Frank asked his daughter Julie what she was carrying, and she responded by saying that these items were her props for her Geometry presentation. She said that the protractor is an instrument that is used for measuring angles and the triangle ruler is often called an Architect scale and that it is usually made in either the 45-degree triangle, or the 30/60-degree triangle. Julie went on to say that Geometry is a branch of mathematics that is concerned with the shapes, sizes, patterns, and positions of individual objects, and the spatial relationships among various objects, and the properties of surrounding space. The Ancient Greeks didn’t invent geometry, they studied what had been discovered by the Ancient Egyptians, Babylonians, and made crucial advancements. It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in surveying, and its name is derived from Greek words meaning “Earth measurement.” Knowing Geometry helps us to recognize shapes and Geometry is often defined as the mathematics of space. Space gives us the geometry terms of distance or length, area and volume and this is sometimes called 1-dimensional, 2-dimensional flat surfaces (plane geometry) or 3-dimensional space (solid geometry), however mathematicians often work in higher dimensional spaces and abstract thoughts and images can also be represented and developed in geometric terms.

Mathematics is an exact science, so not only does the word mathematics need to be defined, but also the word definition needs a definition, and thus a definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Mathematical English is often inconsistent with the normal English language, as mathematicians consider a straight line to be a special form of a curve. Having a grasp on Math lingo is necessary to make formal mathematical statements, and to communicate specific terminology. Math is a highly complex subject, but if you don’t understand the terminology, you will truly be lost. The four labels that are given by mathematicians to statements that can be shown to be true are Corollary, Lemma, Proposition and Theorem. They all basically mean the same thing, that *s*ome mathematical statements are called true, given some starting axioms or previous true statements once they are correctly deduced*. * A proof will show that these statements are true. Even the word “true” can be defined mathematically, as truth is a property of conclusions that are drawn which actually match reality.

An axiom is a statement or proposition that isn’t inclined to have a proof or a disproof because it is regarded as being established, accepted, or self-evidently true. Axioms are rules that give the fundamental properties and relationships between mathematical objects. Axioms are taken as mathematical facts, so deductions can be made from them. An axiom is a basic assumption of a theory that is considered to be true, and it is often used as a basis for proving a theorem to be true.

Living around the time of 300 BC, Euclid of Alexandria was a Greek mathematician, and he is often referred to as the “father of geometry”, because he gathered up all of the knowledge that was developed in Greek mathematics before him and he combined this into a book that he wrote called ‘The Elements’. This book contained thirteen volumes which covered a vast body of mathematical knowledge, spanning arithmetic, geometry and number theory. As a starting point, Euclid established axioms by selecting the most important mathematical statements that were assumed true, but did not have proof, to which he offered his own proofs and demonstrated his thoughts in a clear, skillful, and highly logical manner. Then he developed 465 propositions, and he went on to develop principles to the unknown in a series of steps, a process that he called the ‘Synthetic Approach’. The first six of his books deal with plane geometry and the last seven books deal mainly with the more complicated number theory, irrational numbers, and solid geometry.

A proposition is a declaration that can be either true or false, but not both.* *Using mathematical logic, a proposition can be categorized as either being true or false, although various exceptions may arise that would imply as a consequence or condition that a statement with a simple proof does satisfy given its requirements. Propositions often take the form of implications, because they rely on ‘if’ ‘then’ scenarios.

Euclid proved that the ratio of the area of a circle to the square of its diameter is the same for all circles. Euclid discovered through the method of exhaustion that the area of a circle is something times the square of the diameter and he wrote in his *Elements* Proposition 12.2 that circles are to one another as the squares on their diameters or the ratio of circular area is constant to the diameter squared. The Euclidean plane was created by Euclid a Greek Mathematician and that is what mathematicians used, until Rene Descartes invented the Cartesian coordinate system.

Descartes is considered to be the father of modern philosophy, and he began looking at mathematics philosophically, hoping to find truth by the use of reason, as he believed that mathematics was the only thing that is certain or true, but only after it was proven. Descartes thought that math could be used to reason complex ideas into simpler ideas. Descartes’ study of mathematics resulted in the formulation of analytic geometry, and this is highlighted by the invention of the Cartesian Coordinate System in 1637, which gave us a new way of looking at numbers.

Frank realized that his daughter was very smart and that he would never be able to help her in her studies, so he wished her good luck with her Geometry presentation.

Written for Sadje at Keep It Alive What Do You See #124.