CHAPTER V

The Parallel Theory

THERE are two questions concerning the parallel theory that should occupy our attention. One is the question of the proper definition of parallel lines, which, as we have seen, has been a matter of dispute since Euclid’s time. The other question concerns the manner of treatment which the parallel theory has received from the beginnings of its history to the present day. We shall consider the latter question first.

The whole point and gist of this question comes down to the peculiar form of the geometric propositions of Euclid and their arrangement in his scheme of proof, as exhibited in the Elements. For, although this question has been threshed out for over two thousand years, it has never got beyond Euclid’s presentation of it, and the discussions, if we except the non-Euclideans, have had the sole purpose of bolstering up the one weakness that was discovered in his structure, his fifth postulate. We intend, however, to say a word on his whole treatment of the subject.

What we say here of Euclid, applies with equal force to his successors in Euclidean geometry. Now that a proof has been given, Euclid’s special treatment of the subject no longer concerns geometry itself. Though the order and logic of his method have never been improved upon, that is not saying that he has given us the last word in geometric science. Our purpose is not so much to defend Euclid, as to establish the science with which his name, as that of its greatest benefactor, is usually connected. The blots in his method are his own, and belong to Euclid, the geometer; they no longer concern the correctness or validity of the geometry that bears his name. We are not then so much interested in Euclid’s personal mathematical attainments as such, as in the validity of the traditional science, which has become so much his that very frequently it is simply termed Euclid. We need not hesitate then to take over the defense of the theory from its chief defender, and criticize him where he fails.

That Euclid had difficulty not only in reducing his theorems to strict logical order, but in establishing his theory with scientific validity as far as the parallel question is concerned, no one need doubt, and, as we have before pointed out, the difficulty antedated his time. In proof of the quandary in which he found himself we need only to examine the demonstrations he offered leading up to his conclusion in the thirty-second, particularly that given in the twenty-ninth proposition, and his application of the fifth postulate therein. In this connection his whole order is imperfect, and his argumentation invalid as being circular, or what is technically known as a petitio principii, or a begging of the question. We have seen that his difficulty began with the sixteenth proposition, where he was unable to complete his proof and had to be satisfied with a partial conclusion. But here at least his proof is rigorous. The same, however, cannot be said of the further steps on the road to establishing the parallel theory.

In the first place from our point of view he put the cart before the horse in seeking to establish the more indefinite parallel lines, as lines that never meet, before the more definite parallel lines that are included in the sides of a parallelogram. Yet this is the only way it can be done, and the parallelogram route is simple, easy and clear, and at the same time is the most obvious method of settling the question in a lucid and uncontrovertible way. That the parallelogram furnishes a valid method, there can be no question, as it does not in the least limit the parallel lines to the immediate construction, but these can be extended even to infinity, that is to negative infinity; there need be no limit put to the length of such parallel lines, and that is all that is required.

But there is another point in which Euclid’s mathematical sagacity was still more at fault. It was his lack of perception of the role of the triangle in the proof of the parallel theory. The triangle itself is the simplest of the complete plane figures, and thus naturally becomes an element for the development of those that are more complex, as the parallelogram and polygon. Lines, angles, triangles and polygons would seem to be the natural order of development from the first element, the line, or lines singly and in double, triple, or multiple combination. Euclid’s treatment of the propositions of the first book shows no appreciation of this natural order. It would almost seem that, instead of having a regular plan of development, he had a multitude of propositions and simply placed them in order just as he could fit them best. He shows this lack of insight in his treatment of the sixteenth proposition of the first book, where the complete proof was naturally called for and was at hand by the simple process of completing the figure by joining the vertices of the triangles used in his proof. It is plain from this that geometry is not Euclid’s science, but rather one built up before him to which he held more or less by the normal conservatism of a school. He may have improved the plan in detail and even developed it, but the first book of Euclid certainly does not show the work of a man who had first formed in his mind a complete analytic and synthetic plan of the whole field, before he set about laying it out.

