History of the Euclidean Parallel Postulate

Transcription

1 History of the Euclidean Parallel Postulate

2 History of the EPP: Proclus Diadochus 411 (Constantinople, Turkey) 485 (Athens, Greece) Schooled in Alexandria Commentary on Euclid s Elements A major source of what we know of ancient Greek geometry Last of the classical Greek philosophers Taught Platonism

3 History of the EPP: Proclus Diadochus This [fifth postulate] ought even to be struck out of the Postulates altogether; for it is a theorem involving many difficulties... [T]he statement that, since they converge more and more as they are produced, they will sometime 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 non secant, although it seems improbable and paradoxical, is nevertheless true and fully ascertained with regard to other species of lines [for example curves like the hyperbola that has asymptotes]. Indeed, until the statement in the Postulate is clinched by proof, the facts shown in the case of other lines may direct our imagination the opposite way. And, though the controversial arguments against the meeting of the straight lines should 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 (hypothesis)? 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.

4 History of the EPP: Omar Khayyám Omar Khayyám ( ): First Persian mathematician to call the unknown shiy (meaning thing or something in Arabic) which was transliterated into Spanish as xay and later shortened to x.

5 History of the EPP: Omar Khayyám Wrote Explanations of the Difficulties in the Postulates in Euclid's Elements in Not trying to prove Euclid s V, but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle): "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."

6 History of the EPP: Omar Khayyám Studied what we now call Saccheri quadrilaterals (but which should probably be called Khayyám Saccheri quadrilaterals), and by using his postulate to eliminate the hypotheses of the obtuse and acute angles, derived Euclid s V from his postulate. On the way, he proved many theorems of what is now hyperbolic geometry.

7 History of the EPP: Nașīral Dīnal Țūsī First to treat trigonometry as a mathematical discipline separate from astronomy Wrote Al risala alshafiya'an al shakk fi'lkhutut al mutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250)

8 History of the EPP: Nașīral Dīnal Țūsī Wrote detailed critiques of the parallel postulate and of Omar Khayyám's attempted proof a century earlier. Nașīral Dīn attempted to derive a proof of the parallel postulate by contradiction. He worked with the same (Saccheri) quadrilaterals and attempted to reach a contradiction from the hypotheses of the acute and obtuse angles.

9 History of the EPP: Șadr al Dīnal Țūsī Nașīral Dīnal Țūsī had a son Șadr al Dīnwho for reasons I have been unable to ascertain was sometimes referred to as Pseudo Țūsī. In 1298 he wrote a work based on Nașīral Dīn s later thought, containing yet another attempt to prove the EPP, based on another hypothesis equivalent to EPP:

10 History of the EPP: Pseudo Țūsī Pseudo Țūsī s Hypothesis: Given two lines AB and CD with A*G*B and C*H*D, with GH CD and GH not perpendicular to AB, then the perpendiculars dropped from AB to CD have length greater than GH on the side on which makes an GH obtuse angle with AB and less than GH on the other side.

11 A G B C H D

12 History of the EPP: Pseudo Țūsī Șadr al Dīn s work was published in Italy in 1594 in Arabic, with a Latin title page. It was known to both John Wallis and Girolamo Saccheri, and could well have been part of the foundation for their work on the EPP.

13 History of the EPP: John Wallis Credited with introducing what we now call the number line and our current symbol for infinity: Wallis product:

14 History of the EPP: John Wallis Finally, (supposing the nature of ratio and of the science of similar figures already known), I take the following as a common notion: to every figure there exists a similar figure of arbitrary magnitude. (1693)

15 History of the EPP: Girolamo Saccheri , Italy Jesuit Priest Wrote Euclides ab omni naevo vindicatus (Euclid Vindicated and Freed of Every Flaw) in 1773, shortly before his death. This work was discovered in the mid 1800s by Eugenio Beltrami.

16 History of the EPP: Girolamo Saccheri Attempted to prove the EPP by contradiction, using the quadrilaterals named after him. Likely influenced by the published work of Șadr al Dīn al Țūsī. Eliminated the hypothesis of the obtuse angle, but was somewhat frustrated by his attempts to reach a contradiction from the hypothesis of the acute angle.

17 History of the EPP: Girolamo Saccheri It is well to consider here a notable difference between the foregoing refutations of the two hypotheses. For in regard to the hypothesis of the obtuse angle the thing is clearer than midday light....but on the contrary, I do not attain to proving the falsity of the other hypothesis, that of the acute angle....i do not appear to demonstrate from the viscera of the very hypothesis, as must be done for a perfect refutation.

18 History of the EPP: Girolamo Saccheri Proposition XXXIII: The hypothesis of the acute angle is absolutely false, because [it is] repugnant to the nature of the straight line.

19 History of the EPP: Johann Lambert (Alsace, France/Switzerland) First proof that π is irrational (specifically, showed that if x is a nonzero rational number, then both e x and tan(x) must be irrational. Finished Theorie der Parallellinien (Theory of Parallel Lines) in It was never published.

