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GEOMETRY WITHOUT AXIOMS.
Geometry without Axioms, or the First Book of Euclid's Elements, &c. Fourth Edition. By a Member of the University of Cambridge. London: Heward, 5, Wellington Street, Strand. 1833.
Two thousand years ago, Euclid constructed the mould in which the geometers who succeeded him were to be formed. He adopted the simple plan of proving all he could, and very distinctly stating how much he could not prove: and if he were a prophet, and if our idiom be capable of rendering his ideas, we can imagine him saying to posterity, Try your wits upon this; reason upon the fundamental properties of figure, and take a revolution of the equinoxes, if you please, to settle whether or not it be self-evident, that two straight lines cannot inclose a space; but I will take care that, until you have come to a conclusion, the beginner shall always have satisfactory evidence of geometrical truths; for which, contest it as much as you please, no two of you shall agree upon a substitute.'
His reward has been of a character as peculiar as the merit of his system: medicine is not Galen, nor is history Herodotus; but geometry is Euclid, and many a youth reads six books of the Elements before he happens to be informed that Euclid is not the name of a science, but of a man who wrote upon it.
But to this point the work has not come without much handling scores of commentators have endeavoured to remedy the admitted hiatus which exists in the theory of parallels, each of whom has lived his day as viceroy over Euclid, until he has been pushed from his stool by some other self-elected potentate. Nothing is more singular in the course of this discussion, than the ease with which the reigning, and therefore legitimate commentator, has always been able to destroy the claim of right put forward by his predecessors: and if we come on in our turn, and fit out our little expedition against the powers that be, we can assure our readers it is not with any intention of ruling in their stead.
The improvers of geometry, among whom we place the destroyers of axioms, the squarers of the circle, the trisectors of the angle, &c., may be divided into two classes, in the second and more rational of which we conceive our author must be
placed. The first are ignorant and presumptuous: all they know of geometry is, that there are in it some things which those who have studied it most have long confessed themselves unable to do. Hearing that the authority of knowledge bears too great a sway over the minds of men, they propose to counterbalance it by that of ignorance: and if it should chance that any person acquainted with the subject has better employment than hearing them unfold hidden truths, he is a bigot, a smotherer of the light of truth, and so forth. The members of this class are fond of half-texts only: they think they have all found out things which are hid from the wise and prudent;' but they do not perceive, that neither in humility nor docility, can they fairly be called babes.' The second species contains those who really know what they reason about, and are desirous of employing themselves in giving the thing one more trial, to see whether, with the lights of an older age, the difficulty may either be fairly conquered, or at least demonstrated to be unconquerable. We do not usually find in this class the self-satisfaction, the certainty, the propensity to sneer at others, which characterizes the first; but we must add, that it has become nearly extinct, since the time when most men of science, competent to the attempt, gave it up as hopeless.
The author of the treatise before us, a Cambridge man, evidently well acquainted with geometry, and with the labours of his predecessors on this particular point, (see his Appendix) has made one more attack upon the theory of parallels. Since M. Legendre, no knight of so much prowess has blown his bugle at the gate of this enchanted castle, to try the adventure of the three downright skarts and three cross anes.' The author calls his work an attempt,' expresses himself with perfect propriety as to his own pretensions, and, giving his opinion on the desirableness of getting rid of axioms, in which every one will join him, with the salvo if possible, leaves others to judge of his success. We cannot find, in any part of his treatise, a single positive assertion that he has conquered the difficulty; if therefore we shall show, as we think we can, that there is in his theory an assumed proposition equally difficult with that of Euclid, he may abandon his system, if found to be incorrect, without retracting a word he has written. This is admirable, were it only for its rarity.
When the third edition of this work appeared, we were inclined to examine the part relating to parallels, but were frightened at its length. Twenty-seven closely-printed pages,
*We make it a point to know no more of an author than he is pleased to tell us in his title-page; in another sense, no uncommon practice of reviewers.
with diagrams of unusual complexity, were enough even to scare a reviewer; at least one who was disposed to follow the somewhat obsolete custom of reading his author. We were also deterred by observing that a contemporary, who wrote a laudatory review, appeared to have foundered in the ocean of truth on which we were casting a dubious eye: for, instead of giving any precise information upon this very 'pinch and nip,' (to use our author's words) of the book, he contented himself with observing that, though long, it was not on that account to be rejected, if correct; and helped himself, as well as he could, with some witty invectives against 'royalists' in geometry. So we resolved to let the ocean alone, and to remain contented with our pebbles for the present. When, however, the fourth edition appeared, in which it was announced that the part relating to parallels had been reduced one-half, we left our moorings, which were in a snug position behind the breakwater of Euclid's eleventh axiom, and set sail, resolved to hold on our course so long as it appeared safe. In the second proposition of the series we saw breakers ahead, warning us of the hidden rocks of tacit assumption, whereupon we put about immediately, and here we are, once more on the right side of the breakwater, resolved not to go to sea again in a hurry.
