angle GHD; and they are alternate angles; wherefore A B is parallel (I. 27) to CD.

Again, because the two angles BGH and GHD are together equal (Hyp.) to two right angles; and the two angles AG H and BG H are also together equal (I. 13) to two right angles; therefore the two angles A G H and B G H are equal (Ax. 1) to the two angles B G H and G HD. Take away from these equals the common angle BGH, and the remaining angle A G H is equal (4x. 3) to the remaining angle GHD; but they are alternate angles; therefore A B is parallel (I. 27) to CD. Wherefore, if a straight line, &c. Q. E. D. Scholium 1. The twelfth axiom will now be admitted by the student as a corollary to this proposition; especially when Prop. XVII. and the note added to the twelfth axiom are taken into Scholium. 2: We think it right to introduce our students at this point, to a discussion on the "Theory of parallel straight lines,' which will be of immense advantage to them in their future studies. Our first extract shall be from the Gower-street edition of Euclid.

account.

"

"The theory of parallel [straight] 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 2,000 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 [straight] lines described upon the same plane are related, there are several which characterise two parallel [straight] lines and distinguish them from [straight] lines which intersect. If any one of such properties be assumed as the definition of parallel [straight] 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 the 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 [straight] 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 is 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 [straight] lines has done. Some, it is true, have professed to dispense 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 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 [straight] 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 such [straight] lines intersecting is repugnant to every notion of parallelism. self-evident, or presumed and declared to be so, it ought to be "When the possible existence of the subject of a definition is not proved so. This is the case with Euclid's definition of parallels. How, it may be asked, does it appear that two right [straight] 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 [straight] 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 [straight] line at least may always be drawn parallel to a given right [straight] 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 [straight]line. And here the chain of proof is broken. Euclid was unable to demonstrate, that every other [straight] line except intersect the given right [straight] line if both be sufficiently that which makes the alternate angles equal will necessarily produced. He accordingly found himself compelled to place the deficient link among his axioms."

We now add to this extract, notices of thirty different methods, proposed at various epochs in the history of Geometry, for getting over the difficulty of the Twelfth Axiom of Euclid's First Book. This collection is taken from Col. P. Thompson's "Geometry without Axioms," pp. 137–156.

"The uses of such a Collection are to throw light on the particulars which experience has shown are not to be left unguarded in any attempt at solution, and to prevent explorers from consuming their time in exhausted tracts. To which may be added, that out of so many efforts, some, either by improvement or by a fortunate conjunction with others, may finally be found operative towards the solution desired.

1. The objection to Euclid's Axiom (independently of the objections common to all Axioms), is that there is no more reason why it should be taken for granted without proof, than numerous other propositions which are the subjects of formal demonstration, and the taking any one of which for granted would equally lead to the establishment of the matter in dispute.

2. Ptolemy the astronomer, who wrote a treatise on Parallel Lines, of which extracts are preserved by Proclus, proposed to prove that if a straight line cuts two parallel straight lines, the two interior angles on each side are together equal to two right angles, by saying that if the interior angles on the one side are greater than two right angles, then because the lines on one side are no more parallel than those on the other, the two interior angles on the other side must likewise be together greater than two right angles, and the whole greater than four, which is impossible; and in the same way if they were supposed less. In which the palpable weakness is, that there is no proof, evidence, or cause of probability assigned, why parallelism should be connected with the angles on one side being together equal to those on the other; the very question in debate being, whether they may not be a little more than two right angles on one side and a little less on the other, and still the straight lines not meet.

3. Proclus himself proposes" to take an Axiom of this sort, being the same that Aristotle employed to establish that the world is finite. If from the same point, two straight lines are drawn making an angle, the distance between them when they are prolonged to infinity will exceed any finite distance that may be assigned. He then showed that if the straight lines prolonged from this centre towards the circumference are of infinite length, what is between them is also infinite; for if it was finite, to increase the distance would be impossible, and consequently the straight lines would not be infinite. The straight lines therefore on being prolonged to infinity, will separate from each other by more than any finite quantity assigned. But if this be previously admitted, I affirm that if any straight line cuts one of two parallel straight lines, it will cut the other also. For let A B and C D be parallel, and let E F cut A B in G; I say that it will cut o D also. For since

Ο οὐδέν · γαρ μᾶλλον αἱ αξ γη παράλληλοι ἢ αἱ ζδ ηβ.—Procli Comment. in Primum Euclidis Librum. Lib. 4. It is but right to notice, that Proclus calls this rapaλoyioμòs and deifewor ao0évela; and Barocius the Venetian Translator in 1560, notes it in the margin as Flagitiosa Ptolemai ratiocinatio.

