CHAPTER XIII.

THE PARALLEL POSTULATE.
In the last chapter I considered three classical problems. Another geometrical question, perhaps of greater interest, is concerned with whether the sum of the angles of a plane triangle is exactly equal to two right angles. This is a propo- sition which in ordinary textbooks on elementary geometry is enunciated — and properly so — as if it were undoubtedly true. In one sense this theorem, like the problems discussed in the last chapter, or like the algebraic solution of the general equation of the fifth degree, is insoluble; but the efforts to prove it aftbrd materials for an interesting chapter in the history of mathematics, since many of the demonstrations formerly proposed are fallacious. The fact however that in the reasoning there are pitfalls, logical as well as mathe- matical, adds to the interest of the discussion, and the treacherous nature of the path makes its safe passage the more interesting*.
* The subject has been discussed by numerous writers. A good account of it to the end of the 18th century is given in the notes to J. Playfair's Elements of Geometry, Edinburgh, Ist edition, 1813, and in the Appendix to Geometry without Axioms by T. P. Thompson, London, 4th edition, 1833. For other and more recent researches, see H. Schotten, Flanim^trischen Unterrichts, vol. ii, Leipzig, 1893 ; F. Engel and P. Stackel, Die Theorie der Parallellinien, Leipzig, 1895, 1899; J. Richard, La Philosophie des Mathematiques, Paris, 1903; J. W. Withers, Euclid's Parallel Postulate, Chicago, 1905; M. Simon, Veher die Entwicklang der Elementar-geometrie, Leipzig, 1906. Some of the solutions offered have no interest, and are evidently fallacious. Hence I make no attempt to treat the subject exhaustively, but I mention the more plausible eli'orts.
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Earliest Proof. Thales. We know from Geminus that this proposition was one of the first general results discovered by the Greeks*. From the extant notices the following has been suggested, with considerable probability, as indicating the manner in which it was proved ; at any rate this demonstration involves nothing with which Thales, the traditional founder of the science of abstract geometry, was not acquainted, and it has been conjectured that it is in fact due to him. According to this view, it was, in the first place, stated (or more likely assumed) that in a rectangle the angles were right angles and the opposite sides equal. Hence the sum of the four angles is equal to four right angles. Next, by drawing a diagonal of a rectangle, it will be seen that any right-angled triangle can be placed in juxtaposition with an equal and similar triangle in such a way as to make up a rectangle: this step in the argument may have been suggested by the tiles used in paving floors. Hence the sum of the angles of a right-angled triangle is equal to two right angles. Lastly, any triangle ABC can be divided into two right-angled triangles by drawing a per- pendicular AD from the biggest angle A to the opposite side BG. The sum of the angles of the triangle ABB is equal to two right angles. Hence the sum of the angles B and BAD is equal to one right angle. Similarly the sum of the angles C and CAD is equal to one right angle. Hence the sum of the angles B, G, BAD, and CAD, that is, the sum of the angles of the original triangle ABC, is equal to two right angles.
The only criticism I would make on this proof is that it rests frankly on the assumption that we can construct a rect- angle, and that the opposite sides of the rectangle are equal. This, unless we refer to direct observation or measurement, involves an assumption about parallel lines, which is equivalent to that made in Euclid's postulate.
Pascal's Proof. Another proof, also resting immediately on experiment, to which I may here refer, was discovered by Pascal in the seventeenth century. It is interesting
* G. J. Allman, Greek Geometry Jrom Thales to Euclid, Dublin, 1889, chap. i.
CH. XIIl] THE PARALLEL POSTULATE 809
from its history. Pascal was a delicate and precocious boy, and in order to ensure his not being over-worked his father directed that his education should at first be only linguistic and literary, and should not include any mathematics. Natur- ally this excited the boy's curiosity, and one day, being about twelve years old, he asked in what geometry consisted. His tutor replied that it was the science of constructing exact figures and determining the relations between their parts. Pascal, stimulated no doubt by the injunction against reading it, gave up his playtime to the new amusement, and in a few weeks had discovered for himself several properties of recti- linear figures, and in particular the proposition in question.
