mulas, express there only operations that can be performed, and can only be considered as simple abbreviations.

This table of the properties of the primitive figure being once established, it is necessary to know what modifications the figure, should undergo, in order to represent successively, after the same, manner, the properties of figures correlative to it. The construction of each of these being essentially the same as that of the primitive figure, it is clear that the formulas which express their properties ought to have so much the more analogy with those of the primitive figure, as there is less disparity between them: the correspondentquantities ought to be there combined in the same manner, with re spect to their proper or absolute values. It only remains, then, to express the diversity of positions; and this is done by the change of signs which affect these quantities, or the different terms of the formulas of the table.

In order to discover the changes that in fact ought to take place for such or such a correlative system, I consider it as arising from the primitive system, by virtue of a transformation carried on by insensible degrees; which does not alter the general bases of the first construction, but only modifies the respective positions, in putting, before that which was behind, or in transporting to the right that which was on the left. From this gradual movement, it results that such quantity of the system, which at first was found less than an. other, becomes greater, and respectively. Now, it is from that circumstance solely, and not because the quantities are opposed the one to the other, that we derive the general principle of the change of signs which ought to take place in the formulas of the primitive system; so that they may become applicable to the transformed or correlative system.

In the third section, I delineate, on the different figures, the. tables of which I have at first spoken; viz. those which are proper to represent the collection of the relations existing between the different parts of each; and then I apply to each of these tables that of correlation, by which are known the changes that ought to be made in this primitive table, in order to render it applicable to each of the systems correlative to it.

6

The fourth section contains new applications of the same principles to properties, which in the figures cannot be found without the intervention of linear angular quantities. The linear-angular quan tities are intermediate quantities, serving to connect the lines with the angles, or to establish the relations of one with the other: but, in. this section I examine separately, on the one part, the relations subsisting between the angles solely, and on the other, those which subsist between the lines solely; I there give the notion of the centre of the mean distances; and I remark that this point is the same as that which in mechanics is called the centre of gravity; whence I conclude that the theory of this centre belongs to geometry, and that it would be very advantageous for the progress of that science, to re-establish in this respect the natural order of ideas. ̧

In the fifth section, I apply the principles before established, to: a series of particular questions of the kind of those which form the

subject

Bubject of what is called Application of Algebra to Geometry: whence I take occasion to shew, by many examples, that the theory com monly admitted of quantities called positive and negative is not satisfactory; and that, by the manner of choosing not only the unknown but the given quantities, we oftentimes succeed in making enter, conformably to the idea of Leibnitz, the position in the expression of the conditions of the problem; and in thus diminishing the natural degree of the final equation. These different questions give rise to some remarkable formulas; as the equation of condition that exists between the six angles which are formed by the four faces of a triangular pyramid.

Finally, in the sixth and last section, I apply to curves the forma tion of tables proper to give a general view and collection of the properties of signs: I develope the luminous idea set forth by Godin, in his Treatise of properties, common to all curves, that the art of discovering the properties of curves is, properly speaking, the art of changing the system of co-ordinates; and I give different examples of this operation. My intention was not to write a systematic treatise on the theory of curves, but, solely to vary the application of my principles; and to shew that the formation of tables, proper to represent the whole of a figure, is applicable to lines and curve surfaces, as well as to right lines and plane surfaces. There will be found, in this section, many remarkable properties (I believe, hitherto unknown) of conic sections; and a very curious theory on the points of concourse of several right lines, and of those on the contrary that are ranged on the same right line. Lastly, different properties of curves in general, of which the principal object is to render their equation independent of every point, line, plane, or any fixed object whatever, taken arbitrarily in space, or inherent in the curve.'

Such is the plan of the present publication; which we have given in the writer's own words, because, without particular exertions on our part, he must be deemed most competent adequately to describe it; and from these extracts, the reader may without difficulty obtain a glimpse of the author's meaning and mode of reasoning. We should have considered that meaning as more easily apprehensible, and the reasonings as more perspicuous and level with common capacities, if the size of the work had been less; and if M. CARNOT had avoided those frequent repetitions of the same ideas, in which he seems. to have indulged as if fearing that the abstruseness of his subject would render it difficult to understand him. We confess that we should have understood him much better, had he been more concise; though he is not to be regarded as an obscure writer.

Some time previously to the appearance of this work, the author published another called Correlation of Figures; and he had purposed to give a new and enlarged edition of it: but, in the undertaking, new ideas flowed in, and new views presented

theniselves;

themselves; which insensibly produced, under his hands, the Geometry of Position, containing more than 500 quarto pages.

Though our extracts may be deemed sufficiently large, we have omitted many which we intended to have inserted, because we are alarmed at the bulk to which our criticisms would swell. The work, however, is not to be dismissed without comment. Its chief fault, as we have already observed, is diffuseness and unnecessary repetition; whence it might be inferred that M. CARNOT wrote down his thoughts as they occurred to him, and disdained the labour of compressing and methodizing them. Instead of nine years, this mathematician probably did not keep his lucubrations in his drawer so many months. Throughout a great part of the beginning of the volume, we perceive some ideas with which he seems violently in labour, and of which he is never happily delivered. Another fault consists in the introduction of 'uncouth and scholastic terms, such as correlative systems, direct correlation, inverse correlation, &c. by which neither truths are taught, nor, in our opinion, are properties commodiously classed:-we do not even approve, as we before intimated, of his alteration of the terms positive and negative into direct and inverse.

