SUBJECT:   A New Approach to the Universal Constant "pi"

Dear Sirs,

Attached you will find an original, I would like to believe, research study, concerning «A new approach to the area of a circle» and thus a new approach to the Universal Constant "pi".

The innovative aspect of the new approach is attributed to avoid the insurmountable obstacles presented by the intervention of square roots and infinite series when we try to calculate the circumference as the Upper Limit of the perimeter of a regular Polygon inscribed in the circle, according to the relevant "deceptive" definition, i.e. sin15o = 1/4 {sqrt(6) - sqrt(2)}.

The New Theory considers the above way as an "impasse", which makes us to believe that the so derived number
pi' = 3.14159 2 6535.... is an irrational and trancendental number.

By the new way we approach the area of the circle as the "Lower Limit" of the area of the Superscribed Regular Polygon, coming from the square of side ao = 2R, and we use as main tool the "tangent", the values of which can be easily calculated, with the desirable accuracy, using as algorithm the formula

tan(2q) = 2tanq / {1-tanq} which defines the circle.

By this method we discover that it is impossible for the Lower Limit of the Superscribed Polygon, to be less than the
Limit 4 * 0.7854 = 3.1416.

More precisely the new method meets the Known number
3.14159 2 6535... and proves that in order to reach this number we have violated the sense of the circumference making the arbitrary assumption that: for very acute angles we can consider that
tanq = 1/2 tan(2q) instead of tanq < 1/2 tan(2q), according to the indisputable formula:
tan(2q) = 2tanq / {1-tanq} -> tanq < 1/2 tan(2q) always

and certainly 2sinq < 2arcq < 2tanq always.

However small the side-chord of the inscribed regular polygon, this chord is less than the respective arc and always there exists a minimum area between the chord and the arc, however great is the number N = 4 * 2n of the sides.

The above observation explains the irrational and trancendental character of the so defined circumference, and why "we can not see" the Upper Limit of an inscribed regular polygon.

On the contrary, through the New Theory, which dares the "transgretion", the lower Limit of the regular Superscribed polygon "is visible" and can not be other than 3.1416 R2 exactly.

Also, according to the New Theory, the area of the circle is the first real root of the algebraic equation:
X2 - 4X + 2.69674944 = 0, the second root of which is the 4k = m = 0.8584 R2 that is the area of the 4 equal curvilinear triangles, which lie between the circumference and the perimeter of the superscribed square of side ao = 2R, so that:

Ac + 4k =2R * 2R = 4R2 or 3.1416 + 0.8584 = 4R2.

The above square is the demanded "transgretion" and constitutes the "Reference Base" for the measurement of the circle, since:

I will appreciate any comments. With absolute respect to those who strive to uncover the secrets of Nature,

I remain

Moschos Ath. Karagounis

P R E F A C E

My preoccupation with the under investigation subject, the determination of the number pi, could be considered fortuitous, but by no means accidental in the usual sense we use the word: that is something that by chance and without cause comes to our attention, and whose worth we discover only later.

In the late summer of 1994, from August 6 until September 14, I was able to devote my attention exclusively to this subject under ideal conditions. I was under no obligation to any employer, had no deadline to meet and could, thus, indulge my curiosity, and pursue, as a researcher, my strong intuition that something was not right with the determination of the number pi.

In other words, I was of the opinion that we cannot reach the constant and precise value of pi, for the simple reason that the method used for the determination of the length of the circumference and, by extention, the determination of the value of pi, as a ratio of the length of the circumference to the diameter of the circle, by means of the perimeter of a regular inscribed polygon, was by definition an approximative approach and, consequently, incapable of determing the number pi.

It was impossible for me to believe that Nature would not be absolutely precise on such an important quantity, which had such a determining role on the circle, the sphere, the rotation and on the cyclical and elliptical motion, which justifiably characterize pi as a Universal Constant.

Unavoidably, the method used comes into conflict with the insurmountable obstacles presented by the intervention of square roots and infinite series. Thus, all my research efforts were aimed at circumventing and superceding these obstacles, by the use of legitimate techniques, by obvervation and, sometimes, by the use of intuition and inspiration, which led to the selection of certain spontaneous but decisive choices.

The degree to which all these factors contributed to the final result is inestimable, but no longer of much importance. What is of importance is that they gradually led me to the perception and, eventually, to the certainty that I was on the right track to the solution of the problem, i.e. the determination of the constant and definite value of pi. This in turn led naturally to the squaring of the circle.

I would like to believe that through the New Method, which I shall expose further down, the number pi is not approximated simply by sequential numerical operations, as in the past, but is born, or rather, disclosed, in a classically geometric fashion.

As a research method I consiously chose the method of zero basis, so as not to be influenced by the opinions of others, who might have themselves been misled or mistaken.

Coincidentalty, at this time, I chanced upon Henri Poincarč "Science and Hypothesis". This distinguished mathematician writes in his introduction:

«lt is only to the superficial observer that scientific truth is indisputable; the logic of science is impeccable and, if wise men are sometimes deceived, this is because they ignored its rules».
I adopted this view and considered it fortuitous.

I was considerably fortunate to have had the opportunity to research this topic and to reach certain conclusions. However, the results, whatever their worth, do not belong only to me. They belong to everyone and especially to those who strive to uncover the secrets of Nature, which so skillfully hides from us. In the words of Heracleitus: "Nature is fond of hiding".

Within the spirit of the above and after staling that I have not seen any of the more than 80 -as I am told- attempts to square the circle, I have decided to publicly submit my research, together with its conclusions, so that any interested party may judge it from his viewpoint. I personally have done what I felt to be my duty. It is now up to the free scientific society to judge and determine the truth of the matter.

 Athens, December 6, 1994 Moschos Ath. Karagounis Dipl. Mechanical Engineer

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