CHAPTER IIHistory of the Parallel Theory
That there was some difficulty inherent in the science was already noted by Aristotle, who illustrates the fallacy known as “petitio principii” or “begging the question” by referring to the theory of parallel lines current in his time. He gives as example of such an argument the case of a person who should demonstrate A through B, and B through C, while C was naturally adapted to be proved through A, for it happens that those who syllogize, prove A by itself. “This,” he says, “they do, who fancy that they describe parallel lines, for they deceive themselves by assuming such things as they cannot demonstrate unless lines are parallel.”1 In other words, there was somewhere a vicious circle in the then existing theory, something assumed to prove parallel lines that already demanded the truth of the parallel theory. Philoponus, in an equally obscure passage, seeks to explain the petitio principii alluded to by Aristotle, and Heath2 makes it out to be something in the order of the direction theory of parallels. But this is a critical point on which we need not linger. It would seem then that it was to do away with this vicious circle that Euclid made the disputed point a postulate, the fifth, and by using it in the twenty-ninth proposition of the first book thus got rid of the current petitio principii. The fifth postulate may then, as Heath claims, be original with Euclid, although, as we shall see later, this is by no means certain. Evidently he only made it a postulate because he was unable to make it a proposition to be proved without assuming something else equivalent to it. There can be no question but that Euclid properly appreciated the difficulty in the parallel theory. Besides the fact that we know from Aristotle that the difficulty already existed, the very treatment that Euclid gives the question shows the same. It can hardly be a pure accident that the first twenty-eight propositions are entirely independent of the fifth postulate, and that it is first used in the proof of the twenty-ninth. It would thus seem that he tried to carry out his orderly development of the science without it, but when he finally saw he needed it to avoid a vicious circle he frankly adopted it as a postulate to enable him to proceed with his proof. This will become still more clear if we consider the position of the crucial proposition which is essentially connected with the theory of parallels, that of I, 32, which states that the exterior angle of a triangle is equal to the sum of the two opposite and interior angles; and also the consequence of this, that the sum of the angles of a triangle is equal to two right angles. It was for the proof of this that I, 29 and the assumed fifth postulate were needed. The same point had been taken up in I, 16, where Euclid shows that the exterior angle of a triangle is greater than either of the opposite and interior angles. His difficulty consisted in not being able to prove that the sum of the angles of a triangle is equal to two right angles without having recourse to the parallel postulate, since, if he could, he would have done so directly in I, 16. This latter theorem is the same as that of I, 32, but in an incomplete and indefinite way. The proposition in I, 32 simply resumes it and makes it complete. The reason of this arrangement is that Euclid did not see his way to prove the theorem in I, 32 without the parallel postulate, and he needed the partial proof of I, 16 even to be able to use this. He was, therefore, fully aware of the difficulty and saw no way of getting around it but that of postulating the parallel theory. The proof in I, 32 evidently antedates his time, since there is very frequent allusion to it in Aristotle, and no doubt it was the very same as that given by Euclid, viz., the equality of all the angles about a point being equal to two right angles and to the angles of a triangle. Euclid’s difficulty and that of his predecessors was that he was unable to make this proof independent of the parallel theory. If he had done this, the parallel postulate would have been established, since it would be a mere corollary of this proposition. As we have seen, Euclid cut the Gordian knot of the difficulty by simply and frankly taking the parallel theory and making it a postulate. In this he was considered as more clear-minded than his successors who tried to prove the unprovable. Thus Heath, speaking of his removal of the petitio principii by introducing this postulate, which he considers original with Euclid, says: “This reproach was removed by Euclid when he laid down this epoch-making postulate. When we consider the countless successive attempts made through more than twenty centuries to prove the postulate, many of them by geometers of ability, we cannot but admire the genius of the man who concluded that such a hypothesis, which he found necessary to the validity of his whole system of geometry, was really indemonstrable.”3 But by doing so he merely jumped from the frying pan into the fire. It may be true that he thereby avoided one logical petitio principii, but he fell into another, and did not make the real position of the science of geometry any stronger. He merely displaced the difficulty by introducing what was tacitly assumed before, as an open assumption or postulate. His successors immediately fell upon this difficulty, and their efforts for more than twenty centuries have been towards finding some demonstration for it that would close the gap in the science. That the postulate really should be a theorem and not something to be assumed is evident from the very form of the proposition itself, which is much more like a theorem to be demonstrated than something to be accepted without proof. As a matter of fact Euclid proves many propositions that are in appearance and matter much simpler and of greater evidence than this one. Its statement is such that it differs so entirely from the rest of the postulates both in simplicity and clearness and obvious certitude, that it is evident he made it a postulate only through the force of circumstances, and not because he thought it a postulate of equal right with the others. The same conclusion is also to be derived from the fact that from his day to ours attempts have been continually made to find some proof for it, or to get rid of it by substituting in its stead some other postulate that would serve the same purpose without encountering the same difficulty, and which would therefore be scientifically more acceptable. The consensus of opinion from Euclid’s day to our own, as we shall see, is against accepting the parallel theory as a postulate, and this has constituted a perennial defect which one and all tried to do away with. Our witness for this in ancient times is Proclus (410-485 A.D.), a neo-Platonic philosopher, who wrote a Commentary on the first book, which is preserved. What was written before his time is lost and only a few fragments survive in writers of a later period. Proclus himself not only recognized the difficulty but also attempted to prove the proposition. He states the objection quite clearly: “This ought even to be struck out of the postulates altogether; for it is a theorem involving many difficulties, which Ptolemy, in a certain book, set himself to solve, and it requires for the demonstration of it a number of definitions as well as theorems. And the converse of it is actually proved by Euclid himself as a theorem.”4 He recognizes that it has a kind of prima facie evidence, but that this is far from a real geometric proof. He proceeds: “It may be that some would be deceived and would think it proper to place even the assumption in question among the postulates as affording in the lessening of the two right angles, ground for an instantaneous belief that the straight lines converge and meet.” To such as these Geminus correctly replied: “We have learned from the very pioneers of this science not to have any regard for mere plausible imaginings when it is a question of the reasonings to be included in our geometrical doctrine. For Aristotle says that it is as justifiable to ask scientific proofs of a rhetorician as to accept mere plausibilities from a geometer; and Simmias is made by Plato to say that he recognizes as quacks those who fashion for themselves proofs from probabilities. So in this case the fact that, when the necessary right angles are lessened, the straight lines converge, is true and necessary; but the statement that, since they converge more and more, they will some time meet, is plausible but not necessary in the absence of some argument showing that this is true in the case of straight lines. For the fact that some lines exist which approach indefinitely, but yet remain nonsecant (asymptotoi) although it seems improbable and paradoxical, is nevertheless true and fully ascertained with regard to other species of lines. May not then the same thing be possible in the case of straight lines which happens in the case of the lines referred to? Indeed, until the statement in the Postulate is clinched by proof, the facts shown in the case of the other lines may direct our imagination the opposite way. And although the controversial arguments against the meeting of the straight lines contain much that is surprising, is there not all the more reason why we should expel from our body of doctrine this merely plausible and unreasoned (assumption)? “It is then clear from this that we must seek a proof of the present theorem, and that it is alien to the special character of postulates. But how it should be proved, and by what sort of arguments the objections taken to it should be removed, we must explain at the point where the writer of the Elements is actually about to recall it and use it as obvious.”5 We have quoted Proclus at some length, but we already have here in his words an exact appreciation and a