The municated by the means of torches in the following manner. first letter is Cappa (C); which stands in the second division of the alphabet, and upon the second tablet. The person, therefore, who makes the signal, first holds up the torches upon the left, to signify that it is the second tablet which is to be inspected; and afterwards five upon the right; to show that Cappa is the letter, which he who receives the signal must observe and write. For Cappa stands the fifth in the second division of the letters. Then again, he holds up four torches upon the left, because Rho (R) is found in the fourth division; and two upon the right, to denote that it stands the second in that division. From hence, the person who receives the signal writes Rho (R) upon his tablet, and, in the same manner all the rest of the letters. By this method, an account of everything that happens may be conveyed with the most perfect accuracy. It is true, indeed, that, because every letter requires a double signal, a great number of torches must be employed. If the necessary pains, however, be used, the thing will be found to be very practicable. In both these methods it is principally requisite, that the persons employed should first be exercised by practice that, when a real occasion happens, the signals may be made and answered without any mistake. HAVING given the substance of Mr. Pickering's lecture on telegraphic language, we have annexed a fac simile of Mr. J. R. Parker's Semaphoric Telegraph, as well as the Marine Telegraph Flags, in a plate at the end of the present number, wherein is exhibited a practical exemplification of a telegraph communication, as contained in the United States Telegraph Vocabulary; a second edition of which, we understand Mr. Parker is preparing for the press ; `comprising twenty thousand words, phrases and sentences, applicable to telegraphic purposes, and upon every subject, together with the designating numbers of nearly fifteen hundred American vessels, which have adopted this system of telegraphic communication. We understand that the Navy Department of the United States have lately been furnished with sets of the Marine Telegraph Flags, and a designating number and a signal book for each vessel, which will enable the Government vessels and Revenue Cutters to hold communication with our merchantmen in every section of the Union, as well as to communicate with the several telegraph stations upon our coast. [Furnished for the Tracts and Lyceum.] QUADRATURE OF THE CIRCLE. 'What was once Incontrovertible is overthrown; And what now seems built on the base of truth IN the Tracts and Lyceum for Jan. 15th, I observed an article with the signature of S.' on the quadrature of the circle, wherein some things appear well said, and some, manifestly, for the purpose of burlesque and ridicule. The allegation that the solution of this problem must, in reality, consist in turning a ball into a cube, or a cube into a ball, is a more happy definition than those generally given, which, generally, only go to give us the idea of turning a circle into a square, or a square into a circle of the same area. But assuredly, if the circle may be turned into the square, the ball can also be turned into the cube. But Mr. S.'s following similitudes are rather gratuitous, and do not at all serve as preliminaries to the discovery of the true quadrature; or to make its valuation the every-day amusement of the schoolboy. The writer might have gone still further with his true definitions, and might have said, that the discovery of the quadrature would be nothing short of the ne plus ultra of numbers-the perfection of the admirable method of numbers by fluxions, reducing all flowing quantities, with their maximum and minimum, to a geometric certainty, always resulting in assignable roots and powers. Nay, farther; that by the institution or adoption of a system of nascent ratios, the exertions of nature at extension, without its actual or assignable production, might be as clearly shown to the mind, as when extension is apparent or assignable. Then, may the schoolboy be employed in deducing the true result of the quadrature, in almost an infinite variety of processes; either by chopping off, as Mr. S. would call it, or as M. Legendre would say, by the law of continuation, or by other processes, not in any-wise connected with the approximating modes, but by one grand and agreeable result of the mind, by which means, such school-boy might imbibe quite different views respecting dynamics, and the mechanism of the heavens, (at least, in their explanations,) from those already given. Now Mr. S. seems determined on goading up some external sense of a certain Mr. Thompson, (with whom I am wholly unacquainted, but whose champion I am,) who seems to have declared himself the discoverer of the quadrature, and seems determined on compelling him to come forth with an explanation to the world, and that too, in the Tracts and Lyceum; as though it were to consist of some one or two geometrical diagrams, or an algebraic or fluxionary formula. But let Mr. S. be assured, (in behalf of Mr. Thompson or any other person) that a proper solution of the quadrature can never be given to the world, short