Squaring The Circle

Quadrature

A centuries-old mathematical puzzle, Quadrature of the Circle (also known as “Squaring the Circle”), is thought to indicate the origin of the British system of measurements.

Squaring the circle is “the challenge to construct a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.” (“Squaring the circle”, Wikipedia) But that is only the classical problem’s imposed limitation. Think of a plastic hula hoop (a circle) which you somehow twist and bend into a square: the area of that square would be the same as the previous hula-hoop circle.

It is generally conceded, as of 1882, that the puzzle is impossible to solve because pi (3.141592654…) is “transcendental”, i.e., not the root of any polynomial with rational coefficients. (“Squaring the circle”, Wikipedia)

However, for the purposes of his argument related to the problem of Quadrature of the Circle, J. Ralston Skinner, in 1894, begins with the caveat: the question of value of the quadrature, as to whether it is the expression of exactitude of relation, does not arise. This “work thus is relieved of any necessity of examination into the question of the possibility of what is called ‘the quadrature,’ or ‘the squaring of the circle’…” (Key To The Hebrew-Egyptian Mystery in the Source of Measures, by J. Ralston Skinner. 1894. Republished by www.kessinger.net)

In other words, for Mr. Skinner’s purposes, he need only approach an exact value. And Skinner comes quite close to this.

One Peter Metius, circa 1585 A.D., reportedly arrived at a solution giving the quadrature value of the circle. Later, in the 19th century, a John A. Parker of New York City is said to have found his own quadrature value.

The value found by Metius was 113 for diameter and 355 for circumference. Parker’s value was 6561 for diameter and 20612 for circumference. Notice how both the Metius and Parker values yield an approximation of Pi when their respective circumferences are divided by their respective diameters.

Parker’s process is a little difficult to grasp. The equilateral triangle and the circle are seen to be “opposite to one another in all the elements of their construction.” “The equilateral triangle is the primary of all shapes in nature formed of straight lines, and of equal sides and angles, and it has the least radius, the least area, and the greatest circumference of any possible shape of equal sides and angles.” (Skinner, op. cit.)

The triangle “has the least number of sides of any possible shape in nature formed in straight lines; and the circle is the ultimatum of nature in extension of the number of sides.” (ibid.)

A triangle has sides equal to 1. Totaling the three sides gives circumference 3.

3 squared multiplied by 3 squared yields 81. Two triangles joined together make a square. The perimeter of this square is 81 multiplied by 4 = 324. The area for this square would be 81 times 81 equals 6561.

“Mr. Parker makes use of an element of measure of the equilateral triangle, by which, as a least unit of measure, to express the measure of the elements of a circle in terms of the numerical value of a square: so that, as a conclusion, a square of 81 to the side, or 6561 in area, shall contain a circle whose area equals 5153; or, rectifying the circumference, a diameter of 6561 shall have a circumference of 5153 x 4 = 20612” (Skinner, op. cit.)

Suppose a cube whose total edges add up to 20612. There being 12 edges, 20612 divided by 12 yields 1717.666667. The one-thousandth part of this equals 1.71766… This number, 1.71766…, as a proportion of the British foot measurement is the ancient, Biblical cubit value.

The 20612 value also corresponds to the British inch measurement: There being 12 inches to the foot, 20612 divided by 12 yields 1717.666667.

A measurement system is discerned to be hidden in the Old Testament. In Hebrew, Jared is construed to be “the mount of descent.” Jared, in Hebrew, consists reportedly of the letters Yodh-Resh-Da’leth, which “is literally, in British, Y-R-D; hence, in Jared, is to be found, literally, our English word yard.” (Skinner, op. cit.)

Similarly, with Enoch, the son of Jared, Skinner (op. cit.) finds it noteworthy that Enoch lived 365 years. 365 is, of course, a measurement of days in the year. Skinner somehow intuits an ancient understanding of the relation between space and time.

(A version of the above first appeared at my Melchizedek Communique web site on December 6, 2009)

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About ersjdamoo

Editor of Conspiracy Nation, later renamed Melchizedek Communique. Close associate of the late Sherman H. Skolnick. Jack of all trades, master of none. Sagittarius, with Sagittarius rising. I'm not a bum, I'm a philosopher.
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11 Responses to Squaring The Circle

  1. Pingback: Pi Is Wrong! | Ersjdamoo's Blog

  2. Rod says:

    Re: http://www.aitnaru.org/images/Pi_Corral.pdf

    Having discovered with CAD software that the rPi radius must be drawn at 62.4028873643093955482677952.. degrees (and many more decimal digits), I finally learned that squaring a circle according to the Greek rules would be quite difficult.

    Impossible? Advanced math, focused on the Pi ratio, proves “Yes, impossible”.

