Mechanics Magazine, MUSEUM, REGISTER, JOURNAL, AND GAZETTE. No. 288.] SATURDAY, FEBRUARY 14, 1829. [Price 3d. "If a man love not his country, posterity, and the common good, he will be as a drone in the hive."-RICHARD BAXTER. NEW MECHANICAL MOTION. NEW MECHANICAL MOTION. Sir,-To the curious and inquiring minds of the few who delight in mechanical intricacies, to whom ingenuity of contrivance is the goal for which they run, nothing seems to afford and require such endless resources as that most puzzling thing, verpetual motion; a quality far more inherent in that peculiar class of philosophers themselves, than in any of their specimens of ever-going mechanism. Notwith standing all the assertions of the learned against its practicability, it has many times put on the deceptive appearance of being within grasp. However, the universal complaint has always been "I have thee not, and yet I see thee still." The grand limit to perpetual motion is the wear of materials; but laying this objection aside, a far more important one presents itself, in the apparent impossibility of rendering the perpetual motion in any way beneficial. This would decidedly be the case, were the motive power dependent upon clock work, magnetism, or fluids, or any combination of them. They certainly have been much resorted to; and many strange and wonderful accounts are extant, of what has been executed in this way. Admitting the impossibility of a perpetual motion, and the folly of spending much time or expense upon it, there can be little,-I may perhaps say, no,-doubt, but that, in the same way as chemistry owes much to its forerunner, alchemy; so, in like manner, mechanics is indebted to researches after this motion, for many valuable arrangements. The subject, when looked at in this light, reaps nothing to support its dignity; for these arguments cannot be set down as correct, without carrying its origin to a period when mechanical science did not possess all that geometrical accuracy it now so eminently enjoys. It would be hasty, however, to conclude here, without offering some apology for following perpetual motion schemes. The accuracy of the observation, that all sciences possess certain points, which for ages have remained hid in darkness, can require no proof; it is a fact with which we are all well acquainted. Now, in mechanics, this motion we are treating upon has ever been a matter of debate by many, and a subject of hard seeking after by others; yet after all the machinery that has been made or speculated upon, it is even at the present day clothed in all that mystery, which cannot fail to give it many charms, in the opinion of those who fondly seek after and rake up inscrutable matters. These remarks bring me to what I wish particularly the reader would lend his attention. The unfortunate name "perpetual motion," if changed for "mechanical experiment," would eventually, perhaps, remove the real cause of censuring it, by the different idea given of the object aimed at. Whoever shall look upon it otherwise, considering as practicable the contriving a piece of mechanism to be constantly,in action, even with the limitations before named, will, in the end, find himself grievously disappointed. On the contrary, if merely performed by way of experiment, to solve some new idea of a mechanical motion, its errors may often tend to impress truths upon our minds-its effects may serve to suggest other principles: in either case, we may derive pleasure, if not often improvement, from our pursuits; and they will invariably develope the impossibility of giving that form to matter which can keep it in perpetual movement. From what has preceded, it will be observed, that I am neither an advocate for, nor altogether an enemy against, trials to effect perpetual motion (if moderated by a sense of the fallacies the most plausible experiments must be subject to); because it is matter of curiosity, and excites no little interest to hear speculations elicited concerning it, and to solve, on philosophical prin MR. HALL'S PROPERTY OF NUMBERS. ciples, the query-Will not this or that motion continue? I now, therefore, beg leave to offer some account of a combination of movements, which, from its origin ality, and seeming to possess everything requisite for retaining it in ac. tion, may, possibly, be acceptable before concluding; and for this purpose I proceed to describe tigures 1 and 2. Fig. 1. On the stand A is to be raised the two supports B B, each having a hole cut at c c, of equal heights from the stand, to receive the axles d d; thereby balancing e e and ff, together with their frame ggg; in which frame e e is balanced at h, and ff at i i,-e e being a glass tube, has three brass rings ehe; and ff is a piece of metal, with a ́ groove or gutter cut across it lengthwise, to guide the rolling of a brass ball. The extremities of e e and of ff are connected by slender metal rods klm, moveable at k m, and also at the centre 1, where the extent of its bending is confined by the small prongs at 7, belonging to m. Fig. 2. This diagram shows a side view, with references the same as for fig. 1; and by it we shall explain the movements expected from the machine. In the first place, it is necessary to notice particularly the formation of the rods k l m. It will be seen that I always falls in the direction of the supports B, in consequence of the forked shape of m not allowing the rod to project outwards. On the scale ff there is a ball placed at N; and within the tube e e, at O, a quantity of mercury the weight of the ball. However, O will fall lower (moving the frame fig. 1, g g g), and, raising N, will make the ball reverse its situation, and, by striking the rod m, and thus lengthening the whole