No. II. ON THE THEORY OF PARALLEL LINES. any one of The theory of parallel lines has always been considered as the , reproach of Geometry. The beautiful chain of reasoning by which the truths of this science are connected here wants a link, and we are reluctantly compelled to assume as an axiom what ought to be matter of demonstration. The most eminent geometers, ancient and modern, have attempted without success to remove this defect; and after the labours of the learned for 2000 years have failed to improve or supersede it, Euclid's theory of parallels maintains its superiority. We shall here endeavour to explain the nature of the difficulty which attends this investigation, and shall give some account of the theories which have been proposed as improvements on, or substitutes for, that of Euclid. Of the properties by which two right lines described upon the same plane are related, there are several which characterise two parallel lines and distinguish them from lines which intersect. If such properties be assumed as the definition of parallel lines, all the others should flow demonstratively from it. As far therefore as the strict principles of logic are concerned, it is a matter of indifference which of the properties be taken as the definition. In the choice of a definition, however, we should be directed also by other circumstances. That property is obviously to be preferred from which all the others follow with greatest ease and clearness. It is also manifest that, cæteris paribus, that property should be selected which is most conformable to the commonly received notion of the thing defined. These circumstances should be attended to in every definition, and the exertion of considerable skill is necessary almost in every case. But in the selection of a definition for parallel lines there is a difficulty of another kind. It has been found, that whatever property of parallels be selected as the basis of their definition, the deduction of all the other properties from it was impracticable. Under these circumstances the only expedient which presents itself, is to assume, besides the property selected for the definition, another property as an axiom. This is what every mathematician who has attempted to institute a theory of parallel lines has done. Some, it is true, have professed to dis. pense with an axiom, and to derive all the properties directly from their definition. But these, with a single exception, which we shall mention hereafter, have fallen into an illogicism inexcusable in geometers. We find invariably a petitio principii, either incorporated in their definition, or lurking in some complicated demonstration. A rigorous dissection of the reasoning never fails to lay bare the sophism. Of the 'pretensions of those who avowedly assume an axiom it is easy to judge. When Euclid's axiom is removed from the very disadvantageous position which it has hitherto maintained, put in its natural place, and the terms in which it is expressed somewhat changed, I think it will be acknowledged that no proposition which has ever yet been offered as a substitute for it is so nearly self-evident. But it is not alone in the degree of self-evidence of his axiom, if we be permitted the phrase, that Euclid's theory of parallels is superior to those theories which are founded on other axioms. The superior simplicity of the structure which he has raised upon it is still more conspicuous. When you have once admitted Euclid's axiom, all his theorems flow from that and his definition, as the most simple and obvious inferences. In other theories, after conceding an axiom much further removed from self-evidence than Euclid's, a labyrinth of complicated and indirect demonstration remains to be threaded, requiring much subtlety and attention to be assured that error and fallacy do not lurk in its mazes. Euclid selects for his definition that property in virtue of which parallel lines, though indefinitely produced, can never intersect. This is, perhaps, the most ordinary idea of parallelism. Almost every other property of parallels requires some consideration before an uninstructed mind assents to it; but the possibility of two right lines intersecting, is repugnant to every.notion of parallelism. When the possible existence of the subject of a definition is not selfevident, or presumed and declared to be so, it ought to be proved so. This is the case with Euclid's definition of parallels. How, it may be asked, does it appear that two right lines can be drawn upon the same plane so as never to intersect though infinitely produced ? Euclid meets this objection in his 27th proposition, where he shows that if two lines be inclined at equal alternate angles to a third, the supposed possibility of their intersection would lead to a manifest contradiction. Thus it appears, that through a given point one right line at least may always be drawn parallel to a given right line. But it still remains to be shown, that no more than one parallel can be drawn through the same point to the same right line. And here the chain of proof is broken. Euclid was unable to demonstrate, that every other line except