**(See previous years of eclectic work/thoughts at the Geometry of Mass, Space, Time website - www.geometry-mass-space-time.com )**

**CONTENT**

**Four Methods For Calculating The Area To A Circle**

**Archimedes Proposition 1, Method 5; Area of a Circle + Sixth & Seventh Methods**

**Rational Non-Linear Circle To Rational Non-Linear Diameter**

**Non-linear Geometry of A Natural Circle**

**Calculating The Area Of A Ring**

**From The Cube To The Sphere**

**Twelve steps To The Sphere**

**Areas And Volumes Of Symmetrical Ovals And Ovoids**

**Minus Pi In Black And Yellow**

**Compass And Lines**

**Bi-Radial Arithmetic Of The Circle And Square**

**Context Of A Degree**

**All Seven of the following methods for "Calculating the Area of of A Circle" - Mathematically Concur.**

**Including**

**A. Archimedes Proposition 1; which unlike Pi, does serve to give the Exact Area of a Circle**

**B. Ancient Sumerian Proof**

**C. My Own Proofs**

**Quote**

*Stuart Chase 1888 - 1985*

*For those who believe no proof is necessary -*

*For those who do not believe, no proof is possible.*

**Formulae abbreviations; Radius - R: Diameter - D: Circumferential degree - Cd: Centimetre - cm**

1R = 1 Cd: 2R = 2 Cd's: 2 Cd's = 1D

1D = 2 Cd's x 180 = 360 Cd's

1R = 1 Cd: 2R = 2 Cd's: 2 Cd's = 1D

1D = 2 Cd's x 180 = 360 Cd's

**1cm R = 2cm D = 2 Cd's**

2cm D = 2 Cd's x 180 = 360 Cd's = 360 cm's in length

2cm D = 2 Cd's x 180 = 360 Cd's = 360 cm's in length

**Beginning with a square measuring 120 cm x 120 cm, the perimeter of the square is 4480 cm and its area 14, 400 square cm.**

**Taking one side of the square measuring 120 cm to use as a diameter length, we multiply this length by 3 in order to obtain a length of 360 cm, to the circles circumferential length.**

**Therefore it follows.**

**1. As every circle has 360 degrees to its circumferential length, so each degree of the circle measures 1 cm in length.**

**2. As each side of the square measures 120 cm in length, so the three diameter length of the circles circumference are equal to, three quarters of the squares 480 cm perimeter length.**

**3. As the 360 cm circumference of the circle, measures three quarters of the 480 cm perimeter length of the square, the area of the circle will be three quarters of the square area to the square, 10, 800 square cm.**

*Four Methods For Finding The Area Of A Circle*

1st Method: 14, 400 sq cm divide by 4 = 3, 600 sq cm multiplied by 3 =

1st Method: 14, 400 sq cm divide by 4 = 3, 600 sq cm multiplied by 3 =

__10, 800__sq cm to the circle = 3/4 of the area to the overall square.**2nd Method:**

**60 cm Radius squared = 3, 600 sq cm, multiplied by 3 =**

__10, 800__sq cm to the circle = 3/4 of the area to the overall sq. Which concurs with the result of the 1st method**3rd Sumerian Method:**

**Circumference length of 360 cm, multiplied by itself = 129, 600 sq cm, divide by 12 =**

__10. 800__sq cm to the area of the circle = 3/4 of the area to the overall square.**Which concurs with the result of the 1st & 2nd method**

4th Method as per Diagram

4th Method as per Diagram

60 cm Radius squared = 3, 600 sq cm, divide by 4 = 900 sq cm; multiply 900 sq cm by 3 = 2, 700 sq cm to 3/4 of the area to the square of the radius.

60 cm Radius squared = 3, 600 sq cm, divide by 4 = 900 sq cm; multiply 900 sq cm by 3 = 2, 700 sq cm to 3/4 of the area to the square of the radius.

**Multiply 2, 700 square cm by 4 =**

__10, 800__**sq cm to the circle = 3/4 of the area to the overall square;**

**And each corner of the square, has an area of 900 sq cm.**

*Which concurs with the results of the 1st, 2nd & 3rd methods*

*ARCHIMEDES PROPOSITION 1 - METHOD 5*

*AREA OF A CIRCLE***Proposition 1**.

**The area of any circle is equal to a right-angle triangle in which one of the sides about the triangle is equal to the radius, and the other to the circumference of the circle.**

**5. Fifth Method: Archimedes Triangle for finding the area to a circle**

**In the above diagram using a 12 x 12 square (144 squares) rather than a 120 x 120 cm square; it can be seen that if we give the vertical a circumferential length three times that of the circles 12 square diameter length = 36 squares.**

**And then multiply the 6 square radius of the circle by the 36 square circumferential length we have a rectangle containing 216 squares, which when we then divide by 2, in order to obtain the exact area of the circle which is 108 squares and exactly three quarters of the 12 x 12 squares' 144 square area.**

**Therefore had we had used a 120 x 120 cm square, rather than a 12 x 12 square; the area of the circle would be 1, 800 sq cm**