clear statement of the whole question, such as shows that even at his time the whole attitude of the non- Euclidean Geometers was foreseen and described. Proclus thoroughly recognized the nature of the assumption, that it had not the character of a postulate, but required proof. He acknowledges its plausibility, but perceived that by the very nature of the science of geometry it required more than this, or apodictic demonstration. The plausibility or even practical certainty of the proposition was not sufficient. He even suggests the difficulty brought forward against it from asymptotic lines, and surmises that this might even lead the imagination in the opposite direction, a thing which actually occurred in the case of the founders of non-Euclidean Geometry, Lobatschewsky and Bolyai. It was a keenly critical remark wherein he states that the imagination might be led in the opposite way, thus pointing out that any non-Euclidean Geometry could rest only on imagination, and not on reason. For if the geometry of Euclid met with difficulty from lack of strict demonstration, any opposite assumption would labor under a still greater disadvantage as not having even the plausibility and experimental verification of the Euclidean postulate. In the same paragraph he also speaks of the controversial arguments against the meeting of the lines, showing that even at that early day there were non-Euclidean arguments. We have an example of such an argument given by Proclus himself which seems to show the impossibility of the meeting of the lines described in the Postulate. Here we have already the germ of the idea afterwards worked out by Bolyai and Lobatschewsky and the other neo-Geometers.6 In the passage quoted above Proclus alludes to the fact that before him Ptolemy had in a certain book set himself to solve the difficulty; we shall later recur to his method. Proclus also mentions a certain Geminus, a stoic philosopher, who wrote about 73-67 B.C.7 It is not clear from the context whether or not Geminus attempted to solve the difficulty in the parallel postulate, but at least we find in him the equidistance theory of parallels instead of Euclid’s definition of parallel lines as those which never meet. Proclus also speaks of his own attempt to solve the difficulty, for which, moreover, he felt he had found the solution. Besides these writers mentioned by Proclus as being interested in this question there are some others recalled in the writings of Simplicius. This Simplicius, who lived about 500 A.D., wrote a commentary to the beginning of Euclid’s book including what was called the parallel axiom, and this Commentary is preserved in the Arabic Commentary of an-Nairizi. He states that this postulate is by no means manifest, but requires proof and, concerning this, adds that “Abthiniathus and Diodorus had already proved it by means of many different propositions, while Ptolemy had also explained and proved it, using for the purpose Eucl. I, 13,15, and 16. He also quotes in full the attempt of a certain “Aganis” to prove the parallel postulate. It is not known who these were, with the exception of Diodorus, who was perhaps the author of the Analemma, on which Pappus, also a commentator on Euclid, wrote a commentary. Aganis is some friend or master of Simplicius; Heath conjectures that Abthiniathus, also written “Anthisathus,” might be Peithon the geometer, the friend of Serenus of Antinoeia, also called Serenus of Antissa. Of this Peithon Serenus says that in a work explaining parallels he was not satisfied with what Euclid said. Evidently he had some theory of his own, which he explained.8 There is not much actually left of all these proofs, and those which remain we shall cursorily consider, but there is enough to show that the theory of parallels evoked a great deal of discussion and opposition from the time of Euclid on. Proclus himself in his note on Euclid I, 29, states that before his time a certain number of geometers had classed this postulate as a theorem, and thought it a matter of proof. Simplicius, too, in quoting Aganis, speaks of those who even in ancient times (jam antiquitus) objected to the use of this postulate by geometers.9 Ptolemy. The most ancient attempt of which we have any record to prove the fifth postulate is that of Ptolemy, the astronomer (100-178 A.D.). Ptolemy wrote a special work on this question to which Proclus gives the title, “About the meeting of straight lines produced from angles less than two right angles.” Proclus gives two long extracts from this work, but does not give the whole proof. From what he says we know that Ptolemy used in his own proof many of the theorems of Euclid preceding I, 29. He then attempts a new proof for I, 29, and deduces the fifth postulate from this proof. Ptolemy tries to prove that if a straight line fall upon two parallel
lines it makes the interior angles upon the same side equal to two right
angles. He does this in the following manner.
He takes two parallel lines AB and CD, and lets the transversal FG fall upon them both. Then he makes three suppositions: (l) that the interior angles are greater than two right angles. Then if AFG and FGC are greater than two right angles, then BFG and FGD are also greater than two right angles. But the latter two must be less than two right angles, since the whole four angles are equal to four right angles. The supposition is therefore impossible. (2) In the same way the proof may be adapted to the second supposition where the sum is less than two right angles, and this case also is shown to lead to an impossibility, wherefore, the third case (3) is the true one, viz., that the interior angles on the same side of the transversal are equal to two right angles. He then deduces the fifth postulate of Euclid from this proposition as proved. Suppose the straight lines making less than two right angles with the line falling on them do not meet on the side on which those angles are, then a fortiori they will not meet on the other side. Hence the straight lines will not meet in either direction and are therefore parallel. If this is so, the angles made by them are equal to two right angles, which was proved in the theorem above and hence the impossibility occurs of the interior angles being at the same time equal to two right angles and less than two right angles. Proclus objects to the first argument of Ptolemy that he is not entitled to assume that whatever is true of the interior angles on one side is true of the interior angles on the other. Ptolemy justifies his position by saying that the lines are no more parallel on one side than on the other. Heath objects to this latter statement that it is equivalent to the assumption that through any point only one parallel can be drawn to a given line, which is the equivalent of the very proposition he expects to prove. The objection of Proclus is closer to the point than that of Heath. Ptolemy’s justification is that he is not making a particular assumption with regard to the two angles mentioned, but a universal assumption, that a transversal makes the interior angles on any side either equal to, or less than, or more than two right angles. This is correct as far as the statement goes, and the difficulty is not at all in this proposition; but the same cannot be said of the argument based on it. Ptolemy is here making a disjunctive argument, and his argument will be valuable only in the case that his disjunction is complete, and he is able to reject all hypotheses but the one. But here is just where he fails; his disjunction is not complete. On either side the angles are either equal to, or more than, or less than two right angles; and in combination this makes six possible cases, and of these Ptolemy rejects only two. His argument is therefore logically inconclusive. There is then a point in the objections of Proclus and Heath, but it is not contained in the form in which the objections are put. They are both objecting to the major premise of the argument, which as it stands is quite correct. But Ptolemy is not really arguing from this major, but from another one that is in his mind and is implicit in the argument. It is this: the conditions on either side of the transversal with regard to the parallel lines are the same, or the sum of the angles on either side is the same. This is equivalent to assuming the proposition that the alternate-interior angles are equal, for without this proposition his implicit major is not acceptable. As a matter of fact, the reasoning is only in form an argumentation per impossibile. The assumption its author makes constitutes a direct argument. If he had actually constructed such an argument, instead of hiding behind an incomplete disjunction, he would have discovered his assumption. What he assumes is really the major of a direct argument, and, if the assumption were allowable, this argument would be apodictic, not an argument per impossibile. It should then have this form: Wherever lines are parallel the sum of the angles on either side of the transversal is the same. But the total number of angles is four right angles, therefore there are two right angles on either side of the transversal. The assumption here is no more a legitimate postulate than that of Euclid, and is even less clear. Another weakness in Ptolemy’s argument is his deduction of the fifth postulate from this proposition. His proof again consists in a reductio ad absurdum, but the real fault lies in assuming that lines which do not meet are parallels. It is true that this is the given definition of parallel lines, but nevertheless there is in the argument a fallacia accidentis (para to symbebekos) as the predicate may not have the same extension as the subject. It is one thing to say that parallel lines do not meet; it is quite another to say that all lines that do not meet are parallels; because the question here is precisely to