work in an octavo volume of over 200 pages, which has been gratuitously circulated among scientific men for examination. His principles of reasoning, which necessarily differ from those established by the schools, almost as a matter of course, encounter the sneers and opposition of learned Professors, who are pledged to their old method, and are unwilling to entertain or consider any other. But fortunately for the improvement of the age in other things, and perhaps in this also, among many minds there are always some to be found who are not shackled by error, because it is established by usage. It is a part of our principles to promote inquiry in respect to everything useful, and therefore, without pronouncing any opinion of our own on the work or subject in question, we take pleasure in publishing the following letter, received by the author from a gentleman in Michigan, fully sustaining the truth and value of the alleged discovery. It is selected from among many others which have been received, and which are less full but not less explicit in conceding the truth-that the quadrature of the circle is at last attained. ANN ARBOR, Washtenaw COUNTY, MICHIGAN, January 13, 1852. Mr. JOHN A. PARKER, New York: SIR:-A copy of the "Quadrature" was safely received, and as you doubtless expect some expression of opinion in regard to its merits, and the truth of the principles it contains, I shall take this opportunity for acquainting you with the views at which I have arrived, and at the same time of returning thanks for the opportunity so kindly afforded of its perusal. Nothing, to my mind, can be more satisfactorily conclusive than your preliminary demonstrations of the ERROR OF GEOMETERS in their method of finding the approximation in use. Indeed, this is of itself so palpably evident that it seems hardly to require a demonstration; but from the reasoning you have employed, the error of PRINCIPLE, as well as that of approximation, becomes so manifest that for any one to deny its existence would be equivalent to denying the truth of the most self-evident axiom. For myself, I was never wholly satisfied with the degree of correctness of the approximation in use, more especially as such a mode of measurement does not belong to the logical deductions of geometry, and cannot be ranked among them since it is erroneous and imperfect. For even if this approximation had possessed the degree of correctness assigned it by Legendre and others, the way in which it is obtained is not according to the true basis of geometrical deduction, which should always be the primary and relative properties of the figures considered, and the relations subsisting between them. It is hardly necessary to say that the method by which the ratio in use has been determined is entirely independent of any such relative property, since it is based on the properties of straight lines alone, and since it can lead us only to an approximation, it is therefore imperfect, and to accept it as the basis of any subsequent reasoning is utterly at variance with the true theory of geometrical investigation. I am pleased to discover that, in your demonstration of the quadrature, you have sought first the relative properties of the square and the circle, and the primary relations subsisting between straight lines and curved lines, as the only proper and successful means of investigation, as it is certainly the only way by which the circle can be exactly measured. The propositions and reasoning of chapter 2 of the Quadrature, by which your ratio is determined and proved to be true and exact, are as logical and conclusive as any of Euclid or Legendre; and nothing but the sheerest bigotry can hinder every well-informed mathematician from accepting this as the only true ratio of circumference to diameter of all circles. Your propositions proving the equality of the circle and the square, by the transition and alteration of shapes, and the opposite duplicate ratio of the circle and equilateral triangle, are, I be lieve, entirely new and original, and may be ranked among the most important propositions in geometry; and I cannot help inferring from their discovery that the science of geometry, like every other, is capable of perpetual and almost inconceivable advancement; and that many universal and fundamental principles remain to be determined and proved, and perhaps, also, that some erroneous conceptions of relations and principles, which have held their basis only on the ground of analogies, will be either modified or entirely superseded. Chapter 3, of Practical Questions on the Quadrature, contains the most incontrovertible evidence of your ratio being the true one, from its application in determining the astronomical circles, while at the same time its value in correcting important data is apparent. The greatest merit of your discovery, however, consists, I think, in its capacity for the discovery and development of new principles from its own inherent analogies, and in this way it seems to be specially calculated for enlarging the bounds of mathematical science. The manner in which you have treated the problem of three gravitating bodies, as well as that of the moon's diameter, the sun's distance, &e., is very original and striking, and well worthy of attention, since these are discussed entirely from the application of general and mechanical principles in a way never before attempted, and without the aid of observations. The mind possessed of common intelligence that can, in view of all this, pronounce your ratio incorrect and its theory a delusion, must be either entirely wanting in every attribute of candor, or so misguided by the influence of preconceived prejudice, as to be utterly incapable of judging aright concerning anything not sanctioned by usage, or which has not received the general approbation of professors. Although I have arrived nearly at the end of my sheet, and I fear well nigh exhausted your patience, I cannot pass that division of your work denominated an appendix without some comment, or