Professor Playfair says it is curious to observe in Proclus's account an argument founded on the principle known to the moderns by the name of the sufficient reason (Elem. of Geom. p. 405). If the allusion is to this part, the "sufficient reason" of the moderns must be something very feeble.

from the point G are drawn two straight lines & B, GF, and prolonged to an infinite length, the distance between them will become greater than any assigned magnitude, and consequently than that which may be the distance between the parallels; when, therefore. they are distant from each other by more than this, GF will cut CD." Without disputing that the distance between the straight lines which make the angle will become greater than any assigned magnitude (though the reason given appears to be founded on ignorance of the fact that a magnitude may perpetually increase and still be always less than an assigned magnitude),-the defect is in begging the question, that the distance between the parallels is constant or at all events finite. For the very point in dispute is, whether the parallels (as for instance two perpendiculars to a common straight line, both of them prolonged both ways) may not open out or grow more distant as they are prolonged, and to do this so rapidly, that a straight line making some very small angle with one of them, shall never overtake the other, but chase it unsuccessfully through infinite space, after the manner of a line and its asymptote.

4. Clavius announces that "a line every point in which is equally distant from a straight line in the same plane, is a straight line;" upon taking which for granted, he finds himself able to infer the properties of Parallel Lines. And he supports it on the ground that because the acknowledged straight line is one which lies evenly [ex æquo] between its extreme points, the other line must do the same, or it would be impossible that it should be everywhere! equidistant from the first. Which is only settling one unknown by a reference to another unknown.

5 and 6. In a tract printed in 1604 by Dr. Thomas Oliver, of Bury, entitled, De rectarum linearum parallelismo et concursu doctrina Geometrica (Mus. Brit.), two demonstrations are proposed; both of them depending on taking for granted, that if a perpendicular of fixed length moves along a straight line, its extremity describes a straight line. Which is Clavius's axiom a little altered.

7. Wolfius, Boscovich, Thomas Simpson in the first edition of his" Elements," and Bonnycastle, alter the definitions of parallels, and substitute in substance, "that straight lines are parallel which preserve always the same distance from one another;" by distance being understood the length of the perpendicular drawn from a point in one of the straight lines to the other. Attempts to get rid of a difficulty by throwing it into the definition, are always to be suspected of introducing a theorem in disguise; and in the present instances, it is only the introduction of the proposition of Clavius. No proof is adduced that straight lines in any assignable position, will always preserve the same distance from one another; or that if a perpendicular of fixed length travels along a straight line keeping always at right angles to it, what mathematicians call the locus of the distant extremity is necessarily a straight line at

all.

8. D'Alembert proposed to define parallels as being straight lines "one of which has two of its points equally distant from the other line;" but acknowledged the difficulty of proving, that all the other points would be equally distant in consequence

9. Thomas Simpson, in the second edition of his "Elements," proposed that the Axiom should be, that "If two points in a straight line are posited at unequal distances from another straight line in the same plane, those two lines being indefinitely produced on the side of the least distance will meet one another."

10. Robert Simpson proposes that the Axiom should be, "that a straight line cannot first come nearer to another straight line, and then go further from it, before it cuts it.". By coming nearer or

* We omit the Greek.

"Nam si omnia puncta line AB,æqualiter distant à rectâ Do, ex quo sua interjacebit puncto, hoc est, nullum in eâ punctum intermedium ab extremis sursum, aut deorsum, vel huc, atque illuc deflectendo subsultabit, nihilque in eâ flexuosum reperietur, sed æquabiliter semper inter sua puncta extendetur, quemadmodum recta DC. Alioquin non omnia ejus puncta sequalem à rectâ D D, distantiam haberent, quod est contra hypothesin. Neque verò cogitatione apprehendi potest aliam lineam præter rectam, posse habere omnia sua puncta à rectâ lineâ, quæ in eodem cum illâ plano existat, @qualiter distantia.'-Clavii Opera. In Euclidis Lib. I. p. 50. "la vraie définition, ce me semble, et la plus nette qu'on puisse donner d'une parallèle, est de dire que c'est une ligne qui a deux de ses points également éloignés d'une autre ligne.-il faut ensuite démontrer (et c'est-là le plus difficile), que tous les autres points de cette seconde, seront également éloignées de la ligne droite, donnée."-ENCLYCLOPEDIE. Art. Parallèle.

