In the Search for Beauty

Unravelling Non-Euclidean Geometry

Voldemar Smilga

• Éditeur: WSPC
• Année: 2018
• ISBN:
  9789813274372
• Sujet: Mathematics
• Sous-sujet: Geometry

This is a popular book that chronicles the historical attempts to prove the fifth postulate of Euclid on parallel lines that led eventually to the creation of non-Euclidean geometry. To absorb the mathematical content of the book, the reader should be familiar with the foundations of Euclidean geometry at the high school level. But besides the mathematics, the book is also devoted to stories about the people, brilliant mathematicians starting from Pythagoras and Euclid and terminating with Gauss, Lobachevsky and Klein. For two thousand years, mathematicians tried to prove the fifth postulate (whose formulation seemed to them too complicated to be a real postulate and not a theorem, hence the title In the Search for Beauty ). But in the 19th century, they realized that such proof was impossible, and this led to a revolution in mathematics and then in physics. The two final chapters are devoted to Einstein and his general relativity which revealed to us that the geometry of the world we live in is not Euclidean.

Also included is an historical essay on Omar Khayyam, who was not only a poet, but also a brilliant astronomer and mathematician.

Contents:

• Before Euclid — Prehistoric Times
• Euclid
• The Fifth Postulate
• The Age of Proofs. The Beginning
• Omar Khayyam
• The Age of Proofs (Continued)
• Non-Euclidean Geometry. The Solution
• Nikolai Ivanovich Lobachevsky
• Non-Euclidean Geometry. Some Illustrations
• New Ideas. Riemann. Non-Contradictoriness
• An Unexpected Finale. The General Theory of Relativity
• Einstein

Readership: All students, historians and researchers interested in mathematical physics.
Key Features:

• Presents a very clear description of the foundations of Euclidean and non-Euclidean geometry written in simple terms and accessible to any reader with a high school certificate
• A unique concise narration about the history of mathematics and not just mathematics
• Accompanied by witty funny pictures, which can help the reader better absorb the contents

Chapter 1

Before Euclid—Prehistoric Times

The true beginning of this story goes back to times immemorial.

Where was it, when and how did geometry come into being? Where, how and when did it take shape and become a science? Who was the very first to propose the axiomatic structure of geometry?

We do not know, and most likely never will.

It is generally believed that he was a Greek. But perhaps the glorified priests of Egypt or the renowned chaldean magi are the true fathers of science.

However all that may be, geometry arrived in Greece in the seventh century before the Christian era.

It was there and then that the Greeks, admirers of cold logic and the exquisite elegance of pure intellect, lovingly polished to a brilliance (or perhaps originated) one of the most beautiful creations of human thought— geometry.

Elegance indeed, yet actually the matter was far more involved and intriguing. One thing is certain, and that is that geometry sprang from practical needs.

The development of logic (and consequently geometry as well) was influenced to some extent by the Greeks’ devotion to law and oratory. But in Egypt, too, geometry was important to men of the practical world—very important. And as for endless litigations and court proceedings, the Greeks were far behind the country of the pharaohs.

In a word then, a serious analysis of this question would take us too far astray; let us be satisfied with the fact. Geometry has established itself. This is the start of a gripping, dramatic contest in pure logic that has continued for two and a half thousand years.

The history of the fifth postulate goes back just about as many years. It is as dramatic as it is instructive, a detective story with an unexpected but happy ending.

Now for the story.

Geometry, we believe, began with the Ionic school. To be more precise, its founder was Thales of Miletus who was believed to have lived close to a hundred years (either from 640 to 540 or 640 to 546 BC).

We don’t seem to know very much about him.

We know for sure he had the title of one of the Seven Sages of Greece; we also know that in accordance with the established reckoning he was the first philosopher, the first mathematician, the first astronomer and, generally, the first in all sciences in Greece. We might say that he was to the Greeks what Lomonosov was to the Russians: THE FIRST.

As a young man he most likely made his way to Egypt on affairs of trade, for he began his career as a merchant. Here, the pharaoh Psammetichus had just lifted the “iron curtain” and was beginning to allow foreigners into his country.

Thales remained in Egypt for a good number of years, studying in Thebes and Memphis. Later he returned to Greece and founded a school of philosophy. Obviously, he appeared more as a popularizer of Egyptian wisdom than as an independent thinker.

The view is that he brought with him geometry and astronomy.

