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the same author remarks in another place, that all mathematical investigation must involve two things, the quantities concerned, and the operations performed upon them. Geometry expresses the first by real representations, and the second by words at length. Algebra does both by conventional symbols; and the symbols of operation constitute one very important point of superiority in Algebra.

Of the vast facilities afforded by a symbolical notation, there can exist no question. But as far as the reasoning is concerned, algebraical solutions might be carried on in words at length, and were, in fact, actually so delivered by the earliest writers. The symbolical notation was the gradual result of the necessity for such abbreviations, which was felt in proportion as the operations became more extensive and complex.

Geometrical demonstrations, on the other hand, especially where there are complex figures to be considered, made up of many parts, which are to be added or subtracted, may with material advantage in point of perspicuity, be conducted, by adopting the use of the signs of addition, subtraction, and equality, as employed in Algebra, while the reasoning remains identically the same.

Other writers have put the distinction thus: that in Algebra the reasoning is general, since the symbols are so. In Geometry we select a particular case, on which to prove a general theorem.

But in point of fact, the particular figure, “ the triangle A, B, C," for example, which we take to reason upon, is not a particular case, but merely a name given to the general idea of all triangles constituted under the same conditions, to assist the memory and the conception; the demonstration would be equally valid, only less intelligible, if conducted throughout in general terms, without reference to any diagram, or to any imagined particular figure.

Thus, as to any such distinctions, it is, in fact, evident, that as far as the actual reasoning is concerned, it may be carried on in Geometry without diagrams, and in Algebra without symbols; or in Geometry by symbolical notation, and in Algebra with diagrams (where the subject admitted them), and yet, all the while, the distinction between geometrical and algebraical reasoning might be perfectly preserved.

But the benefits of the algebraic symbols, have by others been considered as counterbalanced by attendant evils; and the absence of them in Geometry, as, in fact, securing it against the difficulties and paradoxes, which, it is alleged, have been introduced into Algebra, from the very comprehensiveness of its symbolical notation.

This, again, is answered by others who will not admit that Geometry has been thus preserved from defects in its chain of reasoning. For if the champions of Geometry have charged upon Algebra the admission of paradoxes and absurdities, and have censured the negative sign and impossible roots, the Algebraists have retorted upon Geometry the mysticism of some of its definitions and axioms, the indirect and circuitous methods of its demonstrations, and the primary deficiency in the theory of parallel lines.

Speaking of reasonings as expressed in particular, or in general terms, Mr. Dugald Stewart observes, “ the former process is analogous to

the practice of Geometers, who, in their most general reasonings, direct the attention to a particular diagram ; the latter to that of the Algebraists, who carry on their investigations by means of symbols *."

He adds, in the course of a note on this passage, “ The straight lines which are employed in the fifth book of Euclid, to represent magnitudes in general, differ from the algebraic expression of these magnitudes, in the same respects in which picture-writing differs from arbitrary characters."

Again, after alluding to the paradoxes into which mathematicians have sometimes been led from want of due caution in the interpretation of algebraic symbols, he remarks, that in Geometry, the use of diagrams effectually prevents the introduction of any of these paradoxes.

Again, let us take the representation of La Place. “ Cependant, les considérations géométriques ne doivent point être abandonnées ; Elles sont de la plus grande utilité dans les arts. D'ailleurs, il est curieux de se figurer dans l'espace les divers résultats et l'analyse ; et réciproquement, de lire toutes les affections des lignes et des surfaces, et toutes les variations du mouvement des corps, dans les équations qui les expriment. Ce rapprochement de la géométrie et de l'analyse répand un nouveau jour sur ces deux sciences ; les opérations intellectuelles de celle-ci, rendues sensible par les images de la première sont plus faciles à saiser, plus intéressantes à suivre.”Syst. du Monde, p. 423.

Another writer, after observing that a demonstration may be strictly geometrical though expressed by algebraical symbols, says, “ Yet when the mind loses the distinct perception of the particular geometrical magnitudes compared, the evidence is similar in its impression to that of algebraical reasoning, in which the previous demonstration of the rules employed is the ground of our assent to the truth of the conclusions, and not the immediate perception of the geometrical magnitudes and their relations."