prove the impossibility of asymptotic lines, which is the point made by non-Euclidean Geometers against Euclid, and which is used as the basis of non-Euclidean Geometry. If the exclusion of asymptotic lines is valid in Ptolemy’s case, so it is in the case of Euclid. In other words, if asymptotic lines are a possible hypothesis with Euclid’s postulate, they are still possible with this hypothesis, and so there is nothing gained. The fifth postulate is even the simpler assumption since it only assumes the lines meet. There is therefore a petitio principii in the whole argument since it assumes just what is to be proved, that lines with which a transversal makes less than two right angles cannot be asymptotic, and that if they do not meet they are parallel. A thing to be noticed in this argument of Ptolemy is that it takes the form of an argument per impossibile or a reductio ad absurdum. This method of arguing remains with this question from this first attempt we have on record down to the present day. All the forms of proof are one or other kind of argument per impossibile. The adoption of this form of argument would seem to show that the early geometers had already tried every form of direct proof, without succeeding; for the presumption is that they would not have had recourse to a reductio ad absurdum while a direct proof was considered possible. The fact that Ptolemy had already adopted this argument, at least in form, would show that the possibility of constructing a direct argument was already considered as beyond their means of proof. For the “proof leading to the impossible” is the weakest of proofs, and is only used where no direct argument is available. Proclus. Proclus makes use of a new assumption to replace that
of Euclid, and with this he proves the fifth postulate. His assumption
is one that is built on an axiom used by Aristotle10
that, if the radii of a circumference are infinite, the interval between
them is infinite. The assumption of Proclus built on this may be thus
stated: “If from one point two straight lines forming an angle be
produced indefinitely, the distance between said lines will exceed any
finite magnitude.” Having established this he gives this proof for the postulate. In the following figure let AB and CD be two straight lines, and let EF falling on them make the angles BEF and DFE less than two right angles. Then AB and CD will meet. Draw HE such that the angle HEB will be the excess of two right angles
over the two above angles. Produce HE to K. But AB cuts HK, therefore it will also cut CD, as shown before. Therefore AB and CD will meet. This proof was criticized later by Clavius and by Saccheri on the ground that the assumed axiom itself required proof. This is true at least to the extent that it is even less simple than the assumption which it is meant to prove. The difficulty with the argument is then supposed to be that it appeals from one postulate to another not one whit more evident, and, if anything, less clear. Clavius and Saccheri argue that as one cannot assume that two lines continually approaching one another will meet, so we cannot assume that a perpendicular let fall from one on another will be greater than any assigned distance. They urge as examples of the former the hyperbola and its asymptote, and for the latter the conchoid of Nicomedes which continually approaches its asymptote, and, therefore, continually gets farther away from the tangent at its vertex, yet the perpendicular from any point on the curve to the tangent will always be less than the distance from the tangent to the asymptote. Without putting any weight on the difficulties drawn from asymptotes and their curves which are after all not two straight lines, the assumption does require proof. But the first theorem is inconclusive and incorrect, not so much because Proclus assumes another postulate, but because the postulate itself does not prove. Aristotle used the assumption legitimately to show the impossibility of a physically infinite universe. It has not at all the same meaning here. There is no such thing as an infinite circumference, and Aristotle only showed the contradiction inherent in such an assumption. Aristotle’s argument is of quite another kind and his use of the principle entirely different. He is arguing the impossibility of an infinite world, and he bases his argument on the supposed fact that the first body which would occupy the outermost sphere of the universe, has a circular motion. His argument, which is one per impossibile, is as follows. If the universe is infinite, then its radii are infinite. If the radii are infinite, then the distances between them are infinite, for the distance between infinite diverging lines must be infinite. But rotation consists in this that one part occupies successively the position of another part. But an infinite distance cannot be traversed. Therefore circular motion or rotation is impossible, because such motion in an infinite body would require that the end of one radius should occupy the position of the end of another radius, or traverse an infinite distance, which is impossible. His argument therefore deals with an assumed actual infinite, and is logically correct. Proclus’ principle deals only with the potential infinite and logically does not conclude. His radii are always necessarily finite and the distance between them is likewise finite, and the assumption does not say which is the greater finite, and hence proves nothing. Here the assumption has no connection at all with the actual infinite or the infinite in actu, but only with the infinite in potentia or possible infinite, which is only an infinity of becoming. Neither distance can ever be made actually infinite, i.e., neither the length of the diverging lines, nor consequently the distance between them. There is, however, this much true in the assumption of Proclus, that the distance between two diverging straight lines drawn from one point can always be made greater than any given distance. But then again we can always give a distance greater than this between the parallel lines. There is no absolute term either on one side or the other. Two diverging lines will diverge farther than any assignable distance, but it is equally true that we can assign a distance greater than any actual divergence. So we are where we started. The assumption therefore proves nothing, or at most only individual cases where the distance between the diverging lines is shown to be greater than the distance between the parallels. It is therefore no geometric proof, since it is not universally applicable. The assumption of Proclus has not then the character of a general postulate, particularly since the distance along the diverging lines is always greater than the distance between the parallels, if the angle of divergence is not a right angle. The distance between parallels is actually a perpendicular. Unless one of the diverging lines is such a perpendicular, the length of the diverging side will always be greater than the distance between the parallels, so that when we assign any given distance to the extent by which the parallels are separated, we shall have to assign a much greater length to the diverging line. If, then, in the supposition of Aristotle, we were to assign an infinite distance as separating the parallels, it would be impossible to have the diverging line intersect them, since this would require to be still greater than the distance from parallel to parallel. Proclus’ assumption is the reverse of Aristotle’s. But of course the idea of the infinite has no place in the theorem. The meaning simply is that we cannot postulate a longer distance as being less than a shorter distance, or a transversal other than a perpendicular so extended that it will necessarily meet another parallel, which does not require even an equivalent distance from the starting point to assure the non-meeting with the transversal. Again, the first step in the proof also requires another assumption. It assumes that parallels are equidistant or at least remain constantly at a set distance, while the converging lines continually decrease this distance. This assumes the very distinction between them which Proclus sets out to prove, which is that parallel lines remain equidistant and do not converge or meet, whereas converging lines meet and diverging lines increase their distance apart farther than any assignable quantity. He therefore assumes what he wants to prove, and is guilty of arguing in a circle. The same fault also appears in the second part of the proof. When he draws the parallel line KH, he draws it in such a way that the sum of the two interior angles is equal to two right angles, and then concludes that the lines are parallel. This may be licit if Euclid’s proofs for the parallel theory are accepted as valid, for they are previous to I, 29, where the fifth postulate is used for the first time. But as a matter of fact they are not. It is arguing in a circle to prove one part of the parallel theory by assuming another. The whole parallel theory stands together, and if the connection of the sum of the interior angles is connected with parallel lines in one instance, it is connected in every instance. If, when the sum is equal to two right angles, the lines are parallel and will not meet, then when the sum is not equal to two right angles, the lines are not parallel, and must meet. But this does not follow, because the first part of the proof has never been demonstrated, as we shall show later on. The whole parallel theory stands or falls together. The argument of Proclus is then much less satisfactory than Euclid’s frank assumption, and furthermore, since he puts aside the possibility of the asymptotic relation of the converging lines, his proof, even granting his assumptions, would not be acceptable to the neo-Geometers. With the eclipse of Greek culture we find much of their learning, including mathematics, taken over by the Arabs. Among other things we also find traces of the discussion of the parallel theory. We have already mentioned an-Nairizi (born at Nazi, died about 922) who wrote a commentary on the first ten books of the Elements. But the only one whose particular parallel theory had any influence on the later development of the question was Nasiraddin attest, who wrote a treatise on the fifth postulate. His work on Euclid was printed in Arabic in Rome in 1594, and a Latin translation of the part concerning the fifth postulate is to be found in the second volume of Walls, which was published at Oxford in 1651. Saccheri in 1733 gave a critical exposition of his proof. Nasiraddin (born at Us, in Persia, 1201-1274) has three lemmas leading up to the proof of the parallel postulate. The content of his proof is as follows. He supposes that when two lines in the same plane on which other lines fall that are perpendicular to one of the lines, form unequal angles with the other, on one side an acute angle and on the other an obtuse angle, then, so long as the two lines do not intersect, they will approach nearer and nearer on that side where the acute angles appear, and diverge more and more on the side where the obtuse angles appear. He also assumes the converse of this. If perpendiculars be drawn on one
