Obrazy na stronie
PDF
ePub

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 a2 will be greater than ax, and their difference, suppose d2, must be negative; we should, on this supposition, then have y=+d√1. 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=±√=d2. 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+BV-1, while 2m cos. xm=(v+u)m, developed, is of the form A-B√-1, and that, therefore, 2m cos. x = (u+v)m+(v+U)TM 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

2

* 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. 429, "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 methodes 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 innume. rable 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.

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=c, 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.

Two straight lines drawn from, or diverging from, the same point, may be conceived to be produced, until their distance becomes greater than any assignable magnitude; as the straight lines AB, AC, diverging from the Point A.

By a straight line, we understand a line that preserves the same direction between its extreme points; as a crooked line is that which varies its direction, and a curve line that which constantly varies it.

H

G D

E

C

F A

B

It is evident, from Euclid, prop. 27, b. i., that if the straight line FD falling on the straight lines HE, FB, makes the alternate angles HDF, DFA equal, these straight lines are parallel; from which, it follows that the strait line GA, drawn through I, the middle of FD, perpendicular to HE, is also perpendicular to FB, (prop. 15 and 26, b. i., E.) It also follows, that if the straight lines EG, BA be each perpendicular to GA, they are parallel, for then the alternate angles EGA, GAF are equal, whence, &c. Now, let AC make with GA an angle less than BAG, in which case, the sum of the angles CAG, EGA, is less than two right angles; Euclid says, in his twelfth axiom, that these lines must then meet. But this not being evident, or being rather a proposition, leaves Euclid's, otherwise elegant system, imperfect.— From the axiom which we have laid down, this proposition evidently follows. For AB and AC, both diverging from the point A, may be produced until their distance becomes greater than any assignable magnitude, and as the distance between the two parallels AB and GE must be an assignable magnitude, AC must, therefore, meet GE, when both are produced far enough. Here then is an evident proof, not only of Euclid's axiom, but also, of what we have asserted in the beginning of this Review, that the simplest truths, and the easiest modes of arriving at them, are, generally, the last perceived.

We have now, we are persuaded, given an impartial review of a subject, which we deem of no little importance, and removed, we hope, a stigma from geometry, which has been long its disgrace. In the great analytic chain, it would be very desirable also, that its broken links could be repaired. To the analytic method, or rather the calculus, we are far from being inimical. We are too sensible of its value. We know that it has enlarged, and will more and more assist in extending that horizon of science, the boundaries of which will for ever recede as we advance; but when, like the parasite that clings to the stately oak, it attempts to destroy that from which it has received its

principal existence; then we think it right, we deem it even necessary, to curtail its lofty pretensions, and to point out its humble origin and dependence. We hope, however, that in this country as well as in the British Isles, and in Italy, both geometry and the calculus will be equally cultivated and encouraged.

ART. V.-1. Sur les Fonctions du Cerveau: et sur celles de chacune de ses parties: avec des observations sur la possibilité de reconnaitre les instincts, les penchans, les talens, ou les dispositions morales et intellectuelles des Hommes et des Animaux, par la configuration de leur Cerveau et de leur Tete. Par F. J. GALL. Paris, 1825. En 6 tomes. 8vo.

Of this work, the first and second volumes, are occupied "Sur l'origine des qualités morales et des facultés intellectuelles de l'homme, et sur les conditions de leur manifestation."

The third volume is, "Sur l'Influence du Cerveau sur la forme du Crane; difficultès et moyens de determiner les qualités et les facultès fondamentales, et de decouvrir le siege de leurs organes."

The fourth and fifth volumes are entitled, "Organologie, ou exposition des instincts, des penchans, des sentimens et des talens; ou des qualitès morales et des facultès intellectuelles fondamentales de l'homme et des animaux; et du siege de leur organes.

The sixth volume is entitled, "Revue critique de quelques ouvrages anatomo-physiologiques ; et exposition d'une nouvelle philosophie des qualités et des facultes intellectuelles."

Such are the general contents of this work; which may be regarded as a supplement to the larger anatomical work of Dr. Gall, entitled, "Anatomie et Physiologie du Systeme nerveux en general et du Cerveau en particulier; avec des observations sur la possibilitè de reconnaitre plusieurs dispositions intellectuelles et morales de l'homme et des animaux par la configuration de leurs tète. 4 tomes en folio, et 4 tomes en 4to. avec atlas de cent planches. Chez l'Auteur et chez N. Maze, libraire Rue Git-le-Coeur, No. 4. Par Gall et Spurzheim."

« PoprzedniaDalej »