20 Aside: The Likeness is Uncanny.

21 History of the EPP: Johann Lambert Undoubtedly, this basis assertion [Euclid s V] is far less clear and obvious than the others. Not only does it naturally give the impression that it should be proved, but to some extent it makes the reader feel that he is capable of giving proof, or that he should give it. However, to the extent to which I understand the matter, that is just a first impression. He who reads Euclid further is bound to be amazed not only at the rigor of his proofs but also at the delightful simplicity of his exposition. This being so, he will marvel all the more at the position of the fifth postulate when he finds out that Euclid proved propositions that could far more easily be left unproved.

22 History of the EPP: Johann Lambert Considered quadrilaterals with 3 right angles, and examined the usual three possibilities for the fourth angle. He was able to reject the obtuse case (as did Saccheri), but had great difficulty rejecting the acute case. He did prove that the truth of the acute case implied that similar triangles must be congruent, which implied an absolute unit of length. Also noted that the defect in a triangle would be proportional to its area. This hypothesis [i.e. the acute case] would not destroy itself at all easily.

23 History of the EPP: Alexis Claude (France) Learned to read from Euclid. A mathematical prodigy, and a focus of great public acclaim. "He was focused," says Bossut, a contemporary, "with dining and with evenings, coupled with a lively taste for women, and seeking to make his pleasures into his day to day work, he lost rest, health, and finally life at the age of fiftytwo." Published text Éléments de Géometrie in Clairaut

24 History of the EPP: Alexis Claude Clariaut Didn t try to prove EPP in neutral Geometry, but suggested an alternative axiom: Clairaut s Axiom: Rectangles exist. Justification: We observe rectangles all around us in houses, gardens, rooms, walls.

25 History of the EPP: Adrien Marie (Paris, France) Best known as the author of Éléments de Géométrie, which was published in 1794 and was the leading elementary text on the topic for around 100 years. This text greatly rearranged and simplified many of the propositions from Euclid's Elements to create a more effective textbook. Legendre

26 Aside: Identity Theft in 1700 s Almost all biographies of Adrien Marie Legendre shows a lithograph which typically also accompanies the biography of an unrelated contemporary politician named Louis Legendre ( ). Visit /answer/record.htm#legendre for the full story, and a link to what is likely the only authentic portrait of Adrien Marie Legendre (which I shamelessly copied onto my previous slide).

27 History of the EPP: Adrien Marie Legendre Believed he could prove EPP in neutral geometry. He was unaware of Saccheri s work, and independently discovered many of Saccheri s main theorems, with different proofs. This includes what is now called the Saccheri Legendre Theorem. In his proofs of the EPP, assumed that every point interior to an angle lies on a segment joining a point on one side to a point on the other side. Unfortunately, this is not true in hyperbolic geometry, and is equivalent to EPP.

28 History of the EPP: Georg Simon Klügel (Who?) 1739 (Hamburg) 1812 (Halle/Saale), Germany Doctoral Thesis: Conatuum praecipuorum theoriam parallelarum demonstrandi recensio. In his text Klügel surveys and criticizes 28 different attempts to prove Euclid's parallel postulate. In particular he gives a thorough and detailed discussion of Saccheri's attempt (1733) almost forgotten at that time and of Wallis's attempt. Other authors considered by Klügel are Proclos, Malezieu, Nașīral Dīnal Țūsī, Segner/Karsten, Koenig, Kästner, Vitale, Hanke, Clavius, Tacquet, Cataldi, Ramus/Schoner and Wolff. In all these attempts, which are classified by Klügel according to the definition of parallelism with which they work, Klügel found points to criticize. So he concluded that nobody did better than Euclid did himself. Remains a valuable scholarly work in which a history of the theory of parallels is given for the first time.

29 History of the EPP: Farkas Bolyai , Transylvania, Hungary. His main work, the Tentamen (Tentamen iuventutem studiosam in elementa matheosos introducendi), was an attempt at a rigorous and systematic foundation of geometry, arithmetic, algebra and analysis.

30 History of the EPP: Farkas Bolyai Much of Bolyai s work was focused on parallel lines, specifically proving Euclid s V. Not surprisingly, he had little success. Although he encouraged his son, János, to pursue a mathematical career, he discouraged him from studying following him in the study of parallelism. Rather strongly.

31 History of the EPP: Farkas Bolyai You must not attempt this approach to parallels. I know this way to its very end. I have traversed this bottomless night, which extinguished all light and joy of my life. I entreat you, leave the science of parallels alone....i thought I would sacrifice myself for the sake of the truth. I was ready to become a martyr who would remove the flaw from geometry and return it purified to mankind.

32 History of the EPP: Farkas Bolyai I accomplished monstrous, enormous labors; my creations are far better than those of others and yet I have not achieved complete satisfaction....i turned back when I saw that no man can reach the bottom of the night. I turned back unconsoled, pitying myself and all mankind.

33 History of the EPP: Farkas Bolyai I admit that I expect little from the deviation of your lines. It seems to me that I have been in these regions; that I have traveled past all reefs of this infernal Dead Sea and have always come back with broken mast and torn sail. The ruin of my disposition and my fall date back to this time. I thoughtlessly risked my life and happiness aut Caesar aut nihil.

34 History of the EPP: Farkas Bolyai For God's sake, please give it up. Fear it no less than the sensual passion, because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.

35 What Mathematics Does to You: