The analysis of construction problems in the dynamic geometry

Abstract

In the 3rd century, the great Greek mathematician Pappus systemized in his ?Collection§ the so called ?analysis§ which Euclid also emphasized but, in his ?Elements§ not appeared. The analysis, the oldest among mathematics heuristics in the history of mathematics, assumes what is sought as if it were already done and inquire what it is from which this results and again what is the antecedent cause of the latter and so on, until by so retracing the steps coming up something already known or belonging to the class of first principles. The synthesis as the reverse of the analysis take as already done that which was last arrived at in the analysis and arrives finally at the construction of what was sought by arranging in their natural order as consequences what before were antecedents and successively connecting them one with another. Greek thought the dialectic integration of analysis and synthesis as a substance of mathematical thought. However, Euclid*s Elements considered synthesis to reduce theorems from the foundation as a way to guarantee the truth of mathematics. This lecture will show that the analysis can provide the new direction for teaching Euclidean Geometry. Traditionally, in teaching Euclid geometry the synthesis to arrange final results by mathematicians is far more emphasized at the expense of the mathematical discovery process by students. It makes normal students think mathematics a very difficult subject because they cannot appreciate easily the deductive or axiomatic proof represented by the synthesis. Furthermore, this lecture shows that it is because of the lack of proper dynamic tools that the analysis known well by Greek mathematicians like Plato and Euclid has not emphasized in schools since Greek era. Also this lecture shows that dynamic geometry like GSP and Cabri developed since the late of 1980s is an almost unique circumstance for the revival use of the analysis which provides an alternative teaching method for normal students to develop their proof abilities in Euclidean geometry.