The eye of Euler was a unique one. It is true that he lost the use of his orbs around the age of 60, but he continued to produce mathematical researches until his death. He was able to continue because his expanded sphere of consciousness made all things revolve around a common central insight: the Sphere.
Before him sir Roger Cotes, under the tutelage, correspondence and general influence of Newton arrived at the same harmonious conclusions about all measurements. The sphere was the standard!
This may come as a surprise to many mathematicians and physicists, who being badly tutored and oftentimes misdirected in their studies, are unaware of the profound underpinning Pythagorean scholarship of their crafts. Indeed, so dark is the light of their teachers that they fein to scoff t what thy know nothing if not vert little of!
Nevertheless, greater minds having toiled all their lives in study and application like Euler, have confirmed the soundness nd veracity of the Pythagorean scholarship. In so doing, they condemn the error introduced through Aristotle, who not finishing his studies at Plato's Acadmy ventured on his own opinion and by his own reasoning to establish a complete categorisation of knowledge by which he made the sphere into a cube, losing, if it were possible the very nature of curvature by measurement!
At Plato's Academy, Eudoxus was usefully employed in bringing Pythagorean credentials of Mathmatikos to all who were enamoured of the adventure. One of these was Euclid of Alexander, who returning to his home city founded a Platonic Acadmy in Egypt, which promulgated the Pythagorean scholarship. As did the Athn's Academy. They were rivalled by the Aristotelian Lyceum, which having imperial backing flourished for a hole, but foundered with the fortunes of the Alexandrian line , eventually being scooped up by Arabian patronage.
The Athenian platonic school and others suffered fates as terminal as the Lyceum, but the underlying Pythagorean model revived and revives. Itself continually in many guises. Th body of Knowledge survives today because of its boldness to follow the communal brotherhood model as opposed to the Mithraistic, Egyptian mystery religious models.
Such models were much in favour by Gnostic, Zoroastrian and many other Cultic assemblies, who obscured their financial and greed motive by a promise of secret wisdom and special access to divinity. Much of what they taught is found in the religions of today. But contrast that with the empirical and pragmatic philosophy of the Pythagorean schools, who studying the stars planets and all around them declared all things divine and worthy of respect, study and modest temperance, balance and moderation.
The talk of secret rites and inner sanctums etc reflect more the Cultic behaviours of those around them than that of the Pythagoreans, who from the outset were a public teaching order, given special patronage to continue their researches in return for public dissemination of their findings. This of course attracted great hostility from rival groups who did not have this patronage or social standing, and frequently resulted in persecutions, infiltrations , political agitations , agent provocateurs. The response was to form a tight knit community who shared the Ideals of Pythagoras and who thus established security models which were claimed to be rites of passage by detractors.
That giving thir knowledge a public airing was ultimately the correct model, is born out by how influential the Pythagorean school of thought is in the public domain. This did not prevent secret cults or conspiracies from going about their business, but it did make the public aware that this was how the world worked.
The Pythagorean school was a model for monastic life in the west, but it derives it's style of teaching, koans and self knowledge from a tradition that goes bck to the Akkadian nd Sumerian Magi, as well as the Egyptian scribal and priestly schools.
The Pythagoreans used a fundamental pattern on the floor and walls of their" temples" to the Musai. These came to be known as mosaics. But these mosaics, often bstract but not always, we're a pragmatic description of what may be known and how it may be known. This is the topic of Epistemology in philosophy. The fundamental point is that we have to record everything , by every means and on every surface. The more permanent the recording medium the longer that record will remain.
But this record is useless without trained interpreters and scholars. So the complete package is a school of scholars nd trainers whose task it is to record phenomenon, and to maintain and interpret the same. The role is to tend to nd grow a vast library and educated elite who could pragmatically, both academically , poetically and Artisanly make use of this growing wisdom for the benefit of the whole of society, not just a select few , who could pay through the nose.
This kind of Utopian ideal has a long history, but is never better expressed than in the mythical life of Pythagoras, and the literary copy redacted into the Jesus myths.
The mosaics are fundamental to the notion of logos and Analogos, on which all Metron theory and thus measurement is based, and from which the notion of a Fractal is mot expertly drawn by Mandelbrot.
Because Mosaics are in the early days mostly abstract patterns, the grammai that form these patterns and the seemeia are ultimately of fundamental significance. In a mosaic, as opposed to a wall painting which has an accurate representational feel, the discrete combination of elements is emphasised, and then overlooked to gather the emergent information.