But the circumstance of the attention being directed to the symbol rather than to the idea for which it stands, would in no way alter the validity of the demonstrations, nor render them less properly geometrical. The sole essential question is, whether with the introduction of algebraic symbols, any processes dependent on assumptions of first principles peculiar to Algebra are introduced ; and then we are, unquestionably, no longer pursuing geometrical proofs in algebraic language, but employing actual Algebra. Still the question remains, What are those principles peculiar lo Algebra ? and are they such as are in any way open to objection, either in themselves or in their application to Geometry?

Philosophy of Human Mind, Vol. I., p. 172.

[To be continued.]



RELATED BY SIR David BREWSTER AND ANOTHER. Iligaly exciting as the “ marvellous” may be to a large proportion of mankind, even in the most advanced state of civilization yet known, it ought never to be drawn from sources whence truth may be obtained by ordinary industry of research, nor furnished by men whose dicta, from their intellectual rank, may be received, without examination or suspicion, not only by the ignorant and unreflecting, but by many who are in the habit generally of requiring proof whenever it is possible to be obtained.

To suborn nature and abuse knowledge, for the vulgar purpose of exciting surprise among the ignorant, can now never acquire more than a very short-lived success, and must, eventually, be productive of great humiliation.

Besides, there is abundantly sufficient among the grand, and even among the minute, operations and productions of nature, to satisfy the most ravenous appetite for the “wonderful" and the “new," without fabricating, or circulating, when fabricated by others, statements at utter variance with all known facts, and ushering them into the world in a manner tending to disturb that confidence in the constant uniformity in the laws of nature, which centuries of investigation have combined to produce, and upon which the philosophic mind reposes with satisfaction and delight.

Without meaning to impute ignoble motives to so eminent a philosopher, and so acute an observer, as Sir David Brewster, it is, at least, surprising to see him expose himself to a charge of this nature, and that, too, in a work whose very intention seemed to be the clearing of the mind's eye, the strengthening of its vision, and the increase, to borrow an astronomical phrase, of its penetrating power, so that it might pierce more thoroughly the mistiness which superstition and knavish cunning, sometimes for base, and frequently for criminal, purposes, envelop some simple, but little known operation of nature, or some refined, but only partially exposed, process of art. Sir David had also an abettor, if not an accomplice in the late Sir Walter Scott, but in him the love of mystification, and the practice of ingenious deception, were so predominant, that we rather wonder he was content to play so second-rate a part in the case which we are about to refer to.

In the “ Letters on Natural Magic, addressed to Sir Walter Scott, Bart., by Sir David Brewster, K.H., LL.D., F.R.S., V.P.R.S.E., &c.," p. 255, &c., is the following passage:

“One of the most remarkable and inexplicable experiments relative to the strength of the human frame, which you have yourself seen and admired, is that in which a heavy man is raised with the greatest facility, when he is lifted up the instant that his own lungs and those of the persons who raise hinı are inflated with air. This experiment was, I believe, first shown in England, a few years ago, by Major H., who saw it performed in a large party at Venice, under the direction of an officer of the American Navy. As Major H. performed it more than once in my presence, I shall describe, as nearly as possible, the method which he prescribed. The heaviest person in the party


lies down upon two chairs, his legs being supported by the one, and his back by the other. Four persons, one at each leg, and one at each shoulder, then try to raise him, and they find his dead weight to be very great, from the difficulty they experience in supporting him. When he is replaced in the chair, each of the four persons takes hold of the body as before, and the person to be lifted gives two signals by clapping his hands. At the first signal, he himself and the four lifters begin to draw a long and full breath, and, when the inhalation is completed, or the lungs filled, the second signal is given for raising the person from the chair. To his own surprise, and that of his bearers, he rises with the greatest facility, as if he were no heavier than a feather. On several occasions, I have observed that when one of the bearers performs his part ill, by making the inhalation out of time, the part of the body which he tries to raise is left as it were behind. As you bave repeatedly seen this experiment, and have performed the part both of the load and of the bearer, you can testify how remarkable the effects appear to all parties, and how complete is the conviction, either that the load has been lightened, or the bearer strengthened, by the prescribed process. “ At Venice, the experiment was performed a much more imposing

The heaviest man in the party was raised and sustained upon the points of the fore-fingers of six persons. Major H. declared that the experiment would not succeed if the person lifted were placed upon a board, and the strength of the individuals applied to the board. He conceived it necessary that the bearers should communicate directly with the body to be raised. I have not had an opportunity of making any experiments relative to these curious facts; but whether the general effect is an illusion, or the result of known or of new principles, the subject merits a careful investigation."

