Welcome to Aquarius, Volume 13 (12/16/2007): __Macro-construction of MQ and inverse value QM__, and (Final) CLOSING ARGUMENTS, including further description of how this work is a *repeatable scientific experiment* that shows how we can begin with a line length of 1 (UN) and then construct a line length that is an exact and constructible multiple of Pi, specifically showing that we can construct a line length of Pi * (phi)^2 * (5/6).

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A) Macro-construction of MQ (and QM)

B) Closing Arguments:

Final argument that my solution is the correct solution to the original riddle.

__A) Macro-construction of MQ procedure__: (Step 1, five parts: 99, 99-1, 99-2, 99-3, 99-4)

First, I hereby name my solution to the square-the-circle riddle, presented here and in Values and Constructions Required, the *Secant of Eighteen to the Fourth Power*, or the "SF solution."

Referring to Values and Constructions Required, pages 27-31, specifically Constructions #9, we see that we can construct a line length that is equal to the secant of 18 degrees raised to the fourth power, or 1.22229123600033648574532213002996... . Later, in Construction #13, on pages 45 and 46, we then construct the ratio (tangent) of that same value, simply by making the Altitude A13 equal to 1.22229123600033648574532213002996... and the Base B13 equal to 1 (UN).

For Macro-construction of MQ (and the inverse value, QM), we see that since we can construct a line-length and ratio of (secant 18)^4, we can just as well also construct a line length equal to the cosine of 18 degrees raised to the fourth power, and a right triangle with a tangent of that same value, as follows: referring to Construction #9, let us perform Construction #99, by reversing the line lengths of A9 and B9. In Construction #99, we construct Altitude A99 as equal to 2.23606797749978969640917366873128... , and the Base B99 equal to [4 * (sine 36)] degrees [2.35114100916989251667482381855629...] . Then we construct the Hypotenuse H99 from the upper end point of A99 to the lower end point of B99. That Hypotenuse for this Right Triangle #99 will have a length of 3.244667016... . Clearly, the tangent of this right triangle #99 is equal to the cosine of 18 degrees [0.951056516295153572116439333379382...], and the angle of interest, G and the Opposite angle OPP for Right Triangle #99 will be the reverse of Right Triangle #9. The trigonometric functions will logically be the "inverses" of those of Right Triangle #9. The facts for Right Triangle #99 are:

__Facts for Right Triangle #99__:

A99 = 2.236067977..., B99 = 2.351141009..., H99 = 3.244667016...

G = 43.5630006..., OPP = 46.4369994...

Sine = 0.689151757..., Cosecant = 1.451059202...

Cosine = 0.72461704..., Secant = 1.38003931...

Tangent = 0.951056516..., Cotangent = 1.051462224...

Following the same algorithm as we did for Right Triangle #9, for #99-1 in Step 1, we reconstruct Right Triangle #99 with the Base B99-1 equal to 1 (UN), and that makes the Altitude A99-1 equal to the cosine of 18 degrees. The facts for Right Triangle #99-1 are:

__Facts for Right Triangle #99-1__:

A99-1 = 0.951056516..., B99-1 = 1 (UN), H99-1 = 1.380039310...

G = 43.5630006..., OPP = 46.4369994...

Sine = 0.689151757..., Cosecant = 1.451059202...

Cosine = 0.72461704..., Secant = 1.38003931...

Tangent = 0.951056516..., Cotangent = 1.051462224...

We then continue to follow the algorithm and in #99-2 of Step 1, we reconstruct Right Triangle #99 and we make the line length of Base B99-2 equal to the length of A99-1 (0.951056516...), and therefore the length of Altitude A99-2 must be (cosine 18)^2.

__Facts for Right Triangle #99-2__:

A99-2 = 0.904508497..., B99-2 = 0.951056516..., H99-2 = 1.312495378 ...

G = 43.5630006..., OPP = 46.4369994...

Sine = 0.689151757..., Cosecant = 1.451059202...

