Via Apollonius we examine the conical sections in a uniform framework. The double cone produces the circle, the parabola, the ellipse and the hyperbola. This geometric approach gives the impression that these sections are all of the same type. Indeed via the equations of analysis we notice that relations exist. The ellipse is the direct generalization of the circle. The hyperbola has an analogous equation with the ellipse, via the change of sign. While the parabola displays a particularity. This means that its extreme angle has no variation only as far as the double section is concerned. However, one way to understand this essential difference is the result of the research of Archimedes as far as the squaring of the parabola is concerned. While he avoided mentally falling into the trap of the circle, Archimedes, without the contemporary help of integration has managed and also in an elegant manner to prove that the parabola can be squared. This is now understood, because of the integration of the function of the parabola, while the equivalent with the hyberbola requires the use of the logarithm which was invented by Neper centuries later. Another result of differentiation of the elements of the family of conical sections, is the problem of the perimeter and the peculiarity of the ellipse, that was revealed by mathematicians like Ramanujan and Abel, both of them centuries later. In reality, with these indications we understand that Archimedes opened the road of differentiation in a field where, apparently a group existed. The understanding of this phenomenon is an actual invention, because he thought that the problem was solvable and he solved it, while his approach towards the squaring of the circle was radically different. This is also a characteristic of his mathematical genius that makes him so special in relation to the other mathematicians of antiquity. He was not just a problem solver, but also an inventor of problems, which were not just utopian for his time and the subsequent ones, but also unthinkable. His approach was not even immediately understood by the experts, as the school of Alexandria demonstrates. His opus was indeed not unknown, however they could not reach its full depth in order to be able to surpass it. We had to wait for the Chinese mathematician Liu Hiu in order to see for example, centuries later, the improvement of the approach of the number π, via an equivalent method, but with the use of areas and of course the notion of the digits of the decimal system. Thus with the conical sections, the innovation of Archimedes resulted into a novelty much later.