public in general, but only for the direction of the governors of the hospitals in question, yet we think it of importance enough to be offered to the notice of our readers. The committee begin their report with considering what were the primary and proper objects of the charity of Bridewell; and they find that arts-masters and apprentices are not even mentioned in the royal grant of Edward VI. but that it was given as a “house for occupations,” 1. For boys of riper years, who were found unapt to learning, and so inexpert in the trades which they had been thought competent to learn, as to be unable to get work. 2. For the fore and fick, when cured and discharged from St. Thomas's, if able to work, that they might not wander about as vagabonds, but have suitable employment. 3. For the lewd, the sturdy beggar, and the idle in general, who would be compelled to labour therein, and so to serve the commonweal ; and lastly, for prisoners discharged at the sessions, that they might have occupations, and not again become thieves or beggars. They farther find that a discretionary power is granted by the charter to alter the rules and regulations as circumstances may require; and they are fully of opinion that, however useful arts-masters and apprentices might have been in the ina fancy of the institution, yet, in this advanced period of the arts of life, they are an useless and a very objectionable part of the charity; being the cause of a great expence, for a species of instruction which would be much better carried on out of the house. They, in consequence, recommend that the numerous apartments occupied by this class should in future be employed in furnishing means of labour to the original objects; among whom none more loudly call for attention than the prisoners discharged at every session from the Old-Bailey, Another point, which the committee, were to examine, was the connexion of the hospital of Bridewell with that of Betlalem; and they answer, in the affirmative, the question, whether the governors are justifiable in applying the revenues of the former to supply the wants of the latter? Of the effential defects in the late system of management with respect to the disbursement of the revenues, the following paragraph affords a moft glaring proof: • Is is suficient here to observe, that, by one of those statements it appears that 59571. us. hath been expended on the apprentices, and 74931. 165.4d. in maintaining the vagrants (the only tivo fuppofd objecis of charity of Bridewell) ; whereas it bas coił, within the jume period, 19,2541. os. 4d. in falities, &c. of the uricer, employed in * Alluding to whai is previoulay tated.; the the management; befide 63411. 6s. 1 d. for their taxes, views of estates, &c. and 32341. gs. id. in feasts ; together with 281. 153. 6 d. and, what seems equally extraordinary, the further enormous fom. of 17,3321. 195.7d. for repairs at the hospital of Bridewell alone." The committee then make their observations under differ ent heads, and propose fuch alterations as fhall improve the utility and economy of the hospitals. The report is concluded by an appendix of the standing rules and orders of the two hospitals, and abridged tables of receipts and expenditures. The public, to whom finally all institutions of this kind muft be considered as belonging, have a right to expect that the labour so ably and faithfully applied by the respectable gentlemen of this committee should not be lost: but that the improvements proposed, and (as we understand,) adopted, should not be suffered, either by neglect or by artifice, to become ineffica-. cious or obsolete. Art. XII. The Destrine of Universal Comparison, or General Proo portion. By James Glenie, Elg: F.R. s. late Lieutenant in the Corps of Engineers. 4to. pp. 45. 5s. Boards. Robinsons. IN a paper presented to the Royal Society in 1777, and pub lifhed in the ad part of the 67th volume of the Transactions, the ingenious author suggested the general plan of those disquifitions, concerning the geometrical comparison of increasing and decreasing magnitudes, which are pursued in the work now before us, and which have employed much of his attention in those intervals of leisure that have occurred amid the various duties of an active profeflion. The investigations which he proposes, and which he has already prosecuted with very confiderable success, are curious and useful; and they are recommended to the mathematician by their novelty as well as by their im. portance. He has shewn how geometrical reasoning, founded on the do&rine of proportion delivered by Euclid in his Elements, may be extended beyond the narrow limits to which it has been commonly restricted both by antient and modern geometers. Their attention has been confined to those relations of magnitudes, which are expressed by the simple, duplicate, and triplicate ratios : but these comprehend only a small portion of that universal comparifon to which geometry may be applied. The author has extended the province of this science, and has investigated a general method of expressing geometrically any combination and variety of ratios that can occur in the compa. silon of inagnitudes. In the paper to which we have referred, he has given, "The demonftration of a general geometrical formula, for finding and exprefsing a magnitude of the same kind with any two ho. mogeneous magnitudes, A and B, which shall have to B any melciplicate ratio of A to B, or a ratio compounded of the ratios of A to B, C to D, E to F, G to H, &c. Supposing A and B, C and D, E and F, G and H, &c. taken two and two, to be magnitudes of the fame, but of any kind; as also of a formula, for finding and expreffing geometrically a magnitude of the same kind with any two homogeneous magnitudes A and B, which shall have to A any multiplicate ratio of B to A, or a ratio compounded of the ratios of B to A, C to D, E to F, G to H, &c. Supposing B and A, C and D, E and F, G and H, &c. taken two and two, to be magnitudes of the same, but of any kind.' In this publication, the author pursues • The geometrical investigation of general formulæ, for finding and expresfing magnitudes of the same kind with any two homogeneous magnitudes A and B, having to B and A ratios arising from the decompolition of the ratios of C to D, E to F, G to H, &c. with the ratios of A to B and B to A separately taken, or from the decomposition of any multiplicate ratio of A to B, with the said ratios of A to B and B to A separately taken.' Having demonstrated the principles which he affumes, in a manner that does not conveniently admit of either an extract or abridgement, the author proceeds to deduce from them a variety of curious and useful theorems, containing expressions which extend equally to geometry and all the abstract sciences in general, and which may be contidered as universally metrical. He observes