Euclid is then perhaps not to blame for the parallel theory as it is actually constructed. This theory first makes its real geometric appearance in I, 27, which is the beginning of the second section of the first book. Up to this he has dealt with triangles and their properties. In this section he introduces the parallel theory, and in the third section he treats of the relation between triangles and quadrilaterals, all of which is built on the parallel theory. Here is where he put the cart before the horse. But the horse was very likely already hitched when he took charge, and all he did was to drive it on as it was actually harnessed.

It is very probable that this treatment of the question antedates Euclid. Some of it was certainly used before his time, for it seems clear that Propositions I, 27 and 28 were known to the science before Euclid. Aristotle alludes to them in two passages; to the one in Anal. Post. I, 5, and to the other in Anal. Prior. II, 17. These propositions may even contain the very petitio principii that Aristotle noticed in the parallel theory as related in Anal. Prior. II, 16. He there speaks of the petitio principii committed by “those who think that they construct parallels for they unconsciously assume such things as it is not possible to demonstrate if parallels do not exist.” In any case the petitio principii is still in Euclid. But let us investigate his development of the whole question.

In the first place Euclid’s whole argument is one of reductio ad absurdum, or a mere argument leading to the impossible. The argument known as per impossibile shows the impossibility of the opposite conclusion, and from that concludes to the correctness of the contradictory; as Aristotle defines it, “All syllogisms concluding per impossibile reach a conclusion that is false, and prove the original proposition by hypothesis, when something impossible results from assuming the contradictory of the original proposition.”¹ This is the weakest of general proofs, since it does not show us why a thing is, but merely that it is. Aristotle tells us this. Affirmative demonstration, he says, is better than negative demonstration, but even negative demonstration is evidently better than that leading to the impossible.²

Proofs per impossibile should therefore be used only in cases where no other is available. Euclid knew this very well, and if he chose it, it was because he could find no other. This, however, would not constitute an insuperable objection to his method, but only shows it a trifle weak. This is especially true of a case where an abundance of middle terms was at hand to prove the conclusion immediately, such as the alternate-interior angles, the two interior angles on the same side of the transversal, etc. The trouble was that Euclid here lacked the necessary geometric sagacity to connect premises and conclusion by a middle term. The matter itself was entirely apt for a real ostensive demonstration.

This is not, then, the great objection to Euclid’s method, although even in this respect, his procedure could easily be improved upon. The objection is much more fundamental. It is not merely that he makes use of a cumbersome reductio ad absurdum, where a clear, direct proof was available, but his argument is a pure begging of the question. In his demonstration he uses both the definition of parallels which has also been called the sixth postulate, and the fifth postulate on converging lines. His reductio ad absurdum consists in opposing the one to the other to prove the reality of the hypothesis that the sum of the interior angles on the same side of a transversal is equal to two right angles, when the lines are parallel. Let us see if he does this validly.

It is in the twenty-ninth proposition of the first book that Euclid uses the famous parallel postulate and pits it against the definition of parallel lines. This proposition states that a straight line falling on parallel lines makes the alternate-interior angles equal to one another, the exterior equal to the interior and opposite angle, and the interior angles on the same side of the transversal equal to two right angles. Here the second and third conclusions depend on the first, and we need only consider them in the order given to see the petitio principii.

Euclid tries to show that, if the alternate-interior angles are not equal, an impossibility will result, and he does this in the following manner. If the two angles are unequal, the interior angles on the same side of the transversal will be less than two right angles. In that case by postulate five, they will meet; but this is an impossibility, since by hypothesis they are parallel, and, therefore, by definition do not meet. From this impossibility he deduces that the alternate-interior angles are equal, and then again from this deduces the third conclusion that the sum of the interior angles formed by the transversal is equal to two right angles.

The petitio principii or vicious circle will perhaps become more clear, if we contrast the three conclusions. What does Euclid wish to prove in this proposition? Three things:

a. That the alternate-interior angles are equal.
b. That the exterior-interior angles are equal.
c. That the interior angles on the same side of the transversal are equal to two right angles.

