There is no royal road to geometry
Euclid of Alexandria said to Ptolemy I, King of Egypt, in 306 BC.
Cape Perpetua rises a thousand feet above the sea, on the rugged Oregon coast, where the powerful waves of the Pacific crash against the jagged inlets below, rhythmically like a functioning clock. Towering above the deep blue ocean, Cape Perpetua is truly unique. A person standing at the very top of this promontory would clearly see the Earth as round. The vast ocean before the observer appears to curve gently downward in every direction visible to the eye. When a ship sets sail, the observer seems to see it gradually sinking beneath the curved surface of the Earth until it eventually disappears behind the enormous blue horizon.
If the Babylonians, Egyptians, or ancient Greeks had lived along the Oregon coast, the history of mathematics and exact sciences might have been entirely different. But these ancient peoples did not live by the Pacific Ocean and never observed the curvature of the space they inhabited. The Babylonians, along with their Assyrian relatives, lived on the flat lands between the Tigris and Euphrates rivers in Babylon, and their world was flat. From the thousands of clay tablets they left behind, detailing every aspect of life in society around 4000 BC, we know that the Babylonians were quite adept at accurately calculating the areas of their fields. They knew how to divide their cultivated land into rectangles so that they could calculate the area of these rectangles by multiplying their two sides. They also understood how to find the area of right-angled triangles by halving the area of the circumscribing rectangle. The Babylonians and Assyrians were experts in this field of flat geometry. The Egyptians also excelled in geometry for surveying, partitioning, and calculating land areas. However, they lived in a flat land and never felt the need to understand an uneven surface. Even the pyramids are masterpieces of geometry consisting of straight lines in three-dimensional space.
In the 6th century BC, Pythagoras and his followers in the school they established in Croton, Southern Italy, discovered abstract theorems based on the applied works of ancient Egyptians and Babylonians. Thus, the Pythagorean Theorem is an extension of the Babylonians’ mathematical descriptions of the real world. This theorem states that the area of a square field whose side is the hypotenuse of a right triangle is equal to the sum of the areas of the two square fields whose sides are the legs of that triangle. The Pythagorean Theorem has significant implications in geometry, as it can be used to determine the shortest distance between two points in Euclidean space. In this space, the distance between two points is the length of the straight line connecting them (today, if we know the difference in the x-coordinates of the two points and the difference in the y-coordinates, the shortest distance between those points is equal to the square root of the sum of the squares of those differences). The Pythagorean followers went even further and discovered irrational numbers. They found that if two sides of a right triangle are equal, then the hypotenuse is an unusual number: the square root of 2. This is an irrational number: it cannot be expressed as a ratio between integers. The discovery of these new, incomprehensible numbers without any corresponding meaning in the real world led the Pythagorean school to the fields of mathematics that have flourished in our time.
Mathematics continued its development, and two centuries after Pythagoras, Euclid of Alexandria wrote the Elements – a 13-book series regarded as the greatest textbook of all time. The volumes of the Elements present the entire theory of geometry – a theory that guided mathematical study for 23 centuries up to our era. Euclidean geometry is an abstraction of concepts about physical space with the goal of using axioms, postulates, and theorems to investigate the essential properties of a space that people in antiquity believed to be the only space.
First, Euclid defines the basic elements of geometry such as point, line, and plane – concepts familiar to anyone who has studied today’s elementary geometry curriculum. Then, Euclid presents 5 fundamental postulates:
1) Through any two points, only one straight line can be drawn;
2) A straight line can be extended indefinitely;
3) A circle can be drawn with any center and radius;
4) All right angles are equal;
5) If a straight line intersects two other straight lines and forms interior angles on the same side that sum to less than 180o, then the two lines will intersect on that side where the angles sum to less than 180o.
The propositions or theorems in the first book of Euclid discuss the properties of lines and the areas of parallelograms, triangles, and squares. While Euclid primarily uses the first four postulates in his proofs, the fifth postulate is not used in any proof. This immediately indicates that his theorems remain valid even if postulate 5 is removed or replaced with another postulate consistent with the first four. Despite the Elements becoming a widely popular book that influenced Western thought for two millennia, the subtlety and mystery of postulate 5 have continued to raise persistent questions in the minds of mathematicians. Even the wording of postulate 5 is somewhat convoluted; while the first four postulates are concise and clear, postulate 5 is too wordy. For many, postulate 5 seems more like a theorem to be proven rather than an obvious truth.