His proof is said* to have consisted in taking a triangular piece of paper and turning over the angular points to meet at the foot of the perpendicular drawn from the biggest angle to the opposite side. The conclusion is obvious from a figure, for
if the paper be creased so that A is turned over to D, as also B and C, we get B = FDB, A=FDE, and C = EDC\ hence A + B^-C = EDF -h FDB + EDG = it. But we can only prove these relations on the assumption that when the paper is folded over BF and ^F will lie along DF, and thus that BF= FA = FB, and similarly that CE = EA = ED ; this assumption involves properties of parallel lines. A similar proof can be obtained by turning over the angular points to meet at the centre of the inscribed circle, and according to some accounts this was the method used by Pascal. I may add in passing that his father, struck by this evidence of Pascal's geometrical ability,
* I believe that this rests merely on tradition.
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gave him a copy of Euclid's Elements, and allowed him to take up the subject for which evidently he had a natural aptitude.
Pythagorean and Euclidean Proof. Leaving the above demonstrations which rest on observation and experiment, I proceed to the classical proof given by Euclid*. This was taken from the Pythagoreans t, and is generally attributed to Pythagoras himself The proof rests on properties of parallel lines, and the Pythagoreans must have prefaced it by some statement of those properties, but there is now no record as to how they treated parallels.
Euclid's treatment of parallels is well known. There is no doubt that he put, at the beginning of his Geometry, certain definitions, axioms, and postulates ; but in the earliest manu- scripts, according to Pe3Tard, the assumption about parallels was not stated there, but was placed in the demonstration of his proposition 29 as a fact conformable to experience, which had to be assumed for the validity of the argument. If this be so, this exceptional treatment seems to indicate that, in Euclid's opinion, the assumption was of a different character to the other postulates, and the difficulty was faced frankly without any attempt to conceal it under a vague phraseology. Unluckily the postulate is often printed in modern school books as an axiom or a self-evident statement. This misplacement may have been due in the first instance to Theon of Alexandria who, about 370 a.d., lectured on Euclid's Geometry. Our modern texts of Euclid are mainly based on Theon 's lectures, and it is only comparatively recently that the commentaries on Euclid's teaching have been subjected to critical discussion.
At any rate Euclid, either at the commencement of his work or more likely in the course of his demonstration, boldly assumed that if a straight line meets two other straight lines so as to make the sum of the two interior angles on one side of it less than two right angles, then these straight lines if continually produced will meet upon that side on which these
* Euclid's Elements, book i, prop. 32.
t Eudemus is our authority for this ; see Proclus, ed. G. Friedlein, Leipzig, 1873, p. 379.
CH. XIIl] THE PARALLEL POSTULATE 311
angles are situated. Accepting this or some similar assumption, the demonstration is rigorous, and was given by him as follows.
Take any triangle ABC. Produce the side BA to any distance AH, and through A draw a line AK parallel to BC. On the assumption that his postulate is true, Euclid showed (Euc. I. 29) that the angle ABC must be equal to the angle HAK, and the angle ACB to the angle KAC. Hence the sum of the three angles of the triangle ABC must be equal to the sum of the angles HAK, KAC, and CAB, that is, to two right angles.
Euclid's postulate and this theorem mutually involve the one the other: if we can prove his postulate this theorem is true, if otherwise we can prove this theorem, then his postulate is true*. Hence the question with which I com- menced the chapter (namely whether the sum of the angles of a triangle is, and can be shown to be, equal to two right angles) comes in effect to asking whether Euclid's postulate is true and can be proved to be true.
Features of the Problem. The postulate, as enunciated by Euclid, has the semblance of a proposition. For many centuries mathematicians believed that it could be directly deduced from the fundamental principles of geometry, and they devoted much labour to trying to prove it. The more notable of these attempts I propose to describe, but I may anticipate matters by saying that about a hundred years ago it was shown
* The demonstration is given by J. Richard, La Philosophie des Mathema- tiques, Paris, 1903, pp. 81—84.