rod klm, will elevate the tube, causing the mercury to flow to the opposite end, and so on ad infinitum. I am, Sir, Yours, &c. advertisement appeared in a country paper, setting forth that the advertiser had discovered a perpetual motion applicable to useful purposes, but wanted pecuniary aid to bring his invention to light. I have heard from one acquainted with a certain Customhouse officer, who occupies every spare moment in cutting and contriving" for perpetual motion, that he has proceeded so far, as to entertain sanguine expectations of perfecting a machine which is never to stop-until worn out. I am informed, too, from a very authentic source, of a gentleman deceased, who had actually hit upon the perpetual motion, and kept a model at work privately, for many years! These few instances show how absurd it is to be led away by the false hope of gaining popularity (the utmost that can be looked for) from the results of a discovery laid claim to by many. Surely those who toil for perpetual motion never call to mind, that so long as that quality is supposed to have been discovered, the most they can expect must be to bear the palm as revivers of a lost invention, and that only after proving three very knotty points : 1st. That it is bonâ fide their own, in opposition to a whole band (that would, doubtless, immediately rise up), and prior to any of them. 2d. That should others appear, theirs has the greatest power 3d. That the perpetuity of its motion be clearly proved. Query-How -by time? D MR. HALL'S PROPERTY OF NUMBERS. Mr. Editor,-Having lately seen some remarks upon the supposed newly discovered property of numbers, by Mr. Hall, as published in your Magazine, No. 283, in which the difference of any series of figures placed in a reversed order, will be divisible by 9, without a remainder; as a demonstration of this proP. S.-In March, last year, an perty may be new, and acceptable HENRY D Finsbury, Feb. 1829. VALUE OF ANNUITIES. (page 444, vol. ix.), was a solution by means of the Tables, which I conceived ought to give a correct result. If money obtained no interest, the value of an annuity of 11. would be equal to the expectation of life. For example, the value of an annuity for a life of 48 years of age, if money bore no interest,would be equal to 19 years' purchase; i. e. 197. in hand for each life would be just sufficient to pay any number of such lives 17. per annum. But when the value of an annuity is capable of improvement, by being put out at interest, the above sum would exceed the latter value. Suppose, for instance, 5 per cent. compound interest was obtained for the purchase money, the present value would be 127, 1s. 84d; if 4 per cent., 131. 2s. 8d. ; and if 3 per cent., 14/. 6s. 54d. For if, at the expiration of the first year, the interest be added to each of these respective sums, and the annual payment of 17. be deducted from each, the remainders will be the principals put out at the respective rates of interest for the second year; at the end of which, if the interest of each be again added, and 17. deducted, the respective remainders will be the principal for the third year. Which process continued to the end of the nineteenth year, each principal and its respective interest together will then amount to exactly 17. for each; at which time the last payment is supposed to be made, and the life to terminate. According to the Bills of Mortality, those lives which exist past the mean period of expectation are compensated by those which fail before it; so that what is lost by one life is gained by another. Hence, admitting the Tables to be computed on the expectation of life, we may, by knowing the present value of an annuity, and its rate of interest, obtain the number of years' purchase, or expectation, by inverting the process. But it appears that the Tables are not computed strictly according to the precise expectation of life; consequently, a correct result cannot be obtained by this method, but may be found as follows: 1st. To find the expectation for the joint lives of two persons, aged 30 and 40 years. The tabular value for the joint lives is 9.576, which corresponds to a single life of 53.95 years. The corresponding expectation for a life of this age is 16.08 years; but as two years are already elapsed, the present expectation is 14.08 years, according to the Northampton Tables. 2d. To find the expectation for the longest of two given lives whose ages are 32 and 42 years, the tabular value for the joint lives is 9.320, which corresponds to a single life of 55 years; the corresponding expectation for a life of this age is 15.42 years; whence -33.86 Sum of the expectations 49.28 Expectation of the joint lives — 15·42 Expectation of the longest of the two lives.... For it is evident, that if the two joint lives are worth 9.320 pounds, the term of the expectation must be equal to that of a single life of the same value. Hence, the correctness of the solution is manifest, substituting the expectation for the value of the annuities. Your correspondent “J. O. B.” is evidently wrong in both parts of his solution to the question of "A. B. C." In the first part of the question, he takes the mean of the expectations for that of the two joint lives. The expectation of the elder, or that of 40 years of age, is 23.08 years, by the Northampton Tables. Now, there is a probability that the younger life may fail before this period terminates; and therefore the expectation must evidently be less than 23 years, instead of 25, as stated by "J. O. B." In his solution to the latter part of the question, he has taken the expectation for the age of the younger life only'; whereas there is a proba. bility that the elder may outlive this · |