Postulate 5 has several equivalent formulations. One is the Playfair postulate[1], which states that through a given point not on a given line, only one line can be drawn parallel to the given line. Another equivalent formulation of postulate 5 is that the sum of the angles in a triangle is always 180o. This formulation is a consequence of postulate 5 and is the easiest to remember.
From the moment the Elements was published, geometers began to doubt the necessity of postulate 5 or even questioned its obvious truth within the entire theory. The first person to make significant comments about Euclid was a geometer, through whom we learned quite a bit about the history of this book. That person was Proclus (410 – 485), a philosopher, mathematician, and Greek historian of the 5th century. According to Proclus, Euclid lived during the reign of the first Roman kingdom in Egypt, namely the Ptolemaic era, and it was this king who wrote a book discussing Euclid’s troublesome postulate 5, attempting to prove postulate 5 based on the other four postulates. This was the first effort, through historical sources, that we know of to prove postulate 5 as a consequence of Euclid’s first four postulates.
When presenting the history of Euclid’s work, Proclus accurately noted that Ptolemy’s proofs actually relied on another assumption equivalent to postulate 5: through a point not on a line, only one parallel line can be drawn to the given line (which is indeed the Playfair postulate mentioned above). Thus, Ptolemy believed his proof had shown postulate 5 to be superfluous. However, his proof was actually flawed (as it used another postulate equivalent to postulate 5).
Arab science flourished during the Middle Ages, after the great civilization of ancient Greece had faded, and before Europe awoke from the darkness for many centuries. Omar Khayyam (1050 – 1122), known in the West for his poetry, was also one of the prominent mathematicians of his time, having written a book titled Algebra (Algebra). In the preceding centuries, two other Arab and Persian scholars also pursued mathematical studies: Al-Khwarizmi (9th century) and Al-Biruni (973 – 1048), both of whom made significant contributions to algebraic theory. When Omar Khayyam died in 1122, Arab science was in decline. However, in Maragha (present-day Iran), there was an extraordinarily talented mathematician: Nasir Eddin Al-Tusi (1201 – 1274), also known as Nasiraddin. Nasiraddin was an astronomer for Hulagu Khan, the grandson of the legendary conqueror Genghis Khan and brother of Kublai Khan. Nasiraddin compiled an Arabic version of Euclid’s works and a treatise on Euclid’s postulates. Like earlier classical mathematicians and the two Arab mathematicians before him, he also doubted Euclid’s postulate 5.
Nasiraddin was the first scholar to recognize the importance of another postulate equivalent to Euclid’s postulate 5: the sum of the angles in a triangle equals 180 degrees. Like his predecessors, Nasiraddin attempted to prove Euclid’s troublesome postulate 5 as merely a consequence of the four preceding postulates. And, like his predecessors, Nasiraddin failed.
The classic book of Euclid has been extensively studied in the Arab world, leading to highly intellectual discussions about the book, including debates over the parallel postulate (i.e., postulate 5), but Europe was unaware of this. In the early 1100s, an English traveler named Adelhard of Bath (1075 – 1160) undertook a journey from Asia Minor to Egypt and North Africa. He learned Arabic along the way, then disguised himself as a member of the Islamic faith before crossing the Strait of Gibraltar to reach Spain, which was under Moorish control[2]. Adelhard arrived in Cordova around 1120 and obtained an Arabic copy of the *Elements*. He secretly translated Euclid’s book into Latin, smuggling it over the Pyrenees into Christian Europe. Through this route, Euclid’s book eventually reached the West. It was copied and came into the hands of scholars and intellectuals, and only at this point did Westerners learn of the foundational principles of geometry that the Greeks had known a millennium and a half earlier. When printing technology emerged, one of the first books to be printed in movable type was the *Elements*. When Euclid’s book was published in Venice in 1482, it was a Latin translation from the Arabic text that Adelhard had smuggled. It wasn’t until 1505, also in Venice, that Zamberti published a version of the *Elements* translated from the Greek text, which had been recorded by Theon of Alexandria in the 4th century.
