From Plato to Legendre : Geometry in Aristotelian Crisis!

It seems funny to write headline like that. It needs some explaining.

Before Plato there was no geometry. Of course there was geo metria, that is land survey, but that was a diminutive version of Astrology, an everyday application of Astrological knowledge and Technique. The Magi, the star knowers and star gazers, developed techniques, tools and instruments and algorithms and records that revealed to them patterns and sequences in the sky. This knowledge and wisdom gave them a power to predict the ways of the Stars and planets. It also gave them power over the earth from the study of these things to determine the patterns on the land, in the agriculture and in commerce.

The early civilisations were governed by this astrological wisdom, not by the measurement of land!

When a great king arose, his astrologers advised him to build observatories! Places where they could study the stars and planets even more accurately. Their astrological skills made the building of great monuments to great precision possible, and wonderous. This wonder reflected on the kings, who by building using a carpenters square came to brknown as Rulers! Their Architecture defined their glory. Thus Astrologers found great employment of their many skills and wisdoms in the designing and building of great monuments for the king, which also served to beneit the subjects culturally and religiously, and commercially.

The great temples were places for worship, trade and tribute.The king sent out his law from there.

The power of the astrologers waxed and waned with the politics of the time, the geopolitical circumstance and whether the king successfully prosecuted their predictions. As the political power of the Kings increased, and the establishment of the priesthood in the temples centralised religion, the Astrologers lost their position of "divine" rule, rulership granted by the Stars, the anunaki. They began to fade, retaining only an advisory role.

Yet they did not fade. They withdrew into societies and secret or mystery schools, passing down their wisdom only to the most ardent students, those willing to live the life of a community of devotees to the astrological arts. They were the first communes and monasteries, their buildings were simple but museum-like. They were Temples to the Musai, the muses, the very repositories of wisdom and tradition, but not dry tomes of writings and dead rituals. Rather spirited and inspirational places that fostered personal creativity. That brought out in each individual that gift which the Musai gave to it alone.

I guess, to bring it home to the modern reader, we could consider these groups as gangs! They were learning gangs or better still Geek Gangs! Most of Academic history up until the establishment of public education retained this geeky clique nature. The gang was more important than any building or territory they might inhabit. Thus gang members frequently on leaving established their own chapters wherever they resided.
The notion of curriculum is therefore not that of pedagoguery but rather that of group dynamics and group discourse. This is why Kant argued that mathematics is discoursive rather than synthetic. However Aristotle was the pedagogue par exemple, so he tended to lay out a curriculum in his gang.

Such was the commune of Pythagoras, and from such communes and priests Thales learned his ways.

The circumstances of the rich wide boy nicknamed Plato, we're such that he had to flee Greece to escape political enemies and assassins. His trip around the Mediterranean countries gave him time to reflect. He wanted Greece democratically to be a utopia, but it was not possible without a republic guided by a strong, wise benevolent leader.

His tutor was the philosopher Socrates. From him he learned how to engage people in a deeply self examining process. While in Alexandria he heard of Pythagoras in Italy. After completing some research in the Alexandrian Library, he came upon a scroll or biblios written by Pythagoras.

He next went to Italy to study as a pythagorean. Gradually he teased out the stories and anecdotes regarding Pythagoras which later he uses in developing a few of the legends surrounding Pythagoras. There is no evidence that he completed his studies with the Pythagoreans, but ample evidence that he had found his spiritual home and concept of Utopia. He returns to his home town and his family estates, with one idea, to establish an Academy. This is a gathering of people( demo) whose purpose was to promote the best (aka-), in fact Plato defined the best in terms of his Utopian ideals based on the Pythagorean model of a commune.

The home of a Pythagorean school is a Mousaion. That is a temple to the Muses. Within it were epipedos, abstract or figurative, and other decorative motifs that silently witnessed to the underlying principles of the school. The Musai, usually about 9 were headed by Apollo, the sun god. So Pythagoreans have been mistakenly described as a mystery cult worshiping pagan gods! Pagan is a wonderful term. At one time Jesus and YHWH were so described!
http://my.opera.com/jehovajah/blog/2012/05/19/arithmos-and-music

The religious backgrounds throughout time has more or less remained the same. Thus we place statues where the ancients may have done the same calling it a shrine. Worshipers still come and admire. We build Cathedrals where the ancients built temples, and we build other buildings for cultural purposes. Animal sacrifice was perhaps more open, not hidden away in abattoirs, but the holocaust was burnt on the altar to create a sweet smell, not a smell of burnt remains. Thus the temple smelt often like a high class restaurant! People's superstitions have hardly changed.

