Euclid’s Elements: a textbook that has lasted over 2,000 years

Open the geometry notebook of any school in Pakistan and one sentence is sure to appear: “Prove that…”. Then comes a figure, a few assumptions and an argument, step by step. About 2,300 years ago, a Greek teacher in the Egyptian city of Alexandria set out this method in an orderly form. His name was Euclid, and his book, the “Elements”, is counted among the most successful textbooks in the world.

Who was Euclid

Strangely, we know almost nothing about the author of so famous a book. According to David Joyce, a mathematician at Clark University in the United States, Euclid lived in Alexandria around 300 BC. Even this estimate rests on a text by the fifth-century AD philosopher Proclus, who lived about 800 years after Euclid.

According to Proclus, Euclid came after the students of Plato and before Archimedes, and lived in the reign of the Egyptian king Ptolemy I. He also recorded the story that the king asked for an easier way to learn geometry. Euclid replied that there was no royal road to geometry.

According to the MacTutor archive at the University of St Andrews, some historians have also asked whether the “Elements” is the work of one man or of a group of Alexandrian mathematicians. For 2,000 years, however, it has generally been accepted as the work of a single author.

What is in the book

The “Elements” has 13 books. According to MacTutor, the first six deal with plane geometry: triangles, parallel lines, squares and circles. Books seven to nine cover number theory, including the method of finding the greatest common divisor of two numbers, now called the Euclidean algorithm. Book ten deals with irrational magnitudes, and the last three with solid figures. The work ends with a proof that only five regular solids are possible.

The book’s real strength is its arrangement. Euclid first gives definitions, then five basic postulates and some “common notions”, for example that things equal to the same thing are equal to each other. Each result is then derived by argument from earlier results. The first postulate says that a straight line can be drawn between any two points. The fifth, the famous “parallel postulate”, was debated for centuries.

According to MacTutor, most of the book’s results were probably not discovered by Euclid himself. It includes the work of earlier mathematicians such as Eudoxus and Theaetetus. But the arrangement and presentation of the material are certainly Euclid’s, and that is his real achievement.

The oldest copies

Nothing in Euclid’s own hand survives. MacTutor, citing the historian David Fowler, says the oldest trace of this material is found on six clay potsherds recovered from Elephantine, an island in the Nile, between 1906 and 1908. They are dated to the third quarter of the third century BC. They carry material from book thirteen, but the wording differs from the text of the book, as if a student were trying to understand difficult material.

A papyrus fragment from the ancient Egyptian city of Oxyrhynchus, now in the Penn Museum in the United States, shows the text and figure of the fifth proposition of book two. Sources differ on its date: some place it in the first or second century AD, others in the third or fourth.

From Arabic to Latin

After the Greeks, Arabic translators played a major part in keeping the book alive. According to MacTutor, al-Hajjaj ibn Yusuf translated the “Elements” from Greek into Arabic. Later, the mathematician al-Nayrizi (died about 922) revised al-Hajjaj’s second translation and wrote a commentary on it, which survives today. In the 13th century, Nasir al-Din al-Tusi (1201-1274) also produced a revised Arabic text of Euclid’s work.

Adelard of Bath, an Englishman born in 1075 who died in 1160, translated the “Elements” into Latin from Arabic sources. According to MacTutor, his first, complete translation was probably based on al-Hajjaj’s Arabic version, and his translations remained the main geometry textbook in Europe for centuries. The book was first printed in 1482, and more than 1,000 editions have been published since.

Euclid in everyday life

According to MacTutor, the “Elements” remained the main source of geometric reasoning until the 19th century. In that century mathematicians dropped the fifth postulate and created new, non-Euclidean geometries, which later proved useful in modern physics.

But Euclid is everywhere in daily life. When a mason uses string and measurement to make a right angle at the corner of a wall, when an engineer draws up a road or bridge, or when a computer finds the greatest common divisor of two numbers, each is using the logic of this book. Above all, Euclid taught that it is not enough to accept a claim; it must be proved. The “Prove that” written in a school notebook echoes that old tradition.