The artistic movement of pointillism expresses this fundamental analysis of vision in the late 19 th century. However the Pythagoreans went beyond visual representation. They researched all senses and concluded hat all knowledge is constructible in the way a mosic is constructed.
Pythagoras is reputedly given an inight into the nure of "Musai" inspired sounds or Music. Again the mosaic pattern of strings reveals itself in the Logos on strings and the Analogos between strings on an instrument? String theorists today are pursuing this kind of mosaic description of reality, unaware tht it has already been done by Eudoxus!
The sphere was absolutely foundational to the Pythagoreans, and thus to Plato. But what was not adduced by scholars of the renaissance was how completely it was studied nd understood. The Arrogant Renaissance Movement took the old knowledges and first promulgated them, and then claimed to advance them! A more sober assessment is that certain Renaissance men made their fortunes by making uch claims hike burying their predecessors works which often were written in Greek or Latin, nd accessible to the few learned men and women of an Aristocratic family.
Ah, but it was ever thus!
The sphere and it's counterpart the disk were well studied and laid out in mosaics on the floor of the Academies of Plato and the Moussaion or temples or Monastic dwellings of the Pythagoreans.
The one Metron that was used for the circular disc was the diameter. Among the many logoi represented in the Mosaicsbwasbthe logos for the semi circular perimeter to the diameter. This was pragmatically laid out by turning a wooden wheel or disc or shield on a lathe. Thn by carefully halving it, and checking that both Semi circle were exact copies of each other by placing one on top of the other.
Then the semicircle was copied into the mosaic by a careful artisan. From this much was learned about the nature of the circle, it's centre , it radii etc. but the diameter was a fundamental, good or right line! Today we say straight, but when analysing the nicest concepts we must be careful not to put our conceptions onto them.
This fundamental good line was investigated. In fact, how "good" it was was defined and redefined by many different constructions over time. Euclid mentions that it is defined in his time by dual seemeia . That means a compass like instrument is used to determine dual points from 2 arbitrary initial centres. The diameter arises from where thes dual points transecting the circle perimeter into x2 equal circumferences or arcs.
Today we even define the circumference incorrectly!
Next by carefully rolling the semi circular dist in a straight line a segment is marked of equal to the circumference of the semicircle, that is the length of half the perimeter. It is an arc length, but more precisely sn arc magnitude. The reason why circumference is defined is to distinguish an arc magnitude from a diameter magnitude!
They needed to be distinguished as different magnitudes because they were both represented by a lineal magnitude!
The ratio or logos of a semi circular circumference to its diameter was defined by Eulet. It is defined as i
This is the same ratio as the quarter arc to the radius, this too is i
The ratio of the whole perimeter to the diameter is defined as Pi. This was also given by Euler. Finally the ratio of the altitude of a point to the radius Was defined as the sine, but the ratio of the change in arc length to that radius was 1 : 1 – 1/x where x was very large. This was the basis of the Napieran logarithms. These were logarithms of the then extant and detailed sine tables, this ratio being ver close to sin (Pi/2 – I/x).
Both Cotes and Ruler recognised in Napiers method the ratio or Logos 1: 1 + 1/x where x was very large, was a significant value for the circle. This tended to the limit that Euler denoted by e. both of these were expressed in terms of the binomial theorem to obtain these results, and thus the basis of ny logarithm is the limit to which such a ratio tends as x grows very large under a binomial expansion of the form
( 1 – 1/ x)^x or (1 + 1/x)^x.
It is therefore simple to see that
Pi = 2*i
What is obscure is why i is defined as _/ -1
Euler defined it as the Sqrt of -1 because no one else before him had bothered to define the negative square area. Although Bombelli had utilised the definition, no one understood that a line or rectilineal form above the diameter had to be given a different sign to one below it . Bombelli did, but did not apparently, to my current knowledge specifically identify squares or quadrature in this way. However, he gave the exact rules for using these types of magnitudes, which he apparently claims was a daring gamble on his intuition. I believe Bombelli came to the same conclusion I did after meditation, that if you are going to define negative magnitudes for accounting then you must also define negative magnitudes for geometrical accounting.
Euler specifically defines the quadrature of the unit circle for positive and negative magnitudes of squares. What his contemporaries failed to pick up on was that the Sqrt of -1 is -1 AND +1 while the sqrtnof +1 is -1 OR +1.
Both Euler and Bombelli used the definition of quadrature in Stoikeioon book 3 to define the new fanged negative quantities. Nobody else really cared enough to tie down this detail, and so confusion reigned for nearly 5 centuries!, resulting in the hybrid we call number thoughtlessly or at best poetically.