Looking at it this way one understands that despite the political motives of the townsfolk around the Pythagoreans they were in fact living rigorous lives, devoted to the study of astrology, culture and tradition! In fact, so,learned were they that the leaders of surrounding towns came to them for advice as well as astrological forecasts. Thus despite being a commune away from the democratic and violent process common in Greece and Greek influence Italy, they nevertheless were drawn into inter town politics and feuds. They therefore needed a powerful protector to continue their studies freely.

The stress of maintaining an isolated community in the face of such suspicion and hostility took its toll on senior members of the community. At some stage Pythagoras had to deal with a powerful challenge to his leadership! The details are screwed up by legend, and confused with the notion of unity! The Pythagoreans sought unity not as some later idea of a " number" but as a group! The unity of purpose, leadership and action under one leader. The square root of 2 had very little to do with it!

The obvious attraction of this legend further served to lampoon the Pythagoreans, whose work on Arithmos and Arithmoi underpin our whole misguided concepts of " number". The so called proof of the irrationality of the square root of 2 can equally be seen as a method for generating right triangles with sides whose Metrons are precise( artios) monads rather than approimate( perisos) monads! The whole concept of artios , perisos and proto Arithmoi has been thoroughly screwed over, and this is because Plato did not complete his studies!

For whatever reason Plato had to leave before completing his studies under the Pythagoreans, he took with him a copy of the writings of Pythagoras to continue his study. Whether he begged, borrowed copied or stole that version is not known, but it seems to be the only source of information about Pythagoras the legend available to Plato and indeed the Neo platonists. Thus studying it on his own lead him into certain misunderstandings which he passed on to his student Aristotle. One of these misunderstandings was over the role of artios, perisos and protos and the two-ness of 2. Aristotle ended up declaring the Pythagoreans( as he knew of them) to be misguided over this issue and he set out a logical argument in defense of his solution.
http://my.opera.com/jehovajah/blog/2012/08/12/aristotles-blunder in which is discussed Aristotles distinction between two and twofold!

Aristotle's defense has never been rebutted, because no one quite knows what he is on about. However it is easy to point out that his entire premise is false, and so his argument collapses.

The Academy is usually presented in the light of a growing democratization of Education. In fact it is best presented as a commercialization of scholastic and monastic communities! though Plato was of a rich family, and devoted his wealth to his vision, he still had to balance the accounts. So not just anyone could come to his academy. They had to be wealthy enough to contribute to the maintenance of the academy.

The second condition or requirement was emblazoned across the lintel, it is said " Ageometrists definitely not into here!"

Because the word geometry is distinguished many have decided this is where geometry as an Academic discipline starts. In fact Ageometrists are those who do not have a qualification in the study of astrology, or any love of the requirements of such a study. It is almost the same as saying "non-pythagoreans not wanted!". Because Plato did not finish his studies he was not qualified as a Mathematikos in Pythagorean terms, he was still a student himself, and geometresei was the practice of developing astrological skills through land surveying. Thus Plato was determined to establish a Pythagorean school. But on a commercial basis with entry requirements that the students be sympathetic to the Pythagorean model.

The Academy, and the idea of the Academy took hold, and quickly attracted young vibrant male thinkers including Aristotle, who were willing to adopt this rigorous training mixed in with cultural and religious traditions. Socrates had taught Plato the necessity of fun and humour in furthering learning goals. Socrates was killed prior to Plato's return, again due to democratic politics. Thus Plato had to be careful and clever. Many of his plays teach Socratic principles, but disguise them as discourses or dramatic descriptions.

Aristotle did not always agree with Plato, but apparently he was loyal. So when it came to the decision about the future of the Academy he advocated its continuance in an non Pythagorean form. While Plato was alive he had been able to fund it and gather funds, but that was becoming increasingly difficult to do in the unstable politics of the time.
http://en.wikipedia.org/wiki/Aristotle

The position and reputation of the Academy was such that it commanded respect even among political rivals, but Plato's successor decided to keep it on strict Pythagorean principles and to keep it in Athens. Aristotle was no longer welcome in Athens for political reasons. Thus he took Plato's academy to wherever he went, but he was not a great fan of the Pythagoreans so he developed the offshoot of the Academy along a different set of principles of his own.