How does he do it? The two latter conclusions are dependent on the first, and receive their validity from its validity. How then does he establish the validity of the first conclusion? By assuming the third. He thus assumes the third to prove the first, and then again proves the third by means of the first. If this is not a real live example of a vicious circle, it would be difficult to find one.

In other words, he proves that the alternate-interior angles must be equal, because if they are unequal, the interior angles on the same side of the straight line will be less than two right angles, whence they will meet; this necessarily supposes the contradictory, that if they are equal to two right angles they will not meet, which is just what he wishes to prove in the third conclusion. He assumes or postulates one contradictory in order to use it in a course of argument in establishing the other contradictory. Euclid, therefore, proves one contradictory by assuming another, which is not a deduction at all, but a pure statement of contradictories. For, if one assumption is true, its contradictory is immediately true, and requires no middle term. This method is then not only logically illegitimate and reprehensible, but what is worse, it is invalid, because it is a circular argument assuming in a different form that which is to be proved. This part of the twenty-ninth proposition is then not a proof at all, since it contains nothing not already implicitly contained in the fifth postulate.

The third part of the proof is therefore invalid, but what about the first and second conclusions? Are they at least valid? Even here correct analysis must answer that the argument is not founded. The vicious circle that we have shown is not the only fundamental invalidity in the reasoning of Euclid in this matter. The proof is invalid for another reason.

Besides the assumption contained in the fifth postulate, the proof requires the existence of parallel lines that do not meet. The mere definitio nominis itself is entirely useless as a matter of proof. But the existence of these parallel lines has not been demonstrated by Euclid, and therefore his proof is of no account. His proof, as we have said, consists in opposing the existence of real parallel lines to the lines described by the fifth postulate, and assumed to be in existence as such. So even granting the reality of the property of the lines described in the postulate, the proof is no proof, because he has neither postulated nor proved the existence of such lines as parallel lines. It is true, he has given what purports to be a demonstration, but our contention is that this demonstration is invalid, as we shall now proceed to show.

Euclid undertakes to show the reality of such parallel lines in propositions twenty-seven and twenty-eight. Since the latter depends on the former, we can leave it aside, and devote our attention entirely to twenty-seven. This proposition states, “If a straight line falling on two straight lines makes the alternate angles equal to one another, the straight lines will be parallel to one another.” Of course, if this proposition is established, then the first part of twenty-nine would follow, if we granted the fifth postulate. The question therefore concerns the validity of this proof.

Euclid’s proof of I, 27 is a reductio ad impossibile, and is developed in this wise. If the lines are not parallel, then they will meet. If they meet they form a triangle. Then in that case the impossible occurs, viz., that the exterior angle of the triangle is equal to the opposite interior angle, which has been proved impossible in theorem sixteen. Therefore they do not meet. The reductio ad absurdum is valid if the disjunctive is complete and legitimate, that is, if all the alternatives are exhausted, and if they are all excluded but one. Here the disjunction is made between two alternatives, and they are given as contradictories. The contradictories are parallel lines and lines that meet. Now it may be quite true in point of fact that these are contradictories, and it is as such that they are employed by Euclid in the whole treatment of this question. But the same thing does not hold with regard to assuming them as such. It is an assumption which he is not entitled to make. They are not logical contradictories. If they are real contradictories, the fact must be shown, not assumed.

Now Euclid has practically assumed non-parallel lines as meeting. But he has not assumed it in this form, but only in the form of lines whose interior angles formed by a transversal are less than two right angles.

Now assumptions, especially those that are postulates, that is, hypotheses assumed as representing existing things, not mere notions of the mind, and as such used for the purpose of proving other propositions, must be strictly interpreted. As far then as this postulate goes, there is no logical connection between parallel lines and lines that do not meet; hence the postulate will not serve, for establishing parallel lines and lines that do not meet as identities.

