Mathematical proofs without letters / Démonstrations mathématiques sans lettres

We all mostly think that mathematical proofs are based on letters and complicated symbols, this is a more recent development. A lot of proofs especially some of geometrical theorems can be based on colored graphic explanations. The book “The first six books of the Elements of Euclid” by Oliver Byrne is an example of this approach.

What distinguishes Byrne’s edition is that he attempts to present Euclid’s proofs in terms of pictures, using as little text – and in particular as few labels – as possible. What makes the book especially striking is his use of color.

Here some theorems and their proofs / voici quelques théorèmes et leurs démonstrations graphiques

If a parallelogram and a triangle are upon the same base and between the same parallels, the parallelogram is double the triangle
Si un parallélogramme et un triangle ont une base commune et si leurs sommets se situent sur des parallèles, alors l’aire du parallélogramme est le double de celle du triangle
 Screenshot 2016-02-02 14.15.05
If a straight line is divided into two any parts, the square of the whole line is equal to the squares of the parts together with twice the rectangle contained by the parts.
Si un segment est divisé en deux parties, le carré de la longueur du segment initial est égal au carré des longueurs des deux parties auquel on ajoute le double du produit des deux longueurs
(a+b)2 = a2 + b2 + 2 a . b
 Screenshot 2016-02-02 14.14.34
Screenshot 2016-02-02 14.14.40
If two circles touch one another externally, the straight line joining their centres passes trough the point of contact.
La droite joignant les centres de deux cercles tangents extérieurement passe le point de tangence des cercles
 Screenshot 2016-02-02 14.13.09

You can find the facsimile version online at or just have a look at some