Because he became a tutor to Alexander his influence and power grew, and he was able to return to Athens, to the Academy of Plato. But he established a rival institution called the Lyceum. This competed for the available funds or wealth in the Greek empire now under Alexander's rule. His students learned by discourse and systematic Aristotelian rhetoric. Because they met in a place, like the Stoics, where there were colonnades called peripatoi when they debated, they were nick named the "peripatetics". These peripatetic Pedagogues, an Aristotelian creation are the main reason there is this mythical crisis in a mythical Geometry!
http://en.wikipedia.org/wiki/Lyceum_(Classical)
http://en.wikipedia.org/wiki/Peripatetic_school

From the establishment of the 2 styles of the Platonic Academy, the Academy based on Pythagorean principles and the Academy called the Lyceum based on Aristotle's principles, there was a rivalry for funds and consequently preeminence. The "curricula" of the 2 institutions also showed a difference in emphasis under the heading of Rhetoric.. The institutions had a different philosophical goal. One devised by Plato, and essentially Pythagorean, the other devised by Aristotle and essentially empirical. They represented 2 responses to the Socratic question of reality.

Socrates asked questions of his students which only they could answer in themselves, coming to their own conclusions. Plato, in his theory or Philosophy of Form/Ideas presented the questions to his students, who took the Pythagorean answer. The reality of the senses is a shadow of a greater reality. Aristotle took the other view. The reality of the senses is the source for an abstracted reality in our minds. These 2 responses have played out to this day.

The 2 academies became the model for higher learning institutions throughout the Greek or Hellenistic world, but the rivalry meant that pedagogues actually were at war with each other for funds, status and subject boundaries! The geopolitical situation in Greece at the time was now built into the educational system! The rivalry of states and regions were played out in the educational arena over subject boundaries. The subject of Astrology was categorized by Aristotle in a way that lead to the subjects of the Quadrivium in later times. His new topic of Aristotelian logic with grammar and rhetoric formed the introductory level for all students (juniors) in the Lyceum.

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Euclid.html
These innovations in the curriculum themselves set off a curriculum war. For this reason the Platonic Academy commissioned and inspired Euclid to produce an introductory level course for their students. This course by Euclid, presented in the Platonic Academy, ran in competition to the Aristotelian logic introduction in the Lyceum . The aims of the 2 introductory courses reflect the goals of their Academies. Euclid's has no need to differentiate Astrology, but the one in the Lyceum had to differentiate into the four second level topics of the Quadrivium. Thus in Aristotle's Academy(with off shoots) we see the beginning of the subject boundary we call geometry.Eudemes of Rodes a close companion of Aristotle is credited with this curriculum development, drawing the Pythagorean traditions into a 3 part categorisation: arithmos(number),gematria (geo metria),astronomy. Unfortunately, non of his works survive in the originals.

These boundary divisions are a feature that Signifies an Aristotelian curriculum: for Aristotle was an inherent taxonomist. His so called "Logic" is really a taxonomic system, which, if accepted, rail roads you into one category or another. Thus a so called logical conclusion is in fact a category in a taxonomic system. Without a rule base, a grammar, logic can not systematically distinguish. But with such a grammar, logic can be constructed as a systematic execution of the rules.

Syntax which is fundamental to all languages including computer languages was studied prior to Aristotle. That he gives no account of it leads to the conclusion that he had no idea of others contributions and that his conception originated from his own experience of learning different languages.

Logic, therefore is merely systematic application of a system of rules. Thus Euclid's Stoikeioon qualifies as a logic as much as Aristotle's system.

In this time, besides Euclid, several other great Astrologers arose in the Greek empire. The 2 most famous are Archimedes and Apollonius. Depending on whether they went to the Lyceum, or came under the influence of the Peripatetic Pedagogues at least , as opposed to going to the Platonic Academy is their actual attitude to astrology and geometry. But certainly by the time Archimedes and Apollonius arrive on the scene Astrology is almost fully mechaniseable into the Antikythera machine, and Euclid's introductory course to the Platonic Academy( a Euclidean offshoot in Alexandria) is being criticised in the light of new developments in Conics .

http://en.wikipedia.org/wiki/Platonic_Academy
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Eudemus.html
It is not until we reach Pappus and Proculus, that a long historical perspective can be taken. By Pappus time their was a group of academicians called geometers as opposed to Astrologers. This type of differentiation suggests that the Lyceum had a greater curricular effect than the Academy. That the Peripatetics persuaded more by their logic than the Pythagoreans, but all saw themselves as Platonists, or neoplatonists.