Cosine = 0.72461704..., Secant = 1.38003931...

Tangent = 0.951056516..., Cotangent = 1.051462224...

We then continue to follow the algorithm and in #99-3 of Step 1, we reconstruct Right Triangle #99 and we make the line length of Base B99-3 equal to the length of A99-2 (0.904508497...), and therefore the length of Altitude A99-3 must be (cosine 18)^3.

__Facts for Right Triangle #99-3__:

A99-3 = 0.860238700..., B99-3 = 0.904508497..., H99-3 = 1.248257282...

G = 43.5630006..., OPP = 46.4369994...

Sine = 0.689151757..., Cosecant = 1.451059202...

Cosine = 0.72461704..., Secant = 1.38003931...

Tangent = 0.951056516..., Cotangent = 1.051462224...

We then continue to follow the algorithm and in #99-4 of Step 1, we reconstruct Right Triangle #99 and we make the line length of Base B99-4 equal to the length of A99-3 (0.860238700...), and therefore the length of Altitude A99-4 must be (cosine 18)^4.

(cosine 18)^4 = 0.818135621484342140063933385739262..., subject to precision of calculator.

__Facts for Right Triangle #99-4__:

A99-4 = 0.818135621..., B99-4 = 0.860238700..., H99-4 = 1.187163222...

G = 43.5630006..., OPP = 46.4369994...

Sine = 0.689151757..., Cosecant = 1.451059202...

Cosine = 0.72461704..., Secant = 1.38003931...

Tangent = 0.951056516..., Cotangent = 1.051462224...

__Macro-construction of MQ procedure__: (Step 2)

Then applying the same algorithm as in Construction #13, pages 45-46 of Values and Constructions Required, we construct a right triangle with Altitude A equal to the line length of Altitude A99-4 [0.818135621...] and Base B equal to 1 (UN). We then have a right triangle with a tangent that is equal to the cosine of 18 degrees raised to the fourth power. Let's call this new right triangle Right Triangle #33, with the tangent being our target value of (cosine 18)^4, being

0.818135621484342140063933385739262... .

__Facts for Right Triangle #33__:

A33 = 0.818135621..., B33 = 1 (UN), H99-4 = 1.292031692...

G = 39.287821321..., OPP = 50.712178678...

Sine = 0.633216372..., Cosecant = 1.579239014...

Cosine = 0.773974822..., Secant = 1.292031692...

Tangent = 0.818135621..., Cotangent = 1.222291236...

Then we move on to the next step for __Macro-construction of MQ (and QM) procedure__:

Whereas SF = (sec 18)^4 = 1.22229123600033648574532213002996... (line/ratio constructed)

and FS = (cos 18)^4 (inverse) = 0.818135621484342140063933385739262... (we now have this line/ratio also constructed), referring to Construction #11 (pages 41 to 45 of Values and Constructions Required), we see that we already have NTP = [__NT * (NT + 9 )__], or 0.999943537369751504522585249236876... [NT]

times 9.99994353736975150452258524923688... [NT+9], __divided by 10__, thereby

NTP = 0.999937891425529516432675419924049...

Working with the value FS [0.818135621484342140063933385739262...], lets "set aside" the divisor of 10. We see that we have the product of [NT * (NT + 9)] (see underlined above) which equals NT^2 + (NT * 9)

But, we also have NT = (FS * [11/9]) and therefore...

NT^2 + (NT * 9) = (FS *[11/9])^2 + ( FS * [11/9] * 9) which also equals

FS^2 * (11/9)^2 + (FS * 11) and that is equal to

[FS^2] 0.669345895141570756309937549130558...

* [(11/9)^2] 1.49382716049382716049382716049383...

(this product = 0.999887077927531623623486956108611... which

equals NT^2 = 0.999887077927531623623486956108611... (confirmed by calculator)

plus [FS * 11] 8.99949183632776354070326724313188...