It is a rule rarely observed, but still it is a good rule, that, in an elementary treatise, an author should not speak a language which can be understood only by considering what is subsequently established. To this maxim, we think, M. CARNOT has by no means adhered :-but, understanding his own system in its several parts, connections, and applications, he speaks as if it were also familiar to his readers; and hence many of his paragraphs, delivered in general terms, cannot, in the regular progress of perusal, be adequately comprehended.

Although, on the whole, M. CARNOT is an acute and sound reasoner, yet instances occur, in this volume, of conclusions that do not consequently follow: for example; he supposes the expression of the conditions of a problem to give him this equation, x2-2 ax+a2—bo; or, says he, (x-a)'=b, whence x-ab. The first -a+b, he observes, is intelligible: but x-a-b, the second, is not so, and becomes so only by transposition. Now the fact is that the equation x2-2ax+ab is not the same as (x-a), except x be greater than a. If x be less than a, then (a-x)'=b; but, since (x-a) and (a-x)2 expanded give the same form, the solution is (x-a)2=b; or (a—x)2=b; or x—a=+√b;_or 4-x+b; which equations are perfectly intelligible. For the sake of analytical commodiousness, they may be comprehended under the abridged form, xa±√b. M. CARNOT terms x—=—√√b, an implicit equation, because there is need

of

of a transformation to make it designate an operation which can be executed: but he does not shew on what principle such a transformation is made; and we deny that it can be made without the intervention of an arbitrary rule.

Notwithstanding the numerous explanations and elucidations of M. CARNOT, it will most probably be suggested to the minds of his readers, that his theory of negative quantities is imperfect, or at least perplexing, from the variety of limitations which restrict their use and application. Sometimes, in the theory of curve lines, they refer to that part of the curve which is on the left of the line of the abscissas; and sometimes they are perfectly ineffective. Again, he observes, that the rules of algebra are subject to exceptions, when the operation is not performed on real quantities; whence he concludes that there is no method of antecedently demonstrating the rules of analysis, which operates indifferently on positive, negative, or imaginary quantities; and that, therefore, its processes cannot be justified, except by the conformity of their results with those of synthesis, and by the assurance afforded by the uniform exactness of the verified results. Now, if this be true, if there exists, no method of antecedently proving the truths of the operations performed in quantities called negative and imaginary, then the use of such quantities is unsafe, and ought to be abandoned; and the proof of the truth of the operations, by comparing their results with results obtained by more rigorous processes, does not extend beyond the particular cases in which the verification has been made :-but, if every particular result is to be verified, why employ negative and impossible quantities, and unnecessarily submit to double labour?

To conclude; we must admit that this treatise well merits. the attention of mathematicians, because it abounds with many just remarks, and interesting discussions. We particularly noticed one contained in the first part, on the true and essential distinction between analysis and synthesis. Of the alterations which the author has made in the names of things, we have already stated our disapprobation, but some of his improvements in notation appear to us very commodious, and deserve to be adopted. His great fault is redundancy; his chief excellence, freedom from what may be called mathematical prejudices. He examines every thing on the score and footing of reason, although all that he establishes is not secure from the assaults of perspicacious and active criticism.

ART.

ART. XI. Marguerite de Strafford, &c.e. Margaret of Straf ford, an Historical Romance; containing many Anecdotes of the Reign of Charles II. and others relative to the Revolution of England. By Madame DE ** 5 Vols. 12mo. Paris, 1803. Imported by De Boffe. Price 15s. sewed.

יT

HE Scene and the story of this work are English, and the author is French:we need state no more to lead the reader to anticipate the tortures which names, places, and facts, undergo in these pages. Why the fair writer intitled her work an Historical Romance, we are wholly at a loss to conjecture; since, if we except the names, it bears no more relation to the event with which it professes to be connected, than it does to the expulsion of Tarquin, or the dethronement of the younger Dionysius. Had she taken a solemn oath, or made a sacred vow, in no particular to conform to the reality of facts, we believe that her five volumes would not furnish the shadow of a suspicion that she had violated her resolution. The Charles II. of this romance carries with him the heart of a man, and is distinguished by the virtues of a monarch; the Strafford of it is the purest of patriots; its Cromwell exhibits no feature but that of a mere ferocious tyrant; and its Albemarle is a generous hero, and an enlightened statesman. Could the truth of history, however, be prevented from beaming on the reader's mind, he would acknowlege himself beholden to the fair novelist for an introduction to characters which call forth the noblest aspirations of the mind, and the finest feelings of the heart; he would be awed and charmed by the dignity and goodness which appear blended in the whole behaviour of a great and virtuous monarch, had not that monarch been designated by the name of Charles II.; he would shed tears over the great pillars of the church and the state falling under the axe of a bloody faction, if they were not called Strafford and Laud; and his veneration would be fixed on the person of the hero who extinguished the domination of mercenary empires, and who restored the throne to the lawful sovereign, did he not bear the name of Monk.

Not contented with subjecting history to this hard usage, the author shews as little respect to the character and manners of the people to whom her tale relates. The heroine of the piece, who is exhibited as a perfect model, admits to her intimate friendship, and lodges under her roof, the mistress of her father; she adopts the crowd of natural children by divers mothers which he leaves behind him; and, in good time, she marries them to persons of the first rank among the English nobility. This may suit continental readers, but must certainly shock the notions of the inhabitants of the British Islands.