Geometry (Historical Context)

Historical Context

Euclidean Geometry

Most principles of geometry upon which mathematicians base their work today — and for the past twenty-three centuries — are related to the theories and methods first recorded around 300 B.C.E. by the Greek writer Euclid. His comprehensive work on mathematical theory, The Elements, was probably heavily based on the work of his predecessor Eudoxus, who had been a student of the philosopher Plato. Euclid refined Eudoxus’s theories, along with geometric principles that were the results of generations of mathematicians. His Elements, written in Egyptian Alexandria, has been a central influence for twenty-three centuries, from the Hellenistic world after the conquest of Alexander the Great to the Roman Empire, to the Byzantine Empire, the Islamic Empire, into the medieval world and on to today.

The Elements is a comprehensive treatise that brings together geometry, proportion, and number theory, tying them all into one complete theory for the first time. It is divided into thirteen books. The first six are about geometry. At the heart of Euclid’s geometry are five postulates. A postulate is a rule that is assumed to be true and does not have to be proved, as opposed to a theorem, which needs proving. Euclid’s first three postulates have to do with construction. For instance, the first one states that it is possible to draw a straight line between any two points. The second and third postulates deal with defining straight lines and circles. The fourth postulate states that all right angles are equal. The fifth postulate was to become a challenge to the mathematical community for centuries to come. It states that two lines are parallel if they are intersected by a third one with identical interior angles. This postulate assumed many facts about parallel lines continuing on for infinity. Euclid himself was said to be uncomfortable with the absolute truth of this statement and declared it to be a given truth only after some hesitation. Its acceptance was a factor that defined a set of geometric theories as Euclidean geometry.

Non-Euclidean Geometry

For centuries, mathematicians tried either to prove Euclid’s fifth postulate right once and for all or to find the overlooked element that proved it to be wrong. In 1482, the first printed edition translating Euclid’s work from Arabic to Latin appeared, stimulating the progress. During the 1600s, various mathematicians rewrote the fifth postulate in ways that helped redefine such concepts as “acute angle” or “parallel” in new ways. By 1767, the French writer Jean Le Rond d’Alembert referred to the problem of parallel lines as “the scandal of elementary geometry.”

In the early nineteenth century, there arose various schools of geometry that rewrote the assumptions, creating whole systems of understanding space without having to accept the fifth postulate. Collectively, these schools of thought came to be known as non-Euclidean geometries. There are two different types of non-Euclidean geometry, each relying on a different understanding of the concept of parallelism. Those that assume that there is no such thing as a “parallel” line that will fail to eventually meet the original one are called “elliptic geometries”; those that assume that there can be multiple lines passing through a point that will parallel the original line without touching it are referred to as “hyperbolic geometries.”

Three mathematicians, working independently of one another, came up with systems of geometry (almost at the same time) in the beginning of the 1800s, all of which left out Euclid’s problematic fifth postulate. Carl Frederich Gauss is credited with being the first of them. Gauss disliked controversy and was unwilling to disagree with the prevailing view that Euclid’s geometry was the inevitable, indisputable truth, so he devised his system in private and did not publish his findings. In 1823, Gauss read the works of Janos Bolyai, a Rumanian whose non-Euclidean theories were hidden in his introduction to a book by his father, who was also a famous mathematician. Though Bolyai could not have known of Gauss’s results, his theories were similar. In 1829, a Russian, Nikolai Lobachevsky, who was himself unfamiliar with the work of Gauss and Bolyai, published his own work of non-Euclidian geometry. These three gave rise to a new way of conceiving of space, changing the assumptions that had been put into place by Euclid more than two thousand years earlier. It is just this sort of advancement of knowledge, of restructuring assumptions that were previously taken to be indisputable truth, that Rita Dove considers in her poem “Geometry.”

Compare & Contrast

• 1980: The United States Department of Education is developed, comprised of a staff of seventeen thousand full-time employees.

  Today: Some people feel that the centralized Department of Education should be disbanded because it cannot adequately understand local issues that affect schools’ environments.

• 1980: A study by UCLA and the American Council on Education finds that college freshmen express more interest in money and power than at any time in the past fifteen years. It is the beginning of a period that came to be known as The “Me” Decade.

  Today: After a long period of economic stability in the 1990s, many students take economic stability for granted. Colleges are seeing renewed interest in careers that are not focused on accumulating wealth, such as mathematics and poetry.

• 1980: Humanity’s understanding of the universe expands with the findings of Voyager I, an unmanned space craft that made new discoveries about Saturn’s moons as part of its three-year, 1.3 billion-mile journey.

  Today: Plans are underway to send two unmanned space crafts to Pluto, the farthest planet in our solar system.

• 1980: The United States Supreme Court finds, in the case of Diamond v. Chakrabarty that a man-made life form — specifically, a microorganism that could eat petroleum in cases of spills — can be patented.

  Today: Biotechnology and genetic technology are growing scientific fields and lucrative sectors of the stock market.