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that this postulate, or any of its equivalent forms, cannot be proved. Thus in every one of the proposed demonstrations there is either a fallacy, or some assumption similar to that made by Euclid.
In order to be able to appreciate the criticisms on some of these attempts it will be convenient to preface the discussion by saying that the postulate and its conclusions do in fact involve considerations of the nature of the space considered. For example, we say that we can draw a line parallel to a given line, and that however far the lines are produced they will not meet. This is not at variance with what we observe, but we have never got to infinity to see what does happen there. Hence, though it is conformable to our experience we cannot say that it is actually true. In fact, it is not certain that the statement is absolutely true of the space we know. An example will show this. If small intelligent beings lived on a strictly circumscribed portion of the surface of a sphere, and evolved a geometry of figures drawn on that surface, they might form a body of propositions similar to those given by Euclid, and resting on the same axioms and assumptions. All their assumptions, except this postulate and the axiom about the impossibility of two straight lines enclosing a space, would be correct. But if the sphere were large enough, and they were confined to a comparatively small part of its surface, they would not be able to find out that this postulate about parallels was incorrect. Accordingly it would be not unreason- able that they should believe it to be true, though in fact it would be false.
What is here said of the surface of a sphere is by way of illustration, but it indicates the possibility of the existence of surfaces such that consistent systems of geometries, closely resembling the Euclidean geometry, might be constructed dealing with figures drawn thereon. This question is men- tioned again later. The above remarks suffice, however, to show that the postulate involves properties of the space in which the figures are constructed. It follows that the best way of stating the postulate will be that which is directly
CH. XIll] THE PARALLEL POSTULATE 313
characteristic of what we may call plane space, as opposed to other kinds of space. Euclid's statement answers this purpose, and it is remarkable that he should have thus gone to the root of the matter. By implication he admitted that the statement could not be demonstrated, and he frankly met the difficulty by telling his hearers that though he could not prove it, they must grant him the postulate as a foundation for his reasoning.
Attempted Deinonstrations of the Postulate. I proceed now to describe a few of these attempts to prove the postulate or the proposition.
Ptolemy s Proof of the Postulate. One of the earliest of these efforts to prove the postulate was due to Ptolemy, the astronomer, in the second century after Christ. It is as follows*. Let the straight line EFGH meet the two straight lines AB and CD so as to make the sum of the angles BFG and FGD equal to two right angles. It is required to prove that AB and CD are parallel. If possible let them not be parallel, then they will meet when produced say at M (or N). But the angle AFG is the supplement of BFG and is therefore equal to FGD. Similarly the angle FGG is equal to BFG. Hence the sum of the angles AFG and FGG is equal to two right angles, and therefore the lines BA and DC, if produced, will meet at M (or N). But two straight lines cannot enclose a space, therefore AB and GD cannot meet when produced, that is they are parallel.
E
Conversely, if AB and CD be parallel, then ^^and CG are not less parallel than FB and GD ; and therefore whatever be the sum of the angles AFG and FGC, such also must be the
* Procius, ed. G. Friedleiu, Leipzig, 1873, pp. C62— 368.
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sum of the angles FGD and BFG. But the sum of the four angles is equal to four right angles, and therefore the sum of the angles BFG and FGD must be equal to two right angles.
This proof is not valid. Apart fix)m all considerations about the nature of space, no reason is given why the sums of the angles on either side of the secant should be assumed to be equal. The whole question turns on whether the straight lines would not meet, even thouo:h the sum of the ansfles on one side is a little more than two right angles, and on the other a little less. It is conceivable that parallels might open out as they are prolonged, and thus that a straight line inclined at a small angle to one of them should never overtake the other, but chase it unsuccessfully through infinite space, just as a curve pursues its asymptote and never catohes it.
Procluss Proof of the Postulate. Proclus, after criticising Ptolemy's demonstration, gave a proof of his own, but in the course of it he assumed that if two intersecting straight lines be produced far enough the distance between a point on one of them and the other line can be made greater than any assigned finite length, and that if two parallel straight lines be produced indefinitely the perpendicular from a point on one of them to the other remains finite. On these assumptions the postulate can be proved. But just as we cannot assume that two con- verging lines {ex. gr. a curve and its asymptote) will ultimately meet, so we must not assume that the distance between two diverging lines will be ultimately infinite.