without expressing regret that the opportunity did not offer for you to discuss more fully those principles and their manifest applicability, which are there set forth. The definitions of the terms "nothing" and "infinity," which are generally accepted by the schools, and on which many demonstrations have been founded, are manifestly absurd and illogical, since we cannot suppose a thing to be so small that it cannot be less, or so large that it cannot be greater, and afterward obtain its true value in numbers, (if these expressions are held synonymous with the meaning usually applied to 'infinity,") for, although numbers are infinite, yet the mind, being in this instance governed by analogy, and necessarily limited in its conceptions to things finite, cannot measure infinity by any cognizable standard. Hence, I fully agree with you in saying that infinity, taken in the sense in which it is usually defined, is what no man can ever comprehend, and that the only way by which it can with propriety be recognized and treated in geometry, is by considering it to be "one ultimate particle of matter, such that, in the nature of the thing under consideration, it cannot be less." There can be no doubt, I think, that the proposition from which this latter definition is taken, demonstrates the only way by which the principle can be received within our comprehension, and at the same time fulfill its uses in mathematical demonstrations. The principle of considering geometry, properly speaking, an abstract science, which is strictly adhered to by the schools, is another delusion, for the principles of geometry are every way connected with the mechanical development of things, and without the existence of material organizations these principles themselves might not have existed; and it is absurd to destroy in our minds the connection, since it often aids us in the discovery of what may be termed abstract truths, for it may be plainly manifest to every reasonable mind that as numbers and things are inseparable, so in the operations of geometry, all our ideas of extension and magnitude presuppose a material medium of exercise and comparison, without which all the propositions and formulas that can be deduced bear about as much relation to the true end and sphere of geometrical research as the most abstract principle in metaphysics. Permit me to bring to a close these hasty remarks, to express my unqualified satisfaction at the peculiar fitness and originality of your methods of demonstrating and illustrating the principles contained in your work. It is quite obvious that new and independent principles often require a mode of demonstration somewhat different from that of conventional formulas, which are often wholly incompetent to determine the existence of a new principle from a lack of a combination of relative properties. Such being the fact, and it being also true, that natural truth is nothing more than the agreement or disagreement of relations with our perceptions, and consequently anything may be considered as self-evident which may be directly referred to this standard, I think mathemati cians should hesitate less to adopt independent methods of investigation whereby important principles may often be brought to light. But I must here close this rambling review of the leading principles contained in your work, and in doing so let me assure you that there is no principle to be found in that work that does not meet my unqualified approval-that has not my most earnest support; and further, that the study of no work on mathematics ever gave me more satisfaction than that of the "Quadrature." Very respectfully, your obedient servant, S. L. WOODRUFF. Art. VI.-OF ARRESTING CONFLAGRATIONS IN COMMERCIAL CITIES. FREEMAN HUNT, ESQ., Editor of the Merchants' Magazine, etc. DEAR SIR:-The subjoined communication on the extinguishment of conflagrations, has recently appeared in the Pennsylvania Enquirer, but as being equally important in its bearings on New York as on Philadelphia, I send it to you for insertion in the Merchants' Magazine. Substituting the Croton reservoirs for those of Fairmount, and North and East Rivers for the Delaware and Schuylkill, all that has been said applies in your city no less than in ours. Indeed, that portion which relates to the employment of large steamboats with aquatic engines, is essentially important in its bearing on New York, because there is vastly more property within reach of the assistance which such a steamboat could afford. The experiment might be tried with some large ocean steam liner, which should be condemned as not worthy to continue in the line. It would be interesting to see how far a jet of water could be thrown by the power of one of the largest engines used in navigation. I am, Sir, with due respect, Your fellow-citizen, ROBT. HARE. PHILADELPHIA, May, 1852. ON THE MEANS OF ARRESTING EXTENSIVE CONFLAGRATIONS. Some time since I corrected an error of an author of a communication in one of the public journals, which ascribed to me the project of employing stationary engines, in extinguishing great fires by means of leather hose. This idea I stated to be impracticable. It never occurred to me that stationary engines could be made to operate on fires in every part of our city, however remote, by means of hose, without making this part of the apparatus too heavy and unwieldy for handling. Even were there an engine for every square, this objection would seem almost insurmountable. The project which I actually suggested, was that of resorting to locomotive fire-engines, each resembling those employed upon railways, with the addition of a powerful forcing pump, and the substitution of high wheels, to enable them to move over pavements. It was assumed that, as in gunnery balls have a range proportional to their weight, when impelled by charges augmenting with their weight, so the distances to which jets of water can be thrown are proportional to the sectional area of the orifice of emission; the pressure in the air vessel