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The circle is a universally understood shape, a perfectly symmetrical round form without corners or angles that form a closed loop. When investigated, it becomes highly intriguing ─ we can observe that a circle is a two-dimensional geometric shape characterized by a set of points, all equidistant from a single central point. It is defined by a continuous curved line, known as the "circumference," forming a closed loop. Moreover, the distance from the center of the circle to any point on its circumference is called the radius. The diameter of a circle, the distance across the center, is precisely twice the length of the radius. In this article, we will explore how this intriguing geometry came to be known to all mankind.

Etymology

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The term "circle" has its origins in Latin, specifically from the Latin term "circus," which means "ring" or "circular line." The Romans employed this term to describe circular arenas used for performances and contests, as well as oval courses for racing. In Greek, the equivalent term is "kirkos," which translates to English as a "circle ring."

Ancient Times

Babylonian clay tablet (Image by Yale Peabody Museum)

The captivating properties of a circle predate recorded history, with early civilizations expressing profound interest and understanding of this geometric form. Over 4000 years ago, both the Egyptians and Mesopotamians demonstrated a grasp of basic geometry, including the properties of a circle, while Babylonians are recorded to have developed a method for roughly calculating the area of a circle.

Euclid's Depiction (Image by Britannica)

Euclid's Elements, Book III (Image by Clay Mathematics Institute)

The first theorems related to a circle are credited to Thales around 650 BC. Centuries later, in Book III of Euclid's Elements, Euclid, a Greek mathematician often regarded as the "father of geometry," delves into the properties of circles and problems related to inscribing and describing polygons. Within this work, Euclid employs a systematic and deductive approach to geometry. Beginning with foundational definitions, he establishes key terms associated with circles, such as radius, diameter, and center. Euclid then presents a series of propositions, including the celebrated inscribed angle theorem, asserting that an angle inscribed in a semicircle is always a right angle. Another fundamental proposition, often termed Euclid's second theorem on circles, illustrates that angles at the center of a circle are twice the angles at the circumference subtended by the same arc. Through precise constructions and geometric reasoning, Euclid lays the groundwork for comprehending intricate relationships within circular figures, making substantial contributions to the foundation of geometric knowledge and mathematical rigor.

Euclid's Elements, Book III Definition (Image by Nicholas Rougeux)

The Roman Period

During the Roman Period, spanning approximately from 753 BCE to 476 CE, engineers and architects in ancient Rome demonstrated a profound understanding and application of geometric principles in their construction endeavors. Renowned for their architectural prowess, the Romans utilized geometric concepts to craft enduring structures, such as the Colosseum, aqueducts, and triumphal arches. The incorporation of circular forms, arches, and domes in Roman architecture highlighted their mastery of geometric symmetry and proportion. This precision in geometry not only enhanced the aesthetic appeal of their constructions but also played a pivotal role in ensuring structural stability and longevity. The application of geometric principles by Roman engineers serves as a testament to their ingenuity, exerting a lasting influence on architectural practices for centuries to come.

Plan of the Roman theater based on the Vitruvian geometry from the 1914 edition of the “Ten Books on Architecture” (Image on Springer)

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*Did you know the Greeks were the first to theorize the Earth is a sphere?

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Some of the key figures:

- Around 570–495 BCE, Pythagoras, the ancient Greek mathematician, suggested Earth's spherical shape, drawing from lunar eclipse observations.

- Parmenides, circa 515–450 BCE, a philosopher, also posited a spherical Earth, guided by logical reasoning and the idea of perfection in symmetry.

- Eratosthenes, 276–194 BCE, a mathematician and astronomer, significantly contributed by gauging Earth's circumference. His calculations, rooted in observations from Syene and Alexandria in Egypt, marked a pivotal advancement.

Meanwhile, in South Asia, a brilliant mind named "Aryabhata," native to India, wrote a mathematical treatise divided into three sections: Ganita, focusing on mathematics, where he introduced decimal notation, named the first 10 decimal places, and provided algorithms for square and cubic roots; Kala-kriya, dealing with time calculations; and Gola, centered around the sphere.

Hellenistic Age

Several centuries after Euclid's era, Apollonius of Perga (about 262 BC–190 BC), renowned as 'The Great Geometer' and an epochal figure in the development of mathematics, conducted further studies on circles. In his book "Conics," Apollonius introduced terms that are now familiar, such as parabola (a U-shaped curve where all points are equidistant from a fixed point [the focus] and a fixed straight line [the directrix]), ellipse (a stretched or compressed circle with two points inside, called foci, and the sum of the distances from any point on the ellipse to the two foci remains constant), and hyperbola (a curve with two separate branches, created by the difference of distances from any point on the curve to two fixed points [the foci], and it has two asymptotes).

Islamic Golden Age

The Islamic Golden Age (from the 8th to the 14th centuries) marked a profound intellectual transformation in the Islamic world. Central to this era were scholars like Al-Haytham and Al-Khwarizmi, luminaries who played pivotal roles in preserving and expanding mathematical knowledge. Al-Haytham, known as the "father of optics," made substantial contributions to understanding light and vision, extending his work to mathematics by exploring the properties of conic sections, including circles. Al-Khwarizmi, often referred to as the "father of algebra," not only laid the groundwork for algebraic principles but also made significant contributions to mathematical advancements. Through meticulous translations and original works, these scholars not only preserved the mathematical heritage of ancient civilizations but also propelled mathematical knowledge forward, paving the way for Europe's later Renaissance and shaping the foundations of modern mathematics.

