A Extent and Condition of the Milky Way,” and two papers by the Astronomer Royal“ on Parallax," which have recently been rewarded by the Parisian Academicians with Lalande's Gold Medal, conclude the list of communications printed by the Royal Society, with the exception of a long French communication on Ship Building by M. Dupin, and the following on Mathematical subjects : paper entitled “ Observations on the Analogy between the Calculus of Functions and other Branches of Analysis,” by Charles Babbage, Esq. M. A. F. R. S. Several singular analogies are pointed out, one of which relates to a method of solving functional equations by a process very similar to that employed by Euler for the solution of differential equations, by means of a factor, which renders the equations integrable. In the application of this, however, to functional equations, some difficulties occur, which still require elucidation. “ Two general Propositions in the Method of Differences.” By T. Knight, Esq. The two propositions here treated of, are to determine the no difference, or nois integral of a function of any number of quantities whose differ. ences are variable. The method pursued by Mr. Knight is sufficiently simple, considering the complicated nature of the formulæ investigated. It might however be rendered still more concise, d.ro + &c. 1 Σ by employing the elegant device of Arbogast, of separating the symbols of operation from those of quantity ; thus the expression: + 4* 'wd" Q(x, y) dr" dy which Mr. Knight denotes thus : (u +w)" d-dy" would be very naturally represented by d (r,y) dy and similarly in other instances. This author does not appear to be aware of the admirable work which contains this artifice of notation, the Calculus of Derivations; in it, if we mistake not, (see Art. 409), he will find a theorem much more comprehensive than the two which he has demonstrated, and which, in fact, contains them, but it is treated by different principles. A paper by T. Knight, Esq.“ on the Construction of Logarithmic Tables.” The author proposes to give instructions for the formation of a table of logarithms to any number of decimals. These will undoubtedly be found useful, when the increased accuracy of experimental inquiry shall have rendered necessary more extensive tables than those in common use; and there is an uniformity in the plan that Mr. Knight has pointed out, which has not been very prevalent in the instructions generally given for this purpose. In the mean time it is much to be regretted that the immense tables calculated under the direction of M. Prony have not yet been given to the world. It is now about twenty years since this vast labour was accomplished, and we have but lately heard of an intention of publishing an abridgement of them to eight places of decimals; and even this appears to depend on the chance of finding a number of subscribers to defray the expence of publication. Even in this reduced state they would be a valuable present to the mathematical world, as from the methods taken to insure their accuracy, they would form a criterion by which the accuracy of the common tables might be ascertained. This volume also contains a note from Mr. Knight, in which he acknowledges that the proof he had given of the binomial theorem, and which the Royal Society inserted in the volume of their Transactions of the preceding year, had been invented and published by Mr. Spence about six years before; and he makes the same acknowledgement with respect to the first theorem in his paper on the construction of logarithmic tables, which we have already noticed as appearing in the present volume. The very learned and ingenious gentleman who has thus anticipated the results of Mr. Knight, is now no more ; he was one of the very few of our countrymen who cultivate the higher departments of mathematical science; intimately acquainted with the present state of that science, he devoted himself with ardour to extend its boundaries; and the successful result of his early efforts furnished abundant cause to regret the premature termination of a career of originality and genius. His mathematcal papers have been examined by Mr. Herschell, and some, which are sufficiently complete, are in the course of publication ; amongst these will be found a paper on the Integration of some Equations of finite Differences, of very considerable difficulty. One of the most important additions to mathematical literature during the past year, is a New Explanation of the Theory of Imaginary Quantities, by B. Gompertz, Esq. It is a subject which has always been considered as involved in considerable difficulty, although the accuracy of the results obtained by their means has never been disputed. The mode of reasoning pursued by Mr. Gompertz is peculiarly delicate and refined, and to those who are sufficiently advanced in these studies to appreciate its force, it is perfectly convincing We shall not attempt to abridge this explanation, which would lose considerably by being deprived of the illustrations with which it is so ably supported. We cannot, however, forbear noticing the singular connection which Mr. Gompertz has shown to exist between the doctrine of imaginary quantities, and one of the most beautiful and interesting branches of geometry, the subject of porisms. The mechanic arts in this country have already reached so high a degree of perfection as to render them, in many instances, almost incapable of improvement; hence, their advances cannot be expected to be so rapid and brilliant as the branches of chemical science, which, as it is developed, is constantly affording new facilities to the man of science, as well as the manufacturing artist. Considerable advances have, however, been made in this department since the first appearance of this work, and the Editors trust that they will not be accused of inattention to this most important branch of British improvement, since they have taken occasional opportunities of introducing accounts of such objects as came to their knowledge, and which seemed most particularly deserving of public attention. Among the most prominent of the mechanical improvements which we have now to notice, is the dry process of preparing and husbanding Flax, of which a particular account is given in our present volume, and which appears to hold out material advantages to this nation. The typographic art has not only been improved and facilitated, but this improvement has been extended to several articles connected with it; and the ingenious application of machinery and steam to the process of printing, as practised at the Times newspaper office, and in a more improved state at the printing-office of Mr. Bensley, cannot fail to excite astonishment at the rapidity with which the work is performed. The new and ingenious printing press of Mr. |