by employing the elegant device of Arbogast, of separating the symbols of operation from those of quantity; thus the expression: d d (x, y) (+)*(x, 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 ander 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 abridgment 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 expense of publication. Even in this reduced state they would be a valuable present to the mathematical world, as from the methods taken to ensure 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 acknowledgment 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 mathematical 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 P. 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 connexion 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 chymical 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. Donkin, the revival and introduction of the lithographic art, or printing from stone, and the improved process of printing in colours without plates, and the use of inks without oil or any material which may discolour the white of the paper, as proposed by Mr. Savage, all possess important advantages. Mr. Didot has likewise materially improved and simplified the process of producing paper in continuous sheets, or cutting it into any size or form. Among the manufacturing processes, that of weaving has received some important improvements, particularly from that most ingenious artist and manufacturer, Mr. Heathcoat, of London, who was the first to contrive effective machinery, which should imitate with precision all the intricate motions of the lace-maker upon the pillow, and produce a fabric in formation similar to that of foreign lace, and scarcely to be distinguished from it. He has since so far improved his machinery, as to introduce the gimp or pattern upon it in the first instance while weaving: nor is his machinery confined to the production of one breadth at a time. He has since directed his attention to that most valuable machine, the stocking frame, and now produces several pairs at once, by a process nearly similar to that which originally produced but one web. Mr. Mersey's new process for weaving coach and livery lace is also highly worthy of notice, as producing a fabric from the |