being in all cases the same per square inch. Hence it would seem possible to throw a jet of water a quarter of a mile, by having sufficient steam-power, and of sufficient size. I also recommended that a steamboat should be provided with a powerful apparatus for throwing water and propelling to any practicable distance by means of hose. Such a boat would be of immense importance to our shipping, and the stores within a certain distance. Nevertheless, it occurred to me, subsequently, that a plan might be adopted, which would enable a great stationary engine to operate on fires throughout the whole ramification of the hydrant pipes by which the city is supplied by water from Fairmount. The plan involves that such an engine, with suitable pumping apparatus, should be made when desirable to take the water from the main, after leaving Fairmount, and return it back to the main, with an increase of pressure sufficient to command the roofs of the highest buildings. Were such an arrangement made, and a branch pipe carried up nearly to the roof in each house, a hose attached to the orifice of the pipe at its upper termination, would command the roofs of the houses in question, and those adjoining it, and thus prevent the access of fire through the usually combustible covering of shingles. Other orifices in the lower stories would enable a hose to command them more conveniently, so that the inhabitants of a mansion, with or without the aid of neighbors, might extinguish a fire or keep it from entering from without. The fire companies might be provided with hose of an extra length, or metallic tubes to be easily attached to each other, so as to be water-tight. By means of such hose water could be carried up to the roof of any house adjoining the one on fire, so as to play upon it with facility. When a great fire should occur, the mains leading to all parts of the city should be closed, excepting that leading to the fire, so as to concentrate the supply of water where it would be most wanted. The additional head might be too much to be borne by some pipes, but these being discovered, should be replaced by stronger pipes. The engine might be erected at the level of the reservoirs at Fairmount, so as to have less work to perform, and a large cistern might be built to receive the water at an elevation sufficient to give the requisite head of water. This being kept full would feed the mains until the engine could be got under way. A large tank of iron would be the best reservoir, probably. I am aware that this plan would be costly, but what is more costly than fires? It was suggested, in a former communication, that if the fire of the 9th of July, 1850, had occurred on the afternoon of the north-east gale of the 18th, the whole of our city to the south-west of the fire must have been destroyed. Our present means are quite incompetent to put out conflagrations of a certain magnitude. It must be evident that a stationary engine would not be necessary to elevate the water to the necessary hight, were there adequate power in the water-wheels at present employed to lift water into a tank sufficiently high. Another modification might be made. A stationary engine might be situated on the Schuylkill or Delaware, or one engine on each river, by which water might be taken from either and forced into the main pipes for the sup ply of water from Fairmount, all the mains being closed which might prevent the concentration of the whole supply upon the district in which the conflagration should exist. I am much more confident of the efficacy of the projects which I have suggested, than sanguine as to my ability to induce the public to adopt measures involving much expense, and which are so much out of the usual routine. I will therefore conclude by proposing a remedy which is not of a nature so adventurous as those above proposed, and though less adequate to arrest conflagrations, is more competent to check incipient fires. The remedy in question is in fact so easy of execution, that it might be put into operation before the other could be gotten under way. It would add much to security against fire if the cocklofts of buildings were furnished with ample reservoirs of wood, lined with sheet lead, to receive the rain-water, which could be collected from the roof. In the vicinity of such reservoirs about a dozen tin buckets should be kept. The ridge of the roof should support a foot-way, with hand-rails on each side. By these means the roof might be kept wet by a few persons; even women might perform this duty if there should not be men enough at hand. Of course a cock with a hose attached might be inserted so as to command the interior of the building, or that of any one adjoining. The presence of a hand-forcing pump would also be advantageous. These precautions might be compensated by a reduction of the cost of insurance. It would be especially important that reservoirs, such as have been mentioned, should be provided in all buildings of more than three stories in hight. In fact, it would seem reasonable that all buildings of an elevation above that of three stories, should be obliged by law to have reservoirs, because their inaccessibility to the jets from fire engines, makes them dangerous to a neighborhood. Well provided, as proposed, with a stock of water, they might acquire the opposite character of affording protection against fires. Houses of not more than three ordinary stories, might have their reservoirs supplied from the public mains, with the hydrant water. Reservoirs of the materials above recommended last for almost any length of time, without giving any trouble. Without any resort to steam, the head of water in our hydrant mains at night would throw the water into reservoirs situated near the roof of any edifice, however elevated, by resorting to water-rams. One portion of the water supplying a house may be thrown up to the roof, while the other may be caught in a tank for ordinary purposes. By these means reservoirs might be supplied independently of rain. R. H. |