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was never so strongly evinced; and the result now obtained, by considering the former and neglecting the latter, is a triumph which the imagination of the most sanguine nominalist could ne ver have anticipated.

But what is the real principle on which such investigations as the foregoing are successful? and what is the precise nature of the evidence that they afford? As to the latter question, we know from experience, that in all the instances where we could compare the conclusions obtained by help of impossible quantities with the results of ordinary investigation, we have found that they agreed perfectly. This agreement cannot be the effect of chance; no man, by tossing about at random the symbols that denote quantities, ever arrived at a true, or even an intelligible proposition. It is therefore clear, that a fixt and determinate principle directs the mathematician, in this case, as well as in those where his understanding accompanies every step of his demonstration. The principle may not be obvious, but its existence is thus rendered undeniable. Many mathematicians, we are convinced, rest here, and carry their inquiries no further; confiding perfectly in the imaginary operations, which, from experience, they have found to lead to truth, whenever a perfect analogy is kept up between them and the real operations of arithmetic and of algebra. But it is certainly reasonable to go a little further, and to inquire what this principle really is. Without knowing it precisely, we shall always be in danger of error. M. D'Alembert, who had bestowed much attention on the subject of impossible quantities, appears, from many passages in his writings, to have been fully aware of the importance of this inquiry, though he has not professedly entered on it, nor given us any ground to conjecture what was the opinion which he enter tained. Maclaurin satisfies himself with supposing, that a cer tain compensation takes place among the impossible quantities, by which they destroy the effects of one another. This, however, presents no clear idea to the mind, and leaves the difficulty of applying the notion of subtraction, &c. to things that cannot exist. More lately, Mr Woodhouse, treating of the same subject, seems to be of opinion that no inquiry of this kind is at all necessary, the identity of the operations performed on the symbols being a sufficient secu rity against error, whatever these symbols denote, whether things real or impossible. We have already stated the reasons that prevent us from acquiescing in this view of the matter, and for thinking that the subject ought to be further investigated. One idea concerning it we must mention, as not destitute of plausibility, founded on this remark; that in all the instances where impossi ble quantities or imaginary expressions have been of use in the in

vestigation

vestigation of mathematical theorems, those theorems have related either to circular arches, or hyperbolic areas; quantities so remarkable for analogical properties, that there is no general affec tion of the one to which there is not a corresponding affection of the other. The theorems then that admit of investigation by an imaginary process, are of such a nature as to go always in pairs, one belonging to the hyperbola, the other to the circle; or, one to the measures of ratios, and the other to the measures of angles. One of these twin theorems can always be investigated by the teal, or ordinary processes of algebra; but when the same me thod is to be extended to the other, the imaginary character makes its appearance; if, however, the symbols be treated in the same way as in the other case, that character disappears, and a theorem emerges perfectly analagous to that already investigated. In this view of the matter, the operations with the imaginary characters are nothing but a mode of tracing, or keeping in sight, the analogy between the circle and hyperbola, and have no more force than any other conclusion founded on that analogy. This may be illustrated by the example formerly given from the circle.

Another question has arisen concerning the investigations carried on by help of imaginary expressions, viz. Whether they ought to be tolerated in sciences that boast, like geometry and arithmetic, of the evidence and clearness of their demonstrations, Among certain Purists in algebraic language, no quarter is allowed to such modes of expression as we have been here treating of; and the investigations that proceed by help of them are considered as delusive artifices, unworthy of the name of science. To this opinion, however, we can by no means subscribe. Whatever has served for the discovery of truth, has a character too sacred to be rashly thrown aside, or to be sacrificed to the fastidious taste of those who make truth welcome only when it wears a particular dress, and appears arrayed in the costume of antiquity. Admitting that imaginary expressions, when applied in the manner we have seen, do nothing more than trace an analogy between two curves related to one another like the circle and hyperbola, and therefore have no force beyond what belongs to analogical reasoning; yet, the simple fact, that the conclusions they have led to have been confirmed by the other less exceptionable modes of demonstra tion (often by the most rigorous synthesis), is reason sufficient for regarding them as valuable instruments for the discovery of truth. The anticipations they afford are of infinite value; and no man who knows the importance even of scientific conjecture, will willingly give up the advantage to be derived from them.

The conclusions of the investigations by help of imaginary expressions, have been so often verihed by other methods, both ana

lytical

lytical and synthetical, that no doubt can remain that they pro ceed on sound and geometric principles, though perhaps not easy to demonstrate with rigour in their utmost generality. The great generality of a proposition often renders the rigorous demonstra tion of it difficult; and though we can apply such demonstration when the proposition is broke down, as we may call it, into par ticular cases, we are yet unable to do so when it remains in its most general form. It is, nevertheless, of great importance to know what that form really consists of.

Though it is true that the investigations which have imaginary expressions for their instrument have been confirmed by other methods, yet the matter was in some instances so difficult, and the result obtained so complete, that the ordinary methods of verifica tion could not be applied, and the method of impossible quanti ties, from its superior facility, was the only one that could be used with success. This has happened in two or three instances of integration given by Euler, where that great mathematician has performed what, one would have supposed, must greatly sur pass the powers even of the most improved analysis. The me thod he employed depended on imaginary expressions, and he appears very much to regret that he had not been able to accom plish the same by any other means. Though it does not appear that Euler ever gave himself much trouble about settling the me taphysical principles of this part of the calculus, his practice was very conformable to the notion we have been endeavouring to enforce; he used the imaginary expressions as the readiest and most expeditious methods of investigation, and those by which great difficulties were most likely to be overcome; but he was always desirous of finding such verifications as are afforded by a more rigorous analysis. We may safely recommend a rule that directed the practice of this profound and experienced analyst.