of two lines to the other, and these perpendiculars become shorter in
one direction and longer in the other, the two lines will approach in
one direction and diverge in the other. Also each perpendicular will make
with the second line unequal angles, acute on one side, obtuse on the
other, and the approach will be on the side of the acute angles and the
divergence on the side of the obtuse angles. From this he proves that if two lines are erected at the extremities of a straight line at right angles to it, and the two lines made equal and their extremities joined, each of the angles thus formed will be a right angle, and the opposite sides will be equal. He proves this by a reductio ad absurdum from the previous lemma. If, for instance, the angle DOA is not right, it must be either acute or obtuse. Suppose it is acute. Then by the first lemma AC is greater than BD, which is contrary to the construction. In this way it is shown that all the angles are right angles and the opposite sides equal. From this he proves that in any triangle the sum of the angles is equal to two right angles. This is proved for the right-angled triangle by the preceding lemma, since all four angles of the quadrilateral are right angles; and also from the right triangle for any triangle, since any triangle can be divided into two right- angled triangles. From this by a roundabout way in which again he distinguishes three cases, he gives us his proof of the fifth postulate of Euclid. Saccheri asserts that the first assumption is not a postulate and should be proved, a thing he claims he himself has done in the second corollary of his third theorem. He finds ambiguity in the second. For why should we not consider the acute angle getting larger all the time as the perpendicular becomes smaller until we arrive at last at a perpendicular that will be common to both lines?11 In this case Nasiraddin’s argument comes to nought. Then he adds with really better sense: “If Nasiraddin can assume as evident that the angles always remain acute on the same side, why cannot one (to speak with Walls) also assume as self-evident the fifth postulate?” In the first question of Saccheri we already see the tendency to non-Euclidean Geometry. Instead of merely urging the lack of proof in the statements of Nasiraddin, he already raises the question of the opposite by inquiring why we cannot make the contradictory assumption. This question would only have to be answered in the affirmative to constitute plain non-Euclidean Geometry. It is this tendency of raising the opposite assumption as a challenge to Euclid’s statements that marked Saccheri’s method, and later culminated in the Geometry of Bolyai and Lobatschewsky. It is not a logical method except as a reductio ad absurdum. There is an immense difference between challenging the demonstrative value of a given proof, and raising the question of the possibility of the contradictory thesis. He is then the patron of those who oppose the contradictory proposition to Euclidean assumptions, although, as we have seen, the same tendency was also found among the Greeks. This is opening the door to non-Euclideanism, and through this door the metageometers entered. Nasiraddin’s lemmas do really require proof. They cannot be assumed as axiomatic any more than the postulate of Euclid. To assume that two lines continually approach if the angles formed by a transversal are less than two right angles, and that the size of the angles will remain constant for perpendicular after perpendicular, is as much of an assumption requiring proof as the thing it is meant to prove. For a continuous acute and obtuse division of angles is not analytically derived from approaching lines. The proof is therefore no more satisfactory than the others. The assumption of Nasiraddin is nearly the same as that of Euclid. Euclid really assumes two things in his postulate. The first is that, if the sum of the angles formed by a transversal with two straight lines is less than two right angles, the lines which the transversal meets, converge. The second is that, if the lines converge, they will meet. Each of these needs proof. The first is proved when it is shown that such lines cannot be parallel or remain equidistant. The second is demonstrated when it is shown that converging straight lines cannot be asymptotic. Of these two assumptions Nasiraddin postulates the first, and most important. His statement of the division of the angles formed by the perpendicular on the other line into acute and obtuse, is really the same as the condition that Euclid poses when he states that the interior angles formed by the transversal on two other straight lines are together less than two right angles. For if one of the angles is a right angle, the other must necessarily be an acute angle, if the sum of the two is to be less than two rights. That the exterior angle is obtuse immediately follows from the fact that the sum of both angles made by the transversal with one side is a straight angle or two right angles. Nasiraddin therefore postulates that when the sum of the interior angles made by a transversal with two straight lines is less than two right angles, the lines converge, which is the first assumption contained in Euclid’s postulate. The proof of the fifth postulate by this postulate is not a proof, but a petitio principii since it already assumes what it is supposed to prove. In the West there was not any critical interest shown in the question till after the fifteenth century. In 1533 appeared the first Greek edition of Euclid’s Elements accompanied by the Commentary of Proclus. This was soon followed by a Latin translation of Proclus himself by Barozzi. Peter Ramus12 and the Jesuit Clavius (1537-1612) called attention to criticism of Euclid’s geometry. The latter edited Euclid, though not as an exact translation, with a vast amount of notes from previous commentators, as well as some good additions of his own. From 1574 to 1738, there were no less than twenty-two editions printed of his work. He changed the Euclidean definition of parallel lines into that of lines that are equidistant from one another, and for this employed a figure that was used later by Giordano da Bitonto and by Saccheri. His own definition of parallels was attacked by Saccheri, but of this when we treat that question. We have already seen how he demanded proof of the assumption of Proclus and objected to it the relation of the conchoid of Nicomedes to its asymptote. From this time on more attention was paid to the theory of parallels. In 1603 appeared Cataldi’s Operetta delle linee rette equidistanti e nonequidistanti, and in 1604 Oliver of Bury’s De rectarum linearum parallelismo et concursu doctrina geometrica, neither of which contained anything new. But it was especially Sir Henry Savile’s lectures that aroused interest in this question. His Praelectiones tresdecim in Principium Elementorum Euclidis Oxoniae habitae, though they do not extend beyond I, 8, discuss the assumptions in the beginning of Euclid. In this work we find: “In pulcherimo geometriae corpore duo sunt naevi, duae labes nec quod sciam plures, in quibus eluendis et emaculandis cum veterum tum recentiorum, ubt postea ostendam, vigilavit industria.” This gave the title to Saccheri’s well-known work: Euclides ab omni naevo vindicatus. Savile founded a mathematical chair at Oxford to which is attached the obligation of holding lectures on Euclid’s Elements. Walls. One of the first Savilian professors was John Walls (1616-1703). We have already mentioned him in connection with the publication of Nasiraddin’s tract on parallel lines. He is further known for his services in the calculus of infinitesimals and in connection with algebra. He also investigated the parallel theory in fulfilling the duties of his chair in the year 1663. This along with the work of Nasiraddin is to be found in the second volume of his works published in 1693. Walls begins his own proof by asserting that those who complained of Euclid’s postulate rested their proofs on assumptions that were in no wise of easier acceptance than the one Euclid postulates, and thus they fell into the fault they wished to avoid. This is very true, and the same thing has been said, and can be said with truth, of his own proof. His system consists of several propositions, and finally in a physical sliding of the two lines forming the interior angles less than two right angles along the base towards one another till they intersect. He proceeds by the following propositions.