**Which concurs with the results of the 1st, 2nd, 3rd, & 4th methods**

**Therefore Archimedes proposition 1, is proven to be correct.**

**However conversely and ironically, relative to his later formulation of the approximation Pi**

**Of**

**22/7 as an improper fraction =**

**3 & 1/7 as whole numbers (Whole diameter/unit lengths) and a 1/7 fraction of a single diameter length**

**22/7 converted to decimal; divide 22 by 7 = 3. 1428591**

**Although as his first proposition shows he was on the right track from the very beginning, the Euclidean and general belief that the exact length of a circle could not be measured, subsequently served to lead him astray.**

6. Sixth Method: "

6. Sixth Method: "

__Applicable only to__" the area to a circle, with a 12 square length diameter

12 square/

12 square/

__right angle__length diameter x 3 = a 36 square length circumference

36 square length circumference x 3 = 108 square area

36 square length circumference x 3 = 108 square area

And each one of the 36 square lengths of circumferential length, is equal to 10 degrees of the 360 degrees to the circles circumferential length

And each one of the 36 square lengths of circumferential length, is equal to 10 degrees of the 360 degrees to the circles circumferential length

**Therefore each 1/10 length; of each of the 36 square lengths, is equal to a 1 degree length of the 360 degree length of the circle.**

**And had we used a 120 square diameter length rather than a 12 square diameter length; the circles area would have been 1, 800 sq cm.**

**Which concurs with the results of the 1st, 2nd, 3rd, 4th, and 5th methods.**

**7. Seventh Method: "**

__Applicable only to__" the area to a circle, with a 10 square (right angle/length diameter.

10 square length right angle x 10 square length right angle = 100 square area to the square, of the circle.

10 square length right angle x 10 square length right angle = 100 square area to the square, of the circle.

10 square length diameter x 3 = a 30 square circumferential length.

10 square length diameter x 3 = a 30 square circumferential length.

30 square length circumference x 2.5 (Not 3) = 75 (%) square area to the circle

30 square length circumference x 2.5 (Not 3) = 75 (%) square area to the circle

In sum

In sum

3 x the length of any straight line or right angle is a circumference.

3 x the length of any straight line or right angle is a circumference.

4 x the length of any straight line or right angle is a

4 x the length of any straight line or right angle is a

**perimeter.**

And given any length of straight line that is used as a diameter; the resultant circumferential length of the circle; will always be three quarters that of the straight lines perimeter length.

And given any length of straight line that is used as a diameter; the resultant circumferential length of the circle; will always be three quarters that of the straight lines perimeter length.

Which concurs with the results of the 1st, 2nd, 3rd, 4th, 5th, & 6th methods for gaining the circles three-quarter area of its square (Square of its diameter).

Which concurs with the results of the 1st, 2nd, 3rd, 4th, 5th, & 6th methods for gaining the circles three-quarter area of its square (Square of its diameter).

**Unlike Pi which is not a measurement nor is it approximate to anything, because unlike scales of measurement which are founded in reality; Pi is an improper fraction of an imperial/natural number, that has been decimalised/decimated into a permanent state of infinite irrationality.**

**And the decimal (10) based number/unit system of weights, measures, and money of itself; is a abstract concoction of "valueless/Abstract numbers" developed in the realms of dishonesty, cunning, and Machiavellian fraudulences**

*(Propaganda - Marketing - Advertising).*

That were inherent to and universally practised 2000 years ago, within the Greco

That were inherent to and universally practised 2000 years ago, within the Greco

*(Merchant)*- Roman*(Thuggish-Military)*Empire*(E.g. Temple of Jerusalem)*; and which have subsequently evolved over time, to become the far more widely spread and complex*(Fine print)*fraudulent practices, of modern day Capitalistic Insatiable Greed.

*Area Of The Square Of The Circle Of The Square*****

**12 X 12 square = 144 squares**

**144 divide 4 = 36 squares x 3 = 108 squares to the Circle**

**66 squares + 24 half squares = 72 squares to the Inner Square**

**Therefore as the square area of the Circle is three quarters that of its Square**

**So the square area of the Inner Square of a Circle, is 1/2 of the square area of the overall square, and 2/3rds of the area of the Circle.**

****

*ARCHIMEDES STRING*

**During the times of Archimedes and extending up unto the present day, it has been believed that it is impossible to measure or calculate the exact length of a circle, and therefore the closet approximation was/is acceptable.**

**Therefore when Archimedes decided to devise a formula for calculating the length to circle, he was happy to accept the closest approximation he could obtain for his formula, using the most accurate, and only method available to him during those times; a cylinder, a length of string, and a straight rule.**

**Based on his formula Pi, which as an improper fraction is 22/7 or 3 & 1/7 as a whole number and a fraction, and given that we now know that a a circles length is 3 x its diameter length; it is a simple matter, to divide 3 into 22 to obtain 3 equal seven measurement unit diameter lengths, or 21/7 measurement units to the circles exact length, with 1/7th of a diameter length remaining over, due to the inaccuracy of using a length of string.**

**One might wonder why Archimedes did not twig the fact, that given 3 equal diameter lengths within the 22/7 measurement unit length he had obtained, that he did not realise that the thickness of the string was causing an excess.**

**However one has to remember that due to his mind set, that it was impossible to find the exact length of a circle; it was also impossible for him to accept, that a circle could be exactly 3 diameter lengths in total length.**

**When measuring the the cylinders circumference with a length of string, he would have first encircled the cylinder with the length of string, and then having brought one end of the string around the cylinder and into contact with the main length of string, he would have marked on the length of string, the point at which the end of the string met the main length; and then measured the length backwards from the marked point on the main length to the end of the string, up against a straight rule.**

**And this is where the error creeps in, because when the string is stretched around the cylinder, it is the inside of the circle of string that is directly in contact with the cylinders curvature, is constricted, while the opposite outside of the circle of string is stretched; and therefore when removing and then straightening the length of string to measure against a straight rule; it is the longer outer side of the stretched length that is being measured, rather than the constricted inner side, which was directly in contact with the cylinders surface.**

And as to Archimedes having transacted the circle to check or confirm his result, and so proceed to develop his formula; refer to "Pi In Black And Yellow, because the linear area of every transaction line drawn, only serves to add to the error.

And as to Archimedes having transacted the circle to check or confirm his result, and so proceed to develop his formula; refer to "Pi In Black And Yellow, because the linear area of every transaction line drawn, only serves to add to the error.

*Rational Non-Linear Circle To Rational Non-Linear Diameter*

*Oxford English Dictionary**Pi symbol of the ratio of the circumference of a circle to its diameter, approximately 3.14159*

FACTS

FACTS

1: Pi is not the symbol of the

1: Pi is not the symbol of the

__"ratio"__of the circumferential length of a circle to its diameter length.

2: Pi is the symbol of the

2: Pi is the symbol of the

__"decimal irratio"__of the circumferential length of a circle to its diameter length; gained by dividing a decimal 7, into the 22 whole measurement units of the__approximate__circumferential length__given to the circle by Archimedes.__**22 divide by 7 = 3.14285714285**

3: Pi is simply the

3: Pi is simply the

__"symbol of the result"__of the decimalisation of the__(approximate)__circumference length of 22/7 as a vulgar fraction, or 3 whole units and 1/7th of a whole unit remaining.

Which in reality equates to three diameters lengths, with each diameter measuring seven whole measurement units in length, and one whole measurement unit remaining

Which in reality equates to three diameters lengths, with each diameter measuring seven whole measurement units in length, and one whole measurement unit remaining

**The original approximation that was made by Archimedes in regard to the ratio of a circles circumferential length, to its diameter length, was in the form of the improper fraction of 22/7; and it was not decimalised until after the decimal system was adopted in France in 1790 during the French Revolution.**

**[However Note: Whereas 22 divided by 3 as decimal gives a result of 7.3333333 diameter length;**

Which when converted/

Which when converted/

__rounded__(*"Intuitively"*) to a fraction, also gives an approximate but empirically rational*(ratio of)*diameter length of 7 & 3/21; which is the same as 7 & 1/7 = the circles diameter length [Regarding 7.3333333 see footnote].

However when 22/7 is divided by the number 7, rather than by the number 3

However when 22/7 is divided by the number 7, rather than by the number 3

It gives a none empirical/irrational based result of, 3.14285714285 recurring; and so does possess any relativity of ratio to the empirical circle whatsoever.]

It gives a none empirical/irrational based result of, 3.14285714285 recurring; and so does possess any relativity of ratio to the empirical circle whatsoever.]

****

Over time similar improper fractions were used by others e.g. Ptolemy (150 AD) who used 377/120, which equates to 360/120 with 17 parts remaining, or 3 17/120. 3 whole units of diameter length, and 17 parts of the 120 diameter length.

Over time similar improper fractions were used by others e.g. Ptolemy (150 AD) who used 377/120, which equates to 360/120 with 17 parts remaining, or 3 17/120. 3 whole units of diameter length, and 17 parts of the 120 diameter length.

**All concur that there are 3 diameter lengths to/**__within__a circles circumferential length**All use the “additional varying thickness's”***(Areas)***of drawn lines to approximate the circle****All give an approximate, that is in excess of 3 diameter lengths to a circles circumferential length**

**Refer to:**

**Some Facts about Pi - Pleacher**

www.pleacher.com/mp/mfacts/pifacts.html

www.pleacher.com/mp/mfacts/pifacts.html

**Therefore as all of the approximations ever made are "in excess of 3 diameter lengths"; and inclusive of the “additional varying thickness's” of the circumferential drawn length;**

So it follows that the diameters actual length; without the “additional thickness” of drawn circumferential line; when multiplied by 3, will give the circles actual circumferential length.

So it follows that the diameters actual length; without the “additional thickness” of drawn circumferential line; when multiplied by 3, will give the circles actual circumferential length.

****

__Actual & Factual Non-linear Circumferential Length__

Although it is only normally possible to approximate the length of a drawn circle to that of its linear diameter length, due to the thickness (areas) of the drawn lines;

Although it is only normally possible to approximate the length of a drawn circle to that of its linear diameter length, due to the thickness (areas) of the drawn lines;

By use of our "non-linear minds eye imaging ability", it is possible to create a completely non-linear circle, and equate it to its diameter length.