Wallis's Proof of the Postulate. I will give next a demonstra- tion oftered hy J. Wallis, Sa\*ilian Professor of Geometry, in a lecture deiiveicd at Oxlord on July 11, 1663. The substance of
CH. XIIl] THE PARALLEL POSTULATE 315
his argijment may be put thus*. It is desired to prove that if two lines AB, CD meet a transversal HACK, so that the sum of the angles BAC, ACD is less than two right angles, then AB and CD must (if produced) meet. One of the angles BAC, ACD must be acute; suppose it is BAC. He first showed that, in this case, fi*om any point B in AB vi^e can draw a line BE which will cut J.C in ^, so that the sum of the angles BEC, ECD is equal to two right angles; hence the angle BE A is equal to the angle DC A. Then if we take the triangle BAE (drawn on J.^" as base and \v4th B as vertex) and construct a similar triangle on J. C as base, he proved that its vertex must be at a finite distance from AC, must lie on AB produced, and must lie on CD (produced if necessary). Hence AB and CD when produced must meet.
The proof is ingenious, but it rests on the assumption that it is possible to construct a triangle on any specified scale similar to a given triangle. This cannot be considered axiomatic and in fact is not true of spherical triangles. The assumption, however, is made explicitly, and it can be used instead of Euclid's pc'stulate if it be thought desirable.
Bertrand's Proof of the Postulate. The following is another interesting demonstration. It was originally given by Bertrand
HA B K
of Geneva "*•. Supp'>5e AX and BY are two lines which meet a third line HABK so that XAB + YBA to show that AX and BY must cut. For simplicity, I draw the figure so that XAB ^ir '2 and therefore YBA
* J. Wallis, Opera, Oxford, 1693, vol. n, pp. 674—678.
t I do not know where ox when it was firs: published. It was given by 9. F. Lacroix in his ElemenU de Gionntrie, Paris, 1802, p. 23.
r)
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but this does not affect the argument. Produce AB indefi- nitely in both directions to H and K. Draw BZ perpen- dicular to AB, and denote the angle YBZ by a. Then the area between BY and BZ is the fraction w/tt of the space round B above UK, that is, oc/tt of the area of the plane above HK. Also the area between AX and BZ is the fraction ABjHK of the plane above HK. Now, however small a may be, w/tt is a definite finite fraction, but AB/HK is indefinitely small. Hence the area between BY and BZ is greater than the area between AX and BZ. But as long as 5 F does not cut AX the area between BY and BZ is less than that between AX and BZ. Hence BY must cut AX.
The objection to this demonstration is that it depends upon a comparison of infinite areas. But we have no test by which we can compare such areas, and to consider the order of infinities involves questions outside the region of elementary geometry. There is also a more fundamental difficulty : the argument assumes that space is infinite, but it is possible that it may be boundless and finite, as, for instance, is the surface of a sphere.
Play/airs Earlier Proof of the Postulate. I will mention next an attempt to prove this postulate by assuming that two lines which cut cannot be both parallel to another line, that is, that through a point one and only one straight line can be drawn parallel to a given straight line. It has been said that this assumption is not axiomatic, for a reason similar to that given above in my criticism of Ptolem3^'s Proof, but to most readers it seems simpler than Euclid's postulate, and as its meaning is easily grasped, some mathematicians prefer it to Euclid's postulate. Like the latter it is characteristic of the space considered. If this assumption is made, it is easy to show* that a transversal meeting two parallel straight lines makes the alternate angles equal, from which the other conclusions of Euclid follow.
* Elements of Geometry, by J. Playfair, Edinburgh, 1st edition, 1813, book i. prop 29. The book is in the same form as Euclid's Elements except for the substitution of this postulate for that given by Euclid.