= 9.99937891425529516432675419924049... or NTP * 10

and that * [SF] 1.22229123600033648574532213002996...

equals 12.2221532123408073944164930824006...

which, divided by 10, = 1.22221532123408073944164930824006..., our value from which we need only to subtract the constructible line value of 0.2222 to get our value MQ.

The procedures described in Values and Constructions Required show how to construct the line lengths and ratios of FS^2, (11/9)^2 and FS*11, also how to construct [NTP*10] times the ratio of SF 1.22229123600033648574532213002996... . The procedures also show the steps for dividing 12.2221532123408073944164930824006... by 10 -- that is perform the Tan Two Twice operation (to divide by 5) and then divide in half. Then the procedures show in detail how to construct a line length of 0.2222 and subtract that value from the longer line value of 1.22221532123408073944164930824006...

THEN, with our value MQ as Altitude A of a right triangle, and the Base B of that right triangle being constructed as 1 (UN), we then have a tangent = MQ and a cotangent = QM, the inverse of MQ, and QM is 0.999984679000655877860501448151795... (subject to precision of calculator), and this value is equal to Pi * (5/6) * (phi)^2, whereby we can then construct that value as a line length and then a straight line length that is equal to Pi exactly. And thereby we can construct squares and circles (as described further in the later part of Values and Constructions Required) of exactly the same area.

(11/14/2007) another "macro-construction" method, which further illustrates the content and meaning of "Proportion" and the unique qualities of (secant 18)^4 or "SF":

SF = 1.22229123600033648574532213002996...

SF * 9 = 11.0006211240030283717078991702696... (SF * 8, + SF)

-1 = 10.0006211240030283717078991702696...

/10 = 1.00006211240030283717078991702696... (Tan Two Twice, then /2)

and 1.00006211240030283717078991702696... (PTN) is inverse of

NTP [0.999937891457407822540348648321735...] subject to precision of computer

SF/ PTN = 1.2222153212... 7304531361559429425797... subject to precision of computer

These "macro-construction" calculations performed by construction also points to interesting realities of proportion. First, the inter-relationships of the (cosine 18) value and NT and (11/9) and MQ, especially the observation that FS^2 * (11/9)^2 = NT^2; and second:

[Pi * (5/6) * MQ] * 10 = 26.1803398874989484820458683436564...

/2 = 13.0901699437494742410229341718282...

/2 = 6.5450849718747371205114670859141... = (cos 36)^2 * 10

/2 = 3.27254248593736856025573354295705... = [(cos 36) +1]^2

/2 = 1.63627124296868428012786677147852... = 26.18033988... / 16

inverse of 1.63627124296868428012786677147852...

= 0.611145618000168242872661065014979...

*10 = 6.11145618000168242872661065014979...

* Phi = 9.88854381999831757127338934985021...

= (Pi * Ho)^2 or (phi * 16)

__B) (Final) CLOSING ARGUMENTS: __ (A "finale" to Volumes 1-12)

__Final Proof that this solution is the correct solution to the original riddle__.

THE DESCRIPTION IMMEDIATELY ABOVE, HEREIN, AND THE ENTIRE WORK OF SQUARING THE CIRCLE EXACTLY AND VALUES AND CONSTRUCTIONS REQUIRED, DESCRIBED ON THESE WEB PAGES OF JMANIMAS, DOES IN FACT CONFORM TO THE FUNDAMENTAL DEFINITION OF SCIENTIFIC EVIDENCE. The constructions presented here are an experimental procedure, with the steps of the experiment fully described so that they can be repeated by any interested scientist, geometer or mathematician and anyone can thereby discover whether they get the same results.

**Claim**: In regard to the **HISTORICAL RIDDLE**, my solution, which I have named the SF solution, meaning **S**ecant of eighteen degrees raised to the **F**ourth power, is the best solution discovered by anyone since the ancient riddle was known, because it is the logical solution that could be found using only the compass and straightedge and verified using only the compass and straightedge. No other mathematical means, such as a digital decimal number system accompanied by computations of infinite series, was available to verify any solution proposed in ancient Egypt or ancient Greece. The number system of those civilizations was the geometric number system, meaning geometry itself was the accepted number system.