By use of our "non-linear minds eye imaging ability", it is possible to create a completely non-linear circle, and equate it to its diameter length.

First we imagine that in front of us we have a perfect sphere

First we imagine that in front of us we have a perfect sphere

*(refer also to [non-linear] From The Cube To It's Sphere, and Twelve Steps To The Sphere)*and that it's measured height/diameter is exactly 7 centimetres;

Secondly we imagine taking a knife, and cutting straight down through the sphere, and its exact centre.

Secondly we imagine taking a knife, and cutting straight down through the sphere, and its exact centre.

We now have two perfect halves of the sphere, and each half of the cross section of the sphere, c

We now have two perfect halves of the sphere, and each half of the cross section of the sphere, c

**an be seen to be in the form;**

Of a perfectly symmetrical non-linear circle; which given that the height of the sphere is exactly 7 centimetres, defines that the diameter to each of the two circles, is exactly 7 centimetres.

Of a perfectly symmetrical non-linear circle; which given that the height of the sphere is exactly 7 centimetres, defines that the diameter to each of the two circles, is exactly 7 centimetres.

We now multiply the 7 centimetre diameter of one of either circle by 3, and the exact non-linear circumferential length of either of the two circles, is exactly 21 centimetres in length.

We now multiply the 7 centimetre diameter of one of either circle by 3, and the exact non-linear circumferential length of either of the two circles, is exactly 21 centimetres in length.

****

Footnote regarding 7.3333333

Footnote regarding 7.3333333

**When the "natural number and improper fraction" of 22/7 is divided by 7, it gives 3 whole units/lengths of 7 smaller units, with 1 smaller unit (1/7) of length left over.**

The reason is because, when any "Natural Number" is divided by itself e.g. 7 divided into 7

The reason is because, when any "Natural Number" is divided by itself e.g. 7 divided into 7

It gives a fractional result of 1/7th of 7, and it does so because, during the process of division the "natural number" 7, has been subdivided into 7 smaller identical proportions of itself;

It gives a fractional result of 1/7th of 7, and it does so because, during the process of division the "natural number" 7, has been subdivided into 7 smaller identical proportions of itself;

7 divided by 7 = 7 7 7 7 7 7 7 and 7 + 7 + 7 + 7 + 7 + 7 + 7 = 7

7 divided by 7 = 7 7 7 7 7 7 7 and 7 + 7 + 7 + 7 + 7 + 7 + 7 = 7

which is confirmed as above, by the fact that e.g the fraction 3/7ths of the "natural number" 7, represents 3 smaller and equal parts of the original "natural number" of 7.

which is confirmed as above, by the fact that e.g the fraction 3/7ths of the "natural number" 7, represents 3 smaller and equal parts of the original "natural number" of 7.

Or e.g. the "natural number" 6 when divided by the "natural" 6 (itself) it = 1/6 1/6 1/6 1/6 1/6 1/6; which when multiplied

Or e.g. the "natural number" 6 when divided by the "natural" 6 (itself) it = 1/6 1/6 1/6 1/6 1/6 1/6; which when multiplied

*(back)*by the original "natural number" 6, is = to 6/6th or one whole unit/natural number of 6, containing 6 smaller identical units of itself.

However when as previously we use decimals (a calculator) to divide the "natural number" and improper fraction of 22/7 by 3:

However when as previously we use decimals (a calculator) to divide the "natural number" and improper fraction of 22/7 by 3:

Instead of obtaining the "natural fractional" value of 3 & 1/7th,

Instead of obtaining the "natural fractional" value of 3 & 1/7th,

We obtain 7.3333333 or 7 & 3/10 & 3/10 & 3/10 & 3/10 & 3/10 & 3/10 & 3/10 recurring

We obtain 7.3333333 or 7 & 3/10 & 3/10 & 3/10 & 3/10 & 3/10 & 3/10 & 3/10 recurring

Rather than the correct answer of 7 & 3/21ths which equates to 7 units & 1/7th of the 7 unit length of diameter].

Rather than the correct answer of 7 & 3/21ths which equates to 7 units & 1/7th of the 7 unit length of diameter].

****

*NON-LINEAR GEOMETRY OF A NATURAL CIRCLE*

**Euclid: Book 1 of The Elements; Definition 17**

Quote: A diameter of the circle is "any straight line" drawn through the centre, and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

Quote: A diameter of the circle is "any straight line" drawn through the centre, and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.

Facts

Facts

1. A "Natural Circle" which a drawn linear circle seeks to "mimic", and which can be empirically confirmed by examining e.g. the cross section circle of a sphere as described in the previous chapter, does not possess a diameter length, only a distance of length extending from its centre of construction.

2. A drawn line cannot be drawn through the centre of a circle, as its width/area of thickness merely occludes and covers over the centre of the circumnavigation of the circle; with a variance of thickness/area to each side of the centre of the circle.

