Euclid’s Elements: Geometry Taught for 2,000 Years

People were trying to make geometry clear to everyone as far back as antiquity. The best-known example is Euclid’s Elements, a book that students learned from for more than two thousand years. Here is how it is built, why it was relied on for so long and how it was turned into a book of color in the nineteenth century. How to help a child come to love geometry is covered in How to Help Your Child Love Geometry: Tips for Parents.

What the Elements Is

The Greek mathematician Euclid wrote the Elements around 300 BC. It runs to 13 books of definitions, postulates, constructions and theorems with proofs. Books I to VI deal with plane geometry, VII to X with the basics of number theory and XI to XIII with solid geometry. In it are Pythagoras’s theorem, Thales’s theorem, the Euclidean algorithm for the greatest common divisor and a proof that the primes never run out. Euclid drew a great deal from earlier mathematicians such as Eudoxus, Theaetetus and Hippocrates of Chios, but he arranged it all in strict order and supplied missing proofs.

How the First Book Works

Book I opens with 20 definitions: the point, the line, the angle, regular polygons. Next come ten starting assumptions, five postulates plus five common notions, and every later theorem is derived from them. The same book holds 48 propositions, and its last two are the oldest proof of Pythagoras’s theorem to survive. Proposition 5, which shows that an isosceles triangle has equal base angles, was nicknamed the bridge of asses in the Middle Ages, as if it divided those able to prove it from those who were not. For centuries people tried to derive the fifth postulate, on parallel lines, from the other four and failed; in 1829 Nikolai Lobachevsky published a geometry that replaces it with a different one.

The Most Successful Textbook

The Elements is often described as the most successful school book of all time. In the Middle Ages it was put into Arabic and Latin, its first printed edition appeared in Venice in 1482, and there have been more than a thousand editions in all, a number that by some estimates only the Bible exceeds. It remained the standard geometry textbook until about the nineteenth century, when newer books began to replace it.

Oliver Byrne’s Color Euclid

In 1847 a version of Books I to VI of the Elements came out, prepared by the Irish engineer and mathematician Oliver Byrne, in which every proposition is explained through colored diagrams. Today the book is admired for a graphic style that was daring for its day and anticipated what the Bauhaus and De Stijl would later try, and in 2010 it was reissued in facsimile. Ways to turn geometry into a game are covered in Geometric Puzzles: Learning with Fun.

How to Get Started

In our geometry course, students get to grips with the properties of figures and with theorems, learn to solve problems on their own and justify the solution, and prepare for tests, exams and olympiads. Lessons are open to students from age 13. A trial lesson costs $8: it is individual and lasts 45 minutes. More about the course is on the Geometry page.