First Mathematician: Pythagoras, Euclid, Archimedes, Or Thales?

Choosing the correct answer from the options—Pythagoras, Euclid, Archimedes, and Thales of Miletus—requires understanding their individual contributions and the timeline of their mathematical endeavors. Let's dive into the history and impact of these mathematical giants to determine who first delved into the study of numbers.

Pythagoras: The Pioneer of Numbers

Pythagoras, often hailed as the father of numbers, stands out as the most influential figure regarding the early investigation of numbers. Born around 570 BC in Samos, Greece, Pythagoras founded a school of thought and a brotherhood that deeply intertwined mathematics with philosophy and mysticism. The Pythagoreans believed that numbers were the essence of all things, influencing everything from music and astronomy to morality and health. Their core tenet was that the universe is fundamentally mathematical, and understanding numbers was the key to unlocking its secrets.

Contributions of Pythagoras:

One of Pythagoras’s most famous contributions is the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²). While the theorem itself was known to earlier civilizations like the Babylonians, Pythagoras and his followers are credited with providing its first deductive proof. This theorem is fundamental to geometry and has countless applications in fields like engineering, architecture, and navigation.

The Pythagoreans also made significant contributions to number theory. They classified numbers into various categories, such as:

• Prime Numbers: Numbers greater than 1 that have no positive divisors other than 1 and themselves.
• Composite Numbers: Numbers that can be formed by multiplying two smaller positive integers.
• Perfect Numbers: Numbers that are equal to the sum of their proper divisors (e.g., 6 = 1 + 2 + 3).
• Figurate Numbers: Numbers that can be represented by geometric patterns, such as triangular numbers (1, 3, 6, 10, ...) and square numbers (1, 4, 9, 16, ...). These numbers were studied for their mystical and geometric properties, linking arithmetic with visual representations.

Furthermore, the Pythagoreans explored the concept of irrational numbers, although their initial encounter with these numbers was unsettling. The discovery that the square root of 2 (√2) cannot be expressed as a ratio of two integers challenged their belief that all numbers could be expressed as rational fractions. This discovery was so disturbing that legend has it that Hippasus, a Pythagorean who revealed the existence of irrational numbers, was drowned at sea.

Pythagoras’s influence extends beyond specific theorems and classifications. His emphasis on the importance of numbers as fundamental building blocks of reality laid the groundwork for future mathematical inquiry. The Pythagorean school's rigorous approach to proof and their dedication to uncovering mathematical relationships set a precedent for the development of mathematics as a systematic and logical discipline.

Euclid: The Father of Geometry

Euclid, who lived around 300 BC, is best known for his monumental work, Elements. This comprehensive treatise is a collection of definitions, axioms, theorems, and proofs that served as the primary textbook for teaching mathematics (particularly geometry) for over 2000 years. While Euclid compiled and organized existing mathematical knowledge, rather than discovering all of it himself, his systematic approach and rigorous proofs established a standard for mathematical reasoning that remains influential to this day.

Contributions of Euclid:

Elements covers a wide range of topics, including plane geometry, solid geometry, and number theory. In the realm of number theory, Euclid proved several important theorems, such as:

• The Infinitude of Prime Numbers: Euclid provided a simple yet elegant proof that there are infinitely many prime numbers. This proof is a classic example of mathematical reasoning and demonstrates the unbounded nature of prime numbers.
• The Euclidean Algorithm: This efficient method for finding the greatest common divisor (GCD) of two integers is still used today in computer science and cryptography. The Euclidean algorithm involves repeatedly dividing the larger number by the smaller number and replacing the larger number with the remainder until the remainder is zero. The GCD is then the last non-zero remainder.
• Fundamental Theorem of Arithmetic: Euclid's work laid the foundation for this theorem, which states that every integer greater than 1 can be uniquely expressed as a product of prime numbers, up to the order of the factors. This theorem is a cornerstone of number theory and is essential for understanding the properties of integers.