    However, the complementary and precise ratio (nicknamed rPi) opens another research door and hints that all possible geometric lines, angles, and objects must be analyzed before we can truly determine that squaring the circle is impossible.

    With this new perspective, we are inspired to ignore the popular transcendental Pi and search anew for geometric complements that balance the squared circle equation.

    Consider this unique scalene triangle (instead of equilateral) and possibilities for squaring the circle seem intuitive, almost jumping “out of the box”. Or focus on the “Transcendence” design that suggests geometry required to prove the square.

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  9. Rod says:

    Re: http://www.aitnaru.org/images/Pi_Corral.pdf (see “This Way Up” design)

    A non-mathematician’s review of the esoteric “This Way Up” palette:

    Consider the longest lines (hypotenuse) of the two green right triangles in the squared golden circle (D = 2,000,000 units). Each triangle is a component of a larger isosceles right triangle having an hypotenuse equal to the square root of Pi; both isosceles triangles are geometrically similar.

    As the hypotenuse line of the isosceles triangle (on the right of the design) rotates 45 degrees counterclockwise toward the center of the design, becoming the side of the isosceles triangle, the side of the other isosceles triangle (at the bottom of the design) rotates 45 degrees counterclockwise toward the center, becoming the hypotenuse of that isosceles triangle.

    The two rotating lines have equal length (hypotenuse of green triangles) when geometric equilibrium is reached. Then, the opposite points of these lines are equidistant (red lines, equal to the square root of Pi). And the two rotated lines (hypotenuse of green triangles) have length equal to the side of a square inscribed in the golden circle.

  10. Rod says:

    Wow! Three years later and I’m still runnin’ in squared circles!
    Here’s some of the the evidence: http://aitnaru.org/images/gEotP.pdf

    Note that the “Cyan Seven Seven 16:1” geometry (near bottom of PDF) clearly invokes the circle-squaring spirit of Pythagoras. Regarding this unique circle-squaring right triangle:

    Circle’s diameter is the hypotenuse of the right triangle.
    2(sqrt(1/Pi)) = 1.1283791670955125738961589031215..
    Diameter / 2(sqrt(1/Pi)) = long side of right triangle
    = side of circle’s square. 😉

    Thus, 2(sqrt(1/Pi)) is a more robust “Pi” constant because it highlights the Pythagorean geometry inherent in squared circles. Quoth the ravin’, “Furthermore …” (or at least once in a blue moon):

    The long side (one side of the circle’s square) forms a scalene triangle having a 45 degree angle and a side having length equal to a side of a square inscribed in the circle.

    What’s a good name for this “robust” Pi constant? Try “rPi” (for “radial Pi”) … since this constant was first explored using a circle’s radius (not its diameter). “Furthermore”, it sounds like “our Pi”, a growing perspective of squared circle aficionados.

  11. Liddz says:

    Creating a circle and a square with equal areas involving 100% accuracy solved.
    From: Liddz.
    It is almost impossible to create a circle and a square with equal areas of measure with 100% accuracy due to the transcendental nature of the modern traditional version of the ratio Pi 3.141592653589793. Remember that there are also 2 other modern values of Pi in addition to the traditional most common value of Pi that is 3.141592653589793. The 2 other values of Pi are: 3.146446609406726 and also 3.14460551102969. 3.146446609406726 is the result of 14 subtract square root of 2 = 1.414213562373095 = 12.585786437626905 and then 12.585786437626905 is divided by 4. 3.14460551102969 is the result of 4 divided by the square root of the Golden ratio = 1.272019649514069. 1.272019649514069 squared is the Golden ratio of 1.618033988749895. Please remember that the Golden ratio can be obtained in Trigonometry through the formula Cosine (36) multiplied by 2. All modern Pi values must be approximated meaning the results that are given to us by the 3 modern Pi values must be reduced to 4 or 5 decimal places resulting in multiple approximations for the 3 main Pi values. There is a debate among scholars regarding which version of the 3 values of Pi are better to use when dealing with circles and geometrical figures. The view of this author is that all 3 of the values of Pi are important and all the 3 values of Pi must be studied as much as possible to achieve accuracy. The 3 values of Pi 3.141592653589793 and 3.146446609406726 and 3.144605511029693 must be compared to each other so the geometrician can determine which value is best suited for the desired task.

    I think that I have solved the problem of creating a circle and a square with equal areas involving 100% accuracy:
    Requirements for creating a circle and a square with equal areas involving 100% include:

    1. The surface area of the circle that is to be equal to the surface area of a square can be derived from the surface area of 2 squares with rational widths of measure but does not always have to be.

    2. The square that has a surface area equal to the surface area of the circle can have a width of an irrational measure or a width of rational measure.