Five hundred years had passed since Nasir al-Din’s work on the fifth postulate, but during these centuries, Western mathematics made very little progress. The Middle Ages were not a prosperous time for mathematics or for science and culture in general. A world embroiled in perpetual conflict and ravaged by plagues was not a conducive environment for the pursuit of knowledge and art. However, in 1733, a small book was published in Milan, written in Latin. Its title was *Euclides ab omni naevo vindicatus* (Correcting all deficiencies in Euclidean geometry). The author of the book was a Jesuit priest named Girolamo Saccheri (1667 – 1733). The book was published in the very year the author died, but this was not the only loss to society: this groundbreaking book could have changed humanity’s understanding of geometry much sooner, but unfortunately, it remained obscured in obscurity for more than a century. It wasn’t until 1889 that it was accidentally rediscovered after three mathematicians published their independent discoveries—discoveries that transformed geometry and its interpretations. These three were Gauss, Bolyai, and Lobachevsky.
While teaching and researching philosophy at Jesuit academies in Italy, Girolamo Saccheri read the *Elements*. Saccheri was deeply captivated by the logical proof method known as reductio ad absurdum (proof by contradiction) that Euclid had employed. This method is widely used in mathematics today, starting by assuming the opposite of what one aims to prove; through a series of logical deductions, one hopes to arrive at a contradiction. The contradiction will demonstrate that the initial assumption is incorrect, thus proving the opposite to be true, which is what needs to be established[3]. Saccheri was well aware of Nasir al-Din’s work from half a millennium earlier and his attempts to prove Euclid’s fifth postulate from the other four postulates. At this point, Saccheri had a brilliant idea: to use the reductio ad absurdum method to tackle the long-standing goal of proving the fifth postulate. He decided to employ his preferred method for this proof. To do so, he had to assume that Euclid’s fifth postulate was not a consequence of the other four postulates, but rather a false postulate. By this time, Saccheri had thoroughly memorized Euclid’s fifth postulate and understood the historical efforts to prove it, evidenced by his own identification of flaws in Nasir al-Din’s proof, as well as mistakes in the 1663 proof by John Wallis (1616 – 1703) at Oxford University.
Indeed, Saccheri assumed that the fifth postulate was false, hoping to find a contradiction. However, he found no contradictions at all; instead, he obtained an unusual result: there could be more than one straight line passing through a given point that is parallel to a given straight line. From this, Saccheri reached three possible conclusions, expressed in equivalent forms to the postulate, regarding the sum of the angles in a triangle. All three statements aligned with the first four postulates of Euclid: the first statement leads to a system in which the sum of the three angles in a triangle equals 180o (Euclidean characteristic, in today’s terms), the second corresponds to the sum of the angles in a triangle being less than 180o, and the third corresponds to the sum being greater than 180o. Today we know that the latter two cases correspond to two different systems of non-Euclidean geometry, each of which is logically coherent and has mathematical validity. They represent perspectives on other worlds. Saccheri achieved significant results within these systems, yet he was unaware that these were indeed new discoveries. The failure to prove the fifth postulate by contradiction stemmed simply from the fact that his assumptions were not wrong—they were mathematically accurate! Ironically, by the time these truths were recognized by mathematicians, Saccheri had long since bid farewell to the world.
Euclid’s fifth postulate, a postulate that challenged and frustrated many generations of mathematicians since Euclid introduced it in his book, actually contained within it the notion that the world is a perfect flat plane. In such a world, straight lines exist, extending infinitely, and no matter how far they are extended, they remain straight, never curving at all[4]. Imagine a perfectly flat surface; on this plane, through a given point not on a given straight line, one could draw a line parallel to the given line. These parallel lines could extend infinitely without ever meeting. On this plane, the sum of the angles in a triangle equals 180 degrees. Now imagine your plane as a flat piece of rubber, with a large sphere pushing it up from below. The rubber surface will curve along the surface of the sphere and gradually transform into a spherical surface. What will happen to the extending parallel lines? They too will curve on the sphere and tend to meet at the extended side. On the sphere, there are no great circles that do not intersect. Here, the sum of the angles in a triangle will be greater than 180 degrees. Imagine a triangle on a globe with one vertex at the North Pole and the other two vertices on the equator. The two sides are two meridians passing through the two equatorial vertices. The angle between each meridian and the equator is a right angle, or 90 degrees. Thus, in the triangle under consideration, the two base angles (on the equator) sum to 180 degrees. Therefore, if you add the angle between the two sides (the meridians), the total of the three angles will exceed 180 degrees.
