mark of perpendicularity. Instead of following that demonstration, it is sufficient for us to show, that in ascribing to √2 its arithmetical meaning, that of a number which, multiplied into itself, gives 2 for the product, the above proportion 1 + √−1 ~~: √2 :: √2:11, is so far from absurd, that it is accurately true, and capable of being understood in the strictest sense. It signifies that two quantities of the form 1+a and I a cannot be found such that their product shall be equal to 2, or, which is the same, that it is impossible to find two numbers such that their sum shall be equal to 2, and that √2 shall be a mean proportional between them. This impossibility is, even without any investigation, abundantly evident; and is, in fact, all that is involved in the preceding proposition, interpreting it strictly, according to the notion, that √1 denotes impossibility, and nothing else. ་ It is easy to prove this impossibility otherwise: Let and y be & the numbers, then x+y2 and xy-2, whence x+2, or, *2 —2—— 2, so that -2x+1=-1, and x-1=√−1, that is, x=-1. The value of x therefore, and consequently of y, is impossible; these values have also the same form that was ascribed to them in the above proportion, $47. affords a very fingular example of the length to which a man may be carried, in the fupport of a favourite theory, without being confcious that he is departing in the leaft from philofophical reasoning. The problem propofed is, to conftruct a triangle, of which the bafe fhall be 2 a, and each of the fides a; a thing evidently impoffible. The algebraic folution, if the perpendicular from the vertex on the bafe be the thing fought, gives that perpendicular equal to an impoffible quantity. This is exactly what ought to -happen, in the ordinary view of the subject. But our author, not content with this, goes on to fhow how the impoffible expref-fion may be interpreted according to his own theory, which makes the fign of perpendicularity. In order to do this, he is forced to fuppofe, that the given lines that are to constitute the fides of the triangle, are not lines without extenfion in any dimenfion but one; that they have, in reality, breadth as well as length; in fhort, that they are rectangles, and rectangles of fuch a magnitude that their diagonals meet in the middle of the bafe 2a. This is certainly to depart from the notions that are moft effential to mathematical science, and that form the facred and indifputable bafis by which the whole fabric of geometry is fupported. In a fcience where all the parts are neceffarily connected, an error can never ftand fingly; a departure from truth in one in ftance, general cast of the diction, yet those argumentative passages are evidently more akin to public speaking than to written composition. Frequent interrogations-short alternative propositions, and an occasional mixture of familiar images and illustrations,-all denote a certain habit of personal altercation, and of keen and animated contention. Instead, therefore, of a work emulating the full and flowing narrative of Livy or Herodotus, we find in Mr Fox's book rather a series of critical remarks on the narratives of preceding writers, mingled up with occasional details somewhat more copious and careful than the magnitude of the subjects seemed to require. The history, in short, is planned upon too broad a scale, and the narrative too frequently interrupted by small controversies and petty indecisions. We are aware that these objec tions may be owing in a good degree to the smallness of the fragment upon which we are unfortunately obliged to hazard them, and that the proportions which appear gigantic in this little relic, might have been no more than majestic in the finished work; but, even after making allowance for this consideration, we cannot help thinking that the details are too minute, and the veri fications too elaborate. The introductory chapter is full of admirable reasonings and just reflections. It begins with noticing, that there are certain periods in the history of every people, which are obviously big with important consequences, and exercise a visible and decisive influence on the times that come after. The reign of Henry VII. is one of these, with relation to England ;-another is that com prised between 1588 and 1640;-and the most remarkable of all, is that which extends from the last of these dates, to the death of Charles II.-the era of constitutional principles and practical tyranny of the best laws, and the most corrupt administration. It is to the review of this period, that the introductory chapter is dedicated. Mr Fox approves of the first proceedings of the Commons ; but censures without reserve the unjustifiable form of the proceedings against Lord Strafford, whom he qualifies with the name of a great delinquent. With regard to the causes of the civil war, the most difficult question to determine is, whether the parliament made sufficient efforts to avoid bringing affairs to such a decision. That they had justice on their side, he says, cannot be reasonably doubted, but seems to think that something more might have been done, to bring matters to an accommodation. With regard to the execution of the King, he makes the following striking observations, in that tone of fearless integrity and natural mildne. of this perf • The ex we have already noticed as characteristic g, though a far lefs violent meafure that that fers to is abfurd. He does fo, as we think, very fuccefsfully; but M. Buëe, who had this paffage before him, and quotes it in his Memoir, feems to get rid of the difficulty by means of the dif tinctions mentioned above. We fhall conclude thefe remarks with fome reflections on what we confider as the great paradox in the arithmetic of impoflible quantities, and as one of the most curious examples of the power of figns which the history of language affords, viz. the purpofe which imaginary expreffions ferve, not for marking the limits of poffibility, but for demonftrating theorems concerning quantities that really exist. The character that denotes impoffibility, as we have seen, appears always in the form of the value of a quantity of fome defcription. It is not a mere abftract note of impotlibility, but of impoffibility attached to fome particular quantity, on account of inconfiftent conditions introduced into the data from which that quantity must be determined. As the character ✔ - 1, or -a, appears, therefore, as the expreffion of a magnitude, it is a fymbol fubject to the fame operations of arithmetic, addition, fubtraction, &c. with other algebraic expreflions of quantity. Hence the imaginary fymbols, confidering them quite abftractly from their fignification, may be treated by the rules ufually employed in thofe operations; and the fame changes may be made on them as if