This and most of what has preceded, is in the Arabic. In a manuscript copy of Euclid in Arabic but in a Persian hand, bought at Ahmedabad in 1817, the editor on the introduction of Euclid's Axiom comments as follows. "And this is what is said in the text. I maintain that the last proposition is not of the universally-acknowledged truths, nor of anything that is de monstrated in any other part of the science of geometry. The best way therefore would be, that if it should be put among the questions instead of the principles; and I shall demonstrate it in a suitable place. And I lay down for this purpose another proposition, which is, that straight lines in the same

going from it, being understood the diminution or increase of the perpendicular from one to the other.

The objection to all these is, that no information has been given on the subject of the things termed straight lines, which points to any reason why the distance's growing smaller should be necessarily followed by the meeting of the lines. It may be true; but the reason why, is not upon the record. On the contrary, it is well known that there exist lines (as for instance the neighbouring sides of two conjugate hyperbolas) where the distance perpetually decreases and yet the lines never meet. It is open therefore to ask, what property of the lines called straight has been promulgated, which proves they may not do the like.

11. Varignon, Bezout, and others propose to define parallels to be "straight lines which are equally inclined to a third straight line," or in other words, make the exterior angle equal to the interior and opposite on the same side of the line. By which they either intend to take for granted the principal fact at issue, which is whether no straight lines but those that make such angles can fail to meet; or if their project is to admit none to be parallel lines of which it shall not be predicated that they make equal angles as above with some one straight line either expressed or understood, then they intend to take for granted that because they make equal angles with one straight line, they shall also do it with any other that shall in any way be drawn across them,—a thing utterly unestablished by any previous proof.

12. Professor Playfair proposes to employ as an Axiom, that "two straight lines, which cut one another, cannot be both parallel to the same straight line" in which he had been preceded by Ludlam and others, and which he says "is a proposition readily enough admitted as self-evident." The misfortune of which is, that instead of being self-evident, a man cannot see it if he tries. What he sees is, that he does not see it. He sees that a straight line's making certain angles with one of the parallels, causes it to meet the other; and he sees that by increasing the distance of the point of meeting, he can cause the angle with the first parallel to grow less and less. But if he feels a curiosity to know whether he can go on thus reducing the angle till he makes it less than any magnitude that shall have been assigned (or in other words whether there may not possibly be some angle so small that a straight line drawn to any point however remote in the other parallel shall fail to make so small a one), he discovers that this is the very thing nature has denied to his sight; an odd thing, certainly, to call self-evident.

13. The same objections appear to lie against Professor Leslie's proposed demonstration in p. 44 of his "Rudiments of Plane Geometry;" which consists in supposing a straight line of unlimited length both ways, to turn about a point situate in one of the parallels, which straight line, it is argued, will attain a certain position in which it does not meet the other straight line either way, while the slightest deviation from that precise direction woul occasion a meeting.

14. Professor Playfair, in the Notes to his "Elements of Geometry," p. 409, has proposed another demonstration, founded on a remarkable non causa pro causa. It purports to collect the fact* that (on the sides being prolonged consecutively) the intercepted or exterior angles of a rectilinear triangle are together equal to four right angles, from the circumstance that a straight line carried round the perimeter of a triangle by being applied to all the sides in succession, is brought into its old situation again; the argument being, that because this line has made the sort of somerset it would do by being turned through four right angles about a fixed point, the exterior angles of the triangle have necessarily been equal to four right angles. The answer to which is, that there is no connexion between the things at all, and that the result will equal to four right angles, Take, for example, the plane triangle just as much take place where the exterior angles are avowedly not formed by three small arcs of the same or equal circles, as in the figure; and it is manifest that an arc of this circle may be carried round in the way described and return to its old situation, and yet there be no pretence for inferring that the exterior angles were equal to four right angles. And if it is urged that these are curved lines and the statement made was of straight; then the answer is by demanding to know, what property of straight lines has been laid down or established, which determines that what is not true in the case of other lines is true in theirs. It has been shown that, as

plane, if they are subject to an increase of distance on one side, will not be subject to a diminution of distance on that same side, and the contrary; but will cut one another. And in the demonstration of this I shall employ another proposition, which Euclid has employed in the Tenth Book and elsewhere, which is, that of any two finite magnitudes of the same kind, the smallest by being doubled over and over will become greater than the greatest. And it will further require to be laid down, that one straight line cannot be in the same straight line with straight lines more than one that do not coincide with one another; and that the angle which is equal to a right angle, is a right angle," We omit the Arabic.

I. 32. Cor. 2.