but this would evidently be shifting the hypothesis, because they are then no longer referred in position, to the line AB. Yet, this is the whole ground of his objection. He makes a similar objection in his "Digression sur la nature des Quantités dites negatives," published at the end of his "Théorie des Transversales," (p. 98) in pointing out a negative secant in the third quadrant, which ought to be positive, when the opposite arc, about which there is question, by the theory, is in the fourth quadrant. The same quantity then can never be negative and positive at pleasure, contrary to what he asserts (p. 100) of the work last quoted, while the same hypothesis is retained. Although Carnot appears to have seen, clearly, that it is to the geometry of position we must have recourse for the solution of the difficulties which the signs give rise to; he has, from adopting these subterfuges, unworthy of his great talents, and therefore making a false step at the beginning, completely failed. Yet, he is entitled to no small degree of praise for having first called the attention of mathematicians to this new and important branch of geometry; a branch, in which whoever succeeds, we venture to assert, will render a service to science little inferior to what Euclid or Newton has rendered. But, unhappily, nature appears very sparing of such men-they are not produced at pleasure.

-

Mr. Buée, in the Philosophical Transactions for 1806, is the next who has considered this subject; and, we believe, the first who has considered it in a rational point of view. He shews (p. 25) that algebra ought not only to be regarded as an universal arithmetic, but as a mathematical language; that the signs + and -, considered as signs of arithmetical operations, indicate addition or subtraction, but considered as signs of operations in geometry, they indicate opposite directions (p. 23); that a quantity with the sign √-1 does not indicate addition or subtraction, nor is it equal to zero, but that it is the sign of perpendicularity. The signs + and -, while taken geometrically, indicate opposite directions from the same point, but taken arithmetically, they destroy each other. He, therefore, shews (p. 65) that an expression may be geometrically real, and arithmetically nothing; that there is no analogy between multiplication and direction, and that we can form no idea of an exponent which indicates the sum of directions, the idea we attach to exponents, being a sum of multiplications (p. 68); that R (√1)" expresses the trace of any arc, described with the radius R; that (p. 81) the circle becomes an equilateral perpendicular hyperbola, and the equilateral hyperbola, a perpendicular circle, when their ordinates become imaginary, with a variety of other curious results. This memoir

consisting of 66 pages, was read June 20, 1805, and published in French.

Mr. J. F. Français, a distinguished Professor, "à l'ecole imperiale de l'artillerie et du genie," in a memoir, dated Metz, le 6 de Juillet, 1813, is the next who has taken up this important subject. He has published his memoir in Gergonne's Annals, vol. iv. and calls it "Nouveaux Principes de Géométrie de position, et interpretation geometrique des Symboles imaginaires." He adopts a notation different from Buee, although his principles are similar. He lays down his definitions, &c. and exhibits the subject in something like a systematic form. He has his ratios of magnitude and of position; his positive and negative straight lines, his positive and negative angles, and a few very important theorems, selected to elucidate his subject. He acknowledges, however, (p. 70) that these ideas were suggested by a letter of M. Legendre to his deceased brother. In the same volume of Gergonne's Annals, (p. 133) M. Argand shews, that in 1806, he published an "Essai sur un maniere de representer les quantités imaginaires, dans les constructions geometriques," the principles of which are analogous to those of Français; that he sent the manuscript to Legendre to have his advice, and thus accounts for the source of communication, of which Mr. Français speaks. In this article, Argand has several interesting remarks. In pages 222-228 & 364, Gergonne gives extracts from letters of M. Français and Mr. Servois, another distinguished professor, full of important remarks relative to the same subject. In vol. v, 1814, p.147, "Annales," Argand has again some interesting reflections, on this new theory of imaginary quantities, shewing, that besides giving an intelligible signification to expressions which we are forced to admit into analysis, it exhibits a method of calculation, or a notation of a particular kind, which employs geometric signs, jointly with the ordinary algebraic signs. In p. 209, of the samne volume, Argand replies to the letter of M. Lacroix to M. Vecten, Professor of Mathematics, &c. at Nismes, vol. iv. p. 367, where Lacroix observes that M. Buée published a memoir in the Philosophical Transactions, for 1806, on the same subject. Argand shews that his Essay was published the same year; but admitting this, we have seen that M. Buée appears to have the merit of priority, his memoir being read in June, 1805. As we cannot enter into any detail relative to these Memoirs, our article having already extended far beyond the limits which we had prescribed for ourselves, all that we wished to notice, are the historical facts, and the importance attached to this new subject in France; whilst in England, and in our own country, efforts are making to abandon geometry altogether. Tempora mutantur. VOL. I.-NO. 1.