In regard to the **GEOMETRIC RIDDLE**, my solution is the best geometric solution discovered by anyone because it produces a value for Pi that is precisely accurate to at least the ninth decimal digital value for infinite Pi, the Pi that is produced by use of the digital decimal number system and a computation of an infinite series or other algebraic method encompassed by the calculus. My SF solution complies with the restriction of "no measuring allowed," better than any other solution either old or new (modern). My SF solution may in fact be numerically correct, and produce infinite Pi exactly, if such a claim could ever be verified that a line length is exactly equal to an infinite series of decimal digits. The key to evaluation of my SF solution is in evaluating the number which I claim to have discovered and have named "M." M is that number not equal to 1, such that M^2 plus (1/M^2) equals 2 exactly. M is the Mother of Phi [1.618033988...), because Phi^2 plus (1/Phi^2) equals 3 exactly. If Phi is not transcendental, then M is most likely also not transcendental, and if we can construct M, then we can construct MQ. If we can construct MQ, we can construct Pi if my claim that Phi^2 = [Pi*MQ*(5/6)] is correct when calculated using our decimal digital number system. Are mathematicians ignoring this proposition? Why? Is it too great a change to contemplate? Do mathematicians resent an outsider attempting to teach? You mathematicians created this situation by claiming that the circle could not be squared and then ridiculing anyone who tried. If you were wrong, it will not be the first time that authoritarianism darkened the record of an institution. I could not have completed the work I performed unless I were genuinely devoted to finding the historical truth that explains how and why the original riddle has a valid solution based on the advanced understanding of the ancients embodied in number values expressed by the lengths of lines.

Closing Arguments:

The evidence shows that there is an (1) historical riddle

and a (2) geometric riddle (not a mathematical riddle).

A) The evidence shows that the SF solution described here is the best possible solution that serves as an equally effective solution to both the historical riddle and the geometric riddle.

B) This is true primarily because the solution presented here, the *Secant of Eighteen to the Fourth Power* (SF) solution, is accurate beyond the ninth decimal digit, and does not require violation of the restriction that there be no measurement or verification by means of a separately defined number system.

C) The other solutions that are known and less precise all violate the restriction against measurement by employing a non-geometrical number system to evaluate the validity of the solution value for Pi by comparing the solution value with another value for Pi that is calculated by means of a non-geometrical number system. To be historically correct, the proposed solution must be one that employs the restricted methods available in ancient times for both the solution itself and for any procedure devised to evaluate the solution. In other words, a solution of historical value that is proposed as being *the solution* that would have been accepted by the ancients cannot be discredited by stating that the line lengths of the proposed solution are not demonstrated to be exactly equal to a number value expressed as an infinite sum or digital decimals. To solve the riddle as a matter of history, we are looking for a solution that would have been both very precise and also acceptable to the ancient geometers as an admirable achievement in geometry, not an achievement in modern mathematics.

D) All accepted values for infinite Pi are generated by acceptance of the Euclidean Circle (EUC) which has an infinite number of sides and is therefore not possible in the real, physical universe.

E) Any number system, other than geometry being its own number system by virtue of proportion, requires that a unit value be designated, and designation of a unit value followed by any comparison of that unit value with any other value is inescapably measurement, and that is a violation of the restriction incorporated in the original riddle. The fact that no measurement is allowed actually means that no number system, other than geometry itself, is allowed. Although this restriction may seem outrageous, it is in fact the meaning of the original restriction. "No measurement" actually means "no number system that serves the purpose of measurement."

The original riddle: "Can we construct a square with exactly the same area as a given circle, using only the compass and straightedge?"

The original **historical riddle**: A teacher is asking a student the question, and the teacher is expecting a specific response which is the correct response and all other responses are incorrect.

