**To see previous years of eclectic work/progressive and developing thoughts go to; www.geometry-mass-space-time.com**

**Which has so far; received some 272, 000 + hits from 136 Nations, of the 195 Nations that go into the international make-up of the World.**

**CONTENT**

**A Single Length Of Degree**

**Simplest Method For Calculating The Area Of A Circle**

**Concurrence 1: Sumerian Method**

**Concurrence 2: Archimedes Proposition 1**

**Concurrence 3: Pythagoras Theorem**

**Methods**

**Concurrence By Rings: The Area Of Rings**

**Concurrence By Symmetry: Symmetry Of The Circle**

**Concurrence By Sphere: From The Cube To It's Sphere**

**12 Steps To The Sphere**

**Concurrence By Ovoid: The Area And Volumes Of Symmetrical Ovals And Ovoids**

**Concurrence By Degree: The Number of Degrees of Longitude and Latitude to a Sphere**

**The Area of 1 Degree of 64, 800 Degrees**

**Symmetry of a Circle**

**Irrationality Of 3.14159**

**Minus Pi In Black and Yellow**

**Phenomenon Of The Pythagoras Theorem**

**EMAIL:**

**universometry@fromthecircletothesphere.net**

**unialphaomega@hotmail.com**

**For the first time in human history; the surface areas and volumes of cylinders, spheres, and ovoids can now be exactly calculated.**

**Galileo**

**“In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.”**

**"Measure what can be measured, and make measurable what cannot be measured"**

**Author: Euclidean linear thinking/application, in regard to our modern day knowledge of physics, is rather as with that of ancient flat earth thinking, in regard to the Earth's Horizon.**

__Circle & Square__: Key**1. Any measure of an alignment, can be used as the radius line of a circle**

**2. Two lengths of radius form a diameter length of a circle.**

**3. A diameter when squared will form a square, with four right angles of the same length as the diameter of the circle; and a perimeter length that is equal to, eight times the length of the radius of the circle..**

*SINGLE LENGTH OF*

*DEGREE***120 Centimetre Right Angle X 4 = 480 Centimetres 120 Centimetre Diameter X 3 = 360 Centimetres**

**1.**

**Use**

**one 120 centimetre Right Angle as a Diameter**

**2. M**

**ultiply this length of Diameter by three**

**3. The Circle is 360 centimetres long**

**4. 360 degrees to a Circle**

**5. Each degree is one centimetre long**

**Therefore it follows**

*SIMPLEST METHOD FOR CALCULATING THE AREA OF A CIRCLE***Diameter 120 Centimetres x Diameter 120 Centimetres = 14, 400 Square Centimetres to the Square**

**1. The area of the square is 14, 400 sq cm's**

**2. The area of the square divided by 4, is 3, 600 sq cm's**

**3. Multiply 3, 600 sq cm's by 3**

**4. 10, 800 sq cm's, to the area of the Circle**

**Concurrence 1**

*SUMERIAN*

*SUMER 4,000 BC***SUMER (Wikipedia)**

Mathematics[edit]

The Sumerians developed a complex system of metrology c. 4000 BCE. This metrology advanced resulting in the creation of arithmetic, geometry, and algebra. From c. 2600 BCE onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[43] The period c. 2700 – 2300 BCE saw the first appearance of the abacus, and a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.[44] The Sumerians were the first to use a place value numeral system. There is also anecdotal evidence the Sumerians may have used a type of slide rule in astronomical calculations. They were the first to find the area of a triangle and the volume of a cube.[45]

Babylonian mathematics (also known as

Mathematics[edit]

*Main article: Babylonian mathematics*The Sumerians developed a complex system of metrology c. 4000 BCE. This metrology advanced resulting in the creation of arithmetic, geometry, and algebra. From c. 2600 BCE onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[43] The period c. 2700 – 2300 BCE saw the first appearance of the abacus, and a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.[44] The Sumerians were the first to use a place value numeral system. There is also anecdotal evidence the Sumerians may have used a type of slide rule in astronomical calculations. They were the first to find the area of a triangle and the volume of a cube.[45]