    3. The Pythagorean theorem is required to construct the square that has an irrational width of measure and a surface area equal to the surface area of the circle. The Pythagorean theorem is also required to test the accuracy of the construction of a circle plus a square that has an irrational width of measure with equal areas involving 100% accuracy. The Pythagorean theorem is based upon a scalene triangle and says that the amount of squares on the hypotenuse is equal to the sum of all the squares that are located on both the adjacent edge of the scalene triangle and also the opposite edge of the scalene triangle. The width of the square with a surface area that is equal in measure to the surface area of the circle must be equal in measure to length of the scalene triangle’s hypotenuse.

    4. If the surface area of the circle is already known and the measure for the radius of the circle is not yet known a solution is to divide the surface area of the circle by Pi and then apply the result to square root. After the surface area of the circle has been divided by Pi and the result applied to square root an approximation of the circle’s radius can be given. The approximation of the circle’s radius must be reduced to 4 decimal points and 1 added to the last 4 decimal places for the approximation of the circle’s radius that has been reduced to 4 decimal places. After the approximation of the circle’s radius has been reduced to 4 decimal places 0.01 can be added to the whole value instead of adding 1 to the last of the 4 decimal digits to determine the correct measure for the circle’s radius. Other values of Pi must also be used to confirm the accuracy of the measure for the circle’s diameter. : https://en.wikipedia.org/wiki/Squaring_the_circle: http://rsjreddy.webnode.com//
    http://www.jainmathemagics.com/
    http://www.iosrjournals.org/iosr-jm/papers/Vol10-issue1/Version-4/C010141415.pdf

    My question is it possible to create a circle with a surface area of 106 equal units because if we can create a circle with a surface area of 106 equal units of measure then we can also create a square with a surface area of 106 equal units of measure by creating a scalene triangle with the longest edge length as 9 equal units of measure taken from the diameter of the circle that has a surface area of 106 equal units of measure, while the shortest length of the scalene triangle has 5 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 9 equal units of measure and the shortest length of the scalene triangle as 5 equal units of measure is equal in measure to the width of a square that has a surface area of 106 equal units of measure. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has 106 equal units of measure.

    Area of circle = 106.
    Diameter of circle = 11.62.
    Circumference of circle = 36.5.
    Traditional Pi approximated to: 3.141135972461274.
    9 squared = 81.
    5 squared = 25.
    81 + 25 = 106.
    Most values of Pi can confirm that if a circle has a diameter of 11.62 equal units of measure then the surface area of the circle with a diameter of 11.62 equal units of measure is 106 equal units of measure.

    Also is it possible to create a circle geometrically on paper with a surface area of 117 square units because if we can then we can create a square with the same 117 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 9 units of measure from the circle’s diameter with an area of 117 units of equal measure. The shortest length of the scalene triangle must have 6 equal units of measure that are also taken from the circle’s diameter with an area of 117 equal square units. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has a surface area of 117 equal units of measure.
    Area of circle = 117.
    Diameter of circle = 12.22
    Circumference of circle is 38.39.
    Traditional Pi is approximated to: 3.141571194762684.
    9 squared = 81.
    6 squared = 36.
    81 + 36 = 117.
    Most values of Pi can confirm that if a circle has a diameter of 12.22 equal units of measure then the surface area of the circle with a diameter of 12.22 equal units of measure is 117 equal units of measure.

    Also is it possible to create a circle geometrically on paper with a surface area of 153 square units because if we can then we can create a square with the same 153 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 12 units of measure from the circle’s diameter with an area of 153 units of equal measure. The shortest length of the scalene triangle must have 3 equal units of measure that are also taken from the circle’s diameter with an area of 153 equal square units. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has a surface area of 153 equal units of measure.
    Area of circle = 153.
    Diameter of circle = 13.96
    Circumference of circle is 43.85.
    Traditional Pi is approximated to: 3.141117478510029.
    12 squared = 144.
    3 squared = 9.
    144 + 9 = 153.
    Most values of Pi can confirm that if a circle has a diameter of 13.96 equal units of measure then the surface area of the circle with a diameter of 13.96 equal units of measure is 153 equal units of measure.

    Also is it possible to create a circle geometrically on paper with a surface area of 365 square units because if we can then we can create a square with the same 365 units by simply constructing a scalene triangle with the second longest length of the scalene triangle being equal to 14 units of measure from the circle’s diameter with an area of 365 units of equal measure. The shortest length of the scalene triangle must have 14 equal units of measure that are also taken from the circle’s diameter with an area of 365 equal square units. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has a surface area of 365 equal units of measure.
    Area of circle = 365.
    Diameter of circle = 21.56
    Circumference of circle is 67.73.
    Traditional Pi is approximated to: 3.141465677179963.
    14 squared = 196.
    13 squared = 169.
    196 + 169 = 365.
    Most values of Pi can confirm that if a circle has a diameter of 21.56 equal units of measure then the surface area of the circle with a diameter of 21.56 equal units of measure is 365 equal units of measure.