that which makes the alternate angles equal will necessarily intersect the given right line if both be sufficiently produced. He accordingly found himself compelled to place the deficient link among his axioms. I have always conceived that the objections which have been urged against the twelfth axiom have arisen from the place assigned to it in the Elements. Standing as it does on the threshhold of the first book, it is contemplated without reference to those propositions with which it is or ought to be immediately connected, which prepare the mind of the student for its reception, and which seem almost indispensable to render the very terms in which it is expressed intelligible. This axiom, as we have stated, is only assumed in the demonstration of the 29th proposition. In the 27th it is proved that if BEF (see fig. Prop. XXVII. Book I.) and D E F be together equal to two right angles E, B is parallel to FD. The axiom declares that no other right line through E can be parallel to FD; for it is plain, that any other line must make with E F an angle which together with DFE is less or greater than two right angles. The twelfth axiom may therefore be expressed in any of the following ways: • Two diverging right lines cannot be both parallel to the same right line.' • If a right line intersect one of two parallel right lines it must also intersect the other.' Only one right line can be drawn through a given point parallel to a given right line.' The axiom expressed by the first of these ways appears to me to be as unobjectionable, as any of those which have been received without dispute. Playfair, in his edition of Euclid, and before him Ludlam, expresses the axiom nearly in this way, but he does not seem to be aware that it is the same axiom as that of Euclid; for he says in his preface that a new axiom is introduced in place of the 12th. In fact, it is the same axiom otherwise expressed. Proclus objects to Euclid's axiom, that it is less entitled to be considered self-evident, because the converse of it (XXVIII. Book I.) confessedly requires proof. Are we hence to infer that Proclus considered no proposition to be self-evident, unless its converse be also self-evident? If this were admitted, I fear it would be fatal to some of the axioms which Proclus himself, in common with the rest of the world, have received. Neither the second nor third of Euclid's axioms respecting equal magnitudes could by this rule be admitted as such; for, so far from their converses being selfevident, they are not even true. Proclus would seem to have attributed to an axiom the quality of a definition, which must always be” what logicians call a reciprocal proposition ; but axioms are universal affirmative propositions not necessarily reciprocal, and therefore notoriously not logically convertible. I shall not attempt to go into the particular details of the various theories of parallels which have been proposed, nor even to enumerate them. I shall, however, mention some of those which, from the eminence of their authors, if from nothing else, must command attention. Clavius rejects Euclid's axiom, and proposes the following as a substitute for it. A line drawn upon a plane from every point of which perpendiculars on a right line in the same plane are equal, is itself a right iine.' From this proposition assumed as an axiom by a most complex and embarrassing process, he shows that the properties of parallels may be deduced. Wolfius, Boscovich, and Thomas Simson, change the definition of parallels and substitute for it the following: Two right lines are parallel when perpendiculars from every point in one upon the other are equal.' This definition is sophistical, and really assumes the axiom of Clavius. D'Alembert, with more acuteness, proposes to define parallels to be right lines, one of which has two points equally distant from the other.' But the difficulty of demonstrating that the other points are equally distant, he acknowledges still to remain undiminished. Thomas Simson, in his second edition, proposes the following axiom : If perpendiculars from two points of a straight line upon another straight line in the same plane be unequal, the two lines will meet if indefinitely produced on the side of the least distance.' Robert Simson proposes the following axiom : “ That a straight line cannot first come nearer to another straight line, and then go further from it, without meeting it.' The meaning of which is, that if from any three points in one right line perpendiculars be drawn to another, they will either be equal, or the intermediate perpendicular will be greater than one of the extreme perpendiculars and less than the other. Varignon, Bezout, and others, propose the following definition as a substitute for Euclid's : Two right lines in the same plane are parallel when they are equally inclined in the same direction to a third right line.' By this either of two things is meant; that the parallels are equally inclined to one particular line intersecting them, or that they are equally inclined to every line intersecting them. If it be taken in the former sense it is insufficient, and if in the