The quest for pi(π), from Archimedes to transcendentals, has been underlying in the study of slopes, quadratic equations, even calculus, as mankind tried to understand the cyclic nature of physical phenomena.

With the name Allah, Merciful Benefactor, Merciful Redeemer.   The above statement may sound provocative, but it can be supported by critical analysis in the history of mathematics.   This treatise is not meant as another chronological account of events. It is rather a commentary on the evolution of mathematical thought, particularly as it pertains to the solution of the problem of calculating the circumference of a circle -or, as it is commonly known as the squaring of the circle.

The problem of the squaring of the circle, has fascinated mankind, from the very early stages of mathematical development. This given the fact that the wheel, the circle, is the only shape that allows for fluid motion. Reflecting on the fact that the level of logic in each society, is mirrored in the level of the society's mathematical reasoning, we also observe that humanity has observed, since its humble beginnings, the cyclic nature of physical phenomena. From the four seasons, to the cycle of life and death, to the description of planetary motion and the structure of the atom, it is apparent that our physical universe maintains its order through cyclical repetitions of phenomenal expression. It is therefore made apparent the need for the understanding of the circle, as a key in understanding and rendering to use, our physical universe.

What may not be so apparent, and this website will attempt to make apparent, to the layman as well as the scientific community, is that the quest for the understanding of the circle, is the underlying and sometimes well hidden reason, for the development of several branches of mathematics, i.e. Calculus. The scholars who disagree with the above statement are welcomed to walk with us through the rest of this presentation and to even communicate and exchange thoughts with us.

Let us also observe that the quest for understanding the circle has been mainly expressed as a quest for the measure of its circumference. The construction of a circle has been classically seen as the employment of a compass on a piece of paper, or other surface, without the consideration of the dynamics during the construction. The “dynamics” should be noted, since there is motion involved in the “drawing” of the shape.

One of the first and most significant calculations of the circumference, is the method used by Archimedes, which we will call, the polygon method. The polygon method uses two polygons: one inscribed to a circle, and one circumscribed to a circle. The sum of the sides of the inscribed polygon, are calculated using successive applications of the Pythagorean theorem, and so is the sum of the sides of the circumscribed polygon. The measure of the circumference is located between the measures of these two sums. We will at this point bring to the reader's attention the necessity of using the Pythagorean theorem, for the above calculations. Without expounding on the problems and the mysterious origin of this theorem (Other books), let us keep in mind that without its usage, the sides of the polygon, could not have been calculated. To satisfy the appetite of the historian, let us also remember that decimals had not been invented yet, neither had the square roots. The calculations that necessitated square roots, were approximated, using fractions.

The polygon method, -and its variations-, has been used over and over throughout the centuries. The number of sides has increased, as the tools for calculating square roots, became more complicated. But there is a basic and fundamental problem in the “logic” of this approach. Euclidean geometry has postulated dimensionless points. (We beg to differ with this premise, since a dimensionless point cannot exist). (Pi - Philosophy) Since the dimensions of the point are undefined, as the sides of the inscribed polygon become smaller and smaller, one does not know , when the side, becomes literally smaller than the point. And this renders the calculation absurd. The same argument applies to the circumscribed polygon. It becomes useless, as the dimensions of its sides -sides that require a beginning point and an ending point- may or may not infringe on the dimensions of the point.

A necessary diversion at this time: Euclid's fifth postulate also disregards the dimensions of a point. Parallelity cannot be defined without dimensionality. Without going in an in-depth discussion on alternative geometries, let us propose that two lines cannot be considered parallel , without each one of them having also a measurable width.

A second diversion: The Greeks required the use of only straightedge and compass for geometric construction. Without going into detail into the fact that the roots of the quadratic equation ax²+bx+c=0, are constructible with straightedge and compass, and without speculating into the reasons that the Greeks had for the requirements if constructibility, let us simply point out the recognition of the importance and fundamental role, of the circle in geometric construction. May the reader reflect on its own.

Several formulas have been used to calculate the circumference of the circle and therefore the value of Pi(π) (the ratio of the circumference to the radius). They are based on the “polygonal” logic, and Pythagorean square roots. Western civilization has proudly developed infinite series for Pi(π), based on the basis of continued fractions. For the reader, not familiar with the subject, continued fractions are part of the “lost”       mathematics, mathematics that are too advanced for High School and too elementary for college. For example, if we assume that the equation

                                                                                            x²-x-1=0, (equ.(1)) is true,

and we solve for x:

                                                                                            x = 1 +1/x = (using successive substitutions for x)

                                                                                          = 1 +1/( 1 +1/x )=

                                                                                            = 1 +1/( 1 +1/( 1 +1/...)