17

As we deem the subject of importance, we wish to be indulged in making a few more remarks, to shew that algebra or the calculus becomes unintelligible, except when confined to numbers, without the aid of geometry. In the equation of the circle given above, if a be less than x, then x2 will be greater than ax, and their difference, suppose d2, must be negative; we should, on this supposition, then have y=+d√. Now, without the aid of geometry, what idea can we attach to these symbols which apparently indicate, and as authors assert, do indicate, an impossibility: Lacroix calls them, (Algebra, art. 115) "Symboles d'Absurdité." There are, however, cases where the square root of a negative quantity, becomes geometrically a real magnitude. Thus, in the preceding example, if DF= +d, and FE-d, their rectangle = −d2, and y=+ √d. We perceive, also, that four squares may exist, as in the figure, about the point F, and in the same plane, the sides of which, respectively, may be +d or -d, and that the rectangle of both these quantities may represent some of these squares, regard being had to their position; or that some of these may be considered positive, and others negative, according to their relative position. We see, moreover, that these negative squares and imaginary roots, indicate operations purely geometrical. Thousands of these squares may exist, if we conceive them to form an angle with the plane of the figure or the paper, and thus angular position is considered. The notation and symbols then, which are calculated to give us a distinct idea of these figures, in their various relations and positions, must be different from those which relate only to number. If similar observations were extended to more complicated figures, we can easily conceive, what an extensive field would then be presented to our view. A new science, as yet in its infancy, and destined, no doubt, to take a distinguished place among the other great branches of human knowledge. It is, however, astonishing, that the hints thrown out by Buée, Français, and Argand, should not be more attended to, and that, as yet, we should not have a sentence in the English language, relative to this interesting subject.

We shall now point out a remarkable instance of the want of this theory, in the application of the binomial theorem, to the arithmetic of sines, this being the foundation of almost all the modern calculus. The developements of the binomials (u+v)" and (v+u)", have been considered equal, whatever the exponent n may signify, by Euler, by Lagrange, and by every writer since Euler, until Poisson, an analyst, perhaps at present, without an equal, first pointed out the difference in the "Correspondance sur l'Ecole imperiale Polytechnique," vol. ii. p. 212.

=

Lacroix notices this, p. 605, vol. iii. 2d edition of the "Traité du Calcul," &c. From Poisson's results, and what we have already shewn, it follows, that 2m cos. xm = (u+v)m, developed, is of the form A+B √1, while 2m cos. xm (v+u)m, developed, is of the form A-B√-1, and that, therefore, 2m cos. xm = (u+v)m+(v+u)m= A. Here then, we have the same quantity equal to three different quantities, respectively, at the same time. It would follow, that these quantities are, therefore, equal to each other, which is absurd. Let the enemies, or those ignorant of geometry, get over this difficulty, legitimately, and, independent of geometry, and then give their opinion about abandoning it.* We