The circumstances under which this narration is given to the public are such, that if the feat, said to have been performed, was not so utterly incredible, they would be amply sufficient to procure for the “wonder,” ready circulation and unhesitating acceptation even among scrupulous observers of truth. Here a Major H. is stated to have performed the experiment successfully in Sir David's presence; and Sir David Brewster, F.R.S., addressing Sir Walter Scott, F.R.S.E., speaks of the feat as one * you have seen and admired,” describes the experiment as one * you repeatedly have seen, and performed the part both of the load and the bearer,” and “can testify how remarkable the effects appear to all parties," &c.

As Sir Walter never in his life-time publicly noticed this appeal nor contradicted the statement, he united his testimony to that of Sir David, and as if even this united evidence could be strengthened, there comes

over sea" another circumstantial account of the same feat, and that from a quarter in which no confederacy could be suspected, unless indeed the American naval officer, who taught it to Major H., at Venice, had carried it across the Atlantic, and up the St. Lawrence. In an American periodical, Silliman's Journal, No. 57, published in April, 1835, there is the following communication to the Editor :

Kingston, Upper Canada, October 31, 1834. “Sir,-As a subscriber to your valuable journal, I take the liberty of asking of some of your scientific readers the rationale of the following experiinent.

“An individual is to place himself on a stool or a table on his back, with his arms and legs crossed, keeping the whole body stiff; four or six others are VOL. I.



then to place themselves at about equal distances, by the sides of the first, say two at the shoulders, two about the middle of the body, and the others by the hips and thighs. Extending the fore-fingers of each hand so as to touch the body, somewhat underneath. At a given signal, the whole party are to take as full an inspiration as possible, and at another given signal, simultaneously to respire very slowly, gently pressing the body upwards at the same time, when it will be found to rise with a very slight effort, and to continue rising until the breath is exhausted, when it will suddenly fall down with great force. The operators must be prepared for this circumstance, and immediately pass their arms under the body to break its fall; it will also be well for one individual to hold a pillow under the head for the same purpose. The experiment appears to succeed best in a closed room, and if the inspirations and respirations are not uniform, it will fail. I first saw it tried about twenty years ago, but have never yet heard or seen any satisfactory explanation of it.

“I am not aware that it involves any principle adverse to the known laws of gravitation, but it certainly appears for a short time to act independently of them. If you deem it (this letter) worthy of a passing notice, I should be glad to see it; if otherwise, let it be deposited in the Archives of the College of Laputa.

“I am, Sir, respectfully yours,

JAMES NICKALLS, Jr." In this account respiration is one of the conditions, and the experiment differs in some other respects from Sir David's, not materially, however; but so far as it does, the feat is rendered still more improbable. This gentleman also states that he“ saw it tried about twenty years ” before, and still, in 1834,“ he had never yet heard or seen any satisfactory explanation of it!"

What shall be believed, then, of this extraordinary fact, so extensively promulgated in the Old Hemisphere, and echoed back from the New? We agree with the editor, in the journal above referred to, that “it is desirable that it should be decided either that the appearance is illusory, or that a reasonable cause should be assigned,” and also with Sir David Brewster, who says,

at the conclusion of the extract given, that “the subject merits a careful investigation."

We have the satisfaction of laying before our readers an investigation and decision, made with a most careful attention to all the circumstances described by Sir David Brewster. We have been permitted by an “Experimental Society," which holds its meetings in London, to have access to that part of their minute-book, in which the introduction and investigation of this very subject, and the final decision of the society, are recorded. In order that the weight due to this investigation and decision may

be properly estimated, we shall state, that though none of the members possess names which are to be compared with the splendour of those of the knight or of the baronet in question, yet some of them have distinguished themselves in the scientific world, and they all have a reputation for veracity, sufficient ability, habits of observation and patient inquiry, quite sufficient to qualify them to form a competent jury to try the question.

Though the members of this society systematically avoid notoriety as a body, the names of the members who assisted in this experiment may, in this particular case, be known, if any person should think it worth while to express the wish for them.

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