__Geometry[edit] Babylonians knew the common rules for measuring volumes and areas. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if__

*π*is estimated as 3.Babylonian mathematics (also known as

*Assyro-Babylonian mathematics*[1][2][3][4][5][6]) was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited.[7] In respect of time they fall in two distinct groups: one from the Old Babylonian period (1830-1531 BC), the other mainly Seleucid from the last three or four centuries BC. In respect of content there is scarcely any difference between the two groups of texts. Thus Babylonian mathematics remained constant, in character and content, for nearly two millennia.[7]

__Sumerian Method__**Geometry: Babylonians knew the common rules for measuring volumes and areas. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, "**

__which would be correct if__*π*__is estimated as 3".__(And__"incorrect"__if

*π*is__estimated__as 3.14159)**1. Diameter 120 cm's**

**2. Circumference 360 cm's**

**3. Circumference squared 129, 600 sq cm's**

**4. Divide 129, 600 by 12 = 10, 800 sq cm's to the circle**

__Concurrence 2__

*ARCHIMEDES***287 - 212 BC**

**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.**

**Archimedes Triangle**

**1. The base right-angle is equal to the radius 60 cm**

**2. The height of the right-angle is equal to the circumference of the circle**

**3.**

**The area of the circle is equal to the above right-angle triangle, which has one side about the triangle that is equal to the 60 cm radius, and the other to the 360 cm circumference of the circle.**

**Check**

**1, The area of the rectangle is 21, 600 sq cm's**

**2. The area of the right-angle triangle is 10, 800 sq cm's**

**3. The three quarter area of the 3 x r squared circle is 10. 800 sq cm's**

__Concurrence 3__**PYTHAGORAS**

**2582 2500 BC**

**THEOREM**

**In any right angle triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.**

**In any right angle triangle, the area of the circle on the hypotenuse is equal to the sum of the area of the circles on the other two sides**

*AUTHOR*

*3 x r squared***A**

**Diameter 120 Cm x 120 Cm = 14, 400 Sq Cm's to the Square**

**1. 60 centimetre radius is squared = 3, 600 sq cm's**

**2. Multiplied by 3 =**

__10, 800__sq cm's to the area of the circle**3. The area of the circle is three quarters that of the**

__14, 400__sq cm square**B**

**The 14, 400 sq cm area of the square, is divided by 4 = 3, 600 square centimetres**

**This is then multiplied by 3 =**

__10, 800__**square centimetres to the area of the Circle**

**Equating to three quarters of the area, to that of the overall square**

**C**

**1.**

**60 cm radius is squared, 3, 600 sq cm**

**2. 3, 600 sq cm is divided by 4 = 900 sq cm**

**3. 900 sq cm is then multiplied by 3 = 2, 700 sq cm**

**4. 2, 700 sq cm is then multiplied by 4, = 10, 800 sq cm's to the circle**

__Absolutes__**The**

__"non-linear" circumferential length__of the area to a circle; is six times that of the circles radius length, and three times that of the circles diameter length.**The**

__"non-linear" circumferential length__to the area of a circle; is three quarters of the length, to the perimeter, of the square of the circles diameter length.**The**

__"non-linear" area__of a circle, is three quarters of the area to that, of the area of the square of the circles diameter length.

**Concurrence By Rings**

*THE AREA OF RINGS***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.**

__Concurrence By Sphere__

FROM THE CUBE TO ITS SPHEREFROM THE CUBE TO ITS 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 SPHERETWELVE STEPS TO THE SPHERE

**CUBE TO CYLINDER**

**Calculating the surface area and volume of a 6 centimetre diameter sphere, obtained from a 6 centimetre cube.**

**Note we use 6 centimetres rather than 120 cm, in order to make the numbers easier to follow.**

**1. Measure cube height to obtain a/its Diameter line = 6 cm**

**2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of perimeter to the square face = Length 24 cm, Square area 36 sq cm.**