    Also is it possible to create a circle with a surface area of 530 equal units because if we can create a circle with a surface area of 530 equal units of measure then we can also create a square with a surface area of 530 equal units of measure by creating a scalene triangle with the longest edge length as 19 equal units of measure taken from the diameter of the circle that has a surface area of 530 equal units of measure, while the shortest length of the scalene triangle has 13 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 19 equal units of measure and the shortest length of the scalene triangle as 13 equal units of measure is equal in measure to the width of a square that has a surface area of 530 equal units of measure. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has 530 equal units of measure.

    Area of circle = 530.
    Diameter of circle = 25.98.
    Circumference of circle = 81.61.
    Traditional Pi approximated to: 3.141262509622787.
    19 squared = 361.
    13 squared = 169.
    361 + 169 = 530.
    Most values of Pi can confirm that if a circle has a diameter of 25.98 equal units of measure then the surface area of the circle with a diameter of 25.98 equal units of measure is 530 equal units of measure.

    Also is it possible to create a circle with a surface area of 612 equal units because if we can create a circle with a surface area of 612 equal units of measure then we can also create a square with a surface area of 612 equal units of measure by creating a scalene triangle with the longest edge length as 24 equal units of measure taken from the diameter of the circle that has a surface area of 530 equal units of measure, while the shortest length of the scalene triangle has 6 equal units of measure. The hypotenuse of a scalene triangle with the second longest length as 24 equal units of measure and the shortest length of the scalene triangle as 6 equal units of measure is equal in measure to the width of a square that has a surface area of 612 equal units of measure. We can use the theorem of Pythagoras: https://en.wikipedia.org/wiki/Pythagorean_theorem to prove that the square with a width equal to the longest length of the scalene triangle also called the hypotenuse also has 612 equal units of measure.

    Area of circle = 612.
    Diameter of circle = 27.92.
    Circumference of circle = 87.71.
    Traditional Pi approximated to: 3.14147564469914.
    24 squared = 576.
    6 squared = 36.
    576 + 36 = 612.
    Most values of Pi can confirm that if a circle has a diameter of 27.92 equal units of measure then the surface area of the circle with a diameter of 27.92 equal units of measure is 612 equal units of measure.
    Also Is it possible to create a circle with a surface area of 36 equal units because if we can create a circle with a surface area of 36 equal units of measure then we can also create a square with a surface area of 36 equal units of measure by creating a square with 6 equal units of measure that are derived from the diameter of the circle with a surface area of 36 equal units of measure. The Pythagorean theorem is not required to construct a square with a surface area equal to the surface area of a circle with 36 equal units of measure. 6 squared is 36. The ratio of both the diameter of the circle divided by the width of the square and also the ratio of the radius of the circle divided by half the width of the square must be 1.13. The hypotenuse of a scalene triangle with its second longest length as 3.39 equal units of measure while the shortest length of the scalene triangle is 3 equal units of measure has a measuring angle of 48.49 degrees when the second longest length of the scalene triangle is vertical and 41.50 degrees when the second longest length of the scalene triangle is horizontal.
    Area of circle = 36
    Diameter of the circle is 6.78 equal units of measure.

    Circumference of the circle is 21.3 equal units of measure.

    Traditional Pi approximated to: 3.141592920353982.

    6 squared is 36.

    Most values of Pi can confirm that if a circle has a diameter of 6.78 equal units of measure then the surface area of the circle with a diameter of 6.78 equal units of measure is 36 equal units of measure.

    Also Is it possible to create a circle with a surface area of 144 equal units because if we can create a circle with a surface area of 144 equal units of measure then we can also create a square with a surface area of 144 equal units of measure by creating a square with 12 equal units of measure that are derived from the diameter of the circle with a surface area of 144 equal units of measure. The Pythagorean theorem is not required to construct a square with a surface area equal to the surface area of a circle with 36 equal units of measure. 12 squared is 144. The ratio of both the diameter of the circle divided by the width of the square and also the ratio of the radius of the circle divided by half the width of the square must be 1.13. The hypotenuse of a scalene triangle with its second longest length as 6.78 equal units of measure while the shortest length of the scalene triangle is 6 equal units of measure has a measuring angle of 48.49 degrees when the second longest length of the scalene triangle is vertical and 41.50 degrees when the second longest length of the scalene triangle is horizontal.

    Area of circle = 144

    Diameter of the circle is 13.56 equal units of measure.

    Circumference of the circle is 42.6 equal units of measure.

    Traditional Pi approximated to: 3.141592920353982.

    12 squared is 144.

    Most values of Pi can confirm that if a circle has a diameter of 13.56 equal units of measure then the surface area of the circle with a diameter of 13.56 equal units of measure is 144 equal units of measure.

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