Measurement and Geometry
Geometric reasoning including parallel lines and
angle sum of a triangle
The incredible constructions of the pyramids and the huge temples of Egypt reveal that the Egyptians must have had a very good working knowledge and understanding of basic geometry, at least at a practical level. However, there is no evidence that they had systematised that knowledge in any formal way. This was left to the ancient Greeks.
The Greek philosopher Thales (c. 624–546 BC) gave the ﬁrst 'proofs' of geometric facts that marked the beginnings of deductive geometry. The Pythagorean School continued this work, and Plato (428–348 BC) is clearly drawing on the work of earlier mathematicians when he mentions geometric facts in his writings. In his work Meno, Socrates gets a slave boy to arrive at a geometric theorem by a series of logical deductions. If the origins of geometry are unclear, the 'ﬁnal product' is not.
Euclid (323–283 BC), writing in Alexandria, produced a remarkable work called Elements, which remained the standard textbook in geometry for more than 2000 years. In this work, Euclid sets out a number of deﬁnitions (such as for points and lines), postulates and common notions. (These days we call them 'axioms'.) From these, he logically developed, in a very carefully chosen order, a great many theorems that we generally refer to as Euclidean geometry. There are a number of other geometric results, such as Pappus's theorem, that were discovered after Euclid, but these are not generally covered in secondary school.
Geometry is used to model the world around us. A view over the roofs of houses reveals triangles, trapezia and rectangles, while tiling patterns in paving and bathrooms use hexagons, pentagons, triangles and squares.
Builders, tilers, architects, graphic designers and web designers routinely use geometric ideas in their work. Classifying such geometric objects and studying their properties are very important. Geometry also has many applications in art.
Just as arithmetic has numbers as its basic objects of study, so points, lines and circles are the basic building blocks of plane geometry.
In secondary school geometry, we begin with a number of intuitive ideas (points, lines and angles) which are not at all easy to precisely deﬁne, followed by some deﬁnitions (vertically opposite angles, parallel lines and so on) and from these we deduce important facts, which are often referred to as theorems.
Thus geometry gives an opportunity for students to develop their geometric intuition, which has applications in many areas of life, and also to learn how to construct logical arguments and make deductions in a setting that is, for the most part, independent of number.