At the secondary school level, geometry is better known as Euclidean geometry and is defined as the study of plane relationships. It begins with a set of ten axioms and common notions upon which is built a comprehensive system that is both deductive and logical. At present, at least 18 axioms or assumptions which cannot be proven but must be accepted as truth are identified. Dozens of theorems have been proved by using the basic axioms.
Although the definitive textbook on geometry was written by Greek mathematician Euclid about 300 before the time of Christ, it remains the basis, not only for geometry but for much of the modern numbers system. Although Euclid did not originate some of the concepts, he was the first to present them in a logical structured format. In addition to geometric functions, Euclid also wrote on the number theory and three dimensional geometry.
Euclid s Elements has a beauty of structure and logical reasoning which has been displayed and followed for 24 centuries and has been translated into dozens of languages. Although the structure of geometry is the main focus of the book, the presentation of principles went far beyond the study of geometry and affected the study of most if not all future disciplines associated with mathematics.
Further studies of the work have revealed that no matter how carefully structured the geometry basis used; it makes some assumptions which are not proven using the format developed. The other problem with Euclidean geometry is that it applies cleanly where the spaces are homogeneous, but does not apply so well where more than three dimensions are existent. Some of the geometry dimensions go as high as Dimension 10 or Dimension 11.
Euclidean geometry as taught in secondary schools is helpful in developing structured thinking and following logical principles. Understanding the concepts of thinking logically and drawing conclusions from given principles or facts can be useful in every field of life. This is the basis of inductive reasoning.
Traditional geometry has used compasses, protractor and ruler as the tools to proceed with the experiments which formed the basis of the science. The interesting aspect of geometry is that it was originally developed from many experiments where certain facts became apparent. After countless measurements with the same results, some generalizations could be drawn. However the basis of geometry is not the measurement, but the inductive reasoning, which starts with the basic concept that a point has no dimensions, a line has one dimension, a plane has two dimensions and a solid has three dimensions.
Inductive reasoning depends on measurement and observation neither of which is necessarily absolutely accurate. Because measurements, no matter how many are taken will never eliminate any possibility of a variance to the concept being developed, a conclusion is usually reached before all possible cases have been studied.
This is the basis of the study of geometry. Based on measurements, inductive reasoning is used to draw conclusions about every possible case. Then by building on basic axioms, statements of probabilities can be defined which are known as theorems.
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