latter it is sophistical. If they be equally inclined to one particular line, it remains to be proved that they are also equally inclined to every other intersecting line, which in fact is, and always has been, the whole difficulty of the question. If it be meant that they are equally inclined to every intersecting line, it is a sophism, in which a theorem is presented in the garb of a definition. Professor Leslie retains Euclid's definition in substance, though somewhat changed in expression. By a singular oversight, he has appended to the enunciation of Euclid's 29th proposition, demonstrations of his 27th and 28th, thus leaving the 29th without a proof. From these specimens the success of the various attempts to mend Euclid's theory of parallels may be estimated. We must, however, make honourable exception of Legendre. His system, considered as a part of the inost elementary mathematical treatise, is certainly liable to objection ; but the objection is of a very different character from those which lie against all the others. Legendre assumes no axiom, makes no change in Euclid's definition, and admits no latent assumption or other fallacy in his reasoning. After his first theorem, which is the 32d proposition of Euclid, his demonstrations are as simple as those of Euclid, and even attended with superior advantages. To comprehend his first demonstration, however, requires a greater familiarity with the language and principles of analysis than students commencing the elements of geometry generally possess. It is to be regretted that a system must therefore be abandoned which is in other respects incomparably the best, and, indeed, the only one which is altogether free from objections on the score of validity. Legendre has given several demonstrations of the 32d proposition ; they are, however, all subject more or less to the objection just mentioned. We shall subjoin that which is most remarkable for its brevity, its elegance, and the importance and extent of its 'applications. By superposition, it can be shown immediately, and without any preliminary propositions, that two triangles are equal when they have two angles and an interjacent side in each equal. Let us call this side p, the two adjacent angles A and B, the third angle C. This third angle C, therefore, is entirely determined, when the angles A and B, with the side p, are known; for if several different angles C might correspond to the three given magnitudes A, B, p, there would be several different triangles, each having two angles and the interjacent side equal, which is impossible; hence the angle C must be a determinate function of the three quantities A, B, p, which we shall express thus, C = 0: (A, B, p). Let the right angle be equal to unity, then the angles A, B, C will be numbers included between 0 and 2 ; and since C=0: (A, B, p), the line p cannot enter into the function d. For we have already seen that C must be entirely determined by the given quantities A, B, p alone, without any other line or angle whatever. But the line p is heterogeneous with the numbers A, B, C; and if there existed any equation between A, B, C, p, the value of p might be found from it in terms of A, B, C; whence it would follow, that p is equal to a number; which is absurd : hence p cannot enter into the function , and we have simply C= 0:(A, B).* This formula already proves, that if two angles of one triangle are equal to two angles of another, the third angle of the former must also be equal to the third of the latter ; and this granted, it is easy to arrive at the theorem we have in view. First, let A B C be a triangle right-angled at A ; from the point A draw A D perpendicular to the hypotenuse. The angles B and D of the triangle A B D are equal to the angles B and A of the triangle BAC; hence, from what has just been proved, the third angle B A D is equal to the third C. For a like reason, the angle D AC = B, hence BAD + DAC, or BAC B+C; but the angle B A C is right ; hence the two acute angles of a right-angled triangle are together equal to a right angle. Now, let B A C be any triangle, and B C a side of it not less than either of the other sides ; if from the opposite angle A the perpendicular A D is let fall on B C, this perpendicular will fall within the triangle A B C, and divide it into two right-angled triangles B AD, DAC. But in the right-angled triangle B AD, the two angles BAD, B D * Against this demonstration it has been objected, that if it were applied word for word to spherical triangles, we should find that two angles being known, are sufficient to determine the third, which is not the case in that species of triangles. The answer is, that in spherical triangles there exists one element more than in plane triangles, the radius of the sphere, namely, which must not be omitted in our reasoning. Let r be the radius; instead of C= Q(A, B, P), we shall now have C = Q(A, B, p, r), or by the law of homogeneity, simply C=0 (A, B, ?). But since the ratio P. is a number, as well as A, B, C, there is nothing to hinder from entering the function , and, consequently, we have no right to infer from it, that C= (A, B). |