Euclid is one of the most prominent mathematicians of ancient Greece and is widely known as the author of Elements, a monumental work that laid the foundation for modern mathematics.
Elements was the first comprehensive mathematical text to systematically present logical proofs. It greatly influenced not only the development of mathematics but also philosophy and the natural sciences.
The name “Euclid” is the English pronunciation; in Greek, it is pronounced “Eukleides” (Εὐκλείδης).
He is believed to have been active around 300 BCE in Alexandria, Egypt. However, little is known about his life or personality, as there are few reliable historical records.
The Historical Context: What Was the Era of Euclid (Eukleides) Like?
The Hellenistic Period and Alexandria
— The Background of Euclid’s Time —
Euclid lived and worked during the Hellenistic period of ancient Greece. In this era, the cultural center of the Greek world shifted from Athens (on the Greek mainland) to Alexandria in Egypt.
Alexandria, ruled by the Macedonian Ptolemaic dynasty, became a cosmopolitan academic city where people from across the Mediterranean, West Asia, India, and the northern Black Sea region gathered.
The city was home to the royal research institution known as the Mouseion, and its adjacent library became one of the largest in the ancient world. This environment fostered the growth of sciences such as astronomy and mathematics.
It is believed that Euclid conducted his scholarly work in Alexandria. Among his contemporaries was the famous scientist Archimedes.
The Author of Elements: A Bestselling Work That Laid the Foundation of Geometry
uclid is widely known as the author of Elements, a monumental work that represents the culmination of ancient Greek mathematics.
Elements was the first textbook to systematically organize the field of geometry, and it became a historically significant masterpiece that profoundly influenced generations to come.
It has often been called “the most widely read book in the world after the Bible,” and in Europe, from the Middle Ages through the early modern period, many mathematicians studied geometry through this text. It was also used for centuries as a standard university mathematics textbook.
The geometric framework Euclid established in this work later came to be known as “Euclidean geometry.” Geometry is a branch of mathematics that deals with the properties of shapes and space through logical reasoning, and Elements can truly be regarded as its enduring starting point.
Euclid’s Theorem on Perfect Numbers
A perfect number is a natural number that equals the sum of its proper divisors—that is, all its positive divisors excluding itself. In ancient Greece, the Pythagoreans regarded perfect numbers as special and even mystical.
For example, the divisors of 6 are 1, 2, 3, and 6. If we exclude 6 itself and add the remaining divisors, we get 1 + 2 + 3 = 6, so 6 is a perfect number. Similarly, 28 and 496 are also known to be perfect numbers.
In his Elements, Euclid proved a theorem about perfect numbers as follows:
If 2ⁿ − 1 is a prime number, then (2ⁿ − 1) × 2ⁿ⁻¹ is a perfect number.
There Are Infinitely Many Prime Numbers — Euclid’s Proof in Elements
The theorem stating that “there are infinitely many prime numbers” was already proven by the ancient Greek mathematician Euclid in his seminal work Elements.
In Elements, Euclid did not use the modern phrase “infinitely many.” Instead, he wrote that “the number of prime numbers is greater than any assigned multitude of prime numbers.”
In other words, no matter how many prime numbers one lists, there are always more beyond them.
To demonstrate that there are infinitely many of something, it is sufficient to show that the quantity exceeds any finite number we might choose.
In fact, Euclid’s proof uses just three prime numbers as an example and then shows that there must be additional primes beyond them.
This simple yet logical argument has been praised by later mathematicians as “one of the most beautiful proofs” and continues to be cited in many mathematics texts to this day.
Euclid’s Famous Saying: “There Is No Royal Road to Geometry”
Among the many sayings attributed to Euclid, one of the most famous is this:
“There is no royal road to geometry.”
According to tradition, when King Ptolemy was studying geometry with Euclid, he asked if there was an easier or quicker way to learn it.
To this, Euclid famously replied:
“There is no royal road to geometry.”
The meaning is clear: even a king must follow the same path of study as anyone else—there are no shortcuts in learning. This anecdote continues to be quoted today as a reminder of the importance of effort and patient understanding in any discipline.
Euclid was said to have taught geometry to King Ptolemy. When the king asked if there was a shortcut to mastering the subject, Euclid responded that no such path existed—not even for a king.
“That Student Seems to Want Money—Give Him Some.”
One day, a student studying under Euclid asked him:
“What use is this proposition?”
Euclid gave no direct reply to the question. But shortly afterward, he turned to a servant nearby and said:
“That student seems to want money—give him some.”
This anecdote has been passed down as a reflection of Euclid’s philosophy: that knowledge is not to be pursued solely for its practical utility. For Euclid, the value of learning went beyond material gain or immediate usefulness—it was an end in itself.
Who Was Euclid? — A Summary
Euclid, or Eukleides in Greek, was one of the leading mathematicians of ancient Greece. He is best known as the author of Elements, a masterpiece that compiled the mathematics of his time and laid the foundation for geometry.
He lived around 300 BCE and is believed to have been active in Alexandria during the Hellenistic period, when the city flourished as a cosmopolitan center of learning under the rule of the Ptolemaic dynasty.
Elements was the first textbook to systematically organize geometry, and it had a great influence on later generations. It has even been called “the most widely read book in the world after the Bible,” and was long used as a mathematics textbook in Europe.
Euclid also proved the famous theorem that there are infinitely many prime numbers and studied perfect numbers, leaving behind elegant and logical proofs.
He is known for the saying, “There is no royal road to geometry,” spoken in response to King Ptolemy’s desire for a shortcut in learning. Another famous episode tells of a student who asked what use a certain proposition had—Euclid instructed his servant to give the student money, implying that true learning is not about immediate utility.