they really fignified things poffible, and actually exifting. Now it fometimes happens, that, by comparing two expreffions thus involving impoffible quantities in both, and combining them by the ordinary rules that would be applicable to them in ftrict logic, if they denoted things really exifting, the impoffible fymbols finally disappear, having been exterminated by fome of the ordinary operations of algebra; so that an equation results that contains nothing impoffible, or involves no quantities but fuch as are real. Thus we have a propofition affirmed concerning real quantities; and it is very well understood, that the propofition thus difcovered is always a truth, a truth that is often very valuable, that in general is fufceptible of demonftration without impoffible fymbols, though without the ufe of them it might never have been difcovered. Now, in this we have certainly a moft extraordinary example of the power of figns, or of algebraic language. A fet of quantities, or of conditions, fome of which are inconfiftent with one another, are thus combined together: no idea is attached to the fymbols; and from the feries of operations, that may be faid to be mechanical, and performed merely by the hand, a truth, applicable to quantities that really exift, emerges at laft. Were we to attempt to reafon concerning thefe pollibilities by any other means, than by thus fubjecting them to to a treatment which belongs only to the figns, no force of head could have brought out any thing, not to fay rational and true, but any thing that poffeffed even the form of a propofition. We come at truth by help of the fymbols alone; by operations that are applicable to them only, and that have no reference to any thing actually exifting. Nothing, certainly, can fhow fo clearly the power of conventional figns in matters of reasoning; and the importance, on many occafions, of neglecting the object, and attending to the fign only. It is indeed no wonder at all, to see men reason, or pretend to reason, in every one of the sciences, by help of imaginary expressions, or words that denote nothing having an actual or even a possible existence; but the wonder is to find, that, by such a process, they are led to the discovery of truth. The history of philosophy is full of instances in which words, having nothing real that corresponded to them, have been combined according to the rules of logic, and, in the forms of syllogism, have been the study of the learned and ingenious of almost every age. If we take the terms which have made the greatest noise in the world from the Quintessence and Entelecheia of the ancients, to the Vortices and Phlogiston of the moderns, and the arguments that have been held concerning them, we shall find a vast deal that has no small resemblance to the operations of our imaginary arithmetic. But, in one thing, the practice of the philosopher was different from that of the mathematician: he had not the secret of exterminating the impossible quantities in the end; so that they remained involved in the conclusion, just as they had done in the premises. The extermination of them he left to his adversaries, or his successors; and, when they accomplished it, the whole system, both the argument and the conclusion, fell to the ground at once, and left nothing behind, but one fact more, to be recorded in the history of error. If the imaginary arithmetic of the algebraist exemplifies the benefit arising from the use of signs, in the highest degree, the imaginary logic of philosophers places the mischiefs that may follow from it in a light hardly less striking. An example of this imaginary arithmetic will explain what has been said. Let us imagine that we have, by some means or other, come to this equation: x being an arch of a circle, (2 sin x) -x-1, in which all the quantities are affected by the imaginary symbol, and to which no direct meaning can be affixed, but that between an arch and its sine; no such --e nx relation as is expressed by the equation (2 sin x) ne -12.0 This fers to is abfurd. He does fo, as we think, very fuccessfully; but M. Buee, who had this paffage before him, and quotes it in his Memoir, feems to get rid of the difficulty by means of the dif tinctions mentioned above. We fhall conclude these remarks with some reflections on what we confider as the great paradox in the arithmetic of impoflible quantities, and as one of the moft curious examples of the power of figns which the hiftory of language affords, viz. the purpofe which imaginary expreffions ferve, not for marking the limits of poffibility, but for demonftrating theorems concerning quantities that really exist. The character that denotes impoffibility, as we have seen, appears always in the form of the value of a quantity of fome defcription. It is not a mere abstract note of impotlibility, but of impoffibility attached to fome particular quantity, on account of inconfiftent conditions introduced into the data from which that quantity must be determined. As the character — 1, or -a, appears, therefore, as the expreffion of a magnitude, it is a fymbol fubject to the fame operations of arithmetic, addition, fubtraction, &c. with other algebraic expreflions of quantity, Hence the imaginary fymbols, confidering them quite abftractly from their fignification, may be treated by the rules ufually employed in those operations; and the fame changes may be made on them as if they really fignified things poffible, and actually exifting. Now it fometimes happens, that, by comparing two expreffions thus involving impoffible quantities in both, and combining them by the ordinary rules that would be applicable to them. in ftrict logic, if they denoted things really exifting, the impoffible fymbols finally difappear, having been exterminated by fome of the ordinary operations of algebra; fo that an equation refults that contains nothing impoffible, or involves no quantities but fuch as are real. Thus we have a propofition affirmed concerning real quantities; and it is very well understood, that the propofition thus difcovered is always a truth, a truth that is often very valuable, that in general is fufceptible of demonftration without impoffible fymbols, though without the ufe of them it might never have been difcovered. Now, in this we have certainly a moft extraordinary example of the power of figns, or of algebraic language. A fet of quantities, or of conditions, fome of which are inconfiftent with one another, are thus combined together: no idea is attached to the fymbols; and from the feries of operations, that may be faid to be mechanical, and performed merely by the hand, a truth, applicable to quantities that really exitt, emerges at laft. Were we to attempt to reafon concerning thefe impoflibilities by any other means, than by thus fubjecting them to |