It would appear that the Quadrivium developed in the Arabic academies under the influence of the Aristotelian form of Platonism, which is a "commercial" extension of the Pythagorean School, which was strictly communal and monastic. Aristotle's rigourous autistic logic and his obsessive compulsivity forced him to Taxonomise everything. This system turned out to be extremely useful for apprehending and developing new knowledge about "reality", and almost impervious to fault finding -both qualities extremely important to an autistic personality suffering with Obsessive compulsive disorder.
Thus Edemus's work on the taxonomy of Pythagorean concerns with "Arithmoi, Gematria (geometeria) and Astrology', under Arabic scholarship become Arithmetic, Algebra(Aljibr) Geometry and Astronomy.

In fact, the Arabs play a key role in this outcome. When the Neo-platonic Academy was destroyed, the material of the platonic academy followed the material of Aristotle into Arabic scholarship. But by then the Aristotelian principles had become enshrined in Arabic thought. Thus the platonic contribution was interpreted in the Aristotelian manner. Later, during the Renaissance the Aristotelian school was first to influence the development of higher learning in the west. Only later did the Platonic Academy' contribution begin to distinguish itself.
https://en.wikipedia.org/wiki/Astrology_in_medieval_Islam

Despite this historical stratification, the subject of geometry still had uncertain boundaries. However, there was a rich collection of texts to learn and teach from, and many were preserved and learned and developed by the Arabs. In the rest of the world, however, this Aristotelian division did not exist. Everything was astrology. The Gunas, and the Ganita sutras formed basic astrological knowledge for the temple building classes in India and Harrapa. In China, the Yi Ching was the fundamental Astrological text in which this kind of knowledge was situated.

So by the time we arrive at the Renaissance, we find the textual confusion still drives an Academic confusion and debate, along with the still ongoing sporadic subject boundary wars! Later, by about 300 years, the Prussian Renaissance finds itself faced with the exact same textual and Academic confusion, but by then Legendre had published his definitive work on what he called Elementary ( as in Euclid's Elements of) Geometry. While this did not clear up the textual and source difficulties to geometry, it did provide a single referent to the concept of Geometry. It defined a subject boundary for the first time in the west!

There are other twist and turns in the story of defining the subject boundary of Geometry. When the Renaissance began, many documents were rediscovered containing the wisdom of the Greek geometers. Many were in ancient Greek. But while this was occurring scholars in Italy and Europe knew that the Arabs had collected all these insights into cosmopolitan centres of learning and translated the worlds knowledge into Arabic. So to obtain a copy of an Arabic translation was a coup de grace. However the dark minds of the western kings and clerics, bitter from defeat demonised all such texts!

Bombelli was therefore the first to bring a lot of Arabic rhetoric into a major work on calculation for engineers. He used the Arabic name Aljibr, to describe a system of thinking geometrically to find solutions. He in fact presented what might be called carpenters set Square "Geometry" under the title Algebra.

Bombelli is quite open about his sources, but DesCartes is not so open, possibly for reasons of social acceptance. In any Case Wallis presents a powerful case for the source of DesCartes method being Herron. The point here is that DesCartes and De Fermat both introduce Algebra to the west as a description of geometry. So DesCartes algebraic work is called La Geometrie.

For a long time scholars have thought that Descartes was algebraising the subject of geometry. In fact, there really was no subject of geometry until Legendre. The Arabs had stylised Rhetoric according to the Aristotelian platonists to lead into the Quadriviium, which was a four part classification of Astrology. Within the Quadrivium, that which might be called geometry was not just the work of Euclid, but an Amalgum of the works of great Geometers since Plato. It could not be separated from the quadrivium, which was an organic whole.

So in the West Geometry was derived from Algebra, but interpreted to western scholars by engineers and Mechanics! This was hardly acceptable as a pedigree for learned gentlefolk! In fact it lead to geometry being unacceptable at academic level in most of Europe.