With these foregoing propositions and assumptions, Walls then proves
the fifth postulate. He lets the two lines be BA and CD and the transversal
making the interior angles less than two right angles be AC. For since aoC is a triangle, we can draw a triangle similar to it in every respect on the base AC, according to the above postulate, or ACP. The sides of the constructed triangle will coincide with AB and CD. But CP and AP meet at P, hence it follows that AB and CD which coincide with them, if indefinitely produced, will meet at P; or the lines meet on the side where the interior angles are less than two right angles.13 This proof is objected to by Saccheri, because Walls assumes that for every triangle we can construct one similar to it in every respect but of any magnitude. Walls’ answer is that although Euclid only proves quantities proportional in the fifth Book, and triangles similar in the sixth, he could just as well have put them at the beginning of the first Book. But of course, as Saccheri observes, this requires proof. While Saccheri’s objection is well placed, as far as it is a question of logic, his criticism has somewhat missed the point. If, as Walls stated, the problems of proportional sides and similar triangles could be proved independently of the fifth postulate, it would be a legitimate method of establishing that postulate. But that is not at all the question, nor the chief difficulty of the proof. The gist of the difficulty is that while Walls might prove that triangles are similar when their sides are proportional and their angles equal, he cannot conclude from this that for every triangle there is another triangle similar to it with any base, without assuming that which he wishes to prove, that the sides meet, if the interior angles are less than the two right angles. He may prove that they meet for the first triangle, but he cannot deduce a similarity until he has a second triangle, and this supposes just what he should prove. He cannot have sides proportional until they meet. Outside of this one point the proof, although rather long drawn out, is an ingenious one. What Walls actually did was to clothe a non-mathematical physical proof in mathematical dress. His proposition is really a practical proof from motion, i.e., a proof to experience and to the sight. All he did was to keep the sides equal and the angles equal and decrease the base till the sides actually intersected. He does however give a demonstration of the necessity of this. What he actually showed is that two lines intersected by a third and having the interior angles less than two right angles, would actually intersect on a certain base, which was so limited that it would be possible to move the two converging lines experimentally till they intersected. His physical proof from motion is clear enough, and even acquires mathematical certainty through propositions 5 and 6. Then he must show that they will intersect on any base, even where it would not be physically possible to realize the intersection by motion. Here is where his special postulate comes in; but, as we have shown, this postulate assumes precisely that which he has to prove, that as the base increases indefinitely, the other two lines will still intersect, if prolonged. His fault lies in attempting to universalize his conclusion; and his adoption of a postulate to enable him to do so, is clear evidence that he had not proved the proposition. If he had given a proof, he could have immediately universalized it, as in all true geometric proofs. If he had proved that the two lines and the transversal formed a triangle in one case, he could immediately conclude that they would do it in all cases. He was aware that he had not proved it, and therefore assumes a further postulate. To universalize his proof, he assumed what he had to prove, and thus, as so many others, made himself guilty of a petitio principii, or begging of the question. His proof is therefore no proof. Walls’ proof is an empiric proof rather than a geometrically apodictic one. It is a physical proof that holds within the limit of experience. For the rest it is only plausible. But its strength lies in its appeal to the imagination. The imagination can easily perceive the meeting of lines that are inclined to one another on a small base, but it boggles when the base becomes so large that it cannot apply ordinary measures. For the intellect this would not hold, and if the proposition were proved necessary geometrically in one case it would hold just as necessarily in all, no matter what the measure. For if we have two lines so conditioned that they will necessarily meet because of those conditions, they will always meet when the conditions are present. It is for this reason that we state that Walls did not have a mathematical proof, but a proof from experience rigged up in mathematical form. His demonstration is more an illustration of a fact, than the deduction of a theory. This is what Walls did; he turned what was indefinite and inform in the imagination into the definite picture of a triangle. It makes the postulate physically certain, but not mathematically or metaphysically certain, as is required. In Italy also the parallel theory aroused discussion. In 1658, Giovanni Alphonso Borelli (1608-1679) published his Euclides Restitutus, in which he criticized Euclid’s parallel theory. His objection to the definition of parallels as lines that never meet, is that it brings in the problem of infinity, and he suggested as the proper definition, coplanar straights with a common perpendicular. This, he contends, is something which the human mind can grasp more easily than the other, which requires the notion of infinity. But as he found this was not sufficient to establish Euclid’s postulate he had to make another assumption, one that had already been made use of by Clavius: “If a straight line fall perpendicularly on another straight line, and is so moved that one extremity always remains on that straight line, if the right angle remain constant, the other extremity will describe a straight line.” Of this Saccheri observes that it must be proved that “the geometrical locus of points equidistant from a straight line is a straight line.” Clavius indeed saw this and tried to prove it, but it was simply assumed by Borelli. From this time on, most of the text books on geometry describe parallel lines as lines that are equidistant, instead of using the old definition of Euclid. The first one that seemed to recognize that such a definition can be applied only when the possibility of such lines is shown, was Giordano Vitale da Bitonto, who also wrote an Euclide Restituto in 1680. At the extremities of a straight line he erects two perpendiculars of equal length. He then attempts to prove that the line joining these extremities is everywhere equidistant from the ground line. His argument is rather involved and inconclusive, and it requires a further postulate to replace the postulate he wishes to prove. His argument tries to show that if the locus of the extremities of the perpendiculars is any kind of a curve, these cannot be of equal length. For this proof he uses the quadrilateral figure and curves already used by Clavius, and further developed by Saccheri. We shall recur later on to this argument. Saccheri. Girolamo Saccheri was born on the fifth of September, 1667, in San Remo, which then belonged to the Republic of Genoa. In 1685 he joined the Jesuits, and after his novitiate was a teacher of grammar in the Jesuit Collegio di Brera in Milan. Later he was a professor of philosophy and polemic theology in the Jesuit college at Turin. In 1697 he came to the college in Pavia, in which city he gave lectures on arithmetic, algebra, geometry, and other mathematical subjects. He died in Milan, October 25, 1733. He wrote several works on different subjects, theological, philosophical and mathematical, among others a Logic entitled Logica demonstrativa theologicis, philosophicis, et mathematicis disciplinis accomodata. The work for which he is chiefly known is the now famous Euclides ab omni naevo vindicatus: sive conatus geometricus quo stabiliuntur prima ipsa universae geometriae principia, published at Milan in 1733. This work consists of two books. The more weighty of these is the first, which attempts the proof of the parallel theory, the first “blot” of Savile; the second, a shorter one, concerns itself with the theory of proportion, the second “blot” of Savile. This work was well known long after his own time. We find mention of it both in France and Germany throughout the eighteenth century. But it was chiefly when his fellow countryman Beltrami in 1889 called attention to the fact that Saccheri was a precursor of Lobatschewsky, that his fame became widespread. The work of Saccheri on the parallel theory has since then been republished in the German work of Engel and Stäckel, Die Theorie der Parallellinien von Euclid bis auf Gauss, 1895; in an Italian version L'Euclide emendato del P. Gerolamo Saccheri, by G. Boccardini, Milan, 1904; and in Latin and English by the translator George Bruce Halstead (Open Court, Chicago, 1920). Saccheri’s proof took the form of considering three possible hypotheses with regard to the truth of the parallel postulate. He takes as a fundamental figure in his proof a two right-angled isosceles quadrilateral ABCD, in which the angles A and B are given as right angles, and the sides AC and BD equal, and also perpendicular to the line AB. It is then a matter of easy proof to show that angles C and D are equal, because the triangles formed by the diagonals are equal. He then considers his triple hypothesis:
From this he starts with the three hypotheses, which he terms respectively (l) the hypothesis of the right angle, (2) the hypothesis of the obtuse angle, (3) the hypothesis of the acute angle, and to each there corresponds a certain group of theorems. His aim is to establish the first hypothesis, by showing that the application of either of the other hypotheses will result in a contradiction. He then successively shows that CD is either equal to, or less than, or greater than AB according to each hypothesis. He next shows that whichever of the three hypotheses is true in one case, the same is true in all cases, and that according as one or other of the hypotheses is considered true, the sum of the angles of the triangle is either equal to, or greater than, or less than two right angles. Accordingly, as either one of these is proved true, the hypothesis of the right, obtuse, or acute angle follows. From there he goes on to show the impossibility of the hypothesis of the obtuse angle. It is false because it is contradictory. For his proof makes the parallel postulate apply in two cases, or shows that two lines intersected by a third and forming interiorly less than two right angles with the intersecting line will meet in both the hypothesis of the right angle and of the obtuse angle. Thence the hypothesis of the obtuse angle is shown false because it is contradictory, since, if the postulate is true, the sum of the angles of a triangle is equal to two right angles, which is contrary to the hypothesis of the obtuse angle. His aim was next to show the hypothesis of the acute angle to be likewise contradictory. It took him much longer to dispose of this hypothesis, and his proof is not so satisfactory. He builds a whole series of propositions on this assumption in an endeavor to work out some contradiction. Finally, after a long and tedious operation, without very much success, he comes to the conclusion that the hypothesis of the acute angle is also false, because it contradicts the Euclidean notion of the straight line. (Prop. 33.) Saccheri’s is really the most unsatisfactory of all the attempts to establish the fifth postulate. It is a long drawn out argumentum per impossibile, by far the longest of all, which, in final reckoning, is no more satisfactory or convincing than any of the others. His attempt would have long ago faded into obscurity, were it not for the rise of non-Euclidean Geometry, and for the fact that non-Euclidean Geometers have come to regard him as one of their precursors. His work was in reality forgotten until it was again hauled forth by Beltrami in 1889 in a tract entitled, Un precursore Italiano de Legendre e di Lobatschewsky. Certain points in their encomiums should be noticed. Saccheri is supposed to have given the whole question a new direction. The quest for a proof up to his time was on the accepted axioms of Euclid, or equivalent ones, but Saccheri introduced three possible hypotheses. His importance then comes from the fact that he was supposed to be the first to contemplate these hypotheses. But this view is not exact. He was by no means the first to use the three hypotheses. As we have seen, Ptolemy already used them in his argumentum per impossibile, but he placed them where they really belong, on the angles formed by base and intersected lines. The same may be said about the fact that in Saccheri’s theorems there occurred for the first time straight lines which are asymptotes of one another.14 As a matter of fact the hypothesis of asymptotes was very clearly stated in ancient times by Proclus. What Saccheri really did was to emphasize certain particular points that had always been more or less in view, and which now came to be used as the basic principles of the new Geometry. He puts them in a much more positive way. What the ancients considered a mere possible hypothesis in order to establish the truth of their theorem, he elaborates and makes more concrete. He pushed his methodic doubt so far as really to construct a mathematics of doubt, and he thus became the Descartes of the new skeptical mathematical movement. It was not of course that Saccheri actually doubted the truth of Euclid’s postulate, but he carried methodical doubt so far practically as to establish a geometry in large part that is akin to the non-Euclidean. We have already seen an example of this. In his critique of Nasiraddin he makes his objection already take a non-Euclidean form. His doubt of the proving power of the argument is changed to expression of doubt of the theory itself and he proposes as an alternative the opposite proposition. In this he was really the precursor of the non-Euclideans. His methodic doubt became actual doubt, then assurance of the truth of the opposite. His real importance, therefore, and his increased modern reputation, lies not so much in the fact that he was a successful defender of Euclid, as that he showed the way to construct a geometry that would be the very opposite of Euclid’s. For in working out his different hypotheses of the obtuse and acute angles, he elaborated certain propositions that form part of the new Geometry, particularly that of Bolyai and Lobatschewsky.15 His reasoning approaches theirs most closely in his theorems 23, 30, 31, and 32.16 These propositions were established anew by Lobatschewsky and Bolyai. But what Sacchari established provisionally, the other two accept definitively as a new Geometry. It was not then the Euclidean elements of his geometry, or his Euclidean leanings that has made him popular with the moderns; for they refuse to accept either of his proofs as having any value, either that of the obtuse angle, which was later adopted by Riemann as the basis of his Geometry, or that of the acute angle where all acknowledge that he failed. They consider that Saccheri is important from the fact that he had a broad and complete non-Euclidean view of the parallel theory in all its generality, more even than Bolyai and Lobatschewsky, who excluded the case of the obtuse angle, which later became the hypothesis of Riemann, while Saccheri has both. He is, therefore, in a sense both the precursor of these two and of Riemann as well. They consider that it was only his prejudice in favor of Euclidean geometry that clouded his supposedly acute geometric vision, and kept him from establishing non-Euclidean Geometry a century before Bolyai and Lobatschewsky. It is this point of view that has made Saccheri renowned among the moderns, and not the value of his work in establishing the fifth postulate. In this there was not much of original. What Saccheri did was to construct
a quadrilateral which was really a parallelogram, and make the three hypotheses
with regard to two of the angles, while arbitrarily determining the other
two. Saccheri had more success in proving the impossibility of the hypothesis of the obtuse angle than he did of the acute angle. His proof here has been generally acknowledged as valid, at least till the advent of Riemannian geometry, according to which it no longer holds, since it is supposed to postulate the infinity of the straight line. Riemann assumes the opposite of this! and hence the proof of Saccheri does not hold for him. When the assumption of the infinity of the straight line is dropped, then Riemann’s hypothesis becomes possible. We shall examine this particular point when we come to it. Saccheri believed also that he had done away with the hypothesis of the acute angle; but none of his successors accepted this point of view. It was supposed to be false, as he says in his thirty-third proposition, because it contradicted the nature of the straight line. If the hypothesis of the acute angle were true, then two lines could enclose space, but by the nature of the straight line two lines cannot enclose space. But this is a postulate that is not admitted and is not valid in the hypothesis of the acute angle, which he actually puts forward. His own hypothesis thus destroys his proof. The reductio ad absurdum of Saccheri is the longest and most far-fetched of all the attempts to prove the fifth postulate. This results from his method of proof. There is no reason why, if the parallel postulate is a regular and natural consequence of our conceptions of space, it should not be as capable of the same kind of demonstration as any other assumption that finds its place in geometry. The reason why there was not more success in finding the necessary proof is that the wrong method had been followed all along, and it could lead to nothing else than to the conclusions of the modern objectors. Such proofs, particularly when failures, made it appear as if the parallel postulate was not a real integral part of the science of space. Still later on interest in this question increased. In the latter half of the eighteenth century Abraham Gotthelf Kaestner drew attention to it in Germany. Kaestner (1719-1800) recognized the difficulty in the parallel theory. In 1758, he published Anfangsgründen der Arithmetik und Geometrie, and in the Preface he states that, after busying himself for many years with the parallel theory, in seeking proofs both in the works of others who had treated the question, as well as by his own researches, he could find no better method than one like that which we have shown Walls to have used. This method