The non-legitimacy of using continuous fractions, is discussed in Pi & Philosophy. At this point lets us say that order forms in a finite universe are finite, and also bring to our attention that if the fundamental equation (1) is wrong, the subsequent structure is also wrong.

The number Pi(π)   has been proven to be transcendental. Let us offer a critique of this point. Transcendental numbers are numbers that do not satisfy an equation of the form:

                                                                                              axⁿ+bx^n-1 +cx^n-2 +......+px+q=0.

Such an equation is called an algebraic equation, therefore Pi(π)   is postulated as not being a solution to an algebraic equation. Let us point out that the criterion of what an algebraic equation is arbitrary, as well as the fact that Pi(π)   not having being calculated with a reasoning other than Pythagorean-derived square roots, does not make such a calculation impossible; it simply means that it has not being achieved yet. The proof of Pi(π) being transcendental, comes from the assertion that the equation e^ix+1=0 (equ.(2))-where i is the imaginary unit-   cannot be satisfied by an algebraic number, and since x=Pi, satisfies equ. (2), Pi(π)   cannot be algebraic. Before the determination of the status of Pi, e was also proven to be transcendental. The number e, (base of natural logarithms) is defined from:

                                                                                lim_n→∞(1-1/n) =1/e

Without further elaboration, suffice to say that the purpose for the original definition of logarithms, was the simplification of calculations. The connection between the number e and trigonometric functions was derived when both e, and the trigonometric functions were written as infinite series: e^ix=cosx+isinx, where i is the imaginary unit.

Let us also keep in mind that imaginary numbers have been defined as results of negative square roots. This formalistic connection between the two concepts:√-n = √i n,   has no conceptual interpretation. For example: 4-dimensional space has as coordinates (x,y,z,ict) where i is the imaginary unit. This application has not enhanced our understanding on neither the concepts of space nor time. Several concepts in mathematics i.e. the concepts of number, space, set, field, group, are also removed from physical reality. If mathematics is reflecting the human system of logic, then the above concepts ought to reflect physical reality and not simply their author's imagination. And some notes on calculus, since the preposition was made at the beginning of this treatise that it has the underlying purpose and also appears to be, an offshoot of the quest for the circle.

The college level definition of a function is:

                                                                                  df(x)/dx = f′(x)=   lim_dx→∞{(f(x + dx) - f(x))   / dx}

and is interpreted as the slope (or the tangent) of the curve which represents f(x), between two points x and x+dx as dx becomes infinitely small.

The definition of an indefinite integral is the reverse of the derivative:

                                                        ∫(df(x)/dx)∙(dx)= ∫f'(x)dx= f(x)

The definition of a definite integral between a and b: _a∫^bf(x)dx represents the area under the curve f(x), between the points a and b. According to the fundamental theorem of calculus, differentiation and integration are processes inverse of one another. From a philosophical point of view (Pi - Philosophy), a curve on the coordinate plane defines the area underneath, and vice-versa, the area underneath defines the curve.

Other than the calculation of Pi, what is not so widely known concerning Archimedes, is that he also calculated the tangent (derivative!) of the spiral (spiral: one of the famous curves that were used to find the quadrature of the circle).

Taking a leap to the time of Newton and Leibnitz -who co discovered calculus- as they were both working on infinitesimals; tangent problems and quadrature problems, depending on differences and sums, were known to be inverses of one another. In Pascal's characteristic triangle, the arc is approximated by a tangent, and similar triangles can be used for its calculation.

It could be argued that differentiation and integration were intended for the study of any curve, and that could have been the case. But what cannot be argued is that the most basic and fundamental curve, on whose properties and methods of study, any other curve is based on, is the circle. It would be foolish to assume that for centuries, scores of scholars in mathematics were oblivious of the aforementioned fact. And even if the scores of scholars were oblivious of this fact, the leaders among them, were definitely not.

The above arguments and observations, are meant to serve as seeds of thought for anyone, the scientist and the layman, the student and the future student, anyone who believes that the right to think is not predicated by class, race, educational background, or any other human limitation, but is a right to the human being by its Creator. A right worth living and fighting for.

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