In Italy it was viewed differently. Geometry was part and parcel of Art and Architecture. Engineers, mechanics and artisans were respectable and respected. Galileo for example, an intellectual and a professor, collaborated with local artisans in the construction of his famous telescopes.

So it really was the discovery of Greek texts and Latin texts that brought the discussion of such topics into the Classics curriculum. The growing recognition of Geometry as a classics subject seems to have been as a result of Barrow returning with Greek and Latin copies of texts by Proculus and Pappas. The translation of these texts into Latin was a endeavour engaged in by classics masters and students over a considerable period. These classical texts were viewed as advanced level philosophy and Astrology, thus they received the Pythagorean stamp of competency called Mathematikos!.

Barrow went on to establish the first chair of Mathematics at Cambridge, and so combined the chair of Mathematics with a chair of Geometry he held at one of the colleges . Thus we see the beginning of the amalgamation of Geometry into mathematics.

Although European institutions differed from the developing British system, it is clear that no institution had a clear , organised system of Geometry or a clear scientific system. So the works of Newton , Kepler , Galileo, Brahe were all fundamental to the evolving of a scientific, physics or metaphysics curriculum, but driving all of this gradual transformation was the logical taxonomy of Aristotle!

The discovery of a fundamentally different type of logic was astonishing. It at once raised the profile of Plato and the Platonic Academy in the west. The Arabic, Aristotelian legacy was facing an age old challenge! And the battleground was not even drawn! No one had a clear idea of what geometry was or what it was not!

The so called 5th postulate exploration is the final piece of the jigsaw, from a western viewpoint. Even this investigation was initially misunderstood in the west.

http://en.wikipedia.org/wiki/Parallel_postulate

History [edit]

For two thousand years, many attempts were made to prove the parallel postulate using Euclid's first four postulates. The main reason that such a proof was so highly sought after was that, unlike the first four postulates, the parallel postulate isn't self-evident. If the order the postulates were listed in the Elements is significant, it indicates that Euclid included this postulate only when he realised he could not prove it or proceed without it.[7] Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by his own axiom.

Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give a postulate which is equivalent to the fifth postulate.

Ibn al-Haytham (Alhazen) (965-1039), an Arab mathematician, made an attempt at proving the parallel postulate using a proof by contradiction,[8] in the course of which he introduced the concept of motion and transformation into geometry.[9] He formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral",[10] and his attempted proof contains elements similar to those found in Lambert quadrilaterals and Playfair's axiom.[11]

Omar Khayyám (1050–1123), a Persian, attempted to prove the fifth postulate from another explicitly given postulate (based on the fourth of the five principles due to the Philosopher (Aristotle), namely, "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."[12] He derived some of the earlier results belonging to elliptical geometry and hyperbolic geometry, though his postulate excluded the latter possibility.[13] The Saccheri quadrilateral was also first considered by Omar Khayyám in the late 11th century in Book I of Explanations of the Difficulties in the Postulates of Euclid.[10] Unlike many commentators on Euclid before and after him (including Giovanni Girolamo Saccheri), Khayyám was not trying to prove the parallel postulate as such but to derive it from his equivalent postulate. He recognized that three possibilities arose from omitting Euclid's fifth postulate; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right angles, we get Euclid's fifth postulate, otherwise, they must be either acute or obtuse. He showed that the acute and obtuse cases led to contradictions using his postulate, but his postulate is now known to be equivalent to the fifth postulate.

Nasir al-Din al-Tusi (1201–1274), in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250), wrote detailed critiques of the parallel postulate and on Khayyám's attempted proof a century earlier. Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate.[14] He also considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them.[13]

Euclidean, elliptical and hyperbolic geometry. The Parallel Postulate is satisfied only for models of Euclidean geometry.

Nasir al-Din's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the subject in 1298, based on his father's later thoughts, which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."[14][15] His work was published in Rome in 1594 and was studied by European geometers. This work marked the starting point for Saccheri's work on the subject [14] which opened with a criticism of Sadr al-Din's work and the work of Wallis.[16]

Giordano Vitale (1633-1711), in his book Euclide restituo (1680, 1686), used the Khayyam-Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. Girolamo Saccheri (1667-1733) pursued the same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case (proceeding, like Euclid, from the implicit assumption that lines can be extended indefinitely and have infinite length), but failing to refute the acute case (although he managed to wrongly persuade himself that he had).

