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elementary examples that can be given of the impossibility of the conditions of a problem, arising from the quantities involved in one of the conditions being too great or too small, in respect of those involved in the other. In the first, the rectangle required to be made has a greater area than the sum of its fides will allow. In the second, the fum of the squares of the two lines is less than is confstent with the sum which the two lines themselves are required to make up. . Though geometry has no character that expreffes impofsibility, it has a sort of negative or indirect expression for it. In the general construction of a problem, the thing to be found is usually determined by the intersection of a curve with a straight line, or of one curve with another. Now, when the conditions of the problem are such, that these lines do not interfect, then the soluzion is impossible; and this incompatibility of the conditions is the fame that algebra denotes by the imaginary symbol V -, or more generally, V.
No part of the language of algebra, it is plain, can be regarded as of greater importance than that in which these imaginary cha. Tacters are employed. It explains the nature of those limits by which the possible relations of things are circumscribed, and marks out the conditions that are capable of being united in the same thing, or in the same system of things. The greatest and the Jeast degrees in which those conditions can co-exist, come in this manner to be determined ; and we arrive at a species of knowledge, which, as it is in itself the most perfect and moft beautiful, is often the most valuable that the doctrine of quantity can supply. The whole of what regards the maxima and minima of quantities, in geometry and in mechanics, and the other branches either of pure or mixt mathematics, is thus effentially connected with the arithsnetic of impoffible quantity.
It is evident, from this account of their origin, that the essential character of imaginary expressions is to denote impossibility : and that nothing can deprive them of this signification. Nothing like a geometrical construction can be applied to them; they are indications of the impossibility of any such construction, or of any thing that can be exhibited to the 'senses. Though this conclusion seems to follow very evidently from what has just been stated, yet there have been more than one attempt to treat ima, ginary expressions as denoting things really existing, or as certain geometrical magnitudes which it is possible to assign,
The paper before us is one of these attempts; and the author, though' an ingenious man, and, as we readily acknowledge, a skilful mathematician, has been betrayed into this inconsistency gy a kind of metaphysical reasoning, which we confess ourselveç
net not always able to understand. He distinguishes between the mark of impossibility, as an arithmetical character, and as a term of algebraic 'language indicating certain operations that have been performed. In the first of these capacities, he considers the Symbol v -1, as really denoting impossibility, in the second, he regards it as expressing something that can be actually exhibited. This distinction, in the very principle of it, seems to us extremely unsound; an expression that, in its most simple and abstract state, has a certain radical and primitive signification, cannot, by being applied to something less abstract, acquire a signification quite opposite, and nowise analogous to that which it had before. We transfer the common arithmetical cyphers, from denoting number in the abstract, to denote; lines or angles, surfaces and solids ; but we never, on that account, think of changing the rules of arithmetic, or supposing 3 times 3 to be 9, in the one case, and not in the other. The same may be said of the signs 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 aud 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 comtroversies and petty indecisions. We are aware that these objections may be owing in a good degree to the smallness of the frage ment 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.
+ and —; they denote opposition of direction when they are applied to the expression of geometric magnitudess but they do not, on that account, lose any of the characters they before possessed : it is from the perfect analogy between opposition of direction in lines, and the opposition of addition and subtraction in numbers, that signs, which were originally appropriated to the latter, are so easily, and so safely transferable to the former signification. Just so, we apprehend, the mark of impossibility cannot be regarded as having one import considered arithmetically, and another quite opposite, when taken as a part of algebraic language, or when applied to geometry.
We do not, indeed, clearly understand what is meant by this distinction; and therefore shall not insist on the general speculation : but shall consider the evidence that is offered by our author for his fundamental proposition, that the square root of expressed perpendicularity. As we must give the reasoning without reference to a diagram, we cannot translate it literally, but we shall do so as nearly as possible. • Suppose three equal straight lines to meet in a point, two of them to be in one straight line, the one to the right of the said point, the other to the left, and the third to be at right angles to them both. If we call the line taken to the right + 1, that taken to the left must be - 1, and the third, which is a mean proportional between them, mast be w-+1', or, more simply, N-1. Thus, N 1, is the sign of PERPENDICULARITY.? ( 10.) Now, we must acknowledge, that though we have read over these few lines very often, and very carefully, we are unable to perceive any force in the argument they profess to contain, or to 'mceive how a man, so dearned and ingenious as the author is on
The iptroductory 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 comprised between 1588 and 1640 ;--and the most remarkable of all, is that which extends from the last of these datęs, to the death of Charles II.- the æra 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 pro ceedings 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 acconimodation. With regard to the execution of the King, he makes the follow ing striking observations, in that tone of fearless integrity and .
' we have already noticed as characteristic of this perf • The ex
ng, though a far less violent measure than