amend his chronology when he next talks of the repeal of the witch-act; and then we shall readily admit that, though we do not hold ourselves to be behind any nation whatever in respect of wisdom and rational freedom, we have not attained to the consummation of either.

ART. X. Géométrie de Position, &c.; i. e. Geometry of Position. By L.N.M. CARNOT, of the National Institute of France, of the Academy of Sciences, Arts, and Belles Lettres, at Dijon, &c. 4to. pp. 530. and 15 Plates. París. 1803. Imported by De Boffe, London. Price 11. 4s. sewed.

I'

Graculus esuriens."

«Omnia novit

we strip this quotation of its contemptuous and sarcastic style, it may be applied to the author of the present work; who, after having wielded "the energies of France," and confounded Europe, sits down quietly to write mathematical treatises.

One of the great objects of this publication is to clear the theory of positive and negative quantities from false and ambiguous notions, and to establish it on a firm basis. The subject is old, but has lately occasioned much controversy. After a system is established, the principles are examined. Thus the step which is first in building is last in science; and, with a deviation from the plan of Palladio and Wren, Newton and Euler have raised beautiful and lasting superstructures, without having previously ascertained the firmness and solidity of the foundations.

The term Geometry of Position will recall to Mathematicians an idea of Leibnitz, the Geometry of Situation. That great man suggested that the diversity of the position of the corresponding parts of complex figures should enter into the expression of the conditions of a geometrical problem; in order that, in separating them, by a properly distinctive character, they might more easily be set apart in calculation. This diversity of positions is often expressed by simple changes of signs; and it is precisely, says M. CARNOT, the theory of these changes that forms the essential object of the researches which I have in view, and which I express by the words Geometry of Position."

The preliminary discourse contains a number of well founded observations on negative quantities, and on the explanations and theories that have been given of them, Yet perhaps M, CARNOT will be thought, in some cases, superfluously to have argued, and to have refuted ideas which nobody now undertakes to defend; we allude to what Euler and Newton have

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said

said concerning negative quantities being less than nothing, D'Alembert, as the present author very properly observes, made many judicious and luminous remarks on the theory of negative quantities. yet this acute writer was more successful in exposing what was absurd, than in exhibiting what was just and true. His argument against negative quantities being quantities less than o is irrefragable: but in what he has said concerning negative quantities being of the same nature as po sitive, but taken in a contrary sense, he is at variance with himself. The work before us affords several proofs, that this latter notion of negative quantities is entirely false. Some of the reasonings, which we should wish to insert, we cannot introduce on account of the diagrams attached to them; but the author's argument may be partly understood from an extract:

This twofold error, of which we have been speaking, is avowed by those who admit the notion of negative quantities; which they express by saying that the calculation rectifies, of itself, the false hypothesis on which it may have been established: but, if the hypothesis from which we set out be false, there is already, an error committed, and if the calculation redresses this error, it can only be by another for when we proceed on a false principle, the more justly we afterward reason, the surer we are of arriving at a result equally false; it is only, then, a new error, made in a sense contrary to that of the first, which can repair it.

For example, we have cos. (a+b)=cos. a cos. b-sin, a sin. b: but this formula, having been established for the sole case in which, a, b, a+b, are angles less than the fourth of the circumference, becomes false when we suppose the contrary. Nevertheless, those who admit the notion of negative quantities regard this formula as general and really applicable to all cases: but, as this supposition is not just, they redress the error by saying that, then, cos. a, cos. (a+b) each becomes negative; and that, in consequence, it is necessary to change the signs of into ; whence we have a result,

cos (a+b) cos. 4. cos. b + sin. a. sin. b;

a true result, but which, from the circumstance of its being true, proves that the supposition of cos. a and cos. (a+b) being negative. is a new error; since, were it not one, nothing could have compensated the result of the false hypothesis made in the first instance, viz. that the formula was applicable to all cases. The proof that it was not is that it is really different, as appears from the new case; and of this we may be assured, in directly investigating it according to the ordinary methods, and by simple synthesis, without employing the notion of negative quantities; a method which is less expeditious than analysis, but the results of which no one disputes.

Thus, these formulas, so frequently used,

cos. (a) sin. a; sin. (2a)=—sin. a; and others similar to them, in which expresses- the fourth of the circumference, are false equations; and can only be employed as

simple algebraical forms, proper, by the very circumstance that they are false, to redress an error previously committed. They in fact redress it in certain cases, by indicating what it is necessary to substitute instead of the real quantities, cos. (+a), sin. (2w+a), when, by a previous error, these same quantities, cos. (+a), sin. (2+a) have been put for cos. a, sin. a, in forms which had only been found for these latter quantities, and which can be applied immediately and without modification to them alone. These expressions, such as sin. a, are what I name the values of correlation of quantities; instead of which, we must substitute them in the primitive formulas. Thus these values of correlation of quantities are nothing else than algebrai cal forms, which, substituted in the primitive formulas instead of the true quantities which they represent, render the formulas applicable to cases at first unforeseen; that is to say, others besides those on which the reasonings had been at first established in the equational statement, or expression of the given conditions. Considered under this point of view, these algebraical formulas are very useful; the only question is, properly to determine the cases in which they can be employed without inconvenience. This remains to be examined.

Having theu already shewn how obscure and false the commonly received notions of quantities called negative are, it remains for me to investigate and establish the true principles of the theory which concerns them.”

