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diately, as according to some accounts, or after having been considerably harassed in their attempts to pass, according to others, quitted the field.' There are also very many instances of unnecessary and cumbrous redundance; as when, instead of saying incalculable advantages, he is pleased to speak of advan tages of an importance and extent, of which no man could presume to calculate the limits. We would object also to such expressions as the then state of Scotland, intelligence which appeared uncertain and provisional, and many others, which, after the specimens we have given, it is needless to enumerate.

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Occupied, indeed, as we conceive all the readers of Mr Fox ought to be with the sentiments and the facts which he lays be fore them, we should scarcely have thought of noticing those verbal blemishes at all, had we not read so much in. the preface, of the fastidious diligence with which the diction of this work was purified, and its style elaborated by the author. To this praise, we cannot say we think it entitled; but to praise of a far higher description, its claim, we think, is indisputable. Independent of its singular value as a memorial of the virtues and talents of the great statesman whose name it bears, we have no hesitation in saying, that it is written more truly in the spirit of constitutional freedom, and of temperate and practical patriotism, than any history of which the public is yet in possession.

ART. II. Memoire fur les Quantités Imaginaires. Par M. Buëë. Prom the Phil. Trauf. for the year 1806. p. 23.

HE language of algebra deferves the attention, not of mathematicians only, but of all philosophers who would study the influence which SIGNS have on the formation of ideas, and the acquifition of knowledge. Other languages have been formed for the purpose of communicating thought from one person to another; and if they have served to make the individual think with more accuracy or extenfion, this effect is a fecondary one, and în fome degree accidental. Algebra, on the other hand, is a language invented exprefsly for the purpose of affifting the mind in the ma nagement of thought: this is its primary destination; and the bufinefs of communicating knowledge, which is principal with respect to other languages, with refpect to it, is fecondary and accidental.

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When, therefore, we would trace the direct influence of signs on the operations of the mind, we must confider the algebraic language as the extreme cafe, or the inftantia fingularis, where the

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extent of that influence, in fome refpects at leaft, is most fully displayed.

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Again, in the language of algebra itself, the part which is most curious, and is, as it were, the extreme of an extreme, is the application of imaginary expreflions to the investigation of theorems, where truth is fometimes difcovered by the help of figns alone, without any affistance at all from the ideas which they reprefent.

In a matter where any thing fo paradoxical occurs as the entire feparation of the fign from the thing fignified, it will not appear furprifing if different opinions have been entertained. The opinions that are fupported in the paper before us, are, accordingly confiderably different from thofe generally received. In or der to judge of them correctly, it is neceffary to confider the manner in which the figns called imaginary, and the correfponding impoffible quantities, are firft introduced into the algebraic calculus.

In the refolution of problems, whether, geometrical or arithmetical, cafes occur when the conditions prefcribed are inconfiftent with one another, and cannot poffibly be united in the same subject. The problem, therefore, cannot be folved, and the quantity that was to be found is faid to become impoffible. Thus, for example, if it were required to divide a line, 10 feet long, into two parts, fuch, that the rectangle under thofe two parts should have an area of 26 fquare feet, it would foon appear that the thing required is impoffible to be done; or that there is no way in which a line only to feet long can be divided, fo that the rectangle contained by the two parts fhall be so great as 26 fquare feet. The fact is, that the rectangle under the two parts of a given line, cannot exceed the fquare of half the line; which fquare, in the prefent instance, is 25; and if we feek for the parts on the suppofition that their rectangle is 26, we find them equal to 51. In like manner, were we required to divide the fame line of 10 feet, fo that the fum of the fquares of the parts fhould be less than 50, that is, less than the fum of the fquares of the parts when the line is bifected, we should find that we were again attempting what was impoffible to be done. If we would have the fum of the faid fquares, for instance, to be 49, the parts would come out

5√, where the impoffibility is denoted as in the former inftauce, by the fquare root of a negative quantity. As no quantity, whether pofitive or negative, when multiplied into itself, can give a negative product, it follows, that no negative quantity can be the product of any quantity multiplied into itfelf; that is, it can have no fquare root; and therefore, when fuch a fquare root appears in the value of any quantity, it expreffes the impoflibility of finding that quantity.