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

**4. Divide the cubic capacity by**

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

**5. Multiply the one quarter cubic capacity by 3. to obtain the 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 of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm. CYLINDER TO SPHERE**

**9. Divide the Cylinders cubic capacity by 4, to obtain one quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm.**

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

**11. Divide the Cylinders surface are by 4, to obtain one quarter of the surface area of the Cylinder = 40 & a half square cm.**

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

**Concurrence By Ovoid**

*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**

__Concurrence By Degree__

**THE NUMBER OF DEGREES OF LATITUDE AND LONGITUDE**

**TO THE SURFACE OF A SPHERE****Square Degree**

From Wikipedia, the free encyclopedia

A square degree is a non-SI unit measure of solid angle. It is denoted in various ways, including deg2, sq.deg., (°)², and ☐°. Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere. Analogous to one degree being equal to

or approximately 41 253 deg2. This is the total area of the 88 constellations in the list of constellations by area. The largest constellation, Hydra covers a solid angle of 1303 deg2, whereas the smallest, Crux covers 68 deg2.[1]

For example, observed from the surface of the Earth, the sun or the full moon has a diameter of approximately 0.5°, so it covers a solid angle of approximately 0.2 deg2 (≈

Assuming the Earth to be a sphere with a surface area of 510 000 000 km2, the area of Northern Ireland, 13 600 km2 represents a solid angle of 1.10 deg2.[2] Similarly, the area of Connecticut, 14 356 km2represents a solid angle of 1.16 deg2.[3]

From Wikipedia, the free encyclopedia

A square degree is a non-SI unit measure of solid angle. It is denoted in various ways, including deg2, sq.deg., (°)², and ☐°. Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere. Analogous to one degree being equal to

*π*/180 radians, a square degree is equal to (*π*/180)2, or about 1/3283 or 3.0462×10−4 steradian (0.30462 msr). The number of square degrees in a whole sphere isor approximately 41 253 deg2. This is the total area of the 88 constellations in the list of constellations by area. The largest constellation, Hydra covers a solid angle of 1303 deg2, whereas the smallest, Crux covers 68 deg2.[1]

For example, observed from the surface of the Earth, the sun or the full moon has a diameter of approximately 0.5°, so it covers a solid angle of approximately 0.2 deg2 (≈

*π*(0.5/2)2 deg2), which is 4.8 × 10−6 of the total sky sphere.Assuming the Earth to be a sphere with a surface area of 510 000 000 km2, the area of Northern Ireland, 13 600 km2 represents a solid angle of 1.10 deg2.[2] Similarly, the area of Connecticut, 14 356 km2represents a solid angle of 1.16 deg2.[3]

**A Circular protractor has 180 Degrees to its "Vertical Height of Diameter Length"**

Which equates to there being 180

Therefore if the Vertical Height of the Sphere is given a rotation of 360 degrees

Which equates to there being 180

__Lateral Circles__of 360 degrees; to the "Vertical Height of a Sphere"Therefore if the Vertical Height of the Sphere is given a rotation of 360 degrees

**180 Vertical Degrees x 360 Degrees of Lateral Rotation = 64, 800 shared Vertical**

*(Longitude)*and Latitudinal Degrees to the surface area of a Sphere.

*THE AREA OF 1 DEGREE*

*OF*

*64, 800 DEGREES***a. Circle of 360**

**⁰**

**b. 6 Diameters 6 Square faces c. Shared Degrees d. 1 Square Face of Cube e. 8 Corners of Cube Removed**

**In the previous chapter in order to gain the number of degrees to the surface of a sphere; we rotated the 180**

**⁰ of**

**height of a protractor by 360**

**⁰, and in so doing defined that there are 180 lateral circles of 360⁰ to the height of a sphere; and therefore there are 64, 800⁰ to the surface area of a sphere.**

**However although this lateral rotational method of finding the number of degrees to the surface of a sphere, does indeed serve to define the number of 64, 000 degrees to the surface of a sphere; it does not serve to give each of those degrees an equal amount of surface area.**