CH. XIIl] THE PARALLEL POSTULATE 317
Personally I agree with those who consider Playfliir's assertion as axiomatic, that is, as being a part of our conception of plane space as derived from experience. This is not in- consistent with admitting that mathematicians can conceive a more general view of space {i.e. non-Euclidean space) and that to them the space of our experience is only, to the highest degree of approximation, Euclidean space*. But many writers do not accept Playfair's assertion as axiomatic. On such an issue no argument is possible.
Attempted Direct Demonstrations of the Proposition. The difficulties connected with the subject of parallelism led to various attempts to prove the proposition directly and thence to deduce some property of parallelism equivalent to Euclid's postulate. Substantially this was the method used by Thales and Pascal. I will mention one or two of these attempts.
Play fair s Rotational Proof of the Proposition. First I will describe an attempt, given by Playfair f in 1813. His argument is as follows. An angle is measured by the amount of turning
H
of a vector. Let ABC be any triangle. Suppose we have a rod A L placed along AB with one end dX A. If we rotate it clockwise round ^ as a pivot through the angle BAO it will move from AL to AK. It will make no difference if we now slide the rod along AK ^o that the end moves from A
* See A. Cayley, British Association Report, London, 1883, p. 9.
t See the notes appended to J. Playfair's Elements of Geometry, p. 432 in the fifth edition. Playfair finds the sum of the exterior angles and thence deduces the sum of the interior angles, but the method is the same as that given above.
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to G. If we now turn the rod, in the same direction as before, round (7 as a pivot through the angle ACB it will move from CK to GH. It will make no difference if we now slide it back along GB so that the end moves from G to B. If we now turn the rod, again in the same direction, round ^ as a pivot through the angle GBA it will move from BH to BA. We can then again slide the rod along BA so that the end B moves to A, when the rod will lie along AU, Thus the rotation, always in the same direction, successively through the three angles of the triangle produces exactly the same effect as a rotation through two right angles.
The demonstration is incorrect. In fact it is assumed that if the angle ABG in the figure above on page 311 is equal to HAK, then BG and AK will be parallel. The fallacy can be seen at once by applying the argument to the case of a spherical triangle or to one whose sides are circular arcs all convex — or all concave — to its median point.
Legendres First Proof of the Proposition. Legendre devoted special attention to the problem and offered various demonstra- tions of it. I give three of them. In one, which appears in the earlier editions of his Geometry, he tried to show that the sum of the angles of a triangle could not be greater than two right angles and could not be less than two right angles, and that therefore it must be equal to two right angles. His demonstra- tion assumes that if any number of equal triangles are placed in juxtaposition along a line it is possible to draw a triangle enclosing them all: the same assumption was made by T. P. Thompson. But unless we assume that space is infinite this is not justified : the insufficiency of the argument is clearly brought out by applying it to spherical triangles.
Legendres Analytical Proof of the Proposition. I think however that the following is the most ingenious of the proofs given by Legendre*. A triangle ABG is completely deter- mined by one side and two angles, say, a, B, G. Given these, the triangle can be constructed, and therefore the angle A de- termined. Now if the unit of length be changed the measure
• Elements de Geometrie, Paris, 12th edition, p. 281.
CH. XIIl]
THE PARALLEL POSTULATE
319
of a will be changed but the triangle, and therefore A, will not be altered. Hence A cannot depend on the value of a; accordingly it must depend only on B and (7,
Now take a right-angled triangle DEF, of which D is the right angle. Draw DG perpendicular to EF. The angle EDG in the triangle EBG is calculated from the other two angles of that triangle, namely E and a right angle, in the same way as the angle F in the triangle DEF is calculated from the other two angles of that triangle, namely E and a right angle. Hence F is equal to EDG. Similarly E is equal to GDF. Therefore the sum of the angles F and E is equal to the sum of EDG and GDF, and therefore is a right angle. Hence the sum of the angles F, E, and D of the triangle DEF is equal to two right angles. Thus the result is proved for a right-angled triangle, and it will follow for any other triangle in the same way as in Thales's proof.