The development of non-Euclidean geometry later essentially repeated what Saccheri had done. If Euclid had had the chance to stand on the peak of Perpetua and see the Earth as a sphere, the development of geometry might have been entirely different (or perhaps he knew the Earth was spherical but did not grasp the importance of this fact).
In the above examination, our original plane was deformed into a spherical shape by a sphere pushing it from below. However, it is also possible to deform the plane in a hyperbolic manner by pressing it down in the middle and stretching the surrounding areas to conform to a saddle surface. On this saddle surface, there are infinitely many “straight” lines parallel to a given line passing through a given point not on the given line. Here, the triangle will take the shape where the sum of its three angles is less than 180 degrees.
Saccheri entered this strange world unconsciously just before he died. But the crucial factor in both cases—the spherical and hyperbolic surfaces—is that the plane has been distorted. Imagine a spacious stone table with three steel rods converging at their ends to form a triangle. If someone starts a fire beneath the table, the heat from the fire will distort the rods on the table, and the triangle will change: the rods will bend due to the heat—the sum of the angles will no longer equal 180 degrees. Albert Einstein, two centuries later, used this example to illustrate the non-Euclidean nature of physical space.
In the early 19th century, Karl Friedrich Gauss (1777 – 1855), a German genius who made extraordinary contributions to science, became a prominent figure in the world of mathematics. Gauss spent decades pondering the problem of Euclid’s fifth postulate. He wrote extensively on various important works but published very little on Euclid’s challenging problem, despite investing much time and effort into it—we only know his thoughts on geometry through his correspondence. From these letters, we know that Gauss understood that the negation of the fifth postulate would lead to non-Euclidean geometries.
While studying at the prestigious University of Göttingen, Gauss befriended a Hungarian mathematics student named Farkas Bolyai (1775 – 1856). Both Gauss and Bolyai devoted much time to attempting to prove Euclid’s fifth postulate. In 1804, Bolyai thought he had found a proof and wrote it in a short manuscript, which he sent to his old classmate. However, Gauss quickly found a mistake in this proof. Undeterred, Bolyai continued his efforts and sent Gauss another proof a few years later. This proof was also incorrect. While being a professor, a playwright, a poet, a musician, and an inventor, Farkas Bolyai continued to study mathematics throughout his life, despite his failed attempts to prove this unprovable postulate. On December 15, 1802, Farkas’s son was born, named Janos Bolyai (1802 – 1860). Farkas wrote a letter to Gauss with great excitement to share the news of his son’s birth: “a healthy and very charming boy with divine gifts: black hair and eyebrows, deep blue eyes sparkling like two jewels.”
Janos grew up under his father’s tutelage in mathematics. He grasped his father’s concerns about Euclid’s fifth postulate and shared the desire to prove it from Euclid’s other postulates and axioms. In 1817, young Bolyai entered the Royal Engineering Academy in Vienna, where he devoted significant time to pursuing his father’s passionate goal of proving the fifth postulate. By that time, despite his frustrating attempts, his father had to write a letter advising him not to waste time on an impossible problem that had consumed so much of his own efforts.
But the son did not waver in response to that advice. He continued to pursue his goal passionately, hoping to redeem his father’s failed efforts over many decades. In 1820, Janos Bolyai reached a stunning conclusion. Instead of being provable as a consequence of the remaining parts of Euclidean geometry, the fifth postulate was the gateway to a marvelous garden: an Absolute Science of Space, as Bolyai called it, in which Euclidean geometry was merely a special case.
Bolyai started from Playfair’s formulation of the fifth postulate, stating that through a given point outside a given line, only one line can be drawn parallel to the given line. He then assumed that this postulate was false. This assumption meant, he concluded, that either there is no line parallel to the given line or there are multiple lines parallel to the given line. The first assumption could not occur because it can be proven that through a given point, there is always at least one line parallel to the given line, and only the second assumption could be considered a viable transformation of Euclid’s fifth postulate. If through a given point not on a given line there are two lines parallel to the given line, then there would be infinitely many such lines.