**The reason being, that if we were to view such a hypothetical sphere from an overhead aspect of view, extending upwards from the spheres equator; each subsequent circle of 360⁰ to that of the circle of the equator, would appear to be smaller than the one below it.**

**Therefore in order to equate all of the degrees to each other, and equally over the surface of the sphere, we have to divide the number of 64,800 degrees into the overall surface area of the sphere.**

**And in order to achieve this via straight linear geometry, we begin by observing that the number of radii/degrees to a circle is the same number for its surrounding square; and relative to the solidarity of a sphere and its surrounding solid cube, this principle does not change.**

**To grasp this rather than think in terms of linear radii; think in terms of radiations extending directly outwards from the centre of a sphere, and continuing outwards through the surrounding body of the cube.**

**Given this aspect of observation it becomes clear, that if we convert the number of degrees to a circle into the same number of centimeters to find the surface area of the sphere; and then divide this surface area by 64, 800 degrees, this will give the area to each one of those degrees.**

**120 cm Height of Cube = 120 cm Diameter Sphere**

One face of the Six faces of a 120 cm cube = 14, 400 square centimeters

Six faces x 14, 400 square centimeters = 86, 400 square centimeters

86, 400 sq cm ÷ by 4 = 21, 600 square centimeters to one quarter of the surface area of the cube

21, 600 sq cm x 3 = 64, 800 square centimeters to the surface area of the ¾ Cylinder of the Cube

64, 800 sq cm ÷ by 4 = 16, 200 sq cm to one quarter of the cylinders surface area

16, 200 sq cm x 3 = 48, 600 square centimeters to the surface area of the ¾ Sphere of the Cylinder

48, 600 sq cm ÷ by 64, 800 degrees to the surface of the Sphere = 0∙75 or ¾ of one square centimeter to 1 degree of the surface, of the Sphere.

__Refer Back To 12 Steps To The Sphere__One face of the Six faces of a 120 cm cube = 14, 400 square centimeters

Six faces x 14, 400 square centimeters = 86, 400 square centimeters

86, 400 sq cm ÷ by 4 = 21, 600 square centimeters to one quarter of the surface area of the cube

21, 600 sq cm x 3 = 64, 800 square centimeters to the surface area of the ¾ Cylinder of the Cube

64, 800 sq cm ÷ by 4 = 16, 200 sq cm to one quarter of the cylinders surface area

16, 200 sq cm x 3 = 48, 600 square centimeters to the surface area of the ¾ Sphere of the Cylinder

48, 600 sq cm ÷ by 64, 800 degrees to the surface of the Sphere = 0∙75 or ¾ of one square centimeter to 1 degree of the surface, of the Sphere.

**Height 180 degrees x 3 1/3rd Rotation's of 120 degrees (360) = 64' 800 degrees**

**Which serves to prove once again; that a circles length is 3 x its diameter length & its area is 3/4 that of its surrounding square.**

**And given that; 48, 600 sq cm ÷ by 64, 800 degrees to the surface of the Sphere =**

__¾ of one "Square Centimetre"__to the area of 1 degree of the surface, of the Sphere.**This also serves to inform us;**

That the three-dimensional, 3/4 square centimetre shape of one degree of sphere surface; is in the shape of a circle.

That the three-dimensional, 3/4 square centimetre shape of one degree of sphere surface; is in the shape of a circle.

Which equates to; the cross section of one

Which equates to; the cross section of one

__radiation__extending from the__centre__of the 120 cm sphere, being circular along its entire radiant length.

__Concurrence By Symmetry__

*SYMMETRY OF A CIRCLE***A circular protractor has**

180 Degrees to its Diameter

360 Degrees to its Circumference

Therefore by giving 180 Centimetres to the length of Diameter

The Area to the "Square" of the Diameter is

32, 400 "Square Centimeters"

Rotate the 90 degree radius 360 Degrees; and the Area to the "Circle" of the Diameter is;

32, 400 "Circumnavigation Degrees"

Proving that the Circumferential length to a Circle has an "Exact Ratio" to its Radius (radii) length, and hence its Diameter length.