1. A "Natural Circle" which a drawn linear circle seeks to "mimic", and which can be empirically confirmed by examining e.g. the cross section circle of a sphere as described in the previous chapter, does not possess a diameter length, only a distance of length extending from its centre of construction.

2. A drawn line cannot be drawn through the centre of a circle, as its width/area of thickness merely occludes and covers over the centre of the circumnavigation of the circle; with a variance of thickness/area to each side of the centre of the circle.

3. The proposition/concept that a diameter is/can be "any straight line" is totally in error. What is proposed in linear terms as being a straight diameter length; is two single lines of radial length, that have been added together to form one length.

3. The proposition/concept that a diameter is/can be "any straight line" is totally in error. What is proposed in linear terms as being a straight diameter length; is two single lines of radial length, that have been added together to form one length.

Whereas in reality; two radial lengths of a circle cannot be added together, as there is a gap in the form of a hole at the centre of the radiated circle; which serves to separate each of the two central ends of the radial lines, from the other.

Whereas in reality; two radial lengths of a circle cannot be added together, as there is a gap in the form of a hole at the centre of the radiated circle; which serves to separate each of the two central ends of the radial lines, from the other.

4. Given the linear concept that two radial lines of a circle do exist, and are in direct opposition to each other; and therefore in effect they are serving as a single line of diameter length bisecting the circle into two halves, this makes sense; because 1 unit

4. Given the linear concept that two radial lines of a circle do exist, and are in direct opposition to each other; and therefore in effect they are serving as a single line of diameter length bisecting the circle into two halves, this makes sense; because 1 unit

*(e.g. circle)*divided by "two ends/terminal degrees" of a single line = two halves of the unit.

And therefore every time a single "natural number of length" is used to divide a circles "natural circular length"; the circle is divided into that natural number of lengths.

And therefore every time a single "natural number of length" is used to divide a circles "natural circular length"; the circle is divided into that natural number of lengths.

****

5. A circle is "perfectly symmetrical in nature", which is why its circumferential length can be sub divided by any number, into "identical equal lengths" of that number

5. A circle is "perfectly symmetrical in nature", which is why its circumferential length can be sub divided by any number, into "identical equal lengths" of that number

*(Proof; hours, minutes, seconds, of a circular clock-face)*.

Therefore given that a circle has been s

Therefore given that a circle has been s

**ub divided by the number 3; into "3 identical equal lengths" of the number of 3.**

And each identical equal length, of the number of 3

And each identical equal length, of the number of 3

Has 7 identical and equal parts to it length

Has 7 identical and equal parts to it length

It follows that the circle will have 21/7 identical equal parts to its length not 22/7, because 22 cannot be subdivided into equal whole units of length, by either the number of 7 or the number of 3.

It follows that the circle will have 21/7 identical equal parts to its length not 22/7, because 22 cannot be subdivided into equal whole units of length, by either the number of 7 or the number of 3.

In Sum

In Sum

The nature of a circle is defined by the extent of area, that is radiated from its centre; not by the thickness/area of a line, that is used to circumvent and enclose the extent of area radiated from the centre of the circle

The nature of a circle is defined by the extent of area, that is radiated from its centre; not by the thickness/area of a line, that is used to circumvent and enclose the extent of area radiated from the centre of the circle

*(Refer to Minus Pi in Black & Yellow)*.

Because a natural circle as with all forms of natural geometry, is a non-linear geophysical and biophysical phenomenon. With the extent and limits of its natural designs and constructs, being subject only to the universal mathematical ratios; that exist between all of the combinations of its "

Because a natural circle as with all forms of natural geometry, is a non-linear geophysical and biophysical phenomenon. With the extent and limits of its natural designs and constructs, being subject only to the universal mathematical ratios; that exist between all of the combinations of its "

*of positive neutral and negative energy/matter; none of which possess a decimal linear based devaluation, into the non-mathematical - non-empirical realm of irrationality/non-ratio.*__3 non-linear forms"__

In conclusion for this chapter

In conclusion for this chapter

We no longer live in the limited age and perspective of Euclid's linear geometry, we live in the age of our developing knowledge of the atom, and its microcosmic universe.

Therefore we know that the centre of a circle cannot be found, as the microcosmic spatial beginnings of it radii disappear into the realm of microcosmic infinity; and equally the same applies to the microcosmic spatial endings of the radii at the circles periphery; which also disappear into the realm of microcosmic infinity.

We no longer live in the limited age and perspective of Euclid's linear geometry, we live in the age of our developing knowledge of the atom, and its microcosmic universe.

Therefore we know that the centre of a circle cannot be found, as the microcosmic spatial beginnings of it radii disappear into the realm of microcosmic infinity; and equally the same applies to the microcosmic spatial endings of the radii at the circles periphery; which also disappear into the realm of microcosmic infinity.

Therefore we can state that although the radius length of a circle, can never be exactly measured;

Therefore we can state that although the radius length of a circle, can never be exactly measured;

The ratio of the circles bi-radial diameter length, to that of its circumferential length;

The ratio of the circles bi-radial diameter length, to that of its circumferential length;

Is dictated by the radial distance that lies between the spatial centre of a circle; relative to the two spatial central points (Degrees), that lie at each end of the bi-radial diameter length.

