I was wondering if Geometry was not created by Descartes?
There is strong historical documentary evidence that Euclid did not write a geometrical text. We also have the platonic inscription which dissuades non geometers gpfrom joining his Academy. Everything points to a pragmatic approach to life, measurement and construction which is Mechanics based not geometry based, ie there was no subject called geometry until Plat defined it in his academy.
Under the rhetorics of the Musai. Geometry actually makes quite a late appearance as a subject, being subsumed in Astrology and notions or ideas of form, the Eidea of Socrates and Plato, and the Arithmoi or epipedos of the temples or academies to the Muses. From The Pythagoreans Plato adopted the term Mathematikos, for those who were competent to calculate the positions of the planets. Thus it was a mark of distinction and graduation. Presumably then, these graduates had to start somewhere, and it would seem that measuring and surveying , rope making and other technical pursuits was the place to start.
Euclid's postulates have received several interpretations, but the simplest is that of entry requirements onto the introductory course on the Platonic theory of Eidea. The student must be able to: draw a straight lie. Extend a straight line, draw a competent circle and extend a line to meet another line on one side of a line that crosses both. If you imagine that this was a course in surveying. You would see the direct relevance of these demands.
However, this is not s course in surveying, but in the theory of Eidea , thus besides the postulates or demanded items, there are some common notions of judgement that must be accepted also, but not demanded. To these Euclid adds the propositions of forms or ideas as flat out definitions.. He then begins to propose.
We have been so schooled in this Euclidean format as geometry, the obvious rhetorical and philosophical style has been ignored! Besides, it seems to have become a runaway best seller among the technically minded of his day , especially the aspiring elite who wished to shape society and the world in line with Platonic ideals.
As an introductory course, it is clear that one cannot graduate to Mathematikos by mastering it alone. Euclid's course went on to include optics, conics and perspectives, all of which would form a good grounding for any aspiring Astrologer. It is the Astrologers who gained the title Mathematikos, and they were few and far between.
Bearing this in mind we have to come to Descartes for a radically different treatment of Euclid, one which in the main ignores the academic development of Euclid's course and imposes Descartes own meditations. In the main, Euclid was not well known in Descartes time, except in Arabia where he was well studied. ButDescartIes, through the Jesuits had access to some "arcane" material, books which elsewhere in Europe were routinely hunted out and burned as "works of the accursed Moors, servants of the Devil!" . The astonishing ignorance of the western clerics is a spot and a stain on western history.
Anyway Descartes had access to a version of Euclid, from which his predilection to meditation allowed him to arrive at a rhetorical style of symbolic notation derived from Al Khwarzim's rhetoric called algebra. Descartes Geometries was a fine algebraic reworking of many of the Eucldean theorems as principles of Geometry and to a very great extent Mechanics. Thus I ask the question : until Descartes was there any real notion of the subject we call Geometry? It is easy to go back, after Descartes and retook at Euclid in geometrical terms, but before Descartes was geometry distinguished from Mechanical construction? Dürer and Bombelli refer to geometry only in the context of mechanics, and Bombelli is one of the first authors to try to shift technical understanding from aritmetic calculations to algebraic problem solving. This is before Descartes. There does not seem to have been a widespread knowledge of geometry as a pure subject until revised and revisited by Descartes.
Mechanics, the building of working constructions and machines was inseparable from geometrical ideas. This is why the Tekne took it to heart. But Mechanics is not Geometry.
Grassmann observes that Geometry has been given a raw deal over the time he learned of it, saying that it was unjustly denigrated by all. He hoped Phorometry would lift Geometry out of the mire, making it of equal importance to Mechanics.
It is a moot point , but of interest perhaps only in passing, and to a few purists of historical subject development. But pragmatically it means that Cartesian Algebra is Geometry, it is the foundational principles of Geometry, and where modern geometry springs from.
It also means that Gauss and Riemann are essentially questioning Descartes about the grounds to his geometry, not Euclid. The fashion of blaming Euclid is entirely misplaced in any case, but definitely on the grounds of geometry or logical rigour of geometry(Russellian critique).