Geometry- A Classical Approach

In this high school textbook, the basic order and presentation of geometry as found in Euclid's foundational work, The Elements, is maintained. The student is introduced to the science of geometry through a rigorous study of the first six books of Euclid's work, a text of which Albert Einstein has said, We reverence ancient Greece as the cradle of western science. Here for the first time the world witnessed the miracle of a logical system which proceeded from step to step with such precision that every single one of its propositions was absolutely indubitable - I refer to Euclid's geometry. This admirable triumph of reasoning gave the human intellect the necessary confidence in itself for its subsequent achievements. If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker. The first objective of this text is that students experience scientific knowledge through the study of geometry. The second is to help students build confidence in their ability to analyze, think logically and reason clearly. The third objective is to help students apply geometric principles to solve a wide range of mathematical problems. To develop critical and analytical thinking skill, the text begins with a unit on logic. Consistent with the classical model of education, students study concepts developed by Porphyry and Aristotle. The next six units follow Euclid's exposition of geometry as found in The Elements. Theorems are preceded by explanations of the concepts employed and followed by problem sets containing applications. Students develop analytical skill by discovering their own proofs for properties not considered by Euclid. They become Einstein's "scientific thinkers" by seeing the development of the science of geometry from its basic definitions, postulates and common notions (axioms). Students see how the figures and magnitudes are constructed from those postulates and they prove a wide range of properties with Euclid as their sage. In the last unit, students apply Euclid's theorems to develop the properties of the Real Numbers and introduce the trigonometric functions.