Is dictated by the radial distance that lies between the spatial centre of a circle; relative to the two spatial central points (Degrees), that lie at each end of the bi-radial diameter length.

*Calculating The Area Of A Ring***Beginning with a square measuring 60 cm x 60 cm, the area of the square is 3, 600 square cm.**

**Therefore as the area of the circle is three quarters of the squares area, the area of the circle is 2. 700 square cm.**

**Therefore we only need to find the area of the central circle and deduct this from the 3, 600 square cm area of the larger circle, in order to give the area of the ring.**

**The diameter of the central circle is 30 cm, therefore the area of the square of the 30 cm diameter is 30 cm x 30 x cm, 900 square cm. And as the square area of the circle is three quarters that of its square, so the area of the 900 square circle is 775 square cm.**

**2, 700 square cm to the area of the larger circle minus 775 square cm; gives 1, 925 square cm to the area of the ring. **

*From The Cube To The Sphere***From The Cube To Its Cylinder**

**Diagrams 1 - 5 serve to depict the potential cylinder within a cube, diagrams 5 - 6 serve to show that when a three dimensional cylinder is rotated, so that its lateral length is face on to us, if we imagine it rather as being as a flat square, with the circle of the potential sphere within the cylinder being visibly apparent; we can see that when the four corners of the cylinder are carved away, the shape of the sphere will be released.**

**The diagram above depicts the three quarter area of the circle of the square relative to the potential cylinder lying within a 16 cm x 16 cm wooden cube. and in order to form the cylinder from the wooden cube; we place it on a wood lathe, and then rotate the cube and shave away the four lateral corners, which are equal one quarter of the cubes mass, leaving the three quarter mass of the cylinder remaining.**

****

**The two diagrams above serve to demonstrate both visually, and via our minds eye, that when the four-dimensional (lateral-Vertical-Angled- Curvature) cylinder is given an angled frontal aspect of view, and then turned towards us to a full face on view, and while at the same time mentally discarding its dimension of the curvature; the front facial view of the cylinder, then in visual effect, becomes a flat square.****Therefore if we use our minds eye, to imagine a circle of the same height and width, relative to the area of the square;****It then becomes apparent, that when we shave off the four corners of the square, which are in reality the corner ends to the cylinder, we will be in reality shaving away one quarter, of the mass of the cylinder. **

**The first diagram above serves to depict the cube fixed on a wood lathe, prior to carving away the four corners to form the cylinder; and the second diagram depicts the cylinder placed lengthwise and laterally away from us, prior to shaving away the corners of the circular face, to form the sphere.**

**This visual perspective then allows us to imagine, that as we use our chisel in a left and right circular motion, acting between and towards each of the two central spindles, we are shaving away the circular dark flat aspect, of the front of the cylinder.**

**And as we do so the round and darker frontal facial aspect of the cylinders length, will gradually move upwards and forwards away from us, and grow smaller, as the final curvature of the sphere, takes its full form.**

**In sum in regard to the mass of wood removed from the cube**

**With our first cut we removed one quarter of the mass of wood from the cube**

**With our second cut we removed one quarter of the mass of wood from the cube**

**Therefore we can say**

**A**

**circle is three quarters of the area of its square**

**A a cylinder is three quarters of its cube**

**A sphere is three quarters of its cylinder.**

**Confirmation by Mass - Weights**

**Given that the cube weighed 160 grams prior to being converted into a sphere**

**The cylinder would weigh 120 grams**

**The wood shavings would weigh 40 grams**

**Given that the cylinder weighed 120 grams**

**The wood shavings would weigh 30 grams**

**Confirming a cylinder is three quarters of its cube, a sphere is three quarters of its cylinder.**

*Twelve Steps To The Sphere*

**Cube To Its Cylinder**

1. Measure cube height to obtain its diameter = 6 cm

1. Measure cube height to obtain its diameter = 6 cm

**2. Multiply 6 cm x 6 cm, to obtain the square area, and the length of the perimeter to one face of the cube; length = 24 cm, area = 36 square cm**

**3. Multiply the square area, by the length of diameter, to obtain the cubic capacity = 216 cubic cm**

**4. Divide the cubic capacity by 4, to obtain one quarter of the cubic capacity = 54 cubic cm**

**5. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the cylinder = 162 cubic cm**

**6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm**

**7. Divide the cubes surface area by 4, to obtain one quarter of the cubes surface area = 54 square cm**

**8. Multiply the one quarter surface area by 3, to obtain the three quarter surface area of the cylinder = 162 square cm**

**Cylinder To Its Sphere**

**9. Divide the cylinders cubic capacity by 4, to obtain the three quarter cubic capacity of the sphere = 40 and a half cubic cm**

**10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter capacity of the sphere = 121 and a half cubic cm**

**11. Divide the cylinders surface area by 4, to obtain one quarter of the surface area = 40 and a half square cm**

**12. Multiply the one quarter surface area by 3, to obtain the three quarter surface area of the sphere = 121 and a half square cubic cm**

