thought and metaphyfics, as the verfes of the original Cowley. The following paffage has little or no refemblance to him: When in a penfive mood I fit, Defcribe thy look, thy step, thy make, Thefe lines are fucceeded by a Monologue to the memory of Chatterton, deploring his fate, and celebrating his genius. Why this Monologue, or the lines in imitation of Cowley, fhould be annexed to the Maid of Arragon, we cannot difcover. Uncommon excellence is not their recommendation. C. ART. XII. Conclufion of our Account of the PHILOSOPHICAL TRANSACTIONS of the ROYAL SOCIETY, Vol. LXIX. Part 1. for the Year 1779. See Rev. for March. MATHEMATICS. Problems concerning Interpolations. By Edward Waring, M. D. F. R. S. and of the Inftitute of Bononia, Lucafian Profeffor of Mathematics in the Univerfity of Cambridge. THE method of interpolating, now fo well known and fo often practifed by aftronomers, was firft invented by our countryman Mr. Briggs, Savilian Profeffor of Geometry in the Univerfity of Oxford, and put in practice by him in the calculation of logarithms. The principles on which he proceeded were afterwards explained by Reginald and Mouton in France. Sir Ifaac Newton, in Lemma v. book iii. p. 486, Phil. Nat. Princip. Mathemat. edit. 1726, gave a moft elegant folution of the problem for drawing a curve line through the extremities of any number of given ordinates; and in the fubfequent propofition applied the folution of this problem to that of finding, from certain obferved places of a comet, the place of it at any given intermediate time. Dr. Waring fays, perhaps a ftill more elegant folution of the problem, in fome accounts, has fince been given by Meffrs. Nichole and Stirling: and he adds, the fame problem is refolved, and rendered fomewhat more general in the paper before us, without having recourfe to finding the fucceffive differences. The paper confifts of two theorems and a problem. In the theorems, the Profeffor demonftrates certain properties which belong to a series of the differences of numbers, or to a series of numbers which have given differences; for both amount to the fame fame thing. In the problem, he fhews how, from these properties, to find certain corrections, which being applied to a feries of numbers, found from certain affumed ones, according to any given law, the fums or differences may be equal to the refults deduced from certain other numbers according to the fame law and he adds, that from thefe theorems, feveral others of a fimilar nature may be easily demonftrated. Art. IX. On the general Refolution of Algebraical Equations. By the fame. In this Article Dr. Waring informs us, that in 1757 he fent fome papers to the Royal Society, which were printed in 1759, and copies of them given to feveral perfons at that time that thefe papers, fomewhat corrected, with the addition of certain properties of curve lines, were published in 1762, with the title of Mifcellanea Analytica; and reprinted, with additions and emendations, in the years 1767, 1768, and 1769, and published in 1770 under the title of Meditationes Algebraica. He farther informs us, that thefe papers contained, among many other inventions, the moft general refolution of algebraical equations yet known; as it contains the refolution of every algebraical equation of which the general resolution had then been given; namely, the refolution of quadratic, cubic, and alfo of M. De Moivre's and M. Hudde's equations; likewife of the equation of which Mr. Berout has fince published the refolution. It moreover difcovers the refolution of an equation of any given number (7) of dimenfions, the fame number (n) of its roots being alfo given; and alfo deduces innumerable equations of any given number (n) of dimenfions, which contain -1 in dependent coefficients. From which the Doctor infers, that it is probable this new method of his contains the most general refolution of algebraical equations that ever has, or perhaps ever will be invented. Having thus given us the hiftory of his publications on this head, he proceeds to lay down the general formula for the refolution of equations, and then illuftrates it by examples in the refolution of equations of particular dimenfions. Dr. Waring's principal motive, in the publication of this paper, appears to be, the vindication of his claim to the invention of this general mode of refolving algebraical equations; which, as we gather from the paper before us, and what he has faid in the preface to his Medit. Analyt. for we have not feen the work which he refers to, has been fince published by fome foreign mathematicians of the firft rank, without fuch acknowledgment, as the Doctor feems to think was neceflary, of his being the firft difcoverer of them. If this be not the cafe, M. M. Euler and Le Grange. We we must own that we cannot account for his giving us, fo often, a chronological hiftory of the times when the books in which they are contained were written and publifhed; namely, twice in the paper before us, and once in the Preface to his Meditationes Analytica. If it be really the cafe, we think the Profeffor might have spoken more plainly without any breach of modefty or decorum. MECHANICAL. Art. XII. Tentamen continens Theoriam Machine fublicarum → An Effay containing the Theory of the Machine for driving Piles. By Thomas Bugge, Aftronomer Royal, and Profeffor of Aftronomy and Mathematics in the Academy of Copen. hagen, and Member of the Societies of Sciences at Copenhagen and Drontheim. Communicated by Sir John Pringle, Bart. Our Author fets out with obferving, that among the numerous advantages which civil fociety have derived from the knowledge of mechanics, the art of driving piles, that is, large oblong beams, into the earth, by repeated blows, is not the leaft. This art was not unknown to the ancients, as may be proved from many paffages in Vitruvius: for although this celebrated author does not defcribe the machine by which they did it, yet their knowledge, in this refpect, is placed beyond all doubt, feeing that without it, it would have been impoffible for them to have built bridges, moles, dams, bulwarks, pyramids, columns, and other edifices, the fize, majefty, firmness and durability of which we admire, but can fcarcely imitate; and all these things require the moft firm and folid foundations. If the foundation of a building is to be laid in a marshy place, large piles must be driven, by means of engines of this kind, to great depths, and the spaces between them