The transformation of Gonu: The Arc of The Angle.

Apollonius in his work on Conics allows us to define the notion of 2 lines in relative rotation whether that be through a fixed point or around a fixed circle. This is rather special and has a bearing on the fifth postulate in Euclid's Elements.
http://en.wikipedia.org/wiki/Parallel_postulate

We can quickly see that the rotation between 2 straight lines has to be 2 right angles if they are to cross at "infinity"!. We can also see that this rotation is relative to the tangents of the inscribed circle. We can use tangency to a circle to define lineal relative rotation@!

However, we can now use tangency to a sphere or a spheroid to define planar or lineal or even volumar rotation! This facility derives directly from the study of the conics, and finds its most general expression in the work of Apollonius the Euclidean Platonic Pythagorean of Perga. It is later revived by the researches of the Grassmanns who sought a Formenlehre like that which the Greeks Euclid, Archimedes and Apollonius had developed.
http://www.omnilexica.com/?q=Apollonius+of+Perga
http://www.nahste.ac.uk/isaar/GB_0237_NAHSTE_P1095.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Apollonius.html

The praxis of Descartes , base don Herron, and De Fermat, was to set out the coordinate structure first, and then define the form in terms of that. This was hardly an advance on Apollonius, who it is claimed started with the form and established a relevant coordinate system. This indeed was the case, but it was revolutionary and innovative. By the time Herron and thus DesCartes and De Fermat learned of it, it had been formalised for some time. It was thus an easy step to set out a relevant coordinate framework first as a given! This was in fact given by Apollonius!

We actually have to come to John Wallis for the key step that distinguishes our modern coordinate system sufficiently from Apollonius' to warrant a significant remark. Wallis made the simple step of fixing the ordinate and the coordinate axes relative to each other through the origin. This simple praxis on his part simplified the formulae for the conics so sweetl that Wallis himself was cock ahoop about discovering the essential form of the conics, thus negating the necessity for wooden, crafted cones to be made ever again! Such cones had been used for nearly 2000 years to teach the conics. Wallis had a purely algebraic formulation that made the cones, as he thought, obsolete. However, good pedagogues kept the cones to explain what algebra could not!

At the same time as Wallis fixed the axes , Leibniz was also trying to free Geometry from the Cartesian Fermat constraints. His study, and genius gave him an intuition that the coordinate or algebraic geometry was going down a difficult and rocky road, somehow less free and intuitive than that followed by the ancient geometers. However, try as he might he could not gain the interest of his fellows in he Royal Societies, or in the geometrical circles he moved in. Frankly war was upon most of them and they did not have time to understand his sentiments.

Thus it is doubly ironic that those at the highest level of Academia should miss what the Grassmanns labours revealed, because: A the Grassmanns were not looking for it, and B the likes of Gauss and Kant and Lobachevsky were, but in the wrong direction.

That Hermann knew he was onto something profound is documented in his Vorreden to Ausdehnungslehre 1844. That many perceived it as profound is also documented. That Gauss redieved a copy of it is also documented. Gauss apparently dismisses it as too revolutionary, requiring too much of his limited time to study. The big question hangs in the air: was this a bit of Gaussian smoothing? was he trying to keep this work under wraps until he fully developed his own thinking? Certainly he had plans for Riemann his student and this would be an ideal culmination of is work in this area,if his student introduced the next big thing in Mathematics.

However the genie was out of the bag. Peano had got hold of a copy of Hermanns work, and inspired by it set in train a whole movement in Italian mathematics that lead to an innovative take on geometry. It is likely that Peano recognised early on that this was an algebraic version of Apollonius's works, something i am sure only gradually dawnwed on Hermann after a considerable amount of restudy and redaction with Robert his brother.

Formenlehre is an apt name for what Apollonius and the ancient greeks did: they chose a form and studieed it. The general form was the cone, from which they derived the circle etc, by cutting with a plane. However, within the cone sat neatly a semisphere/hemisphere. This was the basis of their view of the universe: the earth in relation to the stars, particularly the pole stars.

The ancient greeks were sophisticated enough to know thatthe plane geometry was a projective geometry. What they learned in the plane was a projection, a shadow cast from the reality. Thus the many realisations of the truths in the plane were mere shadows of the rich actual "truths". The study of these truths thus was approached through Astrology.

The Pythagoreans worked out the best route to this wisdom of the Musai, and it was an all encompasing lifestyle choice of study based on the Arithmoi as positioned in the Musaion, among the stimulating representations of the arts inspired by the Musai. All was fluid, all was rhythmical all was Harmonias.

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