Euclid's Elements is not merely a collection of mathematical facts; it is a masterclass in logical deduction. Euclid starts with a set of self-evident axioms and postulates and then uses these as the basis for proving more complex theorems. This axiomatic approach, which emphasizes the importance of clear definitions and rigorous proofs, has had a profound impact on the development of mathematics and other fields of knowledge.

Archimedes: The Mathematical Physicist

Archimedes, who lived from approximately 287 to 212 BC in Syracuse, Sicily, was one of the greatest mathematicians, physicists, engineers, and inventors of antiquity. He made groundbreaking contributions to geometry, mechanics, and hydrostatics, and his work foreshadowed the development of calculus.

Contributions of Archimedes:

In mathematics, Archimedes is renowned for his methods of exhaustion, which he used to calculate areas and volumes of various geometric figures. These methods involved approximating the area or volume by dividing the figure into an infinite number of smaller parts and then summing the contributions of these parts. This approach is a precursor to integral calculus.

Some of Archimedes’s key mathematical achievements include:

• Calculating Pi (π): Archimedes used inscribed and circumscribed polygons to approximate the value of π (the ratio of a circle's circumference to its diameter). He obtained the approximation 3 1/7 > π > 3 10/71, which was a significant improvement over previous estimates.
• Area of a Parabolic Segment: Archimedes found the exact area of a parabolic segment, demonstrating a mastery of geometric techniques.
• Volume and Surface Area of a Sphere: Archimedes derived formulas for the volume and surface area of a sphere, results that he considered among his most important achievements. According to legend, he requested that a sphere inscribed in a cylinder be engraved on his tombstone to commemorate these discoveries.

Archimedes also made significant contributions to number theory, particularly in his work The Sand Reckoner. In this treatise, he attempted to calculate the number of grains of sand that would be required to fill the entire universe, as conceived in his time. To do this, he devised a system for representing extremely large numbers, far beyond those used by his contemporaries. This work demonstrates Archimedes’s understanding of exponential notation and his ability to think about numbers on a cosmic scale.

Thales of Miletus: The First Philosopher and Mathematician

Thales of Miletus, who lived around 624 to 546 BC, is often regarded as the first philosopher in the Greek tradition and one of the earliest known mathematicians. He is credited with introducing deductive reasoning and abstract thought to geometry. Thales was a pre-Socratic philosopher, meaning he lived before Socrates, and his work laid the foundation for the development of Greek philosophy and mathematics.

Contributions of Thales:

Thales is credited with several fundamental geometric theorems, including:

• Thales' Theorem: An angle inscribed in a semicircle is a right angle. This theorem is a basic result in geometry and has numerous applications.
• Congruence Theorems: Thales is believed to have used congruence theorems to measure the height of the pyramids in Egypt. By comparing the lengths of shadows cast by the pyramids and a measuring stick, he could calculate the height of the pyramids using similar triangles.
• Circle Bisection: A diameter of a circle bisects the circle. This simple yet important result is a direct consequence of the symmetry of the circle.

While Thales did not leave behind any written works, his ideas were transmitted through the writings of later philosophers and mathematicians, such as Aristotle and Euclid. Thales’s emphasis on deductive reasoning and his attempts to explain natural phenomena in terms of underlying principles mark a significant departure from earlier mythological explanations.

Conclusion: Who Was the First?

Considering the timelines and contributions of these four mathematical giants, Pythagoras emerges as the first to intensively investigate numbers and their properties. While Thales predates Pythagoras, his focus was more on geometry and the application of deductive reasoning. Euclid systematized and formalized mathematical knowledge, and Archimedes made significant advances in both mathematics and physics. However, Pythagoras and his followers were the first to emphasize the fundamental role of numbers in understanding the universe, delving into their mystical and theoretical properties, making him the correct answer. Therefore, the answer is A) Pythagoras.