The results drawn from this assumption left young Bolyai astonished. His new geometry developed without contradiction or hindrance, as if God intended for spatial geometry to follow this astonishing non-Euclidean path. With special inspiration, he recognized that many propositions arose that had no relation to any assumption about parallel lines, and thus they became universal for all possible geometries, both Euclidean and non-Euclidean. These propositions contained the main substance of the nature of space. In 1823, Bolyai, then just 21 years old, wrote a letter to his father saying, “I have created a strange new universe from the number 0.”
Ultimately, his father expressed support by publishing his son’s pioneering work as an appendix in his own book titled briefly as Tentamen, published in 1832.
After reading the book by the father-and-son duo Bolyai, Gauss commented that he had arrived at similar conclusions during his three and a half decades of contemplation on the fifth postulate. However, there was another mathematician who also reached similar conclusions: Nicolai Ivanovich Lobachevsky (1793 – 1856). He graduated from Kazan University in 1813, which is located 400 miles east of Moscow near the Ural Mountains. Later, he became a professor and, in 1827, the rector of this university. Due to his research, Lobachevsky became renowned as the “Copernicus of Geometry.” Completely independent of Bolyai, Lobachevsky’s geometry also stemmed from the elimination of the parallel postulate, leading to a revolution in geometry. In the early 1800s, when the works of Bolyai, Lobachevsky, and Gauss became known, some mathematicians referred to this new non-Euclidean geometry as astral geometry – the geometry of stars, although the reason for this name remains unclear[5].
In Bolyai-Lobachevsky-Gauss geometry, the sum of the angles in a triangle does not equal 180 degrees. And a circle in this geometry is not the conventional (Euclidean) circle experienced in daily life: here, the ratio of the circumference of the circle to its diameter is no longer Pi.
Einstein’s line of thought began with the “happiest thought” of his life. While still at the Swiss Patent Office, he conducted one of his famous thought experiments: a rotating circle in space. The center of the circle is fixed, but its circumference rotates very quickly. Einstein compared what happens in various reference frames, a standard tool he used during the development of his Special Theory of Relativity. Using his Special Theory of Relativity, Einstein concluded that the circumference of the circle would contract while rotating. A force acts upon the circumference of the circle – the centrifugal force – and this effect is akin to the influence of gravitational force. However, it is the contraction affecting the circumference of the circle that keeps the diameter unchanged. Therefore, Einstein concluded, much to his own astonishment, that the ratio of the circumference to its diameter is no longer Pi. He reasoned that in the presence of a force (or gravitational field), the geometry of space is non-Euclidean.
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[1] John Playfair (1748 – 1819), a Scottish mathematician. His formulation of the postulate became popular and is now used in most geometry textbooks worldwide, including in Vietnam. Thus, Euclid’s fifth postulate is often referred to as the Parallel Postulate.
[2] Spain was invaded by the Moors starting in the 700s and was dominated for several hundred years afterwards. It wasn’t until around the year 1000 that the Spanish people began to rise up to drive the Moors out of their territory, and it wasn’t until 1492 that their fight for independence was fully successful.
[3] A simple algebraic example of proof by contradiction is the problem of proving that the square root of 2 is an irrational number, meaning that the square root of 2 cannot be expressed as a fraction of two integers. To start, assume the opposite, that there are integers a and b such that their ratio equals the square root of 2. Thus, a² = 2b². Without loss of generality, we can assume that these integers are coprime (they have no common factor to simplify). If a is odd, a contradiction arises immediately, since 2b² is an even number (note: the square of an odd number is odd, and the square of an even number is even). If a is even, then a = 2c for some integer c, leading to a² = (2c)² = 4c². According to our assumption, we have 4c² = 2b², which means 2c² = b², implying that b is even, and thus a and b share a common factor of 2, contradicting our assumption that a and b are coprime.
[4] The infinitude of the fifth postulate line in Euclid’s second postulate. At the end of the 19th century, the great German mathematician G.F.B. Riemann (1826 – 1866) argued that Euclidean lines could be considered boundaryless but not infinite. For example, a great circle on a sphere can be viewed as a boundaryless but finite line.
[5] In 1813, Karl Schweikart used this term to describe non-Euclidean geometry to his friend Gerling, a professor of astronomy at the University of Marburg and a student of Gauss.
