Welcome to Aquarius, Volume 7 (August 10, 2006)

The simple fact of asking the question:

What is most fascinating about constructions such as these, in pursuit of "squaring-the-circle," is that the differences between two line lengths are not visible to the naked eye. This search for a solution reveals the simple fact that the original riddle cannot be answered, no matter how much work is done, unless one knows the precise value of Pi. Therefore, this simple fact raises the question as to how and why the ancients would have posed the riddle in the first place, because to ask "Can we square the circle?" is to ask, necessarily, "Do we know the value of Pi?" Without the true value of Pi, there can be no true solution. That the question would be asked, implies that the asker has the value of Pi. This is like someone asking which is the highest of three mountains. The question implies that the asker knows how to measure the height of a mountain.

Postal address only: John Manimas, P.O. Box 536, Bellows Falls, VT 05101

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Citation suggested: Manimas, John. Six Nines, The Rediscovery of Proportion, "Welcome to Aquarius," www.jmanimas.com, Volume 7, August 10, 2006.

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1) Any translation into another language will please carry the following copyright notice and unofficial translation notice in BOTH ENGLISH AND THE OTHER LANGUAGE:

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Part 1: Construction of line A = 0.392836323023349590666713745392125... [ * X] a value that is very near in value to Pi/8.

Step 1: Construct the 36-54 degree right triangle. If you need more information about this right triangle, it is described here in Precision of the Ancients, and labeled as Right Triangle #5. Divide the 36-degree angle in half. Let that new dividing line, which created an 18-degree angle, be the new hypotenuse for our first right triangle. Mark off a length of 1 on the new hypotenuse. From that upper vertex on the hypotenuse, construct a perpendicular down to the new base. You now have an 18-72 degree right triangle.

Initial construction drawing to produce "line A":

You now have a right triangle with the following side lengths and ratios:

A= 0.309016994374947424102293417182819...

B= 0.951056516295153572116439333379382...

H= 1.000...

Sine= 0.309016994374947424102293417182819...

Cosecant= 3.23606797749978969640917366873128...

Cosine= 0.951056516295153572116439333379382...

Secant= 1.05146222423826721205133816969575...

Tangent= 0.324919696232906326155871412215134...

Cotangent= 3.07768353717525340257029057603691...

We see that the cosine, B/H, equals 0.951056516295153572116439333379382...

and the secant, H/B= 1.05146222423826721205133816969575...

Step 2: Our next construction goal is to square the secant value and then double it.

Step 3: Re-construct this right triangle by constructing the horizontal base B and then the 18-degree angle. Mark off the length of 1 on the horizontal side B, and then raise the perpendicular to construct side A. Because side B has a length of 1, and we know that the cosine=0.951056516..., the length of the hypotenuse must now be 1.051462224... .

Step 4: Re-construct the 18-degree right triangle beginning as in previous construction, but mark off the length of 1.051462224... on the horizontal base B, and then raise the perpendicular to construct side A. The secant is again 1.051462224... and the length of the hypotenuse is now 1.051462224... * 1.051462224... or 1.10557280900008412143633053250749... , as follows:

cosine = B = 1.05146222423826721205133816969575...

H = 1.10557280900008412143633053250749...

= 0.951056516295153572116439333379382...

secant = H = 1.10557280900008412143633053250749...

B = 1.05146222423826721205133816969575...

= 1.05146222423826721205133816969575

Step 5: Next construct a sufficiently long straight line and mark off the length of our H twice, now our new line = 2.21114561800016824287266106501498...

|____________________|____________________|

Step 6: The next step is to construct a line length of the square root of 8, which is 2 times the square root of 2, and which is from our previous constructions described in Precision of the Ancients (POA).

sqrt(8) = 2.8284271247461900976033774484194...

Step 7: We now subtract the shorter line 2.211145618... from 2.828427124... and the result is 0.617281506746021854730716383404417... . Our next construction goal is to construct the square root of this line length of 0.617281506... . The resulting line length will then be very near to Pi/4: 0.78567264604669918133342749078425... .

|_________________________|______|

Step 8: Construct a rectangle with a horizontal length of 2 and a width of 0.617281506... . Square this rectangle using the standard squaring-the-rectangle procedure, which is described in POA and in many geometry books. The area of this rectangle and constructed square is 2*0.617281506..., and the side of the constructed square therefore has a length that is equal to [sqrt(2) * sqrt(0.617281506...)] or [(1.414213562...) * (0.785672646...)].

Step 9: Is already completed because we want a square with the side length constructed in our previous Step 8. Our square with a side length of [(1.414213562...) * (0.785672646...)] has a diagonal with length 2 * 0.78567264604669918133342749078425... and this is equal to length 1.5713452920933983626668549815685..., very near to Pi/2. We then double the length of our diagonal for the final length of line A: 3.142690584186796725333709963137... . This length for line A is Pi * 1.00034948216336988124925683259591..., as follows:

sqrt(0.617281506...) = 0.78567264604669918133342749078425...

* 4 = 3.142690584186796725333709963137...

= Pi * 1.00034948216336988124925683259591...

Part 2: Construction of average of (line A plus line B), where line B is our previously constructed line length (described in POA) of 3.14155655389944902222112102645257... .

Step 1: Construct a sufficiently long line and mark off the length of our line A: 3.142690584186796725333709963137... . Then mark off the length of line B:

+ 3.14155655389944902222112102645257... (line B) and divide in half. Repeat 5 times:

-----> = 6.28424713808624574755483098958957... /2 =

|______________________________|______________________________|

|<-----------new line ------------------->| (Step 1)

Step 1: 3.14212356904312287377741549479479...

+ 3.14155655389944902222112102645257...

-----> = 6.28368012294257189599853652124736... /2 =

|______________________________|______________________________|

|<-----------new line ------------------->| (Step 2)

Step 2: 3.14184006147128594799926826062368...

+ 3.14155655389944902222112102645257...

-----> = 6.28339661537073497022038928707625... /2 =

|______________________________|______________________________|

|<-----------new line ------------------->| (Step 3)

Step 3: 3.14169830768536748511019464353813...

+ 3.14155655389944902222112102645257...

-----> = 6.2832548615848165073313156699907... /2 =

|______________________________|______________________________|

|<-----------new line ------------------->| (Step 4)

Step 4: 3.14162743079240825366565783499535...

+ 3.14155655389944902222112102645257...

-----> = 6.28318398469185727588677886144792... /2 =

|______________________________|______________________________|

|<-----------new line ------------------->| (Step 5)

Step 5: 3.14159199234592863794338943072396...

and 3.14159199234592863794338943072396... / Pi = divided by

---> 3.1415926535897932384626433832795...

= 0.999999789519540719278826808398938...

inverse: 1.00000021048050358275423692719123...

Difference: 0.0000006.6124386460051925395255554217116...

Precision to: 3.1415926535897932384626433832795

-3.14159199234592863794338943072396

= 0.000000661 ten millionths of a unit

Part 3: Referring again to Precision of the Ancients, under the Pi and One Set of right triangles, and to The Myth of Exactly Pi, we see that the ratio of our square area to our circle area will be the same as the ratio of our working value for Pi to our infinite value for Pi, and our ratio now is 0.999999789519540719278826808398938... . Does one's ability to get closer with persistence mean that squaring the circle exactly is possible?

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