ART. III. Travels in Turkey, Italy, and Russia, during the Years 1803, 1804, 1805 & 1806. With an Account of some of the Greek Islands. By Thomas Macgill. 2 vol. 8vo. pp. 522. London, Murray. Edinburgh, Constable & Co. 1808.

IN

N our account of Mr Semple's Travels (No. XXI.) we expres sed very great satisfaction at receiving from the hand of a mercantile gentleman a sketch of those foreign countries scarcely accessible to any one else, which he had occasion to visit in the course of his professional pursuits; and we strongly recommended so good an example to the attention of others in similar circum

stances.

stances. The author of the volumes before, us belongs to the same class with Mr Semple, but his work is of very inferior interest and merit in every respect. Mr Macgill, indeed, seems rather to have published his travelling notes as an additional commercial speculation, or a winding up of his accounts, than to have described his tour because it was interesting, or the countries he saw because he had observed them attentively. We do not perceive that he kept any journal or took regular notes of what he saw. He wrote several letters to different friends, sometimes half a year after the anecdotes had occurred which he wished to relate and, finding that the public would read any thing like travels, and every thing called letters, he seems to have considered the manufacture of two volumes as a fit termination to his trading voyage: which having resolved to do, he could have very little difficulty in obtaining the necessary passports of advice of friends to whose inspection they were submitted; their opinion that they would be favourably received; the consent of a bookseller; the assistance of a printer; and all the other encouragements requisite on the occasion. That of employing an author to write for him, we should think, he has omitted entirely; and, however much we may expose ourselves to contempt for so unfashionable a taste, we will own that the omission gives us satisfaction.

Notwithstanding the great inferiority of this work to Mr Semple's, and the slender qualifications, either for speculation or remark, which Mr Macgill appears to possess, we are far from regretting that he has made his letters public. They are by no means devoid of information, although the more important topics are slightly touched, and many things altogether passed over, which an inquisitive and learned reader must greatly desire to find in a work with this title. They contain a number of anecdotes which throw light on the Turkish character, and bring us better acquainted with the present state of their country. They likewise communicate some very useful notices respecting the trade of the Levant and Black Sea, which cannot fail of proving serviceable to mercantile people. It may be added, that the author knows a secret, far from being common with those who have no talent for fine writing,to write plainly and unaffectedly; and while his letters, if not always very instructive and entertaining, are pretty uniformly sensible, and inoffensive both to our feelings and our taste, his modesty, both in ushering them into notice, and in describing what he has done and seen, cannot be passed over? without much commendation. He may, by previously reading some books of general knowledge, and observing more carefully the next country he visits, present us with a more valuable ac-, count of it; and in the mean time, in spite of the remarks we

VOL. XII. NO, 24.

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have

have now made, we prize his present contribution to the stock of public information, infinitely more highly than the feeble and gar rulous quartos of the Stranger Knight, or the pompous inani ties of William Hunter, Esq. We shall, therefore, follow him rapidly in his tour, and point out what may occur worthy of notice, an office which we should scarcely be induced to perform towards Sir John Carr, were he to write a Stranger in Japan; or to Mr Hunter for all the letters he might indite, were his positions, like those of his former volumes, as true and as amusing as Cocker, and his language as glowing as Tom Thumb.

The first letters are from Venice, whither our author retired upon the breaking out of the present inexplicable war. His remarks on that singular place are rather of a gloomy cast. He seems to have found neither mirth nor amusement there; and, after a whole year's residence, was able to discern only indigence and misery. That there may be some truth in this picture, we cannot deny. The nobles certainly suffered greatly from the change of government; that is, from the overthrow of the most tyrannical aristocracy in the world; and the fortunes of some were probably impaired by contributions; but, in general, they sustained far more damage from the loss of those means of extortion which they had formerly enjoyed in secure monopoly. But how the city in general, how the bulk of the people, could have any reason to lament the revolution, we cannot conceive. A few of the chief aristocratic houses are ruined; many of those which survived have deserted Venice, and prefer living on the Terra Firma, where they may still domineer over their vassals, to continuing in the city which they can no longer either frighten or plunder. A few German soldiers parade the streets, with whiskers, and pipes in their mouths; and their officers disfigure the theatres, or insult the audience with talking, and making about a twentieth part as much noise as all ranks of men do in an Engglish theatre during the finest passages. These, we believe, are the chief inconveniences which the people have to suffer in return for the abolition of the state prisons, and other engines of torture; the destruction of secret inquisitors and unknown accusers; the equalization of all taxes and public burdens; and the introduction of the best police known in the south of Europe. Mr Macgill exaggerates even the distresses of the nobility. He may be assured, that whoever told him that above a thousand heads of noble families were begging on the streets in the year 1804, greatly deceived him. A dissolute nobility, like the Venetian, is always sure to number among its ranks many persons too lazy to work, too poor to live idle, and not too proud to beg. These, in the best times of Venice, would thankfully accept of charity; but

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