Is a geometric solution impossible to verify? (Cannot be mathematically verified.)

The original **geometric riddle**: "Can we construct a square with exactly the same area as a given circle, using only the compass and straightedge." The hidden or unstated obstacle to this geometric riddle is: "How can one evaluate a proposed solution?" Investigation of the problem reveals that squares and circles of exactly the same area can be constructed if we can construct both the value of 1 and the value of Pi, in exactly correct proportion, using only the compass and straightedge. However, if any solution is proposed, however simple or complex, the allegedly correct solution will depend upon a straight line value presented as being equal exactly to Pi in proportion to a given or established value for 1. Given that the proposed solution presents us with this concrete, real object, namely a straight line that is described as having a length of exactly Pi, how does one evaluate the length of that line? How do we confirm that the line is in fact exactly Pi in length? How do we show that such a line length it is not exactly Pi in length, and is therefore discredited?

In fact, historically, the same method has been used over and over again to discredit any and all proposed solutions. The accepted value of Pi is designated as a calculation or computation, or sum of an infinite series, that requires measurement, or requires employment of a number system created for the purpose of measurement such as our digital decimal number system, and that asserts either directly or indirectly that the only circle that is a circle is a regular polygon with an infinite number of sides, namely the Euclidean circle. The ongoing discrediting of all proposed solutions is therefore nothing more nor less than the persistent doctrine that Pi is an infinite number, a number value that is an infinite decimal. This doctrine makes it inherently impossible to verify any proposed solution because it is not possible to show that a physical line length has exactly the same value as a decimal number that is an assembly of the digital pieces of a number system that is separate from geometry. What we have done here, over and over again, is assert that Pi is *defined* as a specific infinite, non-repeating series of decimal digits. It is not possible, by any means ever imagined, to prove that a concrete line length is exactly equal to an infinite non-repeating series of decimal digits. Therefore, we have an invalid riddle not because the original riddle was invalid, but because the mathematicians of the past have insisted, perhaps unconsciously or sub-consciously, that the riddle is a *mathematical* riddle, or a *digital decimal numerical riddle*. If it were a mathematical riddle, the correct solution would be mathematical and would comply with our digital decimal number system. The riddle that cannot be solved is not the original riddle, which is a geometric riddle. The riddle that cannot be solved is the distorted mathematical riddle.

This is the distorted and re-invented mathematical riddle: Can we, using only a compass and straightedge and thereby constructing only lines and angles and circles, construct a straight line that has a number value that is exactly the same as a number expressed as an infinite series of non-repeating decimal digits. This is a tough riddle, but not the question that was originally asked by the ancients. Consider this: The diagonal of a square with side S is the square-root-of- two times S, and the square-root-of-two is an endless non-repeating decimal number (1.414213562,,,). We accept that the diagonal of a square with a side of 1 is exactly the square-root-of-two. But, mathematics has failed to point out that the same "disability" that applies to Pi also applies to the square-root-of-two, namely, there is no means to prove that the length of a particular concrete line complies with the decimal digital number equal exactly to S times the square-root-of-two.

F) The fact that the value for Pi expressed in the SF solution may not be equal to our modern infinite Pi value is not sufficient grounds to judge the solution as trivial or to judge the geometry of the ancients as being in some way "inferior" to modern mathematics. When we study the past, the most interesting subject matter that we search for and that is the most difficult to determine is what did the people of ancient times really know. What was their understanding of the universe? The ancient riddle or problem presented as the "squaring-the-circle" problem has actually presented historians with a unique opportunity to address this particular issue as to what did the people of an ancient civilization know. My SF solution is evidence that they knew all of the insight and geometric and proportional numeric knowledge that is required to see and understand this solution, presented here, consistent with the ancient clues and the ancient awareness of the "golden section" or the "golden mean" known also as Phi (1.618033988...) and the sum of the infinite series known as the Fibonacci series.

