Non-Euclidean Geometry: The Forgotten Story

Non-Euclidean geometry is a well-established notion in modern mathematics and science. However, this is a relatively recent development and was not always the case. In fact, the history of non-Euclidean geometry had remained controversial for the majority of its duration.

In this essay, we dive into the origin and the story of how the notion of non-Euclidean geometry established itself in modern mathematics and science. This fascinating story begins with Euclid himself and strings along an assorted list of mathematicians and ends with a physicist. Let us begin.

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The Origins of Non-Euclidean Geometry

Our story begins with Euclid’s Elements. As hard as I have tried, I have found Euclid’s Elements tough to read. Either simplicity was not his strong point or I am not smart enough to “get it” quickly. Regardless, in the context of this essay, we need only to focus on Euclid’s fifth postulate.

Since Euclid’s version of the postulate might be too much for us to handle, I prefer Martin Gardner’s simplified interpretation:

“Through a point on a plane, not on a given straight line, only one line is parallel to the given line.”

In Elements, Euclid states that this postulate is not a theorem, but an axiom. According to him, it has to be accepted without proof. That sounds simple, right? Except, it is not!

Euclid may not be known for his simplicity, but he was a genius. His work was way ahead of his time, and his stance on his fifth postulate was a clear illustration of his genius.

This became apparent when mathematicians for the next couple of thousands of years tried to reduce the axiom to a theorem. Quite a few of these mathematicians thought they were successful, but eventually realized that they had just made an assumption equivalent to Euclid’s own.

We can understand the weight of this issue by looking at a couple of historical examples.

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How NOT to Prove Euclid Wrong?

One of the earliest attempts to prove Euclid wrong is attributed to Thales of Miletus. This proof is based on the assumption that a quadrilateral with four right-angles exists; in other words, that a rectangle exists. It turns out, however, that rectangles cannot exist without Euclid’s fifth postulate.

Fast forward to the 17th century, English mathematician John Wallis thought that he had proven Euclid wrong. His proof was based on the assumption that two triangles could be similar but not congruent. As it turns out, this assumption too was impossible without accepting Euclid’s fifth postulate.

Moving to the 19th century, Hungarian mathematician Farkas Bolyai dedicated almost all of his life to decoding this challenge. He had often discussed the problem with a childhood friend, Karl Friedrich Gauss (also known as the Prince of Mathematics).

In a sad plot twist, Bolyai’s son János Bolyai also became obsessed with this problem and was giving it his all. His father famously begged him to give up on this problem:

“Fear it no less than sensual passions because it too may take all your time and deprive you of your health, peace of mind and happiness in life.”

— Farkas Bolyai to János Bolyai

How Non-Euclidean Geometry Was Born

János Bolyai did not listen to his father and pressed on. Eventually, in 1823, he worked out that Euclid’s fifth postulate was independent of the other axioms. Furthermore, he came up with a new system where an infinity of parallel lines could pass through the same point. This would later become known as (an instance of) non-Euclidean geometric space.

Convinced by his son’s arguments, Farkas Bolyai suggested that János publish his results as soon as possible. The father knew all too well about how discoveries and attributions worked in the world of mathematics. More often than not, it was the first person who published a discovery who got the result, not the one who discovered it first (related to this point, check out my essay on the thrilling story of calculus).

The duo planned to add János’ results in the appendix of a book that Farkas Bolyai was to publish. However, due to various reasons, they did not publish this book until 1832. In the meantime, Russian mathematician Nikolai Ivanovitch Lobachevski had published about it and taken credit.

To add insult to injury, when Bolyai wrote about his son’s findings to Gauss, the Prince replied that he had already worked this result out many years ago. He further mentioned that he had not published the result because he wanted to spare himself from having to convince conservative “crank” mathematicians of the time.

Initially, János Bolyai lost trust in his own father over the issue. However, after seeing ample evidence, János realized that Lobachevski had indeed independently and fairly arrived at the same conclusion.

In an even sadder account, Eric Temple Bell recalls the story of Italian Jesuit Giralamo Saccheri, who had almost completely worked out the concept of non-Euclidean geometry in the early 18th century but essentially dismissed his own discovery in fear of rejection from his conservative peers of the time.