The principles on which the author appears to found his aversion to axioms shall be stated in his own words.
In arguments on the general affairs of life, the place where every man is most to be suspected is in what he starts from as "what nobody can deny." It was, therefore, of evil example, that science of any kind should be supposed to be founded on axioms; and it is no answer to say, that in a particular case they were true.'-Preface, page 8.
Again, speaking of an assumed theorem, the author observes (page 147) as follows, where we have kept the words as nearly as the absence of the context will permit.
'For the sake of removing the argument from vulgar experience, suppose some very great dimension, as for instance equal to the radius of the earth's orbit. If an astronomer should arise, and declare he had found astronomical evidence that this was true,' meaning something contrary to the consequences of axioms derived from immediate perceptions, how would the supporters of the analytical proof,' or any other founded on axioms, the spirit of the passage allows us to add, 'proceed to put him down?'
Again, page 5:
'A cooper knows that, in every instance where he has tried it, the distance that went exactly round the rim of his cask at six times, was the distance to be taken in his compasses in order to describe the head that would fit. But he does not know the reasons
why this will necessarily be the case, not only in the instances which he has tried, but in all which he has not tried also.'
In the first of these extracts, we think our author not so correct in his views as usual. Does he not observe that a politician, or a moralist, or a theologian, generally has his which nobody can deny,' immediately falsified by his opponent? Did he ever hear which nobody does deny' in the same circumstances? If so, has it not always been nobody of any sense,' ' nobody who knows the subject,' or some such little hit at the adversary? That nobody can deny it' is, as times go, the very essence of a controvertible position: that nobody does deny it,' that of an axiom of Euclid.
We agree to the second and third extract, and by them, and his own general principles of reasoning, which are always sound, we proceed to try our author's theory of parallels. No proposition is either to be admitted or denied (page 6) without proof: no evidence is to drawn from our little six-foot perceptions of material objects: it may be that the angles of a triangle are equal to one right-angle and a half, instead of two; there may be an infinite number of parallels to a given line all drawn through one point; nay, if he pleases, the greatest possible equilateral triangle may possibly be that which has the distance between our sun and Aldebaran for its base: and it may possibly be that, though other triangles can be described larger, some freak of nature hinders the equilateral triangle from extending its dimensions. On none of these points will we insist or deny without proof; or any others which can be named: it is but fair, therefore, that among the propositions which are neither asserted nor denied, we should be allowed to place the following.
Let BA and AC be equal lines, making an angle BAC less than two right angles, and at the points Cand B make the angles ABD ACE respectively equal to BAC, and cut off BD and CE respectively equal to AB or AC: then the points D and E will always coincide, whatever may be the angle BAC. This proposition, according to our author's principles, must not be either asserted or denied without proof. We contend that the
'pinch and nip' of the paralogism into which he has fallen, lies in supposing, without proof, that there are or may be supposed to be, cases in which the above is not true. It is certain that our proposition contradicts the senses wofully; but we have agreed to throw them aside, or at least to say we know not what may take place in rectilinear figures as big as the universe. We now proceed to verify our assertion from the author's work: if we have bantered him a little, he will not object, for next to clear general views of logic and perspicuity of language, the art of playing off his predecessors is his own particular forte.
Our author's first proposition (xxviii. A), in which he departs from Euclid, is the following, the letters being altered to suit our figure: If (see the preceding figure) from the ends of a line AB, equal lines BD and AC are drawn, making equal angles ABD, BAC, with AB, and each less than two right angles; then, he says, if the equal lines BD and AC do not meet, a four-sided figure (which he calls a tessera) has been described, of which the angles at D and C are also less than two right angles. This, we admit, he proves. The next proposition requires the construction of the following figure. Let AB and AC be equal lines, making an angle CAB less than two right angles; at the points B and C, make the angles
ABD, ACE, respectively equal to CAB, and BD and CE equal to AC or AB; at D and E repeat the process, and so on. Our author then reasons as follows: take the equal angles ACB, ABC, from the equal angles ACE, ABD, and there remain the angles ECB, CBD, which are therefore equal. Also the sides BD, CE are equal to one another, wherefore DBCE is a tessera.' In proof of the assertion in italics, the author cites the preceding proposition, in which we find the limitation, provided the equal sides do not meet. Here then is our point: if D and E are supposed not to coincide, the proposition about which we insisted on remaining neuter, is contradicted, in one case at least, that is, is declared not to be always true. To make out his enunciation, the author needs the following axiom:-If the sides EC, CA, AB, and BD are equal, and ECA, CAB, and ABD are equal angles, each less than two right angles, then E and D do not always coincide for every value of the angle BAC. Now, of this, where is the proof? The axiom of Euclid, put in its simplest form, is as