At any rate, there is one thing that all philosophers can learn from him—and that is conciseness. Legend has it that his complete works (which naturally were all lost) consisted of only about 200 poems.

We can only conjecture what he accomplished in geometry, though Greek authors attributed a great deal to him.

For instance, Proclus Diadochus (we will be meeting him again) claims that it was Thales—no other—that proved the theorems that:

(a) vertical angles are equal;

(b) the angles at the base of an isosceles triangle are equal;

(0) a diameter divides a circle in half.

And some others.

Assuming even that the historians of science wrote the exact truth, we still do not know whether Thales himself arrived at these theorems or simply repeated ideas of the Egyptians.

Perhaps the only definitely established fact of the scholarship of Thales of Miletus is his prediction of the solar eclipse of 585 BC.

But legends grew up around him in hosts. And this in itself indicates that he was a scholar of stature.

One of the stories is particularly dear to men of learning. It is Aristotle who relates it:

“When Thales was reproached for his poverty, since, as they said, studies in philosophy do not create any profit, it is said that Thales, foreseeing a rich harvest of olives on the basis of astronomical findings, advanced during winter a small sum of money he had accumulated to the owners of all the oil-presses in Miletus and on the island of Chios. He was able to engage the oil-presses cheaply, for there was no competition from anywhere. When harvest time arrived, there was a sudden demand by many people for the oil-presses. Thales then rented out the oil-presses at prices that he himself desired.

“Thales thus accumulated a great deal of money and proved in this manner that it is not difficult for philosophers too to become rich, the only thing is, however, that that is not the subject of their interests.”

We do not know what Thales did with the money he made in this successful practical application of astronomy. We hope he spent it as a true philosopher would.

His pupils and followers apparently paid proper attention to geometry in their philosophical deliberations. However, the central mathematical school of the 6th and 5th centuries BC was the Pythagorean school.

The authentic biographical information about Pythagoras boils down, in essence, to a few stories. In this respect, he is much like Thales of Miletus. The obscurities begin with his origin.

Bertrand Russel sums the matter up by saying that some believe him to be the son of a wealthy citizen named Mnesarch, others the son of the god Apollo, and adds that the reader can take his pick.

It is further believed that Pythagoras lived just as long a life as Thales— something in the vicinity of one hundred years (perhaps 569 to 470 BC).

Again like Thales, he spent some twenty years in Egypt imbibing wisdom, but later (here he surpassed Thales) he lived about ten years in Baby-lonia adding still more to his store of knowledge. It is also claimed that he travelled in India, but nobody seems to believe it.

Boxers claim that Pythagoras took boxing laurels in the Olympic games, but the source of such claims is never indicated. I have nothing to support them either. As in the case of Thales, the exciting thing is the unexpected combination of philosopher, mathematician and boxer.

Pythagoras may not have done much in boxing, but in politics he did, and very actively, though not at all successfully.

The citizens of the Sicilian town of Crotona, where he founded his school after his wanderings in distant lands and also got the town involved in an exhausting war, finally asked him to leave together with his school. Which he did in rather much of a hurry, which was a reasonable thing to do.

As a mathematician and scholar he was a giant, but nevertheless he does not call forth great admiration. His Pythagorean order of philosophers and mathematicians is much too reminiscent of a barracks and Pythagoras himself suspiciously resembles a führer, though much more cultured than any of those of the twentieth century.

It is precisely Pythagoras himself—most likely in a campaign to build up his authority—who built up and popularized the idea that his loving father was the fair-haired effulgent Apollo. Actually he became the true father of the presently popular custom of attributing to himself the scientific results of his pupils. There, the matter was quite official. There existed a fiat according to which the author of all the mathematical studies of the school was to be named Pythagoras.

Though one might repeat that such things are done right and left today, the passage of 25 centuries has greatly softened and civilized the customs. The essence is the same, but the form has become ennobled.

Pythagoras is the unsurpassed leader here because he handled matters so that his faithful pupils claimed him author of work done long after his death. Quite understandably then—that being the state of affairs in the Pythagorean school—that the most cogent of all arguments was a simple reference to The Authority Himself.

That is exactly how the wording went: “He said so Himself”. After which any discussion was totally out of place—even dangerous.

He and his dear pupils also held in secret their methods for solving mathematical problems. Too, he compiled for the members of his order a long list of taboos.

I quote from the rules of good manners of the gentlemen of the Pythagorean Club:

“1. Restrain from using beans in your food.