J. Leslie criticised this proof on the ground that in the corresponding theorem of Spherical Trigonometry, we know that the expression for the value of the angle A involves ajR, where R is the radius of the sphere, and it is conceivable that in plane geometry there might be a length R' (the reciprocal of the space constant) which entered in a similar way in the problem : hence A might involve a and yet not change with the unit of measurement. To this it was replied that the point of Legendre's argument was that the discussion related only to plane geometry : this might, no doubt, be considered as the special case of spherical geometry in which R was inJ&nite ; if so, any term in the expression for A which involved ajR
320 THE PARALLEL POSTULATE [CH. XIII
disappeared, and thus his reasoning was valid ; and to introduce an unknown quantity R' was contrary to all canons of reasoning.
Legendre's Latest Proof of the Proposition. At the end of his life, 1833, Legendre showed* that if we could construct one triangle the sum of whose angles was equal to two right angles, then the sum of the angles of every triangle would be equal to two right angles. All attempts to obtain direct proofs that such a triangle existed failed. He showed, however, that if the sum of the angles of a triangle is not equal to two right angles then linear magnitudes can be determined by angular measurements. Assuming that this latter result is impossible, the proposition is proved. The second part of the argument is in effect a translation into geometry of the analytical proof given above.
Lagranges Memoir. Legendre's great contemporary, La- grange, believed at one time that he had found a solution of the problem. It rested on establishing plane geometry by a generalization from geometry on a spherical surface. He commenced to read to the Institute a paper on the subject, but had hardly begun when he stopped abruptly, put his memoir in his pocket and saying " Gentlemen, I must think further about this," left the room. We do not know what his argument was, but doubtless some flaw in it flashed on him as he commenced his paper f.
Other Parallel Postulates. Euclid's postulate is in accord- ance with experience, and like the axioms and other postulates it rests ultimately on the results of observation, but his statement of the property in question is not easy, and it requires some thought before the point is grasped. For this reason many attempts have been made to put it in other forms which are more likely to be readily granted by an ordinary reader. I enumerate two or three of these.
It will be noticed that the demonstration offered by Wallis and the earlier one given by Playfair rest on alternative postulates about parallels. That assumed by Wallis is sufficient,
* Memoires de Vlnstitut de France, Paris, 1833, vol. xii, pp. 367 — 410. The paper also contains an account of Legendre's earlier investigations. t A. de Morgan, Budget of Paradoxes^ London, 1872, p. 173.
CH. XIll] THE PARALLEL POSTULATE 321
but is not axiomatic, as may be seen by its incorrectness when applied to spherical triangles. It was adopted by Carnot, Laplace, and J. Delboeuf. Playfair's axiom answers the purpose as well as Euclid's : this form was also used by Ludlam. Thales's assumption that a rectangle exists also suffices. This was assumed bv Clairaut.
It has been suggested that Euclid's postulate might be re- placed by assuming that, if at a point ^ in a given line AB a line AX he drawn perpendicular to it, and at another point jB in AB s. line BY be drawn, making with it an acute angle, then AX and BY will cut. But essentially this is only Euclid's form expressed diagrammatically.
Another alternative form which has been suggested is to the effect that through every point within an angle a line can be drawn intersecting both sides (substantially the view of Lorentz and Legendre). This also is sufficient, but the appli- cation is less easy than that of Euclid's postulate.
It has also been proposed that we may reasonably assume that the distance between two parallel lines is always the same (Durer, T. Simpson, R. Simpson), or that a line which is every- where equidistant from a given straight line in the same plane is itself straight (Clavius). Neither of these forms is satisfactory. The conception of distance involves measurement, and this in turn involves a theory of incommensurable magnitudes. Thus before we can rest the theory on such a postulate, other as- sumptions have to be made, and the resulting discussion is neither simple nor clear.
Legendre suggested that it was sufficient to assume that the lesser of two homogeneous magnitudes if multiplied by a sufficiently large number would exceed the greater of them. But to make use of this he had to introduce the assumptions and principles of the infinitesimal calculus, and this can hardly be regarded as permissible in elementary geometry.
