public in general, but only for the direction of the governors of the hofpitals in queftion, yet we think it of importance enough to be offered to the notice of our readers.

The committee begin their report with confidering what were the primary and proper objects of the charity of Bridewell; and they find that arts-mafters and apprentices are not even mentioned in the royal grant of Edward VI. but that it was given as a "houfe for occupations," 1. For boys of riper years, who were found unapt to learning, and fo 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 difcharged from St. Thomas's, if able to work, that they might not wander about as vagabonds, but have fuitable employment. 3. For the lewd, the sturdy beggar, and the idle in general, who fhould be compelled to labour therein, and fo to ferve the commonweal; and laftly, For prifoners difcharged at the feffions, that they might have occupations, and not again become thieves or beggars.

They farther find that a difcretionary power is granted by the charter to alter the rules and regulations as circumftances. may require; and they are fully of opinion that, however ufeful arts-mafters and apprentices might have been in the infancy of the inftitution, yet, in this advanced period of the arts of life, they are an ufelefs and a very objectionable part of the charity; being the cause of a great expence, for a species of inftruction which would be much better carried on out of the house. They, in confequence, recommend that the numerous apartments occupied by this clafs fhould in future be employed in furnishing means of labour to the original objects; among whom none more loudly call for attention than the prifoners difcharged at every feffion from the Old- Bailey.

Another point, which the committee were to examine, was the connexion of the hofpital of Bridewell with that of Bethlem; and they anfwer, in the affirmative, the queftion, whether the governors are juftifiable in applying the revenues of the former to fupply the wants of the latter?

Of the effential defects in the late fyftem of management with respect to the disbursement of the revenues, the following paragraph affords a moft glaring proof:

It is fufficient here to obferve, that, by one of those statements *, it appears that 59571. 11s. hath been expended on the apprentices, and 74931. 165. 4d. in maintaining the vagrants (the only two fuppofed objects of charity of Bridewell); whereas it has coft, within the fame period, 19,2541. Os. 4d. in falaries, &c. of the officers employed in

Alluding to what ispreviously stated.

the

L.

the management; befide 63411. 6s. 1d. for their taxes, views of

d1, 20, 829l. eftates, &c. and 32341. 95. 1d. in fealis ; together - 155. 6 d.

and, what feems equally extraordinary, the further enormous fum of 17,3321. 198.7d. for repairs at the hofpital of Bridewell alone.'

The committee then make their obfervations under different heads, and propofe fuch alterations as fhall improve the utility and economy of the hofpitals. The report is concluded by an appendix of the ftanding rules and orders of the two hofpitals, and abridged tables of receipts and expenditures. The public, to whom finally all inftitutions of this kind muft be confidered as belonging, have a right to expect that the labour fo ably and faithfully applied by the refpectable gentlemen of this committee fhould not be loft: but that the improvements propofed, and (as we understand,) adopted, should not' be fuffered, either by neglect or by artifice, to become inefficacious or obfolete.

ᎪᎥ .

ART. XII. The Doctrine of Univerfal Comparison, or General Proportion. By James Glenie, Efq. F. R. S. late Lieutenant in the Corps of Engineers. 4to. pp. 45. 5s. Boards. Robinsons.

IN

Na paper prefented to the Royal Society in 1777, and publifhed in the 2d part of the 67th volume of the Transactions, the ingenious author fuggefted the general plan of those difquifitions, concerning the geometrical comparison of increating and decreasing magnitudes, which are purfued in the work now before us, and which have employed much of his attention in thofe intervals of leifure that have occurred amid the various duties of an active profeffion. The inveftigations which he propofes, and which he has already profecuted with very confiderable fuccefs, are curious and ufeful; and they are recommended to the mathematician by their novelty as well as by their importance. He has fhewn how geometrical reafoning, founded on the doctrine 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 thofe relations of magnitudes, which are expreffed by the fimple, duplicate, and triplicate ratios: but thefe comprehend only a fmall portion of that univerfal comparifon to which geometry may be applied. The author has extended the province of this fcience, and has investigated a general method of expreffing geometrically any combination and variety of ratios that can occur in the comparifon of magnitudes. In the paper to which we have referred, he has given,

· The

The demonstration of a general geometrical formula, for finding and expreffing a magnitude of the fame kind with any two homogeneous magnitudes, A and B, which fhall have to B any multiplicate 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. Suppofing 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 alfo of a formula, for finding and expreffing geometrically a magnitude of the fame kind with any two homogeneous magnitudes A and B, which fhall 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. Suppofing B and A, C and D, E and F, G and H, &c. taken two and two, to be magnitudes of the fame, but of any kind.'

In this publication, the author pursues

The geometrical investigation of general formulæ, for finding and expreffing magnitudes of the fame kind with any two homoge neous magnitudes A and B, having to B and A ratios arifing from the decompofition 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 feparately taken, or from the decompofition of any multiplicate ratio of A to B, with the faid ratios of A to B and B to A feparately 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 expreffions which extend equally to geometry and all the abstract sciences in general, and which may be confidered as univerfally metrical. He obferves that the binomial theorem which the illuftrious Sir Ifaac Newton derived from induction, and by no means from geometrical reasoning, is only an arithmetical one; and in short nothing elfe than a particular cafe of any one of the geometrical theorems,' which he has propofed, when supposed to become numerical, or to be referred to or unit, as the ftandard of comparison.'

