CLIL CLIL CLIL LANGUAGE IMMERSION MATHEMATICS THE ORIGIN OF GEOMETRY The roots of current computerized drawing technologies go back in time to the most famous and influential mathematical work in classical antiquity: The Elements written by Euclid of Alexandria, now popularly considered as the father of geometry. Very little is known about Euclid s life: we do not know precisely when he was born or when he died. It is assumed that he lived some time around 300 BCE and that he studied mathematics in Athens with the students of Plato. From references in other classical books, it can also be deduced that he was one of the founders of the Alexandrian School of Mathematics. Most of the topics dealt with in The Elements were not discovered by Euclid himself but had been studied previously by other Greek mathematicians such as Pythagoras, Hippocrates and Thaetetus: Euclid is credited with having arranged the theorems about geometry, proportion and number theory into a logical and systematic treatise. The principles of what is now called Euclidean geometry were established on the basis of 5 main axioms, that is, statements whose truth does not require a proof because they are evident: 1 A straight line segment can be drawn joining any two points. 2 Any straight line segment can be extended indefinitely in a straight line. 3 Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre. 4 All right angles are equal. Given a line and a point not on the line, there is only one line through the point that does not meet 5 the other line. The Elements consists of 13 books and is the world s oldest mathematical text still in use. Its Latin version appeared in 1482 while French, German and English translations appeared in the sixteenth century. Until the twentieth century it was regularly used as a textbook of geometry and was the most common reprinted book after the Bible. Book 1 deals with the fundamental propositions of plane geometry: it includes the three cases in which triangles are congruent, various theorems on parallel lines, the theorem regarding the sum of the angles in a triangle and the Pythagorean Theorem. Book 2 is about geometric algebra, that is, geometric theorems with simple algebraic interpretations. Book 3 investigates circles and their properties, including theorems on tangents and inscribed angles. Book 4 examines regular polygons inscribed in or circumscribed around circles. Books 5 and 6 develop the arithmetic theory of proportion and apply it to plane geometry. Book 7 is concerned with the fundamentals of number theory (prime numbers, the greatest common denominator). Book 8 deals with geometric series while various applications of the previous results are contained in Book 9, as well as theorems on the infinitude of prime numbers and the sum of geometric series. Incommensurable magnitudes are classified in Book 10, using a precursor of the integration method called method of exhaustion . Book 11 analyses three-dimensional geometry and Book 12 explains how to calculate volumes of cones, pyramids, cylinders and spheres. Finally, Book 13 investigates the so-called five Platonic solids or polyedrons, three-dimensional figures whose sides are called faces and consist of regular polygons, usually joined at the edges. Other important works by Euclid dealing with geometry are Data and On divisions while Phaenomena is about geometry applied to astronomy and Optica exploits geometrical analysis to develop a theory about vision. 212