In 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyám, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further.[17]

Where Khayyám and Saccheri had attempted to prove Euclid's fifth by disproving the only possible alternatives, the nineteenth century finally saw mathematicians exploring those alternatives and discovering the logically consistent geometries which result. In 1829, Nikolai Ivanovich Lobachevsky published an account of acute geometry in an obscure Russian journal (later re-published in 1840 in German). In 1831, János Bolyai included, in a book by his father, an appendix describing acute geometry, which, doubtlessly, he had developed independently of Lobachevsky. Carl Friedrich Gauss had also studied the problem, but he did not publish any of his results. Upon hearing of Bolyai's results in a letter from Bolyai's father, Farkas Bolyai, Gauss stated:

"If I commenced by saying that I am unable to praise this work, you would certainly be surprised for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty-five years."[18]

The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into hyperbolic geometry (the acute case) and elliptic geometry (the obtuse case). The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868.
Converse of Euclid's parallel postulate [edit]

If the sum of the two interior angles equals 180°, the lines are parallel and will never intersect.

Euclid did not postulate the converse of his fifth postulate, which is one way to distinguish Euclidean geometry from elliptic geometry. The Elements contains the proof of an equivalent statement (Book I, Proposition 27): If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another. As De Morgan[19] pointed out, this is logically equivalent to (Book I, Proposition 16). These results do not depend upon the fifth postulate, but they do require the second postulate[20] which is violated in elliptic geometry.
Criticism [edit]

Attempts to logically prove the parallel postulate, rather than the eighth axiom,[21] were criticized by Schopenhauer, as described in Schopenhauer's criticism of the proofs of the Parallel Postulate. However, the argument used by Schopenhauer was that the postulate is evident by perception, not that it was not a logical consequence of the other axioms.
See also [edit]
For more information, see the history of non-Euclidean geometry

2000 years of puzzlement is a bit rich! Really it was the West that was puzzled. the east was merely extending the range of their astrological thinking. However some later Arabs became confused about the nature of the postulates and did not distinguish them from propositions!

The most important and fundamental of these astrologers seeking to explore astrological thinking was Apollonius. His innovative and inspired combination of Euclid"s Optics and Euclid's Conics revolutionised the power and applicability of Astrology. With it he could explain many astrological behaviours that were visible everyday in nature, from shadows to planets. His work is the fundamental basis of what the Grassmanns called Formenlehre, It is the basis of all Group theoretical structures and the power of Grassmann analytics and synthesis especially as expounded by Norman Wildberger in his Universal Hyperbolic geometry.

But it predates the formal subject area of geometry by 2 thousand years!

Depending on what textual tradition you use one is forced to assume that certain subheadings were devised by Euclid, or placed in by later redactors. It is even possible that Apollonius redacted the original course for his purposes. in any case the subheadings of definitions postulates and ccommon judgements and propositions have been fundamentally misunderstood.

That which are called postulates, in greek are called Aithema. We understand this word as Items today. What we fail to grasp is that Items are requirements usually laid out in list form. Most Academics will begin a course with a few introductory remarks and then list the course requirements. That practice does not seem to have changed at the Academic level for millennia.

The requirements of the course are therefore a set of practical drawing skills, suited for Astrology, but demonstrated by geo-metry! The 5th requirement is not about parallel lines at all! it is about being able to draw straight lines that cross, no matter how far away they cross. `in that respect it is a geo-metrical requirement for doing projective geometry! It is suspiciously like a requirement that Apollonius might introduce to prepare the student for his take on the Euclidean study of the conics or the euclidean study of Optics, if Euclid did not in fact place it there himself.

The idea of a seemeioon , gramme and the notion of iso and euthai are also similarly misunderstood.

i have blogged extensively on these issues so google "jehovajah Euclid" to find out more.

The concept of The Stoikeioon or Stoikeia being a geometrical text is demonstrably misleading. It is what it says in is title, an Introductory course. To what you may ask? To the Platonic Academy's discoursive and educational training in the Pythagorean tradition of Astrological sciences, modified by the Socratic principles or theory of Form/ Ideas.

All Pythagoreans are geometers, but only a few are Mathematikos!

http://www.gresham.ac.uk/lectures-and-events/here%25E2%2580%2599s-looking-at-euclid
http://www.gresham.ac.uk/lectures-and-events/400-years-of-geometry-at-gresham-college

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