cannot be a partizan, therefore the decimal account has been submitted with great propriety to the readers of the "Mechanics' Magazine," (No. 316.) It is asserted, and most truly, that Tea is cheaper in the United States, 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 English accountability, the French par of exchange could be no more than 250 francs. The English exchange is 24 pence farthing; with the decimal reckoning, it should be, 2s. 5d. I believe forty rupees sicca weigh exactly one pound avoirdupois. As it is agreed, to investigate the East India Company's Charter IMPROVEMENT ON THE HYDROSTATIC PRESS. Sir, I have often heard it mentioned that the Hydrostatic Press has not been found to answer by bookbinders so well as simple beating. Though no bookbinder myself, the reason which suggested itself to my mind is this, (the fault complained of being that it does not expel the air) that in beating, the instrument used does not fall with such uniform pressure on the edges and outside of a book, as the press does, and so leaves more room for the air to escape. I propose, therefore, that the press be made in a very trifling degree convex, whereby the air would be gradually pressed from the centre to the openings between the leaves and escape. This to me appears so obvious a remedy, that I am induced to offer it to the notice of bookbinders through the medium of your work. I am, Sir, Yours respectfully, Liverpool, July 6, 1829. QUICK COINING. There are eight presses at the Mint, which, on cases of emergency, can all be put in action, and each press coins 40 sovereigns in a minute, making 320 sovereigns by the whole eight presses in a minute, or equal to 19,200 in an hour. Allowance must, however, be made for the breaking of dies, &c. As many as 150,000 sovereigns per day have been coined. 26 DR. LARDNER'S LECTURES ON MECHANICS. C. A toothed wheel revolving within the brass frame B. D. A rack working into the teeth of C, and of sufficient length to turn C once round; it has a pin 5, working in a slit in B, and has a slit in which a pin 6 on B works, and thus its parallel motion is preserved. F. A ruler, which may be retained in any position by the tightening screws FF, its upper corner being brought to coincide with the line H, of the scale of divisions, and its side with the point K. The distance from H to K, must be exactly equal to the length of the rack, or to the circumference of the wheel C. L. A bar fixed on to a square part of the axis of C, below the frame, and carrying a sliding tracer, and being divided into any number of equal parts. The scale H, must be divided into a similar number of parts, but the length of a division on II, must be to that of a division on K, as 628 to 1, or as the circumference of a circle to a radius. To use the instrument, set the tracer on L, to the diameter of the generating circle, and the bar E, to a division on H, corresponding with the division on L, at which the tracer stands, then bring the angular point of the rack to the point K, when the tracer will be immediately below the axis of C, and slide the point of the rack along the edge of the bar E, and the tracer will describe a cycloid by the time that the point of the rack arrives at the line H. I am, Sir, Your obedient servant, 4, Vittoria-place, Mile-End-road, 2. They are so minute as not to be distinguishable. 3. They are indestructible by human or natural agency. To prove the first it need only be shown that a body is compressible, for it cannot be said that the particles themselves are compressed. Thus, when a piece of metal is contracted by cold, the weight remains the same although the size is different. It is the pores, or spaces between the particles that are contracted. The second observation can easily be imagined, by reference to the description of divisibility before given. The The third property is indestructibility. It has been very justly and truly observed, that "It is as impossible to annihilate one of the parcles, of which a body is composed, as to form one, and there is not one single particle less in the world now than there were at its creation." whole power of man consists in making three changes in matter, viz. by combustion, separation, or evaporation. Thus fuel is changed by combustion, some of the particles escape up the chimney in the form of smoke, (and are dispersed over the earth to conduce to the use or pleasure of man) others remain in the chimney in that of soot, while the rest fall in ashes below, or are dispersed about the room as dust. All substances or materials are divided into three kinds according to the different strengths of cohesion. 1. When the cohesion of a body is so much stronger than its repulsion as to keep it in a firm position, it is called a solid. 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 than the repulsion, the particles will actually repel each other, in which case they are called gases. Every thing in nature may be classed under one of these three heads, with only four exceptions, viz. heat, light, electricity, and magnetism; the study of which is too abstract to be treated of in the present course of lectures. A solid is the only class required to be understood in mechanics; the |