Further, Euclid has defined parallel lines as not meeting, but has not assumed them as existing. In the present proposition he intends to show that they do exist as defined. In order to do this he makes the disjunction, either the lines are parallel or they will meet. The two arms of his disjunction are therefore taken as contradictories. They are not however contradictories in form. The contradictories in this case are either lines that meet and lines that do not meet, or parallel lines and lines that are not parallel. But parallel lines and lines that do not meet, are not logical contradictories. If they are real contradictories, this must be shown, as we said before, and by some middle term. This Euclid neglects to do, and his argument is logically invalid, or does not conclude. To make them real contradictories he would need to have proved that parallels are lines that do not meet, and this is what he set out to do in the very proposition where he needs this for the proof; or he must postulate that they do not meet, and then he would postulate for the proof what he wants to prove in the argument. His conclusion is already contained in his premises. In other words, Euclid is again guilty of the same fault as before; he is arguing in a vicious circle, and his proof is nothing but a begging of the question.

Euclid, or whoever introduced this proof into geometry, only did so because he was unable to see the real mathematical middle term in this question. Therefore he had recourse first to a mere reductio ad absurdum, and in the second place he introduced the vicious circle that very likely was the one that Aristotle had the perspicacity to perceive. Here is the point. Parallels are defined as lines that do not meet. This is only a definitio nominis, and it has no force whatsoever in proving, till its reality is established by a demonstrated construction. Now, in demonstrating the construction, Euclid has actually assumed the definitio realis, or the existence of parallel lines. For he clearly states that, if they are not parallel, they will meet. This is the contradictory of the statement that parallels will not meet. He, therefore, proves the actuality of parallels by assuming the actuality of the contradictory, which is arguing in a vicious circle, and this may be the very one Aristotle saw in the doctrine of parallels.³

Let us now see what the twenty-seventh proposition really proves. It proves clearly that the lines AB and CD which Euclid wishes to prove parallel, do not meet. But this is by no means the same thing as that the lines are parallel. Euclid may have defined parallel lines as lines that do not meet, but he has not proved or even assumed that all lines that do not meet are parallel. The very existence of asymptotes shows that it is not true. The predicate is then of greater extension than the subject and therefore they are not convertible terms. As far then as parallel lines are concerned, they do not exist even after the twenty-seventh proposition. All that really exists are lines not meeting whenever the alternate-interior angles formed by them with a transversal are equal.

As far as one pair of contradictories is concerned, there are only lines that meet and lines that do not meet. As far as the other pair is concerned there are only parallel and non-parallel lines; but as far as the combination of the two is concerned, there are parallel lines, there are converging lines that do not meet, there are converging lines that do meet, and there are diverging lines. Saying therefore that lines do not meet merely leaves the question in the air as to whether the lines are really parallel, or whether they diverge, or whether they converge without meeting. It might be answered that the fifth postulate might take care of one of them, and that converging lines that meet are merely the same ones that diverge. As a fact this is true, but we are not arguing as to what is true or false really, but what is logically valid or not. The fifth postulate is an assumption, and assumptions are to be strictly interpreted, and nothing can be taken as existing except what is actually contained in the assumption, or what can be proved through the assumption. Euclid gives no proof. And strictly Euclid’s postulate is restricted to lines where the interior angles are less than two right angles, and to lines meeting in that condition. But even if this be granted, the further alternative remains, that the lines may converge and not meet, as do asymptotes and their curves. It is quite otherwise of course whenever the matter is proved. For then the determination depends on a middle term, and wherever the middle term extends, the conclusion applies.

As far then as demonstration goes in the twenty-seventh proposition, parallel lines do not exist; hence they cannot be validly employed in the twenty-ninth proposition to establish the truth of the equality of the alternate-interior angles. The proof of twenty-nine is therefore invalid, as is the proof of twenty-seven and twenty-eight; in other words, the whole parallel theory in Euclid is entirely without foundation. For Euclid’s proof is entangled in two different vicious circles, one in the twenty-seventh and the other in the twenty-ninth, that render it wholly useless. Even without the circular argument in the twenty-ninth, the false reasoning of the twenty-seventh would be enough to destroy the whole argument, and the petitio principii of the twenty-ninth adds nothing more than another sign of the confusion of Euclid’s mind on this question.