that the binomial theorem which the illustrious Sir Isaac Newton derived from induction, and by no means from geometrical reasoning, is only an arithmetical one; and in fhort nothing else than a particular case of any one of the geometrical theorems' which he has proposed, when supposed to become numerical, or to be referred to i or unit, as the standard of comparison.' That the principles laid down by Mr. Glenie, in this treatise, are capable of a very extensive application will appear from the following summary of the various subjects to which he proposes to direct his attention, and which, we hope, for his own honour and for the benefit of science, he will have leisure to pursue, viz. riit, To inveftigate the geometrical principles of what is usually called the doctrine of Auxions, or to deliver a method of reasoning geometrically, applicable to every purpose to which the doctrine of Haxions can be applied, without any consideration of motion or ve- ; locicy. • 2dly, To inveftigate the geometrical principles of increments, or to deliver a method of reasoning geometrically on increase and de. Creare, in all the posible degrees of magnitude. Rev. Jas. 179+ D 3dly. • zily, To investigate the geometrical principles of the doctrine of the nicaiures of ratios, or to deliver a method of reasoning geometrically on the quantities of, or the various degrees of, magnitude in ratios. 4thly, To deliver a method of summing infinite series.geometri., cally, 5thly, To deliver the geometrical solutions, by methods as ftri&tly so as any of those made use of in Euclid, of a great number of general problems similar to one' subjoined, which must lay open a new and extensive field in folid geometry, and tend to unfold the great desiderata on that subject, hitherto fought for in vain, both, by antient and modern geometers.' Art. XIII. The Antecedental Calculus; or, a Geometrital Method of Reasoning, without any confideration of Motion or Velocity, applicatlı to every Purpose to which Fluxions have been or can be applied; with the Geometrical Principles of Increments, &c. and the Construction of some Problems, as a few Examples selected from an endless and indefinite Variety of them respeđing Solid Geometry, which he has by him in Manuscript. By James Glenie, Efq. M. A. & F.R.S. 4to. pp. 18. 25. 60. Robinsons. ' 1793. E are happy to find that the author of the Doctrine of Universal Comparison has not been diverted, by the duties of his profession, from prosecuting the plan announced to the public in that valuable work, and we hope that no exi. gence of military operations will detain him long from the peaceful and unmolested pursuit of disquisitions for which he is lo eminently qualified. We have here a very satisfactory specimen of the useful purposes to which his new doctrine may be applied; and, as much remains to be done in the same way, we deprecate any accident that may prevent his resuming a subject which is capable of farther elucidation and improvement. The author's method of reasoning is a branch of general geometrical proportion, or universal comparison. It is founded on principles admitted into the very first elements of geometry, and repeatedly used by Euclid himself; and as it is derived from an examination of the antecedents of ratios, having given confequents and a given standard of comparison, in the various degrees of augmentation and diminution which they åndergo by composition and decomposition, the author (to whom it first osa curred so long ago as the year 1774) has denominated it the Antecedental Calculus. • As it is purely geometrical, and perfeâly scientific, (says Mr. G.) I have since that time always made use of it instead of the fluxionary and differential calculi, which are merely arithmetical. Its principles are totally unconnected with the ideas of motion and time, which, ftriatly fpeaking, are foreiga to pare geometry and abftra& fcience, though in mixed mathematics and natural philosophy they are equally applicable to every investigation, involving the considerațion of either with the two numerical methods just mentioned. And, as many such investigations require compositions and decompositions of ratios extending greatly beyond the triplicate and subtriplicate, this calculus in all of them furnishes every expression in a strictly geonietrical form. The standards of comparison in it may be any magnitudes whatever, and are of course indefinite and innumerable; and the consequents of the ratios compounded or decompounded may be either equal or unequal, homogeneous or heterogeneous. In the Ruxiobary and differential methods on the other hand, 1 or unit is not only the invariable Itandard of comparison, but also the consequent of every ratio compounded or decompounded.' • It appears (continues our author) from the writings of that truly great man, Sir Isaac Newton, that he introduced into geometry the idea of velocity, chiefly with the view of avoiding the exceptionable doctrine of indivisibles, and considered lines, furfaces, and solids, as generated by the motions of points, lines, and furfaces, instead of being made up of them, or formed by the apposition of infinite numbers of indivisible parts. And in his doctrine of prime and ultimate ratios, he has recourse to the idea of time, which however there was certainly no neceffity for. And I am per. fedly satisfied, that had this great man discovered the possibility of investigating a general geometrical method of reasoning, without introducing the ideas of motion and time, applicable to every purpose to which his doctrines of Auxions and prime and ultimate ratios can be applied, he would have greatly preferred it; fince time and motion bave no natural or inseparable connection with pure mathematics. The. Auxionary and differential calculi are only branches of general arithmetical proportion, and the expressions in them are numerical.' The preceding extract will afford the reader a general idea of the nature and exceilence of the method proposed by the author; and none can be unapprized of the extent and utility of its application, who advert to the specimens that are subjoined to this compendious treatise. We regret that they are not more numerous. The actual illustration of the principles which he has established, and of the formulæ deduced from these principles in a greater, variety of instances, pertaining to the fummation of series, the dorine of increments, the meafures of ratios, and the construction of geometrical problems, would have been both pleasing and instructive. This kind of amplification, not at all inconsistent with conciseness and pera spicuity, in establishing his fundamental doctrine, would serve to intereft the attention even of the mathematician, to exempllfy the importance of the new method of reasoning which Mr. $. has discovered, and to render is more popular and more useful. |