That the principles laid down by Mr. Glenie, in this treatife, are capable of a very extenfive application will appear from the following fummary of the various fubjects to which he propofes to direct his attention, and which, we hope, for his own honour and for the benefit of fcience, he will have leifure to purfue, viz:

ift, To inveftigate the geometrical principles of what is ufually called the doctrine of fluxions, or to deliver a method of reafoning geometrically, applicable to every purpose to which the doctrine of Auxions can be applied, without any confideration of motion or velocity.

2dly, To inveftigate the geometrical principles of increments, or to deliver a method of reafoning geometrically on increafe and decreafe, in all the poffible degrees of magnitude.

REV. JAN. 1794.

D

3dly.

3dly, To inveftigate the geometrical principles of the doctrine of the measures of ratios, or to deliver a method of reafoning geometri cally on the quantities of, or the various degrees of, magnitude in

ratios.

cally.

4thly, To deliver a method of fumming infinite feries geometri

5thly, To deliver the geometrical folutions, by methods as strictly fo as any of thofe made ufe of in Euclid, of a great number of general problems fimilar to one' fubjoined, which must lay open a new and extenfive field in folid geometry, and tend to unfold the great defiderata on that fubject, hitherto fought for in vain both by antient and modern geometers.' Re-s.

ART. XIII. The Antecedental Calculus; or, a Geometrical Method of Reafoning, without any Confideration of Motion or Velocity, applicable to every Purpofe to which Fluxions have been or can be applied; with the Geometrical Principles of Increments, &c. and the Conftruction of fome Problems, as a few Examples felected from an endless and indefinite Variety of them refpecting Solid Geometry, which he has by him in Manufcript. By James Glenie, Efq. M. A. & F. R. S. 4to. pp. 18. 2s. 6d. Robinsons. 1793.

W E are happy to find that the author of the Doctrine of Univerfal Comparison has not been diverted, by the duties of his profeffion, from profecuting the plan announced to the public in that valuable work; and we hope that no exigence of military operations will detain him long from the peaceful and unmolefted purfuit of difquifitions for which he is fo eminently qualified. We have here a very fatisfactory fpecimen of the ufeful purposes to which his new doctrine may be. applied; and, as much remains to be done in the fame way, we deprecate any accident that may prevent his refuming a fubject which is capable of farther elucidation and improvement. The author's method of reasoning is a branch of general geometrical proportion, or univerfal comparifon. It is founded on principles admitted into the very firft elements of geometry, and repeatedly used by Euclid himself; and as it is derived from an examination of the antecedents of ratios, having given confe quents and a given ftandard of comparifon, in the various degrees of augmentation and diminution which they undergo by compofition and decompofition, the author (to whom it first occurred fo long ago as the year 1774) has denominated it the Antecedental Calculus.

As it is purely geometrical, and perfectly fcientific, (fays Mr. G.) I have fince that time always made ufe 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, ftrictly fpeaking, are foreign to pure geometry and abftract science, though

in mixed mathematics and natural philofophy they are equally applicable to every investigation, involving the confideration of either with the two numerical methods juft mentioned. And, as many fuch inveftigations require compofitions and decompofitions of ratios extend ing greatly beyond the triplicate and fubtriplicate, this calculus in all of them furnishes every expreffion in a strictly geometrical form. The standards of comparison in it may be any magnitudes whatever, and are of course indefinite and innumerable; and the confequents of the ratios compounded or decompounded may be either equal or unequal, homogeneous or heterogeneous. In the fluxionary and differential methods on the other hand, 1 or unit is not only the invariable ftandard of comparifon, but alfo the confequent of every ratio compounded or decompounded.'

It appears (continues our author) from the writings of that truly great man, Sir Ifaac Newton, that he introduced into geometry the idea of velocity, chiefly with the view of avoiding the exceptionable doctrine of indivifibles, and confidered lines, furfaces, and folids, as generated by the motions of points, lines, and furfaces, instead of being made up of them, or formed by the appofition of infinite numbers of indivifible parts. And in his doctrine of prime and ultimate ratios, he has recourfe to the idea of time, which however there was certainly no neceflity for. And I am perfectly fatisfied, that had this great man difcovered the poffibility of investigating a general geometrical method of reafoning, without introducing the ideas of motion and time, applicable to every purpose to which his doctrines of fluxions and prime and ultimate ratios can be applied, he would have greatly preferred it; fince time and motion have no natural or infeparable connection with pure mathematics. The fluxionary and differential calculi are only branches of general arithmetical proportion, and the expreffions in them are numerical.'

The preceding extract will afford the reader a general idea of the nature and excellence of the method proposed by the author; and none can be unapprized of the extent and utility of its application, who advert to the fpecimens that are fubjoined to this compendious treatife. We regret that they are not more numerous. The actual illuftration of the principles which he has established, and of the formulæ deduced from these principles in a greater variety of inftances, pertaining to the fummation of feries, the doctrine of increments, the mea fures of ratios, and the conftruction of geometrical problems, would have been both pleafing and inftructive. This kind of amplification, not at all inconfiftent with concifenefs and perfpicuity, in eftablishing his fundamental doctrine, would ferve to intereft the attention even of the mathematician, to exemplify the importance of the new method of reafoning which Mr. G. has difcovered, and to render it more popular and more useful.