filled up with great ftones, gravel, fand, and mortar, before the foundation of that building can be laid. The exact form of the machine by which the ancients drove these piles is not now fufficiently known. Several forts have been defcribed by Leopold, Defaguliers, and Belidor. But amongst all thofe, that which was invented by Vauloüe, described by Dejaguliers, and brought into ufe while the foundation of Weftmintterbridge was laying, has greatly the pre-eminence over all others. Its peculiar advantages are, that the weight, ufually called the Ram, may be raifed with the leaft force ;-that when it is raised to a proper height, it readily difengages itfelf and falls with the utmost freedom;-that the forceps are lowered down fpeedily, and inftantly, of themfelves, again lay hold of the Ram, and lift it up on which account this machine will drive the greatest number of piles, in the leaft time, and with the feweft labourers. I Mr. Mr. Bugge next proceeds to fhew that Belider has entirely miftaken the theory of this machine, and then goes on to lay down and explain the true theory of it; in doing which he delivers the following principles: Ift, If the refiftance of the ground, and the maffes of the piles, be equal, the depths to which they will be driven with a fingle blow will be as the product of the weight of the Ram into the height through which it falis. 2d, If the mafles of the Ram and heights through which it falls are both equal, the depths to which the piles will be driven will be in the inverfe ratios of the maffes of the piles into the fuperficies of that part of them which is already immersed in the earth. 3d, If all thefe things be unequal, the depths will be in a ratio compounded of the direct ratio of the heights through which the Ram falls into its mafs, and the inverfe ratio of the. mafs of the pile into its immerfed fuperficies. 4th, If the weights of the Ram be equal, and alfo the weights of the piles; the depths to which they will be driven will be as the heights through which the Ram falls directly, and the immerfed fuperficies of the piles inverfely. Or, because the immerfed fuperficies of the piles are as the depths which they are already driven into the earth, the depths they will be driven are fimply as the fquare roots of the heights through which the Ram falls. From thefe principles, which are in a manner felf-evident, our ingenious Mechanician determines, that the diftante which a pile will be driven by each fucceeding blow will be lefs and lefs, as the fuperficies of that part of the pile which is immerfed in the ground increases; contrary to what had been afferted by M. Belidor: and, confequently, that there is a certain depth, beyond which a pile of a given mafs and fcantling cannot be driven; the mafs of the Ram and the height through which it falls at firft being affigned. He alfo refutes the notion which had been entertained by fome, that the driving of piles is facilitated by loading them with weights: for the depth to which a pile can be driven by any fingle blow (all other things remaining the fame) being inverfely as its mafs, it is manifest that thus loading the pile, and thereby increafing its mafs, will be fo far from accelerating its defcent, that it will abfolutely retard it. He concludes his paper with fome very useful practical hints, and obfervations, relative to proportioning the feveral parts of the machine to one another, the number of men which ought to be employed, examining the ground, and the part of it where the first pile ought to be driven, fo that the others may drive with the greatest ease poffible, W. ARTe (285 ART. XIII. Dedication to the collective Body of the People of England, ee Vol. in which the Source of our prefent political Diftractions are pointed out, and a Plan propofed for their Remedy and Redress. By the Earl 63,143. of Abingdon. 8vo. I s. 6d. Almon, &c. 1780. IS Lordship opens this Epiftle Dedicatory with an ex Hplanation of the reafon why his Thoughts (fee Monthly Rev. vol. lvii. p. 249) are dedicated to the collective body of the people of England at this period of their publication, and not at firft. The public good, fays he, was my object: but whether I had made ufe of the proper means to that end, or no, was not for me to determine. So far indeed as my intentions went, of their rectitude I was confcious: but how far I had fucceeded in ability refted upon the judgment of others. To the judgment of others I appealed, and I called upon the Public, if I was wrong to fet me right. I declared that Truth being my only object herein, I fhould as readily look for it in others as feek it in myself;' and I have waited impatiently for the event: but notwithstanding five editions of thefe Thoughts have been had, and much time has fince elapfed, to this very hour, not the colour of objection, nor the fhadow of argument have been opposed to them. 'These then are the circumstances under which this Dedication now What diffidence had before with-held, makes its appearance to you. acquired confidence hath fince produced; and as, on the one hand, if truth be with me, my reward will be in its ufe to you; so, on the other, if error, my confolation is, that I have been ever ready to retract it. But having faid, that not the colour of objection, nor the fhadow of argument have been opposed to thefe Thoughts; I feel myself obliged to offer a few words in anfwer to one writer, who has been pleafed to honour me with his public correfpondence. This writer is a Mr. Cartwright, and who, in a Letter addreffed to me +, has, fuppofing me wrong in a pofition that I have laid down, called upon me, with great propriety, for my juftification. I rejoice to meet fuch inquiries. They are the avenues to truth. And I am no lefs pleafed with the inquirer. He has written like a gentleman, and what is more than this, like an honest man: for, unlike those anonymous writers, whofe fears are left the infamy of their names fhould increase the infamy of their writings, he has affixed his name to what he has written. It is therefore matter of concern to me to find myself miftaken by this writer: but my hopes are, that to remove his mistake will be equally fatisfactory to him, as to me.' His Lordship then enters upon his vindication, and, as we think, fully proves that the error has arifen merely from a mifconception of his expreffion; and that, in fact, with refpect to + Vid. A Letter to the Earl of Abingdon difcuffing a pofition REV. May 1780. the 144.(Art. |