500 BC-1 AD:

Euclid, alive around 300 BC, completely transformed mathematics. His influence is extremely evident today in the fields of geometry, arithmetic, and optics. Living in a time when questioning thought, trying to understand the changing world, and developing ideas that could help understand the natural world, Euclid was able to identify and create ideas that transformed the way people viewed the world into a more mathematical, logical, and geometric perspective. While many of his works have been lost over time, many of his famous pieces are preserved and widely used today. He was even captured in one of history’s most famous pieces of art!

Euclid is best known for his Elements, which published his ideas on geometry and fundamental algebraic topics. Elements studied geometry, prime numbers, and his Euclidean algorithm. It earned him his title as the Father of Geometry for creating and popularizing geometry. His work, the first complete, coherent review of all geometry theory up until this point in time, organized study so that people could learn by following his work. It is considered one of the most precise and understandable books of mathematics and has helped improve and create the modern mathematics realm. This aspect of Elements allowed it to rise to fame and become one of history’s most translated and studied books despite its overwhelming size (13 gigantic volumes). Euclid is also famous for his logical proofs and deductive methods, one of the main reasons why Euclid’s work served as a pivotal moment of mathematical thinking. Although he did transform the way people saw space, time, and shapes, a huge part of the fame of his book stemmed from his extensive proofs and deductive reasoning starting at basic axioms, or commonly agreed-on concepts. He took common knowledge and used it to build extremely complex theorems and prove old ones. For example, starting with only axioms, he was able to prove the fundamental theorem of arithmetic. By using deductive reasoning, he was able to create nearly perfect theorems. Many later mathematicians, especially Girolama Saccheri, tried to out do the work of Euclid but all eventually gave up when they realized that his theories were flawless due to his extensive proofs. He even adventured into new branches of mathematics and science. First, he published his book, Optiks, which discussed perspective and how people view the world through their eyes. His influence in this realm, although overlooked by most, is extremely influential. He also studied catoptrics, or the mathematical functions of mirrors. He again applied deductive reasoning to understand the principles behind mirrors. He also became an important figure in the study of data, conics, and ratios through his work in arithmetic and geometry. Euclid’s influence on modern mathematics and society are immeasurable. For students studying geometry worldwide, his influence is obvious. As the renowned Father of Geometry, Euclid created the foundation for the field in his Elements. He created a foundation which other mathematicians built off of for the next 2000 years. Without his work, the work of scientists and mathematicians, such as Ptolemy, Brahmagupta, Isaac Newton, Leonhard Euler, and Carl Friedrich Gauss, would not have been possible. Deductive reasoning strategies would also be much less common and popular. Therefore, geometry students would never have the opportunity to use proofs to come to conclusions about various geometric shapes in school. After Euclid, mathematicians developed new theorems about geometry and divided geometry into ‘Euclidean Geometry’ and ‘Non-Euclidean Geometry’, which emphasizes his importance in the field. Lastly, his influence in studies of prime numbers, fractions, ratios, basic arithmetic, optics, and even mirrors is extremely influential in the modern world.

1 AD- 1000:

Brahmagupta...