A different kind of postulate was suggested by Dodgson, who proposed* to replace Euclid's postulate by assuming that 2^ times the area of an equilateral four-sided figure inscribed
• C. L. Dodgsou, Curiosa Mathematica, London, 1890, p. 35. B. R. 21
322 THE PARALLEL POSTULATE [CH. XIII
in a circle is greater than the area of any one of the segments of the circle which lies outside it, where n is any positive integer. Granting that we can inscribe such a figure in a circle, this assumption seems obviously true. But a comparison by the eye of the area of a rectilineal figure with an incom- mensurable area bounded by a curve and a straight line is contrary to all the traditions of classical geometry and to what is usually regarded as permissible in elementary geometry.
Definitions of Parallels, Other writers have tried to turn the difficulty by altering the definitions of parallel lines*. One of the best known suggestions made with this object defines parallel lines as lines which have the same direction, by which is meant lines which make the same angle with a line cutting them (Varignon, Bezout, Lacroix). The phrasings of the proposed definition vary slightly. There is no objection to this if the cutting line is fixed, but then it does not avoid the necessity of our having to assume some postulate. If, however, as is usual, the definition is taken to mean that parallel lines make equal angles with every secant, it involves an unwarrantable assumption. In fact it would seem that the term direction cannot be defined without predicating a theory or properties of parallels f.
Another suggested definition which has met with some favour is to the effect that parallel lines are lines which neither recede from nor approach each other, that is, lines whose distance apart is always the same (Wolf, Boscovich, Bonnycastle). This definition is really equivalent to the postulates laid down by Diirer and Clavius, and to give a definition which involves a disputed assumption is worse than fi^ankly postulating the assumption. The definition, however, agrees with the popular view, but if we take a straight line, and erect at every point a perpendicular of given length, we have no right to assume that the locus of the extremities of these transverses will be a straight line, and even less right to assume that it is a
* See J. Playfair, Elements of Geometry, Edinburgh, 1813; notes to book i, prop. 29.
f W. Killing, Grundlagen der Geome'rie^ Paderborn, 1898.
CH. XlTl] THE PARALLEL POSTULATE 323
straight line perpendicular to them ; and, unless we assume that the distance between the parallel lines is measured by a transverse pei'pendicular to both of them, we cannot use the definition to much purpose. D'Alembert avoided these diffi- culties by saying that one line is parallel to another if it contains two points on the same side of and equidistant from the other ; but this by itself is of no use, unless we assume Euclid's postulate or some smiilar property of parallels.
These definitions, if they are to be useful, involve assump- tions. They slur over the real difficulty, and are less satisfactory than a frank statement of what is assumed.
Non-Euclidean Systems*, It had long been well known that the postulate and proposition were not true in the corresponding geometry on a spherical surface — in fact the sum of the angles of a spherical triangle always exceeds two right angles — and since there were such difficulties in establishing the Euclidean postu- late in plane space, mathematicians began, rather more than a century ago, to consider whether that postulate was true either necessarily or in fact. It required courage, even genius, to make such a conjecture, for though on the one hand the postulate could not be proved, there was on the other no reason to doubt its correctness, and no conclusion inconsistent with observation had been deduced from it, while at first sight nothing seemed to justify the assumption that it was not true.
Saccheri and Lambert raised this question in the eight- eenth century, but their investigations, though intelligent, were incomplete and attracted little attention. Gauss went further, as appears from his correspondence in 1829 and 1831, but even before then he had shown that the proposition and postulate could be proved to be true if it were admitted that a triangle could be drawn with an area greater than a given area; this, however, he rightly regarded as non-axiomatic. Later he discussed some of the properties of hyperbolic geometry. He did not publish his results, and they did not affect the treat- ment of the problem by other writers.
* See R. Bonola, La Geometria Non-Eiiclidea, Bologna, 190G ; and D. M. Y. Somerville, Bibliography of Non-Euclidean Geometry, St Andrews, 1911.
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The credit of first showing that the postulate is not neces- sarily true is due to Lobatschewsky and the Bolyais. They boldly assumed that the postulate was not true and that through a point a number of straight lines can be drawn parallel to a given straight line. On this assumption, they deduced a consistent body of propositions, which is termed hyperbolic geometry.