“2. Do not pick up what has fallen.

“3. Do not touch white roosters.

“4. Do not take a bite from a whole bread.

“5. Do not walk on a highway.

“6. When removing a pot from the fire, do not leave traces in the ashes, but mix the ashes.”

The list could be extended. It was this bunch that rose to power in one Greek town, then in another, implanting the cult of Pythagoras and, accordingly, demanding compliance with their statutes. With melancholy, Bertrand Russel relates that those who were not reborn in the new faith thirsted for beans and so sooner or later rose up in arms.

It is also told that he preached to the animals, for he made little distinction between them and human beings.

But the Pythagorean school advanced geometry and mathematics in general. Very much so, in fact. All of this taken together is not a bad illustration of the danger of idealizing representatives of the exact sciences and of the intellect generally.

Incidentally, to us, Pythagoras is mainly a mathematician. Yet he himself and his contemporaries took the view that his profession was that of a prophet. That was of course their business, they were closer to events. But, as we know, every prophet must be in part magician, demagogue and charlatan.

Pythagoras was apparently past master in each field. The pupils tried hard too. According to one story, one of his hips was of gold, to another that reliable people saw him at two different places at the same time, to a third that when he was wading across a stream, the water overflowed the banks crying “Long Live Pythagoras!”

True, the Greeks had a goodly number of reasonable people.

Xenophanes, the well-travelled, realistic, freethinking and malicious-tongued philosopher and writer, spoke of Pythagoras in a rather different vein. One of his epigrams went: Pythagoras witnessing a puppy being beaten said: “Do not hit him, it is the soul of a friend of mine. I recognized it when I heard it cry out.”

The teaching of transmigration of souls is one of the basic elements in the overall conception of Pythagoras, and Xenophanes, as the reader can see, had a pointed thing or two to say on that score.

Heraclitus was very strict in his portrait of Pythagoras, “multiple knowledge without reason”.

We leave Pythagoras, but before doing so, just one more curious story by one of his honoured admirers. How devious indeed are the pathways of science. Quite naturally, geometry, like all branches of knowledge, was most carefully concealed from the common people by the Pythagoreans. Who knows, perhaps to this day no one would know of geometry (outside the Pythagoreans) if it weren’t for....

But here is the legend as to how the Pythagoreans account for the spread of geometry. One of them is to blame, for he lost the money of the community. After that calamity, the community permitted him to earn the money by teaching geometry, and geometry was given the name “the legend of Pythagoras”.

A curious thing is that there seems to have been a geometry textbook by that name.

As to the story itself, if there is a grain of truth in it at all, then, though I do not consider myself a malicious person, I would be pleased to learn that the truant Pythagorean had not lost the money after all but had spent it in a spree in the local port tavern swilling wine, eating a white rooster with beans, biting a whole roll of white bread and singing drunken songs on the highway.

Another man contributed greatly to geometry, and again to my taste he was an unpleasant character.

His name was Plato (428 to 348 BC).

In his views, in his methods of setting up a school, and in his love of self-advertisement, Plato much resembles Pythagoras. But before I say why I do not like him, let me explain what his most significant contribution to geometry is.

He is considered—and perhaps justly so, for I am not a specialist in the field—one of the greatest philosophers of Greece. Indeed he did a great deal for the development of mathematics and valued it highly. At the entrance to his Academy he had, hewn in stone, the inscription: “Let no one destitute of geometry enter my doors!” The point is that Plato believed that “the study of mathematics brings us closer to the immortal gods”, and educated his pupils in this spirit, adding mathematics where it was needed and where it wasn’t. Some of his pupils became brilliant geometers. Plato had numerous pupils a...

Table des matières

• Cover
• Halftitle
• Title
• Copyright
• Foreword
• Acknowledgments
• Contents
• 1. Before Euclid—Prehistoric Times
• 2. Euclid
• 3. The Fifth Postulate
• 4. The Age of Proofs. The Beginning
• 5. Omar Khayyam
• 6. The Age of Proofs (Continued)
• 7. Non-Euclidean Geometry. The Solution
• 8. Nikolai Ivanovich Lobachevsky
• 9. Non-Euclidean Geometry. Some Illustrations
• 10. New Ideas. Riemann. Noncontradictoriness
• 11. An Unexpected Finale. The General Theory of Relativity
• 12. Einstein
• Name Index