*Areas And Volumes Of Symmetrical Ovals And Ovoids*

**Space, volume, and area, are relative interchangeable commodities, (see space - volume - Area - Mass after the next chapter) therefore and as such;**

**A symmetrical oval is a squashed circle**

**A symmetrical ovoid is a squashed sphere**

**When a straight flat downward pressure is applied to the centre of an inflated sphere/ball, resting on a flat solid surface, the air within the ball is displaced equally in all directions away from the the centre of gravity of the ball; as the passive resistance of the atomic tensile strength, of the solid surface below, causes the air pressure already within the ball, plus the pressure being applied to the ball, to equate in all directions away from the centre of gravity of the ball.**

**Resulting in an equal expansion of the balls envelope, in all directions away from the centre of gravity of the ball.**

**Therefore this same effect can be relatively be applied by the use of the minds eye, to the amount of space that exists within a circle, existent upon a flat surface.**

**Whereby by given the symmetrical confining stricture, to the line of the circles circumference, the simulated downward pressure, can only expand symmetrically, laterally, and geometrically along the increasingly oval length of the former symmetrical circular shape., as per the diagram below.**

**Therefore given that the diameter of the ball in the diagram above is 6 cm, the area of the square of the diameter will be 36 square cm; and the three quarter square area of the circle will be 27 square metres.****Therefore if the diameter length is compressed down to 3 cm****The area of half of the oval will be 13 and a half square cm, and the total area of the oval will be 27 square cm; which will be three quarters of the area to the ovals rectangle.****As the rectangles area would have consisted of 3 rows of one cm squares; totalling 36 square cm of area to the rectangle.****Therefore we can say****As a symmetrical circle is three quarters of its square, and its sphere is three quarters of its cylinder****So a symmetrical oval is three quarters of its rectangle, and its ovoid is three quarters of its cylinder**

*Minus Pi, In Black And Yellow*****

We have two yellow square cards each measuring 120 cm x 120 cm

Card 1. Has a black circle measuring 120 cm high x 120 cm wide, and the square card has been cut into 4 equal squares, each measuring 60 cm high x 60 cm wide.

Each 60 x 60 square card holds one quadrant of the yellow square, and one quadrant of the black circle.

Card 2. Has been cut into 4 equal 60 x 60 cm square pieces, each square has a black circle 60 cm high x 60 cm wide.

We have two yellow square cards each measuring 120 cm x 120 cm

Card 1. Has a black circle measuring 120 cm high x 120 cm wide, and the square card has been cut into 4 equal squares, each measuring 60 cm high x 60 cm wide.

Each 60 x 60 square card holds one quadrant of the yellow square, and one quadrant of the black circle.

Card 2. Has been cut into 4 equal 60 x 60 cm square pieces, each square has a black circle 60 cm high x 60 cm wide.

**A. Black Areas**

**1.**

**All black areas have an equal area to each other**

**2. Any number or type or black area combined will give an equal area**

**3.**

**All black areas combined will give an equal area**

**B. Yellow Areas**

**1. All yellow areas have an equal area to each other**

2. Any number or type of yellow area combined will give an equal area

3. All yellow areas combined will give an equal area

C. All Areas

1. Have an equal area

2. Any number and any combination of black and yellow areas, will give an equal area

3. All black and yellow areas combined, will give an equal area

4. All areas of the two cards combined, will give an equal area (288 squares)

Logic

2. Any number or type of yellow area combined will give an equal area

3. All yellow areas combined will give an equal area

C. All Areas

1. Have an equal area

2. Any number and any combination of black and yellow areas, will give an equal area

3. All black and yellow areas combined, will give an equal area

4. All areas of the two cards combined, will give an equal area (288 squares)

Logic

**1.**

**All of black and yellow areas of circle and square contain an equality of area**

**2. When an equal amount of area is deducted from an equal amount of area, it leaves an equal amount of area remaining**

**3.**

**Pi has a greater sum of inequality to its area, than that of the lesser sum of equality to the area of the circle**

**4.**

**Pi is a mathematical infringement into the area surrounding of a circle**

**In Sum**

Pi represents the physical inequality of the thickness of a length of string, being mathematically inducted into the Archimedes formula.

Pi represents the physical inequality of the thickness of a length of string, being mathematically inducted into the Archimedes formula.

*Compass & Lines*

**Unlike Archimedes, today we use various types of drawing compass to draw our circles and though they are more complex in nature they are far simpler to use and more accurate than any piece of string used by our ancient mathematicians. However though they are more accurate they are still unable to divide/partition the circumference of the circle into exactly equal lengths. The reason for this being that both the drawing compass and the method used for drawing a circle contain inherent physical limitations in regard to obtaining a high degree of accuracy.**

A compass has two arms with a pencil tip at the end of one arm and a steel tip at the end of the other arm, and prior to drawing a circle the two tips of the compass are separated to a distance of measurement that is to equate to the desired radius of the diameter of the circle that is to be drawn. However it is during this process of physically measuring the distance between the two tips of the compass that the first of many inaccuracies in this procedure begin to manifest.