Therefore the "

Which is reinforced by the following; which serve to confirm that a Circles circumference length far from being irrational;

180 Degrees to its Diameter

360 Degrees to its Circumference

Therefore by giving 180 Centimetres to the length of Diameter

The Area to the "Square" of the Diameter is

32, 400 "Square Centimeters"

Rotate the 90 degree radius 360 Degrees; and the Area to the "Circle" of the Diameter is;

32, 400 "Circumnavigation Degrees"

Proving that the Circumferential length to a Circle has an "Exact Ratio" to its Radius (radii) length, and hence its Diameter length.

Therefore the "

*irrational/non-ratio*" formula of Pi cannot be considered or said to be; the ratio of a Circles circumference length, to Diameter length.Which is reinforced by the following; which serve to confirm that a Circles circumference length far from being irrational;

Is perfectly symmetrical

90⁰/cm radius x 360⁰ rotation = 32, 400 circumnavigation degrees

60⁰/cm radius x 360⁰ rotation = 21, 600 circumnavigation degrees

30⁰/cm radius x 360⁰ rotation = 10. 800 circumnavigation degrees

10⁰/cm radius x 360⁰ rotation = 3, 600 circumnavigation degrees

Is perfectly symmetrical

90⁰/cm radius x 360⁰ rotation = 32, 400 circumnavigation degrees

60⁰/cm radius x 360⁰ rotation = 21, 600 circumnavigation degrees

30⁰/cm radius x 360⁰ rotation = 10. 800 circumnavigation degrees

10⁰/cm radius x 360⁰ rotation = 3, 600 circumnavigation degrees

**360⁰/cm**

**÷ 5 = 72 ÷ 5 = 14.4 x 10 = 144 square centimetres ÷ 4 = 36 x 3 = 108 square centimetres to the circle**

And this is why a circles circumference length, can be equally subdivided by any "whole number", into that number of exactly equal, and identical parts of the circle.

And this is why a circles circumference length, can be equally subdivided by any "whole number", into that number of exactly equal, and identical parts of the circle.

**And any part of any circle multiplied by 360, will give 360 degrees to the circle with each degree exactly identical in curvature of length.**

*IRRATIONALITY Of 3.14159*

*Oxford English Dictionary**Pi symbol of the*

__"ratio"__of the circumference of a circle to its diameter, approximately 3.14159

FACTS

FACTS

1: Pi cannot be and is not; the symbol of the

1: Pi cannot be and is not; the symbol of the

__"ratio"__of the circumferential length of a circle to its diameter length; because it is an__"irrational number"__.

2: Pi is the symbol of the

2: Pi is the symbol of the

__"decimal ir-rational"__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.__**3: Pi is simply the**

__"symbol of the result"__of the decimalization of the__(approximate)__circumference length of 22 measurement units, given by Archimedes to the length of the circle; relative to being divided by the 7 measurement units to diameter length.

Which equates to three whole units of diameter length, with each whole unit of diameter length measuring seven measurement units = 21/7. and one measurement unit of diameter length 1/7, remaining.

Which equates to three whole units of diameter length, with each whole unit of diameter length measuring seven measurement units = 21/7. and one measurement unit of diameter length 1/7, remaining.

**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.**

**Until finally, the original fractional value of 22/7 Pi; was**

*(decimated)*after the decimal system was adopted by France in 1790, during the French Revolution.**As decimals**

**22 measurement units divided by 7 = 3.14185714285 diameter lengths**

*(Note not 3.14159)*

**22 measurement units divided**

**3 = 7.3333333 diameter lengths**

**Neither of which unlike the imperial/empirical fractional system, serves to provide an exact ratio of measurement, between the diameter and circumferential length of a circle.**

**And contrary to the claim that the decimal form of 22/7, provides the closest approximation of the ratio of a circles circumferential length, to that of its diameter length; this is totally disproved, by the fact that if we divide 22 measurement units of circumferential length by 7.33 measurement units instead of 7, we obtain the result of;**

