In passing from the hands of the Indian mathematicians to the Arabs, Ptolemy's Almagest underwent a radical change,mbecause the ratio i is a constant Ptolemy's ratios work exactly and more simply on the 1/2 chord. Thus the Indian mathematicians made the tabulation of trigonometric ratios easier. Among these ratios was the unnamed relationship of the arc to the radius. This ratio underpinned everything in the kuklos and fundamentally defined similarity and proportionality in our reality. One other identity is of equal importance: the Pythagorean identity for right angled triangles.
Thales theorem demonstrates this fundamental relationship between i, pi and the right angled triangle on the diameter of a circle.
The Indian mathematicians exploited this relation to calculate the sine tables, and it is this interconnected set of ratios that passed into the meditations of Sir Isaac Newton, and with it the notions of the unit circle. We must not exclude Wallis, who using such relationships derived formal identities for the conic section curves and made far thinking comments on the nature of i, but it is Newton and his students De Moivre and laterCotes who revealed the full extent of the children of Shunya
Much of wallis's teaching passed on to sir Isaac Newton, who wallis found to be a genius of the most extraordinary kind, assimilating rapidly whole sections of mathematical rhetoric and wisdom, and of an adept classical frame of mind. This seems to have occured between 1666 and 1668 when Isaac was sent home to do his studies and to avoid the plague. Isaacs mathematical sketch books are full of verve , vision,trial, error and glimpses of profound insight. By the time he returned to Cambridge he was a proteus in assimilating the classical materials required for the course,min amongst which were the rhetorics of Al Khwarzim, and a prized copy of Euclid's Stoikeioon, which wallis had set about translating into Latin from the Greek. Over the course of time sir John Barrow returned with new definitions of mathematical topics, establishing mathematics as a distinct discipline from the classics in general. Thus Isaac not only had access to prie sources, he had opportunity to contribute to the founding of Mathematics as a subject discipline.
As much as mathematicians may find this interesting, to me it represented a backward step.mmathematics cut off from its cultural life line was bound to become morbid and die eventually. Isaacs genius and subtlety lay not in mathematics, but in philosophy and metaphysics. These mental disciplines informed his keen mind and guided his symbolic reasoning. They gave a framework and a method to work to in making sense of the world,msomething mathematics and mathematicians are singly poor at doing.
Therefore Wallis in his treatises spoke to fellow classicists to whom the ellipsis in his works, and brief references to other avenues were not cryptic, but evidence of a marshalling of the classical paradigms to promote, propose and support a point. This was made the more so clear in that novelty wallis reduced the rhetoric intelligibly to a short hand notation, in order to bring more to bear in a short space. Such a condensed format was new, but universally welcomed as the way forward in philosophical and rhetorical discussion. In every case a page of such abbreviation was meant to be digested steadily and meditatively , without the distraction of voluminous word disposition and ,ultiple page turnings.
By this means philosophical arguments could be closely scrutinised and conclusions succinctly and powerfully drawn within a page or two.
This method was Walls's addition to the notion of algebra and represented a considerable refinement on Descartes notion, and even ob Pierre De Fermat's.
So isaacNewton was well equipped to take Wallis's work forward, in particular his work on the ratio i.mthere is no secret that Wallis had a thing for the circle. He calculated pi regularly in his head when he could not sleep or shut his mind down. His study of the circle identities convinced him that i was a magnitude on the plane! His work on the measurement line concept of number sharpened his understanding, for he it was that reall established the Cartesian ordinate coordinate system. Until him, such a system was used loosely and freely.mthere was np particular origin, and relationships in Conics were derived after the Greek style of a fixed point in relation to a fixed line and relationships between relative measured distances to reference points on a locus of a curve. But wallis changed all that by developing a standard set of axes and a standard form for the conics.
Because the measuring lines used along the axes were all the geometrical quantities known to date, wallis knew tha I could not be measured on these axes, but his understanding of the circle relationships, the measurement ratios convinced him it was a point outside the unit circle somewhere on the plane.
There were many ways to get a handle on this not the least being the "reverse" circle radius. Walls did not fully elaborate, but noted the relation involved ib "rectifying" the circle, which was generally thought impossible to do. Why would any one want to rectify a curve? The reason is that to use a o piece of string is fraught with difficulties of accuracy, but certainly less difficult than finding a solution to the general problem of "rectification", that is straightening out a curve.
The ratio i was only one of the relations Wallis threw light onto. His method of notating roots revealed the logarithmic relations required to solve several differential and integral problems, a precursor to Isaacs formulation of the binomial series and the method of fluents and fluxions.
So it was with great confidence and conviction in the reality of the magnitude of I that Newton and De Moivre went forward to develop the multinomials, and to link the logarithms and the signs to these types of relations. In the course of Which de moivre discovered a curious quantity: the general root of unity! This wasthe first inkling of the family and tribes of Shunya.
The use of i as a constant magnitude was as a fundantal staple of wallis theory of quantities and magnitudes. He simply proposed and demonstrated that any quantity may be created from a ratio of magnitudes, with that ratio being Defined as te unit of that quantity! Thus I was a quantity and it must be the ratio of 2 other magnitudes. In the circle there were many to choose from: comic section relations, Pythagoras, Euclidean,chords,trigonometric.so many to choose from, but none of them fit i. This was because i was being obscured by pi!
De Moivre needed Cotes collaboration to fully pull out what he had discovered, and this was down to Halley informing Cotes tha De moivre had helped him solve the rhumb line problem for pi/4 to the lines of latitude.Thus,musing De Moivre's curious relation Cotes solved the general rhumb line problem and went onto complete his labours on his masterpiece Logathmetica.
They collaborated on the Cote De Moivre equations and theorems in which the children of Shunya are fully revealed and documented. ThenCotes went on to find the most extraordinary relation: the Cotes Euler identity, at least 5 decades before Euler, in which the fundamental harmony of all measurements was revealed as due to the relations of the Children of Shunya, the roots of unity and the factorisation of unity into multiple cofactors.
I think Cotes attempted to define the ratio i in terms of the angles of arc to the radius.mbecause he died before being able to elucidate his collaborators misunderstood the ratio and created the new measure which we now call the radian.mthe ratio of i is the ratio of the quarter arc to the radius,but it is not the ratio of the same units of measurement. Thus angles of arc to the radius is clearly a strange ratio. The radian is a stranger one. It is still a constant but it is not i, however it allows i and pi to be directly related: pi radians to a diameter, pi/2 to a radius,there constants were quantifications of the ratio i, but they are conjugates to the actual ratio i, and the subtle difference makes all the difference.