In first place in regard to the inaccuracies inherent to the process of drawing a circle with a compass is the accuracy of the measurements that are marked upon the ruler that we use in order to measure between the two tips of the compass. If there is even the vaguest shade of a degree of an inaccuracy in the ruler or the measurement taken from it between the two tips of the compass, then this degree of inaccuracy will be multiplied 360 times over by the time that we have completed the circle.

A compass has two arms with a pencil tip at the end of one arm and a steel tip at the end of the other arm, and prior to drawing a circle the two tips of the compass are separated to a distance of measurement that is to equate to the desired radius of the diameter of the circle that is to be drawn. However it is during this process of physically measuring the distance between the two tips of the compass that the first of many inaccuracies in this procedure begin to manifest.

In first place in regard to the inaccuracies inherent to the process of drawing a circle with a compass is the accuracy of the measurements that are marked upon the ruler that we use in order to measure between the two tips of the compass. If there is even the vaguest shade of a degree of an inaccuracy in the ruler or the measurement taken from it between the two tips of the compass, then this degree of inaccuracy will be multiplied 360 times over by the time that we have completed the circle.

**We also have to consider the sharpness (thicknesses) of both the steel and the graphite tips of the arms of the compass that we are using, because the accuracy of the circle is not merely dependent on the accuracy of the measurement of distance between the two tips of the compass, it is also dependent upon**

*that distance/measurement throughout the process of drawing the circle.*__maintaining__

A compass works on the principle of one arm of the compass providing a central fixed point around which the second arm is rotated in order to draw or transcribe a circle.

A compass works on the principle of one arm of the compass providing a central fixed point around which the second arm is rotated in order to draw or transcribe a circle.

**In order to provide a fixed central point (B) the steel tip of the first arm is forced**

*and*__into__*the surface that is to be drawn upon. However as can be seen in the first diagram above at point B wherein the steel tip is forced into the surface to fixate the arm of the compass, the length of the surface radius becomes shorter according to the depth/length of penetration of the steel tip into the central point from which the circle will radiate from.*__below__

After the steel tip has been forced into its central point to fixate its central position, the degree of pressure that has been used and placed on the fulcrum of the compass in order to achieve penetration is lessened. The lessening of the downward pressure on the fulcrum of the compass then allows the pencil tip to be rotated around the central point in order to transcribe the circle.

After the steel tip has been forced into its central point to fixate its central position, the degree of pressure that has been used and placed on the fulcrum of the compass in order to achieve penetration is lessened. The lessening of the downward pressure on the fulcrum of the compass then allows the pencil tip to be rotated around the central point in order to transcribe the circle.

**However if the downward force on the fulcrum is too great, it can cause the arms of the compass to spread apart and so increase the length of the drawn radius.**

There are also many other factors of error that can and do creep into this procedure regardless of, and no matter how careful we are in trying to avoid them, and still further errors creep in if and when we then use the compass to sub divide the circumference of the circle into equal lengths. For example, when using the circles radius to divide the circumference of the circle into six (Supposedly) equal arcs in order to draw a six sided hexagon; the amount of error that was present in the original radius of the circle, is repeated six times on the circumference of the circle; however not exactly, because each time we subdivide the circumference, the steel tip of the compass has to be forced down into the surface; and each time the pressure exerted on the fulcrum, is variant.

There are also many other factors of error that can and do creep into this procedure regardless of, and no matter how careful we are in trying to avoid them, and still further errors creep in if and when we then use the compass to sub divide the circumference of the circle into equal lengths. For example, when using the circles radius to divide the circumference of the circle into six (Supposedly) equal arcs in order to draw a six sided hexagon; the amount of error that was present in the original radius of the circle, is repeated six times on the circumference of the circle; however not exactly, because each time we subdivide the circumference, the steel tip of the compass has to be forced down into the surface; and each time the pressure exerted on the fulcrum, is variant.

Regardless of whether we use a piece of string or whether we use a compass, in order to draw a circle; it is a physical/empirical fact, that each of the minute errors that do occur during the subdivision process, are then amplified over the three hundred and sixty degrees of the circle.

Regardless of whether we use a piece of string or whether we use a compass, in order to draw a circle; it is a physical/empirical fact, that each of the minute errors that do occur during the subdivision process, are then amplified over the three hundred and sixty degrees of the circle.

**Therefore in sum: It is a totally impossible physically reality to be able to draw a perfect circle.**

However this is not the case in regard to our minds eye intelligence, logic, and rationality; which do not suffer from any such physical restraints; and it is therefore only within the environs of our minds, that it is possible to mathematically define/rationalise and draw, the perfectly symmetrical non-linear circle.

However this is not the case in regard to our minds eye intelligence, logic, and rationality; which do not suffer from any such physical restraints; and it is therefore only within the environs of our minds, that it is possible to mathematically define/rationalise and draw, the perfectly symmetrical non-linear circle.

*BI-RADIAL ARITHMETIC OF CIRCLE AND SQUARE*

CONTEXT OF A DEGREECONTEXT OF A DEGREE

*The square of the length of the hypotenuse of a right triangle, is equal to the sum of the squares of the other two sides.*