of a whole new system, or dispensation of the economy of numbers, based upon an appropriate symbolic language; and that even a fair synopsis, would probably contain some hundreds of pages, without the encumbrance of a single diagram, (which, by the way, would be of no possible use in the solution). But Mr. S. has not informed us to what notice of Mr. Thompson he refers: whether it be the one which appeared in 1826, in no less a public print than the paper called Coram's Champion, in which I saw it over the signature of G. H. Thompson; wherein he says, that by the divine assistance,' he was induced to announce to the world his discovery of the quadrature, which he had made by means of the Plane Sliding Sector, &c., and which notice was then answered by D. Gould, Esq. then and now, of Boston-or whether it be a more recent circumstance, when Dr. Wilbur Fisk was called in by Mr. Thompson, and some of his friends and patrons, to examine Mr. Thompson's diagrams upon the quadrature, that Mr. S alludes to, or to some still more recent notice in the public prints, of which I am wholly ignorant. But Mr. Thompson has doubtless learned by this time, that any attempt to give the world his new and better dispensation, is rather an unwelcome task; and that any attempts to introduce it, are only scoffed at and ridiculed; and that the world has taken to itself a faith, which even his better dispensation cannot eradicate. But surely Mr. Thompson is entitled to approbation for his perseverance-his attempts are not ephemeral, like those of Mr. Feldt, Mr. Woodville, the Irish boy, &c. No; he has, as it were, rode out the storm; and is now, doubtless, well anchored in his own assurance-more consolatory, by far, than the applause of a world who know nothing of his vast imaginings: and hence, apprehending that he will not deign to relieve the doubts of Mr. S., by giving explanations, though thick as blackberries;' and in case of his failure, having supposed such explanation might as well devolve upon me as on anybody, I have come forth as his champion: not as an Archimedes redivivus, with brawny limbs and grisly beard, moving the world without a place to stand on, as Archimedes has done for two thousand years, respecting the quadrature, but as the philosopher of the forest, directly from the field of nature; not from the trodden graves of ancient colleges and halls, where all ideas are circumscribed within certain fixed bounds-where all must travel within a circle, the ratio of whose diameter to its circumference is as 1 to 3.1416, &c.— but where our sphere of action (and thinking) is either enlarged or diminished, as fancy may dictate. And although the hill of science is not at present situated wholly in the field of nature, yet I can inform Mr. S. that the field of nature is not destitute of the requisite materials from which the sciences are to be drawn; and who knows but that Mr. Thompson is one of nature's children, and has long cultivated her field, and has seen her vast stores; for, There is her Time, which nothing can control- But to the explanation of the quadrature which Mr. S. asks at the hands of Mr. Thompson, (now devolving on me). And here I will presume that Mr. S. knows as well respecting the symbolic language, by which the true quad rature is to be expressed or determined, as he does respecting the true solution. Hence I shall satisfy myself for the present, by giving him the prime formúla, from the book of Rational Analysis, (which alone contains the true solution) together with such explanation as I think necessary on the present occasion. But, as a preliminary, let D denote the diameter of a circle-F, of the circumference-G, one side of the inscribed square, and I, the radius; let denote the difference between G and F 1 1 1 let denote a rational of the difference between J 2 1 and F-and let 3 denote the rational of the difference between D and F-thus denoting either known or assumed differences between certain linear quantities of the circle. The prime formula is thus: by the aid of which, not only the true quadrature is to be developed, but also those exertions of nature at extension where no assignable extension is produced; and, to this end the formula may be extended, (or rather intended) to any required number of characters we please. And now I will inform Mr. S. that by the help of said formula, in connection with others suitable to the purpose, the true quadrature may be deduced, by a great variety of evolu tions, as he can testify, if he will take the trouble to go through with the proper processes. But after all, he will find the diameter and circumference as incommensurable in the first powers, as are the diagonal and side of a square; whose ratio is not susceptible of a definite expression by our numeral figures. But if Mr. Thompson, or any other person, will show their commensurability in any of the higher powers, they will confer a benefit on mankind far transcending what the most enthusiastic have yet conceived would result from a perfect solution of the quadrature of the circle. Craftsbury, (Vt.) Jan. 28, 1834. A. Y. NOTA BENE.-The compositor cannot find any types with which he can figure Mr. A. Y.'s formula precisely, but he has done |