We cannot state the succeeding remarks exactly in the words of the author, since they have reference to a diagram; but we shall give them as nearly as we can.-Parallel to the diameter of a circle, draw a line; and from points in it, draw perpendicular lines cutting the circle in two points: then, if the distance between the first mentioned line and the diameter be called a, a distance from the same line to the circumference z, and if y be the ordinate of the circle, then the ordinate (y) to the right of the diameter is z-a; to the left, a-z; and some mathematicians have argued that, since a -z is z-a taken negatively, y put za would in effect become ncgitive: but the paralogism (says M. CARNOT) is easily detected.'

In order that y should become negative, when it becomes a―%, it is necessary that should remain greater than a; but z, on the contrary, becomes less than a; a-z is then positive; y, therefore, is positive, to whichever point of the circumference we refer. As in the first case, however, we have y-a, and in the second ya -z, it is plain that, in order to pass from the point D to the point C (two points in the circumference through which the line drawn perpendicularly to the diameter passes), it is necessary to put in the equation,-(-a) instead of +(-a), ory instead of +y This change does not prove that y is become a negative quantity, but only that, of the two quantities a, z, of which it is the difference, that which is the greatest when y answers to D is the least when y answers, to G.

曲

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Thence

Thence I conclude, first, that every isolated negative quantity is a being of reason, and that such as occur in the calculation are only simple algebraical forms, incapable of representing any real and effective quantity: secondly, that each of these algebraical forms, taken abstractedly from its sign, is nothing else than the difference of two other absolute quantities, of which the greatest, in the case on which the reasoning has been established, becomes the least in the case to which we wish to apply the results of the calculation.

This principle answers every objection, and removes every kind of difficulty, without the necessity of introducing those abstract notions on which geometricians cannot agree. In fact, by recurring to simple and intelligible notions, it will naturally present itself to the mind, that there cannot really exist other quantities than those named absolute; and that the signs, by which they may be preceded, do not indicate quantities but operations. Thus these signs, taken collectively with these same quantities, do not form pew quantities, but complex algebraical forms,

After a farther explanation of his ideas and theory, M. CARNOT proposes, instead of the notion of positive and nega tive quantities, to substitute that of direct and inverse. These quantities, direct and inverse, are nothing else than ordinary or absolute quantities, but are considered each as the variable difference of two other quantities that become alternately greater and less, the one than the other. When that which at first was greatest (that is to say, in the system on which the rea soning has been founded) remains constantly the greatest, the quantity that expresses the difference of their absolute values is called a direct quantity on the contrary, when it becomes the smallest, this difference is called an inverse quantity. Thus, says the author, the metaphysics of positive and negative quantities disappears; and there remain only direct and inverse quantities which are absolute equally with all other imaginable quantities. According to the different circumstances in which they are found, we ought to preserve the sign that precedes them in the forms in which they enter, or change it; and it is the theory of these changes that is named geometry of position, since in fact it is by these that the diversity of position of the corresponding parts in figures of the same kind is expressed.

Perhaps it had been as well to have retained the terms negative and positive; for scarcely any thing is added in point of perspicuity and precision by the new terms, inverse and direct. In the first section of the work, it is fully shewn that the ordinary theory of positive and negative quantities is absurd: that the number of positive or negative roots in an equation does not indicate, in any exact matter, either the number of solutions of which the problem is susceptible, or the sense in which they ought to be taken; that, nevertheless, these roots

are

are algebraically exact, and by transformation may be made useful; and that it is precisely and solely in the use which analysis makes of negative or imaginary forms, as if they were true quantities, that it differs from synthesis, and possesses over it such great advantages.

The plan and matter of the succeeding sections may be understood from the author's own words:

Among the different examples which I give of my theory in the second section, is found a general table of the correlation of quantities linear-angular; that is to say, of sines, cosines, tangents, &c. which correspond to the different regions of the circumference. I flatter myself that I have there given the true theory of the varia tions of signs, which these kinds of quantities undergo. "Afterward, I return to the mode practised by the antients, of comparing the ares immediately with their chords, instead of comparing them with the halves of the chords of double the arcs: which, in fact, is the same, but gives means more natural, and oftentimes more simple, of establishing the relations of these quantities. I propose, on this occasion, certain formulas, (hitherto, I believe, not given,) to represent these relations by symmetrical expressions between all the arcs compared.

The other sections are destined to the application of principles developed in the first two: but, on this occasion, I have proposed to myself another object which to me appears at least equally im.. portant; viz. to exhibit a method that shall be capable of representing, by analytical tables, a general view of the properties of any proposed figure whatever; and in some sort to form from them a complete enumeration, as well as of those of all figures that may be related to it. For this purpose, I at first consider this proposed figure, as a term of comparison or primitive figure, and I call those sorrelative figures, which we propose to compare with it.

In the primitive figure itself, I take, among the quantities that compose it, a certain number; such that, they being known, the rest may be determined. These new bases chosen, I express all the other parts of this primitive system in values of these first alone, and I form their general table. This table evidently comprehends all the desired relations of the different parts of this primitive figure, since it affords the means of comparing them, two and two, by the mediation of primordial quantities, taken for the purpose of serving as common terma of comparison.

This primitive figure being supposed to be the real and existing object on which the reasonings have been established, the formulas expressing the relations of its different parts, and composing the general table of which we have spoken, can contain only expressions real and intelligible; and consequently, they cannot indicate any impossible operation, nor any absurd quantity; there cannot, then, occur in them isolated negative quantities, since such a quantity is a being of reason; neither, à fortiori, can there occur imaginary quan tities; that is to say, the signs aud—which enter into these for

mulas,