The Renaissance

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The Renaissance, spanning from the 14th to the 17th century, symbolized a cultural rebirth and intellectual flourishing in Europe. Characterized by a renewed interest in art, science, and intellectual pursuits, this period emerged from the shadows of the medieval era, witnessing a profound shift in human thought that emphasized a revival of classical learning and a newfound curiosity about the natural world. The Renaissance served as fertile ground for polymaths and visionaries, among whom Leonardo da Vinci (1452–1519) stands prominently. Although best known for iconic works like the Mona Lisa and The Last Supper, Da Vinci, an exceptional individual, also immersed himself in geometric exploration.

Leonardo Da Vinci Portrait (Image on Britannica)

Within his extensive notebooks, he delved into the principles of geometry, including the study of circles. Da Vinci's sketches and writings revealed his fascination with the mathematical beauty inherent in the natural world, with meticulous observations of circular forms such as ripples in water or patterns in nature, showcasing his keen interest in the harmonious proportions of circles.

Leonardo Da Vinci's Mona Lisa Exhibiting at Louvre Museum, Paris, France (Photo by Federico Scarionati on Unsplash)

17th Century: Analytical Geometry by Descartes

In the 17th century, René Descartes (best recognized for his famous quote “cogito, ergo sum, translated as "I think, therefore I am," which gave birth to the theory about the duality of mind and body), a visionary philosopher and mathematician, brought about a revolution in the landscape of mathematics by introducing analytical geometry. Descartes ingeniously fused algebraic techniques with geometric concepts, paving the way for the precise description of geometric shapes through algebraic equations. His groundbreaking work laid the foundation for future developments in mathematical analysis, playing a crucial role in the evolution of modern geometry.

18th–19th Centuries: Calculus and Mathematical Analysis

The 18th and 19th centuries witnessed the emergence of calculus and mathematical analysis, led by luminaries like Isaac Newton and Gottfried Wilhelm Leibniz. These pioneering mathematicians independently developed calculus, a powerful mathematical tool that transformed the study of curves, including circles. Calculus and mathematical analysis became indispensable in addressing complex problems in physics, engineering, and mathematics, marking a watershed moment in the history of mathematical thought.

20th Century and Beyond: Advanced Theories in Geometry

In the 20th century and the future, modern mathematicians and physicists have made profound contributions to advanced theories in geometry, particularly in fields such as differential geometry and topology. These disciplines have expanded the study of shapes and spaces, offering new perspectives on the intricate structures that form the mathematical fabric of the universe. The ongoing exploration of geometry in the modern era has not only deepened our understanding of fundamental principles but has also paved the way for groundbreaking advancements in various scientific disciplines.

The historical journey of the circle has traversed a remarkable path, evolving from a simplistic shape understood even by children to a geometric entity with immense intriguing properties. The collective contributions of brilliant minds across various timelines and regions underscore the interconnectedness of humanity in the pursuit of knowledge. As we reflect on the enduring significance of the circle in the annals of mathematics, it serves as a testament to the universal language that transcends cultural and temporal boundaries. In the pursuit of understanding this fundamental shape, humanity's shared intellectual heritage shines through, reminding us of the interconnected tapestry of human ingenuity that spans across centuries and continents.

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Sources:

https://mathshistory.st-andrews.ac.uk/Biographies/Apollonius/

https://www.storyofmathematics.com/circle/

https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/circle

https://www.smithsonianmag.com/travel/world-full-circles-180954529/

https://www.britannica.com/biography/Aryabhata-I

https://www.khanacademy.org/humanities/big-history-project/solar-system-and-earth/knowing-solar-system-earth/a/eratosthenes-of-cyrene

*Did you know the Greeks were the first to theorize the Earth is a sphere?

Some of the key figures:

- Around 570–495 BCE, Pythagoras, the ancient Greek mathematician, suggested Earth's spherical shape, drawing from lunar eclipse observations.

- Parmenides, circa 515–450 BCE, a philosopher, also posited a spherical Earth, guided by logical reasoning and the idea of perfection in symmetry.

- Eratosthenes, 276–194 BCE, a mathematician and astronomer, significantly contributed by gauging Earth's circumference. His calculations, rooted in observations from Syene and Alexandria in Egypt, marked a pivotal advancement.

Story of π

The pursuit of understanding the mathematical constant π, or pi, dates back to ancient times, with Archimedes of Syracuse (287–212 BC) making a pioneering contribution. Archimedes, recognized as one of the greatest mathematicians of the ancient world, sought to approximate the area of a circle. In his ingenious approach, he applied the Pythagorean Theorem to calculate the areas of two regular polygons—one inscribed within the circle and the other circumscribed around it.

Archimedes' method involved inscribing and circumscribing polygons with an increasing number of sides around a circle. By progressively using polygons with more sides, he narrowed the gap between the estimated areas and the actual area of the circle. The breakthrough came as he realized that the limiting values of these calculations converged towards a specific numerical value, which we now recognize as π. In simpler terms, Archimedes' method involved fitting polygons inside and around a circle, refining the approximation with each iteration, ultimately leading to the discovery of the constant π as the ratio of a circle's circumference to its diameter. This foundational work laid the groundwork for understanding circular shapes and contributed significantly to the development of mathematical concepts that continue to shape our understanding of geometry today.