consisted in moving one of the two lines, which together with the base formed two angles whose sum is less than two right angles, along the base toward the other; and when it arrived within a certain distance both lines were found to intersect. To this he adds: “One cannot see why a mere longer base line can make the triangle impossible, but the conclusion should rather be that a longer line would only make longer sides to the point of intersection.”17 He thus shows that he perceived the real argument of Walls, and that it is insight which makes us inclined to accept the proof. It is not a geometric demonstration, but a statement of common sense, made more clear for the imagination to grasp. Kaestner’s interest in this question had caused him to gather together a library of works on this subject, and in 1801 a list of these was published, which contained over seven thousand titles, nearly everything that had appeared on this subject down to 1770. Moreover, he cooperated with Klügel in editing a history of the parallel theory, Conatuum praecipuorum theoriam parallelarum demonstrandi recensio, quam publice examini submittent Abraham Gotthelf Kaestner et auctor respondens Georgius Simon Klügel (Göttingen, 1763). Thirty attempts were criticised and the conclusion arrived at was that all had failed. The general conclusion seemed to be that it is not possible to prove the parallel theory by strict argument, but that we must accept it as a result of experience and ocular proof. For Kaestner assures us that nobody in his senses would really deny the postulate. The dissertation of Klügel was the occasion of calling the attention of Lambert to the question. Johann Heinrich Lambert (1728-1777), also known for his Logic, wrote a work on the parallel theory, Theorie der Parallellinien, which, though written in 1766, was not published till after his death, probably because it did not quite satisfy him. It was given to the public in 1786 by John Bernoulli, a grandson of the well known Basle mathematician of the same name.18 The work is divided into three parts. In the first the author explains what is meant by a proof of the fifth postulate. In the second part he discusses various attempts in proof of the parallel axiom, and shows that in carrying out the demonstration there is always something wanting, and that the really important thing in the proof. In the third part, which is his principal effort, we have the development of Lambert’s own theory of parallel lines. In this part Lambert discusses the same three hypotheses as Saccheri, with whose work he was evidently acquainted. But while Saccheri starts from a quadrilateral with two right angles determined, Lambert assumes one of three right angles, then the fourth is considered by hypothesis to be either a right angle, an obtuse angle, or an acute angle. These three hypotheses he treats separately in a number of theorems, in order to show that the two latter are contradictory, and the first hypothesis the only true one. Lambert follows out the latter two hypotheses, that of the obtuse and the acute angle, even further than Saccheri, especially concerning the area of the triangles in the three hypotheses. He recognizes that the area in the case of the obtuse and acute angle is proportional to the difference of the sum of the angles of the triangle and two right angles. And he remarks that the second hypothesis holds if a spherical instead of a plane triangle is taken, where the sum of the angles is greater than 180 degrees, and the excess in area is proportional to the excess of the angles over 180 degrees. This he claims can be proved independently of the parallel theory. He also comes to the conclusion that the third hypothesis holds of an imaginary spherical surface. We hear of both these theories again in the new Geometry. The spherical surface represents the hypothesis of Riemann, and the imaginary spherical surface the theory of Lobatschewsky and Bolyai. Just as in the case of Saccheri, Lambert is important and has been rescued from oblivion because some of his propositions were those favored by the non-Euclidean Geometers, and not for his success in establishing what he set out to do, to demonstrate the truth of the fifth postulate. Just as he remarked concerning the others that there was always something left in the proof that itself should be proved, so in his own case, that which he assumes without proof, is that a circle can be drawn through any three points in a plane, which became an axiom of Bolyai. It is likely that he was not unconscious of the weakness in his proof, and this may be the reason he did not publish it during his lifetime. Towards the end of the eighteenth century and with the beginning of the nineteenth, there was an ever increasing interest in the theory of parallel lines; hardly a year went by without some attempt at a proof. A partial list of these up to the year 1837 can be found at the end of Engel and Stäckel’s Theorie der Parallellinien von Euclid bis auf Gauss. They did not precisely cease at this period, but the non-Euclidean Geometry began to hold the scene, and they have not had the same publicity or importance since then. A further list leading up to the year 1878 can be found in an article of G. B. Halstead, published in the American Journal of Mathematics, Volumes I and II, entitled, “A Bibliography of Hyper-space and non-Euclidean Geometry.” Bonola brought the list up to the year 1900. This discussion also drew the attention of the French encyclopedists. D'Alembert took up the question in the article, “Parallèle,” in the Dictionaire encyclopédique des Mathématiques, Paris, 1789; in an earlier work in 1759, entitled Mélanges de Litérature, d'Histoire et de Philosophie, in which he wrote an essay on the elements of geometry, he stated that “the properties of the straight line as well as of parallel lines was the rock of scandal of elementary geometry,” and he added that one might define parallels as lines perpendicular to another line, but it would have to be proved that all other perpendiculars were equal to this perpendicular line. Lagrange also attempted to prove the fifth postulate. He recognized that the formulas of spherical trigonometry were independent of the parallel theory, and hoped therein to find a proof for it. All other methods of proof he considered insufficient. Towards the end of his life, Lagrange composed a treatise on parallel lines. He began to read it in the Academy, but he suddenly stopped and said: “Il faut que j'y songe encore,” and put the paper into his pocket. Laplace also concerned himself with the foundations of Euclidean geometry. He attempted to prove the fifth postulate in his Exposition du système du monde en 1834. Like Walls he used the principle of similarity, as also did Carnot in his Géométrie de Position in 1803. But the best known of all the modern attempts were those of the great geometer, Legendre. Adrien Marie Legendre (1758-1833), whose Elements were known and used everywhere throughout the nineteenth century, spent his life in attempting to prove Euclid’s postulate. His attempts appeared in the successive editions of his Éléments de Géométrie from the first in 1794 to the twelfth and last in 1823. Later, in 1833, he published a collection of his different proofs under the title Réflexions sur différentes manières de démontrer la théorie des parallèles. This appeared in the Mémoires de l'Académie Royale des Sciences, XII, pp. 367-410. From the beginning he recognized, as did Saccheri and Lambert, that the fifth postulate had an essential connection with the theorem that the sum of the three angles of a triangle is equal to two right angles, and in his first edition he gave an analytic proof of this, relying as did Walls and Laplace on the principle of similarity, under the form that the choice of a unit of length does not affect the correctness of the proposition to be proved. This is equivalent to Walls’ assumption of similar figures. A similar analytic proof is found in the notes to the twelfth edition. In the second edition of the Elements he proved the postulate of Euclid by assuming that, given three points not in a straight line, there is a circle passing through all three. In the third edition he gives a real geometric proof that the sum of the angles of a triangle cannot be greater than two right angles. He replaced this later by another, which consisted in the continued application of the construction found in Euclid’s I, 16. This proof shows that in any number of successive triangles evolved, while the sum of the angles always remains equal to the sum of the angles in the original triangle, one of the angles increases continually towards two right angles and the other two diminish. This proof is found in the sixth edition. But Legendre found greater difficulty in proving the third of the three hypotheses that the sum of the angles of the triangle is not less than two right angles, and this is just as essential to the whole proof as the other. For this he starts with a given triangle and constructs another triangle that contains the first at least m times. If there is a deficit in the angles of the triangle from two right angles, this deficit will be proportioned to the area of the triangles. Then the deficit of the new triangle which is m times the original triangle, will be m times the deficit of the original triangle, until the sum of the angles of the greater triangle will diminish progressively as m increases, until it becomes