Summary:
The present dissertation aims to present and analyze the progress of the
concept of “tangent” through the classical texts of the ancient Greek maths.
Starting from Euclid’s and his “Elements” where the concept of the tangent
appears only in the circle, we will spot the semantic changes from the verbal
level to the mathematical level through algebraic relationships.
We will proceed to Apollonius who presents a more general view of the concept
of the tangent which expands on all the conic sections. In the second chapter
of the dissertation, where the characterizations given are conform to the
Euclid’s spirit, we will follow the evolution of the concept of the tangent.
In addition, Apollonius proceeds to a passage to more general curves about the
concept of the tangent such as the ellipse, the hyperbola and the parabola
using a more powerful implement of relationships which is the double ratio.
Finally, in the third chapter, the concept of the tangent will be analyzed
according to the spirit of Archimedes, the greatest mathematician of the
antiquity.
This concept is analyzed within the bounds of the perimeter of the spiral of
Archimedes which is thought to be very innovative in comparison with all
previous theories that were developed mainly by Euclid’s and Apollonius.
At the conclusion of the dissertation, we will present several teaching
suggestions in form of exercises that refer to the properties of the tangents
which we examined mainly through the proofs of Apollonius as well as Euclid’s
and which are examined in an analytical way. There are also others that concern
the way of making a tangent in conic sections according to properties of the
conic sections as they are presented by these three great mathematicians. We
also propose ways of fabrication of the tangent in conic sections with
algebraic methods such as the revolutionary method of the use of the harmonious
quartet. Finally, we present ways of trisection of an angle using the spiral of
Archimedes.
Keywords:
Tangent, Cycle, Conic sections, Average ratio, Double ratio