These investigations attracted but slight notice. The writers were almost unknown. N. I. Lobatschewsky, 1793 — 1856, was professor at Kasan, and his works were written in Russian. Wolfgang Bolyai, 1776 — 1856, was an eccentric, simple, rough- clad teacher in Transylvania. He now lies buried at his request under an apple tree, commemorating the three apples which, he said, had so profoundly affected the history of the human race — those of Eve and Paris, which had made earth a hell, and that of Newton which had raised earth again into the company of the heavenly bodies. His son John, 1802 — 1860, had excellent mathematical abilities, and worked out the principles of the new geometry, but he spent his life soldiering, and to him mathematics was only a recreation. Probably he valued his reputation as a musician far above his mathematical abili- ties. He was noted for his fiery temper ; in one of his quarrels he accepted the challenge of thirteen officers of a regiment on condition that after each duel he might play to each of them a piece on his violin. He is said to have vanquished them all, and been, in consequence, retired from the army.
The subject, however, was "in the air," and attracted the attention of G. F. B. Riemann. Riemann was one of the most brilliant German mathematicians of the nineteenth century and, though short-lived, his writings have profoundly affected the development of the subject. His paper on the hypotheses on which geometry is founded was read in 1854. He showed that a consistent system of geometry of two dimensions can be con- structed in which all straight lines are of a finite length. This science, now known as elliptic geometry, is characterised by the fact that through a point no straight line can be drawn which if produced far enough will not meet every other line. The
CH. XIIl] THE PAKALLEL POSTULATE 825
resulting geometry may be compared with the geometry of figures drawn on the surface of a sphere ; in it, space, though boundless, is finite. The discussion of Riemann's paper led to the discovery of the earlier researches of Lobatschewsky and the Bolyais. The subject has since been studied by several mathe- maticians of repute, notably by E. Beltrami and F. C. Klein.
Here then we have three geometries — Elliptic, Euclidean (or Parabolic or Homaloidal) and Hyperbolic — each consistent on its own hypotheses, distinguished from one another according 9-s no straight line, or only one straight line, or a pencil of straight lines can be drawn through a point parallel to a given straight line.
In the parabolic and hyperbolic systems straight lines are infinitely long : in the elliptic they are finite. In the hyperbolic system there are no similar figures of unequal size ; the area of a triangle can be deduced from the sum of its angles, which is always less than two right angles ; there is a finite maximum to the area of a triangle; and its angles can be made as small as we like by making its sides sufficiently long. In the elliptic system all straight lines, if produced, are of the same finite length; any two lines intersect ; and the sum of the angles of a triangle is always greater than two right angles. In the elliptic system it is possible to get from one point to a point on the other side of a plane without passing through the plane ; thus a watch- dial moving face upwards continuously forward in a plane in a straight line in the direction from the mark vi to the mark XII will finally appear to a stationary observer with its face downwards ; and if originally the mark III was to the right of the observer it will finally be on his left-hand.
In spite of these and other peculiarities of hyperbolic and elliptic geometries, it is impossible to prove by observation that one of them is not true of the space in which we live. For in measurements in each of these geometries we must have a unit of distance ; and if we live in a space whose properties are those of either of these geometries, and such that the greatest distances with which we are acquainted {e.g. the distances of the fixed stars) are immensely smaller than any unit natural
326 THE PARALLEL POSTULATE [CH. XIII
to the system, then it may be impossible for our observations to detect the discrepancies between these three geometries. It might indeed be possible for us by observations of the paral- laxes of stars to prove that the parabolic system and either the hyperbolic or the elliptic system were false, but never can it be proved by measurements that the Euclidean geometry is true. Similar difficulties might arise in connection with excessively minute quantities. In short, though the results of Euclidean geometry are more exact than present experiments can verify for finite things, such as those with which we have to deal, yet for much larger things or much smaller things, or for parts of space at present inaccessible to us, they may not be true. Even, however, if our space is only approximately Euclidean, the propositions of ordinary geometry are none the less true of Euclidean space, though that may not be the space of our experience.
I mention later, in Chapter xix, some other problems connected with different kinds of space.
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