**3.00136425648**

**Which is a closer approximation of circumferential length, to diameter length; than either 3.1429 or 3.14285714285.**

****

**MINUS Pi IN BLACK AND YELLOW****Card A Card B**

**Card B Card A**

**Card B Card A Card A Card B**

**120 X 120 cm 120 X 120 cm**

We have two square yellow cards measuring 120 cm's x 120 cm's

Card A. has a black circle of 120 cm diameter, and has been cut into 4 equal quadrants of 60 cm squares

Card B. has four 60 cm diameter circles to each quadrant, and has been cut into 4 equal quadrants of 60 cm squares.

We have two square yellow cards measuring 120 cm's x 120 cm's

Card A. has a black circle of 120 cm diameter, and has been cut into 4 equal quadrants of 60 cm squares

Card B. has four 60 cm diameter circles to each quadrant, and has been cut into 4 equal quadrants of 60 cm squares.

__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

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)

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 Of The Two Cards__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**

Pi represents the physical inequality of the thickness/area of a length of line or lines, having been mathematically inducted into the formula.

__In Sum__Pi represents the physical inequality of the thickness/area of a length of line or lines, having been mathematically inducted into the formula.

*COMPASS AND LINE*

**A compass has a pencil tip at one end, and a pointed steel tip at the other end of its two arms, 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.

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 (thickness) 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.**

**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.

PHENOMENON OF THE PYTHAGORAS THEOREMPHENOMENON OF THE PYTHAGORAS THEOREM

*Geometric Encounter With The Third Dimension*

*In any right angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.*

**References**

www.How many proofs of the Pythagoras Theorem-Maths forum: 370 proofs are given in the book "The Pythagoras Proposition" published by E S Loomis 1940.

www.BabylonianPythagoras: The earliest record of the Pythagoras Theorem appears on the Babylonian Susa Tablet dated between, 1,800 - 1,600 BC.

www.How many proofs of the Pythagoras Theorem-Maths forum: 370 proofs are given in the book "The Pythagoras Proposition" published by E S Loomis 1940.

www.BabylonianPythagoras: The earliest record of the Pythagoras Theorem appears on the Babylonian Susa Tablet dated between, 1,800 - 1,600 BC.

**However as per the above diagram; given a right-angle triangle, with a one to one ratio of 12 squares to each right-angle, it can be seen that the sum of the 17 x 17 square on the hypotenuse, equates to 289 squares; while the sum of the two squares on the other two sides, give a total of one less 288 squares.**

**Therefore given all of the proofs as to the Pythagoras Theorem that have been forthcoming over the previous millennium's; we can only assume. that this phenomenon has been either missed or it has been ignored, by those putting forth their own versions of proof, in support of the Pythagoras Theorem.**

**Regardless of which is true; such a phenomenon should not have been excluded, nor should it be ignored, as this does not serve to facilitate our learning processes in regard to the sciences, or human progress.**

**In the two diagrams above it can be seen, that the phenomenon of an extra amount of area to the square on the hypotenuse, is not isolated to just a 12 x 12 right-angle triangle; it also occurs with the one to one ratio of a 6 x 6 right-angle, where there is an extra 0.25 (1/4) square area, on the 8.5 (8 1/2) square of the hypotenuse.**

**And indeed given a 24 x 24 right-angle triangle, the 34 square length hypotenuse produces a square which equates to 1,156 squares on the hypotenuse; while the sum of the 24 x 24 squares on the other two sides give a total of 1,152 squares on the hypotenuse, which is 4 squares less than that of the 34 x 34 equation.**

**Quite simply given an equal one to one ratio, of a right-angle triangle**

**The area of the square on the hypotenuse**

*"*to an area, greater than the sum of the areas of the squares on the other two sides.__equates__"

__However__**When**

*"*the number of squares, and half squares of the square on the hypotenuse, the number of squares, totals to the same number as the number of squares on the other two sides.__counting and totalling__"

__Intriguing__

**And to be continued**