In the two problems juft mentioned, we have two of the most elementary

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elementary examples that can be given of the impoffibility of the conditions of a problem, arifing from the quantities involved in one of the conditions being too great or too small, in refpect of thofe involved in the other. In the first, the rectangle required to be made has a greater area than the fum of its fides will allow. In the fecond, the fum of the fquares of the two lines is less than is confiftent with the fum which the two lines themselves are requir ed to make up.

Though geometry has no character that expreffes impoffibility, it has a fort of negative or indirect expreflion for it. In the ge neral conftruction of a problem, the thing to be found is ufually determined by the interfection of a curve with a ftraight line, or of one curve with another. Now, when the conditions of the problem are fuch, that these lines do not interfect, then the folution is impoffible; and this incompatibility of the conditions is the fame that algebra denotes by the imaginary fymbol, or more generally, a.

No part of the language of algebra, it is plain, can be regarded as of greater importance than that in which these imaginary cha racters are employed. It explains the nature of those limits by which the poffible relations of things are circumfcribed, and marks out the conditions that are capable of being united in the fame thing, or in the fame fyftem of things. The greatest and the leaft degrees in which thofe conditions can co-exift, 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 most beautiful, is often the most valuable that the doctrine of quantity can fupply. 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 arith metic of impoffible quantity.

It is evident, from this account of their origin, that the essen tial 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 y a kind of metaphysical reasoning, which we confess ourselves

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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-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 + and -; they denote opposition of direction when they are applied to the expression of geometric magnitudes, 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 arithmeticalAy, 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 1 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.

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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, must be 1', or, more simply, -1. Thus, -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 conceive how a man, so learned and ingenious as the author is on

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all hands admitted to be, should have suffered himself for one moment to be deceived by it..

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Any imaginable conclusion, it appears to us, might have been obtained in the same manner; the third line, for example, needed not have been placed at right angles to the other two, but mak ing an angle, suppose of 120 degrees, with the one, and of 60 with the other, It would still be a mean proportional between them, and its square would be therefore, according to the above method of reasoning, equal to + 1 x11, so that the line itself would be 1; and thus -1 would denote, not perpendicularity, or the situation in which a line makes the adjacent angles equal, but that in which it makes one of these angles double of the other. The one of these arguments is just as good as the other; and neither of them, of course, is of any value. Indeed, it would evidently be very unfortunate for science, and could not but be productive of inextricable confusion in mathematical language, if the character which denoted impossibility at one time, should at another signify something actually existing like perpendicularity. It should have occurred to our author, also, that perpendicularity not being a quantity, but a modifica tion of quantity, (for it is neither the right angle nor the perpendicular itself, that the sign-1 is supposed to denote, but the abstract notion perpendicularity), it would be strange indeed if "a character that was applied to quantity, whether as possible or not possible, should pass to the expression of something, of which magnitude or quantity cannot be predicated.

The fundamental proposition which the whole paper is meant to illustrate, being thus, in our opinion, incapable of support, and essentially erroneous, we need not enter much into the consideration of the remaining illustrations. Some of the objections made to the ordinary doctrine of impossible, quantities, are, however, of importance to be considered. That doctrine is certainly not in all respects without difficulty; and it is of consequence to know the objections which are stated by an expert algebraist, who entertains notions on the subject peculiar to himself.

At 35, we meet with the following remarks. 1+1=

(1 + √1) (1-1), or (1 + √ — 1) (1 − √. -—- 1.) — 2,

wherefore 1-1: √2:: √2:1-1; a proposition absurd, if we give to √2 its arithmetical signification; but if we ascribe to 2 its geometric signification, if we make it represent the diagonal of a square, the side of which is unity, the above proportion ceases to be absurd. M. Buee goes on to prove, that in this latter signification of 2, the proportion that has just been stated, is reconcileable with the notion, that — is the

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