In the 3rd Century BCE, following the conquests of Alexander the Great, significant mathematical advancements were taking place on the fringes of the Greek Hellenistic empire.
Alexandria in Egypt emerged as a prominent center of learning during the benevolent rule of the Ptolemies, and its renowned Library soon rivalled the reputation of the Athenian Academy. The patrons of this Library were arguably the first professional scientists, receiving financial support for their dedication to research. Among the most prominent and influential mathematicians who studied and taught in Alexandria were Euclid, Archimedes, Eratosthenes, Heron, Menelaus, and Diophantus.
During the late 4th and early 3rd Century BCE, Euclid played a crucial role as a chronicler of the mathematics of his era and became one of history’s most influential educators. He essentially laid the foundations of classical (Euclidean) geometry as we understand it today. While Archimedes spent most of his life in Syracuse, Sicily, he also pursued studies in Alexandria. He is renowned as an engineer and inventor, but recent discoveries have elevated him to the status of one of the greatest pure mathematicians of all time.
Eratosthenes, who lived in the 3rd Century BCE in Alexandria, was a polymath, excelling in mathematics, astronomy, and geography. He developed the first system of latitude and longitude and accurately calculated the Earth’s circumference. His mathematical legacy includes the “Sieve of Eratosthenes,” an algorithm for identifying prime numbers.
Spherical Triangle
The precise date of the catastrophic fire that consumed the magnificent Library of Alexandria remains uncertain. Nevertheless, Alexandria continued to serve as a significant hub for intellectual pursuits for several centuries. In the 1st century BCE, Heron (also known as Hero) emerged as a prominent inventor from Alexandria. He made substantial contributions to mathematics, including the exploration of Heronian triangles (triangles with integer sides and integer area), the formulation of Heron’s Formula for determining a triangle’s area based on its side lengths, and the development of Heron’s Method for iteratively calculating square roots. He was also the first mathematician to grapple with the concept of √-1, although he lacked the means to fully address it, a challenge that would be resolved by Tartaglia and Cardano in the 16th Century.
In the 1st to 2nd Century CE, Menelaus of Alexandria made pioneering strides by recognizing geodesics on curved surfaces as the natural counterparts to straight lines on a flat plane. His book “Sphaerica” delved into the geometry of the sphere and its applications in astronomical measurements and calculations, introducing the concept of spherical triangles, which he termed “trilaterals.”
Moving to the 3rd Century CE, Diophantus of Alexandria distinguished himself as the first mathematician to regard fractions as legitimate numbers, marking an early foray into what would later evolve into the field of algebra. He tackled complex algebraic problems, including what is now referred to as Diophantine Analysis, which involves finding integer solutions to equations involving multiple unknowns (Diophantine equations). Diophantus’ “Arithmetica,” a collection of problems presenting numerical solutions for both determinate and indeterminate equations, stood as the foremost work on algebra in the realm of Greek mathematics. His problems continued to engage the minds of mathematicians worldwide for the next two millennia.
Conic Sections of Apollonius
Alexandria wasn’t the sole locus of learning within the expansive Hellenistic Greek empire. Another notable figure was Apollonius of Perga, hailing from the city of Perga in modern-day southern Turkey. In the late 3rd Century BCE, Apollonius made significant contributions to geometry, particularly in the realm of conics and conic sections. He not only bestowed upon the ellipse, the parabola, and the hyperbola their familiar names but also elucidated how these shapes could be derived from various cross-sections of a cone.
Hipparchus, a native of Hellenistic Anatolia who lived during the 2nd Century BCE, stands out as perhaps the preeminent ancient astronomer. He revitalized the use of arithmetic techniques originally developed by the Chaldeans and Babylonians and is often credited with laying the foundations of trigonometry. Hipparchus achieved remarkable precision for his time by calculating the distance between the Earth and the Moon. He accomplished this by observing different parts of the Moon from various locations and applying the principles of geometry. Furthermore, he created the earliest table of chords, which represented side lengths corresponding to different angles of a triangle.
By the era of the renowned Alexandrian astronomer Ptolemy in the 2nd Century CE, Greek mastery of numerical methods had advanced to such an extent that Ptolemy could include a table of trigonometric chords in his “Almagest.” This table, while expressed in the Babylonian sexagesimal style, provided accuracy up to approximately five decimal places for circle divisions at increments of ¼°.
However, as the middle of the 1st Century BCE approached and thereafter, the Romans tightened their grip on the former Greek empire. The Romans had little interest in pure mathematics, valuing only its practical applications. The subsequent Christian regime displayed even less enthusiasm for mathematical pursuits. A tragic symbol of this decline was Hypatia, the first recorded female mathematician and a renowned educator who had authored respected commentaries on works by Diophantus and Apollonius. In 415 CE, Hypatia met a grim fate as she was violently killed by a Christian mob, marking a devastating blow to the Hellenistic mathematical legacy in Alexandria.