The fifth postulate is then not the only or even the chief weakness in the whole parallel theory of Euclid; in fact, it is the smallest part of the difficulty, for even if it were granted as legitimate and valid the position of Euclid’s parallel theory would not be improved. There would still be a non sequitur to the conclusions he wishes to establish. The whole proof, even abstracting from the fifth postulate, is entirely invalid, because of Euclid’s inability to connect the parallel theory together by means of its proper middle term. This applies as well to his predecessors and successors. It was the same lack of understanding that brought about the fifth postulate, for if the proper middle term had been applied, the fifth postulate itself would immediately have found its normal place in the structure of the parallel theory.

The strangest part of the whole matter is that this weakness in Euclid’s argument was never discovered. Two thousand years of Euclidean criticism in large part directed to this point, as well as one hundred years of non-Euclidean Geometry, failed to detect it. Non-Euclidean Geometers have not much to boast about as to their possession of geometrical insight and critical faculties, when all their attacks were directed against the sole assumption of what after all was a pretty certain truth, while they failed to see the obvious fallacy on which the whole theory was built. Euclideans, as belonging to the school, and with the conservative faith of a school, might have passed it over, but even then it reflects but little credit on their mathematical penetration. But for the non-Euclideans there was not the same excuse. They set out to tear the theory to pieces, but yet had not the keenness to settle on the real weakness. But if non-Euclidean Geometers had either critical acumen or mathematical insight they would not be non-Euclidean Geometers, trying to hold together the incongruous disarray of ideas that goes by the name of non-Euclidean Geometry.

We have seen where Euclid’s arguments establishing the parallel theory are invalid, and why. We now come to the difficulty in the so-called parallel postulate, and shall see why it is not a legitimate postulate. About its validity there can be no question, at least since we have proved it. Euclid’s assumption in the fifth postulate was valid, but the assumption itself was not legitimate logically. He had already defined parallel lines; he had defined right angles; both of which definitions are legitimate and valid. But in the fifth postulate he connects the two by implication, viz., it follows that where the lines are parallel the interior angles on the same side of the transversal are equal to two right angles, if we assume that where they are not equal to two right angles, the lines will meet. In other words, what he really assumes is that they will meet if the sum is less than two right angles; they will not, if the sum is equal to two right angles. This is not a legitimate procedure. It i9 entirely legitimate to define parallel lines; it is equally legitimate to define right angles. But it is not legitimate to connect parallel lines with right angles, since the terms of one definition are entirely different from those of the other. The connection must be made by some middle term. It is therefore to be shown, not assumed. What we must demonstrate and not assume, is not fit matter for a postulate, which requires the assumption of existence of something that is in itself readily and directly understood, as, for instance, the production of a line, or a circle. There is a double assumption in the fifth postulate; first the theoretic connection between right angles and parallel lines; then the existence of both in connection. This is the real difficulty with the so-called parallel postulate.

The proper middle term for proving the connection was not seen by Euclid and his predecessors, nor by any of his successors. The trouble with his method started in the sixteenth proposition, which proved only in an indefinite and incomplete manner what should have been proved then and there. But he stopped half way, because he could not see any farther. His whole treatment of the question then became confused, and he sought to retrieve himself by illegitimate assumption and circular arguments, until he not only confused himself, but all geometers for the centuries following his time.

Heath supposes that the invention of the postulate was due to Euclid, and was his manner of saving the theory from the petitio principii in it. But this is not so sure. For the petitio principii still remains. The condition then has not been much bettered if this present form is really to be attributed to Euclid as a way of getting rid of the old. At any rate the words of Heath are scarcely applicable. Speaking of the allusion Aristotle made to the petitio principii in the parallel theory, he adds: “This reproach was removed by Euclid when he laid down this epoch-making postulate.”⁴ If indeed he removed it, it was only to place it elsewhere.