G) Let me describe my solution first as the best solution to the **historical riddle**. Then we can examine my solution separately as a proposed solution to the **geometric riddle**. But this geometric riddle is not a mathematical riddle, not a riddle asking us if we can construct a line that matches in value the exact same value as an endless series of decimal digits.

Examination of the historical riddle:

The original riddle: "Can we construct a square with exactly the same area as a given circle, using only the compass and straightedge?"

The original ** historical riddle**: A teacher is asking a student the question, and the teacher is expecting a specific response which is the correct response and all other responses are incorrect.

If we use the description or definition of the historic riddle, as presented here immediately above, then we are not looking for a solution that can be verified by comparison of a line length with a digital decimal number. What the teacher is looking for is a geometric construction that enables construction of circles and squares of exactly the same area, a method that enables construction of a straight line length that is equal to the circumference of a circle. How will the teacher evaluate the work or the solution handed in by the student? If we look at the ancient riddle in the ancient context, we do not have sound evidence that the teacher is going to use our decimal digital system or advanced algebra or calculus, or any digital number system, to evaluate the work handed in by the student. The riddle is a problem in geometry and trigonometry. A proposed solution has to be evaluated by means of geometry and trigonometry. The original situation is comparable to an algebra teacher today, telling students to compute, using the notation of algebraic equations, how many eggs will be available at the end of the fifth week if we begin with twenty-two hens laying five eggs each week, and at the end of each week one of the hens dies. The teacher is looking for one correct solution, showing the algebraic equations. There may be only slight variations in the work that the teacher will accept as correct. All other solutions will be deemed incorrect.

*Therefore, my claim is that my* **SF solution ***to the original riddle is the correct solution, the solution that would have been accepted as correct by the ancient teacher, and all others proposed over the centuries would be found incorrect by the original teacher assigning the original problem of squaring the circle.*

The evidence shows that the acceptable geometric solution, employing only a compass and straightedge to construct, and employing only a compass and straightedge to verify, is pointed to by the steps required for construction of a pentagon. Other clues are pointed to in ancient sources which I describe in "The trail of clues" (Squaring the Circle Exactly, Introduction, Part D) and in even greater detail in the long document "The Precision of the Ancients." My SF solution clearly acknowledges and employs the clues found on that trail.

Proposed negative discredit of my solution.

I propose that if any mathematician wants to prove that my SF solution is mathematically incorrect, they need to prove one of the following three negative assertions:

1) There is no number M, not equal to 1, such that M^2 plus (1/M^2) equals exactly 2.

2) Accepting the definition of M, prove that Phi^2 is not equal to Pi * (5/6) * M^2 .

3) Perform the SF construction, as described in Values and Constructions Required, and show that these constructions do not result in an accurate construction of M^2.

HOWEVER, it is certain already that the M value, as defined, must bring the digital decimal value of Phi^2 extremely close to being equal exactly to the product of [Pi * (5/6) * MQ]. And it is valuable to examine such a reality further. I claim that my work is evidence supporting the conclusion, in regard to the historical riddle, that the ancients understood this equation as being correct.

AND, I make the final claim here, as outrageous as it will be deemed, that the infinite decimal digital value for infinite Pi, accepted by mathematicians throughout the world, is not correct. Based on the accepted physical doctrine that there is a smallest particle of matter, I conclude and claim, as a serious student of physics, that it is impossible for any real object to comply with the Euclidean definition of a circle. Therefore, all real physical objects must have a finite number of sides. And therefore no real, physical object can have a perimeter that complies with a numerical value computed as the sum of an infinite series. The importance of this viewpoint for science cannot be exaggerated. It is not just a philosophical or semantic matter. Our future may depend on our ability to identify the values of M and MQ, and to understand how proportional shapes are the number system of Nature, and whether it is true as I claim here, that Pi is actually a variable value dependent upon the number of sides of a regular polygon. This "crazy" idea has been born. It will grow. Meanwhile, I will set sail on a new ship...

John Manimas, December, 2007.

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