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Two Alternate Approaches to Euclid Emerge

By the end of the 19th-centuty, mathematicians had established that Euclid’s fifth postulate was indeed independent of the other axioms. Furthermore, two alternative approaches to Euclid’s had emerged:

1. An infinite number of parallel lines through the same point in a plane — as established by camp Bolyai, Lobachevski, and Gauss.

2. No line can be drawn parallel to the given line through a point in a plane — established simultaneously by Georg Friedrich Bernhard Riemann and Ludwig Schläfli.

The first approach led to what is now known as a hyperbolic non-Euclidean plane. Such a hyperbolic plane features triangles whose angles sum up to less than 180°. Furthermore, the circumference of a circle on this plane is greater than (π * Diameter), and each line is still a geodesic (shortest path, given end-points).

The second approach led to what is now known as an elliptic non-Euclidean plane. Such an elliptic plane features triangles whose angles sum up to more than 180°. Furthermore, the circumference of a circle on this plane lesser than (π * Diameter), and every geodesic is finite and closed.

Mathematicians had shown that if Euclidean space is consistent, then so are these two alternate spaces. This led to division among mathematicians. This turned into a breeding ground for generations of “crank” mathematicians who kept trying to convince the world that Euclidean geometry was the only acceptable form of geometry there was.

Non-Euclidean Geometry Establishes Itself in Contemporary Science

Eventually, scientists started measuring our reality’s geometrical space to see if it was Euclidean or non-Euclidean. Measurements indicated that our reality was indeed made of non-Euclidean physical space.

“Change” always comes at the cost of overcoming inertia. It was not just the “crank” mathematicians. Mathematical genius, Jules Henri Poincaré held the opinion that even if empirical experiments indicated that our physical space was non-Euclidean, it would be advantageous for us to preserve the simpler Euclidean space assumption.

To give you a tangible example of one of the consequences of this assumption, light rays would not follow geodesics.

Many prominent scientific figures of the time sided with Poincaré’s opinion. But then came Albert Einstein with his relativity theory. After an initial divide among physicists and mathematicians, Einstein’s theory of relativity was proven right.

This relativity theory massively simplified the entirety of our known-physics by modelling a non-Euclidean physical space. All of a sudden, it was not convenient to assume Euclidean space anymore. Most mathematicians and scientists changed their view on non-Euclidean geometry in light of evidence that the relativity theory established.

But a small minority of “crank” mathematicians and physicists continued (continue?) to argue against the notion of non-Euclidean geometry and Einstein’s theory. Having said this, science is built upon proving current knowledge wrong. So, it becomes difficult to tell unintelligent and intelligent apart.

As we trace the eventful history of non-Euclidean geometry, it is clear that it has earned its current place in mathematics and science. As we move forward, it is easy to “take things for granted”, and assume that non-Euclidean geometry is intuitive and has always been the norm. A quick glance at its history proves that this is anything but!

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Reference and credit: Martin Gardner.

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Interesting comment form reader Dr. Rich Spiegel post publishing:

Einstein utilized both kinds of non-Euclidean geometry. This author’s #1 way (an infinite number of parallel lines, etc.) was the basis of Einstein’s thought experiment on how the universe could be imagined as finite: a two-dimensional bug crawling around and around a sphere (like Bolyai’s or Lobachevski’s) would see its cosmos as infinite but we three-dimensional beings can see it’s finite.

An infinite number of geodesic circles constitutes the latitudes and longitudes of our imperfectly circular Earth — they are “parallel” in that sense.

But spacetime, per Einstein, may be closer to Riemann’s non-Euclidean geometry (#2) where NO line can be drawn parallel to a given line — think of a “saddle” shape when a relatively large object squats, like a heavy ball on a trampoline, on the gravitation of other spatial objects in that cosmic neighborhood.

Either way, progress in math and physics required suspending the Parallel Postulate so dear to Euclid.

It is difficult to communicate the almost religious devotion that theorists once had — over millennia — to the Fifth Postulate. Doubting it could end your career.

— Dr. Rich Spiegel

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Further reading that might interest you: How To Really Use The Superellipse For Elegant Designs? and Logarithms: The Long Forgotten Story Of Scientific Progress.

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