In observing this anomaly in a binomial function, it brings to our recollection, the great injustice done to Professor Wallace, who first in this country, in Professor Silliman's Journal, called the attention of mathematicians, to a series published by M. Stainville, in Gergonne's Annals, in which, by the simple operation of multiplication, results the most complicated, and those even of a transcendent nature, have been deduced; and an elegant demonstration of the binomial theorem, as far as numbers are concerned, been given. To exhibit this, was, we are well aware, the view of Professor Wallace, in noticing these interesting results. It appears, however, that his preliminary remarks were suppressed, and the title "New Algebraical Series, by Professor J. Wallace," substituted, and the word new, made the subject of criticism, while they were given as Stainville's, without altering even his notation, and the volume pointed out. It was, therefore, an affair entirely between Stainville and Mr. B. We cannot help, however, observing, that the Northren critic has, on this occasion, exhibited an antipathy to his Southern fellow-citizen, not very becoming, independent of the injustice. Professor Wallace, it appears, did not put his name to the communication, but handed it to a friend, without attaching much importance to it. The editor, Mr. Silliman, in the No. for February, 1825, and also in the No. for June, 1825, acknowledges the error in the title. We should not take any notice of this in our remarks, did we not find, in looking over Gergonne's Annals, that in vol. xv. p. 373, he adverts to Professor Wallace's New Series. Gergonne has not corrected this error in his subsequent numbers. We hope, however, for his own credit, he will correct it. If he had read the piece, he could not have helped seeing the mistake. In the "Revue Encyclopedique," vol. xxvi. the manner in which this communication is noticed, does honor to the impartiality of the distinguished members of the Institute, who conduct this learned work. They observe, p. 439, Au sujet de quelques observations critiques, dont le memoire de M. Wallace, a été le sujet, ce Professeur fait lui-meme plusiers remarques interessantes sur l'histoire des mathematiques, pendant la dernier siecle et dans celui-ci, sur l'ordre des decouvertes, et sur les méthodes des inventeurs." Mr. B. attempts a demonstration of the binomial himself; but among the hundreds we have seen, and those we almost every day see, it is, without exception, the most imperfect attempt. The law of the series assumed by Stainville, is evident; whereas, in B's demonstration, the binomial itself is taken as granted, and then, by a species of tatonnement, he shews that it will hold in pos. whole numbers. Euler, also, assumes the binomial, after all B. has published to shew that his method was identical with Stainville's. What becomes of B's pretended demonstration when applied to Poisson's case; or to the innumerable other cases that may arise, when the properties of numbers, and of extended magnitudes, in general, are considered, as well as hyperbolic logarithms and sines? or what idea must B. form of mathematical demonstration? Thousands of demonstrations have been given of the binomial theorem, not one of which, extend, as yet, beyond the properties of numbers. Can it be said to be demonstrated, even in numbers, when every system and modulus of calculation are considered. M. Poinsot shews in the Memoires of the Institute, for 1819-1820, p.99, &c. that using certain moduli, some of the imaginary expressions that occur with others, will vanish. But it appears, that seeing by intuition, and generalizing too hastily, are among the prerogatives of great analysts.

64

shall add but one example more to shew the dependence of algebra upon geometry. For this purpose, we have selected the 47th of the first book of Euclid; a proposition within the reach of almost every reader. Let a, b, c, represent the base, perpendicular, and hypotenuse of a right angled triangle respectively; by this proposition we then obtain a2+b2=c2, whence a=+√c2-b2. If we now suppose b greater than c, and d the difference of their squares, we obtain a=+√d, an absurdity. But how can this absurdity arise while reasoning on the symbols, according to the strict rules of algebra; or why is there an absurdity in the supposition that b is greater than c. Simple as this inquiry may appear, all the resources of the calculus alone would be found inadequate to give us the required information. For we know, from geometry only, that the greater side of a plane triangle is opposite to the greater angle; and that the angle opposite to c is greater than the angle opposite to b, we know also from geometry; because we know that the three angles of a triangle are equal to two right angles. But this, all the algebra or calculus, ever discovered, could not find out. This truth, itself, depends on the theory of parallel lines and the calculus, in attempting to establish this theory, has completely failed. Remove then those truths derived from geometry from the calculus, and you leave the analyst's superstructure, like Swift's Island, floating in the air. It may however, be urged here, that geometry has also failed in establishing the theory of parallels, and that Euclid, Montucla, Playfair, Leslie, Hutton, and every writer in Gergonne, and every writer whom he quotes, down to the present time, have failed, although many distinguished professors, as well as Gergonne himself, have engaged in the inquiry, and yet, in despair, they have given it up. This we must acknowledge to be the fact. They almost all virtually use Euclid's twelfth axiom, which is, evidently, a propositionexcept in the case where the theory of functions is applied; and this theory, were it even admissible, fails in this instance. With due respect and esteem, for the talents of so many distinguished men, we submit the following demonstration of this theory, being of opinion that it leaves nothing more to be desired on the score of evidence. In place of Euclid's twelfth axiom, we substitute the following, the evidence of which must, we think, be acknowledged the moment it is proposed, which is the criterion of an axiom.