To remove this reproach wholly and entirely, as much as regards the postulate itself, as well as the petitio principii in the whole question as exposed by Euclid, has been our aim. Our method of proof has done away with both. The definition of parallels is established, by showing the construction; the parallel postulate is proved; and the whole argument for the parallel theory is devoid of any petitio principii. We need neither definition nor postulate in our proof; they are rather conclusions than premises.

After our demonstration, the famous fifth postulate, around which so much ink has been spilled during the centuries, is plainly established, as it has become the clearest of corollaries. For, since the sum of all the angles of a triangle is equal to two right angles, the base angles of a triangle, and any side may be taken as base, are less than two right angles. The necessary condition for the formation of a triangle is then that the base angles should be together less than two right angles. Hence, since the base and sides of a triangle is an identical geometric figure with that formed by a transversal together with two coplanar straights having interior angles less than two right angles, these straights are sides of a triangle, and will meet if produced.

We even know the exact angle of meeting, and can tell exactly where they will meet by an easy geometric calculation. The angle of meeting will be the exact angle by which the sum of the interior base angles is less than two right angles. We can thus connect them by measurement with parallel lines, one of which is one of these coplanar straights. The other coplanar straight can then be taken as the transversal falling on the two parallel lines, and the angle of intersection of the transversal with one of the parallels will be equal to the alternate angle formed by the same transversal with the other parallel. Or the angle with the first parallel will be equal to the angle formed by the meeting of the two coplanar straights. The difference between two right angles and the sum of the interior base angles formed by the transversal and the two coplanar straights will be equal to the angle formed by one of these coplanar straights with a parallel to the other. This means that with parallels and a transversal the alternate-interior angles are equal, and one of these equal angles is the meeting angle of the two coplanar straights.

It is evident that in this development we are not seeking to make any definitive arrangement of Euclid’s Elements in this matter, but have merely arranged them here to suit our immediate purpose, and no farther than was required by the necessities of our proof. Our order is entirely determined by this and by no other aim. We are not here reediting Euclid, nor rearranging geometry, but simply proving the parallel theory. Many corollaries could be connected with the proofs given, and a more complete development made, if we had a broader purpose. No one will deny that the order and proofs as given in the First Book of Euclid could be improved; but that is a task we have not undertaken here.

Let us briefly resume the whole question. The parallel theory embraces the existence of parallel lines as well as those which are not parallel, or the contradictory proposition. Parallelism and not meeting are not logical equivalents, and must therefore be compared and identified by means of a middle term. The angles formed by a transversal with coplanar straights give the necessary middle term for all the proofs. If the alternate-interior angles formed by the transversal and the coplanar straights are equal, if the exterior-interior angles are equal, if the interior angles on the same side of the transversal are equal to two right angles, then the lines are always equidistant, and will not meet.

The opposite can also be proved; that is, if the interior angles formed by the transversal are not equal to two right angles the lines will meet. This is a mere corollary from the theorem that all the interior angles of a triangle are equal to two right angles, and hence the base angles, or the interior angles formed by a transversal with two coplanar straights are always less than two right angles, and in such a condition will necessarily form a triangle, if produced; because with straights that is the essential condition in the formation of a triangle.

Having now settled the difficulty in the development of the theory of the parallel question as it was given forth by Euclid and held from his days to ours, and having sought out the petitio principii in this development, as well as the fault in the parallel postulate, we now come to the definition of parallel lines as it was formulated by Euclid and others. There has also been a great deal of discussion on this point, and this we now undertake to clarify by light from the real parallel theory.

References

 

1. Anal. Prior. I, 23.
2. Anal. Post. I. 26.
3. Anal. Prior. II, 16.
4. Heath, Vol. 1, p. 202.