zero or negative. Hence the reductio ad absurdum. But for this he had to assume that we can always draw a straight line from a point on one of the sides of the angle to the other side, when the angle is less than two-thirds of a right angle. This is, however, not an immense supposition, since Euclid assumes in the first postulate that a straight line can be drawn from any point to any point. The difficulty with this proof caused Legendre to abandon it in the ninth and return to Euclid’s method. But again in the final edition he returns to the plan of constructing a series of successive triangles where the sum of the angles of each triangle remains constant, but two of the angles become smaller and smaller, and the third larger and larger, till at the limit the two vanish entirely and leave only a straight line or two right angles. In other words Legendre makes use of the doctrine of infinitesimals and the doctrine of limits; the argument seems pretty conclusive practically, although not a geometric proof. But it has not been accepted.19 In the conclusion to his treatise on parallel lines in 1833, mentioned above, when Legendre claimed that after two thousand years of useless efforts he had at last satisfactorily settled the question, he was certainly not speaking the mind of his critics, who could see neither in his methods nor in his assumptions any improvement on those of his predecessors. His proofs were, therefore, not considered as more conclusive than those of the others. His arguments are founded on assumptions that require proof as much as those of Euclid, and his methods like those of Saccheri and Lambert, are involved and difficult. They are not fitted to find place in an orderly development of the science of the elements of geometry. He follows the beaten path of the argumentum per impossibile, which, as we shall see, is neither necessary nor the proper manner of proof. Legendre ends the more noteworthy attempts to prove the fifth postulate. All the efforts we have examined make use of some other assumption that requires proof as much as the postulate of Euclid. In most of these cases the assumption is tacitly made, but in some there is an open substitution of another assumption for that of Euclid which is supposed not to be subject to the same objections. A number of these alternatives may be found on page 220, of Volume I of Heath’s Euclid. Other efforts also have been made to establish geometry scientifically by attempting to prove the propositions where the fifth postulate is needed by Euclid, without its use. As we saw before, the parallel postulate is first applied in Euclid, Book I, 29, and this proposition is used to prove I, 32, which is then essentially connected with the parallel postulate. Attempts have been made to prove both these theorems without using the fifth postulate of Euclid, and in that case the postulate could be deduced from the proofs, if they are valid. The chief of these attempts is the proof for I, 29, without the use of the fifth postulate, but by using in its stead the so-called “Playfair’s axiom.” This axiom has the following form: “Through a given point only one parallel can be drawn to a given straight line,” or “Two straight lines which intersect one another cannot both be parallel to one and the same straight line.” This axiom is so called because it was used in the Elements of Geometry, containing the first six books of Euclid, by John Playfair in 1795. It was, however, not original with him, but had been stated long before. It is distinctly found in Proclus’ notes to Euclid, I, 31. With this “axiom” assumed, the proof for I, 29 becomes simple by a reductio ad absurdum, and the same is true for Euclid’s fifth postulate, which is deduced in the same way.20 The tendency of most modern textbooks on geometry is to proceed in this manner and deduce I, 29 from “Playfair’s axiom,” and then Euclid’s postulate by means of I, 29. The question is then between the two postulates. As a matter of science the one is no more valid than the other; as between them, Playfair’s postulate may have more the appearance of simplicity, but, as shown by Dodgson,21 Playfair’s axiom involves more than Euclid’s, introduces a negative instead of a positive concept, inasmuch as it postulates lines that do not meet if produced to infinity, and therefore is not as clear to the mind or imagination as positive lines that meet. Another method also was used in the attempt to prove I, 32, independently of the parallel theory. We have seen before that the two propositions concerning the sum of the angles of the triangle and the parallel postulate are essentially connected, and if either is established the other immediately follows. One way to avoid the difficulty in the parallel theory is to establish the theorem concerning the sum of the angles of a triangle without using the parallel postulate. We have spoken of the attempts of Legendre in this line. One other proof also should be mentioned, because it has been considered favorably by several, and that is Thibaut’s method. It brings in a new argument, one based on rotation, and this method of proof has attracted the attention of many mathematicians, among others Schumacher and Gauss.22 This is also the method of Hamilton’s quaternion proof. This proof appears in Grundriss der reinen Mathematik, 1809, by Bernhard Friedrich Thibaut (1775-1832). It supposes a straight line CB produced to D.
But there is also an assumption in this. The rotation is not about one point but about three different points. There is a motion of translation along with the rotary motion. It must, therefore, be assumed that the motion of translation does not affect the rotation and that the motion of translation may be neglected. The assumption is of course true, but the author is not entitled to assume it any more than any of the other assumptions made to prove this proposition, or the fifth postulate itself. It is more an illustration or a proof to the eye that the sum of the angles of a triangle is equal to two right angles, than a real geometric demonstration. This closes the list of those who have attempted in one way or another
to fill the gap in Euclid’s Geometry. To sum all up in the words
of Halstead: “To demonstrate this latter assumption (the fifth postulate)
recourse has been had to many different procedures. All these demonstrations
are without exception false, defective in their foundations, and without
the necessary rigor of deduction.”23 If we put aside the word “false,” we may let
the rest stand as the final judgment of science on the history of these
attempts. Lack of success in proving a thing apodictically does not mean
the falsity of the proposition used, nor of the conclusions reached. It
simply means that they are not sufficient. It is not false to substitute
another postulate for Euclid’s, but simply no improvement. As Euclid’s
position is not false, so neither is the other. This is the chief error
of modern non-Euclideans. But we can readily concur with them in their
judgment of the unsatisfactoriness of all these proofs. The methods employed
alone would suffice to condemn them as not of a kind to find a place in
Euclid’s classic structure. We shall now turn our attention to the
origin and development of non-Euclidean Geometry from this unsatisfactory
state of affairs.
References |
With
this postulate, which he considers an axiom, he proves that if any straight
line cuts one of two parallels, it will cut the other also. Let AB and
CD be parallel. Let EFG cut AB, then it will cut CD also. For, since BF
and FG are two diverging straight lines from the point F, if produced
indefinitely, they will be further apart than any finite distance, and
therefore greater than the interval between the parallels. Whenever then
the distance becomes greater than the distance between the parallels,
the line FG will cut CD. 
Then KH and CD are parallel since the interior angles are equal to
two right angles. 
Then he slides the part BAC along AF, until the point A coincides
with a, and AB assumes the position of ab; it then intersects
CD at o and forms the triangle oaC. Hence also CPA is a
triangle, or the lines meet if indefinitely prolonged. 
In this he uses the method
pointed out to him by Clavius and Giordano da Bitonto. They attempted
to show that the figure is a parallelogram, by trying to prove through
a reductio ad absurdum that the line connecting the extremities
of the two perpendiculars erected at the extremities of the transversal
is a straight line, and further by trying to show that with any kind of
a curve, the perpendiculars from the ground line to this curve could not
be of equal length, which was contrary to their accepted definition of
parallel lines. Saccheri tried the same proof by the three hypotheses
regarding the angles formed by this line connecting the extremities of
the perpendiculars erected on the ground line. 
Then let BD of any length
be rotated about the point B till it arrives at the position BA, then
let it rotate about A into the position AC produced both ways, and finally
about C into the original position CB produced both ways. Then it has
revolved through the sum of the three exterior angles of the triangle.
But since in the revolution it has assumed its original position, it has
passed through four right angles. Therefore the sum of the exterior angles
of a triangle is four right angles. From which it follows that the sum
of the interior angles of the triangle is equal to two right angles.